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LONDON, EDINBURGH, anv DUBLIN
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CONTENTS OF VOL. XLII.
(FIFTH SERIES).
NUMBER CCLIV.—JULY 1896.
Prof. A. W. Witkowski on the Thermodynamic Properties of
Ll. E233 aac ped El) ASRS eb es oleae ee Ae ete re Pa
Messrs. S. W. Holman, R. R. Lawrence, and L. Barr on the
Melting-points of Aluminium, Silver, Gold, Copper, and
eee he ret es Sn ee EA. Mt ks
Prof. J. A. Fleming on a further Examination of the Edison
erprmie Glow Dianips: ss kb ee es eh re we eves ne oe
Dr. Meyer Wildermann on the Degree of Dissociation of
LS EEE DINVIROS UE TU ARCA, AR or ace iO er ear.
ese. Thompson on Hyperphosphorescence .........-..-
Prof. J. V. Jones on the Magnetic Field due to an Elliptical
Current at a point in its plane within it................
Mr. W. Sutherland on High Tensions in Moving Liquids
Notices respecting New Books :—
Mr. H. B. Goodwin’s Azimuth Tables for the Higher
yc asl Sema e nk at ee ea th TAR, Bie ;
Proceedings of the Geological Society :—
Miss E. Aston on an Alpine Nickel-bearing Serpentine
with Fulgurites ; with Petrographical Notes by Prof.
ECITOY og SG SER ONO CRE eRe a rere
Dr. C. 8. Du Riche Preller on the Pliocene Glaciation,
Pre-Glacial Valleys, and Lake-Basins of Subalpine
PPLE WEG el ye le ago Te
Prof. J. EH. Talmage on certain Linear Marks in a Sedi-
GE MMe SSPE MMe PPS, Sra anc Tee as dies eS, aa eS
Mr. A. Strahan on Submerged Land-Surfaces at Barry,
CCIRELC TIER TS NTRS Bieerae Sone ere ae ae ee
Mr. A. Strahan on a Phosphatic Chalk with Holaster planus
Til UGGS. 5 2 ie alae Ion eer ae ore ee
Page
1V CONTENTS OF VOL. XLII.—FIFTH SERIES.
Page
Dr. A. P. Pavlow on the Classification of the Strata
between the Kimeridgian and the Aptian ..........
Mr. Beeby Thompson on the Junction-Beds of the Upper
Lias and Inferior Oolite in Northamptonshire ......
Mr. J. H. Cooke on the Stratigraphy and Paleontology
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Miss M. Crosfield and Miss E. G. Skeat on the Geology
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Ou a Rotational Motion of the Kathode Disk in the Crookes
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NUMBER CCLYV.—AUGUST.
Mr. W. Williams on the Convergency of Fourier’s Series
Mr. Rollo Appleyard on Dielectrics <.52-- 72... see
Mr. A. Campbell on New Instruments for the Direct Measure-
ment of the Frequency of Alternating or Pulsating Electric
Gurremts: 72.223 42 Cae bepinola os x 8 ee en eee
121
122
Dr. 8. P. Thompson on some Experiments with Réntgen’s |
HVAW.S/ banc renee Rite con cee ee Pl ee ope ee ben -
Lord Rayleigh on the Theory of Optical Images, with special
reference to-the Microscope. .2....2. +22.) eee
Dr. G. A. Miller on the Operation Groups of order 8p, p
» aon
being any prime number 4. ..)..5.05...22 2-6 o5
Mr. J. Larmor on the Theory of Moving Electrons and Electric
Charges aie. aa. Ue oa ee lee 2 eee er 201
Notices respecting New Books :—
Mr. R. T. Glazebrook’s ‘James Clerk Maxwell and
Modern Physics’... 2.42. =. oe or 205
Dr. Benjamin Williamson’s Elementary Treatise on the
Imteoral Calculus. 22.9.0. 2e. 5c see 205
Mr. W. C. Dampier Whetham’s Solution and Electrolysis 206
Proceedings of the Geological Society :—
Mr. Clement Reid on the Eocene Deposits of Dorset .. 207
On the Refractive Indices of some Substances for very short
Electrical, Waves, by Dr. Ac Tampa s.- . t.ho) 220 See 207
A Lecture Experiment on meg wae by Dr. Silvio
WMSSAMO es cis. oe Soe ee eee eee as ee ee ee 208
CONTENTS OF VOL. XLII.-—FIFTH SERIES. v
NUMBER CCLVI.—SEPTEMBER.
Mr. F. W. Burstall on the Use of Bare Wire for Resistance-
CFLS 6 Slo ete gitteeka Senha a et 209
Prof. A. McAulay on the Wave-Surface and Rotation of
Polarization Plane in an Aeolotropic Electromagnetic
1 PLEIN E dt cal dos oe, UR cer Pee, ee Ser ae ae 224
Mr. T. Preston on the Continuity of Isothermal Transforma-
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Prof. J. G. MacGregor on the Hypotheses of Abstract Dy-
namics and the Question of the Number of the Elastic
Plas Set ee es es Ss es One Si 240
Sir David Salomons on the Electric Discharge in a Magnetic
ie ee gti. cece hays « se oy a eee oe Cates 245
Prof. G. F. FitzGerald on the Longitudinal Component in
WDE LD? voce EGS <r cane ea 260
Mr. A. Campbell on the Measurement of very large and very
Pa cermanine) Currents.) ..4 3 2).a joe hs as Pls cee eee 271
Dr. A. Goldhammer on the Analytical Peper sen ation of the
Periodic System of the Elements...............0.00004 277
Proceedings of the Geological Society :—
Mr. W. Farnworth on a Head or Gateway driven into
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et Mel lerce ate pot Poet ar Wee ee ee Magda bs Ny ECR 283
Mr. J. W. Spencer on the Geographical Evolution of
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Messrs. 8. 8. Buckman and E. Wilson on Dundry Hill:
its Upper Portion, or the Beds marked as Inferior
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Mr. F. W. Harmer on the Pliocene Deposits of Holland,
and their Relation to the English and Belgian Crags .. 286
On a Damping Action of the Magnetic Field on Rotating
eM AbOES Dive Wir DUAWO Ns 6. ar ain SY ba ea ela eee 288
The Action of Magnetism on Electromotive Force, by Alfred
5. JP UGUIGIGIS ZY aa Bate SE or ean es Pane ie ane ne ar 288
NUMBER CCLVII.—OCTOBER.
ror. 4. Poynting on Osmotic Pressure.........6....%... 289
Mr. F. Bedell on Admittance and Impedance Loci ........ 300
N. Oumoff and A. Samoiloff on Electric Images in the
inield on ay Elittont-s! (Crookes?) Tube... 30.00 enc. oes 308
B. Rosing on the Possibility of explaining the Phenomena of
Magnetism by the Hypothesis of Participation of Matter
in the Motion of the Magnetic Field .......... eee 314
vil CONTENTS OF VOL. XLII.—FIFTH SERIES.
Page
Dr. H. Debus on the Genesis of Dalton’s Atomic Theory .. 350
Notices respecting New Books :—
re: Benjamin's ‘The Intellectual Rise in Electricity—
a ELISTORY fe s.< So Sa; apes RCo Riis et ayer tr 368
Prof. E. W..Brown’s Introductory Treatise on the Lunar
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Proceedings of the Geological Society :—
Messrs. P. Lake and 8. H. Reynolds on the Lingula-
Flags and Igneous Rocks of the Neighbourhood of
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Messrs. 8S. H. Reynolds and C. I. Gardiner on the Kil-
dare Amllice< Sone tecctt aa oss ci ee 372
NUMBER CCLVII.—NOVEMBER.
Mr. W. Sutherland on Thermal Transpiration and Radio-
meter MOON. Goo. 43... sas aye tise Ee ee 373
Prof. J. J. Thomson and Mr. E. Rutherford on the Passage
of Electricity through Gases exposed to Rontgen Rays .. 392
Messrs. J. Frith and C. Rodgers on the Resistance of the
Milectrie Ave; (Plates ILL.—V.): % ..4:..2.5. Se Seo 407
Dr. G. Johnstone Stoney on Microscopic Vision .......... 423
Prof. W. E. Ayrton and Mr. T. Mather on Galvanometers.. 442
Notices respecting New Books :—
Prof. H. Behrens’s Anleitung zur mikrochemischen
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Proceedings of the Geological Society :—
Messrs. J. Horne and EH. Greenly on Foliated Granites
and their Relations to the Crystalline Schists in Eastern
Sutherland. 2% ssc 5 eee oe ok «oe es en 447
Mr. E. Greenly on the Geology of the Eastern Corner
OF AMPIESCY wie. ce si sd ae ee ee eee See 448
M. F. de Montessus de Ballore on Seismic Phenomena
inthe British Hmpire: 204... 635. A 449
Col. H. W. Feilden on the Glacial Geology of Arctic
Europe and its Islands ; with an Appendix by Prof. T.
Gy Bowne 2... s2b<. bases eee 449
Prot. J. P. Iddings on Extrusive and Intrusive Igneous
Rocks as Products of Magmatic Differentiation...... 450
Carbon Megohms for High Voltages, by W. M. Mordey .... 450
Search for Solar X-Rays on Pike’s Peak, by Florian Cajori .. 451
CONTENTS OF VOL. XLII.—FIFTH SERIES. vil
NUMBER CCLIX.—DECEMBER.
Page
Prof. R. Threlfall and Mr. J. A. Pollock on some Experiments
meeerbrami Mens MadiatlOU. —.. . 2. ken See he eee ees 453
Dr. C. Davison on the Diurnal Periodicity of Earthquakes .. 463
Mr. W. Sutherland on Thermal Transpiration and Radio-
meter Motion .......... ral ts Oils SS apenas eee abaieon ee oa 476
Lord Rayleigh on Theoretical Considerations respecting the
Separation of Gases by Diffusion and similar Processes .. 493
ees. Stoney on Microscopic Vision .......5....5+.- 499
Notices respecting New Books :—
J. H. van *t Hoff’s Studies in Chemical Dynamics...... 528
On Experiments with Rontgen Rays, by Prof. Augusto Righi. 530
Volume Measurement of an Air-Thermometer Bulb, by
EERE Lys) ON a eee eh 6 8 oi Goal Pe ea dens. Ties Sle See 530
On the Influence of Temperature on the Refraction of Light
tessa. LeUtlbrichy.. «3 cy0).35 fine os OG hs bee eos «als 532
MaBRMM EERE. f.005 2 Van Petits Ps OA Poe Oe er. oi 533
PLATES.
I. & I. Mlustrative of Prof. A. W. Witkowski’s Paper on the Thermo-
dynamic Properties of Air.
III. to V. Illustrative of Messrs. Frith and Rodgers’s Paper on the
Resistance of the Electric Arc. -
ERRATA.
Page 112 (Mr. Sutherland’s paper), in line 28 from top and again in line 30 from
top, transpose the words “ before” and “ after.”
,, 127, line 17. For “the rate of convergence of the series increases in-
definitely” &c., read “the rate of convergence of the series diminishes
indefinitely ” &c.
5, 240, line 24, for “conservation of natural forces” read “ conservatism of
natural forces.”
THE
LONDON, EDINBURGH, ann DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[FIFTH SERIES.]
in: TBA TERS
heen 11 1896 }
SUL 1896.
ss PATENT ot
a ell
C\
I. Thermodynamic Properties of Air. , Vi
By A. W. Witkowsk1.
[Plates I. & II. ]
Part I[J.*
Specific Heat.
SL. |i order to obtain a full knowledge of the thermo-
dynamic properties of a homogeneous gaseous sub-
stance it is indispensable to perform upon it a twofold experi-
mental investigation: besides its thermometric and dynamic
behaviour the calorimetric properties ought to be studied
independently. It is known that, in order to solve the first
part of the problem, it is sufficient to test the thermal expan-
sion of the gas under different pressures and its compressibility
at one temperature—or conversely, the compressibility at
different temperatures, together with the expansion under a
single pressure. This part of the problem I worked out
experimentally four years ago; the results have been pub-
lished in the 23rd volume of the Rozprawy. In the present
paper I intend to supplement these investigations by the
second, calorimetric part.
In this respect we may make use of the general laws of
* Translated from the 32nd volume (1896) of the “ Rozprawy ” of the
Cracow Academy of Science (Math. Class) and communicated by the
Author. For Part I., see p. 288, vol. xli.
Phil. Mag. 8. 5. Vol. 42. No. 254. July 1896. B
2 A.W. Witkowski on the
thermodynamics. It follows that, in order to disclose the
whole of the calorimetric properties of a homogeneous fluid,
it is sufficient to investigate experimentally the specific heat,
in its dependence on temperature, under a singie constant
pressure, together with its compressibility and thermal ex-
pansion.
§ 2. Any state of thermodynamic equilibrium of the gas
may be defined by means of two independent variables, for
instance by its pressure p, and temperature 6 (=é on the
absolute thermodynamic scale). Consider an infinitely small
change of state, corresponding to the increments dp and dé
of the variables ; then the gas does a quantity of work,
and absorbs a quantity of heat,
dQ =MCp dt == l dp.
We denote by m the mass of the fluid, by ¢, its specific heat
under the constant pressure p, by / a certain kind of latent
heat ; the last two quantities are to be considered as variables,
depending on p and ¢.
The four coefficients - e (thermal expansion and com-
pressibility), cp, and /, when known for every state (p, t) of
the body, give a full account of its thermodynamic properties.
They are not independent; by the general laws of thermo-
dynamics they are subject to the relations
TEpA OI Oly pi) 2 1
J Ot” op... Jdm.ot \. {eee (1)
where J denotes the dynamical equivalent of the heat-unit.
_Amongst other thermodynamic properties we have to con-
sider, in the first place, the specific heat at constant volume
=c,. It may be expressed in terms of the above coefficients
by the known equation
ey
b NOE
oe ne _.. *)
Ov
OG texoey
Sf Tn ao
or else by
It follows from these equations, that the variations of the
Thermodynamic Properties of Air. 3
specific heats, as well as their difference, can be deduced from
the fundamental equation of the body, viz. v = F(y, ¢).
Although we do not possess a sufficiently accurate general
expression for the volume of atmospheric air in terms of the
pressure and temperature, yet some inferences as to the
thermodynamic behaviour of that body at different tempera-
tures, down to the critical, and under different pressures, may
be drawn from the numerical data which have been given in
the first part of the present paper, both in tabular and in
graphical form. These results can be obtained by means of
the graphical calculus. It is true that results arrived at in
this manner cannot be as accurate and convincing as those
obtained by direct experiment. Still 1 thought them worthy
of attention, the more so as direct calorimetric measurements at
very low temperatures would be attended by serious difficulties.
Another reason which induced me to try the indirect method
was the desire to learn something more about the specific
heat of gases, which has been investigated usually only within
very restricted limits of temperature and pressure. Besides
the important work of Prof. Joly*, the object of which was
to determine the effects of increased pressure and temperature
on the specific heat at constant volume, I have to notice a
memoir by M. Margules+ on the variations of the specific
heat of carbon dioxide, determined by calculations based on
Andrews’s and Amagat’s experiments.
§ 3. Integrating (1) along the isothermal ¢, between the
limits of 1 and p atmospheres we obtain
de $2502 070
Ca Clr ar aes
If we assume the thermal expansion to be known, this equa-
tion (Rankine’s) may be used to calculate the values of ¢,
corresponding to ditterent states of the gas.
First of all, however, it is necessary to inquire how far the
quantity e;—a constant of integration with respect to p—
depends on the temperature ; c, denotes evidently the specific
heat under the constant pressure of one atmosphere, at the
absolute temperature ¢.
In order not to depart from the notation adopted in the
first part, I shall denote by p the pressure expressed in atmo-
spheres. The mass m of the gas may be chosen so as to have
v=1 cubic centim. at the temperature of melting ice, under
* “Specific Heats of Gases at Constant Volume,” Phil. Trans. part i.,
vol. clxxxii. (1891); part ii. vol. clxxxv. (1894); part iii, vol. clxxxy.
(1894), ee
+ “Spezifische Warme der comprimirten Kohlensiure,” Wien. Svtzber.
xevil. Ila., 1888.
B2
4 A. W. Witkowski on the
the pressure of one atmosphere (1033° 24 = —E thus it follows
that m=0:'001293 grammes. Assunnags “farthes J = 42,700
gramme-centimetres per gramme- -calorie, the last equation
becomes
PO?u
p=e—18T14t) Sedp 2. 2s
Similarly, instead of (2) we ee
(3)
ge
§ 4. It has been already remarked that the investigation
ought to bear in the first place on the quantity ¢. The
specific heat of atmospheric air under constant pressure has
been repeatedly measured. In the first rank there stands
the fundamental work of Regnault* ; fourteen years later
these experiments were repeated by HK. Wiedemannf+, with
the help of a simpler form of apparatus. Regnault’s experi-
ments cover a range of temperature from +200° to —31°;
air of ordinary density was tried, as well as condensed air,
up to 12 atmospheres. Notwithstanding these relatively
extended limits Regnault was not able to discover any marked
variation of the quantity c,; hence there arose the very
general belief, that the specific heat of air is a constant, inde-
pendent of pressure and temperature—an opinion which, as
will be seen further on, is very far from being correct.
Wiedemann’s experiments, so far as air is concerned, were
intended to control Regnault’s value of the specific heat.
The results obtained by. the two investigators are only in
approximate agreement. According to Regnault the most
probable value “of the specific heat c, between 0° and + 100°
is 0°23741 ; this is the mean value of single determinations
varying between the limits 0°23536 and 0:23890.
Wiedemann’s result is 02389, the limits of smgle determi-
nations being 0°2374 and 0: D414. Although the principal
aim of my own experiments was to find the variations of the
specific heat of air through as wide a range of temperatures
as possible, yet I thought it worth while to redetermine once
more the exact value of this important constant ; this seemed
(5)
* Relation des expériences Sc., tome ii. pp. 41-110, and pp. 206-224,
(Paris, 1862.)
+ “Spez. Warme des Gase,” Pogg. Ann. vol. clvii. (1876), pp. 19-22.
Thermodynamic Properties of Avr. 5
the more desirable, as the thermometric scale is now better
defined, thanks to the investigations of the International
Bureau of Weights and Measures—and some light has also
been thrown on the variations of the specific heat of water.
I may mention now that my result, as regards the specific
heat of air, is nearly identical with that of Regnault.
To detect any variations in the specific heat of air at atmo-
spheric pressure it was desirable to enlarge the limits of the
experiments on the side of low temperatures, because on the
ground of Regnault’s work we may be sure that at high
temperatures such variations do not manifest themselves; in
this respect I have been able to reach the temperature of
—170°.
§ 5. In order to adapt the well-known apparatus employed
usually to determine the specific heat of gases to the particular
problem of low-temperature calorimetry it was necessary to
to modify it in several respects. I wished also to make it
more reliable, so as to lessen as far as possible the limits of
experimental errors, because of the smallness of the expected
variations in the specific heat.
In studying Regnault’s memoir on the specific heat of
gases, one comes to the conclusion that the chief cause of
error in determinations of this kind is to be sought in the
uncertainty as regards the true temperature of the gas at the
moment when it enters the calorimeter, 2. e. the cooler im-
mersed in it. The influence of this cause of error manifests
itself clearly by the fact that the calculated value of the
specific heat depends in a marked degree on the velocity with
which the gas-stream is made to pass through the apparatus :
on increasing this velocity the result tends towards a maximum
value. It is known that Regnault considered only those de-
terminations as valid which corresponded to this maximum.
In my own experiments I used at first an apparatus similar
to that of E. Wiedemann (J. c.). I soon became convinced
that in order to do away with the uncertainty as to the initial
temperature of the gas, and at the same time to diminish as
far as possible the direct influx of heat (or cold) from the
heating apparatus into the calorimeter, it was indispensable
to improve the mode of connexion of these two parts. The
following conditions were to be kept in view in devising this
most important part of the apparatus :—(1) The connecting
piece ought to be perfectly air-tight, and at the same time a
bad conductor of heat ; (2) it ought to transfer the gas from
the heater into the calorimeter with the least possible loss of
heat ; (3) some thermometric arrangement ought to be intro-
duced in it in order to measure directly the temperature of
the gas-stream where it enters the calorimeter.
6 A. W. Witkowski on the
It is known that the determination of the temperature of
a gas is attended by some difficulty. Owing to the great
diathermancy of these bodies, a thermometer introduced in a
gas tends to show rather the temperature of the surrounding
bodies than of the gas itself. To get over this difficulty I
resolved to apply a mode of temperature-measurement which
has proved successful in meteorological observations, and is
perhaps best known in its application in Assmann’s aspirating
thermometer. The thermometer .is placed inside a polished
metal tube, through which a brisk current of air is made to
pass. It has been found that under such conditions the tempe-
rature indicated is nearly independent of the temperature of the
tube. After several trials | constructed upon this principle
the connexion between the heater and the calorimeter ; it will
be described fully in § 7. The only thermometric apparatus
which could be applied under the given conditions was the
thermo-electric couple. The mode of experimenting was
such that, instead of measuring the initial temperature of the
gas, I determined with the aid of the couple only the small
loss of temperature experienced by the gas during its passage
from the heater into the calorimeter. For this purpose one
of the solderings of the couple was placed in the gas-stream,
just inside thé calorimeter, the other in the heater, or in
a separate thermostat of known temperature. This second
soldering was always in contact with the bulb of a hydrogen
thermometer, in order to reduce finally the observed tempe-
ratures to a definite scale. .
§ 6. Another modification concerned the determination of
the mass of air used in every calorimetric experiment.
Usually the quantity of gas has been determined by previous
experimenters in an indirect way. KE. Wiedemann, following
Delaroche and Bérard, measured the volume of water which
replaced the gas, contained initially in a bladder or india-
rubber bag. Regnault used a spacious receiver filled with
compressed gas, and observed the fall of pressure occasioned
by the using up of gas in every experiment. The volume of
the receiver and its temperature being known, the mass could
be calculated on the basis of the law of compressibility, which.
Regnault determined himself expressly for that purpose. I
have found it more convenient and accurate to measure the
mass of air by direct weighing. Accordingly a large
reservoir (about ten litres capacity) was filled with pure
compressed air at a pressure of 80-100 atmospheres: this
was ample enough to provide the gas for some dozen of
calorimetric determinations. From this store I filled small
metal flasks (capacities 136, 220, 208 cubic centim.), as
Thermodynamic Properties of Arr. 7
represented in fig. 1, provided with screw stopcocks. These
were weighed on a large chemical ba- Fig. 1.
lance carrying some 1500 grammes. =r
By gradually opening the stopcocks the
flasks yielded during the calorimetric
experiment a perfectly uniform current
of air. They were then again weighed,
whence, by applying the usual correc-
tions, the mass of gas passed through the
calorimeter could be accurately calcu-
lated. In the three flasks used it was easy
to store up some 90 or 60 litres of gas.
The tightness of the flasks and stop-
cocks was controlled by the weigh-
ings themselves ; even a small leakage
renders an accurate weighing im-
possible *. |
The mode of preparing and compres-
sing the air was exactly the same as
described in Part I., § 3.
§ 7. I shall now describe the calori-
metric apparatus, namely that form of it
which I usedin the low-temperature work.
In my first experiments I constructed
a calorimeter on the lines of that of
Regnault and Wiedemann; the cooler,
1. e€. the system of metallic tubes destined to transmit heat
from the gas to the water, was attached to the calorimetric
vessel by solder, while the exit-tube of the heater was intro-
duced near the bottom of the vessel and fixed there by means
of a cork. This arrangement proved inconvenient, and more-
over it rendered difficult the intended measurement of the
initial temperature of the gas. Accordingly I modified the
apparatus in suck a manner as to make the heater entirely
independent of the calorimeter proper. In fig. 2 (reduced
in the ratio 24:1) DP R’R represents the heater (I shall
continue to call it so, although, when used at low tempera-
tures, it acts really as a cooler); S the cooler ; both are fixed
to a separate support E, which stands on a heavy base EH’.
This part of the apparatus may thus be used in conjunction
with different calorimetric vessels.
* T think that the method described here, of weighing gases in a com-
pressed condition, could be advantageously employed in accurate deter-
minations of the density of some gases under atmospheric pressure. For
that purpose the weighed flasks ought to be discharged into an empty
receiver of known capacity and temperature, provided with a mercury
manometer.
A. W. Witkowski on the
Fig. 2.
x a ee a
@wwrte-
eee ecere ewe a@ eee =
me ew me wm = Ss
1: 2,5
Thermodynamic Properties of Air. 9
The calorimeter K—a vessel of thin sheet silver, capacity
about 250 cub. centims., weight 98°310 grammes—rests on
three pointed glass feet inside a double-walled brass enclosure
Z, protected on the outside by sheets of paper and cotton-
wool; the space between the walls is filled with water.
Through several openings in the double sheet-brass cover of
the enclosure there are introduced into the calorimeter :—
(1) the cooler 8, (2) the stirrer M, the oscillations of which
are maintained by a heavy pendulum (not shown), (3) the
thermometer T.
The thermometer T, an excellent instrument furnished by
G. Fontaine of Paris, was divided in 0°02 degr. (from —0°5
to +0°6 and +115 to +22°3). It has been standardized
with reference to the hydrogen scale by the Central Ortice of
Weights and Measures at Vienna. The corrections proved
so insignificant that it was possible to omit them altogether.
Weight of glass and mercury was determined by the maker.
_h represents a small electromagnetic hammer, destined to
shake the thermometer T, in order to prevent any lagging of
mercury. ¢ and ?¢' are two auxiliary thermometers by means
of which the temperature of the stem of the principal ther-
mometer was determined in order to obtain the well-known
thermometric correction.
The heater was constructed as follows :—A copper tube P,
one metre in length, internal diameter =2 millim., external
diameter =4 millim., is coiled round a vertical brass tube
DF, fixed along the axis of the heater. The outer end of the
copper tube is joined by means of a thick-walled indiarubber
tube with a T-shaped piece. One of its ends P’ is connected
with the flasks filled with compressed air, the other N with an
open mercury manometer, which serves to control the velocity
and steadiness of the current of air. The inner end of the
copper spiral is connected by solder with the brass tube D F,
near its lower end, where a small hole (u, fig. 2a) is provided,
through which the air cooled in the spiral passes into the
brass tube, and thence through the connecting piece C into
the cooler 8. The connexion consists of a thin-walled glass
tube C, some 25 millims. in total length, its upper end being
cemented by means of isinglass into the brass tube D F, the
lower end into the cooler 8 by sealing-wax. The glass tube
serves only to provide an air-tight connexion and to protect
against external heat the inner tube s, which is the real
delivering tube of the gas. The last-mentioned tube s is made
of very thin highly-polished sheet silver ; it is soldered to
10 A. W. Witkowski on the
a silver collar r, fixed tightly in the brass tube DF, just
above the end of the glass tube C.
In the axis of s, some 3 or 4 millims. from its mouth,
there is fixed one of the solderings B of the thermo-electrie
couple, composed of very thin wires of copper and nickel.
The wires pass freely through the brass tube DF’; near its
upper end they are cemented by shellac into a capillary glass
tube, the end of which can be seen in fig. 2, protruding over
the end of DF. The capillary is fixed here by a cork,
tightened by means of marine glue. In order to protect the
cement against the cold the end F of the brass tube is dipped
in water contained ina small glass beaker (indicated by dotted
lines). One of the wires of the couple (copper) leads to the
galvanometer G, the other to the heater, where the second
soldering A is in contact with the bulb W of the hydrogen
thermometer.
This construction of the connecting piece was devised
in agreement with the remarks of § 5. The silver tube s,
being a good conductor of heat, assumes a temperature which
differs very little from that of the heater ; the gas is thus
conducted into the cooler through a channel of nearly its own
temperature. At the same time the polished silver tube is a
bad radiator, and therefore it prevents effectually any inter-
change of heat between the heater and the cooler, with the
exception of that due to the air current.
Secondly, the silver tube s forms part of the thermometric
arrangement, constructed upon the principle of aspiration.
The air passes through the relatively narrow channel s with
a sufficiently great velocity, and it communicates its tempera-
ture very quickly to the soldering B of the couple. In
order to shield the soldering against the radiation of the
cooler 8S, the temperature of which is generally very different
from that of the gas, it is advisable to place the soldering at
some distance from the mouth of s, so as to reduce as far as
possible the cone of rays passing from Bto 8. This radiation
is also intercepted by a small silver bridge, put across the
mouth of the tube s, which serves at the same time to keep in
ae the soldering B, which is tied to it by a bit of cocoon
fibre.
The cooler 8 is made entirely of silver ; it is quite similar
to that employed by E. Wiedemann in his researches on the
specific heat of gases. Four vertical silver tubes are connected
by short pieces of silver tubing, alternately near their upper
and lower ends. They are filled with compressed silver
Thermodynamic Properties of Air. Eh
turnings (fine silver thread-tape) ; the whole weighs 30°024
grammes. I have found that this number of tubes is more
than sufficient to allow perfect equalization of temperatures
of the gas and the water.
The heater is placed vertically along the axis of a rather
wide glass tube R’, which acts as a receiver of the freezing-
mixtures. Near its lower end it is contracted into a narrow
neck, in which the end D of the axial brass tube is fitted
tightly by means of an indiarubber ring. Another glass
cylinder R of larger diameter surrounds the vessel R’; its
bottom a consists of a varnished cork, in a perforation of
which the neck of R’ is fitted. The glass cylinder and the
cork act as insulators against heat.
When liquid oxygen was to be applied as frigorific medium
I found it economic to cool the current of air previously to its
passing into the copper coil P. For that purpose a small
metallic trough f (indicated by dotted lines) was provided,
which was filled with a mixture of solid carbon dioxide and
ether.
In order to determine the specific heat of gases at higher
temperatures, for instance at + 100°, the same form of
apparatus could obviously be used with the glass receiver R!
replaced by some form of steam-heating. In my first experi-
ments, however, I used a slightly different form (fig. 3).
The connexion of the brass tube DF with the cooler, the
thermometric arrangement, &c., are exactly like those in the
apparatus for low temperatures. Instead of the copper spiral
the heating of the gas is accomplished, as in Wiedemann’s
apparatus, in a wide horizontal brass tube P, filled with copper
turnings, and soldered at a right-angle to the tube DF.
Both tubes are placed permanently in a brass trough R, being
soldered to it. The temperature of + 100° was obtained by
boiling water in the trough, with the use of a small gas-
burner G. The soldering A of the thermo-electric couple
was heated in a separate steam thermostat. On this occasion
I found (after having lost much time in search of an error
in all parts of the apparatus) that the wires of the couple
leading into the thermostat ought to be very tine, otherwise
the soldering would not acquire the temperature of +100°,
even in a copious current of steam.
§ 8. The initial temperature of the gas was determined, as
already mentioned, by means of a thermo-electric couple
(copper-nickel) in connexion with a hydrogen thermometer.
It will be useful to say a few words on the mode of calcula-
tion of the temperature.
12 A.W. Witkowski on the
The thermo-electric current was measured by means of a
Despretz galvanometer, resistance about 145 ohms, the sensi-
tiveness of which was very constant and relatively considerable
—163,500,000 scale-divisions per ampere ; this number has
been tested frequently, also by means of a thermo-electric
couple, at temperatures 0° and + 100°.
The calibration of the couple was performed as follows :—
One of the solderings being placed in melting ice, the other
Fig. 3.
was heated or cooled, together with the bulb of the hydrogen
thermometer, to different temperatures : S= +100°, +19°°6,
—79°-0, —104°0, —182°%5. Let us denote by E? the
electromotive forces observed (products of galvanometer de-
flexions and total resistance), and write
E?=5/(9).
Thermodynamic Properties of Air. 13
The experiments furnished the following values of the func-
tion f($) :—
$=+100°, +19°6, —79°0, —104°0, —182°5.
F(S)= 3788, 3429, 2945, . 2804, 2345.
It will be remarked that the law of Tait and Avenarius is not
strictly fulfilled, since these: numbers do not obey the law
of linear variation.
Suppose now one of the solderings to be heated to 5 degrees,
the other to some higher temperature 3’. Then the electro-
motive force will be:—
BE? =B3 — Bp =9'f(9')—3f(9).
If the difference between the temperatures $ and 3’ is a small
one, as it was in my calorimetric experiments, where ¥ —$
corresponded to the small loss of temperature of the gas in
its passage from the heater to the calorimeter, then we may
write
/2) =f) + (3-9) LO
and therefore
Wes (9-8) { 9) 9) +3900)
Using the values of /(3) given before it is easy to calculate
graphically the values of the differential coefficient of an,
and to construct the following table :—
6. oe). Hae ashe
“Jee ae 4-49 4230 28°5
— i aaa 5-22 2533 17-1
= eer 5°62 2220 15-0
= A et 6:06 1242 8-4
The numbers headed Sa have been obtained by division of
14 A. W. Witkowski on the
Bet} by the total resistance of the circuit (148-3 ohms), they
denote the galvanometer deflexions per degree of difference of
temperature at a mean temperature 3. These numbers I have
used to reduce the galvanometer deflexions observed during
the calorimetric work, in order to obtain the losses of tem-
perature of the gas.
§ 9. Every one of the calorimetric determinations was
conducted as follows :—After weighing the metal flasks filled
with compressed air, and testing them for any possible leak-
age, they are connected with the calorimeter. Next the
calorimetric vessel is charged with water from a small distilled-
water bottle, as used in chemical laboratories, and the weight
of the charge determined by observing the loss of weight of
the bottle. Lastly, the heater is filled with the corr esponding
frigorific medium (carbon dioxide, ethylene, or oxygen), the
stirrer started, and during a few minutes (10) the following
temperatures are read at noted times :—temperature of the
calorimeter, of the two auxiliary thermometers ¢ and ¢’, of the
hydrogen thermometer. At the end of the tenth minute a
first assistant starts the current of air, with some determinate
speed, as registered by the mercury manometer, and tries to
keep it as steady as possible by handling the screw valves of
the flasks. At the same time a second assistant begins to
observe the galvanometer deflexions, noting them at each half
minute contemporaneously with the readings of the tempera-
ture of the calorimeter. During these observations the moment
is noted when, the flasks being emptied, the current ceases
and the mercury column in the manometer falls down to zero.
At this time the auxiliary thermometers are read again, as
well as the hydrogen thermometer. During the next ten
minutes the rate of cooling or heating of the calorimeter is
again observed. The experiment closes with the weighing of
the empty flasks.
The whole course of a determination and calculation of
results will be best understood with the help of the following
example :—
Air cooled by liquid ethylene. Weight of air (corrected)
48:8842 grammes. Weight of water =244°410 grs. Redaced
weight of calorimeter =253°829 grammes water. Reading
of manometer =70 centims. of mercury (corresponding to a
current of about 15°3 ers. of air per TSE The following
temperatures were observed :—
Thermodynamic Properties of Air. 15
Galvanom.
i Temp. of | Hydrogen . Auxiliar
Minutes. arieeiace. Ra: deflexion Seta:
mnillim.
oe ee ee 20°954
_ 2 Se ee 868 .
—103°51
2. oe 781
ot See 694
— 103°50
> J ee 601
5 2 517
|. ee 420
— 103-50
> 2 ee 329
: . 240
= =e pa a Fo, 196
10 current on ...... 152 250°4 b—18:2,4/=164
"22 ee 1961 29°6
Lk eee 18°70 PAST
2 ee Eee V9 30-7
a eee 16°95 30°2
_o-eeeeee 16:08 30°4
' 2) ee 15°25 30°3
13m 10s end of eur.!
i ee 14:58 103:0 t=18 0, ¢t’=18-4
“a ee eee 14-523
Nose nawocane 522,
ee os) coscce ee 520
—103°50
Lok eee 510
Se 494
_s, 4:05 Ee 476
2 ek Sa eee 460
2) 440
2. 2 419
Ls {= ee 400
fe 379
2 360
1 2 ee ee 341
With the aid of the numbers recorded in the Ist and 2nd
columns of the above table a curve is drawn, representing the
variations of temperature of the calorimeter. It is composed
of three nearly straight parts (the regular course of which
serves as a proof that the readings are free from mistakes) ;
the inclination of the first and third branches enables us to
determine the influence of external heating or cooling on the
temperature of the calorimeter. We find easily in the present
example :—
16 _ A. W. Witkowski on the
Initial cooling ......... 0:0894 degree per minute.
Pal oS ee 0°0195- - 3 .
Mean ga Pie Sees oe 00544 _—,, =
This mean value I accept as determining the external cooling
influence during the principal period of the experiment, when
the gas current is flowing. I have found that this simple
mode of reckoning gives nearly identical results with the
more elaborate method of calculating the corresponding cor-
rection which has been proposed by Reznault. It seems,
moreover, that a more laborious determination of the external
heating or cooling effect would be useless, on account of the
following anomaly, which presented itself in a more or less
marked degree in every experiment :—The transition from
the rapid fall of temperature, caused by the cold gas-stream,
to the less marked cooling observable during the final period,
is always preceded by a slight depression of temperature (as
in the preceding example) ; in experiments at high tempera-
tures the phenomenon is exactly reversed. Undoubtedly
these anomalous variations of temperature are caused by the
fact that the direct conduction of heat (or cold) from the
heater to the calorimeter is slightly modified by the gas-
current itself. This disturbing cause did not escape the
attention of Regnault (/. c. pp. 83, 214), but he did not sue-
ceed either in getting rid of it, or in taking it into account.
Some uncertainty thus remains in every determination of the
specific heat.
In the foregoing example I considered the movement of
heat as finished at the end of the 14th minute. The initial
and final temperatures of the calorimeter are thus :—20-152
and 14°523 ; when corrected with regard to the temperatures
of the stem (¢ and ¢’) they are :—20°162 and 14°514, whence
the total fall of temperature =5°648. From this there is to
be subtracted the fall of temperature caused by external
influences (0°0544 per minute), the amount of which is :—
4x 0:0544=0°218. Thus we obtain the corrected fall of
temperature =5°430.
As regards the initial temperature of the air, it is equal to
— 103°50 + 2°01=—101°:49, because the mean deflexion
30°15 millims. on the galvanometer-scale corresponds to a
gain of 2°01 degrees. The final temperature of the gas is to
be calculated by taking the arithmetic mean of the tempera-
tures of the calorimeter at the beginning (20°16) and at the
end of the current ; the latter we find on the temperature-
Thermodynamic Properties of Air. Ly.
curve, it is 14°94. Therefore final temp. =17%55, total rise
of temperature of the gas =101°49 + 17°55 =119° 04.
According to Bartoli and Stracciati the specific heat of
water at the temperature of the experiment is 0°9996, the
sp. heat at +15° being taken as unity. Therefore we have
finally :
253°829 x 0°9996 x 5°430=c, x 48°8842 x 119-04,
whence it follows: c,=0°2368 as the mean value of the
specific heat of air between the limits —17° and +100°.
§ 10. In the annexed table I have collected all the data
serving to characterize the different determinations of the
specific heat of air at constant atmospheric pressure. The
columns headed I—XI. contain the following quantities :—
I. Mass of air passed through the calorimeter in grs.
II. Equivalent mass of calorimeter in grs. of water.
Ill. Velocity of current: grs. per min.
IV. External heating of calorimeter, before and after the
| current, in a0 degr. per min.
Y. Time of influx of heat into the calorimeter in minutes.
VI. Initial and final temperatures of calorimeter (corr.).
VII. Total rise of temperature (corrected).
VIII. Ditference of temperatures of heater and gas, as indi-
cated by the galvanometer (degrees).
IX. Initial and final temperature of the gas (degrees).
X. Specific heat of air, under a pressure exceeding slightly
the atmospheric, between temperatures indicated
in LX,
XI. Mean values of the foregoing.
The whole of these results may be summed up as follows:—
isenween -— 20° and +98? ..........5..%. Gr O2an 2
mA Srathibe ad. kre Onin ey con chs ¢, =0°2374
‘5 <= JU) Ree 2 U7 aan net alr =072502
5: pale eter eee PO oe. eeu soe 4: ¢;=0°2427
On the ground of these results we may assert with certainty
that the specific heat c, does not vary in a sensible manner
down to a temperature of about —100°. At the lowest
temperatures, however, there is apparently a small increase
in the quantity c, of about 2 per cent. But I think there is
sufficient reason to conclude that even this small increase is
only apparent, namely, that it ought to be ascribed to the
influence of pressure rather than to that of the temperature.
Phil. Mag. 8. 5. Vol, 42. No. 254. July 1896. C
A. W. Witkowski on the
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20 AGW Watkowsla Guahe
Indeed, we shall see later on that the influence of pressure on
the specific heat at very low temperatures is very considerable;
for instance, at a temperature of —144° a doubling of the
atmospheric pressure brings about an increase of specific heat _
by 0:03. Now the air on entering the heater is clearly under
a pressure exceeding the atmospheric by an amount indicated
by the mercury manometer. Jn the liquid-oxygen series of .
my experiments this was about 70 centims. of mercury. Part
of this pressure-head is absorbed by the resistance of the
heater ; but on entering the cooler there remained still an
additional pressure of about 48 centims. This is amply
sufficient to account for the observed small increase of the
specific heat.
In what follows I shall therefore consider the specific heat of
air under atmospheric pressure asa constant (0°2372) from
+100° down to —170°.
§ 11. We are now prepared to take up the more general
problem proposed in § 2, to determine the dependence of the —
specific heat ¢ on pressure, at different temperatures. That
a dependence of this kind does really exist may be inferred
from the fact that the values of the second differential
2
coefficient a (§ 2, equation 1) are in general different from
9
zero. The values of or can be obtained from the results -
Ot? :
recorded in the first part of the present memoir. Denoting
by v the volume, under the pressure p, at 6 degrees, of a mass -
of air which occupies the volume 1 at 0°, under atmospheric
pressure, we have
v=o (T+e0)5 ... .. 2
a denotes here the mean coefficient of expansion under the
pressure p, between 0° and 6°, y the coefficient of com-
pressibility at 0°, depending on p only. The values of a —
have been given in a table in § 15, those of 4, will be found
in the table of compressibility Ԥ 17, vol. xli. p. 809) in the
column headed “0.”
{n what follows I shall assume
t=273 +0.
2
In order to obtain oe I chose the graphical method of eal-
culation, which seemed more direct and reliable than the use
of empirical formulas such as those of Clausius, van der
Thermodynamic Properties of Air. 21
Waals, &. It is most advantageous to perform the differen-
tiations upon that quantity which has been directly measured,
and the variations of which are most marked, 7.e. upon the
coefficient a, Differentiating twice equation ( (6) we obtain
Orr _ O°Y _ 2m 02 | Mop O°%
02 08 poo p oF
or OF 07a
To obtain 56 and AYE
of values of « arranged in curves of equal pressure, abscisse
representing temperatures ; it is in fact the diagram alluded
to in § 14 (Part I.). These curves were divided in short
pieces, approaching to straight lines; the differences of
(7)
I constructed on a large scale a diagram
ordinates of their ends gave oF the values so obtained
were considered as corresponding to the middle point of each
piece.
Another diagram was then constructed with ee as ordinates.
0
The curves were carefully smoothed, care being taken to keep.
the whole of them in sight in doing so. ‘The process of
differentiation repeated furnished the values of O°»
OG?
Using th Pa ep b , ealeulated by equ:
sing ese values 3 ean ve NOW Calculatec Vv equa-
tion (7). It would be superfluous to reproduce here the
vast number of values thus obtained ; some of them will be
given later on. An idea of the curious shape of the curves
ov
Ya
On
92
* It might be objected that this method of calculation is not capable of
yielding results of any high accuracy. Indeed, it cannot be denied that
in drawing long curves along a comparatively small number of observed
points, the success depends in a large measure on the firmness of eye
and steadiness of the drawer’s hand. Yet on the other hand it is to be
remarked that the curves control one another, gross errors are thus easily
avoided.
Although graphical differentiation, if accurately done, is a very tedious
process, it is unfortunately not possible to perform it by mechanical
means, since, as far as I know, mechanical differentiators (working in
a similar manner to the numerous integrators) do not exist. If Tam
right they are even impossible on dynamical grounds,
69655
can be obtained by inspection of PI. I., where the values of
have been collected on isothermals, from —100° to —144°..
22 A. W. Witkowski on the
The shape of these curves leads forcibly to the conclusion
2
that for small pressures (p=1 atm.) g cannot be zero
throughout the whole range of temperatures. It is very
small indeed for temperatures above —100°; but near the
critical temperature its values cannot be neglected. This is,
of course, an inference obtained by extrapolation (dotted parts
of the curves on Pl. I.), since no experiments are available
on expansion of air at low temperatures under atmospheric.
pressure.
Although this extrapolation does not seem to be doubéfal,
and, moreover, any errors in it do not influence the final
pnr2
result, namely the integral a dp,inamarked degree, yet
1
it seemed desirable to test its probability in an indirect way.
For that purpose I integrated twice the extrapolated values of
0’v :
Oe
nical quadratures, adding to the result 1+ 973 in lieu of
(for p=1 atm.), with respect to temperature, by mecha-
constant of integration. It appears from this calculation
that at a temperature 0= — 140° (hydrogen scale), a constant
pressure (1 atm.) air-thermometer would indicate —140°76.
I know of no experiments to corroborate this result. From
Olszewski’s experiments on constant volume gas-thermo-
meters*) there may be quoted the following results: at a
temperature of —143°-7 (hydrogen scale) a constant volume
nitrogen thermometer indicated —144°-4, a similar oxygen
thermometer —145°:5. This is not inconsistent with the
above extrapolation.
§ 12. In order to obtain the values of the specific heat ¢,
according to equation (4) it remains to calculate the integrals
"0 ap, along the isothermals 2”, » and dp bei
38 py Gg ine Senden tally sp art dp being ex-
pressed in atmospheres. These integrals multiplied by the
respective absolute temperatures ¢ and by the constant factor
18:714 represent the difference between the specific heat ¢,
and the specific heat ¢,, under atmospheric pressure, which is
very nearly a constant, =0°2372.
The integrations have been performed by Simpson’s formula,
2
with the help of a large diagram of o, a reduced reproduction
of which will be found on Pl. J. The results are embodied
in the following tables :—
* “ Rozprawy ” of Cracow Acad. vol. xiv. (Math. Class).
Thermodynamic Properties of Air. 23
=—144, ¢=129. = — 125, 7=148.
: .07u G2 D. 08 co Cy:
P | = 10° aE e as oz
10 1360 0-504: 10 211 0-283
a ’ 20 300 0°352
20 1740 0°866
25 2390 1101 30 456 0°452
| 40 825 0°620
50 1406 0-931
56°9 0 1:106
@=—140, t=188. 60 316 1-087
| 70 — 76 1:040
5 — 5 1:031
p- — 108 Ow Cp. t u
On
10 850 0°408
20 1070 0-640
30 2395 0°993
40 9064 2°607
6=—135, 1= 138.
Pp 108 Ov op
2 ot?
P 1039? | % ‘oe BL
ols 10 151 0-272
————=_ >>| i al |e aia 3) 29%)
| @ |e | | B | & | oe
20 648 0-484 40 382 0-479
30 1110 0-689
0 50 610 0-614
45 8699 2-602 eae 5 Gene
od a 70 —56 0-777
50 —1501 3-004 be ae oe
5D 0 2-848
60 — 295 2°785
65 = Ge 2-758
=—130, t=143.
p 1080” Cp d=—115, t=158
rola
10 311 0-302 Pp —19°9” Cp
20 418 0-397 RYE
30 685 0-536 ee eee =
40 9993 0-873 10 121 0-267
50 2880 1:826 20 135 0-305
53: 0 1-963 30 159 0-348
55 — 303 1-957 40 207 0-400
60 — 357 1-905 50 335 0-476
65 — 168 1:872 60 45] 0-604 |
70 —108 1 855 70 41 0°657
75 —~100 1842 75 29 0-662
24 A. W. Witkowski on the
é=—110, t=163. d= —95, t=178.
30° Cy
P 10 5
10 108 0-264
20 115 0°298
30 119 0:333
40 136 0°370
50 204 0-419
60 444 0-509
70 152 0-607
75 75 0-623
6=—105, t=168. 0=—50, t=223.
Pp. 1020” Co Dp. 1080” Cp
rola Yo Yan
10 89 0-261 10 19 (0-244
20 90 0-288 20 22 0-252
30 93 0-317 30 26 0-262
40 100 0-346 40 30 0-274
50 145 0-382 50 32 0-286
60 279 0:445 60 32 0-300
70 293 0°539 70 30 0312
75 115 0:564 80 29 0-324
90 28 0:336
100 | or 0:347
G=—100, t=1738. 6=0, t=273.
: 108 Q7v Co. Ds ca sv Cy.
P 10 5B P 10 Se P
10 77 0-258 10 9 O20 ss
20 79 0-283 20 9 0-245
30 80 (Q-309 30 9 0-250
40 81 0-834 40 9 0:254
50 107 0-363 50 9 0-259
60 180 0:408 60 9 0 264
70 174 0-469 70 9 0268
80 97 0-512 80 = 0:273
90 32 0-532 90 9 0-277
100 13 0:538 160 9 0-282
Thermodynamic Properties of Azr. 25
The variations of the specific heat cp), as revealed by these
tables, have been represented in « graphical form on PI. II.
It will be remarked that with increasing pressure the specific
heat increases, the more considerably the lower the tempe-
rature of the corresponding isothermal. In the vicinity of the
critical temperature these increments are largest, and in the
critical state itself the specific heat tends to infinity. This
might have been anticipated, on the ground of equation (2), § 2,
because oP =() in the critical state, whilst ¢. and OP vemain
et Ov PLO
finite.
The most interesting feature of the diagram (Plate II.) is
that at temperatures above the critical the specific heat rises
with increasing pressure only to a maximum value, corre-
sponding to a certain limiting pressure (which isa function of
the temperature). Under pressures exceeding this limiting
value the specific heat remains nearly constant, with bat a
slight tendency to decrease. The lower the temperature, the
smaller is this limiting pressure, and the more marked the
transition from increase to approximate constancy of the
specific heat. It would seem as if these pressures marked a
limit between truly gaseous states and a gaso-fluid condition
of matter, in which the intrinsic pressures attain a prepon-
derance against which the external pressure has but little
influence. It is interesting to note that the curves of the
coefficient of expansion a, under constant pressure (Part I.,
plate i.), show similar bends for pressures which are not
much different from the limiting pressures of the specific-heat
eurves. We shall see that neither the curves of the specific
heat at constant volume, nor those of the coefficient of expan-
sion at constant volume, show any trace of bends of this sort.
13. Itisa more difficult matter to calculate the variations
of the specific heat at constant volume. At first sight it would
seem easiest to apply the equation (3), § 2:—
0% t OD
ov) Jim OC
But we shall see that the variations of pressure at constant
volume are so nearly proportional to those of the temperature,
that the calculation of the second differential coefticient
2
or is practically impossible.
In order to find the variations of pressure of air of any
density kept ata constant volume, I shall refer once more to
26 A. W. Witkowski on the
the results obtained in Part I. Through the origin of the
diagram of compressibility (Part I., plate ii.) draw any
straight line. The intersections of it with the isothermals
pv=const. mark evidently a series of pressures corresponding
to the respective temperatures, and satisfying the condition
v=const.; it is supposed that v=1 when @=0°, and the
pressure is atmospheric. Using the original diagram of pv,
{ determined in this manner the constant volume relation of
Se if 1
p and @ for several densities of air, from v= io to P= 900°
The results are given in the following table.
ss Z a a Das = X
a 10 20 25 30 40 50
6. Pressure in atmospheres.
+100 | 13-680 27°41 34°28 41°17 55:08 | 69:09
+ 16 | 10542 21-01 26°20 31:38 41-70
0 9-949 19:80 24-68 29°54 39°20 48 74
— 30d 8640 17-12 21°30 25°44 33°67 41-70
— 785} 7-005 13°76 17°05 20°30 26°64 02°75
—103°5| 6°065 11-82 14:60 17-28 22°48 27°40
—130 5°055 9°71 11-90 14:90 17-99 21°65
—135 4°850 9°26 11:32 13°29 16°98 20°35
—140 4-668 8:87 10°82 12°65 16°14 19°29
—145 4°466 8°43 10°25 11°95 15°10 17:97
iG abet te. ee pie)
mie 60 80 100. |) 120° | 150 |" 200;
0. Pressure in atmospheres.
+100 83°28 | 112°14
+ 16 62°34 83:04 | 10392 | 125-41
0 58°53 77°52 96°80 | 116°40
— 3d 49°70 65 50 81:00 96-40 | 120-00
— 785) 38°65 50°06 60°80 71-02 85°83 | 110-80
—103°5} 32°10 40°88 48°75 56°U9 66:13 81°50
—1380 25:08 31-03 39°90 39°95 44-80 50°32
—135 23°40 28°74 32°90 | 36:18 39°60 43°44
— 140 22 08 26-74 30°35 32°84 35°39 37°80
—145 20°40 24°50 27 35 29-12 30°30 |.
Thermodynamic Properties of Air. 27
In fig. 4 we find a graphical representation of these results
by means of curves of equal density.
Ke
ave On ops
SAIC ACNE =
PETER {Sees ENR es
HAY ye
Some time ago Ramsay and Young announced an important
generalization of the law of Charles, according to which the
constant-volume relation of pressure and temper ature (in the
28 - A, W. Witkowski on the
gaseous and liquid condition of matter alike) would be
a linear one, at any density. It is now known that a law
of that kind is not generally true, or that it holds good
only approximately. Yet it is remarkable how nearly it is
fulfilled in the case of atmospheric air at widely different
temperatures and densities. The curves of fig. 4 depart
only insignificantly from straight lines. But none of them
cuts the axis of abscissee at the point —273°—so often
spoken of (by a curious confusion of ideas) as absolute zero
—except, perhaps, those corresponding to very low densities ;
the pressure of dense gas decreases far more rapidly than
that.
The constant-volume relation p= F (6) willbe perhaps more
clearly expressed by introducing the pressure-coefficient B of
expansion defined by the equation
| p=poll + Be),
po being the pressure exerted at 0° by the gas, when
compressed to a density p=— (unit of p= density at 0°
under atmospheric pressure). The values of $6 are as
follows :—
|
p= 20. 40. | 60. | 80.
100. | 120.
0. 100,000 x B.
+100 386 | 406 426 447
— 785) 387 409 431 452 474 496
—103°5| 389 412 435 457 480 501
— 130 392 416 439 462 484 505
— 140 394 420 444 467 490 513.
—145 396 424 449 472 495 517
The pressure coefficient does not vary much through a
range of 245°, provided the density be kept constant. An
increase of density causes it to augment rapidly. In con-
trast with the tortuous curves representing the coefficient
a, those of @ form a narrow nearly straight bundle, con-
verging approximately to one point, namely, @=0°00367
for p—
14, From what has been just said, it follows that
equation (3) is not suitable for calculating ¢,. I preferred to
Thermodynamic Properties of Air. 29
use for that purpose the relation
(st)
C= Cp + 18'714 LOFF
Op
Ov
z. e. to obtain the values of the specific heat at constant
volume by means of those of cp already calculated. To
simplify the calculation, put pu=7; then it follows that
>)
CDE ae
ov on
a
(30) = 35),
Op :
The values of ry; will be found easily by the preceding
therefore
G@—G—18- 714
section ; those of oa can be obtained by graphical differen-
tiation on the diagram of compressibility (Part I., pl. 1i.).
The necessary data and the results of this calculation are
collected in the following tables :—
O=—140, t=1383.
v. D. Cp- oF. or. Cy. 2.
a nee 4-668 | 0-305 | 0:0392 |—0-:0055] 0-219 | 1:39
= ee 8:87 | 0-385 | 0-083 |—0-0055] 0:278 | 1:38
= les | 1265 | 0-464 | 01363 |—0:0055] 0322 | 144
ao 19:29 | 0624 | 0:239 |—0:0059] 0-483 | 1:44
2: Gtk 22:08 | 0703 | 0306 |—0:0066| 0-457 | 1°54
ples 26-74 | 0-859 | 0-435 |—0-0078| 0-501 | 1:70
...| 80°35 | 1:021 | 0539 |—0-0104] 0535 | 1-91
30 A. W. Witkowski on the
6@=—135, t=138.
v. p- Cp. a of. Cv. 2.
ae 495 | 0-281 | 0038 |—0-0051! 0200 | 1-41
a bt se 926 | 0330 | 0-082 |—0-0051| 0227 | 1-45
= ae 13:29 | 0382 | 0131 |—0-0051| 0-254 | 1:50
= ras 20:35 | 0-490 | 0-234 |—0-0055| 0313 | 1:56
= ie 23-40 | 0546 | 0-295 |—0:0054| 0:334 | 164
= ace 28-74 | 0657 | 0-415 |—0-0086| 0361 | 1°82
1 || 3290 | 0-785 | 0530 |—0:0080] 0388 | 202
100
@= 130, Fe
v. Dp. Cp. or on. Cv. =
a - 5-055 | 0-265 | 00385 |—0:0043] 0-183 | 1-45
= pact: 9-71 | 0299 | 0-081 |—0-0043] 0201 | 1-49
= or 1400 | 0337 | 0-1276 |—0-:0043| 0-220 | 1-53
= Ree 21°65 | 0-416 | 02275 |—0:0045| 0-259 | 1-61
5 ae 25-08 | 0-458 | 0-287 |—00047) 0-270 | 1-70
= ns 31:03 | 0556 | 0-400 |—0-0056) 0306 | 1:82
0366 | 1:89
1...) 35:90 | 0694 | 0521 |—0-0062
100
Thermodynamic Properties of Air. 31
6=—103°5, t=169°5.
OP 07 op
Us : Cpe ——. es Go: on
P Pp 30 Op 4 Cy
1 Patiget “aye De rue
io" 6:065 0:249 0:0376 |—0:0027 1173 1:44
1
2 ae 11°82 0-264 0-077 |—0:0027| 0-180 1:47
ae 17-28 0:278 0-121 |—0:0027| 0-181 1-54
30
- shee 27-40 | 0:305 | 0-213 |—0-:0027| 0-186 | 1:64
= yer 3210 | 0320 | 0265 |—0-0028] 0185 | 1-73
= ee 40°88. | 0345 | 0-364 |—0-0027| 0-189 | 1:83
= ...| 4875 | 0871 | 0-491 |—0:0027| 0-172 | 2-16
@= —78'°5, #=194°5
v p Cp. or ae Cy -
= “eo 7-005 | 0-246 | 0037 |—0-0019| 0-174 | 1:42
= ie 13-76 | 0-256 | 0-076 |—0-0019| 0-177 | 1-45
= 20:30 | 0-266 | 0-118 |—0-0018) 0-178 | 1:50
= a 3275 | 0-286 | 0-208 |-00018| O-181 | 1:58
= ee 38°65 | 0-295 | 0257 |-0:0017) 0-180 | 1-64
= et 5006 | 0316 | 0354 |—0-0016| 0-187 | 1-69
= __| 60°80 | 0:389 | 0:473 |—0-0016) 0-184 | 1:84
32
A. W. Witkowski on the
d= —35, 1=238.
| 1
| OP o7 | Cp
Uv. E Cy. an —— - =e
: rely OP | 2
——}——— 5 ESN |
oo 8-640 | 0-242 | 0-037 |—0-0010! 0-171 | 1-42
1
oo 1712 | 0-248 | 0-075 |—0-0009) 0-174 | 1:43
=~ 25-44 | 0:255 | 0-116 |—0-0009| o-174 | 1-46
= coe 41-74 | 0-270 | 0-204 |—0-:0008! 0-178 | 1:52
Oo
= as 49°70 | 0-278 | 0-250 |—0:0007! o-181 | 1:54
Seta 65°50 | 0294 | 0-342 eee 0-190 | 1:55
30
ae 81:00 | 0-310 | 0-455 |—0-0005! 0-190 | 1:63
100 |
6=0, t=273
v. Dp. Cp. oP | Oo”. | Cy. oe
rely OP Cy
een eee ae
Teena 9°949 0-241 0:037 —0-00051 | 0-°170 | 1-42
1
Tee 19:80 | 0-245 | 0-075 |- 000051! 0-172 | 1:43
a wee 29°54 | 0250 | 0115 |-0:00050 0-178 | 144
ee 48-78 | 0-259 | 0-203 |—0:00036) 0-171 | 1-52
50 |
= 58°33 | 0-264 | 0-244 |—0:00025| 0-175 | 151
oe:
1 77-52 | 0272 | 0-342 |—0-00010! 0-175 | 1:56
ao |
ni 96-80 | 0-282 | 0-446 lo 0177 | 1-60
100 ° |
§ 15. These results prove conclusively that the specitic heat
at constant volume is a variable quantity. For increasing
pressure the quantity ¢, increases at all temperatures, the
more so the lower the temperature.
corresponding isothermals converge approximately to the
value 0°169, which is the specific heat of air of ordinary
density.
For small pressures the
Thermodynanuc Properties of Air. 33
To obtain a graphical representation of the variations of
c, I constructed isothermals of that quantity, considering it at
first as a function of the density 2 . These lines are curves,
turning their concave sides to the axis of abscisse. After-
wards I made the remark that a far simpler law results when
¢, is considered as a function of the pressure p. As shown
in fig. 5 the isothermals of c drawn on that supposition are
Fig. 5,
very nearly straight lines. This means that the increments
of the specific heat at constant volume are proportional to the
increments of pressure caused by increased density, tempera-
ture being kept constant. This empirical relation holds at all
temperatures between 0° and — 140°, and for all densities up
to the hundredfold of the ordinary density.
According to this diagram it is possible to express the values
of c» between the just mentioned limits by the following
linear equations :—
Phil. Mag. 8. 5. Vol. 42. No. 254. July 1896. D
34. A. W. Witkowski on the
6=—140°, ¢,=0°1694+0:0135 (p—1),
= —135°, ¢,=0°169+0-00722 (p—1),
0=—130°, ¢,=0:169 +0:00432 (p—1),
6= —10375,¢ =0°169+0-00056 (p—1),
6=— 78°'°5,¢c,=0°169 +0:00038 (p—1),
0=— 35°, ¢ =0°169 +0°00024 (p—T),
d= 08; ce, =0°169 + 000008 (p—1).
As regards the fact of the specific heat at constant volume
increasing with increasing pressure, these results are in
perfect accord with the direct determinations of Proi. Joly.
A numerical comparison cannot be made properly, because
these experiments have been performed at higher temperatures
(+50° in the mean). Yet the following result of Prof. Joly
may be quoted (J. c. page 99). The specific heat at constant
volume of compressed air, of absolute density =0 0205 (which
corresponds to a pressure of about 19:51 atmospheres), has
been found =0°1721 ; from the same experiments it is con-
cluded, that the value of c, under atmospheric pressure would
be 0°17154. From these numbers there results an increase
of cp, per atmosphere
= (0°1721—0°17154) : 18°51=0-00008.
Considering the difference of temperatures, this agrees well
with my own results.
Prof. Joly has also investigated the influence of the tempera-
ture on the specific heat ¢, of compressed gases. For carbonic
acid of absolute density 0°124 he finds c,=0°1971 at +50°,
and ¢c,=0°1894 at + 90°; this means an increase of specific
heat at low temperatures. He remarks also, that under small
pressures the intluence of the temperature is quite insensible.
All these conclusions are in perfect agreement with what has
been said above with regard to the specific heat of air.
§ 16. By means of the results arrived at in § 14 we may
now calculate the ratio k= ~ of the two specific heats; the
values of & are given in the tables of § 14 in the last column.
The general features of the variations of & will be best under-
stood if we consider & as a function of the temperature, at
constant volume, or at constant density. The corresponding
curves are shown in fig. 6.
With regard to the variability of k our knowledge is ex-
ceedingly scanty. ‘The only fact known till now in this respect
Thermodynamic Properties of Air. 39
is the experimental result of Wiillner, that the ratio k increases
slightly with decreasing temperature (from 1:40289 at + 100°
to 1°40526 at 0°, for air of ordinary density). An increase of
z eee nes
see Aes So
Le bee lel
IC %
\-F aed
7 Be
1
ee a ene
ie
ee Se
|
f
this kind is also shown by the curves of fig. 6. But, more-
over, it follows, that with decreasing temperature & reaches
a maximum value, which is followed by an abrupt fall. At
all densities for which & has been calculated the maximum of
D2
36 Thermodynamic Properties of Air.
k corresponds to a temperature of about —120°. At ordinary
temperatures the temperature-variations of k are exceedingly
minute, for all degrees of condensation of the gas. At
temperatures approaching the critical the increments of the
ratio are the more marked the greater the density ; in the
critical state & is infinite.
With the aid of the diagram (fig. 6) the following table of
values of cp : c, has been prepared :—
a= 10 20. 30 50. 60. 80. 100. |
0. Values of cp: Cy.
@ | 142 | 148 | 1-44) 451 153 | 155 160
= 99 | 1-42 |. 1-43 | 145 1 P51 153 | 1:55 161
249 | 1:42 | 1-43) 1-46 BS Pe aeons 1-65
S60 42 — 144 AG) clases 158 | 161 1-72 |
— g0 | 142 | 145 | 1:50 | 158 | 1:64 <) deouamneee
00 W464 47 sae aes 1-71 1:30 | 2-10
—120 | 1°45 149° | 156 | 1:67") ae79: |e
—140 | 1:38 1-41 1-46 150 | 1:54 | 1-70 | 1-80
Although the general course of the curves of & is very clearly
marked, yet some irregularities manifest themselves in the
diagram of fig. 6, chiefly at the highest and lowest tempera-
tures, where the graphical calculation has been also less
certain. I have not attempted to correct them arbitrarily, but
I drew the curves as near as possible along the calculated
points (marked on the diagram by dots).
§ 17. In concluding I wish to state expressly that the
absolute numerical values of the several thermodynamic
quantities given in §§ 12, 14, and 16, being obtained by a
graphical method of calculation, cannot claim a degree of
exactness comparable with that of direct experimental results:
Notwithstanding this I considered it useful to spare neither
time nor trouble to obtain them, in order to throw some light
on these important and wholly unexplored relations. :
During the preparation of the manuscript of the present
paper I became acquainted with a memoir of Ser. Silvio
Lussano (Nuovo Cimento, 1894, ser. iii. tom. xxxvi. pp. 1, 70,
130 ; 1895, ser. 11. tom. i1., p. 8327; “ Sul calore specifico dei
gas”) on the influence of pressure and temperature on the
specific heat of gases at constant pressure. The results of
Sgr. Lussano, being obtained at higher temperatures, are not
Melting-points of Aluminium, Silver, Gold, Copper, §c. 37
directly comparable with mine. As regards the influence of
pressure Ser. Lussano finds, in the limits of pressures em-
ployed, an increase of ¢, with increasing pressure ; but the
influence of the temperature is just contrary to that which I
have found at low temperatures. It cannot be doubted that
at sufficiently low temperatures the specific heat c, increases
with decreasing temperature, since in the critical state its
value is positive infinity. This point must therefore be
reserved for further researches, the more so, as the increase
of c, with increasing temperature, at higher temperatures, has
been proved long ago in some gases (carbon dioxide &c.)
without any doubt.
Physical Laboratory of the Yaghellonian University,
Cracow, November, 1895.
Il. Afelting-points of Aluminium, Silver, Gold, Copper, and
Platinum. ByS.W. Hotman, with R. R. Lawrence and
L. Barr*.
HE following melting-points are offered as provisional
only, but with the belief that they are more reliable
than previous data. The absolute values depend in part upon
the assumption of 1072° C. as the melting-point of pure gold,
the recent determination of Holborn and Wien at the Reichs-
anstalt. Should that datum, however, be shown to require
revision, the validity of the present measurements would not
be impaired, but new values of the melting-points could be
readily computed from them which would be consistent with
the better value for gold.
Al. Ag. Au. Cu. Pt:
60° SOs tee (LO G2—-C. | 1095° 1760°
Assumed.
The Pure Metals used were of a high degree of fineness,
except unfortunately the platinum.
The gold contained less than 0-01 per cent. total impurities,
these being, if any, probably minute traces of silver and pla-
tinum. It was obtained as part of a specially prepared lot
from the United States Assay Office in New York through
the courtesy of Professor H. G. Torrey, upon whose authority
the above statement is made. The purity was at least as great
as the best “proof” metal used at the United States or London
mints.
* From an advance proof of the Proceedings of the American Academy
vol. xxxi. (n. s. xxiii.) p. 218, communicated by Prof. Holman,
38 . Messrs. Hohnan, Lawrence, and Barr on the
The silver was from the same source and equally pure.
The aluminium was manufactured and given by the Pitts-
burg Reduction Company, of Pittsburg, Penn., and was stated
by Mr. Alfred E. Hunt, President of the Company, to con-
tain but 0°07 per cent. of impurity, consisting entirely of
silicon.
The platinum was the ordinary platinum wire supplied by
Carpentier, of Paris, with his Le Chatelier thermo-electric
pyrometers. It presumably contained 0°5 per cent. or more
of impurity.
The copper was electrolytically produced, and was from
the Lake Superior region. It was kindly given by Mr.
Maurice B. Patch, of the Buffalo Smelting Company, Buffalo,
N.Y., who stated that it showed by analysis 99°99 + per cent.
_of Cu, and contained no Ag, As, or 8, and only 0°0002 per
cent. of Fe.
The Less Pure Metals —Partly for the purpose of testing
the effect of impurities, other samples of gold and copper
were employed with the results stated later. These were :-—
Dentists’ Gold.—This was a gold-foil employed by dentists,
purchased as being of good quality.
Ingot Copper.—This was also from Mr. Patch, of the Buffalo
Smelting Company, who gave its analysis as :—
Cte 4) th ye oe SE ee
AD Song onat ae eee mn ees
Ass irs Yi iicesteeeme ay. mane
dale ae eer ee ct
Be: a Sie Rae eens
Oe is tm eee ae es ali
100-001
This was the company’s “ regular run” of copper.
Commercial Electrolytic Copper.—A sample of commercial
electrolytic rolled sheet copper, furnished by a friend, and not
assumed to be of unusual purity. It was probably Montana
copper.
Commercial Hard-drawn Copper Wire-—This was from a
lot purchased for electrical testing purposes, which showed a —
specific resistance of 0°1440 international ohm per metre-
gram, or an electrical conductivity of about 98-3 per cent.
referred to Matthiessen’s copper.
Methods and Apparatus.—The method consists in measuring
_the thermal electromotive force of a couple ‘composed of one
wire of platinum and the other of a-10-per-cent. rhodo-
platinum alloy. One junction is immersed in the melting or
Melting-points of Aluminium, Silver, Gold, Copper, Sc. 39
solidifying metal, and the other surrounded by ice. The wire
was that furnished by Carpentier, of Paris (through Queen &
Co., of Philadelphia), with the Le Chatelier pyrometer.
The E.M.F. was measured in microvolts (international) by
the Poggendorff null method modified for rapid and con-
venient “working. The disposition of apparatus is shown in
fig. 1. Bisa battery of sufficiently steady E.M.F. (A single
Fie. 1.
Samson-Leclanché cell was entirely satisfactory.) In direct
circuit with this were two water rheostats, W, in series; an
ammeter, A, which was a Weston voltmeter (No. 395) with
the calibrating coil only in use ; and a manganine wire resist-
ance, a,b, c,d, divided into sections, each of accurately known
resistance. T is the thermo-couple connected through a sen-
sitive galvanometer, G, and key to any desired sections of the
coil a, b, c,d. The water rheostats were of about 100 okms
and 8 ohms respectively, and the vertical motion of their
plungers thus served to give a coarse and fine adjustment to
the resistance in the circuit. The current could thus be
promptly and closely adjusted. The voltmeter was one of
the type having a “calibrating coil;”” that is, one having a
connexion by means of which the nsual high resistance series
eoil could be cut out, leaving its resistance about 117 ohms.
Any of the Weston voltmeters with a special connexion made
to effect that res: lt would answer equally well. The volt-
meter was preferred to a mil-ammeter as probably more
reliable. The instrament was carefully and repeatedly cali-
brated throughout its scale by an application of the Poggen-
dorff method, measuring by the Clark cell the drop of potential
ina known resistance through which a current was passing in
40 Messrs. Holman, Lawrence, and Barr on the
series with the ammeter, and at the same instant reading the
ammeter. The calibrations at different times were checked
at the same point, with an average deviation of only a few
hundredths of one per cent. A test for temperature error
showed a change of but O°1 per cent. for a change of
15° C.; so that, as the temperature during the work was
constant within a few degrees, no correction was needed.
The manganine coil, fig. 2, consisted of about 16 feet of
No. 20 wire, had a total resistance Fie. 2.
of about 8°8 ohms, and was divided
into nine sections by copper po-
tential wires leading into different
points along the coil. These sec-
tions were so designed that, by
suitably shiftmg the connexions
along a, b, ce, &c., any thermal
E.M.F. which was to be measured
could be balanced by a current
which would deflect the ammeter
to a point between 90 and 140 di-
visions (readable to tenths)—cor-
responding to currents from 0-006
to 0:009 ampere roughly. The
coil was immersed directly in kero-
sene, and as its temperature-coefficient was but 0:001 per 1°C.,
the correction became very small. The relation and actual
resistance (international ohms) of the whole coil and its several
sections were repeatedly determined against a standard ohm
by the differential galvanometer, and checked by a modified
Wheatstone-bridge arrangement. These data were reliable
probably well within 0 05 per cent. throughout.
In the thermo-couple circuit, the sensitiveness necessary
in the galvanometer to give the smallest E.M.F. to 0-1 per
cent. was easily computed to be only about 7°7.10° (mm. defi.
at 1 m. per ampere ord/c). The instrument as actually used
exceeded this requirement, averaging about5.10". Its resist-
ance, all in series, was 14°3 ohms.
The cold junction ¢ of the thermo-couple was fused together
in an oxyhydrogen flame. The wires, insulated from each
other by having one strung through a very fine glass tube,
were run down another tube of about 4 inch inside diameter
and 8 or 10 inches long. This tube was fused together at the
bottom and top, as well as at some intermediate points, and
when in use was always packed in a double vessel of cracked
ice, as shown in fig. 3.
The intermediate junctions from which the copper leads
ei
i
Melting-points of Aluminium, Silver, Gold, Copper, we 41
went off to galvanometer and key were soldered. They were
kept at an equal temperature by the device of enclosing them
in a stoppered glass tube, which was packed with hair-felt
into a one-inch hole in a five-inch cube of cast iron. ‘This
arrangement was entirely satisfactory, but seems to possess
no mnaterial advantage over making the junction of the copper
leads with the Pt and Pt-Rh serve as the cold junction, and
immersing this in ice as in fig. 3, except that the latter makes
a rather more bulky mass to insert in the ice.
The wires were also fused together at the hot junction
except when this was unnecessary on account of their being
immersed in metal. It may be noted here that, as a null
method was employed, the total resistance of the thermal
circuit, or any variation in it, was without effect other than a
corresponding change in sensitiveness.
As the hot junction was to be immersed in vapour of sulphur
as one of the known temperatures, the following apparatus was
designed for this purpose. It is substantially the sulphur
boiling-point apparatus of Griffiths, and is shown in fig. 4.
Pia 3. Fie. 4.
A glass tube, A, similar to the Victor Meyer vapour-density
tube, 16 inches long and with a two-inch bulb, was provided
with an asbestos jacket and hood, B, B, The upper few
inches of the tube were wound with a spiral wire spring, 8,
which rendered this part efficient as a condenser. The top was
closed with a layer of asbestos, Two overlapping diaphragms
of asbestos were inserted in the tube at Dand E. ‘The couple
passed downward through a glass tube to the asbestos tubular
42. Messrs. Holman, Lawrence, and Barr on the
hood, ©, which served as an umbrella to shed the dripping
cooler sulphur, and as a radiation-screen. The hood, how-
ever, had openings top and bottom for the free circulation of
the vapour. An asbestos diaphragm, H, upon which the bulb
rested, reduced the chances of superheating.
For the melting metals, after trial of several devices, the
one shown in fig. 5 (of exactly half size) was settled upon as
Fig. 5.
oe
proving very satisfactory. The crucible, C (usually of fire-
clay), is supported by clay blocks in the double-walled fire-
clay furnace, F. A carbon block, H, channelled to fit the
crucible, forms its cover, and a carbon diaphragm, D, inside
the crucible serves to support some powdered carbon shown
by the dotted mass. The object of these carbon parts was to
prevent oxidation of molten metals, and they proved very
effective in the case of aluminium, silver, and copper. GG was
an asbestos diaphragm supporting a non-conducting layer of
fibrous asbestos, AA. The temperature was controlled by
the blast-lamp B. The clay crucible was one inch in diameter
outside, and the amount of metal employed ranged from 11
grams (gold) to 385 grams (copper). Larger amounts might
be advantageous, but with 30 to 385 grams it was easily
possible to obtain a constant indication for five minutes during
the melting or solidifying of copper. No difficulty whatever
Melting-points of Aluminium, Silver, Gold, Copper, §c. 43
was experienced with this arrangement with silver, gold, or
copper. Withaluminium, however, a peculiar action ‘occurred,
the cause of which in the time available for investigation
could not be determined beyond doubt. The phenomenon
was that after a few minutes of constant temperature at the
melting-point, the indication of the thermo-couple fell off with
increasing rapidity, On w ithdrawing the couple, cleaning it,
or clipping it off and restoring it to place, the melting metal
meanwhile being untouched, the indications returned fo their
original high value. The apparent explanation was the for-
mation of a 1 slag between the wires; but this was not entirely
satisfactory. ‘The use of a plumbago crucible in place of the
clay and an entirely fresh lot of aluminium did not remove
the phenomenon, and gave the same initial readings, which,
it could not be doubted, were the ones corresponding to the
melting-point. The fusion of the aluminium was, however,
the least sharply defined of all the metals used.
The fusion of platinum was, of course, differently effected.
For this the two wires of the couple were laid close together
on a piece of lime. An oxyhydrogen flame was then directed
upon their ends and the platinum fused into a globule which
with care could be made to travel slowly up the wire. There
was no difficulty in obtaining steady temperatures for a sufh-
cient period to make the necessary readings, and check results
to 0-1 per cent. were obtained on different days.
The galvanometer, keys, coils, and all junctions of dis-
similar metals, were, so far as possible, covered with boxes
of wood, pasteboard, or asbestos to maintain uniformity of
temperature, and thus miniinize local thermo-electric disturb-
ances. With this precaution, no sensible trouble from that
source was experienced.
The procedure is as follows :—To take the observation for
vapour of sulphur, for instance, the hot and cold junctions
are exposed as described. After a sufficient time the main
circuit is closed, the thermal circuit is connected to a suitable
part of a, b, c, d, and the rheostats W are adjusted until on
pressing the key no deflexion occurs in the galvanometer G.
At this instant A is read, which gives the current ¢ in the
main circuit. The adjustment is disturbed and remade a
number of times, and the resulting readings should check to
the nearest tenth of a division of A, provided the metal has
reached a steady state of temperature.
By this adjustment the drop of potential er due to the
current ¢ amperes in the part 7 ohms of the resistance a, 0, ¢, d,
spanned by the thermal circuit is made equal to the total re-
sultant .M.F. in the thermal circuit. The latter, which will be
44 Messrs. Holman, Lawrence, and Barr on the
denoted by =*e or XGe, is the algebraic sum of the thermal
H}.M.F. proper of the junctions, te the Thomson E.M.F. in the
wires, and of any “stray” or local thermal E.M.F. in the
circuit. The last was found to be negligible throughout the
work.
To observe the melting-point, the furnace containing the
metal is heated more or less rapidly until the melting-point is
approached. The blast-lamp is then adjusted to give a slowly
rising temperature. The thermal circuit, w ith the couple
previously fused into the metal, is connected to a suitable
section of a, b, c, d. The rheostats are continually adjusted
for zero deflexions of the galvanometer G, and the corre-
sponding readings of A are taken. These will show gradually
increasing values; but the rise will presently be interrupted
by a series of constant readings, after which the readings will
again steadily increase. This period of constant, or nearly
constant, readings of A is that in which the latent heat of
fusion is being absorbed, and its duration is frequently several
minutes. The temper ature at that time is, of course, that of the
melting-point. The reverse process, starting w ith the metal
ina molten state and cooling it oradually, shows a similar
period of selidification.
No difference was discovered between the ascending and
descending readings when a sufficient amount of the ‘metal
and a slow rate of ‘heating and cooliig were employed. With
small amounts the steady reading was more or less masked by
phenomena which were clearly due to inequality in distri-
bution of temperature throughout the mass of mixed liquid
and solid metal. In the case of aluminium, however, some-
thing more than this irregularity was observed, as elsewhere
stated, but the time at command did not permit a study
beyond the point of satisfying ourselves that the point observed
was unquestionably the true melting-point.
This work was done chiefly as the thesis work of Messrs.
Lawrence and Barr. The efficient assistance of Mr. C. L.
Norton contributed materially to its progress and success
The computation of temper atures t trom the observed electro-
motive forces } ¢ involves a knowledge of the function con-
necting the two, 7. e. of the function
se=ft), or. t= (Se)
This problem has been elsewhere discussed by one of the
authors of this paper *.
In that article two interpolation formulz were developed.
They were respectively of the following forms, applying to the
%* Phil. Mag. xli. p. 465,
Melting-points of Aluminium, Silver, Gold, Copper, §c. 45
case where one junction of the couple is kept at 0° C., and the
other is at any other temperature ¢ C., or Tr=¢+ 273° absolute;
m and n are constants, different for the two expressions;
>i e denotes the resultant thermal E.M.F. of the circuit, viz.
that which is the object of direct measurement. The first,
called the exponential equation, is
yye=mr—B (where B=mr, =m x 273").
The second expression, called the logarithmic equation, is
t n t
yea, of low >,e=—n log t + log mz.
Both formule have been applied to the data of the present
investigation given in Table 1., with results shown below.
The Avenarius formula has also been applied for purposes of
comparison. a
To evaluate the constants m and n of the exponential
equation (for method, consult the paper referred to) it is
necessary to have values of je at three known temperatures.
Of these, however, one may be 3,¢=0, at T= 278°, 2, e. with
both junctions in ice. It therefore remains to fix upon two
other temperatures between which to interpolate, or, in other
words, two other temperatures which shall be assumed as
known. In looking over the ground, it seemed that the
boiling-point of sulphur, being so high and so accurately
determined by Callendar and Griffiths*,
444-53 + 0°082(H —760),
was preeminently one of these points. The other must be
much higher, and the melting-point of pure gold seemed to
be almost, if not quite, the only one upon which reliance could
be placed.
Apart from freedom from oxidation and its conveniently
high point of fusion, gold seemed the more suitable because its
melting-point had recently been so carefully measured by
Holborn and Wien, and because the metal could be obtained
of the necessary purity. Add to these considerations the fact
that its melting-point in a state of at least fairly high purity
has been measured by more experimenters than that of any
other high melting metal, so that it serves as an excellent
connecting link between their work, and we have claims which
no other substances can at present offer. The fusion-point of
gold was therefore chosen as the second reference or calibration
temperature. As to the figure to be assumed as the melting-
* Phil. Trans, clxxxii. pp. 119, 157 (1891).
46 Messrs. Holman, Lawrence, and Barr on the
point of gold, there is room for differences of opinion. The
claims of the work of Holborn and Wien, supported to some
extent by considerations advanced by Barus*, lend much
weight to the conclusion that Violle’s value of 1035° is con-
siderably too low. Granting this, and in the absence of
sufficient basis for the assignment of weights to the work of
divers other investigators, the simplest and best step seemed
to be to adopt provisionally, without modification, Holborn
and Wien’s value, 1072°.
These two points settled upon, the constants m and n could
be computed as elsewhere described, and the equation trans-
posed to deduce other values of ¢ from observed values of S(e.
Representing mtj by 8, a constant, the equation for the tem-
perature as a function of S}e takes the form
ef SFT a)
(ss af 228 _oi30,
m
which is, of course, easily solved by logarithms.
The data given in Table I. yield the values m = 0°3901,
n = 1:488, 8 = 1645, in international microvolts and degrees
Centigrade, so that
Ste=0-3901 71645, or t= 14884 / 20° t 1645 _
03901
From these the temperatures of column 6 have been com-
puted.
The constants of the logarithmic formula have been com-
puted from the same data for sulphur and gold, the method
being sufficiently obvious. The equation becomes
Lie= 249650 1,
The corresponding melting- and boiling-points are given in
Table I. column 7.
Substitution of the same data in the Avenarius equation
yields
dhe=(th—le) {9°7335 + 00048449 (t, + t,)}.
The corresponding melting- and boiling-points are given in
column 5.
Provisional Values of Melting-points.
In the paper referred to it was shown, Ist, that the loga-
rithmic expression fitted the Barus comparisons of the irido-
platinum couple with the air thermometer within the limits
400° to 1200° C. with no sensible systematic error; 2nd, that
* Am, Jour. Sci. xlviii. p. 336,
Melting-points of Aluminium, Silver, Gold, Copper, Se. 47
the exponential equation similarly fitted the Holborn and Wien
comparison of the rhodo-platinum couple with the air thermo-
meter within the same limits; 3rd, that the exponential
equation diverged systematically, although slightly, from the
Barus data, and the logarithmic from the Holborn and Wien
data, by about equal and opposite amounts both inside and
outside these limits, but much more markedly between 0° and
400° than at higher points.
TABLE I.
Melting-points.
[
|
| Temperatures.
De
Date. | Subst. icru- ;
7 ae chy Assumed| From From | From tet, | Provis-
as Cor- | Aven. | Expon. |Log Eq.| — 5 ional
Beck. NetideGac \- gute. ti. Values.
3-2 |H,O | 8853 | 99-64
4-10 |H,O | 890-4 | 10057
888-1 |{100-10}} 874] 9t7 | 1073 | 995
©,,H,| 2213 | 2183
3-23 |C,,H,| 2224 | 218-9
C,,H.| 2216 | 218-2
9218 |[218°5] | 2066 | 211-4 | 2224 | 2169
3-? |S 5287 | 444°7
3-22 |S 5289 | 445:2
3-29 |S 5287 | 4445
5288 | [444-8]
4-24|Cu | 16463
1095- 1095:0 | 1096-5 | 1095-5 1095
4-99 |Au | 16002 | [lo72]| — se = = | (1072)
4-299 | Ae | 14093 - 975: 972 | 969: | 9705 | 970
5-2/3) Pt | 30313 — | 1695 | 1735: | 1783 | 1759 | 1760
ae et |) e638" 665°5 | 6625 | 656-2 | 6594 | 660
Aven. B(¢=(t,—t,) {9°7335 +0:0048449(¢,+¢,)}.
Exp. 2fe=0-3901 7/185 — 1645.
Log. )e=249655 726.
Inspection of columns 6 and 7, Table I., will show that the
computed boiling-points of water and napthalin by the ex-
ponential and logarithmic equations depart widely from the
known temperatures in opposite directions, by about equal
amounts, and in the directions according with the departures
from the Barus and Holborn and Wien data. Also, that the
differences between the computed melting-points intermediate
48 Messrs. Holman, Lawrence, and Barr on the
between sulphur and gold differ but slightly by the two
formule, thus confirming the former conclusions. It is
obvious, therefore, that although either of the two formule
would yield fairly good interpolations for Al, Ag, and Cu,
yet that a mean between the two would probably quite
nearly offset against each other the systematic errors of
the respective equations. This is also true in the dangerous
process of extrapolation for the platinum melting-point, where
the chances of error in the result seem to be probably very
much reduced by averaging. The means of the melting-
points computed by the exponential and logarithmic equations
are, therefore, regarded as the nearest available approxima-
tions, and the round numbers of column 9 are adopted as
provisional values to represent the results of the work.
Comparison of the results of the Avenarius formula,
column 5, will show that they depart widely from the others
in the direction which would have been anticipated from the
conclusions of the previous paper, thus further strengthening
those inferences.
In addition to the foregoing, the melting-points of three
other samples of copper and one other of gold were measured.
The gold was dentists’ gold “ foil,’ purchased in Boston.
This is usuaily classed as “‘ very nearly pure,” but its analysis
was not known. No special interest, therefore, attaches to it
beyond the indication that it gives of the sign and order of
magnitude of the error (about —4°) which would be introduced
by the ase of such gold in the calibration of the Le Chatelier
pyrometer, or in similar ways*. ‘The melting-point was
found to be 1068°.
The four coppers yielded the appended results :—
TABLE II.
Be Melting- | Purity of
microvolts.| poits, C.| Metal, Description.
0
IK
16463 1095-0 99°99+ | Electrolytic. Probably Lake Superior
copper, Buffalo Smelting Co.
16448 10943 99:83 Ordinary ingot. Same source.
16456 10947 Unknown. | Electrolytic. Probably from Montana.
16446 10942 | Unknown.| Commercial hard drawn wire from
Washburn and Moen Co. Sp. Elect.
Conductivity (referred to Matthiessen
value) 98°3 per cent.
* Holman, Calibration of the Le Chatelier Thermo-electric Pyrometer.
See Proceedings of the American Academy, xxxl. (.s, xxiil.), p. 234.
Meliing-points of Aluminium, Silver, Gold, Copper, &c. 49
The concordance of these results on various coppers, together
with the completely satisfactory behaviour of the metal in
fusion, and the ease and cheapness of obtaining the metal of
a very high grade of fineness, suggest the decided availability
of copper in a direct study of high temperatures or melting-
points by the gas-thermometer. A large mass of the metal
could be employed, and a constant and uniform temperature
for a protracted period thus secured for the bulb of the
gas-thermometer, or for other apparatus immersed in the
molten or solidifying material. There are unfortunately too
few substances which fulfil even these requirements. An
added merit lies in the nearness to the gold melting-point,
enabling the two to be satisfactorily connected by some means
of relative measurement.
It also appears that the use of good commercial copper
would introduce sensibly less error (3° less) into the calibra-
tion of the Le Chatelier pyrometer than the use of the
‘“‘dentists’ gold”’ above tested, which is as good metal as
would readily be obtained in the market by most observers.
Reliability of the Results —The points involved are :—
Instrumental errors.
Purity of the metal.
Was the observed point the real melting-point ?
Validity of the interpolation equation.
Error in the assumed melting-point of gold and boiling-
point of sulphur.
The investigation was planned and the apparatus arranged
with the intention of reducing the combined instrumental
errors below one-tenth of one per cent. in the measurement
of Le above 200° C. ‘Tests, check measurements, and a dis-
cussion of the sources of error, unnecessary to detail here,
have given satisfactory demonstration that an even higher
accuracy than this was attained. So far, therefore, as constant
or variable instrumental errors are concerned, it is believed
that no error beyond 0°5 to 1° C. exists in the results, while
probably this estimate is large.
The error from impurities must have been exceptionally
small, as the analysis of the metals indicates. Some impurities
from alloying with the platinum and rhodium of the thermo-
couple must have entered during the experimenting, but
as results at different stages of the work checked those
obtained upon the first use of the metal, and as renewals of
the metal made no difference in readings beyond the limits of
other variations (about 5 parts in 10,000), the error from this
source must have been negligible.
In the case of platinum the metal at command was un-
Phil. Mag. 8. 5. Vol. 42. No. 254. July 1896.
50 Messrs. Holman, Lawrence, and Barr on the
fortunately not of known composition, nor was it possible at
the time to obtain any whose purity was known. An analysis
of the wire used may perhaps be obtained later, and it is
hoped to carry out further measurements with the better
platinum now obtainable through the recent advances made
in its manufacture in Germany and England. |
The aluminium was of a very high grade, but it is thought
that still better may be obtained, and the peculiar occurrence
attending its melting-point measurements should be further
investigated. ,
The actual effect of the small impurities cannot be numeric-
ally estimated, but must have been inconsiderable except for
platinum, where the error probably has the positive sign.
As to the third point, there was no reasonable doubt left in
the minds of the observers that the observed temperatures
were sensibly the melting-points. Except as noted for
aluminium, the readings with rising and falling temperatures
did not exceed about one partin one thousand. Also entirely
independent observations on separate days, and with renewals
of the metals in some cases, were equally concordant. The
average difference was much less than the error of reading
the ammeter. As an example of the concordance, and at the
same time as showing the homogeneity of the thermo wire,
three calibrations in sulphur are quoted in Table ILI.
Tas eE ITI.
Computed :
Date. Le, microvolts. | Temperature of 4 none edie
Sulphur. :
March ?.....c00+ 5287 444-73 5290
March 22 ......... 5289 44518 5287
March 29 ......... 5287 444-53 5293
Between these observations a considerable length of the
wires was necessarily clipped off. Reduced to a common
temperature of 445°, the maximum difference is but six micro-
volts in 5290, or 0-11 per cent., while the average deviation
of a single observation is but 0°02 microvolt, or 0:04 per
cent., and of the mean but 0°02/ 73 =0°012 microvolt, or
0024 per cent. At higher temperatures the discrepancy
was even smaller.
The validity of the interpolation formule has been already
Melting-points of Aluminium, Silver, Gold, Copper, &c. 51
discussed. A statement of the extreme error which may have
been introduced into the results by this source should, how-
ever, be added. ‘This is believed to be for aluminium less than
+29, for silver less than +2°, for copper less than +0°5°,
and for platinum less than + 10°.
Comparison with the temperatures computed by the Ave-
narius equation shows errors by the latter to be about 1°5
times as great for water and naphthalin, and of the same
signs. It is therefore much less reliable, especially for the
platinum temperature, and no weight is attached to its results.
Melting-points by various Authorities—A collection of these
is given in Table IV. Hxcept in the case of the Barus data,
the results are set down directly as given by their authors.
A further discussion of these with reference to the purity of
the metals used, and the characteristic errors of the methods
employed, would doubtless prove instructive, and might partly
remove or account for some of the apparent discrepancies,—
a task which will perhaps be undertaken later.
TABLE IV.
Metals.
Authority. Date. | Method.
Ale) Agee Ae | Cue
|
(e) fe) le} Q
[Els ges i bee 1895 | Th.-el. 660 | 970 | [1072]| 1095
MBM Sire snt0 2 ct cna wime'sc TOPO 1 Sp. EUG. ah. 4c 954 | 1035 | 1054
WiedeWUr: 22.5 s......000- 1884 | Sp. Ht. | ... 960} ... | 1100
Le Chatelier ............ ra Dineel. |) G3on\e 7 035)
MaMlendar (26.4 50se00. 0 ine ee Sieh [945]| L037 |
Erhard and Schertel...| ... ae ue 954 | 1075
Barus, by Log. Eq. *...| 1894 | Th.-el. | 641 | 985 | 1090 | 1095
by Kg 3 oc... re ae Sep 986 | 1091 | 1096
Holborn and Wien ...| 1892 | Th.-el. Bic 968 | 1072 | 1082
Tfean of independent absolute measure-
ments, i.e. excluding H., L. & B.,| 641 | 964 | 1068 | 1083
Le C., and C.
N.B.—Values in brackets [ |] are those assumed by the
observers, and upon them their other values depend to a
greater or less extent.
Rogers Laboratory of Physics,
Massachusetts Institute of Technology,
Boston, October 1895.
* See discussion by Holman, Phil. Mag. xli. p. 465.
Hh 2
[ 52 ]
Ill. A Further Examination of the Edison Effect in Glow
Lamps. By J. A. ¥uemine, M.A., D.Sc., FERS. 2 fo
fessor of Electrical Engineering in University Coliege,
London*.
§1. jae experiments described in this paper had for their
object the further examination of an effect which can
be produced in certain forms of electric incandescence lamps
and to which attention was first drawn by Mr. Edison in 1884.
This effect may be generally described as follows :—A carbon
filament incandescence lamp having the ordinary horse-shoe
loop carbon has a metallic plate placed in the exhausted bulb,
the plate being carried on a platinum wire sealed through the
globe, and fixed so as to stand up between the legs of the
horse-shoe (see fig. 1). If the lamp is set in action at the
Fig. 1. Fig. 2.
usual incandescence by a continuous current of the proper
strength, and a suitable sensitive galvanometer is connected
between the insulated metal plate and the posztive terminal of
the lamp, it will in general be found to indicate a current of
some milliamperes flowing through it. The direction of this
current is from the positive electrode of the lamp through
the galvanometer to the insulated metal plate, or wire. When
the same galvanometer is connected between the negative pole
of the lamp and the middle plate, unless it is very sensitive, it
indicates no current. This effect was very carefully examined
* Communicated by the Physical Society: read March 27th, 1896.
On the Edison Effect in Glow Lamps. 53
by Mr. W. H. Preece in 1885, and he subjected it to a
systematic examination by the aid of a number of lamps
having such metal plates placed in various positions*. By
this observer a number of very interesting facts were col-
lected, the result of which was to point out the general nature
of the phenomenon. A sufficient number of new questions
were, however, suggested by the information so acquired to
invite further inquiry. Whilst confirming and re-examining
the experimental results obtained by Mr. Preece, some facts
that had previously escaped notice presented themselves,
which it is the object of this paper to describe.
§ 2. The first experiments were made with a lamp of the
form shown in fig. 2, similar to some used by Mr. Preece in
his experiments. A metal plate, generally of aluminium, is
supported on a platinum wire sealed through the bulb or
glass receiver, the plate being so fixed that its plane is at
right angles to the plane of the loop of the carbon, and as
nearly as possible midway between the legs. The plate there-
fore projects between the legs of the horse-shoe carbon, and
the carbon conductur arches over it without touching it.
The plate is entirely insulated from the carbon.
The preliminary experiments with this normal type of
middle-plate lamp consisted in determining the effective
potential-difference between the third terminal and one or
other of the two electrodes of the carbon filament when the
lamp was subjected to varying steady electromotive forces
sufficient to raise the temperature of the carbon from dull red
to vivid incandescence, and in determining the magnitude of
the current flowing in a circuit connecting the middle plate
with one or other of the electrodes of the lamp.
§ 3. Haperiment 1.--An ordinary carbun filament electric
lamp, having the horse-shoe shaped conductor, hada platinum
plate (see fig. 3) about 25 centimetres long by 1°5 centi-
metres wide welded to a platinum wire, sealed through the
side of the bulb. The plate was so placed as to project
between the legs of the carbon conductor, having its plane at
right angles to the plane of the horse-shoe, and initially fixed
about halfway between the two legs. ‘This lamp will be de-
scribed in the subsequent paragraphs as Lamp No. 4. Under
a steady electromotive force of 40 volts, this lamp took 3°7
amperes when working at the normal temperature corre-
* “Qn a Peculiar Behaviour of Glow-Lamps when raised to High
Incandescence,” by W. H. Preece, F.R.S. Proceedings of the Royal
Society, 1885, p. 219.
5A Prof. J. A. Fleming on the
sponding to about 3°5 watts per candle-power. When a
milamperemeter having a resistance of 6372 ohms was joined
bet ween the base P of the positive leg of the carbon (see fig. 4)
and the middle plate M, a current was found passing through -
the galvanometer from the terminal P to the plate M. This
: Fig. 3.
LOE,
LAMP NO 4 AK
; TO
BATTERY
Fig. 4.
current had a magnitude of about 3 milliamperes when the
carbon was in the normal state of incandescence.
If the milamperemeter was connected between the negative
electrode of the lamp and the middle plate, no current per-
ceptible by this galvanometer was found. On replacing the
Edison Effect in Glow Lamps. 55
milamperemeter by a more sensitive Hlliott mirror galvano-
meter (resistance 7142 ohms), it was found that a small
current passed through it, when joined in between the negative
electrode of the lamp and the middle plate, but that this
current had a magnitude hardly exceeding ‘0001 of a milli-
ampere when the lamp was at its normal incandescence.
In order to avoid repetition, it may be here said that, unless
otherwise stated, the terminal of the lamp in connexion with
the positive pole of the working battery will be spoken of as
the positive electrode of the lamp; that in connexion with the
negative pole of the battery as the negative electrode. For
brevity’s sake, the half of the carbon filament between the
centre of the filament and the positive electrode will be called
the positive leg, and the other half the negative leg.
§ 4. A preliminary series of experiments was made with lamp
No. 4 by placing the lamp in a photometer and determining
the watts per candle-power and the current taken by the
lamp corresponding to various working electromotive forces,
taken over the whole range of electromotive force from that
necessary just to render the filament incandescent to the
highest the lamp could with safety endure. In any subse-
quent experiments, the simple measurement of the potential-
difference between the electrodes of the lamp enabled the
rate of dissipation of energy in the filament and the watts per
candle-power to be deduced. It may here be remarked that
in the preliminary experiments some difficulties arose from
the occlusion of residual gas by the middle metal plate, but
finally this was overcome, and the vacuum in these experi-
mental bulbs made and preserved as perfect as in good ordinary
commercial lamps. The following results were then obtained
with this lamp No.4. The lamp was raised to various degrees
of incandescence by varying the working volts by the aid of
a rheostat in series with the lamp.
The milamperemeter was employed to measure the effective
potential-difference between the positive electrode of the lamp
and the middle plate and then, tabulating against the working
volts of the lamp the current in milliamperes flowing through
the galvanometer, the potential-difference between the middle
plate and the positive electrode of the lamp was calculated
from these figures. The results are given in the table below
(Table No. 1).
56 Prof. J. A. Fleming on the
TasLeE No.1. Lame No. 4. Milamperemeter.
Table showing the volts between the middle plate and the
positive electrode, and the current flowing through a
galvanometer of 6372 ohms resistance connecting them,
taken for various working voltages of the lamp.
Volts be- | Current in Voltoiee Current in
Working | tween plate milli- Working evesn alee milli-
voltsof and positive| amperes volts of |od eee amperes
lamp. lamp elec- |through gal-|| lamp. Ae Bal! through gal-
trode. vanometer. vanometer.
30 "54 "085 36 LOT 1°69
32 12 "190 37 127 2°01
32 5 1°6 20 38 149 2°36
30 2°8 “44 39 17-0 2°71
BSD. aged “74 40) 18°9 2°99
a4 5:3 “84 41 21-4 3°37
34°5 GL i 4? 23°4 Ee fi!
30 78 1°23 43 25:2 ode
44 26°8 4-25
The results given in table No. 1 are plotted in curve No. 1,
in which horizontal abscissee represent to scale the working
volts of the lamp and vertical ordinates the milliampere
currents through the galvanometer. It will be seen that the
curve representing the current from lamp electrode to plate
takes a rather sharp turn upwards at a point corresponding to
33 working volts, and this occurs when the lamp is working
at about 7°8 watts per candle-power. Beyond this point the
curve is very approximately a straight line. Accordingly, at
and beyond the volts at which the carbon filament becomes.
fairly well incandescent, the effective potential-difference
between the middle plate and the positive lamp electrode is
very nearly a linear function of the lamp voltage ; and at the
normal working volts, viz. 40 volts, this potential-difference
between the middle plate and the positive electrode so deter-—
mined is apparently about half that between the lamp terminals,
the plate being nearly midway between the carbon legs.
The results given in Table 1 are the mean of several obser-
vations, but it was noticed that when the lamp was maintained
at a steady voltage, the potential-difference between the
middle plate and the positive electrode would often jump
Edison Effect in Glow Lamps. 57
suddenly from one value to another. This effect renders it
difficult to obtain the stable values of the plate and positive
electrode potential-difference. Corresponding to any definite
steady voltage on this lamp, the current may have one or
other of two values, but not always permanently preserving
TABLE No. 1.—CuRvE No. 1.
nee
Tee
ee
HEE
ans
Current through Galvanometer in Milliamperes.
2B So Skee BS) 400 c4oe 44 G48
Working Volts of Lamp.
VERE ; a galvanometer deflexion indicating say 10 volts
between the plate and positive electrode of the lamp will
often slowly increase until after a few minutes it is 12 or
14 volts, yet all the time the working volts on the lamp are
58 Prof. J. A. Fleming on the
remaining perfectly constant. It will then often suddenly
jump perhaps to 22 volts, and then slowly decrease to 19 volts,
or so. This tendency of the potential-difference between the
middle plate and positive lamp electrode to jump from a low
to a high value, or vice versdé, is most marked in lamps in
which the plate is about half-way, and symmetrically placed,
between the legs of the carbon. We shall speak of these
two values as the high and low value of the current through
the galvanometer, and defer until later a discussion of some
other causes tending to make the current pass from a high to
a low value or the reverse, as well as its possible explanation.
In Table No. 2 are tabulated a set of observations on the same
lamp No. 4, showing these double values which the potential-
difference and current may have, and it may be here noted
that in the previous Table No. 1, the higher values have been
taken in those cases in which double values exist.
Taste No. 2. Lamp No. 4. Milamperemeter.
Table showing the multiple values of the potential-difference
between the middle plate and positive electrode of the
lamp corresponding to various given working voltages.
Volts be- Current Volts be- Current,
: tween the |through the tween the | through the
Working |niddle plate] graded gal-|| Working |middle plate} graded gal-
voltsiot Grad positive] vanometer || volts of the |and positive | vanometer
the lamp. | electrode of| in milli- lamp. | electrode of} in milli-
the lamp. | amperes. the lamp. | amperes.
30 6 095 39 4-3 “761
32 1-1 174 : 140 | 2-22
34 2°0 317 40 D1 809
35 2-6 412 e 18:1 | 2:87
36 3°2 D07 41 a3 841
37 3:9 618 184 | 2-91
38 43 682 42 6:4 10m
s Ad 714 5 20°0 3°17
< 1HCOF 4] alee 43 6-2 -983
22°0 3°49
. These observations are plotted in Curve No. 2,in which the
abscisse represent the working volts of the lamp and the
Edison Effect in Glow Lamps. 59
ordinates the current in milliamperes flowing through the
galvanometer connecting the positive electrode and the middle
plate. It is seen that corresponding to any working pressure
above 38 volts for this lamp, which is equivalent to 4:2 watts
per candle-power, there are two possible values of the effective
TaBLeE No. 2.—Curve No. 2.
: oe Qe
ee _ bow ial ac
i rome ilk
* att tt :
i Pee eee
i
Current Pees Galvanometer in Pee
Working Volts of Lamp.
potential-difference between the middle plate and the positive
electrode. As the working voltage of the lamp is gradually
raised, the reading of the galvanometer inserted between the
middle plate and positive electrode is also increased, but there
is a great tendency to jump from a certain low value to a
60 Prof. J. A. Fleming on the
higher one, and this occurs when the working pressure of the
lamp is preserved steady. There is also an effect produced
by the presence of a magnet near the lamp bulb. When the
current is at the low value corresponding to any working
voltage, the galvanometer reading does not seem to be per-
ceptibly altered by bringing a magnet near the lamp, but
when it is at its high value, the reading is sometimes increased
for a little, showing a steady deflexion, and then immediately
falls to its low value.
§5. Experiment 2.—The difference of potential between
the middle plate and the positive electrode of the lamp depends
to a considerable extent upon the position of the middle plate.
Supposing the plate to be placed with its plane perpendicular
to the plane of the carbon horse-shoe and then moved to
various positions between the two legs of the carbon, it is
found that the difference of potential between the plate and
the positive electrode will have difterent values according to
the position of the plate. This fact was elucidated by means
of the same lamp No. 4.as used above. By carefully tapping
the lamp, the supporting platinum wire carrying the platinum
middle plate could be bent so as to displace the plate from its
symmetrical position as regards the two carbon legs, and
bring it nearer to one or other of the legs. In several
different positions the current flowing through the milampere-
meter, when connected between the middle plate and positive
electrode, was measured, the lamp being kept meanwhile at
the same working electromotive force.
Iistimating as nearly as possible the fractional distances,
the plate was placed at distances from the negative leg equal
to 7,4, 4, 2, and ,% of the whole distance between the
positive and negative legs, and the lamp being taken through
a definite cycle of volts, the potential-difference between the
middle plate and the positive electrode was measured with
the milamperemeter. The results are collected in the follow-
ing tables. The diagrams in fig. 6 represent the horse-shoe
carbon loop and the middle plate M in various positions, the
galvanometer G being inserted between the plate M and the
positive electrode P. By the phrase “whole distance” in
the following tables is meant the whole distance or width
of the space between the positive and negative carbon
leg :-—
Edison Effect in Glow Lamps. 61
TasLe No. 3. Lampe No. 4. Milamperemeter.
Table showing the potential-difference between the middle
plate and positive electrode of the lamp at various posi-
tions of the plate and at various working voltages.
Volts be- | Current Volts be- Current
Working | tween the [through the) Working | tween the |through the
volts of the|middle plate] galvano- || volts of the middle plate) galvano-
lamp. and positive|meterinmil-|| lamp. and positive/meterin mil-
electrode. | liamperes. electrode. | liamperes.
Middle plate at =, of whole || Middle plate at ? of whole
distance from negative leg. || distance from negative leg.
ol oc "142 dl “) 142
34 2°2 “317 3d4 1°4 "222
37 6:0 "952 37 2°2 "D349
39 10°2 1°61 a9 2°5 "396
4} 14°2 2°24 41 2°9 460
43 16°8 3°0 ‘476
2-66 | 43
Middle plate at + of whole|| Middle plate at ;% of whole
distance from negative leg. || distance from negative leg.
31 9 "142 dl 9 "142
d4 1°8 "285 34 15 "238
a7 D1 809 a7 a4 "539
a9 70 Ill 3g 4°8 “761
4} 8°7 1°38 Al OT 904
43 9°5 1°50 43 6°8 1:07
Middle plate 4 of whole
distance from negative leg.
31 “9 "142
d4 1:7 269
37 3°5 "D00
39 4-3 682
41 5°6 888
43 6:2 "983
62 Prof. J. A. Fleming on the
The results of this Table No. 3 are plotted in the curves
No. 3. These curves are to be interpreted as follows :—The
two vertical lines P and N represent the two legs of the
TABLE No. 3.—CuRvE No. 3.
The middle plate was moved along into different positions between the
two carbon legs indicated by the ‘horizontal distances, and at each
position the current between the middle plate and positive electrode of
the lamp is represented by the vertical ordinate of a curve. The several
curves correspond to different working volts on the lamp.
“UT
\
l
|
|
{
nz
Goat leg of Ghnad
Ourrent through Galvanometer in Milliamperes.
on eps I
= :
+Leg 1/10 1/4 1/2 3 3/4 9/10 —Leg
Position of Plate between legs.
carbon horse-shoe. At various distances on the way from P
to N the milliampere current through a galvanometer con-
nected between the middle plate, placed at that point, and the
Edison Effect in Glow Lamps. 63
positive electrode of the lamp is represented by the magnitude
of the vertical ordinate of each curve. For every one of the
different voltages at which the lamp is worked, there is there-
fore a curve representing by its ordinates this current strength
through a galvanometer inserted between the middle plate,
placed at these positions, and the positive electrode of the
lamp, and it is seen that there isa minimum value for this
current at a position equal to ? of the whole distance between
the legs reckoned from the negative leg.
Imagine the middle plate therefore connected through a
galvanometer with the positive electrode of the lamp, and let
the middle plate be first placed close to the positive leg and
then moved continuously nearer towards the negative leg.
The current through the galvanometer would first fall off as
the plate receded from the positive leg, and after reaching
a minimum at a point about } of the whole distance between
the legs reckoned from the positive leg, would rise up to a
maximum when the middle plate was as nearly in contact with
the negative leg as possible without actually touching it.
§ 6. Haperitment 3.—In order to explore more thoroughly
the action of the different portions of the incandescence carbon
conductor in producing this effect, a lamp was taken having
a horse-shoe shaped carbon, and a pair of small platinum cylin-
Fig. 5.
LAMP NO 3
ders, held on platinum wires sealed through the glass, so
placed as to embrace without touching the carbon conductor.
One of these cylinders, X, was placed so as to embrace the
carbon near the bottom of the leg, and the other, Y, near the
spring of the arch (see fig. 5). These small cylinders had a
64 Prof. J. A. Fleming on the
length of about 12 millims. and a diameter of about 8 millims.
so that the distance from the carbon filament to the inner
surface of the cylinder was about 3 or 4 millims. The lamp
had a rather thick carbon, and at an electromotive force of
48 volts took a current of 1°32 amperes to raise it to its
normal incandescence of 18°8 candles, corresponding to 3°3
watts per candle-power. This lamp will be alluded to as
Lamp No. 3.
It is obvious that there are four possible arrangements in
which a current can be obtained between an embracing
cylinder and a positive electrode of the lamp. ‘These are
illustrated in fig. 6, in which the horse-shoe shaped line stands
Fig. 6.
Position Position Position Position
(1). - (Q): (3). (4).
LAMP NO 3 les
for the carbon filament, X and Y are the platinum cylinders,
P and N are the positive and negative electrodes of the lamp,
and G is the galvanometer.
We will call these arrangements (1), (2), (3), (4), as figured.
It will be seen that if we imagine the carbon filament straight-
ened out, these four arrangements are equivalent to being
able to slide a cylinder along the filament into four positions,
and in each position measuring the potential-difference be-
tween the cylinder and the positive end of the carbon. We
are thus able to place an embracing collecting-plate at four
different places along the carbon conductor, and determine
the potential-difference between this embracing cylinder and
the positive electrode of the lamp. A series of experiments
was made with lamp No. 3, in which the working volis of the
lamp were raised to various values, and in each case the
potential-difference between one of the cylinders X or Y and
Edison Effect in Glow Lamps. 65
the positive electrode of the lamp was observed as before by
means of the milamperemeter. The results are given in
Table 4.
TaBLE No. 4. Lamp No. 38. JMlanperemeter.
Table showing the potential-difference between a platinum
cylinder embracing the carbon and the positive electrode
of the lamp, and the current flowing thr ‘ough the galvano-
meter in milliamperes ; for the four positions shown in
fig. 6.
Volts be- Cay is Volts be- es te
Working | tween cylin- amperes Working | tween cylin- amperes
volts of the| der and through the volts of the; der and through the
lamp. positive galvano- lamp. positive galvano-
electrode. eto electrode. TSG.
Position (1). Position (2).
43 16 "29 43 1-2 18)
45 2°4. 38 45 2-0 roe
46 2°9 “46 46 2°9 “36
Aq a7 58 47 37 08
48 4-2 67 48 42 67
49 A°7 “TA 49 4°9 78
50 D°2 "82 50 a1 Zool
ol a9 “93 aL D3 "84
Position (3). Position (4).
43 2°1 "33 43 Pati "39
45 ay) 62 45 4:0 63
46 49 "78 46 a9 “94
47 6°8 1:08 47 8:0 1°26
48 8:2 1°30 48 Se 5d
49 10°4 1°65 49 12°6 ee)
50 12°3 1°95 50 sa! 2°38
a1 14°9 2°36 ol 13°0 2°85
These observations are plotted in Curve No. 4.
Phil. Mag. 8. 5. Vol. 42. No. 254. July 1896. F
66 Prof. J. A. Fleming on the
This table of observations shows us that taking the lamp at
any definite voltage, the potential-difference between the
positive electrode and a cylinder embracing the carbon
filaments is greatest when that cylinder is as low down near
TaBLE No. 4.—CurvE No. 4.
\
’ E
|
|
|
ay, |
r Th 4) —— |
E ve y
Fe
PosiTijon (1),
positilon(4) |
|
|
rata ee =
us Dn
Current through Galvanometer in Milliamperes.
Working Volts of Lamp.
the foot of the negative leg of the carbon as possible. If the
cylinder is placed near the top of the negative leg that
potential-difference becomes less. If the cylinder is taken
near the top of the positive leg or near the base of the positive
leg itis least, and it seems to be a minimum when the cylinder
Edison Effect in Glow Lamps. 67
surrounds the top of the positive leg, and is as far as possible
away from the foot of the negative leg. We have here a
confirmation of the fact observed with respect to lamp No. 4,
viz.: that the potential-difference between the positive lamp
electrode and the metal plate held somewhere near the incan-
descent carbon conductor is greatest when the plate is as near
as possible to the foot of the negative leg or negative elec-
trode.
In this lamp No. 3, in which in all positions the cylinder
employed is placed very near some point on the incandescing
conductor, the current through the galvanometer joining the
positive electrode and the cylinder has never been observed
to jump or to take double values as in the case of lamp No. 4.
If a sensitive galvanometer is connected to the two insu-
lated cylinders X and Y, and if in addition there js a battery
in series with this galvanometer, then no current can be de-
tected in such an arrangement even when the battery has an
electromotive force of 120 volts, whether the lamp carbon is
incandescent or not. Just as in the case of lamp No. 4, the
current obtained by connecting either cylinder with the nega-
tive electrode of the lamp is excessively small.
In order to obtain curves showing the mode of variation of
the effective difference of potential between one or other of
the metal cylinders X and Y and the positive electrode of the
lamp, a set of observations was made on the lamp when sub-
mitted to various working voltages, and at the same time the
milamperemeter was connected first between the positive
electrode and cylinder X and then between the positive
electrode and cylinder Y, with the following tabulated
results (p. 68).
These results, when plotted out in curves in which hori-
zontal ordinates represent the working volts of the lamp and
the vertical ones the milliampere currents going through the
galvanometer, in both cases have the same general form as
curve No. 1 plotted for the case of lamp No. 4.
F 2
68 Prof. J. A. Fleming on the
TaBLE No. 5. Lamp No. 3. Milamperemeter.
Table showing the potential-differences between the positive
| electrode of the lamp and the two cylinders X and Y
respectively, and the corresponding currents through the
galvanometer.
l
| Volts between
=o Volts between | Milliampere
cylinder X and Milliampere
cylinder Y and current flowing
vee the positive |currentthrough) “the positive | through the
volts of the electrode P of the galvano- | electrode P of galvanometer
lamp. the lamp. | ™eter connect-| the lamp. | connecting
Position (1). ing X and P. | Position (3). Y and P.
| 84 ‘1 015 ‘1 O15
| 85 als 023 a a
5. 086 20 031 ‘3 git
37 pide ese: “4 060 |
38 “6 095 7 aera
39 8 126 9 145 oy
40 1-05. 166 11 apd
Al 13 206 16 25 a
42 1:8 285 2-0 31
43 2-1 333 2°8 44
44 2°6 412 3:5 55
45. 3-0 476 45 17
46 3°6 571 5°8 92
Aq 4-2 659 72 1-12 |
48 4-9 ‘791 8-8 1:39
AO fd Abs 889 10-9 1-72
50 6-2 984 13°6 2°15
Percviel & ydocarten | ecupee 15-1 2°39
a ee eae 18-6 295 |
§ 7. Experiment 4.—The magnitude of the current found
on connecting any galvanometer between one of the cylinders
and the positive electrode of the lamp was found to be de-
pendent to some degree on the perfection of the vacuum.
Lamp No. 3 when first made had not a very perfect vacuum.
A series of measurements was, however, made with it, and the
same repeated after re-exhaustion. The results are tabulated
together below.
Edison Liffect in Glow Lamps. 69
TABLE No. 6. Lamp No. 3. Elliott Galvanometer.
Table showing the relative values of the potential-difference
between cylinder Y and the positive electrode for good
and imperfect vacua in the case of lamp No. 3.
Serr high vacuum in the lamp.’ Imperfect vacuum in the lamp.
Working oa |
ee ere atene Vole taba
P. taken up in and positive taken up in | and positive
carbon. electrode. carbon. electrode.
40 6°75 2°02 eth 4°17
Al eo 2°89 | Ss) 7:25
42 5°30 362 | 65 8°39
43 4-90) 454 | 5 9-80
44 450 608 | 4:3 11:96 |
45 415 7-50 4-0 13-76 — |
46 3°84 es) one TOSOSh 2
47 3:55 10°33 Be | 16 Ot
48 3°30 Pee a2 | 28:28
AQ and 16°3 30 22205)
AQ) 290 138°4 2°8 22599
>I. 2°74 21°8 PA) 23°93
| Sy 2°58 24°4 2°4. 28°26
The imperfection of the vacuum is indicated by the higher
watts per candle-power absorbed at low voltages, and we
see that at any given working pressure the potential-ditference
between the positive electrode and the cylinder Y embracing
the top of the negative leg is greater when the vacuum is
imperfect than whenitis very good. The presence of residual
air tends to bring down the potential of the embracing
cylinder more nearly to that of the carbon at the point adja-
cent to it.
§ 8. Hxperiment 5.—A series of observations was next
made in which the potential-difference between the middle
plate and the positive electrode was determined by the aid of
a condenser. If a condenser of capacity U in microfarads is
charged to a potential of V volts and discharged through a
ballistic galvanometer, we can determine the ballistic constant
of the galvanometer. A second observation of a like nature
70 Prof. J. A. Fleming on the
in which the “throw” of the same galvanometer is observed
when the same condenser is charged by contact with two
points concerning which we require to know the potential-
difference, gives us the means of calculating the electrostatic
potential-difference in volts. A condenser of ‘987 microfarad
capacity carefully determined was charged to a potential of
54 volts and discharged through a certain ballistic galvano-
meter having a needle whose periodic time of vibration was
about three seconds. The resulting “ throw” of the galvano-
meter was 5° 30!. Hence a discharge of 54 x°987=53°3
microcoulombs through the galvanometer produces a “ throw”
of 5° 30’. Neglecting a very small correction for the loga-
rithmic decrement, in this case not of importance, we have
for the ballistic constant R the value
53°3=R sin 4(5° 30’) =R x :04798 ;
hence R=1110.
The same condenser was then connected between the
middle plate and positive electrode of lamp No. 4 and then
discharged through the same ballistic galvanometer. The
lamp was subjected to a working pressure of 39 volts as de-
termined by a corrected voltmeter attached to the electrodes
of the lamp. ‘The charge of the condenser was sent through
the ballistic galvanometer, and a “throw” of 4° obtained.
If vis the potential-difference between the middle plate and
positive electrode of the lamp, we have the following equation
for v in terms of the ballistic constant and angle of “ throw”:
°987 v=1110 sin 2°
=1110 x 0349
4)
ee v= ES = 39 nearly.
The potential-difference between the middle plate and the
positive electrode as determined by this method is therefore
exactly the same as the potential-difference between the positive
and negative electrodes of the lamp.- In other words, when the
filament is brought to full incandescence, the middle metal
plate is brought to the same potential as the negative elec-
trode of the lamp. This observation was repeated with several
other Jamps having middle plates in various positions and of
various forms, and always with the same result, viz., that the
potential of the middle plate when insulated is brought down
nearly to that of the negative electrode.
Edison Effect in Glow Lamps. 71
§ 9. Lxperiment 6.—In order to confirm the results ob-
tained by the condenser method and to eliminate all the
conditions which necessarily exist when we attempt to
measure potential-difference galvanometrically, an electro-
static method of measuring the potential-difference at any
instant between the metal plate and the positive electrode of
the lamp was next used. For this purpose a Kelvin multi-
cellular electrostatic voltmeter was employed to determine
the potential-difference between the positive and negative
electrodes of the lamp and between the positive electrode of
the lamp and the middle plate, with the following results :—
A lamp (No. 4) having the plate fixed between the carbon
legs was raised to various working voltages and the potential-
differences above mentioned taken.
TaBLE No. 7. Lamp No. 4. Kelvin Electrostatic
Voltmeter.
Static potential-difference in
volts between middle plate
Working volts of the lamp. and positive electrode of
the lamp.
Al Al
ord 98°7
61:0 61°95
These observations confirm conclusively the previous re-
sults. The insulated metal middle plate is in this case brought
to the same potential as the base of the negative -leg of the
carbon; and hence, on measuring electrostatically the poten-
tial-difference between that metal plate and the positive
electrode of the lamp, we find it to be the same as the poten-
tial-difference between the two electrodes of the lamp.
§ 10. Haperiment 7.—In order to see if this was the case
when the metal collecting-plate had a very small surface
placed at some distance from the negative electrode of the
lamp, the lamp called No. 1 was employed. In this lamp a
platinum wire threaded through the turns of a double spiral
100 volt carbon lamp (see § 11). The lamp was raised to
various working voltages, and the electrostatic voltmeter
employed to measure at the same time the static potential-
difference between the positive electrode of the lamp and the
platinum wire, with the following results :—
A2 Prof. J. A. Fleming on the
TaBLE No. 8. Lamp No.1. Electrostatic Voltmeter.
Static potential-difference in
volts between platinum wire
Working volts of the lamp.
and positive electrode of
the lamp.
62 D3
79 (ids)
97 | 85
118 | 107
The figures in the above Table No. 8 show that when the
surface of the collecting-plate is very small and is placed
some distance from the base of the negative leg of the carbon
itis brought down only to the potential of some point (probably
the nearest point) on the carbon conductor, and that there-
fore the potential-difference between the plate and positive
electrode of the lamp is somewhat less than the potential-
difference between the working terminals of the lamp. At
the same time, however, the electrostatic voltmeter shows no
measurable potential-difference between the negative terminal
of the lamp and the platinum wire, and the most sensitive
galvanometer between these points gives no indication of any
current.
By means of the electrostatic voltmeter it was, however,
ascertained that in those cases in which the metallic plate
presented considerable surface (several square centimetres)
and was placed so that some portion of it was not removed by
more than’a centimetre or two from the base of the negative
leg of the carbon, it was brought down almost immediately
to the potential of the negative terminal of the lamp. If the
middle plate is placed at a little distance from the carbon
loop then, on testing by the condenser method, it is found
that the plate is not instantly brought down to the potential
of the negative terminal, but that some few seconds haye to
elapse before this is the case. .
§ 11. A series of experiments was then undertaken in order
to determine the effect of varying (1) the surface, and (2) the
position of the metal plate in the bulb, and in these experi-
ments tke plate was sometimes of platinum and sometimes of
aluminium. In all cases the vacuum was a very perfect one,
any occluded gases in the plates being got rid of by special
means. =
Edison Effect in Glow Lamps. (fe
Experiment 8.—A normal 100-volt carbon-filament lamp,
having a carbon filament coiled in a spiral of two turns (see
fig. 7) had a short stout platinum wire (‘024 inch diam.)
Fie. 7:
LAMP NO}
sealed across the bulb so as to thread through, without
touching, the spirals of the carbon. The lamp at 100 volts
took 1°54 amperes and gave an illumination of 40 candles,
equivalent to a power absorption of 3°9 watts per candle-power.
The vacuum was very good. This lamp will hereafter be
called Lamp No. 1. As before, no current could be detected
by a galvanometer when joined up between the platinum wire
and the negative electrode, but when the galvanometer was
connected between the platinum wire and the positive elec-
trode of the lamp a current of some milliamperes was found
passing through it. As in the case of lamp No. 4, this lamp
was characterized by a great tendency to change suddenly
the value of the current flowing through the galvanometer
when the working volts on the lamp were kept perfectly con-
stant. In the first series of observations the milamperemeter
was employed to measure the current flowing between the
positive electrode of the lamp and the platinum wire when it
was connected between these points, and at and beyond a
working-pressure of 90 volts or so the galvanometer would
often jump suddenly from one reading to another, when the
lamp working volts were kept perfectly constant.
In the following table, No. 9, are collected the results when
the working pressure of the lamp was gradually raised from
80 to 100 volts :—
74 Prof. J. A. Fleming on the
TasLe No. 9.. Lamp No. 1. Milamperemeter.
Table showing the potential-difference between the positive
electrode and the platinum wire, and the current flowing
through the galvanometer connecting them, for various
voltages of the lamp.
| |
are ee as Milliampere cae wae
orking platinum current Working _ platinum current
volts of the| yi. ang [through the! voltsofthe) [:1. ang | through the
lamp. | positive eaten? lamp. positive ca
electrode. en electrode. ris
80 4 | p16 99 | 47 270
a 2 | +032 ‘ 2-4 381
| 82 3 | 048 93 Lg 302
83 2 | +0382 “ 2:6 ‘413
84 “4 ‘064 94 3°2 "509
86 “4 064 9-4 1:49
5 7 “114 95 4-9 ‘78
87 9) 080 1. 296 5°8 “92
88 ‘8 127 || 97 71 .| aa
, “9 142 || 98 83 | 1:32
90 1:0 159 | 99 8:9: li tee
1:3 206 || 100 8°6 1:37
91 1-4 229 : 10°3 1-64
These figures show that at any definite working electro-
motive force of the lamp the current between the positive
electrode and the middle plate has very variable values, and
that it suddenly changes from one value to another without
any apparent reason, the working volts of the lamp remaining
constant all the time.
If the surface of the collecting-plate is large, say several
square centimetres, the potential-difference existing between
it and the positive electrode is not found to be so much
reduced by attempting to measure it with a galvanometer
of about 6000 ohms resistance as it is when the collecting
wire presents, as in this lamp No. 1, only a small total
surface of about one square centimetre.
§ 12. Experiment 9.—A horse-shoe carbon filament, taking
1:3 ampere of current at a working-pressure of 42°5 volts,
had a middle plate made of a long piece of platinum wire
bent up in a zigzag shape so as to form a rectangular-shaped
grating (see fig. 8). The object of this was to ascertain
Edison Effect in Glow Lamps. 19
whether a middle plate offering a surface pierced with many
apertures was as effective in producing the current as a solid
Fig. 8.
plate of about the same general outline. Practically it was
found that this was the case.
The magnitude of the currents obtained at various working
voltages are of the same magnitude approximately as in the
case of a lamp like No. 4, that is to say some 3—4 milliamperes
at full incandescence.
- § 13. A set of experiments was then undertaken with the
object of examining the special effect of varying the position
of the middle plate, and a series of lamps was used in which
latinum or aluminium plates held on platinum wires were
placed in the lamp bulb, or in tubes opening into it, in various
positions. These lamps are generally 50-volt lamps of usual
type, and had single horse-shoe shaped filaments.
Experiment 10.—A lamp-bulb had a side tube blown on it
(see fig. 9) and a plate about 6 centims. long and 1°5 centims.
wide welded to a platinum wire was sealed into it. The
platinum plate was placed vertically and edgeways in the side
tube and the side tube was in such a position that the plane
of the platinum plate coincided with the plane of the horse-
shoe filament. This lamp, called henceforth No. 2, when
worked at 48 volts took 1°3 amperes of current and gave a
light of 17°5 candles, equivalent to a power-consumption of
3°55 watts per candle-power. The vacuum was very good.
In the case of this lamp the current between the positive
electrode of the lamp and the platinum plate was found to be
numerically very much smaller at the usual working pressure
of the lamp than was found to be the case in those lamps in
76 Prof. J. A. Fleming on the
which the middle plate was placed between the carbon legs or
in the form of a cylinder embracing the carbon.
The current obtained at any definite working voltage was
considerably greater when the leg of the carbon nearest the
plate was the positive leg than when it was the negative leg.
Fig. 9.
LAMP NOe
A series of observations were taken using the lamp at dif-
ferent voltages and measuring with the Elliott galvanometer
the potential-difference between the platinum plate and positive
electrode of the lamp, and these results were as tabulated
below in Table No. 10.
TasLE No. 10. Lamp No. 2. Elliott Galvanometer. .
Table showing the potential-difference between the platinum
plate and the positive electrode of the lamp at various
working voltages. Positive leg of carbon nearest the
platinum plate.
| Volts | Volts
| between : | between
Working Watts per ee Working | Watts per '
volts of ae | adie. ae eee te of the| candle- ieee
lamp. | power. positive sea ace’ positive
| electrode. electrode.
43 O20 45085 43. 3°59 199
ak 5°80 al es LY) 3°32 "236
45 4°42 "144 50 3°12 "239
AOU > 4 10 OG Pl apd 2°94 "302
| |
!
AT 880 4 172
Edison Effect in Glow Lamps. nd
If we compare together the results obtained with this
lamp No. 2, in which a plate is placed edgeways on and
outside the carbon loop, with the results obtained in the case
of lamp No. 3, in which the plates embraced the carbon in
the form of cylinders, we see the difference produced by the
change of position of the plate. Both these lamps, No. 2 and
No.3,are48-volt lamps when working at normal incandescence.
Referring to Table No. 7 in § 7, we see that for lamp No. 3
at 48 volts the voltage difference of the positive electrode and
the platinum cylinders was respectively 13°2 and 18°3 volts
as measured with the Elliott galvanometer, and this indicated
a current of about 1°3 and 1°8 milliamperes flowing through
the resistances from the positive lamp electrode to the metal
plate ; but in the case of lamp No. 2, at 48 volts the potential-
difference between the platinum plate and the positive lamp
electrode was only *2 volt, and this corresponded to a current
of 03 milliampere nearly. Accordingly the current is greatly
diminished when the collecting-plate is placed edgeways to
and someway outside the loop of the carbon. At normal
incandescence the current between the positive lamp electrode
and the middle plate when joined by the galvanometer is
about 03 or *04 milliampere when the positive leg of the
carbon is nearest the middle plate, but only about °02 or ‘03
milliampere when the negative leg is nearest the plate.
§ 14. Haperiment 11.—In order to compare the previous
results just given with those obtained when the collecting-
plate was placed broadside to and yet outside the carbon loop,
Fig. 10.
f LAMP NO 6
a lamp was made as in fig. 10 in which an aluminium plate
was held on a platinum wire just outside one leg of the
78 Prof. J. A. Fleming on the
carbon and with its plane perpendicular to the plane of the
horse-shoe.
The aluminium plate was 5 centims. long and 1 centim.
wide, and distant from the nearest leg of the carbon about
*5 centim. This lamp therefore differed from lamp no. 4 in
having the plate outside the carbon loop rather than between
the legs. It may be noticed that in this lamp the current
here obtained by joining the positive lamp electrode to the
plate through a galvanometer was slightly greater when the
leg nearest the plate was negative than when it was the positive
leg, whereas in the case of lamp No. 2 it is just the reverse.
This lamp exhibited also the same effect as lamp No. 4, in
that the current flowing between electrode and plate is very
liable to “jump” from one value to another even when the
lamp is kept at constant volts. The following tabular result
of the observations shows this. This lamp is called Lamp
No. 6 and was a 50-volt horse-shoe carbon lamp, taking 1°33
amperes of current at a working electromotive force of
50 volts.
TaBLE No. 11. Lame No. 6. Milamperemeter.
Table showing the potential-difference between the positive
electrode of the lamp and the aluminium plate, and the
current in milliamperes flowing through a galvanometer
connecting them. Vacuum good.
| r
Potential- |,,.,,. Potential- |,,.,,.
difference of |Milliampere difference of |Milliampere
Working the positive |, current Working | the positive | _ Current
volts of the. Hactrods through the | volts of the electrode through the
damp." 7) and plate galvano- || lamp. and plate galvano-
| in volts. er in volts. eee
32 ai 016 ~~ AG 3°8 ‘601
34 3 ‘O47 Oise eRe “555
36 6 ‘095 bln 1aeae oleae
37 8 BBA || Fens 7 a Daeg "602
38 1-0 “Laoe a iH 16°2 2°97
39 1-2 19) #1 | 48 4+] “650
40) Ry Dod Jn Boe 17-7 2°80
41 2-0 Oh fee 42 "666
42 2-3 O64 <1 awe 183 | 2-90
43 2°8 “444 50 19°5 3°03
hea! 371 491 < 20-0 3°17 |
ae 4-0 634 4 4" 650 |
|
Edison Effect in Glow Lamps. 79
The results in Table No. 11 are plotted in curve No. 5.
This table shows that when the lamp is kept at a constant
voltage the current through the galvanometer jumps from
one value to another. The fluctuation of the current takes
place when the negative leg of the carbon is the one farthest
TABLE No. 11.—Curve No. 5.
40
3°0
bo
Oo
Current through Galvanometer in Milliamperes.
Hedpe
eRReeee=
Pop ts Walee eyso] ee
a ete
30 32 34. 36. 38 40: 42) 44 46 48 50
Working Volts of Lamp.
0
from the plate. When the leg adjacent to the aluminium
plate is the negative one then the current is steady at any
definite voltage of the lamp.
§ 15. From the above experiments it is clear that the
current obtained when a galvanometer is connected between a
metallic plate and the positive electrode of the lamp is greater
80 Prof. J. A. Fleming on the
in proportion as the collecting-plate is larger and in propor-
tion as it is brought into close proximity to the base of the
negative leg of the carbon. Also that a plate so placed 1s
brought down to the potential of the negative electrode. It
seemed desirable to see how far the removal of the collecting-
plate to a great distance from the negative leg would influence
these results, and experiments were accordingly tried with a
tube of the form shown in fig. 11.
Fig. 11.
LAMP NO
————— SI NGHES
Experiment 12.—‘n this case a glass tube about eighteen
inches long and three-quarters of an inch in diameter was
attached to a lamp bulb. The end of the glass tube furthest
from the bulb was closed and an aluminium plate welded to a
platinum wire was sealed in near this closed end. The plate
had a length of about 3 centimetres and a width of about
1 centimetre. The tube formed an extension of the bulb-
space, and accordingly this arrangement formed a device by
which a metal plate could be removed to a distance of some
eighteen inches from the incandescent conductor contained in
the bulb. On placing this lamp on a circuit and bringing
the carbon to normal incandescence and connecting the ter-
minals of the Hlliott galvanometer respectively to the aluminium
plate and the positive electrode of the lamp, a very small
current was found to be passing through it, not, however,
exceeding one ten-thousandth of a milliampere. When the
galvanometer was joined in between the aluminium plate and
the negative leg of the carbon no current whatever could be
detected with this galvanometer, which was sufficiently sensitive
to show one hundred-thousandth of a milliampere. We thus
find that the removal of the plate to a distance of some ©
Edison Effect in Glow Lamps. 81
eighteen inches from the incandescent conductors practically
extinguishes the phenomenon.
Experiment 13.—Another similar bulb was provided haying
a side tube blown on it of half the length, viz. about 9 inches
long. At the end of this tube was placed a small aluminium
plate as before, and the tube was bent up about the middle at
right angles (see fig. 12). When the carbon conductor in
Fig. 12,
the bulb, which was that of an ordinary 50-volt 16 candle-
power lamp, was rendered incandescent by being connected
to a circuit of appropriate electromotive force, and the Elliott
galvanometer connected in between the aluminium plate and
the positive electrode of the lamp, a current of not more than
about one twenty-thousandth of a milliampere was detected.
The fact that the ‘“‘ Edison effect ” was extinguished when the
collecting-plate was placed at the extremity of an elbow-tube
was first observed and recorded by Mr. Preece.
§ 16. The effect of position and size of the plate having
been examined, the next step which naturally suggested itself
was to determine the effect of the different portions of the
Incandescent conductor in the production of it.
Haperiment 14.—A lamp like No. 4 was provided, but in
which one leg of the carbon was enclosed in a glass tube of
the size of a quill. The glass tube was sealed on to the
platinum wire and extended nearly up to the bend of the
carbon (see fig. 13). This lamp, called No. 9, was placed on
the circuit in such a manner that the shielded leg was the
Phil. Mag. 8. 5. Vol. 42. No. 254. July 1896. G
82
Prof. J. A. Fleming on the
positive leg, and a series of observations taken as usual of the
current flowing through the milamperemeter when connected
between the middle plate, placed between the carbon legs,
Fig. 13,
LAMP NO 9
NO CURRENT FLOWING
LAMP NO 9
CURRENT FLOWING
and the positive electrode of the lamp. The results are as
tabulated below in Table No. 12. The lamp took 1:25 ampere
of current at a working electromotive force of 42 volts.
TABLE No. 12.
Lampe No. 9. MMilamperemeter.
Table showing the potential-difference between the middle
plate and positive electrode of the lamp, and the current
flowing through a galvanometer connecting them, when
the positive leg of the carbon is shielded in a glass tube.
volts of the
lamp.
36
a8
40
Al
42
43
Potential-
Bt AN Ors difference of| Milliampere
Working [middle plate| current
and positive |through gal-
electrode | vanometer.
in volts.
2 Wz
“2 "048
"95 a ie
Mie 74 19
1°8 "29
oil "49
4°2 "66
44
Working
volts of the
lamp.
—
Ad
46
AT
48
49
50
ol
Potential-
difference of |Milliampere
middle plate] current
and positive |through gal-
electrode | vanometer.
in volts.
6°9 "936
(ar) 109
SET 1°53
12°4 1°96
15-1 2°39
19- 3°03
23°1 3°66
|
Edison Effect in Glow Lamps. 83
So far the results are quite normal, and if the results in the
above Table No. 12 are compared with those in Table 2 for a
similarly constructed lamp with no tubular shield we find that
the magnitude of the current flowing from the positive elec-
trode to the middle plate is in the two cases very much the
same. The lamp No. 9 was then placed on the circuit in such
a manner that the leg shielded by the glass tube was the
negative leg, and a similar series of observations of the current
between the positive leg (now the uncovered leg) and the
middle plate was made. The results were as follows in Table
No. 13 :-—
TABLE No. 13. Lamp No. 9. Milamperemeter.
Table showing the potential-difference between the middle
plate and positive electrode, and the current flowing
through a galvanometer connecting them, when the
negative leg of the carbon is shielded in a glass tube.
Potential-differen 77°
Working volts of | of middle nae me ee ee
the lamp. ne in galyanometer.
AD “10 "016
AT 15 "024
AQ 25 "039
ol 30 044
On comparing the results in Table No. 13 and the previous
one No. 12 we see what an immense reduction in the current
flowing between the positive electrode and the middle plate is
produced by shielding the negative leg. Hence the action in
virtue of which the current is produced is greatly interfered
with by enclosing or covering up the negative leg of the
carbon. In this particular case at 51 volts, the current
between the positive electrode of the lamp and the middle
plate when the negative leg is covered up is only ;5 of that
which it is when the posztive leg is covered up. The results
in Tables Nos. 12 and 13 are plotted together in Curve No. 6.
G 2
84 Prof. J. A. Fleming on the
§ 17. The question remained to be settled whether the
nature of the tubular screen in any way affected the results,
and the glass tube was accordingly replaced by a metal
(aluminium) tube and a lamp (No. 10) taken in which one of
TasiEs No. 12 anp 13.—CurvE Neo. 6.
Current through Galvanometer in Milliamperes.
Boe
pee,
ee
aie
ed
SA REA
34 36 38 40 42 44 46 48 50 52 o4
Working Volts of Lamp.
the carbon legs (see fig. 14) was surrounded by an aluminium
cylinder extending nearly the whole length of the leg, and
also a middle plate of aluminium was placed between the
legs. Both the plate and the cylinder were held on platinum
wires sealed through the glass. The lamp took 1-25 amperes
of current at 41°5 volts.
Edison Effect in Glow Lamps. 85
Experiment 15.—The lamp was placed on a cireuit so that
the leg shielded by the aluminium cylinder was the positive
leg. ‘The milamperemeter was then connected between the
positive electrode of the lamp and the middle plate, and the
usual measurements made. It was found that the current
“jumped ”’ a good deal, and that high and low values of the
Fig. 14,
LAMP NO 10
galvanometer current occurred, even when the terminal
voltage of the lamp was kept perfectly constant.
The lamp was then reversed on the circuit so that the
shielded leg was the negative one, all other arrangements:
remaining the same. The current now between the positive
electrode and the middle plate was practically zero, at any
rate too small to be measured with this galvanometer.
Hence we see that shielding the negative leg, whether by
glass or a metallic cylinder, entirely cuts off the production
of a current between the positive lamp electrode and the
middle plate.
§ 18. Leperiment 16.—Another series of experiments was
made with the lamp No. 10 in which the galvanometer was
connected between the positive electrode of the lamp and the
aluminium cylinder (see fig. 15), the leg inside the cylinder
being either the positive or the negative leg. In this case
the middle plate remained unused and insulated and acted as
a shield between the cylinder and the carbon leg which was
not contained in the cylinder.
It was found that when the cylinder surrounds the negative
leg and its surface is, therefore, as much exposed to it as
possible the current is a maximum, but that when it includes
86 Prof. J. A. Fleming on the
the positive leg the current is greatly diminished, both by
reason of the fact that it opposes less surface to the negative
Fig. 15.
LAMP NO (0 LAMP NO10
leg, and also because the middle plate acts as a shield between
‘it and the negative leg of the carbon.
§ 19. Hxperiment 17.—In order finally to demonstrate
that the negative leg of the carbon loop was the chief active
agent in this production of a current between the middle
plate and the positive electrode, a lamp like No. 4 was taken
having a metal middle plate between the legs, and this middle
plate had attached to it a mica screen (see fig. 16) a little
Fig. 16.
\
Oy
SWS WN
S| A&V7
SSSSSIAAQHQy
\S
\:
SSN
SSS
\.
Ww
=
MIDDLE PLATE SHIELDED MIDDLE PLATE EXPOSED
larger than the plate, and so fixed by a loose rivet that it
could be shaken in front of the plate so as to shield one side
of it, or shaken on one side so as to fully expose the plate..
This device was in fact a removable shield attached to one
Edison Effect in Glow Lamps. 87
surface of the metal middle plate, and when placed up against
it it shielded one surface from, and when jerked on one side
it exposed that surface to, the carbon leg opposite to that
surface. This lamp (called No. 5) was set on the circuit in
the first place so that the leg of the carbon horse-shoe
opposite to the mica-shielded side of the middle plate was
the positive leg. If the carbon was brought up to an incan-
descence corresponding to about 3°5 or 4 watts per candle-
power and the galvanometer connected between the positive
electrode and the middle plate, then it was found that the
effect on the galvanometer current which was produced by
the interposition or withdrawal of the screen of mica between
the positive leg and the plate was not very great. It reduced
the current through the galvanemeter from about -44 mil-
liampere to °38 milliampere. If, however, the current
flowing through the lamp carbon is reversed in direction so
that the mica screen is interposed on that side of the middle
metal plate which faces the negative leg, the result is very
different. When the screen is down, the current flowing
through the galvanometer from the positive electrode to the
middle plate being as before °44 milliampere, the interposition
of the mica screen on the side of the plate facing the
negative leg reduced the current at once to zero. We find
therefore, that the interposition of a mica screen between the
middle plate and the negative leg reduces to zero the current
flowing between the positive lamp electrode and the middle
plate. As in all other lamps with a middle plate set exactly
between the legs, the current through the galvanometer
joining the middle plate and positive electrode is very liable
to “jump ” from a low to a higher value or vce versd. When
the current has its higher value corresponding to any given
voltage on the lamp terminals, the effect of screening is less
marked, and although the interposition of the mica screen on
the side facing the negative leg has an effect of reducing
the current flowing through the galvanometer connected
between the middle plate and positive electrode it is not by
any means reduced to zero.
§ 20. The foregoing experiments afford proof that the
production of the current through a galvanometer joined
between the positive electrode of the lamp and a metal plate
placed somewhere in the vacuous bulb, is an effect due chiefly
~to the negative leg of the carbon, and that shielding the
negative leg by enclosing it in a glass or metal tube, or
covering with a mica screen that surface of the plate which
is exposed to the negative leg, either quite prevents or greatly
reduces the production of this current. The experiments
88 Prof. J. A. Fleming on the
also have shown that the magnitude of the current flowing
through the galvanometer is increased by bringing the plate
near to the base of the negative leg, or, better still, making
the metal plate in the form of a cylinder and making this
cylinder surround the negative leg near its base ; and cor-
respondingly it is diminished by removing it from the
negative leg to a considerable distance, or by shielding this
collecting-plate from the radiation from the negative leg of
the carbon. The experiments with the condenser have also
given evidence that when an insulated metal plate is sealed
into a lamp this plate is brought down either instantly
or in a very short period of time to the potential of the
negative leg near its base or to that of the negative electrode
of the lamp. In looking for an explanation of these facts we
are assisted by our previous knowledge that in carbon incan-
descence lamps, when working at an efficiency equivalent to
‘3 to 4 watts per candle-power, there is a gradual loss of
carbon from all parts of the conductor. We know also that
the carbon molecules which are projected from the conductor
are thrown off into a space so highly vacuous that their mean
free path is of a length comparable with, or greater than that
of the dimensions of the glass bulb. The existence of
‘molecular shadows in incandescence lamps * affords evidence
that from intensely heated portions of the carbon conductor
carbon molecules are projected in straight lines and move
freely forward until they impinge against the glass. Com-
mercial experience informs us that at and above a tem-
perature corresponding to 3 watts per candle-power this loss
-of carbon becomes very rapid and thins away the filament
in one place, or generally reduces the diameter of the carbon
.conductor. Hence we have every reason to believe that
when in a normal state of incandescence the carbon conductor
in a lamp is throwing off im all directions carbon molecules,
and that in the vacuum usually obtained the mean free path of —
‘these projected molecules is comparable with the dimensions
of the vessel containing the conductor. The whole of the
experiments which are detailed here seem to be capable of
consistent interpretation if we may justifiably make the
hypothesis that these carbon molecules or atoms so projected
from the conductor when intensely heated by the current
flowing through it are all negatively charged. Some of the
observed facts seem to point to the conclusion that the
molecules projected from the incandescent conductor, whether
they are portions of the conductor itself or molecules of the
residual gases, respectively carry away negative charges
* See J. A. Fleming, ‘ Philosophical Magazine, August 1885, p. 141.
Edison Effect in Glow Lamps. 89
proportional in magnitude to the potential of the conductor
at the point from which they are thrown off. They may,
therefore, be looked upon as condensers of small but definite
electrostatic capacity charged to the potential (negative)
of that part of the incandescent conductor at which they
separate from it. We have then in addition to explain
how it comes to pass that there are few or no projected
molecules charged positively. Two suggestions may be made
on this point: either the radiation of matter is wholly confined
to that half of the conductor at a negative potential or the
incandescent carbon molecule thrown off from the heated
conductor cannot retain a positive charge. There is much
to lead to the conclusion that from all parts of the incan-
descent carbon conductor there is a constant radiation of
matter carrying a negative electric charge. The nearer down
to the negative electrode of the carbon we select our point
of observation the greater is this molecular charge found to
be. It will be convenient to denote this conveyance of
electric charge by moving charged molecules by the term
molecular electrovection. We can then state the hypothesis
thus—from all portions of the negative leg of the carbon
loop a process of molecular electrovection is going on when
the conductor is incandescent, the molecular charge being
negative, and equal in potential to that of the point on the
conductor from which it is projected.
§ 21. On the assumption that a molecular shower of nega-
tively charged atoms was being projected against the middle
plate when the conductor of the lamp was incandescent. it
was considered probable that a positively charged conductor
connected to the middle plate would be discharged, and this
was found to be the case.
Experiment 18.—A lamp of the form of No. 4, having a
middle metal plate placed between the carbon legs, had its
middle plate connected to one terminal of the Hlliott gal-
vanometer. The other terminal of the galvanometer was
connected to one terminal of a condenser of 5 microfarads
capacity. The other terminal of the condenser was connected
by a wire to the gas pipes of the laboratory. The lamp was
actuated by secondary batteries (see fig. 17) not very well
insulated. If the condenser was charged to a potential of
90 volts so that the plate next the galvanometer was pos?-
tively charged, then this positive charge was instantly dis-
charged when the carbon was rendered incandescent. If,
however, the plate of the condenser in connexion with the
middle plate through the galvanometer was charged negatively
the condenser was not discharged when the lamp was illumi-
90 Prof. J. A. Fleming on the
nated by rendering its carbon incandescent. It is a very
striking experiment to see a condenser charged with this
amount (250 microcoulombs) of electricity instantly discharged
when its positive coating is brought into connexion with the
middle plate of sucha lamp. The discharge may be brought
70’SecY BATTERY To EARTH
about either by joining up the positive side of the condenser
to the middle plate first, and then rendering the carbon of the
lamp incandescent by switching on the lamp, or the lamp
may be first of all illuminated and then the junction of the
condenser effected. In both cases the middle plate when
positively electrified is instantly discharged.
It was found that if the lamp carbon is rendered incandescent
bya highly insulated secondary battery then in order to produce
the discharge, the plate of the condenser not in connexion
with the middle plate, and which is negatevely charged, must
be somewhere connected with the battery circuit. It does
not matter, however, whether the wire from the negative
side of the condenser is in connexion with the positive or the
negative pole of the secondary battery actuating the lamp ;
all that is necessary is that the negative side of the condenser
should be in conducting connexion with the circuit of the
incandescent carbon. ‘The experiment may be interpreted
by considering that this negative charge of the condenser
can escape out of the incandescent conductor and discharge
across the highly vacuous space to the positively electrified
cool middle metal plate; but that a positive charge cannot be
discharged out of the hot conductor, or, which amounts to the
same thing, a negative charge cannot discharge across from
the cool metal plate to the incandescent carbon which is
positively charged. We have then a unilateral conductivity
exhibited by this highly vacuous space bounded by two
Edison Effect in Glow Lamps. 91
electrodes one of which is incandescent and the other of
which is cold. Negative electricity is discharged at once
out of the hot surface but not out of the cold, and a
negative discharge can take place from hot to cold but not
vice versd. When the discharge of a charged condenser is
effected by connecting the positive plate, through a galvano-
meter, with a metal plate sealed into the lamp and the
negative plate with the lamp circuit, and then switching on the
lamp, there is a curious instant of delay betore the discharge
begins. When the metal plate is placed very near the negative
leg of the carbon the discharge of the condenser is complete
in one instant. This the case when a lamp of the type No. 4
(fig. 18) is used. If, however, we employ a lamp of the type
No. 2 (fig. 10), in which the metal plate is at some distance
from the negative leg of the carbon, the discharge of the
condenser is long drawn out and the electric charge in it is as
it were tapped off slowly and notin one short sharp discharge.
Moreover this effect of discharging a condenser takes place
only when the carbon is above a fair red heat. At brilliant
incandescence and when the carbon is above a temperature
corresponding to 3 watts per candle-power, the discharging
power of a lamp of the type of No. 4 is very great. A con-
denser of 10 or 20 microfarads capacity charged to 50 volts
is discharged instantly if its positive plate is connected to the
metal plate placed not far from the negative end of the
incandescent carbon conductor.
The foregoing results were confirmed with lamps of other
types. Using, for instance, a lamp like No. 6 with the
aluminium plate placed outside the carbon horse-shoe and
near the leg, the same discharging power for positive elec-
tricity was found. It was not dependent on the direction of
the current through the lamp carbon, although it seemed a
little more vigorous when the leg nearest the plate was the
negative leg. As above observed, the rate of discharge was
much reduced when employing a lamp having the metal plate
placed edgeways on to the carbon and some way from it, as,
for instance, when employing a lamp of the form of No. 2.
§ 22. Hwperiment 19.—A series of experiments was in this
case also tried to determine the effect of shielding the negative
leg of the carbon. The lamp no. 9 was employed, in which
one leg of the carbon was enclosed in a glass tube connecting
the positive plate of a charged condenser through a galvano-
meter with the middle plate of the lamp, and the negative
plate of the condenser somewhere to the battery circuit ; it
-was found that when the shielded leg of the carbon was the
positive leg the condenser was discharged as before. UH,
92 Prof. J. A. Fleming on the
however, the shielded leg or leg enclosed in the glass tube
was made the negative leg, which could be done by reversing
the current through the carbon conductor, then the condenser
was not discharged when its positive plate was connected
with the middle plate.
The same fact was less perfectly exhibited by employing
the lamp with the middle plate having a removable mica
shield on one side. Weare thus able to assure ourselves that
the active agent in producing this discharging effect upon a
positively charged body connected to the middle plate is the
negative leg of the carbon conductor. The experiments were
varied in many ways, but all pointed to the conclusion that if
a charged condenser is connected to two terminals, one of
which is a metal plate and the other a carbon conductor, both
enclosed in a high vacuum but yet separated from each other
by an inch or so of distance, the condenser is discharged
instantly when the carbon terminal is rendered highly
incandescent, provided that the negative plate of the con-
denser is in connexion with it.
§ 23. If the condenser is left in contact with the middle
plate under some circumstances, not only is it discharged if
previously charged but is charged again in an opposite
direction.
Heperiment 20.—A condenser of 5 microfarads capacity
perfectly discharged has its poles or terminals connected for
one instant, one with the middle plate of No. 4 lamp and the
other with the positive electrode of the lamp (see fig. 18).
Fig. 18.
plu s
uISNIGNOD
N
P
On removing it and testing it with the galvanometer G it is
found that the condenser plate in connexion with the middle
plate of the lamp has received a negative charge and the
other plate of the condenser a positive charge. |
Edison Effect in Glow Lamps. 93
If, however, the condenser is connected between the
negative electrode of the lamp and the metal middle plate of
the lamp, on insulating and testing it we find it has not the
slightest charge.
It is very astonishing to see how instantly a condenser of
very large capacity is charged when one pole of the con-
denser is connected to the middle plate and the other to the
positive elecirede of the lamp.
§ 24. In considering the behaviour of the heated carbon
electrode and the cool metal plate in their respective powers
of discharging the positive or negative charge of the con-
denser, it seemed that the fundamental fact was the power of
the heated surface to discharge negative electricity out of
itself. Hence arose the question, how far the observed facts
would be modified if the middle metal plate itself could be
also heated. One way by which this might have been done
would have been to have rendered this plate incandescent by
heating it by radiant heat concentrated by means of a powerful
mirror or Jens. Some experiments tried in this way were
not satisfactory, and consequently a method was adopted in
which a middle plate of carbon could be rendered incandescent
electrically.
Experiment 21.—A vacuum tube was provided with two
carbon conductors (see fig. 19), one the ordinary carbon
Fig. 19.
twa
INSULATED
SECONDARY
BATTERY
filament L of a 50 volt lamp, and the other the small carbon
S of a4 volt lamp. The smaller carbon was sealed in the
usual way through the glass and placed so as to stand
symmetrically between the legs of the Jar ger carbon loop.
The smaller carbon could be rendered incandescent by an
insulated battery of fifteen secondary cells, appropriate
94 Prof. J. A. Fleming on the
resistance being introduced. The larger carbon also could
be rendered incandescent by the proper electromotive force.
If the smaller carbon was kept cold and employed simply as
a third electrode or middle plate, all the phenomena previously
described as happening with metal middle plates of aluminium
or platinum teok place. If the small (cold) carbon loop is
connected through a galvanometer with the positive electrode
of the larger carbon loop when this last is rendered incan-
descent by a current, we find as usual a current of a few
milliamperes passing through the galvanometer from the
positive electrode of the peer carbon to the small carbon.
If the small carbon (still cold) is connected through the
galvanometer to the negative electrode of the larger carbon
we get no current. This is the normal effect, and it is the
same for a cold carbon conductor used as a middle plate as
for a metal middle plate.
Leperiment 22.—The next experiment consisted in making
this small carbon incandescent by an insulated secondary
battery, appropriate resistance being inserted so that the
carbon was brought to the normal condition of temperature as
indicated by its incandescence. When this was done the
galvanometer was inserted between the positive electrode of
the large carbon loop and one of the electrodes of the small
carbon loop. A current was obtained as before. On con-
necting the galvanometer between the negative electrode of
the large carbon loop and one of the electrodes of the small
carbon loop a current of nearly equal value was now obtained.
In this last experiment it was found to be immaterial whetner
the terminal of the galvanometer was joined to the positive or
to the negative electrode of the small carbon loop. Hence
we find that when the small carbon Joop is not incandescent
and is used as a middle plate or electrode, it is brought down
together with the insulated battery attached to it to the same
potential as the negative end of the large incandescent carbon,
and we get as usual a current through a galvanometer con-
nected between the positive electrode of the large incandescent
carbon and any point on the small cold carbon, vand no current
between the negative electrode of the large hot carbon and
the small cold one. On rendering the smaller carbon loop
incandescent this is all changed. The smaller carbon, now
hot, is not brought down to the potential of the negative ends
of the larger carbon, and we get a current through : a galvano-
meter connected between either positive or negative electrode
of the large hot carbon and any point on the circuit of the
smaller equally hot carbon.
§ 25. Haperiment 23.— With this same vacuum tube having
Edison Effect in Glow Lamps. 95
double carbons, further experiments were performed on the
discharging power of the hot and cold electrodes for positive
and negative electricity. The two carbons could be rendered
incandescent either simultaneously or singly by two sets of
insulated secondary batteries attached to each respectively.
For the sake of distinction we shall speak of the large carbon
loop in this bulb as the L loop and the smaller one as the
S loop. A condenser of 5 microfarads capacity (see fig. 20)
Fig. 20.
S
l
m
ite”
:
‘
BATTERY
g
+ oe CONDENSER
/ | N
was employed, which was charged to a potential of about
50 volts. When the positive plate of this charged condenser
was attached to the carbon L and the negative side to the
earbon 8, then on making L incandescent by its own
insulated battery and keeping 8S cold, the condenser was
found not to be discharged when insulated and tested by a
galvanometer. If, however, the same charged condenser was
connected in the same way to the two carbons and the
carbon 8, to which the negative side of the condenser was
attached, was made incandescent, the condenser was instantly
discharged. If the direction of the charging of the condenser
was reversed the same rule was found to hold good. The
condenser was discharged if the negatively charged plate of the
condenser was connected to the zneandescent carbon loop, but not
if it was connected to the cold carbon loop. Beginning with
the condenser charged and connecting it in between the two
carbon loops, neither of them being incandescent, then the
condenser was discharged instantly if that loop to which the
negatively charged side of the condenser was attached was
rendered incandescent, but not discharged if the loop to
which the positive side of the condenser was connected was
rendered incandescent. If both loops were rendered incan-
96 Prof. J. A. Fleming on the
descent simultaneously the condenser in any case was dis-
charged, but apparently at an accelerated rate. These
experiments show again that if two carbon electrodes are
sealed into a high vacuum, negative electricity escapes very
freely out of either electrode if it is rendered incandescent,
but that the escape or discharge of positive electricity is not
in the same way facilitated by heating the positive electrode.
Accordingly a highly vacuous space bounded by two carbon
electrodes separated by a distance less than the mean free
path of the gaseous molecule at that pressure, presents a
unilateral conductivity when one of these electrodes is cold
and the other highly incandescent. For if the hot electrode
is connected to a negatively charged body and the cold
electrode to a positively charged body, discharge takes place
across the vacuous space, but if the charges are reversed then
no discharge takes place. The negative charge can escape
from the heated electrode but not from the cold one.
§ 26. Hapertment 24.—The question of the apparent uni-
lateral conductivity of the vacuous space bounded by a hot
and a cold electrode was then further examined by the aid of
the lamp No. 6 formerly used.
In this lamp an aluminium plate is sealed into the vacuum
and placed just outside the carbon horse-shoe. If a sensitive
galvanometer (the high resistance Hlliott galvanometer) is
joined up between the metal plate and the negative electrode
of the lamp, then, asin other cases when the Jamp is !n action,
Fig. 21.
N No CURRENT
no current of a magnitude much greater than ‘0001 ofa
milliampere is detected. If a single Clark standard cell is
inserted in the galvanometer circuit (see fig. 21) with its
negative pole attached to the middle plate and its positive
Edison Effect in Glow Lamps. 97.
pole te the galvanometer terminal, the current is barely if at
allincreased. In this case the negative pole of the Clark cell
is in connexion with a cold metal electrode and the positive
pole is in connexion through the galvanometer with the
incandescent carbon electrode, and under these circumstances
the galvanometer detects no current flowing. The position
of the Clark cell is now reversed, and it is joined up so that
its positive pole is in connexion with the middle plate and its.
negative pole in connexion, through the galvanometer, with
the incandescent carbon electrode. It is then found that a
considerable current of some few milliamperes in magnitude
is flowing through the galvanometer. The direction of this
current in the ordinary way of speaking is from the negative
electrode of the lamp through the galvanometer to the metal
plate sealed into the bulb. We thus find that a negative
current of electricity can be made to flow across the vacuous
space between the incandescent carbon and the metal plate
from the hot carbon to the cooler metal plate, but not in the
reverse direction. The space presents an apparently uni-
lateral conductivity.
§ 27. Experiment 25.—The same experiment was repeated,
only using instead of a Clark cell an insulated secondary
battery of 25 small cells.) When the secondary battery
(see fig. 21) was connected with its negative pole to the
metal plate and its positive pole through the galvanometer to
the negative electrode of the carbon, no current greater than
that found with the Clark cell similarly arranged was found ;
but if the secondary battery was reversed and joined up with
its positive pole to the middle plate and its negative pole
through the galvanometer to the negative electrode of the
incandescent carbon, then so strong a current flowed through
the galvanometer that it could not be measured without
shunting-down the galvanometer considerably. The same
experiments were repeated with the lamps having the zigzag
wire as a metal plate, No. 7, and the same general results
obtained. ‘These experiments therefore show that in a circuit
which consists partly of a galvanometer-wire and partly of a
highly vacuous space bounded by two electrodes—one a metal
plate and the other an incandescent carbon surface,—the
insertion of an electromotive force in one direction can produce
a very sensible current, but that if the electromotive force is
reversed then no current flows. The direction of the electro-
motive force must be such as to urge negative electricity from
the hot surface to the cold across the vacuum space. |
§ 28. Laperiment 26.—In order to make use of different
parts of the incandescent conductor as the electrode opposed
Phil. Mag. 8. 5. Vol. 42. No. 254. July 1896. H
98 “Prof. J. A. Fleming on the-
to the metal plate, recourse was had to the lamp No. 3, with
metal (aluminium) cylinders embracing without touching the
carbon at two different places. These “cylinders, as before, we
will call X and Y (see fig. 22). Cylinder X was the one near
the base of the positive leo of the carbon, and cylinder Y was
the one near the top of the negative leg.
W hen the galvanometer was ‘connected between the negative
electrode of te lamp and the cylinder X surrounding the
lower part of the positive leg, no perceptible current was found
to be passing when the carbon was rendered incandescent. On
inserting a single Clark cell in series with the galvanometer so
that the - negative pole of the cell was in connexion with the
cylinder X and the positive pole of the cell through the galvano-
meter in connexion with the negative electr ode of the lamp,
hardly any perceptible current was found to be passing (see
fig. 22). a Clark cell was then reversed, connecting Ve
2 22.
“LAMP NO3
positive pole of the cell to the cylinder X and the negative pole
througn the galvanometer to the negative electrode of the
lamp. On bringing the lamp into. ‘action, a considerable
current of several milliamperes was found to be passing in such
a direction that a current of positive electricity was flowing
across the vacuous space from the metal cylinder to the hot
carbon, or a negative current from the hot carbon to the cooler |
metal aon On switchin g-off the lamp, there was a curious
“kick ” or ‘throw ” of the galvanometer, indicating a sudden
rush of “aan ers in the same direction as the steady current
which the cell had been sending. _ These effects occurred also
when the cylinder Y was employed, and the galvanometer with
or without the cell in series joined in between the negative
Edisin Effect in Glow Lamps. © 99
electrode of the lamp and the cylinder Y embracing the top
part of the negative leg of the carbon (see fig. 22) ; but in the
‘last-case the steady current sent by the cell across the vacuous
‘space between the cylinder and the hot carbon was only about
a quarter as great in magnitude as when the cylinder X was
employed. There was the same kind of “kick”’ of the galvano-
meter on breaking the lamp-circuit. These experiments evi-
dently showed that the highly vacuous space between the hot
carbon traversed by its own current, which rendered it incan-
descent, and the insulated cylinder possessed a sort of unilateral
conductivity, negative electricity from a separate source of
small electromotive force being able to be forced through it
from the hot carbon surface to the cooler metal surface, but
not in an opposite direction.
§ 29. In the above-recorded experiments the carbon con-
‘ductor was rendered incandescent by a unidirectional or con-
tinuous current in a highly perfect vacuum. In seeking for
an hypothesis to connect them together, it became essential to
ascertain how the effects would be modified if the vacuum
was imperfect and if the current was alternating instead of
continuous.
Experiment 27.—The fundamental experiment was therefore
repeated with the normal type of lamp (No. 4), having a middle
metal plate placed symmetrically between the legs of the
carbon. A lamp of this type was set in action by an alternating
current of suitable strength and of which the frequency was
some 80 to 100 per second. On connecting the milampere-
meter between either of the electrodes of the lamp and the
middle plate, a continuous electric current was found flowing
through the galvanometer. The direction of this current was
such that positive electricity was found to flow from either
lamp terminal to the middle plate of the lamp. In other words,
a continuous current of negative electricity flowed out of the
middle plate to one or other of the two terminals of the lamp,
viz. to that terminal to which the other extremity of the gal-
vanometer was joined. Hence, since in this case each leg of
the carbon becomes in rapid succession positive and negative
-when the lamp is. operated with an alternating current, the
‘unilateral effect observed of a current flowing between the
‘middle plate and the positive leg, when the current through
‘the carbon is a continuous current, is here found to exist
equally between the middle plate and both terminals of the
lamp. This is only what might have been expected. The
potential of the middle plate 1s then not the same as that of
the base of either leg of the carbon, but something between
the two depending upon the. ee the plate.
100 Prof. J. A. Fleming on the
» §30. Ezperiment 28.—The effect of lowering the vacuum
was also the subject of experiment. Ina lamp with a highly
perfect vacuum no current greater than about ‘0001 milli-
ampere is observed when a very sensitive high-resistance
-galvanometer is joined up between the negative electrode of
the carbon and the insulated middle plate, and, as we have
seen, the experiments with the electrostatic voltmeter showed
that the plate was brought down under these circumstances to
the potential of the base of the negative leg of the carbon. If,
instead of employing a very perfect vacuum, a bad one is pro-
duced by imperfectly exhausting the lamp, then it is found
that under these conditions the Elliott galvanometer indicates
a not inconsiderable current of something approaching to a
milliampere when joined in between the negative electrode of ©
-the lamp and the middle plate. Hence, when the vacuum is
imperfect the equality in potential between the middle plate
and the negative electrode is not maintained.
The direction of the current in this last case is such as to
show that negative electricity is flowing through the galvano-
meter from the negative electrode of the lamp to the middle
plate. In other words, negative charge is carried over from
the plate to the positive leg of the carbon across the imperfectly
vacuous space ; and the means by which this is effected is the
residual air. This seems to afford proof that the normal effect
of the molecular electrovection of negative electricity from the
negative leg is due to carbon molecules, and that the presence
of residual air exhibits itself, when present beyond a certain
amount, in producing an effect which the carbon molecular
electrovection cannot produce.
§ 31. Laperiment 29.—It seemed very desirable to ascertain
if the effect of molecular electrovection exists in the case of
an incandescent platinum wire rendered vividly incandescent
in a highly perfect vacuum. A bulb was accordingly con-
structed similar in every way to lamp No. 4, but having a
platinum-wire horse-shoe conductor and a platinum middle
plate. When this wire was rendered highly incandescent by
a continuous current, a sensitive galvanometer (the high-
resistance LHlliott) showed a current of about one five-
thousandth of a milliampere when connected between the
positive electrode of the incandescent wire and the middle
plate, but little or no current when connected between the
negative electrode and the middle plate. This molecular
electrovection current was thus very much less in magnitude
than that observed in the case of the carbon filament lamps,
but it is in the same direction. We are, however, enabled to
state that at a condition of vivid incandescence just short of
Edison Effect in Glow Lamps. = ——‘101
fusion a platinum wire zn vacuo exhibits the same effects as a
earbon filament, and that it can disturb the electrical condition
of an insulated metal plate near it sealed into the same vacuum
and tends to bring it down towards the potential of the negative
end of that platinum wire.
On the hypothesis that all these effects are due to a scattering
of negatively charged molecules from the incandescent con-
ductor, we must affirm that the same process goes on in a
platinum wire rendered incandescent in a vacuum, only that
the radiation of matter is far greater in the case of the
incandescent carbon than it is in the case of the incandescent
platinum.
§ 33. Ifa lamp is selected having an insulated plate fixed
between the legs of the carbon filament, it is found that under
certain conditions the electric conductivity of the vacuous space
between the plate and the negative leg is much affected by the
presence of a magnetic field. Ifa galvanometer, preferably a
movable coil galvanometer, having a resistance of about 500
or 600 ohms, is connected between the middle plate and the
negative leg it will show but little current passing when the
lamp is incandescent at normal temperature. If the volts on
the lamp terminal are raised so that the filament is brought
into a state of incandescence corresponding to about 2°5 or
3 watts per candle, then the galvanometer will show a small
current passing through it. If then a horse-shoe magnet is
held so as to create a magnetic field the direction of which is
across the space between the plate and the negative leg, the
current indicated by the galvanometer immediately decreases
considerably. This happens irrespective of the direction of
the field so long as it is across the direction of the line joining
the negative leg and the middle plate. This indicates that
the presence of this transverse field increases the resistance of
the rarefied gas. The galvanometer current responds to the
presence of the magnet in a manner which shows that the
resistance to the flow of the current through the gas is
increased by creating a magnetic field at right angles to the
line of the current. The general fact that gaseous resistance
is increased by such a transverse magnetic field has been
already noted and described by Professor J.J.Thomson. The
behaviour of bismuth as regards electrical resistance in a
magnetic field is strikingly similar.
The “jumping ” of the current from one value to a higher,
which has been already mentioned, appears to be due to some-
thing equivalent to a sudden change in the resistance of the
space between the negative leg and the middle plate when the
lamp is in action and at high incandescence. The fact of
“CC?
102 On the Degree of Dissociation of Electrolytes at 0°.
sending a small current through this space seems to effect-a
change in the qualities of the rarefied gas as a conductor which
makes it conduct better. There are certain aiter-effects in
some cases which are strongly similar to the polarization of
electrodes observed in the case of liquid electrolysis, and which
seem to point to the validity of the view that gaseous con-
duction is effected by a similar process. -
The experiments also confirm the opinion of Professor J. J.
Thomson that gases, or at least certain gases ina rarefied con-
dition, are very good conductors, and they show that the
greatest part of the obstacle to conduction through a vacuum-
tube is at the electrodes and may be largely ‘removed by
heating the kathode to incandescence.
IV. Note on the Deyree of Dissociation of Electrolytes at 0°.
By Meyer WivpErmasn, PA.D.*
N the Philosophical Magazine, February 1896, Mr. Wood
I published a paper, “The Degree of Dissociation of some
Electrolytes at 0°,” in which he, by a curve, illustrates certain
molecular freezing-point depressions, by some misapprehen-
sion attributing them to me. The reader will see from the
paper which he cites (Phil. Mag. July 1895) that, contrary
to his statement, Arrhenius’s generalization finds an excellent
confirmation.
In the interest of the matter I wish to add a few words on
the following point. The good agreement between the degree
of dissociation as obtained from the freezing-point depressions
(at 0°) and from electric conductivity (18° and 25°) has
sien shown that there can only be a little difference
between the degrees of dissociation as obtained from electric
conductivity at 0° and at 25°. Mr. Walden’s inv estigations
(Zeitschr. 7. phys. Chem.) on the electric conductivity of acids
at different temperatures lead to the same conclusion. What
I suggested in my paper was that, as we find some differ-
ences between the degrees of dissociitien of acetic acid at
18° (Kohlrausch) and at 25° (Ostwald), the electric conduc-
tivity of salts and acids, &c. should be investigated near to
the freezing temperature in order to ascertain the small
differences possibly existing. This is of interest because
of the importance of Arrhenius’s generalization ; but it re-
quires accurate work, and must only be carried out on a
broad basis, as has been done by Kohlrausch and Ostwald
at other temperatures. Mr. Wood does not investigate the
* C ommunicated by the Author,
Dr. 8. P. Thompson on Hyperphosphorescence. 103
L., directly but makes inter polations, using the temperature-
coefficients which have been determined a good distance
from 0°, and so runs into the danger of introducing errors
into the calculated degrees of dissociation greater than the
differences he has to determine. His results (at 0°) de-
viate from Prof. Ostwald’s (at 25°) in some cases irregularly
by several per cent., and his investigations are limited to
three acids and two salts. For this reason the investigation
of the matter remains as desirable as it was before.
Christ Church, Oxford,
May 1896.
V5 On Hyperphosphorescence.
By Strvanus P. Toomeson, D.Sc, FAR.S.*
Hf recent researches of H. Becquerelt on the emission
by compounds of uranium and by metallic uranium of
invisible radiations which very closely resemble those dis-
covered by Wiedemannt{ and by Réntgen§, and which yet
unquestionably consist of transverse vibrations, are of so great
importance that any experiments upon the same line, however
incomplete, are of interest to physicists.
In January last the writer and his assistant Mr. Miles
Walker were repeating Réntgen’s now familiar experiments
on the production of photographic shadows by the emanations
from Crookes’s tubes, and were casting about for means. to
shorten the long exposures then necessary, when the idea
occurred to them which has independently suggested itself to
many other experimenters, namely that of employing fluo-
rescent substances in contact with the photographic film to
hasten the photographic action by the emission of rays of a
visible sort when stimulated by the x-rays. Accordingly,
having prepared skeets of paper or of aluminium covered with
fluorescent material, they tried the effect of inserting them in
some cases below the glass plate, in other cases above the
glass plate with the fluorescent surface next the film, and-in
yet other cases above the plate but with the fluorescent sur-
face outside. The materials so tried were sulphide of calcium,
finely powdered fluor-spar, sulphide of zine (natural blende),
sulphate of zinc (artificial), fluoride of uranium and ammo-
nium, and sundry platino-cyanides. -
* Communicated by the Author.
+ Co umptes Rendus, cxxil. pp. 559, 790, &e.
ft Zeitschrift fiir Elektrochemie, i il Ds 159 (Aug. 1895).
‘Ss peengeecr sent der Wi ee Pha iysit-medic. Gesellschaft, 1 1895,
-
104 -Dr. 8. P. Thompson on Hyperphosphor escence.
__ When sheets of paper or aluminium covered with these were
placed face down upon the sensitive film, so that the 2-rays
were compelled to pass first through them, some results were
obtained tending to show that the method might have some
advantages: but the resulting negatives were always patchy
and irregular. The most striking effect, however, was quite
unexpected. Care had been taken to keep these prepared
sheets of fluorescent material in the dark for a sufficiently
long time for all visible phosphorescence or persistent fluo-
rescence to disappear. This, in the case of the sulphide of
calcium, required many hours. The powdered fluor was also
heated beforehand. Nevertheless, though no visible phos-
phorescence was present, the sensitive films were fogged by
rays emitted from these materials. Fluor-spar and the pla-
tino-cyanides did not produce any noticeable fogging, however.
Even after being kept six weeks in darkness the sulphide of
ealcium is very active in the emission of rays that will affect
a photographic plate.
While these experiments were in progress other experi-
ments were begun to ascertain if from any other sources,
such as sunlight or the light of the arc lamp, any rays could
be obtained having, like the x-rays, the power of penetrating
opague bodies. From the are lamp, with an exposure of
about two hours, shadows of pieces of metal were obtained on
a photographic plate through a piece of pine-wood several
millimetres thick ; but aluminium was found to be totally
opaque to everything radiated from the are and to sunlight.
While the experiments on fogging were still in hand there
was published the observation of M. Henry on the effect of
sulphide of zinc in apparently augmenting the transparency
of aluminium to a-rays ; an observation which had an obvious
bearing on that which was under investigation. A number
of small portions of the fluorescent substances with which we
were experimenting were then placed upon the front of a
sheet of aluminium about 0°5 millimetre thick, behind which
was a gelatino-bromide plate (a Cadett’s “lightning ”’ plate) ;
and these were left for several days upon the sill of a window
facing south to receive so much sunlight (several hours as it
happened) as penetrates in February into a back street in the
heart of London. On developing the plate it was found that
behind those spots where portions of uranium nitrate and
uranium ammonium fluoride had been placed, photographic
action had taken place through the aluminium sheet. No
very distinct effect had been been produced by the other
substances. On communicating these observations to Sir G.
G. Stokes he drew the writer’s attention to the similar obser-
‘Dr. 8. P. Thompson on Hyperphosphorescence. 105
vations of M. Becquerel with respect to uranium salts, ob-
servations which have since been so remarkably extended.
‘While agreeing with the Réntgen rays in the property of
penetrating aluminium, zinc, and other opaque materials, and
in exercising photographic actions, the Becquerel rays differ
in the circumstance that they can be refracted and polarized.
Whatever the Rontgen rays may eventually prove to be, the
Becquerel rays consist of transverse waves of an exceedingly
high ultra-violet order.
The circumstance that the strongest fluorescent effects are
found in the compounds of two metals having such heavy
atomic weights as platinum and uranium, when correlated
with the other circumstance that the absorbing power towards
‘x-rays is greatest in elements of the greatest atomic weights,
naturally suggests a new application of the law of reciprocity
between emission and absorption. If that law can hold good
in the phenomena of the Roéntgen rays, or of the closely-
related Becquerel rays, one would argue that the best sub-
stances to employ as emitters of such radiations would be
those substances which absorb them most freely. Now the
property of emitting Rontgen rays has been observed in many
substances, but always under the stimulation of the kathodic
discharge. In Rontgen’s original research glass was the
radiator. Porter and Jackson independently found platinum-
foil to be superior. Roiti has found porcelain and mica also
to serve. The writer has observed Rontgen rays to be
emitted from the following substances exposed to kathode
discharges :—cale-spar, apatite, rubies, sapphires, diamonds,
uranium glass, scheelite, tourmaline, a phosphorescent enamel
containing 6U per cent. of sulphide of calcium, sulphide of
zine (hexagonal blende), zinc, aluminium, copper, iron, mag-
nesium, and platinum. Of the metals in the above list, iron
and platinum appeared to work better than copper, aluminium,
or magnesium. The low melting-points of the last two
render them unsuitable. Metallic uranium would have been
tried had it been possible* to obtain a specimen; but all
inquiries in London proved fruitless. Of the other substances
named, the phosphorescent materials seemed to have some
advantages over ordinary glass, but they are not so convenient
to manage as the metals. Apatite was tried because, consist-
ing as it does chiefly of phosphate of lime, it was thought that
the w-rays emitted from its surface could be more certainly
* [While these lines have been going through the press, a specimen of
metallic uranium has been given me by Mr. C. Vautin. It emits x-rays
freely under the kathode discharge,—S. P. T.]
106 = Dr. 8. P. Thompson. on Hyper ‘phosphorescence.
absorbed by Wee len ace he a-rays emitted from denser
materials such as platinum.
. At an early stage of these investigations the use of a fluo-
rescent screen revealed the fact that the relative transparency
of flesh and bone differed with different materials used as
emitters, and depended also upon the degree of exhaustion.
The necessary inference that the x-rays are not all of one kind,
but are heterogeneous, was announced by the writer ee
the same time* that the same conclusion was drawn by MM.
Benoitt and Hurmuzescu from other causes. To the rays
emitted from apatite, bone was indeed found to be more opaque
than to those emitted from platinum. But apatite, when
subjected to the kathode discharge, continues to give out
gases which after a very few seconds spoil the vacuum ; and
the tube containing apatite as an anti-kathode could not, con-
sequently, be used except attached to the pump. Glass was
found to be more transparent to «-rays emitted from platinum
than to x-rays emitted in the same tube from glass.
The extraordinary property exhibited by the uranium com-
pounds of emitting a persistent invisible radiation that will
pass through aluminium and produce photographic action
would suggest that these rays are identical with Roéntgen’s
were it not that Becquerel’s success in reflecting, refracting,
and polarizing them proves that they are more akin to ultra-
violet light. The latter does not indeed penetrate aluminium:
but it has long been known that ultra-violet rays penetrate
films of silver which though thin are thick enough to reflect
all visible kinds of light. It would seem to be proved, thea,
that Becquerel’s rays differ from the known ultra-violet in
degree rather than in kind, being rays of higher frequency
and shorter wave- length. That their properties are inter-
‘mediate between those of ultra-violet and of the Rontgen
rays furnishes a strong presumption that the latter also differ
only in degree, and are an extreme species of ultra-violet
light. It should not be forgotten that so far back as 1857
M. Niépce de Saint Victor observed many cases in which an
object, an engraving on paper or a figured piece of percelain
or marble, immediately after exposure to sunlight, was found
capable of giving a photographic impression to a sheet of
paper prepared with chloride of silver, with which it was
placed in contact. He even used, after exposure to light,
cardboard imbibed with salts of uranium or with tartaric acid,
and found such to be capable of emitting rays that were
Pe, active. There was no ey made, how-
Comptes Reus @ exxil. p- 807. | ST lid eax, ?. 779...
Magnetic Field due to an Elliptical Curr ent. 107
ever, to: investigate the possibility -of transmitting these
invisible radiations through opaque bodies.
- The phenomenon of persistent emission of these invisible
rays by the uranium compounds long after any electrical or
Juminous stimulus has ceased to be applied would seem,
therefore, to bear the same relation to the transient emission
of them in the. Crookes tube as the persistent emission of
visible light by phosphorescent bodies does to the transient
emission of light by fluorescent bodies. Hence the writer
ventures to give to the new phenomenon thus independently
observed by M. Becquerel and by himself the name of hyper-
phosphorescence. A hyper-phosphorescent body is one which,
after due stimulus, exhibits a persistent emission of invistble
rays not included in the hitherto recognized spectrum.
June 6, 1896.
VI. On the Magnetic Freld due to an Elliptical Current at a
point in tts plane within wt. By J. Virtamu Jonas,
WIA B.Se., FoR. spelt oan and Professor of Physics
in: the University ollege of S. Wales and Monmouthshire,
Cardif®.
§ 1. : a communication presented to Section A of the
British Associationt at Oxford in 1894, giving
an account of measurements made to determine the value
of the International Ohm in absolute measure by the method
of Lorenz, I referred to a small error consequent on the fact
that my standard coil is wound on a cylinder, the section of
which at right angles to the generating lines is not a circle
but an ellipse of small excentricity.
In considering the effect of this ellipticity on the value of
the resistance calculated from the observations, it must be
noted that the ordinary formula implies that the coil is
circular. This formulais ~
R = Ma,
where R=the resistance in absolute measure,
M=the coefficient of mutual induction of the standard
coil and disk circumference,
n=the number of revolutions of the disk per second.
But we are primarily concerned with the balance of the
electromotive force between the ends of the resistance when
* Communicated by the Phy sical Society : read May 22, 1896.
+ Report of Electric Standards ears arene If., Brit. Ass.
Report, 1894.
108 Prof. J. V. Jones on the Magnetic Field
the current is passing through it, and the electromotive force
between the points of contact of the brushes on a radius
of the rotating disk when the same current is passing through
the standard coil; and this balance gives us, in the general
ease, the formula
R = 20n | r Har, oar Lah et
we %
where dp and a, are the distances from the centre of the disk
of the points at which the internal and external brushes are
applied, and H is the magnetic-field intensity at a point
on the radius through these points of contact at a distance r
from the centre when unit current is passing through the
standard coil. [If the coil is circular and coaxial with the
disk, this formula simplifies into the formula first mentioned. |
§ 2. In the case of the coil used in my observations, the
dimensions of which are given below, the excentricity of the
elliptical section is so small that the value of the integral (A)
differs only by a small quantity from the value it would have
for a coil otherwise similar but of circular section with radius
equal to the arithmetic mean of the semiaxes of the elliptical
section ; and to a first approximation we may assume that the
percentage correction to be applied to the value of the integral
for the circular coil to obtain its value for the elliptical coil is
the same as the percentage correction to be applied to its
value for the circle in which the mean plane cuts the cireular
coil to obtain its value for the ellipse in which the mean plane
cuts the elliptical coil. It will be sufficient for our purpose,
therefore, to calculate the latter percentage correction. .
§ 8. Let H, be the value of the field intensity at a given
point.in the disk due to unit current in the ellipse, and H, the
value of the field intensity at the same point due to unit
current in the circle coplanar and concentric with the ellipse
and of radius c equal to the arithmetic mean of its major and
minor semiaxes.
Then we have, in this case,
A=2a a H, dr
QB
= 2a nf ‘ H,dr+2a a| ‘ (H,—H,) dr
° a
ay
=u M,+2en| rede
a
=n (Me+ B),
due to an Elliptical Current. 109
where M. =the coefficient of mutual induction of the circle
and disk circumference, |
and eo —H.,.
$4. To obtain the value of B, we must first find. an
ee for the intensity of magnetic field (H.) due to an
elliptical current at a point in its plane within it in terms of
the semiaxes of the ellipse and the coordinates of the pomt.
Let the equation to oe ellipse be ;
2+5=1,
and let £, be the coordinates of the point in question. 3
The intensity of the magnetic field at the point (&, 7) may
be expressed by the formula le.
2m
mee,
o -P
where p, 8 are the polar coordinates of a point on the ellipse
referred to the point (&, 7) as origin.
Forming the polar equation to the ellipse with the point
(€, 7) as origin, solving for ~, and substituting in the above
equation, we have E ep oo
H = BPE ae |, dO »/f” cos? 6 4+ 2h? cos O sin O +9? sin’ 0,
where f? =P —7’,
f= af,
hi? = En,
Let $*, x’ be determined by the equations
gp? oe x =)" +9
eS ee yeni}
and we have
Jel ab (27 Fey
ee dO / fp? cos 0+ x? sin? 8
tab
= gon Bs x
where
2
Bihan) = { 7 40 VG cos OF x sin’ 6,
0
an elliptic integral of the second kind. The value of K(¢, 7
110 Magnetié Field due to an Elliptical Current.
for any values‘of 6 and y may be readily calculated by finding
their arithmetico-geometrical mean (v. alae : See
pte rals, chap. xili.).
a ‘5. In alte ease of my standard coil
; | 8 a=10°5419 inches,
6=10°5340 inches,
an the angle made by the radius of brush contact ithe iis
major axis is approximately 55°.
I have calculated the values of H- and H, for points on
this radius distant 1 inch, 2 inches, 3 inches, &ie., from the
centre, with the following results : — | ie
He--22. om ro+2e.
r H.+2a
0 094896 094896 — ‘000000 ‘000000
] "095542 "095542 “000000 ‘000000
De 097551 _ 097552 “000001 ‘000002 ....
3 ~*301142 "101144 000002 ‘000006
+ 106764 "106768 “000004 - - 000017
5 *115252 *115260 ~ 000008 “000040
6 "128235 "128250 "000015 ‘000089
on - *149170 "149198. ‘000028 "000199
o=He—He.
~ § 6. We are now in a position to calculate ae value of B.
In the apparatus I used :
ao= "0585,
a= 6°4949.
By numerical integration between tase oe we —
LS eee pe 000163,
and
re rodr = 00643.
Also we have
a= 64949,
and c=10°5377,4 = - =
and hence M,=94:014,
and therefore _ sce
a); B+M,=:0000684,
or the required correction is ‘00684 per cent.
And by the argument contained in § 2 this may ‘be taken
High Tensions in Moving Liquids. 111
to be to a first approximation the correction to be applied in
the case of the coil. Now the coefficient of mutual induction
of the coil and disk calculated on the assumption that the
section of the coil at right angles to the generating lines is a
circle of radius ¢c is equal to 16613°75. Adding the calcu-
Jated percentage correction to this value we have finally for
the apparatus used by me
R=n (16613°75 + 1:14) =16614°89 n.
‘The value of the International Ohm in absolute measure
previously given by me in the paper to which I have referred
was
-99976 x 10° absolute units.
_ The value corrected for the coil ellipticity is
-99983 x 10° absolute units.
VIL. High Tensions in Moving Liquids. X |
To the Editors of the Philosophical Magazine.
GENTLEMEN,
OME years ago, after some holiday enjoyment of making
“ducks and drakes”’ on a calm sea with flat smooth
pebbles, it occurred to me as strange that I had never seen
any theory given of “ducks and drakes”’ ; but convinced
that a phenomenon, so beautiful as these light rebounds of
solid from liquid, and so defiant of those rules of propriety so
concisely laid down in our treatises on hydrodynamics for’
liquids aspiring to be perfect, must have been handled by
some master of fluid motion, I thought it would be a good
exercise to work out a sketch explanation of the phenomenon
and see how far it coincided with the authoritative theory
when found. But without an exhaustive search I have yet
come to the conclusion that there is really no theory of the
phenomenon published, for 1 have not encountered even an
allusion to-an explanation of these most elegant “ducks and
drakes.”” Under these circumstances, and at a time when
hydrodynamicians are making vigorous efforts to break away
from the academic allurements of the perfect liquid and to do
‘some service (hard service it seems to be) in the cause of the
plain liquids of nature with their lamentable imperfection of
an inveterate viscosity, it seems to me that there may be some
justification for the brief publication of my rough sketch of a
theory, because it brings out the possibility of the existence of
high tension in liquids-in motion, so that ina general theory
112 > Mr. W. Sutherland on -
of the motion of natural liquids both viscosity and capacity
for exerting tension have to be taken into account. Of
course one gathers either directly or indirectly that most of.
the great writers on hydrodynamics, beginning with, say,
Poisson and Stokes, saw the philosophical necessity for
recognizing tension in liquids in motion ; and Maxwell, when
introducing his idea of the ‘‘ time of relaxation ”’ of an impul-
sively generated tensile strain, intended it to apply even to
natural gases as well as to liquids. The merit of the “ ducks
and drakes ” phenomenon is that it brings liquid tension from
the region of scientific imagination to that of actual fact, and
demonstrates in a brilliant manner that tension in moving
liquids is no mere subordinate matter only slightly altering
the properties of the liquid from those of the ideal perfect
one, but produces a fundamental change. Of course static
tension, of which Worthington appears to be the latest inves-
tigator, has already received a certain amount of practical
attention. 5
The broad facts of “ducks and drakes” are, that a solid
body having a flat face and made to impinge with this face
on a liquid surface parallel to it tends to rebound from the
surface if the component velocity parallel to the surface
exceeds a certain value; the tendency depending on the rela-
tion between the area of the face and the mass of the body,
and also on the angle the direction of its velocity makes with -
the normal to the liquid surface: the larger the mass of the
body per unit area of flat face the smaller is the ratio of the
normal velocity before impact to that after ; the smaller the
angle that the velocity makes with the normal the smaller is
the ratio of the normal velocity before impact to that after,
until a limiting angle is reached at which the ratio appears
to be zero, so that for smaller angles the rebound ceases
entirely. Of the existence of this limiting angle with ordi-
nary stone, glass, and oyster-shells, and sea or river water, I
have satisfied myself by many trials, and I have thought of
making a quantitative study of the laws of the rebound of
solid from liquid, but see no immediate prospect of the neces-
sary time. But for present purposes the existence of the
limiting angle is an important fact, because it implies also
what has been already stated, namely, that with a given
normal velocity there must be associated a velocity parallel
to the surface not less than a certain value if there is to be
rebound ; and the increasing efficiency of the rebound with
increasing angle implies that the velocity parallel to the
surface is the most important element in the conditions of
rebound. Evidently there is an entire difference between
High Tensions in Moving Liquids. 113
the laws of rebound of a solid from a flat liquid and from a
flat solid surface.
Let us suppose the flat face of the solid to be a square of
side a, and also that it does not pass discontinuously into the
rest of the surface, but by means ofa thin strip of curved
surface passing tangentially into it all round, the sections of
the strip by planes normal to the side of the square being arcs
of radius r. Suppose the solid to move with the flat face
horizontal, and with horizontal velocity u and vertical ve-
locity v at the instant when it encounters a horizontal liquid
surface. To trace the effects of the encounter we had better
for a moment imagine wu to be zero, so that we have first the
simple case of normal impact with velocity v.
The first effect of the impact is to establish both motion and
compression in the water near the solid, and also to compress
the solid and diminish its motion. This goes on till the
instant when the face of the solid and the liquid in contact
with it are moving with the same velocity. But by this time
part of the energy imparted to the liquid by the compression
has changed itself into motion within the mass of the liquid ;
and apparently in all ordinary cases this part of the ener
is usually a large fraction, so that the liquid has only a little
compressional energy left with which to attempt to thrust the
solid body away from it; and thus the rebound fails, the dis-
tinction between the encounter of two solids and that of a
solid and a liquid being that in the latter case a large fraction
of the available energy is soon changed into energy of motion
within the liquid. Under these circumstances it comes to
pass that, in consequence of the motion in the liquid, its sur-
face near the solid takes the form of a curved depression
tangential to the curved edges of the face.
When the vertical velocity v is zero, the effect of impact
with only horizontal velocity can be studied separately. The
first effect is the establishment of intense tensile strain in both
solid and liquid. With viscous communication of motion to
the liquid and from the solid the tensile strain in the liquid
tends to relieve itself rapidly by generating motion within
the mass; but the solid for some little time must tend to restore
the strain as fast as the liquid relaxes it, so that we have a
short period of constant tension in the liquid near the solid.
When both velocities u and v are in existence, a combina-
tion of the states of affairs just described occurs ; by the end
of some short time ¢ the vertical velocity v is destroyed, and
the surface of the liquid contains a curved depression which is
tangential to the curved edge of the flat face of the solid;
the depression moves with a velocity comparable but not
Phil. Mag. 8. 5. Vol. 42. No. 254. July 1896. I
114 Mr. W. Sutherland on
equal to that of the solid, for the surface of the liquid near
the solid has been set in motion by it. Let two sides of the
square flat face be parallel to the direction of wu, then we may
liken the surface of the depression in the liquid to a stretched
membrane, and the front and back edges to portions of
cylindrical surfaces of radius 7, over which the stretched
membrane passes with the tension perpendicular to the axes
of the cylinders, and therefore exerting pressure on the
cylinders ; at the side edges the tension is parallel to the axis
and is therefore devoid of pressure effect.
Let 6 be the width of the strip of curved edge in contact
with stretched liquid, and let the tension near the surface of
the liquid be T per unit width, then the lifting pressure on
each area ab is abl'/r, making an angle 6/2r with the vertical ;
the total lifting force due to tension is therefore
2 cos (b/2r)abT/r.
This result can obviously be extended to the case where the
face of the solid is not flat with rounded edges but is any
eurved surface ; let A be the area of contact between solid
and liquid, T the mean tension, and 1/7 the mean curvature
of the face in the direction of the tension, then each unit of A
is subject to a normal pressure T/r, and the total vertical
lifting pressure is the sum of the vertical components of all
the normal pressures. Thus, then, we can include all cases
in the one general expression sufficient for our purposes if we
say that the lifting force is equal to cAT/r where ¢ is a con-
stant, A is that part of the face of the solid in contact with
the liquid and having finite curvature, the average value of
which is 1/r in the direction of T; c, A, T, and r being also
average values for the duration of the impact.
Now suppose for a moment that the solid has only the
vertical velocity v at the moment of impact, and let h be the
distance below the free surface of the liquid to which its face
penetrates before it is brought to rest, then we may assume
the energy given to the liquid to be proportional to A?, and
then h?=kmv?/2 where m is the mass of the solid and & is a
constant. Then when w is not zero we have to take account
of the fact that the force cAT/r opposes the descent of the
solid, doing work AcAT/r against it, and therefore
1? =k(mv?/2—hcAT/?).
Then the uplifting force cAT/r will in most cases act on the solid
through distance h, and discharge it from the surface of the
liquid with a vertical velocity v’ upwards given by the equation
mv?/2=heAT/r,
vy”? 1
ee hal thr/keAT
fligh Tensions in Moving Liquids. 115
If T is proportional to the excess of the velocity u above a
lower limit U, so that T=K(u—U) where K is a constant,
then v?/=1/{1+hr/KkeA(u—U)},
and this expression corresponds to the general laws of ‘ ducks
and drakes,’”’ for since it holds only for values of u greater than
U there is a certain minimum horizontal velocity required
by a solid which is to rebound from a liquid ; for horizontal
velocities greater than this there is rebound, but the energy
of the vertical motion after impact is always less than that
before impact, although it becomes more and more nearly
equal to it the greater « becomes ; also it appears from the last
equation that with finite velocities the angle of incidence for
which rebound is possible has a limiting value, because
tan z=u/v ; and u having the lower limit U, and v the upper
limit V of experimental possibility, tan 2 and therefore
a has a lower limit. Again, as h increases with m, and A
generally increases with the size of the face of the solid, it
follows from the equation that the smaller the mass of the
body and the larger the face the more nearly does v’” equal
v’, whence the advantage of thin flakes of stone and shell for
getting “ducks and drakes.” It is to be remembered that in
the case of a flat face the area over which there is finite
curvature is only the small strip of transition from the flat
surface to the rest of the surface, and thus A becomes very
small, but so also does 7 become small, and the effect of the
shape of the face is best expressed by making A the product
of an average width a and an average length 6 in the direction
of motion, so that A/r becomes ab/r, which depends on a and
the angle b/r. It should be noticed that the velocities are not
great which are required to produce the phenomena of ducks
and drakes, for at the last rebound of a series of ten or twelve
the motions are very gentle, but the tension called forth must
be remarkably high seeing that the impact lasts so short a time.
We all know the wonderful manner in which Kelvin has
helped us to grasp the coexistence in the ether of apparently
irreconcilable properties by his homely instances of jelly and
pitch, and it seems to me that “ ducks and drakes ” carry in
themselves a suggestiveness only communicated to jelly and
pitch by the sagacity and imagination of a master mind.
WILLIAM SUTHERLAND.
Melbourne, March 1, 1896.
Ranke |
VILL. Notices respecting New Books.
Azimuth Tables for the Higher Declinations. By H. B. Goopwin.
London: Longmans, Green, & Co. 1896.
(THESE tables, extending from 24° to 30° Declination, may be
considered supplementary to those of Burdwood and Davis.
The limits embrace the moon, larger planets, and a belt of bright
stars. The latter are more particularly useful in the Southern
Hemisphere, which contains no practical pole-star.
Unlike the tables of Burdwood and Davis, these have for one
argument the star’s altitude, excepting in the supplementary por-
tion termed Table B, which follows the ordinary usage of having
as argument the star’s Hour-angle. —
The tables are very legibly printed in old-face type, which lends
itself peculiarly to figure work. The omission, however, of the lead-
ing figures excepting at the change of the degree is a very doubtful
advantage, and it is very probable that a table giving degrees
and tenths only with all the figures printed would be a more useful
one in the hands of the navigator. The subdivision of the degree
into minutes possesses no advantages whatever, and only adds to
the labour of differencing and interpolation. We observe a few
figure errors, but on the whole the tables appear to be fairly accu-
rately printed and read, and they should be a valuable addition to
the chart-room of the skilful navigator, and an incentive to the
more general use of star observations in practical seamanship. In
the introduction the author states that Burdwood mentions only
three bright stars between the equator and 23° §. (the limits of his
tables). If we take, however, the stars in the Nautical Almanac
within the author’s limit of brightness (Mag. 2°4), we find ten stars
against four comprised in Goodwin’s tables. Again, the author
states ‘‘for the moon, the tables will be brought into requisition
for approximately one third of the month,” overlooking the fact
that when the moon’s node is between 270° and 90° the declination
of the moon never exceeds the obliquity of the ecliptic. This will
be for a period of over nine years successively, or one-half the
revolution of the node.
IX. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from vol. xli. p. 524.]
March 11th, 1896.—Dr. Henry Hicks, F.R.S., President,
in the Chair.
The following communications were read :—
1. ‘On an Alpine Nickel-bearing Serpentine with Fulgurites.’
By Miss E. Aston, B.Sc. With Petrographical Notes by Prof. T.
G; Bonney, De, LL.D., F-R.8., V.P:G:8.
The specimens described were collected on the summit of the
The Pliocene Glaciation of Subalpine Switzerland. 117
Riffelhorn (near Zermatt) by Prof. W. Ramsay, F.R.S., and J. Eccles,
Esq., F.G.S. As they showed some very well marked ‘lightning-
tubes,’ the former thought that possibly analyses might prove
interesting. These were undertaken by Miss Aston at University
College, London. ‘The rock is a serpentine, somewhat schistose from
pressure, which has been formed by the alteration of a rock chiefly
composed of olivine and augite. One of the analyses gives 4:92
per cent. of nickel oxide and hardly any lime. Prof. Bonney detected
some awaruite under the microscope, but not nearly enough to
account for the analysis. Reasons are given to show that the
nickel oxide probably replaced lime in the pyroxenic constituent of
the rock.
The tubes, about 54; inch in diameter, are round in section,
cleanly drilled, and lined with a very thin film of dark brown
or black glass. The microscopic structure of this is described,
as well as that of glass made by melting the rock with a blow-
pipe (using oxygen). Some fulgurite-glass from the Hornli has also
been examined (much resembling that described by Mr. Rutley
from Monte Viso), and an analysis of this rock is given.
2. ‘The Pliocene Glaciation, Pre-Glacial Valleys, and Lake-Basins
of Subalpine Switzerland: with a Note on the Microscopic Struc-
ture of Tavayanaz Diabasic Tufa.’ By C.S. Du Riche Preller, M.A.,
PED EGS. F:C.8.; A.L-MCLE., M.1EE.
I. The main object of this paper, which is the sequel to one read
last session, was to solve the problem whether the Pliocene glacio-
fluviatile conglomerates of the Swiss lowlands were deposited on a
plateau or in already existing valleys. or the purpose of this
enquiry, the author examined last summer a large additional
number of glacial high- and low-level deposits throughout the Zurich
Valley over an area more than 40 miles in length; and his investi-
gations further led him to important conclusions with respect to the
combination of causes which determined the formation of the lake-
basins lying in the same zone at the foot of the Alps.
II. The author established the true characteristics of the Pliocene
nagelfluh as distinguished from Miocene, purely fluviatile conglome-
rate on the one hand, and from glacio-fluviatile Pleistocene gravels
on the other. With respect to the origin of the Pliocene conglome-
rate, he contended that the material composing the same was not
transported from a great distance, but was, in the main, derived from
the enormous accumulations of Miocene nagelfluh at the foot of the
Alps. Specimens of Miocene nagelfluh-pebbles were exhibited, in-
cluding the so-called ‘ Tavayanaz Sandstone,’ which the author, in
an Appendix to the paper, showed to be diabasic tufa.
III. The author described in detail a variety of glacial exposures,
and showed that Pliocene nagelfluh in situ, of which he exhibited
numerous specimens, occurred not only on the ridges of the hills, but,
at a gradually ascending level, also at and near the floor of the
Ziirich Valley.
118 Geological Society :—
Hence he contended that at the advent of the first glaciation
the Ziirich Valley was already eroded, and that, consequently, the
term ‘ Deckenschotter,’ or plateau-gravel, was not strictly applicable
to the Pliocene glacio-fluviatile deposits of the Swiss lowlands. In his
view, the isolated high-level deposits were formed during the inter-
mittent shrinkage of the Upper Pliocene ice-sheet, while the low-
level deposits were formed during the subsequent recession of
individual glaciers left in the several valleys.
IV. The author reconstructed the pre-Glacial floor of the Ztirich
Valley upon the evidence of the solid rock and of the low-level
Phocene nageltiuh deposits, with the result that the depth of the
lower part of the Valley was approximately that of the present day,
while the floor of the upper part was at a higher level (maximum,
300 feet above present lake-level), and was subsequently lowered
by earth-movements. He further adduced evidence that the Sub-
alpine valleys of the Reuss, Aare, and Rhine were likewise excavated
before the first glaciation. By calculation, he arrived at an estimate
of the time required for the excavation of the Ziirich Valley, and
contrasted the erosive energy of the river with the impotence, on
mechanical grounds, of a glacier 7000 times larger in volume.
V. The author showed that the Lake of Ziirich owes its origin, in
the first instance, to a zonal subsidence (probably between the first
and second glaciation) of about 1000 feet, as evidenced by the
reversed dip of the disturbed molasse-strata between the lakes
of Ziirich and Zug. During the second and third Ice-periods, the
original Jake-basin was gradually filled with glacial and fluviatile
deposits at both ends, and was finally restricted to its present
dimensions by a post-Glacial bar deposited at its lower end by
a tributary river. In the author’s view, the other Subalpine lakes,
extending from the Lake of Constance to Lac Bourget in Savoy,
owe their origin and present limits, in the main, to the operation of
similar causes.
VI. With regard to the main question, the author averred that
the Lower and Middle Pliocene period was, in Switzerland, entirely
one of erosion and denudation on a prodigious scale. Irrespective of
the evidence he had adduced, he was therefore driven to the conclusion
that at the advent of the first Ice-period in Upper Pliocene times,
the principal Subalpine valleys must have been already excavated
approximately to their present depth, and that ever since then the
action of the great Alpine and Subalpine rivers has been, as it is
still in our own day, mainly directed to regaining the old valley-
floors by removing those enormous accumulations of glacial and
glacio-fluviatile material, which are respectively the direct and
indirect products of three successive and general glaciations.
3. ‘Notes concerning certain Linear Marks in a Sedimentary
Rock.’ By Prof. J. KE. Talmage, D.Sc., F.G.S.
The marks described in the paper occur in a fine-grained argil-
On Submerged Land-surfaces at Barry. 119
laceous sandstone referred by the U.S. Geological Survey to the
Triassic or Jura-Trias period, which is found on a low tableland
within 2 miles of the bluffs overlooking Glen Canyon. The marks
commonly appear as straight lines intersecting at right angles, but
some have a pinnate distribution, suggesting engravings of frost-
flowers. A description of the markings is given, and various
experiments made in the Jaboratory to illustrate the effects of
formation of crystals formed over sediment are described.
March 25th.—Dr. Henry Hicks, F.R.S., President,
in the Chair.
The following communications were read :—
1. ‘On Submerged Land-surfaces at Barry, Glamorganshire.’
By A. Strahan, Esq., M.A., F.G.S. With Notes on the Fauna and
Flora by Clement Reid, Esq., F.L.S., F.G.S., and an Appendix on
the Microzoa by Prof. T. Rupert Jones, F.R.S., F.G.S., and
F, Chapman, Esq., F.R.M.S.
Excavations for a new dock at Barry have disclosed a series of
freshwater or slightly estuarine silts with intercalated peats, below
sea-level on the north-eastern side of the island. The site of the
excavation was overflowed by the tide until the year 1884, when the
docks were commenced. The newest deposits seen are therefore
Blown Sand, Scrobicularia-clay, and sand or shingle with recent
marine shells.
These rest on an eroded surface of blue silt, with sedges in
position of growth. Four peat-beds occur in this silt, at 4, 11,
20, and 35 feet below Ordnance datum respectively. The upper-
most peat contains a seam of shell-marl, partly composed of the
shells of ostracoda and partly of Bythinia, Limnea, &c. The
second is a mass of matted sedges. ‘The third is aland-surface, and
in places consists almost wholly of timber with the stools and roots
in situ. The fourth is also an old land-surface, as is proved not
only by the presence of roots in place beneath it, but by numerous
Jand-shells. A fragment of a polished flint-celt was found by
Mr. Storrie embedded in the lower part of the uppermost peat.
By a comparison with the existing maritime marshes of the
neighbourhood, it is shown that the fourth peat indicates a sub-
sidence of not less than 565 feet.
The sea encroached upon the area in consequence of this sub-
sidence. It entered by the lowest of three low cols in the southern
water-parting of the Cadoxton river, thus isolating the portion of
land now known as Barry Island. A slight further movement
would have converted the water-parting into a chain of islands.
2. ‘On a Phosphatic Chalk with Holaster planus at Lewes.’ By
A. Strahan, Esq., M.A., F.G.S. With an Appendix on the Ostracoda
and Foraminifera by F. Chapman, Hsq., F.R.M.S.
This rock, which occurs at the base of the Upper Chalk, at the
120 Geological Society :—
horizon of the Chalk Rock, does not exceed 13 foot in thickness and
persists for a few yards only. In composition and microscopic cha-
racter it presents a close analogy to the Taplow phosphatic deposit,
which, however, occurs at the top of the Upper Chalk. Like it, it
consists of brown phosphatic grains embedded in a white chalky
matrix. The grains include a large number of pellets, attributable
to small fish, phosphatized foraminifera, chips of bone, &c. Fish-
teeth also occur in abundance.
To complete the resemblance, the Lewes deposit rests on a floor
of hard nodular chalk, beneath which is a white chalk traversed by
irregular branching pipes filled with the brown variety. Such
‘floors’ are attributed to concretionary action ensuing upon a pause
in the sedimentation. The piped chalk is compared with the
structure known as Spongia paradowica.
It is concluded that phosphatized deposits may occur at any
horizon in the Chalk ; that the phosphatization is due to small fishes,
attracted by an unusual abundance of food; that they are shallow-
water deposits, and associated with a pause or change in the sedi-
mentation.
Mr. Chapman furnishes a list of 42 species and varieties of
foraminifera and 6 species of ostracoda. The former indicate a
deeper water origin than do those of the Taplow Chalk. He notes
the occurrence for the first time in this country of Gypsina Coete,
Marrson.
3. ‘On the Classification of the Strata between the Kimeridgian
and the Aptian.’ By Dr. A. P. Pavlow, Professor of Geology in the
University of Moscow, For.Corr.G.8.
In this paper the author discusses the new evidence respecting
the paleontology of the Lower Cretaceous and Upper Jurassic
deposits of Russia which has come to light since the publication by
himself and Mr. Lamplugh of ‘ Les Argiles de Speeton et leurs
Equivalents’ (Moscow, 1892). He is now enabled to fix with
certainty the zones of Hoplites riasensis and Olcostephanus hoplitoides
of the provinces of Riasan and Simbirsk, and is thus in a position
to correct and complete his former classification of the Upper
Jurassic and Lower Cretaceous rocks of Russia, and to define more
strictly their relationship equivalent to the strata of other countries.
The whole of the Petchorian Series—that is, the zones of
Ammonites stenomphalus and Amm. Keyserlingi—is now regarded
as Lower Neocomian of a hitherto unknown boreal type, notwith-
standing the affinity of its fauna with that of the underlying
Jurassic (Aquilonian) strata. The author is thus led to carry up
into the Cretaceous the corresponding stages in Western Europe,
including the upper part of the zone of Belemnites lateralis of Speeton
and Lincolnshire, the Upper Berriasian of South-eastern France,
and probably the Hils Beds of Germany, instead of classing these
with the Jurassic as he had previously done.
A table is given in which the detailed correlation of the rocks
Upper Lias and Inferior Oolite in Northamptonshire. 121
between the Kimeridgian and the Aptian of the various regions is
attempted.
The comparison of the beds of England and Germany with those
of Russia is supported by some new evidence based on the Aucelle,
four species of which are described as occurring in the Claxby
Ironstone and Spilsby Sandstone of Lincolnshire.
In conclusion, the author shows that in the period under con-
sideration the shore-lines of Europe have been shifted by slow
progressive movements passing latitudinally through the region, and
that these movements did not affect the whole area simultaneously.
Hence many complicated interchanges of fauna were brought about,
which can only be unravelled by studying the whole course of
events over wide areas.
April 15th.—Dr. Henry Hicks, F.R.S., President,
in the Chair.
The following communications were read :—
1. ‘The Junction-Beds of the Upper Lias and Inferior Oolite in
Northamptonshire.—Part I. Physical and Chemical,’ By Beeby
Thompson, Esq., F.G.S., F.C.S.
The author, while combating the view that a considerable
unconformity exists between the Upper Lias and the Inferior Oolite
of Northamptonshire, brings together much evidence to illustrate
the effects of slipping, and to show that these effects may be mistaken
for those of unconformity. He also applies the evidence which he
has collected to illustrate certain points in the physics of valley-
formation.
After giving details as to the horizon of the springs of the district,
the distribution of water in the Inferior Oolite, and the development
of the springs, be argues that every valley of the district has been
elongated in the direction which it now has by a stream originating
in a spring always at its head, and that the development of channels
towards particular points of discharge has been the chief agent in
initiating the formation and guiding the direction of all the minor
valleys of the river-system within the influence of the same set of
beds. A description of the characters of the slopes follows, and
their significance is discussed, The structure of the hills and valleys
of the district occupies the next portion of the paper, and the
author considers that corresponding to the deepening of a valley by
denudation there is uplifting of the beds below it, and at the same
time an outward and upward thrust along the hillside which lifts
beds there; also, that hills are reduced in height by sinking as well
as by denudation of their upper parts. In discussing the question
of unconformity between the Inferior Oolite and Upper Lias, the
rarity of exposures of true junctions is noted, the junctions which have
been chiefly examined by other observers being obscured by slipping ;
and reasons are given for inferring an absence of unconformity at
the horizon, both on account of the character of the true junctions,
and from other considerations. The author, however, gives reasons
Phil. Mag. 8. 5. Vol. 42. No, 254. July 1896. K
122 Geological Society.
for believing that a slight unconformity occurs in the Upper Lias,
so that the lower part of the juwrenszs-zone is absent, and not its
upper part as has been elsewhere inferred.
2. ‘Contributions to the Stratigraphy and Paleontology of the
Globigerina-limestones of the Maltese Islands.’ By J. H. Cooke,
Ksq., F.L.S., F.G.S.
A bibliography of the Globigerina-limestones is followed by some
remarks on the physical features and general distribution of the
strata. The limestones are divided into nine subdivisions, lettered A
to I, the former being uppermost. Four seams of phosphatic nodules
form the subdivisions B, D, G, and I, and local nodule-bands also
occur in EK. The subdivision G serves as a line of demarcation
between the Langhian Series (Miocene) and the Aquitanian
(Oligocene). Details of the lithological and paleontological cha-
racters of the various subdivisions are given, and the author
concludes that I and the lower part of H were laid down on a
sinking sea-floor, in about 300 fathoms of water; that the upper
part of H and G, F, E, D, composed to a large extent of Globigerina
and other pelagic organisms, were probably deposited in about
1000 fathoms; while C, B, and A were probably laid down, like I
and the lower part of H, in about 300 fathoms of water.
3. ‘On the Geology of the Neighbourhood of Carmarthen.’ By
Miss Margaret C. Crosfield and Miss Ethel G. Skeat.
The area described lies approximately within a 4-mile radius of
Carmarthen. The beds of the district have been subjected to com-
plicated foldings, amongst which an earlier set, giving rise to a
number of small anticlines with north-and-south axes, and a later
more extensive set, due to the series of earth-movements which pro-
duced the great Condrusian ridge, producing anticlines and synclines
having a general east-and-west trend, can be made out. The rocks
forming the subject of the present paper occur in one limb of a
complex anticline produced during the latter set of movements. in
this limb beds of the following ages occur:—Tremadoc Slates,
Lower and Upper Arenig, Llanvirn, Llandeilo, and Bala. These
beds are described in detail. A regular succession of strata from
Tremadoc Slates to Decranograptus-shales is found, while the Bala
beds of Mount Pleasant abut on Arenig strata, aud the reason for
this irregularity has not yet been decided by the authors. The beds
are compared with those of other areas. The Tremadoec Slates are
equivalents of Stage 3.a of the Christiania district ; the Lower Arenig
Beds with Phyllograptus angustifolius, and the Upper Arevig with
Didymograptus nitidus, &c. resemble those of other British areas ;
the Llanvirn Beds contain Didymograptus bifidus and other fossils ;
the Didymograptus Murchisoni-beds are well known elsewhere. The
Llandeilo Limestone is probably represented by sandy beds with
Asaphus tyrannus; and the Dicranograptus-shales are like those of
the Haverfordwest region. The Bala Beds of Mount Pleasant.
Intelligence and Miscellaneous Articles. 123
contain Stygina Murchisone and other fossils found elsewhere in
Bala rocks.
A description of new fossils forms the concluding portion of the
paper.
X. Intelligence and Miscellaneous Articles.
ON A ROTATIONAL MOTION OF THE KATHODE DISK IN THE
CROOKES TUBE. BY FRANCIS E. NIPHER.
T is well known that the equations which represent the pro-
perties of the magnetic field external to a conductor are incon-
sistent when applied to points within the body of the conductor.
Assuming the total magnetic force within to be tangent to a circle
whose centre is at the wire centre, and that the surfaces of equal
potential are radial planes. Assuming the force due to an element
3 2 : 2d ,
of the conductor of infinite length and of section ds to be — it
follows that the force at any point without the conductor varies
inversely, and at an internal point directly, as the distance from
the centre. Jf now within the wire we assume any radial plane
as a datum equipotential plane, and determine the locus of any
other equipotential surface, such that the difference of potential,
measured along the lines of force, is constant, this surface turns
out to be one having as a cross-section a spiral known as the lituus
having the radius as an external asymptote, and reaching the centre
after an infinite number of turns. It is evident that these internal
surfaces of equal potential cannot be both radial planes and spiral
cylinders.
Maxwell disposes of this absurdity to which the equations lead
in the single sentence which closes section 606 of his ‘ Electricity
and Magnetism.’ He says: “ Within the substance of the con-
ductor, there is no such thing as magnetic potential.”
It has long seemed to me that this failure of the equations
must be the result of leaving some elements of the problem
out of the discussion. I have spent a great amount of time in
seeking for some rotational phenomenon hitherto unrepresented in
the equations. Until recently the results were wholly negative.
While recently experimenting with a Crookes tube I observed that
the circular aluminium disk of the kathode became slightly loose on
the aluminium wire, and that it was constantly rocking in rotary
motion on the wire. After several days of use, during which it
had been decided to construct a tube with disks capable of rotation,
the kathode disk suddenly became loosened, and began to rotate
slowly on the wire as an axis.
The bearings were somewhat rough, and the disk was not per-
fectly balanced. It often stopped, but then began to rock against
the obstacle until it again freed itself. The direction of rotation
was contrary to the hands of a clock, when the disk was viewed
124 Intelligence and Miscellaneous Articles.
from the point where the kathode wire pierces the wall of the
tube. All attempts to accelerate or retard the motion by means
of strong bar magnets, as in Barlow’s wheel, were without effect.
Placing the tube at various distances from the induction-coil and
giving the disk all possible positions in the earth’s field produced
no change in the rotation. A more decided rotation was produced
by using the brush-discharge of a 24-inch Holtz machine. No
rotation has been produced as yet when the leading wires were
in metallic contact with the conductors of the Holtz machine,
but when the leads consisted of rods having spherical terminals,
separated by short spark intervals, the rotation was always seen.
When the loose disk was made the anode, no tendency to rotation
was observed. Thus far all attempts to produce the effect in air
of ordinary pressure have failed, but the work in this direction is
not yet concluded.
In the tube used, the tendency to rotation was not observed
until by long use the vacuum had become very high, and it has now
nearly reached the limit where the sparks pass around the tube,
rather than through it.
The leading-in wires are at right angles to each other in the tube
used. Tubes are now in preparation which will have rotary disks
facing each other as well as at right angles to each other, and
various other features, by which it is hoped that many questions
which at onee suggest themselves may be answered. There is
much reason to suspect that the gas particles do not shoot off
normally from the surface of the disk, but in a vortex the axis of
which is in the two dark spots opposite the kathode faces. The
fact that the anode does not respond, and that similar experiments
in open air have thus far failed, seems to point to the kathode dis-
charge as the direct active agent. This view is not easily reconciled
with the result of the experiment made by Crookes with the hemi-
cylindrical kathode (‘ Nature,’ July 3, 1879, p. 229, fig. 3), but the
figure shown does not seem to quite agree with the description of
it. Experiments are now in preparation which will decide this
question. It is possible that the rotation observed is a direct
action and reaction between the current in the disk and the ex-
ternal field due to the current. In this case the rotation apparently
ought to be producible in open air, and on the anode terminal of
the Crookes tube.
Whatever may be the direct agency producing this rotation, it
seems apparent that we now have an experimental basis for impos-
ing a term representing a rotation into the equations representing
the conditions within a conductor.—Transactions of the Academy
of Science of St. Louis, vol. vi. no. 7 (May 8, 1896).
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XI. Onthe Convergency of Fourier’s Series. By W. WILLIAMS,
B.Sc., Royal College of Science, London™.
Me
r4\HE convergency of Fourier’s series is a subject which .
has been so fully investigated by eminent mathemati-
cians that it is necessary to offer some apology for venturing
to discuss it afresh. Itis, however, a subject of such singular
difficulty,—a difficulty which has only been partially over-
come,—and the investigations connected with it are so
laborious and abstruse in character, that any simplification
that may be effected in the method of attack is of value quite
apart from any fresh light that may be thrown upon the con-
vergency itself. The chief difficulty connected with the
investigation is that of assigning the necessary conditions to
be fulfilled by the function which determines the coefficients
of the series, and this difficulty arises from the highly general
manner in which the term “ function ”’ is defined and employed
in modern analysis. Of course, if we confine ourselves to
the comparatively simple functions which occur in the prac-
tical applications of the series, functions, for example, which
are continuous and obey the laws of the differential calculus,
much of this difficulty disappears. But it is necessary that
we should, in such a case, state clearly the limitation which
we make, as otherwise our investigation partakes of a too
general character, and proves too much. For, as we shall
* Communicated by the Author.
Phil. Mag. 8. 5. Vol. 42. No. 255. Aug. 1896. L
©9655
126 Mr. W. Williams on the
afterwards see, even when we limit ourselves to functions
which are finite and continuous, the limitation is too general,
and we cannot determine whether Fourier’s series is con-
-vergent or not until we know something of the nature of the
continuity of the function.
2. The object of the present paper is to simplify the
investigation of the subject, to bring it within the reach of
the student acquainted only with the elements of the Infini-
tesimal Calculus, and to exhibit in an elementary manner the
nature of the difficulties that have to be surmounted and the
principal results obtained. At the same time, in addition to
simplifying the discussion, and rendering it perhaps more
interesting and instructive, it is hoped that some additional
light will have been thrown upon the question of the con-
vergency, and that the limits within which the convergency
holds will be found to be to some extent widened and more
clearly discussed. a
3. The literature of Fourier’s series is very extensive, few
mathematical subjects having, perhaps, been so widely dis-
cussed. A very valuable account, both critical and historical,
of the chief investigations into the subject has been given by
Arnold Sachse (“ Versuch einer Geschichte der Darstellung
_ willkiihrlicher Functionen einer Variabeln durch trigonome-
trischen Reihen,” Gottingen, 1879) in an essay which has been
translated and published in the Bulletin des Sciences Mathé-
matiques, vol. iv. (1880). It is not proposed to enter here
into the history of the subject, or. to discuss the elementary
properties of Fourier’s series, such properties being treated
and illustrated in ordinary text-books. We have here to take
Fourier’s series in its most general form, as it stands, and
determine the conditions under which it is convergent.
4. Fourier showed that if an arbitrary function of 2 can
be expanded into a series of the form |
glee
E(v)=tay+ Sa, cos na + Sp, sin na,
1 1 LSEOE
the coefficients will be determined by the definite integrals —
= =|" F'(v) cos nvdv, a F'(v) sin nvdv,
ca) a TT} ae
v being written for x under the sign of integration. To
investigate the posszbility of the expansion, it is, therefore,
necessary to determine the most general conditions which the
function must satisfy in order that the series thus defined may
be convergent and tend to the limit F(z).
5. Of the different methods that have been employed in
Convergency of Fourier’s Series. 127
this investigation, there are two of particular importance on
‘account of the results to which they have led, and the fact
that they are still the methods most generally employed in
mathematical text-books. These are the methods of Poisson
and Dirichlet.
6. Poisson proceeds * by forming, from the given Fourier
series, another derived from it by multiplying each term of
the latter in succession by ascending powers of a quantity g
less than unity, and then finding to what limit this derived
series tends when g tends to the value 1. This method has
given rise to numerous and interesting investigations. In
particular, the method in the hands of Stokes in England led
to the discevery of the infinitely slow convergence of a
periodic series in the neighbourhood of a discontinuity.
Stokes showed that when a periodic series represents a dis-
continuous function, the rate of convergence of the series
increases indefinitely at the point of discontinuity, or that,
if a certain number of terms is required to represent the
continuous portion of the function to a given degree of
approximation, the number required to represent the function
to the same degree of approximation becomes greater and
greater as we approach a discontinuity. This important
discovery was published, in Dec. 1847, in a paper “On the
critical values of the Sums of Periodic Series”? (Cambridge
Philosophical Society). ‘The subject was independently in-
vestigated, and the same result discovered by Seidel, and pub-
lished in 1848 (Journal of the Bavarian Academy, 1847-49),
another remarkable instance of two investigators proceeding
independently along the same line of inquiry.
7. Dirichlet’s method of proceeding is to form an expres-
sion for the sum of the first n terms of the series taken in
order, and to find the limit to which this tends when n is in-
creased indefinitely. ‘This method was. given by Dirichlet in
1829 (Journal de Crelle, vol. iv. p. 157), in a paper which
contains the first rigorous investigation into the convergency
of Fourier’s series. The method is more direct than Poisson’s,
it enables us to investigate the limitations more simply and
effectively, and it has formed the basis for most of the
researches that have been subsequently made into the subject.
8. Dirichlet starts with the finite series
mF (v) Ov+ = S cos na F'(v) cos nrav
T
1 MS e ~ e
+ ~> sin nw F(v) sin rvdv,
1 aa
* Mémoires de Académie des Sciences, 1828, p. 574.
L 2
128 _ Mr. W. Williams on the
‘which becomes Fourier’s series when n=. Grouping to-
gether corresponding terms in nz, and summing the series
so formed, he gets
ae F(v)dvt+ Ls j F(v) (cos nx cos nv +sin nz sin nv) 2,
277 —7 5 ae ld —t
=| ine ipetes |. (2) cosmiosaaen
277, —T aa 1 ? =7
= a) F(v) ($+ Sen n(v—z) |dv,
—T 1
sordrle sin (2n+1)4(v—2)
Qa Naa sin 4(v—2) Oo
sin (2n+1)4(v—2)
where :
, sin $(v—2)
is the value of [$+ cos n(v—z)],
: |
by a well-known summation in ordinary trigonometry.
This final expression may be called the integral sum of the
series. It involves two variables, or rather it involves the
same variable twice over, namely, once in determining the
coefficients of the series, and then in assigning to the series
its different values. This double use of the same variable is
denoted by the different svmbols employed in the two cases,
namely, vin the one case, and 2 in the other. We may,
therefore, call v the variable of integration, and x the variable
of summation. Denoting the expression by 8,, Dirichlet’s
problem is to determine the limiting value of 8, when n=x
for all values of « between +7. This limiting value we may
conveniently denote by Sa. :
9. Asa result of his investigation, Dirichlet proved that if
the function F is finite, and single-valued between +7, and
has only a finite number of discontinuities and maxima and
minima between those limits, then Fourier’s series is con-
vergent, and tends to the value F(z) for all values of «
“except those which correspond to the discontinuities and the
limits +7 ; the value of the series at a point of discontinuity
being the mean of the values of the parent function on either
side of the discontinuity and infinitely close to it, and its
value at either limit the mean of the values of the parent
function at the two limits. This result has been made the
subject of further inquiry by later mathematicians, notably
Riemann, Heine, Cantor, and P. Du Bois-Reymond, the
inquiry relating to the necessity for the conditions laid down
by Dirichiet. For an account of these investigations, and of
Convergency of Fourier’s Series. 129
the results obtained, reference must be made to Sachse’s paper
already mentioned.
10. The method employed by Dirichlet to determine the
value of 8S, when n=©o is to break up the integral into the
sum of elements which are alternately positive and negative,
that is, into an alternating series with terms of finite magni-
tude. The manipulation of this series is, however, very
laborious, and the method of evaluating 8, by means of it is
long, and highly involved and indirect, and consequently is
not suited to the needs of the average mathematical student.
The investigation given in the following paper is a simplified
form of Dirichlet’s in the sense that it depends upon the
evaluation of the same integral Sp. But the difficulties at-
tending Dirichlet’s evaluation are avoided by breaking up
the integral into three portions, two of which are of finite
range, the limits being —7 to —h, and h to 7 respectively,
while the third portion is taken between +h, A being infi-
nitely small. It is then easy to show in a simple and
straightforward manner that the two first portions vanish
when n=, and that, therefore, the value of the integral
depends only upon the infinitely thin strip taken between
+h. By this means we’are enabled not only to evaluate S,
more easily and directly, but to investigate the limitations to
which the function F(x) must be subjected in a simpler
manner. For, as we shall see, the conditions that have to be
fulfilled by the function F(x) in order that the terms of the
series may be finite and determinate, and that the nth term
may be infinitely small when n=, which are conditions
that have to be fulfilled in the case of every series, are sufficient
to ensure that the two portions of 8, which lie outside the
limits +/ vanish when n=. The difficulties attending the
determination of the remaining conditions to be fulfilled by
the function are thus removed to the infinitely small portion
of it which lies between +A. ‘The investigation is given,
first, for the case of functions which obey the laws of the
differential calculus, this being the only case which occurs in
ordinary analysis. Afterwards, the case of functions in which
this condition is not fulfilled is taken up.
II.
11. Let F(x) be a finite, single-valued, and continuous
periodic function; and where continuous, let it be differen-
tiable. Then, since Fis periodic, and of period 277, the limits
of integration may be shifted through any distance at pleasure,
provided the interval between them remains unaltered and
equal to 27. Hence, whatever may be the value of the sum-
130 Mr. W. Williams on the
mation variable x, we get, by putting ies and inte-
grating between +7, |
Sine nae F(O+e)
. sin Aye 30
= =|" F(@+2)00+ oe cos n008;
5 ee Lis gee) a
so that the function under the sign of integration becomes
infinite only when @=0,
12. In the particular case when F(@+<) has a constant
value ¢ all the terms on the right in Qa) vanish eneeEs the
first, the value of which isc. Hence in this case S. At.
in addition, the limits of integration are from —7 = 0, or
from 0 to z, instead of from —z to am, we get S.=4e.
These results will be required later.
13. Since the function under the sign of integration becomes
infinite when 0=0, we have to break up the integral into
three portions A, B, C, taken respectively between the limits
—7 to —h, —h t0 i and h to 7* We shall now show that
A and C yanish when n=<0 for values of # as small as we
please, and therefore that the value of S. depends only upon
the infinitely thin strip B within which the function integrated
becomes infinite.
14, Consider first the portion C. Let (2n+1)40=4, and
F(@+z4
put +4) ry = (40). Then
oe 1 (Qn+1)37
~ a(2n+1) (2n+1)3 es x(a mri) aOR
Whatever x may be, we can always choose ht so that (2n+1)3 1h,
isa multiple of 7. The integral can therefore be broken up
into a number m of elements in each of which the range
is 27, and one element at the upper limit in which the range
is > or — This latter element will have a finite value a.
For a given value of n let p be the value of the numerically
greatest of the remaining m elements. Then the sum of the
(m+1) elements ae between +mp-+a; and therefore C lies
Beivcen taper
; since is <1. But when z
m
ae +1) 2n+1
* The reasoning ie precisely the same if the limits are —z to he
—h to g, and g to a, h and g being independent small quantities.
T Or, if (2n+1)3/ is not a multiple of mw, each element of range 27
can be broken up into four portions in each of which sin ¢ preserves the
same ‘sign, so ) that the reasoning of (14) is still applicable.
Convergency of Fourier’s Series. 131
increases without limit, p diminishes without limit. For each of
the above m elements can be broken up into two parts of equal
range 7, in one of which sin ¢ is positive, in the other nega-
tive. The value of each element will therefore be of the form
2 (p1—p2) where p; is some value of x (554) taken between the
limits of the first portion of the element, and p, between those
: : oh
of the second. But as n increases, the change in Oil oan i)
when ¢ changes by 27 diminishes; and since y is everywhere
finite and continuous, p, and p, tend to the same value. Hence
by increasing n sufficiently, we can make p,;—p, as small as
we please; and therefore in the limit when n= © it vanishes.
In other words, since as 7 increases x(=55) tends to
remain constant during the integration of any element while
sin ¢ passes through all the values included between +1, each
element tends to the value zero, the value it would really have
if x( mri) remained absolutely constant during the inte-
gration. |
15. This holds for all finite values of 4 however small.
When A is very small, p, and p, will have their greatest
values in the neighbourhood of 6=(2n+1)$h, in which case
(putting «=0 for convenience, the reasoning being applicable
for any value of «)
ey: pew 5, an Oy ul a pny El +) as 1D)
é ‘ 27 ee
where ¢ is some value lying between 0 and Fp te and is in-
finitely small compared with h. (p,—p,) can therefore be
made as small as we please for values of h as small as we
please provided f is so chosen that and ee
are both infinitely small. But since F is everywhere conti-
nuous, and n is to be increased without limit, this condition
can always be satisfied. Hence the limit of p, and therefore
of C, is zero for values of h as small as we please; and in
the same manner we may show that the limit of A is zero.
The value of 8. therefore depends only upon the value of
the infinitely thin strip B of breadth 2A within which the
function integrated becomes infinite, and is independent of
the values of F(0+.x) outside this strip. Consequently, we
132 Mr. W. Williams on the
may, outside the strip B, assign to F(@+.) any continuous
finite values at pleasure.
16. Since within the strip B the range of integration is
infinitely small, we may replace F(@+.x) by F(«)+ 6F'(2).
We then get, putting 36 for sin 40,
tees — sin —— 1)39 4 5 = © ("sin (an-+ 1)1698,
ee ,
2a a - my
which reduces to the first term on the right because the
integral of the term involving F’(x) is zero. The value of
S, when n= is therefore the same as the value it would have
if F(84+ 2) remained constant throughout and equal to F(z).
Hence 8,,=F(z) by (12).
17. If we change the limits of integration in 8, from —7
and a to —7 and 0, or 0 and 7 respectively, we can evaluate
the integral exactly as before. For since the portion taken
between —a and —h, or between h and 7, vanishes when
n=, the value of the integral depends only upon the inf-
nitely thin strip taken between —/ and 0, or 0 and A. Hence,
replacing F(@+ 2) in this strip by F(2) + @F’(), it follows,
as before, that the value of the integral is the same as the
value it would have if F(@+.2) remained constant throughout
and equal to F(z). Hence in this case 8, =3F (x) by the
latter portion of (12).
From this it follows that in the original integral taken
between +7, F(@+.2) may change abruptly in value or
experience a discontinuity when 9=0; for we can break up
the integral into two portions at the point @=0, and evaluate
re AV
each portion by the above as if the other were absent. If
F(@+ 2) is discontinuous when @=0, it will have different
values at that point according to whether @ attains the value
zero from the negative or from the positive side.
Thus, let @ have a small numerical value 6, and lei O0A= —6,
OB=6,AA’/=F(«—6), BB’=F(2#+6). Then when 6 vanishes,
F(xz—6) becomes F(2—0) or OA”, and F(a#+6) becomes
F(#+0) or OB”. If, then, we evaluate each of the above
portions asif the other were absent we get OA” or $F (4—0)
for the first portion, and }OB” or 3F(2+0) for the second.
Convergency of Fourter’s Series. 133
Hence in such a case 8, =3[F(a—0)+F(2+0)]*. F(04+2)
may have such discontinuities for other values of ae as well,
provided their number is finite. For if we break up the
integrals A and C between neighbouring discontinuities into
separate portions, we may show, as in (14) and (15), that each
of these portions vanishes whenn=«. Hence, since there is
only a finite number of them, their sum vanishes, and there-
fore A and C vanish when n=; so that, as before, the
value of S,, depends only upon the value of the infinitely thin
strip which lies between +f. Consequently F(@+.#) may
have any jinite number of discontinuities between +1, the
value of S,, at any discontinuity being the mean of the values
to which F(6+2) tends as the discontinuity is approached
from either side.
18. If F(@+2) is not periodic, we may regard the portion
of it included between +7 as a wave of an arbitrary periodic
function with, in general, finite discontinuities at +7, +37,
&c.; so that when w= +7, S.=3[F(—7) + F(a)] by (1a)t.
* Or thus,
f 2 16
= [| F642 yee ae He+( F(e+2) Oe n+1)36 20]
ea j , sin (2n+1)36
- Ly. [Fe—6)+F (e+e) ntti 9,
Hence, applying to this the method of (16), we get
3[F(#—0)+F(«+9)].
t Or thus :—If F(6+-.2) is not periodic,
ta que sin (2n+1)40
Sn=o- | F(6+2) i ae 38.
—nm—Z
If x lies between 0 and z,
Sn= Bm F(6+2 ye aE oe oak F(6--e— ee (nt 1)a a
sin 40 sin 30
and if x lies between O and — zr,
2 i =T—L 7 . 9 i
= B| Fetetae ) ee oo+9 = F@+2) 39.
In both cases the function under the sign of integration becomes infinite
only when 6=0, and the integration can therefore be effected by the
methods given above.
ae x=r in the former, or x= —z in the latter, we get
¥ sin sin (2n+ 1)30 2" * % sin (2n+ 1); 1)30 6,
LS ae +9) sin 36 ar Qn |, (—wre) sin 36 — °
the limit of eoeeg when 2=00 , is
8. =4(F(—m)+F(n)].
134 Mr. W. Williams on the
19. Hence, finally, if F is finite, single-valued, and con-
tinuous between +7, or, if not continuous, has only a finite
number of finite. discontinuities, and where continuous is
differentiable, then Fourier’s series is convergent, and tends to
the limit F(x) for all values of w except those corresponding to.
the discontinuities and the values +7, +37, &c. The value of:
the series at a point of discontinuity is 4[F («¢—0) + F(#+0)],
the mean of the values to: which the function tends when ap-
proaching the discontinuity from either side ; and its value at
+7, &e., is $[ F(7) + F(—7) ], the mean of the values of the
function at the two limits. .
IIl.
20. The simplification in the above method of evaluating
the integral 8, consists in having first proved that the two
portions A and C taken respectively between the limits —7
to —h, and h tog vanish when n=x however near to the
value zero we take the ordinates +h, so that the value of the
integral depends only upon the value of the infinitely thin
strip B taken between +h. S,, is therefore independent of
the values of F(@+.x) outside the strip B, and consequently
is the same as if F(@+.) remained constant throughout and
equal to its mean value F(x) within B. That is, 8, = F(z).
21. The vanishing of A and C when n= depends upon the
fact that the function integrated, namely y(3@) sin (2n+1)46-
has an énjinite number of finite oscillations (that is, oscillations
of finite amplitude) between —7 and —A, and between h
and 7. Hence, since the number is infinite and the ampli-
tudes finite, neighbouring oscillations differ, infinitely little
from each other, and therefore the area included between the
ordinates —7 and —h, or h and 7, and the portions of the.
function and the axis of @ intercepted by them is infinitely
small. In other words, the mean value of the function from
—7 to —h, and from h to 7 is.zero, and therefore the integral
of the function between the same limits is also zero. But the
function itself is not zero: it is merely indeterminate,—the
oscillations being, as it were, too fine-grained to be traced
individually. The transformation (2n+1)40=¢, however,
resolves these oscillations, however fine-grained they may be,
into oscillations of finite period cutting the axis of @ at equal
intervals a7. We are therefore enabled to deal with each.
individual oscillation instead of with the oscillations as-a
whole, and so to determine the precise effect of each upon the
value of §,,. : :
22. If we break up the portions A and C of the integral S,,
Convergency of Fourier’s Series. 135
into (m+1) elements as above, without transforming the
yariable we can show as before that each element vanishes
when n=. But the sum of the m elements taken in this
form is not determinate when n=x. For as n increases
without limit, m also increases without limit, and therefore
the sum tends to the indeterminate yalue © x0, as in the
case of any definite integral. We have thus no means of
determining whether A and C vanish when n=. But by
means of the transformation (2n+1)40=¢, we see that each
element is Sone of the form.
am (2n+ Seam) oll (5 ae )sin fog.
Here the integral, independently of the facto !
a m(2n-1)’ or
infinitely small when n=, and this multiplied by oo
gives us an infinitesimal of the second order. Hence the
sum of the m elements is ag really (00 x0), but (co x0"),
or @ x 0), eee the form — — when looked sae is found to be
derived from = Daa ee real limit is <1. It is this that
determines the convergence of 8, to its limiting value.
23. It is necessary to remark that in general an element of
| Aqr
Qn+1
vanishes when n= only when @ is, numerically, not less than
h, and h is not less than the value necessary to ensure that
f F(h+t)—F(h)
2 and a me are both infinitely small, ¢ being
the integral S, in which the range of integration is s——
= or <5 =i (see 15). Ofcourse, since ¢ can be diminished
without ei by increasing n without limit, and F(@+z2)
is continuous, this condition can be satisfied for values of h
less than any assignable finite limit, however small. But as
n increases without limit, the two infinitesimals ¢ and A must.
diminish at different rates ; for whereas ¢ tends to the value
zero at a constant rate, h ate be so at a coustonity dime-
h may be oon , &c. The
nishing rate, Thus, ¢ being on = it
consequence of this is that in the integral
(Fete oaks ee 38,
136 ~ Mr. W. Williams on the
although f/ is infinitely small and F(@+~) is corti
between 0 and h, we cannot without a special examination
treat F(@+ 2) as constant in the integral, and write
F( {” sin aE 0.
—h
For, since must be infinitely small compared with 2?,
1
2n+1 . aT
however small A may be, pbuen tha has an infinite num-
2 é
ber of oscillations between 0 and A. In such a case we must
write the integral in the form
F(a le sin aa Cry 208+ | [F@+8)— Fla) Os 1)365 5 7
and determine oe what conditions, if any, the second term
vanishes.
24. Now although the function F (@+.) is continuous
between 0 and A, and therefore F(72+6)—F(2) is infinitely
small between the same limits, it by no means follows that
the second term in the above vanishes when n=. Its
vanishing depends upon the nature of the continuity of the
function F, and we have only proved that it vanishes when
the continuity i is such as to admit of the existence of a derived
function F’. In modern analysis, a function F(z) is said to
be continuous at the point 2 if, 6 and e being positive quan-
tities as small as we please, and @* any positive quantity at
pleasure between 0 and 1, we have for all values of @
F(2+68)—F(z) less in absolute magnitude than e (Cayley,
art. ‘‘ Function,” Lncyc. Britt.). In other words, F(z) is
continuous at a point 2 if a region (a—6) to (+6) can be
found such that the values of the function for all points within
this region (that is, F(2+0@6) for all values of ¢ between 0
and 1) differ from its value at # by a quantity <e, ¢ being
infinitely small: the function may vary in any manner what-
soever within this region provided only the difference between
its greatest and least values is not greater thane. Hencea
function may be continuous according to this definition with-
out admitting of a differential coefficient, for the existence of
a differential coefficient implies, in addition to the above, that
F(#+6)—F(@
5=0 fer] has everywhere a determinate value, —
or, geometrically speaking, that F(2+6) —F (2) is ultimately
* 6 is here the symbol for a positive fraction, and not the variable of
integration.
Convergency of Fourier’s Series. 137
a small straight element inclined ata definite angle to the
axis of wr.
25. A function which is differentiable wherever it is_con-
tinuous is said to possess ordinary continuity. We thus see
that ordinary continuity is only a particular kind of continuity.
It is, however, the kind exclusively dealt with in the Infini-
tesimal Calculus; for the processes of the Differential Calculus
are based upon the properties of the differential coefficient,
and, practically at least, integration is treated as the inverse
of differentiation. While, however, every finite and con-
tinuous function has an integral, only some possess a differen-
teal coefficient. Here, then, the inverse operation is always
admissible (though it cannot always be formally effected),
whereas the dzrect operation is not always admissible. For
this reason Weierstrass, in his lectures, once made the definite
entegral the starting-point for the investigation of the pro-
perties of functions, and especially of the condition for the
existence of a differential coefficient.
26. Hxamples of functions which are continuous and per-
fectly determinate, but not differentiable, were first given by
Weierstrass*. The essential feature in the case of such func-
tions is that the loci consist of an infinite number of infinitely
small zigzags and oscillations (for otherwise the functions
would be ditferentiable). The functions are thus perfectly
lim fpF(e+s6)—F wy
determinate and continuous; but
o=0 | )
cannot anywhere have a determinate value, and the processes
of the Differential Calculus are therefore inapplicable. When
drawn the locus of a function of this kind is indistinguish-
able from that of a function having ordinary continuity, and
whose values at the different points are the mean of the
values of the given oscillating function at the same points.
But we cannot treat the two as analytically the same. Thus,
to borrow an illustration used by Prof. Greenhill, the zigzag
locus CD is indistinguishable from the straight line AB
when the zigzags are infinitely small and infinitely nu-
merous. But we cannot treat it as having the properties of a
straight line. For the length of the zigzag locus is always equal
to the sum of the lengths of C H and E D, however small we
make the zigzags, provided they do notalterinform. If, then,
we treat the zigzag locus as a straight line when the zigzags
are infinitely small and infinitely numerous, it follows that
the third side of a triangle is equal in length to the sum of
* Cayley’s article “‘ Function,” Encye. Britt.
138 — Mr. W. Williams on the
the other two. This illustrates the nature of the difficulties
encountered in dealing with functions of this kind, and the
Lene mem meme mena na mem
i ~S
-~—
=.
PS
~.
=.
~
~~.
~
-~
~
=
re
~
~
=
-—
ae,
~
=~
=
>=
~~ 3 ,
~
=a
~
Soy
=
-
~
~
=.
=
>.
~
~
~
~
~
>
-~<,
=,
~s
.
>
w
danger of applying to them, without a special examination,
processes which have been derived only from the study of
functions possessing ordinary continuity. It is precisely in
the case of functions of this kind that the integral
sin a +1)36
TE(0-+2)—F(a)] 00
becomes indeterminate in value whenn=c. If the function
possesses ordinary continuity we know that the integral va-
nishes ; otherwise the integral may be quite indeterminate.
For the infinite number of oscillations of eed when
n= may conspire with the oscillations of Fle 0) —F (zx) to
produce any value whatever, finite or infinite. In cases of this
kind we can determine nothing as to the value of the integral
until we know something as to the nature of the continuity of
the function; for the ordinary definition of a continuous
function is £00 general, and does not confer upon the function
enough properties to enable us by means of known processes
of integration to evaluate the integral.
27. The conditions under which Fourier’s series has been,
up to the present, proved to be convergent are :—
i. That the function F(z) must not become infinite.
ii. It must be continuous and determinate except at a
finite number of points, where it may change abruptly in
value or experience a discontinuity.
iil. It must, wherever it is continuous, possess ordinary
continuity. .
These conditions are sufficient for all the cases that occur
in ordinary analysis. The third condition, moreover, is
necessary in all such cases, since processes involving differ-
entiation constitute an essential part of the Infinitesimal
Calculus. From the point of view of the general theory of
Convergency of Fourier’s Series. 139
functions, however, it is necessary to consider the cases in
which this condition does not hold.
28. The investigation of Dirichlet involves the first and
second of these conditions, but not the third. The third is
replaced by the more general one that F(z) must not have an
infinite number of maxima and minima between +7. In
Dirichlet’s investigation this condition is applied to the
function throughout the whole extent of the integral §,, that
is for all the values of the variable of integration 6. This,
howeyer, is not necessary. For it has already been shown
that the portions A and C of the integral vanish when n=
if only the function is finite and continuous—the nature of
the continuity being immaterial. The third condition should
therefore apply only to the infinitely small range of values of
F(@+.2) which lie on either side of 2=0. We shall now
show that this condition is sufficient to ensure that the
integral
h l
[feet Fe] SE 6
—h 2
vanishes when n=, and that therefore 8, =F («).
This integral can be put into the form
{90 a Lane
h being infinitely small, while m is infinitely great and 6(6)
infinitely small between 0 and A. Since $(0) has not an
infinite number of maxima and minima, it will ultimately
preserve the same sign, and either constantly increase or
constantly diminish between 0 and h. Let it constantly de-
crease. ‘Then, dividing the variable by m, we get
6G) "an8e
This integral can now be broken up into the sum of a series
of elements which are alternately positive and negative and
]
diminish numerically). Hence the integral becomes an alter-
nating series with constantly diminishing terms, and its value
is therefore less than the first term, which is itself infinitely
small. That is, the integral vanishes. Again, let 6(@) con-
stantly increase between 0 and h. Then its greatest value
will be $(h), and [d(h)—¢(@)] will therefore constantly
constantly diminishing numerically (since cla and (7)
140 Mr. W. Williams on the
diminish. Hence the integral
{ t4e)-69] 800
vanishes by the above when n=. But this integral is
equal to
sca eae | 80 ae 56:
and therefore, since the first term and the difference of the
two are both infinitely small, the second term must also be
infinitely small. Thus in both cases the integral vanishes,
so that S,=I(z). It is interesting to note that the alter-
nating series which appears in Dirichlet’s investigation appears
also here, but in a different manner. For whereas in the
former case it appears with terms of finite magnitude, here its
terms are infinitely small, because the two portions of the
integral 8, which lie outside the infinitely thin strip bounded
by +A have already been disposed of. There is therefore no
trouble in manipulating the series ; for all that we have todo
is to show that the terms decrease numerically, since the
series can then be neglected, the first term being infinitely
small.
29. Functions having an infinite number of maxima and
minima are of two kinds, according as to whether the ampli-
tudes of the oscillations are finite or infinitely small. In the
former case the functions are discontinuous, for they violate
the definition in (24) ; in the latter case they are determinate
and continuous. Dirichlet maintained that all functions
which have only a finite number of indeterminate values, and
are elsewhere continuous, give rise to convergent Fourier
series *; but Du Bois-Reymond and Schwarz have given
examples of functions which are determinate and continuous,
but for which Fourier’s series is divergentt. These functions
are of the class mentioned in (26) for which the integral
{, e+ —F@) Soe
is infinite or indeterminate.
30. The condition that F(@+ 2) must not have an infinite
number of maxima and minima is not a necessary condition
in order that Fourier’s series may tend to the value F(a).
For Lipschitz | has shown that the series may be still con-
* Sachse’s Essay, p. 19. + Ibid. p. 49. t Ibid. p. 21.
Convergency of Fourier’s Series. 141
vergent, and tend to the limit F(x), even when F(@+2z) has
an infinite number of maxima and minima, provided that at
all the points where the function oscillates, the numerical
value of F(vx+0@+6) —F(v@+6@) is always less than Bo*, when
6 tends towards the value zero, B being a finite constant, and
a@a positive exponent: Here again it is really necessary to
apply the condition only to the infinitely small range of
values of the variable of integration which lie on either side
of 6=0; for if the condition is satisfied for these values, the
integral
h oy 1
[F(e+6)—F(a)] Be See
vanishes, and therefore, as before, S,=F(«#). This integral
vanishes under the given conditions because its value cannot
be greater than the value it would have if sin (2n+1)30
were replaced by unity, and all the negative values of
F(2+0)—F (x) made positive. Hence, since for all values
of @ between 0 and h, h being infinitely small, the numerical
value of F(x+0)— F (z) is < BO", the integral cannot be
greater than
2Bh*
h
2B) GaaOU, or b)
0
which is infinitely small, since # is a finite exponent. Thus,
the function may have an infinite number of maxima and
minima of this type, and still give rise to a convergent
Fourier series, whose converging limit is F(x).
31. It is not necessary that F(@+#) should be finite
throughout between +7. It may become infinite at a finite
number of points a, a,... provided that
lim (4te¢
al F(0+ 2)o@
a—U,6
vanishes, 4, and p, being any independent positive fractions.
For if this vanishes, then
lim a tee si
C=O Pa evra,
also vanishes, unless @ passes through the value zero, for it
tends to the value
utp lim: (i es
sea[zeo|, 0+ 2)d2 |,
—pe
Phil. Mag. S. 5. Vol. 42. No. 255. Aug. 1896. M
142 Mr. W. Williams on the
Hence, any element of the form
ee. Es #) sin md 7
es ih oee)
must vanish when e=0, provided F has not an infinite
number of oscillations at the point a, for it cannot exceed the
value it would have if sin m@ were put equal to I all through.
The sum of the finite number of elements of this form which
occur in the integral 8, at the points a,a,... is therefore
zero. Again, since (8+) is continuous up to (a—py,e) and
beyond (a+ p.€), we can always choose for ¢ a value such that
F(a+pe+t) —F(a+pe) isas small as we please, however small
: 20
pe may be, ¢ being = or < Brey? and n=«. Hence, by
(14), the elements p which occur in the neighbourhood of
the infinite values of F(@+ 2) are infinitely small when n=o ,
and therefore, as before, A and C vanish when n=o. HI,
then, (6+) is not infinite when 6=0, 8, =F (a), provided
the conditions relating to the portion B are fulfilled; but if
F(6 +.) is infinite when 0=0, the value of B is «, and
therefore S,=«, or the series is divergent, as we should
expect. Hence, if the function contains a finite number of
infinite values of the above kind, Fourier’s series 1s, ceteris
paribus, convergent for all values of 2 except those corre-
sponding to the infinite values, and for these values of # the
series is divergent.
32. If the function F(@+.) is indeterminate over a finite
range of values of x-—for example, if it has an infinite number
of discontinuities, or maxima and minima of finite amplitude,
over that range—the coefficients of the series and therefore
S, cannot be determinate. But the function may have an
infinite number of discontinuities, or maxima and minima of
finite amplitude, or singularities in the neighbourhood of a
finite number of poznts ; for, since the range within which
these singularities occur in the neighbourhood of one of
these points is infinitely small, and the function is never in-
finite, the elements of the integrals which determine the
coefficients and 8, corresponding to this range must be
infinitely small. Hence, since there is only a finite number
of such points, the sum of the elements corresponding to them
vanishes, so that the values of the integrals are determined
only by the continuous portions of the function. Hence, the
coefficients of the series are finite and determinate, and 8,
tends to a definite limit for all values of 2 except those corre-
sponding to the indeterminate points in the function; and
Convergency of Fourter’s Series. 143
for these points the integrals
('e (a jayne ee 20
0 2
are indeterminate in value.
33. We may therefore summarize the conditions under
which Fourier’s series is convergent as follows, taking first
the case where the function F has no infinite yalues—the
ease of a function having infinite values being discussed
later. In order that the series
me | FO v)dv+ = 00 nel F'(v) cos nv Qu
] e oe ©
i E > sin nae F(v) sin nvdv
I ot
may be convergent when n= for any value of «
i.) The coefficients must be finite and determinate ;
(ii.) The nth coefficient must vanish when n= x
These are conditions that hold in the case of every series,
independently of its particular character. They are therefore
necessary conditions, but they are not sufficient.
34. The first condition is satisfied if the function which
determines the coefficients is not indeterminate or discon-
tinuous over a finite range of values of the variable, but is
continuous and determinate except, possibly, in the neigh-
bourhood of a finite number of points where it may have any
number whatever of discontinuous, indeterminate, or singular
values. The second of the above conditions is also fulfilled
under the same circumstances. For, if we take the coeffi-
cients
+ {Fe COS RV Ov, =| Ga) sin NV OU,
TUS Ns T ) —a
and divide the variable all through by n, we get
So ve (“) COS VAL, a Tey sin vdv.
n n
NT} —nz NT |) —nr
Then breaking up each integral into n elements of range 27
and applying ‘the method of - (14) we can show that the inte-
grals vanish when n is infinitely great.
35. The condition given above to ensure that the coefficients
of the series are finite and determinate (namely, that F(x)
must be determinate and continuous, except in the neighbour-
hood of a finite number of points) is a special case of
Riemann’s general condition as to the integrability of a
M 2
144 Mr. W. Williams on the
function *. Riemann’s condition is as follows :—Consider a
function F(2) between a and 6. Divide (6—a) into intervals
6; 6)... &,, so that (6—a) = (6,4+6,+...4+5,). Let D denote
the numerical value of the difference between the greatest and
least values of F(a) within the interval 5,; similarly D, for
the interval 6,, &. Then D, is called the oscllation of the
b
function within the interval 6,. In order that F(x) 02%
may have a determinate value, 5
(6,D,+6,D,+6;D3+...+6,Dz2)
must tend to the value zero when 6, 6,...6, are diminished
without limit, the necessary and sufficient condition for which
is that the sum of the intervals within which the oscillations D
are greater than a given finite quantity o, however small,
must be infinitely small when the intervals are infinitely
small. If the oscillation within an interval 6 taken on either
side of a given point is always >o when 6 is diminished
without limit, the function is said to be discontinuous at that
point, and the point is spoken of asa point of discontinuity ;
and, on the other hand, if the oscillation is <a, the point is
a point of continuity. If every point within a finite segment
is a point of discontinuity, the function is said to be discon-
tinuous over that segment, as, for example, a function which
has an infinite number of maxima and minima of finite am-
plitude over a finite range of points. If within a given segment
the points of continuity are finite in number, the segment
can be broken up into a finite number of other segments,
over which the function is discontinuous. But if between two
points there are no segments of discontinuity, there may,
nevertheless, be any number whatever, finite or infinite, of
points of discontinuity. In the first case the function is
not integrable, since the sum of the intervals of discontinuity
is finite. In the second case, Hankel, who has investigated
this matter with the view of rendering Riemann’s condition
less indeterminate in character, has shown that the sum of the
intervals of discontinuity cannot be finite t. Hence, the
function is, in such a case, integrable, and accordingly, Rie-
mann’s condition may be more precisely stated as follows :—
A function is integrable between a and b if it ts finite and de-
* “Ueber die Darstellbarkeit einer Function durch eine trigono-
metrische Reihe;” Abhandlungen der k. Gesellschaft der Wissenschaften
zu Gottingen, vol. xiii. This paper has also been translated, and published
in the Bulletin des Sciences Mathématiques, 1873, p. 35.
+ “ Untersuchungen ueber die unendlich oft oscillirenden und unste-
tigen Functionen ;’ Tubingen, 1870.
Convergency of Fourter’s Series. 145
termenate, and ts not discontinuous over a finite range of values
of the variable between aand b. Thus stated the condition is
more general than the one given above, for it does not imply
that the function is continuous: it may have any number
whatever of discontinuous points, but not of discontinuous seg-
ments. The coefficients of Fourier’s series are finite and deter-
minate, and the nth coefficient vanishes when n=0o whenever
Riemann’s condition as to integrability is fulfilled. For, ifthe
function integrated is never infinite, and the sum of the
intervals 6, 6,...6, containing the points of discontinuity can
be made infinitely small, the swm of the elements correspond- -
ing to these intervals in any integral can contribute nothing
to the value of that integral. For thissum cannot be greater
than the product of the greatest value of the function, which
is necessarily a finite quantity, into the sum of the intervals
(0, +6.+... +6,), which is infinitely small. The value of
the integral is therefore the same as the value it would have
if the function were not discontinuous at the given points.
But we have proved that in this case the coefficients are
finite and determinate, and that the nth=0 when n=.
36. Now, the conditions which ensure that the coefficients
of the series are finite and determinate are also the conditions
which ensure that the portions A and C of the integral 8,
vanish when n= , for we have only to replace F(v) sin nv in
the coefficients b, by y(4$0@)sin(2n+1)3@ and apply the
reasoning of (14). Hence, whenever the coefficients of the
series determined by Fourier’s method are finite and determi-
nate, the value of the series depends only upon the infinitely
thin strip
B a ¢ Vk
) F(0+2) on Caer a
= 40
and therefore the remaining condition to be fulfilled in order
that the series may be convergent is that this integral must
have a determinate value when n=. Writing this integral
in the form
F(a) (* sin (2n+1)30 ove
=) Sees 3° 30+ | UFO +2)—F (a) 10
the value of the first term is F(x). Hence, if the series is to
be convergent, the second term on the right must vanish or
tend to a definite limit. In the former case the series repre-
sents the function for the given value of x. In the latter
case it does not.
_ 37. The general conditions under which the second term
in the above vanishes, or has a finite limiting value when
sin (2n+1)40
08,
146 Mr. W. Williams on the
n=0, have not been determined. If the function is con-
tinuous, and
i. If the continuity 1s ordinary continuity ; or
il. Jf the function has not an infinite number of oscilla-
tions 3 or
ili. If the infinitely numerous oscillations satisfy Lipschitz’s
condition;
then the term vanishes, and 8,=F (a). In all other cases
the term must be treated as indeterminate. We may, of
course, investigate its values for different types of continuous
functions, and so widen the limitaticns of the function F. But
we cannot determine the general nature of these limitations
because we cannot evaluate the integrals
(exe ~F(2)] sin ae 1)20. 4
by known methods of integration until we are provided with
conditions other than those involved in the definition of a
“ continuous function,’”—such other conditions, for example,
as 1., ll., and_iii. above.
38. Te is necessary to remark that a series of the form
a, sinz+a,sin 2v7+...+a,sinne.
may be convergent, and tend to a definite limiting value
which we may denote by F(z) for all values of x, and yet it
may be impossible to derive the coefficients by Fourier’s
method from F(a) because F(x) may not be integrable
according to Riemann’s definition. Riemann has given an
example “of such a series in the paper already mentioned. In
a case of this sort, however, since the coefficients are not
determined by Houers meuiod the series is not really a
Fourier series. For a Fourier series is one in which the
coefficients are defined by the definite integrals
7
an=— "F (v) cos nvdv, o.— =| Fe) sin nvgu,
and the object of our investigation is to determine the most
general conditions under which the series thus defined is
convergent. -
Hence, since it is necessary that the function should be
integrable in order that the coefficients to be derived from it
may be finite and determinate, we get when the function has
no infinite values the following necessary and suffictent con-
ditions for the convergency of a Fourier series :—
Convergency of Fourier’s Series. 147
1. The function must not be indeterminate or discontinuous
over a finite range of values of the variable ;
u. Lhe integrals
{ees Fe] OOH 6
must vanish, or tend to definite limiting values, when n=
and h ts infinitely small.
39. This last condition is somewhat analogous to the con-
dition which holds in the case of Taylor’s Theorem when applied
to numerical functions, namely, that the “remainder” after the
nth term must vanish when n is increased without limit ;—a
sort of test to be applied to each individual function dealt
with, because we have no means of determining in a general
manner when the condition is fulfilled. If the first condition
is satisfied, the coefficients of the series are finite and deter-
minate, the nth coefficient vanishes when n= , and the value
of the series alt any point x depends only upon the infinitely
small portion of the function I’ which hes on either side of
that point. If the second condition is satisfied, the series is
convergent, and if, further, the integral involved in this con-
dition vanishes, the converging limit of the seriesis F(z). If
the first condition is not satisfied, the coefficients of the series
are indeterminate and meaningless, and the series cannot
therefore be formed. Whether the function can still be
represented by an harmonic series in such a case—the coefhi-
cients being determined otherwise—is a matter with which
we are not now concerned, nor are we concerned with deter-
mining whether the same function can be expanded harmoni-
cally in more ways than one. We are concerned only with
determining the most general conditions under which Fourier’s
method of expanding functions into harmonic series is appli-
cable. In cases where it fails, we have no general method of
proceeding.
AQ. If the function has infinite values, two cases may arise
according as the function has or has not an infinite number
of maxima and minima where it is infinite. In the former
case, as shown above, the series is convergent (except, ot
course, at the points where the function is infinite) pro-
vided the function becomes infinite only at a finite number
of points, and that its integral vanishes when taken between
limits infinitely near to and on either side of each of these
points. In the latter case, for example in the case of
ont ae At 1
COs) where = is infinite when «=0, and cos a4 has an
c
148 Mr. R. Appleyard on Dielectrics.
infinite number of maxima and minima values, Riemann has
shown in the paper already referred to that this condition is
not sufficient. For, although the integral of the functzon
taken at the point where it is infinite may vanish, this integral
when the function is multiplied by cos nz or sinnxz where
n= may become infinite. In the first case the oscillations
of the function mutually compensate each other, but in the
second case the factors sinna or cosnz may destroy this
compensation when n=, for the oscillations of the two
factors may conspire to produce a resultant function which
is infinite without oscillations. The value of the integral in
such a case is, of course, indeterminate, and so it is not suffi-
cient merely to know that the integral of the functzon vanishes
at the point where it is infinite.
41. The complete investigation of the convergency of
Fourier’s series ultimately resolves itself into a discussion of the
conditions of integrability and the nature of functions. We
thus see that the inquiry leads us to the very foundations of
the Infinitesimal Calculus, and in this respect Fourier’s series
differs essentially from Taylor’s. For in the case of the latter
series the field of investigation is, at the very outset, restricted
for us by the nature of the coefficients, since the process of
differentiation limits us to functions of a comparatively simple
kind.
XII. Dielectrics. By Rotto APPLEYARD *.
oe experiments upon the change of resistance of certain
J dielectrics with the duration of the testing-current, and
with the testing-voltage, were described in a paper f which I
read before the Physical Society two years ago. In continua-
tion of this research some further tests have been made, the
principal object being to determine the effect of temperature
upon the dielectric resistance. For this purpose, mica and
paraffined paper, in the form of condensers, have been
chosen.
The resistances are measured by the “direct deflexion ”
method, and are expressed in megohms pro microfarad. The
testing-voltage is the same throughout all the tests (450
volts), and each measurement is computed from the galvano-
meter-reading noted after the testing-current has been
applied for one minute.
Two paraflin-paper condensers, each of one microfarad,
* Communicated by the Physical Society: read May 22, 1896.
+ ‘* Dielectrics,” Proc. Physical Soc. xili. p. 155, 1895; Phil. Mag.
Oct. 1894, p. 396.
Mr. R. Appleyard on Dielectrics. 149
and eight mica condensers, each of half a microfarad, were
enclosed, separately, in water-tight cases, and submerged in
a tank of water, the temperature of which could be con-
tinuously controlled. The platinum thermometers * used for
determining the temperature within the dielectric of the
condensers have already been described.
Connexion between the condensers and the testing-
apparatus was made by wires insulated with india-rubber ;
these were twenty-five feet long. The connexions within
the condenser cases, where the wires were soldered to the
respective sets of tin-foi] sheets, were carefully sealed with
paraffin wax. The ends of the leads were bared and cleaned
in the usual way, for about six inches. Two such insulated
wires were connected to each condenser, one to each set of
tin-foil sheets.
Surface Leakage.
Suppose a current to enter one of these condensers at one
of its insulated leads, which we will call A ; and the second
lead, B, put to earth. It is clear that there are at least four
paths—(a), (b), (c), (d)—for the current between A and
earth :—
(a) Surface leakage at the end of A where the current
enters the lead.
(5) Surface leakage at the end of A within the condenser.
(c) Surface leakage over the dielectric within the con-
denser.
(d) Through the dielectric of the condenser.
The leads can generally be so chosen that there is practically
no conduction through their dielectrics. When measuring
the resistance of the condenser by the “ direct deflexion”
method, the current to which the galvanometer responds is
that due to the sum of the currents through (a), (6), (c), and
(d). We have to determine what the deflexion would be if
all the current went by way of (d). This can be done,
approximately, by taking a preliminary test, with the end B
disconnected from earth, and carefully cleaned ; noting the
deflexion after the current has been applied at the end A
for one minute. The deflexion thus obtained must be
deducted from the deflexion observed during the second
operation, which consists in putting the end B to earth, and
* A “ Direct-reading ” Platinum Thermometer. Proc. Physical Soe.
March 1896; and Phil. Mag. Jan. 1896.
150 Mr. R. Appleyard on Dielectrics.
again applying the current and noting the deflexion after
one minute.
It sometimes happens that we want to separate (a) from
(5) and (c) ; that is to say, we require to know what part of
the preliminary deflexion is due to the exposed lead, and what
part is due to internal leakage at the condenser. For this
purpose I have applied a suggestion made to me by Mr. W.
A. Price *, and have used a “ guard-wire”’ to eliminate the
leakage at (a). The method, which was used for its first
trial in connexion with these experiments, consists in winding
one or two turns of bare wire around the cleaned end of A,
at some point not very far from the middle of the cleaned
portion, intercepting the path of leakage. This wire is then
connected to the testing-battery, as shown in Fig. 1.
™---—-—-— - -— — -— - - -- -
The efficacy of the “ guard-wire” for the present tests
was proved by the following experiment. A copper earth-
wire was given several turns around the insulated lead A, .
just above the cleaned ends, to facilitate leakage ; and tke :
preliminary test was then made in the usual way. The
observed deflexion, due to (a), (b), and (c), was 4 divs. The
cleaned end was then smeared with a mixture of glycerine a
and blacklead, and the resulting deflexion was far off the
scale; (a) had been increased enormously. A “ guard-wire”
was then applied at the middle of the smeared portion of the
lead, and the deflexion was at once restored to its original
value of 4 divs.
In practice it was always possible to avoid leakage (a) by
giving a little care to the cleaning of the lead ; the “ guard-
wire ” was therefore used only as a check, and if it produced
no change in the preliminary deflexion it was concluded that
the preliminary deflexion was due to the effects of (6) and (ce)
alone, and that there was no leakage (a).
* This method has since been described by Mr. Price. See ‘ Electrical
Review,’ vol. xxxvil. no. 941, p. 702 (1895). In the appendix to the
present paper I have evaluated the possible errors, and shown that they
are negligible.
151
Mr. R. Appleyard on Dielectrics.
Date.
11th March, 1896
12th ” oe)
WGior 5
26th Aug., 1895 ..
17th March, 1896
28th Aug. 1895 ..
2erd_ ,, awe
22nd ,, PROT
Oe ies
19th March, 1896
30th Aug., 1895 ..
20th March, 1896
2nd Sept., 1895 .
20th March, 1896
ord Sept. 1895 ....
TABLE I.
seo eeseessovses
ser eeeseoceoeae
Sere areas en neon
seeeseee peeeeeas
Ce i ey
CC i i ry
Temperature.
132). o°
32:8 0-4
44:25 6:8
54:5 12°5
65°7 Wsse7f
67°4 Isa
68°1 20°1
72°6 22°6
759 24-4
80°4 26°9
84:2 29:0
896 32'0
94:3 34:6
100'1 37'8
107°4 419
109°9 433
Mean values of two condensers.
Paraffin-paper.
Capacity.
mids.
0-98
Resistance. | Elec. p. c.
Megohms between
pro mfd. |1™ and 2”,
17740 oo
11510 30
7216 28
4196 24
3825 25
3622 25
2961 ye
2436 25
1947 25
1659 26
1231 26
1056 27
792 28
611 32
551 32
Mica.
Mean values of eight condensers.
Capacity.
mfds.
0:50
Resistance.
Megohms
pro mfd.
31286
28520
28427
25415
22000
22327
16272
16930
17010
18457
15270
10340
10521
Elee. p. ce.
between
1™ and 2™,
152 Mr. R. Appleyard on Dielectrics.
Results.
The results of the tests are given in Table I., which requires
little explanation. The “ Capacity,” “ Resistance,” and “ Hlec-
trification ”’” of paraffin-paper and mica are there tabulated at
the various temperatures.
Capacity—As standards of capacity two condensers were
used. These were mica of 1 and 0°5 microfarad capacity
respectively. They were submerged i in water and kept always
at the same temperature, 27° C., which is just above the
maximum airtemperature. The object of using two standards
‘ was to avoid shunts and to simplify the work.
In the paraffin-paper condensers there appears to be an
irregular variation of capacity with temperature. It is too
small to affect them in practical telegraphy; but it is suffi-
ciently large to prohibit their use as standards. We must
suppose that the softening of the wax, and the consequent
approach or recession of the tinfoil sheets, i is responsible for
part of these changes, to which must be added the variation of
the specific inductive capacity of the dielectric. It is curious
that with paraffin-paper condensers the capacity diminishes
as the resistance diminishes. With gutta-percha or india-
rubber core the result is nearly always the reverse of this.
Owing to the vibration of the building in which the tests
were made, the capacities could not be measured with a greater
accuracy than 1 per cent.; but within this limit it is safe to
say that throughout the whole range of temperature mica
retains a constant capacity.
Temperature Coeffictents.
It is usual to assume * that the resistance R, of a dielectric
at any temperature 7 is represented by an equation of the form
Reha.
and that similarly, at some other temperature 0,
Ro— he’:
so that
= Ryai=8,
or
log R,=log Re+(7—@) logan. . . . (1)
From this equation, taking the fifteen observations of Table I.
two at a time, in alternate order, and putting them succes-
* See Appendix II.
Mr. R. Appleyard on Dielectrics. 153
sively as R, and R, in (1), we find as a mean value
Icon I OG34 4. | We ee. (2)
The temperature 8 may be given some standard value,
say 20°C.; and from acurve drawn from the fifteen observed
values of resistance and temperature the corresponding re-
sistance Rj» may be obtained. The resistance at 20° C., found
in this way, was, for paraffin-paper,
Ra=3670 megohms;
equation (1) in this case becomes
log R,=3°56467 + (7 —20)(1'96344), . . (3)
from which may be deduced the resistance corresponding to
any temperature t. It is interesting to see how nearly the
results calculated from (3) agree with the observed values, as
can be done by drawing the two curves which represent the
observed and calculated resistances respectively.
The agreement between these two curves is so close, consi-
dering the conventions which always have to be adopted in
tests upon dielectrics, that it has been suggested to me that a
simple logarithmic curve may perhaps represent the true state
of affairs, the observed results being assumed more or less at
fault. This seems improbable for the following reason:—In
equation (1) @ is really the ratio of two resistances at an
interval of 1° C., and it is presumed* that this ratio, 2. e.
R,
R,+ i
is constant for all values of 7. This might be true for some
ideal dielectric whose structure was perfectly definite at all
temperatures, and which made invariable contact with its
electrodes under all conditions ; but there is no reason why a
should be constant for a substance having the instability of wax.
a=
Temperature Tables.
One use of these results is to enable the resistances of
condensers to be reduced to the resistance at some standard
temperature. A similar correction has always to be applied
for gutta-percha and india-rubber cores. It is usual to
draw up a table, assuming @ constant over a given range, and
to supply a correcting logarithm for each degree of tempe-
rature. This may be sufficiently accurate for small ranges of
temperature ; but I would suggest that a better way is to work
from the observed results directly, making no assumptions
* See Appendix IT.
154 Mr. R. Appleyard on Dielectrics.
whatever as to « In order to do this, the “ observed”
results should be plotted as a curve, coordinating resistances
and temperatures. The length of the ordinate corresponding
to standard temperature on the “‘ observed’ curve is then to
be taken as unity, and the lengths of the ordinates at every
successive degree are to be expressed in terms of this unit.
A table can then be drawn up giving acorrecting divisor cor-
responding to each degree. With the aid of a slide-rule such
a table can be completed in little more than an hour; and it
is certainly more expeditious and less arbitrary than the
ordinary tables of dielectric corrections. As an example, I
have given in Table II. the divisors for paraffin-paper, from
33° F. to 110° F., deduced directly from the ‘observed ”
curve.
Tape II. Paraffin-paper.
(To reduce the dielectric resistance at any temperature Fahr.
to the corresponding dielectric resistance at 60° F., divide
the observed resistance by the number opposite the given
temperature.)
Temp. Divisor.|] L°MP-| Divisor.|| Lei"? Divisor. | 7.°y"P’| Divisor.|| f°MP-| Divisor
(eo) e} Oo Oo [e)
33 | 319 || 49 | 1-67 65 | 0775 | et | 0-335 7 | 0-169
34 | 3:07 || 50 | 1:62 66 ‘733 || 82 319 || 98 | -159
35 | 2:96 || 51 | 1:55 67 | -699 || 83 306 || 99 | -153
36 | 287 || 52 | 1-46 68 656 || 84 | -292 || 100 | -148
37 | 2-76 || 53 | 1:39 69 625 || 85 281 || 101 141
38 | 2:66 || 54 | 1:33 70 595 || 86 | -270 |! 102 137
3 2:57 || 55 | 1:27 rl 562 || 87 259 || 103 | -133
40 | 2-48 || 56 | 1-20 72 528 || 88 247 | 104 | -196
41 2-39 57 fae a te 73 ‘501 || 89 234 || 105 | 119
42 | 230 || 58 | 1-09 74 ‘477 || 90 | -225 || 106 ‘114
43 | 219 || 59 | 1-04 75 ‘450 || 91 ‘216 || 107 ‘108
44 | 209 || 60 | 1:00 76 ‘432 || 92 | -207 || 108 ‘105
45 | 2:00 || 61 | 0-944 || 77 411 || 93 | -198 || 109+) c=me
46 1:91 || 62 | 0-901 || 78 389 || 94 189 || 110 | -099
47 1:82 || 63 | 0858 || 79 ‘375 || 95 -180
48 1:76 || 64 | 0820 || 80 | 855 | 96 175
The resistances of mzca condensers at the different tempe-
ratures are also given on Table I.; the curve drawn from
them is not so smooth as that which represents the behaviour
of paraffin-paper condensers ; at the colder temperatures the
- resistance of the mica is so high that the preliminary deflexion,
corresponding to (a), (6), and (c), is by no means small
as compared to the second deflexion, due to (a), (d), (c), and
(d) together. The results given are the best that could he
obtained under the circumstances. In some ways it might
a a SE a Se
MraR, Appleyard on Dielectrics. 155
have been advisable to connect the eight mica condensers in
parallel ; the deflexions to be observed would then have been
greater, but the number of tests would have been fewer, and
there would have been less chance of comparison one with
another.
Electrification —In Table I. the “ Elec. p.c.” is given for
paraffin-paper and mica, at the various temperatures. This
is derived from the galvanometer deflexions taken respectively
at one minute and two minutes after the application of the
testing current, thus :—
m ino—?m 1
Hlec.°/, between 1™ and 2™= Bases gin 2? sadn’ _
1™ reading
100.
Paraffin-paper seems to acquire a minimum Elec. p.c. at
about 20° C.; the reason is not very apparent.
Charge and Ischarge.
The following figures (Table III.) may be useful as indi-
cating the difference between the deflexions obtained in ¢a-
pacity tests, using “charge” and “discharge” deflexions,
respectively, for paraflin-paper, mica, gutta-percha, and india-
rubber. The deflexions represented by “charge” are not so
uniform as the “discharge”’ readings, that is to say, a repe-
tition generally gives a different reading, owing in part per-
haps to some of the last charge having remained. This,
however, is scarcely to be observed with mica. “‘ Discharge ”’
readings can generally be reproduced accurately.
Tarun, TE.
Charge. Discharge. ean
Paraffin-paper ......... 196 196 195°5
ININGAT Sides chan dcltnc ets 223 223 223
207
Gutta-percha’ .,....:.. 208 214 210
210
201 |
Tndia-rubber.....:...... 198 200-5 Fo >
198°5 J
156 Mr. R. Appleyard on Dielectrics. —
Resistance during Melting.
If the temperature of paraffin-wax is raised much higher
than is indicated in Table I., the change of state, as melting is
approached, is characterized by a corresponding decrease in
resistance. In order to examine this effect, a special con-
denser consisting of two horizontal circular plates, 11 in.
diameter, was used. The lower plate was provided with a
projecting rim to retain the wax, and was rather larger than
the top plate. Three small wafers of thin ebonite sheet
(16 mils) were used as distance-pieces. Melted wax was
then poured in, filling the space between the plates.
There are so many disturbing influences affecting the resist-
ance during the change of state, connected with the latent
heat of solidification, the unequal melting, the absence or not
of ‘electrification,’ and the contractions and expansions of
the wax, that only qualitative results can be given.
Heating.—Starting at about 20° C. below the melting-
point, the resistance rapidly diminishes until actual meltiny
begins; there is then a definite fall to something like one
third of what the resistance was just before melting. If heat
is still applied to the condenser, the resistance keeps steady
until the melting is complete ; after which the resistance
again diminishes steadily. So far as I am able to discover
there is no ‘ electrification” while the wax is in the melted
condition.
The spark-resisting power of the melted wax is at least
one third that of the solid. The melted wax in this particular
condenser broke down under 1200 steady volts; the insulation
heals, as in the case of oil, to break down again in a few
seconds.
Cooling.—A converse process occurs during cooling ; and
as the cooling is more uniform than the heating the effects
can be more easily noted. The resistance of the melted wax
gradually increases until the first crystals appear, at which
stage it remains more or less constant until the process is
nearly complete, when it very rapidly increases to about
three times the value it had during the partial crystallization;
it then behaves as solid wax.
APPENDIX I.
When an extra wire (fig. 1) is added from the +testing-
battery to some point near the middle of the cleaned end of
the lead, intercepting the path of leakage (a) at that end, the
circuits can be represented by fig. 2. The (a) leakage is
Mr. R. Appleyard on Dielectrics. 157
here m+n, and the ‘“suard”? wire is supposed to come
between mand n. Let 6, 9, and r denote the resistances of
ae
Io a.
the battery, galvanometer, and the dielectric resistance of the
condenser, respectively ; let EH be the E.M.F. of the testing
battery, and G the current through the galvanometer when
the “ ouard”’ wire is applied. It is clear that m shunts the
galvanometer, and n shunts the battery ; and we have to find
the error which this introduces into the results. From fig. 2,
by Kirchhoff’s laws, we have
G= no a gk @1B)
(6+g+r)+ past (nrg + brm + brg + bmg + nbg)
Let
k= (nrg + brm-+brg-+bmg +nbg), eC)
then « may be regarded as equivalent to a resistance added
to the simple circuit b+ 9+r.
To evaluate x, we will put m=n=r; which means that
the whole leakage, m+n, is assumed to be only twice the
resistance of the dielectric of the condenser, and corresponds
to a worse lead than any I have employed throughout these
tests. In this case, (2) reduces to
3 bg
Be
which has now to be compared with (b+g+7) in (1).
Taking the very lowest observed value of 7, which was
2x 10° ohms, and putting g=8000, and 6=200 ohms, we
have p 1
b+g+r 50,000?
so that « is quite negligible in equation (1), and the error
introduced by the “ guard” wire is practically nl.
Phil. Mag. 8. 5. Vol. 42. No. 255, Aug. 1896. N
K=9tbt
158 Mr. R. Appleyard on Dielectrics.
Apprnpix II...
Change of Dielectric-resistance with Temperature.
Let the resistance at some fixed temperature be R; and
let Ry, Ry, Rs, &c. be the resistances at 1°, 2°, 3°. . abo R.
The assumption is that the resistance diminishes Le, of itself
for every 1° rise of temperature. In this case, ~
tap = R — Lg
nr
R
r= R—R? = ° 5 » . ° ; . (1)
and | |
R= R,— a
Or, putting in the value of n,
Rae
ho = a
Hence
R RY
z =R( 2), R=R(R) 5 hee
and, generally, 3
R R R, : ‘
= a. a
If r and @ are any two values of ¢, we have F
R, wg
R= RR) ee
Ry log R, —log R,
ye bata )=— eg
By putting observed values of R,, R,, 7, and @ in this last
expression, the value of = may be found, and substituted
C:
in (8), which then becomes the working formula. In the
paper, (ie) is represented by a.
Comparison of « for Centigrade and Fahrenheit Scales.—Let
R, be the resistance 1° C. above that at which R is measured.
Then, by the preceding,
aa =
Measurement of Alternating Electric Currents. 159
and (2) becomes
R,= Ret,
But 1° F. may be expressed as (3) C., so that
. Rs) = Ra, uremia eis Serie. ih()
and Ru ) will be the resistance at 1° sh
Hence Rs) =Ra,. ee | enn)
Thus, from (4) and (5),
5
ap a3),
or, log “=(2) log a.
XIU. On New Instruments for the Direct Measurement of the
Frequency of Alternating or Pulsating Electric Currents.
By ALBERT CAMPBELL, B.A.*
Be many experiments with alternating currents it is a very
great convenience to be able to draw the supply from an
ordinary lighting circuit, but the value of such a source of
current is often lessened by the fact that the frequency of
alternation is usually variable, the limits of variation being
in some cases very wide indeed. It was for the purpose of
getting rid of this uncertainty that, about a year ago, I
devised the two instruments described below.
Most methods of measuring frequency depend either on
Impedance measurements or on the production of Resonance
(or Synchronism). The former I avoided, for unfortunately
the wave-form as well as the frequency affects impedance, and
I aimed at an instrument which would give trustworthy
readings for any kind of pulsating current whatever.
Of the Resonance (or Synchronism) methods two are
familiar. The first consists in running a small synchronous
motor and measuring its speed by a speed-counter or indicator.
This is a rather laborious way, and not very accurate if only
a tachometer is used. In the second method+ a stretched
wire carrying the alternating current is placed in a constant
magnetic field; by varying the stretchin g-weight the wire is
* Communicated by the Physical Society: read May 22, 1896.
+ Due to Professors Ayrton and Perry.
N 2
160 Mr. A. Campbell on the Measwrement of the
tuned until it is set into strong vibration, and then the
frequency is calculated from the tension, length, and mass of
the wire. The original arrangement has been simplified, I
believe, by Mr. Alexander Russell, of Faraday House, who
uses a steel wire thrown into vibration by an iron-cored
choking-coil placed near it. In both arrangements some
uncertainty is introduced by the wire pussing over a bridge
or pulley.
In my first type of instrument I developed the last-
mentioned arrangement. The working parts of the instru-
ment are shown in fig. 1:—
Fig. 1.
A steel wire, W, is fastened at one end to a spring, S, and
at the other to a rack, R, sliding in guides. Near the wire
is fixed an electromagnet M with a laminated core; this
magnet is excited by the given alternating current. By
turning the pinion P the wire can be tightened until it is
thrown into vigorous vibration by the magnet. The pointer Q
attached to the pinion then shows directly the frequency ona
suitably graduated scale. Many varieties of this arrangement
may be used ; for example, the pointer may be fastened to 8,
or other kinds of magnifying-gear may replace the rack-and-
inion.
‘ With careful use I believe that this type is accurate to
within less than 0°2 per cent., but it is not so convenient as ~
the second type, which I now proceed to describe. This more
practical instrument is shown diagrammatically in fig. 2.
Here M is a choking-coil fixed near a steel strip A*, which
can be moved back and forward through the clamp B by a
rack-and-pinion, not shown. A’s change of length is
* J find, since writing the above, that in 1889 Professors Ayrton and
Perry suggested the use of a variable magnetic toneue near an alternatin
y suge o 5 g
electromagnet. Their idea, however, was never embodied in a direct-
1ead.ng practical instrument.
Frequency of Alternating Electric Currents. 161
magnified by the rack-and-pinion D and E and the pointer F.
The protruded length of A is altered until the alternating
Fig, 2.
field due to M causes the maximum resonance, and the fre-
quency is read off on the scale G.
By proper choice of the dimensions of the vibrator and the
rack-and-pinion a very extended scale may be obtained. I
have constructed one instrument in which the pointer goes
more than twice round the circumference from 40 periods per
sec. to 150 periods per sec., and at the middle part of the scale
the accuracy of reading is within 0:3 per cent.
The electromagnet M is usually of fine wire, and has a
non-inductive resistance in circuit with it, so that the current
taken is very small.
The reading is usually taken at the point of maximum
resonance. ‘This is observed by means of the sound given out
or by watching the variation in amplitude either directly or
by mirror, light-spot, and scale. Sometimes I fix near the
vibrator an adjustable piece, against which it rattles or jars
when the resonance is sufficient. This jarring piece may be
made part of an electric circuit, including a lamp or an indi-
eating instrument, in such a way that the circuit becomes
broken when the greatest resonance occurs.
The instrument can be used as a speed-indicator for
machinery by attaching to the rotating shaft a suitable com-
mutator to make and break an electric circuit carried to the
instrument, which may thus be at any distance.
If we keep to simple English we might call such an
instrument a ‘ Wave-teller,” but to make the meaning clearer
I propose to name it a ‘‘ Frequency-teller.”’
In conclusion [ may mention that since there is very little
inertia in the moving parts the readings can be taken rapidly,
and thus the instrument can be made to follow fairly quick
changes in frequency.
162 Dr. Silvanus P. Thompson on
{ XIV. Some Experiments with Rintgen’s Rays.
P g )
By Stivanus P. THompson, D.Se., FRS*
HE following isa brief narration of points observed by
me during the past three months, and which are now
brought before the Physical Society.
1. Many experiments have been made to observe rola
tion of ‘v-rays, but no trace has been found. Tourmalines of
several colours, and thicknesses varying from 0°1 millim. to
6 millim., have been used. Andalusite, mica, cale-spar cut
in slices parallel to the axis, epidote, and ripidolite have been
tried without result. The method pursued has chiefly been
to cut the slice of crystal into three parts, laying two of them
upon the third, one of the upper parts having its axis parallel
to the axis of the under part, while the other had its axis laid
at right angles. In this way equal thicknesses of crystal were
compared side by side. Thinking that results might be
obtained from dichroic crystals containing a metal of consi-
derable atomic weight, slices were tried of crystals of sulphate
of nickel, sulphate of nickel and potassium, sulphate ef cobalt
and potassium, sulphate of cobalt and ammonium, fluosilicate
of cobalt, and fluosilicate of nickel, but no trace of polarization
was seen.
Another method consisted in comparing the opacity of
tourmaline in a direction parallel to the axis with that of an
equal thickness in a direction at right angles to the axis.
Not the slightest difference was observed either in the photo-
graphic shadows or with the use of a luminescent screen of
barium platinocyanide.
2. For several weeks in the months of February and March
experiments were made with many different forms of bulb to
determine the source of the x-rays and the form of tube most
favourable to their production. In common with the con-
clusions ef so many other observers, it was found that the
effective source was in every case a surface against which the
kathodic discharge was directed. A form of tube which gave
Fig. 1.
results superior to those of any Crookes tube at the time in
the market is shown in fig. 1, in which the kathode consisted
* Communicated by the Physical Society : read June 12, 1896,
some Experiments. with Rontgen’s Rays. 163
of a number of iron wires spread out from a centre, and the
antikathode, which also served as anode, was a spade-shaped
piece of iron or platinum. Another form of which many
bulbs were tried was provided with an external kathode of
foil, and an internal anode projecting to about the centre,
terminating in a small spade of platinum-foil as antikathode
(fig.2). This also gave good results, but was liable to be pierced
Fig, 2.
at high stages of exhaustion. With this form various experi-
ments were tried as to the influence of the material of the
antikathodic surface. Glass was found to work quite well,
but to be more troublesome than metal. A phosphorescent
enamel made by fusing together two parts of a soft lead-glass
with one part of Balmain’s luminous paint was also tried.
The result of these experiments was to show that, contrary to
the opinion then current that the presence of much phospho-
‘rescence or fluorescence was promotive of the production of
the x-rays, the 2-rays were most freely emitted when the
conditions were such as to waste as little as possible of their
energy in internal fluorescent effects: that in fact a metal
surface was preferable to a surface of glass, enamel, or por-
celain for receiving the impact of the kathodic discharge.
At about this time Mr. Jackson’s perfected form of focus-
tube was brought out, a tube which for photographic purposes
was found superior to any other form, and has not yet been
‘surpassed.
3. Observing in some of these experiments that the metal
spade used as antikathode became red-hot, a special tube was
constructed for me by Mr. Gardiner to test the question
~whether the high temperature of the antikathode was, or was
“not, prejudicial to the emission of w-rays. This special tube
(fig. 3) was furnished with an antikathode of platinum-foil
mounted so that while serving both as antikathode and as
anode it could be heated by passing a current through it
‘from an auxiliary battery. Observing the activity of the
“tube by means of a luminescent screen of platinocyanide of
“potassium, it was found that the heating of the antikathode,
“so far from being disadvantageous, decidedly promoted the
164 Dr. Silvanus P. Thompson on
emission of x-rays, and increased the continuity and brilliancy
of the luminescence. Various amounts of current were sent
through the platinum, the most effective result being obtained
by currents which heated the surface to visible redness.
Whether the effect is a direct one or an indirect owing to the
driving out of occluded gases is not yet determined.
4, When watching with the luminescent screen the emis-
sion of #-rays within bulbs connected with the pump, some
observations were made of direct importance with regard to
the state of exhaustion that is best. The degree of vacuum
which suffices for the production of kathodic shadows is
known not to be sufficient for the production of w-rays. It is
also known that when evacuation is pushed very far the
internal resistance of the bulbs rises very high, so that they
become almost non-conductive. If a bulb is exhausted, and
heated during exhaustion, and the vacuum pushed almost to
non-conductivity, and if a little air is again admitted and the
tube again exhausted, the high degree of vacuum is again
very soon reached, probably because during the first ex-
haustion the gases absorbed upon the walls of the bulb were
mostly removed. After three or four repetitions of this
process the transition from the low state of vacuum to the
high state is exceedingly rapid. Ifa bulb in such a condition
is examined by the luminescent screen while the pump is at
work, scarcely any trace of x-rays can be noticed so long as
the vacuum is such that the resistance is low. A pair of
discharging-points arranged as a shunt to the tube serves as
an approximate gauge. Kathodic shadows can be seen when
the resistance is so low that the discharge- points do not spark
even when placed 3 millimetres apart. When the resistance
rises so that the spark-points must be put 20 or 30 millimetres
apart x-rays begin to be given off ; and are given off both
trom the back and from the front of the antikathode. The
bulb, as seen upon the adjacent screen, shows two pale lumis
some Experiments with Roéntgen’s Rays. 165
nous regions divided by a fine oblique black line which is in
the plane of the antikathode (fig. 4). If the pump goes on
Fig. 4.
(ett PLLITELE Yip
Yy
pe Mba sexrrc2ao
UP yyy rarer sare rrape rane
iy yy
LLY YG)
YY LY
Yy Y thy
ep
Z
LORETO TIE TITLED ILOTEL TT LLLY UY
LLTLELELLE LY
G
working, in two or three seconds, or while only a few cubic
centimetres of mercury pass through the pump, the pheno-
menon changes. The luminosity behind the antikathode dies
out, and that in front of the antikathode increases; so that
there is seen simply a bright anterior region ending at the
oblique plane of the antikathode, beyend which all is dark
(fig. 5). This oblique delimitation can also be seen in the
yellow phosphorescence upon the walls of the bulb. This
sudden transition occurs after the resistance of the bulb has
passed its minimum. The brightest luminescence occurs
when the spark-length exceeds 40-50 millimetres. The lumi-
nosity does not fall off much even at very small angles to the
plane of the antikathode, proving that the emission of w-
rays does not follow Lambert’s law of the cosine by any
means, Hxperiments on this point are still in progress,
166 Some Euperiments with Rontgen’s Rays.
5. The phenomenon of diselectrification by x-rays is very
readily demonstrated. For this purpose I have found a very
convenient instrument to be an electroscope consisting of
two strips of aluminium leaf (which is lighter than gold leaf)
suspended in a thin-glass jar entirely covered with a fine
metal gauze. It is charged with a dry pile, and a metal
cap is then placed over the charging knob, so that it is
entirely electrostatically screened from external electrical
influences. Positive and negative electrifications are both
readily discharged, even at the distance of several feet from
the bulb.
6. On the first announcement of the diselectrifying pro-
perties of x-rays, I attempted to obtain electric dust-figures
as shadows of metallic objects by applying the 2-rays to dis-
charge electrified surfaces of glass or ebonite upon which
mixed powders of red lead and sulphur were then dusted.
These were obtained almost at the first trial; but to produce
them satisfactorily requires a little care.
The object whose shadow is to be obtained—a pair of
scissors, for example—is laid upon a thin sheet of aluminium
placed to stand on four feet at the height of about 20 millim.
over the sheet of ebonite or varnished glass upon which the
shadow is tobe thrown. This sheet of ebonite is first carefully
diselectrified by passing it over an alcohol flame, and then laid
upon an earthed sheet of foil upon the table. The aluminium
tray with the scissors upon it is placed over the ebonite. A
guard-box of lead with a rectangular hole in its top is placed
over all. Then the aluminium tray is charged electrically
by a small influence-machine which has one pole put to
earth and the other connected to the aluminium tray. In
this state of things the ebonite plate hes in an electro-
static field, but is not electrified upon its upper surface.
The x-rays are now caused to fall upon the aluminium tray,
through which they pass save when obstructed by the metallic
object, and, discharging the tray, virtually carry down the
electrification upon the surface of the ebonite in straight
lines, leaving the shadowed portions unelectrified. The
influence-machine is disconnected, the aluminium tray re-
moved, the sheet of ebonite lifted off the table, and the mixed
powders are forthwith dusted over its surface, revealing the
shadow. Both positive and negative shadows can be obtained.
Several alternate dispositions are possible.
These observations were made early in February before the
announcement by M. Righi of some similar cases of production
of shadows by 2-rays.
7. I have also made some observations upon the reflexion
On the Theory of Optical Images. 167
of x-rays. The production of diffuse reflexion by solid bodies
is very easily observed; but hitherto I have no clear evidence
of specular reflexion. Air unfortunately itself sets up dif-
fusion, behaving as a semi-opaque fluid. If ordinary expe-
riments on the reflexion of light had to be carried on in dense
smoke or in miiky water,.a similar diffusion would interfere
with specular reflexion. In one set of experiments a V-tube
made of lead pipes set at right angles, and open at the bottom,
was used, the z-ray source ‘being made to shine down one
limb, while a shielded photographic plate was placed at the
upper end of the other. The surfaces to reflect «-rays were
placed at the open lower ends at 45° to the lines of incidence
and of presumed reflexion. Reflexion of a sort was indeed
‘obtained when surfaces of metal and of glass were placed
across the bottom of the tubes. But an effect was also
obtained even when nothing was placed across the open
bottom. Itseems exceedingly doubtful whether true specular
reflexion has been observed in any case.
XV. Onthe Theory of Optical Images, with Special Reference
to the Microscope. By Lord RayueicH, Sec. Rh. S.*
HE special subject of this paper has been treated from
two distinct points of view. In the work of Helmholtz +
the method followed is analogous to that which had long
been used in the theory of the telescope. It consists in
tracing the image representative of a mathematical point in the
object, the point being regarded as self-luminous. The limit
to definition depends upon the fact that owing to diffraction the
image thrown even by a perfect lens is not confined to a point,
but distends itself over a patch or disk of light of finite dia-
‘meter. Two points in the object can appear fully separated only
when the representative disks are nearly clear of one another.
The application to the microscope was traced by means of a
somewhat extended form of Lagrange’s general optical
theorem, and the conclusion was reached that the smallest
resolvable distance ¢ is given by
e=)/sin a,
X being the wave-length in the medium where the object is
situated, and a the divergence-angle of the extreme ray (the
semi-angular aperture) in the same medium. If Ay be the
wave-length in vacuum,
Sy een Oy SehiD. Cire. tnnes vo owrink Sh)
* Communicated by the Author.
+ Pogg. Ann, Jubelband, 1874.
168 Lord Rayleigh on the Theory of Optical Images,
» being the refractive index of the medium ; and thus
€= SA, /mesinae. +. » -. cen
The denominator wsina is the quantity now well known
(after Abbe) as the “ numerical aperture.”
The extreme value possible for « is a right angle, so that
for the microscopic limit we have
C= ay/f- .« s + «| oer
The limit can be depressed only by a diminution in A», such
as photography makes possible, or by an increase in p, the
refractive index of the medium in which the object is
situated. |
This method, in which the object is considered point by
point, seems the most straight-forward, and to a great extent
it solves the problem without more ado. When the repre-
sentative disks are thoroughly clear of one another, the two
points in which they originate are resolved, and on the other
hand, when the disks overlap the points are not distinctly
separated. Open questions can relate only to intermediate
cases of partial overlapping and various degrees of resolution.
In these cases (as has been insisted upon by Dr. Stoney) we
have to consider the relative phases of the overlapping lights
before we can arrive at a complete conclusion.
If the various points of the object are self-luminous, there
is no permanent phase-relation between the lights of the
overlapping disks, and the resultant illumination is arrived at
by simple addition of separate intensities. ‘This is the
situation of affairs in the ordinary use of a telescope, whether
the object be a double star, the disk of the sun, the disk of
the moon, or a terrestrial body. The distribution of light in
the image of a double point, or of a double line, was especially
considered in a former paper *, and we shall return to the
subject later.
When, as sometimes happens in the use of the telescope,
and more frequently in the use of the microscope, the over-
lapping lights have permanent phase-relations, these inter-
mediate cases require a further treatment; and this is a
matter of some importance as involving the behaviour of the
instrument in respect to the finest detail which it is capable
of rendering. We shall see that the image of a double point
under various conditions can be delineated without difficulty.
In the earliest paper by Prof. Abbe t, which somewhat
* “ Investigations in Optics, with special reference to the Spectroscope.’’
Phil. Mag. vol. vili. p. 266 (1879).
t+ Archiv. f. Mikr, Anat. vol. ix. p. 418 (1873).
with Special Reference to the Microscope. 169
preceded that of Helmholtz, similar conclusions were reached;
bat the demonstrations were deferred, and, indeed, they do
not appear ever to have been set forth in a systematic manner.
Although some of the positions then taken up, as for example
that the larger features and the finer structure of a micro-
scopic object are delineated by different processes, have since
had to be abandoned*, the publication of this paper marks a
great advance, and has contributed powerfully to the modern
development of the microscope f. In Prof. Abbe’s method
of treating the matter the typical object is not a luminous
point, but a grating illuminated by plane waves. Thence arise
the well-known diffraction spectra, which are focussed near the
back of the object-glass in its principal focal plane. If the
light be homogeneous, the spectra are reduced to points, and
the final image may be regarded as due to the simultaneous
action of these points acting as secondary centres of light.
It is argued that the complete representation of the object
requires the co-operation of all the spectra. When only a
few are present, the representation is imperfect ; and when
there is only one—for this purpose the central image counts
as a spectrum—the representation wholly fails.
That this point of view offers great advantages, at least
when the object under consideration is really a grating, is at
once evident. More especially is this the case in respect of
the question of the limit of resolution. It is certain that if
one spectrum only be operative, the image must consist of a
uniform field of light, and that no sign can appear of the
real periodic structure of the object. From this considera-
tion the resolving-power is readily deduced, and it may be
convenient to recapitulate the argument for the case of
perpendicular incidence. In fig. 1 AB represents the axis,
A being in the plane of the object (grating) and B in the
plane of the image. The various diffraction spectra are
focussed by the lens LL! in the principal focal plane, So repre-
senting the central image due to rays which issue normally
from the grating. After passing So the rays diverge in a
* Dallenger’s edition of Carpenter's ‘ Microscope,’ p. 64, 1891.
+ It would seem that the present subject, like many others, has
suffered from over specialization, much that is familiar to the micro-
scopist being almost unknown to physicists, and vice versd. For myself
I must confess that it is only recently, in consequence of a discussion
between Mr. L. Wright and Dr. G. J. Stoney in the ‘ English Mechanic’
(Sept., Oct., Nov., 1894; Nov. 8, Dec. 13, 1895; Jan. 17, 1896), that I
have become acquainted with the distinguishing features of Prof. Abbe’s
work, and have learned that it was conducted upon different lines to that
of Helmholtz. Iam also indebted to Dr. Stoney for a demonstration of
some of Abbe’s experiments.
170 Lord Rayleigh on the Theory of Optical Images,
cone corresponding to the aperture of the lens and illuminate
a circle CD in the plane of the image, whose centre is B.
The first lateral spectrum 8, is formed by rays diffracted from
Fig. 1.
the grating at a certain angle ; and in the critical case the
region of the image illuminated by the rays diverging from
S, just includes B. The extreme ray §,B evidently proceeds
from A, which is the image of B. The condition for the
co-operation at B of the first lateral spectrum is thus that
the angle of diffraction do not exceed the semi-angular
aperture a. By elementary theory we know that the sine of
the angle of diffraction is /e, so that the action of the lateral
spectrum requires that e exceed A/sina. If we allow the.
incidence upon the grating to be oblique, the limit becomes
3A/sin a, as in (1).
We have seen that if one spectrum only illuminate B, the
field shows no structure. If two spectra illuminate it with
equal intensities, the field is occupied by ordinary interference
bands, exactly as in the well known experiments of Fresnel.
And it is important to remark that the character of these
bands is always the same, both as respects the graduation of
light and shade, and in the fact that they have no focus.
When more than two spectra co-operate, the resulting inter-
ference phenomena are more complicated, and there is
opportunity for a completer representation of the special
features of the original grating *.
* These effects were strikingly illustrated in some observations upon
gratings with 6000 lines to the inch, set up vertically in a dark room
and illuminated by sunlight from a distant vertical slit. The object-glass
of the microscrope was a quarter inch. When the original grating,
divided, upon glass (by Nobert), was examined in this way, the lines
were well seen if the instrument was in focus, but, as usual, a compara-
tively slight disturbance of focus caused all structure to disappear.
When, however, a photographic copy of the same glass original, made
with Special Reference to the Microscope. ial
While it is certain that the image ultimately formed may
be considered to be due to the spectra focussed at So, S,...,
the degree of conformity of the image to the original object
is another question. From some of the expositions that have
been given it might be inferred that if all the spectra emitted
from the grating were utilized, the image would be a complete
representation of the original. By considering the case of a
very fine grating, which might afford no lateral spectra at all,
it is easy to see that this conclusion is incorrect, but the
matter stands in need of further elucidation. Again, it is not
quite clear at what point the utilization of a spectrum really
begins. All the spectra which the grating is competent to
furnish are focussed in the plane 8) 8,; and some of them
might be supposed to operate partially even although the part
of the image under examination is outside the geometrical
cone defined by the aperture of the object-glass. For these
and other reasons it will be seen that the spectrum theory *
valuable as it is, needs a good deal of supplementing, even
when the representation of a grating under parallel ugh
is in question.
~ When the object under examination is not a grating or a
structure in which the pattern is repeated an indefinite number
of times, but for example a double point, or when the incident
light is not parallel, the spectrum theory, as hitherto developed,
is inapplicable. As an extreme example of tbe latter case we
may imagine the grating to be self-luminous. It is obvious
that the problem thus presented must be within the scope of
any complete theory, and equally so that here there are no
spectra formed, as these require the radiations from the different
with bitumen, was substituted for it, very different effects ensued. The
structure could be seen even although the object-glass were drawn back
through 14 inch from its focussed position; and the visible lines were
twice as close, as if at the rate of 12,000 to the inch. The difference
between the two cases is easily explained upon Abbe’s theory. A soda
flame viewed through the original showed a_ strong central image
(spectrum of zero order) and comparatively faint spectra of the first and
higher orders. A similar examination of the copy revealed very brilliant
spectra of- the first order on both sides, and a relatively feeble central
image. The case is thus approximately the same as when in Abbe’s
experiment all spectra except the first (on the two sides) are blocked out.
* The special theory initiated by Prof. Abbe is usually called the
“diffraction theory,” a nomenclature against which it is necessary to
protest. Whatever may be the view taken, any theory of resolving
power of optical instruments must be a diffraction theory in a certain
sense, so that the name is not distinctive. Diffraction is more naturally
regarded as the obstacle to fine definition, and not, as with some expo-
nents of Prof. Abbe’s theory, the machinery by which good definition is
brought about.
172 Lord Rayleigh on the Theory of Optical Images,
elements of the grating to possess permanent phase-relations.
It appears, therefore, to be a desideratum that the matter
should be reconsidered from the older point of view, according
to which the typical object is a point and nota grating. Such
a treatment illustrates the important principle that the theory
of resolving-power is essentially the same for all instruments.
The peculiarities of the microscope arise from the fact that
the divergence-angles are not limited to be small, and from
the different character of the illumination usually employed ;
but, theoretically considered, these are differences of detail.
The investigation can, without much difficulty, be extended
to gratings, and the results so obtained confirm for the most
part the conclusions of the spectrum theory.
It will be convenient to commence our discussion by a
simple investigation of the resolving-power of an optical
instrument for a self-luminous double point, such as will be
applicable equally to the telescope and to the microscope. In
fig 2 AB represents the axis, A being a point of the object
and B a point of the image. By the operation of the object-
glass LL’ all the rays issuing from A arrive in the same phase
at B. Thus if A be self-luminous, the illumination is a
maximum at B, where all the secondary waves agree in phase.
Fig, 2.
B is in fact the centre of the diffraction disk which constitutes
the image of A. At neighbouring points the illumination is
less, in consequence of the discrepancies of phase which there
enter. In like manner, if we take a neighbouring point Pin
the plane of the object, the waves which issue from it will
arrive at B with phases no longer absolutely accordant, and
the discrepancy of phase will increase as the interval AP
increases. When the interval is very small, the discrepancy
of phase, though mathematically existent, produces no prac-
tical effect, and the illumination at B due to P is as important
as that due to A, the intensities of the two luminous centres
being supposed equal. Under these conditions it is clear
that A and P are not separated in the image. The question
is, to what amount must the distance AP be increased in
order that the difference of situation may make itself felt in
the image. This is necessarily a question of degree ; but it
does not require detailed calculations in order to show that
weth Special Reference to the Microscope. 1723
the discrepancy first becomes conspicuous when the phases
corresponding to the various secondary waves which travel
from P to B range over about a complete period. The illumi-
nation at B due to P then becomes comparatively small, in-
deed for some forms of aperture evanescent. The extreme
discrepancy is that between the waves which travel through
the outermost parts of the object-glass at L and L’; so that,
if we adopt the above standard of resolution, the question is,
where must P be situated in order that the relative retarda-
tion of the rays PLand PL’ may on their arrival at B amount
to a wave-length (A). In virtue of the general law that the
reduced optical path is stationary in value, this retardation
may be calculated without allowance for the different paths,
pursued on the further side of L, L’, so that its value is
simply PL—PL'. Now since AP is very small, AL’— PL’
is equal to AP.sina, where @ is the semi-angular aperture
L/AB. In like manner PL—AL has the same value, so
that
PL—PL/=2AP.sina.
According to the standard adopted, the condition of resolution
is therefore that AP, or e, should exceed $)/sin a, as in (1).
If e be less than this, the images overlap too much; while if ¢
greatly exceed the above value the images become unneces-
sarily separated.
In the above argument the whole space between the object
and the lens is supposed to be occupied by matter of one
refractive index, and » represents the wave-length zn this
medium of the kind of light employed. If the restriction as
to uniformity be violated, what we have ultimately to do with
is the wave-length in the medium immediately surrounding
the object.
The statement of the law of resolving-power has been made
in a form appropriate to the microscope, but it admits also of
immediate application to the telescope. If 2R be the diameter
of the object-glass, and D the distance of the object, the angle
subtended by AP is e/D, and the angular resolving-power is
given by
DDisinucen piety ei peewee) op (
the well-known formula.
This method of derivation makes it obvious that there is no
essential difference of principle between the two cases,
although the results are conveniently stated in difterené
forms. In the case of the telescope we have to do with a
Phil. Mag. 8. 5. Vol. 42. No. 255. Aug. 1896. ()
174 ~=Lord Rayleigh on the Theory of Optical Images,
linear measure of aperture and an angular limit of resolution,
whereas in the case of the microscope the limit of resolution is
linear and it is expressed in terms of angular aperture.
In the above discussion it has been supposed for the sake of
simplicity that the points to be discriminated are self-lumi-
nous, or at least behave asif they were such. It is of interest
to inquire how far this condition can be satisfied when the
object is seen by borrowed light. We may imagine that the
object takes the form of an opaque screen, perforated at two
points, and illuminated by distant sources situated behind.
- Tf the source of light be reduced to a point, so that a single
train of plane waves falls upon the screen, there is a perma-
nent phase-relation between the waves incident at the two
points, and therefore also between the waves scattered from
them. In this case the two points are as far as possible from
behaving as if they were self-luminous. If the incidence be
perpendicular, the secondary waves issue in the same phase ;
but in the case of obliquity there is a permanent phase-
difference. This difference, measured in wave-lengths, in-
creases tp to e, the distance between the points, the limit
being attained as the incidence becomes grazing.
- When the light originates in distant independent sources,
not limited to a point, there is no longer an absolutely definite
phase-relationship between the secondary radiations from the.
two apertures ; but this condition of things may be practically
maintained, if the angular magnitude of the source be not too:
large. Jor example, if the source be limited to an angle 0
round the normal to the screen, the maximum phase-difference
measured in wave-lengths is esin @, so that if sin @ be a small
fraction of X/e, the finiteness of @ has but little effect. When,
however, sin @ is so great that e sin @ becomes a considerable
multiple of X, the secondary radiations become approximately
independent, and the apertures behave like self-luminous
points. It is evident that even with a complete hemi-.
spherical illumination this condition can scarcely be attained
when e¢ is less than 2. |
The use of a condenser allows the widely-extended source
to be dispensed with. By this means an image of a distant
source composed of independently radiating parts, such as a
lamp-flame, may be thrown upon the object, and it might at
first sight be supposed that the problem under consideration
was thus completely solved in all cases, inasmuch as the two
apertures correspond to different parts of the flame. But we
have to remember here and everywhere that optical images
are not perfect, and that to a point of the fame corresponds
with Special Reference to the Microscope. 175
in the image, not a point, but a disk of finite magnitude.
When this consideration is taken into account, the same
limitation as before is encountered.
— For what is the smallest disk into which the condenser is
capable of concentrating the light received from a distant
point? Fig. 2 and the former argument apply almost
without modification, and they show that the radius A P of
the disk has the value 4)/sina, where @ is the semi-angular
aperture of the condenser. Accordingly the diameter of the
disk cannot be reduced below 2X; and if ¢ be less than » the
radiations from the two apertures are only partially inde-
pendent of one another.
It seems fair to conclude that the function of the condenser
in microscopic practice is to cause the object to behave, at any
rate in some degree, as if it were self-luminous, and thus to
obviate the sharply-marked interference-bands which arise
when permanent and definite phase-relations are permitted to
exist between the radiations which issue from various points
of the object.
As we shall have occasion later to employ Lagrange’s
theorem, it may be well to point out how an instantaneous
proof of it may be given upon the principles already applied.
As before, A B (fig. 3) represents the axis of the instrument,
Fig. 3.
A and B being conjugate points. P is a point near A in the
plane through A perpendicular to the axis, and Q is its image
in the perpendicular plane through B. Since A and B are
conjugate, the optical distance between them is the same for
all paths, e.g. for ARS Band AL MB. And, since AP,
BQ are perpendicular to the axis, the optical distance from
P to Q is the same (to the first order of small quantities) as
from A to B. Consequently the optical distance PRS Q is
the same as ARS B. Thus, if pw, »’ be the refractive indices
in the neighbourhood of A and B respectively, a and 8 the
divergence-angles R A L, 8S BM for a given ray, we have
ie Ae esima— nw BON sim By a" < =” (0)
O02
176 Lord Rayleigh on the Theory of Optical Images,
where AP, BQ denote the corresponding linear magnitudes of
the two images. This is the theorem of Lagrange, extended
by Helmholtz so as to apply to finite divergence-angles*.
We now pass on to the actual calculation of the images to
be expected upon Fresnel’s principles in the various cases that
may arise. The origin of coordinates (E=0, 7=0) in the
focal plane is the geometrical image of the radiant point. If
the vibration incident upon the lens be represented by
cos (27Vt/A), where V is the velocity of light, the vibration
at any point &, 7 in the focal plane ist
= splot 4 ifs | aed, - ne
in which f denotes the focal length, and the integration with
respect to « and y is to be extended over the aperture of the
lens. If for brevity we write
2né)\f=p, 2ry/rAf=q, - -. sees
(7) may be put into the form
oT S 2
sin (Vt —f)— xf 8 (Vif). se)
S={j sin (pa+qy)dzdy, . ...
C=\j cos (pa + qy) da dy. Me
It will suffice for our present purpose to limit ourselves to the
case where the aperture is symmetrical with respect to x and
y. We have then S=0, and
C=\| cos px cos qydedy, . =). es
the phase of the vibration being the same at all points of
the diffraction pattern.
When the aperture is rectangular, of width a parallel to
x, and of width 6 parallel to y, the limits of integration are
from —3a to +44 for x, and from —}% to+4b for y. Thus
___, sin (w7Ea/Af) sin (arnb/Af) s
C=ab mec Se (13)
and by (9) the amplitude of vibration (irrespective of sign)
is C/Af. This expression gives the dittraction pattern due toa
single point of the object whose geometrical image is at $=0,
Te
rf
where
* T learn from Czapski’s excellent Theorie der Optischen Instrumente
that a similar derivation of Lagrange’s theorem from the principle of
minimum path had already been given many years ago by Hockin
(Micros. Soc. Journ. vol. iv. p. 337, 1884).
t See for example Enc. Brit., “ Wave Theory,” p. 430 (1878).
with Special Reference to the Microscope. 177
7=0. Sometimes, as in the application to a grating, we wish
to consider the image due to a uniformly luminous line,
parallel to 7, and this can always be derived by integration
trom the expression applicable to a point. But there is a
distinction to be observed according as the radiations from
the various parts of the line are independent or are subject to
a fixed phase-relation. In the former case we have to deal
only with the intensity, represented by I? or C?/A?f?; and
we get
ee a*b sin? (7Ea/Af)
2 ey
| are 2:
by means of the known integral
St coyene S a ee
i tae= | ee ipa ct (15)
LZ i
—o —e@
This gives, as a function of &, the intensity due to a self-lumi-
nous line whose geometrical image coincides with & =0.
Under the second head of a fixed phase-relation we need
only consider the case where the radiations from the various
parts of the line start in the same phase. We get, almost as
before,
Lee __ sin (7Ea/Af)
yw) __ Cdn=a San ee Ch Pee (16)
for the expression of the resultant amplitude corresponding
to &.
In order to make use of these results we require a table of
the values of sin u/u, and of sin? u/u’. The following will
suffice for our purposes :— |
TABLE I,
4u sin wu sin? w du sin & sin? %
T o yr 7 ue ue
0 + 1:0000 1-0000 9 +:1000 0100
1 -9003 *8105 10 a DAS ‘0162
2 6366 4053 ll 0818 “0067
3 °3001 “0901 12 “0000 “0000
4 0000 “0000 13 — 0692 "0048
| 5 — ‘1801 0324 14 — ‘0909 "0083
6 — ‘2122 -0450 15 — 0600 0036
| 7 ~ +1286 0165 16 0000 0000
8
|
|
| 0000 ‘0000
When we have to deal with a single point or a single line
178
only, this table gives directly the distribution of light in the
image, u being equated to wa/Af. The illumination first
vanishes when w=7, or &/f=X/a.
On a former occasion* it has been shown that a self-
luminous point or line at u= —7ris barely separated from one
at w=(. It will be of interest to consider this case under
three different conditions as to phase-relationship : (i.) when
the phases are the same, as will happen when the illumina-
tion is by plane waves incident perpendicularly ; (i1.) when
the phases are opposite ; and (il1.) when the phase-difference
is a quarter period, which gives the same result for the in-
Lord Rayleigh on the- Theory of Optical Images,
Tass Il.
| ;
sin u sin wu sin 2%
4u uU u a u
‘ie sin (u+7) __ sin (w+7) sin?(w+7))
ut@r Uta (w+ 7 )? |
—4... +1-0000 —1-:0000 +1:000
—3... +1°2004 — 6002 + ‘949
—2... 4+. 1°2732 ‘0000 + -900
ieee +1:2004 + :6002 — + 949
Ozs + 1:0000 +1:0000 +1-000
ha + 7202 +1:0804 + ‘918
Die: + 4244. + °8488 + ‘671
ee + ‘1715 + 4287 + °326
As “0000 ‘0000 “000
Bess — ‘0800 — ‘2801 — ‘206
GO: — 0849 — ‘3395 — ‘247
lees — ‘0468 — ‘2105 — 152
Ore “0000 “0000 “000
9... + -0308 + -1693 + ‘122
LOZ + :0364 + -2183 + -156
1th Peer + :0218 + ‘1419 + -101
ieee “0000 ‘0000 “OGO
tensity as if the apertures were self-luminous. The annexed
table gives the numerical values required. -In cases (i.) and
(iii.) the resultant amplitude is symmetrical with respect to
the point u=—%37 midway between the two geometrical
images ; in case (i1.) the sign is reversed, but this of course
has no effect upon the intensity. Graphs of the three functions
are given in fig. 4, the geometrical images being at the
points marked —7z and 0. It will be seen that while in case
lii., relating to self-luminous points or lines, there is an
approach to separation, nothing but an accurate comparison
with the curve due to a single source would reveal the
duplicity in case i. On the other hand, in case ii., where
* Phil. Mag. vol. viii. p. 266, 1879.
with Special Reference to the Microscope. 179
there is a phase-difference of half a period between the
radiations, the separation may be regarded as complete.
| Fig. 4,
en
ys Cre Ss
PEC ANCEE EEE Eee
SUE ARCRaE ES gas
BUECANECEE Ea
SiMe erein&
he Hee OCecercanced
(EEE EEENEED EER
Bar ceeene ce gate
917]: |
~ In a certain sense the last conclusion remains undisturbed
even when the double point is still closer, and also when the
aperture is of any other symmetrical form, e. g. circular.
For at the point of symmetry in the image, midway between
the two geometrical images of the radiant points, the com-
ponent amplitudes are necessarily equal in numerical value
and opposite in sign, so that the resultant amplitude or illu-
mination vanishes. For example, suppose that the aperture
is rectangular and that the points or lines are twice as close
as before, the geometrical images being situated at w= —}7,
u=0. The resultant amplitude is represented by /(~),
where
sinw sin (wu+47)
— — ° ° e e ie
I (~) U U +4 LD ( )
The values of f(u) are given in Table III. They show
that the resultant vanishes at the place of symmetry w= —}7,
180 Lord Rayleigh on the Theory of Optical Images,
and rises to a maximum at a point near w=47, considerably
beyond the geometrical image at w=0. Moreover, the value
of the maximum itself is much less than before, a feature
which would become more and more pronounced as the points
were taken closer. At this stage the image becomes only a
TasueE III.
Au 4u
— Ft (u). a F (uw).
ose +:00 Ly eRe 2 sie. -— "05
Oe +°36 6.2: eee |
| RCE ye +60 (Pete 23
7 SE AB +:-64 ese eaee —13
Seek 1 +°48 Gr eee +°02
OL tte +21 |
|
very incomplete representation of the object ; but if the forma-
tion of a black line in the centre of the pattern be supposed
to constitute resolution, then resolution occurs at all degrees
of closeness*. We shall see later, from calculations conducted
by the same method, that a grating of an equal degree of
closeness would show no structure at all but would present a
uniformly illuminated field.
* These results are easily illustrated experimentally. JI have used
two parallel slits, formed in films of tin-foil or of chemically deposited
silver, of which one is conveniently made longer than the other. These
slits are held vertically and are viewed through a small telescope, pro-
vided with a high-power eye-piece, whose horizontal aperture is re-
stricted to a small width. The distance may first be so chosen that
when backed by a neighbouring flame the double part of the slit just
manifests its character by a faint shadow along the centre. If the flame
is replaced by sunlight shining through a distant vertical slit, the effect
depends upon the precise adjustment. When everything is in line the
image is at its brightest, but there is now no sign of resclution of the
double part of the slit. A very slight sideways displacement, in my
case effected most conveniently by moving the telescope, brings in the
half-period retardation, showing itself by a black bar down the centre.
An increased displacement, leading to a relative retardation of three
halves of a period, gives much the same result, complicated, however, by
chromatic effects.
In conformity with theory the black bar down the image of the double
slit may still be observed when the distance is increased much beyond
that at which duplicity disappears under flame illumination.
For these experiments I chose the telescope, not only on account of
the greater facility of manipulation which it allows, but also in order to
make it clear that the theory is general, and that such effects are not
limited, as is sometimes supposed, to the case of the microscope.
with Special Reference to the Microscope. 181
But before proceeding to such calculations we may deduce
by Lagrange’s theorem the interval ¢ in the original object
corresponding to that between u=0 and w=7 in the image,
and thence effect a comparison with a grating by means of
Abbe’s theory. The linear dimension (&) of the image cor-
responding to u=7 is given by €=Af/a; and from
Lagrange’s theorem |
eam Gy-sin a, 1 een (Lika
in which @ is the “semi-angular aperture,’ and B=a/2/.
Thus, corresponding to u=7,
66
e=3$)/ sin @.
The case of a double point or line represented in fig. 4
lies therefore at the extreme limit of resolution for a grating
in which the period is the interval between the double points.
And if the incidence of the light upon the grating were
limited to be perpendicular, the period would have to be
doubled before the grating could show any structure.
When the aperture is circular, of radius R, the diffraction
pattern is symmetrical about the geometrical image (p=0,
q=0), and it suffices to consider points situated upon the
axis of € for which 7 (and g) vanish. Thus from (12)
+R
— Jeospe COUT 2) cos pa (R?— 2?) daz . (18)
7 | i
‘This integral is the Bessel function of order unity, de-
finable by
JEG) = a (z cos d) sin’ d dd. ome (19)
Thus, if c=Rcos ¢,
st 251(pR) ¢
ett eanrae ei heipvae py e(20)
or, if we write wu=7& .2R/Af,
(21)*
This notation agrees with that employed for the rectungular
aperture if we consider that 2R corresponds with a.
The illumination at various parts of the image of a double
point may be investigated as before, especially if we limit
ourselves to points which lie upon the line joining the two
* Enc. Brit., “ Wave Theory,” p. 482.
182 Lord Rayleigh on the Theory of Optical Images,
geometrical images. The only difference in the calculations
is that represented by the substitution of 2J, for sine. We
shall not, however, occupy space by tables and drawings such
as have been given for a rectangular aperture. It may
suffice to consider the three principal points in the image due
to a double source whose geometrical images are situated at
u=0 and w= —z7, these being the points just mentioned and
that midway between them at w=—4m. The values of the
functions required are 3
2J,(0)/0 =1:0000 = ¥ {1:0000}.
23 (a) /a4 "1812 = ./{-03283}.
23, (4m) /4or = “7217 = /{°5209%.
- In the case (corresponding to i. fig. 4) where there is simi-
larity of phase, we have at the geometrical images amplitudes
11812 as against 1°4434 at the point midway between.
When there is opposition of phase the first becomes +°8188,
and the last zero*. When the phases differ by a quarter
period, or when the sources are self-]uminous (iii. fig. 4), the
amplitudes at the geometrical images are ./{1°0328} or
10163, and at the middle point ./{1°0418} or 1:0207. The
partial separation, indicated by the central depression in
curve iii. fig. 4, is thus lost when the rectangular aperture is
exchanged for a circular one of equal width. It should be
borne in mind that these results do not apply to a double lzne,
which in the case of a circular aperture behaves differently
from a double poznt.
There is one respect in which the theory is deficient, and
the deficiency is the more important the larger the angular
aperture. The formula (7) from which we start assumes
that a radiant point radiates equally in all directions, or
at least that the radiation from it after leaving the object-
glass is equally dense over the whole area of the section.
In the case of telescopes, and microscopes of moderate
angular aperture, this assumption can lead to no appreciable
error ; but it may be otherwise when the angular aperture
is very large. The radiation from an ideal centre of
transverse vibrations is certainly not uniform in various
directions, and indeed vanishes in that of primary vibration.
If we suppose such an ideal source to be situated upon the
axis of a wide-angled object-glass, we might expect the dif-
fraction pattern to be less closely limited in that axial plane
* The zero illumination extends to all points upon the line of sym-
metry. .
with Special Reference to the Microscope. 183
which includes the direction of primary vibration than in that
which is perpendicular to it. The result for a double point
illuminated by borrowed light would be a better degree of
separation when the primary vibrations are perpendicular to
to the line of junction than when they are parallel to it.
Although it is trae that complications and uncertainties
under this head are not without influence upon the theory of
the microscopic limit, it is not to be supposed that any con-
siderable variation from that laid down by Abbe and Helm-
holtz is admissible. Indeed, in the case of a grating the
theory of Abbe is still adequate. so far as the limit of
resolution is concerned ; for, as Dr. Stoney has remarked,
the irregularity of radiation in different directions tells only
upon the relative brightness and not upon the angular
position of the spectra. And it will remain true that there
ean be no resolution without the cooperation of two spectra
at least. 3 a,
In Table II. and fig. 4 we have considered the image of a
double point or line as formed by a lens of rectangular
aperture. It is now proposed to extend the calculation to
the case where the series of points or lines is infinite, con-
stituting a row of points or a grating. The intervals are
supposed to be strictly equal, and also the luminous intensities.
When the aperture is rectangular, the calculation is the same
whether we are dealing with a row of points or witha grating,
but we have to distinguish according as the various centres
radiate independently, viz., as if they were self-luminous, or
are connected by phase-relations. We will commence with
the former case. at he
If the geometrical images of the various luminous points
are situated at w=0, w= tv, u= +22, K&e., the expressions
for the intensity at any point u of the field may be written as
an infinite series,
Gd) = sin’u sin?(w+v) , sin?(w—v)
ue (Ce) (i—0)*
sin?(u+2v) , sin?(uw—2v)
(w+ 2v)? (uw—2v)? ?
- Being an even function of « and periodic in period v, (22)
may be expanded by Fourier’s theorem in a series of cosines.
Thus
2arru x
K(u) =I, + Tcos m+... +1 cos Baek IG (23)
ee peer (2)
and the character of the field of light will be determined when
184 Lord Rayleigh on the Theory of Optical Images,
the values of the constants Ip, I,, &c., are known. For these
we have as usual
ae AROLe ee ‘f “T(u) 008 a
U 1D)
and it only remains to effect the integrations. To this end
we may observe that each term in the series (22) must in
reality make an equal contribution to I. It will come to
the same thing whether, as indicated in (24), we integrate
the sum of the series from 0 to v, or integrate a single term
of it, e.g. the first, from —o to +0. We may therefore
take
1 (+2 sin?x T
Ih = =| A du= s : 5 5 e = (25)
VU )—a
dU... = an
LS
SO pe am
2 (+ sin?u whe Qrru
v
To evaluate (26) we have
Osim COSISH 9 A a|, i, Vke ae
ie carer ea du =|" du Sie u cos su) du,
and
ae 3 oo
dy SD u COS SU) = — =SiN su
u
2
Boe
=
so that by (15) (s being positive)
+2 gin? =
sin MOOS Myo} — 3 4S 4 }.
u 2
+ sin (2+s)u + Tn (2—s)u;
4 —- 4
the minus sign being taken when 2—s is negative.
Hence
—@
2 TP
lees) or 0). . en
according as v exceeds or falls short of rz.
We may now trace the effect of altering the value of v.
When v is large, a considerable number of terms in the
Fourier expansion (23) are of importance, and the discon-
tinuous character of the luminous grating or row of points is
fairly well represented in the image. As v diminishes, the
higher terms drop out in succession, until when v falls below
%4 only Ij and 1, remain. From this point onwards 1, con-
with Special Reference to the Microscope. 185
tinues to diminish until it also finally disappears when wv
drops below zw. The field is then uniformly illuminated,
showing no trace of the original structure. The case v=7 is
that of fig. 4, and curve ili. shows that at a stage when an
infinite series shows no structure, a pazr of luminous points
or lines of the same closeness are still in some degree
separated. It will be remembered that v= corresponds to
e=+4)/sina, e being the linear period of the original object
and a the semi-angular aperture.
We will now pass on to consider the case of a grating or
row of points perforated in an opaque screen and illuminated
by plane waves of light. If the incidence be oblique, the
phase of the radiation emitted varies by equal steps as we
pass from one element to the next. But for the sake of
simplicity we will commence with the case of perpendicular
incidence, where the radiations from the various elements all
start in the same phase. We have now to superpose ampii-
tudes, and not as before intensities. If A be the resultant
amplitude, we may write
_sinw sin(u+v) , sin(u—v)
A(u) = er, ga aL
2aru Qrru
=A,+Aicos —— +...+A, cos 5 SE ee a - (28)
When v is very small, the infinite series identifies itself
more and more nearly with the integral
1 (+? sin u , Se
— du, viz. —.
u v
Uv J—@
In general we have, as in the last problem,
oon: +0 a
do=5| ne. A=2| secs dus (29)
so that Ay=z/v. As regards A,, writing s for 2rr/v, we
have
fos am SE eas ei +),
—a U Vv ae
=—
the lower sign applying when (1—s) is negative. Accord-
ingly,
A(u)= 24 142 c08=™ +2c08 ==" 4, ie } . (30)
the series being continued so long as 27r<v.
186 Lord Rayleigh on the Theory of Optical Images,
If the series (30) were continued ad infinitum, it would.
represent a discontinuous distribution, limited to the points
(or lines) u=0, u=+v, u=+2v, &e., so that the image
formed would accurately correspond to the original object.
This condition of things is most nearly realised when v is.
very great, for then (30) includes a large number of terms.
As v diminishes the higher terms drop out in succession,
retaining however (in contrast with (27)) their full value
up to the moment of disappearance. When v is less than 27,
the series is reduced to its constant term, so that the field
becomes uniform. Under this kind of illumination, the
resolving-power is only half as great as when the object is
self-luminous.
These conclusions are in entire accordance with Abbe’s
theory. The first term of (80) represents the central image,
the second term the two spectra of the first order, the third
term the two spectra of the second order, and so on. Reso-
lution fails at the moment when the spectra of the first order
cease to cooperate, and we have already seen that this
happens for the case of perpendicular incidence when v=27.
The two spectra of any given order fail at the same moment.
If the series stops after the lateral spectra of the first
order,
LT
$ A(u) = 7 {14200877 i, . jes
showing a maximum intensity when u=0, or $v, and zero
intensity when w=4r, or #v. These bands are not the
simplest kind of interference bands. The latter require the
operation of two spectra only ; whereas in the present case
there are three—the central image and the two spectra of the
first order.
“We may now proceed to consider the case when the inci-
dent plane waves are inclined to the grating. The only
difference is that we require now to introduce a change of
phase between the image due to each element and its
neighbour. The series representing the resultant amplitude
at any point u may still be written
sinw . sin(u+v) pie sin(u—v)
etime
U UtrV U—v
sin(u+2v) _,,
SSS Ce eee ee By
u-+2v i ts (2)
For perpendicular incidence m=0. If y be the obliquity,
e the grating-interval, ’ the wave-len oth, iS
moyenne sity) XN. . = 02) = + ooay
~ with Special Reference to the Microscope. - 187
The series (32), as it stands, is not periodic with respect to
uw in period v, but evidently it can differ from such a periodic
series only by the factor ¢”. ke:
The series —
e~imsinyu emt sin (u+v)
U U+V
—imu—Y)sin(u—v) | ea mt2”)sin(u+2v) © : Be
e | ;
u fos! v + wu =f Qn + ee @ @ ° (34)
is truly periodic, aa may ‘therefore be expanded by Fourier’s
theorem in periodic terms :
(34) =A,+7B,+ (A, +2B,) cos (27ru/v)
+ (C,+7:D,) sin (27u/v)+.....
+(A,+iB,) )eos(2rau/v) + (C,+2D,)sin(2rru/v) +... (39)
As before, if s=2rz/v,
Lo(A_+iB,) =i e~*™sIN U COS § Si
so that B.=0, while
‘s i Se uaa, 7 : ; (36)
U
In like manner C,=0, while
a, D.=|— sin mu ana Set ola (37)
In the case of the zero suffix 3
Bao, chose |" BREED es
When the products of sines and cosines which occur in
(36) &c. are transformed in a well known manner, the inte-
gration may be effected by (15). Thus
cos mu sin u cos su=4F{sin(1+m-+s)u+sin(l1—m—s)u
+sin(1+m—s)u+sin(l—m-+s)u} ;
so that
4v. A =tr{[l+m4+s]4+ [1—m—s]+[1+m-—s]
+[l—m+s]} . (39)
where each symbol such as [1+m-+sj is to be replaced by
at 1, the sign being that of (L+m-+s). In like manner
nD paella os Wl mners| = [l+m-+s]
—[l-—m-—s]} . (40)
188 Lord Rayleigh on the Theory of Optical Images,
The rth terms of (35) are accordingly
sole" ((1+m+s] + [1—m—s])
+e (1 4m—s]+[1—m+s])};
or for the original series (32),
5 emt ™ ([L+m+s]+[1—m—s]) ;
+e"—™ (TL+m—s]+[lL—m+s])} . (41)
For the term of zero order,
amu Timur |
Ave =9,¢ ({l+m]+[1l—m]). . . (42)
From (41) we see that the term in ¢“”*™ vanishes unless
(m+s) lies between +1, and that then it is equal to
m/v.ée"*™ » also that the term in ¢”~®” vanishes unless
(m—s) lies between +1, and that it is then equal to
a/v.é”*, In like manner the term in e’”“ vanishes unless
m lies between +1, and when it does not vanish it is equal to
alv.e”". This particular case is included in the general
statement by putting s=0.
The image of the grating, or row ef points, expressed by
(32), is thus capable of representation by the sum of terms
w/v : A la RE GE TL ERED TE Te ae i : (43)
where s,=277/v, s.=47/v, &., every term being included for
which the coefficient of «u lies between +1. Hach of these
terms corresponds to a spectrum of Abbe’s theory, and repre-
sents plane progressive waves inclined at a certain angle to
the plane of the image. Each spectrum when it occurs at
all contributes equally, and it goes out of operation sud-
denly. If but one spectrum operates, the field is of uniform
brightness. If two spectra operate, we have the ordinary
interference bands due to two sets of plane waves crossing
one another at a small angle of obliquity *.
Any consecutive pair of spectra give the same interference
bands, so far as illumination is concerned. For
wg jefe tea i giulm26e4 Def ae aa ae Te eiulm+2¢r+3)n/0)
v
0) (7)
of which the exponential factor influences only the phase.
In (48) the critical value of v for which the rth spectrum
* Enc. Brit. “Wave Theory,” p. 425.
with Special Reference to the Microscope. 189 -
disappears is given by, when we introduce the value of m
from (33),
; 2a (esiny )=
v ( Xx Seth == a;
or, since (as we have seen) 5 =‘ ~ = : (44)
e(siny+sina)=+ra. . . . . (45)
This is the condition, according to elementary theory, in
order that the rays forming the spectrum of the rth order
should be inclined at the angle a, and so (fig. 2) be adjusted
to travel from A to B, through the edge of the lens L.
The discussion of the theory of a reetan gular aperture may
here close. This case has the advantage that the calculation
is the same whether the object be a row v of points or a grating.
A parallel treatment of other forms of aperture, e.g. the
circular form, is not only limited to the first alternative, but
applies there only to those points of the field which lie upon
the line joining the geometrical images of the luminous
points. Although the advantage lies with a more general
method of investigation to be given presently, it may be well
to consider the theory of a circular aperture as specially de-
duced from the formula (21) which gives the image of a
single luminous centre.
If we limit ourselves to the case of parallel waves and
perpendicular incidence, the infinite series to be discussed is
am Ji(utv) Jy(u—v) Ji(ut2v) .
A(u)= us ke oe
where
1 TEN ee re en (AO)
Since A is necessarily periodic in period v, we may
assume
A(u)=Ay+ A, cos (27ru/v) +...+A,cos (2rau/v)+...3 . (48)
and, as in the case of the rectangular aperture,
+0 a
a Ais Loy ee a) ney ee an)
| ie oe v
These integrals may be evaluated. If a and b be real, and
a be positive *,
—ar t — 1 <a
e~* Jo(ba)da= CET}
* Gray and Mathews’ ‘ Bessel’s Functions,’ 1895, p. 72
Phil. Mag. 8. 5. Vol. 42. No. 255. Aug. 1896. E
(50)
190 Lord Rayleigh on the Theory of Optecal Images,
Multiplying by 6dd and integrating from 0 to 6, we find
1 Ji(or)e™ 1. Ae eS (51)
In this we write b>=1, a=7s, where s is real. Thus
y Ji) eee ee
0
a
If s?>1, we must write z /(s?—1) for /(1—s?). Hence,
if s< i,
J
[SE dex V(l-#), . - 62)
ib: Jee | re
while, if s>1,
| ene
0 X
* J1(x) sin sx .
i da — 4/(s°—1) se
We are here concerned only with (52), (54), and we con-
elude that Ajp=2/v, and that
A,=—-—_———., or0, . -. © {a8}
according as s is less or greater than 1, viz. according as
2r7 is less or greater than v.
If we compare this result with the corresponding one (80)
for a rectangular aperture of equal width (2R=a), we see
that the various terms representing the several spectra enter
or disappear at the same time; but there is one important
difference to be noted. In the case of the rectangular aper-
ture the spectra enter suddenly and with their fu!l effect,
whereas in the present case there is no such discontinuity,
the effect of a spectrum which has just entered being infi-
nitely small. As will appear more clearly by another method
of investigation, the discontinuity has its origin in the sudden
rise of the ordinate of the rectangular aperture from zero to
its full value.
with Special Reference to the Microscope. 191
In the method referred to the form of the aperture is sup-
posed to remain symmetrical with respect to both axes, but
otherwise is kept open, the integration with respect to x
being postponed. Starting from (12) and considering only
those points of the image for which 7 and gq in equation (8)
vanish, we have as applicable to the image of a single lumi-
nous source
C= \y cos pa da dy=2\ycospade . deere (50)
in which 2y denotes the whole height of the aperture at the
point x. This gives the amplitude asa function of p. If
there be a row of luminous points, from which start radiations
in the same phase,-we have an infinite series of terms, similar
to (57) and derived from it by the addition to p of positive
and negative integral multiples of a constant (p,) repre-
senting the period. Thesum of the series A(p) is necessarily
periodic, so that we may write
A(p)=Ao+ .--+A,cos 2rap/p,)+...; . (58)
and, as in previous investigations, we may take
+20
A= C cos Sp dp, ° e ° ° ° (59)
— co
s (not quite the same as before) standing for 2rm/p,, and a
constant factor being omitted. To ensure convergency we
will treat this as the limit of
+0
| on Ceosspd porta’ 24 Gi G0)
the sign of the exponent being taken negative, and A being
ultimately made to vanish. Taking first the integration
with respect to p, we have
e+"? cos xp cos s api= ee + ;
iy P nS P h? (< S)° h? (x s)? F)
and thus
i hy dz hy dx
ES let (+s)? y le (a—s)2’
in which h is to be made to vanish. In the limit the inte-
grals receive sensible contributions only from the neigh-
bourhoods of z= +s; and since
+o dy
{ fae. P F : F alt (61)
iy Pe
192 Lord Rayleigh on the Theory of Optical Images,
we get
A = o(y ey, )= 27 2 ° | oa (62)
From (62) we see that the occurrence of the term in A,,
2. ¢. the appearance of the spectrum of the rth order, is asso-
elated with the value of a particular ordinate of the object-
glass. If the ordinate be zero, z.e. if the abscissa exceed
numerically the half-width of the object-glass, the term in
question vanishes. ‘The first appearance of it corresponds to
$a=2ra/p,=rrf/éi,
in which a is the entire width of the object-glass and &, the
linear period in the image. By (17a),
<P. Suen
S Timesiva, jp esina
so that the condition is, as before,
esina =P.
When A, has appeared, its value is proportional to the ordinate
a x=s. ‘Thus in the case of a circular aperture (a=2R) we
ave
y,., = RY {1—r re sin? a}...
The above investigation relates to a row of luminous points
emitting light of the same intensity and phase, and it is
limited to those points of the image for which 7 (and q)
vanish. If the object be a grating radiating under similar
conditions, we have to retain cosgy in (12) and to make
an integration with respect to g. ‘Taking this first, and
introducing a factor e+*7, we have
+2
i eM cos qy dq = po 6 «+ obras
This is now to be integrated with respect to y between the
limits —y and +y. If this range be finite, we have
5 ee tY Qk dt
Limitiao | a 27, en
independent of the length of the particular ordinate. Thus
+o
C, =| Cdg= 2a \ cos pa diy 2+ oa
the integration with respect to 2 extending over the range for
which y is finite, that is, over the width of the object-glass.
with Special Reference to the Microscope. 193
Tf this be 2R, we have
+a
Cdgq = 47r/p.smpR. . . . (67)
From (67) we see that the image of a luminous line, all
parts of which radiate in the same phase, is independent of the
form of the aperture of the object-glass, being, for example,
the same for a circular aperture as for a rectangular aperture
of equal width. This case differs from that of a self-luminous
line, the images of which thrown by circular and rectangular
apertures are of different types *.
The comparison of (67) with (20), applicable to a circular
aperture, leads to a theorem in Bessel’s functions. For, when
q is finite,
2Si{ Vv (p? +97) R}
= wR? eel
ee se
so that, setting R=1, we get
Son (preys sin p
4 dg=—.. . . . (69
), ae we ay oe
The application to a grating, of which all parts radiate in
the same phase, proceeds as before. If, as in (58), we
suppose
Ap) == Aigcteet, SrtA COSSD tet ses 5. - an (00)
we have
+ 00
=I. CROSS DR tc so 2 (GL)
from which we find that A,is 47? or 0, according as the
ordinate is finite or not finite at e=s. The various spectra
enter and disappear under the same conditions as prevailed
when the object was a row of points ; but now they enter dis-
continuously and retain constant values, instead of varying
- with the particular ordinate of the object-glass which cor-
responds to «=s.
We will now consider the corresponding problems when
the illumination is such that each point of the row of points
or of the grating radiates independently. The integration
then relates to the intensity of the field as due to a single
source.
By (9), (10), (11), the intensity I? at the point (p, g) of
the field, due to a single source whose geometrical image is
*Enc. Brit., “ Wave Theory,” p. 434.
+ This may be verified by means of Neumann’s formula (Gray &
Matthews, ‘ Bessel’s Functions’ (70) p. 27).
194 - On the Theory of Optical Images.
situated at (0,0) is given by
r2f212 = {\\ cos (pa t+ qy) dx dy}? + {MV sin (pe+ gy) da dy}?
= \\ cos (pa’+qy’) da’ dy’ x {V cos (pa+qy) dx dy
+ {\ sin (pa’ + qy’) da’ dy’ x \\ sin (pe+qy) dx dy
= \\\\ cos {p(a! 2) + g(y'—y)} da dy dw! dy’, . (7)
the integrations with respect to 2’, y’, as weil as those with
respect to x, y, being over the area of the aperture.
In the present applic:tion to sources which are periodically
repeated, the term in cos sp of the Fourier expansion repre-
senting the intensity at various points of the image has a
coefficient found by multiplying (72) by cossp and inte-
grating with respect to p from p=—o to p=+x. If the
object be a row of points, we may take g=0; if it bea
grating, we have to integrate with respect also to g from
g=—-H% tog= +o.
Considering the latter case, and taking first the inte-
grations with respect to p, g, we introduce the factors
ettka, the plus or minus being so chosen as to make
the elements of the integral vanish at infinity. After the
operations have been performed, h and k are to be supposed
to vanish*. The integrations are performed as for (6U),
(64), and we get the sum of the two terms denoted by
Qhk
P+ (a a+ 3) + (yyy
We have still to integrate with respect to dady dz’ dy’.
As in (65), since the range for y’ always includes y,
ate 2k dy’
Limitz=o \ ogee = oF 3
(73)
and we are left with
27h dx dy da!
(Wet a
If s were zero, the integration with respect to 2’ would
be precisely similar; but with s finite it will be only for
certain values of x that (#—«#-+s) vanishes within the range
of integration. Unless this evanescence takes place, the limit
when fA vanishes becomes zero. The effect of the integration
with respect to 2’ is thus to limit the range of the subsequent
* The process is that employed by Stokes in his evaluation of the
integral intensity, Edin. Trans. xx. p. 317 (1858). See also Enc. Brit.,
“ Wave Theory,” p. 481. :
Operation Groups of Order 8p, p being any prime number. 195
integration with respect to x. The result may be written
PN da dy) Nae CTS)
upon the understanding that, while the integration for y
ranges over the whole vertical aperture, that for w# is limited
to such values of wv as bring x+s (as well as x itself) within
the range of the horizontal aperture. The coefficient of the
Fourier component of the intensity involving cos sp, or
cos (2rmp/p;), is thus proportional to a certain part of the
area of the aperture. Other parts of the area are inefficient,
and might be stopped off without influencing the result.
The limit to resolution, corresponding to r=1, depends only
on the width of the aperture, and is therefore for all forms of
_ aperture the same as for the case of the rectangular aperture
already fully investigated. -
If the object be a row of points instead of a row of lines,
g=0, and there is no integration with respect to it. The
process is nearly the same as above, and the result for the
coefficient of the rth term in the Fourier expansion is pro-
portional to Vy dx, instead of Vy dz, the integration with
respect to x being over the same parts of the aperture as
when the object was a grating. The application to a circular
aperture would lead to an evaluation of
ea J,?(u) cos su dis
2
we u
XVI. The Operation Groups of order 8p, p being any prime
number*, By G. A. Minter, PA.D.F
CCORDING to Sylow’s theorem these groups contain
kp+1 conjugate subgroups of order p and 6> 1 in the
equation | ae
8p Lay
kp+1_ 2"
Hence they must contain a self-conjugate subgroup of this
order when p>3 and p#7. We shall first consider all the
possible groups that contain such a self-conjugate subgroup.
* M. Levavasseur gives an enumeration of these groups, without ex-
plaining how they were obtained, in Comptes Rendus, March 2, 1896.
His enumeration is not quite correct. He states that there are three
groups which exist only when p—1 is a multiple of 4 without being also
a multiple of 8. We shall prove that there are only two such groups.
{+ Communicated by the Author.
196 Dr. G. A. Miller on the Operation Groups
The few groups which do not contain such a subgroup will
be considered afterwards.
The eight systems of intransitivity* of the given self-
conjugate subgroup are systems of nonprimitivity of the
required groups. Hence each one of these groups must have
a p,1 isomorphism to some group of order 8. As all of
the latter contain subgroups of order 4, all of the former must
contain subgroups of order 4p.
Since the average number of elements in all the substitu-
tions of a group is n—af, n being the degree and a the num-
ber of systems of intransitivity of the group, every subgroup
of a regular group must be intransitive ; and an intransitive
subgroup of half the order of a transitive group must contain
two and only two systems of intransitivity, which are also
systems of nonprimitivity of the transitive group.
Hach one of the groups under consideration must therefore
contain a subgroup of order 4p, which may be formed by
making some regular group of this order simply isomorphic
to itselft. Since the groups of order 4p are known§, our
problem is reduced to the construction of the nonprimitive
groups containing as heads one of the five regular groups of
order 4p in 1, 1 correspondence to itself.
In what follows we shall consider p>2, as the groups of
order 16 are well known ||. We shall first construct all the
groups which contain as heads one of the two commutative
groups of order 4p in 1,1 correspondence to itself. The
cyclical head will be denoted by H cyc., and the non-cyclical
by H.
Groups containing H cye.
Since there are 2( p—1) positive integers which are less than
4p and prime to 4p, H cye. contains 2(p—1) substitutions of
order 4p. The largest group which transforms H cye. into
itself without interchanging its systems transforms these
substitutions according to a regular commutative group (L)
* Every operation group of a given order may be represented by a
regular substitution group of the same order. Cf. Cayley, ‘ American
Journal of Mathematics,’ vol. i. p. 52; also Dyck, Mathematische Annalen,
vol, xxii. p. 84.
+ Cf. Frobenius, Crelle, ci. p. 287.
t Cf. Netto’s ‘ Theory of Substitutions’ (Cole’s translation), § 98.
§ Cf. Holder, Mathematische Annalen, vol. xliii. p. 411; also Cole and
Glover, ‘ American Journal of Mathematics,’ vol. xv. pp. 202-214.
|| Young, ‘ American Journal of Mathematics, vol. xv. p. 160; Hélder,
Mathematische Annalen, vol. xliii. p. 409; Miller, Comptes Rendus,
Feb. 17, 1896.
of order 8p, p being any prime number. 197
of order 2(p—1), containing 3 and only 3 substitutions of
order 2. These correspond to the substitutions which trans-
form those of order 4p in H cye. into their
Frade A 2ptl, . 4p—t
powers. We have therefore to examine four types of tails
that may be added to H cye., viz. those which are commuta-
tive to the substitutions of H cyc., and those which transform
these substitutions into one of the three given powers.
Since half of the substitutions of H cyc. are the squares of
its substitutions there can be only two commutative groups ;
viz. the cyclical group (G,), and the group (G,) obtained by
adding to H cye. a substitution (¢) which simply interchanges
its systems. The squares of the substitutions in the tail of
G, are also the squares of the substitutions of H cye.
We represent the three substitutions* of the second order
which are commutative to ¢ and transform the substitutions
of H cyc. into the three given powers by s1, so, 83. Sit, Sot, Sst
may be used to construct three distinct tails. The first of
these contains 2p substitutions of order 2 and 2p of order 4,
the second contains 4 of order 2 and 4(p—1) of order 2p, the
third contains only substitutions of order 2. We represent
the groups containing these tails respectively by G3, Gy, G;.
Since s; is commutative only with the subgroup of order 4
in H cyc., and half of the substitutions of this subgroup are
squares of its substitutions, there is only one more tail of this
type. This contains only substitutions of order 8. Similarly
we see that there is only one additional tail of each of the
other two types; and that the former of these contains 4
substitutions of order 4 and 4(p—1) of order 4p, while the
latter contains only substitutions of order 4. We represent
the three groups containing these tails respectively by
Ge, G:, Gs.
Hence, when p>2, there are always 8 groups and only 8
that contain a cyclical subgroup of order 4p. In what follows
we need not consider the groups in which such a subgroup
occurs. When p=2, 2p—1 and 2p+1 are not congruent to
1 and 3 respectively with respect to mod 4, as is the case
* That such substitutions can always be found follows from the fact
that we may transform a generating substitution of a transitive cyclical
group into any other generating substitution by a substitution whose
degree is at least one less than the degree of the group. Since the first
power of this substitution which is commutative to the group must be
contained in the group, its order must be equal to the exponent to which
the power into which it transforms the substitutions belongs with respect
to mod «, « being the order of the given cycle.
198 Dr. G. A. Miller on the Operation Groups
for the other values of p. Hence some of our remarks do not
apply to this case. In fact there are only 6 groups of order
16 which contain a cyclical subgroup of order 8.
Groups containing TH.
By adding ¢ to this head we obtain the third and last
commutative group (Gp) of order 8p. There are, therefore,
three and only three commutative groups of this order for
every value of p>2. When p=2 there are five such groups.
The tail of G, contains 4 substitutions of order 2 and 4{p—1)
of order 2p. The squares of these substitutions are clearly
the same as the squares of those of H. It remains to find
the non-commutative groups that contain H.
The isomorphic group of order 8 contains at least three
substitutions of the second order. If this group is commuta-
tive the corresponding tail must transform the substitutions of
H into their —1 powers, as 2p has primitive roots*. We can
easily find a substitution (s’) of the second order which trans-
forms one of the 4 cycles of a substitution of order 2p in H
into its —1 power. By making s’ symmetrical in the ele-
ments of the other cycles of the same substitution we obtain
a substitution (s) which evidently transforms H into itself.
The tail of the group (Gyo) generated by st and H contains
only substitutions of the second order. Since s is commuta-
tive to the substitutions of the second order in H, we may
construct a group (G,) whose tail contains only substitutions
of the fourth order by using, in place of s, the product of a
substitution of the second order in one of the systems of H
into s. The other two groups which may be constructed in
the same way as Gj, are conjugate to it with respect to (s9),
the pth power of two cycles commutative to ¢, these cycles
being contained in some substitution of H whose order is 2p.
It remains to examine the case when the isomorphic group of
order 8 is not commutative.
Since this group of order 8 contains at least 3 substitutions
of the second order and is non-commutative, it is fully deter-
mined. The corresponding tail must interchange two of the
divisions of the head and transform the substitutions of the
other two divisions into their —1 powers. The group (Gy)
generated by H and sszé clearly satisfies these conditions. Its
tail contains 2p substitutions of order 2 and 2p of order 4.
The other possible group is conjugate to Gy. with respect
to a substitution of the second order in one of the systems
of H.
#* Of. Serret’s Cours d’ Algébre Supérieure, vol. ii. p. 82.
of order 8p, p being any prime number. 199
We have now found 12 groups of order 8p which exist for
every value of p>2. As the remaining groups cannot con-
tain a commutative group of order 4p, they must transform the
substitutions of the self-conjugate subgroup of order p accord-
ing to a cyclic group of order 4 or 8. Such groups can exist
only when p—1 isa multiple of 4 or 8. We shall examine
these two cases separately.
Groups which exist only when p—1 1s a multiple of 4.
We shall first consider the case when p—1 is not also a
multiple of 8. The substitutions which are commutative to
those of the self-conjugate subgroup of order p form a com-
mutative group of order 2p. This cyclical group is therefore
also a self-conjugate subgroup of the required groups, and its
four systems of intransitivity are four systems of nonprimi-
tivity of the required groups. Hence we may regard it as
the head (H,) of the required groups.
The tail to these groups contains 4p substitutions which
transform the substitutions of the head into a power e which
belongs to the exponent 4, mod 2p. We can easily find a
substitution s which transforms the substitutions of the head
into their a power, and is of order 4 and commutative to ¢,
t representing a substitution of the 4th order which simply
interchanges the 4 systems of the head cyclically. H, and
st generate a group (G;3) whose tail contains 2p substitutions
of the second order and 4p of the fourth order.
As s is commutative only with the subgroup of order 2
contained in H, there can be only one more group of this
type. This (G4) may be obtained by using the product of a
substitution of the second order in one of the systems of H,
into s in place of s. The tail of G,, contains 2p substitutions
of order 4, and 4p of order 8. It remains to consider the case
when p—1 is a multiple of 8.
The preceding 14 groups are all present in this case. If
there is an additional group it must transform the substitu-
tions of the self-conjugate subgroup of order p according to a
cyclical group of order 8. We can easily find a substitution
(s;) which is symmetrical in the systems of the given selt-
conjugate subgroup and transforms its substitutions into their
B powers, 8 belonging to the exponent 8, mod p.
Hy, and st (¢ being a substitution of order 8 which merely
interchanges the given systems cyclically) generate a group
(Gy5) whose tail contains p substitutions of order 2, 2p of
order 4, and 4p of order 8. As the tail of G,; is not commu-
tative to any substitution in the subgroup of order p, with
200 Operation Groups of order 8p, p being any prime number.
the exception of identity, there can be only one group of this
type. Hence there are always 12 groups of order 8p (p>2)
which contain a self-conjugate subgroup of order p; when p—1L
zs a multiple of 4 or 8 there are respectively 14 or 15 such
groups. It remains to consider the
Groups of order 8p which do not contain a self-conjugate
subgroup of order p.
When p=7, the equation
Sp
Pee re
is satisfied by k=b=1 as well as by k=0, b=8. If the sub-
group of order p is not self-conjugate there must be 8 such
subgroups. These contain 8 x 6 = 48 substitutions besides
identity. The subgroup of order 8 must therefore be self-
conjugate, and its 7 systems of iniransitivity must be systems
of nonprimitivity of the required groups.
Since the substitutions of the group (H,) of order 8 cannot
be commutative to the entire group, they must be trans-
formed according to a group of order 7. Hence all these
substitutions are of the second order, and H, is fully deter-
mined. If we add to H,a substitution (¢) of order 7 which
simply permutes its 7 systems cyclically, we obtain a group
(Gy.) whose tail contains only substitutions of order 7. As
no substitution of H., besides identity, is commutative to ¢,
there can be no other group of this type.
Hence there are 13 groups of order 56; 12 of these con-
tain a self-conjugate subgroup: of order 7. The remaining
one contains 8 conjugate subgroups of order 7 and a self-
conjugate subgroup of order 8. The last group occurs for
the first time as a group of degree 8*.
The only other value of p> 2 for which there can be groups
which do not contain a self-conjugate subgroup of order p
is 8. In this case it is known that there are three such
groupsf. Hence all the groups of order 8p are completely
determined.
Paris, June 1896.
* oh Cole, ‘ Bulletin of the New York Mathematical Society,’ vol. ii
189
p-
+ of Leyavasseur, Comptes Rendus, March 2, 1896.
[ 201 J _¥
) \
XVII. On the Theory of Moving Electrons and Electric
Charges. By J. Larmor*,
1 aS an interesting paper by Mr. W. B. Morton, communi-
cated by the Physical Society to the Philosophical
Magazine for June, there is a criticism of a portion of
my paper on “A Dynamical Theory of the Electric and
Luminiferous Medium’, which if valid would affect its whole
tenor. As, however, the formule of that paper were to a
considerable extent obtained by two independent trains of
reasoning, it would have to be shown that both were wrong
before an error could be fully substantiated. Asa matter of
fact, Mr. Morton’s criticism arises from his reading into the
analysis assumptions which are not there, but which had been
used, with the proper limitations to secure accuracy, in
another place in the previous part of the paper. As the point
is really fundamental, and as the analytical statement in the
memoir is no doubt too brief to convey at once a grasp of the
procedure employed, without somewhat detailed consideration
on the reader’s part, I beg leave to offer the following general
explanation.
The facts of chemical physics point to electrification being
distributed in an atomic manner, so that an atom of electricity,
say an electron, has the same claims to separate and permanent
existence as an atom of matter. The fundamental question
then is, how far the conception of separate isolated electrons,
pervading the ether of free space, can provide an explanation
of electrodynamic and optical phenomena. In the paper re-
ferred to I have gone further back, and have considered the
question how far such a simple underlying scheme is able by
itself to provide an explanation of physical phenomena in
general ; for it will obviously not be permissible to import
into our dynamical notion of an atom of matter more than
simple electric properties, unless these latter prove to be in-
sufficient to include all actual knowledge of its relations.
The conclusion arrived at in the memoir is that there is
nothing in the ascertained laws of general physics which
points to insufficiency in that scheme ; while there are some
experimental results which somewhat militate against the
existence of interatomic forces of any kind other than those
included in it.
The main feature of the theory referred to is that the ether
* Communicated by the Physical Society.
+ Phil. Trans, 1894 (A) , pp. 719-822, and 1895! (A), pp. 695-743.
202 Mr. J. Larmor on the Theory of
is not matter, as ordinarily assumed, nor in any way like
matter ; it is the uniform substratum (analytical basis, if one
is disposed to use that term, for it can never be the direct
object of perception) in which the atoms of matter consist as
permanent configurations of strain and motion. As was to be
expected, the relations of inertia and elasticity of this uniform
medium are simpler than those of matter, which is merely a
molecular aggregate involved in its constitution. In fact,
the only way to arrive at a scheme of the relations between
ether and matter which shall be a complete dynamical
theory and not merely descriptive, is to abolish the apparent
duality in the phenomena, either by taking as here the mole-
cules of matter to constitute singularities (in the mathe-
matical sense of the theory of functions) in the uniform
eether, or else by trying to make out the ether to be ordinary
matter, and so giving up any atiempt to explain why matter
is molecularly constituted. This molecular constitution of
matter is essential to the former theory, just as it is to all
other theories or illustrations, like the vortex theories, which
hypothecate a uniform underlying medium; it is quite
unintelligible—or rather quite unexplained—on the latter
type of theory.
But however these things may be, the point criticised by
Mr. Morton does not involve any considerations so refined,
or—as possibly may be said—so ambiguous. The sections to
which he objects claim to be a reconstruction of ordinary
electrodynamic theory on the basis of permanent electrons
associated with the atoms of matter. Whatever view one
may entertain as to the presence of qualities other than electric
in the atom, all are I think now-a-days agreed that the electron
is there. And whatever view one may have as to the validity
and sufficiency of an ether with simple rotational elasticity,
the formal equations to which that theory leads for free space
are just those equations of Maxwell which Hertz’s experi-
mental work has fully verified. The problem of electrodynamics
is then that of the free zether, whose properties are represented
analytically by these acknowledged equations, disturbed by
the action of the electrons of material atoms moving abcut in
it. The original Amperean electrodynamics, proceeding by
consideration of elements of current, has not proved valid or
sufficient in matters involving electric radiation, or even
ordinary electrodynaimic force. A most successful modification
of it was that proposed by Weber, in which elements of current
were replaced, as the fundamental object of consideration, by
moving electric particles which acted on each other at a dis-
tance according to a law of force involving their velocities.
Moving Electrons and Electric Charges. 203
This theory was, however, shown long ago by Lord Kelvin
and Professor von Helmholtz to be untenable, on account of
its violating the principles of the modern theory of energy ;:
now, of course, direct action at a distance is altogether out of
court. The present question is whether a theory of electrons:
which act on each other, not directly according to a law of
force, but mediately by propagation of the effect across the
intervening ether, suffices to avoid the discrepancies of earlier
theories and give a consistent account of electrical and optical
phenomena ; and it is maintained that the answer is altogether
in the affirmative. This question is, presumably, sufficiently
important and fundamental to justify the present detailed
explanation.
At the end of the first of the two papers referred to,
building chiefly on the analytical results of previous theorists,
the steady ethereal disturbance carried along by a moving
electron had been investigated, and also the law of the force
exerted on each other by two moving electrons through the
intervention of the ether between them. This was on the
hypothesis that each electron carried along with it a steady
trail of ethereal disturbance, but that no sensible derange-
ment of this steady motion ever occurred such as would lead
to loss of the energy of the system by the starting of waves.
If the velocities of the electrons remain always small compared
with that of radiation, then, however their mutual influences
alter their motions, this steady trail will instantaneously adjust
itself to the new conditions without sensible excitation of
radiation, and the theory will apply. But if any of the elec-
trons are moving with velocities comparable with that of
radiation, a change in velocity will involve derangement of
this steady trail of ethereal strain and motion, giving rise to
wave-motion which will carry off some of the energy by
radiation. Accordingly in such a case it is altogether nuga-
tory to speak of laws of action between electrons: the complete
theory must then take account not only of the positions and
velocities of each of the electrons at each instant, but also of
the state of each volume-element of the surrounding eether.
And the theory of mutual actions of electrons as expressed in
the memoir was in fact thus restricted to cases in which their
velocities were small compared with that of radiation : unless
that condition is satisfied there is no such theory at all.
In the second paper (§ 15, segg.) the general problem is
attacked : it is now nota question of a set of electrons by
themselves, each with a definite steady trail, but of the
ethereal medium in general, including such electrons as exist
in it. The analysis there given determines from foundations
204 Theory of Moving Electrons and Electric Charges.
which all who adopt Maxwell’s electrical scheme for free
ether must allow, expressions for the force (P’, Q’, R’) which
acts on an element of volume of the free ether, and for the
force e (P, Q, R), ordinarily called electric force, which acts
on an electron e; and it uses these forces for further develop-
ment of the theory. What Mr. Morton’s computation
virtually does is to assume that the trail of each electron is
steady, and then to transfer to the electron itself the forcive
due to (P’, Q’, R’) acting on this ethereal trail. In the
special case of no radiation, and of velocities small compared
with that of radiation, this forcive can, as above explained,
be transmitted through the ether to the electron itself, and
be supposed there applied. But to so transmit it in general
is to miss the point of the theory, and, as Mr. Morton himself
remarks, to reach the absurdity that the force on a moving
charge depends not only on the state of the surrounding ether
but on the state of the sether at a distance.
As regards the main subject of Mr. Morton’s paper, it may
be of interest to state the following general theorem. Suppose
a system of charged conductors is in steady translatory motion
through the quiescent zther with velocity u, and let v repre-
sent the velocity of radiation in free ether: consider a cor-
relative system of conductors obtained by uniform geometrical
elongation of the actual system along the direction of motion
in the ratio of (1—w?/v’)— to unity, and find the electrostatic
distribution of the same charges on this system supposed at
rest: then the actual distribution of the charges on the
moving system wil] be exactly correlative, viz., equal charges
will exist on all corresponding elements of the two systems.
This proposition is, however, limited to the case in which
none of the bodies of the moving system are dielectrics, but
all are conductors.
Finally, I take advantage of the present opportunity to
draw attention to some special points in which the analysis
of the second part of my memoir is incomplete. In §§ 28, 29,
on the mechanical pressure of radiation, a statical forcive has
been overlooked ; when this is included the result practically
agrees with that given by Maxwell. In §§ 34-6, on the
material forcives in polarized media, the tractions on inter-
faces of transition remain to be developed, and the theory
may be otherwise improved. I hope presently to treat this
subject at length.
Cambridge, June 4, 1896.
er 20), "|
XVIIL. Notices respecting New Books. —
Jumes Clerk Maxwell and Modern Physics. By R.'T. GLAZEBROOK,
FRS. (Century Science Series.) London: Cassell & Co., 1896.,
— fourteen years have now elapsed since Prof. Campbell and
Mr. Garnett published their biography and letters of Maxwell ;
the shock occasioned by his death had then scarcely passed away.
At that time, although all regarded him as one of the founders of
modern physics, few even among physicists realized the magnitude
of his discoveries and research, and probably none imagined that
his ideas concerning the ether would so soon receive the remark-
able development which they have derived from the life-work of
Hertz. For, so far as regards the problem of the ether, Hertz
has been the chief exponent of Maxwell, just as Maxwell had
previously explained and extended the views of Faraday. But
Maxwell contributed by his laboursto many other physical questions,
everywhere not only adding to the stock of knowledge but
furnishing suggestions for future work. Mr. Glazebrook gives, in
a form suited to the general reader, a brief account of Maxwell’s
work in three of these departments of knowledge—the properties
of Cartesian ovals, the theory of compound colours, and the
dynamical theory of gases. He reserves a longer chapter for the
electrical theories, of which a very concise account is given, as
clear as the mathematical nature of the subject will allow in a
non-mathematical volume.
It should not be forgotten, however, that Maxwell was not only
a great thinker and experimenter, but also an organizer. The
present school of physics at Cambridge owes its existence and
much of its efficiency to him, and has served as a model for the newer
provincial colleges. It is fitting that the story of Maxwell’s work
should be told by one so intimately acquainted with his Cambridge
life as Mr. Glazebrook, and more especially by one who has had
the good fortune to call him master and friend. > alo dele
An Elementary Treatise on the Integral Calculus, containing
Applications to Plane Curves and Surfaces, and also a Chapter on
the Calculus of Variations, with numerous Examples. By
BrysaMIn WinLIAMson, /.R.S. (Longmans: pp. xvili+512.)
Tuts is the seventh edition of a work which first saw the light in
1875, in the humble guise of pp. vi+267. It has thus attained
its majority and enormously increased in importance. On the
appearance of the sixth edition in 1891 we noted the great
advances that had been made on former editions, so that in the
case of so well-known a work it is only necessary to point out the
new features. The Calculus of Variations, which in the last
edition formed chapter xv. with some 34 pages only, is now
chapter xvi. and, in two sections—“ Single Integrals and Multiple
Integrals,’—now occupies double the space. Another novelty is
chapter xv. (12 pp.), “on the Sign of Substitution,” inserted
principally ‘with a view to its employment in the Calculus of
Variations.” This symbol Dr. Williamson states was _ first
Phil. Mag. 8. 5. Vol. 42. No. 255. Aug. 1896. Q
206 Geological Society.
introduced into analysis by Sarrus in his prize memoir,‘ Recherches
sur le Caleul des Variations” (1846).
Solution and Electrolysis. By W.C. Dampizr Wueruam, M.A.
Cambridge, University Press, 1895.
In choosing the title of this volume Mr. Whetham has very aptly
indicated the connexion of his subject with both physics and
chemistry ; for while solution is essentially a chemical process
electrolysis is quite as closely allied to purely physical phenomena.
Such borderland subjects, apart from the fact that they interest two
sections of scientific workers, derive additional importance as beg
the meeting-point of two theories, which, although originally
framed to explain phenomena totally different in nature, must
now be made to harmonize. The process of adjustment is not
usually an easy one, and many are the discussions which it
provokes, even when the experimental data have been carefully
ascertained. The subject of solution is passing through this stage
at the present time, but the accumulation of experimental evidence
has received a great stimulus by the adoption of a provisional
theory. According to this theory the molecules of a dissolved
substance move through the solvent independeutly of the latter, in
a manner comparable with the motion of gas molecules through
the space containing them; in the case of electrolytes it is further
assumed that the molecules are dissociated into their constituent
ions to a greater or less degree according to the concentration of the
solution. The reconciliation of this so-called physical theory with
other doctrines of chemistry and physics is, however, not yet
complete. The author proposes to accept the hydrate theory of
solution, according to which solvent and dissolved substance form
a large complex molecule, making the assumption that the chemical
forces acting within the molecule do not interfere with the physical
independence of its constituents.
Mr. Whetham has followed Ostwald in giving an account of the
general properties of solutions, but bis materials for the portion of
the treatise dealing with electrolytes were much more scattered,
the only previous attempts to collect them having been made by
Wiedemann (1883), and by the Electrolysis Committee of the
British Association. The author has selected the more important
parts of these reports and presents them, together with other and
more recent matter, in a form suited to the needs of students.
The book is issued as a volume of the Cambridge Natural Science
Manuals. James L. Howarp.
XIX. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from p. 123.]
April 29th, 1896.—Dr. Henry Hicks, F.R.S., President, in the Chair,
The following communications were read :—
1. ‘ Descriptions of New Fossils from the Carboniferous Lime-
stone.—I. On Pemmatites constipatus, sp. nov., a Lithistid Sponge.
Intelligence and Miscellaneous Articles. 207
Il. On Paleacis humilis, sp. nov., a new Perforate Coral; with
Remarks on the Genus. III. On the Jaw-apparatus of an Annelid,
Eunicites Reidiit, sp. nov.’ By George Jennings Hinde, Ph.D.,
F.G.S,
2. ‘The Eocene Deposits of Dorset.’ By Clement Reid, Esy.,
PiS.; F.G.S.
The new survey of the western end of the Hampshire Basin
shows that the Reading Beds become fluviatile and gravelly in
Dorset (as was already known), and contain, in addition to Chalk
flints, many fragments of Greensand chert. The London Clay thins
greatly and becomes more sandy, but is apparently still marine. The
Bagshot Sands become coarser and more fluviatile, changing rapidly
west.of Moreton Station, till they consist mainly of coarse subangular
gravel. These gravels, formerly referred to the Reading Series, are
now shown to be continuous with the Bagshot Sands, which as they
become coarser cut through the London Clay and Reading Beds to
rest directly on. the Chalk. The Bagshot gravels contain, besides
Chalk flints and Greensand chert, fragments of Purbeck marble and
numerous Paleozoic grits and other stones probably derived from
the Permian breccias of Devon.
Thus there is evidence of disturbance and overlap in Cretaceous
or early Eocene times, causing Reading Beds to rest on Upper
Greensand. Later disturbances allowed the Bagshot river to cut
into Greeusand, Wealden, Purbeck, Permian breccia, Culm Mea-
sures, and granite. Foiding of the strata seems to have taken ~
place during at least four different periods in the district. between
Dorchester and Weymouth, which appears to have been a region of
special weakness.
The Eocene gravels contain all the foreign rocks known to occur
in the Plateau-gravels between Brighton and Dorchester. The
fragments of Greensand chert, so abundant in the Plateau-gravels,
have not been derived, as supposed, from the central axis of the
Weald. They come, as already-formed pebbles, from the Eocene of
Dorset, and originally from the Greensand of Devon.
2. ‘ Discovery of Mammalian Remains in the Old River-gravels
of the Derwent near Derby.’—Part I. By H. H. Arnold-Bemrose;
Esq. M.A. F.G.S. ~
XX. Intelligence and Miscellaneous Articles.
ON THE REFRACTIVE INDICES OF SOME SUBSTANCES FOR VERY
SHORT ELECTRICAL WAVES. BY DR, A, LAMPA.
HE wave-length of the rays of electrical force used was 8 millim.
This number follows from the dimensions of the exciter, but
was also ascertained by means of an interference experiment. In
reference to the arrangement of the experiments, it may be ob-
served that a coherer was used for demonstrating the electrical
rays. Experiments showed that in the experiment on refraction it
is possible to get a pretty sharp adjustment for the maximum
208 Intelligence and Miscellaneous Articles.
action on the coherer; and hence the arrangement was used for
some quantitative deter minations.
The refractive indices of a series of solid and liquid substances
were determined. The following values were obtained for the wave-
length in question corresponding to a value of N=37-500 x 10°.
Para fine 2225 n=1'524 7? =2°32
Ebonite | o- oee 1:739 ra 4
Crown-glass .......... 2°381 5°16
Plinteolisse see os 2°849 8-41
Sulphiary pees. keer: 1:802 3°24
Benzolep eae... TOW ules
Giivieenittemer sr sci: 1843 3:4
Oil of turpentine ...... 1°782 3°17
Vaseline eee ss... : 1°626 2°65
Oil of aimondsee.:, «i... 1°734 3°01
Absolute alcohol ...... 2:568 6°60
Distilled -water-...... . 8:972 80°45
— Wiener Berichte, July 1896.
A LECTURE EXPERIMENT ON DIATHERMANCITY.
BY DR. SILVIO LUSSANO.
A very simple and elegant method of showing toa large audience
the transparency or opacity of bodies for calorific radiations is the
following, which is based on the change of colour some substances
experience by variations of temperature.
Double iodide of silver and mercury is prepared ina fine powder.
At the ordinary temperature it has a beautiful canary-yellow colour,
changing to a purple-red at the temperature of about 49°, and
resuming its original colour when cold. This double iodide is
prepared by mixing in molecular proportions the two substances
Hel, and 2Ag¢I, then adding to the mixture alcohol which partly
dissolves the mercuric iodide; the magma is then well stirred
together in a mortar, alcohol being added from time to time. The
mixture, which at first is red, then changes to orange, and, after
some time, becomes of a canary-yellow colour; the alcohol is
allowed to evaporate, the magma being stirred all the time.
The double iodide thus pr epared i is spread on cardboard, forming
a screen of a canary-yellow colour sensitive to calorific radiations.
If, then, this is placed below a metal ball strongly heated, the
colour of the screen changes, showing the heating due to the
radiations. The same thing takes place if a plate of ebonite is
interposed between the ball and the screen; but if a plate of a
substance opaque to thermal radiations is placed on the ebonite,
the shadow of this plate is projected on the screen, showing a
yellow colour on a red ground. In this way it can be understood
that it is easy to obtain the projections of athermanous substances
enclosed between two ebonite plates.—Communicated by the Author
from ‘Il Nuovo Cimento, May 1896.
THE
LONDON, EDINBURGH, ayn DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.”
[FIFTH SERIES. |
SEPTEMBER 1896.
XXI. On the Use of Bare Wire for Resistance-coils. By
F. W. Bursratt, 1A., formerly Scholar of St. John’s
College, Cambridge*.
I. Introduction.
a is perhaps somewhat surprising that the form of
resistance-coils as used in the ordinary box has, during
the last twenty years, changed less than any other kind of
electrical instrument. The most serious defects of the pre-
sent form of resistance-box are, firstly, the difficulty of ascer-
taining the temperature of the wire forming the resistance,
and, secondly, that the method of short-circuiting any
required number of coils by means of divided brass blocks
and taper plugs leads to contact-resistances which are often
of very variable amount. A resistance-coil, the wire of
which is covered with silk wound very closely and then
covered with paraffin, of necessity conducts heat very badly,
and therefore the temperature, as shown by a thermometer
placed in the box, gives but little information as to the
temperature of the resistance-wire. Ihave endeavoured to
overcome the temperature difficulty by using manganin
wire, but the uncertain nature of this material has forced me
to abandon its use. The suggestion for the employment of a
bare wire immersed in oil is due to Mr. H. H. Griffiths, and
it is with the developments of this suggestion that the
* Communicated by the Physical Society : read June 26, 1896,
Phil. Mag. 8. 5. Vol, 42. No. 256. Sept. 1896. R
210 Mr. F. W. Burstall on the Use of
present paper deals. There can be no doubt that with a
bare wire immersed in oil the heating by the testing current
is much less than with covered wire, and further, the tempe-
rature of the wire can be accurately measured ; but perhaps
one of the most important advantages in the use of a bare
wire lies in the fact that the wire can be annealed in a most
perfect manner by the passage of a sufficiently large current
to heat it to a red-heat for a few seconds. It has long been
known that well-annealed coils are much less liable to change
than coils which have not been so treated. Dr. Lindeck, of
the Reichsanstalt, has, in his various papers, laid great stress
upon the necessity for annealing standard coils, which is
effected by heating the coil in an air-bath to about 140°, the
highest temperature a silk-covered coil can bear without
material injury. The heating of the wire to a red-heat is, of
course, much more efficient than the method adopted by
Dr. Lindeck for freeing the wire from undue stress. |
The wire which I have employed for the whole of the
resistance-box was drawn from one cast of platinum-silver, a
material which experience has shown to be the most perfect
for resistance-measurement.
The two forms of resistance-box usually employed are the
series form and the dial form. In the former a number of
coils are placed in series, and any required number can be
short-circuited. This form, though it requires comparatively
few coils, has the disadvantage that any fluctuation in one
plug causes an error in the final measurement, and if any
considerable number be used, it is almost impossible to pre-
vent variation in the sum of the contact-resistances. For
these reasons it seemed advisable to adopt the second form of
box, the dial pattern, in which the number of plugs employed
is a minimum, and remains the same whatever resistance is
being measured.
It is not essential to make standard resistances in order to
calibrate a resistance-box, as each dial can be measured in
terms of the preceding dial. This method, however, perpe-
tuates any errors that may have been made in the measure-
ment of any of the smaller coils, and it is not possible to
measure the coils in the box with the same accuracy that can
be attained in the determination of the standard coils. If
have therefore constructed standard coils for every dial,
and their values have been obtained from a 1 ohm coil inde-
pendently of the resistance-box.
Bare Wire for Resistance-corls. 211
I]. Construction of the Resistance-box and Standards.
The resistance-box has five dials, each consisting of nine
coils, ranging from 5 of an ohm to 1000 ohms, and four
pairs of proportional arms ranging from 1 to 1000 ohms.
The ;1, ohm coils have a length of 96 millim., the wire
having a diameter of ‘63 millim. JBeing so short, there is no
difficulty in supporting the coils.
The wire of the 1 ohm coils has the same diameter and a
length of 960 millim. These coils were at first wound into
short spirals, and were allowed to hang freely from their
terminals. It soon, however, became apparent that the con-
volutions of the spiral required to be fixed rigidly to the
ebonite top. The method that has been finally adopted is to
screw the spiral into holes pierced in a mica plate ; this
plate being screwed top and bottom to a brass bar which is
attached to the ebonite top.
In the coils of from 10 to 1000 ohms it is not easy to find
a form of bobbin on which to wind the great lengths of bare
wire required in such a manner as to avoid short circuits, and
to bring the wire in contact as little as possible with the in-
sulator. The form I have ultimately adopted is as follows
(fig. 1):—To a central brass rod are attached, for the 10 ohm
and 100 ohm coils, two small brass crosses at a distance of
about 150 millim. apart; to the ends of the arms of these
crosses are screwed serrated strips of mica, the pitch of the
serrations being about 1 millim. The resistance-wire is then
bifilarly wound into these serrations. The 1000 ohm coils
are of the same form, but from their greater length, 250
~ millim., it is necessary to employ a third cross in order to
strengthen the mica, and even then difficulty has been expe-
rienced owing to the mica bending under the coil-tension,
culminating in several cases in the breakage of the wire.
For the 10 ohm coils wire having a diameter of *30 millim-
and a length of 2400 millim. has been used. For the 100
ohm coils, a diameter of *15 millim. and a length of 6200
millim. For the 1000 ohm coils, a diameter of ‘075 millim.
and a length of 14500 millim.
The top is of ebonite, 86 centimetres long, 28 centim. wide,
and 19 millim. thick.
Instead of using divided blocks for the dials, conical plugs,
having a maximum diameter of 9 millim., and a taper of 1 in
12, fitting into conical sockets have been employed*. The
* This method of constructing the plugs is due to Mr. R. W. Paul,
Proceedings of the Physical Society, 1892.
R 2
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SS Vg, re it
‘~e= i
212
fa
i
we ueuunguarteverrrers
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ire
Fn oe iT
On the Use of Bare Wire for Resistance-coils. 213
plugs are connected with the zero socket of the previous dial
by means of heavy flexible leads, having a copper core, 5
millim. in diameter, composed of copper wires of diameter
*2 of a millim. It was found necessary to use comparatively
large wire, as the first set of connecting leads, built up of
wire ‘07 millim. in diameter, parted with the constant
turning and gave rise to a variable plug-resistance. The
use of the conical plug in a solid conical socket is completely
free from errors caused by the change of shape of the insu-
lating material ; further, such forms can be easily and accu-
rately machined, a matter of no small importance to those
who make their own apparatus.
The terminals are of large diameter (25 millim.), and are
arranged with the screw in the moveable part, so that the
top of the fixed part can be readily cleaned.
The proportional arms are fitted with the usual divided
blocks.
The whole of the metallic part is made of phosphor-bronze,
a material which is harder and less liable to oxidation than
ordinary brass.
The containing box is lined with copper for the reception
of the oil.
The form of bobbin and the wire for the standard resistance-
coils are the same in the 1 and 10 ohm standards ; but in
the 100 and 1000 ohm standards the mica is stiffened by
being slipped into a brass back, like the back of an ordinary
tenon saw, which quite prevents any tendency to bend. The
temperature is measured by means of a thermometer placed
in a pierced tube which lies inside the coil. The coils are
inclosed in an outer brass case and fitted with the usual
copper terminals for use with mercury-cup connexions.
The 5 ohm standard is of manganin, of the usual British-
Association pattern.
The oil employed is a pure heavy hydrocarbon oil, obtained
from Messrs. Price’s Patent Candle Company ; it is free
from any trace of alkali or acid, and experiments conducted
by Messrs. Price prove that it has no action on the wire.
Ill. Construction of Bridge and Galvanometer.
The bridge employed was of the usual equal-armed pattern,
having a straight wire of platino-iridium 1°5 millim. in dia-
meter and 500 millim. long. The wire is stretched imme-
diately over the scale by means of two brass split chucks,
these chucks terminating in circular brass rods, having a
diameter of 6 millim. The rods slide in heavy brass blocks,
214 Mr. F. W. Burstall on the Use of
and the wire can be tightened by nuts on the ends of the
rods. Thus the wire is not soldered to the terminals, and can
be removed for repair. The scale is divided into 1000 parts,
and the divisions are then divided into 10 by estimation ;
this is easily done with the aid of a good magnifier. The
slider moves on the scale itself, and is provided with a fine
adjustment. The connexion to the galvanometer is made by
a platinum-tipped spring, resting on a platinum wire which
is laid parallel to the main wire. The equal coils are made
of manganin, and have a resistance of about 10 ohms. They
have several times been adjusted to exact equality, but have
never preserved that equality for a long period. This in-
equality, however, is unimportant, since it is eliminated by
double balancing, which has been done at least once during
each set of determinations. The actual amount of inequality
at the present time is shown by the facts that, with 1 ohm in
each of the outer arms, the bridge centre (¢. e. the mean of
the readings when the coils are interchanged) is 500-25,
whilst with 1 ohm in each arm it is 500°4, and with 10 ohms
in each arm it is 500°5.
The wire is mounted on a stout marble base. To protect
the wire as much as possible from heat radiated from the
observer’s body two blocks of wood have been hollowed in
such a manner that they can be slid over the wire ; the slider
itself being wrapped in cotton-wool. These precautions re-
duce the effect of the thermo-currents to a few scale-divisions
on the galvanometer. The effect of thermo-currents on the
galvanometer was eliminated by the galvanometer-circuit being
kept closed, the battery-circuit being made and broken as
required. In order to connect the box-coils and the standards
to the bridge, three pairs of heavy flexible copper leads have
been constructed ; one pair has mercury-cups, intended for
the reception of the standard coils, at one extremity; a
second pair has conical plugs, so that any particular coil in
the box can be measured alone ; the third pair and the other
six extremities of the leads are provided with heavy copper
spades, by means of which connexion can be made with the
terminals of the box and the bridge. The three pairs of leads
are adjusted to exact equality one with another, and each has
a resistance of ‘00966 ohms at a temperature of 11°°8. The
values of these resistances do not enter into any determination
as the leads were always in pairs, one in each arm of the
bridge.
The galvanometer is a four-coil astatic instrument, the
resistance of each coil being 60 ohms. The magnet system
is supported by a fine quartz fibre 260 millim. long. In
Bare Wire for Resistance-cotls. 215
order to diminish as far as possible the effects of the vibration
caused by London traffic, thick rubber blocks are placed on
the pier supporting the galvanometer ; this precaution has
made it possible to work at any time except when large
masses of iron are being moved in the neighbouring goods
station. The scale is placed 3 metres from the mirror. The
motion of the mirror is observed by means of a small electric
lamp which projects the image of a fine wire on to the mirror,
thence it is reflected on to a ground-glass scale. The observer
~ is at a considerable distance from the galvanometer, a matter
of some convenience since the zero is thus not so liable to be
changed. The galvanometer gives a distinct movement for
an alteration of ‘00001 of an ohm, with 10 ohms in every
arm, the battery being one Hellesen cell, together with an
added resistance in series with it of 10 ohms. The galva-
nometer is connected to the bridge by 20 gauge rubber-
covered high insulation wires ; the rubber covering is neces-
sary as | have found serious errors introduced unless the
insulation of the galvanometer connexion is extremely good,
IV. Determination of Standards.
The temperature difficulties which are most liable to cause
inaccuracy in resistance measurements have been greatly
simplified by the remarkable uniformity of temperature of
the room in which the measurements have been made. The
room is almost entirely underground, and is protected on all
sides by other portions of the building from the heat of the
sun. It is lighted by one small electric lamp, which gives
rise to no appreciable change of temperature. The presence
of one experimenter causes the temperature of the air to rise
about *2 of a degree during the first half hour, and then it
becomes constant ; but the temperature of the resistance-box
and standards, the latter being placed in earthenware vessels
and surrounded by sawdust, takes several hours to rise as
much as *03 of a degree.
The bridge-wire was calibrated by a method due, I believe,
to Mr. Griffiths. In one of the outer arms is placed, by means
of mercury-cups, a coil of low resistance, called the gauge-
coil; on the other hand, the gap can be short-circuited by
means of a thick copper bar. In the opposite outer arms are
placed three resistance-boxes in parallel. The object of this
arrangement is to secure the necessary fine adjustment.
Commencing with the slider at one end of the wire, the
gauge-coil being in place, a balance is obtained by adjusting
the resistance in the boxes, the slider remaining fixed. The
216 Mr. F. W. Burstall on the Use of
gauge-coil is then removed and the copper bar substituted ;
the slider is then moved until a new balance is obtained. Pre-
ceeding thus, the whole wire is divided into a number of parts
the resistances of which are equal. From these observations
a chart can be prepared showing the deviation in resistance
of the wire from a wire of uniform resistance per unit of
length. One of the great advantages of this method is that,
when an abnormal value has been obtained, it is a simple
matter to come back to that particular portion of the wire
for a check measurement. The gauge-coil was equal to the
resistance of 22°293 mean scale-divisions (each 0°5 millim.),
and the greatest error at any point of the wire amounted to
‘4 of a scale-division. Throughout the whole of the com-
parison of the coils this calibration of the wire has been
employed, and several apparently discrepant observations have
been thus brought into line.
The temperature-coefficient of the bridge-wire has been
found by raising the temperature of the room from 10° 4 te
18°1, the bridge having in one arm a coil of known tem-
perature-coefficient, and was found to be ‘071 per cent. per
degree. |
The thermometer employed, which is of hard Jena glass, is
divided into tenths of a degree, each tenth being approxi-
mately *5 millim. in length. By means of a small reading
telescope it can with ease be read to ‘O01 of a degree. It was
tested at the Reichsanstalt and, over the range employed, was
accurate to at least ‘05 of a degree.
The temperature-coefficient of the platinum-silver wire,
over a range from 10° to 25°, was determined by immersing
a coil of about 1 ohm in paraftin-oil at 25°, and allowing
it to cool very slowly. It was found to be ‘000274 per ohm
per degree. .
The 1 ohm coil from which the standards have been built
up is of manganin, and when tested in June 1894 by Mr.
Glazebrook had a resistance of 1°00026 ohms at 13°4, I
was, however, so doubtful of the constancy of this standard
that in December 1895 I constructed a bare platinum-silver
1 ohm coil. This was compared with the manganin standard,
and the mean of eleven determinations gave its value as
1:00000-+:000274 (t—14-96).
No doubt the manganin coil had changed its value in the
eighteen months that had elapsed since it was tested, so that
the above figures were not true ohms.
The 10 ohm standard was obtained by the use of three
Bare Wire for Resistance-coils. 217
approximate 3 ohm coils*, first placed in parallel and
balanced against a 1 ohm coil, and then in series with the
1 ohm coil and balanced against the 10 ohm coil. Instead
of connecting the legs of the coils by mercury-cups, as is
usually done, “the fasible metal made by Mr. F. Thomas, of
Cambridge, was used. This metal, which melts at 60°, gives
very constant contact resistances, and does not seem to i injure
in any way the amalgamated terminals of the coils. I first
determined the value of a manganin 10 ohm coil, which had
been made for nearly a year. ‘The value obtained on
December 20, 1895, was 9°9987 at 9°°3. This coil was used
as a standard to which to adjust the platinum-silver 10 ohm
coil. The value thus obtained for the latter was 9°9895 at
10°-3. On stepping up to the platinum-silver 10 ohm coil,
the value found on December 30 was 9:9905 at 10° 6.
Taking into account the known temperature-coefficient, the
latter at 10°°3 would be 9°9897, which differs by only two
parts in 100,000 from its comme value.
To obtain the value of the ;/5 ohm coil, which was of man-
ganin, a oe was form d 1D two 1 ohm coils, a 10 ohm
coil, and the qo ohm coil, and a balance obtained by shunting
one oF the 1 ohm coils f.
On January 12th, a value of (99934 at 11° 2 was obtained.
bb) bk) 13th, 9° 99 "099932 99 10°°2
The. temperature-coefficient of the wire was known to be
‘000004 per degree. Between these two determinations the
bridge had been dismounted. This coil was again tested on
April 8th, and had then a value of :099944 at 12°1, whereas,
if it had remained unaltered, its value at that temperature
should have been -099938. This change is, however, too
small to cause any error in the box-coils.
For testing coils which are very nearly in the proportion
of 1 to any power of 10 this method is extremely sensitive,
and certainly more accurate than making the comparison by
means of the wire bridge, provided that the resistances of
three out of the four coils for ming the bridge are accurately
known.
In obtaining the values of the 1 ohm and 10 ohm standard
coils, not more than 50 scale-divisions of the wire had been
employed, hence a small error in the total resistance of the
bridge-wire would not cause any inaccuracy in the values so
obtained. The resistance of the bridge-wire was determined
99 >)
_* This method is described in the B. A. Reports, 1888.
- + This method is described by Mr. Glazebrook, B, A. Reports, 1894,
218 Mr. F. W. Burstall on the Use of
in the first instance by placing the standard 1 ohm coil in
parallel with a coil of slightly smaller resistance than the
bridge-wire. This gave the resistance per scale-division of
the wire as ‘00087 at 10°1. After the value of the ~, ohm
coil had been found, the resistance of the bridge-wire was
determined by balancing the 4, ohm coil against a coil of
slightly smaller resistance than the bridge-wire ; the 7/, ohm
coil was then removed, and the gap short-circuited The
actual figures are of interest as showing the value of a cali-
brated wire.
‘lohmin A. Gaugein D. Temperature 10°8.
Reading 394:0.
Correction for calibration —°3.
Short circuit in A. Gauge in D. Reading 967°9.
Correction for calibration, —°2.
Hence, if p be the resistance per scale-division,
"1 ohm coil=2 x 574 p=1148 p.
On interchanging the coils,—
‘l1ohm in D. Gauge in A.
Reading 32:9. Correction for calibration, 0.
Short circuit in D. Gauge in A.
Reading 607°0. Correction for calibration, —°1.
This also gives
‘1 ohm coil=1148 p.
Whence p='00008705 at 10°-8.
A and D being the two outer gaps of the bridge.
The 100 and 1000 ohm standard coils were determined on
similar bridges. The 100 ohm coil,—
On April 14, had a value 99-913, at 10°°8.
in le eigen 3 oe 004, Leos
Whereas it should have been 99:960, according to the
observation on the earlier date. This discrepancy is, I
believe, due to the fact that the shunt required on one of the
10 ohm coils forming the bridge was greater than 12,000
ohms, which was the largest resistance at my disposal : in
order to obtain a balance one of the 10 ohm coils had to be
warmed, by means of a water-bath, about 2° above the tempe-
Bare Wire for Resistance-coils. 219
rature of the room, which led to some uncertainty as to the
actual temperature.
The 1000 ohm standard coil was obtained by forming a
bridge with the 1, 10, 100, and 1000 ohm coils, and had a
value, on April 28, of 1000-24 ohms at 12°38. No other
determination has, as yet, been made. The coils were
adjusted so as to be right at 15°.
V. Calibration of the Resistance-bow.
For the first three of the dials the coils were balanced
against standard coils (fig. 2), and the differences obtained
Fig. 2.
ee ye Oe BO FO): ©
5) Ce) oo
° °© oO ©
oO
oo
.o)
to} fo) ra) Le) (2)
©5°0 @50 oa) 260 O 5 ©
0-1 GHm
STANDARD
WIRE BRIDGE.
by means of the wire bridge. In order to connect the
conical plugs to the arms of the wire bridge, the flexible leads
with conical ends were used; the standard in the opposite
arm being connected by means of the leads with mercury-
cup ends.
For the 1 and 100 ohms in the proportional arms spade-
ended leads were used, and attached to the terminals at the
ends of the proportional arms (see diagram).
For the sake of convenience of comparison, I have reduced
the resistances to their values at 11°C.
The values of the 7/5 ohm dial are as follows :—
No. of
Coil.
Omori HS oar OD
220. Mr. F. W. Burstall on the Use of
Jan.12. Jan.20. Jan. 22. Jan.24. Jan 25. Jan.26. Mar.11. Mar.i4. Mean.
A0007 “LOOTL: : ann .3.0 gee 10065 10065 ‘10073 -10073
10012. 10012 +=-10009 -10008 -10008 wewcss. 7AOOUZ eee
10067 +=:10069 +=-10063 °10061 °10063 _—..... "10050 eae
“‘VOO008 “TOOT4 ci cas- erg apense acess ‘10011 “100103 ae
09976 “09974 --099fo" 2a 09973 09972 -099(S. eee
09982" -O99S(-” -. 22a eee “09985 “~-09985 - 22S eeeeee
“10046. 10046". > ee 10042. -..225. See
“10003, 1LOO002iy Gaara eeaes BP 8s 10003. - .....2. eee
“TOO055 ~~ TOO), Sao cae, 0 bc anes: “10066, =. ...2.4 See
It will be noticed that, in coils 1 and 3, I have rejected
certain values determined on dates before March 11; the
reason being that, when the new flexible leads were attached
in February, these two coils were touched and slightly bent,
and had in consequence to be re-annealed.
The values of the 1 ohm coils are—
No. of
Coil. Jan. 8. Jan. 11. Jan.26. Feb. 12. Feb. 16. Mean.
Wain AS 99929 aS (ad ee a "99931 "99930
Pee anncson 99956 QUOD Oak teat eRe iice ae cathe "99956
Dinas cseaee “99569 GOOD OS Se ere on mt aaa ngy ee waetent 99568
eee “99964 0969) “GOST? =. ea ee a, Pe ‘99968
Dip ssustes ‘99776 Lie Pees. ae Roser namen ees 99975
G cacres Dee cied 100007, 1:00007-—_ +s... 1:00008 1:00007
DRS St Mee ae ‘99976 ‘99979 ‘99978 -99980 “99978
Oieesecenn cela "99878 99875 -99875 -99878 ‘99877
eRe eam! Baca 1:00053 "> 100051... .55-.- 1:00053 100054
Values of the proportional arms are—
April 3. Mar. 8. Mar. 16. Mean.
1 sineXs eee 99863 99860 ‘99860 99861
ieines eee 99645 99641 99642 ‘99643
Considerable trouble was experienced in the determination
of the 1 ohm coils in the proportional arms, owing mainly to
variations in the resistance of the plugs, which were fitted into
the usual divided blocks. The amount of this error is of
little moment when compared with the 10 ohm coil. The
1 ohm coils have not been employed in the determination of
the larger resistances.
The values of the 10 ohm coils are :—
10073
10010
“10050
‘10011
09974
09985
10045
“10003
‘10061
Bare Wire for Resistance-coils. 221
No. of Ooil. March 8. April 3. April 4. Mean.
1. eessGseseesc 9:9765 DO(G4 ye escecaaee 99764
i ian ce weaneee 9-9926 99925. 7. 8 |e) scaeeeaee 9:9925
Siceccaeeace UT Lhe ceeece 10:0033 10-0033 10:0033
es ee SOOO mo) Scavkeca 9:9973 9:9974
Secs anceses 9:9999 10-0000 10-0001 10-0000
Gieearscces 9:9574 DDT focmececes 9:9573
igen 9:9558 D955 0 9 sans. oee 9°9556
Bi sowbcwscaes 9:9863 9:9858 9-9864 9:9862
ee 9°9748 9:9753 9:9758 9°9753
The values of the proportional arms are :—
Mar. 8. Mar. 16. Mar.17. April3. April4. Mean.
Rn eX «....., 99513 9°9506 99513 99507 99509 9:9509
ia V2... 100106 100099 10:0103 10:0098 100101 10-0101
Desiring to obtain a check measurement, I balanced the nine
‘1 ohm coils against the 1 and 10 ohm standards in parallel,
and obtained the value ‘90224 at 13°, while the first five
coils of the dial, when balanced against the two 1 ohin coils
in parallel, gave °50125 at 138°°2. The actual values for the
sums of the nine and five respectively, as found by the
separate determinations, were ‘90259 and 50147. Of course
any contact-error due to the plugs would appear when the
values of the coils are added up. When the separate values
of the nine coils are added up, the contact-error will have
entered 18 times into the result, and for five coils 10 times ;
on the other hand, when the coils are measured in series, the
contact-error will only enter twice in each case, and the
differences between the two sets of values in both cases are
explained by the ascertained result that the contact-resistance
of each plug is ‘00002.
It is somewhat bold to assert that an error of -00004 ohms
has not entered into the determination of each of the 7, ohm
coils. The standard itself is certainly correct, as will be seen
by reference to its values on the various dates ; the only
possible explanation would be that the connecting leads of
the two arms had not equal resistances. A subsequent test,
however, proved that they were equal. I was, therefore,
driven to the conclusion that the contact-resistance as above
found was real, and not due to an experimental error.
The resistance of the box from terminal to terminal with
each plug in zero was obtained in three ways. Firstly, by
short-circuiting the mercury-cups in one of the arms of the
bridge, and measuring the resistance of the leads and plugs
222 Mr. F. W. Burstall on the Use of
on the bridge-wire. This gave a value of ‘00967 ohm at
12°°8. Secondly, the first coil of the 31, ohm dial, together
with the leads, was- balanced against the 745 ohm standard ;
subtracting from this resistance the known resistance. of the
box-coil, we get a value of :00964 ohm for the resistance of
the leads, at 12°-4. Thirdly, a similar method was employed
to measure the first coil in the 1 ohm dial, and this determi-
nation gave the resistance of the leads as -00954 chm at
10°°3. In order that these three observations may agree it
is necessary to assume that the temperature-coefiicient of the
copper leads is about ‘00005 ohm per degree. This is rather
high, but the temperature range is too small for an accurate
determination.
The values of the coils in the 100 dial were obtained on
the box itself. Inthe gap A (fig. 3) the 1 ohm standard coil
Fig. 3.
1 OHM
STANDARD
eo 8 & FO
was placed; this, together with the first three dials, gave a
resistance of about 100°9 ohms; and constituted one arm of a
Wheatstone bridge, the two 10 ohm coils in the proportional
arms formed two other arms, the fourth arm was any one of
the 100 ohm coils, which was connected to the gap C by
means of the two flexible leads with conical ends. The plug
from the centre of the 100 dial was in the zero of the 1000
dial so as to complete the circuit from the gap A to the gap
C. The smallest resistance in the box was 7), of an ohm, so
that the last two figures of the resistance had to be obtained
by interpolation from the swings of the galvanometer. The
battery employed was one storage-cell, and the current was
commuted for each observation. With the coils of the gal-
vanometer placed in series an alteration of ;4, of an ohm in
the box caused a total galvanometer swing of about 116 scale-
divisions.
Bare Wire for Resistance-coils. 223
The values of the 100 ohm coils are: —
No. of Coil. April 5. April 6.
1 ayer 100°511 100°510
Be eee 100°967 COR Man
Deen) Were... 100°266
2 a ae 100-300 100°305
Dogon sts 100°362 100°365
ue 100°671 100°676
Gee 100°603 100°607
Ones occ 100°434 100°435
ars 100°845 100°850
The 100 ohm proportional arms were determined by the
same method as was used for the 100 ohm standard coils.
Their values are :—
April17. April 20, April 21. Mean:
ioetex’ §. 100224 HOO 219 100:220 100°221
roi «100°312 100°314 100°309 100°312
The 1000 ohm coils were also measured on the box itself.
In the gap A (fig. 4) was placed the 100 ohm standard coil ;
Fig. 4.
100 OHM
STANDARD
this, together with the first four dials, formed one arm of a
Wheatstone bridge ; the two other arms were the 100 ohm coils
in the proportional arms. A single flexible lead connected
the last coil of the 100 dial with any required coil on the
1000 dial. Another lead connected the other terminal of the
coil to be measured to the terminal of the proportional arm Y.
The current was provided by two storage-cells.
The values of the 1000 ohm coils are :—
224 Prof. A. McAulay on the Wave-Surface and Rotation
ra April 25.
1 Bee 1023°35
2 Be 1027°30.
Binge neee 942°05.
re ake 102204.
Db aeenent LOTT 2T os
eae Noa 1032-04
TS se 1007-44.
Biabsorce.2 1003°45 _
so Ae 1026-42
The values of the proportional arms are :—
April 27
ILC ine, 7. Ca ee 947-92
NOOO mM Fo. 2.2 e. 994°21
My object in giving the whole of the figures relating to
the resistance-coils is to show to what an extent bare wire
coils are to be trusted. Whether the coils retain their values
as well as the resistance standards of the British Association,
time alone can decide. There is no reason to suppose that,
if the coils are kept continuously in an oil-bath, and the oil
kept free from water, any considerable alteration can take
place.
XXII. On the Wave-Surface and Rotation of Polarization
Plane in an Aeolotropie Electromagnetic Medium. By
Prof. A. McAutay, University of Tasmania *.
N the ‘ Philosophical Magazine’ for June 1885, p. 397 (also
‘ Electrical Papers,’ vol. ii. p. 1), Mr. Heaviside has in-
vestigated in a very instructive manner the properties of the’
electromagnetic wave-surface and index-surface, and the
relations in a plane wave of the electric and magnetic forces
and the displacement and induction.
In noticing that in one respect his investigation can be
simplified, | have been led to some simple connexions between
the electromagnetic surface and Fresnel’s surface. The most
important result is that the former can in two ways by a real
pure strain be converted into a Fresnel surface ; the axes of
the strain being in the two cases those of permittivity and
inductivity. ,
Thus all the peculiarities—such as double sheet, singular
* Communicated by the Author.
of Polarization Plane in an Aeolotropic Medium. 225
point, &c.—of the Fresnel surface, and no more, are repeated
in the electromagnetic surface.
I am not familiar with the details of experimental work in
Physical Optics, and therefore cannot say whether there is
conclusive evidence that the Fresnel surface is accurately
the form of the true optical wave-surface. We see by the
above statement that the mere qualitées of double refraction,
conical refraction, &c. will not. serve to distinguish between
the Fresnel and general electromagnetic wave-surfaces.
The notation and language of Quaternions will be used
below, and Mr. Heaviside’s valuable practice of denoting
(whenever desirable) vectors by Clarendon type will be
followed.
Prof. Hathaway’s (‘Primer of Quaternions,’ Macmillan,
N.Y., 1896) term “ nonion ” for “linear vector function of a
vector’ will be used. [But I should like, in passing, to say
that I think the term a bad one. Some single term is almost
indispensable, and I had suggested “ Hamiltonian.”’ Prof.
Hathaway rightly objected that Hamilton’s name shouid not
thus be appropriated to a minor function occurring in
Quaternions. I therefore prefer “nonion.” But there is
this serious objection to thus indiscriminately extending
the principle which underlies the formation of the word
‘“‘quaternion”’—that there will be many kends of qua-
ternions, many kinds of nonions, &c. Tor instance, a unit
rotor would be a quaternion, and what I have in Octonions
ealled a self-conjugate pencil function would be a nonion.]
If y be a nonion, and y’ its conjugate, the strain corre-
sponding to y’—! may be called the reciprocal conjugate of
the strain corresponding to y. It is physically described as
follows :—If a given strain be effected by first making a
pure strain and: then a rotation, the pure strain may be
called the pure part of, and the rotation the rotation of,
the given strain. The reciprocal of a given pure strain is
naturally defined as the pure strain whose axes are those of
the given strain and whose elongations are the reciprocals of
those of the given strain. The reciprocal conjugate of a
given strain is then one whose pure part is the reciprocal
of the pure part of the given strain and whose rotation
is the same as that of the given strain.
We require the following
Lemma.—If RB be a given surface, and 8 tts polar reciprocal
with regard to a given origin; and if by a homogeneous strain
which leaves the origin unmoved R become W, and by the
reciprocal conjugate strain S become 8’, then S' is the polar
reciprocal of R! with regard to the same origin.
Phil. Mag. 8. 5. Vol. 42. No. 256. Sept. 1896. S
226 Prof. A. McAulay on the Wave-Surface and Rotation
Fix the attention on a tangent plane of R and the corre-
sponding point of S, and on the plane (tangential to R’) and
point (of 8’) into which these are strained.
Let p be the coordinate vector (the origin being the given
one) of any point of the tangent plane of R, p’ the corre-
sponding strained value of p, o the vector of the point on 8,
and o! the strained value of co. Let y be the given strain, so
that y'—! is the reciprocal conjugate. Thus we have
P=; @=7 '¢, .. = er
Spo =—l . . 4. eee
for all values of p (in travelling over the tangent plane). It
at once follows that
and
Spo’ = —1
for all values of p’. This proves the proposition.
The equations that Mr. Heaviside uses in considering the
wave-surface are : :
VVH=cE=D,.'.. 7 ee
—VVE=ywH=B, .... . (4)
where c and w are self-conjugate nonions, viz. permittivity
and inductivity respectively, and where H, E, B, D are as
usual. The medium is immovable, and ¢ and wu have constant
values at all points.
In order to bring these into harmony with the notation of
my paper “On the Mathematical Theory of Hlectro-
magnetism ”” (Phil. Trans. vol. elxxxiii. 1892, A, p. 685:
this paper will be referred to below as “ M.T. E.”’), I prefer
to write them } \
VV'H = cH =D!) . > ee
Wolfe, J aoe
p' being the coordinate vector of an actual point of the
medium.
Suppose now, in accordance with “ M.T. H.,” we write
p = XP). 6 Ae 2 ee
where y is an arbitrary nonion, which, however, has a constant
value. If then we put
H’ == Xie ist Fp’ = a | E,
Baie yb We Do nig aD: |
po =m xu’, of = myo’,
m having the usual meaning with regard to y, (5) and (6)
(3)
of Polarization Plane in an Aeolotropic Medium. 227
become (3) and (4) respectively. This statement can easily
be verified independently of “ M.T. H.”
In the language of “ M.T. H.” this is equivalent to taking
for the standard position not the actual fixed position of
matter, but a homogeneously strained (7!) state of that.
By suitably choosing y we can make either ¢ or wa
constant scalar. I[t will be sufficient to consider only one of
these cases.
Let m/= py! py’ M3’, where py’, Mo’, w3’ are the actual prin-
cipal inductivities (?. ¢., m’ stands towards pw’ as m towards x).
Put
atin Fe gt oy) ee)
Thus
He ee OG OPE aGlO)
and equation (8) becomes
He 5 BY 2H,
Bo =m?y?B, Di =m? p'?D, . ecg Cal)
w=], c= wie pls,
H
=
Since now in eq. (4) w=1, we see at once that in the ideal
space denoted by p,H, D, &c. the wave-surface of propagation
of H and E or Band D will be a Fresnel surface, and the
corresponding index-surface will be its polar reciprocal.
The equation of this ideal wave-surface being expressed in
terms of p, the true wave-surface will be obtained by straining
by the function y or m’—?u because p’=yxp. The ideal
index-surface is the polar reciprocal of the ideal wave-surface,
and similarly for the actual surfaves. Hence, by the lemma
above, the actual index-surface is obtained from the ideal by
straining by y’—! or m? p'-3,
This change of variables from p to p’ thus enables us to
reduce the finding of the wave-surface from the index-surface
to the ordinary Fresnel case, and thus saves us from the only
part of Mr. Heaviside’s investigation which is very complex.
If we notice that in equation (1) above the relation between
o’ and a is the ordinary intensity relation, and remember that
p’ and o” are naturally taken as the coordinate vectors of
points on the wave- and index-surfaces respectively ; and if
we bear in mind the fundamental properties of intensities and
fluxes, we shall find that all Mr. Heaviside’s results for the
actual case can be written down from the corresponding ones
for the simplified ideal case. For instance, if we show in the
simplified case that
o = VBD/3S8(BH + DB),
228 Prof. A. McAulay on the Wave-Surface and Rotation
it will follow that o! = VB'D'/4S9(B'H! +D’E),
because the expression on the right is an intensity.
It can easily be verified directly that Mr. Heaviside’s wave-
surface strains by means of the function «—? into a Fresnel
surface and similarly for the index-surface.
In writing the above it has occurred to me that it is easy by
the methods of “M.T. H.”’ to construct any number of mediums
which shall, according to equations (5) and (6), rotate the
plane of polarization of an electromagnetic plane polarized
wave. Itshould be carefully noticed that the special electrical
theory of “M.T.H.” is not here involved. We deal with
Maxwell’s theory pure and simple, as exemplified by equations
(5) and (6) above. It is only the mathematical methods of
“M. T. EH.” that are about to be used.
In the mediums in question pw’ and ¢’ vary in space but not
in time. I will first describe such a medium, and then
indicate how it is constructed. |
pe’ and c! vary spirally, according to a certain law to be
mentioned directly, about a certain axis fixed in the medium.
For the sake of conciseness, suppose this axis is vertical.
Describe a circular cylinder of any radius R, having this axis
for its axis. Qn this cylinder describe a spiral making an
angle @ (between O° and +45°) with the horizon given by
the equation
tan 20 = 2h/R,-. «ee ee
where / is a given constant length (positive, say, to fix the
ideas, though it must also be possibly negative). This spiral
will be referred to as the first spiral. A spiral on the same
cylinder cutting the first perpendicularly will be referred to
as the second spiral. Thus, through every point of the
medium we have a first spiral and a second spiral.
The principal axes of permittivity and inductivity are the
tangents to these two spirals and the line at right angles to both
(i. e., the perpendicular from the point on the axis).
The first axis will mean that along the tangent to the first
spiral, and the first permittivity will mean the corresponding
principal permittivity, and similarly for the first inductivity ;
similarly also for the second axis, permittivity, and inductivity.
The third axis is of course the remaining one, and similarly
for permittivity and conductivity.
The third permittivity and third inductivity have constant
values cy and fo throughout the medium. The first permittivity
and inductivity are cy cot? 8 and py cot? 0 respectively, and the
second are Cy tan® 0 and py tan” @ respectively.
In this medium a plane polarized wave with normal along
of Polarization Plane in an Aeolotropic Medium. 229
the axis is possible, and will suffer rotation in the direction in
which the first spiral goes round the axis. The rotation is one
radian for every distance h travelled by the wave.
It will thus be seen that for a point infinitely near the axis
the first and second spirals (and therefore the corresponding
axes) are infinitely nearly inclined at an angle of 45° to the
horizon. As we recede to an infinite distance from the axis,
the inclination of the first axis continuously diminishes to
zero, the second axis being, of course, always at right angles
to the first. Also, infinitely near the axis the permittivity and
inductivity are infinitely nearly isotropic, and as we recede to
infinity the first permittivity and inductivity continuously
increase to infinity, while the second continuously decrease to
zero. The geometrical mean of the first and second permit-
tivities is always the constant third, and similarly for the
inductivities.
We thus see that strictly within the four corners of Max-
well’s theory we find room for the explanation of the rotation
of the plane of polarized light in crystals. We may instruc-
tively picture (however far from the real truth the picture
may be) such a substance as quartz to be made up of a bundle
of parallel ropes (as they may be called), each rope being such
a medium as just described. To make the theory strictly
applicable the average diameter of a rope should be large
compared with the wave-length of light. There seems little
doubt, however, that even if the average diameter were com-
parable or even small compared with the wave-length there
would be a rotation of the plane of polarization.
To construct the above medium, first note that for an
immovable medium equations (3) and (4) are precisely equi-
valent to equations (5) and (6). Suppose the permittivity
and inductivity referred to the standard position (¢ and p)
are constant scalars cy and wy. Then many solutions of
equations (3) and (4) are known. But p’ may be taken as
any given function of p. Hence we have corresponding
solutions—which are fully known—for the actual position.
Remembering the connexions of intensities and fluxes with
the position of matter, we see, among other things, that the
line-integrals of EK’ and H’ referred to the actual position are
the line-integrals of H and H referred to the standard position.
In any particular case this fact enables us to see at once how
H’ and H’ are distributed in the actual space when the solution
for EK and H is known. For instance, the above statements
about the rotation of the plane of polarization in the medium
described are seen at once to follow from the following con-
struction of that medium :—
230 Wave-Surface and Rotation of Polarization Plane.
Let the actual position be obtained from the standard
position by the simplest kind of torsion round the axis
(St.-Venant’s torsion-problem for a circular cylinder). Use
columnar coordinates R, ¢, z for the actual position of matter,
the axis being the axis of torsion. Let i, 7, & (¢ and 7 being
functions of the position of a point) be unit vectors in the
directions of dR, dd, and dz respectively. Thus we put
per pe 7 | oe
so that the torsion is a radian per length h along the axis.
For brevity put
al ial
and note that rkr-1=k, Thus
dp’=rdpr-1+2V.Vdrr-', p!
=r{dp+h—'dzVkp}r-}.
Fiionibediag that dz= —Skdp we see that ( being an arbi-
trary vector),
YO=TYOr),. . . se
where
V¥j,O=o—h Vipsok. . >. 2 anne
Now assuming that
C=O) #P=Py. + - 6 =e
where Co and py are constant scalars, we see by eq. (9) § 9 of
“MDE. 2 that
= OoXX', MW =MoNX’, + « - + (18)
since by physical considerations it is obvious that m=1.
Now
XX O=TNOX Tor 7! q
=r{r—or—h"VkpSkr wr —hkSkpr—ar—h-2V kpSkpr- onl os
=o—h"Vkp'Sko— hkSkp’w—h-*V kp/Skp’a,
or putting Vkp’= Ry,
XX O=o—h R(JSlho+kSjo) —h-?RYySjo. . (19)
This gives XNI=t eS ee
and if tan*@+h-'Rtan@d—1=0,. . . 2 (2)
or tan 20=2) R71). cen ey
Xx'o= cot?d where w=jcosO+ksin@. . (28)
On the Continuity of Isothermal Transformation. 231
It will be found that these results together with equation
(18) give the actual medium described above.
It is possible that a torsion other than the simplest (from a
strain point of view) would give simpler electromagnetic
results. The above torsion is the only one I have examined.
Tnstructive results are obtained by considering the wave-
surface and the ray in the p space and their associated wave-
surface and ray in the actual p! space. If a disturbance
emanate from a point on the axis of the medium, its wave-
surface is a sphere both in the standard and actual positions
of matter, but the ray, while straight in the standard position,
is a diverging spiral in the actual position that circulates
round the axis, one complete revolution taking place while
the ray moves a distance 2zcrh in the direction of the axis.
University of Tasmania, Hobart,
May 380, 1896.
XXIII. On the Continuity of Isothermal Transformation from
the Liquid to the Gaseous State. By THomas Preston,
rs a Oe
ANE EN any substance passes from the liquid to the
gaseous state by isothermal transformation, the
relation between pressure and volume is represented diagram-
matically by a curve such as that shown in fig. 1. In this
Bigs
oO Vv
curve, the part AB refers to the condition of the substance
in which it is altogether liquid, and along this part the volume
* From the Trans. Roy. Dub, Soc. n.s. vol, vi. partiv, Communi-
cated by the Author, .
232 Mr. T. Preston on the Continuity of Isothermal
alters only slightly as the pressure is varied. When the
bee is gradually reduced, however (the temperature
eing maintained constant), a point B is reached at which the
liquid begins to boil, and the whole mass may be transformed
into the gaseous state under constant pressure, if heat be
supplied to keep the temperature constant while the volume
is allowed to increase from B to D. The part BD of the
isothermal is consequently a right line parallel to the axis of
volume, and at D the whole mass is in the condition of
saturated vapour. Beyond D the curve DE is approximately
a rectangular hyperbola as it represents the isothermal of a
gaseous substance which approximately obeys Boyle’s law.
Very shortly after Andrews’ celebrated experiments on the
isothermals of carbon dioxide, and on the continuous trans-
formation of matter from the gaseous to the liquid state,
Professor James Thomson, in an ingenious speculation (sug-
gested by the shape of the isothermals immediately above the
critical temperature), proposed an isothermal curve of the
form represented in fig. 2, which embraces the idea of conti-
Fig. 2.
O Vv
nuity of transformation, so much insisted on by Andrews.
Here, in passing from B to D, the substance is supposed to
be homogeneous throughout, and not to be partly liquid and
partly vapour as in the corresponding part BD of the iso-
thermal of fig. 1. The word homogeneous must here, how-
ever, be taken with some reservation, for although the mass,
asa whole, may be apparently homogeneous—that is, one
cubic centimetre may be on the whole the same as another,—
yet when considered in very small portions the mass may be
intensely heterogeneous. For example, small portions may
Transformation from the Liquid to the Gaseous State. 233
approach the gaseous state more nearly than the liquid, while
others may be more decidedly in the liquid condition *.
Since the time of Andrews and Thomson, various attempts
have been made to deduce from dynamical principles a general
relation connecting the volume, pressure, and temperature of
a substance which will apply to the liquid as well as the
gaseous condition of matter, and which will also hold through-
out the transformation from one state to the other. Of these
the most notable examples are those of Van der Waals and
Clausius, both of whom obtained equations (founded on cer-
tain assumptions) for the isothermal curves which, when
traced, presented the characteristics of the curve suggested by
James Thomson, as shown in fig. 2.
A difficulty which presents itself at once to the acceptance
of such a curve as representing a realizable series of trans-
formations, is that the part MN represents conditions of the
substance in which the volume and the pressure increase
together. As a consequence, this part of the curve has been
generally regarded as unrealizable, and experimental evidence
of it has been nowhere found in nature ; yet, the interesting
phenomena of superheating and supersaturation are so well
represented by the portions BM and DN that the whole
curve has been admitted as a possible, if not a necessary,
generalization.
It is to this unrealizable part of the curve that I now wish
to attract attention, and I shall endeavour to show that there
is a conceivable condition of the substance which satisfies the
extraordinary demands of the portion MN, viz., that the
pressure and volume shall increase together, and that through-
out the transformation the substance shall be in equilibrium,
although necessarily unstable.
For this purpose, let us consider the condition of the sub-
stance at any point of the isothermal between B and D,
What really happens in practice is, that bubbles of vapour are
formed in the interior of the liquid mass, and by reason of
the action of gravity these rise vertically upwards, and the
result is that the mass becomes separated into two portions,
the upper part of the containing vessel being filled with
vapour, and the lower part by the remaining liquid. The
action of gravity is thus to separate the vapour bubbles from
the liquid, and it is on this account, as we shall see, that the
part BD of the isothermal is, in practice, a right line as shown
in fig. 1. If, however, we imagine the action of gravity to
be removed, then a bubble of vapour when formed would
* This view has been put forward more than once in the Author's
‘Theory of Heat,’ e. g., p. 896,
234 Mr. T. Preston on the Continuity of Isothermal
remain 2n situ, except in so far as it might drift with currents
in the mass. The formation of bubbles, under these con-
ditions, would cause the mass to swell into a spongy condition
—a heterogeneous mixture of liquid and vapour,—in which,
if the equilibrium could be maintained, the volume and pres-
sure would vary according to laws very different from the
simple law of constant pressure which governs the transforma-
tion of ordinary boiling under the action of gravity (fig. 1).
In order to determine, under these conditions, how the
pressure varies with the volume, at constant temperature, let
us consider the case of a mass of liquid in which a spherical
bubble of the vapour of the liquid has been formed, as shown
in fig. 8. For the sake of clearness, let the mass be enclosed
Fig. 3.—Bubble surrounded by liquid.
aa
Y
in a cylinder by means of a piston, so that the volume and
external pressure can be varied at pleasure, then, if p be the
pressure, applied through the piston (which we may term the
external pressure of the mass, in the ordinary sense), the
pressure at any point in the interior of the liquid will be
p+e, where c is a quantity depending on the surface film, and,
as it arises from the mutual attraction of molecules well
within each other’s sphere of action, may be very large. But,
if a be the vapour-pressure within the bubble, the relation
connecting p and @ is
le amemet e e e e ~ ° (1)
where 7 is the radius of the bubble, and T the surface-tension
Transformation from the Liquid to the Gaseous State. 235
of the surface film separating the liquid and vapour. It is
clear, therefore, that if @ remains sensibly constant, p must
increase as 7 increases, or in other words, the external pressure
and the volume must increase simultaneously, if equilibrium
is to be maintained.
The saturated vapour-pressure @, however, is not quite
constant, but varies at constant temperature with the curva-
ture of the film with which it is in contact, and if a) be
taken to represent the normal saturated vapour-pressure, that
is the pressure of a saturated vapour in contact with a plane
surface of its own liquid, then the saturated vapour-pressure
in contact with a concave spherical surface, of radius 7, is
easily shown to be
where p is the density of the liquid, and p, the density of the
saturated vapour. Hence the relation (1) connecting p and
7 becomes j
9 p1
2) 5 =. sa
By=pt pe bea ps (2)
In this equation all the quantities other than p and r may
be taken as remaining constant during an isothermal trans-
formation, and consequently, within certain limits, the volume
and external pressure of the mass should increase together.
This equation, however, cannot be expected to hold in the
extreme case in which the bubble is so small that the mass
within it ceases to possess distinctly the properties of a
vapour, or in the other extreme case, in which the bubbles
become so large and numerous that the remaining liquid, by
reason of being drawn out into thin films, or otherwise, ceases
to behave as a liquid in regard to the transmission of hydro-
static pressure &c. Within certain limits, however, equation
(2) gives the relation between the external pressure and the
volume of the mass.
Thus, in the case of a single bubble, if the whole mass be
taken as unity, and the mass of vapour within the bubble he
m, then the mass of the liquid portion will be 1—m, and the
whole volume will be
i, Le
pa P1
But, if the radius of the bubble be 7, we have
Vv
— Tp ° e e e e ° 2 (4)
236 Mr. T. Preston on the Continuity of Isothermal
consequently, equation (3) becomes
4 : ( i gag | ) i!
=—T i ===} + — e ° ° 9)
eee 2 Pi ©)
or, denoting the specific volumes of the liquid and vapour by
v, and va, we have, frem equation (5),
at: ae vy
CU Th SS} . a)
Now equation (2) gives
ae Vo
r= : . 2 2 ae
By—P y—Vy
Therefore (6) becomes
(ea \a—p)'= Fem). 8)
Vg—U
Consequently, since the right-hand member of this equation
remains constant, the equation of the isothermal curve assumes
the hyperbolic form
(v—v;) (@o—p)?=constant. . . . . )
This equation holds for a spherical bubble of vapour sur-
rounded by its own liquid, and in this case it is to be noted
that p must always be less than a, or the external pressure
of the mass must be less than the normal saturated vapour-
pressure, and this is what is indicated by the portion MC of
the isothermal lying below the right line BD in fig. 2.
So far we have considered the case of a single bubble,
surrounded by its own liquid, but the foregoing reasoning
will apply when a number of equal bubbles are formed. If
the bubbles are of different sizes, however, the capillary
pressures arising from the curvatures of their surface filnis
will be different, and equilibrium will be impossible—the
larger bubbles tending to expand, and the smaller to collapse.
lt would appear, therefore, that the mass might be gradually
transformed from the liquid to the gaseous condition, by
allowing a system of equal bubbles to gradually increase in
size while the volume increased to v, and the external pressure
to @, and this value would be reached if the bubbles could
be supposed to increase gradually till the whole mass reached
the state of vapour. Long before this final condition could be
reached, however, the liquid portions of the mass, which
interlace the bubbles and fill the spaces between them, would
be drawn out into thin films, and the conditions would be such
that the foregoing reasoning could not be applied.. The
action of the distended surface film, in fact, will be such as to
draw the liquid parts which fill the spaces between the bubbles
Transformation from the Liquid to the Gaseous State. 237
into spherical drops, so that a stage is ultimately reached in
which the mass consists of a system of spherical drops sur-
rounded by their own vapour (fig. 4).
The state of affairs is now reversed, for instead of having
vapour in contact with a concave liquid surface, and therefore
Fig. 4.—Liquid drops surrounded by vapour.
O
O
OO
OO
O
OO
O
O
Oo
exe)
09900
290
H(oyl@) (DVO)
O
O
90.0000
OO
oO
@.er@
C0000
O9 CO
OOO
O
O
O0000
ODO)
O
O50
O 000
O
{e
{fe
10
10
10
iO
O
1 O
O
O
O
ie
O
O
at a pressure less than @,, the normal saturated vapour-pressure,
we have saturated vapour in contact with convex liquid sur-
faces, and therefore at a pressure aw, greater than wp.
Hence, in this limit, we may take the pressure on the
enclosing piston to be that of the saturated vapour, namely a,
the mass will be subject to an external pressure greater than
@), namely p=a, and this brings us into the region CN
(fig. 2) of the isothermal which lies above the normal pressure
line BD. In this it is assumed that the mass is largely in
the condition of saturated vapour, and that the liquid exists as
a system of equal spherical droplets, swimming in their own
vapour.
If the drops were of different radii equilibrium would be
impossible, as evaporation would take place at the surfaces of
the smaller drops, and condensation at the surfaces of the
larger. This instability is made evident by the equation
which shows how the vapour-pressure increases as the radii
of the liquid drops diminish, and when the drops are small, #
may exceed wy by a considerable quantity.
There is a limit, however, beyond which, if the radii of
238 Mr. T. Preston on the Continuity of Isothermal
the drops be diminished, the foregoing equation will cease to
apply, and the pressure a, after reaching a maximum, will
gradually diminish, and finally recede to the value a, when
the drops of liquid vanish. This is the process which takes
place along the falling part ND (fig. 2) of the isothermal.
Similarly, in the initial phases of the transformation here
imagined, namely, when small bubbles are beginning to be
formed within the mass, it is clear that equation (2) ceases to
apply when the bubbles are so small that they cease to possess
the distinctive properties of vapour, and it consequently
follows, that although @a may be very much less than a, at
some part of the branch BMC, yet a condition is attained
with bubbles of a certain diameter in which a is a minimum,
and from which it increases in both directions to the normal
vapour-pressure @p.
Thus, the part BM (fig. 2) of the isothermal is accounted
for, and therefore the whole succession of conditions repre-
sented by an isothermal, such as that imagined by James
Thomson, is rendered conceivable. Such a succession, of
course, cannot be regarded as realizable, for although the con-
dition represented by every point of the curve is shown to be
possible, and one of equilibrium, when the bubbles (or drops)
are all of the same size, yet the equilibrium is essentially
unstable, for when there is any departure from uniformity,
all differences tend to become exaggerated, and the mass may
depart from the condition of equilibrium with explosive
violence.
It is interesting to note that the mass may be transformed
from the condition B to the condition D by two distinct
routes of transformation—one along the right line BD, in
which the condition is stable, and the other along the curved
path BMOND, in which the condition is unstable,—yet the
principle of conservation of energy forces us to conclude that
the work done against external pressure, while the mass
expands from B to D, must be the same in the two cases, and
for this reason it has been concluded that whatever the shape
of the curve AMND may be, the area of the loop BMC
must be equal to the area of the loop CND. At first sight
we might apply the same reasoning to the transformation
from B to C, or from D to C, and rush to the conclusion that
the area of each loop must be zero, or else that we are here
presented with a violation of the principle of conservation of
energy.
But it must be remarked that although at the point C of
the diagram the mass, in both cases, has the same tempera-
ture, pressure, and volume, yet in one case all the vapour is
Transformation from the Liquid to the Gaseous State. 239
collected into one portion of the chamber, and all the liquid
into the other, whereas iu the other case the vapour and
liquid are not distinctly separated from each other, but inter-
mixed in some way so as to occupy the whole space as an
apparently homogeneous mass. Hence the point C represents
two distinct conditions of the mass in which the pressure,
volume, and temperature are the same, but in which the
internal energies may differ very considerably. Thus, although
less external work is done in passing from B to C along the
curve BMC than in passing along the right line BOC, yet in
virtue of the arrangement-of the mass, the internal energy at
C in the former case may be considerably greater than in
the latter.
This, indeed, must be the case if the arrangement of the
mass be of the bubble and drop nature here suggested. For
if a-given mass, existing partly as liquid and partly as
vapour, be arranged in such a way that the liquid is collected
together in one part of the containing vessel, while the
vapour is all collected in the remainder (as ordinarily occurs),
and if we desire to change it from this arrangement into one
like that described above, in which the vapour is disseminated
through the liquid in bubbles, or in which the whole vessel is
filled with vapour and drops, a certain amount of work must
be done in order to effect the transformation—namely, the
equivalent of the surface energy possessed by the enormously
increased surface area of the bubbles and drops in the new
condition. Thus, although less external work is done in
passing along the isothermal BMC than along the rectilinear
path BC, yet the mass in the former case possesses more
surface energy than in the latter, and the excess of external
work in the latter transformation is represented in the former
by an excess of internal work spent in generating the excess
of surface film.
Similar remarks apply to the portion CND, for in passing
along this curve the external work done is greater than that
performed in passing along CD, but this is compensated by
the destruction of the surface film. Thus, along BMC there
is on the whole a creation of surface film with less external
work, and along CND there is destruction of surface film
accompanied by increased external work—the excess in the
former being equal to the defect in the latter.
In conclusion, it may be remarked that the views here put
forward seem to have an important bearing on many interest-
ing questions connected with the boiling-points of liquids, and
the manner in which they are affected by the presence of
dissolved salts. It is sufficient to merely point out, at present,
240 Prof. J. G. MacGregor on Abstract Dynamies and
that obviously any operation which increases the surface-ten=
sion of the film separating a figuid from its own vapour will
also raise the boiling-point, for when T is increased, a greater
vapour-pressure @ within a buljble will be required in order
to enable it to expand against a given external pressure.
This prediction of the theory appears to be in accordance with
the observed facts.
Aes ee
—__ ee
~~
XXIV. Whe Hypotheses of Abstract Dynamics and the ques-
tion of the number of the Elastic Constants. By Prof. J. G.
MacGregor, D.Sc., Dalhousie College, Halifax, N.S,*
ie a fol'mer paper + an attempt was made to formulate the
hypotheses employed in Abstract Dynamics, when bodies
are considered as consisting of particles exerting forces upon
one another at a distance. As these may be expressed in
varlous Ways, Newton’s Second Law of Motion, owing to its
very genelal employment, was selected as one of them, and it
was sought to determine what others are required in order to
establish both the equations of motion and the law of the
conservation of energy. It was shown that the following
indepenglent assumptions are both necessary and sufficient for
this jyarpose, viz., (a) the Third Law in its wide sense, 2. e., as
ass@rting that action and reaction are not only equal and
Opposite but also in the same straight line, and () the law
“of the conservation of natural forces, 2. e., that natural forces
are such that the work done during any change of the con-
figuration of a system depends only upon the initial and final
configurations, or that 2(Xd«+Ydy+Zdz) is a complete
differential. Both (a) and (d) are assertions about natural
forces, (a) referring to magnitude, direction, and action-line,
and (b) to magnitude. They may be combined in one by
noting that when (a) holds, 2(Xdx+Ydy+Zdz) becomes
=Sds, where 8 is the stress between any pair of particles
and s their distance from one another, and that the condition -
that }Sds shall be a complete differential is that each S shall
be a function of all the s’s, or in more precise terms, that the
stresses between the various pairs of particles shall be propor-
tional to the rates of change, with respect to the corresponding
distances respectively, of a function of the distances of all the
pairs of particles of the system.
Thus the requisite hypotheses reduce to two, viz., (1) the
* An abstract (with some additions) of a paper read before the Royal
Society of Canada. Communicated by the Author.
+ Trans. Roy. Soc. Canada, vol. x. sec. iii. (1892) p. 3.
the question of the number of the Elastic Constants. 241
Law of Force—Newton’s Second Law, and (2) the Law of
Stress, just enunciated. To these, however, must be added
a third, viz., (3) the Law of the constitution of bodies, that
bodies may be regarded as consisting of particles acting on
one another at a distance.
In the paper* of which this is an abstract, a similar attempt
is made to formulate the hypotheses employed when, as in
the study of fluids and elastic solids, bodies are considered as.
consisting of elements which exert forces on contiguous
elements only, across surfaces of contact. In obtaining the
equations of motion, in this case, the Third Law is applied,
when, the traction at #, y, zon one end of a parallelopiped
with dz, dy, dz as edges being called P, the traction on the
other end is put equal to — (P =° oe dx ). The Second Law
is partially applied when the quotient of the force on an
element by its acceleration is put equal to pdx dy dz, p being
the density. It is only partially applied, however ; for as
p varies with the time, there is nothing in the resulting
equation to show that the quotient of force by acceleration is
constant, as the Second Law states. Accordingly the equa-
tions of motion thus obtained are insufficient completely to
determine the motion. An additional equation is necessary,
viz., one which completes the application of the Second Law
by expressing in some form or other, that the pdadydz of
the equations of motion is constant. This is the so-called
equation of continuity, which is thus only a partial application
of the Second Law. It was regarded by Rankine as re-
quiring an independent axiom {, and is derived by other
writers by asserting, in a vague kind of way, the impossibility
of the annihilation and the creation of matter, the constancy
of mass, or the continuity of the motion considered.
In order to obtain the law of the conservation of energy,
it is necessary to assume, in addition, that the work done f
by the stress components during a strain, viz., the integral,
between the initial and final states of strain, of
§\\ (Bde+ Qdf+ Rdg + Sda+ Tdb + Ude) dex dy de,
* Trans. Roy. Soc. Canada [2], vol. i. sec. iii. p. 85 (1895).
t+ ‘ Applied Mechanics,’ 9th ed. p. 411.
t I use the term—work done by a force—in its ordinary sense, as
being the product of the force into the component, in the direction of
the force, of the displacement (relative, of course, to a dynamical refer-
ence system) of its point or place of application. The definition of this
term which Newcomb (Phil. Mag. [5] xxvii. (1889) p. 115) proposed to
substitute for the ordinary one would not be suited to the contact-action
conception, See note in the Proc. and Trans. of the Nova Scotian
Institute of Science, vol. viii. (1890-94) p. 460.
Phil. Mag. 8. 5. Vol. 42. No. 256. Sept. 1896. fh
242 Prof. J. G. MacGregor on Abstract Dynamics and
results in the production of an equivalent amount of potential
energy. This is equivalent to the assumption that the above
expression is a complete differential, which is again equiva-
lent to the assumption that the stress components, P, Q, R,
S, T, U, at a point, are proportional to the rates of change,
with respect to the corresponding strain components, e, /, g,
a, b, ¢ respectively, of a function of all these strain com-
ponents.
The Third Law and the hypothesis just enunciated are both
statements partially specifying natural stresses. We may
combine them in one by assuming that natural forces may be
regarded as stresses between contiguous elements of a body
(or medium), the components of the stress at a point having
the relations as to magnitude just specified.
Thus in cases of contact-action also, the purely dynamical
hypotheses reduce to two,—(1) The Law of Force—
Newton’s Second Law, and (2) the Law of Stress, as just
enunciated. In such cases also there is, however, a third
hypothesis, viz., (3) the Law of the constitution- of bodies,
that bodies may be regarded as consisting of elements exerting
forces upon contiguous elements only, across their surfaces
of contact.
The above results have a bearing on the controversy with
regard to the rari-constant and the multi-constant theories of
elasticity. For in order to form an estimate of the relative
probability of deductions from the two theories, accuracy in
deduction being assumed, we must compare the hypotheses
employed.
The multi-constant theorists, in applying the contact-action
conception of bodies, have usually employed as dynamical
hypotheses the Second and Third Laws of Motion and the
Law of the conservation of energy, which together are equi-
valent in hypothetical content to the above Laws of Force and
of Stress, or to the Laws of the conservation and the trans-
ference of energy.
The rari-constant theorists have used the molecular, or
rather the point-atom, conception of bodies, and have em-
ployed as dynamical hypotheses the Second Law and the
assumption that the stress between any pair of particles is a
function of their distance, not of the distances of all the pairs
of particles of the system. Their dynamical hypotheses have
thus a greater hypothetical content than the Laws of Force
and Stress, and therefore also than the Laws of Energy.
It would appear, however, that the discrepancy between
the results deduced from the two theories with regard to the
number of the elastic constants is not due to the additional
the question of the number of the Elastic Constants. 243
assumption which the rari-constant theorists have employed
over and above those equivalent to the laws of energy. If
we take Mr. Love’s account of Cauchy’s deduction of the
stress-strain relations * as being fairly representative of de-
ductions of the kind (I have not access to the literature of
the subject), this seems obvious. For if, in this deduction,
the stresses between particles be regarded as functions of the
distances of all the pairs of particles of the system, not of the
distances between the attracting particles themselves only,
while the expressions for the elastic constants will be changed,
they will still reduce to fifteen. Hence, so far as the number
of independent constants is concerned, the rari-constant
theorists may be said to have employed dynamical assump-
tions equivalent to the laws of energy.
This being so, the apparent discrepancy between the
results of the two theories must be due to the difference in
the assumptions made as to the constitution of bodies. Now
the distance-action conception of the constitution of bodies
involves a larger assumption than the contact-action concep-
tion. This is obvious from the fact that if we assume the
molecular hypothesis, or rather the point-atom hypothesis, it
can then be proved that bodies may be regarded as consisting
of elements exerting forces on contiguous elements only,
across surfaces of contact, while the molecular hypothesis
cannot be thus deduced from that of contact-action. Thus
the point-atom hypothesis may be regarded as consisting of
two parts, (a) that bodies may be regarded as consisting of
elements exhibiting contact-action; and (b) that this is due
to their consisting of point-atoms acting on one another at a
distance. Moreover, in the deduction of the rari-constant
result, the second part of the hypothesis has been employed.
For it is obvious from Mr. Love’s sketch of Cauchy’s reason-
ing, that the possibility of reducing the constants to fifteen
is due to the simplicity, one is tempted to say artificiality, of
the point-atom conception. It follows at once that unverified
deductions from the molecular hypothesis must have a lower
degree of probability than similar deductions from the rival
hypothesis.
While the multi-constant result is thus the more probable
of the two, it cannot be said to be certain. Mr. Love repre-
sents the opponents of the molecular theory as urging
against it, “that the known laws of energy lead to results
which are certainly true whether the molecular hypothesis be
correct or no” t+. Hven, however, if we regard the laws of
* «Treatise on the Mathematical Theory of Elasticity,’ p. 110}
1. Loe: cit, p. 16. fo
244 The Hypotheses of Abstract Dynamics.
energy themselves as certainly true, the results of their
application to the study of elasticity cannot have the same
certainty, because of the additional hypothesis (3) which, as
seen above, is made in applying them.
As the point-atom hypothesis may be expressed in the two
parts given above, it follows, if the conclusion reached above
is correct, viz., that the dynamical hypotheses practically
employed in the two theories are of the same hypothetical
content, that the rari-constant theorist must accept the multi-
constant result. He must hold that with the assumption (a)
only the number of independent constants cannot be reduced
to less than 21, but that with the additional assumption (6)
they are reducible to 15. It is thus obvious that there is no
real discrepancy between the results of the two theories.
According to the one, all bodies which may be regarded as
exhibiting contact-action will be capable of having their
elastic qualities completely characterized by 21 constants, and
unless we have further data with regard to these bodies,
21 constants will be requisite for this purpose. According
to the other, in the case of bodies which may be regarded as
exhibiting contact-action because of their consisting of point-
atoms, the number of the constants may be reduced to 15.
It would seem to be a simple matter, not indeed to devise
and execute conclusive experiments to settle the question of
the existence of relations among the 21 constants, but to
interpret such experiments when made. For if the elastic
constants were found to be reducible to 15, the multi-constant
theory would obviously be proved to be inadequate. If other
relations were found to hold than those deduced from the
point-atom hypothesis, the point-atom conception would be
shown to be erroneous and the contact-action conception to
be inadequate. If it were found that there were no relations
among the 21 constants, the point-atom hypothesis would be
disproved and the contact-action conception would be shown,
so tar as the number of the elastic constants is concerned, to
be adequate. Simple logical considerations of this kind are
sometimes overlooked by writers in the enthusiasm produced
by successful application of the contact-action conception.
Thus Mr. Love says * :—“ Even if the experimental evidence
were all fairly interpretable in favour of the other side, if
there were a general consensus that Cauchy’s relations hold
good, and that Poisson’s ratio is 4, for materials carefully
examined, that would not amount to a proof of the molecular
hypothesis. It would still be open to us to reject that hypo-
thesis as not axiomatic, and in the present state of science we
* Loe. cit. p. 19.
The Electric Discharge in a Magnetic Field. 245
must so reject it. . . . Unless the hypothesis were axiomatic,
there could be no reason to adopt it to-day. Modern Physics
is perfectly capable of deducing a theory of elasticity from
the known laws of energy, without the aid of a subsidiary
hypothesis about intermolecular force, and being in that
position, it is bound to discard the hypothesis. Such a device
is merely a phase in the development of scientific thought,
and, having served its turn as a means of introducing gener-
ality into the subject, it must give place again to a still more
general method.” It is of course quite obvious that the
experimental verification of Cauchy’s relations would not
prove the molecular hypothesis; but it would show the
contact-action conception to be inadequate. It would still
be open to any one with a preconceived idea as to what is
axiomatic to discard the former, but in doing so he would
find himself unable to account for known facts which had
been predicted by the aid of the discarded hypothesis. That
modern physics is capable of deducing a theory of elasticity
from the known laws of energy without the aid of a molecular
hypothesis is surely a mere assertion, if as doubtless is the
case, a satisfactory theory is meant. The experimental deter-
mination of any general relations among the 21 constants
would prove the theory in its present state to be inadequate
and unsatisfactory, and the verification of Cauchy’s relations
would show that, while the molecular hypothesis must of
course be regarded as merely a phase in the development of
scientific thought, it has not yet quite completely served its
turn as an instrument of generalisation.
XXV. The Electric Discharge in a Magnetic F reld.
By Sir Davip Savomons*.
HE study of the electric discharge im vacuo does not
appear to have been followed with the same energy as
has been applied to other branches of electrical science,
until quite recent times. I would venture to divide the
historic period of the subject into four, as follows :—
(1) The time when Messrs. Warren De La Rue, Gassiot,
Spottiswoode, and Moulton were working upon the subject.
(2) When Professor Crookes made his discoveries.
(3) The still more recent investigations of Professor J. J.
Thomson.
(4) The application to photography by employing the so-
called x-rays which are abundantly produced by special forms
of tubes.
* Communicated by the Author,
246 Sir David Salomons on the Electric
I do not intend to deal with the last three periods ; but in
regard to the first, although many interesting facts were
hrought to light, the subject does not appear to have been
systematized, nor am I able to find a great deal published on
the subject before the period of Professor Crookes, perhaps
for the reason that no one has taken the pains to collect all
the information and publish it together. .
Many important discoveries were no duvubt made and the
facts published in some obscure quarter, where they probably
remain to this day. The general interest which now exists
in science and the large technical Press were non-existent
but a very few years ago. |
I now propose to describe briefly a number of points of
interest in connexion with vacuum-tubes and some classes of
work which may be performed by their use, the outcome of
work extending over twenty-five years.
It appeared to me that two points required settling in
regard to these tubes, which, however, are not completely
exhausted as the name would suggest; viz. :—
(1) “That the number of bands produced in a given tube
should be brought under control, that is to say, that the
conditions under which they are formed should be solved.”
(2) “That the reason for their existence should also be
found out.”
In regard to the first point, I believe that this question is
solved and the results given in a paper by me published in
the ‘ Proceedings of the Royal Society,’ in volume lvi.
The origin of the bands has been surmised by many of the
early workers, in fact they have asswmed that they are pro-
duced in consequence of self-inductive effects; but I have
not been able to discover any proof that such is the case. I
shall be able to show by experiments to be described that this
view is correct ; and from the methods adopted, which are
probably the only means available for the purpose, give the
clue why the first investigators did not advance beyond a
state of conjecture. |
The tubes I employed contained rarefied air and various
other gases; but I found for the particular class of experi-
ments that rarefied air sufficed in all cases, and that it was
not necessary to proceed to very high exhaustion.
The great efforts made by earlier investigators in trying to
obtain enormous electromotive force appeared to me wrong
for the class of experiment they were entering upon. I
therefore tried the opposite course—one which has proved so
successful in mathematics, viz. the reduction of the leadin
factor to the smallest quantity possible,—and observed what .
Discharge in a Magnetic Freld. 247
took place. I soon found that a vast number of the pheno-
mena described as fundamental were superposed phenomena ;
and that when exceedingly small E.M.F. was employed the
various phenomena were seen in their purity, and on raising
the E.M.F. the various changes could be seen, up to the
complex appearance described in some of the early papers.
The E.M.F. employed in the following experiments was
sometimes as low as 700 volts and rarely over 2000.
It has been known for a long time that the discharge
through the vacuum-tube is affected by the magnetic field,
but beyond this circumstance I cannot find anything that has
been published upon the subject classifying results.
Having worked to that point when I could control what
took place within the tube, it became a comparatively easy
matter to investigate the discharge in the magnetic field
provided a sufficiently powerful magnet could be obtained.
All the electromagnets in existence were built on the
Faraday-magnet type, and this pattern was found incon-
venient. I therefore had a special magnet constructed
weighing 13 cwt., with a field probably far more powerful
than any which had been made before, with the polepieces
capable of being approached and distanced by means of
screws, and a variety of other details introduced which made
the apparatus a piece of engineering work as well as one
suitable for scientific investigation. This magnet is shown
in, fie... 1,
Fail not repeat what I have already published on the
subject beyond referring to two points :—
Firstly, that the glass of the tube has a considerable influ-
ence in creating the bands; and 7
Secondly, that the bands as generally seen are spurious,
and can only be viewed in their purity at the moment when
the current is so reduced that they disappear from view.
There may be other stages beyond this point which the eye
cannot see, and that such is probable is shown from the fact
that when the tube is made of fluorescent glass the current may
be still further reduced and the bands be visible.
For producing the current I employed Apps’s coils of
various sizes, the contact-breakers being special mechanical
forms devised by myself, and worked by an electro-motor.
I also employed the alternate current from an alternator,
raised to a suitable pressure by means of a transformer.
The induction-coils had primary coils so wound that a 100-
volt current could be put on direct, without the insertion of
a resistance. The number of turns of wire in the primaries
is so great that the self-induction reduces the length of the
248 Sir David Salomons on the Electric
spark from the secondary, because no large amount of current
can traverse the primary coil.
But there is a point in favour of using an induction-coil such
as described which does not appear to have been taken ad-
vantage of. The spark at the vibrator is far more pronounced
than if a lower electromotive force were employed for the
primary current. Indeed, so much energy can be developed
at the vibrator on the “ break ” that it is possible to obtain
the equivalent to a direct intermittent current in the tube, or
at any rate, so far as the eye is concerned this only exists, the
effects due to the current in the opposite direction not being
visible. Consequently two types of current can be produce:
by means of the induction-coil.
Discharge in a Magnetic Field. 249
The phenomena were the same whether the alternate
current from an induction-coil, or the alternate current pro-
duced by the alternator, was used, so far as the visible effects
were concerned.
When a tube, as shown in fig. 2, 300 millims. long and 25
Fig. 2.
(
} - —-— = ee ew ee ee ew eee em em we eew ec ec encore seo Me DoSSprlOTTC OTS COT SST BAT @aTeteUeessesere——=> --l
1 2 i
1 !
ae y
. nN
al
millims. in diameter, containing brush electrodes at each end,
is placed between the poles of the unexcited large magnet (see
fig. 8) and has a very small alternate current traversing it,
Fig. 3.
the bands are produced in the usual way. Now excite the
magnet, the switches having been so adjusted that one pole is
N. and one 8.,a great change is now observed within the
tube, If the E.M.F. of the current is very small, the bands
250 Sir David Salomons on the Electric
will disappear altogether and no current will pass. This stage
will be referred to later. In such an event raise the E.M.F.
until the bands are plainly seen. It will then be noticed that
the bands which at first extended across the tube have now
divided into two columns, and are very much smaller
than before, their position being equatorial. It will further
be noticed that the convex sides are all one way in one
column of bands and in the opposite direction in the other
column of bands, indicating that the two currents have been
completely separated and travel through the tube at opposite
sides, meeting only at the electrodes. Moreover, the bands
are smaller where the field is strongest. Also they will here
be placed closer together and be more numerous. In fact,
the condition of things can be made such, that where the field
is most powerful the bands disappear and a line of light only
is visible. But a low-power microscope will resolve this
line of light into bands, that is to say, they are so close
together that the eye alone does not appreciate their
existence.
Probably the best way to make the experiment is to place
the tube between the poles of the magnet when excited.
Then cut off the exciting current, and watch the tube during
the time that the magnetism is falling. The various stages
can be seen better this way, as they take place slowly. Fig. 4
Fig. 4.
ee a me oe enn + oo +s woe ++ 2 ee se es oe eee eee
mot nn = = a rw + one + - oe rns eens ee eee + 22 oe oe ee ee + eer”
shows the tube with the divided current appearing as two
lines of light. Fig. 5 illustrates one line of bands when the
Fig. 5.
GGGe teu ee
magnetism is less strong. Fig. 6 shows two lines of bands
Fig. 6.
in the tube, when looked at from another point of view.
Fig. 7 shows the bands fillmg the tube, the magnetism
Discharge in a Magnetic Field. 251
having fallen still further. Fig. 8 the next stage. Fig. 9
Fig. 8.
the appearance of the tube when the magnetism ceases to
have influence.
Before proceeding to describe other experiments it is well
to analyse what takes place. My own view is that we have
here shown in a very pretty way Ampéere’s well-known
experiment of the influence of a current flowing in one con-
ductor upon a current flowing in an adjacent conductor.
This experiment is usually shown by wires, one arranged as a
moving conductor. In the instance of the vacuum-tube the
current is passing through a conductor perfectly free to move
or, perhaps, it would be better to say that the current is fr ee
to place itself, without any appreciable opposing resistance,
in the position it tends to take up under the influence of the
magnetic field, which latter may be regarded as a current
travelling in a circular conductor, which may be resolved into
straight line currents parallel to the tube currents.
The experiment described, therefore, shows that the two
currents try, not to move out of the field, as it might be ex-
pressed, but tend to take up definite positions, which naturally
must be in opposite directions, when influenced by certain
powerful currents in their proximity, z.e., the magnetic field.
lt might appear that the resistance of the gas in the centre
of the field, where no bands appear, has been increased,
and therefore the current chooses a path of least resistance,
viz., at the sides of the tube, farthest from the strongest
portion of the field. This may, or may not, be an accessory
in the case. I venture to believe, for the reason that the
alternate current has been separated into its two constituents
on opposite sides, that the true explanation is that of Am-
pere’s theory. When the resistance of the tube is actually
measured it is found to be higher when the magnet is excited,
and this, of course, would be expected, since e the current is
then travel ling through a much smaller sectional area.
Consequently the test of resistance has no value to prove that
the gas contained within the tube has a higher resistance,
because under no circumstances can the resistance of the gas
in the Strongest part of the field be measured, as the curr ent
252 Sir David Salomons on the Electric
refuses to travel that road, though possibly some indirect
method might be devised.
Another conclusion may also be deduced from this experi-
ment. It is that since the bands become more numerous in the
strongest part of the field, they are produced by self-inductive
effects. To put this in unscientific language, the discharge is
comparable to that of a lightning-discharge through a good
conductor, the peculiarities of which were first pointed out by
Professor Hughes; that is to say, the discharge of the current
through the gas is checked, and then proceeds again, checked
again, and so on; and it is fairly evident that at the points of
these various checks heat is developed, and the residual matter
in the tube raised to a high temperature, producing what are
termed bright bands. That the bright bands consist of heated
matter is most probable for the reason that they may be seen
from all points of view.
If instead of employing the alternate current the induction-
coil is so adjusted as to produce an intermittent direct one,
only one line of bands appears, equatorially placed, on one
side or the other, according as the current is in one direction
or the opposite one. The analysis of the column of bands is
the same as that of one of the columns when the alternate
current is employed. Reversing the current through the tube
has no effect upon the appearance of the tube, since the
currents change sides. But if the poles of the magnet are
reversed, then the convexity of the column of bands becomes
reversed ; and this, of course, would be expected.
It has already been stated that when the current flowing
through the tube is exceedingly small, the tube remains dark
and no current would appear to pass. It seems to me that
the probable explanation for this is that the current is, so to
speak, driven into the glass, which has se high a resistance
that no appreciable amount passes. In fact, the whole of the
experiments show that the larger the amount of the current
which flows through the tube, the more nearly do the bands
approach to the centre, or, in other words, the smaller is the
space at the centre of the tube which appears dark.
If the tube, with the current flowing through it, is placed
in an unsymmetrical field, then the lines of bands become
distorted, more or less spiral in form. This is because the
Fig. 10.
VA ee ae eel lee Vad)
oR) lo add hdc dat ha aa
field being unsymmetrical the currents tend to take up dif-
ferent positions in various portions of the tube (see fig. 10).
Discharge in a Magnetic Field. 253
If both poles are made N., or S., and the tube placed
between them, the bands in the tube will cross at the centre
and appear as in fig. 11, the dotted lines representing bands
to avoid confusion.
Fig. 11.
Let us now consider what takes place when the same tube
is placed between the poles N. and S. with an increased
current. The bands now stretch nearly across the tube. In
the strongest portion of the field there stand out very bright
and well defined the magnetic lines of force filling the tube
in a three-dimensional form. The effect is very striking and
ee Besides, the whole tube is filled with a faint
ight.
Hig, 12 is a general diagram showing what takes place ;
and fig. 13 illustrates the tube in greater detail. The pheno-
menon is beautifully shown in the globular tube, fig. 14.
Fig. 12.
Fie. 13.
Of course, what appear to be the lines of force are not really
the lines made visible but the effects due to the form of the
field; since if an effect is due to the field at any given point and
varies with the strength of the field, the result will take upon
itself the form of the lines of force. It is known that the re-
sistance to the passage of the current is less with the lines of
force than across them, which effect may have some influence
in producing the phenomenon. The bright lines are nothing
more than bright bands very closely packed; and the form is
due to self-induction.
254 Sir David Salomons on the Electrie
In fact, all points on similar lines of force in a magneti¢
field lying between the N. and S. poles make up a figure
somewhat like the shell! of an egg ; and the appearance of
what I term the visible field appears to be like a vast num-
Fig. 14,
ber of transparent coloured egg-shells placed symmetrically
one within the other, although, of course, each shell is not
symmetrical with any other, the inner ones being more elon-
gated and the outer ones more spherical, the centre one of
all being a straight line. |
By holding the tube which has been experimented with in
various positions in regard to the poles of the magnet, whether
they be N. and 8., or both N. or bothS8., a variety of appear-
ances are seen in the tube according to the position in which
it is held. All the effects can be traced to what I term the
Ampérian explanation. i
From the experiments which have been described it is
evident that a very important opening suggests itself as to
the practical use of vacuum-tubes, viz. that of exploring the
magnetic field. [have used such tubes for this purpose for
many years past, and have been able to plot out in a few
minutes mentally that which takesa long time by the methods
generally employed. J am ready to admit that the usual
process is more accurate when absolute determinations are
required ; but usually all that is wished for is to ascertain the
general character of a field and the extent of the leakage, and
this can be done at once by employing vacuum-tubes.
Discharge in a Magnetic Feld. ia
The question naturally arises whether in the case of solid
conductors the current becomes displaced when such con-
ductors are placed in the magnetic field. It is well known
that metals, bismuth in particular, offer considerably more
resistance to the passage of the current in the magnetic field,
everything else being equal. It occurred to me that this
increased resistance is possibly spurious, and due simply to
the displacement of the current, the displacement being easier
to effect in some metals than in others. I made a large
number of experiments in regard to this question, and found
that this displacement does take place, although in a very
small degree when compared with that which results in the
case of a conductor consisting of rarefied gas. A few years
ago the displacement was noticed by Mr. Hall, and is gene-
rally known under the name of the “ Hall effect ;”” but the
phenomenon had been observed by myself many years pre-
viously, although I did not publish it, believing it to be a
known fact.
I made a large number of experiments with galvanometers
built on the D’Arsonval-Deprez type, and obtained very vary-
ing results by modifying the magnetic field. By increasing
the field a maximum sensibility was reached, which decreased
on further increasing the field.
The various experiments described no doubt indicate the
cause of this, viz. that the field being made too powerful, less
current passes through the coil, and the sensibility begins to
fall. I had a special galvanometer-apparatus made to fit my
large magnet, converting it probably into the largest galvano-
meter of the type extant; but the sensibility is exceedingly
small when the magnet is fully excited, and increases rapidly
when the excitation is somewhat diminished.
A pretty way to illustrate the sensibility is the following.
A small current is passed through the galvanometer-coil when
the magnet is excited, and the coil-current increased until a
small deflexion is produced. The exciting current is then
turned off to permit the magnetism to fall gradually. The
deflexion will then become greater and greater, until the dot
of light passes the end of the scale, after which the sensibility
again decreases as the magnetism falls.
From all that has been said up to this point, I think it is
worth while for a course of experiments to be made on the
varying resistances of different metals in the magnetic field with
varying currents. It is probable that it will be found that
there is no constant for any given metal ; and if this should
be shown to be the case, my view that the increased resistance
of the metal in the magnetic field is spurious will be proved.
256 Sir David Salomons on the Electric
Returning once more to the vacuum-tube, we observed the
separation of the alternate current within it into two distinet
paths. It occurred to me that it might be possible to com-
pletely separate the two currents. I therefore constructed
tubes as shown in figs. 15, 16, and 17, in order to observe
Fig. 15.
whether the currents would be separated into the tubes, which
on experiment I found to be. the case. It was not neces-
Discharge in a Magnetic Fieid. 257
sary to place the whole tube between the poles. If one of
the electrodes was placed between the poles it was generally
sufficient, provided that the plane of the tube was placed
equatorially.
When such a tube was placed axially, as in fig. 17, the two
columns of bands appeared in each section of the tube.
A tube constructed as in fig. 18 was then employed, the
Fig. 18.
(on
end electrodes being joined together, the currents separated
from the centre to the end electrodes. But in this case the
path traversed by each current was practically equal in
resistance.
Another tube therefore was employed where the distances
were unequal, as in fig. 19, the result being the same, and
Fig. 19.
: =
conclusively showing that the alternate current can be divided
into its two constituents.
A tube as shown in fig. 20 was now taken. The centre
Fig. 20.
bulb being placed between the poles of the magnet, and the
other two electrodes joined together, the alternate current was
divided into two currents, one in each arm of the tube. A
second similar tube was now connected with this one and
Phil, Mag. 8. 5. Vol. 42. No. 256. Sepé. 1896. U
258 Sir David Salomons on the Electric
joined up as shown in fig. 21, the connecting-wires being
of considerable length. ‘The current traversing the second
Fig. 21.
tube was again analysed by another large electromagnet ;
and it was found that only one type of current existed in each
arm of this analysing-tube. Consequently, the current tra-
versing the connecting-wires must of necessity have been an
intermittent direct current.
Another double tube was now taken ; but instead of being
a vacuum-tube it contained water, the analysing-tube still
being a vacuum-tube. The results were the same, although
less marked.
It would therefore appear that we have a magnetic means
of converting the alternate current into two currents of a
direct intermittent type, which two currents can be coupled
up so as to form cne intermittent direct current in one direction.
So far, 1 have only succeeded in producing this result with
very small currents.
When the ordinary electric spark in air (fig. 22) is placed
Fig. 22.
in a powerful magnetic field and the magnet excited, two
additional semicircular displaced discharges appear, as in
fig. 23, the colour of these supplementary discharges being of
a different tint.
Fig. 23.
When the current is reversed, the displacement is reversed,
as in fig. 24.
Discharge in a Magnetic Field. Zou
When the coil is arranged to give an intermittent direct dis-
charge in the secondary only, one seinicircle of light appears,
as shown in fig. 23; and when the current is reversed, it is
as in fig. 26.
: Fig, 25,
These experiments led me to try another, which may have
some bearing on the reason for the zigzag form of the dis-
charge in air. ‘Taking once more the vacuum-tube and
placing it in the magnetic field, 1 permitted a very consider-
able alternate current to traverse the tuhe. First, the two
columns of bands appear; then the lines of force ; and
finally these phenomena remain, with an additional one,
namely, a series of zigzag dischar ges throughout the tube,
permanent in character, but varying their form continually.
It is quite possible that these zigzag discharges consist of de-
formed bands which meet and form zigzag or sinuous lines.
That they should vary in form may possibly be in consequence
of the ever-changing temperature of the gases within the tube,
due to so large a current passing. Hence it is not impossible
that the zigzag discharge in air is due to the magnetic pro-
perties of the current itself. Of course further experiment is
needed to demonstrate the truth of what I have suggested as
a probability.
D2
[ 260 }
XXVI. On the Longitudinal Component in Light. By
Prof. GzorcE Francois FirzGrraup, .A., ScD., F.RS.,
JHE AIS Od ge
| most investigations on the propagation of light attention
has been concentrated on the transverse nature of the
vibration. Longitudinal motions have been relegated to the
case of pressural waves, and investigators have devoted them-
selves to separating the two as much as possible. In Sir George
Stokes’s classical paper on Diffraction, and in Lord Kelvin’s
Baltimore Lectures, the existence of a longitudinal component
is mentioned ; but it is mentioned only to show that it is very
small and that the motion is mostly transverse. Now the
longitudinal component is no doubt generally smali except in
the immediate neighbourhood of a source ; but it by no means
follows that, as a consequence, the actual direction of motion
is transverse at all points ina wave. In every complicated
wave there are points and often lines along which the trans-
verse component vanishes, and at all these places the smali
longitudinal component may be, and often is, of great relative
importance, so that the actual motion is largely in the direc-
tion of wave propagation at these places.
I. The simplest case is that of a simple oscillator whose
theory has been completely worked out by Hertz. There are
two kinds of oscillator, an electric and a magnetic one. They
are exactly complementary, the magnetic forces in one cor-
responding exactly with the electric forces in the other.
If the oscillator be taken as an electric one parallel to z,
we have for the components of the vector potential
PIG o, H=H, Pf;
and the components of the electric force, which are in general
poa¥—-&, Q=aG- 7 R=aHn-S,
where fie dk | dG@ | di
7 da say, aden
become in this case
: @H ; CH 2 sdk aoe
LEIS Oe aoa ~ Oe dy
It is particularly to be observed that P and Q arise entirely
from J, which was dismissed by Maxwell as not coming into
consideration in cases of wave propagation on account of
there being no varying electrification. This is true as regards
-* Communicated by the Author.
On the Longitudinal Component in Light. 261
propagation, but not at all as regards origination. In all
eases of origination we have to do with conduction, or its
equivalent convection, and in most such cases we have
changing electrification which brings in the J term.
The longitudinal component at each point is
eee, e Y ° 2 e
T= 7b +o. Q+-.R
22 Wee af a SRC SE
ps (27 sin pt— gr + 7, COS pt—qr).
This is no doubt very small at a distance from the oscillator
ae 1
compared with the transverse component which involves -
and in consequence the motion is transverse at most places.
On the axis of z, however, the transverse component, which
is proportional to p the distance from the axis, vanishes
entirely. Hence along the axis there is a beam of purely
longitudinal vibration, of no doubt small amplitude, but
nevertheless existing necessarily in order that there may be
no compressions. This all appears on the face of Hertz’s
investigation. He carefully studied the forces as represented
by the above equations, and has plotted them and shown that
they represent a series of whirl rings thrown off from the
oscillator and growing gradually thinner and thinner until at
a distance the rings become nearly plane waves, and the
opposite sides being always a
wave-length apart are the two
opposite phases of the wave.
The accompanying diagram
roughly represents this state of
affairs. It is evident on the most
cursory consideration that these
waves must have a longitudinal
region. The lines of force in
any one wave are up to the axis @ p))
along any one spherical surface
ail round; and if there is not to
be concentration anywhere, 2. e.
if there is no electrification of the
medium, they must turn round
and be continuous with the return
phase of the wave. The reason
why they are so feebly concen-
trated in this return region is
because it is soenormously extended. If the wave-length be
small compared with the distance from the origin, the flows
262 Prof. G. F. FitzGerald on the —
up and down along the equator are very close to one another
and consequently the force is concentrated ; while this same
force which is concentrated within a wave-length has the whole
hemisphere to return in, and so the longitudinal concentration
is quite small, and that is what is represented by the small
value of the longitudinal component at any point. The total
quantity of longitudinal component must be, on the whole,
equal to the transverse component at the equator.
lI. In the ease of several simple oscillators oriented in dif-
ferent directions the resultant vector potential can be repre-
sented by
A=US (pt —qr) 4 ysin Pie
r r
where U and V are vectors at right angles to one another.
The effect is the same as if two opposite electrons were
moving on opposite sides in an elliptic orbit whose plane was
that of U and V and whose axes were these two lines.
It is interesting to observe that this case, coupled with a
slow rotation of the ellipse which would be produced by
almost any small disturbing force in its plane, has been
shown by Dr. Stoney to be a sufficient cause for the double
lines in spectra which are so common and which are familiar
to everyone in the double sodium line.
If the directions of U and V be taken as those of « and y,
and z be taken perpendicular to the plane of this ellipse, we
may take
F=F, cos Sota ace Ee, H20:
and we get a sort of corkscrew wave with a longitudinal
component which can be represented by
SI
ae {Leos (pt—qr+)};
7
where ¢ is the angle between 7 and z, and L and / are func-
tions of Fy, Go, 7, @, and g.
This component vanishes along the axis perpendicular to
the plane of the ellipse, and ig a maximum in this plane. If
’,= Gp, this simplifies to
sin
z
q
CG =
{2q sin (pt—gr— é)+- cos (pt—qr—O)}.
This case is rather interesting, as being the form of magnetic
Longitudinal Component in Light. 263
wave that is thrown off into space by the rotation of each
of the earth’s magnetic poles.
The more complex wave thrown out by the earth with its
two magnetic poles comes under the next head ; but it is
wayes of this type which must be thrown off by the planets
rotating round the sun, if they are electrified, and by their
gravitating property if gravitation be propagated in the same
way as electromagnetic disturbances.
Ill. We can produce any desired combination of complex
doublets by operating on a simple doublet with a function _
S oA =). The typical term of such a function may
(i) -(@):(ae)=*
ne £23 (ptr)
r
be taken as
If we write
we get as a typical case,
hou, (C= lal (0
Also, remembering that A?u+?.u=0, we have for the
electric force corresponding to this typical case of a vector
potential,
: Cpr: du ° du
ere om we Be pe Ns it
Se ha Q . da dy’ K dx dz
da’
Now this operation will introduce all sorts of powers of Z
r
and of g, and I only want to calculate the principal term in
the longitudinal component. In making this approximation
we may simplify the calculation by observing that the largest
terms are always due to differentiations with respect to the
circular part of u, and that differentiation with respect to
2, y, 2, or r lowers a term by one. We may then leave
out all differentiation with respect to coordinates outside the
circular part in terms of the second order, and it is well to
reduce the differentiations represented by 6 so as to produce
: Li aN : ae
terms of the form pe and 6 ee Of
course it very much simplifies calculation to use the typical
form e—” for the circular functions.
We thus get for the values of the components of electric
force to the second order :—
264 Prof. G. F. FitzGerald on the
pe =652 ; 3008 pt— qr
Ue
t—
—14 19/8 +9)— 4 (at B+y— |}
O=, 9 3 5 es
Bs be sine 2h ee
JS Bae Tey a
yg 2% cos pt—gr
R=(.- ee
— 2 [8 4 BF fant + y+ (B—P)are} [SSOP Ee
Uf av a
In this form it is Bart at once that the highest terms
vanish in the longitudinal component
o=P=+Q=+R-
= [ety + 2) {221-8 9) tay? +2) — 20h early + 2)
+7? (a+ 8 +2y— $a + By)
+24 2BFY—Jaty2) | PM
In order to get this we have to observe that when applied
to the circular part only
(2) OFC
ip r is
Any particular typical term of this order vanishes over the
sin pt—qr
1h
quartic cone the coefficient of 6
This is the cone of intersections of the systems of spheres
ee ye 42° 20
with the cubics
04 x*(1L—B—y) +a(y?+ 2°) $= aa? (y +2) +y?) (at B+ 2y—3)a+Byt
+2*4(a+2B+y—3)y +42},
In the particular case of a series of end-on vibrators for
which 8=y=0, this cubic breaks up into the plane r=0
and the quadric cone
y+e= Ae a(y+z).
Longitudinal Component in Light. 265
In every case the quartic cone intersects every plane per-
pendicular to the axis of w in a bicircular quartic.
In the case of a complex oscillator whose components are
not all parallel to the axis of 2, as in the case just studied,
the longitudinal component will vanish to this order over a
quartic cone so long as we confine ourselves to a typical
term 6. ‘This cone is of the form
Uz (x? +y? + 2?) (aya? + diy? + 12”) — faa? (y +2) +by? (e+ a) +c27(a+y)
+ ly22? + m22a? + nx?y?
+ ayz(pet+qy+rz)}.
In general it is quite evident that the motion along the radius
does not vanish.
On considering the general case, we may observe that if
the differentiations involved in 6 are such that for every term
a+ 8++y¥ is either even or odd, then there will be a complex
surface all over which the normal component will vanish to
the second order of small quantities; but that if 2+ 8++y¥ be
even in some terms and odd in others, we shall have o of the
form
o=U cos pi—qr+t V sin pt—gqr,
and this will only vanish over the curve of intersection of
We Vand V=0. 3
IV. If we now consider the case of diffraction through a
narrow aperture, it is simpler to take the case of the electric
displacement of the incident wave as parallel to the edges.
In this case the electric force is everywhere parallel to the
edge, and consequently its longitudinal component every-
where vanishes. On the other hand the magnetic force is
perpendicular to the slit and has a longitudinal component
everywhere except in the plane through the slit perpendicular
to the wave-face. In considering the more complicated case
of the electric displacement being perpendicular to the slit, it
is necessary to take account of the nature of the edges, whether
they are non-conductors or conductors, whether they are
erystalline, and so forth, because their electrification Kc.
must come into consideration. Similarly, in the case of the
electric displacement being parallel to the slit the magnetic
properties of the edges inay be important. In this case, too,
their conductivity influences the effective width of the slit, as
is evidently the case when we are dealing with wire gratings
in the path of the Hertzian radiations. ‘These questions are
involved in a complicated way in the whole discussion of the
effect of a grating on the plane of polarization of the incident
266 Prof. G. F. FitzGerald on the
light. I will take the simple case of a slit bounded by obstacles
which completely stop all action. Although such do not
exist, very close approximations to them do exist.
If we take the slit as parallel to z, and make this axis the
centre of the slit, and assume the phase the same all over the
slit, we have for the vector potential at a point 2, y, 0, due
to any line of the slit at a distance y from its centre,
-* cos pl yr gy
0 r
where
Integrating this for the width of the slit, ¢.e. from +6 to
—b, we get for the complete value of the vector potential
H=2H,| * ‘is C08 pl—=9P dy,
=p) sO If
When we are dealing with the case of 6b being a small
quantity we may take
jute dy=f (yo—y),
:
and we have
ee ee
i SE y=Kyr+0) fob)
a
ae ap 2b? fax
=20(7 ue mee 0
(7) = cos (pt—gr) _
dy/o r
and when y=0 is put in
P= ay tye +P=p +27,
Ss 6? d?u
a = aes
oa y=2(ut os. dip )
If we now integrate with respect to z we get
Haat | | vaya: REL folemnes a:
0 Riner ae (bf £4 og hl nes
Now \" udz is a function of p only, and is a Bessel func-
/ A
But
Longttudinal Component in Light. 267
tion subject to the equation
ot red J 5
dp? == p dp sr) JO,
so that we can write
3 9
H=40h,(J+ aes )
ee ge
i? dd) oe ed) 1 dJ
=ub louie, ae ery cae ses
By means of the differential equation we may of course express
: ‘ ]
all the differentials of J in terms of J and - “ . We may, how-
: p
ever, simplify matters very much in the ordinary case of light
by observing that q is generally a very large number, so that
terms involving its powers are large. Keeping to these we
2 nN
see that ae =—q'J, and that the highest term in si is
dp de
(@Y'5 . Using these terms only we get
; H=4HJ(t— ues (2) +...)
zo) \O
= OGY
ut von sin €
=—4H J - qy — AH Jd 4 €
ao bay. p
p
Without going into the question as to the best series to ex-
press J by it is evident from its integral form and from the
dynamics from which it is derived that it must represent a
wave propagation. In fact by integrating by parts it could
be expanded in the form
J =J, cos (pt—gp) +z sin (pt —gqp).
In any case we can see that for any constant value of p H
passes through a series of values giving the alternate lights
and darks on a screen illuminated by a narrow slit.
Considering now the magnetic force we have
268 Prof. G. &. FitzGerald on the
and hence the longitudinal magnetic component
From this it is evident that in every such case m=O so far as
H is a function of p only. Thus we get
a) =. =
prey isn g @e COS = sin a
Coes
This shows that m2 does not in general vanish but has alterna-
tions of value like H. The tangential component has for its
most important term
It is evident that this longitudinal displacement is necessary
at the edge of the beam in order to prevent any concentration
of the magnetic force. So far as our a priori knowledge of
pure ether is concerned there seems no sufficient reason for
not supposing a concentration of magnetic force just as
probable as one of electric force. It would certainly com-
plicate our equations very much to suppose both. If both
existed we might have two kinds of pressural waves, one a
wave of electric condensation and rarefaction, and the other
a wave of magnetic condensation and rarefaction.
It is quite evident from all these cases and from general
considerations that the edge of every beam of light is bordered
by a region where there are longitudinal vibrations taking
lace.
: V. Asa final example I take the case of a series of slits
forming an optical grating.
In this case the simplest supposition is to assume that the
opacity of the grating varies in a simply periodic manner.
This leads to the same sort of equation for H as in the last
case except that the intensity in each line is proportional to
(1+ cos ly), where /= au and s is the interval between the
lines. $
This leads to the integral
H=2H{ ( Ses Mi) 7
OO
here Sua
* r=xy tyo—y +2’.
Now from general considerations it is evident that it must
be possible to expand this in terms of cos ly by Fourier’s
Longitudinal Component tn Light. 269
theorem, so that
H=h, +h, cos ly+h, cos 2ly+h3cos 3ly+...
Observing then that H being a function of a) and yo only
satisfies the equation
ae 2 Gi.
=e 2
De aie Ny
we get that in general
aN,
dy?
+ (g?—rl)hn=0,
so that
hn rb-cos Ve—nWl? —n'l? . x,
so long as nl is <q; and when nl is >q
hi — ees Vn2—g?e
as the value cannot increase to infinity.
We thus get the general form for H,
H=H) cos (pt —ga) + H, cos ly cos (pt— Vg?—Pa) +...
+H, cos nly cos (pt— fg?—nl?. 2) +...
+H, cos mlye—VwP—¢ cos pt +...
It would appear from this that at the surface of the grating,
where e=0 when ¢=0,
H=H)+H,cosly+...+H, cosnly+...
It would consequently seem that this must-in general
represent the distribution of opacity at the grating, and that
in the case of a simply periodic distribution the general form
of H would be
H=H, cos (pt—qa) + H, cos ly cos (pt— Vq?—/?. x).
We thus get an interesting form for the double integral for H.
The magnetic force to be calculated from this is
dH dH
oe dy’ esa, y=0,
and consequently
a= —1H, sin ly cos (pt— Vg?—F?. 2),
B= qHosin (pt—ge)+ Vy?—PH, cos ly sin (pt— Vq?—P2).
In this a is the longitudinal component of the magnetic
force. This represents a series of waves being propagated
away from the grating, together with a series of elliptic
270 On the Longitudinal Component in Light.
Ts
whirls whose length is = and breadth fae The length
Gir
is the same as the width of the lines of the grating, and the
2
breadth somewhat greater than the length of a wave= ee
is especially obvious in this case that some longitudinal com-
ponent exists.
The existence of the terms depending on e—¥#?—¢x
shows that there may be something analogous to total re-
flexion with its extinction wave in the case of a grating in
respect of the spectra that are of a higher order than can be
transmitted by the grating. It would seem, then, that the
whole energy of the wave might not be distributed over the
spectra unless the variation of opacity in each line be
judiciously made. This may also be connected with the
high absorbing and radiating powers of rough surfaces and
with the action of coherers.
“It is a matter for consideration whether it would not be
worth while manufacturing photographic gratings by causing
the two first spectra on each side of the central image,
together with this central image, or without it, to interfere on
the surface of a sensitive film. We might thereby produce a
grating which had such a distribution of opacity as to repro-
duce only these first order spectra and have all the light that
passed through concentrated in them. Similarly we might
manufacture a grating which would have the light concen-
trated in any desired pair of spectra, though this would
practically come to the same thing as the first proposal, with
the lines closer together. This comes to the same thing as
producing gratings by means of the interference of two beams
of parallel rays of monochromatic light in the manner that
Wiener has shown to be possible.
In all these cuses it is quite evident that a longitudinal
component of either electric or magnetic force is essential
to the existence of waves whose intensity is not constant all
over their surface, and that it 1s a practically universal con-
comitant of all waves of noncondensational type. That in the
case of short waves which vary slowly from point to point,
the intensity of the longitudinal component at any place will
be in general very smal!, because the area is very large over
which the motion along the surface at one place has at its dis-
posal in which to turn and be continuous with the motion
back along the face of the next wave. This does not make it
unimportant, however. Ina great many cases the total flow
along the face of a wave must somewhere flow longitudinally
Measurement of large and small Alternating Currents. 271
so as to be continuous with the flow back along the other
face of the wave. Unless these longitudinal flows are taken
into consideration the whole energy of the wave is not
accounted for. If the rate of variation of intensity over
the surface be comparable with a wave-length, as in the case
of fine gratings, the longitudinal component is a large part of
the phenomenon, and, in fact, represents a large part of the
energy in this case transmitted to the secondary image. This
is all quite obvious in the case of gratings from the ordinary
theory, for the equations given as a solution of this case
represent a series of waves being transmitted in different
directions from the grating corresponding to the directions
of the secondary spectra.
XXVIII. On the Measurement of very large and very small
Alternating Currents. By ALBERT CAMPBELL, B.A.*
a IR-core transformers, although quite inefficient for
ordinary lighting circuits, are yet much more valuable
for testing purposes than most people are aware of. By the
help of such transformers it is possible to extend almost
indefinitely the ranges of many ordinary measuring instru-
ments. If the secondary of an ironless transformer be kept
in open circuit the secondary volts are accurately proportional
to the primary P.D. if the frequency is constant, and hence
by using an electrostatic voltmeter on the secondary we can
transform either up or down, and thus measure voltages above
or below the range of the electrostatic instrument. Of course
the arrangement would have to be calibrated; this might
sometimes be done by taking a reading for which both the
primary and secondary voltages lay within the range of the
voltmeter used. —
The above way of using an air-core transformer was sug-
gested to me some time ago by Mr. Hugh EHrat Harrison, of
Faraday House. I hear since from Mr. Mather that it has
been also used at the Central Technical College f.
If we attempt to measure current (in the primary) by
observing the voltage on an open-circuit secondary, we find
that for a given primary current the readings depend also on
the frequency. It therefore occurred to me that the secondary.
* Communicated by the Physical Society: real June 12, 1896.
+ Since writing the above I have listened with interest to the paper on
the “ True Resistance of the Electric Arc,” by Messrs. Frith and Rodgers.
Their beautiful application of an air-core transformer to measure a small
alternating current superimposed on a large direct current might, I think,
be also employed to separate a darge alternating current from a much
smaller direct current.
272 Mr. A. Campbell on the Measurement of very
circuit should be closed and should besides be highly znductive.
In practice this was found to work with perfect success, the
primary current being proportional to the secondary current,
which latter was measured by an ammeter.
To find the most favourable conditions, let
],, 1, be the primary and secondary currents respectively ;
R, the resistance of the pec circuit ;
N its inductance ;
M the mutual inductance between primary and secondary.
Let p=27n, where n is the frequency of alternation.
Here M and N are constant, while R, may be variable with
temperature.
At first let us assume that the P.D. follows a ss sine
curve. Then I, _ VR2+ pW 41)
| a (
Now let the inductance of the secondary circuit quite swamp
its resistance, so that we may neglect the term R,” in com-
parison with p’N*. In this case we obtain
Ape Nl
ie = WM? Eee ee Se ee Se (2)
i.e. the ratio of the primary current to the secondary current
is independent of the frequency of alternation and the resistances
of the coils. We shall see below to what extent this valuable
result can be realized in practice. It is clear that in the
>
e N e e
above case, since E =o , the secondary current I, is in exactly
opposite phase to I.
Now suppose the primary P.D. to be no longer a pure sine
function of the time but to follow any periodic curve. ‘The
curve of primary current can then be decomposed into a
number of sine curves of various frequencies ; each of these
will produce a secondary of opposite phase, and the ratio of
: N
each primary component to its secondary will = i
Accordingly the total secondary current curve will be of
the same wave-form as the primary and in opposite phase to
it, while the rato of current transformation will be the same as
for a pure sine-curve alternating current*.
* In this I have assumed that the Fourier series for the wave-form
contains no constant term, z. e., that there is no constant component in the
current. If there is such a component, nothing corresponding to it will
appear in the secondary circuit, and the ratio of current transformation
will be by no means the same as if the current were purely alternating.
In fact an air-core transformer affords a good means of separating out the
purely alternating part of any periodic current.
large and very small Alternating Currents. 273
Unfortunately, in practice we cannot quite attain to the
e . N
condition Re =, but we can get near enough to make the
2
error quite small. In the following table are given (approxi-
mately), for different values of 2 , the values of the-ratio 7"
2
for frequencies of n=80 ~ per sec., and n=40 ~ per sec., if
the ratio =1 when n=~.
TABLE I.—Transformation Ratios.
= eee 4 280: poet
166 1-00 1-02 1:08
50 1000 «=| ~—s«1:005 1-02
oF on) Se eto A 00S
10 ~ 1000 1-0002 1-0008
By this it will be seen that “if x2 =0, the error in the
transformation ratio from »=80 to n=c becomes Insignifi-
cant ; and that if = = 25 the ratio is approximately constant
down to n=40. R R
In practice it is easy to make y= 9 but to get N = 25
would require a rather lavish expenditure of copper wire.
Instead of finding the ratio of a particular transformer
(with ammeter) once for all, a simple way is to find it each
time for the frequency to be used, the strength of the primary
current being chosen so as to lie within: “the range of the
instruments available.
The following instance will make the method clearer.
Suppose we want to measure currents up to 1000 amperes
with an electrodynamometer and a Kelvin balance, each up
to 100 amperes. Let the transformer have a “ current ratio ”
of about 10 when the balance is the only load on the secondary.
The electrodynamameter is inserted in the primary circuit, and
a primary current of (say) 98°2 amperes is found to produce
(say) 10 amperes in the secondary. If the electrodynamometer
be now taken out of circuit, any\ primar y current up to LOQU
amperes can be measured ° by reading the balance in the
ness y-and multiplying by 9°82.
‘Phil.-Mag. 8. 5. Vol. 42. No. 256. Sept. 1896. —
274 Mr. A. Campbell on the Measurement of very
In an exactly similar way very small alternating currents
may be measured.
In order to avoid errors due to stray magnetic fields the
transformer may be wound with double coils placed side by
side with their axes parallel and reversed in the well-known
manner.
It might also be well to have the position of one or more
turns of the primary or secondary adjustable with a scale to
show the proper position for each particular frequency.
For the purposes of calibrating at any frequency a second
primary of a different number of turns from the first may be
wound on the transformer, and the relationship of the two
primaries may be determined once for all.
For example, if the first primary is for 1000 amperes the
second may have 100 times the number of turns and be suit-
able for 10 amperes. The correction for any particular
frequency or wave-form may then be found experimentally
by using the second primary only.
I may mention that with very simple instruments I have
used an air-core transformer to measure currents between
1000 and 2000 amperes.
| Added June 12th, 1896. |
In order to get a larger increase of secondary voltage a
Fig. 1.
well-known method is to connect up a transformer with the
primary and secondary in series as in fig. 1. By a somewhat
large and very small Alternating Currents. 275
similar method the ratio of current transformation may some-
times be increased. The connexions in this case are shown
in fig. 2. :
AMPERES
Transformers with Iron Cores.
I have recently investigated the case of transformers with
tron cores whose secondary circuits’were made highly induc-
tive. This was done by short-circuiting the secondary in each
case through a Kelvin 100-ampere balance.
Two transformers were thus tested—an iron-ring trans-
former with very small magnetic leakage, and a small trans-
former with open iron circuit. - The first of these had a
primary of 764 turns of No. 16 8.W.G. wire, and over this a
secondary of 48 turns of 7/16 wire. The core of the second
was a short bundle of iron wires, over which were wound the
primary of 38 turns of No. 16 8.W.G. wire, and over it a
secondary of about 3000 turns of No. 26 8.W.G.
The ring transformer was first tested -with a Kelvin 10-
ampere balance as secondary load, the frequency being 84 ~
per second. ‘The balance had a resistance of about 0°55 ohm
and an inductance of about 0-0016 henry. The results given
in Table IL. show that. with this amount of- resistance in the
Io: Las ead)
secondary the L is by no means constant.
1
X 2
276 Measurement of large and small Alternating Currents.
TABLE II.
Iron-ring Transformer. Secondary Load 10-ampere Balance.
poe ere Ratio :
eae ee eee ey
_ 00764 9:28
mer’ Seo
Aci ae eae: =
Rees Ch aaee ae
TaBLeE LII.
Iron-ring Transformer. Secondary Load 100-ampere Balance.
pumas current. Repo He
Tea Spd)
meee oo
pee one OP Sea
ieee ibaa en
|
Similar measurements were then made with a 100-ampere
balance as secondary load. From Table ILI., which gives
some of the actually observed ratios, it will be seen that the
ratio of current transformation is practically constant.
With almost the same arrangement of secondary circuit
experiments were made to find to what extent the ratio 1,
iF
was affected by change of frequency. Table IV. gives tlic
results of these and shows that the ratio 7s almost BA):
of frequency through the range tried.
Representation of the Periodic System of the Elements. 277%
TasB.e LV.
Ring Transformer.
Frequency, ~ per sec. I, Error from 84
Te ~ per second.
40 | 15-48 0-7 per cent.
57 ch 1552 0:45 per cent.
eS ee eee ee eee
84 15:59
|
The results in Table V. of a similar experiment with the
transformer of open magnetic circuit described above show
how different its behaviour is.
TABLE V.
Transformer with open Iron Circuit.
~~ per second. I, | Error from 84
ae | ~ per second.
44 28:1 | 24-3 per ceut.
84 871 |
It seems clear, therefore, that iron-ring transformers may
in many cases be used in a similar way to that described
above for air-core transformers; but care must be taken to
have the resistance of the secondary circuit sufficiently small.
In conclusion I beg to thank Messrs. Lovell, Macalister,
Sankey, and Norman for their kind help in some of the
experiments.
XXVIII. Remarks upon the Analytical Representation of the
Periodic System of the Elements. By Dr. A. GOLDHAMMER*.
eo attempts have been made in recent times to represent
analytically the periodic dependence of the general
chemical behaviour of the elements upon their atomic weights;
these two researches, entirely independent of each other, and
published at an interval of about eight years, have led to the
same result in a remarkable way. IF’. Flawitzky in Kasan in
1887 f, and J. Thomsen in Copenhagen in 1895 tf, represent
the chemical character of an element e as a function of its
* Translated from a separate impression from the Zedtschr. f. anorg.
Chemie, vol. xii. (1896), communicated by the Author.
+ F. Flawitzky, Verh. d. Naturf.-Ges. Univ. Kasan (1887).
t J. Thomsen, Zeitschr, f. anorg. Chemie (1895), ix. pp. 283-280,
278 Dr, A. Goldhammer on the Analytical Representation
atomic weight p in the form
e=acot wh”, | mer
where a is an unknown constant, and b and ¢ are constants
easily determined for each period.
The-views-of the two authors differ somewhat; whilst
F. Flawitzky, in agreement with L. Meyer *, considers V, Cr,
Mn, Nb, Mo as electronegative or acid-forming, and Cu, Zn,
Ga, Ag, Cd; In as positive, according to Thomsen V, Cr,
Mn, Nb, Mo are electropositive, and Cu, Zn, Ga, Ag, Cd, In
negative; from this it followsat once that Mendeleeff’s eighth
group of positive metals, Fe, Co, Ni, Ru, Rh, Pd, Os, Ir, Pt,
must be left out of consideration by F. Flawitzky, whilst
J. Thomsen takes these metals into consideration.
F. Flawitzky remarks further that in his method of repre-
sentation N, O, F appear just as well as analogues of P, 5S,
Cl, As, Se, Br, Sb, Te, I, as of V, Cr, Mn, Nb, Mo, —, Di,
—, —, Ta, W, which latter analogy is true in fact only in a
remote sense ; according to Thomsen, on the other hand, this
is not the case, since the elements analogous to V, Cr, Mn,
&e., ought to fall into the period Li—F between B and QC,
and in that from Na—Cl between Al and Si.
If we might take these views of J. Thomsen as valid be-
yond dispute, then the question of the analytical representation
of the periodic system of the elements would be nearly solved.
But if we hold with the views of L. Meyer the matter is
somewhat different ; the cotangents can then serve only as a
rough approximation to the truth.
Since I, as a physicist, cannot venture to enter upon details
of chemical views, I permit myself here to show only by way
of experiment how the general character of the chemical
elements as given by L. Meyer may be represented geometri- |
cally or analytically. |
In the system of Mendeleeff we have, as is well known,
six periods : Li—Cl ; K—Br; Rb—I; Cs—?; ? Bi?; ?—?.
After the electropositive alkaline metals Li, Be we have in
the first period a gradual passage through B, C to the electro-
negative acid-forming N, O, F ; from EF a rapid passage to
the positive Na, Mg, and then again after Al, which has some
claim to possess negative properties, a series of more and
more decidedly negative elements Si, P, 8, Cl.
From Cl to the positive K, Ca, Se we have obviously a leap
in-the properties of the elements ; then follow negative V, the
partly negative partly positive Cr, Mn, positive Fe, Co, Ni,
* L. Meyer, Die modernen Theorien der Chemie (4th ed. 1880), i,
p. 167-169, .
of the Periodic System of the Elements. 279
Cu, Zn, Ga, the “ positivity ”’ first increasing then decreasing
(a maximum occurring somewhere between Cu and Zn),
then follow elements becoming more and more negative, As,
Se, Br. Ti and Ge ease the passage through the zero-point.
In the remaining periods we have a complete analogy with
the second ; it will be sufficient, therefore, to consider this
period a little more closely. eRe Meee
Let us express the general connexion between e and p by
e=T'(p), and let us regard e and p as rectangular coordinates
of a point in the plane e, p ; then e=F'( p) is the equation of a
periodic curve. As we do not know any numerical values of e,
it is only possible to form some opinion as to the shape of the
curve by our knowledge of the properties of the elements.
If we assume that e=+o between Cl and K, Br and Rb,
T and Cs, &¢., as F. Flawitzky and J. Thomsen assume, then
we conclude that our curve must consist of as many separate
parts as there are periods of elements. But, as we know, the
periods are unequal ; thus, for the elements of the second,
third, and fifth periods the number 48 (atomic-weight units)
fits, whilst for the ceesium-period this number seems to be
1:5 times too great. Hence we see that the separate portions
of the curve cannot be identical ; only in form do they re-
semble each other.
The curve (fig. 1) drawn with two asymptotes, agrees with
the properties of the elements of the second period ; for the
following periods we have exactly similar curves, each con-
sisting of six portions, AB, BC, CD, DH, HF, and FG.
We remark next that the analogous elements are represented
by the points of the corresponding portions of the curve ; thus
upon the portion AB we have the positive elements
Ke Ca, Se:
Rb; siSr,0 Y,
Cs, Ba, La, Ce,
ee
the transition elements Ti, Zr, Di (?), —, Th (?) corresponding
to the point B.
From the properties of V, Cr, Mn we may further conclude
that these elements are represented by the points of the
portion of the curve CD ; thus upon the portion CD we have
the negative elements
Ver Mn
Nb, Mo, —,
Di 2, Riedy ae
Ta, W, —,
b]
P] P
280 Dr. A. Goldhammer on the Analytical Representation —
In the same way it is easy to see that the portions of the
curve DE, EF, FG correspond to :—
DE +: Fe; "Co, Ni, Cu; EF+: Zn, Ga,
Ru, Rheds A ©, Gd; cine
Os, Ir, Pe, Au; He, Ti,
RG 2.7 Se Be
: “Sb. Te;
iy Pa are:
Bi, Tr La
’] , y] ;
whilst the transition elements Ge, Sn, —, Pb fall at the point
F. In all these periods the portion of the curve BC remains
free from known elements ; but if we compare the portions of
the curve AC, EG, which correspond pretty well amongst
themselves, and if we observe that the elements corresponding
to the portion EF find their analogues (if imperfect) in the
elements corresponding to the portion AB (Ca—Zn, Sr—Cd,
Sc—Ga, Y—In, &c.) itis easy to draw the following con-
clusion:—the elements which may possibly exist between
Ti and V, or Zr and Nb, or Ce and Di ought to appear as
imperfect analogues of the elements As, Se, Br, Sb, Te, I
As to the period Li—Cl, which is much smaller than the
rest (only 30-32) it is easy to surmise that here also the curve
possesses the form ABCDEFG. In fact, Li, Be, evidently
correspond to the portion of the curve AB, the transition ele-
ments B, C, correspond to the point B, but the next following
negative elements N, O, F’, possess exactly the character they
ought to have if N, O, F, are to be represented by the points
of the portion BC of the curve ; N, O, F are imperfect ana-
logues of As, Se, Br, &c. Further, we find in the period
Li—Cl a sufficient place between F and Na for a series of
elements which would fall upon the portion CDE of the
curve ; these would be analogues of V, Cr, Mn, Fe, Co, Ni;
the newly discovered element argon would perhaps correspond
to the point D. ie eee
It is well known that sodium is not a good analogue of
Cu, Ag, Au; this indicates that sodium falls somewhere about
the point E, and is thus much nearer to Li, Rb, Cs. We
have also, exactly as before, the positive metals Mg, Al for
the portion of the curve EF ; for the point F the transition
element Si, and finally the negative elements P, §, Cl, corre-
sponding to the portion of the curve FG.
For the atomic weights smaller than that of lithium we
have only hydrogen and helium; but since H stands nearer
of the Periodic System of the Elements. 281
to Na than to K, and helium appears to be a sufficiently in-
active element, we may infer that in the period ?—Li we
have only the portion of the curve EFG ; then H would fal
about the point E, and helium at the point F.
HHS
EERE EEE Ee
SRGRCRUS ARES CRERES pena
SESEESSEE
sHlse
seaftes
iN
seriesiitusiteses Geimrestins
Sersscerserasesr /arsieni
pT
sanayinestiesiteer’ si
anit
Seuunnem eases UTEETETT STTEET
HH
PEE eee Ht
BEE H
POSH
ageees
s
BoSaeES touneaoaes
Fig. 1.
So far we have left the valency of the elements out of sight.
But, as we know, there exist for each chemical value two
series of elements analogous amongst themselves ; according
to the usual manner of writing we have, for example, the
bivalent and trivalent elements arranged according to in-
creasing atomic weight :
Bey Carenisndin Bay iit on) C3.
Mey 2 Ymi)Cd, =. He
and
282 Representation of the Periodic System of the Elements.
where the elements of the lower series in each set appear to be
more metallic in melting-point and other properties than the
elements of the upper series ; the same holds good also for nega-
tive elements ; thus we have, for example, for the valency 6
O, Gr, Mo, er) i; Gi
5. ste
although here the relations are of a somewhat complicated
character. At all events it is clear that for each determined
value a of the valency the properties of the elements alter
periodically with increasing atomic weight, so that e=/(a, p)
represents for each a a wave-shaped or zigzag curve. The
question now arises whether we really have two different
relationships between e and p in the form e=F(p) and
e=f(a,p); the true answer can, of course, only be furnished
by the theory of the elements to be developed. But if we
recall certain problems of theoretical physics, for example, in
heat-conduction, sound, and light, we perceive at once that we
have there also at least two relations of a similar kind; the
one relationship appears as the integral of a differential
equation, the other as a so-called limiting condition inde-
pendent of the first ; the two contain an undetermined para-
meter, which often appears as a whole positive number (e. g.,
in the theory of the vibration of strings). Hence it appears
not improbable that the chemical theory of the future will
also lead to two relationships,
e=f(a,p), and e=(a,p),. . . . (2)
where a—the valency—a whole positive number plays the
part of an undetermined parameter. It is to be remarked
that so far we always find a<8; but in any case it is not
impossible that certain special conditions of the problem might
exclude certain values of a. If we now eliminate a from both
equations we obtain the relationship of the form
e=F(p),
that is our curve of fig. 1.
Now we are in position to take another step. If the rela-
tionships (2) are really independent, then e may be eliminated
from them ; this leads to the relationship of the form
Alp (G: Pp) 0s Beet ogee ee em
which will give for each a a completely determinate series of
values p, the values of p thus obtained represent, then, the
atomic weights of the actually existing elements. It may
well happen that the number of the real and positive roots of
equation (3) will be finite for each a, hence also the number
Geological Society. 283
of the actually existing elements will be finite and perfectly
determinate. The question is otten‘asked, ‘“‘ Why do there exist
only elements of a few definite atomic weights?” From our
point of view the question is similar to the one, “ Why can a
string give only definite notes ?”’
From the above considerations it appears to result clearly
that an analytical expression for e=F(p) is more complicated
than is the case for the simple cotangent: an expression of
the form
e=a Cos nh t"(A +cotan 7/ )
7
might suit better. But since the periods of the elements turn
out to be unequal (0 to ?, ? to 86, 36 to 84, 84 to 132,
132 to 168 ?, 168 to 216, 216 to 264 ?) the constants 4, ¢, d,/,
A would be themselves independent of p. In such cases
trigonometrical functions offer no special advantage. It
might perhaps be simpler to take
Aineaee alana
é= — aoe
Papi (Paap? Pa 3
- spl PoP Mee p—p) ++
(p—Dp;)( P —P2) (P—P3)---
where 1, P2, P3-.- denote the roots of the equation ==,
p, p’, p'”, ... the roots of the equation e=0, and Ay, Ag,
A;,... and B denote constants.
XXIX. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from p, 207.]
May 13th, 1896.—Dr. Henry Hicks, F.R.S., President,
in the Chair.
HE following communications were read :—
1. ‘An Account of a Head or Gateway driven into the Eastern
Boundary-fault of the South Staffordshire Coal Field.’ By William
Farnworth, Esq., F.G.S.
The author describes certain peculiarities observed during the
driving of a head towards the fault separating the Coal Measures
and Permian rocks, from a pit situated 4 miles east of Walsall, at
the southern extremity of the Cannock Chase Coal Field.
2. ‘On the Geographical Evolution of Jamaica. By J. W.
Spencer, M.A., Ph.D., F.G.S.
The object of the paper is to set forth the physical and geological
284 7 Geological Society :—
characteristics of Jamaica which bear upon the problem of its late
high elevation and former connexion with the continent, and to
trace across the neighbouring seas and islands to the mainland the
evidences of the former linking of Jamaica to North and South
America, The first part of the paper treats of the growth of the
island, and the following events are described. After the formation
of the mechanical sediments, limestones, and igneous rocks which
constitute the basement of the ‘ White Limestones,’ the latter group
was accumulated in later Eocene and early Miocene times to a
thickness of 2000 feet, and they indicate a subsidence of 3000 feet
below sea-level. Their formation was followed by a Pliocene or
Mio-Pliocene elevation in an epoch of long duration ; the uplift was
inferior to a later (post-Layton) one, but sufficient for the removal
of Miocene limestone below sea-level. The period was one of
general elevation, general dislocation of strata, and great erosion.
The formation of the Layton marly beds with loams and gravels
which succeeded this period is referred to the end of Pliocene times.
In early Pleistocene times the ‘ post-Layton’ elevation occurred,
causing an uplift of from 7000 to 11,000 feet above sea-level. The
strata were not greatly deformed, but the epoch was one of enormous
erosion. A subsidence somewhat resembling that of the Layton
formation followed this erosion in late Pleistocene times, and caused
the accumulation of the loams and gravels of the Liguanea forma-
tion. In modern times minor changes have occurred, causing the
formation of terraces, of channels over banks, of coralline limestone,
and of the modern coral terraces.
The second part of the paper treats of the continental connexions
of Jamaica. The author gives details of the submerged plateaux
and drowned valleys which are analogous to those still existing
above sea-level. They indicate that the former altitude of the
West Indian plateau, and some portions of the adjoining continent,
reached 25 miles: ~ but the floors of the Mexican Gulf and Honduras
aud the Caribbean Sea formed low plains draining into the
Pacific Ocean, for at that time the eastern region was high, while
the Mexican area was generally low.
There was a generally high elevation of the Antillean region during
the great Mio-Pliocene period, with probable continental connexion ;
at the close of the Pliocene period a general subsidence flooded the
coastal plains of the continent, and reduced the West Indies to
fewer and much smaller islands than those which now exist. But
the earlier portion of the Pleistocene period was that of the great
continental elevation, when the lately formed Pliocene topography
was deeply dissected by rain and rivers, yet there were apparently
several pauses of terrestrial movements at different altitudes, as
indicated by the various base-levels of erosion. At this time
Jamaica and other islands formed a mountainous tableland bordering
the Mexican and Caribbean plains. Afterwards the later Pliocene
continent was depressed, so as to flood most of the coastal plains of
the continent and reduce the islands to small proportions, and
Dundry Hill: its Upper Portion. 285
since then the minor oscillations have brought the old continent to
the present conditions. While the east was going down, the
Mexican region and western lands were being raised to form high
tablelands.
3. ‘Dundry Hill: its Upper Portion, or the Beds marked as
Inferior Oolite (G5) in the Maps of the Geological Survey.’. By
S. S. Buckman, Esq., F.G.S., and E. Wilson, Esq., F.G.S.
The authors give an account of previous geological work relating
to Dundry Hill, especially that which refers to the correlation of its
strata. Then they describe the different exposures on the Hill,
together with the results of various excavations carried out by
quarrymen under their superintendence for the purpose of the
present communication. Besides demonstrating the sequence of
the strata of Dundry Hill, the authors are able to show as special
results :—
The rapid easterly attenuation of the Freestone.
That there is a non-sequence in the Dundry deposits.
That the chief fossiliferous bed—the [ronshot Oolite—extends
over a very small area.
That the absence of this bed is due to removal by almost contem-
poraneous denudation.
That in the easternmost portion of the Hill this bed and all other
subjacent beds of what is called ‘Inferior Oolite’ have been
removed by this denudation, so that only a thin cap of what
would be called ‘ upper beds of Inferior Oolite’ rests on a thick
clay-bed of the age of the Midford Sands.
That deposits contemporaneous with what are called ‘ Upper
lias’ and ‘ Midford Sands’ in other places are found in some
thickness at Dundry Hill, attaining as much as 65 feet.
That the Lias Marlstone-rock is present at Dundry Hill and
crops out in many places on its flanks, but that this rock-bed is
also wanting from many parts of the Hill.
That the Geological Survey have presumably mistaken this
Marlstone-rock (which is an Ironshot stone) for the Ironshot
Oolite—the chief fossiliferous bed of the Dundry Inferior
Qolite, and formerly called Humphriescanum - zone — beds
nearly 100 feet apart.
That, as a consequence, the map of the Geological Survey shows
round the greater portion of the Hill the boundary- line of the
base of the Inferior Oolite drawn as much below the Marlstone
as would have been correct if this rock had actually been the
well-known Ironshot Oolite.
That, as a further consequence of this, the map of the Geological
Survey shows coloured as Inferior Oolite strata which are
mapped as Lower lias, Middle Lias, Upper Lias, and Midford
Sands in other localities; and that in places the limit for
Inferior Oolite, according to the Survey, is as much as 600 yards
beyond that of the authors.
286 Geological Society :-—
The authors append a map of the strata of Dundry Hill, coloured
on a paleontological basis; and they show how it may be compared
with the map of the Geological Survey, and with a map by Sanders.
May 27th.—Dr. Henry Hicks, F.R.S., President,
in the Chair.
The following communications were read :—
1. ‘On the Pliocene Deposits of Holland, and their Relation to
the English and Belgian Crags ; with a Suggestion for the Establish-
ment of a new Zone “ Amstelien,’ and some Remarks on the
Geographical Conditions of the Pliocene Epoch in Northern Europe.’
By F. W. Harmer, Esq., F.G.S.
The author draws attention to some papers by Dr. J. Lorié, of
Utrecht, describing the strata met with in some deep borings in
Holland, which show that the Newer Pliocene is in that country
nearly 500 feet thick, and that it has been depressed more than
1000 feet below its original position. He enquires whether this
subsidence can be connected with the elevation of the Older
Pliocene in Belgium and Kent, and how far these earth-movements
can be traced in East Anglia and influenced the deposition of the
English Crag.
He gives particulars of the alterations in level which have taken
place during and since the Crag period in England and on the
Continent, showing that the two movements of upheaval and
subsidence have much in common, and especially that they re-
gularly increase in degree to the north and south respectively.
He gives a map showing the extension of the Diestien deposits of
Belgium, and their probable connexion with the Lenham Beds, and
opposes the view of M. Dollfus that the Diestien sea was closed to
the south, though the connexion with it was probably cut off by the
elevation of the southern part of the area at the close of the Diestien
epoch, which also caused the Scaldisien sea to retreat to the north.
At the close of the Scaldisien period the sea retired from Belgium
altogether, no beds equivalent to the Upper Crag of England being
known in that country. A similar alteration of the margin of the
Crag sea can be traced in East Anglia.
He analyses the fauna of the Scaldisien and Poederlien, and shows
its close correspondence with that of the Walton bed and the
difference between it and the Upper Crag, which contains Arctic
shells.
He describes the beds met with in the Dutch borings, regarded by
Dr. Lorié as Diestien and Scaldisien, and their fauna, at some length.
He concludes that a large part of them are altogether newer than
the latter formation, and are equivalent to the Butley Crag, and
he proposes for them the term ‘ Amstelien.’
He doubts whether any deposits of similar age to the Norwich
Crag or Chillesford Beds have been met with in the sous-sol of
Holland, which he considers became at that time a land-area; and
The Pliocene Deposits of Holland. 287
he gives a section to show wherein his classification of the Dutch
strata differs from that of Dr. Lorie.
The distinction between the divisions adopted by the author
comes out more clearly from the consideration of the abundant and
characteristic species only, of each of which he gives lists.
Although the Amstelien beds are more than 400 feet in
thickness, they contain a shallow-water fauna, and were deposited
in a basin which subsided pari passu with their accumulation.
In the map an attempt is made to show the limits of the sea of
the Anglo-Dutch basin during the various stages of the Pliocene
epoch.
"Tt is suggested that the Chillesford Clay was deposited in an
estuary through which the Rhine discharged into the North Sea, its
presence in the western portion of the Pliocene basin being caused
by the elevation of Holland after the deposition of the Amsteline
and a subsidence in Suffolk, which carried the Chillesford Beds over
an area which was not covered by the Norwich Crag sea.
No equivalents of the Weybourn Crag or of the Cromer beds
(Forest Bed series) have been found in the Dutch borings. These
are to be referred to the Pliocene, as pointed out by Mr. Reid, but
possibly some of the unfossiliferous pebbly gravels of Norfolk and
Suffolk may be Pleistocene.
The Weybourn Crag marks a re-invasion of East Anglia by the
sea; but previously to the deposition of the Cromer beds the southern
margin of the Pliocene gulf had again retreated to the north, and an
estuary, similar to that of the Chillesford Clay but situated farther
east, received the waters of the Rhine, which brought down the
drifted remains ot mammalia and some southern mollusca. ©
The newest portion of the Cromer deposits is of an Arctic
character, and seems to show that no great interval separated the
Pliocene and the Pleistocene periods.
A second subsidence of the Dutch area took place in Pleistocene
times; the Glacial and post-Glacial beds being 600 feet thick under
Amsterdam. No Till or Contorted Drift similar to the deposits
occurring in Kast Anglia and in the district north-east of the Luyder
Zee has been met with in these borings. The glaciation of Holland
proceeded from the Baltic and not from Norway, and the Baltic ice
does not seem to have reached the Dutch coast; still less could it
have travelled thence in the direction of East Anglia.
The two prominent physical features of the Pliocene period were
the Rhine and the basin of the North Sea. The hypothesis of a
permanent basin with shifting shore-lines, in contiguity to which
the shallow-water deposits of the Upper Crag were deposited, seems
to agree with all the facts of the case, and to throw light on the
geographical conditions of the Pliocene epoch.
[ 288 |]
XXX. Intelligence and Miscellaneous Articles.
ON A DAMPING ACTION OF THE MAGNETIC FIELD ON ROTATING
INSULATORS. BY WILLIAM DUANE.
ig results from the experiments described by the author, that if
an insulator is made to rotate in a magnetic field about an axis
at right angles to the lines of force, a damping action is exerted
in opposition and nearly proportional to the angular velocity.
If the insulator is paramagnetic, such an action might be ex-
plained on the assumption that the magnetic axis of the insulator
does not coincide with the magnetizing force of the field, but is
displaced in the direction of the rotation. A somewhat more
general assumption sufficing for the explanation is, that for an
insulator at rest the induced magnetism does not vanish at once,
but after an appreciable time. If it vanishes ver y rapidly we get
the proportionality observed between the damping force and the
velocity.
For a diamagnetic insulator with true diamagnetic polarity the
corresponding assumption would give a force accelerating the rota-
tion. Nevertheless, according to several theories of diamagnetism,
even a diamagnetic body has paramagnetic polarity. According
to such theories the explanation given applies also to diamagnetic
bodies.—Wiedemann’s Annalen, No. 7, 1896.
THE ACTION OF MAGNETISM ON ELECTROMOTIVE FORCE.
BY ALFRED H. BUCHERER.
The results of this investigation may be stated thus :—
1. In neutral ferrous salts, if one of the two equal iron elec-
trodes is magnetized, there is no electromotive force produced
which amounts to 0:00001 volt. The currents observed by Gross
and others could not be referred to a change of the electrochemical
potential of the magnetized iron.
2. The forces produced by magnetizing a circuit which contains
magnetic substances as electrodes are to be ascribed to variations
of concentration which the magnetized electrode produces when
dissolved.
_ 8. In the case of ferric salts, the direction of the currents pro-
duced by magnetization depends almost exclusively on their degree
of concentration at the two electrodes.
4. If only ferrous salts are present, the direction of the currents
produced by magnetization depends on the total concentration of
the iron salts.
5. The currents designated by Rowland as primary ones are
agitation-currents.— Wiedemann’s Annalen, No. 7, 1896.
THE
LONDON, EDINBURGH, ann DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF ee aren
[FIFTH SERIES.]
OCTOBER 1896.
XXXI1. Osmotic Pressure. By J. H. Poyntine, Se.D., FP LRS.,
Professor of Physics, Mason College, Birmingham*.
INCE the osmotic pressure of a solution is of the same
KD order as the “gas pressure”’ of the dissolved substance
at the same density, we are naturally tempted to think of it
as an extra pressure produced by the motion of the dissolved
molecules. But if we start from this supposition we soon find .
eurselves surrounded by the difficulties of the dissociation
hypothesis. These are so great that it appears worth while
to examine our ideas of liquid structure in the hope that they
will suggest to us some hypothesis which will free us froin
the necessity of assuming dissociation.
I shall try to show in this paper that osmotic pressure may
be accounted for as an indirect result arising, not from disso-
ciation but from its very opposite, the greater complexity of
the molecules in the solution, due to some kind of combination
between salt and solvent.
The facts of liquid viscosity, diffusion, and surface conver-
sion to vapour may apparently be represented by imagining
a liquid to be, in the main, a solid structure, inasmuch as the
molecules cohere and resist strain of any kind. But the
molecules have so much energy, potential or kinetic or both,
that they are not very far from instability. In a mass of
connected molecules irregularly distributed and irregularly
vibrating, concentrations of energy must occur, and “at the
* Communicated by the Author.
Phil. Mag. 8. 5. Vol. 42. No. 257. Oct. 1896. ¥
290 Prof. J. H. Poynting on Osmotic Pressure.
points of concentration individual molecules may receive so
much energy that they are able to do the work needed to free
them from their immediate surroundings. Such molecules
will travel off, and as they lose their energy will form new
connexions with new surroundings. ‘Thus the solid structure
is continually breaking down and renewing itself. If we
impose a shear strain on the structure, the strain will of
course disappear with the structure in which it is produced.
But the breaking down will always lag slightly behind the
imposition of the shear, and the still surviving shear strain
will be accompanied by a resistance the same in kind as the
resistance to shear in a solid, though in a liquid it is only
recognized as viscosity. This is the view first set forth by
Poisson and developed by Maxwell, and it is to be noted that
it gives an explanation of liquid viscosity entirely different
from the diffusion explanation which so satisfactorily accounts
for gaseous viscosity.
We may obtain an expression for the coefficient of viscosity
by the following method, which is perhaps rather simpler
than that of Maxwell. We must assume that a certain frac-
tion, say A, of the molecules of the liquid get free per second,
and that this fraction remains practically the same when the
liquid is sheared. Hence if s is the strain still existing at any
instant, it is breaking down at the rate As per second. If the
liquid is moving steadily in parallel planes perpendicular to
an axis along which w is measured, and if the velocity is v at
a distance xz from the reference plane, = is the rate at which
shear is being imposed on the liquid. But since the steady
state is reached the rate of imposition equals the rate of decay,
or
dv
Fr se rt fee eee ey (1)
If n is the coefticient of rigidity of the structure, the stress
due to s is ns, and by our supposition this is the viscous
stress, or
— =n, ee Se. oer
where 7 is the coefficient of viscosity. Dividing (2) by (1)
we obtain
fe a) ne
We may compare the liquid breakdown here imagined with
Prof. J. H. Poynting on Osmotic Pressure. 20%
that which must occur in an electrolytic conductor. If D is
the “ displacement” or “induction” in an electrolyte, and if
w is the factor of decay per second, wD is the quantity dis-
appearing per second and dissipating its energy as heat. This
may be equated in the steady state to the new “ displace-
ment” or “induction” introduced per second per square
centimetre, or to the current-density C. Hence
KE
ee (1)
where E is the slope of potential, and K is the specific induc-
tive capacity. But Ohm’s law gives us
4 C= ie ° ° . ° ° ° ° ° (5)
where p is the specific resistance ; whence
At }
Creer ee oh tikog i ( 0)
Returning to equation (3), we see that if n is constant, 7
varies inversely as’. For instance, when the temperature
rises the molecules have more energy, the breaking down of
structure is more frequent, and X is greater. Probably 7 is
not very much altered, though it doubtless tends to decrease.
Hence 7 should decrease, and this is in accordance with
observation. On the other hand, when a salt is dissolved in
a liquid, if, as we are going to suppose, it makes the mole-
cules on the average less energetic by partially combining
the more energetic solvent molecules with the less energetic
salt molecules, they are on the average rather further from
instability, X is less and 7 is greater. This again agrees with
observation.
At the same time the specific electric resistance p is dimi-
nished. This would require that in (6) either w or K, or
both, should be increased, probably both ; and this brings out
a point which must be noted, that the factor of decay X in (3)
is not likely to be the same as yw in (6); for while one relates
rather to the molecules and their relative positions, the other
most probably relates to the atoms and their positions in the
molecules.
Maxwell (Proc. Roy. Soc. exlvili. 1873) gave an account
of some experiments which he made to test this view of liquid
viscosity by shearing a liquid and looking out for double
refraction. He could only observe it in the case of Canada
balsam, in which it had already been found by Mach, and
here the “rate of relaxation”? was so great that he could not
Y 2
C=~D=
292 Prof. J. H. Poynting on Osmotic Pressure.
observe any double refraction after the shearing motion ceased.
Kundt (Ann. Phys. Chem. xiii. 1881) made a series of expe-
riments and found double refraction in many sheared liquids,
notably in olive-oil, but never in a pure liquid with a definite
chemical constitution. The more complex the molecules
apparently the less is A, and the greater is the shear strain
still remaining at any instant of the motion. But in liquids
such as water or glycerine, the decay is so rapid that no
optically appreciable amount remains.
Still it is very possible that olive-oil is only an extreme
case, and that water and other apparently inactive liquids
would show the effect if we could sufficiently increase the
shear, and I think Kundt’s results may be claimed as sup-
porting the hypothesis. Possibly, too, the observation of
Quincke, that double refraction is observed in a liquid close
to a very hot wire, gives further support. The unequal
heating may perhaps be regarded as producing shear strains
in the solid structure which are renewed by the supply of
heat as fast as they break down.
In the case of breaking down of structure near the surface
of a liquid the moving molecules may succeed in escaping
altogether, and may fly off as gas molecules if they are
directed upwards and have enough energy. Of course there
may be many molecules able to move about and yet not able
to evaporate ; for though they may be able to travel when in
the body of the liquid, they may not have energy enough to
get clear away from their neighbours when these are all on
one side and all pull in one direction as they do at the surface.
In the case of practically non-evaporating liquids, such as
mercury at ordinary temperatures, we must suppose that only
a very minute fraction are thus able to do the work needed
to overcome the large cohesion of their neighbours.
It will be convenient to use the term ‘ mobility’ to de-
scribe the number of “free” or ‘‘ mobilised’? molecules
crossing a square centimetre per second in a liquid, where by
“ free” or ‘mobilised’ we mean those which are changing
their surroundings and forming new connexions. Hvidently
we may extend the term to a gas, remembering that then all
the molecules are mobilised, and that the mobility is propor-
tional to the pressure.
When a square centimetre is taken on the surface of a
liquid, the mobility upwards is the rate of evaporation, and
the mobility of the vapour downwards is the rate of conden-
sation. When the two mobilities are equal the pressure of
the vapour is the vapour-tension.
The mobility in the body of the liquid is probably far
Prof. J. H. Poynting on Osmotic Pressure. 293
greater than that at the surface for the reason already given;
viz. that in the one case the neighbouring molecules entirely
surround one which tends to get free, while at the surface they
are all on one side and so tend to pull back and retain a mo-
lecule which may be inclined to move away. If, however,
the internal mobility at a given temperature is altered, say
by the pressure, or by the presence of some substance in
solution, the surface mobility will be altered too. We shall
assume that it is altered in the same ratio as the internal
mobility, an assumption which appears to be justified by the
account which it will enable us to give of the effect of pres-
sure and of solution.
Let us now apply this idea to the familiar case of rise in a
capillary tube standing in a liquid having only its own vapour
above it. Or let us take the more general case of a liquid in
a vessel with tubes which are wet rising above the flat sur-
face, and with tubes which are not wet coming out of the side
Fig. 1.
and turning upwards, and of such diameters that the liquid
does not rise to the top of the tube, as in fig. 1. Thomson’s
294 Prof. J. H. Poynting on Osmotic Pressure.
theorem shows that there is ultimately a balance between
evaporation and condensation at each surface, or that the
vapour-tension is less at the surfaces a and b than the normal
amount existing at c, while at d, e, and f it is greater. In
other words, the surface mobility gradually increases as we
o downwards. This is usually connected with the curvature
of the liquid surface, but, as I have tried to show in a former
paper (Proc. Phys. Soc. vol. iv. p. 271, Phil. Mag. July 1881),
it should rather be connected with the increased pressure of
the liquid just under the surface as we descend ; the curva-
ture of the surface is a non-essential accompaniment.
Taking the pressure of the vapour at the flat surface ¢ in
fig. 1 as w, and the densities of liquid and vapour as p and a
respectively, then at a level h below or above ¢ the hydro-
static pressure is greater or less than at ¢ by gph, =P say,
while the vapour-pressure is greater or less than at ¢ by
oO e e
goh=— ; or the increase in vapour-pressure at a surface as
we descend is proportional to the increase in hydrostatic pres-
sure just under that surface. This is accounted for if we
suppose that the increased hydrostatic pressure results in
increased mobility, and therefore increased evaporation from
the surface. The vapour-pressure increases from @ to
ow + —2 = a(1+ =) :
or the coefficient of increase of its mobility is — per unit of
hydrostatic pressure, and this is the coefficient we must
assume for the increase of internal liquid mobility to account
for the facts on this theory.
We have no direct evidence that increase of pressure does
thus increase liquid mobility. The justification is to be
sought in such explanations of known facts as that just
given*.
It is perhaps worth noting that we obtain the true state of
affairs externally if we picture the liquid in fig. 1 as a kind
* Liquid viscosity should decrease if mobility increases, and should
therefore, in our view, decrease with increase of pressure were mobility
alone concerned. But rigidity also comes in, and we must ascribe to this
complication the result that, in water, pressure lessens the viscosity
while in turpentine it increases it (Cohen, Wied. Ann. No. 4, 1892).
But it would appear fair to seek support for the supposition of increased
mobility in the “flowing” of solids under great stresses, as in the stamp-
ing and wiredrawing of metal, when the molecules undoubtedly change
their positions with very greatly increased rapidity when under great
strain. on
Prof. J. H. Poynting on Osmotic Pressure. 295
of open framework, or as a spongy structure through which
the molecules of vapour pass freely so that they are at the
same pressure within and without the liquid at the same level.
But this conception must be used only to give us the net
result, and not as representing the actual condition.
If, in addition to the vapour, any soluble gas is present in
the vessel, it too will exist both inside and out in quantities
increasing as we descend, and it must be in equilibrium at all
levels. So that if near the flat surface the density of the gas
in solution is x times the density at the same level outside,
the same ratio will hold at all depths. Again the net
external result is the same as if we picture to ourselves a
spongy structure through which the gas passes freely.
As a further illustration of the change of mobility with
pressure, we may take the alteration of the melting-point
which I have discussed in the paper mentioned above. Thus,
in the case of water, water and ice are in equilibrium under
1 atmo. at 0°, and therefore have equal vapour-tensions and
equal surface mobilities. If, however, we put on pressure,
the coefficients of increase of mobility are, as we have just
Oo Oo Oyo
seen, — and ——, where p and p’ are the densities of water
@ @®
and ice, and o and @ the density and pressure of the vapour
respectively. Since p is greater than p’ the water mobility
is increased less than the ice mobility, and so at the surface
of contact the ice sends more molecules to the water than it
receives in return, that is to say, it melts. Below 0° the
vapour-pressures and mobilities at atmospheric pressure are
different, the mobility of water being greater than that of ice.
But if we put on sufficient pressure we may once more equa-
lize the mobilities and so lower the melting-point to the new
temperature. Thus if a and aw’ are the vapour-pressures of
water and ice at —d@, and P is the pressure making the
mobilities equal, or the pressure reducing the melting-point
to —dé,
o(1+ ~2)=0/(1 + =)
ap ‘ Pp
fare
o—o'=Po(5—>) oe s AM ae on area 6
cea (7)
a formula equivalent to that of Kirchhoff deduced by purely
thermodynamic considerations. For using the ordinary for-
mula for lowering of melting-point,
1}! ] dé
ie ~—=)= a e . ° . ° a
te p L @? (7a)
296 Prof. J. H. Poynting on Osmotic Pressure.
we obtain Kirchhoff’s result,
olucdd
a era
Now let us consider the case of a dilute solution of a non-
evaporating salt. We know by direct observation that the
vapour-tension is reduced by the presence of the salt, and we
must suppose, on the hypothesis here advocated, that this
reduction is due to a decrease in the mobility of the liquid.
Let us follow out this idea by imagining that we have in the
same chamber maintained at a constant temperature two deep
vessels, one containing the pure solvent the other a dilute
solution. In this chamber we shall suppose that above the
liquids there is only the vapour of the solvent. To begin
with, we may suppose that each vessel is half full and at the
same level. Then the pure solvent will distill over into the
solution, and will continue to do so until the difference in
level in the two vessels is such that each surface is in equili-
brium with the vapour at its level. The hydrostatic pressure
in the solution at the level of the surface of the pure solvent
will then be the osmotic pressure. If we imagine a number
of non-wettable tubes inserted, as in fig. 1, in the sides of the
two vessels at various depths and turned upwards, the diame-
ters being so adjusted that the liquid does not flow out of any
of them, then in any pair at the same level we realize Fitz-
gerald’s semi-permeable membrane; and at each level the
two liquids must have equal vapour-tensions, which implies
that their mobilities are equal at each level. This also comes
out from our equations. Let a, a’ be the flat surface vapour-
tensions of solvent and solution, p the density of the liquid—
practically the same for each—and H the final difference in
level between the two surfaces, so that the osmotic pressure
P=gpH. If M, M’ be the mobilities at the surface-levels,
Mw
WwW ao”
Now as we descend in the solution the mobility increases, and
the rate of increase is <- per unit pressure. For depth H
this increase is gp a = a or the mobility
P
M’=M'(1+9%5 == M (149 =a
C ~ roy
Vice ee
Bat a=o'+gqHo,
Prof. J. H. Poynting on Osmotic Pressure. 297
whence M’=M, or at the level of the solvent surface the
mobilities are equal. This equality will be maintained if we
descend equal distances in the two liquids below that level.
So that if we now connect the two vessels at any level by a
horizontal tube with a semipermeable membrane in it, the
solvent mobilities on the two sides of the membrane are equal,
and therefore the solvent diffuses through at equal rates in
the two directions.
' We may then explain in the following general terms the
rise which occurs when we place a semipermeabie vessel con-
taining a solution into a solvent. The solvent molecules are
entering the membrane on both sides, but the mobility or
number set free per second from the pure solvent is greater
than the number set free from the solution. The membrane
goes on absorbing the solvent from each side till it becomes
saturated, 2. e¢. holds so much that it returns as many mole-
cules as it receives. It is receiving more from the pure
solvent side, and therefore when saturated for that side it is
supersaturated for the other. Consequently more molecules
are sent into the solution than are received from it, and the
solution grows until the growing pressure so much increases
the mobility that it is equal on both sides of the membrane.
If the solution and solvent are in two vessels separated by
an indefinitely produced vertical and semipermeable membrane,
it is evident that ultimately the two will be in equilibrium at
every level, whether in liquid or vapour.
We may apply the same idea to the change of melting-
point in a solution. In the soiution the solid mobility is
unchanged, but that of the solution is lowered by the fraction
sg
— where P is the osmotic pressure ; and to find the new
melting-point, we must find the temperature d@ below the
normal melting-point at which this is equal to the difference
between the liquid and solid mobilities.
Taking pressures to represent mobilities,
GD ore
But pe. ad?
(OQ) — (ny == 6 )
whence we obtain the ordinary result
Pé
Sat eat MAGE Ios (8)
Comparing the above result with the lowering due to pressure
298 Prof. J. H. Poynting on Osmotic Pressure.
(7 and 7a), we see that the lowering due to a given osmotic
pressure in the solution is greater than that due to an equal
pressure on the pure solvent in the ratio
ie 1
Tah os ey ae.
pp p.
or
v:v—v.
In the case of ice and water the ratio is 1 : 1:092—1
aa fe SES 2
—— om ieee |
It now remains to see if we can give any reasonable
account of the decrease in mobility in a liquid when a salt is
present in solution. If the molecules of salt were simply
mixed with those of the solvent, or if they combined to form
stable non-evaporating compounds with the solvent, which
compounds were simply mixed, then the mixture should have
the same vapour-tension as the pure solvent. For we might
regard the salt or compound molecules at the surface as equally
reducing the effective evaporating and the effective condensing
area, somewhat as a perforated plate or gauze laid on the sur-
face would do. But the salt probably combines with the
solvent to form unstable molecules which continually inter-
change constituents, so that when near the surface they may
serve equally with those of the pure solvent to entangle the
molecules of vapour coming downwards, these descending
vapour molecules. taking the place of molecules attached to
the salt. Probably, however, they are less energetic than the
pure solvent molecules and "do not contribute so much to
evaporation. We shall make the supposition that they do not
contribute at all.
Let then each of the salt molecules combine on the average
with a of the solvent molecules, and in such a way that it
prevents those a molecules from evaporating while the com-
pound molecules formed will entangle returning molecules,
each of the a being replaceable by a vapour molecule. Then
we may regard the solution as solvent, having a number of
molecules simply mixed up and inactive as regards evapora-
tion but active in effecting condensation.
If N is the number of gramme molecules of solvent per
litre, and n the number of gramme molecules of salt added,
the number of solvent molecules left is N—an. Were the n
compound molecules quite active both as regards evaporation
and condensation the mobilities outwards and inwards would
Prof. J. H. Poynting on Osmotic Pressure. 299
be altered in the same ratio and the vapour-tension would
be unaltered. But we are supposing that they are inactive
for evaporation only and that their a molecules of combined
solvent are still active for condensation. So that in the
solution there are only N—an active for evaporation, while
there are still N active for condensation. Hence the vapour-
—an
N
the solvent and solution vapour-tensions,
Or if wand wo’ are
tension is reduced in the ratio N
@oNS an
ao . N
and
If each salt molecule takes one solvent molecule, so that
a==1, we have
Z—al on
coweee INI
which is the usual result deduced for dilute solutions from
the van’t Hoff value of the osmotic pressure. We may, of
course, work backwards from this result, and the work may
be put in the following form :—
If P is the pressure in the solution necessary to restore its
mobility te that of the solvent, z.e. to increase it in the ratio
ase,
a(1 4+ <7 law,
@p
or
aw — a’ op Mage
Bo eer N
and
__ nap
Ne
If M is the molecular weight of the solvent
Fa 7d +at)
oO omy
ee os
ale (1+ at)
300 Mr. Frederick Bedell on
where i is the value for hydrogen at 0°, and this is
o
0/7 H
000896 Where A= i. atmo.
Also NM=1000p.
Substituting these values we obtain
eal? 2 Ap(1+at)
~ N° M‘0-0000896
QnA
0-0896
= 22-3nA(1+ at).
if a has any other value than 1 we must put
P=22:3anA(1+ at),
whence we see that if each salt molecule combines with two
or with three solvent molecules the osmotic pressure is double
or treble the normal value.
The supposition here made is no doubt crude in its
simplicity, but my attempts to introduce other considerations,
such as change in density in the solution, have led to such
complicated results that much more extravagant suppositions
had to be made to reconcile these results with experiments. I
therefore leave the hypothesis in this crude form, in which it
will at least serve to show that it is not necessary to ascribe
osmotic pressure to dissociation but rather to association or
some kind of combination of salt and solvent.
P
1+at)
XXXII. Admittance and Impedance Lect.
By FREDERICK BEDELL*.
fe quantities chiefly considered in the discussion of
alternating currents are electromotive forces and cur-
rents, the values of these being determined for different con-
ditions. Hlectromotive forces and currents are commonly
represented by vector diagrams; and the change in these
diagrams, as some one quantity is varied, is shown by the
loci of the vectors which are altered thereby. What may be
termed electromotive force and current loci are thus deter-
mined. The numerical values for which these are constructed
necessarily depend upon some condition involving an assumed
_ * Communicated by the Physical Society: read June 26, 1896.
Admittance and Impedance Loci. d01
value either of the current or of the electromotive force: thus
we may assume a certain current to be constant (as the pri-
mary current of a transformer), and construct an electromotive-
force diagram with loci showing the changes in the various
electromotive forces as some part of the circuit is changed ;
or we may assume the impressed electromotive force constant,
and ascertain current loci for the same variation. Let us
limit ourselves to the transformer. In the first case above it
will be found that the constant assumed primary current J, is
a factor in the value of every line representing the compo-
nents of the primary electromotive force H,. By factoring
out I, we have an impedance diagram similar to the electro-
molive-force diagram, and without any assumption as to the
value of the current or electromotive force. Similarly each
line in a current diagram, constructed for a constant impressed
electromotive force H,, represents a current which is a mul-
tiple of an admittance (the reciprocal of an impedance) and
the factor H,. By factoring out EH, an admittance diagram
is consequently obtained, similar to the current diagram but
with no assumption as to the current or electromotive force.
Admittance and impedance diagrams accordingly correspond
to current and electromotive-force diagrams respectively,
differing from them only by a factor.
Impedance and admittance loci, or electromotive force and
current loci, for the primary of a transformer will in general
be ares of circles for changes in any one of the constants of
the primary or the secondary circuit.
Some interesting relations avise from the reciprocal nature
of admitiance and impedance. Let us note the following
relations between reciprocal vectors :—
If any vector has an are of a circle for its locus, a vector
proportional to its reciprocal will have an arc of a circle for
a locus. In fig. 1 let p, be any vector from the origin O,
having its locus as shown upon the are of a circle. The
vector p2, drawn in the direction of py and proportional to its
reciprocal, will have its locus upon an arc of a circle, which
may be shown as follows. Let p; and p;/ represent the vector
in any two positions, OA and OA’. ‘The intercepts Oa and
Oa’ will represent the reciprocal vectors p, and p,’; for in the
similar triangles OA’a and OAd’,
Pics Px Pa. > Pos
Hence
Pt P2 =PiP2= 2 constant.
The value of this constant product of p, and pz is OG".
302 Mr. Frederick Bedell on ~
Fig. 1.— Reciprocal Vectors, p, and py.
A
Fig. 2.—Reciprocal Vectors.
By a suitable selection of scale, the loci of p, and its reci-
procal pp may be represented as arcs of the same circle, as in
Admittance and Impedance Loci. 303
1; or they may be represented by different circles, as in
2. In the latter case, Ses
Eh Eh
dg dS
P1 P2=OG,. OG, = a constant.
As the origin O approaches the circle which represents the
locus of p,, the centre of the reciprocal circle becomes more
distant and its radius becomes greater. When the origin O
is a point in the circumference of the first circle, the centre
of the reciprocal circle is at an infinite distance ; that is, the
reciprocal locus is a straight line.
Let us apply these principles to the transformer diagrams.
The locus of the primary impedance, as some particular: quan-
tity is varied, is a portion of a circle. For example, this may
be shown to be the case when the secondary resistance is
varied. Since the admittance of the primary is the reciprocal
of its impedance, the admittance may be represented by the
vector p in the above construction, if the impedance is repre-
sented by p,. These loci may be drawn to scale for actual
values. In a constant current transformer the primary elec-
tromotive force varies directly as the primary impedance.
In a constant potential transformer the primary current varies
directly as the primary admittance. But the admittance is
the reciprocal of the impedance ; hence if we have an arc of a
circle for the locus of the primary electromotive force when
the primary current is maintained constant, we may employ
the above method to obtain the are of a circle which will be
the locus for the primary current when the transformer is
supplied with a constant electromotive force. The converse
operation may likewise be performed.
In fig. 3 let the circle C, represent the locus of the primary
electromotive force Ii, during some particular change of con-
dition, the primary curr ent. meanwhile being maintained
constant and in the direction OA. The difference in phase
between the current and electromotive force is the angle ¢.
The locus of the primary current under the same change of
conditions, if the primary electromotive force is maintained
constant, is the dotted circle C,, which is reciprocal to C,.
If the constant electromotive force is drawn in the direction
OA, the locus of the primary current is the circle C,’, drawn
so that the angles AOC, and AOC,’ are equal.
An application of the method of reciprocal vectors is shown
in fig. 4. Positive rotation is counter-clockwise. The semi-
circle JKN represents the locus of the primary electromotive
force of a transformer, when the primary current is constant
304 Mr. Frederick Bedell on
Fig. 3.
Fig. 4.—Method of obtaining Primary Current Locus by the
Principle of Reciprocal Vectors.
e
—
See
pe Oo
-_
afliz \ N H a
oren\cinculT ty =
\ XK _1 OPEN CIRCUIT
-\ 3 if
\ : / O 0
\ ye
Ne y)
XN 2
Ne ae
Bisa ~~ — eal
and is in the direction of OA, and the secondary resistance is
varied.
Admittance and Impedance Loct. 305
The eiectromotive force has the position OJ on open circuit
and ON on short circuit. OH is the power electromotive
force on open circuit, and includes the effects of primary
resistance and the losses due to hysteresis and eddy currents ;
HJ is the electromotive force to overcome the primary self-
induction. These are in the direction of the primary current,
and at right angles to it, respectively. A line from J to K,
at right angles to the secondary current, would show the
reaction of the secondary upon the primary. It is to be
noted that the line NH represents the effects due to magnetic
leakagé. It is desired to find the locus for the primary cur-
rent when the primary electromotive force has a constant
value, and is drawn in the direction OA. The angle of lag 4,
between the primary electromotive force and cur rent on open
circuit, is JOH. Accordingly, with a constant electromotive
force in the direction OA, the open circuit current I) is laid
off lagging behind the electr omotive force at an angle of
AQO7’=0,=JOH. The open cireuit current I) miy be laid
off to any convenient scale. To construct the locus for the
primary current proceed as follows:—Lay off the line OC,
so that the angles AOC, and AOC, are equal. The point C,
is located so that OC, : OC, :: Oj’: O7. The primary current:
locus is then drawn as the arc of a circle with C, as a centre,
passing through 7’.
The limits of the primary electromotive force locus are the
points J and N. The corresponding limits of the primary
current locus are the points’ and nn’. It will be noted that
these points correspond to the points 7 and n on the circle Cj,
which are reciprocal to the points J and N.
In the absence of magnetic leakage the points N and H
coincide. The point n’ would then lie in the line OA. The
deviation of the primary current locus from the line OA is
produced by magnetic leakage.
An experimental curve showing the primary current locus
for a constant potential transformer, as affected by magnetic
leakage, is shown in fig. 5.
The reciprocal relation between admittance and impedance
vectors gives a simple method for determining the conditions
for consonance and resonance in transformer circuits*, |
Figure 6 is given as a par ticular instance in illustration of
the statement given above that loci produced by the variation
of any one constant are usually ares of circles. The primary
loci are always ares of circies. The diagram shows the
chauges due to a variation in the secondary self induction.
* “ Resonance in Transformer Circuits,” by Bedell and Crehore, ‘The
Physical Review,’ vol. ii. p. 442.
Ponti. oO. Volo 42. No: 257. Oct. 1896. Z
306 Mr. Frederick Bedell on
Fig. 5.—Primary Current Locus for Constant Potential Transformer ;
Determined Experimentally.
OPEN CIRCUIT
O
Fig. 6.—Effect of the Variation of the External Secondary
Self-Induction in a Constant Current Transformer.
Fig. 7 shows the effect of magnetic leakage. The curves
shown are loci for primary electromotive force, when the
primary current is 1,. The primary electromotive force is
composed of the components OH, to overcome ohmic resist-
ance and supply open-circuit losses, HJ to overcome self-
induction, and (with absence of magnetic leakage) JK) to
overcome the back electromotive force of mutual induction.
Admittance und Impedance Loci. 307
The semicircle J KH is the locus for the primary electro-
motive force in the absence of magnetic leakage. The semi-
eircle JK’N is the locus for the primary electromotive force
Fig. 7.—Effect of Magnetic Leakage.
-
OPEN CIRCUIT
Q
when the coefficient of magnetic leakage € is constant from
open circuit to short circuit. In this case we have the
relations
Hes dN et iE
Wi Se wie
ee ea Ne
NE Tie, NOE
JK’ JN
Srl re NP gait N/a
In an actual transformer the magnetic leakage is not con-
stant, but varies with the load. The locus represented by the
dotted curve J K’’N is for such a case in which the magnetic
leakage is zero on open circuit and increases to the maximum
at short circuit. Where the magnetic leakage is variable, it
is determined for any point as K” by the ratio of JK” to J Ko.
TK”
IR
Thus let us suppose that the back electromotive force JK”,
actually induced in the primary by the secondary current I,,
is eighty-one volts ; and that JK, which would be the back
electromotive force in the absence of magnetic leakage, is
100 volts. We then have the equation
81 9
1—t=/ Fa = ip 09
L2
ee)
308 MM. Oumoff and Samoiloff on Electric
which indicates that the mutual induction is nine-tenths of
the value it would have in the case of no magnetic leakage.
The coefficient of magnetic leakage is accordingly found to be
10 per cent. ; thus :—
6¢=1-—0°9=0°10.
Fig. 7 is drawn to represent the values of the various elec-
tromotive forces in the primary circuit of a transformer, for
a given value of the primary current and for different values
of the secondary resistance. If the magnitude of each line is
divided by the primary current I,, fig. 7 represents the
values for the primary impedance (without any assumption
as to constant current or electromotive force) for different
values of the secondary resistance. The effect of magnetic
leakage upon the primary impedance of any transformer is
thus shown for different values of the secondary resistance.
The above construction affords a simple method of studying
the conditions for a decrease of the primary impedance of a
transformer when the secondary circuit is closed.
Curves corresponding to the dotted curve in fig. 7 have
been experimentally determined t by the writer.
Cornell University, May 1896.
XXXIV. Electric Images in the Field of a Hittorf’s ( Crookes’)
Tube. By N. Oumorsr and A. SaMoitorrt.
HE influence exercised by a Hittorf’s tube on electrified
bodies shows that the electric field created in the interior
of the tube extends also to the exterior. Lvidently the
objects brought into this field, whether to explore its elec-
trical properties or, as in the experiments of Lenard and
Rontgen, to produce or to receive shadows acquire a new
electrical condition and by their presence modify the primi-_
tive field. It is impossible to estimate @ priori this modifi-
cation and the part it plays in the production of the pheno-
mena observed. In investigating these questions we finally
adopted an experimental method intended to furnish us with
the means of forming a general idea of the electrical pro-
perties of the field of a Hittorf’s tube and the modifications
which it undergoes.
* Discussed at length before the Physical Society by Mr. E. C.
Rimington in his paper “On Air-core Transformers,” Philosophical
Magazine, vol. xxxvii. p. 394.
t ‘Proceedings of the International Electrical Congress,’ Chicago.
1893, p. 234.
{ Communicated by Lord Kelvin, F.R.S.
Images in the Field of a Mittorf’s (Crookes’) Tube. 309
With this end in view we replace the photographic plate
or the fluorescent screen by a plate of ebonite. The electric
field is maintained for two or three minutes, after which the
action of the tube is arrested and the ebonite plate withdrawn
from its position. By a quick movement the objects on the
plate are thrown to the ground and we proceed to develop the
images by sprinkling over the plate a mixture of sulphur and
minium. As is well known, the sulphur adheres to those
portions which are positively, the minium to those which are
negatively electrified. Thus the colour of the spots shows
the electric condition of the shadows, and the comparison of
their configuration with that of the objects indicates the
modifications introduced in the mode of action of the field.
The tube used in our experiments had the shape of a pear
with flattened base (fig. 1) ; & is the kathode, a the anode.
Fig. 1.
&
WME, 3
In the central part of the base there was a fluorescent spot
about a centimetre in diameter ; and at some distance there
was a less intense fluorescent zone concentric with the spot.
At a few centimetres below the tube is placed a plate or
sereen of ebonite 6, which rests ordinarily on the rim of a
glass vessel 13 cm. in diameter and 21 cm. in depth. The
manner of supporting the plate is quite immaterial provided
that the support is clear of the central portion of the plate.
Thus the screen, or at any rate its central portion, is sur-
rounded by air on both sides. The objects are placed either
above or below the plate ; in the latter case they are held
by projecting arms which are bent over the rim of the vessel.
The-images were developed on both faces of the screen ; we
have also employed two screens placed one on the other and
developed the images on the four faces of the screens.
310 MM. Oumoft and Samuiloft on Electric
We proceed to the description of the experiments :-—
(1) No object being interposed between the tube and the
screen we obtain on the two faces of the latter an intensely
red spot corresponding to the fluorescent spot in the tube ;
the remainder of the plate acquires a reddish tint. The same
effect is obtained on the four faces of two ebonite plates placed
one on the other and making good contact. Thus the fiuo-
rescent spot of the Hittorf’s tube sets up a negative electrifi-
cation on the faces of dielectrics in its neighbourhood, whether
these faces are turned towards the tube or away from it. On
the portions of juxtaposed plates which are not in intimate
contact we find yellow spots and red ones opposite to each
other. The spot formed on the front face of the plate is
sharply defined ; for this reason we must conclude that the
action of the tube is propagated by trajectories emanating
from its surface.
(2) On covering the ebonite plate by a sheet of zine the
plate appears red on both sides. On putting the same sheet
below the plate and in contact with it the face of the plate
turned towards the zine acquires a yellow hue, the opposite
face an indefinite tint. We may say in short, that every body
introduced into the field, if in perfect contact with a dielec-
tric, does not change the negative electrification of the latter
produced by the field, provided the body is between the
tube and the dielectric; when the dielectric is between the
tube and the body, the negative action of the field is replaced
by a positive action... This conclusion is confirmed by the
following experiments :—
(3) Figures cut out of metal (lead, zinc, aluminium), glass,
and paper in perfect contact with the screen give, when they
are placed above it, red images, and when they are beneath,
yellow images. The images are bordered by a neutral band;
the rest of the screen is red.
On placing a glass plate beneath the plate of ebonite on
which the cuttings are arranged the red ground which sur-
rounds the images of the objects changes to an intense yellow
field ; as to the images, their colour appears to tend to black.
Occasionally yellow tufts are seen which come from points
corresponding to projecting points of the objects. The rim
of the glass vessel which supports the ebonite plate always
gives a circle of an intense yellow except at the points which
are not in contact with the ebonite ; to these points corre-
spond red arcs.
(4) A rectangle cut from a sheet of lead and having a
rectangular opening in the middle was placed above the plate
and in contact with it. The image of the metallic portion is
Images in the Field of a Hittorf’s (Crookes’) Tube. 311
red, as usual, the image of the opening is black. On raising
this object to a height of one centimetre above the screen by
a inetal stalk cemented by wax toa glass disk (fig. 2), we
obtain the image fig. 3. The white parts are the neutral
Fig. 2. Fig. 3.
i
portions ; they are black in the image and correspond to the
solid parts of the object which were not in contact with the
screen. The image of the disk, which was in contact with
the screen, is red, which is indicated in the figure by parallel
lines. The image of the opening is markedly larger than the
opening itself and of an intense yellow colour. The rest of
the plate is also yellow; this colour is represented in the
figure by a network of lines crossing one another at right
angles. The rectilinear margins of the object show them-
selves in the image as curved lines: in the image of the
exterior rectangle these curves turn their convexity inwards ;
while in the image of the interior rectangle the curves de-
scribed have their convexity directed outwards. The fact
that the image ot the opening is markedly larger than the
opening itself shows that we could construct the image by
imagining curvilinear rays which, on passing the aperture,
converged towards certain points of the tube.
(5) A rectangle cut from a sheet of lead with a rectangular
opening in the middle was placed beneath the ebonite plate
and in contact withit. The image of the solid parts is intense
yellow ; the image of the opening is black; the rectilinear
margins of the opening are replaced by curves which are
convex inwards. The same object (fig. 4) was lowered one
centimetre beneath the screen. The image is represented by
fig. 5. The solid parts of the object (white in the figure)
are black in the image. The image of the opening is red,
smaller than the opening, and bounded by four curves which
turn their convex sides to the centre of the figure. We
312 MM. Oumoff and Samoiloff on Electric
should therefore obtain this by constructing curvilimear rays
emerging from certain points of the tube in the direction of
Fig. 4. Fig, 5.
the opening. Thus we see generally that transporting the
object from one side of the screen to the other results in the
interversion of the phenomenon.
We may mention that a metallic strip with parallel mar-
gins, curved into a circular are and fixed by its summit on
the screen (fig. 6), gives a black shadow (fig. 7) enlarged at
its two ends and surrounded by a reddish field.
Fic. 6.
(6) Fig. 8 shows the image of a lead band, curved and
fixed vertically on the screen by one of its margins. Fig. 9
shows the change which the image undergoes when a glass
plate is placed beneath the ebonite screen.
(7) Lead strips of equal length but different widths were
bent into circular cylinders of the same diameter (about 2 cm.).
On placing one of these cylinders vertically on the ebonite
screen its base is represented by a red circle ; in the interior
there is a thin black band (neutral) which surrounds a yellow
spot. On placing a glass plate beneath the ebonite the central
yellow spot diminishes markedly and the neutral bands in the
interior, as well as at the exterior, increase in width. The |
Images in the Field of a Hittorf’s (Crookes’) Tube. 313
dimensions of the central spot diminish when the height of
the cylinder is increased. Comparing this phenomenon with
that described in (4) we see that the yellow spot should be
regarded as the image of an aperture. On covering the
cylinder by a metal disk the central yellow spot disappears
and we obtain red traces in the central part of the image.
The diminution experienced by the spot when a glass plate
is brought under the ebonite screen would apparently lead us
io seek the aperture towards which the rays proceeding from
certain points in the tube converge in the optical image of
the circular section of the cylinder nearest to the tube and
obtained by regarding the glass as a mirror.
The central spot of the image of a cylinder placed beneath
the ebonite plate is red ; its size remains the same whatever
be the height of the cylinder provided that its diameter
remains the same. The phenomena are not altered if the
lead cylinders are replaced by glass ones.
The following additional experiments correspond to those
described-above :—on placing on the ebonite brass weights of
500 grams, 200 grams, and 5 decigr., it is only the last,
which represents a thin piece of metal, that gives a red image;
the first two do not give distinct images. In the image of a
coin placed above the screen we find red portions correspond-
ing to the points which were in contact with the screen, and
black and yellowish portions corresponding to the depressions
of the coin. If this last is placed beneath the ebonite, the red
colour is replaced by yellow. We obtain in this manner the
design on the faces of the coin.
In all these experiments the duration of the action of the
Hittorf’s tube has an influence on the clearness and intensity
of the image ; prolonged action imparts a red or yellow colo-
ration (according to the circumstances) to the neutral bands.
By obtaining the images of objects by our method and by
the photographic method we bave proved that the yellow
colour corresponds to the parts of the figure directly attacked
by the w-rays; the red colour to the images of the objects,
and the neutral bands to the shadows which surround, for
example, the images of the cylinders.
Preliminary experiments have shown us that analogous
phenomena are obtained on replacing the Hittorf’s tube by a
metallic point connected to the conductor of an electric
machine. Like phenomena, under slightly different con-
ditions, have been obtained by means of electric discharges by
M. Augusto Righi (Memorie della Accademia delle scienze
dell’ Istituto di Bologna, (4) in. 1881, pp. 291 to 304, and
pp. 461 to 496) ; there is in this memoir a sketch of a
314 B. Rosing on the Participation of Matter
theory. We think that it is indispensable to complete the
experiments above described in order to elucidate the various
questions which present themselves, and we limit ourselves to
a general conclusion that the phenomena observed should be
attributed to electric fluxes proceeding from the Hittorf’s tube
and the objects in its neighbourhood, together with a dielec-
tric polarization ; in this sense the similarity of the electric
fields of a Hittorf’s tube and of an electrified conductor must
be admitted; we must therefore take it into account in the
study of the electric properties of Réntgen’s rays.
April 1896.
XXXIV. On the Possibility of explaining the Phenomena of
Magnetism by the Hypothests of Participation of Matter in
the Motion of the Magnetic Field. By B. Rosine, Fellow
of the Russian Physico-chemical Society®.
a. theories of magnetism, whatever their physical foun-
dations, are founded from the point of view of dynamics
on the supposition of the existence of two principal types of
physical coordinates ; the one fixing the intensity and the
distribution ° of magnetic induction, the other defining the
state of the magnetized matter. But the coordinates, as is
known, can be in general either of positional or of kinosthenic
character}; 2. e. they can occur in the expression for the
energy of a system either explicitly or only through their
differential coefficients. Therefore we may imagine three
combinations of our magnetic coordinates, and consequently
divide all possible hypotheses on magnetism into three cate-
gories. The first category, when both types of cocrdinates
are positional ; Weber’s hypothesis, for instance, of Molecular
Magnets belongs to it. The other, when the one type is
positional and the other kinosthenic ; such is Ampere’s hy-
pothesis of Molecular Electric Currents in Maxwell’s version#:
the latier takes the energy of the electric currents to be kinetic.
A third combination is still possible—when both types of
coordinates belong to the kinosthenic type, 2. e. when it is
supposed that matter when magnetized is put into the same
motion as the surrounding magnetic field. ’é shall take this
third assumption—Have we the right to consider the magne-
* Communicated by the Author.
+ See J. J. Thoinson’s ‘Applications of Dynamics to Physics and
Chemistry,’ p. 10.
t See Maxwell’s ‘Treatise on Electricity and Magnetism,’ 1892,
Vola i. chap: xxi :
in the Motion of the Magnetic Field. 315
tization of matter as a purely kinetic process, and to explain
all magnetic phenomena—such as polarity of paramagnetic
and diamagnetic bodies, magnecrystallic force, hysteresis—as
simple mechanical consequences of the participation of matter
in the motion of zther which takes place in magnetic induc-
tion tubes? Of this we have no positive proof; but this
hypothesis is attractive by reason of its simplicity, and at the
same time does not contain anything improbable. To explain
all phenomena by means of the properties of matter in motion
and to deduce all laws from the laws of kinetic energy—
has not this always been what natural philosophy has striven
to achieve? The idea of connecting the motion of matter and
ether dynamically is also not novel: in this way only was
Helmholtz enabled to explain the phenomenon of anomalous
dispersion. Magnetism also possesses its own anomalous dis-
persion : it is paramagnetic polarity of some bodies in relation
to diamagnetic polarity of others. And in reality, as we shall
see further on, the paramagnetic refraction of lines of mag-
netic foree—to use Faraday’s deep and expressive language—
is just the same mechanical consequence of absorption of
energy as is the anomalous dispersion of lines of the light-
radiation. Diamagnetism of matter, as is known, is explained
very easily by admitting the existence of molecular electric
currents, excited by the surrounding magnetic field and cir-
culating freely and without resistance on the surface of the
particles of matter. Hlectromagnetism* shows that the energy
of a system of spherical currents, referred to one of these
currents, is equal to
Bee) OO. eco et Olle a
w=-3|)2? gas = sy Pd83- - - A)
where the integrals are extended over the surface of the
current in question, @ is the stream-function, and 0 and O«
are the magnetic potentials due to the present current and
the surrounding ones respectively. The sign & refers to all
the surrounding currents. :
Adapting this formula to the case of molecular spherical
currents excited by the magnetic field H, and introducing
the intensity of magnetization I instead of the magnetic
moment of currents, after the transformation the formula for
ae energy of molecular currents, referred to unit of volume,
will be
* See Burbury’s “ On the Induction of Electric Currents in Conducting
Shells,” Phil. Trans, 1888, p. 302.
316 B. Rosing on the Participation of Matter
W=S(2+1)P+1n, . |e
where ¢ is the aggregate volume of the particles in unit of
volume.
At the same time, by applying Lagrange’s equation to the
expression of the energy of molecular currents, we get the
following relation between the magnetization I and the mag-
netic force H :—
An /2
=: +1)1+H=0. > ae
It is evident that this equation can be got by applying
Lagrange’s equation directly to formula (2) and by regarding
I as a velocity. |
Formula (3) represents a case of diamagnetism,
ee = 5)
ae a
3 = +1)
From it we find the coefficient of magnetization « to be
UE 2 eee ae
k= H = EAT ae 5 «cle. Gero annns (4)
ale te 1)
3 \e
and, lastly, that of the magnetic permeability to be
ae =
p= af ts
I+3
The formula (5) of magnetic permeability is found by
assuming the hypothesis of molecular currents, excited by
the magnetic field on the surface of particles which are
themselves absolutely impermeable. It is remarkable that
the same formula can be found by assuming another hypo-
thesis, namely that which takes magnetic induction to be a
flux, propagating through media of different conductivity. In
reality, as is known, the problem of distribution of magnetic
induction in space corresponds exactly with that of electric
eurrents*. But we know from Electrokinematics that the
conductivity of a medium consisting of spheres of conduc-
tivity #2, disseminated through a medium of conductivity py,
is
* See Maxwell’s ‘ Treatise on Electricity and Magnetism,’ 1892, vol. ii.
p. 04.
+ Ibidem, vol. i. p. 57.
in the Motion of the Magnetic Field. 317
2 pb, + fy + 2€(2—
w= —— Le (6)
py + fg — € (My — fA)
Applying this formula to our case of magnetic induction
existing in a space amongst particles of matter absolutely
impermeable, we must take u, to be equal to 1, and py equal
to 0. The formula (6) will then take the following form :
l—e
LE paiea % hhaas Target)
I+5
which is identical with formula (5). Itis easy to see trom this
formula that the coefficient of magnetization « is equal to
—I
k= ea \ sies ot tes nope (8)
3s +1)
and I is connected to H by the formula
a7 (2 41)1+H=0. Le haa (a)
The magnetic energy of unit of volume is, as we know
from electromagnetism, expressed in terms ef the magnetic
induction B and magnetic force H by the formula
Wie opr
or
By transforming this formula and using the relation (9) we
get
mee a foo, t
W= g,, BH = g (H+ 41) i = 3 4 31H
sacl ° pet pel Dae / 2
=, H+1H—5 IH= 5 H’+ 1H + + (< +1) 2.
Here = H’ is the energy of unit volume of magnetic field and
2a (2
the terms LH + = ( iF 1) IP represent the magnetic energy
of the matter. These terms, also formule (7), (8), and (9), are,
as we see, identical with those found before. Such a coinci-
dence in the results of the hypothesis of Magnetic Flux with
those of the hypothesis of Molecular Electric Currents permits
of our concluding that a system of spherical currents excited
without resistance on the surface of particles of matter is merely
318 B. Rosing on the Participation of Matter
a mathematical fiction, representing the conditions of the propa-
gation of lines of tnduction in the space occupied by the particles
of matter, namely the conditions of their reflexion at the surface
of these particles.
Formula (3) or (8) shows that neither of the hypotheses,
when excluding the participation of matter, can explain para-
magnetic phenomena: the first, because the magnetic moment
of induced currents always appears in a direction opposed to
the magnetic field; the second, because the presence of abso-
lutely impermeable matter always lessens the magnetic per-
meability of space. Consequently we are obliged to introduce
a supplementary hypothesis expressing this participation in
some way or other.
We introduce it here by supposing that the matter, when
in a magnetic field, is itself put into some motion ; and con-
sequently, besides the system of coordinates representing
molecular electric currents, coordinates also exist which fix
this magnetic motion for each particle. As these new coor-
dinates, we suppose, are of kinosthenic character, the new
terms, appearing in the magnetic energy of a substance, are
of the form
ALJ and 4yvJ?;
where J is the vector defining in every point the velocity of
magnetic motion of matter, and the coefficients ) and v depend
on the nature of a substance, and denote—the first, r, the
connexion between the motion of magnetic induction and the
magnetic motion of matter, and the second, v, the inertia of
this latter motion.
Thus the magnetic energy of unit of volume wiil be repre-
sented by the following expression :—
W= z(- = 1) P+ IH +A1I + $y ee
By applying to this expression the principle of Least Action,
we obtain Lagrange’s equation in a new form :
for coordinate I
d (Ad 2
5 (a (E+ 1)E+.d +H) =0, .. a
for coordinate J
£ QL) =0: 3 oo eee
Hence, after integrating and putting the initial conditions
1=J=H=0, we have
in the Motion of the Magnetic Field. 319
ae +1 )[+23 +H =0
(13)
AL+vJ =0
and therefore
— H
i 0 eae he ue
gether
or
k= ress x : c 6 e (15)
Formula (15) now contains both cases of magnetization: that
of magnetic and that of paramagnetic bodies. In reality, when
the coefficient X is sufficiently large and v sufficiently small,
« assumes positive values and we enter the sphere of para-
magnetism.
From the equations (13) we obtain besides this
2
a I and oo Pee.
y 2 2»
whence it is evident that—given a comparatively large value
to X and small value to y—the velocity of magnetic motion of
matter and its energy under the same magnetization I are
comparatively greater. We conclude from this, that the
absorption of energy by motion of matter in paramagnetic
bodies is comparatively greater than in diamagnetic ones, as
has been already pointed out, and that in consequence thereof
appears that anomalous propagation of the magnetic induction-
tubes which is observed in paramagnetic bodies. Besides
that, as we already decided to regard the phenomenon of
diamagnetism as the reflexion of lines of induction at the sur-
faces of particles of matter, we must now consider paramag-
netism to be also the reflexion of the induction lines, but
taking place without change of sign. Thus we find here the
same phenomenon of double-signed reflexion with and with-
out change of sign as we also see in other branches of physics,
as for instance in the reflexion of waves of light and sound at
the surfaces separating media of different nature.
In the case of a crystalline substance the magnetic energy
of unit of volume is expressed by a formula which is analogous
to formula (10), but the vectors I, J, H are replaced here by
their components (A, B, C), (L, M, N), and (a, 8, y) re-
spectively. By applying to it Lagrange’s equations we get a
system of equations :
320 B. Rosing on the Participation of Matcer
Atr
ath
Ar
3
3 41)042e. L4 Aye M+ 0..N=—9,
Nez A+ Avy B+Az2 C= — (ver L + vyz M42, N),
Nyc A+ Ayy B+ Aye C= — (ty L+ vy M+, N),
Nz At+AyB+A,C=—(,, L+yzM+vz2N);
¢ +1)A +n Dy M+reN=~a,
2
(2 +1)B+2ry L+ dy M+ N= — =.
which on excluding L, M, N will be reduced to three
equations :
A= Kya + kK + K3y
B=kyat K{B + ky ie
= // // //
C7 at Kk, B+ks ny
where
i fs fea /
HO Coy RC Cg ly Ia
These are, as we know, the fundamental equations for mag-
netization fot a Cry Seliine substance. The coefficients «1, Ky, 3
- here may have both positive and negative values depending
upon the values €), 2, €3) Azz, Avy, Nyy, - ©: Very Veg ee
the different axes of the crystal.
Lastly, by introducing into the formula of energy values of
1
LL, M, N from the above equations, and by adding ge M,
we vet
W= = BH cos BH.
S77
Thus we see that the results found by means of the hypothesis
of mechanical participation of matter in phenomena of mag-
netic induction answer well enough to the fundamental
requirements of the theory of magnetism.
However, these suppositions are not sufficient to explain
all phenomena of magnetism, for instance the phenomena
which take place when iron and such metals are magnetized.
The phenomenon of magnetic remanescence forces us, on the
other hand, to suppose the existence of magnetic deformations
that take place at the magnetic motion of matter, and therefore
to accept a new type of coordinates which would define
them.
Actually, whatever the magnetic motion of matter, it is, of
in the Motion of the Magnetic Freld. d21
course, a periodic motion round the axis of the vector J.
Arising inside matter, it must produce there a kind of
pressure, counterbalanced by the elastic forces of matter,
and must therefore be accompanied by certain deformations.
We suppose that this pressure is kinetic in character, ¢ e., it
passes on bv collisions from particle to particle.
Now it is easy to prove that such a kinetic pressure must
be proportional to the square of the velocity J, and that at
the moment of the change of this pressure forces of reaction
appear which act backward on the vector J.
Actually, the hypothesis of the existence of kinetic pressure
produced by the magnetic motion of matter is dynamically equi-
valent to supposing that a connexion exists between this motion
of matter and the motion arising at its deformations. There-
fore, if we denote the coordinates fixing magnetic deformations
by the letters p,, Pero then the above hypothesis will mean
that terms exist in the formula of Lagrange’s function of the
tollowing kind :—
PaPa t+ Pape + ce og 8 0 : ges (17)
where p_, Pz +++ are coefficients defining the connexion be-
tween the two motions.
Therefore, in Lagrange’s equation tor the vector J, besides
the forces contained already in formula (12), new forces
appear and the equation is :—
d d 4 : fo %
ay ott vy) == Fe el ae gam 50 ee lata)
+ Pgpgd +t -..J=0, . (18)
if we denote by © the coordinate the velocity of which is
1 Cae
dO
Seer AMES ee | a 5 oe eae 9
J ap (19)
Besides this we shall get the fullowing equations relating to
coordinates p_, Pgs e+ 3
ih a 2p - eran)
(PP)aPat GPa = Bp Pe saan |
wee | Op - ols P
Fie ee a — — — s 20)
(PP) 3Pa+ aPp Op, B Ps | \
Phil. Mag. 8. 5. Vol. 42. No. 257. Oct. 1896. 2A
322 B. Rosing on the Participation of Matter
Here the terms (pp) p., (PP) aPp? ... represent forces of
inertia for the coordinates p,, Parr: and the function ¥F
represents the free energy of deformations according to
Helmholtz’s theory. It must be observed, however, that we
simplified the formule by taking the coordinates p_, p grit:
to be independent of each other, and the functions p_, p grt to
depend only on the coordinates © and p,, Parse respectively.
On the supposition of indefinitely slow changes in the coordi-
nates Pp, Pg+++ Wwe can neglect the forces of inertia ;.
further, by giving to the equations the form
dp. ; dJ oF
70? + Po dt = op,
dp dJ oF
B 2 So eS oS eS
ke TPs a OP.
dJ
dJ |
we can neglect the forces Pa? PB ae? °° 7 OB the same
grounds. Lastly, putting
dp
qo Pa =a, (21)
dO ” dO
we get the equations in the case of indefinitely slow changes
to be :—
oJ" = o ) |
ee ee «= Vee
Pia pets or |
B OPg yj
which show the equilibrium bctween the components of pressure
proportional to J* and the corresponding forces of elasticity.
Let us now consider equation (18). After integrating
from the initial moment, when [=J=H=0, this equation
gives :
Al+w+ppItppwt...— "OP a: Jdit — 03 34:
ala BL 8 0 30 4 Jo 3028
Once more assuming the indefinite slowness of change, we
shall have for static magnetization :
in the Motion of the Magnetic Field. 323
Po. Jd
AL+yvJ — ( Pa Jdp,—( , 0°83 Jdp,— He =O),
v9 00 e 0 role)
because we can negiect the terms containing Py Pg ++ and
can accept p_, Pgr+++ as the independent variables under the
signs of integration.
Then, on introducing significations (21), we obtain
M+5-+( ™ x Jdp,,+ o,Sdpgt...=0.. (23)
Jo 0
Thus we have the following system of equations in case of
static magnetization :—Lagrange’s equation (11) for the vector
I, which remains unchanged under the new suppositions,
equation (23), and the system of equations (22) :
a [Ag 62
i LEG Hyrars]=o
AL + yd +{
0
5. ole or
a=, Oa
OP, 26 OPg
Let us integrate the first of these equations and introduce
the J from it into the second term of the second equation.
Then, by making use of quantities o J, o,J,... from the
oe eller | ogJdps +: ive 20,
Jo
oJ
third equation, and by introducing all these quantities under
the signs of integration in the second equation, after denoting
the sum .
oF oF
Le Sek Pe
Op, pat a
by oF,
or or
6k = ont ae l a f eee e e e 24
Op, /4 ae ap, hB* ) (24)
the system of our equations will be transformed into the
following system :
Ar /2
2 I =
3 ie +1) Ae) co cl (0 ene ©)
T=eH—0™| > Sette |S en GeO)
ee
A ee ro)
J? ' Oh wits
c | OP. ond es (27)
2A 2
324 B. Rosing on the Participation of Matter
Here the coefficient «is expressed by the above formula (15),
—e
—— eee
Aq er?”
and oF is defined by equation (24).
This system of equations represents the general conditions
of static magnetization on the supposition of the existence of
magnetic pressure and magnetic deformations. We will pro-
ceed to examine these formule.
Formula
(oF
I=xH—«- | —
eH—#- | —
shows that the process of magnetization is composed of two
processes, one, expressed by term «H, is a process which can
be completely reversible and which represents, as we have
already had occasion to see, a simple reflexion of induction-
lines from matter; the other process is expressed by the
term
_ h(E
Palade:
and arises in consequence of absorption of energy by the mag-
netic deformations. The value of this term wholly depends
on the free energy F of deformations. It is in the nature of
these deformations that the explanation of all the complex and
intricate phenomena which appear in the magnetization of
iron must be sought.
It is evident from equation (25), that in a paramagnetic
substance—given I and H positive—the vector J has the oppo-
site sign to the coefficient A, and « having a positive value, the
product Ke - is greater than nzl. On the other hand,
when I and H are increased, and consequently J too, the de-
formations likewise increase, and with them F. Therefore
the integral
Ses
vj Jd
taken from the beginning of magnetization I=J=H=0 isa
positive quantity.
Thus, owing to the absorption of energy in a paramagnetic
substance, magnetization increases.
To show this still more clearly, we will put the equations
in the Motion of the Magnetic Field. 325
into differential form. Let us suppose that the process of
magnetization takes place at constant temperature. In this
case the free energy IF is a function only of the coordinates
Pay Pp»---+ But these coordinates can be expressed by the
system of equations (27) as a function of J’. Therefore the
energy F may also be considered as a function of the same
quantity J’. In this way we have
ara OF a A BR om
If we now differentiate the equation (26), considering H as
an independent variable and using the formula (28), we have
aT _y_ 2D ad
an ~ vy OJ? dH’
let us replace a by its expression from equation (25);
dJ Lr4c 72 dl
eas ee +0) a +1];
we have, lastly,
20F
dl _ Te aie e
A 2 ee (28)
3 \e OJ?
This is the differential equation of static and isothermic
eee, F
magnetization. Here, as has been said above, = must be
considered as a function of J’, where J, in its turn, is ex-
pressed in terms of I and H by help of equation (25),
3
The form of this function of F is defined by the way
in which F depends on p,, pg. -.. a8 shown in equations (27).
Actually, by differentiating the equations (27) at a constant
temperature,
= (= =e) L+rJ +H=0.
toe or
G,dJ2 = Spe (Pe opdJ* = Opa Pe Mae
and by introducing the quantities dp,, dpg,... from here in
the expression (24) for dF, we shall haye
326 B. Rosing on the Participation of Matter
oF oF
fy, Oper gi Op \ 2,
= | on Shh to Sep te dJ?;
OP, OPS
whence, by comparing this equation with (28), we have
oF OF
oF OP 0
oa=4 o. SR +0357 +. ye
OP, OP
: Semele oF
or, more simply, taking Oh. =, aA: =e eee
oF a3 Apa dpe
QJ? \c, diog P, 2p dlog Pz Raa \ ae - (30)
where @ is the absolute temperature, and Pe, Ps,....are
supposed for simplicity to be only functions of this tempe-
rature and of the coordinates p,, pg, ... . respectively.
It is by this equation, in conjunction with equations (25),
(27), and (29), that the process of maguetization is completely
defined. >F
First of all, it is to be seen from equation (30) that IE
Ap
abe Pid log : eee are, in general,
is a positive quantity, as
greater than nil. dl
Therefore, as shown by equation (29), 7H is for a para-
magnetic substance always larger than « if only
OF sy 3
pst Pee pe ae
4n(= +1)
dl
EK } °. ¥s e
fo) v 7H® infinitely great, and, lastly,
3
aS ni ee RS
od a Aor (= +1 )
€ >F
if with increasing of H, and therefore of J*, iE becomes
Vv
ee
ae tr(=+ 1 )
statical magnetization 1s impossible.
[In the latter case, as is easily proved, the process of mag-
netization represents a kind of free motion at a constant mag-
netic force. Nevertheless, the equations of this process are
When
the equation cannot give real solutions, and
in the Motion of the Magnetic Freld. 327
included in the general equations, and can be shown as
follows :—
EHi=const., (2 qd )I+.J+H=0,
pe »Upeg y a
2 2 a
peeai eee BIA aa Duden? ‘
mee
(pp)eia + Gap, ta0P?'= 5
eee oF
(PP spe + Gepet opel? = Opa
The new terms introduced in the equations appertaining to
the coordinates pa, Dg,.++-3 (PP)oPat Gaps (PP) ape +:Gigpg,---
represent the forces of inertia and viscosity respectively. It
is clear that in the curve of magnetization showing the de-
pendence of magnetization I on the magnetic force H, this
process is shown by the straight-lined parts of it parallel to
the axis of magnetization. |
The same equation shows, further, that at the commence-
ment of magnetization a must be equal to «, because in
these conditions = P =(; and therefore oT =0, if only
= is not ©, which can happen only in particular cases ; for
instance, near the temperature of recalescence. This is the
least value that Ge can have; it is not great, and ¢f the
dH
deformations were not existing, the magnetization of iron would
not differ much from the magnetization of other paramagnetic
bodies. But that in consequence of these deformations,
increases very rapidly, and -—-— increases with it. The
di
quantity FAL exists as long as the deformations change ; when
the deformations cease to change, the differential coefficient
dl e e e
—. again takes small values. This ceasing of change of
dH
deformations must be therefore supposed in order to explain
so-called magnetic saturation. But if we take into considera-
tion that magnetic deformations most likely do not represent
328 B. Rosing on the Participation of Matter
changes of volume or form, but of the structure of matter—
like that which takes place in recalescence, as is most pro-
bable,—then there is nothing impossible in this hypothesis.
The explanation of another very important phenomenon—
magnetic hysteresis—is based on the same phenomenon of
magnetic deformations; we must only suppose that the
same hysteresis exists in the changing of magnetic defor-
mations.
It is also interesting to see to what our hypothesis could
lead in the case of a deamagnetic substance. In this last case,
when «<0, 2. e. the reflexion of magnetic induction from the
particles of matter takes place with the changing of the sign,
the absorption of energy by deformations gives rise to a quite
opposite effect. Formula (29) shows that when «<0, the
absolute value of - cannot be more than K,
—1
SVE eee
(241 )—*
3 \e y
because, as it is very easy to see from this formula, the fol-
lowing inequality always holds here, 3
=e (2 +1)>1.
Further, the same formula shows that s is lessened always
with increasing of oo and therefore in all such cases, when
magnetic deformations increase the most the differential
coefficient a falls to its least value. 7
Owing to this, when we assume the character of the change
of deformations in a diamagnetic body to be the same as in a
paramagnetic one, we shall have for a diamagnetic substance a
curve of magnetization of quite a different form, namely, like
the line O MN shown in the figure. Similarly, when the mag-
netizing force is decreased, the deformations change more
slowly, in consequence of hysteresis, than when it is in-
creased, and the return curve NPQ descends below the
curve OMN and intersects the axis of H before O in the
point P. Again, with a further decrease vf H, magnetization
becomes positive, and the body at H=0 has a paramagnetic
residual. ‘This, as is known, was observed by Messrs. Quintus-
Icilius, Tumlirtz, Lodge, and others.
Besides the diiferential equation of magnetization, it is
tn the Motion of the Magnetic Field. 329
important to examine the signification of the integral
— \ldy,
which represents, as is known, the area of a closed curve of
magnetization.
If we write the equation (11) in the form
7 (= ; )o+ al ded) dH
3 dt dade
Jee ;
the term — ee will represent the force by which the exterior
€
magnetic field acts on the coordinate, the velocity of which
is I. In the time dé it is obvious that this force will per-
form the work
aw=- eae
Therefore during the complete cycle of magnetization the
magnetic field will perform the work
W=— {lH,
where the integral is extended along the curve of magneti-
zation.
On the other hand, if we differentiate equation (26)
25
multiply
it by J, and substitute for J its expression from (2:
»]
), then we
330 B. Rosing on the Participation of Matter
shall have
Jdl—JedH = —«8F;
and therefore
-F(- aa Hdl +
Ar
: 33 +1) «ldH +«HdH
=—K — © oF.
After integrating along the closed curve, we have, in
consequence of equalities which hold in this case,
{HdH=JIdI=0, {Hdl=—f dH,
the equality
— (14 F(E +1) e) | ua = o [aF,
But formula (15) shows that
2
“Lek =( +1)e=0 3
3
consequently
—\IdH= Jj 6F.
Farther,
aap OF
df =6F — Yi] —~ do,
where @ is the absolute temperature.
But the theory of Free Energy tells us that
d¥=dU—ds8,
where U is the complete energy of the body and S@ its bound
energy.
Therefore
be oF
jéF= \dU—fase— 59 d0= JdU — jGdS=\dU—A JQ,
because oe =8, 7. ¢. the entropy, and @dS=AdQ, where Q
is the quantity of heat supplied to the body and A the mecha-
nical equivalent of heat.
In the case of an isothermice process, when the body on
completing the cycle arrives at the initial conditions, we have
jdU=0;
and therefore
in the Motion of the Magnetic Freld. ddl
But in the case of deformations produced by exterior forces
far is, in natural conditions, either nz or a positive quantity.
Therefore \IdH is a negative quantity, and magnetization
proceeds along the curve in a direction opposite to that in
which clock-hands move ; and the work performed by the mag-
netic field is positive, and leaves the body in the form of heat.
Thus, in conclusion, we see that, from the point of view of
the hypothesis which considers the magnetic induction as a
kind of motion of ether communicated also to matter, we can
explain the phenomenon of magnetization as a simple re-
flexion of tubes of induction from matter, where this reflexion
appears with or without a change of sign, with this or
that intensity, depending on the forces of reaction of matter
which exist at the moment of reflexion. These forces of
reaction depend, in their turn, firstly, on the inertia of
matter in relation to the magnetic motion and on the coeffi-
ecient of connexion of ether and matter, which determine the
sion of reflexion; secondly, also on the elastic forces in these
substances in which magnetic motion is accompanied by
magnetic deformations. In those bodies which have a
comparatively large magnetic inertia, the reflexion takes place
with change of sign and the intensity of reflected induction is
greater, the less the inertia and other forces of resistance; this
ts a normal magnetization, and these bodies are so-called dia-
magnetic bodies. In others where the inertia is less than a
certain quantity, the reflexion takes place without change of
sign, and every increase of resistance of matter 1s accompanied
by increase of reflexion. These are so-called paramagnetic
bodies. |
By following in this way the hypothesis treated of here,
one can reduce to the same principles the phenomena observed
at rapid and alternate magnetization and find an answer to the
very important question of the existence of magnetic inertia
and viscosity, and of their réle in the magnetic circuit. This
question, however, necessitates very minute study, and we
shall return to it in the future. At present we will only
remark that the results to which this hypothesis brings us are
directly opposed to those which are arrived at from the point of
view of Weber-Ewing’s theory. In opposition to this theory,
the magnetization of iron here rises with the increase of
resistance offered by the forces of inertia or viscosity. In
this way inertia, when the magnetizing current is closed
rapidly, must involve a greater magnetization; the same is to
be said concerning viscosity; on the other hand, when mag-
netizing current is alternate, viscosity alone increases it;
whereas inertia lessens it; and as in all probability, owing to
332 Dr. G. J. Stoney on Microscopic Vision.
the rearrangement of particles, the viscosity steadily lessens,
this entitles us to suppose that with an alternate magnetization
the maximum of it does not reach that quantity attained by a
static process. And certainly the most recent observations on
the exhaustion of iron in transformers seem to corroborate this.
From the point of view of the present hypothesis, the exhaus-
tion of ircn is nothing but the rearrangement of its particles,
in consequence of which the magnetization is performed with
a smaller absorption of energy, and therefore calls for a less
pronounced magnetic effect.
Physico-chemical Society.
The University, St. Petersburg, Russia,
June 15, 1896.
XXXV. Microscopic Vision. By G. JOHNSTONE STONEY,
WEA TIS. 3 hee
Part I.— FUNDAMENTAL PRINCIPLES f.
1. ISION, whether by the naked eye or with the as-
sistance of optical instruments, may be studied in
many ways; since a correct investigation may start from any
one of the innumerable possible resolutions of the disturbance
which exists throughout the ether in front of and close to the
object. But two only of these will be here considered, viz.:
that most obvious resolution in which the ether in front of
the object is regarded as traversed by undulations of hemu-
spherical waves emanating from each physical point of the
surface of the object, and that other equally general but less
obvious resolution of the disturbance in this portion of the
eether into undulations of uniform plane waves transmitted
forwards in all or some directions from the whole extent of
the objective field.
2. The first of these modes of resolution---that into sphe-
rical waves—is the foundation of Airy’s method of studying
the images formed by telescopes, in which the image is re-
garded as arising from the overlapping and interference of
the spurious disks with attendant rings which in the image
take the place of points on the object. This method has on
this account been sometimes called the Spurious Disk Theory.
The second mode of resolution—that into plane waves—is the
foundation of Abbe’s method of studying the images formed
* Communicated by the Author. rir :
+ Part II. deals with the application of these principles to the mierc-
scope as at present made.
Dr. G. J. Stoney on Microscopic Viseon. 333
by microscopes ; and has usually been called the Diffraction
Theory, because it gives a special prominence to the fact that
when we pass beyond the meagre hypotheses of geometrical
optics, we find that diffracted light* is “the machinery by
which good definition is brought about.” It was undoubtedly
desirable to emphasise this tact, because an error prevailed
and is not yet extinct that diffracted light intervenes only to
impair the image ; and it can scarcely be made any objection
to the name that it runs counter to this error. As, however,
both processes are only methods of investigation, it would
perhaps be desirable to avoid calling either of them a theory.
On this account, and to avoid cavilling about mere names,
the two methods of investigation are in the present memoir
distinguished as the Airy and the Abbe modes of pro-
cedure f.
3. Ina recent paper by Lord Rayleigh the generality of
Abbe’s method seems not to have been appreciated (see Phil.
Mag. for last August, p. 167); and the main object of the
present communication is to offer a fuller account of this
generality than the writer has elsewhere given (see. “On the
Foundation of the Diffraction Theory ” ; ‘ English Mechanic ’
for December 13, 1895, p. 380), and to trace its consequences.
4. ‘Two terms have been used above in the first paragraph
which need to be defined.
(a) By a physical point is to be understood an element of
the volume of the object (if the object be translucent), or of
its superficial layer (if it be opaque), which element of volume
is small enough to justify us in substituting for it in our
investigation a mathematical point regarded as a centre of
an undulation of hemispherical waves. The physical point is
small enough for this use of it, if its linear dimensions are in
any considerable degree less than 4/4, where » is the wave-
length of the light employed. To give definite form to our
conceptions we may suppose its dimensions to be comparable
with 4/10. This is a convenient size; since if an opening of
this size were made in a thin opaque screen, and if a pencil
of light were incident from any direction upon it, the hole is
small enough to ensure that the light which gets through
shall spread on the other side of the sereen in the form of
* Light which advances in other directions than those prescribed by
geometrical optics is called diffracted light.
+ Lord Rayleigh suggests the name Spectrum Theory for the method
of investigation which proceeds by resolving the light into plane waves ;
but will perhaps not press this name on the acceptance of scientific men
when he finds that the limitation which the name implies has no existence.
334 Dr. G. J. Stoney on Microscopic Vision.
hemispherical waves; and at the same time the opening is a
large one when compared with the transversal* of the light
waves, since molecular considerations indicate that this trans-
versal (or rather these transversals, since there are two, an
electric and a magnetic one at right angles to one another)
must be regarded as of a length which is from the thousandth
to the ten thousandth of a wave-length. Hence the directions
of transversals will not be affected in passing through the
opening. On this account, if the incident light be a beam
ot plane waves, whether polarized or not, the intensity of the
light will differ on the various parts of the hemispherical
waves which spread beyond the screen, being a maximum in
the direction of the prolongation of the normal to the in-
cident waves. This must be taken into account in attempts
to apply Airy’s method of investigation to microscopic vision,
since until this is sufficiently done the investigation is too
imperfect for us to be justified in relying on its results except
so far as they can be confirmed by Abbe’s method or some
other which does not involve the above consideration. A
further and more serious imperfection is introduced when
Airy’s method is applied only to the light between the
objective and the image, and not also to the light between
the object and the objective. An inquiry conducted in this
way begins too late, after the more important of the events
that affect the image have occurred. Nevertheless it seems
to be the only one which has as yet been made by Airy’s
method ; see, for example, the investigation on p. 176 of
Lord Rayleigh’s paper. We shall learn in the second part
of this memoir what it is that in this case is being ascertained.
(>) The other term in paragraph 1 that requires definition
is the objective field. By this term is to be understood the
whole of the object and its surroundings of which an image
is formed by the telescope or microscope, or in the eye of the
observer. Accordingly the objective field at and surrounding
the object corresponds to ‘ the field of view’ at and surround-
ing the image of it which is formed in the eye, or at the
focus of an optical instrament. ,
* The word transversal is here and elsewhere used for the transversal
of the displacement under the dynamical wave theory of light.
The dynamical wave theory is that used throughout this memoir,
except where otherwise stated ; since, in the present state of our know-
ledge, it is more easily handled than the electromagnetic wave theory,
and since, except in special cases (as for example the distribution of
intensity over a spherical wave), it furnishes the same results. Besides,
the dynamical theory usually carries us as far as we can go, for, in the
special cases where the electromagnetic theory may yield a different
result, it seldom happens that we yet know that result. d
Dr. G. J. Stoney on Microscopic Vision. 335
5. The following important optical theorem may now be
enunciated, which in its generality compares with Fourier’s
Theorem, of which it is, in fact, in ultimate analysis, an
extension.
PROPOSITION 1.
However complex the contents of the objective field, and
whether vt or parts of wt be self-luminous or illuminated in any
way, however special, the light which emanates from it may be
resolved into undulations each of which consists of unform
plane waves; on the hypothesis that each point of the object
emits continuously the same light: an hypothesis the suffi-
ciency of which will appear in Part IL. of this memoir.
By an undulation is meant a succession or train of waves,
and by a uniform wave is meant one which is at each instant
alike in every part of each wave surface.
6. To prove this theorem we proceed very much in the
same way as in dealing with Fourier’s Theorem. We begin
by positing repetitions of the objective field. For this pur-
pose let a plane be drawn through some point of the objective
field, and preferably perpendicular to the line of sight. This
plane may be called the Objective Plane. Let a square be
drawn in this plane which may be of any size, provided that
it shall include within it the projection upon the plane, from
the point of view of the observer, of the contents of the ob-
jective field: in other words, the square is to be large enough
for the whole of the objective field—the whole of what the
observer can see—to fall within that square, and preferably
well within it. Divide the whole plane up into squares of
this size by two systems of equidistant parallel lines, and
imagine an exact repetition of the contents of the objective
field to occupy the position relatively to each of these except
the first, which is the same as the position actually occupied
by the contents of the real objective field in reference to the
first square. Next suppose light to be emitted from every
point of each of these replicas, which ts at each instant similar
in every respect—t.e. the same in direction, intensity, phase,
and position of transversal—as is the light from the cor-
responding point of the original objective field at that
instant.
Under these circumstances a point p in the original objec-
tive field, along with the corresponding points p! p" &e. in
the replicas of the objective field, form a system of pvints
equally spaced over a plane which is parallel to the objective
plane. Now it is known, from the theory of diffraction
gratings (see the figure on p. 340), that such a system of
336 Dr. G. J. Stoney on Microscopie Vision.
points equally spaced in a plane, and all emitting light which
at each instant is exactly similar, will produce a disturbed
condition of the ether which is resolvable into plane waves
advancing in certain definite* directions. The same is true
of each other point of the original objective plane with its
replicas. Hence, and since by the principle of the super-
position of small motions the total disturbance in the sether
caused by the whole contents of the objective field and of all
its replicas is the resultant produced by a simple geometrical
summation of the disturbances which would be produced by
the several points of the original field and their replicas, it
follows that in ultimate analysis the total disturbance is re-
solvable into the undulations of plane waves into which its
* The luminous effects produced in these definite directions are maxima,
and they are accompanied by luminous effects produced in other directions _
also; but it is legitimate to leave these out of account. We are in fact
investigating the disturbance within a jayer of limited thickness, the layer
between the objective plane and the plane in which the front of the
objective lies; and the luminous energy expended on any effects within
that layer, other than those producing the plane wave dealt with in the
text, can be made relatively as sinall as we please by increasing the
spacing between the replicas.
This will perhaps be inade clearer by considering the analogue in a
Fourier’s expansion. If the first 2 terms of a Fourier’s expansion of any
function be added together, they furnish an approximation to that
function which is nearer the larger m is, and which can be made as close
an approximation as we please by increasing ». Now the sources of
similar light p, p’, p”, &c., furnish a number of fans of undulations of
plane waves, each fan analogous to a limited number of terms of a
Fourier’s expansion, this ihmited number being proporticnal for each fan
to 6/A, where 6 is the spacing of p, p', p’, &c., and A is the wave-length.
They are therefore susceptible of indefinite inciease by increasing 6,
Moreover, the fans which have the smaller number of terms become
rapidly the fainter: see the figure on p. 340, in which the closer the ruling
the smaller will be the number of terms of the corresponding fan.
The outcome of these considerations is that the ztherial disturbance in
fiont of the objective plane may be such that to resolve the whole of it
with absolute accuracy into undulations of plane waves would require
that these undulations shall spread in all (corresponding to 6 being in-
definitely large) instead of some (corresponding to 6 being finite)
directions. But, practically, a very moderate value for 6 is sufficient ;
‘since the approximation is carried far enough when the outstanding
luminous effects are too faint either to be seen by the eye or to affect a
photographic plate sensibly.
Even if the clusest of the replicas were much closer in than we have
‘supposed, they would not sensibly interfere with the vision of the original
object. Two or more diatoms seen together within the same field of
view do not sensibly interfere with the most satisfactory vision of each
of them, nor would they if they all emitted ight from their correspoud-
ing points which was strictly the same at each instant in phase, direction
of transversal, and intensity. Each would still be as fully seen as our
eyes ale capable of seeing, notwithstanding the presence of the others.
Se
7
Ti
Dr. G. J. Stoney on Microscopie Vision. 337
constituent disturbances are resolvable. The number of
these undulations may be reduced wherever any of them
travel in the same direction, since any number of undulations
of plane waves of wave-length A travelling in the same direction
may be combined into a single undulation of plane waves
travelling in that direction. Hence the total disturbance is
resolvable into undulations of uniform plane waves, only one
of which for each value of X travels in each direction.
¢. This valuable optical theorem bears a remarkably close
analogy to Fourier’s Theorem for the expansion of an immense
class of functions. Thus by Fourier’s Theorem a portion of
curve mn along with equidistant repetitions of the same to
the left and right may be expanded in the form
wv 21x
y=Aot Ai cos” +A, cos 2-2 +...
Dare ae
+ B, sin — +B, sin 2 ra +...
a
in which the values of the constants Ay, Ay, A», &e., B,, B,,
&c., depend on what direction has been selecied for the line
over which the repetitions are to be disposed, and on what
interval has been chosen for a (a being mm’, the spacing of
the curves from one another). So in our optical theorem, the
plane waves into which the light emitted by a point p in the
objective field is to be resolved will depend on what plane has
been chosen for the objective plane, and on the intervals at
which p, p’, p”, &c., are to occur in that plane, as well as on
whether the lines joining them lie (as we have placed them
above) at right angles to one another, or in other available
positions. However, just as in a Fourier’s expansion the
original curve is always correctly represented whatever
assumption we may have made as regards the orientation of
the axis of w and the length of the line a, and it is only the
situation of its replicas which is affected by this choice ; so
under our theorem the light in front of the objective field is
always adequately resolved whatever selection we may have
made as regards the optional matters (provided the conditions
laid down in the footnote on p. 336 are observed), and it is
only where its replicas are to be regarded as situated that is
affected by that choice. Moreover, when once we have made
Phil. Mag. 8. 5. Vol, 42. No. 257. Oct. 1896. 2B
n” m n m’” Nn’ ™m 4
338 Dr. G. J. Stoney on Microscopie Vision.
our choice as regards these optional matters, the plane waves
emanating from the whole field into which the light emitted
by the point p is to be resolved under our theorem, become
as definite and unique as do the coefficients of a Fourier’s
series when once we have decided on the direction of the line
mm’ and have selected a value for a. This, however, still
leaves a considerable latitude under our theorem, as to what
the undulations of plane waves shall be, since the objective
field may be variously chosen, and the only conditions which
limit the positions to be selected for the replicas are that they
and the original objective field be equally spaced relatively to
the objective plane, and that the nearest of the replicas shall
lie far enough outside the objective field to ensure that
whether sources of light exist in them or not shall not sensibly
interfere with what is seen by the observer. They are to
him stars below his horizon, whose positions or even existence
in no perceptible degree affect the distinctness with which he
sees the stars that are above his horizon.
8. Principles of Reversal—-A further insight into what it is
that occurs may be gained by a simple expedient. Picture a
portion of the objective plane, of limited but large size—large
enough to have the original objective field near its middle,
and a great many of its replicas disposed round it. If all of
these emit light that is exactly similarly circumstanced, then,
as already explained, it appears that they, acting together,
will produce undulations of very nearly uniform plane waves
which will become more and more disentangled from one
another the farther out they go. It is in fact when thus
disentangled that their consisting of almost quite plane waves
becomes most obvious. The approximation to accurately
uniform plane waves can, of course, be carried as far as we
please by increasing the number of replicas engaged in
emitting the light.
Let now all the zetherial motions be suddenly reversed, and
let at the same time the objective field and its replicas be got
out of the way. The distant undulations which were before
advancing outwards will now travel inwards without ceasing
to be uniform plane waves, and will by simple geometrical
superposition, according as they overlap one another, repro-
duce at each step of their inward journey exactly the same
disturbed state of the ether as had prevailed at the same
stations on the outward journey, except that the directions of
all motions are reversed. Hence plane waves converging in-
wards would by their superposition produce precisely the same
disturbance in the ether, except only with reversed motions,
as that which on the outward journey prevailed close in front
Dr. G. J. Stoney on Microscopie Vision. 339
of the object in the objective field ; and, finally, if the travelling
backwards is continued long enough for the undulations to
reach the positions that had been occupied by the original
object, they will there produce an image of it the most perfect
which the light that had been emitted by the object is capable
of producing. This image thus becomes a standard of per-
fection which may be approached but cannot be exceeded by
the images formed by any optical contrivance from the same
light.
“y rom the way in which the standard image is formed it is
manifest that it is an image of the same size and general
shape as the object or group of objects represented by it.
The further excellence of this image depends upon the amount
of detail upon the object which it is competent to reproduce ;
and this varies, as we shall find presently, with the wave-
length of the light employed, and with the way in which that
light has been supplied to the object. The standard image
may be regarded either as viewed from beyond, or as being
transformed into a picture by being thrown on a screen able
to scatter whatever light falls on it. The screen should not
be flat, but with such prominences and depressions as will
enable it to catch the light everywhere exactly where the
image is formed. Such a picture is entitled to the name of
the standard picture, since it has on it all that part of the
detail on the real object which the light is capable of
showing*
9. Theorem 2. The Standard Image.—Let us consider
somewhat more closely how the standard image is formed.
It is formed by the coalescence and mutual interference of
uniform plane waves. Now when we consider how these
same undulations originated when starting on their outward
journey and remember that the condition of the ether is the
same on their return, except as regards the direction of
the motions; when we further remember that the point p
* Another way of conceiving the standard image which is for some
purposes more convenient, is fo imagine the retreat of the luminous
undulations to be carried farther backwards (the condenser of the
microscope and any other obstruction being of course removed); and
then, at a given instant, to conceive the setherial motions to be again
reversed. ‘Ihe undulations will thereupon travel forwards (2. e. in the
direction in which the light originally moved), will re-form the standard
image when they veach the position that had been occupied by the
object, and will thence proceed to the objective of the microscope in
precisely the same state as was the light that was transmitted to it by
the real object. Jt thus appears that ‘the source of light, the condenser,
and the object may be all removed, and that the standard image emitting
its hight forward may be substituted at them.
2B2
340 Dr. G. J. Stoney on Microscopic Vision.
and its replicas emitted portions of light which at the instant
of starting were exactly alike, and that the undulations which
result from this state of things may be thrown into groups
of undulations, each of which is the same as would have been
emitted by one or other of the uniform rulings of equidistant
lines represented in the accompanying figure, as well as the
2
aN
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a
SM
EIN
SEX
OC
ww
Zax
S
y
S
y
Zz
ee
RE
S
we
Se
BR
cK KL
ps
Sexe
BN
S
WZ
S
V
iS
ZS
IN
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N
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eS
vast number of others that would arise from sufficiently ex-
tending the figure ; when we further bear in mind that every
equal element of any one of the lines in each such ruling
emits the same amount of light, which is in the same state as
that emitted by p except as regards intensity : when all these
things are taken into account we find that the entire of the
standard image may be regarded as built up of such luminous
rulings superposed upon and interfering with one another— |
each of these rulings being due to the convergence and
mutual interference of two or more undulations of the uniform
plane waves which (since the reversal) have been travelling
inwards, and each ruling accordingly being uniform and
extending across and even beyond the whole range of the
objective field.
This is our second theorem. It may be enunciated as
follows :—
PROPOSITION 2.
The standard image may be regarded as resulting from the
superposition and mutual interference of uniform luminous
rulings of equidistant parallel bright lines extending over the
whole field of view; each ruling being produced by the
Dr. G. J. Stoney on Microscopic Vision. 341
convergence upon it, after the reversal, of two or more of
the undulations of uniform plane waves into which the light
emitted by the object may be resolved.
10. Of course other resolutions than the two hitherto
considered—that into spherical wayes thrown off from the
several points of the surface of an object, and that into plane
waves thrown off from the surface as a whole—are possible :
and in fact, if a resolution of the disturbance in the eether
between the object and the objective of a microscope is made
into plane waves, these will become curved while passing
through and after emerging from the objective ; and it is as
curved waves that they reach and produce the microscopic
image. They, in fact, become convex waves that are nearly
spherical. The centres of these nearly spherical waves are
obviously the points of the focal plane (or rather, focal sur-
face, for it is slightly curved) of parallel light incident on the
objective. This focal plane lies between the objective and the
microscopic image, and in all the cases that need to be con-
sidered it lies near the objective, and therefore sufficiently far
from the microscopic image to render the curvature of the
waves where they reach that image but slight.
11. Magnijfication—Let us now return to the standard
image. It is of the same size as the object. If we could by
any contrivance increase the wave-lengths of the light that
forms it—if, for instance, we could make the wave-lengths a
thousand times larger, making them the same fractions of a
millimetre which actual light-waves are of a micron—we
should in this way enlarge the image 1000 times, since the
interference of the longer waves coming in the same directions
as before would produce rulings all of which would be 1000
times coarser than before. This enlarged image would
obviously contain precisely the same amount of detail as the
standard zmage.
This method of enlarging an image is only practicable on a
small scale, since we can but slightly increase wave-lengths
(as when we place the object in a highly refracting medium
and its image in the air); but what is very much the same
result may be brought about in another way, viz., by dimi-
nishing the inclination of the beams of plane waves to one
another, without altering the lengths of the waves ; since the
ruling which results from the interference of two such beams
may be made coarser either by lengthening the waves of
which each beam consists, or by diminishing the inclination
of the beams to one another.
12. Useful work done by the objecttve-—The useful part of
what is accomplished by the objective of a microscope is that
342 Dr. G. J. Stoney on Microscopic Vision.
it diminishes the inclination of these beams to one another.
This brings about two desired results: it enlarges the image,
and it makes it possible for its constituent beams, after they
have passed the focal image, to be collected by the eyepiece
and transmitted through so small an opening as the pupil of
the eye, instead of diverging over the great extent to which
they were spreading when they left the object.
13. Useless work done by the oljective—But the objective
cannot accomplish this useful work without at the same time
producing other effects which are undesirable. Thus, it
transforms the beams of plane waves into convex beams, as
explained in § 10. This somewhat distorts the image. The
image is still more distorted in the direction of the line of
sight, whereby any elevation on the object is shown as unduly
prominent in the image™. Neither of these distortions,
however, would cause the amount of detail in the microscopic
image to fall short of that in the standard image.
That which above all produces this defect, and produces it
however well the spherical and chromatic aberrations of the
objective may have been corrected, is that the angular aper-
ture of the objective falls short of 180°. With the best
immersion-lenses the angular aperture is about 120° or 180°,
so that little more than half the ight would be caught by
the objective, if the light were emitted equally in all direc-
tions. One part of the light thus excluded is that which
in the standard image brings out the finest part of the detail
which that image can reach, since it is the light which pro-
duces the finest of the rulings that form the standard image.
There is another imperfection consequent upon this exclu-
sion of part of the light emitted by the object, viz., the
intrusion into the microscopic Image of intercostal markings,
false resolutions, a general haze of light, and so on—additions
to the image and other alterations of it which have nothing
to correspond to them either in the object under examination
orinits standard image. ‘The following is perhaps the easiest
way of understanding how they arise.
14. The Visual Substitute.—In order to study microscopic
vision, or vision of any kind, with full effect, it is well to
begin with the consideration that what we seem to see with
the naked eye is never the natural object itself, nor is it an
enlargement of it when we examine it through a microscope
or telescope. What we see is, in fact, only a visual substitute
for the real object in the first case, and for an enlargement of
the same when we use an instrument; and the study of
* This distortion may be traced by an elementary investigation in
geometrical optics. _
Dr. G. J. Stoney on Microscopic Vision. 343
vision, whether microscopic, telescopic, or with the naked
eye, is in fact the study of what this visual substitute is and
how it stands related to the real object, 2. e., what alteration
the real object would have to undergo to be transformed into
its visual substitute, which is what seems to us to be the
object presented to us.
The real object, O, sends forward the light which enters the
eye, and, in addition, other light which does not enter the eye,
whereas its visual substitute, S, is to be defined as that other
object from which would emanate the light which enters the
eye and it only. It is evident that objects O and 8 will seem
to us exactly alike, but that whereas we receive the whole of
the light which 8 is competent to dispense, we receive only a
part of that emitted by O. Similarly, when we use a micro-
scope or telescope, what we seem to see is a visual object, C,
which would emit exactly the light which the eye takes in,
and it only ; and this is in all cases less than the light which
an enlargement of the object would emit, and may differ from
it in other respects also. It is, accordingly, to the study of
what these visual substitutes are that we should apply our-
selves.
But as this is a branch of optics which is as yet almost
wholly unexplored *, we must, for the present, be content
with the inferences we can draw from such general considera-
tions as the following :—
15. Proposition 3.—The objective of a microscope has an
angular aperture which is necessarily less than 180°. Hence
the image formed by it is formed by a part only of the light
emitted by the object.
Imagine a hemisphere in front of the object, of so large a
size that the whole object may be treated as though it were
* In one simple case investigated by the writer the visual substitute
fora thin line of light proved to be a double line with a narrow interval
and with very thin appendage-lines on either side. Here we have some
of the phenomena presented by microscopes—a spurious resolution into a
double line, and appendage-lines which are of the same nature as inter-
costal markings. See abstract of communication to British Association,
at p. 583 of the ‘ Report’ for 1894,
344 Dr. G. J. Stoney on Microscopie Vision.
at its centre. The luminous beams* of plane waves, each
emanating from the whole front of the surface of the object,
spread over this hemisphere, and the only case we need at
present consider is where the pupil of the eye (in naked-eye
observations) or the front lens of the objective (when we use
a microscope) takes in only the beams A, viz., those beams of
parallel waves thrown off from the surface of the object which
are directed towards the middle sector of the hemisphere, and
fails to admit the beams B, which are directed towards the mar-
ginal parts of the hemisphere. The excluded beams are partly
Ba, those which, if reversed, form the finer of the rulings that
go to build up the standard image. The rest of these beams,
viz. Bb, are the more oblique members of those fans of beams
which produce the coarser rulings—the whole of the standard
image being made up of these finer and coarser rulings (see
§ 9), whereas the image seen by the observer is made up by the
beams A alone—by those which the front lens of the objective
can catch. :
Let us now define —B to be the same light as + B, except
that all the phases in —B are at each instant the reverse of
what they arein +B. In other words we get the light —B
by adding a to all phases in the light B; hence if the light
+B and the light — B are both present, they exactly cancel
one another.
Now the whole light emitted by the object is A+B; and
it is this light which forms the standard image. Hence, if we
add the light —B to the standard image, and can find what
modification of that image is thereby effected, we thus arrive
at the best image which the light A can form: an image
which the image actually formed on a large scale by the
objective may approach in perfection, but cannotexceed. We
may appropriately name it Standard Image No. 2.
In order to arrive at standard image No. 2, we may add
the portions of light —Ba and —Bd in succession to standard
image No. 1, as these together make up the whole of the light |
—B. The addition of —Ba simply obliterates the finer of
the rulings out of which the standard image is constructed.
* It is convenient to use the word undulation where the waves extend
to an infinite or to an indefinite distance in their plane, and to employ the
terms beam and pencil where we intend the lateral extent of the waves
to be regarded as limited.
Practically luminous beams of plane waves emanating from the
objective field, which is, of course, of limited extent, may be used instead
of the undulations of the theory, which emanate from the entire objective
plane; since the waves of a beam, unless very narrow, do not differ
sensibly from the waves of the undulation, except close up to its bounding
eylinder.
Dr. G. J. Stoney on Microscopic Vision. 345
‘The chief (though not quite the only) effect of this is simply
to render the image incapable of exhibiting some very fine
detail upon the object which before it was able to reach. But
the addition of — Bd has a worse effect. J+ udds to the image
an entirely new set of fine rulings which do not represent any
of the features which exist upon the object, and by this ight
such false effects as intercostal markings, spurious resolutions,
a general haze of light, &., are apt to be, and often are, pro-
duced. Hence we may enunciate Proposition 3 as follows:—
-Propostrion 8.
When, of the light emitted by the olyject, only part is employed
to form the microscopic image, then features may. intrude them-
selves into the microscopic image which are not present in the
standard image, and which do not represent anything upon the
object. | | |
16. Proposition 4. False Colouwration—Another deceptive
effect which is to be referred to the limited apertures of
objectives is the appearance given to uncoloured objects of
being coloured. Only the general principle to be kept in
view will be stated here, as a fuller treatment of this phe-
nomenon can be more conveniently made in connexion with
individual instances which will be dealt with in Part II. of
this memoir. .
The whole light of wave-length X which is sent forward by
the object may be divided into A, which is admitted to the
objective, and B, which is excluded. A similar partition
into these two portions is to be made of the whole light
of each wave-length, but the proportion in which the whole
light is divided between them in general varies from one
wave-length to another. Hence, if the illumination is by
white light and the object uncoloured, there may be a pre-
ponderance of Leht of some colours in A as compared with
others, and an equal deficiency of these same colours in B.
In such cases the image seen in the microscope, since it is
exclusively formed by the light A, has not got the colours
mixed in the same proportions as they are in white light, and
accordingly appears coloured. Hence
PROPOSITION 4.
Under the same circumstances as in Proposition 3, the par-
tition of the light between the portions rececved by and excluded
from the oljective, will in general be different for different wave-
lengths ; and-when the difference is marked a colourless object
will appear to be coloured in the mreroscope.
346 Dr. G. J. Stoney on Microscopic Vision.
17. Proposition 5. The Condenser.—The standard image
admits of being either better or worse. It manifestly admits
of being improved by forming it out of light of shorter wave-
length ; and this may be accomplished either by exchanging
the colour of the illumination employed for a colour of higher
refrangibility, or by mounting the object ina highly refracting
medium. 3
But the degree in which the standard image correctly
represents the object. usually depends even more upon the
condenser. In fact, the disturbance of the ether in front of
the object is determined partly by the features of the object
and partly by the condition of the light which illuminates it.
This is evident because if the reversal spoken of in § 8 were
to take place without removing the object, the light in re-
tracing its steps would first reproduce the disturbed state that
had existed in front of the object ; would next form the
standard image upon the surface of the object ; and would
then pass through the object and form beyond it whatever
disturbed state of the ether had existed between the con-
denser and the object. Hence, that the standard image may
represent the features of the object unmixed with other
appearances not belonging to the object, it is essential that the
light provided by the condenser shall be as nearly uniform
and featureless as possible where it reaches the part of the
object which is being scrutinised. Hence the importance of
thin sections, and of a very well-corrected condenser.
The management of stops, and their function, can be better
treated of in Part II. of this memoir, when we can enter into
details. For the present we content ourselves with a very
general proposition, viz. :—
PROPOSITION 5.
The standard image is the outcome, partly of the features
upon the object, and partly of the state of the light by which the
object 7s illuminated. It may be improved by increasing the
degree in which the first of these factors, and by decreasing the
degree in which the second, contributes to produce, to modify,
or to efface detail in the image.
18. Proposition 6.—When an object is mounted in a more
refractive medium than that in front of the objective, standard
image No. 1, which depends on the wave-length of the light
as it quits the object, is thereby improved ; but standard image
No. 2 is not enabled to grasp any finer detail upon the object
than it would have grasped if the object had been in a medium -
of the same refractive index as that in front of the objective.
Dr. G. J. Stoney on Meroscopic Vision. 347
That is, none of the luminous rulings which form the useful
part of standard image No. 2—none of those that represent
any feature of the “object, excluding those which produce
false effects like intercostal markings—are made any finer by
mounting the object in a medium of extra high refractive
index. But nevertheless an important effect is pr sroduced, Viz.
that the finer of the rulings are made relatively brighter than
they were before, so that, ceteris paribus, the detail which
they portray becomes more conspicuous.
This is evident from the accompanying diagram, in which
ab is the front of the objective and o the object. Both figures
a 6
(a a ee
T
4)
represent the course of one of the more oblique beams of
parallel waves from the whole surface of the object, the first
figure representing what occurs when the object is mounted in
a medium of the same refractive index as the cover-glass and
immersion oil, and the second figure representing what occurs
when the object i is mounted in an optically denser medium.
Ceteris paribus, the ratio of the brightness of the beam that
reaches the objective in the two cases is as cosi/cos 7, which
son );
where n and n’ are the refractive indices in the two media.
348 Dr. G. J. Stoney on Microscopie Vision.
This is a fraction which the more deviates from unity the
greater 7 is, 1.¢. the more oblique the beam. Hence, the
more oblique beams, which bring out the finer detail, are
more increased in brightness than the less inclined, which
deal with the larger features of the object. Hence
PROPOSITION 6,
Mounting the olject in a medium of eatra high refractive
index will, ceteris paribus, increase the conspicuousness of the
jiner detail to be seen upon tt.
Of course other factors, some of which may be even more
potent, have to be taken into consideration, such as the ratio
of the index of refraction of the object to that of the medium
in which it is mounted ; for the farther this ratio is from
unity, the more conspicuous do all the features of the object
become.
19. Proposition 7. Optical Contact—Another proposition
which is of use in interpreting the phenomena presented by
the microscope is a consequence of the condition of the ether
in the rare medium when light is totally reflected from a
surface separating a dense and a rare medium. What then
occurs has been investigated by Sir George Stokes, in his
masterly paper “On the Formation of the Central Spot of
Newton’s Rings beyond the Critical Angle”’ (vol. ii. of
Stokes’s Collected Papers, p. 56). It is therefore only
necessary here to enunciate the result of that investigation in
the form in which it explains optical events which the micro-
scopist has occasion to make use of.
Normally, when a microscopic object is ‘‘mounted dry,”
2.é€. is situated in an air-space between the slip and the
cover-glass, no rays from it can, while traversing the cover-
glass, be more inclined to the vertical than the “ critical
angle.” Now immersion objectives are specially designed to
admit rays that have passed upwards through the cover-glass
in more inclined directions. Accordingly, when an object
that is mounted dry is examined by an immersion objective,
what normally happens is that only a part of the aperture of
the objective is made use of. The event is, however, different
if the microscopic object is excessively close to the cover-
glass, owing to the phenomenon investigated by Sir George
Stokes. |
It follows from Sir George Stokes’s investigation that when
a plane separates an optically dense from a rare medium, then
there is a very thin layer of the rare medium of which the
optical properties are peculiar. In cases of total reflexion, the
15
Dr. G. J. Stoney on Microscopic Vision. 349
ether within this layer is brought into a disturbed condition.
The disturbance in reality penetrates further down, but fades
out by a law so rapid that it is only sensible within a very
short distance (which depends on the wave-length) of the
plane separating the media. ‘The layer of this small thickness,
within which the disturbance is sensible, may be called the
Stokes’s Layer.
It further follows from the investigation, that if light
emanates from a point within the Stokes’s layer, it will be able
to pass up through the dense medium at angles that exceed
the critical angle. It is easy to verify this experimentally.
Take a glass prism—one of the pendants of a glass chandelier
is sufficient. Hold it at the distance of distinct vision from
the eye, and turn it till the light of the sky is seen like
a silvery sheen, totally reflected from the inside of one of its
faces. Then, without moving the prism or the eye, press a
piece of chalk against the outside of that face. A small
portion of the chalk can thereby be brought so close to the
glass that the intervening chink is less than the thickness of
the Stokes’s layer. This small portion of the chalk will then
be seen through the face of the prism, while the rest of the
chalk and the hand that holds it, which are beyond the Stokes’s
layer, are quite unseen. The light from the chalk, by which
it is seen, has obviously passed through the glass at an angle
which is beyond the critical angle. Similarly :
PROPOSITION 7.
Tf a microscopic olject, mounted dry, is so close to the cover-
glass that the chink of air between it and the cover-glass is less
than the thickness of the Stokes’s layer, then light from it can
pass up through the cover-ylass and the owl above it, at angles
both within and beyond the critical angle, and may accordingly
reach any part of the front of an objective whose NA is more
than 1.
20. With the help of these seven propositions, supple-
menting the more familiar Jaws of optics, nearly everything
in microscopic vision may be explained, and useful rules can
be deduced for the manipulation of the instrument. The next
part of this memoir will deal with applications of this kind.
[To be continued. |
frs50 4
XXXVI. The Genesis of Dalton’s Atomic Theory.
By Henry Drsus, PA.D., FRS*
IR H. E. ROSCOE and Mr. A. Harden have lately
published a book f entitled ‘A New View of the Origin
of Dalton’s Atomic Theory,’ with the following introductory
remarks :— 7
‘“‘Jt may seem remarkable that, after the lapse of nearly a
century, since John Dalton first applied the atomic theory of
matter to chemical phenomena, it should be possible to find
anything new respecting the genesis of his ideas. The ex-
planation is to be found in the unlooked for discovery, in the
rooms of the Literary and Philosophical Society of Manchester,
where the whole of Dalton’s experimental work was carried
out, of his Laboratory and Lecture Note-Books contained in
a number of manuscript volumes. A careful study of these
has led us to conclusions concerning the origin of the atomic
theory of Chemistry which differ widely from those which
have been generally accepted. It has hitherto been supposed
that it was the experimental discovery of the law of combina-
tion in multiple proportions which led Dalton, seeking for an
explanation of this most remarkable fact, to the idea that
chemical combination consists in the approximation of atoms of
definite and characteristic weight, the atomic theory being thus
adopted to explain the facts discovered by chemical analysis.
... The actual relations are, therefore, precisely the inverse
of those which are usually accepted. Itwas the theory of the
existence of atoms of different weights which led Dalton to
the discovery of the facts of combination in multiple pro-
portions.”
This view of Roscoe and Harden is not new! Two years
ago 1 published a pamphlett, wherein it is clearly stated that
the atomic theory led Dalton to the discovery of the law of
multiple proportions. It is gratifying to me that Dalton’s
Note-books confirm the view expressed by me in May 1894.
In the same essay (p. 58), I have shown that several years
before Avogadro Dalton had formed the hypothesis that equal
volumes of different gases contain under normal conditions of
temperature and pressure an equal number of molecules.
* Communicated by the Author.
t ‘A New View of the Origin of Dalton’s Atomic Theory, a Contri-
bution to Chemical History,’ by Henry EK. Roscoe and Arthur Harden.
London: Macmillan & Co. 1896.
t ‘Ueber einige Fundamentalsatze der Chemie, insbesondere das
Dalton-Avogadro’sche Gesetz. Hine historische Untersuchung von
Dr. Heinrich Debus.’ Cassel: Gustay Klaunig, 1894, pp. 44-45.
The Genesis of Dalton’s Atomic Theory. ao1
According to Roscoe and Harden this latter statement of
mine is not correct. My argument rests, they assert, on a
confusion between the relative density of the atoms and the
relative density of the gases made up of these atoms.
I cannot allow Dalton’s merits to be set aside in this
manner, and I feel at the same time obliged to show that I
am not guilty of the mistake attributed to me.
The atomic theory and the law of equal volumes (law of
Avogadro) constitute the basis of scientific chemistry. As it
is a matter of importance for everyone to know how the
foundation of his house has been laid, I believe the following
discussion will not be. unacceptable to scientific chemists,
especially on account of some new matter considered in it.
As Dalton’s nomenclature and views differ in several points
from those now in vogue, a few explanations will be necessary.
The elementary gases consist, according to Dalton, of very
small, indivisible particles, called atoms. Hach atom is sur-
rounded by an atmosphere of heat of from one to two
thousand times its own size. Whatever, therefore, may be
the shape or figure of the solid atom abstractedly, when sur-
rounded by such an atmosphere it must be globular ; but as
all the globules in any small given volume are subject to the
same pressure, they must be equal in bulk, and will, there-
fore, be arranged in horizontal strata, like a pile of shot (N.S,
145). The volume of the molecule of a gas is equal to the
volume of the solid nucleus plus the volume of the atmosphere
of heat. Compound molecules result from the juxtaposition
of two or more different atoms. A molecule of oxygen is
composed of only one atom of oxygen, a molecule of carbonic
oxide of an atom of carbon and one of oxygen. Dalton
usually employs the word atom ; he calls a molecule of water
an ‘‘ atom of water ;”’? a molecule of ammonia an “ atom of
ammonia.” I shall express Dalton’s ideas in modern phrase-
ology.
Abbreviations.
N.S. stands for: ‘New System of Chemical Philosophy,’ by
John Dalton. Manchester, 1808-1810.
R. i » ‘A New View of the Origin of Dalton’s
Atomic Theory,’ by H. H. Roscoe and
A. Harden. London: Macmillan & Co.,
1896.
O. i; 5 Ostwald’s Klasstker, No. 8. Leipzig: W.
Hngelmann.
* » Alembic Club Reprints, No. 2. Edinburgh:
W. F. Clay.
352 Dr. H. Dabus on the Genesis of
D. stands for ‘ Ueber einige Fundamentalsitze der Che-
mie, insbesondere das Dalton-Avoga-
dro’sche Oe von Dr. Heinrich
Debus. Cassel: Gustav Klaunig, 1894.
M ‘3 » Molecular Weight. :
S . » Specific oravity.
M/S = 5, Molecular volume.
Division of numbers is expressed thus, M/S:
M
Zz = IS3
(Oz kn) page ... in this paper.
The empirical law that equal volumes of different gases
contain, at normal temperatures and pressure, an equal num-
ber of molecules will be represented by the symbols M/S=C.
I, DALTON’s Atomic THEORY.
The opinion used to be common amongst chemists * that
Dalton had originated the idea of indivisible particles (atoms)
for the explanation of the law of multiple proportions. This,
however, is not the case. The idea that the objects of ob-
servation are aggregates of exceedingly small indivisible par-
ticles is older than science itself; it is attributed to a Phe-
nician philosopher, Moschus, living at Sidon at about 1100 B.c.
(D. 35). His views, developed by the Greeks, were forgotten
after the destruction of the Roman empire, but resuscitated
by Gassendi about the middle of the 17th century. Boyle,
who called the atomic hypothesis the Phenician philosophy 7,
applied it to the explanation of chemical phenomena, and
Newton to the explanation of Boyle’s law. Also the chemists
of the last century employed the atomic doctrine for the
illustration of chemical change (D. 38-40). Dalton, who —
seems to have obtained the idea of atoms from Newton (R. 13,
123), applied it, almost from the beginning of his scientific
career, to the explanation of physical phenomena, such as
diffusion, absorption, and expansion. In 1803 he discovered
a method how to determine the relative w eight of atoms, and
added to the atomic philosophy a series of principles. The
group of principles so added by Dalton I propose to eail
Dalton’s ‘atomic theory.”
The Sale (putas are :—
* Liebig, Handbuch der Cheme, Heidelberg, 1843, P: 65.
+ ©The Pwo of the Right Hon. Robert Boyle.’ London, 1744,
vol. i. p. 228.
Dalton’?s Atomic Theory. Soa
(1) When only one combination of two bodies, A and B,
can be obtained, it must be supposed to be a binary
one, a compound of one atom of A with one atom of
B, unless some cause appear to the contrary.
(2) When two combinations are known, they must be pre-
sumed to be a binary (A+B) and a ternary (A,+ B)
or (A+B,).
(3) When three combinations are obtained, we may expect
one to be a binary and the other two ternary.
(4) When four combinations are observed, we should ex-
pect one binary, two ternary, and one quaternary
(A,+ B) or (A+ Bs). .
(5) A binary compound should always be specifically
heavier than the mere mixture of its two ingredients.
(6) A ternary compound should be specifically heavier than
the mixture of a binary and a simple which would,
if combined, constitute it; &e.
(7) The above rules and observations equally apply when
two bodies, such as (C+D) and (D+E) are com-
bined (N.S. 214). :
The question is: How did Dalton arrive at these principles ?
Roseoe and Harden think that the answer to this question is
contained in some notes which Dalton had written down for
a lecture delivered in London on Jan. 27th, 1810 (R. 138).
The pertinent passages of the notes are as follows :—
“ As the ensuing lectures on the subject of chemical ele-
ments and their combinations will perhaps be thought by
many to possess a good deal of novelty as well as importance,
it may be proper to give a brief historical sketch of the train
of thonght and experience which led me to the conclusions
about to be detailed.
“Having been long accustomed to make meteorological
observations, and to speculate upon the nature and constitu-
tion of the atmosphere, it often struck me with wonder how a
compound atmosphere, or a mixture of two or more elastic
fluids, should constitute apparently a homogeneous mass, or
one in all mechanical relations agreeing with a simple atmo-
sphere.”
In explanation of the foregoing remarks it may be men-
tioned that the molecules of all gases, therefore the atoms of
the constituents of the atmosphere, oxygen and nitrogen, were
at the time assumed by Dalton to be of equal volume: in
other words, equal volumes of oxygen and nitrogen gas would,
under normal conditions, contain an equal number of mole-
cules (N.S. 188). Hence the specific gravity of an atom of
Phil. Mag. 8. 5. Vol. 42. No. 257. Oct. 1896. 2
BD4 Dr.-H. Debus on the Genesis of
oxygen must be greater than the specific gravity of an atom
of nitrogen.
This view, in connexion with the general theory of gases
then in vogue, led to the conclusion that, in a mixture of
oxygen and nitrogen, the heavier atoms of oxygen ought to
form a layer at the bottom of the vessel and the lighter
nitrogen atoms a separate layer at the top.
- Experiments, on the other hand, proved that air, taken
from different heights, was of uniform composition. Dalton
also found that a lighter gas, placed above a heavier gas, will
gradually diffuse downwards (N.S. 151). To explain these
observations a weak chemical affinity was assumed between
the molecules of different gases. The globular molecules
Dalton considered to be arranged like a pile of shot (N.S.
145, 147), and as all the molecules are subject to the same
pressure and exert the same counter-pressure, they must be
all, independent of their nature, of the same size (N.S. 188),
M/S == M’/S’= M”/S”= Ve Se
The air is composed of 77°88 vols. of nitrogen, 21:2 vols. of
oxygen, and 0°066 vol. of carbonic acid. Hence, according
to the above law, for every molecule of carbonic acid there
are 31°8 mols. of oxygen and 118 mols. of nitrogen present in
eur atmosphere.
Now, if a weak chemical affinity is the cause of diffusion,
then compounds of a very complicated nature must result.
This appeared to Dalton very improbable, and accordingly he
rejected the chemical explanation of the diffusion of gases.
Dalton goes on with his notes as follows :—
“In the year 1801 I hit upon an hypothesis (N.S. 153)
which completely obviated these difficulties. According to
this, we were to suppose that the atoms of one kind did not
repel the atoms of another kind, but only those of their own
kind. This hypothesis most effectually provided for the dif-
fusion of any one gas through another, whatever might be
their specific gravities, and perfectly reconciled any mixture
of gases to the Newtonian theorem (R. 15). Every atom of
both or all the gases in the mixture was the centre of repul-
‘sion to the proximate particles of its own kind, disregarding
those of the other kind. All the gases united their effort in
counteracting the pressure of the atmosphere, or any other
pressure that might be opposed to them. This hypothesis,
however beautiful might be its application, had some impro-
‘bable features. We were to suppose as many distinct kinds
-of repulsive power as of gases ; and, moreover, to suppose
Dalton’s Atomic Theory. - 355
that heat was not the repulsive power in any one case ;
positions certainly not very probable.
‘““ Upon reconsidering this subject, it occurred to me that I
had never contemplated the effect of difference of size in the
particles of elastic fluids, or when the expression M/S=C is
of different value for different gases. And if the sizes be
different, then on the supposition that the repulsive power is
heat, no equilibrium can be established by particles of unequal
sizes pressing against each other. .
“This idea occurred to me in 1805. I soon found that the
sizes of the particles of elastic fluids must be different. For
a measure of azotic gas and one of oxygen, if chemically
united, would make nearly two measures of nitric oxide, and
those two could not have more molecules of nitric oxide than
one measure had of oxygen or nitrogen.
“Hence the suggestion that all gases of different kinds
have a difference in the size of their molecules ; and thus we
arrive at the reason for that diffusion of every gas through
every other gas, without calling in any other repulsive power
than the well-known one of heat.
“This then is the present view which I have of the consti-
tution of a mixture of elastic fluids (year 1810). The different
sizes of the particles of elastic fluids under like circumstances
of temperature and pressure being once established, it became
an object to determine the relative sizes and weights, together
with the relative number of atoms in a given volume. ‘This
led the way to the combinations of gases, and the number of
atoms entering into such combinations, the particulars of
which will be detailed more at large in the sequel..... es
(R. 13-17).
From these statements of Dalton, Roscoe and Harden
deduce the following genesis of the atomic theory :—
‘The balance of evidence is, therefore, strongly in favour of
the statement made in London by Dalton himself in 1810, that
he was led to the atomic theory of chemistry in the first
instance by purely physical considerations in opposition to
the view hitherto held by chemists, that the discovery by
Dalton of the fact of combination in multiple proportions led
him to devise the atomic theory as an explanation. It, there-
fore, becomes necessary for us to modify our view as to the
foundation of the atomic theory. There seems to be no
doubt that the idea of atomic structure arose in Dalton’s
mind as a purely physical conception, forced upon him by
his study of the physical properties of the atmosphere and
other gases. Confronted, in the course of his study, with
the problem of ascertaining the relative diameters of the par-
2C2
356 Dr. H. Debus on the Genesis of
ticles, of which, he was firmly convinced, all gases were made
up, he had recourse to the results of chemical analysis.
Assisted by the assumption that combination always takes
place, in the simplest possible way, he thus arrived at the idea
that combination takes place between particles of different
weights, and this it was which differentiated his theory from
the historic speculations of the Greeks. The extension of
this idea to substances in general necessarily led him to the
law of combination in multiple proportions, and the. com-
parison with experiment brilliantly confirmed the truth of his
deduction ” (R. 49-51).
The problem of ascertaining the relative diameters, or rather
the relative volumes, of the molecules of different gases, led
Dalton, according to Roscoe and Harden, to the invention of
the atomic theory. Inthe pamphlet mentioned before (p. 350),
I have stated that Dalton wished to verify the hypothesis
M/S=C, viz. to ascertain whether the molecular volumes of
different gases are of equal or of different magnitudes. For
this purpose he required to know the respective molecular
and atomic weights. This necessity led him to the formation
of the atomic theory (D. 58).
I am much pleased to observe that, with regard to the
problem which originated the atomic theory, Roscoe and
Harden, after a careful study of Dalton’s note-books, arrive
at the same view which I published two years ago, but I
regret very much that I cannot approve the reasons which
have guided their judgment.
The first assertion of Roscoe and Harden, “the balance of
evidence is, therefore, strongly in favour of the statement
made in London by Dalton himself in 1810, that he was led
to the atomic theory of chemistry, in the first instance, by
purely physical considerations,’ I have not met, nor do I
remember to have seen in any of Dalton’s writings, nor am I
able to deduce such an assertion from Dalton’s notes, quoted
before. These notes treat of two distinct subjects, the
theories of gaseous diffusion and the problems of the atomic
theory. The description passes abruptly from the one to the
other without any connecting link. How did Dalton get
over the chasm which separates the two? We want to know
the genesis of Dalton’s methods of determining atomic weights.
This, the essence of the matter, Messrs. Roscoe and Harden
pass over in silence !
Dalton explained the diffusion of gases on the assumption
that the molecules of different gases are not of the same size.
The question, therefore, was, Are the molecules of different
gases really of unequal volume? “J soon found,” says
Dalton’s Atomic Theory. 3957
Dalton,” that the sizes of the particles of gases must be dif-
ferent. Fora measure of azotic gas and one of oxygen, if
chemically united, would make nearly two measures of nitric
oxide, and those two could not have more atoms (molecules)
of nitric oxide than one measure had of azote or oxygen.
Hence the suggestion that all gases of ditferent kinds have a
difference in the size of their molecules ; and thus we arrive
at the reason for that diffusion of every gas through every
other gas....” (R. 16-17). The old view of Dalton, that
in gases the force of cohesion is quite overcome by the force
of repulsion, and that, in consequence, the particles of the
elementary gases consist of single atoms, would lead to the
conclusion which he drew from the volumetric proportions of
nitrogen and oxygen in nitric oxide, without requiring the
atomic theory as described on p. 353.
Passing on to the atomic theory we have the remarks :—
“The ditterent sizes of the particles of gases, under like
circumstances of temperature and pressure, being once estab-
lished, it became an object to determine the relative sizes and
weights together with the relative number of atoms in a given
volume.”
If we remember that these quotations are not from a running
narrative, but from notes intended for a lecture, where the
lecturer could at any moment make verbal explanations and
additions, then we may ask whether, between the notes on the
diffusion of gases and those on the atomic theory, there is, or
there is not, a connexion similar to the one between cause and
effect, as Roscoe and Harden assume? On this point Dalton is
absolutely silent. We wish to know the origin of the methods
by means of which Dalton determined the size, weight, and
numbers of the atoms ; and we cannot discern how he arrived
at these methods from his theory of gaseous diffusion. Messrs.
Roscoe and Harden do not say one word on this subject. May
not Dalton have been engaged in the calculation of molecular
and atomic weights and molecular volumes before he recog-
nized that the size of the molecules of different gases Must be
different? A few facts show that he was so engaged.
The idea of considering the influence of the size of the mole-
cules on diffusion occurred to Dalton, according to his own
statement, in 1805. The result of his consideration was the
abandonment of the theory of diffusion formed in 1801 (de-
scribed on pp. 354, 355), and the formation of a new view,
which ascribed diffusion to difference of size of the molecules
of the diffusing gases. |
This new view is, according to Roscoe and Harden, the
source of Dalton’s atomic theory. If so, then the atomic
358 Dr. H. Debus on the Genesis of
theory cannot have been formed before the year 1805. But
Roscoe and Harden found in Dalton’s note-book a table of
atomic weights dated the 6th of September, 1803. This date
shows that the theory is at least two years older than it ought
to be according to Roscoe and Harden’s view (R. 29). In
order to get over this difficulty, they assume (R. 25) that
Dalton committed a clerical error by writing 1805 instead of
1803 in his description of the formation of the theory of
diffusion based on an unequal size of the particles. Daltcn
does not say that the theory of diffusion led him to the con-
ception of the atomic theory. The statement that it did do so
is only an inference of Messrs. Roscoe and Harden. Therefore
they are not justified in attributing a clerical error to him, and
transferring the conception of the theory of diffusion from 1805
to 1803.
Dalton’s paper, “ Experimental Enquiry into the Pro-
portions of the several Gases or Hlastic Fluids constituting
the Atmosphere,’ was printed in the ‘ Memoirs of the Literary
and Philosophical Society of Manchester,’ second series, vol. 1.
1805, under Dalton’s supervision as Secretary of the Society
(R. 31). We have in this paper the following remarks on the
principle that the elastic or repulsive power of each molecule
is confined to its own kind :—“ This principle accords with all
experience, and, I have no doubt, will soon be perceived and
acknowledged by Chemists and Philosophers in general”
(Al. 5). Now this principle is the principle of the theory of
diffusion formed in 1801 (pp. 354, 355); consequently he must
have still held this theory in 1805, at the time when the paper
was printed, and the theory of diffusion which superseded it
must have been formed at a later period in the year 1805.
Therefore this later theory of the diffusion of gases cannot have
been the origin of the atomic theory of 1803. The conclusion
we arrive at is: “ The opinion of Roscoe and Harden that the
theory of the diffusion of gases of 1805 led Dalton to the con-
ception of his atomic theory is not in accordance with facts.”
II. The Genesis of Dalton’s Atomic Theory.
The notes published by Roscoe and Harden enable me to
render my view of the genesis of this theory, described on
p- 356 (D. 58), more accurate and complete.
In order to explain equilibrium in a mixture of gases,
Dalton had adopted, in the year 1801, the hypothesis M/S=C.
But he was not able for several years to test his conception by
experiments, as no method was known for the determination
of M, the molecular weight. The desire to discover such a
method made him, probably, very observant of all circum-
stances which could promote his wish (N.S. 187-188).
Dalton’s Atomic Theory. ~ 309
Two factors determine every great advance in science—a
fact, and a mind prepared for the full and complete appre-
ciation of the fact. Moreover, the fact must be at hand when
it is wanted by the mind. These conditions were satisfied
when Galileo observed the oscillating lamp in the Cathedral
of Pisa; when Newton saw the falling apple ; when Malus
observed the ray reflected from the window of the Luxem-
bourg ; when Lavoisier was informed of the discovery of
oxygen by Scheele and Priestley (D. 20-21).
The discovery of oxygen is the great fact with which the
history of scientific chemistry commences ; but neither of the
discoverers—neither Priestley nor Scheele—recognized the
significance of the discovery. Lavoisier’s mind was the good
soil wherein the seed bore fruit, because it was well prepared
by a careful repetition of the experiments of Black and Mever.
These experiments prove that carbonates are combinations of
a gas which can be transferred from one base to another.
Quicklime absorbs carbonic acid from the atmosphere and
turns into carbonate of lime. These facts raised in Lavoisier’s
mind the question : Do metals absorb a gas from the atmo-
sphere during calcination? The answer from his own expe-
riments was a decided “ Yes.”? But as he could not prepare
the gas absorbed, the investigation could not proceed. Then,
just at the right moment, Scheele and Priestley informed
Lavoisier of the discovery of oxygen, and the antiphlogistic
chemistry was born (D. 21).
Dalton, the meteorologist, wished to determine the compo-
sition of the atmosphere. Tor this purpose he tried the
absorption of oxygen with nitric oxide. In his note-book
occurs the following memorandum, dated August 4th, 1803:—
‘It appears, too, that a very rapid mixture of equal parts
of common air and nitric oxide give 112 or 120 residuum.
Consequently, that oxygen joins to nitric oxide sometimes
1:7: 1 or at other times 3°4:1” (R. 38, Al. 8).
_ In the Essay on the Composition of the Atmosphere occurs
the remark, ‘‘ These facts clearly point to the theory of the
process ; the elements of oxygen may combine with a certain
portion of nitric oxide, or with twice that portion, but with no
intermediate quantity’ (R.33, Al. 9).
A month after the date of the above note, on September 6th,
1803, the note-book contains a list of atonsic weights (R. 29),
and on the 19th of the same month the diameters of the
molecules of a certain number of gases were written down
R. 41).
. May we venture to conclude, on the strength of these facts,
that the observations of the proportions in which nitric oxide
and oxygen combine over water originated in Dalton’s mind
360 Dr. H. Debus on the Genesis of
the idea that the atoms combine chemically only according to
very small numbers—one atom of A with one, two, or three atoms
of B? ‘There are strong reasons in support of this conclusion.
Dalton, who for some years had held the hypothesis M/S=C
(N.S. 188), who was accustomed to explain expansion, diffu-
sion, solution, &c., in a mechanical way by means of the old
atomic theory, as Boyle did 120 years before Dalton’s time,
was naturally led by the observation of the combining pro-
portions of oxygen and nitric oxide to the view that in simple
compounds one atom of A is united to one or two atoms of
B. His previous speculations and observations (p. 354) had
prepared him for this conception, and once the idea was
formed the transition to the tenets described on p. 353 was
easy. Proust and others had made observations like Dalton,
but to none had this explanation occurred.
My present view of the origin of Dalton’s atomic theory
1s :—
Dalton wished to verify the hypothesis M/S=C;; for this
purpose he endeavoured to find methods for the determination
of the atomic and molecular weights. The observations of the
combining proportions of nitric oxide and oxygen suggested
to him a principle which enabled him to form such methods.
In judging this view, one has to remember that the obser-
vations on the combining proportions of oxygen and nitric
oxide noted on August 4th, 1803, were followed within a
month by calculations of atomic weights and the diameters of
atoms and molecules. (R.38 and 41.)
III. Roscoe and Harden’s Critique.
The first part of my view of the genesis of Dalton’s atomic
theory, described above, is, according to Roscoe and Harden,
not correct. They say :—
““The view expressed by Debus, that the use of the term
specific gravities by Thomson in his account of the atomic
theory implied the acceptance of the law M/S=C by Dalton in
1804, is also seen to be incorrect. Dalton expressly states in
1803 (R. 27) that the specific gravities of different gases and
the specific gravities of their particles are not the same thing.
He never appears to have believed in the law M/S=C, and
this only occurred to him as a possible alternative, at once
shown to be inconsistent with fact, to the statement which he
recognized as the true one, viz. that no two gases agree in
the size of their particles” (R. 47).
In order to forma correct opinion of the assertions contained
in this quotation, we must go back and consider the statements
of Dalton on which these assertions are founded.
_Dalton’s Atomic Theory. 361
Dalton, in his Note-book, i. p. 246 (R. 27), has the following
memorandum :—
“ Though it is probable that the specific gravities of different
elastic fluids (gases) has some relation to that of their ultimate
particles, yet it is certain that they are not the same thing ;
for the ulterior particles of water or steam are certainly
of greater specific gravity than those of oxygen, yet the last
is heavier than steam.”
Instead of building on the dictum contained in this quotation
weighty conclusions, as Messrs. Roscoe and Harden do, I
confess that I do not understand it. What is the difference
between the specific gravity of a gas and that of one of its
ulterior particles or molecules? According to Dalton, the
molecular volume is =M/S (N.S. 226, note). The specific
gravity of a molecule, s, is expressed by the quotient of the
molecular weight and molecular volume :
Molecular weight M _ at
Molecular volume M/S~ °~ *
Hence, 8, the specific gravity of the gas, =s, the specific
gravity of one of its molecules ; or, in other words, both are
the same thing. This result follows from Dalton’s definition
of the molecular volume, and is opposed to his assertion that
the specific gravities of gases and the specific gravities of their
molecules are not the same thing, and also to the conclusions
of Roscoe and Harden, based on this erroneous assertion.
Therefore it does not follow that Dalton did never believe in the
hypothesis M/S =C, or that the first part of my view stated on
p- 360 is incorrect.
Messrs. Roscoe and Harden assert that he (Dalton) “ never
appears to have believed in the law of equal volumes (hypo-
thesis M/S=C), and this only occurred to him as a possible
alternative, at once shown to be inconsistent with fact, to the
statement which he recognized as the true one, viz. that no
two gases agree in the size of their particles ” (R. 47).
What does Dalton himself say on this subject ?—“ At the
time I formed the theory of mixed gases (year 1801) I had a
confused idea, as many have, I suppose, at this time, that the
particles of gases are all of the same size, that a given volume
of oxygen contains just as many particles as the same volume
of hydrogen” (N.S. 188). And with regard to the statement
“that no two gases agree in the size of their particles,’ he
says, ““This idea occurred to me in 1805.” Therefore the
idea that a given volume of oxygen contains just as many
particles as the same volume of hydrogen occurred to Dalton
at least four years before the time at which he recognized
362. Dr. H. Debus on the Genesis of
“that no two gases agree in the size of their particlés.” ~Con-
sequently, the first idea (M/S=C) did not “ oceur to him only.
as a possible alternative, at once shown to be inconsistent with
fact, to the statement which he recognized as the true one, viz.
that no two elastic fluids agree in the size of their particles,”
but was held by Dalton, more or less, from 1801 to 1805, and
only abandoned when he could not bring it into agreement
with his later atomistic views.
It has been shown on p. 361 that the memorandum in Dalton’s
Note-book, i. p. 246, is not correct, and that the specific gravity
of a gas and the specific gravity of one of its molecules are the
same thing. But how, we may ask, could Dalton contradict
his own definition? On p. 260 of his Note-book, i. (R. 42),
occurs a table with the following heading :—“ The molecules
of gases arranged according to their specific gravities ;” and
then follow in arithmetical order the molecular weights of
eleven gases. The specific gravity of the molecules and the
molecular weights are therefore expressed by the same
number. :
If M represents the molecular weight, s the specific gravity,
and v the volume of a molecule, we have
WV ==
As M is put =s, v must be =1; and as this relation is
assumed to be general, all gases have the same molecular
volume, or M/S=C. Hence the molecular weight and the
specific gravity of a molecule are then, according to the table,
‘“‘ the same thing.”
If we now substitute in Dalton’s memorandum quoted on
p- 361, which is not intelligible by itself, for the words “ spe-
cific gravities of molecules” the words ‘‘ molecular weights,”
we obtain :—“ Though itis probable that the specific gravities
of different elastic fluids (gases) has some relation to their
molecular weights, yet it is certain that they are not the same
thing ” (R. 27)—a perfectly intelligible statement. |
Therefore, on p. 246 of Note-book, i., the hypothesis M/S=C
is rejected, and on p. 260 accepted.
~ How are these contradictory statements to be explained ?
On p. 188 of his celebrated work, ‘ New System of Che-
mical Philosophy,’ Dalton says :—“ At the time I formed the
theory of mixed gases, I had a confused idea, as many have,
I suppose, at this time, that the particles of gases are all of the
same size.” Consequently there was a time when Dalton
believed in the hypothesis M/S=C, and his endeavour was
to verify the same by experiment. Accordingly, he lost no
time in calculating M for oxygen and steam, as soon as he had
Dalton’s Atonie Theory. 363
formed his atomic theory. According to this theory, the mole-
cular weight of steam ought to be greater than the molecular
weight of oxygen, and this result in connexion with the hypo-
thesis M/S=C would require the specific gravity of steam to
be greater than the specific gravity of oxygen. Hxperiments
gave opposite resuits; they made the specific gravity of
oxygen greater than the specific gravity of steam. Hence,
one of the two—the new atomic theory or the hypothesis
M/S=C—must be wrong. Dalton rejected the latter (Note-
book, 1. p. 246).
The extension of the investigation to other gases, however,
furnished several examples which gave nearly the same value
for M/S=C. Note-book, i. p. 258 (R. 41), contains a table of
12 gases with their specific gravities and Dalton’s molecular
weights. I have recalculated the specific gravities on the
unit of hydrogen, and in place of the diameters of molecules
calculated by Dalton i have deemed it sufficient to give the
molecular volume, M/S.
Taste of the 19th Sept. 1803 (R. 41).
Molecular Specific
Name of the gas. Weight. Gravity. M/S.
REMORSE My or cri aaieiecrelnalsers aeons 1 1:00 1:00
PPG BOM rere gn ainccat onidcupavaeaacioes 5°66 14°60 0°38
J. 1G REL ss hguse ga Sade Sbeacnic suena 4:00 12°5 0:32
Matrous oxide’ i500) 5. esas tects 13°66 20°9 0°65
INIEPIC: OXTAE 5.520450. 0cscenneetee 9°66 14:3 0:67
Carbonic Oxide 2.2, .wecsacce sox 10:10 13:0 0-77
Carbonic acid ........ Bee ee 15-70 19°5 0-80
Pivdrocarbon 2,52 :2.832..)1 eos 540 8°6 0:63
2 TN STO) 1 cee 5:00 7d _ 066
Sulphurous acid s,2,.:.c.d.0sh++-: 20:00 29:0 0:69
Sulphuretted hydrogen ......... 15°40 14:0 11
SIZE TTL ss ee ee nee ee 6°66 9:0 0-74
The values of M/S are not equal for the different gases, but
may be arranged in three groups :-—
Ee IT. III.
M/S. M/S. M/S,
Nitrogen ... 0°32 Hydrocarbon ......... 0-63 Hydrogen ... 1:0
Oxygen...... 0-38 Nitrous oxide ......... . 0°65 Sulphuretted | Ve
ATMIMONIAs si vc5s>5- oie 0°66 hydrogen j
Nitric oxide............ 0-67
Sulphurous acid ...... 0:69 (SO,)
SHAM ot. arpecued chieces O74
Carbonic oxide ...... O77
Carbonic acid ......... 0°80 (CO,)
364 Dr. H. Debus on the Genesis of |
The values of M/S=C are dependent on the specific
gravities, the chemical composition, and the theories about the
constitution of gases. ‘The composition of water is, according
to Lavoisier, 85 pts. of oxygen and 15 pts. of hydrogen,
according to Gay-Lussac and Humboldt 87-4 pts. of oxygen
and 12°6 pts. of hydrogen. Lavoisier’s numbers make the
atomic weight of oxygen =5°66, Gay-Lussac’s and Hum-
boldt’s =7. The errors of observation in Dalton’s time were
so considerable that he might have assumed for from 50 to
66 per cent. of the gases in the above list the same molecular
volume. The numbers obtained for M/S=C, therefore, le!t
it undecided whether the hypothesis, that equal volumes of
different gases contain the same number of molecules, is true
or not true. The probabilities are, perhaps, slightly in favour
of the hypothesis. This seems to have caused Dalton to
return to his old view, that the particles of gases are all of
the same size, or M/S=C. ‘This is my explanation of the
contradictory statements on two pages, 246 and 260, of his
Note-book, i. (R. 27, 42) (pp. 861, 362). The near agreement
of several of the numbers of M/S in the last table caused a
series of new experimental investigations. Dalton, who up
to this time had done very little practical work in chemistry,
now began with great zeal to determine the composition of
important substances, such as ammonia, marsh-gas, and ole-
fiant-gas. The results of these investigations, as far as they
concern gases, are collected in a table at the end of the second
part of the New System.
Five out of 16 gases, or 31 per cent., and if errors of ob-
servation of 2 per cent. are allowed three more, or, together,
50 per cent. of the gases examined, possess the same molecular
volume. JDalton’s experiments have not explained why
50 per cent. of the gases examined do not conform to the
hypothesis M/S=C, and they could not do so, the chemical
knowledge of the day not being sufficiently advanced. Asa
matter of fact, it has required the practical work of half a
century to convert the hypothesis M/S=C into an empirical
law. .
Thomas Thomson, the author of the celebrated work “ Sys-
tem of Chemistry,’ himself a chemist of eminence, paid a visit
to his friend Dalton in August 1804. On this occasion the
latter communicated to him the principles of the new atomic
theory and the results obtained by their application. In
1807 a new edition of the ‘System of Chemistry’ came out,
in which Thomson, with Dalton’s permission, published a
sketch of the new theory (vol. mi. p. 424). Thomson’s de-
scription is extremely clear and accurate, and Dalton has, as
Dalton’s Atomic Theory. 365
far as I know, never taken exception to any of its statements.
Now Thomson always calls the relative weight of atoms the
‘density of atoms.”” His table of molecular weights is called
table of the density of atoms (molecules) (vol. i. p. 429).
Elementary physics teaches that the weights and densities of
substances are proportional when the substances under con-
sideration are of the same volume. If we say for density
specific gravity, then Thomson’s report of Daiton’s atomic
views, and Dalton’s statement in Note-book, i. p. 260 (R. 42)
are in perfect agreement. Accordingly, Dalton had not, in
August 1804, thrown the hypothesis M/S=C overboard. He
did so in the following year, 1805, when he could not bring it.
into agreement with his atomistic views (p. 851) ; and when
he found it refractory in this respect, he called it “a confused
idea’”’ (N.S. 188).
Messrs. Roscoe and Harden charge me on pp. 10-11 of
their book with a “confusion of ideas.” They say :—“ Debus
further argues that when Dalton communicated his theory to
Thomson, he must have held the opinion that these two
relations, the relative density and the relative weight of the
atoms, were identical, or what Debus appears to consider to
be the same thing, that the relative densities of the gases
were identical with the relative weight of their atoms. .. .
On the strength of this argument, which appears to rest
on a confusion between the relative density of the atoms and
the relative density of the gases made up of those atoms... .”
1 have not asserted that the relative weight and the relative
density of the atoms were IDENTICAL. I have expressed my-
self in the following manner :—“ We must conclude that
Dalton, at the time when he communicated his ideas to
Thomson, considered that the molecular weights of the gases
stand to each other in the same ratio as the densities of their
molecules” (D.51). As for the confusion between the relative
density of the atoms or molecules and the relative density of
the gases made up of those atoms or molecules, I must refer
to p. 361, where it has been shown, if we write for density
specific gravity, that according to Dalton’s own definition,
the specific gravity of a gas is equal to the specific gravity of
one of its molecules. Therefore, the confusion is not on my
side !
The Italian physicist Amadeo Avogadro became acquainted
with Dalton’s atomic theory through Thomson’s report (0.
no. 8, p. 7, note). He puts on Thomson’s statements the
same interpretation as Ido. Dalton’s object is to find the
atomic weights of the elements, and the molecular weights of
366 Dr. H. Debus on the Genesis of
the compounds formed of these elements. The sum of the
weights of the atoms in a molecule is his molecular weight.
Avogadro determines the molecular weights directly from
the specific gravities of the gases. The molecular weight is,
if Imay say so, the end of Dalton’s and the beginning of Avo-
gadro’s work. About the constituents of the molecules, the
atoms, Avogadro is absolutely silent. He knows only mole-
cules and parts of molecules. Whether there is or is not a
limit to the division of a molecule he does not say. Avogadro
treats the subject like a theoretic mechanician, who considers
his particles divisible or not, according to the nature of his
problem. The unit of Dalton is the indivisible atom, the unit
of Avogadro the divisible molecule (D. 70).
It is worthy of notice that Avogadro is not conscious of
these differences between Dalton and himself. He criticises
Dalton’s work as if he and Dalton were engaged with the
solution of the same problems, and calling attention to the
agreement of some of his results with those of Dalton, he
says :—‘ This agreement is in favour of our hypothesis,
which is essentially nothing else but the system of Dalton
improved by a sure expedient* to which we have been led by
the facts discovered by Gay-Lussac” (O. no. 8, p. 22).
- Avogadro could only identify his system with Dalton’s on the
assumption that both systems rested on the same principle.
This principle is the hypothesis M/S=C. As Avogadro received
his information about Dalton’s work from Thomson, it follows
that he put the same interpretation on Thomson’s words as I
have done, and as the latter published his book in the year
1807, and Avogadro his essay in 1811, it appears highly
probable that Avogadro received the first intimation of the
hypothesis M/S=C from Dalton. The great merit of
Avogadro is founded on his proposal to consider the mole-
cules of the elementary gases divisible into two or more parts,
because thereby the hypothesis M/S=C became of practical
importance.
But are the hypothetical molecular weights of Avogadro
the real molecular weights? ‘This question could only be
answered by the synthesis of the molecules from the con-
stituent atoms. The sum of their weights is the molecular
weight. Therefore, logically considered, the atomic weights’
must precede the molecular weights. The determination of
the atomic weights was, accordingly, the task of Dalton and
his generation. This is the real reason why the hypothesis
M/S=C received no attention in Dalton’s and Avogadro’s
time and was soon forgotten. Fifty years later, after
* The division of molecules.
Dalton’s Atomic Theory. 367
thousands of vapour densities had been taken, it reappeared in
the form of an empirical law.
Chemists have often wondered why Dalton did not acknow-
ledge as correct the observations of Gay-Lussac that gases
always combine in simple volumetric proportions. Various
improbable or absurd reasons* have been assigned for his
reluctance to adopt the results of the distinguished French
chemist. The real reason is near at hand. Dalton could
not accept Gay-Lussac’s law of the combination of gases in
simple volumetric proportions, because, if he had done so, he
would have abandoned the chief principle of his atomic theory.
I will illustrate this assertion by an example. The first
principle of Dalton is, that when two elements form only
one compound, the compound is a binary one unless some
cause appears to the contrary.
Only one compound of hydrogen and oxygen and only
one compound of hydrogen and nitrogen were known, hence’
the formula of the first would be HO and of the second HN.
Nitric oxide, on account of its low specific gravity, was like-
wise considered to be a binary compound of the formula NO.
(N.S. 317.) Water, according to Gay-Lussac, is a compound
of one voluine of oxygen and two volumes of hydrogen. If we
assume one volume of hydrogen to contain 1000 atoms
(molecules), then, if Dalton’s formula HO is right, one
volume of oxygen must contain 2000 atoms. eas
Nitric oxide is formed, according to Gay-Lussac, of one
volume of nitrogen and one volume of oxygen. Therefore,
if Dalton’s formula NO is correct, one volume of nitrogen,
like one volume of oxygen, must contain 2000 atoms. Hence
we have :—
one volume of hydrogen 1000 atoms,
one volume of oxygen 2000 _ ,,
one volume of nitrogen 2000 _,,
Now Gay-Lussac asserts that ammonia is a compound of one
volume of nitrogen and three volumes of hydrogen—
one volume of nitrogen 2000 atoms,
three volumes of hydrogen 3000 __,,
and its formula would be N.H, if Gay-Lussac’s observations
are correct.
Dalton’s principle requires the formula NH. Hence the
alternative presented itself to Dalton either to reject his first
principle or Gay-Lussac’s observation. As the atomic theory
hy 47. Grundztige der theoretischen Chemie von Lothar Meyer, 1890,
p- °
368 Notices respecting New Books.
was strongly supported by experience he rejected Gay-
Lussac’s law, and considered his own inaccurate observations
to be correct and the more exact results of the French chemist
to be faulty.
Some of the results of this investigation are :—
(1) Dalton was investigating the state of equilibrium in
mixed gases in the year 1801. This investigation caused him
to adopt the hypothesis M/S=C.
(2) Itis highly probable that the hypothesis M/S=C and
the observations of the proportions in which nitric oxide and
oxygen combine led Dalton to the invention of his atomic
theory.
(3 Atomistic views caused Dalton to abandon the hypothesis
M/S=C in the year 1805. :
(4) If we remember that all theories in chemistry are of a
provisional character, and that they are subject to changes in
course of time, then we cannot deny our admiration to the
great work of Dalton. It was he who first attempted to
weigh molecules and atoms and measure their volumes.
XXXVI. Notices respecting New Books.
The Intellectual Rise in Electricity; a History. By Park BrEn-
JAMIN, Ph.D., LL.B. London: Longmans, 1895.
eke electrical properties of rubbed amber and the phenomena
exhibited by the lodestone have been known since the very
earliest historic times; their discovery was probably coeval with
those of amber and lodestone. While the electrical fact remained
for many centuries isolated and apparently useless, the orientation
of suspended lodestone or of magnets derived from it soon sug-
gested the mariner’s compass and led to important advances in
magnetic science, followed later by theories of attraction and repul-
sion. In the work before us Dr. Benjamin traces in a very inter-
esting manner the development of these facts and theories up till
the time of Franklin, when the recognition of electricity as a
natural force led to its being more universally studied. The
author has searched among the manuscripts and books of many-
lands and all ages in order to find material for his history, and he
has even been assisted by the labours of those who have investi-
gated the ruins and records of ancient civilization in Pheenicia,
Egypt, and Scandinavia. Such a search necessarily occupies many
years, and we owe its satisfactory termination to the author’s
patience and enthusiasm for his subject.
A very plausible theory is put forward to account for the intro-
duction of the lodestone into Europe. It is supposed that the
inhabitants of Central Asia first became acquainted with its pro-
perties ; migrating eastwards as Mongols they carried the know-
ledge into China, and travelling north-westward as Finns and
Notices respecting New Books. 369
Lapps they introduced navigation by compass into Scandinavia.
The subsequent dissemination of the knowledge to various Euro-
pean ports took place through the medium of traders to the Baltic.
“Vixere fortes ante Gilbertum,” if we may be pardoned for altering
Horace; the author has discovered at least two such philosophers,
Neckam and Peregrinus. We cannot agree with Dr. Benjamin
in his criticism of a statement by Peregrinus (p. 174), which he
considers erroneous. Concerning the rubbing of iron against lode-
stone or a magnet Peregrinus writes: “ You will infer what part of
the iron is attracted to each part of the heavens from knowing
that the part of the iron which has touched the southern part of
the magnet is turned to the northern part of the sky. The con-
trary will happen with respect to that end of the iron which has
touched the north part of the stone, namely, it will direct itself
towards the south.” For the thirteenth century this is surely a
very concise and straightforward description, contrasting greatly
with the author’s confusion of true north polarity and north-seeking
magnetism.
The volume contains several reproductions of interesting old
prints and good portraits of Gilbert, von Guericke, and Franklin.
Jie Lig El.
An Introductory Treatise on the Lunar Theory. By E. W. Brown,
M.A., Professor of Applied Mathematics in Haverford College,
Pa. (Cambridge University Press.)
Iv may be said, and indeed it has been remarked by foreign and
English writers alike, that the English student of mathematics is
exceedingly fortunate in the excellence and abundance of the
mathematical text-books at his disposal. It is therefore not a
little curious that the only elementary exposition of either the
Lunar or the Planetary Theory has existed in the form of a
single and, it must be admitted, very inadequate introductory
treatise. At the same time, the field for original research offered
by these particular cases of the general problem of Three Bodies
has been, with notable exceptions, singularly neglected by English
mathematicians; and it is at least doubtful if the contributions of
Lubbock, Airy, Cayley, and Adams are sufficient to relieve English
mathematics of the serious charge of having neglected an im-
portant branch of the science. For this state of things the want
of a good elementary and, at the same time, comprehensive treatise
was doubtless largely responsible, for there was apparently little
to attract the student to take up a subject in which it would be
necessary for him at an early stage to master the contents of a
great number of scattered and exceptionally obscure original
memoirs. {t is therefore with reason that we extend a special
welcome to Prof. Brown’s treatise. The author, who, although
holding an appointment abroad, was formerly Fellow of Christ’s
College, Cambridge, is already favourably known both on account
of his own contributions to the most modern form of the Lunar
Theory and also for his elucidation of the work of the older
theorists.
The work divides itself practically into two distinct parts. the
ial. Mages: 0. Vol, 42. Nov 25¢. Oel. 1896. 2D
370 Notices respecting New Books.
first part forming an introduction to the second, which contains
accounts of the various theories in detail, although it must be
understood that the author adheres strictly to his intention of
giving an explanation of the methods, and not the actual results
obtained from them. After the necessary force functions have
been found, the ordinary simplifications introduced by neglect-
ing the Moon’s mass, and assuming the Sun to describe an
elliptic orbit round the earth according to Kepler’s laws, together
with the consequent corrections, are examined, and a numerical
estimate of the resulting error is given. The equations of motion
are next found as they are required for the methods of de Ponté-
eoulant, Laplace, and Hill, and in addition the ten first integrals
arising from the equations in the problem of Three Bodies. The
third chapter is devoted to a discussion of undisturbed elliptic
motion, the expansions being made with the aid of Bessel’s fune-
tions, and the question of convergence being taken into considera-
tion. The two principal methods of obtaining a solution, namely,
by continued approximation, and the Variation of the Arbitrary
Constants occurring in any orbit which may be taken as “ inter-
mediate,” can now be considered, and the equations for the
variations of the elements in disturbed motion are obtained in an
elementary way and also by Jacobi’s more elegant method. In
this connexion some description is given of Lagrange’s canonical
system with Hansen’s extension, and some theorems of Jacobi,
Hamilton, and Cayley are alsoincluded. The development of the
forms and properties of the disturbing function brings the reader
to the point at which it becomes necessary to study the principal
methods separately. De Pontécoulant’s method is very properly
selected by Prof. Brown as a basis for the elucidation of properties
common to all, and consequently receives the fullest treatment,
the inequalities being grouped according to their origin, and the
consideration of the arbitrary constants, to which a whole chapter
is devoted, being particularly lucid. Delaunay’s method, which is,
next described, is important mainly on account of the high order
of approximation to which the literal developments are carried,
but also because it possesses very wide applications and possibilities
for development which, according to Dr. Hill, have not yet been
fully realized. It should be mentioned that Prof. Brown has suc-
ceeded in simplifying many of the explanations as they appear in
the Théorie du Mouvement de la Lune, as he has also done in
his account of Hansen’s method. The latter has peculiar difficulties
and obscurities, and these the author has taken pains to remove,
by no means without success.
Last of all the theories considered in detail is the one initiated by
Dr. Hill and based on the use of rectangular coordinates referred to
moving axes. It is interesting to note that these were first applied
to the Lunar Theory by Euler, although he had originally used
cylindrical coordinates for the purpose, and it is also remarkable
that their power in the analysis of geometrical as well as dynamical
problems is only now becoming generally recognized. It is in
this most modern method of treatment of the lunar inequalities
that Prof. Brown’s own investigations have been made. An
Geological Society. 371
historical account of the work of other theorists is appended; and
the last chapter is devoted to the inequalities in the moon’s motion,
arising from the action of the planets, the figure of the earth, and
the motion of the ecliptic. Comparative tables of notation, which
will certainly be found useful, together with an index complete
the volume.
The peculiar difficulties of the Lunar Theory arise mainly from
the fact that the original investigators have generally confined
themselves strictly to their own methods, and this independence of
thought among the classical theorists may be fairly said to warrant
Prof. Brown in his peculiar treatment of the subject. Neverthe-
less, while in our present state of knowledge he does well in
recognizing the importance of some acquaintance with the older
methods and the necessity for a separate treatment fer each in
order to show the relations which exist between them, the subject
cannot be considered to be in a perfectly satisfactory state until a
single uniform mode of treatment becomes possible without loss of
rigour or completeness. There can be no doubt that the time for
this will be hastened by Prof. Brown’s work. A highly commend-
able feature of the book consists in the ample references, which
will enable the reader to continue his study of the more recent
theoretical investigations. It was of course impossible to include
all of these developments, and the author has shown excellent
judgment in his selection. If the work of Professor Gyldén is
passed over almost without remark, this course is probably justified
by practical considerations of expediency, although it is to be
reoretted that his work is not better known in this country. The
question of convergence being as it is of the utmost importance
for the legitimate use of infinite series, has scarcely sufficient stress
laid on it, although the absence of certain knowledge is a plausible
excuse. The remarkable contributions of Cauchy to this subject
might, however, with great advantage have been considered. But
with this single exception, we have nothing but praise for the
manner in which Prof. Brown has carried out his task, and we
recognize his book as a most important and valuable contribution
to a highly interesting branch of mathematics. It should. be
mentioned that the volume in the matter of type and general
arrangement is quite up to the high standard we have learnt to
expect in books published under university auspices, and this fact
serves no small part in the lucidity of the explanations.
W. E. PLuMMER.
XXXVI. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from p. 287.]
May 27th (con.).—Dr. Henry Hicks, F.R.S., President,
in the Chair.
2. ‘The Linyula-Flags and Igneous Rocks of the Neighbourhood
372 Geological Society.
of Dolgelly.’ By Philip Lake, Esq., M.A., F.G.S., and 8S. H. Rey-
nolds, Esq., M.A., F.G.S.
The area dealt with in this paper lies south and west of
Dolgelly, between the Arthog road and the hill called Mynydd
Gader, which lies in front of the precipices of Cader Idris. ‘The
stratified rocks belong to the Middle and Upper Lingula-Flags and
Tremadoc Slates. The Middle Lingula-Flags (Ffestiniog Series)
consist of bluish slates with grit-bands containing the usual Lingu-
lella, passing into Upper Lingula-Flags (Dolgelly Series) consisting
of dark slates with Orthis lenticularts, Parabolina spinulosa, ete.,
and containing two andesitic lavas. These pass into the basal
Tremadoc Slates with Dictyograptus flabelliformis, surmounted by
an upper volcanic series with rhyolitic lava. Subsequent intrusions
of diabase occurred, of a laccolitic character, but of such a nature as
to lead the authors to suggest the possible intrusion of the diabase
along a line of unconformity in one case; there is, however, no
newer rock above the diabase to indicate of what date the overlying
beds would be if such unconformity occurred. It is further shown
that the important faults in the area were produced both before
and after the diabase-intrusions, and in one case the movement
appears to have been in one direction before the intrusions, and in
the opposite direction afterwards.
3. ‘The Kildare Inlier.’ By 8. H. Reynolds, Esq., M.A., F.G.S.,
and C. I. Gardiner, Ksq., M.A., F.G.S.
The area described in this paper is occupied by four prominent
hills composed of Lower Paleozoic rocks rising as an inher from
beneath Carboniferous beds. The authors give the foliowing suc-
cession of rocks in descending order :—
. Green and grey micaceous grits and shales of Dunmurry.
. Red and black shales.
Gap: no exposure seen.
. Limestones of the Chair of Kildare.
. Contemporaneous igneous rocks.
. Fossiliferous ash of Grange Hill House.
. Green gritty shales (unfossiliferous).
Nos. 5 and 6 are referred with some doubt to the Llandovery
Series, and perhaps also to higher series. The gap may conceal the
uppermost beds of the Bala succession. The limestones of the
Chair of Kildare are separated by the authors into four subdivisions
of the same general age, and Agnostus trinodus, [Menus Bowmanna,
Remopleurides longicostatus, and Cyphoniscus servalis range through-
out. The contemporaneous igneous rocks of Grange Hill and of the
Hill of Allen are shown by the fossils found in the pyroclastic rocks
to be of Middle Bala age. The lavas consist of basalts and andesites
which the authors separate into four groups distinguished by their .
lithological characters. Petrographical details of these various
rocks are given in the second part of the paper. The age of the
lowest beds which have not yielded any fossils is doubtful.
mre Oo
THE
LONDON, EDINBURGH, ann DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE,
fr =. : 2 1296
[FIFTH SERIES.] ¢ WOV13* «
NOVEMBER 1896.
XXXIX. Thermal Transpiration and Radiometer Motion.
By WILLIAM SUTHERLAND*.
Part 1.— Thermal Transpiration.
| ae comparative neglect into which the radiometer has
fallen is probably the natural compensation for the
exalted interest of its two or three years’ reign over the
scientific imagination twenty years ago. In reading amongst
the papers about it published at that time, one gets an im-
pression of the laboratory of Crookes as of an arsenal where
night and day the equipment of a great expedition into the
unknown was being pushed on under the sleepless eye of a
patriot leader ; but in the answering bustle outside, Stokes,
Schuster, Stoney, Fitzgerald, Pringsheim, Reynolds, and
others soon showed that the new conquest was simply an
outlying part of the Kinetic Theory of Gases. Or, to vary
the figure, Crookes appears as a friendly counsel subjecting
Nature to a passionate and eloquent cross-examination with
his fellow physicists as judge and jury bringing in a verdict
for Kinetic Theory. And then the interest died away rapidly,
perhaps mostly on account of Reynolds’s great paper ‘ On cer-
tain Dimensional Properties of Matter in the Gaseous State ”
(Phil. Trans. clxx.), which was probably held to settle the
essential points of general interest in radiometer motion as
consequences of the kinetic theory of gases, especially as the
same train of reasoning had led him to his discovery of
* Communicated by the Author.
Phil. Mag. 8. 5. Vol. 42. No. 258. Nov. 1896. 2K
Eee, ak: \j4
a Me S. PAT ENN pee
Se
374 Mr. W. Sutherland on Thermal
Thermal Transpiration with the beautiful experimental estab-
lishment of its simple quantitative laws, simple in the illumi-
nation of his theory, but complex enough without it. Unfor-
tunately the mathematical form of Reynolds’s theory is wearily
cumbersome; one gathers that Maxwell found it distasteful,
and Fitzgerald (Phil. Mag. [5] xi.) describes it as inelegant
and unnecessarily elaborate.
A great objection to Reynolds’s mathematics is that it does
not join on naturally with that developed for the general pur-
poses of the kinetic theory of gases ; it has a certain interest
of individuality about it, but. this fails to compensate for the
waste of mental energy to the reader who has to adapt himself
to it. But what appears to me to be the fatal objection to
Reynolds’s mathematical method, is that it takes the mind away
from definite physical concepts of the actual operation of the
causes of thermal transpiration and radiometer motion ; and
the object of the present paper is to construct a theory of
these that will fall into line with the current kinetic theory
of gases and keep the physics of the phenomena to the fore.
The most convenient starting-point is the laws discovered
by Clausius (Pogg. Ann. exv. 1862) for the conduction of
heat in gases. In a vertical cylinder of gas, bounded by a
solid wall impermeable to heat and two conducting plane
ends, the lower at temperature @, and the upper at a higher
temperature 8, when the flow of heat has become steady, the
pressure throughout the cylinder is constant, and the tempe-
rature @ at distance w from the lower end of the cylinder
whose whole length is / is given by the equation
03 = 0,2 + (02!—0,')/l,
and the distribution of density is determined in accordance
with these two results. Now in the establishment of the law
of the temperature, it was shown by Clausius that in a mass
of gas which is not uniform in temperature there is motion of
the gas in the direction of variability ; but it is assumed (as
it can easily be proved) that under ordinary circumstances
this motion can never produce an appreciable departure from
uniformity of pressure, because the rate at which a variation
of pressure throughout a mass of gas is effaced is so rapid in
comparison with the motion which might produce a variation
of pressure, that such a variation can never get itself estab-
lished to an appreciable extent. But when in place of an
ordinary cylinder we consider a very fine tube, we must take
account of the effect of viscosity in reducing the velocity with
which an inequality of pressure along the tube can get itself
effaced ; and if the tube becomes fine enough, this velocity
Transpiration and Radiometer Motion. 375
may be reduced till it is merely comparable with, or even
much smaller than, the velocity with which motion caused by
varying temperature may be tending to establish inequality
of pressure. Thus, then,.in discussing the conductivity of
gas in a nonconducting tube of capillary dimensions, we
could no longer enjoy the convenient simplification which
comes into the problem of Clausius when he writes the pres-
sure constant as one of his fundamental equations, but from
purely kinetic considerations we should have to determine the
laws both of the variation of pressure and of temperature
associated with the steady flow of heat. But in the actual
problem of thermal transpiration if we lose one simplification
we gain another, because we have to do, not with noncon-
ducting walls, but with walls conducting so well and with so
large a thermal capacity compared to that of the gas, that the
law of variation of temperature is fixed entirely by the pro-
perties of the solid; so that the gas, if subject to varying
pressure, is also subject to a fixed law of temperature which
. we are freed from having to find.
In the kinetic theory the molecules which are considered
characteristic of an element are those that have experienced a
collision in it; those passing through without collision are
taken account of in the elements where they do collide. If
the element is a short length of our tube, we do not consider
the molecules rebounding from the solid wall as characteristic
unless they also encounter other molecules in the element,
and thus we might appear to be neglecting the most charac-
teristic molecules of the element. But this is not really so,
because those reflected from the side of the tube and moving
to a cooler element, as a rule collide with those coming from
a still cooler element and including an equal number that
have come from its walls, so that the colliding pairs on the
average possess the qualities that are to characterize the ele-
ment in which they collide. Thus, then, if we do not have
to take account of reflexion from the walls of the tube, we
can consider the gas in it as part of an indefinite mass such
that the temperature throughout a plane perpendicular to the
axis is the same as that in the section of the tube made by the
plane. We wish to find the number of molecules crossing
any section of the tube. This is done by Clausius in his
theory of conduction in gases, and with greater refinements
of accuracy by Tait (Trans. Roy. Soc. Edinb. xxxiii.) ; but
for the sake of clearness we will make the calculation here to
a degree of accuracy suitable for present requirements.
If there are n molecules per unit of volume in a small
element dB, and each has v encounters per second, then the
2H2
376 Mr. W. Sutherland on Thermal
number of molecules colliding in a second in dB is nvdB. It
is not worth while to take account of Maxwell’s law of velo-
cities ; and all molecules will be supposed to have the average
velocity v and travel the mean free path 2% between two
encounters, so that v=v/A. But we must take account of the
variation of % with direction ; for a molecule travelling from
a particular point bas a longer path in the direction of dimi-
nishing density and a shorter path in that of increasing
density, with a maximum parallel to the axis in one direction
and a minimum in the other; while the path at right angles
to the axis is the mean of the maximum and minimum, and
is indeed the mean path 2 of all molecules leaving that point.
Let 2,, be the maximum value there ; then it is equal to the
minimum at distance 2,, along the axis of the tube, and must
therefore be equal to the mean value at distance X,,/2 ; thus,
then, Am=A+AmdA/de2, or Ap»=A+AAdA/de2=AUI +2X7/2).
On the same principle, the free path of a molecule that leaves
the point in any direction so that the projection of its path on
the axis of the tube is x, has a value X+ rd//2. Of the num-
ber nvdB/A of molecules that in unit time have a collision in
dB, the fraction that cross a plane at any distance is found
by drawing from the centre of dB as origin the surface whose
polar equation is p=A+2A‘/2, and estimating the solid angle
subtended at the origin by the segment of this surface cut off
by the plane, supposed to be at distance w, as a fraction of 477.
This is the required fraction, namely {1—#/(A + «d//2) }/2, or
(L—wx/A+ a°r’/ 2X?) /2 nearly.
Thus the number of molecules colliding in dB and crossing
the plane before colliding again is in unit time
nvdB(1—a/A—2°r//2X) /2X,
in which we have changed the sign of 0’ so as to transfer the
origin from dB to the plane. Now dB may be taken as Adz
where A is the area of section of the tube ; the total number
crossing the plane from the tube on one side of it in unit time
is the integral from 0 to A, of Anv(1—w/A—a°n//2d?)da/2n,
where A, is the maximum free path at such a distance from
the plane that a molecule after colliding there and travelling
perpendicular to the plane collides again just at the plane.
Now A=c/n, where c is a parameter depending only on the
size of the molecules ; thus the number is
h |
{ ‘An2o( 1—naz/e—n?wr’ /2c?)daj2e. . . (1)
0
Now if mg and v are the values of x and wv at the plane
Transpiration and Radiometer Motion. ae
we can write n=n,+adn/de=n,+n'e and v=vyt+v'e and
Ay=A,+MA,/2; substituting these values, integrating, neg-
lecting products and squares of n’v’ and 0, and dropping the
suffix 0 as of no more use we get
Anv{l/4+A(v'/u—A//A)/12} ww. 2)
The number crossing in unit time from the negative side of
the tube is obtained from this by changing the sign of n’ and
v’ so that the total gain in unit time from the positive to the
negative side of the plane is (since \’/A= —n’/n)
ie Oo
Anwn(™ += )/6 ee ee
which amounts to the same thing as if the gas had a velocity
mo 6
v=o +> )/ sii rarer yee ai(4)
along the tube ; but the result holds not only for a tube, but
for any space filled with gas and for any direction in it in
which n’ and v’ are the rates of variation of n and v. The
law connecting n and v with position in the general case
must be complicated, but for a gas in contact with a solid the
thermal capacity of the latter is so great as to make v’ and v
for the gas at the surface the same as for the solid there, so
that the problem simplifies to that of getting the law of n.
At a distance z from the solid surface the conditions of n and
v are still such as to tend to produce a velocity like u,so that
in the general case we have to consider the effect of viscosity
in causing these velocities to influence one another. The
friction per unit area parallel to the surface at zis ndu/dz, and
the state of the gas cannot be steady till this is constant.
Returning to the case of a tube, we see that the steady state
will be reached when the velocity v and n are constant
throughout a section, and the velocity wu is therefore also
constant throughout the section. Now under ordinary
circumstances there would be friction between the gas and
the tube over the whole surface, and therefore in this case
there must be an action between the solid and the gas equal
and opposite to the friction, that is to say, that the solid wall
of a tube along which heat is being conducted in constraining
the gas to take its temperature and share in the conduction
of heat exercises a traction on it. The total friction does not
exactly neutralize the total traction, but leaves a small
resultant part of it which we can determine thus: suppose
the tube connects two infinite spaces at the same temperature
378 Mr. W. Sutherland on Thermal
as the ends of the tube, the gas enters at one end with
velocity 0 and leaves at the other with velocity w; in unit
time the mass nmAu passes out with momentum nmAvw?’, and
this therefore is the force exerted by the tube on the gas in
it; this force acts only near the entrance in the part where
the velocity is rising from 0 to u, so that in this part the total
traction exceeds the total friction by nmAu?. In the velocity
wu we have the cause of thermal transpiration, while that of
radiometer motion is implied in the equation
total unequilibrated traction=nmAu%. . . (5)
If the spaces at the ends of the tube instead of being
infinite are finite, the gas will flow till a fall of pressure is
established to arrest it, but we cannot secure that u=0 all
over any section of the tube by an application of pressure,
because the flow established hy excess of pressure at one end
of a capillary tube is not of uniform velocity throughout each
section, but has a maximum velocity at the axis and a
minimum at the surface ; hence to secure that there shall be
no total flow in such a tube we have to establish a difference
of pressure which acting alone would discharge a volume Aw
per unit time in the opposite direction to that of u. Thus,
then, our solution for the motion in a conducting tube when
there is no total flow of gas along it consists of the super-
position of a uniform velocity wand opposite velocities varying
in conformity with the laws of flow in a capillary tube of
uniform temperature, the result being to give a surface of
zero velocity somewhere between the axis and the wall, with
a circulation going up between this surface and the wall, and
backward between this surface and the axis.
According to the theory of the flow of gas in a capillary
tube, if dp/dz or p’ is the rate of fall of pressure along the
tube, where the pressure is p and 7 is the viscosity, then B,
the volume measured at p delivered in unit time from a
circular tube of radius R (O. HE. Meyer, Pogg. Ann. exxvii.), is
B=qp' R/8y re
when the slipping of the gas on the walls can be neglected ;
but if shpping is to be taken account of let its coefficient be
€; then
B=ap'R*(1+46/R)/87. . 2 2 2
As the importance of § depends entirely on its ratio to R, and
as we wish to discuss tubes of any minuteness whatever, a
discussion of slipping becomes of first-rate importance to the
subject in hand.
T ranspiration and Radiometer Motion. 379
Kundt and Warburg (Pogg. Ann. clvi.) showed experi-
mentally the existence of slipping by its effect on the apparent
coefficient of viscosity at low enough densities of the gas in
an oscillating disk apparatus for measuring viscosity, and
they adduced theoretical reasons for the necessity of its
existence and for some of its properties; they also measured
its amount and verified some of its laws, and a little later
Warburg demonstrated the slipping of gas on the walls of
capillary tubes (Pogg. Ann. clix.)
That siipping is a necessary consequence of the kinetic
theory can easily be shown. Consider gas between two solid
parallel planes, one fixed and the other moving parallel to
itself with velocity w; then in the steady state there is a
constant rate of diminution of velocity dw/dx in the gas
between the plates. Suppose the molecules of the solid, like
those of the gas, to be smooth spheres oscillating, but their
centres at the surface having a mean position forming a
plane. Consider a molecule of gas in collision with a
molecule of solid; if its velocity of rebound makes an angle
less than 7/2 with the normal to the plane, the molecule has
little chance of colliding with another surface molecule of the
solid ana is directly reflected; the majority of these directly
reflected molecules of gas must strike the molecules of solid
near their most prominent points, and therefore acquire from
them very little of their velocity parallel to the plane ; thas
a certain fraction 7 of the molecules of gas that encounter the
surface leave it with practically the same velocity parallel to
it as that with which they approached ; the remaining fraction
1—f, or those which at the instant of rebounding from a surface
molecule have directions making an angle greater than 7/2
with the normal to the surface, must each penetrate into the
hollow between two neighbouring solid molecules and suffer
a second encounter with one of them under conditions which
necessitate its taking up on the average any motion that the
surface has parallel to itself.
Now suppose that on the average the molecules of gas
which collide with the solid come a distance /2 since their
last collision with molecules of gas; then their average
distance normally from the surface at the instant of last
collision with their fellows will be the average distance of a
hemisphere of radius \/2 from its base, which is A/4, and thus
the molecules of gas which collide with the solid, which is
fixed, reach it with a relative molar velocity Adw/dzt; but
after the collision only the fraction / retain this, so that the
gas in contact with the solid surface may be said to retain as
a whole the velocity fAdw/dv4, which constitutes a velocity of
380 Mr. W. Sutherland on Thermal
slipping, and shows how slipping arises, but does not give its
amount correctly ; this, however, can soon be obtained. Let
w, be the sudden change of velocity on passing from solid to
gas; then the Adw/dx4 just given must be increased by w,, and
then the average loss of momentum experienced by a molecule
encountering the fixed surface is m(fAdw/dx4 + w,) ; but the
number encountering unit surface in unit time is (2) nv/4,
and therefore the frictional force exerted by unit surface of
solid on the gas is nmv( fAdw/dw4+ w,)/4, which is equal to
ndw/dz, the friction on parallel unit surface in the gas when
the motion is steady : thus
LO CAE PNG
= dane 4)? ore (8)
but 7=°'365 nmva or, working with the same methods of
approximation as we have been using, 7=nmvA/4, and then
w= x17), er
f is a fraction which from its nature is unlikely to exceed
1/2, so that we can write w,=addw/dx with the knowledge
that a isnot much different from unity. At both the moving
and the fixed surfaces there is this discontinuity of amount
w,, so that in the theory of viscosity, instead of writing
dw/dz=w/D for the steady state, we must write
dw w—2uwy, pi yerden w
D? ".de DG an Dye
an or & is called the coefficient of slipping ; under ordinary
circumstances it may be neglected, but when D is comparable
with 2, as it mostly is in connexion with thermal transpiration
and radiometer motion, slipping becomes of fundamental
importance. When D is only a fraction of » viscosity
practically ceases, because the molecules traffic backwards
and forwards between the solids with so few encounters
amongst themselves that they hardly affect one another’s
motion, but they still exercise friction on the solids whose
amount is easily calculated. Suppose that the gas between
two parallel solid planes at rest is also at rest, except of course
for the velocities of agitation, and then let one of the planes
be set moving parallel to itself with velocity w; then, as we
have seen, the molecules colliding with it leave on the average
with velocity fw, and when they reach the fixed plane a
fraction f will have this velocity reduced to zero, while 1— f
will retain it unaltered, so that on the average the molecules
Q
8
Transpiration and Radiometer Motion. 381
leaving the fixed surface after their first encounters with the
moving and with the fixed have velocity (1—/)/w; but
without following up this process any farther we see that it
implies that when the steady state has been reached the
molecules leave the moving plane with velocity w, and the
fixed plane with an average velocity w,, and these must be
connected by the relations
(1—f)w,+fw=w,, and (1—/)w,=w, ;
whence y=w(l—/)/(2—-f), w=u/(2—/),
which give (w,+w,)/2=w/2, as of course they ought.
Hach molecule that encounters the moving plane gains
momentum m(we—w,) or mwf/(2—f7), and nv/4 molecules
encounter unit surface in unit time, so that the friction
between solid and gas is
Be reman fl (mae ire oe ra oe aC)
if f=1/2 this becomes nmvw/12, it is a limiting value of
nw/D(1+28/D) when D is negligible in comparison with ¢,
and it is independent of the distance between the moving and
fixed planes. We see therefore that we can carry the ex-
pression 7/(D + 2¢) into the consideration of cases either where
D is made very small or € very large.
The expression (11) shows that in capillary tubes whose
diameters are only a fraction of the mean free path—that is
with very fine tubes such as the passages of porous plates and
gas at ordinary pressures, or with ordinary capillary tubes
and gas at low pressures, or in any tubes at low enough
pressures—the flow of gas under pressure will not obey
Poiseuille’s laws ; indeed in a line or two we can show that
(11) leads at once to Graham’s laws of transpiration of gases
through porous plates verified and extended by Reynolds.
For if the gas is passing through a fine tube of radius R with
velocity w at distance x from one end, then when the flow is
steady
a Ramvwf/2(2—f) =7Rdp/da,
and taking account of the conditions at the two ends of a
tube of length / by suffixes 1 and 2,
TYwnam=TR’wynym=TRwengm = 2(2—f)\rR(p.—p,)/lv ;
thus the time of transpiration of unit volume measured at the
pressure p, being 1/7 R?w, is $ngnwl/(2—f)7R (p,—p,).
Now Reynolds made some experiments in which p,—p,
was kept a constant fraction of pa, and therefore proportional
to nz, under which conditions the time of transpiration should
382 Mr. W. Sutherland on Thermal
by our last expression be constant for a given gas and all
values of p,, which was the experimental result obtained by
Reynolds ; moreover if we wish to compare different gases,
as at a given temperature v is proportional to 1/m? we see
that the constant time of transpiration for each gas ought to
to be as the square root of its molecular mass, which is Graham’s
well-known experimental discovery verified by Reynolds.
This digression into the properties of a gas in spaces where
the linear dimensions are small compared to the free path has
been made as an appendix to our consideration of slipping in
order to clear up the limiting conditions towards which we
tend in treating of high vacua. We can now return to
thermal transpiration as we left it at (4). To secure no total
flow on account of w along a tube of radius R we are to have
B=ap’R4(14 40/R)/8n = Ru = —7Rvr(n'/n + v'/v)/6; (12) |
but p=nmv’/3, so that p’/p=n'/n + 20'/v, and then
p’R?(1 +46/R) /8n = —vA(p'/p—v'/v)/6. . (13)
Now with the methods of approximation here employed
n=nmvv/4 and p=nmv’/3, so that 7 =3Ap/4v, and then
‘( R2 /
Fi d+4yp) +1} = . ee
As €=aX the coefticient of p’/p is a function of only R/A, and
therefore the controlling influence of the whole phenomenon
of thermal transpiration is this ratio of R to X.
If the molecules are smooth, perfectly restitutional forceless
spheres n=79v/v), Where mn) and vp) are the values of 7 and v
at 0°C.; but with the molecules of the natural gases, on account
of molecular force, the function which expresses 7 in terms of
v is more complicated (see “ Viscosity of Gases and Molecular
Force,” Phil. Mag. [5] xxxvi.). But for present purposes it
will suffice to use the simple relation just given by which we
can express the last differential equation in terms of p and v
as the only variables thus
dp (Gate 3A Rr ld iL sh 15
dv \ 16)70* Nov" ) padv tn o> Ona (15)
which can be written as
dp y Cp Dp a hp
ie vt ge) tee 0.
D , 1 Up, 4
Let a and 8 stand for — XG + 56 (D =a
Transpiration and Radiometer Motion. 383
then the integral of this is
2 ~ log g(p/v? —a) — ia! 2 (p/v? = 8)
log g p/v=constant,
or ati suffixes 1 and 2 to indicate the — of the tube,
: yi ior —8 je EG 5)
© nove? —2 28 p,/v,? abate 7c) 18 pan, oe
But this is a very awkward form of result for comparison
with the experimental data, and we shall be better served if
content with an approximate solution of the differential equa-
tion obtained by putting p’=dp/de=(pe—pi)/l and v=
(%—v)/l, 2p=p2+ pi, 20=%) +), thus
Pepi IR*v,” (ps a pi)? 6aRv (po +7) Opie
i ag ee
Pot Pr Any (v2 + v1) N( vet v,) ar Or
This solution brings out at once the important point that with
v and v, fixed, that is to say, the temperatures of the two
ok constant, ‘there is a certain mean pressure (2+ ,)/2 for
which p,—p, has a maximum value; applying the usual con-
dition for a maximum, we find that Po—p, 1S a Maximum
when po+p,=2(v2+21)7/3Rv. Before proceeding to test
(17) by Reynolds’s experiments, we may remark that if the
mean pressure (7, + ,)/2 is made so small that R/) is negligible
in comparison with unity, then in (14) p’/p=v'/v, that is
Po/P1 = 02/0, a result in accordance with the following common-
sense argument that when the mean path of a molecule is a
large multiple of the radius of the tube, the molecules of the
tube have practically no influence on one another; and the
number that wander in at one end during unit time being
nyv,/4 and at the other n,v,/4, therinal transpiration will con-
tinue till these are equal, fhape is till nyv,=ngv_ or p;/V, = fro/Vo-
So far, our theoretical treatment has related to cylindrical
tubes, while in Reynolds’s experiment the passages through
which the gases transpire are the irregular chains of cav ities
in a porous “plate ; now toa first appr oximation these irr egular
cavities may be replaced by uniform tubes whose sectional
area is equal to the average section of the cavities, but it is
obvious that a better approximation to the natural cavities
would be a succession of frustra of cones of length L and
radii R; and R, at the end sections. The thermal transpiration
through such a frustrum can be readily established from (15),
for taking the origin of coordinates in one end and in the
axis, then the radius at distance « along the frustrum is
R=R,+ ca, where c¢ isa constant: thus for the fall of pressure
Qa
384 Mr. W. Sutherland on Thermal
from one end of the frustrum to the other iz daz, we have
x
from (15)
Re p dv dR
Rp, v dv c(AR?+2BR+1)’
where .
Leo ye es Lae
Soe a Nov
A
y)
which, if we regard dv/dz, p, and v as having constant average
values throughout the short length L, may be written
:
3
:
(OAS 1 ie R,A+B—(B?—A)? R,A+B+ (B?—A)?
» dec 2B?—A) °° R,AA+B+(B’—A)! R,A+B—(B’—A)
or |
epee loess SM log f i me oe
Dude oe 223? —A)s AR,R, + BCR, + R,) + (B?—A)?(R,—R,) + 1 |
The form of this expression suggests that we should expand 3
the log by the approximate relation log (1+z)=z, which
yields
p dv (R,— R,)/e
For a frustrum pointing in the opposite direction we should
have to interchange R, and R, and change the sign of ec,
which would give us our last expression with only the sign
of R,—R, changed in the denominator; thus for a pair of
frustra oppositely directed, we get
;.
P
7
a .
9 P dp ef y
On Ok MG
1
eB)
ARR, + BR +R)+1— (BARR) ARR, +B +R) +1)
or confining our attention to cases where (R,— R,)* may be
neglected, and remembering that (R,—R,)=Le and that
2lidv/dx is equal to the difference of velocities at the two
ends of the double frustrum, we see that an approximate
integral solution of the differential equation of thermal trans-
piration through a series of oppositely directed frustra in
which R, and R, are not very different (R, not to exceed
Transpiration and Radiometer Motion. 385
2R, say) is
oe eg Spel as a) bess apelin CLS)
Potpy e+, A'(p.+pi)?/4+ Bp, +pi)/2+ 1?
where
A’=9R,Ryv9 /Ny (v2 +21)";
B/=6a(R, + Ry) v9/No(ve+ 11)’,
while for a uniform tube we have seen that the solution (17) is
what this becomes when R,=R,=R. The simplicity of the case
of frustra breaks down when R, becomes only a small fraction
of R,, for then we cannot neglect (R,—R,)* as we did above.
To make clear the comparison between (17) and (19)
and experiment a brief description of Reynolds’s arrange-
ments is necessary: imagine a cylinder divided into five
compartments by planes perpendicular to its axis, the middle
one filled by a plate of porous material, those on each side of it
made into small gas-holders connectable with gas supply and
manometers and separated from the end chambers by metal
plates, the end chambers being intended to act as a sort of
jacket to each of the gas-holders, the one having a stream of
steam carried through it and the other a stream of cold water.
When a stationary state of temperature is established along
the cylinder, the two faces of the porous plate come to fixed
temperatures @, and @,, corresponding to the molecular veloci-
ties v, and v,, and the gas transpires from the cooler face of
the plate to the hotter, till the pressures become p, and p;
as given by the equation. The internal diameter of the
cylinder was 38 mm. and the thickness of the porous plates
varied from 1°5 to 14°2 mm., the materials being meerschaum
and stucco. Reynolds gives the temperatures of the two
jacket-chambers, but not those of the faces of the porous
plate, which are the ones we require ; we will show afterwards
how to obtain these approximately, but for the present it
suffices to know that in any one series of experiments v, and v,
remained constant, while the mean pressure (p,+ )/2 in the
passages of the porous plate varied from about 760 mm. of
mercury down to about 4. From any three sets of values of
Po—Pp, and (po+p )/2 for any gas, it is possible by equation
(19) to calculate (vg+%,)/(ve—v,) and A’ and B’, or from the
whole series of measurements mean values of these can be
calculated, and then at all mean pressures p.—p, can be
calculated for comparison with the experimental values.
For Reynolds’s meerschaum plate LI., having a thickness of
6°3 mm. and with the temperature of the steam-jacket at
100° C, and that of the water-jacket at 8°, the values of the
386 Mr. W. Sutherland on Thermal
parameters in (19) are, with the mm. of mercury as unit of
pressure,
A (v2+2,)/2(v2—2,) B' xt
al apne 16 0094 0-0
He kee 12 "0022 0-0
COS cere 16 012 0°0;5 2. e. 000005
5
These give the following comparison :—
(potp,)/2 ... 764 328 217 940 508 231 127
Air4 p,—p,cal. ... 59 50 4-4 31 2-2 1:2 i
PoP, exp... GL: ba 48> 30 oe eaten 9
(p+-p,/2 ... 767 330 199 108 508 254 127
H,1 p,—p cal 1. 238° 160. 112° 738- 38 "ee
p—p exp. s.. 204 “In? - 112 (e 38 2:0 ‘9
(Po+Pp,)/2 ... 764 495 362 267 203 114 508 254 127
CO,4 Ps—Pr al... 37, 38-38 37 35° 29 | 205 ome fi
po—p exp. «. 33 41 41 38 33 28 20 13 10
With the meerschaum plate III., having a thickness of
11-2 mm. and with the jackets at 100° and 17°6°, the para-
meters are :—
(vetv,)/2(v.—2,) B' A!
7a \ aes aC SS 18 “0094 0-0
j 5 Eau beae sn SwAee: 16°6 "00181 0-0
whence the comparison
(pp 2... 77 698 508 317 . 209 - 198) —avaee
Air} Po—p, cal. ... 52 5-1 4:8 4-] 39 38 27
Dap Otp-s DO. DA. 4 pede gee 2:3
(p,+p, 2s. 508 356 (211 102 71L 2S2e ieee
Ried ep tal joy A De Ve ig TD 59 oil Sipe aoe 20
eee exputis.g p16 13 1-0 63 “49 37 ‘39
(potp,)/2 ... 864 762 470 290 193 121 76:2
H,4 PoP cal. ... Py. 812 lies 11-5 8:6 6-0 4-0
p.—p exp... 21 195 159 17 8-0 56 4-0
(act pai2 5. AB 2L (279) 2b GS) las. Sele
H,lp,—pyeal. 3 2A VB «146 59796790) a8
Do—P1 EXP- ++ 27 16 1:68 "82 ‘96 58
With the stucco plate I., having a thickness of 6°3 mm.
and with the temperatures of the jackets at 100° and 18°°4
(17° for H,), the parameters are :—
Transpiration and Radiometer Motion. 387
(%+2,)/2(%,-%,) —-B A’
PCT gah Meee 20 045 "0,9
EI OU sara a 18 00917 0;61
which furnish the comparison
(Po+D,)/2 ... 757 594 564 288 131 88-9
Air} P.—P, cal. ... °59 66 67 “79 "85 "82
Po—P, exp.... °56 63 67 "88 "92 ‘83
(pPo+p,)/2... 59°7 Seg 152 6°35 3°84
Air 4 p,—p, cal.... “77 64 “45 “94 “17
Po—P exp... “T4 58 38 20 17
(Po+p,)/2 ... 858 686 508 406 339 249
H,4 2p, al.... 356 375 S91 394 392 378
| Po—Pp, exp... 340 365 389 389 390 3:82
(p,+p,)/2 .. 203 152 825 508 178 762
Ep peal! 362 S55 255 191 “85°” -40
p,—p, exp... B73 385 264 193 84 40
These comparisons show that the equation (19) represents
the facts of thermal transpiration, for the discrepancies
between calculation and experiment are mostly of the same
order as the experimental uncertainty, as can be seen from a
careful comparison of the experimental data amongst them-
selves.
We have now to consider the relation between the experi-
mental values of the parameters (v2 + )/2(¥2.—v,), B’, and A’
and their theoretical natures, First, as to (vp+ 1)/2(v.—1),
which is equal to (0,2? +6,°)/2(0,;—6,°) where 6, and 6, are
the temperatures of the two faces of the porous plate ; now
the walls between the jackets and the air-chambers are of
thin metal, and the chambers are of the same shape and size,
so that the mean of the temperatures 6, and @, must be
nearly equal to that of the two jacket temperatures 2, and
4,3; thus @.+0,;=%+%, which, with the values given for
(vp + v;)/2(v2—v,), suffices to determine @, and 0,.
ae) He) COs
Gs—21a (48 74
Gi—2is 34 ~ 27 34
@,—273 77 79
Meerschaum III.. . 6,-2738 40:5 39
6,—278 16 77
Siueeoel Uy Re. G = 2g any eA oeain 40
Meerscbaum [I.
388 _ Mr. W. Sutherland on Thermal
The temperatures @, and 6, are determined by the flow
of heat from the thin metal wall of the hot jacket to that
of the cold one, by conduction along the rubber walls of the
cylinder, by radiation across the two gas-chambers, by con-
vection-currents in the gas-chambers, and also by conduction
through the gas of the chambers, but the conductivity of
gases is so small compared to that of even badly-conducting
solids, that the direct effect of gaseous conduction may be
neglected, although the indirect effect of the conductivity of
the gas in determining the amount of heat carried by con-
vection may be appreciable, as would appear to be the case
with hydrogen and meerschaum plate II. It would be
possible to make a rough calculation as to what 0, and 0;
ought to be according to the theory of conduction, but Rey-
nolds states that the condition of the radiating surfaces and
the sizes of the chambers were altered during the experi-
ments, so that it is not worth while to do more than notice
that the values obtained for the temperatures of the faces of
the porous plates are consistent in a general way with what
we should expect from the temperatures of the jackets, the
thickness of the gas in the two chambers which was about
5 mm., and the given thicknesses of the plates.
As to the values of B’, which stands for
6 avp(Re+ Ry)/no(v2 +0)"
since mv? is proportional to 0, and (6,?+0,%)? is nearly the
same in all the experiments, we should expect B’n9/m? to be
constant for different gases with the same plate, and propor-
tional to the mean radius of the passages in the plate; thus, |
using the viscosities as obtainable from Graham’s experi-
ments in terms of that for oxygen as unity, and the molecular
masses in terms of that of hydrogen as 2, we have :—
H,. Air, CO.
WET oD 28°8 44
CT eae take mae “cas ‘90 SLs
which give the following values of 10° B/no/m? :-— |
H.. ° JAiry > acom
Meerschaum Tie 2.2) en68 158 137
Meerschaum III. ... 56 158
Stucco . f 285 TDS
This shows that the values yielded by hydrogen, instead of
being equal to those given by air, are between 4 and } of
them, a discrepancy whose cause will be found immediately;
but it is to be noted that while the results for hydrogen
Transpiration and Radiometer Motion. 389
make the mean radius of the passages in stucco 4:2 and 5:1
times those in meerschaum II. and III., those for air make it
4-8 and 4°8 times, and the agreement of the means 4°65 and
4-8 is close enough to show that our expression
6avy(R, + R,)/no(vs =f v1)
is right enough as regards the occurrence of the mean radius
ot the passages in it ; and moreover our equation (14) showed
that R entered in the form R/A, so that the discrepancy just
found must be due to some considerations being ignored in
connexion with 2X.
Now it is a well known fact established by experiment
that gases are condensed in the passages of porous bodies.
The condensing action exerted by a solid surface on a gas is
easily expressed quantitatively, for near the end of section (8)
of my paper on the Laws of Molecular Force (Phil. Mag.
[5] xxxv.) the attraction of a cylinder of radius ¢, length h,
and density p on a particle of mass m at distance z along the
axis from the nearest end, the law of force being 3Amm'/r*, is
2Ammp[1/z—1/(e+h) -1/(2 +22)? 4+ 1/fe + (e+h)*}"],
whence the attraction on a particle at small distance z from
the surface of a solid may be written 2Ammp/z, and if the
particle is one of the molecules of a gas, the condition of
equilibrium in the gas is
—dp/dz=2Ampnm/z or —dp/p=6Anp dz/zv’;
and if p, is the pressure in the layer nearest to the surface
which is at distance z, from the surface, and p, is the pressure
at a distance z, where the effects of the solid are negligible,
then
6A7rp
vy
log p,/p,.= LOZ a he en (20)
a formula which makes the density of the gas in contact with
the solid nearly proportional to the density where the gas is
free, because with gases 6A7p/v? is a small fraction. This
formula will be investigated a little further in my next paper,
on ‘‘ Boyle’s Law at very Low Pressures.”
A rigorous investigation for condensation in a tube would
be simple enough, but it suffices for our present purposes to
see that in most cases the density at the surface of the tube
will be connected with the density at the axis by the relation
Phil. Mag. 8. 5. Vol. 42. No. 258. Nov. 1896. 2F
390 - Mr. W. Sutherland on Thermal
obtained by putting surface-density and axial density in
place of p, and p, in the last formula.
- It is easy also to obtain an expression for the average density,
but as it is evident that for a given tube at a given tempera-
ture the average density remains proportional to the density
at the axis, which is the same as if there was no attraction
between gas and solid, we see at once that the effect of sur-.
face condensation on our investigation of thermal transpiration
is to multiply the density by a factor which remains nearly.
constant for a given tube or to divide X by the same factor,.
and the effect of ignoring this factor as we have done is to-
produce values of 10°B//m? which ought to be divided by
the factor before they should be expected to be constant for’
any one plate. Now experiment has shown that hydrogen is
much less condensed on solid surfaces than other gases, so_
that with hydrogen the factor will be nearly unity (probably),
and therefore, from the last little table, that for air between 2:
and 3; the factor for CO, ought to be larger still, as this gas”
is much more liable to surface condensation than air, while:
the last table would make the factor to be 137/68 or 2; but
too much reliance must not be placed on the value of B’ for
CQ;, as Reynolds found the trouble caused by the condensa-.
tion of the CO, to be so great as to discourage him from
making any further experiments with it. Thus the apparent
discrepancy in the last table has furnished some new evidence.
in connexion with condensation of gases in the passages of
porous solids.
As to the values of A’, which stands for
DRoRyv9?/No” (v2+ v1)* .
we see that as a is nearly 1 the value of B’?/16A! ought to ba.
nearly equal to (R, + R,)?/4R.R,, and of course the value of the.
ratio B”/16A’ is not affected by our ignoring condensation in
the establishment.of the fundamental equations. In the case of
hydrogen, the values of B’ and A’ for stucco I. lead to an unreal
value of R,/Rj, and thus we see that the formula (19) for conical.
passages is too much of a refinement for present purposes ;
and therefore abiding by the formula (17) for cylindrical
passages we may say that B”/16A’ ought to be not much
different from unity. For stucco I. the values of B’/16A!
are ‘86 for hydrogen and 2:5 for air, while for meerschaum II.
and CO, the value is 1°8; these “dies are near enough to
1:0 to give satisfactory evidence of the general soundness of.
the details in the theory.
Reynolds, g guided by his theory, formulated his experimental
Transpiration and Radiometer Motion. 391
results in a number of laws, but there is no occasion for us to
follow these seriatim as they are all contained in the sym-
bolical statements of equations (17) and (14), which we have
already tested by the experimental results.
An interesting result of thermal transpiration experiments
is that they enable us to calculate the mean radius of the
passages in porous materials, for in the case of hydrogen the
measurements must be but little complicated by molecular
force ; thus the values given for B’ in the case of hydrogen,
if multiplied by 760 to pass from the mm. of mercury to the
atmo as unit of pressure, and then divided by 1033°3 x 981 to
pass to the dyne per sq. cm. as absolute unit of pressure, give
us the values of 12av)R/no(v,+ ,)?; now as vet+v, may be
taken as relating to a mean temperature of 57° C.,
(v2 + v;)?=40,? 330/273,
and %=184400 cm. per see. and 4='00009, while a is
nearly unity ; thus for the mean radius of the passages in
Reynolds’s meerschaum II. we get ‘0000112 cm., in meer-
schaum II]. -0000092, or say 1/10° em. for meerschaum, while
for stucco I.the value is 000047 cm. ‘Thermal transpiration
gives no information as to the number of passages, but this
could be found from a measurement of the volume of air
transpired by a plate in unit time under a measured excess of
pressure on one side, or by other measurements relating to
transpiration under pressure such as those made by Graham
and Reynolds; for the delivery of gas could be calculated as
that due to N tubes of radius R, the discharge of each being
calculated according to O. E. Meyer’ s equation given in (7).
In this way Pherpial and pressural transpiration measure-
ments can be made to yield a measure of the average porosity
of any solid through which hydrogen can pass. By artifi-
cially altering the porosity in a series of preparations, as for
instance by hardening stucco under different pressures and
similar means, so that specific gravity would give a measure
of relative porosity, it might be possible to find a porosity
at which hydrogen just failed to pass, which would furnish
an independent measure of the diameter of the hydrogen
molecule.
[To be continued. |
2F 2
[ 392 ]
XL. On the Passage of Electricity through Gases exposed
to Réntgen Rays. By J. J. Thomson, V.A., FBS,
Cavendish Professor of Experimental Physics, Cambridge,
and H. RutuerrorD, M.A., Trinity College, Cambridge,
1851 Exhibition Scholar, New Zealand University".
fe facility with which a gas, by the application and
removal of Réntgen rays, can be changed from a con-
ductor to an insulator makes the use of these rays a valuable
means of studying the conduction of electricity through gases,
and the study of the properties of gases when in the state into
which they are thrown by the rays promises to lead to results ©
of value in connexion with this subject. We have during
the past few months made a series of experiments on the
passage of electricity through gases exposed to the rays, the
results of these experiments are contained in the following
aper.
s A gas retains its conducting property for a short time after
the rays have ceased to pass through it. This can readily be
shown by having a charged electrode shielded from the direct
influence of these rays, which pass from the vacuum-tube
through an aluminium window in a box covered with sheet
lead; then, though there is no leak when the air in the neigh-
bourhood of the electrode is still, yet on blowing across the
space over the aluminium window on to the electrode the
latter immediately begins to leak.
To make a more detailed examination of this point we used
the following apparatus.
A closed aluminium vessel is placed in front of the window
through which the rays pass. A tube through which air can
be blown by a pair of bellows leads into this vessel: the rate
at which the air passed through this tube was measured by a
gas-meter placed in series with the tube ; a plug of glass wool.
was placed in the tube leading to the vessel to keep out the
dust. The air left the aluminium vessel through another
tube, at the end of which was placed the arrangement for
measuring the rate of leakage of electricity (usually a wire
charged to a high potential placed in the axis of an earth-
connected metal tube through which the stream of gas passed,
the wire being connected with one pair of quadrants of an
electrometer). This arrangement was carefully shielded from
the direct effect of the rays, and there was no leak unless a
current of air was passing through the apparatus; when,
* Communicated by the Authors, having been read before Sectiun A of
the British Association, 1896,
On the Passage of Electricity through Gases. 393
however, the current of air was flowing there was a consider-
able leak, showing that the air after exposure to the rays
retained its conducting properties for the time (about $ second)
it took to pass from the aluminium vessel to the charged
electrode.
We tried whether the conductivity of the gas would be
destroyed by heating the gas during its passage from the
place where it was exposed to the rays to the place where its
conductivity was tested. To do this we inserted a piece of por-
celain tubing which was raised to a white heat ; the gas after
coming through this tube was so hot that it could hardly be
borne by the hand ; the conductivity, however, did not seem
to be at all impaired. If, however, the gas is made to bubble
through water every trace of conductivity seems to disappear.
The gas also lost its conductivity when forced through a plug
of glass wool, though the rate of flow was kept the same as
in an experiment which gave a rapid leak; if the same plug
was inserted in the system of tubes before the gas reached
the vessel where it was exposed to the Rontgen rays, in this
case the conductivity was not diminished. This experiment
seems to show that the structure in virtue of which the gas
conducts is of such a coarse character that it is not able to
survive the passage through the fine pores in a plug of glass
wool. A diaphragm of fine wire gauze or muslin does not
seem to affect the conductivity.
A very suggestive result is the effect of passing a current
of electricity through the gas on its way from the aluminium
vessel where it is exposed to the Rontgen rays to the place
where its conductivity isexamined. We tested this by inserting
a metal tube in the circuit, along the axis of which an insu-
lated wire was fixed connected with one terminal of a battery
of small storage-cells, the other terminal of this battery was
connected with the metal tube ; thus as the gas passed through
the tube a current of electricity was sent through it. The
passage of a current from a few cells was sufficient to greatly
diminish the conductivity of the gas passing through the
tube, and by increasing the number of cells the conductivity
of the gas could be entirely destroyed. Thus the peculiar
state into which a gas is thrown by the Rontgen rays is
destroyed when a current of electricity passes through it. It
is the current which destroys this state, not the electric field ;
for if the central wire is enclosed in a glass tube so as to stop
the current but maintain the electric field, the gas passes
threugh with its conductivity unimpaired. The current pro-
duces the same effect on the gas as it would produce on a very
weak solution of an electrolyte. For imagine such a solution
394 Prof. J.J.Thomson and Mr. Rutherford on the Passage
to pass through the tubes instead of the gas; then if enough
electricity passed through the solution to decompose all the
electrolyte the solution when it emerged would be a noncon-
ductor ; and this is precisely what happens in the case of the
gas. We shall find that the analogy between a dilute solution
of an electrolyte and gas exposed to the Réntgen rays holds
through a wide range of phenomena, and we have found it of
great use in explaining many of the characteristic properties
of conduction through gases.
Thus Rontgen rays supply a means of communicating a
charge of electricity to a gas. To do this, take an insulated
wire charged up to a high potential and surrounded by a tube
made of a non-conducting substance: let this tube lead into
a large insulated metallic vessel connected with an electro-
meter. If now air which has been exposed to Réntgen rays
is blown through the tube into this vessel the electrometer
will be deflected. This proves that the gas inside the vessel
is charged with electricity. Ifthe Rontgen rays are stopped
and the gas blown out of the vessel the charge disappears.
In these experiments we took precautions against dust.
The fact that the passage of a current of electricity through
a gas destroys its conductivity explains a very characteristic
property of the leakage of electricity through gases exposed
to Rontgen rays; that is, for a given intensity of radiation
the current through the gas does not exceed a certain maxi-
mum value whatever the electromotive force may be, the
current gets, as it were, “saturated.’’ The relation between
the electromotive force and the current is shown in the fol-
lowing curve, where the ordinates represent the current and
Fig. 1.
the abscissee the electromotive force. It is evident that this
saturation must occur if the current destroys the conducting
power of the gas, and that the maximum current will be the
current which destroys the conductivity at the same rate as
of Electricity through Guses exposed to Réntgen Rays. 395
this property is produced by the Roéntgen rays. Ifwe regard
the gas as an electrolyte, then the passage of a quantity e of
electricity will destroy e/e of the conducting particles, where
e is the charge carried by one of these particles. Let n be the
number of conducting particles in unit volume of the gas,
q the rate at which these are produced by the rays, an? the
rate at which these disappear independently of the passage of
the current, ¢ the current through unit area of the gas, / the
distance between the electrodes. Then we have
dn ns :
a = q—an = ie 3 ° e ° ° ° e 3)
so that when the state of the gas is steady,
0=g—an?—-, eerie or ere. ence)
le
When the current is small this equation gives
1 Ole ;
and as the number of conducting particles is independent of
the current, the current will be proportional to the E.M.F.
This corresponds to the straight part of the curve.
In the general case the current is proportional to the pro-
duct of n, the number of conducting molecules, and the
potential gradient. If E is the difference of potential between
the plates, U the sum of the velocities of the positively and
negatively electrified particles when the potential gradient is
unity, we have
le
t=neU/I or 2= OB
Substituting this value of n in equation (2), we get
0=q- 2h? ar, ie otesie rs eS eee (3)
We see from this that ¢ approaches the limit gel. Thus the
limiting current is proportional to the distance between the
electrodes ; so that when we approach saturation the current
will increase as the distance between the electrodes increases,
and we get what is at first sight the paradoxical result that a
thin layer of air offers a greater resistance to the passage of a
current than a thicker one. This is, however, easily accounted
for if we remember that the current destroys the conducting-
power, and that as in a thicker layer there are more con-
ducting particles than in a thinner one the current required
-to destroy them all will be greater.
396 Prof. J.J. Thomson and Mr. Rutherford on the Passage
The experiments show that the effect of the distance between
the electrodes (two parallel plates) on the current is very
marked. The following tables show the result of some expe-
riments on this point.
Potential-Difference between Electrodes 60 volts.
Distance between electrodes, Current (arbitrary
in millimetres. scale.)
gi iM «tA Mi ih 5 4
oF Ae” ah Mme tae 8
Nts tout ee eee
Gr Bi ems co teh, Sie 1
it SBOE SE se ep
a ee ee eee
3 ES are Sie OM
8 Pe Selah
With this large potential-difference the current was satu-
rated in all the experiments.
The next table contains measurements with a small potential-
difference.
Potential-Difference between Electrodes 1:3 volt.
Distance between electrodes, Current (arbitrary
in millimetres. scale).
715 adel Mpeg ar 1)
gH aera ee Ee ance 9357
2 SS ie eee
3 Oe isis as ke ee
8 LOIN tai OSes OOS
18 Pp me ere reat: |)
In this case the effect of distance is not so well marked as
in the previous one, where the H.M.F. was sufficient to satu-
rate the current at all distances.
The measurement of the rate of leak when the current is
saturated enables us to form an estimate of the number of
conducting particles present in the gas ; as in this case the
number of conducting particles produced in unit time by the
rays is equal to the quantity of the electrolyte destroyed by
the current in the same time. Let us take the case of hydro-
gen; when the current was saturated, the rate of leak between
two plates each about 10 sq. cm. in area and 1 cm. apart
was about 1 volt per second when a capacity of about 30 em.
was in connexion with the electrometer. Thus the quantity
of electricity passing between the plates in 1 second was
of Electricity through Gases exposed to Réntgen Rays. 397
about 10-! electrostatic units, or 1/3 x10" electromagnetic
units, and this quantity is sufficient to electrolyse all the
electrolytic gas produced by the Roéntgen rays. Now 1 elec-
tromagnetic unit of electricity sets free 10~* grammes of
hydrogen, or about 1 c.c.at atmospheric temperature and
pressure. Hence 1/3 x 10'! electromagnetic units correspond
to about the same number of cub. centim. of hydrogen ; the
volume of the space between the electrodes was about 10 c.c., so
in this experiment the fraction of the gas electrolysed was only
1/3 x 10”, z.e., one three billionth of the whole amount of the
gas. It is not surprising that some experiments we made to
see if any alteration in pressure was produced when a gas
was transinitting Rontgen rays should have given negative
results. The preceding estimate gives the average number of
conducting particles ; if the conducting state is intermittent
there may at certain times be a much larger number of these
molecules present. Itis probable that at all events, when the
current is saturated the conducting power is intermittent.
The action of the coil used to send the discharge through the
vacuum tube is intermittent ; thus, if between the passage of
two sparks the conductivity has time to vanish (and when any
current is passing through the gas the rate at which it
vanishes is very rapid) the gas will be alternately an insulator
and then a conductor. | | ¢
The following experiment is explained by the intermittent
character of the discharge. The gas exposed to the Réntgen
rays was in a piece of lead tubing open at both ends; this
was connected with one terminal of a battery, the other
terminal of which was connected with a wire running down
the axis of the tube. A blast of air was blown through this
tube, and it was found that when the current between the
wire and the tube was small, the blast diminished the current
to a large extent, though a current approaching saturation
was hardly affected by the blast. When the current was
affected the gas blown out of the tube was conducting; when
the current was not affected the gas did not conduct. If the
gas were exposed to steady radiation it would not be affected
by blowing unless the time taken by the gas to acquire the
conducting state under the influence of the rays was com-
parable with the time taken by the gas to pass through the
tube ; this is inconsistent with what we know from other
experiments as to the rapidity of action of the rays. If, how-
ever, the state of the gas is intermittent, then, since the blast
continues when the rays are not acting, it blows out conduct-
ing gas, and so diminishes its average conductivity.
To return to equation (3), if Lis the value of « when E
398 Prof. J.J. Thomson and Mr. Rutherford on the Passaye
is infinite, we may write the equation in the form
2
fa Oe , oa. a
where Lae ie
cal is independent of both E and v.
We have observed the relation between the current aa
the electromotive force for several gases, and for different
intensities of the Rontgen rays. The comparison of the re-
sults of these experiments with equation (4) is given in the
following tables :—
Leakage through Chlorine Gas.
Electromotive Current Current calculated
Force. observed. by equation 4.
eo 65
18 124 116
30 200 180
wi) 245
140 270 279
The observations marked with the asterisk were used to
calculate the constants.
Leakage through Air. -
*9 22
18 ag 38
30 67 67
*70 83
140 90 86
The observations marked with the asterisk were used to
calculate the value of the constants in equation 4.
Leakage through Hydrogen.
5 18 19
9 31
18 53 48
oD 63 58
*70 65
The observations marked with the asterisks were used to
calculate the constants in equation 4.
--of Electricity through Gases exposed to Réntgen Rays. 399
Leakage through Chlorine. (Strong radiation.)
Electromotive Current Current calculated
Force. observed. by equation 4.
5 53 53°4
*1(0) 100 ;
21 189 183
oD 275 255
10 309 ,
140 = aou 405
Leakage through Chlorine. (Weak radiation.)
=5 10
85 16 15
Mecle(, 26 23
- ao aoe,
105 d4 37
Coal Gas (1).
14 10 9°8
2°8 17-3
4:2 22 23
84 32°3 33
16°38 38°3 40
30 43
110 45 44
Coal Gas (2). Weak radiation.
1-4 3°6 4-2
2°8 8
4°2 el 11°2
2°6 14°7 15-2
8-4 Zale Zee
12°6 32 30°4
16°8 38
Hydrogen.
3°4 5
5:1 75 6:9
a) 10 10°1
6 15 13°4
400 Prof. J.J. Thomson and Mr. Rutherford on the Passage
Sulphuretted Hydrogen. (Strong radiation.)
Electromotive Current Current calculated
Force. observed. by equation 4.
15°6 8°7
34 18 iNet
68 30°8 28°5
126 40
Sulphuretted Hydrogen. (Weak radiation.)
15°6 3°8
34 6:3 6°2
68 8 8
136 8°7
Mercury Vapour.
Ses 14:2 14°6
85 23
15°6 35 36°9
o4 55 59
68 75
136 (i) 8:2
As these measurements require the intensity of the radia-
tion to be maintained constant during each series of observa-
tions, a conditicn which it is very difficult to fulfil, we think
the agreement between theory and observation is as close as
could be expected.
We have seen how from the measurement of the limiting
current we could form an estimate of the proportion which
the conducting particles bear to the rest of the molecules of
the gas. We can, in addition, get from the curve represent-
ing the relation between the current and the electromotive
force an estimate of the velocity with which these particles
move. ‘Taking equation (3)
we shall endeavour to express the coefficients in terms of
quantities which our experiments enable us to estimate. Let
I be the limiting current when the electromctive force is
infinite, then
I=dgle.
Let T be the time which elapses after the rays have been
stopped for the number of conducting particles to fall to one
half the number just before the rays ceased, no current passing
through the gas. Then, just before the rays cease to fall on
of Electricity through Gases exposed to Réntgen Rays, 401
the gas, we have from equation (2),
x= {2}
where N represents the number of conducting particles at
this stage ; after the rays have ceased, we have
dn A
Cali
or 1 ]
we
if t is the time which has elapsed after the rays have stopped,
when t=T, n=4N, hence
= —- 8 |e
substituting for N its value, we get
1
T= —,
ag
or 1 le
PG Ph
Substituting for g and a the values just found, equation (4)
becomes
J De cre
fen a ee
or Ee
I(1—.) — Th? Ue? . e ° ° ° ° (3)
Thus in the straight part of the curve, where ¢ is small com-
pared with I, we have approximately
yo HUE
oes ec (8)
Now HU/I is the sum of the velocities of the positively
and negatively charged particles in the electric field. Hence,
equation (6) shows that the current bears to the maximum
current the same ratio as the space described by the charged
particles in time T bears to the distance between the elec-
trodes. In an experiment where / was about 1 cm., the rate
of leak through air for a potential-difference of 1 volt was
about #5 of the maximum rate of leak, hence the charged
402 Prof. J.J. Thomson and Mr. Rutherford on the Passage
particles must in the time T have moved through about 3 of
a centimetre. The time T will depend upon the intensity of
the radiation ; it could be determined by measuring the rate
of leak at different points on the tube through which the
conducting gas was blown in the experiment mentioned at
the beginning of this paper. We hope to make such experi-
ments and obtain exact values for T; in the meantime, from
the rough experiments already made, we think we may con-
clude that with the intensity of radiation we generally
employed, T was cf the order of 3), of a second. This would
make the velocities of the charged particles in the air about
*33 cm./sec. for a gradient of one volt per cm. This velocity
is very large compared with the velocity of ions through an
electrolyte ; it is, however, small compared with the velocity
with which an atom carrying an atomic charge would move
throngh a gas at atmospheric pressure; if we calculate by
the kinetic theory of gases this velocity, we find that for air
it is of the order 50 em./sec.; this result seems to imply that
the charged particles in the gas exposed-to the Rontgen rays
are the centres of anaggregation of a considerable number of
molecules.
The relation between the current and electromotive force
given by equation (4) corresponds to that obtained by experi-
ment for a number of gases; it does not, however, exhibit a
peculiarity which we have sometimes observed, especially
when the radiation was strong, i. e., the existence of a part of
the curve where the current increases faster than would be
the case if Ohm’s law were true ; this is shown by the portion
EF of the curve in fig. 2, which represents the relation be-
~
~
A
~
~
~~
tween the current and electromotive force through sul-
phuretted hydrogen. When the intensity of the Rontgen
rays is altered, the alteration in the current is not the same
of Electricity through Gases exposed to Réntgen Rays. 403:
at different points in the curve. When the intensity of these
rays is diminished, the saturation current is diminished in a
larger proportion than the current for small electromotive
forces. This is shown by the following diagram, which
represents the » and E curves through chlorine gas for dif-
ferent intensities of the Rontgen rays; the weak radiation was
got by interposing a thick aluminium plate. In this diagram
the ordinates for the weak radiation have been increased so
as to make the ordinate for the saturation current of the
weak radiation the same as that of the strong. When this is
done the rest of the ‘‘ weak” curve is above the strong,
showing that the diminution in the radiation has affected the
saturation current to a greater extent than the weaker cur-
rents. The saturation current depends only on the number
of conducting particles produced by the rays ; for the smaller
eurrents the diminution in the number of molecules is to
some extent compensated for by the increase in the time
taken for these to recombine ; thus T is increased when the
intensity of the rays is diminished, so that, as we see from
equation (6), the proportion between a small current and the
saturation current is increased when the intensity of the rays
is diminished.
Whatever is the magnitude of the electromotive force, a
diminution in the intensity of the rays is accompanied by a
diminution in the current, so that the | and B curves for two
intensities of radiation would not intersect if both were
drawn on the same scale.
If, however, instead of keeping the gas the same and
altering the intensity of the radiation, we alter the gas and
404 Prof. J.J.Thomson and Mr. Rutherford on the Passage
keep the intensity of the rays constant, then the I and HE
curves for two different gases may intersect. This effect is
shown in the following diagram, which represents the I and
EK curves for hydrogen and air. We see that for small
electromotive forces the current is greater in hydrogen than
in air, while the saturation current is much greater in air
than in hydrogen. ‘The saturation current depends merely
Fig. 4.
|
|
on the number of conducting particles produced by the rays,
while the current in the earlier part of the curve depends on
the space described by the conducting particles in the time T
(see equation 6), and we infer that more conducting particles
are produced by the rays in air than in hydrogen, but that
the product of U, the velocity of these particles, and T, a
time which is proportional to the time these particles linger
after the rays are cut off, is greater for hydrogen than it is
for air.
In fig. 5 we give the curves for air, chlorine, sulphuretted
hydrogen, and mercury vapour, the curves being drawn on
such scales that the ordinate representing the saturation
current is the same in all these cases. It will be noticed that
the curves for air, for sulphuretted hydrogen, and for chlorine
coincide, mercury vapour falls below, while the hydrogen-
curve would be above. This shows that, using the notation
of equation (6), UT is the same for air, chlorine, and sul-
phuretted hydrogen, and that its value for these gases is
smaller than for hydrogen and greater than for mercury
vapour.
It is remarkable that the shapes of the curves for air,
sulphuretted hydrogen, and chlorine should agree so closely,
of Electricity through Gases exposed to Réntgen Rays. 405
for the absolute values of the current in these gases is very
different, the saturation current in sulphuretted hydrogen
Fig. 5.
be)
ROS
%
NS
~ ™.
XR.
S
BMF
being in some cases three or four times that of air, while that
of chlorine is in some cases as much as ten times that of air.
The value of the saturation current varies greatly in
different gases; of the gases we have tried it is least in
hydrogen, greatest in mercury vapour, the saturation current
in mercury vapour being about 20 times that for air. It does
not seem to depend entirely on the density of the gas, as in
sulphuretted hydrogen it is three or four times what it is in
air, though the densities are nearly equal, while, though the
density of the vapour of CH,I, is greater than that of mercury
vapour, the saturation current in the former gas is only a
small fraction of its value for the latter. The gases which
have large saturation currents are those which contain the
elements which have an abnormally large specific inductive
capacity in comparison with their valency.
We have made a large number of experiments with the
view to seeing whether there is any polarization when a
current of electricity passes through the gas; we have not,
however, been able to satisfy ourselves of the existence of this
effect. The absence of polarization implies, however, that the
ions are able to give up their charges to the metal electrodes.
Experiments on electritied gases show, however, that it is very
difficult to get a charge of electricity from a gas to a metal
unless the metal is exposed to radiation, either by the metal
being sufficiently hot to be luminous, or when it is exposed to
ultra-violet light. But in the case of the passage of electricity
through a gas which has been exposed to Réntgen rays the
Phil. Mag. 8. 5. Vol. 42. No. 258. Nov. 1896. 2G
406 On the Passage of Hiectricity through Gases.
conduction takes place even when the system is not exposed
to the direct radiation fron the exhausted tube; we think it
probable therefore that the gas itself radiates after being ex-
posed to the Rontgen rays.
To test this we tried the following experiment. AB, CD
Fig. 6.
A CG D B
are two concentric cylinders made of thick lead tubing, the
base of the inner one was cardboard, so as to allow Réntgen
rays to pass through the gas in the inner cylinder. A metal
ring was placed between the two cylinders and connected —
with one pair of quadrants of an electrometer so as to allow
the leak from it when raised to a high potential to be mea-
sured. A slit was cut in the inner cylinder in such a place
and of such a size that no rays could pass through it directly
from the bulb. The apparatus was filled with chlorine, as
this gas is one which gives a very rapid rate of leak. When
the slit was left open there was a rapid leak due to the diffu-
sion from the inner cylinder of gas which had been exposed
to Réntgen rays. When, however, the slit was covered up
with a strip of paper the leak wholly disappeared, though the
ring connected with the electrometer was placed at the same
level as the slit and therefore exposed to any radiation that
might come from the gas. ‘This radiation, if it exists, must
therefore either be of very feeble intensity or else it must
differ from the Réntgen rays in not making a gas through
which it passes a conductor of electricity. We are inclined
to think that when Rontgen rays are incident on a metallic
surface the “ diffusely reflected” rays are not of the same
character as the incident ones, and have not nearly the same
power of rendering a gas through which they pass a conductor
of electricity. We base this opinion on the experiments we
have made to detect the existence of electrical effects due to
the “reflected ” rays ; though we have made many attempts
we have never been able to detect the existence of any electrical
On the Resistance of the Electric Are. 407
effects from the reflected rays. Thus we introduced in the
apparatus in fig. 6 a lead plate inclined at an angle of 45° to
the axis of the cylinder, and so placed as to reflect the rays
through the slit, which was covered with a strip of paper ;
the arrangement was so sensitive that if the plate had reflected
-anything like one per cent. of the rays incident upon it, the
leak from the metal ring would have been easily detected ;
there was, however, no trace of a leak. ‘The results of ex-
periments on the photographic effects produced by rays
diffusely reflected from metallic plates seem to show that these
rays are fairly abundant. Taking this result in connexion
with the absence of any noticeable electrical effect produced
by these diffusely reflected rays, we think that the latter differ
in character from the incident rays.
We have not been able to detect any effect produced by a
magnetic field on the rate of leak ; we tried with the lines of
magnetic force parallel and also at right angles to the current,
-and with both small and saturated currents.
The rate of leak through air that had been dried by standing
for three days in the presence of phosphorus pentoxide did
not differ appreciably from the damp air of the room.
In conclusion, we desire to thank Mr. KE. Everett for the
_assistance he has given us in these experiments. The period
during which a bulb gives out Rontgen rays at a uniform
rate is not a long one, and as most of our experiments re-
quired the rate of emission to be constant, they have entailed
the use of a very large number of bulbs, all of which have
been made by Mr. Everett.
XLI. On the Resistance of the Electric Arc. By Jvuuius
Fritu, 1851 FEehibition Scholar, the Owens College, Man-
chester, and CHARLES Ropgers, B.Se., 1851 Le
Scholar, Firth College, Sheffield.
[Plates III. to V.]
HERE seems to be some uncertainty as to what is meant
by the resistance of the are. Any given arc is a phe-
nomenon which exists at a detinite P.D. and current, and
any attempt to measure its resistance must alter the state of
the arc as little as possible or else we are no longer dealing
with the same phenomenon. Hence it seems to us that the
only way in which the resistance of the are can be measured
is by the ratio of a very small increment of P.D. applied, to
* Communicated by the Physical Society: read May 8, 1896.
2G 2
408 Messrs. J. Frith and C. Rodgers on the
the small increment of current produced. It must also be
borne in mind that the change in the current must be of so
short a duration that the form of the carbons is not in any
way altered.
Apparently the only method which fulfils these conditions
is one in which a small alternating current is used, super-
imposed on the main continuous current. This has the effect
-of rapidly increasing and decreasing by a small amount the
current passing through the are. It seems probable that the
effect on the are of each small increase of current is annulled
by the decrease of current immediately following. In fact a
comparatively large alternating current may be superimposed
-on the main continuous current without producing any visible
effect on the are. 3
We are thus led to define the resistance of the arc as the
ratio of a small increment of P.D. applied, to the small incre-
V
dA
It is most important to distinguish this quantity, which we
ment of current produced. This may be briefly written
call the “ instantaneous ”’ from the tangent of the incli-
Vv
TA’
nation of the tangent line of the curve representing the steady
values of V and A, which we will call, for the sake of clear-
(44 = 2) d s
ness, the “ steady DE
We have performed some experiments to exemplify the
difference that exists between these two quantities; and also
to show that in cases analogous with the arc, where, however,
the result can be verified, the instantaneous = found by
superimposing an alternating current, gives correct values
_ for the resistance.
In one of these experiments a glow-lamp, taking 10 am-
peres at about 8 volts, was placed in series with three I.E.S.
50 ampere accumulators, and a current sent through against
the E.M.F’. of the cells. This arrangement is just what is
_ wanted to test the method, namely a resistance in series with
a back E.M.F., both of which are functions of the current ;
and, further, the resistance can be separated from the back
K.M.F. and measured, and the result compared with the value
dV
ebtained for the resistance by the instantaneous re
The result of this experiment is represented on PI, III.
Here are plotted the curves connecting the current and P.D.
between the outside terminals of the arrangement and also
Resistance of the Electric Are: 409
between the terminals'of the lamp and of the battery. This
last gives the back H.M.F., since the resistance of the
cells was small enough to be neglected. The P.D. at the
lamp divided by the current gives the resistance ; this is
plotted on the same diagram ; to make it clearer, however,
the scale of ohms is multiplied by 10. We have measured*
the instantaneous = at various current strengths, and the
values of this are plotted in a dotted line. It is seen that
there is a very close agreement between these two measure-
ments of resistance. On the same sheet are plotted values
for the steady ue This differs considerably from both
the resistance-curves. This leads us to see that the rapid
excursions caused by the alternating current are not along
the curve joining the steady values of V and A, but along
a line which is everywhere more vertical than the tangent
to the curve. This line is formed by joining the point on
the curve to the instantaneous origin, which is distant from
the origin of the diagram by an amount equal to the back
H.M.F. at that particular current. Were the electrical
excursions to travel along any intermediate path, the value
obtained for the instantaneous a would be dependent on
the frequency. As will be seen later, this is not the case
between the experimental limits of 250 and 7 complete alter-
nations per second. If, therefore, the arc, as has been affirmed
by various authorities, consists of a back H.M.F. and a re-
sistance, we feel justified in applying this method for the
measurement of its resistance, which has been found correct
in closely analogous cases.
Now at very low frequencies indeed the electrical oscilla-
tions would travel along the curve connecting the steady
values of V and A ; and this is clearly the meaning of the
critical frequency which we have observed with cored carbons
(see p. 421), namely, that under the critical frequency the
superimposed alternations travel on the steady value curve
and become identical with the “ steady” gy :
Several experimenters have obtained values for the resist-
ance of the arc which agree fairly well amongst themselves,
and which seem to show that the arc has a positive resistance f.
* By method I. below. :
+ An abstract of papers bearing on this subject was given by Mrs
Ayrton in the ‘ Electrician,’ Sept, 18, 1895.
410 Messrs. J. Frith and C. Rodgers on the
It was pointed out, however, by Prof. Ayrton at the
Ipswich meeting of the British Association, that although
there was a marked agreement between the values obtained
by these experimenters, they were not at all in accordance
with the conclusions drawn by himself from consideration
of the curves obtained by Mrs. Ayrton. These curves con-
nect the P.D. between the carbons with the current passing
through the are, for various fixed lengths of arc, and from
them it is seen that for a given are-length an increase of P.D.
is always accompanied by a decrease of current. From this
fact Prof. Ayrton concluded that.if an attempt were made to
measure the resistance of the arc by altering the P.D. between
the carbons and finding the corresponding alteration of cur-
rent produced, the resistance found by taking this ratio must
be negative.
This conclusion was strengthened by some experiments
made by Mr. Mather at Prof. ” Ayrton’? s suggestion. In one
of these experiments two points of equal potential were found
in a circuit consisting of an arc, a battery, and a resistance.
Another battery, consisting of a few ceils, of known H.M.F.
and resistance was applied between these two equipotential
points and the current flowing through the battery was noted.
The resistances of the two parallel halves of the circuit, exclu-
ding the arc, were known, so that the current which, taking
the are resistance as zero, should flow through this battery
~ could be calculated. Now the value of this calculated cur-
_ rent was found to be less than the observed value, no matter
in which direction the P.D. was applied, and this result was
also obtained when an alternating P.D. was used. Hence
the resistance of the arc was apparently less than zero.
The other experiment consisted in running the are ata
steady P.D. and current, suddenly altering the resistance in
circuit by a small amount, and noting the changes in the
ammeter and voltmeter-readings so produced. The new con-
ditions were maintained only long enough to allow of these
readings being taken. The are was then brought back to its
former condition before taking another reading. It was found
that a change of P.D. in one direction was always accompanied
bya change of current in the opposite direction. The results
of both experiments were, however, only qualitative.
All these experiments, together with the consideration of
the curves found by Mrs. Ayrton, lead to the conclusion that
the arc has a negative resistance, while former experimenters
had all obtained a positive resistance.
It was in order to throw some light on this discrepancy
that we undertook a series of experiments to determine with
Resistance of the Electric Arc. “= SEE
som? degree of accuracy the resistance of the arc under various
conditions.
Methods.
A number of methods were tried using alternating currents,
of which the following were most successful :—
Method I. is represented diagrammatically in fig. 1. D is
Jaytes, Ne
fll
B
Ul
the armature of an alternator, the current from which passes
round two circuits in parallel, one of which contains the are
X, and the other an adjustable resistance R. By adjusting R
the alternating currents in the two halves can be made equal.
When this is the case the impedances of the two halves to
-alternating currents must be equal.
In the diagram the continuous-current circuit is shown to
the left. It consists of a battery of accumulators B, the
hand-adjusted arc-lamp X, the resistance K, the ammeter A,
and (with the commutator C as shown) the resistance S
and the alternator D. It will be noticed that the alternator D
carries the continuous current, but this of course does not
prevent its acting as an alternator.
In order to measure the small alternating current indepen-
dently of the continuous current flowing we used the air-
transformer T, the thick wire coil of which was in series with
the alternator D, the thin wire coil being connected with an
412 Messrs. J. Frith and C. Rodgers on the
electrostatic voltmeter E. The reading of the electrostatic
voltmeter is thus unaffected by the continuous current, while
itis, ata given frequency, a measure of the alternating current
flowing.
By means of the commutator OC, the air-transformer T can
be thrown into either circuit, the resistance S being by the
same operation thrown into the other circuit. The resistance
S is equal to that of the thick wire coil of T, so that when 8
replaces T the continuous current is unaffected by the change.
Method of Experimenting.
The are was run at the required current and P.D. by alter-
ing the number of cells in B, K being always kept the same.
The current was kept constant by adjusting the are by hand.
Under normal conditions the current could be kept constant
to within 1 per cent. R was now adjusted till the deflexion
of E was the same when T was in either circuit. When
balanced, a change of 0°01 ohm in R caused an appreciable
difference in the deflexions of KE. It was found to be useless
to adjust more accurately than this, since the small variations
in quality always found in carbons produced differences of
this order in the resistance of the arc.
Let the value of R when a balance is obtained be R;. This
is equal to the resistance, to alternating currents, of the bat-
tery (b;), the resistance (k), the arc lamp and connexions (/),
and the are (2).
R,=k+6,+ 2. .- 7). =e
The carbons are now firmly screwed together and the number
of cells in B reduced till the continuous current is the same
as before. R is again adjusted till the deflexions of E are
equal ; and if R, is the new value thus obtained,
Re=k+betd;... >.<
b, being the resistance of the portion of the battery now used.
The cells are next cut out, the mains leading to them short-
circuited, and a third value R; obtained,
RwSh4 1 2 oy
From (ii.) and (ili.) we obtain the resistance of bj, and by
proportion of any number of cells. Putting these values in
(i.) we obtain the value of 2 in ohms.
The advantages of this method are that no calibration of
the electrometer is required, the speed of the alternator need
not be constant for long periods together, various alternators
giving different frequencies and wave-forms can be used
without materially altering the circuit, and especially that
Resistance of the Electric Are. 413
the resistance of the arc can be obtained directly in ohms as
the difference of two readings of the box R.
Method II. is shown diagrammatically in fig. 2; the arc
Fig. 2.
ES B
Ha
circuit in this case being shown to the right. X is the are,
B the battery of accumulators, K the resistance, A the am-
meter, and V the voltmeter across the arc. B and K were
made large, K being about 1l ohms. The circuit shown on
the left consists of the alternator D, the transformer T, which
together with E is now used simply as a delicate alternate-
current ammeter, a condenser F’, and a commutator C. By
means of C we can put in circuit either the resistance R,
or the arc-lamp X in series with the resistance 8S. The alter-
nating current flowing through K is negligible compared
with that flowing through X on account of the high resist-
ance of K. The condenser F prevents any continuous current
due to the P.D. at the arc from passing through the alter-
nator.
Method of Experimenting.
If L is the self-induction of the circuit and F its capacity, the
impedance I of the circuit is given by Ms R?+ ( = — Lo) ;
where #=27 x frequency. Thisisa minimum when LF'o’=1.
A414 Messrs. J. Frith and C. Rodgers on the’
The alternator was run at a speed corresponding to this fre-
quency, at which the arrangement is most sensitive to changes
of R.
An experiment consists in adjusting R till the deflexions
of E are the same whether R or S+<z is in circuit. Then
R=S+a2.
The experiment was first tried without a resistance at 8,
- but it was found impossible when using solid carbons to
obtain a balance even when R was zero. In order to make
it possible to obtain a balance with R greater than nothing,
a resistance S numerically greater than the negative resistance
of the arc had to be put in series with the arc. In our expe-
riments we kept 8S constant and adjusted R. The experiment
might have been performed by making R=0, and adjusting
S till the readings of E were equal. Then 8+. must be
equal to zero, and therefore z=—S, or the resistance of the
arc is equal to the resistance in the box 8 with its sign changed.
This method gives results which agree well with those ob-
tained by Method I.; but on account of the necessity of
keeping the speed of the alternator constant the arrangement
is more troublesome to manage, and therefore was used only
as a check method.
Description of the Apparatus.
The arc-lamp had a hand adjustment for each carbon with
centering arrangements for the positive. The arc-length was
measured by projecting an image of the arc on to a screen by
‘means of a lens in the usual way.
The resistance K consisted of broad platinoid strips bent
back on themselves so as to avoid self-induction*. It did not
heat appreciably with the largest currents used.
The battery consisted of 50 10-ampere E.P.S. cells arranged
in 4 groups of 10 cells each, 4 of 2, and 2 groups of 1.
These groups could be connected up by means of mercury
cups.
The ammeter and voltmeter were Weston instruments of the
‘horizontal type. ;
The azr-transformer T was made in two sections which were
usually used in series. The constants were as follows :—
Section No. 1:
Fine wire coil.
Diameter of wire (uncovered) =4°3 mils.
£ 5. (covered) “=6S~ ,,
Namber ot turns s4- 2). — tou:
Resistance, . epee 2) =F ohms.
* Described in Phil. Mag. February 1892.
Resistance of the Electric Are. | _ 415
Thick wire coil.
Size of wire No. 12 S.W.G.
Number ofturns 2... « . ==128:
Resissance. . is. = . «. .. =O: l4a°ohm.
Section No. 2:
Fine wire coil. .
Diameter of wire (uncovered) =3 mils.
‘3 seen (covered))iy ==;
Niumaber,of GUEMS 4a). 15. Jo. = 295500.
Resistance... .... « .. = 32,1 20;ohing,
Thick wire coil.
Size of wire No. 12 8.W.G.
Number Of Gunuse 1 taieoy 0 = Gs
IesistiaMmCesey acs. «say.is 47a 02L1 2 ohm.
The thick wire was double-cotton-covered. The thin wire
was silk-covered.
The voltmeter was an Ayrton-Mather reflecting electro-
static voltmeter giving a deflexion of 700 scale-divisions at a
distance of 2500 scale-divisions for 100 volts. This instrument
was admirably adapted to our purpose on account of its very
quick swing and excellent damping.
- The resistance R consisted of a box wound with thick
german-silver wire plugging to 0-1 ohm, in series with which
was a german-silver wire (No. 10 8.W.G.) doubled back on
itself and carrying a slider. Points were marked out at dis-
tances corresponding to 0°01 ohm. These spaces were about
10:7 centims. long.
The Alternator. Several different alternators were used.
The one used while obtaining the curves given was a Pyke
and Harris machine, which was especially suitable, since it
had no sliding contacts, both armature and field being fixed.
The other machines were—tor high frequencies a Ferranti,
and for low frequencies a Gramme alternator. The fields
in all cases were excited by storage-cells.
The condenser had usually a capacity of about 60 micro-
farads.
Variation of Conditions.
The conditions of the experiments were varied as much as
possible. We have studied the effect on the resistance of the
are of variations in the amount, frequency, and wave-form of
the alternating current; the effect of different kinds of
carbons and different P.Ds. and currents ; the effect of using
different combinations of cored and solid carbons, of carbons
cored with substances other than carbon ; and the effect of
the relative size of the carbons.
416 Messrs. J. Frith and C. Rodgers on the
Results.
The amount of alternating current between the limits we
have used does not appear to have any influence on the value
obtained for the resistance of the arc. The largest alternating
current used had a R.M.S. (root mean square) value equal
to about 10 per cent. of the continuous current. The smallest
had about one-tenth of this value. In obtaining the curves
shown, the R.M.S. of the alternating current was usually
about 0:5 ampere.
The frequency also did ae have any effect on the resistance
of the arc between the limits 250 and 7 complete alterna-
tions per second. Frequencies above 150 were obtained
from a Ferranti alternator ; those between 150 and 45 from
a Pyke and Harris; and those between 24 and 7 froma
Gramme machine.
Besides these three machines a Mordey transformer of
ratio 1:1 was sometimes used, with its primary in circuit
with the Pyke and Harris alternator. On account of its
having a closed iron circuit it was not so suitable, but it
gave the same results as the alternators.
We have thus obtained considerable variety in the wave
form as well as in the frequency. Hence the wave form
does not affect our measurements of the resistance of the arc.
The values thus obtained for the resistance of the are
under various conditions are plotted in Pls. [1V.& V. Pl. IV.
shows the relation between the resistance of the are and the
current, the P.D. between the carbons being kept constant.
PI.V. fig. 1 shows the relation between the resistance of the arc
and the P.D. between the carbons, the current being kept
constant. P1J.V. fig. 2 shows the relation between the resistance
of the arc and the arc length, the current being kept constant.
In all these cases the resistance is measured from the central
line marked 0, positive values above the line and negative
values below the line.
Certain makes of carbon have been selected for experiment,
namely, Apostle, Brush, Thomson- Houston, and Carré carbons,
as representative of the various qualities of carbon now in
use commercially.
The diameter of the positive carbon was in all cases
11 millim., the diameter of the negative was 9 millim. in the
case of the Apostle and Brush carbons, 8 millim. in the case
of the Carré carbons, and 11 millim. in the case of the
Thomson-Houston carbons. These sizes were selected as
being most suitable for the currents we were able to take
Resistance of the Electric Are. ALT
from our cells. The sizes of the Apostle carbons were those
used by Mrs. Ayrton in her experiments.
That the diameter of the carbons used does not have any
great effect on the resistance of the arc for the same current
and voltage is shown by the curves D and H, PI. IV. and fig. 1,
P1.V. These are the curves for two solid Apostle carbons—one
set D having carbons 11 millim. and 9 millim. and the other
set E having carbons 18 millim. and 5 millim. in diameter re-
spectively, and it will be seen that they lie very close together.
On the curves each make of carbons is represented by a
particular kind of line as explained in the figure, the Apostle
carbons, for instance, are represented by a dotted line. Curves
bearing the same letter refer to the same combination of
carbons in the different makes. For each make there are four
combinations :— + cored, —cored; +cored, —solid ; +solid,
—cored ; +solid, —solid; represented respectively by the
letters A, B, C, and D.
The general characteristics of these curves are that the
_ordinates of those for +solid —solid are always negative, and
those for +cored —cored always positive, while the other
_euryes all lie between these two extremes, those which have the
positive carbon solid always being more negative than those
-which have the positive cored. The greatest uniformity is met
with in the case of the +solid —solid combination, in which
all the curves lie close together. For this reason the curves for
the Thomson- Houston carbons have been omitted, as they lie
completely on the other curves, and would tend to cause
confusion. In the case of cored carbons, however, the uni-
formity is not so marked, as the material of the core varies
largely with the make. In fact, in a single carbon the core
may vary considerably, and for this reason the curves for
cored carbons are more troublesome to iake.
P]. [V. shows the relation between the resistance of the
are and the current-that the arc is taking, the voltage at the
earbons being kept constant. It will be noticed that in the
case of those combinations which have a solid positive, the
number expressing the negative resistance of the are increases
as the current decreases. With the other two combinations—
those having a cored positive—the curves for Brush and
Carré carbons have the same general characteristics, while
those for the Apostle carbons cut these latter. This is pro-
bably due to peculiarities in the coring of the Apostle
carbons.
Pl. V. fig. 1 shows the relation between the resistance of the
are and the P.D. between the carbons, the current being kept
constant at 10 amperes in all cases. These curves are more
418 Messrs. J. Frith and C. Rodgers on the
instructive than those in PI]. IV., as the character of the are
alters more with change of voltage than with change of
current. As before, the curves for the solid carbons are all
very close together. Jn this case they all show a minimum
(maximum negative) value at about 55 volts. With com-
binations having a cored positive, this minimum becomes
more strongly marked and occurs at a lower voltage.
It has been our practice to examine the image of the are
on the screen and note when any change in the appearance
of the arc takes place. It was first noticed by Mrs. Ayrton
that with cored carbons under certain conditions a dark space
appeared somewhere near the centre of the are, dividing the
purple glow into two parts. This was never noticed in the
‘ease of solid carbons. Our observations have fully confirmed
this, and, moreover, we find that the dark space, when only
one carbon is cored, is nearer this carbon. We have noted
the appearance or disappearance of the dark space, and we
find a remarkable coincidence between this point and the
minimum of the curves. In the curves for cored carbons in
Pl. V., to the right of the minimum point the arc always
shows a dark space which becomes less marked on approach-
ing the minimum, and at the minimum finally disappears. To
the left of this point, although the dark space has quite dis-
appeared, yet the difference in colour between the two parts
of the glow is still observable, that near the negative carbon
being of a redder tint than that near the positive.
P1.V. fig. 2 shows the relation between the length of the are
and its resistance at constant current. The general charac-
teristics of these curves are the same as those of fig. 1, since
the length of the are is roughly proportional to the voltage.
Effect of the Core.
Since the presence of a core in either carbon has sucha
marked influence on the resistance of the arc, we tried the
effect of boring out the core ofa Carrécarbon. The result of
using this hollow carbon as the positive is shown by curve
in Pl. V. It will be seen that this curve follows very closely
that obtained with both carbons solid. The effect was now
tried of filling this hollow carbon first with plaster of Paris,
and then with kaolin. In both cases the resistance of the
arc so produced was positive. Another effect of these cores
was to cause the are to burn at a much lower voltage for
a given current and are length. The are was, however, too
unsteady to allow of a curve being taken.
Resistance of the Electric Are. A19
Particulars of Cores.
he diameters of the various cores are as follows :—
Diam. Diam.
Carbon. of carbon. of core.
mm. mm.
mposties y= Ll Dek
9 Dao
eebrushp 224 an ee hl. Dike
i) 2°18
Carré Sai: he eel tl 2:4
8 1:9
Inverted Ares.
In the experiments hitherto described the positive carbon
has always been uppermost. Hxperiments were now made to
ascertain if the resistance of the arc is dependent upon
whether the positive carbon is the upper or the lower.
- Using solid carbons, the resistance is not appreciably
altered by inverting the arc. With cored carbons, however,
the resistance of the arc is altered, and moreover the physical
character of the arc is changed. As we have pointed out
above, the distinguishing feature of the arc when using cored
carbons is the appearance of a dark space situated near the
centre of the purple glow, which appears to considerably
increase the resistance of the arc. On inverting such anare,
however, we find that this dark space completely disappears,
and, as we should have expected, the resistance is consider-
ably diminished. At low currents or P.Ds., the difference
between the resistances of ordinary and inverted ares dimi-
nishes till the two values become nearly equal. These are
the conditions under which the dark space ordinarily dis-
appears. Hence when the dark space has disappeared,
inverting the arc makes but little difference in its resistance,
which is also the case with arcs using solid carbons, in which
there is no dark space.
By placing the carbons horizontal, a value for the resist-
ance of the are can be obtained which is intermediate between
the two values obtained by placing the positive carbon
respectively above and beneath the negative carbon, the
current and P.D. being the same in the three cases.
From consideration of the part played by the dark space, it
‘would appear that the resistance of the arc is greatly affected
by the state of the contact between the purple glow and the
420 Messrs. J. Frith and C. Rodgers on the
negative carbon. When the inside purple glow is in good
contact with the negative carbon, the resistance of the arc is
most negative. When, however, there is a dark space (caused
by the presence of foreign matter or of carbon in a different
physical state), and hence a lack of contact between this glow
and the negative carbon, the resistance becomes increased.
EMissing Arcs.
An attempt was made, using Method I.; to find the resistance
of a direct-current hissing are produced with accumulators,
but it was found that even with the alternator at rest there
was a large deflexion of the electrometer, showing that the
current through a hissing are ts oscillatory. In order to deter-
mine the frequency of the oscillation, a condenser and
telephone were put across the arc. The sound heard in the
telephone was, however, of no definite pitch.
A method suggested by Mr. Campbell was used to measure
the amount of this oscillatory current. The current taken by
the hissing arc was passed, through the primary of a trans-
former, the secondary of which was put in series with a
reflecting twisted-strip ammeter, the deHexion being noted.
The arc lamp was then disconnected, and a known alternating
current was sent through the primary, of such magnitude
that the deflexion was the same as before. This known
current has the same R.M.S. as the oscillatory current passing
through the are, for if the impedance of the secondary is
large compared with its resistance, the deflexion of the am-
meter is the same for the same R.M.S. current whatever
the frequency. The arc was run at 14°5 amperes and 35
volts, Apostle carbons +9 mm. —9 mm. being used. While
the arc was burning the deflexion of the ammeter varied
hetween 200 and 250 scale-divisions. Primary currents of
0-43 ampere and 0°53 ampere, as measured by another
twisted-strip ammeter, produced these scale-readings respec-
tively. Hence in this case the R.M.S. of the oscillatory
current amounted to about 3 per cent. of the continuous
current.
Low Frequencies.
On comparing our results with the steady sl taken from
the slope of the curves published by Mrs. Ayrton, the agree-
ment as to sign, which held in the case of solid carbons, was
found to break down in the case of cored carbons ; for while
the general shape of the curves obtained by Mrs. Ayrton was
the same for cored as for solid carbons, that is both curves
Resistance of the Electric Are. API
except for short arc-lengths indicated a negative resistance *,
we find that with both carbons cored the resistance is always
positive. With a view to elucidating this point, we repeated
Mr. Mather’s experiments using both carbons cored, and
found the = negative as shown by the following table:—
Brush Carbons. Both cored.
+11 mm. —9 mm.
Arc taking 9 amperes at 43 volts.
Resistance in circuit about 5 ohms.
Resistance
introduced. Volts. éV. Amperes, 8A.
+°5 44°8 +1°8 8°10 — ‘90
+°4 44°2 +1°2 8°30 — “70
+°3 it +10 8°45 — ‘95
+°2 43-7 + 7 8°62 — ‘38
+°1 43°3 + °3 8°79 — ‘25
—l1- 42°8 — 2 9°20 + °20
—2 42°4 — ‘6 9°38 + 38
—'3 42:0 —1:0 9°60 + 60
—'4 41°6 —14 9°85 + °85
—'5 Al-l —1°9 10:08 +1:08
In the curves connecting the P.D. between the carbons and
the current, each point is obtained by taking the readings of
ammeter and voltmeter after the arc has been made to burn
for a very considerable time at constant current and arc-length,
so that one may say that the “frequency ” is zero. In this
case, the resistance, if deduced from the slope of these curves,
would be negative both for cored and for solid carbons. In
Prof. Ayrton’s original method, although the readings are -
taken as quickly as possible, still the “frequency” is small
compared with the lowest frequency we have hitherto used,
and both the carbons and the are may have time to alter before
the ammeter and voltmeter can be read. ‘his method would
lead to the same conclusion as before, viz. that the resistance
of the arc is always negative. Our measurements, however,
with frequencies as low as 7:5 gave a positive resistance for
ares using cored carbons. Hence it seems that there must
be a critical frequency, above which the resistance has a
positive value which is independent of the frequency, and
below which it appears to have a negative value, and further,
that this frequency must lie between 7°5 and 0.
* Mrs, Ayrton does indeed rind that for cored carbons at short lengths
the curves slope slightly upwards, thus indicating a positive resistance,
Phil. Mag. 8. 5. Vol, 42. No. 258. Nov. 1896. 2H
422 On the Resistance of the Electric Arc.
Our former methods were not suitable for working with
a frequency lower than 7, for at this frequency the needles of
the voltmeter and ammeter began to vibrate with the alter-
nating current. This vibration of the needles was made use
of in the following way, to indicate the sign of the pl at
frequencies lower than 7. The Gramme alternator used in -
the previous method was run as a rotatory transformer. By
adjusting the continuous current passing through this trans-
former, and hence its speed, any desired frequency could he
obtained down to as low as one complete alternation in
2 seconds. By an arrangement of mirrors the needles and
scales of both ammeter and voltmeter could be observed
simultaneously. In this way it could be seen whether the
two needles were at any instant vibrating in the same orin |
opposite directions. It is evident that if the needles are
vibrating in phase, that is if an increase of P.D. is accom-
panied by an increase of current, then ay, must be positive ;
dA
while if they are vibrating out of phase, that is if an increase
of P.D. is accompanied by a decrease of current, then it is
negative. With this arrangement, solid carbons at any fre-
quency gave a negative resistance. With cored carbons, how-
ever, we found as we expected, that above a certain frequency
the resistance was positive, while below this frequency the method
gave a negative value for a The critical frequency was
found by observing the point at which the swing of the needles
was uncertain, being sometimes in phase and sometimes out
of phase. At this point the frequency was 1°8, at 1:9 the
sign was certainly positive, and at 1°7 was negative. This
result was obtained with all ares whose resistance as given by
our curves was positive (with one exception mentioned below),
and was not obtained with any whose resistance was negative.
In the case of arcs whose curves cross the zero line, the
transition from positive to negative, as shown by the disap-
pearance of the critical frequency, was well shown and
corresponded with the crossing of the zero lime. As an
example, we ran an-are with Brush carbons +11 mm.
cored —9 mm. solid at 10 amperes, the curve for which is
marked B in fig. 1, Pl. V. It willbe noticed that this curve
crosses the zero line twice, at 55 volts and again at 41°3 —
volts: at 60 volts we found the sign of the = as shown
by the swing of the needles, to be positive at frequencies
Dr. G. J. Stoney on Microscopic Vision. 423
above 1°8 and negative below this frequency, at 45 volts it
was negative at all frequencies ; at 35 volts, however, it was
positive at frequencies above 1°8, and also at frequencies
below 1°8. ‘That the resistance would be positive at all fre-
quencies seemed probable, since the curves connecting the
P.D. and current at constant arc-length for such carbons
show that with small arc-lengths an increase of P.D. is
accompanied by an increase of current.
The experiments have been carried out in the Physical
Laboratory of the Central Technical College, and our best
thanks are due to Professor Ayrton and Mr. Mather for
much kind advice during the course of the work.
Note.—Fluctuation of the Current given by a Continuous-
Current Dynamo.
An attempt was made to replace the cells in Method I. by
a dynamo, but it was found that even with the alternator at
rest the electrostatic voltmeter showed a large deflexion. This
was evidently due to the oscillation of the current owing to
the commutator of the dynamo having a finite number of seg-
ments. The oscillation was found to increase as the brushes
were moved out of the sparkless position. The dynamo was
a 5 kilowatt 2-pole machine, and was giving 10 amperes at
70 volts. By adjusting the brushes to the sparkless position
this oscillatory current, measured in the same way as in the
ease of the hissing are above, could not be reduced below
2°5 per cent. of the continuous current, while by rocking the
brushes out of this position it could be increased to as muchas
9 per cent. without the sparking at the brushes being excessive.
The commutator was in good condition. This oscillation may
introduce errors in the measurement by polarized instruments
of the currents given by dynamos owing to the differences
between the mean and the R.M.S. of such a current.
XLII. Microscopic Vision. By G. JOHNSTONE STONEY, \
MM As D.Sc Tas: L
[Continued from p. 349. |
ADDITION To Parr I.
[The following section should come into Part I. after Theorem 1, be-
tween §§7 and 8.]
7 (a). Extension of Proposition 1.—In the proof of Pro-
position 1 it is assumed that the light emanates from each
individual point of the object in an uninterrupted train of
2H 2
424 Dr. G. J. Stoney on Microscopie Vision.
hemispherical waves; whereas, in consequence of molecular
events, the light really comes off in a broken succession of
such trains, each of moderate duration and, generally, discon-
tinuous from one train to the next. Within each train there
must be such continuity as is manifested by the prolonged
interference effects found in the use of such instruments as
Rowland’s largest diffraction gratings, or Michelson’s Refrac-
tometer. But, to bring about these effects, the average length
of the trains need not be more than some such length as a
metric foot (30 centimetres) which includes about 500,000
luminous waves from the brightest part of the spectrum.
This would correspond to an average duration of trains of
about the thousand-millionth part of a second. But however
brief the duration of each train, nevertheless for that short
time the etherial disturbance it occasions can be resolved in
whatever ways it could have been resolved if the train had
gone on for ever. Hence, in order to take into account the
discontinuity of the trains of waves as actually emitted by the
point p, it suffices simply to recognize that the undulations of
uniform plane waves of Proposition 1 are made up of a suc-
cession of comparatively short sections of uniform plane waves,
which may each include only some few thousand waves and
which may be discontinuous where one succeeds another.
This, however, does not hinder their continuing to be undula-
tions of uniform plane waves: accordingly Proposition 1
extends to the cases where the emission of light from the points
of the object 1s discontinuous.
| Corrections in Part I—In the heading of § 8, p. 388,
change “ Principles of Reversal” into Principle of Reversal.
In the footnote on p. 336, line 7, change “ wave”’ into waves.
In the diagram on p. 337, the repetitions of the curve mn
should have been drawn of the same form as the original
curve. |
Parra
21. The Illuminating Apparatus.—The state in which light
reaches that portion of a microscopic object which is under
scrutiny is determined by the source of light, by the con-
denser, by the use that is made of the iris diaphragm and
stops which are associated with the condenser, and by the
parts of the microscopic object through which the light has
to pass to reach the part which is being specially examined.
The iris diaphragm and stops determine the directions from
which light is allowed to reach the object. These should be
such as will bring into sufficient prominence in the image
Dr. G. J. Stoney on Microscopie Vision. 425
those features upon the object which we wish to examine, and
at the same time such as will exclude or minimise the addi-
tion to the microscopic image of false or confusing effects.
How this may best be aimed at will have to be considered
in Part III., and will be illustrated there.
The source of light and the condenser are to be esteemed
as good in the degree in which they enable the whole of the
light of each wave-length which is brought to bear upon a
point of the object to reach that point at each instant in the
same phase and with accordant transversals. For if this con-
dition could be fully attained the state in which the light would
then leave the object would be complicated by the previous con-
dition of the light only in respects which cancel one another
when averaged over a sufficient period of time. For this average,
300 metres of cosmic time *—the millionth of a second—
would be abundant. There is, therefore, no “ twinkling of an
eye” within which the average has not been struck. Hence
light so supplied will furnish a pure image so far as man can
see—one depending solely on the features of the object.
The efforts to reach this desirable result seem to have been
directed exclusively towards improving the condenser, and
cutting very thin sections; but both theory and experiment
seem to show that when the condenser is good, a further
advance may be made by attending to the source of light,
which apparently ought to be confined to a layer of a thick-
ness small compared with a wave-length, and preferably lying
in a plane, or rather on a slightly concave surface, perpen-
dicular to the optic axis (see § 27 below, p. 435). Of course
the distance of the source of light is not immaterial, since a
well-corrected condenser acts at zts best only when the source
of light is at one particular distance.
However well the light may have been prepared by the
condenser, it is sometimes thrown into confusion before it
reaches the upper surface of the object, by having to enter
through its under surface and to traverse its substance. In
such cases the attempt to ascertain what is on the upper
* Cosmic time means time the portions of which are measured by
lengths—by the distances over which electromagnetic waves in the open
ether, and therefore over which light in vacuo, would travel in those
portions of time. This way of measuring time is convenient in both
optics and molecular physics. The convenience has its foundation in
nature, as it no doubt arises out of the circumstance that this way of
measuring time brings both time and space into the closest possible assc-
ciation in which they can stand to the fundamental units of nature. See
a paper “On the Physical Units of Nature” in the Proceedings of the
Royal Dublin Society of Feb. 16, 1881, or in the Philosophical Magazine
of May 1881,
426 Dr. G. J. Stoney on Microscopic Vision.
surface is embarassed by somewhat similar difficulties to those
we meet with when we try to look through ribbed glass.
Thus, on the siliceous diatom known as Pznnularia nobilis, it
is easy to make out that there are rows of dots on the tongues
that stand inwards towards the median line; but vision is
here mixed up with so many false effects, that it would be
difficult to determine how many of these rows there are and
_where exactly they are placed. All will probably be reduced
to order if the diatom can be mounted between two media, of
which the under one shall have a refractive index of 1:4, the
same as silex, and the upper one a refractive index as much
as possible exceeding 1:4. To mount objects in this way
ought not to be impracticable.
22. The Six Images.—When a particular mode of illumina-
tion has been adopted, all subsequent events are thereby
determined if the optical parts of the microscope are fully
corrected and properly adjusted. To trace these events from
the object on the stage to the image of it delineated upon the
retina of the observer is now our task; and it may be prose-
cuted by beginning with an ideally perfect image and applying
in succession a series of changes which finally transform it
into the image which actually presents itself within the eye
of the observer. Throughout the whole inquiry we may,
without loss of generality, treat only of the light of one wave-
length out of the light employed.
The following is a list of the successive stages of the trans-
eee In reading it the diagram on p. 438 will be found
of use :-—
A. Object A is the actual microscopic object, to which
corresponds Image A, viz.: that ¢deal image reproducing
all possible detail, which only light of infinitesimal wave-
length supplied in a theoretically perfect manner would
be competent to produce.
B. Image B, the standard image, means that image
which the light of wave-length d as actually emitted by
the microscopic object would furnish, if reversed (see
Part I. section 8, p. 838). This image is of the same
size as the microscopic object. To it would correspond
Object B, an object of the same size as Object A, but
differing from it by containing no detail but that shown
in Image B.
C. Image C, standard image No. 2, is that which
the light taken in by the objective would, if reversed,
produce; to which would correspond Object C, an object
containing the same detail as that shown in Image C.
Dr. G. J. Stoney on Microscopie Vision. A27
Some of this detai] may be in excess of any detail shown
in Image B, but, unfortunately, an addition of this kind
does not represent anything on the microscopic object
(Part I., Proposition 3, p. 845).
Image C and Object C, like Image B and Object B,
are of the same size as the microscopic object.
D. Image D, the focal mage formed by the objective,
which lies near the top of the tube of the microscope, is
an enlargement with distortion of Image C. To it
corresponds Object D, viz. that object which the focal
image if seen would appear to be.
HK. Image H, the visual image, is that virtual image
which the eye-piece forms, when it is applied as a mag-
nifying-glass to the focal image. It is an enlargement
and slight distortion of the focal image. It is, therefore,
an enlargement with a somewhat different distortion of
Image C. To it corresponds Object EH, the visual object,
that object which the observer thinks he sees. ne
IF. Finally Image F is the image actually produced
on the retina of the observer.
The state in which the light has been admitted to the object
affects images B, C, D, Hi and the corresponding objects, and
therefore affects Image F. We have now to advance in suc-
cession through these images, beginning with the ideal image
and ending with the image which is actually formed within
the eye of the observer.
28. Transition from A to B.—Image A, an ideal image of
the microscopical object, could only be formed by light of
infinitesimal wave-length supplied in a theoretically perfect.
manner. This light, and this light alone, could form an
image reproducing every detail. By such wave-lengths as
can be used with the microscope—z. e. when we pass from
Image A formed by zmaginary light of infinitesimal wave-
length to Image B formed out of real light—the greater part
of the events that happen in the object are shut out from our
view by being massed together. In fact, when we come to a
numerical example we shall find that a whole squadron of
them, indeed an entire army, occupies that space upon the
object which corresponds to the minutest speck which such
light will show.
Image B, the standard image, as appears from § 8, p. 338,
is formed by the flowing in upon it of beams of uniform plane
waves; and in consequence it may, under Theorem 2, be re-
garded as formed of ]uminous rulings interlacing and inter-
fering with one another : each luminous ruling being formed
428 Dr. G. J. Stoney on Microscopic Vision.
by the convergence of two or more of the above-mentioned
beams of plane waves, or of components of them. The finest
of these rulings are those formed by beams that flow in
horizontally from opposite longitudes. If each of the beams
is resolved into light polarized in, and light polarized perpen-
dicularly to the plane of incidence, then the opposed pair of
each of these components will produce a ruling in which the
brightness varies by the law represented by the expression
1+ cos (47ra/X). This indicates that it is a ruling of which the
spacing is A/2, each space including the width of a line and
of the interval between it and the next. Such a ruling will
be best seen when the intensity of the illumination is such
that the lines and intervals appear of equal width. Hence the
lines of the finest rulings, and consequently any speck in the
image seen by the interlacing of these rulings, will have an
apparent. diameter of X/4, when best seen.
Now the corresponding portion of the microscopic object,
viz. a globe with 2/4 for its diameter, contains within it an
enormous quantity of detail, all of which is massed together
in the standard image and presented to us as one mere speck.
How great a loss is here incurred, and how little any micro-
scope can exhilit to us of nature, may be judged from a
numerical example which will be given in Part ILI.*
24. Transition from B to C.—Vhe important change from
B to C requires to be carefully studied from many sides.
Standard image No. 1 contains everything which the whole
of the light of wave-length d is, in the condition in which it
leaves the object, capable of presenting. Standard image
No. 2 contains the utmost that can be delineated by that por-
tion of this light which the objective is able to take in.
The change from the first of these to the second involves,
in the first place, some obliteration of detail, consequent upon
the exclusion of the inclined beams Ba, which are spoken of
on p. 844, in § 15 of Part I. It also involves a rounding off
* The outcome of the computation here referred to is that with bluish-
green light, which has frahably the shortest wave-length that can be
used with advantage in eye observations, and when its wave-length is
shortened by mounting the object in a medium with as high a refractive
index as immersion oil: under these favourable circumstances, a spherical
portion of the object of the size of the smallest speck which this light is
capable of exhibiting, would still be large enough to contain something
between 800,000 and 800,000,000 of those chemical atoms of which pon-
derable matter is made up. And we should remember that the immer-
sion objective for which the oil is designed cannot exhibit quite so small
a speck as the smallest which light of that wave-length could produce. It
is a limit beyond its grasp. Biologists would do well to ponder what
computations of this kind teach us.
Dr. G. J. Stoney on Microscopic Vision. 429
in the image of any sharp edges that may be upon the object,
owing to the exclusion of the beams Bb. The exclusion of
these same beams also often produces a false glare, or adds
features to the image that are foreign to the object (Proposi-
tion 3, p. 345). We shall have to study of what kinds these
illusory effects are, and how they may be in some degree
controlled by adjustments of the illuminating apparatus.
There is another consequence of the exclusion of the beams
Ba and Bd. It is apt to produce a deceptive colouration of
the images seen in the field of view. (Proposition 4, p. 345.)
The most important part of the study of microscopic vision
is the study of these matters. But before endeavouring to
go more fully into them, which can be best done in Part IIT.,
we must take a glance at the subsequent and less important
steps that intervene between Standard Image No. 2 and the
Visual Image, which last is what is directly presented to the
eye of the observer.
25. Transition from C to D.—Ilt has been proved in the
footnote on p. 339 of Part I., that the illuminating apparatus
and the microscopic object may be removed, and that Standard
Image No. 1 can take their place. Now, for our present
purpose it will be enough to put Standard Image No. 2 in
their place ; since we are only concerned with the light which
is to be made use of by the objective, and this light remains
the same whether it is supplied by standard image No. 1 or
standard image No. 2. In order to be available for this use
of it, standard image No. 2 must be regarded as formed by
subjecting that portion of the light emitted by the object
which can be taken in by the objective to two successive
reversals ; and we shall find it convenient to conceive the
medium in front of the objective, whether oil or air, to be
continued downwards in order that the image may le within
this medium, so that the light may be able to go straight
from the image to the objective. After the second reversal
the light first forms the image and then proceeds on from it
and enters the objective in precisely the same state as the
light actually emitted by the microscopic object would have
done. We may therefore remove the source of light and the
microscopic object, and substitute for them standard image
- No. 2, transmitting the beams that form it on to the objective.
Hach of these beams is a beam of parallel light, and is there-
fore, by passing through the objective, brought to a focus at
2, a situation which.is usually close to the back lens of the
objective. See the figure on p. 433. After passing this
focus the beam diverges in the form of a cone of convex
spherical waves, and it is in this state that it reaches D, the
430 Dr. G. J. Stoney on Microscopic Vision.
place where the objective forms its image of the microscopic
object.
This is what happens to one of the beams. Let us still
farther confine our attention to the course pursued by its
azial ray, i.e. the ray which coincides with the line drawn from
the middle of the objective field perpendicular to those plane
waves which form the beam before entering the objective.
This axial ray starts from the point on the optic axis of the
microscope where this optic axis pierces image C. . When the
course of the ray is traced through and past the objective it
is found a second time to intersect the optic axis, at the point |
where this axis pierces image D, the focal image. Let a and
& be its inclinations to the optic axis before and after it passes .
through the objective. Then by Lagrange’s Theorem
nsina=Mosin 8, <i 2.3).
where M is the magnifying power of the objective, 7. e. the
number of times that the image Dis larger than the image O ;
and where n is the refractive index of the medium between
C and the objective, the refractive index of the air which
intervenes between the objective and D being taken as unity
[otherwise n would have to be the ratio of the two refractive
indices |.
Now if we take two beams, whose axial rays are in the
same meridian plane*, and which are inclined to the optic
axis at angles « and a’, then the ruling in image C, to which
they will give rise, has a spacing
€,=),/(SINa—siIN@), . 6. a eee
A, being the wave-length in the medium which is in front of
the objective. Similarly the spacing of the ruling which |
these same beams produce when they reach image Dis
ég=A,/(sin 8—sin @!),;. « = «eee
where A, is the wave-length in air. Hence we find that the
ratio of €, to €, 1s ;
€p __ A,(sina#—sin 2’) A .
& Ay(sin@—sm6). 7 Ss Bom (4)
and this, when we replace 2,/A, by n, and put in the values
of nsin« and nsin a’ given by equation (1), becomes
a? oa My cy een 9) he vee ee
€y
* A meridian plane means a plane passing through the optic axis of
the microscope.
Dr. G. J. Stoney on Microscopie Vision. 431
The process that has been here followed might in fact have
been employed to prove Lagrange’s Theorem ; but it is here
used for the purpose of showing that the ratio ¢,/e, is inde-
pendent of the values of a anda’: in other words, that all the
rulings of which image C is formed are reproduced on a
larger scale in image D, and that the scale of the enlargement
is the same for them all.
We thus learn that image D is formed of evxactly the same
rulings as image C, only magnified and somewhat distorted—
distorted laterally by the convex form of the beams, and Jon-
gitudinally owing to the decreased inclination of the beams
to one another. Jinage D accordingly contains every feature
which is present in image C, only somewhat distorted laterally,
and still more distorted longitudinally.
Hence the great task we have to set before ourselves is to
find out what image CU, standard image No. 2, contains. This
conclusion is not disturbed by pursuing the course of events
farther.
26. From D to E, and from E to F.—The subsequent
stages need not detain us long. If, as before, we select two
of the beams of plane waves emitted by standard image No. 2,
and if we follow the course of the axial rays of these two
beams, we find that these rays intersect the optic axis where
this axis pierces images C, D, and F (see the figure on p. 483),
and that if the portions of them which lie between the eye-
piece and the eye were produced backwards they would also
intersect the optic axis where that axis pierces the image H,
which is a virtual image. This last is the Visual Image, 7. e.
the image which seems to the observer to be presented to
him. |
Let aa!, 8B', yy’, and 66’ be the angles at which the axial
rays of the two beams intersect the axis at C, D, H, and F.
At each of these images the two beams give rise to a ruling,
and if the spacings of these rulings in the successive images
be designated by ¢, €, €3, €4, we have, by proceeding as in
§ 25,
ég=Me, e=M’e, e=M%e . . (6)
where M, M’, and M” are the number of times that images D,
Hi, and F respectively are larger than the microscopic object ;
and where also, by equation (1),
MW’ =nsina/siny, and is also =n sin @'/sin y’
M”=nsin a/n’ sin 6, and is also=n sin @’/n! sin 8’
M =nsin z/sin 8, and is also =n sin 2’/sin f' ;
ete)
432 Dr. G. J. Stoney on Microscopic Vision.
where n*, 1, and nm! are respectively the indices of refraction
of the medium in front of the objective, of air, and of the
vitreous humour in the eye of the observer.
Hence the ruling to which these two beams give rise in
image C is reproduced in images D, HE, and F. ‘The same is
true of all the rulings by the co-existence of which image C
is formed ; and as within each image the spacings of all the
rulings are modified in the same way, it follows that every
feature which is present in any one of these images is present
in all the others—of course on the supposition that no defect
is introduced by imperfections in the objective, the eyepiece,
or the eye.
Image E appears to the observer like an object presented
to his unassisted eye, and in it the conditions must be fulfilled
for naked-eye vision. These chiefly concern the angular size
which that image must have in order that the minutest detail
in it may be sufficiently large to be satisfactorily seen (see
Section I. of a paper “On the Limits of Vision,” Scientific
Proceedings of the Royal Dublin Society of December 20,
1893, p. 228 ; or, Philosophical Magazine for March 1894,
sol?) .
H ee to what the detail is, and as to the true nature of
that detail, we still have to fall back upona study of standard
image No. 2, and of the adjustments of the microscope which
are available for improving that image. And this can be best
done by the study of individual examples, which is to be the
subject matter of Part IIT.
If any difficulty still hangs over the subject treated of in
the last few sections it will be cleared up by following out
the whole history of one individual beam.
27. Course of an individual beam traced, and Flow to see
7t—It has already been proved that the microscopic object
and all the illuminating apparatus may be removed, and that
standard image No. 2 may be substituted for them, if this
image be formed after two reversals, and if it therefore emits
its light forwards. Let C in the figure be the position of this
standard image and let Cb be one of the beams of uniform
plane waves which it sends forward. Let the dark line
in the figure represent the axial ray, that is the ray which
starting from the middle of image C continues throughout
perpendicular to the waves of which the beam consists. The
* It is the usual practice in England to employ the symbol yp for the
refractive index; but it cannot conveniently be so used in treating of the
microscope, since p is wanted as the designation of the micron. On this
account the continental practice of using ” for the refractive index has
been adopted in the text.
Dr. G. J. Stoney on Microscopie Vision. 433
beam enters the objective at 6, and by the objective is brought
to a focus at x, the focus for parallel rays. Here it forms an
Airy’s spurious disk, or rather a somewhat more diffuse
spurious disk, since the objective falls short of being aplanatic
in its treatment of beams of parallel light incident on its
front surface. Past the focus x the beam advances, and
diverges as it advances. Let D be where the axial ray, the
dark line of the figure, intersects the optic axis of the micro-
scope. Note that the axial rays of all the other beams emitted
from the image C will, like this one, start from the point
where C is pierced by the optic axis of the microscope, and
will all intersect that axis again at D. At D an image of
Cis formed on a somewhat curved surface that stands at
right angles to the optic axis of the microscope. Our
beam, diverging from #, when it reaches this surface con-
tributes its quota towards the formation of the image upon it.
After passing image D the divergent beam continues till
it reaches the front of the eyepiece. It proceeds through
Bigs ts
7
/
I ne ae ee ae ew ak
/
ae
\
Sea ——_— — —
—---
\
the eyepiece and is by it brought to a second focus at y,
where the eyepiece can form an image of 2. If we produce
the portion qy of the axial ray backwards, it will intersect
the optic axis at H, where the backward production of the
convergent beam gy will widen out so as to extend over
the space occupied by the virtual image H, and contribute
its share towards the formation of that image. Accordingly
the light of our beam, which advances in the direction gy,
will enter the eye (the pupil of which should be brought
close to y) in the same state as if it had come direct from the
whole extent of an image at H. After passing y the beam
again diverges, and by the optical action of the front half of
434 Dr. G. J. Stoney on Microscopie Vision.
the eye this divergence is lessened and at the same time the
axial ray is bent in so as a third time to intersect the optic
axis of the microscope, where that axis prolonged pierces the
retina of the observer. Upon the retina, round this point,
the beam of lessened divergence spreads and there assists in ~
the formation of image F, the image of the microscopic object
which is formed by the microscope upon the retina of the
observer.
Twice upon its journey the beam we are dealing with has
been brought to a focus—at x and at y. At both these posi-
tions it is concentrated into a point or, to use more accurate
language, into a spurious disk, and can be seen as such. It may
be seen at w by removing the eyepiece and looking down the
tube of the microscope, taking care to keep the eye central ;
and it may also be seen at y by replacing the eyepiece and
looking with a magnifier at the image y formed in the air
outside the eyepiece *.
Accordingly every visible point of the image seen at «
when we look down the tube of the microscope is the con-
centrated light of one of the beams of uniform plane waves
which has been emitted from the whole extent of the objective
field. We may therefore, by scrutinizing the position and
brightness of the points of this image, learn in what directions
and with what intensity the beams of uniform plane waves are
being thrown off from the whole front of the objective
field.
We must not, however, suppose that by scrutinizing this
image we can learn everything about those beams, inasmuch
as they differ from one another not only in those respects of
which the eye can take cognizance, but also in the form and
position of the elliptic motion in each wave front, and in its
phase at a given instant—particulars which, though the eye
cannot detect them, are of primary importance in determining
what image of the microscopic object they will contribute to
form when they reach the position D.
To complete this-survey, we should trace the course of the
light up to the microscopic object, as well as from that
position forward. The light as it reaches the objective field
would be ideally perfect if it could be resolved into beams of
plane waves, perfectly uniform, coming with equal intensity
* The effect is most striking when the object is such as directs into a
few definite beams each beam of the light supplied to it by the condenser.
Anexcellent objectis the diatom knownas Per istephania eutycha, especially
when the iris diaphragm is nearly closed, so that only a small sheaf
of beams, each extending over the whole objective field, is supplied by
the condenser.
Dr. G. J. Stoney on Microscopic Vision. 435
from all the directions in which the iris diaphragm and stops
allow light to pass, and at the same time in all other respects
—in phase, ellipticity, and so on—such light as would have
been emitted in the reverse direction downwards from a
perfectly featureless self-luminous plane occupying the position
upon the stage of the microscope of the objective field.
This ideal is the more nearly approximated to, the better the
condenser ; and it would appear that the best position of the
source of light is that which would be occupied by the image
which the condenser would form beneath of such a featureless
luminous plane as we have supposed. Hence the suggestion
as to the position of the source of light made in§21. It
further appears that the ideal position for the iris diaphragm
and stops would be at the position which we may call z
(corresponding to x and y) where beams of uniform plane
waves emitted downwards from the supposed luminous plane
would be brought to a focus by the condenser. This is a
position which is usually very close to the condenser ; and it
would be a marked improvement in microscopes if the iris
diaphragm and stops were brought nearer to this ideal position
than they commonly are. -As they are at present placed,
different parts of the field of view are treated differently by
them in an appreciable degree.
We have next to examine into some points the consideration
of which will put us in a better position for interpreting
aright what we shall see in the microscope.
28. Of the Composition and Resolution of Undulations.—We
shall start from the known fact that any luminous undulation
of uniform plane waves of wave-length \ may be resolved
into two undulations of plane-polarized light polarized in
planes at right angles. In order to get their equations in
their simplest form, let the axis of w be placed perpendicular
to the wave surfaces, and the axes of y and ¢ parallel to the
transversals of the two plane-polarized components. Then
the equations of the given undulation, which we may call U,
will be—
n=a sin({ (w—2)+a), . ATA).
f= sin (F(ot—a2) +8), . ee it 69)
in which y and € are the displacements, at the time ¢, in the
two plane-polarized components, a and / are the amplitudes
(7.e. the scalar part of the transversals), v is the velocity of
light, and @ and @ are the initial phases on the plane yz,
436 Dr. G. J. Stoney on Microscopic Vision.
which last are constants that are determined by the ellipticity
of the light of the undulation U, by the position of its ellipse,
and by its initial phase.
Similarly, another undulation V of uniform plane waves
travelling in the same direction will be represented by equa-
tions 38 and 4—
"=o sin (= (ow) +0’), eee
t=)! sin(= (vt—2) +8’). . aes
If the undulations U and V are simultaneously present, the
displacements over the plane w and at the time ¢ will be
n+yn’ and €+¢ ; and these by elementary trigonometry are
found to be
nn =M sin (7 (wea) +A), « . ok Sane
C+C=N sin (= (wa) +B), , ges
where
M?=a?+a?+2aa’ cos(a—a’), . . . . (7)
_ asina+a’ sin a
tana osada cosa’ > nn
N? = 6? + 6 + 260’ cos(8—f'), . ~. see
_ bsinB+0' sin 8’
tan Drip cone Se Gcose? 5 (10)
Hquations (5) and (6) represent an undulation of uniform
plane waves travelling in the same direction as U and V.
We may call this resultant undulation W. Hquations (7),
(8), (9), (10) enable us to determine the constants of any one
of these three undulations, if we know those of the other two.
It appears accordingly that any two of the three undulations
U, V, W being given, the third can be found.
It is an easy inference from this that any number of undu-
lations of uniform plane waves of wave-length A, that travel
in the same direction, may be combined into a single undula-
tion of the same kind travelling in that direction: a proposition
of which use was made above in the latter part of § 6, of
Part [., p. 337.
29. Of elementary sheafs of beams, and of the single beams
which may be substituted for them.—Beams of uniform plane
waves may be emitted in any or all directions from the front
Dr. G. J. Stoney on Microscopic Vision. 437
of the objective field, or from either of its “ standard images.”
Draw what we have called the axial rays of these. They are
lines radiating from the middle of the objective field, and
each perpendicular to the waves of its own beam. Take a
group of these axial rays which lie within a definite cone,
then the corresponding beams may be called a sheaf of beams;
and where the cone within which the axial rays are confined
is a very acute one, the corresponding beams may be called
an elementary sheaf of beams. The whole of the beams
emitted by the objective field, or from either of its standard
images, may obviously be conceived of as divided up into
elementary sheafs of any required degree of minuteness.
If we only have to deal with an image of limited extent
like standard image No. 1, or standard image No. 2 (which
are the same size as the objective field), then we are justified
in substituting a single beam travelling along the axis of the
cone for each elementary sheaf of beams. This may be proved
as follows :—
Let U be one of the beams whose axial ray lies within the
elementary vone, and let @ be the angle between that axial
ray and the axis of the cone. The cone, of course, has its
vertex at the centre of the objective field. Let now V be an
equivalent beam whose axial ray lies along the axis of the
cone, and let the phase of V be such that U and V are in the
same phase at the centre of the objective field. Then, as in
§ 15, let —V mean the same beam as + V, only with wadded
to all its phases. Accordingly, if + V and —V are simulta-
neously present they cancel one another absolutely. We may
therefore add both of these to the elementary sheaf of beams
without altering it. Now —V and U would produce a ruling
which will be the coarser, z.e. with its luminous bands more
widely spaced, the smaller the angle @ is. Moreover, since
+V and U are in the same phase at the vertex of the cone,
which is also the centre of the objective field, it follows that
one of the minima of illumination of the ruling produced by
—V and U will occupy that position. Now by making @
sufficiently small, the spacing of this ruling may be made so
many times larger than the objective field that there is no
appreciable illumination anywhere within the limits of the
objective field. If this be so, we may suppress the beams
—V and U without producing appreciable change within the
limits of the objective field. When this is done, the elemen-
tary sheaf of beams differs from what it was at first by having
--V now in the place of U. By a similar process we may
substitute V', V", &c. travelling along the axis of the elemen-
tary cone for the other beams whose axial rays lie within the
Phil. Mag. 8. 5. Vol. 42. No. 258. Nov. 1896. 21
438 Dr. G. J. Stoney on Microscopic Vision.
cone. And, finally, all these V’s, since they travel'in pre-
cisely the same direction, will, as is proved in the last section,
coalesce into a single resultant beam W travelling along the
axis of the cone, which single resultant may accordingly be
substituted for the elementary sheaf of beams.
The general conclusion is :—The whole of the light emitted
from the objective field may, by Theorem 1, be resolved into
beams of uniform plane waves ; these beams may be divided into
small groups, each an elementary sheaf of beams; and each
elementary sheaf of beams may have a single beam substituted
for it.—In every subsequent step of our investigation we need
only deal with these resultants—these secondary beams as they
may be called—which, though many, are limited in number.
30. Another proof of Theorem 2.—Theorem 2 may be
proved in many ways, and a proof which carries the analysis
of an image down to its simplest elements will be found
instructive. Describe a hemisphere in front of the objective
field and round its centre. Call the point where the optic
Fig, 2.
axis of the microscope pierces this hemisphere, its pole.
Planes passing through the optic axis may be called the
meridional planes ; and the objective plane, being perpendicular
to the axis of the microscope, will be its equatorial plane.
Divide the equator of our hemisphere into seconds of are,
z. e., into 1,296,000 parts, which will afford sufficiently
minute divisions upon which the bases of elementary cones
may abut. Draw parallels of latitude also at intervals of a
second ; and draw meridians as in the figure, marking out in
conjunctiou with the parallels of latitude the bases of the
elementary cones, or rather pyramids. These become
narrower the higher the latitude, and as soon as they have
shrunk to half a second horizontally every alternate meridian
may be omitted, until they have shrunk again till other
meridians may be omitted without any of the little sectors
Dr. G. J. Stoney on Microscopic Vision. 439
being more than a second in width. By this process the
whole hemisphere may be mapped out into these patches
placed in vertical series, each series with a corresponding
series on the opposite side of the polar axis, and the whole
ending up in one patch or in one pair of patches at the pole.
From the middle of each patch draw a radius of the sphere,
omitted from the diagram to avoid confusion, but easily con-
ceived. These are the axial rays of the secondary beams W,
which we are at liberty to substitute for the elementary sheafs
of beams that really exist. 3
In this as in all similar problems it is convenient to begin
by resolving each beam of light into two plane-polarized
beams, one polarized in and the other perpendicular to the
meridional plane in which its axial ray lies) We may then
confine our attentivn first to the light polarized in one of
these ways. Let us then call the secondary beams which are
polarized in one of these ways a, 6, &., and the corresponding
beams opposite to them and polarized in the same way, a’, U,
&c., as in the figure.
In general, beam a and beam a’ will not be alike. How-
ever, heam a’ may by the proposition in § 23, p. 435, be sub-
divided into a’; and a’, of which a’, shall be exactly similar
to a. Accordingly a and a’, produce a definite ruling of
equal and equidistant lines extending over the whole standard
image ; and a’, has still to be disposed of. Resolve 6 into
two beams of which 6, is similar to a’,, and 0b, is the other
component, Then a’, and 5; produce another ruling, and b,
has next to be disposed of. ‘lo do this, resolve 0! into 6’, and
b',, of which 6’; is similar to 62. ‘Then b, and 6’; produce a
ruling and 0b’, is what remains over. By continuing this
process the whole of the vertical columns over a and a’
may be dispose: of ; and in a similar way the other columns
can be treated. All the rulings extend over the whole
of the standard image, and between them use up the whole
of the light polarized one way, except that residium which
is left over in the last patch at the pole. This residium can
be made as small as we please by diminishing the size of
our elementary cones; and of whatever amount it is, it
only represents the limiting case of a ruling produced by
two beams advancing at a vanishing angle to one another,
and of which accordingly the spacing is infinite.
A similar treatment applies to the light polarized the other
way, which also produces its body of rulings; so that finally
the whole of the light emitted by the objective field, however
varied its contents, can be laid down on the standard image in
the form of a number of luminous rulings of uniform parallel
lines, each of them of that optically most simple character
212
ad
440 -Dr. G. J. Stoney on Microscopic Vision.
which ts produced by a single pair of beams that are alike and
polarized, either both in the plane of incidence or both perpendicu-
larly to tt.
A similar treatment applies to standard image No. 2, the
only difference being that to analyse it we are to employ a
sector of the hemisphere in the figure on p. 438 instead of the
whole hemisphere.
Hence Theorem 2 is fully proved.
31. The resolution not unique.—in order to follow the most
simple process when combining the secondary beams in pairs,
we have taken both the members of each pair from the same
meridian. We might of course have combined them lateraily
or obliquely, and we shall find it necessary to bear this in
mind when dealing with some kinds of illumination that are
found useful, e.g. annular illumination. It is obvious that
it is legitimate to combine the secondary beams in any way
which when completed has used up all the light : and our object
should be to combine them in each case in whatever order is
most convenient for the problem in hand. In the practical
use of the microscope it is usually quite easy to see into
what groups it is most advantageous to throw them. Jn
whatever order they are taken the final result is the same;
but one order differs from another in the degree in which it
gives us information that is of use to us.
It is sometimes convenient to think of an optical image as
a kind of picture, and that the rulings are, as it were, suc-
cessively painted in upon the field of view to form it. But
if we conceive matters in this way we must remember that
this luminous paint behaves after a very peculiar fashion.
Where one ruling crosses another or overlies it, they may
obliterate one another in some parts as well as strengthen
one another in others, effects which will depend on the
lengths and positions of the transversals in the two rulings and
upon the relation in which their phases stand to one another.
In this connexion it is very necessary to bear in mind
that two rulings may be seemingly identical—~z. ¢., identical
in position, spacing, intensity, &c., in such matters as the eye
can perceive—and yet these rulings may behave quite
differently towards the other rulings with which they are
associated, owing to differences affecting the transversals and
phases which our eyes are not fitted to take note of. Thus,
what are apparently identical rulings might result from the
interference of two beams little inclined to the optic axis,
and from two others much inclined : or from two beams in
the same meridian and two others in other positions: but
these seemingly identical rulings would all behave differently
towards the rest of the light with which they have to act.
Dr. G. J. Stoney on Microscopie Vision. 44]
32. Criticism of the Abbe and the Airy methods.—The proof
which we have just gone through is instructive in many
ways. In the first place, it carries the analysis of the image
down to rulings of the simplest kind that are known. In the
second place, it makes the flexibility and Protean character
of the whole process conspicuous ; for although we followed
one particular order in combining the secondary beams in
pairs to form rulings, it is manifest that this order was in no
degree necessary, and that the secondary beams might have
been grouped in an infinite number of different ways, the
only condition being that whatever order we adopt we must
take care to use up all the light. This means that the set of
rulings which form an image is not unique: that there are
an infinite number of such sets, any one of which will suffice
to form the image.
But, above all, the proof given in the last section brings
well into view the source of the advantage which is found in
employing Abbe’s mode of procedure as our instrument for
searching out the causes of the phenomena presented by
microscopic vision. By the process we have followed the
light sent forward from the objective field has been analysed
into beams of plane waves. Now this is only one of number-
less possible ways of analysing that light. It is an analysis
which may be made, not which must be made. But it has
the advantage over every other analysis, that the resulting
waves are uniform waves, exactly alike in every respect over
the whole extent of each wave surface and each wave retaining
its form and intensity unchanged during its advance.
This sets us free from a sea of difficulties that embarass
our progress when we attempt to employ any other resolution.
If the analysis be made into curved waves of any kind—into
those of Airy’s method or any others—the resulting waves
are not uniform over each wave-surface ; and as the law of
this want of uniformity is not yet known, we can only
legitimately employ Airy’s method in the cases where this
want of uniformity has an inconspicuous effect upon the
result of our inquiry. ‘This is the case, for instance, in the
treatment of telescopic vision to which Airy applied it. Here
the sector of each hemispherical wave that we have to deal
with is sufficiently small for the want of uniformity within
its small extent to be of negligible amount.
Tt has sometimes been supposed that we can investigate
microscopic vision by applying Airy’s analysis to the light
sent forward to the eye from the image of the microscopical
object which is formed by the objective close to the eyepiece.
This light as it comes from each point of that image is confined
within a cone which is a continuation upwards of the nazrow
442 Messrs. Ayrton and Mather on Galvanometers.
cone from the back lens of the objective to that point of the
image. Accordingly, if that object which is called the wisual
substitute in Part I., § 14, p. 342, could be put in place of this
image, sending forward hemispherical waves from each point
‘of it which might be treated as uniform within the limits of
the aforesaid cone ; then what the Airy method as hitherto
applied has investigated is what minuteness of detail it would
be possible to see in this object. . But such an inquiry does
not. even touch the main points towards which an investigation
of microscopic vision needs to be direcied. It tells us nothing
as to what this visual substitute is, how far it can represent
something on the microscopic object, and how far it consists of
intercostal markings, diffraction-fringes arising from the
mismanagement of the illuminating apparatus, or any other
misleading effects. And of course it gives us no clue as to
how we are either to interpret or control any of these effects.
Every legitimate mode of resolution, and there are number-
less such, must lead to precisely the same result, if we can
succeed in correctly following out its consequences. Where
they differ is in our power of handling them. Abbe’s
resolution into beams of plane waves recommends itself above
all others in regard to this; since it substitutes uniformity
for that want of uniformity which exists in all other methods
of resolution in just those places where in the present state
of our knowledge we are unable to assign the law of this
want of uniformity. And even if we ever come to know
this law, the resolution into uniform plane waves will still
recommend itself in consequence of the law of uniformity
being simpler and therefore more easily handled than any law
of non-uniformity.
(To be continued. |
XLII. Galvanometers. By Prof. W. EH. Ayrton, F.R.S.,
and 'T. MaTHER*,
N an article on “The Electric Discharge in a Magnetic
Field”? contributed by Sir David Salomons to the Phil.
Mag. for September, it is stated on p. 255 :—
“T made a large number of experiments with galvanometers
built on the D?’Arsonval-Deprez type, and obtained very
varying results by modifying the magnetic field. By in-
creasing the field a maximum sensibility was reached, which
decreased on further increasing the field.
‘“‘ The various experiments described no doubt indicate the
cause of this, viz. that the field being made too powerful, less
current passes through the coil, and the sensibility begins to
* Communicated by the Authors.
Messrs. Ayrton and Mather on Galvanometers. 443
fall. I had a special galvanometer-apparatus made to fit my
large magnet, converting it probably into the largest galva-
nometer of the type extant ; but the sensibility is exceedingly
small when the magnet is fully excited, and increases rapidly
when the excitation is somewhat diminished.”
_ The preceding is gravely given by Sir David Salomons as
an illustration of the displacement of the current in a con-
ductor by a magnetic field ‘‘ generally known under the name
of the Hall effect,” in apparant ignorance of the fact that the
“ Hall effect ” is extremely minute. Indeed, had Sir David
Salomons tried to compare the current sent by a given P. D.
through a coil of insulated fine copper wire when placed in
and out of any magnetic field he would have found it practi-
cally impossible to detect any difference, and he would have
- convinced himself that the great falling off in sensibility of a
d’Arsonval galvanometer as the strength of the stationary
magnetic field was increased beyond a certain limit could
have nothing to do with the “ Hall effect.”
- Further, this phenomenon, which is described in the Phil.
Mag. for September as new, has for some time past been
known to electrical instrument-makers, for it was one of the
eauses which prevented the sensibility of the suspended-coil
type of galvanometer being increased beyond a certain limit.
_ In the discussion which took place after the reading of
a paper on a “ Workshop Ballistic Galvanometer”’ before
the Physical Society in June 1892, reference was made—
perhaps for the first time publicly—to the difficulty ex-
perienced in increasing the sensibility of a d’Arsonval
galvanometer by increasing the strength of the field beyond
a certain point. And it was the investigation of the cause
of this phenomenon, and the discovering of a means to over-
come it, that caused the publication of the description
of this instrument to be delayed until the meeting of the
British Association in 1895, as was explained in the account
of this latter instrument given in all the electrical journals
about September 1895.
In the ‘ Electrical Engineer’ for October 5th, 1894, it was
mentioned. that ‘“ great difficulties were, however, found to
arise when the extreme sensitiveness sometimes required
in the laboratory was attempted ... chiefly due to the
traces of magnetism found to exist in the silk and other
parts of the coil usually considered non-magnetic ....
A great number of trials was made at the Central Technical
College which showed that .... the purest wire of electrolytic
copper or of fine silver, specially drawn and covered, still
showed traces, and a method . . . was at length devised whick
has practically solved the difficulty. The non-magnetic pro-
444 Messrs. Ayrton and Mather on Galvanometers.
perty of the new coils allows a magnet to be used whose field
is four times as strong as those ordinarily employed in this
type of instrument.”
Rather more than a year later, in the ‘ Electrical Engineer ”
for December 13th, 1895, further information on this subject
is given :—‘t This phenomenon was first noticed by Mr. Mather
in December 1891 ... The coil to be tested was suspended
in the earth’s field and the period of a complete oscillation
was found to be 25:2 seconds. It was then placed in a field
of 1500 C.G.S. lines and the period was only 3°5 seconds,”
corresponding with a controlling force due to the magnetic
action of the coil fifty times as great us that due to the
suspension itself... “parts of the coil became magnetic and
it can easily be seen that this secondary effect is magnified in
the d’Arsonval type of galvanometer, when, with the object —
of gaining sensibility, the magnetic field is strengthened ...
in fact the magnetic action of the coil will defeat this attempt.”
References are also given in the ‘ Electrician’ and ‘ Hlec-
trical Review’ of the same date to the limitation in sensi-
bility produced by the magnetism of the coil ; and lastly, in
the ‘ Electrician’ for January 31st, 1896, Mr. Fisher, in a
series of articles on the ‘‘ Crompton Potentiometer,” writes:—
‘‘ whilst up to a certain point the deflexions became greater
with increased strength of field, beyond that point the de-
flexion gradually fell off as the field was strengthened, this
being due, as Messrs. Ayrton and Mather have pointed out,
to the presence of iron in the materials used in the construe-
tion of the coils. A remedy for the same was found by
Messrs. Ayrton and Mather, and it is due to Prof. Ayrton’s
kindness in disclosing the process adopted that the sensi-
bility of the instrument is to a great extent due.” :
We may take this opportunity of replying to an article on
‘Galvanometer Design. Waste Space near the Needle,”
published in the Phil. Mag. for December 1895, by Prof. 8. W.
Holman, of the Massachusetts Institute of Technology, as a
criticism of one of the facts brought out in the 1890 paper on
‘¢Galvanometers” by Dr. Sumpner and ourselves. In that
paper it was pointed out that to obtain maximum sensibility
with an ordinary reflecting-galvanometer no wire should be
wound in a certain space close to the needle, or, if wound, it
should be connected up the reverse way to the rest of the coil;
and it was shown that the approximate shape of the space in
question was “an oblate spheroid with a polar axis about 0°72 |
of its equatorial diameter, the latter being, of course, slightly:
larger than the length of the needle.” |
Messrs. Ayrton and Mather on Galvanometers. 445
Prof. Holman admits the accuracy of our reasoning that
there is a certain space in which the wire should either be
left out, or be oppositely connected, but he considers that we
overestimated the volume of this space because we neglected
“the fact that the field over the remainder of the needle is
not the same as at the poles, either in direction or strength.”
And to support his argument he proceeds to consider what
ought to be the boundary of the space in the cise of “a very
thin uniformly magnetized prismatic needle” (the italics are
ours) without observing that this is exactly the case we dealt
with, and, therefore, must lead to exactly the conclusion we
arrived at. For no part of such a longitudinally magnetized
needle, other than its ends, contributes to the deflecting
moment when placed in any weak magnetic field, since no
free magnetism exists except at its ends. Hence his objection
that the field is not uniform throughout the length of the
needle has no weight whatever in the very example he has
himself selected.
We may also call attention to another error into which
Prof. Holman has fallen. He gives as the value of the
deflecting moment, produced by a current in a coil on a very
thin uniformly magnetized prismatic needle, the expression
1
2| m.f cos @.ds,
0
““m being the strength of pole of any thin transverse section
or shell of the needle, ds the thickness of that section, f the
field-intensity at that point, and @ the field-direction angle
with the axis of the coil.” But this expression could only be
correct if every part of the needle were equidistant from the
axis of rotation. And even if allowance were to be made for
this not being the case by introducing s, the distance of a
section from the axis, the expression
l
2| m.fcos@.sds
0
would still only give the correct value for the deflecting
moments in the case of “a very thin uniformly magnetized
prismatic needle” by making m equal to nought for all points,
except at the ends. And when that is done, the conclusions
arrived at by Prof. Holman are profoundly modified.
As to the confirmation by experiment of his conclusion that
the deflecting moment of a coil of diameter about half the
length of the needle is nil when the coil is placed close to the
needle, that merely proves, we think, that the needle he used
in his experiments was not uniformly magnetized. It is, of
course, well known that it is almost impossible to obtain
446 Messrs. Ayrton and Mather on Galvanometers.
magnets uniformly magnetized, and if free magnetism exists
along the length of the magnet as well as at its ends, our
conclusion must, of course, be modified. Indeed, we convinced
ourselves by experiments made at the Finsbury Technical
College as long back as 1884 that with ordinary magnetic
needles it was necessary to place a coil of given diameter
somewhat nearer to the needle than was indicated by the
formula in our 1890 paper before the deflecting moment
changed sign. But since by far the greatest amount of free
magnetism on very thin magnets exists near their ends, we
do not consider that the theoretical deductions contained in
that paper concerning the waste space can have been far
wrong.
We agree with Prof. Holman that uniformity should be
observed in describing the sensibility of galvanometers, and
we were glad to see that the system for denoting sensibility
which was proposed and used by us in our 1890 paper was
adopted in the programme of the Naturforscher und Aerzte
which met this September in Frankfurt. The ¢, however, used
by Prof. Holman in his proposed list of observed quantities
should be the pertodic time, and not, as he states, “the time
of a single swing.”
Further, in addition to the data respecting resistance,
periodic time, and current per millimetre deflexion at given
scale distance, mentioned by Prof. Holman as essential in
descriptions of sensitive galvanometers, we would point out
that it is also important to give the moment of inertia of the
suspended system, for unless this be done it is impossible to
make the comparisons of various instruments complete. As
is well known, the smaller the dimensions and mass of the
moving parts, the more excellent will such an instrument
appear, when excellence is judged entirely by thé deflexion
per micro-ampere at constant scale distance and constant
period. But for certain purposes it is necessary to compare
instruments under conditions of constant controlling moment
per unit angle deflexion, because the stability of the zero and
trustworthiness of the readings depend ou this quantity, as
was pointed out on pages 85 and 89 of our 1890 paper ; so
that to make this comparison, a knowledge of the moment of
inertia of the suspended system is required.
Other useful particulars of galvanometers are the total
volume of the coils, and the decrement, or logarithmic decre-
ment, of the oscillations. And since this latter depends on
the periodic time and on the condition of the galvanometer
circuit, whether closed or open, the damping should be
observed with the instrument under the same conditions as
when the periodic time was taken.
Pu Adgin]
XLIV. Notices respecting New Books.
Anleitung zur mikrochemischen Analyse der wichtigsten organischen
Verbindungen. Vol. Ill. By Prof. H. Brenrens. Hamburg:
Voss, 1896.
HE rapid development of organic chemistry, and more especially
the discovery of substances chemically almost identical but phy-
sically different, has caused the chemist to turn his attention to
physical instruments, with the result that the polarimeter, polari-
scope, and refractometer are to be found in every well-equipped
laboratory for chemical research. Prof. Behrens now seeks to
introduce the microscope to the organic chemist, and in the three
parts of his work which have already appeared he describes the
behaviour of the more important organic compounds when crystal-
lized, or treated with reagents, on the stage of the microscope.
The reactions described result in the formation of crystals, the
size, shape, and optical characters of which are specified. The
present volume deals with the aromatic amines, and is enriched by
77 illustrations of microscopic crystals obtained by precipitation.
lt should prove a useful handbook in the organic laboratory.
Je la. Es
XLV. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
{Continued from p. 372.]
June 10th.—Dr. Henry Hicks, F.R.S., President,
in the Chair.
es following communications were read :—
‘On Foliated Granites and their Relations to the Crystal-
line Schists in Eastern Sutherland,’ By J. Horne, Esq., F.R.S.E.,
F.G.S., and E. Greenly, Esq., F.G.S.
The crystalline schists of Eastern Sutherland are traversed by
great numbers of granitic intrusions, chiefly in the form of lenticular
sills. These generally lie parallel to the foliation-planes of the
schists, but transgressive junctions are also frequent. Thin seams
of granite also occur in such abundance as to constitute with the
schists a banded gneissic series; but these seams can often be seen
to transgress the schistose folia, and even often to proceed from
large masses of granite. The granites contain numerous inclusions
of the schists which they traverse, such inclusions retaining, usually,
the dip and strike of the surrounding rocks,
There are no chilled edges ; and, moreover, the component crystals
of schist and granite mutually interlock along the lines of junction.
The authors give an account of the foliation of the granite. In
some rare cases a foliation parallel to that of the schists traverses
granite-veins. It is generally, however, parallel at once to the
sides of the sill and to the foliation of the schists ; and many of the
structures are the remains of biotite-folia belonging to schists
whose quartzo-felspathic elements have been incorporated with
448 | Geological Society :—
those of the granite. But many sills or veins, traversing the schists
at various angles, are foliated parallel to the line of junction, and so
discordantly to the structures in the schists; and foliated granites may
even be observed to cut each other’s foliation. These can hardly be
anything but original igneous structures; but, if coexistent with
the last-named, would be indistinguishable from it.
The country-rocks are various types of biotite-schist or gneiss,
with quartz-schists at Kildonan, and a scapolite-limestone at Arma-
dale. They are almost all holocrystalline, but it is certain that
sedimentary rocks enter into the complex. The whole series is
powerfully folded.
The granites increase in size and numbers north-westward from
Kildonan: the intimate intrusive relations above described becoming
more highly developed in the same direction. The schists, at the
same time, become more and more highly crystalline, sulimanite
also appearing in them. About Kinbrace they are coarse silliman-
ite-biotite-gueisses, with large striated felspars.
Igneous contact is not held to be the sole origin of metamorphism,
though the cause which brought about the introduction of the gra-
nites has evidently also produced these high types of crystallization,
The evidence of powerful movement which the schists everywhere
present suggests that such movement was the initial cause of the
whole series of phenomena. Movement recurred throughont, though
all cataclastic structures (if such existed) have been wholly effaced
by crystallization ; introduction of granite being the final stage in
the production of the complex, and a high temperature (as shown
by the absence of chilled edges) being maintained to the very end.
With regard to the granites, the authors find it difficult to believe
that they are wholly foreign matter, but remark that it is here
necessary to observe the utmost caution. ~
2. ‘The Geology of the Eastern Corner of Anglesey.’ By E.
Greenly, Esq., F.G.S.
‘I'he notes contained in this paper embody the principal results
obtained during a survey of Anglesey on the six-inch scale.
The schists of the South-east of the island are succeeded uncon-
formably by the slates of Careg Onnan, which appear to be
separated by a strong unconformity from the Ordovician shales.
The Careg Onnan slates appear (pending confirmation from other
sections or direct fossil evidence) to be of pre-Cambrian age, and
the author records the existence of sponge-spicules therein.
The ashy grits and bedded tuffs of Baron Hill near Beaumaris
appear to have been moved somewhat from the E.N.E. along a
thrust-plane. They are traversed by planes of mylonization, and
are much broken and folded.
The Ordovician rocks consist chiefly of sparingly fossiliferous dark
shales and mudstones, but contain a group of volcanic tuffs on the
horizon of the pisolitic ironstone.
The Carboniferous rocks appear to be about 700 feet thick, and
contain conglomerates, sandstones, and shales, with plant-remains
about the middle of the series.
Seismic Phenomena in the British Empire. 449
- The Glacial striz sweep round from 8.8.W. at the north, to S.W.
and W.S.W. at the south end of the district. In the Penmon area
there is cross-hatching with a series running 8.S.E., and it is
suggested that this is due to fluctuations in the power of the Car-
narvonshire glaciers to deflect the ice coming from the north,
combined with the local influence of certain high ground.
3. ‘Seismic Phenomena in the British Empire” By M. F.
de Montessus de Ballore, Captain of Fortress Artillery at Belle-Lle-
en-Mer.
The author gives a brief outline of a plan that he has elaborated
for studying Seismology. He has separated his work into four
parts:—1. The formation of an EKarthquake Catalogue. 2. Refuta-
tion of the empirical laws previously enunciated. 3. Description of
the globe from a seismological point of view. 4. Investigation of
the characters which differentiate stable from unstable regions.
He gives a method by which the relative seismicity (or instability
as regards earthquakes) of regions may be obtained and registered,
and indicates some of the results which he has derived from his study,
including the intimate relationship between instability and surface-
relief, and the independence of seismic and volcanic phenomena,
The main part of the paper is a section of the third division of
the author’s work, and deals in detail with the earthquakes of the
British Empire. In this part of the paper, the recorded earth-
quakes of the British Isles, India, Australia and New Zealand,
British Africa, Canada, and various scattered possessions are de-
scribed.
June 24th.—Dr. Henry Hicks, F.R.S., President,
in the Chair.
The following communications were read :—
1. ‘Notes on the Glacial Geology of Arctic Europe and its
Islands.—Part If. Arctic Norway, Russian Lapland, Novaya
Zemlya, and Spitsbergen.’ By Col. H. W. Feilden, F.G.S. With
an Appendix by Prof. T. G. Bonney, D.Sc., LL.D., F.R.S., V.P.G.S.
The author gives an account of observations made in Arctic
Norway which tend to prove that the shell-bearing terraces are
true marine deposits indicating uplift since their formation, and
that they were not formed by ice-dams. He then describes terraces
recently formed in Kolguev Island, which illustrate the combined
influence of pack-ice, sea-waves, and snow on the formation of
terraces in a risingarea. The glacial geology of the Kola Peninsula
is next considered, and the distribution of the boulders noticed.
There is no doubt that these boulders have been derived from local
rocks, and that no ice-sheet from the North ever passed through
Barents Sea or impinged on the northern coast of Europe.
The author saw no evidence of the former extension of an ice-
sheet over the now frost-riven rocks of Novaya Zemlya. He found
wide-spread deposits of boulder-clay with marine shells in this
region, which he attributes to the action of floating ice. In the
Kostin Schar many of the islands are connected by ridges covered
450 Intelligence and Miscellaneous Articles.
with rounded stones pushed up by floe-ice, with solid rock beneath
glaciated by the floe-ice. Several minor phenomena connected with
the glacial geology of Novaya Zemlya are also described. The
raised beaches of Franz Josef Land are noticed, and immense
deposits occurring in Spitsbergen, which were originally formed
under water in front of glaciers, alluded to. These, as weil as other
submarine deposits of glacio-marine origin seen elsewhere by the
author, show no signs of stratification.
Prof. Bonney describes specimens brought by Col. Feilden from
Norway, the Kola Peninsula, and Novaya Zemlya. From an
examination of the rocks obtained im situ in the latter region,
Prof. Bonney confirms Col. Feilden’s suggestion that the Kolguev
erratics may have come from Novaya Zemlya.
2. ‘Extrusive and Intrusive Igneous Rocks as Products of
Magmatic Differentiation.’ By Prof. J. P. Iddings, For.Corr.G.8.
The author, after pointing out the propositions concerning
differentiation of magmas upon which he is in agreement with
Prof. Brégger, discusses the points of difference, and describes the
relation of the igneous rocks at Electric Peak to all of those which
took part in the great series of eruptions which occupied almost the
whole Teitiary period, and spread themselves over a vast territory
in Montana, Wyoming, and Idaho. In Tertiary times the eruptions
were at first largely explosive, and the accumulation of tuff-breccia
formed a chain of lofty volcanoes, comparable with the Andes in
size as well as in the nature of their material (andesite and andesitic
basalt). After considerable erosion of these volcanoes, gigantic
fissure-eruptions flooded the region west of the denuded volcanoes.
The massive lava-streams which welled from these fissures consisted
at first of rhyolite with an average silica-percentage of about 74,
alternating occasionally with basalt; but the great bulk of the
basalt was poured out immediately after the rhyolite from fissures
still farther to the west and south-west. In the case of these
extrusive rocks, whose volumes are of such magnitude, the evidence
‘drawn from the succession of their eruptions and from their com-
position is of a higher order than that derived from the smaller and
more localized eruptions, and it is upon evidence of this order that
the author ventured to enunciate the principle that in a region of
eruptive activity the succession of eruptions in general commences
with magmas representing a mean composition and ends with those
of extreme composition.
XLVI. Intelligence and Miscellaneous Articles.
CARBON MEGOHMS FOR HIGH VOLTAGES. BY W. M. MORDEY*.
Ss insulation of apparatus and cables used with high voltages
should be tested with high voltages. [or this and many other _
purposes some inexpensive and trustworthy form of high resistance
* Paper read before Section A, British Association, Liverpool, Sep-
tember 23rd, 1896. Communicated by the Author.
Intelligence and Miscellaneous A,'ticles. A5L-
is required. The ordinary carbon megohm is very untrustworthy,
being subject to considerable variation and unsuited for use with
pressures of more than a few volts. The Author having overcome
these defects thinks the simplicity of the plan followed no reason
for withholding a description of it.
A study of some of these carbon megohms, supplied by
instrument-makers, showed that, although the loss is very small,
being oniy one-millionth of a watt per volt impressed, the delicate
conducting film or line of plumbago is too much disturbed by the
heat generated. Experiment showed that on increasing the cross-
section of the conducting film, and correspondingly increasing its
length, it became easy to construct a carbon resistance practically
unaffected by any ordinary variation of temperature, and capable
of being used with pressures of 100 volts per megohm, or as much
higher as may be desired.
- Various ways of carrying this out have been tried. Excellent
results were obtained by the use of long strips of shellaced
cartridge-paper coated on one side to a width of one to two
inches with plumbago, well burnished, the edges of the paper
strip being folded down over the film to protect it. The strip
is then rolled into a loose spiral, secured toa support, and mounted
in a suitable box with terminals.
Examples of these resistances were shown and particulars given
of tests to which they had been subjected. The resistance in one
case was 1:315 megohm at 74° F. It was tested at various
temperatures up to 150° F., and was practically constant throughout
this range, the resistance at the higher temperature being 1°51
megohms.
Another resistance shown, which measured 0°975 megohm, had
been subjected to a pressure of 100 volts for 12 hours continuously
without showing any change. It was stated that these resistances
were being supplied by Mr. Paul, of 44 Hatton Garden, London.
ee eee
SEARCH FOR SOLAR &-RAYS ON PIKE’S PEAK.
BY FLORIAN CAJORI.
Experiments carried on by M. C. Lea* and others have failed
to show the presence of Rontgea rays in solar radiation. If
these rays reach us from the sun, their intensity must be ex-
ceedingly feeble. The suggestion has been made that Roéntgen
rays may exist in sunlight, but are absorbed by the earth’s atmo-
sphere. The fact that Lenard rays are stopped by only a thin
layer of air made it not improbable that Rontgen rays might
be stopped by a thick layer of it. On this hypothesis a mountain-
top is the best locality to examine sunlight for the new rays. The
writer determined, moreover, to expose the photographie plate to
solar rays, not several hours, but several weeks.
* Am. Journ. Sci. [4] i. 1896, pp. 3863, 364.
452 | Intelligence and Miscellaneous Articles.
During preliminary experiments made in Colorado Springs, it
was found that a sheet of aluminium would allow certain solar
rays to pass through. These were not Roéntgen rays, for the
reason that black paper placed between the aluminium and the ~
photographic plate seemed to cast as deep a shadow as did a strip
ot iron. A different mode of exposing the plate was necessary
and a plan similar to Lea’s was finally adopted.
The first 100 leaves of an unbound book were turned over, and a
rectangular trough of the dimensions of the photographic plate
(7-5 by 13 cm.) cut into the next 55 pages. Seven pages above
this trough, thin metallic plates, from 5 to 20 mm. wide, were
placed between two leaves, and held in position by gumming the
two leaves together. Care was taken to let the mucilage dry before
shutting the book. After the photographic plate (Seed, 23) was
placed in the trough, the book was closed, wrapped in black
Ba paper, then in paraffine paper, and finally put into a tin
Ox
The box was prepared as follows :—Its lid was placed externally
over its bottom and a rectangular window, 7°5 by 13°5 em., cut
through them both. Thereupon a sheet of aluminium, 13 by
23 cm., and :29 mm. thick, was placed between the lid and the
bottom, so as to screen the window. The wide margin of the
- ec. lying between the sheets of tin, was united to them
above and below by thin layers of bees’-wax. Externally the
edges of the rectangular window were covered with sealing-wax,
to which a thick layer of paint was finally applied. A new lid
was provided for the open side of the box and both lids were
soldered on. Prepared in this manner, the box could be left
exposed to all kinds of weather without danger that moisture
would reach the photographic plate. To touch the plate, rays had
to penetrate the sheet of aluminium, a few layers of paraffine and
black paper, and 100 pages of the book. Roéntgen rays of intensity
ordinarily met with in the laboratory penetrate at once very much
greater thicknesses of these materials.
Through the kindness of Mr. F. Blackmer and Mr. D. Rupp, of
Colorado Springs, the box was taken to the suinmit of Pike’s Peak
(elevation 14,147 ft.) and fastened by wires upon a roof sloping
southward. The box was left in that position from June 27 to
August 10. When subjected to the usual process of development,
the plate failed to show any action of rays and presented a uniform
surface, without traces of shadows from the metallic strips.
Another plate, similarly exposed from July 7th to August 28th in
Rosamont Park, near Pike’s Peak, at an altitude of 9200 ft., gave
the same result.
Thus even in high altitudes no evidence of the presence of
Rontgen rays in solar radiation was obtained.—Am. Journ. Sct.
[4] 11. 1896, p. 289.
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[FIFTH SERIES.] Kos
DECEMBER 1896.
XLVII. On some Experiments with Réntgen’s Radiation.
By RicwHarD THRELFALL, W.A., Professor of Physics, and
JAMES ARTHUR PoLLocK, Demonstrator of Physics in the
University of Sydney, N.S.W*
| a experiments to be described were performed during
April and May of this year, and were made with the
object of elucidating the nature of the radiation. It was
thought that the following possible explanations should be
tested :—
‘1. The radiation consists of a swarm of material particles
projected through the glass of the generating tube,
Electrical changes taking place at the glass surface are
invoked to explain the differences between Rontgen’s
and Lenard’s rays. 3
2. The radiation consists of an “ sether wind.” Adther is
sucked through the glass towards the source of radia-
tion and then blown outwards. The question as to
whether the radiation observed by Rontgen is the
expression of the motion of sether to or from the source
remaining open.
3. The radiation consists of zther vortices moving to or
from the source.
4, The radiation consists of sther waves: that is waves of
regular or irregular zether motions.
5, The radiation consists of electromagnetic waves of either
* Communicated by the Physical Society: read November 13, 1896.
Phil. Mag. 8. 5. Vol. 42. No. 259. Dee. 1896. 2K
454 Messrs. Threlfall and Pollock on some
yery small wave-length or having longitudinal com-
ponents. This is probably a special case of (4)—at
least if we are to look to the zether to explain electricity
and magnetism dynamically.
6. The radiation is a phenomenon of a new order entirely
unconnected in any way with anything in our past
experience. |
Source of Radiation.
Being thrown entirely on our own resources for means of
production of the radiation—all the Crookes’s tubes in our
possession being almost useless—we arrived at the form of tube
shown in fig. 1. These tubes are easily made; the surface
Fig. 1.
opposite to the kathode being spherical, can be made very
thin, and the electrodes are kept well apart. The kathode
is best made very nearly plane—if concave it will easily fuse
the thin glass against which its rays are projected ; we have
lost many tubes from this cause. We have found that the
bulb may be conveniently about three or four centimetres in
diameter and the main tube as little as 1°5 to 2 cm. in diameter.
~The expansions round the electrodes are intended to obviate
local heating, for it is not always easy to prevent oscillatory
discharges and consequent ‘‘ kathoding” from the “ anode.”
The chief merits from our point of view, however, were that
the tubes were very easy to make out of comparatively small
glass tubing. Their volume is small, so that they can be
exhausted quickly, and they give very intense action. In
fact one tube—the bulb of which ultimately fused under the
Experiments with Réntgen’s Rays. 455
influence of the kathode-discharge—gave quite as intense if
not more intense radiation than a “ focus ”’-tube made in our
laboratory, which appeared to act perfectly, so far as we
could judge.
Experiment to test Hypothesis 1,
When a vacuum-tube is prepared with electrodes of alu-
minium-wire whose ends are about 1 centimetre apart, and
exhausted until the discharge will rather jump across three or
four centimetres of air between balls of 1°5 cm. diameter
than pass in the tube, it is generally noticed that the dis-
charge, when forced to pass by the tube, goes rather more easily
in one direction than the other. By placing a spark-gap
with spherical electrodes in parallel with the exhausted tube
and properly adjusting the distance of the balls from one
another, it is easy to so arrange matters that the sparks pass
mostly by the spark-gap when the current is in one direction
and by the tube when it is reversed. An arrangement of
this kind is exceedingly sensitive to small changes of pressure
in the exhausted tube. In the experiment to be described, the
spark-gap was generally so adjusted that when the coil-com-
mutator was in one position the whole of the discharge
passed over the gap—only the faintest glow being discernible
in the tube in a dark room. When the current was reversed,
however, the. discharge was about equally divided between
the gap andthe tube. No very delicate adjustment of the gap
seems to be necessary, at all events when the discharges follow
each other rapidly.
Having thus obtained a means of testing the vacuum in a
discharge-tube more rapidly and probably much more deli-
cately than by any kind of gauge, we thought it worth
while to try whether Rontgen’s rays would project particles
into the exhausted tube.» If hypothesis (1) be correct,
then particles must be carried into the exhausted tube if
it is thin enough to be transparent; and if in addition it
contains a piece of platinum-foil which stops the radiation,
the particles would also be stopped; also if the particles are
not wholly entangled in the platinum, some change in the
vacuous state of the tube is to be expected.
A tube about 12 cm. long and 1°5 em. in diameter, and
having a bulb about 4 cm. in diameter in the middle of its
length, was prepared of German glass. It was provided with
electrodes fused in from either end, and extending to within
1 cm. of each other in the centre of the bulb. A bit of platinum-
foil lay in the bulb, and the tube was fused on to a Sprengel
pump through about a metre of tubing some millimetres in
2K2
456 Messrs. Threlfall and Pollock on some
internal diameter. A phosphorus-pentoxide tube was included
just above the fall-tube. The whole apparatus, including the
pentoxide tube, was repeatedly heated by a Bunsen burner
and exhausted until the discharge preferred to pass through
3 cm. of air-gap rather than through the tube. So sensitive,
however, was this means of testing the vacuum that for the
first few days, despite frequent heatings and pumpings, the
vacuum would not remain constant for more than some
minutes. After about a fortnight of heating and pumping,
however, the vacuum became so steady that the change in
twelve hours, which was sufficient to entirely stop the discharge
in the spark-gap, could be rectified by the fall of at most
50 drops of mercury in the fall-tube—~. e. by about 30 seconds’
pumping. The change of vacuum occuring during an hour
could still be easily detected by testing with a current in
alternate directions. )
Under these circumstances, experiments were made by
urging Réntgen tubes to their utmost, almost in contact with
the bulb of the exhausted tube, and directing their radiation
on to the platinum-foil. Though many very active tubes were
fused or otherwise destroyed, yet during an hour’s action on
several occasions no change of vacuum in the exhausted tube
could be detected when due allowance for the slight progressive
deterioration was made.
This exper'nent was repeated several times, and a tube
which had successfully exhibited the fluorescence of a screen
of tungstate of calcium through an aluminium plate -7 mm.
thick to an entire audience, was melted down in the operation’;
but no effect whatever was observed.
Professor Wright, of Yale, has given some reasons for
thinking that Rontgen rays when passing through gold-leaf
carry particles (of gold?) off with them ; so that if a positive
effect had been obtained in the experiment described it would
not have been quite conclusive ; neither is the negative effect
observed conclusive against any particles being carried for-
ward ; it only shows that the particles so carried (if any)
either refuse to behave as gaseous particles, or are exceedingly
few in number. : FE
The experiments of Professor Minchin which are now
available have, however, rendered the solid particle theory so
unlikely that it is hardly worth while pushing the investiga-
tion further—at all events until much more powerful appli- -
ances are to hand. In case the experiment appears worthy
of repetition, we would call attention to the fact that it appears
desirable to use specially purified phosphorus pentoxide in
the drying-tube ; for the presence of the vapour of phosphorus
Experiments with Réntgen’s Rays. 457
or its lower oxide fouls the mercury when impure pentoxide
is heated.
Experiments made to test whether Réntgen’s Radiation is
associuted with ether currents in any way.
With regard to hypothesis (2) Michelson has put into our
hands a comparatively simple method of obtaining the inter-
ference of two beams of light which may be used for detecting
the presence of ether currents by their influence on the
velocity of lignt travelling through the moving ether.
' The arrangement employed in fig. 2 was made use of for
Fig. 2.
the purpose of determining whether Réntgen’s radiation is
associated with zether movements.
In this arrangement the light from L is divided into two
beams at A which travel round from mirror to mirror in
opposite directions, eventually reaching the telescope T.
When the adjustment is correct, interference-bands are seen
on looking into the telescope.
In this experiment it was estimated that a shift of the
bands equal to one tenth of the width of a single band, or a
widening of the bands by one fifth of the same amount,
458 Messrs. Threlfall and Pollock on some
could not have escaped our notice. In some experiments the
Ro6ntgen tube was placed so that the line of kathode discharge
made an angle of about 30° with the path BC, and in others
made an angle of 90° with the same line. No disturbance of
the bands could be detected when the coil was started or
while it was working, which shows at once that within the
limits of accuracy imposed by the experimental conditions,
the Réntgen radiation is unaccompanied by ether streams.
This conclusion refers, of course, only to air, and it therefore
appeared desirable to make an additional experiment, replacing
the air by a substance of greater density. Pure benzene
was selected as a suitable liquid, and a glass tank 16°4 cm.
long, and several centimetres wide and deep, was placed in
the path BC. The radiating tube, which was placed above
the free surface of the benzene, was arranged te radiate on to
the path in all directions, and in some cases was actually
immersed in the benzene so that its active surface was in the
field of view of the telescope. ;
No disturbance of the bands commencing when the coil
started, and stopping when it stopped, was ever detected,
though a great many excellent observations were made. Of
course, effects due to the heating and electrification of the
glass can be easily distinguished from those we are in search
of in view of their persistence after the coil is put out of
action. ‘The tube employed gave quite visible fluorescence by
barium platinocyanide at a metre’s distance and through
aluminium *7 mm. thick. We can get an idea of the order
of the minimum ether velocity which could be detected
by this means. The path BC being some 25 cm. long,
we will suppose that only 10 cm. of it are influenced by
the tube and that the radiation is along the path. The sen-
sitiveness of the method was the same whether we employed
air or benzene.
A shift of the bands amounting to a fifth of the distance
from band to band would be produced by an ether velocity
sufficient to change the time of passage of light over the path
by one tenth of a period. As sodium light was employed,
we may call this
5:9/3 x 107"* seconds.
But the time required for light to travel over ten centimetres
in alr is about
1/3 x 10~° seconds ;
or the velocity is not changed by the radiation by more
than six parts in ten million, say. This is about 177 metres
Experiments with Rontgen’s Rays. 459
per second, so that the conclusion to be drawn from the
experiment is that the Rontgen radiation is not associated
with ether velocities greater than, say, one fifth of a kilo-
metre per second, or about a thousand times less than that
of kathode rays as measured in a vacuous tube by Prof. J. J.
Thomson (Phil. Mag. vol. xxxvili. p. 364).
Experiment to test whether Réntgen Rays produce any change
in ether affecting the velocity of light.
An experiment (fig. 8) was arranged on Michelson’s
Fig. 3.
principles. In this case the sodium light from L is divided
at the partially silvered mirror A, one beam travelling to C
and thence to T, the other travelling to B and thence back to
T. The path AC is operated upon by the tube, and the path
AB is screened by a heavy cast-iron screen. The active
460 Messrs. Threlfall und Pollock on some
surface of the tube is brought up to the path CA, and the
Rontgen radiation allowed to traverse it in a variety of direc-
tions from parallelism to normality, and is even thrown on to
the mirror at OC. For experiments on benzene, troughs of
that liquid as similar as possible are inserted in both paths,
the one in the path AB acting merely as a compensator.
This experiment is much more difficult than the one previously
described, and the benzene requires to be well stirred if good
definition is required. In the experiments in air a shift of
the bands by ;!, of the width of a band could be observed,
while in benzene a change of about half the distance from
band to band only could be seen. A widening of the bands
to about half the above amounts could have been detected in
each case. When the benzene was used the fringes were
unsteady, and opportunities for observation had to be waited
for. Of course, when the active surface of the Réntgen tube
dipped into the benzene so as to appear in the field of view
of tne telescope great disturbances due to thermal and electric
changes became visible. These, however, did not appear
instantaneously on starting the coil, nor did they disappear on
stopping it. In no case was any real effect observed.
The chief interest of this experiment lies in tue fact that if
the Roéntgen radiation consisted of longitudinal ether waves,
2. €., waves of longitudinal ether displacement, some effect was
to be expected. If the waves are long compared with the path
AC (the path AB being screened) a widening of the fringes,
or in the extreme case a total disappearance of the fringes, is
to be anticipated. ‘This can easily be realized by blowing an
organ-pipe in the neighbourhood. If the waves are short
compared with the dimensions AC then all will depend on
the azimuth of the tube, or rather on the inclination of the
Roéntgen-ray path to the path of the light rays. This appears
from the fact that wuole waves would produce no effect in
the case contemplated—everything would depend on the
fractional parts of the waves included in or projected on AC.
The conclusion to be drawn from the experiment is that
neither in air nor benzene are the light-transmitting properties
of the ether interfered with. The limits of observational
accuracy are of the same order as in the case of Experiment II.
Action of Réntgen’s Rays on a Selenium Cell.
It so happened that one of us was engaged early in the
year in experimenting with photo-resistance cells made of
selenium which had been laboriously purified and which was
probably as pure as any that has ever been obtained. Con--
trary to expectation, such cells showed a quite normal light-
Experiments with Réntgen’s Rays. 461
sensitiveness whether the electrodes were of platinum or of
aluminium. The cells were made according to the directions
given by Mr. Shelford Bidwell* in his paper read before the
Society last year—the only difference being that purified
selenium was employed.
The result of a good many observations was to show that
a certain selenium cell with platinum electrodes was acted on
to about the same extent and in the same direction whether it
was exposed to the radiation of a “standard” candle ati a
distance of three metres, or to the Rontgen radiation at
a distance of ten centimetres and passing through*7 mm. of
aluminium and about 3 mm. of wood. ‘The tube was
working so as to cause visible fluorescence in a barium platino-
cyanide screen—not of quite the best quality—at a distance
of rather over a metre in a room nearly but not absolutely
dark. This is of course a very rough way of stating the
degree of activity of the tube, but when the experiments were
made Professor Minchin’s work had not reached us, and
consequently the simple scale of tube intensities which it
implies was not available.
In order to test whether the action of the light differed in
kind from that of the Rontgen radiation two experiments
were made—in one the rate of resistance-recovery of the
selenium cell was carefully studied and compared with the
rate of recovery of the cell after exposure to candle-light ;
in the other tests were made in the hope of discovering that a
permanent electromotive force was established by the radia-
tion, and that it persisted after the radiation was cut off.
Neither of these experiments led to positive results. The rate
of change of resistance during the twenty seconds of exposure
to Réntgen’s radiation was, so far as could be seen, exactly the
same as when the candle-flame was substituted at the proper
distance. The recovery curves, extending over about half an
hour, were also very similar on the whole (several tests were
made), though both curves themselves exhibited great
irregularities.
In order to obtain effects as little complicated as possible
by previous history the sensitive cell was kept in circuit on
the bridge, and was traversed by the testing current for two
or three days before the observations.
It was for the same reason that exposures were limited to
20 seconds, for the rate of recovery of resistance with the
cell employed was very slow. On one occasion when an
accidental exposure for several minutes to Rontgen’s radiation
* Phi Mag. vei. xb. p. 233.
462 On some Experiments with Réntgen’s Rays.
was made the cell had by no means recovered four hours
afterwards.
The resistance of the sensitive cell employed was reduced
from 1209 ohms to 1185 ohms in 20 seconds by the radiation
under the conditions mentioned. The testing battery consisted
of two “Obach”’ cells, and the bridge was made up of two equa!
arms of 1000 ohms each—the selenium and the variable arm.
We mention this in case it may ever turn out that the effect
depends on the testing current.
With regard to the electromotive force which it was supposed
might be set up. The cell was kept at rest and undisturbed
for three days before the final trial; it was placed three
centimetres trom the active tube, which was, as before, in a
metallic box, together with the coil. The tube was shut off
from the cell by an aluminium plate *7 mm. thick.
A very sensitive high resistance galvanometer in our
possession, which has been described in a paper read before
the Royal Society but as yet unpublished, was employed to
test for any electromotive force which might be set up. An
exposure to the radiation was made while the cell was in
series with the galvanometer, and it was found that the cell
always exhibited a small electromotive force whether it was
exposed to the rays or not. This prevented the test from
being very sensitive, but inno case was any electromotive
force attributable to the radiation discovered, though if a
a voltage of 10~7 volts had come into operation its effect could
probably not have escaped observation.
At the time these experiments were made we were unin-
formed as to the discharging action of the rays, which has
since been so copiously studied. As soon as we saw an
account of some of this work we felt that the change of
resistance of the selenium cell was no longer an isolated
phenomenon to be worked out by itself, but must be studied
in conjunction with the similar phenomena observable in other
substances, and it is for this reason that the experimental
work was not extended so as to include other cells.
These notes may, perhaps, be summed up as follows :—
(1) It is easy to make a Rontgen tube of great activity by
the most elementary glass-blowing.
(2) The Roéntgen radiation does not consist in the pro-
jection of gaseous matter, or if it does the amount of such
matter involved is extraordinarily small.
(3) The Roéntgen radiation does not consist in the projec-
tion of sether streams having a velocity above a couple of
hundred metres per second: this is true whether the radiation
takes place in air or in benzene.
(4) The properties of ether regarded as determining the
On the Diurnal Periodicity of Earthquakes. 463
velocity of electromagnetic waves are not greatly changed
(2. e. not at all within our experimental limits) by the Rontgen
radiation, and this applies alike together in air and in benzene.
(5) A selenium cell composed of platinum electrodes and
highly purified selenium is affected by Rontgen radiation to
an extent which is comparable with the effect produced by
diffused daylight.
(6) No permanent or temporary electromotive force is set
up in a selenium cell by the Rontgen radiation.
XLVI. On the Diurnal Periodicity of Earthquakes. By
CHaruESs Davison, Se.D., F.G.S., Mathematical Master at
King Edward's High School, Birmingham*.
L W ITHIN the last seven years, two important memoirs
have appeared dealing in part with the diurnal
periodicity of earthquakes. In a paper published in 1889,
M. de Montessus de Ballore + considers the question from a
negative point of view, his object being to show that the
diurnal period is apparent rather than real. More recently,
in 1894, Prof. F. Omori f, in a valuable investigation on the
after-shocks of earthquakes, points out that there are various
periodic fluctuations in their decline of frequency, three of the
periods being a day or less in length. I will first give a brief
summary of the methods and conclusions of these two writers
before proceeding with the immediate object of this paper,
which is to subject the records used by them, or similar
records, to the more rigid process of harmonic analysis.
2. M. de Montessus’s statistical inquiries are based on a
great catalogue of more than 45,000 earthquakes. The
separate entries being of unequal value, he divides them into
seven classes, according to the nature of the district and the
mode of record. The first six classes include all registers
obtained without instrumental aid, the seventh those of the
Italian geodynamic observatories. For every region of each
class he gives the total number of shocks during each hour of
the day. Representing by d the number of shocks occurring
in the twelve day-hours (6 a.m. to 6 P.M.), and by x the
number in the twelve night-hours (6 P.M. to 6 A.M.), he then
evaluates the ratio d/n for each region. Tor the first group,
excluding the fifth or volcanic series, the mean value of d/n
* Communicated by the Author, with some alterations, after being
read before the Royal Society on March 5, 1896.
+ ‘ Etudes sur la répartition horaire diurne-nocturne des Séismes et leur
prétendue relation avec les culminations de la lune.” Arch. des Sc. phys
et nat. vol. xxii. 1889, pp. 409-430 and tables. yi
t “On the After-shocks of EKarthquakes.” Journal of the Collé of
Science, Imp. Uniy. Japan, vol. vii. 1894, pp. 111-200.
ea
AG4 yo Be, Ci Davison on the Diurnal
varies from 0°75 to 0°82, and increases with the scientific
character of the record. Also, the value of d/n being small
for moderate shocks and approaching unity for severe ones,
M. de Montessus infers that earthquakes in reality occur just
as frequently by day as by night.
- The seventh class comprises the records from thirteen geo-
dynamic observatories in Italy, all obtained by means of con-
tinuously recording instruments. The value of d/n varies
from 0°50 for. Corleone to 2:06 for Bologna and 8. luca,
being on an average 1°49. If all shocks are excluded but
those of intensity I. of the Rossi-Forel scale, the mean value
rises to 1:80. This seems to imply the existence of a true
diurnal period, but M. de Montessus interprets the inequality
otherwise, referring the more numerous slight shocks of the
day-time to ‘“mouvements dus a l’homme, roulements de
voitures et de trains de chemins de fer, explosions de mines,
&e.” The suggestion is a useful one and deserves careful
consideration. At the same time, it should be remarked that
the phenomena admit of another explanation, for we might
expect that slight earthquakes would be subject to periods of
greater amplitude than violent shocks*. With regard to the
non-instrumental records, however, M. de Montessus’s analysis
leaves little doubt that the more frequent observation of earth-
quakes at night is due to the conditions being then more
favourable for the detection of weak tremors.
3. Prof. Omori’s investigation is based for the most part on
seismometric records. He makes use of the valuable Tokio
register from 1876 to 1893, that for all Japan from 1885 to
1890 (which is only in part a seismometric record), and
especially the lists of after-shocks at Kumamoto, Gitu and
Nagoya, and Chiran, during the thirteen or fourteen days
following the Kumamoto earthquake of 1889, the Mino-Owari
earthquake of 1891, and the Kagoshima earthquake of 1893.
The shocks are grouped in hourly, two-hourly, and six-hourly
intervals, and curves are drawn, not through the points
corresponding to the numbers so obtained, but by some
process of smoothing which is not explained. The periods
which are brought into prominence by this method are 24
hours, about 8 or 9 hours, and about 4 hours, in length. In
the case of the Mino-Owari earthquakes, while both the eight-
hourly and four-hourly periods are shown on the Gifu and
Nagoya curves, the former is more marked at Gifu and the
latter at Nagoya. The diurnal period for these two stations
appears to have its maximum about 1 a.m., for Tokio in the
évening, and for all Japan early in the morning.
4, The method adopted in this paper is that of harmonic
* See Phil. Trans. 1893 A, pp. 1116-1120.
Periodicity of Earthquakes. 465
analysis. As the absolute frequency of earthquakes in
different districts is extremely variable, the average number
of shocks per hour in each case is represented by unity, so
that the results may be directly comparable. The epochs are
given in the mean local time of the place of observation, with
two exceptions, namely, Japan and Italy, in which the standard
times are those of 135° HE. and 15° E. respectively. In the
Table, however, the figures for Japan refer to mean Tokio
time, for Japanese after-shocks to mean time of 135° H., and
for the Italian stations to mean Rome time.
Japan.
5. Tokio—Japan Seismol. Soc. Trans. vol. ii., 1880,
pp. 4-14, 39; vol. vi., 1883, pp. 82-35 ; vol. vul., 1885,
pee 100-103; yel. x, 1887; pp. 97-995 vol. xv., 1890,
pp- 127-134: Brit. Assoc. Rep. 1886, pp. 414-415 ; 1887,
pp: 212-213 ; 1888, pp. 435-437 ; 1889, pp. 295-296 ; 1890,
pp- 160-162 ; 1891, pp. 123-124; 1892, pp. 93-95 ; 1893,
pp. 214-215.
Duration of record, 1876-1881 and 1883-1892. Number
of earthquakes, 1204 ; in winter, 661 ; in summer, 543.
This valuable record begins in the latter half of 1872, and,
up to the end of 1892, contains 1304 entries. From the end
of 1875, the earthquakes were registered by means of seismo-
graphs, Palmieri’s being in use until April 1885, and the
Gray-Milne seismograph after that date. Towards the close
of 1882 the list is incomplete, owing to the removal of
the instrument to a new station. For general purposes this
would he of little account ; but as I wished to compare the
results obtained from the six winter months (October to
March) with those obtained from the six summer months
(April to September), this year has been omitted.
Whole year. Winter. Summer.
Harmonic
Component. |
Ampl. | Epoch. | Ampl. | Epoch. | Ampl. | Epoch.
h m h m hm
A.M. A.M. tae
Ist (24 hours) ...| 130 | 10 14 093 | 10 39 "176 9 58
2nd (12 hours)...| ‘082 | 10 22 "123 9 26 ‘085 0 12
ord (8 hours) ...! °098 6 28 "086 6 31 ‘111 6 25
| 4th (6 hours) ...) °118 3.7 "148 2 56 ‘096 3 26
5th (44 hours) ...| *030 Ls ‘059 1 49 060 4 2
|
6th (4 hours)...... "024 3 27 ‘097 ott “058 0 58
466 Dr. C. Davison on the Diurnal
6. Japan.—Prof. J. Milne, ‘‘ A Catalogue of 8331 Harth-
quakes recorded in Japan between 1885 and 1892.” Seismol.
Journ. of Japan, vol. iv. 1895, pp. 1-xxi., 1-367.
Duration of record, 1885-1890. Number of earthquakes,
1175 ; in winter, 578 ; in summer, 597.
Prof. Milne’s great catalogue includes all the earthquake-
records collected by the Imperial Meteorological Office at
Tokio. A large number of these were obtained by means of
seismographs, but unfortunately the particular shocks so
recorded are not indicated. For my present purpose, I have
made use only of those in which the time of occurrence is
given in hours, minutes, and seconds: for these, Prof. Milne
informs me, were certainly registered by seismographs. As
many others may, however, be omitted by this mode of
selection, it is obvious that the results will not compare in
value with those obtained from the Tokio record. I have
excluded the shocks occurring during the last two years em-
braced by the catalogue on account of the great number that
followed the Mino-Owari earthquake of 1891 (see §§ 7, 8):—
Whole year. Winter. Summer.
Harmonic
Component. | |
| Ampl. | Epoch. | Ampl. | Epoch. | Awpl. | Epcech.
h m m h m
mi : ) A.M. Ree P.M
Ist (24 hours) ...| °147 11 53 "239 1l 50 ‘061 Op
A.M.
2nd (12 hours)...| -004 9 8 7035 9 48 ‘028 3 58
3rd (8 hours) . 064 6 31 045 6 12 ‘083 6 40
4th (6 hours) "100 2 ao O67 al o2eS "146 2 53
After-Shocks of Japanese Larthquakes. |
7. Prof. F. Omori, Journal of the College of Science, Imp.
Univ., Japan, vol. vii., 1894, pp. 126-188, 157, 178-191, 194,
I am indebted to Mr. K. Nakamura, Director of the Central
Meteorological Office, Tokio, for the hourly numbers of shocks
recorded during each month by a Gray-Milne seismograph at
Gifu from October 1891 to December 1893, and at Nemuro
from March 1894 to February 1895.
Duration of records: Kumamoto, July 81—Aug. 13, 1889;
Gifu and Nagoya*, Oct. 29—Nov. 10, 1891; Chiran, Sept.
8-21, 1893 ; Nemuro, March 1894. Number of earthquakes:
* It should be mentioned that a few of the hourly numbers of shocks
at Gifu and Nagoya given in the table differ by one or two units from those
given by Prof. Omori. The figures in the table are obtained from the lists
of shocks given in Prof. Omori’s tables xi. and xii.
Periodicty of Earthquakes. 467
Kumamoto, 148; Gifu, 1258; Nagoya, 572; Chiran, 238 ;
Nemuro, 345.
_ The Kumamoto earthquake occurred on July 28, 1889 ;
the Mino-Owari earthquake un October 28, 1891 ; the Kago-
shima earthquake on September 7, 1893; and the Nemuro
earthquake on March 22, 1894.
Kumamoto. Gifu. Nagoya. Chiran. Nemuro.
Harmonic
Components. |
Ampl. |Epoch.! Ampl.|Epoch.| Ampl. |Epoch.| Ampl. |Epoch.| Ampl. |Epoch.
h m h m hm hm h m |
AM. A.M. A.M. A.M, ras! AM.
Ist (24 hours) | 623 | 0 3 | 163 | 2 19 | 505 | 0 15 | 096 | 4 35 | 204 | 4 12
2nd (12hours).| 456 | 2 35 | 089 | 5 29} -171 | 1 25 | 069 | 3 58 | 200 | 8 27
pta(s Hours)...| 430 | 2 57 |.-229 | 3 58] 111. | 1 17 | 075 | 6 52 | 327 | 5 53
Ath (6 hours)...| ‘214 | 1 56 | 069 | 1 12 068 | 1 41 | ‘086 | 2 18 | 052 | 4 42
5th (44 hours) .| ‘239 | 1 12 | 051 | 0 50 | 072 | 1 83 | 121 | 0 40 | -148 | 3 10
6th (4 hours)...| 088 | 0 52 | 121 | 0 32 | -210 | 0 37 | -239 | 1 57 | 293 | 0 55
The numerous slight shocks which follow a severe earth-
quake are subject at first toa rapid decline in frequency ™~.
Now, if a simple harmonic series be superposed on a declining
linear series, the harmonic analysis of the compound series
shows that not only are the amplitude and epoch of the
function of the same period changed, but minor harmonic
components are also introduced. Jt is clear therefore that the
epochs given above cannot be supposed to agree exactly with
those of their physical equivalents. At the same time, the
omission of the first one, two, three, and four days in succes-
sion from the Gifu and Nagoya records produces no important
change in most of the components. Moreover, as will be seen
in the next section, the different conditions introduced by the
occurrence of a great earthquake endure even when the decline
in average frequency of the after-shocks ceases to be sensible,
The results for Gifu and Nagoya confirm those obtained
by Prof. Omori by a different method of analysis, the 8-hour
component being more marked at Gifu, and the 4-hour com-
ponent at Nagoya. Prof. Omori gives the time of each shock
recorded at both places until the end of Noy. 10, those at
Nagoya to the nearest second, and those at Gifu generally to
the nearest minute. Out of 1257 shocks recorded at Gifu
* F. Omori, Journal of the Coll. of Science, Imp. Univ. Japan, vol. vii,
1894, pp. 111-126 ; Seismol. Journ. of Japan, vol. iii, 1894, pp. 71-80.
See also Natural Science, vol. vi, 1895, pp. 891-887,
68 Dr. C. Davison on the Diurnal
and 572 at Nagoya there are only 175 whose times of oc-
currence differ by less than a minute; and it does not follow
that even all these are identical, for, during the first day or
two, shocks were frequently felt at Gifu for several minutes
in succession. Though the two stations are less than 14
miles apart, and both are close to the great fault-scarp, it
would seem, from the above analysis, that their after-shocks
do not originate under quite the same conditions.
8. Gifu: Nov. 11-Dec. 31, 1891, number of earthquakes
839; 1892, number of earthquakes, 865; 1893, number of
earthquakes, 272. Nemuro: Apr. 1894—Feb. 1895, number
of earthquakes, 347.
Gifu, ’ ’ : Nemuro,
Nov. 11-Dec. 31, 1891.| “fu, 1892. | Gifu, 1893. |4. 1894 Feb, 1895.
Harmonic =
Com ponent. |
Ampl. Epoch. | Ampl.|Epoch.| Ampl.|Epoch.| Ampl. | Epoch.
hm hm hm hm
4 ea A.M. fag “Sie A.M. |
Ist (24 hours) ...) :205 111 | -007 |5 3{-106 |3 20] -062 0 29
A.M
Qnd (12 hours)..., “119 255 |-171 19 0| -087 |10 43 | -099 4 44
3rd (S hours) ...| °156 029 | +154 | 558] -078 |126| -217 315
4th (Ghours) ... 018 338 | 048 | 033] +188 |149| -053 1 31
5th (44 hours) .... “076 052 | -049 | 438] 085 | 341] 169 2 42
6th (4 hours) .... °187 159 | 133 | 2 33 | -095 | 159 | -205 0 16
Philippine Islands.
9. Manila—P. Miguel Saderra Masd: La Seismologia
en Filipinas (Manila, 1895), pp. 100-108.
Duration of record, 1869-1889. Number of earthquakes,
210*,
Harmonie | Ampl. oe
Component. h m
A.M.
Ist (24 hours) ......... ‘273 10 49
2nd (12 hours) -,..3-..2 136 2 20
ord (8 hours) ......:.. 210 6 38
Ath (6 hours) ......... 310 4 22
* The total number of shocks in P. Saderra’s catalogue is 218, but
three of these the exact time is not given.
Periodicity of Harthquakes. 469
Italy.
10. M. F. de Montessus de Ballore, Arch. des Sei. phys. et
nat. vol. xxii. 1889, tables. |
Duration of record: Acireale to Verona, Dec. 1872-1887 ;
Vesuvius, 1863-1884. Number of earth-tremors: Acireale,
364; Bolognaand 8. Luca, 636 ; Cascia, 505; Corleone, 584;
Rocca di Papa, 388 ; Rome, 2346 ; Velletri, 1491 ; Verona,
700 ; Vesuvius, 547; Italy, 8177.
In the same table M. de Montessus gives the hourly num-
bers of tremors at four other observatories (Alvito, Belluno,
Narni, and Spinea di Mestre), but the total numbers of
tremors are so small (123, 180, 144, and 159) that I have
have not made use of them separately. They are, however,
included in the total for all Italy.
Ist comp. 2nd comp. 3rd comp. 4th comp.
(24 hours). (12 hours). (8 hours), (6 hours),
Observatory. ee 1
Ampl. |Epoch.| Ampl.|Epoch.| Ampl. |Epoch.| Ampl. Thee.
1 m hm hm hm
oa P.M. A.M. A.M. A.M.
CREAN (onc). 40.3632 Hod | O49 (cM, 45) O40) FOG | 143 | 2Por
Bologna & S. Luca .|| ‘705 | 0 8 | -166 | 6 56 | ‘081 | 7 41 | 009 | 5 25
A.M,
IU F210 ihe cidceres 259i 2h O91, Sekt) 166)" 4 Sis -280 eons
@arleone ............ 3957) 0 1d) s0on le 2S 1G OOS esses
Rocca di Papa ...... 523 |11 22 | -159 j11 22 | -342 | 410 | 103 | 2 44
P.M.
lig 2 oe ae 613 | 0 36 | 143 |10 25 | 097 | 1 37 | -150 | 4 6
MeMetrt......cxeeca- 307 | 0 25 | 113 {11 O | 164 | 0 43 | 060 | 2 25
A.M.
WECM BUA a secu csacen 2ooy Li of e270 NOP 130 iO 5S | “s66eiponaG
P.M.
WeamtyIUS: © 6..605.000.. AAO Wiles ie37 lei bob.) :052.7)-5) 59) |2:038) aa
LDL) ener eee ‘o24y | O25 | AIZ IE 14 | O45" | i461 Lis) arag
The nine Italian records thus agree in exhibiting a marked
diurnal period, the epoch in seven cases being about noon,
and in the other two cases not far from midnight. Reference
has already been made to M. de Montessus’s explanation of
the preponderance of slight tremors during the day. There
is much to be said in favour of this explanation. The obser-
vatory on Vesuvius is only a few yards from the main road,
along which there is a great deal of heavy traffic about mid-
day. During the interval embraced by M. de Montessus’s
Phil. Mag. 8S. 5. Vol. 42. No. 259. Dec. 1896. 2.1L
470 Dr. CG. Davison on the Diurnal
catalogue the observatory at Rome was situated close to a
much-frequented street. At the various observatories of the
second order the arrangement of the recording instrument,
I am informed, was not then irreproachable. On the other
hand, at Rocca di Papa, Dr. Cancani tells me that the appa-
ratus could not be in any way influenced by accidental or
artificial movements on the ground outside. He does not,
however, attribute the movements of the tromometers to
seismic causes so much as to the action of the wind either
near to, or at a distance from, the observatory.
This one case, therefore, being free from doubt so far
as artificial disturbances are concerned, it becomes un-
necessary to reject entirely the results obtained from the
other records, especially when the epochs of the principal
harmonic components agree so closely, as some of them do,
with those derived from the Rocca di Papa register. More-
over, while some variability might be expected in the epoch
of the diurnal period if it were due to natural causes, it is
difficult to understand how, according to M. de Montessus’s
explanation, the epoch could ever occur near midnight.
Summary of Results.
11. The following conclusions may, I think, be drawn
from the results of the above analysis :—
(1) The reality of the diurnal variation of earthquake-
frequency seems to be proved by the approximate agreement
in epoch (mean local time) of the first four components for
the whole year at Tokio and Manila, and for the winter and
summer halves of the year at Tokio.
(2) In ordinary earthquakes there is in nearly every case
a marked diurnal period, the maximum generally occurring
between 10 A.M. and noon. The semi-diurnal period, though
less prominent, is also clearly marked, the maximum
occurring, as a rule, between 94.m. and noon and between
9p.m. and midnight. Other minor harmonic components
are also occasionally important—the first maximum of the
eight-hour component probably occurring about 6.30 A.M.
and that of the six-hour component about 3 or 4 A.M.; but in
these two epochs the results are not always concordant.
(3) Though the materials are insufficient for any general
conclusion, a comparison of the results for Tokio and Rocca
di Papa seems to show that the slighter disturbances at the
latter place are subject to a more marked diurnal periodicity.
(4) In the after-shocks of great earthquakes the diurnal
periodicity, as a rule, is strongly pronounced. The maximum
of the diurnal period occurs within a few hours after mid-
night, but the epochs of the other components are subject to
Periodicity of Earthquakes. Aq1
wide variation. A special feature of after-shocks is the
prominence of the eight-hour and four-hour components.
After a year or two there is some return to ordinary con-
ditions; but even when the average hourly number of
shocks is reduced to one-hundredth of that during the first
few days, the characteristics of after-shocks are still per-
ceptible.
Origin of the Diurnal Periodicity of Earthquakes.
12. The pressure of the atmosphere, either at rest or in
motion, has for some time been regarded by seismologists as
a cause of earth-tremors. Prof. Milne, who has made a
detailed analysis of tremors recorded at Tokio, concludes that
they are more frequent with a low than with a high baro-
meter, and with a high than with a low barometric gradient ;
also, that a majority of the tremors were produced by the
action of either local or distant winds upon the surface of the
earth, and possibly by their pressure against a neighbouring
mountain-range *. !
My object is now to inquire how far the larger disturbances
considered in this paper are subject to similar laws. If there
is any intimate relation between the diurnal variation of
earthquake-frequency on the one hand and that of barometric
pressure or wind-velocity on the other, it is evident that the
epochs of their respective harmonic components should not
differ widely ; since any distortion of the earth’s surface
by changes in the distribution of atmospheric pressure must
be propagated, both along the surface and downwards, with
great velocity. |
The records of most value for the purposes of this com-
parison are: (1) those of ordinary earthquakes at Tokio,
Manila, and Rocca di Papa; and (2) those of after-shocks at
Kumamoto, Gifu, Nagoya, Chiran, and Nemuro.
13. Taking, first, the diurnal variation of barometric
pressure and seismic frequency, it will be seen that the only
approximate agreement in epoch is in the second and fourth
components at Tokio and the second at Rocca di Papa. The
epochs of the first component differ by as much as eight or
nine hours at both places.
* “Karth-Tremors in Central Japan,” Japan Seismol. Soc. Trans.
vol. xi. 1887, pp. 1-78, vol. xiii. 1890, pp. 7-19; “ Karth-Tremors and
the Wind,” Roy. Met. Soc. Journ. vol. xiv. 1888, pp. 64-72. It should
be mentioned, however, that P. Camillo M. Melzi has arrived at different
conclusions from his analysis of the tromometric records at Florence.
See especially his memoir, “ Nuove osservazioni sull’ independenza dal
vento nei moti tromometrici nei pendoli isolati,”’ Pontif. Accad. det
Nuovi Lincet, Mem. vol. v. 1889, pp. 3-39.
2L2
In the ease of wind-velocity the agreement in epoch is
A7T2 Dr. C. Davison on the Diurnal
Ist comp. 2nd comp. 3rd comp. 4th comp.
(24 hours). | (12 hours). | (8 hours). (6 hours).
Barometric
Pressure. |
Ampl. |Epoch.| Ampl. |Epoch.| Ampl. |Epoch.| Ampl. |Epoch.
mm. | hm |-mm.| hm | mm. | h m | mme} gee
A.M. A.M. A.M. A.M.
| Tokio. (whole year).| ‘52 | 122 | 54 | 9 5] O7 | 243) 03 | 3 37
Cwinter) ace. ..: 67 | 0 54) 61 8 56) *19 | 1 25 OS raw
» (summer) ...j/ “41 |} 2 2|-48 | 9 22) -03 | 5225 ROR gore
| “Milan (whole year).| ‘18 | 3 7/| °36 9°59) OL 2 40 a: nak
Naples __,, $5 ‘09 | 0.57 | -31 /10 13) 03> | 2-280) ] Oi aaa
Eur: 9; 3 25° |2 3) 38 | 9 59) 04 | 1 34) O02 R tee
1st comp. 2nd comp. ord comp. 4th comp.
(24 hours). (12 hours). (8 hours). (6 hours).
Wind-Velocity.
Ampl. |Epoch. mee Epoch. ee Epoch. ge Epoch.
nm.
per sec.| h m a sec.| h m esate hm were hm
P.M, ice res
Tokio (whole year).| ‘86 | 0 42} -24 | 1 52} 02 | 6 23} 05 | 3 48
A.M.
» (winter) ...... 48 |11 56) -25 |1 7] 04 | 5 23} 06 | 3 34
P.M.
» (summer) ...| 1:25 | 0 59} 26 | 235] 03 | 025| 40 |4 6
Manila (whole sean) 201 | 0 44) 51 | 150] 06 | 1 49] 08 | 3 53
much closer, especially for the third and fourth components
at Tokio. The epochs of the first and second seismic com-
ponents at the same place, however, precede those of wind-
velocity by as much as two or three hours. At Manila
the epoch of the first component of seismic frequency precedes
that of wind-velocity by about two hours, and the epoch of the
second component of the former follows that of the latter by
half-an-hour.
The diurnal variation of barometric gradient between Tokio
and Nagano (112 miles W. 30° N. of Tokio) seems too slight
to produce such important effects, though the epochs of its
harmonic components do not differ widely from those of wind-
velocity *.
* IT am indebted to Prof. Omori for copies of the tables of the hourly
means of barometric pressure at Tokio and Nagano published by the
Periodicity of Harthquakes. 473
Ampl. Epoch.
Barometric Gradient. mm.
per 112 miles. h m
P.M.
Ist component (24 hours) ... 106 0 50
2nd “a (12 hours) ... "057 10 5
3rd = ( 8 hours) ... 010 5 3.
4th = ( 6hours) .. 006 4 0
With regard to the after-shocks of Japanese earthquakes,
the records are of such short duration that satisfactory results
can only be obtained by a detailed comparison during the
intervals over which they extend. It is worthy of notice,
however, that the epoch of the first seismic component occurs
with some persistence early in the day, and does not differ very
greatly from that of the first component for barometric pressure.
It seems evident, therefore, that we cannot attribute the
diurnal variation of seismic frequency exclusively to that of
barometric pressure or of wind-velocity. But it is not im-
probable that it may result from a combination of both
phenomena ; that the diurnal periodicity of ordinary earth-
quakes may be due chiefly to that of wind-velocity, and the
diurnal periodicity of after-shocks chiefly to that of baro-
metric pressure. In support of this suggestion, it may be
mentioned that the amplitudes of the first component of wind-
velocity and earthquake-frequency at Tokio are both greater
in summer than in winter, and that the same amplitudes for
the whole year are both greater at Manila than at Tokio.
In the case of after-shocks, a reason may be given for the
efficacy of barometric pressure if, as seems not improbable,
the principal earthquakes were due to the sudden elevation of
one rock-mass adjoining a fault. In a movement so abrupt
as this must be, the disturbed mass would almost certainly
overshoot its position of equilibrium, and immediately after-
wards would begin to settle back to it by a succession of
minor slips, at first numerous but gradually becoming less
frequent along the whole line of displacement. In such a
condition the elevated rock-mass would, in some part or other,
be constantly on the point of giving way, and a slight increase
of barometric pressure would probably be sufficient to pre-
cipitate a slip resnlting in an after-shock.
Imperial Meteorological Office at Tokio. The hourly means for the
Italian stations are given in Buchan’s ‘Atmospheric Circulation,’
pp. 17,18. The hourly means of wind-velocity for Tokio are obtained
from the ‘ Report of the Meteorological Observations for the ten years
1876-1885 made at the Imperial Meteorological Observatory of Tokio’;
those for Manila from the ‘ Osservatorio Meteorolégico de Manila,
Observaciones verificadas’ (1890-93).
Dr. GC. Davison on the Diurnal
A474
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475
of Karthquakes.
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476 Mr. W. Sutherland on Thermal
Hour. Velletri. Verona. | Vesuvius. Italy. |
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XLIX. Thermal Transpiration and Radiometer Motion.
By WituiaAM SUTHERLAND*.
(Continued from p. 391.]|
Part I].—Radiometer Motion.
I HYNOLDS, in treating of radiometer motion in con-
nexion with his discovery of thermal transpiration,
showed that fundamentally both phenomena are traceable to
the same general cause: the object of the rest of this paper is
to bring out this fact more clearly, and to establish theoretically
the general laws of radiometer motion for comparison with
the experimental results of Crookes and Pringsheim.
ln the theory of thermal transpiration, we have seen that
under suitable conditions the variation of temperature along
a passage through a porous plate can produce a certain
difference between the pressures at its hot and cold ends, and
iherefore the solid wall of the passage must be exerting a
tangential force F from cold to hot, such that, R being the
mean radius of the passage,
P= 7R*(pe—pi),
and, accordingly, the gas in the passage exerts a tangential
* Communicated by the Author.
Transpiration and Radiometer Motion. ATT
force F on the solid from hot to cold. If then a porous plate
had one face heated and was hung on to a string with this
face and the opposite cool one vertical, the tangential force
F acting along all the passages would deflect the string from
the vertical, a case of radiometer motion ; if two such plates
were mounted in a vertical plane and free to revolve round a
vertical axis lying between them, and one face of one was
warmed by irradiation, it would move away from the source,
and thus a continuous rotation could be kept up as in an
ordinary radiometer.
In discussing thermal transpiration we confined our atten-
tion to fine tubes, such as might represent the passages in
porous plates; but as we saw that the phenomena depended
for the most part on the ratio of the radius of a passage to the
mean free path of the gas, it follows that our deductions for
fine tubes will hold for tubes of any size with rare enough
gas to give a free path as large as may be necessary ; thus
with the means of getting high enough vacua and with
delicate manometers it should be possible to demonstrate
thermal transpiration along an ordinary gas-pipe or the
largest gas main ; in the radiometer we have generally to do
with thermal transpiration going on in spaces of ordinary
size.
We have already obtained in (5) an expression for the
traction exercised on the gas in the tube by the whole surface
of a tube along which the temperature varies ; thus
EF =nmu’rR’ ;
a curious result that the traction on the surface should be
proportional to the square of the radius, but it is to be
remembered that the tube is supposed to be long enough in
comparison with its diameter, and of sufficient thermal
capacity, to dominate the temperature of the gas so thoroughly
that the temperature throughout any section of the tube is the
same as that of the wall. ‘his traction has been found for
the case when the motion due to thermal transpiration along
the tube has become steady ; but in connexion with radiometer
motion it is necessary to consider the traction before the
steady state is established. Imagine a solid surface over
which the temperature varies to be suddenly introduced into
a mass of gas at rest and uniform in temperature, and let us
determine the traction which the solid immediately exerts on
the gas. The first effect is to make the layer of gas in contact
with the solid take the temperature of the solid at every point
of the surface, and therefore each molecule that encounters
the surface acquires on the average the velocity u given by
478 Mr. W. Sutherland on Thermal
(4), and the number that encounter unit surface in unit time
being nv/4, the total momentum imparted to the gas in unit
time by a surface S is given by
F=Snnwu/4. . 2. >
This is the initial yalue of the traction ; but as the velocity u
is carried out to the remoter parts of the gas, a molecule
which encounters the surface having come from a region
where it had already acquired a fraction of u does not receiye
the whole of u from the solid, and therefore the traction
diminishes with time. To determine the final value when the
motion of the whole gas is steady we may consider the simple
case of two parallel planes the variations of temperature over
which are such as to produce velocities u, and w in a fixed
direction in the gas in contact with the two surfaces ; then
in the steady state we may suppose the transition from wz to
u, to occur linearly, so that the velocity u at distance x from
one of the planes is wy—(uj—w,)a/D, where D is the distance
between the planes ; then the mass of gas that flows in unit
time along any layer of width 6 and thickness dz is nmbu dz,
and the momentum imparted in unit time to the layer is
nmbu? dz, and therefore the total momentum acquired by the
gas between the planes is
ninbu? da =nmbD (u,? + uta + U2”) /3.
0
Obviously the planes impart the respective fractions
uy?/(uy*+Us”) and wy?/(u,?+us”) of this, so that the traction
per unit area of the first plane, if its length in the direction
of motion is J, is
mmDrey? (ry? + Uytty + Up") |
dL (uy? + uy”) “
but it is really a useless artificiality to consider the traction
per unit surface, as most of the traction is really exerted on
the gas near its entrance to the space between the planes, and
we will therefore confine our attention to the total tractions.
As before in the case of the tube, the result that the traction
should be proportional to the sectional area between the plane
is peculiar, but it is true only when the planes dominate the
temperature of the gas in such a manner that w is a linear
function of the distance from either. Thus the initial total
traction on the first plane is Snmvu/4, which is proportional
to the surface S, that is to both width and length, but inde-
pendent of distance from neighbouring surfaces; and the
final traction in the steady state is
nmbD (iy? + uytts + Ug”) Uy?/3 (uy? + U2”),
Transpiration and Radiometer Motion. 479
which is independent of the length but proportional to width
and distance from neighbouring surface. To bring out the
full signification of these we had better introduce the value
of uw, and let us suppose w, to be 0; then the initial and final
total tractions exerted by a plane of varying temperature at
distance D from a plane of constant temperature are
—Snmv®A(n![n+v'/0) 24, (21)
dnmDv*A%(n//n + v//v)9/108. J
When the conditions are such that the pressure can remain
constant between the surfaces, n//n-+2v’/v=0, and then these
become Snmvav’/24 and bnmDdr?2v2/108, both acting from cold
to hot, and therefore the equal and opposite reaction of gas
on the surface is from hot to cold.
But when the conditions are not such that the pressure is
kept constant, but that a difference of pressure is established
by thermal transpiration which goes on till a steady state is
established, the effect of the difference of pressure may be
much greater than that of the traction, as the following
example will show :—A piston is inserted into a cylinder which
it does not quite fit, and is fixed immovable so as to leave a
clear space of sectional area a between itself and the cylinder,
and the cylinder is closed ; when one end is heated a fall of
temperature gets established along the cylinder, and the gas
at the cold end begins to transpire through the narrow space
into the hot end until the difference of pressure p.—p,
sufficient to stop the flow is established; then the total traction
of gas on the side of the piston is a(p,—p,), while on the area
A of the hot end of the piston there is a total pressure
A(p.—Pp1) in excess of that acting on the cold end, so that the
total force urging the piston from hot to cold is (A + a)(p,—p),
which may be made as much more important than the total
traction as we please by diminishing a. If the piston is freed
it will begin to move under the force (A+a)(p,—), and
become an exaggerated instance of radiometer motion.
This example makes clear the lines on which to formulate
a general theory of radiometer motion; we have only to
adapt our transpiration formulee established for circular tubes
to the case of any space bounded by solid walls over which the
temperature varies. In the general problem of radiometer
motion we have to do with a solid surface over which the
temperature varies and which therefore is subject to a traction
from hot to cold, and also establishes a higher pressure in the
gas towards its hot end than at the cool end; the relative
importance of traction and difference of pressure in producing
motion of the body to which the surface belongs depends
480 Mr. W. Sutherland on Thermal
entirely on the relative position of the movable surface and
the fixed surfaces surrounding it. We have seen that for a
tube the total traction is nmu?mR?, and for a passage bounded
by two parallel planes of width b and distance D apart with
the same variation of temperature along both, it is nmu?)D/2
along each plane, and as wR corresponds to 6 and R to D/2
we see that the traction exerted between the walls of a cylinder
of any section and the contained gas may be written in the
form nmu?sD/2, where s is the perimeter of a right section of
the cylinder, and D is a mean value of the distance between
opposite parts of the perimeter. For the difference of pressure
established by thermal transpiration along a tube of any
section we may use the equations (14) and (17) if in them we
interpret 2R as a mean value of the distance between opposite
parts of the perimeter ; and in the case where there is a
variation of pressure across only a fraction of the perimeter,
as for instance in the case where one plane wall has a varying
temperature and the opposite one a uniform temperature, we
must multiply by a fraction not greatly different from that
fraction (1/3 instead of 1/2 in our example). We can there-
fore state the fundamental equations of radiometer motion as
follows:—If across a length 0 of the perimeter s of any
cylinder a variation of temperature is suddenly established
whose average rate is v’ over a length J, then the initial total
traction between solid and gas is approximately
blnmdoy'/24..- "'. 9 eee
When the steady state is reached the total traction is
approximately
bDamd2v? (n’//n + v’/v)?/108, or bDamar?v?(p'/p —v'/v) 2/108,(23)
and the difference of pressure between the two ends of / is
_ bv — ry 1
PrP see A at pl/t t+ B+ 1/(Pat Ph
where the values of A’ and B’ are those given in (19), with
R,=R,=R. No proof has been furnished here that the
introduction of the fraction b/s rigorously adapts our expres-
sion (19) to the case where only a fraction b/s of the boundary
is operative in producing thermal transpiration, but it is a _
reasonable enough approximation for experimental results at
present available, closer approximation could easily be caleu-
lated if required. If in (23) we write (p,—p1)/l and (ve—v,}/l
for p’ and v’ we can express the total traction in the steady
state entirely in terms of v, and v, which completes the
solution.
» » (24)
- OF
Transpiration and Radiometer Motion. A481
With these results we can now state what ought to be the
behaviour of a radiometer, and as Crookes and Pringsheim
found the best form of instrument for investigating the laws
of the radiometer experimentally to be one in which a single
vane of mica blackened on one side was attached with its
planes vertical to a horizontal arm attached to a vertical
torsion fibre, the whole being suspended in a glass bulb
capable of being filled with any gas at any pressure, we will
discuss the theoretical laws of such a form. Let D be the
mean distance of the edge of the vane from the glass wall
immediately opposite it, b the perimeter of the vane, s—6b the
perimeter of the glass wall opposite, Hi the area of each face
of the vane, H +S the sectional area of the bulb (8 being small
compared to H) in the plane of the vane, and @ the thickness
of the vane; when the black face is irradiated let its tempera-
ture become 6,, that of the clear face being 6,, then there is a
fall of temperature 0,—6, through the thickness of the vane,
and thus the thickness of the vane becomes a surface capable,
along with the surface of the bulb opposite it, of starting
thermal transpiration from the cold edge to the hot, with
elevation of the pressure in front of the hot face to po, and
depression of that behind the cold face to p,;; when a steady
state is established the total traction on the surface of varying
temperature must be approximately equal to S(p.—p,)b/s, and
the excess of total pressure on the black face over that on the
clear face is E(p,—p)6/s, so that the total force deflecting
' the vane whose moment is to be balanced by the torsion
couple of the fibre is (H+8)(p,—p,)b/s. Thus the total
deflecting force is
ay OM 1
(E+8) SU +% 9 D?v9?( po + 71) 3aDv Ste (25)
1679?(v2+ v1)* No(Vet+r;)*> Pot Pr
c/{A’p + B’+1/p}, where p is the mean pressure (p.+ ,)/2.
This equation contains all the theoretical laws of radiometer
motion when 8 is very small compared with H. If everything
is kept constant except the mean pressure (p2+ ;)/2 there is
a value of the mean pressure of the gas in a radiometer for
which the deflecting force is a maximum, a very important
point in radiometer construction. When the pressure is high
enough the last two terms in the denominator may be neglected
and the deflecting force is inversely proportional to the
pressure, and when the maximum is passed and the pressure
becomes small enough the first two terms may be neglected
and the deflecting force becomes proportional to the density
482 Mr. W. Sutherland on Thermal
and dies away indefinitely with increasing exhaustion of the
bulb.
It will be seen that the deflecting force depends on the
dimensions of the apparatus in a somewhat complicated
manner, but that the most important principle is that (as
regards the denominator) it increases with diminishing dis-
tance D between vane and glass wall, except at pressures so
low that the first two terms are negligible ; and as diminish-
ing D means in general an increasing value of b/s, we see
that in general at all pressures the efficiency of the radiometer
is increased by bringing the edge of the vane nearer to the
glass wall. Other things being equal the deflecting force is
proportional to the total sectional area H+S of the bulb.
As regards the effect of the nature of the gas on radiometer
motion the equation shows that at pressures low enough for
neglecting the first two terms of the denominator all gases
give the same deflecting force, a theoretically interesting
result, but not of much practical importance: the practically
important matter is to determine how the different gases com-
pare, each at its maximum effectiveness; now when the
deflecting force is a maximum p+ p,=4%(v2 +v;)7/3DuQ, or
D=2n where A is the mean path at (p.+>p,)/2, and the
deflecting force becomes proportional to 4n9(v,+v,)?/3Dv, so
that the most effective gas is that for which (v,+1,)?/v is
largest, that is to say, for which )/m? is largest ; compare for
instance hydrogen and oxygen, 7 for H, is °44 of that for Og,
while m? is 1/4, and thus at the pressure of maximum efficiency
H, is 1°76 times as efficient as O,, and at higher pressures the
advantage of H, increases till its efficiency is 1:76? that of Ox.
From Rayleigh’s measurement of the viscosity of helium as
‘96 of that of air (Proc. Roy. Soc. Jan. 1896) while hydrogen’s
is about ‘5, and with 4 as the molecular mass of helium as
against 2 for hydrogen, it would appear that helium ought to
be nearly 22 or 1:4 times as efficient in a radiometer as
hydrogen.
The equation (25) contains the laws of the dependence of
radiometer motion on the temperatures of the faces of the
vanes, although as these temperatures have never been
measured experimentally, we cannot verify them as they stand ;
but to a certain extent we can bring them within the range
of experimental verification in the following manner. When
the black face of a vane is suddenly irradiated the tempera-
ture of the black face suddenly rises, while that of the clear
face is unaltered, and the fall of temperature is confined
for the first moment or two to the thickness of the layer of
lampblack ; the first deflexion of the vane takes place in
Transpiration and Rudiometer Motion. 483
accordance with these conditions, but if the vane is steadily
irradiated, conductivity soon establishes a steady distribution
of temperature through the vane from front to back with
permanent temperatures 0, and 6,, so that the first deflexion
alters until it attains the fixed value due to these steady con-
ditions. Suppose the black face to be irradiated by a candle,
and let Q be the amount of heat it absorbs per unit area per
unit time, and E the corresponding amount emitted by the
clear face ; then if we ignore loss of heat by the edge, and
denote the conductivity of the substance of the vane by &,
Oh (Co— Ooi aye ee (ZU)
Now in (25), (v.—v,)/(v2+%) may be written
(v9? — v;7) /(ve +01)’;
(82.—4;) / (03+ 63)’,
in which the denominator is nearly 4(@,+,), since the diffe-
rence of 6, and @, is small ; and as in the experiments (0, + 6,)/2
was always an ordinary temperature it may be taken as con-
stant, so that (v,—¥v,)/(ve+,) was always closely proportional
to 6,—6, and therefore to Q6/k; but Q is inversely as the
square of the distance of the candle from the vane and directly
as the number of candles at that distance (the candles being as
nearly as possible in the normal to the centre of the vane),
thus the deflecting force varies as the number of candles and
inversely as the square of their distance from the vane, which
is the experimental result (Crookes, ‘ Nature,’ xiii.) ; the
deflecting force is proportional to the thickness of the vane
and inversely proportional to the conductivity of its material,
hence the advantaye of using a substance such as mica for the
vane, and the disadvantage of using metal.
To show that this theory of the radiometer is in harmony
with the experimental facts, we will briefly describe the
general results of Crookes’s numerous experiments, and it
will be seen that they accord with the deductions from our
formule.
Crookes obtained his most valuable quantitative results
with an apparatus such as the one of which we have just
considered the theory. The bulb was continued into a
vertical tube for containing a torsion fibre nearly a metre
long, and the rectangular plate of roasted mica was attached
directly to the fibre so that its plane was vertical and its
centre at the centre of the bulb; a continuation of the line of
the fibre divided one face of the plate into two equal halves,
or
484 Mr. W. Sutherland on Thermal
one of which was lampblacked. For all the experimental
niceties reference must be made to the original paper (Phil.
Trans. elxxii.)
It is obvious from the description of this apparatus that it
does not comply with the conditions under which (25) was
established, as the mica plate is probably only a fraction of a
millimetre in thickness and between 5 and 10 millim. from
the glass bulb where it is nearest, so that the length of the
region in which thermal transpiration occurs is much less
than its width, whereas in (25) the contrary is supposed to
be the case. The chief effect of the difference in these con-
ditions will be that thermal transpiration, instead of going on
over the whole distance between edge of plate and bulb, will
extend to a distance from the edge of the plate which will
depend on the conductivity of the gas; in fact, if we move
along the shortest distance between plate and bulb we shall
find the fall of temperature across that line grow less as we
leave the plate and become negligible before we reach the bulb ;
but the better the conductivity of the gas the farther will the
dominating influence of the edge of the plate extend; there-
fore in our formule, when applied to Crookes’s experiments
with the torsion radiometer, D must be interpreted as a
function of conductivity kh’. Then b being the length of the
edge of the black half of the plate, the area S over which
thermal transpiration is effective may be taken to be 6D, over
which at the front and the back of the plate there is an
average difference of pressure pg—p,, which, however, will
not be maintained over the whole front and back of the plate,
because there is so much facility of escape for the gas, but
only near the edge, so that probably H varies as 6D; thus
(H +8)6/s will be replaced by 0K, where K is a function of
k’. Another effect of the fact that thermal transpiration
occurs only to a certain distance from the edge of the plate
will be to reduce the effect of slipping, seeing that the velocity
of transpiration dies away to zero in the gas. To indicate
that slipping has not its full theoretical effect we had better
change B’ to B”, and to remind ourselves that in A’ and B/
the symbol D or 2K now means a function of k’, we will
change A’ to A” and B” to B” and put
bK(v.— v)/(ve o- V1) = Be
then (25) becomes
deflecting force=c'/(A”p+B”+1/p). . . (27)
There is no need to take account of molecular force in
altering density at edge of plate because so small a fraction
of the free path lies in the condensed gas.
Transpiration and Radiometer Motion. 485
The last point to be attended to in applying our equations
to the experimental results is that when one side of the mica
vane is irradiated the glass bulb is also warmed in such a
manner that it is hottest where nearest the candle, and there-
fore there is thermal transpiration along the inner surface
of the bulb tending to raise the pressure near the hottest
point with diminution towards the coldest point ; now we
can afford to neglect the effect of this near the vane until
the pressure gets so small that the mean free path of a
molecule becomes, say, nearly equal to the radius of the
bulb, for then the walls of the bulb, on account of their
much greater area than that of the effective edge of the
vane, must dominate the distribution of temperature and
pressure in the gas even quite close to the vane, and there-
fore at the highest exhaustions the relation between pressure
and deflecting force must tend to a limit determined rather
by the bulb than by the vane. With these explanations (27)
is now applicable to the experiments of Crookes.
With his apparatus Crookes was able to study concurrently
the viscosity of a yas and the forces at play in the radiometer
at pressures from one atmo down to the lowest measurable
by the M‘Leod gauge. The form of his vibrating system
renders the mathematical problem of obtaining an expression
for the viscosity of the gas from the constants of the apparatus
and the observed decrement per vibration of the logarithm
of the amplitude of the vibrations of the mica plate intract-
able ; but it is obvious, from the theory of the vibrating disk
method of measuring viscosity, that the motion of the mica
plate when oscillating must be retarded by the viscosity of
the gas in such a way that the difference of the logarithms of
successive amplitudes is proportional to the viscosity of the
gas, so that although absolute values of viscosity are un-
obtainable with the apparatus, approximate relative ones can
be got with it. Ata number of different densities of the gas
Crookes measured the logarithmic decrement and also the re-
pulsive effect of a candle-flame radiating towards the blackened
half of the mica plate from a horizontal distance of half a metre,
the latter being measured by a reading of the permanent
deflexion of the plate from its position of rest in darkness.
Now from Maxwell’s well known discovery that the
viscosity of a gas is independent of its pressure it follows that
the logarithmic decrement is independent of the pressure so
long as slipping of the gas on the solid surfaces is negligible ;
but, as already indicated, Kundt and Warburg showed
experimentally, with some support from theoretical reasoning,
that slipping ceases to be negligible when the mean free path
Phil. Mag. 8.5. Vol. 42. No. 259. Dec. 1896. 2M
486 Mr. W. Sutherland on Thermal
of the gas beconies comparable with the linear dimensions of
the containing vessels; they did this by pushing the rare-
faction of the gas so high in a vibrating-disk apparatus for
measuring viscosity that the logarithmic decrement diminished
measurably—for example, with air and a distance of ‘11 em.
between the fixed and moving plates the log. dec. at 1 atmo
was 132, at ‘01 atmo it was °129, and at :0008 atmo it was
"111; now at 20° C. and at these pressures the mean free
path in air is about ‘00001 cm., :001 cm., and ‘012 em.
respectively, this last value is nearly 1/10 of the distance
between the plates, so that when the distance between the
plates is only 10 free paths the log. dec. diminishes by 16°
per cent. of its limiting value when the distance is a large
number of free paths. Thus we see how the measurements
made by Crookes of the log. dec. in his apparatus give
valuable information as to the relation between the free
path of the gas and the distance from the edge of his mica
plate to the glass bulb. In the following table the first
row contains the pressures of dry air at 15° C. in terms
of the atmo as unit, the second gives 10* times the log. dec.,
the limiting value of which at higher pressures is 1000, and
the third contains the deflecting force of the candle in an
arbitrary unit :—
IECSS: sein g3i ase 736 495 300 100 72 39
LO oe dees a..25.: 975 966 952 876 824 710
3°5 aes) 10-0 27:0 | S250 eee
def. for. he ne =
eal. 4-0 59 96 250. 3-3
presse falreeGl te. 36 29 19 13. 0
10* log. dec... ..... 605. «657 TT 500 460
OXp. §--5 goo 42-6 38'8 30:9" =27ek
def. for.
Weal. =... 407 40:8 36'8 30°70 =. 268
Thus at a pressure between 36 and 29 millionths of an
atmo the repulsion rises to a maximum, say at 30 millionths,
at which the mean free path is 00001 x 10°/30, that is one-
third of a centimetre. The actual distance between the edge
of the mica plate and the bulb is not given by Crookes, but
from the figure he gives one would imagine that the distance
might be between *5 cm. and 1 cm., and thus the experi-
mental result corresponds to our theoretical one that the
maximum effect is to be expected when D=2); before the
maximum is reached the repulsive effect ought, according to
(27), to vary inversely as the pressure, so that the products
of the numbers in the first and third rows above ought to be
approximately constant, and the first four products are 2600,
Transpiration and Radiometer Motion. 487
2700, 3000, and 2700, which verify the equation. Beyond
the maximum, according to (27), the repulsion is ultimately
to vary directly as the pressure, so that the numbers in the
third row divided by those in the first are to tend towards
constancy : the last three values are 2:0, 2:4, and 2°5, while at
lower pressures the value 3 is reached; but the results at
these lower pressures have not been reproduced in the last
table, because the M‘Leod gauge with air becomes less re-
liable towards 1/10° atmo, and therefore Crookes’s results at
the lowest pressures will be discussed in a separate paper on
the measurement of pressures in the highest attainable vacua.
From Crookes’s experiments we can calculate ce’, A”, B’”
in (27), for with 1/10° atmo as unit of pressure and Crookes’s
arbitrary unit as the unit of repulsion, we have just seen that
e/A” is about 3000 and ¢ about 3:0, so that A”=-001 ; now
the deflecting force is a maximum when p?=1/A”, so that
the maximum value of the deflecting force, namely,
[f(2+B" VA"),
gives a convenient method of finding B’” when A” and c’ are
known ; thus for air, B'’’=-0l and we have all the data for
calculating the deflecting force at any pressure by (27) for
comparison with experiment: the calculated values are given
in the fourth row of the last table, and show that we have
the correct form of equation to represent the experimental
facts. But according to the meanings of A” and B’, B?/16
should be nearly equal to A”, whereas B”’”/16 is only the
1/160 part of it. }
Now the term in B’” expresses the effect of slipping, and
our results for air show that in Crookes’s apparatus the effect
of slipping is only 1/160? or 1/13 of what it would be under
the ideal conditions for (25), indeed (25) with the given
values of A’ and B! stands for one limiting case, and with
B/= 0 it stands for the other where slipping is of no account,
and the conditions of Crookes’s experiments are nearer to
those of the latter limit than of the former; indeed, with
slightly different values of c’ and A” we could put B’/”=0
and get nearly as good a representation of the experimental
results for air as that just obtained. For nitrogen the values
of the repulsive force are about two-thirds of those for air at
the same pressures, except in the case of the small values,
which are somewhat unreliable, thus for nitrogen A’ and B/”’
have about the same values as for air, while ¢’ is about 2:0;
now according to equation (27) c’, as it depends only on the
dimensions of the apparatus and the temperatures of the two
2M 2
488 Mr. W. Sutherland on Thermal
faces of the mica vane and conductivity, ought to have nearly
the same value for two gases so closely alike as air and
nitrogen ; that is of course on the assumption that the value
of c’ for oxygen is not much different from that for nitrogen,
but we had better delay the discussion of this curious point
- until we have considered the data for oxygen.
For CO, and CO the parameters are :—
ce ee ie
COME meee 1°25 000625 0-0
COME tom. 1°32 "000625 0-01
In the case of CO these values give values of the repulsion
or deflecting force agreeing closely with the experimental
over the whole range of pressure, but for CO, the calculated
values are larger than the experimental at the higher pres-
sures; but the matteris hardly worth going into more closely,
especially as oxygen and hydrogen show exceptional behaviour
of the highest interest to which we will proceed.
For oxygen Crookes obtained the following, the pressure
unit being 1/10° atmo, and the unit of repulsion the same
arbitrary one as before :—
Sia SS OUCOORE NEE 1000 803 658 623 613
HOt logsdec. 2a. 1102 1093 1088 1086 1085
def. force ......... 12 12 13 13 13
Dees Sees eee 369 297 190 171 110
1.0% log: dee:)-7.-.. 1070 1058 1038 1033 988
deiiOrce as seee 13 14 20 21 3l
where the deflecting force remains almost constant from a
pressure of 1000 down to 297, after which it rises, and at lower
pressures than those given attains a maximum and then dimi-
nishes. Now Bohr (Wied. Ann. xxvii.) discovered 2 remark-
able discontinuity in the compressibility of oxygen at about
921/10° atmo, which has been corroborated by Baly and
Ramsay (Phil. Mag. [5] xxxviil.), and obviously the above
anomaly must be traced to the same cause as the discontinuity.
These phenomena are so important for the chemistry of oxygen
that I will discuss them in a separate paper on “ Sponianeous
Change of Oxygen into Ozone, and a remarkable type of Dis-
sociation.” Meanwhile we will go on to the region of pressure
in which the repulsion in oxygen is not exceptional; here we
have e’=3'0, A”’=:0007, and B’”=-:0182, which give the
following comparison with the experimental results :—
Transpiration and Radiometer Motion. 489
gate en 297 190 VTE 110 70 48
Basi eee 13 20 22 30 40 44
exp. 14 20 21 Sh 13g Ais
pri Werte 31 28 22 16 12
f cal. 45 44 40 34 29
JES UECGE 44 44 40 35 30
The perfection of the agreement here emphasises the distinct-
ness of the exceptional behaviour at the higher pressures.
In the case of hydrogen we have at the higher pressures
the following values, the numbers in the third row being the
products of repulsion by pressure, which ought according
to (27) to be tending to a fixed limit :—
Beata once 1000 921 526 421 330
def. force ...... 1] 1 3 4 5
1000 921 1578 1684 1650
(i ieee eee 314 234 205 179
def. force ...... 8 ll 14 18
2512 2574 2870 3222
The products do not show the same approach to a limit as was
the case with air, and there is a jerkiness in their variation
which points probably to experimental uncertainty. At the
lower pressures the quotients of repulsion by pressure, which
ought to be tending to a limit, are given in the third row of
the following :—
55 eee 16 14:5 12 8 6:5 5 4
def. force ...... 52 49 45 37 3l 29 26
3-2 3°4 3°7 46 4-8 58 6°5
Here again the convergence to a limit is not satisfactory, a
state of affairs which will be traced in the separate paper on
measurement of low pressures to inaccurate values of the
lower pressures ; and in that paper it will be shown that
hydrogen exhibits a peculiarity which expresses itself in our
equation (27) by dividing A” by (1—ap) where & is another
parameter, thus for hydrogen the deflecting force is
e'/{Ap/(L—ap) + BY” + 1/p}, - « (28)
which makes the deflecting force 0 when p=1/a, a result to be
extended to all higher pressures ; the values of the parameters
ako, —O00G, a——-0016,, BY’ =-01,, and ¢’=4-16.. whien
give the following comparison :-—
490 Mr. W. Sutherland on Thermal
Phe oe 2 eee -4000 - 526 380 914 205 Tape
Foals ae: 0 2 510 “41 > 218 Some
Bek force | eee 1 3 5 8. 14 ogee
pees 59. 4 41. 28265 900 | ee eee
ness ee Le 63 . 70 >) 64 <8 +449) —eaneeeee
exp: tee 64 70 66 58 45 of. ze
in view of the experimental uncertainty already pointed out
at the high pressures and that which is to be proved at the
lowest pressures, this comparison shows that the modified
formula represents the facts for hydrogen about as well as
possible.
We will now compare the theoretical values of the para-
meters c’ and A” with the numerical ones just obtained; ¢
stands for bK(v.—7,)/(ve+1v;), in which K is proportional to D
and is a function of k’; also (v,—v,)/(ve+v,) is the same
for all the gases, so that c’ is proportional to D. But
A” =9 D2u,?/167?(v.+,)4, and v varies as m-?, and therefore
(A’’y?/m)? is proportional to D, which now means distance
from edge of vane to which transpiration extends. The
following table contains 107K’, c’, and (A’’7?/m)? for com-
parison ; 7 in terms of that for O, as 1, and min terms of that
for H, as 2 are appended :—
H,. Air. O.. N,. CO. Cas
Titan a eats 3324 480 563 524 510 317
eee SO 416 3-0 3-0 2-0 1:32 1:25
(A"'n?/m)? 0... ‘762 520 468 “520 ‘411 284
Bete eee 44 90 1-0 ‘87 ‘87; is
Rae ne eY 2 28°8 32 28 28 44
It can be seen that on the whole these numbers confirm the
theoretical conclusion that thermal transpiration near the
edge of the plate is effective to a distance which increases
with the conductivity. The smallness of ¢ for N, as com-
pared to its value for air and the smallness of c’ for CO are
points that require confirmation by experiments with an
apparatus lending itself better to quantitative calculations than
the torsion radiometer of Crookes.
The values of B’’, as they represent only a small amount of
slipping, and are not given very definitely by the experiments,
are not worth further consideration.
So much for what may be called the static form of the
radiometer ; of the results obtained from a great variety of
moving radiometers constructed by Crookes the following are
the most important. In a radiometer containing two flies,
one pivoted above the other, and haying their blackened
Transpiration and Radiometer Motion. 491
sides facing in opposite directions, the radiation from a candle
causes the flies to revolve in opposite directions, which proves
that the driving action is chiefly localised close to the flies ;
this result is of course involved in our theory according to
which the action of the fall of temperature through each vane
is to raise the pressure near its hot face and lower it near the
cool face, but the region of lower pressure of the upper fly
being just above the region of higher pressure of the lower
fly, and, with no obstruction between, ought to produce dissi--
pation ot the driving power of both flies, so that although
they move in opposite directions they hinder one another in
this direct manner as well as through the viscosity of the gas.
Another radiometer contained only one pair of vanes at the
ends of a single arm, and each vane carried opposite to its
black face, at a distance of a millimetre, a large disk of thin
clear mica; the action of a candle on this was to cause
rotation in a direction opposite to the usual, that is, the black
face moved towards the light. When another disk of thin
clear mica was attached opposite the other side of each vane
a candle ceased to have any effect. The theoretical reason for
these facts is clear; in the first case the region of high
pressure set up near the edge of the black face of the vane has
more effect on the clear plate than on the vane and in the
opposite direction, so that there is a resultant differential
pressure driving the vane and its attachments in the opposite
direction to the usual ; when the other clear plate is attached
there is an equal opposite resultant differential pressure due
to it and so there is equilibrium ; in short, when the two clear
plates are attached the whole action is confined to the space
between them, so that there can be no motion of the whole
system.
In another radiometer the four vanes were left clear, but at
the side of the bulb a plate of mica blackened on one side was
fastened in a vertical plane passing through the centre of the
bulb, so that a vane in passing it would leave a clear space of
a millimetre: when light is thrown only on the clear vanes
there is no motion, but as soon as it is allowed to fall on the
fixed plate the fly revolves as though a wind were blowing
from the black surface. ‘This follows from theory at once, as
the edge of the black face becomes a region of higher pressure
and therefore a source of wind.
On replacing the pith or mica vanes by metallic ones
Crookes encountered some new phenomena; perfectly flat
aluminium vanes were found to be much less sensitive to the
light of a candle than mica or pith ; they move in the same
direction, that is with the black surface away from the light,
but when the candle is replaced by a source of dark heat their
492 Thermal Transpiration and Radiometer Motion.
motion is reversed, which is not the case with mica and pith
(of course we are speaking of forms in which two or more
vanes are arranged symmetrically with regard to the pivot) ;
this reversal simply shows that the metal is a better absorber
of dark heat than the lampblack.
But in working with vanes made of gold-leaf Crookes
noticed that while the blackened side of one vane appeared to
be repelled by a candle, that of another appeared to be
attracted, and on examination it turned out that while the
former vane was flat the latter was crumpled and bent in
such a manner as to present a concave surface to the light.
Following up this clue by constructing radiometers with bent
and curved vanes Crookes was able to prove that in radiometer
motion shape of the vane exercises even more influence than
the absorbing power of the surface, so that a convex bright
surface appears to be strongly repelled by a source of light,
while a black surface if made concave to the light is actually
attracted by it.
The theory of curved vanes is simple: consider a convex
vane irradiated by a source on the normal through its middle
point ; then, as the amount of heat that a surface absorbs
depends on its obliquity to the incident radiation, the farther
a part of the convex surface is from the middle the less is it
directly heated, and thus there is a continuous fall of tempera-
ture from the centre of the surface to the edge; conduction,
if allowed time, tends to reduce the amount of the fall but
does not obliterate it, and conduction also establishes a fal! of
temperature along the back from centre to edge; now the
traction of the gas on the solid is trom hot to cold, so that
both on the front and the back of the vane there is a traction
from centre to edge whose resultant effect is to drag the vane
away from the light when the vane is convex to it, so that the
light appears to repel a convex surface ; when the surface is
concave the same reasoning applies, the gas exerts a traction
from centre to edge, and therefore the light appears to
attract it.
There is hardly any need to reproduce any more of Crookes’s
facts or Pringsheim’s skilful experimental analysis of the
parts played in radiometer motion by bulb, vane, and gas ;
enough has been given to show that the kinetic theory can
account qualitatively and quantitatively for all the essential
facts of radiometer motion and furnishes general principles
for the design of apparatus of the radiometer type. An
illustration of the application of these principles will be given
in a separate paper on “Two New Pressure-Gauges for the
Highest Vacua.”
Melbourne, August 1896.
po aoa
L. Theoretical Considerations respecting the Separation of
Gases by Diffusion and similar Processes. By Lorp
RayueieH, Sec. R.S.*
ae larger part of the calculations which follow were
made in connexion with experiments upon the concen-
tration of argon from the atmosphere by the method of
atmolysis +. When the supply of gas is limited, or when it
is desired to concentrate the lighter ingredient, the conditions
of the question are materially altered ; but it will be con-
venient to take first the problem which then presented itself
of the simple diffusion of a gaseous mixture into a vacuum,
with special regard to the composition of the residue. The
diffusion tends to alter this composition in the first instance
only in the neighbourhood of the porous walls ; but it will be
assumed that the forces promoting mixture are powerful
enough to allow of our considering the composition to be
uniform throughout the whole volume of the residue, and
variable only with time, on account of the unequal escape of
the constituent gases.
Let x, y denote the quantities of the two constituents of the
residue at any time, so that —dw, —dy are the quantities
diffused out in time dt. The values of dx/dt, dy/dt will
depend upon the character of the porous partition and upon
the actual pressure ; but for our present purpose it will
suffice to express d y/da, and this clearly involves only the
ratios of the constituents and of their diffusion rates. Calling
the diffusion rates pw, v, we have
dy _vy
eae is oe eee
In this equation z, y may be measured on any consistent
system that may be convenient. The simplest case would be
that in which the residue is maintained at a constant volume,
when 2, y might be taken to represent the partial pressures
of the two gases. But the equation applies equally well
when the volume changes, for example in such a way as to
maintain the total pressure constant.
The integral of (1) is
pee C ar eee!) ys
where C is an arbitrary constant, or
tl Ome at uid iat ane tee (3)
* Communicated by the Author.
+ Rayleigh and Ramsay, Phil, Trans. clxxxvi. p, 206 (1895).
494 Lord Rayleigh on the Separation of Gases
If X, Y be simultaneous values of a, y, regarded as initial,
le <) Sl
ViX= x
so that ao
B/Y— pK
ee x ($e —
In like manner
a/y \v/(u—v)
y= Y (xy) ee
If we write
Xv =
—- =f7T, . A ‘ : - ; - (7)
yr represents the enrichment of the residue as regards the
second constituent, and we have from (5), (6),
ety = x Bi (vB) x iY/e—P)
yy Xey" +xgy? ee
an equation which exhibits the relation between the enrich-
ment and the ratio of the initial and final total quantities of
the mixture.
From (8), or more simply from (4), we see that as 2
diminishes with time the enrichment tends to zero or infinity,
indicating that the residue becomes purer without linut, and
this whatever may be the original proportions. Thus if the
first gas (wz) be the more diffusive (u>v), the exponent on the
right of (4) is negative ; and this indicates that 7 becomes
infinite, or that the first gas is ultimately eliminated from the
residue. When the degree of enrichment required is specified,
an easy calculation from (8) gives the degree to which the
diffusion must be carried.
In Graham’s atmolyser the gaseous mixture is caused to
travel along a tobacco-pipe on the outside of which a vacuum
is maintained. If the passage be sufficiently rapid to
preclude sensible diffusion along the length of the pipe, the
circumstances correspond to the above calculation; but the
agreement with Graham’s numbers is not good. Thus in one
case given by him * of the atmolysis of a mixture containing
equal volumes of oxygen and hydrogen, we have
¥/X=1 50 ye 92-18) G22.
so that r=13 nearly. Thus, if in accordance with the view
* Phil. Trans. vol. cliii. p. 403 (18€3).
by Diffusion and similar Processes, 495
usually held w/v=4, we should have from (8)
: BY
Pals
so that a reduction of the residue to °229 of the initial quantity
should have effected the observed enrichment. The initial
and final volumes given by Graham are, however, 7°5 litres
and °45 litre, whose ratio is ‘06. ‘The inferior efficiency of
the apparatus may have been due to imperfections in the walls
or joints of the pipes. Such an explanation appears to be
more probable than a failure of the law of independent
diffusion of the component gases upon which the theoretical
investigation is founded.
In the concentration of argon from a mixture of argon and
nitrogen we have conditions much less favourable. In this
case
=}x13 °+4x138 = 229;
p/v=V20// 14=:077.
If an enrichment of 2:1 is required and if the original
mixture is derived from the atmosphere by removal of oxygen,
the equation is
ee ee —613 ae
ae Je 2 oe Ol KZ
= 0142 +0029 = 0171,
expressing the reduction needed. The results obtained
experimentally (doc. cit.) were inferior in this case also.
When the object is the most effective separation of the
components of a mixture, it is best, as supposed in the above
theory, to maintain a vacuum on the further side of the porous
wall. But we have sometimes to consider cases where the
vacuum is replaced by an atmosphere of fixed composition, as
in the well-known experiment of the diffusion of hydrogen
into air through a porous plug. We will suppose that there
are only two gases concerned and that the volume inside is
given. ‘The symbols z, y will then denote the partial
pressures within the given volume, the constant partial
pressures outside being a, 8. Our equations may be written
5 ae a AG (9)
dy=v(B—y)dt
or on integration
ta + Ce, y=B+De-, ahr Scare (10)
C, D being arbitrary constants.
After a sufficient time 2, y reduce themselves respectively
to a, 8, as was to be expected.
496 Lord Rayleigh on the Separation of Gases
The constants w, vy are not known beforehand, depending
as they do upon the specialities of the apparatus as well as
upon the quality of the gases. If we eliminate t, we get
y— B=E(z—a)"", . 2
in which only the ratio v/p is involved.
As a particular case suppose that initially the inside volume
is occupied by one pure gas and the outside by another, the
initial pressures being unity. Then in (10)
ae—0,. B=). C= 12 pee
we have .
g=e#, y=1l—e", .. 42
and
ety=lteH—e" . | 2 eee
gives the total internal pressure. When this is a maximum
or minimum, e“—”¥=y/v, and the corresponding value is
es
ae b ITY Dc =
et+y=1+(F) {1 ae i es
Thus in the case of hydrogen escaping into oxygen, p/v=4,
and 2 +y=1—3x 4-#=-528,
the minimum being about half the initial pressure *.
Returning now to the separation of gases by diffusion into
a vacuum, let us suppose that the difference between the
gases is small, so that (v—y)/w=«, a small quantity, and that
at each operation one-half the total volume of the mixture is
allowed to pass. In this case (8) becomes
Nee ve
K
ae ey 7 xy Ss nearly ;
so that roa (a). . » a
This gives the effect of the operation in question upon the
composition of the residual gas. If s denote the corres-
ponding symbol for the transmitted gas, we have
AY Sp (Xe felt es
* (X—a)/K 7 P2a/K~ 1 ax
B (l—r)a/X
anh 1—a/X
* The most striking effects of this kind are when nitrous oxide, or dry
ammonia gas, diffuse into the air through indiarubber. I have observed
suctions amounting respectively to 53 and 64 centimetres of mercury.
=2—r approximately,
by Diffusion and similar Processes. 497
since 7 is nearly equal to unity. Accordingly
ee 1]
‘oo I—r =r nearly,
so that approximately s and 7 are reciprocal operations. For
example, if starting with any proportions we collect the
transmitted half, and submit it to another operation of the
same sort, retaining the half not transmitted, the final
composition corresponding to the operations sr is the same
(approximately) as the composition with which we started,
and the same also as would be obtained by operations taken
in the reverse order, represented by 7s. A complete scheme *
on these lines is indicated in the diagram. Representing the
initial condition by unity, we may represent the result of the
first operation by
ar+3s, or 3(r+8),
in which the numerical coefficient gives the quantity of gas
whose character is specified by the literal symbols. The
second set of operations gives in the first instance
dy? 4 dsr 4+ drs +457,
or, after admixture of the second and third terms (which are
of the same quality),
4 (7? + 278+’) = (
Etsy
2 e
In like manner the result of the third set of operations may
3
be represented by (=): and (as may be formally proved by
' * It differs, however, from that followed by Prof. Ramsay in his
recent researches (Proc. Roy. Soc. vol. lx. p. 216, 1896).
498 On the Separation of Gases by Diffusion.
“induction ”’) of x sets of operations by
(3): , . ae
When we take account of the reciprocal character of ¥ and s,
this may be written
= { oe Ep? SE a ey ea” +o > eds)
the number of parts into which the original quantity of gas is
divided being n+J. If n is even, the largest part, corre-
sponding to the middle term, has the original composition*.
It is to be observed, however, that so far as the extreme
concentration of the less diffusive constituent is concerned
these complex operations are entirely unnecessary. The same
result, represented by (4)",7" will be reached at a single
operation by continuing the diffusion until the residue is
reduced to (4)” of the original quantity, when its composition
will be that denoted by 7”. And evenas regards the extreme
member at the other end in which the more diffusive con-
stituent preponderates, it will be evident that the operations
really required are comparatively simple, the extreme member
in each row being derived solely from the extreme member of
the row preceding f.
If we abandon the supposition, adopted for simplicity, that
the gas is divided into equal parts at each operation, we may
still express the results in a similar manner. If p, o be the
fractions retained and transmitted, then p+oa=1, and in place
of (15) we get yop’. . .04.5 3a
The relation between r and s is
prtos=1;...: . «rn
and the various portions into which the gas is divided after n
sets of operations are represented by the various terms of the
expansion of (prtas)").03 . .
the Greek letters and the numerical coefficients giving the
quantity of each portion, and the Roman letters giving the
quality. But it must not be forgotien that this theory all along
supposes the difference of diffusivities to be relatively small.
* There is here a formal analogy with the problem of determining the
probability of a given combination of heads and tails in a set of 2 tosses
of a coin; and the result of supposing ~ infinite may be traced as in the
theory of errors.
+ Possibly a better plan for the concentration of the lighter constituent
would be diffusion along a column of easily absorbable gas, e. g. CO,.
The gas which arrives first at the remote end is infinitely rich in this
constituent.
P) 499) 0
LI. Meroscopic Vision.
By G. Jounstone Stoney, M.A., D.Sc., FBS.
[Continued from p. 442. ]
[Correction in Part I—Instead of the first line of the
footnote on p. 336, read as follows:—
If the number of replicas is not absolutely infinite, the luminous effects
produced in the definite directions spoken of in the text are each a
maximum of marvima, &e.]
Parr che
CoNnTENTS.
Page
PrP ETON GIS 2 gers Alaa ahaa Alheg, hie ce diab: ea xy ordiey ea ASQ
a Numerical Aperture, or Graspi. cancel. +s aides oe 500
34. Information supplied by Imagex .....,...... 503
Foie LESS CL TEN CID eae ie Uae eR ah Ma i 507
36. Significance of what is seen in the Microscope .. 512
37. Explanation of why, when the focus is changed,
bright specks may become dark, and fine detail
may seem to shift upon an object .......... 513
388. Illustrations of these Phenomena.............. 516
39. illustrations of Illusory Colouration .......... 518
AQ. Illustrations of Fictitious Markings............ 520
Mim Onrrcal COMpaciend. Whit) varias giles deed Geyss ) 524
Ae tlowato See bie EuWlimes) . 1.25. Yas. .ne teat, uy 525
PPE oneniaiie IVCIMATES™.. Jo)... 2 se 2 fc) tee he ee 527
Definitions.—It is convenient to collect here for reference
the definitions of some terms we shall have to use.
Image « is that image which is seen on removing the
eyepiece and looking down the body of the microscope. It
is seen without distortion if the iris diaphragm below the
condenser occupies its proper position. (See p. 435.)
The X scale is one of the radii of image « divided into
a scale of equal parts, with zero at the centre, and on which
the number G (the NA of the objective) is at its outer end.
(See p. 506.)
A punctum means a luminous point in image x. It is what
a beam of parallel light becomes after passing through the
objective. (See p. 903.)
A macula is a spot of light of any size and shape in image
x2, and corresponds to a sheaf of beams emitted from the
object. (See footnote on p. 510.)
If kg, in the figure on next page, be the front of the
objective, and o the middle of image C, then the radiating
lines represent the axial rays of a fan of beams as they emanate
from image ©; and the feet of the perpendiculars are in the
relative positions in which the puncta of those beams will
present themselves in image w. (See p. 504.)
500 Dr. G. J. Stoney on Microscopic Vision.
The X scale may be conceived as lying along the line ép,
its zero being at o, and the number n (the index of refraction
& Z
g O P
of the air or oil that is in front of the objective) being at the
point p on the scale.
The zmage plane means the plane in which image OC lies.
(See p. 504.)
33. Numerical Aperture, or Grasp.—The microscopic
object, the illumination supplied to it, the cover-glass placed
over it, and the media on both sides of this cover-glass—yviz. :
the medium between the cover-glass and the object on the
one side, and the medium between the cover-glass and the
objective on the other (which latter is usually air or oil)—are
what between them determine the condition in which light
enters the objective.
It will be convenient to call the three media between the
object and the objective a, b, c; a being the medium in which
the object is mounted, 6 the glass of which the cover-glass is
made, and ¢ being the oil or air which intervenes between the
cover-glass and the objective. The surfaces of demarcation
between these media are always planes perpendicular to the
axis of the microscope, so that we need only consider this
simple case.
It has already been proved that the light immediately in
front of the object, 2. e., as it exists while in medium a, may be
resolved into beams of uniform plane waves. These beams con-
tinue to be beams of uniform plane waves in travelling across
media } and ¢, since the surfaces separating the three media
are planes. Hence the light immediately before entering the
objective, i. e., as it exists in medium c, may be resolved into
beams of uniform plane waves. We may regard this light as
suffering reversal, as in § 8, and at the same time conceive
medium c¢, the oil or air in front of the objective, to be
extended downwards, all other apparatus beneath * being
removed.
* For convenience of description we suppose the microscope to be
pointed downwards, as it usually is more or less. It should be noted
that what is commonly called the front of the object is what is then its
upper side, the side turned towards the observer, while what is called
the front of the objective is its under end, that end which receives the
light.
Dr. G. J. Stoney on Microscopic Vision. 501
Under these circumstances the light after reversal will
form image OC, 7. e. standard image No. 2, in the extension
downwards of medium c. If the light be allowed to advance
farther down, and then to suffer another reversal, it will in its
subsequent upward journey re-form standard image No. 2,
and ther centinuing past it, will fall upon the front of the
objective in exactly the same state as the light that came
from the microscopic object would have done. We shall
find the state of things which is here sketched te be much
the most conveniert fer our purpose.
We may accordingly remove everything that lies below the
ebjective except medium c¢, which is te be extended down-
wards, and we may substitute image © (¢. e. standard image
No. 2, formed as above described in medium ¢ after two
reversals) for everything so removed. We have thus only to
picture to ourselves image C as present in its proper position
in medium c and as emitting its light upwards. This light
is resolvable into beams of uniform plane waves, each of
which has its own axial ray, viz.: that ray of the beam which
starts from o, the point where image C is pierced by the axis of
the microscope. Accordingly all axial rays diverge from 0,
and after passing through the objective they all converge to, or
rather nearly to, another point s on the optic axis, at D. If
aand 6 are the angles which one of these rays (or one of
these beams*) makes with the optic axis at o and s, then
Lagrange’s theorem states that
ecina— Nisin 6° OF NO) Se eR
where M is the number of times that the linear dimensions of
image D are larger than those of image OC, and where n sin a
is that which Abbe calls (when applied to the most inclined
ray from o which the objective can admit) the “ Numerical
Aperture ” of the objective, symbolizing it by the letters NA.
These—the name and the symbol—are a too limited name,
and an inconvenient symbol. I therefore propose instead to
* We may speak of a and £ as the inclinations either of the axial ray,
or of the beam to which it belongs. With some objectives image x hes
inside the objective, with some eyepieces J) lies inside the eyepiece; and
the diagram &c. will need to be modified accordingly.
Phil. Mag. 8. 5. Vol, 42. No. 259. Dec. 1896. 2N
502 Dr. G. J. Stoney on Microscopic Vision.
speak of the grasp of a beam, and of the grasp of the objective.
This designation is justified, since, as we shall presently see,
the quantity designated by it, when it is applied to an ob-
jective, is the proper measure of its resolving power, 2. e. of
the minuteness of detail which that objective can reach ; and
when applied to an individual beam or to its axial ray, it
indicates the farthest that two rays of that obliquity can go
in the representation of detail, in other words their utmost
grasp. When applied to the most inclined beam in any
meridian plane whose axial ray can be caught by the ob-
jective, it measures the grasp of the objective, and may
conveniently be represented by G, and it may be symbolized
by g when applied to any less inclined beam.
When the beams are in the same meridian but differently
b b’
as
inclined, we may proceed as follows :—Let kg be the front of
the objective, o the middle of image C, and ob and ob’ the
axial rays of two beams lying in the same meridian plane.
These beams, if reversed, will under ordinary circumstances
(i. e. if the transversals are not in altogether discordant
positions) produce a ruling* inimage C. The spacing of this
ruling is given by the formula
=o (sina+sin a’),
where o is the spacing of the ruling, X/ the wave-length in
medium c, and where @ and @’ are counted as positive when
on opposite sides of the vertical. Multiply both sides by n,
the index of refraction of medium c. Then
nN =o(nsina+n sin a’),
* To prevent misapprehension it may be well to call attention to the
circumstance that this ruling may have a very brief existence—lasting for
something like one foot of time in cosmic measure, see footnote, p. 425—
and may be succeeded on the image-plane by other rulings parallel to and
equally spaced with the first, but perhaps shifted in the direction perpen-
dicular to the ruling. Such rulings would be unseen by us. They only
become visible when the two beams have some common origin so that
their phases maintain the same relation to one another in successive
small intervals of time. This prevents the shifting spoken of above.
Dr. G. J. Stoney on Microscopic Vision. 503
which is the same as
Xr
a g+g” (2)
where X (=n)’) is the wave-length in air, and g and 9/ the
*“‘ grasps’ of the two beams. Hence o and g+q’ are in-
versely as one another: in other words, the ménuteness of o
is proportional to g+g’. Of course at least two beams have
to be associated with one another to produce aruling. If,
however, the two beams are equally inclined and on opposite
sides of the vertical, 9’ becomes equal to g,so that in this
case
Xr
Beil ia 2 Kaas
whence the appropriateness of calling g the grasp of beams
that have this obliquity.
If ob and ob’ are the most inclined axial rays on opposite
sides of the vertical that can be taken in by the objective,
formula (3) becomes
o—
Xn
re ee aie al)
where > is the spacing of the finest ruling which can be seen
by that objective transmitting light of wave-length 2.
Whence the appropriateness of calling G the grasp of the
objective.
The case of two beams which are not in the same meridian
plane is dealt with in the next section.
34. Of the Information supplied by Image «. In any of the
cases we have to deal with, the angle @ of the figure on
page 501 is so small that its tangent may be written for its
sine. In fact the two do not differ,in the cases we need
consider, by one part in a thousand, a difference which may
legitimately be disregarded. Now look down the tube of the
microscope. The beams of parallel light emitted from the
objective field are concentrated into the points of the luminous
image # which is then seen. Let us direct our attention to
that one of these beams which is represented in the diagram
on p. 901. Its light is concentrated in the point p of image wz.
It is convenient to have a name for this concentration of a
beam in image a, and we shall call it the punctum of the
beam. Let 7 be the radius from the axis of the microscope
out to this punctum, and let f be the distance from image v
to the focal image of the microscope at D. Then tan B=r/7,
whence finally equation (1), Lagrange’s theorem, may be
2 N 2
504 Dr. G. J. Stoney on Microscopic Vision.
written
oes |
I= Bh. te
and G (called by Abbe NA)= SR, Be. =
R being the radius to the border of the disk of light seen on
looking down the tube. It appears from these formulee that
the grasp of any particular beam is simply proportional to the
radius in image 2 out to the punctum into which that beam
is concentrated ; so that we may put the above formule into
the simple form of words—g zs proportional to r, on the same
scale on which G (the numerical aperture of the objective)
ts represented by Kt (the radius of the disk of hght seen on
looking down the tube of the microscope). This excessively
simple rule will be found of great use in earrying on practical
microscopical work. It may be symbolized by the equation
*
gapr
Image C isin the medium e¢ continued downwards. It
therefore lies in a plane parallel to and close to the objective
plane, but not necessarily coincident with it. Let us call
this plane the image plane, in order to give ita name. We
have spoken of the image as lyzng in this plane, but the phrase
must here be understood in a generalized sense. What is
meant is, not that the image is flat, but that the image plane
is related to image C in the same way that the objective
plane is related to the microscopic object.
Consider now one of the beams that form image C. The
positions in that beam that are in the phase @ at the time #,
are a system of parallel planes transverse to the beam and
separated by intervals of X’ from one another. These planes
intersect the image plane in a system of parallel lines, which
are separated from one another by intervals 6=2/'/sin 2, a being
* In dealing with such matters as are discussed in this section, the
reader should note that g, g', &c., the grasps of individual beams, though
of cypher dimensions are not mere numbers, They are directed quantities,
each standing out in some definite longitude, perpendicularly to the optic
axis. Each accordingly consists of a vector combined with a scalar. It
is thus that they can be fully represented both in direction and magnitude
by the radii v, 7’, &c. in image 2, the radii from the optic axis out to
the puncta of their respective beams. It is otherwise with G, the grasp
or numerical aperture of an objective, the direction of which is immaterial,
and of which, therefore, the scalar part is the only one to which we need
pay attention. G is adequately represented by R, where R is the length
of a radius of image x irrespective of its direction.
Dr. G. J. Stoney on Microscopic Vision. 505
the inclination of the beam to the optic axis. All points of
these lines are in the phase @ at the time ¢.
Proceed in the same way with another beam whose inclina-
tion to the opic axis is e’, and which lies in a plane of incidence
inclined at an angle ¢ to the plane of incidence of the first
beam. It produces on the image plane a system of parallel
lines in the phase @ at the time ¢; which lines stand at
intervals 6’=2!/sin «’, and are inclined at the angle ¢ to the
first set.
Let fig. 1 represent the image plane with the two systems
of lines drawn upon it. They form parallelograms ; and if
we draw the dotted lines of the figure, which are diagonals
Fig. 1.
to these parallelograms, we find that the one beam is always
in the same phase as the other at every point of these dotted
lines—in other words, these dotted lines are the middle lines of
the equidistant luminous bands which constitute the ruling
produced by the two beams on the image plane.
Let o be the spacing of this ruling, 2. e. the interval
506 Dr. G. J. Stoney on Microscopic Vision.
between the dotted lines. Then co, 6, and 6! are the three
perpendiculars of one of the triangles of fig. 1, both in actual
magnitude and in position. Now the lengths of the perpen-
diculars of a triangle are inversely as its sides. ‘Take then
the reciprocals of o, 6, and 6! without changing their positions
and fate ie
GO AO:
and are in the positions of the three perpendiculars of fig. 1.
Draw the triangle of fig. 2 with its three sides parallel to the
perpendiculars of fig. 1. Its sides will then be proportional to
a,b,andc. Again 6=)//sina=)/qg ; and 6/=)//sina’=A/q’ :
therefore
ax a, b; e,
Xr
Ig & 4, be
and are in the positions of the three sides of fig. 2, in addi-
tion to being represented in magnitude by the lengths of
those sides. Observe that A/e, g, and g’ are numbers, z. e.
their scalar parts are of cypher dimensions. Now if p and p!
of fig. 3 are the puncta in image « of the two beams, their
radii represent g and g’ both in position and in magnitude
estimated on the X scale, which means that scale which
prevails throughout image z, and on which the length of the
radius of image a represents G, the grasp or numerical
aperture of the objective. The X scale is one in which
lengths mean numbers. We thus learn that the triangle in
fig. 3 is in the same position as the triangle in fig. 2, and is
similar to it. Therefore d, the line joining the puncta p and
p', represents A/o in magnitude on the X scale, and represents
it also in position. In other words,
Pe d, or more simply o=A/d, + “> eames
if we identify G and R. Hence o 7s equal to X divided
by the number represented by d on the X scale; and, further,
the ruling of which o is the spacing has its luminous bars
perpendicular to the line d.
This is a very important proposition, far reaching in the use
that can be made of it in the interpretation of microscopical
phenomena. We should, however, when employing it,
bear this caution in mind—In every attempt to draw an
inference from image 2, we must recollect that the inform-
ation it gives, though great, is limited. It tells us the
intensities and the positions of the beams of uniform plane
waves into which the light is resolved—beams which are
thrown off from the whole extent of the objective field ; it
also telis us the directions and the spacings of the rulings
produced by these beams ; but it is silent in regard to every-
Dr. G. J. Stoney on Microscopic Vision. B07
thing that is determined by the positions of the transversals
of these beams, or the phases of the luminous waves.
35. Resvlution.—Onur first illustration of the great assistance
which is rendered to the practical microscopist by Abbe’s
theory will be taken from the guidance it gives him when he
wants to set up his apparatus so as to resolve an unusually
difficult “ test-object.”” That which we shall take is Amphi-
pleura pellueda. This cigar-shaped diatom is less than the
tenth of a millimetre in length, and about a tenth of that
again in width. With ordinary objectives the only detail
seen on it is the ridge surrounding it and a longitudinal
midrib which dilates into loops at the two ends. Between
the midrib and the sides of the diatom are what seem to be
two mere uniform plains ; but when adequate arrangements
are made each of these plains proves to be covered by trans-
verse rows of specks so close that Mr. Nelson in one specimen
counted 96 of these rows in the thousandth of an English
inch; and the specks of which the rows consist are somewhat
closer still. Hence, the spacing of the rows, is the 96,000th
part of an English inch, which is the same as o=0°265 p,
where mw stands for the micron, the thousandth part of a
millimetre. This is a good deal shorter than the shortest
wave-length of the most extreme ultra-violet light which can
reach us from the sun. The wave-lengths of the visible part
of light are much larger, ranging from 0°38 uw to 0°76 pw. |
The diatom was one mounted by Professor Van Heurck
in a medium containing arsenic disulphide (As,S,), the
refractive index of which is said to be 2°4; so that the
relative refractive index between it and silex is about 1°7,
thus affording both the advantage described in Theorem 6,
p- 348, and what is in the present case the still greater
advantage described on the same page, in the paragraph
which follows the enunciation of that theorem. Moreover,
the result seems to show that this medium has also a relatively
low dispersive power, which adds to its value. It is unfor-
tunate that a material possessed of these important properties
is so difficult to manipulate, or so risky, that no one seems
to have succeeded in mounting objects in it except Dr. Van
Heurck himself. :
The apparatus employed were an immersion objective
of which the nominal G (numerical aperture or grasp) was
1:35, and a dry condenser of which G’=0°9. ‘To be on the
safe side it was thought well to reckon only on being able
to make 1°25 of the grasp of the objective available, and 0°85
of the grasp of the condenser. Putting these values for
g and g’ into equation (2), viz. :—
AN=a(9+9’), co ol ge oe
508°. De Gee Stoney on Microscopic Vision.
and putting e=0°265 p, we find
A= (0-265) x (1°254 0°85)
=0:'56g;
which is the wave-length of a ray between the lemon-coloured
part of the spectrum and the green. We thus learn that
light of the above wave-length is the least refrangible which
can help us to see the detail we are looking for. We ought
accordingly to exclude all red, orange, and yellow rays since,
if present, light of these colours would throw a uniform glare
over the plains upon which the detail lies. This condition is
best complied with by setting up prisms to limit the range of
wave-lengths that we employ. ‘Two two-inch equilateral
prisms of dense glass brought into their position of mmimum
deviation were found to answer well. The light to be chosen
should be of a colour for which the objective is specially well
corrected, more particularly as regards aplanatism; and for eye
observations it must be in a sufficiently luminous part of the
spectrum to supply brightness enough. It should obviously
consist of the rays of shortest wave-length that fulfil these
conditions ; and on trial the best light was found to be
bluish green of about A=0°52y. This wave-length is
nearly that of the great magnesium triplet in the spectrum.
No collimating-lens was employed to prepare the light for
the prisms, but the slit (a coarse one) was adjusted to
such a distance from the prisms as enabled the condenser to
produce a disk of light in image z of the full size corre-
sponding to its grasp, when the condenser was itself adjusted
at such a distance from the microscopic object as to produce
one uniform colour throughout that disk of light. Upon the
slit the image of an incandescent gas-burner had been
thrown, which was thus the source of light. The mirror of —
the microscope was then turned until the colour seen in
image « was bluish green. With these arrangements we
know that the wave-lengths we are using are in the neigh-
bourhood of A=0°52 p.
We are now in a position to determine what size of stop
it will be advisable to introduce under the condenser, so as to
block out most of those direct beams whose diffracted light
does not come within the grasp of the objective, and which
accordingly would be not only useless but prejudicial. Since
A=0°52 w and c=0'265m we find by equation (2) that
gt+g'=1:96. Hence when g has its largest value, which is
1:25, g’ will have its smallest, which accordingly is
g =0-71.
Dr. G. J. Stoney on Microscopic Vision. 509
We should therefore put in a stop
which will block out all beams that
have a grasp less than this*. The
image at w will then have the ap-
pearance represented on a large scale
in the annexed diagram. An annulus
of bright light will be seen in it
between the edge of the stop, which
lies at g/=0°71, and the limit of Image a, with X scale.
the grasp of the condenser at G’=0°85,
and faint light will be seen farther out to the limit of the
grasp of the objective at G=1°25.
We may, however, with advantage block out still more of
the light. For the specks upon the diatom are numerous,
and although not quite regularly disposed, they are approxi-
mately in straight vertical rows, and less regularly in some-
what wavy horizontal rows. In this description we suppose
c é
75 0 O
| Z
BOQ © On
Image x.
* To get the stop accurately into its place it should be mounted
so that it can be centred relatively to the condenser. Provision for
this adjustment is always useful and sometimes essential for accurate
work ; and Messrs. Watson and Sons, at the request of the writer, made
an adjustable cell for holding the stops which is entirely satisfactory.
510 Dr. G. J. Stoney on Microscopie Vision.
the length of the diatom to be placed horizontally. Now the
nearest of the diffraction-spectra which such sources of light
would produce are disposed round a direct ray z, as in the
first of the diagrams on p. 509. Accordingly, when z is
in the neighbourhood of a’ of the second figure, it produces
one, and only one, diffraction-spectrum, that at a, which comes
within the grasp of the objective. Similarly the direct beams
which have their puncta near 0’ produce diffracted beams
with their puncta near 6. And in the same way the beams
which are concentrated into those puncta of image x which lie
near ¢’ and d’ will produce diffracted light which reaches ¢
and d. But the direct beams that reach the intermediate
positions at e’, f’, g’, and h’ throw all their spectra too far
out—vertically sideways or diagonally—to be caught by the
objective. Hence this intermediate light cannot help to show
the specks, and only produces diffused light tending to veil
them from view. The image will therefore be cleared by
blocking this light out. This is easily done by cutting a
cross out of card, and placing it
over the central stop. What is then
seen at x is depicted in the annexed
figure, for in fact the diffracted
light is so faint, and the direct light
so strong, that to see each macula *
of diffracted light it is best to shut
out all but one of the macule of
direct light; and even then the
diffracted light at ¢ and d is faint and
diffuse enough to be difficult to see.
It now only remains to adjust the draw-tube with extreme
care (since the objective is fully corrected only for one
length of draw-tube) and to apply a compensating eyepiece
of sufficient power. A suitable magnification is 2000, for
eyes that see very minute objects with ease; and if there be
in the observer a slight defect in this respect, the power
3000 + will be found better. The power 2000 causes the
vertical rows—those which stand at right angles to the mid-
rib of the diatom—to appear with the closeness of lines ruled
at intervals of half a millimetre from one another when viewed
* A macula or spot in image z is the representative in that image of a
sheaf of beams emitted from the microscopic object. It may be of any
size, and the sheaf of beams it represents are those beams of which the
puncta are the points of the macula. Bt
+ The objective made use of had an initial magnifying-power of about
80, and the higher power was reached by a 40 compensating eyepiece
which the firm of Carl Zeiss were good enough to make for the writer,
and which he has found to be of great service.
Dr. G. J. Stoney on Microscopic Vision. 511
by the unassisted eye from a distance of ten inches. When
all the foregoing precautions are taken the observer is
rewarded by seeing something like 17 or 18 specks in each row
where the rows are longest, and fewer in other places. The
specks are fainter than the rows, because from their being some-
what closer, the diffracted light at c and d lies farther out than
that at a and 6, and less of it is grasped by the objective. It
is also'more diffuse, and therefore less easily seen in image «,
partly because the specks are disposed in wavy lines longitu-
dinally while they are arranged in nearly straight lines across
the diatom. Another reason why it is fainter is that the longi-
tudinal rows are much less numerous than the transverse.
On all these accounts it is very difficult to see the maculee
at c and d in image a.
Another point is well illustrated by this experiment. The
larger features on the diatom require a certain moderate
aperture to show them well. Now beams so divergent
as those that have their puncta in macule so separated as
a’, b’, ce’, and d’ cannot concur to show these larger
features unless in co-operation with intermediate beams that
have been blocked out. The larger features are accordingly
only seen by the light of these maculz, each acting separately;
and the sheaf of beams that reaches one of them has too small
an aperture to show those features well. They present
accordingly that clumsy blurred-out look familiar to all
microscopists who have worked with annular or oblique
illumination.
Again, the specks themselves look very much like little
molehills viewed from above. This is because they are due
to the intersection of rulings which are of the jirst order,
and which do not differ much from one another in their
spacing. Rulings of the first order mean such as are pro-
duced by interference between only one pair of beams. The
law of intensity in rulings of this simplest kind is that repre-
sented in the accompanying figure, and therefore the specks
which are produced by the intersection of rulings of the first
order must have the same kind of rounded appearance.
Two other appearances are likely to be seen in making
this experiment. One is that the bright specks will become
dark specks on a slight change of focus; the other is that they
may perhaps seem to shift their positions relatively to the
larger features of the object when the focal adjustment is dis-
512 Dr. G. J. Stoney on Microscopic Vision.
turbed. These are both of them phenomena more conspicuous
in others of the experiments which we propose to make, and
will more appropriately be explained in connexion with them.
(See § 37.)
It has appeared desirable to go very minutely through the
successive steps of this example in order to show clearly that
Abbe’s mode of dealing with microscopic vision does success-
fully and in the most instructive manner guide every step
of our preparations, and that it renders a full explanation of
every phenomenon we encounter whether in the course of
the preparations or in the final result.
36. On the Significance of what we see in the Microscope.—
As to the significance of these specks. Their spacing along
each row is somewhat less than the spacing of the rows, and
may be taken to be about 0°24. From the way in which
their image is formed by the intersection of rulings of the
first order, it appears that they, like rulings of this kind, will
appear larger or smaller according to the amount of the
illumination; and that they will be best seen when the
illumination is such that the apparent diameter of a speck,
and the apparent interval between it and the next, are about
equal. Accordingly the portion of the object which corre-
sponds to it when best seen is that which is contained within
a sphere whose diameter is 0°12, and whose radius is there-
fore 0°06. Now if we make the probable hypothesis (see
‘On the Internal Motions of Gases,” § 9, in the Phil. Mag.
for August 1868, p. 140) that the average spacing in solids
and liquids of the chemical atoms of which matter is made up
is somewhere about a tenthet-metre*, which is the same as
0:0001y, then there has needed about 900,000,090 of these
chemical atoms to build up a portion of the object of the
volume of our little globe. If, on the other hand, we suppose
that the spacing of the atoms is about a ninthet-metre, 2. e.
0:001~—and it can hardly be more than this—the number of
atoms in our little sphere becomes 900,000. The actual
number probably lies somewhere between 900 thousands and
900 millions. There is therefore this vast number of chemical
atoms in the tiniest part of an object which can be dis-
tinguished from another part by so good a microscope, when
handled with extreme care. We ought further to reilect that
each of these chemical atoms is itself highly complex, and
that within every one of them all those events are in progress
* A tenthet means a unit in the tenth place of decimals. And as a
quarter pound means the quarter of a pound, so a tenthet-metre means
the tenthet of a metre.
Dr. G. J. Stoney on Microscopic Vision. 513
that are betrayed to us through the spectroscope. In the
face of such facts as these it is in vain for biologists to talk
as if anyone had at any time seen such a thing in nature
as “undifferentiated protoplasm,” or as if any speck of matter
that can be seen by the best microscope is other than a body
of large size from the molecular standpoint, within which
there may be a vast amount of structure and an inconceivable
flow and variety of events continually in progress. The
finest flagellum of a saprophyte, the tiniest rod in karyokinesis,
may, consistently with every lesson taught us through the
microscope and by molecular physics, have quite as elaborate
a structure as that part of the structure of the limb of a
quadruped which can be seen by the human eye.
Coarse rulings are usually produced by a fan of numerous
beams. It is thus that the shape of the bars of which they
consist is brought out. But the finest rulings are of the
first order, 7. e. in their case the fan has been reduced down
to two beams. Now the intensity of the light in rulings of
this kind follows the law [1—cos (27rx(g+9')/X)] *, which is
represented by the diagram on p. 511; and the microscopist
should constantly bear in mind that every speck or band upon
the object which is sufficiently minute to have its image
formed exclusively out of rulings of the first order must
accordingly have the appearance of a little hillock or little
ridge wholly devoid of detail and with blurred outline: and
that notwithstanding this there may be any amount of detail,
variety of outline, and intricacy of motions present upon the
actual object within the limits of the part represented by that
speck or band.
37. Propositions 8 and 9. Cause of bright specks becoming
dark ; and Cause why fine detail often seems to shift upon an
olject.—The finer detail in image C is formed by the inter-
lacing of beams that are inclined at a large angle to one
another. Let wand u’' be two such beams in one meridian
plane, and let the unbroken lines of fig. 1 represent those
wave-surfaces in them which at the instant ¢ are in phase @.
Then it is easy to see that the two undulations reach every
point of the planes represented by the dotted lines in the same
phase. Hence if minute markings are seen by the ruling
produced by these beams cooperating with rulings produced
by other pairs of beams which are but little sloped to w and
* If the two beams are not in the same meridian plane, g+g' should be
replaced by d, i.e. by the number on scale X which is represented by the
length of d, which is the distance asunder in image w of the free ends of
g and g’. (See § 34, p. 506.)
514 Dr. G. J. Stoney on Microscopic Vision.
u'—a case which occasionally happens—then if we put the
object a little out of focus these markings will appear to travei
Pigs:
along the inclined dotted lines and will appear to shift side-
ways if the dotted lines are inclined, which they will be if,
as usually happens, the angles of incidence of u and w’ are
unequal. This shifting need not extend to the larger features
of the object, since they are seen by light that is quite
differently circumstanced. In the case represented by the
figure the markings that are seen do not pass from bright to
dark. Their definition merely fades continuously into a
haze while they are being put out of focus inwards or out-
wards.
But a case which is much more frequently met with is
represented by fig. 2. Here two pairs of undulations wu, u/
and v, v’, both in the same meridian plane, cooperate to pro-
duce one of the rulings by which the markings are seen*. To
avoid complication the wave-surfaces are not represented in
this figure, but the dotted lines sloping up to the left repre-
sent, as in fig. 1, the planes over which uw and w! are in the
same phase. The lines sloping up to the right represent the
same for v and v!. Hence if the objective be focussed upon
the horizontal plane through s everything is in confusion
and the image disappears, whereas on removing the focus a
little farther out to p the ruling reappears, but now dark
* The case where v and wu’ coalesce and form the dioptric beam,
while w and v' are diffracted beams, is that most frequently met with.
Dr. G. J. Stoney on Microscopie Vision. 515
lines occur where bright ones were before. At q all is again
in confusion, to be succeded on drawing the focus farther out
Fig. 2.
Image plane.
by the reappearance of the ruling at 7, with its bright and
dark lines in the same positions as on the image plane.
This is what happens to beams wu, wu’, v, v’, all of which
lie in one meridian plane. If, as usually happens, the corre-
sponding undulations in all the other meridian planes that
contribute to form the minute marking have their dotted
lines (the lines shown in fig. 2) about as much sloped, then at
about the height p they too will produce rulings in all of
which dark lines will now occur where bright ones did on the
image plane. Hence at the height p dark specks will be
produced, by the cooperation of all the beams, where bright
ones were produced on the image plane; and light will be
distributed over the intervening spaces where on the image
plane the darker shades prevailed. The effect depends on the
inclinations of the dotted lines being sufficiently nearly the
same in different longitudes. Usually they approximate
sufficiently for at least one of the alternations from bright to
dark spots to be well seen, and not unfrequently a second or
third may be imperfectly traced.
If, as in fig. 2, v' and v are at the same inclinations as u
and wu’, then the dotted lines are equally sloped to the right
and left, and the dark specks at level p are directly over the
bright ones on the image plane. But this adjustment seldom
happens to be accurately secured, and the dotted lines in
516 Dr. G. J. Stoney on Microscopic Vision.
consequence slope more one way than the other. The dark
specks will then not be vertically over the bright ones, but
will have shifted a little in the direction of the more inclined
dotted lines of the figure. From all which we may enunciate
the two following propositions :—
PROPOSITION 8.
When the image of minute detail is produced by a triplet of
beams, or by two pairs of beams, in each meridian, then the
conditions are usually such, especially when the detail presents
the appearance of round specks, that it will suddenly change
from bright to dark, or vice versa, upon a slight change of
focus ; and under special circumstances which are occasionally
met with more than one of these alternations may occur.
PROPOSITION 9.
The conditions are likely to be such, unless special precautions
have been taken, that on a slight change of focus the minute
detail upon the object will appear to shift its position relatively
to the general position and broader features of the object.
38. Experiments illustrating the last section—All the con-
ditions spoken of in the first paragraph of the last section
can be reproduced if the objective be a good half-inch
apochromatic with G=0°65, and if the object be the large
variety of Navicula lyra which is frequently met with in slides
of diatoms from St. Peter, Hungary. We shall suppose the
diatom to lie horizontally in the field of view, ¢. e. with its
length in the direction sometimes called Hast and West.
Now insert a stop under the con-
denser, which will allow the sheaf of
beams u to pass. The figure re-
presents image «z, which is seen by
taking out the eyepiece and looking
down the tube of the microscope.
Jt is well to use a blank eyepiece to
keep the eye central—that is, the
mounting of an eyepiece without the
lenses, and with a small eyehole.
The transverse rows of specks upon
the diatom, which lie vertically in the field of view, produce
the vivid spectrum v; the longitudinal rows, which are fewer,
closer, and more wavy, produce on these accounts the fainter,
more distant, and more diffuse spectrum w. On replacing
the eyepiece the specks are seen on the diatom with great
distinctness. If the stop is in the position represented in the
Dr. G. J. Stoney on Microscopic Vision. 517
figure, so that 1 and v are at equal distances on either side of
the vertical diameter, then on slightly changing the focus the
specks will not shift sideways. But on account of the
diffuseness and the one-sided lateral extent of w, the adjust-
ment of « and w cannot be the same for different parts of w,
nor can it be the same for different colours, and it may be
found impossible to find any position of u in which there will
not be some shifting of the specks vertically.
In this experiment only two of the macule in image #
(corresponding to two sheafs of beams) lie in the horizontal
direction, and two vertically. Accordingly in each direction
there is (as in fig. 1 on p. 514) only one of the two sets of the
dotted lines in fig. 2 upon page 515. In this case there-
fore the conditions for the passing of bright specks into
dark do not exist, and accordingly on putting the microscope
slowly out of focus the image grows indistinct but no black
specks appear.
The case is otherwise if we use a stop
with a hole in it as at w in the
accompanying figure of image vw. Here
there are three maculz, corresponding
to three sheafs of beams in the horizontal
position and only two vertically. Ac-
cordingly on putting the microscope
out of tocus the vertical rulings (which
are caused by three sheafts of beams) will
change from bright to dark while the
horizontal ones (caused by only two) will not, and the resulting
specks in the microscopic image will become imperfectly
dark.
Diatoms which show the transition from bright specks to
dark in perfection are those known as Actinoptychus, of which
a good example is almost sure to be found ona St. Peter slide.
With the half-inch apochromatic and with the iris diaphragm
below the condenser nearly closed it gives in image w a
ring of strong diffracted light. ‘Two opposite puncta in this
ring and a punctum in the central macula belong to three
beams which produce one of the very numerous rulings which
conspire to form the specks in the microscopic image. Hach
such trio of beams furnishes both sets of the dotted lines in
fig. 2 on p.515. And as the ring is a tolerably circular one
the other similar trios, whether in the same or in other
longitudes, furnish dotted lines in fig. 2 that are not very
far from being equally inclined. These are conditions that
will produce black specks at the height represented by p in
Phil. Mag. 8.5. Vol. 42. No. 259. Dec. 1896. 20
518 Dr. G J. Stoney on Aicroscopic Vision.
fig. 2 on p. 515. Now open the iris diaphragm, insert the
eyepiece, and look at this diatom : it will be found to exhibit
the phenomenon with great distinctness.
A modification of this experiment is to cut a piece of card
of the annexed form and to place it over
the back of the objective. It allows only
two of each trio of beams to pass. These
proceed to form the rulings which are
competent to form specks in the micro-
scopic image ; and accordingly that image
will still exhibit the bright specks, which, however, will now
go out of focus without any dark specks appearing.
Notice that the specks now appear to traverse in a direction
perpendicular to the diameter of the semicircular stop, while
the microscope is being put out of focus. This is because the
dotted lines of fig. 1, p. 514, are here necessarily oblique.
39. Haperiments exhibiting Illusory Colouration. See Pro-
position 4, p. 845.—The half-inch apochromatic, of which
the grasp is 0°65, answers admirably for these experiments.
It should be furnished with a Davis’s shutter, z.¢. a small iris
diaphragm interposed between the objective and the mierc-
scope tube, by which the aperture may be reduced when
desired. We shall also want asmall central stop about 8 mm.
in diameter, which can be put over the back of the ob-
jective, and which may be cut out of card. This wil
enable us, at one stage of our experiment, to exclude the sheaf
of dioptric (2. e. undiffracted) beams, while leaving a free
passage to others.
The object which perhaps most strikingly exhibits the
phenomenon we are now occupied with is the diatom known
as Actinocyclus Ralfsw. 1°. Select a valve of this diatom
which looks blue when the Davis’s shutter is partly closed.
2°, On then opening the Davis’s shutter the colour is for the
most part but not altogether discharged, and at the same time
a quantity of detail comes into view which was not visible or’
was seen imperfectly when the colour was present. It consists
of specks variously distributed over the valve. 3°. Now
introduce the central stop over the back lens of the objective,
and open the Davis’s shutter. The image is thereby con-
verted from biue to red, and there are added to the image
dark grooves and bright lunes distributed over the intervals
between the legitimate specks. 4°. Finally examine tle
diatom with the immersion objective, which has a much
greater grasp than the half-inch. It has now become
absolutely colourless, and new detail has come into view
Dr. G. J. Stoney on Microscopic Vision. 519
consisting of from two to five little points within each of the
specks which are visible with the half-inch.
Now a study of image 2 enables us to trace the cause of
every one of these effects. Focus the diatom with the half-
inch. Take out the eyepiece and look at image w. Nearly
close the iris diaphragm under the condenser, so as to reduce
the incident light to a narrow central sheaf of beams. Then
the macula of this sheaf of beams will be seen in image 2
as a central spot of bright light. Faint light
is visible about it, which is scattered pretty
generally over image [this is light which
helps us to see the larger features of the
object ],.and at about the distance from the
centre where g=0°5 there is a ring of much
more intense light which has been diffracted oe
in all directions to that distance. ‘This ring Bae
of diffracted light is mainly red, owing to the
unequal distribution of colour spoken of in § 16, p. 345.
As so much red has been thrown into this spectrum, there
is an equal deficiency of red in the light which forms the
central macula and the faint diffracted beams by which
the larger features on the object are seen. Hence it is
that when the marginal ring is shut out by partially closing
the Davis’s shutter, the diatom will appear blue: blue
being the colour which lamplight becomes when much of
its red is withdrawn from it. The exclusion of the diffracted
red light has another effect—it prevents the formation of
a number of the rulings which are necessary for the
formation of a good image; and accordingly much of the
detail on the diatom which is visible when the Davis’s shutter
is open, is lost to sight when it is sufficiently closed to render
the image blue.
That the image is not quite colourless with the full aperture
of the half-inch is because there also exist other dittracted
beams which lie beyond the reach of that objective. We
know that they exist and that they are coloured, because the
image seen with the immersion objective is colourless, and
because more detail is seen in it. It has rendered the image
colourless by adding some coloured beams to the slightly-
tinted image which the half-inch, fully open, presented, and it
has brought out further detail by transmitting these additional
beams in directions which furnish new rulings.
Return now to the half-inch objective. When the central
stop is put over its back lens, the central macula of image w
is coveredup. ‘This shuts out the dioptric beams. The image
202
520 Dr. G. J. Stoney on Microscopie Vision.
is then formed by the ring of red diffracted light along with
the fainter and apparently white diffused diffracted light.
Accordingly the image in this case is preponderatingly red.
At the same time the red diffracted light is now obliged to
act without the cooperation of the part of the light of the
dioptric beams with which it before produced rulings. What
was before a triplet of beams producing a ruling has now
become the two extreme beams left to operate without the
middle one, and they produce a ruling that is twice as fine
It is thus that black bands and bright patches are produced
between the legitimate specks. These are very conspicuous
in the image, notwithstanding which they are quite foreign
to the object. They are produced very much in the same
way as intercostal markings, to the illustration of which the
next section will be devoted.
Similar experiments may be made with innumerable other
objects. Thus the familiar diatom, Pleurosigma angulatum,
becomes buff-coloured when viewed with the Davis’s shutter
sufficiently closed to exclude all but dioptric beams and that
inner portion of diffracted light which has its origin exclu-
sively in the larger features of the object ; whereas it becomes
of a delicate blue when seen with the dark-field illumination
obtained by putting a stop of the proper size under the con-
denser. In this case the object is seen by its diffracted light
only, much of which is coloured. Similarly the tubercle
bacillus when stained with fuchsin is intense red when examined
through the half-inch in the ordinary way, but becomes a bluish
white when seen with black-ground illumination. The study
of a variety of cases like the above will be found instructive.
40. Intercostal Markings and allied phenomena. See § 15,
p. 3845.—Light which does not contribute to delmeate any-
thing upon the object is apt to intrude in three forms—
either (a) concentrated into patterns which are superposed —
upon the microscopic image, or (b) scattered in patches over
parts of it, or (¢) spread in the form of a luminous haze
over everything. We shall endeavour to exemplify each of
these.
(a) The first is well shown in the image of Peristephania
eutycha when examined through the half-inch apochromatic
objective. ‘he real detail upon this diatom seems to consist
of divisions into hexagonal cells which make it look like a
honeycomb. Adjust the draw-tube accurately. Now nearly
close the lower iris diaphragm, and look at image z. ‘The
dioptric light makes a central white macula, and round it are
displayed, as in the figure, a beautiful array of coloured
macule, each of which is a spectrum, blue inwards and red
Dr. G. J. Stoney on Microscopic Vision. 521
outwards, To make the following experiment, the lower
iris diaphragm should be opened to so moderate an extent
as will cause these macule just to
come short of touching one another.
When now the Davis’s shutter is
gradually opened, which increases the
aperture of the objective, a succession
of images present themselves. If it
covers up all the maculz except the
central one, the diatom is indeed seen,
but with no detail upon it. In all
subsequent enlargements of the aper-
ture, the honeycomb structure of the Imuge wv.
diatom is visible ; but with the ad-
dition of spurious effects which vary * while the grasp of the
objective is being enlarged. In fact, they depend on what
spectra are permitted to pass and what spectra are excluded.
Let us consider three of these images, 1, 2, and 3, of
which 2 is formed with a larger aperture than 1, and 3 with
a larger aperture than 2. If we start with image 1 and open
the Davis’s shutter so as to pass to image 2, we may regard
2 as being image 1 modified by adding to it the new rulings
which are formed by the newly almitted light cooperating
with a part of the dioptric sheaf. And on the other hand, if
we start with image 3 and close the Davis’s shutter, we may
regard image 2 as ; being image 3 with the rulings added to it
which would result if light equal in intensity but opposite in
phase to the excluded light were allowed to cooperate with
part of the dioptric light. In fact, these rulings fill the gaps
between whatever rulings 3 has and which 2 has not.
Sometimes the one of these processes and sometimes the
other helps us to a better conception of what is going on.
The first process needs no elucidation, and of the second the
following is a very neat illustration.
Keep the same diatom on the stage; then look at image a,
and open the Davis’s shutter till the inner
hexagon of six spectra is seen. On _ re-
placing the eyepiece a few intrusive mark-
ings will be perceived, including a_ bright
pateh of light occupying the middle of each
honeycomb cell. This is image 1. Next
look again at image wv, and open the Davis’s
shutter farther until the succeeding hexagon of twelve
spectra is also allowed to pass. The image. which is then
* With the half-inch apochromatic, two, three, and four intercostal
markings can be produced along each side of each hexagon, the last
number when we employ black-field illumination.
522 Dr. G. J. Stoney on Microscopie Vision.
seen on replacing the eyepiece is image 3. One of its
honeycomb cells is represented in the diagram, from which
it will be seen that there are 19 deceptive markings upon
it. The central one is bright, but by a slight change of
focus becomes dark.
Adjust the focus so as to make it dark, and then close the
Dayis’s shutter into an intermediate position, which shuts out
the whole of the red of the outer twelve spectra, while it
allows their blue halves to pass. This produces a new image—
image 2—which will be seen on replacing the eyepiece. It
will be found to be nearly like image 3, but with a red central
speck instead of a dark one.
The reason of this is that the hexagon of twelve spectra
produces, along with some of the dioptric light, rulings which
formed the dark central speck, as is proved by its being
present in image 3 and not in image 1. Now when we shut
out the red of those spectra, we produce the same effect as if
we added red light in the opposite phases, and half of this with
half the same dioptric light (which has not been excluded)
produces rulings the same as before, except that the bright
and dark bands are interchanged. These accordingly change
the dark into a bright speck at c. The effect is somewhat
startling, since we change the specks from dark to. red by
shutting out red light! This gives one a lively picture of
how it comes to pass that the exclusion of light from image C
can lead to the intrusion of unauthorised markings.
Another matter may here be adverted to. As explained
in § 39, these spectra need not be equally intense for every
wave-length: sometimes the less refrangible light pre-
ponderates in them and sometimes the more refrangible.
Accordingly some of the intercostal markings to which they
give rise will have a preponderance of red, others a prepon-
derance of blue. The first will seem buff-coloured, the others —
whiter than the lamplight we use as our illuminant; and
upon scrutinizing the fictitious markings this difference in
tint between them is plainly visible.
When the dioptric macula is placed excentrically the
resulting intrusive markings become in some degree both
different and differently situated. Hence when we give a
large aperture to the dioptric sheaf of beams by opening the
iris diaphragm below the condenser, its macula becomes
large ; one part of it is central, others are excentric in
different degrees and different directions. Hach part gives rise
to its own set of false markings, and these where they overlap
interfere and may occasion a general illumination at the place
where they are situated instead of distinct markings. This
Dr. G. J. Stoney on Microscopie Vision. 523
is illustrated on our diatom by the whole inside of each cell
becoming full of a nearly uniform sheet of light instead of a
group of definite markings, when the iris diaphragm is
suitably opened. It is easy to see from this that there is no
one opening of the iris diaphragm which will, in all cases,
give the best effect—the etfect most free from intercostal
markings. In each case it depends on the way the spectra
are disposed, and will therefore differ from one object to
another. Its success, so far as intercostal markings are
concerned, depends on the circumstance that when a small
dioptric macula is shifted about in image w, the illusory effects
undergo rapid change, while the image of true detail is but
little affected. Hence the real features of an object are well
seen with a considerable illuminating cone ; and they may
even be better seen on account of the admission of oblique
rays, as these will both add to the visible detail and will
diminish that defect in images which consists in the rounding
off of sharp edges. On the contrary, the false effects produced
by the several small sheafs of beams in the cone are so diverse
that when jumbled together they become undistinguishable.
This is the next best thing to their being got rid of.
It is evident that anything which intensifies the strength
of the more inclined beams wil! give rise to brighter inter-
costal markings. They are, therefore, of exaggerated strength
when the object has been mounted in a medium of extra high
refractive index, owing to the effect which is described in
§ 18, p. 346.
(6) The diatom employed in the foregoing experiments
has the detail upon it disposed with the regularity of a honey-
eomb. It therefore concentrates most of the light diffracted
by it into definite spectra, and this has led to the formation
of intercostal markings of equal definiteness and regularity.
Where, however, less symmetry prevails in the disposition of
the detail upon an object, the light it diffracts is not distri-
buted according to any simple law, and a corresponding want
of regularity ensues among the spurious markings, which
may become shreds, lunes, and patches, black, white, or
tinted, usually twisted about, and sometimes flickering (from
slight movements of the observer's eye), and which are
chiefly conspicuous where there is some flat space in the
microscopic image unoccupied by real detail. Numbers of
diatoms exhibit these phenomena.
(c) Another frequent event is the presence of a haze of
light over everything. It will occur when, from want of a
central stop, there is dioptric light w hose corresponding
diffracted light is abundant and hes beyond the grasp of the
524 Dr, G. J. Stoney on Microscopic Vision.
objective. This has been sufficiently illustrated in § 35. The
remedy is to insert a stop of the proper size. A similar fog
of light will occur if, by opening the iris diaphragm too far,
diffracted light has been rendered too copious in situations
where it is beyond the grasp of the objective. Hach part of
it will then produce its own body of intercostal markings ;
these get massed together and produce a haze. The remedy
is of course to lessen the opening of the iris diaphragm ™*.
A case which should be specially noted is that of a bar on
the object with sharp edges. These produce highly inclined
diffracted beams ; and if the objective cannot take these in,
the beams —Bd of p. 344 which must then be added to
standard image No. 1, round off the edges and add thin
appendage-lines which are often mistaken for diffraction-
fringes though they have a different appsarance and a quite
different origin.
Al. “ Optical Contact.” See § 19, p. 348.—An excellent
illustration of the effect produced by Stokes’s layer—the
effect which microscopists call optical contact—may be
obtained as follows :—
Focus a valve of Pleurosigma angulatum, mounted dry,
under an immersion-lens of, say, G (or NA) =1°3. Adjust
the draw-tube. Then set up the apparatus described in § 35,
p- 12, for furnishing monochromatic light, and select green
light of about wave-length X=0°55 w. Now 0°55 w is also
the value of o upon this diatom, 7. e. it is the spacing asunder
y
LZ»
Image x, with X scale.
of its rows of markings. Hence from the formula
o=)/(¢+ 9’) we find that g+g’/=1, where g’ is the grasp
(or radius in image 2 out to the punctum) of a dioptric
* A haze of light may have a different origin when the section is too
thick. In this case the parts that are out of focus produce it, and it
may often be cleared up without detriment to the observation we want
to make by partially closing the Davis’s shutter.
Dr. G. J. Stoney on Microscopic Vision. 525
beam, and g is the grasp of the associated diffracted beam.
[The regularity of the detail on this diatom throws nearly
all the diffracted light into definite spectra.] We have made
the convention that g and g’ shall be positive when on oppo-
site sides of the centre. Accordingly, when on the same side
we must regard g/ as negative. Now open the iris diaphragm
until the macula of the dioptric sheaf of beams has expanded
so as just to touch those of the diffracted sheafs of beams.
Then what is seen on looking at image 2 is either what lies
within the greater or what lies within the less of the two
large circles of the figure, of which the outer one corresponds
to 1:3, the grasp of the objective, and the inner one to a
grasp=1. If the diatom is “ in optical contact,” that is if the
layer of air between it and the cover-glass is less than the
thickness of Stokes’s layer, then what is seen extends to the
outer circle. Whereas if the chink between the diatom and
the cover-glass is more than the thickness of the Stokes’s
layer, no light can get into the cover-glass except such as
passes up through it and the oil at less than the “ critical
angle,” and this supplies light in image «, only within the
smaller circle which corresponds to G (or NA)=1. The
reason of all this is obvious from what is stated in § 19 ; and
very interesting appearances may be obtained by traversing
the slide sideways and thus bringing diatom after diatom
under the objective. With some the light will extend in its
full intensity to the outer circle. These are they thai are in
good optical contact. With others nothing is seen beyond
the smaller circle. These are they that lie beyond the thin
Stokes’s layer which lies like a varnish on the under side of
the cover-glass. And now and then one may be found in
which the outer ring of light is present but dim. This is
one the interval between which and the glass is nearly the
full thickness of the Stokes’s layer. The whole experiment,
if made with green monochromatic light, is one of exceeding
beauty.
WG. Flow to See the Rulings.—Ilt has been explained that the
microscopic image is produced by the interlacing and mutual
interference of luminous rulings, each of which extends over
the whole image field ; and that each ruling is due to the
concurrence of two or more beams the puncta of which are
situated at equal intervals along some straight line upon
image z. In order, however, to get light enough to see a
ruling, we must be content to use small sheafs of beams
instead of individual beams, and these in image x become
small maculz instead of points. This in practice may be
accomplished by putting a disk of card over the back of the
526 Dr. G. J. Stoney on Microscopic Vision.
mounting of the objective and making a pinhole in it where
we wish a sheaf of beams to pass.
Make one such hole in the centre and examine a slide
containing a variety of diatoms, as for example one of the
show slides made with diatoms from St. Peter, Hungary. In
order to be able to predict the result, we must deal with a
specific case. Let, then, our objective be the. half-inch
apochromatic with which R, the radius of image wz, is about
8 millimetres. Let us further suppose that the hole in the
card is 1 mm. across. Hence the sheaf of beams the puncta
of which lie within this hole, contains beams of which d, the
distance between their puncta (see § 34, p. 506) may on scale
X be as much as one-eighth of 0°65 (the grasp of the objective),
i.e. d is nearly 0°08 on scale X. Put this into the equation
C=n/d;
and put %=0:6h, which is close to the wave-length of the
brightest rays in lamplight. We find then
C= Cop
for the spacing of the finest ruling which can be produced by
the light passing through the small hole. ‘This would admit
of detail upon the object being seen down to about the size of
a speck half the diameter of a disk of human blood.
Accordingly all the large features upon most diatoms can
be seen through this small hole, as may be verified by passing
a slide containing a variety of diatoms under the objective.
Now make another similar hole at some distance from the
centre, suppose in a position corresponding to g=0'6 ; or,
still better, make two holes at that distance on opposite sides
of the centre. ‘This gives us three holes in a straight line at
equal intervals. Then close the iris diaphragm under the
condenser until the sheaf of dioptric beams just fills the
middle hole. Only diffracted light will then appear in the
other two. Now pass the diatoms again under the objective,
and on many of them a ruling will be seen, viz. : on all those
which furnish diffracted light of sufficient strength in the
positions of the two lateral holes. This ruling will have a
spacing calculated by
Here then we have actually in view one of those rulings which
go to build up the ordinary microscopic tmage—that image
Dr. G. J. Stoney on Microscopic Vision. 527
of the object which is furnished by the microscope when the
card is removed.
Note that the ruling as seen does not extend across the
whole field of view, but only a little beyond the boundary of
the diatoms. This is because the ruling we see is made by
sheafs of beams instead of by individual beams. The more
we can reduce the size of the holes without making the ruling
too faint, the more diffuse will the image of the diatom
become, and the farther out will the ruling extend ; until at
the limit it would extend over the whole field of view and be
perfectly uniform everywhere *.
It is instructive to make this experiment with a specimen of
Actinocyclus Ralfsi, selecting one which is blue when seen
through a small aperture. Here we found the diffracted
light to be red, see § 39. This red light cooperating with
some red out of the dioptric light produces a ruling which if
seen alone would consist of alternate red and dark bands.
But there is an excess of dioptric light beyond what is
employed in contributing to form this ruling, and this excess
throws a wash of blue light over everything. Where it falis
on the red it turns it white, where it falls on the dark parts
it turns them blue. Accordingly what is seen is a ruling
of bands which are alternately white and blue. Similarly
on other diatoms the ruling is found to consist of white and
coloured bands instead of merely light and dark. The colour
which takes the place of the dark bands is in each case what-
ever colour the diatom, or the part of the diatom, presents
when seen through a very small aperture.
* What has happened may be clearly apprehended from the following
considerations:—If instead of three individual beams with puncta at a, d, ¢
to produce a ruling, we suppose two such sets as in the figure, and all under
such circumstances that they can interfere, then @ and a’ produce a coarse
ruling with its luminous bands and dark
intervals lying vertically. Rulings in exactly
the same position are produced by 0 and U’',
and by cand c’, so that these all reinforce
one another—they together produce one
ruling ; and the consequence is that all light
from aw’, bb’, cc’ is extinguished at the
middles of the dark bands of this ruling, At
the same time the beams with puneta at a,
b, ¢ produce a fine ruling which lies hori-
zoutally, and, @, 0’, ¢. produce an identical
ruling which reinforces it, and all ight from Imave 2.
ad’, bo, ce’ must disappear at the middles P
of the dark bands of this ruling. Hence we have two rulings that
co-exist, a coarse vertical ruling and a fine horizontal one; and the
outcome is that we see a horizontal fine ruling, which however is visible
only across the bright bands of the coarse vertical ruling, and which
fades out in the intermediate dark spaces,
528 Notices respecting New Books.
43. In our study of microscopic vision no consideration
has been given to the consequences of imperfections in the
mounting of the instrument, or in the objectives and eye-
pieces. To have entered on these branches of the subject
would have been to open new ground, and ground which has
been rendered of less importance by the extraordinary per-
fection both in the stand and in the objectives which may be
secured by a careful selection from among the best that are
available. It is truly astonishing with what accuracy the
chromatic, and especially the even more important spherical
aberrations have been successfully corrected in some speci-
mens from among the best objectives on sale. Objectives
are not unfrequently to be had which, when the draw-tube
is adjusted with sufficient care, will bear an eyepiece mag-
nifying 40 times without observable defect.
However great their degree of perfection, it may he
earried one step further by a skilful use of monochromatic
light; whereby adjustments can be made by the observer
with a completeness which with light of mixed wave-lengtlis
is unattainable. This is indicated “by theory, and abundantly
confirmed in practice.
In whatever branch of microscopic work the obser may
be occupied he will find it of advantage to train himself beth
in the intelligent manipulation of his instrument and in the
interpretation of results, by making a large body of experi-
ments such as those of which a few selected examples have
been described in the foregoing pages; being careful at every
step to understand the reason for everything he does, and to
Jind out the cause of every effect he perceives. To enable him
to do this has been the object of the present memoir.
LIT. Notices respecting New Books. e
*
Studies in Chemical Dynamics. By J. H. van ’a Horr, Revised &
and enlarged by Dr. Ernst CoHEn, translated by THOMAS
Ewan, W.Sc., Ph.D. Amsterdam : Peodenice Muller and Co.;
London: Williams and N orgate.
ne PICs translation of Professor van ’t Hoff’s work will
come as a surprise to many English chemists who are not
aware of the eae and importance of his researches on some
fundamental problems of chemical dynamics. In Britain there are
but few workers in the same field, and, as the translator tells us,
scarcely any text-book in our language deals with the subject. The
reason is, we believe, not far to seek; it 1s, however, a most
lamentable one.
Notices respecting New Books. 529
In this country it is at present one of the disadvantages of any
science which has important industrial applications, that the treat-
ment of it in the universities proceeds upon lines which are more
appropriate to the technical schools. In the course of a few years
the technical schools may afford a partial relief to the universities ;
but we cannot help thinking that the necessity of adapting their
lectures to the student of technology will always tend in some
measure to draw the attention of the university teachers away
from the development of the theory of their subject. Professor
FitzGerald has, on more than one occasion, claimed for the
university teacher increased opportunity for “useless” research,
that is, for investigations of a theoretical character without any
visible practical applications. This claim is worthy of most serious
consideration at the present time. While the future of our higher
technical schools is trembling in the balance, could not a division
of labour be effected which would relieve the university of some of
the teaching at present assigned to it?
In the work at present before us Professor van ’t Hoff does not
tell us anything new concerning the nature of molecules ; he pos-
tulates the existence of reacting molecules just as in ordinary
dynamics force and matter are accepted as facts. But, by assuming
that the rate of decomposition of the substances involved in any
reaction is proportional to the concentrations of these substances,
he shows how a knowledge of the degree of complexity of the
reaction (as measured by the number of different kinds of molecules
taking part in it) can be obtained. The assumption is justified by
experiment in a few cases; in many reactions, however, disturbing
influences cause a departure from this simple law. The author
was thereiore led to investigate the nature and magnitude of the
more obvious forms of disturbance, such as the action of the
medium in the case of liquids and of the walls of the containing
vessel in gaseous reactions. The influence of temperature on
chemical change is next discussed, and is naturally followed by an
inquiry into the conditions of chemical equilibrium, with special
reference to the relations between temperature and equilibrium,
The volume concludes with a chapter on affinity.
The subject is one which affords much scope for research, and
we hope that the appearance of this translation will serve to direct
attention to it. Both the reviser and the translator have per-
formed their duties with care, while the printing of the volume is
a good example of what Holland can produce—very different from
some of the specimens of English and typography which occasionally
reach us from that country.—J. L. H.
Bees
LILI. Intelligence and Miscellaneous Articles.
ON EXPERIMENTS WITH RONTGEN RAYS.
T'o the Editors of the Philosophical Magazine.
GENTLEMEN,
LLOW me to avail myself of your Magazine to make some re-
marks about the Memoirs of Messrs. Oumoff and Samoiloftf
(Phil. Mag. October), and of Messrs. J. J. Thomson and Ruther-
ford (Piil. Mag. November).
The experiments of Messrs. Oumoff and Samoiloff do not, in
my opinion, resemble very closely those of my own experiments on
electrical shadows to which they refer, but they have rather much
likeness with those, concerning the production of Rontgen’s shadows
with the electrical method, which I published in two Notes on the
1st of March (Rendiconti della R. Accad. der Lincei). They have
therefore the same analogy with those afterwards published by
Prof. Silvanus P. Thompson (Phil. Mag. August). No doubt that
Messrs. Oumoff and Samoiloff had no knowledge of the previous
publications above mentioned.
As to the Memoir of Messrs. Thomson and Rutherford, I beg
to observe that the fact proved at pages 395 and 396 (viz. that
a thinner layer of air may offer a greater resistance than a
thicker one) has already been described by myself in a Note on
the 3rd of May (Rend. della R. Accad. det Lrncei). In that
Note I recalled also some experiments I had formerly made, which
proved that an analogous phenomenon may be produced by ultra-
violet rays, and also without the action of any radiation at all.
In the complete Memoir read before the R. Academy of Bologna
on the 3lst May (“On the Propagation of Electricity through
Gases traversed by Rontgen’s rays”), I have described with more
particulars all these phenomena, and also others that I obtained
with Rontgen’s rays. ?
I am, Gentlemen,
Yours faithfully,
Atveusro Rieu,
Professor of Experimental Physics,
in the University.
Bologna (Italy), November 1]th, 1896.
—_—__—_
VOLUME MEASUREMENT OF AN AIR THERMOMETER BULB.
BY WALTER G. CADY.
In all constant volume air thermometry where high tempera-
tures are involved, it is frequently necessary to determine the
volume of the glass bulb used; such a bulb softens at a low red
Intelligence and Miscellaneous Articles. 531
heat sufficiently to have its volume, considerably altered by any
difference between the atmospheric pressure and that within the
bulb. Thus it is important to know at the end of an experi-
ment to what extent, if any, the volume of the bulb may have
changed, as any such change must be taken into account in
calculating the temperature.
The usual method of volumetry by weighing with water is long
and arduous, involving as it does the taking apart of the apparatus.
Below is given a simple and sufficiently accurate method of
calculating the volume at any time without disconnecting the bulb.
In addition to the usual apparatus it is only necessary that the
manometer tube in which the air is confined be graduated for 9 or
10 centim. from the top, so that the volume of air in the tube
may be computed.
In the figure, B is the bulb, PP
a capillary connecting tube, T the
graduated manometer tube, the gra-
duated portion of which is about
1 centim. in diameter, connected in
the customary manner by flexible
rubber tubing with a reservoir.
The scale in question is shown at
T and is graduated in 0-1 centim.
beginning with the fiducial mark.
Assuming the whole apparatus to
be at constant temperature during
the operation, the required volume
V is easily found as
Woes Py seins
2
pep
when v, and v, are volume excesses measured by means of the
scale on T, correspending to pressures P, and P, respectively.
It will be observed that V is not the volume of the bulb alone.
Still, since the bulb is the only variable factor, the above formula
indicates what change it nay undergo.
The following is an example of a number of observations
leading to the volume of a glass bulb used; in each case the
first reading is combined with the third, the second with the
fourth, in findmg the values of P,v,—P,v, and P,—P.,.
Reduced atmospheric pressure =74:76 centim. i
P(cm.) u(c.c.) Vice.)
15:17 0:16 239-0
74°63 1°80 241-1
73°79 4-477
To2l 6°20
d32 Intelligence and Miscellaneous Articles.
The method is equally serviceable when a long capillary of
irregular and appreciable volume joins the buib with the mano-
meter, a condition sometimes unavoidable in practice; the volume
of the capillary may then be separately found by temporarily
plugging up its detached end.—American Journal of Science, (4)
i. p. 341.
ON THE INFLUENCE OF TEMPERATURE ON THE REFRACTION OF
LIGHT BY METALS. BY ©. PULFRICH.
About five years ago LI published a research in the Annalen*, in
which the influence of temperature on the refraction and dispersion
of several transparent solids (glasses &c.) was the subject of an
extended experimental investigation. In this research, I was
able to show by a series of arguments, that the (positive and
negative) variations of the refractive indices observed in those
bodies could be regarded as the result of the combined (or
opposed) action of two causes, change of density and change of
absorption.
This explanation would not at that time apply to the metals
For the encrmous increase in the refractive indices with tem-
perature (about 0°0037 for 1° C.) found by Kundtt, could only be
asclibed to a great change in the absorptive power of the metals.
Observations have, however, so far revealed nothing of such a
change in the absorption of metals. They show rather that the
metals have only a small variation of absorption. It is in agree-
ment with this that the dispersion of metals experiences almost no
alteration with the temperature.
The contradiction arising out of this is solved since Pfliiger , in
a research which has lately appeared, has proved that the value
given above for the variation in the refractive indices of the
metals must be ascribed to a source of error neglected in the
apparatus used by Kundt, and that, as shown by his own measure-
ments made with Kundt’s apparatus and the same form of
experiment, both the refractive index and also the dispersion of the
metals undergo no demonstrable changes with temperature.
The behaviour of the metals can therefore no longer be looked
upon as in disaccord with the explanation I have given.— Wiede-
mann’s Annalen, no. 11, 1896.
* Wiedemann’s Annalen, vol. xly. p- 609 (1892).
t+ Jbid. vol. xxxvi. p. 824.
t Ibid. vol. lviii. p. 493.
[ 333 |
INDEX to VOL. XLII.
ADMITTANCE loci, on, 300.
- Air, on the thermodynamic pro-
perties of, 1.
Air-thermometer bulb, on the volume
measurement of an, 530.
Alternating currents, on the measure-
ment of, 271.
Aluminium, on the melting-point of,
37.
Appleyard (R.) on dielectrics, 148.
Are, on the resistance of the electric,
407.
Aston (Miss E.) on an Alpine nickel-
bearing serpentine with fulgurites,
116.
Atomic theory, on the genesis of
Dalton’s, 350.
Ayrton (Prof. W. E.) on galvano-
meters, 442.
Ballore (F. de M. de) on seismic
phenomena in the British Empire,
449,
Barr (L.) on the melting-points of
aluminium, silver, gold, copper,
and platinum, 37.
Bedell (F.) on admittance and im-
pedance loci, 300.
Books, new :—Goodwin’s Azimuth
Tables for the Higher Declinations,
116; Glazebrook’s James Clerk
Maxwell and Modern Physics,
205; Williamson’s Integral Cal-
culus (7th ed.), 205; Whetham’s
Solution and Electrolysis, 206;
Benjamin’s The Intellectual Rise
in Electricity, 368; Brown’s In-
troductory Treatise on the Lunar
Theory, 369; Behrens’s Anleitung
zur mikrochemischen Analyse der
wichtigsten organischen Verbind-
ungen, 447 ;_ van’t Hoff’s Studies
in Chemical Dynamics, 528.
Phil. Mag. 8. 5. Vol. 42. No. 259. Dec. 1896.
Bucherer (A. H.) on the action of
magnetism on electromotive force,
288.
Buckman (8. 8.) on the geology of
Dundry Hill, 2865.
Burstall (Ff. W.) on the use of bare
wire for resistance-coils, 209.
Cady (W.G.) on the volume mea-
surement of an air-thermoineter
bulb, 530.
Cajori (F.), search for solar X-rays
on Pike’s Peak, 451.
Campbell (A.) on new instruments
for the direct measurement of the
frequency of alternating or pul-
sating electric currents, 159 ; on the
measurement of very large and
a small alternating currents,
ZANE
Carbon megohms for high voltages,
on, 450.
Cooke (J. H.) on the stratigraphy and
paleontology of the Globigerina-
limestones of the Maltese islands,
122.
Copper, on the melting-point of, 37.
Crookes’ tube, on a rotational motion
of the kathode disk in the, 123; on
electric images in the field of a, 308.
Crosfield (Miss M. C.) on the eeology
of the neighbourhood of Carmar-
then, 122.
Current, on the magnetic field due
to an elliptical, at a point in its
plane within it, 107.
Currents, on the measurement of the
frequency of alternating or pulsat-
ing, 159, 271.
Dalton’s atomic theory, on the genesis
of, 350.
Davison (Dr. C.) on the diurnal
periodicity of earthquakes, 463.
yA =
534
Debus (Dr. H.) on the genesis of
Dalton’s atomic theory, 350.
Diathermancity, lecture experiment
on, 208.
Dielectrics, on the effect of tempera-
ture on the resistance of, 148.
Diffusion, on the separation of gases
by, 493. |
Duane (W.) on a damping action of
the magnetic field on rotating in-
sulators, 288.
“Ducks and drakes,” ona theory of,
iBEE
Dundry Hill, on the geology of, 285.
Dynamics, on the hypotheses of
abstract, 240. —
Earthquakes, on the diurnal periodi-
city of, 463.
Edison effect in glow-lamps, on the,
52.
Elastic constants, on the number of,
240.
Electric arc, on the resistance of
the, 407.
currents, on the measurement
of the frequency of alternating or
pulsating, 159, 271.
discharge in a magnetic field,
on the, 245.
images in the field of a Hittorf’s
(Crookes’) tube, on, 308.
waves, on the refractive indices
of some substances for very short,
207.
Electricity, on the passage of, through
gases exposed to Rontgen rays,
392.
Electrolytes, on the degree of dis-
sociation of, at 0°, 102.
Electromagnetic medium, on the
wave-surface and rotation of po-
larization plane in an aeolotropic,
224.
Electromotive force, on the action of
magnetism on, 288.
Electrons and electric charges, on
the theory of moving, 201.
Elements, on the analytical repre-
sentation of the periodic system of
the, 277.
Feilden (Col. H. W.) on the glacial
geology of Arctic Europe, 449.
FitzGerald (Prof. G. F.) on the
longitudinal component in light,
260.
Fleming (Prof. J. A.) on the Edison
effect in glow-lamps, 52.
TN DAR X.
Fourier’s series, on the convergency
of, 125.
Frequency of alternating currents, on
new instruments for measuring
the, 159.
Frith (J.) on the resistance of the
electric arc, 407.
Galvanometers, on, 442.
Gardiner (C. I.) on the Kildare in-
lier, 372.
Gaseous state, on the continuity of
isothermal transformation from the
liquid to the, 231.
Gases, on the passage of electricity
through, when exposed to Rontgen
rays, 392; on the separation of, by
ditfusion, 493.
Geological Society, proceedings of
the, 116, 206, 283, 371, 447.
Glow-lamps, on the Edison effect in,
52.
Gold, on the melting-point of, 37.
Goldhammer (Dr. A.) on the analy-
tical representation of the periodic
system of the elements, 277.
Granites, on the foliated, of Eastern
Sutherland, 447.
Greenly (E.) on the foliated granites
of Kastern Sutherland, 447; on
the geology of Hastern Anglesey,
448.
Harmer (F. W.) on the Pliocene de-
posits of Holland, 286.
Hittorf’s tube, on electric images in
the field of a, 308.
Holland, on the Pliocene deposits of,
286. .
Holman (Prof. S. W.) on the melt-
ing-points of aluminium, silver,
gold, copper, and platinum, 37.
Horne (J.) on the foliated granites of
Eastern Sutherland, 447.
Hyperphosphorescence, on, 103.
Iddings (Prof. J. P.) on extrusive
and intrusive igneous 10cks, 450.
Impedance loci, on, 300.
Insulators, on a damping action of
the magnetic field on rotating, 288.
Isothermal transformation from the
liquid to the gaseous state, on the
continuity of, 231.
Jamaica, on the geographical evolu-
tion of, 288.
Jones (Prof. J. V.) on the magnetic
field due to an ellipt cal current
at : point in its plane within it,
i07.
INDEX,
Kathode disk in the Crookes’ tube, on
a rotational motion of the, 123.
Lake (P.) on the Lingula-flags and
igneous rocks of the neighbour-
hood of Dolgelly, 371.
Lampa (Dr. A.) on the refractive
indices of some substances for very
short electrical waves, 207.
Lamps, on the Edison effect in glow,
52.
Larmor (J.) on the theory of moving
electrons and electric charges, 201.
Lawrence (R. R.) on the melting-
points of aluminium, silver, gold,
copper, and platinum, 37.
Light, on the longitudinal compo-
nent in, 260; on the influence of
temperature on the refraction of,
by metals, 532.
Liquid state, on the continuity of
isothermal transformation from
the, to the gaseous state, 231.
Liquids, on high tensions in moving,
pel.
Lussano (Dr. 8.), a lecture experi-
ment on diathermancity, 208.
McAulay (Prof. A.) on the wave-
surface and rotation of polarization
plane in an aeolotropic electro-
magnetic medium, 224.
MacGregor (Prof. J. G.) on the
hypotheses of abstract dynamics
and the question of the number of
elastic constants, 240.
Magnetic field due to an elliptical
current at a point in its plane
within it, on the, 107.
, on the electric discharge
in a, 245; on a damping action of
the, on rotating insulators, 288; on
the hypothesis of participation of
matter in the motion of the, 314.
Magnetism, on the action of, on
electromotive force, 288; on the
possibility of explaining the phe-
nomena of, by the hypothesis of
participation of matter in the
motion of the magnetic field, 314.
Mather (T.) on galvanometers, 442.
Megohms, on carbon, for high volt-
ages, 450,
Melting-points of aluminium, silver,
gold, copper, and platinum, on the,
o7
——
Metals, on the melting-points of
various, 37; on the influence of
530
temperature on the refraction of
light by, 532.
Microscopic vision, on, 167, 382,
423, 499.
Miller (Dr. G. A.) on the operation
groups of order 8p, p being any
prime number, 195.
Merdey (W.M.) on carbon megohms
for high voltages, 450.
Nipher (F. E.) on a rotational motion
of the kathode disk in the Crookes’
tube, 123.
Operation groups of order 8p, p being
any prime number, on the, 195.
Optical images, on the theory of,
with special reference to the mi-
eroscope, 167, 332.
Osmotic pressure, on, 289.
Oumoff (N.) on electric images in
the field of a Hittorf’s (Crookes’)
_ tube, 308.
Pavlow (Dr. A. P.) on the classifica-
tion of the strata between the
Kimeridgian and the Aptian, 120,
Periodic system of the elements, on
the analytical representation of
UNE, AN The
Platinum, on the melting-point of,
oT.
Polarization plane, on the rotation
of, in an aeolotropic electromag-
netic medium, 224,
Pollock (J. A.) on some experiments
with Rontven’s radiation, 453.
Poynting (Prof. J. H.) on osmotic
pressure, 289, -
ipreller (Dr: C. S. Du R2)jeonsthe
Pliocene glaciation, pre-glacial
valleys, and lake-basins of Sub-
alpine Switzerland, 117.
Preston (T.) on the continuity of
isothermal transformation from
the liquid to the gaseous state,
231.
Pulfrich (C.) on the influence of
temperature on the refraction of
light by metals, 532.
Radiometer motion, on, 373, 476.
Rayleigh (Lord) on the theory of
optical images, with special refer-
ence to the microscope, 167; on
the separation of gases by diffu-
sion and similar processes, 493.
Refractive indices of some sub-
stances for very short electrical
wayes, on the, 207.
36
Reid (C.) on the Eocene deposits of
Dorset, 207.
Resistance of dielectrics, on the
change in the, with temperature,
158.
Resistance-coils, on the use of bare
wire for, 209.
Reynolds (S. H.) on the Lingula-
flags and igneous rocks of the
neighbourhood of Dolgelly, 371;
on the Kildare inlier, 372.
Righi (Prof. A.) onexperiments with
Rontgen rays, 530.
Rodgers (C.) on the resistance of the
electric arc, 407.
Rontgen rays, some experiments
with, 162, 453, 530; on the passage
of electricity through gases ex-
posed to, 392.
Rosing (B.) on the possibility of
explaining the phenomena of mag-
netism by the hypothesis of parti-
cipation of matter in the motion
of the magnetic field, 314.
Rutherford (E.) on the passage of
electricity through gases exposed
to Rontgen rays, 392.
Salomons (Sir D.) on the electric
discharge In a magnetic field,
"245,
Samoiloff (A.) on electric images in
the field of a Hittorf’s (Crookes’)
tube, +08.
Silver, on the melting-point of, 37.
Skeat (Miss E. G.) on the geology
of the neighbourhood of Carmar-
then, 122.
Spencer (J. W.) on the geographical
evolution of Jamaica, 283.
Stoney (Dr. G. J.) on microscopic
vision, 332, 423, 499.
Strahan (A.) on submerged land-
surfaces at Barry, 119; on a phos-
phatic chalk with Ho/aster planus
at Lewes, 119.
INDE X.
Sunlight, on the absence of X-rays
from, 461. ;
- Sutherland (W.) on high tensions in
moving liquids, 111; on thermal
transpiration and radiometer mo-
tion, 373, 476.
Switzerland, on the Pliocene eglacia-
tion, pre-glacial valleys, and lake--
basins of Subalpine, 117.
Talmage (Prof. J. E.) on certain
linear marks in a sedimentary
rock, 118. j
Thermal transpiration, cn, 373.
Thermodynamic properties of air, on
o they le
Thompson (B.) on the junction-beds
of the Upper Lias and Inferior
Oolite in Northamptonshire, 121.
Thompson (Dr. 8. P.) on hyperphos-
phorescence, 103 ; on some experi-
ments with Rontgen’s rays, 162.
Thomson (Prof. J. J.) on the passage
of electricity through gases ex-
posed to Rontgen rays, 392.
Threlfall (Prof. R.) on some experi-
ments with Rontgen’s radiation,
453.
Transformers, on the use of air-core,
for testing purposes, 271.
Vision, on microscopic, 332, 423,
499.
Wave-surface in an_ aeolotropic
electromagnetic medium, on the,
224.
Wildermann (Dr. M.) on the degree
of dissociation of electrolytes at 0°,
102.
Williams (W.) on the convergency
of Fourier’s series, 125.
Wilson (E.) on the geology of
Dundry Hill, 285.
Witkowski (Prof. A. W.) on the
thermodynamic propertiesof air, 1.
X-rays, on the absence of, from sun-
light, 451.
END OF THE FORTY-SECOND VOLUME.
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