UNIV.OF
TORONTO

•K.

PHILOSOPHICAL

TRANSACTIONS

OF T11E

ROYAL SOCIETY OE LONDON.

SERIES A.

CONTAINING PAPEUS OF A MATHEMATICAL Oil PHYSICAL CHAIlACTElf.

VOL. 215.

LONDON:

PRINTED BY HARRISON AND SONS, ST. MAKTIN's LANE, W.C.,

iit ®rbiirarjr to %'is

OCTOBER, 1915.

s I '

\>v

CONTENTS.

(A)
VOL. 215.

List of Illustrations : page v

Advertisement vii

I. Eddy Motion in the Atmosphere. By G. I. TAYLOR, M.A., Schuster Header in

Dynamical Meteorology. Communicated by W. N. SIIA\V, Sc.D., F.K.S.,
Director of the Meteorological Office . 2)a'J(' ^

II. On the Potential of Ellipsoidal Bodies, and /he Figures of k'yiiilibrium of

Rotating Liquid Masses. By J. H. JEANS, M.A., F.R.S 27

III. The Influence of Molecular Constitution and Temperature on, Magnetic.

Susceptibility. Part III. — On, the Molecular Field in Dif (magnetic Substances.
By A. E. OXLEY, M.A., M.Sc., Coutts Trotter Student, Trinity College,
Cambridge, Mackinnon Student of the Royal Society. Communicated by
Prof. Sir J. J. THOMSON, O.M., F.R.S. 79

IV. The Transmission of Electric Waves over (lie Surface of the. Earth. By

A. E. H. LOVE, F.R.S. , Sedleian Professor of Natural Philosophy in the
University of Oxford 105

V. Atmospheric Electricity Potential Gradient at Kew Observatory, 1898 to 1912.

By C. CHEEE, Sc.D., LL.D., F.R.S., Superintendent of Kew Observatory 133

VI. The Lunar Diurnal Magnetic Variation, and its CJiange with Lunar Distance.

By S. CHAPMAN, M.A., D.Sc., Fellow and Lecturer of Trinity College,
Cambridge, lately Chief Assistant at the Royal Observatory, Greenwich.
Communicated by the Astronomer Royal, F.R.S. 161

VII. A Thermomagnetic Study of the Eutectoid Transition Point of Carbon Steels.

By S. W. J. SMITH, M.A., D.Sc., F.R.S., Assistant Professor of Physics, and
J. GUILD, A.R.C.S., D.I.C., Assistant Demonstrator of Physics, Imperial

College, South Kensington 177

a 2

VIII. The Effect of Pressure upon Arc Spectra. No. 5.— Nickel, \ 3450 to \ 5500,
including an Account of the Rate of Displacement with Wave-length, of the
Relation between the Pressure and the Displacement, of the Influence of the
Density of the Material and of the Intensity of the Spectrum Lines upon
the Displacement, and of the Resolution of the Nickel Spectrum into Groups
of Lines. By W. GEOFFREY DUFFIELD, D.Sc., Professor of Physics, and
Dean of the Faculty of Science in University College, Reading. Communi-
cated by Prof. A. SCHUSTER, Sec. R.S. .......... page 205

IX. BAKERIAN LECTURE. — X-rays and Crystal Structure. By W. H. BRAGG, D.Sc.,

F.R.S., Cavendish Professor of Physics in the University of Leeds . . 253

X. Gaseous Combustion at High Pressures. By WILLIAM ARTHUR BONE, D.Sc.

Ph.D., F.R.S., formerly Livesey Professor of Coal Gas and Fuel Industries
at the University of Leeds, now Professor of Chemical Technology at the
Imperial College of Science and Technology, London, in collaboration with
HAMILTON DATIES, B.Sc., H. H. GRAY, B.Sc., HERBERT H. HENSTOCK,
M.Sc., Ph.D., and J. B. DAWSON, B.Sc., formerly of the Fuel Department in
the University of Leeds ................. 275

XL Heats of Dilution, of Concentrated Solutions. By WM. S. TUCKER, A.R.C.Sc.,
B.Sc. Communicated by Prof. H. L. CALLENDAR, F.R.S. ..... 319

XII. Thermal Properties of Carbonic Acid at Low Temperatures. (Second Paper.}

By C. FREWEN JENKIN, M.A., M.Inst.C.E., Professor of Engineering Science,
Oxford, and D. R. PYE, M.A., Felloiv of Neiv College, Oxford. Communicated
by Sir ALFRED EWING, K.C. B., F.R. S. ........... 353

XIII. On the Specific Heat of Steam at Atmospheric Pressure between 104° C. and

115° C. (Experiments by the Continuous Flow Method of Calorimetry
performed in the Physical Laboratory of the Royal College of Science,
London.) By J. H. BRINKWORTH, A.R.C.S., B.Sc., Lecturer in Physics at
St. Thomas's Hospital Medical School. Preface by H. L. CALLENDAR,
M.A., LL.D., F.R.S., Professor of Physics at the Imperial College of Science
and Technology, London, S.W. ............. 383

XIV. Some Applications of Conformal Transformation to Problems in Hydro-

dynamics. By J. G. LEATHEM, M.A., D.Sc. Communicated by Sir JOSEPH
LARMOR, M.P., F.R.S. ................. 439

Index to Volume

LIST OF ILLUSTRATIONS.

Plates 1 to 5. — Dr. W. GEOFFREY DUFFIELD on the Effect of Pressure upon Arc
Spectra. No. 5. — Nickel, X 3450 to X 5500.

Plate G. — Prof. C. FREWEN JENKIN and Mr. D. R PYE on Thermal Properties of
Carbonic Acid at Low Temperatures. (Second Paper.)

ADVERTISEMENT.

THE Committee appointed by the Roycd Society to direct the publication of the
Philosophical Transactions take this opportunity to acquaint the public that it fully
appears, as well from the Council-books and Journals of the Society as from repeated
declarations which have been made in several former Transactions, that the printing of
them was always, from time to time, the single act of the respective Secretaries till
the Forty-seventh Volume ; the Society, as a Body, never interesting themselves any
further in their publication than by occasionally recommending the revival of them to
some of their Secretaries, when, from the particular circumstances of their affairs, the
Transactions had happened for any length of time to be intermitted. And this seems
principally to have been done with a view to satisfy the public that their usual
meetings were then continued, for the improvement of knowledge and benefit of
mankind : the great ends of their first institution by the Royal Charters, and which
they have ever since steadily pursued.

But the Society being of late years greatly enlarged, and their communications more
numerous, it was thought advisable that a Committee of their members should be
appointed to reconsider the papers read before them, and select out of them such as
they should judge most proper for publication in the future Transactions; which was
accordingly done upon the 2Gth of March, 1752. And the grounds of their choice are,
and will continue to be, the importance and singularity of the subjects, or the
advantageous manner of treating them : without pretending to answer for the
certainty of the facts, or propriety of the reasonings contained in the several papers
so published, which must still rest on the credit or judgment of their respective
authors.

It is likewise necessary on this occasion to remark, that it is an established rule of
the Society, to which they will always adhere, never to give their opinion, as a Body,

upon any subject, either of Nature or Art, that comes before them. And therefore the
thanks, which are frequently proposed from the Chair, to be given to the authors of
such papers as are read at their accustomed meetings, or to the persons through wh<
hands they received them, are to be considered in no other light than as a matter of
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And therefore it is hoped that no regard will hereafter be paid to such reports and
public notices; which in some instances have been too lightly credited, to the
dishonour of the Society.

PHILOSOPHICAL TRANSACTIONS.

I. Eddy Motion in the Atmosphere.
By G. I. TAYLOR, M.A., Schuster Reader in Dynamical Meteorology.

Communicated by W. N. SHAW, Sc.D., F.R.S., Director of the Meteorological

Office.

Received April 2,— Read May 7, 1914.

OUR knowledge of wind eddies in the atmosphere has so far been confined to the
observations of meteorologists and aviators. The treatment of eddy motion in
either incompressible or compressible fluids by means of mathematics lias always
been regarded as a problem of great difficulty, but this appears to be because
attention has chiefly been directed to the behaviour of eddies considered as indi-
viduals rather than to the average effect of a collection of eddies. The difference
between these two aspects of the question resembles the difference between the
consideration of the action of molecule on molecule in the dynamical theory of gases,
and the consideration of the average effect, on the properties of a gas, of the motion
of its molecules.

It has been known for a long time that the retarding effect of the surface of
the earth on the velocity of the wind must be due, in some way, to eddy motion ;
but apparently no one has investigated the question of whether any known type
of eddy motion is capable of producing the distribution of wind velocity which has
been observed by meteorologists, and no calculations have been made to find out
how much eddy motion is necessary in order to. account for this distribution. The
present paper deals with the effect of a system of eddies on the velocity of the
wind and on the temperature and humidity of the atmosphere. In a future paper
the way in which they are produced and their stability when formed will be
considered.

It is well known that wind velocity, temperature, and humidity vary much
more rapidly in a vertical than in any horizontal direction, and further that the
vertical component of wind velocity is very small compared with the horizontal
velocity. It has been assumed, therefore, that the average condition of the air
at any time is constant for a given height, over an area which is large compared
with the maximum height considered. If u and v represent the components of
undisturbed wind velocity parallel to horizontal axes, x and y, running from South

VOL. CGXV. A 523. B [Published January 21, 1915.

2 Q. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.

to North and from West to East respectively, and if T and m represent the
average temperature and the average amount of water vapour per cubic centimetre
of air, this is equivalent to assuming that u, v, T, and m are functions of z, the
height, and t, the time ; and that they are independent of x and y.

Vertical Transference of Heat by Eddy Motion.

Let us first consider the propagation of heat in a vertical direction. The ordinary
conductivity of heat by molecular agitation is so small that no sensible error will
be introduced by leaving it out of the calculations. The only way in which
large quantities of heat can be conveyed upwards or downwards through the
atmosphere is by means of a vertical transference of air which retains its heat as
it passes into regions where the temperature differs from that of the layer from
which it started. If T' and «•' represent the temperature and the vertical component
of the velocity of the air at any point, the rate at which heat is propagated

across any horizontal area is j I ptrT'w' (If dij where t> and <r are the density and
specific heat respectively, and the integral is taken over the area in question.

Since there is no vertical motion of the air as a whole pw' dx dy = 0. Hence, in

order that heat may be conveyed downwards, the air at any level must be hotter
in a downward than in an upward current. In order that this may be the case
the potential temperature* of the air must increase upwards. The excess of
temperature in a downward current over the mean temperature at any level will
depend partly on the vertical distance through which the air has travelled since
it was at the same temperature as its surroundings, and partly on the rate of
change in potential temperature with height. If the hot air, after crossing the
horizontal area, continues on its downward course with undiminished velocity and
without losing heat, and if the mean potential temperature continues to decrease
downwards, the rate of transmission of heat across a horizontal area will continually
increase. On the other hand, if the air returns across the area without losing its
heat, there will be no resultant transmission of heat at all. The air must lose its
heat by mixture with surrounding air after crossing the area.

Consider now the transference of heat across a large horizontal area A at a
height z. Suppose that at a time t0 an eddy broke away from the surrounding air
at height z0 and arrived at the point xyz at time t; z0 and t0 are then functions
of x, y, z, and t. Suppose that initially the eddy had the same temperature as its
surroundings. Let 6 (z, t) be the average potential temperature of the air in the
layer at height z at time t ; then since the air preserves the potential temperature

* Potential temperature is the temperature air would assume if its volume were changed adiabatically
till it was "at some standard pressure, say 760 mm.

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 3

of the layer from which it originated, the potential temperature at the point xyz
at time t is 8 (z0, t0).

The amount of heat which passes per second across the area A is therefore

pa- ivQ. (z0(0) dx dy. Now 6 (z0, t0) may be expressed in the form

provided that the changes in Q in the height z — z0 and in the time t — ta are small
compared with 6. Hence

wB (z0, t0) dxdy = 6 (z, t) wdxdy + ^- \\--w (z0-z) dxdy+ ~ \\ w (t0-t) dxdy.

JJA. JJA dZ JJA Ct JJA.

Now £„— t is necessarily negative, and since II wdxdy = 0 it is evident that a
positive value of w occurs as often as a negative one ; hence if the eddy motion is
uniformly distributed w (t0—t) dxdy = 0.

The rate at which heat crosses the plane is therefore

-pa- 7T- jj w (z-z0) dxdy.
The rate at which heat crosses an area A at height z + <5z is

p T ( -~-^-Sz) I w(z-z0)dxdy for w(z-z(>)dxdy

\ dz d Z /JJA JJA

does not vary with z if the eddy motion is uniformly distributed. Hence the rate
at which heat enters the volume A.Sz is

/oo-^-j <5z w (z-Zo) dxdy.
dz JJA

Now since mixtures which take place within this volume merely alter the

distribution of the heat contained in it without affecting its amount, this must be
o/j

equal to pa- — A.Sz. Hence we obtain the equation for the propagation of heat by
ct

means of eddies in the form ^- = — — -,- w (z— z0) dx dy. But -p w (z— z0) dx dy

ct cz A. JJA. AJJA

is the average value of w (z— z0) over a horizontal area, hence it may be expressed
in the form \ (wd), where d is the average height through which an eddy moves
from the layer at which it was at the same temperature as its surroundings, to
the layer with which it mixes, w is defined by the relation \ (wd) = average
value of w (z— z0) over a horizontal plane; it roughly represents the average

B 2

4 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHEEE.

vertical velocity of the air in places where it is moving upwards. The divisor 2
is inserted because the air at any given point is equally likely to be in any portion
of the path of an eddy, so that the average value of z-z0 should be approximately
equal to \ (d).

The equation for the propagation of heat by means of eddies may now be written

30 wclfrf)

The equation for the propagation of heat in a solid of coefficient of conductivity
specific heat a- and density p is

(U pa- z-

It appears therefore that potential temperature is transmitted upwards through the
atmosphere by means of eddies in the same way that temperature is transmitted in
a solid of conductivity K, provided K/pv = ^wd. We shall in future call K the " eddy
conductivity " of the atmosphere.

If we know the temperature distribution at any time (say t = 0), and if we know
the subsequent changes of temperature at the base of the atmosphere we can calculate,
on the assumption of a uniform value for K/pcr, the temperature distribution at any
subsequent time. Conversely, if the temperature distribution on two occasions be
known, and if we know the temperature of the base of the atmosphere at all inter-
mediate times, we can obtain some information about the coefficient of eddy
conductivity, and hence about the eddies themselves.

I was fortunate enough to be able to obtain the necessary data on board the
ice-scout ship "Scotia" in the North Atlantic last year. On several occasions
the distribution of temperature in height was determined by means of kites. The
temperature changes experienced by the lowest layer of the air as it moved up to the
position where its temperature distribution was explored by means of a kite, were
found in the following way. The path of the air explored in the kite ascent was traced
back through successive steps on a chart by means of observations of wind velocity
and direction taken on board English, German, and Danish vessels, which happened to
be near the position occupied by the air at various times previous to the kite ascent.
This method was adopted by SHAW and LEMPFERT in their work on the ' Life
History of Surface Air Currents.' It depends for success on being able to obtain
observations in the right spot at the right time. It frequently happens that no such
observations are obtainable, and in these cases it is impossible to proceed with the
investigation. Owing to this difficulty I was unable to trace the air paths for more
than seven of the ascents.

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHEEE.

Having obtained the path of the air, the next step is to find the temperature of the
sea below it. This is a comparatively easy matter, for a careful watch is kept by the
liners on the temperature of the North Atlantic. The results of their observations are
plotted by the Meteorological Office on weekly charts, on which isothermal lines are
drawn to represent sea temperatures of 80° F., 70° F., 60° F., 50° F., and 40° F. These
charts are published on a small scale in the weekly weather report of the Meteorological
Office, but Captain Campbell Hepworth was kind enough to lend me the originals, and
on them I plotted the air paths.

One of the charts, with the air's path marked on it, is shown in fig. 1.*
It has been found that the temperature of the air rarely differs from that of the
surface of the sea by more than 2° C., and usually the difference is only a fraction of

Path of air and sea temperature for kite ascent of August 4th.

Fig- I-

a degree. The temperature of the "base of the atmosphere at any point along the
air's path has, therefore, been assumed to be that of the surface of the sea. In many
of the kite ascents the temperature of the sea, and therefore of the surface air,
increased up to a certain point along the air's path and then began to decrease.
While the air was moving along the first part of the path its temperature might be
expected to decrease with height at a rate greater than the adiabatic rate.t When

* Others are reproduced in the 'Report of the "Scotia" Expedition, 1913.'

t If the temperature of the air diminishes at the adiabatic rate of 10° G. per kilometre, its potential
temperature is constant, so that no amount of eddy motion can transfer heat either upwards or
downwards.

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.

the air entered the portion of the path along which temperature was diminishing
it might be expected that the cooling effect of the sea would not spread upwards
instantaneously, but that it would make its way gradually into the upper layers.
We might expect, therefore, that, if a kite were to be sent up into the air as it was
passing over the second part of its path, the temperature would increase up to a
certain height ; and that, above that height, it would have the temperature gradient
which it had acquired during its passage along the first portion of its path.

If a curve be drawn to represent the temperature of the atmosphere at different
heights a change from heating to cooling along the air's path will give rise to a
corresponding bend in this curve. The height of this bend above the surface of the
earth will depend partly on the interval which elapsed between the time when the air
was passing over the portion of the path where heating stopped and cooling began

Fig 2

RELATIVE HUMIDITY PER CENT

HEIGHT
IN METRES

1000

600 tc-..

%

T*

3 / ^

TEMPERATURE °C

50

10

12 14

and the time of the ascent, and partly on the eddy conductivity of the atmosphere.
If we know two of these quantities we should be able to calculate the third.

On the right hand side of fig. 2 is shown the temperature distribution at various
heights from the surface up to 1100 metres in the case of the air which had blown
along the path drawn on the chart shown in fig. 1. It will be seen that there are
two bends in the curve. The lowest portion from the surface up to 370 metres
evidently corresponds with the cooling of the lowest strata of the atmosphere which
had been going on ever since the air turned back from the warm water of the Atlantic
towards the cold water of the Great Bank of Newfoundland.

The air explored in the ascent of August 4th turned towards the west at 8 a.m. on
August 3rd and continued blowing on to colder and colder water till the time of the

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 7

ascent, 8 p.m., August 4th. It appears, therefore, that the cooling had extended
upwards through a height of 370 metres in 36 hours. An arrow has been drawn on
the base line to represent the temperature of the sea which, as we should expect, is
slightly less than the temperature of the air which is being cooled by it. The portion
of the temperature curve of fig. 2 which lies between 370 metres and 770 metres is
due to the warming which the air had undergone between the evening of July 30th
and 8 a.m., August 3rd. The portion of the curve above 770 metres to which the
warming of July 30th to August 3rd had not yet reached, is due to the cooling which
the air experienced as it blew off the warm land of Canada on to the cold Arctic water
which runs down the coast of Labrador.

The curve on the left hand of fig. 2 represents the humidity of the atmosphere at
different heights. It is reproduced here for two reasons, firstly, the extreme dryness of
the air at 1100 metres (the humidity being only 20 per cent.) shows that the air really
had blown off the land as is shown on the chart in fig. 1 ; and, secondly, because it
shows that changes in the amount of water vapour in the atmosphere are propagated
upwards in the same way as changes in temperature. Bends in the humidity curve
occur at the same heights as bends in the temperature curve. This is in fact to be
expected, for it is evident that the reasoning which was used to deduce the equation

(l) would serve equally well to deduce an equation (- -r-jfor the propagation

ct jj CZ

of water vapour into the atmosphere.

Temperature-height curves, similar to that shown in fig. 2, were traced for all the
kite ascents which were made from the " Scotia," and most of them did have bends in
them. In all cases in which it was possible to trace the air's path a bend in the curve
was found to correspond, either to a change from heating to cooling (or rice ro-sd) of
the surface air as it moved along its path, or to a sudden change in the rate of cooling
when the air crossed the sharply defined edge of the Gulf Stream.

In most cases the change from heating to cooling was due to a change in the
direction of the wind. Changes in wind direction occur simultaneously over large
areas of the ocean, hence, even if the exact position of the path is not accurately
determined, we may be able to obtain reliable information as to the time at which
heating ceased and cooling began ; and calculations which depend on the interval
between the time of this change and the time of the kite ascent will be more accurate
than those which involve the length or position of the path.

Let us consider the temperature distribution in the atmosphere in an ideal case so
chosen as to represent as nearly as possible the actual conditions of some of the
"Scotia" kite ascents.

Suppose that the initial potential temperature of the atmosphere is taken to be
zero at all heights, and suppose that the surface layers begin to l>e cooled at time
t = 0 in such a way that the potential temperature 6a at the ground, 2 = 0, is a
function <j> (t) of the time, so that 00 = <j> (t).

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHEKE.

The solution of — = — VT which fits these conditions is*

dt 2 3z*

2 r

The two following cases are of interest : —

(a) The surface temperature decreases uniformly as t increases at a rate of p" C.

per second, so that $0 = — pt.

(b) The temperature of the surface layers changes suddenly from 0 to 00 and

afterwards remains constant.

In (n) the integral becomes

9 = -

where £ = z (2w'dt)~* and i/r (£) represents the expression in square brackets.

The curve (a) in fig. 3 represents the values of ^ for values of f ranging from 0 to
1 '2. It will be seen that when £ = '8 the value of ^ is ^h of its value at the
surface, where f = 0.

See ' FOURIER'S Series and Integrals,' H. S. CARSLAW, p. 238.

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.
In (6) the integral becomes

e =

where x(£) represents bhe expression in brackets and f has the same meaning as
before.

The curve (&) in fig. 3 represents the values of x (f) for different values of £ It
will be seen that when £ = 1 '2 the temperature is about TV of the surface
temperature 6a. In actual cases it is not easy to say whether (a) or (/>) is a better
representation of the changes in temperature along the air's path. In most cases
probably (a) is the best, but, in one case, that of the ascent of May 3rd, the bend in
the temperature-height curve was due to the passage of the air across a sharply
defined boundary between the warm waters of the Gulf Stream and the cold arctic
water over the great Bank of Newfoundland,* and then one might expect (/>) to be
a truer representation of the vertical temperature distribution. In either case we
shall not be far wrong if we assume that the height to which the new conditions
have reached at time t is given by f = 1 '0 or

z2 = 2wdt. . . .......... (2)

If we can measure z, the height of the bend in the temperature-height curve, and
if we know t, the interval which has elapsed between the time at which the rate of
change of surface temperature along the air's path suddenly altered its value and the
time of the ascent, the equation (2) enables us to calculate KJpv or ^(wd). The error
in this result may be as great as 30 per cent., but it does at any rate give a good
idea of the magnitude of the coefficient of eddy conductivity and of the amount
of eddy motion which is necessary in order to produce the vertical temperature
distributions which have been observed.

In some of the cases the potential temperature before the change which caused the
bend in the temperature-height curve was not constant at all heights. In the case of
the upper bend in the curve shown in fig. 2, for instance, the potential temperature
increased with height before the warming which produced the upper band occurred.
This, however, makes no difference to the rate at which the bend is propagated

upwards. It is evident that if 0, and 62 be two solutions of — = -— ^-j , then Bl + 62

cC s-i c%

is also a solution. If the initial potential temperature before the change were
6 = T0 + az, and if the surface temperature were to change suddenly to T, at time
t = 0, the temperature at height z at a subsequent time.i would be

It is evident that the term T0 + az does not affect the rate at which the bend in the
temperature-height curve is propagated upwards.

* See ' Reports of the " Scotia" Expedition, 1913.'
VOL. CCXV. — A. C

10

G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHERE.

In Table I. the values of z and t observed in the " Scotia " kite ascents are given.
In the first column is given the date of the ascent, in the second column the height
of the bend in the temperature-height curve, in the third column the interval between
the time of the change in the temperature conditions which give rise to the bend in
the temperature-height curve and the time of the ascent, and in the fourth column
are given values of %(wd) in C.G.S. units, calculated from the equation %(wd) = z*l±t.

TABLE I.

1 1

/

Date of observation. m^s ^

|

} ! i

22
it

in C.G.S. units.

Average
wind force
(Beaufort scale).

|

May 3rd . . . . i 270 15

.3--tx 103

3-3

j 1

July 17th ! 140 24

•57xl03

2

July 25th ...... 610 168
July 29th 170 15

l-5xlO»
1-SxlO8

2 to 3
2-2

August 2nd ... 200 11

2-5 xlO3

3

i

i C 370 :i6
August 4th (two bends

2-6 xlO3

2-5

height curve) [ 77() 12Q

|

3- 4x10"

3-1

It will be seen that the values of ^ (wd) vary through a large range. It is to be
expected that the amount of eddy motion will depend on the wind velocity ; accord-
ingly, a fifth column has been added to show the average wind force during the
time t. The figures in this column are the means of the Beaufort wind-force numbers
recorded at the " Scotia " during a time t before the ascent. In the case of the
ascent of July 25th the necessary observations were unobtainable because the " Scotia "
was in port till July 24th. lu this case the wind force recorded by the steamers, from
whose observations was traced the path along which the air approached the position
of the " Scotia " at the time of the kite ascent of July 25th, varied from 2 to 3 on
the Beaufort Scale.

It will be seen that for .the ascents of May 3rd, August 2nd, and August 4th,
when the wind force was about 3, the values of ^ (wd) are 3'4 x 103, 2'5 x 103, 2'6 x 103

G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHERE. 11

and 3'4xl03; and that for July 17th and July 29th, when the wind force was
about 2, the values were very much lower, being 0'57 x 103 and 1'3 x 103 respectively.
The fact that these figures are so consistent, although t varies from 11 hours to
7 days and z from 140 metres to 770 metres, seems to indicate that the eddy motion
does not diminish to any great extent in the first 770 metres above the surface.

Vertical Change of Velocity due to Eddy Motion.

In the first part of this paper the vertical transference of heat by means of eddies
has been discussed. For this purpose it was necessary to consider only the vertical
component of eddy velocity, but in the questions which are treated in the succeeding
pages it is no longer possible to leave the horizontal components out of the calculations.
It seems natural to suppose that eddies will transfer not only the heat and water
vapour, but also the momentum of the layer in which they originated to the layer
with which they mix. In this way there will be an interchange of momentum
between the different layers. If U. and V; represent the average horizontal
components of wind velocity at height z parallel to perpendicular co-ordinates x and
y, and if?*', t'', w' represent components of eddy velocity so that the three components
of velocity are Uz + w', V,+ r' and «•', then the rate at which a;-momentuiu is trans-
mitted across any horizontal area is

P(Ut+u')w'dxdy, (3)

ff *

and the rate at which ?/-momeutum is transferred is \\p(Vt+v')u/dxdy the integrals

extending over the area in question.

If we were to suppose that an eddy conserves the momentum of the layer in which
it originated so that IL+ H' = U,0 and V;+ v' = V,,,, where z0 is the height of the layer
in question, we could obtain the values of the integrals in the same way that we did
in the case of heat transference. In the case of heat transference, owing to the small
value of the ordinary coefficient of " molecular " conductivity, the only way in which
an eddy can lose its temperature is by mixture ; but in the case of transference of
momentum the eddy can lose or gain velocity owing to 'the existence of local variations
in pressure over a horizontal plane. Such variations are known to exist ; they are in
fact a necessary factor in the production of disturbed motion, and they enter into all
calculations respecting wave motion. We cannot, therefore, leave them out of our
calculations without further consideration, though it will be seen that they probably
do not affect the value of the integral (3) when it is taken over a large area.

Consider a particular case of disturbed motion. Suppose that the fluid is incom-
pressible and that the motion takes place in two dimensions x and z. Suppose that
originally the fluid is flowing parallel to the axis of x with velocity U, and that the

c 2

12 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.

disturbance has arisen from dynamical instability, or from disturbances transmitted
from the surface of the earth. The rate at which x-momentum leaves a layer of
thickness Sz is

But U, is constant over the plane xy and since there is no resultant flow of fluid
across a horizontal plane pUzw'dxdy = Uz\\ pw'dx'dy = 0.

Hence, if we write I for the value of the expression in square brackets
I == j| jj^lL+O w'dxdy = p^u'w'dx dy

The equation of continuity is

<-\ / i~~ f

fill fltii

||+|^=0 (5)

Since the motion is confined to two dimensions

^ / T T /\ ^ /

rt I I J 1 o/ I f' J/i

-- — = twice the vorticity of the fluid at the point (x, y, z).

(J% (J JC

And since every portion of the fluid retains its vorticity throughout the motion, this
must be equal to twice the vorticity which the fluid at the point (x, y, z) had before
the disturbance set in. This is equal to the value of (dUJdz) at the height, zu,* of the
layer from which the fluid at the point (x, y, z) originated. If this value be expressed
by the symbol [dU,/dz~].a we see that the dynamical equations of fluid motion lead to
the equation

d\J, Su' {)?</ _

~ "I" ~^ ^ =

dz

^ f ^N /

Substituting in (4) the values of ^- and ~ given by (5) and (6), we find

The first term integrates and vanishes when a large area is considered ; but the
second term does not vanish.

* «0 is evidently a function of x and z when the motion is confined to two dimensions.

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 13

To find the value of the second term expand [d\Jjdz],a in powers of z0— z.

z (v-z)3 #u,

Hence (7) becomes

So far nothing has been said about the magnitude of the disturbance ; (8) is true
even if the disturbance be large. Let us now suppose that the height, z— z0, through
which any portion of the fluid has moved from its undisturbed position is of such a
magnitude that the change in dUjdz in that height is small compared with dUjdz
itself. In that case (8) becomes

z , / \jj
I = p -7-7^ w' (ZQ-Z) dx dy.

tlz\ J J

The rate at which ie-momentum leaves a layer of thickness Sz is therefore

The effect of the disturbance is therefore to reduce the .(--momentum of a horizontal

asTT
layer of thickness Sz at rate p .," Sz x [average value of ic' (z0 — z)~\ per unit area.

cz"

The same effect would be produced on a layer of thickness Sz by a viscosity equal
to p x average value of w' (z— z0) if the motion had not been disturbed. If, some time
after the disturbance has set in, all the air at any level mixes, no change will take
place in the average momentum of the layer. Deviations from the mean velocity of
the layer will disappear, and the velocity will be horizontal once more and uniform
over any horizontal layer. When, therefore, we wish to consider the disturbed motion
of layers of air, we can take account of the eddies by introducing a coefficient of eddy
viscosity equal to />x average value of IP' (z— za), and supposing that the motion is
steady, z— z0 is the height through which air has moved since the last mixture took
place.

As before in the case of the eddy conduction of heat, we can express the average
value of w' (z— za) in the form jt(wd), where d is the average height through which an
eddy moves before mixing with its surroundings, and w roughly represents the average
vertical velocity in places where w' is positive. It will be noticed that the value we
have obtained for eddy-viscosity is the same as that which we would have obtained
if we had neglected variations in pressure over a horizontal plane, and had assumed
that air in disturbed motion conserves the momentum of the layer from which it
originated till it mixes with its new surroundings, just as it conserves its potential

14 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.

temperature. Whether this result is true when the disturbance takes place in three
dimensions, I have" been unable to discover.

If it is true, there is a relation K/pv = n/p = \ (wd) between K the eddy conductivity
and /at the eddy viscosity ; if any method of deducing /a. from meteorological observa-
tions could be found, it would be possible to verify the relation numerically.

Relation of Observed Velocity to Gradient Velocity.

We may expect to discern the effect of eddy viscosity in cases where the wind
velocity changes with altitude, and where the force due to eddy viscosity prevents the
wind from attaining the velocity which we should expect on account of the pressure
distribution. These conditions arise near the surface of the earth. The velocity and
direction which we should expect on account of the pressure distribution, are called the
gradient velocity and the gradient direction. In general, the wind near the ground falls
short of the gradient velocity by about 40 per cent., and the direction near the ground
is about 20 degrees from the gradient direction. At a height which varies on land
from 200 to 1000 metres the wind becomes equal both in velocity and in direction to
the gradient wind.

Let us consider the motion of air over the earth's surface under the action of a
constant pressure gradient G acting in the direction of the axis of y. The equations
of motions of an imcompressible* viscous fluid aret

Dt p Sx p

Dt p cy p

T)W _ y 1 3»

TA . — £* ,-.

Dt p Sz ,

where u, v, w are components of velocity parallel to the co-ordinates x, y, z ; p is the
pressure, and X, Y, Z, are the components of the external forces on unit mass of the
fluid.

The forces acting are the force due to the earth's rotation and gravity.

Hence

X = — 2wv sin \ ~]

v . I where w is the angular velocity of the

i — Zu>u sin \ > ;

earth s rotation and \ is the latitude.

Z= -g J

The pressure is given by p = constant —gpz + Gy.

* The atmosphere is not incompressible, but compressibility makes no difference in the present
itistance.

t See Lamb's ' Hydrodynamics,' p. 338.

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 15

If we assume the motion to be horizontal equations (9) become

0= -2<«usinX + ^^, (10)

p dz2

G u. d2V / 1 , \

0 = — 2a>u sm X --- h- -r-5 ........ (11)

p ^ rtzj

Eliminating u the equation for v becomes

,--04 i '> wo sin X

^ -- h4B w = 0 where

^
az*

Taking into consideration the fact that v does not become infinite for infinite values
of z, the solution of this is

v = A2e-B/ sin Bz + A4e-B* cos Bz ........ (12)

Differentiating this twice with respect to z we find

f| =2B2(-A,e-B~- cos Bz + A4e-B-- sin Bz).

Cv%

Substituting this value in (ll) we find

G
u = A2e~B" cos Bz— A4e~Bz sin Bz +• — ....... (13)

f '

The quantity or -- : -- is the gradient velocity, so that at great heights,

" sin X

v = 0 and u is equal to the gradient velocity.

The values of A3 and A4 will be found by imposing suitable boundary conditions. If
there is slipping at the earth's surface it seems natural to assume that it is in the
direction of the stress in the fluid. In this case one boundary condition will be

L u Jz=o L v J-=o.

Where the square brackets are intended to show that the values of the quantities
contained in them are to be taken at the surface of the ground, z = 0.

Substituting for u, r, dujdz and dv/dz, and putting 2 = 0, equation (14) becomes .

A G

A2+

A

-^1-4

. .
\ '

where QG represents the gradient velocity.

In order to determine the motion completely one more relation between A2and A4 is
necessary. Let the wind at the surface be deviated through an angle a from the

16 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.

gradient wind in such a way that if one stands facing the surface wind the gradient
wind will be coining from the right if a be positive.
Then

Solving (15) and (16) for A3 and A,

A — tan a (l+tan a) ^

A-? = — — 7 — - ^!G.

1 + tan' a

. —tan a (l— tan a) ,^
l+tan* a

The surface wind, which we may denote by Qs is equal to

QG

— \Xtan5 « ( 1 - tan a)2 + ( 1 - tan a)2,
1 + tair a

or

Qs = Q(i (cos a— sin a) (17)

It is interesting to compare the value given by (17) for the ratio of Qs to Qu with
the value, cos a, given by GULDBEHG and MOHN* for the same ratio, and with the
most recent observations of wind velocity at different altitudes above the surface of
the earth.

Mr. G. M. B. DOBSON of the Central Flying School at Upavon has recently
published! the results of a number of observations made by means of pilot balloons
over Salisbury Plain, which is an excellent place for such observations on account of
its open situation. He finds that a is smaller for light winds than for strong winds,
and he accordingly divides up his ascents into three classes, those which took place in
light winds, when the velocity of the wind at a height of 650 metres is below 4' 5
metres per second, those in moderate winds between 4' 5 and 13 metres per second,
and those in strong winds above 13 metres per second.

The comparison is shown in Table II. It will be seen that the observed deviation
of the surface wind from the gradient direction agrees well with the theory we have
been considering, but not with the theory of GULDBERG and MOHN.

The agreement between theory and observation is, however, more striking in
another respect. The deviation of the direction of the wind at any height from the

* ' Studies of the Movements of the Atmosphere,' 1883-85. An English translation appears in
" The Mechanics of the Earth's Atmosphere," by CLEVELAND ABBE ; ' Smithsonian Miscellaneous Collec-
tions,' 1910.

t 'Quarterly Journal of the Royal Meteorological Society,' April, 1914.

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.

17

gradient direction is due to the retarding of the wind velocity below the gradient
velocity by friction or by viscosity. One might expect, therefore, that the wind
would attain the gradient direction at the same height as the gradient velocity. This
would, in fact, follow from the theory of GULDBERG and MOHN. Most observations
have failed to give reliable information on this point, partly because irregularities on
the surface of the earth have introduced complicated conditions, which cannot be
taken account of, and partly because the observations have not been grouped according
to the wind velocity.* Neither of these objections applies to Mr. DOBSON'S observa-
tions. Salisbury Plain, though inferior to the sea, is as good a place for wind

TABLE II.

Observed value of

Qs

Qo'

Observed angle a.

Light winds.

•72

Moderate winds. Strong winds.

•61

13 degrees

a, calculated from (17) so
as to correspond with
the observed value of
Qs

QG'

a, calculated from GULD-
BERG and MOHN'S
theory so as to corre-
spond with the observed

value of -? .

14 degrees

2 1|. degrees

18 degrees

20 degrees

20 degrees

44 decrees

49 degrees

52 degrees

observations near the surface as one could find on land ; and as has been explained
already, his results are grouped according to wind velocity. Mr. DOBSON finds that
the gradient direction is not attained till a height is reached which is more than twice
the height at which the gradient velocity is first attained. He remarks, in fact, that
the gradient velocity is usually attained at a height of 300 metres, though the gradient
direction is not found till a height of 800 metres has been attained. This is a most
remarkable result, but it might have been expected from the equations (12) and (13).
The height at which the gradient direction is attained is given by putting v = 0 in
(12). If H! be the height in question

A2 sin BHi + A4 cos BE^ = 0

* Owing to the fact that p,/p depends on the wind force we should evidently expect more consistent
results when the observations are grouped according to wind velocity.
VOL. CCXV. — A. D

18

so that

G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHERE.

tan BH, = - -4

A2

Substituting for A2 and A4 their values in terms of a

1— tan a /

tan BH! = - - - = tan a- -

1 + tan a

(18)

Since a is positive and less than ^ the smallest positive value of Ht is given by

(19)

The height H2 at which the wind velocity first becomes equal to the gradient
velocity is given by u2+v2 = QG2. This reduces to

9_BH., _ ( 1 + tan a) cos BH2 — ( 1 — tan a) sin BH2 /2 v

tan a

Equation (20) can be solved so as to give tan a in terms of BH2, and when several
corresponding values of a and BH2 have been obtained BHa can be obtained by
interpolation in terms of a. In Table III. are shown the values of BHi and BH2 and
corresponding to values of a from 0 to 45 degrees.

TABLE III.

a.

BHa.

BHi.

H!
H2'

0 degrees

•78

2-35

3-0

10 degrees

•91

2-53

2-8

20 degrees

1-04

2-70

2-6

30 degrees

1-20

2-88

2-4

45 degrees

1-44

3-15

2-2

It appears, therefore, that Hj/H., varies from 3 to 2'2.

Mr. DOBSON gives 80° metres = 2.g6 ag the observed value of H,/H2, and his values
300 metres

of a all were about 20 degrees. It is probably a coincidence that the observed ratio, 2'66,
should be so very close to the calculated ratio 2'6, but the coincidence is at least
significant.

G. I. TAYLOIi ON EDDY MOTION IN THE ATMOSPHERE.

19

In order more easily to compare the theoretical results with the observations the
curves shown in fig 4. have been prepared. Fig. 4 shows the way in which wind

Fig. 4.
Calculated Curves

•2 -* < -a

WIND VELOCITY IN FRACTIONS OF GRADIENT

BZ

2O* 10°

WIND DIRECTION

Fig. 5.

Observed Curves

METRES

2400

2000

O 2 4 6 8 10 12 14

WIND VELOCITY IN METRES PER SECOND

800

20* 10*

WIND DIRECTION

velocity and direction vary with height in the theoretical case we have been con-
sidering when a. = 20 degrees. Fig. 5 is reproduced by permission of Mr. DOBSON. It

D 2

20 G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHERE.

represents the observed velocity and direction of strong winds at different heights.
In each of the figures the curve on the right represents deviations from the gradient
direction, which is shown as a vertical line. The curve on the left represents wind
velocity at different heights.

It will be seen that there is good agreement between the two sets of curves.
Strong winds have been chosen for the comparison in preference to light winds,
because it is less likely that heat-convection currents will persist through such a
distance before mixing takes place, as to prevent the resistance, due to eddy motion,
from obeying the ordinary laws of viscosity. The observed curves for light winds,
however, agree as well with the theoretical curves as those for strong winds.

Besides the various points of resemblance already noticed between theory and
observation, an inspection of the curves in figs. 4 and 5 reveals yet another.
Above the height at which the gradient direction is attained the wind goes on veering
slightly up to a certain height, when it begins to return again to the gradient
direction. The wind is again blowing along the gradient direction at a height
slightly less than twice the height at which it first attained it. Nearly all the curves
in Mr. DOI?SON'S paper have this characteristic sinuosity, but they are not the only
ones which show it. Mr. J. S. DINES, in his Third Report to the Advisory Committee
for Aeronautics (1912), has published a number of curves which exhibit the same
sort of sinuosity. The theoretical curve, fig 4, has the same characteristic. The
successive heights Hu H'j, H'^, ... at which the wind blows exactly along the gradient
direction are given by the solutions of equation (18).

o

We have already obtained the first solution, namely BB^ = -- + a.

The others are ETL\ = — +a + ,r, BH", = — +« + 27r and .

4 4

The ratio of the first two is—1 = 'fj7?r) + a

When a = 20 degrees this is equal to 2' 16.

In the case of strong winds it will be seen from fig. 5 that the observed values of
H, and R\ are 900 metres and 1750 metres. Hence the observed value of H^/H! is
1'95. The good agreement between the observed and calculated values of H'/H is
possibly a coincidence, but it is interesting to notice that, on theoretical grounds,
we should expect a sinuosity in the curve representing the direction of the wind
at various heights when it blows under the action of a constant pressure gradient,
and that such a sinuosity is actually observed.

The close agreement between theory and observation is evidence that the
assumptions made in the theory are correct. In particular the eddy motion does not
diminish much in the first 900 metres.

G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 21

We have seen that

BH^^+a.

4

If, therefore, we can measure a and Hj we can calculate B. The commonest value
for a on land is 20 degrees, in fact, for all except light winds, it is near to 20 degrees.
In the kite ascents on the " Scotia " the wind usually veered two points (22^ degrees)
in the first 100 or 200 metres and after that remained constant in direction at greater
heights. It appears, therefore, that on the sea also a is about 20 degrees. Assuming
then that a = 20 degrees, we see from Table III. that BE^ = 27.
Substituting for B its value

/
^

sin X

we find the following relation between H^ and the eddy viscosity

fi _ H^w sin X

;= (27)*

But a), the angular velocity of the earth, is 0 '00007 3 ; and in latitude 50 degrees N.,
which is the latitude of the South of England and also of the northern portions of the
Bank of Newfoundland, sin X = 077.

Hence for those regions - = H]2x 077 x 10~5.

P

On land, in the case of the strong winds,* Hj = 900 metres, hence

^ = 62xl03inC.G.S. units;
P

for moderate winds,*

H, = 800 metres and nfp = 50 x 103 ;
and for light winds,*

H! = 600 metres and p/p = 28 x 103.

At sea,t in the regions to which the "Scotia's" cruises were confined, H! commonly
lay between 100 metres and 300 metres so that /m/p lay between 077 x 103 and
6'9xl03.

* See Mr. DOBSON'S paper, loc. at.

t Assuming that the wind had reached the gradient velocity when it had practically stopped veering
with increasing height.

22 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE.

Except for the kite ascent of July 17th, 1913, the values of K/pcr which, as was shown
on p. 14, should be equal to fj.jp, lie between these values.* It is unfortunate that the
lack of skilled assistance in flying the kites from the " Scotia " prevented me in most
cases from being able to get simultaneous values of K/pa- and /j./p. For the kite ascent
of August 2nd, however, I have the following observations : — At 350 feet the wind had
veered one point from the surface wind. At 770 feet the wind had veered two points
from the surface wind, and at all greater heights the veer was two points. It seems,
therefore, that at 770 feet, i.e., 230 metres, or at some less height, the wind had
attained the gradient direction, so the /u//o lay between 0'77 x (23,000)2x 10~5 or
4'0x 10:i and 0'77 x 103. On referring to Table I. it will be seen that the value of
K/pa- on that occasion was 2'5xl03. These results certainly tend to confirm the
theoretical deduction that KJpa- = ftfp, but more evidence is wanted before the point
can be regarded as finally settled.

On p. 13 it was shown that /n/p — ^(wd). The size of the eddies, which produce the
effects we have been considering, are evidently governed by d. We may say roughly
that d is less than the average diameter of an eddy ; if therefore we could measure w,
we should be able to determine the size of the eddies. Now Mr. J. S. DINES has made
a large number of observations of small vertical gusts with tethered balloons. On
p. 216 of the Technical Report of the Advisory Committee for Aeronautics is shown a
trace which represents the vertical component of the wind velocity at any time during
a certain interval of five minutes, on January 19th, 1912. The average wind velocity
during the interval was 7 metres per second ; and I find from the trace, which Mr. DINES
says is typical, that the average deviation from the mean vertical velocity (the mean
wind was not quite horizontal) was 25 cm. per second. We may take this as w.
Assuming that the gradient direction was attained at a height of 800 metres the
value of -%(wd) would be 50 x 103 or wd = 105 approximately.

Hence

105
d = — = 4 x 10s cm. = 40 metres.

aO

The wind was blowing with velocity 7 metres per second so that it would cover
7x60 = 420 metres, or about 10 times d, in a minute. If the vertical and horizontal
dimensions of an eddy are about the same, this would mean (since d is less than the
diameter of an eddy) that rather less than 10 eddies would pass a given spot in a
minute. On examining Mr. DINES' trace it will be found that there are roughly
about 6 peaks per minute on the curve representing vertical velocity.

These calculations are very rough, but they do show at any rate, that actual
observations of eddy motion do not involve anything that is contrary to the
assumptions on which the theory contained in this paper is based.

* See Table I.

G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHERE. 23

NOTE ON THE STABILITY OF LAMINAR MOTION OF AN INVISCID FLUID,

MAY 26TH.

The equation (8) throws a new light on the much discussed question of the stability
of the laminar motion of an inviscid fluid.

Lord RAYLEIGH has considered the stability of a fluid moving in such a way that U,
the undisturbed velocity, is parallel to the axis of x and is a function of z. His method
is to impose a small disturbing velocity of a type which is simple harmonic with respect
to x, satisfies the equations of motion, and contains a factor ewt. He then discusses
the conditions under which n may be complex. If n is not complex the motion is
stable ; if n is complex the motion is exponentially unstable.

Perhaps the most important result of Lord RAYLEIGH'S investigation is the conclu-
sion he arrives at that if d~~U/dz2 does not change sign in the space between any two
bounding planes, unstable motion is impossible. A particular case of laminar motion
in which ePU/cfe8 has the same sign throughout the fluid is that of an inviscid fluid
flowing with the same velocity as a viscous liquid moving under pressure between two
parallel planes. In this case, therefore, unstable motion should be impossible.
OSBORNE REYNOLDS, however, working in an entirely different way, has come to the
conclusion that a viscous fluid moving between parallel planes is unstable if the
coefficient of viscosity is less than a certain value which depends on the distance between
the planes and on the velocity of the fluid. REYNOLD'S result is in accordance with
our experimental knowledge of the behaviour of actual fluids.

It is evident that there is a fundamental disagreement between the two results for,
according to REYNOLDS, the more nearly inviscid the fluid, the more unstable it is
likely to be ; while according to RAYLEIGH instability is impossible when the fluid is
quite inviscid.

Various attempts have been made to find the cause of the disagreement, but none
of them appear to have been very successful.

The object of this note is to show that equation (8) may be used to prove the truth
of Lord RAYLEIGH'S result for the case of a general disturbance, not necessarily harmonic
with respect to x ; and to show also that it may be used to assign a reason for the
difference between RAYLEIGH'S and REYNOLDS' results.

Starting from the principle that when an inviscid fluid in laminar motion is disturbed
by dynamical instability, each portion of it retains the vorticity of the layer from which
it started, it was shown* that the rate at which momentum parallel to the axis of x
flows into a slab of area A and thickness Sz is : —

h^§ JJ

A

the integrals being taken over the area A.

* See p. 13.

24 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHEEE.

This expression is true for all disturbances, however large, but when the distance
z0-z is small the first term only is of importance. Now it is evident that w is related
to z0— z by the relations

""

Hence

D

or

If w, w, and ZL,— z are small, this is equivalent to

Hence

ff w(za-z)dxdy = - U ff £- (s«-z) rZujffy- if (z<1-z)^-(zl)-z)dxdy.

J.JA JJA OX ^ J A C/C

/• p ^

Now when a large area is considered I —(z0—z) dx dy integrates out and vanishes.

Hence

w(z0-z)dxdy= - jjj ^(zt)-z)2 dxdy = - ¥jJ|A (z.-z)2 dxdy.

It appears therefore that the rate at which the x-momentum in the slab A
increases is

Integrating with respect to t we find that the difference between the momentum
in the slab A after and before the disturbance set in is

Lord EAYLEIGH has pointed out that it is difficult to define instability. In the
present case the motion will be held to be unstable if the average value of the square
of the distance of any portion of the fluid from the layer out of which the disturbance

G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHERE. 25

has removed it, increases with time. This evidently includes the case of exponentially
unstable simple harmonic waves.

In unstable motion therefore •=- 1 1 (za-zf dx dy must be positive.

Hence the rate at which z-momentum enters the slab A is positive or negative
according as d2\J/dz2 is positive or negative. In an unstable disturbance of a fluid for
which d21U/dz2 is everywhere positive the momentum of every layer must increase. But
if there is perfect slipping at the boundaries no momentum can be communicated by
them. Hence, as there is no other possible source from which the momentum can be
derived, instability cannot possibly occur. The argument applies equally well if
d2U/dz2 is everywhere negative. Lord RAYLEIGH'S result is therefore proved for a
generalised disturbance. In a case where d2TJ/dz2 changes sign at some point in the
fluid any disturbance reduces the x-momentum in a layer where d2U/dz2 is negative,
while it increases the .r-momentxim in layers in which d2U/dz2 is positive. A type of
disturbance which removes ^-momentum from places where d2\J/dz2 is negative and
replaces it in regions where d-U/dz2 is positive, so that there is no necessity for the
boundaries to contribute, may be unstable.

Now consider what modifications must be made in the conditions in order that
instability may be possible in the case where d2\J/dz~ is of the same sign throughout
(say negative). Suppose that instability is set up so that ./--momentum flows outwards
from the central regions as the disturbance increases. The amount of ^-momentum
crossing outwards towards the walls through a plane perpendicular to the axis of z,
increases as the walls are approached. In order that instability may be set up this
momentum must be absorbed by the walls. There seems to be no particular reason
why an infinitesimal amount of viscosity should not cause a finite amount of momentum
to be absorbed by the walls.

In connection with this two points should be noticed. Firstly, the momentum is
only communicated to the walls while the disturbance is being produced. The time
necessary to produce" a given .disturbance may increase as the viscosity diminishes.
Experimental evidence, however, does not suggest that this is the case.

The second point is suggested by the conclusion arrived at on pp. 11-22, that a
very large amount of momentum is communicated by means of eddies from the
atmosphere to the ground. This momentum must ultimately pass from the eddies
to the ground by means of the almost infinitesimal viscosity of the air. The actual
value of the viscosity of the air does not affect the rate at which momentum is
communicated to the ground, although it is the agent by means of which the
transference is effected.

In any case it is obvious that there is a finite difference, in regard to slipping at
the walls, between a perfectly inviscid fluid and one which has an infinitestimal
viscosity. The distribution of velocity acquired by a viscous fluid flowing between

VOL. CCXV. — A. E

26 G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHEEE.

parallel planes at which there is no slipping is possible for an inviscid fluid when
there is perfect slipping, but is impossible as a steady state for an infinitesimally
viscous fluid which slips at the boundaries.

The finite loss of momentum at the walls due to an infinitesimal viscosity may be
compared with the finite loss of energy due to an infinitesimal viscosity at a surface
of discontinuity in a gas.*

If these views are correct we should expect that Lord RAYLEIGH'S result would
not apply when there are no bounding planes and space is filled with a fluid in which
d'TJ/dz2 is everywhere positive ; for, in that case, there would be nothing to prevent
a positive amount of x-momentum from being communicated to every portion of the
fluid, provided the disturbance increases indefinitely for infinitely great values of z.
In obtaining his result Lord RAYLEIGH assumes that, if there are no bounding planes,
11! — 0 at infinity ;t it does not apply therefore to the case just considered.

The conclusion arrived at is that the discrepancy between RAYLEIGH'S and
REYNOLDS' results is due to the fact that the perfect slipping at the boundaries
assumed in RAYLEIGH'S work prevents the escape of the momentum which is a
necessary accompaniment of a disturbance of a fluid for which d~U/dz2 is everywhere
negative. The complete absence of slipping assumed in REYNOLDS' work enables
the necessary amount of momentum to escape, and so a type of disturbance may be
produced which is dynamically impossible under the condition of perfect slipping at
the boundaries.

* See "Conditions Necessary for Discontinuous Motion in Gases," TAYLOR, 'Boy. Soc. Proc.,' 1910,
A, vol. 84, p. 371.

t 'Phil. Mag.,' vol. 26, 191:?, p. 1002.

II. On the Potential of Ellipsoidal Bodies, and the Figures of Equilibrium of

Rotating Liquid Masses.

By J. H. JEANS, M.A., F.R.S.

Received May 29,— Kead June 25, 1914.

BY an ellipsoidal body is meant, in the present paper, any homogeneous body which
can be arrived at by continuous distortion of an ellipsoid. If/, = 0 is the equation
of the ellipsoid from which we start, and e is a parameter, the distortion of the
ellipsoid may be supposed to proceed by e increasing from the value e = 0 upwards,
and the final figure may be taken to be

For very small distortions the first two terms will adequately represent the
distorted figure, and as we pass to higher orders the remaining terms will enter
successively.

The potential problem, to some extent interesting in itself, derives its chief
importance from its application to the determination of the possible figures of
equilibrium of a rotating mass of liquid. POINCARE,* using his ingenious method
of double layers, has shown how the potential of an ellipsoidal body can be carried
as far as the second-order terms when the distortion is small, but gives no indication
of how it is possible to carry it further, and indeed his method is one which hardly
seems susceptible of being developed further than he himself has developed it.
It is clear, however, that progress with the problem of rotating liquids can only
be made when a method is available for writing down the potential of an ellipsoidal
body distorted as far as we please. I believe the method explained in the present
paper will be found capable of giving the potential of a body distorted to any
extent, although (for reasons which will be explained later) I have not in the
present paper carried the calculations further than second-order terms.

The theory of figures of equilibrium of rotating masses of liquid is at present
in an unsatisfactory state. It has been shown by Lord KELVIN that the Jacobian
ellipsoid is stable at the point at which it coalesces with the Maclaurin spheroid,
and it has been shown by POINCARE to remain stable up to the point at which

* " Sur la StabiliW de FEquilibre des Figures Pyriformes affectees par une Masse Fluide en Rotation,"
' Phil. Trans.,' A, vol. 198, p. 333.

(524.) E 2 [Published February 2, 1915.

28 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

the series of Jacobian ellipsoids coalesces with the Poincard series of
figures. After this point the series of Jacobian ellipsoids must, in accordance \v-itl
POINCARE'S doctrine of exchange of stabilities at a point of bifurcation, lose
stability, but the question of how it loses its stability is in a state of doubt.
DARWIN believed he had proved the Poincare series to be initially stable,* whereas
Li APOUNOFF f has maintained that this series is initially unstable. The importance
of this question to theories of cosmogony is, of course, great, although perhaps
liable to be overrated. A caution of POINCARE'S } may be borne in mind :
" Quelle que soit 1'hypothese [stability or instability] que doive triompher un jour,
je tiens a mettre toute de suite en garde contre les consequences cosmogoniques
qu'on pourrait en tirer. Les masses de la nature ne sont pas homogenes, et si on
reconnaissait que les figures pyriformes sont instables, il pourrait ne"anmoins arriver
qu'une masse hetdrogene flit susceptible de prendre une forme d'equilibre analogue
aux figures pyriformes, et qui serait stable. Le contraire pourrait d'ailleurs arriver
egalement."

The present investigation was started primarily in the hope of setting this
question of stability at rest. I realised that to make a new series of computations
on the subject could be of little value, for whatever the result, there would have
been two- opinions on the one side to one on the other. Moreover, DARWIN has
stated clearly that he does not think the divergence of opinion between
M. LiArouNOFF and himself is one to be settled by new computations §: "I feel
a conviction that the source of our disagreement will be found in some matter of
principle." I had hoped that it might be found possible to discuss the problem
by a purely algebraical method, involving neither laborious computations nor
intricate physical arguments, and that if such a discussion did not give a con-
vincing and satisfying answer to the question in hand, at least it might reveal the
source of disagreement between DARWIN and LIAPOUNOFF. The result arrived at
is one which, as will readily be understood from its nature, is only put forward
with the utmost diffidence, but it is one from which I can find absolutely no escape.
It is that underlying the whole question there is a complication, unsuspected equally
by POINCARE, DARWIN, and (in so far as I can read his writings) LIAPOUNOFF,
which renders nugatory the work of all these investigators on the stability of the
pear-shaped figure. If my method is sound, it appears, as will be explained later,
that it is impossible to draw any inference as to the stability of the pear from
computations carried only as far as the second order of small quantities. The

* " The Stability of the Pear-shaped Figure of Equilibrium of a Rotating Mass of Liquid," ' Phil.
Trans.,' 200 A (1902), p. 251; also papers in 'Phil. Trans.,' 208 A (1908), p. 1, and ' Proc. Roy. Soc.,
82 A (1909), p. 188, all combined in one paper in 'Coll. Scientific Papers,' vol. 3, p. 317.

t "Sur mi Probleme de Tchebychef," ' Memoires de 1'Academie de St. PcStersbourg,' xvii., 3 (1905).

I Loc. eit., p. 335.

§ ' Coll. Scientific Papers,' 3, p. 392.

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 29

materials for an answer to the question are to be found only through the third -
order terms.* Fortunately the method of the present paper admits of extension
to the computation of third-order terms, and so it ought to, and I hope will he quite
feasible to decide as to the stability or instability of the pear, a question reserved
for a subsequent paper.

The reader who is interested in the main conclusions of the paper rather than
in details of theory, method, or calculations, may care to pass directly to § 35.

GENERAL THEORY OF POTENTIAL OF ELLIPSOIDAL BODIES.

2. We proceed to develop a method for writing down the potentials of certain
homogeneous solids ; in particular of ellipsoids and distorted ellipsoids. We are for
the present concerned solely with potential-theory — the discussion of rotating liquids
does not enter before § 19.

As will soon be evident, the problem in potential theory amounts to the following :
to write the equation of the boundary of a homogeneous solid in such a form
F (x, y, z) = 0, that the potential at the boundary is of the form F' (x, y, z) = 0, where
F' (x, y, z) is a function containing the same algebraic terms as Y (x, y, z), but having
in general different coefficients. If this can be done, it only remains to equate
F' (x, y, z) + |-to3 (x2+y2) to F (x, y, z}, and we have at once, on equating coefficients,
a series of equations which will determine the possible figures of equilibrium for a liquid
mass rotating with angular velocity &>.

3. Let F (x, y, z) = 0 be the boundary of any homogeneous solid of density p.
Assuming it to be possible,! let V, be a function of position satisfying V-V; = — 4wp
at all points of space and coinciding with the potential of the solid at all points inside
the solid, and let V0 similarly be a function of position satisfying VaV0 = 0 at all points
of space, except possibly the origin or other infinitesimal region inside the solid, and
coinciding with the potential of the solid at all points outside the solid.

Then Vf must be equal to V0 at the boundary of the solid, and we must also have

c£V d

' = • , ° at the boundary, where -y- denotes differentiation along the normal to the

surface.

* Since writing this paper, I have been surprised to find that this conclusion is quite clearly implied
in a paper which I published in 1902, "On the Equilibrium of Rotating Liquid Cylinders," 'Phil.
Trans.,' A, 200, p. 67. See below, § 36.

t I have not examined in any detail the conditions that this may be possible, because the result of
the paper proves that it is possible in the cases which are of importance. Similarly I have not examined
in detail the difficulties which might arise at the origin or at infinity, because in the final result they
do not arise. We are searching for, and ultimately find, a certain solution of the potential equations,
and after the solution has been obtained it is easy to verify directly that it really is a solution, and
that it involves no complications either at infinity or at the centre of the solid.

30 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Introduce a new function W, defined by

w = v,.-v0

at all points of space, then we must have

V2W = -

at all points of space, and, at the boundary, W = 0 and —: — = 0
These last two conditions are equivalent to

dW dW

dx dy dz

= 0 (3)

at the boundary, together with one further condition. Equations (3) require that W
shall have a constant value all over the boundary ; the further condition is that this
constant value shall be zero.

4. Let F (x, y, z, X) = 0 be the equation of a family of surfaces obtained by varying
the parameter X, and such that the boundary of the solid is the surface X = 0. The
surfaces of this family will divide up the solid into a series of thin shells. There will
be a contribution from each shell to V, and also to V,,. Thus W may be regarded as
the sum of a number of contributions, one from each shell.

Let the thicknesses of the separate thin shells be determined by small increments
in X, say d\lt d\.,, ... , starting from the boundary X = 0. Then we may write

:<&,)+ - (4)

where Y,-(f7Xj) represents the contribution to V; from the shell d\t, and so on.
Similarly

V. = V,(dX1)+V0(dXi)+ : (5)

Suppose that V,-, V0, and W are being evaluated at a point x', y', z' on the shell d!X,
at which the value of X is X'. Then if d\t is any shell inside the shell d\s, the contribu-
tions to V< and V0 from the shell d\t will be the same ; we have V,- (d\t) =- V0 (d\t).
Hence from equations (l), (4), and (5),

. W- V,- V0 = (V.^xO-V^x,)} + {V,.(dx3)- V0(d\a)} + ... + (V^x.)- V0(dx.)}, (6)
or expressed as an integral,

W = ?'<S>(x',y',z',\)d\ .......... (7)

Jo

5. This form for W satisfies automatically the last of the conditions of §3, namely,
that W shall vanish at the boundary. We proceed to determine 4> so as to satisfy the
remaining conditions which are expressed by equations (2) and (3).

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 31

Let -r-be used to denote geometrical differentiation with respect tox — i.e., differen-
dx

o

tiation in space keeping y and z constant — and let ^- be used to denote algebraic

differentiation, i.e., partial differentiation with respect to* keeping y, z, and X constant.
Then

d_ = _3_ + ^X JL &c (8)

dx Sx dx 3x'
and we have

dw aw , d\' aw

dx' 9z' dx' I3X'|

If the point x'. y', z' is on the boundary we have X' = 0, so that from equation (7),
-5—7 = 0, and it appears that equations (3) will all be satisfied if

at all points on the boundary X' = 0.

We notice from a comparison of equations (6) and (7) that

$ (X', y', 2', X') d\. = Vt (d\.)-V0 (dX.),

and the right hand of course vanishes when x', y' , z' is on the shell d\s. Thus we
must have, at all points,

*(x',?/,z',\') = 0 ......... (11)

identically, provided that X' has the value appropriate to the point x', //, z'. This
condition of course imposes more restriction on the value of $ than does equation (10).
Equation (10) was adequate to ensure that the boundary condition (3) should be
satisfied, but the remark just made shows that for (3) and (2) both to be satisfied —
i.e., for W to give the true value of V;— V0, equation (ll) must necessarily be
satisfied. We shall now assume that equation (ll) is satisfied, and proceed to satisfy
the remaining condition expressed by equation (2), namely,

(12)
In virtue of equation (ll), equation (9) reduces to

dx'

32 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

whence on further differentiation (cf. equation (8)),

_

dx'2 " dx'2 dx'

so that equation (12) becomes

From equations (8) and (ll),

dx

__ , , }

' ' M ' y '

so that equation (13) may be written in either of the equivalent forms :

'V , fd\'\a\ ?'$' /1/A

+_— , .... (14)

ax7"

(15)

in which V2 stands for ^-^ + ^ + ^ , * for <f> (,r', ?/', z', X) and $' for 0> (a;', ?/', 2', X').

Thus if 3> satisfies either equation (14) or (15), and also equation (ll), then W, as
given by equation (7), satisfies all the conditions which have been seen to be necessary,
and will therefore give the true value of Vj— V0.*

* Suppose there are, if possible, two solutions to the same problem, say "J? = $\ and 4> = "fv Since
W is determined when the problem is fixed, we must have

so that

where

f\' rV

SvZA. = <f>,f?A,
o Jo

0, or

x (x', y', z, A') - x («f, V, s', 0) = 0 (i)

2

for all values of x', y't z. Thus, if x is such as to satisfy (i) we may add a term ^ to $ and still
obtain a solution of the same problem. A special case in which (i) is satisfied is when

x (x', y', *, A') = /(*', y', z', A') {^ (A') - f (0)},
where/ is any function which vanishes identically for the value of A' appropriate to the point x', y', zf,

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 33

G. As a matter of convenience, involving neither loss nor gain of generality, we
shall write

(16)

in which \[s(\) is any function of X. and f(x, y, 2, X) a quite general function of x,y, z,
and X. Then, from equation (ll), we must have

f(x, V,z,\) = 0 .......... (17)

identically at all points, when X has the value appropriate to the point x, ?/, 2. We
accordingly have

so that equations (14) and (15) reduce to

•A'
— 47TH =

'. (i.)

Moreover the family of surfaces (X = cons.) may now he supposed to he determined
hy equation (17), and the boundary will he given by

f(x,y,z, 0) = 0 (20)

Thus, to sum up, if y and i//- are such that either equation (18) or (1'j) is satisfied,
then the potential of the homogeneous solid of density /> whose boundary is
determined hy equation (20), will be given by

Vf-Vu = W= V(x)/(,;', ,j, z', \)d\, (21)

the value of W being evaluated at the point x', y', z', and X' being determined from
the equation f(x', y', z' ', X') = 0.

7. The boundary X = 0 is of course fixed by the solid whose potential is required,
but we are left with a certain amount of choice as to the disposition of the surfaces

fA
and \p is any function of A whatever. Replacing ^ (A') - \j/ (0) by u (A) d\, we find that if <I> is a

solution then

will also be a solution of the same problem.
VOL. CCXV. — A.

34 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

X = cons. We shall now limit this amount of freedom by assuming that the region
at infinity is made to coincide with the surface X = + °° .

Consider a new function V, defined at any point of space x', y', z', by

(22)

o

then

(23)

by equation (18). Hence, if (as will be .the case in all our applications) the value
of the limit when X' = oo of the term in square brackets is zero, we shall have

V-'V, = - 47TP ........... (24)

At infinity Vu must vanish, so that at infinity, by equation (21),

v, = w = I V wyv, ?/, *', x) d\ = v,

Jo

Thus V; — V, vanishes at infinity, and satisfies V- (V, — V,) = 0 at all points of
space ; whence (except for a possible singularity at the origin, which will be found
not to cause trouble) we must have V^ = V,, so that V,- is the internal potential.
Knowing V{ and W we find Vu immediately by equation (l), and have

,!/,z',\)<lX ........ (25)

)i/>zf,\)d\ ........ (26)

To recapitulate, the condition that these equations shall give the true values of the
potential are

(i) that V,. shall be finite at the origin,

/ 7^ \ 2 ^ f

(ii) that x/, (x) (-=-) ^- shall vanish at infinity.

\CLlli / C\

If these conditions are satisfied, as they will be without trouble in all our
applications, then equations (25) and (26) will give the potentials.

8. If y (x, y, z, 0) is the equation of the boundary, the potential at the boundary
will be

and therefore will contain exactly the same terms in x, y, z as the equation of the

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 35

boundary, but with different coefficients. The method is therefore exactly suited for
the determination of figures of equilibrium of rotating fluid (cf. § 2).

As a method of determining potentials, the procedure is indirect in the sense that
we cannot pass by any direct series of processes from the equation of the boundary of
the solid, as expressed by equation (20), to the general function f (x, y, z, X). We
must first search for solutions of equations (18) or (19), and then examine what
problem is solved.

An obvious case to examine first is that in which f is an algebraic function of the
second degree. In this case V2f is a function of X only, so that the equation for f can
be satisfied if the last term in equation (18) or (19) is a function of X only.

EXAMPLES OF GENERAL THEORY.
I. A Sphere.

9. A quite trivial example may perhaps be taken first, namely that of the sphere
x2 + ys+z2 = a~. It is seen without trouble that any way of forming the function
f (x, y, z, X) will lead to a solution, provided that this function involves x, y, z only
through x2 + y2 + z2, — i.e., provided the surfaces are taken to be concentric spheres.
For instance, we may take

f(x, y, z, X) =
then equation (18) reduces to

-4ir/> = I Ct/r (X) d\ + 2 (x'-n) ^ (X'),

Jo

of which the solution is found to be \js (x) =

-
(X— a)

The potentials are now given by

where M is written for *Trpa?, and r2 = x2 + y2 + z2 — (X — a)3.

II. An Ellipsoid.
10. It is readily seen that a solution of equation (19) can be obtained by taking

l ..... .. . (27)

F 2

36 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

The equation reduces to

')>- ' •'• (28)

of which the solution is readily found to be

These values are found to satisfy the conditions of § 7, so that the potentials are
given by

1 / a? if z*

-*** xJ^jrv, W+x + /TTx + ??x '

T ^
~5 --- r T^ --- r , -------- 1 Ui\.

J 3

III. yl Distorted Ellipsoid.

11. For an ellipsoid distorted in any way, and without any limitation (at present)
as to the distortion being small, assume

4>(x, y, z, \) = + (\)(f+t)

where \// (x) and f have the same meanings as before, being given by equations (27) and
(29), and <jt is any general function of x, y, z, and X. The boundary of the distorted
ellipsoid is of course

Equation (19) becomes

in which x', y', z', \' have been replaced by x, y, z, X now that there is no danger of
confusion, and A, B, C are written for a2 + X, 1r + X, c2 + X. If we further put

•- _

2 Ca ~ 3x '

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 37

the equation becomes

f 3 ,\ / 2 ,\ fA fr> o o

O C0 \ / O C/(7) \ I . / \ \ £ £ Zl V^q

«-r^ __ I — I »rr* ._. I *^. ( A ) • [ I I w^

But from equation (28)

so that on subtraction we find as the equation to be satisfied by <j>,

The equation is too complicated to be attacked directly, but can be effectively
broken up by assuming a solution

</, = w + fc,

in which u and v are general functions of x, y, z, and X, while / is given by
equation (27). On substituting this value for </>, equation (30) reduces, after
considerable simplification, to

4 Ci'-

ox

and this will be satisfied if we satisfy separately the equations

v = 0, ...... (31)

*+/^ =0. . . (32)
c 9aj/

On substituting for /and ^, and writing ABC = A2, equation (31) becomes

A" Jo I " AT Sx % ' \A B (3^] A

in which

-[*o /! l-jl^-Ti ~(1

Jo \A B O/A =Jo axU

38 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Thus equation (31) is equivalent to

x 3t? dv\\d\ 4v /or>v

T5- + ~- U— == — ...... (33)

A fix 3X7 J A A A=»

This must be satisfied for all values of X, so that we must have (as is clear on
putting X = 0 in the equation) v = 0 when X = 0.

It will be remembered that the boundary of the distorted figure is given by

r2 ?/ z2

T + S + T-1+0A.O-0,

a* V c*

and it is now clear that 0A = 0 reduces to «A = n. Thus the generality of the boundary
must be involved in the generality of u, and provided u is kept general, we shall
obtain a general solution of the problem, even if we take the simplest possible value
for r. The most general way of satisfying equation (33) is to take

....... (34)

where x may be any function of x, y, z, and X which vanishes for X or 0, but the
simplest way of satisfying the equation is to take

0 ........ (35)

In each of these equations the sign of identity ( = ) is used in place of the sign
of equality, because in the integrand of equation (33) the value of X is not the same
as the value of X in the upper limit of the integral, which is determined by the values
of x, y, z.

12. To shorten the algebra we may change to a new set of variables X, £ ij, f
connected with the old variables X, x, y, z by the relations

x

Differentiation with respect to the new variable X is given by

_3_ _3_ _, Sx 3

OX new 3X ohl 3X dx

*L

where, since x = (aa + X) f, we have ~ = g = — . and so

oX A

- -

3Xnew 3Xc,d+ A8X

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 39

(32 32 ^2 \

5— a + 5-5 + 5-5 ) , which in the
dx* 3?^ 3zJA = co,,s.

new variables becomes

_LJH JL^., JLJ!!

Aaaf B2av cw '

and this will be denoted by V2f)|f.

Equations (32) and (35) in the new co-ordinates, reduce to

/0-\
°' ..... (37)

(38)

We are assuming that f+<f> = 0 when X has the value appropriate to the values of
£ >i, £ so that in the first equation /"may be replaced by -- </>, but in the second
equation this may not be done.

13. It is convenient, for the purposes of the present paper, to suppose the distortion
to start from the undistorted ellipsoid, and to proceed in powers of a. parameter e.
Thus we assume

u =

and in the equation (37) since f+ (u+fn) = 0, it is clear that, when e is small, f will bo

a small quantity of the smallness of e. As far as e, equation (37) reduces to ~ = 0,

CA

giving •«,. Equation (38) then gives r, ; equation (37) taken as far as e~ will then
give u2, and (38) will give ra; (37) taken as far as e:i will give «3, (38) will give r.(
and so on.

2

As far as e only, equation (37) reduces to •^-L= 0, of which the solution is

(j\

ui = X (f> '/> f) where x is the most general function of f, >i, and f. At the boundary
0 reduces to (wJA = „ or k>x (— , K, —J, so that the generality of the function x enables

\\Jv D G I

us to deal with the most general small displacement possible.

At present we shall consider only solutions for which x is algebraic and of degree
not greater than 3 in f, >;, f. For these solutions equation (38) shows that i\ will
be of degree not greater than 1 in £ ^, f, sa that VJVj = 0, and the equation reduces
to

A^£I _ v ( l ^ i a2 i o2

4 ' f"tWl= " '8

40 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

or

since v must vanish when X — 0.

3

Remembering that -^ = 0, and that f = — eu} + ... , the terms in e~ in (37) now
( X

gve

. en.,

giving on integration

(,,(^,^)! . . . (40)

in which o> is again the most general function possible off,?/, f (enabling us to carry on
the distortion to the second order in any way we please), and the lower limit of the
integral is taken to be zero simply as a matter of convenience.

The addition of a perfectly general function « would be equivalent to the superposition
of a perfectly general distortion (proportional to e2) on to the distortion already under
consideration. The real object of the present analysis is to be found in its ultimate
application to the problem of the rotating fluid, and to solve this problem, it will be
found that » need contain no terms of degree higher than 4 in f, tj, f, this being also
the degree of the other terms in ».,. Hence in what follows it will be supposed that
it., contains no terms of degree higher than 4 in f, >], £.

A value of r2 is obtainable from equation (38), but there are, as has been seen, many
possible forms for r,, and the most convenient is, in point of fact, obtained by going
back to equation (;J4), which in x, y, z co-ordinates is

where x '"!iy be any function of .r, y, z, and X which vanishes (to the power of e we are
now concerned with) both for X and 0.

Let two new functions w and w' be introduced, defined by

rv, /A0\

-Vw" ........ (42)

_iV^ ........ (43)

THE FIGUEES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 41

Clearly since ua is of degree 4 in x, y, z, iv will be of degree 2, so that it/ will be
a function of X only. The term 2-r- ^ — in equation (43) is therefore zero in the
present instance, but is inserted to maintain symmetry. We now have

3 x 3

- .

A oa;/ \3\ A da?

so that after simplification,

Since / vanishes for the appropriate value of X, ^— will vanish for both X and 0

fw'
provided w' is made to vanish when X = 0. Thus •'— - will satisfy the condition to be

satisfied by x in equation (41), and a solution of this equation will be

ra = w+fw' (44)

Since V2 must vanish when X = 0 (§ 11) it appears that both /« and w' must vanish
separately when X = 0. On transforming (4l) and (42) to £ tj, £ co-ordinates
(cf. equation (36)) and integrating, we obtain as the values of w and w' which vanish
when X = 0,

rt px

w = -i V2^u,,d\; w' = -l\ VatvSwd\ ...... (45)

Jo Jo

14. Let us introduce a differential operator D, defined by

D~8 + 3~"l"a~ ' ..... (46)

noticing that, as a function of X, D is purely a multiplier. We have

3D i a2

~~ '~

and when X = 0, D = 0.
VOL. ccxv. — A.

42 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

The value of vt already obtained (equation (39)) is

........... (47)

and the value of u2 (equation (40)) is

(48)

•

Hence from equations (45)

<*

w = -f V2£,f ?{a cZX,

Jo

DV ............ (49)

= -if V

Jo

15. This completes the solution as far as the second order of small quantities. We
shall not attempt to evaluate ?/3 and v3, as the problems discussed in the present paper
require a solution as far as e2 only.

As far as ea, the value of <p has been seen to be given by

0 = u+fv = e(M1+/v,) + e2(w!1+/tt>+/V) ...... (50)

and the potentials can now be found directly from the formula (§ ll)

As in § 7, examine a function V^ defined by

then

}(ZX ........ (51)

Now the value of J + (\) V2fdX is by § 10, equal to -4w/>, while from equation (31)

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 43

we have

f V (X) V (u +fv) d\ = - 4 ( VT (X) v\ ,

Jo

Inspection of the values obtained shows that the limit of >/r(x)v when X = o° is zero,
so that equation (51) reduces to V2V, = — 47r/o, and since V, is equal to the true
value of V< at infinity, and is finite at the origin, V'r must be the true value of the
internal potential. Thus the potentials are given by,

V,- = f V W [/+ * (MI +fvl) + e2 (ua +fw +/ V)] d\ .... (52)

Jo

V,,= f Vr(\)[/+e(?/,+/i'I) + esK+>+/V)]rf\ .... (53)

JA

in which all the quantities must be transformed into x, y, z co-ordinates before
integration.

When X = 0, u., reduces to \<a (£ >/, f) by equation (40),

i Ix y z \ , -, / , Ix y z\

or to -fu> — , fa — while u^ = x (£, *i, <, ) ~ X ( ~> f? ~j '
\a2 fe2 (?) fi? u c J

also Vi, w and w' all vanish when X = 0, so that (cf. equation (50))

a

and the boundary of the distorted ellipsoid is
x2 y2 z2

16. Before proceeding further it will be convenient to examine in detail the
first order solutions which can be obtained from the foregoing analysis, classifying
them according to the degree n of the algebraic function ult and, for brevity, omitting
the continual multiplier e.

n = 0. Solution is u^ = K, vt = 0, <f> = K.

fv\ iyi sy ft tY*'Z

n =; 1. Solution is u^ = pg+gi + r^, vl = 0, <£ —*-r- + ^3 + p-

J\. -D \j

n = 2. u,= af+/3,!+vf +2/rf+%ff+27if,,

+&+ *)

G 2

44 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

A physical interpretation of these first few solutions can readily be found. For
the undisturbed ellipsoid of axes ka, kb, kc, and origin at x0, y0, z0,

, -

~— ~

C

and the special ellipsoid which has been under consideration has been that for
which x0 — y0 = z0 = 0, k = 1. We can change the centre and axes of the ellipsoid
contemplated in equation (55) by varying xa, y0, z0, a2, l>2, c2, and k. If we change

k2 by an amount 3k2 in equation (55), the change in $ is given by M> =

so that f may be regarded as replaced by f+<f> where </> = — Sk~. Thus the solution
n = 0 represents a change from k2 to IC*—K ; physically it represents a change in
the scale of the ellipsoid.

Similarly, if in equation (55) xu is changed by <fcr(), y{l by fy0, and z0 by Sza, we find

that SQ = -2y, (A) (^ + (^f + ^\ so that

A B (

and the solution n = 1 is seen to represent a motion of the centre of the ellipsoid.
Similarly, if we put ,ru = yn = z0 = 0 and k = 1 in equation (55) so that <J> =
and differentiate logarithmically with respect to a2, we obtain

_ _

V/Sa2" 2a2 2A

whence

Clearly, then, the first three terms in the solution n = 2 represent a distortion of
the original ellipsoid produced by a change in the lengths of the axes, and it is
easily seen that the complete solution represents a change of this kind combined
with a small rotation of the axes.

n = 3. There are ten terms in the general cubic function of f, y, f. For the
present purpose it is convenient to regard this general cubic function as made up
of a term e^f, and the sum of three expressions such as

For the solution given by % = efr£, we have V2^Wj = 0, so that vl = 0 and

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 45

It will be shown later that an ellipsoid distorted in this way cannot possibly be a
figure of equilibrium for a rotating fluid (§ 20).
For the solution

«i = £(«f+/V + yf) (56)

we have

and the solution is

x / rj? ^K>y\ ^l i*2 , ?/ , z2 ,\/3«X /3X

= ' -

It will be shown later that this distortion leads to the Darwin-Poincare series
of pear-shaped figures of equilibrium of a rotating fluid.

n = 4. The analysis of §§13, 14 was confined to the case in which ?«, was
supposed of degree not greater than 3 in £ ,,, g. But if, in the solution finally
obtained in § 15, we take u^ = 0, which involves taking also i\ = 0, we are left with
a solution (cf. equation (50))

</> = e?(ua+fw+fau/),

in which u.2, w, and w' do not vanish on account of the occurrence of the arbitrary
function «. And since « has been supposed of the fourth degree, this solution
gives us the solution of degree n = 4 to the first order, the parameter <? replacing
the former e. Thus the solution of degree 4 is

u± = a general function of degree 4 in £ »/, £ (the old e2«2)

This solution is not discussed in detail in the present paper, but is classified here
with the other solutions for the sake of completeness. An ellipsoid distorted in
accordance with this solution would give rise to a series of dumb-bell shaped figures,
which would be figures of equilibrium for a rotating liquid. They would be unstable
for a homogeneous mass, but the corresponding figures might conceivably be stable
for a heterogeneous mass (cf. POINCARE'S remark quoted in § 1 of the present paper).

17. One point of interest must be mentioned here in connection with the potentials
derived from these solutions.

In the potentials arising from the solution of degree n = 2, ^ = a^3, the internal
or boundary potential will be of the form Ix2+my2+nz2, where I, m, n do not
involve x, y, z, or X. Since this must be a solution of LAPLACE'S equation, l + m + n
must vanish, and the potential must be expressible in the form m (y'^—x2) + n (z2—x2).
All the other potentials may be similarly treated.

46 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Making use of this simplification, we arrive at the following scheme for the
contributions to the internal or boundary potentials of the various solutions up to
n = 3. Only typical terms are taken ; <j>b represents the value of tf> at the boundary,
V6 represents the contribution from the typical term to the boundary or internal
potential.

n = 0.
„ = !.

.00

b = K\

Jo

r-\ x v v'x y-^ z-

W +"=^ ~Jo2a2Xl B C

= 3. (i) fc = -^, Vt = '

v r - - ,

-~ "

x v ^x ?-x z- ,

"= -" ~~ 3C

18. In any physical application of this method, and in particular in its appli-
cation to the discussion of rotating masses of liquid, it will be important to know
what changes are produced by the distortion upon the mass (or density), the
position of the centre of gravity, and the moments of inertia of the body. These
changes are given at once by a study of the limiting form of the external potential
at infinity.

The potential at infinity of any mass whatever, taken as far as terms of order —^t
has the limiting form

m + mu .

r r 2r°

where m is the mass of the whole body, x0, ya, z0, are the co-ordinates of the centre of
gravity, and, L, M, N, P, Q, E, are products of inertia defined by px^dxdydz = L,

I pyzdxdydz = P, &c. The moment of inertia about the axis of z is \\\ p (
dxdy dz = L + M, and so on.

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 47

For the solution fa = K of degree n = 0, the limit at infinity of the contribution
to the potential is

f" Trpabc-,

— K I -- - - d\ —

r

+ ...,

showing that the distortion involves a change of mass SM. = —^TrpabcK, accompanied
of course with a change in the inertia terms.

For the solution (j>b — -^ of order n = 1 ,

a

,-rr f* — Trpabc 7. „ 7 a?x i ^mx

00 = X Jx A A = ~ 5 ^ T5" ' ~ ^ ~^'

so that this distortion represents a motion of the centre of gravity by an amount

XT - _l./y2 •

O Ki/0 — 2 ^ •

Solutions of degree n = 2 will clearly involve changes in the moments and products
of inertia. The limiting potentials are found to be as follows :

(i) A- J|,

(ii) 04 = ^j ,

Cv / v /

The first solution does not involve a change in mass, whilst the second does ; both
distortions affect the inertia.

For the solutions of degree n = 3, the limiting values are as follows :

This distortion changes neither mass, centre of gravity, nor inertia.

- - w abc

^

i ^l

/yi'J /y»« /y>

(iii) ^ = - , S Vx = - frp abc f-7 - | TTP abc -^ •

\M I C& I

These two latter distortions move the centre of gravity, but do not affect the
mass or inertia.

It is clear, without detailed examination, that the distortions represented by
solutions of degree n = 4 cannot affect the centre of gravity, but may affect the
mass and inertia.

48 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

FIGURE OF EQUILIBRIUM OF ROTATING MASSES.

The Jacobian Ellipsoids.

19. The condition that a single figure shall be a figure of equilibrium for a
rotation <a about the axis of z is that the centre of gravity shall lie on the axis
of z, and that V(, + i«2 (x2 + y2) shall have a constant value over the boundary X = 0.
In searching for a series of figures of equilibrium we must add a further condition
of constancy of mass.

For the undisturbed ellipsoid, with the notation of § 10, V4+£w2 (*2 + ?/2)

<58)

For this to be constant over the surface, it must be identical with

where x is a constant. If we put

-x

= n,

the equations obtained by comparing coefficients are

T d\ e

-r-r-n = -2>

Jo AA a

r

\
Jo

d\ 9

-r-fj ~n = 7T'
AB 6

AC

Ellipsoids with Distortions of the First Order.

20. We proceed to consider which of the distorted ellipsoids can give rise to
possible figures of equilibrium.

The solutions of degrees 0, 1, 2 lead to nothing except new ellipsoids, so that the
inclusion of these distortions could only represent a step along the already known
series of Jacobian ellipsoids or Maclaurin spheroids.

Consider next an ellipsoid distorted by the addition of a solution of the type (i)

of degree 3 (§ 18), say 0,, = e-^~-a This distortion, as we have seen (§ 18), does

C(f O C

not affect the total mass or the position of the centre of gravity. The boundary
of the distorted ellipsoid is

^.i.^-?2-! ,

a2 b3 c2

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 49

while the additional term which has to be inserted on the right of equation (58) is

Hence the additional equation which has to be satisfied, in addition to equations
(59) to (61), is

A AT>p =^575~2 (62)

Jo ii -rt-Dv-' Ct 0 C

Eliminating 0 from this equation and (61) we obtain, as an equation which must
be satisfied if the distorted ellipsoid is to be a figure of equilibrium,

= 0 (63)

This obviously cannot be satisfied, for the integrand is positive for all values
of A. We conclude that the distortion now under consideration cannot possibly give
rise to a figure of equilibrium.

21. There remain nine terms for consideration in the general cubic function.
Inspection will show, or it will soon become apparent as we proceed with the analysis,
that these fall into three groxips, as in § 16, and that the three terms of any one
group just suffice to give a possible figure of equilibrium when combined with a
term to restore the centre of gravity to its position on the axis of rotation. We
shall accordingly consider a distortion in which the cubic terms are those already
written down in equation (56). These terms are seen (§18) to move the centre
of gravity parallel to the axis of x, and to correct this we shall add a term (<•/. § 16),

KX ,

-T- tO tt1.

Thus, for the distorted ellipsoid now under consideration, the boundary will be

999 / 't O 9

x2 if , z2 .. / x] Oxir xz" x \ ,..N

~«+ih,H — s — 1 +e a — +p ——• , +y — —. +K— -} (64)

a b c \ a ao • arc, a?/

As far as terms in — , the value of the potential at infinity is (cf. § 18)

2 a_
5 a2

so that for the centre of gravity to remain at the origin we must have

(65)

\rOL. CCXV. — A. H

50 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Collecting the terms in VA as calculated in § 18, we find

?/2-Z2 ijN ^ /V«\

(66)

os ,- -T- g -- - -

2c2Ju AC 1 A B Ju A

For brevity in printing, introduce the following notation. Let

dX T r XfZX T

: JoAABCT.

so that, for instance,

And, for the problem immediately in hand, write further

— T ('A T T

Cl ~ -^ABC = ~\~ \~T3f V ^2 = AAC) C3 = -^AAB)

J () AAOU

2

and as before put _ - - = n, tlien equation (66) becomes

x2 (JA — w) +T/2 (JB—

(68)

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 51

The distorted ellipsoid will be a possible figure of equilibrium, as far as the
first order of small quantities, provided the right-hand member of equation (68) is
identical with

- + * + r + . . . (69)

Equating coefficients, we obtain

0

72. Jc = 1' ...... (70)

0 C

(*+*>-«*-*- » ........ (")

(74)

and from these equations, together with equation (G5), we must eliminate or obtain
the coefficients.
If we put

a / ft of 7 ' (>7-\

c?=a' &"* ? = y ....... (7j)

then equations (71) to (73) reduce to

a'(<yH*)-|8V-'/e.= ^?«', ......... (76)

(78)

Cv C

whilst on substitution for K from equation (65) and JA from (70), equation (74)

reduces to

(79)

We have now to deal with four equations (76), (77), (78), and (79), and have
to examine whether, and how, these can be satisfied by values of a, /3', and y .
Since there must, from general physical principles, be an equation of some sort for
points of bifurcation (whether capable of being satisfied by real values or not), we
are led to suspect that these four equations are not really independent.

H 2

52 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Equation (79) was obtained by the elimination of K from two equations (65) and
(74), each of which expressed in effect the condition that the centre of gravity
of the mass should be at the origin ; in fact, equation (65) was only a short way
of arriving at the value of K, which would in any case have been given by equa-
tion (74). We therefore expect that equation (79), derived from (65) and (74),
will prove only to be an identity of which the truth is involved in the three other
equations (76) to (78). And, as a matter of procedure which is entirely at our
choice, we shall elect first to solve equations (76) to (78), and then to verify the
truth of (79).

The elimination of a', /3', y from equations (76), (77), and (78) gives a determinant
which on expansion reduces to

3 _! JL

-(- b2 c:

2) +c3(3a8+6a)} + = 0, (80)

Ct

and this is accordingly the equation giving points of bifurcation on the Jacobian
series of ellipsoids.

22. Two identities of importance are the following :

--f. . • • (so

abc

From equation (70 )

JA = W+|S, ........... (83)

JB = »+p, ........... (84)

Jo=A ............ (85)

O

so that on addition, by the use of (81),

(86)

,

abc \a2 V <?)

giving 0, and hence JA, JB, and Jc, in terms of a, b, c, arid n. We further have

o AAB J0 l(aa-68)AA («2-62)ABjdX = a'-b*
T _ a2JA-c2Jc

IAC- -^-r-' „ -. (88)

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 53

and IAA is given by equation (80) as soon as the values of IAB and IAC are known.
Substituting for JA, JB, Jc, from equations (83) to (85), we obtain

2 c3 20

-n, AA = --

(Jtr -

a i c

IAB = «, IAO = -j— -,n, IAA = -i-ip— -2n--2 . . . (89)

Finally we bave

\d\

f \d\ J_r/I_l\^_ _J __ /T _T \

' " Jo AABC ~ c2-62 Jo \B CJ A A ~ (c2-fc2) ^

— IAC), C3 = 2 „ (IAB — IAA)-

c2-«2V a*-b

With this material it ought to be possible to find the points of bifurcation from
equation (80). As, however, DARWIN'S results are available, it will be sufficient to
make use of his results and merely verify that his quantities satisfy equation (80), as
they are in point of fact found to do.

23. DARWIN'S values, calculated for an ellipsoid such that al>c = 1, are

a = 1-885827, b = 0'814975, c = 0'650659,
n = -£- = 0-1419990,

2-TTp

whence, by equations (86) and (89),

= 0-2068037

a

2 b2

IAB = 0-1419990, IAC = 0-1611871, 1AA = 0'0711382,

Cl = G'07967602, c2 = 0'02874219, c3 = 0'2450100.

With the use of these values, equations (76), (77) and (78) become

0-01216184a' + 0-02450100/3' + 0>02874219y/ = 0 ..... (90)
0'07350300a' + G'1970290/3' +0'07967G02y' = 0 ..... (91)
0'0862266a' +0-07967602/3' + 0-3835217y' =0 ..... (92)

The values of a', ft, y, are, of course, indeterminate to within a common multiplier.
The simplest set of values, obtained by cross multiplication of the coefficients of
equations (90) and (91), is

a! = - 0'003710945, j3' = O'OOll 43630 •/ = 0'000595338.

54 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

If we substitute these values in equation (92), we find

= -0'00031998 + 0-00009111 + 0<00022829 = -O'OOOOOOS.

The fact that the error occurs only in the seventh place of decimals adequately
verifies DARWIN'S calculations, but the tendency of small errors to accumulate in
computation is forcibly illustrated by the circumstance that in the above equation
the final error is as much as one-six-hundredth part of the whole value of the middle
term.

With the values just obtained for a, ft', y , I find

y') n -- -G'00094878,
(So.' lAA + ^LvB + y'lAc) = -0-00094885,

verifying that equation (79) is satisfied, again as far as the sixth place of decimals.

24. With a view to subsequent computations, it is convenient to take a standard
set of values such that a' = — 1. These values are found to be

a' = _l, fjf = G'3081810, y' = 0'1604294,

and with these we have, by equation (65),

3a' + 3' + ' = Q-506278.

These numerical values substituted in equation (64) will give the equation of
POINCARK'S pear-shaped figure as far as small terms of the first order.

The Pear-shaped Figure Calculated to the Second Order.

25. The question as to whether the pear-shaped figure is stable depends upon the
change effected by the distortion upon the angular momentum of the ellipsoid. But
(cf. § 18) the first-order distortion so far considered can be easily seen to produce no
effect at all upon the angular momentum of the figure. It is therefore necessary to
proceed to terms of a higher order, and we now consider terms of the second order.

The first-order terms have been found to be given by

yf + K), ........ (93)

with (cf. equation (47)) ?V= -iDtt^ The value of n2 will be given by equation (48),
in which ur is to be assigned the value (93), and w will be taken to be given by

to = L^ + M^ + Nr + 2/,,2f + 2m^2 + 2wfV + 2(p£2 + g^-Hrf2) + s. . . (94)

It has to be shown that this value for u> makes it possible for the figure distorted
to the second order in this way to be a figure of equilibrium.

THE FIGUKES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 55

It will be noticed that with the value we have assumed for <a, the value of u2
becomes a function of (-, q, £, of degree 4 involving only even powers of £, »;, f ; its
value is

rf) + s. . . (95)

The values of fw and of f*w' are easily seen to be similar in form, so that all the
second-order terms in $ are of this form (cf. equation (50)). Multiplying by i/r (\) dX
and integrating from 0 to o>, we obtain as the terms of the second order in the
potential an expression of the form

- Trpctbce2 (cuo;4 + c22?/4 + o^z4 + cvfx V + c^/Y + o312 V + c^.K2 + c^/y2 + dj? + d4).

If this figure can be a figure of equilibrium at all, it will be for a rotation differing
only by a second-order quantity from that of the original ellipsoid. Let us suppose

that for it - 5— = •>i + e2it," ; then at the boundary, as far as e2,
2-irpabc

+ e2 (cnxl + c2.2y* + c.^ + cvjL?if + f^/V + cmz2x2 + <l}.i-2 + d.jf + c7;(22 + dt
+ third-degree terms in c, the same as before} ...... (96)

At the boundary,

so that for the figure to be one of equilibrium, the right-hand member of equation (96)
must be identical with

f a Ji __a / 2 _2 2

I /•>•* •?/ & I 'Y 1 1 V

1 /\ \ **•' i / i 1 / "^ i O / ft

1 Cf'^ \)^ C \ CJ** Cl'^^ (t (*^ ft

a

. . (98)

56 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Upon equating coefficients of the terms of degree 1 and 3 in x, y, z, we obtain
exactly the same equations as were obtained before when the terms in e3 were
omitted. Thus no alteration is produced in the cubic terms until the solution is
carried as far as e3.

The equations obtained by comparison of the coefficients of x3, y2, and z2 are : —

W A W O 7t- ~T C/ IA/1 VI y I *> /-* . /*

\cr ov

and two similar equations. In these the terms in e2 and the terms independent of e
must be equal separately. The latter terms again give equations (70), while the

terms in e2 give

a^

(99)

(100)
(101)

Filially, upon equating terms of the fourth degree, we obtain the six equations :

(107)

We have seen that equations (70) to (73) must still be true, so that a, b, c, a, /3, y, K
will be the same as before.

26. At present there are eleven quantities to be determined or eliminated, namely,
L, M, N, I, m, n, p, q, r, s, and n". The equations' giving these quantities can be
simplified in the following way.

By an argument already used in § 17, it appears that the terms in e2 on the right
hand of equation (96) must be harmonic. Thus we must have

6cu + c12 + c,3 = 0 .......... (108)

6c22 + c23 + c21 = 0 .......... (109)

6C33 + <%1 + Cg., = 0 .......... (110)

dl+d2 + d3 = 0 .......... (Ill)

These are, of course, not new equations to be satisfied ; they reduce to identities
when the calculated values are inserted for on, c12, &c.

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 57

With the help of (ill), equations (99) to (101) may be replaced by

n" = -4-6 (-£ + £+ r] (U2\

n 4t7L.* + 7.4 + ^ > v11^

a

c

while from equations (102) to (110) we obtain

3L m n

— + - + -=0
3_N l_ in =

These three equations with equations (102) to (104), namely,

^n = iff, (118)

<» = i^, (119)

C33 = i^, (120)

may be used to replace the group (102) to (107).

27. We proceed to evaluate these quantities in detail. The values of u± and i\ from
equation (47) are

(122)

and the value of u2 is already given by equation (95).

For convenience in computation we shall combine all the terms in ?«2 which are
independent of X in the first two lines, with the similar terms in the last line ; we
accordingly write

4 3,2.2

a' * - ' ' ' ' b" " ' c2 7 *

^2+2(pf + qn2 + r^)+s

+sf

= »'(6*f) ..... ............... (123)

VOL. CCXV.— A. I

58 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

The value of ua is now given by

^te » t) . . (124)

whence, by equation (45),

fx
= - V\,,fMad\

Jo

4 ic

(125)

where the K's are constants, to be chosen so as to make ^v vanish when A = 0.

222
The value of V2tf ,, fi (4«r) is derived from that of 4»- by replacing ^3, »/2, f2 by -T-J, ^5, ^2

and omitting all other terms. Thus we have, again from equation (45),

= -i [ V8j,,if(4w)c?X

Jo

__ __
V " a A:t 2 AB2 2 AC3 A8B A2C ABC

+ ++ • • • • (126)

in which K5 is a new constant, chosen so as to make w' vanish when X = 0.
On collecting terms, we obtain

4(w+>') = P,f + P^ + P3i2 + P1 ....... (127)

where

3 . M'A .3 N'A «'A m'

" -

, 03 . .3 , i , I4 ^

8" -^+^-^ +¥+lrTJ" *B

(128)

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 59

= ^-TS + 3*AB + t AfT2 +3¥ A~2

A -£\.J_> -TXl_/ .£\_

L'B

BK,
A

AC

N'B

BK,

L/p M'P XT'

+ ^ ^ + 3. 1V1<^ + 3|. £L . j.i

jrx O

+

'G

A 4 AB

rl^- '35 -"33?

/I/ /' K ]

A2C ABC!,

A B ' C
If the value of ^> is taken to he e(/>i + i'''<j>2, we have as the value of 9^,

B

C

A

A

B G

(129)

(130)

(131)

^4 + M V + NT + 2ZV £8 + 2m1 '£2f + 2n'?n* + 2

(132)

and we have already supposed (§25) that

f °°

-• '»

+ cl2x V + c13a;
I 2

t. (133)

60 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Hence, by comparison, we obtain as the values of the coefficients

Cn =; 4 Jo 'A I A^ r A4! T A*C " A4 ' AV

. i f° <**(_£_ + M.' iVj (135)

4 Jo A \AB4 B4 BV

1 d\ I y L 3) ^1S6^

= ¥.0 ^IAC? + CI cv

! f" rfX /12«/9 6/92 2/9y 2n; PI P2 \
r" ": tJo "A U8B3 A2B:i + A2B2C A2B2 " A2B """ AB2/

Cl3 = *.0 '~& ( A1? + ife + A8BCa + A2!!2 + A% + ICV ' '

-, ............... (143)

Jo A \ A

Evaluation of Certain Integrals.

28. It is clear that before these coefficients can be evaluated, certain integrals
must be calculated of the types JBCA...> IABC... > where the notation is that of §21
(equation (67)).

The values of JA, JB, Jc have already been evaluated in § 23, as also of IAA> IAB,
IAC and IAAB, IAAC, IABC. We also have

ZBC = JB - iJo = T = 0-3916228.

These 10 integrals will form an adequate basis from which to calculate all the

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 61

others. The required integrals can all be obtained by successive applications of
formulfe of the following types, which can be verified without trouble :

(a*-b2) JAB = JB- JA 5 (n'-V) IAB = IB-!A, &c.
3 IBB = 2JB — 1BC— 1AB ; a JAA = JA— 1AA, &c-

2

2n - JAnB - JAn

AnB Anc

(2n+ 1) IAn+i = 2 JA»-IA»B-IA»C-

A great number of the integrals can be calculated in two or more ways, and owing
to this circumstance it is possible to provide very complete checks on the accuracy of
calculation. Two complications- are worthy of mention.

In the first place one must consider the ordinary cumulative error of all prolonged
computation. Since b2 is nearly equal to c3, any error present will be increased when

we evaluate a new quantity of the type^ ' .^ 2 > consequently where a quantity

can be evaluated in two ways, the one in which division by b3—c2 is not involved has
been taken to be the true value, and the one derived by division by I?— c3, has been used
merely as a check, and has generally been found to differ in the sixth or seventh place
(or near the end of the computations even in the fifth place) from the other values.

Secondly, if the 10 integrals used as base were known with perfect accuracy, the
checks ought to be satisfied fully except for the error in the last one or two figures.
But, as has been indicated in § 23, the 10 integrals are not themselves perfectly self-
consistent, so that different methods of computation will lead to a difference of the
final results comparable with the errors in the basic integrals.

The following table gives the values I have selected as the best for the various
integrals required. I have not thought it necessary to record the checks or estimate
the probable errors here, as a much more searching test of the accuracy of the whole
computation can be provided at a later stage.

J = 1'8401326.

JA = 0'2583003, JB = 07G47290, Jc = 0'9769708,

JAA = 0'05262769, JAB = 01751040, JAC = 0'2293883,
JBB = 0-6516017, JBC = 0'8813026, Jcc = 1 '2044842,

JAAA = 0<011873224," JAAB = 0'04234772, JAAC = 0'05641920,
JABB = 0'1647550, JABC = 0'2254075, JACC = 0'3112352,

JBBB = 0'6830283, JBBc = 0'9537991, JBCc = 1'9011148, JCcc = r9011148;

62

ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

JAAAA= 0-0028 1568,
JAABB = 0'04232384,
JABBB = 0*1791995,
JACCC = G'5074646,
JBBCC = 1-611794,

JAAAAA = 0'0006881964,
JAAABB = 0'01099071,
JAABBB = 0'04732648,
JAACCC = 0-1360144,
JABBCC = 0'4340725,

IA= Q'9215282,
IAA= Q'0711382,
IBB = Q'3319454,

IAAA = G'01040244,
IABB = 0-06567633,

JAAAB = 0'01053694,
JAABC = 0'05842981,
JABBC = 0'2518508,
JBBBB = 0'788915,
JBCCC = 2 32179J,

JAAAAB = 0'002669728,
JAAABC = O'Ol 528665,
JAABBC = 0'06687764,
JABBBB = 0'2108169,
JABCCC = 0-6273273,

IB= 1-3322118,
IAB= 0-1419990,
IBC = 0-3916228,

IAAB = Q'02450100,
IABC = 0-07967602,

JAAAC = 0'01421838,
JAACC = 0'08133328,
JABCC = 0-3563870,
JBBBC = 1 '124336,
Jcccc = 3-361221,

JAAAAC = 0'003639559,
JAAAOO = 0'02 142202.
JAABCC = 0'09532252.
JABBBC = 0'3016727,
JACCCC = 0-910874,

Ic= 1-4265252,

IAC = 0-1611871,
Icc = G'4670439,

IAAC = 0-02874219,
IACC = 0-09762467,

IBBB = 0'19794510,

= 0'2478016, IBCC = 0'3131750, ICcc = 0'3996337,

IAAAA = 0'001859707,
IAABB = 0'01423688,
IABBB = 0'04573358,
IACCC = 0-0963965,
IBBCC = 0-2714534,

IAAAB = 0'004874751,
IAABC = O'0 176 1100,
IABBC = 0'05813153,
IBBBB = 0'1590430,
IBOCC = G'3590070,

IAAAO = 0'005853759,
IAACC = 0'02198620,
IABCC = 0-07452920,
IBBBC = 0'2070218,
Icccc = 0-4811801,

Evaluation of the Coefficients cn, cl2, ....

29. It will be noticed that the coefficients cn, cl2, ... are linear in a2, a/3, ... ,
L', M', N', ... , p', q', r', s' so that the various contributions may be calculated
separately and independently.

Contributions from Terms in p', q', r', s'.

As regards the contributions from these coefficients, we may take (cf. equations
(125) and (126))

' K6 = 0,

a

THE FIGUKES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 63

whence we obtain as the contributions to P1( P2, P3, P4 (cf. equations (128) to (131),

P! = P2 = P3 = 0,

J „' „.' I ~J „> ~J \

It now appears that p', q', ?•', s' contribute nothing to the values of cn, cl2, ... , c23
(cf. equations (I34)'to (139), p. 60.) Their contributions to 4d,, 4cZ2, and 4c£3 are as

follows :

, , f °° dX i2p' ^ P4
4fti = I — -^— 4- — -

Jo A VA3 A

= Jo A" \ A7 ~ a2 P ~ 62 AB ~ c2 AC/
so that the contributions are

4c?j = 2p' JAA— 2 IA.\— ^!AB— -T!AC,
' T P' T ^ T *'' T

«2 AB 62 * (?

Since this part of the potential should be harmonic, we ought to have di + d2 + d3 = 0
(cf. equation (ill)). This is clearly the case, in virtue of the identity

2«2JAA = IAA + IAB + IAC-

I have verified that these identities are satisfied by the values in the table opposite,
and the contributions are found to be

= -0141999 )+0'5

Ct

ids= -0161187 ^j-0'39

a

Contributions from Terms in L', M', N1; I', m', n'.
30. As regards these terms, we may take (cf. equations (125) and (126)),

it_ SM; r

a2 62 c2

\a o c /
K4 = 0,

L/ // TT T / 7/

•y I », Jt»-l 3V J_V

a4 62c2 ^ a2 a4 6V

64 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

and the contributions to Pj, P2, P3, P4, are found to be (cf. equations (128) to (131)),

T ' \a /' \2

T> J. / :< v ij A ' _i_ >'

±% - - ¥ \ '* * 7^ Ta^ 1?J «("!

«4 A:
'3I/X . m'X

3M'X n'X

1 a — —

We find as the contribution to d}, (equation (140), p. 60)
d\/l\ PA ^^\J2P, 1/3L/XXX

"~ ++

from which it is easily verified that d! + d2 + d3 — 0, as it ought to be.

In virtue of this relation it is only necessary to evaluate two of the contributions
4cZ1; 4c?3, 4t?3, but I have calculated all three directly from the table on p. 61, so as to
obtain a check on the amount of error involved from the causes mentioned in § 28, as
well as a .check on the accuracy of my own computations. The values I find are

4^ = 0'85375 ^r - 0'073783 ~ - 0'089893 ^r
a4 // c

- 0-054134 ?j-a + 0-072732 -~ + 0'059701 -m ,
be car a b

4d2= -0-041149 ^ + 0-244072^-' -0'194266^;'
a b* c*

V m' ??'

+ 0-051069 ~3 - 0-054134 ~-a - 0'005568 •— »

4tZ3 = -G'044227 ^r - 0'170277 ~ + 0'284157 ^J
a4 b* c4

+ 0-003091 y^ - 0-018600 ~- 0'054134 -— •

c

From these figures the value of d^ + d^ + d^ would be given by

= -0-000001 + 0-000012 -'-0-000002'

+ 0-000026 ~ - 0-000002 4^. - O'OOOOOl n/

j y 2 v vwwv** 22 — WWVA 27

This check gives an idea of the amount of error involved. It illustrates the tendency
of the errors to accumulate in terms where 6's and c's are plentiful, and to be absent

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 65

where as are present. This is a consequence of the method of procedure explained
in § 28.

In calculating the contributions to cu, c]3, ... , I have (unwisely) taken the variables

L' M'
to be I/, M', ... , instead of — , -yj, ... . In virtue of relations (108) to(llO), only three

Ct 0

contributions need have been calculated, but I have calculated five out of the six quite
independently, so as to have two independent checks on the computations.
For the contribution to cn, I find

4cn = — I -7 — | — i

Jo A \A4 A3

-L'/J 27I 3 I ]+w(-> T 3 -

•!• ^ " A A A A ; b A A A A A 1 ~, i A A A T 1U ~T: i \ \ Tt ~TS J

/ q o \ / 1

4. "W'/JiT _T )-L/M__T T

1 8c" AAC 8r2 IAACC/ V46V AAC 4? IAABC

-^-1 -i-T X T «'/ T T 5 T

2 2 1AAA . 2 "-AAAC 2 ^AAAC / "*" ' M ^73 JAAIi 77"-> ^AAAB

B tt 4c6 C &1 U 4';

The other five contributions can be similarly written down. It then can be verified
algebraically that the three relations (108) to (l 10) are satisfied ; this provides a check
on the method and the algebra, but it hardly seems worth exhibiting it here, as the
numerical checks to be given later provide simultaneously checks on the method, the
algebra, and the computations. By independent computations I find for the contribu-
tions to cn, CM, Cgg, c,2, cl3, the following :

4cH =

+ 0'01515472/'-0-0125112W-0'OOG581149///
4c,2 = 0'00044G082l/ + 0'1490178M/ + 0'27801GlN'

-0'390933lZ' + 0'00681885m'-0'00912350/i'
4c33 =

4c12 = -0-003687295L/-0-14G5540M7

4c13 = -0'004468110L' + 0>0698178M'-0'4453435N/

-0'026639GO// + 0'11100470m'-0'02290710?i'.
From these

4(6c11 + c12 + cl3) = -0<000000007L'-0'0000001M' + 0'OOOOOOON/
-0'0000003^-0'0000004m' +

I have not calculated c23 independently, but a check is provided by the comparison of
the values of c23 deduced from the above figures by the use of equations (109^ and (l 10)
VOL. ccxv. — A. K

66 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

separately. For these values I find the following, the upper value being that derived
from equation (109) :

4cw= O'OOIOIOSO , 0747552 1'86947
0-00101078 ~0747550 1'8694G

2-40989 0-004975-3 0'0076531
+ /'— m— n.

2-40992 0-0049754 0'0076515

Contribution from terms in a2, aft, ..., &c.

31. For the computation of these contributions it is convenient to .take the standard
set of value obtained in § 24, namely

a' = 4 = -l, /3' = 75 = G'3081810, y' = \ = 0'1604294,

* v '/ U

K =.- -i (;5a' + /3' + y') = 0-506278.
These give the values

a = -3-556343, ft = Q'2046890, y = 0'06791892,

a" = 12-647573, /32 = 0'041897G, y2 = 0'004612981,

aft = -07279442, ay = -0'2415430, fty = 0'01390225.

Then, as regards the terms in a2, aft, ..., &c., including K, we find (cf. equations
(125) and (126))

a~\a2

5/c2

3ay /3y K K

K K

giving on substitution of numerical values, as regards terms in a2, a/3, ... only

K! = -19-91885, K2 = 0-1102214, K3 = 0'04221664,-
K4 = 0-3603665, K8 = 3'556845.

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. C,7

The values of P!, P2, P3) P4 originating from these terms are

, o3 /^ , 3 y2 , o3 «/3 , 3. «y , ii /3yl
'i «34«243

-4- — J4-Kl

B2 1 A

p - „ r^ + JL -Z- UK ~M ir> — op °y , ')cco L oc*y py

Us AB AC.y 4 4 V " A3 2AB2 2 AC2 A2B A^'C ABC

a ., K, ..

C

3 :i

"

1 have evaluated independently the three contributions to dl} <L,, ds and find for
them

-(26ia2JAAAA + 2i/82JAA

, JAA + ^K2 JAB + £K8 JAC + iK5 JA),

and there are corresponding values for the contributions to 4(7;, and 4'?.. The
numerical values found by independent computations are

4^ = 0-032398, 4da = -O'Ol 1175, 4rZ, = -0'021223.

As a check on the computations, the sum of these contributions ought to be zero ;

we find

= O'OOOOOO.

The six contributions to cu, c,2, ... have also all been calculated separately. The
values obtained are'

4cn = 0-0722711, 4c22 = 0'0271162, 4cffl == 0'0274693,
4c12 = -G'215756, 4cI3 = -Q-217869, 4c23 - 0'053056,

as regards the checks which ought to be satisfied by these values, I find

6cu + cia+c13 = 0'000002, 6c22+c12+c33 = -O'OOOOOS, 6033+^+^= +0'000003.

K 2

68 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Formation and Solution of the Equation.

32. We proceed to form and solve the final equations. It is convenient to deal
with the equations in cu, c12 ... first.

By comparison of coefficients in equation (123), we obtain

L = L'+^-lOm = L' +50-01206. (144)

a

= M'+ 0-01178109 .... (145)

a

2

a2

a

= N'+ 0-001297113 .... (146)
= V + Q'003909145 .... (147)

m = m' + ^f + '^ -5Ky = m'- 0*3538936 ..... (148)

f.t ('

5KjQ =ri- T0060520 ..... (149)
o

Ka = p' - 2-800420 ..... (150)

ct
^ = q' + G'02913935 .... (151)

a

2

= /•' + Q'009668880 . . . . (152)
= s' + 0-07207330 .... (153)

These coefficients can he checked hy comparing the sums of the coefficients in the
two values of «' (equation (123)) ; i.e., taking £2 = ,,2 = £2 = 1. For the difference of
the sums which, ought to vanish, I find 41'8G192-4r86191 = O'OOOOl. Using the
values just obtained,

- L = 0'002585G75L/ + 0-1293149,

8

' + 0-02503914,

-8 N = 12-87541N' + 0-01670087.

Collecting the various contributions to cu, c22, c33, equations (102) to (104) now
become

0'002585675L' + 0-1293149

= 0-001359233L' + 0-01278935M' + 0'04066156N/

+ 0-01515472Z/-0-01251121m'-0>006581149w'+0-0722711,

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 69

2-I25367M' + Q'02503914

12'8754lN' + 0'01670087

while equations (115) to (117) give

L' = -23-52190w'-9-55G72ji'-32-0732G,

M' = -- 0'8204313Z'-0-01162649n'- 0-00329142,

N' = - 0'1354301//-0'00472373m'-0'000154830.
The solution of this set of six equation is

L' = -6172711, M' = -0-01761613, N' = -0'002105706,

V = -- 0-006053622, m' = +0'5865530, n' = + TG59250 . . (154)

No check is needed on the accuracy of these values beyond the fact that they
satisfy the equations. On substituting directly into the equations, I find that they
are all satisfied accurately to the last place of decimals.

The corresponding values of L, M, N, /, m, //- (see opposite, page) are

L= -1171505. M= -G'00583504, N = -0-000808592,

1=-- 0-002144477, m= 0'2326594, n- 0'653I98,. . (155)

and the values of cn, r1L,, ..., calculated directly from equations (102) to (107) are
4en = -0-03029132, 4c,, = -()'() L2401GO, 4c:w = -O'Ol .0410956,
2c2;i = -0-01121809, 2c:u = 0'04245102, 2c,8 = 0'0484230L. (156)

These ought to satisfy the checks afforded by equations (108) to (I 10). In point
of fact, I find

6cn + cia + ci:! = +0-00000007, Gc^ + c^ + c^ = +0'00000012,

'Gc3a+cia+cK = +O-QOOOOOOG.

33. The first use which must be made of these numbers is to determine the
contributions to dlt d2, d3, evaluated on p. 64. We have by separate computation.

L' M' N'

G'085375 ^L-o-073783^—0-089893—.-... =-0'3412367,
«4 l>1 c

-0-041149^+.. =+0-1672652,

a

-0'044227^+... -+0-1739738,

70 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Hence collecting all contributions to 4cZ1( 4d2, 4d3, we obtain as their total values

4d, = 0-303186 ^'-0141999 2^-0161187 -2-0'308839,
a2 b* c'

4(7 = _0141999£ + 0'533622 f-2-0'391623 -2 + 0156090,
a 6 <r

4d, = -0161187 £U-0'391623 ^ + 0'552810 -,+ 0152751.

a* l> c'

From the values already obtained for p, q, r, we can transform equations (112) to
(114) into the following : —

-in" = e(£- + T-+^i} =0'1163013£' + 0-6227300 f2 + 0'9769708 ~0'
\ /v4 7i4 f>* / n. It r.

_ , -I l_l J-J-U^^>J-«-»-'

i4 c4/ a

r

.-3L\ =0-2326027^-1-2454600 ^-0'
a4 6V a b

r r'

4cZ3 ='20-t =1-9539416^+0'04462528.

c c

On substituting the values of 4(Z1; 4cZ2, and 4tZ3, these reduce to

' ' r'

0-212582 ^ + 0'569839 f, + 0'230436 -. = 0'227126,
a o" c

0'161187 ^ + 0'391623 ^+1-401132-.. = 0108126,
a2 b2 c3

0-116301 £, + 0-622730 15 + 0-976971 -2 = 0'0419475-4/j,".
a o c

The solution of these equations is found to be

£ = l-428257 + 32'04689w" ......... (157)

a

p = -0111620-11797935n" ........ (158)

-2 = -0-0559390-0-3891163/i" ....... (159)

C

and this solution has been verified by direct insertion into all the equations.

34. This completes the solution of all the equations, and the determination of all
the coefficients except s'.

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 71

On collecting the values of Kj, K2, K3, K4, K5, from pp. 62, 63, and 66, and inserting
the numerical values already obtained, we find

y±'+^'+m'\ -19-91885 = 28-26823,
aa b2 c2/

K2 = _(™+S£L+I) +0-1102214 = -0-2624723,
\a b cv

K3 =_(5L'+!l+2£L)+Oi04221664 = -0'0986790,
\ct o c /

K4 = -2.-2__^l +0-3603665 = -0'900332-19-85984w",

0^ 0 C

7-2 +3-556845 = -2770998.
a be

To evaluate s' we have to examine the form assumed by the potential at infinity.
The additional terms in the potential, as far as terms in - , produced by the distortion
are readily found to be

and if Sm is the additional mass produced by the distortion, this must be identical

with — . Hence, if s' is determined by the condition of constancy of mass, we have
r

s' = •§K4-12nK5 = -0-230755-13-239893 n"
giving (cf. equation (153)),

s = -0-1586814-13-239893 n".

Discussion of the Figure.

35. The boundary of the pear-shaped figure, as far as the second order of small
quantities has been found to be

a2

a* My4 Nz* 2foV 2mzV 2na?y> 2px2 2qi? 2rz2 \
K — "" T5 — I -- r-+ 7. , — I -- r^~ "I -- TTT- + * , + i" H -- 7- + S =
\ a8 68 c8 Z>4c4 cV a464 a4 i4 c4 /

In this equation all the coefficients have been determined ; the coefficients p, q, r and
have been found to involve n",
the remainder are pure numbers:

2

s have been found to involve n", defined in S 25 by the equation -^- = n + e2n", while

2-jrp

72 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

Let us put

P =

and similarly for q, r, and s, and let us put e2n" = £ ; then the equation may be put in
the form

2n.iV 2^ 2g,)2/2 2rnz2

"IMF' ~^r v ~

For any values whatever of ^ and f, provided only that they are sufficiently small,
equation (id) will give a figure of equilibrium. If we put e = 0, but retain £ the
equation becomes

which is an ellipsoid of semi-axes «,', //, c', given by

or, numerically,

f^ = 1 + 12'71347£ TV = 1-9-20894& ^ = 1-3'50453£
fi~ l> v

It is at once clear that as f varies this ellipsoid coincides with the various Jacobian
ellipsoids near to the standard ellipsoid.

If we put f = 0 but retain e in equation (1G1) we obtain equation (160) with
n" = 0 ; i.e. we obtain a series of figures of equilibrium all having the same angular
velocity as the standard Jacobian ellipsoid from which they are derived.

The two series of' figures obtained by putting e = 0 and f = 0 in equation (161)
may be represented by the two intersecting straight lines POP', QOQ' in fig. 1, the
point of bifurcation being of course represented by the point O. The general figure oi
equilibrium represented in equation (161) is, however, arrived at by assigning values
to both e and f, these values being limited by the condition only that e and f shall be
so small that e3 and f "'•- shall be negligible. Thus the figures of equilibrium given by
equation (161 ) are represented by all the points inside a certain rectangle ABCD
surrounding the point O. They do not fall into two linear series, as it seems to be
tacitly assumed by DARWIN and POINCARE that they will do.

36. The circumstance that the two linear series lose their identity and become merged
indistinguishably into a general area seems to be predicted as a direct consequence of

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES.

73

POINCARE'S general analysis, coupled with the linearity of the equations (V3W = —
&c.) which lead to figures of equilibrium. For the vanishing of the Hessian (A = 0 in
the notation of POINCARE *) which expresses the condition that a point of bifurcation
should exist, expresses also the condition that the two linear series should be merged
with an area of linear series as they approach the point of bifurcation, t

There is, of course, no question that in the neighbourhood of the point O two linear
series do actually exist, such as may be represented by the lines POP', ROE,' in
fig. 1 ; this is abundantly proved by POINCABE'S general argument. \ What is now
maintained is that an expansion as far as e2 only, does not suffice to reveal the

Jacobian Ellipsoids
(unstable)

Jacobian Ellipsoids
(stable)

P

Fig. 1.

directions in which these linear series start out from the point of bifurcation. So
long as our vision is limited to the interior of the rectangle ABCD in fig. 1, we can
know nothing of the direction in which the line OR starts out from 0. And the
whole difficulty is merely one introduced by the artificial method of expansion in
powers of a parameter ; as soon as this artificial method is abandoned the rectangle
ABCD shrinks to an infinitesimal size, and the curves POP' and ROR' become merely
two lines intersecting in the point O without any complications. An exactly

* " Sur Pequilibre d'une masse fluide anime'e d'un mouvement de rotation," ' Acta Math.,' VII., p. 259.
t Cf. footnote to p. 74.
I Loc. tit., § 2.

VOL. CCXV. — A. L

74 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

analogous situation arises in considering the directions in which lines of force start out
from a point of equilibrium in an electrostatic field.*

37. A more interesting illustration of the difficulty will be found in an investigation
of the figures of equilibrium of rotating liquid cylinders which I published in 1902.1
In this paper the equation of the cross-section of a figure of equilibrium corresponding
to a rotation w is supposed to be expressed in the formj

(162)
'&irpl

where £„ f,, £„ £. ... are functions of >j ; g, »; are complex co-ordinates given by
£ = x + iy, >i = x—iy, and x, y are ordinary Cartesian co-ordinates measured from the
axis of rotation. The quantity 0 is a parameter, analogous to the e of the present
paper, measuring distance from the point of bifurcation at which the pear-shaped figure

2

and the elliptic cylinder coalesce. At this point of bifurcation, 1 — ^— = f , so that
£ = £ of which the value is shown to be

* If V is the potential of ,in electrostatic field, the equation of a line of force will be

a a a

T

where /, m, n are direction-cosines. Two lines of force will meet in a point of equilibrium (just as two
linear series meet in a point of bifurcation), and the condition for this is

BV av av

= 5- = 0 ............. (n.)

Cx cy oz

Let ;>,•(,, //„, :„ l)e a point of equilibrium satisfying (ii.), then, if e is a small quantity of the first order, the
point

»ii + Ac, ;//„ + [M, .r0 + ve

will, as e varies from zero upwards, trace out a line passing through :»•„, i/0, % The condition that this
shall be a line of force is, as far as first powers of r,

cV oV 8V

and this is satisfied (analytically) because of equations (ii.) for all values of A, /j,, v. Thus, as far as first
powers of c, there are as many lines of force through the point of equilibrium as there are values of
A, p, v ; an infinite number. But on going as far as «-, it becomes clear that there are only t\vo true lines
of force through this point. The condition that a point of equilibrium shall exist is also the condition
that, if the approximations are not carried far enough, there shall be the confusion of an infinite number
of lines appearing to satisfy the condition for a line of force, and the analogous statement is true for
points of bifurcation and linear series of figures of equilibrium.

t " On the Equilibrium of Rotating Liquid Cylinders," ' Phil. Trans.,' A, 200, p. 67.

\ Loc. cit., equation (71), p. 86

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 75

The calculation of £ presents no difficulty, and equation (162) as far as £ only is
shown to be the equation of a pear-shaped figure. On calculating £2 its value is found
to be of the form (cf. § 22 of the " Cylinders " paper),

180, 2825 4375 _„ , , , /48 } 1984^ |

where S.2 is analogous to the n" of the present paper ; to be exact the rotation for any
value of 0 is supposed given by

2

ZTTP ~

Again, then, as far as O2 there is a doubly-infinite series of figures of equilibrium, not
two singly-infinite series.

In this earlier and simpler investigation, it was an easy matter to carry the
computations to the third, fourth, and fifth orders of small quantities. It was found
that the equations giving £, for a figure of equilibrium could not be satisfied so long
as S2 was kept indeterminate, they could only be satisfied for one special value of S2,

namely <J2 = — . After having determined the value of <^ in this way, it was

448

possible to investigate the stability, and the pear-shaped cylinder was found in point
of fact to be stable. What is important in connection with the present paper is that
it was not possible to determine the stability of the pear until after its figure had been
determined to the third order of small quantities.*

38. The work of PoiNCARE can hardly be compared in detail with the investiga-
tion of the present paper, because POINCARE tacitly assumes the whole point at
issue ; namely, that it is possible to determine the beginning of the pear-shaped series
from an investigation of figures of equilibrium going only as far as second-order
terms. The work of DARWIN admits of detailed comparison, because DARWIN'S
work claims actually to have effected the determination which I am compelled to
believe, after my investigation, to be impossible.

It will be clear that any extra condition in addition to the conditions that the
figure should be one of equilibrium will provide an additional equation which will
reduce the doubly-infinite series inside the rectangle ABCD down to a singly-infinite
series represented by a straight line. For instance, the condition that the angular
velocity shall remain constant would reduce the rectangle to the straight line QOQ' ;
the condition that the angular momentum should remain constant would reduce it to

* The previous investigation on cylinders and the present one on three-dimensional bodies follow
widely different methods ; the present paper is in no sense a translation of the former from two into three
dimensions. The two papers were written at an interval of twelve years, and I had hardly referred to the
former paper in writing the present one until after I had encountered the difficulty of not being able to
determine the stability from the second-order figure ; I then discovered that precisely the same situation
had arisen in my former investigation.

L 2

76 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND

some curve at present undetermined. Hence, if the present paper is sound, we
should anticipate that DARWIN obtained a linear series instead of a rectangle of
configurations by the introduction of some adventitious condition not necessary to
equilibrium.

39. DARWIN supposes his pear defined as far as the second order by the equation

T = c-eS8-2/' S/, ......... (163)

i

where T gives a measure of the surface displacement, e is a parameter analogous to
our e (although not numerically the same), and the quantities f* are coefficients which
must vary as we pass along the linear series, but are constants, as is also e, for any
single figure of equilibrium. The quantities Ss, S," are ellipsoidal harmonics. DARWIN
supposes K to be a small quantity of the first order, and the coefficients f' to be small
quantities of the second order. The energy E of the distorted ellipsoid will differ
from that of the original ellipsoid from which the displacement r is measured by a
small quantity <5E which will involve e, f? and <V, where S<a2 is the increase in the
value of <a2 for the distorted figure.

The first order displacement of the ellipsoid will be represented by the first terms

T = c-t'Ss, .......... (164)

of equation (163), and this is supposed to be a possible figure of equilibrium with
Sta2 = 0. The corresponding value of <?E will be of the form

SE = ae2, .......... (165)

and the coefficient a must therefore vanish if the original ellipsoid was one
corresponding to a point of bifurcation.

If we do not at present assume that a = 0, the value of <5E arising from the second-
order displacement (163) will be of the form

, . . (166)

in which a single term in / has been taken as being typical of all the terms in the
coefficients//. The condition that the displacement (163) together with an increase
$<o" in to3 shall give rise to a figure of equilibrium is that E shall be stationary for all
variations of e and / ; it is expressed by the equations

|(,SE)=;0, |.(<SE) = 0, (/ = /;, &c.) ..... (167)

Expression (166) is the same inform as that given by DARWIN ('Coll. Works/
vol. 4, p. 349), except that he omits the term ae2 from c$E ; and equations (167) are

THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 77

the same in form a$ those from which DARWIN obtains the conditions of equilibrium.

a
The equation ^- (<5E) = 0, written out in full, becomes

0 ......... (168)

If second-order terms are neglected equation (168) reduces to

a = 0, ........... (169)

and if a is taken equal to zero, equation (168) reduces to

0 .......... (170)

DARWIN'S method is in effect to replace equation (168) by equations (169) and (170).
This, in my opinion, introduces one limitation too many on the values of the variables,
and so reduces the doubly-infinite series of figures of the second order to two singly-
infinite series. Equation (169) must undoubtedly be true if second-order terms are
neglected, but we may take

....... (L7l)

where X is a quantity of the second order entirely at our disposal, and still satisfy all
the conditions necessary for equilibrium. I think it will be found that DARWIN'S
procedure in effect introduces the spurious condition X = 0. DARWIN'S equations are
of course sufficient to ensure equilibrium ; what is maintained is that they are not
necessary and so do not disclose all possible figures of equilibrium.

In this way I believe it will be found that DARWIN has limited himself to one linear
series of figures (X = 0) instead of the doubly-infinite series represented inside the
rectangle ABCD in fig. 1. 'If this series should happen to run on continuously at
the edge of the rectangle with the true series ROE/ in fig. 1 , then of course DARWIN'S
investigation would stand. But no reason suggests itself, and certainly DARWIN
(not foreseeing the complication of the doubly-infinite series) gives no reason, why this
should be the case. For some value of X the two series will run on continuously, but
there is no assignable reason why this value should be X = 0.

What, then, will happen if we try to carry DARWIN'S approximation on to third-
order terms fjy his method 1 I think it will be found that his series comes to a dead
end before the third-order terms are arrived at, precisely as if it ran on to the
boundary of the rectangle ABCD in fig. 1, and could get no further. If the displace-
ment goes as far as third-order terms, SE must go as far as sixth-order terms, and
will contain terms of orders 2, 4, and 6, say

78 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, ETC.

The equations necessary for equilibrium are

0) = 0, ifaEa+JE.+JE.) = 0, (/ = /;, &c.), (172)

oe
while the equations provided by DARWIN'S methods would be

= 0, i(,5E6) = 0,

and it will readily be seen that there are more equations than can be satisfied by the
variables at our disposal. The conclusions of this section are put forward with the
utmost diffidence, but to the present writer they seem inevitable.

40. Assuming that no glaring error has been made, the present investigation seems
to indicate that so far the knowledge we have as to the stability of the pear-shaped
figure amounts to absolutely nothing. The required knowledge can only be obtained
by carrying the figure of the pear to a still higher approximation. In the parallel
investigation on cylinders it was found that the stability could be examined as soon
as the figure was determined as far as third-order terms, and doubtless the same will
prove to be the case in the present problem. Fortunately the method of the present
paper is one which lends itself to indefinite extension, limited only by labour of
computation, so that it ought to be possible to proceed to third-order terms and
determine the stability of the pear, and if the pear then proves to be stable, to pro-
ceed to higher orders and so examine the series of pear-shaped forms. In the previous
investigation on cylindrical figures it was found that an expansion as far as fifth-order
terms gave a good approximation to the pear-shaped figure up to a stage where it
was obviously just about to divide into two separate masses. It is only in the hope
that I shall be able to carry the present investigation further, that I have ventured
to put forward the present somewhat lengthy piece of work which has so far led
only to such negative and disappointing conclusions.

III. The Influence of Molecular Constitution and Temperature on Magnetic

Susceptibility.

Part ITI. On the Molecular Field in Diamagnetic Substances.

By A. E. OXLEY, M.A., M.Sc., Coittts Trotter Student, Trinity College, Cambridge,

Mackinnon Student of the Royal Society.

Communicated by Prof. Sir J. J. THOMSON, O.M., F.R.S.
Received June 24, — Bead June 25, — Revised December 2, 1914.

CONTENTS.

Page

(1) Introduction 79

(2) On the local molecular polarization and the local molecular forces within diamagnetic fluid

and crystalline media . 80

(3) The mean and local molecular fields of a diamagnetic crystalline substance 81

(4) On the magnitude of the local molecular field 84

(5) On the stresses and energy associated with the molecular field 90

(6) Additional experiments 96

(7) A relation between the magnetic double refraction of organic liquids and the change of

magnetic susceptibility due to crystallization 97

(8) On the nature of the molecular field 100

(l) INTRODUCTION.

IN the present communication an attempt is made to examine to what extent the
crystalline state of diamagnetic substances (which form a very large proportion of
the whole number of substances known to us) involves mutual actions between the
constituent molecules and to obtain a measure of the forces which hold the molecules
together in a definite space lattice characteristic of the substance. The work is thus
a continuation of that published in 'Roy. Soc. Phil. Trans.,' vol. 214, pp. 109-146,
1914, wherein it was shown how from determinations dealing with the change of
diamagnetic property on crystallization information could be derived concerning the
inner structure of crystalline media and the intensity of the interacting molecular
forces.

In the paper cited (p. 143) it was suggested that the local molecular field within
diamagnetic crystalline media may be comparable with the ferro-magnetic molecular

(525.) [Published February 16, 1915.

80 ME. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

field. In the light of the experimental results on aromatic substances there described
and of those given by DU Bois, HONDA and OWEN, which are concerned with
elementary substances, this suggestion has been found to lead to interesting results.
It is clear that we can easily test whether such an intense local molecular field exists ;
for if so it must be of import in connection with the transition from the liquid to the
crystalline state in general, and the changes of other physical properties which
accompany this change of state must also depend directly or indirectly upon the
molecular field. The object of the present communication is to test the experimental
and theoretical work which has already been completed by applying the results
obtained to a wider range of physical phenomena. Evidence of a change of specific
diamagnetic susceptibility owing to crystallization has been obtained with about forty
substances, but with some substances the change, if it exists, is exceedingly small.
Ten additional substances have been investigated, and the results, in conjunction with
the other physical properties, are used to extend the ideas concerning the molecular
field to diamagnetic crystalline media in general.

PART III.

(2) ON THE LOCAL MOLECULAR POLARIZATION AND THE LOCAL MOLECULAR
FORCES WITHIN DIAMAGNETIC FLUID AND CRYSTALLINE MEDIA.

The passage from a purely mechanical theory of the properties of material media to
the actual discrete molecular theory, as interpreted electrodynamically, involves an
examination of the local polarizations which act in the immediate neighbourhood of
the point considered within the medium. It has been shown by Sir JOSEPH LARMOR*
that the forces exerted at such a point by the surrounding polarized medium can be
separated into two parts, a purely local part and a part due to the rest of the
medium. If we neglect the forces due to the former then those due to the latter are
derivable from a potential due to the combined volume and surface density
distributions of POISSON. With fluid media we can go further and determine, at least
approximately, the forces due to the purely local part. This is attained by imagining
a cavity scooped out round the point and determining the force due to the surface
distribution of polarization over its walls. The removed molecules are now put back
in the cavity and the additional force which they contribute is found by applying the
method of averages. The molecules of a fluid are in rapid motion and the force
at a point in the interior of the cavity due to these replaced molecules averages out
to a small corrective polarization. It can be shown that the force due to the
immediately neighbouring molecules is represented by (i+s) P where P is the

* 'Roy. Soc. Phil. Trans.,' A, vol. 190, p. 233. 1897; and LORENTZ, ' Theory of Electrons,' pp. 137
and 303.

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY. 81

polarization due to the applied field and s is the coefficient of the corrective
polarization — the effect of the replaced molecules.

Within a crystalline structure, however, this method of averages applied to the
molecules within the cavity is not valid, for each molecule is orientated so as to occupy
some definite position with respect to its neighbours. In such a case the local force,
due to a molecule which is close to the point under consideration, is uucompensated,
and we have to resort to indirect deduction from experimental facts to obtain a
measure of the polarization effect due to the molecules immediately surrounding the
point. It is to this uncompensated effect that we have attributed the rigidity of the
crystalline medium.* In order to avoid the complexity introduced by the elasticity
of crystalline media LARMOR confined his investigations to fluids ; but, as we are
attempting to ascribe to these very elastic forces a nature which permits of their
interpretation in terms of a local molecular field binding the molecules together, the
treatment for crystalline substances is the same in kind as that for fluid substances.

The rigidity of gels at low temperatures is of a different nature and is probably
due to an interlocking of the molecules (arranged at random) whose thermal agitation
is sufficiently reduced. This is in accordance with the work of TAM.MANN! who found
that at very low temperatures the power and velocity of crystallization were very
small.

(3) THE MEAN AND LOCAL MOLECULAR FIELDS OF A DIAMAGNETIC

CRYSTALLINE SUBSTANCE.

In a former paperj the author has shown that the change of specific susceptibility,
which accompanies the transition from the liquid to the crystalline state for many
substances, can be satisfactorily interpreted in terms of a mean molecular field which
operates to an appreciable extent only in the crystalline state. The nature of this
molecular field need not be definitely specified, and it will suffice for our purpose to
regard it as a magnetic field limited to such a magnitude as to produce in the
molecules of the substance a distortion, or polarization, such as is equivalent to that
actually produced by the mutual forces of the molecules of the crystalline structure. §

In Part II. of the work referred to, the mean molecular field was represented
magnetically by the term ac . AH, where ac is a coefficient (the constant of the

* 'Roy. Soc. Phil. Trans.,' A, vol. 214, p. 143, 1914.

t WHETHAM, ' Theory of Solution,' p. 44, and references to TAMMANN'S work there given.

I 'Roy. Soc. Phil. Trans.,' A, vol. 214, p. 109, 1914. This paper contains Parts I. and II. of the present
work, and, for brevity, reference is made to these in what follows.

§ A force of electrostatic nature will modify the electron orbits, and therefore the susceptibility, an
effect which we may represent magnetically. If the revolving electrons are controlled by molecular
magnets (as on RITZ'S theory), a magnetic field would be the true interpretation of the molecular field.
Probably both types of molecular field co-exist (vide infra, p. 22, for a discussion of the nature of the
molecular field).

VOL. CCXV. — A. M

82 ME. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

molecular field) defining the extent of the polarization of the molecules in the
crystalline state and depending upon the definite space lattice which the molecules

assume. =— '-, where H is the applied field, defines the distortion produced in an
H

electron orbit by the forces exerted by surrounding molecules when the substance is
in the liquid state. AH is the polarization (proportional to the applied field H) due
to this distortion. As we do not know the actual disposition or the proximity of the
molecules in the structure, the difficulty in attempting to calculate ae is insuperable
and we are compelled to try the more indirect method afforded by experiment. A
rough estimate of the value of ac has been obtained from CHAUDIEE'S observations on
the change of magnetic rotatory power when aniline, benzene and nitrobenzene
crystallize, the order of magnitude deduced being at least 102.* The corresponding
value of a, for the liquid state, deduced from theoretical considerations, is of the order
?{.t Thus in the liquid state the mean molecular field is represented by ^ . AH. The
distortion of the molecules in the liquid state of the substance is therefore insignificant
compared with that for the crystalline state and, neglecting at in comparison with ae,

it was shown that

........ (1)

where H is the intensity of the applied magnetic field, Xc and xi are the specific
susceptibilities of the crystalline and liquid states respectively. Thus we may ignore
the mutual influences of the molecules of the crystalline state providing we supply a
mean molecular field whose magnetic equivalent is ae . AH. Equation (l) shows that
the actual change of x on crystallization (a few per cent, with aromatic compounds)
demands that AH/H shall be of the order 5 . 10~4 ; which implies a small and
reasonable distortion of the molecules of a liquid due to the surrounding molecules.
It should be noted that this is a maximum value of the distortion for the liquid state.
The value will be smaller if, as is probable, ac is of a higher order than 102. At
present no physical interpretation has been put upon the large value of ac, but we see
that it is representative of the local part of the forcive in the crystalline state just as
the smaller constant at is representative of the local part of the forcive in the liquid
state.

On the theory of magnetism developed by LANGEVIN, a diamagnetic molecule
contains oppositely spinning systems of electrons which counterbalance one another
externally so that the molecule possesses no initial magnetic moment 4 When an
external magnetic field is applied the period of one system of electrons is lengthened,
that of the other system is shortened, and the molecule becomes slightly polarized or
distorted. This small differential effect, which is compatible with the Lorentz

* Part II., p. 141.

t LARMOE, 'Roy. Soc. Phil. Trans.,' A, vol. 190, p. 233, 1897.

J This null initial magnetic moment corresponds to the null electric moment possessed by a molecule of
a dielectric which is not subjected to an external electrostatic field.

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY. 83

explanation of the Zeeman effect, is the origin of the small negative moment possessed
by diamagnetic substances when subjected to a magnetic field. The smallness of the
diamagnetic susceptibility in general is due to the self-compensation of the forces
(produced by the oppositely rotating systems of electrons or any other magnetic
system in the molecule) which takes place at points whose distances from the molecules
are large in comparison with molecular dimensions. But quite close up to the
molecule these forces will not compensate and the local force will be comparable with
that due to an unbalanced revolving electron of other magnetic element of the
molecule. It is with this local field that the present communication is mainly
concerned. I shall call it the local molecular field of the diamagnetic substance. In
the crystalline state we may regard the space lattice as defined by the local forces
exerted between a system of electrons in one molecule upon another system in an
adjacent molecule and so on throughout the crystalline structure.

As was shown in Part II., p. 142, the mean molecular field is proportional to the
intensity of magnetization of the substance, and we may write

(2)

where aj is the new constant of the molecular field, of the order 105, N the number of
molecules per gram., AM the diamagnetic moment induced in a molecule by applying
an external magnetic field H, and p is the density of the substance. The local mole-
cular field will have the same constant of proportionality but we must replace the

Fig. 1.

diamagnetic moment AM of the molecule as a whole by the local moment associated
with a pair of molecules. We have no means of measuring directly the value of this
latter quantity, but it is probably very large in comparison with AM, and it was
suggested at the end of Part II. that it may be so large that the local molecular field
is comparable with the molecular field in ferro-magnetic substances.* On account of
bhe null initial moment which has been prescribed for the diamagnetic molecule the
local molecular field will be of an alternating character, the distance over which it is
unidirectional being comparable with the distance between the molecules, as the
diagram above shows.

* It should be noted that this local molecular field will exist in all crystalline substances, whether they
arc subjected to an external magnetic field or not. On the other hand, the mean molecular field is a
differential effect and depends on AM, the diamagnetic moment which is induced by the external field.

M 2

84 MR. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

The small value of at possessed by fluid substances implies a mean molecular field
much smaller than that associated with a crystalline structure. But how much
larger the mean and local molecular fields of the crystalline state are compared with
those of the liquid state has not been ascertained at present. All that has been
shown in Parts I. and II. is that the minimum value of the mean molecular
field in some diamagnetic crystalline media is large compared with that in
liquids. We shall now proceed to obtain an estimate of the intensity of the local
molecular field in crystalline media from the small variation of x which occurs on
crystallization.

(4) ON THE MAGNITUDE OF THE LOCAL MOLECULAR FIELD.

If we apply a strong magnetic field to a diamagnetic liquid whose molecules possess
dissymmetry, LANGEVIN* has shown that the liquid becomes doubly refracting and
that a very small proportion of the unsymmetrical molecules suffer an orientation
under the action of the field. The amount of orientation which can be produced by
the largest magnetic field obtainable is not sufficient to affect the value of the suscepti-
bility appreciably.! When, however, the liquid crystallizes and all the molecules are
similarly orientated, we should expect that the susceptibility will be different in different
directions if the molecules are not symmetrical. But this cannot be the explanation
of the change of x observed in the experiments of Part I. (and in those carried out by
HONDA and OWEN). For in these experiments the crystals were small and their axes
would be distributed at random. In some, cases the crystals were observed to grow in
such a way. The explanation here put forward, and which lias already been advanced
in Parts I. and II., is that each molecule of the Substance is distorted when it forms part
of a crystalline structure by the local forces due to the neighbouring molecules. This
mutual action between the molecules is necessary to account for the rigidity of the
crystalline structure and for the existence of planes of cleavage (see p. 92, infra).
We can therefore find the intensity of the local molecular field by making it of such
magnitude as to produce a change of susceptibility of the same order of magnitude as
that actually observed in the crystallization experiments.

We may write}

AM Hre

M 47TOT

= -10-9H (3)

where AM is the change of moment produced in an electron orbit of moment M by
applying a field H, T is the period of the electron, approximately 10~15 sec., and e/m
the ratio of the charge to the mass of an electron, 17 x 107. The largest field which

* 'Le Radium,' vol. 7, p. 249, 1910.

t Loc. ctt., p. 252.

t LANGEVIN, 'Ann. de Chim. et de Phys.,' s4r. VIII., vol. 5, p. 96, 1905.

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY. 85

can be produced in the laboratory is of the order 105 gauss, and therefore the largest
moment which we can induce is given by

| AM] < 10-4.M.

This induced moment is sufficiently large to account for the diamagnetic suscepti-
bility of the substance.

From equation (3) we have

M,

(5)

ercH

where the subscripts I and c refer to the liquid and crystalline states respectively.

In passing from the liquid to the crystalline state the alteration of M, produced by
the local molecular field Hc is AM'; where

AM', 6T,HC*
M,

MC = M,±AM', ........... (7)

It should be noted that although Hc alternates as we pass from molecule to molecule
of the crystalline structure, still the sign of AM', will everywhere remain the same,
for when Hc changes sign M, also changes sign, so that every molecule of the
crystalline structure suffers a definite distortion due to the action of the local
molecular field. The ambiguity of sign in the above formulae simply implies that the
arrangement of molecules may be such that M, is increased or decreased by the
particular kind of packing which the molecules assume and corresponds to a reversal
of the sign of 3X. (See Part I., p. 133, and Part II., p. 143.)
From equations (6) and (7), we find

and therefore from (5)

M,
Hence

AMC-AM, = erJIA T er,HA ,
M, 47rm\ 47rm/ '

which gives by (4)

AMe-AM, _ rc /, -P er,HA
~AMT "

* Hj, the molecular field within the liquid, is insignificant compared with Hc and is therefore neglected
in equations (6) and (7),

86 MR. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

The electrons which give rise to diamagnetism also give rise to the Zeeman effect,
a slight alteration of their periods accounting for both phenomena.* The order of
magnitude of the local molecular field Hc will not be effected by assuming that the
electrons in a molecule have a mean natural period T,, when the substance is in the
liquid state, which becomes modified to rc = T,±($T, when crystallization sets in. The
change of period Sr indicates the order of magnitude of the displacement of an
absorption band owing to the transition from the liquid to the crystalline state.
Now

Xc = ^j. . AMC) Xl= ^ . AM,, 8X = Xc-x, = |p (AMC- AM,),

where n is the number of electrons per molecule and N the number of molecules
per gram. Therefore

c|x _ AMC-AM, r,L - «-. H \
X! AM, %A'

TI/
The change of period ST is produced by the local molecular field Hc, and therefore

Hence

^i, (12)

This equation gives us a value of the local molecular field Hc when we know the
extent to which x is modified on crystallization. With all the substances investigated
in Part I. and with many of the elements examined by HONDA and OWEN, 9x/X
amounts to a few per cent. Hence either

j ._ 63 . T;2 . H/ 1 ^_ 6 . T, . Hc

TOW — -,c 2 2 °r TOT) — — n — •'

IOTT??! 2-n-m

Taking T,=== 10~15 sec., e/m = 2 x 107, we get either

Hc = 6 x 107 gauss or Hc = 3 x 10" gauss,

and in either case H,. is of the order 107 gauss.

We have no experimental evidence at present as to how far an absorption line is
displaced when a substance passes from the liquid to the crystalline state, but such

* LANGEVIN, 'Ann. de Chim. et de Phys.,' vol. 5, p. 97, 1905.

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY. 87

evidence would be a direct test of the value of Hc.* The above deduction is an
interesting confirmation of the suggestion made at the end of Part II. , p. 143, with
regard to the intensity of the local molecular field in diamagnetic crystalline
substances.

In a crystalline substance which shows natural double refraction we may regard the
molecules as held in position in the crystalline structure by this intense local field, and
in that case the double refraction of the medium would be a consequence of the
orientation of the molecules due to the operation of this field when the substance
crystallizes. Now the magnetic double refraction induced in a liquid is proportional
to the square of the field intensity.! Let us assume that this law holds good for much
larger fields than the largest we can apply in the laboratory 4 If we take 5 x 104 gauss
as the maximum value of this external field then the magnetic double refraction of the
liquid will be represented by Cx2'5xl09, where C is a constant independent of the
applied field. If we could apply a field of 1 07 gauss the magnetic double refraction
would be C x 1011, i.e., 40,000 times as great. This is of the order of magnitude of the
ratio of the double refraction of quartz to liquid nitrobenzene, the latter being
subjected to a field of 3 x 104 gauss.

Conversely, if we assume that on crystallization there is an internal local molecular
field, which if interpreted as a magnetic field is of the order 107 gauss, we can see how
under its influence the molecules would become orientated to such an extent as to give
rise to a crystal possessing the high natural double refraction of quartz.

All elements and compounds which show on fusion a small percentage change of x
must possess a molecular field whose local value, if interpreted magnetically, is of the
order 107 gauss. This value is also supported by observations on the artificially
induced and natural double refractio:is of other liquid and crystalline media.

It is known that most uniaxal crystals have a double refraction comparable with
that of quartz, and of a much higher order of magnitude than that which has been
induced in a liquid by the largest magnetic field at our disposal. § Thus the
abnormally high double refraction of Iceland spar is about 100 times that of quartz.
On the other hand, the double refraction of ice is only about y^o of that of quartz.

* It will be shown later that the mechanical interpretation of the large local molecular field is the
internal stress which accounts for the rigidity of a crystalline medium. Now it has been shown by
HUMPHREYS (' Astrophys. Journal.,' vol. 35, p. 268) that there is a direct relation between the pressure
shift of spectral lines and the Zeeman effect. Hence it is very probable that the shift of an absorption
band on crystallization would be determined by the Zeeman effect of Hc.

t This results from both the theory of HAVELOCK, 'Roy. Soc. Proc.,' A, vol. 77, p. 170, 1906, and that
of LANGEVIN, 'Le Radium,' vol. 7, p. 251, 1910. The law is found to hold experimentally, COTTON and
MOUTON, 'Ann. de Chim. et de Phys.,' ser. VIII., vol. 19, p. 155, 1910.

| There are good reasons for supposing that this law is not accurate for such large fields, for a saturation
effect must come in. But the result of assuming that it holds is suggestive, and probably indicates the
true order of the effects.

§ Vide note, p. 99.

88 MR. A. E OXLEY ON THE INFLUENCE OF MOLECULAR

But COTTON and MOUTON have shown that the magnetic double refraction induced in
water by the largest magnetic field they were able to apply (3 . 104 gauss) is
insignificant compared with that of liquid nitrobenzene subjected to the same field.
In general the natural double refraction of a crystalline medium is of a high order
compared with the artificial double refraction which we can produce in a liquid using
a field of 3 . 104 gauss. The intrinsic molecular field, if interpreted magnetically, must
therefore be large compared with 3 . 104 gauss in all crystalline diamagnetic media.
This result will be used later in the extension of our ideas concerning the
molecular field to diamagnetic crystalline media in general.

In Part II. the change of specific susceptibility due to the crystallization was
represented by the difference, (ac—at) AH, of the mean molecular fields of the

ATT

crystalline and liquid states. -==- , where H is the applied field, defines the distortion

produced in an electron orbit by the forces exerted by surrounding molecules when
the substance is in the liquid state and subjected to a field H. AH is the polarization
(proportional to the applied field H) due to this distortion. at is the constant of the
molecular field for the liquid state and is approximately equal to -j. Now the
molecules of a liquid are moving about rapidly in all directions, and therefore
the polarization defined by AH is an average effect. In the crystalline state, on the
other hand, the molecules are orientated into definite positions with respect to one
another, and therefore the polarization due to the mutual influences of the molecules
in this case (as disclosed by applying an external field H) is large compared with AH.
Let ac . AH be this new polarization. Then ac, which is large compared with ah is a
factor defining the polarization which results from the complete orientation of the
molecules in the crystalline medium. ac will also depend upon the proximity of the
molecules.

Now reverting to equation (8) of Part II., the variation of x on crystallization may
be written

9X / x AH AH

:~ \ae~ al) • -TJ- — ac • ~TT~ >

if ac is large compared with at. The large value of ac implies a correspondingly small
value of AH/H, i.e., a small value for the molecular distortion in the liquid state in
order to account for a given change of x on crystallization.

As in § 3 we may write the mean molecular field ac . AH, which accounts for the
change 3X on crystallization, in the form a'c x (diamagnetic moment per unit volume)
= a! c . N . AM . p, and the value of a'c, the new constant of the molecular field,
is of the order 105 for those substances which show a small percentage change of x on
crystallization.

Now the local molecular field H,. is of the order 107 gauss. We may write
Hc = a'c . I where I is the aggregate of the local intensity of magnetization per unit
volume. a'n the constant of the local molecular field, will be equal to the constant of

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY. 89

the mean molecular field, for the relationship of the molecules to one another in the
crystalline structure determines their common magnitude. Hence Hc . = 107 = 108 . 1
and I = 100.

We can now use these values to form an estimate of the potential energy
associated with a diamagnetic crystalline medium in virtue of the local molecular
field and the local molecular polarization.

But before passing on to this, it will be interesting to compare the above deductions
with regard to the local molecular forcive in diamagnetic crystalline media with those
of WEISS which are concerned with the forcive in ferro-magnetic media.

Imagine a spherical cavity, large compared with molecular dimensions but lying
wholly within the same crystal of (l) a diamagnetic ; (2) a ferro-magnetic medium. On
account of the structure which has been assigned to the diamagnetic molecule, the
force at the centre of the cavity (when there is no external field) will be zero in case (l).
In (2) the force will be ?$ . I where I is the spontaneous intensity of magnetization.
Now let us move our point of observation from the centre of the cavity out towards
the surface. When the point approaches within a range comparable with molecular
dimensions, in case (2), the forcive increases, and when the point is at a distance from
the wall equal to that which separates two molecules of the crystalline structure the
forcive is represented by N I where N is the constant of W Kiss's molecular field, of the
order 104. If we move our point of observation in case (l) (the diamagnetic medium)
from the centre of the cavity to the surface, then as the point approaches to within a
distance comparable with molecular dimensions the local force is due almost entirely
to the molecule which is nearest to the point. This molecule maintains a definite
orientation with respect to the point, and when the latter is so close to the molecule as
to be almost on its surface the polarization is comparable with the saturation intensity
in iron. This is the interpretation of the large values of Hc and Nl which are each
of the order 107 gauss. As H = ac' . I, we may suppose that the large coefficients N
and aj determine the enormous magnitude of the forcives quite close to a molecule in
the respective crystalline structxires. These local forcives we could hardly hope to
calculate directly for we do not know the proximity of the molecules in the structure
or the law of force which holds at such a close range. The local molecular field of a
diamagnetic crystalline substance alternates as we pass from molecule to molecule of
the structure and is therefore localized. If we could take a crevasse between two
molecules of the structure then the induction across it would give us a measure of H,,.
Similarly in the ferro-magnetic case a crevasse of such small dimensions would give us
a measure of WEISS'S field Nl. If we take a crevasse in the ferro-magnetic medium,
which is large compared with molecular dimensions, the force in the gap is H + 47rl and
this is small compared with N I . The difference between these forces must be
attributed to the localization of the intense fields associated with the iron atom. We
then get continuity of magnetic induction while the intense field is still capable of
modifying the structure of a neighbouring molecule.

VOL. ccxv. — A. N

90 MR- A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

The crystallization of a diamagnetic substance may be regarded as accompanied by
the production of a spontaneous local intensity of magnetization. We have up to the
present regarded the molecular field as a magnetic field, and a close analogy has been
found between the phenomena shown by ferro-magnetic and diamagnetic substances.
The question as to how far we are justified in regarding the molecular fields as
magnetic fields will be discussed in § 8.

(5) ON THE STRESSES AND ENERGY ASSOCIATED WITH THE MOLECULAR FIELD.

In the previous work evidence has been given which shows that the forces
associated with a diamagnetic crystalline structure are exceedingly large, and
therefore the potential energy of the crystalline state will be considerable. If 1 1 be
the local magnetic moment which in conjunction with the local molecular field Hc |
binds one molecule to another in the crystalline structure, and if all such elementary
systems (each system consisting of the adjacent parts of a pair of molecules which
are bound together by the local forces) are independent, then the energy possessed
by 1 c.c. of the substance in virtue of a particular crystalline grouping may be
written

& ' -!He|.

Let n be the number of molecules per cubic centimetre. Then

where Hc corresponds to the molecular field in ferro-magnetism. If we put n.
and Hc = a'cl, we find for the energy associated with 1 gr. of the substance

Here a! c is the constant of the molecular field, I the aggregate of the local intensity
of magnetization per unit volume, and pc the density of the substance. This is the
amount of potential energy which the molecules contained in 1 gr. possess in virtue
of their grouping, and is additional to the energy possessed by 1 gr. of the liquid.
"We may treat the crystalline substance as a fluid whose molecules do not influence
one another, providing the energy term represented by (13) is superposed upon any
energy which 1 gr. of the fluid may possess. Therefore if the crystalline structure
be submitted to a magnetic field H, the potential energy associated with 1 gr. may
be written

.r], ......... (14)

'•Pi

where k, and Pl are the susceptibility and mass per unit volume of the liquid. Let us
now compare this with the expression given by LARMOR* for the potential energy per

* ' Roy. Soc. Proc.,' A, vol. 52, p. 63, 1892.

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY. 91

gramme of a diamagnetic liquid the molecules of which have a small mutual influence.
If kt be the susceptibility per unit volume, this energy is shown to be

(15)

where X is a constant approximately equal to iy (our «,). In accordance with our
notation we have TcL . H = n . AM; , where n is the number of molecules per cubic
centimetre, and AM the diamagnetic moment induced in each by H. Therefore,
instead of (15), we may write

] ......... (16)

Comparing (14) with (16) we may identify a'c with X and I with n. AM;. Further,
the applied field H is associated with the moment n . AM, which it produces, in the
same manner as the molecular field Hc is associated witli the aggregate of the local
intensity of magnetization per unit volume, I, which it produces. The analogy is
complete. In (14) we are concerned with the local forces of the crystalline structure
which, on account of the relative fixity of the molecules,' do not average out ; whereas
in (16) we are concerned with the average forces within the fluid, the large local forces
having become averaged out on account of the motions of the molecules round any point.

Prof. LAEMOE has pointed out an interesting case in which the term in X
predominates.* If we suspend a bunch of iron nails from the pole of a magnet, we
find that they adhere to each other endwise and repel one another sideways, while
non-adjacent nails have no action on one another. This is analogous to the result
disclosed by (14) for a diamagnetic crystalline substance. Each molecule may be
considered as possessing two little magnets opposing one another, and these molecules
fit together in such a way that the magnets nearest to one another in adjacent
molecules help one another.

In the case of a fluid X and kt2 are small, so that the predominant term in the

expression for the energy is - — . kt . H2, as is ordinarily assumed. For a crystalline

2ft

medium taken as a whole the corresponding energy term is - — . kc. H2, the expression

*Pc

usually taken in measuring kc.1[ It is only when we are inquiring into the local
forces binding the molecules together into a crystalline lattice, the energy term of

which is — . a'c . I2, that we get the true diamagnetic analogy with the bunch of

2/Jc

iron nails. The molecules of the diamagnetic structure are held together endwise, so
to speak, but we may have attractive forces in a perpendicular direction depending

* Loc. cit., p. 64.

t Strictly speaking, to each of these expressions for the energy should be added a term proportional
to £2, which is very small.

N 2

92 MR. A. E. OXLEY. ON THE INFLUENCE OF MOLECULAR

upon the configuration of the molecule. These latter forces will, in general, be
different from the former, and will give rise to a difference of cohesion which accounts
for the greater ease of cleavage of crystals in certain directions. [*Her'e, again, we
have further proof of the truth of the hypothesis of molecular distortion in crystalline
media. It is true that if to a liquid we could apply such an intense field that all its
molecules are orientated, that liquid would possess a double refraction equal to that
of a crystal, but as long as there is no mutual action between the molecules this
doubly refracting medium could show no signs of rigidity and no preference for
cleavage along certain planes, t Clearly the process at work in the formation of a
crystalline structure is that of a binding force (mutual induction in our case) between
two unbalanced parts of two adjacent molecules. All magne-crystallic properties can
readily be interpreted in terms of such a mutual effect. It is important to note,
however, that unless we recognize the enormous intensity of the local molecular field,
which, together with the large local intensity of magnetization in diamagnetic
crystals, binds the molecules together and by a mutual induction effect distorts them,
we could not account for the rigidity of the crystalline medium, or the extent of its
double refraction. It appears that TYNDALL'S explanation! of the deportment of
diamagnetic crystals when placed in a magnetic field as due to a mutual action
between 'the diamagnetic molecules is sufficient to account qualitatively for the
behaviour observed, but it is difficult to see, on TYNDALL'S view of a simple and very
minute diamagnetic polarity, where such large forces as those demanded for crystalline
media could have their origin. On our view a diamagnetic molecule as a whole
possesses a small diamagnetic polarity, an induction effect of the applied field, and the
force due to it at a point considerably removed from the molecule is small. But in
between a pair of molecules the internal forces are unbalanced, and the intensity of
the local field is comparable with that in ferro-magnetic substances. On the other
hand, quite close up to the diamagnetic molecule conceived by TYNDALL, the force
binding it to a neighbouring molecule is not intense enough to account for the
difference of magnetic property in different directions, as Lord KELVIN pointed out.
The explanation of magne-crystallic action formulated by TYNDALL (this is the theory
of reciprocal molecular induction) accounts qualitatively for the phenomena but
certainly fails from a quantitative point of view. The present conception of a
diamagnetic molecule surmounts the latter difficulty.]

As this question of determining the order of intensity of the local forces and local
polarizations within diamagnetic crystalline media is of the greatest importance, any
additional proof of the correctness of the values assigned to them is valuable. Let us

therefore try to form some estimate of the magnitude of the term — .a'e. P, which

2/>c

* [Added November 12, 1914.]

t Of. the liquid crystalline state.

I TYNDALL, 'On Diamagnetism and Magne-crystallic Action,' 1870, p. 69.

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY.

93

is large compared with the other term of equation (14). Taking the value of the
local molecular field, Hc = a'cl = 107, we find since »'„=¥ 105, that I, the aggregate of
the local intensity of magnetization per unit volume, is of the order 100. This is
comparable with the saturation intensity of ferro-magnetic substances. If pc = I,

which is the amount of potential energy associated with 1 gr. of the crystalline
medium in virtue of its molecular grouping. The thermal equivalent of this will be
of the order 109/4 . 107 = 25 gr. calories, which represents the heat energy required to
destroy the crystalline structure, i.e., the latent heat of fusion. This is of the right
order of magnitude for many diamagnetic substances — organic compounds and
elements.* The above reasoning applies only to the order of magnitude of the latent
heat. It is obvious that until we know the disposition of the molecules within the
crystalline structure the value of a'c is somewhat vague. But the experimental fact
that the latent heat of transformation of iron from the ferro-magnetic to the para-
magnetic state is of the same order of magnitude as the latent heat of fusion of many
diamagnetic crystalline substances is powerful evidence that the local forces and
local polarization which we have assigned to diamagnetic crystalline structures are
enormous, comparable, in fact, as the above and preceding calculations have shown,
with the intense forces and polarization of ferro-magnetic substances.

In the crystalline state we must regard the molecules as orientated into definite
positions with respect to their neighbours by these large intermolecular forces. If at
the higher temperatures the molecules undergo rotational vibrations about their mean
positions, then it would be expected that the value of I2 will be somewhat lessened
by these vibrations, and we should therefore expect that a small fraction of the
energy associated with the grouping would be dissipated as the temperature is raised
towards the fusion point. The effect this would have on the variation of the specific

* The following values of the latent heat for some diamagnetic substances with which we are directly
concerned are taken from 'Recueil des Constantes Physiques,' Paris, 1913, pp. 323-4 : —

Benzene 30

Xylene 39

Chlorobenzene .... 30

Bromobenzene .... 20

Aniline 21

Acetophenone 33

Benzophenone 23

Phenylhydrazine .... 36

Pyridine 22

Nitrobenzene 22

Naphthalene 35

Naphthylamine .... 22

Acetic acid 44

Carbon tetrachloride . . 4

Bismuth 13

Cadmium 14

Lead 5

Silver 22

Tin . . 14

Zinc 28

Gallium 19

Iron (ferro-magnetic) . . 59

94 MR. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

heat with temperature would be to add to the normal variation, expressed by DEBYE'S
theory, the following positive term

^j. <•!.§, - : (17)

where S- is the absolute temperature and J the mechanical equivalent of the calorie.
In a former paper the author has shown that a term of this nature is necessary
to represent the variation of the specific heat of substances in the neighbourhood of
the fusion point.* A corresponding term explains, on WEISS'S theory, the variation
of the specific heat of ferro-magnetic substances in the neighbourhood of the
transformation temperature, on the supposition of a ferro-magnetic molecular field
of the order 107 gauss. t [+The fact that NERNST and LINDEMANN§ have found
experimentally an abnormal increase of the specific heat of diamagnetic substances in
addition to the normal variation due to purely translational vibrations, as the fusion
point is approached, is additional evidence of the importance of the rotational
term (17). DEBYE'S quantum theory of specific heats is concerned with translational
vibrations of the molecules only, and, away from the fusion point, it agrees well with
experiment. Incidentally, in order that (17) may be a measurable fraction of the
specific heat, a'c and I must be large, for, from experimental data showing the
departure from DEBYE'S theory near the fusion point, the interval of temperature
over which the molecules have effective rotational vibrations amounts to several
degrees at least, so that the large value of (17) cannot be attributed solely to a large

•yr

value of the gradient -^r. Unless a'c and I have values of the order we have already

found for them, it would be impossible to account for the measurable departure of the
specific heat near the fusion point from DEBYE'S values. Only a fraction of the

energy term — — . a'c . I2 will be dissipated below the fusion point, the major portion

disappears at the fusion point and corresponds to the latent heat (as described
above).

The departure of the specific heat from the value calculated on DEBYE'S theory is
important in connection with the quantum theory, for if the latter be valid, the
above term, due to the rotation of the molecules, implies that the angular velocities
of the molecules go in definite units. We cannot have the quantum theory holding
for translational motion and not for rotational. The remarkable fact is that the
rotational term (17) is insignificant except near the fusion point. This means that
away from the fusion point the translational motion of the molecules is sufficient to

* A. E. OXLIY, 'Proc. Camb. Phil. Soc.,' vol. XVII., p. 450, 1914.
t WEISS and BECK, ' Journ. de Phys.,' se>. IV., vol. 7, p. 249, 1908.
I [Added November 12, 1914.]

§ 'La Theorie du Rayonnement et lea Quanta,' Paris, 1912; particularly p. 272 and the memoirs of
NERNST and EINSTEIN.

CONSTITUTION AND TEMPEEATURE ON MAGNETIC SUSCEPTIBILITY. 95

account for the observed specific heat, JEANS, in his " Report on Radiation and the
Quantum Theory," published by the Physical Society of London, refers on p. 77 to
the necessity of the rotational term, which was pointed out by the author in ' Proc.
Camb. Phil. Soc.,' vol. XVII., p. 450, 1914. JEANS adds: "The absence of a
noticeable • contribution to the specific heats is accounted for, on the quantum theory,
by supposing that the forces opposing rotational movements of the atoms inside the
solid are so large that the corresponding vibrations are of very high frequency, and
so, normally, possess very little energy. As far as pure theory goes, there is no
question that to the terms in the specific heat contemplated by NERNST'S theory
there ought to be added an additional term of a form exactly similar to the Einstein

term, but having x = =j?L where v3 is the frequency (or average frequency) of the

-L v L

vibrations which depend on the rotations of the atoms.

'' It is worthy of note that sodium and mercury show an increase, beyond that
accounted for by the theories we have considered, in the specific heats as the fusion
points is approached, when, presumably, the intensity of the forces which prevent the
atom from rotating is relaxed, and NERNST and LINDEMANN find that in general the
same is true for the substances they have examined."]

Before passing on to further experimental work and the extension of our results to
crystalline diamagnetic media in general, it will be convenient to collect the results
which have been obtained in the preceding pages. The work contained in Parts I.
and II. has received full support and been confirmed with regard to the enormous
intensity of the local molecular field in about 40 diamagnetic substances which show
a measurable change of x on crystallization. Evidence that the magnitude of this
field is comparable with that of the ferro-magnetic field has been obtained from the
following independent sources : —

(l.) The change of susceptibility observed on crystallization demands a local
molecular field of this order of intensity.

(2.) The natural double refraction of a 'crystalline substance as compared with the
artificial double refraction which can be induced in a liquid by the strongest magnetic
field at our disposal is consistent with the value of the local molecular field implied
by (l) for diamagnetic crystalline media.

(3.) (l) and (2) together imply that the aggregate of the local intensity of
magnetization per unit volume of a diamagnetic substance is comparable with the
saturation intensity of magnetization of a ferro-magnetic substance.

(4. ) The above results lead to a correct estimate of the energy (potential) associated
with the crystalline structure, in virtue of the molecular grouping, as tested by the
magnitude of the latent heat.

(5.) Lastly, unless the forces binding the diamagnetic molecules together were of
the order of magnitude stated, we should not be able to detect a departure of the
experimental value of the specific heat near the fusion point from the value calculated

96 MR. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

on DEBYE'S theory. Every substance investigated by NERNST and LINDEMANN
discloses such a departure.

The above evidence is sufficient to establish the existence of an intense local
molecular field of the order 107 gauss, if interpreted magnetically, in those diamagnetic
crystalline substances (about 40 of which have been investigated) which show a
measurable change of x on crystallization. We shall now pass on to some additional
experiments with the object of extending the above conclusions to diamagnetic
crystalline media in general.

(6) ADDITIONAL EXPERIMENTS.

COTTON and MOUTON have found that aromatic liquids show an abnormally large
double refraction compared with aliphatic liquids when subjected to the same external
magnetic field. According to the theory of molecular orientation, which (in the
opinion of these authors) is unique in accounting for all the observed phenomena of
induced double refraction, the extent of the double refraction is directly proportional
to the degree of dissymmetry of the molecule. Now assuming this to be so, we
should expect that an u asymmetrical molecule, whose electrons are more readily
displaced -in one direction than in another, would have a distortion produced in it,
when subjected to the local field of a neighbouring molecule, this distortion being
characterized by the molecule's own dissymmetry. Therefore those liquids which
show the larger induced double refraction when acted on by a magnetic field should
also be the ones which show a large value of BX on crystallization. All the aromatic
liquids examined in Part I. show an appreciable change of x on crystallization and,
according to COTTON and MOUTON, all these show an easily measurable magnetic
double refraction. With regard to aliphatic compounds, COTTON and MOUTON found
that liquid hexane, chloroforn, carbon-tetrachloride, acetone, hexamethylene, ethyl
and methyl alcohols, had no appreciable induced magnetic double refraction. I
therefore examined some of these for a change in the value of x on crystallization.
The results will now be briefly summarised.

All the experiments were made with the apparatus designed for low temperature
work and most of the substances were investigated three times, the method being
exactly as described in Part I.

Carbon tetrachloride, C.C14.

At the fusion point ( — 30° C.) x passed through a minimum value, as in the case of
benzene, and, on further cooling the crystals, the susceptibility appeared to be the
same as that of the liquid. An effect of the same nature has been observed by
HONDA* with sulphur, x being a minimum at the fusion point (115° C.).

* 'Ann. der Phys.,' vol. 32, p. 10 8, 1910.

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY. 97

Acetone, CH3— CO— CH3.

This substance behaved like carbon tetrachloride.

Acetic acid (glacial), CH3.COOH, and Propionic acid, CH3.CH2.COOH.

Five experiments made on these fatty acids showed that the change of x on
crystallization was less than 1 per cent, in each case.

Aromatic Substances.

Hexamethylene,

Three experiments showed that no change of x takes place at the fusion point
(6° C.).

Chloroform, CH.C13.

The susceptibility of the crystals did not differ appreciably from that of the liqxiid.
A control experiment on nitrobenzene gave the abnormally large value 12 per cent.
(which agrees with the earlier experiments with this substance) for d%.

(7) A RELATION BETWEEN THE MAGNETIC DOUBLE REFRACTION OF ORGANIC LIQUIDS
AND THE CHANGE OF MAGNETIC SUSCEPTIBILITY DUE TO CRYSTALLIZATION.

A Comparison of Experimental Results.

The general facts which have been described with regard to the large value of 3x
during the crystallization of aromatic compounds and the relatively small value which
has been obtained in the later experiments for substances of an aliphatic nature form
an interesting parallel with those relating to the magnetic double refraction of
aromatic and aliphatic liquids investigated by MM. COTTON and MOUTON.* The
parallelism is particularly striking with nitrobenzene and hexamethylene. For the
former the induced magnetic double refraction and the value of 3x are abnormally
large while with the latter both effects are inappreciable. t

I propose to give a table showing the values of the percentage change of
susceptibility on crystallization (3x/x) and the magnetic double refraction (M), each
referred to nitrobenzene as unit, for a number of substances investigated by COTTON

* ' Ann. de Chim. et de Phys.,' ser. VIIL, vol. 19, p. 153, 1910; and ser. VIIL, vol. 20, p. 194, 1910.
t ' Journ. de Phys.,' ser. V., vol. 1, p. 23, 1910.
VOL. CCXV. — A. O

98

ME. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

Substance and constitution.

dx/X-

M.

Acetophenone, <Q \— C<^

0-35

0-49

Benzene, <Q \

0-23

0-23

0

Benzophenone, <Q /~~G ~\ /

0-46

0-57

Bromobenzene, <^ ^ - Br

0-35

0-26

Chlorobenzene, <Q \ -Cl

0-35

0-29

Nitrobenzene;, ( > — "Sf

\Q

1-0

1-0

Toluene, / ~\-CH3

0-38

0-24

Xylene (Ortho), | j;[;J[s

0-37

0-28

Aniline, <^ \— S=ll.

0-38

0-2

.0
Benzoylchloride, / \ of

0-38

0-5

/ — \

0-2

0-25

Clfj Clio

Hexamethylene, Cir2— / \— OH8

small

o-o

CHo CJI2

Carbon tetrachloride, C.C14

,,

o-o

Acetone, CH3.CO.CH3

I!

o-o

Acetic acid, CH3.COOH

,,

o-o

Propionic acid, CH3.CH,,.COOH

„

o-o

Chloroform, CH.C1.,

II

o-o

Water, H3O

0-15*

o-o

a-bromonaphthalene is also an exception. The value of BX/X is 4'5 per cent., while COTTON and
MouTON find that it shows a magnetic birefringence equal to that of nitrobenzene.

It is interesting to note in passing that not only does the Kerr electric effect follow the same laws as the
magnetic double refraction with regard to variation of field strength and wave-length, but also that it
shows a rough parallelism with M for individual-substances.

CONSTITUTION AND TEMPEEATURE ON MAGNETIC SUSCEPTIBILITY. 99

and MOTJTON and myself. From the table it is clear that there is some connection
between these two effects. In each case the benzene nucleus plays an important role.
According to COTTON and MOUTON all liquids whose molecules contain such nuclei have
a magnetic double refraction which is easy to detect experimentally. These liquids
also show a large value of BX on crystallization. If the nucleus be modified by
substitution of a monovalent atom or group for one of the hydrogen atoms the
magnetic double refraction may increase or diminish but always retains the same order
of magnitude. The parallelism in this respect is well represented by the benzene
derivatives investigated in Part I. All these show a value of 2x/X °f the same order
except nitrobenzene. In this substance the substituent — -NOX, which is so active in
augmenting the induced double refraction of the liquid, also contributes an abnormally
large part to BX as would be expected.

Both effects appear to depend also upon the degree of unsaturation of the substance,
and if by any process we can lessen the number of double linkages associated with
the nuclear carbon atoms there is a sudden diminution both of double refraction and
3x- Hexamethylene is particularly interesting from this point of view, for here all six
double linkages are destroyed by the additional six hydrogen atoms. In their later
experiments COTTON and MOUTON have found that most organic liquids and some
inorganic ones show a feeble double refraction in the stongest magnetic fields. These
would be accounted for if the molecules possess a slight degree of dissymmetry, for then
they would become partially orientated by the external field. As regards the change
of x with such substances on crystallization, we should expect that the large molecular
field which comes into play would produce a polarization or distortion in any molecule
characterized by the molecule's own dissymmetry, while the distorting force would
depend upon the degree of unsaturation of the molecule. Although the extent of this
distortion is small, yet the forces which produce it must be large, for even with the
most unsaturated and unsymmetrical molecules 3x amounts to a few per cent, only
when the molecular field is of the order 107 gauss.

With more delicate means of detection and a more intense magnetic field, COTTON
and MOUTON conclude that a double refraction would be shown by all substances.
But this is exceedingly small compared with the natural double refraction of the
crystalline medium, as far as experimental data on this question are available up to
the present,* and this increase of double refraction in passing from the liquid to the
crystalline state demands a value for the local molecular field large in comparison
with the strongest magnetic field which we can produce in the laboratory (§4).

It is expected too, that if we could use a more delicate means of measuring x we

* A large number of ordinary and extraordinary refractive indices are given in ' Recueil des Constantes
Physiques,' published by the French Physical Society. These, however, refer to minerals which occur in
sufficiently large crystals for optical experiments to be readily carried out. Among the few organic
crystals which have been investigated are cane sugar, %t — 1'570, MO = I'USTj benzilc, nf = 1'563,
«o = 1-659.

o 2

100 MR. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

should be able to detect a change of magnetic property when every substance
crystallizes. This again implies an enormous value for the local molecular field, even
if the change of x produced is so small as to be only on the verge of detection by our
present methods, as the first part of § 4 shows.

Among organic compounds aromatic substances show both these effects to an
exalted degree and it is from a study of these that the large molecular forcives in
diamagnetic crystalline media were first recognized. By the argument of the present
section it appears only justifiable to extend these resiilts to diamagnetic crystalline
media in general. Such an extension is warranted also by the values of the latent
heat.

(8) ON THE NATURE OF THE MOLECULAR FIELD.

The large local molecular field which has been recognized in diamagnetic crystalline
substances must have its origin in the individual atoms or molecules. In a liquid the
effects of these forces at any point within the medium average out to a small
resultant effect only as already indicated. When the substance crystallizes the
molecules are fixed in definite positions with regard to one another, and the polarization
or distortion produced in a molecule owing to the forces exerted by its neighbours is
no longer an average value. The modifications of the physical properties of substances
at the fusion point readily fall into line with this view. When the substance is
vaporized we must still assume that the force quite close up to the molecules is very
large and rapidly falls off on account of the compensating action of the other half of
the same molecule. If the pressure of the vapour were sufficiently increased so that
the local fields overlap, then we should expect that any physical property possessed
by the vapour would become modified. Thus HUMPHREYS* has advanced the view
that the pressure shift of spectral lines may be due to the co-operation of intense
magnetic fields located in neighbouring atoms. The pressure shift can be accounted
for if the intra-atomic field, interpreted magnetically, has an intensity of the order
108 gauss. Further, liiTzf has shown how to deduce the expressions of BALMER and
RYDBERG for the representation of spectral series, providing we assume that the
electrons are vibrating under an electromagnetic field whose order of intensity is
108 gauss. For this intensity of the intra-atomic field the frequency of vibration of
the electrons corresponds to that of visible light. If in the crystalline structure the
surrounding molecules exert a local field of the order 107 gauss, the electron in the
particular atom we are considering would have its orbit modified by an amount which
would account for a few per cent, change in the diamagnetic susceptibility. It is not
improbable that the intense intra-atomic fields of RITZ may be identified with the
field due to the magneton, the molecular field being the result of their mutual action

* ' Astrophysical Journal,' vol. 23, p. 232, 1906, and vol. 35, p. 268, 1912.
t 'Ann. der Phys.,' vol. 25, p. 660, 1908.

CONSTITUTION AND TEMPEEATURE ON MAGNETIC SUSCEPTIBILITY. 101

between the molecules of the crystalline structure. In addition it is important from
our point of view to notice that these large fields are required by RITZ and
HUMPHREYS in order to explain certain spectral phenomena, and therefore they
are required apart from the magnetic nature of the atoms or molecules under
consideration.

[*Let us now compare the theory which has been developed above and in Parts I.
and II. with the theory of ferro-magnetism given by WEISS. The intense magnetic
properties of iron below the critical temperature are attributed to the mutual effects
exerted by the molecules of iron. WEISS has shown that if we regard these effects
as purely magnetic, then the large increase of specific susceptibility due to the
transformation from the paramagnetic to the ferro-magnetic state may be represented
by a molecular field Nl, Avhere N is a constant of the order 104 and I is the intensity
of magnetization for a field strength H. For saturated iron Nl has the value
107 gauss (approximately). This internal field is called into action when an external
field H is applied, and its presence determines the abnormally large susceptibility of
iron below the critical temperature.

In a similar way it has been shown that the change of specific diamagnetic
susceptibility when a diamagnetic substance crystallizes can be represented by a
mean magnetic molecular field which is proportional to the intensity of magnetization
(diamagnetic moment per unit volume) and whose constant of proportionality is
comparable with N. On account of the zero moment possessed by a diamagnetic
molecule initially, this mean molecular field is small relative to Nl, but reasons
have been given in prevkms work which show that the local value of this
field is comparable with the ferro-magnetic molecular field. It is this local field which
distorts the electron orbits on crystallization and gives rise to a small change of
diamagnetic susceptibility. This and other considerations have shown that the
intensity of the molecular field in diamagnetic substances is of the order 107 gauss
locally.

Up to the present the representation of this field as a magnetic field has not been
justified except in so far as we have seen that it is in harmony with the important
work of RITZ, whose molecular magnets have moments not inconsistent with that
associated with the magneton, t and whose magnetic field intensity is comparable
with the ferro-magnetic molecular field of WEISS and the molecular field within
diamagnetic crystalline substances. It is important for our purpose to observe that
the intra-atomic fields of RITZ are required to represent the distribution of lines in
spectral series, whether the substance be ferro-, para- or diamagnetic.

Now very good evidence that molecular magnets do exist in ferro-magnetic
substances has been obtained. The fields due to these must contribute to the
molecular field (Nl). Taking the moment of the magneton as 16'5 x 10~22, we find for

* [Added November 12, 1914.]

t P. ZEEMAN, 'Researches in Magneto-optics,' p. 178.

102 MR. A. E. OXLEY ON THE INFLUENCE OF MOLECULAR

16'5 x 10~22x 6'8 x 1023x 11*
the saturation intensity of magnetization of iron - — - — '• - = 1760,

which is nearly equal to the experimental value. Therefore the increase of x on passing
from the paramagnetic to the ferro-magnetic state can be represented by a purely
magnetic molecular field due to the magnetons contained in the iron molecules. This
increase of x is represented on WEISS'S theory by the molecular field Nl = 107 gauss,
and therefore 107 gauss is the magnetic field due to the magnetons when the iron is
saturated. The continuity of magnetic induction demands that this field is to a large
extent localized (see p. 89 supra).

Now reverting to diamagnetic substances we are led, in the light of RITZ'S
theory, to identify the local molecular field (107 gauss) in their case as due to
the existence of molecular magnets, so arranged that for any particular molecule
the moment is zero. • It seems that we could readily account on such a view for
the fact that the chemical combination of two diamagnetic substances can give
rise to a paramagnetic substance possessing magnetons. This is the case for
instance with the union of copper (diamagnetic) and sulphuric acid (diamagnetic)
resulting in the formation of cupric sulphate (which is paramagnetic and contains
10 magnetons to the molecule). The forces which come into play during the
chemical combination we may regard as upsetting the magnetic equilibrium of each
component.

If these molecular magnets do exist in diamagnetic substances then the local
molecular field of the diamagnetic substance will be comparable with the molecular
field in ferro-magnetic substances, and will be represented as far as they are concerned
by a true magnetic field of intensity 107 gauss (approximately). In this case the
localized nature of the field is a necessary consequence of zero moment whicli has
been assigned to the diamagnetic molecule.

In both ferro-magnetic and diamagnetic cases we have represented the molecular
fields as due to magnetic force. Now electrostatic forces, or indeed forces of any other
nature, may distort the configuration of a molecule in the crystalline structure, and
their effect could be represented by a magnetic field for both diamagnetic and
ferro-magnetic substances. Hence part of the molecular field for diamagnetic and
ferro-magnetic substances may be a true magnetic field and part a magnetic
representation of the distortion produced by forces of a different nature. At any
rate we cannot deny that the parallelism which has been found between ferro-
magnetic and diamagnetic phenomena, taken in conjunction with the work of WEISS
on the magneton and that of EITZ on spectral series, points to the conclusion that
the molecular field in diamagnetics and ferro-magnetics is represented in part (at
least) by a true magnetic field. In this connection the introductory remarks of

* An atom of iron has 11 magnetons, and the gramme-atom of iron (56 gr.) contains 6'8 x 1023 atoms,
therefore 1 c.c. of iron (whose density is taken as 8) wUl contain 6'8xl°23 atoms.

CONSTITUTION AND TEMPERATURE ON MAGNETIC SUSCEPTIBILITY. 103

Prof. J. W. NICHOLSON during the discussion on " The Structure of the Atom " (issued
with ' Roy. Soc. Proc.' for July 20) are interesting : —

"I think that Prof. RUTHERFORD has made it clear that the nuclear atom is, as a
matter of fact, the only basis upon which profitable discussion of the constitution of
the atom can really be carried on. The main rival theory, as Prof. HICKS has just
indicated, is the magneton theory. It seems that just recently in the course of a
discussion it has been shown that the magneton theory and the nuclear theory do
probably amount to exactly the same thing."

Future work will test this. But in the meantime it is clear that the magnetic
forces cannot be neglected.]

Note added February 1, 1915.

While this work has been passing through the press, some further extensions have
been made. In particular, there is one which I should like to refer to here because
it bears directly upon the foregoing work,

It can be shown that the magnitude of the change of volume observed on
crystallization may be interpreted as the magneto-stnctioii effect of the local
molecular field. We have seen that the natural double refraction of a crystalline
medium may be represented as due to the complete orientation of the molecules by
the local molecular field. As the magneto-striction effect produced in a liquid and
the induced magnetic double refraction are each proportional to the square of the
applied field intensity, these results mutually support one another.

The further extension of the work is concerned with the validity of the quantum
theory as applied to magnetic phenomena. A discussion of the nature of the magnetic
properties of HEUSLER'S alloys, from the points of view developed in the present
work, will also be given. I hope to publish an account of these extensions in a future
communication.

[ 105 ]

IV. The Transmission of Electric Waves over the Surface of the Earth.

By A. E. H. LOVE, F.R.S., Sedleian Professor of Natural Philosophy in the

University of Oxford.

Received December 19, 1914, — Eead January 21, 1915.

CONTENTS.

§§ Page

1. Introduction and references 105

2. Statement of the problem 106

3. Simple limiting cases 107

4. Analysis of the general problem 108

5. Case of a plane boundary 109

6, 7. Analytical solution for a spherical boundary Ill

8. Historical Note 114

9. Preparation of the series for summation 114

10. Second Historical Note 116

11. Transformation of the series representing the magnetic force 116

12. Series representing the components of the electric force 117

13-15. Approximate summation of the series 118

16. Numerical results 121

17. Criticism of a proposed alternative solution 123

18. Approximate formula; 125

19, 20. Experimental investigations 126

21. Comparison with theory 127

22. Discussion of the experimental evidence 127

23. General conclusions 130

1. EVER since the time, about 1902, when MARCONI first succeeded in sending wireless
signals across the Atlantic the question of explaining the mechanism of such
transmission has attracted attention among mathematicians. The question may be
put in the following form : — The electric waves generated by the sending apparatus
differ from waves of light only by having a longer wave-length, which is, nevertheless
small compared with the radius of the earth ; and the curved surface of the earth
may therefore be expected to form a sort of shadow, effectively screening the
receiving apparatus at a distance. How, then, does it happen that in practice the
waves penetrate into the region of the shadow? Unfortunately, the question has
been investigated by different methods without adequate co-ordination, and the results

VOL. CCXV. A 526. P [Published February 18, 1915.

106 PROF. A. E. H. LOVE ON THE TRANSMISSION OF

that have been obtained are somewhat discordant. In these circumstances it appears
to be desirable to undertake a critical survey of the question.

The various theoretical investigations may be classified as developments of three
suggestions: (l) The imperfectly conducting quality, or resistance, of the material,
generally sea-water, over which the transmission takes place, may cause the effect
observable at a distance to be greater than it would be if the material were perfectly
conducting. (2) Owing to the numerical relations connecting the actual wave-lengths
used in practice, the size of the earth, and the distances involved, the amount of
diffraction, even in the case of perfect conduction, may be greater than would, at first
sight, be expected. (3) Transmission through the atmosphere may be notably
different from transmission through a homogeneous dielectric. We may refer to these
suggestions briefly as the " resistance theory," the " diffraction theory," and the
" atmospheric theory." It may be said at once that the atmospheric theory has
arisen from the alleged failure of the other two, and that it has not yet been
formulated in such a way as to admit of being tested in the same precise analytical
fashion as they can. It is still rather speculative and indefinite. In what follows
I propose to attend chiefly to the first two suggestions, and to investigate the result
that can be obtained by combining them.

To facilitate reference the following list of some of the principal writings on the
subject is prefixed. These will hereafter be cited by the numbers placed before them,
thus : " MACDONALD (')." The list does not pretend to be complete :—

O MACDONALD, H. M., ' Proc. Roy. Soc.,' vol. 71 (1903), p. 251.
(-) RAYLEKJH, Lord, 'Proc. Roy. Soc.,' vol. 72 (1904), p. 40.

(3) POINCARK, H., 'Proc. Roy. Soc.,' vol. 72 (1904), p. 42.

(4) MACDONALD, II. M., 'Proc. Roy. Soc.,' vol. 72 (1904), p. 59.

(5) ZENNECK, J., ' Ann. d. Phys.' (4te Folge), 13d. 23 (1907), p. 846.

(6) SOMMERFELD, A., 'Ann. d. Phys.' (4te Folge), Bd. 28 (1909), p. 665.

('} MACDOXALD, H. M., 'Roy. Soc. Phil. Trans.' (Scr. A.), vol. 210 (1911), p. 113.

(*) POINCARK, H., 'Rund, Circ. Mat. Palermo,' t, 29 (1910), p. 169.

(9) NICHOLSON-, J. W., 'Phil. Mag.' (Ser. 6), vol. 19 (1910), p. 516; vol. 20 (1910), p. 157;

vol. 21 (1911), pp. 62, 281.

(10) NICHOLSON, J. W., 'Phil. Mag.' (Ser. 6), vol. 19 (1910), p. 757.
(u) MACDONALD, H. M., 'Proc. Roy. Soc.' (Ser. A), vol. 90 (1914), p. 50.
H MARCH, H. W., 'Ann. d. Phys.' (4te Folge), Bd. 37 (1912), p. 29.

(13) RYBCZYNSKI, W. vox, 'Ann. d. Phys.' (4te Folge), Bd. 41 (1913), p. 191.

(14) AUSTIN, L. W, 'Bulletin of the Bureau of Standards (Washington),' vol. 7, -No. 3 (1911),

p. 315.

(15) HOGAN, J. L., 'Electrician,' August 8, 1913.

(16) ECCLES, W. H., 'Proc. Roy. Soc.' (Ser. A), vol. 87 (1912), p. 79.

(17) "Report of a Discussion," 'Brit. Assoc. Rep.,' 1912, p. 401.

(18) ZENNECK, J., 'Lehrbuch der Drahtlosen Telegraphic,' 2te Aufl. (Stuttgart, 1913).

2. In order to simplify the problem and render it definite, certain assumptions are
usually made. These may be stated as follows: (l) The earth is taken to be a

ELECTRIC WAVES OVEE THE SURFACE OF THE EARTH. 107

homogeneous conductor, surrounded by homogeneous dielectric, the separating surface
being a perfect sphere. (2) The sending apparatus is represented by an ideal
Hertzian oscillator, or vibrating electric doublet, situated in the dielectric near to the
separating surface, and having its axis directed normally to that surface. (3) The
waves emitted by the oscillator are taken to be an infinite train of simple harmonic
oscillations of a definite frequency. The problem is to determine, in accordance with
these assumptions, the electric and magnetic forces at points in the dielectric, which
are near to the separating surface but not near to the oscillator. If this problem
were solved satisfactorily we should be in a better position for estimating the degree
of success attained by the resistance theory and the diffraction theory ; but it is
precisely in regard to this problem that discordant results have been obtained. This
unfortunate state of things has arisen partly from the attempt to separate the effects
of resistance from those of curvature. With a view to ascertaining the effect of
resistance it has been proposed to simplify the problem still further by treating the
surface of the earth, in the first instance, as plane, and afterwards attempting to
estimate the modification of the results that would be necessary in order to take
account of the curvature. When the effect ot curvature is being investigated it is
usual to regard the material of the earth, in the first instance, as perfectly conducting,
and afterwards to estimate the correction due to resistance. Thus we have a division
of the problem into two : the problem of the imperfect conductor with a plane surface,
and the problem of the perfect conductor with a spherical surface. It will appear in
the sequel that this division of the problem is unnecessary.

3. Current ideas on the subject have been much influenced by the results of two
simple limiting cases of the general problem. In one of these the material is
considered as perfectly conducting, the separating surface as plane, and the
originating doublet as situated on the surface. In this case the waves in the dielectric
are exactly the same as if there were no conductor.* The amplitude is subject to
diminution through spherical divergence only, so that at a distance from the
originating doublet it is inversely proportional to the distance.

In the other limiting caset the distance from the originating doublet is supposed to
be so great that the waves can be treated as plane, and the separating surface is also
taken to be plane. Then, owing to resistance, the planes of the waves are slightly
inclined to the plane boundary, energy being continually supplied from the dielectric
to maintain the alternating currents in the conductor, and thus the amplitude of the
waves in the dielectric is subject to diminution expressed by a factor of the form e~Al,
where x is a co-oi'dinate measured along the plane boundary in the direction of
propagation. The constant A depends on the resistance and specific inductive
capacity of the conductor, and on the wave-length. For low resistances, such as that
of sea- water, and large wave-lengths, such as one or more kilometres, it is nearly

* Cf. J. A. FLEMING, 'The Principles of Electric Wave Telegraphy,' London, 1906, p. 347.

t See ZENNECK(5).

P 2

108 PEOF. A. E. H. LOVE ON THE TRANSMISSION OF

proportional to the resistance and to the inverse square of the wave-length. For
higher resistances and shorter waves the specific inductive capacity of the conductor
affects the value of A sensibly. For sea- water under air, and a wave-length of 5 km.,
the value of I/A, the distance in which the amplitude of the waves is diminished in
the ratio 1 :e, is 478 x 106, lengths being measured in kilometres. From the discussion
of this limiting case of the general problem it has been inferred that increased resist-
ance would be unfavourable to long-distance transmission, while increased wave-length
would be very favourable.

4. Returning now to the general problem stated in § 2, we shall take the axis of
the doublet to be the axis of z. The system being symmetrical about this axis, it is
appropriate to use the function II introduced by HERTZ,* and sometimes called the
Hertzian function.

Let p, z, (f> be cylindrical co-ordinates, the senses of increase of z and <}> being those
of translation and rotation in a right-handed screw, and p denoting distance from the
axis- of z. Let Ep, E_,, E0 denote the components of electric force, measured in
electrostatic imits, and estimated in the directions of increase of p, z, $ ; and let
Hp, H;, H.J, denote the components of magnetic force, measured in electromagnetic
units, and estimated in the same directions. From the symmetry we have the
equations'

E, = H, = H, = 0 ........... (1)

It will be sufficient to consider the case in which botli media are of magnetic
capacity unity, the dielectric is of specific inductive capacity unity, and the specific
inductive capacity of the conductor is neglected. Then one of the electromagnetic
equations is, in both media,

_ _

c & cz " sp '

where C is the velocity of light, 3xl010 cm. per second. The remaining equations
are,

in the dielectric,

p ~u " : ' TT 3.

U ct oz U vt p vp

in the conductor,

4™CEP=-^, 4™CE,=1^(/0H,); ...... (4)

CZ p Cp

where a- is the specific conductivity, measured in electromagnetic units.

Now we are to suppose that Ep, E,, H^,, in so far as they depend upon t, are
proportional to simple harmonic functions of period 2-7T/&C say, where 2Tr/k is the
wave-length, and we may take them to be proportional to e'kct. Then they can be

* H. HERTZ, ' Electric Waves,' English ed., p. 140.

ELECTRIC WAVES OVEE THE SUEFACE OF THE EARTH. 109

expressed in terms of a single function IT, the Hertzian function. The formulae which
hold in the dielectric are

„ ., an F a2ii ~ i 3 / am

H,>=-U,— , &,= -, &2 = -- ^-(p-^-j ..... (5)

Op Op CZ p Cp \ Op /

where II satisfies the equation

0; .......... (6)

and those which hold in the conductor are

P air -p 82ii' v i a / air\

J-l* =-47ro-t--r— , ^ = -5— T- , &.= --- r-U-^— , . ... (7)

up op cz p cp \ cp I

where II' satisfies the equation

(V2 + £'2) II' = 0, .......... (8)

in which

, ........... (9)

and II' has heen written instead of II in the formulae relating to the conductor.

The special form of II which answers to a vibrating electric doublet, situated in the
dielectric on the axis of z, and having its axis directed along the axis of z, is II,, say,
where

„

n«= -ir-, .... . (10)

11 denoting distance from the doublet. We may put

11 = 110+11, ........... (11)

Then II, satisfies the same equation (6) as II.

The conditions to be satisfied by the functions II, and II' are the following : — -

(i.) They are solutions of the equations (V2 + &2) II, = 0 and (V* + k'2) II' = 0 ;
(ii.) II, is free from singularities in the region outside the conductor, and II' is

free from singularities in the region inside the conductor ;
(iii.) II, must represent waves travelling outwards ;
(iv.) The tangential components of electric and magnetic force derived, as above,

from II arid II', nrnst be continuous at the bounding surface of the conductor.

5. When the boundary is a plane, say z = 0, the conditions (iv.) become

k2n = Fir
en air

cz

these equations being satisfied at z = 0. The condition (iii.) requires some modifica-
tion, for both regions now extend to infinite distances. It is now necessary that 11,

110 PEOF. A. E. H. LOVE ON THE TRANSMISSION OF

should tend to zero for large positive values of z, the positive sense of the axis of 2
being directed from the conductor to the dielectric, and that II' should tend to zero
for large negative values of z ; and further, that at great distances from the axis of z,

111 an(i n' should tend to forms which represent waves travelling outwards from
that axis.

The problem in a slightly different form was solved by SoMMERFELD.(6) He took
the doublet to be situated on the plane boundary, so that II', as well as II, has a
singularity on this surface ; and he obtained an exact solution in terms of definite
integrals involving BESSEL'S functions, and devised methods of evaluating the integrals
approximately. His main result is an approximate expression for IIj at a point close
to the boundary, and at a distance p from the doublet, in the form

(12)

This expression gives a valid approximation if kp is not small and not too great ;
but, as p increases, the value of II,/!!,, does not increase so rapidly as this formula
indicates, and for very great values of p it tends to zero. Within the region of
validity of the formula the absolute value of IIj diminishes according to the inverse
square root of the distance ; and the effect represented by II1( being a wave affected by
cylindrical divergence, is described as a " surface wave." This surface wave is the effect
of resistance ; and it appears that, owing to resistance, the electric and magnetic forces
diminish less rapidly witli increasing distance than they would if there were perfect
conduction. SOMMERFELD (") and ('") maintained that this effect of resistance would
probably be intensified by curvature of the surface, and might thus counteract the
tendency of the signals to become enfeebled owing to curvature. With a wave-length
of 5 km. and the conductivity of sea-water (a- = 10~n) the region of validity of the
formula would include distances of 1000 to 10,000 km., and in this region the ratio
II | /| II0 would increase regularly from 1'003 to 1'032.

The value of I I^II,, depends upon the resistance and the wave-length as well as upon
the distance p. It varies directly as the square root of the resistance and inversely
as the wave-length. The formula (12) indicates that in the region within which it
gives a good approximation, increased resistance is favourable to long-distance trans-
mission, increased wave-length unfavourable. These results are directly opposed to
those which were noted in § 3, as derived from the study of the limiting case in
which the waves are treated as plane. On the other hand, the range of values of p
within which the formula is valid becomes narrower as the resistance increases or the
wave-length diminishes. The optical theory of shadows shows that increased wave-
length must be favourable to long-distance transmission, and we see that we cannot
hope to obtain any equivalent result from the solution of the plane problem. It is,
however, quite feasible that resistance also should be favourable within a restricted
range. The only way to settle the question is to solve the problem of the spherical
conductor, supposed imperfectly conducting.

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. HI

6. It appears to be desirable to write out the analysis for the problem of the
spherical conductor rather fully, in order to show how to determine the effect of
resistance, and to criticise the attempts that have been made to determine the effect
of curvature in the absence of resistance.

We denote the radius of the sphere by «, and use polar co-ordinates r, 0, <f>, with
the centre of the sphere as origin, and the radius vector on which the originating
doublet lies as the axis 0 = 0. We write /u for cos 9, and note the formulae

(13)

We denote the components of electric force in the directions of increase of r, 6, </>
by Er, Es, E$, and those of magnetic force by Hr, H9, H^,. In both media we have

E, = Hr = H9 = 0. . ........ (14)

In the dielectric, where r > a, we may put

H il-?'11 V l 3 ( aiM T^ l ?: ( ?IIV

t±+=-ik--} &e = ---U— ), ^r = -—ip—-\-) . . (15)

CP p cr\ op/ r^ dfj. \ cp/

and in the conductor, where r < a, we may put

air „ i a / air\ T, i a / air

H, pi <J 1 1 XT' i. V I Vli \ T.-I L U Vll \ I 1 c\

.'. = — 47TO-O — — , JJij = - -r— I p — — , Ejr = — —\p — — . . ( 1 0 )

(ip p or \ Cp I 7>J d/x \ 3p /

Then II and II' satisfy the differential equations (6) and (8).
The conditions which hold at the boundary r = a are

cp cp

a / aji'\ a / air\ ' ^17^

The function II0 answering to the primary waves is given by (10), and we take the
originating doublet, the origin of K, to be at the point for which r = r(>, 0 = 0.
Then ra > a, but in practically interesting cases (rn—a)/a is small. The functions
K! and II' are to be determined in accordance with the conditions laid down in § 4.

The proper forms for these functions can be expressed as series involving spherical
harmonics, viz. :

-"«

where Bn, B'B are constants to be determined by help of the boundary conditions,
Pn (/u) denotes LEGENDRE'S nth coefficient, or the zonal surface harmonic of degree n.

112

PEOF. A. E. H. LOVE ON THE TRANSMISSION OF

the summation refers to integral values of n, and EB and !/<•„ denote the functions
determined by the equations

EB(2) = (1 ^-Ye— = 2V-V''<-«+w"z-('+lwKll+1/t(w),

\z az/ z

sn

. . . (19)

= 2

Here

2 cos

'Mn+1/2)"J-n-V1(2)-^(-'w^+,/J^)}. • • • (20)

The function II0 can be expanded in a series of similar form. In the region r < ?•„
the series is known (cf. MACDONALD (7)) to be

II0= 2

3*c<, .... (21)

and in the same region we have the known result that

V / O v
• ^ \£l

n = 1

The proof of this result involves the known equations

'*(*), .... (23)

which hold also when \[/- is replaced by E, and the equations

)^5a=IJ (2n+l)P.

of

/*

-=i. (24)

In like manner we may obtain the equations

811' °"

/j -^ — = — 2 (2n+ ]

OP )! = 1

. (25)

where Cn and C'B are new constants, determined in terms of the constants of types
Bn and B',, by the equations

. . . . (26)

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH.

113

The boundary conditions now yield, for the determination of the constants Cn, C',,,
the equations

. . . (27)

In particular we find

0.1 *<*•)£

. (ka)} - E. (ta) {a- V. (k'a)}

. - - (28)

7. We wish to evaluate the components of electric and magnetic force at points in
the dielectric which are close to the sphere. We see that at such points we may put

-» 7T>

= - 2 (2n+l) P'+Vo'-'E,, (kru) a»+' {+„ (^) + CreEn (ka)} (l-V) ~= ^, (29)

n = 1 tl/X

where

,n (Aa) {a-^E. (*a)} -E, (ka) {«»

1-.

(30)

If the material of the sphere were perfectly conducting the second factor on the
right would be replaced by unity. For any good conductor and high frequency j k' is
large compared with k, and we may approximate to the value of the second factor.

Put

jfe" = - im2, m2 = 4TT<rkC, (31)

then we may take

k'a — ^ma */2 (l—i), ik'a — ^ma y/2 (!'+*'),

and, if a is the radius of the earth, ma is a large number, so that the most important
part of sin k'a is — ^ie^k>a. Hence the most important part of ^•n(k'a) is the most

important term of

[ 1 d }n etk'a

(k'a d(k'a)\ Zik'a'
and this is

ineik'a

VOL. CCXV. — A.

2t (k'a)n+l
Q

114 PEOF. A. E. H. LOVE ON THE TRANSMISSION OF

In like manner the most important part of

IS

k'an+l

2t (k'a)n+l
Thus we have the exact result that, as ma tends to become infinite,

k'an+l \}rn (k'a) 3a

tends to i as a limit. Hence the second factor of the right-hand member of (30) may
be replaced approximately by

Fi {«»+%(*»)}

The right-hand member of (29) then takes the form of two series, of which the first
is that which would occur alone if there were no resistance, and the second represents
the correction for resistance.

8. The 'solution in the form of a series for the effect of curvature without resistance
was found as early as 1903 by MACDOXALD(I). He proceeded to sum the series
approximately by substituting approximate values for the BESSEL'S functions which
occur in the definition of the functions \[rn and EB. The amount of diffraction which
he thus found was so great that his result was challenged by Lord R,AYLEIGH(2) and
PoiNCARE(3). In particular Lord RAYLEIGH pointed out that the terms of the series
which would contribute most to the result would be those for which n is large of the
order ka. MACDONALD(') at once admitted the justice of the criticisms, and revised
his calculations so far as to give an independent proof that the amount of diffraction
at a considerable angular distance from the originating doublet must be very small.
He held then, and has always maintained, that the effect of resistance would amount
to no more than a practically unimportant correction.

9. With a view to the approximate summation of the series it is convenient to
introduce the notation

( — )" S" + 1En (z) = \/BW • e~t4" = Vn~iUn>

(-)'*"+V» (*) = v/R. • sin fc, = uu L (33)

If we identify z with ka, we may write RBi0, 0,i0, for the corresponding functions
of kr0. It is easy to prove the identities

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. 115

and we find

+. (ka) {a-% (to)} -EB (In) - {a- V« (*a)}

and

fa"'E.(fa) =

Hence we have at r = a,

311 _ ie**Ct

. . (35)

(36)

, I \ . n.O pt(*.-«..0)T) - iWl _ 2\ n

' " M)<^

ne2-(l-M3)) . . (37)

where . the first line of the right-hand member represents the efiect of curvature
without resistance, and the second line represents the correction for resistance. In
practically interesting cases, z is a large number of the order 104 or higher, arid the
factor r0~2 may be replaced by a~2.

Approximate expressions for Rn, <£„, xn> valid in different ranges of values of n, have
been obtained after L. LORENZ* by H. M. MACDONALD(?) and J. W. NICHOLSON^').
It appears that, if r0— a is very small compared with a, </>„_„ differs very little from <•/>„
throughout the whole range, and Rn 0/Rn is very nearly unity until n exceeds z by
a large number, but when n— z is very great this fraction becomes very small. As
n increases from 1, Rn increases from a number which differs very little from 1, and
ultimately becomes very great, while x« increases continually from a very small value
to ^-TT, passing from a value slightly less than %TT to one slightly greater as n passes
through the integer nearest to z—^r. For large values of n, provided 8 is not near to
0 or TT, we may use the approximate formulae

sin 9

(38)

so that this factor is the product of a large number of the order ^/n and a simple
harmonic function of n. For values of n which are comparable with z, Rn is quite
moderate. These considerations suggest that the most important terms of the series

* L. LORENZ, ' CEuvres Scientifiques,' vol. 1 (Copenhagen, 1898), p. 405,

Q 2

116 PROF. A. E. H. LOVE ON THE TRANSMISSION OF

are those for which n does not differ too much from z, and that, in summing these, the
factors yPvo/v/ftn and el(*""*"'o) may be omitted.

10. For the series, which represents the effect of curvature without resistance,
methods of summation have been devised by MACDONALD(;), PoiNCARE^8), and
NiOHOLSON(9) ; and still another method has been devised by MACDONALD(U). All
these methods depend upon a transformation of the series into a definite integral, and
an approximate evaluation of the integral. POINCARE(S) did not press his method so
far as to tabulate numerical results, but concluded that the expression for the electric
force normal to the surface, at an angular distance 6 from the originating doublet, should
contain a factor of the form e~A*. NIGHOLSON(!') went further in the same direction,
obtained a formula for the magnetic force containing such an exponential as a factor,
and deduced definite numerical results. MACDONALD(T) also obtained definite numerical
results which cannot be reconciled with those of NiCHOLSON(9). The discrepancy
was discussed by NICHOLSON(I(I), who traced it to an alleged flaw in the analysis used
by MACDONALD(T), and it was discussed also by MACDONALD(U), who pointed out a
difficulty in the analysis used by POINCARE(S) and NICHOLSON^). Fresh numerical
results were deduced by MACDONAL,r>(n) from a new method of summing the series^
but they do not agree with those found by NiCHOLSON(9), or with those previously
found by MACDONALD(T) himself.

11. It appears to be desirable to devise a new method of summing the series, to
apply it to obtain definite numerical results in regard to the problem of the perfect
conductor, and to compare these with the results obtained by MACDONALD and
NICHOLSON, further to apply it also to calculate the effect of resistance, and finally to
compare the results with those which have been found by experiment. The method
which I have used is to compute a sufficient number of terms of the series and add
them together. It appeared to be best, in the first instance, to do the work for a
particular wave-length, and even with this limitation considerable labour was
required. The quantity chosen for calculation was the magnetic force H at a point
on the bounding surface.

According to (15), (37), and (38), and the results already cited as known in regard
to the behaviour of the functions Rn and x* , we have

H = • 2 ' • R

where the summation now refers to the relevant terms, and

-z. .......... (40)

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH.

117

This equation (39) may be expressed in a real form. Corresponding to primary
waves given by

n _ cos k(Ct-'R) ,

~ir~

we find

H = -

sn

-z0)-(S21-S22) cos (kCt-z6)

• A/— ?-s • [(S'U-S'12) sin (kCt-ze) + (S'21 + S'22)
v TT sin u

cos

where m = i/(<±ir<rkC), and

Q _ V (H + ±

»„ — z — r

-.Jt^cos Xncos g,,tf- -

7T

T/'

/ 2

V " ~ •

» it/

cos-x.smlg.e- —
cos2 v- tan Y,, cos

and

S'n = 2
S'18 = 2
S'ai = 2 -

— M 2 roq4 v Cl — tin2 v ^ ^in n ft

— . -*-Vfi L/U& VH \ J. — tdll Xtt/ ^m '/?iv7j

A/ -- . R,,2 cos4 x» 2 tan Xn cos qn6,
/Y/-- . Rn3 cos1 Xn (1-tan2 Xn) cos qnO,

V- . Rn2 cos4 xn 2 tan Xn sin qn6.
Iti

(42)

(43)

(44)

n

12. It may be observed here that similar expressions can be obtained for the values
of Er and E9, the radial and tangential components of electric force at a point on the
surface. With the formula (39) for H we should find E« = kH/k', where the second
line of the expression for H may be omitted. This form gives the amplitude of Efl at
any distance as k/m times the amplitude of H at the same distance. In the case of
primary waves given by (41) the value of Er can be obtained from the formula (42)
for H by changing the signs of the coefficients of sin (kCt— z6) and cos*(kGt — z6), and
in the sums such as Su and S'n replacing (w+^)2/i~v'z~s/I by (n + ^) (n+ 1) n'/2z~>;'. Since
z is a large number, and, in the relevant terms, n differs but little from z, it is manifest
that the terms of Er which contain the sine or cosine of (kCt—zQ), are nearly the same,

118

PEOF. A. E. H. LOVE ON THE TRANSMISSION OF

except for a change in sign, as the corresponding terms of H, while those which
contain the sine or cosine of (kCt + zO) are nearly the same and of the same sign.

13. In order to sum the series denoted by S11; ... it was necessary to evaluate Rn
and tan XB. This can be done, when un and vn are known, by means of the formulae

tan XB = un

(W.' + V,*).

(45)

the second of these being obtained from (23). It was therefore necessary to calculate
un and vn. Now, when qn is not more than a small fraction of (£z)'/s, it is known from
the analysis of L. LORENZ already cited, and confirmed by MACDONALD and NICHOLSON,
that vn and un are given very approximately by the equations

. . . (46)

' cos2
1'«z1/' cos -JT sn ,

''• qn

These results were used for the values — |- and |- of qn, corresponding to the values
z and z—l of n, but not for any numerically larger values of qn. For other values of
n the values of un and •<;„ were found from the sequence equations, deduced from (23),

viz. : — •

= 2n-l t _ 2/t -1 u /4*A

n ^^ n — 2 n — 1) n * n — 2 n — 1" \ /

For n increasing beyond z (qn > ^), these were used in the equivalent forms

vn = 2vn_l—vn_2+ <A~ -%'„_,, un = 2un_l-un_.i+ — ^2 iMB_j, . (48)

and, for n decreasing beyond 2—1, (gn < —•&•), in the forms

1, (49)

and the results were verified -each time by substituting in the formulae (47).
The wave-length taken was 5 km., so that ka, or z, was 8000.
The initial results thus found are given in the following table : —

TABLE I.

?».

Un-

vn.

R».

tan x«-

-1

2-56472

4-24281

24-5792

0-5508

ft

2-44958

4-44223

25-7338

0-6040

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. 119

It seems to be unnecessary to print a complete table of the results. Six figures
were retained in computing un, vn, R,n, and this made it possible to compute tan x» to
four figures, as above. The computation was carried from qa = — ^- to qn = ^Mp.

The corresponding values of

Z- . Ra cos2 Xn, (II.) &±& ^z- . EB cos2 Xn tan Xn,

(HI.) t"±R J*. . K,,° cos< Xa (1 -tan' Xn), (IV.) (2±& J *- . Rn2 cos1 Xn 2 tan Xn)

2- ' it/ Z ib

were then computed for the same values of qn, four figures being retained. The
general character of the results can be exhibited to the eye by marking on squared
paper the points which have the above expressions as ordinates and the corresponding
values of n— z, or n— 8000, as abscissae, and drawing smooth curves through the
points. The four curves, numbered I., II., III., IV., in the same order as the four
expressions, are shown in fig. 1, p. 120. In order to keep the figure within bounds,
the values of the expressions III. and IV. have been ' divided by 1 0 before marking
the points representing them.

14. The next thing to be done was to sum the series denoted by Sn, ... for a number
of values of 0. The values chosen were Tr/30, ir/20, v/15, Tr/12, Tr/10 or 6°, 9°, 12°,
15°, 18°. Every term of each of these series is the product of a number shown by an
ordinate of one of the curves I., ... IV., the ordinates of the curves III., IV. being
multiplied by 10, and a simple harmonic function of n, actually a sine or cosine of

or

Thus the terms of the series can be represented graphically by ordinates
corresponding to integral values of n — 8000 as abscissae, and a smooth curve drawn
through the extremities of the ordinates. The curve fluctuates like a curve of sines,
but the amplitude is variable, and the terms may be divided into groups of one sign
by means of the points in which the curve cuts the axis of abscissae, and these again
into halved groups by means of the abscissae for which the corresponding sine or
cosine is a maximum or a minimum. The portions of the curves that correspond to
these groups and halved groups of terms may be described as " bays " and
" half- bays." The method adopted for summing the series is analogous to the method
employed by FRESNEL in problems of diffraction, wherein use was made of the notion
of " half-period elements " ; when the amplitude is nearly constant the terms
belonging to a bay are nearly cancelled by those belonging to the two neighbouring
half-bays.

It seems best to illustrate the method by explaining how S12 was evaluated for
& = 12°. A portion of the fluctuating curve, and the corresponding portion of the

120

PKOF. A. E. H. LOVE ON THE TRANSMISSION OF

Fig. 1.

Fig. 2.

ELECTKIC WAVES OVER THE SURFACE OF THE EARTH. 121

amplitude curve (II. ), are shown in fig. 2, in which the ordinates that separate the
half-bays are also drawn. The greatest ordinates are in the bay between the values
4 and 18 of n -8000 ; with this were taken the two half-bays from —4 to 3 and from
19 to 26. The next similar portion on the left consists of the half-bay from —34 to
— 27, the bay from —26 to —12, and the half-bay from —11 to —4; and the next
similar piece on the right consists of the half-bay from 26 to 33, the bay from 34 to
48, and the half-bay from 49 to 56. The ordinates common to two consecutive half-
bays were halved, and half taken as contributed by each. The sums of the terms
thus contributed to the series by the three pieces of the curve described above were
approximately 61, —14, and —5, which yield as sum 42. It may also be seen that
this is very nearly the sum of the whole series. After the critical middle piece
( — 4 to 26) the series may be broken up into sums of terms contributed by
pairs of consecutive half- bays, e.g., 26 to 33 and 34 to 41, are such a pair, 41 to
48 and 49 to 56 are the next pair, arid so on. The sums contributed by such
pairs have alternate signs and diminish continually in absolute magnitude. It
thus becomes easy to estimate the error made in breaking off the sum at
specified maxima, or minima, of the sine or cosine • involved. In the particular
example of S]2 for 9= 12°, the error made in breaking ofF the sum at —34 and 56
was estimated in this way as about — 1 ; and the value of S12 for 6 — 12° was taken
to be about 41.

15. The value of H expressed by (42) involves eight of these series, but the labour
of calculation can be reduced very much by observing the forms of the two expressions
in square brackets in (42). The terms written in the first lines of these expressions
represent waves travelling outwards from the originating doublet, with a velocity C
along the arc of a great circle drawn from the originating doublet to the point of
observation, the wave having at each point an amplitude and phase depending upon
the values of Sn, ... at the point. The second lines represent a return wave. Now it
is evident that for moderate values of 6, say less than \-K, the amplitude of the
return wave must be very small, and thus we are led to expect that, at least
approximately,

Su = S12, S21 = — S22) S'n = — S']2, S'21 = S'M. . . . (50)

Aii analytical proof that the return wave is negligible in the case of perfect conduc-
tion has been given by MACDONALD(T), arid the above approximate relations have been
verified numerically in a few cases. By assuming them throughout the labour of
calculation could have been considerably reduced.

16. The values computed for S12, S32, S'u, S'22 for the selected values of 6 are
recorded in the following table : — •

VOL. CCXV. — A. B

122

PEOF. A. E. H. LOVE ON THE TRANSMISSION OF

TABLE II.

e.

Eh»

822-

S'12.

S'22-

6°

102

150

6197

3540

9°

73

44

2830

3540

12°

41

6

749

2604

15°

19

-6-5

-182

1550

18°

7-5

-6-5

-456

760

The amplitude of H may now be taken to be

2y/2.^

«v/ '.(TZ sin 0)

(51)

and the value of this quantity for any particular value of 6 may be compared with the
amplitude of H0, the magnetic force at the same point due to the originating doublet
alone. Now, when 6 is not nearly equal to 0, R is large of the same order as a, and
we have the approximate equation

H0 = - Sma
4a sin3

so that we may take the ratio of amplitudes to be

sin 0) /f/g &_g,

•cosJfe(Cfe-B), (52)

m

. . (53)

The ratio of amplitudes is the quantity tabulated by MACDONALD(U) in the case of
perfect conduction. It has been computed anew by the method described above for
the wave-length 5 km. and two values of the ratio k/m, viz., the value zero, answering
to the hypothesis of perfect conduction, and the value 0'001826 answering to the
conductivity of sea-water (a- = lO'11). In the following table the first column gives
the value of 6 in degrees ; the second the corresponding arc-distance D in kilometres ;
the third headed AM, the amplitude ratio as computed by MACDONALD(U) ; the fourth,
headed Aa, the amplitude ratio computed by means of the formula (53) when k/m is
taken to be zero ; the fifth, headed A,, the amplitude ratio computed by means of the

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH.

123

same formula when k/m is taken to be 0'001826. There is some uncertainty about
the third figure in the 6° line in each of the last three columns.

TABLE III.

e.

D.

AM.

A.-

-A.Q..

6°

667

•

1-04

1-05

1-05

9°

1000

0-61

0-61

0-61

12°

1333

0-34

0-34

0-35

15°

16G7

0-18

0-18

0-19

18°

2000

o-io

0-10

0-11

Except as regards the uncertain last figures in the G° line there is complete
agreement between the third and fourth columns of the table ; in other words, my
results confirm those of MACDONALD(U) for the case of perfect conduction. The
methods of summation employed by him are so different from those which I have
adopted that the results may be accepted with great confidence. The correction for
resistance seems to be rather larger than MACDONALD anticipated, but nevertheless
not large enough to have the importance expected by SOMMERFKLD, although his
expectation that it would be increased by curvature is verified.

17. Since MACDONALD'S result for the problem of the perfect conductor is confirmed,
it appears to be unnecessary to enter into a detailed comparison with the results
obtained by PoracARE(8) and NICHOLSON(U), although it may be stated that the
objection raised by MACDONALD(n) to a step in their analysis seems to the present
writer to be well founded. But it is appropriate here to notice a solution of the
problem put forward by MARcn(12) and E,YBCZYNSKi(13). MARCH, apparently acting
upon a suggestion of SOMMERFELD'S (see " Report " (17)), set out to obtain a solution
of the problem in terms of definite integrals, analogous to that obtained by
SoMMERFELD(6) for the problem of the plane boundary. He confined his analysis to
the case of a perfect conductor ; and his final result was that the amplitude of the
waves, received at an angular distance 9 from the originating doublet, should be very
approximately proportional to (0 sin 6)~\ An oversight in his work was pointed out
by POINCARE,* and its correction was undertaken by E,YBCZYNSKi(13) who also
obtained a result differing from that of MACDONALD(U) ; but MACDONALD(n) has

* H. PoiNCARri, 'Paris C.R.,' vol. 154 (1912), p. 795.
B 2

124 PROF. A, E. H. LOVE ON THE TRANSMISSION OF

noted a difficulty in BYBCZYNSKI'S analysis. It seem to me, however, that there is a
serious error in the part of MARCH'S work which is accepted by RYBCZYNSKI. He set
out to determine a function, say \[s, which has the property expressed by the equation

The function ^r satisfies the same differential equation as the Hertzian function IT,
the same conditions of continuity, the same condition. at infinite distances, and similar
boundary conditions ; and the special form of it answering to the originating doublet,

say \'A,, is given by the equation

J v „,•*(«-]<)

*

~ET

In the region outside the sphere \js has the form ^0 + ^,, where ^ in the analogue
of II,, in the region inside the sphere it maybe denoted by i//, analogous to II'. Now
MARCH seeks a solution in which i//^ and i// have the forms

Y,, = L f (a + l) po (cos 6) ?'"Ea (yb-y^'F, (a) da,
r0J-'/,

(a + £) Pa (cos 0) 7-"fa (k'r) e^'F (a) da,
where Pa (cos 0) denotes a certain solution of LI-:GENDRE'S equation

in which a is not in general an integer, Ka (&•>•) and i//-a (k'r) are defined by means or
BESSEL'S functions as in equations (19), but not by means of differential operators, and
Fj (a) and F (a) are functions of a to be determined. He alsd obtains integral formulae
of a similar type to represent \//-,, in the regions >•>•?•„ and ?•„>?•>«. The function
denoted by Pa (cos (?) is free from singularity at 9 = 0.

Now these forms fail to satisfy conditions which must necessarily be imposed on
V',,, Vo, and \l/. It is necessary that \i/ should be finite at r = 0 ; but when r = 0,
and a lies between — ^ and 0, ra i/ra (k'r) is infinite. Further it is necessary that ^0, i^,
and \ff' should be free from singularity on the axis 6 = -w ; but when a is not an integer,
and Pa (cos 9) is finite at 0 = 0, it is infinite at 6 = ir. Hence the form taken for
T// has a singular point at the centre of the sphere, and a line of singular points
extending thence along the radius of the sphere drawn in the direction 0 = TT, and the
forms taken for i/r0 and Y'i bave a line of singular points extending along the
continuation of this radius outside the sphere. It appears therefore that no solution
of the type sought by MARCH exists.

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. 125

This error seems to me to vitiate the whole of the work of MARCH and RYBCZYNSKI.
In particular, it seems to destroy the foundation for RYBCZYJSSKI'S solution of the
problem of the perfect conductor, and his extension of the solution to include the effect
of resistance.

1 8. A result that should admit of being tested experimentally is the law of decrease
of amplitude of the electromagnetic waves with increasing distance. According to
what has been said in §§12 and 15, this law must be very nearly the same for the
electric force in the field as for the magnetic, and it may be assumed that the amplitude
of the received antenna current at any place is proportional to the amplitude of the
magnetic force of the field at that place. The various diffraction theories lead to
approximate formulae for the law of decrease of the amplitude of the magnetic force.
Let H denote this amplitude at angular distance d, Hj the corresponding amplitude at
angular distance Qlt and X the wave-length, measured in kilometres. Then a result
given by MACDONALD(U) leads to the formula

H . _ COS ^

(4r89)X-Main'W,-ainVs»)

a result given by NICHOLSON (") leads to the formula,

H = A A'sinK\ e<*W *-'''<*,-*> ,
H! V Isin ifl,/

and a result given by RYBCZYNSKi(13) to the formula,

H . //fl, sin 0A

dr.-o^.-s)

while ZENNECK(IS) gives a formula equivalent to

jH . //e, sin 6A <„,„ ,-'/,(,,_„ , }

n V e sin

e sin 9

as the result of a correction of the work of MARCH (12). In the exponential factors of
the last three formulae 0 and Ql are measured in radians.

The discrepancies between these various formulas are sufficient to justify the
attempt to obtain an independent solution of the diffraction problem. The con-
firmation of MACDONALD'S result for a particular wave-length affords good reason for
accepting his formula as correct, and consequently rejecting the others as incorrect,
apart from the objections which have been brought against the analytical procedure
of their authors.

MACDONALD has pointed out that the range of validity of such a formula as (54) is
restricted by the condition that neither 9 nor 91 must be too small. For X = 5 it
begins to be valid at about 7°, for X = 2'56 at about 6° 20', for X = 1'22 at about 6°.
If it is desired to compare it with results of experiment, the distances to be considered

12G PROF. A. E. H. LOVE ON THE TRANSMISSION OF

should not be less than about 400 sea miles (6° 40'), or 750 km. (6° 45'). For tne
purpose of testing it experimentally it is necessary to measure the amplitudes of
received antenna currents at various distances exceeding these limits.

19. So far as I know, the only records of quantitative measurement of received
current at sufficiently distant places are contained in the memoir of AUSTIN(U) and
the article by HoGAN(16). AUSTIN'S experiments were performed by transmitting
signals between the U.S. station at Brant Rock and two cruisers in the Atlantic,
measurements being made of the strength of the signals from shore to ship, from ship
to shore, and from ship to ship, at various distances up to 1000 sea miles (1850 km.),
by day and by night, during several months in the years 1909-10. The wave-lengths
employed were 375 km., 1'5 km., and 1 km. HOGAN'S experiments were performed
in the year 1913 between the U.S. station at Arlington, Va., and one of the same two
cruisers. The wave-lengths employed were 3'8 km. for the signals sent from the shore
station, 2 km. for those sent from the ship. The range of distance was 3000 km. for the
shorter waves, and 4250 km. for the longer. The method of observation was to take
shunted telephone readings on the incoming signals, the shunt being adjusted to
reduce the signals to audibility, and the standard of audibility being that strength of
signal which permits a clear differentiation of dots and dashes. AUSTIN'S detectors
were of the " free wire electrolytic type," HOGAN generally used the " Fessenden
liquid barretter," but " some of the readings at extreme distances were taken upon
the heterodyne receiver." HoGAN defines the " audibilty factor " as the ratio
(R + S)/S, where II is the impedance of the telephone, and S that of the shunt which,
when connected across the terminals of the telephone receiver, reduces the signal
intensity to audibility. He states that this ratio is approximately proportional to the
square of the received antenna current. AUSTIN records the results of his experiments
of July, 1910, in tables. He also records all his results, including these, graphically,
by marking on sqxiared paper points whose abscissre are the distances, and ordinates
the received antenna currents. HOGAN records his results graphically in a similar
way, with the difference that his ordinates are proportional to the values of the
audibility factor. Both observers found the results for daylight signalling much more
regular than for night.

20. AUSTIN sought to deduce from his observations the law of decrease of received
antenna current with increasing distance. In this he was guided partly by some
observations taken in 1905 in the Irish Sea by DUDDELL and TAYLOR, who, he says,
found that the received current over water fell off nearly in proportion to the distance.
This was to be expected for comparatively short distances. He was also guided partly
by the ideas of ZENNECK.(6), referred to in §3 above. Accordingly he compared his
results with an expression of the type Ae-°D/D, in which D denotes distance from the
sending station, and A and a are independent of D, but may depend upon the wave-
length. The factor e~°D he described as due to " absorption." The most regular series
of his observations taken in July, 1910, were found to fit fairly well the curves that

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. 127

can be obtained by taking a = (0'0015)x~'/1, and introducing suitable values for A, the
wave-length X and the distance D being measured in kilometres. Similar curves were
drawn by him on the diagrams representing all the series of his observations, and they
show that there is a fair agreement between the daylight observations and the formula.
HOGAN drew on his diagrams the graphs of expressions of the type (Ae~aD/D)3, with
AUSTIN'S values for a in terms of wave-lengths and suitable values for A, and found
that the graphs fitted the daylight observations rather well.

21. When AUSTIN'S formula is expressed in terms of angular distance, so as to
become comparable with those written down in § 18 above, it takes the form

a;

This formula would clearly not show, in a narrow range such as that between 400
and 1000 sea miles, very much divergence from those given by RYBCZYNSKl(13) and
ZENNECK (18) ; and both these writers claim that their formulae represent the results
of AUSTIN'S experiments better than his own. On the other hand, it would show even
in this range a well-marked divergence from MACDONALD'S formula, though not
perhaps sufficient to be conclusive. When the comparison is extended to the wider
range covered by HOGAN'S experiments, the divergence would appear to be decisive.
If the records of the observations could be accepted without criticism, it could be
stated that the law of decrease of amplitude of the electro-magnetic waves with
increasing distance from the sending station, as expressed by the formula (59), has
been tested and found adequate over a wide range of distances and wave-lengths ;
and further, since this formula cannot be reconciled even approximately with
.MACDONALD'S theory, it could be inferred, as in fact it has been, that diffraction cannot
account for the observed facts.

22. I hesitate to draw this inference for two reasons. First, there is some doubt as
to what the observed facts really are. Second, a different interpretation of the
available records leads to precisely the opposite conclusion.

As regards the observations, I wish to make it clear at the outset that I do not
undervalue the work of AUSTIN and HOGAN. The investigations which they
undertook were of such difficulty that to obtain results, so consistent as theirs, must
have required much patient labour and an uncommon degree of skill. Any criticism
that I venture to make is prompted solely by the desire to arrive at a more complete
comprehension of the matter.

The method of observation by means of shunted telephone readings appears not to
admit of great accuracy. AUSTIN states that the errors of such readings on board
ship with good operators in good weather may amount to as much as 20 to 40 per
cent. In stormy weather they are increased. There is some obscurity as to the
meaning of the numbers and ordinates which represent in AUSTIN'S tables and
diagrams the values of the " received current." Many passages convey the impression

128

PROF. A. E. H. LOVE ON THE TRANSMISSION OF

that they are proportional to the values of the square root of HOGAN'S "audibility
factor," but this is not stated expressly. As the received antenna current was
inferred in some way from the value of this factor, it seems unfortunate that the
value of the factor was not recorded. The behaviour of the detectors used for the
shunt readings is not thoroughly understood by electricians. For example, FLEMING
(op. cit. ante, p. 397) states concerning the Fessenden liquid barretter, used by HOGAN,
that its action is probably electrolytic, and due to annulment of polarization, rather
than purely thermal. HOGAN does not state how the approximate law of squares
connecting his audibility factor with the received current was verified, but FLEMING'S
statement would seem to indicate that the action of his detectors would approximate
to that of the detector used by AUSTIN. Now AUSTIN has investigated the action of
his detector. In a series of experiments performed at Brant Rock he measured the
same antenna currents in two ways : (l) by means of the shunted telephone and
detector ; (2) by means of a rectifier, which was " connected in a secondary circuit
coupled to the antenna, and calibrated by means of a thermo-element in the antenna
and an exciting buzzer circuit which could be tuned to the wave-length used."
Apparently the rectifier had to be used because it was desired to measure currents too
weak to be measured by the thermo-element, but it is not clear how the rectifier was
calibrated for such weak currents. The results of these experiments are recorded by
him in a table (p. 319), which does not support the conclusion that the current is
proportional to the square root of the audibility factor. The quantities recorded in the
table are the values of the shunt [SJ in ohms, and the antenna current [I] in
micro-amperes, the resistance [R] of the telephone being 600 ohms. From this table
I find that (R + S)/S is nearly proportional to I2 for large values of I, nearly
proportional to I for small values. This result appears from the following table-
compiled from AUSTIN'S.

TABLE IV.

SI

95

122

150

194

474

568

672

361

363

369

373

374

376

376

10

13

15

18

22

26

29

10

10-8

11-5

11-3

11

10-4

9-7

In view of this result, and the fact that most of the numbers recorded by AUSTIN
as values of received antenna current at distances exceeding 500 sea miles are less

ELECTRIC WAVES OVER THE SURFACE OF THE EARTH.

129

than 29, it seemed to be worth while to see if agreement between MACDONALD'S
formula (54) and observation could be attained by supposing the amplitude of the
electromagnetic waves at great distances, and therefore the values of the received
antenna current, to be proportional to the audibility factor instead of its square root.
I therefore compared the formula in the first instance with HOGAN'S long-distance
observations. The result of the comparison is shown in figs. 3 and 4, in which the
abscissfe of the points marked with crosses represent distances, and the ordinates the
values of the audibility factor at those distances, as observed by HOGAN by daylight.
Everything in these diagrams, except the curved lines, is taken from HOGAN'S
diagrams ; and the curved lines are the graph of H/Hi as given by formula (54) for
wave-length 3 '8 km., the scale being adjusted so that the highest point of the curve
in fig. 3 answers to angular distance 9° (1000 km.). It will be seen that the graph

5000

4000

3000

2000

-5 looo

3

\

60
50
40
30

20
10

\

75O 1250 I75O 22SO

D/sbnce in Kilometres

FIG 3 .

2500 3000 35OO 400O

Distance in Kilometres
FIG 4.

fits the observations rather well. I made a similar comparison for HOGAN'S daylight
observations on the wave-length 2 km., and again found a good fit. Then I compared
the values of the square root of the right-hand member of formula (54) with the values
recorded in AUSTIN'S diagrams for wave-lengths 1 km., 1'5 km., and 375 km., and in
every case found that the daylight observations for distances exceeding 500 sea miles
were in good agreement with the formula thus modified.

[Note added February 6, 1915. — After the paper was read, Prof. C. H. LEES
kindly brought to my notice some more recent experiments of L. W. AUSTIN, which
are recorded, at present in abstract only, in the ' Journal of the Washington Academy

Sciences,' of date December 4, 1914. In these experiments the wave-length used
km., and observations were taken at various distances ranging from 556 km.
to 3705,, km. The method of experiment appears to have been the same as in

VOL. ccxv. — A. s

-130 PROF. A. E. H. LOVE ON THE TRANSMISSION OF

AUSTIN'S earlier work. The values of received antenna current, obtained from the
"smoothed curve of observations," are recorded, and compared with the results that
would be obtained from the author's formula (§ 20 above), and with those that would
be obtained from a formula described as the " Sommerfeld transmission formula,"
which is effectively equivalent to (57). I find that, in this case also, the ratios of the
quantities recorded as observed currents at different distances are nearly the same
as the square roots of the ratios of the magnetic forces at those distances, as
calculated from MACDONALD'S formula.]

23. From this critical discussion I draw the inference that there is nothing in the
experimental evidence, at present available, to compel us to adopt the view that the
diffraction theory fails to account for the facts. On the contrary, that evidence can
be interpreted in such a way as to support the view that the results of the diffraction
theory accord well with those of daylight observations. Until more complete experi-
mental data are available the question of the success or failure of the diffraction
theory must remain open.

However this question may ultimately be, settled, the discussion shows that it is
impossible to accept the hypothesis that the law of decrease of the forces of the
electromagnetic field with increasing distance is a combination of the law of spherical
divergence with a law of absorption, expressed by an attenuation factor of exponential
type. However closely such a law may represent the facts it can have no value
except as an empirical formula. In particular, it is not admissible to draw from it
any inference as to the amount of absorption. Independently of any absorption which
may exist, there must be a law of diffraction expressing attenuation of the field on
account of the curvature of the earth's surface ; and the type of attenuation factor
required to express the effect of curvature Avoukl not differ much from the exponential
type in a moderate range of great distances.

The investigations of SOMMERFELD and those of this paper throw some light on the
effect of the resistance of the medium, over the surface of which signals are trans-
mitted. It appears that, with such wave-lengths as are used in practice, a moderate
amount of resistance, such as that of sea-water, increases the strength of the signals at
great distances. A slightly higher resistance would increase it still more, but no
conclusion can safely be drawn as to the effect of so high a resistance as that of dry
ground. It seems to me, however, to be not unlikely that even a rather high
resistance may be favourable, and I am inclined to regard the known fact that
signals are in general appreciable at greater distances over sea than over land, as an
effect of the broken surface of ground covered with rocks, buildings, or trees. This
question also remains to be settled by quantitative experimental investigation.

Another difficult question presents itself in the known fact that the signals are
generally stronger by night than by day, arid the related fact that the attenuation of
night signals by distance is sometimes less than it would be if they diminished simply

ELECTEIC WAVES OVER THE SUEFACE OF THE EARTH. 131

according to the law of spherical divergence. These facts suggest emphatically that
there is, especially at night, some other cause at work besides diffraction, and that it
may be necessary to take into account the possibilities involved in transmission
through a heterogeneous, and perhaps in parts conducting, atmosphere, as is
maintained particularly by EccLEs(16) and (17). The view that ionization in the upper
and middle atmosphere may be a cause of variations of the medium, large enough to
be favourable to long-distance transmission, inasmuch as the waves may be partially
refracted downwards, and liable, on occasion, to be unfavourable, inasmuch as the
refracted waves may be subject to absorption, demands scrutiny, to be conducted by
means of careful quantitative experiments under varying conditions and controlled by
an adequate mathematical analysis. The adoption of this view at present seems to
me to be premature.

It remains to acknowledge gratefully the valuable help which I have received from
Prof. J. S. TOWNSEND, F.R.S., in the discussion of the experimental evidence as to
the strength of signals transmitted to great distances.

S 2

[ 133 ]

V. Atmospheric Electricity Potential Gradient at Kew Observatory,

1898 to 1912.

By C. CHREE, Sc.D., LL.D., F.R.S., Superintendent of Kew Observatory.
Received December 1, 1914, — Read January 21, 1915.

CONTENTS.

§§

1. Historical 133

2, 3. Absolute observations and apparatus ' 134

4. Standardization of curves 136

5. Variation of potential gradient with height 137

6. Correction to old -values of potential gradient 140

7. Mean monthly and yearly values of potential gradient 141

8, 9. Diurnal inequality of potential gradient 142

10. Variation of earth's surface charge 151

11, 12, 13. Diurnal inequality. Fourier coefficients 152

14. Annual variation 150

15. Comparison of Kew and Edinburgh data 157

§ 1. IN a previous paper,* called E: for brevity, I discussed the results obtained for
the diurnal variation of the potential gradient of atmospheric electricity at Kew
Observatory from 1898 to 1904. The present paper deals with the same subject, but
employs data from the fifteen years 1898 to 1912. The earlier period of seven years,
though longer than that available at most observatories, was too short to give a
satisfactorily representative presentation of some of the phenomena. To obtain results
fairly characteristic of the locality many years data are required of some of the
meteorological elements, especially barometric pressure and rainfall. For the latter
element, in fact, a considerably longer period is desirable than that available even now
for potential gradient at Kew. The same may be true of potential gradient itself, but
various reasons exist for not waiting longer. Owing to building operations, the
electrograph results for 1913 were exposed to special uncertainties. Also the transfer
the electrograph from the position it has occupied since 1898 is now in contemplation.
ThuX1912 may be regarded as ending an epoch.

* 'Phil. Trans.,' A, vol. 206, p. 299.

(527.) [Printed March 4, 1915.

134 DE. C. CHREE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

Another reason requires fuller explanation. The Kew water-dropper — the earliest
it is believed in regular operation — -was erected in 1861 under Lord KELVIN'S personal
supervision. The original electrometer and batteries as they decayed were replaced
by others, but the instrument remained essentially unchanged in its original site until
1896. Of the records obtained prior to that date those of only three years had been
discussed, two years, 1862 to 1864, by Prof. J. D. EVERETT,* and one year, 1880, by
Mr. G. M WHIFFLE.! In both cases the results were expressed in what were really
arbitrary units. The relation between the voltage shown by the instrument and the
true potential gradient in the open was altogether unknown.

In 1896, after a few years' experience, I recognised the expediency of avoiding a
variation in the water pressure which tended to affect the apparent diurnal variation,
and of eradicating a shrubbery in the immediate vicinity of the water jet, which
presumably influenced the annual variation to a small extent.

§ 2. The importance was also realised of securing that the curve measurements should
have a definite meaning. The discharge tube, from freezing of the water or natural
decay, required occasional repairs, and had at intervals to be replaced, so that sensible
variation in the position of the jet was at least a possibility. To provide for this, regular
observations were introduced with a Kelvin portable electrometer in the Observatory
garden, at some distance from the building. At first the electrometer was simply
placed on the top of a convenient stone pillar. Presently it was realised that the stone
pillar, whose diameter is considerable, largely reduced the potential in its immediate
neighbourhood, so that the electrometer had to be placed exactly on the same spot if
the results obtained on different days were to be comparable. Accordingly a special
stand was designed. A rod sliding inside a vertical tube, fixed in the ground, carried
at its upper end a small platform. This bore three short vertical pieces at equal
intervals round its perimeter, forming a stand just large enough to take the portable
electrometer. The sliding rod was designed to admit of observations being taken at
two different heights. It was supposed that the reduction of the field due to the
presence of the stand, electrometer and observer would be roughly the same at the
two heights, so that a fair approach to the true potential gradient in the open would
be obtained from the difference of the potentials observed. To reduce the disturbing
effect of the observer's presence, a device was introduced allowing manipulation of the
electrometer from a greater distance than previously. In practice, however, this
device proved troublesome to work. Also the difference — J metre — in the two levels
at which the potential was observed proved too small, in view of the variability of the
potential and the insensitiveness of the electrometer. Some experiments were supposed
to show that the disturbing effect due to the apparatus and observer was less than
I had anticipated. The outcome was that the special device was laid aside, and
observations were confined to one fixed height, about 1'465 metres above ground level.

* ' Phil. Trans.,' vol. 158, 1868, p. 347.

t ' British Association Report for 1881,' p. 443.

AT KEW OBSERVATOEY, 1898 TO 1912. 135

Time was not available for elaborate experiments, and suspicions were considerably
allayed by the fact that the values obtained for the potential gradient were fully higher
than the average of those obtained elsewhere.

§ 3. It was always hoped that an opportunity would present itself for a fuller
investigation, but this did not arise until 1909, when Mr. J. S. DINES was attached
for a time to the Observatory as student assistant. A number of experiments were
made by Mr. DINES. ' A horizontal bamboo rod, carried in a groove made in a paraffin
block, was supported on a small platform, attached to the top of a vertical rod. The
vertical rod could slide up and down inside a hollow tube, supported by three tripod
legs. The height of the bamboo could be altered by sliding the vertical rod or by
altering the stretch of the tripod legs, and the distance to which it projected from the
centre of the tripod could be altered by moving it in the groove of the paraffin block.
A fine horizontal brass tube fixed to the thinner end of the bamboo held the fuse, and
a fine wire passed from the brass tube to a terminal on the thicker end of the bamboo,
from which another fine wire led to the portable electrometer, which was supported
some yards away on a stand of its own. A second essentially identical apparatus was
constructed, and simultaneous observations were made with the fuses of the two at
the same level, but projecting to different distances from their respective paraffin
blocks. Experiments were also made on the distance the one apparatus had to be
from the other to be without sensible effect on it. In another set of experiments the
fuse holder was carried by a horizontal wire stretched between insulated supports,
borne on vertical poles, a considerable distance apart — a method employed by
Mr. C. T. R. WILSON. The potentials obtained in this way and those obtained
immediately before and after at the same spot with the bamboo apparatus, when the
bamboo projected to the extent finally approved, agreed so closely that one could not
say which was the higher. As between the two methods, it thus appeared wholly a
question of greater or less convenience. It was found desirable that smoke from the
fuse should be blown quite clear of all the apparatus. This was easily secured with
the bamboo apparatus, as the mounting enabled it to be readily swivelled round. It
might also have been secured with the other apparatus by having two wires stretched
in rectangular directions, and using one or the other according to the wind direction.
This promised, however, to be somewhat of a complication, accordingly the principle
embodied in the bamboo apparatus was adopted.

The new apparatus which was then constructed, and which is still in use, resembles the
experimental apparatus in having a vertical rod sliding in a tube, but the latter is
sunk to some depth and rigidly fixed in the ground. The sliding rod has towards its
upper end an enclosing short hollow tube, which projects to an extent depending on
the adjustment of a clamp. This outer tube carries a small platform on which is fixed
araffin block, serving at once to support and insulate a horizontal bamboo rod.
The bamboo has the fuse holder at one end, and at the other a terminal connected by
fine wires with the fuse holder and with a portable electrometer at some distance. A

136 DR. C. CHEEE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

shoulder on the long vertical rod, when resting on the top of the ground tube, brings
the platform and bamboo rod to the lowest height they can assume. The platform
and bamboo can be raised in a few seconds to 1 metre above their original height,
by raising the long vertical rod, slipping a pin through a hole in it, and allowing this
pin to rest on the top of the ground tube. We thus secure two positions of the fuse,
differing in height by exactly 1 metre. To secure that the lower position is exactly
1 metre above ground level, use is made of a vertical rod carried by a flat horizontal
board. A permanent mark on this rod is 1 metre above the bottom of the board.
The board is placed on the ground, and the position of the clamp on the short vertical
tube altered until the axis of the fuse, which projects horizontally, comes exactly
level with the mark on the rod. The rod and board are then removed and the
observation proceeds. The apparatus was designed by Mr. DINES and myself, the
material being supplied and fitted by a local builder. The apparatus, though not a
finished workshop article, seems to have hitherto served its purpose reasonably well.
A recent remeasurement made the 1- and 2-metre intervals each correct to within
CT25 mm.

In actual use larger uncertainties exist in the height, at least in the lower position.
The observations are made over a level piece of turf. But a grass surface, even when
newly rolled, is not a mathematical plane, and though the grass is kept short some
uncertainty necessarily prevails as to the exact level to which zero potential should
be assigned. From the mathematical point of view, it would be better to replace the
grass by wood pavement or a carefully levelled flag area. But it is at least open to
doubt whether on a warm, still day an artificial surface would be as satisfactory as
grass.

§ 4. Absolute observations are made on dry days, usually between 10 and
10.30 a.m. The number of monthly observations is usually from ten to twenty.
Observations taken at times when the action of the electrograph appears faulty are
left out of account. The results, when apparently satisfactory, are entered in three
columns. One contains the potentials observed at the 1 -metre level, a second the
excess of potential at the 2-metre level over that at the 1 -metre level, and the
third the potential as measured on the water-dropper curve. This last is deduced
from the length of the curve ordinate and the scale value, the latter being determined
from time to time by means of the same portable electrometer that is used for the
field observations.

Calling the three potential data for the same occasion Pj, P2, and P3, the value is
found for each month of the ratios 2Pj/2P and 2P2/2P. Eepresenting these by ^
and r2, (r^ + r^lZ is accepted as the quantity by which all voltages as measured in the
electrograms of that particular month are to be multiplied in order to obtain the
corresponding potential gradients in the open.

It seems customary to assume that the factor for converting curve values into true
potential gradient is a constant. It is usually determined once for all from a number

AT KEW OBSEEVATORY, 1898 TO 1912. 137

of observations made under favourable conditions, and not redetermined until some
known change is made in the apparatus. Whether this is satisfactory or not depends
on the special circumstances of each installation. It would not be wholly satisfactory
at Kew, where the discharge tube is long and may develop a sag or have to be
replaced. An incidental advantage of regular absolute observations is the check they
afford on the working of the apparatus. They help to disclose defects such as poor
insulation, and secure more careful attention to the electrograph. One of the reasons
for observing potential at 2- as well as 1 -metre height, was the fact that the discharge
tube of the water-dropper is fully 3 metres above ground level. There is little
experimental information as to the differences that may exist in the potential gradient
at different small heights above ground level, and there is no general agreement as to
the height interval from which the standard potential gradient should be obtained.

§ 5. If V denotes the potential, and z be measured vertically upwards, the potential
gradient (dV/dz) must satisfy the two equations

d?V/<.tf + 4!irp = 0, .......... (1)

and

= 0, ......... (2)

where p is the volume density, and or the surface density of the earth's charge.

Unless p is zero the potential gradient will vary with the height, and if p be known
the extent of that variation can be calculated. Various forms of apparatus exist
which profess to measure p, and one of these — the Ebert apparatus — has been in
operation at Kew, with interruptions, since May, 1911. There are, however, reasons
for accepting the results obtained with some reserve. It is now generally recognised
that the Ebert apparatus takes no account, or only very slight account, of the slow
moving heavy ions discovered by LANGEVIN, though it seems to catch the light mobile
ions on the whole satisfactorily. McCLELLAND has found large numbers of Langevin
ions near Dublin, and Kew, from its proximity to London, may also not unlikely have
large numbers of them, at least with easterly winds. Thus the results derived from
the Ebert apparatus as to p will be incorrect unless the positive and negative heavy
ions are at least approximately equal in number.

In spite of this uncertainty, it seems worth while calculating how the potential
gradient may be expected to vary with height at Kew, assuming no free charge in the
atmosphere except that resulting from the excess of the positive or negative mobile
ions caught by the Ebert apparatus. It will have to be assumed that the distribution
of ions is sensibly uniform within 2 metres of the ground because in general the
ions observed at Kew are from air at a fixed height, about 2 metres above the ground.
The only direct evidence in favour of the hypothesis consists of some experiments
made by Mr. GORDON DOBSON* which showed no certain difference in ionic contents

* ' Proceedings Physical Society of London,' vol. 26, 1914, p. 334.
VOL. CCXV. — A. T

138 DR. C. CHREE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

between air collected at the usual level and air taken from immediately above the
ground.

The Ebert apparatus determines the charges per cubic centimetre of the free
positive ions and the free negative ions separately. The difference between these
gives p. The results obtained at Kew in 1911 and 1912 have been published monthly
in the ' Geophysical Journal ' of the Meteorological Office. The 1911 tables give the
number of ions as calculated from Sir J. J. THOMSON'S original value of the ionic
charge, viz., 3'4 x 10~10 electrostatic unit. The 1912 tables give the charges per cubic
centimetre in electro-magnetic units. For the purpose of calculation it is simplest to
employ electrostatic units, i.e., to multiply the numbers in the 1911 tables by 3'4 x 10~10,
and the charges in the 1912 tables by 3xl010. I have similarly dealt with all the
available data down to the end of July, 1914. There were several gaps, so that only
25 months' observations were available. Allowing equal weight to the months of the
several years, the values obtained for />x 109 for the 12 months, January to December,
were in order

31, 54, 30, 65, 70, 54, 60, 65, 65, 48, 39, 39,

all being plus.

The arithmetic mean of the 12 monthly values gives

P = +52xlO-9.
To utilise this result it is perhaps simplest to regard (l) as equivalent to

0, .......... (3)

where M is the free charge enclosed by the surface S, of which n denotes the normal.
Apply this to a tube of force bounded by 1 sq. cm. of the earth's surface, with the
other end on a plane area 1 metre above the ground. Obviously the upper section
will be sufficiently approximately 1 sq. cm. We get

(dVfdz), = „ - (dV/dz), = , metre = 47T x 52 x 1 0-!i x 1 02

= 0'655xlO-4 E.U.
= 0'656 x 300 x 10-4 volts per c.m.
= 2'0 volts per metre very approximately.

This signifies a decline of 2 volts per metre in the potential gradient, or practically
that the potential gradient deduced from the first metre in the Kew observations
should exceed by 2 volts that derived from the second metre.

As will appear presently, the average value of the potential gradient on rainless
days at Kew is about 300 volts per metre. Thus we should have as an average,
employing r2 and i\ as in § 4,

rjr, = 0'993.

AT KEW OBSERVATOEY, 1898 TO 1912. 139

Data for r.Jt\ were available for 48 individual months. The value was TOO in
4 cases ; in 32 cases it was less, and in 12 cases it was greater than unity. Combining
the months of the same name from different years, the means for the 12 months
January to December in order were

I'Ol, 0'96, 0'95, 1'02, I'Ol, 0'99, TOO, TOO, (T99, 1'02, 0'95, 0'91,

giving as arithmetic mean 0'984.

The observed departure of r.Ji\ from unity is in the direction indicated by theory
and is of the right order, but is in excess of the calculated difference. A close agree-
ment could hardly be expected on account of the experimental difficulties, and there
is the further important fact that the potential gradient observations refer to the
forenoon a little after 10 a.m., while the ionic observations refer to about 3 p.m., and the
diurnal variation in the ionic charges has not been ascertained.

On examining details, however, it will be recognised that some other factor
probably comes in. While the mean value of p for the whole 12 months is+52xlO~9,
its mean value for the 6 summer months, April to September, is + 63 x ICT", and that
for the 6 winter months only + 40 x 10~9.

Thus the decline per metre in the potential gradients in summer and winter should
be nearly in the ratio of 3 : 2. The absolute potentials in these seasons, however, as
will be seen presently, stand to one another roughly in the ratio of 2 : 3. Thus the
departure of rjrv from unity should on the average be fully twice as large in summer
as in winter.

The monthly means recorded above, however, give I'OO for the summer value of
rJr\> as compared with 0'97 for winter. The results for r.J-t\ during the first year of
observation, 1910, fluctuated rather markedly, values in excess of unity being
especially frequent in the first few months, when experience was at a minimum. If
we omit all data for 1910, the summer value of r.Jr\ falls to 0'99, the winter value
remaining 0'97, and the mean for the whole year becomes 0'980.

As before, the summer value of r2/>\ exceeds the winter one. Assuming the
difference to be real, a probable explanation immediately suggests itself. The extent
of level turf in the Observatory garden is limited. It is immediately surrounded by
ground devoted to vegetables, and this in turn is enclosed by a hedge about 4|- feet
high. The vegetation must inevitably have some slight effect in lowering the
potential at small heights, and this effect will naturally be greatest at the season when
vegetation is most exuberant. Also while the grass at the place of observation is
mown more frequently in summer than in winter, its average height is probably
greater at the former season. These disturbing influences will naturally be felt
mainly, if not entirely, by the potential observations at the 1 -metre level.

§ 6. Before concluding his engagement at the Observatory, Mr. DINES took some
observations intended to determine the relation between the potential gradients
obtained in the old way with the old apparatus, and those obtained in the new way.

T 2

140 DE. C. CHKEE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

This was less easy than might appear at first sight. The ground immediately round
.the old stand was gravel, and slightly different in level from the surrounding grass.
The observations with the old apparatus referred to a point directly over the stand,
which was a fixture. There was thus no possibility of taking observations with the
old and the new apparatus at exactly the same spot. The observations, moreover,
were not so numerous as might have been desired. The mean result made the
potential gradient deduced by the old and the new apparatus stand to one another
in the ratio I'OO to 1'65.

When the comparison was made there were several fruit trees near the grass plot,
and the observations made with the new apparatus suggested the expediency of their
removal. It was accordingly decided to cut down the trees at the year's end, and to
start the new year, 1910, with the new apparatus. Unfortunately circumstances
did not allow of any direct determination of the effect of removing the trees. Thus
Mr. DIXES' observations did not suffice for the determination of a factor to be applied
to results obtained prior to 1910 to bring them to what they would have been under
the conditions since prevailing. This, however, can, I think, be done fairly
satisfactorily by considering the values obtained for the mean annual potential
gradient. These have been as follows : —

Old apparatus.

New apparatus.

Year. . . 1898. 1899. 1900. 1901. 1902. 1903. 1904. 1905. 1906. 1907. 1908. 1909. 1910. 1911. 1912

P.O. ... 161 179 141 156 145 162 167 167 156 163 148 164

310 301

300

If we divide the 12 years during which the old apparatus was in use into the period
1898 to 1904, treated in E1} and the later period 1905 to 1909, we find 1587 as the
mean potential for the former period, and 159'6 as the mean potential for the latter.
There is thus no indication of any progressive change in the value of the potential
gradient. Moreover, the mean derived from any three successive years departs but
little from the mean 159'1 derived from the whole 12. We are thus unlikely to be
much in error if we regard the means 159'1 and 3037, derived respectively from the
years when the old and the new apparatus was in use, as representing the same real
potential gradient. This gives 1'91 as the factor to be applied to results obtained
prior to 1910 to bring them up to what they would have been if the new apparatus
had been in use under the conditions now existing.

When preparing Et I did not suspect that so large an underestimate was being
made of the absolute value of the potential gradient, but I quite realised that a sensible
correction was probably necessary, and for that reason, amongst others, a number of
the results were expressed as ratios, not as absolute potentials. Even now it cannot

AT KEW OBSERVATOEY, 1898 TO 1912.

141

be claimed that we are getting an absolutely full measure of the potential gradient in
the open. The influence of the apparatus and observer it is believed is small, but it
may not be wholly infinitesimal. Then the site — apart from the peculiarities already
mentioned — does not altogether adequately represent an infinite plane. There are
now no fruit trees within 65 feet of the ground tube, but there are several low
buildings in the neighbourhood, one coming within about 70 feet of the site. Finally
the main Observatory building, the elevation of whose dome above the garden ground
is fully 60 feet, is only about 60 yards away, and there are four elm trees at distances
of from 55 to 80 yards, still of considerable height though they lost their tops many
years ago. There are sites less open to criticism in the Old Deer Park outside the
Observatory enclosure, but the time required to visit a distant site would be a serious
obstacle to its regular use, even if it were available.

§7. In computing the following tables all potential data for years prior to 1910
have been multiplied by 1'91. The spot to which all absolute values refer must be
regarded as the site of the apparatus for absolute observations, which is about 20 feet
above mean sea level in lat. 51° 28' N, long. 0° 19' W.

Table I. gives the mean value of the potential gradient for each of the 180 months
of observation, with corresponding means for each year and each of the 12 months.
The results are from selected rainless days, free from negative potential, 10 in each
month, except in a few cases where 8 or 9 days only were available.

TABLE I.— Mean Monthly and Yearly Values of Potential Gradient

(volts per metre).

j

h

^

£>

t

£

o

o

Year.

|

ce

O

*

00

S

o>

CD
,a

2

Q

1=

Mean.

i

J
o

1

&

3'

a

P

50

OH

2

CJ

O

0

i-i PM g

•^

^

"3

o

&

o

1898

332 466 413

_
208

185

199

199

239

311

308

477

351

308

1899

544 569 i 388

265

288

201

172

243

159

334

319

626

342

1900

361 348 344

204

183

139

164

199

225

254

300

502

269

1901

455 472 : 269

267

243

218

181

166

223

365

411

309

298

1902

313 443 290

285

178

220

164

212

199

235

340

453

277

1903

309

332 : 365

315

319

298

201

246

258

269

329

477

309

1904

371 363 338

306

248

202

223

225

248

281

495

521

319

1905

418 323

315

311

248

235

258

290

319

309

462

340

319

1906

363 380 351

300

180

241

241

202

273

267

321

466

298

1907

378 435

367

281

275

191

296

220

237

239

432

372

311

1908

527

304

399

357

197

210

168

199

199

216

336

288

283

1909

382 ! 397

309

380

294

166

168

254

248

235

432

477

313

1910

399 i 432 ' 386

262

273

203

232

202

206

330

450

350

310

1911

457

345

378

286

249

219

209

217

250

373

366

269

301

1912

523 337

228

271

204

167

259

231

334

399

316

332

300

Mean .

409

396

343

287

238

207

209

223

246

294

386

409

304

142 DE. C. CHKEE: ATMOSPHEEIC ELECTEICITY POTENTIAL GEADIENT

Even fair weather days in the same month vary immensely amongst themselves
electrically, and there is a good deal that is "accidental" in means derived from 10
days only. When the rainless days available are largely in excess of 10, as is
frequently the case, the natural tendency, at least in summer, is to prefer days in which
the potential is high to those in which it is low. The former usually show the diurnal
variation more clearly, and there is in their case a greater presumption against bad
insulation, a not infrequent occurrence in damp weather, or at seasons when spiders
are most busy. Thus the mean potential in a month when the choice of days is small
is apt to come low. There are, however, individual winter months in which the
contrary tendency prevails. During fog the potential at Kew is often much above the
average. If only part of a day is foggy one prefers to omit it, because the diurnal
variation shown is largely dependent on the accident of what hours the fog was
incident. If, however, the choice of days is very restricted, one is obliged to include
days of intermittent fog. Again the number of days available depends considerably on
how the apparatus has been working. At times there are somewhat numerous
defects, whether from poor insulation, freezing of the jet, accident or lack of
attention, which restrict the choice, leading to the same consequences as frequent rain.
Finally sensible changes sometimes take place between two successive scale deter-
minations, leading to uncertainty in the scale values to be applied. Thus " accident "
plays some part in the individual monthly values in Table 1.

Judging by the fluctuations in the values for months of the same name, a longer
period would be desirable. The smoothness in the annual variation derived from the
15 years is, however, truly remarkable in view of the irregularities visible in the
corresponding variations deduced from either of the sub-periods 1898 to 1904 or
1905 to 1912. It can hardly be doubted that it is a fair approximation to normal
conditions.

The results are shown graphically in the top curve of fig. 4, p. 142, unity representing
304 volts, the mean value for the whole year. The maximum and minimum appear
respectively at about midwinter and midsummer, and so somewhat in advance of
the times of minimum and maximum of temperature at ground level. They are
naturally still more in advance of the times of minimum and maximum temperature
at any depth underground. This suggests that direct solar radiation has more to do
with the annual variation than earth temperature or meteorological conditions in the
lower strata of the atmosphere.

The extent of the annual variation in potential gradient is pretty similar to that in
vapour pressure or density. But the minima of these elements at Kew occur in
February and. the maxima in July, while June and September values are closely alike.
Thus a formula of the type once proposed by ELSTER and GEITEL dV/dn = A/(l + kq0),
where A and Jc are positive constants and q0 vapour density, cannot be made to fit the
Kew data very satisfactorily.

§ 8. In E! conspicuous differences were pointed out between the diurnal inequalities

AT KEW OBSEKVATORY, 1898 TO 1912. 143

derived from 1898 to 1904 and those derived from earlier epochs. As this might arise
either from the periods available having been too short to eliminate accidental features
or form a real change in the nature of the phenomena, data are given here for the period
1905 to 1912 as well as for the whole 15 years. Between 1901'5 and 1909'0, which
may be regarded as the epochs to which refer the data in E15 and those from the years
1905 to 1912, there was considerable growth in Western London, and if the difference
between EVERETT'S results for 1862-4 and mine for 1898 to 1904 were due to
urban extension, then the data for 1905 to 1912 would naturally depart still further
from EVERETT'S. Table II. gives the diurnal inequalities for the whole 15 years, and
Table III. those for the seven years 1905 to 1912. The highest and lowest hourly
values are in heavy type. The hours are really G.M.T., but local time is only
1^ minutes after Greenwich. For purposes of comparison data from Table III., p. 306,
of Ej must be multiplied by 1'91.

Besides the diurnal inequalities for the 12 months, Tables II. and III. contain diurnal
inequalities for the whole year and for three seasons, winter (November to February),
equinox (March, April, September and October), and summer (May to August). The
inequalities for the 12 months in Table II. are shown graphically in fig. 1, p. 146. This
is immediately comparable with fig. 2 of E1; provided the line there marked 50 volts be
regarded as representing 95. To facilitate recognition of the resemblances and differ-
ences between the data from different epochs, fig. 2, p. 147, shows in juxtaposition the
diurnal inequalities for the year and the three seasons derived from 1898 to 1904
(upper curves), the whole 15 years (central curves), and 1905 to 1912 (lower curves) ;
fig. 3, p. 148 contains isopleths for the diurnal variation of the 15 years. For it lam
indebted to Mr. E. H. NICHOLS, B.Sc., professional assistant at Kew Observatory. To
assist the eye, the isopleth curves are drawn thicker or thinner according as they
answer to potentials which are greater or less than the mean for the year, 304 volts
per metre. The two broken lines indicate respectively the times of sunrise and sunset
throughout the year.

A careful comparison of Tables II. and III., or of corresponding curves in fig. 1 and
in fig. 2 of E,, shows differences between the monthly diurnal inequalities derived
from different epochs, but some of these are undoubtedly " accidental." In the seasonal
diurnal inequalities in fig. 2, where "accident" is more completely eliminated, the
differences that exist — except as regards the relative depths of the morning and after-
noon minima — make small appeal to the eye. It requires an auxiliary such as analysis
into Fourier series to show their nature distinctly. It may reasonably be inferred that
no large rapid change is in progress in electrical conditions at Kew, and that the
inequalities derived from the 15 years give a close approach to normal conditions.
Whether the results are sensibly dependent on the particular site selected for the water-
dropper it is impossible to say until adequate data are available from a different site.

A glance at fig. 1 shows a conspicuous difference between midsummer and midwinter.
The time elapsing between the forenoon and afternoon maxima and the depth of the

144

DE. C. CHEEE: ATMOSPHERIC ELECTEICITY POTENTIAL GRADIENT

TABLE II. — Diurnal Inequality

Hour

(G.M.T.).

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

January . .

-51-9

-69-0

-88-5

-104 1

-97-0

-73-3

-31-8

+ 14-9

+ 50-4

+ 61-3

+ 39-4

+ 24 0

February . .

-39-0

-64-7

-87-3

- 96 6

-91-8

-76-4

-32-4

+ 23-4

+ 62-3

+ 77 1

+ 50-7

+ 19-4

March . . .

-19-8

-46-6

-69-0

- T8-1

-68-3

-38-9

+ 4-1

+ 38-8

+ 58-6

+ 45-7

-1-1

-33 3

April . . .

-14-4

-377 -52-4

- 59 0

-46-6

-187

+ 25-2 + 54 -1

+ 50-6

+ 23-0

-17-7

-40-1

May. . . .

-3-4

-21-3 -35-9

- 40-9

-27-2

-69

+ 19-3

+ 37-7

+ 36 -0

+ 16--1

-13-9

-29-7

June . . .

-9-9

-23-8 -33-8

- 35-2

-20-3

- 3-7

+ 19-7 439-2

+ 38-0

+ 24-9

+ 1-4 -18-0

July. . . .

-22-8

-40-4 -42'6

- 43 6

-30 6

-8-2

+ 29-0 + 54 -0

+ 58-2

+ 36-8

+ 3-1 -20-5

August . . .

-25-4

-417 -53-3

- 49-2

-36-4

-5-4

+ 29'3 + 52 '9

+ 56-4

+ 32-4

+ 3-2 -15-8

September . .

-33'6 -49-5 '-5o'l

- 60 1

-50-3

-24-0

+ 6'3 +32-9

+ 36-8

+ 21-5

-4-2 -16-9

October

-50-5 -570 -57 5

- 52 5

-38-4

- 13 -4

+ 19-0 + 54 '4

+ 61 -8

+ 43-2

+ 5-5 -18-2

November . .

-43-8

-59-3 -68-1

- 69 3

-58-0

-31 -4

- G-2 +26-8

+ 41-3

+ 44-5

+ 16-9 - 6-5

December .
Year . . .

-57'3
-31 -0

-76-0
-48-9

-84-4
-GO 7

- 88-9

-82-8
-54-0

-63-7

-30-1

+ 4-3

+ 15-6

+ 47-5

+ 537

+ 29-9

+ 137

- 64 8

-30-9

-t- 37 -1

+ 49-7

+ 40-0

+ 9-4 -11-8

Winter .

-48-0 -67-3

-82-1

- 89 -T

-82-4

-63-0

-25-1 +20-2

+ 50-4

+ 59-2

+ 34-0 +12-7

Equinox

-29-6 -477 -58-5

- 62 4

-50-9

-23-8

+ 13-6 +45-1

+ 52-0

+ 33-4

_ 4-4

-27-1

Summer

-15 '4 -31-8 -41-4

- 42 '2

-28-6

-6-0

+ 24-3 +46-0

+ 4(5-9

+ 27-6

- 1-6

-21-0

TABLE III. — -Diurnal Inequality

1
Hour 1 •' 3

4. 5.

6. 7. 8.

9.

10.

11.

12.

(G-.M.T.).

January . .'-63-2 -82 '8 -108 '0

-125-8-111-3 -84-6 -367 + 20'1

+ 59-5

+ 63 -8

+ 44-3

+ 307

February . . -40 '6 ,-63'0 - 87 '4

- 94 -8 - 90-5 -79 0 -35 '5 +21 '9

+ 59'1

+ 73-9 + 48 '6

+ 13-8

March . . .'-33 '9 -62 '5 - 84 '6

- 93 9 - 83 •;> -51-4 - 1 '0 + 38 '6

+ 62 -4

+ 57-0 +13-4

-20-2

April . . .

-20-6 -51-4 - 70-0

- 80 4 - 63-1

-43 -1 +1-9

+ 36-0

+ 43-4

+ 23-0 -11-8

-307

May. . . . -17-0 -28-5 - 52 '6

-56-8 - 40-0

-17-4 +14-5 +87-4

+ 43-5

+ 26-8 +2-1 -12-5

June . . . -12-8 -2t>-5 - 38 • 4

- 41 -8 - 26-5

-9-9 +15-1 +42-2

+ 40-6

+ 25-4 +3-0 -13-3

July. . . . -27-4 -53-8 - 55 'T

- 52'9 - 40 3 -20-8 +187 +52 '3.

+ 63 4

+ 42-7 + 8-9 -19-0

August. . . -30-8 -46-8 - 55 7

- 47-1

- 35-6

-1-4 i + 33'9 i + 64-O

+ 62-4

+ 36-8 +0-1

-17-2

September .

-48-9 -G3-4 - 71'0

-76-9 - 62-5

-33-9 +0-5 +32-9

+ 37-4

+ 26-4 +4-9

-7-6

October . .

-54-9 -64-6 - 66 T

- 59 -4 - 41 -0

-18-3

+ 11-6

+ 49-3

+ 60-7

+ 49-2 +10-8

-10-3

November . . -54 '2 -71 '4 - 82 '8

- 89 0 - 78-1

-537

-13-8 +31-8

+ 55 '5

+ 5T-0 +80-6 + 27

December . .

-07 -4

-80-0 - 82-4

- 82-6

- 75-8

-59-2

-35-8

+ 18-8

+ 50-2

+ 50-1 +82-3

1

+ 13-0

Year . . .

-39-3

-58-7 - 71-3

- 75-1

- 62-9

-39-4

-1-0

+ 37-1

+ 53-2

+ 44-3

+ 15-6

-5'9

Winter . . .

-56-4

-74-3 ]- 90-2

- 98 1

- 83-9

-69'1

-28-0

+ 23-1

+ 56-1

+ 61-2 + 88 -9

+ 15-0

Equinox . .

-39 -6 .

-60-5 - 73-1

- 77 6

- 64-1

-367

+ 3-3

+ 39-2

+ 61-0

+ 38-9

+ 4-3

-17-2

Summer . .

-22-0

-41 4 - 50 6

- 49-6

- 35-6

-12-4

+ 21-8

+ 49-0

+ 52-5

+ 32 "9

+ 3-5

-16-6

AT KEW OBSERVATORY, 1898 TO 1912.

145

1898 to 1912 (volts per metre).

13.

14.

15.

16.

17.

18.

19.

20.

21. 22.

23. 24. Range.

I

A.I).

+ 8-6

+ 1-8

+ 7'2

+ 21-2

+ 43-1

+ 53-9

+ 63 5

+ 6) -0

|
+ 49-6 +371 '+9-5 -28-3 167 '6

45-5

-14-0

-32-5

-24-9

-11 -0

+ 19-1

+ 48-4

+ 71-6

+ 68-8

+ 67-8 + 51 '3

+ 21-8 -11-4 1737

48-5

-47'9 -50-5

-44-2

-33-0

-7'6

+ 29-3

+ 71-1

+ 85-T

+ 80-4 + 72 -5

+ 43-5 +7'9 1E3-8

44-8

-537 -52-3

-47-0

-38-6 -23-0

+ 9-4

+ 55 -2

+ 82-9

(-83-6 + 69 -5

+ 39 '3 +8'5 142 '6

41 '8

-44-2

-53 0

-45-3

-38-9

-26-8

+ 3-2

+ 40-9

+ 60-6

+ 69-3 +57-5

+ 34-2 +12-8 122-3

32-3

-32-9

-40-3

-41-3 :-35'l

-20-4

+ 2'9

+ 20-9

+ 41-8

+ 52-1 +43-2

+ 23-8 +6'5 93 '4

2G-2

-29-3 -36-9

-36-5 -32-6 -21 '4

- 5-1

+ 18-2

+ 427

+ 57-2 + 4S -3

+ 25-2 - 2-4 101-8 31-1

-26 0

-39 0

-44-4 -38-9 -23-1 + 3 '6 +34 '5

+ 56-6 + 62 1 +49-4

+ 26-5 - 7-3 115-4 33-8

-23-2 -25-9

-21 -4

-11-2 +127

+ 44-0 +66 -4

+ 67 5

+ 5G -4 + 37 '6

49-6 -15-4 127-6

32 -0

1

-3L-3 -337

-20-1 - 0-8 +32-9 +51. '9 +577

+ 48-2 + 31 -3 + 10 '4

-107 -35-9 119-:!

35-0

-14-4

-10-2

+ 7'3 +25-6 +41-2

+ 49-4 + 48 -9

+ 47'3 + 387 + 1G-3

- 0-1 -28 -0 118-7

3 i 9

+ 87

+ 7-2

+ 18-8

+ 37'2 + 49 •!)

+ 59 6

+ 59 6

+ 50-8

+ l!9 -C + 24 'I

+ 0-7 -33-5 148-5

43-1

-25-0

-30-6

-24-3

- 13 -0 , + (5 '6 + 29 '5 + 50 7

+ 59 5

+ 57'3 + 43 ' 1

+ 18-1 -10-G 121-3

33-8

-2-8

-8-5

+ 2-1

+ 18-3 +39-1 +52-8 '+609

+ 57 -0

+ 48 '9 + 32 "2.

+ 6-5 -25'4 150 -0

41 -2

-39-0

-40-6

-33-2

-20-9 +3-8 +34-4 + 62 '6

+ 71 1

+ 62'9 + 47 -5

+ 20-4 - 8-7 133-5

37-2

-33-1

-42 3

-41-9

-36-4 -22-9 +1-2 + 28 '6

+ 50-4

+ 60-2 + 49 -G

+ 27-4 + 2-4 102-5

30-4

1905 to 1912 (volts per metre).

13.

14.

15.

16. 17.

IS.

19.

20.

21.

22.

23.

24.

Range.

A. IX

+ 7'2

+ 8-6

+ 22-0

+ 35 '4

+ 54 7

+ 63 -3

+ 71 -2 + 74 -4

+ 57-5 +36 "2,

+ 2-2

-38-3

200 -2

I
547

-8-2

-29-5

-24-9

-11-4 +19-9

+ 52 -5

+ 75-4 + 70-4

+ 71 -fi + 53 -0

+ 22-4 - If! -6

170-2

4H -5

-33-4

-35-9

-26-5

-20-8 + 3-4

+ 34 -9

+ 73-0 + 80-6

+ 75-5 +67-4

+ 39-4 +0-6

174-5

45-6

-42-5

-39-8

-31-1

-25-9

- 9-1

+ 21 -2

+ 70-5 +102-4

+ 99-0 +76-2

+ 42-3 +8-6

182 -8

43-7

-29-1

-41-1

-34-8

-32-9 -21 -9

+ 9-3

+ 48-6 + 65-6

+ 70-2 +55-3

+ 23 -2-2 -4

127-0

33-1

-25-1

-30-5

-30-6

-28-5 -12-4

+ 9-1

+ 21-9 + 40-4

+ 49-8 +34 -0

+ 14-9 - 0-1

91-6

247

-24-0

-29-7

-26-8

-26-3

-14-1

+ 3-0

+ 27 -0 , + 47 '2 + 56 '6

+ -J9-9

+ 26-6 - 5/4

119-1

33-0

-22-6

-39-0

-44-0

-40-4

-25-9

+ 0-3

+ 34-0 1+57 'I

+ 62-2

+ 44-0

+ 20-0 -13-0

1197

35-0

-14-8

-13-4

-10-6

-0-2

+ 25-4

+ 49-1

+ 70-1

+ T5-8 +60-8

+ 37-0

+ 5-1 -23-4

152-7

35-5

-15-2

-137

- 0-2

+ 12-4

+ 4S-8

+ 61 0

+ 55-2

+ 36-8

+ 13-8

-5-4

-19-6 -38-0

127-7

34-0

-57

-3-2

+ 16-2

+ 32-4

+ 45-2

+ 51-3 | + 52 "3

+ 52-7

+ 40-0

+ 21 -2

-5-6 -SO -7

146-0

40-7

+ 6-2

+ 7-8

+ 23-3

+ 427

+ 55'6

+ 63 8

+ 58-8

+ 50-2

+ 35-9

+ 15-2

-8-3 '-41 -2

146-4

43-6

-17-3

-21-6

-14-0

-5-3

+ 14-1

+ 34-9

+ 54-9

+ 68 7

+ 577

+ 40-3

+ 13-6

-167

137-8

357

-o-i

-4-1

+ 9-1

+ 24-8

+ 43-8

+ 577

+ 64 4

+ 61-9

+ 51-2

+ 31 -4

+ 27

-317

162-5

45-1

-26-5

-257

-17-1

-8-6

+ 17-1

+ 41-6

+ 67-4

+ 73 T

+ 62-3

+ 43-8

+ 16-8

-13-0

151-3

38-3

-25-2

-35-1

-34-1

-32-0

-18-6

+ 5-4

+ 32-9

+ 52-6

+ 59 7

+ 45-8

+ 21-2

-5-2

110-3

31-4

VOL. CCXV. — A,

U

146 DK. C. CHREE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

JAN.

FEB

MAR

APR

MAY •

JUN

JUL

AUG.

SEP.

OCT.

NOV.

DEC

MIOT 2 4 6 8 10 NOON 14 16 18 20 22 M DT
Fig. 1. Diuniiil inequalities, 189S-1912.

AT KEW OBSERVATORY, 1898 TO 1912.

147

MIDT 2 4 G 8 10 NOON !4 16 18 20 22 MIDI

Fig. 2. Diurnal inequalities,
x x 1898-1904. x x 1898-1912. x-- --x 1905-1912.

U 2

148 DR. C. CHEEE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

intervening minimum diminish as we approach midwinter. But the transition is so
gradual that it is difficult to draw a line between summer and winter months. This
is one of the reasons for dividing the year into three seasons.

JAM

FES

MAR APRIL MAY JUNE JULY AUGUST SEPT OCT NOV DEC

APRIL MAY JUNE JULY AUGUST SEPT

Fig. 3. Isopleths.

. Potentials above mean.

„ below „
Times of sunrise and sunset.

OCT

NOV

DEC

The November, December, and January curves are of very similar type. The
February curve resembles the March curve almost as closely as it does the January

AT KEW OBSERVATORY, 1898 TO 1912.

149

curve, and but for the desirability of having the three seasons equal, February might
have been included under equinox and April under summer.

When the type of the diurnal inequality varies but little with the season the range
usually gives a very fair idea of how the intensity of the forces to which the regular

J F M A M' J' j" X S 0 N D

1-5

1-0
05

1-5
1-0

0-5

I -5
1-0
05

Z-0
15

1-0
(J5

1-5

1-0
0-5

M A M' J" j" A' S 0
Fig. 4. Annual variation.

. A.D.

Mean

potential

gradient.

Range

diurnal

inequality.

N D

diurnal changes are due vary throughout the year. An exception to this may arise
when the curves are of a sharply peaked type, unless the times of daily maximum and
minimum happen to fall very close to exact hours. Peaked curves are, however,
unusual in the diurnal inequalities of natural phenomena.

If the type varies much, a better idea of the activity of the forces is derived from

150

DR. C. CHREE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

the quantity termed A.D. (signifying average departure from the mean) in Tables
II. and III. This represents the sum of the 24 hourly differences from the mean —
taken regardless of sign — divided by 24.

The ranges and average departures expressed in terms of their arithmetic means are
represented in the second and third curves of fig. 4. Both curves show a maximum
in February and a minimum in June, with at least a suggestion of a secondary
maximum in Autumn and a secondary minimum in November. But the data from
1905 to 1912 put the principal maximum in January and give a less clear indication
of a secondary maximum and minimum, the latter appearing to fall in October.
Thus while there is clearly a principal maximum towards midwinter and a principal
minimum towards midsummer, the precise times of their occurrence, and the existence
of a secondary maximum and minimum are questions it would require a still longer
period of years to settle satisfactorily.

It should be remembered that when data from a large number of years are combined
there are two distinct causes which lead to large diurnal inequalities, firstly the
persistent occurrence of large amplitudes, and secondly close agreement in the times
of maximum and minimum in months of the same name. If the phase varies much
from year to year, data from different years may to a considerable extent neutralise
one another, and we may form an inadequate idea of the average intensity of the
forces causing the diurnal variation.

§ 9. Our ignorance of the cause of the diurnal inequality renders it difficult to say

TABLE IV.

Month or season.

Ratios borne to mean monthly or seasonal values
of potential gradient.

Night fall
Day fall '

Inequality range.

Day fall.

Night fall.

January

0-41
0-44
0-48
0-50
0-51
0-45
0-49
0-52
0-52
0-41
0-31
0-36

0-15
0-28
0-32
0-38
0-38
0-38
0-4G
0-45
0-25
0-32
0-15
0-11

0-41
0-42
0-48
0-50
0-46
0-42
0-48
0-52
0-52
0-39
0-31
0-36

2-79
1-53
1-50
1-32
1-21
1-10
1-06
1-16
2-04
1-21
2-02
3-19

February

March

April

May

June ....

July

i *
August ....

September
October

November ....

December

Year

0-41

0-38
0-46
0-47

0-26
0-17
0-32
0-41

0-41
0-38
0-46
0-47

1-55
2-22
1-44
I'll

Winter .

Equinox ....

Summer ....

AT KEW OBSERVATORY, 1898 TO 1912.

151

what features are of most importance. Any departure from the mean, whether rise
or fall, is followed sooner or later by a change in the other direction. Which change
is the most fundamental, or whether they possess equal significance, are questions to
which no reply is at present possible. The fact that the two falls shown by the
diurnal inequality occur, the one wholly in night hours the other wholly in day hours,
suggest these falls for consideration in preference to the forenoon and afternoon rises
of potential. They are compared in Table IV. with one another and with the range
of the diurnal inequality. When the principal maximum occurs in the afternoon
and the principal minimum in the morning, as is generally the case at Kew, the night
fall and the range are identical.

The quantities in the three first columns of Table IV. are expressed as fractions of
the corresponding mean monthly values given in Table T. The ratio borne by the
inequality range to the mean value for the month is much less variable than is the
absolute value of the range, and a similar remark applies to the night fall. On the
other hand, the day fall actually diminishes as the monthly mean increases. At
midsummer it is very nearly as large as the night fall, but at midwinter it is only
about one-third as large as the latter.

§ 10. A positive value in the potential gradient implies a corresponding negative
charge on the earth, the surface density being given by (2) of § 5. Any change in
the gradient implies a change in the surface density. The change in the charge on a
sq. cm. of the earth's surface shows only the difference between what enters and leaves
it. The change may represent but a small fraction of the current traversing the
element of surface. The increment to the charge may also arise from a vertical
current or a horizontal current, or partly from the one and partly from the other. As
a matter of fact, we know that usually a vertical air-earth current exists, as well as
so-called earth currents whose direction is mainly at least horizontal. Eartli currents
are much in evidence during magnetic storms, but no certain relation between
potential gradient changes and magnetic storms seems yet to have been established.
The normal diurnal inequality of potential gradient could be connected only with
normal air or earth currents, and information about either of these phenomena is
limited. On rainless days the air-earth current seems normally to be directed down-
wards throughout the whole 24 hours, so it alone cannot account for changes in the
surface charge. It is, however, of some interest to compare the relative sizes of the
air-earth current and the alterations of surface charge calculated from the changes in
potential gradient. February is the month which shows the largest daily changes in
Table V. The hourly changes in it going to the nearest volt are as follows :—

j

1

I

Hour . 0-1. 1-2.

2-3. 3-4.

4-5. 5-6.

6-7.

7-8. 8-9. 9-10.

10-11.

11-12.

I

I !

I 1

Forenoon ... - 28 - 26

-23 - 9

+ 5 +15

+ 44

+ 56 +39 +15

-26

-31

Afternoon . . -33 - 19

+ 8 +14

+ 30 +29

i

+ 23

- 3 - 1 -16

1

-30

-33

152

DE. C. CHREE : ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

When the sign is plus the surface density is increasing numerically. The total
increase in the negative charge per sq. cm. from 4 a.m. to 10 a.m. amounts to
153xlO~18 coulombs, and the most rapid hourly increase — occurring between
7 and 8 a.m. — is 49 x 10~15 coulombs. This is equivalent to a steady current of
l'36xlO~17 amperes per sq. cm. The average value of the air-earth current,
as measured at various stations, is in the neighbourhood of 2xlO~18 amperes
per sq. cm., or about 15 times as big as the above current. Thus in general the
changes in surface density must represent the balance between a much larger air-earth
current and earth currents whether vertical or horizontal.

§ 11. Table V. gives the amplitudes and phase angles of the 24-, 12-, 8- and 6-hour
Fourier "waves" in the diurnal inequalities derived from the whole 15 years, and

TABLE V. — Diurnal Inequality 18(J8 to 1912. Amplitudes and Phase Angles.

Month or

season.

0.

*i.

('.,,

z.,.

fa-

03.

et.

«4.

0

0

0

.

January . . •

57-72

209 • 8

46 • 23

174-4

14-39

22

5-47

261

February

44-43

201-8

61-41

172-4

18-81

5

5-24

238

March . '. .

29-06

156-9

64 • 50

183-1

10-58

45

8-78

301

April ....

23-18

128-1

60-02

187-8

7-04

95

12-52

312

May ....

20 • 82

108-2

46-86

185-9

3-31

100

7-69

317

June ....

9 • 69

97-9

40-52

184-1

3-16

89

4-32

304

July ....

2-95

118-0

47-97

183-9

7-04

110

8-67

285

August . . .

6-29

151-0

52-63

185-5

3-87

123

7-76

317

September . .

26-97

183-8

45-18

196-7

5-63

28

7-15

331

October .

23-89

222-0

48-50

209-4

14-17

30

3-92

290

November .

39-13

211-5

36-60

198-5

13-22

42

5-44

253

December . .

57-52

212-9

37-81

184-5

13-18

33

6-57

241

Arithmetic "1

28-47

49-02

9-53

6-96

means J

' Year ....

23-16

190-4

48-32

186-6

8-01

41-5

6-08

292-7

Winter . . .

49-57

209 • 3

44-82

180-6

14-44

23-7

5-61

248-1

Equinox . .

21-62

172-1

53-77

193-0

8-55

45-1

7-89

310-4

Summer. .. .

9-50

112-8

46-99

184-9

4-26

107-2

6-89

305-6

Table VI. does the same for the period 1905 to 1912. The inequalities are supposed
to be represented by the formula

Cj sin (t + a,) + ca sin (2t + «2) + . . . ,

t being counted from local midnight, and 15° being taken as the equivalent of one
hour. The inequalities in Tables II. and III. referred to G.M.T., and the phase angles
were first calculated in terms of G.M.T., and then corrected to local mean time by

AT KEW OBSERVATORY, 1898 TO 1912. 153

TABLE VI. — Diurnal Inequality 1905 to 1912. Amplitudes and Phase Angles.

Month.

fi.

«!.

<*.

a*

C3-

«8-

C4.

a4.

January . . .

71-06

0

211-6

50-99

0

177-3

15-94

30

7-55

0

271

February . .

46-08

200-1

60-51

173-2

18-16

0

5-62

248

March

34 • 88

183-0

65-09

180-5 12-14

50

8-10

294

April ....

36-18

157-4

61-89

183-4

5-21

91

13-05

309

May ....

15-64

154-5

50-59

183-6 0-97

92

8-28

319

June ....

6-20

148-7

38-31

184-8 4-38

72

6-06

313

July . . • .

9-77

196-7

50-29

182-4

7-8G

09 8-96

279

August . . .

2-44

188-1

54-91

188-2

4-89

131

10-07

319

September . .

38-01

190-9

47-63

195-3

5-02

34

7-18

324

October . . .

35-22

229-9

41-30

212-2

16-58

39

1-60 294

November . .

50-22

216-0

41-67

189-0

15-77

44

7-70

258

December . .

58-54

216-5

36-51

193-0

12-94

31

7-56

230

adding +19' to al5 +38' to aa, and so on. 19' represents the equivalent of the
difference between Kew and Greenwich solar time, viz., about 1-j- minutes.

In the corresponding Tableau E, (Table V., p. 311) the phase angles, it should be
noticed, refer to G.M.T.

The monthly values of c, and <?3 from Table V. are expressed in Table VII., first in
terms of their own arithmetic means, and secondly in terms of the corresponding mean
monthly potential gradient. The former results are shown graphically in the two
lowest curves of fig. 4. The last column of Table VII. expresses Cj as a fraction of ca.

TABLE VII. — Diurnal Inequality. Amplitude Ratios.

Ratios to their own

Ratios to mean monthlv values Mutual

arithmetic means.

of potential gradient. ratio.

Month or season.

f\.

£2-

''i-

r.>. <', /,..

January . . .

2-03

0-94

0-141

0-113

1-25

February. . .

1-56

1-25

0-112

0-155

0-72

March.

1-02

1-32

0-085 0-188

0-45

April ....

0-81

1-22

0-081

0-209

0-39

May ....

0-73

0-96

0-087

0-197

0-44

June ....

0-34

0-83

0-047

0-196

0-24

July ....

o-io

0-98

0-014

0-230

0-06

August . . .

0-22

1-07

0-028

0-236

0-12

September .

0-95

0-92

0-110

0-184

0-60

October .

0-84

0-99

0-081

0-165

0-49

November .

1-37

0-75

0-101

0-095

1-07

December . .

2-02

0-77

0-141

0-092

1-52

Year ....

0-076

0-159

0-48

Winter . . .

—

—

0-124

0-112

I'll

Equinox . . .

—

—

0-074

0-183

0-40

Summer . . .

—

—

0-043

0-215

0-20

VOL. CCXV. A.

154 DR. C. CHREE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

When the phase angle in a Fourier wave varies much throughout the year,
contributions to a diurnal inequality from different months to some extent neutralise
one another. In such a case the amplitude derived from the annual or seasonal
diurnal inequality may give a less accurate idea of the average size of the Fourier
wave than is supplied by the arithmetic mean of the amplitudes derived from the
inequalities of the individual months. For this reason arithmetic means of the
c-coefficients are given in Table V., in addition to the values derived from the mean
diurnal inequality for the year.

The phenomena presented by the 24-hour wave in Tables V. and VI. favour the
division adopted of the year into three seasons. In the four winter months c: is
conspicuously large and the variations in a^ small. In the four summer months, on
the other hand, r, is conspicuously small and a, is distinctly less than in winter, i.e., the
hours of maximum and minimum are later. The equinoctial months represent a
transition from summer to winter.

In the 12-hour wave there is much less seasonal variation either in amplitude or
phase angle. The amplitude of the 8-hour wave is fully as suggestive of the more
usual division of the year into a summer half, April to September, and a winter half;
but the phase angles favour the division adopted into three seasons. In the 6-hour
wave the four winter months resemble one another in the smallness of the phase angles,
but there is no marked difference in this respect between equinox and summer.
There is obviously a good deal that is " accidental " in the amplitudes and phase angles
obtained for individual months. Little significance, for instance, can be attached to
the smallness of the July value of cl in Table V., or of the August value of ct in
Table VI.

§ 12. There are marked differences between the results for cl and «t in Tables V. and
VI., but as these arise from differences between the earlier and later of the 15 years
they are best studied by comparing data for the periods 1898 to 1904 and 1905 to
1912. This comparison is made in Table VIII. It is confined to arithmetic means of
the amplitudes, and to seasonal values of the amplitudes and phase angles, which
suffice to bring out the main features. The data for 1898 to 1904 have been derived
from E! by multiplying the amplitudes by 1'91 and altering the phase angles from
G.M.T. to local mean time. If any hesitation is felt in accepting the multiplier 1'91,
it may be pointed out that it brings the mean values of the potential gradient for the
two periods into agreement. Practically identical conclusions would have been reached
if we had expressed the Fourier amplitudes not in absolute measure, but in terms of
the corresponding mean potential gradients. The multiplier does not affect the phase
angles.

The outstanding feature in Table VIII. is the smallness of the differences between
the results for the 12-hour wave from the two epochs. In the case, however, of the
24-hour wave the differences are obvious. The average amplitude is considerably
larger in the second period than in the first, though summer shows the opposite

AT KEW OBSERVATORY, 1898 TO 1912.

155

phenomenon. In all the seasons, especially summer and equinox, the phase angle is
decidedly larger in the second period, but the seasonal variation in the phase angle
is markedly less in that period. . The data from 1862 to 1864 discussed by EVERETT
made the 24-hour wave much larger relative to the 12-hour wave than the 1898 to
1904 data did. Thus the 1905 to 1912 data show no progressive change, but rather
a reversion.

TABLE VIII. — Comparison of Periods 1898 to 1904 and 1905 to 1912.

Period.

c\.

«!.

?•}.

!
«-'• fa- \ «3-

C4.

a4.

f 1898-1904
Arithmetic means . . <^ ,„„. ,„,-,

27-57
33-69

48-33
49 • 97

— 9-20 —
9-99

6-26

7-64

—

Yea / 1898-1904

16-04

165-4

47-27

187-5 7-30 40-4

5 • 35

293-9

"\1905-1912

31-30

201-3

49-31

185-9 8-61 42-6

6-72

291-5

Wini f 1898-1 904
•11905-1912

42-21
56-14

205-9

211-5

42-46
46 • 96

179-5 13-77 | 22-1
181-6 15-04 ; 25-2

4-22
6-92

238 • 1
253-5

v . ("1898-1904
Equinox . . < -, r,A_ i

16-77

125-0

55-07

195-7 7-60 41-4

8-44

311-8

\ 190o-1912

32-48

191-5

52-84

190-6 9-30 48-2

7 • 35

307-8

s— {ISStlSB

16-85

7-98

86-4
167-6

45 • 34

48-49

185-0 4-37 114-6
184-8 4-21 ! 100-8

5-69
7-97

302-9
307-3

In E1( while recognising a considerable " accidental " element iu individual monthly
values of amplitude and phase angle in the 8-hour and 6-hour waves, I concluded that
the general features were genuine. The results from the two periods in Table VIII.
show a closer resemblance than I had ventured to hope. The accordance in the phase
angles is truly remarkable, considering that the time equivalent of 1° is only
lj- minutes for a3 and 1 minute for a,. The 8-hour like the 24-hour term shows a
well-marked seasonal variation, both in amplitude and phase angle ; but unlike the
24-hour term it has the phase angle largest in summer. The 6-hour term agrees with
the 12-hour term in having the phase angle least in winter, and in showing
comparatively small seasonal variation of amplitude.

§ 13. To ascertain exactly how the difference between the results for the 24-hour
wave from the two periods came in, I calculated the 24- and 12-hour Fourier coefficients
for each individual year from 1898 to 1912. The results appear in Table IX. Fewer
figures are retained than in the previous tables because the diurnal inequalities for
individual years had been calculated only to O'l volt. The 24-hour term data fluctuate
much more from year to year than do the 12-hour term data. The 1911 value of Cj is
fully five times that of 1901, while the extreme values of a, differ by 126°,
representing 8 g hours of time. It is the earliest four years that are mainly responsible
for the smallness of «i for the epoch 1898 to 1904. There is, however, nothing

x 2

156

DR. C. CHREE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

remarkable about c2 or «2 in these years, and so no reason to suspect the genuineness
of the results. The range in a2 during the 1 5 years is only 14°, representing 28 minutes
in time.

It is obvious, so far at least as Kew is concerned, that conclusions as to the relative
importance of the 24- and 12-hour waves, or as to the value of the phase angle in the
former wave, derived from only one or two years data, may depart considerably from
average normal conditions.

TABLE IX. — FOURIER Coefficients from Individual Years.

Year. «j.

BI.

Cj.

as.

1898 21-6 109

54-3

189

1899 23-2

130

58 -G

192

1900 15 -s

150

49-7

186

1901 7-G

90

43-2

190

1902 25-9

197

30-9

180

190:5 21-8

185

50-1

189

1904 30-7

212

37-8

182

1905 33-5

207

53 • 2

183

1900 21 -6

18G

55-5

189

1907 29-1

199

44-3

188

1908 24-9

181

45 • 6

184

1909 35-1

195

4G-G

181

1910 29-5

200

48 -G

18G

1911 42-3

21G

49-3

182

1912 37-0

211

51-6

194

§ 14. The annual variation of an element may be represented by the formula

where M is the mean of the 12 monthly means, while PI, P2 are the amplitudes,
and 0j, 02 the phase angles of the annual and semi-annual terms. The time t is here
measured from the beginning of the year, one month being taken as equivalent to
30°. In the calculations calendar months were treated as if of equal length, but the
errors thus introduced are trifling.

ra

Table X. contains the results obtained from the 15-year period. They are
comparable with the corresponding data in Ej (Table VI.) when the amplitudes in the
latter are multiplied by T91. The results for the mean daily value are in very fair
accordance with those in E1; both as regards amplitude and phase angle. The
variation departs but little from that given by a pure 'sine-wave of 12-month period,
having its maximum early in January.

In the case of the diurnal inequality range the 15-year period gives nearly the
same value of Pj/M as the 7-year period, but a decidedly larger value of P2/M. In
the case of c,, PjM. is greater, and P2/M is less for the 15- than for the 7-year period.

AT KEW OBSERVATORY, 1898 TO 1912.

157

In the case of c2 the reverse holds. The phase angles from the two periods show a
closer agreement than the amplitudes. The most natural inference is that even 15
years is too short a period to give a wholly normal annual variation for i\ and c2. This
conclusion is supported by fig. 4. The c± and c2 curves shown there, while much
smoother than the corresponding curves in El5 present some features which- suggest
abnormality, notably as regards the points representing the September values.

TABLE X. — Annual Variation. Amplitudes and Phase Angles.

Element.

Pi.

0i.

Pi.

Mean value for the day . .

. 108-4

80-8

9-2

Diurnal inequality range .

31-6

56-8

14-1

» ., «i •

23-4

83-0

3-8

,. » «2 •

7-2

346-2

9-6

#•>.

Pi/M.

P./M.

PS/PL

89-9

0-36

0-03

0-085

334-6

0-24

0-11

0-45

110-4

0-82

0-13

0-16

307-8

0-15

0-20

i • :\~>

§ 15. In a recent interesting paper* dealing with atmospheric electricity data
obtained at Edinburgh during 1912, Messrs. CAUSE and SHKAHEII compare a good
many of their potential gradient results with those given for Kew in E,. Apart
altogether from the question of the absolute value of the potentials given in E1; some
doubt may be felt as to how far the results serve to compare the two stations.
Messrs. CAUSE and SHEARER employed all the days whose trace was uninterrupted and
free from insulation troubles. As these numbered 302, in a single year, a considerable
proportion presumably were days when rain fell or negative potential occurred. 1 am
unable to say definitely how results derived from 302 days in one year at Kew
would compare with the results from the 120 selected rainless days. But a recent
paper by Mr. GORDON DoBSON,t which amongst other matters compares results from
different species of days during 1911 at Eskdalemuir, throws some light on the
question. Mr. DOBSON gives three series of diurnal variation results, the first derived
from what are known as 00 days — 101 in number — whose characteristics fairly
correspond with those of quiet days at Kew; the second from all ordinary days, i.e.,
days when the record was complete and conspicuously irregular variations were absent ;
the third from all hourly readings irrespective of whether the day's record was
complete or not.

The ordinary days — which included the Oa days — numbered only 155, so that the
difference between them and the Oa days was very probably less than the difference
one would find between all complete days at Kew and the selected quiet days. The
mean potential gradients for the year derived from the three series of Eskdalemuir
data were respectively 234, 219 and 185. For the four midwinter months, when Oa

* 'Proceedings Royal Society of Edinburgh,' vol. 33, 1913, p. 317.
t 'Meteorological Office Geophysical Memoirs,' No. 7, London, 1914.

158

DR. C. CHREE: ATMOSPHERIC ELECTRICITY POTENTIAL GRADIENT

and ordinary days were relatively few, the corresponding means were 294, 278 and
212. Thus at Eskdalemuir — -and I think the same would prove true at Kew — the
mean potential gradient tends to fall when we include days of negative potential and
of large irregular oscillations. It is thus very probable that the difference between
the mean values of potential gradient at Kew and Edinburgh in 1912 — respectively
300 and 167 — is due in some measure to the difference in the type of days utilised.

Diurnal variation data were obtained by Mr. DOBSON from the three species of days
for individual months, and for the three seasons — winter, equinox, and summer, and
the results are shown graphically in his paper.

The curves for individual months and for the winter season are very irregular, but
those for equinox and summer are less so. The Ott equinoctial curve has larger
ordinates than the ordinary day curve at every hour of the 24, and except for an hour
or two the same is true of the summer curves. The difference in the type of the
diurnal variations derived from the two series of days at these seasons does not seem
to be large, but confirmation would be desirable from a period of years.

Edinburgh has a much dryer climate than Eskdalemuir, and the difference between
all and rainless days there may well be less. In any case, in view of the comparison
instituted by Messrs. CAUSE and SHEARER with the older Kew data, it seemed worth
while to compare their results with those given by the selected days of the same year
at Kew, investigating at the same time how far that year was fairly representative of
normal conditions.

The mean potential gradient at Kew in 1912 differed by only 4 volts from the
mean for the 15 years and so was unquestionably normal.

Expressing monthly means at Edinburgh and Kew in terms of their respective
mean annual values, we obtain the annual variations recorded in Table XI.

TABLE XL — Mean Monthly Potentials as Fractions of Mean for Year.

Month

.

i'

S

0

1

1

1

g

cS

PL,

cs'

O

a
1

1

-2

§

PN

3

1-5

<4

02

O

^

P

Edinburgh, 1912

1-53

no

0-74

1-09

1-09

0-98

0-90

0-86

0-86

0-76

1-16

0-95

Ivew, 1912 . . .

1-74

1-12

0-70

0-90

0-G8

0-5G

0-86

0-77

I'll

1-33

1-05

1-11

! Kew, 15 years .

!

1-35

1-30

1-13

0-94

0-78

0-68

0-G9

0-73

0-81

0-97

1-27

1-35

If, following Messrs. CARSE and SHEARER, we divide the year into a 6-months
summer, April to September, and a 6-months winter, we find for the ratio of the mean
winter potential to the mean summer potential 1'08 : 1 at Edinburgh and 1'46 : 1 at
Kew in 1912, as compared with 1'59 : 1 at Kew in the 15-year average.

£T KEW OBSERVATORY, 1898 TO 1912. 159

The natural inference is that a comparison of winter and summer based on '1912
would not be very wide of the mark, and consequently that the difference between
these two seasons is normally less at Edinburgh than at Kew.

Some individual months, however, of 1912, at Kew, diverged far from the normal.
January, for instance, had an abnormally high potential, while March and December
had abnormally low potentials, and the corresponding figures for Edinburgh are at
least suggestive of a like phenomenon there. On the other hand, while the Kew
October potential in 1912 is abnormally high, Edinburgh presents apparently exactly
the opposite phenomenon.

Coming to the Fourier analysis of the diurnal inequality, we have the following
results in the case of the mean diurnal inequality for the year, employing ca (equivalent
to Messrs. CARSE and SHEARER'S «0) to denote the mean daily value.

TABLE XII. — Comparison of Edinburgh and Kew.

'i 'o.

.'; 0-

'!,'•-'• 'J-\-

a-

Edinburgh, 1912 . . .

0-142 0

102

1 • 39 L>:5:! • 1

18:5

0

Kew, 1912

0-125 0

17"

0 • 7:5 •' 1 1

194

Kew, 15 ye;irs ....

0-076 0

159

0 • 4* 1 c)0 • 1

18G

i;

As Table IX. shows, the value of c2 at Kew in 1912 was fairly normal, though a
little above the average. But the 1912 value of «2 was the largest of the 15-years,
while the 1912 values of cl and a. were exceeded, the former only once, and the latter
only twice.

Thus, so far at least as Kew is concerned, 1912 was hardly a year which one would
have selected as representative of average conditions. Unless, however, there is a
remarkable differeiice between all and quiet day results, there can be but little
doubt that the 24-hour Fourier-wave is of greater relative importance at Edinburgh
than at Kew.

VI. The Lunar Diurnal Magnetic Variation, and its Change with

Lunar Distance.

By S. CHAPMAN, M.A., D.Sc., Fellow and Lecturer of Trinity College, Cambridge,
lately Chief Assistant at the Royal Observatory, Greenwich.

Communicated by the Astronomer Royal, F.R.S.
Received January 27, — Read February 18, 1915.

CONTENTS.

Page.

Introduction 161

The variation of amplitude with lunar distance 1G3

The variation of phase with lunar distance 166

Discussion 1G7

Tables 171

Additional note 175

Introduction.

IN his illuminating article on Magnetism in the ' Encyclopaedia Britannica' (9th ed.,
1882), BALFOTTB STEWART discussed the origin and mechanism of the short-period
magnetic variations, concluding that the only tenable hypothesis was that which
attributed them to currents flowing in the upper atmosphere, under the impulse of
electromotive forces caused by the motion of the conducting atmosphere across the
permanent terrestrial magnetic field. At several stages in the discussion use was
made of the phenomena of the lunar diurnal magnetic variation, as disclosed by
BROUN'S fine study of the subject ;* to these phenomena BALFOUR STEWART
evidently attributed considerable theoretical importance, and an origin similar as
regards situation to that of the solar diurnal variations.

In 1889 ScHUSTERf proved, by the Gaussian potential method, that the solar
diurnal magnetic variations arise mainly from causes acting above the earth's surface,
a demonstration which added much weight to BALFOUR STEWART'S tentative theory.
SCHUSTER also suggested a connection between the magnetic and barometric
variations, an idea which he elaborated and discussed with great cogency eighteen
years later (1907)4 The barometric changes are mainly of thermal origin, and
possible differences between the character of the atmospheric motions at the earth's
surface and in the upper regions may have an important bearing on the theory. In

* BROUN, 'Trevandrum Magnetical Observations,' vol. 1, p. 113, 1874.
t SCHUSTER, ' Phil. Trans.,' A, vol. 180, p. 467, 1889.
J SCHUSTER, ' Phil. Trans.,' A, vol. 208, p. 163, 1907.
VOL. CCXV. A 528. Y [Published April 7, 1915.

162 DR. S. CHAPMAN ON THE LUNAR DIURNAL MAGNETIC VARIATION,

his later paper, SCHUSTER suggested that the lunar magnetic variations might throw
light on these questions : " It is much to be desired that some systematic attempt
should be made to investigate the lunar influence on the magnetic changes, for we
possess at present only the vaguest information as to how the different components
are affected. It is quite possible that the effects may depend on a tidal disturbance
of the upper regions of the atmosphere. If so, we may expect to get a valuable test
of our theory by their investigation " (loc. cit. , p. 181).

BALFOUR STEWART also has said that "it is impossible to refrain from associating
the lunar diurnal magnetic variations either directly or indirectly with something
having the type of tidal action, but in what way this influence operates we cannot
tell" (loc. cit., § 146).

These references to tidal action seem to have been suggested largely by a discovery
of BROUN'K, that the amplitude, of the lunar magnetic variation is greater at perigee
than at apogee in the inverse cube ratio of the moon's distance at these epochs, " as
in the theory of the tides," as he briefly concluded. The semi-diurnal character of the
lunar magnetic variation agrees with this tidal hypothesis.

In 1912 VAX BEMMBLBN,* in an important paper on the lunar magnetic variations,
referred them (by methods similar to those used by SCHUSTER in the two papers
cited) to the lunar atmospheric tide as computed from the Batavian barometric
records; he showed that the electrical conductivity of the atmospheric layers in
which circulate the currents responsible for the production of the lunar magnetic
variations is of the same order as that calculated by SCHUSTER from the solar diurnal
variations. Since the lunar barometric variation is clearly a tidal effect, the
hypothesis of a tidal origin of the lunar magnetic variations is thus supported.

The most direct confirmation of this hypothesis, however, would naturally be
obtained from evidence such as that adduced by BROUN, concerning the influence of
lunar distance upon the amplitude of the lunar magnetic variations. BROUN'S result
coincided remarkably closely with the theoretical value, but further examination
indicates that this was due to a happy, accident ; the ratio of the moon's mean
distance in the half lunations centred at perigee and apogee respectively is I'OO : 1'07,
and the inverse cube of this is 1'23 ; the observed values of the ratio of the mean
amplitudes of the lunar magnetic variation during these epochs varied between 1'15
and 1'34, with a mean value of 1'24.

FiGEE,t making a similar investigation from the Batavian magnetic records,
dissented from BROUN'S conclusion that the amplitudes vary as the tidal theory
would predict.

If a result contrary to that of BROUN could be definitely established, the above
theory of the lunar magnetic variations, which is so attractive in many ways, would
become almost untenable. The matter being of some importance, and BROUN'S and

* VAN BEMSIELEN, ' Meteorologische Zeitschrift,' vol. 5, p. 218, 1912.

t 'Batavian Magnetical and Meteorological Observations,' vol. 26, Appendix, 1903.

AND ITS CHANGE WITH LUNAR DISTANCE. 163

FIGEE'S data alone being insufficient to resolve the doubt, I have made an attempt
in the present paper to decide the question, with the aid of newly computed data.
The result is not so decisive as could be desired, owing to the considerable accidental
errors affecting the determinations of the minute quantities concerned ; but the
evidence, on the whole, is confirmatory of the tidal hypothesis. It can certainly be
stated that, if the amplitude of the lunar magnetic variation is inversely proportional
to an integral power of the lunar distance, this power is the cube and not the square
or the fourth power.

This conclusion, while important, cannot be called surprising. The investigation
very clearly revealed, however, another phenomenon of a remarkable and quite
unexpected nature, viz., that the phase of the lunar magnetic variation at perigee is
considerably in advance of that at apogee (i.e., by about 30 degrees). On referring
back to BROW'S and FIGEE'S papers, it appeared that their data confirmed this result,
though BROUN, not unnaturally, made no remark on what seemed to be nothing more
than an accidental feature of the observations. FIGEE, on the other hand, noticed
and commented on it, but on account of variations in its amount, as determined from
different portions of his material, he expressed doubt as to its reality. He appears to
have gone no further with it, not even examining BROUN'S data for confirmation or
otherwise. The body of evidence in its support, which is here brought forward,
establishes this remarkable phenomenon beyond question.

The Change with Lunar Distance of the Amplitude of the Lunar Magnetic

Variation.

If the amplitude of the lunar magnetic variation varies continuously between a
maximum value at perigee and a minimum at apogee, the ratio of the mean amplitudes
during two periods centred at these epochs must obviously depend on the length of
the periods. The mean eccentricity of the lunar orbit being 0'055, the extreme ratio
of the amplitudes (at exact perigee and apogee) on the tidal theory should be 1'39,
which is the cube of (l'055 -r- 0'945) : for periods of four days centred at perigee and
apogee the ratio should be 1'38, while if the periods each extend over half a lunation
the ratio is 1'23.

BROUN divided his material into the perigee and apogee half lunations, thus utilizing
all his material, but diminishing the magnitude of the quantity to be determined.
His data were ten years' hourly observations of declination at Trevandrum, in India,
and he treated the summer and winter half years separately. The mean inequalities
derived from the four groups of data (summer and winter, perigee and apogee) were
compared according to their mean ranges, mean areas, and harmonic amplitude
coefficients. The perigee-apogee ratios so obtained varied between 1'15 and 1'34, with
a mean value of 1'24 from the whole material. BROUN summed up his result thus :
" The ratio of the moon's mean distance from the earth in the half orbit about apogee
is to that in the half orbit about perigee nearly as 1'07 to 1 ; as the cube of 1'07 = 1'23

y 2

164 DE. S. CHAPMAN ON THE LUNAR DIUENAL MAGNETIC VARIATION,

nearly, we see that the mean ranges of the curves, as well as the mean areas, for the
two distances are in the approximate ratios of the inverse cubes of the moon's distance
from the earth, as in the theory of the tides " (loc. cit., § 405).

FIGEE did not determine the ratio of the mean amplitudes over half lunations, but
only over periods of three days centred at apogee and perigee ; hence his results
should be compared with 1'39 and not with 1'23. This point seems to have been
overlooked by him, for on comparing his figures with BROUN'S, he remarked that
BROUN'S value of the perigee-apogee amplitude ratio (l'24) "is much smaller than
that found for Batavia, 1'68, and therefore the conclusion drawn by BROUN is not
allowed here ' that the mean ranges of the curves, as well as the mean areas, for the
two distances, are in the approximate ratios of the inverse cubes of the moon's
distance from the earth, as in the theory of the tides ' " (loc. cit., § 23). The figure
1'68, here referred to, is the ratio of the areas of the variation curves at perigee and
apogee for declination (winter), the element chiefly affected by the moon at Batavia.
The ratios for the other elements and seasons were also determined, however, though
in some cases (especially that of vertical force) the whole lunar variation is so small
that any effect of the kind sought for would be liable to be masked by accidental error.
But in the case of horizontal force the total amplitudes, while smaller than that
of declination in winter, seem to be sufficiently large to entitle the perigee-apogee
ratios derived from them to some weight. The three values of this ratio which
appear to be the most reliable, amongst FIGEE'S results, are consequently :

Declination (winter) 1'68.

Horizontal force (summer) 1'40.

Horizontal force (winter) I'OO.

The mean of these results* is 1'36, which is very nearly equal to the theoretical ratio
1'39 ; hence, we may conclude that, while it is subject to considerable accidental error,
the mean result from Batavia tends, like that from Trevandrum, to the support of the
tidal hypothesis as to the origin of the lunar magnetic variations.

The newly computed results, next to be described, are derived from the hourly
values of the magnetic elements at five observatories, Pavlovsk, Pola, Zi-Ka-Wei,
Manila, and Batavia (in each case for seven magnetically " quiet" years), treated in
the manner explained in a recent paper on the lunar magnetic variations at Pavlovsk
and Pola.t The amplitude ratio has been determined both from short periods of three

t The other results obtained by FIGEE are as follows : declination (summer) 1 -09, vertical force (summer)
1-97, vertical force (winter) 089. The mean of these values is 1'32, but on account of the smaller
amplitude of these variations, not much weight can be given to this mean value.

t CHAPMAN, 'Phil. Trans.,' A, 214, pp. 295-317 (1914). For the purpose of the perigee-apogee
investigation the work described in that paper is useful up to the end of § 4.

The seven years dealt with are 1897-1903, except in the case of Batavia, where on account of interrup-
tions in the observations the years 1899-1901 were replaced by the correspondingly quiet years of the
previous sunspot cycle, 1888-1890,

AND ITS CHANGE WITH LUNAR DISTANCE. 165

or four days centred at apogee and perigee, and also from half lunations, as in BROUN'S
investigation. In order not to waste labour on computations which were not likely
to afford an accurate result, owing to the relative magnitude of the amplitude change
and the accidental errors, only those elements have been dealt with (and during those
seasons*) for which the range of the semi-diurnal lunar magnetic variation
approximated to at least 2y (2. 10"~5 C.G.S.).

The results as regards amplitude are given in Tables I. and II., the former applying
to the shorter periods about apogee and perigee, and the latter to the half lunations.
The number of days contributing to each individual value of the semi-amplitude
in Table I. was approximately 100, and in Table II., approximately 400. The
theoretical value of the amplitude ratio in the former case is 1'38, and in tin; latter,

1'23. The means of the observed values are as follows : —

Theoretical value,
r Declination (six determinations) . . . 1'33~]

Short periods . < Horizontal force (five determinations) . 1'43 f- 1'38

I Vertical force (four determinations) .

*

f Declination (six determinations) .

Half lunations < Horizontal force (six determinations) . 0'98 ^ 1'23

I Vertical force (four determinations) . .

The individual results in Tables I. arid II., on which these means are based, show
considerable discordances, and better data are much to be desired. The accidental
errors in the determination of the minute lunar magnetic variations, especially from a
comparatively small number of days, are considerable ; in one or two cases, indeed,
some of the computed semi-amplitudes are palpably erroneous, and have been discarded
in taking the means from the tables. The discordance (including even the rather
surprising discordance of the second horizontal force mean value, 0'98) must, I think,
be attributed to the fortuitous effect of magnetic disturbances, and this can be
eliminated only by determining the lunar variation over the average of a considerably
longer period of time. It would be well if observatories would undertake such a
reduction of their own observations, but meanwhile the only available data are those
here communicated.

Apart, then, from the fifth mean value tabulated above, t the present results,
taken in conjunction with BROUN'S and FIGEE'S data as previously discussed, may be
said to confirm with reasonable probability the hypothesis of tidal action. The
material is certainly not sufficient to determine the magnitude of the ratio of
amplitudes at perigee and apogee to within a few per cent., as is desirable. It would

* The year was divided up into three parts, summer, winter, and equinox, comprising May- August,
November-February, and the intervening months, respectively.

t It does not seem probable that the smallness of this result for horizontal force represents a
real phenomenon, especially as the second of the above mean values (1 • 43) would tend to the opposite
conclusion.

166 DE. S. CHAPMAN ON THE LUNAR DIURNAL MAGNETIC VARIATION,

not be unreasonable to conclude from the present data that the effect of lunar
distance may be a little less than the tidal theory would indicate ;* but, at any rate,
it seems clear that we may conclude that if the amplitude of the lunar magnetic
variation is inversely proportional to the wih power of the moon's distance, where n
is integral, then n can have no other value than three, t

The, Change with Lunar Distance of the Phase of the Lunar Magnetic Variation.

In the computations by which the lunar magnetic variation data, discussed here
and in a former paper, were determined, the hourly values of the magnetic elements,
after being freed from the solar diurnal variation, were written out in rows of twenty-
five, the first value on each row corresponding to the civil hour nearest to the moon's
meridian transit on that day.J Hence the origin of lunar time is, in the mean, at
the hour of lunar transit and is independent of the longitude of the moon in its orbit.
We shall represent the lunar magnetic variation by sin (2t + 9), where t, measured
from the origin just mentioned, increases by 2x in a lunar day : then 9 is the phase
of the variation, which is found to vary with the longitude of the moon. We shall
denote its values at perigee and apogee by 0P and $A) and its mean values during the

half lunations centred at these epochs by 0P and 0A. The differences 6P— 0A and 9P— #A
corresponding to the various observatories, elements, and seasons dealt with in
Tables I. and II. are given in Tables III. and IV. respectively ; in practically every
case they are positive, indicating an acceleration of phase at perigee. As a slight guide
to the probable reliability of the determined phase angles, the mean of the amplitudes
given in Tables I. and II. are tabulated in Tables III. and IV. respectively.

The mean values of 6v—6\ from Tables III. (a), (b), and (c), which, being derived
from short periods, should show practically the whole phase change between these

epochs, are

Degrees.

Declination (six determinations) +26

Horizontal force (five determinations) .... +30
Vertical force (four determinations) +18

Similarly the mean value of 6P— 0A are, from Table IV.,

Degrees.

Declination (six determinations) + 8 '2

Horizontal force (six determinations) +9'8

Vertical force (four determinations) +8'9

* A possible explanation of such a conclusion, if it were substantiated, might be framed along the lines
indicated on p. 168, last paragraph.

t The amplitude ratios for "short periods " if n had the values 2, 4 would be 1'23, 1-51 respectively,
and for half lunations, T14 and 1-31 respectively, which seem to be outside the probable limits of error
of the observed results.

I Of. ' Phil. Trans.,' A, vol. 214, § 3.

AND ITS CHANGE WITH LUNAE DISTANCE. 167

These mean results may be compared with individual results from the quoted
memoirs by BEGUN and FIGEE. The values from BEOUN'S paper are

Degrees.

,' October-April .... 0P-<9A = 21
Trevandrum declination •

tion-<

May-September. . . . 0P— 0A = 14

These differences (on which BEGUN made no remark) confirm the phase change
indicated in Table IV., though the amount is larger than the value there determined.

FIGEE gives the phase angles at perigee and apogee only for declination (winter),
dividing his sixteen years' data into various groups as follows : —

Degrees.

Nine maximum sunspot years Qi-—Q\ = 42

Seven minimum sunspot years 33

Eight odd year's of the series 54

Eight even years of the series 11

Whole sixteen years 37

He remarked : " The first formula) reveal the remarkable property that the occurrence
of the maximum of the semi-diurnal wave was accelerated by the increasing magnetic
force exercised by the moon from apogee to perigee, the amount of the acceleration
being more than one hour, with a striking accordance in the two periods chosen
(maximum and minimum). It was to be expected that a subdivision in two other
series (odd and even years) should show a similar accordance, which unfortunately
was not the case, as may be seen from the above figures. By this the reality of an
acceleration of the occurrence of the maximum with decreasing distance of the moon
from the earth is made less probable, though certainly suggested by the above
figures."

Discussion.

The whole body of evidence here collected makes clear the reality of this
remarkable phase change, so that the differences between FIGEE'S various results
would seem to be only accidental, large though some of them are. The magnitude
of the phase change is not very exactly determined, but it appears to be approxi-
mately 30 degrees between perigee and apogee. The value of (0P— 0A)/(#i>— #A)> if
accurately known, should afford information as to how the phase varies between
these epochs. If the phase angle is harmonically periodic during a lunation, so that
the formula for the lunar variation is

sin (2t + c cospt + a + /3)

(where l/p is the number of days in a lunation), then the mean phase angle during a
half lunation should be 2/Tr times the phase angle at the middle point of this period,

168 DR. S. CHAPMAN ON THE LUNAR DIURNAL MAGNETIC VARIATION,

at whatever epoch this middle point may be. Hence the value of (6P— 0^)/(dP— 0A)
should, on this hypothesis, be 2/?r also ; in the present case, if the mean results from
Tables III. and IV. be taken, its value is less than 2/?r — BROUN'S value of 0P— 0A
would satisfy the relation more exactly. Without much more accurate data,
however, it would be very unwise to conclude definitely that the phase angle does
not vary harmonically throughout the lunation.

According to the theory of the lunar magnetic variations which was outlined in the
Introduction to this paper, they are primarily due to a lunar atmospheric tide, which
produces the electric currents responsible for the magnetic variations. It is a
question of ascertainable fact whether or not the changes in phase angle (with
change of lunar distance) which are found in the lunar magnetic variations are
already present in the lunar atmospheric tide, as revealed by the barometric records.
The computations necessary to determine this point have not, however, yet been
made, and therefore in the present discussion it is advisable to consider what
information in the matter may be derived from general theory.

In the case of the atmospheric tides, the tidal theory appropriate £o a uniform
ocean is presumably applicable. A change of phase in the tide appears possible only
if frictional forces are acting, and these would produce a retardation of phase, the
magnitude of which depends only on the period and not on the amplitude of the
tidal oscillation. In the case of the lunar atmospheric tide it is chiefly the amplitude
which alters (over a total range of 40 per cent.), while the period changes" only by
about I per cent., being shorter at perigee than at apogee; this should result in a
retardation of phase at perigee, though of negligible amount.

The matter may be regarded otherwise, as follows : the main lunar semi-diurnal
tides are analysed into M2, an invariable semi-diurnal tide Asin(2£ — a), and N the
principal lunar elliptic tide (see DAKWIN'S ' Collected Papers,' vol. 1, p. 20), the
amplitude of which is approximately one-fifth of M, and which may be described as
semi-diurnal, but with a slowly changing phase angle which increases through 2-n-
during each lunation — it may be written JA sin (2t+pt+/3). For convenience we
may suppose the origin of t chosen so that a = 0. Theory does not definitely predict
what, in the actual case, will be the value of /3, the difference in phase between M3
and N at (say) perigee, but it would be expected to be quite small. The combination
of M2 and N results in a semi-diurnal oscillation, the amplitude of which changes
through a total range of 40 per cent., while the phase varies through approximately
23 degrees, 11^- degrees on either side of the mean. The maximum and minimum
amplitudes occur at the epochs of mean phase. If at perigee /3 = 0, the amplitude
change would coincide with that observed in the lunar magnetic variation, but there
would be no corresponding phase variation ; if /3 = %ir, there would be 110 change of
amplitude between perigee and apogee, but there would be a phase change of 23
degrees between apogee and perigee, in the direction observed in the lunar magnetic

AND ITS CHANGE WITH LUNAR DISTANCE. 169

variation. For intermediate values of /3 intermediate states would prevail — e.g., if
ft = JTT, the amplitude change is reduced to a ratio 1'30, and there is a phase change
of 17 degrees between perigee and apogee. The observed amplitude change in the
lunar magnetic variation might be a little less than 1'39, though hardly so much
below this as 1'30; this would be compatible, on the above view, with a change of
phase of something less than 17 degrees. Possibly in this way a part of the observed
phase change in the lunar magnetic variation might be explained (a more accurate
determination of the amplitude change would decide this point more definitely), but
in no circumstances could the whole of it be thus accounted for. Probably the
explanation which must be sought elsewhere will account for the whole of the
phenomenon under discussion.

Before leaving the consideration of the atmospheric tide, its actually observed
phase, as determined from forty years' barometric observations at Batavia, may be
adverted to. The lunar diurnal variation of barometric pressure was found to be

(TOG28 cos (2t + 05°),

t being reckoned from the local time of the moon's transit. The fact that the tide is
in advance of the moon is difficult to understand, and it is conceivable that the
unknown cause which thus accelerates the tide also has some connection with the
perigee-apogee phase change in the lunar magnetic variations.

So far no real light on the origin of this phase change has been found, since theoretical
reasoning indicates no such change in the atmospheric tide which is supposed to
produce the lunar magnetic variations. Whether the phase of the tide does or
does not so change is, as already stated, unknown ; if it is found to vary correspondingly
with the magnetic variations, the tidal theory of the latter would be strengthened,
though the phase change in the tides themselves would offer a problem demanding
solution. In the present state of ignorance, however, it is natural to consider
whether, if the phase of the atmospheric tide is independent of lunar distance, the
phase change in the magnetic variations could be accounted for in any electromagnetic
way. Self-induction is the only possible cause which suggests itself, and this,
unfortunately, seems as incapable of offering an explanation as tidal friction was
found to be ; the effect of self-induction is to produce a phase retardation which
is independent of the amplitude of a periodic variation, but which diminishes with the
period. In the present case the variation of period would not account in this way for
more than 1 degree change of phase.

What has been said as to the failure of tidal friction to explain the phase change in
the lunar magnetic variation applies also to tides in the substance of the earth,
so that even if (as VAN BEMMELEN supposed in his memoir already cited) such body
tides have a part in the production of the lunar magnetic variation, they do not aid
in the explanation of the phase change under discussion. VAN BEMMELEN in an

VOL. ccxv. — -A. z

170 DR. S. CHAPMAN ON THE LUNAR DIURNAL MAGNETIC VARIATION,

amending paper ('Met. Zeitschr.,' 12, p. 589, 1913) withdrew his earlier suggestion of
a primary internal magnetic field concerned in the lunar variations; it may be
remarked incidentally that such a field is rendered unlikely by the magnitude of the
components other than semi-diurnal in the magnetic variations.* These appear to be
excited by the semi-diurnal atmospheric tide in conjunction with a variable electrical
conductivity of the atmosphere, depending on the solar hour angle ; their phases
change by multiples of 2?r in the course of a lunation, according to ascertained laws.
While only secondary phenomena, they are comparable in magnitude with the
main (semi-diurnal) component of the lunar variation, so much so, in fact, that it
is hardly possible for more than a very small fraction of the latter to be due to
internal causes, since this portion would not account for any part of the secondary
components.

Since perigee and apogee occur at all phases of the moon during a sufficiently long
period of time, no explanation of the perigee-apogee phase change in the magnetic
variations can be looked for in any direct solar action.

Tims far, therefore, the attempt to assign a known cause to this remarkable
phenomena has been unsuccessful. It remains to make one more suggestion, which is
at once very tentative and far from definite. We should naturally suppose that the
tidal effect of the moon is to produce a lunar magnetic variation A cos (2t + a.),
the amplitude (A) of which undergoes a regular variation of the kind observed,
but which shows 110 change of phase ; the observed phase change might be produced
if in some other way the moon produced a semi-diurnal magnetic variation of type
B cos (2t + /3), where ft exceeds a by about 90 degrees, provided that B/A is fairly small
and that B increases more quickly than A as the moon's distance diminishes. The
combination would slightly increase the theoretical variation of amplitude, but this
need only be by a small amount.

Not much importance can be attached to this suggestion, however, since it is
very difficult to conceive of any way, other than tidal, in which the moon could
produce a semi-diurnal variation of terrestrial magnetism. Certainly no appreciable
direct magnetic effect of the moon is at all likely, nor would it, in any case, supply the
desired variation in the present instance, since its effect would be diurnal and not
semi-diurnal. The matter must therefore remain a mystery for the present, but it is
a mystery whose solution may be the clue to some important new fact relating to
magnetism of the atmospheric tides ; it is with this hope that I venture to publish the
preceding very inconclusive discussion.

In conclusion it is a pleasure to acknowledge the assistance which has been placed
at my disposal, in the execution of the computations involved in this paper, by the
Government Grant Committee of the Royal Society.

* Of. 'Phil. Trans.,' A, vol. 213, p. 279, 1913.

AND ITS CHANGE WITH LUNAE DISTANCE.

171

TABLE I. — Ratio of Mean Semi-amplitudes of the Lunar Semi-diurnal Variation of
Terrestrial Magnetism, during a number of periods of three or four days centred
at Perigee and Apogee respectively.

(a) Declination West.

Observatory.

Season.

Semi-amplitude in force units,
10~7 C.G.S.

Ratio.

Perigee.

Apogee.

Pavlovsk .... Summer .

238
225
293
145

191
291

130

159
172
200
170
120
245
(54)

1-50
1-31
1-46

0-85
1-59
1-19

(2-4)

Pola

Summer . . .
Summer .
Equinox .
Summer .
Winter
Equinox . . .

Zi-Ka-Wci . . j
Manila ....
Batavia . . . I

Moan . . . 1 • 33 ± 0 • 08

(6) Horizontal Force.

Observatory.

Season.

Semi-amplitude in force units,
10- C.G.S.

Ratio.

Perigee.

Apogee.

Pavlovsk ....
Pola .... |

Zi-Ka-Wei . . .
Manila ....
Batavia ....

Summer ....
Summer ....
Equinox ....
Winter ....
Winter ....
Winter ....

137
172
1GG
(93)
171
118

72
128
100
175
128
131

1-91
1-35
1-05
(0-53)
1 • 33
0-90

Mean . . .

1-43 ± 0-12

(c) Vertical For.ce.

Observatory.

Season.

Semi-amplitude in force units,
10- C.G.S.

Ratio.

Perigee.

Apogee.

Zi-Ka-Wei . . . [
Manila . . . . J

Summer ....
Equinox ....
Winter ....
Equinox ....

152
142
99
122

101
135
99

72

1-51
1-05
1-00
1-69

Mean . . .

1-31±0-14

z 2

172 DE. S. CHAPMAN ON THE LUNAE DIUENAL MAGNETIC VAEIATION,

TABLE II.— Eatio of Mean Semi-amplitudes of the Lunar Semi-diurnal Variation of
Terrestrial Magnetism, during a number of half months centred at Perigee and
Apogee respectively.

(a) Declination West.

Semi-amplitude in force units,

Observatory. Season.

ID'7 C.G.S.

Eatio.

Perigee.

Apogee.

Pavlovsk ....

Summer .

147

117 1-26

Pola Summer ....

168

138 1-21

rr- ir TIT- • f Summer ....
Zi-Ka-AUi . . [ Equinox. . . .

266
164

198 1-34
159 1-03

161

117 1-38

\ Winter
Batavia ' \ Equinox. . . .

282
111

245

(24)

1-15
(4-63)

I

Mean . . .

1-23 ± 0-04

(6)

Horizontal Force.

Semi-amplitude in force units,

Observatory.

Season.

icr7 C.G.S.

Eatio.

Perigee.

Apogee.

Pavlovsk ....

Summer ....

108

92

1-18

Pola {

Summer ....
Equinox ....

144
88

149

88

0-97
1-00

Zi-Ka-Wei . . .

Winter ....

126

147

0-86

Manila ....

AY inter ....

130

142

0-92

Batavia ....

Winter ....

139

145 •

0-96

Mean . . .

0-98 ± 0-03

(c) Vertical Force.

Observatory.

Season.

Semi-amplitude in force units,
10^7 C.G.S.

Eatio.

Perigee. Apogee.

Zi-Ka-Wei . . . j
Manila . . . . J

Summer ....
Equinox ....
Winter ....
Equinox ....

132
132
101
117

92
116
100
93

1-44
1-14
1-01
1-26

Mean

1-21±0'07

AND ITS CHANGE WITH LUNAR DISTANCE.

173

TABLE III. — Differences in Phase Angle 0 in the Formula sin (2t + 6) for the Lunar
Semi-diurnal Magnetic Variation at Perigee and Apogee, as determined from a
number of short periods centred at these epochs.

(a) Declination West.

Observatory.

Season.

Mean amplitude
(from Table I. (a) )
in force units,
10~7 C.G.S.

Phase difference,

er - 0A.

Pavlovsk

Summer

198

0

21

Pola

Summer

198

36

Summer

246

12

Zi-Ka-Wei >!

Manila

Equinox
Summer

158
156

43

26

Batavia <

Winter
Equinox

268
(92)

22
(15)

Mean . . .

+ 26° ±3° -4

(b) Horizonta

1 Force.

Observatory.

Season.

Mean amplitude
(from Table I. (!>) )
in force units,
lO-7 C.G.S.

Phase difference,
0p - 0A.

j

Pavlovsk

r> i T

Summer
Summer

104
150

0

+ 31
- 2

Pola <^

Zi-Ka-Wei

Equinox
Winter

133

(134)

+ 56

(27)

Manila

Winter

150

28

Batavia

Winter

125

38

Mean . . .

+ 30°±6°-0

(c) Vertical Force.

Observatory.

Season.

Mean amplitude
(from Table I. (c) )
in force units,
10-7 C.G.S.

Phase difference,

0P-0A.

,

Summer

126

0

24-1

Zi-Ka-Wei <

Equinox

138

27-7

., .. C

Winter

99

13-0

Manila ... 4

Equinox

97

7-9

Mean . . .

18°-2±3°-8

174

DE. S. CHAPMAN ON THE LUNAR DIURNAL MAGNETIC VARIATION,

TABLE IV. — Differences in the Phase Angle 9 in the Formula s\n(2t + 6) for the
Lunar Semi-diurnal Magnetic Variation, at Perigee and Apogee, as determined
from a number of half-lunations centred at these epochs.

(a) Declination West.

Observatory.

Season.

Mean amplitude
(from Table II. (a) )
in force units,
10-7 c.G.S.

Phase difference,

0p-0A.

Pavlovsk
Pola ....

Summer
Summer

132
153

8
6

Zi-Ka-Wei {

Summer

232

6

Manila

Equinox
Summer

162
139

17
5

I3atavici . ....

Winter

264

8

Mean . . .

+ 80-2±1°-0

(b) Horizontal Force.

Observatory.

Season.

Mean amplitude
(from Table II. (I,) )
in force units,
10-' C.G.S.

Phase difference,

0P-0A-

Pavlovsk

Summer

100

2

r, i f

Summer

146

-10

Pola <^

Zi-Ka-Wei
Manila

Equinox
Winter
Winter

88
136
136

28
18
10

Batavia

Winter

142

11

Mean

+ 9°-8±3°-5

(c) Vertical Force.

Observatory.

Season.

Mean amplitude
(from Table II. (c) )
in force units,
10-7 C.G.S.

Phase difference
0p - #A-

Zi-Ka-Wei . . j

Summer

112

0

12-2

Equinox

124

7-2

Winter

100

9-3

Equinox

105

6-9

Mean . . .

8° ' 9 ± 0° • 9

AND ITS CHANGE WITH LUNAR DISTANCE. 175

NOTE ADDED MARCH, 1915.

(l) The Seasonal Changes of 6>P— 0A.

If the values of 6P— 0A from Tables III. (a), (b) and IV. (a), (b) be grouped together
according to their season, it will be found that the mean value at the equinoxes is
about double that at the other seasons, the difference being quite noticeable ; this
was kindly pointed out to me by Prof. H. H. TURNER, F.Il.S. In order to examine
this point a little further (since the result mentioned rests on material which is much
more scanty for the equinox than for the other seasons), I caused the results given in
Tables I. to IV. (c) (for vertical force), to be computed. These were not in the
paper as originally communicated to the Royal Society, as I did not then know that
the amplitudes of the variations for this element and these seasons reached the
value 2y.

If now all the values of 6P— 0A in Tables III. and IV. are grouped together
according to season, and the means taken, we obtain the following results :—

Mean Values of 9P— 0A from Table III. (short periods).

Degrees. /

Summer 21 + 3'2 from seven determinations.

Winter 26 ±2' 7 „ five ,,

Equinox 30 + G'G ,, five ,,

Mean Values of 0L,— 0A from Table IV. (long periods).

Degrees.

Summer* G'5±0'9 from six determinations.

Winter 11'2±1'1 ,, five

Equinox 14'7±2'G ,, five

Both groups of mean values show a progression in the phase change from summer
to winter, with maxima at the equinoxes. If this is really the case, the phenomenon
should make success in the search for the cause of the phase change much more
probable ; at present, however, the magnitude of the probable errors are such as to
cast doubt on the reality of the seasonal change, and, until it is more definitely
established, any attempt at its explanation would be premature.

(2) The Lunar Atmospheric Tide.

Further information concerning this is contained in a memoir by WAGNER,! who
has discussed the barometric observations for the years 1903-8 made at Samoa. The
mean result for the semi-diurnal tide is, in millimetres,

0'039 sin (2< + 33 degrees).

* Omitting the negative value - 10 for Pola horizontal force,
t G. WAGNER, 'Gottingen Abh.,' IX., 4 (1913).

176 DR. S. CHAPMAN ON THE LUNAR DIURNAL MAGNETIC VARIATION, ETC.

The seasonal and synodic changes in this variation are discussed, and also the change
during the anomalistic month (from apogee to perigee). But the result in the latter
case is (in the words of the author) disappointing, as no clear progression is made
out, presumably owing to the somewhat small amount of material used for discussion.
Such apparent change as the data show is a retardation from apogee to perigee, but
this cannot be relied on. This first attempt to settle the point mentioned on p. 1G8
is therefore inconclusive, and further work on it is desirable.

The effect of lunar declination on the tide is also dealt with by WAGNER, without
any decided result. Since a tidal effect may be regarded as due to a moon and anti-
moon, one of which will be in Northerly declination when the other is Southerly, and
rice versa, it does not seem probable that any explanation of the perigee-apogee
magnetic phase change is to be found in the variation of lunar declination.

[ 177 ]

VII. A Thermomagnetic Study of the Eutectoid Transition Point of

Carbon Steels.

By S. W. J. SMITH, M.A., D.Sc., F.R.S., Assistant Professor of Physics, and
J. GUILD, A.R.C.S., D.I.C., Assistant Demonstrator of Physics, Imperial
College, South Kensington.

Eeceived December 29, 1913,— Bead February 12, 1914. Keviscd copy received February 15, 1915.

1 . Introduction.

THE manner in which the ferromagnetism of nearly pure iron varies with temperature
has been the subject of many investigations ; but the corresponding and even more
interesting study for steel has been much less complete. Knowledge of the con-
stituents of iron-carbon alloys, acquired in different ways in recent years, makes
it certain that, in the earlier papers upon the change of permeability witli
temperature, salient features of the thermomagnetic curves have escaped notice.
This has happened because of the discontinuous character of the observations upon
which the published curves have been based. An example of the way in which a
striking variation may remain undiscovered was given in a paper published by the
Physical Society in 1912.*

The experiments described in the present paper were made upon a series of steels
containing percentages of carbon varying between 0'15 and 1'53, and our discussion
of them is purposely confined to phenomena observed in the neighbourhood of 700° C.
It is in this region that one of the most important events in the thermal history of
steel occurs. Here steel containing about 0'9 per cent, of carbon changes during
cooling from an apparently homogeneous material into a heterogeneous mixture of
two different substances. One is apparently pure iron and the other is the carbide
Fe3C. This mixture is the eutectoid.

The same change takes place in steels containing other percentages of carbon ; but
it is preceded by the separation, at higher temperatures, of the carbide or of iron
according as the steel contains more or less than the eutectoid percentage of carbon.
In the former — the hyper-eutectoid steels — the eutectoid therefore co-exists below
700° C. with excess of carbide, whilst in the latter — the hypo-eutectoid steels — there

* ' Proc. Phys. Soc.,' XXV., pp. 77-81.
VOL. CCXV. A 529. 2 A [Published April 30, 1915.

178 DR. S. W. J. SMITH AND ME. J. GUILD: A THERMOMAGNETIC STUDY OF

is excess of iron. During heating the reverse changes take place. The eutectoid
transforms first and then the iron or the carbide as the case may be.

Qualitative evidence of these statements is provided by the microscope, and the
theory which accounts for them gives rise to the so-called equilibrium diagram based
upon thermal observations during cooling. It occurred to us that the thermomagnetic
method might throw some useful light upon that part of the equilibrium diagram
which relates to the appearance and disappearance of the eutectoid. This opinion
was based upon the fact that, for reasons which need not be dwelt upon, when the
eutectoid or any portion of it disappears during heating the iron in it should lose
practically all its magnetism. Conversely, when the reverse change takes place during
cooling, that magnetism should be regained.

2. Materials Used and Methods of Measurement.

Most of the specimens of steel examined were cut from materials, described in
earlier papers,* which were given to us by Mr. E. A. WRAIGHT and by Prof. J. 0.
AENOLD, F.R.S., respectively. The analyses supplied, showing percentages of
elements other than iron, were as below : — -

No.

c.

Mn.

Si.

1

0-15

0-20

0-09

'2

0-36

0-20

0-11

3

0-60

0-21

0-17

4

0-71

0-22

0-17

A

0-85

0-06

0-05

5

no

0-23

0-17

S

1-23 0-17

0-15

6

1-53

0-235

0'165

Traces of sulphur and of phosphorus (not exceeding 0'03 per cent, in each case)
were also present. As in the paper last cited, the specimens were cut in the form of
tubes, each 7 cm. long, of which the external diameter was 5 '5 mm. and the internal
3 mm.

The temperature was measured by means of a platinum-rhodium platinum thermo-
couple, calibrated by means of a platinum thermometer, of which the junction was
placed near the centre of the tube under examination. This tube was contained
within a copper tube, about 11 cm. long and 1 mm. thick, of very slightly more than
5 '5 mm. internal diameter, from which it was separated by very thin strips of mica.

The platinum heating coil was wound bifilarly upon a layer of asbestos paper
wrapped round the copper tube. Several thicknesses of asbestos paper were then
wrapped over the heating coil, and the resulting cylinder, about 15 cm. long, was

* ' Proc. Phys. Soc.,' XXIV., pp. 62-69, and pp. 342-349, 1911.

THE EUTECTOID TRANSITION POINT OF CARBON STEELS. 179

pushed into an outer copper tube of similar length, about 2 cm. in diameter. This
latter tube was supported within the magnetising solenoid by means of further layers
of asbestos paper wrapped round it loosely.

Experience showed that this arrangement gave a nearly uniform temperature
throughout the specimen for different steady currents in the heating coil. To obtain
the maximum degree of uniformity the rods might have been made shorter, but they
were required for other experiments in which their length was of consequence.
Moreover, our object was rather to compare the temperatures at which certain changes
took place in the different steels than to find with extreme accuracy the absolute
values of these temperatures.

The rods occupied successively almost exactly the same position in the furnace and
the thermocouple was so mounted, in a small porcelain tube, that its junction when in
use was at a fixed distance from the end of the specimen.

The magnetic measurements were made by means of a sensitive and suitably damped
quartz-fibre magnetometer provided with the usual compensating arrangements.

The data necessary for the deduction of the intensity of magnetisation from the
scale-readings were recorded ; but the magnetisation is expressed arbitrarily in terms
of these scale-readings only, in order to avoid laborious reductions which would have
added little to the value of the conclusions drawn.

The coefficient of self-demagnetisation of the "rods" (roughly 0'09) was determined
by comparing the magnetising solenoid fields required to produce given intensities of
magnetisation in the 0'85 per cent, rod with those required to produce equal intensities
in a ring of the same steel. For the present purpose, however, it is sufficient to give
only the fields due to the solenoid and not the effective fields within the rods.

The procedure in the first series of experiments was as follows. Each of the alloys
in turn was placed in the magnetising solenoid. It was demagnetised and a current
of 1 ampere, producing a field of approximately 50 C.G.S. units, was then passed
through the solenoid. This current was kept constant while the temperature of the
specimen was varied. The latter was raised slowly from air temperature to about
850° C. in each case and then slowly lowered. Corresponding readings of deflection
and temperature were taken over the whole range as the temperature rose and fell.
The results obtained at temperatures above 660° C., which are all that are required for
our present purpose, are collected in figs. 1 and 4. The former contains the observa-
tions made during heating : the latter those made during cooling. The observations
for the different alloys are denoted by different signs according to the scheme shown
on the figures.

The scale of the ordinates differs from one alloy to another. The distances of the
different specimens from the magnetometer were unequal, being such that all of them
gave approximately the same deflection at the air temperature before heating began.

The figures include, for comparison, results obtained with a rod of nearly pure iron
of the same size as the others and examined in the same way. For this specimen the

2 A 2

180 DR. S. W. J. SMITH AND ME. J. GUILD: A THEE MOM AGNETIC STUDY OF

heating and cooling curves very nearly coincide above 700° C. The slight difference
(exhibited in fig. 1 which contains the cooling curve above 730° C. for the iron as well
as its heating curve) is no doubt due partly to a slight excess of the temperature of the
thermocouple during cooling and to a slight defect during heating — not amounting to

660

680°

700'

720° 740°

Temperature.

760°

780°

800°C

more than 1° C. in either case — compared with that of the specimen. The tempera-
ture was not altered quite as slowly in this case as in the others ; but the observations
serve to indicate the magnitude of the error which may arise owing to a temperature
gradient between the thermocouple and the specimen.

THE EUTECTOID TRANSITION POINT OF CARBON STEELS. 181

The differences between the final ordinates seen in fig. 1 are due merely to
differences in the zero readings of the magnetometer in the different experiments,
the ordinates plotted being the actual scale-readings in each case.

3. Comparison of Heating Curves.

The heating curves for the alloys have one feature in common. With so many
curves superposed, in order to economise space, this feature is not perhaps brought
out with the maximum of clearness in fig. 1. But it can be seen that a drop in the
magnetism begins in all the steels at about the same temperature. Actually this tem-
perature is the same within one or two degrees. It is usually quite sharply marked.
For example, the thermocouple temperature was kept steady for several minutes at
733° C., in one case, without alteration in the magnetometer deflection which differed
very little from that at 730° C. The furnace temperature was then raised very slightly,
and, before the thermocouple registered 734° C., a rapid fall of magnetisation set in.

This fall is absent or imperceptibly small in the iron curve and is smallest in that
for the steel weakest in carbon. Its relative magnitude increases with the percentage
of carbon, as fig. 1 shows, until the specimen containing (77 per cent, of carbon is
reached. After that, it is difficult to decide whether the ratio of the rapid fall
(below 740° C.) to the subsequent fall, before the magnetisation becomes too small to
be measurable, depends appreciably upon the percentage of carbon.

The first important fact, then, is that the sudden loss of magnetism begins at the
same temperature in all the steels. This accords with the view that the eutectoid
patches, detected by the microscope, have always the same composition. It might
also suggest that this constant temperature is the true transition point between
eutectoid and homogeneous solid solution.

That this inference would be wrong may be shown in two different ways. Fig. 2
shows one. The curves relate to successive interrupted heatings, of a steel containing
0'85 per cent, of carbon, described later.* At the moment it is only necessary to
call attention to the descending branches of the different curves. It will be noticed
that two (Nos. 4 and 5) proceed vertically downwards and one (No. 2) slopes slightly
outwards from left to right. The significant one, for the present purpose, slopes
inwards, i.e. towards the left. It (No. 3) was obtained with the most gradual rate
of heating and shows that the transition continues at a lower temperature than that
at which it began.

This is a case of a phenomenon which is the converse of recalescence. In the
latter the material is self-warming. Here it is self-cooling. The effect is not very
pronounced and might, at first sight, be attributed to irregularities of heating ; but
the inference that it is due to lag can be justified in another way. This is shown
in fig. 3.

* See § 18. Fig. 2 is printed on p. 199.

182 DE. S. W. J. SMITH AND ME. J. GUILD: A THEEMOMAG-NETIC STUDY OF

Here the heating was interrupted (twice) before the "solution" of the eutectoid
was complete. Cooling was continued until most of the eutectoid had reappeared ;
but heating was begun again while some dissolved eutectoid still remained. Now, the

675

775'

loss of magnetism took place, slowly at first and then rapidly, all at a temperature
below that at which it began when none of the transformed material was present.

If the true equilibrium temperature be that at which iron, carbide of iron, and
solid solution of eutectoid composition can coexist for an indefinite time in the steel
without change, it follows from what precedes that this temperature is below 735° C.

THE EUTECTOID TKANSITION POINT OF CARBON STEELS.

183

Consideration of the results shown in fig. 3 makes it also reasonable to suppose
that the equilibrium temperature is below 730° C., the temperature at which (point P)
the magnetism reaches a minimum during cooling from 735° C. ; but that it is above

660

680

700° 720° 740°

Temperature.

760°

780U

800C

723° C., the temperature at which (point Q) the magnetisation reaches a maximum
during subsequent re-heating. If so, we have a thermomagnetic method of deter-
mining the equilibrium temperature within two or three degrees as the figure shows.
This point is examined more closely later on.

184 DR. S. W. J. SMITH AND ME. J. GUILD: A THERMOMAGNETIC STUDY OF

4. Comparison of Cooling Curves.

Turning now to the results obtained during continuous cooling, collected in fig. 4,
it will be seen that the temperature of rapid return of magnetism in the eutectoid
is no longer constant except perhaps in the hyper-eutectoid steels. In the hypo-
eutectoid steels, the temperature of rapid return appears to become continuously
lower as the percentage of carbon falls. Although ^his is true in general, it would
appear (from the present and other observations) that the behaviour of the different
steels is not quite as regular during cooling as during heating.

The conditions under which the eutectoid forms during cooling are therefore
apparently less simple than those under which it dissolves during heating.

Apart from surface phenomena, which probably exert an appreciable retarding
influence during heating as well as during cooling, it is important to remember that
whereas the transformation of the eutectoid precedes that of the excess iron during
heating the opposite is true during cooling.

Since the solution expands when it changes into the eutectoid (with evolution of
heat) it is easy to see that, especially in the alloys weak in carbon, the pressure
exerted during the transformation, by the enveloping excess iron, may be a cause of
retardation which is present during cooling but is absent during heating.

Experimental evidence concerning the retarding forces operating during cooling is
given below.

5. Comparison of Results Obtained in Different Fields ivith the 0' 15 per cent.

Carbon Steel.

In the experiments which have been described the specimens were submitted
continuously to a constant field of 50 C.G.S. units.

Fig. 5* shows the behaviour of the 0'15 per cent, carbon steel in similarly applied,
but weaker and stronger, fields of about 25 C.G.S. and 200 C.G.S. respectively.

The ordinates represent intensities of magnetisation, in arbitrary units, as before.
The scales for the two fields are, however, very different. Thus the intensity of
magnetisation in C.G.S. units at 680° C. was actually about 13 times greater in the
stronger field than in the weaker.

Observations taken during cooling are represented by crosses in each case.

The effects, upon the magnetisation-temperature curve, of loss and gain of
magnetisability of the eutectoid component of the steel are much less pronounced in
the field of 25 units than they were in the field of 50 units. For example, the return
of magnetisation near 700° C. is now only just perceptible.

* In order to exhibit the parallelism between the time-temperature (" inverse rate ") method of
determining the eutectoid point and the thermomagnetic method, fig. 5 includes curves showing how the
rate of variation of the intensity of magnetisation c/ with respect to the temperature 6 depends upon the
value of 6. The ordinates represent differences between values of ^/ (in arbitrary units) measured at
intervals of 4° • 5 C.

THE EUTECTOID TRANSITION POINT OF CAEBON STEELS.

185

In the stronger field however, the effect of the return of magnetisability, at the
same temperature, is much more pronounced than in the earlier experiments.

550'

800°C

These differences are no doubt due to the fact that in weak fields the susceptibility of
the iron in the eutectoid is much smaller compared with that of the excess iron than
it is in strong fields.

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186 DR. S. W. J. SMITH AND ME. J. GUILD: A THERMOMAGNETIC STUDY OF

The contribution of the carbide to the magnetisability of the material need not be
considered at high temperatures because it is known that it is relatively very feeble
above 250° C.*

6. Elimination of Hysteresis in Strong Fields.

Another significant difference between the curves in weak and strong fields is
observable, particularly below 690° C.

In the former, the falling temperature observations lie increasingly above those for
rising temperatures as the temperature is reduced ; in the latter, the corresponding
observations practically coincide. This difference is no doubt an effect of the same
cause as that to which ordinary magnetic hysteresis is due, namely, the existence of
various molecular groupings which make alignment difficult in weak fields but which
are broken up when intense fields are applied.

Weak fields, although increasingly aided by thermal agitation as the temperature
rises, are unable during heating to break up all of these molecular groupings. But
the magnetism reappears during cooling in a medium in which, after heating to a high
temperature, the molecular groupings are less extensive than before. Hence a greater
intensity of magnetisation will now tend to arise under the same field as before,
because the force required to maintain a given degree of alignment is less than that
required to induce it against already existing groupings.!

In strong fields, on the other hand, the resistance to alignment is determined
mainly by the thermal agitation. This is the same at a given temperature whether
that is approached from above or below. Consequently the coincidence of the heating
and cooling curves in the field of 200 C.G.S. units, except over the region where the
lag in the transformation of the eutectoid occurs, is an indication that the observations
taken continuously in this field are practically free from the effects of hysteresis.
This means that they are practically the same as they would have been if the
material had been demagnetised between successive applications, at different tempe-
ratures, of the field in question.

It therefore appeared that it would be possible to make quantitative use of curves
obtained in strong fields in the way indicated below.

7. A Method of Estimating the Percentage, Composition of the Eutectoid.
Considering fig. 5, we may suppose that the contribution of the iron in the
eutectoid, to the total magnetisation at 697° C. during heating, is represented by the
distance between the point P on the upper curve at that temperature and the
corresponding point Q on the lower curve at the same temperature. In the one case
the eutectoid has not yet begun to lose its magnetism ; in the other it is just about
to regain it.

* Loc. cit. XXIV., p. 63, 1911.

t Such considerations are accentuated by the fact that, although the solenoidal field remains constant,
the field within the specimen increases as its magnetisation falls.

THE EUTECTOID TRANSITION POINT OF CARBON STEELS. 187

We may assume tentatively that the intercept PQ is a measure of the amount of
iron in the eutectoid and that, similarly, the intercept QM is a measure of the amount
of excess iron. The ratio of these two, QM/PQ, can then be compared with the
calculated ratio of the excess iron to the eutectoid iron.

If we assume that the eutectoid contains e per cent, of carbon, then in a steel
containing c per cent, of carbon, the ratio of the amount of excess iron to that
associated with the carbide in the eutectoid is

100(e-c)/c(lOO-15e).

The value of e is not very accurately known. It probably lies between 0'85
and 0'9. Accordingly in a steel containing 0'15 per cent, of carbon, the calculated
ratio lies between 5'35 and 578.

The two horizontal lines A and B, near Q in the figure, are at distances above M
such that the upper one produced would divide PM in the ratio 578 : 1, whilst
the lower one would divide it in the ratio 5 '3 5 : 1.

It appears, therefore, from the experimental position of Q, that, within the limits of
error, the contribution of the eutectoid to the total magnetism corresponds with the
amount of iron which it contains. Viewing this result from the opposite standpoint
it will be seen that, if we accept it, we obtain a thermomagnetic method of
determining the composition of the eutectoid.

In anticipating such a method the only question was as to how intense the field
would require to be in order that the computation might be made with useful accuracy.
Here we could only foresee that the steels would be relatively soft magnetically
at temperatures near 700° C., and that fields of moderate strength might suffice
to produce the necessary approach towards saturation.

The possibility of error in moderately strong fields, owing to the shortness of the
specimen, must of course be borne in mind. For, although the solenoidal field is
kept constant, the field within the specimen will vary appreciably with the intensity
of magnetisation. In the present case the intensity of magnetisation of the specimen,
at the point P, was about 720 C.G.S. units ; while, at the point Q, it was about
600 units.

Assuming a constant demagnetisation coefficient (independent of the distribution
of magnetic material within the specimen) of about 0'09, the demagnetising fields
in the two cases would be about 65 and 54 C.G.S. units respectively. Thus the
effective fields at P and Q would differ by 11 units, being 135 and 146 C.G.S.
respectively, and the comparison made above upon the assumption of their equality
would require a correction for this difference.

The correction required is apparently small. It would of course become negligible
if very strong fields were used. As a step in this direction, we attempted to obtain
curves with a magnetising field of 400 C.G.S. We found, however, that the
compensation between the magnetising solenoid and the particular balancing coil used

2 B 2

188 DR. S. W. J. SMITH AND MR. J. GUILD: A THERMOMAGNETIC STUDY OF

could not be maintained satisfactorily when large currents were passed through them
and were obliged to postpone further experiments in this direction.

8. An Attempt to Apply the Method to Steels Richer in Carbon.

It seemed worth while, however, to examine some of the other steels in the field of
200 C.G.S. units in the same way. Figs. 6 and 7 show the results of experiments

650°

THE EUTECTOID TRANSITION POINT OF CARBON STEELS.

189

made upon the 0'36 and 0'60 per cent, steels. The thicker-lined curves were
obtained during uninterrupted heating and cooling as in fig. 5. The letters P. Q, M
and the lines A and B have the same significance as before.

It will be seen at once that the point Q no longer occupies the position calculated
for it on the assumption that MQ should represent the contribution of the excess iron,

650'

800 C

and QP that of the eutectoid iron to the total magnetisation MP. In each case the
eutectoid iron appears to contribute more than its calculated share to the total
magnetisation. The apparent excess is very marked in the 0'60 per cent, steel.

The error due to the shortness of the specimen, already mentioned, would increase
with the percentage of carbon, and would be much more important in the 0'60

] 90 DR. S. W. J. SMITH AND MR. J. GUILD : A THERMOMAGNETIC STUDY OF

per cent, steel than in that first considered on account of the greater difference
between the ordinates MP and MQ. But the discrepancy between the calculated and
observed values cannot be ascribed to this cause alone if we assume a constant
demagnetising coefficient as before. For, in that case, it would appear that the
calculated position of Q should be below, not above, that observed.

It is not impossible, however, that the demagnetising coefficient is greater when
the material is in the state corresponding with Q than when the rod is more com-
pletely magnetic. Under such circumstances the difference between the demagnetising
fields would be less than a calculation like that in § 7 would give.

Apart from such uncertainties, which only observations in very strong fields could
remove, there are reasons why, even in the strongest fields, the calculated position
of Q should, as in figs. G and 7, be higher than that observed.

To make these reasons clear, it is necessary to consider the conditions under which
the eutectoid forms in the different alloys.

9. The Effects of Incomplete Equilibrium Arising out of the Slowness of

Diffusion.

Suppose that the cooling alloy contains c per cent, of carbon, where c is less than
the eutectoid percentage e. We need consider only the changes which occur below
900° 0. At some temperature between this and the eutectoid temperature, which
is lower the greater c is, iron " crystals " (easily magnetisable below about 780° C.)
begin to separate from the homogeneous " solid solution " of iron and carbide.

As the temperature falls these crystals grow, and simultaneously the remaining
solution becomes richer in carbide.

The conditions of equilibrium at any temperature 6 require that the solid solution
in immediate contact with the separated iron crystals should contain a percentage
of carbon cg which is a definite function of 9, intermediate between c and e, increasing
in magnitude as the temperature falls.

The separated iron crystals grow around nuclei distributed throughout the solid
solution and their growth gradually restricts the regions within which the carbide
is contained. But it will be noticed that the concentration of the carbide within
these regions is not necessarily uniform. It is ce where contact with the separated
crystals of iron occurs ; but it is only by diffusion inwards from the contact layers
that the concentration in carbide can rise throughout the rest of the solution.
Simultaneously with this diffusion, the crystals of separated iron grow.

Thus we see that, unless we suppose a continuous separation of fresh nuclei as the
temperature falls (which surface effects tend to prevent), the rate of crystallisation
of the iron (and of rise in concentration of carbide in the solution remaining) depends
very largely upon the rate of diffusion of the carbide within the solid solution.

Thus also, we see that the amount of iron which has separated at any given

THE EUTECTOID TRANSITION POINT OF CARBON STEELS. 191

temperature tends always to be less than the amount which would be present if the
solution had the uniform concentration corresponding with complete equilibrium.

The calculated fraction of the whole mass which deposits as eutectoid being c/c,
it follows that when c is small the amount of solid solution remaining when the
eutectoid temperature is approached is also small.

Consequently, the effects arising out of the comparative slowness of diffusion are
not likely to be considerable. But when c is larger, these effects may become
important. We shall then have comparatively thick layers of solid solution just
above the eutectoid point, and there may be an appreciable difference of concentration
between their centres and their surfaces.

As the temperature falls, however, the surface concentrations rise and the layers
become thinner. Each of these effects tends to increase the concentration gradients
from the surface inwards. Hence the diffusion tends to accelerate. This acceleration
will be maintained because the iron crystals will grow and maintain the surface
concentration of the carbide in solution as the layers thin down. Hence the trans-
formation of the last of the pre-eutectoid iron will be rapid and difficult to distinguish
thermomagnetically from that of the eutectoid.

Such considerations enable us to see that the amount of magnetisation acquired
after Q in the figures is passed may easily be greater than corresponds with the
amount of iron contained in the eutectoid.

10. Experimental Evidence of the Effects of Diffusion.

In order to confirm the existence of incomplete equilibrium of the kind pictured
above, we performed the experiments indicated by the thinner-lined curves of
figs. 6 and 7.

Instead of allowing the temperature of the material to fall continuously as before,
the cooling was now interrupted repeatedly. After each interruption the temperature
was slowly raised some 10 or 20 degrees, and then allowed to fall slowly to a point
below that at which the interruption took place. This process was repeated several
times in each steel as shown in the curves.

It will be obvious at once that the general result of every interruption and re-heat
is to add to the amount of magnetisation shown by the steel when the temperature
of interruption is regained.

Each cycle of temperature change tends to reduce the concentration differences
within the solid solution and to make the amount of iron set free correspond more
nearly with that required for complete equilibrium. It thus becomes possible to
distinguish more clearly between the real contribution of the excess iron and that of
the eutectoid iron to the magnetism of the material as a whole.

From the positions of the lines A and B with respect to the " interrupted " curves,
it will be seen that there is no reason to doubt (l) that in these steels, as in that of

192 DR. S. W. J. SMITH AND MR. J. GUILD: A THERMOMAGNETIC STUDY OF

fig. 5, the contribution of the eutectoid to the total magnetisation, at say 700° C.,
indicates the amount of iron which it contains, (2) that the eutectoid mixture has
the same composition in all the steels.

It is instructive to consider in detail the forms of the thermal hysteresis loops and
connecting curves shown in the figures, and to examine their significance with respect
to the sequence of changes within the material ; but considerations of space do not
permit further reference to this here.

•

11. Evidence of the Relative Unimportance of the Effects of Diffusion in

Loiv-carbon Steel.

We proceed to describe further experiments of which the results are shown in
fig. 8. These were made upon a steel containing about O'l per cent, of cai'bon. Their
object was to test the inference, from a comparison of fig. 5 with figs. 6 and 7, that
the effects of the slowness of diffusion are relatively unimportant when the layers of
solid solution from which the eutectoid is about to separate are relatively thin. In
the present case the eutectoid patches form about one-ninth part only of the total
mass of the steel.

The thicker-lined curves represent, as before, the results obtained during continuous
slow heating and cooling. The heating was stopped at about 815° C. The
equilibrium diagram indicates that, at this temperature, the solution in contact with
the still undissolved iron contains roughly 0'3 per cent, of carbon. The patches of
transformed eutectoid are therefore bordered by layers in which the concentration of
carbon drops to about 0'3 per cent.

During cooling the iron in these border layers separates out again from the surface
inwards, the surface concentration rising continuously from 0'3 per cent, towards the
eutectoid percentage as the temperature falls. The re-crystallising iron closes in
upon the layers of solid solution ; but the re-precipitation of the excess iron is not
(juite complete at the temperature at which it begins to dissolve during heating.
For it will be observed that the cooling curve US rises slightly, but distinctly, more
rapidly than the heating curve PQ falls, and therefore that, if we assume no solution
of iron along PQ, we must suppose some precipitation along RS.

It is easy to see, as before, how this effect may arise ; but it is also easy to show
how much less pronounced it is in the present case. To do this, we repeated the
experiment, but interrupted the cooling at various temperatures between R and S,
as in the experiments of figs. G and 7.

The results are shown in the centre fig. 8. For the sake of clearness, we have
displaced the observed ordinates downwards. Their true positions are to be found by
raising all of them through the distance between the lower AB and the upper. It
will be observed that the effects of alternation during cooling, although perceptible,
are now very slight, compared with those of figs. 6 and 7, until the temperature
at which the eutectoid appeared during continuous cooling is approached.

THE EUTECTOID TRANSITION POINT OF CARBON STEELS.

193

650

800"

VOL. CCXV. — A.

2 o

194 DR. S. W. J. SMITH AND ME. J. GUILD: A THERMOMAGNETIC STUDY OF

The loops significant of the pronounced effects of concentration gradients are now
absent.

12. Possible Evidence of the Effects of Pressure.

The effects of alternation begun just above the temperature at which the eutectoid
previously made its appearance are noteworthy.

It will be observed that this alternation is sufficient to induce practically the whole
of the eutectoid change at a temperature higher than that at which, in its absence,
the transformation appeared to begin.

It seems to us most natural to explain this phenomenon by supposing that the
transformation of the eutectoid is retarded by the existence of internal pressure.

On this view the eutectoid would form at a higher temperature than that at
which it ordinarily appears were it not for the expansion which accompanies the
transformation and causes pressure to be exerted by the surrounding envelope of iron
immediately after it begins. This pressure cannot increase without limit. The
separation of the eutectoid in bulk begins when the increase in pressure, required
to prevent further transformation as the temperature falls, cannot be supplied. The
effective transition point will thus depend upon the elastic properties of the
enveloping iron.

If, as is likely, the capacity to resist strain is reduced when the temperature is
raised, the transition should simultaneously be facilitated. Fig. 8 shows that what
happens is as if a very slight elevation of temperature at the critical stage induces
that amount of breakdown in the envelope which is necessary before the eutectoid
can appear in bulk.

1 3. Evidence of the Existence of other Effects.

It is apparent that effects of pressure of the kind just contemplated would decrease
along with the amount of free iron present beforehand, and would therefore become
less important as the percentage of carbon in the steel rose towards that contained
by the eutectoid. They might therefore explain the observed gradual rise in the
temperature of reappearance of the eutectoid as the percentage of carbon in the steel
rises towards 0'9 ; but they would not account for the whole of the lag. In particular,
they would not explain the fact that in the hyper-eutectoid steels the temperature
of reappearance seems to be practically constant and higher than in any of the
others.

It is, in fact, obvious that surface effects, considered in conjunction with the
conditions of equilibrium, must play an important part in determining the temperature
at which the eutectoid appears.

We proceed to consider important cases from this point of view.

THE EUTECTOID TRANSITION POINT OF CARBON STEELS.

195

14. The Conditions Preceding the Separation of the Eutectoid in Hyper- Eutectoid

Steels.

Fig. 9 represents the results of experiments upon the hyper-eutectoid steel,
included in the table given in § 2, containing about 1'2 per cent, of carbon. The

800°C

experiments were carried out, in the same way as those of figs. 5, 6, 7, and 8, in a
field of about 200 C.G.S. units.

2 o 2

196 DR. S. W. J. SMITH AND MR. J. GUILD: A THERMOMAGNETIC STUDY OF

Although the eutectoid disappears at the same temperature as in the hypo-
eutectoid steels of those figures, the reappearance now occurs at a much higher
temperature. It is interesting to notice also that there is now no appreciable return
of magnetism prior to the crystallisation of the eutectoid.

The process of crystallisation in such an alloy may be regarded as follows : —
Suppose that the alloy, still at a temperature above that at which the excess carbide
begins to deposit, is cooling down. Eventually it will reach the temperature at
which it could exist in equilibrium with free carbide if the latter were present
in bulk. Precipitation will not begin at this temperature, however, if there is
appreciable surface energy between carbide crystals and solid solution.

Every re-crystallisation is characterised by the fact that, in the minute crystals
first formed, the ratio of surface area to mass is relatively very great. Consequently
the surface energy may be an appreciable fraction of the total energy of these
crystals. This surface energy virtually increases the chemical potential of the
separated material and therefore tends to make the temperature at which crystals
can form, in a given solution, lower than it would be otherwise.

The disturbing effects of surface energy become less important as the crystals
enlarge. • Hence crystallisation, once begun, tends to proceed around the nuclei first
deposited. At the same time the concentration in carbide of the solution in contact
with the crystals approaches the equilibrium value.

This value decreases as the temperature falls, and is attained by deposition from
the solution upon the crystals. As before, owing to the slowness of diffusion, the
equilibrium will not be complete. The pai-ts of the solution remote from the crystals
will be too rich in carbide unless the rate of cooling is infinitely slow.

When the eutectoid temperature is reached the solution in contact with the
crystals will have the eutectoid composition ; but the eutectoid will not form without
lag of a similar nature to that which delayed the appearance of the carbide.

In this case the lag will be due mainly to the iron, since carbide crystals are
already present.

15. An Effect Due to the Excess Carbide.

Indirectly the presence of the carbide crystals will accelerate the crystallisation of
the iron. For when the temperature falls below the eutectoid point the carbide
continues to deposit and the solution in contact with the crystals continues to get
richer in iron. The chemical potential of this iron therefore diminishes less rapidly
as the temperature falls than it would do if carbide crystals were not present. Hence
the amount by which this potential exceeds that of the iron crystals in bulk at any
temperature below the eutectoid point is greater than it would be in the absence of
the carbide. Therefore the resistance to the formation of the iron crystals and hence
of the eutectoid will be overcome at a higher temperature when carbide crystals are
present beforehand, as in the case of fig. 9, than in the eutectoid steel.

THE EUTECTOID TRANSITION POINT OF CARBON STEELS.

197

16. A Working Hypothesis as to the Mode of Crystallisation of the Eutectoid.

We may next consider the eutectoid steel itself and describe some experiments
made upon it.

The surface energy between the solid solution and iron is probably less than that
between it and the carbide. Accordingly, in such a steel, the lag in the precipitation
of the eutectoid is probably terminated by the crystallisation of some of the iron
which it contains. But the sudden precipitation of iron would increase temporarily
to a very high value the concentration of the carbide in solution in its immediate
neighbourhood. Hence, even if the retardations tended to be unequal, the crystalli-
sation of the one constituent would induce the appearance of the other.

It is necessary to attempt to form a picture of the way in which crystallisation
proceeds. For simplicity in description we may suppose that, as frequently happens,
the eutectoid crystallises in grains, each consisting of, approximately parallel,
alternating layers of iron and of carbide.*

It is unlikely that the production of these alternating layers of very different
composition and finite thickness is instantaneous.

We may suppose that the first stage in the crystallisation of each grain results in
the formation of parallel and excessively thin threads at approximately uniform
distances apart, consisting of iron and the carbide in the eutectoid proportions,
separated by layers of still untransformed solid solution in the way represented
diagrammatically in fig. 10 (i.), and that the second stage is the growth of the
threads to their final thicknesses, represented in fig. 10 (ii.), by the diffusion in

Fs, C

Fig. 10.

opposite directions, and deposition, of the constituents of the intervening layers of
the solid solution.

Assuming the first stage, it is easy to see that the kind of diffusion postulated
in the second will occur.

Owing to the lag the temperature is below the eutectoid point. Consequently
the chemical potential of the carbide in solution is higher than in the separated

* Cf. e.g. BENEDICKS, 'Recherches sur 1'acier au carbone,' Photogramme 1.

198 DR. S. W. J. SMITH AND MR. J. GUILD: A THERMOMAGNETIC STUDY OF

crystals. Carbide must therefore precipitate from the solution on the surfaces b in
fig. 10 (i.) until the concentration of that in solution is lowered to the equilibrium value
corresponding with the temperature of the material. Similarly the chemical potential
of the iron in solution is higher than that in the separated crystals, and precipitation
of iron must occur on the surfaces a until the concentration of the iron in solution is
lowered to the equilibrium value.

The solution near the surfaces b thus at once gets weaker in carbide, and that near
the surfaces a weaker in iron. The concentration gradients thus established cause
the iron to diffuse in one direction and the carbide in the opposite. This diffusion in
its turn causes fresh separation of iron on the one side and of carbide on the other.
Hence separation and diffusion will occur continuously until crystallisation is
complete.

The transformation of the eutectoid during subsequent re-heating can be treated
in a similar manner by supposing solution to begin again where the crystallisation
ended. It is not difficult to see how this view of what happens could be extended to
meet other cases in which the structure of the eutectoid is different.

17. Application to the Case in which the Crystallising of the Eutectoid is

Interrupted.

The utility of the above picture of the process of crystallisation can be tested
by considering what, according to it, should happen if the process were interrupted
before completion, and by examining, thermomagnetically, what actually occurs.

Imagine, therefore, that the temperature of the furnace, in which the steel lies,
is raised slowly before crystallisation is complete.

The first effect is to lower the rate of escape of heat from the steel to the furnace,
and therefore to reduce the rate of crystallisation. The diffusion within the solid
solution will continue, however, and will cause the boundary films at a and b (fig. 10)
to become respectively richer in iron and in carbide than they were. This will make
it possible for crystallisation to continue from them at a higher temperature.

Crystallisation need not cease at once, therefore, and may go on for some time after
the temperature of the material has begun to rise. But, unless the diffusion after
interrupted cooling is sufficient to remove the differences of concentration within
the still-untransformed material, solution must begin again before the true eutectoid
temperature is reached.

Fig. 11 is an example of results obtained during interrupted cooling. In the first
case the cooling was not interrupted until the return of magnetisation was nearly
complete. In the third it was interrupted soon after the return had begun.

The curves show that the gain of magnetisation continues until the temperature
has risen appreciably above its value when the cooling was arrested.

They show also that the subsequent loss of magnetisation begins earlier, but

THE EUTECTOID TRANSITION POINT OF CAEBON STEELS.

199

more gradually, than during continuous heating from the atmospheric temperature.
The same effect is shown more strikingly in figs. 12 and 13 below.

It will be noticed that, according to the views already outlined, rapid solution
should not be possible until the temperature of the specimen has risen sufficiently
far above the eutectoid point to make the differences between the equilibrium
concentrations at the boundary films considerable.

680°

700°

720°

B

Fig. 2.

740°

Fig. 11.

700°

720°

760°

7 BOG

740°

0

760°

Fig. 11, together with figs. 12 and 13, shows that what happens is in agreement
with this deduction.

The fact that during heating from the air temperature the solution proceeds
rapidly immediately after it begins is additional evidence of the existence of the lag
already considered in § 3.

18. Application to the Case of Interrupted Solution of the Eutectoid.
Fig. 2 (see this page) serves to show the behaviour of the material after interrupted
heating. To interpret the results we have to consider what will happen when the

78CTc

200 DR. S. W. J. SMITH AND MR. J. GUILD : A THERMOMAGNETIC STUDY OF

temperature of the furnace is lowered after solution has begun. The immediate effect
is to lower the rate of supply of heat to the material, and therefore to lower the rates
of solution of the carbide and the iron at the respective surface films.

Before the lowering, the rates of diffusion of these were sufficiently rapid to keep
their concentrations from rising in the regions where they were dissolving. After it,
they will be more than sufficient, and these concentrations will fall, with the result
that, even when the temperature of the material is lowered, solution may still
go on.

In practice, as will be seen from the figure, the rate of solution soon becomes very
slow, and the temperature has not fallen very far before the magnetisation begins
to rise.

The temperature at which solution ceases will be above the true eutectoid point,
unless the diffusion has sufficed to make differences of concentration negligible within
the solution which remains. It will thus tend to be nearest the true value (and,
incidentally, lowest) when the layers of solid solution are never very broad, i.e., when
the heating is interrupted very soon after solution has begun.

It will be seen that this inference is corroborated by the experimental results.

It remains to explain why, as in the figure, the temperature at which the lost
magnetism begins to be recovered rapidly is lower the higher the temperature at
which the heating is interrupted.

Suppose that re-crystallisation has begun. At one stage, marking the point
where concentration differences produced during solution have been approximately
obliterated by the reverse changes during re-crystallisation, the layers of solution
will be nearly of eutectoid composition. Their thickness will depend upon the amount
of solution that took place before the furnace temperature was reduced.

At this stage the specimen will be at the eutectoid temperature. When it is
passed the difference in composition between one surface film and the other will
change in sign and will become greater as the temperature falls. At the same
time the thickness of the whole layer will decrease. Ultimately, a temperature
will be reached at which the ratio of concentration difference to layer thickness
is sufficient to permit the rapid rate of diffusion which rapid re-crystallisation
requires.

If the layers of solid solution are very narrow when, during cooling, the eutectoid
temperature is passed, the concentration difference required for rapid diffusion will be
relatively small, -i.e., the temperature of rapid re-crystallisation will be relatively high.
If they are very wide, as happens when the solution is nearly complete before heating
is interrupted, that temperature will be relatively low.

The temperature of rapid increase of magnetisation should thus decrease pro-
gressively from its highest value — when solution is arrested almost as soon as it has
begun — to its lowest value when solution is not arrested at all.

This is exactly what happens experimentally as the curves of fig. 2 show.

THE EUTECTOID TRANSITION POINT OF CARBON STEELS.

201

19. Confirmatory Experiments in Weak Fields.

Some points of interest are shown in further experiments upon the eutectoid steel
represented in figs. 12 and 13 and already referred to in § 17.

675

775 C

The field intensity was now only 25 C.G.S. units. The first figure shows the effects
of two interruptions during heating from about 680° to about 800° C. The sequence
will be obvious from the lettering. Thus heating was first interrupted at B, their
cooling was interrupted at C, and so on, alphabetically, to F (about 800° C.).

VOL. ccxv. — A. 2 D

202 DR. S. W. J. SMITH AND ME. J. GUILD: A THEEMOMAGNETIC STUDY OF

These observations may be compared with those of fig. 3, described in § 3, which
•were obtained with the same steel in a field of 50 C.G.S. units. The temperature

675

775"C

of rapid loss of magnetisation is exactly the same as before, but the beginning of the
transformation is much more clearly seen in this weaker field. The temperature

THE EUTECTOID TRANSITION POINT OF CARBON STEELS. 203

coefficient of the magnetisation now. remains positive until the structural change
point is reached.

It would seem, therefore, that by suitable adjustment of the field in any particular
case, the thermomagnetic method can be used to indicate the change point, known
to metallurgists as Ac,, with an accuracy that can scarcely be exceeded in measure-
ments by any other method.

The experiments of fig. 13 followed immediately after those of fig. 12. The
material was cooled from F to G, then heated again to H, cooled to I (about 640° 0.),
then heated to J, cooled to K, and finally heated to L.

The observations from I to J show that the lag during heating (the existence of
which is again clearly shown) returns to its full value when the material is allowed
to cool to about 640° C. before it is re-heated. The series JKL goes a stage further
and shows that the same is also true when the cooling is stopped at about 705° C.

Such observations indicate another thermomagnetic method of determining the
temperature at which, during cooling under given conditions, the eutectoid change is
complete. They also corroborate the hypothesis that the lag during heating is due
to surface effects.

20. The Equilibrium Temperature.

If the interpretation of our results which has been given is correct, they show that
(subject to limitations due to variations of pressure) iron crystals, carbide crystals,
and a solid solution of the two of uniform (eutectoid) composition cannot coexist
without change except at a definite temperature.

According to the results obtained with the 0'85 per cent, steel, shown in figs. 3,
12, and 13, this temperature lies between 725° and 730° O.

Fig. 9A* contains the results of an attempt to find the equilibrium temperature by
means of the steel containing 1'2 per cent, of carbon. The method was the same as
before, except that we used a field of 200 C.G.S. instead of the previous fields of 25
and 50 C.G.S. respectively. The thermocouple had been calibrated more recently and
was probably rather more trustworthy than that used in the earlier experiments.

The point Q on the " interrupted heating " curve is that at which the temperature
coefficient of ss with respect to 0 is the same as at the corresponding temperature 11
on the (uppermost) curve of continuous heating. It is presumably, therefore, to a
first approximation, the temperature at which the change from solution to re-crystal-
lisation began during cooling.

The point P on the " interrupted cooling " curve is similarly that at which the
temperature coefficient of ^ with respect to 6 became the same as at the corre-
sponding temperature on the uppermost curve. It is, therefore, to be regarded as
the point at which change from re-crystallisation to solution began during heating.

It happens that the temperatures corresponding with the points P and Q are

* See p. 195.

204 DK. S. W. J. SMITH AND MR. J. GUILD: A THERMOMAGNETIC STUDY, ETC.

practically the same, viz., 731° C. This is, therefore, according to these measurements,
the equilibrium temperature.

The greatest accuracy cannot be claimed for our measurements of temperature,
although they were made as carefully as the conditions of experiment seemed to
warrant. Sometimes the thermocouple may have been used rather too often for
safety, between renewals and re-calibrations, but it is unlikely that the experimental
error in the measurement of temperature ever exceeded 5° C.

Our general conclusion is that the equilibrium temperature can be measured, and
that it is, so far as our measurements can decide, substantially the same whatever
the percentage of carbon contained by the steel.

21. Possible. Effects of the Presence of a Magnetic Field during Crystallisation.

It is not inconceivable that the temperature at which the eutectoid separates out,
or disappears, should be influenced by the field. There is no evidence of this in the
curves which we have given, nor in others in which we have varied the field from
H = 10 to H = 400 C.G.S.

To apply a severer test, we took " recalescence " (time-temperature) curves, using
a chronograph and a very open scale thermometer, with a specimen of the steel
containing 0'7 per cent, of carbon, placed between the poles of a large Du Bois
electromagnet. We could not detect any material difference between the temperature
of recalescence when the magnet was fully excited, giving a very intense field, and
that when the field was practically zero.

Further, we attempted to determine whether the presence of an intense field made
any appreciable difference in the method of crystallisation of the eutectoid. Using
the rod containing 0'85 per cent, of carbon, we cooled it from above the eutectoid
point, between the poles of the electromagnet, first with its axis parallel to the field
and afterwards with its axis perpendicular to the field. In each case we examined
the subsequent magnetic behaviour of the rod at temperatures below 250° C. ; but
could not detect any difference between the changes accompanying the appearance
and disappearance of magnetism in the carbide in the two cases.* It is, therefore,
unlikely that the field produces any appreciable effect, during crystallisation at the
eutectoid point, upon the orientation of the carbide with respect to the iron.

* Of. loc. at. §§ 1 and 2.

[ 205 ]

VIII. The Effect of Pressure upon Arc Spectra.

No. 5. — Nickel, \ 3450 to X 5500, including an Account of the Rate of Displacement
with Wave-length, of the Relation between the Pressure and the Displacement,
of the Influence of the Density of the Material and of the Intensity of the
Spectrum Lines upon the Displacement, and of the Resolution of the Nickel
Spectrum into Groups of Lines.

By W. GEOFFREY DUFFIELD, D.Sc., Professor of Physics, and Dean of the Faculty

of Science in University College, Reading.

Communicated by Prof. A. SOHUSTER, Sec. R.S.

[PLATES 1-5.]
Received May 28,— Read June 25, 1914.

CONTENTS.

P?ge

1. Preliminary 206

2. Behaviour of the nickel arc under pressure 20G

3. The photographs 200

(1) Region investigated 206

(2) Description of the plates 207

4. The broadening of the lines 208

(1) General features 208

(2) Continuous spectrum 209

5. The reversal of the lines 209

6. Changes in the relative intensities of the lines 210

7. The displacement of the lines 211

(1) The measurement of the photographs 211

(2) Table of displacement 213

(3) The spectrum of the nickel arc under pressures of +155 and + 200 atmospheres . . 218

(4) Displacements towards the violet. .. 219

(5) Displacement diagrams 220

(6) Comparison with previous observations 220

(7) Relation between the pressure and the displacement 224

(8) Relation between the displacement and the wave-length 228

8. Resolution of the nickel spectrum into groups 236

9. Relation between the intensity of a line and its displacement 244

10. Foreign metals in the spectrum of nickel 246

(1) The influence of the density of the material 250

VOL. CCXV. A 530. 2 E [Published May 4, 1915.

206 DK. w. GEOFFREY DUFFIELD ON THE

1. Preliminary. — The apparatus and method of taking photographs of the spectra
of metallic arcs under pressure have been described in previous papers.* The arc
was formed between two poles of the metal (not quite pure) |-inch diameter and
about 6 inches long, which were enclosed in a steel cylinder, the design of Prof. PETAVEL,
F.R.S., capable of resisting a high internal pressure. The light from the arc passed
through a window in the side of the cylinder, and was reflected by a system of mirrors
upon the slit of the 21 g-foot Rowland grating spectrograph in the Physical Laboratory
of the University of Manchester.

As in previous experiments, the spectrum of the second order was used, the
dispersion being 1'3 Angstrom Unit per 1 mm.

An increase in pressure was obtained by the admission of air into the cylinder
from a gasholder, suitable valves and gauges being interposed.

The arc was fed by continuous current from the Corporation mains, which supplied
100 volts ; this was reduced to about 50 volts at the terminals of the arc.

2. Behaviour of the Nickel Arc under .Pressure. — As in the case of the gold arc
the ease with which the arc burned depended both upon the coolness of the poles and
the freshness of the air supply. The arc was maintained for short intervals without
difficulty:

Of electrodes previously used the nickel arc behaved more like that between copper
poles. The intensity increased very markedly with the pressure, but photometric
measurements were again out of the question, owing to the unsteadiness of the arc.

At low pressures the nickel arc was distinctly mauve in tint, but it became whiter
as the pressure of the surrounding air was increased.

3. The PJiotographs : (l) Region Investigated. — -The investigation of the spectrum
extended from A = 3450 to A = 5500 : as this range of wave-lengths extends over
about 162 cms., it was necessary to move the 50-cm. camera into four different positions.
This involved a large amount of labour, as it practically quadrupled the amount of
work which would have been necessary had it been possible to photograph the whole
of the spectrum at once.

The photographs were taken partly during October, 1908, and the remainder during
April, May, and June of 1910, the series being interrupted by the writer's absence in
Australia.

The following table shows the pressures at which photographs have been taken in
different regions of the spectrum : —

* W. G. DUFFIELP, 'Phil. Trans.,' A, vol. 208, p. Ill, 1908 (Iron Arc); vol. 209, p. 206, 1908
(Copper Arc); vol. 211, p. 33, 1910 (Silver Arc); vol. 211, p. 51, 1910 (Gold Arc).

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL.

TABLE I.

207

10 atmospheres
20

10 atmospheres

20

40 „ (2)

60 „ (2)

10 atmospheres
20

10 atmospheres

X = 3450 to X = 4050.

40 atmospheres
60

X = 4050 to X = 4600.

70 atmospheres

80

93

X = 4600 to X = 5120.

40 atmospheres
60

X = 5120 to X = 5500.
20 atmospheres

80 atmospheres
100

110 atmospheres

155

200

80 atmospheres
100

75 atmospheres

The plates used were Imperial Flashlight and the developer Imperial Pyro-Metol
Standard. The exposure varied from five minutes to one hour according to the region
examined, and the difficulty with which the arc burned. It should be noted that
one hour is the total time expended upon the exposure, and includes the time when the
arc was being re-struck. The total time during which the arc burned is a small
fraction of this as a rule. The effect upon the photographic plate is that of the sum
of a very large number of short-lived arcs.

(2) Description of the Plates. — Plates 1 to 5 illustrate the behaviour of the Nickel
Arc under different pressures. Plate 1 includes the region X = 3450 to X = 3740,
Plate 2 the region X = 3740 to X = 4050, Plate 3 the region X = 4030 to X = 4350,
Plate 4 the region X = 4350 to X = 4610, and Plate 5 the region X = 4600 to X = 4900.
The photographs are full-size positive reproductions of the originals, and are arranged
in order of increasing pressure from the top at one atmosphere to the bottom at -1- 100
or +200 atmospheres. The arbitrary numbers enumerated in Table III. have been
affixed to facilitate reference to them.

The central strip in each photograph is the spectrum at atmospheric pressure, and
corresponds to the spectrum at the head of each Plate. Above and below this strip
are the lines as they appear when the arc is subjected to pressure. The central strip
was taken partly before and partly after the pressure exposure, and this provides
a check upon the value of each photographic plate. The shutter which made this

2 E 2

208 DE. W. GEOFFKEY DTIFFIELD ON THE

possible is a modification of that originally used by HUMPHREYS,* and it has already
been described.!

4. The Broadening of the Lines : (l) General Features. — -The general phenomenon
of the broadening of lines under pressure has been described elsewhere. In the Nickel
Spectrum :•—

(a) Some lines broaden nearly symmetrically ; but
(6) Most lines broaden unsymmetrically.

Of the latter class by far the larger number are more extended towards the red end
of the spectrum, but a few are unmistakably broadened more on the violet side.

It is usually possible to distinguish between two classes of lines at atmospheric
pressure : namely those that are sharp and those that are soft or nebulous.

Under pressure some few of the former retain some of their characteristic hardness
of outline at moderate pressures (10 to 20 atmospheres), while others lose their sharp
appearance altogether and quickly become nebulous. Many that start by being
nebulous become more so and disappear.

Measurements of the broadening of the lines have not been made, but some
indication of their relative behaviour in this respect is given in the first row of the
eighth column of Table IX., which classifies them in the following order of increasing
width : slight, s ; moderate, m ; considerable, c ; great, g ; very great, vg ; very very
great, G.

The other columns give some account of the nature of the broadened line, whether
shai-p or nebulous, symmetrical or unsymmetrical. The abbreviation bs indicates that
the broadening is nearly symmetrical (very few are quite symmetrical, there is usually
a slightly larger wing on the red side), br indicates that the broadening is greater
towards the red, and bv that it is greater on the violet side. It is impossible to
examine the broadening of lines in any detail without being impressed by the
inadequacy of the nomenclature of the spectroscopist. Under pressure a "line" is
but a courtesy title for the extended patch of luminosity into which its original sleek
proportions have degenerated, and it is valueless as a description. Nor is the term
band quite appropriate, since a banded spectrum is by common usage something rather
different in appearance from that of a spectrum ordinarily produced by pressure.
With reluctance I have retained the word line.

The terms sharp and nebulous are usually employed to distinguish between the
two well-known types of spectrum lines, but the former does not seem quite
satisfactory when applied to a line whose energy has been diffused over several
Angstrom Units. Nevertheless a distinction is to be discerned even under pressure in
the appearance of the lines, which it is valuable to make, and I have adopted the
usual terms. The origin of the difference seems to lie in the shape of the intensity
curves — these are gradual at the boundaries of those lines which are usually called

* HUMPHREYS, ' Astrophysical Journal,' vol. VI., p. 169, 1897.
t DUFFIELD, 'Phil. Trans.,' A, vol. 208, p. 117, 1908.

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL. 209

nebulous, and steeper in the case of those which are called sharp. The nebulous lines
are not so clean-cut as those described as sharp, indeed the former frequently stand
out against what looks very much like a fogged background.

Many of the strong unsymmetrically broadened lines from 159 onwards are
accompanied by a haziness towards the red. It is doubtful whether it is continuous
with the intensity curve of the chief part of the line or whether the luminous patches
are superposed. It is possible that the different portions are due to light emitted by
different parts of the arc.

A critical comparison between the broadening of the lines of various metals will be
given in a subsequent paper.

(2) Continuous Spectrum. — It has been shown that under great pressure the silver
arc spectrum becomes contimious, and that it is due, at any rate in the region of the
spectrum investigated, to the broadening of the lines of the first subordinate series.*
This phenomenon has been looked for in the nickel spectrum, but though there is a
certain amount of contimious spectrum upon certain photographs (e.g. Plates 3 and 4
at 155 atmospheres), it is generally caused by the hot metallic poles rather than by
the spreading of a line ; as evidence of this we note that the continuous background
is less pronounced at the higher pressure of 200 atmospheres.

There is, however, a very great broadening of some of the lines upon Plate 1, the
spreading of lines 19 (X = 3566'50) and 31 (X = 3619'52) being particularly notice-
able (Plate l). Under 100 atmospheres the wings of 31 extend beyond line 36 on one
side and 26 on the other, and may be responsible for some of the continuous spectrum
which extends towards the region of longer wave-lengths. It is thus in some ways
analogous to the silver lines which belong to the 1st sub-series, and it may be a
member of this sub-series in the nickel spectrum.

5. The Reversal of Lines. — With increase of pi-essure there is at first a greater
tendency for lines to reverse, and these reversals are indicated in Table III., those
in italic (thus 3619'52, 47, 36, 148, &c.) representing a strong reversal, and those in
clarendon type (thus 3670'57, 39, 41, 102, &c.) a weak one. It will be observed that
more reversed lines are found in the region of small wave-length than in other parts
of the spectrum.

In addition to the information conveyed in Table III., further details concerning
the reversal of lines are given in Table IX. , where the width of the absorption line is
indicated, and also the symmetry or dissymmetry of the position of the fine line
upon the broadened emission line.

As in the case of broadening, the widths of the reversals are classified by s = slight,
m = moderate, c = considerable, g = great, vg = very great, G = very very great.
The term rs indicates that the absorption line is nearly symmetrically disposed upon
the emission line ; rv indicates that the absorption line is on the violet side of the
centre of the emission line.

* W. G. DUFFIBLD, ' Phil. Trans.,' A, vol. 211, p. 33, 1910.

210 DR. W. GEOFFREY DUFFIELD ON THE

Where it is stated that a line is nearly symmetrical, but that the absorption line is
slightly to the violet of the centre of the broadened line, it is to be understood that it
is the geometric centre that is indicated ; it is possible that the positions of maximum
emission and maximum absorption are coincident, though this is not necessarily the
case, as was demonstrated for certain iron lines.

We note from the photographs that when a line which is self-reversed is encroached
upon by the wing of an adjacent line, it gives rise to an absorption line upon the
bright wing. (Cf. 32 on 31.)

But that when a line which is not self-reversed at a lower pressure is similarly
encroached upon it does not reverse. (Cf. 35 and 36 in the wings of 31.)

It seems probable that this is due to differences in the distribution of the vibrating
centres responsible for the different lines in the arc itself, which give rise to
self-reversed lines having a different density or temperature gradient from those which
do not produce absorption so readily.

6. Changes in the Relative Intensities of Nickel Lines. — With increase of pressure
the spectrum undergoes a change which involve some lines becoming relatively more
prominent than they were before. It has previously been pointed out that it is very
difficult to assign a value to the intensity of a broadened line, because though the
area it covers is greatly increased by pressure the intensity per unit area is reduced.
The energy due to each line is the important quantity, but unfortunately it could only
be determined by an integration which it would be extremely difficult to carry out.

In the sixth and seventh columns of Table IX. is given an account of the changes
which have been observed in the intensities of nickel lines.

The following are the enhanced lines of nickel given by LOCKYER* : —

384970 4067-30 4245'0 4609'4

3889'80 4187'8 4279'4 46657

4015'76 4192-4 4362'3 4679'4

Only one of these lines has been observed to show a marked change in intensity
under pressure, namely, 3889"80 (line 87 upon Plate 2), which is classified as weakened
with increase of pressure ; this is the result of its great broadening, and does not
necessarily denote any reduction in the total energy emitted by the vibrating centre
responsible for it. One other enhanced line, 4067 "30, appears upon the photograph
at 10 atmospheres pressure, which has had a prolonged exposure, and it is there
strengthened relatively to some of the faint lines near it ; it does not persist at higher
pressures ; as none of the other enhanced lines have been observed under pressure, it is
the only exception to the general conclusion that pressure does not favour the appearance
of enhanced linesf. This is in agreement with previous work upon other spectra.

* LOCKYBR, ' Report,' Solar Physics Committee.

t REECE, in a less conservative list of enhanced lines includes 4368 • 45, here classified as strengthened
under pressure, ' Astrophysical Journal,' vol. XIX., p. 334, 1904 ; also 4231-23, here weakened.

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL. 211

MITCHELL* attributes the occurrence of enhanced lines in the chromospheric spectrum
partly to the reduction of pressure consequent upon the greater altitude and partly
to the presence of hydrogen there. This will be further discussed in a subsequent
paper.

7. The Displacement of the Lines: (l) The Measurement of the Plates. — The bulk
of the measurements were made by Mr. F. E. PEARSE, for whose assistance I am
indebted to the Government Grant Committee. The photographs were placed in the
fixed carrier of a modified Hilger photo-measuring machine in which the movable
part was the microscope which was controlled by a screw whose drumhead reading
could be estimated to the thousandth part of a millimetre. In order that lines of
various breadths could be measured the microscope could be converted into a telescope
and a considerable range of magnification achieved. There were two pairs of parallel
wires of different intervals in the eye-piece, either of which could be set perpendicular
to the direction of travel of the slide and parallel to the spectrum lines. This latter
operation had been found difficult in previous Avork, so the later photographs had been
taken with a shutter, in which short slits had been cut to allow the top and bottom
of the comparison lines to affect the photographic plate above and below the central
strip. In each 50 cm. there were three such extra pairs of slits, each about 2 cm. long,
e.g., lines 15, 16, 17, 18, Plate 1. It was always possible to find one at least of these
in the range of spectrum upon the measuring machine, and the parallelism of the
cross wires was consequently attained with ease and considerable accuracy. Two
readings were taken with the plate placed with the red on the right-hand side, the
setting being first on the upper and then on the lower half of the line under pressure ;
the plate was then reversed and the readings repeated. Four readings were thus
invariably taken. In many instances others were made. The readings were checked
by the writer, who made a point of measuring eacli line at some one pressure. As a rule
these readings were not included in the mean results, because there was some personal
equation in the measurements, and it seemed best to have a homogeneous set of
readings made by one individual because these are then more strictly comparable
with one another. It is interesting to note that even though different observers may
obtain different absolute values for the displacements upon a single photographic plate,
there is usually agreement between the relative values of their measurements of the
displacements of different lines. For instance, the groupings are usually the same and
also the ratios of the mean displacements of the groups.

The order of accuracy obtained is shown by the following, Table II., in which a
few readings taken at random are reproduced. They illustrate the agreement of
Mr. PEARSE'S readings amongst themselves and with those of the writer.

* MITCHELL, ' Astrophysical Journal, vol. XXXVIIL, p. 407, 1913.

212

DR. W. GEOFFREY DUFFIELD ON THE
TABLE II.

Plate 5. — Nickel, 20 Atmospheres.

Line.

\
Readings in thousandths of a millimetre.

Means.

Check readings.
(G.D.).

193

194, 204, 151,

151, 199,

176, 162

176

200, 205

195

129, 123,

154, 153,

128

137

160, 142

196

133, 147, 122, 135, 140, 171, 153, 150

143

160, 200

197

243, 242, 224, 223

233

205, 200

• 201

196, 214, 222, 219, 183, 193, 194, 205

203

195, 210

202

147, 137, 116, 156, 140, 162

143

130, 200

203

125, 131,

147, 134,

152

137

130, 150

Plate 2. — Nickel, 10 Atmospheres.

] ;

Mean readings
in thousandths of an

Mean readings
in thousandths of an

Mean readings
in thousandths of an

A.U.

A.U.

A.u.

Line.

Line.

Line.

PEAKSE.

DUFFIELD.

PEARSE.

DUFFIELD.

PEARSE.

DUFFIELD.

'),{

37

23

62

25

29

70 40

46

64

23

26

63

10

8

71 22

43

55

21

29

84

13

18

72 61

55

56

30

29

60

13

20

73 32

35

57

35

29

66

27

22

74 35

29

59

40

42

67

52

39

78 32

27

00

27

26

68

30

18

79 39

31

61

40

18

69

53

46

83 26

34

Plate D

— Nickel, 10 Atmospheres. The lines are those of short wave-length

and are in 3rd

Order.

Mean readings
in thousandths of an

Mean readings
in thousandths of an

Mean readings
in thousandths of an

A.TI.

A.

U.

A.U.

Line.

Line.

Line.

PEARSE.

DUFFIELD.

PEARSE.

DUFFIELD.

PKARSE.

DUFFIELD.

26

49

43

32

34

17

40 31

29

27

31

41

35

68

30

41 39

17

28

29

43

36

65

47

42 4^

48

29

64

17

38

37

27

43 37

44

SO

44

26

39

M

t»/*

48

44 26

v. small

31

47

— .

|

|

Lines whose displacements are given in italics are reversed at the corresponding pressure.

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL. 213

Except in a few instances the agreement is ^ood. When there were differences
Mr. PEABSE made additional readings ; from these and from the measurements of the
displacements at other pressures it was usually possible to decide upon the more
probable value.

Experience in the measurement of displacements under pressure clearly shows that
the personal equation of a computer is not a fixed quantity ; there is a tendency for a
novice to record values that are too high, and it is found by experience that an interval
may make a considerable difference in one's judgment of a set of displacements. For
instance, the writer measured some lines before and after a voyage of some months'
duration, and found marked differences in the readings, the second set being only
about 70 per cent, of the first. There was, however, excellent agreement between the
rate of displacement with wave-length for lines with the same type of intensity curves.
It is on this account that the writer does not wish to lay too great a stress upon the
absolute values of the displacements for any one metal. The accuracy of the relative
values for different metals depends also upon the shapes of the intensity curves of the
lines ; if these are similar they are more likely to be reliable.

In measuring displacements it is very important that the photographs shall be
illuminated by a source of constant brilliance, and for this purpose measurements
made in artificial light are more constant than those made in daylight of variable
intensity.

(2) Description of Table of Displacements (l t<> 110 Atmospheres}. — Table III. gives
in thousandths of an Angstrom Unit the value of the displacement of each line at the
pressure stated at the top of eacli column. The first column contains a list of the
arbitrary numbers assigned to the lines, the second the wave-lengths of the lines
according to HASSELBERO. The displacements measured for various pressures follow
in successive columns.

Reversed lines are indicated in the manner stated in Section 5, p. 209.

That the displacement increases with the pressure is at once evident.

The second half of the table contains the displacements per atmosphere in
thousandths of an Angstrom Unit, the readings being obtained by dividing
those in the first part of the table by the excess pressure above that of one
atmosphere.

A column is devoted to the Mean Displacement per atmosphere in thousandths of
an Angstrom Unit, to which reference will be made later ; and the final columns
which contain the quotient obtained by dividing the Mean Displacement per
atmosphere respectively by the first power, square and cube of the wave-length of the
line, will also be the subject of subsequent discussion.

VOL. ccxv. — A. 2 F

214

DK. W. GEOFFREY DUFFIELD ON THE

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218

DE. W. GEOFFREY DUFFIELU ON THE

(3) The Spectrum of the Nickel Arc under Pressures of + 155 and + 200
Atmospheres. — The range of spectrum which has been explored at higher pressures
of +155 and +200 atmospheres extends from X = 4050 to X = 4600. The
investigation, which involved a certain amount of risk, showed that there is no
discontinuity in the nature of the pressure-effect between 1 and +200 atmospheres.
As special interest attaches to the effect of such high pressures, and as measurements
of the lines are extremely difficult, a qualitative account of the behaviour of the
nickel lines at these pressures is given in the following table ; when readings have
been attempted they are included, bxit they cannot be regarded as more than
approximate : —

TABLE IV.

Line. Wave-length.

126
133
135
137

141

100

161

163

165
167

4064-55
4104-37
4116-14
4121-48

4142-4

146
147

4184-65
4195-76

148 \
149J
153
(Mn)154

4200-61
4201-88
4231-23
4235-3

(Fe) 158
159

4271-3

4284-83

4288-16

4296-06

4307 • 40

4331-0
4359 • 73

155 atmospheres.

Vanished.
Faint indication only.

Vanished.

Considerably broadened, fairly sym-
metrical. Remains fairly compact
throughout. Resembles 136, which
is due to iron. Displacement
0-170 A.U. to red. Displacement
per atmosphere = O'OOll A.U.
No sign of line, but background is

of increased intensity here.

Faint indication of very broad line.

Vanished.

Merged.

Vanished.

Broadened, but not unduly, remains
fairly compact.

)) )) !>

Greatly broadened and displacement.

Merged into 160.
Same as 200 atmospheres.

Immense broadening and displace-
ment. Unsymmetrically broad-
ened to red.

Considerably broadened.

Vanished.

Greatly broadened and displaced to
red. Broadening unsymmetrical.
Displacement between 0'57 to
0-83 A.U.

200 atmospheres.

Vanished.
Invisible.
Vanished.

Considerably broadened, as at
155 atmospheres. Displacement
0-250 A.U. Displacement per
atmosphere = 0'0013 A.U.

Invisible.

Invisible.

Vanished, though strong at 1 atmo-
sphere.
Merged into very faint hazy band.

Vanished.
Very faint.

Greatly broadened and displaced.

Merged into 160.

Immense broadening and displace-
ment. Unsymmetrical to red, but
curve too flat topped for accurate
measurement.

Very faint.

Broadening rather greater, but now

very faint.

Vanished.

Very broad and diffuse.

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL.
TABLE IV. (continued).

219

Line. Wave-length.

155 atmospheres.

172 4401-70

174 4437-17
176 4459-21

177 4462-59

178 4470-61

182 4520-20

185 4547-30
188 4592-69

189 4600-51

190 4605-15

191 4606-37

Immensely broadened. Unsym-
metrical to red. Displacement
approximately 0-6 A.U. Mean
displacement per atmosphere :
0-0039 A.U.

Doubtful whether the hazy band is
due to this line or a close neigh-
bour.

Immense unsymmetrical broadening.
Displacement approximately

0'69 A.U. Mean displacement
per atmosphere = approximately
4'5 thousandths of an A.U.

Immense unsymmetrical broadening.
Displacement approximately

1 • 1 A.U. Mean per atmosphere
= 0-007 A.U.

Immense unsymmetrical broadening.
Displacement approximately

0 • 97 A.U. Mean per atmosphere
- 0-0063 A.U.

Considerably broadened. Fairly
compact.

Immense unsymmetrical broadening.

Immense unsymmetrical broadening.
Displacement 0'83 A.U. Mean
per atmosphere = 0-0054 A.U.

Immense unsymmetrical broadening.

Immense unsymmetrical broadening.
Displacement 0-91 A.U. Mean
per atmosphere = 0 • 006 A.U.

Merged into 190 or else vanished.

200 atmospheres.

Immensely broadened. Unsym-
metrical to red. Displacement
between 0-71 and 0'91 A.U.
Mean displacement per atmosphere
between 3 • 5 and 4 • 5 thousandths
of an A.U.

Faint hazy patch of luminosity.
Flat-topped intensity curve.

Immensely broadened. Displace-
ment between 0'9 and 1'G A.U.
Mean displacement per atmosphere
= approximately 6 thousandths
of an A.U.

Immense unsymmetrical broadening.

Immense unsymmetrical broadening.
Displacement approximately

1-7 A.U. Mean per atmosphere
= 0-0085 A.U.
Not quite merged in background.

Faint hazy patch of luminosity.

Immense unsymmetrical broadening.
Displacement (1"9) A.U. Mean
per atmosphere = (0'0095) A.U.
Faint patch of luminosity.

Immense unsymmetrical broadening.

Displacement 0-97 to 1-9 A.U.
Mean per atmosphere = 0-005

to 0-009 A.U.

Merged or vanished.

(4) Displacement towards the Violet : — The writer has on previous occasions
chronicled the displacement of a few lines towards the more refrangible part of the
spectrum and so have other observers. In the nickel spectrum similar displacements
have been recorded, and it will be seen from the following table that there is good
agreement between the readings made by my assistant and myself upon these lines.
In accordance with precedent Mr. PE ARSE'S determinations are those which are included
in Diagram 5, where the negative displacements are distinguished by a horizontal line
passing through the dot. It would be of great interest to observe if the displacement
towards the violet increases or decreases with increase of pressure, but unfortunately
reliable measurements of the displacements of these lines were not feasible above
a pressure of 10 atmospheres. The reality of displacements towards the violet

220

DK. W. GEOFFEEY DUFFIELD ON THE

has been .questioned, but in the writer's opinion they are real and not due merely
to unsymmetrical broadening ; it is true that the negatively displaced nickel lines
are more broadened towards the violet than the red, but the broadening takes place
about a negatively displaced position. Several theories can explain qualitatively
how displacements and unsymmetrical broadening towards the red may be accounted
for ; the displacement of a line towards the violet should not be more difficult to
explain than an unsymmetrical broadening in that direction ; the latter phenomenon
is unquestionably true.

The following are the displacements of lines towards the violet measured by
Mr. PEARSE and by myself. The photographs were taken when the pressure of the
air was 10 atmospheres:—

TABLE V.

Line.

i

A.

PEARSE.

G.D.

229

4937-51

-50

232 71-54

- 65

-30

233

80 • 3C

-48

-30

236

5000-48

-47

-30

239 35-55

-56

-40

244

SI -30

-25

0

Lint; 243 A = 508070 is apparently reversed and displaced towards the violet, but
this is found to be due to the strengthening under pressure of a faint line on its violet
edge.

(5) Displacement Diagrams. — In Diagram 1 lines are drawn connecting the
different readings of the displacements at different pressures of a few of the spectrum
lines dealt with, and each line represents the behaviour of one spectrum line. The
proportionality between the displacement and the pressure is apparent from the
diagram, and is approximately linear. In Section 7, p. 224, this point is further
examined. The diagram further illustrates the fact that the lines are capable of
resolution into two groups according to their rates of displacement, a feature which is
more fully treated later.

The diagram includes both reversed and bright lines, but does not distinguish
between them. Without exception the former fall into the group with the
smaller displacement. Lines displaced towards the violet are not included in this
diagram.

(6) Comparison with Previous Observations. — The displacements of certain nickel
lines have been observed by HUMPHREYS and MOHLER* at pressures of 9f, 12^, and

* HUMPHREYS and MOHLER, ' Astrophysical Journal,' vol. III., p. 114, 1896.

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL.

221

cS

s

VOL. CCXV. A.

Displacements in Angstrom Units.
2 G

222

DR. W. GEOFFEEY DUFFIELD ON THE

atmospheres, and by HUMPHREYS* at pressures of 42, 69, and 101 atmospheres. The
results of these two researches are given in full in the following tables.

The displacements per atmosphere have been calculated from their measurements
and are set out in subsequent columns. The mean displacements per atmosphere have
been inserted, and may be compared with the similar values obtained in the present
research.

The agreement is reasonable considering that in the previous investigations the
means for a large proportion of the lines are not based upon a very large number of
observations. The lines 172, 176, 178, &c., are, however, assigned larger displacements
by the writer than by HUMPHREYS.

TABLE VI.

Displacements in

Displacements per atmosphere in

thousandths of an Angstrom Unit.

(HUMPHREYS.)

thousandths of an Angstrom Unit.
(HUMPHREYS.)

Mean
displacement

per

A.

Atmospheres.

Atmospheres.

atmosphere.

(HUM-

PHREYS.)

9f.

1%

l*i

42.

69.

101.

9|.

12$.

14J.

42.

69.

101.

1

3002 • GO

107

1-1

1-1

03-73

103

1-0

1-0

12-10

105

1-0

1-0

38-05

97

1-0

1-0

50-88

32

77

101

0-8

1-1 1-0

1-0

54-40

41

102

1-0

1-0

1-0

57-72

50

90

127

1-2

1-3

1-3

1-3

3161-61

48

1-2

•

1-2

02-00

59

1-4

•

1-4

34-26

60

122

1-4

1-2

1-3

3233-11

49

115

1-2

1-2

1-2

3369-66

77

•

1-9

•

1-9

72-12

48

1-2

1-2

74-35

29

•

0-7

0-7

80-70

96

2-3

2-3

91-21

14

70

1 1

1-7

1-4

93-10

63

1-5

1-6

3413-64

19

1 5

1-5

14-96

19

77

1 5

1-9

1-7

23-80

84

2-0

2-0

33-71

94

•

•

2-3

2-3

37-45

20

34

63

2-0

2-3

1-5

1-9

46-34

•

71

.

.

.

1-7

1-7

1

* HUMPHREYS, ' Astrophysical Journal,' vol. XXVI., p. 36, 1907.

EFFECT OF PRESSURE UPON ARC SPECTRA. -NICKEL.

223

TABLE VI. (continued).

Displacements
in thousandths of an
Angstrom Unit.
(HUMPHREYS.)

Displacements per
atmosphere
in thousandths of an
Angstrom Unit.
(HUMPHREYS.)

Mean
displacement

Line.

X.

per
i

atmosphere.

Atmospheres.

Atmospheres.

(HUM-
PHREYS.)

9f.

12|.

14*.

42.

9f.

12J.

U\.

42.

1

3453-04

62

1-5

1-5

3

58-59

27

29

91

2 "2

2-0

2'2

2-1

4

61-78

16

23

67

1-6

1-8

1-6

1-7

•

67-63

63

1-5

1-5

•

69-64

95

2-3

2-3

5

72-68

80

2-0

2-0

7

93-10

81-

2-0

2-0

8

3501-00

24

35

50

1-9

2-4

1-2

1-8

10

10-47

83

2-0

11

15-17

34

41

2-7

2-8

2-0

2-7

13

19-90

75

1-8

1-8

14

24-65

30

96

2-4

2-3

2-3

15

48-34

80

2-0

2-0

18

61-91

63

1-5

1-5

19

66-50

•

91

2-2

2 - 2

21

71-99

100

2-4

2-4

25

88-08

92

2-2

2'2

26

97-84

102

2-4

2-4

27

3602-41

•

82

2-0

2-0

28

09-44

72

1-8

1-8

29

10-60

•

101

2-4

2-4

30

12-86

80

2-0

2-0

31

19-52

•

65

1-6

1-6

32

24-87

60

1-5

1-5

38

62-10

53

1-3

1-3

Mean
displacement

per

atmosphere.

(Present

Research).

(1-5)

1-4

1-2
0-7
2-1
1-9
1-1
2-1
1-8
1-0
2-2
1-4
(2-6)
0-8
1-1

224

DR. W. GEOFFREY DUFFIELD ON THE

TABLE VI. (continued).

Line.

Wave-length.

Displacements
in thousandths of an
Angstrom Unit.
(HUMPHREYS.)

Displacements per
atmosphere
in thousandths of an
Angstsom Unit.
(HUMPHREYS.)

Mean

displacement
per
atmosphere.

Mean
displacement
per
atmosphere.

Atmospheres.

Atmospheres.

(HUM-
PHREYS.)

(Present
Research.)

42.

69.

101.

42.

69.

101.

39

3664-24

110

.

1-6

. .

1-6

1-7

41

70-57

88

1-3

1-3

2-1

42

74-28

70

1-0

1-0

1-7

43

88-58

68

1-0

1-0

1-8

46

3722-63

111

•

1-6

1-6

2-2

50

36-94

83

.

1-2

1-2

1-6

63 75-71

88

•

1-3

1-3

1-5

65 83-67

58

0-9

0-9

1-3

68

3807 • 30

76

1-1

1-1

1-5

81

58-40

117

1-7

1-7

2-1

101

. 3972-31

75

1-1

1-1

0-8

102

73-70

140

176

2-1

1-8

2-0

1-9

165

4330-85 1
31-78

150 204

2-1

2-2

2-0

2-1

3-0

172

4401-70

480

•

4-8

4-8

12-0

176

59-21

625

6-2

6-2

10-9

178

70-61

580

8-5

8-5

11-0

182

4520-20

120

1-8

1-8

1-7

188

92-69 320

620

7-8

9-1

8-5

10-3

189

4600-51 464

11-3

11-3

9-3

190

05-15

280

600

6-8

8-8

7-8

9-5

193

48-82

270

660

6-6

9-7

8-1

11-4

197

86-39

325

557

7-9

8-2

8-0

11-7

201

4714-59

274

6-7

6-7

11-1

207

56-70

297

•

7-2

7-2

11-3

—

5155-94 24 at 9f atmospheres 2 -5 at

9| atmospheres

2-5

_

(7) Relation betiveen the Pressure and the Displacement. — That the relation between
the pressure and displacement is approximately a linear one is evident from
Diagram 1, in which these two quantities are plotted. But the displacements per
atmosphere are almost invariably greater at low pressures than at high ones
(see Table III.) which seriously challenges the existence of an exact linear relationship.
This is clearly brought out by Diagram 2, in which the mean displacement per
atmosphere form the ordinates and the pressure the abscissae of the curves, each oi
which represents the behaviour of one particular spectrum line whose identity can be
traced from the number assigned to it. There is a general downward trend as the
pressure increases, which is in favour of the rate of displacement decreasing with
increase of pressure. This tendency is apparent in each of the two groups into which

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL.

225

Displacements per atmospheres in A.U./1000.

226

DR. W. GEOFFEEY DUFFIELD ON THE

the lines are obviously divisible, but it is more pronounced in the case of the group
with the greater displacement.

If the decrease of the displacement per atmosphere with the pressure were linear
throughout the whole range, it would lead to an equation between the displacement d0
and the pressure p of the parabolic form d0 = A.p — Bp2 in which the constant B is small.
But though a linear relationship may reasonably represent the graph of djp and p
over the small range of pressure from 20 to 80 atmospheres, there is an indication that
the descent of the graph is more rapid at first and that with increasing pressure it
becomes more gradual, suggesting a curve of an exponential form. This is emphasized
by Table VII., in which are given the ratios of the displacements per atmosphere at

Diagram 3.

Ratio

WAVELENGTH

Ratio of displacements per atmosphere at 10 atmospheres to displacements per atmosphere at

higher pressures.

10 atmospheres to the displacements at higher pressures, the data being taken from
Table III. The former are generally greater than the latter, and the mean value of
the ratio is 1'8. In Diagram 3 these ratios are plotted against wave-length; if the
relationship between the pressure and displacement were precisely linear, the dots
would group themselves about the line marked I'O, but it is very obvious that the
readings at 10 atmospheres are too large for this relationship to hold. The diagram
also shows the curious fact that the departure from a linear relation is much more
pronounced for lines of small wave-length. For large wave-lengths the ratio is nearly
equal to unity. This is partly, but not entirely, due to the fact that the lines of great
wave-length have not been examined over the full range of pressures. There is a

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL.

227

TABLE VII.—

-Katio of Displacement per Atmosphere at 10 Atmospheres Pressure to
Displacement per Atmosphere at Higher Pressures.

Line.

A.

Ratio.

Line.

X.

Ratio.

26
27

3597-84
3602-41

2-33

. 1-72

165

/30-85\
1 31 • 78 /

2-10

28

09-44

2-90

167

59-73

0-92

29

10-60

2-90

172

4401-70

0-96

30

12-86

3-14

174

37-17

0-98

31

19-52

1-80

176

59-21

0-93

32

24-87

4-25

177

62-59

1-29

36

35-10

3-13

178

70-61

1-12

38

62-10

3-36

182

4520-20

1-82

39

64-24

2-82

188

4592-69

1-36

40

69-38

2-38

189

4600-51

1-48

41

70-57

1-85

190

05-15

1-26

42

74-28

2-47

193

48-82

1-22

43

88-58

2-05

195

67-16

1-49

44

94-10

2 -GO

196

67-96

1-52

55

3749-15

1-90

197

86-39

1-57

61

69-58

2-85

199

4701-72

1-57

62

72-70

2 "27

201

14-59

1-54

63

75-71

0-66

202

15-93

1-57

64

78-22

1-44

203

32-00

1-47

65

83-67

1-00

206

54 • 95

1-64

66

92-48

2-70

207

56-70

1-38

67

93-75

2-73

208

62-78

1-31

68

3807-30

2-00

209

64-07

1-35

73

31-82

1-77

211

8G-44

1-58

74

32-44

3 • 50

212

4807-17

1-41

81

58-40

1-85

213

29-18

1-51

87

89-80

2-36

214

31-30

1-34

91

3913-12

2-21

215

32-86

1-31

101

72-31

1-50

217

55 • 57

1-04

102

73-70

2-31

219

66 • 42

1-23

106

95-45

2-15

221

73-60

1-31

112

4019-20

4-33

226

4918-53

1-06

137

4121-48

2-29

228

36-02

0-98

148

4200-61

1-06

238

5017-75

1-24

149

01-88

2 '24

247

5477-13

1-01

159

84-83

1-05

160

88-16

1-48

161

96-06

1-05

Mean value

for Ratio

1-83

163

4307-40

1-50

doubt as to whether this is a subjective or an objective phenomenon. To the general
difficulties of measuring the displacement of spectrum lines reference has already been
made ; in particular it is not easy to compare the displacements of lines of different
width and whose intensity curves are of different shapes, but inasmuch as both
Mr. PEARSE and the writer agree in assigning values to the displacements at high
pressure which are smaller than they should be if a linear relationship exists, there
is good reason for regarding the pressure-displacement relation as not quite linear.
This is in accord with the results of the investigation of the gold spectrum under

228 DR. W. GEOFFREY DUFFIELD ON THE

pressures from 1 to 200 atmospheres, where it was shown that the pressure-displacement
curves were slightly concave to the axis of pressures and where the curves representing
the mean displacements per atmosphere had a general downward tendency as the
pressure increased — -as in this research. Dealing with the spectrum produced by the
copper arc when subjected to the highest pressure, the same features appeared, so that
the evidence favours the general conclusion : " That though the relationship between the
pressure and the displacement is approximately linear, the displacement does not
increase quite as rapidly as the pressure."

(8) The Relation between Displacement and Wave-length : — In Diagrams 4 and 5
each black circle represents the mean displacement per atmosphere of the nickel
line whose wave-length is given by the horizontal scale, the data being derived from
Table 111. Inasmuch as the displacements at 10 atmospheres pressure are dispro-
portionately large they are treated separately in Diagram 5, whereas they are
excluded from the calculations of the mean displacements which are plotted in
Diagram 4. The prominent feature of these diagrams is the increase of the displace-
ments as the wave-lengths increase, but the division of the lines into two groups
is also indicated.

Treating the diagrams critically it is scarcely open to doubt that the displacement
is dependent upon the wave-length, though Diagram 4 alone is perhaps not
conclusive in this respect as there are not many lines in the region X 3900 to X 4200.
Many more lines have been measured at 10 atmospheres pressure than at higher
pressures, so Diagram 5 is able to provide more information about this region of the
spectrum, though on the whole the values are not so reliable since they are derived
from readings at only one pressure.

Granted then that the occurrence of larger displacements in the less refrangible
parts of the spectrum is not fortuitous, it remains to discuss the actual relationship
between these two variables. The diagram at 10 atmospheres points to a steep
descent which might be regarded as approximately linear if it were not that it
would involve the displacements becoming zero and subsequently negative in the
more refrangible regions of the spectrum.

Though negative values for displacements due to pressure have been recorded, the
crossing of the axis has not been observed in any spectrum, and the asymptotic trend
of the black dots in Diagram 4 is contrary to this occurring in the case of nickel.
This conclusion is supported by HUMPHREYS' measurements of lines of smaller wave-
length down to X = 3000 which are included as open circles in the diagram. Though
the majority of these readings are based upon observations at only one high pressure
(either 42 or 101 atmospheres) they are in such good agreement with the results of
the present research as to leave no doubt that the axis is not crossed in this region of
the spectrum. If we make the assumption that the origin is on the curve relating
to displacement and wave-length, we can at least say that it is not contradicted by the
result's of this research. On the assumption that the graph is of the form d = a\" the

EFFECT OF PEESSURE UPON ARC SPECTRA.— NICKEL.

229

s

Displacements per atmosphere in
A.U./1000.

M
cu

SO

4

^-

i*

c
o>

-a

"8

s _

o

\

!

I

10

1

bb

s

I.

Displacements per atmosphere in A.U./1000.

£
v

—

a

o

cj
"E,

03

•3

•s

w

o

"cs

c
o

S -C
<a

VOL. CCXV. — A.

2 H

230 DE. W. GEOFFREY DUFFIELD ON THE

values of d/X, dfx" x d/\3 have been calculated (excluding measurements at 10
atmospheres) ; they will be found in the final columns of Table III.

In the discussion upon the copper arc, a test of the rate of displacement with wave-
length was made by finding the value of n which enabled the quantities djx* to be
most distinctly separated into groups. This did not appear so likely to yield
promising results in the case of nickel, because in Diagram 1 two groups are
remarkably distinct without reference to the wave-length, a point which is further
illustrated in Diagram 6, fig. 1 , in which each number represents the presence in
the spectrum of a line with a mean displacement corresponding to its position upon
the horizontal scale. The distribution resolves itself into a well-defined group with
a mean displacement per atmosphere of about 1'75 thousandths of an Angstrom
Unit and another very diffuse and ill-defined group with a flat-topped maximum
extending from about 9 to 12 thousandths of an Angstrom Unit. It might be
argued that the probable errors of the measurements of the different types of line
are in accord with this grouping, because those lines whose displacements are small
have a smaller probable error than those with larger values, the latter being in general
(but not invariably) associated with a greater width of the lines. But though this is
a consideration which must be given due weight, to regard it as satisfactorily
accounting for the distribution in fig. 1 would be to disregard the significance of the
tendency for the displacement to increase with wave-length which has been
demonstrated in Diagrams 4 and 5.

It is further important to note that lines as 208 (\ = 4762'87), 213 (\ = 4829'! 8),
217 (X = 4856-57), 228 (\ = 4936'02) and 238 (X = 501775) do not clearly fall within
either group, also that the line 24? (X = 5477 '13) is assigned to the group with large
displacement in spite of the fact that it is reversed unsymmetrically, and would not
be expected, in accordance with previous experience of similarly reversed lines in the
iron spectrum, to belong to a group with the highest rate of displacement.

Plotting d/\ against wave-length, fig. 2 is obtained, and again the lines 208, 213,
217, 228 and 247 occupy anomalous positions. There is still less reason for regarding
this distribution as satisfactory. Fig. 3 shows the distributions of the values of d/\2
when the lines 247, 213, and 217 form a group by themselves, though 208 has
attached itself to the outskirts of the first group.

In fig. 4, in which the values of d/X3 are given, these lines have been absorbed by
Group 1 , and the resolution into two groups is clearer. We note also that the second
group is more compact ; this is obviously the most satisfactory diagram, and we may
therefore conclude that as far as values above 10 atmospheres are concerned, the rate
of displacement is not far removed from that of the cube of the wave-length. It may
appear remarkable that any order can be arrived at considering the apparently chaotic
distribution of dots in Diagram 4 in the region of wave-lengths X 4200 to X 4500, and
it is necessary to emphasize the fact that the method consists in the elimination of the
dots with displacements varying from 5 to 8 thousandths of an Angstrom Unit from

EFFECT OF PEESSUEE UPON AEC SPECTEA.— NICKEL.

231

the cloud of dots above them, and that we draw the above conclusion from
the fact that the group of dots representing high values of the displacement in

Diagram 6. Frequency distribution of spectrum Hues.

Fig. i.

5*1
ml

74 68 106

65 I03I6S

'! 75 81

63 «l 6f 77

OS S/ 50 4

<8 «0 *5 1

Fiq.2.

44 30 41 11 I65
fr 58 15 39 19 87!

51 15 36 17 461T1
18 3.615 37 16 39 1C I

I'
HI

Ja

74

fet 9i ,l/ 6

64 65 ioa 7?

1 68

55 51 t5 45 46

46 40 50 41 At

I 4* 56 36 i9 31 i<

ISs 25 JOS? 7 2 _

fii is as is is e 16 is 5

r-N

181 SI

of
73

Fig.4.

67

H

66 61 66 50

87

119

64 55 6342I6546J _J21*^

485! b! 398(41247] f^V *°' W|

44 3840 36 7? 2 J 29 217] _ JM 101 l68l5Ul%e]

32 iB M 15 45 17 262I5J J206«qijaf78 I?/ 143 nij {3ia'72[ y

[ill l»| 1316 15 817 16 19 51 10

|2l6'iOil9i!96'JII 7(1147 146 ''-S 7* '99 1 7* (^ 'V

Fig. 1. According to values of the displacement per atmosphere.

displacement per atmosphere

wave-length

displacement per atmosphere
" (wave-length)-'

displacement per atmosphere
" " " (wave-length)3

(Eeadings at 10 atmospheres excluded.)
[Numbers overlined are of doubtful accuracy. The horizontal scale is proportionate, not absolute.]

Diagram 6, figs. 1 to 4, is more compact when the displacements are assumed to depend
upon a high value of the wave-length. The group might be more compact still if sen

2 H 2

232

DR. W. GEOFFEEY DUFFIELD ON THE

adequate correction could be applied to the measurement of the lines in the neighbour-
hood of X = 4300 to 4400, which appear abnormally high in the diagram, and there is
little doubt that the readings are in fact rather too large. It has been stated that
for the sake of uniformity Mr. PEARSE'S readings have been used almost exclusively
in preparing the tables, my own readings of the lines serving as a check upon them.
The check readings in this region indicate that the measurements of the lines in
region 4800 to 4900 are to those in region 4400 as 5 is to 4. This would improve
Diagram 4, and also make the second group in Diagram 6, fig. 4, more compact by
reducing the displacements of lines 160, 167, 172, 174 and 176.

The above argument is based upon the resolution of the nickel spectrum into only two
groups, but reasons will be given in the next section for distinguishing between three
groups. If this is the case we must consider whether the isolated lines between the
two main groups could form a group by themselves. If the displacement varies with
the square of the wave-length, Diagram 6, fig. 3, indicates that the following lines
would constitute it :— 213, \ = 482918 ; 217, \ = 4855'57 ; and 247, \ = 5477'13.
The last-named is not of the same nature as the first two, and there is a considerable
separation between 213, 217, and the similar lines 87 and 165 which should be
associated with them according to their behaviour and general appearance. Five
more lines, 10, ID, 21, 2<i, 31, which resemble them by being nebulous at atmospheric
pressure, may also be associated with them. It will be seen from what follows that
they form a more compact group when classified according to the cube of the wave-
length. Though the determination is by no means conclusive, the balance of evidence
favours this rate of variation of the displacement.

It is important to see to what extent the readings at 10 atmospheres confirm the
above conclusion.

TABLE VIII. — Displacements at 10 Atmospheres Pressure over Wave-length

Squared and Cubed.

Line.

A.

Displacement at
10 atmospheres.

d (at 10)

d (at 10)

A.2

A.S

26

3597-84

4-9

378

1052

27

3602-41

3-1

240

663

28

09-44

2-9

223

618

29

10-60

6-4

492

1360

30

12-86

4-4

337

933

31

19-52

4-7

358

990

32

24-87

3-4

258

713

36

35-10

4-7

354

976

38

62-10

3-7

276

752

39

64-24

4-8

356

972

40

69-38

3-1

232

628

41

70-57

3-9

290

788

42

74-28

4-2

312

846

43

. 88-58

3-7

271

735

44

94-10

2-6

190

514

EFFECT OF PEESSURE UPON ARC SPECTRA.— NICKEL.
TABLE VIII. (continued).

233

Line.

A.

Displacement at
10 atmospheres.

d (at 10)

d (at 10

A*

A8

52

44-68

10-5

749

1999

55

49-15

2-1

148

398

61

69-58

4-0

282

746

62

72-70

2-5

175

465

63

75-71

1-0

70

186

64

78-22

1-3

91

242

65

83-67

1-3

91

240

66

92-48

2-7

187

495

67

93-75

5-2

361

950

68

3807 • 30

3-0

206

545

73

31-82

3-2

217

568

74

32-44

3-5

238

621

81

58-40

3-9

261

678

82

63-21

5-0

335

867

87

89-80

5-9

390

1000

91

13-12

3-1

203

518

101

72-31

1-2

76

192

102

73-70

4-4

278

702

106

95-45

4-3

269

672

112

4019-20

2-6

161

390

' 126

64-55

(11-9)

(720)

(1772)

133

4104-37

3-0

178

434

135

16-14

(4-1)

242

588

137

21-48

3-9

230

557

140

38-67

7-4

452

1044

141

42J ;^|

10-5

612

1477

143

50-55

5-3

307

741

144

64-82

1-7

98

235

145

67-16

3-1

178

428

146

84-65

8-5

485

1160

147

95-71

6-7

380

910

148

4200-61

8-7

494

1172

149

01-88

(18-6)

1052

2500

151

21-87

9-3

522

1236

153

31-23

18-9

1056

2495

155

36-55

6-9

384

907

159

84-83

10-0

546

1270

160

88-16

16-2

925

2160

161

96-06

15-8

855

1990

162

98-68

15-7

849

1976

163

4307-40

3-9

211

487

164

25 {:£}

8-8

470

1088

165

/ 30-85 1
L31 "78 J

6-3

335

776

166
167
168

56-07
59-73
68-45

11-6
11-5
9-6

611
605
503

1403
1390
1152

170
171
172
174
176
177
178
181

84-68
90-00
4401-70
37-17
59-21
62-59
70-61
4513-20

10-3
9-8
11-6
11-5
10-2
12-3
12-4
10-6

536
508
600
580
514
618
620
520

1222
1158
1360
1305
1148
1380
1390
1153

234

DR. W. GEOFFREY DUFFIELD ON THE
TABLE VIII. (continued).

Line.

A.

Displacement at
10 atmospheres.

d (at 10)

d (at 10)

A*

182

20-20

3-1

152

335

186

51-45

14-6

704

1548

187

60-10

12-8

615

1349

188

92-69

14-1

671

1460

189

4600-51

13-8

650

1410

190

05-15

12-0

569

1222

192

47-47

17-1

792

1703

193

48-82

14-0

648

1390

194

55-85

12-6

581

1248

195

67-16

13-3

610

1306

196

67-96

14-0

642

1378

197

86-39

18-4

835

1788

199

4701-72

21-2

960

2040

200

03-96

5-8

262

558

201

4714-59

17-1

770

1630

202

15-93

17-0

766

1620

203

32-00

13-1

585

1236

204

32-66

20-8

928

1962

205

52-58

(6-3)

(279)

(587)

206

54-95

15-8

700

1470

207-

56-70

15-7

693

1460

208

62-78

6-2

273

573

209

64-07

14-0

617

1290

211

86i-46J

16-8

733

1530

212

4807-17

13-1

566

1180

213

29-18

9-4

404

835

214

31-30

16-3

700

1442

215

32-86

11-6

497

1024

216

38-80

24-7

1055

2180

217

55-57

6/5

275

568

218

57-57

16-0

678

1395

219

66-42

15-1

638

1310

220

70-97

(7-6)

(320)

(657)

221

73-60

1C-8

708

1455

222

87-16

13-2

552

1130

223

4904-56

4-5

187

381

224

12-22

14-9

617

1257

225

14-15

8-1

335

682

226

18-53

14-5

600

1220

227

25-74

15-8

651

1322

228

36-02

15-4

632

1280

229

37-51

231

53-34

17-6

717

1448

232

71-54

(6-5)

(263)

529

233

80-36

4-8

193

388

234

84-30

1-4

56

113

235

98-42

15-7

628

1257

236

5000-48

(4-7)

(188)

(376)

237

12-62

18-4

732

1461

238

17-75

18-5

735

1465

239

35-55

5-6

221

438

243

80-70

(6-5)

(252)

(495)

244

81-30

(2-5)

(97)

(190)

245

5424-85

7-9

266

490

246

36-10

8-9

300

553

247

77-13

7-7

257

479

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL.

235

In Table VIII. the values of the mean displacements at 10 atmospheres have been
divided by the square and cube of the wave-lengths, and in Diagram 7, figs. 1 and 2,
these are shown in statistical form. Many more values are now included, and each
diagram is now very irregular. They are not as trustworthy as the previous
diagrams for the purpose of ascertaining the rate of displacement with wave-length,
but they serve as a guide to the classification of lines which cannot be measured at
pressures higher than 10 atmospheres.

The separation into groups is clearer in the df\2 diagram, which consequently
favours this ratio. If, however, the lines be sorted out from .the dj\3 diagram in

Diagram 7.

Fiq.L

Jli? 247 lot
> 74 245 10!

91 40 It

55 ta<4 66 2? J2 5841 42 3c il 26 67 M]

21518
I48>7&

71 '70 2!2 190 194 164 !'9 m JI3 507 J|4 !
||4C 164 29I68'" IB74I7J 141

_H

IF

Fiq.2

17 M 18 <a 81 30 3^ 87 Til

2002202J5 II

I6S9I 13774101)61 >K[_
\11 14S 66 66 135 +0 81 4S AI J'3|
|l81 55 ;356J4* 75 19 2] 32 :

U7| Jllil tr IS'1 '901'4 IB7I772I4206
67 155 140 !04 176 168 170 WITH 16? 189 l«8
30 31 87 le 171 I4e 151 159 171

~w

15

25

, displacements per atmosphere
Fig. 1. Frequency distribution of spectrum lines according to valu (wave-length)-

2.

(Readings are at 10 atmospheres only.)

:lisplaccmcnts per atmosphere
(wave-length)3

accordance with their grouping at pressures above 10 atmospheres, it becomes much
clearer, as in fig. 2, and we see that the confusion between values 8 and 10 011 the
horizontal scale is due to the overlapping of two frequency curves with rather large
variations. The distributions for d/\\ d^ now appear about equally probable for the
readings made above 10 atmospheres.

The rate of increase of the displacement with the wave-length has been a matter

236 DR. W. GEOFFREY DUFFIELD ON THE

of previous investigation by the writer ; the copper arc spectrum* yielded the result
that the displacement was proportional to a power of the wave-length " at least as
great as the third power and possibly as high as the sixth." The experiments upon
the gold arc under pressuret favoured a dependence upon third power of the wave-
length.

Subsequent important experiments by GALE and ADAMS} upon an extended region
of the iron spectrum under a pressure of 9 atmospheres support this conclusion, and so
do the results of the present research.

The bearing of this upon the spectra of novae has been discussed by the writer in
a paper entitled " The Spectra of Novae and the Pressure Effect. "§ It is there shown
how the Doppler and pressure effects may be distinguished.

An interesting feature of the nickel lines is the gregarious tendency of lines with
large displacements, almost all of which occur within a region of the spectrum
between X = 4200 and X = 5000. Reference to Table X. which gives lists of lines of
similar appearance — reversed lines, nebulous lines, &c., indicates that they occupy only
limited regions of the spectrum. A similar tendency for the iron lines of the same
group to congregate was recorded.

8. Resolution of the Nickel Spectrum into Groups. — -The lines may be grouped in
different ways — according to their displacement, broadening, intensity, reversal, &c.,
but of these the most important is the first-named. It has already been seen in the
previous section that it is possible to resolve the nickel arc spectrum into two main
groups according to their displacement, and that the first of these may be capable of
further sub-division.

The following table has been compiled from the available data to indicate the
general nature of the lines belonging to the two groups.

The line and its wave-length are given in the first two columns, in the third column
" n " denotes that the line was nebulous at atmospheric pressure, in the fourth column
N indicates that it became nebulous or diffuse under increasing pressure ; " sh " in
the next column indicates that the line was classified by HASSELBERG as " sharp " at
atmospheric pressure. The changes in relative intensity are given by " wk " or " str "
in the next two columns. The broadening and reversal are treated in the two following
columns, and the last two indicate the group into which the line falls in the d/\3
diagrams, Diagram 6, fig. 4, and Diagram 7, fig. 2. The first of these is the more
reliable as it includes a larger number of readings, but the second is useful in
supplementing the information supplied by the preceding column ; it is derived from
observations made at 10 atmospheres only.

* DUFFIELD, 'Phil. Trans. Roy. Soc.,' vol. 209, p. 205, 1908.
t DUFFIELD, 'Phil. Trans. Roy. Soc.,' vol. 211, p. 33, 1910.

J GALE and ADAMS, ' Astrophysical Journal,' vol. XXXV., p. 10, 1912. Through a misapprehension,
these authors quote experiments upon vanadium by Rossi as supporting this conclusion.
§ DUFFIELD, 'Monthly Notices, R.A.S.,' 73, p. 631, 1913.

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL.

237

TABLE IX.

Nebulous.

Sharp at
1 atmo-

Changes
in relative
intensity.

Prominent features
as regards

Grouping
according to
displacement.

Line.

A.
<

1 atmo-
sphere,
HASSEL-
BERG).

higher i
pres-
sures.

sphere
HASSEL-
BERG).

Weak-
ened.

Strength-
ened.

Broad-
ening.

Reversals.

Above
10 atmo-
spheres.

At
.0 atmo-
spheres.

'

G

roup .

[.

3

3458-59

n

m b8 r

4

61-78

n

m b8 r

5

72-68

n

c bs r

7

93-10

n

g b. r g

8

3501-00

c bs r .

1

9

02-76

c bs

r

10

10-47

n

g 1'.

r g

1

11

15-17

n

vg b.

'• vg

12

18-80

s bs

13

19-90

c b.

r s

1

14

24-65

n

G b. r vg

15

48-34

s b. : rB m

1

16

51-66

s bs .

1

17

53-63

s br . .

1

18

61-91

sh

s 1>8

rs s

1

19

66-50

n

vg bs r. vg

1

21

71-99

n

c bs ; rB c

1

25

88-08

sh

s bB rs s 1

26

97-84

n

c br rv c

1

27

3602-41

m br- rv s

1

1

28

09-44

m ba rs s

1

1

29

10-60

g bs r, c

1

30

12-86

c br rv m

1

31

19-52

n

G br rv vg

1

32

24-87

sh

s br rv s

1 : l

34

30-04

N

m bv . .

36

35-10

sh

s br .

1

38

62-10

sh

s br

1

1

39

64-24

sh

m br rv s

1

39a

68-35

N

m bv

40

' 60-38

sh

s br rv s

1

1

41

70-57

sh

m bp rv s

1

1

42

74-28

sh

m b.

rv m

1

43

88-58

sh

m br

rv s

1

44

94-10

s bs

l

1

46

3722-63

m bB

rs s

1

47

24-95

N

m bv

1

48

30-88

s

1

50

36-94

sh

m bB

rB m

1

51

39-36

s

r s

1

55

49-15

sh

s br

1

m

1

61

69-58

str

s

.

62

72-70

sh

s

1

63

75-71

g b. r g

1

64

78-22

s

The following abbreviations are used to express the width of bright lines and of reversals m columns 8
and 9 :-s = slight, m = moderate, c = considerable, g = great, vg = very great, G = very very great.

Other abBreviations are:-r = reversed, r. = reversal nearly symmetrical, r, = reversal on violet
of centre of bright line, bB = broadening nearly" symmetrical, br = broadening greater toward
bv = broadening greater towards violet, V = line has vanished under pressure.
VOL. CCXV. — -A. 2 J

238

DK. W. GEOFFREY DUFFIELD ON THE

TABLE IX. (continued).

Nebulous.

Sharp at
1 atmo-

Changes
in relative
intensity.

Prominent features
as regards

Grouping
according to
displacement.

Line.

A.

1 atmo-
sphere

(HAMSEI,-
BERG).

Higher
pres-
sures.

sphere

(HASMEL-

BERG).

Weak-
ened.

Strength-
ened.

Broad-
ening.

Reversals.

Above
10 atmo-
spheres.

At
10 atmo-
spheres.

Group

1
[. (continued).

i

65 3783-67

g b. r g 11

66 92-48

sh

s . . .

1 1

67

93-75

sh

s br

1

68 3807-30

g b. I' g

1 1

73 31-82

;

m bs : rv s

1 1

74 32-44

s . . .

1 1

77 44-40\
44-71 J

u

1 1

81 58-40

vg bs rv s 1 1

82 63-21

N

wk c b, . : [i]

87 89 • 80

N

(sh) wk

g b,. ; . .

89 3909-10

n

wk V

90 .12-44

ii

i wk V

• •

91 13-12

s

1 1

96 44 • 25

n

wk V

98

54-61

n

wk V

99 62-00

n

wk V

100 70-65

n

wk V

m

101 72-31

s

1

1

102 73-70

m br

1

1

103 74-83 n

wk V

m

104

84-18

n

wk V

.

105 94-13

n

wkV

i ,

106

95-45

m br

1

1

110

4017-65 n

wkV

112

19-20

str

s

1

1

133

4104-37

N

str

C \ (I'v)

1

135

16-14

N

wk

m

1

137

21-48

str

m

1

1

140 38-67

N

wk

• br

[1]

143 50-55

N

wk

m br

1

144 64-82

str

s

,

1

145 67-16

n

N

wk

m

.

1

163 4307-40

s

1

1

165

30 • 85 \
31-78J

N

wk

m

1

1

166 56-07 11

N

wk

c b,.

179 4481-30 n

N

c

.

.

180 90-71

n

N

m bv

.

181 4513-20

m b,.

182 20-20

sh

str

s

i i

200

4703-96

n N

wk

m b.

i

205

52-58

N

wk

m bv

i

208

62-78

s

i i

213 4829-18 N

wk

c br 11

The following abbreviations are used to express the width of bright lines and of reversals in columns 8
and 9 : — s = slight, m = moderate, c = considerable, g = great, vg = very great, G = very very great.

Other abbreviations are : — r = reversed, rB = reversal nearly symmetrical, rv = reversal on violet side
of centre of bright line, bs = broadening nearly symmetrical, br = broadening greater towards red,
b, = broadening greater towards violet, V = line has vanished under pressure.

EFFECT OF PRESSURE UPON ARC SPECTRA—NICKEL.

239

TABLE IX. (continued).

Nebulous.

Sharp at
1 atmo-

Changes
in relative
intensity.

Prominent features
as regards

Grouping
according to
displacement.

Line.

A.

1 atmo-
sphere
(HASSEL-
BEKG).

[Highe
pres-
sures.

sphere
[•(HASSEL-
: BERG).

Weak
ened.

- Strength
ened.

- Broad-
ening.

Reversals.

Above
lOatmo
spheres

At
- 10 atmo-
. spheres.

•

1
Group

I. (coi

tinued).

217

4855-57

N wk

c . 11

220

70-97

1

I

1

223

4904-56

N

wk

m liv

I

225

14-15 n

1

229

37-51 n

N

wk

in

230

45-63 n

232

71-54

N

c b..

1

233

80-36

N

c bv

1

234

84-30

N

c 1>, 1

1

236

5000-48 n N

0 bv

1

239

35-55 N

c It..

1

241

42 • 35 n

242

49-01 n

243

80-70 N

wk

c bv r c 1

244

81-30 n N

s

1

245

5424-85

S 1),.

1

246

36-10

sh

s b,.

1

247

77-13

g b,. i\. g

1

1

Group II.

52

3744-68

sh

wk V

9

126

4064-55

m br

2

141

4142-341

42-47 /

c lv

2

146

84-65

m l)r

I

2

147

95-71

wk

g br

(2)

(2)

148

4200-61

sh

g br 2

2

149

01-88

sh

g br

2

2

151

21-87

s b.

2

153

31-23

N

wk

c br 2

155

36-55

N

m br

2

159

84-83

g bt

2

2

160

88-16

g b.

2

2

161

96-06

vtt b

2

162

98-68

g br ;

2

166

4356-07

n

N

wk

c br

2

167

59-73

sh

g br

2

2

168

68-45

str

s b.

2

170

84-68

m br i

1

2

The following abbreviations are used to express the width of bright lines and of reversals in columns 8
and 9 :— s = slight, m = moderate, c = considerable, g = great, vg = very great, G = very very great.

Other abbreviations are :— r = reversed, ra = reversal nearly symmetrical, rv = reversal on violet side
of centre of bright line, b, = broadening nearly symmetrical, br = broadening greater towards red,
bv = broadening greater towards violet, V - line has vanished under pressure.

2 I 2

240

DR. W. GEOFFREY DUFFIELD ON THE
TABLE IX. (continued).

Nebulous.

Sharp at
1 atmo-

Changes in
relative
intensity.

Prominent features
as regards

Grouping
according to
displacement.

Line.

A.

sphere I

1

1 atmo-
sphere '
(HASSEL-
BERG).

ligher i
pres-
sure.

HASSEL- '
BERG).

Weak- !
ened.

Strength-
ened.

Broad-
ening.

Reversals.

Above
.0 atmo-
spheres.

At
.Oat mo-
spheres.

Group II. (con

tinued).

171 4390-00

m br

2

172 4401-70

G br

2

2

174 37-17

wk

g br

2

2

176 59-21

vg br

2

2

177 62-59

vg br

2

2

178 70-61

vg br

2

2

181 4513-20

N

m br

2

186 51-45

m br

2

187 60-10

N1

s br

2

188 92-69

vg br

2

2

189 4600-51

wk

vg br

2

2

190 05-15

vg br

2

2

192 47-47
193 48-82

wk

m br
vg br

!> merged

{,

2
2

194 55-85

s

2

195 67-16

N!

m

2

2

196 67-96

m

2

197 86-39

m bt

2

2

199 4701-72

m br

2

2

201 14-59

g br

2

2

202 15-93

g br

2

2

203 32-00

s br

2

2

204

32-66

N

s b,

2

206

54-95

str

s br

2

2

207 56-70

c br

2

2

209

64-07

s br

2

2

211

86-42\
86-46J

c br

S

2

2

212

4807-17

m bp

2

2

214

31-30

m br

2

2

215

32-86

s br

2

2

216 38-80

N

wk

m br

2

218

57-57

m br

2

219

66-42

c bt

2

2

221

73-60

m br

2

2

222

87-16

t

2

224

4912-22

n

2

226

18-53

sh

m br

2

2

227

25-74

m bt

2

228

36-02

sh

m br

2

2

231

53-34

N

m br

2

235

98-42

m br

2

237

5012-48

sh

m br

2

238

17-75

m br

2

2

The following abbreviations are used to express the width of bright lines and of reversals in columns 8
and 9 : — s = slight, m = moderate, c = considerable, g = great, vg = very great, G = very very great.

Other abbreviations are : — r = reversed, ra — reversal nearly symmetrical, rT = reversal on violet side
of centre of bright line, bs = broadening nearly symmetrical, br = broadening greater towards red,
bv = broadening greater towards violet, V = line has vanished under pressure.

EFFECT OF PEESSURE UPON ARC SPECTRA.-NICKEL.

241

For the purpose of comparing the characteristics of lines belonging to these two
groups Diagram 8 has been prepared. It closely resembles Diagram 6, fig. 4, but
instead of denoting the lines by means of numbers, their characteristics are quoted in
accordance with the abbreviations given below the diagram. It deals only with lines
whose displacements above 10 atmosphere have been determined. It is clear that
Group I. is by no means homogeneous, and that it may be divided into two Sub-
groups IA. and IB. The characteristics of the lines falling into the first part of
Group I. (Group IA.) are : — •

1. Slight broadening. Nearly symmetrical.

2. Sharp at atmospheric pressure.

3. Reversed lines are very nearly symmetrical.

4. Two lines are strengthened, none are weakened.

Diagram 8.

_
laqaaa'b'banNN

10

14

(J 5& 4 6 8

a = slightly broadened.

b = greater broadening (includes moderate, considerable, great, &c.).

s = sharp at atmospheric pressure.

N = nebulous at atmospheric pressure or under high pressures.
/ = broadening or reversal is unsymmetrical.
Lines 8 and 31 not included in this diagram (ef. Diagram C, fig. 4).

The characteristics of lines falling into the second part of Group I. (Group IB.) are :—

1. Greater broadening. [Many are described as moderately broadened, some as

greatly broadened, though several are described as slightly broadened.]

2. Nebulous at one atmosphere or nebulous at 10 atmospheres.

3. The reversed lines are less symmetrical.

4. Tendency to weaken under pressure.

The characteristics of the lines falling into Group II. are :—

1. Great broadening.

2. Their unsymmetrical appearance under pressure. They are all broadened

greatly to the red side.

3. None are reversed under pressure.

242 DR. W. GEOFFEEY DUFFIELD ON THE

If we include in our survey the readings made at 10 atmospheres more lines come
under observation ; among them are several that are nebulous, of these most are
weakened, e.g., 82, 135, 140 and 145, &c., but one, 133, has been classed as
strengthened. We also note that certain other lines which are nebulous under
pressure are included in Table IX., namely, those which are broadened towards the
violet, 232, 233, 239, &c. These nebulous lines all occur in Group L Group II. is
similarly extended by this process, and it now appears to include some lines which
become nebulous under pressure ; these differ from those assigned to Group I. in
appearing to be broadened to the red side, but there is a certain amount of doubt
about the accuracy of their measurements, and therefore of their position in Group II.

Adopting the law that the displacement varies as the cube of the wave-length for
lines of the same group, we see from Diagram 6, fig. 4, that the maxima in the
distribution curves occur at 1075 and 3'0, which gives a ratio of 1 to 3'6 for the
maximum displacements of the two groups, a result very different from that found for
the groups in the iron spectrum. If, however, Group I. be sub-divided into Group IA.
and Group IB., we may take the similar nebulous lines 19, 21, 26, 31, 87, 165, 213, 217,
as forming the more displaced group for which the value of c£/X3 is about 5 x 10~u.
The corresponding value for the remainder is not very different from 2'5 x 10~u.
This would give a ratio of displacement for the three groups 2'5 : 5 : 10'75, which is
close to that found for the iron spectrum, namely, 1 : 2'2 : 4'5, and not far from
1:2:4, but it would only be justified if the division into three groups were an
established fact.

It may be pointed out that the d/X2 diagram (fig. 2), favours this relationship (the
ratios would be 1'3 : 27 : 5 '2), but, as has already been discussed, the diagram labours
under the disadvantage of including 247 in the intermediate group, dissociating it
from lines which it more strongly resembles, and it separates the similar lines 87, 165,
213, and 217. These difficulties are obviated in the d/\3 frequency curves.

We are thus faced with the possibility of the following representations of the
groups :—

Group I.

d = 3'OxlO-14X3.
Group II.

d = 1075xlO-uX3,

in which case the ratio is 1 : 3'5, or if there be three groups : —

Group IA., d= 2'5 xlQ-14X3 or d = 1'3 x 10-10X2,
„ IB., d= 5'0 xlO-14X3 d = 27xlO-10X2,

„ II, d= 1075x10-" X3 d = 5-2xlO-10X2.

In either case the ratio is approximately 1:2:4. The writer inclines to the view that
this is the more probable ratio between the different groups, but the determination is
open to doubt.

EFFECT OF PRESSURE UPON ARC SPECTRA.— NICKEL.

243

The existence of another group is indicated by the occurrence of five lines which are
displaced towards the violet. The amount of their displacement is approximately the
same as that of Group IB. if no allowance be made for the fact that all the measure-
ments were made at 10 atmospheres pressure.

We may designate the nickel groups as follows : —

Group 0, d = - 4'3 xlO-"X3.
IA., d = + 2'5 xlCT14A:!.
IB., d = + 5'0 xlO-14A3.
II, d = +1075xlO-nX3.

In the above groups several lines are included whose displacements under pressure
have not been measured ; their classification is based upon their general behaviour and
resemblance to lines whose grouping lias been ascertained.

In addition to the two main groups, further sub-division is possible as inspection of
the photographs will show. In the following table are given those lines which bear
close resemblances to one another; it may be of service in the resolution of the nickel
spectrum into the usual spectrum scries.

Line.

2

3

4

7

10
11
14
19
31

39
42
43

46

63
65
68
81

A.

3453-64
58-59
61-78
93-10

3510-47
15-17
24-65
66-50

3619-52

I

TABLE X.

All strong linos.

Very greatly broadened under pressure.

Their broadening is nearly symmetrical.

All are strongly reversed under pressure.

The reversals are placed nearly symmetrically upon the bright lines.

2, 3, 4 are very similar.

10, 11, 14 are very similar.

19, 31 are very similar.

It does not appear likely that 10, 11, 14, 19, 31 form a series, because there i
a break in the rate at which the breadth of the reversal increases with wave-
length. In the recognized iron triplets which have been examined there was
a regular decrease with wave-length.

3664 • 24 1 Moderately strong lines.

74-28 I Moderately broadened.

88-58 f Unsymmetrically reversed, the reversal being on the violet of centre.
3722-63 J 39 and 42, 43 and 46 are more closely related.

3775-71
83-67

3807-30
58-40

89

3909-10

90

12-44

96

44-25

98

54-61

99

62-00

100

70-65

103

74-83

104

84-18

105

94-13

110

4017-65

All strong lines.

Greatly broadened under pressure.
Broadening nearly but not quite symmetrical.
Reversals strong and nearly symmetrical.

The continuous increase in separation, in breadth and in width of reversal suggei
that these lines are related in some intimate way.

All nebulous at atmospheric pressure.
}- All weakened under pressure and invisible at high pressure.
100 103 104, 110 have the appearance of converging series.

244

DK. W. GEOFFREY DUFFIELD ON THE

TABLE X. (continued).

Line.

A.

87

3889-80 ]

135

4116-14

140

38-67

Moderately broadened, nearly symmetrical.

143

50-55

Small displacement.

145

67-16

• Diffuse under pressure

147

95-71

All fall in Group L, except 147, whose measurements are doubtful.

165

4331 • —

Weakened under pressure.

213

4829-18

•

"•

217

55-57

149

4201-88 '

159

84-83

160

86-16

167

4359-73

172

4401-70

176

59-21

177

62-59

178

70-61

/ 4547 -14

185

188
189

I 47-44
92 • 69
4600-51

All belong to Group II., are broadened considerably and unsymmetrically
few lines have somewhat the appearance of a converging series.

The last

190

05-15

193

48-82

201

4714-59

J56-70

207

\56-42

211

86 • 46

214

4831-30

219

66-42

221

73-60 j

Line.

A. Line. A.

195
199
201
203

4667-16 196 4667-96
4701-72 — 200 4703-96
14-59 202 15-93
32-00 204 32-66

These constitute " couples " of lines which occur near
another. The members of each couple are dissimilar,
one being slightly stronger than the other, one being
rather more broadened than the other. No simple

205
208
213

62-78 — 209 64-07
4829-18 214 4831-30

relationship has been found between the
differences of the different couples.

frequency

Line.

A.

47

3724-95 ]

200

4703-96

205

52-58

223

4904-56

Nebulous at atmospheric pressure and under high pressure.

232

71-54

Broadened to the violet.

233

80-36 \- Small displacements.

234
236

84-30
5000 • 48

Many of these lines are obviously weakened under pressure.
Lines 232, 233, 236, 239 are recorded with negative displacements.

239

35-55

243

80-70

•^

244

81-30 j

9. Relation between the Intensity of a Line and its Displacement. — In Diagrams 9
and 10 the values of d/\3 for the two groups are plotted against the intensity of each
line at atmospheric pressure. There is a distinct upward drift of the black dots with
increasing intensity, which is made more apparent when the mean value for each

EFFECT OF PRESSURE UPON ARC SPECTRA.-NICKEL.

Diagram 9.

•J45

2

Group 1

o

•

:

Intensity

14

12

10

2-

Group II

10

o

0

I

Intensity

i

VOL. CCXV. — A.

3 4 6

2 K

10

246 I)E. W. GEOFFREY DUFFIELD ON THE

intensity is calculated, these are shown by the circles. Since the high-valued lines
10, 247, 213, 217, and 87 possibly form a separate sub-group, more cogent evidence
is perhaps afforded by the absence of low values for the displacement of lines of great
intensity.

It remains to be proved whether the phenomenon is subjective or objective, whether
it may be explained by a tendency for the computer to read high values when the
density of the silver deposit is great, or whether the amount of energy involved in a
line does actually influence the displacement. One reason which may be urged for
regarding the phenomenon as independent of the observer is that intense lines are
usually broad and consequently more closely resemble lines under higher pressure ;
these have, however, been shown to be relatively less displaced than lines at lower
pressure, hence there would not seem to be any satisfactory reason for regarding the
measurements of the displacements of intense lines as being too large. If the energy
is responsible for the magnitude of the displacement one further enquires whether the
increased intensity is due to a larger number of vibrating particles, or to their
possessing greater amplitudes or to a combination of the two ; thus if the phenomenon
is objective it involves the dependence of the displacement either upon (l) the density
of the particular atom or modification of the atom responsible for each line, or (2)
upon the amplitude of its vibration. In any case the dependence of the displacement
upon the energy involved in a line is not the only factor concerned, since some faint
lines have very large displacements.

The former would seem the more probable, but attempts to show that the
displacement of the line of an element is due to partial pressure have hitherto not
been successful. This will be discussed elsewhere ; the present discussion suggests
that the partial pressure effect due to the density of similar atoms is superposed upon
an effect due to the total pressure.

10. Foreign Metals in the Spectrum of Nickel,. — In addition to the nickel lines
whose displacements have been measured, a large number of lines due to impurities
appear in the spectrum, especially between \ = 3800 and X = 4100. They have been
measured, and their displacements will be found in the following table. As they have
been examined at several pressures, it should be possible to ascertain whether the
" density " of the element has any effect upon the displacement by comparing the
displacements of the lines produced respectively by pure poles and by small traces of
the element.

EFFECT OF PRESSUKE UPON AKC SPJICTRA.— N ICKEL.

247

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250

DK. W. GEOFFKEY DUFFIELD ON THE

(l) The Influence of the Density of the Material in the Arc. — In Table XII. the
measurements made during the present research upon the lines due to traces of iron
in the nickel poles are compared with the displacements of the same lines produced
in an arc between solid iron rods.

At the top of each column will be found the pressure at which the examination was
made, and opposite each spectrum line the ratio of the displacement of the diluted to
that of the undiluted material. It will be seen that if the readings at 10 atmospheres
be left out of account (they are unduly high in nickel as we have already shown) there
is a smaller reading when the material is diluted.

TABLE XII.

Ratio of

di-ph-emenf iron as imPuritv in Ni

iron in pure iron-arc

Line.

Wave-length.
10 20

30

1

60 70 80

100

atmo- atmo-

atmo-

atmo- atmo- atmo-

atmo-

spheres, spheres.

spheres.

spheres.

spheres, spheres.

spheres.

121

4045 • 90

[1-6 '

0-48

0-72

0-78

0-91

0-66

125 4063-63

1-2 "

0-84

0-96

0-82

0-88

127 4071-79

1-36'

1-03

0-93

0-90

0-70

136 4118-62

'1-52

0-66

0-62

0-67

0-89 0-82

142

4143-96

1-31'

0-37

0-59

156 4250-9

[1-77^

1-90

157

4260-64

"1-45'

158

4271-93

'1-67'

1-02

1-28

0-72

0-91

173

4415-27

"0-73]

183

4528-78

;i-63

Mean values [1"42] 0'90

0-85

0-78

0-89

0-84

0-66

I

The measurements of the displacements of the lines due to pure iron were made
some years ago both by the writer and by assistants, while those due to a small trace
of iron in the nickel poles were made Mr. PEARSE. In order to see if a comparison
between these readings was legitimate, Mr. PEARSE measured one of the original pure
iron arc photographs (at 70 atmospheres pressure), and his readings agreed so
excellently with the earlier ones that there was no hesitation in regarding the two sets
as strictly comparable.

The ratio of the displacements with diluted to those with pure iron is, with the
exception noted above, less than unity, suggesting that the density of similar atoms
influences the displacement ; if the displacement depends upon the proximity of
similar vibrating centres it is to be expected that the displacements would be more

EFFECT OF PKESSURE UPON ARC SPECTRA.— NICKEL. 251

marked at high pressures, since increasing the pressure reduces the mean free path
between similar molecules just as increasing the total number of molecules does.

The evidence thus favours the amount of the displacement being dependent in part,
at least, upon the amount of material present, a conclusion which is in keeping with
that arrived at from the consideration of the variation of the displacement with the
intensity of the line.

The chief sources of contamination of the nickel poles are iron, cobalt, and
manganese. There is some doubt in one or two instances as to the origin of the lines ;
lines 136 and 137 for instance are very much alike under pressure, both in amount of
displacement and in their intensities, but one is ascribed to nickel and the other to
iron.

The feature of the lines due to impurities is that they remain fairly compact even
at very high pressures, and do not spread out to the same extent or become so foggy
as the nickel lines.

It is interesting to note that all the lines of the highest intensity due to iron do
not appear in the nickel spectrum. Some lines seem to characterize the spectrum due
to only a small quantity of material. For example, of the two iron lines (JJ?) 4143'50
(10) and 4143'96 (10), only one, the latter, appears in the nickel spectrum. Of the
two iron lines (150) at 4250'2 (10) and 425CT9 (10), both classed as of intensity 10,
only the latter appears, similarly, only the last-named of the two iron lines 4271 '30
(10), 4271'23 (10) is visible upon the plate. It is important to note that it is the line
which is self-reversed which most readily shows itself, it is the less refrangible line in
the pairs quoted.

It has always been surprising to the writer that no dependence upon the density
of the material manifested itself in previous experiments. On one view one might
expect the general magnetic field of the surrounding atoms to influence the frequency
of any particular atom, and since, presumably, this general field depends upon the
nature of the atom the amount of material present or the nature of the surrounding
gas should have some effect, but nothing definite has hitherto been observed. On
another view, the specific inductive capacity of the medium in which the atom under
consideration is immersed should similarly affect the frequency, but this has teen
tested without positive result.

The explanation seems to be that in the arc the isolation of a molecule is never
actually accomplished, that the vaporization of the metal, even if only a trace of it
be present, involves the liberation of an immense number of atoms of that element all
in close proximity, so that this incandescent mass behaves very much as though it
were isolated from the other materials in the arc, rendering it very difficult to influence
the immediate environment of an atom, since only the few atoms on the outskirts
of this mass are affected by the inductive properties of the surrounding gas (or
other metallic vapour produced by the vaporization of the poles), consequently
the predominant frequency is that of the atom surrounded by similar atoms.

252 DK. W. GEOFFREY DUFFIELD ON ARC SPECTRA.— NICKEL.

Decreasing the amount of impurity present would make the outer portions of its
incandescent vapour relatively more important, and increase of pressure would achieve
the same end by decreasing the average distance of each vibrating centre from the
molecules of the surrounding gas which might then exert an influence which is
negligible when the pressure is low. We thus expect to find a reduced displacement
when the material is greatly diluted and a relative reduction with increased pressure.
The former is supported, and the latter suggested, by Table XII. We have assumed
that the surrounding atmosphere is less effective in producing displacement than an
atmosphere of similar molecules, but it should not be impossible for the opposite to be
the case, according to the nature of the substances employed, unless it can be proved
that similar molecules alone have influence upon a radiating molecule.

Approaching the problem of the structure of the arc from a different direction, the
writer has been impressed by the importance of the part played by the surrounding
gas in maintaining the arc ; it would seem to require that each metallic atom is, for a
brief interval at least, associated with one of the atoms of the surrounding gas so
that something akin to chemical action takes place between them (or at least
involves the influence of what may be called chemical affinity). If this is the case it
would appear contrary to the view of the density effect just put forward, since that
does not contemplate the commingling of the individual atoms of the metal with those
of the surrounding gas. But it is further possible that the spectrum line is due, not
to single atoms but to a system, such as one consisting of a metallic atom combined
or interacting with one atom of the surrounding gas (the function of the latter being,
by its interactions with the atom, to excite it to emit its characteristic radiation),
and that such systems form the aggregate already alluded to, and that the frequency
of the resultant spectrum line is characteristic of this particular atom system in its
environment of similar systems. These systems would not be interfered with by
foreign systems until either the reduction of the amount of material below a certain
minimum amount or the increase of pressure made the proximity of foreign atomic
systems relatively more important.

Possibly this consideration is responsible for the decrease in the rate of increase of
displacement with pressure when the latter is high.

The writer is indebted to Dr. SCHUSTER, at whose suggestion the series of experi-
ments upon the effect of pressure upon spectra were begun some years ago, and to Sir
ERNEST RUTHERFORD in whose laboratory the photographs were taken, and expresses
his thanks to them for having placed the necessary apparatus at his disposal.

The photographs were measured by Mr. PEARSE in a careful and thorough manner.
Part of the expense of this research was defrayed by a grant from the Government
Grant Committee.

Outfield

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Phil. -Trans., A Vol. 215.PI.1.

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[ 253 ]

IX. BAKERIAN LECTURE. — X-rays and Crystal Structure.

By W. H. BRAGG, D.Sc., F.R.S., Cavendish Professor of Physics in the University

of Leeds.

Lecture delivered March 18, — MS. received April 7, 1915.

THE method of investigating crystalline structure by the use of X-rays has already
been explained in papers read before this Society. It will be convenient nevertheless
to re-state its principle very briefly in order to introduce some further considerations
which I propose to lay before you.

The statement of the principle may be made in the following way. Let a train of
waves of length X be passing through a medium in which are particles having the
power of scattering the radiation. Suppose, further, that the scattering power is not
distributed evenly through the medium, but that directions can be found along each
of which there is a periodic variation of the scattering power of the material contained
in strata perpendicular to the given direction, strata being, of course, taken of equal
thickness for comparison. Let the distance of recurrence or spacing be called d.
Let 0 be the angle between the rays and the strata. Then there will be a
" reflection " of the radiation by the medium of n\ = 2d sin 8, where u is any integer.

For instance, the Lippmann process of colour photography produces such a
distribution of scattering power in the sensitive film through the agency of stationary
waves.* If light is incident on the film it is strongly reflected when X = 2d sin 6 ; if
the light is white the film selects the appropriate wave for reflection, and this is the
origin of the colour manifestation.

f*, ^y_ ^y-^ •*•' ' y P

P ^ —

Fig. 1.

The formula is readily explained by aid of the figure, which shows a set of
regularly spaced planes p, p, p, ••., each reflecting a minute fraction of the incident

* 'Physical Optics,' WOOD, p. 149.
VOL. CCXV. A 531. 2 L (Published July 13, 1915.)

254

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

radiation, and transmitting the remainder. All the reflected portions which unite to
travel along BC will be in the same phase if

n\ =

-AB = A,D-AB = DN = 2d sin 0,

where n is any integer. In the figure n = 1. Only when this relation is satisfied is
there any sensible reflection. The greater the number of planes concerned in the
action the more abruptly does the effect disappear if 6 is slightly varied.

We speak here of reflecting planes and represent them as surfaces. But the
general result is exactly the same if we replace a plane by a thin layer containing
scattering particles ; and it still holds exactly if we suppose a continuous succession
of thin layers of varying density forming the periodic structure to which we have
referred.

The atoms of a crystal are distributed in an orderly manner. They can be thought
of as arranged on series of parallel planes ; and this can be done in many ways. A
natural face is always parallel to a series of this kind.

An atom possesses the power of scattering X-rays to an extent which appears to
depend mainly upon its mass. It is not quite clear how this power is distributed
within the atom ; one would, expect that both nucleus and electrons share in it, and
that it extends more or less over the whole atom. As the diameter of the atom is of
the same order of magnitude as the spacings of the crystal planes this last consideration
is of importance, and I propose to return to it later. For the present we may observe
that if the scattering power were confined to one central point in each atom we
should be able to represent the "reflection" by a plane containing atoms as if it
occurred in a reflecting surface, as in fig. 2. If, on the other hand, we suppose the
scattering power to be distributed through the atom, and if we take a section through
the crystal perpendicular to the direction x (see fig. 2). the scattering power of the

substance in the layer Sx will be a periodic function of x. In either case the formula
n\ = 2d sin 6 gives the only values of Q at which reflections can occur.

Let us take a numerical instance. The atoms of rock salt are now known, as the
result of the investigations we are considering, to be arranged as in fig. 3. The
plane of the paper is parallel to a cube face. The atoms are represented as circular,
and as having a definite boundary ; but that is merely because we must give some
form in a drawing. In reality we know very little about such things.

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

255

C

Fig. 3. Arrangement of atoms
in any plane in rock salt
which is parallel to a cube
face. Blank circles repre-
sent sodium atoms ; the
others chlorine, or fin vtrsA.

The atoms of the crystals may be considered as arranged in a series of planes which
are perpendicular to the paper and cut it in lines parallel to AB. These planes all
contain equal numbers of Na and Cl atoms and are all
alike. The spacing is 2'81 A.U. (10~8 cm.).

On the other hand, we may think of the atoms as lying
on sets of planes which are perpendicular to the paper
and intersect it in lines parallel to BC. These planes
are also all alike, containing equal numbers of Na and Cl
atoms, although the atoms lying along the lines of inter-
section with the paper are all of one or of the other kind.

0 / —

The spacing is very nearly 2 A.U., being \/2 times less

than the former spacing.

Again, if we proceed in a direction parallel to a cube-
diagonal of rock salt, we encounter alternately planes
containing nothing but Na atoms, and planes containing
nothing but 01 atoms. The periodicity now has a spacing

o

3 '24 A.U. ; extending, say, from one Cl plane to the next.

If we pass a pencil of X-rays of wave-length X through a piece of rock salt, there
will be a reflection in the planes parallel to a cube face if the rays make with those
planes an angle 9 where sin 6 = A/5 '62, in the planes perpendicular to a face-diagonal
if the angle made with these planes is given by sin 9 = X/4, and in the planes
perpendicular to the cube-diagonal if the angle made with such planes is given by
sin 9 = X/6'48. Of course it is extremely unlikely, though quite possible, that more
than one of these conditions can be satisfied at one time by the same set of
homogeneous rays. But if the pencil of X-rays contains rays of a great range of
wave-lengths various constituents may be found to satisfy the condition of reflection
not only by these but by many other planes, or by many others and not by these.
In this way a number of different constituents of the original pencil may be reflected
in different directions, which will have an ordered arrangement dependent on the
symmetry of the crystal. This is the explanation of the photographs first obtained
by FRIEDEICH and KNIPPING as the result of the brilliant suggestions of LAUE.*

The X-rays usually penetrate such a little distance into a crystal that the reflection
seems to occur at the face of the crystal in the ordinary way. It is usual though not
always possible to cut the crystal so that the surface is parallel to the set of planes
to be considered, unless a natural face is available. - If the crystal is cut imperfectly
so that the face is not parallel to the planes, the angle between the incident and
reflected rays is not affected thereby, because the reflection is truly related to the set
of the planes and ignores the face. Moreover it is just as sharp where the face is
rough as when it is smooth, so long as the crystal is uniform.

* The explanation was given in . this simple and complete form by W. L. BRAGG, ' Proc. Camb. Phil.
Soc.,' vol. xvii., Pt. I., p. 43.

2 L 2

256

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

In practice a fine pencil of X-rays is allowed to fall upon a crystal face, natural or
prepared, and the crystal is gradually turned so that the angle between rays and
face, called the glancing angle, steadily increases from zero onwards. Reflection
takes place whenever the formula is satisfied, the reflected pencil being detected by
the ionisation which it produces in a chamber containing a heavy gas. The chamber
can be made to turn about the same axis as the crystal.

If the incident rays are heterogeneous and contain pencils of various wave-lengths,
each is reflected at the proper angle, and the rays are analysed, forming a spectrum.
The figure shows the analysis of the rays issuing from an X-ray bulb in which the

XlHXfw*

//? Ca/c/fe

I'M)

.'"*

X?

Secono

order.

*..•. '

v .

* • • »

,

• • * • *

. •

t

4-0' ||° 20' 4-0 ' (2° 20' <K>'

Fig. 4. Spectrum of Rh X-rays. Each dot records a separate measurement.

anti-cathode is made of rhodium ; the spectrum is therefore characteristic of rhodium.
The abscissa are the various values of the glancing angle, the ordinates are the
measured intensities of the reflected rays. It may be mentioned that glancing angles
can be measured with little difficulty to a minute of arc. Intensities can be measured
to one or two parts in a hundred if the circumstances are not varied. If the experi-
ment is repeated under fresh circumstances the agreement is not quite so good.

It will be observed that the rhodium spectrum contains four " lines." In other
words, when stimulated by the impact of the cathode stream rhodium emits four
pencils of different wave-lengths. These have been calculated* to be 0'534, 0'545,
0'614 and 0'619 A.U. respectively. The crystal planes here used are the ( 1 1 1 ) planes
of calcite, those which are perpendicular to its axis. The spacing is 2'83 A.U. The
doublet au a2 is very convenient for crystallographic work.

If we always make use of the same homogeneous X-rays, for example of the line «2
(or, which is much the same thing in most experiments, the doublet a1; a2) we can
compare the spacings of different sets of planes, whether of the same or of different
crystals. This is one step towards the determination of crystalline structure. In
order to make its significance clear it will be well to consider certain elementary
principles of crystallography.

However complicated may be the arrangement of the atoms in a crystal with
respect to each other they must allow of being grouped into units which are repeated
regularly throughout the crystal. Every unit resembles every other exactly as
regards both the internal disposition of its members and the external disposition of
group to group. The grouping may, however, be effected in more than one way. If

* 'Phil. Mag.,' March, 1914.

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

257

a point in each group is chosen to represent that group — it must be similarly
situated in each group — the points will lie at the intersections of such a set of lines
as are shown in fig. 5. If they do not do so the unit chosen must be incomplete and
is indeed not a unit at all. The figure may be regarded as composed of rhomboidal
cells, and in the most general case three unequal edges of each cell meet at every
corner and include three unequal angles.

The determination of the form of this fundamental lattice is a necessary preliminary
in any investigation of crystal structure. It involves the measurement of the three
edges and three angles of the elementary rhomboidal cell.

But the form of the crystal structure is not fully known until we have also deter-
mined the disposition of the atoms about the representative point in each unit.

There are thus two stages in such an investigation, and the X-ray method operates
in entirely different ways in the two.

^ ^

7 / ^^7 / _^1 I _
_^1 / >7 / ,>? / ,>/

i/iu/rm

I

LK / /^/ /

v / / // / / 7, . .

/ ~Z'-77 / /

Fig. 5.

Fig. 6.

Let us first consider the determination of the fundamental lattice.

If we knew what faces of the crystal, if any, were parallel to the faces of the cell,
we should merely have to measure the spacings perpendicular to the faces and our
work would be done. It might not even be necessary to make more than one such
measurement because the relative values of the spacings might be known from con-
siderations of crystalline form. The same knowledge of form which told us the faces
to measure, would give us also the angles of the cell, and so all the quantities would
be known.

Although, however, a knowledge of crystalline form gives the most valuable
suggestions as to the dimensions of the elementary cell, its indications are not always
definite. A very interesting and important example occurs in the case of cubic
crystals. If the elementary cell were a cube, we ' should certainly expect the form of
the crystal to be cubic ; but there are also two other forms of lattice which may lead
to the cubic habit and one of these is of the greatest importance to us. If the three
edges are all equal, and the three angles are all 60 degrees, we have the cell shown in
the dotted lines of fig. 6, and the distribution of points which it gives may also be

258

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

represented, as crystallographers have shown us, by the " face-centred " cube drawn
in full line. The cubic habit may therefore be acquired in this way also. By the X-ray
method we can easily tell one from the other. For the spacings of the (100), (110) and

(ill) planes are, in the case of the simple cube, in the ratio 1 : —7= : — - — . Consequently

v 2 v 3

the series of the corresponding glancing angles are as 1 : \/2 : \/3. We have no real
example of this lattice, though potassium chloride (sylvine) accidently gives it. The
potassium and chlorine atoms are nearly of equal weight, and weight is all that seems

Fig. 7.

to affect the scattering of X-ra3rs, so that the arrangement of the atoms shown in
fig. 7 is effectively equivalent to the simple cubic arrangement.

On the other hand, it may easily be calculated that the spacings of the face-centred

1 2
cube are in the proportion 1 : — = : —7= ; and the series of the corresponding angles

\/2 v3
of reflection are as 1 : \/ 2 : \/3/2. A very good example occurs in the case of copper ;

COPPER
(100)

^

(no;

-&

-"•--

:(.i.)

A

-IS'

A-

stw

49-

30'

)" 10* 10' lii' W 50' 6(

Fig. 8. Spectra of Pd rays given by certain planes of a copper crystal. The angles recorded

refer to the a. ray.

the angles are given in fig. 8, and the relations can be readily verified. The copper
atoms are therefore arranged on the lattice in which the elementary cell has three
equal sides meeting at a point and including three equal angles of 60 degrees.

PKOF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE. 259

Rock salt, diamond, zinc- blende, fluor-spar and other cubic crystals are all built
on the same fundamental lattice. In all cases the sines of the angles of the first
order spectra of the (100), (110) and (ill) planes are in the ratio 1 : v 2 : \/3/2 ;
except by accident, the manner of which will be explained presently.

The prevailing occurrence of the face-centred form suggests that even in calcite,
which is not cubic, the representative points may be conceived of as being placed at
the corners and the face-centres of a rhomb of the same form as the calcite crystal
itself. Calculating the spacings of various planes on this basis, we find that the
corresponding angles of reflection are correct to the minute when determined by
experiment. It may be interesting to give part of the calculation.

Let the side of the rhomb be a. The angles of the rhomb being known, it may
easily be calculated that the volume is a8 x 0*925. Since the specific gravity of the
crystal is 2712, the mass in the rhomb is «3x (T925 x 2712.

But there are four molecules to each rhomb,* the weight of each being 100 times the
weight of the H atom, which is l'64x 10~24 gr.

Hence

rt3xO'925x2712 = 4x I00xl'64x 10'2'
or

a = 6'40xlO-8

= 6'40 A.U.

From this other linear dimensions of the rhomb may be found. For instance, the
spacing of the planes parallel to the face is 3 '02 5 A.U., and the reflection of the a2
rhodium ray should occur at an angle 0 given by

0'614 = 6'05 sin 0.

Hence 0 = 5° 48', and this is exactly verified by experiment.

It is easy to find the dimensions of the elementary cell of the lattice. The three
equal edges join the blunt corner of the rhomb to the middle point of the adjoining
faces. The length of each is 4'03 A.U., and the angle between any two, 75° 54'.

In similar ways we may determine the space lattice of any crystal. It is to be
remembered that the data on which we depend are the positions of the spectra of the
various faces.

Let us now proceed to consider how we may determine the arrangement of the
atoms about the representative point. We now work on an entirely different plan,
building in fact on a determination of the relative intensities of the reflections of
different order and different sets of planes. As has already been said, measurements
of this kind are much more difficult than measurements of position ; and moreover
their exact interpretation has been by no means clear. • But in many cases we already
know so much from considerations of crystal symmetry, and from measurements of

* ' Roy. Soc. Proc.,' 89, p. 280.

260

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

the angles of reflection, that the very roughest measurements of relative intensity
give us all that we want. This is at any rate the case with some of the simpler
crystals.*

Let us take one or two examples. The spectra of the (100), (HO) and (ill) planes
of rock salt are shown diagrammatically in fig. 9, the positions of the vertical lines
showing the relative magnitudes of the sines of the angles at which the various
reflections occur and the heights the relative intensities on an arbitrary scale.

The sines of the angles of the first order reflections are as 1 : ^/2 : \/3/2. Hence
we know that the fundamental lattice is that which gives rise to the face-centred
cube.

The actual values of these sines agree with the supposition that one atom
of Na (or Cl) is associated with each point of the face-centred cube. The calculation
has been already published, t and it will not be necessary to repeat it.

100

no

111

vz

2/3

Fig. 9.

We can imagine a face-centred lattice of sodium atoms, and an exactly similar
lattice of chlorine atoms which is at first coincident with it, but is then moved out
from it in a direction and to an extent which we must gather from the intensity
observations. Each chlorine atom is then related to the sodium atom from which it
came, as regards distance and orientation, in identical fashion throughout the
crystal.

Considering the diagram we see that there is a rapid and steady decline in
the intensities of the spectra of the (100) and (110) planes as we proceed to higher
orders. From experience of many cases we believe this to be normal, that is to say,
it always occurs when all the reflecting planes are similar and equally spaced. We
conclude that the chlorine atoms have so moved that they still lie in the (100) and
(110) planes of the sodium lattice. But the (ill) spectra are not of the same type.
A marked alternation of intensity, odd orders weak and even orders strong, is

* W. L. BRAGG, ' Roy. Soc. Proc.,' 89, p. 478.
t 'Roy. Soc. Proc.,' vol. 89, p. 246 or p. 276.

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

261

superimposed upon the normal rapid decline. This implies an alternation in the
strength of the reflecting planes.

For, suppose the lines a, a, a, in fig. 10, represent a set of reflecting planes,
giving the spectra A ; and suppose also the lines b to represent a set of reflecting

A

\ \

X

a

a

£

\

h 6

)

b

*)

C

\ \

a+b,

Fig. 10.

planes which have half the spacing of A, and therefore give spectra twice as far
apart as in B.

If we combine the two sets of planes, we compound the reflections as in 0.

Hence we learn that the chlorine atoms have so moved away from the sodium atoms
that their (ill) planes lie half-way between the (ill) planes of the sodium lattice,
for in this way is formed a set of planes of alternating strength like the a + b, b,
a + b, b, of the figures.

The structure shown in fig. 7 is in accordance with these conditions.

Let us take the case of zinc blende, the spectra of which are given in fig. 11.

(100)

(no)

(111)

Here again the relative positions of the first spectra of the (100), (110), and (ill)
planes are those of the face-centred lattice ; and the actual angles of reflection show
that one zinc atom is associated with each point on the lattice.

Again, we may start with a zinc lattice and suppose a similar sulphur lattice to be
at first coincident and then to move away to its proper place. The (110) planes are

VOL. CCXV. — A. 2 M

1

r

1

/2
1

2V2
1

v'S/a S3 3^3/2

Fig. 11.

262

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

normal, so that the sulphur atoms remain on the (HO) planes of the zinc lattice.
But the (100) planes show the same sort of alternation as we observed just now
in the case of the (ill) planes of rock salt. The sulphur lattice must have moved so
that its (100) planes lie half-way between the (100) planes of the zinc lattice. Again,
the (ill) spectra show a new peculiarity in that the second order spectrum is
abnormally small. Such an effect is readily explained by supposing that the
(ill) sulphur planes divide the intervals between the (ill) zinc planes in the
ratio 1:3.

All these conditions are fulfilled by the structure shown in fig. 12, in which the
sulphur lattice lias moved away from the zinc lattice in the direction of the cube

*;

M

• Zn
O S

/*

Fig. 12.

diagonal, the distance of the movement being one quarter of the length of the cube
diagonal.

Let us take yet one more illustration from the case of the diamond ; the spectra are
given in the figure.

100

no

111

2V2

I

a/2

V3/2 3V3/2

Fig. 13.

5^3/2

In this case it might appear that the relative positions of the first order spectra are
not those of the face-centred lattice, but this should be regarded as purely accidental.

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE. 263

The spectra resemble those of zinc blende, except that the odd orders of the (100)
spectra and the second order of the (ill) spectra are not merely abnormally small,
but have entirely disappeared. These facts are very simply explained. The structure
is obtained in exactly the same way as that of zinc blende ; but the two lattices are
alike, both being of carbon, and when their influences interfere, the interference is
complete.

Finally, consider the case of fluor-spar (CaF2) for a special reason.

As regards relative positions and intensities the spectra are exactly the same as
those of diamond. If we begin with a face-centred lattice of calcium atoms, we must
so move out from it two similar lattices of fluorine atoms as to obtain the observed
relative intensities. This we do by moving them along the cube diagonal, but
in opposite directions, the distance moved being one quarter of the length of the
diagonal. This gives a highly symmetrical structure. A fluorine atom lies at the
centre of eacli of the eight cubes into which the large face-centred cube can be
divided. The (100) planes of fluorine now lie half-way between the (100) planes of
calcium, and we explain the disappearance of the first order (100) spectrum by
supposing that they have equal reflecting powers. We assume, in fact, that two
fluorine atoms of weight 38 are equivalent, within errors of experiment to one atom
of calcium of weight 40. This is only one instance out of many ; it seems certain
that two planes containing equal weights per unit area are equivalent in reflecting
power, no matter how the weight is made up nor how it is distributed in the plane. As
regards distribution, this is what we should expect : as regards the effect of weight
the result is not so obvious, though it cannot surprise us if we suppose the scattering
to be due to the cumulative effect of the electrons and nuclei of the atoms.

In the simple cases which we have been considering, the considerations of crystal
symmetry, though unable of themselves to determine crystal structure, come so near
to doing so that a few plain hints given by the new methods have been sufficient for
the completion of the task. The exact positions of the atoms are then known.

But this is not the case with more complicated crystals. As an example we may
take the case of iron pyrites. There is a fundamental face-centred lattice of iron
atoms, and sulphur lattices are displaced from it in a manner similar to that which
has already been described, yet more complicated. The main point, however, is that
the extent qf the movement cannot be determined from symmetry considerations.
In the cases described above the exact movement could be told from these con-
siderations, as soon as the X-ray method had revealed its nature. But in iron pyrites,
and probably in the vast majority of crystals, the movement must be calculated from
determinations of the relative intensities of the X-ray spectra.

We require, therefore, to determine in the first place how they should be measured,
and in the second how they are to be interpreted. In what follows I propose to
consider these two points, particularly the latter.

The method of measurement has already been described ('Phil. Mag.,' May, 1914).

2 M 2

264

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

A fine pencil of X-rays falls upon a crystal, and the ionisation chamber is set to
receive the reflection. The crystal is first set at an angle which is just so far from
being correct that no reflection takes place. It is then made to revolve steadily, or
rather by uniform small movements at regular intervals ; it passes through all the
positions in which it can reflect and when the angle has again become such that no
reflection takes place, the total movement of the electroscope leaf is observed. In
this way the whole reflection effect is integrated. This method is fully discussed in
the paper quoted. It is enough to say here that such measurements are really

-AU.

10

(Ill)

1

1

6

O 6

T

3

4O

60 6|iT6°12
5in0m=0-I08

^=2-83

(211)

37

.5

6

0

7

c

6

OjCa C
i 68 €

ooloo

I6I6D6I6

IJiLL

OjCa

8

COj

(100)

36

1

i

Cd

6

CO CaCO CaCO

3 68

16 16 |l6
1 ill

68 0I055°48'

•

1 . IO°
1 1

(no)

149

L8

1

15

|

6-7

i

Gat

f

0 o ocacoo

Z 52 5
I6If)If>|l6If>If.

II MM

OCa£0
' Sin 6^0-124

(iii)

1

12-5

I

3-3
1

Z-t

Ca

1

COOOCaCOOOf

4O

3 1 28

flfll

AC Oar*"32'
sin 6rt,=oo8i

40 <C=3'82

29

' ?J

34

(no)

3

4

21

1

0
<

tt 0 CaO COO
3 68 68

16 16 16 16

MIL

CaO CO

61

6-4 z-8 CaCOO 0 0
CaCO

0
CtCO

Fig. 14.

comparable with each other, and in particular that the effect of crystal imperfections
(sometimes serious) are avoided. A small allowance has to be made for the reflection
and scattering of general radiation which enters the slit with the reflected pencil
actually measured. This is easily done.

Let us now consider the interpretation of these measurements of intensity. Very
useful information on this point is to be gathered from a study of the spectra of various
calcite planes. The positions and intensities of the spectra of six of the most
important planes are shown in fig. 14. The diagram also shows the spacings of

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

265

these planes and the angles at which the first order spectra occur. The heights of the
vertical lines represent intensities, and their magnitudes are also indicated in figures
underneath, in arbitrary units. Their abscissae represent the sines of the angles of
reflection. The numerals placed above the lines on the right of the diagram show
the percentages of the molecular weight lying in the various planes.

The structure of calcite has already been investigated by W. L. BRAGG,* and fig.. 15,
is reproduced from his paper. In its determination use was made, as in other cases,
of a knowledge of the crystal symmetries. The fact that the (ill) planes contain
alternately calcium atoms only, and groups of CO3 only was inferred from a
comparison of the (ill) spectra of the calcite series of crystals. In all cases there is

Fig. 15.

clearly an alternation of composition because, as in NaCl, the odd order spectra are
abnormally small. In chalybite FeCO3 the odd order spectra disappear altogether,
implying that the alternate planes are now equally weighted. If one plane contains
Fe only, and the other C03 only, this equality does actually occur. The series being
isomorphous we must suppose this structure to be common to them all.

Considerations of symmetry show that the arrangement of the carbon and oxygen
atoms must be as shown in the lower line of drawings in the figure.

But there is one variable which symmetry cannot determine, viz., the distance
between the oxygen and the carbon atoms. This can only be found by interpreting
the intensity measurements. In the original paper the best measurements available

* ' Roy. Soc. Proc.,' A, vol. 89, p. 486.

266 PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

were only rough, and the mode of interpretation was not very certain. Nevertheless
it was possible to say that x/d (see fig. 15) was about 0'30.

The measurements here shown are much more complete and accurate. They
indicate that the value of x/d should be put a little lower, say 0'25 to 0'27. Perhaps
this appears most clearly from the (ill) planes. These are of somewhat peculiar
structxire, as shown in fig. 14. Whatever may be the value of x/d, planes of
calcium and of CO alternate with each other ; but oxygen planes interleave them.
If x/d = 0'25 they bisect the distances between the Ca and CO planes. This would
account for the strength of the fourth order spectrum (21). The smallness of the
second order (100) also is in accordance with this view. There can be little doubt
that the 0 atoms are very nearly a quarter of the way from one carbon atom to the
next. The figure shows the arrangement of atoms in the planes on this supposition.

Now it is remarkable that in two cases we have different sets of planes with the
same or at least commensurable spacings. The second order (ill) occurs at almost

exactly the same angle as the first (211). Tins enables us to compare the effects of

differing arrangements of atoms apart from the effect of differing angles of reflection.

The first order (ill) is due to the full spacing of the (ill) planes, viz., 2'83 A.U. ;

o

but the second order is due to half this spacing, viz., 1'41 A.U. In the former the
Ca and the C()3 in each molecule are in opposition, in the second in concurrence.
The whole weight of every CaCO3 unit (40 + 12 + 48 — 100) is thrown into the scale
in the formation of the latter spectrum. Its intensity is 58.

The first order (211) is due to the same spacing. In this case the planes may be

o

divided into two groups : first, a set of CaCO planes with a spacing 1'44 A.U. ; and,
secondly, a set of O planes with a regular spacing of 0'72 A.U. The latter have no

effect at all at the point where the first (211) spectrum occurs. The whole intensity
of the latter, viz., 38, is due to the CaCO in each molecule, that is, to a weight of 68
out of the 100. The intensities of the two spectra are proportional to the weights
contributing to them for 68/100 = 0'68 and 38/58 = 0'66.

It is to be observed that the conditions for this comparison are excellent ; the
angles of reflection are the same, so that there is no need to consider anything that
depends on this angle. Also the number of molecules swept over by the X-rays is
the same in each case. The only difference is that in one case the whole molecule is
in action, in the other, part of it.

A second opportunity of comparison occurs in the (ill) and (110) spectra. The
first order spectrum of the latter is due to the CaCO in each molecule, as in the

— 0

(211) case, and to a spacing 1'90 A.U. ; its intensity is 61. The second spectrum of
the (ill) is due to the same spacing. The Ca and CO planes may be considered to
act together, with the O planes in opposition. The mass 68 is acting against the
mass 32 ; the balance 36 may be considered as the origin of the second (ill)

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE. 267

spectrum of intensity 34. Here, again, the intensities are very nearly proportional to
the amounts of the masses concerned.

Assuming that these cases illustrate a general principle, let us apply it to a case
where spacings differ as well as plane composition. Consider the various orders of
the (100) spectra, The first order corresponds to a spacing 3 02 A.U. and is due to
CaCO planes only ; the remaining oxygen planes are regularly spaced at an interval
0'151 A.U. and contrihute nothing to the first order spectrum. The second order
spectrum is due to CaCO planes with the O planes in opposition. The third order is
due to CaCO planes only, and the fourth to CaCO and O planes acting in conjunction.
Thus the effective masses are in the ratio 68 : 36 : 68 : 100. The normal rate of
decline is about 100 : 20 : 7 : 3. When these two causes of variation of intensity are
superimposed we. obtain the series 68x100 : 36x20 : 68x7 : 100x3, which may be
expressed as 149 : 16 : 10'5 : 6'6, the first being put equal to 149 for the sake of easy
comparison with the experimental figures. It will be seen that, on the whole, there
is a good agreement. The third order does not compare very well ; but the
explanation of the small second and large fourth, as compared with a normal series
of spectra, is very clear.

It may be pointed out that the (211), (100), and (110) planes must have exactly
similar spacings and distributions of weight no matter what value we adopt for x/d.
This is obvious when we consider that they all intersect the (ill) planes in successive
lines such as nm, ore, pq of fig. 15, their inclinations to the (ill) planes being the
only variables. In the figure the plane nopqcrn is a (ill) plane, nfiam a (100) plane,

ngbm a (llO) plane, and EoCe a (211) plane. The last contains the axis and is
perpendicular to the (ill) planes. *

Now if we compare the intensities of the higher orders of these planes, we find
they are not quite alike. For instance, 149 : 18 : 15 is not the same as 61 : 6 '4 : 2'8.

The differences are accounted for in part at least by the effects of temperature
which always affect the higher orders more than the lower. The normal decline of
100 : 20 : 7 : 3 is only a rough average for a moderate spacing; if the spacing is
small and the spectra are consequently found at higher angles, their decline in
intensity is more marked. We should expect the (110) spectra to fall off more
quickly than the (100) spectra, although the two sets of planes are exactly similar.

Probably, however, there are other minor causes at work ; we shall presently come
to a suggestion of one at least. The point is that putting aside these small, but real
and perhaps important discrepancies, we do succeed in accounting very satisfactorily
for the large differences in the intensities of the different orders. Perhaps this is
best seen from fig. 16. Here all the intensities are grouped together in one diagram ;
the crosses indicating the amounts as transferred from fig. 14. In each case
the intensity has also been divided by the fraction of the molecule which contributes
to it. For instance, we divide 149 by 68, 18 by 36 in the (100) series ; the first 34

268

PKOF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE.

in the (ill) series by 12 ; the second 34 by 36, the 37 by 12, and the 21 by 100, and
so on. The results are indicated by the dots, which, it will be observed, follow closely
a regular curve.* We may be said to have allowed for differences in the masses
contributing to the various spectra, and to be free to attack the question of the cause
of the rapid change in intensity as we proceed to higher orders.

As far as regards the investigation of crystal structure an explanation is not
immediately pressing. We know that it exists, and the law of the decline is not
very different in different cases. We can assume it to be expressed by the
series 100 : 20 : 7 : 3 ; and we can interpret the variations from the normal values by
means of the principle we have discussed above. Finding in this way the distribution
of the atomic masses in the various planes, we have data sufficient for the determina-
tion of crystalline structure. But it is of course unsatisfactory from a physical point
of view to leave the question at this point.

;

f

t

1

Uli

*

1 1

v I

t t « i

III 10

) ill lib no IQ

o 111 lib ic

o iii no K

to in no no

no

Fk. 16.

I think that an ample explanation of the rapid diminution of intensities is to be
found in the highly probable hypothesis that the scattering power of the atom is not
localised at one central point in each, but is distributed through the volume of
the atom.

We may take the analogy of a diffraction grating. If the transparent portions are
very small compared to the opaque, the different orders of diffracted spectra are all
equal in intensity. (KAYLEIGH, 'Phil. Mag.,' 1874, XLVII, pp. 81 and 193.) But
if the transparent portions are not relatively small, the higher orders tend to be
weaker than the lower. And again, if the transparent portions are not sharply
defined, but have " fuzzy" edges, the higher orders are much diminished in intensity,
as A. B. PORTER has shown by actual experiment (' Phil. Mag.,' Jan., 1905).

* Note, Jurw 7. — I have not been able to account by this principle alone for all the relations between
the intensities of the spectra of every crystal hitherto examined : for instance, iron pyrites. But it is
most likely that different atoms have different distributions in space and we have not yet taken account
of that. Moreover, no case has yet occurred in which the opportunities for comparison are so direct as in
calcite, with its different sets of equally spaced planes.

PKOF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE. 269

We should expect the scattering centres of the atom to be not only diffused
through its volume, but also to be less dense at the edges than at the centre, thus
producing exactly those conditions which would reduce greatly the intensities of the
higher orders of the spectra.

I have attempted in what follows to make a quantitative estimate of the operation
of this principle, but it must be regarded as no more than approximate and provisional.
It is based on methods used by RAYLEIGH, SCHUSTER and A. B. PORTER.

Let the density of the scattering centres in a stratified medium be given by

PO+/J! sin 27T . ^, where d is the spacing. We might call this a simple harmonic
ct

stratification, or for brevity a harmonic medium.

As we know already, reflection of radiation of wave-length X cannot occur except
at angles given by n\ = 2d sin 0.

Let us consider the amount of the reflection at these different angles. The amplitude
of the wave reflected to any point P by the stratum dx may be written as

proportional to

x\ • f 2x sin 6\ j

(p + Pi sin 2-n- -j ) sin ( <j> — %TT . - -)dx,

\ (*/ X /

where account is taken both of the density of the scattering centres in the layer dx,
and of the loss of phase due to the depth of the stratum below the surface.

Fig. 17.

It seems to me that we do not contradict our principle that the intensity of the
reflected pencil is proportional to the mass concerned when we here take the amplitude
of the reflection by the layer to be proportional to the number of scattering centres or
the mass of the layer, although the intensity of a vibration is proportional to the
square of its amplitude. The amplitude, here considered, of the reflection is no doubt
only one of the factors which determine the intensity. The reflection is not directed
strictly in one direction, though it is more and more nearly so the more centres there
are in a layer. If we imagine the number of scattering centres iii a layer to be
gradually increased, the maximum amplitude is increased proportionally, but the
intensity of the whole reflection does not increase at the square of this rate, because
it is being continuously limited in its divergence from the true direction of reflection.
As we know from our experiments, the intensity measured, as we have measured it, is
proportional to the mass concerned, other things being the same. In our present
discussion we estimate the amplitude of the reflection disturbance at its maximum.

VOL. CCXV. A. 2 N

270

PEOF. W. H. BEAGG ON X-EAYS AND CEYSTAL STEUCTUEE.

As will be seen presently, we are examining the circumstances when this is zero, and
we are at liberty to conclude that when this is zero there is no reflection at all, which
is all that we require.

The amplitude of the wave at P, as made up of reflections from all the strata within
one period, x = 0 to x = d, is

I"

Jo

• o • .

! Sin 2-7T -r Sill rf> — ZTT .

dj

7

ax.

Thus

Remembering that 2 sin $/A = n/d, we see that this vanishes unless n = 1.
a harmonic medium can reflect at one angle only, not at a series of angles.

If we know the nature of the periodic variation of the density of the medium
we can analyse it by FOURIER'S method into a series of harmonic terms. The
medium may be looked on as compounded of a series of harmonic media, each of
which will give the medium the power of reflecting at one angle. The series of
spectra which we obtain for any given set of crystal planes may be considered as
indicating the existence of separate harmonic terms. We may even conceive the
possibility of discovering from their relative intensities the actual distribution of the
scattering centres, electrons and nucleus, in the atom ; but it would be premature to
expect too much until all other causes of the variations of intensity have been
allowed for, such as the effects of temperature, and the like.

There is no harm, however, in ignoring these considerations for the present, in order
that we may examine the working of the principle. We believe these disturbing
causes to have no great effect.

Let us imagine then that the periodic variation of density has been analyzed into a
series of harmonic terms. The coefficient of any term will be proportional to the
intensity of the reflection to which it corresponds. This may be justified in the
following way.

Suppose the density of the medium to be represented by the ordinates of the
harmonic curve in fig. 18 ; we may look on all the matter represented by the
area below the dotted line as inoperative. It is only the part represented by the

1

\ /

\ f

,'

\J

\J

A
7)

4 d \

if
c

Fig. 18.

Fig. 19.

corrugations above that are effective. The number of effective centres is proportional
to the area above the dotted line. If we compare the effect with that of a deeper
corrugation as in fig. 19, where the amplitude has been increased, the number of

PKOF. W. H. BRAGG ON X-KAYS AND CRYSTAL STRUCTURE.

271

effective centres is increased in the same proportion. We suppose the same number
of corrugations to be acting in each case.

This is practically the very condition of the comparisons we made in certain cases of
calcite.

So also if we compare figs. 18 and 20, where the amplitudes are the same but the

C

Fig. 20.

spacings are narrower, we may expect the intensities to be proportional to the
spacings, when the number of corrugations is the same, because the number of
effective centres varies in that proportion.

Suppose that the density, or the number of the effective centres, in various strata
of an atom perpendicular to a given direction is given by the curve in fig. 21, where
QOP is drawn in the given direction, and MN represents the stratum density at the
distance ON from the centre.

Consider a crystal formed of atoms of one kind, arranged in regularly spaced
planes, which all contain the same number of atoms to the unit of area. One atom
from each plane will suffice to represent that plane. If, therefore, the density
distributions of atoms from the different planes are represented by the curves
QCP, Q'C'P', &c. (fig. 22), the periodic density of the whole crystal is represented by

Fig. 22.

the upper wavy curve formed by adding together the curves of the separate atoms.
The distance from crest to crest is what we have called d, the spacing.

Suppose this curve were known and a harmonic analysis made of it. We should
obtain a series of harmonic terms having periods d, d/2, d/3, ....

2 N 2

A

A'

272 PEOF. W. H. BEAGG ON X-EAYS AND CEYSTAL STEUCTUEE.

Let the curve in fig. 23 represent one of these terms, in fact, the second. The
existence of this term implies that the medium, so far as regards the reflection
of a wave in the second order, may be looked on as a harmonic
medium of amplitude AA'. The rise and fall of the curve above
the line AB implies that the medium has a certain distribution
of density which will give a second order spectrum if the rays
fall on the medium at the proper inclination to the planes. The
area ABCD represents the mass of one atom (there being one
jj,.(r L atom to eacli spacing) ; and of the whole, the portion A'B'CD is

ineffective so far as this reflection is concerned. The ratio of

AA' to AD represents the fraction of the atom which is effective. Now the X-rays,
however they strike the medium, always traverse the same number of atoms. The
ratio of AA' to AD therefore represents that portion of the medium which is effective
in the production of the spectrum, and must be proportional to its intensity.

If the distance apart of the planes is not varied the amplitude of the harmonic,
that is to say, AA' must be proportional to the intensity. Experiment shows that
after allowing for temperature effects the intensity is nearly proportional to the
inverse square of the order. The periodic function which represents the density of
the medium must therefore be of the form

cos lirxld cos 4TTX/d cos nirxld

constant + - —^ -^-i - + ...- —^ + ...

(Q_ \3

= constant + ' — — , where 6 is put for 2-n-x/d,

that is to say, a series of parabolas arranged as in the upper curve of fig. 22.

We have, therefore, to find such a form of density curve for the individual atom,
that when it is combined with others the resulting curve is a series of parabolas, or
something very near to it.

Suppose the density of the atom to be represented by y = ke~" for x positive, and
y = JceCI for x negative. Of course, this implies that the atom is of infinite extent, but
it will appear that in practice no real disadvantage arises.

The ordinate of the compounded curve is given by

y = k{e-cz + e-ct-'Jae + e-cz-'af + ... e<*-*" + eei-'M + ... }

e-cz + ac + ^x-ac

= k — - , where 2a = a.

e™_e-ac

This is very nearly parabolic in form, as can be found on trial, except when the
individual exponential curves of fig. 22 do not overlap sufficiently.

PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE. 273

Proceeding in the usual way, we find that this function from 0 to 2a can be expressed
by the Fourier series

2k

ac I ac +TT a ac + n TT a

If a2c2 is small compared to «V2, the harmonic coefficient is nearly proportional to
l/n2, so long as a is unchanged. That is to say, the intensities of the various orders
for a given spacing are inversely as the square of the number of the order, which
means no more than that the exponential curve we chose to express the variation of
atomic density was sufficiently correct. It gives a compound curve whicli is nearly
a series of parabolas.

The quantity ac is really a measure of the overlapping of two atoms, because the
stratum density of an atom in the plane containing the centre of the next is ke''2ac.
If, for example, e"2™ = O'Ol, ac = 2" 3 ; if e-2nc = O'l, ac = 1'15. Overlappings of this
magnitude are quite to be expected, because unless the planes are very widely
separated, the atoms of one plane penetrate some distance into the interstices
of the next.

If ac = 1'15 the values of the coefficient a2c2/(«2c2 + wV2) for increasing values of n
are 1 .* , , ^-^, 9 \} 4 , -, .-,%.,» or very nearly as the inverse squares of 1, 2, 3, 4.

If ac = 2 '3 the departure from the inverse square law is still not very great. The
numbers are then Y^~», 4^T»> T^l.> 1613.,.

Now let us consider the effect of altering the spacing. As we bring the planes
together, let us say, the area ABCD of fig. 23 must always represent the weight of one
atom because there is one atom to each spacing. The amplitude of any order, say
the nth, is

OZ. ri'*f£

—.A Lv L/

ac ' a2 "

The effective portion of the atom is therefore

2k aV

AD x ac '

of the whole. But AD . x a = the mass of the atom. Also the mass of the atom

2k

r

= 2 lce~"
Jo

-"dx

c

Hence, finally, the fraction of the atom which is effective is aV/(a2c3 + nV3).

Now we have to explain two things : in the first place, the fact that for a given
spacing the intensities of the various orders fall off as the inverse square, in the second
the fact that the intensities of two spectra of the same order belonging to different
spacings are proportional to the squares of the spacings. . For in practice we find

274 PEOF. W. H. BRAGG ON X-EAYS AND CRYSTAL STEUCTURE.

that the intensity of any spectrum is proportional to l/sin20, where 6 is the glancing
angle, no matter whether the spectra belong to one or to several spacings of the planes.
This involves both the facts just stated.

We chose such a form of the density curve for the single atom that we could
account for the first fact. The important point is that without further hypothesis
we explain the second. The formula we have obtained shows that when a2c2 is small
compared to nV the intensity varies inversely as n2 when a is constant, and as a2 when
n is constant.

Thus our hypothesis is self consistent. It does not seem unreasonable. If it turns
out to be true, though there is much to be done before its truth can be considered
proved, it seems to offer an excellent means of determining the distribution of
electrons in the atom.

One or two subsidiary points may be considered very briefly.

I have considered the case of a crystal in which the atoms are all alike and the
planes are spaced regularly. It is easy to make the proper changes when more
complex cases are considered.

When a\r is not small compared to «V2 which would be most likely to happen
when n = 1, and a or c, or both a and c are large, that is to say for reflections at small
angles when there is little overlapping of atoms, then the intensity should be smaller,
in comparison with other intensities, than is indicated by the inverse square law. It
may be a misleading coincidence, but certainly this effect sometimes appears, for

example, in the first (ill) of calcite : perhaps, too, in the first (100) of zinc blende.

An atom for which c is large — a " condensed " atom — should on that account give
relatively stronger reflections in the higher orders. The high order intensities can be
measured with considerable accuracy, and their comparison should be interesting.

L 275 j

X. Gaseous Combustion at High Pressures.

By WILLIAM ARTHUR BONE, D.Sc., Ph.D., F.R.S., formerly Livesey Professor oj
Coal Gas and Fuel Industries at the University of Leeds, now Professor oj
Chemical Tecltnology at the Imperial College of Science and Technology,
London, in collaboration with HAMILTON DAVIES, B.Sc., H. H. GRAY, It.Sc.,
HERBERT H. HENSTOCK, M.Sc., Ph.D., and J. B. DAWSON, R.Sc., formerly of
the Fuel Department in the University of Leeds.

Received December 22, 1914,— Read February IS, 1915.

CONTENTS.

Page

PART I. Introduction, description of apparatus and methods 275

„ II. Explosion of methane with less than its own volume of oxygen (with Messrs. HAMILTON

DAVIES and H. H. GRAY) 288

„ III. Experiments showing the relative affinities of methane, hydrogen, and carbon

monoxide for oxygen in explosions, and conclusions arising therefrom as to

the mode of combustion of hydrogen (with Messrs. HAMILTON DAVIES and

H. H. GRAY) 290

,, IV. Experiments on the relative affinities of methane and carbon monoxide for oxygen in

explosions (with Messrs. HAMILTON DAVIES and H. H. GRAY) 304

,, V. Experiments -with mixtures of ethylene, hydrogen and oxygen of the type

C..,H4 + 0.2 + a:H2 (with Messrs. H. H. GRAY and J. B. DAWSON) 308

„ VI. Experiments on the explosion of ethane with its own volume of oxygen (with

Dr. H. H. HENSTOCK) 310

,, VII. Pressure experiments (with Messrs. H. H. GRAY and J. B. DAWSON) 313

INTRODUCTION.

IN continuation of my previous work upon the mechanism of combustion,* I was
enabled, in 1906, with the aid of funds provided by the Government Grant Committee
to instal at Leeds University a complete apparatus for the study of- gaseous

* 'Trans. Chem. Soc.,' 1902, vol. 81, p. 535; 1903, vol. 83, p. 1074; 1904, vol. 85, pp. 693 and 1637;
1905, vol. 87, pp. 910 and 1232 ; 1906, vol. 89, pp. 652, 660, 939 and 1614.

VOL. CCXV. A 532. 2 O [Published July 31, 1915.

I

276 PROF. W. A. BONE AND OTHERS ON

explosions under high initial pressures. The present memoir contains a description of
the installation, together with the principal results obtained therewith up to the time
of my leaving Leeds in 1912, when the apparatus was removed to the Imperial
College of Science and Technology, London.

My previous work (loc. cit.) had enabled me to put forward a new theory of the
mechanism of hydrocarbon combustion, based on an experimental study of the whole
range of conditions between slow combustion at relatively low temperatures and
explosive combustion (including detonation) at initial pressures of between 270 and
1180 mm.

It was important, from several points of view, to carry the matter still further by
examining the behaviour of certain explosive mixtures (which may be regarded as
"crucial mixtures" in reference to the various theories of hydrocarbon combustion
recently under discussion) when exploded under high initial pressures.

In addition to the above, important information has been gained as to the
distribution of oxygen between methane and hydrogen or carbon monoxide, respectively,
when mixtures containing insufficient oxygen for complete combustion are fired under
high initial pressures ; from such facts can be deduced the relative affinities of these
gases for oxygen in flame combustion, as well as certain conclusions concerning the
mechanism of the combustion of hydrogen and of carbon monoxide.

Finally, a study of the pressure curves obtained in a series of experiments in
which methane, hydrogen, and carbon monoxide, respectively, were separately
exploded with sufficient air to complete the primary oxidation in each case, has
definitely proved that there is no direct (if any) relation in such cases between the
relative times required for the attainment of the maximum pressure and the relative
affinities of the respective combustible gases for oxygen.

The points adumbrated in the foregoing paragraphs will be discussed more fully in
the paper. — W.A.B.

PART I. — APPARATUS AND EXPERIMENTAL METHOD.

The apparatus was specially designed by Prof. J. E. PETAVEL, F.R.S., for the
accurate investigation of the mechanism of gaseous combustion under high initial
pressures, using mixtures of known composition whose constituents have been
prepared on a laboratory scale in a considerable degree of purity. The complete
installation includes, therefore, means and apparatus (l) for preparing, purifying, and
storing fairly large quantities (25 to 50 litres) of the various gases ; (2) for separately
compressing each gas into steel cylinders up to pressures of between 50 and 100
atmospheres; (3) for exploding accurately prepared mixtures under known initial
pressures in steel bombs of different shapes and capacities so as to obtain reliable data
for the interpretation of the chemical changes involved ; and (4) for obtaining complete
graphs of the pressure changes involved in the explosions of simple gaseous mixtures
of known chemical composition.

GASEOUS COMBUSTION AT HIGH PRESSURES. 277

(A). Preparation and Purification of Gases.

The gases employed in the experiments were prepared in quantities of from 25 to
50 litres at a time, briefly as follows : —

Hydrogen by the action of pure dilute sulphuric acid upon the electrolytic
"crescent" zinc (guaranteed 99'98 per cent, purity), supplied by Brunner Mond
and Co., Ltd. ; the gas was thoroughly washed through a series of Erlenmeyer flasks
containing a hot alkaline solution of potassium permanganate, which treatment
reduced hydrocarbon impurity to a negligible point.

Carbon Monoxide by dropping commercial formic acid (95 per cent.) into warm
sulphuric acid and subsequently scrubbing the gas in a coke tower with a spray of
strong caustic soda solution.

Methane* by the action of dilute hydrochloric acid upon aluminium carbide, and
subsequent liquefaction of the washed and dried gas in a cylindrical glass receiver
immersed in liquid air. The liquid hydrocarbon was finally fractionated, the first and
last tenths being rejected.

Ethane* by the decomposition of zinc ethide (supplied by K.AHLBAUM) with water,
and subsequent liquefaction of the washed gas and fraetionation of the liquid, as in
the case of methane.

Etliylene* by the interaction of ethyl alcohol and syrupy phosphoric acid, of specific
gravity 175, at 200° C. to 220° 0. (NEWTH'S method. 'Trans. Chem. Sue.,' 1. 901,
vol. 79, p. 915), and subsequent liquefaction and fraetionation, as in the cases of
methane and ethane.

Oxygen by gently heating recrystallised potassium permanganate and washing the
resulting gas with strong caustic potash solution.

The purified gases were collected over a mixture of equal volumes of glycerine and
water in large glass holders each of 15 to 20 litres capacity, from whence they were
subsequently drawn into the cylinder of the compressing pump.

(B). The Compression Cylinder and Pump.

This part of the installation provides for the separate compression of each gas in
a steel cylinder over glycerine and water. Fig. 1 represents the cylinder, A (internal
capacity about 2|- litres), which was machined out of a mild steel forging and closed
at either end by a special type of valve. The upper valve, B, serves for the admission
of the gas from the holder, and is so constructed that as soon as the cylinder is full
of gas the holder may be shut off by a movement which simultaneously opens
connection with one or other of the high-pressure storage cylinders into which the
gas is sent after compression. The lower valve, C, is provided with fairly wide

* The purity of the above-mentioned hydrocarbons was in each case established by a careful explosion
analysis with an excess of pure oxygen.

2 o 2

'278

PROF. W. A. BONE AND OTHERS ON

passages to facilitate the outflow of the compressing liquid into a special graduated
reservoir underneath the experimental table during the" inflow of the gas from the
holder ; the same valve on being closed to the said reservoir simultaneously opens
connections with the compressing pump.

TO £0 LITRE GAS HOLDER

TO STORAGE CYLINDER

mo SAUGE THROUGH

3- WAY CONED VALVE.

FROM OPPRESSING PUMP

TO KSCKVOIK FOR COMPRESSING U9UIO •

Fig. 1.

The compressing pump (fig. 2) is constructed out of an iron casting into which is
fitted a bronze valve body and piston. It has two valves— inlet, A, and outlet, B -
and also a third valve, C, for releasing any gas which may accidentally become
imprisoned in the head of the pump. The valves are all of bronze and easily
accessible for cleaning purposes. The piston, which is worked by means of a,

GASEOUS COMBUSTION AT HIGH PRESSURES.

279

lever about 12 inches long, is 0'5 inch in diameter with a stroke of 1'5 inches, and it
is easily capahle of giving a pressure of 100 atmospheres, if required.

The function of the pump is to raise the mixture of glycerine; and water from the
reservoir below the table (into which the liquid is discharged by gravity flow during
the admission of the gas into the compression cylinder) and to force it into the
compression cylinder, whereby the gas is compressed and also transferred under
pressure into its proper storage cylinder. The mixture of glycerine and water is thus
kept in constant movement during the cycle of operations involved in the compression
of the gas, being alternately run out from the bottom of the compression cylinder as

/7

Fig. 2.

Fig. 3.

each successive charge of gas is being drawn into the system from the large storage
holder, and afterwards pumped back into the compression cylinder during the
compression of each charge and its transference under pressure to its small storage
cylinder. This cycle of operations is controlled by the two valves of the compression
cylinder and the entrance valve to the storage cylinder.

(C). The Storage Cylinders.

The storage cylinders, of which there are four in the installation, have each a
capacity of about 500 c.c. ; the body of each cylinder (fig. 3) is constructed out of a
mild steel forging, and is fitted with a bronze-coned valve, A, provided with a union

280

PROF. W. A. BONE AND OTHERS ON

joint, B, and controlled by the handwheel, C. Each cylinder is held by two studs in a
cast-iron base which is bolted down to the experimental table. The joints and valves
of the cylinders are so accurately made that even hydrogen may be stored in them at
a pressure of 100 atmospheres for months together without appreciable loss.

(D). The Explosion Bombs.
Two explosion bombs are included in the installation, the one of cylindrical bore

TO HIKING WLVC MO
STHNDHRD BOi/fOON
PtCSSVKC

Fig. 4.

and approximately 103 c.c. internal capacity, and the other having a spherical cavity
of approximately 275 c.c. capacity, as follows :—

(l) The Cylindrical Bomb (Bomb A, fig. 4) was machined from a solid forging of
open-hearth steel and accurately bored along its axis so as to make an explosion
chamber, A, 1 inch in diameter and 8 inches long (capacity circa 103 c.c.). It is

GASEOUS COMBUSTION AT HIGH PRESSURES.

281

fitted with two (upper and lower) coned valves, B and C ; by means of the upper
valve, B, connections are made, through the union joint, D, and an external four-way
"mixing valve" (shown in plan in fig. 8 and in elevation in fig. 7) with (l) a
standard Bourdon pressure gauge and (2), each of three storage cylinders. The lower
valve, C, serves as a convenient outlet through which samples of the various gaseous
mixtures and products were collected for analysis in glass tubes over mercury.

The shell of the vessel is bored, about an inch below the top valve, for the reception
of a special joint, E, carrying the ignition plug. This plug is insulated by means of
an ivory cylinder through whose axis there passes a steel spindle, the end of which
forms one electrical pole, the body of the explosion vessel forming the other.

Fig. 5.

Ignition is effected electrically by the fusion of a short length of thin platinum wire
bridged between the spindle and the outer steel screw of the plug.

All joints in the explosion vessel are " metal to metal " and are so accurately
machined as to be absolutely tight at a test pressure, of 1000 atmospheres. The bomb
itself is securely held in a vertical position by four bolts in a massive iron casting,
which in turn is bolted down to the experimental table. In order to ensure the
perfect mixing of the constituents of a given explosive mixture before firing it, the
explosion chamber contains a perfectly smooth bronze sphere of £ inch diameter
which, after the gases have been introduced, is caused to roll up and down the

282

PKOF. W. A. BONE AND OTHERS ON

chamber about 200 times by subjecting the vessel when in a horizontal position to a
slight rocking motion. This is an important precaution in experiments of this
kind, the omission of which may vitiate the results owing to imperfect mixing of the
charge and stratification effects.

(2) The Bomb with Spherical Explosion Cavity (Bomb B, figs. 5 and 6) was like-
wise machined from a solid forging of open-hearth acid steel, out of the centre of which
was cut the spherical cavity, A, 3 inches in diameter (capacity approximately 275 c.c.).
The body, B B, of the bomb is cylindrical in outline, its axial length being 10 '2 5 inches,
and its diameter 8 inches ; it is mounted on a cast-iron stand by means of ball bearings

B

Fig. 7.

Fig. 6.

Fig. 8.

which permit of a rapid rotational motion of the bomb on its axis in order to ensure an
effective mixing of its gaseous contents before an explosion. C, C, are steel stops
which may be inserted against the plugs, D, D, when it is desired to keep the body
of the bomb at rest.

The bomb is fitted with the coned valve, E, and a plug, F, and also with a special
joint, G, carrying an insulated ignition plug, H, of similar design to that used in the
case of the aforesaid cylindrical bomb A. All joints are metal to metal, and capable
of holding up a pressure of 1000 atmospheres, which was the testing pressure to which
the bomb and all its fittings was submitted.

GASEOUS, COMBUSTION AT HIGH PRESSURES.

283

(E). The Standard Bourdon Pressure Gauges.

Three standard pressure gauges, of the Bourdon type, made by Messrs. Schaffer and
Budenberg, Ltd., of Manchester, registering up to 30, 60, and. 90 atmospheres,
respectively, were used during the research to measure the pressures of the original
mixtures fired, and of the cooled products after an explosion. These gauges were
re-standardised at frequent intervals during the research by direct comparison against
a mercury column at the makers' factory. The installation also includes three other

A ZO LITRE qAS HOLDEf

B. COMPetSSINS CVLINOEff

C. STORAGE VESSEL

D. 3 WAV CONED VALVE

L 100 ATMOSPHERE BOURDON

F. COMPRESSING PUMP

AND WATER-

Fig. 9.

Bourdon gauges, each registering up to 100 atmospheres, for ascertaining the pressures
in the various storage cylinders at any time.

(F). Diagram and Plan of the Installation.

The general arrangement of the installation will be understood from figs. 9, 10, and
11, respectively, which show diagrammatically, the relation of its various parts, and
the plan of the experimental table and its connections. Fig. 9 shows, in elevation,
the arrangement for compressing the gases into their respective storage cylinders.

VOL. CCXV. — A. 2 P

284

PROF. W. A. BONE AND OTHERS ON

Figs. 10 and 11, show, in plan and elevation respectively, the general arrangement
and connections of the storage cylinders, valves, mixing valve, and explosion bomb on
the experimental table.

C STORAGE VESSELS

D. 3 WAY CON£O VALVC3

E tOO ATMOSPMtKE flOUWOON QAUQE3

H MIXINq VALVE

K STANDARD BOi/f?DON

(-- BOMB. A.

Fig. 10.

Figs. 12 and 13, show, in plan and sectional elevation, one of the coned valves
employed in connecting up the various parts of the installation.

C STORAQE VESSELS

0. 3 WAY CONED VALVES

E. 100 ATMOSPHERt BOURPON

H MIXINq VAWC.

K 377VNOU)0 80t/«OON

L. BOMB A.

Fig. 11.

(G). The Recording Manometer.

For the pressure experiments described in Part VII. of the paper, the spherical
bomb, B, was fitted with a recording manometer designed by Prof. J. E. PETAVEL, F.K.S.*

* ' Phil. Trans., 1905, A, vol. 397, pp. 361 to 364 -

GASEOUS COMBUSTION AT HIGH PEESSURES.

285

(fig. 14). Full details as to the experimental procedure in such a case will be found
in the section of the paper referred to.

(H). Gas Sampling and Analysis Arrangements.

Samples of the orginal mixture fired, as well as of the exploded products in each
experiment were drawn off, through one of the coned valves of the particular
explosion bomb used, into tubes over mercury, and were subsequently analysed over
mercury in an apparatus embodying the Franklancl-McLeod principles specially
designed by Prof. BONE for rapid and accurate work.*

The whole of the installation described under the preceding Sections A to D,
inclusive, as well as the recording manometer and the chronograph used in the pressure
experiments described hereafter in Part VII., were made by Mr. C. W. Cook, of

C

Fig. 12.

Fig. 13.

Manchester, from designs kindly furnished by Prof. J. E. PKTAVEF,, F.H.S., to whom
the authors desire to express their obligations. Figs. 12 and 13 (showing the 3- way
coned valve), 14 (the manometer), and 15 (the chronograph), appeared in the memoir on
" The Pressure of Explosives : Experiments in Solid and Gaseous Explosives," published
by Prof. PETAVEL in the ' Philosophical Transactions' in 1905, t and are only reproduced
here for the sake of completeness. Fig. 2 (the compression pump) also appeared in a
later paper published by Prof. PETAVEL in 19084

The two bombs, A and B, used by the authors (figs. 4 and 5) are similar in design to
those shown on pp. 368 and 370 of Prof. PETAVEL'S first memoir (lac. cit.), but differ
from his in dimensions, internal capacities, the arrangements of the valves, and the
position of the firing plug.

* 'Proceedings Chemical Society,' 1898, vol. 14, p. 154.
t ' Phil. Trans.,' A, vol. 205, pp. 357 to 398.
I ' Physikalische Zeitschrift,' vol. IX, p. 75.

2 P2

286 PROF. W. A. BONE AND OTHEKS ON

Experimental Method.

Before each experiment the bomb was thoroughly tested by filling it with either
air or hydrogen up to from 30 to 50 atmospheres pressure, and in no case was an
experiment proceeded with unless this pressure was held without appreciable loss for
a period of twelve hours. At frequent intervals, also, during the investigation, each
bomb was tested by exploding in it mixtures of oxygen with excess of hydrogen (the
initial partial pressure of the oxygen varying usually from 3 to 10 atmospheres). In
the rare event of the observed diminution in pressure exceeding the partial pressure
of the oxygon multiplied by 3'0, a circumstance which might be due either to a
slight outleakage at one of the valves during the explosion or to the occurrence of
rust on the walls of the explosion chamber, the test was repeated until a satisfactory
result was obtained. As an indication of the degree of accuracy obtainable when
such mixtures are exploded in the apparatus, the following summary of the first
22 tests carried out with bomb A, are given below : — •

Limits.

(1) Initial partial pressure of the oxygen varied

between 3 and 10 atmospheres.

(2) Tlatios between partial pressures of the

hydrogen and oxygen in the mixture

fired varied between 3'0 ,, 8'75 „

(3) Ratio of pressure fall on explosion to initial

partial pressure of oxygen varied

between 2'85 „ 3'07*

After a successful preliminary test with either hydrogen or air, and a thorough
drying out of the explosion chamber by means of a current of dry air, the bomb and
its connections, right up to the standard gauge and the storage cylinders, were
exhausted by means of a Geryk pump. It was then filled up to about 5 atmospheres
pressure with one of the constituents (usually the oxygen) of the particular explosive
mixture under investigation, and after blowing off the excess of pressure, the bomb
and its connections were once more exhausted. The first constituent of the proposed
mixture was thereupon slowly admitted to the bomb from its storage cylinder until
the standard Bourdon gauge indicated the desired pressure. The admission valve of
the bomb was then closed and the excess pressure in the outside system between it
and the gauge and storage cylinders blown off by momentarily unscrewing the union
joint nearest to the bomb. The connections up to the bomb were then repeatedly
swept out and finally filled at high pressure (exceeding that of the gas already in the
bomb) with the second constituent of the proposed mixture, which was immediately
afterwards slowly admitted to the bomb, up to its desired partial pressure, by suitable
manipulation of the proper valves. The contents of the bomb were then submitted to

* In 14 out of the 22 experiments this ratio was 3'0 exactly.

GASEOUS COMBUSTION AT HIGH PRESSURES. 287

a thorough mixing process, after which a sample of the mixture was withdrawn
through the bottom valve for subsequent analysis. In cases where the mixture under
investigation was one of three constituents, the first two (one of which was always
the oxygen) were first of all mixed in the bomb in their proper proportions, and this
first mixture was always checked by analysis before the third constituent was added.
Moreover, in all such cases, the special mixing device was brought into play after the
addition of each constituent so as to ensure perfect homogeneity and accuracy in
composition of the final mixture exploded.

Immediately before firing the mixture, the connections between the admission valve
of the bomb and the standard gauge were filled up with oxygen to a pressure just
below that of the mixture in the bomb ; on momentarily opening the admission valve
a little of the mixture in the bomb would pass outwards into the connections, and the
gauge would within two or three seconds record the exact pressure (i.e., to within O'l
atmosphere) of the mixture remaining in the bomb. On again closing the admission
valve a moment later the pressure indicated by the gauge plus one atmosphere (since
the gauge readings were pressures ftborc the atmospheric pressure) would be the
" observed initial firing pressure."

After firing the mixture, the excess pressure in the connections between the
admission valve of the bomb and the gauge was blown off, and then, after opening the
valve of the bomb, the gauge reading was once more recorded. This reading was
subsequently multiplied by a factor (the numerical value of which had been previously
accurately determined by actual trial) representing the ratio of the whole volume of
the bomb plus outside system, up to and including the tube of the gauge (now
occupied by the products of explosion), to that of the bomb ; the reading so multiplied
plus 1'0 atmosphere would thus accurately represent the actual pressure of the cold
gaseous products in the bomb after the explosion.

In cases where the mixture fired in the bomb contained constituents which would
deviate from BOYLE'S law at high pressures more than the diatomic gases hydrogen,
oxygen, and carbon monoxide (which for all practical purposes may be considered as
deviating equally within the limits of pressure covered by the experiments included in
this memoir) the " observed initial firing pressures " were subject to a certain
correction on this account. In the cases of methane and ethylene, the values for PV
as determined by POUILLET for pressures between 1 and 40 atmospheres were used
in calculating the " correction " to be applied in any particular case. In the case of
ethane, however, for which no authentic data could be found in scientific literature, a
set of values were determined by direct comparison of the actual volumes (l) of
hydrogen and (2) of ethane, respectively, measured at 15° C. and 760 mm., required
to fill up bomb, A, to certain definite pressures, e.g., 5, 10, 15, 20 and 25
atmospheres, respectively. The various pressures tabulated in the subsequent part of
this paper represent the experimental pressures after due " correction " for deviations
from BOYLE'S law in each particular case.

288 PROF. W. A. BONE AND OTHERS ON

PART II. — THE EXPLOSION OF METHANE WITH LESS THAN ITS OWN VOLUME OF

OXYGEN.

(With Mr. HAMILTON DAVIES, B.Sc.)

It has been known since DALTON'S time that at ordinary pressures methane cannot
be exploded with much less than its own volume of oxygen. DALTON thus described
the behaviour of a mixture of equal volumes of methane and oxygen on explosion
(New System, Part II., p. 446) :— " If 100 measures of carburetted hydrogen be mixed
with 100 measures of oxygen (the least that can be used with effect], and a spark
passed through the mixture, there is an explosion without any material change of
volume : after passing a few times through lime water, it is reduced a little,
manifesting signs of carbonic add. This residue is found to possess the characters
of a mixture of equal volumes of carbonic oxide and hydrogen."

In the light of my work on the slow combustion of methane* this result is best
expressed by the following equations :—

or, in ol

, = [CH,O + H,0] = CO + H3+H3O.
CO+OHa— ±C03+H3,

>ther words, tlie immediate result of the interaction of the hydrocarbon and
oxygen is the formation of formaldehyde and steam — probably due to the thermal

H

decomposition of di/iydroxymethane H • C • OH. The formaldehyde at once decom-

OH

poses, yielding equal volumes of carbon monoxide and hydrogen, and afterwards, as
the system CO + H2 + HaO is cooling, the reversible change CO + ( )HS ~ ^ CO., + H2
comes into play. It is to be noted that there is no deposition of carbon in such an
explosion.

By employing such high initial pressures as were at our command with the new
apparatus, it is possible to explode mixtures of methane with even as little as half its
own volume of oxygen. The behaviour of a series of mixtures between the limits
2CH4 + Oa and CH4 + Oa has been studied with results of considerable interest.

First Series. (Bomb A.)

Taking the experiments in their proper sequence, "the results obtained by firing
mixtures of methane with as nearly as possible half its own volume of oxygen in
bomb, A, at initial pressures gradually increasing from 8 '5 up to 31 '2 5 atmospheres,

* BONE and WHEELER, 'Trans. Chem. Soc.,' 1902, vol. 81, p. 535; 1903, vol. 83, p. 1074.

GASEOUS COMBUSTION AT HIGH PRESSURES. 289

will first of all be considered. So far as could be judged from the character of the
sound emitted, the explosion at initial pressures up to 16 '5 atmospheres was not very
violent; at 21'69 atmospheres the sound was distinctly more audible, and at 31'25
atmospheres a sharp metallic click, indicative of detonation, was heard. Carbon was
deposited and steam abundantly formed in all experiments ; there was always a
permanent increase in pressure (20 to 30 per cent.) on explosion, the ratio, pjpi,
increasing regularly with the initial pressure. It is also noteworthy that, although
much of the original methane remained unchanged, there was never the slightest
trace of either acetylene or ethylene in the products.
The principal observations are given in Table I.

Second Series. (Bomb B.)

In the next series of experiments (Table II.) the same mixture was exploded, at
initial pressures between 10 and 20 atmospheres in the large spherical bomb, B.

Third Series. (Bomb A.)

In this series of experiments (Table III.) the effect of gradully increasing the
proportion of oxygen between the limits 2CH4 + O2 and CH, + 02 was investigated.
The initial pressure was nearly the same (12 '6 to 1275 atmospheres) throughout
the series. The most noteworthy fact brought to light was the total cessation of any
separation of carbon after the proportion of oxygen exceeded the limit 3CH, + 202.

In interpreting the results given in Tables I. to III. it is important to distinguish
between (l) the primary oxidation of the hydrocarbon, which is an exceedingly rapid
process, and is probably completed during the short interval between ignition and the
attainment of maximum pressure ; and (2) certain probable secondary interactions
whose influence may extend far into the subsequent long cooling period. For it is
only these latter which would be affected by variations in the rate of cooling down
from the maximum temperature.

With regard to the primary oxidation, the results with the mixture 2CH4 + 02 are
obviously inconsistent with any idea of a preferential burning, whether of carbon
or of hydrogen. If, however, it be supposed that the oxygen initially enters
the methane molecule, forming unstable " hydroxylated " molecules, which then
decompose, it becomes necessary to consider the probabilities of the direct formation
of CH3.OH and CH2(OH)2, respectively, and the nature of their respective
decomposition products at high temperatures, together with the magnitude of the
corresponding heat changes.

Incidentally, the " hydroxylation " theory, which the author's previous researches
has shown to be consistent with all the known chemical data concerning hydrocarbon
combustion, is also indirectly supported on the physical side by Prof. W. M.
THORNTON'S recent observations on the electrical ignition of the paraffin hydrocarbons in

290

PROF. W. A. BONE AND OTHERS ON

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GASEOUS COMBUSTION AT HIGH PRESSUEES.

293

air.* Prof. THORNTON'S experiments on the ignition by the continuous current of such
hydrocarbon-air mixtures have proved that if I is the least igniting current and p the
percentage of gas in the mixture, the ratio I/p is nearly proportional to the number
of hydrogen atoms in the molecule, a fact which suggests that the initial attack of
the oxygen is directed upon the hydrogen atoms in the hydrocarbon molecule ; and
inasmuch as any supposition of a preferential burning, in the old sense, of hydrogen
is clearly inadmissible on chemical grounds, the " hydroxylation " theory seems to
be the only satisfactory substitute.

With regard to the thermal decomposition of methyl alcohol, CH3.OH, it now is
definitely established that at high temperatures it is resolved ultimately into carbon
monoxide and hydrogen, without any deposition of carbon or steam formation, in

accordance with the following scheme : — t

H

CO + H2.

And with regard to dihydroxymethane, CH2(OH) 2, there is little doubt but that
at high temperatures it is ultimately resolved into carbon monoxide, hydrogen, and

steam, as follows : —

H H

OH

CO + H2.

If now, the above facts be combined with the corresponding thermal changes,
expressed in kilogram-centigrade units per gram-molecule, the scheme for the primary
oxidation of methane may be represented as follows : —

A. B.

oxidation

wSHa:C:(OH)a
• H3 : C • OH > H2

C.

oxidation

CH4

+ 30

I

+ 59

-22'8

a
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a

a
8

Pi

CO + 2H2
B'

-13'4

CO + H2

C'.

* 'Roy. Soc. Proc.,' 1914, A, vol. 90, pp. 272-280.

t BONE and DA VIES, 'Trans. Chem. Soc.,' 1914, vol. 106, pp. 1691-1696.

2 Q 2

294 PROF. W. A. BONE AND OTHERS ON

From which it is evident that the passage from A to B involves an evolution of
30 heat units, of which 22 '8 would be absorbed if the transition B to B' supervened,
whereas the further passage from B to C involves an evolution of 59 heat units, of
which only 13 '4 would be absorbed during the transition C to C'. Hence, it might
be expected that, whenever circumstances are favourable or permit, the transition A to
B would be immediately followed by B to C, rather than by B to B'. Or, in other
words, there would usually be a strong tendency for a " non-stop " run from A to C
through B, that is, without any thermal decomposition in passing through B. The
last-named, although not precluded, would only occur exceptionally. Moreover, it
may be pointed out that the thermal decomposition of CH3OH(B to B') involves a
slightly greater heat absorbtion than the resolution of methane itself into its elements
(namely, — 22'8 for methyl alcohol, as against —227 for methane) and that, therefore,
heat liberated by oxidation (A to B or B to C) would be just as likely to decompose
any residual methane as it would the methyl alcohol formed in passing through B.
This consideration is of importance in connection with the course of events when
methane is exploded with considerably less than its own volume of oxygen.

Any theory must, however, be capable of accounting for the following outstanding
facts, namely (l) the ultimate formation of large quantities of both carbon and steam
in the explosion of the mixture 2CH4 + 02; (2) the disappearance of carbon from the
products when the proportion of oxygen in the mixtures exploded exceeds 40 per cent.;
and (3) the occurrence of oxides of carbon and steam in the products of all the
mixtures fired.

There is no difficulty in explaining, on the lines of the hydroxylation theory, the
facts comprised under (l) and (3), especially in view of the strong tendency there
would always be for a non-stop passage from CH4 to CH2(OH)2, without any breaking
down at the intermediate CH3.OH stage.

In the case of the equimolecular mixture, CH4 + O2, the primary oxidation may thus
be represented as a single transaction,

2 = [H2 : C : (OH)J = H2 : C : 0 + H30

involving the formation of steam but no separation of carbon. The facts observed
with the mixture 2CH4 + 03, may also be explained on the supposition that in the
primary oxygen attack half of the methane molecules are directly transformed into
CH2(OH)2, and that the heat so liberated is sufficient to decompose part of the other
half of the methane into its elements, the remainder being found intact after the
explosion.

To explain the non-separation of carbon when the mixture 3CH4 + 2O2, was exploded,
it must be supposed that whilst one-third of the methane is direc.tly transformed into
CH2(OH)3, the other two-thirds cannot, for lack of oxygen, get beyond the CH3. OH

GASEOUS COMBUSTION AT HIGH PRESSUEES. 295

stage, and that, therefore, the subequent thermal decompositions are restricted to
those of such hydroxylated molecules as CH3.OH and CH,(OH)2 which do not involve
any separation of carbon.

The above view of the matter is also consistent with another notable. fact, namely,
that in the case of mixtures intermediate between 2CH4 + 02, and CH4 + 02 (vide
Table III.) the minimum proportion of the oxygen appearing as steam in the products
is found, not with 2CH4 + O2, as might at first appear likely, but with 3CH4 + 202.

During the relatively long cooling period, which follows the attainment of the
maximum explosion temperature, the following secondary interactions may come into
play, namely: (a) the reversible change CO + OH2^Z^lC02 + H2 ; and, in the case of
mixtures containing less oxygen than 3CH4 + 202, (fr) the interaction of steam and
carbon C + OH2 = CO + H2; or, possibly (c) some slight interaction between methane
and steam.

In this connection the recently published work of G. W. ANDREW on the " Water
Gas Equilibrium in Hydrocarbon Flames,"* may be cited as proving that in a system
containing only CO2, CO, H2, and H20, rapidly cooling down from the high

temperatures prevailing in hydrocarbon flames, the equilibrium ratio -=— ^ ~ adjusts

itself automatically with the temperature until a point between 1500° C. and
1600° C. on the cooling curve is reached (corresponding to a value K = 4'0, approxi-
mately), after which no further re-adjustment occurs. He also found that the
adjustment of the equilibrium is not greatly influenced even when relatively large
quantities of methane and carbon are found in the final products.

The results of our experiments, 1 to 3 inclusive, with the mixture 2CH4 + O2, where
the initial pressures, 8'5 to 16'5 atmospheres were not sufficient for detonation, and in

which the ratios j-2 in the final products varied between 4'45 and 376 only,

_tI2

confirm ANDREW'S conclusions. But in experiments 4 and 5, where with initial
pressures 2T69 and 31'25, respectively, the mixtures detonated, the much lower ratio

CO x OH

= 2'56 and 2'24, respectively, indicated some intervention of the separated

x -

carbon in the chemical interaction during the cooling period ; there is nothing,
however, in the results suggestive of any appreciable intervention of methane.

This conclusion is also borne out by the following figures relating to the mixture
2CH4+02, which show the ratio "B," which the sum of the oxygen as H2O,
plus half the oxygen as C02 in the products (assuming the C02 to have arisen wholly
from the interaction CO + OH2 ~^I C02 + H2) bears to the total oxygen originally
present in the mixtures fired in our experiments. The deviation of this ratio from
0'5 may be regarded as a measure of the participation of separated carbon in the
secondary interactions during the respective cooling periods.

* 'Trans. Chem. Soc.,' 1914, TO!. 105, pp; 444-456.

296

PKOF. W. A. BONE AND OTHERS ON

EXPERIMENTS with Mixture 2CH4 + 0, in Bomb A.

Experiment No.

Initial pressure in
atmospheres.

k.

R.

1

8-53

4-45

0-473

10

12-74

3-65

0-487

2

13-62

3-76

0-472

3

16-45

3-78

0-450

4

21-69

2-56

0-420

5

31-25

2-24

0-400

In the case of experiments with the mixture 2CH4 + O2, in bomb B, where, owing
to the much smaller wall-surface per unit volume of mixture exploded, the maximum
temperature would be somewhat higher, and the cooling period longer, than in the
corresponding experiments in bomb A, the ratios "R" varied between 0'400 and
0'346.

PART III. — EXPERIMENTS SHOWING THE RELATIVE AFFINITIES OF METHANE AND
HYDROGEN FOR OXYGEN IN EXPLOSIONS, AND CONCLUSIONS ARISING THERE-
FROM AS TO THE MODE OF COMBUSTION OF HYDROGEN.

(With Messrs. HAMILTON DAVIES and H. H. GRAY.)

The observations made during the preceding experiments that there is no
deposition whatever of carbon when mixtures of composition intermediate between
3CH4 + 202 and CH4 + O2 are exploded under pressure, opened up the possibility of
comparing the relative affinities of methane and hydrogen for oxygen under the
conditions of homogeneous flame combustion, in the manner hereinafter to be
described.

The theoretical importance of such a comparison is enhanced by the fact that it
provides not only a basis for the discussion of the question whether or not the known
rates of flame propagation through explosive mixtures of different combustible gases
are governed by their relative affinities for oxygen, but also affords decisive evidence
as to the mode of combustion of hydrogen in explosions.

Ever since Sir HUMPHREY DAVY'S experiments on flame, the combustibility of
hydrogen has been considered superior to that of methane, and, as a matter of fact,
not only is the ignition temperature in air of hydrogen (580° C. to 590° C.) lower
than that of methane (650° C. to 750° C.),* but also the rates of inflammation (i.e., of

H. B. DIXON and H. F. COWARD, 'Trans. Chem. Soc.,' 1909, vol. 95, p. 519.

GASEOUS COMBUSTION AT HIGH PRESSURES. 297

propagation of flame by conduction) of mixtures of hydrogen and air are much
higher than those of methane-air mixtures. Thus, according to LE CHATELIER.*

Hydrogen- Air Mixtures.

Hydrogen per cent. . . 10 20 30 40 50 60 70
Metres per second . . . 0'60 1'95 3'30 4'37 3'45 2'30 I'lO

Methane-Air Mixtures.

Methane per cent 6 8 10 12 14 16

Metres per second .... 0'03 0'23 0'42 0'61 0'36 O'lO

Moreover, also, H. B. DIXON has shown that the rate of detonation of electrolytic
gas, 2817 metres per second, is greater than the fastest rate for any mixture of
methane and oxygen, namely 2528 metres per second for the equimolecular mixture
CH4+02.t

But inasmuch as ignition temperatures and rates of explosion are known to be
governed chiefly by physical factors, such as heats of combustion, specific heats and
molecular weights of products, and (possibly also) ionisation effects, nothing can be
deduced from such data as to the relative affinities of the gases for oxygen. On the
chemical side, however, certain experiments carried out, in the year 1856, by
LANDOLT} in BUNSEN'S laboratory, in which the partly burnt products of a coal gas
flame were sucked off for analysis through a fine platinum tube at different vertical
heights (0, 10, 20, 30, 40, 50 mm.) along the vertical axis of the flame (100 mm.
in height) have been cited as proving the vast superiority of the. affinity of hydrogen
over that of methane for oxygen in flames. LANDOLT, it is true, concluded that with
regard to the different constituents of coal gas : — " Der Wasserstoff ist unter alien
Gasarten diejenige, welche am leichtesten verbrennt, es nimmt daher derselbe auch in
der Flamme am schnellsten ab ; etwas langsamer verschwindet das Grubengas, und
zuletzt kommen die schweren Kohlemvasserstoffe, deren Verbrennung hauptsachlich
erst in der oberen Halfte der Flamme vor sich geht."

The circumstance that LANDOLT employed a platinum tube of narrow bore, fixed
along the vertical axis of the flame, for the withdrawal of the partly burnt products,
is in itself sufficient to vitiate his conclusion, inasmuch as recent researches in my
laboratory have proved that in " surface combustion " (e.g., in contact with firebrick
at 500° C.) the usual order of the affinities of various combustible gases for oxygen
in flames are entirely reversed.

* ' Le9ons sur le Carbone,' p. 279.

t Bakerian Lecture, 'Phil. Trans.,' 1893, vol. 184, pp. 177, 181.

t ' Habilitationsschrift, Breslau,' 1856 ; 'Pogg. Ann.,' vol. 99, 1856, pp. 389 to 417.

298 PROF. W. A. BONE AND OTHERS ON

On the other hand, it has been proved in my previous researches upon hydrocarbon
combustion : ( 1 ) that in slow combustion in borosilicate glass bulbs at temperatures
between 300° C. and 400° C., methane, ethane, ethylene and acetylene are all oxidised
at a much faster rate than is either hydrogen or carbon monoxide,* and (2) that on
exploding such mixtures as C2H4 + H2 + O2 or C2H2 + 2H2 + 02, the hydrocarbon is
burnt in preference to hydrogen, t facts which are at variance with LANDOLT'S
conclusion.

The possibility of deducing from our bomb experiments a direct comparison between
the relative affinities of methane and hydrogen in explosions arose from the fact that
the primary oxidation of methane involves a direct transition from CH4+Oa to
CH2(OH)2, which latter breaks up into, ultimately, CO + H2 + H2O, without any
deposition of carbon. Whence it follows that if mixtures, CH4 + 02 + a;H3, be exploded,
the division of the oxygen between the methane and hydrogen during the extremely
short period of actual combustion (i.e., direct oxidation) may be deduced from the
proportion of the original methane found intact in the final products, provided always
that there is no separation of carbon, which in fact is never observed in such
circumstances.

The experimental method consisted, therefore, in exploding a series of mixtures
CH4 + O2+a?H2, in which the hydrocarbon and oxygen were initially present in as
nearly as possible equimolecular proportions, but in which x (the volume ratio of H2
to CH4) was varied between 2 and 8, and determining from the percentage of the
original methane remaining intact in each case (l) the oxygen distribution when
x = 2, and (2) the influence upon such distribution of successive equal increments of
x up to 8. And in order to determine the possible influence of the walls of the
explosion vessel upon the results, parallel series of experiments were carried out in
each of the two bombs, A and B.

In each case the mixture exploded with a distinctly audible sound, which
diminished in intensity as the proportion of hydrogen x increased, until with the
mixture CH4 + O2 + 8H2 only a faint puff" could be heard. In no case was there any
separation of carbon.

Before discussing the bearing of the results upon the matter under investigation,
it will be convenient to detail and summarise them in the following tabulated form.
(Tables IV. to VIII. inclusive.)

* ' Trans. Cham. Soc.,' 1902, vol. 81, pp. 538 and 539 ; 1904, vol. 85, p. 694.

t Ibid., 1906, vol. 89, pp. 669 and 670, also 'Proc. Royal Institution,' 1908-10, vol. 19, pp. 73
to 87.

GASEOUS COMBUSTION AT HIGH PRESSURES.

299

TABLE IV. — Explosion of Mixture CH4 + O2 + 2H2 in Bomb A.

Experiment No

14

15

16

17

v\

18-004

18-335

19-625

48-85

Vi

18-0136

18-2473

19-454

48-149

Pz/Vi

1 • 0005

0-995

0-991

0-986

{CII
U 4
O~ '

atmospheres.
4-469
8-863

4-672

atmospheres.
4-706
9-007
4-622

atmospheres.
4-834
10-030
4-761

atmospheres.
11-80
24-93
12-12

rco., . .

Partial pressures in gaseous I CO . .
products in atmospheres \ IL> .
lCH4. .

atmospheres.
0-3672
4-2960
13-1300
0-2204

atmospheres.
0-3367
4-2960
13-410
0 • 2046

atmospheres.
0-394
4-470
14-320

0-270

atmospheres.
1-164
10-910
35-520
0-555

CO x OIL

4-01

4-16

3-6

3-0

C02 x H2

Per cent, distribution of"! , pry
oxygen deduced from L° TJ 4 ' '

v, . ntr 1 to Ho .
unburnt CH4 J

95-31
4-69

95-70
4-30

94-74
5-26

95-6
4-4

Per cent.

rtoCH4 = 95-34.
Mean distribution of oxygen <

I to H2 = 4- fao.

VOL. CCXV. — A,

2 R

300

PROF. W. A. BONE AND OTHERS ON

TABLE V.— Explosion of Mixture CH4 + 03 + 4H2 in Bomb A.

Experiment No

18

19

20

21

22

23

i).

20-50

21-206

29-39

31-36

37-07

71-6

jfi

Vt . ...

18-5258

19-3907

26-280

28-14

34-387

67-70

fi

Vt /»!

0-9038

0-9147

0-8942

0-8973

0-9276

0-9456

atmo-

atmo-

atmo-

atmo-

atmo-

atmo-

spheres.

spheres.

spheres.

spheres.

spheres.

spheres.

Partial pressures in original J JT '
mixtures in atmospheres j ^.~

3-51
13-41
3-58

3-888
13-61
3-708

4-79
19-75
4-85

5-20
20-80
5-36

6-26
24-71
6-10

12-1
47-1
12-4

atmo-

atmo-

atmo-

atmo-

atmo-

atmo-

spheres.

spheres.

spheres.

spheres.

spheres.

spheres.

rco,. .

0-1432

0-2011

0-270

0-32

0-358

0-87

Partial pressures in gaseous I CO . .

2-836

2-889

3-780

4-21

4-976

10-86

products in atmospheres j H2 . .

14-840

15-450

21-38

22-68

27-770

53-55

L^-^4

0-7066

0-8506

0-85

0-93

1-283

2-42

CO x OH2

[5-471

3-90

3-23

3-65

3-38

2-93

CO, x H2

Per cent, distribution of"] pry
oxygen deduced from ^ „ 4
unburnt CH4 J t(

80-83
19-17

78-47
21-53

82-65
17-35

83-0

17-0

81-0
19-0

81-0
19-0

Mean distribution of oxygen t

Per cent.
toCH4 = 81-0.
toH2 = 19-0.

GASEOUS COMBUSTION AT HIGH PKESSURES.

301

TABLE VI. — Explosion of Mixture CH4+02 + 6H2 in Bomb A.

Experiment No 24

25

26

27

pl 22-15

41-767

40-923

32-967

p-2 . . 18-20

34-021

33-393

27-229

Pi/Pi 0-8217

0-8145

0-8160

0-8260

atmospheres.
T> <-• i • f"CH4 2-84

atmospheres.
5-448

atmospheres.
5-030

atmospheres.
4-080

Partial pressures in „

30-910

30-800

24-770

original mixture | Q- 9.8,

5-409

5-093

4-117

atmospheres.
{CO., .... 0-11

atmospheres.
0-246

atmospheres.

0-172

atmospheres.
0'152

CO . . 1'45

2-772

2-724

2-340

Ho . . 15 '33

28-28

28-050

23-03

CH4 1-31

2-723

2 • 447

1-707

COxOH,

3-11

4-23

4-01

CO, x H2

Per cent, distribution of oxygen f to CH4 54 • 3
deduced from unburnt CH4 (_ to H2 . 45-7

52-57
47-43

54-2

45-8

59-3
40-7

Mean distribution of oxygen I

Per cent.
toCH4 = 54-9.
toH2 = 45-1.

2 R 2

302

PEOF. W. A. BONE AND OTHEES ON

TABLE VII. — Explosion of Mixture CH4 + O2+8H3 in Bomb A.

28

29

30

?>i

39-135

49-656

50-09

29-9723

38-6634

38-6622

#->/7?i .

0-7659

0-779

0-772

atmospheres.
3-859

atmospheres.
5-038

atmospheres.
5-02

Partial pressures ml TT

31-390

39-580

40-16

original mixture \ r\"

3-886

5-038

4-91

rco2

Partial pressures in I CO
gaseous products | H2

atmospheres.
0-1243
1-104
25-870

atmospheres.
0-1724
1-551
33-570

atmospheres.
0-1592
1-512
33-170

(_GRi

2-874

3-370

3-821

CO x OH2

2-42

2-35

2-16

CO2 x H2

Per cent, distribution of oxygen J to CH4 .
deduced from uhburnt CH* |_ to H-> .

30-0
70-0

33-8
66-2

30-4
69-6

Mean distribution of oxygen -I

Per cent.
toCH4 = 31-4.
to H2 = 68 • 6.

GASEOUS COMBUSTION AT HIGH PKESSUEES.

303

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304

PROF. W. A. BONE AND OTHERS ON

The mean results of the foregoing experiments, so far as the distribution of oxygen
between methane and hydrogen is concerned, are summarised as follows : —

TABLE IX.— Mixtures CH4+0

X.

2.

4.

6.

s:

{ 0., to OIL

Per cent.
95-34

.
Per cent.
81-0

Per cent.
54-9

Per cent.
31-4

BombA {oltoH,
f 0> to CEL

4-66

97-1

19-0
91-0

45-1
72-6

68-6

Boml)B {(itoH,

2-9

9-0

27-4

It is at once evident from the results with the mixture CH4 + 02 + 2H2 that the
affinity of methane is at least twenty to thirty times greater than that of hydrogen
for o.i'i/gen in explosions. The actual distribution of oxygen when a particular
mixture i,s exploded is undoubtedly influenced to some extent b}'' the walls of the con-
taining.vessel but not by the absolute initial pressure. The influence of the containing
walls would presumably disappear after a certain -limiting ratio of area/volume
is attained, and, had the resources at our disposal permitted, it would have been
interesting to have made further experiments with a still larger bomb than B.
An examination of the summarised results in Table IX. leads to the important
conclusion that the influence of successive increases in x, the volume ratio of H2 to
CHi in the mixture exploded, upon the actual oxygen distribution is for a given vessel
proprotional to ,r3. This can hardly mean other than that in explosion flames
hydrogen is directly burnt to steam as the result of trimolecular impacts,

2 = 2H20.

PART IV.- — EXPERIMENTS ON THE KELATIVE AFFINITIES OF METHANE AND
CARBON MONOXIDE FOR OXYGEN IN EXPLOSIONS.

(With Messrs. HAMILTON DAVIES and H. H. GRAY.)

The success of the experiments described in the preceding section led us to make
a similar attempt to determine the division of oxygen between methane and carbon
monoxide when mixtures of the general composition CH4 + O2+o;CO are exploded
under pressure. In this we were not completely successful, owing to a slight
separation of carbon in the explosions whenever x exceeded 3'0 or thereabouts. Pro-
vided, however, that x did not exceed this limit, no carbon was deposited during the
explosion, and such experiments may be given in detail as quite reliable so far as the
question of the oxygen distribution between the two inflammable constituents is
concerned.

GASEOUS COMBUSTION AT HIGH PRESSURES.

305

Explosion of Mixture CH4 + 02 + 3CO in Bomb A.

The results of the following two very concordant experiments in which the mixture
was exploded, in bomb A, at an initial pressure of about 24 atmospheres, may be
recorded as typical of the series. All that could be heard of the explosion was a faint
squeak, and although on afterwards opening the bomb there was a trace of carbon on
the firing wire, no carbon at all had been deposited on the bomb itself during the
explosion. Condensation of moisture from the products could be detected, and the
explosion was accompanied by a small increase of pressure. The results showed that
between 8 and 9 per cent, only of the original oxygen had, during the initial
"oxidation" stage of the explosion, combined with the carbon monoxide; the
remainder had reacted with the methane. The results are tabulated as follows :-—

TABLE X.— Explosion of CH4 + (X + 3CO in Bomb A.

Experiment No.

Pressures in atmospheres < * '
Pi/l'i

atmospheres.

23-0

25-2
1-068

fCH,
Partial pressures in original mixture < CO .

LO, .

rC02
Partial pressures in the gaseous products < TT

LciX

atmospheres.

4-32

14-00

4-68

atmospheres.

2-23

16-85

5-78

0-34

CO x OH,
CO., x II,

Per cent, distribution of oxygen f to CH4
deduced from unburnt CH4 |_ to CO

3-17

92-0
8-0

41

atmospheres.
23-25
24-70

1-063

atmospheres.

4-70

13-85

4-70

atmospheres.

2-29

16-39

5'57

0-45

3-34

91-0
9-0

Explosion o/CH4 + O2 + 6CO in Bomb A.

In three other experiments (Nos. 42 to 44, inclusive), mixtures approximating to
02 + 6CO, were exploded in bomb A, but on account of there being a slight
deposition of carbon during the explosion, which might conceivably have somewhat
affected the accuracy of the determination of the oxygen distribution, we do not
propose to publish them in detail, but for purposes of comparison with experiments

306

PEOF. W. A. BONE AND OTHEES ON

40 and 41, we may give the following summarised figures, from which it will be seen
that there was practically no change in pressure after explosion and that approximately
20 per cent, of the original methane remained intact in the final products : —

TABLE XI.

Experiment No

42

43

44

p\ atmospheres

35-25

38-25

38-95

35 • 30

37-95

38-60

7'9/Pl .

TOO

0-995

0-991

Percentage of the original methane
unburnt

23-9

17-5

20-3

Explosion of Mixture CH, + 0, + 2CO in Bomb B.

The following two experiments with a mixture CHi + 02+2CO, were carried out in
bomb B,. for purposes of comparison with the corresponding experiments with the mix-
ture CH, 4- (X + 2H., ( ride Table VIII., p. 303). There was no deposition of carbon during
the explosion, but some moisture condensed on cooling, and the final pressure was
approximately (J per cent, greater than that of the original mixture. Taking the mean
of the two experiments, 47 per cent, of the original methane remained intact in
the products, as compared with an average of 2 '9 per cent, in the corresponding
experiment with the mixture

TABLE XII.

Experiment No

45

46

p\ atmospheres

20-90

19-44

pi atmospheres ....

22-66

21-33

»,//?! .

1-085

1-097

,-

CH4

5-184

4-870

Partial pressures in the original 1
mixture, atmospheres . . . |

CO
02

10-470
5-248

9-831
4-743

Partial pressures in the gaseous I
products, atmospheres . . ]

C02
CO
H2

1-963
14-110
6-320

1-758
13-260
6-106

CH4

0-266

0-207

CO x OH3

3-64

3-56

C02xH2 '

Per cent, distribution of oxygen 1"
deduced from unburnt CH4 \

toCH4 . . .
to CO . . . .

94-9
5-1

95-7
4-3

GASEOUS COMBUSTION AT HIGH PRESSURES.

307

Explosion of CH4 + O2 + 6CO in Bomb B.

Two experiments were carried out with this mixture in bomb B, but again a slight
deposition of carbon occurred during the explosion, and therefore only the following
summarised results need be given. There was practically no change in pressure in
the cooled products, and approximately 16 to 17 per cent, of the original methane
survived the explosion, as compared with an average of 27'4 per cent, in the corres-
ponding experiments with the mixture CH4 + 02 + 6H2 (vide Table VIII, p. 303).

TABLE XIII.

Experiment No

47

48

pi atmospheres

43-13

42-75

po atmospheres

43 • 50

43-26

IJo/jUi .

1 '009

roi2

Per cent, of the original CH( unburnt .

16-3

17-1

Review of Results.

It is important to compare the results obtained with the foregoing CH.1 + 0:!
mixtures, not only amongst themselves, but also with the results of the corresponding
experiments with the CH4 + O2 + o;H2 mixtures.

In the first place, it is evident that the affinities of either hydrogen or carbon
monoxide are greatly inferior to that of methane for oxygen in explosion flames ;
thus it may be inferred from the experiments with CH, + O2 + 2H2 that the ratio
of the affinities CH4/H2 is of the order 20 or 30 to 1 at least, and probably
higher, if the influence of the walls of the containing vessel could be entirely
eliminated.

Owing to the uncertainty of our knowledge as to the precise mechanism of the
combustion of carbon monoxide in explosions, that is to say, as to the extent and
character of the intermediary action of steam, it is perhaps difficult to assign, even
approximately, any numerical relation between the affinities of methane and carbon
monoxide for oxygen in flames. Nevertheless, it may be pointed out, without laying
undue stress on the fact, that when the mixtures, initially containing methane, hydrogen
or carbon monoxide, and oxygen in stochiochemical proportions (i.e., CH4+ O2 + 2H2 and
CH4+O2 + 2CO), were exploded under similar conditions in bomb B, the carbon
monoxide was apparently more effective than hydrogen in pulling away oxygen from
the predominating affinity of the hydrocarbon. From the theoretical standpoint it

VOL. ccxv. — A. 2 s

308 PROF. W. A. BONE AND OTHERS ON

would probably be well worth while to undertake a further extended study of the
matter in larger explosion vessels than we have employed, although it would be both
a costly and a laborious enterprise.

Whereas, in the case of hydrogen, the influence of successive increments in x,
the volume ratio of the other combustible constituent to methane in the mixture
exploded, upon the actual oxygen distribution is proportional to a;2, in the case of
carbon monoxide, it is more nearly proportional to x. This points to some fundamental
difference between the modes of combustion of the two gases in flames ; thus whilst
the evidence is strongly in favour of the supposition that hydrogen is burnt directly
to steam as the result of trimolecular collisions 2H2 + 02 = 2H2O, the results with
mixtures CH4+Oa + CO seem to require some different supposition, such, for instance,
as an intermediary action of steam.

PART V. — EXPERIMENTS WITH MIXTURES OF ETHYLENE, HYDROGEN AND
OXYGEN OF THE TYPE C2H4+O2+xH2.

(With Messrs. H. H. GRAY and J. B. DAWSON.)

It has long been known that when ethylene is exploded with its own volume ol
oxygen there is a doubling of the volume in the cooled products, with the formation
of principally carbon monoxide and hydrogen, but without any separation of carbon or
condensation of steam, substantially in accordance with the empirical equation

It has also been shown that the addition to such a mixture of hydrogen
sufficient to bring its composition up to C2H4 + 02 + H2, does not cause any separation
of carbon on explosion, although there results a slight condensation of steam on
cooling.*

That the above facts do not really imply a preferential combustion of carbon (as
some have supposed) was proved by a study of the behaviour of a mixture 3C2H4 + 2O2,
which on explosion gives rise to large quantities of both carbon and steam,
together with methane, acetylene, hydrogen and oxides of carbon. Indeed, the
facts harmonise very well with the hydroxylation theory, which would require the

H . C . OH

intermediate formation of monohydroxyethylene , a substance which on

H.C.H

thermal decomposition would yield H20 and j CH residues, the latter subsequently
either (i.) undergoing hydrogenation to CH4 in an atmosphere sufficiently rich in

* BONE and DRUGMAN, ' Trans. Chem. Soc.' 1906, vol. 89, pp. 669 to 671.

GASEOUS COMBUSTION AT HIGH PRESSURES. 309

hydrogen; or (ii.) uniting with each other forming acetylene; or (iii.) decomposing
into carbon and hydrogen, according to the experimental conditions.*

In the presence of a sufficient oxygen supply, however, there is a " non-stop " run
through the monohydroxy to the dihydroxy stage which, on subsequent thermal
decomposition, would give rise, first of all to formaldehyde, and then to carbon
monoxide and hydrogen. The scheme of the " oxygen attack " is as follows, with the
proviso that no decomposition occurs in " running through " stage A when there is
sufficient oxygen to complete the transition to stage B.

A. B

H.C.H H.C.OH H.C.OH

H.C.H H.C.H H.C.OH

i A

2 = (

)H + H2O 2CH,O

x^X

2CO + 2H,

/ i

2CH4 C2H2

It occurred to us that it would be of great interest to study the behaviour of
mixtures C2H4 + O2 + »H2 in the bomb apparatus at initial high pressures, because
successive additions of hydrogen would, according to the above scheme, operate in two
distinct ways, namely (i.) by participating, more and more, in proportion to its active
mass, in the initial oxygen distribution, it would prevent some of the hydrocarbon
completing the A to B transformation, and so bringing about some decomposition at
A; and (ii.) by participating in any "A" decomposition in such a manner as to
" hydrogenise " the • CH residues, and so counteracting, and perhaps even suppressing
altogether, separation of carbon.

Such a twofold influence would be marked by (i.) an increasing steam and methane
formation in the products with successive initial additions of hydrogen; and (ii. ) a
restraint or even total prevention of carbon deposition.

On actually putting matters to the experimental test it was found possible to add
as much hydrogen as would correspond to C2H4 + 02+8H2 without causing any
deposition of carbon whatever on explosion, and is it probable that an even larger
proportion of hydrogen could have been added without producing such a result.
Progressive additions of hydrogen did, indeed, have the anticipated effects in regard
to steam and methane formation as the following results will illustrate ; there was,
however, never any trace of acetylene found in the products.

* See also BONE and COWARD on " The Thermal Decomposition of Hydrocarbons," ' Trans. Chem.
Soc.,' 1908, vol. 93, pp. 1198 to 1225.

2 S 2

310

PEOF. W. A. BONE AND OTHERS ON

TABLE XIV.

Experiment No. . . .

49

50

51

52

Mixture

C2H4 + 02 + 2H2

C2H4 + 02 + 4H2

C2H4 + 02 + 6H2

C2H4 + 02 + 8H2

p\ atmospheres

20-84

26-00

40-67

49-0

pz atmospheres

P-2/Pl

29-40
1-41

29-43
1-132

46-2
1-135

45-0
0-918

Per cent, of original carbon"!
appearing as CHj in pro- >
duets j

7 -Go

22-4

26-9

43-5

Per cent, of original oxygen"!
appearing as li.,0 in pro- >
ducts (approximately) . . \

4 • 0 to 6 • 5

«

18

27 to 42

36 to 46

PART VI. — THE EXPLOSION OF ETHA.NE WITH ITS OWN VOLUME or OXGYEN.

(With Dr. HERBERT H. HENSTOCK.)

Owing to its crucially important bearing on the theory of hydrocarbon combustion,
the behaviour of an equimolecular mixture of ethane and oxygen on explosion has
been the subject of much investigation and discussion in recent years.

According to the now discarded theory of a preferential burning of carbon, such a
mixture should, on explosion, give rise to carbon monoxide and hydrogen without any
separation of carbon or steam formation, in accordance with the equation,

GaH9+03 = 2CO+3H

2 i

and, inasmuch as carbon monoxide and hydrogen are, to all intents and. purposes,
mutually inert in flames, the final result should be unaffected by variations in initial
pressure, or by the rate of cooling of the products for the maximum pressure of
explosion.

It has, however, been shown that the behaviour of the said mixture is not at all in
accordance with the requirements of the theory in question, inasmuch as it always
gives rise on explosion to large quantities of free carbon, methane, steam, and
aldehydic vapours, as well as oxides of carbon and hydrogen, just as would be
expected from the standpoint of the hydroxylation theory, according to which there
would be a "non-stop" run, through CaH5.OH, to C2H4(OH)2, which would then
decompose, yielding first acetaldehyde and steam, the acetaldehyde then breaking

GASEOUS COMBUSTION AT HIGH PRESSURES.

311

down into carbon, hydrogen, methane and carbon monoxide, in accordance with the
scheme.

H

H.C.H

H.C.H

H

H

H

H-C H

H-C H

v

H.C.OH

s~

H.C.OH

H

OH

"(1) CH4+CO
(2) C + 2H. + CO

4H.O

H.O + ILC.CHO

This conclusion was endorsed, in general terms, by Prof. H. B. DIXON in his
Presidential Address to the Chemical Society in 1910, as the result of his own
independent observations on the rate of detonation of a mixture C2H6 + 02.*

The modes of decomposition of dihydroxyethane and acetaldehyde at high tempera-
tures, and the nature of the resulting products are such as would permit of the play of
several secondary interactions, such for instance as (i.) CO + OH2^__1C02 + H,, and
(ii.) C + OH2 = CO + H2, during the cooling period of an explosion. And therefore it
is to be expected that the final result in any particular case will depend upon (i.) the
maximum flame temperature and the rate of the subsequent cooling therefrom, which
conditions are in turn governed by such factors as initial pressure, and (ii.) the phase
attained by the explosion (i.e., " inflammation" or " detonation"), and (iii.) the area
and nature of the cooling surface per unit volume presented by the walls of the
containing vessel.

The dependence of the final result upon such conditions is well illustrated by the
following tabulated records (Table XV.) of the behaviour of a mixture C2H6 + Oa (i.)
when inflamed at less than atmospheric pressure in glass vessels presenting very
different areas of cooling surfaces to a given volume of the combining gases, and (ii.)
when detonated in a leaden coil at initial pressures of 900 to 1200 mm.

It is thus seen (l) that the rapid cooling of the exploded gases in (a) favoured the
survival of steam, aldehydes, and unsaturated hydrocarbons ; (2) that the longer
duration of the flame in (b) greatly increased the separation of carbon, at the expense
of the saturated hydrocarbons and aldehyde, whilst it also favoured the secondary
reduction of steam by carbon ; and (3) that the much higher temperatures and
pressures attained in detonation, whilst militating against the survival of aldehyde
and unsaturated hydrocarbons, was chiefly effective in promoting the secondary
reduction of steam by carbon. Nevertheless, even in detonation, as much as 17 to 20
per cent, of the original oxygen appeared as steam in the final products.

The foregoing considerations led us to study in detail the behaviour of the same
mixture when exploded in bomb A, under various initial pressures between 10 and 40

* ' Trans. Chem. Soc.' 1910, vol. 97, p. 665.

312

PEOF. W. A. BONE AND OTHERS ON

atmospheres, with results which are set forth in Table XVI. In all cases there was a
marked deposition of carbon, and condensation of steam on cooling, but, as was

TABLE XV. — Explosion of Mixture C2H6 + O2.

Inflammation in

i.

(a)

(6.)

Detonation in lead coil.

Experimental conditions.

Long glass tube.
1 • 5 metre long.

fflass

Internal

globe.
diameter

20 metres long.
36 mm. internal diameter.

20 mm. internal diameter.

= 8-

5 cm.

Volume = 3100 c.c.

Volume = 470 c.c.

Volume =

- 320 c.c.

Area of walls = 940 cm.2

Area of walls = 226 cm.2

mm.

mm.

mm.

mm.

mm.

V\

701

448

685 906

1180

Pi

1018

724

1187 1700

2240

Ott/f)

1-45

1-61

1-73 l-RR

1-90

E 'o 6 { CO, .

4-20

4-00

3-40 1-65

1-80

8 ft CO ...

34-80

34-10

36-10 39-00

39-10

« § 3 C,Ho , .

5 • 00 T

{1 -20

0-90

g £ i •§ 1 C,H4

2.65]7-65 2-25

0-50

0-50

t- a wo CH* ' ' '

8-85

6-85

7-25 6-65

7-70

PH [H2 . . .

44-50

52-80

53-05

51-00

50-00

Per cent, of original

carbon deposited .

7-6

19

17

5-3

3

Per cent, of original

oxygen appearing as

H20, &c., in con-

densable products

37-8

32-8

27-5

20

17

Remarks . . .

Luminous flame travelled Lurid flame filled the

Coil filled cold ; then its

at approximately 0-5 globe and was of

temperature was raised

metre per second. longer duration than

to 98° C. before the

Each successive layer of in (a).

mixture was deto-

the burning gas re- Products gave faint

nated ; afterwards

mained incandescent

aldehydic

reaction,

apparatus cooled by

TrV to ^ second.

and steam condensed

immersion in cold

Products gave strong

on cooling.

water.

aldehydic reaction ;

much steam condensed

_

on cooling.

anticipated, an increase in the initial pressure tended, on the whole, to diminish the
proportion of both these results. The ratios pjply in the different experiments varied
between 1'85 and 2'05.

GASEOUS COMBUSTION AT HIGH PRESSUEES.

313

Another very notable and significant feature was the complete absence of both
acetylene and ethylene from the final products, which, however, always contained con-
siderable quantities of methane, even at the highest pressures. Such facts point to
a sufficient violence in the explosion to shatter completely any unsaturated hydro-
carbon or acetylene, if indeed such were momentarily found in the flame, and afford
also another striking proof of the great stability of methane at the highest explosion
temperatures.

TABLE XVI. — Explosion of Mixture C2H6 + O2 at High Pressures in Bomb A.

Experiment No

53

54

55

56

57

PI atmospheres

10-9

14-77

18-55

25-21

39-65

p.> atmospheres

20-56

27-93

37-22

51-72

73-4

1-88

1-89

2-0

2-05

1-85

Partial pressures in original f GjHc . . .
mixture in atmospheres \0-> ...

5-54
5-36

7-55
7-22

9-20
9-35

12-67
12-54

20-15
19-50

rco, . . .

Partial pressures in gaseous I CO
products in atmospheres ] H-> . . .

LCH, . . .

0-570
7-525
10-830
1-635

0-95

9-80
14-50
2-68

1-04

14-14
19-76
2-28

1-31

19-49
27-25
3-67

3-15
27-35
34-7

8-2

Per cent, carbon deposited

11-7

10-9

5-0

3-4

4-0

Per cent, of original oxygen appearing"!
as H20 in the final products ...._)

19-0

19-0

13-1

11-8

13-8

PART VII. — PRESSURE EXPERIMENTS.

The foregoing experiments (Part III.) having proved the absence of any direct
connection between the relative affinities of methane, hydrogen and carbon-monoxide,
respectively, for oxygen, in homogeneous flame combustion, and the rates of flame
propagation through explosive mixtures, in which the combining gases are present in
proportions corresponding to the effective primary reactions concerned, it now remained
to prove whether or not such chemical factors determine the rates of attainment of the
maximum pressures in explosions.

It had been our original intention to obtain optical records of the complete pressure
curves, both before and after the attainment of maximum pressure, when the
" primary " mixtures CH4+02, 2H2+02, and 2CO + 02, respectively, are exploded at

314

PKOF. W. A. BONE AND OTHERS ON

high initial pressures. But in this we were not entirely successful, owing to the
excessively short intervals between ignition and the attainment of the maximum
pressure, more particularly in the case of the fastest burning of the mixtures referred
to, namely, 2H2 + O2. It was decided, therefore, to limit the investigations to the
study of corresponding " air " mixtures, whose composition may, for all practical
purposes, he written — •

2Ha+03+4N3,

.+4N2.

The experimental method consisted in firing such mixtures at initial pressures
of from 45 to 50 atmospheres in the spherical bomb, B, to which was attached
a Petavel recording manometer with its optical accessories.

The action of this manometer depends on the principle that the time period 0 of a

Fig. 14.

vibrating system may be expressed in terms of W the weight of its moving parts, and
A the force required to produce unit deflection therein, as follows,

Vw
-,
Kg

and in order to make Q as short as possible, consistent with a sufficiently great
strength (W) to withstand the sudden development of extremely high pressures, the
controlling force (A) brought into play per unit length of motion is made as great
as possible, by using the stiffest spring obtainable. This means, of course, that the
actual motion produced in the spring by the explosion is exceedingly small, but for
recording purposes this is magnified by a special optical device, which focuses a point
source of light on to a rapidly revolving drum, to which is attached a sensitive photo-
graphic film.

The following description of the instrument (fig. 14) used in our experiments is

GASEOUS COMBUSTION AT HIGH PRESSURES.

315

taken, with slight modifications, from Prof. PETAVEL'S memoir on "The Pressures of
Explosions."""

The gauge is screwed into the explosion chamber by means of the thread, U. A
gas-tight joint is formed by the ring, D, on the manometer pressing against a flat
ledge in the enclosure. The end of the gauge from D to E fits well in the. walls of
the explosion chamber, and the joint is thus protected from the direct effect of the
explosion. The spring, S, about 5 inches in length, is tubular in shape, and to
prevent any buckling it is made to fit closely the cylinder, in which it is contained,
in two places el and c.2. The spring is fixed at its other end, Z, being held in position
by the nut, K, whilst at its other end it is free and supports the piston, P.

The mirror, M (which in our experiments was of silvered glass with a focal length
of about 2'5 feet), is attached to a lever, and at the outset of an experiment is kept in
the zero position by means of the stretched piano wire (gauge 25), W ; adjustment of

Fig. 15.

the zero is effected by means of the screw, V. The distance between the knife edges
of the mirror holder in our experiments was one thirty-second of an inch.

The sources of light employed by us was the crater of an electric arc in which the
carbons were at right angles, and the beam was reflected by the mirror on to a
sensitive photographic film fixed on to the drum, A, of the chronograph (fig. 15), also
designed by Prof. PETAVEL (loc. cit.}.

This drum (diameter = 8'8 cms.) which was usually rotated at a constant speed of
between 200 and 600 revolutions per minute by means of an electric motor, was
anclosed in a light-tight aluminium casing, BB, provided with a longitudinal slit, the
width of which could be adjusted to suit the particular experimental conditions, and
which could also be closed when required. An electrically-controlled tuning fork,
giving 100 to 200 vibrations per second, was used as a time indicator ; the fork being
so placed that each vibration momentarily interrupted the light falling on to the
photographic film, thus producing a dotted record on the resulting pressure curve.

VOL. ccxv. — A.

'Phil. Trans.,' 1905, A, vol. 205, p. 363.
2 T

316

PEOF. W. A. BONE AND OTHERS ON

The results of two typical experiments with each of the three mixtures are recorded
in Table XVII., whilst figs. 16, 17 and 18, reproduce the corresponding pressure
curves, from the moment of ignition until far into the cooling period after the
attainment of maximum pressure, in each case, for one experiment with each mixture.

TABLE XVII. — Pressure Experiments.

1
Experiment No. . . 58 59 60

61

62

63

Mixture fired .... 2H2 + 02 + 4N2 CH4 + 02 + 4N2 2CO + 02 + 4N2

j
Per,, off.H2 = 29-6 H2 = 29-7 CH4 = 16-7 CH4 = 16-8 CO = 28-1

m x reTeT 1 0'-' = 14'5 °* = 14'3 °* = 17-1 °* = 16'8 0., = 14-0
mixture hred . . • \ Nj M B5-9 N, = 66-9 N2 = 66'2 N, = 66-4 N, = 57'9

.;

j

CO = 28-03
0., = 14-60
N2 = 57-37

^osp-h"6/; in} 53-° ; ™-» ; «•»

49-8

Maximum explosion")
pressure in atmo- > 425'0 400 '0 240 '0
spheres^,,, . . . . J

280-0

423-0

420-0

Final pressure of cold"!
products in atmo- > 30 '5 28 '74 47-1
spheres p,- . . . . J

51-7

42-6

41-46

Eatio Pmjp: . . . j 8'0 8-0 5-35

5-6

8-3

8-4

Ratio pf/pi ' 0'575 0'575 1'049

1-034

0-832
0-10

0-837

Time required for attain- "]
ment of maximum > O'Oll O'OIO 0-05 '0-08 about
pressure in seconds . J

.

o-io

It is evident, from the above records, that the hydrogen-air mixture is by far the
" fastest burning " of the three mixtures investigated, the maximum pressure being
reached in approximately one-tenth of the time taken in the case of the slowest
burning carbon-monoxide mixtures (O'Ol as compared with 010 seconds).

The time required for the attainment of maximum pressure in the case of the
methane air mixtures (0'05 to 0'08 seconds) was at least five times as long as that
required in the case of the hydrogen-air mixtures, notwithstanding the fact that the
affinity of methane is something like twenty times as great as that of hydrogen for
oxygen in flames. Apparently, therefore, there is no direct relation between the
actual rate at which the potential energy of an explosive mixture is transferred on

GASEOUS COMBUSTION AT HIGH PKESSUEES.

317

explosion as sensible heat to its products, and the magnitude of the chemical affinity
between its combining constituents.

Fig. 16.

Attention may be drawn to the extreme slowness of the cooling after the
attainment of maximum pressure in such explosions, a circumstance which has been

I

CH4+CL+4N,

•*•<

-

t

/

/

/

S

>;

-^

T^

3

\

.

i

i

i

0 15 2

0 25 !

0 3

TIMC IN -fei, SECONDS

after Firiru

Fig. 17.

observed by previous workers in other similar cases of gaseous explosion. This was
particularly marked in the case of the methane-air mixtures, in which there was

PRCSSURE JN ATNOSPHCFfS
SO O O O

o o o o

2CO + Oat4.N2

— <

/

'

/

f

/

y

i

i

f

^

*

0

2

468

10

12

14 16 18 20 25 30 35 4
TIME IN Toi SECONDS
Offer Rrv«j

Fig

318 PKOF. W. A. BONE AND OTHERS ON GASEOUS COMBUSTION, ETC.

hardly any appreciable cooling during an interval of 0' 22 seconds after the attainment
of maximum pressure. This circumstance may possibly be attributed to (l) the
combustion taking place in well defined chemical stages, and also to (2) the operation
of the exothermic secondary interaction between carbon monoxide and steam during
the cooling period. The cooling is rather more rapid in the case of the carbon
monoxide-air mixture, and still more so in the case of the hydrogen-air mixture.
The last named would appear to be a simple cooling curve uninfluenced by chemical
combination.

Our thanks are due to the Government Grant Committee of the Society for liberal
grants out of which the cost of the expensive apparatus employed on the work has
been defrayed.

r 319 i

XI. Heats of Dilution of Concentrated Solutions.

By WM. S. TUCKER, A.RC.Sc., B.Sc.
Communicated by Prof. H. L. CALLENDAR, F.R.S.

Received March 23,— Read April 22, 1915.

CONTENTS.

Page

(1) Review of previous work on heats of dilution 319

(2) Description of apparatus employed for determinations of specific heat and heat of dilution

of solutions .321

«

(3) Detailed observations for hydrochloric acid solutions 324

(4) Study of the results obtained 329

(5) Heat of dilution of lithium chloride solutions 334

(6) Heat of dilution of sodium hydroxide solutions 334

(7) Heat of dilution of calcium chloride solutions 338

(8) Variation of heat of dilution with temperature for hydrogen and lithium chlorides . . . 338

(9) Examination of THOMSEN'S results 344

(10) General conclusions ... 349

(l) REVIEW OF PREVIOUS WORK ON HEATS OF DILUTION.

HEATS of dilution have been exhaustively studied by Prof. JULIUS THOMSEX for a
very large number of aqueous solutions.* His method chiefly consisted in taking
some concentrated solution and diluting it considerably. The total amount of heat
generated or absorbed in this process was thus found and quoted against the final
concentration expressed in molecules of water to one molecule of solute. It is to be
noted that during the process of dilution most of the thermal change occurs in the
early stages, and that after the first ten molecules of water are added the total heat
generated or absorbed increases but slightly.

The probable reason for this procedure is, that the experimenter finishes the
operation with a dilute solution, so that starting with various initial concentrations,
he may need only a few specific heats of certain dilute solutions. It must be

* ' Thermochemische Untersuchungen,' Bd. III.
VOL. CCXV. — A 533. 2 U [Published August 25, 1915.

320 MR. W. S. TUCKER ON HEATS OF

remembered that only the final specific heat of the solution need be known in order
to measure the heat generated.

This process of largely diluting a solution, i.e., in all cases using a larger bulk of
solvent than solution, has been employed also by BERTHELOT,* STEIN WEHR,t and
BISHOP.}

Very few experimenters quote results sufficiently numerous to establish any
relation between heat of dilution and concentration. THOMSEN, from results on
solutions of nitric acid and sulphuric acid, deduces a hyperbolic relation between the
total heat of dilution and the final concentration N in molecules of water per

«N
molecule of solute. It is of the form Q = ^ — - where a and It are arbitrary

constants. The equations admit of no simple interpretation.

For hydrochloric acid solutions, lie employs the different type of equation

THOMSEN obtained a large number of readings for sodium hydroxide solutions. At
low concentrations the heat of dilution changes sign, and no attempt was made to
give a relation between Q and N.

Acetic acid also changes the sign of its heat of dilution. It is negative for strong
solutions and positive for weak ones. A great number of results were published by
THOMSEN for many salts, but not sufficient in any one case for the purpose of
deducing a Q and N relation.

RuMELIN§ has more recently found heats of dilution for phosphoric acid solutions,
but only for four concentrations. LEMOJNE|| also examined lithium chloride
solutions, but quoted results in which no high order of accuracy was suggested.
These methods differ from those previously quoted, for here the diluting was done in
stages.

Summing up all the results and the conclusions derived from them by the
various authors, there seems to be no simple law which can be applied to all
satisfactorily.

The author in previous work with concentrated solutions^ has obtained evidence
in certain cases of those solutions being simpler in character than weaker ones. The
present paper deals almost entirely with the thermal effects in what may be called
concentrated solutions, and a sufficient number of observations are taken to get a
good concentration relation with heat of dilution. Heats of dilution were found for

* ' Annales de Chimie et de Physique,' vol. IV., p. 468, 1875.

t ' Zeitschr. fur Phys. Chem.,' vol. 38, p. 185, 1901.

I ' Physical Review,' vol. 26, p. 169, 1908.

§ ' Zeitschr. fur Phys. Chem.,' vol. 58, p. 458, 1907.

|| ' Comptes Rendus,' vol. 125, p. 604, 1907.

U 'Proceedings of the Physical Society,' vol. XXV., Part II., p. 111.

DILUTION OF CONCENTRATED SOLUTIONS. 321

the concentrated solutions without making them dilute in the process ; thus about
20 gr. of water were added to 300 gr. of solution. The heat generated
or absorbed in this case divided by the number of gr. molecules of water added,
gives the differential heat of dilution sufficiently accurately for a concentration which
is the mean of the initial and final concenti^ations.

The differential heat of dilution d Q/rZN may be defined as the quantity of heat
liberated or absorbed per molecule of water added to an infinite quantity of solution,
whose concentration will thereby be unaffected. This condition can only be
approximated to, but the error in the above estimate will be small.

THOMSEN added large quantities of water and so obtained a total heat

Q =. [ ' Sg . 3N. This involved a large thermal equivalent and a small rise in

Jy,

temperature.

By the method here described the thermal capacity is kept sufficiently low to
give a fair rise in temperature on dilution, and the relatively small quantity of
water employed can be placed in a vessel completely immersed in the solution.
Temperatures can thus be well equalised before dilution takes place.

The following record describes work extending from October, 1912, to December,
1914. The accurate determination of heat of dilution is necessarily a tedious business,
owing to the fact that only one observation can be taken in an ordinary working day
of six hours.

(2) DESCRTPTFON OF APPARATUS. (Diagram T.)

Measurement of Temperature,

In most of the previous researches on heat of dilution a mercury thermometer was
employed. Here temperature changes were measured with certainty by a platinum
thermometer with fundamental interval of 12 -8 ohms — mounted in a very thin-
walled cylindrical bulb, whose length is not far short of the depth of the solution.
Long experience with the working of this thermometer in other researches and
careful application of all the corrections peculiar to it, enable the author to guarantee
its readings. (Diagram I., T.)

Importance of Stirring.

One of the most important features of the experiment is the stirrer. Different
types were tested, and the most efficient one was found to be of the screw propeller
type turned by a high speed motor. The process of mixing was rapidly performed
even with some of the very viscous solutions employed, and a careful experiment was
performed to see if any appreciable temperature rise was indicated during the process.
After long periods, less than xth}° C. rise was noticeable. (Diagram I., S.)

2 u 2

322

MR. W. S. TUCKER ON HEATS OF

1

The Choice of a Calorimeter.

Kadiation is another factor which becomes important when very small temperature
changes are produced. A very well made Dewar cylinder was employed. (Diagram
I, A A.) Its walls were well silvered, and during the experiments its contents,
together with stirrer thermometer and other adjuncts, were shut in by a well-fitting
cork stopper, which again Was covered with a layer of cotton wool.

Curves were plotted when the contents had
varying thermal capacities, and these were found to
be perfectly smooth and enabled one to estimate
with certainty the rate of cooling at any tempera-
ture. The cubical capacity of the vessel was about
400 c.c., and in no case did it contain more than
300 c.c. of solution. For 20 degrees excess of
temperature in the contents over that of the room,
a rate of cooling of about 0'038° C. occurred per
minute.

Each series of observations was accompanied by
its own radiation experiment and complete consis-
tency was given throughout.

The Mixing Pipette.

It is highly important that the water added
should have the same temperature as the solution.
A thin walled glass pipette was employed of the
form shown in the diagram. (I., P.) Its bulb
passes throughout the length of the cylindrical
mass of solution, and parallel to the bulb of the
thermometer. Two such pipettes were employed—
one with a capacity of about 25 c.c., the other about
45 c.c. The water was weighed out in these, the
former being used when the solutions were strong,
the latter when weak. The walls were of thin glass, and the syphon tube was of such
length as to introduce the water at the bottom of the vessel containing the solution.
The water, being of course less dense than the solution, would rise, and the process of
mixing would be assisted in consequence.

Before mixing, the glass bulb with its contents was left immersed in the solution
for some hours, and the stirrer was worked continuously. In this way error due to
temperature difference was brought to a minimum. The upper end of the pipette is
then connected by a thin rubber tube with the rubber ball, such as is used for a scent

8

6

^

iH

^•>

it

1

- i

•:••

&

P

H

«:

X

5

Diagram I.

DILUTION OF CONCENTKATED SOLUTIONS. 323

spray, and the water is projected out. (Diagram I., P.) The process of mixing is
then carried out, and after a sufficient time the temperature change is noted. Some
estimate is made of the water in the pipette which has not been driven out, viz., that
wetting the walls of the tube. For this purpose, the tube being removed, dried
externally and weighed, allowance is now made in the amount of water quoted.

Thermal Capacity of Vessel and its Contents.

The thermal capacity of the vessel and its contents must be very carefully
determined, a process which is always difficult. When a glass mixing vessel is
employed special care has to be taken in the determination, as glass lias a high specific
heat. Moreover, its low conducting power involves the temperature not being so
nearly equal throughout its mass. Thus it was found that the thermal capacity
differed with the amount of solution taken. In all cases quantities of solution taken
were of either 250, 275 or 300 c.c., the amounts being measured by standardised
measuring vessels.

Thermal capacities were found for the vessel and its contents also, when the mixing-
pipettes were present or absent. To ensure accuracy experiments were repeated.
Consistent values were obtained. "Roughly speaking the thermal capacity was of the
order of one-tenth that of the total thermal capacity.

Mode of Heating.

The method of determining the thermal capacity of the vessel under the varying
conditions, was that of heating by a known electric current a weighed quantity of
water contained in the vessel. Some difficulty was experienced in the construction of
a satisfactory heating coil. The difficulty was solved by selecting a piece of glass
tubing of very thin walls, and elliptical in section. (Diagram I., H.) Manganin
wire was wound upon a flat strip of mica and inserted into the tube, one end of which
was sealed. Good thermal contact between the coil and the walls of this tube was
ensured by filling up the tube with the best lubricating oil, and the upper end was
lightly close with glass wool. The coil so made was thus perfectly insulated, and yet
presented a large heating surface. There was found to be very little lag in the
taking up of the heat by the solution. The length of the coil was so taken as to
extend throughout the depth of the solution.

The heat generated was measured by the products . where C is the current, E

4 J. o 0

the volts, t the time in seconds, and 4'185 the mean value of JOULE'S equivalent. A
Weston ammeter and a Weston voltmeter were employed having ranges from 0 to 5
in both cases. It was found possible to read to O'OOS ampere or volt with fair
accuracy. Both instruments were calibrated against standards and the observations
quoted are those observed, to which the requisite correction is applied. The

324

MR. W. S. TUCKER ON HEATS OF

instruments employed showed an accuracy to within 0'02 of a volt or ampere almost
throughout their whole scale. The heating coil was of about 0'9 ohm resistance, and
such a current was employed as to give readings nearly at the middle of the scales in
each instrument. The steadiness of the current was determined by the condition
of the secondary cells employed. Such variations as there were, however, were
eliminated by readings of each instrument during each minute of heating.

To find the Thermal Capacity of the. Apparatus.
One typical set of observations may be quoted.

May 8, 1913.

250 gr. of water were heated for 940 seconds, and the following readings of voltmeter
and ammeter were taken alternately — at minute intervals.

Corrected Readings.

Ammeter.

Voltmeter.

Ammeter.

Voltmeter.

2 • 945

2 • 595

2-910

2-570

'2 • 935

2-590

2-910

2-570

2 • 930

2-590

2-910

2-570

2 • 930

2-590

2-905

2-560

2 • 940

2-595

2-920

2-580

2-940

2-600

2-925

2-580

2-940

2-595

2-925

2-585

2-920

1

2-570

2-915

2-575

The mean value of the current is 2'925 amperes and of the P.D. 2'582 volts. The
heat supplied is 169G'4 calories. Initial temperature 15'37° C. Final temperature
21-22° C.

The mean radiation at the final temperature is 0'0072° G. per minute, and the
correction for the whole period 0'06° C.

Hence, the corrected temperature rise is 5'91° -C., and the water absorbs 1477-5
calories. The heat absorbed by the apparatus is thus 218 '9 calories, giving a water
equivalent of 37 '0 gr.

For 275 gr. of water, the water equivalent was found to be 42 '5 gr. When the
mixing pipettes were employed, values of 4475 and 4 5 '6 gr. were obtained
respectively.

(3) DETAILED OBSERVATIONS FOR HYDROCHLORIC ACID SOLUTIONS.

1. Experiments at Air Temperature.

Four typical strong solutions were studied :— Hydrochloric acid, lithium chloride,
sodium hydroxide, and calcium chloride.

DILUTION OF CONCENTRATED SOLUTIONS.

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326 ME. W. S. TUCKER ON HEATS OF

In order to obtain their values of heat of dilution a series of specific heat values
must be found for varying concentrations. As all the experiments were to be
performed at air temperatures, varying from 15° C. to 20° C., the method of heating
was adopted as described above.

Only a short range of temperature rise was permitted as the specific heat may
change rapidly with temperature. For this reason, such heat was supplied as to keep
this rise. of temperature to about 5° or 6° C. The time chosen for the heating process
was 1000 seconds, for some solutions, but in the case of hydrochloric acid solutions
which have a low specific heat at high concentrations, periods of 10, 11, and 12 minutes
were employed.

In order to show the manner in which the observations were taken the full details
for hydrochloric acid solutions are given. Eight solutions are taken and are
distinguished by numbers, the strongest being solution 1. The acid employed was
the purest obtainable, and the more dilute solutions were obtained from the stock
solution by adding distilled water. Observations of specific heat determinations were
taken from May 15 to June G, 1913, and are quoted in Tables I. and II.

Some preliminary experiments with concentrated solutions showed that the gas was
driven oft' to an appreciable extent when the temperature exceeded air temperature.
For this reason the solutions were cooled about 8° below air temperature, so that their
mean temperature is about 5° below that of the air. Later experiments show that the
specific heat alters very little with temperature for these solutions, hence no appreciable
error is introduced into the heats of dilution as calculated from these specific heats.

Table I., p. 325, shows the relation between the specific heat and the concentration,
expressed as molecules per 100 molecules of water, the molecular weight being taken
as 36*5.

The concentration was obtained from density tables supplied by LUNGE and
MARCHLEWSKI. The final temperature is corrected for radiation, and the mean
temperature of these is nearly that of the solution for the given determination.

275 c.c. of the acid were employed throughout, and the water equivalent is 42'5 gr.
The first three solutions were heated for 600 seconds, the fourth for 660 seconds, arid
the remainder for 720 seconds.

For the purpose of interpolation many attempts were made to get an equation
connecting specific heat and concentration. No simple equation appears to satisfy the
condition for dilute as well as strong solutions. The author's interest, however, is
chiefly in the concentrated solutions, and for solutions whose strengths lie between
the limits of 7 '7 and 22 '8 molecules of the solute to 100 molecules of water it was
found that an equation of the type sn" = b was suitable, where s and n are specific
heat and concentration respectively, while a and b are constants.

In the case of hydrogen chloride the equation takes the simple form

,m'"= 1-277.

DILUTION OF CONCENTRATED SOLUTIONS.

Table II. shows the degree of accuracy obtained.

TABLE II.

327

Concentration

Specific heat

Specific heat by

(»)•

(«)-

calculation.

22-8

0-588

0-58-4

20-23

0-599

0-602

17-80

0-619

0-622

13-80

0-662

0-662

10-95

0-702

0-702

7-70

0-766

0-767

5-40

0-815

0-838

The last value in this table shows the failure of the relation for dilute solutions, but
since heats of dilution at this concentration are so small, very little relative error is
introduced by interpolating graphically.

The value of the constant b changes with the temperature.

Heat of Dilution.

All the observations were taken with the initial temperature of the solution and
the water that of the air. No radiation correction need be applied, since the process
of mixing was very rapid. In the first six experiments the smaller pipette was
employed, and the water equivalent 44'75 gr. ; for the next two experiments, the
larger pipette was used with water equivalent 4 5 '6 gr. The last two experiments
were performed by a pipette which was not immersed in the solution. The solution
and distilled water were brought to the same temperature (that of the air) by enclosing
the latter in a vessel immersed in the former. Quantities of each were measured out
in turn— in experiment (IX.), 200 c.c. of solution to 100 of water, and in (X.), 150 c.c.
of each. The mixing process was performed in the same vessel, and the water
equivalent was 46 '0 gr.

As a check on the concentration the densities of the solutions were obtained before
and after mixing, and the new value of concentration after the process of mixing was
found by application of the expression

P, 1800 Qcd

(1800 + cm) (Qd + x) - Qdcm
where c is the initial concentration in molecules of acid per 100 molecules of water,

Q the volume of solution taken in c.c.,

d its density at the temperature employed (gr./c.c.)

m the molecular weight— here 36 '5,

x the number of gr. of water added.

VOL. CCXV. — A.

2 x

328

MR W. S. TUCKER ON HEATS OF

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DILUTION OF CONCENTRATED SOLUTIONS.

329

Table (III.) contains all the observations; the strongest solution is placed first in
the series.

NOTE. — In obtaining the specific heats of the solutions, a slight correction is made
for the change of specific heat with temperature. A later series of experiments were
performed in which the change was found for different concentrations.

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Concentration :- -Molecules per 100 molecules of Welter.
Diagram II.

Diagram II. includes the curve showing the relation between heat of dilution and
concentration for hydrogen chloride.

(4) STUDY OF THE EESULTS OBTAINED.

In the above table a column gives THOMSEN'S values and the results are calculated
out from the relation,

^ = 17,340-12,000^

n N

where Q is the heat generated per n gr. molecules of solute dissolved in N molecules
of solvent.

The heat of dilution is given by

5N

2x2

330

ME. W. S. TUCKER ON HEATS OF

It will be seen that the results derived from his equation lie considerably below
those obtained by experiment.

The discrepancy between THOMSEN'S values and those here obtained does not appear
from the value of total heat of dilution quite so obviously. Thus on diluting a
solution from HC1 . 2'6H20 to HC1 . 10ILO, nearly the same quantity of heat appears
to be generated. The reason for this is that THOMSEN'S dQ/dJS exceeds the values
here obtained for solutions of strength exceeding HC1 . 4H2O.

In order to obtain a measure of the total heat generated, values of dQfdN can be
plotted against number of molecules of water per molecule of solute, i.e., the abscissae
would read " dilution " instead of concentration. (See Diagram III.)

16

18

2 4 6 8 10 12 «4

Dilution : — Molecules of water to 1 molecule of solute.
Diagram III.

The area intercepted between two ordinates separated JN would be ^ §N, and

20

between limits of concentration Nj and N2.

It will be seen that the curves cross at N = 4, and that the total areas from the two
curves are nearly equal.

The curve connecting heat of dilution and concentration for hydrochloric acid,
should, by THOMSEN'S theory, be parabolic. The results of the above experiments show
that with concentrations exceeding HC1 . 10H20 the values lie very nearly on a
straight line, which however, for lower concentrations, bends round and approaches
the origin. If the straight portion be produced it cuts the axis of concentration at

DILUTION OF CONCENTRATED SOLUTIONS.

331

6'66 molecules per 100 molecules, which is HC1 . 15H20. For concentrations above
10 therefore — when all concentrations are reduced in such a manner as to assume
HC1.15H20 as the solvent — a proportional relation is obtained between heat of
dilution and concentration.

This may be best expressed by the equation

= a

— c

If c is expressed in gramme-molecules of solute per molecule of solvent, and a is the
tangent of the angle of slope of the portion referred to (4082), taking N as 1,

= a(n-0'066G).

If expressed in terms of dilution

n = ^ and ^S = *f -0'0666a,

N c£N N

and the total heat generated, when one molecule of solute combined with N\ molecules
of solvent is diluted till combined witli N2 molecules, is expressed by

N -
N,

x,

Giving Nj the value 2'6, which is about THOMSEN'S maximum concentration, and
N2 the value 10, THOMSEN'S values give 3415 calories, while those obtained above give
3427 calories. For concentrations below 10 molecules of water the two curves nearly
coincide, so that the total heat obtained by diluting very largely would be nearly
the same.

The following results (Table IV.) show the degree of accuracy with which the
equation

^ = 4082 (£-0-0666 ,

dN VN

fits the results for concentrations above 10 molecules per 100 molecules.

TABLE IV.

Concentration
molecule
per 100 molecules.

Results of
experiment.

Results by
calculation.

Percentage error.

22-17

635

632-9

-0-33

19-62

527-5

528-7

+ 0-23

17-40

439-3

438-2

-0-25

15-37

353-8

355-3

+ 0-43

13-72

289-2

287-9 '

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11-39

186-2

192-8

+ 3-01

9-99

128-3

135-7

+ 5-81

332

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(5) LITHIUM CHLORIDE. (Molecular weight 42'5.)

The solutions of lithium chloride were treated in precisely the same way as those of
hydrogen chloride. The solutions when concentrated, possessed a much greater
viscosity and greater care had to be taken in stirring. Also the solutions were very
hygroscopic and the experiments were performed, as far as possible, in an air-tight
vessel.

The following tables (Tables V. and VI.) summarise the observations. As the same
mixing vessel was used and the same quantities of solution, the same water equivalents
were employed. In the specific heat determinations the current and voltage employed
are the mean of sixteen observations for each solution. The specific heats were
obtained for nearly the same range of temperature as that employed for the heat of
dilution determinations, hence no temperature correction for specific heat need be
applied. Seven specific heats were found and 300 c.c. of each solution were taken.

The curve connecting heat of dilution and concentration shows a more gradual
increase of the former with increase of the latter. The same peculiar property,
however, is shown for concentrations exceeding LiCl . 6H2O above this value to
concentration LiCl . 3H20, the truest representation for this portion is a straight line
which cuts the concentration curve at LiCl . 8HaO. (Diagram II.)

For this range the equation may be written

Jg = 6927

dN

-0125J-

A few experiments have been performed by DUNNINGTON and HOGGARD with
lithium chloride ('American Chemical Journal,' vol. 22, p. 210, 1899). Their method
was only approximate, as they did not attempt high accuracy. The following few
results show good agreement with those obtained above : — •

Mean concentration.

Results by
interpolation.

By DUNNINGTON and
HOGGARD.

22'22
18-33
13-39

672 -<J
403-8
180

678
422
183

LEMOINE* has determined heats of dilution for some concentrations of lithium
chloride, but his results are not expressed to give a high degree of accuracy sufficient
to compare with the above results.

(6) SODIUM HYDROXIDE. (Molecular weight 40.)

With solutions of sodium hydroxide the same processes were employed. It was
found more convenient to take 250 c.c. of the solutions for the specific heat
determinations.

'Comptes Rendus,' vol. 125, 1897, p. 603.

DILUTION OF CONCENTRATED SOLUTIONS. 335

The gilded brass stirrer with its glass insulated shaft was replaced by a glass one,
the shaft and the propeller blades being fused together. Some slight modification of
the water equivalent was made to suit the new conditions. The strong solutions were
very viscous and the rate of stirring was increased.

Some slight change in the heating coil due to a short circuit of a few turns altered
the resistance of the heating coil — but errors were eliminated by taking readings
of both ammeter and voltmeter each mimite. The heating occupied a thousand
seconds in all cases.

Table VII. gives the data from which the variation of specific heat with concentra-
tion was obtained.

Table VIII. gives the data for the heats of dilution of the solutions at various
concentrations, and at the temperature of the air.

One of the curves in Diagram II. shows the nature of the relation between heat of
dilution and concentration. The points do not lie so evenly on the curve as in the
case of hydrochloric acid, and this is doubtless due to the greater difficulty of experi-
menting with a highly viscous solution. It is, however, interesting to note that the
points for higher concentrations lie most nearly on a straight line, which passes
through the axis of concentration at 12'5 molecules per 100 molecules of water. This
would correspond to a concentration of NaOH . 8H20.

The equation most nearly expressing this part of the curve is

d® = 784ofe -0-125

VN

and this may be considered to hold while the concentration changes from
NaOH . 6H2O to NaOH . 3H20 ; within the same range the heat of dilution
changes to five times its value at the lower concentration.

A curve (fig. 7) may be plotted connecting heat of dilution and the inverse of
concentration, and from it the total heat of dilution as found by THOMSEN can be
estimated from the relation

The relation between Q and N for concentrations below NaOH . 6H2O is very
uncertain. Both THOMSEN and BERTHELOT found the heat of dilution to become
negative for high dilutions, and this also was observed in the above experiment for
concentrations below those recorded above. The effect, however, was very small.

THOMSEN gives the number of calories evolved in changing the composition of the
solution from NaOH . 3H2O to NaOH . 5H2O as 2131 calories. Integrating the above
equation between these limits we get Q = 7840 log, f — 7840 x 0125 x 2, which
gives 2083 calories. Thus although a quite different interpretation of the experi-
mental results is given, the results are in good agreement.

By reference to the curve it is seen that a further dilution of four molecules of

VOL. ccxv. — A. 2 Y

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338 MR W. S. TUCKER ON HEATS OF

water, i.e., from NaOH . 5H8O to NaOH . 9H2O, 979'2 calories are produced. For the
same range THOMSEN gives 962. This agreement is as good as the difficult nature of
the experiment will allow, under the conditions chosen.

(7) HEAT OF DILUTION OF CALCIUM CHLORIDE SOLUTIONS.

All the above solutions yield two ions to the molecule. A fourth solution, that of
calcium chloride, was chosen, not only because of its existence in a high condition of
concentration, but also because it yielded three ions to the molecule. The con-
centrations were obtained from . the density table given by PICKEBING, and the
molecular weight is taken as 111.

Tables IX. and X. give the data obtained in experiments in which the procedure
was very much the same as in the above solutions.

The curve for calcium chloride is of a different character. A small change of slope
occurs for all concentrations. For a limited distance at high concentrations the curve
appears to be nearly a straight line which would cut the axis of concentration, if
produced at about 7 '69 molecules per 100 molecules. This would make a limiting
concentration of CaCl2 . 13H30.

Many attempts were made to get an equation connecting cZQ/c?N and n/N, but
without satisfactory result. The total heat of dilution can, however, be obtained
graphically by plotting dQ/c?N and N/n, as in the case of hydrochloric acid, when the
total heat is measured by area.

The recorded temperature changes on dilution were smaller than in the above
solutions, and the degree of accuracy was consequently not so high.

(8) VARIATION OF HEAT OF DILUTON WITH TEMPERATURE.

The above heats of dilution were determined under nearly isothermal conditions
and at air temperatures.

Very little work has been done to determine the variation of heat of dilution with
temperature, COLSON* lias obtained a so-called " dead point " for various solutions.
He found certain temperatures at whicli the heat of dilution vanishes for certain
solutions. Taking one of these solutions he found the dead point the same for all
concentrations. Later on, however, when experimenting with sodium sulphate, he
found the dead point to vary with varying concentration. He concluded that those
solutions giving a constant dead point contain solutes in which the state of aggrega-
tion is independent of the concentration, and cited sodium sulphate as a substance
whose composition varies with concentration.

MAGiEf found a change in heat of dilution for barium chloride from positive
to negative, giving a zero effect at temperatures which decrease as the dilution
increases.

* 'Comptes Kendus,' vol. 134, pp. 1496-1497, 1902.

t ' Amer. Phil. Soc. Proc.,' vol. 51, 1912.

DILUTION OF CONCENTRATED SOLUTIONS.

339

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XI

DILUTION OF CONCENTRATED SOLUTIONS. 341

No record is given of a systematic study of the variation in heat of dilution by
either of these investigators.

In order to obtain numerical results, the manner in which the specific heat varies
with temperature must first be found.

The temperature effect on the heats of dilution of hydrogen chloride and lithium
chloride were investigated.

The experiments were more difficult to carry out owing to radiation corrections
which now become a greater consideration. Also the range of temperatures for which
the heat of dilution experiments were possible is limited. With concentrated
hydrochloric acid no temperature above 20° C. could be attempted, and below 0° C.
the water employed would, of course, freeze.

With lithium chloride solutions the maximum concentration employed at air
temperatures could not be employed at 0° C. owing to the much smaller solubility of
the salt. Since, therefore, only moderately strong solutions could be taken much
smaller temperature changes were observed on dilution. Attempts were made to get
the heat dilution of these solutions at temperatures above 30° C., but without success,
owing to the small heating effect and the large radiation.

To eliminate the radiation loss the heating coil employed for the specific heat
experiments was tried, but it was found impossible to regulate the temperature of the
coil with sufficient delicacy. Observations were made, but failed to give consistent
results on repetition.

In each specific heat and heat of dilution experiment, separate radiation-loss
experiments were 'made, and these were checked in the following manner.

The quantity of solution to be experimented with was introduced into the vessel
and its temperature raised from about —10° C. (to which it had been previously
cooled), by means of the heating coil, by steps of about 5° C. to about 50° or 60° C.
The rate of cooling was then plotted against temperature, and gave a curve which
closely approximated to a straight line, within the range 10° C. on either side of
air temperatures. The points all lay perfectly on the curve, and the curve cut
the axis of temperature at a point which was found to coincide exactly with air
temperature.

The regularity of this effect, and the certainty with which it could be measured,
made experiments with a relatively large radiation loss quite satisfactory. Thus, in
the specific heat determination of lithium chloride solution at a temperature of about
70° C., a rise in temperature by heating of 10 '21° C. is observed, while the radiation
loss is about 2° C.

For lithium chloride only one concentration of solution was attempted, as
the results only showed a small change in heat of dilution ; and moreover
with the concentrations less than that employed the heating effect is so
small, since in both solutions it falls off very rapidly as the concentration is
diminished.

342 ME. W. S. TUCKER ON HEATS OP

The following specific heats were obtained for hydrochloric acid :-

Concentration
molecule

Temperature.

Specific heat.

per 100 molecules.

23-75

- 3-87

0-589

11-94

0-583

17-05

0-585

14-9

- 4-03

0-657

1-30

0-647

16-25

0-651

10-44

- 3-92

0-722

5-89

0-715

19-78

0-718

24-01

0-732

28-07

0-751

The above table shows that at about air temperatures very little change occurs in
the specific heat of concentrated hydrochloric acid solutions.

Even allowing for possible errors of experiment there is evidence of a minimum
value whose position lies between 5° C. and 15° C.

The above results are consistent with those previously obtained at an average
temperature of 9'92° C. Thus for concentration 2375 the specific heat by inter-
polation is 0'583 ; for concentration 14'9 the value is 0'648 ; and for concentration
10'44 it is 0711.

The values employed in the subjoined table of heats of dilution were derived from
the above tabulated values by interpolation.

The process of equalisation of temperatures before mixing water with the solution
was performed as nearly as possible before the pipette was immersed in the solution.
After immersion the pipette was left for some hours and the solution was continuously
stirred by the propeller stirrer, driven at a high speed.

The process of mixing of water and solution was allowed five minutes to complete,
and during this interval the radiation correction was applied. It was always found
that after that period the temperature changes were always those which would be
foretold by the previous radiation experiments.

DILUTION OF CONCENTEATED SOLUTIONS.
The following values of heat of dilution were obtained for hydrochloric acid : —

343

Concentration.

Temperature.

Heat of dilution
per molecule
of water added.

22-23

°C.

2-12
9-61
20-00

510-5
551-6
581-3

18-85

3-57

20-00

455 • 9
498-7

In both cases, therefore, the heat of dilution rises with temperature, and the mean
coefficient of increase with temperature is greater for the concentrated solution than
for the dilute one, between the extreme ranges of temperature.

Lithium Chloride.

The specific heats of a solution of lithium chloride were found in the same manner
as those of hydrogen chloride. It is possible, however, to employ higher temperatures
than in the above case. On the other hand, the strongest solutions coidd not be used
at the low temperatures owing to the solution becoming saturated.

At the highest temperatures radiation becomes far more prominent, and larger
heating currents were employed. The solutions were heated for 1000 seconds.

The following table gives the specific heats of a solution of lithium chloride
(concentration 18 '8) at various temperatures :—

Temperature.

Specific heat.

"C.

- 0-99

0-688

12-65

0-718

21-67

0-739

35-79

0-780

41-19

0-792

The specific heat increases rapidly with temperature, the relation between them
being nearly a linear one. Heats of dilution for a solution of concentration 14 '2
molecule per 100 molecules are here quoted.

Temperature.

Specific heat.

Heat of dilution.

°C.
2-13

5-98
22-04

0-734
0-742
0-800

221-6
218-5
221-5

VOL. CCXV. A.

2 z

344 ME. W. S. TUCKEE ON HEATS OF

It is evident that although the thermal capacity of the solution may change
rapidly with temperature the heat of dilution suffers very little change.

(9) EXAMINATION OF THOMSEN'S KESUI/TS.*

In most of THOMSEN'S experiments the ranges of concentration are not such as
make the results comparable with those quoted above.

There are, however, a few substances such as sulphuric acid, nitric acid, and
hydrobromic acid, other than the above substances, in which a series of results are
obtained at high concentrations.

THOMSEN finds a hyperbolic relation between total heats of dilution and con-
centration. With the first two acids named this relation is of the form

Q =

where a and b are constants and N — number of molecules of water to one molecule
of acid..

The heat of dilution at any given concentration then becomes

ab

This form of relation cannot be applied to other substances such as hydrochloric
acid and sodium hydroxide. In the latter case the heat of dilution changes sign
when the solution is very dilute. The varying thermal changes observed at higher
dilutions for several substances, suggests that these solutions are more complex than
the stronger ones.

^ Moreover experimental difficulties prevent the measurement of these quantities
with sufficient accuracy to warrant their use for deriving a general equation to
represent the thermal changes for all concentrations.

An attempt is therefore made to analyse the results for stronger solutions only, and
it will be found that the relation Q = a logeN + Z>N + c holds well under these
conditions.

(l) Sulphuric Acid.
From THOMSEN'S values of Q for N == I, N = 2 and N = 5 respectively, the equation

IS

Q = 4666 log, N-196 N + 6595.
' Thermoehemische Untersuchungen."

DILUTION OF CONCENTEATED SOLUTIONS.

345

The accuracy with which this can be applied is comparable with that of THOMSEN'S
formula — in fact the accuracy is greater if the most dilute solution is ignored.

Concentration

By calculation.

molecules of water to

Results by experiment.

1 molecule of acid

(Q).

(N).

THOMSEN'S equation.

New equation.

1

6,379

6,383

6,379

2

9,418

9,428

9,418

3

11,137

11,108

11,114

5

13,108

12,840

13,108

9

14,952

14,883

15,061

19

16,256

16,315

16,587

49

16,684

17,270

14,931

The equation for heat of dilution now becomes

dQ_4666

= ~~

Plotting dQ/cZN and 1/N thus gives a straight line which cuts the axis at N = 23'8,
which is nearly H2S04 . 24ILO. (Diagram II.) The suggestion might be made, there-
fore, that accepting the linear relation between Q and 1/N, the heats generated were
such as if the true solvent were H2S04 . 24H2O instead of pure water.

(2) Nitric Acid.

THOMSEN gives six results for Q with solutions diluted down to five molecules of
water.

Taking three of the values

N = i N = 2'5,

and

N = 5,

the equation derived is

Q = 2099 log, N-18 N + 3437.

By reference to the preceding equations the low value of the coefficient of N
corresponds to ^ff^ molecules of water in the limiting case where cZQ/dN vanishes.
A further equation derived from N = £, and N = 5, which assumes the coefficient
of N to vanish, gives quite as satisfactory a relation, if allowance for experimental
errors is made.

Here

Q = 2020 log, N + 3406.

2 z 2

346

MR. W. S. TUCKER ON HEATS OF

Values of Q derived from these two equations are shown in the table (columns 3
and 4). THOMSEN'S results derived from the relation

NX 8974

N + 1737

are shown in column (5).

Values of Q.

Concentration

By calculation.

molecule of water

to 1 molecule of By experiment,
acid (N).

New

New

THOMSEN'S

equation (1).

equation (2).

equation.

(1)

(2)

(3)

(4).

(5)

0-5

2005

2005

2005

2008

1-0

3285

3419

3406

3285

1-5

4160

4153

4225

4160

2-5

5276

5276

5258

5301

3-0

5690

5641

5625

5710

5-0 6655 6655

6655

6668

THOMSKN quotes results for concentration below HNO:i . 10H20, but for them
neither of the above formulae holds. Referring to the simpler equation here adopted
good agreement is obtained except for N = 1 where an error of about 3 '5 per cent
is shown.

The heat of dilution at any given concentration is now given by

dQ 2020

dN '" N

i.e., with slight error the heat of dilution is proportional to the concentration.
(Diagram II.)

Developing a similar argument to that employed for sulphuric acid, there will be
no limiting hydrate formed.

(3) Hydroibromic Acid.

THOMSEN gives six values for total heat of dilution for various concentrations
of hydrobromic acid. The last of the series is for a weak solution, and is ignored.
The equation is derived from the concentrations N = 3 and N = 6, is

Q = 3356 logeN + 12,222.

DILUTION OF CONCENTRATED SOLUTIONS.
The following table shows the good agreement obtained : —

347

Concentration

Values of Q.

molecules of water to

1 molecule of acid.

By calculation.

By experiment.

2

13,817

13,860

3

15,910

15,910

5

17,622

17,620

6

18,250

18,250

10

19,949

19,100

100

• —

19,910

THOMSEN does not quote an expression for this acid. It is seen as before that the
above type of equation gives good agreement over those ranges in which heat is most
generated, and therefore most accurately measured. As with nitric acid, no limiting
hydrate appears to be formed since

dQ _ 3356

cZN '' N

The graph for this relation is shown in Diagram II.

(4) Acetic Acid.

This solution differs from all the others since the heat of dilution for the highest
concentrations is negative, acquires a minimum value, rises to zero and for dilute
solutions becomes positive.

The results do not lend themselves to either of the forms of equation used by
THOMSEN. Results are quoted for concentrations between CH3 . COOH . ^H,O and
CH3 . COOH . 8H20. The greatest measurable thermal changes occur in this range.

An equation is developed by the method of least squares, and found to be

Q = -63'49 loge N + 40'84 N-194'09,

and the table here given shows the order of agreement between experimental results
and those calculated.

Concentration

Values of Q.

molecules of water to

1 molecule of acid

(N).

By calculation.

By experiment.

0-5

-129-7

-130

1-0

-153-1

-152

1-5

-158-6

-165

2-0

-156-4

-156

4-0

-118-6

-111

8-0

+ 0-37

- 2

348 MR. W. S. TUCKER ON HEATS OF

THOMSEN must have experienced great difficulty in measuring accurately such small
quantities of heat, and the variation of the calculated from the experimental values
are well within the limits of his possible experimental errors.

The limiting concentration is in this case well within the range over which the
linear relation holds. Thus the heat of dilution

dN • \N 63'49/

The second term within the brackets has very nearly the value f and corresponds
2CH3 . COOH . 3HA or CH3 . COOH . 1'5H2O.

Values ofdQ/JN are obtained from concentration 0'5H2O to 8H20. The negative
values are obtained while there is excess of acid added to the critical hydrate
2CH3 . COOH . 3H20, and the positive values when there is excess of water.

DUNNINGTON AND HOGGARD's RESULTS.*

A series of determinations of heats of dilution for different strong solutions, were
made by the above investigators. They made saturated solutions of various salts, and
then added water until some whole number of molecules of water was associated with
each molecule of the salt. Then one or two molecules of water were further added,
and the heat generated or absorbed quoted.

A solution of ammonium acetate was taken in three molecules of water. Heat was
generated throughout the process of dilution. The results quoted show a distinct
break in continuity between solutions containing from six to eight molecules of
water.

Lithium chloride is the only other solution which generates heat to any extent on
dilution. These results have already been compared with those obtained above.

One solution only is given with values extending over a range of concentrated
solutions, in which heat is absorbed. This is ammonium nitrate solution. The initial
and final concentrations are quoted in columns I. and II. in the subjoined table,
expressed in molecules of water (N) to one molecule of solution. The heat generated
was quoted as in column III., using the same type of relation as that previously
adopted for this heat.

Values of a and 6 are thus formed for the range of solutions between N = 3 and
N = 12. They are

a = -1172 I = +4877.

* ' Amer. Chem. Journal,' vol. 22, p. 211.

DILUTION OF CONCENTRATED SOLUTIONS.

349

Column IV. shows the calculated value, which it may be noted compares most
favourably with Column III.

I.

II.

III.

IV.

Initial

Final

Experimental

concentration.

concentration.

values.

Calculated values.

Ni.

N2.

Q.

3

4

- 290

- 288-9

3

5

- 495

- 502-4

3

6

- 667

- 668-9

3

8

- 911

- 914

3

10

-1105

-1075-4

3

12

-1241

-1201

The equation for heat of dilution is

ai

1172

~N"

and applies most accurately for concentrations from NH4NO:! . 3H2O and
NH4N03 . 10HaO, after which it shows a similar divergence to that exhibited by
the other concentrated solutions.

Here we have a case of a solution giving negative heats of dilution ; yet the same
relation may still be applied. The extrapolated value of dQ/dN vanishes when
N = 24'01, suggesting a solution of limiting concentration NH4N03. 24H20. This
may be interpreted in a similar manner suggesting that the first effect of adding
water is to produce this solution, and that we are really diluting solutions, with
another weak solution, not water, of strength NH4N03. 24H2O.

(10) CONCLUSION AND SUMMARY.

To sum up, the following equations can be quoted for heats of dilution for strong
solutions.

(1) Hydrochloric acid (N < 10)

(2) Nitric acid (THOMSEN) (N < 6)

_ 4082
~ ~

2020

N

(3) Sulphuric acid (THOMSEN) (N < 15)

dQ _ 4666

dN '- ~W

350

ME. W. S. TUCKER ON HEATS OF

(4) Hydrobromic acid (THOMSBN) (N < 10)

3356

N

(5) Acetic acid (THOMSEN) (N < 10)

(6) Sodium hydroxide (N < 6)

dQ 7840 q80

~N~ "

(7) Lithium chloride (N < 6)

(8) Ammonium nitrate (N < 10)

6968

N

1172

N

- 871.

+ 4877.

The following table of heats of dilution can thus be drawn up : —

Molecules of water
to 1 molecule
of acid.

N.

Values of r^L

(/iN

HC1.

H2S04.

HN03.

HBr.

CH3 . COOH.

1

2

• —

4470
2137

2020
1010

(3356)
1678

-22-7
+ 9-1

3

4

(1089)
749

1359
970

673

505

1118
839

19-6
25-0

5

544

737

404

671

27-1

6

7

408
311

581
470

(336)

559
479

30-2
31-8

8

238

387

•

419

32-9

9 182

322

373

—

10

271

336

—

The quantities within the brackets are extrapolation values.

The following general conclusions may be drawn from the above facts :—

(l) That, in spite of the varying thermal effects obtained on diluting different

solutions, these effects may be expressed in a simple manner for the range over which

the solutions may be considered concentrated ;

DILUTION OF CONCENTRATED SOLUTIONS. 351

(2) That when the solutions become more dilute they, at the same time, become
more complex ;

(3) That those thermal changes occur in strong aqueous solutions, which suggest
that the solute is dissolved, not in water, but in some definite hydrate of the solute,
in general containing a large number of molecules of water.

In special cases, however, such as that of acetic acid, the water molecules may be
few in number, while with nitric and hydrobromic acids the solvent becomes pure
water.

(4) That calcium chloride does not show the effect in so marked a degree, probably
because it is impossible to work with solutions of the same relative strength.

Full facilities for the execution of this research have been most kindly afforded by
Prof. CALLENDAR. The author wishes to thank him, and to express his appreciation
of Prof. CALLENDAR'S advice and encouragement throughout the work.

VOL. CCXV. — A. 3 A

[ 353 ]

XII. Thermal Properties of Carbonic Acid at Loiv Temperatures.

(Second Paper.}

By C. FKEWEN JENKIN, M.A., M.Inst.C.E., Professor of Engineering Science, Oxford,
and D. R PYE, M.A., Fellow of New College, Oxford.

Communicated lij Sit-- ALFKED EWINC;, K.C.B., F.R.S.

[PLATE 6.]
Keceived March 13,— Read May 13, 1915.

CONTENTS.

Page

Introduction 354

The use of throttling experiments 355

Description of the experiments —

1. Measurement of the total heat of CO-.; gas 356

2. Measurement of the total heat of liquid C0> 360

3. Throttling experiments on superheated gas 362

Construction of the !</> chart 364

Appendices —

I. A method of plotting pressure curves in the superheated area of the ti<l> chart 369

II. A method of plotting I lines in the superheated area of the $</> chart 369

III. Modifications produced in the #</> chart by a correction in the values accepted for the total

heat of the liquid CO,. 370

IV. A method of plotting the results of throttling experiments 371

V. A method of plotting the gas-limit curve from the results of a line of throttling

experiments 372

VI. A method of correcting the results of a throttling experiment for a small departure from

standard conditions 373

VII. Allowance for the small drop of pressure in the 1-inch calorimeter when used to measure

the total heat of the gas 375

List of papers referred to 375

Tables—

A. Observations on the total heat of CO-2 gas 376

B. Mean specific heats of CO2 gas at constant pressure for 10° C. intervals 377

C. Specific heats of CO;, gas at constant pressure 378

D. Observations on the total heat of liquid C02 , 378

E. Throttling experiments 379

F. For plotting throttling experiments 381

VOL. CCXV. A 534. 3 B [Published September 1, 1915.

354 PROF. C. FEE WEN JENKIN AND MR. D. R. PYE ON THE

INTRODUCTION.

L\ a former paper (l) the authors described a series of measurements on the thermal
properties of C03 and the construction of a #<•/> chart embodying those results. It
was pointed out that the superheated area of the chart was incomplete, the
constant pressure lines being only approximately accurate and the total heat (I) lines
omitted. It was also pointed out that throttling experiments on the superheated
gas would form a valuable check on the accuracy of the chart, and the construc-
tion of an L/> chart was postponed till more accurate measurements should have
been made on the superheated gas and the whole had been checked by throttling
experiments.

The present paper describes the additional experiments required to complete and
check the 9</> chart, including a re-measurement of the total heat of the liquid for
which some extrapolated values had been used before, and finally the construction of
the L/> chart. This chart lias been constructed graphically, as the 6(j> chart was,
directly from the observed data and its accuracy checked in various ways by tliermo-
dynamic equations. These equations apply quite generally to all Irf> charts and are
independent of the particular properties of carbonic acid.

The value of the !</> chart in all calculations connected with refrigeration was
originally pointed out by MOLLIKR, and has recently been emphasized in the Report of
the Research Committee of the Institution of Mechanical Engineers (1914).

The authors hope that the new chart may be of practical use, as it extends and
corrects the original one prepared by MOLUKK.

The experiments were made, as before, in the Engineering Laboratory at Oxford.
The apparatus employed was generally similar to that used in the former experiments,
but all the heat measurements were made in anew calorimeter, no;. 1, which eliminates

' O J

radiation and conduction losses.

The measurements of the total heat of the gas for each pressure were made at
several temperatures, so that the variation of specific heat with temperature was
determined. The values of the specific heat found by these experiments were used to
plot the new constant-pressure lines on the 9<p chart. The method used for plotting
these lines is explained in Appendix I. The re-measurement of the total heat of
liquid C02 showed, as was expected, that the values previously used required
correction ; the liquid-limit curve of the 6<f> chart was corrected accordingly. The
throttle experiments, which provide an independent check on both the gas-limit curve
and constant-pressure curves, confirmed the accuracy of the pressure curves and of
the limit curve above -8° C., but showed that a small correction was required
below that temperature. This correction having been made, the chart was completed
by drawing the I lines in the superheated area, (See Appendix II., fig. 6.) The
combined corrections of the two limit curves leave the values of the latent heat
practically unaltered ; the alteration at -50° C. is only 0'3 per cent.

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 355

THE USE OP THROTTLING EXPERIMENTS.

In a throttling experiment gas flows through a throttle valve while the
pressures and temperatures are measured on each side of the valve. No other
measurements are required. It is convenient to make a number of experiments
starting from the same initial conditions, expanding the gas to a series of
lower pressures. Such a group of experiments may be called a line of experi-
ments, since it corresponds to expansion along one I line on the fi<j> chart. It
is convenient to make a number of such lines of experiments all starting
from the same initial pressure but from successively higher temperatures. The
series of final pressures to which the gas expands must be the same in all the
lines of experiments, if the method described below is to be used for plotting the
results.

Throttling experiments cannot, without some other data, be used to plot any part
of the 6<j> chart, but with a single constant-pressure curve they can be used to plot all
the other pressure curves in the superheated area, and also the gas-limit curve. The
resulting curves are not affected appreciably by errors in the total heat of the liquid
(see Appendix III.), and are quite independent of errors in the latent heat, so that
throttling experiments form an independent check on the accuracy of the gas-limit
curve and all the constant-pressure curves.

A method of plotting the throttling experiments so as to check the constant-pressure
curves is explained in Appendix IV. By this method a series of constant-pressure
curves is drawn which are to be compared with the pressure curves already obtained
from the specific heat measurements. If the curves coincide, the throttle experiments
confirm the accuracy of the chart ; if the curves are parallel but not coincident, the
throttling experiments confirm the accuracy of the specific heat measurements, but
indicate that there is an error in the position of the gas-limit curve, which may be
shifted so as to make them coincide. If they are not parallel, there is a disagreement
between the throttling experiments and the specific heat measurements and one or
other must be wrong. Since shifting the gas-limit curve will make the curves, if
parallel, coincide, a new gas-limit curve may be plotted derived directly from the
throttle experiments — a method of doing this is described in Appendix V.; whether the
new gas-limit curve is more reliable than the original depends on the reliability of the
different experiments.

In order to use the methods we have adopted the throttling experiments must
be arranged exactly in lines, and the series of lower pressures must be exactly the
same in all the lines, as already stated. When actually making the experiment
it is hardly possible to adjust the initial pressure and temperature and the final
pressure exactly to the standard values, so that the observed results have to be
reduced to standard conditions. A method of making this reduction is explained in
Appendix VI.

3 B 2

35 6

PROF. C. FREWEN JENKIN AND MR. D. R. PYE ON THE

DESCRIPTION OF THE EXPERIMENTS.

Measurement of the Total Heat of CO2 Gas.

The method employed in these experiments was the same as that used in the former
series (Series III., p. 73), but the apparatus was modified in several details. The gas
used was, as before, supplied by Messrs. Barrett and Elers, Limited ; the same instru-
ments were used for weighing the gas, measuring its pressure and temperature, and
the electric heat given to it. The instruments were re-calibrated from time to time as
a check, in the same manner as before.

The weighing flasks, condenser and drying flasks were the same as before, but
calcium chloride was used in the drying flask instead of phosphorus pentoxide, which
occasionally gave trouble. The single-stage pump was replaced by a larger two-stage
pump lent for the purpose by Messrs. J. and E. Hall, of Dartford. With this pump
the rate of working could be more easily controlled, and the attainment of low
pressures was facilitated. The large calorimeter (No. I.) was discarded, and the
evaporation and warming of the gas was done in calorimeter No. II. with the addition
of a simple heating vessel made of a length of 2^-inch steel pipe enclosing an electric
heating resistance.

SCALE

INCHES

Fig. 1.

All measurements of heat were made in a new 1-inch tubular calorimeter, fig. 1.
This calorimeter consists of a central body with a thermojuriction fitting at each end.
The central body is made up of three concentric pipes held between two vulcanite
ends ; the outer pipe is made of steel, the intermediate pipe of vulcanite, and the inner
pipe of fused silica. Inside the inner pipe is a coil of nickel-chrome wire forming the
heating resistance. Folded round the outside of the silica tube is a thin strip of
copper which serves as the return lead for the current flowing through the heating
coil. The whole is held together by two external bolts. The gas enters at one end,
passes through the silica tube, and then returns outside the silica tube and inside the
vulcanite one, and finally returns again outside the vulcanite and inside the steel tube.
At each end are the two thermojunction fittings ; they are the same fittings as were
used with the throttle valve.

THEEMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 357

The muff was then

It will be seen that the temperature of the entering gas is measured just before it
enters the body of the calorimeter, and the temperature of the issuing gas just after
it leaves the body of the calorimeter ; in neither case can the temperature of the gas
in the calorimeter be affected by conduction along the pipes. Most of the experiments
were made with the issuing gas at atmospheric temperature, so that the outer steel
shell was at atmospheric temperature and could neither give out nor receive heat. To
make this doubly sure the calorimeter was wrapped in cotton wool and placed inside
a muff filled with water kept at approximately the same temperature as the shell of
the calorimeter. The only remaining path by which heat can enter the calorimeter
is through the left-hand vulcanite end ;-the amount flowing in by this path must be
extremely small. There is a Hoskins alloy thermocouple clamped to the outside of
the steel t\ibe and connected to a very sensitive galvanometer which gives a deflection
of 20 mm. per degree change of temperature of the junction. The electric power was
regulated during an experiment so as to keep the temperature indicated by the
galvanometer as steady as possible.

A few experiments were made with the issuing gas at +30° 0.
heated to about the same temperature, but it is
possible that a little heat leaks out at the ends
of the muff, so that these experiments may be
slightly wrong.

The passages through this calorimeter arc
rather small, so that there was an appreciable
drop of pressure in the gas as it passed through
it. The difference of pressure between the inlet
and outlet was measured by means of a differen-
tial pressure gauge, the construction of which
is shown in fig. 2. An inner glass tube dips
into mercury contained in the outer steel tube.
The difference of level of the mercury inside and
outside the glass tube, which is proportional to
the difference of pressure is measured by the
change of resistance of a wire stretched down
the centre of the glass tube. Several sorts of
wire were tried ; some made uncertain contact
with the mercury, but a bright hard steel banjo
wire answered very well. The resistance of the
wire was measured with a simple Callendar and
Griffith bridge. The gauge was calibrated by
direct comparison with a glass U-gauge containing mercury. The arrangement of
the valves enabled the pressure to be applied and released quickly, so that the zero
and pressure readings could be repeated several times as a check on their accuracy.

HIGHER

"PRESSURE

LOWER
PRESSURE

SCALE

1 INCHES

Fig. 2.

358

PROF. C. FEEWEN JENKIN AND MR. D. R. PYE ON THE

The general arrangement of the apparatus is shown in fig. 3. The liquid CO2 from
the weighing flask expands through the throttle valve V, and is evaporated in the
2^-inch heater. From there it passes in the condition of nearly dry vapour into
No. II. calorimeter where the evaporation is completed and the gas is warmed to the
desired temperature. It then passes into the 1-inch calorimeter where its temperature
is raised through an accurately measured range by the application of an accurately
measured electric power. From this calorimeter it passes on through, regulating
valves and a drying flask to the pump, which compresses it into the condenser and so
back to the weighing flasks.

The object of the 2^-inch heater was to facilitate the regulation of the temperature
of the gas. By adding most of the heat necessary for evaporation in the 2^-- inch

STXI/WVWO
G«AG£ (7)

V6 Y*

DRYING

FLASK

OIFF-

GUAGE CMJORIMETEfl

Fig. 3.

(-24

heater, the temperature of calorimeter No. II. could be kept only a few degrees above
the required gas temperature, so it acted as a steadying reservoir of heat. Without
the 2^-inch heater the arrangement was unstable, for then the calorimeter No. II. had
to be kept much warmer in order to transfer sufficient heat to the vapour. The
point where the vapour became dry was then near the top of the coil, and the least
variation in the rate of flow made this point move up or down and so left a varying
amount of surface for superheating the gas, the temperature of which, therefore,
varied widely. For experiments at low pressures, viz., 150, 200, 300 and 400 Ibs.
per square inch, the speed of flow was controlled by the valve Vj while V2 and V3
were open. For the experiment at 500 Ib. V3 was also partially closed. For the
experiments at 600 and 700 Ib. V3 was open and V2 was partially closed. For the
700-lb. experiment the weighing flask had to be slightly warmed by a gas ring.
During an experiment the valve V5was kept open and V6 closed, so that the standard

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 359

pressure gauge read the gas pressure pit and the differential pressure gauge was in
equilibrium, which avoided any danger of blowing the mercury out of it. Just before
the end of each experiment the valve V5 was closed and V4 and V6 opened and
readings taken on the differential gauge.

The quantities measured in an experiment were : —

The rate of flow of the gas.

Its initial pressure and the drop of pressure in the calorimeter.
The temperatures of the gas entering and leaving the calorimeter.
The electric power supplied to the calorimeter.

-30

30

The following quantities were also noted :—

The temperature of the muff and of the atmosphere.
The galvanometer readings.

The results are given in Table A (p. 376), arranged in the same way as in Table IV.
of the former paper. The correction for the small drops of pressiire in the calorimeter
is explained in Appendix VII. The results are plotted in fig. 4.

300

PROF. C. FREWEN JENKIN AND MR. D. R. PYE ON1 THE

The values of the mean specific heat of the gas for 10° C. intervals have been
measured from the curves and are given in Table B. The pressure curves in the 00
chart were plotted with these values, by the method explained in Appendix I., after
the limit curve had been corrected by the throttle experiments.

The values of the specific heat for each pressure at temperatures varying by 50° C.
intervals from —30° C. to +30° C., as measured from the curves, are given in Table C.

Measurement of the Total Heat of Liquid C02.

The experiments described in the former paper on the total heat of the liquid CO2
were not completely satisfactory for two reasons.

(i.) The lowest temperature reached was — 39 '1° C., so extrapolated values had to
be used for drawing the 0</> chart between —40" C. and —50° C.

(ii.) The measurements made from — 39'1° C and — 35'2° C. (the two lowest),, both
had considerable corrections for " fall of bath temperature " and " radiation," amounting
together to 8.V per cent, and 5^ per cent, respectively of the measured heat.
Subsequent experience has shown that the correction for the " fall of bath

70 PUMP

FROM
WE/Cff/HG FLAbK

CAL? N°1L

temperature " is liable to considerable error. It was assumed to be equal to the
water-equivalent of the bath multiplied by the fall of temperature during the
experiment. But the value of the water-equivalent of a composite body such as the
calorimeter, consisting of a good heat conductor surrounded by an insulator, depends
on the quantity of heat absorbed by the lagging, and this depends on the rate of
change of temperature ; as this may be very different in the actual experiment from
what it was when the water-equivalent was measured, the value assumed for the
water-equivalent may differ considerably from the effective value.

The effect of conduction along the pipes appears also to be somewhat uncertain ;
the correction for this was included in the radiation correction.

These considerations made it desirable to repeat the measurements of the total heat
more accurately. The method employed for this purpose was the same as before
(Series II., p. 72), but improved apparatus was used. By using the new 1-inch calori-
meter described above, radiation and conduction losses and the correction for the
change of temperature of the bath were eliminated, and the introduction of a heat

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 361

interchanger enabled the temperature of the liquid to be carried down to — 50'6° C.,
eleven degrees lower than before. The interchauger is made of two concentric copper
pipes, 10 feet long, coiled into a ring 9-inch diameter. It is the same piece of
apparatus which is described in the former paper on p. 92. By means of this

e

•K) --05

ENTROPY

interchanger the liquid CO2 was cooled before passing the throttle valve by the cold
gas leaving calorimeter II. , an arrangement which considerably increases the available
refrigeration ; it was only used for the last experiment at the lowest temperature.
The arrangement of the apparatus (including the interchanger) is shown in fig. 5.

VOL. CCXV. A.

3 c

362 PROF. C. FKEWEN .TENKIN AND ME. D. E. PYE ON THE

The liquid from the weighing flask passes through one coil in calorimeter II. and then
through the 1-inch calorimeter, then through the inner coil of the interchanger and
expands through the valve V into the second coil in calorimeter II. ; from there it
passes through the outer coil of the interchanger, through the drying flask to the
pump. The liquid is cooled to the desired starting temperature in calorimeter II. It
is then warmed through an accurately measured range of temperature in the 1-inch
calorimeter by the accurately measured electric power. The warm liquid is cooled in
the interchanger and then, by expanding through the valve into the second coil, cools
the calorimeter II, thus supplying the refrigeration required to cool the liquid in the
first coil. The cold gas passing out of calorimeter II. serves finally to cool the liquid
in the interchanger and then passes to the pump.
The quantities measured were :—

The rate of flow of the liquid CO2.

The rise of temperature of the liquid in the 1-inch calorimeter.

The electric power supplied to the 1-inch calorimeter.

The results of the experiments are given in Table D, arranged in the same way as
in Table. IIT., of the former paper. The new results have been used to correct the
liquid-limit curve on the 0</> chart (see fig. G), where the full line is the new and the
dotted line the old curve. The maximum correction, at —50° C., amounts to 0'0054
carnots and decreases to zero at —20° 0.

A correction of the limit curve produces a very small correction on the I lines, as is
explained in Appendix III. The I lines have been shifted 0'0005 to the right at
— 50° C. — too small a distance to be shown in the figure.

Throttling Experiments on Superheated (if as.

Throttling experiments are simple and quickly made. The only four quantities to
be measured are the pressures and temperatures before and after the throttle valve.
The rate of flow is immaterial ; it is only necessary to adjust the apparatus till the
conditions are steady and then to take the four readings. The throttle valve used
in the new experiments was the same one that was used for the experiments on
throttling the liquid C02 described in the former paper (Series IV., p. 91).

The arrangement of the apparatus is shown in fig. 7. The liquid C02 coming from
the weighing flasks expands through the valve Vj into the 2j-inch heater where it is
evaporated and then passed into calorimeter II., where it is warmed to the required
initial temperature. It then passes through the special throttle valve where the
pressure falls to the required lower pressure, and so through the drying flask to the
pump. The valve V2 serves to regulate the lower pressure when necessary.

The results are given in Table E where each line of experiment is denoted by a
letter (column l).

Columns 2 and 3 give the standard initial and final pressures.

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 363

Columns 4, 5, 6 and 7 the observed gauge pressures and corresponding absolute
pressures.

Column 8 gives the observed initial temperature.

Column 9 gives the same temperature + dti (see Appendix VI.).

Column 10 gives the standard initial temperature.

Column 11 gives the observed final temperature.

Column 12 gives the same temperature +dt2 (Appendix VI.).

Column 13 gives 02 which is the observed final temperature +dt2 + dT2
(Appendix VI.).

The lines of throttle experiments do not all extend over the same range of
pressures.

Lines H, J, K, L, Q and E extend from 600 to 200 Ib. (J misses two lines.)

M extends from 500-200.

N „ „ 400-200.

O „ „ 300-200.

M, N and O were limited at their upper ends by the maximum temperature to
which the experiments could be carried. Lines S, T, U and V start at 600 but

Fig. 7.

observations were only taken on the 200 and 150 curves. They were made to link on
the 150 curve, which had been omitted from the earlier experiments. Line P
extended from 700 to 600 and 500. This was the only line taken up to 700-lb.
pressure.

These results were plotted on the 6<f> chart in the way explained in Appendix IV.
The results for the 700, 600, 500 and 400 pressure curves were found to coincide with
the curves already plotted, and the results for the 300, 200 and 150 pressure curves
were parallel to the curves already plotted, thus confirming the values of the specific
heats used to plot these curves, but indicating that the limit curve required a small
shift at pressures below 400 ; the limit curve was accordingly redrawn below the 400
curve in the way explained in Appendix V. (see fig. 6), in which the full line represents

3 c 2

3G4 PROF. C. FREWEN JENKIN AND MR. D. R. PYE ON THE

the- new curve and the dotted line the old curve. The maximum shift, at -50° C. is
0'0042 carnots. The three pressure curves, 300, 200 and 150 were then redrawn to
start on the new limit curve, and the throttle experiments for these curves replotted.
The calculations, made as explained in Appendix IV., for this second plotting are set
out in Table F, which gives the co-ordinates 9 and <£ for the points on each line.
The co-ordinates are arranged in columns ; each column contains the results for one
curve, the value of the pressure being stated at the top of the column. The second
line in each column gives the value of I on the limit curve for that pressure.

The 400-pressure curve was assumed to be correct in plotting the results of all
lines which had readings on that curve, i.e., H, J, K, L, M, N, Q and R The O
line was plotted from the 300-pressure curve and S, T, U, V, from the 200 curve. P
was plotted from the 500 curve. The results, with the three bottom curves replotted in
this way, should all fall on the pressure lines : the agreement is remarkably good, the
errors being too small to show in fig. G. The alteration in the two limit curves which
are shown to be necessary by the new experiments leave the width of the diagram
between the limit curves practically unaltered, so that the values previously obtained
for the latent heat are hardly changed.

The reasons for accepting the modification of the bottom of the gas-limit curve are
as follows : the extreme simplicity of the throttling experiments make errors improb-
able. The close agreement between all the specific heat experiments and all the
throttle experiments over wide ranges of temperature makes any systematic error in
the throttle experiments improbable. The close confirmation of the accuracy of the
upper part of the limit curve also supports the accuracy of the throttle experiment.
The temperatures readied in the throttle experiments were lower than ^hose given by
the uncorrected chart. Had there been errors in the throttle experiments due to
conduction or radiation the difference would have been in the other direction.

Construction of tlie I^> Chart.

The chart (Plate 6) is plotted on skew co-ordinates, the angle between the axes being
arc-tan ( — 0'3 degrees). The vertical I scale is 1 inch = 5 Th.U. ; the horizontal 0 scale
is 1 inch = O'OOS carnots. The whole chart is divided into a skew graticule, the distance
between horizontal lines being 1 Th.U. (J, inch), and the horizontal distance between
the sloping lines being O'Ol carnots (2 inch).

The part of the chart from — 50° C. to -I- 23° G. (and the super-heated area up to + 30° C.
for pressures below 800) represents the results of the authors' experiments. The rest
of the chart has been completed by using MOLLIEB'S (2) limit curve (approximately)
and AMAGAT'S (3) results for the relation between pressure, volume and temperature.
These data alone are not sufficient — some data on heat quantities are necessary ; we
have therefore used JOLT'S (4) experiments on the specific heat at constant volume,
which cover a considerable range (between the lines J^ and J4 on the chart), and for

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 365

the rest of the area we have assumed a value for the specific heat at constant volume,
viz., Cv = 0'214. This assumption is probably close to the truth, since this value,
found by JOLY for the line J4 is exactly equal to the value deduced from our own
experiments for the line A. MOLLIEK assumed that Cv was constant and equal to
0'182 for the whole of the area outside the limit curve.

It would appear at first sight that most of the 10 chart might be constructed by
simply replotting the 0<p chart on the new co-ordinates, and our first chart was plotted
in this way, but when it was checked it was found to be inaccurate. Investigation
showed that the errors were due to three causes ; the difficulty of reading the I values
accurately enough from the $<•/> chart; the insufficient accuracy of MOLLIER'S (5)
table of AMAGAT'S results which we had used ; and lastly a small error in the limit
curve which was greatly magnified when transferred to the new co-ordinates, where
the (/> scale is about five times as large as in the 6</> chart. MOLLIER'S table gives
AMAGAT'S results in a very convenient form ; it is accurate enough for most purposes,
but not for plotting the volume lines of the !</> chart, particularly near the critical
point. We found it necessary to plot AMAGAT'S results and interpolate graphically
to obtain the required data with sufficient accuracy. The errors in the first chart led
us to devise a number of ways of checking it, many of which turned out to be most
useful in reconstructing it. For this purpose the following equations connecting the
total heat I with the other variables, p, v, 6 and <j> are useful.

(l) The slope of any constant-pressure curve in an !</> chart is equal to the
temperature, or

(2) The slope of any constant-temperature curve in an !</> chart is equal to the

absolute temperature minus - — : — , or

the dilatation

'dl\ fde\

— = Q—V[— (ll)

df/t \dvlp

\

Cor. 1. At the critical point the limit curve, the constant- temperature curve, and
the constant-pressure curve are mutually tangent to one another, and their slope is
equal to the critical temperature. There is a point of inflexion in the constant-
temperature curve, and the curvature of the constant-pressure curve is zero at this
point.

Cor. 2. Constant-pressure curves do not change their direction on crossing either
limit curve.

Cor. 3. There is no point of inflexion in any constant-pressure curve ; curves of
the shape shown by MOLLIER (7) for the 80 and 90 kg. curves in the CO2 chart are
therefore impossible.

366 PROF. C. FEE WEN JENKIN AND MR. D. R. PYE ON THE

Cor. 4. The slopes of all the constant-pressure curves are the same at the points
where they are cut by any one temperature curve. Thus every constant-pressure
curve on the CO2 chart has the same slope at the point where it is cut by the 35° C.
temperature curve, viz., 308.

Cor. 5. The slope of the constant-temperature curves is a convenient measure of
the " imperfectness " of the gas, for the slope is equal to the difference between the
reciprocals of the dilatation of a perfect gas and of the gas represented, i.e.,

slope = -

dilatation of perfect gas dilatation of actual gas
CM. \ _ fl idj

jUi

(4) The slope of any I curve in a Q<f> chart may be expressed in any of the following
forms : —

(1Q\ = A l~1 _0 AMI
di/Ji ~~ C;, L v W#/ J'

p v

j.0 (de\ ,. .

. -J-) ......... (ivc)

Equations (iv6) and (ivc) are particularly useful in the saturated area, where they
reduce to : —

'de\ ._ e2 Vs-V,

and

fde\ J0 de

, , / - • --, --^- (ivc?)

d(f>/i V Lj-l!

7 J i • . , .......... (ive)

d(j>/i v dp

of which the first is the more convenient.
Cor. 1. Equation (ivd) may be written :—

dj?\ = e2 V3-V1

~'

where x is the dryness fraction. Thus the slope of I lines varies with x across the <ty
chart from one limit curve to the other.

Cor. 2. The forms (iva), (iv6) and (ivc) become indeterminate at the critical
point, but (ivd) or (ive) may be used. For CO2 the slope of the I line at the critical
point is about 3730.

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 367

Cor. 3. The horizontal distance S<f> between two constant-temperature lines in the
saturated area of the 10 chart which differ in temperature by §0 is

The distance may be found for any value of <j> by using the corresponding value of
V (depending on the dryness).

This expression was used in plotting the 10 chart to find the distance of the 30° C.
line from the 23 '2° C. line (900 Ib.) and the distance of the critical point from the
30° C. line.

Using the above equations as checks a second 10 chart was drawn, plotting from
the original data as far as possible instead of copying from the 9<j> chart. During
construction it was constantly checked by the above formulae and small corrections
made ; in this way we finally arrived at a result which represents all the experimental
data and also complies with the above general theorems. The only part of the
diagram which does not fit in quite satisfactorily is the short piece of the liquid-limit
curve between-f 10° C. and + 23° C.

In order to make sure of the remarkable form of the temperature curves on the
left of the diagram, these curves were extended some distance further than they are
shown. There was no difficulty in doing this, since AMAGAT'S results go to much
higher pressures, but as the pressures increased we found it impossible to make the
slopes of the pressure curves comply with equation (i). This discrepancy appears to
point to some variation in the specific heat at constant volume in this region. As
there are no data available we have omitted the curves beyond 1800 lb.,up to which
pressure the discrepancy, is very slight.

If the chart be examined it will be seen that JOLY'S experiments and ours overlap
and may therefore be compared. JOLY measured C^ and we measured Cp ; to compare
the results it is necessary to obtain an equation connecting the two. The most
convenient equation is

a form which may be derived directly from the fundamental thermodynamic relations,
and is quite general.

The familiar relation for a perfect gas — -

°>-c- =>*>©,

does not give even approximate results. Comparing JOLY'S results with ours by
equation (i) and using AMAGAT'S figures to obtain the differential coefficients, we found
that they agreed within the limits of accuracy of JOLY'S experiments.

The first lines drawn on the 10 chart were the 700-lb. line from zero down to

368 PROF. C. FREWEN JENKIN AND MR. D. R. PYE ON THE

— 50° C. and the 900 line from zero up to + 23'2° C. where it meets the limit curve.
The starting points for these curves are the points

I = + 0'64, </> = - 0'0024 for 700-lb. line.
I = +0'44, </> = - G'0049 „ 900-lb. „

The limit curve from —50° C. to + 23 '2° C. was then set off from these curves;
it passes through the zero temperature point where <p = 0, I = + 0'9. The sloping
straight lines corresponding to pressures in the saturated area from 900 Ib. to 150 Ib.
were then drawn from the limit curve at slopes equal to their temperatures. The
gas-limit curve was then determined by marking points on each of these pressure
lines at values of <-/> taken from the 0</> chart. The apex of the limit curve was then
plotted by means of equation (iv) Cor. (3).

The gas-pressure and temperature curves up to + 20° C. were th'en plotted from
the throttle experiments ; and starting from the 20" C. line JOLY'S four constant-volume

lines J1; Jn, J3 and J4 were plotted, ten degrees at a time, by means of the expressions

/j
(51 = Ct,M + Anty> and §</> — OJog, — using JULY'S values for Cu. Several more constant-

PI

volume lines were then drawn between J4 and 0° C. in the liquid area, assuming that
Gv = 0'214. The values of ftp were in all cases calculated from AMAGAT'S data. The
points on all these volume lines were then marked where they are cut by the
constant-pressure curves — again using AMAGAT'S data. The constant-temperature
curves for 10° (J. intervals were then drawn through the points already marked on
the volume lines. The constant-pressure curves were then drawn by drawing an
envelope of tangents passing through the points already marked on the volume lines.
Any error in the chart was shown up to this stage by the envelope of tangents missing
the marked points on some of the volume lines. This check is a very rigorous one.
An error of ±0 thermal unit in the position of the liquid-limit curve will throw a
pressure curve 1 inch away from the point it has to pass through.

AMAGAT'S data do not go below zero, so a different procedure was necessary for the
liquid area below zero. Constant-temperature lines were drawn at —25° C. and — 50° C. ;

the first is horizontal because (— ) =0 at — 25° C. as is shown in fig. 9 of the former

paper ; the second slopes upwards at an angle which can be calculated from the
formula

(dl\ =_(dl\ (d6\
\dplt \dejf ' \dp)i

= Gpx throttle drop* per 100 Ib.
using the data given in Table VIII. of the former paper, extrapolated to — 50° C.

^ The throttle drop per 100 Ib. is given in fig. 9 of the former paper, where it is called the Joule-
Thomson effect.

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 369

The spacing of the pressure curves on the horizontal —25° C. temperature line is
given by the expression

the values of ( ~ ) were calculated from fig. 1 1 of the former paper.

\('l6/p

The spacing of the pressure lines on the —50° C. temperature line was assumed to
be uniform, which must be close to the truth as may be seen from fig. 11 of the former
paper.

The pressure curves were then extended from the 0° C. volume line through the points
marked at — 25° C. and— 50°C.; keeping them similar in form to the 700-lb. line which
was drawn from experimental data. The lower ends of the high-pressure curves have,
however, been omitted from the final chart, because the data for plotting them were
considered to be hardly sufficiently accurate.

Additional constant-temperature curves were interpolated using the above formula
to give the initial slope, those above —25° C. being slightly curved to match those at
higher temperatures whose form was known. These complete the !</> chart.

APPENDIX I.

A Method of Plotting Constant-pressure Curves in tJte Superheated Area of

the 6<l> Chart.

Each pressure curve starts from a point on the gas-limit curve whose temperature
is given by the pressure-temperature curve for saturated vapour. Above this point
it is plotted in 10° C. steps. For each vertical step of 10° C. the horizontal step, d<j>, is

given by the equation : —

6d(j> —

in which 6 is the mean temperature of the step, dO = 10° C., and a- is the mean specific
heat of the gas for the step (see Table B).

APPENDIX II.

A Method of Plotting I Lines in the Superheated Area of d<f> Chart.

The I lines have already been drawn in the saturated area for values differing by
5 Th.U., i.e., for I = 0, 5, 10, 15, &c. The last of these is I = 55. It is required to
plot the lines I = 60, 65, &c., which fall in the superheated area. The line is plotted
by finding the points where it crosses the pressure curves, Let ALP be any

VOL. ccxv. — A. 3 D

370

PBOF. C. FEEWEN JENKIN AND ME. D. K. PYE ON THE

pressure curve, cutting the I = 55 line in A., and the limit curve in L. We have to
find the point P on this curve where I = GO.

1 = 55

With the notation in the figure

Again .
and also
Therefore

0md,j, =

do = Ii

Having found dO, the point P on the pressure curve is marked off. Similar points
are found on the other pressure lines and joined up, thus forming the required I curve.
Points on all the curves after the first are plotted, each from the one before, by means
of the equation

,18 = 5 Th-U- .

APPENDIX III.

Modifications Produced in the 9<f> Diagram by a Correction in the Values
Accepted for the Total Heat of the Liquid C02.

If the line AB represents the constant-pressure curve, p = 700. Ib. per sq. inch,
then the area MABN under this curve down to absolute zero represents the heat
required to raise the temperature of 1-lb. of CO2 at 700-lb. pressure from 0A to 0B, i.e.,
the total heat from 0A to 0B. If the total heat were assumed to be 5 per cent, less,
then A would have to be shifted to A', where A A' is 5 per cent, of AQ, i.e., about
0'005 carnots at -50° C. The limit curve, which is set off from the constant-pressure

THEEMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 371

curve, would be shifted almost exactly the same distance to the right, pivoting round
the point B' which has the same I as B on the pressure curve.

I

ENTROPY

The constant I lines would hardly be shifted. They are set out from points on the
limit curve in the way explained on p. 80 of the former paper, and calculation shows
that they would only be shifted about one-tenth as far as the limit curve is shifted.
Thus on the —50° C. line they would only be moved about O'OOOo carnots to the ri^ht.

-27-3°

M

N

APPENDIX IV.

A Method of Plotting tlie Results of Throttling Experiments.

In the following method the results of the throttling experiments are used to draw
all the constant-pressure curves but one ; the position of that one is assumed to be
known. The pressure curves drawn in this way should coincide, if the chart is
correct, with the curves drawn by means of the observed specific heat of the gas.

Let CHAA' A" be the pressure curve _p,, assumed to be correctly drawn, on which
A, A' ... represent the starting points of a number of experiments throttling down to
a second pressure pa. Let A be the starting point of the experiment beginning at the
lowest temperature, and let the observed final temperature, when throttling down to
p.2, be 0K. Draw a horizontal line at this temperature. It is required to find the
point B on this line representing the final condition.

3 D 2

372

PEOF. C. FKEWEN JENKIN AND ME. D. E. PYE ON THE

We have

IB = I,

(i)

and IA = Iji + a-i(0A— 9C), where o-j is the specific heat of the gas at A, and In, on the
limit curve, is known. Let K be the point on the limit curve corresponding to p2 and
let d<f> be the horizontal distance between K and B.
Then

= -IK H

0_L A*
B -T "K

-

_ T

A'

an equation giving d^ and thus fixing B.

Choosing the next experiment, starting at A', we find the point B' by equating
IA. — IA = IB< — IB- and so on.

Joining up the points B, B', &c., we have the constant-pressure curve corresponding
to the pressure p2 derived from the throttling experiments. A similar construction is
used to plot the throttling results corresponding to all the other pressures. If the
chart is correct the curve B, B', B" will coincide witli KL, the constant-pressure curve
plotted through K by means of the observed specific heat of the gas.

APPENDIX V.

A Method of Plotting the Gas-Limit Curve from the Results of a Line oj

Throttling Experiments.

The limit curve is plotted by finding the points where it cuts successive pressure
curves. Let A (Oipi) and B (02p2) represent the initial and final conditions of the
gas in one of the lines of throttle experiments. COA and DPB are the pressure

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 373

curves p^ and p2 through A and B, OP is the limit curve, and CD the last I curve in
the saturation area.

We assume that the points O and A are known and we require to find P where the
limit curve cuts p.2. Draw a horizontal line DP at 03. Then P will be fixed when
0i = 0p — 0n is found.

Let — -3 = 9m and JI be the difference of total heat between D and B.

Zt

Then

But

therefore

_SI-o-2S9.

(i)

SI = 0.^

(ii)

Similarly, selecting other pressure lines, we find a series of points P which all lie on
the limit curve. The same method is applicable for finding points P on pressure
curves above O, if O is not on the highest pressure curve. For checking our 6<f> chart
the point O was chosen on the 400-lb. pressure curve.

APPENDIX VI.

A Method of Correcting the Results of a Throttling Experiment for a Small
Departure from Standard Conditions.

Let A (pA) be the standard starting point and p2 the standard final pressure.
P and E are the actual starting point and finishing point. P and R lie on an I line
which cuts P! in Q and p2 in S. The I line through A cuts the p2 line in U. We

374

PROF. C. FREWEN JENKIN AND MR. D. R. PYE ON THE

have to find the temperature of U, i.e., the temperature which would have been
observed if the conditions had been standard.

With the notation in the figure

Similarly,
and

dta = (^)i dp.

7T1 "*1 7"P

ala = — alj.

The required temperature of U, 9a = 6K + dt.2 + dT2.

The values of ( -. - ) were measured from the 00 chart and plotted against tempera-
\dpji

tures, and it was found that for the range covered by the experiments the values
were roughly constant for any one temperature (i.e., independent of pressure). They
may, therefore, be given with sufficient accuracy as follows : —

Mean temperature

•c.

30

20

10

0

-10

-20

-30

Id0\

vWi'

0 C./lb.

0-074

0-079

0-085

0-093

0-103

0-116

0-135

THERMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 375

APPENDIX VII.

Allowance for the Small Drop of Pressure in the \-inch Calorimeter ivhen used to

Measure the Total Heat of the Gas.

Let P be the point representing the condition of the gas entering the calorimeter,
and Q the point representing the gas leaving the calorimeter. PR is the constant-
pressure curve meeting the I curve through Q in II. Through Q draw a horizontal
cutting the pressure line in S. Let the drop of pressure in the calorimeter he dp.

The amount of heat given to the gas = I(i — IP = IH— Ip-

If there had been no drop of pressure in the calorimeter, this amount of heat would have
raised the gas to 0U instead of fl(i. The actual heat given is IK-!,., but to raise the tem-
perature to 0^ (or 0S), if there were no throttling, we should have needed only Is — IP.

The heat may therefore be corrected by subtracting Iu — Is = rrdO = a(~7~) &P- '^'ie
" throttle correction " (Table A) is calculated in this way ; it is in all cases very small.

PAPERS, &c., REFERRED TO.
Reference.

(1) " Thermal properties of carbonic acid at low temperatures " by JENKIN and PYE,

'Phil. Trans., Roy. Soc.,' Series A, vol. 213, p. 67 (1914).

(2) MOLLIER, ' Zeit. fur die gesamte Kalte-Industrie,' Heft 4 and 5, pp. 65 and

85, 1895.

(3) AMAGAT, ' Annales de Chemie,' tome 29, 6th Series, 1893.

(4) JOLY, 'Phil. Trans.' A, 1894, p. 943.

(5) MOLLIER, ' Zeit. fur die gesamte Kalte-Industrie,' Heft 4, p. 65 (1896).

(6) EWING, "The Mech. Production of Cold," 1908. The reference is to p. 195.

(7) MOLLIER, ' Zeit. des Vereins Deutsch. Ingenieure,' 1904.

376

PROF. C. FEEWEN JENKIN AND MR. D. R. PYE ON THE

(M lO

CO <M to
-•^ CO O O O

<M t- in t

00 OO r-H O CO C

t— IM m o in
s o to ^ o *^

GO Oi iO OO TH C
i— ( <M i— < C

•s O OO CD O CO

GO O CO CM O

« IM CM O
D T(t CO t~ O t-

I-H I-H f-n GN7 GN i~

H CO

CD in CD CO C

os os o in co -^

s CD i— OO O oo
H O •* CO O CO

CO
CN 1^- tM 1O t

OS C^ t*»

"> oo o» in o in

r-H I r-H i — 1 r-
O

O

H CO

O i— < lO O ^ C

i— 1 i— I i—( CO i-H C

s o m •**< o *^

o
o

IO

GO O CO ^ i-

m m

CM CO r-H (M

•I CO 00 CM O CM

oo

1^ O CD CD i-

o> 05 in in co if

OO (M r-H i—l

* to t- in o in

J O 1— 1— O I—

< CD

m

<M ^ O Oi -rft C

i — 1 r— ( i — 1 i — I i-H C

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1 CO

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1

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r-H CM r-H O r-H

i— 1 •** CO O CO

Ol

CO O i-H ^ O

OS CO to
S (M t- CO O to

CO
O 0 O r-H C

0

— < >-< O O

CO -HH -HH O ^

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^H i-H i-H. ^H r-

1 +

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IM i — 1 i — 1 r-H r—

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1 CO

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CM t— CS O CS

CO -TtH r-H 1~ C

CM CO r-H in
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CM CM r-H i — 1 i—
1 +

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I-H os o in CD ir

r-H r-H r-H r-H r-
1 +

5 o m •* o -*

o

CM

CM m I-H CD c

OS in r-H -^H
1 CO CM OO O TO

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r-H CM CO CO t-

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r-H CM r-H I — 1 i — 1 r-

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CD oo r-H os in o

> o 10 ^ o "^

1 +

+ +

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OO <M CO to C

0 0

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-*

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i — 1 I-H j i-H r-H r-

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US <£> <N 00 C-

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> O -* CO O CO

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1 (M

r-H -rH T* CO -* C

1 I-H CO t- O l^

1

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5f ^£ .S O K ' a

p 1 s =

cr1

OT

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~ a •>

i 1 • •

j - oa:| ';| :'!!!

J

; | ; « j | : 1 •

: o '•"! |S :i_;

Pressure ....

8 :|:|:.g£|

pJ^ "^3 ' — ' „ fl, °2 P-i

« i 1 1 J i 1 1 ^ j

' co * i ' "3 '

§ ' ~s • b ' "S

' ft ' « 5 0 ' *" '

'•« '^ ^ S "S '

in! iliii

iinw^; ^rd-Co-J^
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02

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PH

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CD

CD

CO

P9

THEKMAL PROPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 377

TABLE B.— Mean Specific Heats of COa Gas for 10° C. Intervals.
(Measured from the curves, fig. 4.)

Note. — This table is used for plotting constant-pressure curves, see Appendix I.

Pressure.

Range.

Mean specific heat at
constant pressure.

Ibs. per sq. inch.

700

12-5 to 20

0-535

20 „ 30

0-430

GOO

0 to 10

0-46

10 „ 20

0-415

20 „ 30

0 • 365

500

0 to 10

0-38

10 „ 20

0-345

20 „ 30

0-313

400

- 10 to 0

0-34

0 „ 10

0-32

10 „ 20

0-29

20 „ 30

0-28

300

-20 to -10

0-291

- 10 „ 0

0-282

0 „ 10

0-272

10 „ 20

0-262

20 „ 30

0-253

200

- 30 to - 20

0-262

-20 „ -10

0 • 255

- 10 „ 0

0-248

0 „ 10

0-241

10 „ 20

0-235

20 „ 30

0-228

150

Constant

0-2315

VOL. CCXV. — A.

3 E

378

PEOF. C. FREWEN JENKIN AND MR. D. R. PYE ON THE

TABLE C. — Specific Heats of CO2 Gas at Constant Pressure.
(Measured from the curves, fig. 4.)

Pressure, Ibs. per sq. inch.

Temperature.

150.

200.

300.

400.

500.

600.

700.

-30

0-2315

0-270

-25

0-2315

0-266

—

—

-20

0-2315

0-262

—

— .

73

—

-15

0-2315

0-258 0-292

— . — .

43

—

-10

0-2315

0-253 0-287

/-)

- 5

0-2315

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0-342

.

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f-l

0

0-2315

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0-330

0-396

.2

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0-2315

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0-319

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M

10

0-2315

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0-308

0-364

0-450

15

0-2315

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0-296

0-347

0-421

0-547

20

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0-285

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0-490

25

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0-251

0-274

0-314

0-363

0-434

30

0-2315

0-221

0-246

0-263

0-298

0-334

0-378

TABLE D. — Observations on the Total Heat of Liquid C02

Weight of CO, . . .

Ibs

13

6

8

10

Duration of experiment .

minutes

27-87

34-4

26-33

20-65

Initial temperature ....

° C.

-50-6

-40-5

-38-0

-33-0

Final temperature ....

+ 4-6

+ 5-8

+ 4-0

+ 3-4

Atmospheric temperature

))

8-2

11-8

13-1

14-1

Electric heat (total heat) . .

Th.U.

365

145-1

17.7-5

194-7

Heat per Ib. .

Th U /Ib

°8-04

24-2

22-19

19-47

THERMAL PEOPERTIES OF CARBONIC ACID AT LOW TEMPERATURES. 379

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[ 383 ]

XIII. On the Specific Heat of Steam at Atmospheric Pressure between

104°' C. and 115° C.

by the Continuous Flow Method of Calorimetry performed in the
Physical Laboratory of the Royal College of Science, London?)

By J. H. BKINKWORTH, A.R.C.S., B.Sc., Lecturer in Physics at St. Thomas s Hospital

Medical School.

Preface by H. L. CALLENDAR, M.A., LL.D., F.R.S., Professor of Physics at the
Imperial College of Science and Technology, London, S.W,

Received April 1, 1915,— Read May 13, 1915,

uJ

Q-

<

UJ

I-

THERMOMETER

PREFACE.

THE conclusion of Mr. BRINKWORTH'S experiments on the specific heat of steam at
atmospheric pressure in the neighbourhood of 105° C. marks a definite stage in an
investigation which has been in progress with
varying success for the last twenty years. It
may therefore be of interest to review the
situation in the light of collateral evidence which
has been accumulating from various quarters
during the progress of the research.

The investigation originated from some experi-
ments undertaken in conjunction with the late
Prof. J. T. NICOLSON, at McGiLL College in
1895, " On the Law of Condensation of Steam"
(published in ' Proc. Inst. C.E.,' 1898), in the
course of which it was necessary to observe
repeatedly the wetness of the steam employed.
The special form of throttling calorimeter, shown
diagrammatically in fig. 1, was employed for this
purpose. The sample of steam, taken from the
steam-pipe, was throttled through a thin tube of

EXIT
THROTTLE

EXHAUST

Fig. 1.

small bore, in order to eliminate the error due to conduction of heat through the
throttle, to which most forms of throttling calorimeter then in vogue were liable.

VOL. CCXV. A 535. 3 F [Published September 21, 1915.

384 PROF. H. L. CALLENDAR, PREFACE TO

After circulating round the thermometer pocket, the throttled steam, before leaving
the apparatus, was made to circulate twice round the calorimeter to minimise external
loss. The whole apparatus, including the steam-pipe and the throttle tube, were
also well lagged in the usual way. The pressure after passing the throttle was
generally atmospheric, but an exit throttle and gauge were provided for raising the
pressure and observing its value when required. The exit could also be connected,
if desired, to a condenser vacuum.

The wetness of the steam (l— q) was determined in the first instance by means of
the usual formula with REGNAULT'S coefficients, namely,

(l-g)L' = 0-305 (*'-100)-0'475 («"-100), (l)

where L' is the latent heat of dry saturated steam at the initial temperature, t' and
t" is the temperature observed after throttling to atmospheric pressure. In spite of
all precautions suggested by previous experience in calorimetry, the wetness of the
steam persisted in coming out negative (which was obviously impossible) to an extent
beyond the probable limits of error of observation. Since many at that time suspected
REG:STAULT'S value 0'475 of the specific heat, and had proposed other values, ranging
from 0''380 (GRAY) to 0'5G8 (ZEITNEK), it was decided to make a direct determination
of the specific heat in the neighbourhood of 100° (J. by the continuous electric method,
which had been devised some years previously, for the determination of the mechanical
equivalent of heat.

The first experiments on the specific heat by this method were made by passing a
current of slightly superheated steam at atmospheric pressure over an electric heating
coil, and observing the rise of temperature with a pile of several thermojunctions in
series. The electric energy supplied was measured with standard Weston instruments,
and the steam current by the usual method of condensation and weighing. This was
a comparatively rough method, because the steam current could not be very accurately
controlled, or the external loss exactly determined. Some difficulties were also
experienced with the insulation of the thermojunctions. The results obtained showed
discrepancies of 1 or 2 per cent., but indicated a value of the specific heat falling from
0'50 to 0'49 over the range of temperature from 110°C. to 160° C. covered by the
experiments. This was regarded as a decisive confirmation of REGNAULT'S value
0'475 over the range 125° C. to 225° C., since a slight diminution of the specific heat
with rise of temperature was to be expected owing to the existence of the Joule-
Thomson cooling effect.

Having thus obtained by direct experiment the values of the mean specific heat at
atmospheric pressure over the required range of temperature, it was possible to deduce
the values of the total heat of saturated steam at temperatures above 100° C., from
the observations already taken with the throttling calorimeter, by means of the
equation,

(2)

MR. J. H. BRINKWORTH ON SPECIFIC HEAT OF STEAM. 385

where H, is the total heat of the saturated steam at the initial temperature and
pressure, and S is the mean specific heat at atmospheric pressure from 100° C. to t",
the temperature observed after throttling to atmospheric pressure. The curve of
variation of total heat found in this way, instead of being straight like REGXAULT'S,
with a constant coefficient dRfdt = 0'305 for saturated steam, showed a decided
curvature, the rate of increase falling from 0'40 calorie per 1° C. at 100° C. to 0'30 at
160° C. This indicated that REGNATJ L,T'S coefficient 0'305 might be a fair average of
the rate of variation over the range 100° C. to 200° C., where his experiments were
most concordant, but that the coefficient probably increased at temperatures below
100° C., approximating to the value 0'475 (as it should according to RANKIXE'S theory)
at low pressures and temperatures, where the vapour should behave as a perfect gas.
This view, while disagreeing materially with REGNATJLT'S observations of the total
heat at low temperatures, Avhicli were very discordant, gave good agreement with
those of DIETERICI at 0° C., and of GRIFFITHS at 30° C. and 40° C. But in order to
determine the complete variation of the total heat, H, from the thermodynamical

formula

dR = 8 fZT-SC rfP, ......... (3)

and to deduce the form of the characteristic equation from the relation,

V, ......... (4)

it was first necessary to determine more completely the variation of the cooling effect,
C, over as wide, a range of T and P as possible.

The Differential Throttling Calorimeter.

The single throttling calorimeter previously employed was well adapted for
determining the variation of the total heat of saturated steam in terms of the specific
heat at atmospheric pressure, but it was not well suited for obtaining values of the
cooling effect, C, at a particular point. Values of C at a particular temperature and
pressure could be obtained only by successive observations with the single calorimeter,
during which the initial state of the steam might vary. Since a change of 1 per cent.
in the wetness of the initial steam produced a change of about 10° C. in the observed
temperature after throttling, it was most important to employ a differential method
in which such effects were automatically eliminated. This was successfully accom-
plished by connecting two exactly similar calorimeters to the same steam supply, and
adjusting the terminal pressures P' and P" by exit throttles to a suitable difference
P'— P" read on a differential gauge. The corresponding difference of temperature
T'— T" could be obtained by a single reading with a pair of differential platinum
thermometers, from which the value of the cooling effect at the mean temperature
and pressure, C = (T'-T")/(P'-P"), is directly obtained, with almost complete

3 F 2

386

PEOF. H. L. CALLENDAR, PREFACE TO

elimination of errors due to variation of wetness or external heat loss. The apparatus
is shown diagrammatically in fig. 2. The calorimeters were made of very thin steel
tube, and each was well lagged and drained, and doubly jacketed with its own
exhaust, so that a steady state was reached within five minutes of turning on the
steam. The throttle tubes were made interchangeable, so that the external loss could
be determined by using throttles of different sizes simultaneously in the two
calorimeters. The elimination of the heat-loss, in the usual method of employing the
differential instrument with equal throttles, proved to be nearly perfect, but the
observations were very useful in correcting the previous results with the single
calorimeter. The apparatus was completed early in July, 1897, and was exhibited to

SUPPLY

GAUGE Pn

II!

THROTTLES

Fig. 2.

several members of the British Association on the occasion of their visit to McGill
College in August. But owing to pressure of other work the results could not be
completed in time for the meeting at Toronto. About thirty observations were taken
by Prof. NICOLSON and myself during the vacation, which were supplemented
by others taken by the Senior Demonstrator, Mr. H. M. JAQUAYS, and the fourth
year students during the ensuing session. These confirmed the previous conclusions,
but the work was unfortunately interrupted, before any observations had been
obtained at temperatures above 180° C. or pressures above 130 Ibs., by the unexpected
translation of Prof. NICOLSON and myself to posts in England early in 1898.
Prof. NICOLSON intended to set up a duplicate of the apparatus at Manchester as soon
as possible, while I continued the investigation at University College. But facilities

MR. J. H. BRINKWORTH ON SPECIFIC HEAT OF STEAM. 387

were lacking, and the necessity of fresh equipment caused inevitable delays. The
experimental part of the work was still incomplete when the appearance of
GRINDLEY'S paper, " On the Cooling of Saturated Steam by Free Expansion " (' Phil
Trans.' A, 1900) made it necessary to publish a preliminary account of the theory,
which appeared in ' Roy. Soc. Proc.' for June, 1900.

GRINDLEY'S observations of the cooling effect agreed remarkably well over the
whole range with those given by the differential throttling calorimeter, although his
experimental method was very different. But he deduced the values of the specific
heat by assuming REGNAULT'S formula dH/dT = 0'305, for the rate of variation of the
total heat of saturated steam, and obtained results for the specific heat at atmospheric
pressure increasing from 0'387 at 100° C. to 0'665 at 160° C., which could not be
reconciled with direct experiment. Owing to the uncertainty of REGNAULT'S formula
for the total heat, it seemed better to deduce the variation of the total heat from the
measurements of the specific heat as already explained.

The theory of the relation between the total heat and the specific heat was worked
out by assuming a modified form of the characteristic equation employed by RANKINE
and by JOULE and THOMSON for a similar purpose. This equation was put in the

form,

V-/; = RT/P-c, .......... (5)

where b is the molecular volume of HIRN, a small constant of the same order of
magnitude as the volume of the liquid, and c represents the diminution of volume of
the vapour due to coaggregation or pairing of the molecules. JOULE and THOMSON
deduced the values of c from their empirical expression C = A/T3 for the cooling effect
by integrating equation (4) on the assumption that S was constant. This method was
shown to give unsatisfactory results in the case of CO2, and was quite inadmissible in
the case of steam. The opposite procedure was accordingly adopted ; an expression
was assumed for c, and the resulting equation for SC, namely,

-b, .......... (G)

(7)

was compared with the experimental values obtained for S and G.
The complete variation of S with pressure was directly given by

where S0 is a function of the temperature only, which appears from experiment to be
very nearly constant over the experimental range for steam. The value of the index
n was taken as being S0/R — 1, because it had been found by experiments with a
sensitive platinum thermometer on the adiabatic compression of steam that the
pressure temperature relation was of the form P/Tn+1 = constant, with n constant,
which was shown to be thermodynamically exact in spite of variation of S, provided
that S0/R = n+l.

388 PEOF. H. L. CALLENDAR, PEEFACE TO

The numerical values for steam were deduced in the first instance with MAXWELL'S
theoretical value 3 '5 for the index n in the case of a triatomic molecule, because it
happens, owing to the form of equation (6), that the value of n cannot be determined
with great accuracy from observations on the cooling effect. The value n = 3 '5
repi*esented most of the properties of steam satisfactorily, including the saturation
pressure, but did not agree very well with the observed values of the adiabatic index
and the specific heat, which required a value of n more nearly approaching 3'3. This
value was confirmed by experiments on the specific heat and the adiabatic index,
which were then in progress, but could not be completed in time for inclusion in the
original paper. The results of these experiments were first published in the article
"Vaporisation," in the ' Encyclopaedia Britannica/ 1902, where revised values for
steam were given. The methods and apparatus employed were also described and
exhibited at meetings of the Physical Society of London in October and November,
1902, but no detailed account of the work was published.

Continuous F/oir Vacuum Calorimeter.

The principal improvements introduced in these experiments on the specific heat
were as follows. The flow tube of the calorimeter, containing the heating coil and
the thermometers, was enclosed in a vacuum jacket to minimise external heat-loss,
which in turn was protected by an external steam jacket maintained at a pressure
slightly above atmospheric. The electric energy was measured very accurately by
means of a potentiometer and Weston cells, and the temperatures were observed with
platinum thermometers in place of couples. The chief difficulty encountered at first
was that of securing a perfectly steady temperature for the steam jacket, and a
perfectly steady current of steam, which had not been obtainable in the earlier
experiments in which steam had been taken from an ordinary boiler. This difficulty
was finally overcome by employing a boiler heated by gas, the supply of which was
regulated by the steam pressure itself acting on a mercury column. The regulator
could be set to cut off at any desired pressure within certain limits, and worked with
almost incredible perfection, maintaining the temperature in the jacket as nearly
constant as the barometric fluctuations of pressure would permit. This regulator was
subsequently employed by MAKOWER in experiments on the adiabatic index, and is
fully described and illustrated by my collaborator in the following paper.

The method of obtaining a steady current of slightly superheated steam will be
readily understood from the accompanying fig. 3, which illustrates one of the many
arrangements adopted in measuring the specific heat.

The steam at a steady pressure from the regulator, after passing through the
double-walled jacket surrounding the calorimeter, entered a separator at the top of
the jacket where any entrained moisture was drained off. A glass tube of small bore
fitted in the exit from the separator to the interior of the jacket, served as a throttle

MR. J. H. BRINKWORTH ON SPECIFIC HEAT OF STEAM.

389

SEPARATOR
LAGGING

THROTTLE

to determine the magnitude of the steam current, which could be varied in any

desired ratio, within certain limits, by fitting throttles of different sizes. Apart from

barometric variations, which were generally small, any desired current could be

maintained constant to about 1 in 1000. After passing the throttle, the steam at

atmospheric pressure, already slightly superheated, was raised very nearly to the

temperature of the jacket by passing through a

number of gauze discs. In the earlier experiments,

the temperature of the steam on entering the vacuum

jacket was taken by a platinum thermometer before

reaching the heating coil. In many of the later

experiments this thermometer was omitted, because

the measurement was found to be superfluous, owing

to the extreme steadiness of the regulator. The

heating coil was of platinum, wound on a mica frame,

having a resistance of about 5'5 ohms at 110° C. It

was provided with current and potential leads, and

connected in series with a 5-ohm manganin standard

for measuring the current in the usual way. The

leads to the heating coil were insulated by glass

tubes (not shown) round which the steam, after

passing the heating coil, was made to circulate

spirally by a rubber coil or metal spiral nearly fitting

the flow tube. The thermometers consisted of coils

of silk-covered platinum wire, each having a funda-

mental interval of 1 ohm, and capable of sliding up

or down in small bore tubes, 3'5 mm. diam., with the

object of interchanging thermometers, and varying

the immersion when desired. The bulbs were gener-

ally fixed at a distance of 10 cm. from the heating

coil, to allow time for the mixing of the steam to

a uniform temperature. The coils were wound on

glass, and the copper leads were insulated by glass

capillaries.

It would be tedious and unnecessary to describe
in detail all the different arrangements adopted for
measuring the specific heat, and the steps by which

VACUUM
JACKET

INLET

OUTLET
HERMOMETER
TUBES -

Fig. 3.

improvements were effected, but the following summary of observations taken with the
apparatus above described, shortly after the reading of the paper on June 21st, 1900,
is instructive as illustrating some of the difficulties of measurement in the immediate
neighbourhood of the saturation point. The temperatures tabulated under tf, &' are
the initial and final superheats reckoned from 100° C, The rise of temperature

390 PROF. H. L. CALLENDAR, PREFACE TO

tf'—tf, denoted by dO for brevity, is the observed change of temperature of the flow Q
due to the supply of electric energy EC in watts. Q is given in grammes per second.
The heat supplied per gramme per degree rise EC/QcZ0 is the most convenient
quantity to calculate in the first instance. The observed value has to be reduced, as
subsequently explained, to allow for slight variations in the temperature limits
Q' and 0". The reduced values corresponding to the range 103" C. to 113° C. are
denoted by X and are employed in calculating the results for S, h and k in the usual
formula for the continuous-flow method, which may conveniently be expressed by the

following equation : —

X =

The greater part of the heat-loss, represented by hd9 watts, appears to be independent
of the flow and proportional to the rise dd, but there is usually a small term kdO/Q
varying inversely as the flow, due to conduction and other similar causes. X may be
regarded as the unconnected value of the specific heat, which always appears to increase
with diminution of the flow, but tends to a limit coinciding with S as the flow is
increased. From the values X15 X2, for any pair of flows Q1} Q2, reduced to the same
range dQ if necessary, we obtain immediately by eliminating h,

S = (QiXj— Q2X2)/(Q1— Q2)-(-AyQjQg = Sia + &/QiQ2,

where S12 is the value of S obtained by neglecting k as is often done. If three flows
are available, designated by the suffixes 1, 2, 3, in descending order of magnitude, we
may eliminate k by finding S,3, and obtain,

S = Sia + (Sia-Saa)Q:,/(Q1-Q3) = (SiaQ1-Sa8Q3)/(Q1-Q8).
The values of k and h are easily obtained from the relations,

k = Q2Q3(S-SS3), h = (Xs-S)Q3-Jfe/Q3,

in which the smaller flows are preferably employed as giving the largest differences.

It will be observed that the complete expression for the specific heat, S, as deduced
from the values of X for three flows by the above method, may be put in the following
symmetrical form,

S = X1Q12/(Q1-Q2)(Q1-Q3) + X2Q22/(Q2-Q3)(Q2-Q1) + X3Q32/(Q3-Q1)(Q3-Qa),

which is useful in estimating the effect of errors of observation. Thus if the flows
Qi» Qa, Qs, are in the ratio 4:2:1, the terms depending on X1; Xa, X3, are approxi-
mately in the ratio 8:6:1, which shows that the accurate determination of X3 is much
less important than that of Xx or X2. Further, since each product XQ is independent
of Q, the effect of percentage errors in the measurement of the flows are in the same
proportion as the corresponding X terms in the expression for S. Thus an error of

MR. .1. H. BRTNKWORTH ON SPECIFIC HEAT OF STEAM.

391

3 in 1000 in flow 1, 2, or 3, will produce an error in S of 8, 6, or 1 in 1000 respectively.
The flows could generally be measured to O'OOOl gr./sec., so that the error due to any
one flow could not in general exceed 1 in 1500 on the value of S, and would be much
less for the smallest flow, which was the most difficult to measure.

Since the values of h and_& depend on relatively small differences, the values of S,
calculated by this method from each single set of three flows, are very susceptible to
small accidental errors in X, due to slight variations in the conditions, especially with
flows taken on different days. Much better agreement may be obtained by taking a
large number of observations under identical conditions, and employing constant mean
values of h and k in the reduction of the series. This method has been adopted with
most satisfactory results by Mr. BRINKWORTH, but could not be applied to the
observations given in the following table, because the conditions were varied widely from
day to day. The number of observations taken with each calorimetric arrangement was
insufficient to enable the variations of // and k to be interpreted with certainty at the
time. But Mr. BRINKWORTH has since shown that the values of heat -loss observed in
these experiments, when plotted according to his method in fig. 10, show a most
remarkable agreement with the results of his analysis. This is a very severe test,
because the maximum heat-loss is less than 4 per cent, and each individual result is
plotted separately. The agreement is so close that it leaves little doubt of the
substantial accuracy of the observations. The only difference is that, owing to the
arrangement of the leads, the heat-loss in my apparatus was somewhat greater than
in his under similar conditions.

SUMMARY of Observations with Vacuum Calorimeters.

Temperatures.

EC/Q ,t

t> = X.

Date and

Rise.

Flow.

Watts.

Results.

flow.

(10.

Q.

EC.

Oh

S, It, and /.'.

ff.

D".

\j it-
served.

Reduced.

Unsilvered jacket. D :

= 7 cm.

between

i heater

and thermometer.

June 25

(3)

4-030 17-257

13-227

0-1577

4-774

2-289

2-309

-N S = 2-062

(2)

3-826

10-228

6-402

0 • 3302

4-513

2-135

2-135

h = 0-0110

2

3-831

13-121

9-290

0-3288

6-495

2-126

2-136

> k = 0-0044

1

(1)

3-432
3-434

12-472
12-494

9-040
9-060

0-6072
0-6067

11-467
11-462

J2-088

2-092

J

Silvered jacket. D =

7 cm. between heater and thermometer.

June 28

(1)

3-382

12-544

9-162

0-6035

11-463

2-073

2-077

1 8 = 2-052

\ /

(2)

3-803

13-083

9-280

0-3304

6-450

2-104

2-114

\ h = 0-0085

(3)

4-012 13-599 9-587

0-1566

3-387

2-256

2-269

J /„• = 0-0040

VOL. CCXV. — A.

3 G

39'2

PROF. H. L. CALLENDAR, PREFACE TO

SUMMARY of Observations with Vacuum Calorimeters (continued).

Date and
flow.

Temperatures.

Rise.

dB.

Flow.
Q.

Watts.
EC.

EC/Q,

16 = X.

Results.
S, h, and /•.

ff.

6".

Ob-
served.

Reduced.

New thermometers and heating coil, refitted to reduce k D = 7 '5 cm.

July 9

(2)

3-902

13-278

9-376

0-3427

6-378

1 • 985

Gauze only, no spiral

(1)

3-551

12-510

8-965

0-0328

11-281

1-988

Insufficient mixing.

New silvered jacket

. Mixing spiral refitted. D = 8 cm.

July 10

(1)

3-528

1 2 • 350

8-822

0-6200

11-401

2-084

2-089

S = 2-070

Pressure raised by altering regulator.

(2)

4 • 703

12-566

7-863

0-3910

0-440

2-093

2-108

h = 0-0110

Pressure readjusted

to previous value.

(2)

3-749

13-119

9-370

0-3272

0 • 435

2-099

2-108

k = 0-0005

(••5)

3-954

15-424

11-470

0-1568

3 • 850

2-144

2-160 |

Repe;

,ted under slightly different conditions.

Julv 12

(3)

4-032

12-056

8-624

0-1506

2 • 906

2-152

2-163 ] S = 2-064

(1)

3-401

12-554

9-153

0-0042

11-532

2-085

2-089 h = 0-0137

July 13

(1)

3-350

12-502

9-140

0 • 0043

11-515

2 • 083

2-086 : /„• = 0-0003

(2)

3-712

13-045

9-333 ]

0-3315

6-492

2-099

2-108 J

Till spiral

in place of rubber for mixing.

D = 8 cm.

July 16

(1)
(2)

3-376
3-800

13-226
14-307

9-850
9-707

0-5620
0-3217

11-488
6-491

2-075
2-078

2-081
2-091

} S12 = 2-068

(2)

3-800

11-320

6-720

0-3213

4-508

2 • 088

2-090

J

(2)

3 • 800

8-877

4-277

0-3218

2-889

2-098

2-092

(2)

3-800

C-991

2-391

0-3223

1-628

2-112

2-091

Unsilvered

jacket.

Rubber mixing spiral.

D = 9 cm.

July 17

(1)
(2)

3 • 343
3-777

12-102
12-930

8-759
9-153 i

0-6238
0-3338

11-465
6-463

2-098
2-115

2-100
2-124

| S12 = 2-072

MR. J. H. BR1NKWORTH ON SPECIFIC HEAT OF STEAM.

393

SUMMARY of Observations with Vacuum Calorimeters (continued).

Temperatures.

EC/Q./0 = X.

Date and

Rise.

Flow.

Watts.

Results.

flow.

dO.

Q.

EC.

OK

S, //, and k.

ff.

0".

\jit-
served.

Reduced.

Same jacket.

Single thermometer in place of

pair. D = 9 cm.

July 18

(1)

3 • 357

12-101

8-744

0-6267

11-494

2-097

2-099

] S = 2-063

(2)

3-796

12-962

9-166

0-3348

6-493

2-116

2-125

//. = 0-0242

Lowered pressure by 2 cm.

(3)

3-372

12-407

9-035

0-1478

2-897

2-170

2-173

j /.• = -0-0012

Altered separator and fitted

new condenser.

D = 7'5 cm.

July 19

(3)

3-396

12-667

9-271

0-1452

2 • 896

2-152

2-150

T S = 2-055

(1)

2-710

12-428

9 • 688

0-5710

11-533

2-085

2-079

1 It - 0-0132

(2)

3-178

13-168

9-990

0-3103

6-500

2-097

2-100

J /; = 0-0002

Rearranged

leads of heater.

D = ]

1 cm.

July 24

i

(1)

2-747

12-400

9-653

0-5543

11-437

2-139

2-133

1 S 2-061

(2)

3-205 !

12-567

9 • 362

0-3163

6-460

2-183

2-184

L /t = 0-0413

(3)

3-400

11-812

8-412

0-1488

2 • 884

2 • 304

2 • 305

J /„• = -0-00074

*

New silvered jacket. New rubber mixing spiral. D =

= 8'5 cm.

July 28

(3)

3-845

12-917

9-072

0-1451

2-943

2-236

2-246

1 S - 2-064

(2)

3-535

14-524

10-989

0-2794

6'575

2-142

2-153

I h = 0-0230

(1)

3-214

16-174

12-960

0-4255

11-636

2-110

2-121

J /„• = 0-0005

Same distance

Test for variation with range of temperature.

July 30

(1)

3-207

16-242

13-035

0-4223

11-590

2-105

2-116

(1)

3-262 j

13-154 ;

9-892

0-4243

8-869

2-113

2-117

Altered distance to D

= G cm.

(1)

3-337

13-509

10-172

0-4240

8-857

2-054

2-060

1 S = 2-054

(2)

3-667

14-721 ,

11-054 .

0-2842

6-487

2-065

2-077

It = -0-0046

(2) 1- 3-665

8-604 !

4-939

0-2846

2-931

2-085

2-075

k = 1-0-0032

(3)

3-890

13-076

9-186

0-1455

2-890

2-162

2-172

J

3 G 2

394 PROF. H. L. CALLENDAR, PREFACE TO

SUMMARY of Observations with Vacuum Calorimeters (continued).

Date and

flow.

Temperatures.

Rise.

dO.

Flow.
Q-

Watts
EC.

EC/Qy/0 = X.

Results.
S, h, and k.

ff. 9".

Ob-
served.

Reduced.

Altered distance to D = 10 cm. Same

conditions.

July 31
(3)
(1)

i

3-812 ; 12-687 8-875 0-1437
3-107 12-813 9-706 0-4287
3-527 13-015 9-488 0-2840

|

2-915 2-286
8-889 2-136
5-847 2-170

2-295
2-137

2-177

1 S = 2-060
}, h = 0-0327
J /„• = 0-0001

Mean S = 2-060, 103' C. to 113" C.

The value found for h increases, as one would naturally expect, when the distance
between the heating coil and thermometer is increased. Conversely h may be
diminished by reducing the distance, but it is necessary to allow sufficient space
for the mixing of the steam to a uniform temperature, and it was found
that D could not be reduced below 6 cm., even with the most effective spiral
mixing of the flow. Mixing the fluid by passage through wire gauze, which has
been commonly employed by other observers, was found to be very ineffective as
compared with the turbulent flow induced by the spiral. The maximum value of Ji
observed at a distance D = 11 cm. was only 3'5 per cent, of the heat supply for the
largest flow. The average was about 2 per cent., but none of the jackets maintained
a good vacuum when heated in steam, and they all cracked after a few days use. The
term &/Q2 was generally less than 1 in 1000 for the large flow, but increased to nearly
1 per cent, when the distance was reduced to 6 cm.

Since the measurement of the electric energy is made in watts, it is convenient to
calculate X and S in joules or watt-seconds per gr. degree. The values of X are
always a little greater than 2 for steam, and since the formulas involve only differences
between corresponding values of S and X, it saves a good deal of trouble in calculation
to work only the differences from the round value 2000 in each case. It is important
for the same reason to keep the differences between the flows as large as possible.
The observed values of X: were generally consistent to 1 in 2000 or better, but an
error of 1 in 2000 in the value of X: or X2 would make an error of 3 or 2 in 2000 in
the result calculated from 3 flows. The order of agreement of the results, which
range_ from 2'052 to 2'070 is, therefore, as good as can be expected, considering the
range of variation of the conditions of the experiments.

The results of this particular series of observations were reduced in the firsi instance
by employing the theoretical formula (7) for the variation of the specific heat over

MR. J. H. BRINKWORTH ON SPECIFIC HEAT OF STEAM. 395

different ranges. The mean result found in this way was S = 2'074, corresponding to
S = 0'496 calories per 1° C., which agreed as closely as could be expected with my
previous experiments by other methods, giving S = 0'497 over the same range. A
systematic variation depending on the initial temperature was subsequently noticed,
which requires further explanation, and which formed the starting point of fresh
investigations.

Effect of Impurities in the Steam.

Steam in the immediate neighbourhood of the saturation pointl is liable to' carry
small particles of water in suspension, which cannot be evaporated completely by a
moderate degree of superheat if any impurities, such as salt in solution, are present.
Since 1 mgr. of water requires more than half a calorie to evaporate it, and the heat
required to raise the temperature of 1 gr. of steam 10° C. is only 5 calories, it is
necessary that the initial steam should not contain more than 1 in 100,000 of water,
if the specific heat is to be found correct to 1 in 1000 over a range of 10° C.

The rise of boiling point 6 produced by .?• gram-molecules of salt per gram of water
is approximately lOOO.r ° C. The proportion of suspended water remaining un-
evaporated at any degree of superheat 0' will be 1000x/0'. The quantity evaporated
in heating the steam from 6' to 6" will be lOOO.r (9"-9')/9'9". This will produce an
apparent increase of the mean specific heat of the steam over the range 9" — 9'
equivalent to 1000L,r/0V, where L is the latent heat of evaporation. It was found
that this extremely simple and convenient reduction formula fitted the results
obtained over different ranges of temperature with extraordinary precision, and
reconciled apparent discrepancies which had previously been attributed to errors of
observation. A few examples of this reduction are given in the preceding table, e.g.,
the observations with flow (2) for different ranges of temperature on July IGtli, which
are seen to agree extremely well when reduced to a common range 103° C. to 1 13° C. by
this formula. The numerical value of the constant lOOOLx in the reduction formula was
found to be 173 joules, which would be equivalent to the presence in the steam of an
impurity of about half a millionth of a gram-molecule of salt per gram. It is quite
possible that an impurity of this order of magnitude might have been produced by the
continued use of ordinary water in the boiler, and the passage of the steam through
considerable lengths of rubber and glass tubing. Small variations in the impurity
might also account for some of the differences in the results obtained on different
occasions, but as the same boiler and regulator were employed in all cases, the
variations in the quality of the steam supplied would probably be unimportant.

The effect of this reduction is evidently greatest when the superheat is small. Any
small change in the initial superheat 9' was found to produce a relatively large effect
on the results. The length of the steam jacket cannot allow sufficient time for
raising the steam exactly to the temperature of the jacket before entering the
calorimeter. There is always a systematic fall of initial temperature with increase of.

396 PEOF. H. L. CALLENDAE, PEEFACE TO

flow, which is clearly shown in the table of observations. The application of this
correction has accordingly the effect of reducing the mean value of the specific heat
from the value S = 2'074, originally obtained, to 2'060 joules per gr. per 1° C., or from
0'496 calories to 0'493, over the range 103° C. to 113° C.

The value so obtained has still to be corrected for the water present in the steam,
which was probably of the order of 2 parts in 10,000 at the initial temperature of
103° C. The correction can be obtained from the reduction formula, regarded simply
as expressing the results of experiment, without requiring an exact knowledge of the
nature of the impurity present. Assuming that the formula applies up to a
temperature of 113° C., which was within the range of the experiments, and that all
the water would probably be evaporated by the time the steam had reached this
temperature, the value of the specific heat at 113° C. given by the formula of
reduction, should be approximately that of dry steam at this temperature. If the
correction to the mean specific heat over the range 9' to 0" is given by the formula
173/0' 0", the correction to the specific heat at any temperature 6 is given by 173/02.
The value of the specific heat at 113° C. will be less than the mean over the range
103° C. to 113° C. by 173/3 x 13-173/132, which is OU342 joules per 1° C. If the
value from 103° 0. to 113° C. is 2'060, the value at 113° C. will be 2'026 joules, which
agrees very well with the value 2'030 at 105° (J. obtained by my collaborator, when
special precautions were taken to secure dry steam free from impurities. Briefly
enumerated these precautions include (l) a higher initial temperature ; (2) a much
longer steam jacket ; (3) pure water for the boiler ; (4) tubes of pure tin in place of
glass or rubber, for conveying the steam to the calorimeter; (5) special attention to
the arrangement of the separator and drain tubes for removing water of condensation.

Indirect, Verification.

The results obtained for steam with the apparatus above described were indirectly
verified in various ways, so that there could be little doubt that they represented cor-
rectly the specific heat of the steam actually employed, although there might be a slight
uncertainty as to the quality of the steam. The first method of verification applied
was to use the same apparatus for finding the specific heat of air. There was in this
case no difficulty with regard to wetness or insulation, and it was possible to employ
thermometers of bare wire in addition to the tube thermometers. The results found
for the specific heat of air at atmospheric pressure lay between 0'240 and 0'241 gr.
calories per gr. per 1° C. and were about 2 per cent, larger than those given by
KEGNAULT, but agreed very well with the value deduced from JOLY'S experiments at
constant volume, and with the known value of the adiabatic index for air. The chief
difficulty encountered was the regulation and measurement of the air current. This
was overcome by constructing a delicately balanced and compensated gasometer for
.maintaining a steady pressure, and passing the air current through a resistance

MR. J. H. BRTNKWORTH ON SPECIFIC HEAT OF STEAM. 397

consisting of 20 small-bore tubes in parallel, immersed in a water bath to keep the
temperature steady. The current was measured and adjusted by observing with an
oil gauge the difference of pressure between the ends of the resistance. The same
apparatus was subsequently employed by SWANN (' Phil. Trans.' A, 210, pp. 199-238,
1910) in a more accurate and extended investigation of the specific heats of air and
CO2. SWANN found the value 0'2414 for air at 0° C., which corroborated the
conclusion that REGNAULT'S value was appreciably too low.

A more direct verification of the result for steam was obtained by measuring the
adiabatic index n+l in the equation P/TB+1 = constant, at 108° C. in the neighbour-
hood of atmospheric pressure, by means of a compensated platinum thermometer of
0 '001 -in. wire, similar to those employed in the experiments with Prof. NIOOLSON on the
law of condensation of steam in 1895. The resulting values of the index lay between
4'26 and 4'30, giving, according to the foregoing theory, values of S at atmospheric
pressure and 108° C. lying between 0'488 and 0'493 calories per gr. deg. These
experiments were performed by W. MAKOWER at University College under my
direction in 1901-2, and were, published in the ' Phil. Mag.' for January, 1903.

Jt<wtltn of otlcr CM.srnvr.s-.

The experiments of HOLUORN and HENNTNO (' Ann. Phys.,' 18, p. 739, 1905),
extending over the range 110° C. to 820° C. at atmospheric pressure, indicated that
the specific heat of steam was constant within the limits of error of their observations,
but threw no light on the variation with pressure in the neighbourhood of saturation,
as their Icnvest range extended from 110° C. to 270° (J. Their results gave most
accurately the ratio of the specific heat of steam to that of air, which was found to
be 1'940 over their lowest range. Their value for air being 0'2315 over the same
range, that for steam would be 0'449, which is evidently too low, on account of the
error of the value 0'2315 for air. SWANN'S value (loc. cit.) for air over this range,
namely, 0'2443,* with the same ratio T940, would give the value 0'474 for steam,
agreeing closely with REGNAULT'S value from 125° C. to 225° C. If equation (7) is
adopted for the variation of the specific heat, this result would correspond to a value
0'482 for the specific heat at 108° C. and 1 atmosphere. The agreement with the
value 0'485 found for dry steam at 108° C. is fairly close, but rests on the assumption
that the specific heat at zero pressure is accurately constant, which may be to some
extent in error. The results of explosion experiments indicate a considerable increase
in the specific heat at high temperatures, which, if represented by a linear formula,
according to the common practice, would require an increase 0'00034 in the specific
heat per 1° C., or 0'027 between 110° C. and 190° C., which is more than sufficient to
compensate for the diminution O'OOS over this range represented by the term

* This value has quite recently been confirmed by HOLBORN and HENNING.

398 PEOF. H. L. CALLENDAR, PREFACE TO

n(n+ l)cP/T in equation (7). It is probable that tbe variation of S0 with temperatur
is not linear, but is more rapid at high temperatures, as indicated by the radiation
theory of the variation of specific heat suggested in the Report of the B.A.
Committee on Gaseous Explosions (B.A. Rep., 1908, p. 340). At the same time there
is probably an appreciable increase over the range 100° C. to 200° C., since the
equation of saturation, pressure appears to require a mean value of S0 = 0'477 over
this range, but a lower value from 0° C. to 100° C.

The observations of KNOBLAUCH and JACOB (' Forsch. Ver. Deut. Ing.,' 35, p. 109,

190G) extending from 2 to 8 kg./cm2 and from 150° C. to 350° C., showed a variation

of S with pressure of the same type as that indicated by equation (7). The absolute

value of the change with pressure found by these observers agreed with that given

by formula (7) at a temperature of 210° C., but increased more rapidly below this

point, mid diminished more rapidly at higher temperatures. On the other hand, the

value of the specific heat S0 at xero pressure, obtained by extrapolation, showed an

increase from 0'447 at 100° (I. to 509 at 400° C., giving nearly the same rate of

variation between these limits us that found by LAXGEX in his explosion experiments

at much higher temperatures. This variation could not possibly be reconciled with

the experiments of HOLBOKN and HENNTNG. A repetition of the observations by

KNOBLAUCFI and H. MOLLIKU (' Zeit. Ver. Deut, Ing.,' 1911, p. GG5), raised the results

at 120' C. by 0'02(J, and lowered the results at 400° C. by 0'020, reducing the

increase with temperature over this range from 0'OG2 to O'OIG, which is quite a

possible value. The magnitude of the correction, amounting to 10 per cent, on the

specific heat, illustrates the difficulty of determining the variation of the specific heat

by the methods which they described, since the variation is of the same order as the

systematic errors of experiment. This correction brings the results of KNOBLAUCH

and MOLLIEE into very fair agreement with those given by equation (7), except that

their results for the variation with pressure near the saturation line, obtained chiefly

by extrapolation, appear to be excessive, and cannot be reconciled with experiments

on the cooling effect.

Although it is probable that there is some variation of S0 with temperature, the
amount of the variation is so small and uncertain over the range required in steam-
engine practice that it seems best for the present to take a mean value of S0 in
equation (7) for practical purposes. It is almost certain that the variation of S0
cannot be linear. No exact theory of the variation has yet been proposed, and
any attempt to represent it by an empirical formula would be highly speculative, and
would introduce endless complications in the theory, without offering any material
advantages so far as the steam-engine is concerned. The mean value of S0 adopted
in 1902 as giving the best general agreement over the required range was 13R/3 or
0-478. The present investigation appears to show that the actual value of S0 at
100° C. is G'463, but since it is very difficult to obtain perfectly dry steam near
saturation (especially in the throttling experiments), and since Sa probably increases

MR. J. H. BRINKVVORTH ON SPECIFIC HEAT OF STEAM. 399

with temperature, it appears likely that a constant mean value, such as 0'477, over
the range 100° C. to 300° C., will agree better than a variable value with actua
practice, in addition to simplifying the theory.

Later Experiments.

The differential throttling calorimeter was redesigned for higher pressures and
temperatures in conjunction with Prof. DALBY in 1906, and was set up in the
laboratory of the City and Guilds Institution (now part of the Imperial College of
Science), at South Kensington. The late Prof. ASHCROFT assisted the writer in taking
a series of observations on superheated steam. A differential platinum thermometer
was employed for reading the temperature difference, and a mercury gauge for the
pressure difference. The observations extended to an upper limit of 376° C. and
agreed so closely with the variation of the cooling effect predicted by equations (6)
and (7), that it was not considered worth while to publish them in detail, as it was
hoped to obtain a more complete series at much higher pressures as soon as a suitable
boiler and regulator could be installed. Owing to the great increase in the number
of students in the engineering department, which necessitated a continual rearrange-
ment of laboratories and equipment, it was found to be impossible to proceed with
this work at the time. But the opening of the new engineering building is expected
to relieve the pressure, and permit the work to be resumed at an early date. The
apparatus is designed for simultaneous measurements of S and C under the same
conditions, which is particularly advantageous, since both are required in the
equations.

About the same time, preparations were made for attacking the problem of the
variation of specific heat with temperature from the theoretical side, by a quantitative
investigation of the absorption and emission spectra of gases in the infra red, which
must, according to the theory of radiation, be intimately related to the variation of -
the specific heat. The elaborate apparatus required for this purpose was made in the
physics workshop of the Imperial College, and took some years to construct, in
consequence of the great pressure of other work. It was practically completed two
years ago, and gave promise of supplying very useful information, but the work has
now been brought to a standstill for want of funds to supply the necessary assistance
and materials for carrying it on. It is hoped that something may be done towards
the solution of this important problem when the assistants now serving at the front
are set free at the conclusion of the war, but the work requires two or more observers,
and nothing can be done without highly trained assistants, owing to the great
difficulty and delicacy of the observations.

H. L. C.

VOL. CCXV. — A. 3 H

400 MR. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

On the Specific Heat of Steam at Atmospheric Pressure between

104° C. and 115° C.

By J. H. BRINKWORTH, A.R.C.S., J3.Sc., Lecturer in Physics at St. Thomas's Hospital

Medical School.

»•.

Introduction.

With one exception, all the experimental determinations of the specific heat of
steam at atmospheric pressure have been made with steam considerably superheated.
The first reliable investigations were made by HEGNAULT, and the value he obtained,
when corrected for various errors, reduces to 0'475 ; this being the mean specific heat
over the range 125° C. to 225° C. Prof. CALLENDAR, using the continuous flow method
of calorimetry, has made the one determination previously quoted, in which the
specific heat at atmospheric pressure was measured at a temperature not far removed
from that of saturation. The value experimentally obtained, 0'493 over the range
103° C. to 113° C., is somewhat higher than that deduced by extrapolation from the
results of other observers, and requires reduction (as above explained, p. 396) in order
to allow for a minute fraction (about 1 in 10,000) of water present in the steam. In
consequence, Prof. CALLENDAR suggested to me that a redetermination of the specific
heat of steam near 100° C. was desirable. My other duties are such as to allow only
a certain amount of time for research, hence this investigation has been rather
protracted ; but through the whole course of the work I have been encouraged by
Prof. CALLENDAR, and here I wish to record my great indebtedness for the advice he
has always given so readily. This investigation has been carried out in the Physical
Laboratories of the Imperial College of Science, and I desire to acknowledge the
kindness of the Board of Governors in furnishing me with the necessary equipment
and with laboratory accommodation for so long a period.

Outline of the Method.

Steam is generated in a boiler of about 15 litres capacity and thence led to one
limb of a U-tube pressure regulator, to be described in detail later. The pressure of
the steam forces the mercury down in this limb of the U-tube and up in the other
limb in which the adjustment of the supply of gas to the large ring burner, used for
heating the water in the boiler, is made. After passing the regulator, the steam, now
maintained at a constant pressure, is led between the walls of a jacket surrounding
the calorimeter proper, thence through a separator and a throttle into the space
enclosed by the double-walled jacket, whence it passes down the calorimeter flow-tube
to a condenser. During the passage of the steam through the flow-tube it is heated
by means of an electric current passing through a platinum heating coil, and its
temperature is measured on a platinum resistance thermometer. Another temperature

ATMOSPHEEIC PRESSURE BETWEEN 104" C. AND 115° C.

401

measurement is made when the supply of electrical energy is cut off, and the difference
between these two temperatures gives the rise in temperature d6 of the steam. If
Ct is the electric current and E! the potential difference between the ends of the heating
coil, we have, when the temperature conditions through the apparatus are steady,

where h^dO is a term representing the heat-loss, and S is measured in joules per gr.
degree C. If now, in similar experiments, the rates of flow, Q, of the steam are about
one-half and one-quarter of the above value respectively, and if the electric current is

To Fnm

BURKtR BOILfK

Fig. 4.

adjusted in each experiment so that the rise in temperature of the steam is the same

as before, we have p ^,

and

If the heat-loss is independent of the flow, a linear relationship will exist between the
values of CE/QcW and the corresponding values of 1/Q, but if this is not so, another
term depending on the flow must be inserted in the fundamental equation, which then

1)60011168 CE . 8Qd6+(h+k/Q) de.

By means of three equations of this form we can obtain the values of S, h, and k.

3 H 3

402

ME. J. H. BRINKWOKTH ON THE SPECIFIC HEAT OF STEAM AT

The arrangement of the apparatus is represented diagrammatically in fig. 4. The
gas, supplied from the mains, was led into the gas holder, R, from which it passed
through a Wolff's bottle containing sticks of KOH, into the limb, G, of the pressure
regulator and thence through the tube, B, to the burner. The steam generated in the
boiler was led through block tin tubes to the other limb, DS, of the regulator, thence
through E into the calorimetric apparatus and separator from which it passed down the
flow-tube to the condenser.

Measurement of Electrical Energy.

An insulated battery of 60-ampere-hour accumulators supplied the steady current
to the heating-coil through a standardised resistance of approximately the same value
and a rheostat. The method adopted for connecting the heating-coil in the circuit is
indicated in fig. 5. The four leads coming from the heating-coil are led into four

0

•wvw

co

C3
u3

DC

r»
1

» 1

— •
i <

K

i

'

VvAA,

\

Hearing Coil

cc.

LU

CO

mercury cups, and the main current leads are similarly led into two other cups cut in
the same block of paraffin wax. With the aid of two copper connectors either pair of
leads may be made the " current-leads " to and from the heating-coil, the other pair
functioning as " potential-leads." By means of this arrangement it is also possible to
send the current through either pair of leads connected to one end of the heating-coil
and thus to investigate the heating effect 'due to the passage of the current through
the leads. The potential differences across the heating-coil, across the standardised
resistance and across a known number of cadmium cells in series, were obtained in
terms of the readings on a Thomson- Varley 2-dial potentiometer. If X,, X2 and X0
are the corresponding readings when the conditions are steady, the energy per second,
in watts, supplied to the heating-coil and used in heating the steam, is

Where

X02R

R = resistance of the standardised coil.
e = E.M.F. of a cadmium cell.
n = number of cadmium cells employed.

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C. 403

In order to maintain the same rise of temperature for different values of the steam
flow, it was necessary to alter both the number of accumulators and the rheostat.
With the flows used and for a rise of temperature of about 8° C., 9, 6 and 4 accumu-
lators were required for the maximum, medium and minimum flows respectively ; the
final adjustment of the current, to give a constant value of d9, being made by means
of the rheostat. This rheostat, which is also shown in fig. 5, consisted of pieces of
manganin wire (No. 20) each about 35 cm. long. By means of the arrangement of
mercury cups and the trough, indicated in the diagram, any number of these wires
could be connected in series or in parallel until the requisite current was obtained.

The Standardised Resistance.

This consisted of a coil of No. 22 manganin wire wound on a wooden frame which
was placed in oil contained in a double- walled vessel. A current of cold water was
passed through the space between the double walls, and the temperature was taken
by a mercury thermometer dipping into the oil ; this temperature was assumed to be
that of the coil itself. This secondary standard was compared with a 10-ohm primary
standard. The same current was passed through both, and the potential differences
across the ends of the resistances were obtained by the potentiometer. The results of
several comparisons, full details of which have been kept, show that the resistance of
the secondary standard is 5 '01 36 ohms at 17° C., and the temperature coefficient of
resistance 0'000020.

The, Cadmium Cells.

The cadmium cells used were of the H -pattern and were some of a set of twelve cells
which I made in 1906, the instructions given by JAEGER and WAOHSMUTH being
accurately followed. The mercurous sulphate and cadmium sulphate were supplied
by KAHLBAUM. Only seven of these cells have been used, Nos. 1 to 7 inclusive, in
those experiments in which seven cells were required, and Nos. 2 to 6, or 3 to 6 for
those experiments in which either five or four cells were required as a standard of
electro-motive force. During the experiments the cells were immersed in an oil-bath,
the current of water used for cooling the standardised resistance being first led round
the nest of cells through a compo-tube immersed in the oil. Cell No. 7 was sent to
the National Physical Laboratory to be tested in 1908, and again in 1913, and the
two reports give the values 1 '01875 volts at 17° C. in 1908 and 1 '01860 volts at 17° C.
in 1913. The N.P.L. comparisons were made with Weston normal cells, the E.M.F.'s
of which have been directly determined in terms of the international ohm and the
international ampere (10"1 C.G.S.), and found to be 1'0183 international volts at 20° C.
During the 1913 standardisation the cell was put through a temperature cycle of
10° C., from 15° C. to 25° C., and though examined daily over a period of three weeks,
no anomalous changes of E.M.F. with temperature were recorded. All the cells used

404

ME. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

in these experiments have been compared with No. 7 at various dates, and the
differences between this cell and any individual cell have remained small and approxi-
mately constant. The differences expressed in hundredths of a milli-volt, between
any cell and No. 7 are given in the following table, the negative sign indicating the
amount to be subtracted from the E.M.F. of cell No. 7 in order to obtain the E.M.F.
of the cell in question.

Number of cell.

Date.

Tempera-

'

1. 2. 3. 4.

5.

6.

7.

°C

July 21, 1908 . . .

21

- 1 -17 - 5 -16

-2

0

standard

September 22, 1908 .

17

•1 - 8 1 -15

+ 1

standard

at N.P.L.

August 2, 1910. . .

20 +10 0 4 -33

+ 8

+ 13

standard

May 5, 1913 . . .

17-5 +18 -IS -12 -24

+ 3

+ 21

May 11, 1914 . . .

17

+ 22 -20 -10 -31

— 2

+ 25

"

From the last three sets of observations it will be seen that the cells used in these
experiments have remained constant to 1 part in 10,000 over a period of nearly four

years.

The Potentiometer.

This was of the well-known Thomson- Varley pattern with two dials, its total
resistance being 100,000 ohms. The current for the potentiometer was supplied by
Leclanche cells, the number employed being adjusted in accordance with the magni-
tude of the electric current in the main circuit, to make the potentiometer readings
greater than 5000. The galvanometer used was of the Thomson pattern, having a
resistance of 7000 ohms, and the sensitiveness was such that a change of unit amount
on the vernier-dial of the potentiometer produced an alteration of 7 scale-divisions in
the galvanometer deflection. A calibration of the potentiometer showed that no
error amounting to 1 in 10,000 could be introduced by any arrangement of the
sliders.

The Pressure Regulator (fig. 6).

This was of the form designed by Prof. CALLENDAR, and exhibited by him at a
meeting of the Physical Society of London in 1902. It consists, essentially, of a
U-tube the limbs of which, lettered G and S respectively, are partially filled with
mercury. The gas after passing through a Wolff's bottle containing sticks of KOH,
to remove impurities in the gas, is led into the limb, G, through a tube, A, the lower
end of which is ground so that its cross-section is elliptical. The gas then passes out

ATMOSPHERIC PRESSUEE BETWEEN 104° C. AND 115° C.

405

through B to the burner. In the tubes, A and B, there are T- connections which are
joined by a rubber tube, C, on which is a screw-down stop-cock, whereby the amount
of gas going through this bye-pass can be adjusted.

Steam, generated in the boiler, is led directly through the tube, D, into the other
limb of the U-tube and passes thence 1 through E into the calorimetric apparatus. Tf

CAS
SUPPLY

Fig. 6.

the connecting tap, H, be open, so that the columns of mercury in the two limbs, G
and S, are connected, it is obvious that, as the steam pressure rises, the mercury will
be forced downwards in S and upwards in G, until eventually the main supply of
gas passing through the lower end of the tube, A, is more or less cut off. When the
gas supply to the burner is thus reduced, the rate at which steam is generated is

406 MR. J. H. BEINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

lessened, and in a few seconds equilibrium is attained ; the current of steam keeping
its own pressure constant. This pressure is measured on a manometer connected with
the tube, M. Perfection in regulation is not obtained until sufficient water has con-
densed in the limb, S, to fill the length of tube between the top of the mercury and
the exit, E, but to hasten this condition the tap, J, may be opened, thus allowing
water which has condensed in the bulb, F (which is also connected to the tube, M) to
run into the limb, S, until the latter is filled to the requisite height. The pressure of
the steam can be altered either by varying the amount of mercury in the regulator,
or by raising or lowering the end of the tube, A. At the end of every experiment
the main gas supply was cut off and the tap, H, closed, so that when the regulator
was put into action at the commencement of the next experiment, the limb, S, con-
tained sufficient water to ensure perfect regulation at once. It is necessary for the
gas supply to be constant, since variations in the pressure in the gas main produce
corresponding variations in the quantity of steam generated. Such effects were not
noticeable on the thermometric readings of the " cold temperature," but they were
quite evident when the heating current was on. Sudden and violent fluctuations
were prevented by passing the gas into the bell of a small gas-holder before it entered
the regulator. The accuracy of these experiments depends chiefly on the constancy
with which the pressure can be maintained, and the wonderful steadiness in the
temperature of the steam is exemplified in the extended tables of observations which
are given.

From these it will be seen that the " cold temperature," observed after an interval
of about an hour, agrees to 0'03 cm. of bridge-wire or to about one two-hundredth of
a degree Centigrade, with that determined before the electrical energy was supplied.

The Calonmetric Arrangements (fig. 7).

The calorimeter proper consisted of a glass tube, Y, about 50 cm. long, in which
the heating coil, 0, and the thermometer sheath, N, were fixed. This tube was
jacketed by another glass tube, S, which enclosed the length occupied by the heating
coil and thermometer. In the following experiments three different types of calori-
meter were used, each of which will be described later in connection with the results
obtained.

The calorimeter flow-tube and its surrounding glass sheath were carried on a split
rubber cork wound with omega tape and fixed, making a steam-tight joint, into a
space enclosed by a double-walled brass jacket, 45 cm. long and 5 cm. internal
diameter. From the lower part of this jacket there was a double-walled side-tube
60 cm. kmg, packed tightly with gauze discs, through which the steam had to pass
before it reached the calorimeter proper. The whole external jacket hence consists of
two L-shaped vessels, one inside the other, the space between the double walls being
continuous.

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115' C.

407

This space was provided with an inlet, E, connected directly on to the pressure
regulator, and the steam entering here, passed between the double walls of the
jacket and out at F into the steam separator, G, whence it entered, through the
throttle, T, into the side-tube.

On its passage through the tightly packed gauze discs in the side-heating tube
the steam was heated up to the temperature of the jacket. It then entered the
space surrounding the calorimeter, whence it finally passes down the flow-tube, Y,
to the condenser fixed on to Z. In fig. 7, which represents diagrammatically the
arrangement thus described, it will be noticed that the cylindrical space inside the

K

n

i /

1

I

k

*l

3
5

Q

Y

o

i

m

<r\

d

Fig. 7.

double walls of the main jacket is divided into two compartments by a brass disc, K.
This was soldered round the inner tube about 5 cm. from the upper end. The
function of this disc is to prevent the steam impinging on the rubber cork closing
the upper end of the tube, and thus being cooled.

Any slight cooling due to the steam striking the lower split cork is of no importance,
since the steam would be warmed again during its passage up between the calorimeter
and the surrounding jacket. The whole of the jacket, the separator, and the con-
necting tubes were heavily lagged with felt. The main jacket containing the
flow-tube was fixed vertically. The side-heating tube was thus slightly inclined

VOL. ccxv, — A. 3 i

408

ME. J. H. BKINKWOKTH ON THE SPECIFIC HEAT OP STEAM AT

to the horizontal, consequently, any water condensed between the double walls
gravitates towards the drain-tube, d2, whence it is expelled. The other drain-tubes,
dl and ds, serve to remove condensed steam from other parts of the apparatus.

The drains d1 and d2 consisted of lengths of about 2 metres of small-bore compo-
tubing. The drain dA consisted of tubing of the same bore, but the lengths fixed

J

f \

v y

Y

Y

s

P

n

) r

(a)

(b)

Fig. 8.

/

here were altered according to the magnitude of the steam-flow in such a manner

that there was always a small steady current of steam passing out from this drain.

e dram-tube, d3, was connected to a brass tube, 3 mm. diameter, carried by the split

ffk to which reference has been made, the end of the brass tube being flush with

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C. 409

the inner surface of the cork. A water manometer, Q, was also attached to the
brass tube so that an approximate estimate of the pressure in the space surrounding
the calorimeter could be made.

The, Heating Coil.

This consisted of about 130 cm. of 0 '004-inch platinum wire, bent double, the
two strands being wound, in parallel, on a serrated mica frame. The loop end was
gold-soldered to a piece of 0'008-inch platinum wire which was fixed centrally along
the mica framework. A length of 12 cm. of No. 22 silver wire, bent double, was
joined at its loop to the free end of this thicker platinum wire, and the two other
free ends of the 0'004-inch platinum wire were likewise soldered to the loop end of
a similar piece of silver wire. These silver wires were bound to the mica frame,
which was always immersed to a depth of 4 or 5 cm. in that part of the current
of steam which was actually heated ; thus any error due to conduction of heat from
the heating coil was prevented. The four free ends of these silver wires were then
soldered to the main current and potential leads. Each of these leads was a length
of No. 20 copper wire, to the upper end of which was hard soldered about 1 cm. of
No. 27 platinum wire. A thin lead glass tube enclosed each conductor and was fused
round the platinum wire. The leads thus enclosed were brought down symmetrically
around the glass jacket, S, surrounding the flow-tube, Y, and were carried by the
split cork. The general arrangement is indicated in figs. 7 and 8, P P being the
potential and 1 1 the current leads.

The Thermometer.

This consisted of about 120 cm. of 0'004 inch platinum wire, wound in a screw
thread of 40 turns per cm., cut on an ivory cylinder. This was carried on a long
lead of No. 24 copper wire, the latter passing axially through to the far end of
the cylinder. The cylinder was fixed in its position on the wire between two
small lumps of solder. The ends of the platinum wire were soldered, one to the
lower end of this lead, the other to the end of a length of No. 28 copper wire
which only reached as far as the near end of the ivory cylinder. Another pair of
leads, consisting of equal lengths of Nos. 24 and 28 copper wire, which again
only reached as far as the near end of the cylinder, where their ends were
soldered, formed the compensating leads of the thermometer. All these leads
were enclosed in fine glass capillary tubes firmly bound together with silk. Before
winding the platinum wire, the ivory was baked for some time at a temperature
of about 120° C. in order to shrink the cylinder. The over-all length of the
thermometer was 35 cm., the platinum wire being wound over a length of
3'6 cm. of ivory. The thermometer and leads were enclosed in a glass sheath, N
(fig. 7), 2 '5 mm. in diameter, which was carried by a rubber cork fixed tightly

3 I 2

419

ME. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

into the lower end of the flow-tube. This thermometer sheath reached right up
the flow-tube to within a few millimetres of the lower end of the heating coil,
so that by moving the thermometer up or down in its sheath, the distance
between the end of the heating coil and the middle of the thermometer could be
easily varied between wide limits. This thermometer was made by Mr. W. J.
COLEBROOK, Superintendent of the Physics Workshop at the Imperial College,
and here I desire to record my indebtedness to him for the assistance I have
obtained during the construction of the apparatus used in this investigation.

The Platinum Th.ermometer Bridge and the Measurement of the Fundamental
Interval and Stem-exposure of the Thermometer.

The well-known form of bridge designed by Prof. CALLENBAR was used, the
plug contacts being replaced by mercury cups. Three calibrations of the coils and
bridge-wire were made during the course of this investigation and corresponding
readings in these different calibrations agreed to within 0'005 cm. of bridge-wire.
An astatic galvanometer, with a long quartz fibre suspension, was used with the
bridge, and the sensitiveness in the final experiments was such that by moving
the contact maker 01 cm. on the bridge-wire, an alteration of from 10 to 15
divisions in the deflection was produced on reversing the current. Observations
of the sensitiveness were made during each experiment, and no attempt was
made to adjust the contact to obtain an exact balance. The usual apparatus was
employed for the determination of the fixed points, and these were re-determined
after every removal of the thermometer from its sheath. A series of observations
of the freezing- and boiling-points, with various lengths of the thermometer stem
exposed, was made. These values were plotted, and from the curves obtained
the variation of the fundamental interval with stem-exposure could be deduced.
The values obtained on September 12, 1912, were as follows :—

Exposure.

Freczi

cm.

0

138

2

138

4

138

6

138

8

138

10

138

12

138

14

138

15-5

138

Boiling-point.

Fundamental interval.

1919-435

537-785

1919-43

537-775

1919-42

537-755

1919-415

537-74

1919-405

537-705

1919-40

537-67

1919-38

537-58

1919-305

537-355

1919-11

536-93

. These results show that the fundamental interval only falls by about 1 part
' when the stem-exposure is changed from 0 to 16 cm. In the calculations,

ATMOSPHERIC PEESSUEE BETWEEN 104° C. AND 115° C.

411

the values taken for the boiling-point and fundamental interval are those corre-
sponding to the given amount of exposure. A few determinations of these
constants obtained on different dates are included in order to show the constancy
with which they could be repeated.

Date.

Freezing-point.

Boiling-point.

Fundamental
interval.

Exposure.

1912.
April 18

1381 -G5

1919-46

537-81

cm.
3-5

September 12 . . . . <
November 14 ...

1381-66

1381-71
1381-89

1919-42
1919-40
1919-58

537-76
537 • 69
537-69

3-5

8-7
8-7

1913.
March G ....

1381-85

1919-62

537-77

3-5

It will be noticed that there is an increase in the values of the freezing- and boiling-
points but that the fundamental interval remains remarkably constant. The increased
resistance is easily explainable. It is due to the slight squeezing of the wire against
the side of the glass sheath, whereby a gradual reduction in the cross-section of the
wire is produced after many movements of the thermometer in its sheath. During
the autumn of 1913 the thermometer was broken ; after mending, by fusing on a
short length of the same kind of platinum wire, the constants were : —

Date.

Freezing-point.

Boiling-point.

1914.

1383-45

1921-74

May 1 .

1383-50
1383-55

1921-72
1921-76

Fundamental
interval.

538-29
538 • 22
538-21

Exposure.

cm.

2
8

8

Methods used to Obtain a Uniform Temperature Distribution in the Steam Floiv.

By far the most troublesome source of error to eradicate was that due to imperfect
mixing of the steam. When it is remembered that the velocity with which the
steam is flowing may be as great as 80 cm. per second, it is obvious, that in order to
ensure a uniform radial distribution of temperature throughout the steam, in the
short time of its passage from the heating-coil to the thermometer, the mixing
arrangement must be extremely good. The only method of testing the excellence of
the stirring arrangement is to conduct experiments with different distances between

412 MR. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

the heating-coil and the thermometer. Until agreement in the calculated values of
the specific heat obtained under the different experimental conditions, and a definite
relationship between the values of the heat-loss and the distances separating the
heating-coil from the thermometer are obtained, it cannot be assumed that the mixing
is adequate. The methods of mixing employed may be conveniently referred to as—
(a) gauze mixing ; (b) spiral mixing. The method (a) was employed by SWANN in
his experiments on the Specific Heat of Air and Carbon Dioxide, and has been used by
other observers in their determinations of Specific heats by the continuous-flow method.
Circular discs of copper gauze are cut with a diameter slightly greater than the
internal diameter of the flow-tube. These discs are then bent just to fit the flow-
tube, each disc having a small flange. The space between the heating-coil and the
end of the thermometer tube is then filled with these discs, and since any one disc
only touches the flanges of the discs on either side of it, longitudinal conduction from
disc to disc is necessarily very small. At the same time, in virtue of the high thermal
conductivity of copper, a uniform distribution of the heat throughout the gas, as it
passes from the heater to the thermometer, might be expected. This method of
mixing was employed in a large number of experiments extending over a period of
more than two years. In some experiments, discs were also fitted over that part of the
thermometer tube surrounding the ivory cylinder ; in other experiments the ther-
mometer previously described was replaced by a bare-wire thermometer consisting of
a single loop of 0'001-inch platinum wire. Although, in most cases, very good agree-
ment between the results of experiments under any given conditions was obtained,
no such agreement was found between these results and those obtained under different
conditions. A further consideration of the results obtained with this form of mixing
will be found later.

(6) Spiral mixing. — A very efficient method of mixing the steam consisted in
tightly wrapping around the thermometer tube a spiral of small bore compo-tubing,
. the walls of which were then pinched together to form a screw thread closely fitting
the flow-tube and with a pitch of about 1 cm. The only apparent disadvantage of
this was that conduction along the spiral might be considerable, so another form of
spiral was designed in which this error was minimised.

The mixing arrangement as used in the final experiments is indicated in fig. 8. A
number of circular discs, each one having a diameter equal to that of the flow-tube,
were punched out of thin copper sheet. These were then clamped one above the
other and a hole of diameter equal to that of the thermometer sheath was drilled
eccentrically through all of them, fig. 8, D. Each disc was then cut along the
diameter common to the disc and the hole, and was slipped on to the thermometer
sheath. By slightly bending the discs in the manner shown, they, when soldered
together, form a kind of spiral which mixes the steam most efficiently ; moreover,
since the discs are only connected by small solder junctions, longitudinal conduction
is greatly reduced.

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C. 413

Determination of (he Weight of Condensed Steam.

Prof. CALLENDAR has shown the importance of obtaining steam free from impurities,
hence only distilled water was used in the boiler. The steam passed to the calori-
metric apparatus through block tin tubes, and the various joints were made in such
a way that the steam came in contact with as little rubber as possible. After its
passage through the calorimeter, the steam was led into a brass condenser more than
1 metre long. The water was collected in half-litre flasks, placed in such a manner
that the end of the condenser was always 2 or 3 inches inside the flask. In all
experiments the flasks were closed immediately after their removal from the condenser
and then weighed. The question of the evaporation of the water during its conden-
sation and collection was carefully considered. The air in the flasks was always fully
saturated, since a certain amount of water remained in them from previous experi-
ments, so that one would not expect any appreciable amount of evaporation during
the actual collection of the water. Experiments were made in which, immediately
after weighing a flask, it was uncorked, placed near the end of the condenser for a
known time, and then weighed again. In no case \vas there a change in weight
amounting to 0'03 gr. in 15 minutes. In some experiments the collecting flasks were
fitted with corks, through which passed (a) a drying tube filled with calcium chloride ;
and (6) a tube which was only just wider than the nozzle of the condenser. Again
no systematic differences were observed and, since the rapid adjustment of a flask
under the end of the condenser was made less facile, in most of the experiments the
flasks have been used without drying tubes. The weights were obtained by means
of a large Oertling balance, sensitive to O'Ol gr. with 500 gr. on the scale-pan. It
was sufficiently accurate to note the weights to 0'05 gr., care being taken that the
total error on the difference between the weights (i.e., flask alone and flask + water)
was not cumulative. The maximum error on the weight determinations is not likely
to be greater than 0'05 gr., or 1 part in 3000 on the minimum flow. The weights of
water have been reduced to weights in vacuum ; the air displaced being always fully
saturated, its density under average conditions, viz., 760 mm., and at 15° C., has
been taken as 0'00121 gr. per cubic centimetre.

Adjustment of the Flow.

The amounts of steam flowing through the calorimeter were adjusted by means of
throttles, fig. 7, T. Each throttle consisted of a short length of glass tube which
was tightly fixed, by means of a rubber tube and wire, into the outflow tube
connecting the steam separator with the side-heating tube. The diameters of the
throttles were such that the flows used were of the order of 0'8, 0'4, and 0'2 gr. per

second.

Method of Making an Experiment.

The tap, H, fig. 4, between the two main limbs of the regulator was opened slightly,
so that the level of the mercury in the tube in which the gas current is adjusted was

414 ME. J. H. BEINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

lowered sufficiently to allow a full stream of gas to pass to the burner ; this tap was
then closed. As soon as the steam was generated it passed into the outer annular
jacket surrounding the calorimeter, and a small current of steam escaped through the
drain-tube, d2. The pressure gradually rose until the difference between the levels on
the gauge attached to M was the same as that between the levels of the mercury in
the regulator, when the tap, H, was again opened and the regulator put into action.
These conditions were maintained for about half-an-hour, in order that the copper
discs in the side-heating tube and the calorimeter itself might be warmed up before
passing steam into the inner jacket. At the end of this period the tap, X, fig. 4, was
slowly opened, thus allowing steam to enter the separator and to pass through the
throttle into the calorimetric apparatus. It was possible to obtain an approximate
reading of the temperature of the boiling-point of steam under the given barometric
conditions whilst this gradual heating up was taking place. For during this initial
warming of the calorimeter by the inflowing current of steam a small amount of
condensation necessarily took place in the flow-tube, this moisture being evaporated
at a constant temperature after the passage of the steam for a few minutes.

At the end of about 40 minutes the thermometer reading was quite steady, and
after making allowances for changes in the barometric height, agreed to within
0'03 cm. of bridge-wire with the cold temperature reading taken after the heating.
In some of the earlier, and in all the final, experiments, these " initial cold tempe-
ratures " were recorded until the indicated temperature was steady for about ten
minutes, but since it was found that the cold temperatures obtained after the
heating (i.e., three-quarters to one hour later) agreed so well with the initial values,
in many experiments these initial readings were not recorded but only noted, so that
it was known when the calorimeter was heated up, and the conditions practically
constant. The electric current was then started through the heating coil, adjusted
to give the desired rise of temperature, and approximate balances noted on the
potentiometer. After another interval of from 20 to 40 minutes, depending on the
flow, the balance-point on the bridge again became steady to about 0'02 cm. or
0'004° C., and the main calorimetric observations were commenced. At a definite
instant a weighed half-litre flask was put under the end of the condenser, left there
for exactly ten minutes, in the case of the maximum flow, and then taken away.
Ten seconds afterwards it was replaced by another flask, and a second observation
of the flow during another ten-minute interval was taken. The duration of the
experiments with the medium and minimum flows was 15 minutes. In most cases
the amounts of steam condensed in two consecutive experiments, under as nearly as
possible identical conditions, agree to about 1 in 1000, but when the difference in
the observed flows was greater than this, due to some real change in the current of
steam, the variation had often been expected, since a change in the temperature
readings, not due to a change of barometric pressure nor to a change in the rate of
supply of electrical energy, had pointed to such an effect. The readings of the

ATMOSPHERIC PKESSUKE BETWEEN 104° C. AND 115° C. 415

thermometer were taken at every even minute, the contact on the bridge-wire being
set at the nearest millimetre, and the deflection of the galvanometer noted on
reversal of the current.

During every experiment several readings were taken with the contact made
at the next millimetre reading on the bridge which gave a galvanometer deflection
in the opposite direction. The height and temperature of the barometer were read
at intervals of about seven minutes, unless they were varying very rapidly, when
readings were taken more frequently. The approximate value of the current was
noted on the ammeter ; a knowledge of this was useful when adjusting the current
in the determination of the heating effect of the leads. The potential balances across
the standard cells, across the secondary standard resistance, and across the heating
coil were taken at intervals during the experiment. In addition, the temperature of
the standard cells and that of the oil bath in which the secondary standard was
placed, were noted. Since seven Leclanche cells were required in the maximum flow,
and only five in the medium and minimum flows, a small steady change took place in
the potentiometer reading for the standard cells during the short interval following
the change in the number of Leclauches used. The variation was only 3 or 4 parts
in 7000 in about half-an-hour, and after that period of time had elapsed the reading
remained quite constant. The potential balances across the heating coil and across
the secondary resistance were wonderfully steady, especially during experiments
with the two smaller flows. In the case of the maximum flow the irregular
fluctuations only amounted to about 30 divisions on each side of the balance-point on
the galvanometer scale, and a change in the potentiometer reading of 4 (in about
8000) was sufficient to throw the balance out by the same amount. At the end of
the second experiment with any flow, the electric current was switched off and the
weights of condensed steam determined. As soon as the temperature was again
nearly steady, another flask was put under the end of the condenser and a " check "
flow obtained. Meanwhile temperature readings were taken until the recorded
values were constant. This temperature reading is tabulated as " cold temperature
before." A current equal to that used in the main experiments was then passed
through each pair of current and potential leads, and the heating effect of the current
in each pair of leads measured. Then the cold temperature was again determined :
this value is tabulated as " cold temperature after." I have included specimen
tables showing all the observations made on one day, for thereby the remarkable
steadiness of the conditions is illustrated very clearly.

Reduction of the Observations.

In each experiment there are as many observations of the " hot temperature " as
there are minutes of time in the duration of the flow, and, in addition, there are the
two " cold temperatures," the barometric readings, the various temperature observa-
tions and the potential balances. The mean value of each set of readings is then

VOL. CCXV. — A. 3 K

416 ME. J. H. BKINKWOETH ON THE SPECIFIC HEAT OF STEAM AT

obtained, thus correcting them to' the middle of the interval in order to correspond
to the time of average flow. The potential readings were not taken as frequently as
the thermometric readings, this being unnecessary, since the latter were irregular
whilst the former gradually changed in one definite direction ; they generally showed
a slight fall. The electrical energy supplied per second was determined from the
mean values of the potentiometer readings ; the necessary temperature corrections
for the resistance of the standardised coil and the E.M.F. of the cadmium cells having
been previously made. The mean thermometric readings give the temperatures
of the steam when hot and when cold, and from other measurements the rise of
temperature due to the heating effect of the current in the leads connected to the
heating coil can be obtained. All these temperatures are expressed in terms of
centimetres of the bridge-wire, the calibration corrections of the bridge itself being
included.

The rise of temperature due to the passage of the current through the leads is
subtracted from the observed "hot temperature," the value thus obtained and the
mean " cold temperature " are then reduced to platinum temperatures, and these
again to temperatures measured on the air scale, the latter reduction being made by
means of the well-known parabolic formula t—pt = 1'50 x I0~4t (t — 100). The flows
given are corrected to correspond to weighings in vacuo.

The number of flows taken for a complete experiment was always three, each flow
being repeated twice. The concordance between the values of CE/Qc£0, calculated
from two consecutive flows of the same magnitude, gives a criterion of the accuracy
of the experimental measurements. Whenever possible, three flows were taken on
the same day, as it was thought that the heat-loss might vary from day to day ;
also the order of the flows was usually medium, minimum, maximum, in order to
reduce the time taken between experiments made with different flows. For when
the flows are taken in this order, the initial heating up is fairly quick, and less time
is lost in refilling and reheating the boiler between successive flows. Hence there is
no long interval during which steam is not passing round the outer jacket, and the
calorimeter does not cool to any very great extent between consecutive experiments.
Even when taking the flows in this order it was impossible to complete the three
experiments in less than eight hours.

Preliminary Experiments.

A long series of preliminary experiments was made in order to find the best
method of mixing the steam, to investigate the relationships between the heat-
loss and the flow, and between the heat-loss and the distance separating the
thermometer from the heating coil. Several different arrangements of the
apparatus were made, and various methods of stirring employed. It will suffice
to summarise these experiments in the order in which they were carried out, by

- ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 105" C. 417

referring to the main details and arrangement of the calorimetric apparatus, and
to the results obtained. All the results quoted have been calculated in the same
way as in the final experiments.

(a) A few experiments were made with the apparatus used by SWANN for air
and described by him in his paper. In this arrangement bare wire thermometers,
wound on mica frames, were used. No satisfactory results were obtained owing
to the condensation of steam upon the mica frames and the consequent impairment
of their insulating properties.

(6) The apparatus was dismantled and set up again with a single thermometer
(that used in the final experiments) enclosed in a glass sheath passing centrally
along the flow-tube and carried by a rubber cork fixed in the outflow end of the
tube. The stirring arrangement consisted of the pinched spiral of compo-tubing,
to which reference has been made. The main flow-tube was fixed vertically, as
is indicated in fig. 7, with the non-vacuum jacketed sheath.

The current and potential leads entered the flow-tube in the manner indicated
in this figure, but they were carried by a cork (see Sw ANN'S paper, fig. l) fitting
into the top of the outer brass jacket, and were not bent round and carried
down outside the flow-tube as in fig. 7. The disc, K, had not been soldered
across the jacket at this time, but its function was performed by a rubber bung
which fitted tightly to and was carried by the current and potential leads. The
initial temperature of the steam was about 103° C., and the three flows varied
from 0'57 to 0'14 gr. per second. The values of the constants deduced were

Sc at 103° C. = 2'021, h = G'0400, k = 0'00400.

In these experiments the heating coil had a resistance of about an ohm, and
the standard resistance used as a comparison was 1'039 ohms at 19° 0. The
temperature variation of the specific heat was about 0'0035 per degree, a value
three times as great as that found in the later experiments. Later, the separator
was re-designed, and in all the subsequent experiments it was of the form shown
diagrammatically in fig. 7. A new heating coil and a new standardised coil of
about 5 ohms resistance were made and used in all the experiments made
afterwards. In addition, the regulator was altered to give a " cold temperature "
of 104'5°C. instead of 103° C. The effect of these changes was to reduce the
temperature coefficient of the specific heat to about O'OOIO joules per degree.

(c) Many experiments were made in which the temperature of the steam was
measured by means of a bare wire thermometer consisting of about 3 cm. of
0'001-inch platinum wire. Spiral mixing was impossible, and in consequence the
gauze method of mixing was employed. Measurements of the heating effect of the
current in the thermometer were made. It was found that although this heating
effect was slightly different in the different flows, it was the same on both hot
and cold temperatures observed with any one flow, and no systematic error was

3x2

418 MR. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

thereby introduced. The apparent values of the specific heat obtained with
various amounts of gauze and with the above arrangement of apparatus ranged
about 1'95, gradually increasing as more gauze discs were inserted.

(d) Further experiments with gauze mixing were carried out with the calorimeter
and jacket, arranged as in fig. 7, and with the enclosed thermometer used in the
earlier and in all the later experiments.

That part of the thermometer sheath surrounding the ivory bulb of the platinum
thermometer carried gauze discs, a variable number of similar discs being interposed
between the end of the heating coil and the end of the thermometer tube. With
4'5 cm. length of discs on the thermometer tube, and a length of 7 cm. of the total
distance between the end of the thermometer and the heating coil filled with similar
discs, the value of Sc calculated from the experimental observations increased
(approximately proportionately) with the actual distance between the heating coil
and the thermometer. Again, with a distance of about 14 cm. between heating coil
and thermometer, the calculated value of the specific heat increased with the length
of tube filled with gauze, the latter variation being from 1'965 with 4'7 cm. of gauze
to T990 with 10 cm. of gauze. These results show that, in order to mix the steam
thoroughly by means of gauze discs, a much greater distance must exist between the
thermometer and heating coil than was possible in this arrangement without having
a great length of the thermometer-stem exposed. Further experiments on gauze
mixing were not made for the above reason, and also because, in order to change
either the position of the thermometer or the amount of gauze, it was necessary to
dismantle the calorimeter.

The Final Experiments.

These may conveniently be divided into two groups : one consisting ot the
experiments made with the non-vacuum jacketed calorimeter, the other group
including all the experiments performed with the different types of vacuum vessel.
Abridged tables of the experimental measurements are to be found below (p. 425).
Complete records of all the observations are preserved in the Archives.

Group 1. Experiments with the NoTi-vacuum Jacketed Calorimeter. — Fig. 7 shows
the general arrangement of the calorimetric apparatus during the course of these
experiments. The thermometer was fixed eccentrically in the spiral constructed of
discs of copper foil. The conditions under which the experiments in this group were
made were varied considerably. The group itself consists of four series of experiments
which only differ in that the heat-loss has been progressively augmented by increasing
the distance between the heating coil and the thermometer by sliding the latter in its
enclosing sheath. In each series three flows were used, and at least two different
values of the rise of temperature.

Group 2. Experiments with Vacuum-jacketed Calorimeters.— The general arrange-
ment was as in the last group of experiments and as indicated in figs. 7 and 8.

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C. 419

There are two series of experiments in this group : one series (a) in which the
vacuum-jackets were exhausted before their insertion in the double-walled brass
jacket, and a later series (b) with the silica calorimeter, in which case the vacuum
was formed when the calorimeter was in situ. In this case the space between the
double walls of the calorimeter was connected with a Geissler tube, a charcoal tube,
and a pump. Three sets of experiments were made with different pressures in this
jacket.

Calculation of the Results.

The values of CE/Q dd, obtained with three different flows, but with the same rise
of temperature dd, were calculated from the reduced tables of observations, and then,
by means of three equations of the form

the values of Sm, h, and k were obtained. S.m is the mean specific heat over the
range 104 '5° C. to (I04'5 + c£0)° C. Similar calculations were then made from the
results obtained under the same experimental conditions but with other values of dO,
and it was found that, although the values of h and k were constant within the limits
of experimental error, the values obtained for Sm decreased witli an increase in the
value of d9. This relation is represented by the equation

Sc = S^ + O'OOlOdfl.

Se = specific heat at the cold temperature, i.e., 104 '5° C.

This means that, for any given flow with a given calorimetric arrangement, each
quantity CE/Q d6 depends on the magnitude of d9, hence by applying this tempe-
rature correction to the observed values of CE/Q d9 the latter are brought to the
values they would have had if the rise of temperature had been zero. Then, by
taking the mean of all the maximum, all the medium, and all the minimum values of
CE/Qdf0, thus corrected, and substituting these in the three above equations, the
most probable values of Sc, h, and k can be deduced. The values of Sc thus obtained
show a distinct variation with the barometric height. This variation of the specific
heat at constant pressure with pressure can be calculated by means of equation (7),
p. 387. In the case of steam, at a temperature of 104 '5° C., if S is expressed in
joules/gr. deg., and 1 cm. of mercury is taken as the unit of pressure dS/dp = O'OOIS,
this means an increase in the value of S of 0'065 per cent, for an increase of pressure
of 1 cm. of mercury. Experimentally, the value of dS/dp is found to be about twice
as great as this, but, taking into consideration both the smallness of the correction
and the fact that the barometer only varied by about 2 cm. during any one series of
experiments under identical conditions, the agreement is very satisfactory. A pressure

420

MR. J. H. BRINKWOETH ON THE SPECIFIC HEAT OF STEAM AT

correction of O'l per cent, per centimetre has been applied to the results calculated as
above in order to reduce them to the values corresponding to a pressure of one
atmosphere. The variation of the specific heat with temperature, indicated by the
experiments made with different values of dO, was directly measured in the case of
each calorimetric arrangement in the following manner. With a constant value of
the flow a series of .experiments was made in which only the value of d6 was
changed. The observations made in the case of two such experiments with the
medium flow are given (fig. 9).

In the results quoted, the observed values of CE/QcZ0 have been corrected to
correspond to an infinite value of the flow and the pressure correction has been applied.
These corrected values of CE/Q dO are the values of the mean specific heat Sm. The

2030^

2020

ZOIO

2000

1995

Fig. 9.

relationship existing between the values of Sm and dd is well represented by the
equation previously given. Similar experiments have been made with both the
maximum and minimum flows and the observed temperature variation found in each
case agrees with that given above. In some experiments a more rapid variation of
the mean specific heat with temperature, in the neighbourhood of the cold temperature,
has been indicated, but with small values of dd, the experimental errors are magnified,
and in consequence, the straight line relationship has been applied to all the observa-
tions. The temperature variation of the specific heat at the constant pressure of one
atmosphere as calculated from equation (7) gives dS/de = -O'OOIO or S« = Sc-0'0010 d6
where Se is the actual specific heat at the temperature d. Hence the observed
coefficient is double that deduced from theoretical considerations. The experimental

:' ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C. 421

value has been used in reducing the mean specific heats to the value corresponding to
zero rise of temperature.

Throttling Effects in the Calorimeter Flwv-Tube.

During its passage between the heating coil and the thermometer the steam
expands adiathermally and its. temperature is lowered. This effect depends on the
actual temperature of the steam and, in consequence, the cooling will be different
on the hot and the cold flows ; it is greater on the cold flow. However, it is only the
change in the amount of cooling which affects the results systematically ; the correc-
tion is such that it raises the observed value of the specific heat. The pressure
drop between the heater and the thermometer was estimated from the indications
of the manometer, Q, and the variation of the cooling effect was calculated from
equation (6), p. 387. With the straight flow-tube calorimeters the difference in the
cooling effects is about 0'0035° C. for a rise of temperature of 10° C. in the case of the
maximum flow, and for the medium and minimum flows the differences are still
smaller. Hence the throttling effects in the straight flow-tube calorimeters are quite
negligible. An additional proof that this is the case is afforded by the results of the
experiments made with these calorimeters, with different distances, I), separating the
thermometer1 from the heating coil. The mean values of the specific heat show no
systematic variation with D greater than O'l per cent, and this is of the order of
accuracy of the experimental measurements. With the spiral flow-tube calorimeter
the cooling effects were larger, for they depend on the throttling occurring in the
spiral, and the pressure differences between the heating coil and the thermometer were
greater than in the case of the experiments made with the straight flow-tubes. An
approximate calculation showed that the systematic error introduced, due to the
neglect of these throttling effects, could not amount to more than O'l per cent, on the
specific heat. Only one set of observations was made with the spiral flow-tube
calorimeter and these experiments are, in comparison with the others, unimportant,
and as the throttling correction is small and rather uncertain it has not been made.

(I designed the spiral flow-tube to be of such dimensions that the throttling effects
would be negligible, but in the piece of apparatus supplied by the makers the spiral
tube was constricted along its entire length.)

Specimen Tables of Observations.

A complete set of the observations made on May 1st, 1914, with the silica calori-
meter, is included in order to show the steadiness of the experimental conditions.
The vacuum jacket was exhausted by means of charcoal in liquid air, and throughout
the three experiments no discharge could be produced in the Geissler tube connected
with the jacket. The order in which the observations were taken has been described
on p. 414. The values of Sc, h, and k calculated from these observations,- after the
necessary corrections have been made, are 2'0309, 0'00470 and — 0'00005 respectively.

422 ME. J. H. BEINKWOETH ON THE SPECIFIC HEAT OF STEAM AT

BLACK VACUUM. May 1st, 1914. Maximum Flow.
Temperature Observations.

Initial cold temperature.
Box coils 1940-94.

Hot temperature (C = 1 • 48 amps.)
Box coils 1990-98.

Cold temperature (before).
Box coils 1940-94.

„,. Baro- -rj -yj7 Deflec-
meter. tion.

Time.

Baro-
meter.

B.W.

Deflec-
tion.

Time.

Baro-
meter.

B.W.

Deflec-
tion.

h. m. ° C.

h. m.

°C.

h. m.

°C.

i

3 15 4-70 8-

3 35

770-3

0-30

1 +

4

9

4-70

23 +

17 5-

36

19

2 +

10

12 +

19 770-15 6-

37

0

11

6 +

19

38

1 +

12

3 +

20

7-

39

4 +

13

0

22

7- 40

3-

16

770-4

1-

23

4-60 7+ 41

9-

18

19

3-

25

7 +

42

9-

20

2-

43

8-

22

2-

44

770-35

7-

45
46

19

i

4 -

7-

47

7 -

i

48
49

5-
5-

Leads (C = 1 • 48 amps.).

j

50

7-

25

4-80 4 +

01

6 -

27

5+ (1)

53

o -
4-

29

770-35
19

5 +

54

770-35

4-

31

6 +

55
56

19

5-

X.

33

6 +(2)

57

2-

35

j 5 +

58

2 +

59

770-4

2 +

4 00

19

1 +

j

1

2

0
3 -

Cold temperature (after).

3

4

0-20

4-

39

! 4-70

0

5

770-35

11 +

42

770-4

0

I
1

19

45

1

19

0

Weight Determinations.

gr. h. m. s. gr. h. m. s. gr. h. m. s.
527-65 3 45 0 530-30 3 55 10 540-45 4 5 20
54-00 3 35 0 56-70 3 45 10 66'90 3 55 20

gr. h. m. s.
528-00 4 22 0
54-10 4 12 0

473-65 473-60 473-55

473-

90

Q = 0-7894 gr./sec. Q = 0-7893 gr./sec. Q = 0-7893 gr./sec.

0- 7898 gr /sec.

Potentiometer Readings.

5 Cadmium cells.

Temperature.

Time. Standard

resistance.

Temperature.

Time.

Heater.

Time.

v °c.

h in

o p

4937
4937

16-3

3 34 7261
40 7261

31-0

. m.

3 35
41

8965
8964

h. m.
3 36

41

4937
4937

16-3

7260
54 7259
4 3 7258

30-8

47
53
4 5

8964
8963
8962

46
54
4 4

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° 0.

BLACK VACUUM. May 1st, 1914.

Temperature Observations.

Medium Flow.

Initial cold temperature. Hot temperature (C = 1 '04 amps.). ' Cold temperature (before).

Box coils 1940-94.

Box coils 1990-98.

Box coils 1940-94.

Time.

BT" B.W.
meter.

Ton" Time'

Baro-
meter.

B.W.

Deflec- rr-
, . .urn
tion.

Baro-
meter.

B.W.

Deflec-
tion.

1

h. m.

°C.

h. m.

°c.

h. m. " C.

9 40

5-80

6+ 10 30

1-80

4- 11

1

6-30

9 +

42

12+ 31

768-9

4- 8

5 +

48

6-10

0 32

17-5

1- 9

2 +

50

6 +

33

2+ 10 769-0

0

52

11 +

34

5+ 1

2 18

2-

53

6-20

0 35

3+ 14

3 _

57

768-75

5 + 36

1+ 16

4-

58

17-5

3+ 37

1+ 19

3 -

10 2

5+ 38

1 +

3

6-30

8- 39

1-90 13-

5

9 _

40

768-9

12-

Leads (C =

1 -04 amps.).

41

17-5

1-80

2 +

42

3+ 23

6-10

3 +

43

3- 25 769-2

6 +

44

4- 26 18

12 + 0)

45

4- 27

6-50

1 -

46

4- 29

2-

47

3- 30 709-25

6 • 50

3 -

48

2- 31 18

4 -(2)

49

2- 3

o
t)

3-

50

768-9

o _

51

17-5

1-

52

1 - Cold temperature (after).

53

1-

54

0 3

6

G-30

8 +

oo

0 3

7

4 + 0)

56

1- 3

9 769-35

5 +

57

1- 4

1 18-5

5 +

58

2+ 4

4

6 +

59

769-0

1- 4

0 769-4

i

7 +(2)

11 00

17-5

1-70 12+ 4

8 18-5

7 +

Weight Determinations.

gr. h. m. s.

gr. h. m. s. gr.

h. m. s.

393-85 10 45 0 397-15 11 0 10

408-85

11 23 0

54-15 10 30 0

57-35 10 45 10

69-20

11

8 0

339-70 339-80

339-65

Q = 0-3774 gr./sec. Q = 0-

3776 gr./sec. Q =

0-3774 gr./sec.

Potentiometer Readings.

5 Cadmium cells. T<fPeril-
ture.

Time.

Standard
resistance.

Tempera-
ture.

Time.

Heater.

Time.

°C.

h. m

0 C.

h. m.

h. m.

6946 16-6

10 29

7119

"25

10 30

8668

10 30

6946

36

7118

37

8668

37

7117

44

8667

44

6946

48

7116

49

8666

50

6946 16-5

59

7115

25

58

8665

57

VOL. CCXV. — A.

424

ME. J. H. BEINKWOKTH ON THE SPECIFIC HEAT OF STEAM AT

BLACK VACUUM. May 1st, 1914. Minimum Flow.
Temperature Observations.

Initial cold temperature.
Box coils 1940-94.

Hot temperature (C = 0 • 74 amps.)
Box coils 1990-98.

Cold temperature (before).
Box coils 1940-94.

Time.

Baro-
meter.

B.W.

Deflec-
tion.

Time.

Baro-
meter.

B.W.

Deflec-
tion.

Time.

Bare- B w
meter.

Deflec-
tion.

h. m. ° C

h. m.

0 C

h. m.

•

C

12 22

7-10

7 +

i

5

4-30

4 +

1 44

7-30

21 +

25

769-

7

12 +

6

6 +

45

9 +

26

18-

o

7-20

2 +

1

7

769-75

3 +

46

4 +

28

4 +

8

18-5

7 +

47

770-0

1 +

33

769-

75

4 +

9

3 +

49

19

4-

35

3 +

10

3 +

52

7-

37

3 +

1

1

4 +

54

g _

38

769-

75

7-30 11- | 1

2

769-8

1 +

56

769-9

9-

40

18-

0

11- 13

18-5

2 +

59

19

10-

14

2 +

15

2 +

16

4-40

11-

*

17

4-30

2 +

Leads (C = 0-74 amps.).

18

3 +

19

4 +

2 4

7-40

0

20

769-8

5 +

6

2 + (l)

21

18-5

5 +

8

2 +

22

5 +

23

4 +

9

4 +

24

4 +

10

770-0

5 + (2)

25

3 +

13

19

4 +

26

3 +

27

3 +

28 !

5 +

1 29

4 +

30

769-9

4 +

Cold temperature (after).

i 31

18-5

4 +

32

6 +

16

7-30

2-

33

7 +

18

770-0

6-

34

8 +

20

19

7-

35

770-0

7 +

24

7-

38

18-5 4-40

7 _

27

6-

Weight Determinations.

214-90

h. m. s.
1 20 0

217-95

h. m. s. gr.
1 35 10 228-

h. m. s.
95 200

54-40

1 5 0

57-50

1 20 10

68-

70 1 45 0

160-50

160-45

160-

25

Q = 0-1783

gr-/sec- Q = 0-1783 gr./sec. Q = 0

1781 gr./sec.

Potentiometer

Readings.

5 Cadmium
cells.

Temperature. Time.

Standard
resistance.

Temperature.

Time.

Heater.

Time.

6945
6945
6945
6945
6945

° C h. m.
16-5 1 4
11
16
27
16-5 33

5010
5009-5
5009
5009
5008-5

21
21

h. m.
1 5
11
18
26
33

6075
6074-5
6074
6074
6073-5

h. m.
1 6
12
18
26
34

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C. 425

Abridged Tables for the Calorimctric Experiments.

The following tables contain a summary of the observations. The original tables,
from which these have been abridged, are preserved in the archives with the reduced
tables referred to on p. 416. The second column gives the rise of temperature dd,
corrected to the absolute scale. The third column gives the flow in grammes per
second. In column four is each value of CE/Q dd as obtained from the experimental
observations, this quantity being expressed in joules per gr. deg. C. The next
column gives CE/QcZ# reduced to the standard temperature 104 '5° (J., and to a
barometric pressure of 760 mm. Column 6 gives the heat-loss expressed in joules per
gr. deg. C., i.e., II/Q, the quantity H being the sum of the terms h and &/Q. In the last
column are the experimentally determined values of the specific heat of steam under
atmospheric pressure at a temperature 104 '5° C. After the completion of the
experiments with the non-vacuum jacketed calorimeter it was thought advisable to
design a vacuum calorimeter in order to determine whether the magnitude of the k
term could be reduced. In consequence I designed the spiral calorimeter which was
used as soon as it arrived. After these experiments I designed the straight vacuum
jacketed flow-tubes which were exhausted by the makers. In experiments made with
these calorimeters the value of k was not so small as in the previoxis experiments with
the spiral calorimeter, and six more glass calorimeters were obtained which I hoped to
evacuate in situ. Unfortunately, each one cracked during the initial heating and no
experimental results were obtained until the silica calorimeter was set up.

3 T, -2

42G

MR. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

NON-VACUUM Jacketed Calorimeter. D = 11 '5 cm.

(1.)

(2.)

(3.)

(4.)

(5.)

(6.)

(7.)

1912
April 23

5-057
5-057

0-8187
0-8196

2-0689
2-0651

2-0718
2 • 0680

0-0566

i

2-0152
2-0114

April 25

5-316
5-308

0-7891
0-7894

2-0683
2-0687

2-0727
2-0731

0-0591

2-0136
2-0140

April 29

8-521
8-499

0-7938
0-7940

2-0647
2-0653

2-0721
2-0727

0-0587

2-0134
2-0140

April 25

5-332 0-3829 2-1492
5-330 0-3826 2-1488

2-1534
2-1530

0-1400

2-0134
2-0130

April 29

9-133 0-3802
9-109 0-3804

2-1477
2-1479

2-1565
2-1567

0-1411

2-0154
2-0156

April 25

5-538
5-534

0-1798
0-1794

2-3899 2-3948
2-3948 2-3997

0-3839

2-0109
2-0158

April 29

9-058 0-1793 2-3930 2-4016
9-035 0-1794 : 2-3940 2-4026

0-3847

2-0169
2-0179

— —

Mean St = 2-0143, h = 0-0400, k =

0-00520.

D = 87 cm.

April 30

5-210
5-208
5-210

0-7956 2-0422 2-0457
0-7961 2-0413 2-0448
0-7961 2-0414 2-0449

0 0298

2-0159
2-0150
2-0151

May 2

8-970
8-966

0-7913 2-0338 2-0425
0-7911 2-0346 2-0433

0-0301

2-0124
2-0132

April 30

5-157
5-153

0-3841
0-3837

2-0905 2-0941
2-0923 2-0959

0-0801

2-0140
2-0168

May 2

8-935
8-931

0-3784
0-3783

2-0859 2-0949
2-0870 2-0960

0-0818

2-0131
2-0142

April 30

5-078
5-074

0-1819
0-1818

2-2618
2-2628

2-2656
2-2666

0-2497

2-0159
2-0169

May 2

8-984
8-982

0-1836
0-1835

2-2538
2-2544

2-2630
2-2636

0-2482

2-0148
2-0154

' _

Mean Sc = 2-0147, h = 0-0172, k =

—

0-00520.

t

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C.

427

NON- VACUUM Jacketed Calorimeter. D = 6'2 cm.

0-)

(2.)

(3.)

(4-)

(5.)

(6.)

(7.)

1912
May 8

9-230
9-223

0-7928
0-7931

2-0193
2-0179

2-0266
2-0252

0-0125

2-0141
2-0127

May 9

5-147
5-143
5-141

0-7972
0-7976
0-7980

2-0236
2-0230
2-0221

2-0275
2-0269
2-0260

0-0123

2-0152
2-0146
2-0137

May 16

7-703
7-703

0-7891
0-7891

2-0194 2-0285
2-0199 2-0290

0-0126

2-0159
2-0164

May 6

9-000
9-000

0-3801 2-0506
0-3800 2-0497

2-0595
2-0586

0-0447 2-0148
2-0139

May 9

4-975
4-969

0-3847
0-3850

2-0554
2-0546

2-0590
2-0582

0-0437 2-0153
2-0145

May 16

6-157
6-157

0-1823
0-1824

2-1842
2-1825

2-1916
2-1899

0-1744

2-0172
2-0155

May 16

10-311
10-307

0-1826
0-1826

2-1744 2-1861
2-1748 2-1865

0-1742 2-0119
2-0123

Mean Sc =

2-0145, h = 0-0033, k =

0-00520.

D = 13 '8 cm.

May 23

9-460
9-458

0-7841 2-0829 2-0937
0-7840 2-0827 2-0935

0-0811 2-0126
2-0124

May 30

9.463
9-457

0-7861 2-0845 2-0949
0-7861 2-0850 2-0954

0-0808

2-0141
2-0146

June 6

5-606
5-595

0-8031 2-0846 2-0911
0-8026 2-0873 2-0938

0-0789

2-0122
2-0149

May 24

8-693

8-684

0-3901 2-1870 2-1945
0-3902 2-1862 2-1937

0-1800

2-0145
2-0137

May 30

9-156
9-150

0-3881 2-1828
0-3882 2-1820

2-1932
2-1924

0-1811

2-0121
2-0113

June 6

5-217
5-208

0-3883
0-3886

2-1900
2-1885

2-1967
2-1952

0-1810

2-0157
2-0142

May 23

9-254
9-250
5-940

0-1838
0-1838
0-1840

2-4633
2-4631
2-4649

2-4739
2-4737
2-4722

0-4635
0-4627

2-0104
2-0102
2-0095

May 30

9-268
9-264

0-1846
0-1847

2-4650
2-4630

2-4756
2-4736

0-4607

2-0149
2-0129

June 6

5-261
5-257

0-1866 2-4653 2-4721
0-1865 2-4665 2-4733

0-4544

2-0177
2-0189

Mean Sc =

2-0135, h = 0-0569, k =

0-00520.

428 MR. J. H. BEINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

SPIRAL Vacuum Calorimeter.

Length of path between end of heater and middle of thermometer,

approximately 30 cm.

(!•)

(2.)

(3.)

(4.)

(5.)

(6.)

(7.)

1912.
November 28

J

7-803 0-C457
7-799 0-6457

2-0779
2-0789

2-0861
2-0871

0-0599

2-0262
2-0272

November 30

7-7G8
7-7G6

0-6372
0-6366

2-0774
2-0779

2-0867

2-0872

0-0607

2-0260
2-0265

December 2

10-341
10-341

0-6427
0-6423

2-0791
2 • 0792

2-0892
2 • 0893

0-0601

2-0291
2-0292

December 9

7-203
7-211

0-6249
0-6237

2-0836

2-0837

2-0897
2 • 0898

0-0619

2-0278
2-0279

November 21

7-944
7-940

0-3502
0-3501

2-1313
2-1324

2-1380
2-1391

0-1140

2-0240
2-0251

November 25

10-170
10-170

0-3472
0-3467

2-1297
2-1314

2-1405

2-1422

0-1151
0-1156

2-0254
2-0266

December 9

7-106

7-112

0-3535
0-3529

2-1347
2-1351

2-1415
2-1419

0-1130

2-0285
2-0289

November 21

7-805
7-805

0-1665
0-1665

2-2791
2-2798

2-2858
2-2865

0-2601

2-0257
2-0264

November 25

10-119
10-117

0-1655
0-1655

2-2752
2-2745

2-2863
2-2856

0-2619

2-0244
2-0237

December 9

7-020
7-018

0-1677
0-1673

2-2738
2-2789

2-2806
2-2857

0-2577
0-2586

2-0229
2-0271

Mean Sc = 2-0265,

h = 0-0370, k = 0-00105.

ATMOSPHERIC PKESSUKE BETWEEN 104° C. AND 115" C.
STRAIGHT Vacuum Calorimeter, No. 1. D = 10 '8 cm.

429

(1.)

(2.)

(3.)

(4.)

(5.)

(6.)

(7-)

1913.
March 18

8-653
8-642

0-7816
0-7820

2-0343
2-0341

2-0462
2-0460

0-0223

2-0239
2-0237

March 31

8-482
8-480

0-8171
0-8163

2-0346
2-0356

2-0441
2-0451

0-0212

2-0229
2-0239

April 3

7-838
7-834

0-8111
0-8111

2-0368
2-0361

2-0439
2-0432

0-0214

2-0225
2-0218

April 3

7-921
7-917

0-8160
0-8161

2-0369
2-0362

2-0439
2-0432

0-0212

2-0227
2-0220

March 18

8-511

0-3855

2-0677

2-0794

0-0025

2-0269

March 31

8-282
8-286

0-3919
0-3916

2-0640
2-0642

2-0735
2-0737

0-0515

2-0220
2-0222

April 3

7-794
7-786

0-3915
0-3916

2-0661
2-0664

2-0736
2-0739

0-0515

2-0221
2-0224

March 18

8-585
8-578

0-1819
0-1818

2-1547
2-1561

2-1669
2-1683

0-1449

2-0220
2-0234

March 31

8-328
8-331

0-1839
0-1837

2-1604
2-1598

2-1702
2-1696

0-1425

0-1428

2-0277
2 • 0268

April 3

7-707
7-700

0-1853
0-1848

2-1549
2-1608

2-1625
2-1684

0-1409
0-1415

2-0216
2-0269

Mean Sc = 2-0235, h = 0-0148, I; =

0-00210.

Same Calorimeter. D = 15 '3 cm.

April 7

8-159 0-7602 2-0541
8-157 0-7598 2-0556

2-0626
2-0641

o-o tn

2-0213
2-0230

April 8

7-588 0-7796 2-0572
7-586 0-7794 2-0565

2-0640
2-0633

0-0400

2-0240
2-0233

April 9

7 -438 0-8039 2-0523
7-429 0-8037 2-0531

2 • 0585
2-0593

0 • 0387

2-0198
2-0206

April 14

7-458 0-8063 2-0527
7-449 0-8062 2-0541

2 • 0595
2-0609

0-0386

2-0209
2-0223

April 7

8-084 0-3877 2-0979
8-084 0-3875 2-0990

2-1069
2-1080

0-0872

2-0197
2-0208

April 9
April 14

7-452 0-3903 2 '1044
7-441 0-3904 2-1054

7-455 0-3920 2-1001
7-449 0-3921 2-1016

2-1110
2-1120

2-1070
2-1085

0-0866
0-0861

2-0244
2-0254

2-0209
2-0224

April 8

7-640 0-1814 2-2357
7-638 0-1815 2-2343

2-2431
2-2417

0-2196

2-0235
2-0221

April 9

7-437 0-1843 2-2290
7-439 0-1842 2-2285

2 • 2358
2-2353

0-2151

2-0207
2-0202

April 14

7-467 6-1812 2-2336
7-467 0-1810 2-2354

2-2409
2-2427

0-2199
0-2201

2-0210
2-0226

Mean Sc = 2-0220, h = 0-0285, k = 0-00206.

430

ME. J. H. BRINKWOKTH ON THE SPECIFIC HEAT OF STEAM AT

STRAIGHT Vacuum Calorimeter, No. 2. D = 13 cm. (approximately).

(1.)

(2.)

(3.)

(4.)

(5.) (6.)

(7.)

1913.
June 30

7-076
7-078

0-7790
0-7785

2-0560
2-0556

2-0612 0-0383
2-0608

2-0229
2-0225

July 3

7 • 203
7-201

0-7644
0-7642

2-0568
2-0560

2-0636 0-0391
2-0628

2-0245
2-0237

June 28

7-032
7-040

0-3885
0-3882

2-1032
2-1006

2-1090 0-0835
2-1064

2-0255
2-0229

June 30

7-028
7-032

0-3888
0-3891

2-1043
2-1007

2-1097 0-0833
2-1061

2-0264
2-0228

July 3

7-043
7-043

0-3807
0-3872

2-1008
2-0968

2-1074 0-0839
2-1034

2-0235
2-0195

Juno 30

7-077
7-077

0-1800
0-1803

2 • 2337
2-2324

2-2392 0-2141
2-2379 0-2137

2-0251
2-0242

July 3

7-105
7-105

0-1806
0-1804

2-2275
2-2296

2-2345 0-2131
2-2366 0-2135

2-0214
2-0231

Mean Sc =

2-0235,

h = 0-0272, k = 0-00204.

Silica Calorimeter.

Black Vacuum. D = 9'0 cm.

1914.
February 25

8-810
8-808

0-7758
0-7759

2-0261
2-0255

2-0354 0-0044
2-0348

2-0310
2-0304

February 26

8-741

8-741

0-8048
0-8045

2-0233
2-0233

2-0318 0-0042
2-0318

2-0276
2-0276

March 2

8-286
8-277

0-8119
0-8111

2-0231

2-0247

2-0308 0-0042
2-0324

2-0266
2-0282

February 25

8-851
8-849

0-3760
0-3760

2-0305
2-0299

2-0401 0-0101
2-0395

2-0300
2-0294

February 26

8-708
8-712

0-3768
0-3766

2-0312
2-0303

2-0401 0-0101
2-0392

2-0300
2-0291

February 25

8-831
8-831

0-1770
0-1772

2-0465
2-0435

2-0562 0-0265
2-0532

2-0297
2-0267

February 26

8-676
8-672

0-1800
0-1803

2-0463
2-0431

2-0554 0-0259
2-0522 |

2-0295
2-0263

Mean . Sc =

2-0287,

h « 0-0030, jt = 0-00030.

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C.

431

SILICA .Calorimeter. Pressure in jacket about 1 mm. D = 9'0 cm.

(1-)

(2.)

(3.)

(4.)

(5.)

(6.)

(7.)

1914.
March 2

8 • 504
8-502
8-506

0-8002
0-8002
0-8006

2-0187
2-0179
2-0153

2-0265
2-0257
2-0231

0-0136

2-0129
2-0121
2-0095

8-518
8-512

0-3782
0-3779

2-0447
2-0464

2-0529 0-0430
2-0546

2-0099
2-0116

8-539
8-533

0-1793
0-1793

2-1576
2-1581

2-1659 0-1544
2-1664

2-0115
2-0120

Mean S, = 2-0114,

h = 0-00600, 7; = 0-00390.

Silica Calorimeter. Black Vacuum. D = 12'5 cm.

March 5 8-482
8-467

0-7999
0-7995

2-0243
2-0244

2-0348 0-0061 2-0287
2-0349 2-0288

March 7

8-562
8-558

0-8201
0-8198

2-0209
2-0215

2-0312
2-0318

0-0060 2-0252
2 • 0258

March 18

7-184
7-178

0-7990
0-7986

2-0243
2-0251

2 • 0346
2-0354

0-0001 2-0285
2-0293

March 5

8-542
8-540

0-3778
0-3773

2-0305
2-0319

2-0405
2-0419

0-0140 2-0265
2-0279

March 7

8-601
8-595

0 • 3806
0-3807

2-0336
2-0331

2-0441
2-0436

0-0139

2-0302
2-0297

March 9

7-180
7-171

0-3808
0-3811

2-0312
2-0310

2-0416
2-0414

0-0139

2-0277
2-0275

March 18 8 -481

8-475

0-3775
0-3768

2-0314
2-0301

2-0435
2-0422

0-0140

2-0295
2-0282

March 5 8-598
8-000

0-1764
0-1761

2-0514
2-0536

2-0621
2-0643

0-0351

2-0270
2-0292

March 7

8-652
8-641

0-1756
0-1761

2-0573
2-0533

2 • 0680
2-0640

0-0353

2-0327

2-0287

March 18

8-367
8-367

0-1749
0-1749

2-0526
2-0515

2 • 0647
2-0636

0-0355

2-0292
2-0281

Mean Sc = 2-0284,

h = 0-0045, k = 0-00030.

VOL. ccxv. — A.

3 M

432 MR. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM

SILICA Calorimeter. Black Vacuum. D = 16'0 cm.

AT

(1-)

(2.)

(3.)

(4.)

(5.)

(6.)

(7.)

1914.
March 21

6-804
G • 795

0-7862
0-7867

2-0287
2-0295

2-0390
2 • 0398

0-0102

2-0288
2-0296

March 28

8-217 .
8-20G

0-8051
0-8051

J

2-0325

2 • 0325

2-0401
2-0401

0-0100

2-0301
2-0301

March 21

6-713 0-377.'!
6-711 0-3772

2-0390
2-0391

2 • 0500
2-0501

0-0223

2-0277
2-0278

March 26*

7-258
7-254

0-3663
0 • 3GG3

2-0345
2-0355

2-0458
2-0468

0-0229

2-0229
2-0239

March 28

0-083
9-077

0-3731
0-3732

2-0433
2-0428

2-0519
2-0504

0-0226

2-0283
2-0278

March 23

7-129 0-1792
7-136 0-1788

2-0801
2-0732

2 • 0900
2-0831

0-0519

2-0381
2-0312

March 28

9 • 233
9 • 235

0-1748
0-1717

2-0738
2-0738

2-0831
2-0831

0-0531

2-0300
2-0300

Mean S, = 2-0290, // = 0-0076, /• = 0" 00030.

Silica Calorimeter. Pressure in jacket about 1 mm. D = IG'O cm.

March 30

8 • 327
8-316

0-7853 2-0627
0-7855 2-0615

2 • 0700
2 • 0688

0-0612

2 • 0088
2-0076

April 2

8 • 300
8-341

0-7873 2-0626 2-0710
0-7866 2-0649 2-0733

0-0611

2-0099
2-0122

April 3

8 -509
8-547

0-8081 2-0633
0 • 8080 2 . 0628

2-0714
2-0709

0-0593

2-0121
2-0116

April G

8 • 6G5
8-663

0-7908
0-7908

2-0572 2-0677 0-0607
2-0564 2-0669

2-0070
2-0062

March 30
April 0

8-132
8-128

9 • 005
9-007

0-3782
0-3778

0-3547
0 • 3550

2-1419
2-1428

2-1580
2-1565

2-1493 0-1429
2-1502

2-1692 0-1542
2-1677

2-0064
2-0073

2-0150
2-0135

March 30

8-025
8-021

0-1768 2-3840
0-1769 2-3821

2-3914
2-3895

0-3783

2-0131
2-0112

April 3

8-573
8-542

0-1816
0-1819

2-3666
2-3637

2-3751
2-3722

0-3655

2-0096
2-0067

April G

8-997
8-989
8-978

0-1751
0-1756
0-1753

2-3854
2-3795
2-3853

2-3967
2-3908
2-3966'

0-3837

2-0130
2-0071
2-0129

Mean Sc = 2-0101, h= 0-0425, £ = 0-00435.

* Vacuum deteriorated.

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C. 433

SILICA Calorimeter. Pressure in jacket about 1 mm. D = 9'0 cm.

(1.)

(2.)

(3.)

(4.)

(5.)

(6.)

(7.)

19U.
April 24

9-086
9-082

0-8257
0-8261

2-0206
2-0206

2-0280
2-0280

0-0169

2-0111
2-0111

9-075
9-069

0-3566
0-3566

2-0624
2-0634

2-0698
2-0708

0-0589 .

2-0109
2-0119

9-196
9-192

0-1755
0-1751

2-1967
2-2020

2-2045
2-2098

0-1943

2-0102
2-0155

Mean Sc

= 2-0118, h = 0-0085, A- =

0-0045.

Silica Calorimeter. Pressure m jacket 760 mm.

D = 9'0 cm.

April 27

9-170
9-170

0-8467 2-0189
0-8465 2-0189

2-0256
2-0256

0-0184 2-0072
2-0072

April 28

8-800
8-809
8-813

0-8144 2-0240 2-0311
0-8132 2-0245 2-0316
0-8129 2-0238 2-0309

0-0195 2-0116
2-0121
2-0114

April 30

8-706
8-714
8-727

0-8094 2-0223
0-8098 2-0192
0-8081 2-0200

2 • 0308
2-0277
2-0285

0-0196

2-0112
2-0081
2-0089

April 27

9-245
9-247

0-3569
0-3570

2-0672
2-0662

2-0739
2-0729

0-0634

2-0105
2 • 0095

April 28 8-785
8-785

0-3792
0-3788

2-0628 2-0698
2-0642 2-0712

0-0580 2-0118
2-0132

April 30 8-583
8-591

0-3768 2-0568 2 '0655
0-3765 2-0565 3-0652

0-0582 2-0073
2 • 0070

April 27 9 • 054
9 • 052

0-1780 2-2025 2-2094
0-1779 2-2043 2-2112

0-1961 2-0133
2-0151

April 28 8-814
8-816

0-1781 2-1935 2-2011
0-1780 2-1935 2-2011

0-1961 2-0050
2 • 0050

April 30

8-742
8-742

0-1780
0-1780

2-1966 2-2056
2-1958 2-2048

0-1961

2-0095
2-0087

Mean S(. =

= 2-0097, U = 0-0105, k =

0-00135.

Silica Calorimeter. Black Vacuum.. D = 9

'0 cm.

May 1

8-603
8-597
8-601

0-7903
0-7902
0-7902

2-0298
2-0309
2-0295

2-0365
2-0376
2-0362

0-0062

2-0303
2-0314
2 • 0300

8-590
8-588

0-3778
0-3780

2-0368
2-0351

2-0440
2-0423

0-0140 2-0300
2-0283

8-884
8-884

0-1785
0-1785

2-0562
2-0562

2-0639
2-0639

0-0346

2-0293
2-0293

Mean Sc

= 2-0298, h = 0-0045, k =

0-00030.

3 M 2

434

MR. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

Summary of the Results.

The following table has been drawn up to show how the observed values of S,., h,
and k vary with the experimental conditions. In column (2) the calorimetric
arrangement is briefly described. The dates in column (l) show the period of time
over which the experiments with each calorimetric arrangement extended. Columns
(3), (6), and (7) give the values of Sc, h, and k. The total number of observations
giving the mean value Sc is stated in column (4). In column (5) are the distances D
separating the middle of the thermometer bulb from the end of the heating coil.
The last two columns give the value of the heat-loss per deg. per cm. (8), and the
ratio of this quantity to k (9).

(3.)

S,.

1912.

April to

June

Non-vacuum and
silvered jacket

non-

f 2-0145
2-0147

2-0143

1912.

Nov. to Spiral flow-tube with sil-
Dec. vered jacket 2-0265

1913.
March to Straight flow-tube with sil

April

June to
July

vered vacuum jacket, •
No. 1 [_ 2-0220
Ditto, No. 2 2-0235

1914.

Feb. to Silica flow-tube, silvered
May non-vacuum .... 2 '0097

Pressure in jacket about/
1 mm.

L 2-0101

Black

vacuum

I" 2-0287
2-0284

2-0290
(J 2-0298

(4.)

(5.)
D.

(6.)
h.

(7.)
k.

(8.)

(9.)

15

G-2

0-0033

0-00520

13

8-7

0-0172

0-00520

0-0074

1-4

14

11-5

0-0400 0-00520

(0-0390) '(0-00538)

19

13-8

0-0569

0-00520

20

i
— 0-0370

0-00105

19

10-8

0-0148

0-00210

0-0030

1-4*

20

15-3

0-0285

0-00206

14

13-0

0-0272

0-00204

20

9-0

0-0105

0-00435

13

9-0

0-0072

0-00420

0-0051

1-2

19

16-0

0-0425

0-00435

14

9-0

0-0030

0-00030

20

12-5

0-0045

0-00030

0-OOOGs

2-2

H

16-0

0-0076

0 • 00030

7

9-0

0-0045

0-00030

Extrapolated value for the specific heat Sc at 104'5° C. and 760 mm., corre-
sponding to zero heat-loss = 2'0300 joules per gr. deg. C.

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C.

435

With any definite calorimetric arrangement the values of Sc and k are constant
within the limits of experimental error, while h increases linearly with D, fig. 10.
In the set of experiments made with the non- vacuum jacket the values of h and k
calculated in the manner described on p. 419 are given in the table in brackets. The
most probable value of k, viz., 0'00520, has been used in the determination of H in
the abridged tables, h being slightly altered in the case of the experiments with
D = 11 '5 cm. In the experiments made with the silica calorimeter, with as
perfect a vacuum as possible in the jacket, the values of k calculated from the

•09

0 5 10 D.cms. 15

Fig. 10.

X Prof. CALLENDAR'S experimental values. Unsilvered calorimeters.
® „ ,, ,, Silvered „

G> BRIXKWORTH. Mean values.

individual measurements varied between — O'OOOl and 0'0007, and in the calculation
of H/Q for these observations the mean value 0 '00030 has been given to k.

From the observations of May 1st, 1914, it can be seen that the difference in the
values of the specific heats as calculated (l) from the experimental values of h and k,
and (2) from the adjusted value of h corresponding to k = O'OOOSO, is only 0'05 per
cent. In the last column of the above table are given the values of the ratio h per
cm/k. Within the limits of error this ratio may be considered constant, and

436 ME. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM AT

probably of the order 1'5. The two values of this ratio obtained with the silica
jacket are not reliable, since in the case of the value 2'2 when k = O'OOOSO, a change
in k of 100 per cent, will only cause the value of h/cm to be altered slightly, while
when h/cm/k = 1'2 the experiments were carried out with a partial vacuum in the
jacket, and there is a likelihood that the vacuum was not exactly the same through-
out the period of these experiments, though practically constant for all the
experiments made on one day.

Discussion of the Observed Variation of the Heat-loss with the Flow.

After completing the experiments with the non- vacuum jacketed calorimeter, the
actual temperature gradients in the flow-tube, both when "hot" and when "cold,"
were observed in the case of each flow. The supply of electrical energy was adjusted
so that witli a distance of about 14 cm. between the end of the heating coil and the
thermometer the latter indicated an oiitflow temperature about 9 '3° 0. above the cold
temperature. The curves showing the temperatures at various points along the flow-
tube are straight lines, provided that these temperatures are measured with the
middle of the thermometer, not nearer than 6 cm. nor farther away than 16 cm.
from the' end of the heating coil. The actual gradients fur the maximum, medium
and minimum flows were of the order 0'044°, 0'090°, and 0'170° per cm., but it is
necessary to point out that the third figure, though given, is not to be relied upon
since the total fall of temperature is so small, only amounting to 0'35° C. over a
distance of 8 cm. in the case of the maximum flow. The product of the temperature
gradient over the straight line part of the curve and the value of the corresponding
flow is in eacli case about 0'033. This number when multiplied by the specific heat
and the product divided by the mean temperature of the flow-tube, gives 0'0072, a
number which represents the heat-loss per cm. per degree and which is in good agree-
ment with that deduced from the results of the calorimetric experiments, 0'0074.
The gradient for a medium flow of (T365 gr./sec. is shown in fig. 10 (dotted curve.)
Another point worthy of notice is that the gradient ceases to be linear in that part of the
flow-tube near the heating coil, and the shortest distance which must exist between the
heating coil arid the thermometer in order to obtain complete mixing, is indicated, both
in the gradient observations and in the calorimetric experiments, to be about 6 cm.
The agreement between the values of h/cm , calculated from the calorimetric experiments
and from the gradient observations in the lower part of the flow-tube, indicates that the
&/Q term in the heat-loss represents the loss from the upper portion of the flow-tube.
A possible explanation of this is as follows. The steam in the flow-tube loses heat to
the upward current surrounding it, and heat is lost from this upward current to the
surrounding brass jacket. However, it is only the heat given to the jacket which is
actually lost. The heat lost from the steam in the flow-tube to the up-current is
proportional to the rise of temperature de. The temperature of the up-current is
consequently raised by an amount proportional to de/Q, so the actual loss to the

ATMOSPHERIC PRESSURE BETWEEN 104° C. AND 115° C.

437

jacket IB ¥ d6/Q, where kl is some constant. If, on the other hand, there had been no
counter current, the heat-loss would have been due to radiation alone and would have
been represented by a term hdQ. With the counter current there will still be
radiation, and the term representing the total heat-loss will be of the form (h + kfQ) d6.
If we assume that the heat-loss h^ d9, by direct radiation from the upper part of the
flow-tube to the jacket, is of the same order as that fraction of the heat-loss h2 d6
gained by the up-current from the lower part of the flow-tube, which is returned
into the flow-tube, and if this equivalence exists when the upper part of the flow-tube
includes a length extending to 6 cm. below the heating coil, the fi d6 term will
represent the total direct radiation loss from the lower part of the flow-tube to the
jacket, and the k/Q,d9 term, the heat lost from the upper part of the flow-tube to the
jacket. If this assumption is correct we should expect to find k constant in all the
experiments made with a definite calorimetric arrangement ; moreover, k should be
proportional to the value of the heat-loss per centimetre. Since the thermal exchanges

••050

2-000

•001

Fig. 11.

D Silica calorimeter.
O Other calorimeters.

which occur between the two currents of steam, and between these and the outer
brass jacket, are undoubtedly very complex, the above explanation must only be con-
sidered as approximately describing what is taking place. It has been put forward
mainly in order to justify, to some extent, the adoption of the type of equation used
in the calculations. The values of Sc obtained show a distinct variation with the
magnitude of k, i.e., with the value of the heat-loss per degree per centimetre (fig. 11).
The extrapolated value of the specific heat, corresponding to zero heat-loss, is only
0'05 per cent, higher than the mean deduced from the results obtained under the
best experimental conditions, hence this value has been taken as the final result of

these experiments,

S = 2'0300 joules per gr. deg. at 104'5° C.

S = 0'4856 calories per gr. deg. at 104 '5° C.,
this latter result being in terms of the calorie at 20° C.

438 MR. J. H. BRINKWORTH ON THE SPECIFIC HEAT OF STEAM, ETC.

Comparison of the Result with Theory.

The values of the specific heat at constant pressure can be deduced from equation
(7), and this relation has been used in order to compare my value of the specific heat
with values obtained by other observers. Assuming the linear variation with
temperature, as experimentally determined, my own observations give the value 0'4878
as the specific heat at 100° C. Substituting this value in the above equation we
find S0 = G'4634.

The values of the specific heat at atmospheric pressure and at various temperatures
can then be calculated and the mean value over any desired range deduced. Over
the range 110°C. to 230° C. the mean value is thus found to be 0'47G. From
equation (5) the following'expression showing the variation in the total heat of steam

can be derived

F-F0 = So(0-00)-(n

Assuming 640'3 calories as the value of the total heat at 100° C. we find the latent
heat at 0° C. to be 5967 cal./gr. which is in good agreement with that found by
DIETERICI. The above values of S0 and L0 may be used to calculate the saturation
pressures of steam. The agreement found between the calculated values and those
determined experimentally is exceedingly good, especially at low temperatures when
the value of S0 is approximately constant. For example, the calculated pressures at
0° C. and at 60° C. are 4'585 mm. and 149'23 mm. respectively, the experimental
values at these temperatures are at 0° G. 4'600 mm. (!{EGNAULT) and at 60° C.
148'80 mm. (REGNAULT).

Comparison of this Result with the Work of other Observers.

On pp. 397-399, Prof. CALLENDAR has discussed the work of other observers. The
value of the specific heat at 108° C. deduced from my experimental measurements is
2'0230 or 0'484 with the calorie as the unit.

In fig. 10 I have plotted the values of /^obtained by Prof. CALLENDAR against the
corresponding values of D. The points line about a straight line which practically
coincides with that obtained from my measurements with the non- vacuum, non-silvered
calorimeter. Apparently there is no systematic difference in the values of hfcm
obtained with the silvered and unsilvered calorimeters respectively. When con-
sidered, together with the results of my own experiments with various pressures in
the vacuum jacket, this concordance indicates that in the many calorimeters, evacuated
by the makers, which have been used, the vacua produced were far from perfect. It
is again of interest to note that in Prof. CALLENDAR' s experiments similar absurdly
low values of the specific heat were obtained when attempts were made to mix the
steam by means of wire gauze.

[ 430 ]

XIV. Some Applications of C (informal Transformation to Problems in

Hydrodynamics.

By J. G. LEATHEM, M.A., D.Sc.
Communicated by Sir JOSEPH L ARMOR, M.P., F.R.S.

Received May 13, 1915.

CONTENTS.

§§ Page

1-.'!. Introduction 439

4-12. Conformal curve-factors 441

13-20. Hydrodynamical illustrations 447

21-22. Inflexional curve-factors 456

23-26. Curve-factors of semi-infinite linear range 458

27-29. Double curve-factors 461

30-33. Synthesis of curve-factors from elementary types 463

34-41. Liquid motions with free stream-lines 467

42-43. General remarks on curve-factors 475

44-48. Curve-factors regarded as the limits of products of Schwarzian factors 477

49-51. Transformations involving both variables explicitly 481

52-55. Supplementary note on curve-factors 485

INTRODUCTION.

1. THE problem of the confonnal representation of the part of the plane of a variable z,
which is bounded by a rectilineal polygon, upon the half-plane of a variable w bounded
by the real axis, is solved (save for an integration) by the well-known transformation

of SCHWARZ

dz = CU(w~ar)-"'l"dw,

•

where C, «1; a2, &c., are real constants, and TT — OL^ TT— «2, &c., are the internal angles
of the rectilineal polygon.

A more difficult problem is that of the confonnal representation upon the half-plane
of w of a region in the z plane whose boundary is partly curved ; it is with this
problem that the present paper is concerned, always however with a view to inter-
pretation of results in terms of the two-dimensional flow of liquid in regions having
particular types of boundary.

VOL. COX V. A 536. 3 N [Published September 22, 1916.

440 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

At the outset something of a suggestion might seem to be found in consideration
of the fact that, if the intrinsic equation of a curve in the z plane is x = F (s), the
vectorial element of arc dz is the same as ds exp (?x) or ds exp {t'F(s)}, where i as
usual represents y^(— l). From this it appears that the transformation

where F and f are functions having real values for all real values of their arguments,
makes the real axis in the w plane correspond to a curve in the z plane whose intrinsic
equation is x = F(.s'), the relation between w and s being s=f(w), where f is
an arbitrary function. But transformations of this type, though they settle the
correspondence of prescribed boundaries, oft'er no guarantee against zeros, infinities
or singularities at points where w is not real, and so do not necessarily give conformal
representation. The geometrical idea of the transformations is nevertheless useful.

2. Some problems of conformal transformation with partially curvilinear boundaries
were worked out by Mr. W. M. PAGE in a paper* published by the London
Mathematical Society a few years ago. Mr. PAGE definitely rejects, on account of
an apparent indeterminateness, the method of treating a curve as the limit of a
rectilineal polygon and seeking the limit of the product of the corresponding
Schwarzian factors. Instead he has recourse to the empirical but useful device of
writing down the Schwarz transformation as it would be for a rectilineal polygon
having the same angles as the prescribed curvilinear one, and then introducing into
the expression for dz/div a further factor which has the effect of making the relevant
side of the z polygon curvilinear while leaving the other sides straight. There is no
question of prescribing the form of the curve in the z plane, one simply takes such
conformal transformations as one can succeed in formulating in the above manner
and tries to find out what sorts of curve they yield.

Mr. PAGE, acknowledging a suggestion by Mr. H. W. RICHMOND, takes factors
which occur in well-known problems, and by associating them with new sets of
Schwarzian factors obtains a new series of results. The familiar case of a semi-
circular boss on a straight line, namely

(1)

supplies the factor w+ (^-c2)1'-', and the case of a semi-elliptic boss on a straight line,
namely

}, ..... (2)

supplies the factor w sinh a+ (w2-c2)l/> cosh a. These two factors and their square
roots ^are the factors which Mr. PAGE associates with other factors of the Schwarzian
type in order to obtain new results.

" Some Two Dimensional Problems in Electrostatics and Hydrodynamics," ' Proc. Lond. Math. Soc.,
Ser. 2, vol. XL, 1912-13, p. 313.

CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 441

3. One object of the present paper is to try to extend the range of transformations
available for the conformal representation upon a half-plane of a region bounded by
a polygon some of whose sides are straight and some curvilinear. Aspects of the
subject will be discussed which have the theoretical advantage of being to some
extent free from the empirical character of Mr. PAGE'S method, but the application
of these to particular cases presents great practical difficulty. It is therefore proposed
to begin by studying those functions of w which, when introduced as factors into
formulce of transformation which are otherwise of the Schwarzian type, lead to
conformal representations of the required general character. Every increase in the
range of such functions made available, and every advance in knowledge of the theory
of such functions, diminishes the empirical element in their employment for the
solution of particular problems.

The main object of the paper is the extension of the range of solvable problems
in two-dimensional liquid motion, and cases of such motion will be used as illustrations
of the pure mathematical theory.

CONFOKMAL CURVE-FAOTORS.

4. Definition and Characteristics. — -Consider a transformation

(w-ar}-^div, ......... (3)

wherein A, a1; a2, ... are real constants, «,, a,, ... real constants in ascending order of
magnitude, and ^ a function of u\ If ^ were absent this would be a Schwarzian
transformation giving a conformal representation upon the half-plane of w of the
region in the z plane bounded by a rectilineal polygon of external angles a1( a,, a3 ... .
Suppose ^ to be such that the transformation as it stands gives a conformal
representation upon the half-plane of w of the region in the z plane bounded by
a polygon having the external angles a]; «2, a3, ... as before, all its sides but one
straight, and that one side (say the side corresponding to ar+l > w> ar) curvilinear.
A function ^ of w having this property may be called a " conformal curve-factor," or,
for brevity, a "curve-factor." In view of the possibility of our having to consider
functions which, when thus introduced into a Schwarzian transformation without
spoiling its conformal character, replace the rectilineal polygon by a polygon having
the same angles and not one only but several curved sides, it may be well to
distinguish between "simple," "double," "triple," &c. conformal curve-factors,
according to the number of curved sides which they introduce into the polygon.

If a function ^ of w is a simple conformal curve-factor it must satisfy the following
requirements : —

(i.) The vector angle of the complex <$ must have a constant value for all real
values of w greater than ar+1, and a constant value for all real values of w less than

3 N 2

442

DE. J. G. LEATHEM ON SOME APPLICATIONS OF

ar. For real values of w intermediate between ar and ar+l the vector angle of *&
must change continuously with w.

(ii.) ^ must not be zero or infinite for any definite value of w which is real, or for
any definite complex value of w corresponding to a point on the relevant (positive)
side of the axis of w real. Such zeros or infinities would destroy the conformal
character of the transformation, and would, if occurring on the boundary, interfere
with the prescribed arrangement of corners. The word " definite " is here used so
that the restriction may be understood not to apply to w infinite. Certain
singularities at definite points might be capable of interesting interpretations in the
application to hydrodynamics, but it seems best at present to exclude them.

(iii.) *$ must not have any definite branch points on the relevant side of the axis
of w real. It may however have branch points on the axis of w real, since this axis
serves as a barrier preventing such circulation round a branch point in it as would
make the function many- valued.

(iv.) The form of ^ for w infinite must conform to conditions depending on the
nature of the particular problem to which the transformation is to be applied.

5. Linear and Angular Ranges. — Tlie range on the axis of w real corresponding
to values of w for which the vector angle of ^ is variable may be called the " linear
range" of the curve-factor. The difference between the vector angles at the
extremities of the linear range may be culled the "angular range" of the curve-
factor, this being reckoned as positive when the greater angle corresponds to the
smaller value of w.

In the case of a simple curve-factor the linear range consists of the real values of
•w between those typified by «,. and «r+1) and the angular range is the angle between
the tangents at the extremities of the corresponding curved side of the polygon, the
standard case being that of convexity towards the relevant region.

A curve-factor may of course have its angular range zero ; in that case there will
be an inflexion on the corresponding curved side of the polygon.

It is readily seen that if ^ be a curve-factor, and if n be a constant, then #* also
is a curve-factor, having the same linear range as ^, but its angular range equal to
n times that of ^. It is important to notice that this statement is valid even if n is
negative, for as ^ has no zeros or infinities in the relevant region (except possibly
for w infinite), the raising of ^ to a negative power does not introduce any fresh
infinities or zeros. But if 9 has a positive angular range and if n is negative the
angular range of V* is negative. Thus, if «* gives a curve which is everywhere
convex to the relevant region, positive powers of « give curves having the same
characteristic, but negative powers of ff give curves which are concave to the
relevant region.

In connexion with requirement (iv.) above, the form of # for w infinite is
sometimes important. If, for w^oo, V + ku,« where k is a constant, it will be
convenient to speak of m as the "order at infinity" of the curve-factor. Clearly

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 443

there may be curve-factors which do not possess an order at infinity in this
sense.

It is important to bear in mind that the form of the curved side of the polygon
which corresponds to a particular curve-factor depends not merely on the analytical
form of the curve-factor, but also on the Schwarzian or other factors which appear in
the formula of transformation. Thus the same curve-factor leads to different curves
in different transformations. But these curves will have some features in common,
provided the linear range with which the particular curve-factor is associated is not
also in whole or in part included in the range of some other curve-factor ; for example
if (0> in any such simple connexion represents a cm-ve everywhere convex to the
relevant region, then in other such connexions both ^ and positive powers of *$ will
always represent curves convex to the relevant region. This appears from considera-
tion of the vector angle of $*, which, when added to the vector angles of the other
factors, the latter being constants for the linear range of" *$, gives the angle of
direction of the tangent to the curve.

6. Convention as to Fractional Power*. — In the course of the work expressions
involving square roots and other fractional powers of quadratic expressions in tr will
occur frequently. The convention must therefore be made at the outset that when
X is fractional an expression such as (vf — «)x shall be interpreted as having a positive
real value for real values of w greater than a, and for other values of w corresponding
to points on the relevant (positive) side of the axis of w real such value as corresponds
to continuous passage from a point where w is real and greater than a without
departure from the relevant half-plane, the branch point if = a being avoided when
necessary by a detour in the relevant region. With this convention an expression
such as (w— «)A will be free from ambiguity, and in particular the passage round the
point a from a real value greater than a to a real value less than a will lead to the
form (a—wY exp(iX-Tr) for real values of w less than a.

7. The Curve-factor of Semi-circular Type. — It is natural to expect that the
study of the two curve-factors employed by Mr. PAGE may lead to suggestions for
the construction of other curve-factors. That which occurs in the transformation (l)
above may be called the curve-factor of semi-circular type ; it is

^ = w + ^'-c2)1'', (4)

and has linear range from +.c to -c, and angular range •*.

It is to be noted that the vector angle of ^,, when c > w > -c, is tan"1 [(ca-wa)v'/w]»
and it is only because the denominator of the fraction in the square bracket vanishes
for a point in the linear range that the angular range is TT instead of being zero.

The reason why (@l satisfies the requirement of having no zero is easy to see.
Let Sft be the surd conjugate to tflt that is w-(iv*-c*}\ Every zero of ^ must
(since 9l has no infinities) be a zero of the product V&, but tf^ = c2 and has no
zeros, therefore <^>1 has no zeros.

444 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

If the attempt be made to construct other curve-factors of the type of a rational
function plus a square root, it appears that for a simple curve-factor the root sign
must cover only a quadratic in w, with real factors. Other factors, if imaginary,
would introduce branch-points ; if real they would introduce fresh corners of angle TT,
and so effectively yield a multiple curve-factor instead of a simple one. Thus the
type must be

where f and g are rational functions. If this is to be free from zeros in virtue of
neither it nor its conjugate surd having zeros, it is necessary that

{f(iv}}2— {g (w)}2 (iv2—c2) = const.

If g(w] is of the first order the rationality of f(w) requires g (w) to be simply w,
and then the curve-factor is

which has the angular range 2?r. This is not really a new result, for ^ = ffli-
Similarly, if g (w) be assumed of the second order, the rationality of f(w) restricts to
the form

which is not a new result, since ^ =

It seems therefore safe to conclude that the semi-circular curve-factor is a very
special type which does not admit of generalisation.

8. The Curve-factor of Semi-elliptic Type.— 'The factor which occurs in trans-
formation (2) above may be called the curve-factor of semi-elliptic type. It is

^4 = ?rsinh «+ (^3 — c2)''2 cosh a (7)

This has, for c > w > -c, the vector angle tan'1 [(c2-«;2)'« cosh a/w sinh a], and its
angular range is -w because the rational part of ^4 vanishes and changes sign at a
point in the linear range.

^4 is free from zeros, but the reason of this is different from that which holds in
the case of ^,. Denoting the surd conjugate to <^4 by Sft, it is seen that

^A = c2 cosh2 a— w3, (g)

and this has two real zeros. It has to be shown that both these zeros belong to 9te
so that ^4 has none.

For w real ^4 has three different forms, namely

(\ \
>v -^ c)> w sinh a+ (w2— c2)1'2 cosh a,

(c > w >-c), w sinh a + i (c2-iv2)^ cosh a,
(-c > w), w sinh a- (tv2-c2)'1' cosh a.

CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 445

The first of these is the sum of two real positive quantities, and so cannot vanish ;
the second is a complex whose real and imaginary parts have no common factor, and
so cannot vanish ; the third, because the real part has changed sign in the linear
range, is the sum of two real negative quantities, and so cannot vanish. Thus $"4
has no real zeros, and accordingly, as no imaginary zeros are possible, has no zeros at
all. The change of sign of the real part of ^4 is an important element in the
argument.

9. Modified Semi-elliptic and Semi-circular Types. — Within the limits of those
properties by which the freedom of $"4 from zeros is secured, namely that the product
of the curve-factor and its conjugate surd has only real roots and that both terms of
the curve-factor have the same sign for real values of w outside the linear range,
there is room for modification of the type ^4 by the introduction of an additional
parameter. Considering

, ....... (9)

it is seen that, provided c > k > — c, both terms of ^5 are positive for real values of iv
greater than c, and both negative for real values of w less than — c ; thus ^"5 cannot
vanish for any real value of w. If the conjugate surd be ^6,

#B0S = (c2 + /;2shih2a) cosh2 a- (ic + k smh" a)3, ..... (10)

which has only real zeros. Hence ^5 has no zeros, and is a curve-factor. Its angular
range is TT. On the other hand, if k > c, ^ is positive for w = + <x> and negative for
iv = c, so it must have a zero and is not a curve-factor. The same applies when
k < — c, since then ^5 is negative for w = — co and positive for w = —c.
Similarly ^ may be modified to the form

Vt = w-k+(w'-G')11' ......... (11)

with the restriction c > k > — c. The only zeros that might be possible are those of

......... (12)

and this has only one zero, which is real. Thus ^ has no zeros, and is a curve-factor
of angular range TT.

It may be noted that

-«B ........... (13)

The utility of the adjustable parameter k will be seen in the working of particular
examples,

446 DR. J. G. LEATHEM ON SOME APPLICATIONS OF

10. The form of ^6 suggests another possible curve-factor,

^7 = («;-&) cosh a +(w'-c')1/!Isinh a (14)

Any zeros which this may have must be zeros of

#7#7 = (w-k cosh2 «)2+sinh2 a (c2-F cosh2 a) (15)

If F cosh2 a > c2, the zeros of ^7^7 are both real, and if c2 > /P, <$1 cannot have
real zeros. Hence ^7 is a curve-factor if c2 > k2 > c2 sech2 a.

If F cosh2 a < c2, which involves a fortiori that P < c2, then ^7^7 has imaginary
zeros, and the question arises whether ^7 can have any imaginary zeros at points on
the relevant side of the axis of w real. This may be tested by means of the theorem
that, if a function f(w) be free from infinities in a given region, the sum of the

orders of all its zeros in that region is equal to (l/27rt) \f'(w)/f(w)dw taken round

the boundary ; a simpler enunciation of the theorem is that the sum of the orders of
all the zeros of ,/'("') 'ln the region is 6/2-Tr, where B is the algebraic sum of all the
changes, abrupt or gradual, which take place in the vector angle of f(w) as w makes
a complete circuit of the boundary in the positive sense.

In the present application ^ is put for /(«;), the region is the half- plane of w on
the positive side of the real axis, and the boundary consists, in the main, of the whole
of the real axis and a semi-circle of radius 11 which tends to indefinite greatness.
The points w = +c, being branch points, have to be avoided by semi-circular detours
of infinitesimal radius on the positive side of the axis. If there were zeros on the
axis they would have to be avoided by similar detours, but in this case there are
none. The points w — ±c not being zeros or infinities, the detours round them do
not result in any alteration in the vector angle of <@-. From - oo to —c this vector
angle is -w ; from — c to +c it continually diminishes (because c > k >— c) from -w to
zero; from -t-c to + co it is zero; along the semi-circle from + oo round to — oo it
increases from zero to TT. Hence 9 = -TT + TT = 0, and so ^ has no zeros in the
relevant half- plane.

Therefore, if k3 < c2 sech2 a, <$„ is a curve-factor.

11. Relation let-ween Angular Range and Order at Infinity. — In the test for the
presence or absence of imaginary zeros employed in the previous article it will be
noticed that the sum 6 is made up of two parts, one corresponding to the linear
range and equal to minus the angular range, the other corresponding to the infinite
semi-circle and equal to T multiplied by the order at infinity of the curve-factor.
From this it is seen that if an expression of the type

f(w)+g(w)(w'-c?)11' (16)

is free from infinities and real zeros, and has its angular range equal to TT times its
order at infinity, it will also be free from imaginary zeros in the relevant region, and
is a conformal curve-factor.

CONFOEMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 447

For example consider

(^s = A(w-kl)(w-k2) + E(w-i)(w3-c^, ..... (17)

where A > 0, B > 0, c > kt > I > k2 > -c.

For w real and greater than c, or for w real and less than — c, ^8 consists of the
sum of two positive quantities. Hence ^8 has no real zeros. The angular range of
^ is 2-7T, the vector angle being zero for iv ±: c, increasing, as w diminishes, to \v for
w = ki, then to ?r for w — I, then to |TT for w — k2, then to 2?r for w = — c, and
remaining at 2x for w < — c. The order at infinity is 2, so that 6 = 0. Hence
^8 has no zeros in the relevant half-plane, and is a conformal curve-factor.

If the angular range wei'e different from TT times the order at infinity 0 would not
be zero, and the function would not be free from zeros. Tims if in ^ the order of
magnitude of the constants were c > I > L\ > /-2 > — c the angular range would be
zero and ^ would not be a curve-factor.

It may therefore be taken as an established theorem for all curve-factors having
a definite order at infinity that the angular range is equal to TT times the order at
infinity.

12. Curve-factors having Branch Points of Unequal Orders at the- Extremities
of the Linear Range. — Consider the function

ff9 = A(u<-£) + B(«;-a)<'(tt--&)1-") ....... (18)

where A > 0, B > 0, a > k > 1>, and 1 > a > 0.

For w real and greater than a this is real ; for w real and between a and b it takes

the form

...... (19)

having a vector angle which is zero for w = a and increases to x for w = b ; for
w real and less than b it takes the form

-A(k-w)-~B(a-wY(b-w)l-a ....... (20)

Thus if* ^9 is a curve-factor it has linear range a to b, and angular range TT. Since
both terms of (18) are positive, both terms of (20) negative, and the terms of (19)
one real and one complex, ^9 has not any real zeros. Since the angular range is TT
and the order at infinity unity, the theorem of the previous article shows that ^fl has
no imaginary zeros in the relevant half-plane. Hence ^0 is a conformal curve-factor.

HYDRODYNAMICAL ILLUSTRATIONS.

13. The utility of the conformal curve-factors so far considered may be exemplified
by employing them in the specification of some cases of two-dimensional liquid motion.
In such applications the interpretation of w will be that implied by the relation

w =

VOL. COXV. - A. 3 O

448 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

where 0 and i/>- are the velocity-potential and stream-function of the liquid motion,
so that the velocity components u, v are (when z = x + iy)

v = -fy/dy = ty/dx ..... (22)
With this interpretation it is known that

dz/dvv = —q~l exp(ix), ......... (23)

where q is the resultant velocity, and x the angle which its direction makes with the
axis of x.

When a transformation is such that for w real and tending to + °° there is a
definite limit for dz/dw, it will be convenient to denote this limit by V"1. This
means that the limiting velocity at points indefinitely distant in the direction of the
axis of x is a velocity V parallel to the negative direction of that axis.

The interpretation of other constants in the transformation formulae might be
emphasised by explicitly indicating a factor V in every constant which is associated
with w by addition or subtraction ; for example (w—cj1' might be replaced by
(/c— c'V)''-', where c' has the dimensions of a length and depends only on the
geometrical configuration in the z plane. But, in order to avoid an unnecessary
appearance of heaviness in the formulae, this device for emphasis will not be employed ;
it will be easy and sufficient to remember that the velocity of flow corresponding to
a given transformation can be altered everywhere in the ratio of V to V, without
changing the geometrical configuration, by replacing V where it occurs explicitly
by V, and c, in an expression such as (w — c)>!'\ by cV/V.

14. Flow round a Semi-infinite Barrier in the Form of a Wedge with Smoothly
Rounded Apex. — A transformation in which a curve-factor is not accompanied by
any Schwarzian factor is typified by

.......... (24)

where C is a constant and ^ a curve-factor of definite linear range (say — c to.c) and

definite angular range y. For example *& may be
any one of the curve-factors already enumerated.
The boundary in the z plane corresponding to
w real is free from corners. As the point w moves
along the negative direction of its real axis the
point z moves backwards parallel to the axis of x
until w reaches the value c ; then z describes a

_. curve whose tangent (with the direction corre-

sponding to w increasing) makes a continually

increasing angle with the axis of x until that angle attains the value ny ; after that,
for w still decreasing, the boundary is straight. The configuration is of the general

CONFOEMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS.

449

character of fig. 1 , wherein the arrows indicate the direction of w increasing, and the
relevant region is that on the left of the arrows. The angle ny must be positive
since that is TT times the order at infinity, and a negative order at infinity would
imply infinite velocity at infinity ;* the angle ny must not exceed TT as otherwise the
boundary would intersect itself.

The form and dimensions of the curved part of the boundary depend on the form of
^ and on the constants C and n. By equating
dx and idy respectively to the real and the
imaginary parts of C ^n dw, when w is real and
between +c and — c, the problem of expressing
x and y in terms of a real parameter w is reduced
to one of quadrature. The degree of arbitrariness
in the form of the curve corresponds to the extent
of the range of known forms of *& and the
adjustable parameters which these contain.

A simple case is got by taking n = 1, and giving to
^5 and ^7 ; this leads to

dz — \A(tc — k) + ~B('t(>2—c2)'l\

where

A > 0, B > 0, and c > k > -c.

Fig. 2.

a form

representative of
(25)

The boundary in the z plane consists of two parallel lines extending to infinity on
the right, and smoothly joined by a curve on the left, as in fig. 2, the relevant region
being to the left of the arrows. Here z can be integrated, and if the origin of z be
taken at the point on the boundary where w — k, the result is

z =

-<? log {w+(u? -<

<? lo

], . . . (26)

whence, for points on the curved part of the boundary,

x =

_w (C8-

)1'' -c" cos-1 (tc/c) +c2 cos-1 (A/c)].

The co-ordinates of the extremities of the curve are readily deduced, and it is seen
that the distance between the straight parts of the boundary is JBcV.

If v0 be the velocity of the liquid at the point z = 0, V,,-1 is the value of dz/dw |
for w = k. Hence v^1 = B(c2-F)1/2, and the distance between the straight parts of
the boundary may be expressed in the form Jirc!V1(c8-*')-1/1. In checking the
dimensions of this expression it should be remembered that c and k are both of the
dimensions of a length multiplied by a velocity.

* See, however, § 42, infra.
3 o 2

450 DR. J. G. LEATHEM ON SOME APPLICATIONS OF

15. Longitudinal Motion of a Ship with Curved Sides terminating in a Pointed
Bow and Stern. — -The problem now to be considered is that of the disturbance
produced in a liquid stream, which if undisturbed would be uniform, by a stationary
object which is bounded by two curves and is symmetrical about a line parallel to the
stream. The curves meet at one end at an angle 2p-7r and at the other end at an
angle 2qir, p and q being each less than a half. The configuration in the z plane is of
the general character indicated in fig. 3.

As there is a line of symmetry which is obviously a stream line, only half the
configuration need be dealt with, and a transformation is required which will make
the boundary shown by a thick line in the z diagram correspond to the axis of w real,
and conformally represent the region to the left of the arrows.

The angles in the boundary must be taken accouiit of by Schwarzian factors in the
transformation ; if c and — c be the values assigned to 10 at the corners, these factors
are (iv — c}~p and (»; + c)~?. The curved side must be represented by a curve-factor <@
of linear range c to — c.

Let the angular range of a transformation be defined as the angle between the
limiting directions of the vector dz for w real and -> + oo and for w real and -^ — oo ,

-c

Fig. 3.

measured in the positive sense from the former to the latter. In the present
instance the angular range of the transformation is clearly zero. But, if y be the
angular range of the curve-factor, the angular range of the transformation is made
up of -p-TT contributed by one Schwarzian factor, -q-w contributed by the other, and
y contributed by the curve-factor. Hence y- ( p + q) •* = 0, and the curve-factor
must have an angular range of (p + q) TT. This can be secured by taking the (p + q)th
power of a curve-factor of angular range TT. Thus the type of transformation is

dz = C(w-c)-p(w + c^-"^(P^dw, ....... (28)

where ^ has the linear range c to -c, and an angular range TT.

Of the curve-factors so far considered the most general is #,. The use of this, in a
slightly modified form, leads to the transformation

w , ,

where X > 0, c > k > - c, and it has been arranged that | dz/dw \ + V"1 for w ->• + oo .

CONFOEMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 451

It must be remembered, however, that this transformation has been constructed
solely with a view to the angular configuration, and that it affords no guarantee that
the stream-lines corresponding respectively to w > c and w < — c will not be merely
parallel instead of being in the same straight line.

There is, in fact, the possibility of a configuration of the character indicated in
fig. 4, and it must now be shown that this possibility can be provided against by a
suitable adjustment of the parameter k.

16. Significance of the Early Terms in the Expansion of w for Great Values
of z. — If in a transformation such as that of formula (29) the variable w be supposed
to have its modulus large, and an expansion be made in ascending negative powers
of w, the result is a formula of the type

dz = V-1{l + Sw-1 + VDw-s}dw> ' . . (30)

where negative powers of w beyond the second are neglected, and 8 and D are
constants.

Omitting in the first instance the term in w~2, it is seen that the first approxi-

Fig. 4.

mation to z is V~'w, and that to the next degree of approximation the formula is
equivalent to

= (V-8z-1)dz,

whence

w = Vz-Slogz ........... (31)

In this expression for w the first term represents uniform flow V in the negative
direction of the real axis, and the second term represents a source of strength S at
the origin. If there be superposed a uniform motion V in the positive sense of the
real axis there results a liquid motion due to the uniform motion of the internal
boundary with velocity V, the liquid being "at rest at infinity"; and the first
approximation to this motion, for z great, corresponds to a complex velocity and
stream function w' consisting of a " source-term," namely,

w' = -S log z ........... (32)

Now 2x times the strength of the source represented by the " source-term " must
equal the rate at which liquid is being displaced by the intrusion of the internal

452 DK. J. G. LEATHEM ON SOME APPLICATIONS OF

boundary, and if the boundary is of the character shown in fig. 4 the rate of displace-
ment is 2lV, where 21 is the breadth of the semi-infinite straight portion on the left.

Accordingly

............ (33)

If the boundary is as in fig. 3, I = 0 and so S = 0 ; it is in any case clear that the
motion of a finite rigid internal boundary cannot give rise to a " source-term " in the
expansion of w' for z great.

Thus in order to secure that the transformation (29) shall correspond to a
configuration like that of fig. 3, the constants must be arranged so as to make
S zero.

17. The Doublet-Term, and the Coefficient of Inertia. — If S is zero the formula (-30)
takes the form

which is equivalent, to the same degree of approximation, to

dw = (V -Dz-2) dz,
and so to

w^Vz + Dz-1 ........... (34)

When by superposition the liquid is brought to " rest at infinity " and the internal
boundary is moving to the right witli velocity V, the first approximation to the
motion at great distance consists of a " doublet-term," namely,

w' = Dz-1 .......... (35)

Considering for a moment the translational motion, with velocity V parallel to the
axis of x, of a finite solid or " ship " which is not necessarily symmetrical about a line
in the direction of motion, it is seen that the full expression, in terms of polar
co-ordinates, for the velocity-potential <j> of the surrounding liquid may be written
in the form

0' = r-'(D cos 0 + E sin 0) + Z«mr-'ncos(m0-f a,,,) (m = 2, 3, ...). . (36)

Let GREEN'S- theorem be applied to the functions 0' and x in the region bounded
internally by the solid moving boundary, the " ship," and externally by any curve.
It then appears that

where ds is the element of arc, and cm the element of outward drawn normal, has the
same value for the internal solid boundary as for any surrounding curve. In
particular the surrounding curve may be taken to be a circle whose radius tends to
indefinite greatness, and for this it is seen that the terms of type r~m cos(m0 + «m)
make no contribution to the value of the integral, so that the total value is 27rD.

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 453

At the inner boundary, if I be the x cosine of the outward normal, dx/dn = I, and
d<f>'/dn =—lV; so the integral is equivalent to

If A be the area of the section of the ship,

\lxds = A,
and there results the equality

cfo .......... (37)

If/) be the density of the liquid, and ra the impulsive pressure required to generate
the motion of the liquid instantaneously from rest, nr = p<\> ; consequently

(38)

•where X0 is tlie x component of the resultant impulsive pressure exerted by the ship
on the surrounding liquid at the instant when the motion was set up. If the mass
of the ship M be equal to that of the liquid which it displaces, as would normally be
the case, A = /r'M. Hence (37) is equivalent to

X,,) .......... (39)

Now if (X, Y) be the impulse required to set up the whole motion, both the motion
of the ship and the motion of the liquid, X = MV + X,,; consequently (39) may be
written in the form

D = X/27r/> ............ (40)

By using y instead of x in the application of GREEN'S theorem it may be shown
similarly that

E =

Thus the components D, E of tlie doublet represented by the doublet-terms in the
expansion of w' or 0' at great distance are proportional to the components of the total
impulse which would generate the motion from rest.*

In the case of a ship which is symmetrical about the line of motion E and Y are
zero, and the effective inertia of the system for longitudinal motion is XV"1 or
27r/oV-1D.

* The corresponding theorem in three dimensions was given by the writer in a paper entitled " On
Doublet Distributions in Potential Theory," ' Proc. Royal Irish Academy,' vol. XXXII, Sec. A, No. 4,
1914, § 14.

454 DK. J. G. LEATHEM ON SOME APPLICATIONS OF

18. If the transformation (29) be reduced to the approximate form (30) it
appears that

'

2(p + q) p + q (p + q
Thus the condition for the absence of the source-term is

. . . (42)

........ (43)

X/

and the effective inertia of the ship (S being zero) is

OT/ f 2P<L _ o« /I _

2"

The values of the parameters must of course be such that c > k > — c, and it is clear
that there is a wide range of values of a, 73, g, and X which permit of this being true.
It may be well, in connexion with (44), to recall the fact that c is of the dimensions
of a length multiplied by a velocity, and that for a given configuration in the z plane
C/V is independent of V.

Though the formula (44) does actually contain the law of the dependence of the
inertia of the ship (having a certain type of shape) upon the angles 2p?r and Iq-w at
bow and stern, it must not be assumed that the functional law corresponds to the
explicit appearance of p and q in the formula. The other parameters are to be
regarded not as data, but as functions of what ought properly to be the data, for
example the length and breadth or the length and area of the ship. If X and cV"1
were expressed in terms of such data the formula would show explicitly the required
law, and would probably be a much more complicated function of p and q.

The specification of the shape of the ship can without difficulty be reduced to
quadratures by putting w real on the right-hand side of formula (29), and equating
the real and imaginary parts respectively to dx and idy. This gives x and y as
integrals of certain functions of a real parameter w, but the functions are generally
so complicated as to make evaluation of the integrals difficult.

In the special case of a — £, q = p, if dz be equivalent to da exp (ix) for w real,
it is seen that

X = -pir + 2p tan'1 {(c2-w2y''/\w},

ds = {(\2-l)w2 + c2}
so that, if the radius of curvature be R,

E = _ 2* = {(^
dx

c\'ooB1-»(x/2p)

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 455

This formula shows that when p < |- the value of R for w = ±c is zero. It also
shows that in the neighbourhood of the point where x = 0 the curve can be made
as flat as may be desired by taking A sufficiently small.

19. It is natural to expect that the parameter a of the transformation (29) will
particularly affect the shape of the curve at the extremities of the linear range.
In order to study the form of the curve in the immediate neighbourhood of the
point w = c it is convenient to put

-(x-x0+iy) exp (ipir) = g+i>,,

where x0 is the value of x for w = c ; this makes f and »/ co-ordinates referred to the
point w = c as origin, the axis of £ being tangential to the curve in the direction
of w decreasing. In the expression on the right-hand side of the transformation in
is put equal to c—e, and an approximation made on the supposition that e is small.
This leads to

d(g+i,,) = Ke

where

__ _
X+l J V(2c)''

Integration gives

£ =

K

l-p

>i —

l+a-p

sin a- . e

— I+°-P

so that in the immediate neighbourhood of the corner

(46)

(47)

The velocity of slip near the corner is de/d£, and consequently is proportional to the
{p/(l-p)}ih power of £

20. Motion of a Ship with Pointed Bow and Flat Stern. — When the internal
boundary in the z plane is symmetrical about a line in the direction of flow, and

Fig. 5.

consists on either side of a curve departing from the line of symmetry at an angle p-v
and rounding smoothly into a straight line which cuts the line of symmetry at right
angles, the configuration has the general character of fig. 5.

VOL. CCXV. — A. 3 P

456 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

The transformation must have angular range zero, and if b > c a suitable type is

* dw, .. ...... (48)

where ^ is a curve-factor of linear range cto-c, and angular range TT. For example
if the factor ^10 be adopted the transformation is

- + -cw (49)

v(\+iy+l'*(w->'

When this is expanded, for w great, in the form

dz = V-1(

it is found that

S=pC-ib

and

As S must vanish, the value of /,; must be

to, ........ (50)

and the other constants must be such that this /• is between — c and c.

The inertia-coefficient, for longitudinal motion, of the system consisting of the ship
in liquid at rest at infinity, is

INFLEXIONAL CURVE-FACTORS.

21. In seeking for curve-factors whose angular range shall be zero it will be
observed that as w decreases through real values from +c to — c the vector angle
of the complex

¥(w) = \(w-k) + (wt-<?)\ ........ (52)

wherein X > 0, will increase from zero through I-TT to TT, or will change from zero
to ± tan"1 {cX-1 (k2— c2)"1'2} as a maximum or minimum and then relapse to the value
zero, according as X (w—k) does or does not change sign for a value of w in the linear
range. Hitherto the angular range -K has been secured by postulating c2 > k2 ; it may
now be enquired whether, when c2 < k\ the function F (iv) can constitute or help to
constitute a conformal curve-factor.

CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 457

The product of F (w) and its conjugate surd is a quadratic expression obviously
possessed of real zeros, and so the only zeros for which F (iv) need be tested are real
ones. If the case of k > c be first considered, it is seen that F(- oo), F(-c), F(c)
are negative, while F (k) and F ( + co ) are positive. Hence F (w) has just one zero,
namely, between c and k, the corresponding value g of w being

0 = [X%-{x*P-(xs-l)cT'']/(\s-l); ...... (53)

the quantity g' derived from this by changing the sign of the square root is a zero
of the expression conjugate to F (?/;).

If k<-c, say k = -k', it is seen that F(- co) and F(-&') are negative, while
F(-c). F(c), and F(+oo) are positive. Thus F(?r) has just one zero, namely,
between — k' and — c, and its value —h is given by

\W-(\'-l)W]/(\>-l); ..... (54)

the quantity —h' derived from this by changing the sign of the .square root is a zero
of the expression conjugate to F (ir).

In each of these cases the zero of F(»?) is of the first order. Hence if in the
alternative cases attention be directed to ¥(w}/(iv—g) and F(w)f(iv+Ji) respectively,
it is seen that these are functions free from zeros and infinities in the relevant region.
Thus for k > c, k' > c, and X > 0,

(55)

w + h

are conformal curve-factors. In each case the linear range is c to —c, the angular
range is zero, and the order at infinity is zero. Within the linear range the modulus
of «„ is v/{(X2-l) (g'-u')/(g-ic)} and that of «\, is v/{(X2-l) (»- + l/)/(u- + !>)}. It
is to be noticed that, if A > 1, g' > k and h' > kf ; if\<l,g'<—c and // < — c.

Any power of *$u or ^12 is a curve-factor of inflexional character, and the greater
the power the more pronounced is the variation in the direction of the tangent at
different points of the curve. ^n corresponds to a curve which, as w decreases from
c to — c, is first concave and afterwards convex to the relevant region ; ^ corresponds
to a curve which is first convex and afterwards concave. But any negative power of
either corresponds to a curve in which, as compared with the original, the order of
convexity and concavity is reversed.

22. Hydrodynamical Example. — The specification of a liquid stream disturbed by

3 P 2

458 DK, J. G. LEATHEM ON SOME APPLICATIONS OF

the presence of a pear-shaped body of the general character shown in fig. 6 may be
got by combining a Schwarzian factor, a curve-factor of angular range %TT for the

Fig. 6.

forward curve, and a power of ^12 for the inflexional curve. Such a transformation
would be

, U (™-6) + {(w-a)(w-c)Yiy* C dw , .

vCu+iyMx+iy^-a)''*

where a > l> > c. The condition for the second straight stream-line being in the

production of the first is

V

/u. +

CURVE-FACTORS OF SEMI-INFINITE LINEAR RANGE.

23. For a range extending from w = 0 to w — — oo the simplest type of curve-
factor is

»„ = «.'«•+«,* ......... (59)

where a''- > 0. Its angular range is \ir and its order at infinity is |-. The trans-
formation

gives a boundary consisting of a straight line and a curve intersecting it at an angle
p-n-. In the special case of p = % the curve is a semi-parabola.

It may "be noted that when w is negative | ^13| = (a—wf!\

24. If a more general expression be considered, namely

V = w-k+ (cw)11*, ...... ... (61)

where c > 0, it is found that a distinction has to be drawn between the case when
Jc > 0 and the case when k < 0.

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 459
Let k > 0, and let ® be the surd conjugate to <&. Then

= (w—kf—cw,

which is positive for iv = 0, negative for w — k, and positive for w = + oo. Hence
<&& has real roots, of which one (g) is between 0 and k, and the other (g') is greater
than k. The values are

g = k+^c— (kc + JfC2)'1'*, g' = k + ^c+ (kc+^c^)'1' (02)

Now ^ is negative for w = 0, positive for w = k and for w = +00. Hence & has
one zero, which must be g. But

wherein & >• 0, has no zeros and is a conformal curve-factor. Its order at infinity is
zero, and therefore its angular range ought to be zero. This is readily verified, for if
X be the vector angle of ^u when ^v is real and negative

— tan = — cw'/2 — w + &

As — w increases from zero the denominator of this expression diminishes from
+ oo to a minimum value corresponding to —w = />', namely 2k'1', and then increases
again to infinity. Hence x diminishes from zero to a minimum value — tan 1(c1/s/2i1''),
and then increases again to zero.

When w is negative the modulus of ^M is {(g'—w)/(g—w)}ll\

When k < 0 let it equal —k''; then <$ takes the form

(cw)ll> .......... (64)

This can have no real zeros, since the two terms when both real are also both
positive. There can be no imaginary zeros in the relevant region because the vector
angle is x = tan"1 {(—cw)ll-/(w + k')}, which increases from zero (for w = 0) through
\-w (for w = —k') to TT (for w = — oo) and so gives an angular range TT, while the order
at infinity is unity. Thus ^15 is a conformal curve-factor.

When w is negative

which has real or imaginary factors according as c is greater or less than 4&'. In the
former case c may be put equal to 4F cosh2 ft, so that ^15 takes the form

......... (65)

460 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

and it appears that when w is negative

(66)

The utility of a knowledge of the moduli of curve-factors will be exemplified later.
24a. Curve- Factors involving Powers other than the Square Root. — A more general
type of curve-factor having semi-infinite range is obtainable as follows. Consider

.......... (67)

where |- > a > 0, and both b and b" are positive. This has no real zeros, since both
terms are positive when w > 0. The linear range of curvilinearity is from zero to
— oo, and on this range w" is represented by ( — w)a exp (ionr), so that the vector

angle x of ^17 is

— w)asin
- - -

= tail

• — - - -
\ba+(-w)acos

Tlie denominator in this expression cannot vanish, so, as w decreases from zero,
X increases to cnr ; thus the angular range is aw. The order at infinity is a, so there
are no imaginary zeros in the relevant region. Hence ^17 is a curve-factor.

The case of 1 > a > ^ has been excluded because it would give an angular range TT,
and so introduce imaginary zeros into <$ll.

25. Consider also

#]8 = «> + // + &l-««* ......... (68)

where k' > 0, 4- > a > 0, and both b and bl~" are positive. This has no real zeros
since all the terms are positive when ir > 0. On the range of curvilinearity the

vector angle x is

bl~a ( — w)asin
X = tan l

l-a

The denominator in this expression is negative for w =— <» and positive for w =—k'
and w = 0, and so has a zero for a value less than —k'. Therefore as w decreases
from zero x increases through |TT to TT, and the angular range is TT. The order at
infinity is unity, so there are no imaginary zeros in the relevant region. Hence ^18
is a conformal curve-factor.
26. The function

........ (69)

wherein X, /*, v, and a are all positive, is real for w real and positive. For 0 > w > — a
it has the vector angle

tan-1 [^{(w + a) (-«;)}'/= + „ (-w^/X,

and for w < — a the vector angle

tan-1 -w

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 461

so that it has an angular range TT corresponding to the linear range 0 to — oo . The
order at infinity being unity there are .no imaginary zeros in the relevant region,
and so $",<, is a confbrmal curve-factor. It has something of the character of a double
curve-factor.

DOUBLE CURVE-FACTORS.
27. An example of a double curve-factor is afforded by the function

»-6»)}lS ..... (70)

where b > a, > 0. The form of this must be considered for different parts of the
axis of w real.

For w > b the function is real ; moreover w2— ^ (a3 + b3) is positive, so %>.M has no
zero in this part of the axis of w real.

For b > w > a the form is

of which the real part changes from positive to negative as ir diminishes through the
value +{i (as+62)}1''. The vector angle accordingly ranges from zero to tr.
For a > w > — a the form is

of which both terms are real and negative.
For —a > w > — b the form is

of which the first term changes from negative to positive as w diminishes through
the value -{£ (as + 62)}1/2. The vector angle increases from TT to 27r.
For w <—b the form is

of which both terms are positive.

Thus the factor corresponds to two curves separated by a straight line, and over
the whole range from + b to -b the angular amplitude is 2*-. The order at infinity is
2, so there are no imaginary zeros in the relevant region. It appears therefore that

^20 is a curve-factor.

28. It is interesting to enquire what sort of a configuration is given by a trans-
formation which involves #„ and also Schwarzian factors introducing corners at
points ±a, ±b. Such a transformation is

,

=

dw (71)

'

462

DE. J. G. LEATHEM ON SOME APPLICATIONS OF

and it gives a configuration of the character indicated in fig. 7, the second and third
straight lines being inclined to the first at angles (n-p-q) •*• and (2n-p-q-p'-gf) TT

respectively.

In order that the three lines should be

— > parallel it would be necessary to have

n = p + q =

(72)

Fig. 7.
Hence the transformation

If it were further desired that the first and last
lines should be parts of the same straight line
the constants would have to be so adjusted
as to make the term in iv~l vanish in the
expansion of dzfdiv for w great. This condition
is equivalent to

which, when combined with (72), gives

p' =• p = n— q = n— g'. . . (73)

dz =

n 7
on CLW

'20

2"V (ifl8- a8)1-' (w* -

(74)

gives a configuration of the character of the thick line in fig. 8, which, in the hydro-
dynamical application, may be duplicated by reflexion.

Fig. 8.

The particular case of q = ^ gives a dumb-bell shaped boundary as in fig. 9,
and the case of n = q gives a ship with straight sides and pointed ends, as in fig. 10.

29. A double curve-factor containing a greater number of arbitrary parameters
than ^o is

, . . . (75)

where X is positive, and a, g, b, c, h, d are real constants in descending order of
magnitude. The use of this factor instead of ^20 in the applications of the previous
article would give greater variety in the possible shapes of boundary. The constants
could probably be adjusted so as to make all three straight portions of the boundary

CONFORMAL TRANSFORMATION TO PEOBLEMS IN HYDRODYNAMICS. 463

parts of one line, and so lead to a specification of liquid flow past two objects of
oval form.

Fig. 9.

Another double curve-factor, corresponding to two curves separated by a straight
line, is

where X > 0, and a>b>c>d. It lias a total angular range of TT, made up of fr
from each of the two curved portions.

Fig. 10.

It would be interesting to know whether a double curve-factor is necessarily the
product or 'other simple function of simple curve-factors.

SYNTHESIS OF CURVE-FACTORS FROM ELEMENTARY TYPES.
30. A simple function which has not yet been considered as a possible curve-factor is

F («>) = £-».'+ («'3-c2)'/J.

The product of this and its conjugate surd has only one zero, a real one, so F (?r)
need only be tested for real zeros.
It is to be noticed that

jL;w_(V-c2VH - i w

dw i

which is negative both for w real, positive and greater than c, and for w real,
negative and less than — c, due regard being paid to the significance of the square
root. Thus {w— (w2— c2)1'2} is positive for w real and greater than c, and continually
decreases as w increases, having the limit zero for «>->• + «> ; for w real and less
than — c, {w— (w3— c2)1'2} is negative and continually increases to the limit value zero
as w->— oo. It follows that F(w) has no stationary values for w real and w2 > c2 ;
the values of F(w) for w = — «, w = —c, w = c, and w = + oo} are k, k + c, k—c, and

VOL. CCXV. — A. 3 Q

464 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

k respectively. The signs of these four values of F (w) are all positive if k > c,
and all negative if k < — c. But if c > k > 0 the signs are +, + , — , +, and if
. 0 > k > — c the signs are — , +, — , — .

Thus, if k2 < c2, F (w) changes sign either between w = c and w = + oo or between
w = — c and w = — °° ; accordingly ~F (w) has a zero and is not a curve-factor. But,
if F > <-3, Y(w) does not change sign on either of the above-named ranges of w, and
so lias no zeros. Thus for1 F > c2 there is a curve-factor

VM = *-w+(«>»-e»)* ......... (77)

having the linear range +c to — c, the angular range zero, and the order at infinity
zero.

31. On examination it will be found that some of the curve-factors already
discussed are resolvable into factors of simpler type. If in ^ the parameter k be
replaced by the introduction of ft, such that

k sinh a = c sinh fi, (a > ft > —a),
it is readily verified that

^;, = ir sinh a — c. sinh ,8+ (ir — c2)'1'1 cosh a

= (2c)-le*{w + ce-^ + (w3-c*)l/«}{ce"-*-w+(w>-c!iyi>}, • • (78)

so that ^5 is equivalent to the product of two factors, one of the type of ^6) the
other of the type of ^.^.

Similarly if in ^7 a parameter /3 be introduced such that

k cosh a. = ±c cosh /3, (a > /3 > 0),
it appears that

^7 = w cosh a- (±) c cosh p+ (tf-c*)'!'- sinh a

- ^}t . (79)

so that ^7 is equivalent to the product of two factors, one of the type of 9t, the
other of the type of r$23.

Vn and #„, reclassified according as X is greater or less than unity, are equivalent
to

<g>24 _ w cosh a + c cosh /8+ (tg'-c8)'^ sinh a

w + ccosh(/3-a) ' ..... ' '

in which 0 > a, and

^26 _ w sinh q-c sinh 0+ (w*-<?}l!* cosh at

w-ccosh(/3-a) ......

CONFORMAL TRANSFOEMATION TO PROBLEMS IN HYDRODYNAMICS. 465
in which fi> > «2. It can be readily verified that

I'l

25

Consequently both these types of inflexional curve-factors are equivalent to simple
fractions having curve-factors of the type of ^ in numerator and denominator.
In the case of a semi-infinite linear range it is seen that

'It =

1'»)) . . . (84)

so that ^]6 is equivalent to the product of two curve-factors of the type of ^13. Also
(with the notation of article 24)

_w-k+ fc/r)1" _/''-• + 1 (;''-
w-0 V'+w1''1

in which fraction both numerator and denominator are of tlie type of $*,;,.

32. Of the curve-factors which have been studied some are such that, for real
values of w which render them complex, the squares of their moduli have real factors ;
others have not this property. To the latter class the theorems of the previous
article do not apply, but as regards the former class the results suggest a useful
method of generalisation. Within the limits of this class of curve-factors it seems
possible to regard the types ^6 and "^ as fundamental for the finite linear range — c
to + c, and the type ^13 for the semi-infinite range -- o° to zero. Bearing in mind
that any power of a curve-factor, whether positive or negative, is itself a curve-factor,
one may take any number of curve-factors of the types ^ and <$.^, raise each to any
power, and multiply all together, so producing a resultant curve-factor for the
assigned finite linear range. In the same way one may take the product of arbitrary
powers of any number of curve-factors of the type <$w, so getting a resultant curve-
factor for the assigned semi-infinite range. These curve-factors have a quite arbitrary
number of adjustable parameters, and so correspond to an endless multiplicity of
curves in the plane of z.

g

If, for example, the ship problem already discussed in article 1 5 be again considered,
it is legitimate to take a set of parameters kl} k.2, ... kr, ... intermediate in value
between — c and +c, and a set of corresponding indices n^ n3, ... nr, ... ; also a set
of parameters ll} 12, ...I,, ... greater than c or less than — c, and a set of corresponding
indices mlt m2, ... m,, ... . These combine to give the curve- factor

c3)'l>}»'U{l.-w+(iv3-c>)ll'}'»; .... (86)
3 Q 2

466 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

leading to the transformation

The parameters must comply with the condition for zero angular range, namely

2wr = p + g,

and with the condition for the vanishing of the source-term in w. Subject to these,
the constants are arbitrary and are in theory adjustable to meet a corresponding
number of possible requirements.

The range of adjustable parameters in *@.M seems, however, to be unexpectedly and
embarrassingly great, and there suggests itself a doubt whether the degree of
generality of the formula can be really so high as it appears to be. It is noticeable
that the ms parameters do not enter into the condition for zero angular range, and it
seems worth while to study the ^33 type of curve-factor more closely.

When w is real and c > w > — c,

which is not formally a product or quotient of rational integral functions of w. So
far as this is true, ^,3 could not be expressed as the product or quotient of curve-
factors the squares of whose moduli are rational integral functions of w. But it must
not be forgotten that there is a curve-factor, namely ^ = w + (i(?— c2)''', whose
modulus is a constant. The use of this gives a fractionalisation of ^23, for it is readily
verified that

2~l 2— c2)1''

wherein, as k2 > c\ (c2k l)2 < c2, so that ^23 is a fraction whose numerator is of the
type of ^6 and whose denominator is ^,. Moreover, it is to be noticed that <^l is only
a particular case of ^6.

There is therefore no loss of generality in discarding ^23 from the category of
fundamental types, and in forming new curve-factors by combining arbitrary powers
of different forms of ^ only.

Thus the curve-factor

^ = n{w-kr+(iv2-c2)1/'}"' (88)

f

is no less general than ^26. It may be substituted for ^ in the transformation (87)
for the problem of the doubly pointed ship, with the same condition for zero angular
range.

33. The formula just obtained obviously suggests the further step of establishing a
functional relation between nr and kr, and letting k, range over all real values subject
to the limitation c > kr > — c.

CONFOEMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 467

The process would be to take a real variable Q representing kr and to put f(6)d9
for nr, f(0) being a real function not involving w. It would generally be legitimate
to replace the limit of the product in formula (88) by the exponential of the limit of
the sum of the logarithms of the various factors, which last limit is an integral.
There results the very general curve-factor

w-6+(iV2-cT'}d6, ..... (89)

in which there is no d priori restriction of f(6) to continuity, beyond the exclusion
of such infinities as would prevent convergence of the formula. There may, for
example, be such infinities, for particular values of 9, as would make *$.& include as
factors definite powers of the corresponding forms of ^8.
For values of w on the linear range the modulus of ^28 is

-

and the vector angle

P f(d) tan-1 {(J-w^'Kw-e) }d6.

J -c

The angular range is

f(e)de.

LIQUID MOTIONS WITH FREE STREAM-LINKS.

34. In a liquid motion the characteristic of a stream-line which is free, or is a
possible line of discontinuity in the motion, is that along it the resultant velocity q
is constant. Now q-1 = \ dz/dw , and therefore a transformation of the type
dz/dw =f(w) will give as part of the boundary a possible free stream-line if there
is a part of the range w real for which f(w) \ is constant.

The simplest example of this is presented by the curve-factor ^ ; since the modulus
of this, within its linear range, is constant, a transformation in which /(«•) consists
solely of a power of <@l gives a free stream-line as part of the boundary. In fact, the

transformation

dz = ^{w+(w3-c2p}"dw (90)

gives a configuration of the general character indicated by fig. 1, with this special
feature that on the curved part of the boundary there indicated the velocity is
constant. Thus, so far as the ordinary theory of discontinuous fluid motion is
concerned, the curved part of the boundary may be a free stream-line.

35. Other cases of free stream-lines across finite gaps in boundaries which are
otherwise rectilineal may be built up from curve-factors of the types ^« and #„.
These being taken in the forms

468 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

it is seen that their respective moduli (on the linear range) are

{2ce~Y(ccoshy— w)Yk and {2ceY(c cosh y— w)}1/a,

and it is to be remembered that the angular range of ^23 is zero. Thus the
transformation

j u • ww asmnb^-m /m \*

dz= , . , — — - ^. ^ , (91)*

U (2c)p (w— c cosh y)p

representing a configuration of the kind indicated in fig. 11, gives \dz\dw\ = U"1 along
the curved part of the boundary, so that the curve may be a free stream-line along
which the velocity is U. If m be taken equal to p the angular range of the whole
transformation is zero, so that the first and last straight stream-lines are parallel.

The extension to boundaries with a greater number of corners is obvious.

36. Another simple example, leading to well-known results, is afforded by the use
of^13. As the modulus of a!- + w!'! for ?r negative is (a— «;)'''-, the combination of

fl K/\c ccshf

Fig. 11.

a power of ^13 with a Schwarzian factor consisting of a suitable power ofw—a will
give a free stream-line.

In fact, the transformation

gives a liquid flow of the character indicated in fig. 12, there being a fixed obstacle
consisting of two planes meeting at an angle Zp-n-, and stream-lines extending to
infinity and tending to parallelism with the undisturbed stream.

The case of p = ^ is that of an obstacle consisting of a single-plane wall at right
angles to the stream.

37. The previous transformation suggests a method of building up a transformation
applicable to symmetrical flow past an obstacle in the form of an open polygon. The
method consists in employing for the range of w negative, (the range which is to

* An equivalent form of this transformation is

dz =

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 469

correspond to a free stream-line), the product of powers of a number of curve-factors
of the type of ^13, with different parameters.

Suppose the configuration of liquid flow, (halved by the line of symmetry), to be of

Fig. 12.

the character indicated in fig. !.'>, the fixed obstacle being polygonal witli corners
A, B, C, and corresponding external angles p-n-, q-n-, r-jr. Let values «, /;, c be assigned
to w at A, B, C, these being in descending order of magnitude, and let the value
of iv at the end D of the polygon be zero. Consider the transformation

this gives all the corners as specified, and has a constant modulus for ii> negative.
Hence the curve DE is a free stream-line.

Fig. 13.

The parameters are adjustable so as to make the dimensions of the polygon such
as may be desired.

470 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

It need hardly be said that a reentrant angle of the polygon would, through the
corresponding Schwarzian factor, lead to a zero of dzfdw, and so to an infinite velocity.
In such a case some different transformation would apply, namely one which would
include a free stream-line starting from the reentrant corner, at least one of the
subsequent corners being in the region of still water.

38. Free, Stream-lines when the Fixed Boundary includes Curves. — The method
of the previous article suggests a step towards generalisation which consists in taking
part of the polygonal fixed boundary to have an indefinitely large number of corners
each of indefinitely small external angle, introducing a suitable curve-factor and
Schwarzian factor for each corner, and taking the limit of the product.

The parameters b, c, ... are replaced by a real variable 9 which ranges between the
values assigned to w at the extremities of the curved part of the fixed boundary, and
the indices q, r, ... of the corresponding Schwarzian factors are represented by C^X/TT,
where x is the angle between the tangent, drawn in the sense of w increasing, and a
fixed direction.

Thus there is obtained the resultant curve-factor

«* - Liin II {(iv

winch is generally equivalent to

the integration being extended over a suitable range of real values of 6.

-V

Fig. 14.

For example if a liquid flow be interrupted by a symmetrical and symmetrically
placed obstacle consisting of finite curves CA, AB meeting at A at an angle 2pir, and
the values of w at B and A be taken as zero and a respectively, the transformation
would be of the form

(95)

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 471
where K is a real constant, and

........ <«>

Clearly the formula (96) does not become definite until a functional relation is
specified between x and 6. There are two ways in which the attempt may be made
to utilise the formula. One is the purely empirical device of assigning arbitrarily
such relations between x and $ as would make the definite integral susceptible of
precise evaluation, and then ascertaining the corresponding configurations of fixed
and free boundaries. The other is to seek the particular functional relation which
shall correspond to an arbitrarily assigned form of fixed boundary.

If x =/(0), ^ takes the form

wherein it may be noted that the logarithmic infinity, which would correspond to
0 = w in the particular case of w real, does not destroy the convergence of the
integral. On substitution of this in (95), dz being represented by cfo exp ($x), it
appears that when w is real and between zero and a,

x = _p7r_ [j»(8)dO=f(w),

Jw

f(a) being taken equal to — £>TT, and

7 TT- 7 / «/ T I* V

as = K aw -n n- exp -

\ah—w''/ TT

Suppose the intrinsic equation to the curved part of the fixed boundary to he
prescribed, so that

where F is a prescribed function. Then it appears that

This relation specifies the property which the unknown function /(w) is required to
satisfy. It is an integral equation in/' (w), of by no means encouraging appearance.

39. It is clear that the practically useful method of applying formula (97) is to
assign a convenient form to/(0) and to enquire what sort of problems can thereby be
solved. To this end it is convenient to separate ^29 or V^ in the first instance from
any transformation such as (95) and to study it as a double curve-factor having two
contiguous linear ranges, namely from — oo to 0, and from 0 to a, with the special
property that on the former of these ranges it has its modulus independent of ^v.

VOL. ccxv. — A. 3 R

472 DR. J. G. LEATHEM ON SOME APPLICATIONS OF

It is convenient to remove the logarithm from the integral by integration by parts,

thus

By way of illustration, let /(0) = X01'3, then indicating the particular case by a
fresh suffix,

TT IQO- ^.n = X0''a log( — ,1 irj + Xw1" log \W—(

and

#;ii = ["'';;+«'j^(w-«j^ (100)

For v real and greater than a, ^ is clearly real. For a > n~ > 0 it takes the
form

.' '/-.. i r/.V Ar''fe la _ >r A" ' i/., i;.,

and its vector angle ranges from zero for »• = a to — Xa"' for ?r = 0.
For »• real and less than xero it takes the form

A''-L^'>_ ,„

w - ik/i '

which has modulus unity, and vector angle

\x~1^2o1'*tan~1(— -) +( — w)1'2 log

a

wliich ranges from —Xa1- for ?r = 0 to zero for w = -co.

The assigning of different values to the parameter X is equivalent to taking
different powers of the factor corresponding to X = -K.

If ^3, be employed in the transformation (95) the form of the curved part of the
fixed boundary depends on both the parameters X and p.

40. There is another method, slightly different from that just discussed, of building
up a product whose limit is a curve-factor of a type suitable for dealing with problems
of the class represented by fig. 14. In § 38 important factors of the product were
curve-factors having all the same semi-infinite linear range of curvilinearity but
different moduli ; in the present instance curve-factors having different semi-infinite
linear ranges, but the same modulus, are multiplied together.

CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 473
A type of ^13 is

which has the linear range — co < w < a— K, angular range %-ir, and modulus, in the
range, (a— w)1'2.

Thus <$/Md* has angular range ^TT/ (/<•) d/c and modulus (a — iv)'l'f'K)'lf.

It follows that

Lira "iT «'• + i0

has a double range of curvilinearity consisting of (i) --Gc<?r<0, on which the
modulus is Lim II («— «')''2/(")l'", and (ii) 0 < w < a — e, on which the modulus lias a
less simple form. The angular range is Lim ^ir2,f(ic) die.
In general the limit form is

f" f

with angular range ^TT ./(/c)c&c and modulus (a — »')'/!J-/( <(".

f"
If an angular range pw be desired f must be chosen so that I J (K) '^K = J>-

The transformation

dz = (w-a^^dw ......... (102)

would give a configuration different from fig. 14 in this respect that the fixed
boundary would be straight from w = a to w = a — e, and would then round smoothly
into a curve from w = a — e to tv = 0.

It is to be remarked that <@K is not a curve-factor for K = 0, though it is so for any
positive value of K however small. This might seem to preclude the putting of e = 0
in formula (101 ). However there is no real difficulty, for the subject of integration
has no discontinuity at K = 0, provided due precautions have been taken in the choice
of/; and in any case the integral with lower limit zero is concerned with (amongst
others) vanishingly small values of K, but not with the actual value K = 0. If the
form of ^,2 for e = 0 be denoted by *$.&, the transformation

gives a configuration like that of fig. 14.

A simple example is afforded by giving to f(K) the value 2pa~l. It is readily
verified that

(a
log {K'
. 6

dK

3 R 2

474 DR. J. G. LEATHEM ON SOME APPLICATIONS OF

so that, denoting this particular case of ^3., by ^M2p,

11 i(i+w\ 11 ( w —tt\ n it

lt(— >(w-a)'t(~ > e-*1*'- !'J* ..... (104)

By way of checking this result it is to be noticed that, for w real and 0 < w < a,

u takes the form

a + 1D fi^—iji 1 1 __ ] , . / n — in \

(a^+wl'')aa (a-w) 4" e-1'"' °°~ 'e ^ la ' ,

iii which the vector angle is ill ) TT and increases from zero to XTT as w diminishes

4 \ a I

from a to /ero.

For •»' real and negative the form of ^:M is

whose modulus is ('6 — iv)'!l, and vector angle x, where

, , / —w\la i/' — »''''-
-tan-1

2« \ rt / \ a / \ a/

it is readily seen that, as in decreases from /ero to — =», x passes from \ir to the limit
value -OTT.

Thus ^ is a double curve-factor of angular range ^TT equally divided between the
two parts of its linear range — oo to 0, and 0 to a.

On the former part of the linear range the modulus is (a — w)1'', so that the
transformation

dz = A.(w-aY1>f^-Mpdw ........ (105)

gives a configuration as in fig. 14, with a free stream-line tending to parallelism with
the undisturbed stream.
In general

flog {«''.+ (w-a+K)J'*} F (K) ch = [F (K) log {*'''+ (,,-a + ^'^T-i f - .F^^x
-e Je ~ Je/c(2(w— a + /cj'2

An example, simpler than the former, is got by putting F (*) = (*/a)'/;!, so that

/W = *M-lfc.

This makes the integral with lower limit zero equal to

log (a1/. -HP'/,) -a-1'' {w1"- (w-a)1''},
and gives the curve-factor

-a->l'{ivl"-(w-a)m ..... (106)

with the same sort of properties as <&M.

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 475

^as is really the product of two simpler curve-factors, one of the now familiar type
of ^13, the other

a double curve-factor whose linear ranges are a > w > 0 and 0 > w > — co , having
total angular range zero, and modulus unity for w < 0.

The formula of integration by parts suggests that other forms of ^M could be
factorised in a corresponding manner.

41. A configuration of the type of fig. 15 is given by the transformation

, A (a/-+ w''')*-» (6'/» W')" {(«-&)•/*+ (jt,-7,yH»
(w-aY(w-V)"

where a > b > 0, n is arbitrary, and PTT, qir are the angles indicated in the figure.
The boundary corresponding to w negative is a free stream-line. Greater generality

can be obtained by introducing factors of the typo {(>.<> — c)''-+ (»' — c)'/2i •, wliere
6> c> 0.

GENERAL REMARKS ON CURVE-FACTORS.

42. Note on the Relation between Angular Range and Order at Infinity. — It has
been already shown that the angular range of a curve-factor is x times its order at
infinity ; the same is obviously true for a Schwarzian factor. Hence all the trans-
formations which have been considered are characterised by the property that the
total angular range is TT times the order at infinity. Thus if the angular range be
/3— TT, where £TT S ft > 0, the first approximation to the form of the transformation
for values of w having very great modulus is

dz = AttA'-1 dw,

where A is a real constant, and the limit form of the boundary consists of lines
constituting the arms of an angle /3.

476 DR. J. G. LEATHEM ON SOME APPLICATIONS OF

Integration leads to the approximate relations

w = Bz"''^ + C, dw/dz = DTrft~ z ~ ,

where B and C are real constants.

Now if w be expressible, for great values of z, as a series of terms of the type cz\
since iv has- to be real for z real, c must be real ; and since w has to be real for z of
the form r exp (ift), it is necessary that X/3 = mr, where n is a positive or negative
integer. Thus the admissible terms in the expansion are of the type cnzmlft. When
ft = TT the expansion may also include a term in log z, but it must be remembered
that z, when great, is of a higher order of greatness than log z.

The real part of w being represented by <p, it is known that

taken over any area in the z plane is equal to <f> (30/3»/) ds taken round the boundary,

J

?v representing an element of outward normal. If the boundary be made up of the
locus of ir real and an arc of a circle with centre at the origin and radius r, the
subject of the line-integration is zero on the former part of the boundary. On
the latter part Or — rr and cZ.s = rdQ ; thus

I liC<l>\2 , /C0\ 1 77 I"

U ^- + Uf • dx dy =
JJ l\3ay ^ oy] Jo

where a -> /3 as r is increased indefinitely. If the area-integral is to cover the whole
of the relevant region the limit of the right-hand side must be taken for r -> <».

A term cnznnjf> in w would give in the limit of the line-integral a term ^mrcn2r2nir^,
and this tends to zero if n is negative and to infinity if n is positive. If the area-
integral vanishes w must be constant throughout the region. Hence, for an assigned
value of ft, there can be no w of any significance unless the most important term
in iv for z great has a positive index (or, as a possible alternative in the case of ft = TT,
is a multiple of log z). For this condition the least admissible value of n is unity.

Thus generally the transformations made up of curve-factors and Schwarzian
factors are such that the most important term in w, for z great, is of the least possible
order of magnitude that is consistent with w being other than a mere constant. It
is true that, in the case of ft < TT, div/dz is infinite for z infinite, but the conditions
of the problem do not then admit of any w free from this objection.

In the hydrodynamical applications it may be said that the transformation gives,
for any specified region in the z plane, an irrotational continuous motion which is as
free from singularity at infinity as the nature of the geometrical configuration allows.

CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 477

Adverting for example to §14, it is seen now that the provisional exclusion of a
negative value of ny was an unnecessary piece of caution. For ny negative the flow
would take place in a region like that on the
left of the arrows in fig. 16. It is true that
in this flow the velocity at infinity would be
infinite, but then there is no irrotational con-
tinuous flow possible in the region which does
not have infinite velocity at infinity. The / ~c
flow is, in fact, no more impossible than is the
region in which it is supposed to take place.

43. Curve-factors not having a definite order
at infinity. — -An example of a curve-factor of a Fig. 10.

kind not likely to be directly useful in physical applications is

/M+grM^-c8)'''}, (H,y)

where f and g are rational algebraical functions, or any functions which have no
infinities other than for w infinite. The vector angle, on the linear range <• > ir > — c,
is X = 9 (w) (ca— to2)1'-, and the angular range is zero.

^..,7 has usually no definite order at infinity, and so the proposition of ijll dues not
apply to it.

The possibility, however, of *$m having a definite order at infinity may be
illustrated by an example which is in one respect a little more general than ®'37,

namely,

w-c-(w-aY(w-by-*}. (110)

For iv great this tends to the limit exp{a« + (l— a) b — c}, so that the order at
infinity is zero.

Of somewhat similar character are the curve-factors

ay(w-by-a-(w-aY(w-by-1'}, .... (ill)
l-a)b + c\ w-(w-c](w-aY (w-b}1-]. . (112)

CURVE- FACTORS KEGARDED AS THE LIMITS OF PRODUCTS OF SCHWARZIAN FACTORS.

44. The Schwarz-Christoffel transformation being so widely known and of such
proved utility, the most natural way of trying to obtain a transformation for the
conformal representation of a region whose boundary is partly curvilinear would
seem to be to treat the curve as the limit of a rectilineal polygon and to seek the
corresponding limit of the product of the Schwarzian factors appropriate to the
corners that have to be smoothed out into a continuous curve.

478 DR. J. G. LEATHEM ON SOME APPLICATIONS OF

If x be, as usual, the angle between the axis of z real and the tangent at any point
of the curve corresponding to w real, the sense of the tangent being that of tv
increasing, the external angle of the polygon whose limit is to be the curve may be
taken to be dx, so that, if Q be the value of w at the corner, the Schwarzian factor is
(^—fl)-''*/". The parameter 9 varies continuously along the curve, and the curve-

factor

lira IL(w-0)-d*lr

can generally be replaced by the expression

the integral extending over the whole curve.

It is to be noticed that in this expression there may be an infinity in the subject of
integration at a point corresponding to a real value of ir. This infinity, being less
powerful than an infinity of the type (ir— 0}~\ does not interfere with absolute
convergence of the integral. Formal proof of this statement is considered unnecessary
here, being of a character readily suggested by familiar treatments of convergence
tests. It may be illustrated, in the hydro-dynamical application, by the fact that,
while a convex angle in the boundary gives rise to an infinite velocity, a convex
smooth curve does not.

But, while there is no divergence of the integral, the formula (113) is nevertheless
indefinite in the absence of any specified functional relation between the variables 6
and x- The Schwara transformation takes account only of the angles of the
configuration to which it is applied, and leaves the adjustments of all lengths to
be dealt with after transformation by the assigning of suitable values to the
various parameters associated with in in the factors. It is therefore not surprising
that a transformation containing only Schwarzian factors and limits of products
of such factors should ensure only that the directional and angular aspects of
the configuration are properly dealt with, leaving all settlement of correct linear
dimensions to be adjusted subsequently by the assigning of suitable values to isolated
parameters and suitable functional relations to parameters which vary continuously.

It is therefore proper to assume a functional relation between x and ®, say
X = vf(6), with the reservation that the nature of the function f is determined
by the configuration which is being dealt with, and depends not merely on the curve
to which ^4! corresponds but on the whole angular and linear configuration of the
prescribed boundary.

45. Before attempting to. formulate the condition which f must satisfy in order
that ^41 in a transformation of suitable type may represent a curve of assigned form,
it is convenient to note the usefulness of an alternative method of applying formula
(113) which consists in assigning such arbitrary forms to f as lead to simple

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 479

integrations, and so generating curve-factors whose geometrical significance in typical
transformations can be examined. As the geometrical interpretation has to do with
real values of w, it is to be remarked that for w real and b > w> a

-[ Iog(w-8)f'(et)d6=-\blog\w-6\j'(e)d6+iir{f(w)-f(b)}, . (114)

J a J a

so that the vector angle of ^41 is •>r{f(w)—f(b)} and its modulus

expj- "iog\w-e\f'(e)de}.

I Ja

A simple example is got by taking/(0) = —p9 + q, where p and q are constants, so
that dx = —TrpdQ. The argument of the exponential in the curve-factor is

= p{a—b+ (w—a)log(w—a)—(w—b) \og(w—b)}.

After omission of a constant factor this yields the curve-factor

^42 = (,r-rt)><<--«> (,,<-/,) -*<'<'-" (115)

It is readily seen that the angular range of ^ia is irp(b — <i), and that for w great
it approximates to the form e(b~"} wp(l'-"\ so that there is a definite order at infinity,
namely jo (6— a). The curve-factor does not tend to zero as w approaches a or b.
By putting f(0) = 92 one gets the curve-factor

^4:j = (vf— />)w;-"-2)(7t'-«)('"2-"J)e1Ma-'')("+''+2";), (116)

with angular range 7r(b2— «-) and order at infinity (b2— «2).

There would probably be little difficulty in finding a considerable number of forms
of f(6), which would permit of evaluation of the integral in formula (113) and so
yield types of curve-factor.

46. The problem of formulating the condition which f(6} must satisfy, in order
that the resulting curve-factor (in combination with suitable Schwarzian factors) may
be applicable to a prescribed curve, may be exemplified by taking the case of the
doubly-pointed ship. Suppose a configuration like fig. 3 has to be dealt with, the
form of the curve in the linear range — c < w < c being prescribed. The trans-
formation will be of the form

-f f(e)]og(v-e)de\dip,. . . (117)

and the question is what is required of f(0) in order that the curve corresponding to
— c < w < c may be as prescribed.

VOL. CCXV. A. 3 S

\

480 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

It seems simplest to deal with the curvature of the curve. When dz is put equal
to dsexp(ix), it appears that, for w real and between — c and c,

)-f(c)}

(118)

the assumption — pir = vf(o) being equivalent to the previously implied assumption
that the curve-factor of the type ^41 has no latent Schwarzian singularity at either
end of the range, so that the transformation dz = <$4l div would give a boundary
without corners. It appears also that

ds = (c—\

so that

- P f (6) \og\w-6\d9\dw, . . (119)

J -c

, expj-f /'(0)log w-0 de\

i (120)

Now on the prescribed curve ds/dx is a known function of x, say E,(X/TT); hence/
must satisfy

expj- f f'(e)log\w-8 dO\
U - •*-»

which can be regarded as an integral equation inf.

A. less complicated-looking form of the condition can be got by taking logarithms
of both sides of (121) and differentiating with respect to w. In carrying out this
operation it is useful to note that the differentiation of the definite integral is effected
by differentiation under the sign of integration and taking the Cauchy principal
value (indicated by P.V.) of the resulting integral ; proof of this is omitted as the
result is probably well known. The resulting expression of the condition which /
must satisfy is

_

f (w) c-w

(IB)

47. Transformation for any Prescribed Boundary. — Instead of confining the
application of the method of § 44 to a single curved portion of the boundary of
the region which is to be represented conformally on the half-plane of w, it is
legitimate to deal with the whole boundary in a single formula, namely

dz = exp\--\log(w-0)dx\dw, (123)

I TT J J

where 6 ranges from +00 to -co, and there is an unknown and possibly
discontinuous, but definite, functional relation between 0 and X- A straight part of
the boundary, along which dx is zero, makes no contribution to the formula. A

CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 481

corner, where x undergoes finite change, say pir, while 6 is at a standstill say at a,
contributes to the integral an amount — p-ir log (w— a), and so leads to the ordinary
Schwarzian or corner factor. The curved portions of the boundary contribute
distinct curve-factors of the type ^41.

For any one continuously curved stretch of the boundary, say a < w < b, let /t (0)
be the suitable expression for x/^, and let Fj (w) be the modulus of all that part of
the right-hand side of formula (123) which corresponds to values of 0 outside
the range a < Q < b. Then the condition that /i has to satisfy is of the type

. . . (124)

To other curved parts of the boundary correspond other functions, /2, /-,,..., &c.,
which satisfy similar conditions involving F2, F3, ... , &c., but F! depends on f2,fz, ••• ,
and F2 depends on/,, f:t, ... , so that the complete expression of the conditions which
the/'s must satisfy is a complicated set of simultaneous integral equations with the
function log | w—Q \ as kernel.

48. Free Stream-lines. — As an alternative to the formulation of the previous
article it may be assumed that along the curve x = ^(s)> and s = g (w), where g is
the unknown function. Then

-O dedw,

{ .'a

so that

= o, .... (125)

an integral equation (or equations) in g'.

The characteristic property of a free stream-line is that along it the velocity
is constant, say unity, so that w = s and g' = 1. Thus if one portion of the
boundary, instead of being of prescribed shape, is to be a free stream-line, the
corresponding function \js must satisfy the condition

-8\d6 = Q, ..... (126)

an integral equation in >// with kernel \og\w— 0\.

The solution of this equation for the case in which the fixed boundary is a
rectilineal polygon is derivable from the results of §37.

TRANSFORMATIONS INVOLVING BOTH VARIABLES EXPLICITLY.
49. The study of conformal representation by means of transformations of the type

would obviously present even greater difficulty than the kind of transformation

3 S 2

482 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

hitherto considered; but in connexion with the problem typified in fig. 3, the
problem namely of obtaining shapes of doubly-pointed ships whose hydrodynamical
effect in longitudinal motion can be exactly specified, something can be done with the
more general transformation.

It is easy to verify that the transformation for a semi-circle and the productions of

a diameter, namely

dz = G (w2-c2}-^ {w+ (w2-cs)l/2} dw,
is equivalent to

d* _ A?- dw /,27\

*"(5C^K!l

and that the transformation for a semi-ellipse, (Cc cosh a, Cc sinh a), and the
productions of its principal diameter, namely

dz = C (nf-cY1'1 {w sinh «+ (w'-c2)1'- cosh a] dw,
is equivalent to

an intermediate variable f being for convenience introduced in each case.

In these forms it is to be noticed : (i) that the factors in the denominators on the
left-hand side do not lead to zeros of dz/dw or to corners because the points in the
z plane where they have zeros are not in the relevant region, and (ii) that the only
factors left on the right-hand side are the Schwarzian or corner factors. Thus the
effect of the (z, f) transformation is to straighten out the curved part of the boundary
without alteration of the corner-angles, and the effect of the (£, w] transformation is
to smooth out the corners.

In both cases the (z, c) transformation is of the Schwarzian type, making the axis
of z real correspond to a rectilineal open polygon in the f plane, but these polygons
do not constitute the boundaries of the regions which are relevant to the present
problem. A straight line joining points in the first and last arms of the open
polygon in the £ plane screens off all the original corners from the relevant region, as
it were short-circuiting a part of the boundary including all the corners ; and the
corresponding line in the z plane is a curved line which screens off from the relevant
region all those points on the real axis which corresponded to the corners of the
Schwarzian transformation.

This aspect of the (z, f) transformation suggests generalisation by the introduction
of any number of corners on the broken line in the £ plane which corresponds to the
part of the axis of z real which is screened off from the relevant region.

The desired configuration in the z plane being as shown in fig. 17, the values pv,
q-ff of the marked angles being prescribed, and the values c, -c being assigned to z at

CONFOKMAL TRANSFOEMATION TO PROBLEMS IN HYDRODYNAMICS. 483

the ends of the curve, any number of real parameters «„ a2> ax, ... may be taken,
subject only to the condition c > a, > a2 > «3 > ...> — c, and another set of para-
meters K!, K2, KS, ... associated with them, subject to the condition

... =p + q ......... (129)

Then the transformation

(130)

makes the axis of z real correspond to an open polygon in the £ plane such as that in
fig. 18. The external angles K^TT, K^TT, ..., so that the angle between the directions of
the arrows on the first and last sides is (KI+K.,+ ...) -w. The straight line in the
£ plane joining the points ±c must correspond to a convex curve in the z plane.

The representation of the f configuration upon the half plane of w is of the
Schwarzian type, taking account of the corners in the boundary of the relevant

Fig. 18.

region. Thus if a, /3 be the values of w corresponding to the corners, the full (z, w)

transformation is

dz dw M3l)

(«-aj)* (*-«,)*(«-«•)*... ~ (w-*)p(w-W

subject to the condition (129).

Still further generalisation can be secured by making some of the screened-off sides
of the polygon in the £ plane curved. This can be done by introducing curve-factors

484

DK. J. G. LEATHEM ON SOME APPLICATIONS OF

(expressed now in terms of z) into the left side of the transformation. For example
if one factor & of linear range ar to ar+l and angular range yir were introduced, the
transformation would be of the type

with the condition

.—y = p + q.

(132)
(133)

A practical inconvenience of the present method of constructing transformations
applicable to the ship problem lies in the fact that, while the condition (129) or
alternatively (133) secures that the angles PTT and q-rr shall have any prescribed sum,
it does not suffice to secure prescribed separate values for these angles. When this
is desired there is a further condition to be satisfied in the form of a relation between
the parameters —c, a}, a2, ..., c. What is wanted is that the vector angle of the
complex f (c) — f ( — c) shall be — pir, and this is equivalent to

pc

Vector angle of f(z)dz = —p-jr.

Generally the fulfilment of this condition cannot be arranged for without previous
evaluation of the indefinite integral of f(z). Thus the problem of integration, which
must inevitably be faced in any case in the detailed interpretation of a transformation,
presents itself here at an earlier stage.

50. The method of the previous article may be further modified by associating
with dz such Schwarzian factors, powers of z±c, as shall remove the corners at ±c
and make the first and last portions of the boundary productions of the straight line
that corresponds to the curve in the z plane. In this case there are no Schwarzian
factors to be associated with dw, so f and w become identical save for a constant

to-

(nn

multiplier, and the configuration in the w plane is as shown in fig. 19, the marked
angles being called rmr and mr. The transformation is of the form

~ ,

= C dw.

In comparing the z and w configurations as represented by figs. 17 and 19, it is to
be noted that corresponding angles at c, though not equal, are proportional. Hence

CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 485

: IT = pir : ( 1 — p) TT, so that m=p/(l—p); similarly n — q/(l—q). Thus the

transformation is

p _2_

a 7'.c/z- ^y/l'r dz\* = c ****» (i34^

with the conditions

c > at > a2 > a3... > — c, (135)

and

.... (136)

the latter being the condition that the transformation have zero angular range. In
the application to ships there must also be the condition for vanishing of the source
term in the expansion of w for z great.

Clearly it is possible, with obvious precautions, to introduce on the left-hand side
of (134) curve-factors in z having a linear range inside the range — c to c, due
account being taken of their angular range in (136).

A particular example (as regards the w configuration rather a limit case) of the
transformation (134) is got by putting/) = f, q = |-, KI = 1, /ra = 1, «i = ft, «2 = — «.
This gives

Gdw=~~dz, (137)

Z — (Jb

wherein c > a. The curves in the z plane corresponding to this transformation are
KANKINE'S " oval neoids."*

51. The transformation for the semi-ellipse (c cosh a, c sinh a) and the productions
of its minor axis, written in the form

(z2— c2 sinh2 a)dz 7 /ir,0\

aWl , r- / 2 aw • i . = dw> (138)

c ) " {z cosh a+ (z +c ) " sum a}

exemplifies the possible presence in the denominator on the left-hand side of (i) a
function having imaginary zeros which are not in the relevant region, (ii) a function
entirely free from zeros, and possessed of branch points outside the relevant region.
Doubtless it would be possible to effect some generalisation of a formula including
these features.

SUPPLEMENTARY NOTE ON CURVE-FACTORS.

*

(Added 15th June, 1915.)

52. It should be noticed that the product of two curve-factors, of which the linear

range of one is contained in the linear range of the other, is itself a curve-factor for

the greater linear range. The resultant curve-factor is not, however, simple in the

sense specified in § 4, for though a curve represented by the curve-factor in any

* RANKINE, " On Plane Water-Lines in Two Dimensions," ' Scientific Papers,' p. 495.

486 DE. J. G. LEATHEM ON SOME APPLICATIONS OF

transformation might have no discontinuity of direction in the full linear range there
would be discontinuities in the analytical form of the equation to the curve at
the exlremities of the inner range.

This illustrates the fact that a curve-factor, which, for its total linear range
of curvilinearity, is not necessarily simple, may have branch-points not only at
the extremities of the linear range but also at points within the range. If the
general character of the types already chiefly considered is to be maintained in such
a multiple curve-factor, the net effect of proceeding along the real axis of w with a
suitable detour round each branch-point must be simply a change of sign when the
whole range has been traversed. Thus the formula

^,i = w-k+(u--a)"(w-c1)^(w-c2)^...(w-bY,. . . . (138A)

where a > C; > c., > ... > cr > k > cr+1 > ... > b, gives a curve-factor provided all the

indices are positive, and

1 ......... (139)

The function has no real zeros, and there can be no imaginary zeros in the relevant
region unless the angular range is different from IT. On writing down the expression
for the tangent of the vector angle x in any sub-range c, to c,+l it can be verified
that, except in the case of s = r, infinities of tan x, if they occur at all, must occur in
pairs which are not separated by a zero. Thus all the sub-ranges have zero angular
ranges, except one whose angular range is TT, and so the total angular range is IT.

The complete curve corresponding, in any transformation, to a factor ^44m consists
of two undulatory curves of zero angular range extending from a to cr and from cr+1
to b, joined by a curve of angular range m-n- corresponding to the interval cr to cr+l.

Of course no one of the c's may be equal to k, as if w—k were a factor of ^44 there
would be a corner at w = L

53. The multiple character of the curve-factor ^44 arises from the discrete
distribution along the linear range of branch-points of definite order. This feature
can be eliminated by substituting a continuous distribution of branch-points, each
factor having an infinitesimal index except the factors at the ends of the range.
Thus to the factor w — 9 is assigned the index f(9)dd, and the limit is taken for
vanishing of dd in each case. The form thus suggested is

. . . (140)

So long as f(Q) is free from infinities in the linear range, the second term of this
expression has no zeros in that range, and it will be supposed that f(Q) is thus
restricted. An infinity of f(9) at 6 = k might introduce a power of w— k as a factor
iii ^ and so introduce a corner; infinities of f(d) at other points of the range, if
their effect were to introduce factors of the type (w— c)n into the second term, would
leave ^ still a curve-factor but not a simple curve-factor.

CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 4H7

The undulations which characterise ^44 do not appear in curves represented by W^,.
The relation corresponding in this case to formula (139) is

(141)

.ft

An example of ^15, got by putting f(0) equal to a constant y. is

ll«-»u-M ..... (142)

subject to

-b] = 1 ......... (143)

54. A generalisation of *$ ^ is

#17 = (*/•-*,)"'( «•-*..)-... +(w-a)'(w-c1)ni(w-c.t)^...(iv-b)9, . (144)

where the k's lie between a and b but do not coincide with any of the c's, the //<'s are
all positive, and

It is readily verified that, for real values of w between n and b, the imaginary part
of ^.17 does not vanish, while the real part vanishes an odd number of times. Hence
the angular amplitude is TT.

Treatment- on the lines of article 53 leads to the form

48 = exp lF(0)log(-»r-0)'M + exp /

J i; J it''

where F and f are functions of 0 free from infinities between a and />.

55. The forms obtained in the last three articles suggest the possibility that the
addition of Schwarzian factors or the products of Schwarzian factors to one another.
or of curve-factors to one another, or of members of the one class to members of the
other class, may generally lead to results which are themselves curve-factors, provided
all the terms so added have the same angular range. Probably further limitations
would have to be introduced into the enunciation of such a theorem to make it valid,
but it seems clear that the method of addition is a useful means of obtaining fresh
forms of curve-factor.

The following examples suggest themselves : —

^49 = (w-a)~ + («--c1)m + (w-c2)m + ... + (ir-b)'", . . . . (147)
where 1 > TO > 0, and a > c^ > ca > ...>/;; and

^50= [f(e)(w-6)"de, ....... (us)

Jfc
where J (0) is positive for values of 0 between a and b.

VOL. CCXV. — A. 3 T

.

[ 489 1

INDEX

PHILOSOPHICAL TRANSACTIONS

SERIES A, VOL. 215.

A.

Arc spectra, effect of pressure on ; No. 5, Nickel (DUFFIELD), 205.

Atmosphere, eddy motion in (TAYLOR}, 1.

Atmospheric electricity potential gradient nt Kew (CuREE), 133.

B.

BAKERIAN LECTURE (BKAOO), 253.

BONK (W. A.). Gaseous Combustion at High Pressures, 275.
BRAOO (VV. H.). BAKERIAN LECTURE. — X-rays and Crystal Structure, 2~>:i.

BRINKWORTII (.T. H.) and CALLENOAR (11. L.). On the Specific Heat of Steam at Atmospheric Pressure hclweeii 104° C .
and 115° C., 383.

C.

CALLENDAR (H. L.). See BWNKWORTH and CALLENDAR.

Carbonic acid, thermal properties of, at low temperatures (JE.VKIN and PYE), 353.

CHAPMAN (S.). The Lunar Diurnal Magnetic Variation, and its Change with Lunar Distance, 161.

CUREE (C.). Atmospheric Electricity Potential Gradient at Kew Observatory, 1898 to 1912, 133.

Conformal transformation applied to hydrodynamics (LEATHEM), 430.

Crystal structure and X-rays (BflAao), 253.

D.

Diamagnetic substances, molecular field in (OXLEY), 79.

Dilution, heats of (TUCKER), 319.

DDTFIELD (W. OK). The Effect of Pressure upon Arc Spectra. No. 5.— Nickel, \ 3450 to A. 5500, 205.

E.

Eddy motion in atmosphere (TAYLOR), 1.

Electric waves, transmission of, over surface of earth (Lovs), 105.

Ellipsoidal bodies, potential of (JEANS), 27.

Equilibrium, figures of, ot rotating liquid masses (JEANS), 27.

VOL. CCXV. A. 3 U [Published October, 1915.

490 INDEX.

G.

Gaseous combustion at high pressures (BONE), 275.
GUILD (J.). See SMITH (S. W. J.) and GUILD.

H.

Hydrodynamics, application of conformal transformation to problems in (LBATHKM), 439.

J.

/

JEANS (J. H.). On the Potential of Ellipsoidal Bodies and the Figures of Equilibrium of Rotating Liquid Masses, 27.
JENKIN (0. F.) and PYE (D. R.). Thermal Properties of Carbonic Acid at Low Temperatures (Second Paper), 353.

L.

LEATHEM (J. G.). Sonic Applications of Conformal Transformation to Problems in Hydrodynamics, 439.
LOVE (A. E. H.). The Transmission of Electric Waves over the Surface of the Earth, 105.

M.

Magnetic susceptibility, influence of molecular constitution and temperature (OXLKY), 79.
Magnetic variation, lunar diurnal, and change with lunar distance (CHAPMAN), 161.
Molecular constitution, temperature, and magnetic susceptibility (Oxi.EY), 79.

0.

OXLEY (A. E.). The Influence of Molecular Constitution and Temperature on Magnetic Susceptibility. Part III. — On
the Molecular Field in Dianiajiuetic Substance*, 79.

1'.

1'YE (I). R.). See JKNKIN and Pvi:.

S.

SMITH (S. W. J.) and GPILD (,!.). A Thennomagnetic Study of the EutectoM Transition Point of Carbon Steels, 177.

Solutions, heats of dilution of concentrated (TUOKKR), 319.

Specific heat of steam (BiuNKWOimi and CALLENDAR), 383.

Spectra, arc, effect of pressure upon. — Nickel (I)i'KFiELD), 205.

Steam, specific heat of, at atmospheric, pressure between 104° C. and 115° C. (BRINKWORTH and CALLBNDAB), 383.

Steels, thermomagnelic study of eutectoid transition point of (SMITH and GUILD), 177.

«

T.

TAYLOR (G. I.). Eddy Motion in the Atmosphere, 1.
TUCKER (W. S.). Heats of Dilution of Concentrated Solutions, 319.

X.

X-rays and crystal structure (BBAG&), 253.

HARBISON AND SONS, PRINTERS IN ORDINARY TO HIS MAJB8TY, ST. MARTIN'S LANE, LONDON, W.C.

Q

U

L82
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