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.
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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 civility, in return for the respect shown to the Society by those communications. The like also is to be said with regard to the several projects, inventions, and curiosities of various kinds, which are often exhibited to the Society ; the authors whereof, or those who exhibit them, frequently take the liberty to report, and even to certify in the public newspapers, that they have met with the highest applause and approbation. 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