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CONTENTS.
(A)
VOL. 213.
Advertisement page v
I. On the Refraction and Dispersion of the Halogens, Halogen Acids, Ozone, Steam,
Oxides of Nitrogen and Ammonia. By CLIVE CUTHBEKTSON, Fellow of
University College, London University, and MAUDE CUTHBERTSON. Commu-
nicated by A. W. POSTER, F.R.S. page 1
II. On a Cassegrain Rejlector with Corrected Field. By Dr. R. A. SAMPSON,
F.R.S. 27
III. The Thermal Properties of Carbonic Acid at Low Temperatures. By C. FBEWEN
JENKIN, M.A., M.Inst.C.E., frofessor of Engineering Science, Oxford; and
D. R. PYE, M.A., Fellow of New College, Oxford. Communicated by Sir
J. ALFRED EWING, K.C.B., F.R.S. 67
IV. The Capacity for Heat of Metals at Different Temperatures, being an Account
of Experiments performed in the Research Laboratory of the University
College of South Wales and Monmouthshire. By E. H. GRIFFITHS, Sc.D.,
F.R.S., and EZER GRIFFITHS, B.Sc., Fellow of the University of Wales . 119
V. On the General Theory of Elastic Stability. By R. V. SOUTHWELL, B.A., Fellow
of Trinity College, Cambridge. Communicated by Prof. A. E. H. LOVE,
F.R.S. 187
a 2
VI. Some Phenomena of Stmtpots and of Terrestrial Magnetism.— -Part II. By
C. CHRER, Sc.D., LL.D., F.R.S., Superintendent of Kew Observatory
. . page 245
VIL On the Diurnal Variations of the Earth's Magnetism produced by the Moon
and Sun. By S. CHAPMAN, B.A., D.Sc., Chief Assistant at the Royal
Observatory, Greenwich. Communicated by the Astronomer Royal . . 279
VIII. A Critical Study of Spectral Series.— Part HI. The Atomic Weight Term
and its Import in the Constitution of Spectra. By W. M. HICKS,
F.R.& ....... ."*.' . ".' 323
IX. On the Self -inductance of Circular Coils of Rectangular Section. By T. R. LYLE,
M.A., Sc.D., F.R.S. 421
X. A Method of Measuring the Pressure. Produced in the Detonation of High
Explosives or by the Impact of Bullets. By BERTRAM HOPKINSON,
F.R.S. . 437
XI. Gravitational Instability and the Nebular Hypothesis. By J. H. JEANS,
M.A.,F.R.S. 457
Index to Volume 437
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PHILOSOPHICAL TRANSACTIONS.
I. On tli,' Ift'frtirfioii <uul Dispersion of the Halogens, Halogen Acids, Ozone,
Steam, Oxides of Nitrogen and Ammonia.
Hit ( 'I.IVK ( YTHIIKRTSON, l''<'il»ir <>f I ' ,' > r, ,-^ity College, London Univerm/ >/.
n,td MAUDE CUTHBERTSON.
Communicated by A. W. PORTEK, F.H.*.
Received October 18, 1912,— Read January 16, 1913.
CONTENTS.
Page
INTRODUCTORY.
Scope of research
Expression of results : formula and its interpretation . . .
Summary of results
EXI'KKIMENTAI..
Measurements of refraction and dispersion —
Chlorine
Standard conditions
IT
Bromine
Iodine
Hydrochloric acid
Hydrohromic acid
determination of density of I3
•I n
Ilydriodic acid
„ „ determination of density of ; •
Steam . . . .' • •
Ozone
Ammonia
AO
Nitric oxide
no
Nitrous oxide
98
Errors of experiment
05
Calculation of constants
IT has long been well known that the refractive indices of simple gaseous compounds
do not obey the additive law so closely as those of solids or liquids. From the study
of these last GLADSTONE and DALE, and their followers, succeeded iu obtaining
refraction equivalents for a large number of the elements which were fairly constant
for the same class of compound. But in gases the discrepancies were found to be
VOL. CCX1IL— A 497. B Published separately, April 4, 1913.
Ml:. CLIVE COTHi:m:i>"N *N" MAVM COTffi*ftfBOU ON THE
much wider and this api-eared the more surprising since, in other fields of research,
the gaseous state has Pn,v,-d peculiarly favourable for the discovery of simple
relations.
Accumulation of Data.— The investigation, of which the present paper forms part,
was designed to throw light on the cause of these anomalies.
The first step was to'enlarge the field of the enquiry by the accumulation of data,
and with this object we have, either together or in collaboration with others, deter-
mined and redetermined the refraction and dispersion in the gaseous state of fourteen
elements and ten compounds within the limits of the visible spectrum, and the
refraction of six elements and four compounds for a single wave-length.
The numU-r is still far too small. Many interesting compounds remain to be
invest igated. But, as the present instalment of work has occupied nearly two years
and has led to certain definite conclusions, it seems better to publish it rather than to
await the addition of more difficult and, perhaps, less instructive examples.
( '/,,./.-. ,•/' ( '••ntiHtundx. — In selecting the compounds to be examined we have been
guided by the principle that the molecule should consist of as few atoms as possible,
and that the refraction and dispersion of each constituent should be measurable in
the gaseous state. It is unfortunate that the list of compounds which comply with
these requirements is very short. All compounds of carbon are excluded. Of those
substances which are dealt with in the present paper, the most important are the
halogen acids, which form a regular series of simple diatomic molecules. Steam, S02,
and HjS form an interesting cycle, and the two oxides of nitrogen are very instructive
since the eonstituents are the same in both cases.* Ozone is remarkable as an
example of the effect of polymerization on the refractivity.
l-'.j- 1 >,•• .<>•/••// of Results. — The choice of a formula for the expression of results is of
fundamental importance. Previous workers on the subject of gaseous refractivities
have almost invariably used that of CAUCHY, with two terms or three, according to
the degree of accuracy of their figures. But this formula is not based on modern
physical theory. Moreover, we have shown in a previous paperf that, when only two
constants are used, it is inadequate to express the experimental results, even in the
\ isible spectrum ; while if a third constant and term involving I/A4 is introduced, the
shape of the dispersion curve cannot be easily grasped on inspection of the figures.
For these reasons we have abandoned this formula, and have used, tentatively, a
formula of SKI.LMKYKKS type, £ l = M-i (approximately) = N
« yl LJl^ ff"
It is unnecessary to defend the adoption for gaseous indices of this expression,
been widely used for solids, and is in general outline established on theory
• The exwnin.tiou of NO, and N A haa been postponed on account of its difficulty, but promises to be
1 important owing to the association which characterises it
Hspersion of Air, Oxygen, Nitrogen, and Hydrogen and their Relations,"
, Mid M. GtTHBwmtoN, • Roy. Soc. Proc.,' A, vol. 83, p. 151, 1909.
KKFKACTION AND Dlsl'KltSION OF T1IK HALOGENS, HALOGEN ACIDS, ETC. 3
and confirmed by experiment. But it is desirable to emphasise the fact that the
calculation of the constants given in this paper from a formula containing only one
tt-nn on the right-hand side is only provisional, since the main conclusion of the
authors is that, Ixitli for i-lcim-iils and compounds, a single term is inadequate, except
in the case of monatomic gases. The simple form of the formula is, however, useful
for indicating at a glance the direction and magnitude of the changes in refractive
and dispersive power which take place when elements combine to form a compound.
It' /<, n-, are the refractive indices of a substance for two wave-lengths for which the
frequencies an- //, H.J, we have
I -Ma _ ni-nj ._
approximately,
since »/ is usually small compared witli H{IJ. The left side of the equation expresses
the dispersive power of the substance, which is thus seen to be inversely proportional
ion,3.
Chiiniji-x <>f /f,'fi-'i<-/ir<- Power. — Let yuA — 1 = 3 A »'• MB~! = » IL~l be the
^0 A — ft "•' B — ft
formuko which express the refractivity of two gaseous elements in the region of the
visible spectrum, and let MAB— 1 = 3 ** a express the refractivity of the compound
~~
wliich they form. If the molecules of the two elements are diatomic, and one
molecule of the compound is formed of one atom of each, then the change of refractive
jx>wer on combination is
. f NA ( NB 1 NAB
' In.^v-w8 «»'B— n*J WAB-W*'
<>f />/.<.•/ /r/-.s//v I '< ,!<•<• r.— If the additive law were strictly followed the
dispersive power of a compound, measured in a region remote from free frequencies,
would lie between the dispersive powens of its constituents, i.e., HO'AB would lie
Ix-tween nj^ and H^B. For it can be shown that, for a short region of the spectrum,
it-mote from free frequencies, so that n0a is large compared with ;ta and n*jn* can be
neglected,
A n A "o
and this expression lies between these limits.
Hence, if the experimental value of /iuaAB differs from this value, the variation must
be due to the changes in one or more of the four qiiantities NA, NB, n/x, n^a
consequent on combination. It is evident that all four unknown quantities cannot
be determined from a knowledge of NAB> ft,iaAB, which is all we obtain from a deter-
mination of the dispersion of the compound. But two cases should IK' distinguished.
The value of MU'AB mav vary owing to changes in u/v and ti^B consequent on
combination, i.e., to real modifications of the free periods of the vibrators ; or it may
B 2
4 MI: n.m: rrniMi-.KTsoN \M> \i.\n.i: ITTIIKKKTSON ON Tin-
be ih- t.. tin- introduction of a new free period, or the elimination of an old our,
previously wrap|»-«l up i" "A •'«"'' "..V
^ nnvtry qf HemJtf. — Before proceeding to give the experimental work it will }>e
eonvi-nii'nt t<> Niimmaris.- tlie results obtained :—
(1) In hydrochloric, hydrobromic, hydriodic acids, sulphur dioxide* and sulplm-
n-tt.-d hydrogen* tin- refractivity of the compound is less than thr sum of
tin- ivtra<-tivitit-.s of tin- tlt-im-nts, and the dispersive power of the compound
lie* Ix-twccii those of its constituents ;
rJi In nitrous oxide, nitric oxide, ammonia, and ozone the refractivity of the
compound is greater than the sum of those of its constituents, and the
dispersive power is greater than that of either constituent ;
(3) In steam the refractivity of the compound is less, and the dispersive power
greater, than those of its constituents ;
(4) In all cases the change in dispersive power is great relatively to the change in
refractive power.
We hav«- framed a hypothesis which, in our opinion, would account for these
changes in a qualitative manner, and we hope to puhlish it elsewhere.
CHLORINE.
-Tlie refractive index of gaseous chlorine has only been measured
IH-|...Mst found M-l== "000772 for white light with gas, prepared from
MnO,, whose density was 2' 47 (air l).
MASCART} found „-! == "000768 for white light. He worked at low pressures at
tin- temperature 12° C. and compared the refractivity with that of air under the
vuiie condition.-,.
ITie dispersion of the gas has not previously been measured.
-The gas we used was prepared by dropping strong hydrochloric
» potassmm jK-rmanganate. After washing in water and drying by sulphuric
«™fe»ed in a hath of acetone cooled with solid C02 and then allowed to
the air ,„ tl,, oonn.cting tubes had been displaced. The refractometer
My ,-va,uated, was then placed in connection with the chlorine by a
tube so fi,lt- that the gas entered the tube sufficiently slowly for the
Wndn to be ,ount,l. When the pressure of gas in the refLtometer
-*
;:
l;| ;i -I: ACTION ANI> IHSI'KKSIOX OF TIIK HALOGENS, HALOGEN ACIDS, ETC. 5
After each experiment was over tl..- -as was iiUirlied over soda lime and only
experiments in which the in.|..iriti.-s were negligible were used in determining the
refractive index,
The light used WM that ..f tin- green mercury line, A =
Refractivity. The following figures were obtained in fire experiments, the
experimental values U-iiig reduced to <>' < '. ami 7C,n nun. liy the formula :-
^ T 760
» -J73-T-'
Ou-OlO7. . . . 797«i, 7UK5, 7'J6«. 7'JKo, 7981. Mean 7980.
St.n.'i'ir'l ( 'onditiona. 'I'he practice of reducing observations of refractivity to the
standard temperature of .1 < '. and the pressure of 760 mm., dates from a time when
deviations fn-m the laws of BOYI.K and GAY-Lrss.vr were alike unknown. As
accuracy improved and the field of research was extended to vapours, these ,-/•//.
became insufficient and sometimes meaningless. MASCART, the volume of whose
work entitles him to l>e considered the leading authority on the subject, at first
adopted the old conditions, and even in the OBM of SO, expressed the refractivity as
it would lie at 0° and 760 mm. But, in his later work, when dealing with chlorine
..,„.! bromine and with some organic compounds f,,r which the coefficients of thermal
expansion and compressibility were unknown, be contented himself with determining
the refractivity at pressures as low as possible and comparing it with that of air at
the same temperature (12° C).*
LK Roux, in bis experiments on sulphur, mercury, phosphorus and arsenic
expressed the ratio of the refractivity to the density, and LORENZ and PRYTZ adopted
the same system. It is evident that this principle is the most convenient for those
who wish to compare the refractivity of equal numbers of atoms of different elements,
or of molecules of different eomixmnds. Accordingly, in the present work, we have
reduced all ref inactivities to the values which they would have had if the gas or
vapour had the density of hydrogen at 0° C. and 76 cm., ('000089849) gr./(cm.)-
multiplied by the ratio of the theoretical molecular weight of the substance in
question to that of hydrogen. But, in order to avoid confusion, we shall denote this
value by the syinlx>l (M-l) rP^r , where D denotes the standard density as here
defined, and (<!„-,,,} the density at ()" C. and 7»i cm
The density of chlorine at the temperature and pressure of t he at mospliere has
determined recently by TI;I: M.WKI.I. and CnRI8TIE.t They found that at "J(f C., the
molecular volume W»fl L'-jn:!-.! and liUOaO'O at 10° C.
* MASCAKT, 'C. R.,' v.,1. s6, pp. :ii!l and lls-j.
t F. P. TRKALWELL and W. A. K. CHKISTIK, 'Zcits. aiiorg. Chem.,' Vol. 47, p. 446, 19(
Mi:. CLIVE CUTHBKRTSON AND MAUDE CUTHBERTSON ON THE
Tli-- average temperature of our experiments was 19°'4 C. and at that temperature
the molecular volume of chlorine would be 22038'9 c.c. That of hydrogen is
•_'L' 1^8'8 c.c. Hence the refractivity observed must be multiplied by the ratio of these
immlwrs. We thus arrive at the number '00078412; and since the accuracy of the
i-\|M-rinu*ut8 is not greater than 1 part in 1000 we may accept '000784 as the
n-t'ractivity of chlorine for the green mercury line.
Dispersion. — Assuming this value, the dispersion was measured at seven other
points of the visible spectrum by the method described in previous papers.*
The following table shows the results : —
TABLE I. — Dispersion of Chlorine.
Ax 10".
D
•<*»)
Observed.
Calculated.
Difference.
6707-85
77563
77556
_ 7
6438-5
5790-5
5769-5
5460-7
• c'09-1
5085-8
4799-9
77703
78121
78135
78400
78651
78791
79166
77697
78123
78139
78402
78655
78792
79156
- 6
+ 2
+ 4
+ 2
+ 4
+ 1
-10
The numbers shown in the column headed "calculated" are derived from the
formula
(M_l) D 7-3131x10*
Kw) 9629 '4 xlO»-«»'
„ i8 the frequency of the light, i.e., = X - _jxl°lu
XX (in cm.)
M^mW ,o -W^W-The absorption spectrum of chlorine has
-nveafgated by Ifa. LA.RD.t She describes the spectmm aa con8isting of
» of general abeorpt.on which extends, in a column of gas 60 cm long
.50 to A 2599, and lengthens in hoth directions with increasing preZa but
* .L7 L8 , ^rangibk 6"d' reaChi"« X 4"° "' ^ 25Z?2
.pectrum ly.ng between A 4799 and X 5350 which consists of * of da,
* fce «Roy. Soc. Proe.,' vol. 83, p. 152, 1909.
'Aatrophyi,. Journal,' 14, p. 85, 1901.
INFRACTION AND DISPERSION OF THE HALOGENS, HALOGEN ACIDS, ETC. 7
The region of which the refractivity was measured by us extended from \ 6708 to
X 4799'9 and thus covers the whole range of the line spectrum and 200 A.U. which
are affected by the general absorption.
It is, therefore, of interest to find that the observed values of the refractivity lie on
a smooth curve. But it is significant that the calculated curve cuts across the
experimental, in which the curvature is greater. This appears to indicate that a
single term formula is inadequate to express the results, and that a second term is
required, in which both N and n* are small, to represent the influence of the
absorption band.
In the region of the line spectrum it was to be expected that if any variations of
refractive index accompanied the variations of absorption they would be found either
in the immediate neighbourhood of each dark line or possibly affecting the whole
breadth of each group of lines forming a fluting. In order to investigate this point
the following test was made : —
The paths of the two interfering rays of light were equalised, so that when the
wiive-length of the light employed was continuously changed from red to violet no
rh.mge was observed in the position of the interference bands in the field of view.
Chlorine was then admitted into one tube till the path of that beam had been retarded
by 450 bands (\ = 5461). Next, by means of the compensator,* the same beam was
accelerated by an equal amount. If, now, the wave-length of the light be changed
from red to violet any movement of the bands would be due to the difference
of refractivity of glass and chlorine for the particular wave-length which is in the
field of view. The bands can easily be read to 1/10, so that when 450 bands have
passed a difference of refractivity of 1/4500 can be detected.
The slit was then narrowed till the interference systems due to X 5790 and X 5769
were clearly separated in the field of view : i.e., till the light composing any particular
part of the image varied by less than 20 A.U. On changing the wave-length
continuously from red to violet no sudden change in the bands could be detected. It
may therefore be concluded that between X 6708 and X 4799 any sudden change of
refractivity exceeding '000784/4500 = '0000017 must te confined to a breadth of less
than 20 A.U. and probably to less than half that amount. It is not possible to detect
small changes in the refractivity in a narrower section of the spectrum than this, since,
if the light is sufficiently dispersed, it becomes too feeble to read tenths of a I >;i i H I.
BROMINE.
Previous Work. — DUFKT records MASCART'S value n = 1 '001 125 for the D line.
The dispersion of the gas has, apparently, not been attempted.
Preparation. — The purest bromine obtainable from Kahlbaum was used. Before
every experiment the bulb containing the liquid was cooled to —80° C. and exhausted,
h This compensator, of special construction, retards all wave-lengths equally except in so far as
dispersion affects them.
MI; n.ivK r.rrm:i:i;is"V .\\i» MATDI: CPTFIBERTSMN ox THK
K> as i" irt ri.l "I' ' '' H Br which might have formed since the previous
experiment. Sim-.- it was neressary for the gas to pass through greased taps more
.•lal.T.it. f.ivr.-niti.Hiis f',,r purification would have l>een useless. The grease used
s.. illv prep I firoi ; . paraffi md oil. \tter each experiment the
uir \\liirh entered the refractometer tube was admitted to contact with the
iiry and alworhed. Any admixture of HBr would have been measurable as
hydrogen. In the experiments on which we rely to obtain the index the residue was
negligible.
/V...-.-I/I//V. — In order to reduce the observed refractivity to standard conditions it
was necessary to measure the density of vapour employed. For this purpose the
bulb containing tin- liquid was connected with a density bulb in parallel with the
refractomettT tulx- and a determination of the density of the vapour accompanied
each experiment. The atomic weight of bromine was taken as 79'97 (O = 16).
Owing to the great al>sorption of the vapour in the green it was necessary to use red
light for the determination of the absolute index. With X = 6438 as many as 80 or
90 bands could be read, whereas at X 5461 the band system was no longer readable
after 25 bands had passed.
The absolute index was determined from the following five experiments which were
well corroborated by several others not quite so trustworthy :—
TABLE II. — Refractive Index of gaseous Bromine, X =*6438.
Experiment.
(n l\ 10« ^
Number of bands
counted.
\r l) lvr ,1 ~Ci
(«o<6)
1
1158
55
•J
1156
65
3
1154
89
4
1159
50
;>
1157
cso
Mean ....
1157
Owing to the strong absorption of the vapour in the visible region
th- dispersion were difficult, and the accuracy attained was much
|t -how,, n, the case of chlorine. The following figures show approxi-
l«u,ds winch were readable at different points of the spectrum :-
Number of bands
readable.
• . More than 120.
Ax 10*.
. 7' J
,-,
RM
It
120.
115.
90.
60.
A x 10«.
5600
5461
5209
5085
Number of bands
readable.
. Less than 30.
• ,, „ 25.
„ „ 20.
10.
REFRACTION AM' l>lsri;i;sl<>N OF Till. 1 1 A LOGENS, HALOGEN ACIDS, ETC. 9
Beyond 5461 the number of Iwinds read was not sufficient to ensure trustworthy
values.
The following table gives the experimental values in column 2 : —
TABLE III. — Dispersion of gaseous Bromine.
ILL \\ 107 P
? (4,76)
Ax 10".
Observed.
Calculated.
Difference.
6708
11525
11518
- 7
6438
11570
11571
+ 1
6000
11662
11675
+ 13
5800
11735
11731
- 4
5750
11741
11746
+ 5
5700
11762
11762
0
5600
11796
11767
-29
5461
11849
11842
- 1
The experimental numbers fall approximately on a smooth curve which is given by
D 4'2838 x 10*7
(M-!)
(t/,,76) 3919'2 xlO*7-™3'
The figures calculated from this equation are given in column 3.
Relation of Dispersion to Absorption. — As in the case of chlorine a test was made
for a rapid change of refractivity affecting a narrow section of the spectrum, but none
was detected.
In this case also the change of refractivity is small compared with the increase of
the absorption as we pass from the red to the green.
IODINE.
Previous Measurements. — The only determination of the refractive index of iodine
on record is that of HURION,* who gives M = 1 '00205 for the red and 1 '00192 for the
violet. He employed a prism and heated the iodine to 700° C.
It was hoped that with a refractometer a higher degree of accuracy could be
obtained, but the results of experiment were disappointing. The absorption band
which has its maximum at X 5000 extends so far into the red that, with the
faint light available in a Jamin apparatus, the band system was very soon
obliterated.
In the red (X = 6438) as many as 21 bands could be observed with difficulty, but
VOL. ccxm. — A.
* 'Journal de Physique,' I., VII., p. 181.
C
10
MR CLIVE CITHRERTSON AND MAUDE CUTHBERTSON ON THE
at X = 5600 it was not possible to read more than three, and on the violet side of the
region of absorption no measurements were possible. In attempting to measure the
dispersion tin- . -\IH-I im.-nt.T has to choose between a small number of bands read over
a sli^'htlv wiili-r range and a larger number read over a small range. In either case
the errors of observation are relatively large.
Procedure. — A weighed quantity of iodine was introduced into the refractometer
tube which was evacuated and sealed off.*
The tube was then heated till the solid had all sublimed and the bands observed.
Refraction. — The wave-length selected for the absolute determination was 6438,
and this was obtained from white light of a Nernst lamp by means of a fixed deviation
spectroscope. The volume of the refractometer tube was 49' 1 c.c. and the weight of
iodine which it contained was '00473.
The beet experiments gave for /u-1 the value '00210, and this is probably correct
to 1 or 2 per cent. It agrees well with HURION'S value, which was probably for a
longer wave-length than 6438.
Dwpemon. — Assuming this value the following numbers were obtained for the
refractivity in the red-orange, the number of bands read being 97 for X = 6438.
TABLE IV. — Dispersion of gaseous Iodine.
Ax 10s.
"-"""(W
G438
2100
G280
2100
G150
2150
6100
2180, 2170, 2140
In another set of readings the number of bands read was only 2'1 in the red, and
the following readings were taken :—
Xx, 0- 6708, G438, 6215, 6,80, 5COO. 5250, 5,00, 5005, 5000
0.-0.0- . ,970, 2100, 2,30, 2,30, 2170, 2250, 2210, *1M, 2120.
r r at i~t th«
, ,ug an absorption
"ch seems to
„,
... A, vol. 204,
REFRACTION AND DISPERSION OF THE HALOGENS, HALOGEN ACIDS, ETC. 1 1
It was not considered worth \vhilr to s|x'ii<l further time in multiplying observations
which could never command great confidence, owing to the fewness of the bands
read. We hope to return to this rlnnent, using the method of crossed prisms, which
is more suitable than that of the interferometer.
HYDROCHLORIC ACID.
Previous Determinations. — DULONG obtained 1*000447 for white light, and
MASCART 1 '000444 for the D line. The dispersion has never before been
attempted.
Preparation. — The gas was prepared by dropping sulphuric acid on pure sodium
chloride. After passing through two drying bulbs filled with sulphuric acid it was
condensed in liquid air and allowed to boil off. When the gas had flowed through
the connecting tubes for 15 minutes so as to displace the air, it was admitted to the
refractometer and allowed to flow till the pressure was atmospheric. The following
table gives the experimental values found, reduced to 0° C. and 760 mm. by the
formula
/ .x Tx760
}
Experiment . 1, 2, 3, 4, 5, G, 7.
(,u-l)l07 . . 4514, 4513, 4508, 4512, 4510, 4509, 4510.
Mean 451 1.
This value requires correction for the density of the gas. GRAY and BURT* found
that the volume of hydrogen from two volumes of hydrochloric acid is 1'0079.
LEDUC gives the mean coefficient of expansion of the gas at constant pressure as
•003736.
The average temperature of our experiment was 16° C.
Hence the experimental value must be multiplied by
lx(l + 16x'003736)
1'0079 x(l + 16 x '00366)'
whence we obtain
Dispersion. — Assuming this value the dispersion was determined from six experi-
ments. The following table shows the results: —
* 'Trans. Chemical Society,' 95, II. of 1909, p. 1604.
C 2
!•_• \||;. i I.IVI-: <rTlli;i.KTSON AND MAUDE CUTI1BERTSON ON TIIK
TABLE V. — Dispersion of Hydrochloric Acid.
tlL }}•* 10» D
(P i) x iu • 7«\
AxIO*
Observed.
Calculated.
Difference, 3-2.
6707-85
44375
44367
- 8
6438-5
44444
44437
- 7
5790-5
44656
44661
+ 5
5769-5
44666
44670
+ 4
5460-7
44800
44803
+ 3
5209-1
44930
44933
+ 3
5085-8
45007
44994
-13
4799-9
45187
45191
+ 4
Using the SELLMEYER form of equation the refractivity is expressed by
/ _!\P _ 4-6425 xlO27
K76) 10664 xlO27-?^'
Hie values calculated from this expression, in which the constants are calculated
from the observations by the method of least squares, are shown in column 3 above,
and the differences between columns 3 and 2 are given in column 4.
HYDKOBROMIC ACID.
^ Prevwu* Determinations.— M.ASGA.KT obtained M- 1 == "000570 for the D line The
dispersion has not been attempted.
/'reparation -The gas was prepared by dropping the purest aqueous solution of
. phosphorus pentoxide. After passing through tubes containing red
>rus and phosphorus pentoxide, it was condensed in liquid air, sometimes twice
>met,mes once only. !„ successful experiments the acid was obtained as a pure
and a colourless liquid. After an experiment the gas was absorbed over a
xia hme ,» r^o. Only those experiments in which the impurity
ras^n. ?lig,t B were used for the determination of the index.
adopted fo
Experimnit . .
(M-l)lO'x-£-
of llydrobromic Acid. X = 54G1.
l> 2. 3, 4,
6167, 6153, 6151,
Mean 614y.
6141,
5,
6139,
6.
6141.
REFRACTION AND DISPKRSION OF THE HALOGENS, HALOGEN ACIDS, KTC. 13
l>i*persion. — Assuming this value the following values of the dispersion were
obtained from eight
TAHLE VI. — Dispersion <>f Jlydrobromic Acid.
Xx 10".
/„ i\io» D
W«)
Observed.
Calculated.
Difference, 3-2.
6707-85
60752
60751
- 1
6438-5
60878
60873
- 5
5790-5
61245
61245
0
5769-5
61256
61260
+ 4
5460-7
61490
61490
0
5209 • 1
61704
61710
+ 6
5085-8
618-j)
61830
+ 6
4799-9
62160
62149
-11
Using SELLMEYEK'.S formula the refractivity can be expressed by the equation
_.v D 5'1446 xlO-'7
''(da76) 8668-4xl027-N,a'
Calculated values are shown in column 3 and differences in column 4.
Density of Itydrobromic Acid. — As the density of the gas at temperatures higher
than 0° C. does not appear to have been previously measured the following values are
perhaps worth recording. The degree of accuracy was not carried beyond one part in
a thousand, since errors in reading the refractivity were not less than this amount.
The gas was weighed at atmospheric pressure and the temperature of the room,
which averaged 19° C., and the values were reduced to 0° C. and 760 by the formula
T 76
D' = D*X273XP-
Three experiments gave, for the weight of a litre, on these assumptions, 3'648,
:i '('• 17, and 3'650 gr., the mean of which is 3'6484.
The theoretical weight is :n>l(i:'.:!.
HYDRIODIC Acm
Previous Determination*. — MASCART found /x— 1 = '000906 for the D line. The
dispersion has not been attempted.
Preparation. — The gas was prepared by slowly dropping pure aqueous solution of
14
Mi: CLIVE CUTHBERTSON AND MAUDE CUTHBERTSON ON THE
the acid on phosphorus pentoxide and proceeding as in the case of hydrobromic acid.
The solid obtained on free/ing was colourless, but the liquid was usually a pale pink,
owing to a trace of dissolved iodine. As the boiling point of HI is far below that of
imline the quantity of iodine subliming, at the boiling point of HI, from this mixture
was negligible.
Tests for impurity, similar to those in the case of HBr, were equally satisfactory.
Rtfractinty, \ = 5461. — In this case also measurements of refractivity had to be
supplemented by those of density as this has not previously been determined carefully.
In three trustworthy experiments the following figures were obtained for the
refractivity at the green mercury line : —
a076
. . . . 9237, 9277, 9260. Mean 9258.
The mean is taken as the best value.
Dispersion. — From seven experiments the following values were obtained for the
dispersion : —
TABLE VII. — Dispersion of Hydriodic Acid.
AxlO8.
(n 11 x 10s ^
(476)
Observed.
Calculated.
Difference, 3-2.
6707-5
6438-5
5790-5
5769-5
5460-7
5209-1
5085-8
4799-9
— •
91087
91334
92087
92106
92580
93015
93257
93900
91089
91335
92080
92109
92572
93016
93259
93905
+ 2
+ 1
-7
+ 3
-8
+ 1
+ 2
+ 5
Using SELLMEYER'S equation the refractivity can be expressed by the formula
Qu-1) D 57900 xlO27
K76) 6556-4 xlO"-n''
Fhe calculated values are shown in column 3 and the differences in column 4.
** of Hydnodic ^.-The density of this gas also has not been accurately
Umlated m the same way as in the case of hydrobromic acid three
i gave for the weight of a litre 5789, 5791, 5793, mean = 5791 gr
we,ght is 57151, taking H == 1'008 and I = 126-97, and the weight
ot a litre of oxygen as 1 '4290 gr.
REFRACTION AND DISPERSION OF THE HALOGENS, HALOGEN ACIDS, ETC. 15
STEAM.
Previous determinations on the refractivity of water vapour are given by DUFET
as follows : —
Observer.
Light
(/x-l)10«.
FlZKAU
D
254
JAMIN ....
D
257-9
MASCART
D
257
LORENZ
D
250
No one appears to have attempted the dispersion.
Procedure. — A weighed quantity of distilled water, sealed up in a thin capillary
tube, was introduced into the refractometer tube, which was then evacuated and
sealed off. On breaking the capillary by a jerk the tube was filled with vapour.
After adjusting the tubes between the mirrors of the interferometer the centre of the
tube containing the water was first cooled to a known temperature and then the tube
was heated till the whole of the water present had evaporated. To the number of the
bands read was added a proportionate number for the vapour present at the initial
temperature.
In order to eliminate the errors of drift other experiments were made in which the
ends of the tube were kept near the maximum temperature required (about 140° C.)
and the centre of the tube gradually cooled to the temperature of ice.
Refractivity. — Experiments were made with four charges of water. The results
are given below : —
Kefractivity of STEAM.
Experiment.
^-1)10T^6'
Approximate numltcr of
bands read.
Remarks.
(D
(2)
(3)
W
2523
2491
2534
2524
178
379
130
300
Mean of 3 experiments.
» 4 „
4
» »
>i 2 „
Mean of 1, 3, and 4 . .
2527
It will be seen that the second charge yielded results considerably lower than the
other three. The cause of the discrepancy was found to be the unequal distribution
of vapour between the main portion of the refractometer tube aiid the small
" appendix " left when the side tube is sealed off. When the temperature of the
ends is markedly higher than that of the middle (as it was in this series), the error
becomes considerable. Neglecting this experiment we take the mean of the other
I6 MR n.lVK PBTHBBMSON AND M AM-.: CUTIIHERTSON ON THE
thw, M the value for the*** n.ercury line. The variations of these experiments
t.^— ADBU1U nif, > „ , j ]
from seven experiment* with the largest charge of water, the number of bands read
being about 380 for X = 5461 :-
TABLE VIII. — Dispersion of Steam.
AxlO».
; o-1)101^-
Observed.
Calculated.
Difference, 3-2.
6707-85
6438-5
5790-5
6769-5
5460-7
5209-1
5085-8
4799-9
25028
25069
25191
25195
25270
25345
25380
25495
25027
25068
25191
25196
25272
25345
25384
25490
-1
-1
0
+ 1
+ 2
0
+ 4
-5
Using the SKI.KMKYEK equation the refractivity can be expressed by
, ,s_D_ 2-62707 x 10a;
'd076 ~ 10697 xlO27-?!2'
The numljers calculated from this expression are shown in the third column and
the differences between column 3 and 2 are given in column 4.
OZONE.
/'/•,/•/« I/.N- II'.. rk. — No previous work on the refractivity of ozone is recorded in the
usual books of reference. The difficulties are considerable. It is impossible to
prepare ozone even approximately pure, and if it were possible it would be inadvisable
to do so, since the decomposition of the molecules during the time necessary to
measure the refraction and dispersion would introduce fruitful sources of error.
fVponhtre. — Of the two best methods of preparing the gas, electrolysis of a
solution <>f sulphuric acid has produced the highest percentages of ozone, FISCHER
and MASSENEZ* having obtained over 28 per cent, by weight. But the objections to
this method seemed to us to outweigh its advantages. It was necessary that the gas
used should be absolutely pure oxygen, for the smallest trace of moisture, air or
hydrogen would introduce large errors; and in the electrolytic process the gas is
v f O
produced wet and is mixed with air in the connections. For these reasons the method
selected was that of ozonising by means of the silent discharge in a vessel of the type
* ' Zeit. fur Anorg. Chemic,' vol. 52, p. 229, 1907.
REFRACTION AND DISPERSION OF THE HALOGENS, HALOGKN ACIDS, ETC. 17
used by BERTH ELOT. The average yield was 6 per cent, by volume, but on one or
two occasions it reached 10 per cent. We failed to identify the causes which
produced these higher yields, and were unable to rejx-iit them, hut succeeded in
obtaining between f> ;tnd 7 per cent, with fair regularity.
.Methods. — As in the case of other gases, tin- work was <li\'ul.-<l into two part*
(l) the determination <>f the refractivity for a single wave-lengtli (tin- green mercury
line), and (2) the measurement of the dispersion in the visible spectrum relatively to
this value.
For the measurement of the refractivity two methods were employed. In the first
of these the quantity of ozone present was estimated by destroying the ozone by heat,
and measuring the increase of the gas in volume. In the seco/id, the ozone was
estimated chemically by bubbling the mixture of gases through a solution of potassium
iodide, and titrating with thiosulphate of soda.
As the results of the enquiry were remarkable the following details may be of
interest : —
Dry oxygen, prepared by heating permanganate of potash, and stored in a gas
holder over mercury, was led through an ozoniser into the interferometer tube, whirl i
was previously evacuated. The interference bands which crossed the field were
counted till atmospheric pressure was reached. The pressure was then read by con-
necting the apparatus with a mercury manometer filled with oxygen and separated
from the ozonised gas by a long capillary tube. The temperature of the water bath
was observed and the tap which led to the refractometer tube turned off. Having
again evacuated the connections the gas in the refractometer tube was allowed to
flow slowly into the pump, passing through a spiral of fused silica heated to redness,
which effectually destroyed the ozone. From the pump it was transferred to another
gas holder over mercury and thence again allowed to flow into the refractometer tube,
where its temperature and pressure were again measured. If V, is the volume of
the ozonised, and V2 that of the deozonised oxygen, the percentage of ozone is given
by V,— Vi = afVj/200. In the present case V, was about 150 c.c., so that if x = 8 the
total increase of volume is 6 c.c. In order to determine the value of the refractivity
to 1 per cent, it is therefore necessary that the total error in pumping the gas round
the cycle should not exceed '06 c.c. In practice this accuracy was not quite attained.
It was necessary to grease stopcocks with a mixture of pure paraffin and vaseline,
which will not hold a vacuum indefinitely ; while, in order to destroy the ozone, the
gas had to be pumped through a spiral 12 inches long of fine capillary bore, which
made it difficult to evacuate the last traces from the connections. It was also
necessary to know the refractivity of the oxygen very accurately, since an error in
this figure is multiplied in the ratio of 100 fx. After a sufficient number of trials had
been made to prove that our oxygen was approximately pure, its refractivity was
assumed to be that previously determined by us,* viz., /u— 1 = '0002717. X 5461.
* C. and M. CUTHBERTSON, ' Roy. Soc. Proc.,' A, vol. 83, p. 151, 1909.
VOL. OCX 1 1 1. — A. D
I8 ME. CLIVE CUTHBERTSON AND MAUDE CUTHBERTSON ON THE
The ftfewbg are the details of a typical experiment by this method, in which the
refractivit y of the deozonised gas was separately determined :-
Part I Refractivity of the ozonised oxygen-
Bands (A = 54607) 489-9. Length of tube 99786 cm. Pressure difference
742-2 mm. (corrected). Temperature 16°'25 C.
^T9xJUfi0'7 x 289-25 x 760 x 1Q-" _ .0002909.
99786 x 273 x 742'2
Part II. Refractivity of the deozonised oxygen-
Bands 474-2. Pressure difference 764'33 mm. Temperature 14° 7 C. ; whence
M_l = '00027193.
Part III. Percentage of ozone—
V, = 764-33x289-25 = ro95±
V, " 742-2x2877
Thus percentage of ozone = 3'54 x 2 = 7"08.
Part IV. Refractivity of pure ozone—
The refractivity of the mixture is the sum of the refractivities of its components.
Let /KO,— 1 denote that of pure ozone, then
7*08 xta-l) + 92-92 x '00027193 = 100 x "0002909,
whence ^,-1 - '000539.
By this method the following results were obtained : —
TABLE IX. — Refractivity of pure Ozone. (First Method.)
Experiment.
Percentage of ozone
by volume.
Refractmty,
(IL TUO*
Remarks.
(W
1
9-5
508
2
7-68
543
3
7-08
539
4
6-24
511
6
6-24
545
6
6-5
560
7
3-5
[585]
Not reliable, percentage of ozone
too small.
8
8-72
502
9
7-14
497
Mean . . .
525
REFRACTION AND IMsl'KKSlON OF THE HALOGENS, HALOGEN ACIDS, ETC. 19
Second method. To check these results a second set of experiments was made, in
which the quantity of ozone was estimated by chemical testa This method was
found to give more concordant figures.
TAISI.I: X. — Refractivity of pure Ozone. (Second Method.)
K\|)t;riment.
Percentage of ozone
by volume.
Refractivity,
Remarks.
1
G-09
515
2
5-47
521
3
4
4-86
6-40
[495]
522
Not very trustworthy.
5
6-36
530
G
7-08
516
Mean . . .
516-5
The results obtained by the two methods are tolerably concordant, and would be
even better if the third experiment were omitted.
Their mean is 5207, but having regard to the smallness of the proportion of ozone
present it would be unsafe to rely on this number beyond the second significant
figure, and we therefore conclude that the refractive index of pure ozone for the
mercury green line is
/x = 1'00052.
Comparison with the Refractive Index of Oxygen. — It will be observed that this
result is remarkable.
The refractivity of oxygen is '0002717, and if the third atom of oxygen on joining
the molecule had the same refractive effect as in the normal gas we should expect a
refractivity (M-l) 107 of $ x 2717 = 407 '5.
The experimental value 520 is very largely in excess of this, and indicates the
existence of interesting peculiarities in the molecule which may probably be ascribed
to the linkage.
Dispersion of Ozone. — Nine experiments were made on the dispersion of mixtures
of ozone and oxygen. In each of these the refractive index of the mixture for the
green mercury line was separately determined, and the other seven refractivities
were calculated 'with reference to it from the observations as previously described.
The first experiment, being a trial, is omitted, and the refractivity of ozone
calculated from the remaining eight as follows : —
D 2
20 MR. CLIVE CUTHBERTOON AND MAUDE CUTHBERTSON ON THE
TABLE XI.— Dispersion of Mixtures of Ozone and Oxygen.
. •
Experiment.
(
/»-!)*
lȣs-
4,76
Bands read.
6708
6438
DTM
5770
5461
5209
5085
4800
1
28264
—
—
—
28533
—
28664
—
400
28143
28194
28332
28339
28420
28505
28544
28667
491
28509
28554
28698
28707
28789
28879
28926
29061
250
28500
28555
28683
28702
28789
28879
28927
29054
491
28499
28543
28692
28701
28789
28870
28919
29050
499
i>:t7 1
28416
28556
28562
28648
28735
28785
28915
527
•28'M>->
28410
28553
28559
28648
28737
28782
28910
533
8
28368
28412
28549
28552
28648
28726
28781
28909
538
9
28624
28667
28785
28810
28898
28988
29038
29172
238
Means
28423
28469
28606
28616
28703
28790
28838
28967
RefractiTitiesofOs. .
26952
26988
27098
27102
27170
27237
27272
27366
Ref [-activities of ()s. .
50764
50968
51514
51624
52000
52375
52621
53290
Adding together all the values of the refractivities for each wave-length separately,
and dividing by the number of experiments, we obtain the refractivities for the
average mixture of ozone and oxygen, which are given as "means." Assuming
[MM^I"!!), = '000520 the percentage of ozone in this mixture is found as follows : —
52Qx+(lOO-z) 2717 = lOOx 287*03, whence x = 6'1764.
To find the refractivities for the other seven wave-lengths we have only to use this
value and the refractivity for the corresponding wave-length of oxygen which we
take from our previous determinations, vide loc. cit. supra p. 2. Thus
U-1JO.X6-1764 + 93-8236XU-1],, = [Mx- 1]
mixture.
Tin- numbers obtained in this way are shown in the next line.
It is at once noticeable that the dispersive power of ozone is much greater than
that of oxygen. And here again, as in the case of chlorine, we find that the curvature
of the experimental curve is greater than that calculated. Using M(J708- 1 and M4800- 1
we obtain the formula M-l = • 2 0414x 1Q!" whence we find ,
4221'3 x 10*7— ?t*' /*6
whereas the experimental value is 52000.
As in the case of chlorine, the inference is that a second term is required.
KF.FRACTION AND DISPERSION OF THE HALOGENS, HALOGEN ACIDS, ETC. 21
AMMONIA.
Previous Determination.*. — Previous determinations of the refractivity of ammonia
are us follows : —
Observer.
Light.
(M-l)10«.
Corrected for density.
BIOT and ARAOO . . .
DULONO
White
381
383
MASCART
D
377
376 1
LORENZ
Li
371
D
373
373
G. W. WALKER . . .
>i
379'3±-5
374-3
It would occupy too much space to analyse the causes of these discrepancies, which
are chiefly due to differences in the standard conditions assumed and in the coefficients
of thermal expansion and compressibility adopted. But the figures in the last
column give approximately the figures corrected for the theoretical density.
Preparation. — Our gas was prepared by warming a mixture of ammonium chloride
and calcium oxide in a flask. After passing over red hot lime and cold dry lime it
was condensed at —80° C. and allowed to boil off, the middle fraction being collected.
Three samples were used.
Calculation of Jtesitltx. — In reducing the results the figures given below were used,
following GUYE* :—
Coefficient of thermal expansion (l + '003914<).
Coefficient of compressibility
l_£l£l = A (/>,-»„), A = -0002(1- -0000030.
Po»o
Weight of a litre of ammonia at 0° C. and 760 mm., "7708 gr. Theoretic
density, '7605 gr.
Thus the equation for reduction is
D
(dJG)
NX 7605
L 7708
x 76
W3914*,
1 + -003914*.,
where N is the number of bands observed, X the wave-length, L the length
of the tul>e, and ptp.» ttt.2 the initial and final pressures and temperatures.
Refraction. — The determinations for X 5461 were, as usual, made by pairs of
experiments, witli pressure rising and falling.
The mean of nine such experiments, whose extremes were 1 '0003782 and 1 '0003790,
Me"m. Soc. de Phys. et d'Hist. Nat. de Geneve,' vol. 35, 1908.
22 MR. CLIVE CUTHBERTOON AND MAUDE C0THBERTSON ON THE
was 1-OU.M786. Seven of these were at room temperature and two at 0° C. We
adopt -0003786 as the refractivity for the green mercury line.
/L*r«on.-F.ve experiments were made to determine the daemon.
Thefbllowing table gives the mean results and compares the observed values with
those calculated from the formula
)_ 2-9658x10"
~
8135-3
which was, as usual, calculated from the observations by the method of least
squares : —
TABLE XII.— Dispersion of Ammonia.
D
(/* 1) llr ^ .
XxlO».
Observed.
Calculated.
Difference.
6707-85
37376
37374
-2
6438-5
37455
37456
+ 1
5790-5
37701
37700
-1
5769-5
37707
37710
+ 3
5460-7
37860
37861
+ 1
5209-1
38002
38006
+ 4
5085-8
38083
38085
+ 2
4799-9
38300
38295
-5
NITRIC OXIDE. (NO.)
Previous Work. — DUFET gives the following : —
Light.
Oi-l)10».
Observer.
White
D
302
297-1
DULONG.
MASCART.
Mr. K P. METCALFE, in collaboration with one of us,* obtained 293 '9 for \ = 5893.
The gas used by MASCART had 10 per cent, of impurity.
Preparation. — Following the third method described by GUYE! we prepared the
gas by the action of of dilute sulphuric acid (10 per cent.) on dilute nitrite of soda
(6 per cent.) in a vacuum. After bubbling through concentrated sulphuric acid and
pawing over P,0t it was condensed in liquid air and fractionally distilled. The gas
employed, tested with ferrous sulphate, showed less than 1 part in a 1000 of impurity,
1 CtJTHBiBTsoN and E. P. METCALFK, ' Roy. Soc. Proc.,' A, vol. 80, p. 406, 1908.
t Gtmt, • Mta. Soc. de Phys. ct d'Hiat. Nat. de Geneve,' vol. 35, p. 547, 1908.
REFRACTION AND DISPERSION OF THK HALOGENS, HALOGEN ACIDS, ETC. 23
probably nitrogen. As the refractivity of nitrogen is almost identical with that of
nitric oxide the results were not modified by the impurity. The observations were
T^ 7 f*
reduced by the m-dinary formula /*— 1 = (v— 1) -—- x — .
«/ t)
Refraction. — Six can-fill double experiments (i.e., pressure rising and falling) gave
Experiment .... 1, 2, 3, 4, 5, 6.
(M_1)K)' -2959, 2957, 2952, 2955, 2951, 2956. Mean 2955.
We adopt this mean '0002955 as the value for the green mercury line.
Calculating the value for the I) line from this value and the dispersion formula
obtained below we find '0002944, which agrees well with 2939 found in 1908.
Diapernon. — From five observations the following values for the dispersion were
obtained : —
TABLE XIII. — Dispersion of Nitric Oxide.
XxlO».
*•-«-*.
Observed.
Calculated.
Difference.
6707-85
29306
29302
-4
6438-5
29344
29344
0
5790-5
29468
29469
-1-1
5769-5
29474
29474
0
5460-7
29550
29553
+ 3
5209-1
29622
29628
+ 6
5085-8
29666
29668
+ 2
4799-9
29776
29776
0
Using SELLMEYEK'S formula the results are expressed by
D 3*5210 x 10"
' K76) 122161
NITROUS OXIDE. (N,O.)
Previous Work. — DUFET gives
Light.
/*•
Observer.
White
1-000507
DULONO.
Red
1-000507
JAMIN.
6439-2
1-0005132
MASCART.
—
1-0005152
it
5378-9
1-0005192
«
5086-1
1-0005207
i*
4800-2
1-0005230
H
M
Mi; U.IU. (TTHBKRTSON AND MAUDE CUTHBERTSON ON THE
M ,„ VKTS gM wm pnfMd from ammonium nitrate Mad itaim-,1 10 per cent,
of impurity.
The gas we used was obtained from two sources: (l) The commercial gas,
obtained in cylinders, condensed and fractionated at the temperature of liquid air, and
(2) gas prepared by the action of ammonium nitrite on hydroxylamine hydrosulphate.
It was bubbled through strong potash and dried with sulphuric acid and phosphorus
pentoxide.
/,' t'l-.i.-tir,: Index.— Three sets of experiments on different samples gave
n
Series.
w*m\
Source.
1
5092
Commercial.
5102
2
5097
»
5098
5099
3
5087
From hydroxylamine.
5091
Mean . . .
5096
In reducing these experiments the coefficient of thermal expansion used was
•00371.
The purity of the gas was tested by absorption in an excess of water boiled in
vacua. The bubble of gas left unabsorbed was not so great as 1/2000 of the whole ;
and even this was probably due to the error of the test experiment, which is not very
easy. But as traces of air were probably present we think 5100 a more trustworthy
value than the exact experimental mean, and probably correct to 1/500 at least.
Dispersion. — From five experiments the following values were obtained for the
dispersion : —
TABLE XIV.
A x 10«.
(in 1HQ8 ^
\r '/ *" ,. -„,•
(Oo76)
Observed.
Calculated.
Difference.
6707-86
6438-5
5790-5
5769-5
5460-7
5209-1
5080-8
4799-9
50544
50616
50848
50857
51000
51145
51208
51415
50540
50616
50848
50857
51003
51142
51215
51420
-4
0
0 .
0
+ 3
-3
+ 7
+ 5
REFRACTION AND DISPERSION OF THE HALOGENS, HALOGEN ACIDS, ETC. 25
The refractivity can be expressed by the formula
i\D 5-6685 x 10".
'
Thr calculated values are shown in column 3 and the differences in column 4.
ERRORS OF EXPERIMENT.
Refraction. — In the determination of the refractivities for the green, mercury line
the principal source of error is the impurity of the gas, and, in the case of vapours
which absorb light, such as the halogens and sulphur, the limitation of the number of
bands which can be read before the light fails.
It will be seen that experiments of a series generally agreed to 1 part in 500, and
the mean is probably within 1 in a 1000 of the truth. In iodine and ozone, however,
the errors may amount to 1 or 2 per cent.
Dispersion. — It will be observed that the values of the refractivities for the other
seven wave-lengths are relative to that found for the green mercury line.
The degree of accuracy attainable depends on the number of bands read and the
dispersive power of the gas. As an example take nitric oxide.
Here O-l]*.^ = '00029306, Gu-l]A = 4«o = "00029776.
The dispersive power is 297.^~^3<)6 = 2*™6 ; and if 400 green bands are counted,
the number which represents the effect of dispersion is 12ai,87Q7ua° = 6'3 bands. We
consider that 1/15 of a band can be read; so that the value of the dispersive power
should be correct to 1 part in 95. It may be assumed that by determining the
constants from eight independent values of the refractivity instead of two the
accuracy is at least doubled, and the error should not exceed 1/200 of the effect
itself.
This claim is supported by the experimental results. Thus, in the six experiments
from which the dispersion of HC1 was determined, the values of (/*&&— /u^oo) * 108 were
816, 815, 813, 818, 805.
In eight experiments on HBr they were 1370, 1388, 1393, 1356, 1372, 1376, 1368,
1363.
CALCULATION OF THE CONSTANTS.
The calculation of the constants N and n£ of the formula /*— 1 = N/(«0a— «2) by the
method of least squares is very laborious if carried out in the ordinary manner. The
following modification was, therefore, adopted. Using subscripts to denote the eight
refractive indices and their frequencies we have eight equations of the fonn
_1_ -n:'
M,-l " N
.. VOL. CCXIII. - A. E
RKFRACTloN AN1» I>!SN.I;>!<>\ Off TIM- HAl.ncl-VS, IIALOCKN ACIDS. KTC.
Suhtnii-tini,' tli.' (/, f 4)"' fn.ni tin- /<"' equation, we obtain four equations similar to
._!_— LI {»,•-»,•}.
Mj-l /u,-l IN
Ix.t — --- — be expressed by x, and (nf-nf) by y, and similarly for the other
M»— 1 MI — 1
three equations. Then it can be shown that, applying the method of least squares,
N = 2 (.r. i/)/2 (x2), and hence n»' = *N 2 - + 2
We have much pleasure in recording our deep obligations to many friends. To
Prof. TKOUTON and Prof. A. W. PORTER we owe most grateful thanks for their
unwearied patience in assisting, guiding, and encouraging us. To Prof. N. WILSMORE
and Dr. WHYTLAW-GRAY we are indebted for instruction and invaluable help in the
whole of the chemical side of the work. To the Royal Society we owe our grateful
acknowledgment for the assistance of pecuniary grants.
II. On a 'V/.s.yYY//y/w Reflector with Collected Field.
Dr. R. A. SAMPSON, F./t.S.
Received Decemlwr 28, 1912,— Raul February 13, 1913.
THK great advantage enjoyed by the reflecting telescope is its equal treatment of
rays of all colours, and tin* geometrical defects or aberrations of its field are less
than those of many of the older refractors. The most serious of these defects is
coma, owing to which different /.ones of the object i\e d<> not place tlie light which
they receive from the s'une object |>oint symmetrically around-any common centre n
the image area, but arrange it in a radial fan or Hare, the light from the outer /.ones
being most diffused : besides spoiling the image this tends to neutralise, for any
except narrow fields, the value of extended a|)erture in the objective as a light-
collector. In the refractor this can be and is now always met by adjusting the
curves of the two lenses, for when achromatism, as far as possible, and spherical
aberration are allowed for, there still remains one unused datum ; in old forms this
was often used to make the inner curves contact curves that might be cemented
together if it was convenient to. do so, but it is properly employed to extinguish
coma. But with the reflector the case is different. In the Newtonian form there is
only one available surface, and when this is made a paralxiloid to cure spherical
aberration, nothing is left to adjust. In the Gregorian or Cassegrain forms there are
two curved surfaces and, theoretically, these would offer means to correct two faults.
An illuminating study of the possibilities of a system of two mirrors has been made
liy SCHWARZSCHILD in his ' Untersuchungen zur Geometrischen Optik';* I shall
i leal with its outcome below. Its general tenor is comprehensive and exploratory
rather than detailed, and it remains doubtful whether any of the forms which he
indicates for the reflector, at the point at which his research stops, could actually be
made successfully upon a scale that would show their advantages. My own purpose
in the present'paper is essentially a practical one. I have in mind throughout a
telescope of large aperture and considerable focal length, and seek to devise a
correction for the faults of its field which shall leave its achromatism unimpaired,
which can really be made and which shall effect its purpose without employing any
curves and angles outside those that are already known to work well. It has been
* ' K. Gesell. d. Wissenschaften zu Gottingen, Abhandl. Math.-Phys. Claase,' Neue Folge, Bd. IV., 1905.
VOL. CCXIII. - A 498. E 2 Published Miparmtely, April 18, 1918.
!>K. !,'. A. SAM I Si )X M\ A
said that "an ol.ject -glass cannot be made on paper," but the possibilities of new ami
•.,n,|ili.-ated .-onst ructions must in all cases first be demonstrated on paper,
.,;„ never conveniently vary more than a single factor at a time.
Study is directed N the Cassegrain because of the great advantage which this design
possesses in shortening the tube of the instrument for given focal length, and in
placing the observer at the lower, in place of at the upper, end of it.
The best introduction to the subsequent work will be in the form of a few remarks
upon SCHWARZSOHII-D'S results. These are not meant as a complete criticism or
estimation of it but are merely such as arise naturally in relation to the points with
which I deal afterwards. The traditional form of Cassegrain telescope consists of a
great concave mirror faced by a small convex one, which is placed between the great
mirror and its princijxd focus, and throws the image out through a hole cut centrally
in the great mirror. The small mirror increases the effective focal length in the
ratio of its distances respectively from the final principal focus and from the
principal focus of the great mirror. This ratio for example is 5 '4 in the great
Melbourne telescope, 3 J to 4$ in the Mount Wilson 60-inch when used as a Cassegrain,
and it can hardly fall much below 2£ unless the small mirror is to cut off a dispro-
portionate amount of the area of the great mirror. The Cassegrain is, therefore,
generally speaking, a long focus instrument. From all these features SCHWARZSCHILD'S
forms ditt'er widely, except that they place the small mirror between the great mirror
and its principal focus. His small mirror is concave in place of convex, and shortens
the effective focal length, bringing the beam to a focus between itself and the great
mirror. The effect of this change in design is to render possible a flat field. Spherical
aberration and coma are removed from the image by modifying the spherical figures
of the two mirrors into definite hyperboloidal and ellipsoidal forms. To confine
reference to the case which he considers generally the best (loc cit., II., §11), the
necessary deformations are given respectively by bt = -13'5, &2=+r97, where
- 1 would deform a sphere into a paraboloid. The image-surface for this case
would In- very nearly flat, and the images of points would be very nearly circles,
.vhich r.-ached a diameter of 8 seconds at an angular distance of about 1 degree
IVum the centre of the field. This may seem somewhat large but it is a quantity
proportional to the aperture-ratio, which in this case is large also, namely 1 : 3'5.
is in brief a very rapid instrument of short focus and of field about
1-le to that of a good long-focus refractor. The chief objection to it is found in
that it requires. Until some one turns such curves out, it must remain
>l>"tl>er it is feasible at all to make the construction a practical success.
VMWAK/.S, HILD'S analysis is the use of a concave small mirror. This
> destroy coma, which may equally be removed in the Cassegrain
<".s of Ike mirrors, and those indeed of less pronounced degree than
n..ds necessary. But as will be shown below there then remains a
That severe and irremovable curvature of the field
MUM
CASSEGRAIN REFLECTOR WITH CORRECTED FIELD.
Tl»- general conclusion which I «lraw from S, ,,u AR/SCHILD'B investigation is that
unification of the two mirrora is in itsrlf not enough to give a practical solut.on .
tl,,- problem. We have to d.-:d with spherical aberration, coma, curvature of
field and astigmatism. Distort!,,,, may be set aside, Wause in itself it does not
vitiate the image of a point, and errors which it introduces into relative d.stano
nrny be computed and allowed for. We have at our disposal the figures of
min-i-s ,.,..1 their separation and curvatures. The last are so lock.,1 up with 1
kind ,,f t.-l.-s,-one which we wish to produce that they are hardly available
...Ijnstment-if we want a short-focus instrument we have to take SCHWARZS, ....
choice and for a long-focus one the Cassegrain form. It turns out that the former
of these may have a flat field and the latter must have a curved field and we have
to rest content with that. And with respect to the figures of the mirrors ,
within our control to say whether they shall offer themselves in our equations i
favourable form for removing undesired terms ; it appears from the reward
they appear somewhat unfavourably entailing the use of surfaces decidedly far 1
the sphere. It is my object to obtain a workable solution and not merely a theor
one and therefore I have recourse to a more complicated system, by passing the
beam through a definite set of lenses, the curvatures of which are more or
completely at our disposal. It might, at first sight, appear that this would impa
the achromatism of the reflector, but if a system of not less than three separa
lenses be made of the same glass, the two conditions for achromatism at a give,
plane may be completely satisfied, equally for all colours. With such a system we
can produce deviation in a beam, but more emphatically we can produce aberrat,
The details at which I arrive are given on p. 66, and need not be repeated
generally the plan is to replace the convex mirror by a weak convexo-concave
silvered at the back, and about two-thirds of the way between this and
of the great mirror to place a system which I call the Corrector, being a pair ,
lenses of nearly equal but opposite focal lengths, of which the first IB double
with the lesser curvature first, and the latter nearly plano-convex.
Choosing the curvatures properly a telescope is thus produced which gives,
strictly in the focal plane, an image free from chromatic faults, except for minute
chromatic residues of aberration, from spherical aberration and from coma, and
which points of the object are represented in the image by spots strictly circu
reach a diameter of 2'2 seconds at a distance of 1 degree from the centre of the
The givntrst angle of incidence upon any of the surfaces is 11 degrees, o
than alx>ut two-thirds of what is customary upon the anterior surface of
lens of the object glass of a refractor ; all the surfaces are spherical except
the great mirror which is intermediate between the sphere and paraboloid, and
cannot see that anywhere any serious constructional difficulty is introduced,
effective aperture-ratio is 1 : 14'05, or, say, about 1 : 15, allowing that 5 per .
more light will be lost in this construction than in other possible ones.
30 DR. R A. SAltfPSON ON A
Tin- iiictliinls which I employ are those of a memoir recently published.*
SCHWARZSCHILD usetl the Characteristic Function. Our methods thus differ, but
since aberrations of the third or any other order are the same things, no matter how
they are obtained, where we occasionally touch the same matter the differences are
at most those of notation, and occasionally these are slight ones. I have not
attempted to remove them because it seems to me that an investigation is easiest
to read if expressed in notation that grows naturally out of its own processes. I
shall therefore adhere strictly to the notation of my Memoir, amplifying its results
88 occasion requires.
We may take for reference the following specifications of the faults of an optical
field at its principal focus in terms of the coefficients <?,G, &c. :—
a = semi-aperture.
f = effective focal length.
/3 = tangent of inclination of original ray to axis.
Position of least circle of spherical aberration . . . Sf = + f / V^G.
Angular radius of this circle ........ 25783" x ^- xa*K G.
t/
Comatic radius ............ 1 03133" x ± x «/W,G.
Secondary focal line after principal focus ....
Primary focal line after secondary ..... f'ffS. H
Radius of focal circle ..... 103133"x ^
Curvature of field (convex to ray if positive) . . . (l/^
Distortional displacement ...... 103133"x ( 1 //') x
...... (1)
With respect to these it may be explained that the Comatic Radius is the radius of
> around which rays from a zone of radius a are distributed, the centre of the
circle being displaced from the normal image-point by an amount equal to its
the "secondary" focal line is the line in the plane of the axis; the word
ana after, in the order in which light reaches the points ; the focal circle
cle half way between the two focal lines, through which, in the absence of
the zone would pass ; the curvature of the field refers to the field
the focal circles of all object-points.
Now, if we secure a field for which
= 0, ...... (2)
"A New Treatment of Optical Aberrations," 'Phil. Trans.,' vol. 212, pp. 149-185.
CASSEGRAIN REFLECTOR WITH CORRECTED FIELD. 31
it will be free from spherical al>erration and from coma, and the images of points will
be circles in the plane through the principal focus, the radii of which are given by
1031 3.3" x (a //') x /S^H. If <J,H, which by (2) is made equal to -S3G, is not zero, the
instrument will be successful for such values of the angular radius of the field as keep
this down below desired limits. These conditions give the objects which I aim at
attaining. Given the general design of the instrument as regards apertures and focal
lengths, it will be found that the lens which is used as a mirror, or the Reverser as I
shall call it, is completely determined in its curvatures by the conditions for
achromatism, and the quantities available for adjustment are the figure of the great
mirror and the curvatures of the two lenses of the corrector. These are used to
satisfy rigorously equations (2), and the essential difficulty of the problem is to find a
case among the great number of those that- are open for trial, the solution of which
shall prove to be of a practical kind, not involving excessive curvatures. Once an
approximate solution is obtained, to refine it only requires patience, but to arrive in
the neighbourhood of a solution is a problem in which trial needs some guide. In
this connection I would draw attention to the theory given below of the Thin
Corrector. This is an optical system of two or more thin lenses in contact, null as
far as deviation and colour are concerned, and introducing aberrations only which are
available for correcting existing aberrations. Thus simplified, it is manageable
algebraically, and its indications will show the possibility or otherwise of any projected
arrangement.
If we denote by 3) the curvature of the field and by $ PETZVAL'S expression
being the curvature of the surface (2r), as in the Memoir, p. 162, we have
Sa
at the principal focus ; hence <$aH which gives the amount of astigmatism is
determined by
.......... (3)
a result which can also be deduced at sight from known expressions for astigmatism
and curvature of field according to SEIDEL'S theory. In the special case of a flat field,
or 9J = 0, it becomes
........ . (3A)
and this may be taken in place of the third of equations (2) as one of our necessary
conditions. We notice that it is only possible to control the astigmatism through the
value of *JJ, and the value of $ depends only in small degree upon the distribution of
curvatures between the two faces of a lens. It is a matter then of the general design
of the instrument to keep SaG down to a suitable magnitude. This presents no
difficulty. I have been content to keep it small enough for my purpose. If a field of
DR R. A. SAMPSON ON A
radius greater than 1 <l.-^r«t- \v«-iv desired, it could be made even smaller, but it would
aeem t» involve tin- sacrifice of some other conveniences.
Tli.- v.-dii.-s uf tin- quantities <J,G, &c., for the combined system are built up step by
step by proceeding from surface to surface or from lens to lens by the sequence
••qua ti. His (17), p. 160, of the Memoir referred to above. For making these steps
it is not convenient to lay down any one procedure as being the best for all cases,
but two methods may be mentioned, one or other of which is frequently suitable.
t we can proceed from conjugate focus to conjugate focus, the first focus being
the principal focus of the first or great mirror, and each successive conjugate focus
being the principal focus of the whole combination which precedes it. That is to say,
at each stage we have
g = 0, hk= -1, h' = 0,
so that the equations we require to consider are
. (4)
In these fir', ... refers to the new or added element, g, ... to the combination from
the beginning up to this element, and G, ... to the resulting combination including
this element. We thus notice that S,g contributes to S,G simply by multiplying by
flr', which is the magnification of the new element between its conjugate foci under
consideration. We notice, too, that so long as we confine ourselves to §,G, the only
coefficients which it is necessary to find for each added element are SJi', calculated
between the same conjugate foci. If the aberrations of the second element are given,
referred to some other origins, they must be transferred to the conjugate foci in
question by means of the equations for change of origin (22), p. 164. A case will
present itself that requires a modification of this process, namely, when one of the
conjugate foci belonging to an element introduced by one of the steps described is at
a great distance ; to meet this case we may take this element together with the next
following m,e and combine them into one before adding them to the combination, or
we may take a second completely different method as follows : —
Let Ow O. be the initial and final origins ; O0, O0. the origins to which the known
aberrations of a part of the system are referred. Calling {g', h' ; k', I'} the subsequent
normal system O.- to O., transfer the aberrations to origins Oa...OB by use of the first
part of ,.,,,,ati,,,,H (17), p. 160, viz., S,G = g'Slg + h'Slk ..... Then caUing {g, h ; k, 1}
• pi-.-.-,- ling MMr.ual scheme O0 to Oa, transfer the so-found coefficients from O.....O,,
( >. by using the forms of the second part of the same equations. An example
«•»' this method will be found on p. 55.
We now study the formulae for thin lenses. It will be pointed out later how to
make use of these when the lenses are thick.
CASSEGRAIN REFLECTOR WITH CORRECTED FIELD. 33
Thin tenses.
The atarration coefficients for a single surface are given in the Memoir, p. 161 ;
iff = (l-n) B3, *# = 0, r^/ = 0, V = ( I -n) B, SJi = 0, 3Ji = 0,
,V = (l-n)(-l+n-na)Ba, <V = -n"(l-n)B, 3J = -n(l-n'), . . (5)
where I have written e = 1 — e, so that e = 0 for a spherical surface, and « = 1 for a
paraboloid.
Both origins are at the surface, and
g = l, h = 0, k = (n-1) B, / = n, ) = £, n = ju_,/ix+,.
The case of the thin lens, with origins at its surface, is derived from this hy an
application of equations (17), p. 160.
Write
then
* nl ' " \ n
&& = 0, ^ = 0,
i^h = -kn = -<), SJi = 0, ^ = 0,
V = 0 ........... ... ...... .... (6)
It may be mentioned that B, the curvature, is positive when the convex face is
presented to the ray.
It seems unnecessary to give the algebra leading to these expressions in all cases.
It is quite straightforward, and that for <J,&, which is relatively long, may be taken
as a model. From the Memoir, p. 160, we have, taking <5,K to refer to the joint effect
of the two surfaces
f + P3JV} + k{$/
VOL. CCXIII. — A. F
I)R R A. SAMPSON ON A
It i. ** that the ten,,, in ,, i come to the value, Biven. Leaving these aside
n n
This appears, multiplied by -(1 -») B + B', and added to Wtf + W,* which is
n
the whole is
n
~ n? n9
+ B* x -3(l-n)(\-n + n3).
This may be written
= K'+KX
where
X = nBM-rr
This is the given expression if finally we write small letters for capitals.
It will be noticed that q, which contains the reference to the distribution of
curvatures, apart from their effect upon focal length only presents itself in the forms
in which it is introduced by Slk, Stg. It is somewhat remarkable that the same is
true when we have any number of thin lenses in contact.; thus, if we have a system
of thin lenses in contact, giving a set of coefficients S&, ..., and add a single thin lens
to it for which we have Stf, ..., then, noticing that
CASSEORAIN KKFLKCTOR WITH CORRECTED FIELD. 35
we have
**»', V* = J.G = 0,
J,H = Sfi + Sfi = -)>-»>' = -#, A.H = J,H = 0,
where E is the sum of terms in e, e' for each of the lenses ;
^,L = 4,K-K$ = K'-K'P-i.G,
J,L = ^K-9 = K, ^3L = 0 ............. (8)
Thus, to form the coefficients cS,G, ... for any system of thin lenses in contact, we
require to know only the forms for ^,G and <5,K. I add the forms of these for three
lenses,
-k") tf + l(2k+2V + U')(k + V)tf'. . (9)
From these, if necessary, the general case may be written down by analogy without
much difficulty, e.g., in (5,K the coefficient of Jj>'</' '8 three times the k of the
preceding system minus the X: of the following system ; but I shall not require more
than three.
We may employ these equations where we require to obtain algebraically rough
but reliable indications of the properties of a given actual system. Thus, consider
the aberrations of any set of thin lenses in contact, at their principal focus, that is, at a
distance — K"1 beyond their common surfaces. We must form S{T = <5,G— K~M,K, ...
where ^G, ... are the quantities just found which refer to the surfaces of the lenses as
origins. Hence for example, referring to p. 30, we see that the radius of the focal
F 2
DR. R A. SAMPSON ON A
.......
of vature i. a.ways al»ut two-fifths of the focal
IWS,'''o.,,,.lition for d-.» of coma, .bid. » ™,ally ftiw as ABBB'S Sine Condition,
"^ to P1" 0 = Kl,r = KWJ-a,K = ,1,0-K' ;
in-thb the right-haml member, apart from the foeal length,,, is a linear function of the
quantities q.
The condition fur absence of spherical aberration is
0 =
which is a quadratic function of q, ... .
A numerical example of the use of such approximations will be given later.
It is necessary to deal with express care with the case of the mirror. t may 1
treated as a single surface for which n = -1, and then
^ = -2(3-e)B8, SJc-- -2B2, ^- = 0,
A,/ = -SEP, .y = -23, .y = o,
but this leaves the positive axis after reflection opposite to the direction of the ray.
It is better to reverse the direction of the axis, and this may best be done by
multiplying by the scheme {g, h; k,l} = {!,* ; *, -1}, and gives the following set
to represent the mirror : —
0 = 1, A=0, * = 2B, Z=+l, J>=-2B,
i0 = 2BJ, ^ = to = °. M = 2B, V* = XJt = 0,
i,Jt = 2(3-e)B», ^fc = 2B», ^Jfe = 0,
SJ = 6BS, «V = 2B, ^ = 0, ........... (10)
the signs of all terms in k, I being reversed by this step, while g, h, \> remain
unchanged. Notice that the convention for the sign of B has not been altered, so
that, e.g., for the concave mirror B is negative, and the new value of k = (1— w)B is
negative also.
If we write J,Jb = Jfc*+fcx+E, we must put x = —it*.
Besides the simple mirror I shall have also to deal with the system consisting of a
meniscus, silvered at the back. Such a system 1 shall call a lleverser. For neglected
thickness the coefficients follow readily from the case above (p. 35), of the juxta-
position <.f thmr thin lenses, replacing the middle lens by a mirror, and taking for
CASSEGRAIN REFLECTOR WITH CORRECTED FIELD. 37
the third lens the original lens with the surfaces in reversed order. This reversal
of order will replace B, B' respectively l>y — B', — B. Hence k, ) will equal k", J>"
respectively, but q + q" = 0.
in the • ions (9), using ' to denote the mirror surface
• . (11)
Thr same expression is true of a more complicated reverser of any numl>er of thin
lenses with the last surface silvered. Also
(12)
To conclude this preliminary discussion of systems of thin lenses in contact I shall
introduce a system which consists of two thin lenses in contact, of equal and opposite
focal length and of the same glass, and therefore a null system in every respect
except for aberrations. The use of such a system will l>e illustrated hereafter. Its
simplicity is such that its aberration-coefficients reduce to very easy forms, and can
therefore be handled algebraically in an experimental investigation, in order to
discover what system will correct the aberrations of a proposed system ; it will
supply a useful approximation to a solution when any less idealised system is too
complicated to manage.
From the expressions (8) we have for the Thin Corrector
K = k + k' = 0, $ = kn + kn = 0,
k (k-k')k'n
..... (13)
and all the rest of the coefficients run in agreement with p. 35, so that
38
|iK. K. A. SAMPSON <>\ A
<J,K = <5,L = -iiG and the rest are zero. These are the values at the surface of the
corrector. \V«- notir.- that all are zero when qjk + q'jV = 0, that is, when the
curvatures of the two surfaces in contact are the same.
In onli-r t<> illustrate the manner of using these, for example, let it be proposed to
timl the curvatures of a corrector, which when interposed at a given point of an
aherrant beam shall produce assigned changes in it. Let this place be at a distance
v befon> the beam comes to its focus. After passing through the corrector it will
still come to a focus at the same place, so that applying the formulae of the Memoir,
p. 164, (22), we have for the distances from the first conjugate focus to the
corrector d = r, which is negative, and from the corrector to the second conjugate
fo<-us d' = —r, and transferring from the surface of the corrector to these conjugate
foci, we have
where <J,y, J,* are written for the values of ^G, <5,K given in (13).
We must now apply the formulae (4) of p. 32. For the corrector g' = 1. Let the
assigned changes be, say,
A,, = 4G-4gr, A, = 89G-S#,
so that the equations (4) of p. 32 give
therefore
J.A3. . (14)
From these equations the values of the curvatures of the two lenses may be found
with the help of equations (13). An example of their use will be found below, on
p. 44.
In connection with the question of assigning a system which will produce definite
may I* remarked that it is not difficult to solve the equations (17) of
. Menu,,,- so aa to give explicitly either S>g, ... or *J, ... so that we have
>-s.r,d either the antecedent set or the consequent set which combine to
.il«rat,on coefficients J.G, ... . The former are obviously obtained by
J ' Mi '^L' -^«G+^K' -WH-AL, which give respec
^.»U»Un'V For the latter coefficients V,.- we form
CASSEGRAIN REFLECTOR WITH CORRECTED FIELD. 39
which give <$,«/, t\h' ; and similarly we have $}tf, <?,/'. Form also
-/<M,G -I- (gl + f,k) S,G-gk<\G = ...+«»
-hlSt 1 1 t . . .
and
AM.H-...
\\ itli similar equations in it.K, iT.L. These equations, for example, answer the question
df what al>errations are shown when a known system is reversed and presented with
the opposite face to the beam, the unit-points Iwing simply interchanged so that the
normal effect as shown in the position of the focus is the same as before. For if an
unaherrant heam originating ;it <• is Drought to a focus at ()' and shows there
aberration ooaffioienta <\y, ... ; or, what is the same statement, an alwrmnt )>eain with
coefficients Stg, ... emerging from (V and passing through the system in the opposite
direction is brought to an unaberrant state at O, then if Stgf, ... are the coefficients
introduced by the reversed passage we have the joint effect of A,^, ... superposed to
•\{j, ... is null, or <^,G, ... are all zero. But it must lie noted, as was pointed out for
the mirror, that as the direction of the axis is reversed the signs of c$,£...<y must be
reversed before they are brought into the equations with <?,0r', ... ; further, since
G = 1, H = 0, K = 0, L = 1, we have gf = /, It' = -li, k' = -k, I' = g, and n = 1.
The whole question has some general interest, but I shall not pursue it further at
present, because it is somewhat beside our mark, and I return to considerations that
bear upon the main problem.
Coming now to the immediate object of my paper, which is the Cassegrain
telescope, I shall first consider what can be effected with two mirrors simply, which
will give opportunities for writing down useful expressions of various forms relating
to mirrors.
A mirror with both origins at its surface, and the reversal included, gives the
scheme (10) p. 36, or say
g = 1, h = 0, k = k, 1=1, p = —k,
where k = 2B, together with the aberration coefficients
#», 0, 0; *, 0, 0; i(2 + e)i3, &*, 0 ; $k>, kt 0 ..... (16)
With the surface for one origin and the principal focus for the other, these become
g = o, h = -k~l, k = k, 1=1,
with the coefficients
-bk3, -££, 0; -i*. -1, 0; ibid. ; ibid ..... (17)
It by the formulae of the Memoir, p. 164 (22), we transfer the origins to two
40 DR. R. A. SAMPSON ON A
conjugate foci, P, P respectively, say at distances PO = u, OP' = v along the ray
from tli«- surface, so that
u+v + kuv = 0
—where it IB to I* noted that the positive direction for both u and v is the direction
of the ray, which is reversed at the surface, so that if P, P' are found upon the same
side of the mirror u and t' will have the same sign — we have the scheme
g = l+hv, h = 0, k = k, 1=1 +ku,
\sitli tli.' roftHcifiita
ie)]. (18)
To obtain the system for a Cassegrain telescope, we must combine two systems,
(gfi...), (grT...), as in the Memoir, p. 160 (17), of which the former gives the great
mirror at its principal focus, by (17) above, while the latter gives the second mirror
between two conjugate foci, by (18). Let AC, e refer to the great mirror, and /, e' to
the second one. If we confine attention to spherical aberration, coma, curvature, and
astigmatism, it will suffice to form ^G, (5aG, <$;,G for the compound system, deriving
<J,H with the help of the equation (5;,G— ^H = H$. The resulting expressions are
(53G = -K'V/^I-K'U O + sVI + ieY'ttV [l -KuJ/K, (19)
with
The quantities e-l, e'-l are what SCHWARZSCHILD calls the deformations of the
mirrors, from spherical figures; when e = 0, or the deformation = -1, we have a
paraboloid ; if we choose them so as to annul coma and spherical aberration we have,
from the equations S3G = 0, «J,G = 0 respectively,
while if we eliminate e from S3G, we get
Curvature of field = -K
= llu + K'u{l+K(-u+v)}/v,
and
(20)
.CAS8EGRAIN KKFLECTOR WITH CORRECTED FIELD. 41
These expressions are identical, except for notation, with results given by
SCHWARZSCHILD ; they contain the complete theory of the Cassegrain combination,
corrected by figuring for coma and spherical aberration, except as regards distortion,
and this could i-asily I"1 added by calculating •>'. II.
\\. read IV. mi <-<\\i:i\ i»ns (L'n) tliat l'»i a ^i\.-n d.-i^n ••!' in-t ruiii.-nt , as -|..-.-iti.-d in
the values of *, *', u at r, we can adjust the figures of the two mirrors so as to annul
spherical aberration and coma at the principal focal plane, and then the curvature of
the field and astigmatism amount to determinate quantities. Coma is annulled only
f«>r the purpose of getting a larger field for photography, and there is very little use
in annulling it if the field possesses pronounced curvature, or in less degree, if the
focal circles are not reasonably small. Hence the practical questions are : can the
design be made such that curvature is nearly absent and astigmatism small, and
can the corresponding values assigned to the deformations be realised in practice ?
All these questions are treated more or less explicitly by SCHWAIIZHCHILD, and I
shall traverse the ground again only in order to connect the problem with its
subsequent development and bring out the points which I require.
Itegarding the expression for curvature, v— u is the positive distance from the
principal focus of the great mirror to the principal focus of the combination. In
the Cassegrain form the latter point is, as a rule, not far beyond the surface of the
^n-at mirror, so that v— u is not far from the focal length of the great mirror and
I+K (—?* + »') will be a small fraction ; also K'U is numerically less than unity. Hence
the curvature of the image will differ very little from 1/w, the reciprocal of the
distance from the second mirror to the principal focus of the great mirror, a distance
which would seldom be more than one-third or one-fourth of the focal length of the
great mirror, or one-tenth to one-twentieth of the focal length of the combination.
The common Cassegrain is subject to the same objection. The values of its errors
may l>e read from the equations (19) on p. 40, if we have the means to determine e, e'.
As an illustration we may take the great 60-inch reflector of Mount Wilson
Observatory, which can be used either as a Newtonian, with a focal length of
25 feet, or in three different forms as a Cassegrain ; taking the form designed for
direct photography, it has an effective focal length of 100 feet, so that v/u = —4.
If we take the final focus at the great mirror, which is nearly the case, we have
u = — 5, v = +20, and K = +3/20. Now since the telescope is corrected as a
Newtonian, the great mirror is parabolic, or e = 0 ; and therefore taking it as
corrected for spherical aberration as a Cassegrain, ^-eVw = 1, or e' = —16/9, which
is a hyperboloidal form, the deformation from a sphere being nearly three times that
which would produce a paraboloid. Substituting ^eVttu = 1 in the equation for SaG,
we have, after some reductions, ^aG = £ru/r = — ^K, or the coma of such an
arrangement is the same as for a simple mirror of the same focal length. Also we
find (?:1G = —15, $3G— <SSH = — 11, so that the radius of curvature of the field is
one-nineteenth of the focal length or about 5^ feet only. As to the astigmatism
VOL. CCXIU. — A. O
42 DR. R. A. SAMPSON ON A
w« have <J,H = -4, wlii«-li may be compared with (53H = -1 for a Newtonian,
l»ut since the aperture ratio aff' is diminished in the ratio 1 : 4 by the increase of
effective focal length, the radii of focal circles at all distances from the centre of
the field will have the same angular amount that they had in the Newtonian form,
neitluT more nor less. There remains then only the above-found curvature of the
field to notice. Taking as a convenient mark a distance 34''3 from the centre of the
field, namely where /3 in the formulae of p. 30 equals one-hundredth, we should have
at this point the field curved back from the plane through the principal focus by
more than one inch. In spite of this pronounced curvature, exquisite photographs of
the Moon, as well as of small objects like Mars, have been obtained with this
telescope in Cassegrain form. The photograph of the Moon (R.A.S. photographs,
No. 214) appears to me second only to the Yerkes photographs with the 40-inch
refractor and colour screen ; but technically it would lie more instructive to examine
a photograph of a wide field of stars.
It is worth while to demonstrate that curvature of the field cannot be removed by
replacing the second mirror by a set of lenses in contact, used as a reverser, as
explained on p. 37. By such a replacement we introduce the quantity $ which, for a
given focal length of the reverser, is adaptable by throwing different proportions of
the deviation of the rays upon the lens system and silvered surface respectively.
Then using the formulae (4) of p. 32, in which we may put hk = -1,1= l,k now
referring to the great mirror and *• to the reverser x
where, if Jjy, ...-* refer to the reverser at its surface,
by (11), p. 37,
Thug
m
Eliminate ft,*...) by forming
CASSEGRAIN REFLECTOR WITH CORRECTED FIELD. 43
Now && = —%k, 3& = 0 ; and if by figuring or otherwise we annul coma, so that
G = 0, we have
Also
so that
also K = kj(f = —ku/v ; so that the curvature is
lv ...... (21)
If we compare this with the expression given in (20) above we see that the sole
effect of the change is to replace the reciprocal of the focal length of the second
mirror by (2/c + ir) for the reverser, and, since its factor in u, v is small, this change
will not allow any considerable modification of the curvature of the field.
To meet the difficulty of curvature SCHWARZSCHILD considers a design of instrument
fundamentally altered. Thus in (19) the curvature of the field will vanish if
K = -v/ua{l+K(-u+v)}
and this may be secured if K' is negative as well as K, or if the second mirror is concave ;
but in order that the curvature of the mirror may not be too great we must then
take I+K(— u + v) sensibly different from zero, and also v/u the magnification of the
second mirror, not too large. The system to which SCHWARZSCHILD is led as
generally the best to be found under such conditions has been already described
(p. 28). It is so different from anything that has yet been made that it must be
regarded merely as an interesting exploration of the possibilities of the theory until
an attempt is made to realise it. In particular it is utterly different from the long-
focus Cassegrain which I have in mind, and therefore I shall not require to refer
to it further.
Returning to the question of the Cassegrain proper we see that if an improvement
is to be made it must be by inserting a corrector of some form in the course of the
beam. Hence we come to the system which I have indicated on p. 29. To get an
approximation to what is required, suppose that the reverser is merely a convex
mirror, that the corrector consists of a pair of thin lenses of which the theory is given
on pp. 37 and 38, and that all the surfaces are spherical except that of the great
mirror which is figured so as to annul spherical aberration. To fix ideas I shall
suppose that the unit of length is 100 inches, and that with this unit the aperture of
the great mirror is 0'40 and its focal length 2'0000, also that the separation of the two
mirrors is 1'3333, that the magnification of the second mirror is 2 '4, from which it
results that its focal length is l/'875 = T1429, and the principal focus of the combina-
tion is thrown beyond the great mirror by '2667, at a distance T6000 from the
a 2
]>i; R. A. SAMPSON ON A
id mirror. It will be seen from the expressions (14) that it is desirable that the
corrector should lie as far as practicable from the principal focus if its aberrations are
to be as small as possible, that is to say, if its curves are to be as shallow as possible.
It cannot be too far forward or it will cut off some of the rays coming from the great
mirror to the revereer. It appears that a convenient distance is O'OOOO from the
revereer, or 07000 from the principal focus. That is to say, in the formula; (19) of
p. 40,
*=-'5000, «'=+-8750, w=-'6667, v=+l'6000,
so that, with e' = 1 , for a spherical reverser,
3.jg = + '3383, S.& = -3-0301.
Now we have to make
S3G = Q, $,G+4H = 0,
and we have
<J,G-4H = H^ = +4'8000 x -'3750 = -1'8000.
Hence the changes A,, A3, which the corrector must introduce, are respectively,
A2=-'3383, As =+2-1301.
These are the quantities so denoted in (14) p. 38. In the same equation, the
values of k, I to be used come from the scheme resulting from the combination of the
two mirrors, viz.,
g= , h =+4-800, £=-'2083, I = +2'1667,
and v giving the position of the corrector with respect to the principal 'focus,
v = -7000.
Hence
A-'A, = + 1 -6238, ko A3 = + -3 1 06,
H*-+-S160, (!-«,)-» = 1-4620, (-2 + Mv)/(l-Uv)
(-3 + 4klv)/(l-klvY = -37107,
Ay= -1-6238- '4541 = -2'0779,
A* = -17472-1-1525 = -2'8997.
** for • thin
] = +2-0779
CASSEGRAIN REFLECTOR WTTH CORRECTED FIELD. 45
In order to secure shallow curve-, the quantities t//i, </'/*' should be as ninull as
possilile. It is therefore evident that *rr should be taken negative, that is * positive.
The actual value of K the reciprocal of the focal length of each member of the
corrector has now to be chosen. By increasing K, q, q1 will l>e made smaller but at
the Hum R time the lenses emplnynl will be shortened in focus. As a reasonable trial,
take K = +1*4286, so that KV = — 1, and the focus of the combination of the two
mirrors is also a focus of either lens of the corrector ; then taking, say,
M = 1*5200, n = '6579, 7i(l+2n)/4(l+n)a = '13857,
we have
q/K+q'/*' = +6-3168,
9/*-tf/*' = -2*1809,
or the equations give
q/K = +2-0680, qf/K' = +4'2488.
The curvatures of the lenses are now found from
, = l-lW-B'4) = +1-4286,
q = 1 +± (B4 + B'4) = +2-9543,
\ nl
or
B4= -7875, B'4 = +1-9597,
and
/ i\
-- (Bg-B',) = -1-4286,
nl
q' = l+ + B-.) = -6-0697,
\ nl
or
B« = +'1698, B', = -2-5779.
These results are a very fair approximation. The final solution, when the thick-
nesses and consequent separations of all the lenses are allowed for, as well as the
introduction of a third weak lens in the reverser to preserve achromatism, with
resulting change in the focal length of the second lens of the corrector, is
B4 = -'6930, B'4 = +2-0482,
B, = -'0242, B', = -2-6120.
The first lens is a double concave, the radii of its two surfaces being 1'270 and
0*510 respectively; the second is double convex, with radii 5*907 and 0*388. The
remaining astigmatism is measured by the value of (5jH, which by p. 44 is +0"9000,
which is about the same as the residual amount present in the focal plane of a
46
DR. K. A. SAMPSON ON A
refracting d..ul»lrt. These are all reasonable amounts, so that we are now in
possession of a good approximation to a workable solution which corrects coma and
. urvature of the field, and leaves the figure of the great mirror to correct spherical
aberration.
It only remains then to adapt this solution to include consideration of all the
secondary factors that have been left on one side.
We must now turn to the question of achromatism in general. A thin corrector,
such as is contemplated on p. 37, is, among other properties, achromatic ; but when
the lenses are made thick and their unit points separated, as must be, to make the
system real, this property is lost in greater or less degree. With two lenses only it
is not possible to restore it completely. Reserving the quantities q, q' for adjusting
aberrations, we may alter the ratio k : kf from the value —1, but this gives only one
adjustable element, whereas there are two necessary conditions for achromatism
for any specified position of the object, namely, identical position for the image and
identical magnification. It is true that in the ordinary achromatised refractor,
consisting of a doublet, results are obtained with satisfaction of only a single
condition, but the achromatism secured is necessarily very imperfect for another
reason — the imperfect rationality of the dispersions of the two kinds of glass —
and this masks the neglect of the second condition. For the reflector, where we
aim at perfect achromatism, we must add a third lens to supply an additional
adjustable element. I shall now give the theory of complete achromatism at a
chosen point with three lenses of the same glass, separated by given distances. To
make all the lenses of the same glass secures achromatism for all colours if it is
attained for any two. The lenses are supposed thin, and the results must therefore
be considered merely as approximations, since the thickness will alter the positions
of their unit points as well as their focal lengths when a ray of different refractive
index is considered. But the approximation will be generally close, and an
illustration of how to make a complete adjustment will be given later.
> lenses be placed at O,, 04, 0, and produce images in succession at P3, P8, P7
" ln the ^ Then the
CASSEGRAIN REFLECTOR WITH CORRECTED FIELD. 47
Write
O,O4 = <*3, O4O. = d4,
PA = "i, 0,Ps = 7i3; P,04 = v,, 04P6 = u&; ¥fi. = vtt O.P7 = ult
also
F,), x4 = l-
then we have the equations
0,
XjVjttj = 0,
= 0;
and the linear magnification is equal to
-(ttj/W, ).(«,/''»)• (W"*)-
Varying the system with respect to 1/n, the refractive index, and making a
condition that v,, M7, and the magnification are unchanged, we have
= 0,
= 0,
and
/M6— Arj/t's = 0 ;
eliminate A?<3, Ac& and this gives
Ai», (!/«,+ !/»,)
eliminate Av3, A7<6 and we have the two equations
A-c,
finally
or .... (23)
thus, knowing c£3, fi6, r,, and choosing, say, *„ we determine in succession «, the value
of the ratio, Ug, «6, x4, va, «7, xa. But this choice and order is open to modification.
For example, if we take, as on a subsequent page,
t= +'9261, dj= -K01G94, tt3=+r6000, x4=+l'4286,
4g pi:. R A. SAMPSON ON A
„ = -'01056, *= -1-3704.
This is an illustration of the simple corrector (,,-* in contact) modified by a slight
separation of the two lenses and completed by the addition of a weak lens „ at a
considerable distance, and adjusted for a point which is nearly at the principal focus
, ,f t he middle lens. The exact solution on pp. 51-53, gives
,,= -'01152, *.= - 1-3459;
the differences are considerable ; this must be expected because the thicknesses of the
lenses are of the same order as the separation d, of the unit-points ; but in all cases the
solution will be close enough to supply a good approximation that will allow the actual
case to be adjusted.
The general process, suitable for use when we have obtained an approximation by
the method just explained, will be the following. Let the standard scheme and that
of the varied refractive indices be
{G, H; K, L} and {G+AG, H + AH; K+AK, L+AL}
respectively. Then the conditions for complete achromatism at the principal focus
are simply
AG = 0, AK = 0,
for these imply that the focal length is unchanged and also the distance — G/K from
the origin to the principal focus for either way. Then using the approximation
already supposed found, calculate the values of AG, AK which it shows. Vary the
focal length of the first lens and recalculate them. Vary also the third lens and
recalculate them. We then have means for interpolating the correct values of the
first and third lenses requisite to give an achromatic system in conjunction with the
middle lens.
This will be illustrated by the calculation of the actual system which I set out to
tii id and to which I now come. It will be understood that it was obtained by steps
of approximation.
It is unnecessary to give details regarding all these steps, which were unnecessarily
circuitous, owing to numerical mistakes and ill-judged processes. I shall therefore
give the final stage only.
The notation is slightly varied from the standard notation of SEIDEL, 00 is the
vertex of the great mirror, B0 its curvature, Oa the vertex of first surface of the
reverser, O', the vertex of the second or silvered surface, 0".,, which is the same point
as O» is the last surface of the reverser ; Bs, B'.,, B"2 = -B2, are the corresponding
curvatures ; 04, (y4 are the vertices of the first and second surfaces of the first lens
of tin- corrector, with curvatures B4, B'4; O«, 0'6 with curvatures B6) B'6 refer to
the second lens of the corrector. For the thicknesses of the lenses I employ here
CASSEGRAIN REFLECTOi; WITH CORRECTED FIELD. 4fl
even suffixes, thus ta = OSO'., = < >'J >" .... /, = <>,O'4, /„ = OgO',, ; for the separations,
d, = 000,, d, = 0",04, d6 = O'A,
H,, H"a are the unit points of the reverser ; H4, H'4 and Ha, H',,, those of the two
lenses of the corrector. Similarly F, is the principal focus for the great mirror,
F3, F'3) F"3 for tin- ditferent surfaces of the reverser, and so on, the final focus of the
whole combination bein<j F"7.
Writing, as above,
* = (l-i)(B-B'), 7 = (l + i)(B + B'), q = 9/K,
we find by considering the scheme
, -in, or ,, • i r«. H-
.(n-l)B, nj L* U Un^-lJF, n-'J [K, L.
that for any thick lens
K = K-n(l-n-1
and
. (24)
For the reverser we have the scheme, including reversal of the ray at the reflection,
i-i)B,, »Jl* ij l.2B'z, ij 1* ij -(W-'-OB,, w-'J [K,, L,
whence
K2 = 2n
= -O",H"a ............. (25)
Write (Ka) for the part of K2 which is due to the lens of the reverser, namely,
(K,) = (l-«-1
By methods essentially the same as those exposed below I was led to the following
approximate values as a system corrected for aberrations :—
B0 =-• '250000, eu= +'16502, a0= +'200000,
d, = +1 '320 133,
Ba = -B"2 = +'469009, B'a = +'450653, tt = +'020000,
d3 = +'906760,
B4= -'697845, B'4= +2'043309, tt = +'012500,
c?5 = +'002500,
Bj = +'003705, B'6 = -2'610677, tt = +'012500.
VOL. ccxiir. — A. H
5{) |i|!. R. A. SAMPSON ON A
The initial **lUp«tlM, *, does not enter the calculations, Imt is carried through
at ite wl..pt«l vain.-, which is recorded here for reference.
It follows that
K, = +'875000, (K,) = - '010297,
0,H, = -0",H", = + '013200,
K4= +1-428571, 04H4= +-006116, O'4H'4= -'002089,
K«= -1'359456, 0^,,= + -008211, 0',^.= -'000012,
and that
0,,H2= +1-333333, H"2F", = +1'600000, (26)
and the power of the combination of great mirror and reverser is the same as in the
preliminary solution. The achromatism of the system proved also satisfactory, but
the numbers had to be recast because of the following defect. As will be seen on
p. 63, the semi-aperture of the lenses of the corrector is about a = +'0615. Hence
the separations of the vertices of the surfaces which are next to one another must be
at least ^*(B'4— Bg) = +'00387. Hence enough separation has not been allowed,
since we have taken d& = + "00250. I therefore increased db to the value of + '005000.
At the same time I decided to increase the thickness t6 also to £6 = +'015000. To
change rf&, /„ means upsetting the balance of achromatism between the lens of the
revereer and the lenses of the corrector. All the quantities then will require adjust-
ment. The first step is to re-establish the achromatism. In doing so I keep the
first lens of the corrector unchanged, and two trials at least will be requisite to get
material for a proper adjustment of the other two as explained on p. 48. I found
by inspection and by previous trials that an alteration of the second lens of the
corrector produces its effect almost solely upon the coefficient K of the final scheme,
ami hardly at all upon G ; hence- 1 first adjust the lens of the1 reverser so as to make
A< I = 0 for variation of refractive index, and then the second lens of the corrector so
as to make AK = 0 also. .Since the system 0"2...06 from the last face of the
ivvereer to the first face of the second lens is unaltered throughout I take it in one
o
pier.-, taking the lens (4) with the data of p. 49, and
da = +'906760, d&= + '005000,
and taking in succession
fi = n-' = 1 "520000, n = "657895
and
' = 1"()1 xrr1 = 1"535200. n + Sn = '651381,
CASSEGRATN REFLECTOR WITH CORRECTED FIELD. 51
sve have then for the piece O",...O6,
1, +'906760'! f 1, ] fl, +'012500'! f 1, *
1 J [+'238736, + '657895 J[* 1 J[ +1 '062521, +1'520000
1, +'005000] f + 1'010127, + -929210]
«< I k [»],
1 J [+1-428571, +2-304108 J
and
"1, +-9067601 f. 1, * I f1' +'012500l f L * 1
1 _ +'243282, +'651381 J 1 ^ + T093579, +1'535200J
"1, + '005000 "I f + 1'010393, + '929370]
*< > [n + Sn].
1 J [+T470392, +2-3421 97 J
Now, taking the reverser first as given on p. 49, we have for the system
<>„... O",,
1, *1 fl, +1'320133"1 f 1, * 1 f1' •f'020000l
-'500000, lj [* 1 J [-'160450, h '657895 J [* 1 J
1, *] fl, +'020000] f 1, * ]
•
+ '901306, lj 1* 1 J [-'243885, +1'520000/
+ '330582, +1 '36 1934"!
-'208333, +2'16667 J
and
1, *1 fl, +1-320133"! f 1, * 1 fl> +'020000
•500000, lj [* 1 1_--163505, +'651381 J [* 1
1, *| fl, .+-020000"!
+ '901306, lj [* 1 [-'251014, +1*535200
1, I i, • T VXlUVW I i,
X
+ '330672, +T 36 1508]
> [n + Sn],
-'208522, +2'165574j
H 2
DR. R. A. SAMPSON ON A
alao the second lens of the corrector 0....(V. with curvatures, M given on p. 49, but
increasing the thickness to '015000, is
r i, 1 ft, +-oi5oool J i,
[-'001267, + -657895J [* 1 J [-r357552, + T520000.
+ '999981 + '009868 |
-T359452 +'986604j
!IM<1 !, * 1 Jl, + '0150001 f 1, * 1
-'001292, +'651381 J I* 1 J L-1'397234, l'535200j
+ '999981, + -009771]
|> [n+Sn],
-1-399190, + '986348 J
Hence the whole combination gives
•330582, +1-3619341 r + 1'010127, + '9292101 f+ '999981, +'0098681
-'208333, +2-166667 J 1+1-428571, +2'304108 J [-1-359452, + '986604 J
+ '140265, +3'457413l
-'198450, +2'237713 J
and
'330672, +1-3615081 f + 1'010393, + '9293701 f+ '999981, +'009771 1
-'208522, +2165574 J[ + 1-470892, +2'342197 J [ + 1'399190, +'98634sj
+ '140291, +3'45733G]
[n+Sn].
-'198480, +2'236730J
Hence for n + Sn there is an excess in the coefficient G of 26 units ; to correct this,
guided by previous experiments, I made a trial change in (K2), which refers to the
lens of the reverser, of —1220 units, so that
(Ka) = -'010297-'001220 = -'011517.
This gives, to redetermine the reverser, supposing its power is to remain unchanged,
-lW,(l + tJca)* = + '875000,
fcs = (n-l)B2, *,= +'020000,
B,= +'472584, B',= +'451898,
CASSEORAIN REFLECTOR WITH CORRECTED FIELD. 53
and these give for the system 0,,...O"a the schemes, liuilt up just as on the
previous page,
{ + '330582, +1-3619341 f + '380671, + T361508]
> and [n + Snl •{ }•
-'208333, + 2-166667 J [-'208547, +2'165477j
of which the first is the same as we had before, supplying a verification of the
solution of the equations for B,, B',.
Substitute these in the schemes Ou...O',, in place of the values already used;
[n] is, of course, unchanged, and we find for
f + '140266, +3-457242"!
[n+*»], I-
[-•198505, +2'236634 J
Hence G has now the same value in lx>th schemes and it is unnecessary to make a
further trial or change of the reverser, but there remains an excess in K of —55
units ; to deal with this, try reducing the curvature of each face of the second lens of
the corrector by one-hundredth part. This will give the schemes
1, "1 fl, +-015000"! f 1,
-'001254, + '657895 J\* 1 J \-r343976, + T520000_
+ '999981, + '009868]
-1-345856, + '986738 J
and
1, 1 fl, + -015000] f 1,
-'001279, + '651381 J \*
+ -999981, + -009771"]
[n + Sn].
-T385200, + '986484 J
Substituting these in the combination O,,...O'S we get
f + -140265, +3-457414") f + '140266, +3'457242]
[»J, -j > and [n + Sn], 4
-'196543, +2-284718J [-'196543, +2'284996j
(27)
Hence both G and K are now identical, and, in consequence, both schemes indicate
the same principal focus and the same focal length ; in other words, complete
achromatism at the principal focus.
5000"! f 1, * 1
1 J [-1-383262, + 1-535200 J
l>l;. |;. \. SAMPSON ON A
\\ . LOW return t.. tin- .-ilHMT.-iti.mH ; we have replaced the numbers of p. 49 by the
following: —
B, = - -250000,
,/, = +1-320133,
B, = -B", = +'472584, B'3 = +'4 5 1898, t, => +'020000,
,lt = +'906760,
B« = -'697845, B'4 = +2'043309, tt = +'012500,
(i, = +'005000,
B,= +'003667, B',= -2'584570, ^=+'015000, ...'.. (28)
;unl in these changes the aberrations calculated for the lenses of p. 49 will be changed ;
\\ «• now require to find new values for qt, qt, which will restore the disturbed correction.
It may be remarked that the chromatic correction depends very little upon the
distribution <>f the curvatures between the two faces which is indicated in the value of
q, and it might have been reflected that as the surface (6) is nearly plane, and the
U-.-im meets it nearly at right angles, while the surface (6') produces almost the whole
deviation of the beam for which the second lens is answerable, it would have been
better to keep B'6 unmodified while the second lens was adjusted for achromatism, but
this was not noticed until the solution which follows had been made, and was found to
reproduce almost exactly the value of B'8 of p. 49.
The aberration coefficients for a thin lens at its surface are given by (6), p. 33. I
have not so far succeeded in supplementing these by any algebraic expression containing
ivt'crence to thicknesses or separations of lenses, which are simple enough to be useful.
IliMice the procedure for finding qt, qe, e0 must be by approximation, and the following
is the method adopted. Calculate at the principal focus of the complete combination
tjivt'n by (28) the numerical values of the aberration coefficients, or at least the
essential ones <5,G, SaG, <5gG, in three parts, namely, first, the great mirror and reverser
together in which e,, is easily included as an unknown ; second, the first lens of the
corrector ; and third, the second lens of the corrector. The conditions for a corrected
system are then
these are not satisfied we must bring in corrected values of g4, q», e0 to satisfy
I asKiim.- lor the purpose of approximate correction that the quantities q, q'J
.Iculated aberrations with the same coefficients as if the lenses were thin ;
this Hup,M«iti..i. I calculate the algebraic values of the aberrations, carrying them
the surfaces of the lenses forwards to F'7 and backwards to O0 by a double
tl..- formula. (17) of p. 160 of the Memoir. Assuming that these
W* ••,« tl,.. adjustable parameters ,„, qtt qt account for the discrepancies
10118 to determine e0, qt, qt> and in consequence amended values of the
CASSEGRAIX KKFLKCTOlt WITH CORRECTED FIELD. 55
curvatures of (28), that is to say, the material to repeat the approximation, if
required, and finally to pro\.- that no further change is necessary.
The numerical calculation of all aberration! follows the model given in the Memoir,
pp. 172 ct ?«•</. , and it will l>e unnecessary as a rule to give details of the working
here, though I may mention thai I have found a noteworthy abbreviation of it.
The great mirror and reverser together, the former treated as parabolic, contribute
atF,
J,G = ... + '057176, SyG = +' 354860, SjGr = -3'297523.
\Vc must also introduce the deviation of the mirror from a paraboloid, viz., we
have at the surface of the mirror the additional term Stk = ...+2e0B08 = — '0312500e(l,
and all the others unaffected. To find the effect of this in the final set StG. ... , by
(17) of the Memoir we must take h'itk in <J,G merely, where h' belongs to the scheme
<>....F'7 and is simply equal to the final focal length, which comes out +5*087942;
hence we must supplement the numerical values above by the unknown term
J,G = ... -'i58998e0, £,G = *, <J3G = *.
Next we find that the other two lenses contribute together at F;
J,G = ... -'0301G7, 33G = ...- '342260, ^G = ... + 2'494419.
Further, for the three sections
$ = - '417950+ '937763-'885449 = -'365636,
H = +5*087942, ^H = -'930167,
and the three equations to satisfy being
<5,G = 0, «*2G = 0, <J,G = -'930167,
we find the actual numbers leave residuals in the left-hand memliera of the values
+ '027009, +'012600, +'127057 ....... (29)
These are to be brought to zero when supplemented by the proper expressions in
*•» ?4» <?«» and «o is dealt with above.
Now referring to the expressions for a thin lens and writing q = qjk so that for the
system just computed q4 = +2'3734, qfl = +4'8325, and confining attention to the
forms in which q is introduced at the surfaces of the lenses, these are respectively : —
First lens —
Sl7<= ... -'671321q4,
<V4 = ••• +'959031il,+ >JC.5793q/, S^ = ... +'671321q«,
<J,A4 = ... +'67i:i21q4,
and the rest zero ;
5fi DR. R. A. SAMPSON ON A
Sin=... --59583H,
.- - '8019054,- -222246q,', -V* = ••• + '595833qB,
... + '595833^
and the rest zero.
For the second lens, the subsequent normal scheme OV-.F7 is
{gf,hf; VI'} = {I, +713667; * ,1}
and by (17) of the Memoir, tbis gives for the second lens from Oa...F'7 the terms
in q:—
Coefficient, q(-,.
-1-168126
Coefficient, q02.
-'158610
+ '425226
*
*
+ '425226
*
*
#
*
#
The preceding normal scheme O0...Ort is
{g,h; k,l} = { + '140346, + 3'389017 ; -'007760, +6'937860}.
We see, by referring to the equations (17) of the Memoir already quoted so
frequently, that in order to get <S,G, S2G, S3G, we must form gS,y6 + kS,^e(s - 1,2,3)
with these values of g, h, k, I, and multiplying them respectively by
g1 = + '019697, 2gk = - '002178, k2 = + '000060,
gh = + '475635, gl+hk = + '947402, kl = - '053838,
= +1T485436, 2M = +47'025051, P = +48'133901,
take the sums. The values of gS,y^ + kS^ are
Coefficient, q0. Coefficient, qa2.
* = 1 • -'167242 -'022260
2 ..... +'059679 *
3 ..... * *
the resulting values are
Coefficient, q6. Coefficient, qa2.
<!iG=- -'003424 -'000438,5
-'023006 -'010587,6
+ '885540 -'255667,4. . . . (31)
CASSKGHAIN KKH.I-.rHMl WITH CORRECTED FIELD. 57
In the same way, for tin- fir-t l.-ns ..(' the corrector, the subsequent normal scheme
()',.. .O, is V, h'\ /•'./' ={+'(>: , 7U-J68; - 1 '345856, +'980009}, which
• rivi-s for the first system. !>••( \\.-.-n O4...F7
Coeffificnt, q,. Coefficient, q4*.
s +-65H4!)<; + -1898 1 7
= + -479503
* *
+ •479503
* *
* *
The preceding normal aclicnu- < > < >, is
(gj, ; /-,/} = { + '141675, +3-326581 : -'208333, +2'1GGGG7},
wliich gives
g* = + '020072, 2gk = - '059031, F = + '043403,
gh = + '471294, gl + hk = - '386074, kl = '451388,
A' = +11-066161, 2/i/= +14-415196, I3 = +4'694446,
so that with the values of gr«Vy4 + W.«»i, which are
Coefficient, q4. Coefficient, q4».
s=l - -006604 +'026897
2 +'067934
8 ..... * *
we find the contributions of the first lens ()„... F'7
Coefficient, q4. Coefficient, q4a.
«$,G = . . . . -'004143 + '000539,9
^G = . . . . -'029340 +'012676,2
S3G= .... +'906201 +'297642,1. . (32)
With the values q4 = +2'3734, q« = +4'8325, the joint contribution of the two
lenses in respect to the terms q, qa would be, from these expressions,
J,G= ...-'033578, JaG= ...-'356618, ^G = ... +2'136186.
Hence if new values of «,, q4, q, are to satisfy the conditions exactly, these are
determined by the equations
Coefficient, ««. Coefficient, q4. Coefficient, q4». Coefficient, q«. Coefficient, q«*. Constant.
0 = -'158998 -'004143 +'000539,9 -'003424 -'000438,5 + '060587
0= * -'029340 +'012676,2 -'023006 -'010587,6 +'369218
0= * +'906201 +'297642,1 +'885540 -'255667,4 -2'009113 . (33)
VOL. COXIII. — A. I
58
PR R. A. SAMPSON ON A
the solutions of which are
4, = +2-390547, q,( = + 4'936038,
eu = +'164675.
. (33A)
1C with these values of q4, q* we calculate the curvatures of the two lenses from the
t'«>rmul>u (24) of p. 49, we find that the completed approximation directs us to replace
the numbers of p. 54 from which we set out by
B4 = -'693009, B'4 = + 2'048193,
B«, = -'024163, B'« = -2-612025, (34)
together with the value of e0 just written down.
Turning back to p. 49, where these data from a previous approximation are set
down, we see that the chief effect of the step is to restore B'g to the value given on
p. 49, throwing the change in focal length which is demanded for achromatism, in
accordance with p. 53, almost exclusively upon B6, which is a surface that contributes
very little to these aljerrations. The changes are thus in reality smaller than they
appear. Following now strictly the plan given on p. 54, the next step is to take
the new system as a whole and calculate exactly its numerical aberrations at its
principal focus; it is unnecessary to give the details of this step, which contains
nothing new ; the following numbers show first the normal schemes from the surface
Oa up to each other point, and then the contribution of each surface to each of the
coefficients <5,G.. .<S3H at the principal focus F'7.
Surface O0
Oa to O2
Otf to Oa and surface O2
00 to O'a
O9 to O', and surface 0'2
00 to O",
0B to O", and surface 0",
O0 to O4
O, to O4 and surface O4
O, to 0'4
O, to O'4 and surface O'4
O0toO8
O0 to ()„ and surface O,
00 to O',
O. to O7. and surface 0'.
00 to F-
M = 1
['5200.
Normal Schemes, {g, h ; k, I}.
{ +
1 '000000,
*
j
-'500000,
+ 1'000000},
{ +
'339933,
+ T320133 ;
-'500000,
+ roooooo},
{ +
'339933,
+ 1'320133 ;
-'383906,
+ -444465},
{ +
'332255,
+ T329022;
-'383906,
.+ '444465},
{ +
'332255,
+ T329022;
-'083615,
+ 1-645630},
{ +
'330583,
+ T361935;
-'083615,
+ 1-645630},
{ +
'330583,
+ 1-361935;
-•208333,
+ 2-166667},
{ +
'141675,
+ 3-326584;
-'208333,
+ 2'166667},
{ +
'141675,
+ 3'326584 ;
-'103472,
+ 2'214112},
{ +
'140382,
+ 3'354260;
-'103472,
+ 2'214112},
{ +
'140382,
+ 3-354260;
-'007762,
+ 6-937938},
{ +
'140343,
+ 3-388950;
-'007762,
+ 6'937938},
{ +
•140343,
+ 3'388950;
-'003947,
+ 4'592448},
{ +
•140284,
+ 3-457837;
-'003947,
+ 4-592448},
{ +
•140284,
+ 3*457837 ;
-'196540,
+ 2'283904},
{
*
>
+ 5-088015;
-'196540,
+ 2-283904}.
(35^
CASSEGHAIN REFLECTOR WITH CORRECTED FIELD.
M = T5200.
Aberration Coefficients at F'7.
50
Surface.
Lateral.
Obliquity.
Lateral.
Obliquity.
Lateral.
Obliquity.
0
+ '29182
-'31800
*
+ '63600
*
*
2
-'01094
+'06023
-'04251
-'19332
•16509
+ '47099
2'
-•((4216
+'08433
-' 16866
-'01984
'67464
T50801
2"
-'01195
-'02233
-'04922
+ '19242
•20279
1'21803
4
+ '00227
+ '00722
+ '05341
-'04912
+ I' 254 14
'04592
4'
+ '00550
-'00212
+ '13140
+ '03502
+ 3*13954
+ 8-63910
6
'00000
'00000
'00000
-'00029
+ '00006
+ '21537
6'
-'02836
-'01552
-'69898
+ '17309
-17*22893
+ 6-37874
+ -201; is
-'20619
-77456
+ 77396
-13-87771
+ 12-93224
v
j
v
,
V
j
40-
-'00001
S,G =
-'00060
OgvJI ^—
-'94547
Surface.
Lateral.
Obliquity.
Lateral.
Obliquity.
Lateral.
Obliquity.
0
- T.3600
+ 1 '27200
*
-2'54401
*
*
2
+ '05238
•28821
+ '20342
+ '92507
+ '79000
2-25380
2'
+ '18850
•37701
+ 75400
+ '08871
+ 3-01601
+ 674169
2"
+ '04992
+ '09328
+ '20564
- '80386
+ '84719
+ S'08862
4
+ '00103
+ '00327
+ '02417
- '02222
+ '56746
'02078
4'
+ '27 142
•10500
+ 6-48513
+ 172840
+ 154*95440
+ 426'39006
6
- '00006
- '00021
•00153
+ '17490
•03694
-132-40118
6'
-'33991
• '18598
-S'37835
+ 2'07474
-206'51664
+ 76-45958
-'41272
+ '41214
70752
+ T62173
• 46'37852
+ 380-00419
V
,
v
/
V
= -'00058
SM = +'91421
SaR = +333-62567
. (36)
From these results we read the particulars of the field from the data given on
p. 30. We have, by pp. 58, 49,
/' = +5'088015, a =+'200000.
Hence, from <S,G, we find for the remaining spherical aberration a circle of radius
0""0004 at distance '000001 before the axial focus.
I 2
,„._ ,;. A. BAUFBOfl
O v
,,., t:iui,s that d^nd "M "Equity, I shall take as standard
0 = -01 = tan'1 34' 22"'6,
,,,„ sl,,l! ,1s,, ,iv,- the -suits for 0 = tan 30', and = tan «(»'. We have then for the
radius of the <;»nat!c circle
fl- tan 30'. 0 = tan 34' -4. £ = tan 60'.
_0"-0042 -0"X)048 -O'"0083.
I-',,,- tli,- /•./</;»* of the focal cir»-l'-
0-taiiSO1. 0 = tan 34' -4. 0 = tan 60'.
+ 0'"282 + 0"'370 + 1"'127.
P« the m<//»* '!/'/"• wnwtore ,,f the field, -1G2'817 ; and hence for the displace-
ment ,,f the f.K-al circl. IV. .11 1 the plane through the axial principal focus
/3 = tan30'. 0 = tan 34' -4. £ = tan 60'.
- -000006 -'000008 -'000024.
Finally. i«>r the distortional displacement
ft = tan 30'. ft = tan 34'-4. ft = tan 60'.
+ 4"'48 4-6"75 +35'"89.
It will he recalled that the linear unit is supposed to be 100 inches.*
We conclude that spherical aberration, coma, and curvature of the field are now
completely insensible, and that stars would be represented by strictly circular images
of diameter 0'56 seconds at a distance of 30 minutes from the centre of the field, and
2'25 S.M-.. in Is at 1 degree distance. No images at present obtained with any telescope,
at the middle of the field, where all obliquity-faults are absent, are sensibly less than
1 Beoond in diameter. Hence this also is completely satisfactory up to a diameter of field
of l£ degrees, or even more. There remains distortion, which requires examination.
This can !*• calculated precisely and applied as a correction to measures made, along with
difl'i'tvntial ivt'raotion and other unavoidable corrections. Hence, even if its amount
is very considerable it can be dealt with in a way that will not vitiate the use of the
telescope. It is possible, indeed, that a correction for distortion requires to be applied
toother 1'-l«-M-"|x-s now in use, especially those in which the lenses of the object-glass
are aepurati-d. It is instructive to look into the contributions of the different surfaces
t<» tin- t,.tal <•(', Ml. Tin- most remarkable is — 132'4 units from the surface (6) which
is nearly a plain- surface. This is an obliquity-constituent, and would be present if
tin- surface were a perfect plane. We see by examining the normal scheme next
preceding the surface (6) that the original obliquity, /3, of the ray is increased nearly
•iililnl Mmrh .v, IHI.i. — It is of interest to add that these conclusions have been checked by
trigonometrical calculations also, made by Mr. A. E. CONRADY at the instance of one of the Referees.]
CASSEGKA1N KKFLECTOU WITH CORRECTED FIKLD. 61
seven fold • before impact upon this surface. It is this that produces the large
It might be possible, with these numbers before one, to rearrange the general plan
of the surfaces so as to produce a smaller value of <J3H, but as explained above, it is not
essential to do so in a telescope which is not likely to I* used for exact measures over
a field of more than 30 minutes radius.
We now return to the question of achromatism. We shall first verify that as fur
as the normal scheme goes, the achromatism which was secured for the scheme of
p. 54, has not been sensibly impaired by the changes since made in the distribution
of curvatures between the surfaces. Writing down only the surfaces, we have
1*5352.
Surface < >„
Surf:uvs ( >,„ ( ).,
,, ' '0' * *2> ' ' 'J
„ o,,... <>"„
„ 00...04
,, 0....0",
„ o0...o,
„ IV..O'.
Normal Srhi-im-s for
{ +1-000000,
{+ '339933, +r320i:i:<
{+ '332299, +1*328811
{+ '330671, +1-361508
{+ '141569, + '3325076
{+ '140298, +3-352749
{+ '140287, +3-388119
{+ '140283, +3'457665
* , +5-088015
--500000, +r oooooo},
-•381G9G, + '433886},
-'08 1365, +1-634860},
-'208547, +2-165477},
-'101641, +2'213876},
-'002245, +7'074002},
-'000281, +4-636412},
-'196540, +2'284156},
-•196540, +2-284156}.
(37)
liy comparing this with the schemes (35), p. 58, it will be seen that the rays of
different refractive index separate decidedly in the course of their passage through
the instrument before they are brought together at their common principal focus.
The filial agreement was to be expected as it was within our control, as far as the
normal schemes were concerned, but it now remains to be considered whether there
is any sensible chromatic difference of aberrations ; this is found by recalculating the
aberration coefficients with refractive index 1*5352 in place of 1*5200. The results
are as follows : —
n + Sn.
<J,G = -'00018, (J.G = +''00846, J,G = - '45888.
.5,H = +'00844, 4H = + 1-39705, <?3H = +351*826. .
(38)
DR. R. A. SAMPSON ON A
Iiit.-rpivtiiig these, as on p. 59, we conclude that for
fi = 1-5352.
of Least Circle of Alwrration, -0"'007.
Comatic radius .
Focal radiiiH
Distortion
DisplaceiiK-n
Radius of curvature of field, — 5"423.
Tlie effect of the distortion at j8 = '01 will be to draw out the image into a small
spectrum of length 7"' 13— G""75 = 0""38. The radius of curvature of the field is
decidedly changed ; but the effect of the change as shown in the corresponding
displacement of the image-circle is not considerable.
ft = tan 30'.
ft = tan 34' -4.
P = tan 60'.
IS
+ 0"'0599
+ 0"'OG86
+ 0"-1197
+ 0"'431
+ 0"'5GG
+ 1"724
+ 4"74
+ 7"' 13
+ 37"'91
of focal circle .
- '000159
- '000240
'000638
AXIS •» 6
Fig. 2. Whole instrument. Scale 1 : 30.
It will be remarked that all these numbers run in the sense of ^increasing the
uU-rrations ; as there is no minimum property about the original index 1'52, we
conclude that the aberrations for smaller indices would be proportionately diminished,
and we see that it would have been better to have secured exact agreement for the
larger index in place of the smaller one. In estimating the effect we may, for
instance, take the following values, which are the indices for CHANCE'S hard crown
glass:—
0, D, F, G,
Indt-x
T5150, T5175, 1-5235, T5284 ;
that » to say with such a glass two-thirds of the excesses shown in the table above,
"•Hulls of p. 60, would cover all chromatic differences. There appears to
H»lBg „, any ,,f them that calls for a revision of the calculations
<»", to the question of the actual sizes and places of the mirrors and
CASSKCK.MN KI.IU-rTOK WITH O >i;|;K(TU • FII.I.H. C,:;
lenses in respect to the paasji^'- of ;i ray through the instrument. Calculate from
the normal schemes, p. 58, for b= +'20, and ft = —'01,0, +'01 respectively, the
value of // at each surface ami also at the focal plane F'7 ; this will give the
necessary ajH-rtnn-s for complete inclusion of all rays I'mm the great mirror, up
to these, limits of ohli(|iiity. We find ax follows:—
Value of Semi-aperture.
Surface.
/8 - - -01.
0- -00.
P = +-01.
0
+ '200
+ '200
+ '200,
2
+ '055
+ '068
+ '081,
2'
+ '054
+ '067
+ '080,
2"
+ '053
+ '066
+ '079,
4
-'005
+ '028
+ '061,
4'
-'006
+ '028
+ '062,
6
-'006
+ '028
+ '062,
6'
-'007
+ '028
+ '063,
7'
-'051
0
+ '051.
Hence if the great mirror is 40 inches in diameter, the reverser requires to be
16'2 inches, the first face of the corrector 12'2 inches, and the last face 12'6 inches;
the diameter of the image at the focal plane would be 10 '2 inches.
It is necessary to verify that the corrector does not cut out any rays coming from
the great mirror to the reverser. By the data on p. 54, the first face of the
corrector is at a distance +'413750 beyond the surface of the great mirror.
Calculating the value of i/ along the ray y1 = fiz' + b', for this value of x', where
b', ft are taken from the normal scheme for the ray Ixstween the surfaces O0 and O,
we have
Value of b. ft = - -01. P = '00. /8 = + -01.
+ '200 +'154 +'159 +'163,
+ '081 +'061 +'064 +'068. . . . (40)
Thus the ray which just cleared the reverser on its way to the great mirror
would clear the corrector on its return.
Allowing that '085 of the radius of the great mirror is unavailable the effective
aperture-ratio is reduced from 40/508'8 = 1 : 1272 to 36'28/508'8 = 1 : 14'05.
The following table shows the inclinations of the ray to the axis of the telescope
between the various surfaces : —
,; i M;. I; A. SAMPSON ON A
Inclination of Kxtreme Kay to Axis.
For b =+-200. 0---01. 0 = -00. /i=+-OI.
I: Inn- siuC.r,. ()„ -0'6 0'0 +0'6,
i <)„ and Oa -6'3 -57 -5'1,
O, „ O'2 -4'6 -4'4 -4'1,
O', „ O", -T9 -TO 0'0,
O", „ O4 -3'6 -2-4 -n,
o, „ O'4 -2'5 -T2 -0%
<>', „ O, -4-1 -01 +3'9,
<>„ „ O'« -27 0'0 + 2'6,
<>'« „ F'7 -3'6 -2-3 -0'9. . (41)
The inclinations of the ray to the normals of the surface (.,„) are given by
. + 'l*B;h,. /^'a. + hj.Bj, which may be calculated at once from the normal schemes;
but note that as these include reversals for the case of a mirror we must then take
in place of the latter ftf.Jn— b^E^ : —
Inclination of Extreme Ray to Normal to Surface.
For6=+-200. ft = - -01. ft = -00. /3 = + -01.
r-3'4
|_3'4
-2'9 - 2'3
-2'9 - 2'3
0 r-4'8 -3'9 - 3'01
1-8-2 -2'6 - 1'9 J
(V f-3'3 -27 - 2-n
1-3-3 -27 - 2-1 J
O", f-3'3 -2'8 - 2'31
1 _5'0 -4"2 - 3'3 J
O f-3'4 -3-5 - 3-6-1
-2'3 - 2'4J
0'4 /-3'1 +2'1 + 7-3-1
ll-Oj
-47 +3'2 -Hll-
-4'1 -O'l
-27 -0-1
Ort - -O'l + 3-81
+ 2-5J
O'. l'-17 -4'2 - 6-81
L-2'6 -6'4 -10-2J . . (42)
Thus the greatest angle of incidence is ll'O degrees upon the second surface of the
the corrector. This is much below what is permitted in the construction
t glass of a refractor; we find, for example, in STEINHEIL and VOIT'S
J an aperture ratio of 1 : 12, the angle of incidence of extreme rays,
Jly parallel to the axes, upon the first surface if the flint-lens exceeds 1 5 degrees
CASSKUKAIN BEFLEOTOR WITH OOBBEOTBD FIKI.D.
I would add a few remarks upon the problems presented by the construction of
such a telescojM', or at. any rate, i>t' its optical parts. It requires the production of a
great mirror and three lenses which shall be in due relation to one another. None
of the sizes or curves go outside what has already been made ; and whenever a
refractor is made, three of the surfaces must be turned out in agreement with the
fourth. Hence there is no new difficulty in making and the problem is essentially a
question of testing. The testing must be optical and not mechanical, for the former
far outruns the latter in delicacy — it is said ten times. And because there are so
ninny surfaces it would 1» essential to test them independently of one another. In
the lenses, four out of the six surfaces are concave and spherical and can be tested
AXIS
Fig. 3. Koverser and corrector. Scale 1 : 3.
with reflected light. The great mirror is neither a sphere nor a paraboloid, but its
radius of curvature for different zones can be laid down, and each zone tested for
agreement with this, just as in making a paraboloid. There remain then two convex
surfaces, and the question of figuring the lens-surfaces to allow for inequalities of
refractive index within the glass. These are matters for the skill of the maker and
it would seem a not unreasonably difficult task.
I add a plan of the whole instrument and, upon a larger scale, of the reverser and
corrector, and also the final specification, collected from pp. GO, G2, but making the
unit 1 inch. For comparison the field of a Newtonian of the same aperture and focal
length is added. It may be recalled that the displacement of the centre of the comatic
circle is twice the comatic radius. For an uncorrected Cassegraiu the field would be
very much the same as for a Newtonian of the same aperture but of focal length equal
to that of the great mirror, except in respect to curvature and distortion, see p. 41.
I would express my acknowledgments to Mr. R. W. WKIGLEY who helped me to
perform many of the calculations.
VOL. CCXIII. A. K
66 DR. R A. SA MI-SUN: A CASSRUAIX REFLECTOR WITH CORRECTED FIELD.
Final Scheme.
(Jivat mirror-
Aperture .... -«« = 40'
Radius of curvature R0 = -400'000,
e,,= +' 16468.
c?, = +132'013.
Revereer —
Aperture ...... 2«2 = 16<2»
First surface ..... R, = +211 '603,
Silvered surface R'2 = +221'289,
Thickness ... <2= 2-000.
d,= +90-676.
Corrector, 1st lens-
Aperture 2«4 = 12'2,
First surface R4 = -144'298,
Second surface R'4 = +48 "824,
Thickness . tt = 1'250.
d6 = +0'500.
Corrector, 2nd lens-
Aperture 2a6 = 12'6,
First surface R6 = -4138'559,
Second surface R'6 = -38'285,
Thickness te — 1'500.
d7= +71-377.
Focal length /'. = +508'802.
Distance of principal focus beyond
surface of great mirror +33"290.
Whole length of instrument .... 167'3.
Specification of Field at /8 = "01 = tan 34'"4.
H = T5200. /x = 1-5352. [Newtonian.]
" // //
Radius of least circle of aljerration . . O'OOO — 0"007 O'OO
Radius of comatic circle . . . -0'005 +0'069 +0'80
Radius of focal circle , +0'370 +0'566 -0'41
Distortional displacement +6'75 +7"13 O'OO.
Curvature of fi.-l.l -1/16282 -l/542'3 -1/508'8.
[ 67
III. Tin- 'I'lirriiHil I'ro/ierties of Carbonic Acid at Loiu Temperatures.
ll,i ( '. Ki: i :\v i:\ .1 1 \ KIN, M.A., M.Inxt. C. A'., /'rufensor of Engineering Science, Oxford;
and D. R. PYK, M.A., Fellow of New College, Otford.
< 'niitmnnicnti'il hi/ Sir J. ALFRED KWINO, K.C.B., F.R.S.
Heceiveil January 27,— Read February 27, 1913.
CONTENTS.
Page
PART I. — Object, scope, and theory of the experiments . 68
PART II. — Detailed description of Series I., II., and III 82
PART III. — Detailed description of Series IV. and V 90
PART IV. — Discussion of the results 94
Summary 102
References to literature 104
Tables I. to XVIII 104
LIST OF TABLES.
Table I. — Pressure-temperature observations.
„ II. — Series I. Observations.
HI.- „ II.
IV.- „ III.
V.- „ IV.
„ VI. — Latent heat calculation.
„ VII. — I and Cp from smooth curves.
„ VIH. — &/> between pressure curves and limit curve.
• „ IX. — Collected results. T, P, I, </>, L from smooth curves.
„ X. — Comparison of pressure-temperature observations.
„ XI. — Sample record, Scries I. (Pressure, heat, weight, Sic.).
„ XII. — „ „ „ I. (Temperatures).
„ XIII.— „ „ „ II. (Heat, weight).
„ XIV. — „ „ „ II. (Temperatures).
XV.- „ „ „ III. (Heat, weight).
„ XVI. — „ „ „ III. (Temperatures).
„ XVII. — Comparison of #</> diagrams.
„ XVIII. — New data, arranged as in MOU.IKK'S paper and KWINO'S ' Mech. Prod, of Cold.'
VOL. < 'CXI I [. A 490. K 2 PublUhed separately, July 28, 1913.
Pi:uK. c. FKi:U'KN .IKNKFN AND Ml;. l>. R, PYE OX THK
PART L— OHJECT, SCOPE, AND THEORY OF EXPERIMENTS.
THK experiments described in the following paper were originally undertaken to
determine the Latent Heat of Liquid C0a and the Specific Heats of the liquid and
of the gas at temperatures below —30° C., which is the lowest temperature for which
MOLLIER has calculated them, and also to check MOLLIER'S Entropy-Temperature
diagram by direct experiment, as it appeared likely that the calculated results might
be appreciably wrong near the limits of their range. The results of the first
experiments confirmed this expectation, and it became apparent at the same time that
MOI.I.IKU'S 6<f> diagram could not be modified to agree with the experimental results
without some further data. The investigation was therefore extended so as to
include the measurement of all the quantities required for the construction
de novo of a 6<f> diagram for saturated gas at low temperatures. Finally, by
Sir ALFRED EWING'S suggestion, the range of the experiments was further extended
to higher temperatures, to enable the diagram to be constructed nearly to the
critical point.
The experiments made to carry out this programme were : —
1. The determination of the Pressure-Temperature Curve for Saturated Vapour ;
2. Three series of heat measurements, called Series I., II., and III., to determine
the Latent Heat L, the Total Heat I of the liquid, and the Specific Heat of
the gas ;
3. A series of throttling experiments, called Series IV., to determine the Joule-
Thomson effect for liquid COa ;
4. A series of direct volumetric measurements, called Series V., to determine the
Dilatation and Elasticity of liquid C0a.
Fhe experiments also supply data from which may be calculated : Specific Volume
'saturated vapour (or its reciprocal, the Density) ; relative Densities of liquid CO2
at saturation pressures ; Specific Heat of liquid CO3 either at constant pressure or at
saturation pressure.
The pressure-temperature curve has been often observed during the last 50 years ;
ific volume of saturated vapour and the specific heat of the liquid at
. pressure have not been observed before below -25° C. ; the latent heat,
heat of the gas, and the dilatation and elasticity of the liquid have not
rved before below 0° C. The total heat of the liquid, the specific heat
onstant pressure, and the Joule-Thomson effect have never been
The latent heat has often been calculated, but the specific volume
ted vapour, on which the calculations are based, ha* not been observed
... and only once below 0° C., so that all calculations below -25° C are
baaed mi axtrapplatiaaa.
TIIKHMAI. PBOPEBTBB OF CAl.T.ONir ACID AT LOW TEMPERATURES. (i«»
l'Y.>in the results of these six sets of experiments all* the data were calculated for
constructing the fy> diagram ('mm +20° C. to —50° C. The diagram is shown in
fig. 12, p. 79, and some of the results are given in Table IX.
Tin- experiments WIT.- i-.m iud mil in tin- Kngineerinjr I,alx>ratory at Oxford with a
vapour- compression free/ing machine, presented to the Laboratory by BrasenoM
College. In addition to the usual parts the apparatus includes a pair of suspended
flasks by \\liii-li the rate of flow of the CO., round the circuit can be measured. The
following additional apparatus was made for these experiments: — Two electrically
heated calorimeters, one of which always replaced the refrigerating tank of the
free/ing machine; several thermo-junctions for measuring the temperatures of the
calorimeters and of the CO, at various points in the circuit ; a graduated glass
capillary tul>e, with regulating valves, for measuring the changes of volume of liquid
( K )3 under varying pressures and temperatures in Series V. experiments ; and a
s|ie.-i:d throttle- valve for Series IV.
The gas used was commercial CO2, supplied by Messrs. Barrett and Elers, Limited,
of London, who have kindly informed the authors how it is made. Coke is burnt in
a furnace and the products of combustion, after being washed with hot and cold
water in scrubbers filled with fragments of limestone to eliminate any SO3, are passed
through absorbing towers filled with coke over which a stream of potash lye flows
which absorbs the CO.,. The enriched lye is then heated in iron boilers and the CO,,
driven off by the heat, is compressed in compound pumps into the steel flasks in
which it is sold. It is dried by passing over calcium chloride between the first and
second stage of compression. The gas made in this way is said by the makers to
contain no impurities, except possibly £ per cent, to £ per cent, of air and traces
of SOa. These traces of S02 were probably eliminated with the moisture in the
special drying appliances used in these experiments, so that the oidy impurity left
was air.
To estimate the amount of air present the gas was analysed in a modified form of
HKMPEL'S apparatus specially arranged for this test. About 100 c.c. of gas was
measured over mercury in a burette and then passed into the potash absorption bulb.
The residue of undissolved gas (air) was then drawn back into a small burette and
measured over potash solution. The apparatus was arranged so that the test could
!»• repeated as often as desired while the residue accumulated in the small burette.
In this way a sufficient quantity could lx- analysed to allow of an accurate deter-
mination of the small amount of air. The amount of air found was only O'll per
cent, by volume ('073 per cent, by weight). ANDREWS,! in his classical experiments,
never was able to reduce the air in the gas he used to less than Boo t° roloo- The
h This is not strictly accurate. In working out Series V. the density of liquid CO... at one temperature
is needed. BEIIN'S result has IHJCII used. Any possible error in thia has no appreciable effect on the
result. With this exception every quantity needed has been measured.
t '1'hil. Trans.,' 18G9, p. 381.
70 PROF. • FIMUIN ..KNKIN AND MR D. R. PYK ON THE
pn^nce of thiB small amount of air has introduced a small error into the pressure-
!.', „,,,,„„. ,urv,., hut does not appreciably affect the results of any of the other
experiment*
Fig. 1.
The general arrangement of the apparatus connected for normal working is shown
diagrammatically in fig. 1. The gas enters the pump at pressure p2 and is compressed
to a higher pressure />,. It is then condensed in the condenser at the corresponding
saturation temperature fy. From the condenser it flows as liquid through the
weighing flasks to the throttle valve on the calorimeter. In passing the throttle
valve the pressure falls to p.2, some of the liquid evaporates and the temperature falls
to Oj. The mixture of liquid and gas then enters the calorimeter, where the rest of
the liquid evaporates, taking up heat at the constant temperature 6a. From the
calorimeter the gas passes back to the pump. In order to make sure that all the
liquid has evaporated, the gas is warmed (superheated) a few degrees above 02 before
it leaves the calorimeter. The approximate 6<f> diagram* for this cycle is shown
in fig. 2.
The line AB represents the expansion of the liquid through the throttle valve from
j>,0, to p£r AB is a line of constant total heat I. The line BE represents the
evaporation of the liquid at constant pressure p.t and temperature Q.2. The line EC
represents the small amount of superheating from f)2 to 03 at constant pressure p2.
The line CD represents the adiabatic compression in the pump from p2 to p,. The
* Cf. EWING (4), p. 80.
TIIKRMAL PROPKRTIKS OF CMJimMC ACID AT LOW TKMI'KK ATURKS. 71
line DFA represents the cooling ami oondeoMtion in the condenser, l»oth at constant
pressure pt.
The ])iv.s.surr-tiMn|MT:ituiv mm- was iL-tcnniiicd with the apparatus working in
this way : A pressure gauge was rumu-rted to the pipe immediately after the throttle
A.
IA const:
B Pi
ft If
Approximate 6t diagram
for freezing machine cycle.
Entropy
r Q.
Fig. 2.
valve and a thermo-jiinction inserted at the same place, as shown in fig. 1. The gas
is always saturated at this point so that the temperature is unaffected by radiation
or conduction along the pipes. By varying the adjustment of the throttle valve a
series of readings of corresponding pressures and temperatures was obtained ; a
Fig. 3.
summary of the ol«ervations is given in Table I. The olwervations were plotted and
a smooth curve drawn through them. Figures taken from the smooth curve are
i^iven in Table IX. The smooth curve is copied in fig. 18, p. 95, for comparison with
previous observations.
7-
,.I;|1|. ,.
.'FN-KTN AND MR. 1>. R.
ON THE
Serie* I. mea8urement8 were also made with the apparatus working in the normal
way described above, two thermo-junctions and a pressure gauge being connected as
sh^n in fig 3. In this series the principal quantities measured were the rate of
hW ,,f CO, and the electrical power supplied to the calorimeter to balance the
refrigeration. From these data the refrigeration, i.e., the heat absorbed per lb.,
represented by the area NBECQ (fig. 2) was calculated. This is the heat required
to evaporate the liquid part of the CO, and to superheat it all from 6, to 08, the heat
used in superheating being represented by the area PECQ. A series of experiments
was made with different values of 6r A summary of the observations is given in
Table II. ; the results are also given in column a, Table VI.
Experiments were not made at temperatures above 20° C. owing to the increasing
difficulties of manipulation. At the higher temperatures the weighing flask had to
he heated to keep the pressure above the evaporation pressure. At the same time its
capacity fell off rapidly owing to the great expansion of the liquid. The condenser
and pump had to be run at correspondingly higher temperatures and great care
exercised lest the condenser and flasks got over-full of the expanded liquid.
CALORIHf TEH 1 CALOaiHf TEK
Fig. 4.
Series II. — For this series the normal arrangement of the apparatus was slightly
modified, as shown in fig. 4. The liquid C02, before reaching the throttle valve, was
led first through the second coil in calorimeter I., so that its temperature was reduced
to any required temperature &„ and then through calorimeter II., in which it was
warmed again at constant pressure to any desired temperature Qy. The quantities
measured were the rate of flow, the rise of temperature of the liquid, 6y—6z, and the
electric power supplied to calorimeter II. From these data the change of total
heat I of the liquid at constant pressure for the range 9X to 6y was calculated. Two
sets of experiments were made, one at 700 Ibs. per sq. in. pressure and one at 900 Ibs.
per sq. in. pressure. A summary of the observations is given in Table III. The
observations were plotted and smooth curves drawn through them. Figures taken
from the smooth curves are given in Table VII. This series of experiments does
not determine the absolute values of I, but only differences ; the zero of the I scale
was determined later, see p. 80. The slope of the I curve is the specific heat of the
liquid at constant pressure. Values of the specific heat deduced from the slope of
the curves are given in Table VII.
Tlll.KMAI, PROPERTIES OF GAUBONIC ACID AT LOW TEMPERATURES. 73
111. •--!•'. ,r (Ms SLM'lrs tin- liorni.'il :irr:iii^<Miii-iit "f tin- ;ip|i;ir;ilus w:is
slightly modified, as is shown in fig. 5, by inserting calorimeter II. between calori-
meter I. and the pump. The gas leaving calori-
meter I. at a temperature 63 was warmed to any
desired extent in calorimeter II. The quantities
measured in this series were the rate of flow of
COa, the rise of temperature of the gas in calori-
meter II., and the electrical power supplied to
calorimeter II. From these data the specific
heat of the gas at constant pressure was calcu-
lated. A series of measurements was made with different values of Qr The results
are shown in fig. 6, where the specific heat of the gas near the limit curve* is plotted
against the pressure.
CALOHIHC ren I CALO*i*e re KM
Fig. 5.
0-6
0-5
0-4
0-3
0-2
0-1
Spec : heat of C0£ jas
at const: press.
Near the .saturated condition
IOO
zoo
500
400 500
Lba/iq.incn.
Fig. 6.
600
700
600
900
Combining the results of Series I., II., and III., an approximate value of the latent
heat may now be calculated. If we neglect the complications introduced by the
changes of volume of liquid COa (or what is equivalent, if we assume, as a first
approximation, that the limit curve coincides with the constant-pressure curves)
then the difference of total "heat I8— I, from 63 to 0, is represented in fig. 2, p. 71,
by the area RGAM.
k The limit curve is the boundary of the area on the diagram representing saturated vapour, separating
it on the one side from the area representing liquid and on the other from the area representing super-
heated gas. The two sides of the curve »re called the " liquid-limit curve " and the " gas-limit curve " ;
they meet at the critical point.
VOL. CCXIII. — A. L
74 ,.|;OF. C. FKKWKN JKXKIN AND MR. D. R. PYE ON THE
Alao, by a well-known property of constant-pressure lines,
Area RGBN = RGAM
which can be read off the I curves.
Taking any experiment of Series I., we have
Area BQ, given by Series I. (column a, Table VI.) ;
„ EQ, which may be calculated from the specific heat of the gas obtained by
Series III. (column y, Table VI.) ;
„ GN = I,— ID read off the I curves obtained by Series II. (column S, Table VI.).
Whence the latent heat, L = BQ-EQ + GN, may be found.
On the same assumptions an approximate 6<j> diagram may be constructed.
Starting at the zero-point on the 6<f> diagram (9 = 273° C. abs., 0 = 0) plot, step by
step, the constant-pressure line corresponding to the curve of I, remembering
that on the 6<j> diagram the area under each element of the curve is equal to
the corresponding difference of I ; this is quickly done since the curve is almost
$ T R
ft JV
Corrected $t diagram
for freezing machine cycle.
Fig. 7.
Since we are neglecting changes of volume of the liquid, this curve will
» with the liquid limit curve (as it practically does in the Steam diagram).
the gas-limit curve, mark off the values of L/0 for various temperatures,
the hquid-limit curve and measuring to the right ; joining up the points
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES.
75
so found we have the gas-limit curve. The diagram might be completed by adding
the I lines, &c., but it will be convenient to consider first the modifications required
to allow for the changes of volume of the liquid CO, which have so far been
neglected.
Fig. 7 represents the same diagram as fig. 2 with the addition of some lines to
show the effects of the dilatation and elasticity of the liquid.
Let AHK be the constant-pressure line through A, in the liquid area,
Let GA represent the limit curve as before,
Let GK be the constant I line through G, meeting AHK in K.
The heating of the liquid CO, in Series II. experiments is now represented by the
line KA, instead of by GA.
The true value of L is found as follows : —
L = GN + BQ— EQ as before; also BQ and EQ are given, as before, by Series I.
and III. experiments, but GN is no longer I,— I,. We now have
GN'= IB-IG)
= IA— IK, since AB and GK are constant I lines,
= (I, — I,) + (I,— 14), using suffixes to refer to the temperatures #„ 0,, and 04.
I,— I;, is read off the I curve as before, but I,— I4 can only be read off the curve
when 04 is known. The quantity I3— 14 may be regarded as a small correction to be
applied to the approximate value of L to allow for the elasticity and dilatation of the
liquid.
Series IV. experiments were made to determine the difference of temperature
da— Ot between H and K, i.e., the Joule-Thomson effect for the pressure drop Pi~p3.
It was observed directly by measuring the change of temperature as the liquid passed
through a throttle valve. The arrangement of the apparatus is shown in fig. 8.
Fig. 8.
A summary of the observations is given in Table V. The observations are plotted in
fig. 9 and a smooth curve drawn through them. The values of I,— 14 calculated from
L 2
, c. M:K\\TN .M:\KIN AND MR. D. R. PYE ON THE
this curve for each of Series I. experiments are given in column S of Table VI., and
tli.- c,,rn-.-t«-.l v.ilu.-s of L in the last column. These values of L were plotted ;m<l
values taken from the smooth curve are given in column 5 of Table IX.
+•3
4-2
+•1
O
— I
—2
— 3
—4
—5
— 6
—7
-8
-•9
-6
^*^
^\°
^Z~
^^
--^
~^^S.
-Ss^
^V
^5
k
\
Ser
ies W.
\
\
i^se
£nd ..
; of obse
M
rvations
it
\
\
\
0 -50 -40 -30 -20 -10 0 +10 +2C
Temp: °C.
Fig. 9.
If we proceed now to the construction of the true 6<j> diagram a further difficulty is
met with. If a constant-pressure curve is drawn as before, using the values of I
already obtained, the result is the curve KA. The difficulty is to draw the true
limit curve GA, which was previously assumed to coincide with KA. The authors
have been unable to devise any direct experiment to fix the position of the limit
curve relatively to the constant-pressure curve KA, and an indirect method had to
be used. The fundamental thermodynamic equations give the well-known equation*
(**} - (**\
\dpJ0 \dO/P'
Let H, fig. 7, be the pointed.,; then
HG = ** - r(i) *•
Jpi \O,u,ip
Thus the distance fy between the limit curve and the constant-pressure curve can
* Cy. PMSTON'S -Theory of Heat,1 p. 740, 2nd edition.
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 77
be calculated from the temperature coefficient of the liquid. This coefficient* does
not appear to have been determined below 0° C., so the experiments called Series V.
were undertaken to measure it. AMAGAT'S (5) results give a few values above 0° C.
Fig. 10.
Series V. — The general arrangement of the apparatus is shown in fig. 10. A
capillary glass tube, closed at the bottom, was partly filled with liquid COa ; on the
top rested a mercury indicator. The upper end of the tube was connected through a
valve to a flask of C0a gas at high pressure. A release valve was also connected.
By adjusting these two valves any desired pressure could be obtained in the glass
tube. The glass tube was placed upright in the calorimeter, so that it could be kept
at any desired temperature, while the volume of the liquid CO3 was read directly
by noting its length in the glass tube as shown by the position of the mercury
indicator.
A series of measurements of volume were made at different temperatures and
pressures. The results are plotted in fig. 11, with pressure and volume as co-ordinates.
The slope of the curves is the elasticity (dr/dp)t, and the distance between the curves
divided by the temperature difference is the dilatation (dr/dS)p.
To calculate $<f>, however, it is not necessary to evaluate these functions, for the
area between any two adjacent curves at temperatures Ql and 6y is
* The specific volume of the liquid at low temperatures for points on the limit curve is known. This
enables the rate of change of volume with temperature, along the limit curve, to be calculated, but what is
required is (dv/dtyp, which is equal to
dv/dO + (dvjdp)t <lp/d9,
where dvjdO and dp/dO are taken along the limit curve.
IM;oF. C. FREWEN JENKIN AND MR. D. R. PYE ON Till;
P.V. isothermal curves
for liquid C02.
(Mean results)
Area, scale :-
I sq. cm = .00167 th':u-
Points: A "and dotted curves show Amagats
observations.
ZOO 3OO 400 5OO 600 7OO 8OO 9OO I,OOO 1,100
Pressure, Ibs. per sq: inch.
Fig. 11.
If the area between the 0t and 0a curves is bounded at the side by the ordinates at
pressures pt and pa, then the area is
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 79
EN TROP Y - TEMPERA TURE
01 -05
and therefore
area P* idr\ , . , ,
j- I dp = 3<f> between pressure curves pl and p.f
"i — "a Jpi \d6/f
Similarly, if the area is bounded by the limit curve and the ordinate at j»,, then
area
= S<f> between limit curve and pressure curve pt.
corresponds in each case with the mean temperature
80
PROF. c. FI:K\\KX JENKIN AND MR. D. R. PYE ON THE
V-,hi, of fy between the 700-lb. and 900-lb. pressure curves and the limit curve
were calculated in this way and are given in Table VIII. From this table we see
that the 700-lb. and 900-lb. pressure curves cut the 0° C. temperature line at
0 = --IM.-J4 and 0 = -'0049 respectively. These two points serve as the starting
fur plotting the two constant-pressure curves (700-lb. and 900-lb.) on the 00
diagram. .
The corrected 00 diagram may now be constructed. Ihe result is shown in
fig. 12. This diagram was drawn as follows :— Starting at the points just found,
the 700-lb. and 900-lb. pressure curves were drawn as before, remembering that the
area under each curve on the 00 diagram between any two temperatures equals the
difference of I for the same temperature range. The liquid-limit curve was then set
off on the right of the pressure curves by plotting the small values of $<f> given in
Table VIII.
The gas-limit curve was then plotted by measuring off the values of L/0, taken
from Table VI., to the right of the liquid-limit curve.
To plot the constant I lines it is first necessary to find a starting point. At the
origin (0 = 273, 0 = 0) the value of lo is given by the equation I0 = Apv, where
p is the saturation pressure at 0° C. = 508 Ibs. per sq. in. = 73,200 Ibs. per sq. ft.,
v is the volume of 1 Ib. of liquid = "0173 cub. ft.,*
A = T^jVo J therefore Io = '905 thermal unit.
The point on the 700-lb. curve having the same I was then calculated as follows :—
The change of temperature at 0° C. is taken from the curve, fig. 9, p. 76, viz., '24° C.
per 100 Ibs. difference of pressure. The difference of pressure is 700 — 508 = 192.
Therefore the temperature of the required point is '24x1 '92 = '46° C. In other
words, I = '905 at a point + '46° C. on the 700-lb. pressure curve. Similarly,
I = '905 at a point + '94° G. on the 900-lb. pressure curve.
Having found in this way the true value of I for one point on each of the pressure
lines, we can mark the true zero on the scale of the I curves, so that they shall give
absolute values of I instead of only differences (see p. 72). Using the new scales,
the points on the 700-lb. curve corresponding to I = 0, —5, —10, —15, —20, and —25
were marked off on the 00 diagram; also the points on the 900-lb. curve corre-
sponding to 1 = 45, +10, and +15. The corresponding points on the limit curve
were then calculated from the difference of pressure multiplied by the rise or fall
of temperature, given in fig. 9. Having found these points, the rest of the I curves
within the saturated area are easily constructed. Draw a horizontal (constant
temperature) line through one of the points, say, where I = 0, on the limit curve.
This is at temperature -1° C. = 272° abs. Along this line, starting at the limit
curve, mark off a series of points, equally spaced, at distances S<j> = ^ apart. These
will be points on the +5, +10, &c., I lines.
* B£HN's(6) value of the density of liquid C02 at 0° C. is -925.
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES.
81
Five horizontal lines were divided in this way and the I lines drawn through them.
The values of I for the liquid, given in Table IX., were found by interpolation. The
values of I given for the gas were obtained l>y adding the corresponding values of L
to the values of I for the liquid.
Approximate constant-pressure lines in the dry area may be drawn, if the specific
heat is assumed to be constant. They are logarithmic curves given by the equation
d<f> _ a_
d6~ 6'
where a is the specific heat of the gas at constant pressure.
40 — . . .
20
<J
o
-IO
-20
-30
-40
\
Comparison of
.tropy-tempera.tu.re diagrams.
Dotted lines are plotted from
Mollier's figures.
-•1O -08 -oe -o* -ot O
•ot -O4 «e -oe -1O
Entropy .
Fig. 13.
•16 -1C -2O -ti •£•» -U
These curves are drawn in the diagram as straight lines, since for the short length
shown the curvature is imperceptible. The values of o- are taken from the curve,
VOL. CCXIII. A. M
82 PROF. C. FREWEN JENKIN AND MR. P. R- PYE ON THE
fig. 6, p. 73. They agree closely with MOLLIEE'S curves, as may be seen in fig. 13 (p. 81),
where a few of each (at different pressures) are drawn for comparison.
Tim four dryness curves on the diagram were drawn by dividing the distances
between the limit curves into quarters. This completes the construction of the
diagram.
PART II.— DETAILED DESCRIPTION OF THE APPARATUS AND METHOD OF CARRYING
our THE EXPERIMENTS SERIES I., II., AND III.
The compressor is a single acting pump made by Messrs. J. and E. Hall, of
Dartford. It is driven by a variable speed electric motor. The piston-rod gland is
formed of a pair of cup-leathers, between which oil is forced by an auxiliary piston,
thus no leakage of CO., takes place, but a little oil enters the cylinder, and is pumped
over with the CO3 ; it is mostly caught in an oil separator, but a trace of oil is carried
round the whole circuit with the CO-j. Under ordinary conditions the pump runs
cold, but when working under the abnormal conditions of some of the tests it ran hot
and gave trouble till a water-jacket was fitted round the cylinder.
The condenser is a coil of pipe in a tank through which cooling water flows.
The drying flask is a steel flask containing a little phosphorus pentoxide. The gas
is led in by a pipe leading nearly to the bottom of the flask and leaves by a pipe from
the top. A few ounces of PaO5 were put in the bottom of the flask and renewed from
time to time. When the apparatus was first tried great trouble was experienced
with moisture which collected and plugged the throttle valve with ice. The whole
apparatus had to be thoroughly dried out and all the gas dried by passing it through
calcium chloride in the drying flask before the difficulty was got over. After this all
fresh charges of gas were passed through a small flask filled with calcium chloride
before they entered the apparatus, and the above described drying flask was kept in
circuit to eliminate any traces there might be left. Some of the oil carried round by
the CO;, collected in this flask.
The u-etghing apparatus (see fig. 1, p. 70) consists of two steel flasks, each capable of
holding -in 11 is. of liquid CO.,; both were originally hung on spring balances. Each
flask has valves at the top and bottom so that they may be alternately filled and
emptied. The connections to the flasks are made of coils of copper pipe, flexible
enough to allow of a small vertical motion. The spring balances were calibrated to
allow for the stiffness of these coils. This arrangement had certain defects and was
subsequently modified. In order to be sure that no CO2 passed unweighed, it was
necessary to stop the supply of C0a from one flask before starting it from the other ;
this inevitably caused a momentary variation in the rate of flow. There was also
some doubt as to the effect of the weight of C08 in the coils of pipe connected to the
flasks, which might be full or empty at the moment of weighing. The spring-
balances were divided in pounds, and tenths of a pound could be roughly estimated.
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES.
83
cent. No calibration
copper pipes is necessary.
The readings, however, had to be made in haste, and the probable error in the weight
of COa might amount to nearly 1 per cent. The balances had to be frequently
recalibrated, as the stiffness of the pipes was found to change gradually. Only three
of the 17 observations in Table VI., viz., those at — 26'1, —14*9, and — 8'6, were made
while working in this way.
To get over these defects the following modification was made : — One flask was
hung on a steelyard, the arm of which was allowed only a very minute movement.
When the arm fell it made an electric contact and rang a bell. This arrangement
was found to be so sensitive that it would turn with '01 Ib. When the arm fell it
was raised again by hanging a weight (usually 1 Ib.) on the flask. When another
pound of COa had passed out of the flask the bell rang again and another weight was
hung on, and so on. The increased sensitiveness of this arrangement made it possible
to record the rate of flow of CO, accurately at
short intervals and to complete the whole test
with one flask full or less of CO,, for as small
a quantity as 10 Ibs. could be weighed to -^ per
No calibration or allowance for the
A simple dash-pot
made of a disc of tin in a vessel of oil got over
all difficulties due to vibration without reducing
the sensitiveness.
The calorimeters are tin-plate tanks contain-
ing coils of copper pipe and electric heaters;
the tanks are lagged on the outside to prevent
the inflow of heat. The larger calorimeter,
fig. 14, contains two copper coils and the smaller
one a single coil. The coils can be connected in
different ways for the different series of tests.
Calcium chloride brine was used to fill the tanks
for the first experiments, but was replaced
later by methylated spirits, which answered
much better. Special care was taken in the
ilfsign of the calorimeters to provide a definite
path for the circulation of the liquid, which was maintained by a screw propeller
driven by an electric motor. The large calorimeter was originally lagged with
2 inches of slag wool ; this was found to be insufficient and 2 inches of felt were
added, covered by varnished calico to keep out the moisture. A wooden top was
fitted, covered by felt and calico. The small calorimeter, which was completed after
experience had been gained with the larger one, was wrapped in 2£ inches cotton
wool surrounded by about 2 inches slag wool, all contained in a wooden box. The
cover was formed of 3 inches of wood. Several sorts of heater were tried and failed ;
M 2
Fig. 14.
PROF. C. FRKWEN JENKIN AND MR D. R PYK ON THE
"" t
liM,lly coils of No. 16 S.W.G. Eureka wire, insulated with vulcanized indiarubber
Lu d directly on to the coils of the evaporating pipe, were tried; these
perfectly Each coil was about 50 yards long, had a resistance of about
To InT L Juld absorb 1000 watts, taking 10 amperes from the 100-volt power
m,ins There are three such coils in the larger and two in the smaller calorimeter.
Jfcomtmtflt of the Electrical lfa*-The electrical power entering the calon-
meters was calculated from the measured resistances of the heating coils and the
ol*erved E M F across their terminals. The E.M.F. was measured by means of a
Siemens millivoltmeter with fine pointer and mirror, which was calibrated against a
cadmium cell with N.P.L. certificate. The scale is divided in single volts and ^ vol
can be accurately estimated. Two of the coils in the large calorimeter were always
connected, when in use, on the full supply voltage (100), and the third was used in
series with an adjustable resistance. The voltmeter has a two-way switch so that
the two E.M.Fs. could be read successively. Read-
ings were taken every minute throughout the tests.
At the full voltage of 100 an error of 1 volt means
an error of '2 per cent, in the power. At the lowest
readings, 30 volts, an error of '1 volt means an error
of '6 per cent, in the power. The resistance of each
heating coil was measured before the tests were
begun by a bridge which was checked against a
standard ohm with N.P.L. certificate. The resist-
ances were measured again after the tests were
completed and had not altered appreciably. No
temperature correction was made as the coils were
made of Eureka, but corrections were made for the
resistance of the leads.
The temperature measurements were all made
with Eureka-copper thermo-couples. The couples
were made of No. 22 gauge double cotton and india-
rubber-covered Eureka and copper wire, all cut from the same coils, soldered together
at the ends. The couples used in the baths for measuring the temperature of the
circulating liquid were put into rubber tubes, the bare ends projecting about ^ inch.
The couples used for measuring the temperature of the CO2 were held in the special
fittings shown in fig. 15, so that the bare wires projected into the CO2 about
lj inches. The wires were carried through the gumnetal plugs in fine rubber tubing
(bicycle valve tubing). The holes in the gumnetal were tapered and small brass
beads were soldered on the wires, so that when the wires were drawn back the beads
jammed in the holes. This simple device made an insulated joint which was gas-tight
under the highest pressures used (1100 Ibs. per sq. inch).
Preliminary calibrations showed that the relation between the E.M.F. of the
Fig. 15.
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 85
thermo-junctions and temperature could not be satisfactorily represented by the
empirical formula often used, viz., log E = nlog t + m. A. calibration curve was
therefore drawn, points on it being obtained as follows. For temperatures between
+ 100° C. and 0" C. the junction was compared with a standard mercury thermometer
with N.P.L. certificate.
At —20* C. it was compared with a mercury thermometer which had Ix'en vi-ritii-il
at that temperature by the N.P.L. The melting-point of mercury was then ol»erved.
The mercury was purified by dropping it through nitric acid, washing it in wat«T, drying
it at 120° C., and finally distilling it in vacuo. About l£ Ibs. of the purified mercury
was put in a glass vessel and frozen by packing it with COa snow. The mercury
was then gradually melted and the temperature of the melting-point observed. There
was no difficulty in keeping the mercury half-melted and half-frozen for any length of
time desired. The melting-point was assumed to be — 38°'80 C.* A calibration curve
was drawn through these points and extrapolated to —50° C. As this extrapolation
was open to doubt a further point at —50° C. was afterwards obtained by comparison
with a platinum-resistance thermometer which had been carefully calibrated against
the standard thermometer. The new point obtained in this way fell within -,1,,0 C. of
the curve, thus confirming it satisfactorily. The curve is believed to be correct to
A'c.
To maintain the other junctions of the wires at a steady known temperature they
were immersed in a large tin of paraffin oil, well jacketed with slag wool, fitted with
a calibrated thermometer and lens so that the temperature of the " cold junction " (in
our case usually the warmer of the two) could be read to T,\0° C. The oil was stirred
by blowing air into it. An incandescent lamp was placed in the oil, so that the oil
could be warmed to approximately atmospheric temperature. The temperature of
the oil was read several times during each test and rarely varied more than t\y° C.
The E.M.F. produced by the thermo-couples was measured against the standard
cadmium cell by a potentiometer with twenty 1-ohm coils and a gilt manganin slide-
wire, 1 m. long, and of just over 1-ohm resistance. Special precautions were taken
to avoid thermo-electric effects. The various thermo-junctions could be switched on
in turn to the potentiometer by a two-pole six-way selector switch, designed to avoid
thermo-electric effects. The potentiometer was sensitive enough to measure tempe-
rature differences of 7^° C. Such accuracy was of use when measuring the small rise
or fall of the temperature of the bath during the run ; also in Series IV. experiments,
which depend on small differences of temperature, and in measuring the slow
temperature rise during radiation tests.
The pressure of the gas was measured by steel tube Bourdon gauges made by
Messrs. Schaeffer and Budenberg with specially fine needles and fine scale-divisions.
They were calibrated by means of a dead-weight testing machine, in which a dead-
* CHAPI-UIS, 1900, quoted KAYE and LABY, p. 48. Dr. J. A. BARKER, of the N.P.L., has informed
the authors that 38° -86 C. is probably a more accurate figure.
86 PROF. C. FREWEN JENKIN AND MR. D. R. PYE ON THE
weight rests on a plunger in an oil cylinder. There was some doubt as to the
effective area of the piston, since it was not exactly uniform in diameter, and there
was a small clearance between it and the cylinder. In order to clear up this point
the testing machine was checked against a mercury column about 1 m. high. The
value of the effective area of the plunger was found in this way ; it only differed by
Bio from the maximum measured area. The testing machine only gave pressures up
to 400 Ibs. per sq. inch. Up to this pressure the gauges showed a practically
constant error, and it was assumed that the error remained the same at the higher
pressures.
The gauge used for the pressure-temperature curve, and for experiments where
accurate high-pressure readings were needed, was subsequently calibrated for its
whole range by the N.P.L. ; the results agreed closely with those obtained by the
authors. In a few of the experiments, where accurate low pressures were needed, a
low-pressure gauge was used, which was calibrated over its whole range. The
pressures are believed to be correct to about 1 Ib. per sq. in.
Adjustments. — Before beginning any test the apparatus was run for a considerable
time while the conditions were adjusted to what was required ; the test was not
begun until a steady regime had been attained and all the parts had reached steady
temperatures ; moreover, unless a test was completed without anything more than
trifling changes of any of the conditions, the results were discarded. The conditions
were adjusted by regulating the speed of the pump, opening or closing the throttle
valve, and switching on more or less electrical power to the calorimeters. While the
adjustments were being made the flask B was emptied and the flask A filled, so that
before the actual test began the C02 was circulating through A, which was full. The
potentiometer and the temperature of the cold junction were also adjusted.
As soon as everything was ready, the valve on the top of flask A was closed and
that on B opened, and the regular readings of all the instruments was commenced.
In most tests these readings were made every minute. Each time the weighing bell
rang the time was entered to the nearest second — the first ring marking the time of
start of the test, which usually continued till the flask was almost empty.
In Series I. the apparatus was connected as shown in fig. 3, p. 71, the object being
to measure the heat represented by the area NBCQ (fig. 7, p. 74) for a series of tempe-
ratures 0,, ranging from the highest to the lowest attainable. To keep the rate of flow
of CO, within convenient limits the pump was run as slowly as possible for the higher
values of 0, and as fast as possible for the lower values of 02. The temperature 6a
was not directly measured, but was deduced by the pressure-temperature curve from
tin- pressure shown by the gauge connected to the pipe leaving the calorimeter ; the
tlm.ttle valve was adjusted so as to keep this pressure steady at the figure selected
The electrical power was adjusted so as to keep the temperature of
I the percentage of air present was not always approximately constant, this method of estimating
9t it liable to error.
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 87
the bath a few degrees above 6a, so as to make sure that the gas leaving the calori-
meter was slightly superheated. This was checked by the direct measurement of the
final gas temperature ft, by a thermo-junction in the pipe.
During the test the following observations were made : —
Times when the weighing l>ell rang ;
K.M.F. on each heating coil, observed every minute ;
Pressure of gas leaving coil, „ „ „ ;
Temperature fy of liquid OCX, before the throttle valve, every three minutes ;
„ of gas 63 leaving calorimeter, every three minutes ;
„ ,, bath, every three minutes ;
„ ,, cold junction, several times during test ;
„ „ atmosphere, „ „ „ „ .
During the test small adjustments of the throttle valve and of the electric power
were made so as to keep the pressure and bath temperature as constant as possible.
As an example, the complete records for one experiment are given in Tables XI.
and XII. The times when the weighing bell rang were plotted as a check on the
uniformity of the rate of flow of CO., during the experiment. The initial temperatures
of the liquid COa were plotted to obtain the true mean.
In Series II. the apparatus was connected as shown in tig. 4, p. 72, the object being to
measure the total heat I of the liquid, i.e., the heat represented by the area SKAM,
fig. 7, p. 74, for a series of ranges of temperature. The speed of the pump was settled
as in Series I. The electrical power entering the large calorimeter was adjusted so as
to keep its temperature a few degrees below 6t, so that the liquid C02 might be cooled
to the desired temperature 64. The electrical power put into the small calorimeter
was adjusted so as to keep it at the selected temperature 0,. The throttle valve was
adjusted so as to keep the evaporation temperature a little below the temperature of
the large calorimeter, but this temperature was of no importance in this series.
During the test the following observations were made : —
Times when the weighing bell rang ;
E.M.F. on each heating coil in calorimeter II., observed every minute ;
Temperature 6t of liquid COa entering calorimeter II., every three minutes ;
>, Qi » „ leaving „ „ „ „ „ ;
„ of bath, calorimeter II., observed every three minutes ;
„ ,, cold junction, several times during test ;
„ „ atmosphere, „ „
No observations were entered for the large calorimeter, which was only used as a
cooler for the liquid, but the power was adjusted as required to keep its temperature
constant.
As an example, the complete records for one experiment are given in Tables XIII.
and XIV.
88 PROF. C. FKKWKN .IKNKIN AND MR. D. H. PYE ON THE
In Series III. the apparatus was connected as shown in fig. 5, p. 73, the object being
to measure the specific heat of the gas at various temperatures near the saturation
points. Tin- .•iil.jiistiiHMits of the pump, throttle valve, and electrical power for the
large calorimeter were made exactly as in Series I. The electrical power put into
the small calorimeter was adjusted so as to keep it at a steady temperature, a
moderate amount above 6* It was not possible to start heating the gas exactly
at 0,, but 0, was kept as close to 0., as possible, so that the range through which the
gas was heated began only a few degrees above the saturation temperature. The
actual ranges are shown in lines 4 and 5, Table IV. The last line gives the mean
specific heat for this range of temperature.
During the test the following observations were made :—
Times when weighing bell rang ;
E.M.F. on each heating coil in calorimeter II., observed every minute ;
Temperature 93 of gas entering calorimeter II., every three minutes ;
,, ,, » leaving ,, ,, ,, ,, „ ,
„ „ bath, calorimeter II., every three minutes ;
„ „ cold junction, several times during test ;
„ „ atmosphere, ,, „ ,, ,, .
As an example, the complete records for one experiment are given in Tables XV.
and XVI. The times when the weighing bell rang were plotted as a check on the
uniformity of the rate of flow of CO., during the experiment.
Corrections.
Before making use of the data obtained in the tests, it is necessary to consider the
effects of differences between the actual and theoretical cycles and also the corrections
for radiation, conduction, and change of temperature of the calorimeter during
the test.
Differences betiveen the Actual and the Ideal Cycle.
(i.) Friction in the evaporation coil produces a small difference of pressure between
the two ends. The evaporation, therefore, should not be represented by the constant-
pressure line BE (fig. 7), but by a curved line starting a little above B and falling
to E. It is easy to show that this has no effect on the heat absorbed, which is
always IB-IA, and is accurately represented by the rectangle NBEP.
(ii.) The vapour is moving in the pipe with some velocity and consequently
possesses kinetic energy. A simple calculation shows that the kinetic energy is
always small enough to be neglected.
(lii.) The compression in the pump is not adiabatic and there are other deviations
in the condenser, but as this part of the cycle is not included in the measurements
they have no effect.
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 89
(iv.) Change of Temperature of the. Hath during a Test. — The temperatures of the
bath at the beginning and end of a test were accurately determined by taking several
observations of temperature at short intervals before and after the actual moment of
the start or finish and plotting them. The temperature at the actual moment
of start or finish was then road from the graph. Any change of temperature of the
bath showed that the electric heat supplied had been slightly too much or too little.
The balance of heat (excess or shortage) is simply the temperature rise or fall
multiplied by the water equivalent of the calorimeter. The water equivalent was
determined by a simple heating experiment when the bath was at approximately the
atmospheric temperature so that radiation could be neglected. The results obtained
in this way are not quite accurate since the heat capacity of the lagging varies with
the rate of heating.
Radiation.
A number of experiments were made to determine the rate at which heat
entered the calorimeters from the surrounding air before concordant results were
obtained. In the end good results were arrived at and the radiation was found to be
proportional to the difference of temperature between the calorimeter and the
atmosphere. The rate for the large calorimeter was '83° C. per hour with a
temperature difference of 40° C. The water equivalent being 97 '5 this radiation
corresponds to *83x97'5 = 81 thermal units per hour for 40° C. difference of tempe-
rature. The small calorimeter rate was 1°'92 C. per hour or I'92xl5 = 28'8 thermal
units per hour for 40° C. difference of temperature. These figures include the
mechanical work put in by the stirrer motors, which in the small calorimeter was
measured and found to be about 2 '2 thermal units per hour.
Conduction along the Pipes.
The calorimeter coils, and most of the connecting pipes, were made of copper,
£ inch external diameter, J inch internal diameter. Such a pipe would conduct about
11 thermal units per hour with a temperature gradient of 40° C. per foot. To
minimise conduction, the small calorimeter had a piece of thin-walled steel pipe"
inserted in both the ingoing and outgoing connections which would conduct heat at
about one-sixth of the above rate. All the pipes were well lagged. When the
apparatus is working it can easily be shown that there can be only a very small
temperature gradient beyond the thermo-junctions so that no appreciable conduction
can take place. Temperature measurements along the pipes confirmed this. When
the vapour is not circulating the conditions are not quite the same and a small
amount of conduction takes place. This conduction makes the apparent radiation
rather too large, so that the corrections applied for radiation are a little too large.
An approximate estimate shows that the error introduced is not greater than
'3 thermal units in the value of L at the lowest temperature and that it will not be
appreciable above —30° C.
VOL. ccxni. — A. N
90 PROF. C. FREWKN JENKIN AND MR. D. R. PYE ON THE
I'M; i III -DETAILED DESCRIPTION OF THE APPARATUS AND METHOD OF
CARRYING OUT THE EXPERIMENTS SERIES IV. AND V.
In Series IV. the apparatus was arranged as shown in fig. 8, p. 75, the object
being to measure the Joule-Thomson effect, i.e., the change of temperature 0,-04
(fig. 7), corresponding to a change of pressure from p, to the pressure on the limit
curve p,. Iii the actual tests the pressure could not be allowed to fall quite down to
the limit curve for fear of introducing errors due to the commencement of evaporation.
To get the total 03-64, the observed difference of temperature was increased in the
proportion of the observed drop to the total drop of pressure. In fig. 8 the pressure
drop is measured by the gauges a and b and the temperature change by the thermo-
juuctions A and B. The liquid C0a from the weighing flask first passes through the
inner coil of the calorimeter and is there cooled to any desired temperature. It then
flows in order past : —
Gauge a and thermo-junction A.
Throttle valve Vj.
Thermo-junction B and gauge b.
Throttle valve Vr
Outer coil in the calorimeter and gauge c.
The C0a is liquid up to the second valve, Vj.
The pressure of the liquid up to the valve Vt is pv The valve Vt is adjusted so as
to allow it to drop to a pressure pt a few pounds above p2.
The valve V, is adjusted so as to allow it to drop from px to p2, which is the
saturation pressure corresponding to the temperature of the liquid. The difference
of temperature 0A— 9B between A and B is the Joule-Thomson effect corresponding to
the drop of pressure Pi—p^ Therefore
In the first series of experiments the valve Vt was a large bronze hydraulic valve,
and the thenno-junctions were inserted in the gunmetal fittings shown in fig. 15,
which were connected to the valve by pipes about 6 inches long. The whole
apparatus was well lagged, but the amount of heat which leaked in when the
temperature was low was sufficient to raise the temperature of the C02 to an extent
which was large compared with the small temperature change which had to be
determined ; this leakage was allowed for by observing the temperature change due
to the leakage only, when the valve V, was full open, and subtracting this from the
temperature change when V, was throttled, the rate of flow being kept the same in
the two experiments. Conduction along the pipe between A and B has no effect.
Each observation was repeated a number of times and the means taken. It was not
TIIHIiMAL I'ROl'KIJTIKs (i| CAIHIONIC ACID AT LOW TEMPERATURES. 91
found possible to make observations at temperatures below —30° C., and even at this
temperature the results were open to criticism as being small differences between
large measurements, see Table V. (first part).
To get reliable results it was therefore necessary to design a special apparatus,
combining two thenin ^junctions and a throttle, which should not allow of appreciable
radiation. Experience with the original apparatus showed that the throttle had to
be adjustable to enable the required pressure ranges to 1*3 obtained. After one or
two experiments the apparatus shown in fig. 16 was made and answered perfectly.
H. Steel tube.
K. Ebonite tube.
L.
M. Copper jauze plug.
N. Inlet.
Outlet.
Iron yoke.
Vulcanite block for
control expc.s.
Ebonite valve body.
Packing.
Brass gland.
vaJve spindle.
" valve.
Ebonite distance piece
Brass cap.
Fig. 16.
The body of the valve is vulcanite, the gland and screw are brass, but the screw is
insulated from the passage through the valve by an extension rod of vulcanite. A
minute brass valve is inserted under the extension rod, and is the only metal
encountered by the CO., in passing through the valve. The two thermo-junctions are
held in vulcanite plugs inserted in the steel tubes on either side of the valve. They
project inside the inner vulcanite tubes, shown in the figure. The outer space forms
a jacket of CO3 at approximately the same temperature as the inner space — which is
entirely protected by the vulcanite tubes from any external influence. Vulcanite is
one of the best thermal insulators — the specific conductivity is given by KAYE and
LABY as '00042. The whole apparatus is held together by an iron yoke, and can be
taken to pieces in a moment by slackening one of the set screws at the end. The
N 2
,.|;<)F. f. FKKWEX JENKIN AND MB. D. R. PYE ON THE
,;„„. j,,i,,te between the steel tubes and the vulcanite pieces were perfectly tight
UII1|,.r tl..< maximum pressure used, 900 Ibs. per sq. in. The bore of the valve ,s
,', in,-!, diameter and the passages are t)ell-mouthed oa both sides. A plug formed
of rolled copper gauze serves to dissipate the kinetic energy of the jet issuing from
the valve. The time occupied in the passage from one thermo-junction to the other,
at the slowest rate of flow, was only a fraction of a second.
The thermo-junctions were connected to the selector switch (p. 85) in such a way
that the temperature of either junction, or the difference between the two, could be
measured. The latter arrangement was convenient, since it was the difference of
temperature which was being investigated.
Before using the new apparatus tests were made to ascertain whether the heat
insulation was in fact perfect, and also what difference, if any, there was between the
readings of temperature given by the two thermo-junctions. The construction of the
apparatus makes these tests very simple.
To avoid the slightest drop of pressure between the thermo-junctions, a simple full-
bore block of vulcanite (also shown in fig. 16) was substituted for the adjustable valve,
then liquid COj was passed through the apparatus, first with the thermo-junctions in
their normal positions, and again when the thermo-junctions had been interchanged,
end for end. The apparent difference of temperature between the thermo-junctions
in the first case represents the sum, and in the second case, the difference, of the two
quantities to be determined, viz., the actual change of temperature of the liquid and
the difference of the thermo-junctions.
Experiments were made with liquid C0a at temperatures ranging from + 20° C. to
— 50° C. They showed that there was no change in the temperature of the liquid,
but that there was a very slight difference between the thermo-junctions. This was
allowed for in subsequent measurements.* As a further check, similar tests were
made with wet CO, vapour instead of liquid C02. The difference between the
readings of the thermo-junctious was confirmed. There could be no difference of
temperature of the vapour in this case, since the pressure was the same at the two
junctions.
As the calorimeter could not be cooled much below —40° C., and readings were
wanted at —50° C., an " infra-cooler" was inserted in the pipe between the calori-
meter and the throttle valve, by which the liquid could be cooled to any extent
desired. By this means readings were obtained down to -55° C. The " infra-cooler "
consisted of two concentric copper pipes 10 feet long, the inner one £ inch external
diimu-ter, the outer one f inch internal diameter. The liquid C02 passed through the
inner tube, while a separate supply of liquid C02 was admitted to the outer tube and
evaporated at a pressure of about 80 Ibs. per sq. in., escaping through a throttle valve
ato the atmosphere. The concentric pipes were bent into a coil about 9 inches
diameter and well lagged with cotton wool The arrangement worked well, though
* See note on Table V.
THKliMAL PROPERTIES OF CARBONIC ACID AT LOW TKMPKKATURES.
it was not possible to keep the temperature of the liquid CO, aljsolutely constant
dm -ing an observation. Slight variations of temperature account for the irregularities
of the observations below —30° C. ; above that temperature the " infra-cooler" was
not used. The results of the tests with the original and with the new apparatus are
summarised in Table V., and both are plotted in fig. 9, p. 76. The two sets of results
are in good agreement, considering tin- smnllness of the quantities being measured.
Scries V. — The apparatus used for measuring the elasticity and dilatation of
liijiiid C()a is shown in figs. 10 and 17 : —
a is a capillary glass tube, the lower end of which is closed and the upper end
thickened and blown into a thistle funnel. A centimetre scale was etched along its
whole length, and it was carefully calibrated by measuring the
variation in the length of a thread of mercury in different
positions.
It is a gunmetal socket, shown in detail in fig. 17, turned to
hold the glass tube, the joint being made by a thin rubber
sleeve. The top of the socket is closed by a screw plug.
c is a fine copper pipe, J-inch bore, 12 feet long, connecting
the socket to the regulating valves, pressure gauge, and, through
a drying flask, to the CO., flask. This pipe is sufficiently
flexible to allow the glass tube to be moved about as required
while under pressure.
The glass tul>e was charged as follows : — The tube was laid
in a nearly horizontal position and a small quantity .of mercury
poured into the thistle funnel, where it lay without obstructing
the entrance to the tube. The plug was then inserted in the
gunmetal head. The air was removed by means of an air pump,
successive charges of CO3 gas being admitted and exhausted.
The lower half of the glass tube was then surrounded with ice,
and COj, gas was admitted up to a pressure slightly above the
saturation pressure at 0° C. The gas then condensed in the
glass tube, and the meniscus could be seen travelling up the tube. The rate of
condensation could be regulated with ease by modifying the pressure. When
sufficient liquid was condensed, the tube was raised into a vertical position so that
the mercury in the funnel flowed into it, on to the top of the liquid CO^. The tube,
kept vertical, was then lowered into the calorimeter, the temperature of which had
been adjusted to a few degrees below 0° C. The amount of liquid used and the
length of the mercury column were chosen so that the whole of the liquid CO., was
below the level of the bath in the calorimeter, and therefore at a uniform tempera-
ture, and the top of the mercury showed above the lid of the calorimeter, so that
its position could be observed with a cathetometer.
17
94 |.,J,,F. f. FKKtt'EN JENKIN AND MR. D. R. PYE ON THE
\Vl.rn making an .-xprriment, the bath was first cooled to the desired temperature.
Thru rra.lings were taken with the cathetometer as the pressure was varied step by
>t, •). IP.III tlir iiiaxinniin available to a pressure just above the saturation pressure
corresponding to the temperature of the bath. These readings give a constaut-
temperature curve on the p.v. diagram. A series of experiments was made with
different temperatures, so that a number of constant-temperature curves were
obtained, as shown in fig. 11, p. 78.
The apparatus appeared to work satisfactorily, but the results obtained are not in
perfect agreement amongst themselves ; the cause of this has not yet been ascertained.
The curves shown in fig. 1 1 have been arbitrarily constructed to represent the mean
results of several sets of observations and to agree amongst themselves ; they must be
taken as only approximate, but they are sufficiently accurate to determine the small
correction 3, Table VI., for which they are required. AMAGAT'S (5) curves have been
added to fig. 1 1 for comparison.
By extrapolation the curves may be extended to the left to the saturation pressure,
which is only just below the lowest pressure observed. This has been done in fig. 11.
By joining up the ends of all the curves, the new curve (named "limit curve" in the
figure) is obtained which shows the change of volume along the limit curve. A curve
giving BEHN's(6) observations is also drawn for comparison. The agreement is fairly
good.
It is not necessary to describe the lengthy and rather complex calculations required
to reduce the data obtained in these experiments. Allowance was made for the
changes of volume of the glass and of the mercury indicator at different temperatures,
and corrections applied to allow for the variation in the bore of the tube. All these
corrections are small compared with the changes of volume of the C02 due to
temperature. Since the actual weight of the column of liquid CO3 was not measured,
the density at some one temperature had to be assumed ; BEHN'S value, viz., 0'925
at 0° C., was used.
The elasticity and dilatation may be derived from the curves directly. The
elasticity (dv/dp)t is the slope of the constant temperature lines. The dilatation
(dv/dt), is the distance apart of the constant temperature lines divided by the
difference of temperature.
These quantities have not been worked out because of the known inaccuracy of the
observations. The authors intend to repeat the experiments and hope to obtain
accurate results.
PART IV. — DISCUSSION OF RESULTS.
Pressure-Temperature Cuwe. (Fig. 18. Tables I. and X.)
Tl,« n-lation between the pressure and temperature of saturated CO2 vapour has
by RKUNAULT (7), CAILLSTET (8), AMAGAT (5), KUENEN and KOBSON (9),
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES.
95
ZKLENY and SMITH (10), and others. The observations are tabulated in parallel
columns in Table X. and all plotted in fig. 18 besides the authors' curve.
The most accurate determinations are probably KUENEN and ROBSON'S below zero
and AMAGAT'S above zero. Our curve agrees closely with KUENEN and ROBSON'S,
900
- :,,
700
600 -too
500
400
306
200
10(1 Id
>
1C
LATENT HEAT
-T CURVE USED BY MOLLIER
. THE AU1
PRESSURE- TEMPERATURE
HORS' P-T CUXVE
KUENEN AND ROBSON •
ZELENY AND SMITH ---o
CAILLETCT - X
-REGNAULT- •- --+•
AMAGAT A
CAILLETET AND MATH/AS - •»
MATH/AS -J
MOLLIER --e
CHAPPUIS
-60
-50
-40
-30
-20 -10
Fig. 18.
\
DEGREES CENTIGRADE
10
20
30
only differing by 3 Ibs. at 510 Ibs. pressure. It also agrees fairly closely with
AMAGAT'S, differing by G Ibs. at 830 Ibs. Both these differences are almost exactly
equal to the calculated effects of the presence of the observed '11 per cent, of air (by
volume) under the special conditions of our test.
96
PROF. C. FRKWEX JENKIN AND MR. D. R. PYE ON THE
an,l SMITH'S points lie Hose to our OW?e, bat i <-iir\r tlm.ii^li tli.'ir points
would be definitely flatter than ours. There is a difference of 10 Ibs. at their highest
pivssmv, 410 Ibs.
CAILLETET'S figures only go up to -34° C. They do not agree very closely in
position or slope.
KEONAULT'S figures lie well above the others. He states that his experiments did
not satisfy him.
MOLLIER adopted a composite curve, using AMAGAT'S figures above zero and
REGNAULT'S below zero, modified so as to make them fit together. His curve is
shown dotted in fig. 18.
An accurate determination of the pressure-temperature curve is important because
its gradient, dp/dS, is one of the factors in CLAPEYRON'S equation, which may be used
to calculate the latent heat or the vapour density.
The gradient of MOLLIER'S curve is clearly too small, particularly at low tempe-
ratures. It is remarkable how large a difference in the gradient results from a very
small divergence between the curves. The values of dp/dd, used by CAILLETET and
MATHIAS(H), MOLLIER, and KUENEN and KOBSON for calculating L, and by the
authors who only used it for calculating the specific volume of the gas, are given
below : —
VALUES of dp/d9.
T.
C. anclM. (11).
MOLLIER (1).
-K. and R. (9).
Authors.
°c.
+ 20
lbi./in.5 ° C.
20-4
Ibs./in.2 ° C.
19-4
Ibs./in.2 ° C.
lbs./in.2 ° C.
19-6
+ 10
17-18
16-6
_,
16-3
0
14-12
13-18
13-46
13-6
-10
11-32
10-69
11-01
11-1
-20
8-84
8-55
8-73
8-9
-30
—
6-79
6-87
7-0
-40
—
—
5-37
5-3
-50
"
4-06
4-0
The Total Heat I and Specific Heat Cp of the Liquid.
Our observations were plotted and values read from the smooth curve are given in
No experimental determinations of these quantities appear to have been
Flgures for comparison might be deduced from MOLLIER'S (3) U diagram
this is beyond the range of this paper.
The probable errors in the values of I do not exceed about £ per cent., from +20° C
a, but rise to 1 per cent, at -50° C. The presence of '073 per cent, of air
has no appreciable effect on the results.
Experiments have been made by DIETERICI (15) and by MARGULES (16) on the
TIII.KMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 97
difli'ivnces between the total heats at O8 C. and at a few other temperatures. These
quantities refer to heating along the limit curve and not to heating at constant
pressure.
Specific Heat of the Gas. (Fig. 6.)
No experimental determinations of this quantity appear to have been made at
temperatures below 0° C., though much work has been done at high temperatures.
The specific heat is believed to vary considerably near the limit curve. Our
measurements give a mean value for a moderate range of temperature, starting in
each case a few degrees above the limit curve ; they are probably not correct for
higher temperatures than those at which they were measured. They are only used
for the small correction given in column y, Table VI., and for plotting the pressure
lines in the superheated area.
Latent Heat of Liquid CO.,.
Our ol>servations were plotted and values read off the smooth curve are given in
Table VI. The figures are probably correct to about £ per cent, from +20° C. to
— 30° C. and to 1 per cent, at —50° C. One point, at —8° '6 C. lies 1 per cent, off
the curve. This was the result of the first experiment made, when the spring
I >a lances were still used for weighing the CO.). The trace of air present only
produces a proportionate error in the value of L, i.e., '073 per cent., which is
negligible.
No experimental determinations of L below zero have been published. REGNAULT*
made a single determination at +17° C. ; CHAPPUIS (13) a single determination at
0° C. and MATHlAs(l2) made a series of measurements between +6° C. and +31° C.
CAILLETET and MATHIAS (ll) calculated L from their own determination of the
liquid and vapour densities and REGNAULT'S pressure-temperature curve. KUENEN
and ROUSON (9) calculated L from AMAGAT'S (5) vapour density (extrapolated) and
BERN'S (o) liquid densities and their own pressure-temperature curve. MOLLJER(I)
calculated L from AMAGAT'S (5) vapour and liquid densities (extrapolated) and the
compound pressure-temperature curve mentioned above, based on AMAGAT and
REGNAULT'S results.
All these results are plotted in fig. 18 beside our curve. If the drawing is examined
it will be seen that the mean of all the previous determinations lies above our curve.
We have investigated the causes of this divergence in detail as it appeared to cast
some doubt on the accuracy of our results. The investigation has shown, first why
the previous results tend to agree amongst themselves, secondly why they differ from
ours, and, finally, has resulted in an indirect confirmation of our results.
* Recalculated by MATHIAS (12).
VOL. CCXIII. — A. O
yg ,.,;,„.-. , H;I;\\I:N .IKNKIN AND MR D. R. PYE ON TJII-:
Tli,- HI- <>(' tin- n.u^l, a-rniement among the observations is that the sain.- data
|*en n'1-.-.t.-.lly "s.-.l. Thus CAILLETET and MATHIAS' densities were used by—
MATHIAS | .^ work- ollt tjiejr observations,
CHAPPUIS J
CAILLKTKT . , ,.
>• in their calculations.
and MATHIABJ
AMAUAT'R densities were used by —
KUENEN and ROBSON j ^ ^ calculations.
MOLLIEB J
REONAULT'S pressure-temperature curve was used by—
CAILLETET and MATHIAS "1 . . , , .
> m their calculations.
MOLLIER (in part) J
The difference between the various results and ours may be traced to the inaccuracy
of REGNAULT'S (and consequently MOLLIER'S) pressure-temperature curve and to the
error in CAILLETET and MATHIAS' gas densities below —20° C, and to the uncertainty
of the extrapolated values of the densities used by MOLLIEB and KUENEN and
ROBSON.
All the calculated values of L were obtained by means of CLAPEYRON'S equation
where
V = specific volume of saturated vapour,
v = „ „ liquid at saturation temperature.
Their accuracy therefore depends on the accuracy of the two factors dp/dd and
(v-v).
The differences in the pressure-temperature curves and their gradients have already
been discussed, so that it is only necessary to consider now the other factor (V—v).
The specific volume of the liquid v is much smaller than of the vapour, and the values
obtained by different observers do not differ much ; it is, therefore, not necessary to
discuss it here. The specific volume of the saturated vapour, V, is much more
doubtful. In fig. 19 are plotted —
(1) AMAUAT's(5) smooth curve from his observations of V.
(2) CAILLETET and MATHIAS' (ll) smooth curve from their observations of V.
(3) KUENEN and RoBSON's(9) extrapolation (by the Law of Corresponding states)
of AMAOAT'S curve.
(4) MOLLIER'S (1) extrapolation of AMAGAT'S curve.
(5) Our values, calculated from our observed values of L by CLAPEYRON'S equation,
using BERN'S (6) liquid densities.
Values taken from our smooth curve are given in Table IX.
TIIKKMAI. I'linl'KKTIKS o| CAIM'.ONIC ACID AT LOW TKM I'M; V I I 1:1 S. <i;i
\ »
\ \
\ \
\ I
\ \
\ V
\ 1
\ \
50
45
40
35
30
25
20
15
10
3
\\
\ \
\ \
\ \
\\
\\
\\
\\
SPECIFIC
VOLUME OF
SATURATED
1
VAPOUR
DE
VS/TY
\
\
^^
M
\
V
Y CALCULAl
\C AUTHORS,'
\
c or/ton THI
OBSCKVATIOHS
i
\
LI
1
EXTRAPOLATION
BY KUCNCHAMD
10BSON
\
EXTRAPOLATH
NBYHOLLICR
\^
-CAILLETCT A*o
L
WMUSJI^m
OBSIWATIO
MS ^^\.
^
f LIMIT Of AHA.
'•AT'S OBSCRVAT,
o/vs.
^
AH,
^^ oasta
^<J
IGAT-S
VAT IONS
TCHP- "C
^
CAM
^^
0 -40 -3O -20 -10 0 10 20 30
Fig- 19-
0 2
100 PROF. C. H;i:\vi:X JKNKIN AND MB. P. R. PYE ON THE
Tin- ..Iflervations on which CAILLETKT and MATHIAS' smooth curve is founded lie
erratically and at considerable distances from the curve. The curve is clearly
rrn •Hi-nils below —20° C.
It will be noted that our values agree very closely with CAILLETET and MATHIAS'
from -20° C. to + 10" C. Our curve also lies almost exactly parallel with AMAGAT'S
curve and KUENEN and ROBSON'S theoretical extension of it. The agreement between
our curve, the observations, and the theoretical curve is a confirmation of the accuracy
of the shape of our L curve.
The inaccuracy of our pressure-temperature curve, due to the presence of '11 per
cent, of air, referred to on p. 69, introduces an error into our values ofd p/dO which
probably does not exceed 2 per cent. If this were allowed for, it would raise our
specific- volume curve by 2 per cent., and bring it closer to KUENEN and ROBSON'S.
Tin- value of V does not enter into the construction of our 00 diagram.
Joule-Thomson Effect. (Fig. 9.)
No experiments on the Joule-Thomson effect for liquid C02 appear to have been
published. Figures for comparison might be deduced from MOLLIER'S I<f> diagram (3),
but that would be beyond the range of this paper. It is not easy to say what effect
the presence of the trace of air may have on these results.
Dilatation and Elasticity of liquid C02.
As has been explained, the results of the Series V. experiments were not sufficiently
concordant to warrant the publication of values of the dilatation and elasticity
derived from them, though they are accurate enough to determine the values of $<f>
between the constant-pressure curves and the limit curve on the 6<f> diagram. The
confirmation of BEHN'S densities, shown in fig. 11, must not have much weight
attached to it for the same reason.
The only results previously published are a single curve at 13°'l C., given by
ANDREWS (14), and three curves given by AMAGAT(5) at 0° C. +10° C., and +20° C.
We have failed to fit ANDREWS' curve on our figure, but AMAGAT'S curves are shown
for comparison in fig. 11. ANDREWS suggests that the curvature of the lines near
the saturation pressure may be due to the presence of air.
0$ Diagram. (Fig. 12.)
As stated at the beginning of this paper, the primary object, for which all the
quantities already discussed were measured, was the construction of a 00 diagram for
comparison with MOLLIER'S.
It may be useful to recapitulate here the steps in the construction of the two
diagrams. Fig. 12 was drawn as follows :—
TIIKKMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 101
The starting points of the 700-lh. and 900-lb. constant-pressure curves (0 = 273° C.,
^ = —'0024 and —'0049), found on p. 80, were first marked, and the two constant-
pressure curves were then drawn in segments of 10 degrees each.
The liquid-limit curve was then set off from these pressure curves at the distances
fy given in Table VIII.
The gas-limit curve was then set off from the liquid-limit curve at the distances
fy = L/tf (Table VI).
A constant I line was then drawn through the origin, at a slope determined by the
Joule-Thomson effect. Thus the value of I at the origin (calculated on p. 80)
was transferred to the two constant-pressure curves. The points 1 = 0, +5, +10,
+ 15, and — 5, —10, —15, —20, —25 were then marked on these curves and trans-
ferred back to the limit curve by drawing I lines as before. Thus the starting points
of the I lines on the limit curve were determined. Measuring from these, a number
of points were then marked off at distances $<f> = 5/6, 10/6, 15/6, &c., and I lines
drawn through these points. The space between the limit curves was then divided
into quarters, thus determining a few dryness lines. The constant-pressure curves in
the superheated area were then drawn, starting at the corresponding saturation
temperatures at slopes
=
where the values of Ql— 6a were the actual temperature ranges in the experiments,
Series III., and a- was the corresponding specific heat of the gas at constant pressure,
given in fig. 6.
The maximum probable error in the liquid limit curve is <ty> = '0008 at —30° C.
and '0032 at —50° 0. At higher temperatures the error is probably not more than
'0005.
MOLLIER'S ti<f> diagram was constructed in a very different manner. He assumed
that the characteristic equation of the gas might l>e expressed in VAN DER WAAL'S
form
K0
V-a. (
and determined the constants by means of AMAOAT'S observations. He also assumed
li-—}
that f(6) was of the form f(6) = e* '•', and by means of these equations obtained a
general expression for the entropy of the gas, which would hold down to the limit
curve. With this he plotted the gas limit curve on the 6<f> diagram.
He then found an empirical mathematical formula for the slope of the pressure-
temperature curve, dp/d6, which, on integration, corresponded well with the pressure-
temperature curve constructed from AMAGAT'S and REGNAULT'S observations. With
this and AMAGAT'S values for the vapour and liquid densities (extrapolated) he
calculated the values of L, and set off the liquid limit curve to the left of the gas limit
curve at distances $<f> = L/0.
102 PROF. c. n.i:\\i N U:XKI.\ AND MR. P. R. PYE ON THE
To construct the constant-pressure lines in the superheated area he assumed (2)
that the specific heat at constant volume, Cw was constant, which he points out is
not ,,.iit,- tin,-. ( )n this assumption the constant- volume lines on the 6<f> diagram are
the same as those for a perfect gas (logarithmic curves). The constant-pressure
curves were then determined from these constant-volume curves by the p.v. 6 curves
observed by AMAUAT. MOLLIER does not draw I lines on his Q$ diagram. His
values of I are given in his 10 diagram (3), which is outside the range of the present
discussion.
The agreement between the two diagrams, constructed by such widely different
methods, is remarkable, the more so because MOLLIEH had no data to go on below
0" C., except REONAULT'S very imperfect pressure-temperature curve. The following
brief comparison between the two methods shows how widely they differ.
Our diagram is based in the most direct manner possible on experiments at the
temperatures and pressures represented, whereas MOLLIER'S is based on a mathe-
matical equation obtained from experiments at higher temperatures. Our diagram
is constructed from the left, his from the right hand. Our diagram is based on
measurements of heat, his on measurements of density and pressure.
The two diagrams are superposed in fig. 13, and values of 0 and L are given in
Table XVII. for comparison. In Table XVIII. our data are arranged for direct
comparison with the tables given by MOLLIER (l) and EWING (4).
The differences between the diagrams are due to the differences in the various data,
which have been already discussed. The authors take this opportunity of expressing
their great admiration for the judgment and skill by which Dr. MOLLIER has selected
the most reliable data and devised mathematical methods capable of giving results
which direct experiments have confirmed so closely.
SUMMARY.
The authors have reconstructed the d<j> diagram by a new method and, at the same
time, extended it from -30° C. to -50° C. The reconstruction is based on direct
heat measurements, and the results are believed to be more accurate than those
arrived at indirectly by MOLLIER.
The observations include the direct measurement of the following quantities : —
The latent heat ;
The total heat of the liquid ;
The specific heat of the gas ;
The dilatation and elasticity of the liquid ;
The Joule-Thomson effect for the liquid.
From these direct measurements the following quantities have been calculated : —
The specific volume of the saturated vapour ;
The specific heat of the liquid at constant pressure.
TMKKMAI. l'l;o|'Kl;TIKS OF CARBONIC ACID AT LOW TEMPERATURES. in:;
Most, "t'tlifsc iiii'asiin'iiiriit^ an- new ; the details of what has IM-CII dime Ix-fure
(in p. 68 and in Part IV., p. '.»!.
In addition to the ordinary data given in #•/> diagrams and tables, the authors have
given the values of I, so that a complete I<f> diagram might be constructed from the
data supplied in the paper. The authors have made this diagram, but before
publishing it they intend to make a series of throttling experiments on superheated
gas to check the constant-pressure curves, which have so far only Ixjen approximately
determined.
P.S. — Since this paper was completed the authors have commenced the gas-
throttling experiments referred to above. These may be used as an independent
check on the accuracy of the Q<f> diagram, ('housing an experiment as close to the
gas-limit curve as possible — to avoid possible errors in the approximate constant-
pressure lines — the confirmation obtained is remarkably good.
Starting at the point " V," tig. 12, on the 700-lbs. pressure line, the gas was
throttled down to the 150-11)8. pressure line. The point reached is marked " W."
Calculated from the 6<j> diagram, the point W should have fallen exactly on the limit
curve;, •/>., at " E." If the limit curve be moved '0001 to the left, W and E will
coincide. A shift of '0001 corresponds to an error of '3 per cent, in the value of L.
This is the accumulated error in the whole set of measurements for the complete
cycle KAVWEGK shown in fig. 20.
Fig. 20.
The authors desire to express their special gratitude to Brasenose College for the
gift of the freezing machine with which the experiments were made. They also have
to thank Mr. D. H. NAGEL, of Trinity College, and Mr. H. B. HARTLEY, of Balliol
College, for advice on chemical questions and for having generously placed the College
Libraries at their disposal.
104
. ( Fi;i:\Vi:\ JENKIN AND MR. D. R. PYE ON
PAPKRS, &c., REFERRED TO.
K- Terence.
(1) MOI.LIKR, 'Zeit. fur die ges. Kalte-Industrie,' 1895, Nos. 4 and 5, pp. 66
and 85.
(2) MOLLIER, 'Zeit. fur die ges. Kalte-Industrie,' 1896, No. 4, p. 65.
(3) MOLLIER, 'Zeit. des Vereines Deutsch. Ingenieure,' 1904.
(4) EWINO, 'Mech. Production of Cold,' 1908.
(5) AMAGAT, ' Annales de Chimie et de Physique,' 6th Ser., 1893, vol. 29, p. 68.
(6) BEHN, 'Annalen der Physik,' 1900, Ser. IV., vol. 3, p. 733. Quoted in
ABEOO and AURBACH, p. 154.
(7) REONAULT, 'M^m. de 1'Acad.,' vol. 26, p. 335 (1862).
(8) CAILLETET, 'Arch, de Geneve,' 1878 (?), quoted by ABEGG and AURBACH.
(9) KUENEN and ROBSON, 'Phil. Mag.,' Ser. 6, vol. III., p. 149 (1902) and p. 622.
(10) ZELENY and SMITH, 'Phys. Zeit.,' vol. VII., p. 667 (1906).
(11) CAILLETET and MATHIAS, ' Journ. de Phys.,' Ser. II., vol. 5, 1886, p. 549.
(12) MATHIAS, 'Theses & la Facultd des Sciences de Paris,' No. 687 (1890).
(13) CHAPPUIS, ' Comptes Rendus,' vol. 106, p. 1007.
(14) ANDREWS, 'Phil. Trans.,' 1869.
(15) DIETERICI, 'Annalen der Physik,' 1903, vol. 12, p. 154.
(16) MARGULES, 'Wieu. Akad. Sitzber.,' 1888, vol. XCVIL, Abth. 2a, p. 1385.
TABLE I. — Pressure-Temperature Observations.
Pressure.
Temperature.
Pressure.
Temperature.
lb..,in.5.
°C.
Ibs./inA
°C.
96
-51-0
354-5
-13-0
128
-43-2
387-5
- 9-8
167
-35-8
407
8-15
185
-33-3
429-5
- 6-2
206
220-5
243
270
295
310
316
331
-31-5
-28-0
-25-0
-21-6
-18-8
-17-1
-16-6
-14-9
474-5
511
514
612
711
812
898-5
- 2-6
+ 0-2
+ 0-4
+ 7-15
+ 13-15
+ 18-8
+ 23-05
TIIHRMAI, I'UOI'KKTIKS OF CARBONIC ACID AT LOW TKMI'KRATURES. 105
1
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to
ep ej»
cj S
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Total heat
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VOL. CCX1II. — A.
106
,.|;<>F. C KKEWEN JENKIN AND MR. D. R. PYE ON THE
rs
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Fall of bath temperature
Atmospheric temperature
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Radiation.
Total heat
. 43
£ 2
U.S
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11
Wco
THERMAL PROPERTIES OF CARBONIC ACID At LOW TEMPERATURES. 107
fS
i
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CO 00 O 00 f)
• O O O CO CO O O
o «o o «e • • •
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1 1 1
*- S««
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1 + +
o o o <n
s£387«?r r
t- >* « o O O O
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£
•, • -^JS 8
8
•is 1J: !
(—1
Initial temperature . .
Pressure l>efore throttle va
Pressure after throttle val
Drop of pressure at valve
Change of temperature, th
Change of temperature, nc
Change of temperature, di
Change of temperature pe
Initial temperature, " C.
Initial pressure, p\ . .
Final pressure, y/-_. . .
Fall of pressure . . .
Change of temperature*
Change of temperature pe
P 2
I'M
G FRKWKN JENKTN AND MR. D. K. PYE ON THE
TABLE VI. — Calculation of Latent Heat.
a.
0.
7-
8.
Latent heat.
Evaporation
temperature, "...
Refrigeration.
Series I.,
NBCQ*
I, -I,.
Series II.,
THAM.*
«(8t-0t\
Series III.,
PECQ.*
Is - 1*
Series IV.,
SKHT.*
L = a + 0-y + S,
RGER*
°C.
Th.U.
Th.U.
Th.U.
Th.U.
Th.U.
-53-4
47-0
37-9
4-4
+ 0-69
81-2
-6P6
47-75
35-7
3-5
+ 0-51
80-45
-48-3
48-2
34-8
3-72
+ 0-45
79-75
-43-2
49-5
31-2
3-7
+ 0-35
77-35
-40-9
48-5
28-6
1-5
+ 0-3
75-9
-35-6
50-1
24-7
1-49
+ 0-18
73-5
-30-0
50-0
22-8
1-3
+ 0-08
71-6
-26-1
50-7
20-6
1-87
0
69-4
-21-4
50-8
17-9
1-7
-0-07
66-9
-14-9
51-3
14-0
1-7
-0-1
63-5
-11-6
50-0
14-4
2-0
-0-21
62-2
- 8-6
53-0
11-0
3-1
-0-2
60-7
- 8-0
50-67
14-0
4-22
-0-25
60-1
- 1-7
47-75
10-0
2-27
-0-32
55-2
+ 6-6
46-1
6-7
3-61
-0-37
48-8
+ 13-4
44-5
4-5
5-2
-0-55
43-25
+ 20-05
39-35
1-7
4-2
-0-35
36-50
These letters denote the areas shown in fig. 7, p. 74.
TABLE VII. — Total Heat I and Specific Heat Cp of Liquid C02 at Constant Pressure.
Taken from Curves.
Temperature.
I, 700 Ibs.
I, 900 Ibs.
(V
-60
-25-9
0-47
-40
-21-0
,__
0-49
-30
-16-0
0-515
-20
-10-7
0-54
-10
- 5-2
- 5-5
0-57
0
+ 0-6
+ 0-3
0-60
+ 10
+ 6-8
+ 6-5
0-C4
+ 20
~~~
+ 13-1
0-68
THKKMAL PROI'KI.TIKS OF CARBONIC ACID AT LOW TEMPERATURES. 109
TAHLE VIII. — $<f> between Pressure Curves and Limit Curve.
Mean temperature.
>•«/> between limit
curve and
700 lbs./8q. inch
line.
o./. between 700 and
900 ll)8./8q. inch
lines.
<ty between limit
curve and
900 lbs./sq. inch
line.
"C.
-31-5
0-00289
0-00104 0-00393
-21-7 0-0029
0-00122
0-0041
-10-6 0-0031
0-00157
0-0047
+ 0-5 0-0024
0-0025 0-0049
+ 8-2
—
0-0033
0-0049
+ 13-2
—
—
0-00485
+ 17-9
—
—
0-0031
NOTE. — 700 Ibs. curve meets limit curve at + 12° -5 C.
900 Ibs. + 23°-2C.
TABLE IX. — Collected Results.
Tempera-
ture.
Pressure,*
Authors.
Liquid.
Latent
heat.
Vapour.
Vapour
density.
Pressure,*
K. and R. and
AMAOAT.
•
I.
*
L
I.
*•
•c.
lb../«.«
gr./c.c.
!*./!„.*
atmospheres.
-50
98
-26-4
-0-1064
79-9
53-9
0-2515
0-0181
'.'7
6-60
-45
120
-23-8
-0-0958
77-8
54-2
0-2458
0-0220
119-5
8-12
-40
145
-21-2
-0-0850
75-7
54-5
0-2403
0-0263
144-5
9-H2
-35
174
-18-6
-0-0742
73-6
55-0
0-2350
0-0316
173-5
11-8
-30
206
-15-96
-0-0635
71-3
55-35
0-2300
0-0378
206
14-0
-25
242£
-13-3
-0-0528
69-0
55-7
0-2253
0-0448
242-5
16-5
-20
284
-10-5
-0-0422
66-5
56-0
0-2207
0-0528
284
19-3
-15
331
- 7-7
-0-0318
63-8
56-1
0-2154
0-0621
329-5
22-4
-10
384
- 4-9
-0-0211
GO-9
56-0
0-2100
0-0730
382-5
26-0
- 5
443
- 2-05
-0-0105
57-7
55-6
0-2042
0-0855
441
30-0
0
508
+ 0-91
o-o
54-1
55-0
0-1981
0-100
505
34-35
+ 5
580
+ 4-05
+ 0-0110
50-3
54-35
0-1919
0-117
—
—
+ 10
658
+ 7-25
+ 0-0223
46-2
53-45
0-1850
0-138
653
44-4
+ 15
743
+ 10-4
+ 0-0331
41-9
52-3
0-1775
0-163
—
—
+ 20
835
+ 13-45
+ 0-0435
36-55
50-0
0-1682
0-196
829
56-4
* The pressures given in the last column, taken from KUENKN and RORSON'S figures from - 50° C. to
0° C. and from AMAOAT'S from 0" C. to + 20° C., are probably more accurate than the authors', for reasons
explained on p. 95.
no
PROF. C. FKEWEN JKNKIN AND MR. D. R. PYE ON THE
TABLE X. — Comparison of Pressure-Temperature Observations for Saturated CO2
Vapour.
Pressure in Atmospheres (7fiO mm. Hg).
Temperature.
Authors.
REGNAULT.
CAII.LETKT.
AMAGAT.
K. and R.
Z. and S.
1912.
1862.
1878.
1891.
1902.
1906.
-60
4-35
4-30
4-35
-56
5-24
—
—
—
5-19
-55
5-44
—
—
—
5-35
-54
5-64
—
5-46
-50
6-66
—
6-8
6-GO
6-73
-46
7-86
—
—
—
7-89
-45
8-16
—
8-12
-44
8-50
—
8-72
-40
9-86
—
10-25
9-82
9-88
-35
11-84
—
—
11-8
11-92
-34
12-25
—
12-7
-30
14-0
—
14-0
14-21
-25
-20
16-52
19-3
17-12
19-93
—
—
16-5
19-3
16-74
19-52
- 15
22-5
23-14
—
22-4
22-46
- 10
26-1
26-76
—
—
26-0
25-83
- 7
28-5
—
—
27-80
- 5
30-1
30-84
30-0
0
34-55
35-40
34-4
34-3
+ 5
38-43
40-47
+ 10
44-75
46-07
44-4
+ 15
50-5
52-2
+ 20
56-8
58-85
56-4
+ 25
—
66-1
+ 30
I
73-85
—
70-7
—
—
IIII.KMAL PROPERTIES OF CARBONIC ACID AT LOW TlvMi'KKATURES. Ill
TABLE XL— Series I. Test I. on March 25, 1912.
C.F.J. Record.
Time.
( lunge
pressure.
Voltmeter.
Time of
ring.
Weight on
flask.
Cold
junction
tem-
perature.
Atmospheric
tem-
perature.
Coil A.
Coil C.
h. in.
lb«./in.1
rolb
roll*
m. -.
lb».
•C.
"0.
12 18
266
98-2
99-3
18 21
5
—
Start.
19
3
8-3
9-4
—
7
12-47
13-4
20
3
99-4
100-3
20 2
21
4
8-3
99-8
—
11
22
2
8-0
9-4
23
3
8-1
9-2
23 17
•
24
4
8-1
9-3
—
15
25
5
6-1
:••-
—
—
12-475
13-4
26
5
6-1
9-8
26 34
27
4
6-2
9-7
—
19
28
3
5-7
9-6
29
3
90-0
100-1
29 57
30
2
88-8
99-6
—
23
31
3
89-1
99-3
32
5
89-2
9-2
33
5
89-3
9-3
33 18
34
5
9-9
9-9
—
27
35
3
9-5
9-5
36
2
9-4
9-3
36 45
37
2
9-7
9-8
—
31
38
3
9-9
9-9
39
3 95-6
9-9
40
2
4-9
9-5
40 9
•
41
1
5-0
9-5
41 50
33
42
3
5-1
100-2-
35
43
3
5-8
0-2
43 34
—
12-51
13-50 Stop.
Mean 1 „.,„.,
pressure /
Duration, 25-217 mins.
{97-5 for !i •»•>:> in in.v
89-48 „ 10-0 „ 99-646for 25-22 mins., Coil C.
95-28 „ 5-57 „
{97-14 for 9-65 mins.
89-06 „ 10-0 „ 99-106 for 25-22 mins., Coil C.
94-80 „ 5-57 „
Mean absolute pressure, 272.
Mean 0* -21 -4.
Heat = 1484 Th.U.
ML-
PROF, c i I;I:\\I:N .IKNKIN ANI> MK. D. R. PYE ON THE
TAHLB XII.— Series I. Test I. on March 25, l!M2.
D.K.R liecord.
T.J. No. 1.
T.J. No. 2.
T.J. No. 5.
Bath temperature.
Exit gas temperature.
Liquid temperature.
Time.
Potentiometer.
Time.
Potentiometer.
Time.
Potentiometer.
h. m. f.
ohms cm.
h. m. s.
ohms cin.
h. m. -.
ohm cm.
12 10 0
-5 74J
12 12 0
-5 56J 12 14 30
+ 0 19i
16
71
16
+ 0 19|
18 30
72J
19
20
+ 0 15
Start I2h. 18m. 47s.
22
7l|
23
24
-0 5
26
70-
27
28
-0 29
30
70
31
52
32
-0 52£
34
71
35
53
36
-0 78
38
73
39
54
40
-1 14
41
75
42
75
43
55J
44
-1 43J
Stop 12h. 44m. Os.
45
75
46 30
76
Note.— D.R.P. watch
48
n\
[
was 26s. fast on |
C.F.J. watch.
Means. . .
-5 73J
-5 53
Plotted.
Temperature
1
from cali-
bration curve
| -16"
•1C.
-15°-OC.
+ 9"-7C.
ohms cm.
Hath
temperature at start -5 72-4
„ finish .... - K 7fi • 7R
fall .
n° . i n r<
1- IF.
Correction for change of cold junction temperature . . . = - 0° • 05 ,
Therefore nett bath fall
0°-14
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 113
TABLE XIII. — Series II. Test III. on March 26, 1912.
C.F.J. Record.
Time.
Voltmeter
coil D.
Time of
ring.
Weight on
flask.
Cold
junction
temperature
Atmospheric
tem-
perature.
h. m.
T01U.
m. i.
DM.
°C.
°C.
4 46
63-6
—
14
14-51*
16-3*
* At 4h. 42m.
47
4-2
46 32
—
—
—
Start.
48
4-4
—
16
49
4-1
49 29
50
5-9
—
18
51
4-2
52
5-2
52 25
53
4-7
—
20
54
4-1
:,.-,
3-9
55 17
t
56
4-3
—
22
57
3-9
58
3-9
58 9
59
3-8
_
24
5 00
4-0
1
4-1
1 1
2
3-9
—
26
3
4-9
3 54
4
4-6
—
28
5
3-9
6
4-0
6 48
•
7
4-1
30
8
3-2
9
3-3
9 45
10
2-5
—
32
11
1-8
12
3-2
12 43
13
3-0
—
34
14
3-8
15
4-1
15 43
—
14-51
16-7
Slap.
36
Duration, 29* 18 in ins.
Mean . . 63 '96
Corrected! AQ.™
volts I 63 70
(63-7)»x29-18m.x3-189 ,
77-5Th.U.
10»
VOL. CCXIH. A.
II I
PROF. C. l'i;i:WKN JENKIN AND MR D. It. PYE ON THE
TAHLE XIV.— Series II. Test III. on March 26, 1912.
D.R.P. Record.
i
T.J. No. 1.
T.J. No. 3.
T.J. No. 4.
Bath temperature.
Initial temperature.
Final temperature.
Time.
Potentiometer.
Time.
Potentiometer.
Time.
Potentiometer.
h. m. a.
ohm cm
h. m. e.
ohms cm.
h. m. s.
"Inn cm
4 42 0
-1 26;
1
4 43 0
-8 8{
4 44 0
-1 24
Start 4h. 47m. 29s.
45
26,
46
8
47
24
Slop 5h. 16m. 39s.
48
26J
;
49
6f
50
23f
51
25
52
4
53
22|
Note.— D.R.P. watch
54
24
55
6
56
22^
56s. fast on C.F.J.
57
25
58
5
59
23J
watch.
500
26
5 1 0
4f
520
24|
3
27i
4
5
5
25
6
27,
7
6£
8
25^
9
27-
10
7
11
25j
>
12
29
13
8J
14
15
30
16
8|
17
26j
17 30
29}
18
30
Means. . .
-1 27-
5
-8 6-6
-1 24-
3
Temperature
+ 8°
•5C.
-25°-6C.
+ 8"
•3C.
4
ohm cm.
Bath temperature at start .... - 1 26-3
» i. finish ... -1 29-85
Correction for cold junction . n° • n
Nettfall . . .
"
0°-195 „
TIIKIiMAI, I'KOPKKTIES OF CARBONIC ACID AT LOW TEMPERATURES. 115
TABLE XV.— Series III. Test IV. on March 28, 1912.
C.F.J. Record.
Time.
Voltmeter,
coilD.
Time of
ring.
Weight on
flask.
Cold
junction
temperature.
Atmospheric
tem-
perature.
h. m.
TOIU.
in. ».
)b*.
°C.
°C.
5 19
_
12
14-32*
15 -Of
*At5h.9m. tAt5h.l(im.
20
46-9
20 32
—
—
—
Start.
21
•9
—
14
22
•9
22 23
23
•8
—
18
24
•9
25
•7
26
47-1
26 7
—
14-34
27
•3
—
22
28
•4
29
•5
29 46
30
•1
—
26
31
•2
—
—
—
15-3
32
•o
33
4G-9
33 31
34
•9
30
14-36
35
47-0
36
•1
37
•0
37 18
,
38
•1
—
32
39
•2
39 12
—
14-365
15-2
Slop.
Duration, 18 '67 mius.
Mean . . 47-15
Volts . . 46-96
Hoat (46-96)'xl8-67m.x3-189 ,31.oThU
Q 2
in;
PROF. C. I u:\VI.N JENKIN AND MR. D. R. PYE ON THE
TABLE XVI.— Series III. Test IV. on March 28, 1912.
D.RP. Record.
T.J. No. 1.
T.J. No. 3.
T.J. No. 4.
Bath temperature.
Initial temperature.
Final temperature.
Time.
Potentiometer.
Time.
Potentiometer.
Time.
Potentiometer.
h. m. •
ohm cm.
h. m. s.
ohm cm.
h. m. s.
ohm cm.
5 17 0
-0 57
5 17 30
-5 49'
1
5 18 0
-0 54f
19
57J
20
50;
[
21
55-
22
58
23
51
24
55;
25
58f
26
48,
27
55;
28
58}
29
47]
30
56
31
59
32
45
33
55f
34
59i
35
47
36
55|
37
59
38
51;
39
55£
40
59
Means . .
-0 58-4
-5 49
-0 55-7
Temperature
+ ir-ic.
13°
•oc.
+ 11°-2C.
olim
cm.
Bath temperature at start .... = - 0
57-75
„ „ „ finish .... = -0
59-0
„ fall
1 • 9R nrr> f>° • f>7 P
Correction for change of cold junction temperature . . . - 0° • 03 „
Nettfall ....
.... Q°-04
THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 117
TABLE XVII. — Comparison of 0^ Diagrams.
4> liquid.
Latent heat.
<f> vapour.
Temperature.
Authors'.
MOI.UER'S.
Authors'.
M'-l Mill:'-.
Authors'.
MOI.I.IKR'H.
•o.
+ 20
+ 0-0435 +0-045
36-55
36-93
0-1682
0-171
+ 10
+ 0-0223
+ 0-021
46-2
47-74
0-1850
0-189
0
0
0
54-1
55-45
0-1981
0-203
-10
-0-0211
-0-019
60-9
61-47
0-2100
0-215
-20
-0-0422
-0-036
66-5
65-35
0-2207
0-226
-30
-0-0635
-0-053
71-3
70-4
0-2300
0-236
-40
-0-0850
—
75-7
—
0-2403
-50
-0-1064
—
79-9
—
0-2515
—
TABLE XVIII. — New Data arranged as in MOLLIER'S paper (l) and EWINO'S
' Mechanical Production of Cold ' (4).
Tempo ra-
l^rAaoiiff*
BKUN'S
Volume of
Latent heat,
*•
ture.
1 1 DHoUI < .
volumo
liquid.
vapour.
1-
r.
Liquid.
Vapour.
°0.
kg./om.*
o.o./gr.
o.c./gr.
-00
6-82
0-866 55-2
-26-6
79-9
-0-1064
0-2515
-45
8-41
0-881 45-4
-24-0
77-8
-0-0958
0-2458
- 40
10-16
0-896 38-0
-21-45
75-7 -0-0850
0-2403
-35
12-20
0-912
31-65
-18-9
73-6 -0-0742
0-2350
-30
14-49
0-930
26-45 16-3
71-3 -0-0635
0-2300
-25
17-05
0-958
22-30 -13-75
69-0 -0-0528
0-2253
-20
19-96 0-970
18-95 -11-0
66-5 -0-0422
0-2207
-15
23-16 0-993
16-10 8-3
63-8 -0-0318
0-2154
-10
26-9
1-02
13-70
- 5-6
60-9 -0-0211
0-2100
- 6
31-0 1-048
11-70
- 2-85
57-7
-0-0105
0-2042
0
35-5 1-08
10-00
o-o
54-1 0-0000
0-1981
+ 5
40-5 1-119 8-55
+ 3-00
50-3
+ 0-0100
0-1919
+ 10
45-9 1-163
7-25
+ 6-05
46-2
+ 0-0223
0-1850
+ 16
51-8 1-222
6-13
+ 9-10
41-9
+ 0-0331
0-1775
+ 20
58-3
1-295
5-10
+ 11-95
36-08
+ 0-0435
0-1682
-
IV. The ('a parity for Heat of Metals at Different Temperatures, being an
Account of Experiments performed in the Research Laboratory of the
i/// Cn//ii/i a/ ' Sunlli II till* a,, <l Monmouthshire.
By E. H. GRIFFITHS, Sc.D., F.M.S., and EZER GRIFFITHS, B.Sc., Fellow of the
University of Wales.
Received April 1,— Read May 1, 1913.
CONTENTS.
Section. Page
I. Introductory 119
II. Outline of apparatus 122
III. Measurement of temperature 126
IV. Measurement of the resistance of the heating coil 131
V. Measurement of potential difference 137
VI. The thermal capacity of accessory substances 138
VII. Measurement of mass 140
VIII. Measurement of time 141
IX. Temperature control of the baths 141
X. Methods of experiments—
(1) Total heat 143
(2) Intersection .! '..'.'.'. 147
XI. Experimental results —
(1) Cu, (2) Al, (3) Fe, (4) Zn, (5) Ag, (6) Cd, (7) Sn, (8) Pb 159
XII. Summary of results 170
XIII. NERNST'S observations at low temperatures 175
APPENDIX I. Discussion of the results 178
APPENDIX II. The relation between melting-points and atomic heats 183
APPENDIX III. The soldering of glass and quartz tubes to metal 184
SECTION I.
Introductory.
A STUDY of the published determinations of the capacity for heat of the elements
leads to the conclusion that further information of an accurate nature is desirable. It
will be found that, in most cases, the values are deduced by observations of the heat
absorbed or given out when the changes of temperature are large, and the conclusions
VOL. CCXIII. A 500. Published separately, July 28, 1913.
120 DR. K. H. GRIFFITHS AND MR. EZER GRIFFITHS ON THE
derived therefrom are based on the assumption that the relation between the specific
heat and the temperature is of a linear order.
Again, some, in fact a large majority, are comparative determinations and
dependent on the capacity for heat of other bodies, as, for example, those which
assume REGNAULT'S values for the capacity for heat of water at ordinary tempe-
ratures — values which we now know to be inaccurate.
The experimental difficulties connected with the method of mixtures are considerable
and that method has probably been pushed to its extreme limits of accuracy. The
agreement between the results obtained by different observers, and also between those
resulting from repetition by the same observer, is rarely satisfactory. The tempe-
rature changes have, as a rule, been measured by means of mercury thermometers,
without a proper appreciation of the difficulties attendant upon the use of those
instruments for accurate work.
A further possible source of uncertainty is the effect of the sudden chilling of a
metal when rapidly cooled from a high temperature.
The experimental conditions have not been varied sufficiently to demonstrate the
absence of unsuspected causes of error and, according to the chemists, sufficient care
has not been devoted to the detection and elimination of the impurities present in the
samples used.
It is true that there are determinations of a high order of accuracy which may not
justly be subject to this criticism, but such examples are few and it is difficult,
when comparing the evidence, to assign to each determination its due weight.
As an illustration of the divergences which exist, we append the values given by
leading authorities in the case of copper, a metal which does not appear to present
any peculiar difficulties and one in which the values obtained by different observers
are, on the whole, in better agreement than is the case with other metals.
All the observers agree with the conclusion that the capacity for heat of copper is
a function of the temperature, but they differ markedly as to the value of the function.
For example TOMLINSON ('Roy. Soc. Proc.,' 1885) gives
S, = G'09008 + 0'0000648£.
LOREXZ gives values at 0° C., 50° C. and 75° C., from which the Mowing expression
is obtained : —
S, = G'0898 + 0'
an expression which denotes that the specific heat increases more rapidly than the
temperature.
GAEDE ('Phys. Zeitschr.,' 4, 1902) gives values at various temperatures between
J. and 92" C., from which the following expression can be deduced :—
S, = 0'
This would give a maximum capacity for heat at a temperature of 341° C.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES.
121
If we consider the endeavours to ascertain the mean capacity over the range 0° C.
to 100° C., the same lack of agreement is evident. For copper we have —
Temperature range.
°0.
15-100
0-100
At 50
17-100
23-100
15 100
At 50
Specific heat.
G'09331
G'09332
0-09169
G'09333
0-0940
0-09232
0-09261
Observer.
I '.U.K.
TMMI.INSMV.
LORENZ.
NAOCARI.
TROWBRIDGE.
TILDEN.
GAEDE.
The methods described in this paper, although they doubtless present their own
peculiar difficulties, are, we believe, free from many, if not all, of the sources of error
above referred to. The method is briefly indicated in the following numbered
paragraphs : —
1. The energy was supplied electrically and the conclusions are not dependent
upon any assumption concerning the capacity for heat of other bodies than those
under consideration.
2. The substances were raised across a given temperature through very small ranges
of temperature (extreme limit of range, about 1°'4 C.).
3. These temperature changes were measured by means of' differential platinum
thermometers, for which purpose these instruments are admirably adapted.
4. Large masses of the substances were used, ranging from 1 to 4 kgr.
5. The apparatus was constructed with all its parts duplicated. The metals
examined were suspended by quartz tubes in similar air-tight brass cases whicli were
placed side by side in a large tank containing rapidly stirred water or oil. This tank
was electrically controlled with great constancy at any given temperature, 00.
One of the metal blocks remained at the tank temperature throughout an experi-
ment while the other, having been previously cooled below 00, was raised to a
somewhat similar temperature above it by a supply of heat electrically developed in
the centre of the block, the difference in temperature between the two blocks being
determined at regular intervals by means of the differential platinum thermometers.
All changes in the surrounding conditions would therefore affect both blocks equally ;
hence, by measuring the difference of temperature only, many possible causes of error
were eliminated.
6. The equation connecting the various quantities is
where M = total mass, S its specific heat ; 0, the initial temperature, and 03 the
final temperature ; E, the potential difference at the extremities of the resistance
VOL. ccxin. — A. R
122 DR- K. H- GRIFFITHS AND MR. EZER GRIFFITHS ON THE
coil R ; *J = 4'184x 10' ; and Q, the number of thermal units lost or gained during
time t from sources other than the electrical supply.
In these experiments the values of fy and 0a were so arranged that Q was in every
case small or negligible, and, if necessary, could be estimated with sufficient accuracy.
7. With two exceptions, the samples of metals used were supplied by Messrs.
Johnson and Matthey, to whom we wish to express our sincere thanks for the trouble
they have taken in the matter. Their certificate concerning the degree of purity is
in each case appended. Information regarding the remaining metals (Cu and Fe)
will be found in the sections dealing with those two elements.
8. Experiments on identical samples at the same temperature were repeated under
very varied conditions, in order to enable us to detect unsuspected sources of experi-
mental error. Two separate methods of experiment, involving different data and
methods of reduction, were employed. Three different sets of differential platinum
thermometers were used. The rate of heat supply was varied in the ratio of 9 : 1.
The determination of S at a given temperature with a particular sample was in several
OMOO repeated after the lapse of some months ; the quartz tubes and cover were
replaced by others of different masses &c. We were thus enabled to ascertain causes
of error which would otherwise have remained undetected (see p. 139).
9. The results of our observations have been deduced from the actual experimental
numbers and in no case from " smoothed curves."
The most serious difficulty presented by this method of experiment is that of
determining the mean temperature of the block of metal when its temperature is
altering. Temperature gradients must necessarily exist, since equalisation of
temperature by stirring is an impossibility. The manner in which this difficulty was
surmounted is described in later sections.
When embarking on this investigation we proposed to extend our range of tempe-
rature to the lowest point obtainable by means of liquid air, limiting the inquiry to
the study of two or three metals only. Owing, however, to delay by the contractors
in the delivery of the liquid-air plant, we were compelled to postpone that portion
of our investigation dealing with temperatures below 0° C. to a later date, and
therefore enlarged the scope of our inquiry so as to include the following metals,
namely, Aluminium, Iron, Copper, Zinc, Silver, Cadmium, Tin and Lead.
As the data already accumulated concerning the capacity for heat of these metals
over the range 0° C. to 100° C. may be useful to other obsorvers, we see no reason for
delaying the publication of the work already completed.
SECTION II.
Outline of Apparatus and of the Method of Experiment.
A diagrammatic sketch of the apparatus within the tank is indicated in fig. 1.
f FK this value of J' see P- no
of Energy,' by E. H. GRIFFITHS (Camb. Univ. Press).
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 123
\
*— Dry Air
(while cooling
Cube in °
position)
1-J4 DR. R II- CKIFFITHS AND MR. EZER GRIFFITHS ON THE
As the left-hand portion is a replica of the right, it will suffice to describe the
latter only.
The metal (A), whose capacity for heat was to be determined, was cast and then
" turned" into the form of a cylinder 15'2 cm. long and 57 cm. in diameter.
This cylinder fitted accurately into a thin copper case (C) of mass, apart from the
lid, of 149 grms. Thus the actual radiating surface surrounding the metal blocks was
similar throughout all experiments.
Two small copper pins attached A in its proper position to the copper lid to which
the case was fastened by a copper ring bearing a screw thread.
Three quartz tubes passed through the brass pipes fixed in the lid of the external
case, and supported the copper case and block within the outer brass cylinder.
These quartz tubes having been previously platinized were soldered at their lower
ends into short copper ferrules which formed parts of the copper lid and at their upper
extremities to the top of the brass tubes which, for 7 cm. of their length, were washed
by the tank liquid.
Between the case-lid and the outer brass lid a mica disc of nearly the diameter of
the brass cylinder was placed and through it passed the three quartz tubes.
After the metal block had been fixed in its case and pins and ring firmly screwed
home, the case and contents were lowered into the outer brass vessel, the edges of the
mica disc (which were slightly padded with cotton wool) resting on a projecting
circular ring (H) about 3'5 cm. above the top of the copper lid. Thus the effect of
the flow of convection currents from inner to outer case, or vice-versd, was
diminished. The brass lid was firmly screwed down over a lead ring by eight bolts.
After the parts were assembled, the air-tightness of the apparatus was ascertained
both by pressure and exhaust tests. The lateral clearance between the inner and
outer cases was 2 cm. ; the vertical, between the lids, 6 '5 cm. ; and between the
bases, 6 '5 cm.
The volume of air contained in the brass case after insertion of the metal block was
about 1500 c.c. A pressure gauge containing a light oil was connected by means of
a 3-way tap with one of the tubes leading from this case to the exterior of the tank.
Observations of the air pressure within the case were taken immediately before and
after an experiment. The air being slightly warmed, the pressure rose during that
interval, and thus the presence of any leakage could be detected.
The annular air space could be regarded as the bulb of a constant volume _
thermometer, and from the change in pressure during an experiment, the change i
the average temperature of the enclosed air could be deduced.
It was found that this change was about one-sixth of that of the contained block.
The approximate magnitude of any correction rendered necessary by the capacity
for heat of the contained air could thus be ascertained (see Section VI.).
In our preliminary experiments the copper case was placed within specially
11 vacuum vessels," the exterior walls of which fitted closely into the
air
in
CAPACITY FOR HEAT <>F MKTALS AT DIFFERENT TEMPERATURES. 125
surrounding brass vessels. The reasons for discarding their use are given on
p. 159.
A cylindrical hole (K), !) nun. in diameter and 14 cm. in length, was bored down
the centre of each metal block, co-axial with the central quartz tube.
Into this was fitted t h« " heating coil," the wire of which was wound on a light mica
frame of the X section used for platinum thermometers. The edges of the frame were
deeply serrated to prevent any possible contact between the wire and the surrounding
metal \v;ills. This hole was filled, at low temperatures, with liquid paraffin (previously
boiled and placed in vacuo to drive off volatile constituents), and at the higher
temperatures with a heavy hydro-carbon oil.
Small mica " baffle plates " were inserted at intervals into the triangular sections
of the mica rack, in such a manner as to deflect the convection currents outwards.
The hole O (depth 10 '3 cm., diameter I'l cm.), co-axial with the left-hand quartz
tube, contained one of the differential thermometers, the other being inserted into the
corresponding hole in the left-hand block.
The position of the hole O was such that about half the total mass of metal was
contained in the annular ring whose outer surface was in contact with the copper
case, and whose inner passed through the centre of the hole. The various precautions
taken to secure accuracy in the use of these differential thermometers will be described
in Section III.
The third hole (G) was used for the purposes of cooling the block below the
surrounding temperature, by the insertion of a thin-walled glass tube containing ether
and connected with a water pump.
When the bath temperature was high the cooling process was a rapid one, but
somewhat tedious at lower temperatures.
To prevent the entrance of laboratory air within the brass cases during cooling—
which, by the deposition of moisture, might have had a serious effect, especially
when the tank temperature was 0° C. — a current of well-dried air was passed by
a branch tube into a larger one (F) which formed a continuation of the quartz
tube leading to the cooling hole. This rapid up-flow of dry air was continued until
the cooling tube had been withdrawn and replaced by a glass stopper, the lower end
of which reached within 3 cm. of the inner copper lid, and thus prevented convection
currents.
Our methods of experiment involved measurements of the following quantities : —
(1) Temperature;
(2) Resistance of heating coil ;
(3) Potential difference at ends of heating coil ;
(4) Mass ;
(5) Time;
(6) Thermal capacities of such bodies as oil, quartz, &c., whose temperature
changed with that of the metal blocks.
126 DR. E. H. CHIFFITHS AND MR. EZER GRIFFITHS ON THE
The validity of our final conclusions is dependent upon the accuracy with which
th.-s,- ,,u.mtitirs NV.TO determined, and in the following sections will be found a
description of the methods adopted for their measurement. An error of 1 in 1000
in Nos. 1, 2, 4 and 5 supra, and an error of 1 in 2000 in No. 3 would affect our final
results by O'l per cent. The thermal capacities of the bodies mentioned in (6),
however, were so small, as compared with the capacities of the blocks, that the effect
of an error of 1 to 5 per cent, in their valuation would fall below the O'l per cent.
referred to.
We have, however, no reason to suspect that errors approaching such limits exist
in any of the measurements above enumerated.
SECTION III.
Measurement of Temperature.
The platinum thermometers were of the standard form, thick platinum leads and
compensators connecting the coil with the heads. All connections, both to the
thermometers and the bridge, were made by means of small cups hard soldered to the
ends of the leads and containing a fusible metal which expanded on solidification.
The electrical connection thus formed was a perfect one and easily disconnected and
re-made.
In our earlier experiments two thermometers, labelled AB and CD, were used. The
constants of these thermometers have been previously published, and as far as we
can detect, show no signs of change over a lapse of 1 5 years.
Their resistance was ascertained by means of a Callendar and Griffiths " self-
testing " bridge,* containing bare Pt-Ag coils immersed in rapidly stirred oil.
Thus, the temperature of the coils could be ascertained with great accuracy.
The bridge was carefully calibrated at the beginning of this work and all its coils
and bridge wire divisions expressed in terms of the mean box unit ; the absolute value
of which, for the purposes of temperature measurements, was of no consequence.
The slight inequality (but 27 parts in 1,000,000) of the " equal arms" (s, and s2)
was ascertained in the usual manner by observations of the apparent alteration in the
resistance of a platinum thermometer immersed in ice, caused by exchanging the
positions of s, and s.j.
The resulting correction has been applied to all our measurements of resistance
taken with this box.
All the precautions previously published by one of us were observed, and we do not
* Hereafter referred to as " Box A." This bridge wan last used in 1900, and it was then observed that
one of iU larger coila was showing signs of change. That coil was replaced by another one, and hence a
rccalibration of the whole bridge was necessary.
CAPACITY KOI; IIKAT (»!•' MKTAF.S AT I 'IFFKKKNT TKM I'KKAT! I;KS.
127
tliink it necessary to encumber the paper with a full table of the results. By the
introduction of the new coil, the mean bridge unit (approximately Ti0 of an ohm)
suffered alteration. Hence the values of R, and R,, for AB and CD differ somewhat
from those previously published. The alteration, however, is in the unit employed,
rather than in the thermometers themselves, and the value of R|/R0 and $ may be
regarded as unchanged.
One addition to the bridge, however, is worthy of mention, as it may be found
useful in other cases. To obtain good contact by means of plugs, considerable pressure
lias to be exerted. As the insulating surface holding the
brasses (in this case marble) is always somewhat yielding, the
tendency of the brasses is to gape and thus contact is only
made round a small horizontal section of the plug.
We therefore affixed to each pair of brasses an additional
contact maker of the kind shown in fig. 2. The spring of
the strip AB caused the wedge fastened on its lower surface
to spring clear of its twin wedge on the block C when the
screw-head E was sufficiently raised. The upright holding
the screw passed through a slot in the strip without making
contact therewith. The plug being firmly inserted in its hole,
the head E was tightly screwed down ; thus the wedge
surfaces were firmly pressed together, and that with a sliding
movement. The strain on the insulating surface was thus
relieved, excellent contact was made by the wedge surfaces,
also by the screw E connecting the strip with C and also by
the plug itself. By this arrangement consistent results were obtained and one of the
troubles of exact resistance measurements eliminated.
The constants of AB and CD were found to be as follows : —
Fit
Thermometer.
V
B.
AB
CD
2449-201
2449-044
1763-279
1763-308
The difference at 100° C. resulting from the separate standardisations is therefore
AB-CD= +0-157.
The thermometers were then coupled up differentially and together immersed in the
hypsometer. The differential reading was then found to be +0'158.
' The barometer used when observing R! was one whose scale coefficients, &c., had been determined by
companion with a French Standard of the Bureau International in 1896. Owing to an accident we had
to refill it prior to this work : this was done with all the usual precautions.
I-J"
DR. K. II GRIFFITHS AND MR. EZER GRIFFITHS ON THE
In the same manner when in ice : —
Difference when determined separately
Difference when determined differentially
+ 0'028.
+ 0'027.
In the course of such a comparison, eight connections had to be undone, re-made
separately in sets of four, and then replaced in the first position .i.e., 12 removals ;m<l
replacements. In the one case, a large number of the box coils were in use ; in the
other, the bridge wire only. The identity of the results is sufficient proof of the
accuracy of the methods employed. These thermometer coils were surrounded by
very thin walled tubes of Jena glass which fitted closely into the holes in their
rwpective blocks and thermal connection between these tubes and the surrounding
walls was assisted by a thin film of oil of known weight.
Although the observations with AB and CD appeared satisfactory, it was decided,
after a considerable number of determinations of specific heats had been made by
their means, to alter the conditions and replace them by two other thermometers
labelled AA and BB. In these, the platinum leads were fused through glass heads,
while the protecting tube was cut off just above the top of the coils, leaving about
ij cm. to 2 cm. of glass projecting into the hole in the metal block.*
These thermometers were standardised by temporarily surrounding them with thin
tubes containing sufficient oil to completely cover their coils, as we proposed to
immerse them similarly when in the blocks. We afterwards found, however, that
the effect of the oil was to increase, rather than decrease, the temperature " lag."
Their coils were therefore freed from any traces of oil by washing with ether.
Their temperature then very rapidly responded to changes of temperature in the
walls of their cavities, their heat capacity being very small as compared with their
The constants of AA and BB were as follows : —
areas.
Thermometer.
Ri.
Bo.
RI -Ro-
AA
2582-983
1863-367
719-616
BB
2582-341
1863-307
719-034
After a considerable number of experiments had been performed, the glass head of
AA was fractured. It was replaced by another thermometer of the same type which
had been in our possession for the last 13 years.
Its resistance, however, was slightly less than that of BB. It was necessary,
therefore, to reduce the latter until the two became approximately equal. These
thermometers were labelled AA' and BB'. Their constants were :—
' The tube* being wrapped round with threads of asbestos to prevent the passage of convection currents
from the canty.
CAPACITY FOK HEAT OF MKT.M.S AT DIFFERENT TEMPERATURES.
I '_'!•
Thermometer.
IM.
Ro.
Ri-Ro-
8.«
AA'
BB'
2576-422
2576-984
1859-579
1860-052
716-843
716-932
1-50
It may here be stated that the results deduced from experiments performed with
different pairs of thermometers were in excellent agreement, and afforded strong
evidence of the accuracy of the temperature measurements.
The twin thermometers were connected in the usual manner, i.e., the coil of the
first in series with the compensator of the second on one side of the bridge, the
compensator of the first and the coil of the second being placed on the opposite side.
For the remaining sides of the bridge, several forms of approximately equal arms
were used. All our later experiments were performed with two Pt-Ag coils, wound
together on a mica rack and placed in a brass tube containing oil, the tube itself being .
immersed in a constant temperature tank. Their resistances at 0° C. were
S, = 1533-618, Sa = 1533-685.
Their resistances could be taken separately, and were determined in ice and also
differentially, both at 0° C. and at higher temperatures. Their continued equality
with change of temperature was remarkable.
The galvanometer contact with the junction of S, and Sa was made on a Pt-Ag wire
connecting their ends and situated near the bridge wire. As both ends of the
galvanometer circuit were connected with similarly situated Pt-Ag wires, the
magnitude of the thermo-electric effect having its origin in these contacts was
diminished. The Pt-Ag wires were further shielded by the massive casting of brass
which carried the contact maker. During all standardisations and experiments the
current through the bridge battery circuit was maintained at 0"013 ampere.
Fig. 3 shows the general arrangement and the approximate resistance of the
various arms is indicated when the thermometers are at 0° C., the resistances being
so arranged as to give nearly the maximum sensitiveness for a given current.
The current through the thermometer coils was less than 0'006 ampere ; its heating
effect was so small that it could be disregarded, both thermometers and, therefore,
both blocks of metal being equally affected.
The battery key was of the type described in ' Phil. Trans.,' vol. 184, p. 398, and
re-established the galvanometer circuit after the battery one was broken. Hence, the
* The value of 8 was determined in the vapour of boiling aniline (184° -13 C.), as the resistance of
thermometer AA' in sulphur vapour would have exceeded the total range of our standard marble top box A.
The value of 6 was of secondary importance, as we were concerned only with the value of d6/dpt at vaiioiu
tank temperatures.
VOL. CCXIH. — A. 8
130
DR. E. H. GRIFFITHS AND MR. EZER GRIFFITHS ON THE
position of the galvanometer spot when the battery was disconnected was that due
to any thermo-electric currents existing in the bridge and its connections, and thus
any movement viable on establishing the battery circuit was attributable to that
circuit only. The key, however, presented some novel features. Brass segments were
Pt-Ag
r-~
Pt-Ag
Fig. 3.
fixed on a vertical spindle in such a manner that when the pointer was at 0, the
galvanometer circuit alone was complete, when rotated through 120° both galvano-
meter and battery circuit were established, and on a further rotation through 120°
the battery was reversed. The segments were so devised that induced currents
during the " makes " and " breaks " would not affect the galvanometer. The whole
series of operations could thus be performed very rapidly by one turn of the spindle.
The galvanometer was one of the original Paschen type. Its four coils were wound
with wire whose diameters increased with their distance from the centre.
Four coils, each of about 5 ohms and separately adjustable, were used. It was
desirable for the present work to obtain a system whose period of oscillation would be
small, which would rapidly settle to its final position and yet have great sensitiveness.
For such a purpose it is advisable to use a system whose moment of inertia is
reduced as far as possible. The type, constructed by ourselves, consisted of two
groups, each containing 18 magnets astatically arranged. The extreme length of
the longer magnets was about 1 ^ mm. The whole system, together with the mirror
and the connecting glass fibre, weighed less than 11 mgf. It was suspended by a
quartz thread about 17 cm. long and between 3 and 4 /x diameter. The clearance
allowed by the ovals in the coils was but a fraction of a millimetre and the faces of
the coils were almost in contact, these faces being coated with tinfoil, to promote
the damping of the oscillations by electro-magnetic induction. Reckoned on the
usual scale, the sensitiveness of this galvanometer could have been easily raised
beyond 10"1", but by exterior magnetic control we reduced it until by one reversal of
battery a deflection of 1 mm. indicated about 1B[j,00° Pt as we found that, owing to
the wandering of the zero point, a higher degree of sensitiveness detracted from,
rather than increased, the accuracy of our observations.
The galvanometer had to be placed at a considerable distance (about 13 m.)
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 131
from the tank, as, if nearer, it responded to the Hashing on and off of the heating
lamps, the changes in the magnetic system of the chronograph, &c. It stood on the
top of a massive pillar of masonry which passed through the laboratory floor without
contact, and whose foundations were embedded deeply below the base of an under-
ground rhamlx-r. Tlu- traffic in ( 'anliil' is heavy, but by taking special precautions,
the galvanometer in these circumstances was but little affected. We found it
necessary, however, carefully to guard the system against convection currents.
Every small opening near the suspension was blocked with slips of mica, and the
whole galvanometer was enclosed within two separate chambers. Many of the
oscillations usually attributed to earth vibrations are, we believe, due to insufficient
attention to the effect of convection currents.
As all our temperature measurements were observed in terms of lengths of the
bridge wire, it is evident that the accuracy obtainable was dependent upon the accuracy
of the calibration of that wire. We have notes of a calibration made some 14 years
ago. Before these observations were begun a careful re-calibration was made in terms
of the "mean unit" of the marble-top box (A) previously referred to. The d'Arsonal
galvanometer used on that occasion, however, was not sufficiently sensitive to enable
the determination of the smaller inequalities.
The calibration was made in terms of 3 coils in Box A, of the approximate value of
1, 5, and 10 hundredths of an ohm. Near the conclusion of our present work, a very
careful re-calibration was conducted, with the object of ascertaining the accuracy of
the earlier one and also of ascertaining if the bridge wire had suffered any alteration
through use. The Paschen galvanometer was employed ; two separate and inde-
pendent calibrations were conducted by the two observers and the results were in
remarkable agreement.
It appeared that the calibration over the longer intervals on the former occasion
was correct, thus showing that the wire had not suffered in the interval.
Each unit of the wire was then expressed in terms of a " mean Box A unit " (the
same \init as that used in the standardisation of the thermometers), and a table was
formed showing the value of a bridge wire unit at regular intervals, in terms of one
Pt degree of each pair of thermometers.
It should here be stated that until the final steps in the reduction of our results,
all temperatures are expressed in the platinum scale.
SECTION IV.
Resistance of Heating Coil.
Our methods of reduction demanded a knowledge of R under the actual conditions
prevailing during an experiment. As it was impracticable to stir the oil in which the
coil was immersed, a wire of small temperature coefficient was chosen to reduce to its
smallest limits the correction for the heating effect of the current on the wire.
8 2
132 I>R E. H. ORIFFITHS AND MR. EZER GRIFFITHS ON THE
For preliminary experiments a 10-ohm coil of constantan wire was used, but was
replaced in the final form of apparatus by a 20-ohm coil of bare manganin wire, as it
was essential to eliminate, as far as possible, sources of thermo-electric forces in the
potential circuit.
The diameter of the circle in which the wire was formed was approximately 7 mm. ;
the number of turns being 59. The upper end of the wire coil was situated about
20 mm. below the surface of the block ; two straight manganin leads (l mm. diameter)
projecting from the coil terminated at their upper extremities at the junctions with
the current and potential leads.
Both potential and current leads were of manganin, the latter being 1 mm.
diameter, and to further diminish the heating effect of the current, two leads were
connected in parallel. Thus six leads extended up the central quartz tube to a
distance of 30 cm. These leads were insulated by perforated mica discs. A solid wad
of such discs was fixed between the top of the rack and the junctions to the current
and potential leads, in order to diminish the passage for convection currents.
As the resistance of the coil had to be observed in situ at each temperature and at
frequent intervals, four brass cups, amalgamated inside and containing mercury, were
soldered on the current and potential leads outside the apparatus ; plugs enabled us
to isolate these circuits, when a resistance had to be taken, from the various
connections to battery, &c. Heavy leads from a dial resistance box terminated in a
pair of brass cups alongside those above referred to.
If
R be resistance of coil,
r, and ra „ „ „ current leads,
»a and r4 „ „ „ potential leads from cups to coil,
then, if
we have
N, = rl + r2,
N< = r3+r4,
As the absolute value of R was required, the resistance of a reference heating coil
f the same construction and about the same value as the one used) was determined
by means of the dial box, and then forwarded to the National Physical
tory, where its value was determined in international ohms ; on its return we
d our previous determination by the dial box.
This enabled us to reduce our determinations of the resistances of the heating coils
used in the work to international ohms.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES.
133
RESISTANCE OF REFERENCE COIL.
National Physical laboratory Report.
( 'OIL Immersed in Unstirred Paraffin Oil cooled to 0° C.
Resistance in international
oh ma.*
Testing current through
resistance.
20-1360
20-1362
20-1370
•mpere
0-025
0-060
o-ioo
Resistance in legal ohms in
terms of our dial box.
Testing current through
resistance.
20-2010
ampere
o-ooi
Hence, the factor to convert our box readings to international ohms = 0'99G78.
As the same ratio arms and plugs in the thousand and hundred dials were used in
the determination of both coils, any change in the relative values of the dial box
20-65
20-64
20-65
Temjo»
10° 20° 50° 40° 505 60* 10^ So3 so5 wcT uo°
Fig. 4. Variation of resistance of heating coil with temperature and time.
Curve ABC, July 18 to October 26, 1912.
„ EF, December 4 „ December 30, 1912.
„ GHK, January 19 „ February 10, 1913.
LM, February 17 „ „ 19, 1913.
NO, 23 „ March 2, 1913.
i.v
since calibration (' Phil. Trans.,' vol. 184, p. 409) would not appreciably affect OUF
results. Fig. 4 shows the variation with temperature of the heating coil and the
permanent change by use and exposure to temperatures of about 100° C.
" Probable error of resistance values is not greater than 2 parts in 100,000."
,„; ,.;. ||. <;i;imTHS AND MR. EZKR GRIFFITHS ON THE
The value of R so determined requires a small correction, as it includes the entire
resistance of the two straight leads, previously mentioned, connecting the upper ends
of the coil with the potential junctions.
Of the heat generated in these leads, a portion is lost by radiation, &c., owing to
their projecting 24 mm. above the surface of the oil.
The resistance of the leads could not be diminished beyond a certain limit, on
account of thermal conduction along them of the heat from the hot oil. A diameter
of 1 mm. was decided upon, as the thermal conductivity of a manganin wire of this
size would be negligibly small.
A certain amount of heat was developed in these short leads by the current.
That generated in the 20 mm. below the surface of the oil would undoubtedly be
absorbed by the oil. Of the heat generated in the 24 mm. above the oil-surface, it is
probab'e that about one-half would pass into the block, &c., by conduction and by
radiation to the ferrules and quartz tubes.
Taking the actual figures : — Resistance per millimetre of the wire = 0 '000642.
Hence, resistance of portion above the oil surface = 0'0154 ohm.
On the above assumption only the heat generated in half of this was, in the case
of either lead, effective in heating the block. We confess that this is merely an
assumption, but, with our knowledge of the actual conditions, it appears to be a
reasonable one ; moreover an error of 10 per cent, therein would only affect the
absolute value of our results by less than 1 part in 10,000.
Change of Resistance due to Change in the Current.
If R' is the resistance of the heating coil when a certain current (defined later) is
passing through it and R is the value determined in the usual manner by the dial
resistance box, then we define SE, by the relation
The effect of the temperature rise (produced by the heating current) on the
resistance of the wire was of course very small in the case of an alloy like manganin ;
the resulting correction, however, could be determined with considerable accuracy in
the following manner (see fig. 5).
A series of observations was made in which the current was measured by the
ordinary potentiometer method. Included in the circuit was a 3-ohm coil ( W) of bare
manganin wire immersed in stirred paraffin oil. It consisted of 4 strands of 0'4 mm.
diameter, in parallel, wound on eight projecting mica plates fixed longitudinally on a
wooden drum. The passage of the maximum current (0'45 ampere) for intervals of
several minutes did not produce any appreciable change in the temperature of the oil.
One observer adjusted the current in the circuit until the potential difference at
CAPACITY Foi; IIKAT OF MKTAI.S AT IMFFKKKNT TKMPKIIATMJKS.
135
t lie i-mls i .f tin- liriiting roil I!' was lial.'inrril a^iinst tliat «\' :i mi!iil«T <>(' <-:i<liiiimn
cells, as in the ordinary method of experiment.
Tin- siTuinl nlisiT\ IT inr.-isiin-il tin- [Hitrntiiil dill'Ti-iK'c ;il tln-i-iids of tin- :; olmi
coil, by means of a Thomson-Varley potentiometer (P), the readings being taken to
about 1 part in 50,000, by interpolation by galvanometer swings.
Jin
R'.
W.
P.
S and 8'.
B.
Cd.
GI.
Gj.
Fig. 5.
Heating coil in metal block.
Manganin coil (about 3 ohm) immersed in oil.
Thomson- Varley potentiometer.
Rheostats to adjust current.
Main storage battery.
Standard cells balanced at ends of R'.
High-resistance galvanometer.
Paschen galvanometer.
Observations were taken when the potential difference at the ends of the heating
coil was varied in steps from that of three to eight standard cadmium cells. The
temperature of the block was maintained approximately constant by cooling with the
ether tube.
Calculation o
If to is the resistance of the 3-ohm coil, W, and the current in the circuit is caused by
a potential difference of nE at the ends of the heating coil, and R' is the resistance
n
136 DK. K. H CKimTHS AND MR. EZER GRIFFITHS ON Till!
of the heating coil for that value of the current, then
r _ nE
R''
Potential difference at the ends of 3-ohm coil is
— x a
If Si, *2, ..., *„ be the potentiometer readings corresponding to 1, 2, 3, 4, ...,
number of Cd cells balanced at the ends of the heating coil, then
•p
SH = K • ^ x a. ; where K is a constant.
R
Hence,
By plotting n3 horizontally — since the heating effect is proportional to the square
of the electromotive force — and the quantity n/sB vertically, but reduced in such pro-
portion that for n = 0 it is unity, we get the relation between SB,, the increment of
resistance, and n3. The resulting points fall (within the limits of experimental error)
on the straight line, <SR = ten*. These observations were repeated when the tank
temperatures were 0° C. and 97° C., and for both the copper and iron blocks.
In the locality of 0° C. the temperature coefficient of manganin is positive, as shown
by the relation
At 0° C., SR = 0-0552n2.
About 100° C. manganin has a negative coefficient and it was found on reducing
the results that SR was negative.
At 97° C., SR = -0'0586n2.
These equations represent the extremes of SR in our range, for at intermediate
temperatures, owing to the locus of R being concave downwards, the factor k was
smaller and vanished altogether between 50° C. and 60° C.
It may be pointed out that for the highest rate usually employed, viz., that due to
8 cells, the correction on account of SR amounted to only 3 parts in 10,000, corre-
sponding to a temperature change in the wire of 10° C. As the values of SR at both
0° C. and 97° C. indicated that the rise in temperature of the wire depended on n2 only
and was independent of all other conditions, we could, from the curve giving the
relation between temperature and resistance, calculate the relation between £R and
rt1 for any temperature within our range.
The value of R was the one directly determined by the dial box when the heating
effect was insignificant, the current through the coil being in that case 0'0015
ampere.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 137
8l • ri"V V.
The potential difference at the ends of the heating coil was always balanced against
an integral number of standard Weston cells in series. A batch of 25 cells was
constructed for use in this investigation, according to the method descril>ed by
G. A. HII.KTT ("The Construction of Standard Cells, &c.," 'Physical Review,'
Vol. XXXII., 1911).
The glass work was of the usual H form, the platinum leads not being sealed
through the glass. The two limits extended about 15 cm. above the cross tube, and
were closed by corks ; through these corks passed the electrodes sheathed by capillary
tubing, the fine platinum wires projecting 5 mm. beyond the sealed ends.
This mode of construction admits of the cells being directly immersed in water,
with the limbs projecting about 7 cm. above the surface.
The water tank containing them was of considerable capacity and well lagged, the
temperature rarely varying by one-fifth of a degree Centigrade per day.
The leads from the cells passed to a switchboard across well insulated supports.
The cells were frequently compared by means of a Thomson- Varley potentiometer
with two standard Westou cells constructed by Mr. F. E. SMITH of the National
Physical Laboratory. (We take this opportunity of thanking Mr. F. E. SMITH for
presenting us with these cells.) Table I. gives the values of the cells in terms of the
N.P.L. standard. All our results are expressed in terms of these standards.
From an examination of the comparisons at various times during the course of
fifteen months, we can find no change greater than that which might be attributed
to the experimental errors.
TABLE I.
Temperature 17° C.
The National Physical Laboratory Standards are denoted by symbols BC— 1, BC— 2.
No. of cell.
E.M.F. in
international volte.*
No. of cell
E.M.F. in
international volts.
BC-1
•0184
10
1-0184
BC-2
•0184
11
1-0185
1
•0183
12
•0184
3
•0183
13
•0184
4
•0183
14
•0185
5
•0182
15
•0183
6
1-0183
16
•0183
8
1-0183
17
•0184
9
1-0184
1 The tables used in the reduction of the ohiervations express the E.M.F. of our cells in terms uf the
standards correct to 1 part in 20,000.
VOL. OOXIII. — A. T
PR. K. II. CKIFFITHS AND MR. EZER GRIFFITHS ON THE
Statement received with Weston normal cells BO-1, BC-2 :—
" E.M.F. = 1'01830 international volts at 20° C.
T.-mperature coefficient: —
E, = Eao-0'0000406(<-20) -0-00000095 («-20)3+0'00000001 (/-20)3."
While an experiment was in progress, the current in the heating coil was
continually adjusted to keep the balance exact. This was effected by two rheostats
in parallel ; the shunt being of fairly high resistance. After the preliminary
adjustment, the potential balance could be maintained by use of the shunt alone.
The sensibility of the high resistance Thomson galvanometer (7000 ohms) in the
standard cell circuit was such that a deflection of 1 mm. on the galvanometer scale
corresponded to a change in the potential difference of 1 part in 20,000.
During the course of an experiment, the potential balance could be maintained
with great steadiness, the slight oscillations rarely amounting to more than 1 part
in 10,000.
SECTION VI.
Minor Collections for the Thermal Capacity of Accessory Substances.
In order to facilitate transmission of heat from the heating coil to the metal block,
the central hole was, as previously stated, filled at low temperatures with boiled
paraffin oil, and at higher temperatures with a heavy hydrocarbon.
The quantity of liquid thus inserted varied slightly with different blocks, the
average volume being about 7 c.c. It was therefore necessary to ascertain the
thermal capacity of the oils and their approximate variation with temperature.
In the construction of the apparatus 15 '06 gr. of alloy were used in fixing the
quartz tubes to the lid of the copper case first employed, and 8'25 gr. of solder in
the second and somewhat heavier lid.
Again, the thermal capacity of the portions of the glass sheaths of the thermo-
meters which entered the block had to be allowed for; the mass .thus inserted
amounted, when thermometer CD was in use, to 3'1 gr., and, in the case of AA and
AA', to T67 gr.
Lastly, allowance had to be made for the heat absorbed by the lower ends of the
quartz tubes which supported the apparatus. This was a most difficult correction to
determine, as it was not possible, (I priori, to specify what mass of the quartz could
be regarded as raised through the same temperature as the copper lid.
1. Specific Heat of Glass and Oil.
The mean specific heat of the paraffin oil between 0° C. and 100° C. was determined
with a Bunsen's calorimeter, by the introduction of about 2'8 c.c. of oil sealed in a thin
glass bull).
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 139
Tim mercury drawn into the Bunsen was directly determined from the loss in weight
of a small capsule ; in the first experiment, 27347 gr. of mercury ; in the second
experiment, 27335 gr. of mercury.
The constant assumed was 15'44 mgr. mercury per calorie.
The resulting value of the mean specific heat for this paraffin was 0"491, its density
l>eing 0'818.
The mean specific heat of glass was determined in a similar manner and was found
to be 0'194.
The specific heat and temperature variation of the hydrocarbon oil had already l)een
determined ('Phil. Trans.,' vol. 186, p. 338), viz., S, = 0'466 + 0'0009£, its density
being 0"865. As both oils were paraffins, the temperature coefficients were assumed
to lie the same, to a first approximation.
2. Alloy and Solder.
The alloy-fixed lid was used only in our earlier experiments and when determining
the specific heats of certain metals at 0° C.
The mean specific heat of this alloy over the range 0° C. to 47° '6 C. (its melting
point being 97° C.) was also ascertained by the Bunsen calorimeter and was found to
be G'0348.
The soldered lid was used over the range 0° C. to 125° C., hence, both its specific
heat and its temperature variations were required.
A block of the sample of solder (3 kgr.) was cast and machined to the same size as
the other metal blocks. Its specific heat at 0° C. and 97° C. was determined in our
apparatus in the same manner as copper, &c.,
S, = 0*0422 + 0-000038*.
On analysis, the composition of the block was found to be 537 per cent, tin and
46'0 per cent, lead, with bismuth and antimony as impurities. The density of the
solder block was 8 "7 7. The platinum deposit on the ends of the quartz tubes was
negligible, the weight of three coats being only 0"0105 gr.
3. Heat Absorbed by the Supporting Quartz Tubes.
As previously indicated, two different copper lids were used. The copper alone in
the first weighed 51 "6 gr. and thin quartz tubes were fixed into its ferrules (which
were 15 mm. in depth) by fusible alloy. The copper alone in the second lid weighed
68 "5 gr. ; the tubes were of much heavier make and fixed with solder. The masses
of the quartz tubes per unit length in the first lid were only three-fifths of those
used in the second ; advantage was taken of this difference to determine the effective
capacity for heat of that portion of the quartz which might be regarded as rising
through the same range as the block of metal
T 2
U0 I)K. F, H. «1MFFITHS AND MR. FJ5ER GRIFFITHS OX THE
Two series of experiments with copper at 0° C. were performed under precisely the
same conditions, except that the lid with the lighter tubes was employed in the first,
and that \\ith the heavier, in the second, series.
•ra the differences in the capacity for heat resulting from these two series, the
dim-rence in the amount which had passed into the quartz tubes could be determined,
the capacity of the alloy, solder and copper, being known. Let m^, and m.^ be the
respective quartz capacities. Then, from the experimental results we found
ml», = l'37; T»A - 2'28.
No doubt, a temperature gradient existed along these tubes, but the value of sm
thus obtained gave the " effective" capacity, i.e., the number of calories which flowed
into the quartz tubes as the temperature of the block was raised by 1° C.
A comparison of other experiments at 0° C., where the conditions were similarly
altered, indicated that the accuracy of this " quartz correction " was sufficient.
4. Heat Absorbed by the Air Within the Brass Vessel.
As stated in Section II. we were able to measure the average increase in pressure,
and therefore in temperature, of this air during an experiment. The volume being
approximately 1500 c.c., the average rise of pressure was 07 mm. Hg, indicating an
increase of temperature of 0°'2G C. The number of calories thus expended equals
0'08.
As the average heat supply during an experiment was about 400 calories, it is
evident that this correction would not amount to more than 1 part in 5000 ; we did
not consider, therefore, that the accuracy of our experiments necessitated the inclusion
of this correction, especially as it could be only roughly determined.
SECTION VII.
Mass and Density of the Metals.
The masses of the blocks varied from 1 to 4 kgr. The balance used for the larger
masses was capable of weighing to O'Ol gr. ; masses under 100 gr. were determined
by a Verbeek short-beam balance. The method of double weighing was used and a
correction was applied for the displaced air.
A calibration of the box weights proved that, relatively, they were correct to a
high degree of accuracy; as, however, the absolute values were required, the 2 kilos.
(from another box), and the 1 kilo, and 100 gr. from this box were forwarded to the
National Physical Laboratory for standardisation. The kilogram and the 100 gr.
weights of our box were found to be exact and the correction on the 2 kilos was
given as + 0'29 gr.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 141
Densiti**.
The densities of the metuls were found by weighing in air and water, the usual
precautions being observed. The values were checked by calculation from the
dimensions and weight.
SECTION VIII.
Measurement of Time,
The only records made during the course of an experiment were the times of transit
of the temperature of the thermometer in the block past bridge-wire divisions, and as
these were effected mechanically by the depression of a key, the observer's attention
could be concentrated on the transits.
The time-recording arrangements may be briefly described as follows :—
An electrically driven seconds pendulum bob, suspended by an Invar rod, at each
swing tilted over an exhausted tube about 2 inches long by ±-inch diameter, fixed in
a frame capable of oscillating about an axis perpendicular to the length of the tube.
As the carriage bearing the tube was unstable about this axis, a slight impulse
sent it over from one stop to the other, causing a small mercury pellet to run down
the tube and make momentary connection between two platinum wires fused midway
into it, thus completing the electro-magnet circuit in the chronograph.
A series of equally spaced dots about 2 cm. apart on the tape indicated seconds,
while the marks of the respective observer's keys were recorded on opposite sides of
the tape. By counting and measuring the fraction, the times of transit could be
obtained to .2\^ second.
Although the seconds pendulum kept a fairly constant rate— being fitted with a
cut-out device to keep its amplitude to a definite limit — the absolute rate was
determined for each experiment by comparison with a rated chronometer.*
This comparison of the total time also afforded a check on the accuracy of the
reading of the tape, always a somewhat laborious process, the lengths of tape used
varying from 20 to 60 metres.
SECTION IX.
Temperature Control of tfie Huthn.
The absolute steadiness of the bath temperature was of prime importance, as our
conclusions were based on the assumption that the temperature of the walls
surrounding the blocks remained constant throughout an experiment.
• Wo are indebted to Mr. T. J. Williams, 63, Bute Street, Docks, Cardiff, for the loan of this
chronometer, and for kindly checking its rate from time to time.
142 DR. E. H. GRIFFITHS AND MR. EZER GRIFFITHS ON THE
When the values of the specific heat at 0° C. were required, a special tank of
15 gallons capacity and lagged with asphalt was used. Two screws— protected by
strong metal cages— caused a rapid circulation of water through the powdered ice.
For work at higher temperatures this tank was replaced by an oil or water bath, of
capacity about 20 gallons, the heat being supplied by immersed electric radiator
lamps.
Some difficulty was encountered in insulating the 200-volt leads of these lamps
when the tank-temperature was above 50° C., owing to the softening of the stretched
rubber tubing by continued exposure to hot water. The cement fastening the caps
of the lamps frequently broke down and entailed the loss of several experiments.
The most satisfactory method of insulation hitherto tried was by clipping a
discarded motor tyre tube over the end of the " radiator " lamp, the leads being
separated within the tube by lengths of glass tubing.
The " lux " lamps used for the purposes of fine adjustment, were insulated by
fixing glass tubing of slightly larger diameter over their ends, the joint being closed
by a short length of rubber tubing well covered with adhesive tape.
JTiermostats.
We tried a considerable variety of thermostats which proved defective from
one cause or another. The two most satisfactory ones may be briefly described. The
first was composed of thin solid-drawn copper piping f inch internal diameter and
16 feet long.
This tube was wound into an oval spiral, so as to surround the two brass cases in
the tank (see fig. 1). Two glass tubes were soldered into reduction pieces at each end,
one terminating in a tap, the other in a U-tube containing mercury. As the method
of constructing these soldered joints is both simple and effective, we have given in
Appendix III. a brief description of the process.
Another form of thermostat used in a considerable number of determinations
consisted of a large branchwork of glass tubes fused together and so distributed as to
take the mean temperature of the tank. The thermostat was filled with toluol,
which however proved unsatisfactory at high temperatures, and was replaced by
commercial aniline, which in every way seems to be a suitable liquid for thermostats.
It has a high coefficient of expansion, low viscosity and a high boiling point (184° C.).
We found it necessary to keep the tap closing the thermostat well greased,
otherwise slow leakage and consequent drift of temperature took place.*
The motion of the mercury in the U-tube operated a relay, which in turn switched
on and off the lamps in the tank. Both the make and the break in the main circuit
> We have some reasons for suspecting that, owing to neglect of this precaution, the temperature of the
bath was not maintained with its accustomed steadiness during some of the group of observations about
67' C.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 143
and the relay circuit had a pair of aluminium plates in water as shunts, to diminish
sparking.
A considerahle fraction of the heat necessary to maintain the tank at the required
temperature was given by a constant supply, while the relay operated the fine
adjustment.
The intermittent lamps were placed close to the stirrer, and thus the whole
arrangement tended to keep the oscillations of the temperature within narrow limits.
At some temperatures we had thermometers by which we could detect changes
«•(' -.00° C., but when the apparatus was working satisfactorily we at no time observed
oscillations of this magnitude.
Another circumstance which perhaps assisted in diminishing the oscillations was
the fact that the stirring was sufficiently vigorous to cause a continual vibration of
the U-tube of the thermostat and so prevent any adhesion of the mercury to the
platinum point which established connection with the relay.
SKCTION X.
(1) The Total Heat Method.
The metal under examination was cooled to a temperature lower than that of the
tank (00) and the fall being observed by means of bridge-wire olraervations, it was
stopped when it had passed below the range of the bridge.
The contact-maker was then set at a certain reading, which, for clearness, we will
specify as —9.* Meanwhile, the "heating" current was adjusted on an auxiliary
coil enclosed in a tube containing oil. This auxiliary coil was a duplicate of the
coil in the metal block and the change-over from the one to the other could be
effected by the depression of a recording key. Before the transference of the current,
the temperature of the block rose very slowly by radiation, &c., and could be followed
by the gradual approach of the galvanometer spot to its zero mark.
The rate of rise was of the order of 0°'043G Pt per second, consequently the
temperature throughout the block was practically uniform.
The moment the temperature had reached the bridge reading —9, as indicated by
the transit of the spot across its zero mark, the heating current was switched over,
the keyt at the same time recording the time on the chronograph tape. A slight
readjustment of the rheostat was usually required to maintain exact potential balance
when the change-over was effected. The contact-maker of the bridge was then set
at the next integer, —8 (the temperature interval from —9 to —8 being roughly
* This was the customary starting point.
t The key was so constructed that any time lag between the marking of the tape and the actual switch
on was compensated for during the operation of switching off.
U4 ,„.. E. n. (;RIFFITHS AND MR EZER GRIFFITHS ox THE
,'4 I't), .-.IK! after slight adjustment* of the galvanometer spot to its zero mark, the
• r1.llv..lM..iii.-i«T k.-y was tunied so as to re-establish the bridge current (0'013 ampere).
Tlie rise of temperature, as indicated by the movement of the spot, was uniform,
:md its transit was recorded by a tap on the chronograph key. The cycle of
operations was repeated; the transits of the temperature across each bridge-wire
nM<liiig being recorded in succession until the temperature had risen to +8 bridge-
wire reading, when the current was switched over to the auxiliary coil.
After the current had ceased to impart any heat to the metal, the observed
temperature continued to rise, on account of excess of heat in the oil, the gradient
from the interior to the surface of the metal and the temperature lag of the thermo-
meter.
The metal would, however, after its temperature had risen to a maximum, part
with its heat by radiation, &c., only, the resulting fall being slow and regular. This
" rise above," as we termed it, could be accurately determined by the following
procedure: —
The bridge contact-maker was set above the switching-off point by an amount such
that the galvanometer spot moved to near the centre of the scale before the regular
cooling l>egan. The galvanometer deflections on reversal of the bridge current were
noted, and also (on the chronograph tape) the times of the observations, until the
deflections had increased beyond the range of the scale. The value of 1 mm. scale
deflection in terms of a bridge-wire unit being known, f the rate of fall in temperature
could l)e determined, and the temperature time-curve P... ABCD could be constructed.
One of the resulting diagrams for the " rise above" is shown in fig. 6.
If P is the point at which the current was switched off, the slope of the line CD, i.e.,
the rate of uniform cooling, gives the data required for the determination of the
horizontal line EG, and thus the temperature which the metal would have attained, in
the alwence of radiation, &c. , can be ascertained.
If GE l>e produced liackwards to meet the temperature ordinate at F, then it will
be evident that F falls on DC produced.
Thus PF, the rise due to the residual heat in the block, could be determined at the
close of each experiment with considerable accuracy. The value rarely exceeded
0°'l Pt and could be measured to 1 part in 1000, that is, about 1 in 15,000 of the
whole range.
It may be mentioned that the " rises above " for a series of experiments with the
same metal under the same conditions were proportional to na, i.e., to the rate of supply.
The galvanometer system generally required this slight readjustment between each observation of
transit in order to maintain the spot on the scale zero when the bridge current was broken. This was
effected by the movement of a small subsidiary control magnet on the table by the observer, and about
1 J m. distant from the galvanometer. The changes of zero were chiefly those due to variations in the
thermoelectric effects in the circuit, and with considerable attention to shielding the various junctions w
succeeded in diminishing such changes to small dimensions, but could not altogether eliminate them.
t Ascertained for each get of experiments.
we
CAPACITY FOR HEAT OF METALS AT DIKKKKKNT TKMPERATURES.
145
We may hen; point out that the two fundamental observations which determined
the temperature range were taken when tin- temperature of the metal was steady and
practically uniform, the only ch.-mgr taking place being that due to the very small
rate of rise or fall consequent on radiation, &c.
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Fig. 6. "Rise above" for Experiment IV., June 2, 1912 (7 standard cells).
We have next to consider any other necessary correction for the effect of radiation
during an experiment. When the current was established at —9 bridge- wire reading,
the oil had first to rise in temperature, then a gradient established from centre to
surface of block and, when the temperature began to rise, the thermometer would
undoubtedly lag behind the temperature of the surrounding walls.
For these reasons, the time of rising through the first bridge-wire division would
considerably exceed the times of passing over succeeding equal divisions.
It was found that when the temperature had reached the end of the first bridge-
wire division, the conditions had become practically steady, as shown by the fact that
in subsidiary experiments in which the current was switched off at the end of the
first bridge-wire division, the " rise above " was found to be very nearly the same as
when the experiments were completed in the customary manner.
We also investigated, with the smaller currents, the curve showing the rate of rise
of the thermometer throughout this first interval and it appeared that, during the
first hah0 of the time of passing through the interval, the thermometer only rose from
— 9 to —87 bridge-wire. Consequently, this reading —87 bridge-wire may be regarded
as approximately the mean temperature throughout the time of the first interval
VOL. CCXJII. — A. U
U ,-, hi; K. II iiKIFFlTHS AND MR. EZER GRIFFITHS ON THK
Tlie temperature ranges above and below 00 were so selected that, excluding the
first interval, the two ranges were equal and as they were small (about 0°'6 C.), the
times over these ranges were so near equality that the losses and gains due to radiation
illicit be neglected.* Hence, the only radiation correction required was that which
expressed the heat thus received as the metal rose through the first bridge-wire
interval. The true time, however, over that interval was less than the time recorded
between the switching on the current and the first transit, owing to the causes of lag
above referred to. As this lag was known in terms of temperature, by the " rise
above," it was possible to obtain from it an expression involving time.
If
0, is the " rise above " in degrees Pt,
<, = the average time of rising through 1° Pt when the temperature of the block
is rising steadily on account of the heating current,
then 6, x £, would be the approximate time, at any part of the range, of moving through
the " rise above " ; this we term the " time lag " = T.
T was found to be practically the same for all rates of heating for the same metal.
For example, in the case of copper, 36 seconds ; of silver, 40 seconds.
Hence, the actual time over the first interval was equal to the observed time
diminished by T.
The rate of rise per second due to radiation alone was obtained by two distinct
methods, namely : —
I. By subsidiary experiments in which rate of rise due to radiation alone over the
range —9 to —8 was observed ;
II. From the observations of the transits taken during the actual experiment when
the conditions were settled.
For, if
M = mass of the substance,
S = specific heat a,t #0,
ms = thermal capacity of oil, copper case, &c.,
.7 = rate of rise due to radiation alone for a difference of 1° Pt between the
block and the surroundings,
t = time in seconds,
E = E.M.F. of a standard Weston cell,
n = number of cells balanced,
R = resistance at this temperature corrected for heating effect of the current,
In a previous communication (• Phil. Trans.,' vol. 184, p. 500) it was shown that if f, is the time of
iny temperature below that of the tank, and (, is the time from the lower temperature
time ^+t ' the 8Um °f ^ 1088e8 and ^ dUC to radiati°"' &c" is
' the error due to the assumption that the
CAPACITY FOR HEAT OF METALS AT DIFFERENT TKMPKRATURES.
147
and
de\ #,_
/ur
JR(MS+ww)
(»E)»
JR(MS+nw)
-#)=(%} -(%
\dtUt \dt
Hence
The table below shows the values of <r deduced from the two methods, and their
agreement affords strong evidence of the accuracy of the resulting correction.
Metal.
Method I.
Method II.
Silver
0-000078
0-000079
Copper .
0-000057
0-000053
Cadmium
0-000093
0-000096
Hence, if 0, is the range below 6a, corrected for radiation, and 0a the range above 00,
and 6, the " rise above " after correction for radiation, then
nrE?t
JR
Table II. (p. 148) represents a typical series of experiments by the "total heat"
method, the metal being copper at 0° C., and the thermometers AB and CD.
(2) The Intersection Method.
The metal having been cooled a considerable distance beyond the limits of the
bridge, the current and potential balance were established from five to ten minutes
before the temperature came within the bridge range. This preliminary heating up
under the normal conditions of the experiment was essential, as it ensured a steady
state of gradient, lag, &c., being established before the commencement of the
observations. The time of transit of the temperature across each bridge wire division
was recorded on the chronograph tape, as in the " total heat " experiments.
The current was switched off and the " rise above " taken in the usual way.
Similar experiments over the same range were performed with various values of n
(the number of standard cells balanced at the end of the heating coil).
From these observations the value of SO/St at the centre of each scale unit of the
u 2
its
DR. E. H. GRIFFITHS AND MR. EZER GRIFFITHS ON THE
TABLK II— " Total Heat " Method.
Copper at 0° C. Thermometers AB, CD.
February 24
„ 24
„ 25
„ 25
„ 25
,. 26
28
March 10
10
10
11
12
II.
5
6
1
-
9
4
7
8
9
7
4
i
III.
1374-8
9«5-5
3743-5
557-4
446-8
2125-9
717-8
557-6
446-9
718-3
2126-0
2123-8
IV.
109
85
231
60
52
150-5
70-5
59-9
52-2
70-3
150-9
150-2
V.
0-0027
0-0018
0-0071
0-0009
0-0006
0-0042
0-0012
0-0009
0-0006
0-0012
0-0042
0-0042
VI.
0-0355
0-0510
0-0141
0-0843
0-1036
0-0239
0-0662
0-0841
0-1039
0-0658
0-0241
0-0234
VII.
1-3356
1-3519
3095
3863
4059
3224
1-3678
3861
•4062
•3674
•3227
1-3219
VIII.
0-09070
0-09062
0-09068
0-09070
0-09073
0-09066
0-09064
0-09073
0-09073
0-09072
0-09070
0-09060
IX.
+ 0-02
-0-06
0
+ 0-02
+ 0-05
-0-02
-0-04
+ 0-05
+ 0-05
+ 0-04
+ 0-02
-0-08
Mean . . .
0-09068 +0-00016*
rhere
M = 3395-80,
ms = 6-489,
R = 20-599,
E= 1-01843(17-0.),
Column I. = date of experiment,
„ II. = n (number of standard cells),
„ HI. = / (seconds),
„ IV. = time over first interval,
., V. = radiation correction on range,
„ VI. = " rise above " in degrees Pt,
,, VII. = true range,
„ VIII. = specific heat,
„ IX. = percentage difference from mean.
* The value 0-09068 for S is obtained on the assumption that 00 coincides with 0 bridge- wire reading;
if, however, the balancing point was at + 0 • 1 bridge wire reading, and the range from - 9 to +8 bridge
wire, then, on account of radiation gain, the above value of S requires a correction of + 0 • 00005.
At the time these experiments were made we did not realise the importance of this correction and,
conaequently, did not determine the balancing point on the bridge with sufficient care (see p. 151).
In our rough notes made at the time we have values ranging from +0- 1 to +0'4 bridge wire.
Applying the "intersection method" (see Section X. (2)) to the above experiments, omitting the first
two or three transits in each case, we find from the calculated value of 0N that the balancing point should
be at +0-33 bridge wire. If we assume this value, the correction on S is +0-00016.
Hence
8 = 0-09084.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 149
bridge-wire division could be calculated, and also tan <f>, the slope of the resulting
straight line obtained by plotting dd/dt against 6.
If there were no losses or gains by radiation, the resulting lines would be
horizontal
As the rate of rise due to radiation depends solely on the difference of temperature
between the metal and the surroundings, the lines representing the observed values
of SO/St for the various rates of electrical supply have all the same inclination to the
horizontal, within the limits of experimental error.
The equation of the line representing an experiment, where n standard cells are
Iwlanced at the ends of the heating coil, is seen to be
* +<r(8 + 8 -8)= n>E>
where
$8/&t is the observed rate of rise,
8 is the temperature indicated by the thermometer,
80 is the temperature of surrounding envelope,
8, is the lag of the observed temperature for the particular rate behind the
temperature of the " radiating " surface.
(The determination of this lag is discussed below.)
Hence, by dividing throughout by n3, we have
:3-? + 5<«H-4)-
JR(MS+m.)'
The right-hand side would represent the rate of rise due to the electrical supply
with a potential difference of one standard cell
i $6
Hence, if we can determine the particular value of — . — at the temperature which
n ot
we denote by 0N, when the second term of the equation vanishes, we have the rate
of rise due to the electrical supply only.
1 88
Plotting — . — against the observed temperature due to the various values of n,
n 6t
we obtain a series of straight lines whose tangents vary inversely as n*.
Now, for each experiment thus plotted, there is a certain point on the line where
1 Xfi V*
- . -- represents TTt ... — r alone, and this would correspond to the temperature
n" St JR(MS+mx)
0N at which there are no losses or gains by radiation, i.e., when the mean temperature
of the surface subject to radiation is coincident with the temperature of the
surroundings. As the co-ordinates of this point are the same for all rates, the lines
would intersect at one point if either the observed 6 was the actual temperature
of the " radiating " surface, or the lag was constant for all.
, -„, DR. E. H. GRIFFITHS AND MR. EZER GRIFFITHS ON THE
The dotted lines in fig. 7 represent a typical case— that of copper at 0° C. with
thermometer AA.
It will l» noticed that the lines representing the higher rates of supply are
markedly to the left of those obtained from the lesser values of n, indicating that the
" lag " increases with the rate of supply, as might be expected.
\
I
-•1°
'x \ XK, **•< 389-
\ \
\ \
\ \
\ \
\ \
v \
s \
Fig. 7.
A. study of the " total heat " experiments led us to the conclusion that the " rise
above " was intimately connected with this " lag." Although the entire " rise above "
on switching off could not be solely due to thermometer lag, yet, as a first
approximation, it represents the superior limit.
Hence, by shifting each line parallel to itself to the right by the value of 6t
determined at the close of the experiment, we obtained the figure shown in full lines,
1 ^/l _^
the result, of course, being the same as if — . — had been plotted against 6 + Ot.
72- of
Owing to observational errors, the lines do not intersect in a single point, but
enclose a small area. In cases where a really satisfactory series of observations has
been obtained, however, the area of the triangle (when three experiments are
considered) is vanishingly small even when the results are plotted on such a scale
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 151
i SB
that 1 cm. vertically represents a change of 1 in 2000 in — . — and there are many
instances where the ordinatea of the vertices do not differ from the mean by more
than 1 part in 5000.
This, in our opinion, is the strongest evidence in support of the assumption that the
" rise al>ove " is practically equal to the " lag " to the degree of accuracy to which the
hori/.untal scale is required. We may state here that when plotting the results, we
used a scale such that 5 cm. abscisste represented 0°'l Pt, the vertical scale, of course,
Ix'iiig considerably greater, enabling the fifth figure in the value of -j . - to be
n ot
determined. In our earlier reductions, we ascertained the mean ordinate by reading
the ordinates of all the points of intersection ; for example, for 4 values of n, we
obtained 6 intersections. In cases, however, where the angle ^ resulting from two
experiments differed but by a small amount, as in the case of n = 7 and n = 8, a
slight error in the inclination of either line might cause a large displacement in the
point of intersection. We therefore adopted a method of reduction* which enabled
us to calculate the co-ordinates of the point such that, measured along the ordinate
passing through this point, the sum of the moments of inertia of the points of
intersection of the several lines with this ordinate is a minimum about this point.
Or, stated otherwise : — The point so calculated gives, by the method of least squares,
the most probable value of the ordinate of the point of intersection of all the lines
(for a typical example see p. 157).
A large number of determinations of the specific heats of Cu at 0° C. were made
by both the " total heat " and the " intersection " methods (see Section XI.).
The correspondence between the final results obtained was remarkably close
(the differences in no case exceeding 1 in 1000), and indicated the validity of both
methods. Having satisfied ourselves on this point, we adopted the latter method for
all our remaining experiments, as it avoided the following cause of difficulty and
delay which was xmavoidable in the former.
The removal of our metal block and its replacement by another was a lengthy
business, requiring considerable care, as all the soldered joints in the various electrical
circuits had to be separated and remade, the brass case removed and opened, Ac.
It was not possible to complete the operation in less than several hours, and the
temperature of the tank necessarily suffered some alteration in the process. On re-
establishing the system, small consequential changes in the balancing point on the
bridge might have occurred, or, at all events, the absence of any such changes had to
be ascertained. Thus, it was necessary to allow time for the newly inserted block to
settle to the tank temperature, and, as its approach to that temperature was slow and
asymptotic, at least a day or two had to elapse before the " zero " point could be
ascertained with certainty. The im]x>rtance of this matter is indicated by the fact
* For this suggestion we are indebted to Mr. G. M. CLARKE, M.A.
132 DR. K H CKIFFITHS AND MR. EZER GRIFFITHS ON THE
that an error of O'l bridge-wire division (= 0°'007 Pt) in the estimation of the zero
I>oiiit would affect conclusions derived from an experiment of average length by the
total heat method by (in the case of Cu, for example) 5 parts in 9000. In our
earlier " total heat " experiments we had not realized the importance of this zero
reading, and this no doubt is the cause of certain discrepancies.
The position of the zero point was, however, of little importance when the
intersection method was adopted, for so long as the temperature of the reference
block remained unchanged, the effect of any alteration in the zero point was self-
eliminated.
The method of reduction is shown by one example, namely, that of copper at 0° C.,
with thermometers AA, BB.
The only reason which has guided us in the selection of this out of the 48 similar
groups, is that it happens to be first of the groups given in Table XI. The large
amount of arithmetic involved in the reduction of our observations is well illustrated
by this example.
Explanation of Tables.
n = number of standard cells balanced on heating coil.
Column I., bridge readings. — The successive points on bridge wire across which
transits were taken.
Column II. (t ). — Times of transit from chronograph tape.
Column III. (St). — Interval between successive transits. (If transits observed every
l£ bridge wire, as in Experiment IV., then St for 1 bridge wire calculated.)
Column IV. (Se). — Value in Pt degree of bridge- wire division corresponding to St.
Column V. (^ x 107).
\ot I
Column VI. (0). — Temperature at mid-point of SO, measured from centre of bridge
wire.
Column VII.— The letters denote the values of Sd/St taken in pairs, for the purpose
of obtaining the slope of the line.
Column VIII. —Change in S8/St for equal intervals of temperature.
CAPACITY FOR IIKAT OF MKTALS AT DII-TKKKNT TKMI'KKATrKKS.
153
TABLE HI. — Experiment I., June 3, 1912. Number Standard Cells, 5.
I. II.
III.
IV.
V.
VL
VII.
VIII.
Bridge
readings.
/.
&l.
86x10*.
gxlO,
e.
8*A MA.
"S~ ~ fc~'
«A «V
215-25
8-5
286-90
71
7-2820
10163
-0-657
A
7-6
369-45
7 -2 •;..-,
7-2845
10041
-0-584
B
A -A'
357
6-5
48S'M
M-75
7 • 2882
10018
-0-511
C
5-5
504-95
72-75
7-2901
10021
-0-438
D
B-IT
381
4-5
578-75
73-80
7-2874
M7B
-0-365
E
3-5
652-75
74-00
7-2742
9830
-0-292
F
c-cr
357
2-5
726-55
73-80
7-2736
9994
-0-219
Q
1-5
801-20
74-65
7-2902
9766
-0-146
H
D-IX
353
0-5
875-60
74-40
7-2968
MQ6
-0-073
A'
+ 0-5
951-15
75-55
7-2978
9660
0
B-
E-ff
311
1-5
1026-70
75-55
7-2990
9661
+ 0-073
G
2-5
1102-20
75-50
7-3018
1I66M
+ 0-146
V
F-P
304
3-5
1178-80
76-60
7-3-.'30 !i. •><;:!
+ 0-L'l'.i
K
4-5 1255-90 77-10
7-34-J- 95L>6
+ 0-292
r
<;-({'
413
5-6 • 1332-50 76 -CO 7-3390 '.<:,* \
+ 0-365
G'
6-5
1409-90 77-40
7-3106 9446
+ 0-438
H H-H'
320
f
350
Mean . . .
9789 at -0°-110 C.
Mean . . <
for difference
I
of 0'-584C.
TABLE IV. — Experiment II., June 1, 1912. Number Standard Cells, 4.
I
L
II.
III.
IV.
V.
VI.
VII.
VIII.
Bridge
readings.
/.
s/.
80 x 10s.
£*'<"•
</.
S0A S0A.
*A *A '
-9-5
0-75
-8-5
111-30
110-55
7-2820 6587
-0-657 A
-7-5
222-00
110-70
7 • 2845 6580
-0-584 B
A -A'
372
-6-5
334-00
112-00
7-2882 6507
-0-511 C
-6-5
447-10
113-10
7-2901
6446
-0-438 D
B-B'
327
4-5
561-50
114-40
7-2874
6370
-0-365 E
3-5
675-20
113-70
7-2742
6398
-0-292 F
C-C'
337
2-5 789-75
114-55
7-2736
6350
-0-219 G
1-5 906-10
116-35
7-2902 6266
-0-146 H
D-D'
361
0-5 1023-50
117-40
7-2958 6214
-0-073
A'
+ 0-5
1140-20
116-70
7-2978 6253
0
B'
E-E'
280
1-6
1258-50
118-30
7-2990 6170
+ 0-073 C'
2-5
1378-60
120-00
7-3018 6085
+ 0-146 D'
F-F
332
3-5
1498-75
120-25
7-3230
6090
+ 0-219 E'
4-5
1619-80
121-05
7-3422 6065
+ 0-292 F
G-G'
329
5-5
1741-70
121-90
7-3390 6021
+ 0-365 G'
6-5
1864-00
122-30
7-3106 5978
+ 0-438 H'
H-H'
288
r
328
Mean . . . 6290 at -0°-110C.
Mean . .<
for difference
I
of 0'-584C.
i
VOL. ivxiii. — A.
154 |,!;. K. H. GRIFFITHS AND MR. EZER GRIFFITHS ON TIIK
r
TABLE V.— Experiment III., June 2, 1912. Number Standard Cells, 6.
L II.
III.
IV.
V.
VI.
VII.
VIIL
Bridge
tt
80x10*
%*™-
e.
se. _ SOA,
readings.
5
«A "SI.
-9-5
151-65
-8-5
202-00
50-35
7-2820
14463
-0-657
A
-7-6
252-90
50-90
7-2845
14311
-0-584
B
A -A'
513
-6-5
303-90
51-00
7-2882
14291
-0-511
C
-5-5
354-80
50-90
7-2901
14322
-0-438
D
B-B'
358
-4-5
406-40
51-60
7-2874
14123
-0-365
E
-3-5
457-90
51-50
7-2742
14125
-0-292
F
C-C'
348
-2-5
509-20
61-30
7-2736
14178
-0-219
G
-1-5
561-00
51-80
7-2902
14074
-0-146
H
D-D'
427
-0-5
613-30
52-30
7-2958
13950
-0-073
A'
665-60
52-30
7-2978
13954
0
B'
E-E'
148
1-5
717-95
52-35
7-2990
13943
+ 0-073
C'
2-5
770-50
52-55
7-3018
13895
+ 0-146
D'
F-F
337
3-5
822-90
52-40
7-3230
13975
+ 0-219
E'
4-5
876-15
53-25
7-3422
13788
+ 0-292
F
G-G'
344
929-20
53-05
7-3390
13834
+ 0-365
G'
m
6-5
982-30
53-10
7-3106
13768
+ 0 • 438
H'
H-H'
306
r
348
Mean . . .
14062 at -0°-110C.
Mean . .<
for difference
I
of 0°-584C.
TABLE VI— Experiment IV., June 2, 1912. Number Standard Cells, 7.
I.
II.
III.
rv. v.
VI.
VII.
VIII.
Bridge
•i
\
f\
0SA S0A,
readings.
.
°2' ^X
V.
W^'-st'
-9-6
75-55
8-5
112-75
37-20
7-2820
19575
-0-657
A
7-5
150-50
37-75
7-2845
19297
-0-584
B 1 A- A'
526
6-5
188-00
37-50
7-2882
19435
-0-511
C
5-5
225-70
37-70
7-2901
19337
-0-438
D
B-B'
292
4-5
263-70
38-00
7-2874
19177
-0-365
E
3-5
301-55
37-85
7-2742
19219
-0-292
F
C-C'
278
2-5
339-55
38-00
7-2736
19141
-0-219
G
1-5
377-70
38-15
7-2902
19109
-0-146
H
D-D'
469
0-5
416-00
38-30
7-2958
19049
-0-073
A'
+ 0-5
454-40
38-40
7-2978
19005
0
B'
E-E'
255
1-5 492-50
38-10
7-2990
19157
+ 0-073
C'
2-5
3-5
531-20
569-90
38-70
38-70
7-3018 18868
7-3230 18923
+ 0-146
+ 0-219
D'
E'
F-F
295
4-5
5-5
608-70 38-80
647-60 39-90
7-3422
7-3390
18923
18866
+ 0-29.' F
+ 0-365 G'
G-G'
275
6-5 686-65 39-05
7-3106
18721
+ 0-438 H' H-H'
388
19113 at -0°-110 C.
347
Mean . .
for difference
of 0°-584C.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 155
TABLK VII. — Experiment V., .Inn.- •_', l<>12. Number Standard Cells, 8.
I
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
Bridge
readings.
/.
$St x 10*.
80 x 10s.
£«'<"•
e.
S0A S0A.
S/A «A. '
-9-0
86-20
7-5
l-J'.i -50
43-30
7-2836
25228
-0-602
A
6-0
I7l>-80
43-30 7-2890 L-.VJ50
-0-493
B
A -A'
269
4-5
216-60 43-80 7-2892 24963
-0-383
C
3-0
260-10 4:1 -50
7-2718 25075
-0-274
D
I: l;
367
1-5
303-85
43-75
7-2875 24985
-0-164
E
0
347-70
43-85
7-2964 24959
-0-055
A'
C-C'
372
+ 1-5
391-70
44-00
7-2985 24882
+ 0-055
B'
•
3-0
436 • iT>
44-55
7-3034 24591
+ 0-164
cr
D-D'
166
!•:.
480-45
11-20
7-3400 24909
+ 0-274
D'
6-0
569-50
11-65
7-3320 24631
+ 0-383 E'
E-E'
354
f
306
Mean . . .
24947 at + <T-110C.
Mean . .<
for difference
of Oe-547C.
In Experiment L, 5 cells, jp~ ^/0A-0A' == 350/0*584 = 59'9x 10~7 for 00>1 C.
II., 4 „ „ „ „ = 382/0-584 = 56'2 „ „
HI., 6 „ „ „ „ = 348/0-584 = 59'G „ „
IV., 7 „ „ „ „ = 347/0-584 = 59'4 „ „
V., 8 „ „ „ „ = 306/0-547 = 55'9 „ „
Hence mean difference for change of 0°'l Pt = 58 X 10~7.
Reducing the mean S6/St in each experiment from — 0°"110 C. to 0° C. by this mean
tangent and then calculating the values at — 0°'l C. and +0°'l C., we obtain the
following results : —
Experi-
80
80
1 80
ment
„. j- at 0° C.
— at*- 0 ' 1 C.
-r— at +0 "1C.
—( j- ] x 10".
—1 5— I x 10".
No.
of
«
-,c.
I.
1
5 9725
'.'7-:;
9667
39132
38668
II.
4 6226
6284
6168
39275
38550
III.
6 13998
14056
13940
39044
38722
IV.
7 19049
19107
18991
88994
38757
V.
8 24883
24941
24825
38970
38789
x 2
156
Ml;. K. II. CKIFFITHS AND Ml;. KXKK GRIFFITHS ON THE
T\\" • •< >rrections are necessary to make the values off- -) comparable —
\n3 St /
(1) The change in resistance by change of" current ;
(2) The departure of the mean E.M.F. of the group of standard cells used from
the standards.
1 $0
The correction to — — for these is designated by Cd cell and £R, the experimental
/. . /
results are now arranged in order of n.
Experiment
No.
n.
Cd cell.
SR.
i(I) *10-
n^ot /»t -Wi C.
«a\^/at +o°-ic.
II.
i
+ 10
+ 3
39288
38563
I.
5
+ 10
+ 5
39147
38683
III.
6
+ 9
+ 7
39060
38773
IV.
7
+ 6
+ 10
39010
38773
V.
8
+ 5
+ 13
38988
38807
|
Before proceeding with the next step — the determination of the mean point of
intersection either graphically or by calculation— we require the value of the " rise
above " in each experiment.
As an example of the method, we quote that of Experiment IV., June 2, 1912.
Number standard cells, 7.
Contact maker of bridge set up 0°'088 Pt beyond switching-off point.
TABLE VIII.
Time (from I ,,.,,.
instant of : Mllllmetr<>
switching , Jca'?
Ojj.v ' deflection.
0 x 10s.
0 after
radiation
correction
xlO3.
Time (from
instant of
switching
off).
Millimetre
scale
deflection.
0 x 10s.
0 after
radiation
correction
xlO».
149 27
160 30
172 32
184 33
195 29
209 27
223 23
234 17
245 11
257 9
267 4
275 1
283 - 5
291 - 8
302 - 14
313 -19
2-0
2-2
2-3
2-4
2-1
2-0
1-7
1-2
0-8
0-6
0-3
0-1
-0-4
-0-6
-1-0
-1-4
7-4
8-0
8-5
9-0
9-1
9-5
9-7
9-6
9-6
9-8
9-9
10-0
9-8
9-9
9-9
9-9
324
335
348
362
375
390
405
422
439
453
467
483
498
512
528
546
- 24
- 28
- 35
- 43
- 52
- 58
- 66
• 74
- 82
- 91
- 97
-104
-111
-118
-128
-135
-1-8
-2-0
-2-6
-3-1
-3-8
-4-2
-4-8
-5-4
-6-0
-6-6
-7-1
-7-6
-8-1
-8-6
-9-3
-9-8
9-9
10-1
9-9
9-9
9-7
9-8
9-8
9-8
9-8
9-7
9-7
9-8
9-8
9-8
9-7
9-9
— — — — — ^
— — — ^~—. __
CAPACITY FOR HEAT OF MKTAI.S AT DIFFKKI.NT TKMI'ERATfKKS.
157
Hence
"Total rise above" = 0'088 + 0'0099 = 0°'0979 C.
'I'lie figure on p. 145 represents the above data.*
TABLE IX. — Calculation of the Co-ordinates of the " Most Probable Common
Point of Intersection."
Experiment
No.
n.
(.J^^xlOi.
i($_xH,.
Shift
xlO.
Equation of line after
shift applied.
II.
4
39288
38563
0-334
y - 39046 - 362^
I.
5
39147
88683
0-512
y = 39034 - 232-e
III.
6
39060
38773
0-729 >i - 39021 - 144/
IV.
7
39010
38773
0-979
y = 39008 -118*
V.
8
38988
38807
1-254
y = 39011- 9Lr
Equation of mean line u = 39024 — 189jr. (\\
Multiplying each term by the coefficient of x in the same equation
3620+ 131044s = 141346,
2320+ 53824o- = 90559,
1440+ 20736Z = 56190,
13924J? = 46029,
8281-r = 35500,
Mean
0+240sc = 39031 (2)
Solving equations (l) and (2) for x and 0 we have the co-ordinates of the required
point
x= +0-137,
0 = 38998.
Expressing x and y in absolute measure we have
• fa: x O'l = degrees Pt,
0x10-"=-,^.
>M* ft
Hence
n
0N = +0°-0137 Pt,
I^N = 38998 XlO-9.
n1 tit
* This figure indicates how the " rise above corrected " can be obtained more simply by the prolongation
of n straight line.
158 I'll. V- II. «;i;IFFITIIS AND MR. K/F.R GRIFFITHS ON THE
\ ^iniill uncertainty in tho value of 0S has but little effect on —j-^', for example,
an error of (VI in x would only produce an error of 1 in 2000 in the above value of y.
A correction of —3 has to be applied to y for the clock rate, which was a losing one
of 0'05 sec. per 1000.
The distribution of the results of the individual experiments about the " most
probable point of intersection ' may be determined by solving the equation of each
line for its intersection point with the ordinate through x = +0'137.
TABLK X
I.
II.
III.
IV.
V.
VI.
Experiment
No.
«.
.'/
(mean).
.'/
(calculated).
Difference.
(Difference)2.
II.
4
f 38993
-2
4
I.
III.
IV.
V.
5
6
7
8
,
38995
38999
38998
38989
38996
+ 4
+ 3
-6
+ 1
16
9
36
1
Total . . .
66
Probable observational error = ± § */-. 2 (difference)2 = +g
V No. experiments —1
Coefficient of variation per cent. = ±0'01.
In Tables XL to XVIII., pp. 161 to 169, we give the value of y (calculated) for
h group ,„ Column VII. ; the error per cent. (Column VIII.) being the coefficient
of variation obtained as above.
^ig. 7, p. 150, represents the above group of experiments ; the most probable point
tion obtained by calculation is shown by a large cross.
Attention may be drawn to the fact that no " smoothed curves" have been used in
the above reduction.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 159
r •mjrcil
[ms + MSJ —
1 CiHv fj
n» dt JR/
•
M = mass copper block -f case = 3392'37
ms = thermal capacity of oil, quartz, glass and solder = 4'898,
K = resistance of coil = 20'599,
E = E.M.F. standard cell at 15° (!. == 1 '01 848,
f=d (U)(d (Ft) at 0° C. = 0'98480.
Hence
S = 0-09094.
NOTE. / = 3 (#)/? (Pt). The values of 30/3 Pt at temperature fl are obtained from
('IIAITIMS Mini HAKKKU'S tables, ' Phil. Trans.,' vol. l!)4, p. 114. Assuming S = T54.
SECTION XI.
Experimental Re.mlts. Preliminary Experiments.
A considerable number of preliminary experiments were performed with a view of
testing the apparatus employed and deciding on the most suitable conditions. Some
were carried out with a constantan heating coil of 10 ohms resistance, which was
replaced by a manganin coil of wider section and greater resistance.
A large number of experiments were performed with silvered vacuum vessels
interposed between the metal blocks and the brass cases. The results obtained with
different rates of energy supply were discordant. The faster the rate of rise, the
lower the value found for the resulting specific heat. These differences were roughly
proportional to the duration of the heating ; the range being practically the same
in ail.
The source of this error we traced to the effect of radiation, &c., on the inner walls
of the vacuum vessel. This surface received heat by radiation from the block and as
it parted with the heat but slowly, its temperature rose with that of the block to an
extent dependent on the rate of increase of temperature of the metal.
After the removal of the vacuum vessels, the loss or gain by radiation was
dependent on 6— 6a only, as the surrounding walls were now those washed by the tank
water and remained at a constant temperature. Our anxiety to minimize loss or gain
of heat from external sources by the interposition of these flasks had led us, when
designing the apparatus, to regard the insertion of the non-conducting walls as
: this precaution, however, was a cause of much loss of time and labour.
160 DR K. H. UKIFFITHS AND MR. EZER GRIFFITHS ON THE
Explanation of the Tables.
Column I. — The temperature at which experiments were performed.
During our experiments at 0° C. we changed both thermometers and lids ; we have,
therefore, in this column indicated the thermometers and lid used.
Letters AA and AA', indicate the thermometers referred to in Section III.
Letters L,, the lighter, and L,, the heavier lid (see Section VI.).
Where no indication is given, the thermometer used was AA', and lid, La.
Column II. — The dates on which the series were performed is given to indicate the
results obtained on repetition after lapse of time.
Column III. — The number of transits denotes the number of observations of $8/St
obtained during the experiment.
Column IV. — No. Cd Cells. — The number of standard Weston cells in series, whose
E.M.F. was balanced at the ends of the heating coil.
Column V. " Rise Above." — This was determined at the close of each experiment.
The line representing an experiment was shifted horizontally by this amount.
1 <W
Column VI. Tangents (Abscissae 0°'l Pt). — The slope of the line — • -j- with tempe-
IV 0*'
rature H as abscissa.
Absolute value = number in Column x 10~9.
Column VII. — 'The points of intersection of the lines of various rates with the
ordinate through 0N (see p. 158).
Absolute value = number in Column x 10~9.
Column VIII. — The probable observational error per cent, of the group.
Column IX. — The data required for the reduction. It will be noticed that the
mass of the metal block has in some cases changed during the course of the experi-
ment, owing to certain alterations such as enlarging holes, &c., which were found
necessary.
ms denotes the capacity for heat of copper case and the group of subsidiary
substances, or of the latter only when the block itself is copper.
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPER ATUKKS.
161
TABLB XI. — Copper.
L
n.
III.
IV.
V.
VL
VII.
VIII
IX.
Tem-
perature
tank.
Date.
No.
transits.
No.Cd
cells,
n.
Rise
above
•Pt
Tangents
(abtehM
0*-1 C.).
i dev
n?~ST
Error
per
cent
•o.
0
AA
L,
1/6/12
3/6/12
2/6/12
2/6/12
2/6/12
16
16
16
16
10
4
5
6
7
8
0-033
0-051
0-073
0-098
0-125
362
232
144
118
91
38993
38999
38998
38989
38996
0-01
R - 20-599
E- 1 -01848 (15' C.)
M = 3392-37
nu - 4-898
— n-nonaj
38995
0
A A'
L*
25/7/U
23/7/12
24/7/12
25/7/12
23/7/12
24/7/12
16
16
16
16
10
10
5
6
6
7
8
8
0-039
0-057
0-057
0-075
0-097
0-098
237
165
16!)
121
93
93
38755
38754
38743
38737
387 1 1 mean of two
38741
0-03
R- 20-609
E- 1 -01838 (18° C.)
M = 3409-18
ms = 5-577
8 = 0-09079
0
AA'
L*
3/12/12
3/12/12
3/12/12
3/12/12
If,
16
16
12
4
5
6
7
0-037
0-058
0-081
0-107
449
244
169
124
38631
38682
38654
38642
0-05
R- 20-620
E= 1-01842 (17* C.)
M = 3409-05
ms = 5-533
8 = 0-09098
38652
0
AA'
L!
5/12/12
16
16
12
10
6
5
6
7
4
4
0-059
0-085
0-114
0-040
0-040
235
164
120
367
367
39052
38964
38931
38969 mean of two
0-08
R = 20-620
E = 1-01842 (17° C.)
M = 3409-05
ms = 4-789
80-09088
38979
28-42
2/8/12
16
16
10
6
7
8
0-053
0-071
0-091
189
138
105
37916
37877
37846
0-06
R= 20-627
E= 1-01847 (15° C.)
M = 3409-18
>„, 5-819
8 = 0-09230
37880
63-5
1/9/12
Set of three experiments of little
value owing to leakage of tank-heating
circuit affecting galvanometer.
36763
8 = 0-09365
67-32
13/9/12
13/9/12
16/9/12
16/9/12
13/9/12
13/9/12
20
20
20
10
13
13
4
6
7
8
7
8
0-027
0-064
0-086
0-110
0-084
0-114
427
191
139
108
139
108
36632
36655
36630 mean 7
36629 mean 8
36636
0-02
R- 20-635
E- 1-0185(14"-5C.)
M- 3409 -10
ms = 6-313
8 = 0-09387
97-4
11/10/12
16
16
12
8
5
6
7
8
0-042
0-057
0-077
0-103
330
229
151
100
35802
35774
35825
35783
35796
0-04
R= 20-621
E= 1 -01840 (17'-5C.)
M = 3409-05
nu = 6-598
8 = 0-09520
VOL. ccxin. — A.
I'K. K H. GRIFFITHS A XI) MR. KZER GRIFFITHS ON THE
TABLE XII. — Aluminium.
I.
II. III.
IV.
V.
VI.
VII.
vm. ix.
Tem-
perature
tank.
Date.
No.
transits.
No. (k
cells,
ft.
1 Rise
above
"Pt.
Tangents
(abscissae
0°-1 C.).
i dev
n* ilt
Error
per
cent.
•c.
30/6/12 19
6
0-111
244
54722 R = 20-599
0 19 7
0-148 178
54747 E = 1-01845 ne'e.}
AA
12
8
0-191
137
54720
0-02 M = 954-342
ms = 23-478
L,
54730
8 = 0-20937
14/7/12
19
6
0-076
243
54022
19
7
0-102
177
54005 R = 20-610
0
19/7/12
12
12
8
8
0-131
0-130
137
137
54041 0-07 E = 1-01838 (18° C.)
54139 M = 954-342
AA'
19
7
0-102
177
54144
ms = 25-764
19
6
0-077
243
54080
L*
20/7/12
13
7
0-102
177
54144
13
7
0-102
177
54143
8 = 0-20957
19
5
0-054
349
54108
Jl/7/12
13
8
0-130
137
54095
21
6
0-077
243
54114
54094
28-36
6/8/12
19
12
12
6
7
7
0-075
o-ioo
0-100
239
188
188
52345
52344
52350
R = 20-627
E = 1-01845 (16° C.)
0-02 M = 954-342
19
5
0-053
344
52353
ms = 26 • 207
52348
8 = 0-21471
51
25/1/13
25/1/13
26/1/13
18
12
9
18
18
12
6
7
8
5
6
7
0-073
0-100
0-130
0-056
0-073
0-100
254
187
143
366
254
187
51058
51076
51006
51140
51039
51130
R = 20-643
E = 1-01842 (16°-7 C.)
0-04 M = 954-00
ms = 26-772
18
5
0-056
366
50989
8 = 0 21842
51064
3/10/12
18
4
0-034
670
48936
97-48
18
12
9
13
10
5
6
7
6
7
0-052
0-073
0-101
0-073
0-101
431
299
220
299
220
48939
48973
48828
48959
49018
R= 20-621
E= 1 -01838 (18° C.)
0-08 M = 954-00
ms= 27-632
8 = 0-22482
-
—
~
48942
CAPACITY KOI; UKAT OF MKTAI.S AT DIFFKKKNT TK.MI-KKATI u>
I.;:;
TABLK XIII.— Iron (Ingot).
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
Tem-
perature
tank.
Date.
No.
transits.
No. Cd
cells,
n.
Rise
aliove
*Pt.
TangenU
(uWissa-
0°-1 C.).
1 </0s
n* (// '
Error
per
cent.
"
°0.
12
7
0-112 131
38405
R = 20-620
0 24/11/12
16
6
0-083
175
38370
E- 1-0184(17'-5C.)
24/11/12
16
5
0-057
377
38402
0-05
M = 2798-67
25/11/12
20
6
0-083
175
38339
tw- 25-767
18
4
0-037
397
38354
8 — 0-1045
•
38374
12/2/13
17
5
0-026
257
38502
R = 20-625
0
18
6
0-046
177
38525
0-03
E = 1-0184 (17° -5 C.)
18
7
0-062
131
38494
M = 2781-22
m» = 25-727
38507
8 = 0-1046
23/2/13
18
7
0-059
128
37958
R = 20-634
10
18
6
0-044
173
37980
0-02
E = 1-01854 (13° C.)
18
5
0-032
251
37959
M = 2781-22
m* = 25-889
37966
8 = 0-1059
24/2/13
18
6
0-032
257
37948
9-9
18
6
0-047
177
37807
All same as 23/2/13
10
7
0-060 131
37923
0-11
12
8
0-081 100
37938
8 = 0-1061
37904
31/1/13
17
6
0-040 171 371C7
R = 20-638
20-5
31/1/13
12
7
0-066 126 37168
0-06
E= 1-01843 (16'-5C.)
1/2/13
38
4
0-018 379 37176
M = 2781-22
12
8
0-074 96 37182
ms = 20-638
37171
8 = 0-1078
4/2/13
12
7
0-060
149
37170
Constanta same as for
21-5
4/2/13
17
5
0-029 287 37159 0'03
20°-5C
4/2/13
18
6
0-042 269 37192
— 0-1077
37174
Y 2
164
DR. E. H. GRIFFITHS AND MR EZER GRIFFITHS ON THE
TABLE XIII. (continued).
I.
II.
1
HI.
IV.
V.
VI. VII.
VIII.
IX.
Tem-
perature
tank.
Date.
No.
transits.
No. Cd
cells,
H.
Rise
above
°Pt.
Tangents
(abscissae
O'-IC.).
1 d0s
tfW
Error
per
cent.
•c.
24-5
2/3/13
18
6
0-045
213
37067
R = 20-642
18
18
5
7
0-028
0-055
307
156
37073
37080
0-02
E = 1-0185 (15° C.)
M = 2781-22
ms = 26-231
37073
8 = 0-1080
19/1/13
18
7
0-057
128
35705
R = 20-643
50-3
18
18
6
5
0-042
0-029
175
254
35729
35701
0-03 E = 1-0184 (16° C.)
M = 2798-67
ms = 26-938
35712
8 = 0-1105
66-3
7/12/12
18
18
18
4
5
6
0-029
0-048
0-065
756
483
339
35363
35167
35410
0-24
R = 20-635
E = 1-0184 (17° C.)
M = 2798-67
ms = 27-299
35313
8 = 0-1112
97-5
9/10/12
16
16
16
12
4
5
6
7
0-027
0-037
0-051
0-069
883
565
395
289
34248
34226
34238
34243
0-02
R = 20-622
E= 1 -01838 (18° C.)
M = 2798-67
ms = 27-798
I
34239
8 = 0-1137
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES.
165
TABLE XIV.— Zinc.
L
II.
III.
IV.
V.
VI.
VIL
VIIL
IX.
Tem-
perature
tank
Date.
No.
transits.
No.Cd
cells,
n.
Rise
above
*Pt
Tangents
(ah* !--.i-
0--1C.).
\de,
«,~3T
Error
per
cent
"C.
27/11/12
16
5
0-062
300
47389
R = 20-620
0
27/11/12
12
6 0-090 209
47314
E- 1-0184 (17* C.)
27/11/13
10 7 0 120 164 47367
0-07
M = 2538-40
27/11/12
16 4 0-040 471 47307
•V* 25-644
28/11/12
12
7 0-120 154
47271
8 = 0-09150
47330
9/2/13
18
7 0-075
143
47219
I! L'o-61!!
0
18
6
0-056
195
47170
0-04
E- 1 -0184 (17' C.)
18
5
0-039
281
47209
M = 2538-4
nu = 25-582
47199
8 = 0-09180
6/2/13
37
5
0-039
295
4G335
R - 20-638
18
C 0-055
100
46383
E= 1-0184 (16* C.)
21-6
12
7 0-075
149
46348
0-03
M = 2538-4
9
8
0-099
116
46330
w« = 26-231
46349
8 = 0-09265
18
6
0-055
205
45210
R = 20-643
22/1/13
18
7 0-077
150
45149
E = 1-0184 (16' C.)
50-5
12
8
0-100
115
45196
0-04
M = 2538-40
18
5
0-038
295
45194
ma = 26-756
18
7
0-077
190
45215
14
8
o-ioo
145
45224
8 = 0-09412
*
45198
21/10/12
17
4
0-029
599
43944
R = 20-621
21/10/12
18
5 0-038
382 43982
E = 1-01836 (18° C.)
97-4
21/10/12
20
4
0-029
599
43926
M = 2538-40
21/10/12
12
6
0-055
265
43980
0-06
nu = 27-607
21/10/12
8
7
0-074
194
43947
12
6
0-055
265
43876
8 = 0-09534
43942
16/2/13
17
5
0-039
606
44064
R = 20-6266
97-4
18
6
0-065
420 44105
0-05
E= 1-0185 (14° -5 C.)
18
7
0-073
308
44055
M = 2538-40
ing = 27-488
44076
8 = 0-09507
18/2 '13
18
6
0-051
505
43401
B= 20-609
18
5
0-038
725
43446
E= 1-0185 (13-5' C.)
123-4
18
7
0-070
370
43582
0-18 M- 2538-40
12
8
0-088
283
43305
»w = 27-886
43434
8 - 0-09570
166
DR. K. H. GRIFFITHS AND MR EZER GRIFFITHS ON THE
TABLE XV.— Silver.
I.
II.
j m
IV.
V. VI.
VII.
VIII.
IX.
Tem-
perature
tank.
Date.
No.
transits.
i
No. Cd
cells,
n.
Rise
above
*Pt.
Tangents
(abscissae
O'-IC.).
1 d0s
1? dt'
Error
per
cent.
•o.
27/5/12
18
4
0-048
496
52897
R = 20-599
0
AA
18
8
6
8
0-106
0-183
220
124
52926
52887
0-02
E = 1-0185 (14° C.)
M = 3733-10
ms = 23-443
L,
52903
S = 0 • 05560
3/8/12
19
5
0-054
339
51302
R = 20-627
28-4
19
12
6
7
0-078
0-102
235
172
51273
51304
0-02
E= 1-0184 (16° C.)
M = 3733-10
ms = 26-188
51293
8 = 0-05613
18/9/12
18
4
0-042.
613
50001
18
6
0-092
272
50005
R = 20-635
67-41
12
9
19/9/12 9
7
8
8
0-123 201
0-158 153
0-159 153
50017
49908
49986
E = 1-01848 (15° C.)
M = 3733-10
ms = 27-056
12
7
0-122 201
50145
18
6
0-090 273
50024 0 • 07
20/9/12 20
21/9/12 10
6
8
0-090 273
0-159 153
49963
50016
8=0-05680
10
8
0-159 153
50013
10
7
0-123 201
50054
10
7
0-123
201
50009
50011
97-4
13/10/12
18
12
18
18
6
7
5
4
0-082
0-111
0-057
0-037
307
227
442
692
49098
48970
48991
49018
0-07
R = 20-621
E = 1-0184 (18° C.)
M = 3733-10
ms = 27-564
49019
8 = 0-05737
CAPACITY FUK HKAT OF METALS AT DIFKKKKVT TKMl'KUATI KKS. 1«7
TABLK XVI.— Cadmium.
I.
II.
ni.
IV.
V.
VI.
VII.
VIII.
IX.
Tem-
[>erature
tank.
Date.
No.
transits.
No. Cd
cells,
Rise
alx>ve
' Pt.
Tangents
O'-l C.).
*£
Error
per
cent.
•o.
12/5/12
19
4
0-053
595
63811
R - 20-599
0
19
6
0-117
266
63852*
0-03
E- 1-0186 (15° C.)
AA
10
8
0-202
149
63795
M - 3070-71
nu - 23-325
L,
63819*
8 = 0-05468
8/8/12
19
6
0-080
284
61660
R - 20-627
28-4
11
7
0-107
209
61626
E= 1-01847 (15* C.)
14
7
0-107
209
61629
0-02
M = 3070-71
19
5
0-068
410
61633
nu = 26-054
61637
8 = 0-06564
29/12/12
18
5
0-058
415
60193
R = 20-638
12
6
0-082
289
60254
E= 1-0184(17*0.)
54-5
8
7
0-113
213
60214 (mean)
M = 3070-71
30/12/12
18
3
0-019
116
60226
*•- 26-718
17
4
0-036
648
60229
0-02
10
7
0-113
213
8 = 0 05616
60223
16/10/12
18
4
0-036
784
58408
R = 20-621
97-64
18
5
0-061 502
58483
E = 1-01836 (18° -50.)
.
16
5
0-061 491
58303
M = 3070-71
11
6
0-085 349
58413
0-07
»w = 27-437
13
6
0-085 349 58404
58402
8 = 0-05714
* Omitting 6 cell experiments on the date 12/5/12, we hare from other two experiments
8 = 0-05475.
168
DR. E. H. CRIFFITHS AND MR. EZER GRIFFITHS ON THE
TABLE XVII.— Tin.
I.
II.
m.
IV.
V.
VI.
VII.
VIII.
IX.
Tem-
perature
tank.
Date.
No.
transits.
No. Cd
cells,
n.
Rise
above
Tangents
(abscissie
i do*
n* dt '
Error
per
cent.
.»
17/5/12
18
IK
5
4
0-085
0-055
466
728
75254
75209
R = 20-599
E= 1-01851 (14° C.)
U
5
0-085
466
75309
M = 2591-49
19/5/12
8
8
6
6
0-123
0-122
323
323
75166
75186
ms = 23 • 434
7
6
0-122
323
75136
0
AA
20/5/12
21/5/12
9
19
9
7
4
8
0-162
0-056
0-216
237
728
182
75250
75126
75240
0-05
•
14
5
0-086
466
75225
S = 0-05363
L,
22/5/12
9
9
8
7
0-216
0-163
182
237
75134
75240
26/5/12
9
9
8
8
0-217
0-216
182
182
75105
75233
75201
28-4
9/8/12
12
12
7
6
0-118
0-087
235
318
72122 (mean)
72081
R= 20-627
E = 1-0185 (16° C.)
19
5
0-062
459
72107
0-02
M = 2591-49
14
7
0-118
235
ms = 26-169
72103
S = 0-05465
2/1/13
16
4
0-038
793
17
4
0-038
793
70292 (mean)
R = 20-638
18
5
0-059
508
70308 ( „ )
E= 1-0184(17°-5C.)
53-9
12
6
0-089
352
70296 ( „ )
o-oi
M = 2591-49
3/1/13
9
7
0-120
258
70317
ms= 26-798
13
6
0-089
352
18
5
0-059
508
80-05549
70303
19/10/12
10
5
0-065
541
67819
K = 20-621
12
5
0-065
541
67809
E = 1-01836(18°-6C.)
97-6
9
6
0-088
375
67704
0-09
M = 2591-49
20/10/12 10
6
0-088
375 67603
ms = 27-580
18
4
0-042
844
67645
SO- 05690
67716
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES.
169
TABLE XVIIL— Lead.
I.
n.
III.
IV.
V.
VI
VII.
VIII.
IX
Tem-
perature
tank.
Date.
No. of
transits.
No. (M
cells,
«.
UN
above
•Pt.
Tangents
•te IMSJ
O'-IC.).
1 dO,
»*~3T
Error
por
. , !.I.
"0.
0
20/6/12
23/6/12
23/6/12
29/6/12
29/6/12
18
12
12
9
9
8
18
12
8
4
5
5
6
6
7
7
3
7
8
0-062
0-093
0-094
0-135
0-132
0-178
0-175
0-034
0-176
0-229
813
521
521
355
362
266
266
1443
291
222
84253
84462
84294
84577
84461
84504
84324
84550
84406
84679
0-09
R- 20-599
E- 1-0184 (16* C.)
M - 4016-56
mt - 23-442
S 0 03020
84441
28-38
5/8/12
19
12
19
19
5
6
4
5
0-060
0-074
0-038
0-052
508
371
835
508
81440
81231
81334
81238
0-08
R. 20-627
E - 1-0184 (16" C.)
M = 4016-56
ww- 26-148
81311
8 = 0-03053
51
23/12/12
17
18
18
12
20
12
5
4
3
6
5
6
0-065
0-041
0-026
0-093
0-065
0-093
B01
949
161
280
553
280
80008 (mean)
80118
79838
79808 (mean)
0-19
R - 20-639
E= 1-01841 (17" C.)
M = 4016-56
•»- 26-786
8 = 0-03073
79938
57
13/12/12
18
18
3
4
0-020
0-037
151
931
Intersection of
the two lines.
R = 20-638
E= 1-01840 (17-5° C.)
M = 4016-56
ms = 26-743
79710
8 = 0 03078
67-4
22/9/12
18
12
18
18
20
20
13
5
6
4
5
5
4
6
0-070
0-105
0-045
0-070
0-070
0-045
0-105
562
390
879
562
562
879
390
78743
78681
78561
78515
78579
78715
78603
78629
0-07
R = 20-635
E = 1-01846 (15-5' C.)
M - 4016-56
m* = 27-056
S 0 03102
97-45
15/10/12
18
18
12
9
10
13
S
4
5
6
6
5
0-025
0-044
0-067
0-097
0-097
0-067
177
990
633
441
441
633
77252
77532
77632
76991
77298
77448
77358
0-19
R - 20-621
E - 1-01838 (18* C.)
M = 4016-56
w.« = 27-587
8 = 0 03127
VOL. COXIII. — A.
,„;. ,.;. II. CIMFFITHS AND Ml!. KXKIJ CKIFHTHS ON THE
SECTION XII.
Summary of Results. Copper.
The validity of our methods was rigorously tested by the determinations of the
specific heat of copper at 0° C.
We have already, in the previous sections, discussed the various changes mad.
during the course of these experiments, and the table below summarises the results.
TABLE XIX.
Method.
No.
of experi-
ments.
Observed
error
per cent.
Remarks.
Specific heat.
12
0-03
Thermometers AB, CD (see
0-09084
Table II.)
3
0-03
Thermometers A A, BB
0-09095
" Intersection method " . .
4
0-05
Thermometers AA', BB' ; heavy
lid; Decembers, 1912
0-09098
" Intersection method " . .
5
0-08
Thermometers AA', BB' ; light
lid; Decembers, 1912
0-09088
" Intersection method " . .
6
0-03
Thermometers AA', BB' ; heavy
lid ; July 23-25
0-09079
" Intersection method " . .
5
o-oi
Thermometers AA, BB ; light
lid ; June 3
0-09094
" Intersection method " . .
12
0-04
Thermometers AB, CD. Inter-
section method applied to
1st group
0-09081
Giving equal weight to each group, we have
S0 = 0*09088 ±0'000047, i.e., probable error = 0'05 per cent.
In Tables XX. to XXVII. we summarise our final conclusions.
Messrs. Johnson and Matthey state that the previous treatment of all the metals,
except copper and iron, was as follows :—
''The cylinders in every instance were cast, and then allowed to cool, sub-
sequently being turned in a lathe, they were, not annealed."
CAPACITY FOR HEAT OF MFTAI.S AT DIFFERENT TEMPEKATI'IM >
171
The data supplied by the manufacturers indicate that the physical condition of the
iron is probably distinct from that of the other metals, and this may to some extent
account for the marked difference in the rate of change of its S and 0 curve over the
range 0" C. to 100° C., as compared with the remaining curves.
We are desirous of maintaining the iron in its present condition until we have
invest i<,';i»f<l its l*'li;i\ imir ut low temperatures, but we hope eventually to ascertain
the effect of careful annealing on this specimen.
TABLE XX. — -"Copper.
Weight, 3392 grms. Density, 8"922.
Temperature
0*C
28°-42C
63"-52C
67"-32C
97*-4C.
S (experimental value)
0-09088
0-09230
0-09366 t
0-09387
0-09521
1 MfTerence from curve per cent. . .
0
+ 0-09
-0-13
-0-07
0
S, = 0-09088 (H-0'0005341^ - 0'00000048(!!').
This copper was electrolytically deposited.
Mr. C. T. HEYCOOK writes as follows :—
" Cu = 99*95 per cent. Remaining 0"05 per cent, consists of Pb, Fe, and a very
little SiOj. You will 1x3 correct in stating that it is of high purity."
TABLE XXI. — Aluminium.
Weight, 954 grms. Density, 2704.
Temperature
0"C.
28°-35C.
Bl'-OC.
97'-48C.
S (experimental value)
0-20957
0-21471
0-21842
0-22482
Difference from curve per cent. . .
0
0
0
o
S, = 0-20957 (l + 0-0009161*-0'00000m3).
Messrs. Johnson and Matthey state : —
" Aluminium we have reason to believe to be exceptionally pure, say 99'90 per
cent., with traces of iron."
With the exception of one group of three at 0° C., these experiments were extremely
satisfactory, so much so that the fifth figure appears to have some real significance.
The perfect agreement of the experimental and the curve values is very noticeable.
z 2
172
DR. F, II. r.KIFFITHS AND MR. EZER GRIFFITHS OX THK
TABLE XXII. — Iron (Ingot).
Weight, 2798 grms. Density, 7 '8 58.
Temperature ....
O'C.
10° -OC.
and
9°'9C.
(mean)
20°-5C.
21°-6C.
24°-6C.
50"-3C.
66°-3C.
97°-5C.
S (experimental value) .
0-1045
0-1060
0-1078
0-1077
0-1080
0-1105
0-1112?
0-1137
Difference from curve"!
per cent. . . . . J
0
0
+ 0-22
+ 0-10
0
-0-22
-0-99
+ 0-10
S, = G'1045 (l+0-001520<-0-000006m2)-
This specimen was obtained from the American Rolling Mill Company, who
state : —
" Material rolled from an ingot into a billet (4 inches by 4 inches), on ' Blooming
Mill ' ; billet forged into round section at blacksmith's shop. Same had no
further annealing nor additional heat treatment, other than when rolled
and forged."
Specimen turned down to size in laboratory workshop.
" Sample from which material was taken and forged shows following analysis :—
Cu = 0'040 per cent.
O = 0'015
N = 0-0026
' S = 0'021 per cent.
" P = 0-005
" C = 0-012
" Mn = 0-036
H = O'OOOS
" Silicon, trace ; Fe (by diffi), 99'87."
Our sincere thanks are due to the American Rolling Mill Company, Middletown,
Ohio, U.S.A., for presenting us with this sample.
TABLE XXIII.— Zinc.
Weight, 2538 grms. Density, 7-141.
Temperature ....
0° P,
01°. K p
C/\° . fr /I
OU 0 U.
97 4 O.
•4 L.
S (experimental value) .
0-0917R
0-ftQOfiK
0. AQ,4 T O
09521
09i)70
Difference from curve per cent. .
0
-0-14
+ 0-19
-0-01
+ 0-08
S, = 0-09176
CAPACITY FOR HEAT OF METALS AT DIFFERENT TEMPERATURES. 178
Messrs. Johnson and Matthey state :—
" Approximately, 99'95 per cent. Zn."
The agreement between the results on repetition at the same temperature was less
satisfactory than usual, the extreme difference from the adopted value at 0° C. being
0'3 per cent, (see Table XIV.).
TABLE XXIV.— Silver.
Weight, 3733 grins. Density, 10'45G.
Temperature
0°C.
28°-41 C.
67'-40C.
97°-44C.
S (experimental value)
0-05560
0-05613
0-05680
0-05737
Difference from curve per cent. . .
0
0
0-07
+ 0-07
S, = 0-05560 (I + 0-000339W -O'OOOOOOHIf*).
Messrs. Johnson and Matthey state :—
" Better than 999 '9 fine."
At 0° C. two series of experiments by " total heat " method were performed—
With thermometers AB, CD, ten experiments ; probable error,
±0'05 per cent. ; S = 0'05551.
With thermometers AA, BB, six experiments ; probable error,
±0'04 per cent. ; S = 0'05575.
TABLE XXV.— Cadmium.
Weight, 3070 grms. Density, 8T>52.
Temperature
0°C.
28°-34C.
54-5'C.
97'-64C.
S (experimental value)
0-05475
0-05554
0-05616
0-05714
Difference from curve per cent. . .
0
0-02
-0-04
0
S, = 0'
+ 0>000520<-01000000725J:I).
|7) ,„.. ,.;. n. GRIFFITHS ANI> MR. EZER GRIFFITHS ON THE
Messrs. Johnson and Matthey state : —
" Fully 99-75 per cent, pure, with very ' slight traces of iron and zinc.' '
Series of four total heat experiments at 0° C., with probable error of ±0'08 per
M,,t. gave S = 0-05468. TiBLE XXVI._Tin.
Weight, 2591 grms. Density, 7 '292.
0"C.
28*'4C.
53°-9C.
97"-6C.
S (experimental value)
0-05363
0-05465
0-05549
0-05690
Difference from curve per cent. . .
0
+ 0-02
-0-02
+ 0-02
S, = 0'05863 (1 + 0-0006704<-0'000000458«3).
Messrs. Johnson and Matthey state : —
" Probably analyse to 99'80 per cent., with trifling quantities of arsenic, lead,
and iron."
TABLE XXVII— Lead.
Weight, 4016 grms. Density, H'341.
O'C.
28°-38C.
51°-OC.
67°-4C.
97°-45C.
8 (experimental value)
0-030196
0-03053
0-03073
0-03102
0-03127
Difference from curve per cent. . .
0
0
-0-16
+ 0-19
-0-03
S, = G'030196 (1 + 0'000400<- 0-00000036^).
Messrs. Johnson and Matthey state : —
" Approximate to 99'90 per cent., with inappreciable traces of arsenic and
bismuth."
The "probable error" of the various groups with this metal is higher than in the
case of other metals. This is probably an effect of the low conductivity of lead and
the consequent steepness of the thermal gradient within the cylinder.
Fig. 8 represents the increments in specific heat over the range 0° C. to 100° C., on
the assumption that the specific heat at 0° C. for each metal is represented by unity.
We had hoped to present curves showing the actual values of the specific heat over
this range, but the scale required was so large, that we found that, if reduced to the
size necessary for reproduction, they were of little value.
CAPACITY F()l; IIKAT <>F MKTAI.S AT MFFKKKNT TKMI'KKATPKKS. 175
I-IO
1-09
J
20°
60 •
• Temperiture.
80°
ioo»
120"
Fig. 8. Variation with temperature.
(Assuming specific heat at 0* C. aa unity.)
A noticeable decrease in the increment of the specific heat of zinc is observable at
temperatures above 50° C., which may have some connection with its change in physical
properties, as zinc becomes malleable about 120°C. For this reason we pushed our
examination of this metal up to a temperature of 123° C.
Our thanks are due to the Court of the University of Wales for a grant towards
the purchase of the specimens of metals.
We are greatly indebted to Mr. EDGAR A. GRIFFITHS, of this College, for his help
in the construction of apparatus and in the conduct of the experiments.
SECTION XIII.
Discussion of the Results of NERNST'S Observations at Low Tempi- ratures.
[After the preceding paper was written, it was suggested to us that we should
discuss the relation between our experiments over the range 0° C to 100" C. and
those of N ERNST at lower temperatures. We feel, however, that a discussion of this
kind would carry more weight after the completion of our own work at temperatures
below 0° C.
17'J
DR. K. H. GRIFFITHS AND MR. EZKR GRIFFITHS ON TIN-:
IM-..MI tlie brief description of the method published* by NKRNST, it is impossible to
estimate the magnitude of any errors arising from the neglect of the loss or gain by
iM.liatiun, &c. It must be remembered that boiling liquid air is not at a steady
temperature and therefore the metal block suspended within the envelope could not
settle to the temperature of the surroundings ; hence, observations of the temperature
after switching off" the heating current, afford little information concerning losses or
gain by radiation.
NERNST'S experiments, however, had one great advantage over those of other
observers at lower temperatures, inasmuch as the ranges of temperature employed
were small, e.g., 27° C. When we consider the curvature of the specific heat curve,
it is evident that changes of temperature of the order of 100° C. and upwards can
give little accurate information as to the value at the centre of such ranges. Two
metals, only, appear to have been examined by N ERNST, namely, lead and silver. For
lead he obtained the values of the atomic heat given in column II. below ; column III.
gives the values calculated from the modified EINSTEIN'S formula
(A)
= 3K
where 11 is the gas constant, equal to 1'985 gr.-calories.
For lead
a = 58, & = 7'8xlO-5.
In column IV. we give values obtained by extrapolation of the parabolic formula
representing the locus of our specific heat curve 0° C. to 100° C., (see p. 174 supra).
ATOMIC HEAT. — Lead.
I.
II.
III.
IV.
Absolute temperature.
NERNST'S observed
value.
Calculated from
formula (A).
Calculated from
GRIFFITHS' parabolic
formula.
62
66
79
93
5-63
5-68
5-69
5-76
5-58
5-63
5-75
5-84
5-62
5-64
5-68
5-73
Mean . . 75
5-69
5-70
5-67
* 'Journal de Physique,' tome he., 1910, p. 721.
CAPACITY FOR HKAT OF METALS AT I>Il IT.K'KNT TEMPERATURES.
177
Tt will be seen that for the purpose of representing the experimental results, there
is little to choose between the two formulae, the greatest difference from our parabolic
formula being less than 1 per cent, which NKRNST states to be the probable
experimental error in his observations.
The greatest divergence between NERNST'S results and the modified EINSTEIN'S
formula amounts to 1'4 per cent., and it must be remembered that the empirical term
bT3" in that formula was added as a consequence of these experimental numbers.
Thus it appears that, in the case of lead, the simple parabolic formula holds over
the range 62° C. to 373° C. absolute.
In the case of silver, NERNST records five observations (column II., iitfrci).
ATOMIC HEAT. — Silver.
L
II.
IIL
IV.
Absolute temperature.
NEKNNT'S observed
value.
Calculated from
formula (A).
Calculated from
GRIFFITHS' parabolic
formula.
64
3-72
3-61
84
4-43
4-44
—
86
4-40
4-60
200
5-73
6-78
5-84
208
6-92
5-81
5-86
If we consider the group about 200° C., we have the following results :—
At 204° C. absolute-
Mean observed value 5'83
Calculated (EINSTEIN'S modified formula) 5 '80
(GRIFFITHS' parabolic formula) 5 '84
Here, again, the conclusions of the different observers are in close agreement.
At the still lower temperatures, the decrease in the observed values is so marked
that, assuming the validity of NERNST'S values, the parabolic formula cannot possibly
hold good, and we can only conclude that some marked change takes place in the
nature of the curve below 200° C. absolute.
We hope to investigate the values of the capacity for heat of silver at some
intermediate points in the large gap between the groups determined by NERNHT.
In conclusion, it is notable that, with the exception of three observations upon
silver taken at closely adjacent temperatures, all the values obtained by NERNST fall
(within the margin of probable experimental error) upon the loci of the parabolas
which express our experimental results at higher temperatures.]
VOL. ocxili. — A. 2 A
178
ni; K. H. GRIFFITHS AND MR. EZER GRIFFITHS ox THE
APPENDIX I.
The hypothesis of DULONG and PETIT has undoubtedly been of great service to
chemists ; nevertheless, it is acknowledged that, at best, it is but approximately true
and that whatever value of the constant is assumed, the number of exceptions at
ordinary temperatures, especially in the case of elements of small atomic weights,
entitles us to regard it as an indication of a probability rather than as a valid
generalization.
Let us consider the values it would yield, at 0° C., if we apply it to the metals
whose specific heats we have dealt with in this communication, arranged in order of
their atomic weights, assuming that
Atomic weight x specific heat = 6'25.
TABLE XXVIII.
I.
II.
III.
IV.
6-25
Element.
Our value at 0° C.
Column II. - Column III.
atomic weight
Al
0-2306
0-2096
+ 0-02100
Fe
0-1119
0-1045
+ 0-00740
Cu
0-09832
0-09088
+ 0-00744
Zn
0-09561
0-09176
+ 0-00385
Ag
0-05794
0-05560
+ 0-00234
ca
0-05560
0-05475
+ 0-00085
Sn
0-05252
0-05363
-0-00111
Pb
0-03018
0-03020
-0-00002
The increase in the numbers in column IV., as the atomic weights diminish, is very
noticeable.
If we plot the experimental values (column III., supra) as ordinates and the atomic
weights as abscissae, the points lie very evenly about a smooth curve of an exponential
type ; Cu being rather low, Zn rather high, and Sn decidedly high.
In order to obtain an expression for the curve, assume (column III.) the following
values : —
Al = 0'209G ; mean of Cu and Zn = 0'09132 ; and Pb = 0'03020.
Then the curve drawn through these three points will be found to follow closely a
mean path through the above experimental values.
The expression for this curve is
S = 4'804xa-u'(*,
CAPACITY FOR HEAT OF METALS AT IMFFF.KF.NT TEMPERATURES. 17'.)
TAIU.B XXIX. — (The elements arranged in the order of their atomic weights.)
I.
II. III.
Observer and data indicating how values in
Column III. were obtained.
I.
•Uement.
s,
8 = 4- 804 x a"9*, experimental
values at 0* C.
Element.
H
4-8(i|
2*402 .Toi.Y, at constant volume
H:
He
1--J90
0 • 762 Deduced from (1) C,, - C. - j ; (2) £* = 1 • 652
He
Li
0-763
0-778 BERNINI, 0° to 19° - 0-837; 0* to 100° = 1-098
Li
B
0-492
0*251 Mean of values deduced from Koi'i* and Mois B
SAN and GAITIKR (amorphous)
C
0-453
0-113
As diamond, WKBKM, at 11*
C
N
0-391
0*175 PIER, at constant volume
N
0
0-344
0*170
HIM. BORN and AUSTIN (Reichsanstalt)
O
C» - 0 • 2320 ; y = 1 • 400 (Lummer)
Na
0-244
0-291 1 BERNINI, at 10°- 0-297 ; at 128* - 0-333
Na
Mg
0-232
0-234
VOIOT, 18° to 99' -0-246; STCCKKR at
oor»* A. owl
Mg
Al
0-2092
£ift*j » \r fuj i
0-2096 I'KIFFTTHS'
Al
Si
0-201
0-1771
WEBER, at 57* = 0-183; at 232* - 0-203
Si
(Cryst.)
P
0-184
0-188
(Yellow), RBT.NAULT, -78° to +10° - 0-17 ;
P
KOPP 13° to 36" = 0-202
Cl
0-162
0*08591
STRECKKR gives Cff = 0*1155 over 16° to 343°
Cl
and y = 1 * 322 ; CB reduced to 0"
K
0-148
0-1671
Data uncertain. ScHttz, 0*166 at -27°;
K
REONAULT, 0- 165 at - 39"
A
0-145
0-0738
DlTTENBERGER, C,, = 0*123; NlKMEYER,
A
y- 1-667
Fe
0-1052 0-1047
GRIFFITHS'
Fe
Ni
0-1003 0-1004
BEHN, -186" to +18" = 0-086; +18° to
Ni
+ 100° «= 109 ; TILDKN at 0° = 0- 1007
Co
0-1000 0-099
TII.DKN, -182° to +15° = 0-082; +15° to
Co
100° = 0-103; 15° to 630° = 0'123
Cu
0-0929 0-0909
GRIFFITHS'
Cu
Zn
0-0905 0-0917
Zn
As
0-0795 0-0778
BKTTENDORFF (Cryst.), 21° to 68° = 0-0830;
As
amorphous, 21° to 65° = 0'076
Kr
0-0723 0-0359
Cp-C, = ?; £*= 1-666
Kr
Pd
0-0569 0-057 BEHN, -186°~"to +18* =0-053; 18" to
Pd
100° - 0-059
Ag
0-0563 0-0556
GRIFFITHS'
Ag
(M
0-0541 0-0547
Cd
Sn
0-0512 0-0536
Sn
Sb
0-0608 0-0499
GAEDK, 17° to 92° = 0-0508; temperature co-
Sb
efficient from NACCARI, = 0-000016
Cs
0-0462 0-0482
ECKARDT and GRAEFK, at 13*
Cs
Pt
0-0319 0-0314
BKHN, -186° to +18* = 0-0293; +18° to
Pt
100° - 0-0324
Hg
0-0313 | Liquid by BARNKS ; for solid see WATSON
Hg
Tl
0-0307 0-0308
SCHMITZ, -192° to +20° = 0-0300; 20 to
Tl
+ 100° = 0-0326
Pb
0-0303 0-0302
GRIFFITHS'
Pb
Bi
0-0302 0-0300 GIEBK, -186° = 0-0284; WATERMAN +22° to
Bi
100° - 0-0304
U
0-0265 0-02741 BLUMCHE, at 49° (0° to 98°). Assume decrease
U
like Pb
2 A 2
I Ml
|>K. K. (I. CKIKK1TIIS AND MR. KZKK GRIFFITHS ON TIIK
Hence, if a = 1, we obtain S = 4'804, that is, just twice the value found by JOLY
for the specific heat of hydrogen at constant volume.
We have endeavoured to ascertain how nearly the values obtained from this
expression are in harmony with the conclusions of other observers in the case of
elements not included in our list. It is difficult, however, in regard to the majority of
the elements, to consider any conclusion thus arrived at as decisive. The determinations
in the case of the rarer elements have been made with such small quantities that the
results are open to suspicion, and, but few investigators have so arranged their tempe-
rature ranges as to include 0° C. Where values of S for different values of 6 have been
given, we have, on the assumption that the changes are of a linear order, deduced
the probable values at 0° C., and in Table XXIX., p. 179, we have indicated the
authority and the temperature ranges from which those Values were deduced. Where
no data for such a reduction can be found, we have inserted any values which fall near
0° C., together with the mid- temperature and the experimental range. For example,
Cs 13° C. (EcKARDT, 0° C. to 26° C.). We have given all the information we have
TABLE XXX.
Column I.
Column II.
Per cent.
Per cent.
differences.
differences.
H
0
He
f 15-0
Li
+ 2-0
N
-10-0?
0
-1-2
Cl
+ 6-0
5?
+ 0-9
0
Na
K
+ 16-0?
+ 11-0?
P
+ 2-2
Sn
+ 4-6
A
+ 1-9
Cs
f 4-01
Fe
-0-5
Ni
0
Co
-1-0
Cu
-2-2
Column HI.
Zn
+ 1-3
As
-2-3
Kr
-0-7
Pd
0
-1-2
+ 1 . i
Calculated. Experimental values.
Sb
i i
-1-6
Pt
Tl
Pb
Bi
-1-6
+ 0-3
-0-3
0-6
B
C
Si
0-492 0-251
0-452 0-113
0-201 0-177
U
+ 3-01
Hg
0-0313 0-0335 (liquid)
0-0314 (solid)
Sum of per cent. \
differences / !
CAPACITY FOR HEAT OF MKTAI.S AT DIFFERENT TEMPERATURES. 181
iible to gather, ooooeraing the specific heats of the elements, which appeared to
us to carry sufficient weight to render the deduced values of any service in such an
enquiry ; many of those included should, tor the reasons given in the introduction to
this paper, he regarded as rough approximations only. No element has l>een omitted
in connection with which any satisfactory evidence concerning the specific heat at 0° C.
was obtainable.
In column I. of Table XXX., p. 180, we enumerate those elements in which the
agreement between the calculated and the experimental values may be regarded as
close (i.e., within 3 per cent.), and we have in each case indicated the percentage
difference and its sign.
In column II. we place those in which the differences vary from 3 to 16 per cent.,
including some in which the probable error may be of like dimensions ; in column III.,
those experimental results which differ so greatly from the calculated as to exclude
the possibility of agreement. In the case of gases, the experimental values are
multiplied by 2.
Remarks on Columns.
Column I. — The sum of the differences ( — 0'5) and the distribution of the signs
show that the experimental values are very evenly distributed about the locus of the
curve.
Column II. — The experimental values of N, Na, and K do not appear to be
sufficiently established to lend much weight to the results.
The experimental value of the specific heat of tin at 0° C., as compared with that
of other metals examined by us, is high. It is a significant fact that tin, at
temperatures below 0° C., tends to revert into the grey powder form.
[Since the above was written, we have made determinations of the specific heat of
sodium at 0° C. Two different samples were used, and the results were in close
agreement, giving the value 0'2863 for the specific heat.
The few experiments at higher temperatures (50° C.) indicate that the increase in
specific heat with temperature is considerably greater in the case of sodium than in
the other metals examined by us, and is of the order of O'll per cent, per I8 C.
In this connection it should be remembered that sodium has the lowest melting-
point of all the metals considered in the above table.]
Column III. — Two curious coincidences present themselves. The calculated value
of C is almost exactly four times that of the diamond.
The mean experimental value' for amorphous B is closely half of the calculated one.
It has been shown (see, for example, Al and Pb supra) that the rate of change of
SS/S9 as 6 changes, varies markedly for different elements ; hence, any relation such as
that denoted by the equation S = 4'804 x a~°'*, which holds true for any given tempe-
rature, cannot be valid at other temperatures. There are, therefore, serious difficulties
in the way of accepting any definite connection between " S " and " a " at an arbitrary
I>R E. H. CRIFFITHS AND MR. KZER GRIFFITHS OX THE
temperature such as 0° ('., although it is probable that a large majority of
are in a stable condition at that temperature.
It is, however, evident that the curve S* = 4'804xa~"'!l5 yields throughout the
whole range of atomic weights values of S (of 2S in the case of gases) which, in the
large majority of cases, are within 2 per cent, of the most probable values.
We prefer to postpone any expression of our views on this matter until we are able
to ascertain the results of our experiments at low temperatures.
The Relation between S and 6.
The curves given in fig. 8 show that the curvature from 0° C. to 100° C. is far
more marked in the case of Fe and Al than in any of the remaining metals, with,
perhaps, the exception of Zn at the higher temperatures. If we produce backwards
the paralwlas whicli have been found to represent the mean paths over the above
range, it is found that the curves of Al and Fe (if they continue of the same character)
must cross those of the remaining metals before the temperature falls to absolute
zero.
If we venture to extrapolate, in order to ascertain the values of S given by the
respective parabolic equations at -273° C., we obtain the numbers given in column II.,
Table XXXI.
TABLE XXXI.
Element.
S at - 273° C.
Atomic heat at - 273° C.
Al
0-1306
3-540
Fe
0-0131
0-730
Cu
0-07438
4-728
Znt
0-06554
4-294
Ac
f^
0-04986
5-378
Cd
0-04402
4-948
Sn
0-04199
4-997
Pb
0-02186
4-527
Mean of all but Al and Fe . . .
= 4-813
It appears possible that the values of the specific heats of the last six metals may
ontinue to follow the parabolic paths as the temperature falls to -273° C., for an
1 The expression, atomic heat = 4 • 804 x <.<><* is obviously an alternative manner of expressing the
same relation.
t In the case of Zn, the equation is deduced from values found from 0° C. to 50-5 C., for reasons
given.
CAPACITY FOR HEAT OF MI.TAI.S AT DIFFERENT TEMPERATURES. 183
exceedingly small error in their coefficients over the range 0° C. to 100' C. would
account for considerable discrepancies in the values of their Atomic Heats at absolute
zero, the more especially as the resulting values of S are multiplied by factors ranging
IV. mi 63 to 200.
Experimental errors, however, could not account for such divergent results as those
given by Al and Fe, hence either the atomic-heat curves of these two metals undergo
change at low temperatures, or their values at absolute zero must be lower than that
of the others in the above list.
If we assume the continuity of the paths of the six metals above referred to and
deduce their respective specific heats at -273°C. from the mean atomic heat (4'813),and
for the other two points on the parabola employ the values at 0° C. and 100° C., we
obtain the following equations (t being expressed in the absolute scale)* :—
Cu . . . . S = 0'0758 (1+0-0008352«-0'00000039JJ),
Zn . . . . S = 0-07374 (l+0-0011155«-0-000000807t3),
Ag . . . . S = G'0447 (l+0-00122A-0-00000122<a),
Cd . . . . S = 0-0429 (l+O-OOlSSSGf-O'OOOOOHSG^),
Sn . . . . 8 = 0-0405 (1 + 0'0014514<-0-0000009665**),
Pb ..... S = 0-02327 (1 + 0-001544<-0-00000166«*).
If the values of S at the various temperatures at which it was determined by us are
now tleduced from these equations, it will be found that the differences between the
experimental and the calculated values are very small, in, no instance exceeding 0'3 per
cent., and in most cases much less.
The remarkable approximation between the hypothetical value of the atomic heat
at 0° C. (4'804) of a body with atomic weight 1, and the likewise hypothetical value
of the atomic heat of this group of metals at al«olute zero (4'813), is probably a
coincidence, but may possibly be of some significance.
APPENDIX II.
An inspection of the atomic heats of the metals investigated by us indicates that
those of low melting-points have high atomic heats. This is true throughout the
range 0° C. to 100° C., if the values at any given temperature within that range are
* Many equations of MI exponential nature, and also of the forms suggested by Professor PKRRY
(•Phil. Trans.,' vol. 194, pp. 250-255) have been investigated, but none of them fitted the experimental
results so closely as the parabola.
184
DR. E. H. CRTFFITHS AND MR. EZER GRIFFITHS ON THE
considered. It thus appears as if there was some relation between the temperature
..f the melting-points and the atomic heats.
In rig. 'J the atomic heats at 50° C. have been plotted as onlinates, and the
melting-points as abscissae.
This temperature was selected for comparison as the most reliable data given by
other observers have been obtained over temperature ranges including 50° C. as a
mean.
8-0
Li
.
\ *
g ^
1
? \
«,
\
1 3T>0
\
\
M
\
| 6-5
yrv
>^
+Fb
1 6-0
Bt
M^
Zn
Sb
Ca
^
Au
'Al
S
POO
100° 20O° 300°
400° 500° 600° 700°
»• Melting Points.
Fig. 9.
800° 900° 1,000° IJOO0
[Determinations made by us since the communication of this paper to the Society
give the value of the atomic heat of sodium at 50° C. as 7 '01, in place of the value
7 '37 shown in the diagram, this latter number having been based on the values of
NORDMEYER and BERNOUiLLi between -185° C. and +20° C. (5'38), and BERNINI'S
at +10° C (6'83) and +128° C. (7'66).]
APPENDIX III.
Soldering Glass to Metal.
The process is identical for glass, quartz and, no doubt, for porcelain.
The end of the glass or quartz tube is painted with a solution of platinum chloride
in a volatile oil. (Solution is sold under the trade name of Liquid Platinum, No. 1.)
The coating is very gently heated at first, and the temperature slowly increased,
until all the volatile matter has been driven off and a brilliant film of platinum
obtained. The higher the temperature to which the tube is raised, the better the
adherence of the film. The tube should glow with a dull red light, before being
CAPACITY FOi; Hi AT OF METALS AT DIFFERENT TEMPERATURES. 185
allowed to cool. If a thick film is desired, additional coatings can be given. Care
should be taken to prevent contamination by flame-gases ; if this occurs the surface
should be brightened by means of ordinary metal polish.
The next step is to " tin " the surface, and this requires care. The tube is gently
heated and rubbed with a lump of resin ; the solder melted on with a clean soldering
iron which should be only sufficiently hot to just melt the solder. With care the
entire platinised surface may be coated with an irregular coating of solder. Vigorous
rubbing of the surface with the soldering iron should be avoided, as it would probably
tear the film away from the glass.
The tube is then ready to be soldered into the metal ferrule which should be
" tinned " on the inside.
VOL. CCXIII. — A. 2 B
V. On the General llieory of Elastic
I '.a R. V. SOUTH \VI-.M,. Ji.A., Fellow of Trinity College, Cambridge.
Communicated by Prof. A. E. H. LOVE, f'./f.N.
«
Received January 4, — Read January 30, 1913.
CONTENTS.
Page
Introduction and summary of paper 187
EQUATIONS OF NEUTRAL EQUILIBRIUM IN RECTANGULAR CO-ORDINATFA
Method of derivation 190
Example in rectangular co-ordinates. Stability of thin plating under edge thrust 198
EQUATIONS OF NEUTRAL EQUILIBRIUM IN CYLINDRICAL CO-ORDINATES.
Derivation of the equations 202
Kxamplea in cylindrical co-ordinates. Stability of trailer flues and tubular struts 208
Solution for boiler flue without end thrust 217
Comparison with experimental results 222
Validity of investigation by the theory of thin shells 224
Comparison with existing formula! 225
The " critical length " 226
Solution for tubular strut. Special case 227
Validity of investigation by the theory of thin shells 230
Solution by the theory of thin shells. General case 231
Comparison with existing formula1 236
Stability of tubes under combined end and surface pressure 236
GENERAL THEORY OF INSTABILITY IN MATERIALS OF FINITE STRENGTH.
The practical value of a theory of instability 237
Stability of short struts 242
Need for further research. Conclusion 244
Introduction and Summary of Paper.
PROBLEMS which deal with the stability of bodies in equilibrium under stress are so
distinct from the ordinary applications of the theory of elasticity that it is legitimate
to regard them as forming a special branch of the subject. In- every other case we
VOL. COXIII. A 501. 2 B 2 Published Mpantcly, August 6, 1913.
188 MR R V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY.
are concerned with the integration of certain differential equations, fundamentally
the same for all problems, and the satisfaction of certain boundary conditions ; and
by a theorem due to KIKCHHOFF* we are entitled to assume that any solution which we
may discover is unique. In these problems we are confronted with the possibility of
two or more configurations of equilibrium, and we have to determine the conditions
which must be satisfied in order that the equilibrium of any given configuration may
I »• stable.
The development of both branches has proceeded upon similar lines. That is to
say, the earliest discussions were concerned with the solution of isolated examples
rather than with the formulation of general ideas. In the case of elastic stability, a
comprehensive theory was not propounded until the problem of the straight strut had
been investigated by EuLER.t that of the circular ring under radial pressure by
M. LEVY} and G. H. HALPHEN,§ and A. G. GREENHILL had discussed the stability of
a straight rod in equilibrium under its own weight, || under twisting couples, and when
rotating.^
In a paper which has become the foundation of the theory in its existing form,**
G. H. BRYAN has brought these isolated problems for the first time within the range
of a single generalization. Examining the conditions under which KIRCHHOFF'S
theorem of determinacy may fail, he was led to the conclusion that instability is only
possible in the case of such bodies as thin rods, plates, or shells, and in these only
when types of distortion can occur which do not involve extension of the central line
or middle surface, so that it is legitimate to discuss any problem in elastic stability
by methods which have been devised for the approximate treatment of such
bodies. He showed, moreover, that the stability of the equilibrium of any given
configuration depends upon the condition that the potential energy shall be a
minimum in that configuration.
A closer examination of BRYAN'S theory suggests that some of the conclusions
which have l«en drawn from it are scarcely warranted. The contention that no
closed shell can fail by instability, because any distortion would involve extension of
the middle surface, will be discussed later.ft For our present purpose it is sufficient
to remark that the whole theory is based upon the assumption that the strains
occurring previously to collapse must be kept to the extremely narrow limits within
which, in the case of ordinary materials, HOOKE'S Law is satisfied. This assumption,
of course, expresses a restriction necessarily imposed upon the range of practical
A. E. H. LOVE, 'Mathematical Theory of Elasticity ' (second edition), §118.
t ' Hist. Acad. Berlin,' XIII. (1757), p. 252.
} 'LiouviLLE's Journal,' X. (1884), p. 5.
§ 'Comptes Rendus,' XCVIII. (1884), p. 422.
II 'Proc. Camb. Phil. Soc.,' IV. (1881), p. 65.
H 'Proc. Inst. Mech. Eng.,' 1883, p. 182.
** 'Proc. Camb. Phil. Soc.,' VI. (1888), p. 199.
tt Cf. pp. 222, 236.
MR. R. V. SOUTH WKI.L ON THK GENERAL THEORY OF ELASTIC STABILITY. 189
problems which can he treated by the ordinary theory of elasticity ; but it is not
legitimate to conclude that instability is only possible, even if its conditions were only
calculable, in the case of materials which obey HOOKE'S Law, and there is no warrant
for tin- rmployment of "crushing formulae" in the design of short struts and thick
boiler flues.*
A more serious weakness in the existing theory of elastic stability, when regarded
from tin- mathematical standpoint, is the fact that the methods which it employs are
admittedly only approximate. The higher the elastic limitt of the material under
consideration, the less adequate are these methods to deal with the whole range of
problems which should come within its scope. In fact, we are faced with the
anomaly that, while in its ordinary applications the theory of elasticity is not
concerned with the conception of an elastic limit, in questions of stability the
existence of finite limits is an essential condition for the adequacy of its results. In
an ideal material, possessing perfect elasticity combined with unlimited strength,
types of instability could occur with which existing methods would be quite
insufficient to deal.
The theory of elastic stability is thus in much the same position as that of the
ordinary theory of elasticity before the discovery of the general equations, and oue
aim of the present paper is to remedy its defects by the investigation of general
equations, which may be termed " Equations of Neutral Equilibrium," and which
express the condition that a given configuration may be one of limiting equilibrium.
These equations are universally applicable only to ideal material of indefinite strength,
and the possibility of elastic break-down must receive separate investigation ; but
they are also applicable, even with materials of finite strength, to any problem which
comes within the restrictions imposed by BRYAN'S discussion, and therefore enable us
to test the accuracy of his treatment of problems, such as that of the boiler Hue, for
which the ordinary Theory of Thin Shells has been thought insufficiently rigorous.^
In every problem of this paper it is found that the Theory of Thin Shells gives a
solution which is correct as a first approximation, and the practical advantage* of the
new method of investigation are, therefore, not immediately apparent. But it must
be remembered that the approximate theory of thin plates and shells has not as yet
been rigorously established, and that much work has recently lx;en undertaken with
the object of testing it by comparison with accurate solutions of isolated problems.§
Now in finding conditions for the neutrality of the equilibrium of any given
configuration we are at the same time obtaining the solution of a statical problem ;
for a configuration of slight distortion from the equilibrium position will also be one
* W. C. UNWIN, ' Elements of Machine Design ' (1909), Part I., p. Ill ; S. E. SI/XJUM, "The Collapse
i >f Tubes under External Pressure," ' Engineering,' January 8, 1909.
t By "elastic limit" is intended, here and throughout this paper, the limit of linear elasticity.
J I'f. pp. 210, 224.
§ LOVE, op. tit., Introduction, p. 29, and Chapter XXII.
100 MR. R. V. SOUTHWELL ON THE GENERAL THEOKY OF ELASTIC STABILITY.
of equilibrium. Hence every solution which we can obtain will add to the number
of these " test cases," which has not hitherto included solutions for any but plane
plates.
A far more important advantage of the new method, from the practical point of
view, is the accuracy with which it follows the actual " stress history " in a body
which fails by instability under a gradually increasing stress. In cases where
instability precedes elastic break-down this difference of method is not important ;
but for the discussion of instability in overstrained material, where the stress-strain
relations are intimately dependent upon the previous stress history, its introduction
is absolutely necessary.
The extension of EULEB'S theory to struts of practical dimensions and materials,
which forms the conclusion of this paper, suggests a large and new field for
investigation. The number of similar cases which can be treated, in the existing
state of our knowledge of plastic strain, is very small, and indications are given below
of the questions which still require an answer ; there is reason to believe that the
requisite experimental research would not present insuperable difficulties, and that
we may hope in the future to obtain an adequate theory of experimental results
which are at present very little understood.
EQUATIONS OF NEUTRAL EQUILIBRIUM IN RECTANGULAR CO-ORDINATES.
Method of Derivation.
The question of stability arises in regard to any system in which there is a
possibility of slight displacement from the configuration of equilibrium. This possi-
bility may be afforded either by a more or less limited degree of mechanical freedom —
in which case the problem is one of statical stability, and practically unaffected by
Fig. 1.
the tendency, which any actual body displays, to distort under the influence of
applied forces ; or it may be due, more or less entirely, to this tendency. In the
latter case the problem is one of elastic stability, and must be treated by distinct
methods. There is, however, no essential difference between the two types of
ME. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 191
instability, and a general discussion of the elastic type may be very conveniently
illustrated by reference to a mechanical example.
In this connection we may consider the system illustrated by fig. 1, in which a
uniform heavy sphere rests in equilibrium within a hemispherical l>owl, under the
action of its own weight and of the pressure exerted by a pointed plunger, which is
free to move in a vertical line through the centre of the bowl. This system has been
chosen for the illustration which it affords of collapse under a definite " critical
loading." In this it bears an unusual resemblance to examples of elastic instability —
the stability of most mechanical systems being dependent solely upon the relative
dimensions of their members. In the absence of friction, we find that the equilibrium
will Income unstable as the load on the plunger is increased through a critical value
given by Wr
P' = R^'
W is the weight of the sphere,
r is the radius of the sphere,
R is the radius of the bowl.
The above solution rests upon the assumption that the sphere, bowl and plunger
are absolutely smooth and rigid, and the possibility of slight displacement is afforded
by the freedom of the sphere to take up any position of contact with the bowl. To
discuss the equilibrium of the sphere in the position illustrated we must consider the
forces which act upon it in a position of slight displacement. These include two
systems, one tending to restore the initial conditions, the other tending to increase
the distortion, and stability depends upon the relative magnitude of the two effects.
We may investigate the problem by three methods, fundamentally equivalent, which
are described below : —
(1) TJie Energy Method. — We may derive expressions for the potential energy of
the system in a position of slight displacement from the equilibrium position.
The condition of stability requires that the expression for the potential energy
shall have a minimum value in the equilibrium position.
(2) The Method of Vibrations. — We assume that the slight displacement has been
effected by any cause, and investigate the types of vibration possible to the
system when this cause is removed. The condition of stability requires that
all such types shall have real periods.
(3) The Statical Method. — We confine our attention to the special case in which
the stability of the equilibrium position is neutral. In this case there must
exist some type of displacement for which the collapsing and restoring
effects, discussed above, are exactly balanced, so that it may be maintained
by the original system of applied forces. We have, therefore, to find
conditions for the equilibrium of a configuration of small displacement, under
the given system of applied forces.
192 MB. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY.
Any of these methods is valid for the investigation of elastic stability, and all have
in fact been employed, the displacement considered being that of the central-line or
mi. Idle-surface of the rod or shell, and the resultant actions over cross-sections being
derived in terms of this displacement, by the approximate theory first suggested by
KIROHHOFF. The third method is generally found to be preferable, and is the basis
of the investigation to be described below, but the actual procedure will be found to
possess one or two novel features.
In the first place, an endeavour will be made to dispense with the assumption that
elastic break-down occurs at very small values of the strains ; instead, we shall deal
with an ideal material possessing perfect elasticity combined with unlimited strength.
Such a material could not fail, unless by instability, and our problems will no longer
be confined to thin rods, plates, or shells. It follows that we can only obtain sufficient
accuracy in our conditions for neutral stability by deriving them with reference to a
volume-element of the material.
Further, since instability will in some cases not occur until the strains in the
material have reached finite values, we shall have to introduce an unusual precision
into our ideas of stress and strain. The discussion of finite strain is merely a problem
in kinematics, and has been worked out with some completeness* ; but the corre-
sponding stress-strain relations in our ideal material are necessarily less certain, since
they must be based upon experiments in which only small strains are permissible.
For example, if we assume that HOOKE'S Law is satisfied at all stresses, we must
decide whether our definition of stress is to be
T . FTotal action over an element of surface"!
or
Original area of that surface J
T . [" Total action over the surface ~l
' LArea of that surface after distortionj '
For the ordinary purposes of elastic theory the two definitions may be regarded as
equivalent, and the distinction is too fine to be settled experimentally. In the
absence of any generally-accepted molecular theory which might indicate the correct
result, it seems legitimate to make the simplest possible assumptions which do not
involve self-contradictions, and which yield the usual results when the strains are
very small.
It may be shown t that in a distortion of any magnitude three orthogonal linear
elements issue from any point after distortion, which were also orthogonal in the
unstrained configuration, and that these linear elements undergo stationary (maximum
or minimum or minimax) extension. Hence an elementary parallelepiped constructed
at the point, with sides parallel to these linear elements, undergoes no change of
angle in the distortion. It is clear that only normal stresses will act upon its faces
For a discussion of the theory, with references, see LOVE, op. at., Appendix to Chapter I.
t LOVE, op. cii., $ 26, 27.
MR. i;. V. sorTinvKU, ON THK (JKNERAL THKORY OF ELASTIC STABILITY. 193
after distortion, and that if tli.-se stresses be expressed in terms of the extensions of
the sides we have complete relations between stress and strain.
We shall therefore assume that these principal stresses and principal strains,
irtottever their magnitude, arc <-n,,n. •<•/,-,/ I,,/ //,,• ,„•,//„„,•,, i-rfimf,',,,,.^ ../' I[O<>KK'S L«\i-;
that is to say, if the extensions in the principal directions are elt e» est and the
corresponding stresses are R,, H3. Tt:i, then
where E is YOUNG'S Modulus, and — is POISSON'S ratio for the material under
m
consideration.
These relations may be written in the form
20
....... (2)
m-2L
where C is the Modulus of Rigidity.
In these relations the measure of extension is assumed to be
Increase in length of linear element
Length of the element tafore strain '
and of stress*
Total action over an element, of surface
Area of the element before strain
We have then the usual expression! for the energy of strain, per unit volume of the
unstrained material, in terms of the principal extensions, viz.:—
i
(3)
The above assumptions yield sufficient data for the calculation of the stress system
in any configuration of equilibrium, even when the strains are not small. Assuming
that the calculation has been effected, we have to show how conditions for the
stability of the system may be obtained.
We must distinguish three configurations : the unstrained configuration, in which
the co-ordinates of any point are given by x, y, z ; the configuration of equilibrium
under the stress-system, the stability of which we are investigating ; and a configu-
' This assumption is open to the objection that it would render possible the compression of a material
to zero volume by means of a finite stress. It will not, however, introduce any serious error, and has the
advantage, which more probable assumptions do not possess, of leading to a definite energy -function.
Tin- definitions of stress and strain given alwve are generally employed in the construction of " straw-
~tr:iin diagrams" from a tension test, the extensions of the specimen being taken as abscissa;, and the
total loads as ordinates of the plotted curve,
t LOVE, np. ci/., § 68.
VOL. CCXIII. — A. 2 C
Mi: I,'. V. sol Tl I \VKLL OX Till. HKXKRAL THKOKY OF KLASTIC STABILITY.
ration of sli^lit distortion from the equilibrium position, which can be maintained
ivithout the introduction of additional stress at the boundaries, if the equilibrium of
the second configuration is neutral. We shall consider first a stress-system which is
such that the principal stresses in the second configuration have the same magnitudes
and directions throughout the body ;* and we shall take these directions as axes of
x, y and z. We may then define the second and third configurations by saying that
in them the co-ordinates of the point (x, y, z) become
and
respectively. We shall not limit the values of et, e3, ?3, although in practical cases
they must be small : u', i/, «/ are infinitesimal. In the second configuration the axes
Ox, Oy, Oz are directions of principal stress, and the stresses are
2C
X, = — gKni-lJcj + Cj + eJ, ...,&c (2.) bis
In the third configuration we shall find that lines which in the first configuration
were slightly inclined to Ox, Oy, Oz become directions of principal stress and strain.
The final extension of a line which originally had direction-cosines /, m, n is
It may be shown that e' has a stationary value when
(!+«,)
m = m, =
and
(i+«i)=!
» I/ *S—
n = n, =
(5)
to terms of the first order in n', v', w/.t
In some cases, such as GRRENHIU/S problem of the stability of a heavy vertical rod (p. 188, footnote),
i necessary to allow for variation in one or more of the principal stresses ; the necessary alterations are
easily made, and as they are not required for the examples of this paper their consideration would involve
unnecessary complexity.
A,l,M May 1.— The approximation of these expressions is insufficient if any two of the principal
(«i, «*, «s) in the second configuration are equal ; in this case additional terms must be retained in
denominators. The equilibrium under hydrostatic stress (,, = e, = «,) is necessarily and obviously
stable.]
Mi;, i; V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC SI ABII.ITY. 195
Thus the line initially given by the direction-cosines
1, m,, n},
liecumes a direction of principal stress in the final configuration. Its direction-cosines
(referred to Ox, O#, and Oz) are then
1,
or
,ai/
,<'"'
1,
: (6)
which we shall write as 1, m',, »', ; and its final extension, to terms .if the first order
in ?/', /•', /'•', is
' ' "' lv\
In the same way we find that the other directions of principal strain in the final
configuration are given by the direction-cosines
-
-
and
_
-
(1 +,>,)»-(!
(which we may write as — wi',, I, u'2, and — u',, —>*'», 1), and that the final extensions
in these directions are
and
— .
The stresses in these directions, which we shall call the directions of JT',
referred to the original areas of the faces on which they act,* are therefore
, and zf,
, v ^ ax, s«' . axr ar' . ax, a«/
. i — Ar + — — . -r — T - . -T— + -r . -r ,
8<>, ?a: ce, dy oc, dz
(.0)
* ('/. the assumption of p. 193.
2 O 2
MR. R. V. sorrmVKI.I. ONT THE GENERAL THEORY OF ELASTIC STABILITY.
to the new areas of the faces on which they act, they are
Xi V' , 7' ,
•V/ * y "/' _ ** i
X» _
» -
Y' =
1 »
(1+^,
and to the required degree of approximation we may write
x' x, r i a*/ i a*//]
•"• * i \~n \ ^ — r — • "^ — ~ • ~T —
( 1 + c2) ( 1 + <'3) L 1 +^i oy 1+ 6» vz J
20
m—2 f/ -,\dit' . dv' , dw'~\ ,
+ 7— — : (m-l)-T— + -T-+ -T- ..., &c. . . . (11)
(l-fp2)(l+«-:!)L 3* 3«/ 3« J
Then if a;, y, z denote the co-ordinates in the final configuration, referred to the
original axes, of the point which was originally at (x, y, z), so that
x =
t/, ..., &c.,
we may find the stress components in the third configuration, referred to the original
axes, and to the strained areas of the faces upon which they act, by the scheme of
transformation
X
y
x
y1
z'
The following expressions are thus obtained (to the required order of approxima-
tion) : —
X; = X'X, Y- = Y',, Z; = Z'.,,
(1
«,)
, ..., &c.
(12)
Now the stress-components (12) must satisfy the ordinary equations of equilibrium
which are three of the type
_ ^ , j
(13)
MR. R V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 197
and since the co-ordinates of the point which ultimately goes to (x + Sx, y, z) were
we
have
1 8 9a: a aa; _8 &(,
'
It follows that (13) may be written (to our approximation) as follows :—
20 \av, av av
,
i+e a2
Substituting for m',, n\, we have finally
m-iav av av m /av av
(- 1 h I ^ — :r~ T ^ — ^~
m — 2 fa3 dw1 cz m—2\cxcy czcx
+
/ay av \ xI+z7 /av av\_0
8^ *»w 4ci + ^±^^82!
(15)
X
and two similar equations. In any ordinary problem we may neglect e^... in
comparison with -j=f ... .
The equations thus obtained may also be written (with LAMP'S notation for the
elastic constants) as follows : —
= 0,..,&c.( . (16)
2
where
., aw' dv' , du/
A = ^— + -— + -— »
T7.=- 4..+
= fa* + w*dz>'
a«/ a«' „ , a«'
==-~'' • •~"'
198 MR. K. V. sulTIIUKl.L <>N THK UKXKIJAL THKOKY OF ELASTIC STABILITY.
.mil in this form they may be conveniently compared with the ordinary equations of
elasticity.*
The three equations of the type (15) we shall term Equations of Neutral Equilibrium.
The equilibrium of the stress-system Xz, Yy, Z. will be neutral, provided that solutions
for u', r', w>' exist which satisfy certain boundary conditions. These boundary
conditions are peculiar to each problem, but usually express the condition that the
additional stresses involved by u', v', «•' shall vanish on certain boundary surfaces.
T/iey never determine the magnitude of u', tf, w' , so that our solution gives the farm
only of the distortion which tends to occur in the body under consideration when its
equilibrium becomes unstable. It gives a definite relation between the stress-system
X, ... and the dimensions of the body, which must be satisfied in order that any
distortion may be permanent ; but if this relation be satisfied, no limits are imposed
by the equations upon the magnitude of the distortion which may occur, t
Example in Jiectangidar Co-ordinates. Stability of Thin Plating under
Edge Thrust.
It seems advisable, before we employ a new method on problems which have not as
yet received satisfactory treatment, in some degree to test its validity by the result
to which it leads in a more familiar example. For this purpose we may consider the
stability of an infinite strip of flat plating under edge thrusts in its plane. The
accepted formula}: for the thrust necessary to produce instability, per unit length of
edge, is
where
2t = thickness of plate,
/ = breadth of plate,
and the opposite edges are simply supported. If the edges are built in, the thrust
required has four times this value.
To investigate this problem by the new method we take axes Oa; and Oz in the
middle surface of the pkte, in the direction of its breadth and length respectively, and
Oy perpendicular to the middle surface. The initial stress-system is then given by
X, = const. = G (say), ~
* LOVK, of. at., § 91, equation (19).
are> hoWeVOr> rig°rous only in the case ^ infinitesM displacements; ef. footnote,
, 40
. LOVK, op. cti., § 337 (a), whence the above expression may be obtained.
MI;, i; v. SOITTHWKI.I. ON THK KKNKKAL TIIKOUY OF ELASTIC STABILITY. i«jy
:ind we may assume that the system i.f strain which is introduced at collapse will lie
two-dimensional, so that
^ = ^=0, - = const. (19)
z
The third equation <>f utmtnil stability (for the direction Oz) is then satisfied
G*
identically, and the other two equations become (if we neglect terms of order -^u' ...J
rn-2
and
m-2
m-2
m-2
4C
G_
4C
k . . (20)
Let us assume a solution of the form
/ = 2[V. cos a
"1
J '
(21)
where U. and V. are functions of y only. It is easy to show that this assumption as
to the phase-relation of u' and v' is justified. We have then
and
-. . . (22)
The solution of these equations is of the form
U. = (P// + Q) sinh ay + (R// + S) cosh ay,
G
3»i-4 G\
m-2 4C \ R
__G la
- > cosh a?/,
. . (23)
»-2 4C /
where P, Q, R, S are constants.
The boundary conditions now demand attention. It is clear that the stresses
introduced by «', t1', w' must vanish at the surfaces of the plate. Hence these
surfaces will still be planes of principal stress, and, moreover, the normal stress upon
200 Mi: i: V. SOUTHWKl.L ON THE IJKNKKAL THEORY OF ELASTIC STABILITY.
(licin must vanish. Hut, as we have already seen, the line which becomes a direction
< )//' of principal stress has initially the direction-cosines
— m,, 1, n, :
it follows that at the surfaces of the plate the expression for mt and na must vanish
identically ; moreover, at these surfaces, Y'y must vanish. These conditions may be
written in the form
identically, when y = ±t. . . (24)
i i\ v , i . «
(m-l) — + — + — = 0,
cy tiz dx
The first condition is already satisfied. The other two give (if we neglect terms of
and
or
rf//
- when y = +t,
(25)
2m-
G
; E
m+
m-2 40
.GL\
/. _
shaj/
and
(m-2) ay sinh ay-2 (m- 1 )
4G
cosh ay
when
___ G_
m-2 40
(m-2) ay cosh ay-2 (m-\) ^— sinh ay
m G
m-2 40
y = ±t . . >
+ Q(m— 2) a sinh ay
+ S (m— 2) a cosh ay = 0
(27)
Mi; R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 201
Thus we obtain
aS= P
1-
2m- 1 <i
/ m GVi i m-1 —}
-\rn-2 4C/\ m+l l< '
•2 m-\ G
-2)'2C ., .
at cotli at
) = It
( m G \ / , m-} G \
ZT>> if V I , i ' T7i /
at tanh at
aS = P
WI-
TH—2 \ wt
40
— at tanh at
Vm-2 4C'
aQ = R
»n —
4(,1
wi G
- I — at coth at
^w— 2
(28)
There are two solutions of the equations (28). Either
2m -1 G
1-
(m+l)(m-2) 2C , aO .,T«-I/ 4(- \ ,,
at tanh at = - p = 2 - — T I —at coth at,
G\7, . m-1 (• R m-2\ wt G I
and
or
1-
ro+1 4(J/
2m -1 G
-2 4C/
P = S = 0,
(29)
-2 2C
aS m-\
Ijn _G
and
F *\-S A Al A C*kJ ^ III I
-PTT —at coth at = -=^- = 2 -
-1 G\ P wi-2
, _
4C
. m . ^
f mTT ' '"
m (•
—at tanh at,
(30)
The criterion for neutral stability is in the first case
r-1 G-
1 +
,/ .1 k A
at (coth at -tanh at) =
and in the second case
1 +
1 +
4. v
at (tanh at-coth at) =
1 +
2»i(m+l) C
2m"- 1 G_
2>n(wi.+ l)' C
1 <
2wi(m + l) C
Vi >L. CCX1IJ. A.
2 D
•_•,.•_• MK. IL V. SOrTIIWKLI. OX THE GENERAL THEORY OF ELASTIC STABILITY.
so Unit tlu- \aln. - <>f <;. for which collapse by instability may be expected to occur,
G l-2oc< cosech
~ 2m (m+l ) V ~ 2m"- \-2ett cosech 2at
and
G _ 1 + 2ai cosech 2ett
~ 2m(m + l)C ~ 2m3 '- 1 + 2at cosech 2at '
respectively, the total thrust, per unit length of edge, being
JJ = -2*G ........... (33)
The first approximations to a solution, in terms of t, are
G=_$.-™£-.aV' . (34)
m — 1
and
=-E ......... (35)
m
respectively. Since the complete wave-length of the corrugations into which the
plate distorts is
A = ^, (36)
we see that (34) is equivalent to (17), and that the latter formula is therefore
supported by our investigation as a first approximation. The second solution (35) is
without practical interest, owing to the magnitude of the thrust required to produce
collapse. It refers to a type of distortion, theoretically possible for _an ideal material
without limits of elasticity, which is approximately realized in actual specimens of
ductile material, when tested to failure under compressive stress. Since Q = R = 0,
we see from (23) that in this type the middle surface remains plane. In the first
tyjxj of failure, where P = S = 0, we find that U. = 0 when y = 0, so that the middle
surface undergoes no change of extension in the distortion given by «', v', w'.*
EQUATIONS OF NEUTRAL EQUILIBRIUM IN CYLINDRICAL CO-ORDINATES.
Derivation, of the Equations.
The equations (15) of neutral equilibrium are expressed in a form which is
unsuitable for the investigation of problems concerned with the stability of thin
tubes, and we have next to obtain the corresponding equations in cylindrical
* Besides the harmonic solutions to (20) we may have
«*' = gx, v = hy, ?/•' = fc;
but g, h, and k vanish in virtue of the boundary conditions.
MR. R. V. BOOTHWBL ON THE GENERAL THEORY OF ELASTIC STABILITY. 203
co-ordinates. We Hlmll limit our discussion to stress-systems which produce a
displacement symmetrical about an axis, up to the instant at which the equilibrium
becomes unstable and distortion occurs : in PKARSON'S notation, the principal stresses
in the equilibrium configuration an- /•/•, rt#, and zz, and these quantities are functions
of r only.
The new equations are derived by a method very similar to that which has already
been explained. The co-ordinates of a point in the unstrained configuration are
r, e, z;
in the second configuration (of equilibrium) they are
r+it, 6, z+ir,
and in the third configuration (of slight distortion from the position of equilibrium)
they are
r+ »' + "', 0
(the radial, tangential, and axial displacements u't r', iS being ultimately taken as
infinitesimal).
The extension of a line-element joining the point (r, B, z) to the point (r+ir, 0 + 30,
z + $z) is
*> /I I I •" **
/• r >T0 / ?z
GIP
T I T 1" ^T Till I +
L cr r < rt \
where
m = r — , and n — -^ ;
cwAl'l "I
TF/J !'
and this has a stationary value for a line very slightly inclined to the radius, given
by
where <•„ c,, c3 are written for dli/2r, w/c, ami < </•/< : ivnpectivt'ly.
2 D 2
MI: i: v SIHTHNVKM. ox TIII: CKNKK.VL THEORY OF ELASTIC STABILITY.
The extension of this line-element in the third configuration is
(39)
and its inclination to lines issuing from the point (in its final position) in the null..!.
tangential, and axial directions is given by the direction cosines
where
mill
1, m\, »'„
l+c,
(40)
-
We find also that the other directions of principal stress in the final configuration
are initially inclined to radial, tangential, and axial lines through (r, 6, z) at angles
whose direction cosines are
and
-m,, 1, na,
W], Wa, Ij
and that in the third configuration they have corresponding inclinations to radial,
tangential, and axial lines through (r, 6, z) — in its final position — which are given by
and
where
and
_r/ — «' 1
'<• i > "• a> * >
«, =
?l , =
r d(
(41)
The extensions of the corresponding line-elements in the final configuration are
respectively
r r
(42)
Mi;, i: V. SOUTHWELL ON Till. CKNKiai. THKORY OF ELASTIC STABILITY. 205
so that the principal stress.-s in tin- third configuration, referred to the fined areas of
the faces on which they act, are
t "
I H! + 1 ^ ay i
*• r a0 a?
!+<•, ~1+«J
20
I dtl/ cV 1
« !__^__^L
(l+f3)(l +f,) I 1 +e, l+e,J
2C
m-2
r
[a«' «; IrV I
f> r r ?0
1--; n
1 + r, !+«-, J
2C
m-2 r, . v a«^
+ 71 - rr; - d (»i-l)^^
(l+e,)(l+r,)l - .
Then by the scheme of transformation
«' i a
— + - ==-
r r cflj
• (43)
tf z'
1 -m', -n',
0
m', 1 -n',
i
n\
n',
we find for the stress components in the third configuration, at the point which was
initially at (r, 6, z), — referred to axes in the radial, tangential, and axial directions
through the final position of the point, and to the final areas of the faces upon which
they act — the expressions
.
rr = rr', 00 = 00', zz = z/,
= 07- =
required approximation), <fec.
(to the
(44)
206 MR. R V. SOUTHWELL OX THE OKNKHAL THEORY OF ELASTIC STABILITY.
The equations of equilibrium, to be satisfied by the stress components (44), are*
1 ZrO . far . ri'-ee
<> r $6
czr 1 cQz czz zr _
?r r 20 ?2 r ' '
where
and
Moreover, since
9 = 0+-!—,
r + u
z =
ay i -f
•e, »/ a?t?'
1
a ar i -j
•ea ?• i a a/- a
cu' 3r (!+,,,)(
l+ej )• cQ (\ +e.)(\ +e ) a2 '
1
a»-
_1
av
i a»-'
as a
i it) *• ae
(45)
and
St/
i a
the equations (45) may be expressed in differentials with respect to >-, 0 z The
erms which do not involve »', t,', „•' vanish in virtue of the equilibrium conditions
the second configuration, and only terms of the first order in these quantities
need Ije retained.
In general, ,„ ,„ <>:t may all be functions of ,-, but in this paper we shall only
onsider problems in which ,„ and ?z have constant values. Moreover, in all problems
of practical importance we may neglect terms of order j£«'... in comparison with
\^
* LOVK, «/>. fit., § 59 (i).
Mi;
, R. V. SOUTHWELL OX THE GENERAL THEORY OF ELASTIC STABILITY. 207
terms of order 2 »'... . We then obtain, as the equations of neutral equilibrium in
\j
cylindrical co-ordinates,
irv i3i«' «r\ i av av 3/11-4 i fv »* /i av av\
2^2^ + 7Tv"~ ?' + ^?^i"f "a?" m-2 •75aT"'"m-2'rarad azaw
/5+^\I A/1 F^-^-^U/SiS^ -( — - — ) = 0, .... (46)
l\ icT/rwVrafl a/- r/ 4C /az\az
?„' PV 1 Sr' r' -1 1 3V . aV . m I 3V
V
av
-4
4C
^ + ^-i^-'N)l = o> . .-. (47)
*
and
av
av
m-2
/
1 ? /I a«''
4C
L«l_^')l = 0. . . (48)
Equations (46-48) represent the conditions for neutraUtability in the equilibrium
of a body subjected to a stress-system »T, W, K, where n is constant, and ri- and 60
are functions of r only, which satisfy the condition of equilibrium
,
_ j- -
?»• r
(49)
For comparison with the ordinary equations ot equilibrium in cylindrical
co-ordinates they may be written (with LAME'S notation for the elastic constants) in
the forms*
and
where
. (50)
i a / ,x , ia»-'
=;a7.(nt) + 7.aT +
* Cf. LOVK, oj>. r»<., § 199.
80 tliilt
208 MI: K. \ soi rnui.i.i, ON TIIK I;KXKIJAL THEORY OF ELASTIC STABILITY.
and
?/• ' r I9r 90 J '
r9rv"~r/ ' r 90 f IT
Examples in Cylindrical Co-ordinates. Stability of Boiler Fines and
Tnbular Struts.
The Equations of Neutral Equilibrium in Cylindrical Co-ordinates enable us to deal
successfully with some difficult problems connected with the stability of cylindrical
tubes. Two examples of considerable importance will be discussed in this paper — the
collapse of boiler flues and the strength of tubular struts. It should be noticed that
neither of these problems has been quite satisfactorily treated by the ordinary theory
ooo
Fig. 2.
of thin shells, which requires the assumptions that the middle surface of the sbell is
unextended, and the inner and outer surfaces free from applied tractions* ; hence their
solution is a problem of considerable interest, even apart from practical considerations,
and has attracted a great deal of attention. It will be convenient at this point to
review the work which has already been done.
The question of the stability of tubular struts is important, owing to the frequency
of their employment in practice. In economy of material the cylindrical tube
possesses an advantage over struts of solid cross-section, and both the theory of
* Cf. A. B. BASSET, 'Phil. Trans. Roy. Soc.,' A, CLXXXI., p. 437 ; and RAYLEIGH, 'London Math.
Soc. Proc.,' vol. XX., p. 379.
MR. R. V. SOUTHWELL OX THE <:i NKKAL THEORY OF ELASTIC STABILITY. 209
l-'.c I. F.I;' .-iii'l LM.I: \\«.i> ;in<l tin' more |>r;n-t I'M! f-rniul.i ..(' I;\\MM. VQg^e«l thai
this advantage increases without limit as the thickness of the tube is reduced. Such
a conclusion is, however, inaccurate, for types of distortion are possible in the case of
a tube which do not involve flexure of the axis, and when the tube is thin these
types, of which some practical examples are shown in fig. 2, may be maintained by a
smaller thrust than would be required to produce failure of the kind discussed by
EULER. Moreover, the natural wave-length, for these symmetrical types of distortion,
is in general small, so that distortion can occur without hindrance in quite short
tubes. Hence, for a considerable range of length the strength of a tube to resist
end-thrust is practically constant, and is not given by any of the usual formulae for
struts.
The determination of the strength of tubes to resist these symmetrical types of
distortion is obviously a problem of the highest practical importance, and has
attracted a great deal of attention in recent years. Illustrations of collapsed tubes,
showing symmetrical types of distortion, have been published by A. MALLOCK| and
R. LORENZ,§ and a great deal of experimental work has been carried out by
W. E. LILLY. || Theoretical discussions, by approximate methods, have been proposed
by A. GROS,1f W. E. LILLY,** S. TiMOSCHENKott and R. LORENZ.}}
The problem of the boiler flue seems first to have been suggested by the experi-
ments commenced by FAIRBAIRN in 1858.§§ These showed that the collapse of tubes
under external pressure was in some degree analogous to that of straight columns
under end-thrust, and a discussion of the phenomenon, based on EULER'S theory of
struts, was given by W. C. UNWIN,|||| who assisted FAIRBAIRN in his research. The
similar problem of a circular wire ring subjected to radial pressure has been
discussed by M. BRESSE^H and M. LEVY,*** and rational theories of the boiler-
flue problem have been given by G. H. BRYAN.ttt A. FoprL.jjJ P. FORCHHKIMER§§§
* " Sur la force des colonnes," ' Hist. Acad. Berlin,' XIII. (1757), p. 252.
t " Sur la figure des colonnes," ' Miscellanea Taurinensia,' V. (1773).
J ' Koy. Soc. Proc.,' A, LXXXI. (1908), p. 389.
§ ' Physikalische Zeitschrift,' XII. (1911), p. 241.
|| ' Proc. Inst. Mech. Eng.,' 1905 ; 'lust. Civ. Eng. Ireland,' 1906 ; ' Engineering,' January 10, 1908.
f 'Comptes Rendus,' CXXXIV. (1902), p. 1041.
** 'Inst. Civ. Eng. Ireland,' 1906.
tt 'Zeitschrift f. Mathematik u. Physik,' LVIII. (1910), p. 337.
Jt ' Zeitschrift des Vereines Deutscher Ingenieure,' October 24, 1908 ; 'Physikalische Zeitschrift,' XII.
(1911), p. 241.
f§ ' Phil. Trans. Roy. Soc.,' CXLVIIL, p. 389.
HII ' Proc. Inat. C.E.,' XLVI. (1875), p. 225.
fH 'Cours de M&anique Appliquee,' 1. Partic, Paris, 1859.
*** ' LIOUVILLE'S Journal,' X. (1884), p. 5.
ttt 'Proc. Camb. Phil. Soc.,' VI. (1888), p. 287.
JJt 'Resistance des Materiuux ' (1901), p. 286.
§§§ 'Zeitschrift des Oesterreichischen Ingenieur- mid Architekten-Vereines,' 1904.
VOL. COXIII. — A. 2 E
210 MK. K. V. siMTHNVKI.L (>X THK CKXF.RAL THEORY OF ELASTIC STABILITY.
and R. LORENZ.* W. E. LiLLYt has indicated the correct form of the result for
an infinitely long flue, and A. E. H. LOVE{ has discussed the strengthening effects of
constraints which keep the tube circular at its ends.
A. B. BASSET§ has given a very clear exposition of the difficulties which are
encountered in an attempt to construct a theory of flue collapse by usual methods.
To obtain sufficient equations we must assume that the middle surface undergoes no
extension ; and the existence of pressure on one or both surfaces of the tube not only
makes this assumption very improbable, but violates an essential condition upon
which the theory of thin shells is based. When one surface only is subjected to
pressure, there is reason to believe that BKYAN'S solution is substantially correct ;
but no treatment can be looked upon as rigorous which neglects the cross-stresses in
the material.
The experimental researches of A. P. CABMAN|| and R T. STEWART^ have revived
interest in this problem, since they offer the first information which has been obtained
as to the behaviour under practical conditions of tubes which in circularity, uniformity
of thickness and homogeneity are fair approximations to the ideal tube of theoretical
analysis.**
We commence our discussion by considering the stability of a thin cylindrical tube,
subjected to the combined action of end and surface pressures. We shall thus be
able to derive the required solutions for the thin tubular strut, and for a boiler flue
without end-thrust, as particular cases, and from the general solution we may obtain
indications of the way in which end-thrust tends to promote the collapse of a
boiler flue.
In the most general form of the boiler-flue problem, as enunciated by BASSET,tt
pressures are acting on both surfaces of the tube, and we shall therefore investigate
conditions for neutral stability in a tube subjected to the following system of
stresses : —
(i.) An end-thrust of total amount ,&, uniformly distributed ;
(ii.) An external hydrostatic pressure, of intensity $! ; and
(iii.) An internal hydrostatic pressure, of intensity $2.
* ' Physikalische Zeitschrift,' XII. (1911), p. 241.
t 'Inat. Civ. Eng. Ireland,' 1910.
| ' Proc. Lond. Math. Soc.,' XXIV. (1893), p. 208.
§ ' Phil. Mag.,' XXXIV. (1892), p. 221.
"Resistance of Tubes to Collapse," 'Bulletin of the Univ. of Illinois,' No. 17, 1906.
" Collapsing Pressures of Bessemer Steel Lap- Welded Tubes," ' Trans. American Soc. Mech. Eng.,'
1906, p. 730.
1 The experiment* of FAIRBAIRN were restricted to tubes which were constructed from sheet metal,
with brazed and riveted seams,
ft Lot. eit., p. 223.
Mi; It V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 211
We shall consider a tube of indefinite length, of which the inner and outer radii are
a±t (so that the thickness is 2t),
and we shall write
for the ratio -•
a
The com-spoiid'nig stress-system, for the position of equilibrium, is easily obtained.*
We have
TV = —
DO = -v-
and
zz = —
&
. . . (51)
It can also be shown that e3 is constant, and equations (46-48) may therefore be
taken to express the conditions of neutral equilibrium. The degree of approximation
to which these equations have been obtained (p. 20<i) will be maintained for the rest
>-s
•»•»•*
of this paper, i.e., terms of order -^ «'... will be neglected. They may also be written
as follows : —
m
r
2
A\ 1 3V /3m-4 A\ 1 dv'
• / *N r\ A ' I «1 »> rt
rn-2 2/r3r80 \rn-2 2 / r9 SB
V =
I m A)I
W-2 2/ r
-2 4 \" r8/ ' 4
A\ 1 3V . /3m— 4 A
m-2 2
Bl
(52)
m-1 1 3V
7H-2V 39s
m A A
B 1 1 3V
-
and
f m
m-2 4 \
4j 82 8r lm-2 4
3z
m A/ .aa\ B\l 8V
hlm-2 4V r»r 4
_
4\ 1*
B118V m-1 3V
_
-
* LOVE, ' Mathematical Theory of Elasticity ' (2nd edition), § 100.
2 E 2
_•!•_' Ml; K. V. SOITTHWKLL ON THE GENERAL THEORY OF ELASTIC STABILITY.
where
A= -
4C
B =
and
<T = —;
(55)
We may assume a solution for equations (52-54) of the form*
u' = 2 T U*. , sin k (6 + 6,) sin 2 (z + Zg) J
= 2 W4. sin i-
cos
2 (2 + Zq) ]
where Jt must be integral, and UM, VM, Wti, are functions of r only, which satisfy
the differential equations
A
• f/ w A\l d /3m- 4 A\ l"lv
A"L\m-2 2/rdr Vm-2 " 2/f*J fc
2.\^ — Afi+^r\+iid w 0
aLm-2 4 \ r*/ 4_Ur
(57)
and
(68)
aLlm-2 4\ i» ) 4 J dr \rn-2 4
r A / 2\ TJ ~n i
aLm-2~T\ "^) + T\r k-q
L A
4 *
r
m-2
r2 / 4 J \r dr rV
^< . = 0.
(59)
It is easy to show that the phase-relations assumed in equations (56) are necessary.
Mi;. R. V. SOUTHWELL ON Till! (IF.NKKA!, THKOKY OK HI. ANTIC STAHILITY. 213
The boundary conditions now require investigation. From the consideration that
the cylindrical Ixmndary surfaces of tin- tulx- must continue to be tangent to principal
planes of stress, in any possible type of distortion, we deduce the conditions
m, = 0
n, =0.
•, identically, when r = a±t.
(60)
The other boundary conditions are more complex. Since the pressures acting on
the surfaces of the tube are hydrostatic, it is clear that the radial stress, as defined
on p. 193, is increased at points on the boundary surfaces of the. tube where the
distortion involves positive extension. In the notation employed above, we have
rr1 = — |Ji, when r = a+t,
= —$;i» when r = a—t,
and from (43) we deduce the following equations, which must be satisfied identically,*
m— :
u
v _,_ i at/ _,_ a
+ + ' wheu
_, , ,
C [r r26
r = a—t,
.. . . (61)
Substituting from (56) in the identities (60) and (61), we finally obtain, as the
required boundary conditions in U*.,, VM, and WAi?,
and
. (62)
when r = a±t.
* In obtaining these equations it should be noticed that before distortion occurs - 5, and - 1)3 are the
values at the boundary of
^~s /"^
Bid not of rr, if we retain the significance for rr which was assumed on p. 193. The distinction is not
really needed for the approximation of the following work, but it may lead to confusion if neglected.
L'l I \1K I: V. SOUTH WKLL ON THE GENERAL THEORY OF ELASTIC STABILITY.
The differential relations (57-59), with the boundary conditions (62), are
theoretically sufficient for an exact solution of our problem : we shall, however,
content ourselves with approximate solutions for ($,— $2) and <§, correct to terms
in r3. To obtain these, we assume solutions for UAt g, VAi r Wkit, in series of ascending
powers of the quantity (r—a). Thus we write
(63)
\a/ 2!W "J
where r = a+h.
We may now derive, from equations (57-59), any required number of relations
between the undetermined coefficients £,...»;<,...£,..., and the boundary conditions (62)
take the form of equations in series of ascending powers of the small quantity T, in
which the sums of the odd and of the even powers must vanish separately. If we
neglect in these equations terms of order higher than some definite power of T, we
may obtain corresponding approximations to the values of A and B, by the elimination
of the undetermined coefficients.
The approximate boundary conditions, correct to terms in r3, are*
(64)
(65)
-i:t.fly (67)
in\ _
2fi-?-|-& = 0,. . . . (68)
In deriving these boundary conditions it is to be noticed that a- is to a first approximation equa
to - 1, 80 that to our approximation - T* may be written for <n-2.
MR. R. V. SOUTHWELL ON THE GENERAL THEORY OP ELASTIC STABILITY. 215
and
^-gf3 = 0, . . (69)
and these equations involve the fifteen coefficients
£o •••£!> in • ••i?i> M"-MO
From equations (57-59) we may obtain nine other relations between these
coefficients, as follows : —
m-
+ -Z-)+T<r —
— 1 m—
-8 A
-4 A
A-B
i-2 4
— a- •
• (72)
216 MI; i; v. sorrrnvKi.!. ox TIIH CKXKKAL THEORY OF ELASTIC STABILITY.
_ A)],,
4 / J
2 m— 2
0, . (74)
(75)
A-B A 1 . m A-B A
2
A-B A
(70)
Mil. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 217
m A— B . A~| . i m A— B
- f kqrr A.;,, -f ^kqtr^ - $kf/ - + <T —
[_m— 2 4 4 J
+ <T — ,
(78)
We may now eliminate the coefficients from equations (64-78), and obtain a
detciinin.uital equation, of fifteen rows, which gives a relation between A, B and the
dimensions of the tube. This relation is the condition for neutral equilibrium of the
initial stress-system, and is clearly correct to terms in T* ; but by further consideration
of the terms involved we may show that the labour which would be required for its
complete evaluation is. unnecessary, and as the fifteen-row determinant may be
written down directly from the above equations it will not be given here.
Solution for Boiler Flue tvithout End Thrust.
We shall begin by deriving a sufficiently approximate expression for the difference
of pressure required to produce collapse of a tube, when there is no resultant end
thrust or tension ; and in the first case we shall deal with a form of collapse possible
only in the case of tubes of infinite length. That is to say, we make B and q equal
to zero in the fifteen-row determinant, which may then be reduced to one of ten
rows.
In the latter determinant we may treat A as a quantity of order T* ; for if A be
put equal to zero, and the determinant be expanded, the terms which are independent
of T vanish identically. Hence —A may be written for o-A, and AT* may be
neglected.
The ten-row determinant, simplified by these and other obvious modifications, is
given on pp. 218 and 219. Expanding it from the top row, with the neglect of terms
of order higher than r2, we obtain
m— 2 \m— '
whence
VOL. CCXIII. — A. 2 F
218 MR. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY.
0, - £ 0,
0,
o,
-1,
(m-2)A,
m-1
m— 2
n
— 2
m-2
m-2'
m-1
m-2'
m-1
1,
o,
1,
0,
m,
m— 2
m-
m-
m
A
— »
m— 2 2
37>t-4 , A
m-2 "2*
3m-4 A
m-2 "2
0,
i,
0,
1,
(m-2) A«r,
3tn-4 A
m— 2
m-2
\ 2
0,
m A
m-2 2
3m- 4 A
0,
0,
0,
0,
0,
m— 1
m-2'
n
U,
A
-1,
o,
0,
0,
0,
0,
-,
0
1
m-2'
U,
3m- 4 AN
m-2 " 2 ''
m— :
-1,
o,
Mi;. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 219
A T' T1
AT,
P(m-2)A,
-4 A
m-2 " 2
u /3m-
3m-4 , A\
^2- + T/1'
3p/3mJ,4A\
\m— 2 2/
A
"F
-1,
.l+(m-2)A,
2 + A,
-(2 + A),
2 + A,
-(
"i
m-2
m— 1
-2
1+y+2»-2
<,j,
-(
i+v+2- I*1'-
2 m— 2
m— 1
'm-2'
0,
0,
-tf
A<r,
0,
0,
0,
m-2
2
m AN
0,
i- ACT,
0,
0,
o,
2
1+A,
o,
o,
_^/_^_A),
\m-2 27
5m-4 A
"i
-* 1 + =
2 m-2
0,
0,
o,
o,
A(l+<r)|, 0,
o,
0,
o,
o,
m
0,
'+*
i-±\
o,
o,
i-*A(r
AO-,
0,
+«-; .
-2 *'
0,
_A(,
m-2 4
0,
o,
'•
0,
0,
0,
o,
0,
0,
0,
0,
0,
0,
0,
2 F 2
Mi; i;. V. SOITIIU KI.L ON TIIK CKNKKAT, THEORY OF Hl.ASTIC STABILITY.
But to a first approximation
so that we have
(80)
which agrees with BRYAN'S result.*
To complete our discussion of this problem we must consider types of distortion in
which the axial wave-length is finite, and thus obtain a theoretical estimate of the
strength of short flues with fixed ends. A solution giving A correctly to terms in ra
may be derived from the complete fifteen-row determinant ; but we may show that
for practical purposes the labour which this evaluation would entail is quite
unnecessary.
We find first of all that those terms in the expression for A which are independent
of T contain q* as a factor. Now 2AC being approximately equal to the mean hoop
stress in the tube before collapse, it is clear that A must in all cases of practical
importance be a very small quantity. It follows that in the expanded equation the
terms in A are of primary importance, and A2 and higher powers may be neglected ;
further, since q must also be small, that terms in <f and higher powers of q may
be neglected in comparison with terms in g4, and that of the terms in r2 those which
involve q are negligible in comparison with the terms already found.
In accordance with these principles we may derive the terms which are required to
complete our solution from a nine-row determinant, obtained by omitting terms in r2
from the general determinant. This simplified determinant is given on pp. 218
and 219. Further, we may neglect A2 in the expansion, and in the coefficient of A
retain only those terms which do not involve q ; we thus obtain the equation
But, by equations (55),
and therefore, to the approximation of equation (81),
* Cf. footnote, p. 209.
Mi:, i: V ^HTTIIWKLL OX THK (IKNKKAL THEORY OF ELASTIC STABILITY. 221
Combining this result witli (80) \v<> have, us ,,ur final fxpn'ssion for the pressure-
'lill'crence which can produce collapse of the flue,
(83)
In this equation t/a is the ratio of the thickness to the diameter of the tube, and k
is the number of lobes in tin; distorted form of its cross-section. The quantity q is
connected with the axial wave-length X of the distortion by the relation
q\ = 2ira.
(84)
We may imagine a flue subjected at its ends to constraints which merely keep the
ends circular, without imposing any other restrictions upon the type of distortion.*
In this case the end conditions may be written in the form
when z =
(85)
and from (56) it is clear that I, the length of the flue, is equal to
c.
|M|
1
*
•*8
\B
2
TIIBOKETICAL COLLAPBIKG PBRHBI-BIS
&\
l^
— L
Vn
I
T "
K :t x KV" pound* )>er >q. inch,
i
= 2 -07 x 10'- d.vnes per «q. cm
1
\
.S*^
\
\
Tliickncu : diameter = 0 "01.
•j 2
H
i \
POISSOK'B n
it in. 0 '3.
Pressure it
K) Zf.
^^*^—
\
£>~
^^^>>ii>
}
I'nJues if Z £. 4 6 8 K> II
Fig. 3.
[* Added June .?. — Thin circular discs, inserted into the tube at its ends, but not fixed to it, would
approximately realize these conditions.]
Mil. K. V. ROITTH\VI-LL ON THE OKNT.K'AL THEORY OF ELASTIC STABILITY.
In practice, the end constraints will also tend to maintain the cylindrical form at
the ends of the flue, and this effect will strengthen the tube, by an amount which is
not easy to determine exactly. In any case we may say that
UJ.
a q
and we may illustrate the way in which the end effects die out by plotting the
pressure differences ($.-&,) against the quantity q~\ To do this we must take
some definite value of the ratio tfa, and plot different curves for the values 2,3, ....
Ac., of k. The result is shown by fig. 3, in which the following values have been
assumed for the constants : —
E = 3 x 107 pounds per sq. inch,
= 2-07 x 10" dynes per sq. cm.
m = J^, -
From an inspection of the different curves we see that long tubes will always tend
to collapse into the two-lobed form, since the curve for k = 2 then gives the least value
for the collapsing pressure, but that at a length corresponding to the point A the
three-lobed distortion becomes natural to the tube, and for shorter lengths still, of
which the point B gives the upper limit, the four-lobed form requires least pressure
for its maintenance. Thus the true curve connecting pressure and length is the
discontinuous curve CBAE, shown in the diagram by a thickened line.
Whatever lie the relation between q and the length of the flue, it is clear that
instability is theoretically possible in cases where the distortion involved is not even
approximately " inextensional." For if T is sufficiently small, the collapsing pressure,
as given by (83), need not involve elastic break-down in the position of equilibrium,
even though the first (or " extensional ") term in (83) be equal to, or even greater
than, the second. Of course, elastic break-down will occur by reason of the extension
very soon after the commencement of the distortion. Nevertheless, failure in such a
case must be regarded as due entirely to instability ; for if this source of weakness
were removed, effective resistance could be offered for an indefinite period to pressures
which actually result in collapse.
Comparison with Experimental Results.
Although, as we have just remarked, it is theoretically possible for failure to occur
by true elastic instability in comparatively short tubes, yet the relative dimensions of
the tubes must be such as it would be quite impossible to test experimentally. In
any practical case, instability will not occur until the properties of the material have
been altered by overstrain, and the value of the pressure at collapse is therefore very
much less than the foregoing theory would suggest.
MR. R. V. SOUTHWELL ON THE GENKKAL THEORY OF ELASTIC STABILITY. 2S8
It is, however, of interest to compare the general shape of the theoretical curve
CBAE (fig. 3) with the results of experiment, and fig. 4 has been constructed for this
purpose. It represents a number of tests conducted by the author upon seamless
steel tul>e (0'028 inches thick and 1 inch in external di;mn'ter), and shows the relative
amounts of resistance to external pressure offered by different lengths of tube. In
these experiments (selected for fig. 4 from a more comprehensive series which is still
Collapsing Pressure in. Poiuuls per Square fiifh,.
1000 ZOOO 3000 4000
*
^
•Oo
o
COLLAPSING PB18S0BI8 OF SKAMLKI
STKKL TUBIB.
(Tested by hydraulic pressure.)
lobes in diitorted crow-section.
il
0
4-f Two
t
OO Three „ „ „
•*•$• Four; „ „ „
1
+
*
* +
•f
h
•f
Unsupported. 5 Length of 10 Tube in, 15 fnrheg. 20
Fig. 4.
in progreas) the ends of the tube were gripped by means of slightly conical plugs and
sockets, the interior being kept in free connection with the atmosphere, and no
attempt was made to balance the axial thrust due to hydrostatic pressure on the
plugged end of the tube. Other experiments have shown that the existence of this
thrust is not seriously important.
It will be seen that the general shape of the theoretical curve is well reproduced,
as well as the changes in the number of lobes which characterize the distorted cross-
section. Similar results to those of fig. 4 have been obtained by CARMAN,* but his
experiments were not sufficiently numerous for a satisfactory comparison with the
theoretical curve of fig. 3, his object in conducting them being merely to discover
what is the limit of length beyond which the strength of a tube may be taken as
Of. footnote, p. 210.
MI!. R. V. sorrmVELL ON THE GENERAL TIIKOKY OF KLASTIC STABILITY.
sensibly the same for all lengths. The main interest both of CABMAN and of
- i \\ \KT* was confined to tubes in excess of this limit, experiments on which may
fairly lx- compared with the theoretical formula (80); their results showed that this
formula gives a satisfactory estimate of the strength of very thin brass and steel
tulx-s. but must not be taken as a basis for design throughout the whole range of
dimensions employed in practice.
The experiments of FAIRBAIRN,! on the other hand, were restricted to tubes of
such relatively small length that he failed to realize the existence of a definite
minimum below which the strength of a tube, however long, will not fall. He also
neglected the possibility of discontinuities in the curve of collapsing pressure at points
where there is a change in the form of the distorted cross-section. In the light of
these facts, figs. 3 and 4 help to explain his well-known formula, by which the
collapsing pressure is given as inversely proportional to the length of the flue ; for a
curve of hyperbolic form will represent as well as any other single curve the scattered
points of fig. 4, and trial shows that the hyperbola
(86)
is very closely an envelope of the discontinuous curve CBAE in fig. 3, in each case
doivn to the point of least collapsing pressure.
Validity of Investigation l»y the Theory of Thin Shells.
One important result of our investigation, which is apparently new, is shown by
equation (83). It may be seen that collapse is practically dependent upon the pressure-
difference alone, and that the absolute values of the pressures are immaterial. In
view of this result, the objections raised by BASSET against BRYAN'S treatment of the
problem { require further consideration.
These objections are : first, that the ordinary expressions for the stress-couples in a
plate or shell, in terms of the curvature of its middle surface, are not valid when the
surfaces are subject to pressure ; and secondly, that it is not legitimate to assume, as
we must if sufficient equations are to be obtained, that the middle surface is
unextended in a configuration of slight distortion. Hence the theory of thin shells
is not applicable to this problem.
The above difficulties may be almost entirely overcome by a change in the method
of investigation which is employed. It is customary to derive equations for the
equilibrium of the distorted shell directly, and without reference to the position of
equilibrium. Such procedure renders it necessary to make BRYAN'S assumptions, that
the middle surface is unextended, and that the usual expressions for the stress-couples
* Of. footnote, p. 210.
t Cf. footnote, p. 209.
t See footnote, p. 210.
MR. R. V. SOUTHWELL ON TIIK cl-.NI-.l: \l. TIIKORY OF ELASTIC STABILITY. 225
are valid. But we may also proceed, aa in the foregoing discussion, by first deter-
mining the stress-system for the equilibrium position, and then deriving equations
for an infinitesimal displacement. The stress-couples which appear in these equations
will be dm* to the ndilitionnl ••</ rcsses introduced />// tin- distort inn, u,,d since these, to
a first approximation >it /<•".< ,-nnixli nt the surfaces of t/n tnl,,-. ///«// //•/// lie. given
with sufficient accuraci/ lit/ tlie usual expressions. Moreover, when the distortion is
two-dimensional (as in BASSET'S problem), the change in the " hoop " stress-resultant
\\ ill be of an order which is negligible, so that the middle surface may be regarded as
undergoing no extension relatively to the equilibrium position, even though its area
may be sensibly changed in comparison with the unstrained configuration.*
The method of investigation just described, which follows the actual sequence of
occurrences in the material, is suggested as in every way preferable to existing
methods, for the investigation of any problem in elastic stability. For the present
'•\iimple, in particular, it leads to the same results as the more rigorous methods of
this paper.
Comparison uith Existing Formula.
Previous discussion of the boiler-flue problem by analytical methods have, without
exception, dealt with a tube subjected to pressure on one surface only, and almost all
of them have been restricted to the case of an indefinitely long flue. Their results
have, therefore, to be compared with our equation (80), when $., is zero. It will be
found that this equation agrees with the formula obtained by BRYAN! and BASHKT::}:
FOPPL'S formulat omits the factor — ; , which measures the increased resistance to
m — 1
flexure of a long tube as compared with a circular ring.
The more general formula may be compared with that of LORKNZ,* if $., IKJ put
equal to zero. It will be found that there is a serious want of agreement in regard
to both terms in the expression (83). In support of the latter result, it may be
urged that LORKNZ' solution gives for the indefinitely long flue a result which does
not agree with equation (80) (and, as we have just noticed, this is supported by
previous investigations), and which vanishes, not when k = 1, but when k = 0. Now
the value 1, in the case of an infinitely long flue, corresponds to translation of the
tube as a whole, without distortion, and the value 0 to a change in the diameter
of the tube, without any departure from circularity. It is clear that the applied
pressures can have no tendency to maintain such a form of distortion, so that LOUDTZ'
formula can hardly be correct.
[* ;t<ttle<l June 8. — The arguments of this section are more fully developed in n paper by the author
" On the Collapse of Tubes by External Pressure," published in the ' Philosophical Magazine ' for May,
1913 (pp. 687-698).]
t Cf. footnote, p. 209.
\ Cf. footnote, p. 210.
VOL. CCX1II. A. 2 O
._...,; Mi; I: V ->"! Tl I XVKI.l, ON THE GENERAL THEORY OF ELASTIC STABILITY.
The " Critical Length."
A. E. H. LOVK* has investigated the rate at which the strengthening effect of
circular ends falls off when the length of a lx>iler flue is increased. His result
suggests that at a distance which is great compared with the quantity ^/at the
influence of the ends becomes negligible, and the flue collapses under sensibly the
same pressure as a tul>e of infinite length ; hence, in order that " collapse rings " may
have any appreciable effect, their distance apart must not exceed some experimentally-
determined multiple of this quantity.
The greatest length of tube over which the ends exert any appreciable strengthening
influence, or the least length for which collapse is possible under a pressure sensibly
equal to the critical pressure, has been called by Prof. Lovst the "critical. length."
It is a conception of great importance in experimental work; for, as we have seen,J
tests on any length of tube in excess of this limit may be taken to give the strength
of an infinite length of the same tube, and their results compared with the theoretical
formula (80)§: but as a basis for the spacing of "collapse rings" it is superseded by
the theory of this paper, which yields an expression for the greatest length of tube
consistent with stability, when the thickness and diameter of the flue, and also the
collapsing pressure, are given ; and Prof. LOVE has suggested to the author that it
would be better now to employ the term " critical length " in this more general
significance. As we have seen (p. 222), the length of the tube is some multiple of the
quantity a/q, and we may therefore obtain from (83) the following formula :—
Critical length = — Ma
where M is a constant, depending upon the type of the collapse ring, and k has
that integral value which gives the least value for the right-hand expression of
equation (87).
Before this subject is dismissed, it should be noticed that the theory of this paper
does not support Prof. LOVE'S estimate, mentioned above, of the rate of decay of end
effects. The term in equation (83) which depends upon the length of the tube may
be regarded as negligible, compared with the constant term, when the ratio
' Proc. Lond. Math. Soc.,' XXIV. (1893), p. 208.
t ' Theory of Elasticity ' (2nd edition), § 337 (b).
Page 224.
f
i In this sense the term » critical length » has also been employed by CARMAN, who began his research
-mg the strengthening effects of the end plugs with which he sealed his tubes for test.
MR. R. V. SOUTHWELL OX THE OKNEKAL THEORY OF ELASTIC STABILITY. 2'27
has some sufficiently small value ; and -2. being inversely proportional to the length of
0
the tube, we deduce for the " critical length," in the original sense of the term, an
equation of the form
where / is constant. Prof. LOVE, as has been said, has obtained an equation of the
form
which is very different ; but he has informed the author that in the light of the above
investigation (pp. 210-222) he does not regard his method as adequate.*
Solution for Tubular Strut : Special Case.
We may obtain another simplification of the general determinant to ten rows by
taking a zero value for k. This corresponds to a type of distortion, possible in the
case of a tubular strut, in which the axis remains straight and the cross-sections
circular, the diameter varying in a sinusoidal manner.
The ten-row determinant for this case is given on pp. 228 and 229 ; the factor — q*
has been cancelled from the sixth column, and terms in A have been omitted, so as to
yield a result for tubes collapsed by end pressure alone. The expansion is only
correct to terms of order r2, and for a first approximation we may also neglect the
square and higher powers of B, which must be small in any case of practical
importance. Investigating first the terms which are independent of r3, we obtain
We may now employ the substitution
. . . . (88)
m q*
m q
in the determinant, and expand it from the top row, neglecting terms of higher order
than T*.
A considerable amount of unnecessary lal>our may be avoided by a preliminary
examination of the relative importance of the various terms involved. It will be
[* Added Mail 4- — -An argument in favour of the new formula may IKJ drawn from physical considera-
tions. The resistance offered by a tube to any given form of distortion is due partly to the extension
and partly to the flexure which such distortion entails ; and it is clear that the relative importance of the
extcnsional part increases as the thickness is reduced. Hence, other things being equal, the effects of the
ends, which necessitate extension of the middle surface, are more important in a thin than in a thick
tube ; that is to say, they are sensible over a greater length.]
2 O 2
228 MI: i: v sorrnu 1:1.1. n\ TIII-: CKNKKAI. TIIKOKY OF KI.ASTIC STABILITY.
-[•
I. 0,
0 1»
1, m— 1,
0, m,
m~1 \<f(\ BY|, 9*-l
J- 0,
»- •••••-/-.'f
»»— 1, (3m— 2)
m—2 4/J "m— 2'
.m— 1 m— 1 R\~l
2 m-2' ' °'
2^" A rt ?/l~~ 1
m-2 L m-2 ' * \ 4/J'
fi»»-l. m-1
m-2 "m-2
L 1 c m — 1 i 2/1 B \ 1 m — 1
m-2 °m_2' -1
-Z2-.4.I1. m B
*• " hy ( 1 — 1 L
' L m—2 \ 4/J m—2
0, 0,
m B
m-2 T' °'
»»-Z 4 OT_2 ' 4 '
0, »( m i BN)
Vm-2 4 /
0,
o,
rn-2 4
rn-2 4
MR. It. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 229
o,
0 1- — , 0,
•y/2'
0,
B
/ B\T*
o,
o, o, i-f.
o,
v"T/ sf
o,
I, -?V, -q*±,
o,
o,
T^
1, -<7", -<?^>
' "-*-*> '
o,
6
b
o,
OrP \ - -L J O
o,
o,
o,
0,- 0, • : -*fc*I + f)>
0,
o,
1
0, 0, 0,
,/ m B\
o,
m-2'
a^ \m— 2 4 /
0,
Om-l j B j B
0,
o,
"m— 2 4 4
0,
Om-l Om-l, B\
1 B
o,
m— 2 m— 2 \ 4 /
4
0,
Om-l , m-1 ,
a/t B\
1 B
S\ 4/
4 '
230 MK. K. v. vH-TinvKu. 0* TIIK aiffeAii THEORY OF ELASTIC STABILITY.
r<..u..I that B' contains terms in 7' and L, «* weU as terms independent of q. Thus
the complete expression for B is of the form
f\ m
and it is clear that B has a minimum value when the axial wave-length has a finite
value, given by
v =
m
This minimum value, which alone is of practical importance, is given, to a first
approximation in terms of T, by the equation
(89)
so that the determination of a and /8 is not required.
By expansion of the determinant we find
£ m
y~~ 3m-l'
and from (55) we deduce, for the minimum thrust required to produce collapse,
......... (90)
This expression is correct to terms in t".
Validity of Investigation by the Theory of Thin Shells.
A complete investigation of the tubular strut problem must deal with lobed forms
of deformation, since it is possible tbat one of these may require a smaller end-pressure
for its maintenance than the circular form treated above. We have, therefore, to
obtain a general expression for B (when A is zero) in terms both of k and q.
The derivation of this expression, if we employ the rigorous methods of the present
paper, will entail nothing less than the evaluation of the complete fifteen-row
determinant ; for the existence of a " favourite type of distortion," of finite axial
wave-length, which we have noticed in the particular case (k = 0), is found by
practical experiment to be equally a feature of the lobed forms of distortion, and
shows that the terms in TZ are important. Now it will be shown that the value of
Smh,., when k = 0, may be obtained, correctly to terms in t2, by the ordinary theory
of thin shells ; and as there is no reason to believe that the latter theory will lead to
MR. R. V. SOUTHWELL OX THE GENERAL THEORY OF ELASTIC STABILITY. '231
less accurate results when k has a finite value, it does not seem necessary to employ
iiur more rigorous method, with the very laborious calculations which it entails. We
shall therefore rely upon the approximate theory for the treatment of the tubular
'strut problem in its gcnrr.il f'«>rm. Slight modifications in method will be introduced,
as suggested alx>ve (pp. 224-225), and only the more important steps will be given
here.
Solution by (tie TJteot-y of Thin Shells : General Case.
We consider tin- stability of an element of the tube, originally bounded by the
planes
6, 9+ S6, and z, z + Sz,
as shown Ixslow—
The other dimension of the element is the full thickness of the tube, denoted in
this paper by 2t. The radius of the middle surface is a.
The initial stress system is
P, = const. = - - = [PJ (say).
In the distorted position this system produces a radial force on the element, of
amount
. -
where B is the radius of curvature of a section of the distorted element by an axial
plane (see fig. 5).
It also produces a tangential force, in the direction of 6 increasing, of amount
where -^ (see fig. 5) = - — (l + «„) Sz.
a 06
\IK i; v MUTHWI i.i. ON TIM: <;KM:RAL THEORY OF ELASTIC STABILITY.
The above system of distorting forces must l>e exactly balanced by the restoring
system shown in the upper part of the figure. Hence we obtain the following
equations of neutral stability :—
;
R 9z a 80 a
8^ T, ISP, au,_
a 80 a a 30 82 '
82* + a 80'"0' '
v\Jl m 1 Oil ,.
~Sz ' l~ a~SO '' '
(91)
Now R and e,, may be expressed in terms of the displacements of the middle
surface, as follows : — •
1 8V
R " 8z> ' : 82 ;
(92)
and the restoring system of stress resultants may also be expressed in terms of this
system, as follows* :—
p 3DT8M/ , 1 / , a-jAI
-ti = -j-\ T~ + ~~ (u + ^77 .
r L 82 m«\ 80/J
«
- —
m a \.8082 82;
. . (93)
where D is the quantity
mr
m2-!
(94)
<Mathc"iatiCa
°f Etatid^' <80cond editio«). Chap. XXIV. u', „', W' have the
of thu
MR R. V. SOUTHWELL OX THE CENEKAL THEOUY OF ELASTIC STABILITY. 233
Eliminating T, and T, from equations (91), and substituting from (92) and (93), we
have
u' ^ 3V ( 1 3t/ t 1
a* 32* a2 30 ma 3z
t*[l 3V 1 8V 2 8V 3V 1 3V tii-l 1_3V_~|
3 La4 30" a4 S04 a* 30*32* 82* m«* 32* " m «* 3032*.] =
I P^ 1 aV m-1 8V . /m+1 . \ 1 3V
a» 30 a1 36s 2m 3z» \ 2m ' /a 303z
_^n<V ,13V 1 3V m- 1 1 3 VI
~ 3 L<? 30 a4 dP a* 30 82* m a* 82* J *
J_ <to/ m + 1 1 8V m-1 1 8V 3V =
o«. o™ „» 2tf* a.** " '
82 " 2m a 3032 2m a* 30* " 82*
(95)
where
(96)
Assuming a solution of the type (56), we find, as the criterion for neutral stability,
m a
m — 1 „» i i w»— 1 ,
m ' a"
m+1 ;,
2m
m
m + 1
2m
m— 1
= 0.
(97)
This equation, in its expanded form, is
t4 + 3/ty + 2
^=1 g4]} = 0.
nr
(98)
Taking first the terms which are independent of t1, and neglecting the square of
(which must be small), we find, as the first term in our solution,
m*-l
m
(99)
VOL. CCXIII. — A.
2 H
234 MR. R- V. SOrrmVKLL ON THE GENERAL THEORY OF ELASTIC STABILITY.
When k = 0, this becomes
»ia-l 1 ..... (100
m* q3'
which shows that q must be great, if * has a value possible in practice. Similarly,
when k >0, we see that q must be small, and the approximate expression m this
case is
4,_^!nl __ 2l_ ..... (101)
m* F(*3+l)
We may now determine sufficiently approximate expressions for the terms in t2/a3,
by treating q as great when k = 0, and as small when k> 1. That is to say, we
retain only the highest and the lowest powers of q in the two cases*
Thus, when k = 0, the important terms are
and we have
a >
a?
or
(102)
When k > 1, the important terms are
i94 + i-^4(^-1)2 = 0' • • • • (103)
771 t*
whence, to terms in t^/a3,
m'-i
_m8-! q3 iFjF-l)^2
m2 F(P+1) *ga F+l a2'
with sufficient accuracy, when q is small.
This leads to the result
For practical purposes only the stationary values of S are important. It is readily
seen that the minimum value obtained from (102) agrees with (90), and is therefore
* In every case it is legitimate for practical purposes to neglect the term in ¥2.
Ml;. K. V. SMITH WELL ON THE GEXKKAL THEORY OF ELASTIC STABILITY. L'.'if)
accurate as far as terms in t3; we shall assume that (104) gives the same approxi-
mation, which for practical purfxises is quite sufficient. We then find, for values of k
other than 0 and 1, the expression
When k = 1, the axis does not remain straight after distortion of the tube has
occurml. This is the type of distortion (sometimes called " primary flexure ") which
was discussed by ?]UI,KK, and it is easy to see that his result is identical with that of
equation (104), which Incomes in this case
(106)
The exact expression for the length of the tube, in terms of q, is not a matter of
great importance in the present problem, because the wave-length corresponding to a
minimum value of the collapsing pressure is in all cases small, and the strength of
iiny strut of ordinary dimensions will therefore be given by equations (90) or (105),
into which the li-n^th does not enter. As in the case of the boiler-flue problem, we
\Murs ofZj 4 6 8
Fig. 6. Strength of Tubular Struts.
10
12
may illustrate the effects of length upon the collapsing thrust by plotting the
intensity of stress, or Sfj^at, against q~\ For this purpose we must take some
definite value of the ratio t/a, and draw separate curves for different integral values
of k. The result is shown by tig. 6, in which the following values are assumed :—
m =
E = 3 x 107 pounds per sq. inch.
= 2 '07 x 10" dynes per sq. cm.
2 H 2
Mi; I, v sorTHWKLL ON THE GENERAL THEORY OF ELASTIC STABILITY.
From an inspection of these curves it is easily seen that as the axial wave-length
increases the type of distortion which involves the least value for the collapsing
thrust (and which the tube therefore tends naturally to assume) changes. For very
short lengths we shall expect the circular type (k = 0) ; then, as the length increases,
lobed forms of distortion, in which the value of k becomes less as the length increases.
The limit is reached when k = 1 ; hence, the tendency of very long tubes is always to
collapse in the manner discussed by EULER.
It is also to be noticed that those parts of the different curves which lie to the
right of their lowest points have no practical significance. The actual curve, which
shows the effect of length upon the value of the collapsing thrust, will approximate
to the form shown in thick lines, since the wave-length (which varies as q~l) will
naturally not increase beyond that value which involves the least collapsing thrust.
Comparison with Existing Formula.
The formulje of equations (90) and (104) may be compared with the results of other
discussions of this problem. Equation (90) has been obtained by LORENZ,* and
LILLY* has given the same result, except that the factor A/ -m—- is omitted. t
The only existing solution for lobed forms of distortion is due to LORENZ,* and this
is not in agreement with equation (104). In support of the latter formula it may be
urged that LORENZ' formula does not agree with EULER'S result when k = 1.
It may also be remarked that the foregoing results for the tubular strut problem
contradict BRYAN'S theorem, that a closed shell cannot fail by instability, because
distortion would involve extension of the middle surface; for although the first
terms in equations (102) and (104) are due solely to extension of the middle
surface, yet the compressive stress at collapse, as given by (90) or (105), may be
insufficient to produce elastic breakdown in the position of equilibrium, if the ratio
- has a sufficiently low value.
Stability of Tubes under Combined End and Surface Pressure.
We shall not treat this case in any detail, but it requires notice in connection with
the " localization of collapse " which is observed in experiments conducted upon long
tubes tested under hydrostatic pressure, the permanent distortion being generally
confined to a portion only of the length of the tube. This result is not predicted by
the theoretical formula (83), which suggests a steady fall in the value of the collapsing
pWMure as the wave-length increases ; and a partial explanation may possibly be
found in the fact that the method of test has generally left a wholly or partially
* Cf. footnote, p. 209.
t For a similar omission in a solution of the boiler-flue problem cf. p. 225.
MR. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 237
ii n lialanced end-thrust, due to the water pressure acting upon the closed ends of the
t ul>e.
It is clear that the expansion of the general fifteen-row determinant will give an
equation of the form
... = 0,
where a, ft, y... depend upon the dimensions of the tube and the type of the
distortion. But in any practical case, as we have already observed, A and B must be
very small quantities. It follows that an approximate solution may l» obtained from
the terms
0 .......... (107)
]) .-ui'l S I"- ill'- \.-ilui-s ..r the extent*] pn-^mv .-UK I <>f il ..... ml- thrust, r.-idi <.f
which, acting alone, could produce collapse into the assumed type of distortion. Then
equation (107) may clearly be written as follows :—
(108)
where $, and S are the values of the external pressure and end-thrust which can
produce collapse when acting in conjunction.
It may be seen from this equation that $, can have a minimum value for some finite
value of the axial wave-length when, and only when, <& exists. If the end-thrust be
entirely unl>alaneed, we have
,, ......... (109)
and the collapsing pressure may, in this case, be determined from equation (108).
GENERAL THEORY OF INSTABILITY IN MATERIALS OF FINITE STRENGTH.
The Practical Value of a Theory of Instability.
In the concluding section of this paper an attempt will be made to estimate the
practical value of a theory of elastic instability ; to suggest ways in which we may
hope to increase this value ; and to indicate the questions to which answers must be
found in order that further advance may be possible.
The first point which must be noticed is the non-realization in practice of our
conception of a " critical loading," owing to imperfections which always exist, and
which violate our ideal assumptions. In any actual example the displacement of the
system increases continuously with the load, and the system collapses at a smaller
value of the load than our theory would dictate. It is necessary to inquire whether
serious discrepancies are to be expected.
In some mechanical problems the effects of imperfections may be calculated. We
may take, as an example, the system illustrated in fig. 1, and consider any one of the
Mi:. U. V. sol TIIUKI.I. ON Till: OIWEEAL THEORY OF ELASTIC STABILITY.
many ii,,rrfections which occur in practice. For simplicity, let us assume that the
sph.-'r.-. l"wl, ami pluiitf-r are still smooth, rigid, and accurately formed, but that
tli.- line of thrust of the plunger is eccentric by an amount S. It is easy to see from
ti- 7 that the displacement of the sphere from the line of thrust of the plunger, when
the system is in equilibrium under a load P, is
d = r sin 6 = A+(R-r) sin 0,1
where L . (110)
_P _ tan 0 .
W ~~ tan 6- tan $
and these equations enable us to trace the steady increase of the sphere's displacement
as the load on the plunger is increased from a zero value.
o Values of d.
i-or
Fig. 7.
Fig. 8.
Thus in fig. 8 curves are drawn to connect P and d, for a value 3 of the ratio R/r,
when the initial displacement S has the values 0, O'Olr, and O'lr respectively. At
the points on these curves for which P has a maximum value, " collapse " will occur,
since the equilibrium then becomes unstable. The locus of these points is shown in
the figure by a broken line, and a dot-and-dash line shows the connection between S
and the maximum value of P. From the latter curve it is evident that a small
initial inaccuracy may cause a material reduction in the " collapsing load " ; never-
theless the " critical load " gives a limit which will be more and more nearly attained
as our experimental accuracy is improved, and its investigation is by no means useless
for practical purposes.
When the problem is one of elastic stability, the discussion of imperfections by
analytical methods will, in general, be beyond our power ; but it is clear that similar
remarks will apply. An " exchange of stabilities " at some " point of bifurcation "*
must be regarded as a purely ideal conception, and in practice there will always be a
steady increase of distortion as the load is increased, owing principally to practical
imperfections of form. A strut, for example, may be very accurately loaded, if
suitable methods are employed, but its centre-line will never be quite straight ; the
initial deflection which characterizes it may be regarded as composed of a series of
* H. POIXCARE, ' Acta Mathomatica,' 7. (1885), p. 259.
Mi:. R. V. SOUTHWELL ON TIIK « iKNT.i: AL THEORY OF ELASTIC STABILITY. 239
harmonic terms, and when the load is applied one of these harmonics will be
developed very much more than the others, just as one constituent harmonic may be
developed by " resonance " in an alternating current wave of irregular shape. In the
ordinary strut problem this in.i^nifi.-il liannonir is sin-b that one-half wave occupies
the length of the strut, but in other problems, such .'is that of the tubular strut,
though there is always a " favourite " or " natural harmonic " which is especially
magnified, its relation to the dimensions may l>e more complicated.* In any case the
effects of practical imperfections of form might be studied, if the analytical difficulties
could t)e surmounted, by investigating the rate at which the amplitude of this
" natural harmonic " increases with the load, when its value in the initial configu-
ration is given ; and the results of the investigation might be shown graphically by
curves of distortion, similar in character to the curves of fig. 8, in which the aljscissae
represented the amplitude of the natural harmonic, and the ordinates represented the
magnitude of the applied stress-system, or " load."
These " curves of distortion " are of considerable utility for the study 01 problems
in elastic stability, even though their true form can only be guessed. They help us,
for example, to explain, and in some degree to remedy, the serious discrepancy
existing between EULEK'S theory and the results of experiments on short struts.
The discrepancy has often been attributed to practical imperfections of form ; but it
should hardly be necessary to point out that practical imperfections are likely to
diminish rather than to increase in importance, as the dimensions of an elastic solid
become more nearly comparable, so that they will never be more effective as causes of
weakness than in struts of great length, which, as a matter of fact, give results in
close agreement with EULER'S formula.
A more satisfactory explanation of this, and of similar discrepancies in other
problems, may be found in the fact that the ordinary theory of elastic stability
neglects the possibility of elastic break-down. If we attempt to draw " curves of
distortion " for any single problem, we shall find that, apart from the other data of
the problem, three possible cases exist, depending upon the elastic limit of the material
under consideration : —
(1) The material may be of infinite strength ;
(2) Its elastic limit may be so high that the critical load, as determined by the
theory of instability, is not sufficient to cause elastic break-down in the
configuration of equilibrium ;
(3) Elastic break-down may occur, even in the position of equilibrium, at a load less
than the critical value.
In the first case (which is, of course, purely ideal), the distortion due to loading
will vanish when the loading is removed, and in this sense we may say that the
* In the problem of the tubulur strut, the " favourite harmonic " is, of course, defined by that value of q
which corresponds to a minimum value of J in equations (102) or (104).
240 MB. R. V. SOUTH \\KI.I, ON THK CKNKKAL THKOKY OF ELASTIC STABILITY.
material will never fail The " curves of distortion," if we could determine their true
sli:.|*', would probably be approximately of the form shown in fig. 9. The theoretical
methods of this paper enable us to fix the position of A, the " point of bifurcation,"
but give no information as to the form of AB, beyond the fact that it cuts OA at
right angles.* The other curves of the diagram will approach more and more closely
the limiting form OAB as the initial value of the amplitude is decreased.
In the second case, we have the additional complication of elastic break-down
under finite stress, which reduces the resistance of the material and causes the new
"curves of distortion," shown by thick lines in fig. 10, to begin at certain points to
fall away from the corresponding curves of fig. 9 (reproduced in fine lines for com-
parison) ; these points will lie on some line such as CD, cutting OA at a point
above A, and it is clear that to the right of CD the curves of distortion refer to
displacements which do not wholly vanish when the load is removed. Total collapse
of the system will obviously occur at the points of maximum load on the curves of
distortion, and the locus of these points, which is shown on the diagram by the
dot-and-dash line EF, may lie termed the " line of final collapse."
Amplitude
Fig. 9
Natural
Fig. 10.
Harmonic.
Fig. 11.
A knowledge of the true form of EF would enable us, when we are given the initial
value of the amplitude, to predict the load at which the system will collapse ; and
these quantities could be connected by another curve AG, which would show
at once whether the resistance of the system to collapse is seriously reduced by
practical inaccuracies of form. A complete theory of any problem in elastic stability
must yield information on this very important point, as well as an expression for the
" critical load " ; but in most cases more powerful methods would be needed for its
derivation than are at present available. The investigation of the " critical load" is
therefore not without utility, for although never realized in practice, this forms a limit
which should be fairly closely approached when considerable accuracy is possible.
In our third case the " critical load," as deduced by theoretical methods, is more
than sufficient to cause elastic break-down. We may proceed as before to draw
hypothetical curves of distortion. The line CT)' (fig. 11), which corresponds to the
* It must not be assumed that AB is a horizontal straight line; in general, since the distorting
effect of the applied stress-system, which varies as the deflection, increases less rapidly than the
resistance, which varies as the curvature, AB will tend to rise from A.
MR. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 241
\\iw CD of fig. 10, will intersect OA at a point below A, and the other curves of
distortion at correspondingly lower points. We have seen that the effect of local
elastic breakdown upon fig. 10 was to deflect the curves of distortion from the forms
which they would have assumed if the material had possessed indefinite strength ;
and it is clear that this deflection will begin at lower values of the loads in the
present case. We may therefore expect curves of the type shown in thick lines in
fig. 11, where the curves already obtained are reproduced in fine lines for comparison.
As before, we may draw a line A'F "of final collapse" through the points of
maximum load on the curves of distortion, and connect the collapsing load with the
initial value of the amplitude by another curve A'G'.
It is clear that the curves of distortion must tend to a limit which is no longer
OAH, but some other curve OA'H', where OA', the critical load under the new
conditions, is more than sufficient to produce elastic breakdown, but less than OA.
We can see further that the curve A'G' is not likely to fall away from A' much more
steeply than AG from A in fig. 10. The great weakness of short struts in practice,
compared with EULER'S theoretical estimate, is now explained. Whereas long struts
come within the conditions of fig. 10, the failure of short struts will be repre-
sented by fig. 11, and occurs at comparatively low stresses, not because practical
imperfections have a greater effect upon the strength, but because OA', the true
value of the critical load, is less than OA, the value which EULER'S theory would
dictate.*
It is the rule, rather than the exception, that the critical load, as found by the
ordinary theory of elastic stability, is more than sufficient in practice to produce
elastic break-down. This may be readily seen in reference to any particular example.
In the case of the tubular strut, fig. 10 is only applicable when the ratio of diameter to
thickness is greater than 560 (for an average quality of mild steel), and for thicker
tubes the critical load falls, apparently by a very considerable amount, t below the
theoretical estimate. The determination of the critical load, in cases where this is
more than sufficient to produce elastic break-down, is thus a problem of great
importance, since it forms a limit which can never, under any circumstances, be
exceeded. In the ordinary strut problem the determination can be effected without
difficulty, and an apparently new field is thus indicated for research. The distin-
guishing feature of its problems is the dependence of the stress-strain relations upon
the past history of the material, rendering absolutely necessary a method which
follows the actual cycle of events up to the occurrence of collapse.
[* Added May 11. — Since this paper was written, the author's attention has been drawn to a
dissertation by T. VON K A KM AN (' Untersuchungen uber Knickfestigkeit,' Berlin, 1909), in which the
forms of these " curves of distortion," for solid struts of practical dimensions, are deduced both from theory
and from experiments. KARMA'N also gives a relation equivalent to that of equation (112).]
t Experiments conducted by the author upon seamless steel tubes showed failure under loads which
were in every case little more than sufficient to produce " permanent set."
VOL. CCXIII. A. 2 I
MS MH. R V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY.
Stability of Short Struts.
This problem has been discussed elsewhere by the author,* and it will be noticed
here only at sufficient length to indicate the directions in which further research is
needed. We have to derive an expression for the collapsing load of a straight strut,
when this is more than sufficient to cause elastic break-down of the material ; and we
proceed as before by considering three configurations of the strut : (l) before strain ;
(2) in a position of neutral equilibrium under uniform end-thrust; and (3) in a
position of infinitesimal distortion from the second configuration.
For a first approximation we may say that cross-sections remain plane in the third
configuration, so that the diagram of longitudinal compressive strain for any cross-
section is as shown in the upper part of fig. 12 ; the horizontal line fg shows the
f Diagram F of G Strra K
Strain.
Fig. 12. Fig. 13.
uniform strain of the second configuration. Then, if fig. 13 be the stress-strain
diagram for a compression test of our material, and this uniform strain corresponds to
a stress p which is represented by the point B, we see that to the right of the point
F in fig. 12 the longitudinal compressive stress in the third configuration must be
greater, and to the left less than p.
Now it is a well-known property of metals that if at any point B on the stress-
strain diagram, beyond the elastic limit, we begin to decrease the load, the diagram
is not retraced, but that we obtain a line BC which is parallel to OA.t It follows
that the ratio decrease of stress
decrease of strain
is still given by E, YOUNG'S Modulus for the material. On the other hand, the
diagram shows that if we increase the load beyond B by an infinitesimal amount, the
r:itl" • P
increase of stress
increase of strain
is a smaller quantity E', which may be found from the slope of the diagram at B.
* ' Engineering,' August 23, 1912.
t A. MORLEY, 'Strength of Materials,' § 42.
MR. K. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 243
We are considering an infinitesimal distortion in the third position, so that if we
represent the increase of strain in fig. 12 by \z, the increase of longitudinal com-
pressive stress to the right of F may be taken as E'Xz, and the decrease of this stress
to the left of F as EXz. Hence we obtain, for the section under consideration, the
diagram of longitudinal stress which is shown in the lower part of fig. 12. The
uniform stress of the second configuration is shown by the horizontal line In, and it is
a condition for neutral stability in the second configuration that no increase of thrust
shall be required to maintain the distortion. If the cross-section of the strut is
rectangular, of dimensions a x 2t, it follows that the triangles Imk and qmn must be
equal in area, or
19? -E (111)
PF~E''
This relation fixes the position of F on the cross-section, and in terms of the
dimensions shown in fig. 12 we may write for the moment of resistance about G,
M =r+*E'a\2(z -6)^2+1° Ea\z(z-b)dz
But if y is the deflection of the strut at the point G, in the infinitesimal
distortion, we have, as in the ordinary theory of bending,
where x denotes the distance of the section from one end ; and for equilibrium in the
third configuration M must be equal to the bending moment due to thrust, or 2atpy :
hence,
Equation (112) shows the modification which must be introduced, to take account
of elastic break-down, into EULER'S equation
(113)
and it is easy to see that if I is the length, calculated from (113), of a strut which
can just support the stress p, and I' the length as calculated from (112), then
But, from (111),
t + b _ /E
t^b~ V W
2 I 2
•j»4 MK. IS. \. SiMTWVKLL ON THE GENERAL THEORY OF ELASTIC STABILITY,
so that, finally,
T-— ^ <1U>
This result leads to a simple method by which the collapsing loads of short struts
may be obtained graphically from the compressive stress-strain diagram. A full
explanation is given in the paper to which reference has already been made, and a
comparison, showing satisfactory agreement, is made with the results of experiments.
One conclusion of some practical importance may be noticed : the curves of collapsing
stress show that great ductility of material is by no means desirable in struts,
the primary requisite being a high elastic limit.
Need for Further Research. Conclusion.
With slight modification, the theory just given for short struts might be applied to
the problem of circular rings under radial pressure ; but these appear to be the only
cases in which we can at present discuss the stability of overstrained material. In
any problem dealing with plates or shells distortion from the equilibrium position
must introduce new stresses, in directions perpendicular to that of the stress which
has caused elastic failure. The circular type of distortion in a tubular strut, for
example, will introduce " hoop " stresses, and at present we have no knowledge of the
corresponding stress-strain relations when " set " has occurred.
This and many other stability problems may be regarded as special cases of a
general problem, viz., the determination of the changes of strain which occur when an
infinitesimal stress-system, defined by principal stresses g, r, s, is impressed upon a
material already overstrained by a simple stress p. The problem is not simple, and
its solution would probably entail much theoretical and experimental work ; but this
would be justified by the importance, both for theory and practice, of its applications.
In conclusion, the author desires to express his indebtedness to Profs. LOVE and
HOPKINSON, for valuable criticism and advice ; to Mr. L. S. PALMER, for the photo-
graphs reproduced in fig. 2 ; and to Messrs. H. J. HOWARD and D. P. SCOTT, for
assistance in the prosecution of the experiments described on p. 223. He also takes
this opportunity of thanking Messrs. Stewarts and Lloyds, Ltd., of Glasgow, for
gifts of very accurate steel tube for experimental purposes.
[ 245 ]
VI. Some Phenomena of Sunspots and of Terrestrial Magnetism. — Part II.
liy C. CHREE, Sc.D., LL.D., F.R.S., Superintendent of Kew Observatory.
Received June 10,— Read June 26, 1913.
CONTENTS.
§§
1. Introductory 245
2. International magnetic " character " figures 246
3-6. 27-day period in international " character " figures 247
7-8. Subsequent and previous associated " pulses " 250
9-11. Estimates of length of period 255
12-13. Relations between primary and associated pulses 258
14-15. Comparison of previous and subsequent associated days in 1890 to 1900 261
16. Annual variation in 27-day period 263
17-18. Relation of magnetic " character " figures to sunspot areas 267
19. Sunspot areas, projected and corrected, faculse and WOI.FER'S sunspot frequencies . . 271
2Q. Amplitude of 27-28-day period in projected sunspot areas 274
21-22. Concluding remarks 275
§ 1. IN a previous paper, described here for brevity as S.M.,* whicb referred to
sunspots and terrestrial magnetism, I had occasion to enquire into the existence of
any relation between the magnetic character of individual days and that of days
separated from them by a given interval of time. References to previous work
bearing on the subject will be found in S.M., p. 97.
The material of which I made principal use consisted of magnetic " character "
figures — on the international scale 0 (quiet), 1 (moderately disturbed), and 2 (highly
disturbed) — assigned by myself to all the days of the eleven years 1890 to 1900, from
consideration of the Kew magnetic curves.
In each of the 132 months of the eleven years the five days were taken which gave
the largest daily range to the magnetic horizontal force. In default of any more
satisfactory means of selection, the 660 days thus obtained were taken as representative
of disturbed conditions. Regarding any one of the selected days as day n, the
" character" figures for the 41 successive days n— 5ton + 35 were written down in a
row. This was done for each of the 660 selected days in succession, so that there
were in all 41 columns of figures, each containing 660 entries.
* 'Phil. Trans.,' A, vol. 212, p. 75.
VOL. CCXIII. A 502. Publiihed »epar»telj, Augurt 8, 1913.
,J46 PR. C. CHREE: SOME PHENOMENA OF SUXSI'OTS
The " character " figures in each column were then added up as if they were purely
arithmetical quantities, and an arithmetic mean was taken. This was regarded as a
measure of the disturbance existent on the representative day of the column. Thus
the means for columns n, n-1, and » + l represented respectively the amount of
disturbance on the typical selected disturbed day, and on the days immediately
preceding and following it. These mean " character" figures showed in the clearest
way the existence of a period somewhat in excess of 27 days, but no shorter period
was disclosed. This implied that if any day were considerably more disturbed than
the average day of the month, then the day 27 days subsequent to it was likely to be
also more disturbed than usual
The acceptance of the arithmetic mean of a number of " character" figures as itself
a measure of magnetic disturbance is open to criticism on several grounds. There is no
strict line of demarcation between the three classes of days. There are in reality an
infinite variety of grades intermediate between the extremely quiet day, which cannot
get less than " 0," and the extremely disturbed day which cannot get more than "2."
Some days to which " 2 " is allotted represent disturbances whose energy on any
conceivable view must be immensely more than twice — possibly more than twenty
times— the energy of disturbance on the average day of character " 1." The
procedure was suggested by the practice followed at de Bilt, where the " character "
figures supplied by the different observatories are dealt with. Supposing data to be
supplied by, say, 40 observatories, the 40 figures assigned to any one day are summed
and the mean taken to the nearest O'l, and the result is accepted as an international
measure of the amount of magnetic disturbance on the day in question.
§ 2. The " character" figures in S.M. were based on the curves of only one station,
Kew ; they were assigned by a single individual, myself ; and they referred to one
period of years, 1890 to 1900. I have thus thought it desirable to repeat the
investigation for a second period of years, 1906 to 1911, making use of the inter-
national "character" figures published at de Bilt. 1906 was the earliest year for
which international figures existed, and 1911 was the latest for which these figures
were complete when the present enquiry commenced. As before, five days were
selected for each month ; but they were selected solely by reference to the international
lists, being the five days of highest " character " figures in each month. When, as
occasionally happened, there was a possible choice between two or more days for the
last place on the monthly list of five, the criterion applied was that the selected days
should, if possible, be consecutive. I had had occasion some years ago, before the
present enquiry was even thought of, to select the five most disturbed days of each
month of the years 1906 to 1909, and had made use of the above criterion. There
seemed no reason to discard the old list, or to follow a different principle when dealing
with 1910 and 1911. My experience when forming the first list had led me to regard
five as a happy choice for the monthly total of disturbed days. A considerably
smaller number, such as one or two a month, gave too few days to eliminate
AND OF TERRESTRIAL MAGNETISM. 247
accidental features, unless a much larger number of years were available. On the
other hand, if one took as many as ten days, there would in most months be several
days competing for the last place on tin- list, and during magnetically quiet times
m.iiiy of the days occurring in tin- monthly choice would have represented quiet
rather than disturbed conditions.
The present p:i|»-r is not confined to the period 1906 to 1911, but utilises as well
my original data for 1890 to 1 900 for the investigation of various points not considered
inS.M.
^ :;. Tin- first step was to make sure that the period of approximately 27- days was
confirmed by the international "character" figures from 1906 to 1911. The mean
results obtained for the individual years from 5 days before to 30 days after the
representative day n of large disturbance are given in Table I. The entries represent
the mean international " character " figure The last column gives for comparison the
mean "character" figure for all days of the year. In the case of 1911, December
was excluded, so as to keep all the days dealt with within the six years. The results
were really taken out to three decimal places, and these more exact values were used
in calculating some of the later results in the paper.
§ 4. Before discussing the main question, some phenomena in Table I. call for
remark. The entries in column n and the means from all days show but little
variation from year to year, and the natural inference would be that the six years
were almost equally disturbed. The phenomena, however, is I believe largely due to
another cause. The international data are published quarterly. Thus the man
whose duty it is to assign " character " figures at any observatory naturally deals with
the curves of not more than three months at a time. In most cases, doubtless, he
has a desire to maintain something like a uniform standard ; but unless his verdict is
based on the exact measurement of some definite quantity, such as the daily range,
he is inevitably much influenced by the accident of whether the months he is dealing
with are quiet or disturbed. One of the leading objects is the discrimination
between the days of each individual month, and if " O's " are given to nearly all the
days of a very quiet month, there is no adequate discrimination. The natural
tendency is thus to assign a " 1 " in quiet months to days which in highly-disturbed
months would naturally get a "0."
§ 5. Another point to bear in mind is that highly disturbed conditions are seldom
confined to a single day, and not infrequently extend over three or four consecutive
days or even more. Not infrequently three or even four of the five most disturbed
days of the month were consecutive. In February, 1907, the whole five were con-
secutive days, and in March and April, 1910, seven of the ten selected disturbed
days were consecutive. This explains why the " character" figures for days n—l and
n+1 in Table I. invariably are next in magnitude to those for days n. But the next
highest figure, it will be seen, occurs on day n + 26 (once), n + 27 (four times), or
(once).
L'4-
DR. C. CHREE: SOME PHENOMENA OF SUNSPOTS
TABLE I.— Mean " Character " Figures from Selected Disturbed Days and from
Previous and Subsequent Days.
Year.
n-5. n-4. n-3. n-2. n-1.
n.
n+1. n + 2. n + 3. n + 4. n + 5.
1906 ....
1907 . . .
0-63 0-56 0-59 0'66 0'92
0-59 0-60 0-60 0-64 0'95
1-31
1-32
0-99 0-63 0-55 0-57 0-58
0-96 0-72 0-65 0-56 0-53
1908 ....
0-55 0-49 0-52 0'68 I'Ol
1 34
1-08 0-81 0-64 0-57 0-57
1909 ....
0-61 0-66 0-55 0-66 0-91
1-32
0-99 0-74 0-60 0'69 0-62
1910 . . .
0-66 0-64 0-66 0-77 0'97
1-31
1-04 0-90 0-84 0-81 0'73
1911 ....
0-67 0-58 0-53 0-64 0'95
1-32
1-07 0-83 0-70 0-71 0'72
Mean . .
0-62 0-59 0-57 0'67 0'95
1-32
1-02 0-77 0-66 0-65 0-63
n + 6. n + 7. n + 8. n + 9. n +
10.
n +
11. m+12. n +
13. n+H. Ti + 15.
1906 ....
0-60 0-66 0-63 0'62 0-63
o-
58 0-60 0-67 0-69 0-61
1907 ....
0-57 0-57 0-59 0'62 0-56
o-
61 0-62 0-68 0-67 0-65
1908 ....
0-64 0-73 0-77 0-78 0-69
o-
58 0-54 0-53 0'49 0-51
1909 ....
0-55 0-56 0-53 0'50 0-51
o-
49 0-52 0-48 0'49 0-52
1910 ....
0-74 0-72 0-68 0-66 0'60
o-
65 0-71 0-72 0-68 0'68
1911 ....
0-66 0-67 0-61 0-56 0-57
o-
63 0-57 0-51 0-49 0'45
Mean . .
0-63 0-65 0-64 0-62 0'59
o-
59 0-59 0-60 0-59 0'57
n+16. n+17. n+18. n+19. n + 20.
n+21. n+22. n + 23. n + 24. n+25.
1906 ....
0-65 0-68 0-65 0'69 0-
68
o-
60 0-56 0-57 0-58 0'62
1907 ....
0-62 0-58 0-60 0'60 0-
64
o-
64 0-65 0-
61 0-59 0-65
1908 ....
0-55 0-67 0-73 0'72 0-
63
o-
62 0-58 0-
61 0-72 0-78
1909 ....
0-55 0-53 0-53 0'53 0-
52
o-
58 0-68 0-
73 0-65 0-51
1910 ....
0-70 0-68 0-67 0'65 0-
69
o-
65 0-64 0-
63 0-64 0-72
1911 ....
0-54 0-51 0-54 0-58 0-
60
o-
68 0-65 0-
60 0-56 0-63
Mean . .
0-60 0-61 0-62 0-63 0-
63
o-
63 0-63 0-
62 0-62 0-65
n + 26. n + 27. n + 28.
Ti + 29.
n+ 30.
Mean from all days.
1906 .
0-71 0-73 0-69
0-67
0-63
0-65
1907 ....
0-72 0-77 0-72
0-71
0-75
0-66
1908 ....
0-90 0-89 0-85
0-70
0-66
0-68
1909 ....
0-55 0-73 0-75
0-72
0-69
0-62
1910 ....
0-83 0-92 0-86
0-85
0-80
0-72
1911 . . . .
0-79 0-99 0-98
0-83
0-70
0-65
Mean . .
0-75 0-84 0-81
0-75
0-70
0-66
AND OF TERRESTRIAL MAGNETISM.
24'J
Taking the means from the six years, the mean "character " figures for days n + 27
and n + 28 considerably exceed all others, that for day n + 27 being decidedly the
larger. The dost- ivsrmlil.mrr to the results for the epoch 1890 to 1900 in S.M. will
be readily recognised on consulting fig. 1.
o-5
In 1890 to 1900 the mean character figures for day n, for day n + 27, and for the
mean day of the period were respectively 1'51, 0'94 and 070, so that the excess of
the " character " figure for day n + 27 over that for the average day was 30 per cent,
of the excess for day n. In 1906 to 1911 the corresponding percentage is 27.
It is unlikely that my personal standard for disturbance when assigning "character"
figures to the days of 1890 to 1900 agreed with that of the international list, which
represents a compromise of most diverse standards from some forty observatories.
Thus the fact that the mean " character " figure for the selected disturbed days of
1906 to 1911 was only 87 per cent, of that for the selected disturbed days of 1890 to
1900 does not necessarily imply that the second epoch was the quieter of the two.
Such, however, was actually the case on the whole, though no year of the later period
was as quiet as 1900.
The two curves of fig. 1 agree in showing no decided trace of any period shorter
than 27 days. Other points of resemblance are that the fall subsequent to the
maximum during days n + 28 to 71 + 30 is decidedly slower than the rise during days
n + 25 to n + 27, and that the pulse centering about day /i + 27 is spread over more
days than the primary pulse centering at day n. The latter phenomenon would
obviously tend to happen if the period had not always the same length but oscillated
slightly about a mean value.
§ 6. With a view to following up this last idea, I took from the selected disturbed
days of the six years all those whose "character" figures were not less than 1'5, the
VOL. CCXITI. — A. 2 K
250 DR. C. CHREE: SOME PHENOMENA OF SUNSPOTS
group thus representing a specially high grade of disturbance. There were in all 103
,,f tli.-sr days, the annual number varying from 15 in 1907 to 20 in 1906. The
following were the mean character figures found for the primary day and the
subsequent days indicated :—
Day
n.
n + 25.
n + 26.
n + 27.
n + 28.
?! + 29.
«+30.
"Character"
1-683
0-564
0-689
0-821
0-871
0-842
0-748
This gives a period if anything in excess of 28 days, and so suggests a slight
increase in the length of the 27-day period as the intensity of the primary disturbance
is increased ; but a considerably larger number of days, and so a considerably longer
period of years, would be required to establish the result.
The mean "character" figures given above for days n + 28 to n + 30 are dis-
tinctly larger than the corresponding figures in Table I., but the excess in these
days is relatively less than that on day n itself. Thus the excess in the
"character" figure given above for day n + 28 over the average day of the
six years (i.e., 0'871 — 0'663 = 0'208) is only 20 per cent, of the excess on day
n(l'683— 0'663 = 1'020), while the corresponding percentage from Table I. was 27.
§ 7. If individual magnetic storms are directly due to individual sunspots, as
various writers have suggested, it is, of course, a natural inference that when the
sun's rotation has brought a spot round to the position it occupied relative to the
earth when a magnetic storm occurred, a second storm will be experienced. This
seemingly is what led HARVEY and MAUNDER independently to suggest a 27^-day
period for magnetic storms.
Our previous investigations show a period of about 27 days, which, however, is not
confined to what are usually termed " magnetic storms," but belongs equally to
moderate disturbances, which are frequent events. If, then, magnetic storms are due
to sunspots, equally so it would seem must be the minor disturbances ; and if
magnetic storms sometimes recur, as Mr. MAUNDER and the Rev. A. L. CORTIK
believe, at several reappearances of one and the same sunspot, the same thing is to
be expected of minor disturbances. This implies that " character " figures should
show a pulse near day n + 54, as well as near day n + 27.
This conclusion, however, seems a natural one apart from all theory. The
impression left on my own mind after a study of the " character " figures was that a
tendency existed for the magnetic conditions, whether disturbed or not, to be in some
way related to or — as biometricians would say — correlated with the magnetic conditions
prevalent 27 days earlier or kter. Tbe days forming columns n + 26 to n + 30 in
Table I., or in the corresponding table for the years 1890 to 1900, are disturbed
sensibly more than the average day, and we should thus expect more than average
disturbance on days n + 53 to » + 57, with a culmination about days n + 54 and n + 55,
AND OF TERRESTRIAL MAGNETISM.
251
as the period seems in excess of 27 days. As the expected effect appeared likely to
be small, it seemed best to utilise the data from the longer period of years 1890 to
1900. Calculations in that case had previously extended to day n + 35, and they
were now extended to day n + 60. " Character" figures were assigned to the earlier
days of 1901, so as to utilise all the 660 selected disturbed days of the 11 years.
The mean "character" figure from all days of the 11 years was 070. The mean
" character" figures up to day n + 35 are given in S.M. (Table XL, p. 101) ; those for
days 7i + 36 to n + 60 are given in Table II.
TABLE II. — Mean " Character" Figures for Days n + 36 to n + 60, n being the
Eepresentative Disturbed Day of the 11 Years 1890 to 1900.
Day ....
n + 36.
n + 37.
n + 38.
n+39.
n + 40.
n + 4I.
» + 42.
n + 43.
"Character" .
0-63
0-68
0-68
0-66
0-66
0-63
0-64
0-66
Day ....
n + 44.
n + 45.
n + 46.
n + 47.
n + 48.
n + 49.
n + 50.
n + 51.
"Character" .
0-65
0-66
0-67
0-64
0-64
0-66
0-63
0-65
i>»y
n + 52.
n + 53.
n + 54.
n + 55.
» + 56. n + 57. n + 58. n + 59. n + 60.
"Character" .
0-72
0-78
0-84
0 85
0-81 0-76 0-71 0-68 0-64
As shown in S.M. (Table XI.), the " character" figure lay between 0'61 and 0'66
from day n + 5 to day n + 24, and exceeded 070 only from days n— 2 to n+3, and
days ?i + 25 to n+31. There is thus clear evidence in Table II. of a pulse from day
n + 52 to day n + 58, or possibly n+ 59. The figures for days n + 54 and n + 55 distinctly
overtop their neighbours, that for day n + 55 being slightly the higher.
§ 8. Reasoning in the same way as before, we should now expect an excess in the
" character" figures for days n + 79 to n + 84, and so on. It will probably have been
realised ere this that carrying the investigation up to day n+60 entailed exceedingly
heavy arithmetical labour, and, as the time at my disposal was limited, it was
important to economise effort. It was anticipated that the successive pulses would
diminish rapidly in .magnitude, and that they would spread themselves over an
increasing number of days, so that the distinction from neighbouring days would be
more and more difficult to establish. Further, there is the possibility that normal
conditions at the time, which includes days which follow the selected disturl>ed days
after a long interval, may differ sensibly from normal conditions answering to the
selected days themselves.
2 K 2
.J5._, DR. C. CHREE: SOME PHENOMENA OF SUNSPOTS
Eventually a practical and economical plan suggested itself. Before adopting it I
had assured myself that the 27-day phenomenon applied to quiet days. It then
became clear that if one selected 5 quiet days for each month, and considered the
days which followed them after any given interval, as well as the days following the
selected disturbed days after the same interval, it was necessary to consider only a
comparatively few consecutive days near the date when the pulse was expected to
appear. For instance, days from 79 to 84 days subsequent to the 5 selected
disturbed days of January, 1906, are practically contemporaneous with days from 79
to 84 days subsequent to the 5 selected quiet days of the same month. If there is an
appreciable pulse with crest (or hollow) about 81 days subsequent to the represen-
tative disturbed or quiet days, this will be rendered manifest by the differences
between the two sets of subsequent days, irrespective of what the appropriate
average character figure from all days might be.
By this time I had also discovered that the 27-day period is as clearly recognisable
in days which precede as in those which follow selected disturbed days. It was thus
decided to consider days before as well as days after the selected days, and to go
equally far in both directions. It was also decided to take the later period, 1906 to
1911, so as to have an international basis for the selected days, whether quiet or
disturbed. The quiet days were those actually selected at de Bilt.
The final mean results of the investigation are given in Table III., p. 254, and are
shown graphically in fig. 2. But for considerations of time, it would have been desirable
to take more than six days near the epochs where the pulses were expected.
The columns headed D and Q respectively in Table III., refer to the days associated
with the selected disturbed days and to those associated with the selected quiet days.
The number of selected days used was always the same for the disturbed and the
quiet days, but varied, as shown in the second line, because only parts of the first
and last years of the series could be utilised. For example, when dealing with the
days which were from 84 to 79 days prior to selected days, April 1906 was the
earliest month whose selected days one could employ. For that particular quest the
15 selected days of the first 3 months of 1906 had to be omitted, leaving only 345
selected days. Similarly, as no data subsequent to December 1911 were to be used,
the last 15 selected days of 1911 had to be omitted when dealing with the days
79 to 84 days subsequent to selected days. January 1, 1906, was a selected quiet
day, and December 31, 1911, a selected disturbed day. Thus the earliest and the
latest of the selected days, both quiet and disturbed, were omitted from the central
group of days n-3 to n+3, leaving 358 available.
The "character" figures in the third line of Table III. relate to the periods
covered by the corresponding selected days. Thus 0'659 given for the group of days
84 to n-79 is the mean for the period commencing April 1, 1906, and ending
In some ways it would have been better to have replaced this
by a mean applicable to the period containing the days which preceded the selected
AND OF TERRESTRIAL MAGNETISM.
253
days by an interval of from 84 to 79 days, but complications would have ensued,
tacause a day 80 days, for instance, prior to a selected April day may fall in January
or in February.
A general idea of the phenomena disclosed by Table III. will be most easily
grasped by consulting fig. 2. The central vertical line in the figure applies to the
representative days, disturbed and quiet. Abscissae, measured from this line, represent
the interval in days from the representative day, time previous being measured to
+ 0-5
+0-5
ta
•
27
•
8280
84 •'»
/111
£2 r
h
3 -A -A r
LJB. Last f &*
\f Normal
81
V "V
3 * . i Normal „,
25 i 52 •'
9 ! o ^ ^
• If 5
! i \l
55 K
1 ^ V
1 •
27
I/ 5*
27
i
-0-5
6
Fig. 2.
the left, and time subsequent to the right. The numeral attached to any particular
point on a curve signifies the interval in days from the representative day, whether
previous or subsequent. The ordinate represents the algebraic excess of the
" character " figure over the corresponding normal " character " figure in the third
line of Table III.
The representative disturbed day had a "character" 1'321. Its excess, 0'664,
over the corresponding normal value (0'657) is represented by the positive ordinate
marked 0. The representative quiet day, on the other hand, had a "character" of
254
DR. C. CHREE: SOME PHENOMENA OF SUNSPOTS
TABLE III.—" Character " Figures on Specified Days preceding or following
Selected Disturbed and Quiet Days n, of Years 1906 to 1911.
Days . . .
it - 84 to ft - 79.
it - 57 to n -
52.
71 -
30 to n - 25.
7t-3 to
u + 3.
Number of ~|
disturbed or I
quiet days j
used J
345
350
355
358
Mean ~j
"character" 1
from all days f
of period. J
0-659
0-660
0-663
0-657
•W—
" Character."
Day.
" Character."
Day.
" Character."
"Character."
Dav
uay. -
D.
Q.
D.
Q.
D.
Q.
D.
Q-
n-84 (
«-83 1
n-82 1
n-80 1
n-79 1
3-704
3-705
3-718
3-718
3-719
3-702
0-655
0-662
0-627
0 606
0-632
0-646
n- 57
7t-56
ft-55
n-54
« -53
n -52
0-701 l
0-716 (
0-753
0-784
0-755 l
0-729
3-638
3-599
3-547
3-579
3-611
3-637
71 -3(
7i - 2£
n-Vi
71-21
n-2f.
1 0-647
1 0-677
( 0-783
0-836
. 0-813
> 0-754
0-626
0-584
0-566
0-515
0-561
0-660
n-3 0-572
7i-2 0-667
n-1 0-949
n 1-321
7t+l 1-016
7i + 2 0-767
n + 3 0-659
0-661
0-543
0-347
0-135
0-409
0-664
0-746
Days . . .
71 + 25 to n + 30.
Ti + 52 to 7i + 57.
Ti + 79 to T! +84.
Number of "j
disturbed or 1
quiet days f
used J
355
350
345
Mean ~|
"character" 1
from all days f
of period. J
0-663
0-666
0-667
Day.
"Character."
Day.
" Character."
Day.
" Character."
D. Q.
D.
Q.
D.
Q.
n + 25
n + 26
n + 27
» + 28
n + 29
n + 30
0-662 0-591
0-748 0-490
0-830 0-486
0-806 0-535
0-746 0-614
0-704 0-661
n + 52
n + 54
Ti + 55
tt + 57
0-683
0-732
0-776
0-767
0-735
0-697
0-593
0-551
0-527
0-570
0-600
0-649
w + 79 0-662
7i + 80 0-692
7i + 81 0-717
7i + 82 0 • 718
n+83 0-706
7i + 84 0-706
0-614
0-599
0-602
0-602
0-589
0-638
AND OF TERRESTRIAL MAGNETISM. 255
only 0'135, and its deficiency, 0'522, is represented by the negative ordinate
marked 0. The algebraic difference of these ordinates, T186, represents the difference
in " character " between the representative disturbed and quiet days.
It will be seen that the day which is three days prior to the representative
disturbed day is decidedly quieter than normal, and is less disturbed than the day
which precedes by three days the representative quiet day. On the other hand,
the day which is three days subsequent to the representative quiet day is decidedly
more disturbed than normal, and is less quiet than the day which is three days
sulwequent to the representative disturbed day. The latter result especially was
quite unexpected, in view of the frequent occurrences of sequences of disturbed days,
and still more of quiet days. A sequence of five, or even ten, successive O's in the
returns from an individual observatory is not unusual in months of minor disturbance.
The natural inference is that the proverb " the calm precedes the storm " has some
claim to recognition even in terrestrial magnetism.
It may create surprise that the representative quiet day had so large a " character"
figure as 0'135. Days, however, of international "character" O'O are very rare.
There were only four, for instance, during 1906. The phenomenon is considerably
due to a few observatories where O's are assigned to only exceptionally quiet days.
On the other hand, if latitudes over 55 degrees were adequately represented, 0'0's
would be still rarer.
A glance at fig. 2 will show that the 27-day period is just as prominent for quiet
as for disturbed characteristics, and that it can be traced backwards as readily as
forwards. The corresponding patches of curve associated respectively with the
disturbed and the quiet days, as it were, repel one another. This would probably
serve to prove the existence of pulses considerably beyond the range covered by
Table III. and fig. 2.
§ 9. One of the principal objects originally in view was to obtain a more exact
estimate of the length of the period by measuring the interval in days between the
crests of pulses remote from one another. But even in the 79- to 84-days' pulses —
i.e., the third subsequent pulses — the difference between the ordinates answering to
successive days has become very small, so that trifling accidental irregularities are
prejudicial to accurate time deductions. This difficulty will naturally tend to
disappear as the number of years for which international data are available increases,
and the power of the method will thus continually develop.
In § 6, it will be remembered, we obtained a result which suggested that the length
of the period increased with the amplitude of the selected disturbance. If, however,
this were the case, one would expect the interval between successive subsequent
pulses associated with the selected disturbed days to gradually diminish, and the
intervals derived from pulses associated with quiet days to be shorter than those from
pulses associated with disturbed days. These tendencies are not apparent in fig. 2.
§ 10. The fact that the rise in the " character " figure in the two days immediately
._,5(; DR. C. CHREE: SOME PHENOMENA OF SUNSPOTS
preceding the representative disturbed day exceeds the fall in the two immediately
following days has been already noticed. This peculiarity is a prominent feature in
all the associated pulses in fig. 2, except the third previous, where the exact day of
incidence of the maximum is not clearly indicated. In the case of the selected quiet
days, on the other hand, the fall in the " character" figure in the two immediately
preceding days is less rapid than the rise in the two immediately succeeding days,
and the same peculiarity is reproduced in the first previous and the first and second
subsequent pulses. The second previous pulse shows the opposite phenomenon, but
this may arise from the same disturbing cause which has brought the maximum to
day —55 instead of day —54. In the third previous and third subsequent pulses the
shape of the curve is irregular.
Speaking generally, in the case both of the disturbed and the quiet days, while
corresponding pulses respectively to right and left of the central line 00 are very
similar, the curves are not images of one another with respect to 00. The character
of the primary (i.e., central) pulse seems to be impressed on the associated pulses
which precede it, as well as on those which follow it.
The curves for days —30 to —25 and for days +25 to +30 will have a much closer
fit if we cut the paper along the line 00, and bring the lines answering to days —27
and +27 over one another by sliding the one half sheet over the other, than if we
effect this superposition by folding the paper about the line 00.
If the curves had been images of one another, by adding " character " figures for
days n+m and n— m — where n denotes the representative disturbed or quiet day —
we might have got as smooth results for day m as if we had been able to use 12 years'
data while confirming ourselves to days following the selected days. The want of
symmetry makes the conditions somewhat less favourable for evaluating the length
of the period, supposing that not to be an exact number of days. The maxima at
days —54, —27 and +27 in the associated disturbed pulses are sufficiently prominent
to fairly justify the view that the true maxima lie within half a day of the apparent
maxima. This gives for the time of three periods 81 ±1 days, or for one period
27 ±0'3.
The ordinates answering to days +54 and +55 differ but little, while those for
days +81 and +82 are practically equal. Thus the values deduced for the period
from these summits and that at day -54 are respectively 108'5/4, and 135'5/5 days,
or both approximately 271 days.
On the curves associated with the selected quiet days, the maxima at days -81,
27, and +54 are the clearest. From-81 and +54 we get 27'0, and from -55
and + 54 we get 27*25 days.
The associated disturbed curve for days -30 to -25 and the associated quiet curve
for days +25 to +30 both suggest slightly under 27 days for the period.
11. If instead of treating the "character" figures from the disturbed and the
quiet associated days separately, we combine them, we obtain results of much greater
ANI> OF TKKi;l-:sTi;lAL MAGNKTISM
257
symmetry. This has been done in Table IV., the entries in which represent the
differences of corresponding D and Q results in Table III. To save decimals, the
results are expressed in terms of O'OOl "character" unit as unit. As day 0—
i.e., what is called day n in Table III. — is neither previous nor subsequent, but
fundamental for both previous and subsequent days, it appears in both the first and
second lines of Table IV. The entry 1186 ascribed to it represents of course
( 1 '321 -0*135) x 1000. The algebraic sign when omitted is plus. The "character"
figure for the associated disturbed day was invariably the larger, except for the third
days before and after (he selected days.
TABLE IV. — Differences Disturbed less Quiet Associated Days (Unit = O'OOl of
"Character "Unit).
0.
1.
2.
3.
25.
26.
27.
28.
29.
30.
Previous . .
Subsequent . .
Sum ....
1186
1186
602
607
124
103
- 89
- 87
94
61
252
258
321
353
217
271
93
132
21
43
2372
1209
227
-176
155
510
674
488
225
64
52. 53.
54.
55.
56. 57.
79.
80.
81.
82.
83.
84.
Previous . .
Subsequent . .
Sum . . . .
92 144
90 181
205
249
206
197
117 63
135 48
56
48
87
93
112
115
91
116
43
117
49
68
117
182 325
454
403
252 111
104
180
227
207
160
The accordance between the results for the previous and the subsequent days
1, 2, and 3 in Table IV. is quite extraordinarily close. In other words, the primary
pulse obtained by taking the excess of " character " figures for selected disturbed
and adjacent days over the corresponding figures for selected quiet and adjacent days
is almost perfectly symmetrical as between time previous and time subsequent. We
cannot hope to see equal symmetry in the associated pulses, whose form is necessarily
more dependent on accident, but there is at least no marked a-symmetry in the second
and third associated pulses. If curves were drawn to represent these, they would
not be markedly steeper on one side of the maximum than the other. This suggests
adding the two sets of results, as has been done in the last line of Table IV., and
applying the sums to the evaluation of the period. The most orthodox way probably
would be to fit an algebraic curve to each of the successive sets of figures, and
calculate the abscissa of its maximum ordinate. But as there is nothing to guide one
as to what the theoretical shape of such a curve should be, rougher methods may not
VOL. CCXIII. — A. 2 L
25R DR. C. CHKKK: SOME PHKNO.MKNA OF SUNSPOTS
,,,,lik,-ly be iiuite as satisfactory. As an example of the methods actually used, take
tin- data f..r days 52 to 57 in Table IV. The maximum obviously comes between
days 54 and 55, say at 54+ a1. Assume the slopes from the maximum down to days
;,1 and 55 to be the same, and to be the arithmetic means of the slopes from days
:>:! and 54 (129 per diem), and from days 55 to 56 (151 per diem).
Then we have
454 + 140Z = 403 + 140 (1-x)
or
x = 0'318.
Thus twice the period is 54 '3 18 days, i.e., the period is 27 '16 days.
If we take the same days, but assume the slope on the two sides of the maximum
to Ixj the mean of those from days 52 to 54 and from days 55 to 57, the only difference
is that we replace 140 in the above calculation by 141, and again find for the single
period 27' 16 days.
Treating the data for days 79 to 84 in the same way, taking first the arithmetic
mean of the slopes from days 80 to 81 and 82 to 83, and then the arithmetic mean of
the slopes from days 79 to 81 and 82 to 84, we get as estimates for the triple period
81*29 and 81*31 days, both giving 2710 days for the single period.
§ 12. An inspection of fig. 2 suffices to show that the ratio borne by the maximum
ordinate of the first associated pulse — whether for disturbed or quiet days — to the
maximum ordinate of the primary pulse is notably less than the ratio borne by the
maximum ordinate of the second associated pulse to that of the first. These ratios
and those between the maximum ordinates of the several associated pulses are fairly
alike, whether we take subsequent or previous days, and whether we take disturbed
or quiet days. Thus the most accurate information on the subject is probably that
derivable from the data in the last line of Table IV. The ratios between the successive
maximum ordinates deduced from the data in question are as follows : —
Primary.
First associated. Second associated.
Third associated.
1 :
0-284 1 : 0-191
: 0-096
The maximum ordinates of the first, second, and third associated pulses stand to
one another almost exactly in the ratio 3:2:1. It is easily seen in fact in fig. 2
that the summits of corresponding first, second, and third associated pulses lie nearly
on straight lines, which, if produced, would cut the zero line at points answering
roughly to days ±110. This linearity in the summits cannot well represent the true
phenomenon exactly, because it would imply that no finite associated pulse existed
except those shown in fig. 2, whereas there can be but little doubt that if data existed
for a really long series of years, pulses could be recognised considerably beyond the
AND OF TERRESTRIAL MAGNETISM
259
range of the figure. At first sight, one might have expected to find the maximum
on [mates in successive pulses decreasing after an exponential law. But two things
have to be remembered. First, the breadth of successive pulses increases as the height
diminishes, representing a distribution of energy over a greater and greater numl>er
of days ; and secondly, as has been already remarked, the true maxima do not
seemingly fall on exact days, so that the true maxima are not available. We should,
for instance, accepting the figures in Table IV., put the true maximum for the second
associated pulse between days 54 and 55, and the numerical value corresponding
would thus naturally be in excess of 454, the value found for day 54. A similar
remark applies to the other associated pulses, so that the ratios given above are at
best only approximations to the truth.
§ 13. Evidence that the results of §§ 8 to 12 are not confined to the period 1906 to
1911, nor due to any peculiarity in international " character " data, was derived from
a study of data for 1890 to 1900. The results of this investigation are summarised in
Table V. They were derived from days associated with disturbed days. Only the
TABLE V. — Primary Disturbance Pulse and Associated Pulses, Years 1890 to 1900.
(Unit = O'OOl " Character" Unit.)
Day . . .
-30.
-29.
- 28. - 27. - 26.
-25.
-3.
-2. -1.
0.
+ 1. +2. +3.
>
33
105
197 262 202
89
-56
76 348
812
411 167 77
Day . . .
+ 25.
+ 20.
+ 27. +28. +29.
+ 30.
+ 53.
+ 54.
+ 55.
+ 56. +57.
11
129
242 223 145
95
85
142
148
114 62
Day . . .
+ 80.
+ 81
. + 82. + 83.
+ 84.
+ 107.
56
+ 108.
+ 109.
+ 110. +111.
64
91
94 50
35
39
71
105 67
first previous pulse was considered, but the investigation extended to the fourth
associated subsequent pulse. The entries in the table are the excesses of the mean
"character" figures for the days stated over the normal figure 0'G97 derived from
all days of the 11 years. To avoid decimals the unit employed is O'OOl of the
" character " unit, as in Table IV. The associated disturl>ed day had a " character "
figure in excess of the normal, except in the one case in which a negative sign appears
in the table. The representative disturbed day is descril>ed as day 0, as in Table IV.
The maximum for each pulse is in heavy type.
Uncertainties arising from variations in the normal " character " figure appropriate
2 L 2
•v.,,
Dl{. C. CHfcEE: SOME PHENOMENA OF StJNSPOTS
at timrs corresponding to the several groups of subsequent days, naturally become
less the longer the period of years dealt with. The fact that the commencing months
of both 1890 and 1901 were all very quiet is also to the advantage of the 11 -year
group, as compared with the 6-year group. Still, I should have preferred, but for
considerations of time, to have included quiet as well as disturbed day data for the
1 1 years, employing the Astronomer Royal's quiet days for the former.
The data for the previous associated pulse, and the first, second, and third
subsequent associated pulses in Table V. are very fairly smooth ; but those for the
fourth associated subsequent pulse seem unduly affected by " accidental " phenomena,
which depress the entry for day 108 and raise that for day 110. The eleven years
were dealt with in four groups—
(A) Surispot minimum years, 1890, 1899, and 1900;
(B) Sunspot maximum years, 1892, 1893, and 1894 ;
(C) Highly disturbed years, 1891, 1895, and 1896 ;
(D) Other years, 1897 and 1898.
The largest " character " figure for the five days 107 to 111 occurred on day 111
in group (A) and day 110 in group (B), but on day 107 in groups (C) and (D) ; while
the lowest figure occurred on day 108 in group (A), and on day 111 in groups (C)
and (D). Considering this variability, much weight cannot be attached to details in
the results for the fourth associated subsequent pulse. The fact, however, that the
figures for all five days 107 to 111 are so decidedly in excess of the normal seems
clear evidence that this pulse is by no means negligible.
The primary pulse in Table V. shows the two characteristics noted in the discussion
of fig. 2. The third day prior to the representative disturbed day is decidedly quieter
than the average day. The rise to the maximum in the primary pulse is considerably
more rapid than the subsequent fall. This a-symmetry is also clearly shown by the
first and second associated subsequent pulses.
The ratio borne by the excess of the maximum " character " figure for the primary
pulse over the normal to the corresponding excesses for the associated pulses are as
follows : —
Primary pulse.
First associated.
Previous.
0-323
Subsequent.
Second associated
subsequent.
Third associated
subsequent.
0-298
0-310
0-182
0-11C
These ratios are fairly similar to those derived in § 12 from the combined disturbed
and quiet day data of the 6-year period. In the present case, however, we have
AND OF TERRESTRIAL MAGNETISM.
261
very nearly for the ratios of the amplitudes of the three associated subsequent
pulses : —
First : Second : Third : : 1 : 0'62 : (0'62)8.
Thus the amplitudes of the successive associated pulses do, in this instance, decrease
nearly in geometrical progression. At this rate we should have had the amplitude of
the fourth associatx-d subsequent pulse in Table V. about GO.
The remarks made on the sources of uncertainty affecting corresponding data in
§ 12 apply here equally.
§ 14. It seemed desirable to make sure that no period shorter than 27 days was
indicated by days previous to the selected disturbed days. Mean "character"
figures were accordingly calculated for all days up to the 35th prior to the selected
disturbed days of the 11 years 1890 and 1900. The "character" figures thus
deduced appear in the first line of Table VI. The second line supplies for comparison
TARLE VI. — " Character " Figures on Previous and Subsequent Days associated
with the Selected Disturbed Days of the 11 years 1890 to 1900.
1.
2.
3.
4.
5.
6.
7. 8.
9.
10.
Previous -days • ...
1-05
0'77
0-64
0-61
0-63
0-61
0-60 0-59
0-61
0-62
Subsequent days ....
1-11
0-86
0-77
0-70
0-66
0-62
0-63 0-64
0-62
0-61
11.
12.
13.
14.
15.
16.
17. 18.
19.
20.
Previous days
0-63
0-64
0-67
0-67
0-65
0-63
0-63 0-65
0-64
0-62
Subsequent days ....
0-63
0-63
0-63
0-64
0-63
0-63
0-63 0-61
0-61
0-62
21.
22.
23.
24.
25.
26.
27. 28.
29.
30.
Previous days
0-64
0-69
0-68
0-72
0-79
0-90
0-96 0 89
0-80
0-73
Subsequent days ....
0-64
0-64
0-63
0-65
0-71
0-83
0 94 0-92
0 84
0-79
31.
32.
3
3.
34.
35.
Previous days
Subsequent days ....
0-64
0 72
0-67
0-70
o-
o-
63
67
0-63
0-64
0-59
0-61
the corresponding figures for the 35 days subsequent to the selected disturbed days,
as given in S.M. The "character" of the representative disturbed day was 1'51.
Figures in excess of the normal value 0'70 are in heavy type.
2g2 1,1;. <•. rmiKE: SOME PHENOMENA OF SUNSPOTS
Tl „•.••• is .1 faint suggestion of a period of about 13£ days, but if it exists its
amplitude is very small.
'I'll,, tirst subsequent pulse is not clearly shown in Table VI. before day 25, while
the first previous pulse clearly persists until day 24 if not day 22. Also the previous
pulse is not clearly shown until day 30, while the subsequent pulse obviously extends
until day 32.
The differences arise undoubtedly in the main from the fact already noticed in
connection with the 6-year period, that the first previous and subsequent pulses both
follow the primary in having the rise to the maximum more rapid than the subsequent
fall. The primary pulse itself in Table VI. is not clearly manifest until the second
.lay I M -fore the selected disturbed day, while it clearly persists until the fourth day
thereafter. But, in addition to this, there is at least a suggestion that the interval
lx-t wen the crests of the primary and the first previous pulse is shorter than that
between the crests of the primary and the first subsequent pulse. This result is also
suggested by the 6 -year data in Table III.
Even if we accept the figures as mathematically exact, a real difference in period
does not necessarily follow. The phenomenon may be a consequence of the diurnal
variation which undoubtedly exists in disturbance. Analysing the list of Greenwich
magnetic storms between 1848 and 1903 given by Mr. MAUNDER,* I found that
accepting the times of commencement assigned, 60 per cent, of the storms began
between noon and 8 p.m., leaving only 40 per cent, for the remaining 16 hours.
Again at Potsdam, where individual hours have their disturbance "character"
classified, 55^ per cent, of the hours counted as disturbed from 1892 to 1901 fell
between 4 p.m. and midnight. The natural inference is that the disturbances which
give the "character" to the day at Kew occur in the majority of instances in the
afternoon. Thus, supposing the period to be somewhat over 27 days, the occasions
when the associated subsequent disturbance falls on the 28th day following would
naturally be more numerous than the occasions when the associated previous
disturbance fell on the 28th day previous. This marked diurnal variation of
disturbance is a difficulty, whatever plan is adopted. It might seem at first sight
that the international "character" data would be unaffected. This might be so if
the stations were uniformly distributed in longitude, but in reality there are but few
stations in the hemisphere whose central meridian is 180° from Greenwich, and
European stations largely predominate.
§ 15. The same mean "character" figure may be arrived at in many ways. For
example, in the case of the 1 1 years, when 660 selected days were dealt with, a mean
"character" TOO might arise from a 1 on each day. or from a 2 on 330 days and a 0
on the remaining 330 days ; or, more generally, from p cases of 0, p cases of 2, and
660 -2p cases of 1, where p may be any positive integer not exceeding 330. It thus
appeared desirable to ascertain whether there was an essential difference between the
* ' Astron. Soc. Month. Notices,' vol. 65, pp. 2 and 538.
AND OF TERRESTRIAL MAGNETISM.
263
ways in \\ hich subsequent and previous associated pulses were made up. The enquiry
\\.-is confined to the first of tin- previous and suksequent pulses associated with the
distu rlx*d days of the 11 ye;n<. That period was preferred because a greater
ilefiniteness attached to the individual "character" figures. When international
data are taken, the figure assigned to any individual day may be built up in a large
variety of ways.
Table VII. shows the results of the enquiry ; only the days containing tin- main part
i»C tin- pulses were considered. The data for the subsequent days were derived from
S.M. The representative disturbed day is counted as day 0.
TAHI.K VII. — Analysis of "Character" Figures during the First Previous and the
First Subsequent Pulses associated with Selected Disturbed Days of 1890 to 1900.
T\n\ro
Previous pulse.
SulisfijuiTit pulse.
isays.
- 30. - 29. - 28. - 27. - 26. - 25.
+ 25. +26. +27. +28. +29. +30.
Number of "2V. .
" 1 't "
n n P • •
99 105 139 155 126 80
284 319 312 323 341 359
79
302
96 120 148 132 101 90
275 305 324 343 354 343
Disturbed days . . .
Quiet „ ...
.
383 424 451 478 467 439
277 236 209 182 193 221
381
279
371 425 472 475 455 433
2*9 235 188 185 206 227
Disturbed days in Table VII. include all of " character "2 or 1, those of " character"
0 being called quiet ; so that the sum of the disturbed and quiet together necessarily
amounts to 660. The distribution one would have had in 660 average days appears
under " normal." As regards the number of 2's, days +27 and —27 decidedly over-
top their neighlxnirs. The incidence of 2's in the pulses is more alike if we invert
the order of days in the previous pulse, i.e., regard days —25 and +25, &c., as
corresponding. But in both pulses the marked tendency is for days of moderate
disturbance to follow the crest. No significance probably attaches to the fact that 2's
are slightly more numerous in the previous than in the subsequent pulse ; because,
while the highest " character " figure in the first previous pulse exceeds that in the
first subsequent pulse in the case of the 11-year period, it does not do so in the
6 -year period.
§ 16. Table VIII., p. 265, represents the results of an enquiry into the possible varia-
tion of the 27-day period throughout the year. The 11-year and 6-year periods were
treated separately. The 660 selected days of the former period gave 55 January
days and so on. These 55 January days and the subsequent days associated with
them are treated as a separate group in Table VIII. The first two columns give the
mean character figures for the selected disturbed days of the 12 months, for the two
periods. Columns 3 to 8 give the mean character figures for days 25 to 30 subsequent
..,;4 DR. C. CHREE: SOME PHENOMENA OF SUNSPOTS
to the selected disturbed days of the 11 years; columns 9 to 14 do the same for the
6 years. The largest " character" figure found in days n + 25 to n + 30 is in heavy
type, and the ratio borne by this maximum to the character figure on day n (i.e., the
ratio' of the maximum for the first subsequent pulse to that of the primary pulse) is
given for the two periods separately in columns 15 and 16. Column 17 gives the
arithmetic mean of the ratios in the two previous columns.
Investigations by Mr. W. ELLIS and Mr. E. W. MAUNDER, covering a very long
series of years, showed that whether one considers magnetic storms— averaging about
13 a year or days of large and moderate disturbance— averaging about 77 a year—
the frequency of occurrence of disturbance at Greenwich is above the average in the
4 equinoctial months, and below it in the 4 summer months, May to August ; the
numbers in the equinoctial months standing to those in the summer months roughly
in the ratio of 8 to 5.
A preponderance of disturbances in the equinoctial months has been noticed at
many other stations, but there is reason to doubt whether it is universal.
Dr. W. VAN BEMMELEN'S lists of disturbances at Batavia, averaging about 60 a year,
showed but a very slight excess in the equinoctial months, and the records of Captain
SCOTT'S expedition in the Antarctic during 1902 to 1904 indicated a marked maximum
of disturbance at midsummer. Still the equinoctial months are undoubtedly the
most disturbed at Kew, or at the average station on which the international figures
depend.
In both periods of years the order in which the months come as regards disturbance
is not quite the same when one takes the mean character figure of the selected
disturbed days, given in Table VIII., as when one takes the mean character figure of
all days of the month, or when one takes the number of days of character " 2."
In the 6-year period the months of March, September, February, and October
appear to have been distinctly the most disturbed. In the 11 -year period, March
and February were clearly the most disturbed, and judging by the number of days
of " character " " 2," October came next. Thus both periods manifested the usual
tendency to an increase of disturbance towards the equinoxes, but that season was
less prominent than in ELLIS and MAUNDER'S lists. Also the want of smoothness in
the sequence of the figures in the first two columns of Table VIII. suggests that
a considerably longer series of years would be required for the elimination of
" accidental " features.
All months in Table VIII. show the first subsequent pulse clearly, the. crest
generally falling on the 27th day itself. The maximum in the subsequent pulse is
considerably larger in some months than others, but the months in which it is
largest, or smallest, are not the same for the two periods. In both, the maximum
figure is above its average in January, February, March, August, and September ;
but these months represent Winter, Summer, and Equinox.
Judging by the differences between the two periods, and between successive
AND OF TKRKKSTKIAL MACXKTFSM.
265
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VOL. COXIII. — A
M
266 PR- C. CHRKK: SOME PHENOMENA OF SUNSPOTS
months, a high value in the maximum for the subsequent pulse is in considerable
measure accidental, but even if accepted as a physical fact, it might have more than
one interpretation.
When we took from amongst the selected disturbed days those whose " character "
figure exceeded 1'5, the amplitude of the associated pulse was decidedly larger than
that associated with the full choice of 5 days a month. Consequently, the amplitude
of the subsequent pulse increases with that of the primary pulse. Thus a large
maximum in columns 3 to 8, or 9 to 14, of Table VIII. is naturally regarded as due
at least in part to a large corresponding value in columns 1 or 2. But it might also
arise from a greater potency of the 27-day period at one season of the year than
another, or simply from a large average amount of disturbance during the month in
which the subsequent pulse falls. If the principal cause of a large amplitude in the
subsequent pulse is large amplitude in the primary, then, apart from accident, one
would expect only minor variations in the ratios of these two quantities given in the
three last columns of Table VIII. If, on the other hand, the 27-day period is
markedly more potent at one season than another, one would expect the values of
the ratio to show a marked annual variation, and this to be at least approximately
the same in columns 15 and 16.
In column 15 the highest value exceeds the lowest by 0'28, or 44 per cent, of the
mean value 0'64. In column 16 the corresponding excess is 40 per cent, of the mean
value. Thus the fluctuations are considerable. But the variations, especially in
column 16, do not suggest any regular law, and they do not follow a parallel course
in the two columns.
It will be found that there is a distinct tendency for the figure in column 16 to be
high or low, according as the corresponding figure in column 2 is less or greater than
the figure for the immediately subsequent month. In January and July, for instance,
the ratio given in column 16 is very high, while the January and July figures in
column 2 are considerably less than those for February and August. The same
phenomenon may be traced in columns 15 and 1.
To see the extent to which this phenomenon prevails, the values were calculated of
the ratio borne by the maximum figure in any month in columns 3 to 8 to the figure
assigned to the next subsequent month in column 1, and the same calculation was
repeated for the 6-year period. The twelve monthly ratios thus obtained for the
11-year period had the same mean value 0'64 as the ratios in column 15, but they
ranged only from 075 in September to 0'55 in May. Their average departure,
irrespective of sign, from their arithmetic mean was only 0'040, as compared with
0'053 for the ratios in column 15. In the case of the 6-year period, the corresponding
figures were respectively 0'054 and 0'076.
'he days which are from 25 to 30 days subsequent to a given selected disturbed
day fall, in the majority of instances, in the subsequent month. Thus the natural
inference from the previous figures is that the amplitude of the first subsequent pulse
AND OF TERRESTRIAL MAGNETISM. 267
depends more on the character of the month in which that pulse falls than on the
amplitude of the primary dtsfcnrlMHMM with which it is associated.
On the whole, Table VIII. suggests no special development of the 27-day period at
any particular season. If, tin- example, we take the three months clustering round
each equinox (i.e., hVhnuuy to April, and August to October), the mean of the ratios
in column 17 is 0'653 as compared with 0'635 from the other six months. A very
similar conclusion follows if we take the ratios in which the second member is the
character of the representative disturbed day in the month subsequent to the primary
pulse.
When a sufficiently long series of years is available, it will be possible to replace
the ratios in columns 15 to 17 by others sufficiently smooth to show the real nature
of the annual variation, if such exists. The investigation might then be extended to
the second and third subsequent pulses, and to the previous pulses. When this is
done, in the case both of selected disturbed and selected quiet days, results of interest
may be expected.
§ 17. The primary object of S.M. was to investigate the nature of the connection,
if any, between sunspots and the daily range of H (horizontal force). Use was made
of the Greenwich projected sunspot areas. The 5 days of largest spot area in each
month of 1890 to 1900 formed the selected days, and the mean H ranges at Kew
were found for days previous and subsequent to the selected days. Denoting by n
the representative selected day of large sunspot area, the H range showed a marked
pulse with its crest at day rt + 4. Moreover, when curves were drawn having time
for abscissae, the ordinate being in the one case sunspot area and in the other H range,
the rise of the latter curve to a maximum and its subsequent decline closely resembled
the course of the former curve, but .with a lag of about 4 days.
If we take the H trace for an individual highly disturbed day, it may be difficult
even for an expert to recognise the influence of the regular diurnal variation. But if
a number of such days are combined, a regular diurnal inequality emerges, which in
the case of H differs little from that characteristic of quiet days, except in being of
larger amplitude. Even on days of character " 2," the H range owes an appreciable
amount to the regular diurnal inequality, and on the average day — especially in a
quiet year — the regular diurnal inequality is the principal contributor. Thus there
were strong d priori reasons for regarding the relation described above as involving
the regular diurnal inequality rather than magnetic disturbance. This view was
supported by an examination of the Kew "character" figures for days previous and
subsequent to the selected days of the 1 1 years. The mean " character " figure of
each column was derived from 5x12x11, or 660 days. Of the 660 days occurring
in column » + 4, where the crest of the pulse in the H ranges appeared, 86 were of
" character " " 2." Out of 660 average days of the 11 years, 82 had a "character" "2" ;
thus the excess of days of " character " " 2 " in column n + 4 was only 4, and of the
31 columns from n— 15 to n+15, 10 showed an excess larger than this, the excess
2 M 2
-I-,-
I.j; c. C'HK'KE: SOMF I'llKM 'MKNA OF SUNSPOTS
N-iiiU in one case 14. The pulse in the H range curve owed its crest at day n + 4
almost rntirely to the frequency of days of " character " ' 1." The columns containing
most "2's" were n— 12 with 94, n— 11 with 96, and n— 10 with 94. The concentra-
tion of 2's in these columns proved to be the chief, if not the sole, cause of a subsidiary
pulse in the H range curve, to which there was no corresponding feature in the
sunspot curve.
H range data were not available for 1906 to 1910, so no further comparison of tlu-m
with sunspots was possible. But a comparison was made between sunspots and
international " character " figures, taking the same fundamental days as in the
previous part of this paper. In the present case, then, the basis of selection was the
"character" figure, whereas in S.M. it was the sunspot area. The research was limited
to the 5 years 1906 to 1910, as Greenwich spot areas for 1911 were not published at
the time. There were thus 5x12x5, i.e., 300, representative days n. Spot areas
were entered in 32 columns, n— 20 to w+11, and the columns were summed. The
TABLE IX. — Projected Sunspot Areas on Days of Largest International " Character "
and on Previous and Subsequent Days, as Percentages of the Mean Area for
the Five Years 1906 to 1910.
Day ,
n-20
n- 19
71 18
n 17
1) If,
n IK
m 14
_ -lo
n — id.
Percentage
91-7
90'1
90 '6
88-8
00. A
M- 7
QQ . O
1
JO 9
Day .
«- 12
71 11
;/ 10
Q
., O
n
n — 1 .
n — D.
71-5.
Percentage ....
91-3
90-<>
01 .Q
Ql' ^
O7 • 7
OO . 7
Jl 1
yy i
102 6
104 8
Day ,
n 4
•n 3
no
„ 1
n — 1.
n.
»+ 1.
ii + 2.
n + 3.
Percentage .
108-4
108-5
m. 9
m-a
m. i
1
112*2
•Q
109-1
Day .
,,4.4.
M _1 £
n + o.
71+7.
n + 8.
n + 9.
71+10.
w+11.
Percentage .
105-0
10S-2
1H9- 1
m. t\
o
100 '0
99-5
97-8
94-5
mean projected areas for the years 1906 to 1910 were in order 1047, 1453, 952, 941
157, the unit being the one-millionth of the sun's apparent disc. Thus the total
average days, 60 from each year, comes to 285,000. The figures
appearing in Table IX^epresent percentages of this number. The three last selected
52, 28 and 29, 1910; thus two days in each of the columns
AND OF TERRESTRIAL MAGNETISM.
269
u + 4 to 7i + 9, and three days in columns n+10 and w-f II, fell in 1911. As no
siins|Hii data for 1911 \vciv availalili-, \\liil.- sutis|x»t areas in December, 1910, were
veiy small, it was decided to treat the few days specified as spotless. The percentage
figures in columns u + 4 to n+11 may thus !*• slightly too small, hut the error* is
unlikely to exceed <>'•_'.
The highest and lowest percentage's in the table are in heavy type.
Table IX. appears at first sight to demonstrate a very definite relationship between
contemporaneous suns]x>t area and magnetic disturbance. It shows a regular pulse
in siinspot area whose crest ataolutely synchronises with that in magnetic " character."
The form however of the two pulses is widely different. This is readily seen on
consulting fig. 3, which represents graphically the sunspot figures in Table IX., and
130
Swnspot Area,
o MagrieCic Character.
n-20
zoo
ta
o
150
o
o
1
<3
H-
O
100
w-no
the corresponding " character " figures expressed as percentages of the mean
" character " figure for the 5 years. The ordinate scale, it should be noticed, is
five times as open for the sunspot areas as for the magnetic " character." The
" character " percentage is above 100 only on 5 days, rising from 87'6 on day n— 3 to
T.iS'2 on day n. The sunspot area, on the other hand, is above its mean from days
n— 6 to » + 8 inclusive, and the change from column to column is very gradual
* July 2«, 1913. — The correction required is +0'1 from day n + 7 today n+ 11.
270 !„;. Q niKKK: SOME PHENOMENA OF RUNSPOTS
throughout. Thus there is nothing in the observed sunspot variation to account for
the rapidity of the variation in magnetic " character."
Taking the individual years, the largest sunspot area occurred in 1906 in column
n-7, in 1907 in column »+3, in 1908 in column n + 10, in 1909 in column n-2, and
only in 1910 — a year of small sunspot area — did it occur in column n. Thus the
occurrence in column n of the highest percentage met with in Table IX. is a fact of
somewhat doubtful significance. A considerably longer series of years would be
required to give a result whose representative character could be relied on.
§18. In S.M., in the comparison made between sunspot area and magnetic
" character," the representative, days n were the days of largest spot area. On the
average of the 11 years 1890 to 1900, magnetic "character" was below its mean
from days n— 7 to n inclusive, and above its mean from days n + l to w + 11. The
highest " character " figures appeared in columns n+4 to n + 6, that in column n + 4
being slightly the highest. In this case the sunspot area (primary) pulse was much
more concentrated than the " character " (secondary) pulse, and there was a marked
"character" crest in columns n-12 to n-W, but little inferior to that in columns
n+4 to n + 6 to which nothing in sunspot areas corresponded. Thus the apparent
connection between magnetic " character " and sunspot area was much more
ambiguous than that between H daily ranges and sunspots. Still the 11 -year mean
" character" figure in column n + s was very decidedly in excess of that in column
n—s, for all values of s from 1 to 7, and the natural inference was that in the
average year there is a distinct tendency for maxima in magnetic disturbance to
follow maxima in sunspot area. Thus one would have expected to find in Table IX.,
not an array of figures symmetrical about column n but a decided excess of the figure
in column n— * over that in column n + s for small values of s, the largest value
occurring prior to day n.
It was obviously desirable to ascertain whether the departure from the result
anticipated represented a real difference between the two periods dealt with, or arose
from the difference in the procedure followed. Accordingly a second investigation
was made, adopting the same procedure for 1906 to 1910 as had been followed in the
case of 1890 to 1900, the selected days n being now the 5 days of largest projected
spot area in the month.
The calculations were made for the " character " figures assigned at Kew alone, as
well as for the international choice at de Bilt, in view of the possibility that the
results for 1890 to 1900 in S.M. might have been influenced by some peculiarity in
the choice of Kew " character" figures. This contingency could be provided for only
in part, because the date at which "character" figures were assigned to the years
1890 to 1900 was subsequent to 1910, and undoubtedly 2's were more freely given
than in dealing with the years 1906 to 1910. During the latter 5 years 2's were
given only 48 times at Kew, as compared with 37 times at Greenwich ; whereas in
1911 the number of 2's was 38 at Kew, as against 6 at Greenwich. The Kew
AND OF TERRESTRIAL MAGNETISM. 271
standard was intentionally changed in 1911 ; whereas the Greenwich standard has, I
believe, remained nearly uniform, a " 2," these Ix-in^ roughly equivalent to the
" magnetic storm " of ELLIS and MAUNDER. The number of magnetic storms in
Mr. MAUNDKK'S list averaged about 13 a year, while the number of 2's awarded to
the years 1890 to 1'JOO at Kew averaged about 44 per annum.
Tin- investigation irf'ciml to ;ilx>vc was confined to days n— 2 to n + 4, except that
•lay n— 11 was added for the Kew data. The results appear in Table X. The
absolute values are given of the mean " character " figure for the stated days of the
individual years. Values above the normal — or mean value from all days — are in
heavy type. The percentage figures for 1906 to 1910 express the arithmetic means
of the " character " figures in column n— 2, Ac., as percentages of the corresponding
mean of the normal day values. The two last lines give comparative percentage
results for the 11 years 1890 to 1900, and the last five years of that period
respectively.
Table X. confirms the physical reality of the difference between the two periods
1890 to 1900 and 1906 to 1910, but the percentage figures obtained for the later
period in columns n— 2 to » + 4 bear a remarkable resemblance to those applying to
the five years 1896 to 1900.
In the 11-year period, 1890 to 1900, it was the contribution of the sunspot
maximum years, 1892 to 1894, which mainly determined the excess of the " character"
figures in columns n + s over those in columns n—s. Since 1900 sunspot development
has been somewhat poor and irregular, and the results derived from the shorter period,
1906 to 1910, would naturally be less representative than those derived from 1890
to 1900. Still, it would be desirable to have results from several 11 -year periods
before dogmatising on this point.
In the case of the Kew "character" figures for 1906 to 1910 there were thirteen
occurrences of " 2 " in days n + 3 and » + 4, as compared with eleven occurrences on
days n— 2, n— 1, and n + 2, and nine occurrences on days n. But the number of
disturbed days (i.e., days of " 2 " and " 1 " combined) was most numerous on day n,
being greater by one for that day than for day n + 3.
Day n— 11, in 1906 to 1910, had only five occurrences of " 2," or nearly three below
normal, and occurrences of " 0 " were five above normal ; whereas in 1896 to 1900, as
in 1890 to 1900, day n— 11 had fewer occurrences of " 0" than normal. Taking the
whole 11 years, 1890 to 1900, day n— 11, it will be remembered, had more 2's than
any other. This was the reason for including it in Table X.
§19. The Greenwich volumes of heliographic results give "corrected" as well as
" projected " areas of sunspots. The corrected areas allow for foreshortening, and
take as unit the one-millionth of the visible hemisphere. Projected and corrected
areas are also given for faculse. It was decided to replace the projected spot areas of
the investigation in § 17 by corrected spot areas, projected faculie, and WOLFER'S
sunspot frequencies in turn. The fundamental days n, as in § 2, were the 300 selected
DR. C. CHKKK: SO.MK PHENOMENA OF SUNSPOTS
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disturl>cd days of the five years 1906 to 1910. The investigation was
restricted to the seven days n — 3 to n + 3. The results for the several years appear
in Tahle XL, the figures being expressed as percentages of the normal value of the
quantity concerned for the year in question.
TABLE XI. — Relation of Sunspot Areas and Frequencies and of Faculee to Magnetic
DisturlKince (?/, Ix-ing Representative Day of Large Disturlmnce).
Year.
Greenwich whole spot areas.
Projected.
Corrected.
w-3.
»-2.
»-l.
«. n+1.
» + 2. n + 3.
» - 3. n - 2.
n-1.
n. n+1.
« + 2.
fi + 3.
1906
1907
91
108
113
ll'l
120
83
114
112
124
134
84
120
114
123
139
85 87
124 128
114 106
121 112
141 137
88
133
97
104
137
88
134
93
100
137
100 94
107 1 1 2
109 114
118 118
117 127
91
113
110
111
128
88 86
118 119
114 106
114 111
134 129
88
124
99
109
132
87
128
98
103
130
1908
1909
1910
First mean . . .
Second mean . .
111
108
113
110
116
113
117 114
114 112
112
111
110
109
110 11."
109 111
111
108
114 110
111 108
110
109
109
108
Year.
WOLFER'S frequencies.
Greenwich faculaa projected areas.
n~3.
n-2.
n-1.
n. n+1.
W + 2.M + 3.
n-3.n-2.
«-l.
n. n+1.
n + 2.
n + 3.
1906
90
109
108
110
114
92
113
111
111
125
92
108
106
107
129
91 89
114 113
101 96
105 104
119 119
92
112
95
104
115
93
113
93
101
119
98 97
101 101
105 112
112 99
116 114
101
93
111
101
104
96 95
92 97
103 95
103 108
110 101
90
98
93
105
102
94
103
96
102
101
1907
1908
1909
1910
First mean . . .
Second mean . .
106
105
110
108
108
105
106 104
104 103
104
102
104
102
106 105
105 104
102
102
101 99
99 98
98
96
99
96
Of the two sets of mean values given in the last two lines of Table XL, the first
are arithmetic means of the percentages for the individual years ; the second were
obtained by summing the area or frequency figures for the 300 days in each column,
and expressing the mean as a percentage of the corresponding mean derived from all
days of five years.
Table XL shows that, at least for the years considered, it does not much matter
whether projected or corrected spot areas are taken for comparison with magnetic
disturbances. If anything, the projected percentages are a trifle the larger. We
VOL. CCXIII. A.
2 N
.,-, DR. C. CHREE: SMMK I'HKNaMKNA OF SUNS POTS
,.\| ..... t.-.l .. marked difference if the proximity of the spot — supposed to
••• disturbance — to the central meridian had been an important element.
Win. n:i:'s frequencies give results of (he same general character as spot areas, but
tin- percentages are decidedly smaller. Also the percentages in the last two lines
derived from the Wolfer frequencies are less symmetrical with respect to column n,
IKMIIJJ distinctly larger for the previous than for the succeeding days. This a-symmetry
is still more developed in the percentages based on facuhe.
On the average of the five years, the maximum magnetic disturbance was
preceded by two days by the Wolfer frequency maximum, and by at least four days
by the maximum faculse area.
On the average highly disturbed day, the faculae area was almost exactly normal.
Whether we take spot areas or frequencies, 1906 shows a markedly diminished
solar activity for days »— 2 to n + 3 ; and 1910 — -a year of small solar activity and
very quiet magnetically — is the year which most strongly suggests a parallel variation
between magnetic disturbances and solar activity.
§ 20. It appeared desirable to ascertain the extent to which a 27-28 day period of
the type here considered manifests itself in sunspots themselves. The selected days
of the investigation were the five days of largest projected spot area in each month
of the five years 1906 to 1910. Projected spot areas were entered in the columns for
days n-30, n-28, n-27, n-25, n, n + 25, n + 27, n + 28, and n+30. That seemed
likely to be a sufficient choice of days to show the nature and amplitude of the
anticipated phenomenon. The results obtained are given in Table XII. In the first
five lines the projected spot areas are expressed in terms of the Greenwich unit.
The five subsequent days associated with the five selected days of December 1910 had
to be omitted, so the entries for columns n + 25 to n + 30 in that year were based on
55 days oidy.
The results for the 300 (or 295) days included in each column were summed, and
each sum was expressed as a percentage of that for the normal day.
The last line in Table XII. gives for comparison corresponding results calculated for
the first previous and first subsequent pulses in " character " figures, the selected
days ?i in this case being those of maximum " character" for the sixty months of the
five years.
If we take a mean from the previous and subsequent pulses in Table XII., the
largest excess above the normal in the first subsidiary pulse bears to that in the
primary pulse the ratio 27 : 122, or 0'221 : 1, for the spot areas, and 21 "5 : 98, or
1, for magnetic " character." This is a very striking resemblance. It did not,
however, extend to individual years. Thus the previous and subsequent sunspot area
curves were better developed in 1907 than in the other years, but the development
the previous "character" pulse was best in 1908, and that of the subsequent
" character" pulse was better in 1908, 1909, and 1910 than in 1907.
A noteworthy difference is that the crests of the subsidiary sunspot area pulses in
AND OP TERRESTRIAL MAGNETISM.
•.'7..
Table XII appear on days n— 28 and n + 28, and not on days n— 27 and n-t-27 as in
the case of magnetic " character." It is also curious that the spot area on day n — 28
should so largely exceed tint on day //+28. As this phenomenon, however, is not
shown in 1906 it may be "accidental." The sunspot uiv;i pulses, both primary and
TAHLK XII. — The 27-28-Day Period in Projected Sunspot Areas (n being the
I!f|.i-.-.i'iii;iti\ r I i.iy • •!' Large Spot AI-.-.-I ).
Year.
n - 30. n - 28. n - 27. « - 25.
n.
»» + 25. » + 27. n + 28. n + 30.
Normal
day.
1906
968 1069 1058 938
2234
954 1118 1170 1137
1047
1907
22H 2398 2339 2018
3136
1685 1991 2001 1891
1 4.r>3
1908
1170 1416 1466 1 •«:;.•:
2110
1155 1130 1114 1117
952
1909
1306 1086 941 7.'.7
3137
1006 1058 1074 1099
941
1910
431 394 401 371
933
394 340 320 286
357
Percentage of normal \
(sunspots) . . . J
Percentage of normal 1
("character") . . J
128 134 131 116
98 114 121 111
822
198
109 119 12.0 117
99 122 116 106
100
100
secondary, appear considerably rounder than those in magnetic " character," and this
is probably responsible for the greater variability in the position of the crest in the
subsidiary pulses of sunspot area than in those of magnetic " character." Thus in
190!) and 1910 the largest spot area in the subsidiary pulses appear on day n — 30 ;
while the spot areas on day n— 25 in 1909, and on days n + 27, « + 28 and « + 30 in
1910 are actually below the normal.
§21. The results obtained in S.M. and in the present paper put it Ixjyond a doubt
that there is in terrestrial magnetism a period of alxiut 27 days, in the sense that if
day TI is either decidedly more or decidedly less disturbed than the normal day, then
days n± 27 show a distinct tendency to differ from the normal day in the same
direction as day n. The characteristic is just as clearly shown by quiet days as by
disturbed days. The phenomenon appears in disturbed years and in quiet years, in
years of many and in years of few sunspots. It was particularly prominent in 1911
when sunspots were few, and it was also well developed in 1910, a year in which only
one day \vas award.- 1 character " 2 " at Greenwich.
Prof. SnnsTKii, as is well known, has adduced arguments which appear fatal to the
view that a magnetic sit. mi on the earth can l>e due to any limited jet of electrified'
particles emanating from the sun. It may thus seem a waste of time to consider
other difficulties, in the way of jet theories, suggested by the present enquiry. There
an-, however, physicists, with whom I to some extent sympathise, who have a feeling
that demonstrations of the impossibility of some physical hypothesis may prove in
the long run less conclusive than was at first supposed. Fresh physical discoveries
may remove what seemed at one time insuperable barriers. Thus it may not be
2 N 2
in:, c. CHI;I:I: s..\n: nil NO.MHNA OF SUNSPOTS
(1 effort to direct attention to the difficulty which seems to be raised by tin-
oonspininiis nature of the 27-day period in quiet days. The rapidity of the decline
in disturlKim-e and the rapidity of its resuscitation after the representative quiet day
are prominent facts. It will hardly, I think, be suggested that there are limited solar
I -similar to sunspots in dimensions — whose direct presentation to tin- earth
exerte a soothing or damping influence on magnetic disturbance on the earth, removing
or diminishing disturbances which otherwise would have made their presence felt.
§ 22. A serious difficulty in the way of an exact determination of the period is that
magnetic storms, and magnetically quiet times, are events usually covering a large
iiumW of hours. A magnetic storm is seldom confined to a single day. Successive
magnetic st»rms do not as a rule present closely similar features, nor are they usually
of closely similar length. There is thus as a rule no such thing as a definite interval
between them. In the majority of cases opinions would differ — often by hours — as to
when a magnetic storm logins, and still more so as to when it ends. The uncertainty
is least about the time of commencement, and that is presumably the reason why
Mr. MAUNDER calculated his intervals from the times of commencement. If, however,
one could assign exact intervals for the beginning and ending, the natural interval
would seem to be, not the time from beginning to beginning, but the time from centre
to centre. If wo accept a jet theory, then if the second of two magnetic storms is
shorter than the first, the jet and so presumably the corresponding solar area has
contracted. In the absence of definite knowledge to the contrary, the most natural
hypothesis would seem to be that the jet has contracted uniformly about its centre.
If successive magnetic storms were of roughly equal duration, and if in a number of
instances they both had what are termed " sudden commencements," much less
uncertainty would attach to the interval. As I pointed out, however, in a review of
Mr. MAUNDEK'S first paper, Nature but seldom presents this simple case. Of the
276 magnetic storms which Mr. MAUNDER'S list gave for Greenwich between 1882
and 1903, only 77 had "sudden commencements." Of the 91 storms which he
regarded as showing a 27-28 day period during these 22 years, only 15 had "sudden
commencements," and there were only four cases in which two successive storms of
his sequence groups had both " sudden commencements."
The definition of a magnetic storm is purely arbitrary. A striking example of this
is afforded by the Kew and Greenwich lists of " character " figures supplied in 190G,
1907 and 1908 to de Bilt. In both lists the days of character " 2" — i.e., magnetic
storms according to Greenwich standard — numbered 29, but only 19 of these days
were common to the two lists. Both lists gave eleven 2's in 1907 ; but the
Greenwich list gave nine 2's in each of the years 1906 and 1908, while the Kew
list gave five in the former year and thirteen in the latter. Thus if attention is
confined to "magnetic storms," where one man gets a sequence of approximately
the right period, another gets no sequence at all. If, on the other hand, one takes
disturbances moderate as well as large, the number is so great that it does not require
AND OK TKREE8TBIAL MAGNETISM. 277
any great skill to find between pairs of them intervals of 27 flays, or of any other
imml>er of days which the individual desires.
The dithVulties in the way of treating disturbances individually exist in at least
equal measure in the case of quiet days. On some occasions a quiet time ends with
great precipitancy, hut t" say exactly when it commences would usually prove an
impossible task.
I have referred t<> this aspect of the problem l>ecause it is rather a fashion amongst
experimentalists to regard statistical enquiries such as the present with suspicion.
They an- unable wholly to purge themselves of the popular superstition that statistics
can prove anything which the statistician desires. In the present case, however, the
popular view is the exact opposite of the truth. The statistics employed are in large
part international data, published before the enquiry commenced, and based on
estimates of magnetic " character " made independently, at ol«ervatories Mattered
IIMT the world, by individuals none of whom had any suspicion of the purpose to
which they would be put. The observational data, on the other hand, are usually of
so complex a nature, and so influenced by the latitude and longitude of the station,
that the observer does not know what to regard as essential and what to consider
secondary. Moreover, the record is in nearly all cases photographic. Except in a
few of the better staffed observatories, the fact that a magnetic storm has occurred is
not known until a day or two afterwards, when the photographic sheets have been
developed. If a continuous succession of solar pictures and contemporary magnetic
changes could appear side by side during the actual progress of a magnetic storm, an
olwerver would have a better chance of framing the right guess as to the nature of
the solar link, provided corresponding events on the sun and earth are nearly
simultaneous, or are separated by a constant small interval of time. In the case,
however, of sunspots and the amplitude of the daily H range at Kew, during the
eleven years 1890 to 1900, the results reached in S.M. indicated a clear lag of aliout
four days in the magnetic range, and they were at least consistent with a similar lag
in magnetic " character." The results of the present paper do not suggest a lag in
magnetic " character," but the rate of change of sunspot area near the time of
maximum " character," as shown in Table IX. arid fig. 3, is slow, so that the question
of lag in " character " is still an open one. If there is a lag, and especially if the lag
is of variable amount — as might well be the case if cathode rays or electrified
particles are concerned — the difficulties in the way of direct observation will be
materially increased.
We have seen that magnetic " character " and sunspots have both periods of from
27 to 28 days. In some years the phenomena are, so to speak, in phase, in other
years not in phase. The period seems better developed in some years than in others,
and the years in which it is l)est developed do not seem to be necessarily the same for
tin- two sets of phenomena.
[ 279 ]
VII. On the Diurnal Variations of the Earth's Magnetism produced by the
Moon and Sun.
By S. CHAPMAN, B.A., D.Sc., Chief Assistant at the Royal Observatory, Greenwich.
(Communicated by the Astronomer Royal.)
Received April 24,— Read June 19, 1913.
CONTENTS.
Page
Introduction 279
PART I. — General Discussion 281
PART II. — Mathematical Theory 288
PART III. — The Observational Material 307
Introduction.
§ 1. WHILE the observational study of terrestrial magnetism is receiving ever more
and more attention, and being rewarded with success by the acquisition of new and
important data, the theoretical side of the subject shows a much less rapid advance.
The search for a physical theory of the earth's magnetism and its changes is
fascinating but elusive. Perhaps in one case only — that of SCHUSTER'S important
theory* of the diurnal variations of the magnetic state of the earth — has there been
put forward a clearly outlined theory which promises to explain the real mechanism
of any magnetic phenomenon.
On this theory, the solar diurnal variations are attributed to the action of electro-
motive forces produced in masses of conducting air in the upper atmosphere, by their
motion across the permanent magnetic field of the earth. The magnetic field of the
resulting electric currents is identified with that which produces the observed diurnal
changes. SCHUSTER has shown that if the motion of the air is taken to be sub-
stantially that which is indicated by the barometric variations, the atmosphere being
supposed to oscillate as a whole, the conductivity required by the theory is not
unreasonable, considering the ionization of the tenuous upper atmosphere by ultra-
* ' Phil. Trans.,' A, vol. 208, p. 163.
VOL. CCXIII. — A 503. Published separately, August 22, 1913.
280 DR S. CHAPMAN ON THE DIUENAL VARIATIONS OF THE
x iol,-t radiation from the sun.* The fundamental assumptions are in accordance with
S. • HI-SIKH'S demonstrationt that the magnetic variations are principally due to a
system of currents above the earth's surface. In order to explain the relative
magnitudes of the diurnal and semi-diurnal terms in the magnetic potential, it is
necessary to suppose that the conductivity of the atmosphere varies with the solar
hour angle, which is certainly a priori probable : the great excess of the summer
variation over the winter variation is unexplained, however, as the usual rapid rate
of recombination of ions makes it difficult to believe that the solar ionization is slowly
cumulative.
There is at present much uncertainty as to the numerical constants of the potential
of the magnetic field responsible for the solar diurnal variations, as the only two
calculations yet madej show serious disagreement. A new determination of this
potential is now in progress at the Royal Observatory, Greenwich. Whatever be the
result of this calculation, however, there will remain several important features of the
phenomenon which require explanation — in particular, the seasonal changes. By
the elucidation of these difficulties, terrestrial magnetism may throw light on the
ionization of the upper atmosphere. The variables at disposal in the theory are,
unluckily, too numerous to get very definite knowledge of any one of them from a
single source, and therefore it is peculiarly fortunate that there is a kindred but
independent set of phenomena, produced by the moon jointly with the sun, which
promises very valuably to supplement the knowledge furnished by the solar diurnal
variations. It should be specially instructive to compare the seasonal changes of the
two sets of phenomena.
§ 2. The general outlines of this paper may be briefly indicated here. The principal
known facts regarding the lunar magnetic variation are first summarized, and it is
shown that, so far as they go, they seem most easily explicable in the manner
proposed by SCHUSTER for the solar diurnal variations. Nothing in the nature of a
proof is yet possible however. Some new facts, deduced by harmonic analysis of
existing material for the lunar variation at the separate phases of the moon, are then
described, and it is pointed out how they confirm the hypothesis of the variable
conductivity of the atmosphere in a very direct way, and provide a powerful means
of quantitatively investigating the changes of the conductivity. The details of the
calculation of these new harmonic terms in the lunar variation, and the actual tables
of results, are collected in Part III. of the paper. In order to discuss the bearing of
these observational results on the theory, it is necessary to extend SCHUSTER'S
calculation of the effect of an atmospheric oscillation, under the influence of the
earth's radial magnetic forces and the variable conductivity of the air, in producing
* That there is a highly conducting layer in the upper atmosphere is also indicated by the bending of
electric waves round the earth.
t ' Phil. Trans.,' A, vol. 180, p. 467.
J SCHUSTER, ' Phil. Trans.,' A, vol. 180, p. 467 ; and FRITSCHE, St. Petersburg, 1902.
EARTHS MAGNETISM PRODUCED BY THE MOOT* AND SUN. 281
magnetic diurnal variations. The calculations are given in Part II., in a very general
funn; the work is in some respects simpler and more direct than in SCHUSTER'S
investigation, owing to the adnptinn of the resistivity, instead of the conductivity,
as the variable. The formal results (which as yet, however, are at a somewhat
incomplete stage) are reduced to numerical form and compared with the observed
data. The whole of the discussion is collected in Part I., and it is shown that the
fourth harmonic component of the lunar variation favours the assumption that the
atmospheric conductivity may fall to a very small value during the night hours.
The question of the seasonal variations, as affecting both the solar and lunar effects,
is barely touched on, since though it arises naturally from the calculations in Part II.,
better observational material is necessary to reali/e the proper use of the theoretical
work. A fuller discussion is reserved therefore till the new determination of the
potential of the solar variation, already mentioned, is completed.
I'AliT I. —
§ 3. The magnetic elements show regular periodic changes depending on the lunar
hour angle,' just as on the solar hour angle : the latter variations are considerably the
greater of the two, and .almost entirely mask the lunar variations. KREIL,* of
Prague, in 1841, first established the existence of these changes, and since then a
very limited number of investigators! have confirmed and extended KREIL'S
discovery. Owing to the nearly equal length of the solar and lunar days, the
separation of the two effects involves considerable rearrangement of the observed
data as usually tabulated, and the smallness of the lunar variation renders it
necessary to deal with a large quantity of material in order to eliminate accidental
errors. The determination of the lunar diurnal variation for the three magnetic
elements at a single station is therefore a laborious undertaking, and hardly any
observatory, as yet, includes such an examination of its observations in its scheme of
work. If the potential of the magnetic field producing these variations is to be
found, however, they must be computed not merely for one, but for several stations,
well distributed on the earth's surface. This formidable task would be much
* Bohemian Society of Sciences, 1841.
t BROUN, 'Trevandrum Observations,' I., 1874; CHAMBERS, 'Phil. Trans.,' A, voL 178, p. 1 (1887) ;
' Batavian Observations," BERGSMA and VAN DBR STOK, vols. I., III., IX., X., XVI., also • Proc. Roy.
Acad.,' Amsterdam, IV., 1887, and ' Archives Nderlandaises," XVI. ; FIGEE, ' Batavian Observations,'
XXVI., 1903; LAMONT, 'Sitz. d. K. Akad. d. Wiss.,1 1864, t. 11, 2, Munich. SABINK, 'Phil. Trans.,'
1863, 1856, 1857 ; 'Roy. Soc. Proc.,' X., 1859-1860.
Also the published ol>scrvations at St. Helena, Toronto, Hobarton, and Cape of Good Hope (edited by
SAIIINK), and at Melbourne, Dublin, and Philadelphia. Also AIRY, 'Greenwich Observations,' 1859
and 1867.
Also Moos, ' Bombay Magnetic Observations,' 1846-1905, vol. II. (1910) ; and VAN BEMMEI.EN, ' Met.
Zoitschr.,' May, 1912.
VOL. COXIII. - A. 2 O
DR. S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
expedited if various observatories would undertake the reduction of their own data
on a unif'Tiii plan, and it is partly in the hope that some may be induced to co-operate
in this work that the present preliminary paper has been written.
§ 4. When determined from the mean of a number of whole lunations, the lunar
diurnal variation is found to be always of the same character, for every element and
at every station : it consists solely of a very regular semi-diurnal oscillation. Other
harmonic components of relatively small amplitude may be present, but their lack of
regularity and consistency proves them to be accidental inequalities which are no real
part of the phenomenon. This simplicity makes it probable that the lunar diurnal
variation will be easier to explain than the solar diurnal variation.
SCHUSTER'S theory of the latter naturally suggests that the former is due to the
lunar tidal oscillations of the atmosphere. These oscillations have very little effect
upon the barometer, the ordinary diurnal barometric variation being a thermal and
not a tidal effect ; but a lunar barometric tide does exist, and has been evaluated
with a considerable degree of accuracy at some tropical stations (St. Helena,
Singapore, and Batavia).* The explanation gains weight from the fact that at
perigee the lunar magnetic variations are of distinctly greater amplitude than at
apogee,t and there is some evidence that the ratio of the amplitudes at the two
seasons is that which would be predicted by the tidal theory (1'23), though the
observational results do not suffice, as yet, to establish this definitely.
§ 5. Dr. VAN BEMMELEN, at Batavia, has recently collected all the existing
determinations of the lunar magnetic variation for different stations, and has
examined this material, together with newly computed data for other stations, to see
whether the magnetic field which produces these effects has a potential, and whether
the latter has its source above or below the earth's surface.J He finds that most of
the field, at any rate, has a potential, and that this arises partly above and partly
below the earth's surface, but that the internal field is too great to be merely a
secondary induction effect. This result should be accepted with some reserve, at
present, not only on account of the imperfections of the data, but also because the
seasonal change of the variations was disregarded ; in certain elements at some
stations the summer and winter variations are of opposite sign, and this renders it
unsafe to take the mean variation for the whole year. At many stations,
unfortunately, the data so far computed apply only to the whole year, so that if this
material was to be used, no course was possible save to adopt the mean of the year
for all One important result of VAN BEMMELEN'S work was to show that the
principal term in the potential of the lunar variation field was of the form Q33 (in the
usual language of harmonic analysis, a tesseral harmonic of the second kind and third
* SABINE, ' Phil. Trans.,' 1847 ; ' Batavian Observations,' 28 (1905).
t See ' Trevandrum Observations' (BROUN), vol. I., p. 137, and SABINE'S and FIGEE'S discussions
already cited
t ' Met. Zeitschr.,1 May, 1912.
EARTH'S MAGNETISM PRODUCED BY THE MOON AND SUN.
degree). This is in accordance with the theory that the lunar atmospheric tide is the
main cause of the phenomenon, although, of course, it does not prove this to be
the case.
§ 6. So far reference has been made entirely to the lunar variation as determined
from a number of whole lunations, as has been generally done (the exceptions are
Trevandrum, Bombay, and Batavia). It will be remembered that SCHUSTER'S theory
of the solar diurnal variation involved the hypothesis of a variable conductivity
depending on the sun's hour angle. This should, of course, also affect the electric
currents which arise from the lunar atmospheric tide, and so make the lunar magnetic
variations depend on the sun as well as on the moon. In the course of a lunation,
however, the angle between the sun and moon, viewed from the earth, changes from
0 to 2-jr, and the mean lunar variation for such a period cannot be expected to show
any special dependence on solar time. At any particular lunar phase, however, the
solar day hours, during which (over a given part of the earth) the atmospheric
conductivity is greatest, occur at a definite part of the lunar day, this part changing
with the lunar phase ; and it has, in fact, been found* that the lunar variation
determined from the mean of a number of days all at the same lunar phase is not of
the semi-diurnal form. The variation curve goes through a regular cycle of change
with lunar phase, in such a manner as to leave the mean variation over a whole
lunation of the simple form already described. The magnetic needle is most mobile
during the day hours : at certain seasons of the year, BROUN found that the
amplitude of the lunar diurnal variation of magnetic declination at Trevandrum was
five times as great during the solar day hours as during the night hours. t These
facts clearly show that the conductivity of the medium in which the electric currents
flow to produce the lunar magnetic variation depends on the position of the sun ; and
since it is unreasonable to suppose that the mechanisms concerned in producing
the lunar and solar diurnal magnetic variations are materially different, the
assumption of variable conductivity in SCHUSTER'S theory is confirmed in a very
definite and independent way ; in SCHUSTER'S discussion two barometric oscillations,
diurnal and semi-diurnal, were concerned, and it was necessary to explain why the
resulting magnetic variations, deduced on the assumption of uniform conductivity,
did not bear the proper ratio to one another. This might be because the conductivity
was not uniform, or because the ratio of the two oscillations was different in the
upper regions of the atmosphere from that indicated by the barometer. This latter
uncertainty is absent in the case of the lunar variations, where there is only a single
barometric oscillation, from which arise magnetic variations of other periods,
depending on the solar hour angle.
§ 7. In order to examine the effect of this variable conductivity, it is natural to
determine the harmonic components of the lunar diurnal variation for different lunar
* By BROUN, CHAMBERS, FIGEE, and Moos in the investigations already cited,
t 'Trevandrum Observations,' vol. I., p. 121.
2 o 2
284 I 'If- S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
phases, but (rather strangely) this has only once been done hitherto, and then
without result.* CHAMBERS! obtained an analytical expression for the variation and
its dependence on phase, which satisfactorily represents the observations, but it is
not of a simple character. His formula was
/,,(*.) cos 2 +/..,* sn
where h is the hour of the solar day, P is the mean period of a lunation in solar days,
and t is the age of the moon in solar days; fe.2(h) and f,,2(h) are the observed
variations at new moon and one-eighth phase respectively. This formula, it will be
noticed, expresses the lunar variation as, in reality, a solar diurnal variation (h, the
solar time, being the variable) which merely runs through a cycle of change depending
oil the age of the moon. This, in fact, was CHAMBERS' view — he termed the variation
" luni-solar." It will be seen later, however, that there is a true lunar semi-diurnal
variation which remains unchanged throughout the course of ar lunation, as well as
luni-solar components governed by the position of both bodies. As to CHAMBERS'
expression for the variation, while it is numerically correct, it does not aid in
interpreting the phenomenon, because it depends on two complex curves fc,-2(ti) and
f,,t(h\ for which no analytical expression was obtained; these two curves are not
independent, as will appear later.
§ 8. FIOEE determined the harmonic coefficients of the diurnal and semi-diurnal
components of the variation at each lunar phase, and came to the conclusion that " a
regular variation of the movement of the magnetic needle with the moon's phases is
not indicated by the observations at Batavia."J It will be shown, on the contrary,
that the Batavian observations agree with those made at other places in manifesting
considerable regularity of change with lunar phase.
§ 9. Moos§ has made the valuable suggestion that the luni-solar variation may be
regarded as a simple lunar variation the amplitude of " part of which goes through
a series of wave-like changes in the course of a lunation." He multiplies each hourly
value of the mean lunar variation determined from a whole month by 1 + cos (t + »),
where t is the lunar time reckoned from upper culmination (one hour equalling
15°), and v is the angular measure of the moon's age, reckoned as 0° at new moon, and
changing through 360° in the course of a month. Curves showing the results of this
calculation are exhibited for comparison with the observed curves, for the eight lunar
phases, for the element of declination. The general similarity of the two sets of
curves is sufficiently striking to show that the suggestion is in the right direction.
It will be seen that this idea is, formally, much akin to SCHUSTER'S idea of variable
* ' Batavian Observations,' XXVI., Appendix, p. 195, §44.
t CHAMBERS, 'Phil. Trans.,' A, vol. 178 (1887).
t ' Batavian Observations,' XXVL, Appendix, § 44.
§ 'Bombay Magnetical Observations,' 1846-1905, vol. II., §526.
KAUTH'S MAGNETISM PRODUCED BY THE MOON AND SUN. 2B5
conductivity, and is most naturally interpreted in that way. Moos, however, seems
in >t to have thought of the matter in this simple light, hut speaks of changes in the
radio-activity of the, earth's crust, due to a tidal action, as possibly responsible for the
luni-solar changes, perhaps by ionizing the atmosphere indirectly; and also of the
reflection by the moon of ionizing radiation from the sun.*
§ 10. Since the mean variation of any element over a whole lunation is almost
exactly a semi-diurnal wave, Moos'tf expression is equivalent to
'2 COS (2<-M0)[l + COS (< + •;)] = C08(t + t0-»)+2cOB(2t + t0) + C08(3t + t0 + v), . (A)
though he did not himself write it out formally thus. The examination of the data
by harmonic analysis, which is effected in the third part of this paper, is the best
means of numerically testing Moos's suggestion, being preferable to a mere comparison
of two sets of curves by eye. The desire to apply this test partly occasioned the
present re-examination of the existing data, which also has in view the comparison
of the results of these past determinations of the lunar magnetic variation (on which
enormous lalxnir has been spent) to see how far they confirm one another, and gauge
the possibility of obtaining accurate information from them.
Moos's suggestion implies the presence, in the lunar diurnal variation at a
particular lunar phase, of first and third harmo'nic components of amplitude equal
to half that of the semi-diurnal component, and with phase angles which respectively
decrease and increase by 45° with each change of lunar phase, the epoch of the
second component remaining constant. No other relations or components would
satisfy the above equation.
§11. The calculations from the observational data show that while first and third
harmonic components possessing the alxrve phase relations are present, the amplitudes
are not generally in accordance with Moos's equation. Moreover, a fourth harmonic
component, which was calculated in the first instance merely because to do so involved
scarcely any trouble after the other components had been computed, was also found
to be present, of quite appreciable amount, and obeying an unexpected phase law ;
its phase angle increases during each lunation by 4?r, twice the amount of change in
the phases of the first and third components.
There is considerable accidental error in the determinations of the phase angles and
amplitudes at each lunar phase, as, of course, the material is much subdivided. While,
however, the phase angles go through an easily recognizable monthly cycle, the
amplitudes show no regular variation with lunar phase (the mean of a number of
lunations is dealt with, of course, so that perigee and apogee occur at different phases
during the period). The mean of the amplitudes at the separate phases gives,
therefore, the best determination of the amplitudes of the first, third, and fourth
* 'Bombay Magnetical Observations,' 1846-1905, vol. II., §527. It may be mentioned that earlier
investigators had regarded the lunar variations as possibly due to the direct or indirect action of induced
magnetism in the moon, arising from solar or terrestrial magnetism, or both.
286 DR. S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
components, as well as of the second ; and similarly, by correcting the separate phase
angles by the amount indicated by the regular phase law, and taking their mean, the
accidental error of the determined phase angle at any particular lunar phase can >>e
much reduced. In this way, as described more fully in § 27, the expression of the
lunar variation at every period of the lunation, complete as far as the fourth harmonic
term, is obtained. It is found that the amplitudes of the first and third harmonics
are often unequal ; sometimes their amplitude exceeds that of the second component,
but generally they are less, down to about half this amount. The determined values
of C and ta in the formula
C, cos^-Ko'-^ + C, cos (2« + £0") + Q»cos (S« + <o/" + ") + C4cos (4t + t0"" + 2»), (B)
which has been found to fit the observations, are given in Tables XL, XII., and XIII.
for all the stations and elements for which data were available. Moos's representation,
it is seen, though it pointed in the right direction, is of too simple a character to
represent the phenomenon ; the solar excitation which it indicates is a matter which
concerns the whole earth, and this action cannot be represented by a simple harmonic
factor at each individual station.
§ 12. SCHTJSTKR* has calculated the effect of an atmospheric oscillation with a
velocity potential Q/ (which is also the main component of a lunar diurnal tide)t in
producing, under the influence of a variable conductivity of amount
P = A>(l+ycoso>), y£l
(where a is the zenith distance of the sun from each particular point on the earth's
surface), magnetic variations of one, two, three and more periods in the solar day.
Adopting the rather more general expression
p = A,[l+y' cos 0+y sin0 cos (X + «)] ........ (C)
where B is the colatitude, X is the longitude, and \ + t is the local time, he finds that
the resulting magnetic potential (apart from a constant factor) is of the form
}, . (D)
where QWT sin {T(\ + t)-a} is the velocity potential. The coefficients p' and qn" are
numerical constants which depend on v and „' ; their values are tabulated in the paper
referred to.
b is shown in Part II. of the present paper that the above equation (D) holds
good, whatever be the functional relation between P and «, and this calculation is
* 'Phil. Trans.,' A, vol. 208, p. 163.
t Q»' represent* the tesseral function sin' Od'PJdv. where P« is the zonal harmonic of degree n.
O
EARTH'S MAGNETISM PRODUCED BY THE MOON AND SUN. 287
adapted, in J; L':;. |.> n,\ t-r t In- case ..(' t In- luiii solar ina^m-t ic \ ariat inn*. It istlirn-
shown that the equivalent expression to (I>) is in thin case (apart from a constant
factor)
This expression, it should be noticed, consists of series of harmonic components of
one, two, three, and more periods in the lunar day, with phase angles which depend
on the age of the moon. In the second series the phase angles increase by 2 (<r+2) •*
per lunation ; this phase change is very rapid, even for the diurnal term, and with the
lunation divided up into not more than eight parts, hardly comes within the range of
observation, even if the coefficients q* were of the same order of magnitude as the
p£ coefficients. The theoretical values of qm' are, however, much less than those of
the important members of the pn' set of coefficients, and therefore this part of the
magnetic potential can be neglected. The other part consists of terms of period 2iw,
whose phase angles increase by 2 (a-— 2) IT per lunation ; thus the phase of the first
harmonic decreases by 2-ir each lunation, that of the second component remains
constant, while the third, fourth, and higher components increase by amounts 2ir, 4ir,
6-r, and so on. This, however, is exactly the law of phase change which is indicated
by the formula (B), whicli was determined empirically from the observations.
At new moon, when v = 0, the formula indicates that all the harmonic components
should have the same phase angle, or differ by 180 degrees exactly (since the
coefficients may be of different sign). The data obtained in this paper show a very
satisfactory agreement with this conclusion, when the extreme smallness of the whole
phenomenon is considered.
§ 13. The amplitudes must next be considered. The actual calculations necessary
for the comparison of theory and observation are given in § 25, and only the results
obtained will be cited here. It appears that as regards the relative magnitudes of
the first three components in the lunar variation, there is tolerably good agreement
with the results derived either from SCHUSTER'S simple theory p/pi = l + cosw, or
from the more general theory of Part II. of this paper. The numerical constants
(p/p<>= 1 + 3 cos o> + Jco8*w) might be altered to fit the observations better, but it
seems hardly worth while to do this till better observational material,, is available.
The given constants were chosen to represent a function which should have a large
maximum at midday, and should be small and nearly constant during the night
hours.
§ J 4. The deciding factor between the two expressions for pfp^ is found to be the
amplitude of the fourth harmonic component. Three tables are given in §25 to
illustrate this. They give the ratio of the amplitudes of the four harmonic compo-
nrtits to that of the second component, for the three elements X, Y, Z. The first
288 DR. 8. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
table is that calculated on the hypothesis />//>„ = 1+cosw, the second that calculated
tV.iin p/pt = l+3co8a>+f cosaw, and the third gives the observed values. The simple
f or 1 1 1 of/> gives altogether too small a value for GjCa, while the second expression for
p gives values of the right order, at any rate. Perhaps the detailed calculations
in Part II. have not been carried to a sufficient degree of approximation, as the
expressions for p' do not converge very rapidly. When better data are available,
this point must receive consideration. Enough evidence, however, has been brought
forward to show that the fourth harmonic component of the lunar variation favours
the hypothesis that the conductivity during the night hours is small compared with
its value during the daytime,* and that the rate of recombination of ions in the
upper atmosphere (assuming this to be the seat of the effect) is rapid, as would
naturally be expected.
The proper discussion of the observations, whether of the lunar or solar magnetic
variations, can only be made on the basis of a reliable determination of the numerical
coefficients of the various tesseral harmonics in the potential, derived from a number
of observatories properly distributed over the globe. The significance of the lower
harmonics in the lunar variation makes it desirable to obtain the terms in the
potential down to those of the fourth type (Qn4) — not only for the lunar variation,
but also for the solar variation ; its fourth harmonic shows a sufficient degree of
constancy, at most observatories, to entitle it to respect as having definite physical
significance.
PART II. — Mathematical Theory.
§ 15. The problem in hand is to determine the current function of the electric
currents induced in a spherical shell of fluid by its quasi-tidal motion across a radial
magnetic field of force, the electric conductivity of the fluid at any point being a
known function of the angular distance between that point and another (that with
the sun at its zenith) which uniformly rotates round the axis of the sphere. The
velocity potential t/r of the motion will be expressed as the sum of a number of terms
such as
where Q_T is a surface harmonic of degree m and type T, and X is the longitude
measured towards the east from some standard meridian, at which the local time is t.
The colatitude and zenith distance of the sun will be denoted by 6 and o> respectively ;
* It is not aaserted that any observational evidence has been brought forward in favour of the particular
numerical constants here chosen for P, but only that the observations indicate the presence of an
ible term in P depending on cos 2o>, and that this term, if present, may be expected, on general
physical grounds, to be of such a sign as to diminish the value of P at night as compared with the value
by day.— June 11, 1913.
MAONI-TISM PRODUCED BY THE MOON AND SUN. 289
if <f is the dtvlinutinn of the sun, ev'ulmtly we have
cos« = sin & cos 6+ cos tJsin
\vliriv
x = sin S cos 0, 1y = cos ^ sin 0, /u = cos (\ + 1).
The conductivity and resistivity at the point (0, \) will be denoted by p and «'
respectively, p«' being, of course, equal to unity. For the present we shall suppose
that p and / are finite and continuous functions of to, so that they can be expressed
as FOURIER'S series in cos nu> over the range 0, v ; p will certainly satisfy this
condition, and the case of p = 0, K = °° will be considered later. Further, it will be
assumed possible to express K' as a TAYLOR'S series in cos w, and it is in this form that
we shall suppose the resistivity to be given, as one of the data of the problem.
Theoretically this is a limitation of the problem, as there are some functions which
cannot be expressed in the form stated ; for instance, if the conductivity were
proportional to cosco in that hemisphere on which the sun is shining, and zero or
constant over the other hemisphere, K' could not be so expressed. But in reality
nothing of value is lost, as any continuous function can be approximately expressed
in the form of a TAYLOR'S series to any desired degree of accuracy.
[Some further explanation of this use of series may be desirable. The series used
in the analysis are all written as infinite ones, for the sake of formal simplicity and
theoretical completeness. In the detailed execution of the work, however, only a
finite number of these terms can be utilized, as workable general expressions for the
coefficients in the current function II cannot be obtained. The actual procedure,
therefore, must be to take a finite number of terms of the FOURIER'S series for />,
transform this into a polynomial in cos u> (this also, of course, will have only a finite
number of terms), and work out the coefficients of R in terms of the coefficients of
this polynomial to as great a degree of accuracy as is practicable and desirable.
This is the course of the work in §§ 18-20, where the terms (a + b cos u> + c cos 2o>) of
the FOURIER'S series for p are taken, and the expression for R is worked out as far as
concerns the terms in a, b, />", and c. The resistivity ifp is introduced into the
calculations for purely mathematical reasons, on account of certain analytical
advantages which it seems to offer. The results obtained in this way, in terms of
the coefficients of p, might be got otherwise by an extension of the method used by
SCHUSTER. This identity of results is clear from the fact that if the FOURIER
coefficients of p are small enough the TAYLOR'S series for l/p is absolutely convergent,
and the legitimacy of the use of l/p is in this case immediately evident ; the formal
results, however, do not depend on any property of convergence, so that the results
obtained by using l/p remain equally valid with those obtained in any other way,
even though the series for l/p should become non-convergent. This is one of many
instances in which it is possible and advantageous to use expressions which may
VOL. ccxin. — A. •_' i1
290 DR. S. CHAPMAN ON THE DIURNAL VARIATIONS OF THK
become non-convergent to obtain results which can be got less simply in other ways.
Whether the final result is convergent depends in this case only on p, and not on tl it-
processes of analysis used to deduce II from p. — Added June 11, 1913.]
We shall write, therefore,
00
K = Cae 2 df cos7* <D, = COCK,
o
where C, e, and a are constants (introduced for convenience) whose meaning will be
explained later, and the coefficients dp are given numbers. Expanding cosp w in terms
of n, we have
where
0>
and (since
2*~V* = 2m~1cosra(x + «) = co8m(\ + t)
fp is given (for jo = 0) by
00
/,= s
4 = 0
B>
Here we have written
In virtue of the definitions of ep and apq, we have
Next we consider the differential coefficients of *. We have
~
We shall write ff for ; evidently we have
cos - sn
D
KARTH'S MAGNTTISM PRODUCED BY THE MOON AND SUN. 291
!; li!. Following SCHUSTER'S notation and treatment, the earth will be regarded as
;i uniformly magnetized sphere of radius «, whose magnetic potential may be resolved
into the zonal harmonic of the first degree and the tesseral harmonic of the first degree
.ind type. The former harmonic is much the larger of the two, as the inclination (0)
of the magnetic to the geographical Jixis is small. The radial force can be expressed as
V = C cos ti + G tan 0 sin 6 cos X,
where C is a constant not differing much from — $ (the force being measured positive
outwards), and X is now the longitude measured from the meridian (68° 31' west of
Greenwich) containing the magnetic axis.
The components of electric force, X and Y, measured towards the south and east
respectively, are
Xa = -.
am 0
\js being the velocity potential.
If we express X and Y in the form
dS J dli dS
x = y_
~ '
dd e sin 0 d\ ' siu 6 d\ e dd '
where K is the known resistivity and e the thickness of the conducting atmospheric
shell, the function R will be the current function of the electric currents produced by
X and Y (neglecting electric inertia). The function S is the potential of a system of
electric forces which in the steady state are balanced by a static distribution of
electricity revolving round the earth, and causing a variation in the electrostatic
potential which is found to be too weak to affect our instruments.
To determine R we shall eliminate S, thus obtaining the equation
dx deV "/-BinedxVdx/1 de
Instead of using the resistivity it', SCHUSTER worked with the conductivity p (using
the special form 1 + k cos w), in order to avoid the difficulties introduced by " the high
and possibly infinite values which K would take when the conductivity sinks low or
vanishes."* These difficulties, however, are found not to be serious, and the work
is greatly simplified by the use of K, which enables R to be determined directly,
without first evaluating S, as is necessary when p is kept as the variable quantity.
The investigation can also be made much more general, without formal complexity,
when K' is used.
* SCHUSTER, 'Phil. Trans.,' A, vol. 208, p. 190.
2 P 2
292 DR- S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
lini,' tin- i-xpressions for X, Y in terms of the velocity potential \Js, the left-
hand side of the last equation, after division by Ga sin 6, may be written*
—
27»+l
2(2m + l)
+ {- (m-1) (ro+ 1) (ro + r) (ro + r-l) Q,,,-!7-1
The right-hand side of our equation for R, after division by Ga sin 0, becomes
equal to
K ! d* ad a d
^ ysl y
We suppose R to be expressed as the sum of a number of tesseral harmonics
pH"tyn° sin (o-V— a'), where pn" is a numerical coefficient, X' has been written for X + £,
and & i^anges (possibly) from — oo to +00. The contribution of each such term to the
total value of the last expression is easily seen to be the product ofpn" into
-n (n+ 1) Q/ sin (erX'-a') {/„ + 22/p cosj>X'},
where we have inserted the values of K and its differential coefficients, and have
transformed the first line by means of LAPLACE'S equation
iSo far ff and fp have been defined only for positive and zero values of p ; we now
extend the definition by the equations
JP J -pi J p = j -p-
* Ibid., pp. 188, 189.
EARTH'S MAGNETISM PRODUCED BY THE MOON AND SUN. _'.' ,
This enables us to write expression (3) in the form
= 2 Il/(/>)sm(<r+jo.\'-a'),
where
R/(p) = (-n(n + l) Q/-«?9£k+.
Whenp is positive, substituting our expressions for/p and/',, we timl
/(p) = 2
q = 0
* r/O ff
2 (p + 2?) «,,7^ . £ cos ,* cos 0 ^ . JT1,
fl = 0
/C080^--^,)+g
Since, in the original equation for R,*(/)), a change in the sign of p only affects the
term <rQn'/sin 8 in the first term, from our last expression we may at once write down
the value of B.*(— jo), p being positive, by changing the sign of (rQ.'/sin 6. Thus,
§ 17. For convenience and clearness, some well-known formulae of transformation
will now be set down. These have been much used by SCHUSTER in his papers, and
294 OR- 8. CHAPMAN ON THE DIUKNAL VARIATIONS OF THE
the equations, and some formulae derived from them, will be denoted by the same
Roman letters which lie uses.*
i<r ......... (A)
= (n+<r)(»+a-l)Q._1*-1-(n-«r+2){»-«-H)Qt+1(r-»i (C)
1'+1) ...... (D)
81110
1'-1, . . . . . (E)
. . (F)
sin
+ (w+ 1) (n + cr) (w + o— 1) Q,,,/-1},
Making use of these equations, we obtain the following expressions :—
In II.' (— p), the second term remains the same, while the expression in square
brackets in the first term becomes
These expressions for RB* ( ±p) are of the type
Now by equations (B) and (C), it is evident that Q/+ V~l, Q/'V'1, and Q/y' can
be expressed as the sum of a number of tesseral harmonics all of type <r+p or all
of type <r-p (at will), and of degrees ranging, by steps of 2, from v±(p-l),
v±(p-l) and f±p respectively. Further multiplication by y* can be so arranged as
* Ibid., pp. 187-189.
r.AKTII'S MACNKT1SM I'KODITKD BY THE MOON AND SUN. 295
to leave the type unchanged, while extending the range of the degrees by 4q. Also
by equation (A), the coefficient a, which, it will be remembered, is a power series in
cos 0, leaves the type unchanged while it increases the range of the degrees of the
resulting tesseral harmonics. In every case, therefore (p positive or negative), It/
can be expressed as the sum of a number of terms such as Q,**'. Therefore if we
write ^ for the sum of all the expressions (2) resulting from each term QmT sin (T\'— a)
in the velocity potential \}s, the fundamental equation (l) for R takes the form
(4) ^ = 2 { 2 kn'-pQn'*f sin (<r+p . \'— a')},
where kn''p is a coefficient whose value can be determined in terms of pn* and the
coefficients dp in the TAYLOR'S series for K. By equating the coefficients of harmonics
of the same degree and type, on the two sides of the equation, we obtain equations
to determine the coefficient pm' in terms of the dp» and the known constants of
the velocity potential. In practice this must be done by a process of successive
approximation. Knowing, from the form of the above equation, which is linear in
p" and dp, that every coefficient p," can be expressed as a TAYLOR'S series in
-^ , -^ , -j2 , and so on, we can determine this series by successively assuming that
all save one particular variable -f are zero, and considering this variable alone, it
may easily l>e seen that the phase angle of every term in R arising from a particular
term in Sk is the same as that of the latter.
§ 18. SCHUSTER has worked out the values of pn' for the special form of conductivity
already mentioned, and for the two terms Q,1 sin (\'— a) and Q/sin(2\'— a) in the
velocity potential, to the fourth order of approximation, and he finds that the
numerical coefficients of the terms are such that only the first order term (depending,
in our notation, on dt/da) are large enough to be detectable by olwervation. The
present calculation will not be carried so far, therefore, and will not include terms of
higher order than d.Jd0 or (di/d0)3. Also, since in the expression for ^ the term
depending on the inclination of the magnetic to the geographical axis is multiplied
by the small factor tan <f>, the part of It depending on this term will only be calculated
as far as the first order dtfd0. Further, since the actual atmospheric oscillations seem
to be mainly performed in the simplest mode possible, so that m = r for the principal
terms, the second order terms will be neglected for the smaller harmonics in the
velocity potential \js, for which in ^ T.
We therefore consider the terms in R which depend upon a term AM'TQmT in "b,
where m' and r are quite general, except that in the terms depending on d-Jd^ we
shall suppose m' = r'+l (since Q,/ = 0 when T > m', the term in (2) depending on
Q*-iT vanishes when m = r).
It will first be necessary to write out the developed expressions for R/(o),
R/(±l)> IV (±2) as far as the terms in da. No other values of p in R,*(p) give
296 DR. S. CHAPNfAN ON THE DIURNAL VARIATIONS OF THE
terms containing dtt> d» d* It is convenient to write down first of all the values of
/„,/„/„/'«./'., A omitting d, and higher terms.
cos S cos 0-el sin ^ sin 6,
fi = &i cos S cos 0-2e^ sin ^ sin 6,
fa = e^ cos S cos 0.
R/(0)= -n
Q/]. '
As already stated, in the last two lines we have substituted a- = n— I in working
out the numerical coefficients, as also in all other terms with d2 as factor.
w ,
n(n+8)(2n-l)(n-l)
+ 3 (n- ij !(n-2) (»+ 1) (2n+ 1)} Q/
If in the above expressions the term Q,,-1 occurs, it may be replaced by
EARTH'S MAGNETISM PRODUCED BY THE MOON AND SUN.
•297
§ li). Consider now the terms .-irising ('nun a term AJ..TQm.T' sin (r'X'—ct) in >k. The
only term of corresponding type involving da, on the right-hand side of equation (4),
\\hich does not vanish, is
-pjm' (m'+ 1 ) d&S sin (r'X'-a').
Consequently, to order
1>
, and of = a,
and no other term is of order l/d0. Next, taking terms of order d,/o?0, it is evident
that these can only arise from ~Rm-T (0), Rm-T (±l), which involve d1Qm'±,'/, C^Q-»*i*'*1.
Eqtiating the sum of the coefficients of the terms containing these harmonics, with
factors d0 and dl} to zero, we get the following general expressions for pm'±/, pm'±\*1,
to order dt/da : —
(5)
T _
PM'~I
/i sin S m' (m'—r'+l)
da (m'
ro'(2ro'+l)
Pm
i cos S
m'
m'+l
V',
' I 1 \ tCl^l I 1 \ P"' '
T-_, _ i dl cos 3 (mr + 1 ) (mf + r) (m' +r'-l) T
* '/ /(
If the type of any of these coefficients exceeds the degree, it must be set equal to
zero. No other terms are of order djd^.
So far no restriction has been placed on m' and T. In making a further approxima-
tion, we shall not write out general expressions, but shall consider the effect of the
second order terms (d.Jd0 and df/df) on two specific terms in the velocity potential
of the atmosphere, viz., Q,1 sin (\'— a,) and Q23 sin (2\'— 04), which give rise in ^
to terms
-Q^sinx'-a, and -'«
The terms of the proper order on the right-hand side of (4) are (a) those involving d.t
from Rm.T'(±2), and (b) those involving df/dj from Kw±1' (p) and Rm.41T'*' (p)
where p = 0, ±1.
VOL. ccxill. — A. 2 q
DR S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
Considering the diurnal variation first, the terras (a) are found, from the formulas
at the foot of p. 296, to be
)<W [W-13Q,1) cos2 S sin (X'-«1)-(12Q41+ 16Qa') sin2 S sin (x'-a,)
-{(4Q4'+3Q33) sin (2X'-a1)-(48Q10-6Q2°) sin (-a,)} sin S cos S
-£Q<3 cos" Jain (3X'-a,) + 3 (Q^-Qa1) cos2 i sin (-X'-a,)],
and the terms (l>) are
MCMHQ/- W) sil1 (2X'-a,) + ( W-¥Q,°) si" (-«i)l cos i
-(*W+ ¥Q.1) sin <* sin (x'-«0],
sin (2X'-ai) + Q2° sin (-"')} cos ^-Qs1 sin <S sin (x'-a,)],
'-aOl cos J
Q22) sin S sin (2X'-a1)],
21) {sin (X;-a,) + sin (-X'-a,)} cos S
2°) sin 5 sin (-a,)],
i°^i[{-W sin (X'-a^-iQ,1 sin (-X'-a,)} cos ^-2Q2° sin S sin (-a,)].
The sum of the coefficients of any particular term Q/ sin (o-X'-a^ in the above,
must be equated with pn'n (n + 1 ) Q." sin (o-X' - a,). The values of p.' thus calculated
are given in Table I. It should be remarked that Q^1 has been replaced by
-Qn/n(n + l), and the coefficient of Q,1 sin (X' + a,) will be denoted by qn\
§ 20. The second order terms arising from the semi-diurnal atmospheric oscillation
are similarly written down as follows : —
(«) P.M[{(W-4Q/) cos2 <H¥Q52 + 4Q32) sin2 S} sin (2X'-a3)
+ {(- W-Q/) sin (3X/-a2) + (Y-Q51-6Q31) sin (x'-aa)} sin S cos S
- iW cos3 S sin (4X'-ai!) + (l2Q30-ArQQ50-^Q1u) cos2 S sin (-0,)],
(l>) pa'di[{-W sin (2X'-a8) + (VQ3u-SQiu) sin (-as)} cos S
-(-15<iQ31 + IQi1) sin S sin (X'-a,)],
in (2X'-«2) + (-mu-¥Q3°) sin (-a,)} cos 3
-(¥QB1 + ¥Q81) sin S sin (x'-aa)],
sin (3X'-oa) + (gQ31- VQ,1) sin (x'-a,)} cos S
i[{(-5Qo3+tQ33) sin (3X'-a2) + (lGQ51-25Q31) sin (X'-a,)} cos S
-(8QB3+ 10Q3a) sin S sin (2X'-a3)],
+P<*di [{~^Q84 sin (4X'-a2) + (8Q52-35Q33) sin (2X'-aa)} cos S
-(¥Q53+-m3)sm«r].
The values of p.* calculated from the above expressions are given in Table II.
EARTH'S MAONKTFSM I'lJOMCKI) BY THK MOON AND SUN. 299
T\I:I.I: I. Velocity Potential Qt1 on
, _ _J_ 5 d* cos3 S _ 13 cos'i! / , _ d,j
P- '' Gdt> 432 d* 252 ' da* l ' ; '/.,
3 dl cos
20
»'- 15
cos
ft1-
3 ntt sin
20 rC
ftl-
d} sin
^
45
ft'-
1 c?i cos S
90 d0a '
ft' =
1 ti," sin S cos $ 1 sin 8 cos
72" "^7" "42" dua
2 BJU J cos S / , _ <V \
35<C "T" ci,/'
3 _ sin S cos S d£ _ sin S cos ^ /
v>? = ' — TT- [d* — j- ) ,
•^4 o m w ^ \ •* /7 .' '
COS'
« 7 _
V ' do/'
7a - -
cos" S d* coe" S ij d*\
144 ' d* 84d^\ * 0
OOtfS
ZSOdJ V1**
2 Q 2
300 DR. S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
TABLE II.— Velocity Potential Q2»sin (2\ + t-a2).
,
-1 __ i!_ 2 .
'
i _ H> r/, cos i$
PJ G3~d7~'
i _ 3 dt cos (?
p4 ' 70~d^T'
J= _8_ dl sin (?
63 d,,2 '
3 _ J_ f?i sin (?
35 rf
„ .
.t _ 1 r/i cos
"
140 df
d2
o=.._CC08 __
^
36
o= _ __, ,
105 ^ r-^j'
„, i _ _§_ sin J cos S I, d^\
35" df -TJ>
i _
sin ,? cos J sin <$ cos c5 / , d 2
" - -
-o . _ 16 sin 3 cos S ,
525 2"'
2 = _
525
sin a cosj _ sin ^ cos S I, d
~--
« a = sn
' ll
,= _
EARTH'S MACNF.TISM PRODUCED BY THE MOON AND SUN. 301
As this paper is primarily concerned with the lunar diurnal variation of the earth's
magnetism, the numerical values of the coefficients pn* arising from expression (2)
owing to the inclination of the magnetic to the geographical axis will not be written
down here. This can at once be done, when necessary, from the equations (5), as
also the terms in the current potential arising from diurnal and semi-diurnal
atmospheric oscillations of degree higher than the type.
§ 21. The main general result of our investigation is the same in form as that of
SCHUSTER'S more special calculations, viz., that the current function R of electric flow
induced under the action of the vertical force C cos 6 in a shell of air oscillating with
a velocity potential AmTQmT sin (r. \ + t— a), under the influence of a variable resistivity
depending on the zenith distance («) of the sun, is
(6) AMT[ 2 p/Q/sin {<r(X + t)-a}+ 2 g.'Q.' sin {<r(\ + *)
9=0 V= 1
In order to obtain the magnetic potential of the variation caused by the flow of air,
a factor — 4ir(/t+l)/(2n+l) must be inserted before each term Q,'.
We have considered only those terms in the resistivity which depend on cos u> and
cos* u>, though the general theory has been given for any numl)er of terms. If then
cos w+dj cos* u>),
we have for the conductivity p, to the same degree of approximation,
If we put
Caed0 *"" d0 p0' c
this becomes
p = p0 + pl COS o> + pa COS2 w.
In SCHUSTER'S calculation, the last term was omitted, so that p, was taken equal to
zero, while p, cos $ and p, sin S were written p9v and p0v respectively. If we make
these substitutions in Tables I. and II., it is readily verified that the present results,
as far as they go, reduce to those obtained by SCHUSTER. The extra terms depending
on d3— -j- give the effect of the term cosa <a in p.
dQ
§ 22. Finally, a word must be said with regard to the legitimacy of our analysis,
considering the fact that if p falls to zero, K', the resistivity, must become infinite.
Regarding the matter physically, it is evident that an infinite resistivity is not likely
to introduce spurious terms into the current potential, and an examination of the
equation (l) for R will show that an actual infinity in K would only lead to a
/.fro term in R. But such an infinite term should not occur in the analysis, and it
302 I>R. R- CHAPMAN ON THE DIURNAL VARIATIONS OF THE
is clear that by altering the constant term in p, so that p never falls to zero, the
;I|K>V<- calculations become formally and really legitimate ; when we wish to return to
the actual case we must appeal to the " law of continuity," and the fact that our
mathematics is applied to an ordinary physical problem, to allow us to pass to the
limiting value of da in the final result. The latter is expressed as a power series
in \jd0, and if d0 is sufficiently diminished, this series might become non-convergent.
But the actual results do not indicate any such behaviour, and are, as we have seen,
identical with those obtained by SCHUSTER'S method (in which the conductivity only
was considered), so far as the scope of the two calculations is the same.
§23. So far the calculations have been kept quite general, in that no relation
between the causes of the variable conductivity and of the atmospheric oscillation
has been assumed. Thus they may both be caused by the sun, in which case the
mathematics is that applicable to the theory of the solar diurnal variations of the
earth's magnetism. Without much modification, however, they may equally well be
adapted to the case of the lunar diurnal variations. We shall consider it sufficient,
for our purpose, to regard the solar and lunar periods as equal at any one time,
allowing for the slow cumulative effect of their inequality by introducing a variable
phase angle v into the expression for cos w, the quantity on which p and K depend.
Thus
cos to = sin <5 cos 0 + cos<?sin0 cos (\ + t' + v),
where tf is now the local lunar time of the standard meridian (measured from upper
culmination), and v measures the lunar phase, increasing from 0 to 2-rr from one new
moon to the next. The velocity potential will be Q22 sin (2\ + t'—a). The calculations
will be formally the same if we now change the meaning of X' to X + t'+v, so that the
velocity potential becomes
Q/sin(2X'-a-2i>).
Thus by equation (6) the current function obtained is
2 J>/Q/sin{«rA'-a-2,-} + 2 qn'QS tun(a\'+a + 2v)
» = 0 <r=l
2 g/Q/sin {<r(\ + «') + « + (<r+ 2)*}.
<r=l
The terms on the left of the last line change in phase through an angle 2(o— 2)v
each month, viz., -2* for the diurnal term, zero for the semi-diurnal term,
+ 2ir for the third component, and +4ir for the fourth component, as the observa-
tions indicated. The terms on the left change phase by 2(^+2)^ each month,
a change so rapid that it would be difficult to detect in the observations,
affected as these are by accidental error. The coefficients qn', moreover, are very
small, so that altogether these terms are negligible.
One interesting result of the analysis may be noticed here, viz., that the main
EARTH'S MAGNETISM PRODUCED BY THE MOON AND SUN.
303
lunar term in the magnetic variation, Qa*, has a coefficient p' which does not (to the
order of accuracy of our calculations) show any dependence on solar declination. Thus
any seasonal change in this term of the magnetic potential cannot be referred to the
effect of the varying declination of the sun. This is not quite the case with regard
to the main diurnal term in the solar diurnal magnetic variation.
§ 24. We will now consider what are likely values of p,/p0 and p.Jpa to substitute in
our formuLe, in order to get a comparison with the observed data. The conductivity
should rise to a maximum during the daytime and fall to a minimum about midnight.
It cannot actually be less than zero, though it is not so clear that it is better to have
the least value of p zero than to have it slightly less, in order to make the mean
nightly conductivity small in amount. However, we will keep to this condition, and
make pmln. = 0 ; it is found that the following is a very satisfactory expression for the
representation of a function of 6 which is large for values of 6 up to -, and much
2i
smaller, while never negative, from 6 = £ to -IT : —
The following table and figure gives the value of 4p/p0 for every 10°. It is seen
that the mean of the nine day values is '24' 1 times that of the nine night values.
The function has a physically false maximum at midnight, but this is of very small
amount, and some such feature cannot be avoided with so simple an expression
for p : —
(D.
0°.
10°.
20°.
30°.
40°.
50°.
60°.
70°.
80°.
90°.
4p/*>
25-0
24-5
22-2
21-1
18-5
15-4
12-2
9-2
6-4
4-0
ID.
100°.
110°.
120".
130°.
140°.
150°.
160°.
170°.
180°.
4p/Po
2-2
0-9
0-2
o-o
0-1
0-4
0-6
ro
1-0
* I might remark here that in working out Part II. of this paper I had not contemplated the possibility
of the coefficients of p/p0 being greater than unity, as seems to be necessary if the atmospheric conductivity
is small and nearly constant at night. The size of these coefficients makes it necessary to carry the
calculations some steps further than I have already done, before a sufficient degree of approximation is
arrived at. The present work suffices, however, to establish the point with which I am most immediately
concerned, viz., that the size of the fourth harmonic in the lunar variation is inexplicable with the form
a + fccosw for p, while the addition of a term ccos3ui introduces a fourth harmonic in the theoretical
result, which agrees, as to order of magnitude, with the observed quantity. Better olwerved data are now
lifing obtained, and concurrently I shall proceed to carry the theoretical calculations further, in order to
test the exact numerical agreement between theory and observation. — June 11, 1913.
|.K. S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
Diagram illustrating the assumed form for the atmospheric conductivity
P = po (1 + 3 cos <o + j cos2 <o).
§ 25. Substituting the values
in the expressions for pf in Table II. (the table which relates to the lunar variation),
we get the following values for p0Caepn°. The terms for which o- = 0 are omitted, as
they merely produce a monthly change in the mean magnetic elements.
V.
n.
1.
2.
3.
4.
5.
1
ij sin 8 cos 8
J|cos8
}£ sin 8 cos 8
-VffCOsS
- j1,^ sin 8 cos 8
2
5", sin 8
5
A s>» 5
-Tr!ff(3co828-l)
3
jYo sin 8 cos 8
TTT cos S
T 1 f sin 8 cos 8
4
TAuCOS^S
EARTH'S MACM 1IS.M l'l:oi>(JCEP BY THE MOON AND SUN. ::<'..
The following are the values of the corresponding tesseral harmonics:—
Qi'ssind, Q,1 = 3 sin 6 cos 0, Q,1 = i sin 0(5 cos' 6-1),
Q,1 = i sin 0(70 cos3 6- 15 cos 0), Q*1 = g sin 0(63 cos4 0-84 cos* 0+3),
Qa3 = 3 sina0, Q,s = 15 sin*0 cos 0, Q4S = \* sin'0(l4 cos8 0-1),
Qs» = J-lp sin* 0 (3 cos3 0-2 cos 0),
Q3S = 15 sin" 0, Q<» = 105 sin8 0 cos 0, 06' = H'A «n* 0 (9 cos* 0-2),
Q/ = 105 sin4 0, Q6« = 945 sin4 0 cos 0.
Since all the stations for which we have observational data, in Part III., are
tropical, we shall consider the values of X, Y, and Z for such stations only. Hence,
in our expression for V, the magnetic potential (which we must now use instead of
the current function), all terms containing coss0 may be neglected, and will be
omitted. Thus we get
•CT /. - ,, « sin 0 cos 0 ,, . . »sin0\ /., , \
V = [£H COBS. - • + TgJ08>"<*C08 S— — )c08(\' + »<-a),
\ r r I
,//,., ., - —j-i — Txsina0cos0 l7] • . sin* 01 /„ , ^
+ {(¥ + i I • 3 cos" S- 1) - - - + -fts sm S . -pp-j cos (2X'-a),
• . ,sin30\ /„./ \
sindcosd — — )cos(3\ — v— a),
2 . sin40 CO80 / .. , 0 \
cos o • - ~j -- OP6 (4X — 2c— a).
In the above expression, the terms depending on sin S represent the main seasonal
effect. Since
Y v
aX = -r— , aY = .
'
-r— , . n . , -- =-,
df) ' BmOdX dr
it is evident that when cos0 is put equal to zero after the differentiation, only the
terms in V which do not contain cos 0 will contribute any result to Y and Z. But
the above equation shows also that these terms always contain sin S, so that at
equatorial stations Y and Z change sign in passing from summer to winter.
Tables XL and XIII. corroborate this sufficiently well, especially when it is
remembered that the stations are not quite equatorial, and that the obliquity of the
magnetic axis also produces a disturbing effect. A further interesting consequence
of the above equations is to indicate that at the equator the terms in X which
depend on sin <?, i.e., the seasonal terms in the horizontal force variation, vanish.
This agrees with the known fact that at tropical stations the X variation hardly
'•lianas throughout the year. Table XII. illustrates this, especially for the most
nearly equatorial observatory, Batavia (6° S.).
Km comparison with observation we shall write down the values of the ratios
of the amplitudes of the first, third, ami fourth harmonic components to that of the
vol.. i . AMI. — A. 2 R
SOfi
1,1;. s. CHAI'MAX ON THE DIURNAL VARIATIONS OF THE
second; for X we take the mean value of cos S in our equations, and neglect the
seasonal changes ; for Y and Z the terms in cos 6 and sin S are taken separately.
The values of the amplitudes of the second component in the several cases are
also given. It should be remarked that our calculations have not been carried
sufficiently far to give the seasonal variation of the fourth component, but it is less
important than the term in cos 6, for such stations as Bombay. We thus obtain the
following table : —
die,
(VC*
C4/C2.
C2.
x
0-61
0-47
0-13
2-61
Y(cos0)
(sin S)
Z (cos 0)
(sin 8)
0-31
0-04
0-41
0-05
0-70
0-55
0-62
0-49
0-27
0-22
5-22 cos 6
1-OlsinS
7 -83 cos 6
1 • 52 sin 8
From SCHUSTER'S calculations, taking pjpo = 1+cos w, the following table of values
of C/C,, in which the seasonal changes are disregarded, is obtained : —
eye*
Ca/Cj.
C4/C2.
X
0-G7
0-38
0-002
Y (cos 6>)
Z (cos ff)
0-33
0-62
0-58
0-46
0-003
0-0025
Our observational data only allow us to make the roughest possible comparison
with these calculations, and the following table is enough to give an idea of what
agreement is present. It is got by taking the mean amplitudes at Bombay, Batavia,
and Trevandrum (as many as afford data in each case) for the whole year, combining
the columns April to September and October to March together by simply averaging
the amplitudes regardless of phase.
C,/C2.
Ca/Cj.
C4/C2.
X
0-94
0-42
0-28
Y
0-50
0-64
0-23
Z
0-85
1-06
0-47
The size of the fourth harmonic shows that the term cos3 a> in p/p^ has distinct
importance, for without the presence of such a term, as the second of the above
tables show, there should be no appreciable fourth harmonic at all. As regards the
other harmonics, there is little to chose between the two expressions for p/p0, though
KAIMirs MACNKTISM 1'KOIHVKH i:Y TIIK MOON AND SUN.
807
the more complex one might bo made to fit better than the above figures indicate, if
the constants of the formula were altered a little. This, however, is not worth while
doing till better observational material is to hand.
PART III. — Tlw Ol>*< r fit tonal Material.
§ 2G. The following are the data which were available for examining the dependence
of tin1 lunar magnetic variation upon lunar phase : —
Station and period.
Sub-division of month.
Seasonal ( li vision.
Trovandrtim (1854-64) ....
Bombay (CHAMBERS) (1846-71) .
(Moos) (1872-89) . .
Batavia (1883-99)
Bombay (CHAMBERS) (1846-73) . .
(Moos) (1872-89) . . .
„ (1873-79, 1881,
1883-85) . . .
Batavia (1883-99)
DECLINATION.
Four quarters of month
Eight phases
HORIZONTAL FORCE.
Eight phases
Bombay (Moos) .
Batavia (1883-99)
VERTICAL FORCE.
Eight phases
Separate months of year.
Nov.-Jan., Feb.-April, May-July,
Aug.-Oct., April-Sept., Oct.-March.
Nov.-Jan.
April-Sept., Oct.-March.
As for declination.
Nov.-Jan.
May-July.
April-Sept., Oct.-March.
As for declination,
it >> •
For purposes of comparison, the Trevandrum results for the separate months of the
year have been combined into the four quarters and the two half years (as for
Bombay) ; also the 25 hourly values have been reduced to 24.
The separate tables of the 24 hourly values will not be repeated here, nor the a
and b, and C and 0 coefficients of the first four harmonic components which have
been calculated from those tables. The harmonic formula used has l>een
a, cos t + &, sin t + Oy cos 2t + 62 sin 2t + a3 cos 3t + b3 cos 3< + «4 cos U + 64 sin U,
In the case of all the coefficients a, b, 0, the adopted unit is 10~7 G.G.S. units of force
(the declination results were also reduced in terms of force), and this was reckoned
positive towards the North, West, and upwards (II, D, V).
$-7. The tables of harmonic coefficients showed that they were subject to an
accidental error of amount small in itself, but quite a considerable fraction of the
2 R 2
308 DR. S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
whole effect. This is not surprising when the minuteness of the lunar variation is
considered. The tables showed some outstanding features, however, in particular
the constancy (within reasonable limits) of C2 and 0a ; CIt Cs, C4 are generally smaller
and rather more irregular in amount for the eight phases. The phase angles 0,
showed a fairly regular increase through 2-* with the moon's age, while 0, showed a
less regular decrease of the same amount. No regular change in 04 was noticed,
partly because C4 is small and 04 therefore not well determined, till the Batavian
summer declination results were considered ; in this case the fourth harmonic
happened to be exceptionally large, and the phase therefore better determined.
This clue having once been obtained, the same feature, viz., a monthly increase of
4ir in the phase angle 94, was verified to be present in most other cases, where C4
was not too small. The examination of the phase laws followed by the harmonic
components was first undertaken by means of vector diagrams, and independently of
the theoretical considerations which suggested themselves later, and which are
embodied in §§ 12, 23.
The real test of the phase laws suggested by the vector diagrams was made, of
course, by correcting the phases by the amount through which the law indicated they
had changed from the period of new moon. The corrected values, tf (where
V2 = &i>
v being the moon's age, in angular measure, at the particular lunar phase considered)
should then all be the same (for the same value of the suffix and different
values of v), apart from accidental error. The Tables III. to X. show that this
is the case, generally, as far as we have any right to expect, though, in some
instances, the agreement is not very apparent. Even in these cases, however,
the mean value of & frequently agrees so closely with the mean value of tfa as to
show that the phase law is acting, though its manifestation is obscured by large
accidental error. This agreement between tflt tf2, &3, and 0'4 is a noticeable feature,
of which, as well as of the monthly changes of phase, the theory of the lunar variation
gives a satisfactory account (§ 23). On general grounds, too, it is to be expected
that if any simple relation exists at all between the phase angles of the four harmonic
components, this relation should assume the simplest form (which proves to be
equality) at new moon, when the sun and moon are on the same meridian. The
equality of the phase angles at new moon points to a single exciting cause (the lunar
atmospheric tide being suggested) of the four components.
The regular monthly change in the values of 0,, 03, and 04 results in the
i:\HTIIS MAUNETISM PRODrci-.l» UY THE MOON AND SUN. 309
disappearance of th(> corn-sponding harmonic components from the lunar variation,
as (I'-ti-i -inined from the mean of a whole number of months. It is found, indeed,
that any such component still remaining is of purely accidental character.
As to the amplitude of the various components, this appears to be independent of
the lunar phase, the irregularities l>eing accidental. The mean of the amplitudes at
the separate phases has therefore been taken as the best value of the true amplitude,
except that a correction has l>een applied to allow for the fact that the instantaneous
amplitude is greater than that deduced from the mean of a few days, during which
the phase is varying. Thus, if we tabulate a function c cos(n6 + kv), where Q is the
lunar hour angle (one hour = 15°) and v the age of the moon in angular measure, in
lunar hours for successive days over an interval of the month t>l to i/2, the mean result
may be taktm as
c (cos nd cos kv— sin nd sin kv)
where cos kv, sin kv are the mean values of these functions over the range vl to v.2.
This equals
showing that the phase of the mean wave is equal to the true phase at the mean
time, but that the amplitude is reduced in the ratio
sin
2 ("•-"*)
The corresponding factors to counterbalance this are for Trevaudrum, where va— vt
is one-quarter of a month, or ^,
and l'57(C4),
and at other stations, where va— vt = -",
4
1'02(C,, C,) and l'll(C4).
The values of the mean amplitudes, thus corrected, and of the phases of the
four components, reduced to new moon, for all the stations, are summarized in
Tables XI.-XIII.
Tin* resolved parts of the amplitudes in the direction of the mean phase (where
the separate values of tf depart much from the mean) might have been taken, but
this would not have altered the mean amplitude greatly, and seemed hardly worth
810
DR. S. rilU'MAN ON THE DIURNAL VARIATIONS OF THE
while in view of the large accidental variations of the determined amplitudes. In
Tables XI.-XIII. the values of the mean phases tf have been characterized by
affixes 1, 2, 3, 4, 5, representing (in descending order of merit) the reliability of the
mean as judged from the accordance of the separate values of tf. Only the numbers
marked 1 to 3 can be considered at all satisfactory.
As regards the accordance of the results from the same or different stations, the
best feature is the extremely good agreement between CHAMBERS' and Moos's values
for Bombay, for different periods of time and for different instruments.*
* VAN BEMMEI.EN, in his paper in the 'Met. Zeitschr.,' May, 1912, already referred to, remarks that
the two computations do not agree at all, but this must evidently be due to a mistaken reduction of
CM AM units' results, which he quotes at three times their proper value.
TABLE III. — Trevandrum. Declination West.
Lunar phase.
c,.
ffi.
oy,
0-2-
C3.
0'3.
C4.
<?i.
New moon
First quarter
Full moon
Last quarter
96
36
75
45
0
294
270
220
261
Noveml
204
162
162
162
0
>er-Janu{
285
266
263
277
iry.
96
90
66
75
0
306
252
296
300
15
15
12
9
0
259
313
254
246
Mean . . .
63
261
172
273
82
288
13
268
February-April.
New moon
First quarter
Full moon
Third quarter
84
54
36
57
5
3
-114
- 29
90
105
102
114
291
310
318
353
45
60
39
51
339
331
367
363
11
3
11
10
43
6
9
86
Mean . . .
58
326
~ i —
103
~
313
49
350
9
31
The unit of force in the tables of amplitude is 1Q-* C.G.S.
unit.
KAKTII'S MAGNETISM PRODUCED BY THE MOON AND SUN.
311
TM.I.I: III. (continued).
Lunar phase.
c,.
">
( Ot.
C*
r»
C4.
r*
•
•
May-July.
•
«
New moon
First quarter
Full moon
57
72
54
115
72
43
72
63
39
84
90
102
60
30
36
70
130
93
15 62
12 272
8 27
Third quarter
13
98
42
71
45
96
21 - 37
Mean . . .
49
82
54
87
43 97
14 81
*
August- October.
New moon
39
39
36
134
18
149 4
141
First quarter
Full moon
36
81
92
43
42
63
174
139
16
39
214 16
160 15
167
324
Third quarter
33
140
39
184
15
187
17
203
Mean . . .
47
78
45
158
22
177 13
209
April-September.
New moon
51
73
69 88 48
82 13 59
First quarter
Full moon
45
66
50
37
45 95 24 116 5 - 56
30 106 36 100 6 - 23
Third quarter
12
6
24
67 21 96 10 - 66
I 1 '
Mean . . .
43
42
42
89
32
98
8
341
October-March.
New moon 60
32
162
282
75
306
6
253
First quarter 12
- 3
135
276
78 296
14
330
Full moon 51
-141
135
273
48
301
10
294
Third quarter
54
- 57
120
289
51
316
5
228
Mean . . . 1 44
318
138
280
63
305
9
276
i i
[
The unit of force in the tablet of amplitude is 10~7 C.G.S. unit.
;: l •_'
DR. S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
TABLE IV. — Bombay (CHAMBERS). Declination West.
Lunar phase.
CL
*i.
ft.
ft.
cs.
ff*
C4.
**
0 • •
Augusts-October.
0
1
73
125
Ill
112
63
108
18
108
2
19
160
68
150
32
110
19
31
3
68
93
92
150
39
152
23
211
4
41
104
62
131
38
116
12
241
&
33
96
89
124
78
99
20
49
6
53
126
68
127
22
128
6
(82)
7
44
48
83
104
68
89
30
45
8
23
71
93
138
65
117
19"
135
Mean . . .
44
103
83
129
51
115
18
117
April-September.
1
71
97
106
100
55
107
13 -
- 30
2
45
105
96
117
58
109
18
29
3
55
90
76
119
33
108
14
277
4
52
120
64
91
57
86
11
9
5
33
75
71
108
75
114
17
95
6
60
135
63
124
38
108
20
197
7
17
64
68
90
58
107
8
20
8
31
82
70
120
45
111
14
248
Mean . . .
45
96
66
109
52
106
14
106
October-March.
1
15
374
83
240
15 257
3
2
22
207
62
239
13 433
15
400
3
23
314
82
250
36 239
26
151
4
45
169
62
229
11 148
10
360
6
47
196
36
256
3
,
7
270
6
31
225
54
247
8
297
8
180
7
31
247
66
239
8
209
17
156
8
27
240
62
228
19
176
8
107
Mean . . .
30
246
63
241
14
251
12
232
The unit of force in the tables of amplitude is 10~7 C.G.S. unit.
EARTH'S MAGNETISM rkODUCED ItY THE MOON AND SUN.
:U3
T\I:LE IV. (continued).
Lunar phase.
o,.
r,.
C2.
**
c,.
"
C4.
v 4*
November-January.
1
21
256
104
L't5
29
235
14
168
2
19
304
79
239
13
212
21
424
3
33
327
111
217
62
229
34
209
4
51
184
114
235
36
202
5
(136)
5
79
192
93
•J55
48
252
32
226
6
66
227
110
229
45
248
12
98
7
>;,
236
119
240
56
209
33
190
8
60
249
86
237
49
220
7
(266)
Mean . . .
52
247
102
241
42
228
20
219
February-April.
1
72
392
32
238
19
45
4
_
2
32
283
13
222
42
76
11
32
3
29
265
54
263
18
- 30
13
53
4
43
183
23
318
42
56
18
- 60
5
34
288
32
355
33
17
7
—
6
13
170
22
293
51
48
14
0
7
37
304
13
296
22
55
28
127
8
26
378
30
218
9
40
18
- 26
Mean . . .
36
270
27
275
30
38
17
21
May-July.
1
71
96
93
98
25
104
34
296
2
77
103
109
106
63
101
17
19
3
58
79
92
109
36
93
17 260
4
93
121
88
93
67
64
24
7
5
43
>:>
78
102
72
64
28
102
6
57
149
79
113
46
100
29
- 29
7
14
138
66
91
48
96
25
- 55
8
28
18
64
92
51
104
29
240
Mean .
55
99
86
101
51
93
25
105
The unit of force in the tables of amplitude is 10~7 C.O.S. unit.
VOL. CCX1II. — A. 2 8
314
DR. S. CHAI'M AX ON THE DIURNAL VARIATIONS OF 11 IK
TABLE V. — Bombay (Moos). Decimation West.
Lunar phase.
C,.
r,,
0»
fc
C3.
r»
C4.
f*
0
Novemb
0 0
er-January.
e
1
55
210
143
236
61
233
18
268
2
63
285
58
234
33
178
29
93
3
52
252
145
240
81
217
38
238
4
5
115
96
227
48
205
37
166
5
21
167
129
234
69
233
11
252
6
46
156
125
222
75
215
25
210
7
28
244
80
216
38
230
4
223
8
43
195
96
225
61
204
8
95
Mean . . .
39
203
109
229
58
214
21
193
TABLE VI. — Batavia. Declination West.
Lunar phase.
0*
*l.
C2.
o*
C3.
9*
C4.
r*
g
0
April-September.
o
o
1
38
38
40
104
14
173
12
263
2
59
147
49
167
35
312
20
240
3
20
99
41
115
17
251
10
239
4
43
295
36
13
13
321
14
232
5
1
—
12
6
30
269
28
264
6
16
251
33
119
17
206
20
235
7
18
203
29
255
34
257
21
256
8
11
189
10
270
14
225
28
227
Mean . . .
26
175
31
131
22
264
19
244
October-March .
1
85
280
238
278
153
291
16
299
2
44
177
172
266
126
290
61
300
3
34
346
175
268
131
283
62
237
4
83
171
130
257
104
289
51
395
5
84
276
237
288
148
296
68
320
6
68
178
154
264
144
283
68
301
12
347
181
264
131
282
32
304
8
53
173
148
260
97
284
23
323
Mean . . .
58
243
179
268
129
287
48
310
The unit of force in the tables of amplitude is 10~7 C.G.S. unit.
KAl.TM's MACNKTISM l'l;i MUVF.I) I!Y TIIK MOON AND SUN.
315
TABLE VII. — Bombay (On \MKI:IW). Horizontal Force.
Lunar phase.
c,.
ffi.
Ca.
#2- Cg.
r»
C4.
**
November-January.
1
124 180 178
188
50
183
28 192
2
68
173 107
179
52
220
23 125
3
103
123 102
152
53
183
14 340
4
155
166 133
178
14
168
3 60
6
78
180 123
172
63
208
29 178
6
161
196 151
173
53
235
29
806
7
136
212 135
195
54
177
47
207
8
191
200
149
177
63
198
38
229
Mean . . .
127
179
136
177
50
198
26
204
February- April.
1
68
185
98
164
18
190
15
34
2
96
191
90
165
47
202
52
190
3
78
184
110
156
42
172 8
(135)
4
97
148 93
182
27
285 64
222
5
77
145 48
179
11
(397)
24
178
6
143
181 45
155
21
199
12
124
7
24
(25) 30
(250)
30
344
29
136
8
189
181
70
176
54
225
52
95
Mean . . .
97
174 73
168
31
231
32
140
May-July.
1
63
225
50
275
20
158 18
46
2
71
157
88
163
9
103
9
60
3
62
214
83
174
47 135
12
-12
4
33
151
50
101
20 82
18
- 8
5
35
126
16
184
17 164
16
82
6
50
283
60
243
9 25
11
68
7
46
237
16
297
13
66
38
66
8
152
152
60
172
39
269
38
-24
Mean . . .
64
193
53
202
22
125
20
34
The unit of force in the tables of amplitude is 10~7 C.G.S. unit.
2 8 2
316
]>i; s. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
TABLE VII. (continued).
Lunar phase.
c,.
ffi.
c,.
fe
C3.
*»
C4.
0Y
August-October.
• 0
1
67 195 24
211 12
309 1
2
99
254
76
178 24
127
38
100
3
102 250
41
198 41
283
8
45
4
72 145
92
186 38
207
26
184
5
43 139 48
184 10
246
23
206
6
93 180
36
147 18
326
8
325
7
63
101
27
146 35
33
8
35
8
93
178
72
148
32
2
13
127
Mean . . .
79
180
52
175.
26
192
16
146
April-September.
1
82
211
43
239 9
237
20
54
2
74
195
91
163 8
123
18
141
3
77
234
76
181 8
162
20
- 15
4
72
160
53
158 24
219
9
80
5
35
162
33
197 15
151
17
110
6
33
212
35
197 12
127
7
26
7 23
224
18
278 3
22
68
136
118
61
171
16
264
11
- 31
Mean ... 66
190 51
198
12
183
15
54
October-March.
1
78
173
115
177
34
176
15
171
2
58
201
92
181
51
206 27
139
3 74
140
93
149 50
187 10
138
4
101
153
115
179
8
180 29
208
5
71
147
93
171
37
231 32
205
6
149
185
93
172
39
253 14
287
7
52
186
85
187 30
201
30
183
8
170
193
113
169
52
201
13
166
Mean . ...
94
172
100
173
38
204
21
187
The unit of force in the tables of amplitude is 10"" C.G.S. unit.
KAKTHS \I.\CNKTISM l'i;< H '((!.!' IIV Till'. MOON AM' SUN.
317
TABLE VIII. — Bombay (Moos). Horizontal Force.
Lunar phase.
c,.
9+
c,.
A
C,.
r»
c.
r.
November-January.
1
129 126
145
170
71
169 43 177
2
162
232
151
177 95
183 11 207
3
164
143
129
179 79
210 36 241
4
86 201
140
190 85
183 44 207
5
101 171 166
180 80
198 14
212
6
161 175 133
151 66
192 69'
241
7
112 156 89
162 59
167 17
189
8
113
178
107
172
15
119
38
60
Mean . r— r1
128
173
132
171
69
178
34
192
t
May-July.
1
85
185
48
185
58
306
53
301
2
122
229
22 263 46 6 28
358
3
76
267
67 194 8 261 13
304
4
68
195
28
180 22
51 22
292
5
87
227
72
117 14
210 15
291
6
75
112
40
228
40 192 13
254
7
86
229
18
172
10 186
4
234
8
139
216
88
179
20
165
24
409
Mean . ...
92
207
48
190
27
172
" 22
305
The unit of force in the tables of amplitude is 10~T C.G.S. unit.
318
DR. S. CHAPMAN ON THE DIURNAL VARIATIONS OF THE
TABLE IX. — Batavia. Horizontal Force.
Lunar phase.
Ci.
ffi.
C,.
0+
Cs.
*i.
C«. fft.
•
1
April-September.
1
118
215
99
250
18
251
23
- 34
2
64
126
62
222
36
143
3
3
11
72
46 184
18
242
14
93
4
70
169
68 200
24
155
9
22
5
122
221
70 279
29
288 15
-136
6
11
144 78 197
50
128
1
^_
7
43
105 19 173
22
408
14
30
8
55
107 64
213
32
121
32
130
Mi -.in . . * .
62
145
63
215
29
217
14
21
October-March .
1
68
203
86
249
52
230
26
163
2
41
36
52
198
36
166
19
277
3
70
190
115
231
75
205
13
191
4
8
—
80
190
34
149
29
209
5
45
207
74
252
35
204
34
280
6
50
92
68
197
47
315
51
275
7
92
100
98
194
62
200
13
270
8
65
148
95
200
23
125
33
220
Mean . . .
55
122
83
214
45
199
27
236
TABLE X. — Bombay (Moos). Vertical Force (upwards).
Lunar phase.
c,.
ffi.
C2.
0*
cs.
9»
C4.
9+
0
Novemb
•
«r-JanuE
try.
o
I
1
23
230
66
272 56 259
27
259
2
10
438
32
289 35
221
19
159
3
22
149 33
246 32
240
7
310
4
12
169
52
259
48
233
20
205
5
32
220
58
262
50
259
18
284
31
261
67
276
38
235
10
229
16
449
25
279
35
234
20
250
12
353
32
273
37
206
16
195
Mean . . .
20
284
46
270
41
236
17
235
The unit of force in the tables of amplitude is 10~7 C.G.S. unit.
EARTH'S MAGNETISM PRODUCED BY THE MOON AND SUN.
TAI-.LK X. (continued). — Batavia. Vertical Force.
Lunar phase.
ft.
fl.
c,.
fc
c*
9+
C*
r*
• • • *
April-September.
1
66
64
157
172
17
200
13
227
2
17
21
95
199
29
284
4
248
3
42
33
86
192
11
176
15
267
4
70
74
50
178
20
345
7
247
5
80
42
94
173
11
421
17
340
6
32
-20
62
193
54
311
22
293
7
28
7
82
194
21
286
8
264
8
59
-12
33
181
41
321
9
311
Mean . . .
49
26
82
185
26
293
12
275
October-March.
1
47
349
36
385
113
12
38
15
2
43
293
44
335
101
0
26
17
3
29
367
17
351
99
6
34
3
4
27
304
26
393
74
- 3
28
9
5
33
427
37
368
88
- 4
33
27
6
94
321
33
293
75
9
59
9
7
47
286
23
327
91
4
20
45
8
14
333
8
306
64
0
42
-24
Mean . . .
42
335
28
345
88
3
35
13
The unit of force in the tables of amplitude is 10 7 C.G.S. unit.
820
Pit. S. CFIAI'MAN OX THE DIURNAL VARIATIONS OF THE
TABLE XI.— Declination West.
Nov. -Jan.
Feb.-April.
May-July.
Aug.-Oct.
April-Sept.
Oct.-March.
C.
ff.
C.
9.
C.
ff.
C.
ff.
C.
9.
C.
ff.
1
•J
3
4
70
172
91
20
2612
273'
2888
869
64
103
54
14
3263
3131
350'
312
Tre
54
54
48
22
vandru
822
871
971
815
m.
52
45
24
20
78a
1581
1772
2098
48
42
36
13
422
891
98l
3412
49
138
70
14
318s
2801
3051
2762
Bombay (CHAMBERS).
1
2
3
4
53
102
43
22
247»
2411
2282
215<
37
27
31
19
2704
275s
382
21s
56
86
52
28
992
101'
931
1055
45
83
52
20
1032
1291
1151
1178
46
66
53
16
962
1091
1061
1065
31
63
14
13
246s
2411
2515
2325
Bombay (Moos).
1
2
3
4
40
109
59
23
203"
2291
2141
193s
Batavia.
1
2
3
4
27
31
23
21
175*
131*
2648
2441
59
179
132
53
2433
2681
2871
3102
The unit of force in the tables of amplitude is 10~7 C.G.S. unit.
K \l; HI'S MACNKTISM 1'ltODUCED BY THE MOON AND SUN.
TABLE XII. — Horizontal Force.
321
Nov. -Jan.
I'Vlt.-April.
May-July.
Aug. -Oct.
*
April-Sept
Oct. -March.
C.
ff.
C.
"
C.
".
C.
ff.
C.
ff.
C.
ff.
E
•ornba^
f (CHA
MUER8
)•
1
130
179'
99
174*
66
193s
81
I-"
68
190"
96
172*
2
135
177'
73
168'
53
202s
52
175*
51
198'
100
173*
3
51
198"
32
231<
23
1254
27
1924
12
1833
39
204s
4
29
2044
36
140s
22
34*
18
146<
17
54s
23
1873
Bombay (Moos).
1
131
173s
94
207s
2
132
171'
48
190s
3
71
178*
28
172s
4
38
1923
24
305»
Batavia.
1
64
145«
56
122«
2
63
215*
83
214s
3
30
217s
46
199«
4
16
21*
SO
236»
TABLE XIII.— Vertical Force.
Nov.-Jan.
Feb.-April.
May-July.
Aug.-Oct.
April-Sept.
Oct.-March.
C.
e.
C.
e.
C.
e.
C.
e.
C.
e.
C.
e.
Bom
bay(M
oos).
1
20
284s
2
46
2701
3
42
236'
4
19
235»
Batavia.
1
50
26*
43
335*
•2
82
185'
28
345*
3
27
293s
90
V
4
13
276*
39
13s
The unit of force in the tables of amplitude is 10"7 C.G.S. unit.
VOL ••••Mil. — A.
2 T
[ 323 J
VIII. A Critical Study of Spectral Series. — Part HI. The Atomic Weight Term
and its Import in the Constitution of Spectra.
By W. M. HICKS, F.R.S.
Received June 7,— Read June 26, 1913.
CONTENTS.
Page
Abbreviations 323
The " oun " 324
Collaterals 336
The diffuse series —
General 338
Table of denominator and satellite differences 341
Satellite separations 352
Order differences 353
The D (2) term 372
The S and P series 377
The F series 379
Closer approximation to the value of the oun 400
Ag and Au 403
Summary 407
Appendix I.— Sc 408
„ II. — Wave-lengths and notes 411
ABBREVIATIONS.
[I.] and [II.] denote the two previous parts of this discussion published respectively in the 'Phil.
Trans.,' A, voL 210 (1910), and vol. 212 (1912).
The formula for a line is » = N/D^ - N/Dm*.
N/Di2 is the limit or value when m «= o>.
£ denotes the correction to be added to any limit adopted to give the true value.
N/D,«Z is referred to as the V part (variable).
Dm is referred to as the denominator of the line.
"Separation " of two lines is the difference of their wave numbers.
" Difference " of two lines is the difference of their denominators.
"Mantissa" is the decimal part of the denominator.
v denotes the separation of two lines of a doublet.
A is used for the denominator difference of the two lines which produces v.
v\* "fc Aj, A2 are similar quantities for triplets.
VOL. C'CX I II. A 504. 2 T 2 Publithed •ep.raUly, October 22, 1913.
;<•_•( DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
\V .lenotes the atomic weight, w = W/100.
8, denotes the "oun" = 90-47^.
S, » n&i, but S is used for &4 = 361 • 89w*.
0 - C is used for the difference in wave-length between an observed line and its calculated value.
O denotes the maximum possible error of observation.
In gtnrrnl, figures in brackets before lines denote intensities, and in brackets after, possible errors of
observation.
THE doublet and triplet separations in the spectra of elements are, as has long been
known, roughly proportional to the squares of their atomic weights, at least when
elements of the same group of the periodic table are compared. In the formulae
which give the series lines these separations arise by certain terms being deducted
from the denominator of the typical sequences. For instance, in the alkalies if the
ja-sequence be written N/Dra2, where Dm = m + n + a/m, the ^-sequence for the second
principal series has denominator D — A, and we get converging doublets; whereas
the constant separations for the S and D series are formed by taking
S, ( oo ) = D, ( oo ) = N/D,2 and S2 ( oo ) = D2 ( oo ) = N/(D, - A)2. It is clear that the
values of A for the various elements will also be roughly proportional to the squares
of the atomic weights. For this reason it is convenient to refer to them as the
atomic weight terms. We shall denote them by A in the case of doublets and A,
and A2 in the case of triplets, using v as before to denote the separations. Two
questions naturally arise. On the one hand what is the real relation between them
and the atomic weights, and on the other what relation have they to the constitution
of the spectra themselves ? The present communication is an attempt to throw some
light on both these problems.
The Dependence of the Atomic Weight Term on the Atomic Weight.
The values of the A can be obtained with very considerable accuracy, especially in
the case of elements of large separations, i.e., of large atomic weight. If, therefore,
the definite relation between these quantities can be obtained, not only may it be
expected to give some insight into the constitution of the vibrating systems which
give the lines, but it may afford another avenue whereby the actual atomic weights
of elements may be obtained, and the solution of the problem is therefore of importance
to the chemist as well as to the physicist.
t may be interesting to note the steps which first led the author to the solution
vhich follows, and incidentally may add some weight to the formal evidence in its
has long been known that in the case of triplets the ratio of A, : A2 is
slightly larger than 2. It was natural, therefore, in an attempt to discover
ion to the atomic weight to consider the values of A,-2A2. These were
for several cases, A, and A3 being expressed in terms of the squares of the
weights. It was at once noticed that in several cases these differences were
s of the same number, in the neighbourhood of 360, e.g., Ca 1, Sr 3, Ba 8,
DR. W. M. HICKS: A CRITICAL STUDY" OF SPECTRAL SERIES. 325
Hg 19, and, further, that in many cases A, and Aa were also themselves multiples of
the same number. As, however, Mg with a difference of 450 and Zn of 543 could not
possibly be brought into line with the others, this line of attack was given up. But
later the case of Zn, which at first had seemed to stand in the way of an explanation
on these lines, gave cause for encouragement. The series for Zn are well defiued, the
measures good, and the formulae reproduce the lines with great accuracy.* Great
confidence can thus be put in the values for A, and A2, and it was noticed that they
were both extremely exact multiples of the difference Aj — 2Aa. In fact, the values
are A, = 31 x 543'446?03 and A., = 15 x 543'476t0*. This relation could hardly be due
to mere chance, especially when it was also noticed that 543'44 is very close to 3/2 the
former 360, and, further, the 450 of Mg is about 5/4 tin- same. In other words, with
the rough values used 360 = 4 x 90, 450 = 5 x 90, and 540 = 6 x 90. This looked so
promising that a systematic discussion of all the data at disposal with limits of
possible variation was undertaken. The theory to be tested then is that the A of
any element which give its doublet or triplet separations are multiples of a quantity
proportional to the square of the atomic weight. We will denote this by $ = qu?.
It will be convenient, in general, to deal with the 360 quantity, and S will be used to
denote this. If other multiples are dealt with as units a subscript unit will be used
giving the multiple of the 90. Thus <?, denotes the smallest, St = 542710*, and so on.
The results are given in Table I. below.
The value of A is obtainable as the difference of two decimals with six significant
figures. It is convenient therefore to tabulate the values of 10"A. The exactness of
the calculated value depends on (l) the correctness of the adopted value of S(o°),
(2) the exactness with which v is measured, and (3), when expressed in terms of 70*,
the exactness of the value of W or the atomic weight. In the case of the latter the
value W/100 = w is used and the values of ItfA/iv3 are tabulated. The method
adopted may best be seen by taking an actual example, say that of calcium. The
values of •»,, va as found by least squares from the S-series are 105'89, 52'09. The
value of S(oo) as given in Table I. of [II.] is 33983'45, and the correct value is
supposed to be £ larger. The numbers 33983'45, 34089'34, 34141'43 are then thrown
into the form N/Da, and the denominators are 1796470, 1793679, 1792310, giving
for differences A, = '002791, Aa = '001369, which are tabulated as 2791, 1369. The
corrections for the error £ are found to be — '2f and — '!£ Moreover, the last digits
of 10*D may be '5 wrong and the value of the A be ±1 out. In cases where the v
are known to three decimal places, the calculations are carried out with 9-figure
logarithms, and the values of A determined without this ambiguity. The values of v
may be wrong by dv, i.e., \05'89 + dv, &c. This will produce; ;i variation in A, of
26'Sdv — in general dv is a fraction <'l. Thus
A, = 2791±l-'2£+26'3cZ»<.
* See Table I. of Part II.
326
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
The atomic weights are supposed to be those given by BRAUNER,* +x, where x is
a number to be added to the fourth significant figure in BRATTNER'S value. BRAUNER
gives for Ca 40'124. A, is then divided by (-40124)".
The result is
A, = (17336-1 ±6-22-l'24£+ 163^,-8'64a;)ty2.
= 48(361>169±-13--025£+3-4cZ»'1-'180a;)wa.
Table I. gives the values for those elements in which the series have been
established. The second column contains the atomic weight as given by BRAUNER,
except for the few belonging to volumes of ABEGG'S ' Handbuch ' not yet published,
with estimated possible error beneath. In the third column the top number gives
v and the second 10'A. For triplets there are therefore two sets. The fourth column
gives lO^A/MT1, and the multiples of which it is composed. In general the 360 ratio
is taken, but in several cases it is necessary to take 2x90 or 180 and 1 x 90. The
second line of columns 5, 6, 7, 8 gives the coefficient of the error corrections to be
applied to this number 360, or 180, &c., as the case may be, and the upper line of
figures gives the maximum errors estimated, which have, in general, been less than
those permissible by the observations. The last column gives the difference between
361 '8 and the factor given in the .fourth column, except that when it is not the
4 x 90 term it is brought up to it by multiplying by 2 if it is 180 and 4 if 90. The
maximum errors are also attached.
In many cases it will be seen that the number of significant figures in the
calculated numbers is larger than in the data from which they are derived. In these
cases the number of significant figures in the data must be supposed to be made up
to the proper number by the addition of zeros. This enables new calculated values
to be determined when the data are improved without the trouble of recalculation.
TABLE I. — Evaluation of S and of m.
Notation. — W = atomic weight ; qw* = § with w = (atomic weight)/ 100 ; £ error
in n,, ; dv, error in v ; x, error in w on the fourth significant figure.
W.
v, 10«A.
gw2, mS.
±1.
-£
dv.
-x.
361-8 + .
Na
22-99*
17-175
14027-96
0
i
0
•2
"2
2
743-0
155x90-50
0
•021
5-21
•078
•141
K
39-097
57-87
19224-86
1
•5
•3
•92
3
2939
53x362-72
•012
•037
6-26
•142
3-22
* ABEGG, ' Handbuch der Anorganischen Chemie.'
I>i; W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SKHII.S.
327
TABLE I. (continued).
W.
v, 10«A.
yw*, roS.
±1.
-f
dr.
-X.
361-8 + .
Rb
85-445
237-54
17715-86
1
•3
•5
-•40
5
12935
49x361-40
•003
•022
1-52
•084
•56
Ce
132-623
553-80
18449-48
1
•4
•07
-•06
7
32551
51x361-74
0
•025
•65
•96
•33
Cu
63-55
248-49
18075-8
5
•4
1
•04
1
7311
50x361-84
•05
•02
1-44
•115
•80
Ag
107-88
920-61
23879-34
10
•02
•
0
•01
27791
66x361-81 '006
•017
•39
•33 '2
Au
197-20
3815-52
.
7
113961+ Sly
8 1*36 1-80
Mg
24-362
40-90
14389-05
16-91
3
•02
5
•19
2
854
159x90-497
•423
•010 8-79
•075
1-00
19-89
6992-33
•04
1-78
415
77x90-89
•219
•003
4-57 -075
3-12
A!+A, =
59x362-36
Ca
40-124 105-89
17336-1
6
•1
5
- -63
5
2791
48x361-169
•129
•025
3-4
•180
1-52
52-09
8503-4
•1
- -24
1369
47x180-923
•129
•013
3-48
•090
1-00
Aj + A, =
143x180-696
Sr
87-65
394-35
15401-6
4
•2
3
•59
3
11835
85x181-195
•015
•008
•46 -041
•72
186-93
7200-4
•1
-1-78
5533
20x360-02
•065
•019 1-95 -083
•585
A!+A, =
125x180-82
Ba
137-45
878-21
15528-2
lOt
•2
•6
- -68
6
29328
43x361-121
•012
•019
•41
•526
•600
370-33
6340-96
1 '2
+ -54
11976
35x181-170
•015
•017
•49
•263
•882
Ai + A, -
121x180-74
Ra
226-4 2050-26
18077-16
lot t
t
- -257
92658
50x361-543
•004
•023 -023
•32
2t
832-001
6709-3
•86
34390
37x181-33
A,+A, -
137x180-92
DR W. M. HICKS: A OWTIOAl STUDY OF Sl'KCTKAL SKIMKS.
TABLE I. (continued).
W. v, 10"A.
jw2, rofi.
±1.
-£
dv.
-x.
361-8 + .
Zn
65-40
3
388-905
7204-42
190-093
3486-20
16843-68
31x543-334
8150-74
15x543-383
0
0
1
•015
•015
0
0
3
•135
•135
•423
•420
•456
•420
Ai + A2 =
46x543-356
Cd
\\-l-3
1
1170-848
23105-56
541-892
10368-54
18321-33
lOlix 180-504
8221-64
91x90-348
0
0
•3
•007
•003
0
•15
•16
1
•321
•160
- -8
•646
- -408
-•644
AJ+AS =
49x541-50
Eu
151-93
3
2630-5
51223
22191-06
123 x 180-41
101
•007
3
•068
•3
•236
- -98
•68
1004
7940-57
1
- -94
18329
22x360-93
•013
•136
•472
•39
A, + A2 =
333x90-485
Hg
200-3
3
4630-648
87814-99
21888-03
121 x 180-892
0
2
•013
0
•078
3
•354
- -015
2-12
1767-01
7478-05
•002
•3
-1-41
30002-3
83x90-096
•0026
•05
•091
1-15
Aj + As =
54x543-816
Al
11-10
112-15
23884
3
•02
5
•079
5
1754
66x361-879
•21
•012
3-21
•266
1-635
Ga
69-9
826-10
27715
1
•10
30
-1-87
3 13498
77x359-93
•026
•Oil
•435
•103
3-17
In
IU'8
2212-38
28593-88
1
•25
5
•147
5
37684
79.x 361-947 -01
•117
•165
•630
3-32
Tl
204-04
7792-39 32223-62
0
•03 -5
•263
5
134154
89x362-063
•002
•012
•047
•355
•18
Sc
44-7
320-80 36714
1
50
- -086
5
7140
101^x361-714
•05
•014
1-136
•165
9-45
or 6404
91x361-89
He
3-99
1-007
20860
0
5?
•35
33-377
58x361-45
0
0
•355
•180
1
I)H. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SKRIKS.
TABLE I. (continued).
vv.
v, 10«A.
„,*
±1.
-t
dv.
-x.
361-8 + .
0
16
:! • 65") r
6692
•03
0
- -01
171-3 ln,,,l 18$ x 361 -79
2-11
1-13
100
:.
2-O.S p 3714-9
•03
0
1-0
95 J I
41 x 90-20
•95
•34
25
2
'62\0"
34 ;(
145,
s
32-07
17-96
10150-85
10
•1
•34
1044
28x362-14
•35
•019
20-14
•226
3
11-21
6329-7
•46
651
35x180-67
•21
•009
10-05
•113
1-5
A1 + A,-
91 x!81-l
Se
79-2
103-70
10192
•2
201 2-20
6392
28x364-00
•057
•034
3-r.
•092 31
44-69
4363-4
1-81
2737 12x363-61
•057
•04
4-08
•092 31
A1 + A,=
161x90-40
Data on which the Table is based.
Na. The limit is 24476-11. It is the limit found in [I.] corrected by the result of ZiCKENDRAirr's
measurements of the high orders of NaS. The value of v adopted is that deduced from FAIIKY and
I'KHOT'S interferometer measurements of the D-lines using 9-figure logarithms. Consequently, the results
are more reliable than would otherwise bo expected from such a low atomic weight. But on this point,
see below (p. 331).
The limit for K is 21964-44 — corrected from the value in [I.] by addition of 1-06 as indicated by
ZICKKNDRAHT'S observations. The value of v is very ill-determined. A value of v = 57-73 would make
q = 361 • 80. SAUNDERS' results for S (3) give v = 57 • 75, and K.R.'s for S (4) give 57 • 60 ± • 30. The
value in the table is that used in [I.] 57'87±1. The limits for Rb and Cs are those given in [I.] for
S ( oo ), viz., 20869-73 and 19671-48.
Cu. S ( oo ) = 31515-48.
Ag. D ( oo ) = 30644 • 60, found from first three lines. FABRY and PKROT have measured by the
interferometer Dn (2) and DJI (2) and K.R.'s are extremely close to these. They have been taken as
correct to -001 A.U. The lines DU and Dn are so close that their difference of wave number as given
by KAYSER and RUNOK are prok-ibly of the same order of exactness. We may regard, therefore,
K. and R.'s DU - DU and F. and R.'s D^ - DH as quite exact up to the second decimal place. This gives
v — 920 -61. It cannot be uncertain to more than a few units in the first decimal. A more correct value
is obtained below (p. 404).
The limits of the 2nd and 3rd groups of elements are those given in [II.]. In the cases of Zn, Cd, Hg,
the interferometer measurements of FABRY and PEROT are used, except v-t for Cd and Hg, with 9-figurc
logarithms in order to get an extra significant figure, their readings being reduced to ROWI.AMI'S si-.-ili-
by HAKTMANN'S factor 1-000034. In these cases the values of v may be taken as practically correct
to -001 A.U.
For Sc, see Appendix I.
VOL, COXIII. — A. 2 U
:; .-.,. DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
He. Limit given in [I.J which is practically exact, v = 1 -007 is given by PASOHEN as a result of all
hia readings and is probably not more than '002 in error. Consequently, the numbers for He have weight
in spite of its low atomic weight.
0. The limit is 23204-00. Although v}, v2 are known with fair accuracy, the possible proportional
errors are considerable, so that the data have small weight. The limit for the doublet series is 21204 with
v = • 62. The values are still more indefinite.
S. Limit 20106. The D series give 20110. This gives a considerable range of uncertainty.
Se. Limit 19275-10. The atomic weights for 0, S, and Se are those of the International Committee
of 1910.
The table shows at once that the two groups which give doublet series agree in
giving the A as multiples of a number close to 861'Sw1. Group II., giving the triplet
series, require in several cases multiples of 9CH02 or 180w2. It is curious that the
groups which first indicated this relation do not show it so markedly and with so
little doubt as the doublet series, in which by themselves it would probably never
have been noticed. There seems to be some kind of displacement with the middle
lines of the triplets. If, for consideration, the values of A^Aa be taken, this
irregularity disappears, and, moreover, with the larger observed quantities, the
proportional errors will be less.
If we agree to look upon the 361 as the normal type, and for numerical comparison
multiply the 90 by 4 and the 180 by 2, and, if further, the results are supposed to be
weighted by the estimated limits of variation assigned in the last column of the
table, the method of least squares gives for the value of q = S/w2 —
Group 1 361-900,
,, II 361720,
» HI 362-051,
All three groups 361'890.
In Groups II. and ILT. it is possible too much weight has been given to Hg, va, and
Tl. We will take as the preliminary value for q that of silver, viz., 361 '81, which is
practically that of the general weighted mean. The true value cannot vary much
from this— probably less than '2. With this, the subsidiary values become 180'90± '1
and 90'45 ±'05.
It is seen that in the doublet groups all the elements can come within this limit.
In fact, with the exception of K and Ga, they come extremely close. Ga is
spectroscopically uncertain as well as in its atomic weight, and the uncertainty of K
s due to the uncertainty in its value of „. In the triplet groups also, all calculated
from A, + AS have possible variations which will bring them within, although the
not so marked as for the doublet elements. The sequence formula 'are
itablished in Groups I. and III., but there are uncertainties in Group II. which
•equire clearing up. In this relation also, the table shows slight regular variations
as, e.g., A, and As err from the general mean in different directions, but in these cases
DR W. M.' HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 331
the values of A, + A3 come much closer to it. The values of (A, + A;,)/t^ are therefore
added to the table. It is clear, however, that when the spectroscopic observations
are good, the relation here established will enable very accurate measures of the
atomic weight to be obtained. In fact, with the possible accuracy attainable in
spectroscopic measurements, it may be hoped to obtain far more reliable values of these
constants than by weighing, except in those cases where they are small. The table,
for example, affords considerable support for BRAUNER'S estimates, except, possibly,
in the Mg group, where the irregularities are due to spectral causes. The case of Zn
may be taken as an example here. Its spectral values are very good, it shows with
w = 65'40 the multiple 543'357 instead of 542'70±'30. If the excess is due to the
value of the atomic weight, it should be '048 larger, which would be allowable within
BRAUNER'S estimates to bring it to the adopted value of q, i.e., w = 65*448. This is
more fully considered below. The numbers for Se also seem to show that 79'2 is too
small for its atomic weight. 79'40 would make q for A, = 362'16 and for Aa = 36177,
and the spectral uncertainties would account for the outstanding differences.
If $i is written for ^3, it may be noticed that the values of the A for the first of
each sub-group may be written —
i. ii. iii. vi.
Na. Cu. Mg.* Zn. Sc. Al. S.
1553, 50 x43, 32x53, 31x63, 52x73, 33x83, 8x143,
and, moreover, the same multiples of 3, recur in several of the same group, e.g.,
A, + A2 for Zn, Cd, and Hg, and A, for Eu are all multiples of 63,, also the 53, occurs
in Mg, Sr, Ba, and Ra. Analogy would lead to a corresponding 33, for Na. The
values of the atomic weight and the doublet separations of Na are known with great
accuracy, and no possible value given to £ could change the multiple from 155 to 156
or 153. The only loophole for an explanation may be that the value of v as found by
FABRY and PEROT comes from the Principal series, and that VP,(l) is not really
S(oo). This latter point has been discussed in [I.] and also in [II., p. 38]. It is
equivalent to a considerable change in S ( o° ). To obtain a value 156 or 52 x 3 would
require an increase of '07 in v, i.e., to 17 '25, Such a value would be quite well in
consonance with the measures of SAUNDERS and of K. and R. for other doublets, e.g.,
D(2) 17'30 (S.), S(3) 17'22±'26 (K.R), S(4) 17'05±'38 (K.R.), P(l) 17'20 (K.R.).
But FABRY and PEROT'S values for P(l) — independently verified by Lord RAYLEIGH —
would seem conclusive against this value, unless F. and P.'s apply only to VP (l)
and 17'25 to S ( » ). This would correspond to a lateral displacement of 3, (see below)
between VP (l) and S ( oo ).
S and Se both give 8 x 143, which falls in line with the other sub-groups. In fact,
1 This is the value first deduced when the international system of atomic weights was used. It is
'^i more than that in the table. The question is considered below.
2 T7 2
,„;.
M NICKS: A CRITICAL STUDY OF SPECTRAL SEKIKS.
,1,.,,,,,,. tin- *kf of the group, the sub-groups would be based on (2n + l),\ :.ml
(2n + 2) t, This would leave J, and 2J, for group 0. He, as is seen, may IK- cither.
The foregoing evidence is, I think, conclusive that the atomic weight terms are
multiples of a quantity very close to one quarter of 361'8w». Before attempting ....
existing knowledge to obtain a closer value to this quantity, it will be desirable to
consider certain other ways in which the atomic weight plays a part, and which will
provide further data for its more exact determination. As it will be convenient to
have a name for these quantities which seem to have a real existence, the word " oun "
(u>v)* is suggested.
The curious irregularities in the value of the oun noticeable in the elements of the
2nd group in connection with the separate A! and A2 values, whilst the values found
from A, + A2 are normal is worth examining in closer detail. The values of Vi + va
given in the table are deduced from the sums of vlt vy, each determined independently
by least squares from the best observations. If the values of »i + va are determined
directly the values are slightly different, which is natural as they are found from
selected pairs. The old values and the values thus found are collected here, and with
them the values of S/iv*.
New .
Old .
New .
Old .
New .
Old
Mg.
6079(362-36)
Ca.
158'Gl (361-45)
St.
581-21(361-60)
6079(362-36) 157'98 (361'39) 58T28 (361*64)
Ba. Ra. Zn.
1248'85(361-56) .2882'26(361I84) 578'998 (362'23)
1248-54(361-48) 2882'26 (361'84) 578'998 (362'23)
Cd. Eu. Hg.
1712-84(362-41) 2634'5(36r94) 6397'53 (362'46)
171274(362-39) 2634'5 (361'94) 6397'66 (362'57)
It will be shown later that spectroscopically Mg belongs rather to the Zn sub-group
than to the Ca. The same tendency is exhibited here. The more probable values of
fi + v-j, have brought the oun more closely to equality with 36T60W2 for the Ca
sub-group, and with 362"4 for the Mg and the Zn sub-group. The value of vl for Eu
may be 2633*5 instead of 2630'5, and if so, its value of the oun would come to
362*34. If the variations in the value of the oun had been more irregularly
distributed, it might have been natural to assign the variations (small as they are) to
errors in the value of the atomic weight. But this does not seem justified unless
there are chemical reasons whereby atomic weights in any particular group have a
liability to be all over-estimated or all under-estimated. In view of the latter
' The pronunciation of oun will be the same in the chief European languages.
I>K. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SEltlES. :; ::::
|K»s.sil)ility it may be well to determine tin- amount of such error required t<> bring,
Kjiy, the valur 'ICi'J't to ;{<>rK, ami :;i;rCi to 3G1'9, as it is prol table the true value of
the ratio lies I tetween 361'8 and 3G1''J. The former requires an increase in atomic
weight of raW» an(l the latter a decrease of ^faf of the accepted values. The
following would be the changes in atomic weight required : —
Ca. Sr. Ba. Mg. Zn. CM. Hg.
-'025 -'036 -'052 +'02; +'04 +'09 +'2
According to the estimates of accuracy given by BliAUNEK the changes for Mg and
Ca are quite impossible, for Zn just possible, and for the others possible. In the case
of Mg and Ca, however, small errors in vl-\-v3 are considerable proportional errors
and the deviations may be caused by these. It is necessary to have these estimates
before us. Notwithstanding them, the close agreement of the numbers in each set,
and the difference between the two sets must produce the conviction that the
differences are real, and are not due to errors either in the spectroscopic measure-
ments or the atomic weight determinations.
In the table the multiples given are those which give the oun most closely. An
inspection, however, shows that in each element there is some disturbing influence
affecting the A, and A3 in opposite directions. Moreover, the sum of the multiples
chosen are in certain cases not the multiple taken for A, + A.,, and this should clearly
be so. This happens in Cd, Eu, and Hg. There is apparent a general rule that c, is
too small and va is too large, the deviation increasing with the atomic weight. The
discrepancy is equivalent to a transference from the true A, to the true Aa.
Evidently the transfer in Cd, Eu, and Hg has been so large as to increase A2 by
more than <$,, so that the closest multiple now appears to be too large by unity. If
the multiples in Aa be diminished by unity, the sum is equal to that for A, + A3, and
the discrepancy between the ouns from A, to Aa increases in a regular order. A
similar change has occurred in Sr, only here while the multiple of A, has apparently
increased, that of A! has apparently decreased. If the ratio A;, : A, be taken as
79 : 171 in place of 80 : 170 the discrepancy again falls into order with the others.
With these changed ratios the values become
Sr 171x90'OG8 79x91*144
Cd 203x90-252 90x91*351
Eu ...'.. 246x90*205 87x9T27
Hg 242x90-446 82x91'17
This transference must take place in the Da(oo) = Sa(o°) term. The values are
given in the following table in which the first column gives the value of Aa : A,, the
second the value of the transference, the fourth the transference in vt to vtt the fifth
the new value of A2, and the third and sixth are as explained later : —
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERll.s.
A,: A,.
*
«.t
*
A2.
SN.
9
Sr
Ba
77 : 159
47:96
79:171
35:86
37 • 100
•96
1-9
44-4
29-0
77-76
•180
•130
•639
•1698
•1678
•05
•05
1-49
•90
1-84
414-0
1367-4
5488-9
11950-3
34312-2
0
0-16
5-20
3-34
8-13
Zn
Cd
Eu
Hg {
15:31
90 : 203
87 : 246
41:121
82 : 243
0
87-81
157-8
182-7
274-46
0
•7714
•7577
•5030
•7563
0
4-53
8-53
10-47
15-72
3486-20
10282-7
18171-2
29817-7
29725-6
0
11-86
21-76
25-64
38-50
What is the nature of the modification ? Perhaps the simplest explanation to test
is that a fraction of J, is transferred. The third column gives the fraction of ^ which
is equal to the transfer. It is noticed at once that the two groups fall into two
separate sets. With the exception of Sr, the fraction in the first is about '17. Mg
and Ca can both fit in with this, for the values are so small that they depend on
decimals in the value of A1} A3, and therefore beyond our significant figures. In Ca,
indeed, evidence is given later that A2 is somewhat higher and would bring the ratio
close to '18. But Sr is quite out of step with the others. Zn has no transfer, Cd
and Eu are equal, but Hg is 5030. If the ratio in Hg be taken to be 82 : 243
however, the fraction agrees with those of Cd and Eu. The Hg oun is then 361'43w*
in place of 362'54 and closer to the mean value, and as will be shown later there is
evidence for the new value of A3 (see p. 397). If this explanation is valid it must be
possible to bring Sr into the scheme with a transfer of 11 '7, but it is difficult to see
how this can be done. '639 is about four times too great, in other words, where the
others are modified by a fraction of Slt Sr is modified by the same fraction of S. The
above arrangement brings Mg into the Ca group and upsets the law whereby its first
A, should be a multiple of 5^. As this law seems to have a considerable weight of
evidence in its favour, and moreover, as will be seen shortly, Mg tends to go
spectroscopically with the Zn group, it may be well to see the result of keeping
A, = 405 and the ratio A3 : A, = 19 : 40. This will require a transfer of about 6*3
with a considerable uncertainty owing to the small values of A3 and A,, and
A3 = 4087. With this the fraction of Sl is T1727. To bring to the same fraction as
in Cd the transfer should be about 4, which the uncertainty in 6 '3 is not great
enough to permit. As the fraction 77 is of the order 1 — "215 it suggests that the
modification is produced by adding Sl to the atomic volume term in the sequence
of the P series, viz. (atomic weight term
l-
The question must be left
open at present. It has been noted that the arrangement which gives A, = 159^ for
Mg throws it out of the rule that the first members of the different groups are
successive multiples of ^. When the calculations were first made, the values of the
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 335
international atomic weights wen- used, ami {'or Mg it is 24'32 in place of BRAUNER'S
24-362. This clearly gave A, = 160J, and A. + A, = 237<J, with S = (362'09±'65) w*
the uncertainty '65 not including that of atomic weight and being chiefly due to
uncertainty in vi + va. The transference required now is 270, and the fraction of <J, is
'502, again clearly not that of the Ca group, but when account is taken of the
uncertainty in vi + va quite possibly agreeing with that of the Zn group. The
assumption that the international value of w is more correct than BRAUNER'S certainly
gets over the difficulties mentioned above. But we are not justified in choosing the
values from the particular systems which best suit our theories. The discrepancy
between the international and BRAUNER'S is very great — from 10 to 15 times
BRAUNER'S indication of his possible error.
Another suggestion as to a. possible explanation may be given. There have been
various indications in [I. and II.] that small variations in N may occur. If so it is
possible to produce the changes observed by a small change SN in the middle line of
the triplet. The necessary changes to do this are given in the sixth column. The
changes clearly depend on the squares of the atomic weight, for if they are expressed
in the form xw* they are
Sr . . . 5'20 = 6767t^ = 4x l'564i^ Cd . . . 11'86 =
Ba . . . 3'34 = 1777W2 Eu . . . 2176 = 9'426w*
Ha ... 8'13 = 1-58610* Hg. . . 38'50 = 9'596w*
in which it may be noticed that 9 '426 = 6x 1'571. Again multiples of a quantity
depending on the square of the atomic weight enter, and it is especially interesting
to note that the Zn group are affected with the multiple 6. If Ca and Zn show
similar displacements, Ca would require <5N = '25 in place of '16 and Zn 4'03. Zn
is clearly 0, i.e., is unaffected, but considering the small numbers involved in Ca and
conseqtiently large proportional errors, Ca might well show '25 instead of '16. The
question naturally arises, do these quantities depend in any way on the oun ? Now
any change in N may be supposed to arise either as a real change in N itself or an
apparent change due to the introduction of a factor in connection with the 1/D*. In
other words, the quantity VD is
(l+/)a
N or N
• -
Looked at from this point of view, 9'426tpa requires N (l + '000859^) or
N (1 + '000429 iv*y. Now 55j would give <000452«rl, but if the present explanation
is the true one, this is not a likely value since it will not include the alkaline earths.
A value 6<$i = <0005428wa would be expected. The Ba value 1777 would give
(T000088J02), or practically (!+<$,)". It rather looks as if this explanation is a part
of the truth. If more exact measures were at disposal it might be well to assume
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIKS.
;:,,.,,. ivsiilts ;is holding, rccaJflBklia l;»i the deaOfeiltttON ;ni<l dismiss (In-
ment now required. It may be noted, however, that in the Zn sub-group a factor
(1 +xitY in D, ( oo ) would reduce the calculated oun below 362'4w* and (l -y<\)3 raise;
it in the Ca sub-group above 361'GOw8 and at the same time increase the factors in
the numbers above towards (l+6J,)'and (l +<?,)'. The factors may of course enter
«-it her as (l+x$Y or (l-
Collaterals.
The first set in doublet or triplet S or P series is always the stronger. The others
may l>e considered as receiving a sort of lateral displacement, by the atomic weight
term, in the recognised way, and may be called collaterals. This kind of displacement
is, however, not confined to the series generally recognised, but is of very common
occurrence, and, indeed, depends not only on the A but also on other multiples of S.
In fact, the doublet and triplet series are only special cases of a law of very wide
application. Some evidence of its existence will be given below. It will be sufficient
now only to refer to certain points connected with the law, and to a convenient
notation to represent it. This kind of relation was first noted in the spectra of the
alkaline earths,* and as the lines are both numerous and at the same time strong and
well defined, and, therefore, with very small observation errors, any arguments based
on them must have special weight. Moreover, there are long series of step by step
displacements involving large multiples of A between initial and final lines, so that
we may feel some certainty that these large multiples are real and not mere
coincidences.
As a compact notation is desirable the following has been adopted. In general t
the wave number of a line is determined by a formula of the form N/D^-N/D,,,2, and
lateral displacements may be produced by the addition (or subtraction) of multiples
of S, say xS or xA, to D, or Dm. This is indicated by writing (xS) to the left of the
symbol of the original line when it is added to D,, and to the right when added to Dm.
Thus CaS1(2) is 6162'46. So far as numerical agreement goes G439'36 is a collateral
of this represented by (2A, + 10A2) CaS, (2) ( + A2). This means that whereas, see [II.],
Wave number of CaS, (2) = 7 -^- J*_
(1796470)2 (2-4S4994)2
Wave number of 6439'36 = -, - — _ __ ^ _ ,
(1796470 + 2A. + 10A,)2 (2'484994 + Aa)2
N N
(1-815732)2 (2'48G362)2<
» A note on this relationship was given at the Portsmouth meeting of the British Association, see
'Report, B. A.' (1911), p. 342.
t Though not always, as I hope to show in a future communication.
DR. W. M. HICKS: A CRITICAL STl'hY OF SPECTRAL SERIFS. 337
going further it is desirable here to consider tin- nature of the cumulative
effects produced by errors in the values of «5, or of the limits, in the course of a
succession of step 1 1\- sti-p displao-ni'Mits. There may I*- a small ermr in the starting
point, !-.</., S(o&) in the alwve example, or in the value adopted for i. We will
consider these separately, taking the case where the displacement is on the left, or
t he lirst term.
1. The limit correct, but . s/i</l,tti/ too large. — Then S calculated from this is also
slightly too large. It will, however, serve to identify a large series of steps in
succession, i.e., to reproduce the successive difVen-nr.-s of the wave numbers of the
lines. But the errors will all be cumulative, and if the last line of a set be calculated
direct from the first, its denominator is too large and its wave number too small. Tn
this case a more correct value of S can l)e obtained by using these extreme lines, and
this corrected value must satisfy all the other lines. In general a new correction will
only affect an extra significant figure in the value of S.
2. S correct, but limit /n-otig. — In this case a slight error in the limit will be of no
importance unless the S and its multiples are considerable; and, as a rule, the limits
are known with very considerable accuracy, except possibly in the alkaline earths and
a few others. Let us suppose the limit adopted (say S(»)) is too large, that is, its
denominator too small. If the second line is due to a positive displacement, its
denominator is larger than that of the first, and the wave number less. Suppose D,,
D2 the denominators for the two lines, D2 > I), if the displacement is positive, the
separation is N/D,a— N/D/. If the limit is chosen too large D, and Da are chosen too
small, although D,— D, is correct since S is supposed correct. If D, becomes D,— a;,
the error in the separation is 8N«/Dj'— SNor/D,*, which is positive since Da is
supposed > D,, i.e., the calculated separation is too large. If the displacement is a
negative one, Da < D,, the true separation is now 2N/D/— 2N/D,2 and the error
2NX/D!3— 2NX/D/, which is now negative since D2 < D,. The effect would be that
in any series of step by step displacements S would appear to require continual
decreases, and at the end the " corrected values " would not at all fit the initial
• -rises. If, then, it is found that when S is corrected as in Case 1 the corrections
tend to alter the former corrected one, and not to produce additional significant
I inures only, it may be surmised that the limit has been wrongly chosen. It is
dear, then, that where there are a number of successive collaterals with a large
multiple of S between the extreme ones, we have at disposal a means whereby
much more accurate values for S and the limits are obtainable. Cases are given
below, e.g., in BaD.
For low atomic weights 5, is always a small quantity and except for orders where
«i = 1 or 2, the alteration in wave number is small. For the present purpose which
is to obtain proof of the existence of the displacements here indicated, no evidence can
be admitted in which the change in wave numl>er produced by a displacement £<J,, is
comparable with the possible error of observation. The evidence, therefore, is of
VOL. CCXIII. — A. 2 X
SS8
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
greatest weight when derived from the spectra of elements of high atomic weight, or
from cases in which the displacements are due to multiples of A.
It is possible for a line to be simultaneously displaced to right and left, as for
instance CaS, (2) given above. Such lines exist, but since there is a very considerable
scope for adjustment of values by a proper choice of say x and y in (xS) X (y$), and
specially so in y when m > 2, such cases cannot be considered as established unless S is
very large, or the A enter only, or unless there is independent evidence by the existence
of intermediate steps.
When these collaterals were first found it was noticed that in general a positive
displacement seemed in the majority of cases to increase the intensity of the lines, and
a negative to decrease it. This is clear when the displacements considered are those
from the 1st to the 2nd set of a doublet series where the displacement is a negative
one and there is always a decrease in intensity. It is also evident in the satellites of
the D series. Apparently, as will be shown, the typical line of the series is the
satellite. The strong line is a positive collateral of this and always shows a great
increase of intensity. Although these facts are obvious the connection was not
recognised, until the relation showed itself first in a series of collaterals. It is, I think,
safe to say that a positive displacement produces a tendency to increase of intensity ;
there may be other causes acting so as sometimes to mask the effect, but in general,
where the rule appears to be broken, the suggested displacements should be regarded
with some doubt. In so far as I have used this rule in the following, the results are
biassed and of course the evidence for the rule to that extent weakened.
It would be possible to give here long lists of collaterals. As, however, the present
communication has reference chiefly to the discovery of general laws as a necessary
preliminary to the more thorough examination of special spectra, it will be sufficient
to refer for evidence to the cases which arise in the succeeding discussion. This seems,
however, a natural place to refer to certain cases discussed in Parts I. and II., where
unexpected deviations occurred between the calculated and observed position of a line
in the middle of a series in which for the other lines the agreement was especially
good. As special instances, the cases of TlSj (4) and CaSj (5) [II, p. 39] may be taken.
The suggestion that TISj (4) may be due to a transcription error is not valid, and was
occasioned by an oversight in confounding d\ with dn. If the normal line be denoted
by T1&! (4), the observed is the collateral TlSj (4) (154) giving 0-0= - '01 in place of
- 1'21. Similarly, the observed Ca line is CaS! (5) (-6A2) with O-C = - '03 in place
of '61. There are many examples of such sudden jumps which are certainly not due
to errors of observation. Several instances will be found below in the D series.
The Diffuse Series.
To the question what is the positive criterion of a Diffuse series no clear answer up
to the present has been given. We find in general three sets of series associated
together. Two of these have the same limits, the other a limit peculiar to itself.
I'i;. \V. M. HICKS: \ riMTK'AI. STI I»V OF M'KCTKAI, SKI: IKS. 339
Tin- Lilt' i is the Principal series, and the difference between the wave numbers of its
first line and of its limit gives the limit of the other two. Of the other two series,
< int- shows a Zeeman effect of the same nature as that in the Principal. This is called
the Sharp series — or (by KAYSER and RUNOE) the 2nd associated series. The third
series is called the Diffuse — or the 1st associated series. It has in fact a negative
kind of criterion. The preceding definitions apply to the three series in all elements,
including such elements as Li, He, and others which show singlet series. When
doublets and triplets appear, we have a simple physical criterion for the Principal
series in that it is that series in which the doublets or triplets converge with increasing
order. This criterion can be applied even when the 1st line has not been observed.
In certain elements the constant separations are shown between satellites. In these
< MSOS the series is certainly a D-series, at least in those recognised up to the present —
but further knowledge may show that in certain cases such satellites may appear in
other seri« s. • If, passing beyond the mere physical appearance of the series or their
visible arrangement in the spectrum, we attempt to represent their wave numbers by
formulae of the recognised types, we have further criteria for the Principal and Sharp,
viz., that the 1st line of the Principal may also, very nearly at least, be calculated
from the formula for the Sharp — or vice versd — and that the denominators in their
formulae differ, roughly indeed but sufficiently closely for use as a criterion, by
a number not far from '5. But when an attempt is made to deal in the same way
with a line of the diffuse series, no general type of formula has, at least as yet, been
found. In the alkali metals, as was seen in [I.] all the D-series take a positive value
for at — in other words, the fractional parts of the denominators decrease with
increasing order, and the general conclusion might be drawn that this was a common
feature of all diffuse series. But the opposite occurs in the triplet spectra of the
2nd group of elements, whilst a similar rule of a positive value of a recurs in the
3rd group. This suggests that the series giving doublets have a positive and triplets
<x negative, but this is contradicted by the triplet series of O, S and Se, which behave
in the same way as the doublets of Groups 1 and 3. The question naturally arises,
is there a typical D-sequence with a positive, and the diffuse series in the 2nd group
do not really belong to this type, or is there no actual D-sequence, i.e., no regular
type, of formula to which the D-series conform. The difficulty of finding formulae to
accurately represent any particular D-series would point to the latter supposition,
a supposition also which is strengthened when we study comparatively the series of
numerical values of the denominators found directly from observations as is done
below. In the case of the alkalies the formulas given in [I.] (as well as those in l/m3)
do not reproduce well the high orders and are probably only within the limits of error
because the lines are so diffuse that the observation errors are very large. In fact
one of the few excessive deviations found in [I.] was that of NaD (6), in which it is
* K.g., in ScS., see Appendix I.
2x2
..,,, |,K. w. M. HICKS: A CIMT1CAL STUDY OF SPECTRAL SKRIKS.
not prok-il>l.- that the error is one of observation. In Group 2 the Zn sub-group can
be repn«|iirr<l fairly well with a formula in <x/(2w-l) in which a is negative. M^
can H!HO be reproduced within error limits by a formula of the same kind, but it is
impossible to do so for Ca, and Sr and Ba require additional terms in l/ma. In
Group 3 Al is quite intractable, and if really depending on a formula, appears to
require complicated algebraic or circular functions. In and Tl also are not amenable
to formulae in «//» only or a/m + P/m". Nevertheless, the general build of the series
is so similar to that of the others that it would seem probable that the wave numbers
should also be of the form S(oo)— N/(wi + f?m)3. If so it is possible to calculate dm
from the observations and a comparative study may throw some light on the origin of
the different lines. The attempt to deal with these series from the formulae point of
view, however, brought out the fact that the satellites are related to the strong lines
in a similar way to that in which the Principal line doublets are, viz., by a constant
difference in the denominators and that their differences probably depend on
multiples of the " oun," as is the case in the Principal series. As the evidence
depends also on a comparison of the numerical values of dm, this point will also be
considered now.
The actual values of dm will depend on the accuracy of the value S ( oo ) (or D ( <x> ))
of the limit. In the calculations below the most probable value has been used (see
note under each element) and the true value has been taken to be that + £ In order
to be free from mental bias these have been in general taken to be the same as S ( oo),
which involves the theorem that D(o°) = S(oo). But of this little doubt can be felt.
The true values of dn can then be given in the form dm + kg where k is small. For
high orders of m, k is comparatively large and can only be used when £ is very small.
It is however generally the case that errors made in this way are only a fraction of
the observational errors.
As in the normal type where there are no satellites VDj = VD3 = VD3, and where
there are satellites VD1:J = VDal, VD13 = VD.,,, it is only necessary to tabulate the
values of dm for the case of VD, or VDU, VD12) VD,:, respectively. When this is done
certain regularities are clearly apparent, which can be made more exact by allowing
small observational errors and giving a small permissible value to £ It would cumber
the space at disposal to give both sets of values, especially as it is possible to easily
indicate the differences on the one set of tables. Table II. then gives the values of Dm
with the modified value of £ with the maximum errors attached in the usual way
in ( ), and the calculated value given as a correction to the selected value. Thus for
NaD (3), D (3) = 3-986626 (l33)-289£-104, 3'986626 is the selected value, 133
possible change in last three digits in this, -289 is change for £ = + 1 , and the observed
value is 104 less than the selected. The values of the errors of observed wave-length
over calculated (O-C), and of possible observed errors (O) are given in each case on
the right. The tables for the different elements are collected together and discussion
of each is given later when considering the ordinal relations of the denominators.
1)11. W. M. HICKS: A riMTlCAL STUDY OF SPECTKAL SKIMKS.
a 1 1
TABU II.
Na.
A = 743. $= 19'I7.
D.
0-C.
0.
S.
2 -988656 (36) -121^
2030 = 3A-200 '
K.I;.
3 -986626(1 33) - 289£ -104
4A
•02
•03
»»
4 • 983654 (452) - 565£ + 35
44
- -01
•2
M
5 -980682 (2248) - 977£ -81
17A
•02
•5
'/..
6'9G8051(l)-1545£ + 23
-28A
0
1
7 • 988855 -2329£- 528
8 -988857-3324$ -850
9 -98886-456-; -520
10-990-6-1^
ll-992-7-9£
12-999-10-1^
13-986 -12-6|
14-951 -15£
K.
A = 2939. S = 55-45.
D.
0-C.
0.
p.
2-853302(38)- 100$
57936 - 20A-844
s.
3 -795366 (224) -249$ +167
- -30
•4
9A
K.I;.
4 -76891 5 (74) -494-; -61
•04
•05
„
5 • 754220 (452) - 869$ + 348
- -11
•15
3A
„
6-745786(1050)- 1400^+381
- -07
•2
2A
>»»
7-739527 (8462) - 2115$ -I- 867
- -10
1-0
s.*
8-723519
L.D.
9-731179
L.D.
10-686 ID line
* S. gives v = 61-25; L.D. give 59-15; both give
D2 (8) the same. If we take this as correct and make
v - 57-87, the denominator = 8-733756. L.D.'s value
=. 8-729879.
Na. NaP(oo ) ia given in [I.] = 41446- 76 ± 1 '69, but WOOD'S measurements of the high orders require
a value alxnit 1-48 larger, say, close to 41448-24. Also FABKY and BUISSON'S interferometer measure of
NaP(l) give, when referred to HAKTMANN'S It scale, n = 16972-85. Whence
VP(1) = 41448-24-16972-85 = 24475-39
and this should be !>(-/.) and S(<x). Further, S(<») is given in [I.] as 24472' 11 ±3-84 and
XICKKNOKAIIT'S measures of high order require alwut 3 or 4 more, or, say, 3'5, which is within allowable
limits. This would give S(oo) = D(oo) = 24475-61. Thirdly, D(oo), calculated from m = 3, 4, 5,
gives 24475-20. ZICKK.M>I:AHT'S measures, however, if exact, require about 2 larger. The three
combined appear to point to a value close to 24475*40, and this was taken for calculation. In the
modified table above, it was found Ixjtter to take D( oo) about 1 larger, in the direction of ZICKKNURAHT'S
results, and the table is therefore based on 24476-40. For Nal)(6) ZICKKNDRAHT, as well as K.K., gives
an abnormally great separation. LEHMANN'S value of D\('2) gives 243 greater, making 1st ordinal
difference = 3A. K.R's value of D(6) gives 6-965755.
K. Kl',(oo) from [I.] = 35006-21 ± 1 -55 and agrees well with BBVAN'S measures of high orders,
possibly slightly less. For P,(l) K.R give n= 13041-77 and S 13042-96. These, then, give for
D(») «= S(») = 21964-44 or less (K.R.) and 21963-25, or less, (S.). The value 21964 has been taken
PR W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SEKIKS.
TABLE II. (continued).
Kb.
A = 12935. <& = 26377.
D.
0-C.
0.
RAN.
2-766216 -96£
60669 = 2308
S.
3- 705547 -232£ + 0
•00
f
8S8
4 -6837 18 (234) -468£ -26
•05
•20
388
K.R.
5 -673688 (382) -833£ + 22
- -00
•15
198
))
6-668673 (676) - 1352£ - 118
•01
•15
158
R. or S.
7-664713-2053£-68
0
1
148
S.
8-661017 -2963£- 70
0
1
9-661017-4114^ + 391
0
10-6428-55^
ll-6464-72|
1)
12-635-9£
ce) = 33687 -50 ±2 [I.], and probably greater. ' SEVAN'S observations show slightly larger, say,
about -5, ,'.«., P ( OD ) = 33688. Using SAUNDERS' for P! (1), VP (1) = 20871 • 29 = D ( « ).
According to SAUNDERS, RbD shows satellites for m = 3 and 4, giving Dn (3) - Dj2 (3) = 2 '63 and
D,, (4) - D,2(4) = 2-02 with uncertainties of 1. These give differences in the denominators respectively
of 610 and 946, and 9Si = 593, HSj = 923.
Cs.
A = 32551.
S = 638-22.
DH. D12.
0-C.
0.
0-C.
0.
p.
2 -554329 (228) -76^-43 46^ -546989 (226)- 97
- -5
3
-1-3
3
308
tl
3 -535183 (200) -201^ + 40 5^ -526567 (200) + 9
•2
1
•o
1
108,
S.
4 -533588 (160) -424^ + 1 148 -524635-161
•00
•5
•5
1
n
5 -5331 10 (400) -768£ +22 148 -524175-26
R.
6-532631-1264^ + 77 148 -523696-428
M
7 -532631 -1945£- 158
D
8-532631-2826^ + 56
M
9-525411-3975^
II
10-52533-530^
n
11-5326-70^+11
DR. VV. M. HICKS: A CRITICAL STUDY OF SPECTRAL SEKIKS
TAIU,K 1 1 . (continued).
Since the publication of [I.] RANDAU.* has measured P(l) with considerable accuracy. This, with
P(2, 3), gives PI(OO) •= 31401-78, and RANDAI.I.'S v;ilue for PI(!) gives VP(1) - 19673-12. BKVAN'S
observations show PI(») about 2 larger. Probably, however, this value for VP(1) is close to the true
value for D( <x ), and the calculations are based on D( oo ) — 19673-00.
For D, (3) LKHHANN gives denominators 548 larger for DM and 548 less for D!2. If we allow S. twice
the weight of L. the value of O - C would come out about zero.
Cu.
A = 7311. &= 146-22.
D,,. Dn.
0-C.
0.
0-C.
O.
2 -979076 (43) -120^ -6 228, -978272 + 4
•01
•10
-•01
•05
Ml
3 -984047 (173) - 288£ + 24 278, -983060 + 30
- -01
•10
-•01
•20
108
4 -985509(1) -565^ + 35
-•01
1
D(oo) = 31515-48 found from K.R.'s value for D(2, 3, 4).
A = 27791.
S = 421-07.
D,,. D18.
0-C.
0.
0-C.
0.
2-979583(19) -120^ + 3 238, -977150(19)+ 12
- ;01
•05
-•03
•05
418,
3-983898 (1967) -288^+23 108, 1982891 (175)- 27
- -01
1-00
•01
328,
4 -987280 (7776) -565^ -2 158, f9«5701 (2170)
•00
2-00
•00
98,
5-988130 (3720) - 979£ + 46
•00
•50
t Calculated from D* - v.
D( oo ) found from first three by formula = 30644-66, modified to 30644-76.
The limit uncertain, see special discussion (p. 403).
The observation errors after the first are so large that the satellite differences might be also 238,, or
larger, us in Cu.
* " Zur Kenntnis ultrarotcr Linieiispektra," ' Ann. d. Phys.' (IV.), 33, p. 743.
N4
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
TABLE II. (continued).
Mg.
A, = 854. A, = 413. S = 21'48.
I).
0-C.
0.
p.
1-822169 -27 -5$+ 12
-1
1-5
16AZ
K.R.
2-828774 (20) -103$ -12
•02
•03
3-831255 (79)- 256$ + 59
•02
•03
3A.,
*
4-832494(190)-514$ + 61
•01
•03
2A,
5-833320 (1808) - 904$ - 24
•00
•15
t
6 • 833320 (42 10) - 1 452$ - 583
•02
•20
* Calculated from D^i - n-
t The observed value gives v, = 42 '87 in place of the normal value 40-92. If this be corrected to
40-92, giving equal weights to D3, D3, the value would be 6 -833809.
D(oo) = S(oo) = 39752 -83 ±2 -73, as given in [II.], from the formula in 1/m2. This is modified in
the above to 39751-08.
Ca.
A, = 2791. A, = 1369. S = 58'14.
Dn. DJ2. DIS.
0-C.
0.
0-C.
0.
0-C.
0.
1-947172 (8) -33$ -4 138 -946417 (25) + 5 88 -945952 (25) + 20
•5
1-0
-•6
2-0
-2-4
3
99A.,
3- 082696 (20) -133$ +14 138 -081941 (20)+ 11 88 -081476 (20) - 14
- -02
•03
- -01
•03
•02
•03
1558
4-091707 (104) -312$+ 17 148 -090893(104) - 17 *-090428-21
0
•05
0
•05
•01
•05
0
t5 • 09 1 707 (538) - 598$ + 326
-•06
•10
0
16-091707 (4856) - 1012$ - 1400
•14
•50
0
t7-091707 (7732) -1546$ + 161
-•02
•50
* Calculated from D^ and Dsl) treating each as of equal value.
t Collaterals (see text). The values calculated direct from the observations are respectively 5-082736,
6-056500, 6-976528.
D(oo) = 33981-85, being 33983- 45 ±5 -8, as given for S(oo) in [II.], with 1/m2 modified by putting
- -1-6.
DR. W. M. HICKS: A CKITICAL STUDY OF SI'KCTUAL
345
TAHI.K II. (< tinned).
St.
A, = 1 1835. Aa = 5533. 3 = 277 '89.
D,,. D,* D,j.
0-C.
O.
O-C.
0.
O-C.
0.
|-'J931S4(7)-36£+l 138 • 989572 (8)- 7 88 *• 987349 + 45
- -3
1-5
1-7
2-0
t
T
6538
3- 174741 (17) -146£ + 15 128 • 171407 (17) - 7 88 -189184(29) + 16
- -02
•03
•01
•03
- -02
•05
848
I Hi8084(101)-337£ + 41 158 • 1*9919 (100) -83 88 -191693 (?) + 597
- -02
•Of)
•01
•05
•30
t
198
5 -203364 (963) - 642£ +112 158 t ' 199 196 (482) + 300 88 t' 196973(1) -84
- -02
•20
-•06
1
28
6-203919 (2539) -1088£- 40 158 *• 199751 (?) + 210
•00
•30
- -02
?
0
7 • 203919 (2838) - 1702£ - 1819
•12
•20
0
8-203919 (20174) -2522^ + 5971
- -3
1-0
* Calculated from D3i - v\ - vt. The difference might be 7,o = 318,.
t Calculated from l)Si - vt and D*. - »•,.
J Calculated from DJI - v\.
D(oo) == 31027-25, being 31027 '65 + 4, as given for S(oc) in [II.], with !/;«-, modified by putting
A, = 29328.
Ba.
= 11970.
= 683'2.
D,,. !>!* DW-
O-C.
O.
O-C.
0.
O-C.
0.
1 -825551* \
2-080655 /
3-114613(12)-137£-ll Itf 105049(20)+ 19 118 -093194 (20) - 11
•02
•03
- -04
•05
•02
•05
648
4-158338(8l2)-325f+24 108 • 151506 (325) - 154 298 • 13169» (160) + 101
- -01
•50
•09
•20
- -06
•10
-148
5-148774-49
1
t
6-1517
7-0414
* Possibly not BaD (see text).
l)(oo) = 28610-63, being 2S642-63+ t, as given for S(oo) in [II.], with 1/ro, modified by £
as explained in U'xt, p. 358.
VOL. CCXIII. — A. 2 Y
-32
DR W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
TABLE II. (continued).
Ua.
A, = 92658. S = 1853'IG.
Dn. DI* Dl3'
0-C.
4-065042 -306£ 28, -064116 + 10 28 -060410-46
358
5-081257 -598£ + 26 23, -080331 + 115
218,
6-090986-1028^ + 22
•00
-•01
•00
- -01
-•05
•05
D(oo) = 22760-19, being 22760-09, as given for D(oo) in [II.], modified by £ -
The numbers are calculated from the problematic data given in [II., p. 65].
Zn.
A, = 7204*42. A, = 3486 '20. S = 154'93.
D,,. Di2- DIO-
0-C.
o.
0-C.
0.
0-C.
0.
2 -905892 (28) -112^-20 158, -905311 (49) + 36 98, -904963(100) - 15
•02
•03
-•03
•05
0
•10
3 -90751 9 (108) -272^ + 6 138, -907015 (108) + 54 108, -906627 (?) + 43
- -00
•03
- -01
•03
- -01
68,
4 -908913 (404) -539^-8
•00
•05
46,
5-909843(:>961)-940£-40
•00
•20
•6 • 909593 (4068)
•20
•7-910072(11320)
•30
Collaterals (see text).
D(<x) = 42874-17 being 8 (OD) = 42876-42 + ^'j^, as given in [II.], modified by £ = -2'25.
DU, DIS are calculated from the more accurate D.>I, DJ-J by the use of the exact value of >'i = 388 • 905.
DR. \V. M. HICKS: A CRITICAL STUDY OF SI'KCTRAL SERIES.
347
TABLE II. (continued).
(M.
A, = 23105'56. A., = 10368-54. S = 455'28.
D,,. DM. D,,.
0-C.
0.
O-C.
0.
O-C.
0.
2-902039(28) -111^-13 188, -899990(44) + ? US, -898748(83)+ 16
•01
•03
-•01
•05
-•02
•10
756,
3-910576(91)- 272£- 11 196, -908413(630)- 22 1M, -906706(t) + 2
•00
•03
•00
•20
0
1
315,
4-914104(216)-541£- 19 196. -911941 (70K) + 31 156,
•00
•03
•00
•10
128,
5 -915470 (2656) -942^-29
•00
•20
118
6 -920598 (6648) -151 If -60
•00
•30
D(oo) = 407 10 -85 being S( oo ) = 40710'60 + 2.'ig , as given in [II.], modified by ( - -25
Eu.
= 51223. Aa = 18329. S = 833'04.
Du. D,,. D,8.
O-C.
2-930707 -114f 295, -924667-25 335, -917794 + 26
57«,
3-942578-279^-5
205,
4 -946742-552$ +17
136,
5-949450-960^ + 94
0
0
0
0
•03
- -03
D ( oo ) - 40363 • 1 9 being S ( oo ) in [II., p. 73]. No estimated possible errors given.
2 Y 2
848 DR. W. M. MICKS; A CRITICAL STUDY OF SPECTRAL SK1MKS.
o>
.1
!
CO
o
o
o
eo
00
OO
0
8
10
«— e~
1
S
o
0 O
0 O
o
i
1
0
•8
o
-
o
o
o
o
o
o
'
6
m
O
o
m
«— c—
o
1
iH
O
8
8 S
o
1
0
«
8
S -
0
1
o
o
I-H
O
8 8
o
1
IO
00
+
l~ 1O
+
0&
1 +
4
r-H
•***
CO •— I
0
CO
Cl
03
-* r-c
<M IM
•
— -
O3 O3
(M
• •
03
p
OS
T-l
_I_
00
•Uji
>.
•8
«B
1
e
r-^
a
I
be
o
8
s
c
J
2
o
II
. W. M. HICKS; A CKITICAI, S'lTMY ' 'I SI'KCTUAI. SKKIKS
340
TABU II. (continued).
AL
A = 1754. t = 26-57.
DII- DIJ.
0-C.
0.
0 «
0.
2-631287(25)-83£ + 0 4* -6311*1 (•_>">)+ 12
•00
•OS
- -01
•03
117A
:! • 426069 (82) - 1 83£ - 5 30* • 425272 (82) - 8
•00
•03
•00
•03
Ma
4 -261 194 (200) -353£ + 20 528 -2598 12 (200) + 2
•00
•03
•00
•03
Ma
5 -166498 (620) -629^ -130
•01
•05
<>• 115632(2088) - 1044£+ 160
•00
•10
16A
7-087568(3431) -1626^ + 666
- -02
•10
7A
8-0753 + 40O
•2
9-0604
•2
10-0523
•2
48164-12 being S( oo ) = 48161 -46 ±2 -49, as given bi [II.], modified by £ = 2 '66.
In.
A = 37684.
= 477-01.
Du. D,,.
0-C.
0.
0-C.
0.
2 si»3978(48)-102£+10 W -821593 (48) - 10
- -01
•05
•01
•05
37ft 58rt
3-806329 (167) -251^ -76 28.^ -793927 (167) - 154
•02
•05
•04
•05
62<"> 62-
4- 776755(31(2) -497^ -20 26o -764353(784) - 287
•00
•05
•03
•10
1 i 50
5 -755767 (2954) -869^ -63 32^ -740503(7343) - 98
•00
•20
•00
•50
50ft
f, 696300(4791) - 1369^ *-716653- 179
•20
•0«
•30
7- *-71.V.-7
•80
8- *-717621
•30
9- '-717556
•30
10-
•30
11-
1
* Collaterals (2.5,) D.,(m).
D(w) = 44454-76 ±2-48 being S ( <x ) of [II.]
350
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIKS.
TABLE II. (continued).
Tl.
A -134 154. S= 1507-34.
D,,. »»•
0-C.
0.
O-C.
0.
2-897392 1 *• 888344
»2- 899520 (80) -111$+ 10 / 248, *• 890476 -16
- -01
•03
•02
•03
28, 58i
3 -898764 (89) -270$ -30 278, »• 888590 + 53
•01
•03
- -02
•03
28, 38i
4 -898010 (213) -535$ -26 288, -887459 + 53
•00
•03
•00
•03
38, 3'i
5 -896880 (41 4)- 935$ +1 288, -886329-116
•00
•03
•01
•03
38, 38i
6 -895750 (2287) -1495$ -220 288, -885199 + 39
•01
•10
•00
•10
38,
7-894620 (3431) - 2242$ - 1 197
8 • 894620 (51 34) - 3209$ + 301
9-894620 (7067) - 4417$ - 591
10-8946 + 33
•03
•00
•01
- -03
•10
•10
•10
•20
11-8946 + 31
- -02
• -20
12-8946 + 70
- -04
•30
13-8946 + 278
- -14
•30
14-89464406
- -16
•30
* Collaterals (see text).
D(o>) = 41470-53± 1-72 being S(o>) of [II] modified by £ = '3.
O.
A, = 172. A2 = 95. $ = 9
33.
D'".
O-C.
0.
D".
O-C.
0.
2 -972467 (40) -120$ + 16
34A,
3 -966621 (14) -284$ -12
14A,
4-964213(18)-558$+15
- -12
•015
- -008
•3
•018
•01
2 -980383 (93) -121$ -3
9A,
3 -978835 (72) -287$ + 7
9A,
4 -977287 (33) -562$ + 28
•03
- -026
- -017
1-00
•26
•02
5 -962837 (38) -966$ + 20
5\
6 -961977 (67) -1538$ -15
2A,
7 -96 1633 (309) -2301$ + 20
- -005
•005
- -001
•01
•01
•03
5 -97677 1(68)1-974$ -33
6 -976599 (21 5) -1547$ +38
7 -986756 (626) -2321$
8 -985948 (6610) -3305$
•010
- -007
•02
•04
•07
•5
8-961285 (951) -3281$ -63
9 -965800(1) -4521$
•003
•06
9-976599 (?)- 4523$+ 1042
- -06
J
D'"(w) = 23204-1.
D' (<x) = 21204-2.
The observed D" corrected to give v = -72, treating the observed line as the mean of the doublet.
I»K. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
.351
TABLE II. (continued).
S.
A, = 1044. A, = 651 7. •* = 37'28.
0-C.
0.
4 -553453 (47) -430^ + 7
14A,
5-544330(21)-776£-7
6A,
6 -5391 10 (38) -1273£ -29
3A,
7;535978(194)-1944£ + 173
2A,
8 • 533890 (396) - 2828£ - 202
4A,
9-529714 (1141) -3936^-129
- -01
0
•007
- -02
•02
•01
•05
•01
•01
•03
•04
•08
D(o>) = 20084-2.
Se.
A, = 6407. Aa = 2722. S = 226'8.
Calculated from Observed Values.
4 -629262 (54) -452^
5-621963(56)-810£
6 -615643 (105) -1320^
7 -601 450 (240) -2002^
8 -61 1058 (728)- 291 1£
9- 592507 ( 643) -4024|
10- 57831 (270) -540^
*• 626 1 33 (108)
•622797
•617055
•608778
D(<x) = 19274.
* Calculated from I);., - D..J.
Modified Table.
0-C.
0.
0-C.
O.
4-629285 + 22 558, -626167 + 11
- -02
•06
- -01
•09
5-624976 + 41 38*, -622822-56
- -02
•03
•03
t
198
6-620667-5 558, -617549-362
•00
•03
•10
1
7-616358 + 156 558, -613240-531
- -02
•04
•09
•09
198
8-612049-700
•076
•08
198
9-607740 + 91
-•00
•05
A. + A,
10-59C60
•15
,,K. U. M. HICKS: A CRITICAL STUDY OF SPKITUAL SERIES.
Tin- >••»/••///'. S.ri,--iiinM.— As the values of the satellite differences are practically
Jndewodanl <-!' ih.- exact value of D(«), their consideration may be taken up at
,„„,. ..,,„! th-- details of the calculations respecting the tables postponed until the
consideration of the older differences. An examination of the tables will show
,,,,,,.hisiv,-ly that these differences are multiples of the "oun." Dealing first with
tli,- differences for the first lines, the following figures, contained implicitly in th.-
tables, will show how closely this is the case. The nearest multiples of the oun are
appended, as calculated from the first approximations of Table I. The possible errors
are those of the D,, lines (except Zn).
Cs.
Cu
&g
Ca.
Sr.
Ba.
Zn.
7394 ±228 46*, = 7340
7 94 ±43 22*, = 804
2424 ±19 23*, = 2422
f 746±8 51*, = 741
'1 450±25 31*, = 451
j"3620±7 52*, = 3596
•\2170±? 31*, = 2154
. . Not observed, as deduced
below (p. 388), 15*, 9*,
or 60*,, 36*,
T 525±77 IS*, = 581
' 'I 369 ±28 9*, = 350
Cd .
f2029±28
'1_1234±44
18*.
= 2049
= 1248
Eu .
J6065±?
' ' L6822±?
29*,
33*,
= 6040
= 6873
Hg -
r3980±25
. A 6909 + 38
L 346±38
11*,
19*,
= 3991
= 6894
= 363
Al. .
. . 94±35
14*i
= 93
In . .
'. . 2405 ±48
20*!
= 2385
Tl. .
. . 9048 ±27
24*,
= 9044
Se.
3129±54
55*,
= 3118
The only case of " failure " is the first difference for Sr in which the estimated
possible error is extremely small. If the possible error be the sum of those of each
line, the value is 15 in place of 7, and if *, be calculated from the most probable
value of the oun it should be about ¥iD greater, i.e., 52*, = 3600. It will be noted
that where triplets enter, the two satellite differences, and consequently the two
satellite separations, are extremely close to the ratio 5 : 3. This ratio seems to
persist also in Hg where the separations are in reverse ordtir, and we find a ratio of
3 : 5 in place of 5:3. The law for this ratio is in fact much more closely obeyed
than the corresponding one for the ratio 2:1 for the triplet separations. It is
therefore of great assistance in searching for the lines of F series whose limits are
VI ) ('!}, and which consequently possess constant triplet separations in this ratio. Its
explanation should be expected to throw some valuable light 011 the constitution of
tin- atom. The general dependence of the differences on the small "oun" J, should
also be noted.
Passing now to the consideration of the satellite differences for orders beyond the
first (m>'J), it is seen that they still depend on multiples of the oun, but different
from those of the first order. In a large number of cases the multiples are the same
DR. W. M. HICKS; A CRITICAL STUDY OF SPFXTTRAL SERIES. 353
for different orders within limits of errors, especially in the doublets and differences
'•I' tin- second and third satellites. Thus we find Cs, 14<J; Ca, 13<J, 83; Sr, 12& for
m = 3 and 15<5 for m>3 for first separations, and 8& for all orders in the second ; Ba
and Zn show too few for comparison (see discussion below); Cd, 19<J,, 15^, ; Hg is
ii regular, Al is anomalous ; In 26$, Tl, 27 <Ji for m = 2, and 28 J, for m>3 ; Se, as
amended later, shows 55(5, for m = 4, 6, 7, and 38(J, for m = 5, the lines for m<3
Ix-iiig outside region of observation. The evidence points to a normal rule that the
differences for the orders beyond the first in any spectrum are the same, but different
from — in general greater than — that of the first.
The Order Differences. — The order differences change very considerably with a
change in the value taken for the limit, i.e., in the value given to £ No doubt with
unlimited choice of £ it would be possible to arrange a set of denominator differences
all multiples of the oun within error limits, for a series of values of £ could be found
making the first difference a multiple. Out of these one or two would probably give
the second such a multiple. After the second the error limits as a rule come to be
very large, in fact larger than half the oun itself, except in case of very high atomic
weight. No conclusions could be drawn from any such arrangement. But in the
present cases the choice of £ is bounded by very narrow limits, for the relation
D(oo) = S(oo) is supposed to hold, and, as a rule, the values of S(») are known
with very considerable accuracy, and the possible limits of variation are known.
They were given in [I.] and [II.]. Before proceeding to draw general conclusions
from Table II., it will be well to consider in more detail the data for the different
elements on which the table is based.
Na.
Although the readings for NaD are very inexact, the peculiarity of the large
depression shown for m = 6, as well as the large recovery afterwards to raantissse
close to unity, must be real effects. It is of course possible that NaD, (6) is a
collateral from the normal type. If D, (6) be calculated from D.,(6)— v, the mantissa
becomes '989054, in other words, the D2 line begins to show the rise to the large
final value at m = 6, whilst D, does not do so until m = 7. The D, lines would seem
to succumb to the disturbing effects sooner than the D,. It was pointed out in
[I., p. 83] that in the Na the D series apparently belongs to the F type, in which
the mantissa is '998613. It would almost seem that the peculiar rise shown is due
to the fact that it reverts to this F sequence. Here, as we shall see in other cases,
the values of the first members of these series appear to be subject to some kind of
displacement which affects their (supposedly) normal relations to other lines. If now
tin- first mantissa be supposed normally to be this 998G13, it is 9691 above that in
tin' modified table, and this is 13A + 32, thus completing the order differences as
multiples of A. But in any case the data for Na are of small use for the present
purpose, as the errors are so large, and A so small. The arrangement in the table
VOL. CCXIII. — A. 2 Z
,„;. w. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
• riven the values of O-C least. If ( be taken about '6, the order differences can run
3A . 4A, 4A within error limits.
K.
It is seen that the numbers, with the exception as in other cases of the first
difference, fall into multiples of A quite naturally without a change in D ( »), though
possibly a' small change in £ might make the values of O-C less. A is so large, that
the theory of the dependence on multiples receives considerable support. The first
also is close to 20A. This element is one in which the value given in the first
discussion above for the oun is 862'68tc», which is presumably too large by '8 to "9.
If it be 361'8, A should be ¥b less and = 2933. This would scarcely effect the other
intervals, but it would make the first one = 20A-704. Again there is a sudden fall
(at m = 8), It doubtless corresponds to a real effect, for SAWDERS as well as
LIVEINO and DEWAR make » anomalous here. S. observes * = 61'25 and L.D. 59'15,
hut both give D,(8) the same. If this be taken as correct and the normal Dt(8)
found from D-,-57'87, the mantissa is 733756, giving the same difference 2A. This
shows that the Da set have not participated in the sudden fall — at least to the same
degree as D, — a result analogous to what happens in Na.
Eb.
In Rb there is some doubt whether a satellite series exists. The question has
already been discussed in [I., pp. 71, 86]. SAUNDERS has given for D (3) lines whose
wave numbers are 12883'93, 12886'56, and 13121'19, with normal separations 237'26,
and satellite separations 2 '6 3. Also D (4) is a doublet having a separation 23 5 '52,
which certainly points to an unobserved satellite about 2'4±'l. But RANDALL'S
observations for D (2) show only a doublet of normal separation — that is clearly with
no satellite. Moreover the F series, which depends on Dn (2) and DJ2 (2) for its
limits, is a singlet series and not a double one. In the table the series is taken as if
it were without a satellite, the reading for Dt (4) being corrected from D2 (4)— v. In
other words it represents the satellite lines if they actually do exist. In the latter
case the strong lines would show denominators about 609, and 1100 above those in
the table for Du (3) and Dn (4). The first is about 28 and the second of the order of
4cJ, whilst if normal, judging from other cases, they should be equal. Moreover, in
all other cases (In and Tl excepted) the satellite separations are practically the same
multiples of the oun in the same group. Cs shows a difference of 14^, so that the
supposed ones here are far too small, as well as irregular. The observed separations,
moreover, are equal within errors of observation, which would rather point to an
alteration in D, ( t» ). Now a lateral displacement of 2S on D ( o> ) would produce a
separation of 2 '41, which is about the observed value.
The table shows a stationary point at m = 8 and 9 and then the large fall shown
in the other elements. They could be accounted for by a lateral displacement
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 355
(2<T,)P(<»), the mantissa of m = 8 being at the same time subject to the fall of
multiples of S which would scarcely stop at 14<J. In the table, however, the errors
are inserted on the supposition that the mantissa? remain constant.
Cs.
The mantissre appear to run down by equal intervals from m = 4 to 6, are equal for
6, 7, 8, then a large drop of alxmt US to the same value for 9, 10, and return to the
value at m = 8 for m = 11. The possible errors are so large that the regularity is
curious. It is possible that they might run down at equal intervals of 3(5, to the last
one for m = 10. Or, if there are very small observational errors, the drop for 9, 10
may be due to a lateral displacement, about ( + 7<J,) D ( oo). It should l)e noticed that
with Cs all the order differences but one are multiples of 3*,, or the group multiple.
Cu.
The two first doublets of CuD are strong. The third is much weaker than would
be expected. Moreover, it gives a separation between Dn and D21 of 252*14, whereas
it should be somewhat less than v = 248'28. This (X = 3688 '6) can therefore scarcely
be the normal chief line of this doublet. Now EDER and VALENTA give a spark line
at 368775 which gives a separation with D2, of 245'91, leaving a satellite separation
of 2 '37, which is within limits in fair order with the corresponding separations in the
two previous doublets, viz., 6'60 and 3'39. Moreover, the satellite separation of 2'37
gives a satellite difference of 1317 and 9S = 1315, so that the normal satellite
differences would run 22e?,, 27<5,, 36<$,. This, then, would seem to be the wanting
normal chief line, and it is then interesting to note that the line usually accepted as
Du (4) is a collateral of this. The denominator difference of the two is 2474 and
17* = 2484 (S = 146). Hence the K.R line 3688 is the collateral D,,(4)(-17J), and
apparently the small intensity is due to the usual decrease produced by a negative
lateral displacement. The modified table is taken on £ = — '08. It is remarkable
how close the observed differences come to multiples, but little reliance must be
placed on deductions from them both on account of the large possible errors and the
smallness of <V
Ag. ' /••
Unfortunately the D series in Silver is poorly developed — only the first satellite
has been observed, and the three chief lines after the first have very large possible
observation errors. Nothing, therefore, can be learnt as to the march of the satellite
differences beyond the fact that the observed — 2436 — is very close to 23<S, = 2421.
In the modified table £ = '1, the differences are very close as is seen to multiples of
J,, but there can be no certainty with such large possible errors.
2 Z 2
356 DR- W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
In the D series of Magnesium, as arranged by KAYSER and RUNGE and as generally
accepted, there are clearly certain abnormalities. D! (4) is more intense than we
should expect, and its separation from D.j(4) is 45'39 in place of 40'92, whilst that of
Da(4) and D3(4) is very close to the exact value. This cannot be due to observational
error, for this is very small ('03). Either, therefore, the true line is hidden by this
bright one, which can scarcely be the case, or it is a collateral. In the former case
the true line ought to be that found by deducting vl and v1 + va from the satellites.
In the second case it would require the addition of 19^ to the denominator of D ( o°),
and the addition would explain the increased intensity. The two results agree, the
wave numbers resulting being respectively 35054'80 and '71. The former would give
a denominator '832041 in place of that in the table, but its observational error would
be that of D3, viz., 945, while that of the collateral depends on the observed Dls and
is 190. D(5) gives normal separations within limits. D (6) gives v = 46 '87 and
22*15, but normal within the observation errors (2'8) [see Note 1 at end].
But there is another question which arises in connection with Mg. In the Ca
sub-group the first lines have a denominator about 1'9, i.e., with m = 1. In the Zn
sub-group the lowest value of m is 2. In the MgD series, as generally accepted, the
first line is X = 3838, which requires m = 2. If there is a line corresponding to
m = 1 it should be in the neighbourhood of 14900. Now PASCHEN has observed a
strong line at X = 14877'!, but there is no triplet, which would be decisive against
the allocation if we could be certain all the lines must exist. But there are cases
where normal lines are observed weaker than we should expect, or are not seen at all.
The well-known case of KD(3) is one example, and it is curious that if 14877 be
taken as MgD(l) the denominator comes out as given in the table in a very natural
order with the other denominators. The question is considered later under the F
series, and the evidence there adduced is rather against the present suggestion
(p. 398).
Ca.
The value of S is calculated from A2 as 58 '14, A^A;, would give 58' 18, practically
the same. To bring the differences of the first three denominators to multiples of S
it is necessary to diminish the limit given from the consideration of the S series by
1'6 (variation limits given in [II.] = 5'8). The values can then be arranged as in the
table. One result of this is to increase the value of At (for the given Vl = 105'89) to
from 2791, which gives S = 361'30^ in place of 361'17, and thus closer to the
adopted value 361 '80. A noticeable peculiarity in this series is the very rapid falling
the denominators after m =-- 4. It is so large and at the same time so
that they cannot be brought into line with the others without diminishing
the limit by a large amount and by different amounts. It clearly points to the existence
DR. VV. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 357
of collaterals, formed by the addition of ouns to the limit D(oo). As such increase
tends to increase intensity it may account for these surviving when the typical ones
are either too faint or are destroyed to form the collaterals. It is useless to attempt
to determine these multiples, because the observation errors are so large themselves
as to be a large multiple of the oun, and at the same time we have no knowledge <>!'
what the typical VI) (m) should be. In general in the 2nd group the successive
denominators are formed by the successive addition of smaller and smaller multiples
of the oun until probably a constant value is reached. In the present case, with the
quantity 155(5, that limit is certainly not reached. But it may be instructive, in
order to illustrate the nature of the suggestion, to find what the collaterals ought to
be if the denominators of VD(»t) remain the same for TO > 4, viz., "091707. The
multiples are found to be 7S, 15<S, 33d The series of the observed D,, lines may then
be exhibited by the following scheme, where d stands for '091707 :—
D ( oo)-VD (2+d-99A8-155<J),
D(oo)-VD(4+d),
The line D12(2) is interesting. PASCHEN gave it as 19859'9 with the remark
" Wahrscheinlich doppelt 1<)856'9, 19864'6," and he allotted 19864'6 to D13 and
19856 '9 to the Principal series. But in [II., p. 56] reasons were given against the
latter allocation. In fact the line is very close (probably within observation errors)
to the collateral formed by adding one oun to D12. The wave-length of such would
be 19857'S = D,., (2) ( + S) [see Note 2 at end].
Sr.
The value of S is calculated as 277 '8 9 from A, + Aa=125x4, which gives
$ = 361'64?^. The differences as shown in the table are extremely close to multiples
of S. Moreover, the limits of variation for the first two are so small that the
variations of ROWLAND'S standards from the correct values for his scale may become
of importance. For D(3) the values should be 2 less, whilst for D(2), failing direct
observations for reduction to vacuo, recourse must be had to extrapolation on
KAYSER and RUNGE'S formula,* which has been done. In order to bring the
differences for Du(3) and Da(4) and of Du(4) and Du(4) to multiples of S within
error limits, it is necessary to take £ about — '4 or D(oo) = 31027 '25. When this is
done the denominators can be arranged as in the table. The difference of the two
* RANDALL appears to have done this for Dn but not for DU, which also makes his value of v\ »= 392 • 6
instead of 394-42, which is close to the true value.
358 1>R- W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
first denominators = 181557 = 653x278'03. It is possible the real errors attached
to D (2) by RANDALL may be greater and the difference slightly less ; but if we
suppose 278-03 to be the real value of S it makes $ = 361'82t^, and, therefore, very
close to the adopted value. It would appear that D12(4) has been displaced from its
normal value, judging from the irregularity introduced into the separations. If so
the separations might be 14<5 in place of 15S.
Ba.
Starting with the uncertain value of S(<») = 28642'63, as given in [II.], the value
of S as calculated from A, + A3 is 68270, and from Aa is 684'34, both being near the
most probable values but on opposite sides. The value 683 is taken at first as a rough
approximation. Apparently, the first set of lines have not been observed. RANDALL*
has observed two lines, 29223'4, 23254'8, which give a separation 878'27, which is vlt
but no signs of satellites — or, rather, if there are satellites, the separation observed
should be much smaller. If, however, the satellites have gone here, and this pair
denote the first two lines of the first triplet, they depend on VD13, and the value of
the denominator is 2'085331,t which would range well with those of Ca and Sr, viz.,
1'946, 1'987, but the second lines of these give 3'082, 3'169, and of Ba 3'093, which
would rather point to a less value than 2'085 for the first line. But if VD (2) is
larger than D(oo), the lines would be -23254'8 for the first and -29223'4 for the
second, giving a denominator for the first of 1 '825551. The differences of the
denominators of the D13 lines for m = 2, 3, 4, will then be 267632, 38612 instead of
6528, 38612, and are therefore more in agreement with the type of the other elements
of this group. Moreover, the former, as we shall see, is a multiple of S, whilst the
other (6528) is as far out as it can be. Both values, however, are inserted in the
tables (see also p. 389).
The satellite differences for D (3) are 9492±32 and 7516±40. The values of US
and US are respectively 9562±5 and 7513±4, and hence the first cannot be 14<S within
limits of error, although it is so close as to produce a conviction that it really is so.
Now for small variations of the limit D ( oo ) the separation differences are scarcely
affected, but, as we saw in [II.], there was evidence to show that the limit S ( oo ) was
considerably less than that found, and, in that case, the separation differences would
be slightly changed. A decrease of D ( oo ) would increase those differences. If,
however, it is so large as to bring up the first to 14<$, the second is increased so much
that it is not IIS within limits. Consequently, the two conditions confine the choice
of D(oo) within very narrow limits. It is found to be close to a decrease of 32,
i.e.t D(») = 28610'63. This again changes the values of A1} A2, with the given
values of Vl and v, to 29379 and 11997, giving from A,, S = 683'2. The table is
* 'Ann. der Phys.,' 33 (1910), p. 745.
t The values in this and in the table are calculated from the limit as modified below.
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 359
drawn up on this basis. With these data, the suggested allocations for D, (2) give
I IK- following differences between tli.-ir denominators and that of D13(3), viz. : —
With D,3(2) = 29223 6528 = 9'5 x 683
Dlg(2) = -23254 267632 = 392 x 683.
The first, therefore, cannot be a multiple within error limits.
The values shown for D (4) agree very well, but the regularity is upset. Also the
actual lines have changed in appearance, and their intensities are not normal. The
intensities for D! (3) are 10, 6, 4, those for D, (4) 4, 4, 6. We should expect Dn (4) to
be stronger, and certainly D,:,(4) to l)e much weaker. It would seem that collaterals
must displace the normal lines. We have, in the foregoing pages, been led to expect
that an addition of an oun increases the intensity and a deduction diminishes it. If
so, we should expect a deduction in Dn and an addition in D13. To bring Du 14<5
above D13 requires the deduction to be made in VD. This would make the typical
value of the denominator greater by 2733, viz., 4'161071. In the case of D13, to
bring it closer to D,a, i.e., distant US, the addition would have to be to D(oo), and
if so, the value of 29<5, given in the modified table, would have been a mere coincidence.
But no such addition of a multiple of S (nor of <Jj) will do this. If, however, 2S be
added to the denominator in D ( »), it is brought to separation of 10<5+36, giving an
error in X for D13(4) of —'02 in place of —'06. If, then, S be also deducted from VD,
the separations will be 11(5. The separations would then take the form
423(5
US US
66S
US US.
This arrangement is to be preferred in that (l) it explains the abnormal intensities
of D (4), (2) brings the separations into line with other elements. The arrangement
suggested may be stated thus: if Du(4) and DI3(4) represent the typical lines, the
observed Dn is Du(4) ( — 2<5) with decreased intensity, and D,3 is ( + 2S) D,3(4) (— S),
the increased intensity due to the + 2S in D ( oo ) being greater than the decrease due
to -S in VD.
The results for orders > 4 are similar to those of the other elements of their groups,
probably collaterals of additions to D ( <» ).
Zn.
On account of the small values of <$„ and the considerable observation errors, the
satellite separations in Zinc do not give decisive resulta If we take the observed
values for the D12 and D13 sets, the denominator differences are 489, 419 for m = 2,
.•mil 584, 399 for m = 3. Now the second sets, the D.,, have much smaller observation
:;,;.• DR. W. M. HICKS: A CKITICAL STUDY OF SPECTRAL SElilKS.
errors than the satellites of the first — see remark in the table. If the latter be
calculated from them, using the accurate value of »l = 388 '90, the differences come
out to be 525, 369 and 456, 399, quite reversing the order of magnitude for the first
satellite of the lines for m = 2, 3, and, moreover, their differences are larger than S}.
The differences best consonant with the measures — using the derived values from Da-
are 13<5, = 503 and IOS1 = 388, giving a ratio of 1'30. Those entered in the table,
however, are 15^ = 581, 9(5, = 348, both within the limits of observation, and adopted
because they give a ratio 5 : 3, the same as the other elements of Group 2. The
satellite separations for m = 3 may be the same as the latter within limits, but not
necessarily so.
The order differences do not work in well with the above when £ = 0. If, however,
£ be put = —2 '2 5, they fall into line with extreme accuracy, as shown in the table.
It is of interest to notice that the differences are multiples of Sv = 6^, which seems to
be specially associated with zinc.
The denominators for m = 6, 7 are now G'915855 (4068) and 7'924179 (11320),
showing too large a rise to be due to error observations. Treated as collaterals
with -2J, and -3^ they become 6'909593, 7'910070, clearly near the probable
limit.
Judging from analogy, we should not expect the differences to stop at 4^6. The
series (7, 6, 4) <S6 would probably be continued, but the errors are too large to settle
the question. If the series were continued, e.g., by 2Se, 0, the denominators would be
6-910288-G95 and 7'910288-218. But the best agreement is to take them as they
are. This would also be in analogy with Ca, in which D ( oo ) begins to change when
VD stops changing.
Cd.
In the table £ is taken '25 above S(o>), as given in [II.]. It is seen that the
arrangement fits in with great accuracy, and as ^ is as large as 114, the arrangement
may be considered to have some weight. The denominators calculated from the
D, and D3 lines (more accurately determined) do not agree with the observed Dt
satellites. It would therefore appear that the second and third members of the
triplets may also be subject to special displacements. Here, for instance, the lines of
order 3 are brought into line if the observed VD21 (3) is VD12 (3) (-5,). The value of
D,,(4), calculated from K.R.'s D31(4), cannot be the normal one, even when his
extreme possible error is allowed. This shows again that VD31 (4) must be displaced
from VD13(4), in this case by lOc?,.
^ The denominator for m = 6 shows the sudden large rise after a slow change which
Zu also exhibits, but it cannot be explained, as in that case, by treating it as a
llateral due to a modification of D ( oo) alone. The closest collateral of this type
due to J, and this changes D (oo) by 5'66 producing in the denominator
a change too far in the opposite direction. In fact it becomes 6'912000, 3471 below
DR W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
th.it of m — 5 in place of 50(50 above. But the observation error in this line is so
l.-ir^i- thai us it stands it may correspond really to a denominator equal to that of
/" - ^. If further the series of differences 75<?,, 31»f,, 12<?,, be continued diminishing
further as l<x>ks probable, it would come more nearly in line. For instance, as each
term is about '4 the previous, suppose the next is 5<J, = 569. The value of D, (6) as
thus calculated differs from the observed by —'19 while the possible error is '30.
This would make the constancy of the denominator begin at an order one higher than
in Xi i. The case of Hg will be seen to support this tendency.
There are two sets of doublets, 364974, 3500'09, and 3005*53, 2903*24 with
separations 1171*13 and 1171*95 which are clearly associated with the D series.
If we write down the observed satellite separations in the D,(2) and the D, (3)
lines find of D,3 with the above we get the following scheme: 18*23, 11*10,
267*13 and 7*98, 6*18, 262*40. At first sight it makes the new lines appear as
collaterals by the change of about £A, on D(oo), but this cannot be the case, because
a change of this amount would very considerably diminish the doublet separations
below 1171. If the D1:l— D13 separation be deducted from that of the lines in question
there results 267*1 3 — 11*10 = 256*03 and 262*40-6*18 = 256*22. Now the separations
ll'lO, 6*18 depend, as is seen above, on 11<J,, 15<J,, so that the new lines may be
written *P(oo)_VDI3(2)(-llJI)-A, D ( oo)_VD13(3)(-15<J,)-A, where A is a
constant which on more accurate calculation is found to be 255*80 ±'2. In other
words, the VD of the new lines is derived from VI)1:, in the same way as that is from
VI), a. This formula is of a type of which there has been no example hitherto. If it
remained there the evidence, in spite of the curious connection with the other
satellites, would scarcely be weighty enough to cause the introduction of a new
departure. I hope however to show in a future communication that this expresses a
very common relation between sets of lines, the constant A being in reality a com-
posite one. The question naturally arises do the new terms give rise to an F series
in the same way as the ordinary D ? It should be at a distance about 2(57*13 in wave
number above that of F. The line 15713*3 (u = 6362*25) is 266*90 above that of the
line 16401*5 (n = 6095*35) which is allotted by PASCHKN to F3(3) and is clearly the
first of the lines in question. There is an F(4) line at 11630'8 (n = 8595*57) and
another at 1 1268*4 (n = 8872*01) is 266*44 above this. This may lie the corresponding
line sought for, but if so the line 11630 must be F3(4) and the lines F, (4), Fa(4)
would then Ixi absent. These lines were at firstt assigned by PASCHEN to a new
doublet set of series, but later}! to combinations of his new singlet series with D, ( oo ),
D., ( w ). This question will be considered as a whole later, but the suggested
explanation given above points rather to the fact that we have to do with a triplet
series in which the third number is too faint to be observed.
* DC00) stands as usual for DI ( oo ) or Dj ( oo ).
t ' Ann. d. Phys.,' 29, p. 650 (1909).
| 'Ann. d. Phys.,' 30, p. 749 (1909).
VOL. CCXIII. A. 3 A
362
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
Hg.
The D series of Mercury shows a marked divergence from those of the other
elements so far considered in that (l) the separations of the satellites increase as tli.-y
go from the chief line, (2) the satellites do not seern to correspond in the different
orders, and (3) there are a larger number. The increased tendency which this element
has shown to break up into collaterals appears also here. One is led to infer that
with varying conditions of the production of the light different collaterals appear.
The dependence on the oun is, however, here clearly shown, and the evidence is all
the stronger because the magnitude of <*, itself is large (.363) and because all the
apparently unconnected differences come within close multiples of <V This is clearly
seen in the following table where the denominator differences are exhibited
together : —
3980 = 11^-11 6909 = 19^-15 346= ^-17
6201 = m, + 32 6530 = 18^+ 2 5398 = 15^-45
5782 = 16^-20 6262 = 17^ + 93 9750 = 27^-48
7989 = 22<S,+ 6 18516 = 51^-10.
It is still possible within errors that the differences for the first satellites shall be
the same as for the second and succeeding, viz., 17<51( but it is very unlikely. For the
second satellites this cannot hold. It is clear that the regular law is not contradicted,
but is upset by the formation of new configurations or aggregations in the oscillators.
The table is drawn on £ = '8. This brings in the best agreement and, moreover,
brings S ( » )— supposing it and D ( °° ) are the same — closer to the value found from
the first three lines. The value given in [II.] being one modified slightly to bring all
calculated (even for m = l) within limits. The agreement is seen to be remarkably
close. It is to be expected that the differences for the satellites of the same lines
will be more accurate than the differences between the chief lines themselves, and
this is exemplified in the table. The observation errors after m = 6 are too con-
siderable to draw certain conclusions from. Apparently the denominators increase
by small multiples of S to about m = 8 and then remain constant.
Al.
In Al, the satellite differences deviate from the ordinary rule in that they increase
with increasing order for m = 2, 3, 4. They are 94, 800 and 1380, and by no
stretching to the extreme possible errors can the two last be made equal. The
inequality is certain. Moreover, the observed differences are very close to multiples
If the first satellite position be calculated from D21, its difference is 110, the
observed is 94 and 4<S = 106, 94 can be 106 within limits. D21 gives in the same way
= 3, the same as observed and 1468 for m = 4 instead of 1380. The last
may be the same as the observed within limits, but as 52^ = 1381 '6 and 553 = 1462,
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SEKIIX
it is possible there may be this difference and VDS1 (4) is not VD,,(4) as in the typical
cases. The real difference may be any multiple between 52<J and 55i. In fact 3 is so
small that there is not absolute certainty.
AID has proved itself the most intractable series to bring into any simple formula
of the ordinary kind. It was, in fact, the difficulty with this element which first led
mi- to seek another solution — on the lines now being considered.
It will be seen that it lends strong support to the theory suggested. The table is
arranged with £ = 2 '66. The exactness of the relations there shown is very
remarkable, and when it is remembered that A is a large number like 1754, the
practically exact multiples referring to the first five lines must carry very great
weight in the argument that AID at least is subject to a modification of successive
denominators by multiples of certain units. The objections to the arrangement are
two : (l) that £ = 2'66 is outside the error limits of S ( o°), and (2) the denominators
appear to go on diminishing without reaching a limit. A slight alteration, however,
in A will get over the first. For instance, if A = 1754 — '5, £ would be about '5 less,
D(oo) would be within limits of 8(00), and the same arrangement would also hold,
but it could not be much more diminished because with m = 5 and 6 the changes
introduced into the denominators by £ would upset the multiples 54 and 29. A
change of A by —'5 would change the ratio to w* from 361'88 to 36178. D(oo) is,
therefore, probably very close to 48163'62.
If £ = —10, the denominators tend to a limit about '107 for m = 7 and beyond.
But this is far outside permissible limits of D(oo), and, moreover, the striking
arrangement with multiples of A is quite upset. We must therefore conclude either
that the limit is not reached until an order m = 10, or beyond, or collaterals enter.
If the former, multiples of A can enter, but the observation errors are too large to
give certainty. If collaterals based on (9<5)D(oo) are used with £ = 2' 16 or
D(oo) = 48163'62 the mantissas for 7, 8, 9 come respectively to '11:3569, '113590,
'113956, and for 10 for a VDU = VTX,,, 113700. The separation observed for m - 10
is 107'96 instead of 1 12'15, 'either an observation error or a displacement of Dn or Da,.
A displacement of D,, by + 2$ on D(o°) brings the separation very nearly correct,
although the observation error in the wave number of these two lines is as large as
4 '4, it is probable their difference is much more exact and that the defect shown by
v = 107 is real. (For m = 10, Dn is practically = Dia.) The results therefore go to
show that the D, and Da lines for m = 7, 8, 9, 10 are collaterals, (9c$)D(oo), except
that for D(lO) an extra displacement of 2<5 is added. Although the numbers above
are so nearly equal we must not place too much reliance on them, as the observation
errors have a very large effect on the denominators for such high orders. If the
suggested arrangement is correct it must mean that K. and B.'s measurements must
have been of a very high order of exactness, which would further mean that the
measures for the D., lines would not be so exact since the observed values of the v for
m = 7, 8, 9 are respectively —'68 (possibly real on Da), '7, '13 in error.
3 A 2
:;.;i DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
I am inclined therefore to think that the exact equality for ( + 9^) is a coincidence,
especially as the difference for m = 6 and 7 is not a multiple of A like the others.
Taking the corrected D(w) = 48163'62, the mantissa for m = 6 is '1 1(5154, giving
with M 13569 a difference 2585 = (l+£) A-46, which is as far as it can l)e from a
multiple of A.
If + 7S be taken for the collateral, and 5A for the difference, the limiting
denominator is '107384 and the corresponding O — C are —'03, +'03, '07. Now these
make, as against the observed D3 lines, the values of vl about correct, which gives a
certain amount of weight to this arrangement of collaterals based on ( + 7S).
In the foregoing the conclusions up to m = 6 may be taken as well based. No
definite answer can be given to the question of what happens beyond m = G, although
the balance of evidence perhaps points to the last, viz., collaterals based on ( + 7^),
and this is strengthened by considerations which follow.
MANNING* has recently observed under diminished pressure certain groups of lines
which by their look suggest doublets and satellites related to the D type. The
strongest set, apparently a Dn and D2 doublet, are 4260'05, 4241 '25, giving a
separation of 104'05 ('5). If 4260'05 be treated as having the same limiting term as
the D series, the denominator of the VD part comes out to be 2107364 (21). Now
this mantissa has the limiting value according to the supposition of the preceding
paragraph, viz., 107384. If this is not a mere coincidence, the connection should
throw a great deal of light on the relations of these series, and would warrant a more
searching discussion.
This would, however, lead too far from the immediate point at issue. It will be
sufficient merely to indicate somewhat more clearly the connection. With the limit
(7<J)D(oo) the mantissa of D1(7) is 299A below that of m = 2 (see Table II.),
and it should therefore be (with D (oo) = 48163'62) 631328 (25) -83£-299A,
whilst that of MANNING'S 4260 is 107364(21) -42^. If these are the same
299A = 523964 (46) -41f
A = 1752'388--14£±-15,
and this gives S = 361'55< which is too small. But in this D(7) of the accepted
series is referred to (7$) D ( oo ), whereas 4260 is referred to D ( o> ). If it be referred
also to (7<5)D(oo), its mantissa is 107868 (21) -
and 299A = 524468 (45) -41£
A = 1754-073±'15--14£
and therefore of the right order with $ = 36T89W2.
MANNING'S new lines suggest, on a superficial glance, a series of bands, but there
-tie doubt of their connection with the Diffuse series. Their relations to one
another can be discussed either on the basis of the old D ( oo ) or of (7$) D ( oo ). With
Observed in my laboratory, since published in ' Astrophys. Joura.,' May, 1913.
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 365
f) as D,, (intensity 10) goes 4241'25 as D.,, (intensity 9). D,,-112'15 should
give DIt s 4261 '58. This lias not been observed, but it is D,,(l)(— 13<5). In fact,
the error Ix-tween this calculated value and that deduced from 1).,, is only
dX = '02, and a satellite difference of 13(5 is more in accordance with that of
other elements than the small one of 14(5, in the accepted series. Amongst the other
linns are the collaterals 4280'4 (intensity 9) = (A) (4260) with O-C = -'04, and
436.V7 (intensity 2) = (5A) (4260) with O-C = '1.
In.
As in the case of Al, so In shows an increase of satellite differences with the order.
The first three, 58, 26(5, 26(5 may be considered as certain, but the next, 32(5, although
it is close to the observation, may, as in the case of aluminium, be the same as the
others (268) within error limits, owing to the large error in D,,. K.R. gives the
iliU'erence in wave-lengths as 1'04 A.U., whilst HARTLEY and ADENEY in the spark
give it as '4, i.e., closer. In the table it is entered as 32(5 as being closer to the
olwervations, but if it really is 26(5, the O— C is +'20 against O = '50. It is possible
that many cases of diffuseness may be due to the simultaneous existence of several
collaterals based on differences of St, which for lines where m is large or for small
wave-lengths give differences in X too small to resolve. In this case, for instance,
with m = 5 a displacement by (5, produces collaterals differing by about '006 A.U.,
and several would give the impression of a nebulous line, broadened on one side or
the other. For m = 6 there is clearly some collateral change different in Dn and Dai.
For if Dj2 be calculated from D.^ it gives a position for D12 of longer wave-length
than Dn, or the inverse of the typical order. No conclusions therefore can be drawn
as to the satellite differences for m = 6, except that D21 is probably of the form
(zJ) Dj,,. Beyond this it is curious that the D2 lines persist while the D, lines do not,
which may l)e accounted for by their being also like m = 6 additive collaterals.
Again, also, the order differences show themselves as close multiples of S. The
table is based on £ = 0, but it may be brought into still closer agreement by taking
£ a small negative number, about —'2 to — "4. The difference between 5 and 6
becoming suddenly so large (59463 order 124(5) and the entrance of the peculiarity
mentioned above, suggest that some collateral influence comes in. Further, if we
regard the denominators of D.,, or of D,, calculated from D.,, — v, after a small
difference of 8824, the differences begin again to increase. This has always in the
previous cases pointed to a collateral displacement in D(oo). The first object is to
see by what displacement the denominators may be brought to a limiting uniform
value. If £ be put - 13, the denominators for m > 5 become 715818,71 4394, 7 1 6 1 98,
715743, 7222, 7319. Omitting the values for m = 10 and 11, in which the probable
errors are very large, it is clear that, allowing for quite reasonable observation errors,
the denominators are in the neighbourhood of a limiting value. Now, a collateral of
( + 2(5,) in D(oo) produces a displacement of — 13'48, and this makes the denominator
366 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
for m = 6 to be 716474, and this is 23931 below the observed denominator for D2(5).
Now, 50S = 23850, which is the same as the difference for m = 4 and 5 instead of
being less, as analogy with others would lead us to expect. But the observation
errors are large (maximum of order 18<5, about) so that room is allowed for this.
There would then be one step from m = 5 to 6 of something less than 50(5 combined
with a collateral displacement of ( + 2^) on D ( oo ). To indicate the explanation this is
entered in the table, a difference of 50S making the denominator for D (6) = 6716653.
The observed abnormality as between Du(6) and D21-i/ = D12(6) is that the wave
number of Dn is 1'26 less than D12, whereas it should be 8 or 9 greater, with a
denominator about 32 S greater instead of 1721 less. There is, in fact, a further
defect beyond the normal value of about 36d Thus, if the difference for D2(5, 6) is
xS, that of DI (5, 6) is (x + 36) S. The lines would then be represented as follows :—
d, = 755767, d2 = 740503,
D,(m) =
The collateral addition intensifies D2 and explains its continuance, but in Dj the
increase, owing to addition of 2^ to D ( oo ), is overweighted by the diminution of the
excess 36S in the VD part, and so, after the first, the rest are too faint to observe.
At least, that is a suggestion of a possible explanation.
Tl.
In Thallium the satellite separations appear to be the following multiples of <^ :
24, 27, 28, 28, 28. But so far as limits of error permit, they might be 24, 27, 27,
27, 27. A peculiarity appears in the D lines in that the doublet separation for the
first set is 7793'08 (48), whereas the normal value is very close to 7792'39. The
difference is therefore real and not attributable to observation errors. Moreover, the
next four show a gradual diminution, although still remaining normal within extreme
permissible errors. The doublet values beyond this depend for the measurements of
the second line on measurements of CORNU. They give separations 15 to 20 less
than normal, but little reliance can be placed on deductions from them, for COBNU'S
results may err possibly by several units in the first decimal place, and with these
small wave-lengths any error in X is multiplied by 22 to 23 in the wave numbers.
I have, therefore, not brought them into the discussion.
The table is based on £ = '3, though no attempt has been made to find the best
value. The mantissa of the first line is abnormal, since it is less than the second
instead of greater, and, moreover, its difference from it cannot be a multiple even of S^
Since the satellite difference is very close to such a multiple (9048 = 6<S + 4) it is
probable that the abnormality affects both in the same manner. Now the arrangement
may be made normal by regarding the first line as a collateral (<!,) Dj (2). The
I)K. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SKRIKS. U67
addition of 5, to the denominator of D(oo) produces a change — 19"19, and this
changes the denominator of VD to those ' given in the table, and as is seen now,
produces a difference of 2^, tatween it and the next. Now, this alteration in D ( » )
diminishes the value of v by 5'65, whilst, as we have seen alx>ve, it is apparently '69
too much — or VDa(2) is 5'65 + '69 = 6'34 below the value of VD,,(2). Now, this is
just the change made by deducting 2<\ from the denominator of VD,,. The exact
value is 6"74, which is within the limits. The way in which, witli the considerable
numbers involved (i$, = 377), all the different abnormalities are simultaneously made
to fit in with a normal scheme gives some confidence that this is the real explanation.
The scheme of actual lines may be represented thus :—
Actual Du (2) = ( + £,) DM (2),
„ D13(2) = ( + <*,) D,,(-24,M,
Contrary to the case in other elements the successive differences are equal after the
first, and the limiting value of the denominator is reached at m = 7. They can all
from 7 to 14 be, within limits, equal; but there is an apparent rise with the high
orders.
O.
Two series— one of doublets and one of triplets— have been recognised in oxygen.
The table shows that the D lines of both sets fall into line quite naturally with
multiples of A, closely except in the case in the doublet sets of m = 7, 8
these cases the denominators are equal within limits, but much larger than those for
m = 6 instead of being less, and the deviation is real since the difference is more
than 15 times the probable error for m = 7, and T5 times that for m = 6, which
latter has a very considerable probable error "5 as against '07.* The divergence for
m = 8 can be accounted for, as it is probable that there are two close lines here due
to different series, viz., that for this series m = 8 and the other for a parallel
for which m = 5, and may therefore be stronger. As it throws some light
subject it may be well to say a few words about it here. RUNGE and
three lines at 626478, 6261'68, 6256-81, with separations 7'83, 12"43, and mtens
1, 3, 1, so that the centre is the strongest. There is a corresponding s
5408-80 5405-08, with the same separations within error limits and mtensit:
again with the centre strongest. The strongest lines of these two triplets forn
series with the observed value of D" (8). They are of a diffuse type and
* These are not to be confounded with K.R.'s possible errors. The po«ible errors are probably
larger.
368 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
come between the D'" and D" series. The limit taken is 22926' 11 and the scheme is
as follows : —
O - C. O.
3'969545-8 .... '01 '03
6A,
4-968512 + 7 .... 0 '04
6A!
5-967480 + 24. ... 0 '5
Moreover, the difference between the first denominator and the corresponding one for
D'" is 2924, and this is 17Aj. It is of course understood that the digits "11 in the
limit have been chosen so that the l7Aj, 6A1; come very close.- The argument
depends on the possibility of doing this. In fact RYDBERG'S tables give the limit
22926, so that the modification by '11 is extremely slight. Thus the observed line is
the line corresponding to m = 5 of this series, and it probably hides the weaker line
of D" (8). This accounts for the deviation noted above between calculated and
observed in D" (8). I have no explanation to offer for the corresponding deviation
for m = 7. All the others come so close that it is difficult to imagine that this does
not fall in with the rule. It is equivalent to an error in X of about 1'2 A.U. The
doublet separation for D" is "62 very closely, and the corresponding doublet difference
is 15(5j = A say. A lateral displacement of 7 A on the limit would just make the
change, but that explanation seems out of place here. The separations 7' 8 3, 12 "4 3
of the new lines require denominator differences in the limit of 373 and 473, and
4A2 = 380 and 5A2 = 475. There is another line at 6267'06, showing a separation of
5 "81 ("3). If this has the same VD as 6261 it requires a denominator difference in
the limit of 277 and 3A2 = 285.. The four lines aj-e therefore (-5A2) (6261),
(-3A2) (6261), 6261, and ( + 5A2) (6261).
S.
If RUNGE and PASCHEN'S estimates of their errors are valid the value of the limit
of the S series is determinable very accurately. It is 20085 '46 (l'34), but to bring
m = 7 as calculated within limits it is necessary to take S ( oo ) more than 1 less.
Accordingly the D lines have been calculated on the supposition that D ( oo ) = 20084 '5,
and it cannot be far from this. To bring the differences within multiples it has been
necessary to diminish this limit by putting £ = -'3. The multiples then come in
partly as multiples of A2 and partly of A,. The value of A2 given in the first part
of the paper is 651, but this gives A2 = 35 x 180'67w2, whereas it should be, if
the rule there established is correct, in the neighbourhood of 180 '9, or -^ larger,
say 6517. This value has been adopted in the table, although the old one can be
made to fit in though not so well. The agreement is good, especially when it is
remembered that K. and P.'s estimates are less than possible errors,
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
Se.
The lines allotted by RUNGE and PASCHEN to the D series present a quite different
appearance from the normal, although there can be little doubt but that they form
the SeD. The weak satellite lines after the first appear on the violet side of the
strong lines, whereas in all other yet known cases they lie on the red side. Moreover,
the strong lines instead of standing by themselves are each the first members of
complete triplets for m = 4, 5, 6, 7 (m = 4 is the first set observed). The numbers in
the table are calculated with D(oo) = 19274. The value of S(<») calculated from
(4,5,6) is 19275*10 (2'4), but it requires, as in S, a further diminution of over 1
(i.e., within error limits) to bring in the fifth line. Hence D(») cannot be far from
19274. The value of S is calculated from A, + Aa = 161 x90-40w*, and consequently
must be considered very exact. A, and A2 are calculated by transferring 15w* from
calculated A2 to A,, making the values 28ix36r62ws and 12 x 361*61^. The
numbers calculated fi-om the observed values are given in a separate list. A glance
shows that the usual regularity is here quite upset, and one feels convinced that
some disturbing influence must have been at work. If we examine the wave numbers
of the first four sets as exhibited in the following table, we notice that for the first
J~ (14149-27)? (3)14252-84 (3) 14300'31
:41(5) 14156-19 (3)1425976
f(6) 15803-95 (3) 15907'80
5 1(1) 15804-95 (1)15908-67 (4) 15953'89
J(5) 16768-10 (1) 16872-15
31(1) 16769-17 (1)16872-67 (3) 16917"33
f(7) 17375-92 (l) 17482'33 (3) 17523'15
= 7 1(2) 17379-58 (7) 17483'00 (4) 17527'20
set, we should expect a weak satellite about 6'92 behind 14156, which is not likely
to have been observed in that region far in the red. Its difference, 3129, is close 1
55^ = 3118, and provisionally we may regard this as normal The next two triplt
(ro = 5,6) give separations respectively TOO and 1'07, corresponding to a lateral
displacement in D(») of S, (S, actually gives '925 displacement),
separation is 3'66 corresponding to a displacement of S (4*. gives 3*7).
for this line the intensity has increased from 5 to 7 and gives suspicion of a d
ment by addition. If we suppose that the chief lines have a lateral displa
( + J)D («) it means adding 370 to their wave numbers, i.e., they are now
and 16771-80, and they come 27, 2'63 in front of their satellites, which, allowing i
errors in observation, is in fair order with the first separation 6'95.
only is the strong line abnormally more intense than for m -
VOL. ccxur. — A. 3 B
370
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
also — which suggests they are both displaced — a suspicion increased by the abnormal
increase of the difference shown in the table of denominators between 6 and 7.
Provisionally the least change is to suppose the faint line displaced by $ and the
strong line by 2<S, as it must, as was noted above, be $ more than the faint line.
For m = 8 it is cxirious that only one line occurs and no triplets. This suggests
that there is no intensification by lateral displacement, and that provisionally it should
be taken as normal. The table of difference shows an abnormal increase instead of a
decrease, but this may be due to observation errors. If we now calculate the
denominators for m = 4, 5, 6, 7, on the above suppositions, displacing the lines for
m = 9, 10 also by S, we get
•626133(108)
•622797 (?)
•617055 (?)
•612509(480)
4'629262 (54) 3129
4326
S'624936 (56) 2139
4406
6'620530 (105) 3475
4216
7'616314(240) 3805
5256
8-611058(728)
3629
9-607429 (643)
9090
10-59833
Thus the changes indicated by the appearance and arrangement of the lines have
brought the denominators and satellites into greater accordance with the general
rule. The practical constancy of denominator differences is exhibited also in Tl. The
only outstanding irregularity appears to be the satellite difference for m = 5.
A lateral displacement of — <\ in D ( oo) would decrease the denominator by 743, and
increase the difference from 2139 to 2882. It is better to leave the difference
without an attempt of explanation at present.
The second list has been drawn up on this basis, taking £ = — '1 as the errors are
somewhat smaller with this value. The denominator for m = 10 is left without
further change. Another displacement of 2^ would bring it also 19S below that for
m = 9, but the observation errors render any deductions quite unreliable. The
DR. W. M. HICKS: A ClilTICAL STUDY OF SPECTRAL SERIES. 371
suggested scheme of actual lines may therefore be represented as follows where
Dn, DH stand for the normal type, and D,,(m) = Dn (m) (-55J,) :-
D., (4), D13(4)
( + ^)DU(5), Du(5)
( + <*)DU(G), Dw(6)
( + 2J)Dn(7),
Dn (8)
The order (4) of the first line is so large that the error limits are too wide for absolute
certainty. In fact better agreement on the whole for the satellites would be
obtained by taking the difference as 56^, i.e., 4<?u, <JU being specially associated with
this group (see p. 331). The line 6269'28 is separated from G2G6'36 by 6'44, and is
therefore possibly the lateral ( + 2<S) D,, (5).
The table shows of course the known essential difference between the liehaviour of
the elements of Group 2 and that of Groups 1, 3, G, signified by the signs of a in the
formula. It consists in the fact that in Group 2 the orders are formed in succession
by the addition of multiples, whilst in the others it is by subtraction, with the
exception that Cu and Ag of Group 1 are additive. But there are certain other
features which appear between the different sub-groups when higher orders are
looked at. The alkalies all show a gradually decreasing decrement with a sudden
dive. Na then shows a sudden rise continued for several lines, and Cs has a similar
indication. Cu and Ag with only a few lines observed show decreasing incrementa
The alkaline earths show decreasing increments and a sudden dive (Mg excepted).
The Zn sub-group shows decreasing increments and then a sudden ascent. The Al
Sub-group 3 show decreasing decrements (Sc decreasing increments). ( ) with S and
Se show decreasing increments. In fact, were it not for the very clear behaviour of
Zn, Cd, and Hg, the evidence would rather point to the conclusion that in each
group, the low melting-point sub-group show subtraction (at positive) and the high
melting-point addition (a negative). If this series depends on a formula sequence, it
is difficult to see how it can be any simple algebraic one — the mantissa would rather
seem to depend on a term similar to sin ma or tan ma. In the detailed discussion
above, however, it is seen how these changes of direction can be explained by lateral
displacements. It is noticeable that where the irregularity observed in the first lines
as compared with the others in the satellite differences appears, a similar irregularity
exists also in connection with the first order differences. This is evident especially in
the alkalies, where the first differences are so close to exact multiples of A or & as to
cause the conviction that they really are so.
3 B 2
372 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
It is a remarkable fact also, and one which will probably be of importance in
throwing light on molecular constitution, that all those elements which do not
exhibit satellites have order differences depending on multiples of A, whereas all the
others (Al excepted) depend on multiples of the oun, S or <\. The elements without
satellites are Na, K, Mg, possibly Al, both series of O, and S. All these depend on
multiples of A2 or At. None of the others do so, and it may be regarded as an
argument in favour of Kb possessing satellite series that its differences do not depend
on A directly. It would appear that Rb only begins to show them for m — 3. For
m = 2 the line is not split up into a chief line and satellite, the doublet separation is
normal, and it is instructive to observe that the order separation between the first
set and second line is close to 5 A, and only deviates from it in the same way that is
mentioned in the previous paragraph. Also Ba seems to have in the same way no
satellite for m = 2, the separation is quite normal, and this also shows a first order
difference very close to Aj.* But Ca, on the contrary, which has a first difference
= 99A2, possesses satellites.
It is noticeable also that the high atomic weight elements appear to follow more
regular and simple rules. Thus both Cs and Tl show descent by equal steps in both
cases = 3<^.
The result of the discussion would seem to be that there can be no doubt but
that satellite differences as well as the doublet and triplet differences depend on
multiples of the oun. For the other supposition, viz., that in the Diffuse series the
order differences also depend on differences of the oun, it can only be said that a case
has been made out. The supposition in all cases fits conditions, but the conditions
are not all sufficiently definite to give certainty. After the first two or three orders
the observation errors are larger than the Slt and even for these the value of Sl for
the low atomic weights is comparable with the errors. In some of these cases,
however, multiples of A which is much larger enter and strengthen the argument.
The strongest examples are those of the alkaline earths (small errors and large A or
Sj}, first lines of Cd, and Hg, Al (series in A), In, Tl, and the A series of O and S.
The D (2) Term. — If the foregoing theory of the constitution of the Diffuse series
is correct, it is further necessary, in order to complete the discussion, to determine
the origin of the first term. The apparently close relation of the F series to the D
series, and the several cases of collaterals of the former which had been noted with
large multiples of A, suggested a trial to see if the denominators were multiples of
this quantity. As in cases where satellites are present, the separations depend on
them and not on the strong line, it is natural to expect that the satellite is a normal
line and the strong line a collateral, and this is found to be justified by the calcu-
lations on this theory. In Table III. the first column of figures gives the value of
the denominator taken from Table II. The second column gives the factors together
with possible variations. Thus the denominator of KDU(2) = '853302. This has
* But see discussion of BaF below.
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 373
TABLE III.
Na
Dn '988656
•
K
Dn -853302 = 291 (2932 -27 ±-130- -3640 = 291A 361-944
362-68
Rb
1D12 -766216(t)-8S = 59(12950'84±2-74- 1-44$) = 59A 361-991
-108 = 59(12942-06) a^.?^
361-40
Cs
D12 • 54698 9 = 857 (638 • 260 ± • 233 - • 0887 ft = 8676 36 1 • 786
Dn -554286 = 17 (32605-06± 13-41 - 4-47?) = 17A-10S
361-74
Cu
Ag
> Theory of constitution uncertain.
Mg
DI -828688
362-36
Ca
Dig -945972 = 691 (1368-99 ± -03 - -047£) = 691Aj
361-84
361-84
Sr
DU -987349, not a multiple.
D12 -989572 = 178 (5559-39 ± -25- -20£) = 178A.2 361-738
360-02
Ba
D13 -825511 = 69(11963-9± 1) = 69A2 361-968
*D18 1-041954 = 87(11976-4) = 87A., 362-352
362-34
Ra
Not observed. = 31A8.t
Zn
D]3 -904978 - 260 (3480- 68 ± -38- -430£) = 260Aj
361-682
362-26
Cd
Dn -902039 = 87 (10368 -26 ± -26-1 -276£) = 87 ^
DIB, not a multiple.
361-382
361-392
Eu
D1S -917794 = 50(18355-88± 1-2-28£) = 50A,
361-44
360-93
Hg
D13 -921662 = 31(29731-0±l-22-3-80£) = 31A,
361-50
362-46
Al
Dn -631287 = 360(1753-575+ -069 - -230£) = 360A
D12 -631181 = 360 (1753- 280 ± -069- -23o|) = 360A
361-777
361-717
361-879
In
D12+16S = 22(37676±2-18±4-619£) = 22A 361-871
361-947
Tl
D12 = -888344 = 590(1505-667 + -136 - 1 -881£) = 590S 361-650
Dn coll. = -899520= 597 (1506-73+ • 136 - 1 -881£) = 5978 361-913
362-063
0
D'" -972483 (40) -120^
46(171-66±?) = 46A,
D" -980380 (92) -121£
63(172-063) = 63A,
NewD -969543
362- 46 ±
363-308
361-79
S
Di(4; -553446 = 530 (1044- 24 ± -088 - -811^) = 530A,
362-113
362-14
Se
Di(4) observed -629262 = 99 (6356- 18 ± -64 - 4'565£) = 99A,
361-893
364-00
* See below under discussion of BaF — two different triplets in question,
t See below under discussion of RaF.
374 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
observation errors and also possible error due to incorrectness of D(<»), i.e., £
These give the denominator as 291 (2932'27±'130-'364£). Now 2932'27 is very
close to 2939, which is given as the approximate value of A in Table I. The
denominator is then written 291 A, and with this new value of A the corresponding
value of the oun is calculated as 361'944wa instead of the old value 362'68w2, which
for comparison is entered next to it.
Notes on the Tables.
Na. A is so small that several multiples of it might be taken for D within limits.
Eb. DI does not give a multiple of A although close to it. If, however, Rb has
satellites, the denominator for D12 will be a few multiples of S less than that of Dn.
That for Cs is 1 l^S less in the corresponding case. The values in the table are given
for 8(5 and 10<1 Judging from the value of the oun it is probably near 8<J. In any
case the multiple would be 59A. This is a very strong argument that lib does
possess satellites.
Cs. Neither Dn or D12 give multiples of A.
Mg. As in Na, A2 is too small, and- the denominator too large to give anything
definite.
Sr. Dj3 gives the oun clearly too small, although better than in the original table.
D12> however, gives a value 361738 quite close to the probable value. A similar
result is shown also by Cd which occupies an analogous position in the next
sub-group. If D13 behaves in what appears to be the normal manner, it would
appear necessary to take the atomic weight to be '10 less than BBAUNER'S value, viz.,
87 '56 in place of 87 '66, which is probably too large a change to be acceptable.
Ba. In Barium the first set is doubtful. That taken above shows no satellites.
The denominator is therefore that for D13, and this is a multiple which gives a value
of the oun much nearer the probable value than that in Table I. Evidence will be
given later however under BaF that there is a normal satellite triplet, outside the
region of observation, where D13 has the denominator 2 '04 19 54, which, from analogy
with the other elements of this group, has a " mantissa" 1 '04 1954, and this is again a
multiple of A2. This, therefore, is probably the correct value, and the other set will
be collaterally displaced from this by 18A2.
Ra. The first line should be far in the ultra red, and has not been observed. The
multiple 31A2 is determined indirectly (see BaF below).
Cd. This element shows the same irregularity as in Sr, in that DJ3 does not give
an exact multiple of A2, although one close to it. Here, however, we have to go
back to Dn before finding the exact multiple, and Dn gives almost the precise value
of the oun as in Table I., which was itself very exact.
Eu. D13 is '917794, which is 50A2+1344, A2 having the value 18329 of Table I.
If it is 50A2 exactly, A2 would be 18355'88-2'28£ making the oun (361'46-'04£) in
place of 360 '93 of Table I., a great improvement.
DE. W. M. HICKS: A CRITICAL STUDY OF SPFXTTRAL SERIML J175
D13 gives denominator = 31 x 29731 '0, and the latter factor is
7410-87«r' = 82x90'375wa,
which is much closer to the probable value of the ouu, and moreover 82 is the correct
rmiltiple to give 54 x 543'816w3 for A, + A^ which has been taken as a basis for 3.
This value of A is supported by the discussion of the F series below.
Al. As the order differences are all multiples of A, and there may therefore be
some doubt as to the real existence of satellites the values for D,t and D,, are
inserted. The denominators for the two only differ by 4<J = 108, or the olwerved by
96. As the A differences can only refer to the Dn set, it would seem that these
should be taken as the normal lines giving 361777 as the value of the oun.
In. Neither I),, nor D1:J are exact multiples of A although they are very close to
22A. D,2 is 1722 x 477'1 1, or 1723 x 476'83. If these be taken as multiples of the
oun, they give the oun as 362'01 and 36T80 in place of 36T94 of Table I., but the
multiples are too large to found any conclusions upon. It would rather seem that
there is some displacement from a typical multiple. Using A as given in Table I.,
viz., 37684, 22A = 829048. So that Dw = 22A-7455 and 7455 = 16f$-177. If it
is 22A — 16(5 exactly, A becomes 8'04 less and the oun 361 '87 lw* in place of
36T947. The value of D,.j+16«5 is therefore inserted. If the typical term were IG&
higher, the order differences would run 72<5, 62S, 508, 50<S, in place of 58$, 62<5, &c.,
and hence more in line with others.
Tl. Neither the observed nor the supposed collaterals are multiples of A. They
are expressed as multiples of S. Although they are large multiples, th«ir values are
quite definite provided we know a priori that the denominators are multiples as
a fact. If the multiple be altered by unity, the resultant quotient cannot come
within the limiting values of the oun.
If the normal D,., (2) = 7A = 939078, the order difference over DIi((3) would be
939078-888643(89) = 50435±89, and 33^=134^ = 50495, so that the order
difference would come out as usual a close multiple of <V All this group seein to
show the same kind of irregularity.
O. There are three separate series, see data for Table II., differing by multiples of
A!, just as iii the order differences. A, is too small to test the multiples of the
denominators themselves.
S. The D(2), D(3) lines for S and Se are beyond observed regions. Sulphur
however shows no satellites, and we may surmise therefore in analogy with others
that the differences for D (2), D (3), D (4) are like the others multiples of A, or A,.
As a fact, D, (4) is a clear multiple of A,, and the surmise is justified so far as A, is
concerned. The value of £ is not very certain.
Se. Se apparently has satellites, and the order differences are only multiples of S.
It should not therefore be expected that D, (4) or D3(4) should be a multiple of A,.
Nevertheless D! (4) is clearly such a multiple and is entered in the table.
376 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
The table shows that where triplets occur the multiples are those of A2 and not of
A,, except in the case of the oxygen group of elements, in which A! clearly takes the
place of A2. If the law of multiples is correct, the values of the A obtained in this
way must clearly be far more exact than those obtained direct from the separations.
A glance at the deduced values of the oun compared with the former values shows
how much closer to the mean value 361 '9 the new ones are than the old, and to some
extent this adds to the weight of the evidence. The cases where the multiples do not
appear to enter are those of lib, Cs, In and Tl. The case of Rb has been considered
above and a natural explanation offered. Cs, In and 'I1! have all large values of S, in
which case we have already seen a tendency for the spectra to depend on smaller
multiples of the oun than the A. In the case of Cs, the oun is smaller than the
multiple and it can give no evidence nor data for the oun. The case is different
however for Tl. If the oun enters, the multiple can be no other than that given,
and as is seen the value of the oun is improved. All the elements of the Al group
show a deviation from the normal type in that the first satellite separations are much
smaller for the first order lines than for the second, and seem to point to some
displacement. As the Al orders differ by multiples of A, any irregularity in the
multiple between the first and second orders does not alter the dependence of the
denominator on the multiple of A. In In and Tl, however, the differences go by
multiples of § or Sl} and any irregularity on them will throw out the dependence of
the first denominator on a multiple of A. As was shown above the addition of 16(5 in
In not only produces the multiple, but at the same time shows a more usual march of
differences for the orders. In Tl the observed denominator for D12 (2) is less than that
for D,2 (3) and quite abnormal. The other anomalies occur in that in Sr, D12 appears
to take the place of D13, and in Cd, Du. RaD (2) is in the ultra red and has not been
observed. The elements Na and Mg must be left out of account because the
ratio denom./A must be so large that a number of multiples can be found all giving A
within observation limits. Cu shows a multiple, but the theory of the constitution of
the series of Cu and Ag is doubtful and must also be left out.
With the above doubtful cases the values for K, Ca, Sr, Ba, Zn, Cd, Eu, Hg, Al, S
and Se, are clearly exact multiples, and the large values of A in Ba, Cd, Eu and Hg
show that these multiples are real. This rule, exhibited as it is in so many cases, and
in by far the majority of the elements comparable, must correspond to a real relation
and cannot be due to mere coincidence. Against the reality of the relation is the
antecedent improbability that those elements with the smallest value of A should
have the largest values of the denominator, as e.g., in the case of Na and Mg. A
possible explanation is that the mantissa is the nearest multiple of A2 to some group
constant. But see also under discussion of the F series. It might however have
been expected on this ground that the denominators would be of the form l-M(A).
But the case of Na is clearly against this. Its denominator "988656 = 1 — '01 1344
and 11344 is 15'26A and cannot be a multiple. It would seem conclusive that the
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SRKIKS.
377
denominators of the extreme satellites of the first line are multiples of A, or A, and
that explanations should be sought for apparent exceptions.
The S and P Series.
The relationships between the doublet and triplet sets of the P series and between
the S and P series were discussed in [II., p. 51] by comparing the differences between
the corresponding denominators. It is now possible to see how, if at all, these
differences are related to the oun.
The P Series. — In the alkalies the differences between the corresponding
denominators of the two sets were found to be constant within error limits and
of course equal to A. In the other elements in which the P series have been
allocated, there was always a drop in the difference, which in several cases then
remained constant for the succeeding orders. The values were given on [II., pp. 51-53].
They are reproduced here, and it is seen at once how they proceed on quite analogous
lines with successive satellite differences of the D series considered above. The
possible eiTors of the single lines from which they are deduced are given in brackets.
Thus ZnP(l) are 1'599352 (2), F592143 (3), 1'5886G9 (4), and the differences are
given as 7209 (2, 3), 3474 (3, 4). The higher orders, in which the possible errors are
so large as to be themselves multiples of Slt are not included. The value of <$, is
given with the symbol for the element.
It will be noticed that the more accurate the observations the closer are the
differences to the multiples of the oun. But the observed variations from true
multiples in the case of the large separations would seem to point to a difference
in the a as well as in the M- Iu any case it would seem that n must alter
per saltum from order to order, unless the sequence formula is a complicated
function of m.
Zn (S1 = 3875).
, = 113-8).
7209 (2, 3)
128-5
5355 (19, 28)
8 + 9
5191 (9, 9)
8,-!
5154 (30, 30)
3474 (3, 4)
68 + 20
2525 (28, 36)
38! -5
2414(9,9)
«i
2375 (30, 30)
87815 (?)
118-118
71967(?)
28, -122
71364(1)
23109(4,5)
508, -5
17423(25,25)
18269 (922)
17109(26, 28) J
38, + 27
10368 (5, 5)
308, -26
6980 (25, 25)
6461 (1837)
6407 (28,25) t
58, +4
VOL. CCXIII. — A.
Hg («*, = 362-87).
30002 (1)
148, + 143
24779 (1)
38, — 126 or 28, + 237
23817 (t)
3 c
378 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
Note how Zn still affects <?6. The variations from multiples in Hg seem to have
relation to the transference properly noted above, viz., from Aj to A2.
Al (J, = 26'57). Tl (,?, = 376-835).
1751 (1) 134154 (?)
16S,-3 5382 + 191
1329(78,78) 94018(1)
-25t + 5 23-191
1377(23,23) 91195(?)
0-3 2S,-173
1380(48,48) 90615 (?)
In Al the value after the first is 1381 within limits of error for all.
The S and P Connections. — The differences of the corresponding denominators in
the S and P are also given in [II.]. The values are, however, subject to uncertainties
due to uncertain limits in both S and P, in which the £ are not the same for both.
In the case of the alkalies there seems a very clear connection with the A, except in
Cs, where as often before S enters. In the other elements it was shown that the
sequences are inverted and the differences are to be taken between the first of the S
and the second of the P. In Al, Tl, and Zn, there is again a clear relation, but it is
now to the denominator differences of the Pj (2) and P2 (2), or 1329, 94018, 2525
respectively, say A' for each. In the case of Cd and Hg no clear relation is apparent,
although they behave approximately like Al and Tl. This want of exact agreement
may be due to the effect of the transference inequalities considered above (p. 333)
in connection with the oun. The relations indicated above are shown in the following
scheme, in which the differences for the S and P are taken from [II., p. 51-53].
Na . . . . '490162 = '5-13A,
K . . . . '464597 = '5-12A,
Rb . . . . '487501 = '5-A,
Cs .... '491944 = '5-14(5 (roughly).
Al .... "489330 = '5-8x1334 = '5-8A',
Tl . . . '594887 = '5 + 94887 = '5 + A'.
' Zn .... '528306 = '5 + 11x2573 = -5 + 11A',
Cd . . . . '526358 = '5 + 26358,
. . . . '603628 = '5+103628.
DR. \V. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 9ft
The F Series.
In Part I. the symbol F was used to denote the series whose limit depends on the
values of VD (2) in a similar way to that in which the limits of the S and I) series
depend on VP(l). Where the D writ's show satellites tin- F series in ronsei|ueii(v
consist of doublet or triplet series with constant separations. They comprise some of
the strongest lines in the respective spectra, but as in general they occur in the
ultra-red region they have not received the same attention as the other better known
ones. In the alkaline earths, however, they come well within the visible regions, and
show strong sharply defined lines. They are related also to other strong lines by
collateral and other displacements depending on considerable multiples of A, and so
naturally come under discussion in the present communication. As will be seen later,
the discussion gives the means of obtaining very accurate determinations of the A—
and consequently of the oun — as well as of settling other questions. I propose,
however, not to attempt an exhaustive discussion in the present communication,
partly because the main object now is only to illustrate the influence of the oun, and
partly because it would seem that a large number of lines which clearly belong to the
F cycle are related in a manner neither ordinal nor collateral, nor according to Rm's
combining theory.*
For convenience of reference the wave-lengths of these lines are given in the
Appendix, together with short historical notes.
The Alkalies. — The table below gives the denominators for the two first lines in
each as calculated from PASCHEN'S and from RANDALL'S results. BERGMANN'S measure-
ments for other lines are too much in error for the present objects. The limits used
are the calculated values of VD, using the limits D(oo) given in Table II. above and
the values of D (2) in the Appendix.
Na. K. Bb. Cs.
3-997919 (169) - 219& 3 -992817 (252) -290& 3 • 987849 (433) - 289£, 3-977334(146)-
4 -997267 (2845) - 569£, 4 -989237 (696) -566$, 4 • 983697 (846) -564£, 4-9698
5-9710
6-9642
The question that first arises is, tlo these refer to actually the first lines of the
series? If, like I), the lowest value of m were 2, the wave number* of
would be somewhat above, Na, 0 ; K, 1200; Bb, 2060; Cs, 4450.
be outside, but the others come within regions observed by PASCHEN.!
He gave for K lines at wave numbers 1346'3 and 118:
* The relation is extremely common in certain type, of spectra, e.g., the rare gase. other than Ho. I
hope to return to this in a future communication.
t "Zur Kenntnis ultraroter Linien-spektra III.," ' Ann. d. Phys.,' 33 (1910), p. 1
3 o 2
380 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
denominator 3'007542 and the latter 2'987479, the one apparently too large and the
other too small to fit in with the progression of the lines for m = 3, 4. But the
mantissa for the latter is within limits 2(5 below that for m = 3 [see Note 3 at end].
In Rb there is a line at wave number 2129'0 which would require a denominator
2'997805, well in step with the other two. PASCHEN identifies it as Di2(3) — P2(4),
assuming the existence of satellites in RbD. It would seem to be more probably the
F(2) sought for. There is another line given as 2156'! or 2164'4. If the former is
the more correct it gives denominator = 3 '00 11 38. In Cs no line appears with wave
number near 4450. There are two lines, however, with wave numbers 3409 '93 and
3321 '37 which differ by 88'56±'6, and certainly suggest the doublet series depending
on the D(2) satellite. This requires a separation of 97 '96, and if they belong to the
F series there must be a satellite with a separation 9 '40 ±'6 which we should not
expect to observe as being too faint. The lines give a mantissa 2 '8 5 1708 with a
satellite difference 1003. The latter may be, within limits, 2<5, a value which in the
alkaline earths seems to be closely associated with F satellites. But the mantissa is
less than that for m = 3, when a larger value should be expected. Even if not F (2)
itself it may be related to the F cycle in a similar way to certain displacements found
in the alkaline earths (see pp. 383, 413), and it should be noted that if so there
seem to be lines in corresponding positions in K and Rb. They are (in wave numbers)
the 1182'9 referred to above for K and 1911 '05 in Rb. The latter requires a
denominator 2'971391. In this connection it is interesting to note that PASCHEN
makes the remark that this line at times shows itself double. The separation
calculated from his numbers is 1'12, giving a denominator difference of 107 for
F](oo) and F3(«>), i.e., for VDn (2) and VD]2(2). This would indicate a sort of
incipient satellite in RbD. These considerations seem to show that there is some
likelihood that m = 3 does not give the first line of the F series, and they will be felt
to have greater weight when the curious irregularity in the F(2) of the alkaline
earths to be noticed immediately is taken into account. The question is further
discussed on p. 397 in connection with the other elements.
The next question is, is there any indication of F satellites in the accepted lines ?
If so we should only expect to find it in Cs. Now RANDALL gives weak lines 8080'9
close to 8083'!, Fj (4), and 8018'9 close to 8020'6, F2(4). They look like satellites
only on the wrong side. The first changes the denominator by 2000, which is within
easy limits of 1914 = 3^. It will be shown that this is a common satellite difference
in the alkaline earths. Further, it makes the denominator 4'9718, thus bringing the
values for m = 3, 4, 5, 6 in order, which is not the case in the table above. There is,
therefore, something to be said in favour of taking the normal F (4) doublet to be at
8080'9, 8018'9, and that that is then collaterally displaced by 3<5 to the stronger lines
8083'!, 8020'6. It is also quite in keeping with analogy in the alkaline earths that a
similar displacement is not shown in the case of the first lines F(3) (if F(3) are the
first lines).
DR. W. M. HICKS: A CRITICAL STUDY OF 8PFXTTRAL SERIffi.
381
Group II. The Alkaline Ea/rths. — The series are most fully and regularly developed
in Ca and Sr. In Ba and Ra the configurations which give rise to the nonnal type
seem to be so modified that displaced lines become common, and in cases the normal
line has disappeared. On the other hand, Mg seems to range itself with the Zn
sub-group. It will be best therefore to deal with Ca and Sr first, and as they are
built on a precisely similar plan to consider them together.
The following table gives the wave numbers of the series together with certain
others which are clearly similarly related in the different elements. The separations
are indicated by thick figures. The wave-lengths are given in Appendix II.
Ca.
16203-40 21-75 16225-15
16204-72
21799-02 21-13 21820-15 13'58 21833-73
24391-49 21-50 24412-99 13'66 24426-65
25793-67 21-64 25815-31 13 "55 25828-86
26634-00 22-43 26656-43 14-29 26670-72
27177-76 21-58 27199-34 15-09 27214-43
Sr.
14801-57 101-73 14903-30
15046-98 61-70 15108-68
20530-76 69-78 20590-54
20432-18 100-64 20532-82
20435-10
23045-78 99-26 23145-04 58'54 23203-58
24457-06 100-05 24557-11 59 25 24616-36
25303-32 100-47 25403-79 58 '61 25462-40
25850-61
Analogous Sets in Ca and Sr.
17847-46 21-94 17869-40 13'86 17883-26
17887-55 21-81 17909-34
18968-53 21-58 18990-10 14'01 19004-11
Ba.
13089-79 260-60 13350-39
13471-69 259-76 13731-45 157'55 13889-00
13477-44
18686-80255-17 18941-97
21308-19 252-51 21560-70 148'32 21709-02
18061-87 100-20 18162-07 56-61 18218-68
18239-35 100-62 18339-97
19016-64 100-34 19116-98 59'75 19176-73
Ra.
17793-09
17300-80 699-93 18000-73
22037-41 456 64 22494-05
21350-92 701-89 22052 -«1
22706- 84'"
23667-07™
22979-57
23919-27°" 23995-83
"> F12(5)(A.2). *F,(6)(9A,).
382 DK. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
It is clear that these lines also show satellites. Also, it is curious that the first
sets of triplets apparently have the lines corresponding to the second separation
displaced below those forming the first. Thus in Ca the second set (giving i/2 = 13'5)
have not been observed, in Sr the two (Vl = 101, i>2 = 60) are separated by a gap of
143, and a similar effect will be found later in Ba. Owing to this fact, the formulae
constants are calculated from the 2nd, 3rd, and 4th sets. They give for Fn —
Ca 28934-93-N/(m + ''
Sr 27612'37-N/(m + '875560 +
•37-N/(
m / '
•100548V
\ m
These give the following values of O— C : —
m. 2. 6. 7.
Ca . . . -1'18 '18 '22
Sr . . . • '42 '05 '02
The agreement is good, except for m = 2, and in this case the agreement is sufficient
to show that the allocation for m = 2 is correct.
The limits are close to those of VDU (2), which is not known with great exactness
because the values of S ( oo ) or D ( oo ) given in [II.] for the second group are subject
to possible errors of some units. With formulae in ] /m the values ofD(oo) = S(oo)
are given [II., p. 36] as 33994'85 for Ca and 31037'27 for Sr, whilst with formulae in
1/m2, the respective limits are 33983'45, 31027'64. The values of VDn deduced
from these are- respectively 28939'93, 27615'65 with 1/m and 28928'53, 27606'02
with 1/m2. The limits, therefore, found above for Fn ( oo ) lie each between their
corresponding values as deduced from the D series direct. Assuming that the F ( oo )
are more accurate, the values of D (oo) deduced from them are 33989 '8 5 for Ca and
3 1033 '99 for Sr, in both cases close to the mean of those in [II.]. If the series
depend on formulae sequences, these limits may be taken as close to the correct
values. If, however, the different orders proceed by multiples of S or A in the way
illustrated in Table II. for the D series, the limits may require modification by a
few units.
As the separations of the F series depend on the separations of the satellites of the
first lines of the D series, and these depend on displacements by definite multiples
of S, as given in Table II., it is possible to calculate the values of the former with
extreme accuracy. Table II. gives 13(5 and 8S as the multiples in question for both
the elements Ca and Sr. Using the values of S and of the denominators of Dn there
given, the separations in question calculate out to 22'49, 13'75 for Ca and 100'34,
62 '01 for Sr. These may be regarded as exact to the 2nd decimal place and
independent of any possible variation of £
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 383
In the case of Ca the observed lines, with differences somewhat less than 22'49 and
1375, seem to indicate the presence of close satellites. If 16203'40 is really Y^(2),
the separation of 'the first satellite is 1'32, with possible errors ('26±'2G), which form
a very considerable proportion of the total amount. A displacement of 3<5 produces a
separation of T51 and it may be this. But 16203'40 has an excessive intensity for a
satellite line, viz., 6, as against 4 for Fn, and, moreover, it may possibly be the
collateral Si (2) (- A2) which gives O-C = '03 with 0 = '10. If the latter allocation
is correct, it would hide FJ2, which should be 16225'15('26)— 22'49 = 16202'6G ('26),
giving a separation of 2'06 ('52) due to 4<$ which gives 2'02. The same considerations
applied to the second set give a separation of 1'36 for the first satellite, in which
again 4<! gives 1"26, and '22 for the second, S giving '32. The separations are so small
that no certain conclusions can be drawn as to their origin. The actually observed
numbers may be due to 4(5 and S, but 3S and 2$ are just possible [but see Note 4].
For Sr the first doublet is useless, as the line is due to the early measurements of
LEHMANN, which are affected with considerable errors. The observed separation for
Fn and FJ2 gives 10173 instead of something less than 100'34. The second triplet
gives 2 '92 for the separation of the second satellite from the first and 2 '06 for the
separation of the satellite of the second line of the triplet, and from analogy with
other satellite series, this would be the separation of the first and second satellites of
the first line. Differences of 3(5 and 2<5 give separations of 3'OG and 2'04, so that it
may be concluded that the satellites depend on these differences, a conclusion
supported by the fact that a similar result is indicated as possible for Ca.
Returning to the curious fact noticed above that the first triplets of the series
seem to be dislocated, the second fragment in Sr is found at a distance 143'68 below
its normal position. For the present we note this can be explained by one of two
possible collateral displacements, viz. (-18^) F (2) or F(2) (3A2), where F stands for
the normal F2 or F:). The case of Ba below will give evidence in favour of the latter
explanation.
In addition to the lines of the series itself, there are two sets of triplets and a
doublet which are clearly analogous in the two elements. They are given in the list
above, following the series lines. The first triplets in each are curious as having the
middle line the strongest.* They are also related to others in the way indicated in
the following scheme :—
(8) 17847*46
21-94
Ca (8) 17842'52 13'98 (8) 17856'50 12'90 (10) 17869*40
13-86
(8) 17883-26
* A similar peculiarity has already been noted in the associated OD series.
384
DR. W. M. HTCKS: A CRITICAL STUDY OF SPECTRAL SERIES.
(6) 18061-87
100-20
Sr (6)1804471 59'69 (8) 18104'40 57'67 (10) 18162'07
56-61
(8) 18218T>8
(10) 18968-53
21-57
Ca (8) 18985-31 4'79 (6) 18990'10 7'15 (6) 18997'25
14-01
(4) 19004'H
(10) 19016-64
100-34
Sr (10)19083-27 33'71 (8) 19116'98 15'25 (8) 19132'23 10'70 (8)19142-93
59-75
(4) 1917673
The first lines of the triplets give as denominators, supposing the true limits to
1st triplet
2nd
Ca.
3'145123(22)-141-8£
3-317300 (30)-166'4f
172177 + 30p-22g-24'6£
Sr.
3-388848 (28)-177'4£
3-572008 (23)-207'8£
= 33
= 126 (136878- '20£)-5c5
where p, q lie between ±1. Clearly the differences are the multiples 126A2, 33A2,
for the two elements respectively.
The first lines of the doublets give for Ca 3*150824 (22)- 142"6£ and for
Sr 3-420693 (18)-182'5£
These differ from the denominator of Fn (3) by
769567 + 123j9-22g-132'6^= 562 (l369'33 + '22p--04g- -236^)
= 88
Ca.
Sr .
That is, they apparently differ by 562A2, 88A2 respectively. We shall see shortly
that the best value for £ makes the relation for Sr very exact, whilst that for Ca is
more doubtful.
DK. W. M. HICKS: A CRITICAL STUDY OF SI'KCTKAL SKRIKS
The following list contains the wave numbers of certain lines related to the P series
in the two elements with the denominators appended for the chief lines :—
Ca.
a.
-15380-80
13-92
-1539472
-15447*35
21-75
A-15469'10
1-571602- 177£
B- 5011
•98]
2-141121
501474 j>22'71 2'141244- 447£
5034-69 j
10-06
504475
C
6171-18
21-12
6192'30
D 13133-01
20-04
13153-05
E 15580-59
20'66
15601-25
VOL. CCXIII. - A.
2-194987- 48'2£
2-634565- 23'3£
2-865778-107-7^
-31237-30
100-84
-31338-14
-31345-01
68-93
-31413-94
99-99
-31513-93
3 n
8894'66
1527172
99-09
15370-81
Sr.
1-363986-
1-361954-
2-420625-
2-981157
386
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
Ca.
Sr.
G
H
17847-46
21-94
17869-40
13-86
17883-26
17887-55
21-81
17909'34
18968-53
21-57
18990-10
14-01
19004'H
3-145123-141-8^
3-150824-142-6^
3-317300- ?
18061-89
100-18
18162-07
56-61
18218-68
18239-35
100-82
18339-97
19016-64
100-34
19116-98
59-75
19176-73
3-388848-177-4^
3-420693-182-5^
3-572008-207 '
I 21022-14 4-048761
59-91 or
21082'OS 1-499227
From these we find the following differences, m denoting the mantissa only : — •
Ca. Sr.x
a2—a1 3(5 within limits
mofF-mof«a '. . 5 (5378'80-33'2£)
mofD-wofA .... 46(l36875-l'43f)
mofG-mofB .... 7(136871-14^) 68-118^=0
m of Fi:!(3)-m of F . . . 562 (l368'82-'236£) 88(5549'81-r02£)
H-F 126(1366'48--20£) 33 (5550'30-'91£)
126(l36878-'20£)-55
D-C 32l(l369-40 + -51d\--110£)*
G-F13(2)t 158(1368'89--173^) 89 (5558'90-77f)
G-F12(2)t 158(1368'16--173£) 89 (5552'66-77^)
E-D 169(1368'12--144^)
Also the first triplet A in Ca shows the same kind of dislocation as in F (2) of the
other elements. The dislocation is 52'63, corresponding to a denominator difference
of 931, and 163 is 930.
* dXon 13133 maybe >1.
t FIS (2) as calculated from the formula.
J Allowing 28 for the satellite difference Fi2 - F]3.
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. -;
The number of the cases where multiples of A2 enter, as well as their a]i|N>aranou in
the corresponding position in the two elements where corresponding lines are olwerved
must produce a conviction that they represent real and not chance relations. In the
case of Ca it makes Aa close to 13687 corresponding to $ = 3617710*. If £ has any
but a very small value, the first two multiples are upset, but these may be due to
chance. If £ be made — 6'5 as suggested below, Aa will be about 1370'2 with
<£ = 362'15tt>3, which is considerably greater than the most probable value. The
probability is that £ can only be a small ± quantity, £ = ± 1 changing J/if* by ± '06.
A similar reasoning applied to Sr rather tends to show that here the value —6 '5 is to
be preferred. It makes the first multiple = 5x5594'60, and the other values of Aa
become 5556'44, 56'21, 57'66, and 5556 gives $ = (36 1 '52 ± "24) IP*, the uncertainty of
this being due to a possible error in the atomic weight of 87'66, with £= — 5'5 the
first relation gives A3 = 556T40.
Again the most probable values of the denominators of F13(2) are
Ca = 934539*-115'2£ = 937277-115'2f-2A:i,
Sr = 925946t-H4'0£ = 937060 -114'0£- 2 A^
The numbers on the right are practically equal. If analogous relations are found in
Ba and Ka it points to the existence of a group constant alxmt 937300. On the
other hand it would seem that the denominators of VFU (2) are, like those of
VD13 (2) multiples of A2 also, for
denominator of CaF,,(2) = 934539 + 5<S = 683 (136871-'16£),
„ SrF,,(2) = 925946 + 5,5 = 167 (55527<J-'68£), .
and £ = -6, 5 in Sr makes A,, = 5557'21 in line with those above.
The denominator of CaFn (2) is 8Aa less than that of Cul)w (2),
„ SrFa(2) ,,11A3 „ „ „ „ Sri) (2).
Which of these two interpretations is the more likely must l>e left until the cases of
Ba and Ka are considered. It should however be noted that there may be some
uncertainty as to what lines really represent Fu, Fw, or F13, i.e., as to which of them
the multiple law is to be attached.
There remains to consider the question of the real limits.
supposing them to be Dn, D]S> Du are so strong that it is necessary to see whether
the values obtained direct from the F series, and those required in Table
be brought into agreement.
If the F series possess what has been called in [II.] a formula sequence, t
obtained for F( °°) above cannot be more than a few units in error, and in this case
* Calculated from formula.
t The observed is probably Fls (2) since the separation with Fs(2) is the f
J5 D 2
388 DR. W. M. HICKS: A CEITICAL STUDY OF SPECTRAL SERIES.
must be possible to raise the limits for D ( oo ) to agree with those calculated from
F(oo). That is, to raise that for Ca from. 33981 '85 to 33989'85 and for Sr from
31027'25 to 31033*99, or Ca by 8'00 and Sr by 674 or thereabouts. It may probably be
possible to find numbers near those which would still make the order differences of
Ca and Sr multiples of S, but only by supposing that the successive mantissa-
differences in the D series after rising begin to decrease with higher orders, which is
against the rule in other cases. This is so far an argument against this way of
reconciling the different values of the limits. If however the order differences in the
F series behave in a similar manner to that considered above for the D, i.e., by
multiples of $ or A, the exactness of the F (oo) found by means of a formula is no
longer so close, and the question becomes one of seeing if, when they are made 8 less
for Ca and 674 for Sr, it becomes possible to arrange the denominators in the
same way.
If the attempt be made to reduce F ( oo ) by 8 in CaF, a similar objection to that
raised above will enter, viz., the successive mantissa-differences after falling begin to
rise after m = 5. If however a reduction of about 6 '5 be made, reducing the limit to
that found in [II.] for S ( oo ), the order mantissse differ successively within observa-
tion limits by 10A2, 4A2, A2, A3, 0. Further, in the case of Sr a fall of 675 produces
a similar fall and rise in successive denominators. If however £ be put — 1'33,
the mantissas differences become within limits 3A2, A2 + 9(5, 16<5, 11$, 4$. If this is
justified, it is curious that as in the D series where there are no satellites, the
differences proceed by multiples of A2 the same rule should hold for CaF, where
satellites are at least not certain. The difficulty can only be stated and the solution
left open. It is possible that the order differences must be compared from the Fj2 of
one line to the Fu of the next, for which there is evidence in Ba and Ra.
Barium. — In discussing Ba we start under the disadvantage that the lines
belonging to D (2), with the corresponding satellite separations have not been
observed, for the ultra-red doublet treated in the discussion on the D series does not
seem to belong to the normal D (2). Moreover, the observed lines which are clearly
related to the F series are so dispersed by collateral displacements that it is
questionable whether it is possible to arrange a series proceeding by an algebraical
sequence as in the other cases. The lines exhibited in the table above run on parallel
lines with the corresponding lines in Ca and Sr, and are clearly closely related to the
successive orders of the series, even if they are not the typical ones themselves. An
attempt to obtain a formula from the first three gives a limit = 259067, and gives
a value of the wave number for m = 5 of 22729'52 close to the strong line 22706'84.
It is 250'05 behind the strong line 22979'57, which indicates that the last is probably
the normal F21(5), and makes the normal Fn(5) about 250 behind. This is in fail-
order with the march of the others. We may therefore feel justified in settling that
the limit of F^oo) js near 259067. The F separations are close to 260 and 157,
they are therefore based on satellite differences in the D series of 15$ and 9<?; S is so
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
389
large that there can be no doubt. These numbers are in analogy with the values
for Ca and Sr viz 13* and M, are in the usual ratio 5:3, and stand to the
observed values for BaD (3) given in Table II. in a similar relation to those in Sr
.
,™ oT }. mU8t ^ '3 (2)> "^ lf the general rale found above that ^e mantissa
of VDU (2) is a multiple of A, holds, it is possible to obtain a very accurate value As
a fact, with F, (.) = 25906 the mantissa of F,(«) » very nearly the multiple of
87 A, fit is made so exactly, taking A, = 11 960, then F,(«) becomes about 25922
This value, with the D satellite differences of 15*, 9*, give
F,(oo) = 25922
260-17
F2 ( oo ) = 26182-17
157-95
F3(oo) = 26340-12
and it is seen how close the separations come to those observed. If we put
F, ( co) =• 25922 + £ the mantissa of
D13(2) = l-040539-387£=87(H960-2--45£) = 87A,.
The D(2) lines calculated from these and D^oo) = 28610"G3 found above give the
following scheme in wave-lengths on ROWLAND'S scale in vacuo : —
D,.
D,.
D*
44042'97
31758'94
2841675
41178-36
30241'91
37193'67
The last one only comes within the region in which RANDALL'S ultra-red lines lie,
his longest wave-length being 29223, belonging to the doublet treated as a possible
D line above. If the rules employed are valid, these values can only err by a few
units. The denominator of VFn (2) is 2-923500-113'9f In the cases of Ca and
Sr there is apparent the existence of satellites, viz., Fw(2) = F,,(2) ( — 3*) and
Fi3(2) = Fn(2)( — 5$), and in both cases the mantissa of one of the F, (2) sets thus
found are multiples of A.? If this is general the mantissa for BaF,., (2) will be
920085-113-9^. This is 77 (I1949'l-r48£), sufficiently near to give some weight
to the allocation. If £ = -10 this is 77 x 1 1963*9 and the value for D13 given above
becomes 87 x 11 964 "7, giving a value of A2 = 11964 within limits of error A, = 11964
makes the oun 683'66 = 3Gl'98^ with W = 137'43. The F satellites thus constituted
would, if existing, have separations for m = 2 of 18 "02 and 12 "03, and for m =• 3 of
390 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
7 '63 and 5 '09 — but they have not been observed for F(2) — -as indeed is the case in
Ca and Sr the first set in which show separations of the full amount. The curious
dislocation of the second half of the triplet from the first seen in Sr shows itself here
also. The analogue appears to be shown in the triplet coming next in the list which
appears to have kept its first member and satellite. The displacement is 381'06±.
This cannot be due to a displacement in F ( oo ), for if so the separations due to
15(5, 9$, would be considerably larger. If it is treated as a displacement on VF the
denominator difference is 43403-5'3£ while 3A2+ lie? = 43414. It is probably
therefore this. The corresponding displacement in Sr was found to be 3A2 The
separation between the first line and the satellite is 575, the satellite being due to
LEHMANN whose measures are not very accurate, it may well be 6 '01 corresponding
to a satellite difference of S. The lines may therefore be represented
FU(2)(3A2+11<$), F21(2)(3A2+11<S), Fm (2) (3A2
Fn(2)(3A2
If the next two lines are correctly allocated, 18686 should have an unobserved
satellite with a difference 2§. This would make 18941 '97 or F21 260'26 ahead
of the satellite, so that this supports the allocation. The line 21308"19 = Fn(4)
corresponds to a satellite with 58. This makes, on the supposition of satellite
differences of 2S, BS, 2156070 or F22 (4) 260'89 ahead of the satellite F13(4), the
satellite F12 (4) being absent. The line for m = 5 appears to be displaced to 22706.
The value calculated from the rough formula gives a line 250 behind the strong line
22979'57, clearly showing that the latter is a F2 (5) line, and 22706 is very close to a
displacement of A2 on the calculated. If this be made exact the undisplaced line
would be at 22719'97, or 259'60 behind 22979'57. This is within error limits of 260'17.
Hence F13 (5) has been altogether displaced to 22706'84 = F18 (5) (A2), and 22979 '57
is F2(5). For m = 6 the formula gives Fx (6) = 23582'83. There is a doublet at
23667'07 ('28), 23919*27 (114) with a separation 252'20, and no others in the neigh-
bourhood. If these are the displaced F (6), the normal F (6) would be 23 59 5 '82 and
23855-93 and the observed lines 23667'07 = Fx (6) (9A2) and 23919'27 = F2 (6) (8Aa).
The calculated normal lines have separation 260'11, or practically 260'17. A line at
23995-83 is 413 = 260+153 ahead of the calculated F, (6). It is therefore the
undisplaced F3 (6).
There are a large number of other lines clearly related to the F type. Their
complete discussion would require a more searching investigation than can be given
now. Several sets are related in a manner which is quite common in spark and rich
arc spectra, indicated by the fact that a number of lines may differ in succession by
nearly the same separation — a kind of relation which cannot be due to collateral
displacement by equal denominator differences. There are a few also which seem to
be attached parasitically to S and D lines. There may be uncertainty also as to
DR. W. M. HICKS: A CRITICAL STUDY OF SPFXTTRAL SKRIKS.
whether the separations shown which differ from 2GO and 158, differ through a
satellite effect, or by successive collaterals of 15$ and 9cJ. For instance, putting
158 = A', F1(oo)-F1(oo)(A') = 264>10, F,(oo) (-A') -F,(o>) = 256'39,
Fa(oo)(-2A') -Fa(oo)(_A') = 252-35,
all which separations occur. In the lines now to be referred to, however, the
separations will he supposed to owe their defect from 260 to the satellite effect, and
thus treated it is clearly seen what an important r61e the Aa term plays.
Amongst the ultra-red lines observed by RANDALL* appear the following in wave
numbers :—
(70) 9387-55 (4'4)
(60)9547'08(l-82)
1. <! (60) 9771-54 (-95)
(5)9804787(1-5).
. (60) 9964*527 (3)
159-93
257-70
159-76
The figures in brackets before the numbers give intensities and those after the
estimated maximum errors. They clearly l>elong to the F cycle, and show within
error limits the normal separations. The run of the intensities would point to
negative values, with the first four respectively for /31, /„, /„, /,„ but also the 2nd,
4th and 5th might be fl}f3,f3, whereby 9771 would not come in and the small
intensity for fa would be abnormal. On the first supposition, /„ = -9771 gives a
denominator 1752908 (23)-24'5£ ' and /„ = -9804 =/„ (-3). On the second,
/„ = 9547 gives a denominator 2'588000 (143) -79f Now
or denominator of /12 = 63 (ll96ri±-36--36£) -2S = 63Aa-2A
In the following the wave numbers of some sets of lines with their separations are
given. The low frequencies have been observed by LEHMANN and by HERMANN and
HOELLER. LEHMANN gives many weaker ones not observed by HERMANN and vice
versd. LEHMANN'S observations were earlier but are not nearly so accurate as those
of the others.
260-17 12896-53
260-94 14402-09 158'63 1456072
254-37 15157-63 157-88 15315'51
2. 12636-36
3. 14141-15
4. 14903'26
14934'33
5.
16669-61 259-69
16921-93
16929-30
156-03 17077-96
'Ann. d. Phys.,' 33, p. 745(1910).
392
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
G. 17219'30 262-78 17482'08
7. 17508-93
8. 18577'96
261-16
258-12
17770'09
18836-08
144-15 18980'23?
9. 18585-56 255'92 18841'48
10. 18632-62 261-69 18894'31
11. 23667*07 252-20 23919'27
The following numbers give the corresponding denominators calculated from
Fj ( oo ) = 25922. Where the separation differs from 260, the satellite value — or,
which is the same thing, the denominator for F21 — is inserted as well, but, in order to
distinguish it, it is printed further to the right — the changes due to £ being the same
for both.
(2) 2-873178-108-1^"
Fn(2) 2-923500-113-9^
F'12(2) 2-967999-119-2^
8+3
F'u(2) 2-968684-119'2£_J
-3-051164-129-5^"
•051267
•154087
(4)
3'154916-143'1£
3'159373-1437f.
'441229
(5)-j -441583
2S+5
L3-442954-186'0^
3-549987-203-9^
(•)-]
•550517 _
3-610577-214-5^
(7)<j Sx+43
•610791
95506 = 8(11938-3-1-39^)
83165^ =
108209J =9(12023-2-1-58^)
11952
11895(11969)
12039(11982)
28667 = 24(11944-6-176^)
11962
•107563 = 9(119517-1-99^)
60060 = 5(12012-2-08^)
11971-6
11964-6
11932
11962
DR. W. M. HICKS: A CRITICAL STUDY OP SPECTRAL SERIES.
(" '863911
(8H 28,+H
[.3-864437-263^ '
f '865855
L3'866436-263^
-3-878900-266-1^
'879299
Fla(3) '892050
Fn(3) 3-893395-269-0^"
F12(4) -871538
Fn(4) 4-875553-528-3^"
FIS(5)(-Aa) 5-840534-908-3^
A.-8-14
F12(5) 5-851797-908-3^
?Fn(5) '852480
f '961761
(ioH A,
[6'974086(432)-1546'4£
The differences are given in thick figures. The last column gives the corresponding
value of A2 without regard to observation errors when £ = -10. In the case of (3)
if the differences be referred to a hypothetical F,,, displaced 3<? from 14141, the two
abnormal values come to 11969 and 11982. It will l>e remembered that we had an
indication, above of £ = -10 with A2 = 11964 in treating both VI),, (2) and VF13(2)
as depending on multiples of Aa. It would seem, therefore, that the value of Aa is
close to 1196413 and the value of F,(o>) = 25912.
The actual differences of successive denominators in the normal series may thus lie
represented :—
14463 = 11929 + 15J,-28-3-l£
14495 = 11933 + 15^-2-9^
•978143 = 82 (11928-5-3-16^) 11970
— 968996 = 81 (l 1 962 "9 15 '93 — 4 "6.3£)
964981 = 81(11913-3+ -4'63£) 11959'6
976927 = 82(119137- 4'63£) 11960
F,i(2), F13(3) = 8l(ll957-4-l'91£) = 81
Fu(3), F,8(4) = 82(11928-5-3-16^) =
Fn(4), Fn (5) = 82(ll9137±3'6-4-63^)
119597,'
, 11960-1,
, 11960-013-6,
in which the last column also gives the value of A2, where £ = —10.
The first multiple of 81 with addition of 2$ suggests (l) a real F,,(2), displaced li<5
from 13089, or (2) that there is a normal type Fu about A2 behind. The latter may
well not l>e a typical Fu line since it makes the exact separation 260 with F^. There
VOL. ccxin. — A. 3 E
394 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
is a line by LEHMANN at 13096'55, but its collateral displacement cannot be 2<? within
any likely limits of even LEHM ANN'S measurements. As to the second supposition,
there is a line at n = 12992'53 by LEHMANN which gives denominator 2'892050.
This gives a difference with F12(3) of 82 (11943>5-1'90^), which with £= -10
would again make A2 = 11962'5. It would thus appear that the normal Fn (2)
line is 12992, and the system receives a double displacement, first to 13089, and
again to 13471. The mantissa is 912483-112'6£ The addition of 2A2 makes it
936403 — 112'6£ which with £ = —10 is 937529, well within error limits of the same
quantity in the case of Ca and Sr. Again we are met with the apparent
simultaneous existence of two explanations which cannot be compatible. Is the true
explanation that the typical first line is 937300 — 2A2, but that the corresponding
configuration is not very stable and transforms to one depending on the nearest
complete multiple of A2. Certainly such instability is indicated in Ba.
Radium. — -The discussion for radium is rendered even more uncertain than that
for barium, in that the ultra-red region has not been observed, the process of
disintegration and re-aggregation has proceeded further, and, in addition, there is
some uncertainty about A2 = S7S2 adopted.
RUNGE and PRECHT'S plates were only sensitive up to 6500 A.U., and EXNER and
HASHEK give only two lines above this, 664273 and 6641 '38, both of which belong
to the F cycle. The number of lines, however, coming within this cycle is very large,
but a complete discussion would involve the consideration of the new kinds of
relationships referred to under barium, and cannot therefore be undertaken here. It
will be sufficient to deal only with some generalities, specially bearing on the series
proper, which will also give some further light on the general D series.
There are a large number of triplets with 'separations in the neighbourhood of 692
and 432, which are roughly in the proper ratio 5 : 3, allowing for the fact that the
actual separations must be larger. Those in the table of F lines above are roughly
parallel to the BaF, and give a limit somewhere about 24520. VDn (2) would
therefore be about this, and VD13(2) more than 692 + 432 = 1124 larger. The
denominator of VD13(2) should be a multiple of A2. Using the most probable value
of A2 = 374, it is found that the denominator comes out very close to a multiple of
31A2. If this be made exact it is found that VD13(2) = F3(oo) = 2575275. The
value of F3(oo) is then taken 25752 7 5 + £ The values of S are so large that there
can be no ambiguity about the multiples to be chosen to give the separations, viz.,
16(5, 1QS. These multiples march well with those for Ca, Sr, Ba. The separations
resulting are 705'93, 456'69, with F^oo) = 24590'! 3 and F2(oo) = 25296'06. If we
apply the rule shown in the preceding elements for F13(2), the denominator is
2'937300-2A2 = 2'868676. Satellites depending on 38, 28 would give separations
51 '44 and 34 '20, and the fact that these separations occur in connection with
n = 17300 renders the identification of that for F(3) rather doubtful, a doubt which
is increased when we test the allocation by the law indicated above that the
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 395
differences of the denominators of Fn(m)-F»(m+l) is always the same multiple of
A2, as ,s done below. The line 19897 has separation 689-33, and therefore should
have a satellite (too faint) 16"60 above it. Allowing for observational errors on
197 this is 5* on the denominator. The following scheme will then illuHtrate the
law of formation : —
Calculated F13(2) 2'868676
Satellite
17165-94
17236T.8
17300-80
Satellite
19897-05
21350-92
3-837028
3^8
3*843521 -258'8£
S'861964-262-6^
98
3'878913-266M£
4*825655
4' 8 30 49 2
28
4-834202-515^
5'818813-898'2£_
-993288 = 29(34251 ±14-9-05^), 34296± 14
993464 = 29 (34257 -4- 8 -8£), 34301
— 993158 = 29(34246'8-13-2£), 34312
In the above the first is the denominator calculated from 937300-2Aa. It is
affected with an uncertainty of about 400 on the 937300. The line n = 17165'94
has a separation 680'84 with 1784678, and therefore should have a satellite 25'09
above it. Its denominator difference is 6493 behind and 3£<? = 6492. The line
17236 '68 is associated with 17300. Its denominator is 16949 behind that of 17300
and 9<5 = 16695— the same within limits. The satellite of 19897 is displaced 4$<J.
There is no evidence of a satellite 2S behind it, but the difference of 29Aa is made
with this suppositions one. It is seen that a value of £ about —5 makes these the
same within limits. The corresponding values of Aa are appended in the last column.
It may be taken that the discussion has established, that the satellite differences
in the lines EaD(2) are 16(5 and IOS. This is the only result of which there can be
certainty.
The Zn Sub-group. — Using the limits given in Table II. above and the corresponding
values of Dl (2), the limits F, ( » ) = D ( oo )— Dn (2) come as follows : —
Zn
Cd
Hg
12988*37, with separations 4'88, 374.
13022-83 18-23, 1T10.
12753-07
34'68, 62-04.
3 E 2
396 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
PASCHEN allots the following for Zn and Cd, viz., in wave numbers :—
6059-50
6065'51
ZnF(3) 6062-55
3'05
2-43
6064-98
CdF(3) 6083-37
6095-35
CdF3(4) 8595-57
17-86
11-98
The line 8595 must be allotted to F3 because it makes the difference 266 with
8872, so that the two are F lines connected with DJ3(2) and iiie companion to the D13
line 267 above it. This relation has already been discussed under the D series above.
For Hg he assigns 5814 to Fj (3), 5843 to F2(3), 5908'68 to F4(3), 8316'40 to F, (4),
8409'85 to F3(4). Now 5843 is double with dn =4"8, corresponding to a displace-
ment S, and may well be F21 . F22, whilst 5908 is F3(3). Again, 8316'40 and 8409'85
are separated by 93"45, which is so close to 34'68 + 62'04 as to indicate that they are
Fj (4) and F3(4) and that F2(4) has not been observed.
With these allocations VF(3)* = 6939'07 and VF(4) = 4436'66 and the denomi-
nators for the lines calculated from the first lines (except CdF3 (4)) are
Zn.
3'978529-287'lf
Cd.
3'970387-285'3^
4-960809-556'4^
Hg.
3'975605-286'5^
4-971938-560-3^
The differences of the mantissae of the two orders in Cd and Hg are
Cd.
9578 + 271£ 213 = 9561 ;
Hg.
3667 + 274£
= 3629;
in which the probable variations of £ are small fractions. In fact, the greatest
uncertainties are due to observation errors.
There is not much material to throw light on the origin of the F term here, nor in
fact is there evidence that the fundamental lines, or the first lines, of the series are
* On the basis of RITZ'S combining theory PASCHEN gives the following allocations (lines in wave-
lengths) : —
3011-17 = S1(oo) -VF (3) 2799-76 = Si (oc)-VF (4)
2642-70 = S2(oo)-VF(3) 2478-09 = S2 ( a,) - VF (4)
2524-80 = S3 ( 00) - VF (3) 2374-11 = S3(oo)-VF(4)
There can be little doubt about the correctness of this allocation. Using the value of Si ( oo) in [II.]
40139-55 and the wave numbers -33200-19, 35707-02 of the first lines of each set there results
VF (3) = 6939-36, 4432-53, which are practically the same as those found direct.
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SKRIffi. 397
for m = 3. Any lines -corresponding to m = 2 would have wave numbers about 500,
and to m = 1 negative wave numbers m the neighbourhood of 16000. Now PASCHKN
has noted lines which may be treated as the actual lines in question. They depend
on terms S' ( °° )- VD (2) where S' ( » ) is the limit of his singlet series, and of course
VD(2) is the F(oo) of the above. Using sequences for the 8' series of the form
ft. = 1 +f, the limits of the series are
Zn. Cd.
29019-96 28843'40
The lines in question are
Zn. Cd.
A = 0238-21 G325'40
n = 10025*87 15804'98
Hg.
30114-33
Hg.
5769-45
17327-96
If these wave numbers be added to Fa(oo) m each element, there results 29018'62,
28846-14 and 30115-71, i.e., the value of the S' ( oo) above. The corresponding lines for
F! ( co ) do not seem to exist. There is no d priori reason to take F,( oo ) rather than
Fj ( co ) for Zn. In Cd, however, the case is settled in favour of F2, as the other lines
exist, viz., -15520'84, -15793'05, -15804'98, giving the differences 266'21, 1T93
corresponding therefore to the companion series to D13(2), to D13(2) and Dla(2), D,,
not appearing. But in Hg -17327'9G, -17264'98, -17223'97, with differences
62 "98, 41 "01 would seem to assign 17327 to the F3 term. Nevertheless to get the
limit of PASCHEN'S S' series it is necessary to take Fa ( oo ).
If these be regarded as the first lines of the F series, the denominators are
Zn, 1-943072-33-5^; Cd, l-949840-33'9^; Hg, T908346. In Hg the line
n = —17121 '30 would seem to stand in a normal relation to the F,, as it comes into
line with the others as is seen below. With this the apparent limit with F, would
be 29874*37, giving denominator 1 "916040 — 32"0£ The question now is, are these
denominators related in any way to those for m = 3. The differences of their
mantissas are, using our new Hg line
Zn.
34557-254£
10(34557-25-4^)
= 10A2
Cd.
20547-252£
2(10273-126$
Hg.
59565-254*5^
2(29782-127-2^)
2A,
well within errors, it being also remembered that £ can only be a fraction. The value
of A2 for Hg adopted is the corrected one 29765, from $= 36r85w*. This is a
striking connection. It shows that the limits for PASCHEN'S singlet series are either
VF(1) or are formed from VF(3) by deducting 10A3 for Zn, 2A, for Cd, and
apparently 2A2 for a normal type in Hg which then receives some displacement.
31)8 I>i;. W. M. HICKS: A CEITICAL STUDY OF SPECTRAL SERIES.
Magnesium. — We are now in a better position to take up the consideration of the
place Mg is to occupy in the second group of elements, viz., whether it is allied with
the Ca or the Zn sub-groups. In the discussion of MgD (p. 356) there was evidence
in favour of either view. If it belongs to the former, then the line X = 14877 is
D^l); if to the latter we have PASCHEN'S allocations of 14877 to F(3) and 10812'9
to F (4). Take first the supposition that Mg is analogous to the earths. In this case
F(») = 39751-08-6719-95 = 33031*15, 6719 being the wave number of 14877. If
the F series is formed on the type of the Ca set the denominator of the first line will
be 2*937300-2A2 = 2'936474. This gives a line n = 20312. No line has been
observed sufficiently near to this to be identified with it. In the other case
F(oo) = D,(2) = 39751*08 - 2604'4*99 = 13706'09. PASCHEN'S allocations then
give denominators 3*962183-283*5£ 4*958710-5557£ with a mantissa-difference
= 3473 + 272£ With £ = '9 this is 3717 or 9A2. The value £ = '9 will upset the
difference in Table II. between D (2) and the supposed D (l) which in this case does
not exist. It still leaves the difference between the denominators of D (2) and
D (3) = 6A2. If Mg is completely analogous with the Zn set the combination lines
S(oo)-VF(3), S(oo)_VF(4) should exist. They should be at n = 32764'92 ('67),
+ !/!, +v2, and 3529072 ("87), +»i, +v.2. Now EDEB and VALENTA give two spark
lines of weak intensity at 3050*75, 3046'80, and SAUNDEBS a weak arc at 3051.
The wave numbers in vacuo are 32769*48 and 32811*95 separated by 42*47, which
is clearly Vl = 40"90. These are therefore the looked for 81( oo)-VF (3), and
S2 ( °° ) - VF (3), the third S3 ( oo ) - VF (3) not having been seen. As to the other set,
SAUNDERS has observed a line at 2833 giving n = 3528819 which is clearly
Sj ( co) — VF(4). The existence of these combination lines seems to settle the question
in favour of Mg belonging spectroscopically to the Zn group of metals rather than
the alkaline earths. It is possible that as a transition element it belongs to both
types. Judging from PASCHEN'S various readings it might well be that X = 14877 is
double so that one might be D (l) and the other F(3).
Group III. — In Al and Tl alone have the ultra-red lines been observed, and
here the F lines are found in a similar position to those in the Zn groups, and with
them EITZ'S combinations S(oo)— VF(3) and S(o°)-VP(3). Using the values of
D(co) of [II] the values of F(oo) = VD (2) are 15837*92 for Al and 13064*21
with separation 81*98 (*24) for F2(oo) for Tl. For Aluminium PASCHEN gives
n = 8882*19 (*80) and 11392*8(3*90) for F(3) and F(4), from which result
VF (3) = 6955*73 (*80) and VF (4) = 4445*12 (3*90). The combination
-- 41204*14 (3*39) = S, ( oo )-VF (3) gives VF (3) = 6957*32 (3*39).
For Thalium PASCHEN gives n = 6118*19 (75), 6200*67(77) for F, (3), F2 (3) and
8622*47 (*37), 8706*78 (l*5l) for F1(4), F3(4). These give separations 82*48 ±75 ±77,
and 84*31±*37±1*51 instead of 8 1*9 8 ±'24, but the same within limits. He also
gives n = 34526*21 (179), 42321*40(2*69) for S(oo)-VF(8) and 37022*23(6*85) for
PR W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
S, (o°) — VF(4). The possible errors of the latter, however, are so large that they
cannot be used to improve the values found from the direct lines. The limit
calculated from D( oo) of [II.] and D,, (2) is 18064-21, but there is some uncertainty
owing to the abnormality of D(2) as explained above under the dinrassinn of the
D series. The lines Ft (3) and F, (4) give 6946*02 (73) for VF(3) and 444174 ('37)
for VF (4).
In the case of In no ultra-red lines have been observed. In K.R.'s list there
appears a doublet X = 2720"10, 2565"59, which shows a separation 2213"32, the true
doublet separation being about 2212'38. Its relative position in tin- s|»-.-tiuin
compared with that of Al and Tl point it out as the Ititz combination S( oo)— VP, (3).
K.R. also give a line at X = 2666'33 or n = 3749377 (2'8l), which from its pnsitim,
might be S,(oo)-VF(3). If so, the value of VF(3) is 6960-99 (2'80) and clearly
in line with those of Al and Tl. We shall adopt it provisionally. K.R. mark all
these lines as doubtful, but the existence of the doublet separation points to their
real existence- as In lines. Collecting these give the following :—
Al. In. Tl.
VF(3) . . G95573 + '80p + £ 6960'99 + 2-80p + £
Denom. . . 3'970842-228jJ-285£ 3'969340-801p-285£
VF(4) . . 4445-12 + 3'90g + f 444174 + '
Denom... 4'967208-2177?-558£ 4'969095-207qr-559£
It is seen that Al and In may be the same within limits. In Tl the uncertainty in
D (2) referred to above is such as to raise the limit — and by 19 '19 — if the explanation
there given is correct. A rise of 10 would make the denominator for m =: 3 the
same as for Al and In.
One of the most striking results of this discussion of the F series is the distinct
divergence in type between the spectra of the high melting-point elements and those
of the low melting point, and at the same time the close resemblance between the
individual elements in each division. So close indeed is the resemblance l*;tween all
the low melting-point elements of Groups II. and III. that the differences between
them appear to be almost wholly due to the difference of the limits, or the value of
VD (2) and the values of VF (3) are almost the same. To see how closely they agree
the denominators to four places of decimals are collected here, and for comparison
those of the alkalies.
Mg 3-9621,
Al 3'9708,
Na 3-9979,
Zn S'9785,
Ga ? ,
K 3-9928,
Cd 3-9703,
In 3'9693,
Rb 3-9878,
Eu ? ,
?
Ca 3-9773,
Hg 3-9756,
Tl 3'9736,
It is seen how closely the elements in each group agree in spite of a very wi
difference in atomic weight, and moreover the mantisase in all are very close
400 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
It was shown in [I.] that the F series of the alkalies could be represented by a series
of the form m+l— a(l — 1/m). The same is the case with the Al and Zn groups.
As a is so small and varies so little it can scarcely be a function of the atomic weight.
The atomic volumes of the elements are much more even and it may be a function of
them as is the case with the p-sequence. In fact, in the case of the alkalies, the a
are not far from being proportional to v, 2v, 2v, 3v, for the four elements considered,
but the data are so inexact and uncertain that it seems not worth while to undertake
an exhaustive numerical discussion.
We do not know that the chief lines of these sets are those depending on m = 3.
If lines exist depending on m = 2 they would all be in extremest reel, in fact with
wave-lengths comparable with those of electro-magnetic waves capable of being
experimentally excited, and it is possible that VF(2) might be the same for all low
melting-point elements and as for He (see [I.]). For m = 1 we should expect the
lines of negative wave number in regions which have been observed and in which no
such lines have been seen.
The F series in the high melting-point elements, on the contrary, are profoundly
influenced by the atomic weight term. Either the lines observed belong to a different
type from those of the others, or they are based on a normal type of aggregation
which is modified by collateral and other types of displacement due to the splitting
up of the typical aggregations, or to a more complex system of new aggregations.
The notation F for these series was adopted in [I.] under the idea that the
sequence for it was of a more fundamental nature than the others, and that
impression is rather strengthened by the present discussion. It has been seen, for
instance, how the limits of PASCHEN'S singlet S' series in the Zn group depend on it.
It would be interesting to know whether similar series appear in the alkalies and
aluminium group.
The Value of the Chin.
The further knowledge now gained as to ways in which the oun — or the A — enters
in the constitution of spectra, enables a much closer approximation to its actual value
to be obtained than was possible from the consideration of the doublet and triplet
separations themselves. Amongst the principal aids are (l) the separations themselves,
(2) the dependence of the first D denominator on a multiple of A, (3) in the triplet
elements, on the collateral relations between associated lines, (4) the satellite
separations in the D series, (5) the order separations in the D series which show
no satellites, (6) collaterals depending on A. Of these, Nos. (l) and (4) have the
great advantage that the values depend only slightly on the exactness of the limit
(the value of £), but (4) has the disadvantage that only small multiples of S are
involved, and (l) only A itself. There are also various uncertainties which show
themselves when a high order of accuracy is desired — chiefly in the elements of the
3rd group. No. (2) has the great advantage of giving considerable multiples of A,
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 401
but they depend to a larger extent than (l) and (4) on the exactness of the limit.
This inexactness is, however, in general more than compensated by the largeness of
the quantities dealt with. Collateral relations also are capable of giving very exact
values, but always subject to uncertainty as to the actuality of the relations indicated
by the numerical coincidences. This is less apparent in the F series of tlir high
melting-point elements in Group III., where the relations are largely «M.-ililishr<l by
analogy between the different elements involved. No. (5) is affected by the exactness
of the limit, and is only useful when the separation is taken between the first two
orders and it is a considerable multiple of A, as, for instance, 117A in Al.
For the special purpose of obtaining as exact a value as possible of the ratio A/ir3 it
will be better to exclude from consideration Na, Ga, He, Sc, 0, S, and Se. Na is
excluded on account of the uncertainty as to whether F. and P.'s interferometer
measures of the P (2) lines are to be taken as giving the value of v for the S and
D series, in which a somewhat larger value is indicated by observers using ordinary
methods. Ga is omitted on account of its poor spectroscopic data. He because its v,
although very accurately determined, is so small that slight errors are very large
proportionate ones. 0 because v is small and the observations not so exact, and
Sc, S, and Se because their spectra have not been sufficiently discussed. There
remain 17 elements for consideration. In the following the case of each element is
considered first, with estimates of its possible error. Then using these possible errors
as probable errors, the most probable value of ^ is deduced by least squares. The
ratio S/w2 is denoted by q.
K. The observations determining v are very bad. The v adopted gives A =
D12(2) = 261A and gives A = 2932'27±-130--364£ W == 39'097±-003, and £ is
about ±1. The value of q from this is 361'944± '11. This is adopted with probable
error = '1.
Rb. The only source is from v, since there is no light from the
satellites are doubtful. The value in Table I. is 36r40±'5G.
taken = 'GG.
Cs. Table I. gives 361 74 ± "33. Dn (2) is so close to 17A that
to adopt it. The observations seem to show that £ should 1*
denominator is subject also to an observation error of 228.
the consequent value of q is 362'24± "30, but this value of ( makes the former
much less. The relation may be a coincidence, as it ought to be ne*
therefore be safer to take the first adopted value, 36174±':
Cu and Ag as in Table L, viz., 36r84f8, 361'81±'2.
Mg With £=2,dv =•• -06, W == 24'362±-002, ,, + * gives q =
The actual first D line has been seen to be uncertain, and in any case A2 u
that the actual multiple cannot be obtained. There is an order di:
between m = 2 and 3, but the observation errors, and those Jue to f, give A v
less exactness than from „, + „,. Value therefore adopted, 3
VOL. CCXIII. — A.
402 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
Ca. From Vl + Vat g= 6, dv = *1, W = 40'124±*005, q = 361'56 + '60.
The denominator of D13(2) = 691 A2 gives q = 361*84 ±'1.
From the discussion of CaF, q = 361*77 ±*2.
Mantissa difference of CaD (2) and CaD (3) = 99A2 = 133542 + 100£±28,
q = 361*870 + 1*76, the great uncertainty being due to g.
The most reliable appears to be that from D13(2), and is included in the others.
Value adopted, 361*84±*1.
Sr. From Vl + va> g = 10, dv = *2, W = 87*66±*03, q = 361'63±-56.
The denominator of D13 (2) does not appear as a multiple of A2, whereas that of
D12 = 178A2. If this is a real relation, q = 361735 + '33.
From the F collaterals and the denominator of F (2), q cannot be far from 361 '77 + *2.
Adopted value, 361 77 ±'2.
Ba. From v1 + v2, with g = —32, as modified in Table II., and +5 allowed, dv = '2
and W = 137'43±*06, q = 362'07±'53.
From the D13(2) collateral = 69A2, q = 361'968±'3
From D13(2), as found from the F series, q = 361'856±*36.
From the F discussion, q = 361*971 ±*39.
The most reliable is probably the mean of those depending on D13. Adopted
value, 361*913 ±'4.
Ka. From Vl + v2, g = 1, dv = '2, W = 226*4 + '02, q = 361'846 + '66.
From the F discussion, q = 361'94 + 'll, but as there is some uncertainty in the
F theory, the limits of error should be greater. Adopted value, 36 1*94 + '33.*
Zn. From VI + VK with g = 3, dv = 0, W = 64'40±'03, q = 362'238 + '36.
From Dj3(2), q = 361*682±*47, and from the F values lying between, 362'15 and
361*87. Value adopted, 362*01 ±'25.
Cd. From Vl + va, with g = 2, dv = '1, W = 112'3 + *1, q = 362'36±*66.
In the D13(2) theory Cd appears to occupy a similar position to that of Sr in the
other sub-group, in that the multiple of A is carried back to Dn or D12. The most
accurate is that from i/!+'(/2. Adopted value, 362'36±'66.
Eu. From Vl + Va, with g = 10, dv = 4, W = 151'93±'03, q = 361*94±'8.
From D13(2), q = 361*44±1. Adopted value, 361'94±'8.
Hg. From v1 + v2. There is some uncertainty as to the ratio of A2 : At. A2 = 41$,
best agrees with the transference value from Ax to A2 discussed at the commencement,
and it gives a value of A2 = 29725'65 which is in close agreement with the value
found in the F discussion, viz., 297 82 -127 '2g.
* HONIGSCHMID ('Sitz. d. k. Akad. Wiss. Wien,' November, 1912) has recently made a careful
determination of W, and gives 225-97 in place of 226-4. This would make the oun 363- llw2— a value
quite inadmissible if the spectroscopic data are reliable. Although they are not good, they can hardly be
so uncertain as this value of q would indicate. For what it may be worth, the spectroscopic data would
seem, therefore, to weigh against the acceptance of the new atomic weight.
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 403
The former gives with £ = 4, d» = 0, W = 200'3±'3, q = 361*423± 1*61, and the
latter 3G2'09 with a large uncertainty owing to 127 "2£
D,,(2) gives q = 361*50±1*28. The large possible error is due to the uncertainty
in the atomic weight of Hg. Adopted value, 36T50±r33.
Al. In Al also there is a large possible variation due to the uncertain atomic
weight. W = 27'10±-05. From v, q = 361*88 ±1*5. There is an order difference
117 A between m = 2 and 3 for the D series (see Table II). This gives
q = 361*871 + 1-30. The denominator of D (2) gives q = 361777±1'92, or if there
is a satellite D12(2) gives 361717±1'92. Adopted value, 361'871±1'33.
In. From », with £ = 1, dv = '25, W = 114*8±*5, q = 361*947±3*31.
From D,,(2), if 22A-16<S, q = 361-871, but the theory is uncertain. Adopted
value, 36 1'947 ±3'33.
TL From v, q = 362'00±'20. From D,,(2), q = 361 '913 ±'6, but with somewhat
doubtful theory. Value adopted, 362'00 ±'20.
These values for the 17 elements weighted according to the possible errors now
give q = 361*890. This is the same as our first approximate value, but its probable
error is much less. If the determination of the value depended only on questions of
errors of calculation and of observation in spectral and atomic weight data, the above
number would probably be extremely close to the actual one. .It must lie remem-
bered, however, that our theory of the constitution is not yet complete. For
instance, in [II.] it was seen that the supposition that N was not constant for the
^-sequence, but that the value for the first line was slightly larger explained the
introduction of a term in the denominator. A similar explanation might explain the
fact that the value of q appears to deviate from the mean by alwnt the same amount
in each group of elements, and if it were justified, the value of q calculated as above
would receive a slight modification. I believe it will be found ultimately that the
true value will lie within the limits given by 361'890±'05 or 90*4725±*0125.
If the existence of the oun as a definite proportion of the (atomic weight)3 be
considered as established, the best and most direct method of determining the value
of the factor q would be from the discussion of an element in which the spectroscopic
data are good and in which the atomic weight has been determined with great
accuracy. For this purpose we naturally turn to silver. Regarded as the ultimate
standard of atomic weight determinations, no error in the atomic weight enters — the
value of q is determined in terms of W = = 107*88. Moreover its separation is large,
so that any error of measurement is a small fraction of its total value, and in addition
the actual error is extremely small. It is therefore tantalising to find that the lines,
D (2) excepted, are not susceptible of such exact measures as in many others, that
the typical series are not well developed, and that in fact there may be a doubt
whether the lines generally accepted as the P, S, D series follow laws altogether
analogous to those in other groups. In KAYSER and RUNGE'S measures four lines are
assigned to Da (2-5) and three to D, (2-4), the possible errors for DM being much less
3 F 2
404 DR. W. M. HICKS: A CRITICAL STUDY OF Sl'KCTlfAL SKKIKS.
than for Dn. Using D21 — v for DJ2 and calculating the formula constants from the
first three, there results for the D12 series
n = 30644'60-N/L + '994354-
m I
with the large possible variation in D(oo) of £= 12'23. Using this value of the
limit and calculating the formula for Dn, the line Dn (5) is reproduced with
O — C = '22, O being '5. If, however, the S lines be used with /u. =f the limit comes
to S(oo) = 30614'60±3"60 and the fourth line is reproduced within limits. S(oo)
and D ( QO ) cannot be the same within error limits. If /a = 1 +f, S ( co ) = 30633 '47 ± 3,
and the limits can be the same, viz., with £ = — 973±2'50 on D(oo)5 but /a. cannot
be 1 +y* if the doublet usually assigned to P belongs really to a P series. I had
intended to supplement the direct determination of A from v by a discussion of
collaterals, of which both Ag and Cu afford a large number. The doubt however about
the form M = 1 +f, and the presence of the numerous collaterals, gives a suspicion
that the series are related to the F series with limits based on the typical S ( co ), that
they are analogous to the F terms in the high melting-point elements of Group II.,
and that the doublet usually allocated to the P series is really analogous to the
F triplets with negative wave numbers found in Ca and Sr. That discussion is
therefore held back for the present.* But certain points not open to doubt and
forming a portion of the work of the accurate determination command a place here.
The D (2) lines are sharp. F and P have measured the lines Dn (2) and D21 (2)
with their interferometer. Their measures give a separation of 900'3419±'0070.
In order to get full advantage of their accuracy, and to avoid the uncertainty due to
the last significant figure it is necessary to use logarithmic tables with more than
7 figures. This has been done on the supposition that F and P's errors are not
larger than '001 A.U., i.e., unity in the seventh significant figure. The old measures
are sufficient to show with certainty that the satellite difference is 23^, and
the old approximation to A will give 23^ with an inappreciable error, whence D12
can be found. Taking D ( GO) = 30644'6000 + £ the mantissa of Dn (2) = 5465'671 is
979596'44-120'59£ and 23^ = 2421'18-'15£ Hence the mantissa of D12 is the
difference or 977175'2G-120'44f VD12 calculated from this is 12373'6789 + 1'0045£
giving with VD21 the value v = 920'4431 + 0045£ in which the correction for £ is only
effective if £> 10. The value of A calculated from this is 27786'80-l'473f±'20, in
which the uncertainty of '20 is due to the uncertainty '001 in A. This value is 4
less than that of Table I. obtained by supposing K.Il.'s values for D12 and D21 had no
errors. It gives S = (361'754±'0026-'0152f )iv2. With the mean value £ = -975
suggested above this becomes S = (361'902±'0026)w2.
The foregoing is interesting also because it shows how the application of the laws
developed in the present discussion can help towards more accurate determinations of
* The value obtained for q from the collaterals was 361-708 ± -0026 - 0169£, which with £ = - 10, as
indicated in the text, gives a value surprisingly close to that deduced from all the elements combined.
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 405
quantities involved. For instance, it has enabled UB to obtain a value of * correct
to about a unit in the sixth significant figure. In the case of Au, the knowledge
is still more fragmentary than in Ag and the value of A has not been determined.
By the application of our new laws, however, it is possible to obtain a good deal
of information based on evidence of weight, and it will be int»M-«sting to consider
it shortly here. Although the spectrum of Au shows many analogies with those of
Cu and Ag, no lines have been assigned to the S or D series. Tin-re is a strong
doublet in the ultra-violet 2676*05, 2428'06 (v = 3815*28) analogous to the linen
allocated to the P series in Cu and Ag. There is only one other doublet in K.H.'s
list with the same separation, viz., 6278'37, 506475 (» = 3815'54). This is clearly
analogous to the doublets 5782*30, 5700*39 in Cu and 5545*86, 5276*4 in Ag, which
have the respective doublet separations but which do not belong to the S or D series.
E and H however give an arc line at 4811*81, which gives a separation of 3815*57
with K.K.'s line at 4065'22. This has the appearance of a D set, D,, being at
4792*79 with a satellite separation of 82*47. But if so it is quite out of step with
the progression of the Dia lines for Cu and Ag, viz., 5220 (Cu), 547l(Ag). But
5837'64 gives with the above 479279 a separation 3733*43 the same as that between
4792 and 4065, and they are in step with Cu and Ag as D,, (2) and D«(2), the
fainter satellite D12 being unobserved. This would seem the more probable allocation.
In any case, the curious doubling of a D type would have to be explained. There is,
however, here not sufficient data to determine the limits, or the other formulas
constants or the value of A. But it is possible to arrive at a probable estimate by
the following considerations. The limit D ( ») will probably be in step with those of
Cu and Ag, viz., 31515, 30644, i.e., will l>e in the neighbourhood of 30000. Now A
must give v = 3815*54 and must itself be a multiple of the oun, in fact if it is similar
to Cu and Ag of <54. Now W == l'J7'20 with an uncertainty of a few units in the
second decimal place. The ratio q = 361'80* + y, where y is probably not greater
than 1 in the first decimal place and it will be regarded as a correction on the *l
From this it follows that S = 1406'930±*097 + '38y. The uncertainty *097 due to
the uncertainty in W produces so small an effect that it may be neglected here.
Now A must tie a multiple of S and must give with the proper value of D(oo),
v= 3815*54 + *30s, '30 being the maximum error of v and therefore s between ±1.
This condition gives the following sets of possible limits in the neighbourhood of
D(co) = 30,000:- 30819-15 + r57*-6y with A = 76<*
about 30542 ,. 77f5
30266 ,, 785
„ 29994 7:'
„ 29724
29465*18 + l'45.<r-57y „ 8lJ
* The actual calculations were made before the last most probable 361-890 was obtained, but nothing
is to lie gained by recalculating to it.
406 DK. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
If now the lines 5837'64, 479279 lie taken as Dn (2), D21 (2), their wave numbers are
17125-54('14), 20858'97('22), giving for the wave number of D12(2), or D21-i/, the
value 17043'43 ('22q— '30s). In this '22g is the possible observation error in D2] (2).
The satellite separation of 82'11 must therefore be caused by a denominator difference
which is also a multiple of St. If this be tested, it is found at once that only the first
and last of the above set can satisfy this condition, 3.0819 taking 24^ or GS, and
29465 taking 28^ or 78. The corresponding values for Cu and Ag are both 23^.
The differences between the values calculated from the lines and from the multiples
of S1 are (p between ±1 giving the observation error in Dn (2)).
'Oy+14p-22g with 24^ = 8441
= 9869.
It is clear that either can easily be made to vanish well within possible errors,
more especially the latter. The limit 30819 is higher than that of Ag instead of
lower as might be inferred from the fact that the limit of Ag is lower than that of Cu.
The limit 29465 is 1179 below that of Ag, which is itself 931 below that of Cu.
This seems a probable order of magnitude, especially when it is remembered that there
is a gap in the Periodic Table between Ag and Au. But there is further evidence in
favour of the latter. If the lines 6278'37, 506475 are collaterals of D (2) as the
corresponding lines in Cu and Ag appear to be, 6278'37 should be Dn(2)(xSl).
With the limit 30819 this cannot possibly be the case. The oun Sl = 351 is so large
that there can be no doubt. If however the limit is 29465, the line is Dn (2) (A+ 15^).
Further, with neither limit is the mantissa of D (2) a multiple of S, and as this is also
not the case with Ag or Cu, it may be regarded that in this group either these lines
are not of the D type, or possibly like the high melting-point elements of Group II.
the first lines correspond to m = 1 and not m = 2. The actual values of the
denominators as found are so close to the same value for all three elements as to
suggest the existence of a group constant. If the limit 29465 is used the denominators
are as given below, and as is seen they differ from such a constant by very small
multiples of S.
Cu. Ag. Au.
Density. . . 978276(21)' 977162(19) 971409(26)
146 = S 1263 = 3i 7034 = 53
978422(21) 978425(19) 978443(26)
or say a group constant 978430. Whether this apparent equality corresponds
to a real relation or not must be left for further evidence. In any case a limit
D ( o°) = 30819 would throw this relation quite out.
DR. W. M. HICKS : A CRITICAL STUDY OF SPECT1CAL SEKLES. 407
As a final result the evidence would seem conclusive that D(oo) for Au is
29465'18±7, that A = 81* = 1139Gl±31'5y, and that the satellite separation is
produced by 28^.
Summary.
It must be confessed that much of the foregoing discussion is of a problematical
nature, and that, in fact, some of the suggestions offered are incompatible with one
another. This is no objection in a preliminary search for general principles, as the
raising of questions is only next in importance to answering them. Nevertheless,
some results appear to be well established and others to have considerable evidence in
their favour. Amongst the first are —
(1) The dependence of the spectrum of an element on its oun, a quantity
proportional to the square of its atomic weight and which prolwbly does not differ
from S1 = 90'4725M>3 by more than '013102 where w denotes one-hundredth of the
atomic weight ;
(2) The direct dependence of the ordinary doublet and triplet separations on
multiples of the oun ;
(3) A similar dependence of the satellite separations in the Diffuse — or the 1st
associated — series on multiples of the same quantity ;
(4) The existence of collateral displacement, whereby new lines are formed by the
addition or subtraction of multiples of the oun. Until, however, the laws which
govern the formation of collaterals are more fully known, it is not safe to assume that
any displacement indicated by mere numerical coincidence corresponds to the physical
change such collateral indicates. Nevertheless, many cases of clear displacement of
this kind, involving considerable multiples of A, especially in the F series, are given
which serve to give more accurate values of the oun.
The conditions which govern the various multiples of the oun which enter in the
various separations have not been determined. It is probable, however, that the
multiple for the doublet, or first two of a triplet, in the two sub-groups of the
nth group of elements contain 2/H-l and 2w + 2 respectively as factors.
It is probable that the mantissa of the normal first line of the Diffuse series, the
last satellite, when such exists, being considered as the normal, is a multiple of A,
and it is possible that its magnitude has some general relation of approximation to
that of the corresponding F series, which again may depend directly on a group
constant.
It is possible that the wave numbers of the lines in the Diffuse and F series may
not depend directly on a mathematical function of the order m of the line, and it is
probable that this is the case when there are no satellites, the differences now being
multiples of the A themselves.
In the discussion of the material it has been attempted to keep the mind as free as
408 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
possible from any preconceived theories as to the origin of the vibrations which give
the lines. The aim has been to discover relations, which it must be the object of
theories to explain. Nevertheless, the way in which multiples of a quantity depending
directly on the element enter, and indeed multiples of these multiples, irresistibly
suggests that each line is due to a special configuration built up of aggregates of the
same kind. Thus, in the Zn group appear multiples of 6^, in Mg, of 5(5,, &c. These
smaller aggregates peculiar to a group then appear to enter like radicals into more
complex aggregates, e.g., in Zn Aj = 31<V A2 = 15^B, and again, multiples of A2 occur
in collaterals. In cases, a certain aggregate, normally to be expected, appears to be
affected with instability, a certain number of ouns are expelled or added and we get
a stable collateral. In the case of rich spectra and of spark spectra, a very large
proportion of the lines appear to be collaterally connected. It suggests systems in
which a greater freedom of aggregation is permissible. But there is another way in
which the matter may be looked at. The actual multiples may be determined by the
number of electrons taking part in the vibrations, and the quantity enters into the
formula as the product of this number by a fundamental quantity of the atom. But
it is difficult to see how this quantity should depend on the square of the mass. It
would almost look as if the gravitational pressure of two atoms always at the same
distance produced some change in the configuration of the surrounding aether
proportional to the pressure, and that the vibrations were conditioned by this change
and by definite numbers of electrons. In any case, the existence of the oun, and the
extent in which its influence is shown in a spectrum, point to the conclusion that the
positive atom plays an essential part in at least those vibrations emitted which are
slow enough for us to observe.
APPENDIX I.
The Value of A in Scandium.
The value of A as a multiple of $ in Scandium is of importance in connection with
the evidence as to the curious relation, that the A's of the first elements of the two
sub-groups in the nth group are multiples of (2n+l) ^ and (2n + 2) S^ The lines in
the visible part have been measured by FOWLER,* and lines both in visible and ultra-
violet by EXNER and HASHEK.f
I do not altogether feel full confidence in the allocation suggested below, but it
gives related series, even if not the typical ones, and so will serve to determine A.
The doublet separation is 320+ a small fraction. There are over 34 doublets with
this separation — two, 3613'96, 357271 and 3576'52, 3535'88 containing some of the
strongest lines in the spectrum. The lines suggested for the S series appear to show
* ' Phil. Trans.,' 209, p. 66.
t ' Spektren der Elemente : Bogenspektren.'
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIKS.
109
satellites in the first two sets. Further, it appears that the P series take the
s-sequence and the S series the p-sequence as is the rule outside the alkaliea
m.
4.
5.
6.
7.
8.
9.
10.
2.
3.
1.
2.
The P Lines.
(Figures in brackets give intensities.)
P,.
(5)5717-51
(5)5258-49
(2)5021-67
(1)4880-90
(2)4791-69
(2)4728-95
(0)4682-16
1. (6)6413-54
*V n,.
(0)5721-20 17485-35
1901170
19908-26
20482-40
2086372
21140-54
2135178
S Series.
(3)6284-66 1558778
84-35
11-27
319-65
(6)6379-02
(15)3646-46
(50)3642-93
(2)3139-98
(3)5146-43
(6)5392-30
(\b) 3603-1
15672-13
2742371
25-81
27449-52
322-39
(1)310870 31838-31
Parallel S Series.
(0)5323'1)4 18457-31
320-37
320-65
82-60
18539-91
(2) 3273-76
17474-08
15907-43
27746'K)
32I58T.H
1K777'96
30537-33
TJie S Series— For a reason to be seen shortly, it is necessary to regard S,, ( 1 ),
Sn(2), S,(3) as the typical series, S,(3) as not displaced, but the doublets corre-
sponding to 1 and 2 are displaced collaterally to S». In other words, we have t« do
with bodily displacements of the first two doublets and not true satellites. The
mean of the doublets gives „ = 320'80, and the formula calculated from the three
lines is
37949-90-N/(m+ 1-244902- ™™
VOL. CCXIII. — A. J G
410 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
With the value of v above the value of A calculates with this limit to 7140 =
with <^ = 70 '34. The denominator differences of Sn and S13 for m = 1 and 2 are
respectively 4188 and 3964, i.e., close to 59S = 4150 and 57^ = 4010, with errors in
d\ of '31 and 0, divided between the two sets of lines. It is possible they may be
the same within limits of error (59<?), when the value of d\ in the second would
be -'16.
The formula for P, calculated from the first three lines, is
n = 22281'97-N/(m + '828585-
.'97-N/l
m
This formula gives the following values of O — C for the lines for m = 7, 8, 9, 10,
viz., —'55, '85, '26, — 1'86. If the denominator be treated in the same way as
Al,* i.e., deducting the group constant '043761, it may be put into the form
-'043761 + '872346(l- 21480)) which reproduces the '215 constant. Also
m
m
VS(l) = 22277'80 = P( QO) within error limits.
If 8^3) had been taken as S12(3) this would not have been the case. But VP(l)
extrapolates to 40717'20 with a denominator 1'641213, whilst S ( <*>) = 39749'90
with a denominator 1 '699998, and they cannot be the same even approximately if
the typical formula holds. The extrapolated value of VP(l) requires A = 6428 to
give v = 320'8 or 91<5, again giving a multiple of 7<^ and at the same time more in
line with other elements as being a multiple of S itself, and a multiple more in step
with them. It points to the likelihood that the series chosen for S is a parallel series
to the true S, i.e., the VS (m) is correct. If so, using the values of VS(l, 2) with
S ( oo ) = the extrapolated limit 40717, we should expect doublets with the first lines
at 18440 and 30217. There is a doublet at 18457'31, no line observed at 30217, but
the doublet companion expected at 30537 is found at 30537 '33. Also 18457 appears
as a satellite to a stronger line 18539'91 in a corresponding position 82' 60 a-head.
The second set of lines above is, therefore, probably the true S series, the first being
a parallel one. With limit 40717'20 and v = 319'45, the mean of all the doublets,
the value for A is 6404 = QlS with S = 361'89it'2. The value of 361'89 is subject to
considerable uncertainty owing to uncertainties in the limit value and the atomic
weight, and its agreement with the final estimate for the oun is a mere coincidence.
The spectrum of Sc is a most interesting one, but its discussion must be postponed.
The object of touching upon it here is to obtain some indication of the nature of its
oun as Sc occupies the first place in its sub-group. It would clearly appear that the
multiple of the oun in A contains 7 as a factor, viz., 13 x 7^ = 52 x 7(V
The separation of PI(±) and P2(4) is 11 '26, corresponding to a denominator
difference of 5607 = 80<5. This is in fair agreement with the case of other P series in
which the differences for orders below the first are about '8 A.
* [II., p. 46.]
DR. W. M. HICKS: A CItlTICAL STUDY OF SPECTRAL SERIES.
411
APPENDIX II.
The I) Scries.
Na.
K.
S.
(2)8196-1
8184-5
P. (2)
1177173
1168976
K.R.
5688-26
5682-90
S.
6966-3
Kl * 1 * 1 1 O LJ • t>
. It. tt'MM o
»
4983-58
4979-30
K.R.
5832-23
5812-54
»
4669-4
4665'2
,,
5359-88
5343-35
"
4500-0
4494'3
„
5112-68
509775
Z.
4393-5
4390'!
„
4965-5
4952-2
"
4324'S
4321-3
S.
487T3
S. 4856'B
"
42767
4273'6
1,1).
4808*8
L.I). 4796-8
»
424 1'8
4239-0
M
4759-8
,.
4215'S
4213-0
„
41957
4192-8
„
4180-2
4177-2
4168
S.
K.R.
RE.
K.U.
Rb.
Cs.
5290-3
7759-5 1
14754-0
7619-2
P. (2)361277 -i
34892-5 J
30099-9
7757-9 J
6298-8
K.R. 6206'7
9208-3 -»
9172-5 J
876 1 '5
5724-41
5648-18
S. 6983-8 I
K.R. 6 7 23 '«
54:51-83
5362-94
K.R. 6973-9 J
5260-51
519576
RE. 62 17 T) -i
6010-59
515T20
5089-25
K.R. 62KT4 J
5()76'3
5023
5017
K.H. 4963
RE. 5847 "86 i
5845-3 1/
5664'U
4983
„ 4926
5635*44
5466*1
4953
4892
5503-1
534T15
5414-4
5256-96
5351
5199
5304
5154
H. 5118
3 o 2
412
DK. \V. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
(2) 3838-44
3097-06
2852-22?
Mg.
3832-46
3093-14
2848-53
3829'Sl
3091-18
2846-91
(2) 19917
19864
19777
M
•6
•4 J
Ca.
19507-1 1
19452-9 J
19310-6
2736-84
2733-80
2732-35
4456
•81"]
4435-86 1
4425-61
2672-90
2669-84
2668-26
4456'OS j>
4435-13/
2633-13
2630'52
.
4454
•97 J
2605-4?
3644-861
3644-50 J
3631-101
3630-83 J
3624'IS
3361
•92
3350-22
3344-49
3225
74
3215-15
3209-68
3150
'85
3140-91
3136-09
-
3101
•87
(2) 301107 1
29225'9 J
Sr.
27356-2 1
26915-4 J
26024-5
-
Ba.
4971-85"]
4968'H I
4876-351
4872-66.J
4832'23
(3)5819-291
5800-48 \
S. 5536-071
5519-37/
5424-82
4962-45 J
5777'
84J
4033-25"]
3970-151
3940'91
4506'
"1
4333-041
4264-45
4032-51 I
3969-42J
4493'
82 I
4323-15J
4030-45 J
4489'
50J
3705-88 J
3653'90l
3653-32 J
362915'
3628-62-
1 S. 4087"
53 L
S. 3947'6 1
„ 3946'6 J
3889'45
3547-92
3499'40
3477'33
„ 4084-
94 J
3457-70
341T62
3390'09
„ 3895
2
„ 3767-5
3400-39
„ 3787
(2) 3346-041
3345-62 I
3345-13J
Zn.
3303-031
3302-67 J
3282-42
(2) 364974
3614-58
3613-04
Cd.
-j 3500-09 1 3403-74
3467'76 [•
3466-33 J
1
2771-051
2756-53
3610-66
-
2801-17 >
2770-94 J
3005-53
•} 2903-24 "] 2837'01
2801-00 J
2982'Gl
2881-34 i-
2608-65
2582-57
2570-00
2981-46
2880-88 J
2516'GO
2491-67
2479-85
298075
_,
DR
Zn
2463-47
243074
W. M. HICKS: A CRITICS STUDY OF
(continued).
2439-94
2407-98
Eu.
(2) 3637-84
3638-22
3629-94
362270
3004-9
3001-48
2683*29 -I
.268272-1
2564-27
(see [II., p.
3212'89
2596-49
2708-91
2455-03
2423'OS
2401-15
2387-41
60] for details)
2764-29
2763-99
2660*45
260T99
SPECTRAL SERIES. 41.3
Cd (continued).
2677-65 2639T>3
(2) 368074
3663'46
3663-05
3654-94
3650'Sl
3027'62
302579
302371
302 T64
2806-844
2805-422
2804-521
2803-69
2700-92
2699'503
2698-885
2639-92
2603'IG
2578-34
256T15
2548-51
253174
2525-90
2521-27
2517-57
2514-48
2580-33
2525-57
Hg,
3144-61
313T94
313T68
312578
2655-29
2653*89
2652-20
2483-871
2482763
2482-072
2400-570'
2399'HIU
2399-435
2352-647
2323-0
2544-84
2967-64
2967-37
253672
2534-89
}
.
2380-06 n
2378 '392 J
23(12-16/1
2258-871
414
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
Al.
Ga.
(2) 3092-951
3092-84 J
3082-27
See Part II.,
p. 71.
2575-491
2568'OS
2575-20J
2373-451
2373-23 J
2367-16
2269-20
2263-52
2210'IS
220473
2174-13
2168-87
2150-69
2145*48
2134'Sl
2129-52
2123-44
2118-58
In.
Tl.
(2) 3258-661
3256-17/
3039-46
(2) 3529-581
3519-39/
2767'97
2714'05l
2710-38/
2560-25
2921-631
2918-43/
2379-66
2523-081
•2521-45J
2389-64
2710771
2709-33/
2237-91
2430-8 1
2306-8
2609-861
2168-68
2429 76 J
2609-08 J
237974
2260'6
2553-071
2129-39
2230-9
2552-62J
2211-2
2517-50
0. 2105'!
2197-5
2494-00
„ 2088'S
2187'S
2477-58
„ 2077-3
2180'G
2465-54
„ 2069-2
2456-53
„ 2062-3
2449-57
„ 2057-3
2444'OG
„ 2053-9
2439-58
„ 2050-6
„ 2048-4
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
II ,
P. (2) 9264-28
R.P. 6158-415
5330-835
„ 4968-94
„ 4773-94
4055*54
„ 4577-84
452370
I)'".
615G'993
5329774
4968-04
4773-07
405474
O.
6165-198
5329-162
4907-58
477272
4654-41
D".
P. (2) 11287'G
R.P. 7002-48
595875*
5512-92
513070
4973-05
4576-97
4522-95
New D.
R.P. (3) 626478 626 T68 6256'81
5410'97 5408'80 5405'08
5037"U
s.
R.P. (4) 6757-40 6749-06 6743'92
6052-97 6046-23 6042'17
5706-44 5700-58 5G97'02
5507-20 550178 5498*38
5381-19 5375-98 5372'82
5295-86 5290-89 5287'88
Se.
7062
7014-25 1
7010-86/
J 6325*81
6284'Sl
1 6325-4
6284-19
J" 5902-08
159617
5925-31
5925-13
J 5753*52 5718-5
1 5752-31 5718*28
5618'OS
5528-64 5497'OG
5464-82
0990-90'
5909-49
5703-86
Double.
416
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
The F Series.
Na.
P. 18459-5
„ 12677'G
Eb.
P. 4G960 ?
EL. 134437
„ 1008T9
BN. 8872
8271
Ca.
K.E.
6169-87
6161'60
L.
5)
6169-36
))
Rs.
4586-12
4581*66
4578'Sl
K.E.
5)
4098-66
4095'OS
409276
E.H.
)»
3875-85
3872-60
3870-57
K.E.
) )
3753-56
3750-40
3748-39
?)
»
3678'46
3675'53
3673-49
F.
K.E. 5601-51
„ 5588-96
5270-45
5594-64
5582-16
5264-46
5590-30
5260-58
K.
P. 84520
|
„ 15165
'8
„ 11028
•o
BN. 9590
„ 8908
„ 8500
Cs.
P. 30099-9 ?
293W3 ?
EL. 10124-0
10025'S
„ 8083'! -1
8020'6
8080'9 J
S. 7280-5
7228-8
„ 6872-6
6826-9
„ 6630-5
6588
„ 6475
6434
Sr.
L. J6754-21
5) *•
6708-10
6644'OS 6616'92
K.E. r
4869'41 4855'27
E.H. < 4892-90 K.E. 4868'92
K.E. [4892-20
4338'OG
4319-39 4308-49
F. 4087'67
4071*01 4061-21
3950-96
3935-33 3926-27
S. 3867-3
i in Ca and Sr.
K.E. 5535-01
5504-48 5486-37
„ . 5481'IS
5451-08
„ 5257-12
5229-52 5213-23
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
417
Ba.
l;,
7488'38
HR. 7280-58
K.R.
HR. f 7637 '47
L. < 7420-96
HR. [7417-80
HL. 534971 K.R. 5277'84
469174 4636-80
440275(1) 4350-49
4224*1 1(2) 4179'57(3>
<»F12(5)(AS).
Mg.
P. 14877-1
10812-9
L. 7197-99
R.&P. 5778-5 5553-81
4605'H
4333*04*
4166-24
» F,(6)(9A.).
* Dss (4) and Fs (5) not resolved.
4444-4
4682-359 4533'327
P. 16498-6
Zn.
16490-3
164837
Cd.
P. 16482-2 16433-
)>
Al.
P. 11255-5
8775-1
16401-5
11630-8
Hg.
P. 17195 17110-05 16919-84
„ 1202T28 1188771
TL
P. 16340'S 16123-0
„ 11594-5 11482-2
Na. The first seven doublets were allocated by RYDBERO, using the measurement*
by ANGSTROM, THALEN, and LIVEING and DEWAR. They were, also so given by
K. and R. The remainder were given by ZKJKENDRAHT. The actual measures
given in the list are by the observers indicated by the letters.
K. Both RYDBERG and KAYSER and RUNGE interchanged the S and D series,
allocating those in the list to the S series. This was first corrected by Rm.* The
mistake is repeated in KAYSER'S ' Spectroscopie,' Bd. V. The line 6938 "8 covers both
KD2(3) and KSt (2). The pair at 4871 were first observed by L.D., but the more
recent measurements by S. are inserted.
Rb. RL. refers to RANDALL, RE. to RAMAGE. The lines 3-6 were allocated by
RYDBERG, and by K.R. The first doublet was observed and allocated by RANDALL.
The second doublet has raised the question of whether Rb possesses satellites. The line
7757 has only been observed by SAUNDERS, who allocates it to RbDu (3), with 7759'5 as
VOL. CCXIII. — A.
• Ann. d Phys.,' 12, 1903, p. 444.
3 H
418 DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES.
satellite. But EDER and VALENTA observe 7 7 59 '5 and not 7757, whereas the former
if a satellite should be fainter than the latter. The question is discussed in [I.] and
also in the present communication, and the weight of evidence would appear to be
against the existence of satellites in Rubidium. I have entered the third line as
6298 '8 instead of K.R.'s value of 62987, because the former value agrees with
independent observations by E.V., S., and E.H., and gives a better value for v.
Cs. The first were observed and allocated by PASCHEN, the second first by
LEHMANN, but the measures are those of PASCHEN. SAUNDERS was the first to draw
attention to the satellites.
The last line, 5118, was observed by HARTLEY, and is clearly the D2(12) line. The
corresponding D! (12) would be in the neighbourhood of 5256'96, or D2(9).
Mg, Ca. The lines were all assigned by RYDBERG. He also wrongly assigned two
lines about 12000 to MgD(l). The measures are by K.R., except the ultra-red in
Ca due to PASCHEN (' Ann. d. Phys.,' 29, p. 655).
Sr. Assigned and observed by K.R., except ultra-red due to PASCHEN.
Ba. Assigned by SAUNDERS (' Astrophys. Journ.,' 28, p. 223). The measures are
those of K.R., except those with S. attached.
Zn. The lines down to D2 (6) with the exception of the satellites were assigned by
RYDBERG. The measures, as well as allocations of the others are by K.R.
Cd. Lines to D2 (4), satellites excepted, assigned by RYDBERG. The remainder
allocated and all the measures by K.R.
Eu. The Eu spectrum gives evidence of much collateral disturbance, and the
unobserved D lines are possibly displaced in this way.
Hg. The Dn lines to Dn(4) assigned by RYDBERG, to D3(5) satellite by K.R.
D! (6) to Dj (16) by MILNER (' Phil. Mag.,' (6), 20, p. 636).
Al. RYDBERG gives the first two doublets without the satellites and assigns 11280
wrongly to D (l). Further he assigns D lines to the S series, but the observations at
his disposal were too inexact. The measures and allocations are by K.R.
In. RYDBERG down to Di(5) with satellites of first two. The remainder and all
the measures by K.R. As in the S series it should be noted how the D2 lines are
more persistent than the Dj.
Tl. RYDBERG gives all except from D^IO), and he gives D2 down to m = 15.
The measures given are by K.R. except those below 2105, which are due to CORNU.
The F Lines.
BERGMANN in 1908 (' Z. S. f. Wiss. Phot.,' 6, see also ' BEIBL.,' xxxii., p. 956)
measured lines in the ultra-red spectra of the Alkalies and observed in Cs a number of
doublets which clearly formed a series, and a few lines in Na, K, Rb which were
evidently analogous. It was, however, RUNGE ('Astrophys. Journ.,' 27, p. 158;
' Phys. Z. S./ 9, p. 1) who pointed out the dependence of the limit of the series on
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERO& 419
D (2). The lines in the far ultra-red have since been observed by PABCHKN (' Ann.
d. Phys.,' 33, 1910, p. 717). BERGMANN'S lines began with the order m == 4, but in
the lists above more recent and more accurate measures by others have been inserted.
BN. refers to BERGMANN, RL. to RANDALL ('Ann. d. Phys.,' 33, p. 741), 8. to
SAUNDERS (' Astrophys. Journ.,' 20, p. 188).
In 1905 FOWLER ('Astrophys. Journ. ,' 21, p. 84) discovered the lines in the F
series for Sr, beginning with K.R.'s 4892, adding two sets of his own observations
and "also the two connected triplets given in the table. But he attempted to combine
them all in one formula. The disarranged triplet for m = 3 is assigned in the text
above from K.R. and LKHMANN (' Ann. d. Phys.,' 8, p. 647). In the same volume
SAUNDERS (' Astrophys. Journ.,' 21, p. 195) added the last line and gave a similar
series for Ca, commencing with 4586. The measures inserted in the table, however,
are from later observations by BARNES (Bs.) (' Astrophys. Journ.,' 30, p. 14).
SAUNDERS also suggested a corresponding series for Ba in which the separations are
larger than those of the series assigned in the text. It is possible they may form a
similar series connected with the enhanced series. The lines for RaF and allied sets
are assigned also in the text above.
The F lines for Mg, Zn, Cd, Hg, Al, and Tl are all due to PASCHEN (' Ann. d.
Phys.,' 29, p. 651, and 35, p. 860). In Hg, however, he assigns 17195 to a combination
line, whereas in the text above it is assigned to F,.
[Notes, September 2, 1913.]
Note 1 -Since the present communication was read Messrs. FOWLER and REYNOLDS ('Roy Soc. Proc.,
89 p. 137) have published more accurate and extended measurements of the series, and other linen
The limit of the D or S series appears to be somewhat higher than that adopted in thu Table I
and the table will require the limit to be slightly raised to bring in multiples of A for the order d
With FOWFER and REYNOLD'S limit the mantissas show rising values for the first few o
decreasing an effect which has been explained in the text in similar cases by a collateral d.splaccracn
the limit after a particular order. If the limit be taken to be 39757-78 (i.e., 6-7 higher than in t
there results an order difference of 4A3 between m = 2 and 3, and equal mantissa, nfwrwar
» = 9, when there is a sudden change to a rising. If the limit is now raised by unUy for t
mantissa can again be all equal. Now, a displacement of 3 on the limit makes a change o
3 H 2
420 DR. W. M. HICKS : A CRITICAL STUDY OF SPECTRAL 'SERIES.
the change can be explained l>y a collateral change of ( - 3) D ( *) for m = 10-12. The scheme is then as
follows : —
2-828063 (20) + 10
*\
3-829715 (79) -36
4-829715 (190) + 9
5-829715 (362) + 58
6-829715 (626) -580
7-829715 (945) -173
8-829715 (2300) + 63
9-829715 (3200)+ 1373
f 10 -8297 (44) -16
Collaterals with j 1 1 • 8297 (57) - 8
(-S)D(oo) 1 12-8297 (78) -5
[13-8297 (92) + 8
The arrangement is seen to be exceedingly simple. The value for m = I has been omitted as the
evidence seems strong that Mg conforms to the Zn type and has no D(l). If, however, it is retained, it
is now 15A2 below that for m = 2. The values of Ao are so small that Mg can give no positive evidence
for any arrangement. This is evidenced by the changed arrangements called for by more accurate values.
The measures of FOWLER and REYNOLDS for the D lines are here adduced in order to complete
Appendix II. : —
3097-03 3093-09 3091-19
2851-76 2848-54 2846-88
2736-63 2733-64 2732-16
2672-53 2669-66 2668-24
2632-98 2630-14 2628-73
2606-73 2603-98 2602-59
2588-37 2585-63 2584-32
2575-02 2575-30 2570-96
2565-00 2562-30 2560-96
2557-29 2554-70
2551-22 2548-56
Note 2, p. 357. — The effect of positive collaterals on D ( ex) for m = 5, 6, 7 is to diminish the separations
of the triplets, so that from m = 5 onwards they would show diminishing values. It is interesting to note
that the observations bear this out.
Note 3, p. 380. — The F series of the alkalies. In K. the denominator 3-007542 has its mantissa
1-007542, and this is 14725(250 + ?) above that of K.F. (3), and 5A = 14700. This suggests that if
K.F. (3) has a denominator 3 + d, that of Fu(2) is 2 + d, and the 1182 -9 (W.N.) is Fi2(2) with a satellite
difference 28 and 1346-3 is the collateral Fn (2) (5A). In Rb 3-001138 has its mantissa 13289(430 + ?)
above that of RbF (3) and A = 12935, so that should have a similar arrangement to that in K. and 2156
would correspond to Flt (2) (A).
Note 4, p. 383.— CaF (2). The line with wave number 16203-40 is possibly the Fn and F12 combined.
If so, FIS is 16202-66 and is -74 behind 16203-40, and the satellite might have displaced the observation
towards itself from 16203-66. The actual separation would easily be 1-00 corresponding to a difference
2S, a usual F satellite difference. 16024-72 would be 1-06 on the other side and would correspond to a
collateral forming on the violet side.
[ 421 ]
IX. On the Self-inductance of Circular Coil* of Rectangular Section,
Ity T. H. LYLE, M.A., Sc.D., F.K.S.
Received February 27,— Read March 13, 1913.
As an approximate formula for the calculation of the self-inducfcince of a coil of
rectangular section,
L = 47r«najlog^-2}
was first given by MAXWELL,* where a is the mean radius and >• the geometric mean
distance of the section of the coil from itself, the current being supposed to be
uniformly distributed over the section.
In the following paper it will be shown that the same formula will give the self-
inductance to any order of accuracy when in it are substituted for a and r the mean
radius and the G.M.D. respectively, each suitably modified by small quantities which
depend on a and on the section of the coil, provided, of course, that the series for L is
convergent.
Tables will be given by means of which the modified values of a and r for any coil
of rectangular section can be found, and which, when substituted in the above formula,
will give L correct to the fourth order, uniform current density over the section being
assumed.
1. For the purpose of this paper the mutual inductance of two coaxal circles can
best be obtained after WEiNSTElNt by substituting in MAXWELL'S exact elliptic
integral formula!
M = Waft {(§-*) F- fE
I \ « / "•
the series expressions for F and E in terms of the complementary modulus V.
Thus we obtain
which is rapidly convergent when /•', the ratio of the least to the greatest distance
between the circles, is small.
* ' Elect, and Mag.,' vol. II., §706.
t ' WiED. Ann.,' 21, p. 344, 1884.
J ' Elect, and Mag.,' vol. II., § 701.
VOL. cexiir.-A 505. Pubiuhed -I—*' J""* 31' 101*-
422
DR. T. R. LYLE ON THE SELF-INDUCTANCE OF
Let these two circles be filaments A and B in the rectangular conductor whose
section is PQRS, fig. 1. Then, if x and y be the co-ordinates of B relative to axes
through A parallel to the sides of the rectangle,
and on substitution in the above we obtain, as ROSA and COHEN* have done,
y
2a 16aa
= M0 say,
where a is the radius of the circle A and i^ =
48«*
6144a4
p
2
A
B-
x y
>
,
;..
j
V*
4
-x-
XT-
r>
Fig. 1.
If the co-ordinates of the A circle, referred to axes through the centre of the
rectangle, be X and Y, then a in the above expression for M becomes a + Y where a
now and in what follows is the mean radius of the coil.
This substitution is most easily carried out by aid of TAYLOR'S Theorem. Thus the
complete expression for M is given by
M = M0+ Y + 7+&c.
da 1.2 da2
1 Bull, Bureau Standards,' vol. 2, p. 364, 1900,
CIRCULAR COILS OF RECTANGULAR SECTION. 423
2. It is well known that the self-inductance of a single circular conductor with
rectangular section for uniform current density is given by
i ffc el* ri«-Y ri*-x
~ I I
Mr J -jc J -a* J -S«-Y J -ifc-x
and for a coil of 71 turns is n" times this if MAXWELL'S correction* for space between
the wires be neglected, l> being the breadth and c the radial depth of the rectangle.
The evaluation of this definite integral, even for the second order terms, has
presented considerable difficulties. The first correct result to this order was that
given by WKiNSTEiN.t So far as I am aware no one has published a determination of
it to the fourth order. By a method indicated in the appendix to this paper the
integration can be carried out to the fourth order without difficulty and to still higher
orders if desired.
Thus the following expression for L has been obtained
211 . 3 . 5 . T.
23 6fr4— 7&V ,
or H . 2 . w
28 21
3659064-20356V-11442(
23 .3.5.7
in which
a = mean radius, 1> = breadth, c. = depth, d = jV + c1 = diagonal of the rectangle.
3. If r be the G.M.D. of the rectangle from itself, it is well known that
log r = log d— <{>
where
* -Elect, and Mag,' vol. II, § 693. ROSA (see •Bull. Bureau Standards/ vol. 3, p. 37, 1907) ha,
greatly improved on MAXWEr.L's correction.
' t ' WiED. Ann,' 21, p. 329, 1884.
424 DR. T. R. LYLE ON THE SELF-INDUCTANCE OF
Substituting for d in terms of r and </> in the above expression for L we obtain
aT, 8a , d2 \ Sa 1 d4 \ 8a ]~|
L = 47r«n2 log-- -2 + -5 ^2log - - +g,[ + J jo.log-- + ?4 ,
L " it l ? j ct (. ) JJ
where
=
P* 25.3
4-2085^V2- 11442c4
2-15.7
4. If
A = a( l+ml^7.+m,2
\ a a
d*
:!
a" a a /
!, »(2, v(1; ?ta> Jls can t*e determined so that
shall differ very little from the value for L given in § 3.
After substituting for A and R in the above and expanding in a series in d*/a3, the
first three terms of the expansion are identified with the corresponding terms of L in
the usual way, and in addition, the coefficient of the fourth term of the expansion, that
is the coefficient of def(t*, is equated to zero. The n3 term in R enables this to be done,
with the result that a closer agreement is obtained between the proposed formula and
that in § 3.
Thus
=p4, n2 = -
Hence, when A and R have been so determined the formula
will give the self-inductance of the coil correct to the fourth order.
CIRCULAR COILS OF RECTANGULAR 8FX7TION. 425
It will be seen that in the application of this formuk the coefficient », neecl rarely
xL It ^becomes important, however, when the mean radius is leas than the
diagonal of the section and especially in this case when the section is square or
1 1 < '.' M ly so.
It is obvious that in a similar way to the above, series for A and R could be obtained
which, when substituted in the proposed formula would make it practically equivalent
to L, no matter to what order the integration, if performed, had been carried out
5. In order to render convenient the practical application of the above formula to
termination of self-inductances the following tables* have been prepared.
TABUS I.
G.M.D. = r. rf» = A^e2.
c I
T or -•
.A __ ]{x(f
r
o c
r
•b + e
o-oo
1-5
0-223130
0-025
1-474734
0-223328
0-05
1-451005
0-223455
o-io
1-407566
0-223599
0-15
1-368975
0-223664
0-20
1-334799
0-223686
0-25
1-304680
0-223686
0-30
1-278284
0-223675
0-35
1-255312
0-223658
0-40
1-235461
0-223639
0-45
1-218448
0-223619
0-50
1-203998
0-223601
0-55
1-191853
0 • 22358 1
0-60
1-1817G8
0-223570
0-65
1-173516
0-223558
0-70
1-166888
0-223548
0-75
1-161691
0-223540
0-80
1-157752
0-223534
0-85
1-154914
0-223530
0-90
1-153034
0-223527
0-95
1-151987
0-223525
1-00
1-151660
0-223525
Table I. contains (l) values of ^>, that is of log,—, for different values of the ratio
r
b/c or c/b (2) values of the ratio r/b + c for different values of b/c. It will be noticed
how nearly r the G.M.D. of a rectangle from itself is proportional to the sum of the
* All the tables given in this paper have been calculated with the greatest care by the aid of a
" millionaire " calculating machine. Each separate series of numbers, not only the final series but every
intermediate series that had to be determined, was calculated at least twice, the end terms and one or two
intermediate terms of each series were carefully re-checked, and each series then examined by taking
successive differences.
VOL. CCXIII. — A. 3 I
426
DK. T. R. LYLE ON THE SELF-INDUCTANCE OF
sides. These figures will enable the G.M.D. for values of c/b or b/c, intermediate to
those given in the table, and consequently the first or important term of L for such
intermediate values to be obtained with great accuracy.
Table II.- contains the values of the coefficients m,, m2, nl} n.it nx, for thick coils, that
is for ones in which b is greater than c for different values of the ratio c/b, and
Table III. contains the values of the same coefficients for thin coils, that is for ones in
which b is less than c for different values of the ratio b/c,
TABLE II.
c
b'
10'2Wi.
10%,
10%.
10%,.
10"%.
o-oo
3-12500
-9-766
0-78125
-8-647
- 6-9
0-025
3-12370
-9-746
0-69934
-8-179
- 8-2
0-05
3-11980.
-9-688
0-61606
-7-663
- 9-7
o-io
3-10437
-9-461
0-44541
-6-505
-12-6
0-15
3-07916
-9-094
+ 0-26934
-5-202
-15-7
0-20
3-04487
-8-604
+ 0-08919
-3-795
-18-9
0-25
3-00245
-8-011
-0-09342
-2-326
-22-1
0-30
2-95298
-7-340
-0-27664
-0-838
-25-2
0-35
2-89764
-6-614
-0-45856
+ 0-630
-28-0
0-40
2-83764
-5-857
-0-63754
+ 2-045
-30-6
0-45
2-77417
-5-090
-0-81198
3-378
-32-8
0-50
2-70833
-4-332
-0-98060
4-610
-34-7
0-55
2-64166
-3-596
-1-14240
5-727
-36-1
0-60
2-57353
-J2-895
-1-29662
6 • 725
-37-2
0-65
2-50622
-2-237
-1-44274
7-600
-37-8
0-70
2-43988
-1-626
-1-58048
8-356
-38-2
0-75
2-37500
-1-066
-1-70975
8-999
-38-2
0-80
2-31199
-0-556
-1-83060
9-536
-37-9
0-85
2-25115
-0-097
-1-94321
9-978
-37-5
0-90
2-19268
+ 0-314
-2-04787
10-333
-36-8
0-95
2-13672
+ 0-680
-2-14491
10-611
-3o-9
1-00
2-08333
+ 1-004
-2-23473
10-824
-35-0
1-05
2-03255
+ 1-289
-2-31773
10-978
-33-9
It will have been noticed that the coefficient mY and ma are algebraic and can be
easily calculated for any value of c/b. Those in the tables are given for convenience.
6. If the formula for L given in § 2 be written in the form
a [7, d2 , d*\, Sa , , , <P , , cZ4l
L = t-ran* f l+m, -,+m, -Jlog -r-^+Z, -2+4 -, L
L \ d \JU I 'ji C* C* _J
tables giving mlt m2, Z0, llt and 12, for different values of c/b would also render easy
the computation of the self inductances of coils. Such tables have been computed from
WEINSTEIN'S formula by STEFAN,* but he is in error in thinking that the second order
coefficient has the same value for a given value of b/c in a thin coil as it has for the
WIED. Ann.,' 22, p. 113, 1884.
.CIRCULAR COILS OF RECTANGULAR SFXTTION.
127
same value of c/b in a thick coil. The second order coefficients he gives are correct for
thick coils.
In using the above formula with tables for the computation of L for valum of r/h
or b/c intermediate to those given in the tables, the value of /0 which is part of the
large or first order term will have to be obtained by interpolation, whereas in the
TABLE III.
b
c
lOtai.
Win*.
10*11,.
10*11*
lOS,.
o-oo
1-04167
2-387
-3-21180 6-073
+ 0-5
0-025
1-04297 2-391
-3-23737 6-134
+ 0-6
0-05
1-04686 2-403
-3-25967.
6-214
+ 0-5
o-io
1-06229
2-451
-3-29420
6-430
+ .0-1
0-15
1-08751
2-524
-3-31479
6-724
- 0-6
0-20 1-12179
2-614
-3-32107
7-090
- 1-7
0-25 1-16421
2-711
-3-31313
7-513
- 3-2
0-30 1-21368
2-806
-3-29150
7-978
- 5-1
0-35
1-26902
2-886
-3-25703
8-463
- 7-3
0-40
1-32902
2-943
-3-21091
8-949
- 9-8
0-45
1-39250
2-969
-3-15447
9-414
-12-4
0-50
1-45833
2-960
-3-08918
9-845
-15-0
0-55
1-52551
2-912
-3-01651
10-226
-17-7
0-60
1-59313
2-824
-2-93794
10-548
-20-3
0-65
1-66044
2-697
-2-85483
10-805
-22-8
0-70
1-72678
2-534
-2-76844
10-994
-25-2
0-75
1-79167
2-337
-2-67990
11-115
-27-4
0-80
1-85468
2-111
-2-59020
11-171
-29-3
0-85
1-91552
1-861
-2-50019 11-164
- 31 • 1
0-90
1-97399
1-590
-2-41057
11-100
-32-6
0-95
2 • 02995
1-303
-2-32192
10- 98.-!
-33-9
1-00
2-08333
1-004
-2-23473 10-824
-35-0
1-05
2-13412
0-696
-2-14940
10-624
-36-2
method previously given, the whole of the first order term can be easily got with great
accuracy, by making use of the nearly constant ratio of r to b + c indicated by the
figures given in the third column of Table I.
The coefficients /„, /„ /,,, occurred in the computation of Tables I., II., and III., m,
and ma are the same as in these tables, and are in any case algebraic as
m, =
nt« =
2". 3». 5
Table IV. gives the values of /,„ /„ and /, for different values of the ratio c'b for
thick coils, and of b/e for thin coils.
3 I 2
428
DR. T. R. LYLE ON THE SELF-INDUCTANCE OF
TABLE IV.
For both thick and
thin coils.
For thick coils.
For thin coils.
c b
T OT~-
1) C
k.
e,
V
MWh
10«/2.
b
c
10%
10422.
o-oo
0-500000
o-oo
0-78125
6-510
o-oo
3-73264
4-167
0-025
0-525266
0-025
0-78358
6-490
0-025
3-73250
4-161
0-05
0-548995
0-05
0-79098
6-427
0-05
3-73181
4-143
o-io
0-592434
o-io
0-81983
6-184
o-io
3-72716
4-058
0-15
0-631025
0-15
0-86679
5-794
0-15
3-71605
3-897
0-20
0-665201 '
0-20
0-93023
5-283 '.
0-20
3-69664
3-655
0-25
0-695320
0-25
1-00821
4-677
0-25
3-66784
3-336
0-30
0-721716
0-30
1-09841
4-010
0-30
3-62925
2-951
0-35
0-744688
0-35
1-19836
3-313
0-35
3-58103
2-516
0-40
0-764539
0-40
1-30570
2-614
0-40
3-52385
2-050
0-45
0-781552
0-45
1-41799
1-940
0-45
3-45866
1-572
0-50
0-796002
0-50
1-53310
1-311
0-50
3-38668
1-099
0-55
0-808147
0-55
1-64911
0-740
0-55
3-30919
0-648
0-60
0-818232
0-60
1-76440
+ 0-238
0-60
3-22752
+ 0-231
0-65
0-826484
0-65
1-87761
-0-191
0-65
3-14294
-0-143
0-70
0-833112
0-70
1-98767
-0-546
0-70
3-05662
-0-468
0-75
0-838309
0-75
2-09376
-0-829
0-75
2-96960
-0-740
0-80
0-842248
0-80
2-19532
-1-044
0-80
2-88278
-0-959
0-85
0-845086
0-85
2-29194 -1-197
0-85
2-79693
-1-127
0-90
0-846966
0-90
2-38342 -1-294
0-90
2-71265
-1-245
0-95
0-848013
0-95 2-46966
-1-342
0-95
2-63045
-1-318
1-00
0-848340
1-00 2-55069 -1-349
1-00
2-55069
-1-349
1-05
0-848044
1-05
2-62659 -1-320
1-05
2-47369
-1-344
7. The only available means of testing the above methods of computing self-
inductances and of finding the limit outside which they are practically reliable is to
compare the results they give with those given by LOBENZ'S* exact elliptic integral
formula for the self-inductance of a current sheet solenoid, which is
m
3 d? ( F k3
where a is the radius, d the length of the solenoid, and
.
— 1
Thus consider the case of a solenoid whose length is twice its radius.
Here
* ' WIED. Ann.,' 7, p. 161, 1879,
CIRCULAR COILS OF RECTANGULAR SECTION. 429
and from Table II.
A = (1 + 4xOU3125- 16x0-0009766) a,
= ri09375a.
R = (1 + 4 x 0-0078125-16 x 0-0008647-64 x 0-0000069) r,
= l'016973r.
so
,= log.
.
"1 '016973
(where <f> is given in Table I.)
= log, 4 + ^ + 0-086965,
= 2-973259,
and
(8A
log^-2 = 4 irax T07970.
L\
LORENZ'S exact formula gives
L = 4Tax T08137.
Thus the error in this case is 1 part in 650.
When the comparison is made in less extreme cases we find the agreement with the
Lorenz formula very close.
Thus when the length of the solenoid is equal to its radius («) either of the methods
of this paper give
L = 207453a,
while LORENZ'S formula gives
L = 207463a,
showing an error of 1 part in 20,000, and when the length of the solenoid is half the
radius we obtain
L = 28'85332rt,
as against the Lorenz value
L = 28'85335a,
showing an error of only about 1 part in 1,000,000.
430 DR. T. R. LYLE ON THE SELF-INDUCTANCE OF
APPENDIX I.
In order to determine L to the fourth order we have seen that it is necessary to
evaluate the definite integral
fie rift /•'•"-* rjA-x
MdxdydXd\T
J -!JC. I _?.A J-Jc-Y J-^A-X
where
M = P + QY+RY'+SY-'+TY4,
P, Q, R, S, and T being functions of x and y.
If we proceed in the ordinary way by putting in the limits after each integration
the expression becomes very cumbrous on account of the nature of some of the
functions (log and tan"1) with which we have to deal.
By the method to be explained below all the integration will be carried out first
and the limits introduced in an easy and symmetrical way at the finish.
1 . Dealing first with P, the term independent of Y, if
y = 6 (xy),
the result, with limits introduced, of the integrations with respect to x and y will bo
where
y = IC_Y y. = — fc— Y.
We have now to evaluate four definite integrals of which the first is
Changing the variables to xt and «/i and the limits accordingly, this integral is
equal to
e(xly1)dxldyi
Jc •>!>
where
<j> (xy] = |j 6 (xy) d.c dy = Jjjj P
Dealing in the same way with the three remaining integrals
-f" T 0(xM)dXdY, -\* ['' 9(x.4h)dXdY, and T f"* 6(x2y2
J_i<:J_iJ J_JcJ_i6 J_;CJ_JJ
CIRCULAR COILS OF RECTANGULAR SECTION. 431
we find that they become
} ^efayJdXidfo, £j" ^(x^d^dy, and f j" 6(sj/,)d*tdi/,
respectively, which are equal to
and
*(0,0)
respectively, where
0 (xy) has the same meaning as before.
Hence, if
fk r}6 rk-Y pj*-x
J-Sc J-}t J-Je-Y J-Jt-X
is equal to
This expression, obtained from the function ^>(a;«/) by substituting in it b, c, —h, —c,
0, 0 in the way indicated, will, in what follows, be designated by
2. As an illustration of the above I will indicate the process as applied to the simplest
term in P involving log (a^+y2).
Thus to obtain
fk (•»* rk-Y rJA-x
J_JC J_i» J-k-Y J-J6-X
we find by simple integrations that
By inspection it is seen that
0(00) =
2*(±6, ±c) =
0 (0, c) = * (0, -c) = - |- log c".
432 DR. T. R. LYLE ON THE SELF-INDUCTANCE OF
Hence the definite integral above is equal to
s , -\ V i &2 + e2 <? ^ &a + c" 25 III. , <• <• .l\~]
+<r)- 6? log IT - w log — -T + a (c tan i + b tim ;)j
The above is the well-known definite integral used for determining the G.M.D. of
a rectangle from itself.
3. To determine
between the given limits.
If
6(xy) = II Qdxdy,
the result of the integrations with respect to x and y will now be
Y{o(xM)-e\
where xu yl} x2, y% have the same significations as before.
We have now to evaluate four integrals of the type
T T Y
J -\c J -|i
Proceeding as in § 1, these, affected by their proper signs, become
(ic-f/OH^O^i^-f f ($c+ya) 6(x,y2}dx^dy.,
•Ic Jb J —c Jli
+ (^c-t/i)0(x2y1)dx2di/l-\ \
•>c J -b J -c J -b
so that, if in this case
*' M = J V dy jjj Q dx- dy,
and
-<!> (-6, -P) + 20 (0, -c)-20 (0, c),
then
between the given limits is equal to
where 2 has the signification given to it in § 1 .
CIRCULAR COILS OF RECTANGULAR SECTION. 433
4. In :, similar way it can 1* 8|1OWII t|,at
tetween the given limits is equal to
where, in this case
and that
between the given limits is equal to
where, in this case,
The result of integration can now be easily written out for integrations involving
higher powers of Y.
5. Before proceeding with the integrations it is advisable to have prepared
beforehand a table giving
and
faftan-'^f/ic
.' x
from n = 0 to it , = 7.
If this be done, and the method indicated above followed, the work presents little
difficulty and is not veiy tedious.
VOL. CCXIH. — A. 3 K
434 DK. T. It LYLE ON THE SKLF-IXDUCTAWK OF
APPENDIX II.
(Added October 1, 1913.)
Since writing the above I have determined the sixth order term of the series for L.
In order to do this it was necessary to extend to the sixth order MAXWELL'S series
formula (see § l) for M, the mutual inductance of two unequal coaxal circles which
ROSA and COHEN* had already extended to the fifth order.
Thus
r { 2a 2*. a? 25. a'
15,r4-42xy-17?/4 4 5.x '//- 3 (.).*•-//
210.a4 2". a6
35s8-345a!V+ 45igy + 89/1
214. a"
2. a 24.a2 24.3.a:i
93a:4-534xV-19?/4
2u.3.a* 212. 3. 5. a5
1235x6-17445a:y+12045.ry-7371j/r|
When in M we substitute, as explained in § 1, a + Y for a, the term of the sixth
order in the variables x, y, and Y becomes equal to U, where
U = p + q Y + ?-Y2 + SY3 + * Y4 + M Y5 + t'Y",
in which
p = ^a |- log
1235X-"- 17445./V + 12045^y-7371i/>i
2*. 3. 5.0*
g = 4^ai XU+V r/ -log-* + : X!/<>ii\ f'% ~f
^.Cc / A * D • v • Ci J
4 -. ,
»1) g "
2».«1) g r " 2". a6 J'
?• 25.3.a6
* I 24. a8 °ff 7 26.3.a« J
n = 4-rra .
= 47T« .
2 . 5 . a«
1
2. 3. 5. a"'
* ' Bull. Bureau Standards,' 2, p. 364, 1906.
CIRCULAR COILS OF RECTANGULAR SECTION. 435
The term of the sixth order in the series for L is the value of the integral
between the specified limits, and I have found it to be equal to,
2". « *5. 7 a5 [(525y-161OfrV + 7706V + 108c») log ^
+ (^jjf- V-3220&V+2240&V4) ti,
OKQ /c 7
•Ijj. c«r _ 2" ( | ft" _ 46 V + 1 6V j IT,
2161453 ,, 617423 M, 8329 M 4 . 4308631
" 23.3.5.7 ~ 271^5 6 ~2T3~56 + 2'.3.5.7 j
in which w, c, w, and ti have the significations assigned to them in § 2.
The method of integration indicated in this paper renders the determination of L in
series form comparatively easy for the special cases of a solenoid (c/b = o), and a flat
circular ring coil (b/c = 0), uniform current density being assumed.
Thus COFFIN'S formula* for a solenoid can be easily obtained, and KAYLKwnt and
NIVEN'S formula for a coil whose axial dimension (b) is zero can be extended to the
sixth order, giving
[7. c3 lie4 103«* 8a
L = 4^'a 11 + j—tf + 2».3..5.0« + 2'«.3.5.7.
43cr c4 429H57i><-'1
7. 3*. a3 2s. 3 . V. a4 2". 3'. 5'. 7'. «"J '
which can also be obtained by putting b = 0 in the general formula obtained above
for L, and remembering that when b/c = 0, o = 1, and w' = 1.
* 'Bull. Bureau Standards,' 2, p. 113, 1906.
t RATLKIOH, ' Collected Papers,' 2, p. 15.
3 K 2
[ 437
X. A Method of Measuring th,- Pressure Produced in the Detonation of
Explosives or by the Impact of Bullets.
/>'// BERTRAM HOPKINSON, F.R.S.
Received Octoter 17,— Read November 27, 1913.
THE determination of the actual pressures produced by a blow such as that of a
rifle bullet or by the detonation of high explosives is a problem of much scii-ntitir
and practical interest but of considerable difficulty. It is easy to measure the transfer
of momentum associated with the blow, which is equal to the average pressure
developed, multiplied by the time during which it acts, but the separation of theae
two factors has not hitherto been effected. The direct determination of a force acting
for a few hundred-thousandths of i» second presents difficulties which may perhaps be
called insuperable, but the measurement of the other factor, the duration of the blow,
is more feasible. In the case of impacts such as those of spheres or rods moving at
moderate velocities the time of contact can be determined electrically with pomndambk
accuracy.* The present paper contains an account of a method of analysing experi-
mentally more violent blows and of measuring their duration and the pressures
developed.
If a rifle bullet be fired against the end of a cylindrical steel rod there is a definite-
pressure applied on the end of the rod at each instant of time during the period of
impact and the pressure can be plotted as a function of the time. The pressure-time
curve is a perfectly definite thing, though the ordinates are expressed in tons and the
abscissae in millionths of a second ; the pressure starts when the nose of the bullet
first strikes the end of the rod and it continues until the bullet has been completely
set up or stopped by the impact. Subject to qualifications, which will l)e considered
later, the result of applying this varying pres-
sure to the end is to send along the rod a wave
of pressure which, so long as the elasticity is
perfect, travels without change of type. If the
pressure in different sections of the rod be
plotted at any instant (fig. l) then at a later
time the same curve shifted to the right by a distance proportional to the time
will represent the then distribution of pressure. The velocity with which the wave
travels in steel is approximately 17,000 feet per second. As the wave travels over
any section of the rod, that section successively experiences pressures represented
* SEAUS, ' Proc. Camb. Phil. Soc.,1 vol. xiv. (1907), p. 257, and references there given.
VOL. CCXIII.— A 506. Publisl"d "P"*"1* J"""ry "• 19U
Figl
438 MR. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
by the successive ordinates of the curve as they pass over it. Thus the curve also
represents the relation between the pressure at any point of the rod and the time,
the scale being such that one inch represents the time taken by the wave to travel
that distance which is very nearly .,00*000 of a second. In particular the curve
giving the distribution of pressure in the rod along its length is, assuming perfect
elasticity, the same as the curve connecting the pressure applied at the end and the
time, the scale of time being that just given.
The progress of the wave of stress along the rod is accompanied by corresponding
strain and therefore by movement. It is easy to show that the same curve which
represents the distribution of pressure at any moment also represents the distribution
of velocity in the rod, the scale being such that one ton per square inch of pressure
corresponds to about 1'3 feet per second of velocity. Until the wave reaches any
section of the rod that section is at rest. It is then, as the wave passes over it,
accelerated more or less rapidly to a maximum velocity, then retarded, and finally left
at rest with some forward displacement. In this manner the momentum given to the
rod by the application of pressure at its end is transferred by wave action along it,
the whole of such momentum being at any instant concentrated in a length of the rod
which corresponds, on the scale above stated (one inch = .,00^)OU second), to the time
taken to stop the bullet completely. Consider a portion of the rod to the right of any
section A (fig. l) which lies within the wave at the moment under consideration.
The pressure has been acting on this portion since the wave first reached it, that is
OA
for a time represented by the length OA and equal to -y^- where V is the velocity of
propagation. The momentum which has been communicated to the part under
consideration is equal to the time integral of the pressure which has acted across the
section A, that is to the shaded area of the curve in the figure. The portion of the
rod to the right of the section is continually gaining momentum at the expense of the
portion to the left while the wave is passing, the rate of transfer at any instant being
equal to the pressure.
When the wave reaches the free end of the rod it is reflected as a wave of tension
which comes back with the same velocity as the pressure wave, and the state of stress
in the rod subsequently is to be deter-
mined by adding the effects of the direct
and °f the reflected waves. Now suppose
that the rod is divided at some section, B,
near the free end (fig. 2), the opposed
surfaces of the cut being in firm contact
and carefully faced. The wave of pressure travels over the joint practically
unchanged and pressure continues to act between the faces until the reflected
tension wave arrives at the joint. The pressure is then reduced by the amount
of the tension due to the reflected wave and as soon as this overbalances at
INTONATION OF HIGH EXPLOSIVES OK BY THE IMPACT OF BULLETS. 439
section B the pressure of the direct wave (which is the moment shown in the
figure) the rod, being unable to withstand tension at the joint, parts there and
the end flies off'. The end piece has then acquired the quantity of mmm-ntum
represented by the shaded area in the figure, equal to the time-integral <>f th-
pressure curve from 0 to B, less that of the tension wave during the time for whirl,
it has been acting, that is from O' to B. The piece flies off with this um»unt ..f
momentum trapped, so to speak, within it. If it be caught in a ballistic pendulum
and its momentum thus measured we have the time integral of the pressure CIIIA.-
between the points B and B' on the pressure-time curve which are such that they
correspond to equal pressures on the rising and falling parts of the curve, while th->
time-interval between them is equal to that required for a wave to travel twice the
length of the end piece. By taking end pieces of different lengths and measuring tl it-
momentum so trapped in each the area of the pressure-time curve over corresponding
intervals can be obtained. In general the precise form of the curve itself cannot he
deduced tecause the points of commencement of the several intervals are not known.
Thus a given set of observations would be consistent with
any one of the three forms shown in fig. 3 which can be
derived from one another by shearing parallel to the base
so that the intercept of any line such as AA' is the same
i>n all. But the maximum pressure and the total duration
of the impact can always be obtained, and these are the
most important elements. The maximum pressure is the limiting value of the
average acting on a piece when the piece is very short, and the duration corresponds
to twice that length of piece which just catches the whole of the momentum leaving
the rod at rest. If the circumstances of the impact are such that the pi-emu re is
known to rise or to fall with great suddenness, the curve assumes the form I. or III.
and its form may l)e determined completely from the observations.
This is the basis of the method described in the present paper. A cylindrical rod
or shaft of steel is hung up horizontally by four equal threads so that it can swing in
a vertical plane remaining parallel to itself. A short piece of rod of the same diameter
is l.uttc.l up against one end being held on by magnetic attraction but otherwise free.
A rifle bullet is fired at, or gun-cotton is detonated near, the other end-; the short
piece flies off and is caught in a box suspended in a similar manner to the long rod.
Suitable recording arrangements register the movement both of the long rod and of
the box, and the momentum in each is calculated in the usual way as for a Iwllistic
pendulum. Sufficient magnetic force to hold the end-piece in position is provided by
putting a solenoid round the rod in the neighbourhood of the joint. The slight force
required to separate the piece from the rod under these conditions may lie neglected
in comparison with the pressures and tensions set up, since these amount to several
tons on the square inch, and, practically speaking, the joint will transmit the pressure
wave unchanged but will sustain no tension.
440 MR. P. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
A~
-, i-, e
Pressure Produced by tlie Impact of Lead Bullets.
The pressure which should be produced by the impact of a lead bullet can be
predicted theoretically, and the study of this pressure was made rather with a view
to checking the method than in the hope of discovering any new facts. At velocities
exceeding 1000 feet per second lead behaves on impact
Pjg.4. against a hard surface practically as a perfect fluid.
The course of the impact is shown in fig. 4. The base of
the bullet at the moment of striking is at A ; a little later
it is at B. Assuming perfect fluidity the base of the
bullet knows nothing of the impact at the nose and
continues to move forward with unimpaired velocity.
Hence the time elapsing between the two positions shown
in the figure is -^. The momentum which has been destroyed up to this time is to
a first approximation that of the portion of
the bullet which has been flattened out,
namely that portion shown shaded in the
dotted figure. Knowing the distribution
of mass along the length this is easily ^
calculated. This simple theory is subject ^
to some qualifications due partly to want ^
of perfect fluidity, and partly to the fact ^j-
that the sections of the bullet are not s
brought right up to the face and there
stopped dead, as is assumed in the theory,
but are more or less gradually retarded or
deflected in the region of curved steam-lines
at C. These corrections are, however, most
conveniently introduced when comparing
the theory with the experimental results.
The bullets used were of two patterns,
one the ordinary service form (Mark VI.)
and the other a soft-nosed bullet supplied
on the market for sporting purposes. Both
are of lead, encased in nickel. Sections of
the bullets are shown in fig. 5.* Sample
bullets were sawn into sections, and the
sections weighed. The distribution of weight along the length thus determined
The soft-nosed bullet (lower figure) has four longitudinal saw-cuts in the nickel casing ; the section is
taken through two of these cuts.
•5 -75 I-O
Lengths, Inches.
DETONATION OF HIGH EXPLOSIVES OR BY THE IMPACT OF BULLETS. 441
is shown in the curve fig. 5. The hullets were almost precisely alike both in regard
to total weight (0'0306 Ibs.) and distribution of weight along the length.
Most of the experiments were made with the service cartridge, in the service rifle,
giving an average velocity of 2000 feet per second. These cartridges were very
uniform, the range of variation in velocity being under one per cent. Some
experiments were also made with cartridges giving velocities of about 1240 feet per
second and 700 feet per second respectively.
The rod against which the bullet was fired was in most cases of steel containing
C, 0'4 per cent. ; Mn, 1'05 per cent. Its breaking strength was 37 tons per square
inch with 24 per cent, elongation over 8 inches. The end of the rod was heated to a
white heat in the forge and quenched and would then stand a large number of shots
without serious damage. In some cases tool steel hardened, and tempered blue, was
used, but it was found difficult to get the temper exactly right. The pieces butted to
the end of the rod were usually of mild steel. For recording the movement of the rod
and of the box in which the piece was caught each was fitted with a pencil which
moved over a horizontal sheet of paper and the length of the mark was measured.
Assuming that the bullet strikes the rod fairly in the centre, and that the
fragments are shot out radially, the total momentum recorded in rod and piece should
be equal to the momentum of the bullet, which at 2000 feet per second is 61 "2 Ib. feet
per second units. In fact, considerable variations were found in the total momentum.
For instance, in 110 shots fired at a 1-inch rod, the maximum total recorded was 76,
100
60
40
Fig.6.
Length of piece , Inches
the minimum 50, and the average 63. With a rod of l£ inches diameter, the
variation was less ; 61 shots showed a minimum of 59, a maximum of 70, and a mean of
62-5 High values are probably due to fragments being thrown back by irregularities
in the surface of the rod, low values to slight errors in aiming. It was found,
however, that with a piece of given length, the total momentum was shared between
the piece and the rod in a nearly constant proportion, though the
VOL. CCXIII. — A. 3 L
442 MR. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
might vary widely. This is to be expected if the explanation just given of the
irregularities is correct. For instance a cup-shaped cavity in the rod such as is
formed after a large number of shots will give a high value for the momentum, but if
not too pronounced it will not seriously affect the form of the relation between
pressure and time.
The results have accordingly been reduced by taking in every case the percentages
of the total momentum found in the piece. The following table gives details of one
set of experiments. It was found that there was no systematic difference between the
service bullets and the soft bullets, and the results for both types are included in
the table : —
ROD, 1 inch diameter, 43 inches to 50 inches long. 2000 feet per second.
Percentage of total in piece.
Total momentum in rod and piece.
Length of
Number of
piece.
shots.
Maximum.
Minimum.
Mean.
Maximum.
Minimum.
Mean.
inches
0-5
19
'11-6
9'8
10-9
63
58
60
1-0
25
24-0
20-4
22-1
66
58
62
2-0
8
46-0
40-6
43-2
73
60
65
3-0
26
63-0
58-0
61-0
66
59
62
3-5
6
71-0
69-0
70-4
71
65
67
4-0
6
82-0
79-0
79-7
67
50
62
5-0
11
93-0
94-5
93-5
76
59
67
6-0
9
99
• — •
97-6
69
63
66
The mean percentages given in the third column of the table are plotted against
length of piece in fig. 6. As the wave travels 2'04 inches in 10~5 seconds, 1 inch
length of piece represents 0'98 x 10~8 seconds.* The slope of this curve represents
pressure, and as already explained the maximum pressure is represented by the slope
at the origin. This is 22 per inch, and assuming an average total momentum of
61 '2 units the corresponding pressure is
0'22x61'2xlQ5
32'2xO-98
= 42,600 Ibs. or 19'0 tons.
It will also be noticed that the impact is practically complete in 6xlO~5 seconds,
97^ per cent, of the total being then accounted for in the piece.
According to the simple theory, which regards each element of the bullet as coming
up to the end of the rod with its velocity v 0 unimpaired and there suffering instant
t The value of E for the mild steel of which the pieces were made was found to be 3 '00 x 107 Ibs. per
square inch. The density was 482 Ibs. per cubic foot. Both determinations are probably right within
/TJT
1 per cent. The velocity of propagation A/ -- is 17,000 feet per second.
DETONATION OF MICH KXPI.OSIVKS <>l; I!Y TIIK IMPACT <»K P.l'l.l I I :
stoppage, the pressure at any time is \v0* where X is the mass per unit length at the
section which is undergoing stoppage at the time. The pressure-time curve, calculated
in this way, is shown in fig. 7, in which the ordinates are proportional to the values of X.
This is the same curve as that giving the distrihution of mass along the length of the
bullet, the abscissa scale being such that the length OF within which the impact is
.MIDI III
40000
^3001)0
zo.ooo
10,000
Rg.7
-
Time , M's3a».
t
•• F
complete is equivalent to the time required by the bullet to travel its own length
(1'25 inches) at a velocity of 2000 feet per second. This is 5'2x lQ-» seconds. The
maximum pressure corresponds to the maximum value of X (0 35 Ibs. per foot) and is
Pj35_x2000x200p
32'2
= 43
This difference is
which is 2£ per cent, in excess of the value found by experiment,
no more than can be accounted for by errors of observation.
The momenta which should according to theory be taken up by various lengths
piece are readily calculated from this curve. For instance, that corresponding
3-inch piece is the area ABODE. The following table shows the results a
with the corresponding observed values. The momenta are reckonec
of the total :—
Percentage momentum in piece.
Length of piece.
Calculated.
Observed.
inches
Q
65
61
1
84
80
98-5
93-5
6
100
97-5
3 L 2
444 MR. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
The differences between the calculated and observed figures in this table are
probably rather outside experimental errors. Especially is this the case as regards
the 5-inch and 6-inch pieces. The impact seems to last appreciably longer than it
ought.
The Effect of the Rigidity of the Bullet.
In the simple theory it is assumed that the bullet is absolutely fluid. In fact, it
possesses a certain rigidity, partly because of the nickel casing and partly because of
the viscosity of the lead the effects of which may be quite appreciable at such high
speeds of deformation. The general effect of rigidity may be represented by saying
that any section of the bullet requires to be subjected to an end-pressure P before it
begins to deform at all, and this pressure must act across the section CC (fig. 4) where
deformation is just beginning and where, if the bullet were really fluid, there would be
no pressure. To a first approximation, P will be proportional to the area of the cross-
section of the bullet which is undergoing deformation, that is to X the mass per unit
length in the plane CC. The pressure P is added to that due to the destruction ot
momentum, making a total pressure P + Xv2 where X is the mass per foot of the section
of the bullet in the plane CC, v the velocity of that section. Further, the part of the
bullet behind CC is being continually retarded by the pressure P, with the result that
the hinder parts do not come up with unimpaired velocity va, as they would if the
bullet were quite fluid, but with a diminishing velocity.
The general effect of this is obvious. In the early stages of the impact there has
not been time for much retardation, and the pressure will be increased above the
theoretical value1 by nearly the amount P. As the hinder parts come up, however,
with less and less velocity, the fluid pressure term diminishes until the pressure falls
below the theoretical value in spite of the rigidity term P. Applying this correction
to a pressure curve such as that in fig. 7 in which the maximum pressure occurs
somewhat late in the impact, it will be seen that the general effect will be to reduce
that maximum, and also to make it flatter. Furthermore, since the tail of the bullet
takes longer to reach the end of the rod, the impact will be prolonged beyond the
theoretical time.
It is easy to get a rough idea of the magnitude of these effects. Assume that the
bullet is cylindrical and of mass X per unit length and that the deforming pressure is
constant. Let x be the length of the bullet behind the plane CC (fig. 4). This
portion is moving as a rigid body with acceleration x and its equation of motion is
\xx = — P,
which integrates in the form
p
?x2 = log x + const.
X
If I be the length of the bullet and 1 0 its velocity on striking, and if we neglect the
DETONATION OF HIGH EXPLOSIVES OR BY THE IMPACT OF BULLETS. 445
small distance between the plane CC and the end of the rod, the constant of integra-
tion is
p
bt'> + -
\
and we have
x* 2P, /
1 -- , = — -log-.
t'u8 At'* " ./•
From this x can be plotted in terms of x, and thence in terms of t. 'IV total pressure
P + Xcc3 is then plotted in terms of the time.
As an example, take X = 0'35 Ibs. per foot, I = 1'05 inches which correspond to a
bullet having the same mean density diameter and total mass as those used in the
experiments. The pressure required to stop such a bullet at 2000 feet per second, if
fluid, would be constant and equal to 43,500 Ibs. If P be taken as ^ of this, or
2170 Ibs., and the curve plotted as described, it will be found that when x = 0'3/ the
hydrodynamical pressure Xca has dropped 12 per cent, making, after allowing the
addition of 5 per cent, for the rigidity, a nett drop of 7 per cent. Furthermore,
the momentum still left after a fluid bullet would have been completely set up is about
4 per cent, of the whole.
If corrections of this amount were applied to the calculated figures in the last section.
the effect would be to make the observed maximum pressure about 4 per cent, too
high, while the observed time of impact would" be still slightly too long. It was found
that to crush the cylindrical part of the service bullet in a testing machine required
an end pressure of about 1800 Ibs., but the nickel casing failed by buckling, where!*
in the impact it apparently bursts and is torn into strips along the length of the
bullet. The pressure required to deform the bullet in the latter case, after rupture is
once started, is probably less than 2000 Ibs. Thus, while the difference between the
observed and calculated times of impact may undoubtedly be referred in part to
rigidity, it is unlikely that the whole can be accounted for in this way.
Discussion of Errors Inherent in the Method of Experiment.
In calculating the pressure from the momentum in the piece which is tin-own off
the end of the rod it is assumed that the pressure wave transmitted along the rod
represents exactly the sequence of pressures applied at the end, that it travels along
the rod and through the joint without change of type, and that it is perfectly
reflected at the other end. These assumptions are correct if the wave is long
compared with the diameter of the rod, and if the pressure is uniformly distributed
over the end, but are subject to certain qualifications in so far as these conditions are
not fulfilled.
(a) Effect of Length of the Rod— The mathematical theory of the longitudinal
oscillations of a cylinder shows that a pressure wave of simple harmonic type is
propagated without change, but the velocity of propagation depends on the wave-
length. Because of the kinetic energy involved in the radial displacements, which is
446
MR. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
negligible when the wave is long compared with the diameter, the velocity diminishes
with the wave-length. If the wave-length be -- , and if the radius of the cylinder
/E 7
be a, the velocity is A/ — (1— £<r2y2a2) correct to the square of ya* In a wave of
P
any form, the simple harmonic components move with different velocities, and the
wave accordingly changes its form as it progresses.
Rough calculation of this effect on waves generally similar in form to that produced
by the blow of the bullet, but of periodic character, showed that the change should
not be very serious with rods of the lengths and diameters used in these experiments.
It was, however, thought advisable to check this inference by direct experiment, and
trials were therefore made with a rod 15 inches long and 1 inch diameter. The
small mass of this rod precluded its use as a ballistic pendulum suspended in the
ordinary way, it was therefore arranged to slide in bearings and to compress a spring
buffer. Difficult questions arose as to the precise allowance which should be made
for the kinetic energy given to the spring (which was of considerable mass) by the
rod, and no attempt was therefore made to get an accurate measure of the total
momentum. Instead of taking the fraction of this total which was trapped in the
piece, the absolute values of the momenta so trapped were taken in a series of shots,
in each of which, from the accuracy of the aiming and the absence of cupping in the
end, it might be assumed that the total momentum was approximately equal to the
average. The results are shown in the following table and are compared with the
corresponding figures obtained with the long rod : —
ROD, 1 inch diameter. 2000 feet per second.
Momentum given to piece.
Length of piece.
Number of
shots.
Short rod (15 inches).
Long rod.
Mean.
Maximum.
Minimum.
Mean.
inches
i
7
6-5
6-8
6-4
6-7
1
5
13-3
13-9
12-8
13-5
2
2
26-5
26-8
26-2
26-4
4
6
49-3
51-2
48-6
48-8
5
2
60-2
61-3
59-1
57-2
It is clear from these figures that there is no systematic difference between the
results obtained with the two rods. The change, if any, between the forms of the
wave when at 15 inches and at 45 inches from the end consists in a shearing of the
* LOVE, ' Mathematical Theory of Elasticity,' 2nd edition, p. 277.
DETONATION OF HIGH EXPLOSIVES OR BY THE IMPACT OF BITLLETS. 447
whole curve as iu the manner illustrated in fig. 3. Such a change of form — analogous
to the change preparatory to breaking which a wave experiences as it advances into
shallower water — would not be detected by these experiments, and it is not impossible
that it occurs to some extent.
(b) Reflection and Effect of the Joint. — The simple harmonic pressure-wave which
is propagated without change of type is accompanied by a distribution of shearing-
stress across the section. This shearing-stress depends on the square of the ratio yo,
and is small. That it plays no important part in these experiments is shown by the
fact that if there be a joint in the long rod the results are unaltered. Such a joint
transmits the pressure, but stops the shearing-stress part of the wave. As might be
expected, it was found that the faces of the joint must be a carefully scraped fit if
the wave is to pass it unaltered.
The small magnitude of the shearing-stress is the foundation of the assumption
that the wave is perfectly reflected at the free end. Strictly accurate reflection is
not possible. A reflected wave which is exactly the same as the incident wave, except
that the signs of all the stresses are reversed, will when combined with the incident
wave give no normal force over the free end. The shearing-stresses corresponding to
the two waves do not, however, neutralise each other, but are added, hence accurate
reflection can only be brought about by the application of a distribution of shear over
the free end. The shear required is, however, of the order -/a* and the experiment
with the joint shows that its effects may be neglected.
(c) Effect of the Diameter of the Rod.— The pressure exerted by the bullet is
confined to a comparatively small area in the centre of the end ; whereas the pressure-
wave travelling without change of type implies a nearly uniform distribution of
pressure over the section. The question of the nature of the wave developed under
such conditions seemed quite intractable mathematically, but from general
considerations it appeared probable that it would not differ greatly from that of
wave originated by a uniform pressure distribution. In order to te* this poi
2000 feet per second.
Percentage of momentum in piece.
Length of piece.
} inch.
1 inch.
H inch.
inches
0-5
1-0
10-8
21-1
10-9
22-1
42-2
10-35
22-0
40-5
•0
3-0
4-0
5-0
61-3
79-5
92-5
61-2
79-7
93-5
97-5
60-2
78
88
89
•o
448 MR. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
comparative tests were made with rods off inch, 1 inch, and l|- inch diameter. The
lengths of the rods were roughly 48 inches, 43 inches, and 30 inches, respectively.
The results are exhibited in the table on p. 447, in which the figures for the 1-inch
rod are the same as those already given.
It will be seen that the diameter of the rod has no appreciable effect up to a
length of 4 inches, but that for greater lengths the large rod gives appreciably lower
values. In other words the apparent maximum pressure is not much affected by the
diameter, and is presumably correctly given by all three rods, while the duration of
the blow is largely overestimated by the 1^-inch rod, and presumably somewhat
overestimated by the other two, though as they are in substantial agreement on
this point the error cannot be very large. It may be surmised that some at any
rate of the difference between the observed and calculated times of impact is due to
this cause, though, as already pointed out, the rigidity of the bullet is competent
to account for part of it.
Experiments at Lower Velocities.
Measurements were also made with cartridges giving velocities of about 1240 feet
per second and 700 feet per second respectively, the same types of bullet being used.
The results for the 1240 feet per second cartridges are exhibited in the following
table, which corresponds to that already given on p. 442 for the 2000 feet per second
cartridges : —
ROD, 1 inch diameter, about 40 inches long. Velocity of bullets 1240 feet per second.
(Mean of 5 shots : maximum 1257, minimum 1229.)
Percentage of total in piece.
Total momentum in rod and piece.
Length of
Number of
piece.
shots.
Maximum.
Minimum.
Mean.
Maximum.
Minimum.
Mean.
inches
0-5
1
6-5
37-9
1
8
12-9
12-3
12-7
38-5
36-8
37-7
2
7
26-7
25-8
26-5
35-9
31-5
34-0
3
4
38-4
37-5
38-1
39-4
37-7
38-4
4
5
51-6
50-6
51-1
39-1
38-3
38-6
5
3
63-0
61-5
62-1
40-4
39-1
39-7
6
4
67-7
67-5
67-6
37-2
35-9
36-6
9
5
89
81-5
85-8
39-0
35-8
37-0
The mean total momentum registered (37 shots) is 37'7 units ; the calculated total
is 1240 x 0'0306 = 38 units.
The percentage figures are plotted in fig. G (curve marked " 1240 feet per second").
DETONATION OF HIGH EXPLOSIVES OR BY THE IMPACT OF BULLETS. 449
The percentage of momentum trapped by short pieces is 13 per inch, and the
corresponding maximum pressure for the normal velocity of 1240 feet per second is
0-13x38
32;2xO-98xlO-» =
The maximum pressure which should be exerted by a perfectly fluid bullet having the
same mass and velocity is
The time taken by the bullet to travel its own length is 8'4 x 10'* seconds. Thus
if the bullet were perfectly fluid, the whole momentum should be trapped in a piece
9 inches long, whereas in fact only 86 per cent, is so trapped. The errors inherent
in the method of experiment, which have been discussed in the last section, will all be
less at the lower velocity. On the other hand the rigidity of the bullet will IKJ
relatively more important and probably suffices to account for much of the difference
between the theoretical and observed times of impact.
The 700 feet per second bullets showed a maximum pressure of 5450 Ibs., as
compared with 5320 Ibs. calculated. 54£ per cent, of the momentum was trapped by
a 9-inch piece. It was not possible to experiment with longer pieces, so that the time
of impact in this case could not be determined.
It should be observed here that just after the piece has been shot off it tends to
pull the rod after it by magnetic attraction, which of course still continues after the
joint is broken, though it diminishes rapidly as the distance between piece and rod
widens. The effect of this is to give more momentum to the rod and less to the piece
than they would respectively possess as the effect of the blow alone. By measuring
the amount of the magnetic pull when the piece is held at different distances from the
rod, the current in the solenoid being the same as that used in the impact experiment,
it is possible to estimate the amount of this effect. With 2000 feet per second bullets
it is quite negligible, but when the velocities are lower particularly with long pieces,
it necessitates a correction. This correction has been applied in the figures given
above for the 1240 feet per second and 700 feet per second bullets.
Detonation of (run- Cotton.
It is well-known that a charge of 1 Ib. gun-cotton will shatter a mild steel plate
1 inch thick or more, if it be detonated in firm contact with it. The fracture is quite
" short," like that of cast-iron, though the broken pieces are usually more or less
deformed. Typical fractures of this kind obtained on plates of very good mild steel
are illustrated in figs. 8, 9, 10, and 11. Figs. 8 and 9 are photographs of a plate
1± inches thick originally quite flat. It was broken by a slab of gun-cotton weighing
1 Ib. which covered the section of the plate AB and was detonated in contact with
VOL. CCXIII. — A. 3 M
450 MR. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
that which became the convex face (lower face in fig. 9). Fig. 10 is a view of the
broken edge of one of the two fragments. The plate shown in fig. 1 1 was a flat piece
of boiler plate l£ inch thick. A slab of 1 Ib. of gun-cotton was detonated against
that which is the under side in the figure and the two pieces subsequently fitted
together again and photographed. Thinner plates — e.g., I inch thick — are usually
B
A I \B
i \
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
cracked in two places, one at each edge of the gun-cotton slab, and the portion
covered by the slab is blown out of the plate, sometimes whole and sometimes
shattered into pieces. The fact that no tamping is necessaiy suggests that the
duration of the process of detonation is of the same order as the time taken by sound
to travel an inch or less in air, so that during the conversion of the cotton into gas
there is not time for much expansion.* If this be so, the maximum pressure
h The velocity of detonation of long trains of gun-cotton has often been measured and is variously
estimated at 18,000 to 20,000 feet per second. If the same velocity obtained in the small primers they
would be completely converted into gas in about 2 x 1Q~6 sees,
DETONATION OF HIOH EXPLOSIVES OR BY THE IMPACT OF BULLETS. 451
developed must be that which would l,e reached if the cotton were fired in a closed
chamber of a volume not greatly exceeding that of the slab. The pressure is then
d,8sipated with great rapidity by the expansion of the gas, which is resisted only !,y
its own inertia and that of the surrounding air.
Experiments on the detonation of gun-cotton have been made by the method
described m this paper. It has only been possible hitherto to use quite small charge
and the results are a very rough approximation, but as they thmw light .,., a matter
Fig. 12.
of which little is known I have thought it worth while to give them. Briefly,
the conclusion is that the pressure at a point distant £ of an inch from the surface of
one ounce of dry gun-cotton (a cylindrical " dry primer " about 1^ inch diameter and
l£ inches long), when detonated with fulminate, has fallen to less than J of the
maximum value within 2 x 10~8 seconds. At least, 80 per cent, of the blow has been
delivered within that time. Over an interval of 10~6 seconds round about the time
of maximum pressure the average pressure is alxmt .'30 tons per square inch, and tlu-
3 M 2
452 MR. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
actual maximum is probably of the order of 40 tons per square inch. At a point on
the surface the maximum pressure is at least twice as great, 80 tons per square inch.*
The arrangements are shown in fig. 12.
The gun-cotton cylinder A is fixed by short splints of wood opposite the end of the
shaft B, which is of mild steel l£ inches diameter and from 15 to 30 inches long. This
shaft is suspended as a ballistic pendulum with a pencil and paper for recording its
Total nett
Percentage
Average
Length of piece.
momentum shaft
and piece.
of
total in piece.
percentage in
piece.
inches
[
40-7
/ 86-6
"*
46-2
93
38-8
93
3-85
50-0
92
90
41-4
90
t31-2
90
•
145 6
88
-
c
60-0
89
"j
'
57-3
50-9
88
89
S 89
I
74-3
90
J
f
36-7
81
1
2 {
38-7
42-9
87
84
l> 83
I
42-0
79
J
f
40-8
57
•
42-3
57
0-95 -j
1
44-1
55-4
55
60
57
t44-C
57
I
t50'8
58
\
movement. The end piece C, from |- to 6 inches long, is held on by magnetic
attraction. The faces of the joint are a scraped fit. In line with the shaft is the box
D, which is also suspended as a pendulum and provided with a recording pencil. Some
part of the momentum given to the box is due to the blast from the gun-cotton ; this
* The pressure developed by the explosion of gun-cotton in a vessel which it completely fills does not
appear to have been measured. From measurements made with changes of lower density Sir ANDREW
NOBLE estimates that it would be about 120 tons per square inch ('Artillery and Explosives,' p. 345).
Allowing for the partial expansion during the process of detonation, this agrees fairly well with the
pressure here determined.
t In these cases the air space between the gun-cotton and the end of the shaft was 1 inch. In all the
others it was f inch.
DETONATION OF HIGH EXPLOSIVES OR BY THE IMPACT OF BULLETS. |ft|
was estimated from experiments in which there was no piece on the end of the shaft.
Separate experiments were also made to determine the effect of the blast on the
supports of the shaft. The momentum accounted for by the blast is in each cane
deducted from the total recorded momentum to get the nett momentum due to the blow
on the end of the shaft. This correction in the case of the box amounted to about
8'3 units with a 15-inch shaft, and 1'2 units with a 30-inch shaft. The correction for
the blast on the supports of the shaft was 5 units.
The table on p. 452 gives the results of all the trials made with the gun-cotton about
f inch from the end of the shaft.
The total impulse of the blow when the air space is f inch varies from almut 35 to
70 units, the average being about 46 units. The percentages abHorlx.*! by the different
end-pieces are, however, more nearly constant, and from them a rough approximation
to the pressure wave transmitted by the rod in an average case may be constructed.
As already explained the precise form of this curve depends on the way in which the
pressure rises, but it may be assumed in this case that the pressure reaches its
maximum in a time that is short even in comparison with the duration of the blow.
Assuming an average total momentum of 45 units, fig. 13 has l>een constructed. The
K>
Pressure, io4lbs pereq.1
_t;»*tnO»)*3
| \
Pressure at a distance of \ inch
FljJ.KJ. from surface of
one ounce "dry primer."
\
X
j
\
1
1
1 1
1
I
I
X
X
X.
2
3 -— V.,
1 2
Time, /o~asecs
area of the parallelogram marked 1 represents the momentum given to a 1-inch piece,
the width of this parallelogram is 10"' seconds and the height is the average pressure
acting during the first 1Q-5 seconds. The parallelogram marked 2 represents the
excess of the momentum given to the 2-inch piece over that given to the 1-inch piece
and its height is the average pressure acting during the second 10'* seconds
dotted curve gives the same average pressures over the successive intervals of time.
It is obviously largely conjectural, but it gives a rough idea both of the max!
pressure and of the duration of the blow.
The chief difficulty experienced hitherto in measuring by this method the pn
developed in the detonation of gun-cotton has been the permanent defonnation of the
end of the rod by the blow. No steel has yet been discovered which will stand, wn
flowing or cracking, the detonation of gun-cotton in contact with it, and even wh.
454 MB. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE
cushion of air f inch thick is interposed some flow takes place.* In consequence of
this, the pressure wave which emerges and is propagated elastically cannot he quite the
same as the wave of applied pressure. It is easy to see that the general effect of the
setting up of the end must be to deaden the hlow, that is to reduce the maximum
pressure and prolong its duration. In fig. 14, A is the (conjectural) curve representing
Fig.14.
Time io'5 sees.
the pressure applied to the end of the rod. If the rod were perfectly elastic, the.
pressure across a section 2 inches from the end would be represented on the same time
base by the curve B, which is the same as A, but moved 10~5 seconds to the right.
The momentum in the end 2 inches at any time is the difference between the areas of
the curves up to that time. For instance at 2 x 10~5 seconds it is represented by the
shaded area under curve A. But if the end be not completely elastic, the higher
pressures developed over section B will be less than those acting on the end at
corresponding times. Thus the record of pressure over the section 2 inches from the
end will be a curve such as B' and the momentum in the end two inches at any time
will be greater than it would be if the end were elastic by the difference between the
areas of curves B and B' which is double shaded in the figure. This extra momentum
is transferred to the remainder of the rod later on, causing the curve B' to rise above
B. The curve B' represents the wave of pressure actually sent along the rod. It is
this curve which is determined by the method which has been described, and it is
evident that that method under-estimates the maximum pressure and over-estimates
the duration of the blow.
A few experiments were made with the gun-cotton touching the end of the shaft.
The average total momentum given to the shaft and piece in this case is about
90 units or roughly twice as great as that transmitted through f-inch air-space. Of
this total about 80 per cent, is caught in a piece 4 inches long, and about 50 per cent,
in a piece 1 inch long. When the gun-cotton is at a distance of f-inch these figures
are 90 and 60 respectively. The apparent duration of the pressure is therefore
rather greater at the surface of the explosive. The setting up of the end of the
shaft is, however, much more marked when the gun-cotton is in contact and it may
h This is when the steel is in the form of a shaft, so that there is no lateral support of the part
subjected to pressure. It is, of course, possible to make a plate with hardened face which will withstand
the attack of gun-cotton on a portion of the face.
DETONATION OF HIGH EXPLOSIVES OR BY THE IMPACT OF BULLED). 455
be that the distribution of the pressure in time is not materially different in the two
cases. If that were so, the maximum pressure developed on the surface of the gun-
cotton would be 80 or 100 tons per square inch.
It is hoped that by the use of special steels it may be possible to give greater
precision to these estimates of the amount and duration of the pressure produced by
the detonation of gun-cotton in the open. Meanwhile the iiitbrmutioii already
obtained as to the order of magnitude of these quantities is sufficient to throw some
lightvon the nature of the fractures produced. The general result obtained may be
expressed by saying that a gun-cotton cylinder 1^ inches x !$• inches produces at ite
surface, when detonated, pressure of the average value of 100,000 Ibs. per square
inch lasting for TToTooo second. Probably figures of the same sort of magnitude will
describe the blow produced by the detonation of a slab 1^ inches thick, one of whose
faces is in contact with a steel plate. It may be that the pressure is greater and the
duration correspondingly less, but this does not affect the point that the pressure is
an impulsive one in its effect on the plate. That is, the effect of the pressure is to
give velocity to the parts of the plate with which the gun-cotton is in contact but the
pressure disappears before there has been time for much movement to take place.
For instance, if the plate be 1 inch thick (mass 0'28 Ibs. per square inch) a pressure
of 100,000 Ibs. per square inch acting on it for 5^jo second will give a velocity of
about 230 feet per second, and while the pressure is being applied it will move
0'028 inches.
The parts of the plate not covered by the gun-cotton are left behind and the strain
set up by the forced relative displacement is the cause of the shattering of the plate.
The magnitude of this strain, and of the consequent stress, depends (speaking
generally), on the relation between the velocity impressed on the steel by
the explosion and the velocity of propagation of waves
of stress into the material. For instance, if the sec-
tion AB (fig. 15) be given instantaneously a velocity
of 200 feet per second and this velocity be maintained,
the state of the plate after the lapse of 10o!ooo
second will be that represented diagrammatically by
fig. 15. The section AB has moved forward rela-
tively to the remainder by 0'002 feet. As soon as
this section started moving a wave of shear stress
started out from A into the parts of the plate to
the left which had been left at rest by the blow. This wave travel
11000 feet per second and will therefore in 1001OU» "OOOd get
AC = O'll feet. To the left of C the metal has not moved, the wave not hu
reached it ; therefore the average shear in the section AC is -—
forces of this durationl even mild steel has nearly perfect elasticity up to very
A
B
I
SteeL
Gun-ooUiwt.
23
_ 1
1 ' :
Vetoc&on/aec
'Atn*t]
— \oa
C- A
~ Fig.L
5.;
456 MR. B. HOPKINSON ON MEASURING THE PRESSURE, ETC.
high stresses.* If it maintained its elasticity and continuity the shearing stress
would be of the order O'OIS x 1'2 x 107, or say 220,000 Ibs. or 100 tons per square inch.
This illustration is of course very far from representing the actual effect of suddenly
giving velocity to a portion of a plate, the real distribution of stress would be far
more complicated, but it gives an idea of the magnitude of the stresses which may be
expected to arise. In static tests on mild steel, the material begins to flow as
soon as the shearing stress exceeds about 10 tons per square inch and no stress
materially greater than this can exist. But when the metal is. forcibly deformed at a
sufficiently high speed the shearing stress is increased by something analogous to
viscosity and the tensile stress which accompanies it may be sufficient to break down
the forces of cohesion and tear the molecules apart. Thus the steel is cracked, though
in ordinary static tests it can stretch 20-30 per cent, without rupture, just as pitch,
which can flow indefinitely if given time, is cracked by the blow of a hammer. The
essence of the matter is the forcible straining of the substance at a velocity so high
that it behaves as an elastic solid rather than as a fluid, thus experiencing stresses
which are measured by the strain multiplied by the modulus of elasticity. The effect
of gun-cotton on mild steel shows that in this material a rate of shear of the order
1000 radians per second is sufficient to cause cracking.
The most probable account of the smashing of a mild steel plate by gun-cotton is,
then, that the plate is cracked before it has appreciably deformed, the cracks being
caused by relative velocity given impulsively to different parts of the plate. Bending
of the broken pieces occurs after the plate has cracked and the pieces have
separated from one another. It is due to relative velocity in different portions of each
piece which still persists after the initial fracture, and is taken up as a permanent set
in each piece. In this connection the fracture shown in fig. 11 is instructive. It
will be noticed that the general bend of the plate, after the pieces have been fitted
together, is opposite to that which might at first sight be expected as the result of
the blow in the middle. Inspection of such fractures leads to the conclusion just
stated as to their history. The experiments on gun-cotton pressures described in this
paper, though lacking in precision, supply I think the missing link in an explanation
which is otherwise probable, namely, sufficient evidence that the blow may be regarded
as an impulsive force communicating velocity instantaneously.
Most of the experimental work described in this paper was done by my assistant,
Mr. H. QUINNEY. I also received valuable help in the earlier stages from
Mr. A. D. BROWNE, of Queens' College, and from my brother Mr. K. C. HOPKINSON,
Trinity College. To these gentlemen I wish to express my obligation for aid without
which it would hardly have been possible to carry out a research of this character. I
have also to thank Sir EGBERT HADFIELD, Mr. W. H. ELLIS, and Major STRANGE for
providing steel plates and shafts.
* HOPKINSON, 'Roy. Soc. Proc.,' 74, p. 498.
[ 457 ]
XI. Gravitational Instability and the Nebulir / / i//tothesi*.
x
By J. H. JEANS, M.A., F.R.S.
Received October 22,— Read Novemlwr 27, 1913.
Introduction.
§ 1. A CONSIDERATION of the processes of cosmogony demands an extensive knowledge
of the behaviour of rotating astronomical matter. What knowledge we have is
based upon the researches of MACLAURIN, JACOBI, POINCARE, and DARWIX. These
researches refer solely to matter which is perfectly homogeneous and incompressible,
although it is, of course, known that the primordial astronomical matter must be far
from homogeneous and probably highly compressible as well. The question of how
far we are justified in attributing to real matter the behaviour which is found to be
true for incompressible and homogeneous matter is obviously one of great importance.
§ 2. There are d priori reasons for expecting that there will be wide differences
between the two cases. Consider first a sphere of homogeneous incompressible
matter devoid of rotation. This will be stable if every small displacement increases
(or, at least, does not decrease) its potential energy. The sphere has a number of
independent possible small displacements which can be measured by the number of
harmonics which can be represented on its surface. The spherical configuration is
known to be stable because it can be shown that every one of these displacements
increases the potential energy.
Contrast this case with the corresponding one in which the matter is compressible.
The number of possible small displacements in this latter case is measured by the
sum of the numbers of harmonics which can be represented on all the sphencal
s^lrfaces inside the sphere. Let R be the radius of the outer surface ; let r, r1, r", ...
be the radii of all the spheres which can be drawn inside this outer sphere,
and let r», r'n, /'„ ... R, be the number of independent harmonics which can be
represented on these spheres. To prove that the sphere is stable it is now necessary
to prove that every one of the rB +?', + /',,+ ... R. possible displacements increases
the potential energy. If we argue by analogy from the case of an incompressible
sphere we are, in effect, merely considering R. of these displacements and neglecting
the much greater number rn+ 1\ +r"n+ .... Furthermore, in these neglected dis-
placements, the nature of the displacement is essentially different from that in the
VOL. CCXIII.-A 507. 3 N ™>ii.h.d «p.»t.i7. rebm.rr a, 11.14.
458 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
RB displacements, so that there appears to be no justification at all for an argument
from analogy.
In each of the neglected displacements, the change in the potential energy will
consist of two terms. There will be a change in the elastic energy of the
compressible material, and this can be easily shown to involve an increase in the
potential energy. There will, in addition, be a change in the gravitational energy,
and this can be shown* to involve invariably a decrease in the energy. If W, E, G
denote the total, the elastic, and the gravitational potential energies,
in which SG is invariably negative. The condition for stability is that for every
possible displacement £E shall be numerically greater than SG.
It might naturally be thought that by considering a system in which the matter
was, so to speak, very gravitational or very little elastic we could have JE small or
SG great, and so should have instability of the spherical configuration. But it must
be remembered that the gravitation and the elasticity of the matter are not
independently at our disposal. The action of the gravitational forces tends to
concentrate the matter and so involves that the elasticity becomes large in the
equilibrium configuration. If we consider a system in which the elasticity is
artificially kept small, as, for instance, by adding an additional repulsive field of force
to annul, or partially annul, the gravitational field, we can easily construct systems
for which a spherical configuration is unstable,! but, short of this, it appears to be a
general law that the elastic and gravitational agencies must march together in such a
way that £E is always numerically greater than <5G,| so that every natural spherical
system is stable.
The nearest approach in nature to the artificial repulsive field imagined above is
found in the influence of rotation. This influence may be represented by the super-
position of the usual repulsive field of centrifugal force of potential — %w2(x2 + y2).
The field is not spherical, and so the figures of equilibrium obtained under its influence
cannot be spherical. But it can be regarded as made up of a spherical part — §wV
and a superposed harmonic disturbance fyufPtf*. The first term is certainly a
spherical repulsive field, and will, of course, tend to annul the concentrating influence
of gravitation. The problem which requires study is that of how far, or in what
circumstances, the presence of rotation can disturb the otherwise general law that SIS,
is always greater than SG.
The problem is one of enormous complexity and great generality. It will hardly
be expected that the present paper will contain anything approaching a general
* Of. J. H. JEANS, "The Stability of a Spherical Nebula," 'Phil. Trans.,' A, vol. 199, p. 1.
t Of. J. H. JEANS, "On the Vibrations and Stability of a Gravitating Planet," 'Phil. Trans.,' A,
vol. 201, p. 157.
J Cf. below, §§11, 22.
AND THE NEBULAR HYPOTHESIS. 459
solution, and it may as well be stated at once that it does not. All I have been able
to do is to grope after general principles by solving a problem here and a problem
there as seemed needful to illuminate a possible path towards a general theory, and
the present paper is confined to a very few of the special problems I have considered,
but I have selected those which seemed to have most bearing on the general question
in hand.
v
Medium in which the Pressure is a Function <>f ///'• 7>c//.v/V//.
§ 3. In the most general astronomical medium the pressure is, of course, not a
function of the density. The relation between pressure and density varies from
point to point, partly on account of inequalities of temperature and partly on account
of variations of chemical constitution. But no general theory can l>e expected to
apply to the most general heterogeneous mass of matter possible, and before any
general theory can be deduced we must have material from which to deduce it.
§ 4. The simple system from which we shall start will be a system in which the
matter is homogeneous as regards its properties, so that at all points the pressure
and density will be connected by the same relation. It will be seen later (§ 1 5) how
it is possible, in at least one important respect, to escape from this limitation.
For the present we assume the pressure and density to be connected by the
relation
0)
at every point. We take the centre of gravity of the rotating mass to be the origin,
and the axis of rotation to be the axis of z. The equations of equilibrium are
= + i(f
r
in which V is the potential of the whole gravitational field of force. In virtue of
relation (l), these equations have the common integral
......... (2)
in which «> stands for *+* and ,(,) for j<fe, which is by hypothesis a function of
3 N 2
460 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
p only. There is further the relation of POISSON,
V2V = -4^, ........... (4)
so that on operating on (3) with V2 we obtain
2w\ .......... (5)
the differential equation which must be satisfied by p in any configuration of
equilibrium under a rotation w.
§ 5. In general a solution of equation (5) will involve negative and zero values of p-
In the physical problem p will be limited as to values, and this limitation will
determine the physical boundary of the rotating mass.
Let Vm denote the gravitational potential at any point in space of the finite mass
determined in this way. We have found a configuration of equilibrium under a
potential V, the potential of the mass is Vm, so that for equilibrium we require an
additional field of potential V— Vm. We can say that the configuration found will
be a true configuration of equilibrium under an external field of force of potential V0
such that
(6)
And, inasmuch as V2Vm = —4^/0 = V2V, it is clear that V2V0 = 0, so that the
external field has poles only at the origin or at infinity. The condition that any
solution shall lead to a configuration of equilibrium for a mass rotating free from
external influence is, of course, V0 = 0.
§ 6. The simplest solution of equation (5) is obviously that in which p is a function
of r only, but it is clear from (3) that this cannot give a free solution except when
w = 0.
§ 7. The next simplest form of solution is that in which p is a function of z and w
only, and this can give a free solution. It includes, of course, as a particular case
the system of Maclaurin spheroids. For this class of solutions every section at right
angles to the axis of z is circular, and in any such section the lines of equal density
are circles. The density at any point is of the form p =/(&, z).
Let O denote colatitude measured from Oz, and let \}r be azimuth measured from
the plane of xz. The most general configuration which can be obtained by displace-
ment of that just considered will have a law of density of the form
GO
P=fo (w, *)+ 2/5 (w, z) COS S\fr.
It is easily seen that the separate cosine terms lead to independent displacements,
and we shall for the moment only consider the displacement of the first order, for
which the law of density is
P =/o(w, z)+/(w, z)cos^, ........ (7)
where/! (ra, z) is a small quantity of the first order.
AND THE NEBULAR HYPOTHESIS. 461
The boundary being a surface of constant pressure must also be a surface of
constant density, say o-. The equation of the boundary is accordingly
, z)ao6\l, = <r ..... .... (8)
The whole mass inside this boundary may be regarded as composed of coaxial rings
of matter as follows. Inside the figure of revolution /0 (m, 2) = <r, we suppose there
to be a series of rings of density given by (7), while the surface inequality can be
regarded as represented by the presence of rings on this figure of revolution of
density proportional to cos \[r.
On integration the potential Vm at any external point is seen to be of the form
osi/' ........... (9)
where xo> Xi are functions of w and 2 only.
Suppose now that the surface is so nearly spherical that spherical harmonic analysis
may be used with reference to it, then, since Vm is a solution of LAPLACE'S equation at
all external points, and is also of the form (9), it must be of the form
(10)
where M = cos 6, and P,1 (/x) is the usual tesseral harmonic — P, (/u). Moreover, since
t*\j
the centre of gravity of the mass is supposed to coincide with the origin, AI must
vanish.
We have, from equation (3), if V = Vm,
v =
»
, z) cos + +' {/„(«, z)} -
at all internal points. Equating these two expressions, we must have at the
boundary
or, neglecting small quantities of the second order,
Hence either /, (w, z) vanishes at the boundary or is of at least the second order of
harmonics.
It follows that if there can be a configuration of equilibrium which dil
figuration of revolution P =f.(w, 2), by a displacement proportional to the first
con
462 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
harmonic, this configuration must be one in which /J (w, z) vanishes at the boundary,
so that the boundary must be a figure of revolution about the axis of rotation.
It now follows from (3) that V must be a function of w and z only on the
boundary, and hence also (since V is harmonic) at all external points. It follows
that -7— , and hence also -£- , are functions of is, z only at the boundary. Whence
rin en
again, by equations (4) and (5), it follows that — - and -^ are functions only of TS
and z at the boundary. And, by successive differentiation of equations (4) and (5),
it is seen that all the differential coefficients of V and p are functions only of rs and z
at the boundary.
It can be seen from this* that the configuration must be one of revolution through-
out. In other words, there can be no configuration of equilibrium which differs from
the configuration of revolution by first harmonic terms only.
LAPLACE'S Law.
§ 8. I have not found that any progress worth recording can be made with the
general relation^? =f(p), so that progress can only be hoped for by examining special
cases.
The case that suggests itself as most important is that of the gas law p = Kp,
satisfied in a perfectly gaseous nebula at uniform temperature. The difficulty is that
such a nebula extends to infinity in all directions, and so cannot rotate as a rigid
body. Or rather, when it is caused to rotate, it throws off its equatorial portions
and the remainder rotates in the shape of an elongated spindle of infinite length. In
this connection I have worked out the purely two-dimensional problem of a rotating
gaseous cylinder of infinite length. The results are too long to be worth printing ;
it will, perhaps, suffice to record that the analysis bears out in full the conclusions
arrived at in this paper.
The law which is most amenable to mathematical treatment is LAPLACE'S law
or, as it is more convenient to write it,
(12)
in which c, p, K, and a are constants, <r being the value of the density at the free
*
I have not succeeded in obtaining a rigorous proof of this. It might be objected that
nothing in the above argument precludes first harmonic terms proportional to such a function as
« ~f(w, :) , where /(cr, g) = 0 is the equation of the boundary. The pure mathematician may not,
although the astronomer will, be influenced by the consideration that such functions never occur in natural
problems. If such a function did occur, it would involve an extremely fantastic relation between p and p.
AND THE NEBULAR HYPOTHESIS. p; ,
surface. This law has the merit that the case of an incompressible fluid is covered
by the special value c = oo or K = 0, the density now having the value a throughout.
There is the d priori objection to the kw that its form precludes first harmonic
displacements (rf. below, §11). This objection would be fatal were it not that we
have seen that first harmonic displacements are in any case of no importance. This
being so, the objection falls to the ground, and I have thought it worth working out
this-law as far as possible.
§ 9. Using the relation (12), we have in place of the more general equations (3), (4),
and (5), the particular equations
(13)
VV = -4*7>,
On putting
this last equation reduces to
(17)
No Rotation.
*
§ 10. When there is no rotation w = 0, x = p and the equation becomes
(V'nV) /> = ().
The general solution is
P-ZA.r-'tf.+.fcGtr)^.. •
while the particular solution giving a spherical boundary is
the last being, of course, the well-known solution which occurs in LAPLACE'S theory
of the figure of the earth. It will now be shown that this configurat,
8 a be the free surface corresponding to the simple solution (19).
Consider an adjacent solution
and let the corresponding free surface be
464 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
On substituting this value for /• in (20), neglecting squares of b and equating
corresponding harmonic terms, we obtain
A '/ T / \ A,. SHI Kd lnc^\
<r = AOGT M./. (irf*) = -7=- -, ....... (22)
- {«-'/' J,/2(*«) I = *A0/,>a-'M,/2 (Ka), . . . (23)
whence
6y = A.a-1^J»^MJ'/.M ...... (24)
* J»/, (* «)
By integration, the value of Vm at a point on the sphere r = a is found to be
v- -
A _+8/2 JB+y, (<ra) Jy, (<ra) o
"
while the value of V, as given by equation (13), is
V = ^a-'/'J 1,M+^a-1/2J»+-,MS1,+ cons.
AC AC
If we put
V-Vm = v = w0 + v,S«
we obtain, after some reduction,
/ \ J,+./.MJ./.M
l
In general, this gives the value of An which determines the tide raised by a field of
/r\*
potential vn(-j 8, proportional to SB. We notice that when n = 1, vn = 0 or AB = oo
\a/
independently of the values of /c, a. This merely expresses the obvious fact that
there can be no equilibrium at all so long as the fluid is acted on by a force
proportional to a harmonic of the first order.
If it is possible for there to be a configuration of equilibrium when vn = 0, other
than that given by AB = 0, this configuration will of course determine a point of
bifurcation in the series of symmetrical configurations. The points of bifurcation are
accordingly given by vn = 0, or by
Jn-'/g (KO) _ Jy, (KO)
•*»+'/,(*«) J'/2(*«)'
For brevity in printing, let us introduce the function un defined by
AND TIIK M.r.i I..M;
ir, ,
Near KO, = 0, ?<„ = r-^— ; the value of w, is cot *a, and successive u's satisfy
itflt ~|~ \. ^ ' £
the difference-equation
.T- ' ' W
K<t
With the help of these properties it is easy to draw approximate graphs of the
curves y = un. Such a graph, for values of KO, up to the first zero of «, («« = 4'49) is
represented in fig. 1, in which the vertical scale is 2^ times the horizontal scale.
Fig. 1. Graphs of the functions u,,.
In terms of these functions, the points of bifurcation are given by «„ =
once evident that there is no root of this equation for values of*, less than ,
therefore (cf. equation (19)) no point of bifurcation at all so long as , is nrtric
being always positive. It follows that the spherical configura
displacements.
Small Rotation.
§ 12 When the fluid experiences a slight rotation w, the spherical configuration is
of course slightly flattened. The appropriate solution of equal
-•M«.(«r)P,. (29)
x =
VOL. CCXI1I. — A.
3 o
466 ME. J. H. JEANS ON GRAVITATIONAL INSTABILITY
where P2 is the second zonal harmonic about the axis of rotation as 6 = 0. Assuming
the free surface to be
r = a + bP2, ...... ..... (30)
the equations analogous to (22), (23), and (24) are found to be
or-I^Aoa-'/'J^M, ..'.-.• ...... (31)
A2a-'/8Js/2(fc«) = K^la-^S^Ko), ........ (32)
** . (33)
l W \T / \
1- — «M««)
-
Let v be given by
v = V+ty>w'-Vm = V-V,,I + ^V(1-P2),
then, instead of equation (25), we have
(34)
in which constant terms are omitted, and the value is taken on the sphere r = a.
For a configuration of equilibrium under no external field of force we must have
V = Vm, and therefore v in equation (34) equal to -^wVP2. Neglecting squares
of w2, and therefore omitting the factor 1 - ^— in the denominator, the equation
ZTTIT
becomes
(35)
giving A2 in terms of w2, when w2 is small. It will be readily verified that this
equation is identical with that obtained by THOMSON and TAIT ('Nat. Phil.,' §824,
equation (14)).
§ 13. We next examine the solution
w2
X = /'-^ = A0r-^J1/2(^-) + A2r-^Js/2(^)P2 + ABr-1/-JB+1,(/cr)SB, . . (36)
which is appropriate to a mass of fluid having a rotation w given by equation (35),
and acted on by a field of force of potential vn$a. By analysis exactly similar to
that just given, we obtain at r = a
AND THE NEBULAR HYPOTHESIS. 457
This gives A, for the general tide raised by the field vnSm. The condition for a
point of bifurcation is ?>„ = 0, or
....... •- ..... <38>
Thus the points of bifurcation, if any, are still determined by the intersections of
the graphs in fig. 1, except that the graph of «, must be supposed decreased vertically
o
in the ratio 1 -- — to 1 .
27T<7
We may, if we please, imagine that we start with very small rotation, and allow the
rotation progressively to increase, this increase being accompanied in imagination by a
greater and greater flattening of the graph of u,.
It is clear that under all circumstances the curve which will first be intersected by
the flattened graph of ?<, will be the graph of u3. It is further clear that the
requisite value of iv* is least when *a = 0, and progressively increases as *a increases, .
at any rate up to KO, = TT.
This means that in the first place the circular vibration will invariably become
unstable through a vibration proportional to a second harmonic, so that the first point
of bifurcation reached will be one such that the spheroidal form gives place to an
ellipsoidal form. If the rotation is so small that the problem may be treated as a
statical one, there will be no question as to there being an actual exchange of
stabilities at the point of bifurcation, for clearly vn changes sign at this point. Thus
for rotation greater than that at the point of bifurcation, the spheroidal form will be
definitely unstable, and the ellipsoidal form definitely stable, at least until the next
point of bifurcation is reached.
Our result shows, in the second place, that the masses which become ellipsoidal for
the smallest values of it? are those for which *a is smallest. To put it briefly, the
mass which is most unstable when it begins to rotate is the incompressible mass — a
somewhat unexpected result.
For any value of *•«, the value which iff1 must have for the spheroidal form to
become unstable is (cf. equation (38))
and when Ka = 0, the value of u-Ju^ = I (cf. § 4).
Thus our equations would make the spheroidal mass of incompressible fluid first
become unstable when £- = "400, but these equations have only been obtained on
Zww
the supposition that — is so small that its squares may be neglected, a supposition
2x<r
which is now seen d posteriori to be hardly admissible. Probably the result
obtained are qualitatively true, but quantitatively unreliable. In point of fact the
3 O 2
468 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
first point of bifurcation for an incompressible mass, instead of being given by
o 2
= '400, is known to be given by the widely different value ^— = '1871.
2?rfr
Our analysis has nevertheless proved rigorously the point which is really most
important, namely, that there can be no point of bifurcation at all for quite small
Q
values of -t — . At the same time, since the question of when and how a rotating mass
first becomes unstable is one of considerable importance, I have attempted to obtain
a more reliable investigation than the preceding. I have found that the accuracy is
not greatly improved by taking the analysis as far as squares of w2/27r<7, while the
labour of working with a general power series would be appalling. I have, therefore,
reluctantly been compelled to give up hopes of carrying the rigorous solution of the
problem further in this direction, but have thought it worth while to examine the
analogous problem for rotating cylindrical masses. All the essential physical features
of the natural three-dimensional problem appear to be reproduced in the simpler
cylindrical problem, so that it seems legitimate to hope that an argument by analogy
may not lead to entirely erroneous result.
Cylindrical Masses in Rotation.
§ 14. The fundamental equations are, of course, the first two of the equations
already written down in § 3. The third equation does not occur, since -'- = 0. The
oz
equations have, as before, the integral (13) leading to the differential equation (15)
for p.
The most general solution possible will be
= -+ Z AnJn(^)cos(^-e), ....... (40)
0
in which r now stands for \/(x2+ya). No matter how great the rotation, there is
always a special circular solution
«r), ......... (41)
this being analogous to the spheroidal figures of equilibrium investigated in § 12.
Let us examine the deformed solution
•
p = ^-+AoJ0(/cr)+ABJB(/cr)cosn0, ....... (42)
/TT
in which AB is supposed small, but there are no restrictions on the value of — . If
2?r
the free surface p = or is supposed given by (cf. equations (21) and (30))
r = a + b cosn6,
AND THE NEBULAR HYPOTHESIS. ,,, .,
then, as in equations (31) and (32), a and b must satisfy
m. . . . ....... (43)
AAOta) = -'
whence
b-A.
The potential of the mass, Vm, can be regarded as arising from a distribution of
density P inside the cylinder r = a, together with a surface density b<r cos n0 spread
over the surface of the cylinder.
The first part of the potential, evaluated at R, 6, is
G~ jj[log {7J + R2-2rft cos (0-6)}] [j£ + Vo (<r) + A.J. (*r) cos n01r drde
ri°r ™ ^ ~i r «*
= 0-2 logR-2 - rcoss(0-0) + A0J0(*r)+A
JJ|_ i slv JL2»
f" 2?r 7*"
= A.n ) - r-j JB (AT?-) r dr cos n6 + terms independent of 0
- ABJn+1 (*«) — — cos n0+ terms independent of 0.
The potential of the surface distribution is
so that, at r = a,
v. = {
I Kn
while, by equation (5),
If, as before, we express the tide-generating potential V-V. in the form
Vt>+vn cos «0, we obtain for the value of vn, at R = a,
ABJB+1 («o)+ 2* cos n6+ terms independent of 6,
H )
11
(45)
- J,M
2x«r/
It will be seen that this equation is exactly analogous to the former equation (37),
470
MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
but with the important difference that the present equation is true for all values of
w3, without limit. The points of bifurcation are given by vn = 0, or
'W>'\- I \ (AR\
••/,(*«), I4°)
«.-'/, M^1-
which again is exactly analogous to the former equation (38). The graphs of the
functions •%„ M>/2, . . . will be found to lie as in fig. 2, and we may again imagine that
0-8 -
0-6 -
0-4 -
0-Z -
JC=0
123
Fig. 2. Graphs of the functions
points of bifurcation are sought by flattening the curve u\tt until it intersects the
other curves.
It is clear that, under all circumstances, the first curve to be intersected will be
the curve u>tl, corresponding to a displacement proportional to cos 20. Thus, as
before, when the circular form becomes unstable, it gives place to a form of elliptic
cross-section, which is stable. Moreover, the smaller *a is the lower the value of
•M^JTTO- for which the circular form becomes unstable.
These results are true without any regard to the value of w2, so that they confirm
the results stated, but not rigorously proved, in § 5. The numerical calculations
which follow will make the matter clearer.
If p denotes the mean density of the rotating mass, the total mass per unit length
is given by
Pr dr =
4-7T
+ AO
AND THE NEBULAR HYPOTHBBB.
giving, on substitution from equation (43),
471
whence, for the ratio of p to o-, we have the general formula
a- J0(*a)\ 2wr/
For the particular con%uration which occurs at a point of bifurcation,
<ta
so that
whence we obtain
In the following table I have calculated the values of u^/2ir<r and of »»*/2irj5 for
which cylinders of different radii (a) and compressibility (*) first become elliptical 111
cross section : —
V*
1C*
KOi.
u.. .
u...
'/>
'/,
2™-'
2rp'
0
o-oooo
o-ooo
0-500
0-500
0-1261
0-063
0-503
0-600
0-2582
0-126
0-510
0-502
0-4040
0-192
0-525
0-506
1
0-5751
0-261
0-546
0-511
2
2-575
0-612
0-762
0-554
2-4048
00
0-829
1-000
0-593
3
-1-304
1-433
2-099
0-687
3-8317
o-ooo
00
00
1-000
It will be seen that the general result is fully confirmed, that incompressible
masses are the first to become uustable, and that the more compressible the mass is,
the greater is the rotation required for it to depart from a symmetrical configuration.
Rotating nearly Spherical Mass n-ith High Internal Temperature.
§ 15. We now leave the artificial two-dimensional problem and return to the real
problem in three dimensions discussed in § 13.
The coefficient K was there assumed to have the same value throughout the mass,
as of course it would if the matter were homogeneous and of uniform temperature
472 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
throughout. But to represent natural astronomical conditions there is no question
that K ought to increase on passing from the centre to the surface, thus representing
a mass in which the temperature is highest inside and falls towards the surface.
We are in this way led to study the question of stability when K is a function of r.
It would be difficult to say precisely what function ought to be chosen if we were
trying to represent natural conditions as faithfully as possible. It appears, however,
that no continuous function will lead to equations which admit of integration. The
only case which appears to be soluble is that in which the matter, before rotation,
may be treated as if formed of a series of different layers, each being homogeneous
and at a uniform temperature in itself, but the temperature varying from layer to
layer. To represent this we take different values of K in the different layers, K being
smallest in the interior.
There is no limit to the number of layers which can be treated analytically, but
the assumption of a great number of layers naturally leads to highly complicated
formulae which are capable of conveying their meaning only after laborious numerical
calculations. Both in order to obtain comprehensible results and to simplify the
argument, the layers will, in what follows, be supposed to be only two in number.
They may conveniently be referred to as the core and the crust. It will be found
possible to generalize the results obtained so as to apply to any number of layers.
§ 16. We accordingly suppose that there is an interior core of radius a, in which
the coefficient of compressibility has the uniform value K, and that outside this is the
crust of external radius c, in which the coefficient is K. It is again necessary to
suppose the rotation to be so small that w2 may be neglected.
As in § 3, the density /> must satisfy
o ... ...... (47)
throughout the core, and the same equation with the appropriate value of K throughout
the crust. The most general solution of equation (47) is
_l^(«r)}£ ..... (48)
ZTT o
Iii the former problem all the terms in B71 could be omitted because p had to be
finite at the origin. In discussing the solution for the crust these terms must be
retained. The solution can, however, be put in a more concise form.
Let the constants An, Bn be replaced by new constants Cn, 6n given by
A,, = Cn cos 6H, Bn = C» sin dn,
and let us introduce a function JB+i/a (x, 9) defined by
J»+v.(*. 0) = •*.+•/,(«) cos 0 + J_(n+1/2)(x) sinB ...... (49)
AND THE NEBULAR HYPOTHESIS. 473
Then the solution (48) may be replaced by
(50)
which is formally analogous to (7).
The following properties of the function J.+.,,(a-, 6) may readily U< verified, an.l
will be required later : —
.
(52)
(5:1)
There is a ready rule for writing down the values of these functions. In the first
place, we have
T , , sinx cos x
J •/,. W = /T — . J -v, W =
V^TJ-
so that
. sin(.r + 0)
J,,,*, 0 = -
V ^Tr./'
Now let ^(.e + fl) be used to denote a general function made up of circular functioiiH
of x + 6 and of algebraic functions of x. Then J>/,(-r> 6) is of the form <f>(j: + 6), and
any number of differentiations with respect to x, or of multiplications by powers of x,
will still leave it in this form. It follows from (52) that •!•/,(•'', — 0), Jv,(ar, 0), Ac.,
will l)e of this form. Hence we have the general law
<54)
in which the functional form of $ is at once given by
#(*) = •!.+'/, (-4
For instance,
/o \ 3
J,, (x) = Pj - 1 ) sin x- - cos x,
\3t / X
so that
J.,,(ir, 0) = *(* + <>) = (^-l)sin(a- + 0)-|co8(z+0). |
§ 17. We proceed to carry out analysis similar U> that of § 13. Suppose that under
a tide-generating potential rnSn, and a rotation w, the core assumes a configuration
such that its boundary r = <t becomes deformed into
. . • • I55)
VOL. (10X111. — A,
474 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
while the cmter surface becomes
XP, ........... (56)
Let us suppose the densities in the two layers to be
r-'''Ji,(,r)P. . . . (57)
wT
in the core, and
,» = ^C°r~l/Jjl^ . . (58)
in the crust. The boundary between the two layers must clearly be an equipotential,
and therefore a surface of constant pressure and density. . Let a-, </ be the densities
at this boundary in the core and crust respectively.
On replacing r by a + frSB + /3P2 in equations (57) and (58), the values of p must
become a- and or respectively. This leads to the relations
* ~ 5 - = A0a-1/s JVi (Ka), .......... (59)
(60)
(61)
h (_ _ = c -.h.+.<fa..<,a.«- (62)
\ 27T/ (cJ^(/ca, —a)
From similar analysis applied to the outer boundary, if <r0 is now the density at
this Ixnmdary,
<fi-il>j,l,(Ktc,*),. . . . •.-;.-' :;:. •.' . (63)
-
K J3/., (KC, —at.)
(64)
Similar equations, of course, connect the coefficients which depend on the rotation.
The value of Vm at a point on the sphere >• = c can now be written down, as in
11, and is found to be
V = 4-7r
'" " c
~ 7T + ^ {e1'' J.,,^, -«)-a '^J3,(/«, -a)}
AND THK NEBULAR HYPOTHESIS. 475
in which the rotational terms proportional to the second harmonic are omitted and
UB is given by
TT C r *./ • A
U = — " {(' '' .1 i IrV ftl n* + ll T / ' O\"\ "• »*•' T / \
' I »+ /i \* *-') M/ "»+•/, \f «, ~F*){ T fl J 11 (»'()
•
The value of V at /• = r is, from equation ( l .;).
,. (67)
whence, evaluating V-VW and picking out the coefficient of 8., we find as the value
of vn at r = c,
As before, the points of bifurcation, if any, are given by i\ = 0.
§ 18. It is now necessary to consider the boundary conditions which must be
satisfied at the junction of the two layers. The condition of continuity of material,
•i.e., that the inner surface of the crust shall coincide with the outer surface of the
core, has been expressed in equation (55), b and ft being the same for both core and
crust. There is an equation of continuity of pressure expressed by
13 a
& ~<rau
(69)
which <rix) is now used to represent the density associated with zero pressure.
Finally, there is a condition of continuity of normal force and this requires careful
discussion.
Let M! denote the mass of matter actually forming the core and let V, denote it«
potential at any point outside the core. Let M., denote the mass which would
replace the core if the solution (58) for the crust were extended to the centre and let
V2 denote the potential of this mass at any external point. It will Ixi noticed that if
solution (58) were extended to the origin, it would give an infinite density p at the
origin and also an infinite value of V in virtue of equation (o). On the other hand,
it is readily found, by direct integration, that V, the potential of the mass M,, in
finite at every point, including the origin. It follows that V can only be the
potential of this imaginary arrangement of matter when it is acted on by certain
external forces of which the potential becomes infinite at the origin. Let V, represent
the potential of these forces. The value of V, is readily found, for it must satisfy
VaV, = 0 and must coincide with V or with -^ to within an additive constant at
K
the origin. Thus V3 is the limit of the right-hand side of equation (67) when r = 0,
3 P "2
476 MR. J. H. JEANS ON GRAVITATIONAL INSTABILIT
/• replacing c. This is found to be
v 4«TC0(frr)'/'an«
s~ '2
r
The condition now to he satisfied is clearly that
at all points on the boundary r = a + 6Sn. This requires that Vl— V2 — V3 shall vanish,
to within a constant, at all points outside this boundary, and therefore, in particular,
at r = c. It will be readily seen that
V —V — V — (V }
v 1 » 3 - vm V ' m/n = 0
while V3 is exactly the value of the terms in Vm that involve a, when a is put equal
to zero. The conditions sought are, therefore, simply that all the terms in a which
occur in Vm shall vanish at every point of the sphere.
§ 19. We may now equate the coefficients of the separate harmonics, and obtain
On T I i \ Afl T / \ ^7fA
fr-£U • • (71)
On account of the simplifications made possible by these eqxiations, equation (68)
may be put in the form
<-„ = ac-'^v, (KC> /3)-
n > ..... (72)
The elimination of AO and C0 from (59), (60), and (70) gives
- ^\ Js/2 (K'a' -ct) - /„ _ «^
"
K' Jv7(?aT a) " 2W IT Jv, (*a) '
while similarly the elimination of An and Cn from (61), (62), and (71) gives
i 4. J,,^/.,(^«, -$)3*,.(K'a, -a)\ __ i _ uf\ L J,+./. (K«) J./, (*»)!
J.+Vi (K'a, ft) J,, (,'a, a) I ' 2W I J.+.fc M J-/3 M/ '
For brevity we introduce a function un(x, 9), a generalisation of the un of § 11, the
AND THE NEBULAR HYPOTHESIS.
new function being defined by
H.(X,0)= -Alog{x-'^J..lJX,0)} = J^i^l. ,
J.-'/.u, fl)
Equations (73) and (74) now become
<r'~2~ ff~~
K
1 \ / n+l\*^*/!« • \'*/
while (72) becomes
so that points of bifurcation are given by
Ul(A-,a). . '. ..... (78)
Again, if the rotation may be treated as small, there will invariably be a change of
stability at these points of bifurcation, since vjCn changes sign on passing through
one of them.
§ 20. It is at once clear that the method can be extended to a mass consisting of
any number of layers — the only difficulty occurs in the numerical computations at
the end. At each boundary between two consecutive layers there will be equations
of continuity precisely similar to (75) and (76), while the final value of »'. will be
given by an equation exactly similar to (77), which it will be seen involves only
quantities associated with the outer boundary.
The procedure in any particular case will be to start, so to speak, with the
innermost core of the system. Equation (75) is linear in cos a and sin a, so that
tan a is uniquely determined. Leaving out of account systems in which the densitv
is, in any part, negative, this will be found to l>e adequate to determine a uniquely.
Equation (76) now becomes a linear equation in cos ft and sin ft, from which ft can l»e
determined uniquely. Tn this way. passing from layer to layer, we can determine
the various values of a, ft for the different layers. Finally, the a's and ft'n la-ing
known, equation (78) can be regarded either as an equation for w* or as an equation
for c, i.e., it can be regarded either as determining the highest rotation for which
the symmetrical configuration is stable for a given value of »•, or as determining the
largest value of c for which the mass is stable under a given rotation. If the value of
-^— obtained by the first method is not small, the result will be inaccurate ; if the
2*xr(
value for <• obtained by the second method is so great that the density is in places
478 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
negative, the result will be of no interest except as proving stability for smaller
values of c.
§ 21. It may be well to take a general survey of the equations before giving special
calculations. For simplicity we again consider two layers only, core and crust. From
(75) and (76) it is clear that, when a = 0 or K = K (involving a- = </), the values of «
and ft vanish. Broadly speaking, the more distinct the core is from the crust, tl it-
larger a and ft are. Equation (78), of course, differs only from the corresponding
equation previously found, by the presence of the terms a. and ft. The effect of these
terms is seen ou noticing that, in the notation already used, «, (K'C, a.) is of the form
<f> (K'C— a). Thus, to allow for the effect of the core on the term M, (K'C, a), we have to
leave the algebraic part of the function unaltered, but to change all the trigono-
metrical arguments from K'C to K'C— a. Speaking very broadly, the general effect on
the graph of «, (cf. fig. l) is a compromise between leaving the graph unaltered and
moving it bodily a distance a. along the axis. Similar statements apply to the graph
of »„. Thus, while rotation as before is represented by flattening the graph of ut in
fig. 1, the presence of a core is represented by a distortion of the graphs which may,
with some truth, be thought of as bodily movements parallel to the axis. These
bodily movements may cause new intersections between the graph ut and the other
graphs, and the points of intersection will represent points of bifurcation at which
the symmetrical configuration will become unstable.
No Rotation.
§ 22. The case that may properly be inspected first is that of no rotation. The
equations reduce to
^ul(K'a,a) = ^ul(Ka),. ..... . . (79)
, (K'a, oL)um+,(K'a, ft)} =0- {!+«, (**)«,,+,(<**)}, . . . (80)
and, the equation for points of bifurcation,
un(K'c, -ft) = Ul(K'c,a) ..... • . . . . (81)
When n = 1, it is seen that ft = —a. is a solution of (80), and must therefore (§ 20)
be the only solution. To verify that ft = —a is a solution, replace ft by —a in (80)
and it becomes
. . . .... . (82)
which is seen to be identical with (79) (cf. equations (53) and (74«)). Equation (81)
now reduces to an identity, so that every configuration is formally a point of
bifurcation. The interpretation is, of course, the same as that of § 11, the displace-
ment for which n = 1 is a rigid body displacement, and so requires no force to
AND THE NEBULAR HYPOTHESIS
maintain it. There is, of course, nothing of the nature of a change of stability, for
?<„/(}„, instead of changing sign, remains permanently zero. The consideration of
n = 1 is of no value except that it provides a check on the result of a rather involved
series of analytical processes.
When there is homogeneity hetween core and crust the non-rotating system has
l>een found to he stable for all displacements. To examine whether this is altered by
the presence of the crust, it is natural to test first the extreme case in which (In-
difference between the core and crust is as great as possible. Let us make th«- core
so hot that its density is zero, so that * has to be zero in order that the internal
pressure may be maintained (cf. equation (12)).
Putting a- — 0, equations (79) and (80) reduce to
= 0.
or, by equation (74a),
J.fc(*'«, -a) = 0, J.+.fc(«'o, ft) = 0
whence (equation (49)) a, /3 are given by
= ; tan ft = -
(83)
(84)
(85)
The values of a, /3 corresponding to a few values of *a are given below—
KCI.
ft-
n = 2.
-
1 • '
0 0
• >
0
1 12 1«
- 0 59
2 51 9
15 7
3 100 20
- 48 12
4
153 13
- 91 54
5
- 207 48
- 140 45
6
263 12
192 21
7
319 8
-245 33
9
432 0
-354 51
12
602 21
-521 54
n = 3.
0
0 2
•2 21
16 39
46 •_>•.'
S6 9
The case which is most favourable to the occurrence of points of bifurcation with
positive values of P is when <ru falls to zero at the outer boundary. Let us accordingly
examine this case. We have (equation (63))
J1/3(/c,a) = 0 (86)
so that .
KC = r-a.
480 MR. .1. H. JEANS ON GRAVITATIONAL INSTABILITY
And in virtue of (86), the equation giving points of bifurcation (going back to
equation (72)) is
. ^TTC ' On y / / n\ IQQ\
"2n+l *' " ''l
so that points of bifurcation of order w are given by
J._Vf (ir-a, -^8) = 0 (89)
When n = 2, this becomes
tan (TT — 0. + /3) = TT — OL ;
when n = 3, it is
3 (TT — a
% / „•>
tan (,-«_# =
On treating these equations numerically it is found that they can never be satisfied
We conclude that the non-rotating mass is stable for all displacements, subject, of
course, to the condition that the density shall be everywhere positive.
Slow Rotation.
§ 23. We consider next the stability of a rotating mass of the type under
2
consideration, in which we are limited to - - being small compared with the density of
^7T
the main mass. If we suppose that a-, the density of the core at its outer boundary, is
a
equal to — , we shall have a case — somewhat artificial of course— in which the density
2?r
of the core is very small compared with that of the crust, and in which the equations
are not too complex to admit of treatment.
o
We accordingly assume that a- = — , and the equations (75), (76), and (77) (or (72))
2?r
reduce to the same equations as in the case of no rotation (equations (83)). Thus a, ft
have the same values as before, being given by the table on p. 479.
If we suppose that at the outer boundary of the crust the density falls to the small
o
value <r0 = — , then the value of c, the radius of the outer boundary, is, as before, given
2tTT
by equations (86) or (87), and the value of vn is still given by equation (88). Thus the
analysis is exactly the same as in the case of no rotation, and there are no points of
bifurcation.
It follows that, when <r0 does not have this special value assigned to it, the only
hope of finding points of bifurcation rests upon the gravitational tendency to instability
which arises from the presence of the small layer of crust in which p has a value less
o
than — . Let us pass at once to the examination of the extreme case in which o-g = 0,
iTT
AND THE NEBULAR HYPOTHESIS. ig
Denoting as before the density of the crust at its inner surface (r = a) by </, aud
putting <TO, the density at the outer surface (r = c) equal to zero, we have
P «-'/» T / >„ \ i W1
\j0a Ji/f (K a, a) = <r ,
whence, on elimination of C0,
Equation (72) still gives
-''. J,
_ j _ c sin
a sin (*'«
J..V, (*V, -ft) = 0,
(90)
as the condition for points of bifurcation, and when n = 2 (the only case which
appears to be worth examining), this reduces to
tan (<c'e+/8) = K'C,
(91)
in which /3 is given from the table on p. 479. The procedure is to find /<• from equation
(91), and hence calculate the value of utfeiro-' from equation (90). The results for a
few values of K'a are given in the table following (the last column is explained
later) :—
t
/ca.
K'C.
c/a.
tr*/2™-'.
W*/2»0.
0
C 1
4-489 = 257 27
00
00
0-40
1
4-475 = 256 54
4-475
0-2221
0-45
2
4-713 = 269 21
2-356
0-2269
0-60
3
5-380 = 308 16
1-793
0-2159
0-52
4
6-153 = 352 37
1-538 0-l*i:i
0-51
5
7-026 = 402 40
1-405
0-1570
0-50
6
7-944 = 455 7
1-324
0-1365
0-49
7
8-886 = 509 9
1-269
0-1198
0-48
The obvious remark must at once be made that prol>ably all the values for
are too large for results obtained by the neglect of w* to be accurate. But apart from
absolute accuracy there is an obvious tendency for the value of K^/STO-' to fall off as *'u
increases — for lower values of c/a the symmetrical configuration becomes unstable (• >r
lower and lower values of w"/2»w'. For /a = 100, the value of w*/2T<r' is 0'0104.
§24. Against this, it must be noticed that the value of w'/'2ir<rf is of very slight
importance ; what we are concerned with is the ratio of w'/'Z-ir to the mean density of
the whole mass. For a very rough calculation, we may assume the mean density of
VOL. ccxni. — A. 3 Q
482 MR. J- H. JEANS ON GRAVITATIONAL INSTABILITY
the crust to be £</, whence it follows that the mean density of the whole mass will be
roughly equal to a density 0 defined by
and the value of w^flirQ will be approximately the same as the quantity w2f'2irp which is
computed from observations of binary stars. Values of w2/2v8 are given in the last
column of the table on the preceding page ; the value 0'40 corresponding to tc'a = 0
(no core) being inserted from the result of the previous analysis (§ 13). As before,
the numbers are not numerically accurate, but their general trend may be expected to
reveal the general trend of the true series of numbers. It at once appears that the
values of w^fe-rd are surprisingly steady : there is certainly no rapid decrease in their
amount as //« increases.
Summary and Conclusion.
§25. The problem we have had under consideration has been that of testing whether
the behaviour of a rotating mass of compressible heterogeneous matter differs very
widely from that of the incompressible homogeneous mass which has been studied by
MACLAUEIN, JACOBI, POINCARE, and DARWIN. The result obtained can be summed
up very briefly by saying that the ideal incompressible mass has been found to supply
a surprisingly good model by which to study the behaviour of the more complicated
systems found in astronomy. The problem especially under consideration has been
that of determining the amount of rotation at which configurations of revolution
(e.g., spheroids) first become unstable. In so far as we have been able to examine the
question, it appears probable that the compressible mass will behave, at least up to this
point, in a manner almost exactly similar to the simpler incompressible mass, and
results obtained for the latter will be nearly true, both qualitatively and quantitatively,
for the former. The compressible mass, set into rotation, will apparently pass through
a series of flattened configurations very similar to the Maclaurin spheroids ; it will
then, for just about the same amount of rotation (as measured by w2jp), leave the
symmetrical form and assume a form similar to the Jacobian ellipsoids. Beyond this
stage our analysis has not been able to deal with the problem. Indeed, strictly
speaking, our analysis has hardly been able to carry this far. A question of
importance has been whether the quasi-spheroidal form for a compressible mass does
not become unstable for a much smaller value of iv* than the incompressible mass, and
whether the instability does not set in in a different way. These questions we have
been able to answer, with, I think, a very high degree of probability, in the negative.
The whole matter is of necessity one of probability only, and not of certainty, for the
general heterogeneous compressible mass is not amenable to analysis until a great
number of simplifying assumptions have been made.
It was first pointed out that a compressible mass has an infinitely greater number
AND THE NEBULAK HYPOTHESIS. 483
of vibrations than an incompressible one, and as the mass is only stable when every
vibration individually is stable, it might be thought that a compressible mass had more
chance of being unstable — or would become unstable sooner — than the corresponding
compressible mass. This has on the whole been found not to be the case, and ->n
looking through the analysis the reason can be seen.
A vibration in a compressible mass may be regarded loosely as a system of waves ;
the distance from one point of zero displacement to the next may be regarded as a sort
of wave-length of the vibration. The stability or instability of a vibration depends on
which is the greater — the gain in elastic energy or the loss in gravitational energy
when the vibration takes place. But as between a vibration of great wave-length and
one of short wave-length there is this important distinction : for equal maximum
amplitudes the gravitational disturbance caused by the disturbance of great wave-
length is much greater than that caused by the disturbance of short wave-length, since
the elements of the latter very largely neutralise one another. Thus the change in
gravitational energy is enormously the greatest for disturbances of great wave-length,
while it is easily seen that the changes in elastic energy are approximately the same.
It follows that if the mass becomes unstable it will be through a vibration of tin-
greatest possible wave-length, i.e., a wave-length about equal to the diameter of the
mass. This general prediction is amply verified in the detailed problems that have
been discussed. When we reflect that the vibrations of greatest wave-length are
exactly those which are common both to compressible and incompressible masses, we
see readily that, in this respect at least, compressibility is likely to make but little
difference.
The vibrations of greatest wave-length are put in evidence, both in the compressible
and incompressible mass, by the displacement of the surface. A vibration in which
the displacement is proportional to a zonal harmonic P, may be thought of as having
a wave-length approximately equal to -a-a/ii. In accordance with the principle that
vibrations of great wave-lengths are most effective towards instability, we should
expect the lowest values of n to give the vibrations which first bacome unstable, and
this is, in fact, found to be the case. But here a very real distinction enters between
the compressible and the incompressible mass. In the incompressible mass vibrations
of order n = 1 are non-existent, the displacement being purely a rigid tx»dy displace-
ment ; in the compressible mass vibrations of order » = 1 can certainly occur, and so
might reasonably be expected to be the first to become unstable.
It is in point of fact known that the incompressible mass becomes unstable through
vibrations of orders 2, 3, ... in turn; it is found that the compressible mass also
becomes unstable through vibrations of orders 2, 3, ... in turn, the vibrations of
order 1 failing completely to produce instability. The reason for this apparent
anomaly can, I think, be traced in the following way. In a displacement of order 1
any spherical layer of particles will after displacement be spread uniformly .
another sphere excentric to the first. The gravitational force produced
3 Q 2
484 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY
sphere of particles both before and after displacement is exactly nil at a point inside
the sphere. Thus the gravitational field set up by a displacement of order 1
neutralises itself in a way not contemplated in the general argument outlined above.
Also the vibrations of order 1 and of greatest wave-length in the interior are not
available, for they represent solely a rigid body displacement.
The question of vibrations of order 1 is treated in §§ 3-7 ; it is shown that they
may be disregarded, and we pass to the consideration of vibrations of orders 2, 3, ...,
expecting (as, in fact, is found to be the case) that instability will first set in through
a vibration of order 2.
It is only possible to discuss special cases, and the one which is most amenable to
analysis is that in which the pressure and density are connected by LAPLACE'S law,
p = c(p*— <r2). It is first proved (§§ 8-11) that, for a mass of such matter at rest,
the spherical form is stable for all displacements. Later (§§ 15-22) it is shown that
this is true when c varies inside the mass ; it is true even up to the case which is the
most likely to be unstable, in which the matter in the interior is of negligible density
and the main part of the mass is collected in a surface crust — a sort of astronomical
soap-bubble.
We proceed next to examine for what amount of rotation these figures will become
unstable, treating first the case in which c is the same throughout the mass.
Imagining c and a- to vary we can get a variety of types of mass. The surprising
result is obtained (by something short of strict mathematical proof) that the figure
which is the first to become unstable (as w2/2?rP increases uniformly for them all) is
the perfectly incompressible one — gravitational instability appears to act in the
unexpected direction, at any rate when the degree of rotation is measured by iv^feirp,
p being the mean density. As it was not possible to obtain strictly accurate figures
in this case, the result was checked by considering the artificial, but physically
analogous, problem of rotating cylinders of Laplacian matter, in which it was possible
to obtain perfectly exact results (§14). The result was confirmed, and the additional
information was obtained that the value of w*/2-7rp remains surprisingly steady
through quite a wide range of compressibility (vide table on p. 471).
The physical reason for this can, I think, be understood as follows. The more
compressible the matter is the more it tends to concentrate near the centre, i.e., in
just those regions where the " centrifugal force " obtains, so to speak, least grip on it.
Incompressibility neutralises the gravitational tendency to instability, but tends to
compel the matter to place itself so that the rotational tendency to instability can
act at the best advantage.
The similar problem is next investigated (§§ 23, 24) when c varies inside the mass ;
in particular, the limiting case of a soap-bubble-like mass is considered. Again the
surprising result emerges that the value of nffe-wp needed to establish instability of
the symmetrical configuration is just about the same as before (vide table, p. 481).
The matter is now constrained to remain, so to speak, on the rim of a fly-wheel where
AND THE NEBULAR HYPOTHESIS. 485
the centrifugal force can act at the best advantage and gravitational instability has
full scope. If p is the mean density of the crust, w*/2irp must obviously be le»» than
before. But if p is the mean density of the whole mass, p/p is also much smaller.
These two quantities march with approximately equal steps, so that i<*l'2*]t remains
almost unaltered.
Thus we have the general result that for all the varied types of mass that have
been considered the spheroidal or quasi-spheroidal form always becomes first unstalil.-
for just about the same value of t^/2irp. If, from the point of view of discover! ng
new processes in nature, the present investigation has been somewhat l>arren, at least
we may reflect that the work of DAKWIN and POINCAR£ has been shown, to some
extent, to have an enhanced value, in that it seems to apply to the real bodies of
nature and not merely to mathematical abstractions.
[ 487 ]
INDEX
TO THI
PHILOSOPHICAL TRANSACTIONS
SERIES A, VOL. 213.
c.
Carbonic acid at low temperature*, thermal properties of (JssitiN and Pn), 67.
Cassegrain reflector with corrected field (SAMPSON), 27.
CHAPMAN (S. C.). On the Diurnal Variations of the Earth's Magnetism produced by the Moon and Sun, 279.
CHRBB (C.). Some Phenomena of Sunspote and of Terrestrial Magnetism.— Part II., 246.
CUTUBBBTSON (C. and M.). On the Refraction and Dispersion of the Halogens, Ilalogen Acids, Oione, Steam, Oiidt* of
Nitrogen und Ammonia, 1.
O.
Gravitational instability and nebular hypothesis (jBANs), 457.
GRIFFITHS (E. H. and £.). The Capacity for Ilent of Metals at Different Temperatures, being an Account of KiperimenU
performed in the Research Laboratory of the University College of South Wales and Monmouthshire, 119.
II.
Heat, capacity for, of metals at different temperatures (GRIFFITHS), 119.
HICKS (W. M.). A Critical Study of Spectral Series.— Part III. The Atomic Weight Term and iU Import in tlw
Constitution of Spectra, 323.
HOPKINSON ( 1!.). A Method of Measuring the Pressure produced in the Detonation of lligh Kxplo*ives or by the Impact
of Bullets, 437.
J.
JEANS (J. H.). Gravitational Instability and the Nebular Hypothesis, 467.
JKNKIN (C. F.) and PTB (D. R.). The Thermal Properties of Carbonic Acid at Low Temperature*, 67.
LY.LB (T. R.). On the Self-inductance of Circular Coils of Rectangular Section, 421.
VCL. CCXIII. A 508. 3 R Published separately, February 6, 1914.
488 INDEX.
M.
Magnetism, diurnal variations of earth's, produced by moon and sun (CHAPMAN), 279.
Magnetism, terrestrial, and sunspots, some phenomena of (CHBKE), 245.
Metals, capacity for heat of, at different temperatures (&BIFFITHS), 119.
N.
Nebular hypothesis and gravitational instability (JEANS), 457.
P.
Pressure produced in detonation of explosives or by impact of bullets, method of measuring (IIopKiNSON), 437.
PYE (D. R.). See JENKIN and PIE.
R.
Refraction and dispersion of halogens, halogen acids, ozone, &c. (CCTHBEBTSON), 1.
S.
SAMPSON (R. A.). On a Cassegrain Reflector with Corrected Field, 27.
Self-inductance of circular coils of rectangular section (LYLE), 421.
SOUTHWELL (R. V.). On the General Theory of Elastic Stability, 187.
Spectra, constitution of, and the atomic weight term (HiCKs), 323.
Stability, general theory of elastic (SOUTHWELL), 187.
Sunspots and terrestrial magnetism, some phenomena of (CHBEE), 245.
HABBI80X AlfD SOUS, PBIKTBBS IN ORDINAEY TO HIS MAJESTY, ST. MABTIN's LANE, LONDON, W.C.
W Roya^ . of London
41 Philosophical tr*nB»etiona.
L62 Ser nautical and
v.213 ph ijs.
Se..
Seh.k
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