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S30.4 .TI3S

C.I

SdenUAc papers, by

StanfOfd Un^mlty U&raries

lllllllllllllll

3 6105 046 472 580

'^>^^. \

SCIENTIFIC PAPEES.

lonDon: 0. J. CLAY and SONS,

CAMBBIDGE UNIVERSITY PRESS WAREHOUSE,

AVE MARIA LANE.

€rIasgotD: 50, WELLINGTON STREET.

ILof^is : F. A. BROCKHAUS.

fi^ jgork: THE MAGMILLAN COMPANY.

iSombag: £. SEYMOUR HALE.

SCIENTIFIC PAPEKS

PETER GUTHRIE TAIT, M.A., Sec. R.S.E.

HONORARY FELLOW OF PETERHOUSE, CAMBRIDGE,
PROFESSOR OF NATURAL PHILOSOPHT IN THE UNIVERSITY OF EDINBURGH.

VOL. II.

CAMBRIDGE:

AT THE UNIVERSITY PRESS.

1900.

Pi'escr V e^'hc, X - /?/

imm OF THE
LELAND STANFORD JR. UmERSITY.

SEP 6 1900

l^ambnDge :

PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.

PREFACE.

'T'HIS volume contains, in addition to a further selection from my
-^ scientific papers, a few articles reprinted from the last edition of the
Encyclopaedia Bfntannica ; and an Introductory Lecture to my Ordinary
Class, devoted mainly to the question of how Natural Philosophy ought, as
well as how it ought not, to be taught. For pennission to reprint these
I am indebted to the courtesy of Messrs A, & C. Black, and of Messrs
Isbieterj respectively.

I have been assured by competent judges that my remarks on Science
Teaching, as it is too commonly conducted, are not only in no sense
exaggerated, but are even now as appropriate and as much needed as they
seemed to me twenty years ago.

To the short article on Quaternions I was inclined to attach special
importance, of course solely from the historical point of view ; for (in
consequence of my profound admiration for Hamilton's genius) I had spared
neither time nor trouble in the attempt to make it at once accurate and
ajs complete as the very limited space at my disposal allowed. Yet, aa
will be seen from the short note now appended to the article, the claims
of Hamilton to entire originality in the matter have once more been
challenged : — on this occasion in behalf of Gauss, [It is noteworthy that
Hamilton himself seems to have had at one time a notion that, if he had
been anticipated, it could have been only by that veiy remarkable man.
But he expresses himself as having been completely reassured on the subject,
by a pupil of Gauss who was acquainted v^ith the drift of his teachers
unpublished researches. See Hamilton's Life, Vol. in. pp, 311 — 12, 326.]

VI PREFACE.

It is therefore with much regret that I allow this volume to be issued
before full materials are available for the final settlement of such an
important question in scientific history. But it is reasonable to conclude
that the so-called anticipations had at least no very intimate connection
with a subject at once so novel and so unique as Quaternions. For Gauss,
though he survived their (hitherto supposed) date of birth for about twelve
years, certainly seems to have made no (public) claim in the matter.

The arrangement of the contents is, as nearly as possible, that adopted
in the former volume : — all papers on one large subject, such as the
Kinetic Theory of Gases, Impact, the Linear and Vector Function, the
Path of a Rotating Spherical Projectile, &c., being brought into groups
in relative sequence. I have reprinted only the later of my papers on
the Kinetic Gas Theory. The earlier were numerous, but fragmentary, and
a great part of their contents (often in an improved form) had been
embodied in the later ones.

I have again to thank Drs Knott and Peddie for their valuable help
in reading the proofs.

It is intended that a third volume shall contain some later papers
together with a complete list (including those not re-published) and a
general Index.

P. G. TAIT.

College, Edinburgh,

Janua/ry Ibth, 1900.

CONTENTS.

PAGE

LXI. Report on some of the physical properties of fresh water

and of sea-water 1

From the " Physics and Chemistry " of the Voyage of H.M.S.
Challenger; Vol. ii. Part iv., 1888. (Plates I, II.)

LXII. Optical notes 69

Proceedings of the Royal Society of Edinburgh, 1881.

LXIII. On a method of investigating experimentally the absorp-
tion of radiant heat by gases ..... 71

Nature, 1882.

LXIV. On the laivs of motion. Part I. 73

Proceedings of the Royal Society of Edinburgh, 1882.

LXV. Johann Benedict Listing ...... 81

Nature, 1883.

LXVI. Listing's Topologie 85

Philosophical Magazine, 1884. (Plate III.)

LXVII. On radiation 99

Proceedings of the Royal Society of Edinburgh, 1884.

T. II. b

VUl CONTENTS.

PAGE

LXVIII. On an eqiuition in quaternion differences . . . 101
Proceedings of the Royal Society of Edinburgh, 1884.

LXIX. On vortex motion . . 103

Proceedings of the Royal Society of Edinburgh, 1884.

LXX. Note on reference frames 104

Proceedings of the Royal Society of Edinburgh, 1884.

LXXL On various suggestions 05 to the source of atmospheric

electricity 107

Nature, 1884.

LXXII. Note on a singular passage in the Prindpia . . 110
Proceedings of the Royal Society of Edinburgh, 1885.

LXXIII. Note on a plane strain . 115

Proceedings of the Edinburgh Mathematical Society, Vol iii.,
1885.

LXXIV. Summation of certain series 118

Proceedings of the Edinburgh Mathematical Society, Vol. iii.,
1885.

LXXV. On certain integrals 120

Proceedings of the Edinburgh Mathematical Society, Vol. iv.,
1885.

LXXVI. Hookers anticipation of the kinetic theory, and of

synchronism . . . . . . . . 122

Proceedings of the Royal Society of Edinburgh, 1885.

LXXVIT. On the foundations of the kinetic theory of gases . . 124

Transactions of the Royal Society of Edinburgh, Vol. xxxiii.,
1886.

CONTENTS. IX

LXXVIII. On the foundations of the kinetic theory of gases. IL 153

Transactions of the Royal Society of Edinburgh, Vol. xxxiii.,

1887.

LXXIX. On the foundations of the kinetic theory of gases. HI. 179

Transactions of the Royal Society of Edinburgh, Vol. xxxv.,
1888.

LXXX. On the foundations of the kinetic theory of gases. IV. 192

Transactions of the Royal Society of Edinburgh, Vol. xxxvi.,
read 1889 and 1891.

LXXXI. On the foundations of the kinetic theory of gases. V. 209

Proceedings of the Royal Society of Edinburgh, 1892.

LXXXII. Note on the effects of explosives 212

Proceedings of the Royal Society of Edinburgh, 1887.

LXXXIII. On the value of A'*0'"/n'", when m and n are very large 213

Proceedings of the Edinburgh Mathematical Society, Vol. v.,
1887.

LXXXI V. Note on Milners lamp . . . . . . . 215

Proceedings of the Edinburgh Mathematical Society, Vol. v.,
1887.

LXXXV. An exercise on logarithmic tables 217

Proceedings of the Edinburgh Mathematical Society, Vol. v.,

1887.

LXXXVI. On Glories 219

Proceedings of the Royal Society of Edinburgh, 1887.

LXXXVIL Preliminary note on the duration of impact. . . 221
Proceedings of the Royal Society of Edinburgh, 1888.

X CONTENTS.

PAOB

LXXXVTIL On impact 222

Transactions of the Royal Society of Edinburgh, Vol. xxxvi.
Revised 1890. (Plate IV.)

LXXXIX. On impact. II. 249

Transactions of the Royal Society of Edinburgh, Vol. xxxvii.
Read 1892.

XC. Quaternion notes 280

Proceedings of the Royal Society of Edinburgh, 1888.

XCI. Obituary notice of Balfour Stewart . . . . 282
Proceedings of the Royal Society of London, 1889.

XCII. The relation among four vectors . . . . . 285
Proceedings of the Royal Society of Edinburgh, 1889.

XCIII. On the relation among the line, surface, and volume.

integrals 288

Proceedings of the Royal Society of Edinburgh, 1889.

XCIV. Quaternion note on a geometrical problem . . . 289
Proceedings of the Royal Society of Edinburgh, 1889.

XCV. Note appended to Captain Weirs paper ''On a new

azimuth diagram'' 292

Proceedings of the Royal Society of Edinburgh, 1889.

XCVI. On the relations between systems of curves which, together,

cut their plane into squares 294

Proceedings of the Edinburgh Mathematical Society, Vol. vii.,
1889.

XCVI I. On the importance of quaternions in physics . . 297

Philosophical Magazine, 1890.

CONTENTS. XI

PAGE

XCVIII. Olissettes of an ellipse and of a hyperbola . . . 309
Proceedings of the Rojal Society of Edinburgh, 1889. (Plate V.)

XCIX. Note on a curioTis operational theorem . . . 312

Proceedings of the Edinburgh Mathematical Society, 1890.

C. Note on ripples in a viscoics liquid . . . . 313

Proceedings of the Royal Society of Edinburgh, 1890.

CI. Note on the isothermals of ethyl oodde . . . . 318
Proceedings of the Royal Society of Edinburgh, 1891.

CII. Note appended to Dr Sang's paper, on NicoVs polarizing

eyepiece 321

Proceedings of the Royal Society of Edinburgh, 1891.

cm. Note on Dr Muir's solution of Sylvesters elimination

problem 325

Proceedings of the Royal Society of Edinburgh, 1892.

CIV. Note on the thermal effect of pressure on water . . ^7
Proceedings of the Royal Society of Edinburgh, 1892.

CV. Note on the division of space into infinitesimal cubes . 329
Proceedings of the Royal Society of Edinburgh, 1892.

CVI. Note on attraction 333

Proceedings of the Edinburgh Mathematical Society, Vol. xi.,
1893.

CVII. On the compressibility of liquids in connection with

their molecular pressure . . . . . . 334

Proceedings of the Royal Society of Edinburgh, 1893.

CVIII. Preliminary note on the compressibility of aqueous

solutions, in connection with molecidar pressure . 339

Proceedings of the Royal Society of Edinburgh, 1893.

xii CONTENTS.

PAOB

CIX. On the compressibility of fluids 343

Proceedings of the Royal Society of Edinburgh, 1894.

ex. On the application of Van der Wa^ils' equation to the

compi^ession of ordinary liquids .... 349

Proceedings of the Royal Society of Edinburgh, 1894.

CXI. Note on the compressibility of solutions of sugar . . 354
Proceedings of the Royal Society of Edinburgh, 1898.

CXII. On the path of a rotating spherical projectile . . 356

Transactions of the Royal Society of Edinburgh, Vol. xxxvii.,
1893. (Plate VI.)

CXIII. On the path of a rotating spherical projectile. II. . 371

Transactions of the Royal Society of Edinburgh, Vol. xxxix..
Part IL Read 1896. (Plate VII.)

CXIV. Note on the antecedents of Clerk-MaxwelVs electro-
dynamical wave-equations 388

Proceedings of the Royal Society of Edinburgh, 1894.

CXV. On the electro-magnetic wave-surface .... 390

Proceedings of the Royal Society of Edinburgh, 1894.

CXVI. On the intrinsic nature of the quaternion method . 392

Proceedings of the Royal Society of Edinburgh, 1894.

CXVII. Systems of plane curves whose orthogonals form a

similar system 399

Proceedings of the Royal Society of Edinburgh, 1895.

CXVIII. Note on the circles of curvature of a plane curve . 403
Proceedings of the Edinburgh Mathematical Society, 1895.

CONTENTS. XUl

PAGE

CXIX. Note on centrobaric shells 404

Proceedings of the Royal Society of Edinburgh, 1896.

CXX. On the linear and vector function 406

Proceedings of the Royal Society of Edinburgh, 1896.

CXXI. On the linear and vector function 410

Proceedings of the Royal Society of Edinburgh, 1897.

CXXII. Note on the solution of equations in linear a.nd vector

functions 413

Proceedings of the Royal Society of Edinburgh, 1897.

CXXIII. On the directions which are most altered by a homo-
geneous strain 421

Proceedings of the Royal Society of Edinburgh, 1897. (Plate
VIII.)

CXXIV. On the linear and vector function ..... 424
Proceedings of the Royal Society of Edinburgh, 1899.

CXXV. Note on Clerk- MaxwelVs law of distribution of velocity

in a group of equal colliding spheres . . . 427

Proceedings of the Royal Society of Edinburgh, 1896.

CXXVI. On the generalization of Josephus problem . . . 432
Proceedings of the Royal Society of Edinburgh, 1898.

CXXVII. Kirchhoff 436

Nature, Vol. xxxvi., 1887.

CXXVIII. Hamilton .440

Encycloptedia Britauuica, 1880.

XIV CONTENTS.

PAGE

CXXIX. Quaternions 445

Encyclopaedia Britannica, 1886.

CXXX. Radiation and convection 457

Encyclopsedia Britaimica, 1886.

CXXXI. Thermodynamics . 469

EncyclopsBdia Britannica, 1888.

CXXXII. Macquom Ranhine ....... 484

Memoir prefixed to Bankine's Scientific Papers, 1881.

CXXXIII. On the teaching of natural philosophy .... 486
Contemporary Review, 1878.

LXI,

REPORT ON SOME OF THE PHYSICAL PROPERTIES OF
FRESH WATER AND OF SEA-WATER.

[From the *' Physics and Chemist^ ** of the Voyage of HM.S. Ohallengm- ;
' Vol. It Part IV., 1888,]

INTRODUCTION,

As I had taken advantage of the instruments employed for the detemiination
of the Pressure Errors of the Challenger Tftermameters^ to make some other physical
investigations at pressures of several hundred atmospheres, Dr Muiray requested me to
repeat on a larger scale such of these as have a bearing on the objects of the
Challenger's voyage. The results of the inquiry are given in the following paper The
circumstances of the experiments, whether favourable to accuracy or not, are detailed
with a minuteness sufficient to show to what extent of approximation these results
may be trusted* My object has been rather to attempt to settle large questions about
which there exists great diversity of opinion, based upon irreconcilable eKpedmental
results, than to attain a very high degree of accuracy* My apparatus was thoroughly
competent to effect the first, but could not without serious change (such as greatly
to affect its strength) have been made available for the second purpose. The results
of Grassi, Amaury and Descaraps, Wertheim, Pagliani and Vincentini, &c., as to the
compressibility of water at low pressures, differ trom one another in a most distracting
manner; and the all but universal opinion at present seems to be that, for at leaat
five or six hundred atmospheres, there is little or no change in the compressibility,
the explicit statement of Perkins notwithstanding. My experiments have all been
made with a view to direct application in problems connected with the Challenger
work, and therefore at pressures of at least 150 atmospheres, so that I have only
incidentally and indirectly attacked the first of these questions ; but I hope that no
doubt can now remain as to the proper answer to the second. The study of the
compressibility of various strong solutions of common salt has, I believe, been carried
out for the first time under high pressures; and the effect of pressure on the
maximum-density point of water has been approximated to by three different experi-
mental methods, one of which is direct.

1 N&rr. Chall, Esp.^ vol ii., App. A., 1SS2. (Anii, No. LX.)

T, IL

[lxi.

CONTENTS.

PAOB

IVTBODUCTION 1

C0MPBE88IBILITT OF Watbb, Glass, and Mercury —

L (General Acootmt of the Investigation • 3

n. Some former Determinations 7

m. The Piezometers — Reckoning of Log. Factors — Compressibility of Mercury . 14

IV. Amagafs Manom^tre k Pistons Libres 19

y. Compressibility of Glass 22

YI. R^snm^ of my own Experiments on Compression of Water and of Sea-Water . 25

Vn. Final Results and Empirical Formulae for Fresh Water ..... 29

Vm. Reductions, Results, and Formulae for Sea-Water 37

IX. Compressibility, Expansibility, tfec., of Solutions of Common Salt ... 40

Associated Physical Questions —

X Theoretical Speculations 44

XI. Equilibrium of a Vertical Column of Water 46

Xn. Change of Temperature produced by Compression 48

XIII. Effect of Pressure on the Maximum-Density Point 52

SUMMART OF RESULTS 57

Appendix A. On an Improved Method of measuring Compressibility .... 59

R Relation between True and Average Compressibility 60

CL Calculation of Log. Factors . 61

D. Note on the Correction for the Compressibility of the Piezometer . 61

E. On the Relations between Liquid and Vapour 62

F. The Molecular Pressure in a Liquid 66

0. Equilibrium of a Column of Water 67

uo.]

COMPRESSIBIUTT OF WATER, GLASS, AND MERCURY,

L General Aocomrr of thi iKTESTtcsATiON.

I WILL £i^ give a geoend acecvunt of the siibjeetB ti«fttad^ of the mode of cxm-*
the experimeDts, and of tjie difficulties wMcli I have mor^ or leas oompli^tely
w^efcome in the eouiBe of Beyeral years' work. The reader will then be in a poaitioti
lc» foUow the (nil detidk of each branch of the inqiiir)%

The exprnm^its were for Uie most part carried on in the lai^ Fraser gan
hUj deacribed and figured in my previous Report^. But it was found to be tm-
ptactieilile to maintain this huge mass of metal at any steady temperature, exoept
tlia:t of the atr of the cellar in which it is placed. The great thickness of the
Colk^ walk, aided by the comparative mildness of recent winters, thus limiled till the
b^inmntg of the present year the available range of temperature for this instrument
lo thai from 3^ CL to about l^"" C. As I did not ocmdder this nearly sufficient* and
as comparatiYe experiments at the higher and lower of these temperatures could only
be made at mtervals of about six months, I procured (in Hay 1887) a much less
unwieldy apparatus. It was made entirely of steel, so as to be of as small mass as
poaible, with the necessar)' capacity and strength : and could at pleasure be used at
the temperature of the air, or be wholly immersed in a large bath of melting ice.
As this apparatus was mounted, not in a cellar but» in a room sixty feet above the
ground and facing the south, it enabled me to obtain a temperature range of 0° C
to 19^ C» with which I was obliged to content myself A great drawback to the
use of this apparatus was fouud in the smallne^ of its capacity. Not only was I
limited to the use of two, instead of six or seven, piezometers at a time ; but the
preesure could not be got up so slowly and smoothly as with the large appamtus,
and (what was still worse) it could not be let off so slowly. In spite of the&e luid
other difficulties, to be detailed later, I think it will be found that the observations
made with this apparatus are not markedly inferior in value to those made with the
great gun.

In the piezometers I have adhered to the old and somewhat rude method of
recording by means of indices containing a small piece of steel, and maintained in
their positions (till the mercury reaches them and after it has left them) by means
of attached hairs. These indices are liable to two kinds of deceptive displacement,
upwards or downwards, by the current produced at each stroke of the pump, or by
that produced during the expansion on relief of pressure. The first could almost
always be avoided, even in the smaller apparatus, provided the pressure was raised

» Pre&BiiTe EmjM of the Challengar Thenuoawters. 2!«f^, Kg, LX,

T O

4 REPORT ON SOME OP THE PHYSICAL PROPERTIES OF [lXI.

with sufficieDt steadiness, and the index brought down to the mercury at starting.
But the instantaneous reaction, partly elastic, partly due to cooling, and on rare
occasions due to leakage of the pump or at the plug, after a rash stroke of the
pump, sometimes left the index a little above the mercury just before the next stroke.
If another rash stroke followed, the index might be carried still farther above the
point reached by the mercury. Practically, however, there is little fear of my estimates
of compression having been exaggerated by this process. They are much more likely
to have been slightly diminished by a somewhat sudden &11 of pressure which, in
spite of every care, occasionally took place at the very commencement of the relief.
Once or twice the experiments were entirely vitiated by this cause; but, as we had
recorded the sudden outrush before the plug had been removed in order to take out
the piezometers, we were fully warranted in rejecting the readings taken on such an
occasion : — and we invariably did so, whether they agreed with the less suspicious
results or not.

Another and very puzzling source of uncertainty in the use of these indices
depends on the fact that the amount of pressure required to move them varies from
one part of the tube to another, sometimes even (frt)m day to day) in the same part
of the tube: — and the index thus records the final position of the top of the mercury
column in different phases of distortion on different occasions. The effect of this will
be to make all the determinations of compression too small, and it will be more
perceptible the smaller the compression measured. And in sea-water, and still more
in strong salt-solutions, the surface-tension of the mercury changes (a slight deposit
of calomel (?) being produced), while the elasticity of the hairs also is much affected.
But, by multiplying the experiments, it has been found possible to obtain what
appears a fisdrly trustworthy set of mean values by this process.

I discarded the use of the silvering process, which I had employed in my earlier
experimentsS partly because I found that the mercury column was liable to break,
especially when sea-water was used, partly from the great labour and loss of time
which the constant resilvering and refilling of the piezometers would have involved.
This process has also the special disadvantage that the substance operated on is not
necessarily the same in successive repetitions of the experiment.

And the electrical process* which I devised for recording the accomplishment of a
definite amount of compression could not be employed, because it was impossible to
lead insulated wires into either of my compression-chambers. This was much to be
regretted, as I know of no method but this by which we can be absolutely certain
of the temperature at which the operation is conducted.

My next difficulty was in the measurement of pressure. In my former Report I
have pointed out the untrustworthiness of the Bourdon gauges, and the uncertainty
of the unit of my external gauge. This gauge was amply sufficient for all the
purposes of my investigation of the errors of the Challenger thermometers, where the
inevitable error of a deep-sea reading formed, according to the depth, fix)m 5 to 20
per cent, of the pressure error; but, besides the uncertainty as to its unit, it was
on so small a scale that an error of 1 per cent, in the reading, mainly due to

^ Proc, Roy. Soc. Edin., vol. xiu pp. 1I28» 224, 18S3. * Appendix A to this Report.

lxl]

FRESH WATER AND OF SEA-WATER,

capUlarj effects at the surface of the mei^ury column, was quite possible when the
pressure did not exceed 150 atmospheres. Fortunately I was informed of the great
improvement made by Amagat on the principle of the old ManomUre DesgoffeB, — an
improvement which has made it an instrument of preci&ion instead of an ingenious
scientific toy. M* Amagat was so kind as to superintend the construction of one of
his instruments for me (it will be a surprise to very many professors of physics in
this country to hear that the whole work was executed in his laboratory), and to
graduate it by comparison with his well-known nitrogen gauge* My measurements of
pfessure are thei^fore only mie remove from Amagat's 1000 feet column of mercury.

The change of temperature produced by compression of water is one of the most
formidable difficulties I have encountered. During the compression the contents of
the piezometer, as well as the surrounding water, constantly change in temperature ;
and the amount of change depends not only on the initial temperature of the water,
but also on the rapidity with which the pressui-e is raised. It was impossible to
ascertain exactly what was the true temperature of the water in the piezometer at
the instant when the pressure was greatest, and a change of even 0*^1 C. involves a
displacement of the hair index, which is quite easily detected even by comparatively
mde measurement. Any very great nicety of measurement was thus obviously super-
fluous. My readings, therefore, were all made directly by applying to the tube of
the piezometer a light but very accurate scale. The zero of this scale was adjusted
to the level of the upper surface of the mercury of each piezometer the instant it
was removed from the water- vessel, in which it was lifted from the pressure-chamber,
and the position of the index was afterwards read at leisure. As the same scale was
employed in the calibration of the piezometer tubes, it^ unit is, of course, of no
consequence. The expansibility of water at atmospheric pressure is so small^ at least
up to %° C*, that no perceptible displacement of the mercury can have been intro-
duced before the zero of the scale was adjusted to it. The effects of the raising of
temperatore by heating are two: a direct increase of the volume (provided the tem-
perature be above the maximum-density point, and the pressure be kept constant),
and a diminution of compressibility (provided the temperature be under the minimum
compressibility point). These conspire to diminish the amount of compression produced
by a given pressure. At IS'' C, or so, the first of these is, in the range of my
experiments^ the more serious of the two, especially in the case of the solutions of
common salt.

The water in the compression apparatus, even when the lai^ one was used,
slowly changed in temperature from one group of experiments to the next: — some-
times perceptibly during the successive stages of one group. The effect of this source
of error was easily eliminated by means of the rough results of a plotting of the
ttneorrected experimental data. From this the effect of a small change of temperature
on the compressibility at any assigned temperature was determined with accuracy far
more than sufficient to enable me to calculate the requisite correction. This correction
was therefore applied to all the experimental data of each group, for which the
tempemture differed from that at the commencement of the group. The corrected
numbers were employed in the second and more complete graphical calculation^ I

6 REPORT. ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

endeavoured to raise the pressure in each experiment as nearly as possible by 1, 2,
or 3 tons weight per square inch: — having convinced myself by many triab that this
was the most convenient plan. The cure for any (slight) excess or defect of pressure
was at once supplied by the graphical method employed in the reductions, in which
the pressures were laid down as abscissae, and the corresponding . avei^Eige compres-
sibilities per atmosphere as ordinates.

When this work has been fully carried out, we have still only the apparent
compressibility of the water or salt-solution. The correction for the compressibility of
glass, which is by no means a negligible quantity, — being in &ct about 6 per cent,
of that of water at 0'' C, — involves a more formidable measurement than the other ;
but I think I have executed it, for two different temperatures, within some 2 per cent
or so. The resulting values of the true compressibility of water may therefore err,
on this account, by O'l per cent. This is considerably less than the probable error of
the determinations of apparent compressibility, so that it is fSeu: more than sufficient.
With a view to this part of the work the piezometers, whether for water or for
mercury, were all constructed from narrow and wide tubes of the same glass, obtained
from one melting in Messrs Ford s Works, Edinburgh ; while solid rods of the same
were also obtained for the application of Buchanan's method \

My results are not strictly comparable with any that, to my knowledge, have
yet been published, except, of course, those which I gave in 1883 and 1884. The
reason is that the lowest pressure which I applied (about 150 atmospheres, or nearly
one ton weight per square inch) is far greater than the highest employed by other
experimenters, at least for a consecutive series of pressures. I must except, however,
the results of Perkins and some remarkable recent determinations made by Amagat*.
Perkins' results are entirely valueless as to the a^tud compressions, because his pressure
unit is obviously very far from correct. They show, however, at one definite tem-
perature, the rate at which the compressibility diminishes as the pressure is raised.
Amagat's work, on the other hand, though of the highest order, is not yet completed
by the determination of the correction for the compression of the piezometer.

The extension of my formulae to very low pressures, though it agrees in a
remarkable manner with some of the best of accepted results, such as those of
Buchanan and of Pagliani and Vincentini, is purely conjectural, and may therefore
possibly involve error, but not one of the least consequence to any inquiries connected
with the problems to which the Challenger work was directed.

The piezometers, which had been for three years employed on water and on sea-
water, were, during the end of last summer, refilled with solutions of common salt of
very different strengths, prepared in the laboratory of Dr Crum Brown. The deter*
minations of compressibility were made at three temperatures only, those which could
be steadily maintained, viz. 0° C, 10° C, and about 19° C, the two latter being the
temperature of the room, the former obtained by the u^e of an ice-bath. Here great
rapidity of adjustment of the scale to the mercury was requisite, even in the experi-
ments made near 0° C, for the salt solutions (especially the nearly saturated one)

1 Trans. Roy. Soc. Edin., vol. xxn. pp. 689-598, 1880.
' Comptes Rendus, torn, oiii., 1886, and torn, oiv., 1887.

LXl.] FRESH WATER AND OF SEA-WATER. 7

show considerable expansibility at that temperature. In these salt solutions, however,
the hair indices behave very irregularly; so that this part of my work is much
inferior in exactitude to the rest.

Besides the determinations briefly described above, there will be found in this
Report a number of experimental results connected with the effect of pressure on
the temperature of water and on the temperature of the maximum density of water.
Though I afterwards found that the question was not a new one, I was completely
unaware of the fact when some experiments, which I made in 1881 on the heat
developed by compressing water, gave results which seemed to be inexplicable except
on the hypothesis that the maximum-density point is lowered by pressure. Hence . I
have added a description of these experiments, since greatly extended by parties of
my students.

And I have appended other and more direct determinations of the change of the
maximum-density point. I also give, after Canton, but with better data than his,
an estimate of the amount by which the depth of the sea is altered by compression.
Also some corresponding inquiries for the more complex conditions introduced by the
consideration of the maximum-density point, &c.

An Appendix contains all the theoretical calculations, the results of which are
made use of in the text; as well as some speculations, not devoid of interest, which
have arisen in the course of the inquiry.

II. Some former Determinations.

There seems now to be no doubt that Canton (in 1762) was the first to establish
the iact of the compressibility of water. But he did far more; he measured its
apparent amount at each of three temperatures with remarkable accuracy, and thus
discovered (in 1764) the curiously important additional &ct that it diminishes when
the temperature is raised. As his papers, or at all events the second of them, seem
to have &llen entirely out of notice S and as they are exceedingly brief and clear,
1 think it well to reproduce some passages textually from the Philosophical Trans-
actions of the dates given above.

" Having procured a small glass tube of about two feet in length, with a ball
at one end of it of an inch and a quarter in diameter; I filled the ball and part
of the tube with mercury; and, keeping it, with a Fahrenheit's thermometer, in
water which was firequently stirred, it was brought exactly to the heat of 50 degrees;
and the place where the mercury stood in the tube, which was about 6J inches
above the ball, was carefully marked. I then raised the mercury, by heat, to the
top of the tube, and sealed the tube hermetically; and when the mercury was
brought to the same degree of heat as before, it stood in the tube -^ of an inch
higher than the mark.

1 PerhapB the reason may be, in part, that by a printer's error the title of Canton's first paper is given
(in the Index to vol. ui. of the PhiL Tram,) as "Experiments to prove that Water is not oompressible."

8 REPORT ON SOME OP THE PHYSICAL PROPERTIES OF [lXI.

'The same ball, and part of the tube being filled with water exhausted of air,
instead of the mercury, and the place where the water stood in the tube when it
came to rest in the heat of 50 degrees, being marked, which was about 6 inches
above the ball ; the water was then raised by heat till it filled the tube ; which
being sealed again, and the water brought to the heat of 50 degrees as before, it
stood in the tube -^ of an inch above the mark

''Now the weight of the atmosphere (or about 73 pounds avoirdupois) pressing
on the outside of the ball and not on the inside, will squeeze it into less compass*.
And by this compression of the ball, the mercury and the water will be equally
raised in the tube; but the water is found, by the experiments above related, to
rise ^ of an inch more than the mercury; and therefore the water must expand,
so much, more than the mercury, by removing the weight of the atmosphere.

''In order to determine how much water is compressed by this, or a greater
weight, I took a glass ball of about an inch and ^ in diameter which was
joined to a cylindrical tube of 4 inches and -^ in length, and in diameter about j^
of an inch; and by weighing the quantity of mercury that exactly filled the ball,
and also the quantity that filled the whole length of the tube; I found that the
mercury in ^ of an inch of the tube was the 100,000 part of that contained in
the ball ; and with the edge of a file, I divided the tube accordingly.

"This being done, I filled the ball and part of the tube with water exhausted
of air; and left the tube open, that the ball, whether in rarefied or condensed air,
might always be equally pressed within and without, and therefore not altered in its
dimension& Now by placing this ball and tube under the receiver of an air-pump,
I could see the degree of expansion of the water, answering to any degree of rare-
faction of the air; and by putting it into a glass receiver of a condensing engine,
I could see the degree of compression of the water, answering to any degree of
condensation of the air. But great care must be taken, in making these experiments,
that the heat of the glass ball be not altered, either by the coming on of moisture,
or its going ofif by evaporation ; which may easily be prevented by keeping the ball
under water, or by using oil only in working the pump and condenser.

" In this manner I have found by repeated trials, when the heat of the air has
been about 50 degrees, and the mercury at a mean height in the barometer, that
the water will expand and rise in the tube, by removing the weight of the atmo-
sphere, 4 divisions and -^ ; or one part in 21,740 ; and will be as much compressed
under the weight of an additional atmosphere. Therefore the compression of water
by twice the weight of the atmosphere, is one part in 10,870 of its whole bulk*.

> **See an aooonnt of experiments made with glass balls by Mr Hooke (afterwards Dr Hooke) in Dr Birch's
HUtory of the Royal Society, vol. i. p. 127."

* **If the compressibility of the water was owing to any air that it might stiU be supposed to contain,
it is evident that more air most make it more compressible; I therefore let into the baU a babble of air
that measored near ^ of an inch in diameter, which the water absorbed in about four days; but I found
upon trial that the water was not more compressed, by twice the weight of the atmosphere, than before.'*

**The compression of the glass in this experiment, by the equal and contrary forces acting within and
without the baU, is not sensible: for the compression of water in two balls, appears to be exactly the same,
when the glass of one is more than twice the thickness of the glass of the other. And the weight of an

LXI.]

TREBH WATER AND OF SEA -WATER.

"The famous Florentine Experiment, which so many philoaophical writers have
tnenlioned ns a proof of the incompreeaibility of water, will not, when cfiLrefttUy
considered, appear sufficient for that purpose : for in forcing any part of the water
cotitmned in a hollow globe of gold through its porea by pressure, the figure of the
gold must be altered ; and consequently, the internal space containing the water,
diminished ; but it was impossible for the gentlemen of the Academy del Cimento
to determine, that the water which was forced into the pores and through the gold,
was exactly equal to the diminution of the internal space by the pressure/'

**By similar experiments made since, it appears that water has the remarkable
property of being more compressible in winter than in summer; which is contrary to
what I have observ^ed both in spirit of wine and oil of olives; these fluids are (as
one would expect water to be) more compressible when expanded by heat, and leas
so when contracted by cold. Water and spirit of wine I have several times examined,
both by the air*pump and condenser, in opposite seasons of the year : and, when
Fahrenheit*!:? thermometer has been at 34 degrees, I have found the water to be
compressed by the mean weight of the atmosphere 49 parte in a million of its
whole bulk, and the spirit of wine 60 parts ; but when the thermometer has been
at 64 degrees, the same weight would compress the water no more than 44 parts
in a million, and the spirit uf wine no less than 71 of the same parts* In making
these experiments, the glass ball containing the fluid to be oompreaaed must be kept
under water, that the heat of it may not be altei-ed during the operation,

''The compression by the weight of the atmosphere, and the specific gravity of
each of the following flnids, (which are all I have yet tried,) were found when the
barometer was at 29 1 inches, and the thermometer at 50 degrees.

Millionth parte.

Specifia gravity.

Compression

of spirit of Wine,

846

j»

Oil of Olives,

918

Rain- Water,

1000

Sea-Water,

1028

Mercury,

13595

These fluids are not only compressible, but also elastic : for if the weight by which
they are naturally compressed be diminished, they expand ; and if that by which
they are compressed in the condenser be removed, they take up the same room as
at first. That this does not arise from the elasticity of any air the fluids contain,
is evident; because theii* expansion, by removing the weight of the atmosphere, is
not greater than their compression by an equal additional weight : whereas air will
expand twice as much by removing half the weight of the atmosphere, as it will
be compressed by adding the whole weight of the atmosphere,

*' It may also be worth observing, that the compression of these fluids, by the
same weight are not in the invei-se ratio of their densities or specific gravities, as
might be supposed. The compression of spirit of wine, for instance, being compared

ftiaioHpbere, which I found would uoEiipinesB meroutj in one of these bftlk but | part of & diviaioD of the
lube, oompreAiefl water m the aam^ ball 4 diviiiona And ^/'

T, II. '2

10 REPORT ON SOM£ OF THE PHYSICAL PROPERTIES OF [lXL

with that of rain-water, is greater than in this propcnrtion, and the oompreaBion of
sea-water is less.**

With the exception of the mistake as to the non-effect of oompreasibility of glass,
and its consequences (a mistake into which Orsted and many others haTe &Uen
long since Canton's day), the above is almost exact The argument from the &ct
that thick and thin vessels give the same result is unfounded; but the diaoovery of
the fact itself shows how accurate the experiments must have been. The formula (A)
below (Section VIL), if extended to p = 0, gives for the value of tiie apparent
compressibility of water at lO"" C. (50'' F.), which is what Canton really measured, the
number

0-0000461,

exactly the same as that given by him 126 years ago !

The next really great step in this inquiry was taken by Perkins in 1826. He
showed beyond the possibility of doubt that in water at 10"" C. the compressibility
diminishes as the pressure is increased, quickly at first, afterwards more and more
slowly\ This was contested by Orsted, who found no change of compressibility up to
70 atmospheres. Many other apparently authoritative statements have since been made
to the same effect Unfortunately Perkins' estimates of pressure are very inaccurate,
so that no numerical data of any value can be obtained from his paper.

Colladon* is sometimes referred to as an authority on the compression of liquids.
But, referring to Canton, he states that there is no difference in the oompres^bility
of water at 0"* C and at 10"" C. His words are : " Nous avons trouv6 que Teau a la
m^me compressibility ji 0"* et ii -h 10^ Nous avons Ai}k CEut observer les causes
d'eneur qui ont Ait alt^rer les rdsultats des experiences de Canton." There can be
no doubt whatever that there is a difference of 6 per cent., which is what Canton gives !

In Regnault's experiments' pressure was applied alternately to the outside and
to tiie inside of the piezometer, and then simultaneously to both. From tiie first
Appendix to my Report on the Pressure-Errors, &c, it will be seen that tiie three
measurements of changed content thus obtained are not independent, the third giving
the algebraic sum of the first two; so that, unless we had an absolutely incom-
pressible liquid to deal with, we could not employ them to determine the elastic
constants of the pieaometer. For the compression of the liquid contents is added to
the quantity measured, in the second and third of the experiments. Thus B^pault
had to ML back on the measurement of Young's modulus, in order to obtain an
additional datum. In place of this, Jamin afterwards suggested the measurement of
the change of external volume of the piezometer; and this process was carried out
by Amaury and Descamps. But there are great objections to the employment of
external, or internal, pressure alone in such very delicate inquiries. For, unless the
bulbs be truly spherical, or cylindrical, and the walls of perfectly uniform thickness

^ The earefiiUy dzmwn pUle which illnstratoe hit paper is oim of the 107 heat euAj nTtnipliwi of the use
of the grmphie method. Phil, Trans., toL cti. p. Ml, 18S6.

* M€wL, hut. Savans itrang,, torn. t. p. 296, 183S.

* Jf^a. Acad, Sei. Paris, torn. xxi. pp. 1 et seq,, 1847,

i-xl]

FRESH WATER AND OF SEA-WATER,

mod of perfectly iiniforni material, the theoretical conditions will not be fulfilled : —
and the errors may easily bo of the same order wa is the quantity to be measured,

Finding that he could not obtain good results with glass vessels, Regnault
employed spherical shellB of brass and of copper* With these he obtained, for the
compressibility of water, the value

0 000048 per atm,

for pressures fit>m one to ten atmospheres* The temperature, unfortunately, is not
specially stated.

Graasi\ working with Regnault's apparatus, made a number of determinations of
compressibility of different liquids, all for small ranges of pressure.

The following are some of his results for water ; —

Temperature,

r-5

10°-8
18'0

53°0

CompreBsibilUy per atm.
OW00503
515

480
462
456
45,'!
441

These numbers cannot be even approximately represented by any simple formula ;
mainly in consequence of the maximum corapressibility which, they appear to show,
Hew somewhere about V"^ C* No other experimenter seems to have found any trace
of this maximum*

Grassi assigns, for sea-water at 17**5 C*, 0*94 of the compressibility of pure water^
and gives

000000295

as the compressibility of mercury* He also states that the compressibility of salt
solutions increases with rise of temperature* These are not in accordance with my
refinlts* But, as he further states that alcohol, chloroform, and ether inarease in
compressibility with rise of pressure (a result soon after shown by Amagat to be
completely erroneous), little confidence can be placed in any of his determinations.

A very complete series of measurements of the compressihiUty of water (for low
pressures) through the whole range of tempemture from 0^ C. to 100** C», has been
made by Pagliani and Yincentini^ Unfortunately, in their experiments, pressure was
applied to the inside only of the piezometer, so that their indicated results have to
bi* diminished by from 40 to 50 per cent. The effects of heat on the elasticity of
glass are, however, careiuUy determined, a matter of absolute necessity when so large
a range of temperature is involved The absolute compressibility of water at 0" C.

^ Ann» d> VkimU^ s^r* '6^ iom, xjexl p* 437, 1851.
^ Sulla CojHpretaibilitH dei LiqaiiHt Tontio^ 1SS4.

2—2

12 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

is assumed from Qrassi* The following are some of their results, showing a much
larger temperature effect than that obtained by Grassi: —

Temperatnre. CompressibUity per atm.

0^0 C. 0-0000503

2°-4 496

15"-9 450

49^-3 403

61^0 389

66°-2 389

77°-4 398

99^-2 409

Thus water appears to have its minimum compressibility (for low pressures) about
63° C.

My own earlier determinations^ will be given more fully below (Section VI.).
I may here quote one or two, premising that they were given with a caution (not
required, as it happens), that the pressure unit of my external gauge was somewhat
uncertain. They are trmt not average, compressibilities. See Appendix B.

At 12"0 C.

Fresh water 0-00720 (1 - 0034p)

Sea-water 0-00666 (1 - 0034^)

Ratio
1 : 0-925

At 15°o C.

Ratio
Fresh water 0-00698 (1 - 005^)

Sea-water 000645 (1 - 005p) '

In all of these the unit of pressure is one ton-weight per square inch (152-3 atm.).
The diminution of compressibility with increased pressure was evident from the com-
mencement of the investigations. I assumed, throughout, for the compressibility of

0-000386 per ton,
which, as will be seen below, is a little too small.

By direct comparison with Amagat's manometer, I have found that the pressure
unit of my external gauge is too small, but only by about 0-5 per cent. This very
slight underestimate of course does not account for the smallness of the pressure term
of the first expression above. As will be seen later, the true cause is probably to
be traced to the smallness of the piezometers which I used in my first investigations,
and to the fact that their stems were cut off " square " and dipped into mercury.
Allowing for this, it will be seen that the above estimates of compressibility agree
very fairly, in other respects, with those which I have since obtained. The sea-water
employed in the comparison with fresh water was collected about a mile and a half
off the coast at Portobello, and was therefore somewhat less dense (and more com-
pressible) than the average of ocean-water. In my later experiments, to be detailed

1 Proc. Roy. Soc, Edin, 18S3 and 1884.

LXI.]

FRESH WATER AND OF SEA- WATER.

below, the aea-water operated on was taken at a point outside the Firth of Forth,
considerably beyond the Isle of May*

Ab stated in my Report on the Pressure Errors/ &c., the unit of my external
gauge was determined by the help of Amagat's data for the oompressioii of air. As
the piezometer contaixiing the air had to be enclosed in the large gun, the record
was obtained by silvering the interior of the narrow tube into which the air was
finaUy compressed :— and the heating of the air by compresaioii, as well as the un-
certainty of the allowance for the curvature of the mercury, alone would easily
account for the underestimate. Besides, it is to be remembered that the reading <jf
the external gauge for 152 atm, ia only about 22mm.j so that a slight variation of
surface'Curvature of the mercury would of itself explain a considerable part of the
half per cent, deficit. It is, however, a matter of no consequence whatever, as reganls
the conclusions of that Report.

Buchanan, in the paper already cited, gives for the compressibility of water at
2^-5 a the value OO000516; and at 12''%5 G, 0WO0483. The empirical formula, which
is one of the main reaults of this Report (Section VIL below), extended to p = 0,
l^ves 0 0000511 and 00000480 respectively. The agreement is very remaj-kable,

Amagat^s' investigations, which were carried out by means of the electric indicator
already alluded to (which informs the experimenter of the instant at which a given
ammmt of conip^^eBsion is reached), have been extended to pressures of nearly 20 tons
weight on the square inch (3000 atm,V As a preliminary statement he gives the
average apparent compresi^ion (per atmosphere) of water at XT'^Q C* as follows: —

V From 1 to 262 atm 0 0000429,

,, 262 to 805 „ 00000379,

„ 805 to 1334 „ 00000332.

And he states that, at 3000 atmospheres, water (at this temperature) has lost about
1/10 of its original hulk. But Araagat has not yet published any determination of
the compressibility of his glass, so that the amoant of compression shown by his
experiments cannot be compared with the results of this paper. The rate of diminution
of compressibility with increased pressure, however, can be (very roughly) approximated
to ; and Amagat appears to make it somewhat less than I do. He operated on
distilled water, thoroughly deprived of air. My experiments were made on cistern
water, boiled for as short a time as possible. The analogies given in the present
paper appear to show that this difference of substance operated on may perhaps
suffice completely to explain the difference between our results,

I am indebted to a footnote in the recent great work of Mohn^" for a hint
which has led me to one of the most singular calculations as to the compressibility
of water which I have met with. As it is given in a volume" whose very rawon d*itre
is supposed to be the minutest attainable accuracy in physical deterrainations, I con-
sulted it with eagerness* The reader may imagine the disappointment with which I

1 Connie* Rendiu.tom. ctiu p. 429, 1886, and torn, ciy, p. 1159, 18S7.

* Iksi Horake NordhftVi-Ei^ped., Nordkaveta Dybder, ^c, ChmtioiLia, ISST*

* Travanx et Mimo%r£$ du Bureau Inttmaimial des Poidt ct MeMureSt tonii It. p* BSO^ P&rii, 1883.

REPORT ON SOME OF THE PHYSICAL PROPERTIES OF

[lxl

found that, as regards compressibility of water, its main feature is the amazing
empirical formula, —

501-53 - l-58995f - 0-003141113<» !

This formula represents a parabola which is everywhere convex upvrards, and thus
cannot possibly be consistent with the existence of a minimum compressibility. Instead
of representing the results of new experiments, it is based on data extracted fix>m
the old and very dubious results of Grassi (two data being wrongly quoted), Descamps,
and Wertheim, which differ in the wildest way from one another. What method of
calculation has been employed upon this chaotic group we are not told. The result
is a smug little table (D. IX.), in which no single entry can be looked upon as
trustworthy! Plate II. fig. 1, shows some of the materials, as well as the final
extract or quintessence derived fix>m them.

nL The Piezometers — Reckoning of Log. Factors — C!ompressibility of

Mercury.

The annexed sketch shows the form of piezometer employed. Six of these instru-
ments, three filled with firesh water and three with sea-water, were simultaneously
exposed to pressure. The upper end of the bulb at B was drawn out
into a very fine tube, so that the instruments could be opened and
refilled several times without appreciable change of internal volume.
They were contained in a tall copper vessel which was let down into
the pressure cylinder, and which kept them (after removal from it)
surrounded by a large quantity of the press water till they could be
taken out and measured one by one; each, after measurement, being
at once replaced in the vessel. Large supplies of water were kept in
tin vessels close to the pressure apparatus; and the temperatures of
the contents of all were observed fix)m time to time with a Kew
Standard.

The stems, AC, of the piezometers were usually from 30 to 40 cm.
in length, and the volumes of the cylindrical bulbs, CB, were each
(roughly) adjusted to the bore of the stem, so that the whole displace-
ment of the indices in the various vessels should be nearly the same
for the same pressure. At ^, on each stem, below the working portion,
the special mark of the instrument was made in dots of black enamel
{e.g. .'., .., :, &c.), so that it could be instantly recognised, and a£Sxed
to the record of the index in the laboratory book. Above this enamel
mark a short millimetre scale was etched on the glass for the purpose
of recoi-ding the volume of the water contents at each temperature before
pressure was applied. The factor by which the displacement of the index
has to be multiplied, in order to find the whole compression, varies
(slightly) with the initial bulk of the water-contents. This, in its turn,
depends on the temperature at which the experiment is made. Practi-
cally, it was found that do correction of this kind need be made in experiments on

LXI.] FRESH WATER AND OF SEA-WATER. ' 15

firesh water between 0"" and 8"" C, but for higher temperatures it rapidly came into
play. In the case of the stronger salt-solutions it was always required.

As an example of the general dimensions of the piezometers, I print here the
details of a rough preliminary measurement of one only ; and employ these merely
to exhibit the nature of the calculation for the compressibility of the contents.

Measurements for (:).

21/12/86. At temperature S"" C. (:) filled with Portobello sea-water gave for

413 of gauge (about 150 atm.) 131*2 of displacement for index
834 „ „ 300 „ 256

1254 „ „ 450 „ 373-6

Before pressure, mercury 20 mm. from enamel.

This experiment is selected because its data were taken for the approximate
lengths of the columns of mercury used to calibrate the stem of (:).

22/6/87.

Length of col. of meroory in stem. Weight, meroary and diah.

End 18 mm. from enamel 130*8 mm. 12*567 grm.

„ 45 „ 130-8 „ Dish 9387 „

„ 72 „ 130-9 „

„ 100 „ 130-9 „ Hg. 3180 „

„ 140 „ 1311 „

Another column of Hg. : —

End 18 mm. from enamel 261 mm. 15*712 grm.

„ 36 „ 2611 „ 9*387 „

„ 57 „ 2611 „

„ 75 „ 261*1 „ Hg. 6*325 „

„ 94 „ 261*3 „

Again another: —

End 18 mm. from enamel 372*6 mm. 18*407 grm.

„ 43 „ 372-4 „ Dish 9*387 „

Hg. 9*020
Weight of dish with Hg. filling bulb and stem to

599 mm. from enamel, 517*63

Weight of dish, 37-69

Hg. in piezometer, less 599 of stem, 479*94

Hg. in 599 of stem, 14*56

Whole content to enamel, 494*50

„ 20 from enamel, 494*0

REPORT ON SOME OF THE PHYSICAL PROPERTIES OF

[lxi.

The calculations are as follows, — the Gauge log will be explained in Section IV.: —
the formula is given in Appendix C, and the mantissse only are written: —

log 494 = -69373
log 130-8 =11661

(Sum)
log 318 =

(Difference)
Gauge log

(Sum)

•81034
•50243

•69373
•41664

■69209
•43856

•13065 =

: log factor for pressures near 150 atm.

•69373
•57124

•11037
•80106

■26497
•95621

•69069
•43856

•69024
•43856

• 1009R <

•19fifin Av« ARA Bfm

Hence apparent average compressibility of Portobello sea-water per atm. at 3° C. as
given by (:) on 21/12/86 is,

For first ton 11793 = log 1312

■61595 = log 413

•50198
log factor -13065

•63263

first two tons 40824

•92117

■48707
■12925

•61632

first three tons 57240

•09829

•47411
•12880

•60291

Antilog= -00004292

AntiIog = -00004134

Antilog = ^00004008

LXI.]

FRESH WATER AND OF SEA-WATER,

A few larger iustnxiDents were made for very accurate comparisons, of fresh water
and sea-water at about I ton weight per square inch, and at different temperatures.

The niercuty contents of their bulbs, &c., were over 1000 grm. The content of
250 mm. of stem in mercury was about 7 grm. ; and the log factor, for pressures about
150 atm., nearly ==0*8.

For the compressibility of mercury, the annexed form of piezometer
was employed, as in this case the recording index could not be put in
contact with the liquid to be compressed. The bulb A and stem to B
contain mercury, and so does the U-tube CD. Between B and C there
is a column of water, whose length is carefully determined. The recording
index rests on the mercury column at C Thus, obviously, its displacement
is due to

Compression of mercury jlB + Compreasion of water -BC — Compression
of voL of glass vessel from A to C

The measure men ts of this apparatus are: —

Mercury Piezometer. 25/7/87.

^g. and vessel.
Vessel

1100 gnu.
37-7 „

Weight of mercurj' whose compression is me^isured... 1062'S

Hg. and dish
Dish ...........

Weight of mercury in 210 mm, of tube BC .
Length of water column BC* . ,.,,.

14-412 ,
9386 ,

5026 ,
286 mm.

The obser\'ations made with this apparatus were as follows, the results calculated
being added, enclosed in square brackets : —

22/6/86. Kew Staudai^. 12"-75.

Alteration of Index, 17 mm.

Gauge pressure, 811

[Apparent compressibility, 0 00000102]

24/6/86.

25/C/86.

K.S. 12='3.

Index, 18'5

260

Pressure, 834

1252

[0-0O000109]

[102]

T. II.

K. a 12 4,
Index, 17
Pressure, 833
[000000098]

260
1257
[101]

18 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

23/7/87.

K. S. l°-2.

Index, 7-3

17-3

Pressure, 436

866

1264

[000000074]

[94]

[93]

25/7/87.

K. S. 16°-5.

Index. »

16-6

Pressure, 459

866

1271

[000000093]

[92]

[95]

The range of temperature is quite sufficient to allow a change of compressibility
of the water column to be noted; but the experiments unfortunately do not enable
us to assert anything as to a change in that of mercury; though, were it not for the
last set of experiments, there would appear to be a decided increase of compressibility
of mercuiy with rise of temperature. The experiments are only fairly consistent with
one another; but this was noted at the time as the fault of the index, which, of course,
tells more as the quantity measured is less. It may be as well to show how to
deduce the compressibility of mercury from them at once, assuming the requisite data
for water and for glass from subsequent parts of the Report.

Take, for instance, the first result of 25/0786. 834 of gauge is about 305 atmospherea

Also shortening of 286 mm. of water column (in glass) at I2'''3 C. by 305 atm. = 3*7 mm.

nearly: — so that the compressed mercury apparently loses about the content of 14*8 mm.

of narrow tube » bulk of 0*354 grm. Hg.

0*354
Apparent compressibility = ^^ — . ^^ = 000000109.

The average of all the normal experiments gives 0*000001 very nearly.
Add compressibility of glass =» 0*0000026,
Compressibility of mercury =0*0000036.

It is well to remember that though Grassi, working with Regnault's apparatus,
gave as the compressibility of mercury

0*00000295,
which Amaury and Descamps afterwards reduced to

000000187,
the master^ himself had previously assigned the value

000000352.
Had Qrassi's result been correct, I should have got only about half the displacements
observed; had that of Amaury and Descamps been correct, the apparent compressibility
would have had the opposite sign to that I obtained, so that the index would not
have been displaced* In such a case the construction of the instrument might have
been much simplified, for the index would have been placed in contact with the mercury
at B, and the bent part of the tube would have been unnecessary.

^ Relation des Ezp^rienoes, &e,, Mini. Acad, Sei. Parish torn. xxi. p. 461, 1S47.

lxl]

FRESH WATER AND OF SEA- WATER.

IV. AMAGAT's MANOMfeTRE A PiSTOMS LIBRE8,

The annexed sketch of the instrument (in which the large divisions shown on the
manoraetric scale correspond to decimetres), with the section given below, will enable

the reader to uoderstand its size and construction without any detailed description
beyond what is given in the instructions for setting it up. [The window FF, whose
{K^eition is nearly immaterial, occupies diflFerent positions in the sketch and in the
section.]

As already stated, the principle on which this instrument works is the same aa
that of the ManomUre Desgoffes, a sort of inverse of that of the well-known Bf^anmh

3—2

REPORT ON SOME OF THE PHYSICAL PROPERTIES OF

[lxi.

Press. In the British instrament pistons of very different sectional area are subjected
to the same pressure (that of one mass of liquid), and the total thrust on each is, of
course, proportional to its section. In the French instrument the pistons are subjected
to equcd total thrusts, being exposed respectively to fluid pressures which are inversely
proportional to their sections. The British instrument is employed for the purpose of
overcoming great resistances by means of moderate forces; the French, for that of
measuring great pressures in terms of small and easily measurable pressures.

Amagat's notable improvement consists in dispensing with the membrane, or sheet
of india-rubber, which was one of the main features of the old Desgoffes manometer,
and making his large, as well as his small, piston, fit all but tightly the hollow cylinders
in which they play: — a very thin layer of viscous fluid passing with extreme slowness
between each piston and its cylinder. The adjustment is very prompt, even in winter
when the viscosity of the fluids is greatest: — but it is made almost instantaneous by a
simple but ingenious device, which enables the operator to give the pistons a simul-
taneous motion of rotation. The following directions which accompanied the instrument
will enable the reader fully to understand its coDstruction and use. I have given an
accurate version, not a literal translation, of them : —

*' Process of setting up the Apparatus.

LXi.]

FRESH WATER AND OF SEA -WATER,

" 1, Screw in, at E, the manometer tube, and at H the regulating pump.

" 2. Pour in the layer of mercury, and on it that of castor oil. Fill the pump
with glycerine^ and insert it^ piston, taking care to exclude air-bubbles,

"3. InBert the gun -metal part K, Its bearing (at s) on the rim of the cast-iron
l^ae-piece must not be made with leather, but with a ring of india-rubber, or of very
uniform cardboard. The fixing down of this part, by means of the (six) screws^ must
be done with great exactness: — otherwise (thick as it is) it might suffer a very sUght
distortion, and the piston PP would not work in it*

" 4". After pouring in, if necesijary, some more castor oil, insert mry cautiously
the piston PP, carefully wiped, and then anointed with castor oil. To put it in, it
is to be held by means of A, which, for this purpose, is screwed into the middle of it.
Daring the insertion of the piston the hole b is left open to allow of the escape of air
and (possible) excess of castor oil. Close b by means of its screw, the piston being held
at the desired height. Take out A, and screw B into the piston in place of it,

*'5. Put on the part MM — after inserting in it the small piston pp, with its
cylinder nn— in such a way that the rod co may pass between the two studs d on
the piston PP, opposite to the opening FF.

"6* Pour a little treacle over the small piston at aa\ screw on the piece JVJV,
and fill it with glycerine ; then adjust to NN the coupling-tube of the compression
apparatus, which should be filled with glycerine or with glycerine and water,

" Observiditym.

" It is not necessary that the whole space between the mercury and the piston
PP should be filled with castor oil A layer of glycerine and water may be placed
o?er the mercury, then a thin layer of the oil. In fact, the regulating pump is full
of glycerine and water,

"'The rod cc is placed aa shown to give a simultaneous rotation to the two pistons^
so as to overcome stiction,

" It should be moved slowly, and in such a way as to exert no vertical force
upon the piston PP. It ought to he pushed by a vertical straight-edge, moved
horizontally. One can judge of the delicacy of the apparatus by the displacement of
the mercury column when the slightest vertical pressure is exerted on the rod.

** I will again call attention to the scrupulous care which must be bestowed on
the pistons and on the cylinders in which they work v- — the slightest scratch, due to
dust, would make it necessary to retouch these surfaces ; and after several retouchings
they will become too loose.

"The manometer tube, which is to be cemented into the iron piece which screws
into E, should be chosen of small enough diameter to prevent sensible change of level
of the mercury in the reservoir, and yet not so narrow as to prevent free motion of the
mercury.

"Importani Remark. — During the successive operations the large piston should
always, by means of the regulating pump, be kept at such a height that the rod cc
shall not come in contact with the wall of the opening FF, and not high enough to
make the wide lower part of the small piston come against the piece M (this, of course^

22 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [l«XI.

when the smaller of the two upper pistons is used: — that whose lower part is
thickened).

"There are two pistons pp for this manometer. The ratio of the section of the
larger to that of PP is 1/61-838, and the reading per atmosphere is 12*290 mm.

**For the smaller, the ratio of the sections is 1/277*75, and the reading per atmo-
sphere is 2*736 mm.

"The former serves for the measurement of lower pressures, up to the point at
which the oil passes visibly round the large piston. For higher pressures the latter
must be used.

" The treacle must be changed from time to time ; first, because, after a while,
some of it passes the small piston ; second, because it gradually dissolves in the glycerine,
and at last becomes hardened round the small piston, so as to make the friction too
great. The small piston and its cylinder should occasionally be cleaned with the
greatest care, and anointed with neats-foot oil."

In all my later experiments I have used exclusively the smaller of the two small
pistons. The scale which I fitted to the manometer tube was a long strip of French
plotting paper. It had shrunk slightly, so that 752'5 divisions corresponded to 750 mm.
Neglecting the difierence in the values of gravity at Lyons and at Edinburgh, the
number of scale divisions per atmosphere is 2*736 x 752*5/750 ; and its logarithm, %.e,
the Gauge Log. above spoken of, is *43856.

V. Compressibility of Glass.

Buchanan's process, already referred to, consists simply in measuring the fractional
change of length of a glass rod exposed to hydrostatic pressure, and trebling the linear
compressibility thus determined. The only difficulty it presents is that of directly
measuring the length of the rod while it is under pressure. I employed a couple of
reading microscopes, with screw-travelling adjustment, fixed to the ends of a massive
block of well-seasoned wood. This block was placed over the tube containing the
glass rod, but quite independently, — the two distinct parts of the apparatus being
supported separately on the asphalt floor of a large cellar. No tremors were per-
ceptible except when carriages passed rapidly along the wooden pavement of the street,
and even then they were not of much consequence.

The ends of the tube containing the rod must, of course, be made of glass, or
some other transparent material. In the first apparatus which I used, tubes of soda-
water-bottle glass were employed, their bore being about 0*2 inch, and the thickness
of the walls about 0*3 inch. The image of the small enamel bead at the end of the
glaw rod was very much distorted when seen through this tube, but the definition
wtm greatly improved by laying on it a concavo-plane cylindrical lens (which fitted
the external curvature), with a single drop of oil between them. I found, by trial,
timi, bad it been necessary to correct for the internal curvature also, the employment
of w'mieT'green (or OauUheria) oil as the compressing liquid would have effected the
purf^f^ completely: — the refiuctive index being almost exactly the same as that of
Ae gfeen gbuw.

LXI.] FRESH WATER AND OF SEA-WATER. 23

As the constructioD and mode of support of this apparatus did not enable us
completely to get rid of air from its interior, there were occasional explosions of a
somewhat violent character when the glass tubes gave way ; and the operators who
were not otherwise protected (as by the microscopes, for instance) were obliged to hold
pieces of thick plate glass before their eyes during the getting up of pressure. The
explosions not only shattered the thick glass tube into small fragments, but smashed
the ends of the experimental glass rod, so that a great deal of time was lost after
each. Only on one occasion did we reach a pressure of 300 atm., and an explosion
occurred before the measurement was accurately made. On these accounts, after four
days experimenting (the first being merely preliminary), we gave up working with this
apparatus: — and the results obtained by means of it cannot be regarded as wholly
satisfactory, though they agreed very well with one another.

As a sudden shock might have injured the Amagat gauge, all the pressures were
measured by the old external gauge, whose unit is now determined with accuracy.
Hence the readings are in tons-weight per square inch (152'3 atm.), which are below
called "tons" as in the vernacular of engineers. Three of us at least were engaged
in each experiment, one to apply and measure the pressure, and one at each micro-
scope. Pressure, in each group of experiments, was applied and let oflf six or seven
times in succession, readings of the two microscopes being taken before, during, and
after each application of pressure. To get rid of the possible effects of personal
equation, the observers at the microscopes changed places after each group of experi-
ments (sometimes after two groups), so that they read alternately displacements to the
right and to the left.

The values of the screw-threads were carefully verified upon one of the subdivisions
of the scale which was employed to measure the length of the experimental rod; these
subdivisions having been since tested among themselves by means of a small but very
accurate dividing-engine of Bianchi's make.

These experiments were made in July 1887, when the day temperature of the
room was nearly 20"" C. In the last two groups the compression tube was surrounded
in great part by a jacket containing water and pounded ice. We had no means of
ascertaining the average temperature of the glass rod, but it cannot have been more
than some 5 or 6 degrees above 0° C. This was done merely to ascertain whether
glass becomes less compressible or no as the temperature is lowered, not the amount
of change. The question appears to be answered in the afiirmative.

Early in the present year Mr Buchanan kindly lent me his own apparatus, which
is in three respects superior to mine. (1) A longer glass rod can be operated on.
(2) The air can be entirely got rid of from the interior, so that when the glass
tubes give way there is no explosion. (3) The glass tubes are considerably narrower
in bore (though with equal proportionate thickness), and consequently stronger. I used
my own pump and external gauge, but the necessary coupling pieces were easily
procured ; and the reading-microscopes were fastened to a longer block of seasoned
wood than before. These experiments have been made near one temperature only,
but it is about the middle of the range of temperatures in my experiments on
water and sea-water.

KKFOKT OX 80MB OF THE PHYSICAL PBOPEKTIES OF

[iJU.

It is not Dcccaaiy to ptint the details of the experimaitB in fblL I give below
put of a page of the bibotmt<MT bo(A for a single day's wofk, to show bow fiu- the
ezpetiments of one group agree with <Mie another. I porposelj choose one in which
the glass rod was somewhat diqdaced in the ajqnratos daring the oonise of the
measoremoita: —

23 2 8&

Kew Studard. 9-1 C.
(Length of glass rod. 75'75 inchesL)

Bxtenal Gwge
(Liater).

41-51

63-5

41-51

41-5 1
63-5
41-5 •

41-5 1

63-5

41-51

22 = lt(«

*»

42 )

64 ^22

42 i

42 ]

64 - 22

42 )

42 \
64 22

42 I

Lift IfienMM
(Peddie).

in.

^4570
o 475
570

0-3377
3
7

0-4571
473
572

0-3377
3
6

0-4572
473

572

03376
2
6

(Prfaie.>

0-4566

469

574

(Saed.)

0^3380

0-4575
475
574

0-8373
68
73

0-4574
475
574

^3374
70
73

Mean.

00099
00099

O0102
O0102

O0103
00103

00100
OOlOl

00105
00104

O0103
O0102

O0102

The mean thus obtained otMndded tot dosely with the mean of all the experi-
ments. Hence the avenge linear compressibility per atmosphwe (or the first ton is,
at 9=1 C

0O102
152-3 X 75-7»

whence the compreBsilHlity of glass is

OO0000265.

The two soies of experiments agreed biily with one another, and appeared to
show an increase of compiessilHlin' with rise of tonpaatnre, and a diminntion with
rise of presBore, hot these aie not made cotain. Considerably greato- ranges, bodi
of pressure and of temperKtore, are necessaiy to settle so4^ questions^

As I cannot tmst to a unit or two in the last {Jace {Le. the seventh |Jaoe of
decimals) my results for the apparent oom|»essibility of water, and as an ennr of

LXJ.]

FRESH WATER AND OF SEA-WATER.

reading of the external gauge may easily amount to 1 per cent, of the whole ton
applied, I have taken from the above experiments the number 0"0000026 as expressing
with sufficient accuracy the compressibility of the glass of the piezometers tlivoughout
the rang© of temperature 0"" to lo"" C, and of pressure from 150 to 450 atm. This
number is simply to be added to all the values of apparent compressibility. Had I
pushed the pressures farther than 450 atm*, this correction would ha%'e required
reduction, as shown in Appendix D.

VI, RfiSUMfi OF MY OWN EXPERIMENTS OK CoUFEBaSION OF WatEB

AND OF Sea- Water,

The following details are, where not othei-wise stated, taken from my laboratory'
books. I was led to make these experiments by the non-success of an attempt to
determine the exact unit of the external gauge (described in my former Report).
Not being aware of the great discover)^ of Canton (in fact, having always been accus-
tomed to speak of the compressibility of water as 1/20,000 per atm<), I imagined that
I could verify my gauge by comparing^ on a water piezometer, the effects of a
pressure measured by the gauge with those produced by a measured depth of sea-
water, without any reference to the temperatures at which measuremeuts were made;
provided, of course, that these were not very different. The result is described in the
following extract': —

**To test by an independent process the accuracy of the unit of my pressure
gauge, on which the estimated corrections for the Challenger deep-sea thermometers
depend, it was arranged that H.M,S. * Triton' should visit duiing the autumn a region
in which soundings of at least a mile and a half could be had, A set of mano-
meters, filled with pure water, and recording by the washing away of part of a very
thin film of silver, were employed. They were all previously tested, up to about 2^
tons weight per square inch, in my large apparatus. As I was otherwise engaged.
Professor Cbrystal and Mr Murray kindly undertook the deep-sea observations ; and
I have recently begun the work of reducing them.

'*Tbe first rough reductions seemed to show that my pressure unit must be
somewhere about 20 per cent, too small. As this was the all but unanimous verdict
of fifteen separate inst rumen ts, the survivors of two dozen sent out, I immediately
repeated the test of my unit by means of Amagat*s observed values of the volume
of air at very high pressures* The result was to confirm, within 1 per cent., the
accuracy of the former estimate of the unit of my gauge. I then had the mano-
meters resilvered, and again tested in the compression apparatus. The results were
now only about o per cent, different from those obtained in the * Triton.' There
could be no essential difference between the two sets of home experiments, except
that the first set was made in July, the second in November, — while the tempera-
tures at which the greatest compressions were reached in the * Triton * were at least
3° C. lower than those in the latter set. Hence it seems absolutely certain that

^ Ftqc^ MffU' ^^' EditL, ToL xu, pp. 15, m, 1882.

T. n.

26 RKPOBT ON 80MB OF THB PHYSICAL PBOPSBTIE8 OF [lXI.

wmter becomes ooosideFaUy more compressible as its temperatme is lowered, at kast
as &r as 3° C (the ' Triton ' tempoatore). This seems to be ccmnected with die
lowering by presBore of the maTimnm density point of water ^ and I intend to w<^
it oat. It is dear that in fntore triak of soch manomet^s some liquid less anomaloas
than water mnst be employed.

"Another preliminaiy result, by no means so marked as the above, and possibly
to be ex|dained away, is that by doabling (at any ooe temperatore) a high pressore
we obtain somewhat less than doable the compression. This, however, may be dae
to the special constraction of the manometer, which renders the exact determination
<^ the fiducial point almost impaesible.**

In the winter of 1882 and the succeeding spring, I spent a great deal of time
in trying to get definite results from the records of the "Tritcm" trials, and in
making farther experiments on those of the specially prepared fnezometers which had
not been broken or left at the bottom of the sea. But this work led to no result
on which I could rely. I then directly attacked the {woblem of the compressibility
of water at diflRanent temperatures and pressures, having once more verified the unit
of my pressure gauge by comparison with Amagat's data for air. Results for ooe
tempoature were published, as below, in the Proe. Boy. Soe. BdxK^ vd. xn. f^ 223,
224, 1883. [The mercury content of the bulbs of the new piezometers was about
200 grm., and that of 100 nmi. of stem about 2*6 grm.]

''The apparatus employed was of a very simple character, similar to that which
was used last autunm in the 'Triton.'

"It consisted of a narrow and a wide glass tube, forming as it were the stem
and bulb of a large air-thermometer. The stem was made of the most uniform
tube which could be procured, and was very accurately gauged; and the weij^t of the
ccmtent of the bulb in mercury was determined. Thus the fraction of the whole
cmtait, corresponding to that of one millimetre of the tube, was found.

"This apparatus had the interior of the narrow tube very carefully silvered; and
while the wlnde, filled with the liquid to be examined, was at the temperature of
the water in the ccMupression apparatus, the open end was inserted into a small
vessel containing clean mercury. Four instruments of this kind were used, all made
of the same kind of glas& [They were numbered, as in the headings of the ocJumns
below, 1, 2, 3, 4, respectively. 20 6 88.]

"The following are the calculated apparent average changes of volume per ton
weight of pressure per square inch (t>. about 150 atmospheres) : —

Fresh Watkr, at 12 C.

SBBie

Meui.

000670

•

663

666

0O0667

0O065T

•

646

656

0D0653

2^5

01)0651

650

640

648

OO0647

&O0641

633

636

OO0636

KoiB. — Tlw first tvo cr|Wfiinent» vith No. S Iftiled in eonHqnenoe of a Med in tiie sihcriBg.
1 [The imsoo for this remnik wiU b» seen in tht saeond eztnel in Seetion ^SIL hilov. S0^SS8l]

lxl] fresh water and of sea-water. 27

The compressibility of glass was not directly determined. It may be taken as approxi-
mately 0'000386 per ton weight per square inch.

"From these data, which are fairly consistent with one another, we find the
following value of the true compressibility of water per ton, the unit for pressure (jp)
being 1 ton-weight per square inch, and the temperature 12° C,

00072 (1 - 0034jp) ;
showing a steady falling off &om Hooke's Law.

Sea- Water, at 12° C.

Msore

Mean.

0-00606

611

615

627

000615

000595

607

598

601

000600

2-5

000600

600

694

590

0-00594

000588

593

586

0-00588

Note. — The sea- water employed was ooUeoted about 1} miles off the coast at PortobeUo.

These give, with the same correction for glass as before, the expression

0 00666(1 -0034P).

Hence the relative compressibilities of sea and &esh water are about

0-925 ;

while the rate of diminution by increase of pressure is sensibly the same (3^ per
cent per ton weight per square inch) for both.

" With the same apparatus I examined alcohol, of sp. gr. 0*83 at 20° C.

Alcohol, at 12° C.

Pressnie

Mean.

001202

1193

•

0 01200

2-5

0-01040

1052

1050

1056

0-01049

0-01043

1050

1043

1058

001048

These experiments were not so satisfactory as those with water. There are peculiar
difficulties with the silver film. I therefore make no- definite conclusion till I have
an opportunity of repeating them."

It will be observed that the diminution of compressibility as the pressure is
raised is here brought out unequivocally for all the three liquids examined.

In the course of another year I had managed to obtain similar results for a
range of temperature of about 9° C. They were described in Proc. Roy. Soc. Edin.,
vol. XIL pp. 757, 758, 1884, as follows : —

" I had hoped to be able, during the winter, to extend my observations to
temperatures near the freezing point, but the lowest temperature reached by the large
compression apparatus was 6°'3 C. ; while the highest is (at present) about 15° C.

4—2

28 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

From so small a range nothing can be expected as to the temperature effect on the
<5ompre8sibility of water, further than an approximation to its values through that
range.

" The following table gives the mean values of the average compression per ton
weight per square inch: —

Pressare in Tons

6°-3 C.

0-00704

692

684

672

...

y^-e

. • .

682

...

670

660

ll°-3

684

670

. . •

654

• • •

13°1

• • •

666

• • ■

648

• • •

15°-2

673

654

• « •

633

■ • •

637

" These are all fairly represented by the expression

000743 - 0000038^ - OOOOlSp,

where t is the temperature centigrade, and p the pressure in tons weight per square
inch. This, of course, cannot be the true formula, but it is sufficient for ordinary
purposes within the limits of temperature and pressure above stated. It represents
the value of

Vq — V
pVo

"With a new set of compression apparatus, very much larger and more sensitive
than those employed in the above research, I have just obtained the following mean
values for the single temperature 15°'5 C. : —

Pressure in Tons 1 1^ 2 8

Fresh water 0K)0678 663 657 638

Sea-water 000627 618 609 593

" These are the values of -^ , and they give, for the true compressibility

(1 dv\*
— -j-j at any pressure, and temperature 15°*5 C, the formulae.

Fresh water 0*00698 (1 - 005p)

Sea-water 0"00645 (1 - 0-05jp)

"The ratio is 0*925, %,e. the compressibility of sea-water at the above temperature
is only 92*5 per cent, of that of fresh water."

The new and larger piezometers referred to were made when Mr Murray requested
me to write this Report. They are those whose form and dimensions have been
detailed in Section III. above. The former piezometers had no capsule containing
mercury, but had the stem simply cut off flat at the end, and when filled with
water were merely dipped in mercury. I had felt that to this was probably due

* [See Appendix B to this Report.]

lxl]

FRESH WATEE AND OF SEA- WATER.

the fact that my experiments gave a value of the compressibility at 0"* C. somewhat
smaller than that usually accepted It will be seen that the very first data givea
by the new instrumeuts at once tended to set this matter right. For while the
formula representing the results of the smaller instruments gave the compression of
water at l5^'o Q as 0W678 for one ton weight per square inch, that for those of
the new instruments gave 000698, t.e, about l/34th more, which is much nearer to
the result of my later experiments.

For two T^inters after this period the apparatus was kept in working order in
the hope that I niight be enabled to employ temperatures between 6"" and 0"" C-
But a single day's work at V7 G, and a few days at temperatures between 3" and
5" C, were all I got Hence the reason for procuring the smaller compression
apparatus, as stated in Section I. But, as yet, my measurements of pressure were
not satisfactory.

In the spring of 1886 I obtained the Amagat gauge, and after a careful com-
parative trial determined to employ exclusively the lesser of the two small pistons*
Some time was spent upon a comparison of the indications of this instrument with
tbose of the external gauge, with the result that single indications of the latter
could not be trusted within about 1 per cenL, though the mean of a number of
observations was occasionally very close to the truth. I therefore put aside all the
compre-ssion observations already made^ and commenced afresh with the same piezo-
meters as before, and witfi the Amagat gauge exclusively,

In the summer of 1886 I obtained a long series of determinations at about
11**8 C, and others at l^''^ and lo"" C. In December of the same year I worked
for a long time between 3" and S^'o C. All of these were with the large Fraser gun.

In June 1887, with the new compression apparatus, I secured numerous deter-
minations at 0''4 C.

In July the piezometers were filled with solutioDs of salt of various strengths,
and examined at temperatures near 19° C, and 1° C» In November these were again
examined, this time in the large gun at about 9^^ C* ; and the piezometers were again
filled, some with fresh water and some with sea- water.

During the winter complete series of observations in the large gun were obtained
at about 7^ 5°, S''^2, 2"'3, Vl ; and; finally (on March 16, 1888). at 0°^5 C.

The piezometers were^ once more, filled with the salt solutions, as I considered
that I had obtained sufficient data for fresh water and for sea- water ; except in the
one important particular of the exact values of the ratio of their compressibilities at
one or two definite temperatures and pressures.

These were finally obtained in May and June 1888, with piezometers considerably
latger and more delicate than the former set.

Final Results and Empirical Formula for Fresh Water,

Although my readings and calculations were throughout carried to four significant
figures, I soon found that (for reasons already sufficiently given in Section I.) only three
of these could be trusted even in the average of a number of successive experiments.

30 REPORT ON SOME OF THE PHYSICAL PROPERTIES OP [lXI.

and that the third might occasionally (especially with sea-water) err by an entire
unit or two ; at most ^ per cent, of the whole quantity measured. Of course, now
and then there occurred results so inconsistent with the rest as to indicate, without
any doubt, a displacement of the index by upward or (more frequently) downward
currents.

This was made obvious by comparison of the indications of any one piezometer
in successive experiments at the same temperature and pressure; but it was even
more easily seen in the relative behaviour of a number of piezometers which were
simultaneously exposed to exactly the same temperature and pressure several times in
succession. A single page of my laboratory book, taken at random, sufficiently illus-
trates this. To avoid confusion, I give the records of two of the ordinary instruments
(with fresh water) alone, leaving out the records of those with sea- water, and I insert
[in brackets] the pressures and the average apparent compressibilities calculated from
the data. The water employed was that of the ordinary supply of Edinbuigh, and
was boiled, for a short time only, to expel air: —

11.

III.

IV.

23/7/86.

KG.
25-0
46-4
25-0

A.G.

419

2 c.
28-0

.-. 136-2

[Pressure 0 983 tons]
[4333]

K. S. (in gun) 14''-9 C.

251
47-0
251

423

K. S. 15°

280

.-. 137-7
.. 122-5

[0-993]
[4339]
[4342]

251
68-1
251

841

K. S. 15°

56-0

.'. 269-0
.. 256-6

[1-992]
[4218]
[4214]

25-2
68-4
25-2

844

56-0.

.-. 269-8
.. 2581

[20]
[4216]

[4224]

25-2
90-0
25-5

8
1261

K S. 15°

85-0

.-. 393-7
.. 376-9

[2-997]
[4092]
[4116]

25-6

900

1263

25-5

LXI.] FRESH WATER AND OF SEA-WATER. 31

[3002]

850 /. 394-4 [4093]

.. 376-9 [4110]

The left-hand column gives the readings of the external gauge, the next those of
Amagat's gauge, before, during, and after the application of pressure. The third gives
the pressure as read by one of the internal gauges described in my previous Report.
The fourth column gives the readings of the two piezometers selected ; the fifth the
pressure (in tons) for each experiment, and the compressibility calculated. The latter
numbers are multiplied by 10®.

Notice that, in the first experiment (..) failed to give a reading. Also in the
fifth and sixth the indications of the two instruments do not agree very closely. The
character of the results, however, points apparently to an error in gauging one or other
of the instruments. It was the unavoidable occurrence of defects of these kinds that
led me to make so many determinations at each temperature and pressure selected.
The above specimen contains less than 1 per cent, of my results for fresh water, and
I obtained at least as many reduced observations on sea-water.

To obtain an approximate formula for the full reduction of the observations, I first
made a graphic representation, on a large scale, of the results for different pressures
at each of four temperatures, adding the compressibility of glass as given in Section VI.
above. From this I easily found that the average compressibility for 2 tons pressure
(at any one temperature) is somewhat less than half the sum of those for 1 and for
3 tona Thus the average compressibility through any range of pressure falls off more
and more slowly as that range is greater. And, within the limits of my experiments, I
found that this relation between pressure and average compressibility could be fairly
well represented by a portion of a rectangular hyperbola, with asymptotes coincident
with and perpendicular to the axis of pressure; Hence at any one temperature (within
the range I was enabled to work in), if Vq be the volume of fresh water at one atmo-
sphere, V that under an additional pressure p, we have

Vo — V A

pvo n -{-p

very nearly, A and 11 being quantities to be found.

I had two special reasons (besides, of course, its adaptability to the plotted curve)
for selecting this form of expression. Firsty it cannot increase or diminish indefinitely
for increasing positive values of p, and is therefore much to be preferred in a question
of this kind to the common mode of representation by ascending powers of the
variable, such as two or more terms of

Bo + B,p + B,p^'^&c,

or the absolutely indefensible expression, too often seen in inquiries connected with
this and similar questions,

REPORT ON SOME OF THE PHYSICAL PROPERTIES OF

[lxi.

Second, it becomes zero when p is infinite, as it ought certainly to do in this physical
problem. It appeared also to suggest a theoretical interpretation. But I will say no
more about. this for the present, as it is simply a matter of speculation. See the latter
part of Section X., below. But there is a grave objection to this form of expression,
in the fact that small percentage changes in the data involve large percentage changes
in A and 11, though not in the ratio AjTl, This objection, however, does not apply to
the use of it in the calculations preliminary to the full reduction, as in them it is -4/11
only which is required.

Next, on calculating from my data the values of A and 11 for different temperatures,
I found that, within the recognised limits of errors of the observations, 11 might be
treated as sensibly constant. Thus I was enabled easily to make graphic representa-
tions of the average compressibility at each pressure, in terms of temperature. Again
I obtained curves which could, for a first trial at least, be treated as small portions
of rectangular hjrperbolas, with the axis of temperature as one asymptote. Hence

T + t'

where T is a constant; and B also may for a time be treated as constant.
Thus I arrived at the empirical expression

(n+p)(T+t)

whose simplicity is remarkable, and which lends itself very readily to calculation. As I
required it for a temporary purpose only, I found values of the constants by a tentative
process; which led to the result

0-28
(36 +/>) (150 + 0'

This gives the average compressibiliti/ per atmosphere throughout the range of additional
pressure p, the latter being measured in tons' weight per square inch.

The following brief table shows with what approximation the (unreduced) experi-
mental results (multiplied by 10') are represented by this formula. The nearest integer
is taken in the third place: —

1 ton.

2 tons.

8 tons.

Temp.

Obs.

Calo.

Oba. Cala

OlB. Calo.

0°'4

503

489 490

-1

477 477

3°-2

492

494

-2

479 481

-2

466 469

-3

ll-'S

467

468

-1

454 455

-1

441 444

-3

is^-o

459

448 447

+ 1

436 435

+ 1

The agreement is tolerably close, so that the empirical formula may be used, without
any great error, in the hydrostatic equations, so long as the temperatures and pressures
concerned are such as commonly occur in lakes.

But the columns of differences show that the form of the formula is not suitable.
The pressure fiEu^tor seems appropriate, but it is clear that, at any one pressure, the

.XL]

PEESH WATER AND OF BEA-WATER-

curve representiog the compression in terms of the temperature baa greater curvature
than the formula assigns. Still the formula amply suffices for the reduction of the
obser\^at!ons of any one group when the pressures or temperatures were not precisely the
same in ail. It was, however, not much required, for the pressure could be adjusted
with considerable accuracy, and (especially when the large gun was used) the changes
of temperature were very slow*

The next step was to enter, aa shown in Plate IL fig. 3, ail the results obtained
from the various piezometers at each definite temperature and pressure, with the view
i>f selecting the most probable value. The amount of discordance was in all cases very
much the same as that shown in the plate for the series of experiments at two tons'
] pressure and the one temperature 5° C. It will be observed that the extreme limits of
divergence from the mean ai^e not more than about two units in the third significant
place. For a pressure of one ton this corresponds to about half a millimetre in the
[Kisition of the indices, so that after what has been said about their peculiarities of
behaviour it may obviously be treated as unavoidable error. Thus the ordinary process
4)f taking means is applicable, unless the observations themselves show some peculiarity
which forbids the use of this method.

All the results of observations made up to June 1887 (with the help of the Amagat
gauge) having been treated in this way, the following mean values of apparent average
compressibility (multiplied by 10*) were deduced from them : —

Apparmyt Cotupremlnlity of Cistern Water^ bailed for a short time.

Temp. C.

1 Con.

0'-4

4770

3°-2

4670

3'4

4671

ll°-8

4416

l*°-2

4330

H°-4

4344

is'-o

4338

3 tonn.

4617
4527
4521
4276
4220
4217
4219

3 tona.

4510
4402
4395
4163
4115
4105
4102

[I think it extremely probable that the small irregularities amoag the last three
immbers in each pressure column may be due to want of uniformity of temperature
throughout the column of water in the pressure chamber. The day-temperature of
the cellai' is, in summer, always a good deal above that at night, so that in the
forenoon (when the experiments were made) the gun and its contents were steadily
growing warmer. Thus the column of water was not at a uniform temperature. The
assumed temperature was the meau of the readings before the vessel containing the
piezometers was inserted, and after it was taken out. While it was in the chamber,
the contents could not be properly stirred except by raising and depressing the vessel
itself,]

The points thus determined were laid down (marked with a *) aa in Plate I.,
aud smooth curves were dmwn l-^erd manu among them. From these curves the

T- II. 5

REPORT ON SOMB OF THE PHYSICAL PROPBRTIES OF

[lxi.

following values were taken at intervals at 5'' for the sake of ease of calculation, 260
being added to each for the compressibility of glass : —

0°

10°

16°

1 ton

5044

4874

4723

4594

2 tons

4898

4733

4584

4466

3 tons

4776

4608

4468

4360

The fact that water has a temperature of minimum compressibility led me to try to
represent these numbers by a separate parabolic formula for each pressure. The follow-
ing were easily found : —

504 - 3'60t + 004^ j

490-3-65« + 005««[ (A),

478-3-70« + 006«M

for 1, 2, and 3 tons respectively. [The terms independent of t belong to the formula
620 — 17jp+|)*. This will be made use of in future sections.] The utmost difference
between the results of these formulae and the numbers from which they were obtained
is less than 1/lOth per cent. No closer approximation could be desired, much less
expected, especially when we consider the way in which the « points (on which the
whole depends) were themselves obtained. These are represented as follows: —

Obfl. Calo.

603 502-5

487-7 488-5

477 476-5

8°-2
ObB. Calo.

493 493
478-7 479
466-2 466-8

IP'S
Obs. Calo.

467-5 467-2
453-6 453-9
442-3 442-7

14«>-4

Obs. Calo.

460-4 460-5

447-7 447-8

436-5 437-1

16°0
Obs. Calo.
459-8 459
447-9 446-5
436-2 436

In one instance only does the difference reach unit in the third significant place. [It
must be remembered that all these numbers commence with the fifth digit after the
decimal point.]

In spite of some remarks above as to uncertainty about temperature, I am con-
vinced that the mode of experimenting employed is calculated to insure considerably
greater accuracy in the comparison of compressibilities at different temperatures for
any one pressure, than in that of compressibilities for different pressures at any one
temperature. The displacement of the indices by the expanding water is likely to be
more serious the higher the pressure, as the difficulty of effecting the relief quietly is
much greater. Probably all the values for the higher pressures are a little too small
for this reason.

The results given above are represented with a fair degree of accuracy by the
simple formula

0-001863/
36 +p V

3t_
400 "*■ 10,000,

;)•

which will amply suffice for ordinary purposes In this form, however, some small but
highly expressive and apparently important features of the formulae (A) for the separate

LXI.] FRESH WATER AND OF SEA- WATER. 35

pressures are, of course, lost. The statement above, as to the greater uncertainty of
the values the higher the pressure, renders it probable that, in the pressure factor in
this formula, both the constants ought to be somewhat larger. It is clear that very
small changes in the relative values of the compressions for 1, 2, and 3 tons would
loake great changes in these constants. In fact, an error of 1 per cent, at 3 tons
involves an error of some twenty per cent, nearly, in each of the constants of the
pressure fietctor.

Again, this last formula would give, for aU pressures, minimum compressibility at
about 37° C; while the former three give 45° C. at 1 ton, 86°-5 at 2. and 80°'8 at
3 tons: — these minima being 423, 423'4, and 421 respectively.

If we venture to extend the formula (A) to atmospheric pressure, we are led to

520-3-65« + 003e*.

I have already shown ^ that this is in close accordance with Buchanan's results at 2''*5
and 12°'5 C. Buchanan's pressure unit is thoroughly trustworthy; for it was deter-
mined by letting down the piezometer, with a Challenger thermometer attached, to a
measured depth in the ocean. It would thus appear that the extension of my formulae
to low pressures is justified by the result to which it leads.

This formula gives 415 for the minimum compressibility of water at low pressures,
the corresponding temperature being about 60° C. This accords remarkably with the
determination made by Pagliani and Vincentini, who discovered it, and placed it at
63° C.

On Plate II. I have exhibited graphically a number of known determinations of
the compressibility of water for very low pressures at diflferent temperatures. The line
marked Hypothetical is drawn from the formula above, the authors of the others are
named in the plate. It will be seen at a glance that, if Pagliani and Vincentini
had taken Grassi's value of the compressibility of water at 1°*5 C, instead of that at
0° C, as their single assumption, their curve would have coincided almost exactly with
my Hypothetical curve !

So far matters seemed to have gone smoothly enough. But when I came to reduce
the observations made since June 1887, I found "^ that they gave a result diflfering,
slightly indeed but in a consistently characteristic manner, from that already given.
The processes of reduction were carried out precisely as before; and the points deter-
mined by the second series of observations are inserted in Plate I., marked with a 0.
Curves drawn through them as before are now seen to be parallel to the former curves,
but not coincident with them. And the amount of deviation steadily diminishes from
the lowest to the highest pressure. These curves, of course, are very closely represented
by the formulae (A) above, provided the first terms be made 499, 488, 477, respectively,
i.e. provided 5, 2, and 1 be subtracted from the numbers for 1, 2, and 3 tons re-
spectively. Thus, while the amount of the compressibility is reduced, it is made to
depend on temperature precisely as before, but the way in which it depends on
pressure is altered. The rate of diminution of compressibility with increase of pressure

* See p. 18, above.

5—2

REPOBT ON 80HE OP THE PHySICAL PEOPERTrES OF

[lxl

is now made constant at any one temperattire, instead of becoming slowly less as the
pressure is increased. This is incompatible with the results of all of the first series
of experiments. The total amount of the compressibility is likewise dimiuished, by
1 per cent, at 1 ton, by 04 per cent, at 2 tons, and by 02 per cent, at B tons.

Small as these differences are, their regularity struck me as very remarkablep and as
pointing definitely to some difference of conditions between the two «ets of experiments.
Now there were undoubtedly many circumstances in which the series of experiments
differed : —

First The observers were not the same. All the readings in the first series were
made by myself; but (in consequence of an accident which prevented me from working
in the cellar) I was unable to take part in the second series, and the readings for
it were all made by Mr Dickson, Thus there may be a difference, of personal equation,
in the mode of applying the scale to the stem of the piezometer, or in the final
adjustment of the manometer. Such an explanation is quite in accordance with the
results, as a constant difference of reading would tell most when the whole quantity
measured is least, i.e. at the lowest pressure. But a difference of a full mi!limetre
in the piezometer readings may be dismissed as extremely improbable.

Second, It is possible that, during the second series of experiments, less care niay
have been taken than in the first series to let off the pressure with extreme slowness.
Thus the indices may have been slightly washed down, and the record of compression
rendered too small. Even with the greatest care, this undoubtedly occurred in some, at
least, of the experiments of the first series; and the screw-tap may have been altered
for the worse during the second series.

Third, It is recorded in the laboratory book that, during the second series of
observations (which were made for the most part in the exceptionally cold weather of
last spring) the oil and treacle in the manometer had become very viscous, so that
it was difficult to make the pistons rotate. As artificial cooling, of the pressure apparatus
alone, was employed in the first series, this objection does not apply to it. A constant
zero error of 4 mm. only in the gauge would fully explain the discrepancy. And there
was another cause which may have tended to produce this result, viz. the oxidation of
the mercury in the manometric column, which had soiled the interior of the lower part
of the tube, and thus made it very difficult to read the zero,

FQurth. The piezometers had been twice refilled, and of course slightly altered in
content, between the two series, and the hair-indices had necessarily been changed.
The former cause could have produced no measurable effect ; but if the indices were
all somewhat stifier to move in the second series than in the first, the discrepance
might be fuDy accounted for.

Fifth, Between the two series all the piezometers had, for several months, been
filled with strong salt-solutions. Imperfect washing out of these solutions may have
had the effect of rendering the second series a set of experiments on water very
slightly salt-

SimtL To make my observations applicable to natural phenomena, I purposely
did not employ distilled water. The ordinary water supply of Edinburgh ia of very
fair quality, and I took care that it should not be boiled longer than was absolutely

LKI.]

*RB9H WATER AND OF SKA- WATER"

necesBary to prevent air- bubbles from forming in the piezometers. But it comei» from

dififerent sources, and is supplied as a mixture containing these in proportions which
vary from time to time. From this cause also the substance operated upon may have
been slightly different in the two aeries of experiments.

As will be seen in next section, I have obtained direct proof that the tirst seines of
observations is to be preferred to the second,— though I have not been able to ascertain
definitely which of the above causes may have been most efficient in producing the
discrepancy.

It will be observed that this discussion has nothing to do with the important
question, Does the compressibility of water diminish from the very first as the pressure
increases, as was asserted by Perkins ? The first and rudest of my experiments sufficed
to answer this definitely in the affirmative ; though the contrary opinion has been
confidently advanced, and is very generally held to this day.

The discussion deals with a much more refined and difficult question, viz. Is the
diminution of average compressibility simply proportional to the pressure for the fixet
few hundred atmospheres, or does the compreasibility fall off more slowly than that
proportion would indicat6| as the pressure is raised ?

VIII. Reductions, Results, and Formul-*: foe Sea- Water.

As already stated, thi'ee of the six piezometers employed were filled with fresh
water and three with sea-water, so that simultaneous observations wei*e made on the
two substances. The accordance among the various observations made with sea-water,
at any one temperature and pressure, was not so good as it was with fresh water;
especially when the smaller compression apparatus was used. There is some curious
action of salt upon the hairs attached to the indices, which has the effect of rendering
them too loose, however stiffly they may originally have fitted the tube. Treating the
obeervatious of the first series exactly as described in the preceding section, I obtained
the points marked • in Plate I. Drawing smooth curves through these, I obtained
parabolic formulae for the apparent compressibility. These gave the following results
when compared with the data from observation ; —

Apparent Compressibility of Sea- Water,

ton.

tons.

3 tOlM.

Oba.

Cale.

Oba.

rule.

Oba. Calc.

o-i

435

420

410 410

3'-0

427

413

402-5 403

ir-8

4r04

404

392

383-5 384

U°-2

398

399

389

388

380 380

IS'O

398

397

387

378 378

REPORT ON SOME OF THE PHYSICAL PROPERTIES OF

[lxl

Adding the correction for glass, the formulaB became, for 1, 2, and 3 tons
respectively —

462 - 8-2W + 0-04^ )

447-5 -3-05e + 0-06<«i (B),

437-6- 2-95^ + 0-05«» I

which may be compared with (A) for fresh water; and which may be approximately
expressed in the form (very nearly correct for jp = 2) —

0-00179
38+jp

V 150"^ 10,000/'

with sufficient accuracy for most purposes of calculation.

Of course it is easy to deduce from formulsB (B) the points of minimum com-
pressibility, etc., for diflferent pressures; but the data are scarcely accurate enough to
warrant such a proceeding. We may, however, extend the formuUe tentatively to the
case of very low pressures, for which we obtain

481-3-4^ + 0-03<«.

[The term independent of t in the formulae (B) is of the form

481- 21-25;) +2-25p«.]

The second series of observations gave, when reduced, the points marked 0 on
the plate. The curves which I have drawn, and which evidently suit them very
closely, are parallel respectively to the curves drawn through the ♦ points. The
interval between them is throughout about 7 for 1 ton, 4 for 2 tons, and 3 for 3 tons,
which must be subtracted fix)m the first terms of (B) respectively. The corresponding
intervals for the fresh water curves in the two series were 6, 2, 1. The differences
of corresponding intervals between the sets of curves are 2, 2, 2; the same for all the
groups of four curves each.

This seems to throw light on the question raised in last section, and to show
that the main cause of the discrepancy between the first and second series of obser-
vations is not due to a difference in the substance operated on. The constant
difference of the differences is due to such a cause, being at once traceable to the
fact that the sea-water put into some of the piezometers for the second series of
experiments was taken from the same Winchester quart bottle as was that with which
they had been filled two years before. During these two years the sea-water had
probably, by evaporation, become slightly stronger, and, therefore, less compressible.
The change of compressibility is less than 0*5 per cent, of the whole, and is there-
fore practically (as it is in the third significant figure) the same for all three pressures.
If we now look back to the suggested explanations in last section, we see that the
above remarks entirely dispose of the fifth and sixth so far as fresh water is con-
cerned, though the sixth, in a modified form, has to do in part with the discrepancy
between the two series of observations on sea-water.

To decide between the two series I made a new set of observations, employing
the two pieaometers of lazge < ^^^^ "^okeii of at the end of Section IIL These

[0-997 ton]

310-9

[4466]

234-7

[4080]

;•

1260

[4463]

LXI.] FRESH WATER AND OF SEA- WATER. 39

are called Mi and M^, On the first day of experimenting Mi held sea-water from a
Winchester quart filled at the same time with the first, but which had remained
unopened, if, had fresh water. On the second day M^ held sea-water, and Mi fresh
water. The object of this was to discover, if such existed, errors in the calibration
of the piezometers, and then to eliminate them by a process akin to that of weigh-
ing with a false balance.

One of the ordinary piezometers (•.*), filled with fresh water, was associated with
the others as a check. I quote the results of one experiment only, made on the
second day : —

5/6/88

5 9°-4

422
5

Thus we have the following comparison of estimates of true average compressibility
for the first additional ton : —

Fresh Water. Sea- Water.

I 1st Series 474 434

2nd „ 469 427

New „ 473 434

A few of the experiments were not thoroughly decisive ; none were in favour of
the second series. This seems (so far as the first ton is concerned) to settle the
question in favour of the first series.

The formute (A) and (B) may therefore, for one ton at least, be regarded as
approximations to the truth, probably about as close as the apparatus and the method
employed are capable of furnishing.

They show that the ratio of compressibilities of sea-water and fresh water varies
but little from

0-92

throughout a range of temperature fix)m 0** to 15° C.

[The doubts as to the behaviour of the indices, which have been more than
once alluded to above, have just led me to make a series of experiments (at one
temperature but at diflferent pressures) by the help of the silvering process. The
results with fresh water were not much more concordant than when the hair-indices
were used. When means were taken, exactly as before, it was found that the results
for 1 ton were almost identical with the former. For 2 tons the average value was
usually greater than before by a unit (and in some cases two units) in the third
place. For 3 tons it was also greater, but now by one or two (and sometimes three)
units. Hence it is probable that the hair-indices do behave as I suspected, but that
the effect is small, — not at the worst {i,e. at the highest pressure) more than about
0'5 per cent, of the mean value found. With sea-water there was a complex reaction,
which made it difiicult to read the indications of the silver film. The ratio of the

40 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

true compressibilities of sea-water and fresh water was now found to be about 0*925,
the value which I gave from my earliest experiments. 30/6/88.]

Dr Gibson has furnished me with the following data regarding specimens of aea-
water taken from two of the Winchester quarts filled oflf the Isle of May. One of
these had remained unopened; the other had been often opened, and not closed with
special care. These correspond (at least closely) to the materials used in the first
and second series of experiments respectively: —

Density.

er c. 12'' c.

1-026745 1-025834

1-027405 1-026462

Taking the reciprocals in the last three columns, we have

Volume.

Peroentage of CI.

o°c.

1-8649

1-027286

1-9094

1-027941

0»C.

6°

0-973439

0-973951

0-974816

0-972818

0-973326

0-974220

Expressing these volumes as parabolic functions of the temperature, we find, for the
maximum density points, — 5°*7 and - 4°*9 respectively.

IX. Compressibility, Expansibility, etc., of Solutions of

Common Salt.

This part of the inquiry was a natural extension of the observations on sea-water,
but it was also in part suggested by the fact that an admixture of salt with water
produces effects very similar to those of pressure. Thus it appeared to me that an
investigation of the compressibility of brines of various strengths might throw some
light on the nature of solution; and also on the question of the internal pressure
of liquids, which (in some theories of capillary forces) is regarded as a very large
quantity.

The solutions experimented on contained, roughly, 4, 9, 13'4, and 17*6 per cent,
of common salt. The piezometers used for the experiments already described were
filled with these solutions in July 1887 ; one, for comparison, being left full of
firesh water. I obtained a large number of results at temperatures about I"*, Q"", and
19"* C, and at 1, 2, and 3 tons weight per square inch. Unfortunately these were
still more discordant than those made with sea- water; so much so, in fact, that an
enor of 1 or occasionally even 2 per cent, was not by any means uncommon. However,
bj 'pknUing all the observations exactly as described in the two last sections, I found
Aftfc thej oonld be fairly represented by the curves shown in Plate L In most cases
of the three points for each curve were fairly determinate; one of these
MM, within a degree or so of 10** C. For this was obtained by

lxl] fresh water and of sea- water. 41

experiments in the large gun, where the difficulty of relieving the pressure without
jerks is much less than in the smaller apparatus. Of the general accuracy of these
curves I have no doubt. Thus, for instance, it is certain that the compressibility at
any one temperature and pressure diminishes rapidly as the percentage of salt increases.
And the rate at which the compressibility (for any one range of pressure) diminishes
as temperature increases, becomes rapidly less as the solution is stronger. My obser-
vations do not enable me to settle the more delicate question of the variation of
the rate at which the compressibility (at any one temperature) falls oflf with increase
of pressure in the various solutions. For the limits of error in the various deter-
minations, especially with the more nearly saturated solutions, are quite sufficient to
mask an eflfect of this kind unless it were considerable. An attempt, however, will
be made in next Section.

There is little to be gained by putting the results of the inquiry in a tabular
form ; for they can be obtained from the plate quite as accurately as is warranted
by the limits of uncertainty of the experiments. See p. 44.

I am indebted to Dr Gibson for the following determinations, which have a high
value of their own as showing the connection between the strength of a salt-solution
and its expansibility: —

Density.

Percentage of NaCl.

o°c.

6=C.

12° C.

3-8845

1-029664

1-028979

1-027935

8-8078

1-067589

1066144

1-064485

13-3610

1101300

1-099341

1097244

17-6358

M38467

1136040

1133565

From Dr Gibson's numbers, with the help of a table of reciprocals, we have the
following data as to volume instead of density : —

Percentage of NaCl.

o°c.

6°

12°

3-88

-97119

•97184

•97282

8-81

•93669

•93796

•93942

13-36

•90802

•90963

•91137

17-63

•87837

•88025

•88217

Next, to find the maximum density for each solution, and the corresponding
temperature, we must represent these volumes by parabolic functions of t. Thus the
first three numbers are closely represented by

y = 0-97083 + ^:^\9 + <)'.

so that the first solution has its maximum density (1*030) at — 9"" C, and its coeffi-
cient of expansion is

00000093 (9 + e).

Such formulae, of course, must be taken for no more than embodiments of the data,

and any application of them considerably beyond the temperature limits 0*" — 12° 0. is
purely hypothetical.

T. IL 6

42 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

For the second solution —

y = 0-93306 H-^-:5^\37-2 + *)..

SO that (under the reservation just made) the maximum density is 1*0717, at — 37°'2,
and the coefficient of expansion is

0-0000056 (37-2 + e).
For the third —

y = 0-89884 + 00000018 (72 + tf.

The maximum density is 1-1125, at —72** C. ; and the expansibility

0000004(72 + 0.

The numbers for the volume of the fourth solution are so nearly in arithmetical
progression that we can hardly use them to approximate, even roughly, to the position
of the maximum density point, or the corresponding density. The expansibility has
practically (from 0** to 12"* C.) the constant value

0-00036.
Thus we have for the various salt solutions: —

Peroentoge
MaCl.

Max. DenHity
Point.

Max. Density.

Density at 0° C.

ExpansihiUty.

+ 4°

099986

- 0-000068 (l - 1)

3-88

-9°

1030

1-02966

+ 0-000084^1+1)

8-81

-37°

10717

1-06759

0-00021(1 + ^)

13-36

-72"

11126

1-10130

000029 (1 + ^)

17-63

1-13847

000036

As a good illustration of the analogy at the beginning of this section, let us
deal for a moment with fresh water at such a pressure that its maximum density
point is — 9° C, that of the first of the salt solutions. It will be seen later that
the requisite pressure is about 4 tons. At that pressure (A) gives

468- 3-75^ + 0-07e^

Hence as the unit of volume at 1 atm. and 4*" C. becomes 1000136 at 1 atm. and
0** C, it is reduced at 4 tons and 0° C. to

(1-000136) (l - ^^~^) = 1 - 0-0284,

so that the density has become

1-0292.

AVERAGE APPARENT COMPRESSIBILITIES for 1,2, A 3 TONS,
nmltiplied by lOT

?k*,e :.

CI Solutioi

IS,

\
\

..^

«%_

—

o*c

10*

ZQO

Plate n.

Observations of Apparent Average
Compressibility of Water for 305

Atmospheres at 5®C, multiplied
\>ylB^. 29 & 30/12/87.

Tq foUaM "SV^^N.^

IiU.]

FBESH WATEE AND OF SEA-WATBR,

At the same temperature, and at 1 aim., the density of the salt solution, which baa
the same maximurn density point, is

10297.

If we assume the forma Ise (A) to be applicable to temperatures ao far as 9° below
zero (a somewhat precarious hypothesis, inasmuch as water at 4 tons has its freezing
point about —4s^'5C.)f the ma Kim urn densities alike of the compressed water and of
the salted water are closely represented by

1-030.

[In obtaining the first of these numbers, I assumed from Despretz that the
density of water at 1 atra. and — 9'' C. is 0'9984?,] Of course it would be vain to
attempt similar calculations for the stronger solutions, as the indicated maximum
density points are so widely outside the limits of ray experiments. But the example
jiist given seems to show that if fresh water be made, by pressure, to have ife
maximum density point the same as that of a common-salt solution under atmo-
spheric pressure, the densities of the two will be nearly the same at that point, and
will remain nearly alike as temperature changes.

NOTE,

In all that precedes it has been tacitly assumed: —

L That the pressure is the same outside and inside the piezometer.

2, That the pressure measured by the gauge is that to which the contents of

the piezometer were exposed.

3, That the pressure was uniform throughout the contents.

None of these is strictly truSi so that cause must be shown for omitting any
consequent correction.

The third may be dismi^ed at once, as the height of the piezometer bulb is
only a few inches.

The diflference of levels between the upper end of the gauge and the bulbs of
the piezometers, when in the pressure-chamber, was about three feet, so that on this
account the pressure applied was less than that in the gauge by one-tenth of an
atmosphere* But as differences of pressure alone were taken from the gauge, this
cause merely shi/is (to a small extent) the range through which the compression was
measured- But the rise of mercury in the piezometer stem made a reduction of the
range of prcijsure as metisured, which for 3 tons pressure might amount to about
0'5 atm* The error thus introduced was, at the utmost, of the order 0*1 per cent,
of the compressibility measured. Thus the second cause, also, produces only negligible
effects.

I preferred to settle the first question by experiment rather than by calculation,
as the obtaining of the data for calculation would have required cutting up of the
piezometer bulbs. The 0'5 atm. spoken of above represented, in extreme cases, the

6—2

44 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

excess of external over internal pressure in the piezometers. By direct experiment
on two of the instruments themselves, it was found that their internal volume was
diminished at most 0*00002 of the whole by 0*6 atm. of external pressure. This would
involve as a correction the adding of O'l per cent, only to the results at 3 tons, so
that it also is well within the limits of error of the measurements above.

ASSOCIATED PHYSICAL QUESTIONS.

X. Theoretical Speculations.

If instead of the percentage of NaCl in the solutions we tabulate the amount of
NaCl to 100 of water, and along with it the compressibility at zero, we have —

s=amoant of Average compressibility at 0° C. x 1(F.

NaGl to 100 of water. For first ton. First 2 tons. First 3 tons.

0-0 503 490 477

40 449 438 428

9-6 396 386 378

15-4 354 345 338

21-4 321 313 306

The relation between these numbers is very fairly represented by the formula —
Average compressibility for first p tons = ^7j .

It is remarkable that if we put ^ = 0 in the formula of Section VII., we have —
Average compressibility of fresh water for first p+s tons = ^^

which presents an exceedingly striking resemblance to that last written.

Though these formulae are only approximate, we may assume the true constants
to be at least nearly the same in both, and make the following statement as a sort
of niemoria technica in this subject : —

At 0*" C. the average compressibility, for p tons, of a solution of 8 lbs. of common
salt in 100 lbs. of water, is nearly equal to the average compressibility of fresh water
for the first p + 8 tons of additional pressure.

The numerical coincidence above is, of course, accidental; because the formulae
are taken for the special temperature 0° C, and the special unit of pressure 1 ton
weight per square inch.

But a coincidence of a much more striking character, and one which does not
depend upon special choice of units, is suggested by the common form of the
expressions compared.

It appears from the Kinetic Theory of Gases, in which the particles are treated
as hard spheres, whose coefficient of restitution is 1, and which exert no action on

lxl] fresh water and of sea- water, 45

one another except at impact, that the pressure and volume of the group at any
one temperature are connected by a relation approximately of the form

p{v — a) = constant.

The quantity a obviously denotes the ultimate volume, i,e, that to which the
group would be reduced if the pressure were infinite.

I have pointed out^ that this expression coincides almost exactly with the results
of Amagat's experiments on the compression of hydrogen. The introduction of an
attractive force between the particles, sensible only when they are at a mutual
distance of the order of their diameters, merely alters the constants in this expression.
Let us see what interpretation it will bear if, for a moment, we suppose it roughly
to represent the state of things in water.

The average compressibility of such a group of particles, between the pressures
vr and «r H-p, viz.,

Vq — V
pVo

where Vq is the volume at «r, and v that at -er+p, is easily shown to be

Compare this with the empirical expression above for the compressibility of water
say at 0° 0. (per ton weight on the square inch) —

152-3 X 0-00186 ^ 0'283
S6+P "S^+p

and we see that they agree exactly in form. If, then, the results of the kinetic
theory be even roughly applicable to the case of a liquid, we may look upon the
36 in this expression as the number of tons weight per square inch by which the
internal pressure of water exceeds the external pressure. And the corresponding
empirical expression for the compressibility of a solution of common salt may be
interpreted as showing that the addition of salt to water increases the internal
pressure by an amount simply proportional to the quantity of salt added.

That liquids have very great internal pressure has been conjectured from the
results of Laplace's and other theories of capillarity, in which the results are derived
statically from the hypothesis of molecular forces exerted intensely between contiguous
portions of the liquid, but insensibly between portions at sensible distances apart. A
very interesting partial verification of this proposition was given by Berthelot^ in 1850.
By an ingenious process he subjected water to external tension, and found that it
could support at least fifty atmospheres. The calculation was made on the hypothesis
that a moderate negative pressure increases the volume of water as much as an equal
positive pressure diminishes it.

^ Trafu. Roy. Soc, Edin., vol. xxzin. p. 90, 18S6. ^ Ann, de Chimie, torn. xxx. p. 232.

46 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXL

I wss led to the conclusion that the internal pressure of a liquid must be greatly
snpefior to the external, as a consequence of the remarkable results of Andrews'
expenmoits on carbonic acid, and of the comments made on them by J. Thomson
azxl Clerk-Maxwells It was Prof E. Wiedemann who, while making an abstract of
my paper (Appendix E) for the Beibldtter zu den Ann, d. Phyeik, first called my
attention to Berthelot s experiment.

In Appendix F a short account of Laplace's calculations is given, and it is shown
diat the work required to carry unit volume of water, from the interior to a distance
from the sor&ce greater than the range of molecular forces, is

2K X 1 cub. inch,

There K is the internal molecular pressure per square inch. The speculation above

toqU make this work

72 inch-tons.

Bat, in work units, the heat required to vaporize 1 cub. inch of water at 0° C. is

1728 ^^^ ^ ^^^^ foot-pounds,
or 163 inch-tons.

The two quantities are at least of the same order of magnitude, and it is to be
remembered that what has been taken out in the one case is very small particles
of water; in the other, particles of vapour. This raises another extremely difficult
<{iieetioD, viz., — What frttction of the whole latent heat is required to convert water,
in excemively small drops, into vapour?

The comparison above, if it be well founded, would seem to show that the utmost
reduction of volume which water at 0° C. can suffer by increase of pressure is 0*283;
tje. that water can be compressed to somewhat less than 3/4ths of its original bulk,
but not further.

Of course the whole of this speculation is of the roughest character, for two
retHon^, The kinetic gas formula has been proved only for cases in which the whole
volume of the particles is small compared with the space they occupy. The com-
preemou formula is only an approximation, and was obtained for the range of pressures
from 150 to 450 atmospheres; while we have extended its application to much higher
preHdures.

XI. Equilibrium of a Vertical Column of Water.

Irj Canton's second paper we have the following interesting statement: —
**The weight of 32^ feet of sea-water is equal to the mean weight of the atmo-
*(/fc*rr*r: and, as far as trial has yet been made, every additional weight equal to
iitSiX fd the atmosphere, compresses a quantity of sea- water 40 millionth parts; now
it thii» a^nstantly holds, the sea, where it is two miles deep, is compressed by its
//wii i%tf;ight 69 feet 2 inches; and the water at the bottom is compressed 13 parts

J Tluory of Heat, chap, vi., LondoD, 1S71.

LXI.] FRESH WATER AND OF SEA-WATER. 47

Either Canton overestimated the density of sea-water or he underestimated the
amount of an atmosphere, for undoubtedly 33 feet is a much closer approximation to
the column of sea-water which produces 1 atmosphere of pressure. He does not give
his process of calculation, but it was probably something like this: — The pressure
increases uniformly from the top to the bottom (neglecting the small effect due to
change of density produced by compression), and eveiywhere produces a contraction
proportional to its own value. Hence the whole contraction is equal to that which
would have been produced if the pressure had, at all depths, its mean value, ie.
that due to half the whole depth. This process, with Canton's numbers, gives nearly
his numerical results.

If, then, a be the depth, and po the original density, gp^/2 is the mean pressure.
If « be the compressibility, the whole contraction of a column, originally of length a,
is egpifiL*/2. Now, a mile of sea- water gives nearly 160 atmospheres of pressure, so
that the loss of depth of a mile of sea (supposed at 10° C. throughout) is

160 X 0-000045 X 5280/2 = 19 feet, nearly.

For other depths it varies as the square of the depth; so that for two miles it is
76 feet, and for six miles 684 feet nearly.

This, however, is an overestimate, because we have not taken account of Perkins'
discovery of the diminution of compressibility as the pressure increases. The investi-
gation for this case is given in Appendix G, where the change of depth is shown to be

«r being the pressure at the bottom in tons weight per square inch, and H (by
Section VIII.) being 38 in the same units.
For six miles of sea this is, in feet —

684 (l - ^ -h ^ - &c.) = 620 nearly.

In the Appendix referred to I have given a specimen of the hydrostatic problems
to which this investigation leads. Any assigned temperature distribution, if not
essentially unstable, can be approximately treated. But the up- or down-rushes which
result from instability are hopelessly beyond the powers of mathematics.

One remark of a curious chai-acter may be added, viz. that in a very tall column
of water (salt or fr^sh), at the same temperature throughout, the equilibrium might
be rendered unstable in consequence of the heat developed by a sudden large increase
of pressure. For, as will be seen later, the expansibility of water is notably increased
by pressure; and thus the lower parts of the column will become hotter, and less
compressible, than the upper. This effect is not produced in a tall column of air,
for the expansibility is practically unaltered by pressure. And the opposite effect is
produced in bodies like alcohol, &c., where the compressibility steadily increases with
rise of temperature.

48 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXL

XII. Change of Temperature produced by Compression.

The thermal eflfects of a sudden increase or relaxation of pressure formed an
important element in my examination of the Challenger thermometers, and were
practically the origin of this inquiry; one of the most unexpected of the results I
obtained being the very considerable compression-change of temperature of the vulcanite
slabs on which the thermometers are mounted. Thomson's formula for this heating
effect, in terms of the pressure applied, and of the specific heat and expansibility of
the body compressed, is given in Appendix C to my former Report. My first direct
experiment on the subject was described as follows^: —

"When... the bulb of one of the thermometers was surrounded by a shell of lard
upwards of half an inch thick, the total effect produced by a pressure of ^ tons
weight was 5° F.; while for the same pressure, without the lard, the effect was only
1 '-8 F. The temperature of the water in the compression apparatus was 43° F^ so
that the temperature effect due to the compression of water was less than 0**'2 F."

On May 16 of the same year I read a second note on the subject, fix>m which
I extract the following*: —

"I have examined for a number of substances the rise of temperature produced
by a sudden application of great pressure, and the corresponding fidl of temperature
when the pressure was very suddenly relaxed. The copper-iron circuit is, however, too
little sensitive for very accurate measurements; as, from the nature of the af^Muratus,
the wires must be so thin as to have considerable resistance, and the thenno-electric
power of the combination is not large.... I content myself, for the present, with a
general statement of the results for cork and for vulcanized india-rubber, whiidi aie
apparently typical of two classes of solids quite distinct from one another in their
behaviour.

" In the case of india-rubber the rise . of temperature was found to be about
1°*3 F. for each ton-weight of pressure per square inch; and the &11 in relaxation
was almost exactly the same.

''With cork each additional ton of pressure gave less rise of temperature than
the preceding ton; and the fall on relaxation of pressure was, for one or two tonsc
only about half the rise. For higher pressures its ratio to the rise became greater.
Two tons gave a rise of about V'6 F., and a fall of O^'Q F.

"With the same arrangement, the fell of temperature in water sudd^j rdieved
from pressure at a temperature of 60"" F. was found to be for

One ton-weight per square inch 0''25 F.

Two „ „ 0^-56 „

Three „ „ 0=-93 ^

Four „ „ l"-85 ^

"These numbers give the averages of groups of hdrlj concordant nesiih& I
employed cooling exclusively in these experiments, because one of the valres of my

» Proe. Rof. Soc. Edin., voL xl p. 51. 1881. « Proe. Bay. Soc. Editu, toL XL pv^ 117. tl«L 1«1.

LXI.]

FRESa WATER AND OF SEA-WAtEE-

pump was out of order, and the pressure could not be raised at a uniform rate.
The effects obtained for successive tons of pressure are thus, roughly, O'^'Sd, 0°'31, 0**'37,
and 0 -42 F.

"If these results may be trusted, they probably indicate a lowering of the maximum-
deDsity point of water by pressure'/'

In the next exti*aet it will be seen that I deduced from these data a lowering
of the maximum-density point amounting to about 3" C, per ton.

The experiments on water were carried further in the following year by Professors
Marshall and Michie Smithj and Mr Omondl The second of "their papers contains

the annexed graphic representation of the results, which is alluded to in the following
extract. The final result of these experiments, as assigned by the authorSj was a
probable lowering of the maximum-density point of water by 5° C. for one ton pressure.
To this paper I added the following note {Lc, p. 813): —

**If wo assume the lowering of the temperature of maximum-density to be pro-
portional to the pressure, which is the simplest and most natural h^^othesis, we may
write

C = ^p — ^Pt

where p is in tons weight per square inch,

" Now Thomson's thermo-dynamic result is of the form

St^A{t~U')Sp.

''This becomes, with our assumption,

Bt^Ait-t^ + Bp)Sp.

■'As the left-hand member is always very small, no sensible error will result from
integrating on the assumption that ( is constant on the right (except when the

^ [See footDote to p. 20. J ^ Proe. Eoff^ 8oc, Edin., yaL xi. pp. 620 and 809| 1882.

T. IL 7

50 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXL

quantity in brackets is very small, and then the error is of no consequence). Inte-
grating, therefore, on the approximate hypothesis that A and B may be treated as
constants, we have for the whole change of temperature produced by a finite pressure p—

''I have found that all the four lines in the diagram given [from Messrs Marshall,
Smith, and Omond, on last page, where y is the heating eflFect of p tons at temperature
t] can be represented, with a fair approach to accuracy, by the formula

y = 0-0095 (t - 4>)p + 0017j9»,

where p has the values I, 2, 3, 4 respectively. Hence, comparing with the theoretical
formula, we have the values

4 = 00095, 5 = 3°-6C.

"B expresses the lowering of the maximum-density point for each ton weight of
pressure per square inch.

"It seems, however, that all the observations give considerably too small a change
of temperature; for the part due to the first power of the pressure is from 30 to
40 per cent, less than that assigned by Thomson's formula and his numerical data.
One obvious cause of this is the small quantity of water in the compression apparatus,
compared with the large mass of metal in contact with it. This would tend to
diminish all the results, whether heating or cooling; and the more so the more
deliberately the experiments were performed Another cause is the heating (by com-
pression) of the external mercury in the pressure gauge. Thus the pressures are
always overestimated; the more so the more rapidly the experiments are conducted.
A third cause, which may also have some effect, is the time required by the thermo-
electric junction to assume the exact temperature of the surrounding liquid.

"Be this, however, as it may, the following table shows the nature of the agree-
ment between the results of my original experiments [antef p. 48] and the data
derived from the present investigations. The gauge and the compression apparatus
were the same as in my experiments of last year; the galvanometer, the thermo-
-electric junctions, and the observers were all different. The column MSO gives the
whole heating or cooling effect at 15*''5 C, calculated for different pressures fit)m the
results of the investigation by Professor Marshall and his coadjutors. The column T
<x>ntains the results of my direct experiments at that temperature: —

p (tons)

MSO

Thomson.

0131 C.

0139 C.

01 77 G

0-294

0-311

0-355

0-465

0-516

0533

0-665

0-750

0-711

"It will be noticed that there is, again, a fair agreement; though the results are,
as a rule, lower than those calculated from Thomson's formula. My own agree most
nearly with Thomson's formula, probably because they were very rapidly conducted.
As they stand, they give about 3° C. for the effect of 1 ton on the maximum-density

LSI.]

FRESH WATER AND OF SEA-WATER.

point. It is to be observed that if we could get the requisite corrections for con-
duction and for compression of mercury, their introduction would increase (as in fact
is necessary) the constant A above, but would have comparatively little effect on the
value of B, which is the quantity really sought."

The experiments on other substances were carried out for me by Messrs Creelman
and Crocket, from whose important paper^ I extract the following results, which have
some connection with the subjects of this and of my former Report: —

Cork, at 15° C.

PTenanre. Rise per ton. Fall per ton.

1 0°-75 C-Sl

2 0°-65 0°-45

3 0°-59 0°-42

Glass, at 15° C.

1 0°12 0°12

2 0°13 0°14

3 0°13 0°14

Gutta Percha, at 16° C.

1 c-es 0°-67

2 0°-60 0°-64

3 0°-58 0°-63

Solid Paraffin, at 14° C.

1 0°-56 0°-57

2 0°-56 0°-59

3 0°-54 0°-61

Chloroform, at 17° C.

1 l°-44 l°-45

2 l°-34 l°-45

3 1°-31 l°-47

"Challenger" Vulcanite, at 16° C.
Fressare. Rise per ton. Fall per ton.

1 0°-33 0°-33

2 0°-31 0°-33

3 0°-28 0°-32

India-rubber, at 16° C.

1 0°-74 0°-79

2 0°-70 0°-79

3 0°-70 0°-80

Beeswax, at 15* C.

1 0°-83 0°-83

2 0°-79 0°-86

3 0°-78 0°-89

Marine Glue, at 15°'6 C.

1 0°-91 0°-98

2 0°-86 0°-90

3 0°-82 0°-91

Sulphuric Ether, at 21° C.

1 l°-8 l°-9

2 l°-74 l°-8

3 l°-7 l°-7

As was to be expected from the fact that the getting up of pressure requires a
short time, while the relief is practically instantaneous, the heating effect is generally
a little smaller than the cooling effect for the same change of pressure.

These experimenters thus completely confirmed my statements as to the curiously
exceptional behaviour of cork, but they found no other substance, in the long list of
those which they examined, which behaves in a similar manner.

It is to be remarked that as, in all the experiments described or cited in this
section, the temperature-changes were measured by a thermo-electric junction which
was itself exposed to the high pressures employed, there may be error due to the
compression of the materials forming the junction. The wires were, for several reasons,

Proe. Roy. Soc. Edin., vol. xm. p. 311, 1886.

7—2

52 REPORT ON SOME OF THE PHYSICAL PROPERTIES OP [LXL

Tcn* dim; so that the error, if any, is not due to changes of temperature in them,
hsn to (poGsible) change of relative thermo-electric position, due to pressure. This is
a Tdj insidioos source of error, and it is not easy to see how to avoid it.

XTTT Effect of Pressure on the Maximum-Density Point.

Though the lowering of the maximum-density point of water by pressure is an
hnmfrfiare consequence of Canton's discovery, that the compressibility diminishes as
the tempentore is raised, it seems to have been first pointed out, so lately as 1875,
by Poschl^ I was quite unaware of his work, and of that of Van der Waals', when
(9B Aawn in Section XII. above) I was led to the same conclusion by the differences

between theory and experiment, as to the heat
developed by compression of water.

This can very easily be shown as followa
Let the (vertical) ordinates of the curve ABC
represent the volume of water at 1 atm., the
abscissse the corresponding temperatures, B the
maximum-density point. Let the dotted curve
abc represent the same for a greater pressure,
say two atmospheres. Then, by Canton's result,
- the vertical distance between these curves (the

1^ ^ difference between corresponding ordinates) di-
minishes continuously from A to C; 8o long, at
UoiMt, tm the temperature at C is under that of minimum compressibility. Hence
the inclination of abc to the axis of temperatures is everywhere greater than that
<A the corresponding part of ABC, Thus the minimum, 6, of the dotted curve
^wbere it« tangent is horizontal) must correspond to a point, fi, in the full curve,
wherti tb« inclination is negative — i.e. a point at a lower temperature than B.

To calculate the amount of this lowering, by the process indicated, we must
kAT/w the ((^rm of the curve abc. This, in its turn, can be calculated firom a know-
UA^e ol tbie fixrm of ABC, and of the relation between compressibility and temperature.
h^Ah of thfi authors named took their data as to the latter matter from the experi-
m^iso^ tA Qrwmi; and, as was therefore to be expected, gave results wide of the
truth. Puijcbl calculates a lowering of 1° C. by 87*6 atm., which is certainly too
^aijUl ; Vbh der Waals, 0^78 C. by 10*5 atm., as certainly much too large.

To obUkiu a good estimate in this way is by no means easy, for authorities are
iMA quite agreed as to the form of the curve ABC. If we calculate firom the datum
^A lJe«pr<?te which ban been verified by Rossetti', namely, —

vol at 0° a J.
vol. at 4 C.

^ HiUunrfth. d. math.-naturw. CI. d. k. Akad. d. Wist. Wien^ Bd. lxzii. p. 2S3, 1875.
* Arehivet N4erl., torn. xn. p. 467, Haarlem, 1877.
'* Pogg. Ann., Krgilmungthand, v. p. 260, 1871.

LXl.]

7RE9H WATEB AND OF SEA-WATEB.

we obtain for the volume of water at 1 atm*, in terms of teraperaturei

1 + 00000085 (e- 4)" ..., (1).

[This refers only to the part AB of the curve, which is what we want. There geema
general agreement that the curve is not symmetrical about the ordinate at £.] Now,
by (A), the factor for reduction of volume by 1 ton of additional pressure is

1-0^007676 + 0^0055^ -0^00000061^= „,. ,..(2),

The product of these factors, (1) and (2), is a minimum when

0-000017 (^ - 4) ^ - 0-000055 + 0W000122i;

,.*-f»^-4-m.

ThuSj according to these data, the maximum-density point is lowered by S"*'!? C.
per ton of pressure. It will be observed that this is not much less than the result
I calculated from the data of Professor Marshall and his comrades, but it agrees
alm*mt exactly with that which I derived from my own.

The following description of the results of my earlier attempts to solve this
question directtt/, is taken from the Proc, Roy, Soc. Editu, vol XII. pp. 226-228, 1883: —

"I determined to try a direct process analogous fco that of Hope, for the purpose
of ascertaining the maximum-density point at different pressures. The experiments
presented great diflSculties, because (for Hope's method) the vessel containing the
water must have a considerable cross section ; and thus I could not use my smaller
compression apparatus, which was constructed expressly to admit of mesisurements of
temperature by thermo-electric processes. I had therefore to work with the huge
Fraser gun employed for tlie Challenger work, and to use the protected thermometers
(which are very sluggish) for the measurement of temperatures. It was also necessary
to work with the gun at the temperature of the air, — ^it would be almost impossible
to keep it steadily at a much lower temperature, — so that I had to work in water
at about 12^ C,

"The process employed was very simple. A tall cylindrical jar full of water had
two Challenger thermometers (stripped of their vidcanite mounting) at the bottom,
and was more than half-filled with fragments of table-ice floating on the water, and
<:onfined by wire-gauze at the top This was lowered into the water of the gun, and
pressure was applied,

"It is evident that if there wm*e no conduction of heat through the walls of the
cylinder, and if the ice lasted long enough under the steadily maintained pressure,
tho thermometers would ultimately show, by their recording minimum indices, the
tnaximum-density point corresponding to the pressure employed: — always provided that
that temperature is not lower than the melting point of ice at the given pressure,

•* Unfortunately, all the more suitable bad conductors of heat are either bodies
like wood (which is crushed out of shape at once under the pressures employed) or
like tallow, &c, (which become notably raised in temperature by compression). I was
therefore obliged to use glass. The experiments were made on successive days, three

54 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [LXI.

each day, with three diflferent cylindrical jars. These had all the same height and the
same internal diameter. The first was of tinned iron; the second of glass about ^ inch
thick ; the third, of glass nearly an inch thick, was procured specially for this work.

"With the external temperature 12°'2 C, the following were the results of 1^^ tons
pressure per square inch, continued in each case for 20 minutes (some unmelted ice
remaining on each occasion). The indications are those of two difierent Challenger
thermometers, corrected for index-error by direct comparison with a Kew standard: —

Tin Cylinder.

Thin OlasB

Thick OlMS.

4°C.

2°-67

0°-83

4°

2''-61

0°-83

The coincidence of the first numbers with the ordinary maximum-density point of
water is, of course, mere chance. When no pressure was applied, but everything else
was the same, the result was —

Tin. Thin. Thick.

5°-7 C. 5° 4°

It is clear that the former set of numbers points to a temperature of maximum density,
somewhere about 0° C, under 1| tons pressure per square inch. But still the mode
of working is very imperfect.

*'I then thought of trying a double cylindrical jar, the thin one above mentioned
being enclosed in a larger one which surrounded it all round, and below, at the
distance of about f inch. Both vessels were filled with water, with broken ice floating
on it, and had Challenger thermometers at the bottom. By this arrangement I hoped
to get over the difficulty due to the temperature of the gun, by having the inner
vessel enclosed in water which would be lowered in temperature to about 3** C. by
the application of pressure. The device proved quite successful. The result of IJ tons
pressure per square inch maintained for 20 minutes, some ice being still left in each
vessel, was from a number of closely concordant trials —

Temperature in outer vessel . . . V'7 C.
Temperature in inner vessel . . O'^'S C.

The direct pressure correction for the thermometers is only about — 0°1 C, and has
therefore been neglected.

"The close agreement of this result with that obtained (under similar pressure
conditions) in the thick glass vessel leaves no doubt that the lowering of the maximum-
density point is somewhat under 4° C. for 1^ tons, or 2'''7 C. for 1 ton per square
inch. It is curious how closely this agrees with the result of my indirect experiments."

Further work of the same kind led me to the conclusion that even the double
vessel had not sufficiently protected the contents from conducted heat, and to state in
my ffetU (p. 95| 18S4) that ''a pressure of 50 atmospheres lowers the maximum-density
point Iqr r a~

LXI.] FRESH WATER AND OF SEA- WATER. 55

During the next two years I made several repetitions of these experiments, with
the help of thermometers protected on the Challenger plan, but very much more
sensitive. These experiments were not so satisfactory as those just described. The new
thermometers caused a great deal of trouble by the uncertainty of their indications,
which I finally traced to the fact that the paraflSn oil which they contained passed,
in small quantities, from one end of the mercury column to the other. I was occupied
with an attempt to obtain more suitable instruments, when the arrival of the Amagat
gauge turned my attention to other matters.

So far as I can judge from the results of the three different methods which I
have employed, the lowering of the maximum-density point of water by 1 ton of
pressure is very ne&rly, though perhaps a little in excess of, 3"^ C.

It is peculiarly interesting to find that Amagat, by yet another process, — viz.
finding two temperatures not far apart at which water, at a given pressure, has the
same volume, — has lately obtained a closely coinciding result. He says: "A 200 atm.
(chiffres ronds) le maximum de density de Teau a rdtrograd^ vers z6ro et la presque
atteint; il parait situ^ entre z6ro et 0°-5 (un demi-degr^)^" This makes the effect
of 1 ton slightly less than 3"" C.

As the freezing point is lowered, according to J. Thomson's discovery, by about
1°'13 only per ton of additional pressure, — ^and has a start of but 4°, — ^the maximum-
density point will overtake it at about — 2°*4, under a pressure of 214 tons.

The diagram 2 of Plate IL shows the consequences of the pressure-shifting of
the maximum-density point in a very clear manner,— especially in its bearing on the
expansibility of water at any one temperature but at different pressures. The curves
in the diagram are for atmospheric pressure, and for additional pressures of 1, 2 and
3 tons respectively. They are traced roughly by the help of Despretz's tables of
expansibility at atmospheric pressure, and the compression data of the present Report.
The quantity of water taken in each case is that which, at 0° and under the
particular pressure, has unit volume. Thus all the curves pass through the same
point on the axis of volumes. How, in consequence of the gradual lowering of the
maximum-density point, the expansibility at zero, which is negative at atmospheric
pressure, and even at 1 ton of additional pressure, becomes positive and then rapidly
greater as the pressure is raised, is seen at a glance.

I have to state, in conclusion, that my chief coadjutors in the experimental work
have been Mr H. N. Dickson and my mechanical assistant Mr T. Lindsay. Mr Dickson
also reduced all the observations, about half of them having been done in duplicate
by myself.

In the compression of glass I had the assistance of Mr A. Nagel, and occasionally
of Dr Peddie.

Mr A. C. Mitchell assisted me in the graphic work, and checked the calculations
in the text.

» Comptet Rendus, torn. civ. p. 1160, 1887.

56 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

I have already acknowledged the density determinations and analyses of sea-water
and salt solutions made by Dr Gibson.

And I have again been greatly indebted to the very skilful glass-working of
Mr Kemp.

[7/9/88. — The following analysis of the glass of my piezometers is given by Mr T.
F. Barbour, working in Dr Crum Brown's Laboratory: —

SiO, = 61-20
PbO = 20-94
AlA + FeA = 0-82
CaO = 2-20
MgO = 0-26
K,0 = 1-93
Na^O = 11-72.]

ADDENDUM (8/8/88).

The reader has already seen that I have, more than once in the course of the inquiry^
found myself reproducing the results of others. A few days ago I showed the proof-sheets
of this Report to Dr H. du Bois, who happened to visit my laboratory, and was informed
by him that one of Van der Waals* papers (he did not know which, but thought it was a
recent one) contains an elaborate study of the molecular pressure in liquids. 1 had been under
the impression, strongly forced on me by the reception which my speculations (Appendix E.,
below) met with both at home and abroad, that Laplace's views had gone entirely out of
fashion; — having made, perhaps, their final appearance in Miller's Hydrostatics, where 1 first
became acquainted with them about 1850. In Van der Waals' memoir "On the Continuity
of the Gaseous and Liquid States," which I have just rapidly perused in a German trans-
lation, the author expresses himself somewhat to the following effect : If I here give values
of K for some bodies, I do it not from the conviction that they are satisfactory, but because
I think it important to make a commencement in a matter where our ignorance is so
complete that not even a single opinion, based on probable grounds, has yet been expressed
about it.

Van der Waals gives, as the value of K in water, 10,500 atmospheres; and, in a sub-
sequent paper, 10,700 atm. ; while the value given in the text above is about half, viz.
5480 atm. So ^ as I can see, he does not state how these values were obtained, though
he gives the data and the calculations for other liquids. It is to be presumed, however,
that his result for water was obtained, like those for ether and alcohol, from Cagniard
de la Tour's data as to any two of the critical temperature, volume, and pressure. Van
der Waals forms, by a veiy ingenious process, a general equation of the isothermals of a
fluid, in which there are but two disposable constants. This is a cubic in v, whose three
roots are ,real and equal at the critical point. Thus the critical temperature, volume, and
proMore oan all be expressed in terms of the two constants, so that one relation exists
tbem. Two being given, the equation of the isothermals can be formed, and from it
bo afe ODoe found.

LXI.] FBESH WATER AND OF SEA-WATER. 57

My process, as explained above, was very different. I formed the equation of the iso-
thermal of water at 0"" C. from the empirical formula for the average compressibility under
large additional pressures; and by comparing this, and the corresponding equation for various
salt solutions, with an elementary formula of the Kinetic theory of gases, I was led to
interpret, as the internal pressure, a numerical quantity which appears in the equations.

I have left the passages, in the text and Appendix alike, which refer to this subject
in the form in which they stood before I became acquainted with Van der Waals' work.
I have not sufficiently studied his memoir to be able as yet to form a definite opinion
whether the difficulty (connected with the non-hydrostatic nature of the pressure in surface
films) which is raised in Appendix E. can, or cannot, be satisfactorily met by Van der Waals'
methods. Anyhow, the isothermals spoken of in that Appendix are totally different from those
given by Van der Waals' equation, inasmuch as the whole pressure, and not merely the external
pressure, is introduced graphically in my proposed construction.

SUMMARY OF RESULTS.

It is explained in the preceding pages that the pressures employed in the experi-
ments ranged from 150 to 450 atm., so that results given below for higher or lower
pressures [and enclosed in square brackets] are extrapolated. A similar remark applies
to temperature, the range experimentally treated for water and for sea-water being
only 0° to 15° C. Also it has been stated that the recording indices are liable to
be washed down the tube, to a small extent, during the relief of pressure, so that
the results given are probably a little too svialL

Compressibility of Mercury, per atmosphere, . . . 0*0000036
„ Glass 00000026

Average compressibility of fresh water: —

[At low pressures 520 . 10"' - 355 . IQ-^t + 3 . lO"*^']

For 1 ton = 1523 atm. 504 360 4

2 „ =304-6 „ 490 365 5

3 „ =456-9 „ 478 370 6

The term independent of t (the compressibility at 0° C.) is of the form

where the unit of p is 152*3 atm. (one ton-weight per sq. in.). This must not be
extended in application much beyond p = S, for there is no warrant, experimental or
other, for the minimum which it would give at p = 8-5.

The point of minimum compressibility of fresh water is probably about 60° C. at
atmospheric pressure, but is lowered by increase of pressure.

As an approximation through the whole range of the experiments we have the

formula : —

0-00186 (._3t_ t^ \ .
36 +i? \ 400 ■*" 10,000; '
T. II. 8

58 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

while the following formula exactly represents the average of all the experimental
results at each temperature and pressure: —

lO-' (520 - 17p + p«) - 10-» (355 + 5|)) e + 10-» (3 +i)) <".

Average compressibility of sea-water (about 0*92 of that of fresh water): —

[At low pressures 481 . IQ-' - 340 . IQ-^t + 3 . lO"^]

For 1 ton

2 „

3 „

Term independent of t : —

10-^(481 - 21-25p + 2-25jo»).
Approximate formula:—

462

320

447-5

305

437-6

293

0-00179 /- t

3,ooor

38+2) V 150 '^'m

Minimum compressibility point, probably about 56** C. at atmospheric pressure, is
lowered by increase of pressure.

Average compressibility of solutions of NaCl for the first p tons of additional
pressure, at 0° C: —

0-00186
S6-^p + 8
where a of NaCl is dissolved in 100 of water.

Note the remarkable resemblance between this and the formula for the average
compressibility of fresh water at O"* C. and p + 8 tons of additional pressure.

[Various parts of the investigation seem to favour Laplace's view that there is a
large molecular pressure in liquids. In the text it has been suggested, in accordance
with a formula of the Kinetic Theory of Oases, that in water this may amount to
about 36 tons- weight on the square inch. In a similar way it would appear that
the molecular pressure in salt solutions is greater than that in water by an amount
directly proportional to the quantity of salt added.]

Six miles of sea, at 10° C. throughout, are reduced in depth 620 feet by com-
pression. At 0° C. the amount would be about 663 feet, or a furlong. (This quantity
varies nearly as the square of the depth.) Hence the pressure at a depth of 6 miles
is nearly 1000 atmospheres.

The maximum-density point of water is lowered about 3° C. by 150 atm. of
additional pressure.

From the heat developed by compression of water I obtained a lowering of 3** C.
per ton-weight per square inch.

From the ratio of the volumes of water (under atmospheric pressure) at 0° C.
and 4° C, given by Despretz, combined with my results as to the compressibility, I
found 3°-l7 C. : — and by direct experiment (a modified form of that of Hope) 2°'7 C.
The circumstances of this experiment make it certain that the last result is too small.

Thus, at ordinary temperatures, the expansibility of water is increased by the
application of pressure.

LXI.J FRBSH WATER AND OF SEA-WATER. 59

In consequence, the heat developed by sudden compression of water at temperatures
above 4° C. increases in a higher ratio than the pressure applied ; and water under
4° C. may be heated by the sudden application of sufficient pressure.

The maximum density coincides with the freezing-point at — 2*''4 C, under a
pressure of 214 tons.

APPENDIX A.
On an Improved Method of Measuring Compressibility ^

"When the compressibility of a liquid or gas is measured at very high pressures, the
compression vessel has to be enclosed in a strong cylinder of metal, and thus it must be
made, in some way, self-registering. I first used indices, prevented from slipping by means
of hairs. Sir W. Thomson's devices for sounding, at small depths, by the compression of
air, in which he used various physical and chemical processes for recording purposes, led me
to devise and employ a thin silver film which was washed off by a column of mercury.
Much of my work connected with the Challenger Thermometers was done by the help of
this process. Till quite recently I was unaware that it had been devised and employed by
Cailletet in 1873, only that his films were of gold.

" But the use of all these methods is very laborious, for the whole apparatus has to
be opened for each {individual reading. Hence it struck me that, instead of measuring the
compression produced by a given pressure, we should try to measure the pressure required
to produce a given compression. I saw that this could be at once effected by the simplest
electric methods; 'provided that glass, into which a fine platinum tmre is fused^ were capable
of resisting very high pressures without cracking or leaJeing at the junctions. This, on trial, was
found to be the case.

"We have, therefore, only to fuse a number of platinum wires, at inter\'als, into the
compression tube, and very carefully calibrate it with a column of mercury which is brought
into contact with each of the wires successively. Then if thin wires, each resisting say about
an ohm, be interposed between the pairs of successive platinum wires, we have a series whose
resistance is diminished by one ohm each time the mercury, forced in by the pump, comes in
contact with another of the wires. Connect the mercury with one pole of a cell, the highest
of the platinum wires with the other, leading the wires out between two stout leather washers;
interpose a galvanometer in the circuit, and the arrangement is complete. The observer
himself works the pump, keeping an eye on the pressure gauge, and on the spot of light
reflected by the mirror of the galvanometer. The moment he sees a change of deflection he
reads the gauge. It is convenient that the external apparatus should be made to leak slightly ;
for thus a series of measures may be made, in a minute or two, for the contact with each of
the platinum wires. Then we pass to the next in succession."

M. Amagat' remarks on the use of this method as follows: — "Le liquide du pi^zom^tre,
et le liquide transmettant la pression dans lequel il est plough (glycerine), s'^chauffent con-
sid^rablement par la pression; oette circonstance rend les experiences trds longues: il faut

1 Proc, Roy, Soc, Edin., vol. zm. pp. 2, 3, 1SS4. ' Compte$ Rendtu, torn. cm. p. 431, 1SS6.

8—2

60 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

on temps considerable pour €quilibrer la masse qui est peu conductrioe ; il faut r^pdter les
lectures jusqu'4 ce que I'indication du manom^tre deviemie constante au moment du contact.
Les series faites par pressions d^croissantes produisent le m^me effet en sens inverse; oH
prend la moyenne des r^ultats, dont la concordance montre que Tensemble de la m^thode
ne laisse r^llement presque rien k d^irer.

" On voit par Ik quelles grossi^res erreurs ont pu ^tre commises avec les autres artifices
employ^ jusqu'ici pour la mesure des volumes dans des conditions analogues.'*

It must be remembered that M. Amagat is speaking of experiments in which pressures
rising to 3000 atmospheres were employed.

APPENDIX B.

Relation between True and Average Compressibility.
The average compressibility per ton for the first p tons of additional pressure is

/Wo

where Vq is the initial volume, and v is the volume at p additional tons.
The true compressibility at p additional tons is

dv
vdp'

Hence, if one of these quantities is given as a function of p, it may be desirable to find
the corresponding expression for the other. The simplest example, that on p. 28, will suffice
to show the principle of the calculation. Let

^ = e{l-/p) (1);

PVq

where e is, in general, a much smaller quantity than /. We have

where the expansion may be easily carried further if required.

If the terms in the second and higher powers of p are to be neglected, (1) and (2)
as written show at once how to convert from true to average compressibility, or vice verad.

LXI.] FRESH WATER AND OF SEA- WATER. 61

APPENDIX C.

Calculation of Log. Factors.

Let W be the weight of mercury which would take the place of the liquid in the piezo-
meter, w that of the mercury which fills a length I of the stem. Then a compression read as
X on the stem is

X w

This assumes the stem to be uniform; in general it must be corrected from the results of
the calibration : — unless, as in the example given on p. 15 of the text, I be chosen very nearly
equal to a;, as found by trial for each value of the pressure.

Also if y be the reading of the gauge, and if a on the gauge correspond to an atmosphere,
the pressure is

- atm.
a

Hence the average apparent compressibility per atmosphere is

X wa
y'TW'

Its logarithm is log a - log y + (log 117 — log fT - log /) + log cl

The last four terms, of which log a is the ''gauge log," form the log factor as given
in the text.

APPENDIX D.
Note on the Correction for the Compressibility of the Piezometer.

The usual correction neglects the fact that when the compressibility of the liquid is
different from that of the walls, the liquid under pressure does not occupy the same part
of the vessel as before pressure.

Let V be the volume of the part of the vessel occupied by liquid; a that of the tube
between the two positions of the index, both measured at 1 atmosphere; e, c, the average
absolute compressibility of liquid and vessel per ton for the first p additional tons. Equate
to one another the volume of the liquid, and the volume of the part of the vessel into which
it is forced, both at additional pressure p. We have thus —

F(l-ep) = (r-a)(l-cp),
whence e=c(l-|.) + ^.

62 REPORT ON SOME OF THE PHYSICAL PROPERTIES OF [lXI.

As ^ if) usually small, this equation is treated as equivalent to

i.e. the absolute compressibility of the liquid is equal to its apparent compressibility, added
to the absolute compressibility of the envelop.

One curious consequence of the exact equation is that, if the compressibilities were both
constant, or were known to change in a given ratio by pressure, it would be possible
(theoretically at least) to measure absolute compressibilities by piezometer experiments alone,
without employing a substance whose absolute compressibility is determined by an independent
process. For the additional term in the exact equation makes the coefficients of e and c
numerically different; whereas in the approximate equation they are equal, but with opposite
signs, and therefore can give e-c only.

In my experiments described above, a/V rarely exceeds 0*02, so that this correction
amounts to (0*02 x 26 in 500, or) 5 units in the fourth significant place; and thus jtist
escapes having to be taken account of. When 4 places are sought at lower pressures than
3 tons, or 3 places at pressures of 4 tons and upwards, it must be taken account of.

APPENDIX K

On the Relations between Liquid and Vapour.

In connection with the present research a number of side issues have presented them-
selves, some of which come fairly within the scope of the Report. I commence by reprinting
two Notes, read on January 19 and February 2, 1885, to the Royal Society of Edinburgh^: —

ON THE NECESSITY FOB A CONDENSATION- NUCLEUS.

''The magnificent researches of Andrews on the isothermals of carbonic acid formed, as
it were, a nucleus in a supersaturated solution, round which an immediate crystallization
started, and has since been rapidly increasing.

"They gave the clue to the explanation of the paradoxical result of Regnault, that
hydrogen is less compressible and other gases more compressible, under moderate pressure,
than Boyle's Law indicates; and to that of the companion result of Natterer that, at very
high pressures, all gases are less compressible than that law requires. Thus they furnished
the materials for an immense step in connection with the behaviour of fluids cibove their
critical points.

'* But they threw at least an equal amount of light on the liquid-vapour question, i.e. the
behaviour of fluids at temperatures under their critical points. In Andrews' experiments
there was a commencement, and a completion, of liquefaction; each at a common definite
pressure, but of course at very different volumes, for each particular temperature.

''In 1871 Professor J. Thomson communicated to the Royal Society a remarkable paper
on the abrupt change from vapour to liquid, or the opposite, indicated by these experiments.
He called special attention to the necessity for a 'start,' as it were, in order that these

1 Proc. Roy, Soc. Edin,, vol. xm. pp. 78 and 91, 1886.

I^I.]

FEESH WATER AND OF SEA- WATER,

€haiig€s migbt be effected. [It Ib to this point that the present Note is mainly directed,
but I go on with a brief analysis of Thorns on *s work*] He pointed out tbat tber© were
nudefOUB experiments prov^ing tbat water couJd be beatedj under certain conditions, far above
its boiling point without evaporating; and tbat, probably, a team might be condensed iso-
therm ally to supersaturation without condensing. Hence he was led to auggeat an isothermal
of continued cnrvaturej instead of the broken line given by Andrews, as representing the
eof^numts passage of a fluid from the state of vapour to that of liquid ; the whole mass
betng supposed to be, at each stage of the process, in the same molecular state,

"In Clerk-Max weirs Treatiie an Heat^ this idea of J. Thomson's was developed, in con-
nection with a remarkable speculation of W. Thomson', on the pressure of vapour as depending
on tbe curvature of the liquid surface in contact with it. This completely accounts for the
deposition of vapour when a proper nucleus is present. Maxwell showed that it could also
account for the ^ singing * of a kettle, and for the growth of tbe larger drops in a cloud
at the expense of tbe smaller ones.

**The main objection to J. Thomson's suggested isothermal curve of transition is that,
as Ma^cwell points out, it contains a region in which pressure and volume increase or diminish
flimultaneously* This necessarily involves instability » inasmuch as, for definite values of pressure
at constant temperature within a certain range in which vapour and liquid can be in equi^
librium^ Thomson's hjrpo thesis leads to three different values of volume : two of which are
stable ; but the intermediate one essentially unst-able. According to Maxwell, the extremities
of this triple region correspond to pressures, at which, regarded from the view of steady
increase or diminution of pressures, either the vapour condenses suddenly into liquid, or the
liquid suddenly bursts into vapour.

"If this were the case, no nucleus would be ahsohudy requisite for the formation either
of liquid from vapour or of vapour from liquid. All that would be required, in either ease,
would be tbe proper increase or diminution of pressure \ — temperature being kept unaltered.
The latent heat of vapour, which we know to become less as the critical point is gradually
arrived at, would thus be given off in the explosive passage from vapour to liquid. It is
difficult to see, on this theory, how it can be explosively taken in on the sudden passage
froDi liquid t<^ vapour

**Aitkens experiments tend to show, what J. Thomson only speculatively announced, that
possibly vapour may not be condensed (in tbe absence of a nucleus), when compressed
isotliermally, even at ranges far beyond the imtximum of pressure indicated in Thomson's
ligures. Hence it would appear that the range of instability is much lees than that given
by Thomson's figures, and may (perhaps) be looked on as a vanishing quantity ; the corre-
sponding part of the isothermal being a finite line parallel to the oxiB of pressures, corre-
sponding to tbe sudden absorption or ^ving out of latent heat."

037 EVAPORATION AND CONDENSATION.

"While I was communit^ting my Note on the NecessUy for a Condensaiwn NxiclmM
at the last meeting of the Society, an idea occurred to me which germinated (on my way
home) to such an extent that I sent it off by letter to Professor J. Thomson that same
night

"J. Thomson^s idea, wbich I had been diBCuasing, was to preserve, if possible, physical
(as well as geometrical) c(miinuiiy in the isothermal of the liquid-vapour state, by keeping

1 ftw, Bq^. Sqc. Edin^, vcl. vn. p. 63, 1S70.

REPORT ON SOME OF THE PHYSICAL PROPERTIES OF

[lxi.

the tohole mass of fluid in one state throughout. He secured geometrical, but not physical,
oontinulty. FoFj a8 Clerk-Maxwell ahuwed, one part of his curve makes pressure and volume
increase simultaneously, a condition essentially unstable. The idea which occurred to me wa^f
while preserv^ing geometrical continuity, to get rid of the region of physical instability^ not
(aa I had suggested in my former Note) by retaining Thomson's proposed finite majtimum
and minixnuni of pressure in the isothermal, while liringiiig them infinitely close together so
far as volume ia concerned, and thus restricting the unstable part of the isothermal to a
finite line parallel to the pressure axis ; but, 61/ inakini^ both iJie Tfnasiiivtuni and fiiinimuni irtfinite.
Geometrical continuity^ of course, exists across an asymptote parallel to the axis of pressures ;
so that, from this point of view, there is nothing to object to. On the other hand, there
is essentially physical discontmuity, in the form of an impassable barrier between the
vaporous and liquid states, so long at least as the substance is considered as homogeneous
thruughout.

"It appeared to me that here lies the true solution of the diiHculty. As we are dealing
with a fluid mass essentially homogeneous throughout, it is cle-ar that we are not concerned
with cases in which there is a molecular etirface-hlm*

"Supposej then, a fluid mass, somehow maintained at a constant temperature (lower than
its critical point)^ and so extensive that its boundaries may be regarded as everywhere intinitely
distantj what will be tlie form of its isothermal in terms of pressure and volume 1

"Two prominent experimental facts lielp us to an answer,

^* FirgL We know that the interior of a mass of liquid mercui^ can be sulxjected to
hydrostatic temimi of considerable amount without rupture. The isothermal must, in this
case, CTQSii the line of volumes ; and the limit of the tension would, iu ordinary language,
be called the cohesion of the liquid. I am not aware that this result has been obtained
with water free from air; but possibly the experiment has not been satisfactorily made.
The common experiment in which a rough measure is obtained of the foroe necessai*y to
tear a glass plate from the surface of water is vitiated by the instability of the concave
molecular film formed.

**6ecort£i Aitken has asserted, as a conclusion from the results of direct experiment^
that even immensely supersaturated aqueous vapour will not condense without the presence
of a nucleus. This may be a solid body of finite size, a drop of water, or fine dust particles.

" Both of these facts tit peHectly in to the hypothesis, that the isothermal in question
has an asymptote parallel to Uie axis of pressure ; the vapour requiring (in the absence of
a nucleus) practically inflnite pressure to reduce It, without change of state or of temperature,
to a certain finite volume j while the liquid, also without change of state or temperature,
may by sufficient hydrostatic tension be made to expand almost to the same limit of voluma

^* This limiting volume depends, of course, on the temperature of the isothermal ; rising
with it up to the critical point.

"The physical, not geometrical^ discontinuity is of course to be attributed to the latent
heat of vaporisation. The study of the adiabatics, as modified by this hypothesis^ gives rise
to some curious results.

" It is clear that the experimental realisation of the parts of the here sugge-sted curve
near to the asymptote, on either side, will be a matter of great difficulty for any substance.
But valuable information may perhaps be obtained from the indications of a sensitive thermo^
electric junction immersed in mercury at the top of a column which does not descend in
a barometer tube of considerably more than 30 inches long, when the tulje is suddenly
placed at a large angle with the vertical ; or from those of a similar junction immersed

UCI,]

FRESH WATER AND OF SEA-WATER,

in wsteTp when il h»& a concaTe surface of graat curvature from which the atiiioi|>h6nc
ii removed*
Kotldtig of what is said above will necessarily apply when we have vapour and Iiqaid in
of one another, or when we consider a small portion of either in the immediate
me^bonrbood of another body. For then we are dealing with a state of stress whidi cannot,
Mke hjdrost&tic pre^ure or tension, be characterized (so far as we know) by a single number.
Thm «ttist in these molecular films is probably one of tension in all directions pai^allel to
the film, aad of pressure in a direction perpendicular to it. Thus it is impossible to represcmt
iodi a stttle prtvperly on the ordinary indicator diagram. This question is still farther com-
plicsled bj the possibility that the difference between the internal pressure^t, in a liquid and
its Tapour in thermal equilihriam, may be a very large quantity*"

As soon as I beard of Berthelot's experiment, I had it successfully repeated in my
tabormtofy; and I considered that it afforded very strong confirmation of the hypothesis
advBiioed in tlie last preceding ejc tract.

Bat anoe I have been led to believe that there is probably truth in Laplace's state-
nCDl as to the ver}' great molecular pressure in Liquid;^, I have still further modified the
ftp^jTBlnliffii I now propose to take away the new asymptote, and make tbe two branches
of the isothermal join one another by what Is practically a part of that asymptote ; — thus
making the liquid and the vaporous stages continuous with one another by means of a
portion very nearly straight and parallel to the pressure axia^ Somewhere on this will be
found one of the points of inflection of the isothermal, the other being at a somewhat
saialler volume, and at a pressure which in moderate for tempeiutures close to, but under, the
"critical point,'^ but commences to increase with immense rapidity as the temperature of the
isotbermal is lowered. All the Lsothermals will now present the same general featuresi
dependent on the existence of two asymptotes and two points of inflection, whether they be
above or below the critical point; but their form will be modified in different sens^ above
and below it. The portion of the cun*e which is convex upwards will be nearly horizontal
at the critical point, and ¥rill become steeper both aboi^e and below it; but pressure and
voiume will nowhere increase together. This suggestion, of course, like that in the second
extract abovei, is e«entially confined to the case of a fluid masei which is supposed to have
no boundaries; for their introduction at once raises the complex difficulties connected with
the surtace^skin. Thus it ml! be seen that the conviction that water has large molecular
prenure has led roe back to what is very nearly the first of the two hypotheses I proponed.

A practical application of some of the principles just discussed is described in the following
Utile paper:—

out AH APPLIOATtOK OF THE ATMOUETER^

**T%e Atroometer is merely a hollow ball of unglazed clay, to which a glass tube is
luted. The whole is filled with boiled water, and mverted so that the open end of the
tube stands in a dish of mercury. The water evaporates from the outer surface of the
clay (at a rate depending partly on the temperature, partly on the dryness of the air), and
in consequence the mercury rises in the tube. In recent experiments this rise of mercury
has been carried to nearly ^5 inches during dry weather. But it can be carried much
farther by artificially drjing the air round the bulb. The curvature of the capillary surfaces

1 FroiL Boy. Soe^ Edin., vqI mn. pp. 116, 117. 1835.

T. IL

66 REPORT ON SOME OP THE PHYSICAL PROPERTIES OF [lXI.

in the pores of the clay, which supports such a column of mercury, must be somewhere
about 14,000 (the unit being an inch). These surfaces are therefore, according to the curious
result of Sir W. Thomson {Proe, Boy. Soc, Fdin., p. 63, 1870), specially fitted to ah&arh
moisture. And I found, by inverting over the bulb of the instrument a lai^ beaker lined
with moist filter-paper, that the arrangement can be made extremely sensitive. The mercury
snifsoe is seen to become flattened the moment the beaker is appHed, and a few minutes
suffice to give a large descent, provided the section of the tube be small, compared with the
surface of the balL

**I propose to employ the instrument in this peculiarly sensitive state for the purpose of
estimating the amount of moisture in the air, when there is considerable humidity; but in
its old form when the air is very dry. For this purpose the end of the tube of the
atmometer is to be connected, by a flexible tube, with a cylindrical glass vessel, both con-
taining mercury. When a determination is to be made in moist air, the cylindrical vessel
is to be lowered till the difference of levels of the mercury amounts to (say) 25 inches, and
the diminution of this difference in a definite time is to be carefully measured, the atmo-
spheric temperature being observed. On the other hand, if the air be dry, the difference
of levels is to be made nil, or even negative, at starting, in order to promote evaporation.
From these data, along with the constant of the instrument (which must be determined for
each clay ball by special experiments), the amount of vapour in the air is readily calculated.
Other modes of observation with this instrument readily suggest themselves, and trials, such
as it is proposed to make at the Ben Nevis Observatoiy during summer, can alone decide
which should be preferred."

APPENDIX F.
The Molecular Pressure in a Liquid.

Laplacs's result, so ^ as concerns the question raised in the text, may be stated thus.
If MM'if>(r) be the molecular force between masses M, M' of the liquid, at distance r, the
whole attraction on unit mass, at a distance x within the surface, is

X = 2ir/3 / rdr I ^ (r) dr,

where p is the density of the liquid. The density is supposed constant, even in the surface-
skin. As we are not concerned with what are commonly called capillary forces, the surface
is supposed to be plane.

The pressure, p^ is found from the ordinary hydrostatic equation
Hence the pressure in the interior of the liquid is
where a is the limit at which the molecular force ceases to be sensible.

LXI.] FRESH WATER AND OF SEA-WATER. 67

But the expression for K is numerically the work required to carry unit volume of the
liquid from the interior, through the skin, to the surface. It is easy to see that the further
work, required to carry it wholly out of the range of the molecular forces, has precisely the
same value. Thus the whole work required to carry, particle by particle, a cubic inch of
the liquid from the interior to a finite distance from its surface is

2K X 1 cub. in.

This investigation assumes p to be constant throughout the liquid, and thus ignores the
(almost certain) changes of density in the various layers of the surface-skin; so that its
conclusions, even when the question is regarded as a purely statical one, are necessarily
subject to serious modification. With our present knowledge of the nature of heat, we cannot
regard this mode of treatment as in any sense satisfactory.

APPENDIX G.

Equilibrium of a CJolumn of Water.

FiBST, suppose the temperature to be the same throughout Let a be the whole depth,
Po the density, on the supposition that gravity does not act. Then, if p be the density at
the distance £ from the bottom, when gravity acts, we have by the hydrostatic equation

dp 1

if we adopt the rough formula of Section VII. for the compressibility. The integral is

Now the conditions are —

(1) f = ^^ (the altered depth), jt? = 0;

(2) ^ = 0, p = gpoa = vr suppose.
So that ^, = a(i-^),.^IIlog5j«

„a-.i.^%(i*!).

Since, even in the deepest sea, m/II is not greater than 1/6, we may expand the logarithm
in ascending powers of this fraction. We thus obtain

H /w vr txr

XT'

9—2

OH Hicyour tm mmz or the pktocai. PBorcnxB or watkb. [uu.

m^^O-S"^^^)-

An Uift fn/H/ir y</fl fUoiii for wlmt k eaflttl # in dbe ted, Urn fint ters is Urn remit
givmi in tim Utxi ; «nd ttM» r/tben dum horn it m maSoAA \ff tddag: aeeout of tfe diminirfied

ooiiiprdNMihility »! ih« hiij^ber

Of rnxLvm WH uiif^tt hftre emplojcd the mofe exact ionralc, (A) or (B) as the case may
Im, but for M pracriical applkationa the roogb foranila

Tt might Im) inU^raittng to stodj the efleet en the mean levd of a lake due to the
indireoi hm waII an ttu» direct remilt* of diao^ of temperatine. Healang of the water throa^KKit,
if ihara )m) a cime fd the kind, woold increase the d^ith not onlj in consequence ci expansion
(providad tlie tem[)eratare were nowhere under the mazimnm-densitj pcMntX but also in con-
M^|uanoA of the dimination of compressibility which it prodnces. Thus there would be an
efiioiant cauHi) of variation of depth with the seasons, altogether independent of the ordinary
queHtionM of supply from various sources and loss by evaporation.

If the temperature be not constant for all depths, p,, p, and A are functions of $. Sub-
atituting their values in the hydrostatic equation, we must integrate it and determine the
oonitont by the name conditions as before.

The condition for stable equilibrium is merely that dp/di shall not be anywhere positive.
Until some definite problem is proposed, no more can be done with the equation.

[39/10/88. — At Dr Murray's request I have calculated, from the data given in his paper:
** On the Height of the Land, and the Depth of the Ocean " (ScoUish Geographical Magazine^
vol. IV. pp. 1 — 41, 1888), that the whole depression of the ocean level, due to compression, is about

116 feet only.

If water ceased to be compressible, the effect would be to submerge some 2,000,000 square miles
of land, about 4 per cent, of the whole.]

Lxn,]

LXII,

OPTICAL NOTES.

[Proceedings of the Royal Society of Edinburgh, 16 January^ 1881.]

1* On a Singular Phenonisnon produced by some old Window-Panes,

A FIGURE, ilkistrating the action of a cylindrical lens, which was inserted in a
recent page of these Proceedings ^ has reminded me of my explanation of a phenomenon
which I have repeatedly seen for more than twenty years in the College. When
sunlight enters my apparatua-room through a vertical chink between the edge of the
blind and the window-frame, the line of light formed on the wall or floor shows a
well-marked Idnh Similar phenomena, though not usually so well marked, are often
seen in old houses, when the sun shines through the chinks of a Venetian blind.
They are obviously due to inequalities (buirs-eyes) in the glass which wag used more
than a generation ago for window-panes. It is evident that the focal length of
successive annuli of such a piece of glass, treated as a lens, increases from the central
portion to the circumference, whei*e it becomes infinite. For an approximate study of
its behavionr we may assume that the focal length of an annulus of radius r is
6^/(a — r), where a is the extreme radius, at which the aides of the pane become
parallel. Suppose sunlight, passing through a narrow slit, to fall on such a lens at
a distance e from its centre, and to be received on a screen at a distance c from
the lens. It is easy to sea that the polar equation of the illuminated curve on the
screen is (the pole being in the axis of the lens)

e sec 9 f 1^ ax

p = vj— (ac~b^^ce sec 0),

This curve can be readily traced by points for various values of the constants. In
fiw3t, if r be the radius vector of a straight line, the vector of any one of these
curves (drawn in the same direction) is proportional to t(A — r), and the curve can
therefore be constructed from a straight line and a circle* Here the value of .^ is
(oc — b^jh ; i*e*, it is a fourth proportional to c, a, and the distance of the screen

70 OPTICAL NOTES. [lXII.

fix)m the focus of the central portion of the lens. When A is small compared with

the least value of r, the curve has a point resembling a cusp, but as A increases

the kink appears. This is easily observed by gradually increasing the distance of the

screen from the lens; and the traced curves present forms which are precisely of the
general character of those observed.

2. On the Nature of the Vibrations in Common Light,

One of the few really unsatisfactory passages in Airy's well-known "Tract" on
the Undulatory Theory of Optics is that which discusses the nature of common light.
To explain the production of Newton's rings in homogeneous light to the number of
several thousands, it is necessary that at least several thousand successive waves
should be almost exactly similar to one another. On the other hand, we cannot
suppose the vibrations (which will in general be elliptic) to be similar to one another
for more than a small fraction of a second; if they were so, we should see colour
phenomena in doubly refracting plates by the aid of an analysing Nicol only.

And, moreover, the nature of the vibration can have no periodic changes of a
kind whose period amounts to a moderate fiuction of a second. Nor can it have a
slow progressive change. Either of these would lead to its resolution into rays of
different wave-lengths. Airy suggests, as consistent with observation, some thousand
waves polarized in one plane followed by a similar number polarized in a plane at
right angles to the first. But no physical reason can be assigned for such an hypothesis.

The difficulty, however, disappears if we consider the question from the modern
statistical point of view, as it is applied for instance in the kinetic theory of gases.
We may consider first a space average taken for the result due to each separate
vibrating particle near the surface of a luminous body. When we remember that,
for homogeneous light, of mean wave-length, a million vibrations occupy only about
one five hundred millionth of a second; it is easy to see that the resultant vibration
at any point may not sensibly vary for a million or so of successive waves, though
the contributions frt)m individual particles may very greatly change. But when we
consider the time average of about a hundred millions of groups of a million waves
each, all entering the eye so as to be simultaneously perceptible, — in consequence of
the duration of visual impressions, — we see that the chances in favour of a deviation
from apparently absolute uniformity are so large that, though possible, such uniformity
is not to be expected for more than a very small fraction of a second. The im-
probability of its occurrence for a single second is of the same nature as that of
the possible, but never realised, momentary occurrence of a cubic inch of the air in
a room filled with oxygen or with nitrogen alone.

[Added; May 1, 1882. — I am indebted to Professor Stokes for a reference to his
paper "On the Composition and Resolution of Streams of Polarized Light from
Diflferent Sources" (Camb. Phil. Trans., 1852), in which the nature of common light
is very fully investigated. I find I was not singular in my ignorance of the contents
of this paper, as the subject has quite recently been proposed as a Prize Question
by a foreign Society.]

LXIII.]

LXIII.

ON A METHOD OF INVESTIGATING EXPERIMENTALLY THE
ABSORPTION OF RADIANT HEAT BY GASES.

(Read by Sir W. Thomson at the B. A, Meeting at Southampton.)
[Nature, October 26, 1882.]

There are grave objections, which have been only partially overcome, to almost
all the processes hitherto employed for testing the diathermancy of vapours. These
arise chiefly from condensation on some part of the apparatus. Thus when rock-salt
is used, an absorbent surface-layer may be formed; and, when the pile is used with-
out a plate of salt, the eflFect of radiant heat may be to cool it (the pile) by the
evaporation of such a surface film. The use of intermittent radiation is liable to
the same objection.

Some time ago it occurred to me that this part of the difficulty might be got
rid of by dispensing with the pile, and measuring the amount of absorption by its
continued eflfects on the volume and pressure of the gas or vapour itself

Only preliminary trials have, as yet, been made. They were carried out for me
by Prof. MacQregor and Mr Lindsay. Their object was first to find whether the

8TEAM

method would work well, second (when this was satisfactorily proved) to find the best
form and dimensions for the apparatus.

72 ON A METHOD OF INVESTIGATING THE ABSORPTION OF HEAT. [lXIIL

The rough apparatus is merely a double cylinder, placed vertically. Cold water
circulates in the jacket, and steam can be blown into the double top. The changes
in the presHure of the gas are shown by a manometer U tube at the bottom, which
contains a liquid which will not absorb the content& This apparatus was 4 feet long,
with 2 inches internal radius. The results of a number of experiments show that
it should be shorter and much wider. The former idea I was not quite prepared for,
the latter is obvious.

The effects on the manometer are due to five chief causes: —

1. Heating of the upper layer of gas by contact with lid.

2. Cooling „ „ „ „ „ sides.

3. Heating of more or less of the column by absorption.

4. Cooling of do. by radiation.

5. „ „ contact.

(1) and (2) only are present in a perfectly diathermanous gas, and in a perfectly
adiathermanous gas or vapour.

All five are present in a partially diathermanous gas or vapour.

The preliminary experiments show that the manometer effect is only very slightly
less for dry defiant gas than for dry air, while moist air shows a markedly smaller
effect than either of the others.

This is conclusive as to the absorption of low radiant heat by aqueous vapour,
but it shows also that the absorption is so small as to take place throughout the
whole column.

Even with the present rude apparatus I hope soon to get a very accurate
determination of the absorbing power of aqueous vapour, by finding in what pro-
'portions defiant gas must be mixed with air to form an absorbing medium equivalent
to saturated air at different temperatures.

I have to acknowledge valuable hints fix)m Prof. Stokes, who, before I told him
the results I had obtained (thus knowing merely the nature of the experiments)
made something much higher than a guess, though somewhat short of a prediction,
of the truth.

In these preliminary trials no precaution was taken to exclude dtist. The results,
therefore, are still liable to a certain amount of doubt, as Mr Aitken's beautiful
experiments have shown.

The paint of the method is that there can be no question of surfiewse-layers.

[Since the above was written, Messrs MacQregor and Lindsay have made an
extended series of experiments with dry and moist air, and with mixtures of dry air
and olefiant gas in different proportions. The cylinder employed was 9 inches in
nuJiuh. The results will soon be communicated to the Royal Society of Edinburgh.]

LXTV.]

LXIV.

L ON THE LAWS OF MOTION. PART I.

{Proceedings of the Ra^al Society of Ediiiburgk, December Id, 1882,]

The substance of part at least of this paper was given in 1876 as an evening
lecture to the Britieh Association at its Glasgow meeting. [Antdf No, XXXVII.]

While engaged in writing the article " Mechanics '* for the Enci/. Brit.y I had to
consider carernlly what basifl to adopt, and decided that the time had not yet come
in which (at lea^t in a semi-popular article) Newton's laws of motion could be
modified. The article was therefore based entirely on these laws, with a mere hint
towards the end that in all probability they would soon require essential modification.
It is well, however, that the question of modification should now he considered; and
this should be done, not in a popular essay but, before a scientific society.

The one objection to which, in modem times^ that wonderfully complete and
compact system is liable, is that it is expressly founded on the conception of what
is now called *' force " as an agent which " compels '' a change of the state of rest
or motion of a body. This is part of the first law, and the second law is merely a
definite statement of the amount of change produced by a given force.

(Next comes a digression as to what was Newton s expression for what we now
mean by the word force, when it is used in the correct signification above,}

There can be no doubt that the proper use of the term for<^ in modem science
is that which is implied in the statement — Force is whatever changes a body's state
of rest or motion. This is part of the tirat law^ of motion. Thus we see that force
is the English equivalent of Newton^s term vu impressa. But it is also manifest
that, on many occasions, but ofnltf tvhere his meaning admitted of no douhty Newton
omitted the word impressa and used vis alone, in the proper senae of force. In
other cases he omitted the word impressa, as being implied in some other adjective
Buch as centripeta, gravitans, &c„ which he employed to qualify the word im. Thus
(Lemma X,) he says: — Spatia, quw corpus urgente qudcunque vi finitd desaihit, i&c.

IE LAWS OF MOTION.

[lxiv.

It is needless to multiply examples. But that this is the true state of the case is
made absolutely certain by the following ^ —

Dejijiitio IV, Vis impreaaa est actio in corpus exerdta, ad tmdandum ejm staium
vel qttieseendi vel fnovendi uniformiier in directum.

Contrast this with the various senses in which the word vis is used in the
comment which immediately follows, viz. :—

Constitit hsec vis in actiooe sola, neque post actionem permauet in corpore.
Persevemt enira corpus in statu omni novo per solam vim inertiie. Est autem via
iuipressa diversarum originum, ut ex ictn, ex presaioue, ex vi centripeta*

These passages are translated by Motte as below: —

*' Definition IV, An impressed force is an action emrted upmi a bod^, in order
to change its slate, either of real, or of moving umformlg fojward m a right line."

'* This force consists in the action only, and remains uo longer in the body when
the action is over For a body maintains every new state it acquires, by its vis
inertim only. Impressed forces are of different origins; as from percussion, from
pressure, from centripetal force,**

The difficulty which Motta here makes for himself, and which he escapes &om
only by leaving part of the passage in the original Latin^ is introduced solely by
his use of the word force as the equivalent of the Latin vis.

If we paraphrase the passage as follows, with attention to Newton's obvious
meaning, this diflSculty disappears, or rather does not occur : —

" This kind of vis consists in/' &a For the *^ body continues . , . • by the via
of inertia/' &c. However, we may quote two other passages of Newton bearing
definitely on this point

Definitio III, Materiw vis insita est potentm rmistendi, qud corpus timimquodque,
quantum in se est^ perseverat in statu suo vel quiescendt vel movendi uniformiter in
directum.

It is perfectly clear that, in this passage, the phrase vis insita is one idea, not
two, and that vis cannot here be titans la ted by force. Yet Motte has

**The ms insita, or innate force of matter, is/' &c.

Definitio V, Vis centripeta est, qud cojyora vet^stis punctnm aliquod, tanquam ad
cmtrum, midique traJiuniur, impellmitur, vel utcunique t&ndunt

It is obvious that the qualifying term centripeta here includes the idea suggested
by impressa, defining in fact the direction of the vis, and therefore implying that its
origin is outside the body.

After what has just been said, no farther comment need be added to show the
absurdity of the terms accelerating force, innate force, impressed force, &c. All of
these have arisen simply from mistranslation. Vis, by itself, is often used for force ;
but vis acceleratf^y vis impressa, vis insita, and other phrases of the kind, must be
taken as wholes; and, in them, vis does not mean force.

The absurdity of translating the word vis by force comes out still more clearly
when we think of the term vis tftVa, or living force as it is sometimes called ; a
name for kinetic eneig^y, which depends on the unit of length in a different way
from force. It must be looked upon as one of the most extraordinary instances of
Newton's clearness of insight that, at a time when the very terminology of science

LXIV.j

ON THE LAWS OF MOTION,

was only as it were shaping itself, he laid down with such wonderful precision a
system absolutely self-coosistentp

From the passages just quoted, taken in conjunctioD with the second law of
motion, we see that (as above stated) in New ton *a view —

Force is whatever causes (but not, or tends to oattse) a change in a imitf^s state
0f rest or motuni,

Newton gives no sanction to the so-called statical ideas of force. Every force,
in his view, produces its effect. The effects may be such as to balance or compensate
one another; biit there is no balancing of forces.

(Next coraes a discussion as to the objectivity or subjectivity of force. Ajd
abstract of this is given in ^ 288 — 296 of the article above referred to, and therefore
need not be reproduced here,)

But, just as there can be no doubt that force has no objective existence, so
there can be do doubt that the introduction of this conception enabled Newton to
put his Aimomata in their exceedingly simple form. And there would be, even now,
no really valid objection to Newton*a system (with all its exquisite simplicity and
convenience) could we only substitute for the words *' force *' and " action," &c., in the
statement of his laws, words which (like rate or gradient, &c.) do not imply objectivity
or causation in the idea expressed. It is not easy to see how such words could be
introduced ; but assuredly they will be i-equired if Newton's system is to be maintained.
The word stress might, even yet, be introduced for this purpose ; though, like force,
it has come to be regarded as something objective. Were this possible, we might
avoid the necessity for any very serious change in the /or^ti of Newton's system. I
intend, on another occasion, to consider this question. How complete Newton's
statement is, is most easily seen by considering the so-called ** additions '* which have
been made to it.

The second and third laws, together with the scholium to the latter, expressly
include the whole system of " effective forces," &c. for which D'Alerabert even now
receives iu many quai^ters such extraordinarily exaggerated credit. The ** reversed
effective force "^ on a particle revolving uniformly in a circle is nothing but an old
friend—" centrifugal force." And even this phantom is still of use, in skilled Imnds,
in forming the equations for certain cases of motion.

The chief arguments for and against a modem modification of the laws of motion
ai-e therefore as follows — where we must remember that they refer exclusively to the
elementary teaching of the subject, and have no application to the case of those
who have sufficient knowledge to enable them to avoid the possible dangers of
Newton's method : —

I. For. Is it wise to teach a student by means of the conception of force, and
then as it were to kick down the scaffolding by telling him there is no such thing?

IL Against. Is it wise to give up the use of a system, due to such an
altogether exceptional genius as that of Newton, and one which amply suffices for
nil practical purposes, merely because it owes part of it^ simplicity and compactness
to the introduction of a conception which, though strongly impressed on us by our
muscular sense, corresponds to nothing objective ?

10—2

ON THE LAWS OF MOTION*

[lxiv.

Everyone must answer these questions for himself, and his answer will probably
be deterrained quite as much by his notions of the usefulness of the study of
natural philosophy as by his own idiosyncrasies of thought. To some men physics is
an abomination, to others it is something too trivial for the human intellect to waste
ifcB energies on. With these we do not reason. To others again all its principles
are subjects of Intuitive perception. They could have foreseen the nature of the
physical world, and they kn^m that it could not have been otherwise than they
suppose it to be. Many minds find delight in the contemplation of the three kinds
of lever; others in the ingeniously disguised assumptions in Duchayla*s ''proof" of
the parallelogram of forces ; some, perhaps, even in the wonderful pages of Vis
InertuB Victa! The case of these men is only not more hopeless than that of the
former classes because it is impossible that it could be so.

But those who desire that their scientific code should be, as far as possible,
representative of our real knowledge of objective things, would undoubtedly prefer to
that of Newton a system in which there is not an attempt, however successful, to
gain simplicity by the introduction of subjective impressions and the corresponding
conceptions.

In the pre-sent paper simplicity of principle, only, is sought for; and the mathe-
matical methods employed are those which appeared (independent altogether of the
question of their fitness for a beginner) the shortest and most direct, A second part
will he devoted to simplicity of method for elementary teaching,

(1) So far £^ our modem knowledge goes there are but two objective things
in the physical world^ — matter and energy* Energy cannot exist except as associated
with matter, and it can be perceived and measured by us only when it is being
transferred, by a *' dynamical transaction/* from one portion of matter to another. In
such transferences it is often *' transformed " ; but no process has ever been devised
or observed by which the quantity, either of matter or energy, has been altered.

(2) Hence the true bases of our subject, so far as we yet know, are^ —

1. Conservation of matter,

2. Conservation of energy*

3. That property (those properties ?) of matter, in virtue of which it is the
necessary vehicle, or as the case may be, the storehouse, of energy,

(S) The third of these alone presents any difficulty* So long as energy is
obviously kinetic, this property is merely our old friend iiwrtia. But the mutual
potential energy of two gravitating masses, two electrified bodies, two currents, or two
magnets, is certainly associated (at letist in part, and in some as yet unknown way)
with matter, of a kind not yet subjected to chemical scrutiny, which occupies the
region in which these masses, &c., are situated. And, even when the potential
energy obviously depends on the strain of a portion of ordinary matter, as in
compressed air, a bent spring, a deformed elastic solid, &c., we can, even now, only
describe it as due to " molecular action/' depending on mechanism of a kind as yet
unknown to us, though, in some cases, at least partially guessed at.

LXIV.]

ON THE LAWS OF MOTION,

(4) The necessity for the explicit assumption of the third principle* and a hint
at least of the limits within which it must be extended, appear when we considt^r
the very simplest case of motion, viz., that of a lone particle moving in a region
in which its potential energy is the same at every point For the conservation of
energy telb ua merely that its speed is unaltered. We know, however, that thie is
only part of the truth : the velocity is constant. It will bi* seen later that this ha.s
most important dynamical consequences in various directions.

(The remarkable discussion of this point by Clerk-Maxwell is then referred to, in
which it is virtually shown that, were things otherwise, it would be possible for a
human mind to have knowledge of absolute position and of absolute velocity*)

(5) But MaxwetFs reasoning is easily seen to apply equally to any component
of the velcx*ity. Hence, when we come to the case in which the potential energy
depends on the position, the only change in the particles motion at any instant is
a change of the speed in the normal to the equi potential surface an which the
particle is at that instant situated. The conservation of energy assigns the amount
of this change, and thus the motion is completely determined* In fact, if w be
perpendicular to the equipotential surface, the equation

im {if=* + f + i^) + V= const.

gives

mx = —

for fjf and i are independent of ^. Generally, in the more expressive language of
quaternions,

In fact, this problem is precisely the same as was that of the motion of a luminous

corpuscle in a non-homogeneous medium, the speed of passing through any point of
the medium being assigned.

(6) It is next shown that the above inertia -condition (that the velocity parallel
to the equipotential surface is the same for two successive elements of the path) at
on<^ leads to a '' stationary " value of the sum of the quantities t:c^ for each two
successive elements, and therefore for any finite arc, of the path. This is, for a
single particle, the Principle of Lemt Action^ which is thus seen to be a direct
consequence of inertia,

(It is then shown that the results above can be easily extended to a particle
which has two degrees of freedom only,)

But it is necessary to remember that, in these cases, we take a partial view of
the circumstances ; for a lone particle cannot strictly be said to have potential
energy, nor can we conceive of a constraint which does not depend upon matter
other than that which is constrained* Hence the true statement of such cases
requires further investigation.

(7) To pass to the case of a system of free piotiolfis we requirs dome quasi*
kinematical preliminaries. These are summed up in the following self-evident

78 ON THE LAWS OP MOTION. [lXIV.

proposition: — If with each particle of a system we associate two vectors, e.g., ©i, ^i,
with the mass m,, &a, we have

2me4> = 2 (m) . ©o^o + 2m5^,

where 0 = 00 + ^,

and 2m0 = 2 (m) . ©o,

so that Bo fti^d 4>o are the values of B and ^ for the whole mass collected at its
centre of inertia; and 0, <f>, those of the separate particles relative to that centre.

(8) Thus, if e=P = Po-f p be the vector of m, ^ = e=P = Po + p, its velocity,
we have

2mPP = 2 (m) . PoPo + Impp,

the scalar of which is, in a diflferentiated form, a well-known property of the centre
of inertia. The vector part shows that the sum of the moments of momentum
about any axis is equal to that of the whole mass collected at its centre of inertia,
together with those of the several particles about a parallel axis through the centre
of inertia.

If e = * = P,

we have SmP* = S (m) . PJ + ^mp^,

t.e., the kinetic energy, referred to any point, is equal to that of the mass collected
at its centre of inertia, together with that of the separate particles relative to the
centre of inertia.

If we int^rate this expression, multiplied by dt^ between any limits, we obtain
a similar theorem with regard to the Action of the system.

Such theorems may be multiplied indefinitely.

(9) From those just given, however, if we take them along with 3 above, we
see at once that, provided the particles of the system be all free, while the energy
of each is purely kinetic and independent alike of the configuration of the system
and of its position in space,

1. The centre of inertia has constant velocity.

2. The vector moment of momentum about it is constant.

3. So is that of the system relative to any uuiformly moving point.

4. ^fmvds is obviously a minimum.

(10) The result of (7) points to an independence between two parts of the
motion of a qrstem, ic, that relative to the centre of inertia, and that of the
whole mass supposed concentrated at the centre of inertia. Maxwell's reasoning is

LXIV.] ON THE LAWS OF MOTION. 79

applicable directly to the latter, if the system be self-contained, %.e., if it do not
receive energy from, or part with it to, external bodies. Hence we may extend the
axiom 3 to the centre of inertia of any such self-contained system, and, as will
presently be shown, also to the motion of the system relative to its centre of inertia.
This, though not formally identical with Newton's Lex III., leads, as we shall see,
to exactly the same consequences,

(11) If, for a moment, we confine oiu* attention to a free system consisting of
two particles only, we have

WiPi + m^3 = (r/ii + mj) a,

or rwi/h + 7n^a = 0 (1).

This must be consistent with the conservation of energy, which gives

i(m^J + tw^J)=/(r(p,-p,)) (2),

since the potential energy must depend (so far as position goes) on the distance
between the particles only. Comparing (1) and (2) we see that we may treat (2)
by partial diflferentiation, so far as the coordinates of mi and ?n, are separately
concerned. For we thus obtain

^Pi = ^Pi •/=/'• 0^(pi-p2),

Each of these, again, is separately consistent with the equation in § 5 for a
lone particle. Hence, again, the integral /(rwiVicbi + rMiVacfoa) has a stationary value.

Hence also, whatever be the origin, provided its velocity be constant,

lmVp}} = 0,

Thus, even when there is a transformation of the energy of the system, the
results of § 9 still hold good. And it is to be observed that if one of the masses,
say m^, is enormously greater than the other, the equation

?nipi4- ^2^2=0

shows that p^ is excessively small, and the visible change of motion is confined to
the smaller mass. Carrying this to the limit, we have the case of motion about a
(so-called) "fixed centre." In such a case it is clear that though the momenta of the
two masses relative to their centre of inertia are equal and opposite, the kinetic
energy of the greater mass vanishes in comparison with that of the smaller.

These results are then extended to any self-contained system of free particles,
and the principle of Varying Action follows at once. It is thus seen to be a general
expression of the three propositions of § 2 above.

(12) So far as we have yet gone, nothing has been said as to how the mutual
potential energy of two particles depends on their distance apart. If we suppose it
to be enormously increased by a very small increase of distance, we have practically

80 ON THE LAWS OF MOTION. [lXIV.

the case of two particles connected by an inextensible string — as a chain-shot. But
from this point of view such cases, like those of connection by an extensible string,
fall under the previous categories.

The case of impact of two particles falls under the same rules, so &r as motion
of the centre of inertia, and moment of momentum about that centre, are concerned.
The conservation of energy, in such cases, requires the consideration of the energy
spent in permanently disfiguring the impinging bodies, setting them into internal
vibration, or heating them. But the first and third of these, at least, are beyond the
scope of abstract dynamics.

(13) The same may be said of constraint by a curve or surface, and of loss of
energy by friction or resistance of a medium. Thus a constraining curve or surface
must be looked upon (like all physical bodies) as deformable, but, if necessary, such
that a very small deformation corresponds to a very great expenditure of energy.

(14) To deal with communications of energy from bodies outside the system, all
we need do is to inclvde them in the system. Treat as before the whole system
thus increased, and then consider only the motion of the original parts of the
system. This method applies with perfect generality whether the external masses be
themselves free, constrained, or resisted.

(15) Another method of appljdng the same principles is then given. Starting
from the definition dA = l,mSpdp, the kinematical properties of A are developed. Then,
by the help of § 2, these are exhibited in their physical translations.

(16) The paper concludes with a brief comparison of the fundamental principles
of the science as they have been introduced by Newton, Lagrange, Hamilton, Peirce,
Kirchhoff, and Clerk-Maxwell, respectively; and also as they appear in the unique
Vortex-system of Thomson.

'txv.]

LXV.

JOHANN BENEDICT LISTING.

[Naiwre, February/ 1, 1883.]

One of the few remaining linkB that Htill continued to connect our time with
that in which Gauss had made Gottingen one of the chief intellectual centres of the
civilised world has just been broken by the death of Listing.

If a man's services to science were to be judged by the mere number of his
publiBhed papers, Listing would not stand very high. He published little, and (it
would seem) was eyen indebted to another for the publication of the disco^eiy by
which he is most widely known. This is what is called, in Physiological Optics,
Listings Law. Stripped of mere technicalities, the law asserts that if a person whose
head remains fixed turns im eyes &om an object situated directly in front of the
face to another, the fioal position of each eye-ball is such as would have been pro-
duced by rotation round an axis perpendicular alike to the ray by which the first
object was seen and to that by which the second is seen. '* Let us call that line
in the retina, upon which the visible horizon is portrayed when we look, with upright
head, straight at the visible horizon, the horizon of the retina. Now any ordinary
person would naturally suppose that if we» keeping our head in an upright position,
turn our eyes so as to look, say, up and to the right, the horizon of the retina
would remain parallel to the real horizon. This is, however, not so. If we turn our
eyes straight up or straight down, straight to the right or straight to the left, it
is so, but not if we look up or down, and also to the right or to the left* In
these cases there is a certain amount of what Helmholtz calls 'wheel -turning* {Rad-
drehung) of the eye, by which the horizon of the retina is tilted so as to make
an angle with the real horizon. The relation of this 'wheel-turning* to the above*
described motion of the optic axis is expressed by Listing*s law, in a perfectly simple
way, a way so simple that it is only by going back to what we might have thought

82 JOHANN BENEDICT LISTING. [lXV.

nature should have done, and from that point of view, looking at what the eye really
does, and considering the complexity of the problem, that we see the ingenuity of
Listing's law, which is simple in the extreme, and seems to agree with fact quite
exactly, except in the case of very short-sighted eyes." The physiologists of the time,
unable to make out these things for themselves, welcomed the assistance of the
mathematician. And so it has always been in Germany. Few are entirely ignorant
of the immense accessions which physical science owes to Helmholtz. Yet few are
aware that he became a mathematician in order that he might be able to carry out
properly his physiological researches. What a pregnant comment on the conduct of
those "British geologists" who, not many years ago, treated with outspoken contempt
Thomson's thermodynamic investigations into the admissible lengths of geological periods !

Passing over about a dozen short notes on various subjects (published chiefly in
the Qottingen Nachrichten), we come to the two masterpieces, on which (unless, as
we hope may prove to be the case, he have left much unpublished matter) Listing's
fame must chiefly rest They seem scarcely to have been noticed in this country,
until attention was called to their contents by Clerk-Maxwell.

The first of these appeared in 1847, with the title Vorstudien zur Topologie, It
formed part of a series, which unfortunately extended to only two volumes, called
Gottinger Studien, The term Topology was introduced by Listing to distinguish what
may be called qualitative geometry from the ordinary geometry in which quantitative
relations chiefly are treated. The subject of knots furnishes a tjrpical example of these
merely qualitative relations. For, once a knot is made on a cord, and the free ends
tied together, its nature remains unchangeable, so long as the continuity of the string
is maintained, and is therefore totally independent of the actual or relative dimensions
and form of any of its parts. Similarly when two endless cords are linked together.
It seems not unlikely, though we can find no proof of it, that Listing was led to
such researches by the advice or example of Gauss himself; for Gauss, so long ago
as 1833, pointed out their connection with his favourite electromagnetic inquiries.

After a short introductor)- historical notice, which shows that next to nothing
had then been done in his subject, Listing takes up the very interesting questions
of Inversion {Undcehrung) and Perversion (Verkehrung) of a geometrical figure, with
specially valuable applications to images as formed by various optical instruments.
We cannot enter into details, but we paraphrase one of his examples, which is par-
ticularly instructive: —

**A man on the opposite bank of a quiet lake appears in the watery mirror perverted,
while iu an astronomical telescope he appears inverted. Although both images show the
head down and the feet up, it is the dioptric one only which : — if we could examine it : —
would, like the original, show the heart on the left side; for the catoptric image would
show it on the right side. In type there is a difference between inverted letters and
per\-erted ones. Thus the Roman V becomes, by inversion, the Greek A; the Roman R
perverted becomes the Russian H; the Roman L, perverted and inverted, becomes the
Greek r. Compositors read perverted type without difficulty: — many newspaper readers in
England can read inverted type. ^ ^ ^ The numends on the scale of Gauss' Magnetometer
must^ in order to appear to the observer in their natural position, be both perverted and
in\*erted, in consequence of the perversion by reflection and the inversion by the telescope/'

LXV^.]

JOHAIJN BENEDICrr LISTING,

Liatitig next takes up helices of various kinds, and discusses the qaeatioii as to
which kind of screws should be called right-handed. His examples are chiefly taken
from vegetable spirals, such as those of the tendrils of the convolvulus, the hop, the
vine, &c.| some from fir-conea, some from snail-shells, othera from the "anair' in clock*
work. He points out in great detail the confusion which has been introduced in
botanical works by the want of a common nomenclature, and finally proposes to
found such a nomenclature on the forms of the Greek S and X,

The consideration of double-threaded screws, twisted bundles of fibres, &c., leads
to the general theory of pmradromic winding. From this follow the properties of a
large class of knots which form "clear coils/* A special example of these, given by
Listing for threads, is the well-known juggler's trick of slitting a ring-formed band up
the middle, through its whole length, so that instead of separating into two parts, it
remains in a continuous ring* For this purpose it is only necessary to give a strip
of paper one kalf-tmBt before pasting the ends together If three half-twists be given,
the paper still remains a continuous band after slitting, but it cannot be opened into
a ring, it is in fact a trefoil knot. This remark of Listing's forms the sole basis of
a work which recently had a large sale in Vienna: — showing how, in emulation of
the celebrated Slade, to tie an irreducible knot on an endless string !

Listing' next gives a few examples of the application of his method to knots.
It is greatly to be regretted that this part of his paper is so very brief; and that
the opportunity to which he deferred farther development seems never to have arrived.
The methods he has given are, as is expressly stated by himself, only of limited
application. There seems to be little doubt, however, that he was the first to make
any really successful attempt to overcome even the preliminary difBcultieeS of this
unique and exceedingly perplexing subject

The paper next gives examples of the curious problem: — Given a figure consisting
of lines, what is the smallest number of continuous strokes of the pen by which it
can be described, no part of a line being gone over more than once? Thus, for
instance, the lines bounding the 64 squares of a chess-board can be drawn at 14
separate pen strokes. The solution of all such questions depends at once on the
enumeration of the points of the complex figure at which an odd number of lines
meet

Then we have the question of the "area" of the projection of a knotted curve
on a plane ; that of the number of interlinkings of the orbits of the asteroids ; and
finally some remarks on hemihedry in crystals. This paper, which is throughout
elementary, deserves careful translation into English very much moi'e than do many
Gennan writings on which that distinction has been conferred.

We have left little space to notice Listing's greatest work, Der Genatis rdumliche)*
Cumplewe (Gottingen Abkandlungen, 1861). This is the less to be regretted, because,
as a wholcp it is far too profound to be made popular; and, besides, a bir idea of
the nature of its contents can be obtained from the introductory Chapter of Maxwell's
great work on Electricity. For there the importance of listing's Cycloais, Periphractic
Regions, &c., is fully recognised.

One point, however, which Maxwell did not require, we may briefly mention.

In most works on Trigonometry there is given what is called Eulers Theorem

11—2

84 JOHANN BENEDICT LISTING. [lXV.

about polyhedra: — viz. that if 8 be the number of solid angles of a polyhedron (not
self-cutting), F the number of its faces, and E the number of its edges, then

8 + F=-E+2.

The puzzle with us, when we were beginning mathematics, used to be "What is this
mysterious 2, and how came it into the formula?" Listing shows that this is a mere
case of a much more general theorem in which comers, edges, faces, and regions of
space^ have a homogeneous numerical relation. Thus the mysterious 2, in Euler's
formula, belongs to the two regions of space: — ^the one enclosed by the polyhedron,
the other (the Amplexum^ as Listing calls it) being the rest of infinite space. The
reader, who wishes to have an elementary notion of the higher forms of problems
treated by Listing, is advised to investigate the modification which Euler's formula
would undergo if the polyhedron were (on the whole) ring-shaped: — as, for instance,
an anchor-ring, or a plane slice of a thick cylindrical tube.

LXVI,]

LXVI.

LISTING'S TOPOLOGIE.

INTRODUCTORY ABDRESS TO THE EDINBURGH MATHEMATICAL SOCIETY^

November 9, 1883.

[^Philosophical Magnsiney Janimry, 1884 J

Some of you may have been puzzled by the advertised title of this Address. But
certainly not more puzzled than I was while seekiug a title for it*

I intend to speak (necessarily from a very elementary point of view) of those
space -relations which are independent of rneamt^re^ though not always of number ^ and
of which perhaps the very best instance is afforded by the various convolutions of a
knot on an endless string or wire. For, once we have tied a knot, of whatever com-
plexity, on a string and have joined the free ends of the string together, we have an
arrangement which, however its apparent form may be altered (as by teazing out,
tightening, twisting, or flyping of individual parts), retains, until the string ia again cut,
certain perfectly definite and characteristic properties altogether independent of the
relative lengths of its various convolutions.

Another excellent example is supplied by Cnim Brown's chemical Graphic FormukB.
These, of course, do not pretend to represent the actual positions of the constituents of
a compound molecule, but merely their relative connection.

From this point of view all figures, however distorted by projection &c., are con-
sidered to be uuchanged. We deal with grouping (as in a qidncunx), with motion by
starts (as in the chess- knight's move), with the necessary relation among numbers of
intersections, of areas, and of bounding lines in a plane figure ; or among the numbers
of comers, edges, faces, and volumes of a complex solid figure, &c.

For this branch of science there is at present no definitely recognized title except
that suggested by Listing, which I have therefore been obliged to adopt GeonistHe
der Lage hm now come, like fche Giotndtrie de Positwn of Carnot, to mean something

LISTINGS TOPOLOGIE.

[lxvi.

very different from our present subject; and the Geom^na sit'As of Leibnitz, though
intended (as Listing shows) to have specially designated it, turned out| in its inventor s
hands, to be almost unconnected with it. The subject is one of very great importance,
and has been recognized as such by many of the greatest investigatorB, including Gauss
and others ; but each, before and after Listing's time, has made his separate contri-
butions to it without any attempt at establishing a connected account of it as an
independent branch of science.

It is time that a distinctive and unobjectionable name were found for it; and
once that is secured, there will soon be a crop of Treatises. What is wanted is an
erudite, not necessarily a very original, mathematician. The materials already to hand
are very numemus. But it ia not eaay {in English at all events) to find a name
for it mthout coining some altogether new word from Latin or Greek roots. Topology
has a perfectly definite meaning of its own, altogether unconnected with our subject.
Position, with our mathematicians at least, has come to imply measure. Situation is
not as yet so definitely associated with measure; for we can speak of a situation to
right or left of an object without inquiring how far off. So that till a better term
is devised, we may call our subject, in our own language, the Science (not the GeomMry,
for that implies measure) of Situation.

Listing, to whom we owe the first rapid and elementary, though highly suggestive,
sketch of this science, as well as a developed investigation of one important branch
of it, was in many respects a remarkable man. It is to be hoped that much may
be recovered from his posthumous papers ; for there can be little doubt that in con-
sequence of his diBinclination to publish (a disinclination so strong that his best-known
discovery was actually published for him by another), what we know of hia work is
a mere fragment of the results of his long and active life.

In what follows I shall not confine my illustrations to those given by Listing,
though I shall use them freely; but I shall also introduce such as have more
prominently forced themselves on my own mind in connection mainly with pure physical
subjects. It is nearly a quarter of a century since I ceased to be a Professor of
Mathematica ; and the bnvnches of that great science which I have since cultivated
are especially those which have immediate bearing on Physics, But the subject before
us is 80 extensive that, even with this restriction, there would be ample material, in
ray owu reading, for a whole series of strictly eleraentar)" lectures.

I ought not to omit to say, before proceediog to our business, that it is by no
means creditable to British science to find that Listing's papers on this subject — the
VoTStudien zur Topulogie (GoUinger Studien, 1847), and Ber Cmisiis rdundicher Complete
(Gottingen Abhandlungen, 1861) — have not yet been rescued from their most undeserved
obscurity, and published in an English dress, especially when so much that is com-
pai-atively worthless, or at least not so worthy, has already secured these honoui^. I was
altogether ignorant of the existence of the VGrstudien till it was pointed out to me
by Clerk -Max well, after I had sent him one of my earlier papers on Knots; and
I had to seek, in the Cambridge University Library, what was perhaps the only then
accessible copy.

(I) Down and Up are at once given us by gravity. They are defined as the

LX?I,]

LISTINGS TOPOLOGIE,

directioE in which a stone fallF, or in which a plummet hangs, and its reverse. Even
below-decks, when the vessel is lying over under a steady breeze, and we ** have our
sea-legB on" we instinctively keep our bodies vertical, without any thought of setting
ourselves perpendicular to the cabio-floor. And this definition holds in a?ery region
of space where the earth's attraction is the paramount force. In an imaginary ca\ity
at the earth's centre the terms would cease to have any meaning.

East, in the sense of " Orients' is the quarter in which the sun rises ; and iAis
distinction is correct all over the earth except at the poles, where it has no meaning*
But if we were to define South as the region in which the sun is seen at midday,
our definition would be always wrong if we were placed beyond the tropic of Capricorn,
and at particular seasons even if we were merely beyond that of Cancer. Still there
is a certain comensas of opinion w*hich enables all to understand what is meant by South
without the need of any formal definition.

But the distinction between Right and Left is atill more difficult to define. We
must employ some such artifice as " A man*s right side is that which is tinned east*
wards, when he lies on his face with his bead to the north/* For, in the lapse of
ages of development, one may perhaps he right in saying, with Molifere's physician,
^'Nous avQTis changi tout cete"; and mens hearts may have migrated by degrees to the
other side of their bodies, as does one of the eyes of a growing flounder. Or some hitherto
unsuspected superiority of left-handed men may lead to their sole survival ; and then
the definition of the right hand, as that which the majority of men habitually employ
most often, would be false*

I will not speak further of these things^ which I have introduced merely to show
how difficult it sometimes b to formulate precisely in words what every one in his
senses knows perfectly well; and thus to prepare you to expect difficulties of a higher
order, even in the very elements of matters not much more recondite*

(2) But there is a very simple method of turning a man's right hand into his
left, and vice veisd, and of shifting his heart to the right-hand side, without waiting
for the (problematical) results of untold ages of development or evolution. We have
only to look at him with the assistance of a plane mirror or looking-glass, and these
extraordinary transformations are instantly effected. Behind the looking-ghiss the world
and every object in it are perverted {verkehrt, as Listing calls it)* Seen through an
astronomical telescope, everji-hing is inverted merely (utngekehrt). Particular cases of this
distinction, which is of very considerable importance, were of course known to the old
geometers. For two halves of a circle are congruent ; one semicircle has only to be
made to rotate through two right angles in its own plane to be superposable on the
other But how about the halves of an isosceles triangle formed by the bisector of
the angle between the equal sides ? They are equal in every respect except congniency ;
one has to be turned over before it can be exactly superposed on the other-
Listing gives many examples of this distinctioUj of which the following is the
simplest : —

Inversion: — (English) V and (Greek) A*

Pervemon:— (English) R and (Russian) H.

Inversion and perversion: — (English) L and (Greek) V.

88 uausG'B TOPOLOGiE. [ljlyi.

He abo giTes an elaborate discnsBion of the different relatire situations of two
dice whose edges are parallel, taking account of the pamis on the Yarioos side&

When we Jlype a glove (as in taking it off when Teiy wet, or as we skin a hare),
we perform an operation whidi (not describable in English by any shorter phrase than
*" turning outside in '^ changes its diaracter firom a right-hand glore to a left. A
pair of trousers or a so-called revenMe water-proof coat is» after this operati<xi has
been performed, still a pair of trousers or a coat, bat the l^;s or arms are inter-
change ; unless the garments, like those of " Paddius k Corko," are buttoned behind'.

(3) The germ of the whole of this part of the subject lies in the two ways in
which we can choose the three rectangular axes of x, y, z; and is intimately con-
nected with the kinematical theory of rotation of a solid.

Thus we can make the body rotate throu^ two ri^t angles about one axis, so
that each of the other two is inyerted. Such an operation does no! change their relative
situation.

But to invert one only, or all three, of the axes requires that the body should
(as it were) be puUed through itsdf, a process perfectly ccmceivable firom the
kinematical, but not firom the physical, point of view. By this process the relative
situation of the axes is changed.

When we think of the rotation about the axis of x which shall bring Oy where
Oz was, we see that it must be of opposite character in these two cases. And it
is a mere matter of convention which of the two systems we shall choose as our
normal or positive one.

That which seems of late to have become the more usual is that in which a
quadrantal rotation about x (which may be any one of the three) shall change Oy
into the former Oz (%.e. in the cyclical order x, y, z), when it is applied in the
sense in which the earth turns about the northern end of its polar axis. Thus we
may represent the three axes, in cyclical order, by a northward, an upward, and an
eastward line. So that we change any one into its cyclical successor by seizing the
positive end of the third, and, as it were, unscrewing through a quadrants

The hands of a watch, looked at firom the side on which the face is situated,
thus move round in the negative direction ; but if we could see through the watch,
they would appear to move round in the positive direction. This universally employed
construction arises probably firom watch-dials having been originally made to behave
as much as possible like sun-dials, on which the hours follow the apparent daily
course of the sun, i.e. the opposite direction to that of the earth's rotation about
its axis.

(4) This leads us into another very important elementary branch of our subject,

> When a Treatise comes to be written (in English) on this sdenoe, great care will have to be taken
in exactly defining the senses in which such words as inversion, reversion, perversion, &e, are to be employed.
There is much danger of confusion unless authoritative definitions be given once for aU, and not too late,

^ These relations, and many which follow, were iUustrated by models, not by diagrams; and the reader
who wishes fully to comprehend them will find no reason to grudge the little trouble involved in constructing
such models for himself.

lxvl]

LISTING S TOPOLOGIE.

one m which LiBting (it is to be feared) introduced oomplication rather than
simplification, by his endeavours to extricate the botanists from the frightful Dhaos
in which they had involved themselves by their iiTeconcilable descriptions of vegetable
spirals. [He devotes a good many pages to showing how great was this confusion.]

When we compare the tendrils of a hop with those of a vine, we see that
while they both grow up wards » as in coiling themselves round a vertical pole, the
end of the hop tendril goes round with the mn (secundum 8oleni}t that of the vine
tendril ogauiM the mm (contra s&leiu).

Thus the vine tendril forms an ordinary or (as we call it) right-handed screw,
the hop tendril a left-handed screw.

Now, if a point move in a circle in the plane of ys in the positive direction,
and if the circle itself move in the direction of w positive, the resultant path of
the point will be a vine-, or right-handed screw. But if the circle's motion as a
whole, or the motion of the point in the circle, be reversed, we have a left-handed
screw ; while if both be reversed, it remains right-handed. Every one knows the
combination of the rotatory and translatory motions involved in the use of an
ordinary corkscrew; but there are comparatively few who know that a screw is the
same at either end — that it has, in fact, what is called dipolar sf/mmetrtf.

With a view to iissist the botanists, Listing introduced a fancied resemblance
between the threads of the two kinds of (double-threaded) screws and the Greek
letters X and S, for the latter of which he also proposed the long / used as a sign
of integration ; thus XXXk and 2888, or JfJJ.

The first, which is our vine- or right-handed screw, he calls from his point of
view (which is taken in the axis of the screw) laeotrop, the other dmiotrop. He
also proposes to describe them as lambda- or delta-Windimgeti, But it is clear that
all this " makes confusion worse confounded." Every one knows an ordinary sciHiW.
It b right-handed or positive. Hence he can name^ at a glance, any vegetable or
other helix.

(5) A symmetrical solid of revolution, an ellipsoid for instance (whether prolate
or oblate), has, if at rest, dipolar symmetry. But if it rotate about its axis, we can at
unce distinguish one end of the axis from the other, and there is dipolar aaymmietry.

This distinction is dj^amical as well as kinematical, as every one knows who is
oonveraant with gyroscopes or gyrostats*

A Hat spiral spring, such as a watch- or clock-spring, or the gong of an
American clock, if the inner coils be pulled out to one side, becomes a right-handed
screw; if to the other, a left-handed screw. In either case it retains the dipolar
symmetry which it had at first, while plane.

But when we pass an electric current round a circle of wire, we at once give
it dipolar asymmetry. The current appeai-s^ from the one side, to be going round
in the positive direction; from the other, in the negative. This is, in fact, the point
of Ampfere s explanatiori of magnetism.

A straight wire heated at one end has dipolar asymmetry, not only because of
the diflferent temperatures of its ends, but because of the differences of their electric
|>oteDtial (due to the "Thomson effect").

T. II. 12

90 listing's topologie. [lxvi.

The same is generally true of every vector (or directed) quantity, such as a
velocity, a force, a flux, an axis of rotation, &a

(6) An excellent example of our science is furnished by the Quincunx, which is
the basis of the subject of Phyllotdxis in botany, as well as of the arrangement of
scales on a fish.

A quincunx (from the scientific point of view) is merely the system of points
of intersection of two series of equidistant parallel lines in the same plane. By
a simple shear parallel to one of the two series of lines, combined (if necessary)
with mere uniform extensions or contractions along either or both series, any one
quincunx can be changed into any other. Hence the problems connected with the
elements of the subject are very simple; for it follows from the above statements
that any quincunx can be reduced to square order. The botanist who studies the
arrangement of buds or leaf-stalks on a stem, or of the scales on a fir-cone, seeks
the fundamental spiral, as he calls it, that on which all the buds or scales lie. And
he then fully characterizes each particular arrangement by specifying whether this
spiral is a right- or lefb-handed screw, and what is its divergence. The divergence
is the angle (taken as never greater than tt) of rotation about the axis of the
frindamental spiral from one bud or scale to the next.

(7) It is clear that if the stem or cone (supposed cylindrical) were inked and
rolled on a sheet of paper, a quincunx (Plate III. fig. 1) would be traced, consisting
of continuously repeated (but, of course, perverted) impressions of the whole sur&ce.
Hence if A, Ai, he successive prints of the same scale, B a scale which can be
reached from -4 by a right-handed spiral, AB, of m steps, or by a left-handed
spiral, AiB, of n steps, these two spirals being so chosen that all the scales lie on
n spirals parallel to AB and also on m spirals parallel to AiB, we shall find a
scale of the fundamental spiral by seeking the scale nearest to AAi within the
space ABAi.

Here continued fractions perforce come in. Let fi/p be the last convergent to
m/n. Then, if it be greater than m/n, count fi leaves or scales from A along AB,
and thence v leaves or scales parallel to BAi, and we arrive at the required leaf or
scale. If the last convergent be less than m/n, count v leaves along AiB, and
thence fi parallel to BA, If the leaf, a, so found in either case, be nearer to A
than to Ai, the frindamental spiral (as printed, i,e. perverted) is right-handed; and
vice versd. Thus the first criterion is settled.

To find the divergence, take the case of fi/p greater than m/n; and a, so found,
nearer to A than to ^i. Draw ac perpendicular to AAi, and let the spirals through
a, parallel to BA and BAi respectively, cut AAi in d and e,- Then the divergence
is 2irAc/AAi. This is obviously greater than 2irAd/AAi (i.e. ^irv/n), and less than
2irAe/AAi (i.e. 2'n'fi/m); and can be altered by shearing the diagram parallel to AAi,
or (what comes to the same thing) twisting the stem or cone. To find its exact
value, draw through B a line perpendicular to AA^ (i.e. parallel to the axis of the
stem or cone), and let C, the first leaf or scale it meets, be reached from fi by r
steps along BA, followed by s steps parallel to BAj. Then the divergence is easily

I^VI.]

LISTINGS TOPOLOGIE.

seen to be 2^ {pkS + pr)/{nis + nr) ; and we have the complete descnption of the object,
BO far as our, science goes.

In the figure, which in taken from an ordinaiy cone of Finns pinastei*, we have
til = 5, « = 8; wheiKe /i = 2, v = Z. Aim r=3, it = 2; and the fundamental spiral
(perverted) is therefore right-baoded, with divergence 27rl3/34*

Should m and n have a common divisor p, it is easily seen that the leaves
are arranged in wkork] and^ instead of one fundamental spiral, there is a group of
p such spirals, forming a multiple-threaded screw. Each is to be treated by a process
similar to that above.

(8) The last statement hints at a subject treated by Listing, which he calk
jparodromic mindimf. Some of his results are very curious and inatrtictive.

Take a long narrow tape or strip of paper. Give it any number, m, of half-
twists, then bend it round and paste its ends together.

If VI be zero, or any other even number, the two-sided surface thus formed has
two edges, which are paradroniic. If the strip be now slit up midway between the
edges, it will be split into two. These have each m/2 full twists, like the original,
and (except when there is no twist, wheu of course the two can be separated) are
mj2 times linked together.

But if m be odd^ there is hid one mrface and ojie edge; so that we may draw
a line on the paper from any point of the original froDt of the strip to any point
of the back, without crossing the edge. Hence, when the strip ia slit up midway, it
remains one, but with m ftdl twists, and (if m > I) it is knotted. It beeomea, in
fact, as its single edge was before slitting, a paradromic knot, a double clear coil
with m crossings.

[This simple result of Liating*8 was the sole basis of an elaborate pamphlet
which a few years ago had an extensive sale in Vieana; its object being to show
how to perform (without the usual conjuror's or spiritualists deception) the celebrated
trick of tjing a knot on an endless cord.]

The study of the one-sided autotomic surface w^hich is generated by increasing
indefinitely the breadth of the paper band, in cases where ni is odd, is highly
interesting and instructive. But we must get on,

(9) I may merely mention, in passing, as instances of our subject, the whole
question of the Integral Carvattire of a closed plane curve ; with allied questions such
as *'In an assigned walk through the streets of Edinburgh, how often has one
rotated relatively to some prominent object, such as St Giles* (supposed within the
path) or Arthur's Seat (supposed external to it) ? " We may vary the question by
Buppoeing that he walks so as always to turn his face to a particular object, and
then inquire how often he has turned about his own axis. But here we tread on
Jellinger Symonds' ground, the nofj-rotation of the moon about her axis I

But the subject of the area of an autotomic plane curve is interesting. It is one
of Listings examples. De Morgan, W. Thomson, and others in this country have also
developed it as a supposed new subject But its main principles (as Muir has shown
in PhiL Mag. June, 1873) were given by Meister 113 years ago. It is now so well
known that I need not dilate upon it

12—2

92 listing's topologie. [lxvi.

(10) A curious problem, which my ooUesgue Chijstal leomllj mentioiied to me,
appears to be capable of adaptation as a good example of oar sobjeck It was to
this efiFect: —

Draw ike circle of least area fMek imdudes fomr giwem poimis im ome flame.

In this form it is a qnesticm of <»dinazy gecMnetiy. Bat we maj modify it as

follows : —

Given three points in a plane; dimde the wkole emrfaee imto regiome suck that
wherever in any one of tkoee regUme a fburik point be ckaeen^ tke rmle for construct-
ing the least circle surrounding tke four ekall be tke acune.

There are two distinct cases (with a transitaon case iriucfa maj be refierred to
either), according as the given pcnnts A^ B, C (suppose) fonn an acate- or an
obtuse-angled triangle.

(a) When ABC is acute-angled (fig. 2). Draw from the omIs of each side
perpendiculars towards the quarter where the triangle Ues, and prodoce each of them
indefinitely firom the point in which it again int^sects Uie circamscrilMng cirdcL

The circle ABC is itself the required (me, so long as D (the fourth point) lies
within it.

If D lie between perpendiculars drawn (as above) firom the omIs of a side, as
ABy then ABD is the required cirde.

If it lie in any other r^ion, the required circle has D for one extan»ni^ of a
diameter, and the most distant of A, £, C for the othor.

03) When there is an obtuse angle, at C say (fig. 3V Make the same con-
stmction as befi^e, but, in additaim, describe the circle whose diamet^- is AK All
is as before, except that AB is the circle required, if D lie within it; and that if
D lie within the middle pcHtion of the laiger of the two lunes formed the required
circle is ABD,

[In figSL 2, 3, 4, which refer to these two cases in ordo-, and to the intomediate
case in which the triangle is right-angled at C, eadi r^;ion is denoted by three or
by two letters. When there are three, the meaning is that the required circle passes
through the c<»Tesponding points; when there are but two« these are the &ais of
a diameter. The separate r^ons are, throughout, bounded by foil lines; the dotted
lines merely indicate ccH[istruction&]

(11) A very celebrated question, directly connected with our subject, is to make
a Knight (at chess) move to eadi square <m the board once only till it returns to
its original positi<ML From the time of Eal^ onwards numinous solutioiis have been
given. To these I need not refer forther.

A much simpler question is the motion of a Rook, and to this the lately
popular American ^'lo-punle" is easily reduced. For ang closed path of a rn^
contains an even number of squares, since it must pass finom white to black altematelg.
(This furnishes a good instance of the extreme simplicity which often characterises
the solutions of questions in our subject which, at first sight, appear formidable]
And in the American puzzle every piece necessarily moves like a rook. Haoce if an

LXVT,]

LISTINGS TOPOLOOIK

even number of interchanges of pieces will give the requii'ed result, the puzzle can
be solved ; if not, the artftngemeot is irreducible.

(12) A few weeks ago, in a rail way- train, I saw the following problem proposed; —
Place four sovereigns and four shillings in close altet^nate order in a line. Required,
in four moves, each of two contiguous pieces (without altering the relative position
of the two), to form a coniimious line of four sovereigns followed by four shillings.
Let sovereigns be i^presented by the letter B^ shillings by A.

One solution is as follows: —

Before starting:— , . ABABABAS

1st move ......... B A A B A B A , . B

2nd „ B A A B . . A A B B

3rd , B . . B A A A A B B

4th ,, , BBBBAAAA . .

If we suppose the pieces to be originally arranged in circular order, with two
contiguous blank spaces, the law of this process is obvious* Operate always with the
penultimate and antepenultiniatej the gap being looked on as the end for the time
being. With this hint it is easy to generalize, so as to get the nature of the
solution of the corresponding problem in any particular case, whatever be the number
of coins. It is also interesting to vary the problem by making it a condition that
the two coins to be moved at any instant shall first be made to change places,

(13) Another illustration, commented on by Listing, but since developed from a
different point of view in a quite unexpected direction, was originated bj a veiy
simple question propounded by Clausen in the Astronmnische I^achricldefi (No. 494),
In its general form it is merely the question^ "What is the smallest number of
pen-strokes with which a given figure, consisting of lines only, can be traced ? " No
line is to be gone over twice, and every time the pen has to be lifted counts one.

The obvious solution is : — Count the number of points in the figure at each of
which an odd number of lines meet. There must always be an even number of such
(zero included). Half of this number is the number of necessary separate strokes
(except in the zem case, when the number of course mtisi be unity). Thns the
boundaries of the squares of a chess-board can be traced at 14 separate pen-sti*okes;
the usual figure for Euclid L 47 at 4 pen-strokes; and fig* 5 at one-

(14) But, if 2n points in a plane be joined by 3n lines, no two of which
intersect, {i.e. so that every point is a terminal of 3 different lines), the figure requires
n separate pen-strokes* It has been shown that in this case (unless the points be
<livided into two groups, between which there is but one connecting line, fig, T) the
3n lines may be divided into 3 groups of n each, such that one of each group ends
at each of the 2w points. See fig- 6, in which the lines are distinguished as a, ^,
or 7. Also note that ajSctjS &c., and ayjj &a form entire cycles passing through all
the trivia, while ^97^7 &c. breaks up into detached subcjclea

Thus, if a Labyrinth or Maze be made, such that every intersection ot roads is

94 listing's topologdsl [lxti.

a Trivium, it may always be arranged so thai tlie seTeiml loads meetiiig at each
intersection may be one a gras$-path» coie grmTd, and the olho* paTement. To make
sure of getting out of such a Labyrinth (if it be possibleX we most select two. kinds
of road to be taken alternately at each socceesire tnTium. Thos we may elect to
take grass, gravel, gras6^» gravel* &c.» in whidi case we arasf ^ther come to the exit
point or (without reaching it> return lo oor staitii^pouil^ to tir a new eombiDation.
For it is obvious that, if we follow our rale, we eaaiiol possiblT pass through the
saino triviuni twice before retiuming to owr starting^-poini.

(15) This liN^ds to a very simple solntko of the pfohlem of Map^akmrimff wHk
fimr Ci^QHTS, originally prc^>06ed by Guthiie^ and smce treated bj Caytey^ Kempe,
nnd others.

The boundaries^ of the eountki^ in a nap genetaffy nsei m threes. But if four,
or u\iut»» meet at certain points^ let a saaall coostr be insened sanroodi]^ each
i*\ioh (Hviut ; aiui there will then be trivia of buMuadbties edhr. TlMse lanoos boozidaries
mivy» by our last reej^ulti. be dinded v^*:^*"*^ ^ Hway dtJieKBS wmys> iMto three
tmtt^gories, 9l ti% y suppose^ swcli thai e^i^ii triTvua t$ fccwKd bf the Beedag ef one
ftxvm each cate(^4rY. Now take four coKmhcSv ^. & C IK awl apfty doa^ acoxdiBg to
v\\\%\ an II^Ulow«^ 9o that

« ijt^paialiw^ Jl awl Jl or C aad It
;» ^ JaadO^JIawill
t .. Jlaadl^^JlaiaC

aud Ihe IhU^ is \W«h\ IfVw iW snaatt vvtuMitSk whkk w^t^ vuBiwiKsd fir tik^ sake
ol' Ibi^ \^^^tlr\H^i^^ in^ h^w bi^ aiaiAe to iciNiLttafit wtd^u Eaaa iS the bomHiKKs
l^HH^oe a»^ Ib^y >if^Nr¥ at ttowt.

1^0 w^ii^^iv'^ Wlw^Mi tWir tw^ iftieiKeiui^ ^^«^ jh^ esee&u ShnssettML ef the

K^^O)^ ba» pk>4¥l^ out iIm fo^MT v>{iMU»^ ^ wa tR gsmMat «dk« Sm a aap
dm>^^ v^K^H a i^^hx^i^^vNNaapKW :$i«rfkN^ :$iidk a» thttit nr^ a ainr or andkir-Bv^
11vU y%^^ v^^ v^i^^ iftow for yv^ME!$iiix^ ^ <(«$aaMbhwt ^mt ;Tim[iii iiwtMn il iTMs^

V^l^^ ^V^M iW xv<>k ^MtMiw^ ^^f «/<» ^*M».Vv thiir :^csQi»t^ sir »«aa. si^ wi; isiGdWii
iK^^w ^w $ l*v ^Mi^v W ^?^Ba^M at^ ^ia«^ JfcM:%J*£ jpiaair jryMMai Jt fuBfitirm mkmi
hs^¥¥ h>W^\^ ^^Mf<aM^ «/iM^ YVk^ ;My two v)J^«t»Mt:$^ v»tt6)set^ ^^ eacej^i&iWv w&ndk wUL

\\<\\ \\^ #%^N| /«««^ v»* ^W ^:^^ra^ >ftKt!^ srf ciasit^ w^^ca I i»»*t arf «wiiB» aif

lxvl]

LISTINGS TOPOLOGIE.

each of the angles. That is^ in another form, every guch polyhedron may be projected
in a figure of the type shown in fig, 9, where the dotted lines are supposed to lie
below the full lines. But, in the words of the extraordinary mathematician Kirkman,
whom I consulted on the subject, "the theorem,..**. has this provoking interest, that
it mocks alike at doubt and proof'.** Probably the proof of this curious proposition
has (§ 11) hitherto escaped detection from its sheer simplicity. Habitual stargazers
aire apt to miss the beauties of the more humble terrestrial objects.

(17) Kii'kman himself was the first to show, &o long ago as 1858, that a "clear
circle of edge^" of a unique type passes thmngh all the summits of a pentagonal
dodeeahedroru Then Hamilton pounced on the result and made it the foundation of
his Icomin QmnBy and also of a new calculus of a very singular kind. See figures
9, 10, 11, which are all equivalent projections of a pentagonal dodecahedron.

At every trivium you must go either to right or to left. Denote these operations
by r and I respectively. In the pentagonal dodecahedron, start where you will, either
r* or ? brings you back to whence you started. Thus, in this case, r and I are to
be regarded as operational symbols — each (in a sense) a fifth root of + 1. In this
notation Kirkmans Theorem is formulated by the expression

rbirrrlUrlrlrtTlU = 1 ;

or, as we may write it more compactly,

[{rlfr'l^]^=h or [{lryrH^f=h

It may be put in a great many apparently different, but really . equivalent, forms; for,
so long as the order of the operations is unchanged, we may begin the cycle where
we please. Also we may, of course, interchange r and I throughout, in consequence
of the symmetry of the figure.

It is curious to study, in such a case as this, where it can easily be done, the
essential natui*e of the various kinds of necessarily abortive attempts to get out of
such a labyrinth. Thus if we go according to such routes as (riyirU^^ or r^li'^ (sequences
which do not occur in the general cycle), the next step, whatever it be, brings ub
to a point already passed through. We thus obtain other relations between the symbols
r and L We can make special partial circuits of this kind, including any number of
operations from 7 up to 19.

All of these remarks will be obvious from any on© of the three (equivalent) diagi-ams
9, 10, or 11,

(IS) As I have already said, the subject of knots affords one of the most typical
applications of our science, I had been working at it for some time, in consequence
of Thomson's admirable idea of Vortex-atoms, before Clerk-Maxwell referred me to
Listing*s Essay; and I had made out for myself, though by methods entirely different
from those of Listing, all but one of his published results. Listing's remarks on this
fascinating branch of the subject are, unfortunately, very brief; and it is here especially,
1 hope, that we shall learn much from his posthumous papers. In the Vorstudiej^ he

^ 'Bspdni of Math. Papers from tlie Ed. TimSB,' laSl, p. 113^

96 listing's topologie. [lxvi.

looks upon knots simply from the point of view of screwing or winding; and he
designates the angles at a crossing of two laps of the cord by the use of his \ and
8 notation (§ 4). Fig. 12 will show the nature of such crossings. Figs. 13, 14, and
15 show what he calls reducible and reduced knots. In a reducible knot the angles
in some compartments at least are not all \ or all 8 (the converse is not necessarily
true). In a reduced knot, each compartment is all \ or all S.

(19) My first object was to classify the simpler forms of knots, so as to find to
what degree of complexity of knotting we should have to go to obtain a special form
of knotted vortex for each of the known elements. Hence it was necessary to devise
a mode of notation, by means of which any knot could be so fully described that
it might, from the description alone, be distinguished from all others, and (if requisite)
constructed in cord or wire.

This I obtained, in a manner equally simple and sufficient, from the theorem which
follows, one which (to judge from sculptured stones, engraved arabesques, &c.) must have
been at least practically known for very many centuries.

Any closed plane curve, which has double points only, may be looked upon as
the projection of a knot in which each portion of the cord passes alternately under
and over the successive laps it meets, [The same is easily seen to hold for any number
of self-intersecting, and mutually intersecting, closed plane curves, in which cases we
have in general both linking and locking in addition to knotting.]

The proof is excessively simple (§ 11). If both ends of one continuous line lie
071 the same side of a second line, there must be an even number of crossings.

(20) To apply it, go continuously round the projection of a knot (fig. 16), putting
A, B, Cy &c. at the first , thirds fifth, &c. crossing you pass, until you have put letters
to all. Then go round again, writing down the name of each crossing in the order
in which you reach it. The list will consist of each letter employed, taken twice over.
-4, B, 0, &c. will occupy, in order, the first, third, fifth, &c. places; but the way in
which these letters occur in the even places fully characterizes the drawing of the pro-
jected knot. It may therefore be described by the order of the letters in the even
places alone ; and it does not seem possible that any briefer description could be
given.

To prove that this description is complete, so far as the projection is concerned,
all that is required is to show that from it we can at once construct the diagram.
Thus let it be, as in fig. 16. EFBACD. Then the full statement is

AEBFGBDAEGFDjA &c.

(21) To draw fix>m such a statement, choose in it two apparitions of the same
letter, between which no other letter appears twice. Thus AEG FDjA (at the end of
the statement) forms such a group. It must form a loop of the curve. Draw such
a loop, putting A at the point where the ends cross, and the other letters in order
(either way) round the loop. Proceed to fill in the rest of the cycle in the same
way. The figures thus obtained may present very dififerent appearances; but they are
all projections of the same definite knot. The only further information we require for
its full construction is whicli branch passes over the other at each particular crossing.

LXV1-]

LISTINGS TOPOLOGIE.

ThiB can be at once supplied by a + or — sign attached to each letter where it occurs
in the statement of the order in the even places.

(22) Famished mth this process, we find that it becomes a mere question of
skilled labour to draw all the possible knots ha%TJig any assigned number of crossings.
The requbite labour increases with extreme rapidity as the number of crossings is
increased. For we must take eveiy possible arrangement of the letters in the even
places, and trj^ whether it is compatible with the properties of a self-intersecting plane
curve. Simple rules for rejecting useless or impracticable combinations are easily formed.
But then we have again to go through the list of survivors, and reject aU bat one of
each of the numerous groups of difierent distortions of one and the same species of knot.

I have not been able to find time to carry out this process further than the
knote with seven crossings. But it is very remarkable that, eo far as I have gone,
the number of knots of each class belongs to the series of powers of 2. Thus:

Number of crossings 3, 4, 5, 6, 7,

Number of distinct forms... I, 1, 2, 4, 8,

It is gr^itly to be desired that some one, with the requisite leisure, should try to
extend this list, if possible up to 11, as the next prime number. The labour, great
as it would be, would not bear comparison with that of the calculation of 7r to
600 places, and it would certainly be much more useful. [But see No3, XL, XLI, which
are of later date than this Address, 18U9.]

Besides, it is probable that modern methods of analysis may enable us (by a single
"happy thought*' as it were) to avoid the larger part of the labour. It is in matters
like this that we have the true ''raison d'etre'' of mathematicians*

(23) Thei^ is one very curious point about knots which, so £sLr as I know, has
as yet no an^ogue elsewhere. In general the perversion of a knot (ie. its image in
a plane mirror) is non- congruent with the knot itselC Thus, as in fact Listing points
out, it is impossible to change even the simple form (fig* 14) into its image (tig. 15),
But I have shown that there is at least one foraij for every even number of crossings^
which is congruent with its own perversion. The unique form with four crossings gave
me the fii^t hint of this curious fact. Take one of the larger laps of fig. 17, and
turn it over the rest of the knot, fig, 18 (which is the perversion) will be produced.

We see its natui*e better from the following process (one of an infinite number)
for fomting Amphicheiral knots. Knot a cord as in fig. 19, the number of complete
figures of " eight " being at pleasure. Turn the figure upside down, and it is seen to
be merely its own image. Hence, when the ends are joined, it forms a knot which is
congruent with its own perversion.

(24) The general treatment of links is, unless the separate cords be also knotted,
much simpler than that of knots — i^. the measurement of bdinkedness is far easier
than that of beknottedness,

I believe the explanation of this curious result to lie mainly in the fact that it is
possible to interweave three or more continuous cords, so that they cannot be separated t
and yet no one shall be knotted ^ nor any two linked togethei*.

T. U. IS

98 listing's topologib. [lxvi.

This is obvious at once from the simplest possible case, shown in fig. 20. Here the
three rings are not linked but locked together.

Now mere Unkings and mere lockings are very easy to study. But the various
loops of a knot may be linked or locked with one another. Thus the full study of a
knot requires in general the consideration of linking and locking also.

(25) But it is time to close, in spite of the special interest of this part of the
subject. And I have left myself barely time to mention the very interesting portion
of the Topohgie which listing worked out in detail. Tou will find a brief synopsis
of a part of it prefixed to Clerk-Maxwell's Electricity and Magnetism, and Cayley has
contributed an elementary statement of its contents to the Messenger of Mathematics
for 1873; but there can be no doubt that so important a paper as the Census rdwm-
licher Complexe ought to be translated into English.

To give an exceedingly simple notion of its contents I may merely say that Listing
explains and generalizes the so-called Theorem of Euler about Polyhedra (which all of
us, whose reading dates some twenty years back or more, remember in Snowball's or
Hymers' Trigonometry), viz. that "if 8 be the number of solid angles of a polyhedron,
F the number of its faces, and E the number of its edges, then

5 + ^=^ + 2."

The mysterious 2 in this formula is shown by Listing to be the number of spaces
involved; is. the content of the polyhedron, and the Amplexum, the rest of infinite
space.

And he establishes a perfectly general relation of the form

where V is the number of spaces, 8 of surfaces, L of lines, and P of points in any
complex; these numbers having previously been purged in accordance with the amount
of Cyclosis in the arrangement studied. But to make even the elements of this
intelligible I should require to devote at least one whole lecture to them.

Meanwhile I hope I have succeeded in showing to you how very important is
our subject, loose and intangible as it may have at first appeared to you ; and in
proving, if only by special examples, that there are profound difficulties (of a kind
different altogether from those usually attacked) which are to be met with even on the
very threshold of the Science of Situation.

Rat.- n:

Fig.l.

Fig. 7.

Fig. 12. Fig. 13 Fig. 14.. Fig. 16.

Fig.n.

Fig. 18.

"^^jV-c^-Vi"

Lxvn.]

LXVII.

ON RADIATION.

[Proceedings of the Rayal Society of Edinburgh^ February 18, 1884,]

The first part of this commimication was devoted to a recapitulatioQ of the
advances in the Theory of Exchanges made by Stewart in 1858, and published in
the Transactions of the Society for that year* Such a recapitulation it will be seen
is necessary \ as Stewart's papers seem either to have fallen into oblivion or to be
deemed unworthy of notice. It was pointed out that Stewart showed in these papers
that the radiation within an impervious enclosure containing no source of heat must
ultimately become, like the pressure of a n on -gravitating fluid at rest, the same at
all points and in all directions; but that this sameness is not, like that of fluid
pressure, one of mere total amount; it extends to the quantity and quality of every
one of the infinite series of wave-lengths involved. For, as one or more of the bodies
may be black, the radiation is simply that of a black body at the temperature of
the enclosure. Any new body, at the proper temperature, may be inserted in the
enclosure without altering this state of things; and must therefore emit precisely the
amount and quality which it absorbs. This remark containd aU that is yet known
on the subject. For we have only to assume for the purpose of reasonings the
existence of a substance partially, or wholly, opaque to one definite wave-length, and
perfectly transparent to all others; or with any other limited properties we choose;
and suppose it to be put (at the proper temperature) into the enclosure. If we next
assume that its temperature when put in differs from that of the enclosure, the
experimental fact that, in time, equilibrium of tempei'ature is arrived atj shows that
the radiation of any particular wave-length by a body increases with rise of temperature.
And so forth*

Yet in the latest authoritative work on the subject, Lehrbuch der Spektralanalyse,
von Dr H, Kayser (Berliuj 1883), though historical details are freely given > the name
of Stewart does not occur even once ! There are in the same work other instances of

13—2

100 ON RADIATION. [lXVII.

historical error nearly as grave. Thus the physical analogy, by which Stokes in 1852
first explained the basis of spectrum analysis, is given in Dr Kayser's work; but it

is introduced by the very peculiar phrase " woUen wir versuchen, eine mechxinische

Erkldrung der Erscheinungen zu geben, welche auf unsere Anschauungen liber das
Leuchten begriindet ist "; and the name of Stokes is not even mentioned in con-
nection with it!

The second part of the paper deals with the question of the limits of accuracy
of the reasoning which led Stewart, and those who have followed him, to results of
such vast importance. Dr Eayser, indeed, announces his intention ''in aller Strenge
mathematisch zu beweisen" the equality of emissive and absorptive powers. But the
mere fact that phosphorescent bodies, such as luminous paint, give out visible radiations
while at ordinary temperatures, shows at once that there are grave exceptions even
to the fundamental statement that the utmost radiation, both as to quantity and as
to quality, at any one temperature, is that of a black body: — and very simple con-
siderations show that all the reasoning which has been applied to the subject is
ultimately based on the Second Law of Thermodynamics (or Camot's principle), and
is therefore true only in the sense in which that law is true, i.e, in the statistical
sense. The assumed ultimate uniformity of temperature in an enclosure, which is
practically the basis of every demonstration of the extended law of exchanges, is
merely an expression for the average of irregularities which are in the majority of
cases too regularly spread, and on a scale too minute, to be detected by our senses,
even when these are aided by the most delicate instruments. The kinetic theory of
gases here furnishes us with something much closer than a mere analogy. For the
very essence of what appears to us uniform temperature in a gas is the regularity
of distribution of the irregularities of speed of the various particles. And, just as in
every mass of gas there are a few particles moving with speed &r greater than
that of mean square, so it is at least probable that a black body at ordinary
temperatures emits (though, of course, excessively feebly) radiations of wave-lengths
corresponding to those of visible light. Effects apparently or at least conceivably
due to this cause have been obtained by various experimenters.

If we could realise a dynamical system, analogous to that of a gas on the kinetic
theory, but such that none of the particles could have any but one of a certain
limited number of definite speeds, and if there were still a tendency to the nearest
statistical average, we should have something capable of explaining phosphorescence at
ordinary temperatures.

lxviil] 101

LXVIII.

ON AN EQUATION IN QUATERNION DIFFERENCES.

[Proceedings of the Royal Society of Edinburgh, February 18, 1884.]

When the sides of a closed polygon are bisected, and the points of bisection joined
in order, a new polygon is formed It has the same number of sides, and the same
mean point of its corners, as the original polygon. In what cases is it similar to the
original polygon? In what cases will two, three, or more successive operations of this
kind produce (for the first time) a polygon similar to the original one?

Take the mean point as origin, and let q^a, q^ ... qna be the n comers. Here
a is any vector, which, if the polygon be plane, may be taken in that plane; and
qu ••• ?n are quaternions, which in the special case just mentioned are powers of one
quaternion in the same plane. We obviously have, if i)jr = Sr+i, for the plane polygon
two conditions: — the first,

(l + Z) + 2)»+... +Z)^0?ra = O,
depending on our choice of origin ; and the second

^(l+nrqra^QD'qra.

depending on the similarity of the mth derived polygon to the original In this last
equation, Q is a scalar multiple of an unknown power of the quaternion of which the
q's are powers, expressing how the original polygon must be turned in its own plane,
and how its linear dimensions must be altered, so that it may be superposed on the
mth derived polygon. Also « is an unknown integer, but it has (like Q) a definite
value or values when the problem admits of solution, r has any value from 1 to n
inclusive, as may be seen at once by operating by any integral power of D, and
remembering that we have necessarily

The solution of this case is easily effected, and gives the well-known results: — the
general solution involving all equilateral and equiangular polygons, where m may have

102 ON AN EQUATION IN QUATERNION DIFFERENCES. [LXVni.

any integral valua Besides this, there are special solutions for the triangle, and for the
quadrilateral reduced at one operation to a parallelogram. In the former of these m
may have any value; in the latter (unless the figure be a square) m must be even.

But, when the polygon is gauche, the second of the above conditions becomes

and the solution is somewhat more difficult. Its interest consists in its leading to a new
and curious question in quaternions.

APPENDIX.

Theorem relating to the Sum of Selected Binomial-Theorem Coefficients.

[Messenger of MathemaMcSy Febrtbory, 1884.]

Let equal masses be placed, two and two together, -at the comers of an m-sided polygon.
Slide one from each end of a side till they meet at its middle point. They now form a
new, and smaller, m-sided polygon, but their centre of inertia has not been disturbed. Repeat
the process indefinitely, and the masses will ultimately be collected in the centre of inertia.

Now if the distances of the comers of the original polygon from a fixed plane be

Uly t*8, ..., ti„,

those of the first derived polygon will be

These are aU included in the expression

i(l + 2>)w„

with the proviso that Z>"*w,. = u,..

Similarly, the first comer of the nth derived polygon is

2-"(l + 2>)«t^.

Now let ^r^, where r is not greater than w, be the sum of the rth, (r + m)th, (r + 2m)th,
&c. coefficients of the binomial (l+o;)*; the above expression becomes

2-» (iV^rwi + N^'^th + ... + ^/^Ur + ... + N^^'u^),

But, when n is infinite, its ultimate value is (as above)

-(Wi + W2+...+M„).

Hence Z„.. (2-iVr,-) = i ;

and it seems remarkable that the limit is independent of r.

LXix.] 103

LXIX.

ON VORTEX MOTION.

[Proceedings of the RoycU Society of Edinburgh, February 18, 1884.]

This paper contained a discussion of the consequences of the assumiptian of continuity
of motion throughout a perfect fluid; one of the bases of von Helmholtz's grand in-
vestigation, on which W. Thomson founded his theory of vortex-atoms. It is entirely on
the assumed absence of finite slip that von Helmholtz deduces the action of a rotating
element on any other element of the fluid, and that Thomson calculates the action of
one vortex-atom or part of such an atom on another atom, or on the remainder of itsel£
The creation of a single vortex-atom, in the sense in which it is defined by Thomson,
involves action applied simultaneously to all parts of the fluid mass, not to the
rotating portion alone.

104 [lxx.

LXX.

NOTE ON REFERENCE FRAMES.

[Proceedings of the Royal Society of Edinburgh, July 7, 1884.]

As I understand Prof. J. Thomson's problem (Proc. iJ. 8. E. xn. p. 668) it is equi-
valent to the following: —

A set of points move, (Jalilei-wise, with reference to a system of co-ordinate axes ;
which may, itself, have any motion whatever. From observations of the relative positions
of the points, merely, to find such co-ordinate axes.

It is obvious that there is an infinitely infinite number of possible solutions ;
because, if one origin moves Galilei-wise with respect to another, and the axes dra¥ni
from the two origins have no relative rotation, any point moving Qalilei-wise with
respect to either set of axes will necessarily move Qalilei-wise with respect to the
other. Hence any one solution suffices, for all the others can be deduced fit)m it by
the above consideration.

Referred to any one set of axes which satisfy the conditions, the positions of the
points are, at time t, given by the vectors

^i + A^ for A, 02 + fi^t for B, &c., &c.

But it is clear, from what is stated above, that we may look on the pair of vectors
for any one of the points, say aj and /8i for il, as being absolutely arbitrary: — though,
of course, constant We will, therefore, make each of them vanish. This amounts to
taking A as the origin of the co-ordinate system. The other expressions, above, will
then represent the relative positions of B, C, &c., with regard to A.

The observer on A is supposed to be able to measure, at any moment, the lengths
ABy AC, AD, &c. ; the angles BAG, BAD, CAD, &c. ; and also to be able to recognise
whether a triangle, such as BCD, is gone round positively or negatively when its comers
are passed through in the order named. What this leaves undetermined, at any particular

LXX,]

NOTE ON REFERENCE PRAMES,

105

instant, is merely the absolute direction of itny one line (as AB), and the aspect of
an^ one plane (as ABO) parsing through that line. Thase being assumed at random,
the simultaneous positions of all the points can be constructed from the j^ermissible
observations. But it is interesting to inquire how many obaervations are necessary;
and how the ^'s depend on the a's.

Thus, at time t, whatever be the mode of measurement of time, we have equations
such as follow^: —

- a ^ a,' + 2Sa^^ . t + ^./f.

For any one value of t we have n equations of each of the lat and 3rd of these
types, and «(«— l)/2 of the 2nd, n-^-l being the whole number of points. lo all^
n{n + iy2 equations.

The scalar unkno^Tis involved in these equations are (1) the values of ^; (2) a^\ ti^\
&a; (3) 0^\ 0f. &c.i (4) iSa^, &c,; (5) S/3A. &c.; (6) SaA. SaA. &c.; and
(7) jS(cg3;i + /9M &e. Their numbers are. for (2), (3), (6). n each; for (4), (5), (7),
n (n — l)/2 each ; in all 3w (n + 1) 2. Suppose that observations are made on m suc-
cessive occasions. Since our origiu, and our unit, of time are alike arbitrary, we may
put t = 0 for the first observation, and merge the value of t at the second observation
in the tensors of 0^, ^g, &c. This amounts to taking the interval between the first
two sets of observations as unit of time. Thus the unknowns of the form (1) are
m — 2 in number. There are therefore

mn {n + l)/2 equations and 3n (n + 1)/2 H- m — 2 unknowns.

Thus m = 3 gives an insufficient amount of information, but m = 4 gives a superfluity.

In particular, if there be three points ooly, which is in general sufficient^ 3 complete
observations give

9 equations with 10 unknowns;

while 4 complete observations give

12 equations with 11 unknowns.

Thus we need take only two of the three possible measurements, at the fourth instant
of observation.

The solution of the equations, supposed to be eSected, gives us among other things,
Oj*, a,', aud Sa^^^t^ Ant^ direction may be assumed for o,, and any plane as that of
a, and «». From these assumptions, and the three numerical quantities just named,
the co-ordinate system can be at once deduced.

This solution fails if (Sar^o,)^ = Oa^i/, or TVa^i^^O; for then the three points A, B,C,
are in one line at starting. But this, and similar cases of failure (when they ai*B
really cases of failure) are due to an improper selection of three of the points. We
need not further discuss them*

T- II. 14

106 NOTB ON BEFERENCE FRAMES. [lXX.

But it is ixiteresting to consider how the vectors fi can be found when one position
of the reference frame has been obtained. Keeping, for simplicity, to the system of
three points, we have by the solution of the equations above the following data: —

where c, e\ /, g, g\ h are known numbers; which, as the equations from which they
were derived were not linear, have in general more than one system of values. The
second, third, and sixth of these equations give

i8:rS . OsOj/S, = AFoA + (/- 5^8,0,) Faw8, + ^F/8a.

Provided jS^ is not coplanar with Oj, a,, this equation gives, by the help of the fifth

above, a surface of the 4th order of which /8, is a vector. But ^8, is also a vector

of the plane ScLfii = e, and of the sphere T/S^^g. Hence it is determined by the
intersections of those three surfaces.

But if 8 . OjO^a vanishes, the equation above gives (by operating with S . Va^)

0 = A (F(Vs)' - (/- S0,(x,) S.jSiV. OaFoA + e'S . 0,r. a^Va^,

which gives a surface of the second order (a hyperbolic cylinder) in place of the surface
of the fourth order above mentioned. This may, however, be dispensed with: — for ^a
is in this case determined by the planes 802^2="^ ^^d 8 .ou^Xaff^^O, together with the
sphere Tfi^ — g-

LXXI.]

107

LXXI.

ON VARIOUS SUGGESTIONS AS TO THE SOURCE OF
ATMOSPHERIC ELECTRICITY ^

[Nature, March 27, 1884.]

We have seen that, taking for granted the electrification of cIoudB, all the
ordinary phenotnena of a thunderstorm (except globe lightning) admit of easy and
direct explanation by the known laws of statical electricity. Thus far we are on
comparatively sure ground.

But the case is very different when we attempt to look a little farther into the
matter, and to seek the source of atmospheric electricity. One cause of the difficulty
is easily seen. It is the scale on which meteorological phenomena usually occur; so
enormously greater than that of any possible laboratory arrangement that effects,
which may pass wholly unDoticed by the most acute experimenter, may in nature
rise to paramount importance. I shall content myself mth one simple but striking
instance.

Few people think of the immense transfonnations of energy which accompany an
ordinary shower. But a very easy calculation leads us to startling results. To raise
a single pound of water, in the form of vapour, from the sea or from moist ground,
requires an amount of work equal to that of a horse for about half an hour! This
is given out again, in the form of heat, by the vapour when it condenses ; and the
pound of water, falling as rain, would cover a square foot of ground to the depth of
mther less than one-fifth of an inch. Thus a fifth of an inch of rain repre^sents a
horse-power for half an hour on every square foot, or, on a square mile, about a million
horse-power for fourteen hours ! A million horses would barely have standing room on
a square mile. Considerations like this show that we can account for the most violent
hurricanes by the energy set free by the mere condensation of vapour required for the
concomitant rain.

^ Bead at the meeting of the Seottieh Meteorological Society on Uarch 17r And oommuDieated by the Society^

14—2

108 ON VARIOUS SUGGESTIONS AS TO THE SOURCE OF [lXXI.

Now the modern kinetic theory of gases shows that the particles of water-
vapour are so small that there are somewhere about three hundred millions of millions
of millions of them in a single cubic inch of saturated steam at ordinary atmo-
spheric pressure. This corresponds to j^ or so of a cubic inch of water, i.e. to
about an average raindrop. But if each of the vapour particles had been by any
cause electrified to one and the same potential, and all could be made to unite,
the potential of the raindrop formed from them would be fifty million million times
greater.

Thus it appears that if there be any cause which would give each particle of
vapour an electric potential, even if that potential were far smaller than any that
can be indicated by our most delicate electrometers, the aggregation of these particles
into raindrops would easily explain the charge of the most formidable thundercloud.
Many years ago it occurred to me that the mere contact of the particles of vapour
with those of air, as they interdiffuse according to the kinetic theory of gases,
would suffice to produce the excessively small potential requisite. Thus the source
of atmospheric electricity would be the same as that of Volta*s electrification of dry
metals by contact. My experiments were all made on a small scale, with ordinary
laboratory apparatus. Their general object was, by various processes, to precipitate
vapour from damp air, and to study either (1) the electrification produced in the
body on which the vapour was precipitated; or (2) to find on which of two parallel,
polished plates, oppositely electrified and artificially cooled, the more rapid deposition
of moisture would take place. After many trials, some resultless, others of a more
promising character, I saw that experiments on a comparatively large scale would be
absolutely necessary in order that a definite answer might be obtained. I commu-
nicated my views to the Royal Society of Eldinburgh in 1875, in order that some
one with the requisite facilities might be induced to take up the inquiry, but I am
not aware that this has been done.

I may briefly mention some of the more prominent attempts which have been made
to solve this curious and important problem. Some of them are ludicrous enough,
but their diversity well illustrates the nature and amount of the difficulty.

The oldest notion seems to have been that the source of atmospheric electricity
is aerial friction. Unfortunately for this theory, it is not usually in windy weather
that the greatest development of electricity takes place.

In the earlier years of this century Pouillet claimed to have established by
experiment that in all cases of combustion or oxidation, in the growth of plants,
and in evaporation of salt water, electricity was invariably developed. But more
recent experiments have thrown doubt on the first two conclusions, and have shown
that the third is true only when the salt water is boiling, and that the electricity
then produced is due to friction, not to evaporation. Thus Faraday traced the action
of Armstrong's hydro-electric machine to friction of the steam against the orifice by
which it escaped.

Saussure and others attributed the production of atmospheric electricity to the
condensation of vapour, the reverse of one of Pouillet's hypotheses. This, however,
is a much less plausible guess than that of Pouillet; for we could understand a
particle of vapour carrying positive electricity with it, and leaving an equal charge

LXXIJ

ATMOSPHEEIC ELECTRICTTy.

109

of negative electricity in the water froDi which it escaped But to account for the
separation of the two electricities when two particles of vapour unite is a much less
promising task,

Peltier (followed by Lamont) assumed that the earth itself has a permanent
charge of negative electricity whose distribution varies from time to time, and from
place to place. Air, according to this hypothesis, can neither hold nor conduct elec-
tricity, but a cloud can do bath ; and the cloud is electrified by conduction if it
touch the earthy by induction if it do not. But here the difficulty is only thrown
back one step. How are we to account for the earth's permanent charge ?

Sir W. Thomson starts from the experimental fact that the layer of air near
the gi'ound is often found to be strongly electrified, and accounts for atmospheric
electricity by the carrying up of this layer by convection currents. But this process
also only shifts the difficulty,

A wild theory has in recent times been propojsed by BecquereL Corpuscles of
some kind, electrified by the outbursts of glowing hydrogen, travel firora the sun to
the upper strata of the earth's atmosphere,

Muhry traces the source of electricity to a direct effect of solar radiation falling
on the earth *s surface,

Luddens has recently attributed it to the Miction of aqueous vapour against dry
air. Some still more recent assumptions attribute it to capillary surface-tension of
water^ to the production of hail, &c»

Blake, Ealischer, &c,, have lately endeavoured to show by experiment that it is
not due to evaporation, or to condensation of water. Their experiments, however,
have all been made on too small a scale to insure certain results. What I have
just said about the extraordinary number of vapour particles in a single raindrop,
shows that the whole charge in a few cubic feet of moist air may altogether escape
detection.

And so the matter will probably stand, until means are found of making these
delicate experiments in the only way in which success is likely to be obtained, viz,
on a scale far larger than is at the command of any ordinary private purse. It is
a question of real importance, not only for pure science but for the people, and ought
to be thoroughly sifted by means which only a wealthy nation can provide.

110 [lxxii.

LXXIL

NOTE ON A SINGULAR PASSAGE IN THE PRINCIPIA.

[Proceedings of the Royal Society of Edinburgh, January 19, 1885.]

In the remarkable Scholium, appended to his chapter on the Laws of Motion,
where Newton is showing what Wren, Wallis, and Huygens had done in connection
with the impact of bodies, he uses the following very peculiar language: —

"Sed et Veritas comprobata est a D. Wrenno coram Begid Societate per experimentum
Pendulorum, quod etiam Glarissimus Mariottus Libro integro exponere mox dignatus est."

The last clause of this sentence, which I had occasion to consult a few da}rs ago,
appeared to me to be so sarcastic, and so unlike in tone to all the context, that I was
anxious to discover its full intention.

Not one of the Commentators, to whose works I had access, makes any remark on
the passage. The Translators differ widely.

Thus Motte softens the clause down into the trivial remark "which Mr Mariotte
soon after thought fit to explain in a treatise entirely on that subject."

The Marquise du Chastellet (1766) renders it thus: —

" mais ce fut Wrenn qui les confirma par des Experiences faites avec des

Pendules devant la Soci6t6 Royale: lesquelles le c^l^bre Mariotte a rapport^es depuis dans
un Traits qu'il a compost exprfes sur cette matifere."

Thorps translation (1777) runs: —

" which the very eminent Mr Mariotte soon after thought fit to explain in a treatise
entirely upon that subject."

Finally, Wolfers (1872) renders it thus:—

"der zweite zeigte der Societat die Richtigkeit seiner Erfindung an einem Pendel-
versuche, den der beriihmte Mariotte in seinem eigenen Werke aus einander zu setzen,
fur wiirdig erachtete."

Not one of these seems to have remarked anything singular in the language
employed. But when we consult the "entire book" in which Mariotte is said by
Newton to have "expounded" the result of Wren, and which is entitled Traits de la

lxxil]

NOTE ON A SINGULAR PASSAGE I3T THE PEINCIPIA,

Percussion on Choc dm Corps ^ we find that the Dame of Wren is not once mentioned in
its pages ! From the beginning to the end there is nothing calculated even t€ hint to
the reader that the treatise is not wholly original.

This gives a clue to the reason for Newton's sarcastic language; whose intensity
is heightened by the contrast between the Clarissivttts which is carefully prefixed to
the name of Mariotte, and the simple D. prefixed, not only to the names of Englishmen
like Wren and Wallis, but even to that of a specially distinguished foreigner tike
Huygens.

Newton must, of course, like all the scientific men of the time (Mariotte included),
have been fully cognizant of Boyle's celebrated controversy with Linus, which led to
the publication! in 1662. of the Defence of the Doctrine touching the Spring and Weight
of the Air. In that tract, Part II. Chap. 5, the result called in Britain Bot/les Law is
established (by a very remarkable series of experiments) for pressures less than, as well
as for pressures greater than, an atmosphere; and it ia established by means of the very
form of apparatus still employed for the purpose in lecture demonstrations* Boyle, at
least, claimed originality, for he says in connection with the difficulties met with in
the breaking of his glass tube : —

"...,,. an accurate Experiment of this nature would be of great importance to the
Doctrine of the Spring of the Air, and has not yet been made (that I know) by
any man "

In Mariotte s Discours de la Nature de VAir, published fourteen years later than
this work of Boyle, we find no mention whatever of Boylet though the identical form
of apparatus used by Boyle is described. The whole work proceeds, as does that on
Percussion^ with a calm ignoi'ation of the labours of the majority of contemporary
philosophers.

This also must, of course, have been perfectly well known to Newton : — ^and we
can now see full reason for the markedly peculiar language which he permits himself
to employ with reference to Mariotte.

What was thought of this matter by a very distinguished foreign contemporary,
appears from the treatise of James Bernoulli, De Gravitate JStheris^ Amsterdam, 1683,
p. 92.

" Veritas utri usque huj us regulEB manifesta fit duohus curiosis experimentis ab
Illustr, Dn. Boylio banc in rem factis, quse videsie in Tractate ejus contrd, Linumf
Cap. K» cui duaa Auctor subjunxit Tabulae pro divergis Condensationis et Rarefactionis
gradihus.'"

In order to satisfy myself that Newton's language, taken in its obvioua meaning,
really has the intention which I could not avoid attaching to it, I requested my colleague
Prof. Butcher to state the impression which it produced on him. I copied for him the
passage above quoted, putting A for the word Wrenno, and B for Mariottus; and I
expresfily avoided stating who was the writer. Here is his reply : —

"I imagine the point of the passage to be something of this kind (speaking without
ferther context or acquaintance with the Latinity of the learned author): —

"A established the truth by means of a (simple) experiment, before the Royal
Society; later, B thought it worth his while to write a whole book to prove the
same point.

112 NOTE ON A SINGULAR PASSAGE IN THE PRINCIPIA. [lXXII.

"I should take the tone to be highly sarcastic at £s expense. It seems to suggest
that B was not only clumsy but dishonest. The latter inference is not certain, but
at any rate we have a hint that B took no notice of A*8 discovery, and spent a deal
of useless labour."

This conclusion, it will be seen, agrees exactly with the complete ignoration of
Wren by Mariotte.

When 1 afterwards referred Prof. Butcher to the whole context, in my copy of
the first edition of the Principia, and asked him whether the use of Clarissimus was
sarcastic or not, he wrote —

" I certainly think so. Indeed, even apart from the context, I thought the
Clarissimus was ironical, but there can be no doubt of it when it corresponds to
D. Wren."

In explanation of this 1 must mention that, when 1 first sent the passage to
Prof. Butcher, 1 had copied it from Horsley's sumptuous edition; in which the Ds
are omitted, while the Clarissimus is retained.

Alike in France and in Germany, to this day, the Law in question goes by the
name of Mariotte. The following extracts, from two of the most recent high-class text-
books, have now a peculiar interest. 1 have put a word or two of each in Italics. These
should be compared with the dates given.

"Diese Frage ist schon friihzeitig untersucht und zwar fast gleichzeitig von dem
franzosischen Physiker Mariotte (1679) und dem englischen Physiker Boyle (1662)."
Wiillner, Lehrbuch der Experimentalphysik, 1882, § 98.

"La loi qui r^git la compressibility des gaz k temperature constante a ^t^ trouv6e
presque simultanem^nt par Boyle (1662) en Angleterre et par Mariotte (1676) en France;
toutefois, si Boyle a public le premier ses experiences, il ne sut pas en tirer T^nonc^
clair que donna le physicien franfais. C'est done avec quelque raison que le nom de loi
de Mariotte a pass^ dans Tusage." Violle, Cours de PhysiqiLe, 1884, § 283.

On this I need make no remark further than quoting one sentence from Boyle,
where he compares the actual pressure, employed in producing a certain compression in
air, with "what the pressure should be according to the Hypothesis^ that supposes the
pressures and expansions to be in reciprocal proportion." M. Violle has probably been
misled by the archaic use of " expansion " for volume.

It must be said, in justice to Mariotte, that he does not appear to have claimed
the discovery of any new facts in connection either with collision or with the effect of
pressure on air. He rather appears to write with the conscious infallibility of a man for
whom nature has no secrets. And he transcribes, or adapts, into his writings (without
any attempt at acknowledgment) whatever suits him in those of other people. He seems
to have been a splendidly successful and very early example of the highest class of what
we now call the Paper-Scientists. Witness the following extracts from Boyle, with a
parallel citation from Mariotte of fourteen years' later date at least. The comparison
of the sponges had struck me so much, in Mariotte's work, that I was induced to search
for it in Boyle, where I felt convinced that I should find it.

"This Notion may perhaps be somewhat further explained, by conceiveing the Air
near the Earth to be such a heap of little Bodies, lying one upon another, as may be
resembled to a Fleece of Wooll. For this (to omit other likenesses betwixt them)

LXXII.]

NOTE OK A SmOULAR PASSAGE IN THE PRINCIPIA.

1X8

consists of many slender and fleKible Hairs; each of which, may indeed, like a little
Spring, be eaaily bent or roiiled up ; but will also, like a Spring, be still endeavouring
to stretch itself out again. For though both these Haires, and the ^Slreal Corpuscles to
which we liken them, do easily yield to ex tern all pressures; yet e^ch of them (by virtue
of its structure) Ls endowed with a Power or Principle of Selfe-Dilatation ; by virtue
whereof, though the hairs may by a Mans hand be bent and crouded closer together,
and into a narrower room then suits best with the Nature of the Body. yet. whils't the
compression lasts, there is in the fleece they corapoaeth an endeavour outwards, whereby
it continually thrusts against the hand that oppoeeth its Expansion* And upon the
removall of the external pressure, by opening the hand more or less, the compressed
WooU doth, as it were, apontaneously expand or display it self towards the recovery
of its former more loose and free condition till the Fleece hath either regain 'd its
former Dimensions, or at least, approached them as neare as the compressing handi
(perchance not quite opened) will permit. The power of Belfe-Dilatation is somewhat
more conspicuous in a dry Spunge compressed, then in a Fleece of WoolL But yet we
rather chose to employ the latter, on this occasion, because it is not like a Spunge,
an intire Body; but a number of slender and flexible Bodies, loosely complicated, as the
Air itself seems to be."

And, a few pages later, he adds : —

'*.,,,,, a Column of Air, of many miles in height, leaning upon some springy
Corpuscles of Air here below, may have weight enough to bend their little springs,
and keep them bent : As, (to resume our former comparison,) if there were fleeces of
Wooll pil'd up to a mountainous height, upon one another, the hairs that compose the
lowermost Locks which support the rest, would, by the weight of all the Wool above
them, be as well strongly compress*d as if a Man should squeeze them together in his
hands, or imploy any such other moderate force to compress them. So that we need not
wonder, that upon the taking off the incumbent Air from any parcel of the Atmosphere
here below, the Corpuscles, whereof that undermost Air consists, should display them-
selves, and take up more room than before/'

Mariotte (p, 151). "On pent com prendre k peu pres cette diffi^rence de condensation
de TAir, par Texemple de plusieurs Sponges qu*on auroit entass^es tes unes sur les
autres. Car il est Evident, que celles qui seroient tout au haut, auroient teur entendue
nature! le: que celles qui seroient immddiatement au dessous, seroient an peu moins
dilat^es ; et que oelles qui seroient au dessous de tontes les autres, seroient trfes-serr^e^
et condens^es, II est encore manifeste, que si on 6toit toutes celles du dessus, celles du
dessous reprendroient leur dtendue naturellc par la vertu de ressort qu'elles ont, et
que si on en 6toit eeulement une partie, elles ne reprendroient qn'une partie de leur
dilatation/'

Those curious in such antiquarian details will probably find a rich reward by making
a careful comparison of these two works; and in tracing the connection between the
Liber int€gm\ and its fons et origo, the paper of Sir Christopher Wren.

Condorcet, in his Eloge de Mariotte, says :■ — ** Les lois du choc des corps avaient 6td

trouv^ par une m^taphysique et par une application d'analyse, nouvelles Tune et Tautre,

et ei subtiles. que les demonstrations de ees lois ne pouvaient satisfaire que les grands

math($mattciens. Mariotte chercha k les rendre, pour ainsi dire, populaires, en les

T. II* 15

114 NOTE ON A SINGULAR PASSAGE IN THE PRINCIPIA. [lXXII.

appuyant sur des experiences, &c." t.e., precisely what Wren had thoroughly done before
him.

''Le discours de Mariotte sur la nature de Fair renferme encore une suite d'exp^-
ences int^ressantes, et qui dtaient absolument neuves." This, as we have seen, is
entirely incorrect.

But Condorcet shows an easy way out of all questions of this kind, however delicate,
in the words: — "On ne doit aux morts que ce qui pent Stre utile aux vivants, la v^rit^
et la justice. Cependant, lorsqu'il reste encore des amis et des enfants que la v^rit^ pent
affliger, les ^gards deviennent un devoir; mais au bout d'un si^cle, la vanity peut seule
Stre bless($e de la justice rendue aux morts. **

Thus it is seen that even the turn of one of Newton's phrases serves, when rightly
viewed, to dissipate a widespread delusion: — ^and that while Boyle, though perhaps he
can scarcely be said to have been *'bom great," certainly "achieved greatness''; the
assumed parent of La Loi de Mariotte (otherwise Mariotte' aches Oesetz) has as certainly
had "greatness thrust upon" him.

Lxxin.] 115

LXXIII.

NOTE ON A PLANE STRAIN.

[Proceedings of the Edinburgh Mathematical Society, February 13, 1885. VoL ni.]

The object of this note is to point out, by a few remarks on a single case, how
well worth the attention of younger mathematicians is the full study of certain
problems, suggested by physics, but limited (so far as that science is concerned) by
properties of matter.

In de St Venant*s beautiful investigations of the flexure of prisms, there occurs
a plane strain involving the displacements

xy f_^

Physically, this is applicable to de St Venant's problem only when x and y are each
small compared with D. But it is interesting to consider the results of extending
it to all values of the coordinates. This I shall do, but very briefly.

1. The altered coordinates of any point are given, in terms of the original co-
ordinates, by

Hence &c' = &r(l + J) + 8y J,

15—2

116 NOTE ON A PLANE STRAIN. [lXXHI.

From these we see at once that, so far as an indefinitely small area is concerned,
the strain is a mere extension in all directions in the ratio

^/('-fi■-5■■■

combined with a rotation through an angle whose tangent is

2. Hence elementary squares remain squares ; and any two series of lines, dividing
the plane into little squares, will continue to do so after the strain.

One simple case is furnished by sets of lines parallel to the axes. Thus y ~ 6
becomes the parabola

^ =--^D- [y -^^W) <^>'

and x=^a becomes a parabola

^-¥(^-^) <=^

These groups of parabolas, (1) and (2), must evidently be orthogonal, and if the
simultaneous small increments of a and b be equai, must divide the plane into
little squarea But, as it is clear from (2) that the sign of a is immaterial, the
two lines

« = a, « = — a

are both deformed into the same parabola. Hence it appears that every part of the
area becomes duplex. This will be examined by another and more suitable method
later.

Having thus obtained another set of lines which divide the plane into squares,
we may begin again with it and obtain a third set, &c.

3. A line, y = mx, passing through the origin, becomes the parabola

The orthogonal trajectories of all such parabolas are the curves into which the circles

a;» + y« = c»
are deformed. Their equation may be put in the form

where y" is written instead of y' + ^t) •

LXXIII.] NOTE ON A PLANE STRAIN. 117

These curves have the property that, at every point, the sum {or difference) of the
distance from a given point, and of a multiple of the square root of the distance from
a given line, is constant

4. But, if we express the new rectangular coordinates of a point in terms of its
original polar coordinates, we have

0^ = 7008 5 + ^cos (^^""o)'

Thus the deformed circles, above spoken of, are seen to be epicycloids of the cardioid
series. Their orthogonal trajectories are the parabolas just mentioned.

5. Another curious set of questions is, as it were, the reverse of these: — i.e., what
were the curves, in the unstrained plate, which became the system

a; = a, y = 6,

or the other (also orthogonal) system

y = mx, a^ + i/^^^c^l

6. But a diflferent transformation is still more explicit in the information it gives.
Shift the origin to (0, — D), and we have

^ " D ' y - 2D
If we put X = psimf), y = /» cos ^, these give

Hence a circle, of radius p, surrounding the new origin, becomes a circle of radius

^ surrounding the point (O, — o) half-way between the new and old origins. The

<t> of any point in the circle becomes 2(f>.

Hence the whole surfiwse is opened up like a fan round the new origin, every
radius through this origin having its inclination to the axis of y doubled. Thus the
parts of a diameter, on opposite sides of the centre, are brought to coincide; and an
infinitely extended line, through the centre, becomes limited at the centre. Thus what
was a single sheet becomes duplex, as was said above.

7. It suffices to have indicated, by a partial examination of some of the curious
features of a single case, the stores of novelties which are thus easily reached. See
esjKicially, for additional materials of the same kind, the investigation in §§ 706-7 of
Thomson and Tait's Natural Philosophy,

118 [lxxiv.

LXXIV.

SUMMATION OF CERTAIN SERIES.

[Proceedings of the Edinburgh Mathematical Society, June 12, 1885. Vol. iii.]

[Abstract'.]

The attempt to enumerate the possible distinct forms of knots of any order,
though unsuccessful as yet, has led me to a number of curious results, some of which
may perhaps be new. The general character of the methods employed will be obvious
from an inspection of a few simple cases, and any one who has some practice in
algebra may extend the results indefinitely.

Take, for instance, the series

r^-nir + s)"^ + ^ ' ^ (r + &)•" - &c.

where the coeflScients are the terms of (1 — 1)", and the other factors are the mth
powers of the terms of an arithmetical series: — m being a positive integer. The
well-known properties of exponential series give us an easy method of summing all
expressions of this form. For we have

(gP* — €9»)« = e^p» — ne <~--^^+9> * + ^'^ "" £(«^+a9) « — &c.
which may be written in the form

((i'-?)«+^'^+^'^+&c-)"

= 2 - , Inp*^ - n (np + q — p)"* + ' ^ (np + 2q — p)^ — &c. J a:*".

> This abstract is part of the paper read in June, entitled **0n the detection of amphioheiral knots, with
special reference to the mathematical processes involved." I have unfortunately mislaid the MS. — P. G. T.

LXXIV.] SUMMATION OF CERTAIN SERIES. 119

Make np — r, g — ;) = «; and p and q are known.
The required sum is then the coefficient of a;"* in the expansion of

It vanishes therefore, so long as m < n ; and for m = n its value is

ml (p — g)"* = (— )"*m! s^.

When the coefficients in the given series are the cUtemate terms of (I — I)'*, we
have only to treat, as above, the expression

Such results may be varied ad libitum, by introducing two or more quantities in
place of X, and comparing coefficients of like terms: — e,g., as in finding, by the two
methods, of expansion, the term in afy* of the quantity

But it suffices to have called attention to processes which can give endless varieties
of results, some of which may have useful applications.

120 [ixxr.

LXXV.

ON CERTAIN INTEGRALS.
[Proceeiimgg of Ae Edimbwyk Maikemtaiieal Society, DeuwJber 11, 1885. ToL nr.]

This paper was bssed mamlj chi the results of an iiiTesdgatiao which wiU i^ypear
in fan in the TramaaetUmM of the Bojal Sodetj of Edinbozgh. Incidentally, however,
it kd to a diacaason of the question : — Fimd the Itnr of deMsitjf of a plamets atmo-
epktre, emppommg Bogles law to he trme for all preteurm, and tie temperatwre to be
wnform tkmmgkouL

Bojie's law gives p = kp, where p is the density at distanoe r firom the planet's
centre.

The Hydnetatic condition is -^ — — pB, where R is the attraction on unit of

k^^-p =-^^_ ^ where n i

Hence k^^ — p ' ^* ^ , where r^ is the radius, and M the mass of the

planet.

Write this as

^S— "-.C^'*

and difTerentiate ; and we obtain the carious equation

iGT:)-^"" "^

LXXV.] ON CEBTAIK INTEQRAU9. 121

A special value of p (compatible with the absence of a solid nucleus) is

but this cannot be generalised.

The finding of the integral of (1) in a form convergent for all values of r
greater than Vq presents novel and grave difficulties; but it is clear from the physical
question on which the wholie is based that such a solution exists.

If we change the independent variable to 8, where rs = 1, (1) becomes

or, if logp = w, ^ = ^,

cP\ogp_ ^ p .

^ £.«

This seems to be the simplest form into which the equation can be transformed.

[See a paper by Sir W. Thomson, " On the Equilibrium of a Oaa u/nder its oum
Gravity onlyr Proc. R. S, E, Feb. 21, 1887 ; or Phil. Mag. 1887, I., 287. 1899.]

T. II. 16

122 [lxxvl

LXXVI.

HOOKE'S ANTICIPATION OF THE KINETIC THEORY, AND OF

SYNCHRONISM.

[Proceedings of the Royal Society of Edinburgh, March 16, 1885.]

While collecting materials for a Text-book of the Properties of Matter, the
author had occasion to consult the very curious pamphlet by Robert Hooke, entitled
Lectures de Potentia Restitutiva, or of Spring (London, 1678).

In this work there is a clear statement of the principle of Synchronism, which
was applied by Stokes to the explanation of the basis of Spectrum Analysis. There
is also a very remarkable statement of the elementary principles of the modem
Kinetic Theory of Gases, the first mention of which is usually fixed sixty years later,
and ascribed to D. Bernoulli in his Hydrodynamica (Argentorati, 1738).

[Here is the chief passage referred to: —

"In the next place for fluid bodies, amongst which the greatest instance we have is
air, though the same be in some proportion in aU other fluid bodies.

''The Air then is a body consisting of particles so small as to be almost equal to the
particles of the Heterogeneous fluid medium incompassing the earth. It is bounded but on
one side, namely, towards the earth, and is indefinitely extended upward, being only hindred
from flying away that way by its own gravity, (the cause of which I shall some other time
explain.) It consists of the same particles single and separated, of which water and other
fluids do, coDJoyned and compounded, and being made of particles exceeding small, its motion
(to make its ballance with the rest of the earthy bodies) is exceeding swift, and its Yibrative
Spaces exceeding large, comparative to the Yibrative Spaces of other terrestrial bodies. I
suppose that of the Air next the Earth in its natural state may be SOOO times greater
than that of Steel, and above a thousand times greater than that of comuKm water, and pro>
portionably I suppose that its motion must be eight thousand times swifter than the former,

Lxxvi.] hookb's anticipation of the kinetic theory, etc. 123

and above a thousand times swifter than the latter. If therefore a quantity of this body
be inclosed by a solid body, and that be so contrived as to compress it into less room,
the motion thereof (supposing the heat the same) will continue the same, and consequently
the Vibrations and Occursions will be increased in reciprocal proportion, that is, if it be
Condensed into half the space the Vibrations and Occursions will be double in number: If
into a quarter the Vibrations and Occursions will be quadruple, isc,

"Again, If the conteining Vessel be so contrived as to leave it more space, the length
of the Vibrations will be proportionably inlaiged, and the number of Vibrations and Occur-
sions will be reciprocally diminished, that is, if it be suffered to extend to twice its former
dimensions, its Vibrations will be twice as long, and the number of its Vibrations and
Occursions will be fewer by half, and consequently its indeavours outward will be also weaker
by half.

"These Explanations will serve muUUia mutandis for explaining the Spring of any other
Body whatsoever." 1898.]

16—2

124

[lxxvu.

LXXVII.

ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES.

[Transactions of the Royal Society of Edinburgh, May 14, 1886, Vol. xxxiii.]

INDEX TO CONTENTS.

PAOE

Introductory 124

Part I. One Set of Equal Spheres, §§ 1-5 . 126
„ II. Mean Free Path among Equal

Spheres, §§ 6-11 .... 129
„ III. Number of Collisions per Particle

per Second, §§ 12-14 . . .134
„ IV. Clerk-Maxwell's Theorem, §§ 16-22 135
„ y. Rate of Equalisation of Average

Energy per Particle in two Mixed

Systems, §§ 23, 24 . . .140

PAGE

Part VI. On some Definite Integrals, §§

25-27 142

„ VII. Mean Path in a Mixtiu^ of two

Systems, § 28 . . . .144
„ VIII. Pressing in a System of Colliding

Particles, §§ 29, 30 . . . 144
„ IX. Effect of External Potential, §§ 31,

32 . . . . ' . .149
Appendix 152

The attempt to account for the behaviour of gases by attributiug their apparently
continuous pressure to exceedingly numerous, but nearly infinitesimal, impacts on the
containing vessel is probably very old. It certainly occurs, with some little develop-
ment, in Hooke's tract of 1678, Lectures de potentid restittUivd, or of Spring; and,
somewhat more fully developed, in the Hydrodynamica of D. Bernoulli, 1738. Traces
of it are to be found in the writings of Le Sage and Provost some 80 or 90 years
ago. It was recalled to notice in 1847 by Herapath in his Mathematical Physics,
and applied, in 1848, by Joule to the calculation of the average speed of the particles
in a mass of hydrogen at various temperatures. Joule expressly states* that his results
are independent of the number of the particles, and of their directions of motion, as
also of their mutual collisions.

* The paper is reprinted Phil. Mag. 1S57, II. See especially p. 215.

Ixxvil] on the foundations of the kinetic theory of gases.

125

In and after 1857 Clausius greatly improved the treatment of the problem by
taking account not only of the mutual impacts of the particles but also of the
rotations and internal vibrations which they commuuicate to one another, with the
bearing of this on the values of the specific beats ; at the same time introducing
(though only to a limited extent) the statistical method. Id this serie^s of papers
we find the tiiiat hint of the length of the mean free path of a particle, and the
explanation of the corapamtive slowness of the process of diffusion of one gas into
another. But throughout it is assumed, so far as the calculations are concerned, that
the particles of a gas are all moving with equal speeds. Of the Virial, which Clausius
introduced in 1870^ we shall have to apeak later.

In the PhiJmQphical Magazine for 1860 Clerk- Maxwell published his papers on
the " Collisions of Elastic Spheres/' which had been read to the British ABSOciation
in the previous year* In this very remarkable investigation we have the first attempts
at a numerical determination of the length of the mean free path. These are
founded on the observed rate of dififnsion of gases into one another; and on the
viscosity of gases, which here first received a physical explanation. The statistical
method is allowed fr^ee play, and consequently the law of distribution of speed among
the impiuging particle^^ is investigated, whether these be all of one kind or a mixture
of two or more kinds. One of his propositions (that relating to the ultimate partition
of energy among two groups of colliding spheres), which is certainly fundamental, is
proved in a manner open to very grave objectiatis i — not only on account of the
singular and unexpected ease with which the proof is arrived at, but also on account
of the extraordinary rapidity with which (it seems to show) any forced deviation
from its cx>nchision3 will be repaired by the natural operation of the collisions,
especially if the mass of a particle be nearly the same in each system* As this
proposition, in the extended form given to it by Boltzmann and others, seemed to
render the kinetic theory incapable of explaining certain well-known experimental facts,
I was induced to devote some time to a careful examination of Maxwell's proof
(mainly because it appears to me to be the only one which does not seem to evade
rather than boldly encounter the real difficulties of the question*), with the view of
improving it, or of disproving the theorem, as the case might be. Hence the present
investigation, which has incidentally branched off into a study of other but closely
connected questions. The variety of the traps and pit-falls which are met with even
in the elements of this subject, into some of which I have occasionally fallen, and
into which I think others also have fallen, is so great that I have purposely gone
into very minute detail in order that no step taken, however slight, might have the
chance of escaping criticism, or might have the appearance of an attempt to gloss
over a real difficulty.

* Compare anotlier inveatigatioD, al&o by Clerk -Maxwell but baaed on BoltsGmaim^B procefliea. which is given
in Naturt, vni. 537 (Ool. 23, 1B73), Some remarks on this wUl be miide at th& end of tho paper. Meanwhlie
it i£ flufflcient to point oat tliat this^ like this, (lesa elaborate) tnveiitigatiii^ufi of Meyer and Wataont m^relj
attempts to ehow that a certain state, once attained, is permarient. It gives no indiaation of the rate at
which \% would be restored if diaturbed. As wiU be Been later, I think thai this ^^rate*^ ia an element of
Tery great importance on account of ttie reasons for oonMence (in the general reaults of the investigiation)
which it so etrikingly furnishes.

126 ox THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVII.

The greater part of the following investigation is concerned only with the most
elementary parts of the kinetic theory of gases, where the particles are regarded as
hard smooth spheres whose coefficient of restitution is unity. The influence of external
forces, such as gravity, is neglected; and so is that of internal (molecular) forces.
The number of spheres is regarded as extremely great (say of the order 10* per
cubic inch): but the sum of their volumes is regarded as very small in comparison
with the space through which they are free to move; as, for instance, of the order
10~' or 10~^. It will be seen that several of the fundamental assumptions, on which
the whole investigation rests, are justified only by reference to numbers of such
enormous magnitude, or such extreme minuteness, as the case may be. The waUs of
the containing vessel are supposed simply to retene the normal velocity of eveiy sphere
impinging on them.

I. One set of Equal Spheres.

1. Very slight consideration is required to convince us that, unless we suppose
the spheres to collide with one another, it would be impossible to Bpplj any specieB
of finite reasoning to the ascertaining of their distribution at each instant, or the
distribution of velocity among those of them which are for the time in any particular
region of the containing vessel But, when the idea of mutual oollisicHis is intro-
duced, we have at once, in place of the hopelessly complex question of the behaviour
of innumerable absolutely isolated individuals, the comparatively simple statistical
question of the average behaviour of the various groups of a community. This di^
tinction is forcibly impressed even on the non-mathematical, by the extraordinary
steadiness with which the numbers of such totally unpredictable, though not uncommon,
phenomena as suicides, twin or triple births^ dead letters, &c., in any populous country,
are maintained year after year.

On those who are acquainted with the higher developments of the mathematical
Theory of Probabilities the impression is still more forcible. Every one, therefore,
who considers the subject from either of these points of view, must come to the
conclusion that continuous collisions among our set of elastic spheres will, provided
they are all equal, produce a state of things in which the percentage of the whole
which have, at each moment, any distinctive property must (after many collisions)
tend towards a definite numerical value ; frx>m which it will never aftierwards markedly
depart

This principle is of the utmost value, when legitimately applied; but the present
investigation was undertaken in the belief that, occasionally at least, its powers have
been to some extent abused. This appears to me to have arisen frt>m the difficulty
of deciding, in any one case, what amount of completeness or generality is secured
when the process of averaging is applied in successive steps from the conmiencement
to the end of an investigation, instead of being reserved (as it ought to be) for a
single comprehensive step at the very end.

Some of the immediate consequences of this principle are obvious without calcu-
lation : such as

LXXVII,] ON THE FOUNDATIONS OF THE KINETIC THEORY OF OASES.

127

(a) Even diatribution, at any moment, of all the particles throughout the space
in which they move,

(b) Even distribution of direction of motion among all particles having any one
speedj and therefore among all the particlGs.

(c) Definite percentage of the whole for speed lyii^g between definite limita.
These apply, not only to the whole group of particles but, to those in any portion

of Bpace sufficiently large to contain a veiy great number of particles.

(d) When there are two or more sets of mutually colliding spheres, no otm
of which is overwhelmingly more nmnerom than another^ nor in a hopeless viinority as
regards the sum of the others, similar assertions may be made as to each set
separately.

2, But calculation is required in order to determine the law of grouping as to
speeds, in (c) above. It is quite clear that the spheres, even if they once had equal
speed, could not possibly maintain auch a state. (I except, of course, such merely
artificial distributions as those in which the spheres are supposed to move in groups
in various non-intersecting seta of parallel Hues, and to have none but direct impacts.
For such distributions are thoroughly unstable ; the very slightest transverse impact,
on any ofie spheref would at once upset the arrangement.) For, when equal smooth
spheres impinge, they exchange their velocities along the line of centres at impact^
the other components being unchanged j so that, onlg when that liue is equally
inclined to their original directions of motion, do their speeds, if originally equal,
remain equal after the completion of the impact. And, as an extreme case, when
two spheres impinge so that the velocity of one is wholly in the line of centres at
impact, and that of the other wholly perpendicular to it, the first is brought to rest
and the second takes the whole kinetic energy of the pmr. Still, whatever be the
final distribution of speeds, it is obvious that it must be independent of any special
system of axes which we may use for its computation. This consideration, taken
along with (6) above, suffices to enable us to find this final distribution.

3. For we may imagine a space -diagram to be constructed, in which lines are
laid off from an origin so as to represent the simultaneous velocities of all the spheres
in a portion of space large enough to contain a very great number of them. Then

(b) shows that these lines are to be drawn evenly in all directions in space^ and

(c) that their ends are evenly distributed throughout the space between any two
nearly er^ual concentric spheres, whose centres are at the common origin. The density
of distribution of the ends {Le., the number in unit volume of the space-diagram) is
therefore a function of r, that is, of Ja^ + y" -f ^. But the argument above sbowa,
further, that this density must be expressible in the form

/(«)/(y)/(^)

whatevef rectangular axes be chosen, passing through the origin. These joint conditions
give only two admissible results : viz», either

/t^)=i!, or f{x)=^Bt^\

128 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVII.

The first is incompatible with the physical problem, as it would make the per-
centage of the whole particles, which have one definite speed, increase indefinitely with
that speed. The same consideration shows d fortiori that, in the second form of
solution, which is the only one left, C must be negative. Hence the density of the
distribution of " ends " already spoken of is

If n be the whole number of particles, *.«., of " ends," we must obviously have

^irB* I €""***ryr = n,
Jo

The value of the integral is 2\/ h»^

so that the number of spheres whose speed is between r and r-^dr is

^ J^^ n^-^'^r^dr (1).

This distribution will hereafter be spoken of as the " special " state.
The mean speed is therefore

V^Jo VTrA

while the mean-square speed is

This shows the meaning of the constant h. (Several of the results we have just
arrived at find full confirmation in the investigations (regarding mixed systems) which
follow, if we only put in these P for Q passim : — i.e., pass back from the case of a
mixture of spheres of two different groups to that of a single group.)

4. Meanwhile, we can trace the general nature of the process by which the
"special" arrangement of speed expressed by (1) is brought about from any initial
distribution of speed, however irregular. For impacts on the containing vessel do not
alter r, but merely shift the particular "end" in question to a different position on
its spherical locus. Similarly, impact of equal particles does not alter the distribution
of velocity along the line of centres, nor along any line perpendicular to it. But it
does, in general, produce alterations in the distribution parallel to any line other
than these.

Hence impacts, in all of which the line of centres is parallel to one common
line, produce no change in the arrangement of velocity-components along that line,
nor along any line at right angles to it. But there will be, in general, changes
along every other line. It is these which lead gradually (though very rapidly) to
the final result, in which the distribution of velocity-components is the same for all
directions.

LXXVII,] ON THE FOUNDATIONS OF THE KINETIC THEOKY OF OASEa.

129

When thk is arrived at, collisions will not, in the long run* tend to alter it.
For then the umformity of distribution of the spheres in apace, and the symniGtry
of distribution of velocity among them, enable us (by the principle of averages) to
dispense with the only limitation above imposed; viz,, the parallelism of the Unea of
centres in the collisions considered.

5. In what precedes nothing whatever has been said as to the ratio of the
diameter of one sphere to the average distance between two proximate spheres, except
what is implied in the preliminary assumption that the sum of the volumes of the
spheres is only a very small fraction of the space in which they are free to move.
It is probable, though not (so far as I know) thoroughly proved, that if this fraction
be exceedingly small the same results will ultimately obtain, but only after the lapse
of a proportionately long time ; while, if it be infinitely small, there will be no law,
as there will be practically no collisions. On the other hand, if the fraction be a
large one (t.e,, as in the case of a highly compressed gas), it seems possible that
these results may be true, at first, only a6 a very brief tinie-average of the condition of
the spheres in aoy region large enough to eoutain a great number; — that, in fact, the
distribution of particles and speeds in such a region will be for some time subject to
considerable but extremely rapid fluctuations. Reasons for these opinions will be seen
in the next section of the paper. But it must also be noticed that when the particles
fill the greater part of the space in which they move, simidtaneoKs impacts of three
or more will no longer be of rare occurrence; and thus a novel and difficult feature
forces itself into the question.

Of course with infinitely hard spheres the probability of such multiple collisions
would be infinitely small It must be remembered, however, that the investigation ie
meant to apply to physical particles, and not to mere mathematical fictions ; so that
we must, in the case of a highly compressed gas, take account of the possibility of
complex impacts, because the duration of an impact, though excessively short* is
essentially finite.

II. Mean Free Path among Equal Spheres.

0. Consider a layer, of thickness Sa?, in which quiescent spheres of diameter s
are evenly distributed, at the rate of 7ii per unit volume. If the spheres were opaque,
sudi a layer would allow to pass only the fraction

1 — n,Tr3^&c/4

of light falling perpendicularly on it. But if, instead of light, we have a group of
spheres, also of diameter 5, falling perpendicularly on the layer, the fi-action of th^e
which (whatever their common speed) pass without collision will obviously be only

1 — %7rfi*&c ;

for two spheres must collide if the least distance between their centres is not greater

than the sum of their radii. It is, of course, tacitly understood when we make such

a statement that the spheres in the very thin layer are so BccMered that no one

T, n, 17

130 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GA8E8. [lXXVH.

prewntM anoiker from doing its full duty in arresting those which attempt to pass.
Thus the fraction above written must be considered as differing very little from
miity. In tact^ if it differ much from unity, this consideration shows that the estimate
of the number arrested will necessarily be exaggerated. Another consideration, which
ahoald also be taken into account is that, in consequence of the finite (though veiy
small) diameter of the spheres, those whose centres are not in the layer, but
within one diameter of it, act as if they were, in part, in the layer. But the
cofrectioiis due to these considerations can be introduced at a later stage of the
mrestigatioiL

7. If the spheres impinge obliquely on the layer, we must substitute for Sx the
tUckneflB €f the lay^ in the direction of their motion.

If the particles in the layer be all moving with a common velocity parallel to
the layer, we must substitute for Sx the thickness of the layer in the direction of the
rdaii9e velocity.

If the particles in the layer be moving with a common velocity inclined to the
plaiie oi the layer, and the others impinge perpendicularly to the layer, the result will
be the same as if the thickness of the layer were reduced in the ratio of the
lelative to the actual speed of the impinging particles, and it were turned so as to
be perpendicular to the direction of the relative velocity.

8. Now suppose the particles in the layer to be moving with common speed
«^, bat in directions uniformly distributed in space. Those whose directions of motion
are inclined at angles between fi and fi + dfi to that of the impinging particles are,

in number,

nisinfidfil2;

and, by what has just been said, if v be the common speed of the impinging
particles, the virttuil thickness of the layer (so tax as these particles are concerned) is

where v© = Vv" + Vi' — 2tn;j cos fi

m lYkH reUUioe speed, a quantity to be treated as essentially positive.

ThfM the fraction of the impinging particles which traverses this set without

^Almfm in

1 — niW^BxVo sin I3dfil2v.

T/> And the fraction of the impinging particles which pass without collision
th-ff^nffh th45j layer, we must multiply together all such expressions (each, of course,
mitn»f>Jy nearly equal to unity) between the limits 0 and ir of 0. The logarithm
/v< RhA, fffAnct in

Iv Jo

LXXVII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF OASES. 131

Making Vo the variable instead of fi, this becomes

If v be greater than Vi, the limits of integration are v — Vi, and v + Vi, and the
expression becomes

but, if t; be less than Vi, the limits are Vi — v and Vi + v, and the value is

These give, as they should, the common value

- 4w,7r«»&r/3
when v = Vi.

9. Finally, suppose the particles in the layer to be in the "special" state. If
there be n in unit volume, we have for the number whose speed is between the
limits Vi and Vi + dvi

ni = 4mvi^dvi ^ / — €■*•>*.

Hence the logarithm of the fraction of the whole number of impinging particles, whose
speed is v and which traverse the layer without collision, is

The value of the factor in brackets is easily seen to be

_dV J_^ (2v l\ _j^
dit Sv' dh* '*' \W^ 2h?v) ^ '

where V

and thus it may readily be tabulated by the help of tables of the error-function.
When V is very large, the ultimate value of the expression is

1 /tt
4V A''

which shows that, in this case, the "special" state of the particles in the layer does
not affect its permeability.

17-2

132 ON THE FOUNDATIONS OF THE KINETIC THEORY OF QASE8. [lXXVU.

10. Write, for a moment, — c&r

as the logarithm of the firaction of the particles with speed v which traverse the
layer unchecked. Then it is clear that

represents the fraction of the whole which penetrate unchecked to a distance x into
a group in the "special" state. Hence the mean distance to which particles with
speed V can penetrate without collision is

0 I

e'^dx

This is, of course, a function of v\ and the remarks above show that it increases
continuously with v to the maximum value (when v is infinite)

t.0., the mean path for a particle moving with infinite speed is the same as if the
particles of the medium traversed had been at rest

11. Hence, to find the Mean Free Path among a set of spheres all of which
are in the special state, the natural course would appear to be to multiply the
average path for each speed by the probability of that speed, and take the sum of the
products. Since the probability of speed v to v-k-dv is

the above definition gives for the length of the mean finee path,

or, by the expression for e above,

1 /• €-*-Vrfr

l^hiH may without trouble (see § 9> be transformed into the simpler expiessicm

1 r* U^-^dr

wliioh iuliuit-s of easy numorical approximation. The numerical w<^ would be
NiinpliluHl by dividing aK>vo and below by c"**. but we prefer to keep the preaent
form on aoiHumt of its dinvt applicability to the case of mixed systemsL And it is
(MiriouA to noto that 4«~^ is the third difieiential coefficient of the denominaUK:

LXXVII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF OASES. 133

The value of the definite integral (as will be shown by direct csomputation in
an Appendix to the paper) is about

0-677 ;

and this is the ratio in which the mean path is diminished in consequence of the
motion of the particles of the medium. For it is obvious, from what precedes, that
the mean path (at any speed) if the particles were quiescent would be

[The factor by which the mean path is reduced in consequence of the " special "
state is usually given, after Clerk-Maxwell, as 1/V2 or 0*707.

But this appears to be based on an erroneous definition. For if n, be the
fraction of the whole particles which have speed v, p^ their fi*ee path; we have
taken the mean free path as

2 (ThPvX

according to the usual definition of a " mean."

Clerk-Maxwell, however, takes it as

S (n^v)

^(nvvlpv)'

i.e., the quotient of the average speed by the average number of collisions per
particle per second. But those who adopt this divergence from the ordinary usage
must, I think, face the question '' Why not deviate in a diiSerent direction, and
define the mean path as the product of the average speed into the average time of
describing a free path?" This would give the expression

S (ihv) . S (fhpt/v).

The latter factor involves a definite integral which differs from that above

solely by the factor V^/a? in the numerator, so that its numerical determination is

easy from the calculations already made. It appears thus that the reducing factor

would be about

>- X 0-660, = 0-734 nearly ;

i,e,, considerably more in excess of the above value than is that of Clerk-Maxwell.
Until this comparatively grave point is settled, it would be idle to discuss the small
effect, on the length of the mean free path, of the diameters of the impinging
spheres.]

134 ON THB FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVH.

III. Number of Collisions per Particle per Second.

12. Here again we may have a diversity of definitions, leading of course to
different numerical results. Thus, with the notation of § 11, we may give the mean
number of collisions per particle per second as

2(n„v/pt).

This is the definition given by Clerk-Maxwell and adopted by Meyer; and here the
usual definition of a " mean " is employed. The numerical value, by what precedes, is

Meyer evaluates this by expanding in an infinite series, integrating, and summing.
But this circuitous process is unnecessary; for it is obvious that the two parts of
the expression must, from their meaning, be equal ; while the second part is integrable
directly.

13. On account of its bearing (though somewhat indirectly) upon the treatment
of other expressions which will presently occur, it may be well to note that a mere
inversion of the order of integration, in either part of the above double integral,
changes it into the other part.

Otherwise: — we may reduce the whole to an immediately integrable form by the
use of polar co-ordinates; putting

v — r cos 6, Vi = r sin 0,

and noting that the limits of r are 0 to x in both parts, while those of 6 are 0
to 7r/4 in the first part, and 7r/4 to 7r/2 in the second. [This transformation,
however, is not well adapted to the integrals which follow, with reference to two
sets of spheres, because A has not the same value in each set.]

14. Whatever method we adopt, the value of the expression is found to be

2
and, as the mean speed is (§ 3) -j== ,

we obtain Clerk-Maxweirs value of the mean path, above referred to, viz.,

1
mrs'^2 '

LXXVIl] ON THE FOUNDATIONS OF THE KINETIC TEEORY OF GASES,

135

Bot (in illustmtion of the remarks at the end of § 11) we might have defined
the mean number of colliiiioBa per particle per Becond as

2{n^tr)

, or as ^

; &Cm &o.

The first, which expresses the ratio of the raeaa speed to the mean free j^th^ gives

2 wn^

and the secondp which m the reciprocal of the mean value of the time of describing

a free path, gives

J. mtfl*

The three values which we have adduced as examples bear to one another the
reciprocals of the ratios of the above-mentioned determinations of the mean free path.

IV, Clerk- MaxwelVs T/ieorem.

15* In the ardour of his research of 1859*, Maxwell here and there contented
himself with very incomplete proofs (we can scarcely call them more than illustrations)
of some of the most important of his results. This is specially the case with the
investigation of the law of ultimate partition of energy in a mixture of smooth
Bpherical particles of two different kinds. He obtained, in accordance with the so-
called Law of Avogadro, the result that the average energy of translation is the
same per particle in each system; and he extended this in a Corollary to a mixture
of any number of different systems. This proposition, if true, is of fundamental
importance. It was extended by Maxwell himself to the case of rigid particles of
any form, where rotations perforce come in. And it appears that in such a case
the whole energy is ultimately divided equally among the various degrees of freedom*
It has since been extended by Boltzmaun and others to cases in which the individual
particles are no longer supposed to be rigid, but are regarded as complex systems
having great numbers of degrees of freedom. And it is stated, as the result of a
process whichp from the number and variety of the assumptions made at almost every
stage, is rather of the nature of playing with symbols than of reasoning by consecutive
steps, that in such groups of systems the ultimate state will be a partition of the
whole energy in equal shares among the classes of degrees of freedom which the
individual particle-systems possess. This, if accepted as true, at once raises a formidable
objection to the kinetic theory. For there can be no doubt that each individual
particle of a gas has a very great number of degrees of freedom besides the six
which it would have if it were rigid : — the examination of its speotrum while
incandescent proves this at once. But if all these degrees of freedom are to share
the whole energy (on the average) equally among them, the results of theory will no
longer be consistent with our experimental knowledge of the two specific heats of a
gas, and the relations between them.

* FhiL Maff., 1860.

136 ON THE FOUNDATIONS OF THE KINETIC THEORY OF 6A8B8. [ULXVIL

16. Hence it is desirable that Clerk-MazweU's proof of his fondamental Theorem
should be critically examined, and improved where it may be found defective. If it
be shown in this process that certain preliminary conditions are absolutely necessary
to the proof even of Clerk-Maxwell's Theorem, and if these cannot be granted in the
more general case treated by Boltzmann, it is clear that Boltzmann's Theorem must
be abandoned.

17. The chief feature in respect of which Maxwell's investigation is to be
commended is its courageous recognition of the diflSculties of the question. In this
respect it far transcends all other attempts which I have seen. Those features, besides
too great conciseness, in respect of which it seems objectionable, are : —

(a) He assumes that the transference of eneigy from one system to the other
can be calculated from the results of a single impact between particles^ one from
each system, each having the average translational eneigy of its system.

Thus (so &r as this step is concerned) the distribution of energy in each
system may be any whatever.

(6) In this typical impact the velocities of the impinging spheres are taken as
at right angles to one another, so that the relative speed may be that of mean
square as between the particles of the two systems. The result obtained is fidlacious,
because in general the directions of motion after impact are found not to be at
right angles to one another, as they would certainly be (on account of the perfect
reversibility of the motions) were this really a typical impact

(c) Clerk-Maxwell proceeds as if every particle of one system impinged upon
one of the other system at each stage of the process — Le^ he calculates the trans-
ference of energy as if each pair of particles, one from each system, had simultaneously
a typical impact. This neglect of the immensely greater number of particles which
either had no impact, or impinged on others of their own group, makes the calculated
rate of equalisation far too rapid.

(fit) Attention is not called to the fa^t that impacts between particles are
numerous in proportion to their relative speed, nor is this consideration introduced in
the calculations.

(e) Throughout the investigation each step of the process of averaging is
performed (as a rule) before the expressions are ripe for it

18. In seeking for a proof of Maxwell's Theorem it seems to be absolutely
essential to the application of the statistical method to premise : —

(A) That the particles of the two systems are thoroughly mixed

(B) That in any region containing a very large number of particles, the particles
of each kind separately acquire and maintain the error-law distribution of speeds —
ie., each set will ultimately be in the ** special" state. The disturbances of this
arrangement produced in either system by impacts on members of the other are

lxxvil] on the fouhbations of tbe kinetic theory of qasbs.

137

regarded as being promptly repaired by means of tlje internal collisions in the system
iti^elf. This is the sole task assigned to these internal collisions* We assume that
they accomplish it, so we need not further allude to them.

[The warrant for these assumptions is to be sought as in 1 4 ; and in the fact
that only a small fraction of tbe whole particles are at any instant in collision ;
i.e., that each particle advances, on tbe average, through a conaiderable multiple of
it8 diameter before it encounters another.]

(C) That there is perfectly free access for collision between each pair of particles,
whether of the same or of different systems ; and that, in the mixture, the number
of particles of one kind is not overwhelmingly greater than that of the other kind,

[This is one of the essential points which seem to be wholly ignored by Boltzmann
and his commentators. There is no proof given by theui that one system, while
regulating by its internal collisions the distribution of energy among its own members^
can also by impacts regulate the distribution of energy among the members of
another system, when these are not free to collide with one another. In fact, if (to
take an extreme case) the particles of one system were so small, in comparison with
the average distance between any two contiguous ones, that they practically had no
miUual collisions, they would behave towaixis the particles of another system much as
Le Sage supposed his ultm-mundane corpuscles to behave towards particles of gross
matter. Thus they would merely alter the apparent amount of the molecular forces
between the particles of a gas. And it is specialty to be noted that this is a
question of effective diameters merely, and not of masses : — so that those particles
which are virtually free from the self-regulating power of mutual collisions, and
therefore form a disturbing element, may be much more massive than the others*]

19* With these assumptions we may proceed as follows : — Let P and Q be the
masses of particles from the two systems respectively ; and when they impinge, let
u, V be their velocity -components measured towards the same parts along the line
of centres at impact. If these velocities become^ after impact^ u', v' respectively, we
have at once

P(u'-u)--

2PQ

an immediate consequence of which is

(u-v)--Q(v'-v);

P(n'*-u*) = -.

4PQ

[Put _ Qyi ^{P^Q)nv]^~Q (v'« - v%

{P^Qf

Hence, denoting by a bar the average value of a quautityp we see that transference of
energy between the systems must cease when

Pu^^Q?-(P^Q)^ = 0 ,.,, ,..., (1),

and the question is reduced to finding these averages.

[I thought at first that uv might be assumed to vanish, and that u^ and v*

might each be taken as one- third of the mean square speed in its system. This set

of suppositions would lead to Max well's Theorem at once. But it is clear that, when

two particles have each a ^veti speed, they are more likely to collide when they

T. JI. 18

138

ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVU.

are moving towards opposite parts than when towards the same parts. Hence uv
must be an essentially negative quantity, and therefore Pu* necessarily less than Qv\
if P he greater than Q, Thus it seemed as if the greater masses would have on
the average less energy than the smaller. These are two of the pitfalls to which
I have alluded. Another will be met with presently.]

20. But these first impressions are entirely dissipated when we proceed to
calculate the average values. For it is foimd that if we write (1) in the form

Pu«-uv-Qv»-uv = 0,

the terms on the left are equal multiples of the average energy of a P and of a
Q respectively. Thus Maxwell's Theorem is rigorously true, though in a most unexpected
manner. There must surely be some extremely simple and direct mode of showing
that u* — uv is independent of the mean-square speed of the system of 0*s. Mean-
while, in defiault of anything more simple, I give the investigation by which I arrived
at the result just stated.

21. Suppose a particle to move, with constant speed v, among a system of
other particles in the "special" state; the fraction of the whole of its encounters
which takes place with particles, whose speed is from t^ to Vi-^dvi and whose
directions of motion are inclined to its own at angles from /8 to /8 + d/S, is (§ 8)
proportional to

€-*''Vi*rft;,t;o sin /8dy8,

or as we may write it for brevity

PiVo sin fidfi.

This is easily seen by remarking that, by § 8, while the particle advances through
a space &r, it virtually passes through a layer of particles (such as those specified)
of thickness Vo&r/v. Here (§ 3) 3/2i is the mean-square speed of the particles of the
system.

Let the impinging particle belong to another group, also in the special state.
Then the number of particles of that group which have speeds between v and v-^dv
is proportional to

as we will, for the present, write it.

Now let F, Fj, Fo, in the figure, be the projections of v, »j, », on the unit

sphere whose centre is 0; (7 that of the line of
centres at impact. Then VOV^^fi, Let V.OV^a,
FoOF, = flt„ FoOC=7, and VV.C^^. The limits of 7
are 0 and 'n'/2 ; those of ^ are 0 and 27r. Also the
chance that C lies within the spherical surface-
element 3mydydif>, is proportional to the area of
the projection of that element on a plane perpen-
dicular to the direction of ««, i.e., it is propor-
tional to

cos 7 sin yd^d^

LXXVII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF OASES. 139^

But by definition we have

u = v cos VOC = V (cos a cos 7 + sin a sin 7 cos ^),

V = Vi cos FiO(7= Vi (cos ai cos 7 + sin «! sin 7 cos ff>)\

and by the Kinematics of the question, as shown by the dotted triangle in the figure^
we have

t; cos a — Vi cos aj = v©,

v sin a — Vi sin «! = 0.

Thus, as indeed is obvious from much simpler considerations,

u — V = Vo cos 7,

I w/jVo sin fid^M (u — v) cos 7 sin ydydtf)

so that u* — uv =

/■

jwiVo sin /3dl3 cos 7 sin yd^d<l>
WiVq sin fidfiv (cos a cos 7 + sin a sin 7 cos <f>) Vo cos' 7 sin ydyd<f>

I i/i/iVo sin /Sd/S cos 7 sin ydyd<t>

where each of the integrals is quintuple.

The term in cos^ vanishes when we integrate with respect to ^: — and, when we
further integrate with respect to 7, we have for the value of the expression

^ jwiVo sin fidfiwo cos a

I vviVo sin fid/3
where the integrals are triple.

Now 2tWo cos a = v" + V — Vi\

and WiBinfidfi^^Vodvo,

so that the expression becomes

/* Vq* dvo

\PVi-

It will be shown below (Part VI.), that we have, generally.

2i»-l

/■

Wi 2n + l 4 (AJfc)«+> '

and that it is lawful to differentiate such expressions with regard to A or to A;. Hence

d d

i'-i'-{i-i}'-i' 1

"'-"'-J 7:73 *•

Thus Clerk-Maxwell's Theorem is proved.

18—2

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ram JL "ill* ?QtiGsL ssLSd. im "Ste AT«fni^ aitfrxTr >«r i«ztii.-{t; *« >» atfHr*nu in a* 7**:*

LXXVII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 141

referred to, we put (hky^ for A*, and also put k for A in the exponentials where the
integration is with respect to v,, it becomes

according to the notation of § 21. This is the average number of impacts per second
which a P has with Q's.

Hence, if w be the whole energy of the P's, p that of the Qs, per unit volume, the
equations of § 19 become

w = —

16 PQ ^ M{h + k). .

3 {P + QY * V -Jl— ^"•^ - '""> = -P'

from which we obtain, on the supposition (approximate enough for our purpose) that
we may treat l/h + l/k as constant,

am —vip = (7e~''^,

, I le PQ ^, ^ . /7r(A + &)

-here j,-= ^ ^.^-^—^(,n^n)^-^^ .

The quantity nv — vip = inn (v/m — p/n)

is 7nn times the difference of the average energies of a P and a Q, and (since
c*<^= 100 nearly) we see that it is reduced to one per cent, of its amount in the time

t-ir(^T- ^^'^ ^^^Q>' /.^^ seconds.

24. For a mixture, in equal volumes, of two gases in which the masses of the
particles are not very different, say oxygen and nitrogen, we may assume as near enough
for the purposes of a rough approximation

?/i = /i=^xlO»,

whence m + n (per cubic inch) is double of this,

1^ = ~ = (12 X ICOO inch sec.)«,

s=^Sx 10-» inch,

so that ti = r^; — ^ — - — T^A^- - lo - i">aa \/ a =0" i riu sGconds, nearly ;

16 x9 X 3 X 10** X 12 X 1600 V ^TT 3x10* ^

and the difference has fallen to 1 per cent, of its original amount in this period,
i.e., after each P has had, on the average, about four collisions with Q*s. This calculation
has no pretensions to accuracy, but it is excessively useful as showing the nature of the
wan-ant which we have for some of the necessary assumptions made above. For if
the rapidity of equalisation of average energy in two s)'stems is of this extreme order

142 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVII.

of magnitude, we are entitled to suppose that the restoration of the special state in anj
one system is a phenomenon taking place at a rate of at least the same if not a
higher order of magnitude.

Clerk-Maxwell's result as regards the present question is that, at every typical
impact between a P and a Q, the difference of their energies is reduced in the ratio

(P-Q

(P-Q\\

so that, if the masses were equal, the equalisation would be instantaneous.

VI. On some Definite Integrals,

25. It is clear that expressions of the forms

I e'^^afdx I e'^'ifdy and / e^^afdx I e'^y'dy,

where r and s are essentially positive integers, may lawfully be differentiated under the
integral sign with regard to h or to k. In fact they, and their differential coefficients,
which are of the same form, are all essentially finite.

As, in what immediately follows, we shall require to treat of the first of these forms
only when r is odd and s even, and of the second only when r is even and s odd,
it follows that their values can all be obtained by differentiation fix>m one or other
of the integrals

fe-^xdxl' e-^»'dy= '^'^
Jo Jo ^

^h'Jh + k'

and I e"***cJa? / e~^ydy^

Jo J X

These values may be obtained at once by noticing that the second form is integrable
directly; while, by merely inverting the order of integration, it becomes the first with
h and k interchanged.

26. In ^ 21, 22 we had to deal with a number of integrals, all of one form, of
which we take as a simple example

I./S^fw^'^dvo

' J Wi

= g J e-^xdx - j €-** ydy {{x + yf -{x- yf) +j e'^^ydy ((y + x^ - (y - xfyt .

lxxvil] on the foundations of the kinetic theory of gases. 143

From the remarks above it is clear that this can be expressed as

2V7r
3 4

IV dhdk ^ dk^J h ^/h+'k \ dkdk "^ dhV k VATikJ
3 4 2U«(A + A)*"^A;»(A + A:)»/

= ^^ (A;'-f 3^-'A)-^(3M'-^A»)

^ V;w; (A + fc) *
4 (AA;)« •

The peculiar feature here shown is the making up of the complete cube of A: 4- A
in the numerator by the supply of the first half of its terms from the first part of the
integral, and of the remainder from the second*. On trial I found that the same thing
holds for /o and /j, so that I was led to conjecture that, generally, as in § 21

2w + l 4 {hky-^^ '

After the preliminary work we have just given, it is easy to prove this as folio wa
We have always

{{X + y)«+^ "{x- y>r^') {{x + y)» + (a: - yf)

= (ar -f y)«»+» - (a: - y)«^+» + (aJ» - y^^ {{x + y)^^ - (« - yf^').

Operate on this by / e~^^xdx\ e~^^ydy i J,

and on the same expression, with x and y interchanged (when, of course, it remains
true), by

I e-'^'xdx I e-^^^ydy ( j ,

and add the results. This gives at once

which is found on trial to be satisfied by the general value given above.

* Prof. Cayley has called my attention, in connection with this, to the foUowing expression from a Trinity
(Cambridge) Examination Paper: —

(a+6)«»=(a + 6)»(a»+6»)

+ (a + 6)*"^ (na* h + nab*)

+ („+^)-.(»•-5±?a.6.^.».^?i+la»6-)

+ (a+6) —^-2^ — ~{n-l) (* '^ +« ^ )•

144 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVII.

27. Partly as a matter of curiosity, but also because we shall require a case of
it, it may be well to mention here that similar processes (in which it is no longer
necessary to break the y integration into two parts) lead to the companion formula

^7rl.3.5...(2n-l)(A-f fc)^-'

And we see, by Wallis' Theorem, that (when n is increased without limit) I^ is
ultimately the geometric mean between /jn-i and /gn+i-

VII. Mean Path in a Mixture of two Systems,

28. If we refer to § 10, we see that, instead of what was there written as — eBx,
we must now write —(e + ei)Bx; where Ci, which is due to stoppage of a particle of
the first system by particles of the second, differs from e in three respects only. Instead
of the factor 4»*, which appears in e, we must now write («4-«i)'; where Si is the
diameter of a particle of the second system. Instead of h and n we must write Ai
and rii respectively.

Hence the mean fr^e path of a particle of the first system is

y ttJo e + Si '

which, when the values of e and Si are introduced, and a simplification analogous to those
in §§ 9, 11, is applied, becomes

4f€''^X!^dx

-«• + (!+ 2a?)j%-'^dx + ^ (i^'J (a:.€-«.'+ (1 + ^*) ]"%-«* dx^

Thus the values tabulated at the end of the paper for the case of a single system
enable us to calculate the value of this expression also.

VIII. Pressure in a System of Colliding Particles.

29. There are many ways in which we may obtain, by very elementary processes,
the pressure in a system of colliding particles.

(a) It is the rate at which momentum passes across a plane unit area; or the
whole momentum which so passes per second. [It is to be noted that a loss of
negative momentum by the matter at either side of the plane is to be treated as a gain
of positive,]

in whicli

LXXVII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES,

145

In thb, and the other tnvestigatiOQf^ which follow, we deal with planes suppcxsed
perpendicular to the axis of ^; or with a thin layer bounded by two such platiea.

The average number of particles at every instant per square unit of a layer, whoge
thlGkness i» Ss^, is it 8a;, Of these the fraction

-v^

f-A»Vrfv

have speeds from v to v+dv. And of these the fraction

are moving in directiona inclined from ff to ff-\-dff to the axis of ^. Each of them,
therefore, remains in the layer for a time

and carries with it momentum

ix/v COS /3,

parallel to a?. Now from /9 ^ 0 to fi= x we have positive momentum passing towards jt

IT*

positive. From ^ ~a to ^^w we have an eqiicd amount of negative momentum leaving

X positive. Hence the whole momentum which passes per second through a plane unit
perpendicular to x is

2>c

I Pn j pifTcm*^ sin 0d0 = I Pnv\

where the bar indicates mean value. That h

Pressure ==jO= 5 (Kinetic Energy in Unit Volume)*

(b) Or we might pi-oceed as follows, taking account of the position of each particle
when it was last in collision.

Consider the particles whose speeds are from v to v + dv, and which are contained
in a layer of tMckneas Sap, at a distance ts from the plane of yz. Each has (§ 10) on
the average m collisions per second. Thus, by the perfect reversibility of the motions,,
from each unit area of the layer there start, per second ^

such particles, which have just bad a collision. These move in directions uniformly
distributed in space ; so that

sin ffdfffi

of them are moving in directions inclined ^ to ^ + d)8 to the axis of m. Of these
the fraction

m—4mm!C$

T. II.

146 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVU.

(where a; is to be regarded as signless) reach the plane of yz^ and each brings
momentum

Pt;cos/8

perpendicular to that plane. Hence the whole momentum which reaches unit area
of the plane is

ix\nP{ vr^ f cos 13 sin fidfif edxe"^^^
^ Jo Jo Jo

= wP/ i^f cos^fiainfidfi,
Jo Jo

the same expression as before.

30. But this result is approximate only, for a reason pointed out in § 6 abova To
obtain a more exact result, let us take the virial expression itself. It is, in this case, if
N be the number of particles in volume F,

\pif^ = lpr+\x(Rr),

where B is the mutual action between two particles whose centres are r apart, and is
positive when the action is a stress tending to bring them nearer to one another. Hence,
omitting the last term, we have approximately

P = 3F^^'
which we may employ for the purpose of interpreting the value of the term omitted.

[It is commonly stated (see, for instance, Clerk-Maxwell's Lecture to the Chemical
Society*) that, when the term ^li(Rr) is negative, the action between the particles is in
the main repulsive: — "a repulsion so great that no attainable force can reduce the
distance of the particles to zero." There are grave objections to the assumption of
molecular repulsion; and therefore it is well to inquire whether the mere impacts, which
must exist if the kinetic theory be true, are not of themselves sufficient to explain the
experimental results which have been attributed to such repulsion. The experiments
of Regnault on hydrogen first showed a deviation from Boyle's Law in the direction of
less compression than that Law indicates. But Andrews showed that the same thing
holds for all gases at temperatures and pressures over those corresponding to their
critical points. And Amagat has experimentally proved that in gaseous hydr^o^en, which
has not as yet been found to exhibit any traces of molecular attraction between its
particles, the graphic representation of pF in terms of p (at least for pressures above an
atmosphere, and for common temperatures) consists of a series of parallel straight lines.
If this can be accounted for, without the assumption of molecular repulsion but simply

* Ckem. Soc, Jour., xm. (1S75), p. 493.

LXXVn.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 147

by the impacts of the particles, a real diflSculty will be overcome. And it is certain that^
at least in dealing with hard colliding spheres if not in all cases, we have no right to
extract from the virial, as the pressure term, that part only which depends upon impacts
on the containing vessel; while leaving unextracted the part depending on the mutual
impacts of the particles. The investigation which follows shows (so far as its assumptions
remain valid when the particles are not widely scattered) that no pressure, however great,
can bring a group of colliding spheres to a volume less than four times the sum of
their volumes. If they were motionless they could be packed into a space exceeding the
sum of their volumes in the ratio 6 : 7rV2, or about 1*35 : 1, only.]

In the case of hard spheres we have obviously r = 8; and, with the notation of
§ 19, remembering that Q = P, k = h, we have

i2 = -P(u-v).

Hence we must find, by the method of that section, the mean value of the latter
expression. It is easily seen to be

^ p fvviVo^ sin 13 d/S cos'y sin ydrfdif> 2P fppiVo^dVo/ Wi

fvPiVosia 13 d/S cosy siaydyd(f> " 3 JwiVdVo/tWi

" 3 /,/3" V:

2h'

But, § 14, the average number of collisions, per particle per second, is

2 N

V^i

Hence, for any one particle, the sum of the values of R (distributed, on the
average, uniformly over its surfiM^) is, in one second,

Thus it would appear that we may regard each particle as being subjected to the
general pressure of the system; but as having its own diameter doubled. It is treated^
in &ct, just as it would then be if all the others were reduced to massive points.

The value of the term in the virial is

\nst{R)

because, though every particle suffers the above average number of collisions, it takes
two particles to produce a collision. This is equal to

— np!rf^ = — 6p (sum of volumes of spheres) ;

so that the virial equation becomes

— 3

nPv*/2 = q1> {F — 4 (sum of volumes of spheres)},

which, in farm at least, agrees exactly with Amagat's* experimental results for hydrogen.

* Armalei de Chimie, xxn. ISSl.

19—2

148 ON THE FOUNDATIONS OF THE KINETIC THEOBT OF GASES. [lXXVH.

These results are closely represented at IS"* C. by

p(F- 2-6) -2781;

mi at 100" C. by i>(^- 2-7) « 3518.

The quantity subtracted fix)m the volume is sensibly the same at both temperatures.
Tlie light-hand members are nearly in proportion to the absolute temperaturea The
preasure is measured in metres of mercury. Hence the volume of the gas, at 18^ C.
and one atmosphere, is (to the unit employed)

2-6 + 2731/0-76 = 3696 nearly.

Thus, by the above interpretation of Amagat's results, we have at 18" C.

tiw-«» = 3-9/3596.

CSeik-Maxwell, in his Bradford Lecture^, ranks the various numerical data as to
gases aooHding to ''the completeness of our knowledge of them." The mean fines path
appears in the second rank only, the numbers in which are regarded as rough ap-
proximations. In the third rank we have two quantities involved in the expression
fx the mean fi-ee path, viz., the absolute diameter of a particle, and the number of
partidcg per unit volume (s and n of the preceding pages).

To determine the values of 8 and n separately, a second condition is required.
It has usually been assumed, for this purpose, that the volume of a gas, "when
reduced to the liquid form, is not much greater than the combined volume of the
nwlemlefC* Maxwell justifies this assumption by reference to the small compressibility of
Hqnidw,

Bat, if the above argument be, even in part, admitted, we are not led to any
andi conclusion, and we can obtain n^ (as above) as a quantity of the second rank.
We have already seen that n^ is inversely proportional to the mean firee path, and
is Urns also of the second rank. From these data we may considerably improve our
apprcadmations to the values of n and of 8,

Taking Maxwell's estimate of the mean free path in hydrogen, we have (to an
inch as unit of length)

5:^ = 380.10-.

From these values of n^ and n^ we have, approximately, for O"* C. and 1 atmosphere,

n = 16.10», « = 6.10-*.
The values usually given are

»=3.10»», 5 = 2-3. 10-«.
It must be recollected that the above estimate rests on two assumptions, neither
* PMU Mag., 1878, ii. 458. See alM Nature, Tin. 298.

LXXVII.] ON THE rOUNDATIONS OF THE KINETIC THEORY OF GASES.

149

of which is more than an approximation, (a) that the particles of hydrogen hehave
like hard spheres, (b) that they exert no mutual molecular foroes. If there were molecular
attraction the value of ns^ would be greater than that assumed above, while m^ would
be unaltei^ed. Thus the particles would be larger and less numerous than the estimate
shows.

^H [Of course, after what has been said, it is easy to see that F should be di-

^^ minished further by a quantity proportional to the surface of the containiug vessel

I and to the radius of a sphere. But though this correction will become of constantly

i greater importance as the bulk occupied by a given quantity of gas is made smaller,

I it is probably too minute to be detected by experiment]

f co:

I un

IX. Effect of Ea^srmil Potential (Added Jtine 15, 1886.)

SI, Another of MaxwelVs most remarkable contributions to the Kinetic Theory
consists in the Theorem that a vertical column of gas, w*hen it is in equilibrium
under gravity, has the same tempeiuture throughout* He states, howeverj that an
erroneous argument on the subject, when it occurred to him in 1866, "nearly upset
[his] belief in calculation/'* He has given various investigations of the action of
external forces on the distribution of colliding spheres, but all of them are complex.
The proc€6s of Boltzmann, alluded to in a foot-note to the introduction {ant^j p, 125),
and which Clerk- Maxwell ultimately preferred to hia own methods, involves a step of
the following nature.

An expression, analogous to the f of 1 3, but in which B and C are undeter-
mined functions of the coordinates ar, y, $ of a point, is formed for the number of
particles per unit volume, at that point, whose component speeds, parallel to the
axeSj lie between given narrow limits. I do not at present undertake to discuss the
validity or the sufficient generality of the process by which this expression is obtained,
though the same process is (substantially) adopted by Watson and others who have
written on the subject However obtained, the expression is coirect It can be
established at once by reasoning such as that in ^ 2, 3, 4 To determine the forms
of the aforesaid functions, however, a most peculiar method is adopted by Boltzmann
and Maxwell The number of the particles per unit volume at x, y, z whose cor-
responding " ends " occupy unit volume at m, v, w in the velocity space-diagram (§ 3),
is expressed in terms of these flmctions, and of ti* + »* + tef. The variation of the
logarithm of this number of particles is then taken, on the assumption that

&e = uSt, &c., Sa = - -^ Bt, &a,

where U is the external potential; and it is equated to zero, becniise the iitmiber of

* Nature^ mu, Maj 29, 187 S. M»iweire iiAme doen not oecnr is the Index to thla volame, Ibotigh he hu
made it least five eootribatios* to it, most of which bear on the preceat subject:— Ti2. at pp. B5, 298, S$l,
527, fi37.

150 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVH.

particles is unchangeable. As this equation must hold good for all values of u, v, w,
it furnishes sufficient conditions for the determination of B and C. The reasons for
this remarkable procedure are not explained, but they seem to be as below. The
particles are, as it were, followed in thought into the new positions which they would
have reached, and the new speeds they would have acquired,, in the interval 8t, had
no two of them collided or had there been no others to collide with them. But this
is not stated, much less justified, and I cannot regard the argument (in the form
in which it is given) as other than an exceedingly dangerous one; almost certain to
mislead a student.

What seems to underlie the whole, though it is not enunciated, is a postulate
of some such form as this : —

When a system of colliding particles ha^ reached its final state, we may assume
that (on the average) for every particle which enters, and undergoes collision in, a
thin layer, another goes out from the other side of the layer precisely as the first
would have done had it escaped collision.

32. If we make this assumption, which will probably be allowed, it is not
difficult to obtain the results sought, without having recourse to a questionable
process of variation. For this purpose we must calculate the changes which take
place in the momentum, and in the number of particles, in a layer; or, rather, we
must inquire into the nature of the processes which, by balancing one another's
effects, leave these quantities unchanged.

Recur to § 29, and suppose the particles to be subject to a potential, U, which
depends on x only. Then the whole momentum passing per unit of time perpen-
dicularly across unit sur&ce of any plane parallel to yz is

3^^Jo^=2A'

where n (the number of particles per cubic unit), and h (which involves the mean-
square speed), are functions of x.

At a parallel plane, distant a from the first in the direction of x positive, the
corresponding value is

But the difference must be sufficient to neutralise, in the layer between these planes,
the momentum which is due to the external potential, i,e.,

ax
dx " n dx h dx ^

LXXVII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 151

Again, the number of particles which, in unit of time, leave the plane unit
towards the side x positive is

1 f" f* 1 r*

^n j m; I COS /8 sin fidfi = t ^ I vv.
Hence those which leave the corresponding area at distance a are, in number,

But, by our postulate of last section, they can also be numbered as

where 5'^ = 2a--^.

This expression is obtained by noting that none of those leaving the first plane
can pass the second plane unless they have

t;»cos«/8>2a-^ .

All of the integrals contained in these expressions are exact, and can therefore give
no trouble. The two reckonings of the number of particles, when compared, give

-2A'^^ = l^-i^ (2)

From (1) and (2) together we find, first

^ = 0
dx ^'

which is the condition of uniform temperature; and again

which is the usual relation between density and potential.

[In obtaining (2) above it was assumed that, with sufficient accuracy,

e-*^* = l-A?«.

To justify this : — note that in oxygen, at ordinary temperatures and under gravity,

^ = 1550* in foot-second units,

''^-32
80 that, even if a = 1 inch, we have approximately

It is easy to see that exactly similar reasoning may be applied when 17 is a function
of a?, y, 2r; so that we have, generally.

152

ON THE FOUNDATIONS OF THB KINETIC THEORY OF GASES. [lXXVU.

where A is an absolute constant. And it is obvious that similar results may be
obtained for each separate set of spheres in a mixture, with the additional proviso
from Maxwell's Theorem (§§ 20, 21) that Pjh has the same value in each of the sets.

APPENDIX.

The following little table has been calculated for the parposes of g 11, 28, by Mr J. B. Clark,
Keil-Amott Scholar in the University of Edinbuigh, who used six-place logarithms: —

Here -r, = a;V*' and X^rra^c"*", whUe Tj = a;c-*' + (2a^+ 1) fc-^cfe.

jr,/jr,

JT,

XJX,

•1

•000099

•200665

•00049 +

•000990

•00493 +

•2

•001537

•405312

•00379 +

•007686

■01896 +

•3

•007420

•617838

•01198 +

•024676

•03994-

•4

•021814

•841997

•02691 -

•054637

•06477 +

•5

•048675

1-081321

•04501 +

•097360

•09003-

•6

•090418

1-339068

•06762 +

•160698

•11264-

•7

•147091

1-618194

•09089

•210130

•12986 +

•8

•215978

1921318

•11241-

•269973

•14051 +

•9

•291870

2260723

•12968 +

•324301

•14409-

1-0

•367879

2608351

•14104-

•367879

•14104-

•436590

2995826

•14572 +

•396900

•13249 -

1-2

•491380

3-414479

•14388 +

•409409

•11990 +

1-3

•627004

3865384

•13633 +

•405388

•10488-

1-4

•641119

4-349386

•12441 +

•386514

•08887-

1-6

•533581

4867132

•10962 +

•355721

•07309-

1-6

•506619

6-419114

•09348 -

•316637

•05843-

1-7

•464174

6-006696

•07729 -

-273044

-04546 +

1-8

•409127

6-627149

-06203 +

•228404

•03447-

1-9

•352643

7-283668

•04840-

•185549

•02647 +

•293040

7976369

•03674 +

•146520

•01837 +

•236390

8-702340

•02715 +

•112567

•01294-

•186224

9-464667

•01966-

•084193

•00889

2-3

•141065

10-262360

•01373 +

-061333

•00598-

2-4

•104541

11^095474

•00941 +

-043559

•00393-

2-5

•075390

11-964016

•00630 +

•030166

•00252 +

2-6

•052962

12-867980

-00411-

•020370

•00158 +

2-7

•036242

13-807388

•00262 +

•013423

•00097 +

2-8

•024155

14-782249

•00162 +

•008627

•00068 +

2-9

•016700

15-792549

•00099 +

•006414

•00034 +

•009963

16-838302

•00057 +

•003321

•00019 +

The sum of the numbers in the fourth column is 1*69268, so that the approximate value
of the integral in § 11, which is 04 of this, is 0-67707.

The sum of the numbers in the sixth column is 1*62601, so that the value of the int^pral
in [the addition to] § 11 is about 0*6504.

Lxxviii.] 153

LXXVIII.

ON TBtE FOUNDATIONS OF THE KINETIC THEORY OF

GASES. II.

[Transdctions of the Royal Society of Edinburgh, 1887, Vol. xxxiii.]

PAGE

Introductory and Preliminary . . 163

Part X. On the definite Integrals, I —

Jo «

'^ r7&'5§33, 34. . 158

INDEX TO CONTENTS.

PAOB

Part XI. Pressure in a Mixture of Two

Sets of Spheres, § 35 . .159
„ XII. Viscosity, §§ 36, 37 . . . 161
„ XIII. Thermal Conductivity, §§ 38—44 162
„ XIV. Diffusion, §§ 45—56 . . .167

Appendix. Table of Quadratures . . 178

In the present communication I have applied the results of my first paper to
the question of the transference of momentum, of energy, and of matter, in a gas
or gaseous mixture; still, however, on the hjrpothesis of hard spherical particles,
exerting no mutual forces except those of impact. The conclusions of §§ 23, 24 of
that paper form the indispensable preliminary to the majority of the following in-
vestigations. For, except in extreme cases, in which the causes tending to disturb the
" special'' state are at least nearly as rapid and persistent in their action as is the
process of recovery, we are entitled to assume, from the result of § 24, that in every
part of a gas or gaseous mixture a local special state is maintained. And it is to
be observed that this may be accompanied by a common translatory motion of the
particles (or of each separate class of particles) in that region; a motion which, at
each instant, may vary continuously in rate and direction from region to region;
and which, in any one region, may vary continuously with time. This is a sort of

T. II. 20

134 ox THE FOrXDATIOXS OF THE KIKEnC THBORY OF GASES. [lXXVUI.

g€iiendisatkm of the special state, and all that follows is based od the aaramplaoii
that such is the most general kind ot modon which the parts €i the srstem can
have, at least in anj of the qnesdons here treated. Of eooise this trBiwJatinnal
speed is not the same for all particles in anj smaU part of the sjrstem. It is ma^j
an arerage, which is maintained in the same rooghlT approximate manner as is the
''special state." and can like it be assumed to hoM with sofficieiit aocmcr to be
made the basis of calculation. The mere htt that a ''steadr" state, saj of difinsioii,
can be reatixed experim^itallr is a soffident warrant far this asBampdoQ; and there
jeems to be no reason (or supposing that the irregolarities ot distiibutkm of the
czaosIatorT TeI«xitT among the partides of a groop should be mare serioas fcr the
kigher than for the lower speeds, or rice ictkL For each particle is sometimeB a
•|TCDdL sometimes a slow, moving one: — and exdianges these slates manj thousand
TEoiiss per seocMftd. AU that is reallj required by considerations of this kind is aUoved
J:-r hj kkt waj of looking at the m^an free paths for different speeds.

I may take this opportunity of answering an objection whidi has been raised
jel crjrrespjodenoe by Professor XewoMnb, and by Messrs Watson and Burbory, to a
pasBftge in § 3 of the first P^rt €i this paper*. The words objected to are put in
Iia&cs: —

"Bat Ae arjMtmetd abort <Aok«, furth^, that this doisity must be expressible in
the form

vhasexer nectangnlar axes be chosen, passing through the origiiL''

The staxem^it itself is not objected to, but it is alleged that it does not follow
from the premises assumed.

This part of my paper was introduced when I reTised it for press, some months
afta- it was read; the date of reTision, not of reading, being put at the head. It
was written mainly for the purpose of scringing together what had been a set of
detached fragments, and was in ccMisequence not so fiiUy detailed, as they wei«L
I made some general statements as to the complete rerincation of these preUminaiy
pn>positi^>ns which was to be obtained frv>m the more com|Jex investigations to which
they led: thus showing that I attached comparatiyely tittle weight to sudi intro-
ductory matters If necessaiy. a detailed proof can be given on the lines of § 21
of the paper. The *^ argument^ in question, however, may be given as below. It is
really involved iu the italicised w<»ds of the following passage of § 1: — ^*in place of
the iM^less question of the behaviour of innumerable absolutely isolated individuala,
the comparatively simple statistical querstion of tMe arerape hdiawiomr of Ae mfiont
mmpf of a cotHmnmity/

Suppose two ideal planes, parallel to x = 0. to move with conmNO speed, x»
ihroogfa the gasL The portion of gas between them wiU consist of two quite distinct

* Is the PhxL 3l£^^ fvY Ainl 1SS7. the sum objectkn » imwd Vr Ptot BohmuA: vhD i
XX to the E&^rosh timzKlKHB of his pinpcr FraKBt^r to he ic^nrcd lo. B«t be con
v^tfoeton. &Dd *ecuw* me cf reaaooiBg in a dmie.

LXXVIIl] on the FOU^^DATIONS of the kinetic theory O^GiSl

155

classes of particles : — the greatly more numerous class being mere fleeting occupants.
the niinority being (relatively) as it were permanent hxlgers. These are those whose
speed perpendicular to the planes is very nearly that of the planes themselves. The
indtviduids of each class are perpetually changing, those of the majority with extra*
ordinary rapidity compared with those of the minority; but each class, as such, forms
a definite '' grtrnp of the community/' The method of averages obviously applies to
each of these classes separately ; and it shows that the minority will behave, so far
as y and s motions are concerned, as if they alone had been enclosed between two
material planes, and as if their lines of centres at impact were always parallel to
these. The instant that this ceases to be true of any one of them, that one ceases
to belong to the group ; — and another takes its place. Their behaviour under these
circumstances (though not their number) must obviously be independent of the speed
* of the planes. Hence the law of distribution of components in the velocity space-
diagram must be of the form

/(*).J'(y. «);

and symmetry at once gives the result above.

[{Inserted March Btk, 1887*) Another objection, but of a diametrically opposite
characterj raised by Mr Burbury* and supported by Professor Boltzmannf, is to the
effect that in my first paper I have unduly multiplied the number of preliminary
assumptions necessary for the proof of MaxweU's Theorem concerning the distribution

of energy in a multure of two gase**. In fonn, perhaps, I may inadvertently have done
80, but certainly not in ^ibstance^

The assumptions which (in addition to that made at the commencement of the
paper (| 6) for provision against simultaneous impacts of three or more particles,
which was introduced expressly for the purpose of making the results applicable to
real gases, not merely to imaginary hard spheres,) I found it necessary to make, are
(I 18) as follows; briefly stated,

(A) That the particles of the two systems are thoroughly mixed,

(B) That the particles of each kind, separately, acquire and maintain the "special
state."

• The Foandfttioaa of the Klnttio Theory of Gas6a, Phit. Mag, ISSe, I, p, 461.

f Ub«r die scum theoretisoben BoweiAu d<is Avogadro^iicheii Gesetzea erforderlidien Vorfttissetstitig^ii, SiUh.
der kititi. A had., xcivv. 1886, Oct. 7* In tbia article ProC Boltzmuim ^tate^ that I have nowhere expreesly
pointed out that my te^ulU are applicable odIj to the oa«e of hard spheres. I mi^jhl plead that the article he
jBi6i9 to in a brief AbMlraet onlj of my paper; bnt it tsontaiDfl the foUowiDg atateiiient«, wbkh are iiu«lj
cxpUoit enough an to the object I had i& view : —

" Thifl is specially the case with his [Maiwell'sJ investigation of the law of ultlmale partition of energy
in a mixture of smooth spherical particles of two di£feren| kinds/'

**It has sin£» been extended by Dolt^mann and others to eases in whioh the partiolea are no longer
supposed to be hard nmooth spheres/*

"Heuce it is desirable that Maxwell's proof of his fundamenUl Theorem should be critically exammed/*
Then I proceed to examine it, not Professor Boltzmann^a extension of it. In mj paper itself this limitation is
most expressly inaisted ou.

20—2

156 ON THE FOUNDATIONS OF THE KINETIC THEOBT OF GASES. [lXXYHI.

of the same kind or of different systems; and that the number of particles of one
kind is not overwhelmingly greater than that of the other.

Of these, (A) and (B), thoagh enunciated separately, are regarded as consequences
of (C), which is thus my sole assumption for the proof of Clerk-Maxwell's Theorem.
Professor Boltzmann states that the only necessary assumptions are: — that the particles
of each kind be uniformly distributed in space, that they behave on the average
alike in respect of all directions, and that (for any one particle?) the duration of
an impact is short compared with the interval between two impacts. His words are
as follows: — ''Die einzigen Yoraussetzungen sind, dass sowohl die Molekule erster als
auch die zweiter Gattung gleichformig im ganzen Raume vertheilt sind, sich durch-
schnittlich nach alien Richtungen gleich verhalten ond dass die Dauer eines Zusam-
menstosses kurz ist gegen die Zeit, welche zwischen zwei Zusammenstossen vergeht."

He farther states that neither s}\stem need have internal impacts: and that
Mr Burbnr}' is correct in maintaining that a system of particles, which are so small
that they practically do not collide with one another, will ultimately be thrown into the
"special" state by the presence of a single particle with which they can collide.

Assuming the usual data as to the number of particles in a cubic inch of air,
and the number of collisions per particle per second, it is easy to show (by the help
of Laplace's remarkable expression^ for the value of A"0*/n"* when m and n are very
large numbers) that somewhere about 40,000 years must elapse before it would be
so much as even betting that Mr Burbury's single particle (taken to have twice the
diameter of a particle of air) had encountered, once at least, each of the 3 . 10* very
minute particles in a single cubic inch. He has not stated what is the average
number of collisions necessary for each of the minute particles, before it can be knocked
into its destined phase of the special state; but it must be at least considerable.
Hence, even were the proposition true, aeons would be required to bring about the
result. As a result, it would be very interesting: but it would certainly be of no
importance to the kinetic theory of gases in its practical applications

I think it will be allowed that Professor Boltzmann's assumptions, which (it is
easy to see) practically beg the whole question, are themselves inadmissible except
as consequences of the mutual impacts of the particles in each of the two systems
separately. Professor Boltzmann himself, indirectly and without any justification (such
as I have at least attempted to give), assumes almost all that he objects to as redundant
in my assumptions, with a good deal more besides. But he says nothing as to the
relative numbers of the two kinds of particles. Thus I need not, as yet, take up
the question of the validity of Professor Boltzmann's method of investigation (though,
as hinted in § 31 of my first paper, I intend eventually to do so); and this for the
simple reason that, in the present case, I cannot admit his premises.

* TMorie Analytique det ProbabiliUs, Livre ii. chap. n. 4. [In QBing this formalA, we miut make sure thai
the ratio mfn is safficiently large to justify the i^proximation on which it is founded. It is found to be
80 in the present case. At my request Professor Cayley has kindly investigated the correct formula for the
ease in which m and n are of the «am€ order of large quantities. His paper will be found in Proe. R.S.E^
April 4, 1887.]

LXxrni.] OJf THE F0UNDATIOK9 OF THE KINETIC THEORY OF GASES.

157

Mr Burbory assumes the non-colliding particles to be in the "special state" and
proceeds to prove that the single additional particle will not disturb it. But» sup-
posing for a moment this to be tme. it does not prove that the solitary particle
would (even after the lapse of ages) reduce any non -colliding system, with positions
at any instant, iipeeds, and lines of motion, distributed absolutely at random (for here
there cannot be so much as plausible gi^ouuds for the introduction of Professor
Boltzmann's assumptions) to the " special state*" If it could do so, the perfect re-
vembility of the motions, practically limited in this ease to the reversal of the motion
of the single particle aloiw, shows that the single particle would (for untold ages)
continue to throw a system of non-colliding particles further and furthor out of the
"special" state; thus e!cpressly contmdicting the previous proposition* In this oonse-
quence of reversal we see the reason for postulating a very great number of particles
of each kind. If Mr Burbury's sole particle possessed the extraoi-dinaiy powers
attributed to it, it would (except under circumstances of the most exact adjustment)
not only be capable of producing, but would produce, ^absolute confusion among non-
colliding particles already in the special state. 0>usidering what is said above, I do
not yet see any reason to doubt that the assumption of collisions among the particles
of each kind, separately, is quite as essential to a valid proof of Maxwell's Theorem
as is that of collisions between any two particles, one from each J^ystem. I have
not yet seen any attempt to prme that two sets of particles, which have no internal
collisions, will by theii' mutual collisions tend to the state assumed by Professor
Boltzciann. Nor can I see any ground for dispensing with my iarther assumption
that the number of particles of one kind must not be overwhelmingly greater than
that of the other. A small minority of one kind must (on any admissible assumption)
have an average energy which will fluctuate, sometimes quickly sometimes very slowly,
within very wide and constantly varying limits.

De Morgan* made an extremely important remark, which is thoroughly applicable
to many investigations connected with the present question. It is to the effect
that "no prlnmry considerations connected with the subject of ProhiihiUty can be.
or ought to be, received if they depend upon the results of a complicated mathe-
matical analysis/* To this may be added the obvious remaik, that the purely
mathematical part of an investigation, however elegant and powerful it may be, is
of no value whatever in physics unless it be based upon JLidniissible assumptions.
In many uf the investigations, connected with the present subject, alike by British
and by foreign authors, the above remark of De Morgan has certainly met with scant
attention.]

In ray first paper I spoke of the errors in the treatment of this subject which
have been introduce<l by the taking of means before the expressions were ripe for
such a process* In the present paper I have endeavoured throughout to keep this
danger in %'iew ; and I ho|>e that the results now to be given will be found, I'ven
where they are most imperfect, at least more approximately accurate than those which
have been obtained with the neglect of such precautions.

* Encyc, MtlropQUUma. Art. Th&.>ry of I^obftbiUties.

158 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVIII.

The nature of Clerk-Maxwell's earlier investigations on the Kinetic theor}% in
which this precaution is often neglected, still gives them a peculiar value; as it is
at once obvious, from the forms of some of his results, that he must have thought
them out before endeavouring to obtain them, or even to express them, by analysis.
One most notable example of this is to be seen in his Lemma (Phil, Mag. 1860, II.
p. 23) to the effect that

J ^r vi + 2 dx^ '^

where TJ and r are functions of x, not vanishing with x^ and varying but slightly
between the limits — r and r of x\ — and where the signs in the integrand depend
upon the character of m as an even or odd integer. This forms the starting point
of his investigations in Diffiision and Conductivity. It is clear frt)m the context why
this curious proposition was introduced, but its accuracy, and even its exact meaning,
seem doubtful.

In all the more important questions now to be treated, the mean fr^e path of
a particle plays a prominent part, and integrals involving the quantities e^ or e + ^i
(as defined in §§ 9, 10, 28) occur throughout. We commence, therefore, with such a
brief discussion of them as will serve to remove this merely numerical complication from
the properly physical part of the reasoning.

X. On the Definite Integrals,

Jo e Jo ei-\-ze2

33. In the following investigations I employ, throughout, the definition of the
mean free path for each speed as given in § 11. Thus all my results necessarily differ,
at least slightly, from those obtained by any other investigator.

By § 11 we see at once that

^ '''^*' Jo //"*'^'(^»' + ^iV3t;»)^*'i + /*^"*'^'(W3 + ViVt^)rft^i
/•« 4af^^€'^dx

««VA'' I

^^. .(2a- + l)|%-

a:€-*"+{2a;» + l) e^^^d^

The finding of Cr is of course a matter of quadratures, as in the Appendix to
the First Part of this paper, where the values calculated are, in this notation, (7_i
and Co; and Mr Clark has again kindly assisted me by computing the values of

IXXVIII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GA8BS.

159

Cip C„ Cj, which are those required when we are dealing with Viscoaity and with
Heat- Conduction in a single gas. The value of (7^ has also been found, with a view
to the study of the general expression for C^, These will be given in an App^ndu- to
the present paper.

-34, When we ct>me to deal with Diffusion, except in the special case of equality
of density in the gases, this numerical part of the work becomes extremely serious,
even when the assumption of a "steady" state is permissible, As will be seen in § 28
of my first paper, we should have in general to deal with tables of double entry, for
the expressions to be tabulated are of the form —

Jo ei + Z€i nw^^h'' I

4^+*€'**dar

X€

= '®'^n^?Vr''^^PP^^'

For the second gas the corresponding quantity will be written as JB^

Hei-e

and

so that they are numerical quantities, of which the first depends on the relatife
masses of particles of the two gases^ while the second involves, in addition, not
only their relative size but also their relative number. It is this last condition which
introduces the real diflScuIty of the question, for we have to express the value of the

integral as a ftmction of z before we can proceed with the further details of the
solution, and then the equation for Diffusion ceases to resemble that of Fourier for Heat-
Conduction.

The difficulty, however, disappears entirely when we confine ourselves to the study
of the "steady state'* (and is likewise much diminished in the study of a variable
state) in the special case when the mass of a particle is the same in each of the
two gaseous systems, whether the diameters be equal or no. For, in that case, we
have k^^h and aji = a;, so that the factor 1/(1 + ^) can be taken outside the integral
sign- Thus, instead of i®r, we have only to calculate Cr of the previous section.

XI, Pressure in a Mi^ttre of Two Sets of Spheres^

35. Suppose there be ii, spheres of diameter % and mass Pi, and n^ with s^, P-,
per cubic unit. Let s^(Si + s^)l2.

Then the average number of collisions of each P| with PjS is, per second,

vt "••■••

160 ox THE i>OrXDATIOX8 OF THE KINETIC THEORY OP GASB. [lXXTUL

The impulse is. oo the avenge (as in § 90l,

-'"Va-

SimikriT. eauch P^ encoonters, in each second (| 23^ the arenge Dumber
of P^. and the aT«n2« impact is

P.P, A-<*. -»-*»>

Th.-^ tbe aT«ta0r 5wn vf impacts on a P, is, per seoond.

Is tbe Vzial expre^oD ^^^i^rw § 9d , r most be taken as <■ for tiie fiist of
ibs» pcnxns. and as « for tlie second. Hence ve hare

1^ i> ^ P ^. 3 p.p. <*,-!-*»> ^ i*t . /

4-'^'=-2;r">-'--.A;p.>M,"-'^-]^"-^^

where « = ii-, -s- »j.

In ihe spcinal ca&e #.=«, = «. this becomes^ as in | 30^

To oiKain an »>» as 70 hov ^be "uliiixtase Toiumer spoken oif in uiai section.
2> aSected br ihe d:Ser«xe of siae of ihe pardcIeiSw suppose Sr^ai. The vaLiKs <rf
ibe aK>Te qua&iiii^e arv

— -p *-.*-r f**-^*.* and --rtiw*:
<o ifcas iis we mighi hare expected* oispiii^ of s£ae. wi^h The same mean of

T^::s. if *,:*:*,:: 1 : f : :l.

•i«r nii:' cf :he expoK^saocs aK^re is 11 : X The Z4a»?5 ril:2e it ^sla hax^e iwhen
* *i is iridriie or is evanescent « is 5 : i

LXXVIII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 161

XII. Viscosity.

36. Suppose the motion of the gas, ds a whole, to be of the nature of a simple
shear; such that, relatively to the particles in the plane of yz, those in the plane
J- have a common speed

[jarallel to y. V, even when x is (say) a few inches, is supposed small compared
with the speed of mean square. We have to determine the amount of momentum
parallel to y which passes, per second, across unit area of the plane of yz.

In the stratum between x and x + Bx there are, per second per unit surface, nvevhx

collisions discharging particles with speed v to v + dv (distributed uniformly in all

directions) combined, of course, with the speed of translation of the stratum. The

number of these particles which cross the plane of yz at angles 0 to 0 + d0 with

the axis of ^ is

€'^^*Bin0d0/2.

[Strictly speaking, the exponent should have had an additional term of the order
eBaf/v; but this is insensible compared with that retained until x ia s, very large
multiple of the mean free path. See the remarks in § 39 below.] Each takes with it
(besides its noimal contribution, which need not be considered) the abnormal momentum

PBx,
relatively to yz and parallel to y.

Hence the whole momentum so transferred from x positive is

PBn

f(t Jo

-^l wl sin0d0 \ e'^^^exdx,
^ Jo Jo Jo

PBn r wr- ,^ . ^ ,. PBn r w

2 Jo ejQ 6 Jo e

Doubling this, to get the full differential eflfect across the plane of yz, it becomes (§ 33)

PBnC, PBn OSSS
37ms^y/h 37rns*^/h

The multiplier of B, i.e. of dV/dx, is the coeflScient of Viscosity. Its numerical
value, in terms of density and mean path, is

-''> 0-412.

Clerk-Maxwell, in 1800, gave the value

^0-376.

which (because / = 707X/677, as in §11) diflFers from this in the ratio 20:21. In
T. II. 21

162 OS TBR vomsDAnoss of the Kixsnc thbobt of gasBu [lxxtih.

ihK 6He the sbart cuts empfered hare obrioodT entailed little mmKnciJ error.
fX k fvynafiT for mnj one gm&. the Visooatr «as MarveJl pomted ovti is
of the decaXT.

37. Both exjvBGEiaBS are propartiaBal to the mfaan-Toot ol the afaBohite

We maj see at once that, od the hjpoiheBS w hare adofited^ such most
be the case. For, if we saoppoeie the speed of ererr qihere to be soddenhr iiMiiiiwd
■I ibUL the openxkns viD go oo piotiaclT as befcre, onlr m times frster. But the
aibaohzte leaipentiire wiQ be iimca^d as ■i' : 1. Similar anticipatiQiis mar be
i& the eases of IKffbsMii and of Thermal GoDdiictiTitT.

Vaxveil vas led br his expenmenial measures cf Viscosin'^ vhich aee^^ to
Aam* that it increases nearir in profwrdoB to the fiist pover of the ahBohite
leBoperamre. to discard die nodoo of hard sphere, and to introduce the hypothesis of
panides repelling one another viih ferce inxenelj as the fifth pover of the distmcg,
I hare abeadr ssaSed that there are Terr gnre obyectioiis to the introdnctian of
rfpmhmm into this subject, except of comse in the fenn of elastie leatiimiop. That
the putides cf a gas hare Ait fpopenj is plain trcok their rapahHrtr of lifaratin^
9D that thej must lose enogv of translatioii hr impact: and I inienL in the nea:t
of this inxestigatkn. so &r lo modi^ the fondamental asBsmptaon hitherto
IS to deduce the effgets cooespoiiding to a coe&aent of lestiiuiion 1b
and afao to take aecovmt of moSecftiar ottraefMi. specianT limited in is
not moch greaier than ihe diaokescr of a sphene.

XHL ntrmal C^mimdiniy.

3S. We mitst conieni ccu^xes viih the companx^Tehr smpie case of the steadjr
flc'W of heax in v<he dii>Ecdv4i: sit paraDe-I lo ihe axis of x. Tliis viH be aasiuncd
to lie TerticaL the i^mpiEsaSQie in ibe gas incresfesiLg vpvaidsL <o as lo prerent
omv^ectien cvrrcni& No attcntScci neied. v-sberaisic. be paid to the e&css of giaiiii.

H^ice the ibik-ving ocodiiioos m:2<3 he saxssSed: —

la* Each bocin?ciiaI lajrrr of ibe gas is in the specsaZ state. eocbpo«nded vith
a deoniie muos^azkn wrtioM jr.

%i* The pressure is cooscani Thnxigboci ihe ga&

\c* Tifecfe isc on ihe vbove, 3>i> passa^ of g»s ai:z>Kk> anv hocinrntal piane.

««» Eij-sal as>wXin:^ cf c£!er«:nr an^^ oq ihe Th'.^. tra&sc^rned (in the same diiectaon)
as\>^ ;::ni4 aiva o£ all soc& piane&

^. Le^ « he ihe n^isat^K- oc pan^r-I'ets fKr is:nit ^-nme in tM larer betveen
X and s^ie: j i^-e »co.>c oc ^besn vb.itse sceetl xv-iji^T>r> tc^ the aeij^baars as a

LXXVIII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 163

The number of particles which pass, per unit area per second, from x positive
through the plane x^O, is the sum of those escaping, after collision, from all the
layers for positive x, and not arrested on their way: — viz.,

Iff I'

^ .' 0 J 0 J 0

ri ra -aoc»f«te . ^,^vcostf-a

Si II "Ki/^e J"" Sin Odd ^-(

2./o Jo Jo vcos^

Here a, though in any ordinary case it need not be more than a very small
fraction of an inch, is a quantity large compared with the mean free path of a
particle. Its value will be more exactly indicated when the reason for its introduction
is pointed out.

The last factor of the integrand depends on the fact that the particles ai'e
emitted from moving layers: — involving the so-called Doppler, properly the Romer,
principle.

We neglect, however, as insensible the diflference between the absorption due to
slowly moving layers and that due to the same when stationary.

Because a, the range of x^ is small we may write with sufficient approximation

n = Wo + no'a?, &c., &c.

Introducing this notation, the expression above becomes

2J0 JoJa ^{ ^r?o Po ej ) VQO&0

Now, to the degree of approximation adopted,

I oia? = CoO? -f CoV/2.

The second term of this must always be very small in comparison with the first,
even for an exceptionally long free path. But, if we were to make

a: = 2cp/eo',

the second term would become equal to the first. Hence a, the upper limit of the
X integration, must be made much smaller than this quantity. Thus we may write

^-•ec»/;edx^ ^.^sec*(l « ^o V SCC tf/2 + . . . )•

We said, above, that e^^aj -

is a large number, say of the order 10*. It appears then at once that terms in

€-^^ = €-'~ = 10-*» nearly

may be neglected. Such terms occur at the upper limit in the integration with
regard to x above, and what we have said shows, first why a had to be introduced,
second why it disappears from the result.

21—2

164 ON THB FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVUI.

Writing now only those factors of the above expression which are concerned in
the integration vrith respect to x, we have

l{cos^ + l(=^+?^)co^4.

The terms in e^' are found to have cancelled one another, a result which greatly
simplifies the investigation.

Had we complicated matters by introducing Oo + flEo^a; in place of a, the term in
Ot^ (which, if it exist at all, is at least very small) would have been divided on
integration twice by Bq, a quantity whose value is, on the average, of the order 5.10*
(to an inch as unit of length).

The expression now becomes

We have omitted the zero suffixes, as no longer required; and, as the plane x^O is
arbitrary, the expression is quite general.

Omitting the product of the two small terms, and integrating with respect to 0,
we have

\l?'{"'-'*{i*7M'

The corresponding expression for the number of particles which pass through the
plaae from the negative side is, of course, to be obtained by simply changing the
mgDM of the two last terms. Thus, by (c) of § 38, we have

I'M- it ^^H'"'
'-[<i^^)'i^ (')•

40. The pressure at the plane x^O, taken as the whole momentum (parallel to
X) which crosses it per unit area per second, is to be found by introducing into our
first integrand the additional factor

P (v cos tf — a),
where P is the mass of a particle. There results

P
2

/;„.{«./3-..+(=:.^'),./4.}.

LXXVIII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES.

165

We must take the sum of this, and of the same with the signs of the two last
terms changed ; so that the pressure (which is constant throughout, by (6) of § 38) is

p-J?^^^ <2)-

Thus n/h is constant throughout the gas.

[If a very small, thin, disc were placed in the gas, with its plane parallel to
yz, and the steady state not thereby altered, the difference of pressures on its sides
would be

or (see § 42 below) p ^-^^ J' {^S- (| (7. - (7,) - 1 (7. 4- (?.} .

For ordinary pressures, and a temperature gradient 10° C. per inch, this is of the
order 10""'' atmosphere only.]

41. For the energy which passes per second per unit of area across a? = 0, we
must introduce into the first integrand of § 39 the additional factor

2 (t;«-2t;ocosd);

and the result of operations similar to those for the number of particles is

^=-fr"'^te'+r)A-M <«)•

This expresses the excess of the energy passing from the negative to the positive
side of x = 0, over that passing from positive to negative ; and, by (d) of § 38, must
be constant.

42. To put (1) and (3) in a more convenient and more easily iutelligible form,
note that because

1/ = 4 A /~ e'^^t^dv,

we have
But, by (2),
Thus, by (1),

i/^3 a;

V 2 h"

n h '

AV.

Similarly (3) becomes

166 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVIII.

43. The only variable figwjtor {h'jh^) in these expressions for a, and for E, \a the
same in both. Hence, as E does not vary with x, Kjh} is constant, and so also is
o. Thas since, if t be absolute temperature, we have

hr = constant ;

we find at once, i^ = J. + Bx,

Thus the distribution of temperature, and therefore that of density, is determined
when the terminal conditions are given. The formula just given agrees with the
result first obtained by Clausius in an extremely elaborate investigation^, in which
he showed that Maxwell's earliest theory of Heat-Conduction by gases is defective.

The general nature of the motion of the gas is now seen to be analogous to
that of liquid mud when a scavenger tries to sweep it into a heap. The broom
produces a translatory motion of the mud, which is counteracted by gravitation-sliding
due to the surfjEU^ gradient: — just as the displacement (by translation) of the whole
gas, from hot to cold, is counteracted by the greater number of particles discharged
(after collisions) from a colder and denser layer, than from an adjoining warmer and
less dense layer.

44. The results of calculation of values of Cr given in the Appendix enable us
to put the expressions (!') and (3^ into the more convenient forms

- = |.'70O6 (1"),

^ = ;^P'^0-45 (3").

where it is to be remarked that the product p\ is independent of the temperature
of the gas.

The Conductivity, fc, is defined by the equation

"dx ^'
and thus its value is A:= - /— r 47-;0'45,

V V V V

where Tq, Ap are simultaneous values of t and A.

At O"* C. {%,e. T = 274) this is, for air, nearly 3 . 10~' in thermal units on the
pound-foot-minute-Centigrade system: — i.e, about 1/28,000 of the conductivity of iron,
oi 1/3600 of that of lead+. Of course, with our assumption of hard spherical particles,
we have not reckoned the part of the conducted energy which, in real gases, is due
to rotation or to vibration of individual particles.

• Pogg. Ann., cxv. 1862; Phil. Mag., 1862, I.
t Tram. R.S.E., 1878, p. 717.

lxxviil] on the foundations of the kinetic theory of gabes.

167

XIV, Diffusion.

45. The complete treatoient of this subject presents difficulties of a very formidable
lcJiid> several of which will be apparent even in the comparatively simple case which
18 treated below. We take the case of a uniforrn vertical tube, of unit area in
section, connecting two vessels originally filled with different gases, or (better) mixtures
of the same two gases in different pix)portions. both, however, maintained at the same
temperature ; and we confine ourselves to the investigation of the motion when it
can be treated as approximately steady* We neglect the effect of gravity (the denser
gas or mixture being the lower), and we suppose the speeds of the group-motions to
be very small in comparison with the speed of mean square id either gas. [In some
of the investigations which follow, there are (small) parts of the diffusion-tube in
which one of the gases is in a hopeless minority as regards the othen Though one
of the initial postulates (d of § 1) is violated, I have not thought it necessary to
suppress the calculations which are liable to this objection ; for it is obvious that
the conditions, under which alone it could arise, are unattainable in practice.]

Clerk-Maxweirs Theorem (§ 15), taken in connection with our preliminary assump-
tion, shows that at every part of the tube the number of spheres per cubic unit,
and their average energy, are the same, Hence^ if w,, n^, be the numbers of the
two kinds of spheres, per cubic uniti at a section s of the tube

ji, + fh— ti = constant *.•.,**. ,.* (1).

Also, if Pi J Pat be the masses of the spheres in the two systems respectively, A|
and ki the measures (§ 3) of their mean square speeds, we have

FJh, = P,/K = {niP.lh, -h 7i,P,/h,)ln = 2p/n .(2),

where p is the constant pressure*

Strictly speaking, the fact that there is a translational speed of each layer of
particles must affect this expression, but only by terms of the first order of small
quantitiea

46. The number of particles of the P, kind which pass, on the whole, towards
positive X through the section of the tube is (as in § 39)

where a, is the (common) translational speed of the P/s, and I/Bj the mean free
path of a Pi whose speed is v. We obtain this by remarking that, in the present
problem, ^ is regarded as constant, so that there ia no term in vi\

Hencei if <?i be the mass of the first gas on the negative side of the section,
divided by the area of the section, we have

dff,

--PiMi-n/,aCJ3),

.(3).

168 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVUL

If Gs be the oorresponding mass of the second gas, we have (noting that, by (1),

^' = -P,(iiA+ih',C,/3) (4).

From the definitions of the quantities Gi, Gt, we have also

(5).

47. We have now to form the equations of motion for the layers of the two
gases contained in the section of the tube between x and x + &r. The increase of
momentum of the P, layer is due to the difference of pressures, behind and before,
caused by P/s; minus the resistance due to that portion of the impacts of some
of the P,'s against P,'s in the section itself, which depends upon the relative speeds
of the two systems, each as a whole. This is a small quantity of the order the
whole pressure on the surfaces of the particles multiplied by the ratio of the speed
of translation to that of mean square. The remaining portion (relatively very great)
of the impacts in the section is employed, as we have seen, in maintaining or
restoring the "special state" in each gas, as well as the Maxwell condition of
partition of energy between the two gases. If /2 be the resistance in question, the
equations of motion are

^(P«.S.,-1^(^)S.H.«S.J
where d represents total differentiation.

48. To calculate the value of R, note that, in consequence of the assumed
smallness of Oj, a,, relatively to the speeds of mean square of the particles, the
number of collisions of a P, with a P„ and the circumstances of each, may be
treated as practically the same as if a, and a, were each zero: — except in so fiu* that
there will be, in the expression for the relative speed in the direction of the line of
centres at impact, an additional term

(fli - a,) cos V^,

where -^ is the inclination of the line of centres to the axis of x. Thus to the
impulse, whose expression is of the form

-pTq(^-">'

as in § 19 of the First Part of the paper, there must be added the term we seek, viz.,

2P,P, , , .

LXXVIII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 169

This must be resolved again parallel to x, for which we must multiply by
cos'^. Also, as the line of centres may have with equal probability all directions,
we must multiply further by sin '^rf'^/2, and integrate from 0 to ir. The result
will be the average transmission, per collision, per P^ of translatory momentum of
the layer parallel to x. Taking account of the number of impacts of a Pi on a P,,
as in § 23, we obtain finally

where 8 is the semi-sum of the diameters of a Pi and a P*.

49. To put this in a more convenient form, note that (2), in the notation of
(5), gives us the relation

hi dx Aa dx ^'

whence Oilhy^-GJK^^px (8).

We have not added an arbitrary constant, for no origin has been specified for x.
Nor have we added an arbitrary function of t, because (as will be seen at once
from (3)) this could only be necessary in cases where the left-hand members of (6)
are quantities comparable with the other terms in these equations. They are, however,
of the order of

and cannot rise into importance except in the case of motions much more violent
than those we are considering.

From (8) we obtain ^/^^ + ?^«/a, = 0 (9),

which signifies that equal volumes of the two gases pass, in the same time, in opposite
directions through each section of the tube. This gives a general description of the
nature of the cases to which our investigations apply.

But, by (3) and (4), we have for the value of

PiP^WiWa (a, - Oj)

the expression - P,w, (~ * ~ ^ Piw/iCTi ! + P^n, (^' + \ P,n/,«D,) ;

or. by (9), (2), and (5)

-2^*Kf-3n^<^-«^^ + ^-«^))-

T. II. 22

170 ON THE FOUNDATIONS OF THE KINETIC THBOBT OF 6ASBS. [LXXVm.

Sabstituting this for the oorrespcmding fiu^tors of jR in the first of equmtions (6X
and n^ecting the left-hand side, we have finaUy

or, somewhat more elegantly,

A^=(8n^Vijg?"3«^'^^«' + -«'<^^^ <^«>-

50. Tlus equation resembles that of Fourier for the linear motion of heat; bot,
9A already stated in § 34, the quantities fS^ which oocor in it render it in general
intractable. The first part of what is osoally called the digMtUmrCoefieimit (the mul-
tiplier of ^PGifda^ above) is constant; but the second, as is obvious from (5) and (8X
is, except in the special case to which we proceed, a function of dGJdx\ ue, of the
percentage composition of the gaseous mixture.

51. In the special case of equality, both of mass and of diameter, between the
particles of the two systems, the diffusion-coefficient becomes

^=.s!^x/.

2 . ^.

rA 3nir»' %'A '

where X is the mean firee path in the system. Hence the diffusion-coefficient among
equal particles is directly as the mean free path, and as the square root of the
absolute temperature. Fourier's soluti«>ns of (10> are of course applicable in this
special case.

If we now suppose that our arrangement is a tube of length / and section S,
connecting two infinite vessels filled with the two gases respectively: and, farther,
assume that the diffusion has become steady, the equation (10"^ becomes

where the left-hand member is constant. Also, it is clear that, since dGi/dx must thus
be a linear function of jt, we have

dO,

f.p.,.p...-f),

LXXVIII.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 171

SO that the mass of either gas which passes, per second, across any section of the
tube is

SDpH,

where p is the common density of the two gases.

For comparison with the corresponding formula? in the other cases treated below,
we may now write our result as

Also, to justify our assumption as to the order of the translatory speed, we find
by (3)

1'38X

Hence, except where Z — a? is of the order of one-thousandth of an inch or less, this

is very small compared with A~'. And it may safely be taken as impossible that ?t,
can (experimentally) be kept at 0 at the section x—l.

If the vessels be of finite size, and if we suppose the contents of each to be
always thoroughly mixed, we can approximate to the law of mixture as follows. On
looking back at the last result, we see that for p we must now substitute the
difference of densities of the first gas at the ends of the connecting tube. Let gi, g^
be the quantities of the two gases which originally filled the vessels respectively;
and neglect, in comparison with them, the quantity of either gas which would fill
the tube. Then, obviously,

dO, ^ _ SDp (G, _ yy -Ga
dt I \g, ~ g^ 1'

whence (?. = -^i'^- )^i+/-^^4

This shows the steps by which the initial state (^i, 0) tends asymptotically to the
final state ( — -— gti, -~ — gA, in which the gases are completely mixed. When the
vessels are equal this takes the simple form

G,-i(

iSDfitx

1+ € "^' J .

52. In the case just treated there is no transmission of energy, so that the
fundamental hypotheses are fully admissible. In general, however, it is not so. The
result of § 41, properly modified to apply to the present question, shows that the
energ}' which, on the whole, passes positively across the section x is, per unit area
per second,

22—2

172 ON THE FOUHDATIOXS OF THE KHTETIC THECttT OF GASBL [ULXVm.

This, of coarse, in generml differs firom secUoo to seetum, and tbns a diHtiirlMuice of
temperature takes place. In soeh a case we can no iaoger ammiie that i| and i.
are absolute constants; and thos terms in C^ woold eome in; past as a term in (7.
appeared in the expression for energy condocted (§ 42X Thus, in order that oar
investigation may be admissible, the proeesa moat be coodiicted at constant tempera-
tore. This, in general, presopposes conditions external to the appaiaiosL

53. Thoagh it appears hopeless to attempt a g«ieral sofaition of equation (lOX
we can obtain from it (at least approximatdy) the conditions fix- a steady state ot
motion such as must, we presume, finally set in betweoi two infinite tcskIs filled
with diffident gases at the same temperature and preaBoreL For the left-hand member
is th^i an (unknown) constant, a second constant is introduced hj integrating once
with respect to x; and these, which determine the complete solution, are to be found
at once by the terminal conditicms

And, hj a sli^t but obvious modification €it the latter part of § 51 aboTe, we can
easify extend the process to the case in which the tcsbcIs are of finite sixe: —
always, however, on the assumption that their contents may be r^arded as promptly
assuming a state of unifi)rm mixture. The considomtion of § 52, however, shows that
the whole of the contents must be kepi at constant temp»atnre, in order that this
result may be strictly applicable.

54. Recurring to the ^tedal case of § 51, let us now suppoae that, while the
masses of the particles remain equal, their diameters are different in the two gasea
Thus, suppose #i><s. Then it is clear that

*,«-«», and j»-j,«,

are both positive. In this case, infinite terminal vessels being supposed, (10) gives for
the steady state

whose integral, between limits as in (11) above, is

iniVAl4«»V 2^ 3 V«»-V «i«-«»^(«i«-«»)"^#^(«»-*,«f^ W) •

Here A is the rate of passage of the first gas, in wuus per seccxid per unit area
of the section of the tube.

If now we put Sj^s+a, «, = # — a,

then, when a is small compared with s, the multiplier of (7,r/3 is

(l+a«/3s*)/«». neariy.

LXXVIII,] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES.

173

When a is nearly equal to a, i.e. one of the seta of particles exceedingly small
compared with the other, it is nearly

1-283/s*.

Thus it appeals that a difference in size, the mean of the diameters being unchanged,

favours diffusion.

« : «„

3r2:l,

Suppose, for instance,
and we have A = - -j^^ |^ ^^ + -g- (j^ + ^^Iog^ + ^Iog-jj,

II yM' 1085} = -^^ 1-24.

7rI«*VA

vm.

Compare this with the result for equal particles (| 51), reniemberiiig that X
BOW stands for the mean free path of a particle of either gas in a space filled with
the other: — m\A we see that (so long at least bm the masaes are equal) diffusion
depends mainly upon the mean of the diameters, being but little affected by even
a considerable disparity in size between the particles of the two gases. Thus it
appears that the viscosity and (if the experimental part of the inquiry could be
properly carried out) conductivity give us much more definite information as to the relative
sizes of particles of different gases than we can obtain from the results of diffusion.

Equation (12) shows how the gradient of density of either gas vmies, in the

stationary state, with its percentage in the mixture. For the multiplier of -j-^ is

obviously a maximum when

1 1

in which y = fti/^» is so. This condition gives

Hance the gradient is least steep at the section in which the proportion of the two
gases is inversely as the ratio of the diameters of their particles; and it increases
either way from this section to the ends of the tube, at each of which it has the
same (greatest) amount. This consideration will be of use to the full understanding
of the more complex case (below) in which the masses^ as well as the diameters, of
the particles differ in the two gases,

55. Let us now suppose the mass per particle to be different in the two gases.
The last terms of the right-hand side of (10), viz.,

1 , ^ ^vd*Gi

174 OK THE FOUXDATIONS OF THE KINETIC THEORY OF OASES. [uCXVin.

may be written in the form

P^ dn, f(n-n,)h^ r /(y)dy

'^^ jo (n - «,) A.V^(y) + nMF (y>y/^)/
where the meanings of / and F are as in § 34.

If we confine ourselves to the steady state, we may integrate (10) directly with
respect to x, since dGJcU is constant. In thus operating on the part just written,
the integration with regard to x (with the limiting conditions as in (11)) can be
carried out under the sign of integration with respect to y: — ^and then the y inte-
gration can be effected by quadratures.

The form of the x integral is the same in each of the terms. For

;»ilri, + fi(n-n,)"j,il(n-wO + fin, A--B\^A^B^A]'

This expression is necessarily negative, as A and B are always positive. When A
and B are nearly equal, so that B = (l'^e)A, its value is

A\2 3^ /'

so that, even when A and B are equal, there is no infinite term.

It is easy to see, from the forms of F(y), and of its first two differential
coefficients, that the equation

can hold for, at most, one finite positive value of y.

56. As a particular, and very instructive case, let us suppose

P, :P^::h,:h^ :: 16 : 1,
the case of oxygen and hydrogen.

(a) First, assume the diameters to be equal. Then the integral of (10), with
limits as in (11), taken on the supposition that the flow is constant, is

1^3=^ p,

dt irt^t/hi

3 /,-T7~ 1

F(y)-16F[l)

^(y)/(y)-16'^(|)/(|) 16f'(|)Y

LXXVIII.] ON THE FOUNDATIONS OP THE KINETIC THEOBY OF 6ASBB.

175

As remarked above, the definite integral is essentially negative. For so is every
expression of the form

a --^ Aa-Bb B
A^B'^{A^Bf^''^A

provided A, B^ a, and h be all positive. When A and B are equal its value is

I have made a rough attempt at evaluation of the integral, partly by calculationt
partly by a graphic method. My result iSj at best, an approximation, for the
various instalments of the quadrature appear a^ the relatively small differemes of two
considerable quantities. Thus the three decimal places^ to whichi from want of
leisure, I was obliged to confine myself^ are not sufficient to give a very exact
value. The graphical representations of my numbers were, however, so fairly smooth
that there seems to be little risk of large error. The full curve in the sketch
below shows (on a ten-fold scale) the values of the integrand (with their signs

' 1

_ _.

/ /
^ /

' '*•. \. *

-i. ' ^ ~x

D ^

changed), as ordinates, to the values of ^ as absclssfe. The area is about — 2'HI5«
Hence we have

176 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVIH.

(6) Suppose next that the diameter of a Pi is three times that of a Pj, but
the semi-sum of the diameters is « as before. The definite integral takes the form

/(.V)

16/(9

The corresponding curve is exhibited by the dashed line in the sketch, and its area
18 about — 3'167. !Sence, in this case,

f'^|. = __^ 3-793.
at 7r«" y/hi

(c) Still keeping the sum of the semidiameters the same, let the diameter of a
P, be three times that of a Pi. The integral is

Mil

16/(1)

\Fiy)Ay) 64f(|) 576J'(|)/(|) seF{^

The curve is the dotted line in the cut, and its area is about — 1*713. Hence we have

If we compare these values, obtained on such widely dilBferent assumptions as to the
relative diameters of the particles, we see at once how exceedingly difficult would
be the determination of diameters from observed results as to diffusion. (Compare § 54.)

But we see also how diflfiision varies with the relative size of the particles, the
sum of the diameters being constant. For the smaller, relatively, are the particles
of smaller mass (those which have the greater mean-square speed) the more rapid is
the diffuHion.

And further, by comparison with the results of §§ 51, 54, we see how much more
quickly a gas diffuses into another of different specific gravity than into another of
the same specific gravity.

lxxviil] on the foundations of the kinetic theory of oases. 177

When the less massive particles are indefinitely small in comparison with the
others, the diameter of these is s; and for their rate of diffusion we have

When it is the more massive particles which are evanescent in size, the numerical
bictoT seems to be about 3*48. Hence it would appear that, even in the case of
masses so different, there is a minimum value of the diffusion-coefficient, which is
reached before the more massive particles are infinitesimal compared with the others.

[At one time I thought of expressing the results of this section in a form
similar to that adopted in the expression for i) in § 51. It is easy to see that the
quantity corresponding to X would now be what may be called the mean fi:^e path
of a single particle of one gas in a space filled with another. Its value would be
easily calculated by the introduction of hi for h in the factor v of the integral

I-

Joe

while keeping e in terms of h. This involves multiplication of each number in the
fourth column of the Appendix to Part I. by the new factor €~<*i"*^*'Ai*/A*. But, on
reflection, I do not see that much would be gained by this.]

T. II. 23

178

ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXVIII.

APPENDIX.

The notation is the same as in the Appendix to Part I.

xXJX,

^XJX,

o?XJX,

afXJX,

•000049

•000005

•000001

•000000

•2

•000758

•000152

•000030

•000001

•003694

•001078

•000323

•000029

•4

•010364

•004146

•001658

•000265

•5

•022505

•011252

•005626

•001407

•6

•040512

•024307

•014684

•005250

•7

•063623

•044636

•031175

•015276

•8

•089928

•071942

•057564

•036834

•9

•116712

•105041

•094537

•076575

1-0

•141040

•160292

•176321

•193953

•234683

1-2

•172656

•207187

•248624

•358019

1-3

•177229

•230398

•299517

•506184

1-4

•174174

•243844

•341382

•669108

1-5

•164430

•246645

•369968

•832427

1-6

•149568

•239309

•382894

•980209

1-7

•131393

•223368

•379726

1^097407

1-8

•111654

•200977

•361758

11 72098

1-9

•091960

•174724

•331976

1 •I 98432

•073480

•146960

•293920

1175680

•057015

•119731

•251435

M08829

2-2

•043032

•094670

•208274

h008046

2-3

•031579

•072632

•167054

•883714

2-4

•022584

•054202

•130085

•749288

2-5

•015760

•039375

•098438

•615234

2-6

•010686

•027784

•072238

•488332

2-7

•007074

•019099

•051667

•375926

2-8

•004536

•012701

•035563

•278812

2-9

•002871

•008326

•024145

•203063

3-0

•001710

•006130

•015390

•138510

•001071

•003320

•010294

•098925

3-2

•000629

•002014

•006445

•065997

3-3

•000361

•001192

•003935

•042852

3-4

■000211

•000689

•002344

•027098

3-5

•000111

•000389

•001361

•016671

3-6

•000066

•000240

•000865

•010004

3-7

•000037

•000136

•000505

•005839

3-8

•000229

•003307

3-9

•000118

•001798

4-0

•000062

•000985

2095244

2^954862

4630593

14624154

Thus the values of C,, C„ C„ and C, are respectively 0-838, M82, 1-852, and 5-849.

LXXIX.]

179

LXXIX.

ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. III.

[Transactions of the Royal Society of Edinburgh, Vol. xxxv.]

INDEX TO CONTENTS.

Introductory
Part XV.

„ XVI.

PAGE
179

XVII.

Special Assumption as to Mole-
cular Force . . .181

Average Values of Encounter and
of Impact .... 182

Effect of Encounters on Free
Path 186

PAGE

Part XVIII. Average Duration of Entangle-
ment, and consequent Average
Kinetic Energy ... 188

Appendix —

A. Coefficient of Restitution less than Unity 189

B. Law of Distribution of Speed . . .189

C. Viscosity 190

J). Thermal Conducti%'ity .... 191

I HAVE explained at some length, in my "Reply to Prof. Boltzmann*," the
circumstances under which the present inquiry originated and has been pursued. Of
these I need now only mention two:— ^r5^, the very limited time which I can spare
for such work; second^ the very meagre acquaintance I possessed of what had been
already done with regard to the subject. My object has been to give an easily
intelligible investigation of the Foundations of the Kinetic Theory; and I have, in
consequence, abstained from reading the details of any investigation (be its author who
he may) which seemed to me to be unnecessarily complex. Such a course has,
inevitably, certain disadvantages, but its manifest advantages far outweigh them.

In August 1888, however, I was led in the course of another inquiry "(• to peruse
rapidly the work of Van der Waals, Die Continuitdt des gasformigen und flilssigen

* Proc, R. S. £., January 1888 ; PhiL Mag., March 1888.

t ** Report on some of the Physical Properties of Water,'* Phy$, Chem. Chall, Exp., Part IV. [LXI. above, p. 66.]

23—2

180 ox THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXIX.

Ziutandes. This shows me that Lorenz had anticipated me in making nearly the siune
correction of the Virial equation as that given in the earlier part of § 30 of my first
paper. His employment of the result is a totally dilBferent one from mine; he uses
it to find a correction for the number of impacts. The desire to make, at some time,
this investigation arose with me when I was writing my book on Heat, as will be
seen in the last paragraphs of § 427 of that book. [First edition, 1884.] It was caused
by my unwillingness to contemplate the existence of molecular repulsion in any form,
and my conviction that the elBfects ascribed to it could be explained by the mere
resilience involved in the conception of impacts.

The present paper consists of instalments read to the Society at intervals during
the years 1887-8. The first of these, which is also the earliest in point of date, deals
with a special case of molecular attraction, on which, of course, depends the critical
temperature, and the distinction between gases and vapours. Here the particles which,
at any time, are under molecular force have a greater average kinetic energy than the
rest. Mathematical, or rather numerical, difficulties of a somewhat formidable nature
interfered with the exact development of these inquiries. I found, for instance, that
in spite of the extreme simplicity of the special assumption made as to the molecular
force, the investigation of the average time between the encounter of two particles and
their final disengagement from one another involves a quadrature of a very laborious
kind. Thus the correction of the number of impacts could not easily be made except
by some graphic process.

One reason for the postponement of publication of the present part was the hope
that I might be enabled to append tables of the numerical values of the chief integrals
which it involves, especially the peculiarly interesting one

yrsc"*' I €^dx.

Want of time, however, forced me to substitute for complete tables mere graphical
representations of the corresponding curves, drawn from a few carefully calculated values.
These are not fitted for publication, though they were quite sufficient to give a general
notion of the numerical values of the various results of the investigation; and enabled
me to take the next step: — viz. the approximate determination of the form of the Virial
equation when molecular attraction is taken account o£ Part lY. of this investigation,
containing this application, was read to the Society on Jan. 21, 1889, and an Abstract
has appeared in the Proceedings. It appears that the difference of average kinetic
energy between a free, and an entangled, particle is of special importance in the physical
interpretation of the Virial liquation.

An Appendix is devoted to the consideration of the modification which the previous
results undergo when the coefficient of restitution is supposed to be less than 1. This
extension of the investigation was intended as an approximation to the case of radiation
from the particles of a gas, and the consequent loss of energy. But, so far as I have
developed it, no results of any consequence were obtained. I met with difficulties of
a very formidable order, arising mainly from the fact that the particles after impact

I.XXIX.] ON THE FOUNDATIONS OF THE KrNETIC THEORY OF GASES.

181

do not always separate from one another. The full treatment of the impact of a single
particle with a double one is very tedious ; and the conditions of impact of two
double particles are so complex as to be totally unfit for an elementary investigation
like the present*

The remainder of the Appendix is devoted to two points, raised by Professors
Newcomb and Boltzmann, respectively : — the first being the problem of distribution of
speed in the "special" state; — the other involving a second approximation to the
estimates of Viscosity and Thermal Conductivity already given in Part IL

XV, Special Assumption as to Molecular Force.

57. To simplify the treatment of the molecular attraction between two particles,
let us make the assumption that the kinetic energy of their relative motion changes
by a constant (finite) amount at the instant when their centres are at a distance a
apart. This will be called an Encoufder* There will be a refraction of the direction
of their relative path, exactly analogous to that of the path of a refracted particle
on the corpuscular theory of light. To calculate the term of the virial (| 30) which
corresponds to thiSp we must find

(a) The probability that the relative speed before encounter lias between u and
tt + di*,

(b) The probability that its direction ia inclined from 0 to ff-hdff to the line of
centres at encounter.

(c) The magnitude of the encounter under these conditions, and its average value.
Next, to fijid the (altered) circumstances of impact, we must calculate

(d) The probability that an encounter, defined as above, shall be followed by
an impact.

(e) The circumstances of the impact.

{/) The magnitude of the impact, and its average value per encounter.

In addition to these, we should also calculate the number of encounters per
fiecond, and the average duration of the period from encounter to final disentangle-
ment, in order to obtain (from the actual speeds before encounter) the correction for
the length of the free path of each* This, however, is not easy. But it is to be
observed that, in all probability, this coiTection is not so serious as in the case when
no molecular force is assumed. For, in that case the free path is always shortened ;
whereas, in the present case it depends upon circumstances whether it be shortened
or lengthened. Thus, if the diameters of the particles be nearly equal to the en-
counter distance, there will in general be shortening of the paths* and consequent
diminution of the time between successive impacts : — if the diameters be small in
comparison with the encounter distance, the whole of the paths will be lengthened

182 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXIX.

and the interval between two encounters may be lengthened or shortened. Thus if
we assume an intermediate relation of magnitude, there will be (on the average) but
little change in the intervals between successive impacts. Hence also the time during
which a particle is wholly free will be nearly that calculated as in § 14, with the substi-
tution, of course, of a for s.

XVI. Average Values of Encounter and of Impact

58. The number of encounters of a v, with a Vi, in directions making an angle
)8 with one another, is by § 21 proportional to

pviVo sin 0dfi,

where Vq^^v^^ v^ — 2vvi cos )8.

Hence the number of encounters for which the relative speed is from u io u-^-du
is proportional to

u'duf^ (1).

The limits of Vi are v±u, or u±v, according as v<u, and those of v are 0 to x , so
that the integral is

Jo vJv-u Vi Jo 2At;^
The first term of this integral may be written as

and the second as —I Ix—^jdxe'^^^.

2
u

Together, these amount to I xdx e'^'^ '\' u j ctee"^**".

The first term vanishes, and the second is

u lir
2V 2A-

Thus the value of (1) is ^^"*'*'^\/j3 •- ^^^'

LXXIX.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 183

But, on the same scale, the whole number of encounters in the same time is

Thus the fraction of the whole encounters, which takes place with relative speed u to
u + dii, is

whose integral, from 0 to x, is 1 as it ought to be.

59. Now these relative motions are before encounter distributed equally in all
directions. Let us deal therefore only with those which are parallel to a given line.
The final result will be of the same character relative to all such lines; and there-
fore the encounters will not disturb the even distribution of directions of motion.

Refer the motion to the centre, 0, of one of the encountering particlea Let
A be the point midway between the particles at encounter, B that of impact, the

encountering particle coming parallel to CO. Let OA = a/2, OB (as before) = 8/2. Let
0, (f> be the angles of incidence and refraction at encounter, yfr that of incidence at
impact, u and w the relative speeds before and after the encounter. Then

a sin 0=io sin <f> ;

and, if Pc^ represent double the work done in the encounter by the molecular forces,

ti^ cos' O-^-C'^iv^ cos* <^,

80 that u^^cr= lu^.

Also it is obvious from the diagram that

s sm vr = a sm 0 = — sni r.

184 ON THE FOUNDATIONS OF THB KINETIC THEORY OF GASES. [lXXIX.

Hence the eDCOunter will Dot be followed by an impact if

8in 0> — .
au

60. We must next find the average value of an encounter, and also of an
impact; in the latter case taking account of all the encounters whether or not they
involve an impact.

The numerical value of the encounter-impulse in the above figure is evidently

P{w cos <^ — u cos 0)1%

which must be doubled to include the repetition on separation; and the average value,
when the relative speed is u, is

2P I * sin 5 cos 5 (w cos ^ - 1« cos 5) d^

op .

= Q;:iKc'+u«)*-c-u«} (3).

3u'

The value of the subsequent impact is

— Pw cos y^,
and the average value

- 2Pw /cos dsin ^ a/i - ^sin'^cW.
When sfU) > an, the limits are 0 and ^ , and the value is

-i^»S{-(-^)') w-

But when 8w < au, the limits are 0 and sin~* — , and the value is

-1''"% w

By (2) and (3) we find as the average value of the encounter, taking account of all
possible relative speeds,

+ g AM u du€'^'^ {(c« + w')» - C - u%

LXXIX.] ON THE FOUNDATIONS OP THE KINETIC THEORY OF GASES. 185

or, if we write for simplicity, e* = Ac"/2,

=Th\^^^l?''''^'^"^^-^i} • («>•

The expression obviously vanishes, as it ought to do, when e = 0. And it is
always positive, for its differential coefficient with respect to e is

In a similar way (4) and (6) give, with (2), as the average impact per encounter,

^=-f*l/f*'«'«<'«--«'('-('-S)V/V-«'"-'"''}

The first integral we have already had as part of the encounter. To simplify the
second, let s/a = cos a, and it becomes

reoota •

I ndue-^^'^f^ (u^ + c* - u^ sec'a)*,
Jo

which, with c* - w' tan'a = ^^

gives cot^aj'^d^e^^^^-^^^*

or m tan»a6-*^'<»*V2j ^^ ^^^^

The whole is now

= ~ '^ V/f " {^ a/J ■*" "^^^ ~ 'J'^^f^'^^y + -^^c tan' a - V2e -'"'»»*• tau' a f '"* V<irl

'"' ~ ;;7a *'**^' " I**" a/I "*" '^^^ ^®*'' " ~ "^^ /'«~''*'y - ^2e-«'~*'« tan* o r V<irl ,
T. n. 24

186 ON THE FOUNDATIONS OP THE KINETIC THEOBY OF GASES. [lXXIX.

which, when 6 = 0 and cos a = 1, becomes

as in §30.

It would at first sight appear that the value of the impact is finite (=~-P^v r)

when there is no nucleus \i,e, a^-^\. But, in such a case, we must remember that

the second part of the first expression for R above has no existence. In fact the value
of the second of the two integrals is V2 tan* a . e cot a, when 6 cot a is small; and this
destroys the apparently non-vanishing term.

XVII. Ejfect of Encounters on the Free Path.

6L If two particles of equal diameters impinge on one another, the relative path
must obviously be shortened on the average by

I 27rsi

.•'0

sin 0 cos* 0d0 «
2s

27r sin 0 cos ^(20
Jo

But if V, Vi be their speeds, and Vq their relative speed, the paths are shortened
respectively by the fractions v/vq and Vi/vq of this. The average values must be equal,
so that we need calculate one only.

Now the average value of v/vq is obviously

jvPiVQml3dl3

jvviVoainffdff

where fi is the angle between the directions of motion, so that

tWi sin ^d^ = Vodvo,
Hence the average above is

[vv.v^dv, I,/3 ^/27r J V2'
J Wi is

LXXIX.] ON THE FOUNDATIONS OF THE KINETIC THBOBY OF GA8BS. 187

Hence the mean of the free paths during a given period becomes

1 V2g,

V2n7r«» 3 '

that is, it is shortened in the ratio

l-g7iir«» : 1,

or 1—4 (sum of vols, of spheres in unit vol.) : 1 = 1 — ^r : 1 say.

Hence the number of collisions per second, already calculated, is too small in the same
ratio.

Thus the value of X (i2) in § 30 must be increased in the ratio 1:1 — ^, and the
virial equation there given becomes

nP^I2 = lp/v-

('-^)

If this were true in the limit, the ultimate volume would be double of that before
calculated, %.e. 8 times the whole volume of the particles.

62. Another mode of obtaining the result of § 61 is to consider the pcui^icles as
mere points, and to find the average interval which elapses between their being at a
distance 8 from one another and their reaching the positions where their mutual distance
is least. The space passed over by each during that time will have to be subtracted
from the length of the mean free path calculated as in § 11 when the particles were
regarded as mere circular discs.

The average interval just mentioned is obviously

8cos0,8in0co8 0dO
0 2«

3ii'

2 ;
sin 0 cos 0d0

Hence the average space passed over in that interval is

2* f— . ^i^»*^ //8_y2«

2» [

^' w, ^"Z 3 " 3

If we put a for « in this expression we have the amount to be subtracted from
the average path between two encounters in consequence of the finite size of the region
of encounter.

24—2

Ill

188 ON THE FOUNDATIONS OF THE KINETIC THEOBY OF GASES. [lXXIX.

XVIIL Average Duration of Entanglement, and consequent Average Kinetic Energy.

63. We have next to find the average duration of entanglement of two particles: —
i.e., the interval during which their centres are at a distance less than a.

The whole relative path between the entering and leaving encounters is

2 (a cos ^ — « cos -^X

or 2a cos <f>,

according as there is, or not, an impact.

Hence the whole time of entanglement is the quotient, when one or other is divided
by w. And the average value, for relative speed w, is

r = --zj (a Jw^ - u^ sin» 0 - Jv^^ - a*u* sin* 0) cos 0 sin 0d0

when W8>au;

and ~"j]| aVw* — ti*sin"tf cos^sin ^d^— / Jv/*^ — a^u* eixi^ 0 cos 0am 0d0> ,

Su^W^ ^ a^u^ /

when W8 < au.

These must be multiplied by the chance of relative speed t^, as in § 58, and the
result is

or, with the notation of § 60,

3 Je w ^ ^ ■ ' 3 Jo &-z^

As the value of this expression depends in no way on the length of the free
path, it is clear that the average energy of all the particles is greater than that of
the free particles, by an amount which increases rapidly as the length of the free
path is diminished.

LXXIX.] ON THE FOUNDATIONS OF THE KINETIC THEOBY OF GASES. 189

APPENDIX.

A. Coefficient of Reatitutiati leas them Unity,

Let us form again the equations of § 19, assuming e to be the coefficient of restitution.
We have

P(u' - u) = - ■'^I^Cu - V) =- e (v'-y),
80 that P(u'»-u') = -^^|^^«-v){(2P + <2r:-«)u + C(l+«)v},

C(v''-V)= :^|il±f)(u-y)[i'(l+«)u+{2e + /'(l-«)}y].

The whole change of energy in the collision is half the sum of these quantities, viz.,

_\PQ(l-ey

2 p+Q y^ ^i-

With the help of the expressions in § 22, we find for the average changes of energy of a i*
and of a Q, respectively,

IP (^» - ?) = - ^.^±±^{2 (Pk -Qh)+Q{l- e) (h + k)l

iQ(V^-^)= 2^^^,{2(>A- Qh)-P{l-e){h^k)}.

The first term on the right is energy exchanged between the systems; and, as in the case
of e=:l, it vanishes when the average energy per particle is the same in the two systems.
The second term (intrinsically negative for each system) is the energy lost, and is always
greater for the particles of smaller mass. The average energy lost \)er collision is

It is easy to make for this case an investigation' like that of § 23. But we must
remember that there is loss of energy by the internal impacts of each system, which must
be taken into account in the formation of the differential equations. This is easily found
from the equations just written, by putting Q= P : — but the differential equations become
more complex than before, and do not seem to give any result of value. [Shortly after
Part I. was printed off. Prof. Burnside called my attention to the fact that the equations
of interchange of energy in § 23 are easily integrable without approximation. But the
approximate solution in the text suffices for the application made.]

B. The Law of Distribution of Speed.

In addition to what is said on this subject in the Introduction to Part II., it may be
well to take the enclosed (from Proc, R. S. E., Jan. 30, 1888).

190 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXIX.

**The behaviour parallel to y and z (though not the number) of particles whose velocity-
components are from x to x-^dx^ must obvumsly be independent of x, so that the density
of 'ends' in the velocity space diagram is of the form

/(«) Fii,, z).

The word I have italicised may be very easily justified. No collisions count, except those
in which the line of centres is practically perpendicular to x (for the others each dismiss
a particle from the minority; and its place is instantly supplied by another, which behaves
e^cactly as the first did), and therefore the component of the relative speed involved in the
ccllisUma which we require to eonaider depends wholly on y and z motions. Also, for the
same reason, the frequency of collisions of various kinds (so far as x is concerned) does
not come into question. Thus the y and z speeds, not only in one x layer but in all,
are entirely independent of x\ though the number of particles in the layer depends on
X alone.''

C. Viscosity.

In my "Reply to Prof. Boltzmann" I promised to give a further approximation to
the value of the coefficient of Viscosity, by taking account of the alteration of permeability
of a gas which is caused by (slow) shearing disturbance. I then stated that a rough
calculation had shown me that the efifect would be to change my first, avowedly approximate^
result by 11 or 12 per cent. only. I now write again the equations of § 36, modifying
them in conformity with the altered point of view.

The exponential expression in that section for the number of particles crossing the plane
of yz, must obviously now be written

-iec9 I ev,
c Jo

sin edOI2,

where Vq is the velocity relative to the absorbing layer at f, and e also is no longer
constant. But we have at once

1^0 = i; + ^( sin ^ cos ^,

sec^

sec 0 n^
that the exponent above is I {ev + (ev)' B$aia6coB ^} d(.

Thus the difierential of the whole ^-momentum which comes to unit sur&u^e on a; = 0 from
the layer x, x + dx^ is

Integrating with respect to ^ from 0 to 2ir, to x from 0 to oo , and to $ from 0 to
^ , and doubling the result, we have

'H-ih-'s^'

The first term expresses my former result, viz.

BPC,
Stts^ Jh '

LXXIX.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 191

But the whole is ^ f^(l-i|')= ^ f vA^= ^J^

15 jo \« « / 15 Jo e \bir^ Jh

e 15ir«* ^h
The ratio is 2C,/5Ci = 3-704/419 = 0-882.
It is worthy of remark that the term

15c«
has the value --^ — - — J- ,

and that 4/5ths of the Ci term are due to e'.

D. Thermal ConductivUi/,

Applying a process, such as that just given, to the expressions in § 39, we find that
the exponential in the integral for the member of particles must be written

c *» 2 =€ (1- Co o^ sec $/2v + -^^ j

to the required degree of approximation. [Properly, the superior limit of the 0 integration
should be cos"* - ; but this introduces quantities of the order a' only.] Thus equation
(1) becomes

In the same way equation (3) of § 41 becomes

Thus equations (1') and (3^) of § 42 become, respectively,

h' P /5 \ PC^'

, p h' P /25^ -^ _\ 3PCfi' 5P C^'

Thus we have finally to deal with the new forms of (1") and (3") of § 43, viz. : —

„=4_e^oo6-e^oi2,

7A' p ph

When similar methods are applied to the difiusion equations, they become hopelessly
complicated.

IVt [lxxx.

LXXX.

r>N THK FOUNDATIONS OF THE KINETIC THEORY OF

GASES^ IV.

{frmmuitiom of ih0 Hoj^ul Soci^^ of Kdinbwyk, Vol xxxvi. Read Jan. 21, 1889,

mA April 6» 189L]

INUKX TO CONTENTS.

PASS

fHM^mifftA^^ l^

l^4f^f i|U 'IV |iH«U(i»vuml KHVU^tivMutA ol Yau

PiUtT XXL Relatioci between Kinetie Energj-

and Testpecatore . 201

UkI' Wtu^U )4uU rUmuuM . . 190 I ^ XXII. The l^uafckm of Isodiennab . 203

%K Thtt VUiHlK\4VVHtUiua««illn^U^^ . «» XXIIL Compwaon wzOi Ezpenment . 204

M|ihm'ii>4^l (W^ivlv« . . . It^

I A ¥^y( WiMHU mv UiKS>«ie«M7 W ^xplnau wbt^ thie po^tsent paper has iiititerto been
|iHMMi(l iM Al»Mtmv^t K^\y. tUKi %%> i^kow what moJiticatiiHii^ it hais ondergone ance it

|u t'lu^ (u^l^^v. t4^ vt w«v^ tirtitt pvvt^ut^ to tth^ ^Society. I coatented myself with
|Im( Ut«u^l (M^^aiw v'f v»\^iw.'tkug tivm tht> viml a m^gative term {^ffp) to lepres^it
Mt- h'Hui. is ji^vUva^ vvt' ^Kv> )Mi't du^ W %hf^ iih>iecular repolstoa at impact. But, as will
|(M MM\'i^ i\V U^v* VkKiUHolj pi'iutvil at th^ time \,t^>K\ &;^. ^Soc. Edin., 21. I80X I ^^tated
\Ui\[> Uu^Ui^h iK^ ^4\K\^luiv in vviifvct wht^Kl molecular attractioa b not taken into
tUir>uuiii'. il UH^uiVv^M vH>4^^HK'mMv> uivditivatiou when such attractions are introduced. I abo
'iMMul OuU iU uu^U4 vlHvt >Kv>uId be to alter one of the dispo^sable i{aancitiea {A) in
(UV <'(|UuUiu\ I Khnv >iU4vv H\vu that the deduitioti, of what we are now to nndeistuid
\{y U'iii(H'i^Uii\\' vvKu'h I ifhvu lUtivducoU* leads uaturally and diiecdy tso the writtn^

LXXX.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES.

193

where E is proportional to the absolute temperature and to the average energy of
a free particle. This remark really substitutes the new undetermined quantity e for
the ^ which occurred in niy former expression. But the equation in its new form,
though containing as many arbitrary constants as before, is considerably more simple
to deal with, a« p occurs only in the term pv^ in which both factors are directly
given by experinieDt. The term p{v — ^) was a source of great trouble in the
attempt to determine the proper values of the constants. It was recognised by
Van der Waals, even in his earliest papei*, that the quantity ^ suffers large changes
of value J with changes of volume of the gas, so that no formula in which it is treated
as a constant could suffice to represcDt more than a moderate volume-range of the
isothermals with any consistent degree of accuracy.

When I first read my paper, I had made no serious attempt to attack the
formidable numerical problem of determining values of the constants which should
adapt my main formula to Andrews' experimental data. I contented myself with
obviously (and professedly) provisional assumptions, which showed that it was well fitted
to represent the result;^ ; but I also gave the relations among the constants of the
formula and the data as to the mass, and the critical values of the pressurei volume,
and temperature of the substance.

Later, having carefully reduced Andrews* data to true pressures (by the help of
Amagat's determinations of the isothermals of air at ordinary temperatures), I proceeded
to try various assumptions as to the values of the quantities ir, p^ a in jny form u lee,
on which (as i^=30^'9C. was already given by Andrews with great precision) all the
constants can be made to depend, I at first endeavoured to adjust these so as to
make ^ — 00017, in consequence of a statement by Amagat {Ann, ds OhimtBt 18SI,
xxji. p. 397) as to the ultimate volume of COa. But I failed to get results giving
more than a general accordance with Andrews' experiments; so that I made further
guesses without taking account of this datum. I had, however, become accustomed to
the employment of it, as a quantity of the order 10"* of the volume of the gas at
0^ C. and 1 atm., so that I was much surprised to find that one of my chance
assumptions, which gave ^ ^ 000005, led to a formula far more closely agreeing with
Andrews than any I had till then met with. The reason for this agreement is now
obvious: — The term — ^p is not the proper expression for the part of the virial which
it is intended to represent \ and the true mode of iittroducing that part is, aa pointed
out in my Ahstract, to alter the value of A firom isothermal to isothermal, and from
volume to volume.

In January last I happened to ask M. Amagat if he could give me the value of
|w for CO3 at 0"* C. and 1 atm., which is wanting in his remarkable table (in the
Ann. de Chimw, above referred to). In reply he kindly furnished me with a new
and extremely complete set of determinations of pv, in terms of p, for CO^ ; tlie range
of pressures being I to 1000 atm., and of temperature 0"" to 100° C, some special
isotherm alg up to 258° being added. My first step on receiving these data was to try
how far they agreed with Andrews' results, which I had carefully plotted (tcf true
pressures) from SVl to 4 1'' C, and for volumes from ^03 to *002. My object was to
discover, if possible, by comparison of the results of two such exceptionally trustworthy

T. ir. 25

194 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXX.

experimenters, whether any modification of the behaviour of COj is (as some theoretical
wiiteis have asserted) produced by the molecular forces due to the walls of the very
fine tubes in which Andrews' measurements were made. I could find nothing of the
sort. The isothermals, plotted from Amagat's numbers (which in no case were for any
of Andrews' temperatures), took their places in the diagram almost as if they had
been an additional part of the work of one experimenter. The slight discrepancies at
the smaller volumes were obviously due to the trace (1/500) of air which, as Andrews
pointed oat, was associated with the carbonic acid in his tubes.

But, although I have got from them only negative information as to the molecular
eflRnrts said to be due to glass, Amagat's isothermals are so regularly spread over the
diagnun as to be far more readily available for calculation than are those of Andrews.
I have not, however, the leisure requisite for anything like an exhaustive treatment of
tbem ; and all that I have attempted is to obtain values of the constants in my formula
which make it a fair representation of the phenomena in the experimentally investi-
gated range of the gas region of the diagram; and, more especially, that portion of it
wh«« the volume exceeds the critical volume. It appears to me that to try to push
the j^iproximation further at present would be waste of time; it cannot be attempted
with any hope of much improvement until certain points, referred to below, have been
properly investigated. These may lead to modifications of parts of the formula which,
thoagh unimportant in the regions now treated, may greatly improve its agreement
with the hct8, in the remaining portions of the diagram. Besides, there is in the data
the uncertainty due to the presence of air, which was not wholly removed (though
reduced to 1/2500) even in Amagat's experiments. This, as above remarked, begins to
tell especially when the volume is small.

It is very much to be regretted that Clausius did not avail himself of Amagat's
data in reducing Andrews' scale of pressures. He expressly 8a,ys he rejected them
because they were not consistent with those of Cailletet. Hence the formula which
he obtained after great arithmetical labour, though it is in close, sometimes in almost
startling, agreement with the data through the range of Andrews' work, is not
properly a relation among p, v, and t. If we make it such, by putting in the
correction (in terms of v) for the pressures as measured by the air-manometer, a
new t;-&ctor is introduced into the equation, and its simplicity (which is one of its
most important characteristics) is lost. I tried to obtain hints for the values of the
constants in my own formula by making this change in that of Clausius. But
I found that the factor 1/t which Clausius introduced into the virial term (in order
to approximate to the effect of the aggregation of particles into groups at the lower
ranges of temperature), made his formula inapplicable to the wide regions of the
diagram which Andrews did not attack, but which have been so efficiently explored
by Amagat. There are, no doubt, traces of this systematic divergence even in the
special Andrews region, but they become much more obvious in the outlying parts.

It is certainly remarkable that my simple formula, based entirely on the behaviour
</f mnofyih spheres, should be capable of so close an adjustment to the observed facts;
ariri I think that the agreement aflfords at least very strong testimony in favour of
tb/j jirojK^Hed mode of reckoning the temperature of a group of particles. When this

LXXX.] ON THE FOUNDATIONS OP THE KINETIC THEORY OF GASES. 195

is introduced, it appears at once that the term of Van der Waals* equation, which
he took to represent Laplace's K, is not the statical pressure due to molecular forces,
but (approximately) its excess over the repulsion due to the speed of the particles.
And hence the (external) pressure is not, as Clausius put it, ultimately the difference
between two very large quantities, but the excess of one very large quantity over
the very large difference between two enormously great quantities; and thus the
whole phenomena of a highly-compressed gas, or a liquid, are to be regarded as
singular examples of kinetic stability. 28/5/91.]

Preliminary.

In the preceding part of this paper I considered the consequences of a special
assumption as to the nature of the molecular force between two particles, the
particles themselves being still regarded as hard, smooth, spheres. My object was to
obtain, by means of rigorous calculation, yet in as simple a form as possible,
a general notion of the effects due to the molecular forces. My present objects are
(1) to apply this general notion to the. formation and interpretation of the virial
equation (in an approximate form), and (2) to apply the results to the splendid
researches of Andrews and their recent extension by the truly magnificent measure-
ments of Amagat.

Passing over some papei*s of Him, and others, in which the earliest attempts
were made (usually on totally erroneous grounds) to form the equation of the isother-
mals of a gas in which molecular forces are prominent, we come to the Thesis of
Van der Waals*, who was the first to succeed in representing, by a simple formula,
the main characteristics of Andrews' results. His process is based upon the virial
equation, and his special object seems to have been an attempt to determine the
value of the molecular constant usually called "Laplace's K" Though the whole of
this essay is extremely ingenious, and remarkably suggestive, it contains (even in its
leading ideas) much that is very doubtful, and some things which are certainly
incorrect. One of these was specially alluded to by Clerk-Maxwell +, who, in reviewing
the essay, said: — ^" Where he has borrowed results from Clausius and others, he has
applied them in a manner which appears to me to be erroneous." It will conduce
to clearness if I commence with an examination of the equation which is the main
feature of Van der Waals' Thesis, and the modifications which it underwent in the
hands of Clausius.

XIX. The Isothermal Equations of Van der Waals and Clausiits.

64. The virial equation (§ 30, above) is

iS(mi^»)=fpt; + i2(iir);

where, to save confusion, we employ t* to denote the speed of the particle whose
mass is m. From this Van der Waals derives the following expression : —

(i) + 5)(f-/9) = i2(mu»);

* Over de eontinuiteit van den gat- en vloeiitoftoettand. Ldden, 1873. t Nature, Oct. 15, 1874.

25—2

196 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXX,

and he treats the right-hand member as a constant multiple of the absolute tempera-
tore. (This last point is of extreme importance, but I shall discuss it farther on;
Ml fresokt I confine myself to the formation of the equation.)

It is certain (§ 30) that, when there is no molecular force except elastic resili-
eooe, die term

in the viiial equation takes, to a first approximation at least, the form of a numerical
multiple of

"IT"'

and thus that, if this term be smali in comparison with the other terms in the equation,
we may call it

Thus the virial equation becomes /> (v — ^) = J2 (mu*).

(So fiu*, all seems perfectly legitimate; though, as will be seen later, I think it has
led to a good deal of confusion: — at all events, it has retarded progress, by intro-
ducing what was taken as a direct representation of the "ultimate volume" to which
a substance can be reduced by infinite pressure. When this idea was once settled
in men's minds, it seemed natural and reasonable, and consequently the left-hand
member of the virial equation is now almost universally written p(V'-l3); although,
even in Van der Waab' Thesis, it was pointed out that comparison with experiment
shows that 0 cannot be regarded as a constant. But its introduction is obviously
indefensible, except in the special case of no molecular force.]

Van der Waals* next step is as follows: — Although p, in the virial equation, has
been strictly defined as external pressure (that exerted by the walls of the containing
▼eflsel), he adds to it, in the last-written form of the equation (deduced on the
express assampiion of the absence of molecular force), a term a/v*, which is to
represent Lflpiace's K. Thus he obtains his fundamental equation

(p + |)(*^-^) = i2(mu«).

or, as it is more usually written (in consequence of the assumption about absolute
temperature, already noticed),

_ kt a

iHbere i; is an absolute constant, depending on the quantity of gas, and to be deter-
mined by the condition that the gas has unit volume at 0"" C. and 1 atmosphere.

I do not profess to be able fully to comprehend the arguments by which
Van der Waals attempts to justify the mode in which he obtains the above equation.
Their nature is somewhat as follows. He repeats a good deal of Laplace's capillary
work; in which the existence of a large, but unknown, internal molecular pressure
is established, entirely from a statical point of view. He then gives reasons (which

LXXX.] OK THE FOUNDATIONS OF THE KINETIC THEORY OF GASES.

197

aeem, on the whole, satisfactory from this point of view) for assuming that the
magnitude of this force is as the square of the density of the aggregate of particles
considered. But his justification of the introduction of the term o/i?^ into an account
already closed, as it T?ere, escapes me. He seems to treat the surface-skin of the
group of particles as if it were an additional bounding-snrface, exerting an additional,
and enormous, pressure on the contents. Even were this justifiable, nothing could
justify the multiplying of this term by (v — ff) instead of by v alone. But the whole
procedure is erroneous. If one begins with the virial equation, one must keep strictly
to the assumptions made in obtaining it, and consequently everytkinff connected with
molecular force, whether of attraction or of elastic resilience, must be extracted from
the term X (Rr),

It is very strange that Clausius*, to whom we owe the virial equation, should
not have protested against this striking misuse of it, but should have contented
himself with making modifications (derived from general considerations, such as aggre-
gation of particles, &c.) which put Van der Waal«' equation in the form

_^ kt a

65. Van der Waals' equation gives curves, whose general resemblance to those
plotted by Andrews for CO, is certainly remarkable : — and it has the further advantage
of reproducing, for temperatures below the critical point, the form of isothermals
(with physically unstable, and therefore experimentally unreal isable, portions) which
was suggested by James Thomson, as an extension of Andrews* work* For a reason
which will presently appear (§ 67), Van der Waals curves cannot be made to coincide
with those of Andrews.

The modified equation of Clausius, however, seems to fit Andrews* work much
better:— but the coincidence with the true isothermals is much more apparent than
real, because Clausius* work is based on the measurement of pressures by the air-
manometer, as they were originally given by Andrews, who had not the means of
reducing them to absolute measure.

But a further remark of Clerk-Maxweirs (in the review above cited) is quite as
applicable to the I'esults of Clausius as to those of Van der Waals, viz»: — *' Though
this ^reement would be strong evidence in favour of the accuracy of an empirical
formula devised to represent the experimental results, the equation of M. Van der
Waals, professing as it does to be deduced from the dynamical theory, must be
subjected to a much more severe criticism."

66. Before I leave this part of the subject, 1 will, for the sake of future
reference, put the equations of Van der Waals and Clausius in a form which I have

found to be very convenient, viz.; —

{v — vY \ . k

''-K'-<:^[^)-.-%<'-''-

(A)

Armakn der Fky$ik, ix. ISSO.

198 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXX.

In these equations p^ v, t belong to the critical point, determined by the conditions that
at such a point j) is a minimax in terms of v. The special advantage of this mode of
representing the isothermals depends on the fact that the first part of the value of p
belongs to the critical isothermal; so that by comparing, at any one volume, the
pressures in dififerent isothermals (as given experimentally) we have a comparatively
simple numerical method of calculating the values of some of the constants in the
equation.

67. But, even if we were to regard the formula of Van der Waals as a purely
empirical one, there is a fatal objection to it in the fact that it contains only two
disposable constants. Thus, if it were correct, the extraordinary consequence would follow
that there is a necessary relation among the three quantities, pressure, volume, and
temperature, at the critical point: — so that, no matter what the substance, when two
of these are given the third can be calculated from them. I do not see any grounds
on which we are justified in assuming that this can be the case. Certainly, if it were
established as a physical truth, it would give us views of a much stronger kind than
any we yet have as to the essential unity of all kinds of matter. Van der Waals
seems to have taken his idea in this matter from one of Andrews' papers, in which
there is a hazardous, and therefore unfortunate, speculation of a somewhat similar
character. Anyhow, it would seem that, at least until experiment proves the contrary,
we are bound to provide, in our theoretical work, for the mutual independence of at
least the three following quantities: —

1. The diameters of the particles.

2. The range of sensible molecular force.

3. The maximum relative potential energy of two particles.

Besides these, there is the question of the law of molecular force, which we are certainly
not entitled to assume as necessarily the same in all bodies. This has most important
bearings on the formation of doublets, triplets, &c., at lower temperatures.

The modified formula of Clausius has one additional constant, and is therefore not
so much exposed to the above objections as is that of Van der Waals. Still I think it
has at least one too few.

XX. The Virial Equaiion for attracting Spherical Particles,

68. What is required is not an exact equation, for this is probably unattainable
even when we limit ourselves to hard spherical particles. To be of practical value the
equation must (while presenting a fair approximation to the truth) be characterised by
simplicity. And, should the experimental data require it, we must be prepared to give
the equation of any one isothermal in two or more forms, corresponding to various
ranges of volume. It is exceedingly improbable (when we think of the mechanism

LXXX.] OK THE FOUNDATIONS OF THE KINETIC THEORY OF OASES,

199

involved) that any really simple expression will give a fair agreement with an isothermal
throughout the whole range of vtilutnea which can be experimentally treated.

From the general results of Part IIL of this ]^per we see that the tenn

in the virial equation must, when molecnlar forces are taken into account, contain a
term proportional to the number of particles which are at any (and therefore at eveiy)
time within molecular range of one another. Hence if^ when the volume is practically
infinite, we have for the mean -square speed of a particle

^ mn

(where n is the whole number of particlesX we shall have, when the volume is not
too much reduced, no work having been done on the group from without,

J2(mu^) = ^ +

1^ + 7

where C and 7 may be treated as constants, the first essentially positive if the
molecular force be attractive, the second of uncertain sign. Even if the volume be
very greatly reduced it b easy to see, from the following considerations, that a similar
expression holds. The work done on a particle which joins a dense group is, on
account of the short range of the forces, completed before it has entered much beyond
the skin, and is proportional, ceteris pmibus, to the skiu-density. Hence the whole
work done on the group by the molecular forces is (roughly) proportional to

the first factor expressing the number of the particles, the second the work done on
each. But, as we are dealing with a definite group of particles, the first factor is
constant, so that the whole work is directly as p,), or inversely as (say) t^ + 7, because
Pa < p. But the work represents the gain in kinetic energy over that in the fi-ee state,
BO that this mode of reasoning leads us to the same result as the former for the
average kinetic energy of aU the particles^

In so &r as i£ depends on the molecular attractianj the term

is evidently proportional, per unit volume of the group, to the square of the deusity:^
for the particles, in consequence of their rapid motions, may be treated as occupying
within an excessively short time every possible situation with regard to one another.
ThuSj as regards any one, the mass of all the rest may be treated as diffiised uaifomily
through the ^pace they occupy* In volume v, therefore, the amount is as vp\ But,
in the present case, the quantity vp is constant, so that, again, the approximate value
of the term is directly as p, or inversely as v. But, once more, we must allow for
the bounding film (though not necessarily to the same exact amount as before), so we
muy write this part of the term as

» + a

200 ON TBB FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXX.

But there is another part (negative) which depends on resilience. This is (§ 30)
proportional to the average kinetic energy, and to the number of particles and the
number of collisions per particle per second. The two last of these factors are practi-
cally the same as those employed for the molecular attraction. Hence the whole of
the virial term may be written as

A-e\E + C/(v + y)]
v-\-a
Thus if we write again A and C for

A + and C +

a — 7 a — 7

respectively, the complete equation takes the form

^ . G A-eE

pV = E-\- — ; ,

which is certainly characterised by remarkable simplicity.

69. We must now consider how far it is probable that the quantities in the
above expression (other than p and v) can be regarded as constant. E, of course,
can be altered only by direct communication of energy; but the case of the others
is diflferent. Generally, it may be stated that there must be a particular volume
(depending primarily upon the diameters of the particles) at and immediately below
which the mean free path undergoes an almost sudden diminution, and therefore we
should expect to find corresponding changes in the constants. In particular, it must
be noted that some of them depend directly on the length of the free path, and
that somewhat abrupt changes in their values must occur as soon as the particles
are so close to one another that the mean free path becomes nearly equal to their
average distance from their nearest neighbours. For then the number of impacts per
second suffers a sudden and large increase. Thus, in consequence of the finite size of
the particles, we may be perfectly prepared to find a species of discontinuity in any
simple approximate form of the virial equation. From this point of view it would
appear that there is not (strictly) a "critical volume" of an assemblage of hard
spheres, but rather a sort of short range of volume throughout which this compara-
tively sudden change takes place. Thus the critical Isothermal may be regarded as
having (like those of lower temperature) a finite portion which is practically straight
and parallel to the axis of volume. That this conclusion is apparently borne out by
experimental facts (so far at least as these are not modified by the residual trace of
air) will be seen when we make the comparison.

In fact we might speak of a superior and an inferior critical volume, and the
portions of the isothermals beyond these limits on both sides may perhaps have
equations of the same form, but with finite changes in some at least of the constants.

Another source of a species of discontinuity in some, at least, of the constants
is a reduction of E to such an extent that grouping of the spheres into doublets,
triplets, &c., becomes possible. Thus we have a hint of the existence of a "critical
temperature."

LXXX.]

ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES.

201

It must be confessed that, while we have only an approximate knowledge of the
length of the mean free path (even among equal non-attracting spheres) when it
amounts only to some two or three diameters, we practically know almost nothing
about its exact value when the volume is so much reduced that no particle has a
path longer than one diameter.

[It might be objected to the equation arrived at above, should it be found on
comparison with experimeTit that a and 7 are both positive, that it will not make
p infinite unless v vanish. To this T need only reply that the equation has been
framed on the supposition that the particles are in motion, and therefore free to
move. What may happen when they become jammed together ia not a matter of
much physical interest, except perhaps from the point of view of dilataucy. If the
equation represents, with tolerable accuracy, all the cases which can be submitted to
experiment, it will fully satisfy all lawful curiosity.]

XXI. — Relation between Kinetic Energy and Temperature.

70. Before we can put the above virial equation into the nsual form of a relation
among jo, v, and t, it is necessary that we should consider how the temperature of
an assemblage of particles depends upon their average kinetic energy.

Yan der Waals and Clausius, following the usual custom, take the average kinetic
energy as being proportional to the absolute temperature. Clerk- Max well is more
guarded, but he says : — *' The assumption that the kinetic energy is determined by
the absolute temperature is true for perfect gases, and we have no evidence that
any other law holds for gases, even near their liquefying point/*

On this question I differ completely from these great authorities, and may err
absolutely. Yet I have many grave reasons on my side, one of which is immediately
connected with the special question on hand. To take this reason first, although it
is by no means the strongest, it appears to me that onli/ if E above (with a constant
added, when required, as will presently be shown) is regarded as proportional to the
absolute temperature, can the above equation be in any sense accumtely considered
as that of an laothermaL If the whole kinetic energy of the particles is treated as
proportional to the absolute temperature, the various stages of the gas as its volume
changes with E constant correspond tu changes of temperature without direct loss or
gain of heat, and belong rather to a species of Adiabatic than to an IsothermaL
Neither Yan der Waals nor Clausius, so far as I can see, calls attention to the fact
that when there are molecular forces the mean-square speed of the particles necessarily
increases with diminution of volume, even when the mean-square speed of a free
particle is maintained unaltered; and this simply because the time during which each
particle is free is a smaller fraction of the whole time. But when the wliole kinetic
energy is treated as a constant (as it must be in an Isothermal, when that energy
is taken as measuring the absolute temperature), it is clear that isothermal compression
must reduce the value ot E, It further follows that the temperature of a gas might
be enormously raised if its volume were sufficiently reduced by the process (capable
T. II. 26

202 ON THE FOUNDATIONS OF THE KINETIC THEOBY OF GASES. [lXXX.

of being carried out by Clerk-Maxwell's Demons) of advancing, at every instant, those
infinitesimal portions of the containing walls on which no impact is impending. This
is certainly not probable. If, on the other hand, we were to look at the matter from
the point of view of intense inter-molecular repulsion (such as, for instance, Clerk-
Maxwell's well-known hypothesis of repulsion inversely as the fifth power of the
distance, which was so enthusiastically lauded by Boltzmann), we should be led to
the very singular conclusion that such an assemblage of particles might possibly be
cooled even by ordinary compression; certainly that the Demons could immensely cool
it by diminishing its volume without doing work upon it.

If this mode of reasoning be deemed unsatisfactory, we may at once fall back
on thermodynamic principles; for these show that a gas could not be in equilibrium
if either external, or molecular, potential could establish a difference of temperature
from one region of it to another. For it must be carefully remembered (though it
is very often forgotten) that temperature-differences essentially involve the transference
of heat, on the whole, in one direction or the other between bodies in contact: —
80 that if there be a cause which can produce these temperature-differences, it is
to be regarded as a source of at least restoration of energy. Let the contents of
^ual volumes at different parts of a tall column of gas under constant gravity be
compared. In each the pressure may be regarded, so Sbu: as it is due to the external
potential, as being applied by bounding walls. But the temperature is the same in
each, and the only other quantity which is the same in each is E. For, as the
particles are free to travel from point to point throughout the whole extent of the
group, the average value of E must be the same for all; and, therefore, in regions
where the density is small, it must be that of free particles: — t.e., absolute temperature.

71. For the isothermal formation of liquid, heat must in all cases be taken from
the group. This must have the effect of diminishing the value of E. Hence, in a
liquid, the temperature is no longer measured by E, but by E+c, where c is a
quantity whose value increases steadily, as the temperature is lowered, from the value
aero at the critical point. Thus, since of course we must take the physical &ct of
the existence of liquids as a new datum in our calculations, and with it the agglo-
meration into doublets, triplets, &c. (whose share of the average energy differs in
general from that of their components when free), we see that the state of aggre-
gation which we call liquid is such that, as it is made colder and colder, a particle
which can escape frt>m it requires to have more and more than its average share of
the non-molecular part of the energy.

We might be tempted to generalise fruther, and to speculate on the limiting
conditions between the liquid and the solid states. But these, and a host of other
curious and important matters suggested by the present speculation, prominent among
which is the question of the density of saturated vapour at different temperatures
(with the mechanism of the equilibrium of temperature between the liquid and the
vapour), must be deferred to the next part of this paper. It is sufficient to point
out here how satisfactorily the present mode of regarding the subject fits itseLf to
the grand fiBM^ts regarding latent heat, and to its steady diminution as the pressure

LXXX.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. 203

ander which ebullition takes place is gradually raised to the critical value. What
we are called upon to do now is to justify, by comparison with experiment, the
hjrpothesis which we have adopted as to the proper physical definition of temperature,
and the form of the virial equation to which it has led us. If we have any measure
of success in this, we may regard the main difficulty of at least the elements of these
farther problems as having been to some extent removed.

What has been said above leads us, in the succeeding developments, to write (so
long at least as we are dealing with vapour or gas)

where t is the absolute temperature, and R (whose employment is now totally changed)
is practically the rate of increase of pressure with temperature at unit volume, under
ordinary conditions.

XXII. — The Equation of Isothermals.

72. Assuming the definition of temperature given in last section, the virial equation
of § 68 becomes

•^ \ v + aj t> + yv + a

For the minimax, which occurs at the critical point, we must have simultaneously

„ ^ dp , A- Ret C

But v-f- + p =

rfr -^ (v + ay (v + yy

^ dp_ A-Ret

dti*^ dv '' (v + ay^(v+yy

Denoting by a bar quantities referring to the critical point, these equations give

_ A-Rd C

0 =

(v + ay (v + fy
A-Ret C

{v + ay (v + yy

whence A-Rei=^^^±^. (7 = ^^^±^.

0—7 a— 7

26—2

204 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXX.

But the first equation of this section can be written as

By the help of the values of il — Rei^ and C, just found, and the further condition
that j^, v, ~i satisfy this general equation, we can easily put it in the form

i>=^(i- .i^r. 0+^(1 + 4-)— (C)-

-^ -^V v(« + a)(v + 7)/ V v + a/ v

There are seven constants in this equation : — viz., j5, v, \ a, 7, e, and R ; but there
•are two relations among them, one furnished by the usual condition that the gas
treated has unit volume at 0°C., and 1 atm.; the other (from the conditions of the
minimax) being

Su 4- a + 7 = ^=- .
V

73. If we compare (C) with the corresponding forms of the equations of Van
der Waals and Clausius ((A) and (B) of § 66 above) we see that all three agree in
a remarkable manner as to the form of the equation of the critical isothermal. In
fact, the only difference is that in (C) the divisor of (v — vf contains three distinct
£Eu;tors, while in each of (A) and (B) two of the three factors are equal. It is quite
otherwise with the term which expresses the diflference of ordinates between the
critical isothermal and any other of the series: — so that even if all three equations
agreed in giving the correct form of the critical isothermal no two of them could
agree for any other.

XXIII. — Comparison with Experiment.

74. We must now compare our formula with experiment. And here I have been
exceptionally fortunate, as the kindness of M. Amagat has not only provided me with
a complete set of values of pv in terms of p for CO, between the limits 1 to 1000
atm. and 0° to 100° C, but has further replied to my request for a set of values
of p, at different temperatures, for certain special values of v. This important table
I give in full, inserting columns of differences. It is very much better adapted than
the former to numerical calculation, as the form of the virial equation requires that
V should, for this purpose, be treated as the independent variable.

LXXX,]

ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES*

205

Presmre of COi in terms of Volume and Temperature (Amagat),

At 0° C. and I atm. the volume is unity. Alter the eiperimenta were cotDpleted the COj waa tested,
and left 0QO04 of its volume when aljsorbed by potash.

The interpolated columns oje diflferaDcea (or average differenoea, if in bracketi) of preasuii) for

10'' at eonstaut volume

Tol.

^02365 -oiBae

■013

'01

■00768

■00578

'00428 -00316

0035

*0Q2

•00187

844

344

307-6

7-4

"lO

418

44-4

...

».i

404

a-3

'ii-s

118

45^1

51-1

56-3

66^4

■ + ■

66-4

3O0

520

3-a

5 +

11-9

14-3

14 3

131

1073

48*3

56-5

62-8

68-3

70-7

71^5

109

384

627-5

37-4

56-4

64-1

73-7

74-6

49-9

67-6

65-8

72-6

773

79-5

84-7

5-i

8-3

12-4

171

Sfl-6

m-3

51'4

59^7

68'6

76-6

83-1

878

165

470-5

750

8-2

11 -e

170

27-5

891^

lOflS

40-9

64^5

63-8

74-6

84-8

94^7

104-8

125-3

201

560

8565

1-9

3-1

11-5

171

28*

»1

J*7

42-8

57-6

67-8

80'3

92-8

106-2

121-9

153^8

260-£

651

963-6

i-e

ni>

^■4

44-7

60-6

71-8

86*8

100-6

117-6

138-9

183-2

396-6

745

l-ft

ai»

ira

174

S8^

473

8§-S

46-6

63-6

75-7

91-3

108-2

i28^e

166-3

2U-5

346

832-5

3-9

5-4

11-4

17'2

48*&

83 3

4e^5

66-5

79-6

96-7

116

140-2

173*3

240-5

3945

918

3fl

4 0

U'l

I7d

305

100

60*5

69'5

83-6

102-3

123-8

151-3

191-1

271

443-5

998

[ITS]

(a-8j

[3-73

[fill

[7-2J

[lo^J

[l«-4]

(2SJ

[4(J-81

137-6

97-6

1216

151

191

262-6

376

619

tisi]

[4-81

137]

E53]

[711

[lO-t]

[1711

f29'4]

t48J

120

153^5

195

267

366

664

909

tltai

tas]

[aa)

(4^1

[flBl

[98J

[10-«J

78-5

112

140

181

234-5

316

4496

It is obvious, from a glance at the columiiB of differeDces, that the change of
pressure at constant volumei while the COs is not liquid, is almost exactly proportional
to the change of temperature, M. Amagat expressly warned me that the three last
temperatures in the table are only approximate, as they were not derived from air-
thermometers, but simply from the boiling-points of convenient substances^

They appear to indicate a slow diminution of dpjdt (tf constant) as the temperature
is raised above lOD^'C., but this is beside our present purpose*

Leaving them out of account, v^e find that in the range 31° to lOO^^C. the
fiuctuations of the changes of pressure per IC (at constant volume) are very small,
and do not seem to follow any law. These fluctuations besides are» especially when
the volume of the gas is small, well within the inevitable errors of observation in a
matter of such difficulty. Hence we take a simple average in each column \ and thus
we have the following table : —

206 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXX.

Average Change of Pressure per 10"* of Temperature a;t Constant Volume.
V -02386 01636 013 -01 00768 00578 00428 -00316 0025 -002 -00187

1-93

4-0 66

7-9

115

17-2

285

47^8 87-7

108?

vAp

•046

•049

•052 056

•061

•066

•074

•090

•120 175

•20?

Calc.

•046

•049

•062 056

•061

•068

•077

•087

•061

•073

•093

•122 175

•20

The numbers in the fourth row are the values of
and those in the fifth row are from

It is clear that these formulse give &ir approximations to the data, the first for
volumes down to 0*005 or so. the second for smaller volumes.

Comparing with formula (C) of § 72, we see that the values of R, Re, and a
are respectively

000371, 0000021, and 0001

for the larger volumes, and

000371, 0000011, and -0-0012

for the smaller. The values of y and v can now be determined by the relation in
§ 72, and a few experimental data. After a number of trials I arrived at

v = 00046,

as most consonant with the data for larger volumes; and I have provisionally assumed
the value

t; = 0004

for the lower range of volumes, in agreement with what was said in § 69 above as
to the probable existence of a short, horizontal, portion of the critical isothermal. The
value of 7 for the first portion of the curve is found to be 0*0008; and I have
assumed it to be —0*0008 for the rest, thus ignoring the condition for the minimax
at the commencement of this part of the curve. I consider this course to be fully
justified by the arguments given in § 69 above. Thus, taking from the assumption
below the value 73 atm. for the critical pressure, we arrive at the following equations
for the parts of the critical isothermal which lie on opposite sides of the short,
approximately straight, portion: —

P \ v(v + 0001) (w + 0-0008)/ '

J TO ft (v-0004)» \

*°^ P^^^V-viv- 00012)(« - 00008)j •

LXXX.] ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES.

207

In a careful plotting of the isothermals of CO^ from tiie whole of Amagat's data
(including, of course, those given above), I inserted, by means of diflFerences calculated
from the preceding formulae for dpjdt, the probable iiw^t hernial of 3F C, This is only
0°^1 higher than the critical temperature as given by Andrews, which is certainly a
little too low in consequence of the small admixture of air. The experimental data
in the following table were taken directly from the curve so drawn. They are, of
course, only approximate : — especially for the smaller volumes, for there the curves are
so steep that it is exceedingly diflScult to obtain exact values of the ordiuates for
any assigned volume. It is also in this region that the effects of the slight trace
of air are most prominent.

Appromrnate Isothermal of SV C.

The third line is calculated from the tirst of the above formuliBj the fourth line from the aecond*
V 1 024 -02 '015 0125 01 0075 006 005 0045 004 0035 003 0025 002

|>(exp,) 112 371 42-4 516 57 2 63 4 69*6 72 4 729 73 73 73 2 768 114 392
[1^3 37-2 42-5 5V4 57-0 63*3 69'6 72*3 72'95 73 73-16 74'4 79^6 96*4 149

730 73'2 791 117 6 377

p (calc,)

For volumes down to 0"0035 the agreement is practically perfect. The remainder of
the data, even with the second formula, are not very well represented* The value of
p for volume 0*003 has given much trouble, and constitutes a real difficulty which
I do not at present see how to meet. It is quite possible that, in addition to the
defects mentioned above, I may have myself introduced a more serious one by assum-
ing too high a value for the lower critical volume, or by taking too low a temperature
for the critical isothermal Had I selected the data for the isothermal of 31 ^"3 or
so, it is certain that (with a slight change in v) the agreement with the formula
would have been as good as at present for the larger volumes, and it might have
been much better for the smaller. But I have not leisure to undertake such tedious
tentative work. As it is, the formulas given above represent Amagat's results from
31 ** to 100° C. for volumes from 1 to 00035, with a maximum error of considerably
less than 1 atmosphere even at the smallest of these volumes. And, even with the
least of the experimental volumes, the approximations to the corresponding (very large)
pressures are nowhere in error by more than some 4 or 5 per cent. This is at
least as much as could be expected even from a purely empirical formula, but I hope
that the relations given above (though still extremely imperfect) may be found to
have higher claims to reception*

[Since the above was put in type it has occurred to me that this remarkable
agreement, between the results of experiment on a compound gas, and those of a
formula deduced from the behaviour of hard, spherical, particles, may be traced to
the &uit that the virial method is applicable, not only to the whole group of ptrticles

208 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASE8. [LXXX,

but (at every instant) to the ^ree particles, doublets, triplets, &c., in so far as the
interyicd relations of each are concerned. Hence the terms due to vibrations, rotations,
and stresses, in free particles, doublets, fee,, will on the average cancel one another
in the complete virial equation. How far this statement can be extended to particles
which are not quite free will be discussed in the next iDStaiment. 5/6/31.]

' [Some of the above remarks on Van der Waals* treatment of the virial equation
were objected to by Lord Rayleigh and by Prof. Korteweg. The correspondence will be
found in Nature (Vols. XLIV. and XL v., 1891—2). I quote here a few sentences of my
own which, had I been rewriting instead of merely reprinting my paper, might have
been m part at least incorporated in it.

** I had not eiamined with any particular care the ojjening chapters, to which your letter chiefly
refera; probably haviog supposed them to contain nothing beyond a sUttemenf and proof of the Virial
Theorem (with which I way alreatly familiar) along with a reproduction of a good deal of Lapliice's work

Of course yotu' account of this earlier part of the pamphlet (which 1 have now, for the first
time, read with care) is correct. But I do not see that any part of my statementa (with perhaps
the single except ion of the now italicized word in the phrase * the whoU procedure is erroneoiia ') Is
invalidated by it. No doubt» the sudden appearance of aj^ in the fonnula above quoted is, to some
extent at least, accounted for; but is the term correctly introduced?"

"I think that the mere fact of Van der ^^aals'a saying (in a passage which is evidently applicable
to his own proceijaeij, though it is applied only to that of Lorentz) *die gan^e Bechnung dock nur
bis auf Oroasen der ersten Ordnung (wie hji>) genau iat' throws very grave doubt on the whole
investigation. For in the most interesting part of the critical isDthermal of CO^ the fraction bfv
cannot be looked upon aa a small quantity of the first order* In lact, without raising the question,
either of Van der Waals^s mode of interpreting the term JS (w F^) or of the paucity of constants in
his equation, the above consideration would of it^nelf render the results untrustworthy. Van der WaaJa
has most opportunely and eftectivcly called attention to an exceedingly promising mode of attacking
a very difficult problem, and his methods are both ingenious and suggestive; but I do not think
that his results can be regard ed^ even from the most favourable point of view, as more than * GumstM
ai TruihJ

For, if we take the experimental test, there can be no doubt that (as I have stated in § 65
of my paper) *Van der WaaWs ciu'ves cannot be made to coincide with those of Andrews/ Atid
I think I have given reasons for believing that *the term of Van der Wiials's equation, which he
took to represent Laplace's -fi", is not the statical pressure due to molecular forces but (approximately)
its excess oyer the repulsion due to the speed of the particles,^ Of course I mean by this that,
when Van der Waals, comparing his equation with ei:j>eriment, assigns a numerical value to his
terra a/i'*, he is not justified in regarding it as the value of Laplace^s A"; though that quantity
was, he tell& us, the main object of hL^ inquiry J'

"I do not agree with Pttjf, Korteweg's statement that Van der Waals's method, if it oould bt
worked out with absolute rigour, would give the J*ame result sua the direct method* There iB but
one way of dealing with the virial equation : — if we adopt it at starting we mitst develop its terms
one by one, and independently. In this connection I may refer to Lord Rayleigh^ statement
(Mature^ 26/11/91): *It thim appears that, contrary to the assertion of Maxwell, p is subject to
correction.' I cannot admit that p is 'corrected*; it ie not even changed either in meaning or in
value. It is introduced as^ and remains (at the end of any legitimate transformations of the equation)
the value of the pressure on the containing vessel. This^ of course^ depends upon what is going on
in the interior* Other terms in the virial equation, which happen to have the same factor, may be
associated with p for convenience; they assist in finding its value, but they do not change its meaning,
nor do they * correct' it." 1899,]

LXXXI.]

2oy

Ijjfi^ AJlI .

ON THE FOUNDATIONS OF THE KINETIC THEORY OF

GASES. V.

(Abstract)
[Procmdirigs of t/te Ruyal Society of Edinburgh, February 15, 1892.]

The first instalment of this part of my paper deals mainly with the theory of
the behaviour of mixtures of CO. and N, for which some remarkable experimtmtal
results were given by Andrews about 1874. His full paper, so far as he had drawn
it up for press, was published posthumously in the PhiL Tram, for 1886, and ib
reprinted in his Scientijic Papm^St No. L. One special reason for the introduction of
this question at the preisent stage of my work was the desire to attempt a cor-
rection of Amagat's numbers, for the (very small) iid mixture of air with his COa.
The filial equation for a mixture is formed on the same general principle m that
I employed for a single gas. There are, of course, more undetermined constants: —
and, unfortunately ^ the data for their determination are barely adequate. The general
results, however, agree in character with those described by Andrews: — the particular
phenomenon which is most closely studied being the increase of volume^ at constant
pressure, when the gases (originally separated by the liquefaction of one) were allowed
to diflfuse into one another.

Since Part IV, of this paper was printed, M. Amagat has published {Comptes Rendus,
October 12, 1891) additional data of a most valuable character bearing on the iso-
tbermals of OOt; — especially the very important isothermal of 32"* C. ; and he has
given the pressure of the saturated vapour at 0"", IQ^, 20°, and 30** C. I have
endeavoured to utilise these, as far as possible, not only for my present main object : —
the examination of the relation between temperature and kinetic energy: — but alao,
incidentally, for the determination of the latent heat of the saturated vapour at
various temperatures, and the relative densities of the liquid and vapour when in
equilibrium. These data have also enabled me to obtain more exact approximations

210 ON THE FOUNDATIONS OF THE KINETIC THEORY OF GASES. [lXXXI.

to the values of the constants in my formula, and thence to improve my determin-
ations of the critical temperature, pressure, and volume.

In §71 of Part IV. I arrived at the conclusion that "in a liquid the temperature
is no longer measured by E [the part of the kinetic energy which is independent
of the molecular forces], but by E-\- c, where c is a quantity whose value increases
steadily, as the temperature is lowered, from the value zero at the critical point."
For numerical data to test this conclusion, I study a cycle formed from the critical
isothermal and any lower one, and two lines of equal volume, corresponding to those
of the liquid and the saturated vapour when in equilibrium at that lower temperature.
The change of energy in passing from one of these limits of volume to the other
is found to be less for the critical isothermal than for any lower one. Thus the
mean specific heat at constant volume, for the range of temperature employed, is less
in the vapour than in the liquid. But from the equation, which is found to satisf}'
very closely the data for the isothermals of the gas for some 70 degrees above the
<;ritical point and of the vapour for 30 degrees below that point, it appears that
the specific heat at constant volume is sensibly constant within these limita [At

100° C. and upwards, it appears that -2 falls oflF; so that ^ is negative, and the

specific heat at constant volume is therefore, even in the gas, greater for smaller
volumes. But this does not seriously affect the above statement.] Hence, at any
volume less than the critical volume, more heat is required to raise the temperature
1 degree when the substance is wholly liquid than when it is gaseous. This com-
pletely justifies the statement quoted above, provided that we assume the properties
of the liquid and gas to merge continuously into one another at the critical temperature;
but, unfortunately, the data are not sufficient to give more than very rough estimates
of the value of the quantity c there spoken o£

I am at present engaged in endeavouring to obtain more exact values of the
constants in my equation, in order to improve my estimates. Thus the numbers which
follow may have to undergo some modifications, but there seems to be no reason for
thinking that these are likely to be serious.

If Vi, v, be the respective volumes of the saturated vapour, and of the liquid,
at absolute temperature t, we know that the latent heat is expressed by the formula —

From Amagat's data I find for the values of this quantity, and for the ratio
of the densities of the liquid and vapour: —

Temperature C.

»,/»,

0°

4-369

9-023

3-788

6-200

2-882

3-823

1-460

1-906

lxxxl] on the foundations of the kinetic theory of gases. 211

Taking the density of COa at 0° C. and 1 atm. as 0002, it is easy to see that
the values of \ must be multiplied by

to reduce them to ordinary heat units. Thus the latent heat at 0° C. is about 53,
while at 30° C. it is only 178.

In the following table P represents the gain of energy from the liquid state to
that of saturated vapour, at the indicated temperature : — i.e.,

while <i=f^('i-Py'

is the corresponding gain of energy, in the critical isothermal, between the same limits
of volume.

Tompeiatnte C.
0"

P
3-747

Q
3-577

3-244

3-113

2-45!)

2409

1-233

1-203

The difference, P ~ Q, is (when multiplied, as above, by 12*2) nearly equal to the
excess of the heat required to raise the temperature of the liquid (at constant volume)
to the critical point, over that required to raise the temperature of the vapour, from
saturation, through the same range, the volume remaining unaltered.

It appears that COa, when passing through the range of volume spoken of in
§ 69 of Part IV. of my paper, has about half the density of water.

[The paper, of which the above is an Abstract, was never fully written out for
press. Further papers of M. Amagat soon led to (slight) modifications of the curves
which had been employed in my calculations; so that the numbers given above require
some change, and we have now the data necessary. I find, for instance, that I had
long ago noted (as an improved version of the last column of the table opposite) the
figures

9-52
6-45
4-08
1-79

which are very nearly the same as those given by M. Amagat in the Comptes Rendus
for March 13th of this year. 1899.]

27—2

212 [lxxxii.

LXXXII.

NOTE ON THE EFFECTS OF EXPLOSIVES.

[Proceedings of the Royal Society of Edinburgh, February 21, 1887.]

Many of the victims of the dynamite explosion, a year or two ago, in the London
Underground Railway, are said to have lost the drum of one ear only, that nearest to
the source. This seems to point to a projectile, not an undulatoryy motion of the air
and of the gases produced by the explosion. So long, in fact, as the disturbance
travels faster than sound, it must necessarily be of this character, and would be capable
of producing such effects.

Another curious fact apparently connected with the above is the (considerable)
finite diameter of a flash of forked lightning. Such a flash is always photographed
as a line of finite breadth, even when the focal length is short and the focal adjust-
ment perfect. This cannot be ascribed to irradiation. The air seems, in fact, to be
driven outwards firom.the track of the discharge with such speed as to render the
immediately surrounding air instantaneously self-luminous by compression.

Such considerations show at once how to explain the difference between the effects
of dynamite and those of gunpowder. The latter is prepared expressly for the purpose
of developing its energy gradually. Thus while the flash of gunpowder fired in the
open is due mainly to combustion of scattered particles, — that produced by dynamite
is mainly due to impulsive compression of the surrounding air, energy being conveyed
to it much faster than it can escape in the form of sound.

lxxxiil] 213

LXXXIII.

ON THE VALUE OF A«0~/n~, WHEN m AND n AEE VERY

LARGE.

[Proceedings of the Edinburgh Mathematical Society, Vol. v., 1887.]

I HAD occasion, lately, to consider the following question connected with the Kinetic
Theory of Osaes : —

Given that there are 3.10^ particles in a cubic inch of air, and that each has
on the average 10^^ collisions per second ; after what period of time is it even
betting that any specified particle shall have collided, once at least, with each of
the others?

The question obviously reduces to this: — Find m so that the terms in

which contain each of the n quantities, once at least, as a &ctor, shall be numerically

equal to half the whole value of the expression when Xi^X2= =a^=sl. Thus

we have

Z"»-2(Z-a?^)"» + 2(Z-a?,-a?.)~- =iZ"»

or A«0^/n"*=i.

It is strange that neither Herschel, De Morgan, nor Boole, while treating
differences of zero, has thought fit to state that Laplace had, long ago, given all that
is necessary for the solution of such questions. The numbers A"0^ are of such im-
portance that one would naturally expect to find in any treatise which refers to them
at least a statement that in the Thiorie Analytique des Prohabilitis (livre IL, chap, n.,
§ 4) a closely approximate formula is given for their easy calculation. No doubt the

214 ON THE VALUE OF A'*0"*/n'*, WHEN m AND n ARE VERY LARGE. [lXXXIII.

process by which this formula is obtained is somewhat difficult as well as troublesome,
but the existence of the formula itself should be generally known.

When it is applied to the above problem, it gives the answer in the somewhat
startling form of "about 40,000 years." [Ante, No. LXXVIIL, p. 166. 1899.]

P./S. — April 4, 1887. — Finding that Laplace's formula ceases to give approximate
results, for very large values of m and n when these numbers are of the same order
of magnitude, I applied to Prof. Cayley on the subject. He has supplied the requisite
modification of the formula, and his paper has been to-night communicated to the
Royal Society of Edinburgh. [Cayley*8 Mathematical Papers, Vol. XIL, No. 853. 1899.]

Lxxxiv.] 215

LXXXIV,

NOTE ON MILNER'S LAMP.

[Proceedings of the Edinburgh Mathematical Society, Vol. v., 1887.]

This curious device is figured at p. 149 of De Morgan's Budget of Paradoaes,
where it is described as a "hollow semi-cylinder, but not unth a circular curve*'
revolving on pivots. The form of the cylinder is such that, whatever quantity of oil
it may contain, it turns itself till the oil is flush with the wick, which is placed
at the edge.

Refer the " curve'* to polar coordinates, r and 0] the pole being on the edge,
and the initial line, of length a, being drawn to the axis. Then if 0o correspond to
the horizontal radius vector, /8 to any definite radius vector, it is clear that the
couple due to the weight of the corresponding portion of the oil is proportional to

[ r»dd|acosdo-Qrcos(d-^o)|.

This must be balanced by the couple due to the weight of the lamp, and of the oil
beyond /8; and this, in turn, may be taken as proportional to

cos (a + do).
Thus the equation is

acosdoj T^de-^Uosdoj r»cosdd^ + sindof r» sin dd^^ = 6» cos (a + ^o).

Differentiating twice with respect to do, and adding the result to the equation, we
have (with 0 now put for 0q)

2ar»sin d- 2ar ^ cos^ + 2r«^ = 0.

216 NOTE ON MILNBB's LAMP. [lXXXIV.

Rejecting the &ctor r, and integrating, we have

r»=2arco8d + a
This denotes a circular cylinder, in direct contradiction to De Morgan's statement!

As it was clear that this result, involving only one arbitrary constant, could not
be made to satisfy the given differential equation for all values of 6, a, and fi, I fancied
that it could not be the complete integral I therefore applied to Pro£ Cayley, who
favoured me with the following highly interesting paper. It commences with the
question I asked, and finishes with an unexpectedly simple solution of Milner's
problem. [Caylej/s Mathematical Papers, Vol. xiiL, No. 889. 1899.]

It appears clear that De Morgan did not know the solution, for the curve he has
sketched is obviously one of continued curvature — and he makes the guarded state-
ment that a fiiend "vouched for Milner's Lamp."

Lxxxv.] 217

LXXXV.

AN EXERCISE ON LOGARITHMIC TABLES.

[Proceedings of the Edinburgh Mathematical Society, Vol. v., 1887.]

In reducing some experiments, I noticed that the logarithm of 237 is about 2*37... .
Hence it occurred to me to find in what cases the figures of a number and of its
common logarithm are identical : — i.e., to solve the equation

logioa? = a?/10*",
where m is any positive integer.

It is easy to see that, in all cases, there are two solutions; one greater than, the
other less than, e. This follows at once from the position of the maximum ordinate
of the curve

y = {\ogx)jx.

The smaller root is, for m = l, a? = l'37l288

m = 2, a: = 1023855

For higher values of m, it differs but little fi:om 1, and the excess may be calculated
approximately fixjm

y- y«/2 + ... =(1 +y)log. lO/lO"*.

Ultimately, therefore, the value of the smaller root is

100 0230258

where the number of cyphers following the decimal point is m — 1.

T. II. 28

218 AN EXERCISE ON LOGARITHMIC TABLES. [lXXXV.

The greater root must have m-^-p places of figures before the decimal point ;
p being unit till m = 9, then and thenceforth 2 till m = 98, 3 till m = 997, &c.* Thus,
for example, if m > 8 < 98 we may assume

a- = (m+l)10'"-f y,

so that log., "^ + log„|l + ^^/i)io4 = ll» •

which is easily solved by successive approximations.

But it is simpler, and forms a capital exercise, to find, say to six places, the greater
root, by mere inspection of a good Table of Logarithms.

Thus we find, for instance.

182,615.

,10"

192,852

.10"

979,911.

,10"

989,956.

,10"

* [It is easy to see that the indices, of the integral powers of 10 which satisfy the original equation,
are themselves of the form 10*, where q is snoh that

m=10«-g.
Thos, with 9=0, we have 10 itself as the greater root when m=l. 1899.]

Lxxxvi.] 219

LXXXVI,

ON GLORIES.

[Proceedings of the Royal Society of Edinburgh, Jidy, 1887.]

When Mr Omond was appointed to the Ben-Nevis Observatory I requested him
to take every opportunity of observing what are called Glories — especially noting, when
possible, their angular diameters and the order of their colours, so that it might be
possible to decide upon the exact mode in which they are produced.

Young, while attributing to their true cause the spurious (or supernumerary)
rainbows, proceeds to say: — "The circles, sometimes seen encompassing the observers
shadow in a mist, are perhaps more nearly related to the common colours of thin
plates as seen by reflection." — [Lectures, ii. p. 645.]

Now from Mr Omond's observations it appears that the mists to which the
glories are due produce coronae of, say, 2"* or 3° radius; — from which it follows that
the diameter of the particles is somewhere of the order y^ inch. It is thence
shown that, were Young's explanation correct, the radii of the rings would varj*
with great rapidity in passing from one kind of homogeneous light to another. This
is altogether irreconcilable with Mr Omond's observations.

That the glories are not of the nature of spurious rainbows is shown very simply
by the fact that they are more intense as their radii are smaller.

Hence, the only possible explanation is diffraction depending on the form of the
vertex of the reflected wave. The form of an originally plane wave, once reflected
inside a drop of water is, roughly, when the central ray has just emerged, a portion
of an hyperboloid of revolution, doubled back cusp-wise round its border. An
approximate calculation is given, based on this assumption.

28—2

220 ON GLORIES. [lXXXVI.

A simple first approximation to the theory of glories is given by the behaviour

of a plane wave incident normally on a screen pierced with a great number of very

small circular apertures of nearly equal size. They are thus, to a certain extent,
analogous to coronae.

APPENDIX.
On Mr Omond's Observations of Fog-Bows.

[Proceedings of the Royal Society of Edinburgh, January, 1888.]

The author remarked that one of the constituents of the double fog-bow described
in some of Mr Omond's recent observations*, is obviously the ordinary primary rainbow,
diminished in consequence of the very small size of the water drops. But the other, having
nearly the same radius but toiih its coUywrs in the opposite order, appears to be due to
ice-crystals in the fog. This is quite consistent with the record of temperatures. Just as
small drops of water may remain uufrozen in air below 0"* C, small ice-crystals may remain
unmelted at temperatures above that point.

• Proceedings R.S.E. xiv, p. 814.

Lxxxvii.] 221

LXXXVII.

PKELIMINARY NOTE ON THE DURATION OF IMPACT.

[Proceedings of the Royal Society of Edinburgh, Feb. 20, 1888.]

The results already obtained were got by means of a roughly made apparatus
designed for the purpose of testing the method used, so that only a single instance,
to show their general character, need now be given. When a wooden block of 10 lbs.
mass fell through a height of 18J inches on a rounded lump of gutta-percha, the
time of impact was found to be somewhere about 0*001 sec., and the coefficient of
restitution was 0*26.

As the principle of the method has been found satisfactory in practice, new
apparatus is in course of construction, which will enable me to use a fall amounting
to 10 feet at least. It is proposed to make a series of experiments on different
substances, with great varieties of mass and of speed in the impinging body.

222 [lxxxviii.

LXXXVIII.

ON IMPACT.

[Transactions of the Royal Society of Edinburgh, ToL xxxvl Revised Nov. 8, 1890.]

The present inquiry is closely connected with some of the phenomena presented
in golf: — especially the fauct that a ball can be "jerked" nearly as £eu- as it can be
''driven." For this, in itself, furnishes a complete proof that the duration of the
impact is exceedingly short. But it does not appear that any accurate determination
of the duration can be made in this way. Measurements, even of a rude kind, are
impracticable under the circumstances.

In 1887 I made a number of preliminary experiments with the view of devising
a fonn of apparatus which should trace a permanent record of the circumstances of
impact. I found that it was necessary that one of the two impinging bodies should
be fixed: — at least if the apparatus were to be at once simple and manageable.
This arrangement gives, of course, a result not directlj- comparable with the behaviour
of a golf-ball. For pressure is applied to one side only, both of ball and of club;
but when one of two impinging bodies is fixed it is virtually struck simultaneously
on both sides. Even with the altered conditions, however, the inquiry seemed to be
worth pursuing. I determined to operate, at least at first, on cylinders of the elastic
material; so fixed that considerable speed might be employed, while the details of
Keveral successive rebounds could be recorded. It is not at all likely that this will
be found to be the best form for the distorted body; but it was adopted as, in
many respects, convenient for preliminary work. For the main object of the experi-
ments was to gain some information about a subject which seems to have been left
almost entirely unexplored; and it is only by trial that we can hope to discover
the best arrangement Messrs Herbertson and Tumbull, who were at the time Neil-
Aniott Scholars, and working in my Laboratory, rendered me great assistance in these

lxxxviil]

ON IMPACT.

preliminary tnald, whose result was the construction of a first rude apparatus on the
following plan.

A brick -shaped block of hai*d wood wajs dropped endwise from a measured height
uj^KJu a i^hort cylinder of cork, vulcanized india-rubber, gutta-percha, &c.j which was
imbedded to half it-s length in a mass of lead, firmly cemented to ao asphalt floor.
The block slid freely between guide-rails, precisely like the axe of a guillotine* In
front of the block was a massive fly-wheel, fitted on one end of its axle, and carrying
a large board (planed true) on which was stretched, by means of drawing-pi na, a
sheet of cartridge-paper. The sheet was thus made to revolve in its own plane, A
pencil, projecting from the block, was caused by a spring to press lightly upon the
paper; and it was adjusted so that its plane of motion pa^taed as exactly as possible
through the axis of the paper disc. To prevent breakage of the pencil on the edge
of the disc, it was pushed into its bearings, and released by a trigger only after it
had, in its fallj passed the edge. The block, having fallen, i*ebounded several times
to i^apidly diminishing heights and, after a second or two, came to rest on the eork
cylinder. The pencil then tmced a circle aud, as soon as this was complete, the fly-
wheel (previously detached from the gas-engine) was at once stopped by the application
of a very powerful brake. The circle thus described was the datum line for all the
subsequent measures; since the tracings which passed beyond it were obviously made
during the impact, while those within it referred at legist mainly to the comparatively
free motion between two successive impacts. The dui-ation of the impact was at once
approximately given by the arc of the cii*cle intercepted between the tracings of the
pencil as it passed out and in, combined of course with the measured angular velocity
of the fly-wheel It is not yet known at what stage during the recovery of form
the impinging bodies go out of contact with one another. In the present paper we
are content to assume that contact commences and terminates at the instants of
passage across the datum circle. This is certainly not rigorously true as regards the
commencement, but the assumption cannot introduce any serious error; while of the
temiination we have no knowledge. It may be remarked, in passing, that the error
at commencement will necessarily be greater the larger the mass of the tailing body.
It will also be greater for soft than for hard bodies, and especially for those of the
former class which most depart from Uooke's Law,

In the winter 1887-8, and in the subsequent summer, some very curious results
were obtained by Messrs Herbertsoii and TurnbuU with this rough apparatus* Several
of these were communicated to the Society at the time when they were obtained.
Thus, for instance, it was found that although the mass of the block was over 5 lbs.,
the time of impact on a cork cylinder was of the order of 0"'01 only, while with
vulcanite it was of the order 0**00L Also, for one and the same body, the duration
was hss, the more violent the impact, [The golf result mentioned above was now
at once explained ; for, as the mass of a go If- ball is less than ^ of that of the
block, under equal forces its motions will be fifty times more rapid. Thus, even if it
were of cork, the time of impact would be of the order of about one five-thousandth
of a second only; and the shorter the more violent the blow,] Taking the coefficient
of restitution as 0"5 on the average, the time-average of the force during impact after

224 ox IMPACT. [lxxxviii.

a fieill of 4 feet was, for these classes of bodies respectivelj, of the orders 400 lbs.
weight and 4000 lbs. weight This result is of very high interest from many points
of view.

The values of the coefficient of restitution (or impacts of different intensity were
obtained by drawing tangents to the fidl-curve at its int^sections with the datum
circle corresponding to the assumed commencement and end a( each impact, and
finding their inclination, each to the corresponding radius of the circle. The co-
efficient of restitution is, of course, the ratio of the tang^its of these anglea The
results of these graphical methods could easily be checked by forming the polar
equations of the various branches of the fidl-curve (ascending and descending) and
obtaining the above-mentioned tangents of angles by direct differentiation. If we
assume the friction (whether of rails or pencil) to be approximately constant, it is
easy to see that the equation of the part of the tracing made during a fall, or
during a rise, can be put in the very simple finrm

r = 4 + £^.

Here the centre of the disc is the pole, and the initial line is the particular radius
which was vertical when the block was at (me of its successive highest positions.
This radius separates the rise, fit>m the £EdI. part of each branch of the curve. A
is of course the same for both parts, but B (being directly as the acceleration of
the block, and inversely as the square of the angular velocity of the disc) is larger
for the rise than for the fall; because friction aids gravity in the ascent and acts
againitt it in the descent. A number of sets of corresponding values of the polar
corirdinatcN were measureil on each part of the curve, the angles being taken from
an lipproximately assumed initial line. Three of these sets determined A, B, and the
irna \Hm\iUni of the initial radius; and the others were found to satisfy (almost
i^xwaily) th() oqiiation thus formed. This shows that the assumption, of friction nearly
(umnitiui throughout tho whole trace, is sufficiently accurate. B is always positive in
th« ^tquiitioh, hut A Ih negative or positive according as the block does, or does not,
rfi\f(tui\(\ to u hoight grt)Utor than the radius of the datum circle.

It \H not noeowiary to tabulate here any of the very numerous results of these
mr\i$^r <iX|ioriiuontii, While the work ^\'as in progress many valuable improvements of
th^ ft|i|mmtUN HUggoHtod themsolvos, and I resolved to repeat the experiments after
thtiHtt Uiul hotin intiHHhuHHl The whole of these subsequent results are tabulated
Ul(/W, Tlio fi»ll<»wiug wort^ found to be the chief defects of the earlier arrangement,
mt fM' ttl. Imi^i WW tht»y woiv not absi>lutely inherent in the whole plan. These have
h^u NifMMi rMiniidioil; and iwults obtained with the improved apparatus have been,
trfffii lIlfM* Ui liino, <u»unuuuioatod to tho Society.

I Tlin lino of u \mm\ i« objectionable from many points of view. Serious worry
Mt4i itiiU^U lotti» nf tiiun aro iuourrtHl in consequence of the frequent breaking of the
Utt$4t, i<v*«M wliMM fivory poHHiblo pivoaution seems to have been taken. Then the
rnpUi wtiM'Um iliiwii of tho juunt by the cartridge-paper causes the later-traced portions
fft fifptiU tUnKmiu (IholiHlihK tmpooially tho datum circle, which is of vital importance)
ht hi Hrttwn in browl linni*, whono exact point of intersection can be but roughly

LXXXVIII

ON IMPACT.

225

guesstKl at. The friction, also, was (mainly on account of the roughness of the paper)
ao large that the yalues of S, for the oacending and descending parts of any one
branch of the curve, differed from the mean by a large fraction of it^ sometimes as much
as 20 per cent This is approstiniately the ratio which the acceleration due to friction
bears to that due to gravity; m that the friction was, at least occasionallyp as much
as one pound weight. This, of course, seriously interfered with the accurate measure
of the coefficient of restitution. Instead of the board and cartridge- paper I intro-
duced a specially prepared disc of plate-glasa, which ran perfectly true. It was
covered uniformly with a thin layer of very fine printers' ink, which was employed
wet. For the pencil was substituted a needle-point, so that this part of the apparatus
was rendered exceedingly lights strong, and compact. The lines traced could easily
be made as hne as those of an etching, but it was found that a slightly blunted
point (giving a line of about 0'005 inch in breadth) produced probably less friction,
at all events less irregularity, than did a very sharp one. The diflference of either
value of B from the mean rarely amounted to more than 1*5 or 2 per cent, of the
moan. When the ink was dry, which happened after about a day, photographic printi^
were taken by using the disc as a negative. [In the later experiments it was found
that, when proper precautions were taken, no delay on this account was necessary.]
To test whether the paper of the positives had been distorted, in drying after tixing,
a number of circles were described on the glass disc at various places before the ink
was dry. They were found to remain almost exactly circular on the dried photograph.
All the subsequent measurements were made on these photographs. In a subsequent
paper I hope to give the results of careful micrometric measures, made on the glass
plate itself, of the form of the trace during impact. This may lead to information
which could not be derived from the photographs themselves with any degree of
accuracy. My first object was to obtain a number of separate experiments, so as to
get the general laws of the phenomena^ and for this purpose the glass plate had to
be cleaned and prepared for a new series of experiments as rapidly as possible. The
micrometric measui-es cannot be effected in a short time.

2. In the earlier experiments the fly-wheel continued in connection with the
gas-engine until the fall was completed. Hence the rate of rotation was irregular,
and the mode adopted for its measurement gave an average value only. In the later
experiments an electrically- con trolled tuning-fork, furnished with a short bristle, made
its record on the disc, simultaneously with the fall of the block; and the gas-engine
belt was thrown on an idle pulley immediately before the experiment commenced.
The angular velocity of the disc was sensibly different in different experimeota,
according as the engine was thrown off just before, or just after, an explosion. But
the fact that its fly-wheel is a gigantic one made these differences of small importance.
They were, however, always taken account of in the reductions. The disc, when left
to itself, suffered no measurable diminution of angular velocity during a single turn.
In the earlier experiments one rotation of the disc occupied about 0**3; but I was
afraid to employ so great a speed with the glass plate^ so its period was made not
very different from one second. I found it easy to obtain on the glass disc the
records of four successive falls^ each with its series of gradually diminishing rebounds,

T. n. 29

226 ON IMPACT. [lxxxviii.

and along with these the corresponding serrated lines for the tuning-fork. These
records were kept apart from one another by altering the position of the fork, as
well as that of the needle-point on the block, immediately after each fall. The latter
adjustment alters, of course, nothing but the radius of the datum-circle, and the
corresponding values of the quantity A. As soon as the four falls had been recorded,
the glass disc was dismounted, and all the necessary details of the experiment — e.ff,,
date, heights of fell, substance impinged on, mass of block, &c. — were written (backwards)
on the printers* ink, with a sharp point, and of course appeared on the photograph.
The changes of mass, just alluded to, were occasionally introduced by firmly screwing^
on the top of the block a thick plate of lead of mass equal to its own.

3. A very troublesome difficulty was now and then met with, but chiefly when
the elastic substance employed was a hard one, such as vulcanite or wood. For the
block was occasionally set in oscillation during the impact, and especially at the
instant when it was beginning to rebound. The trace then had a wriggling or wavy
outline, altogether unlike the usual smooth record. Sometimes the wriggle took place
perpendicularly to the disc, and the trace was then alternately broadened and all but
evanescent. After some trouble I found that the main cause was the slight dent
(produced by repeated falls on hard bodies) in the striking part of the block, which
had originally been plane. The wriggling always appeared when this dent did not
fit exactly upon the (slightly convex) upper end of the hard cylinder. To give free
play at the moment of impact, the lower part of the guide-rails had been, by filing,
set a very little further apart than the rest, and thus small transverse oscillations
of the block were possible. I hope to avoid this difficulty in future, by fixing a
hard steel plate on the striking part of the block, and making all the remaining
experiments with this. Of course a few of the former experiments must be repeated
in order to discover whether the circumstances are seriously, or only slightly, modified
by the altered nature of the striking surface. There can be no doubt that the
distortion, as tabulated, belongs in part to each of the impinging bodies; but it is
not easy to assign their respective shares.

The general nature of the whole trace of one experiment will be obvious from
the upper figure in the Plate, which is reduced to about 0*3 of the actual size. The
lower figures (drawn full size) show the nature of the trace during impact: — the first
series, some of which exhibit the "wriggles" above described, belonging to the pencil
records of the old apparatus; the second series containing some of those obtained
with the improved form just described.

In the earlier work, with the cartridge-paper, falls of 8 and even of 12 feet
were often recorded. The results of the later work have been, as yet, confined to
fells of 4 feet at most. But I intend to pursue the experiments much further, after
fitting an automatic catch on the apparatus; such as will prevent the block fit>m
descending a second time if it should happen to rebound so far that the needle-point
leaves the glass disc.

What precedes is of course designed to furnish only a general notion of the
nature of the apparatus, the principle on which it works, and the results already
obtained with it. Some further remarks, on the physical principles involved, will be

lxxxviil]

ON IMPAOT.

227

made after details of dimensions, and of numerical data have been given. But it
must be stated here that with the later form of the apparatus it was found necessary
to have a party of at least three engaged in each experiment ; one to attend to the
driving-gear, a second to the falling block, and a third to the tuning-fork. My assist-
ant, Mr Lindsay, took the first post; 1 usually took the second myself; and the
fork was mauaged by Mr Shand, to whom I am besides indebted for the greater part
of the subsequent measurements and reductions. These, of course, involved an amount
of work which, though not perhaps more difficult than the rest, waa incomparably
longer and more tiresome.

Descriptioii of the Apparatus.

Two beams nearly 12 feet long, and 6 inches by 2^- inches cross section, are rigidly
fixed, vertically, and at a distance of 8^ inches from each other, to a massive stone
pillar. To them the rails, which act as guides for the falling body, are screwed, the
distance between them being 6f inches- At the base, between the rails, is a cylinder
of lead, 6 inches by 6 inches, firmly imbedded in a mass of concrete, and having on
its upper end a hole, | inch deep and 1{ inch diameter, for holding the lower end
of the substance experimented on. This consists of cork, india-rubber, vulcanite, &c.,
m the case may be, cut into a cylinder, l{ inch diameter, and 1{ inch long, with
the lower end flat and the upper slightly rounded. It thus projects about | inch
after being thrust home into the hole in the leaden cylinder, in which it rests on
a thin disc of gutta-percha. This was found etfectualiy to prevent the cylinders
being displaced hx the lead*block. Before it was introduced, the cylinder was occasion-
al ly left not in contact with the bottom of the hole, so that the record of the next
impact was vitiated. Sometimes, indeed, the cylinder had jumped entirely out of the
hole before the block red esc ended.

In a plane, parallel to that which contains the guides and nearly 2| inches from
it, a massive fly-wheel, 28| inches diameter, whose moment of inertia is 102"6 in lbs*
aq, ft., is placed. The iron frame supporting it is fixed to the concrete floor by means
of bolts, so that the whole can be rigidly fixeci in position or lifted back at pleasure
A thick wooden board is firmly attached to the front of this wheel, and on it is
laid a sheet of felt. On the top of the felt, an octagonal plate of glass, about {
inch thick, the edges of which are bevelled, is placed, and then firmly pressed to the
board by means of bevelled metal plates, covered with felt, and screwed down on
four alternate edges.

The mass of the glass is 28 lbs,, its moment of inertia 25*21. For the wood
these are 21*5 and 2419 respectively. The total mass (including the fly-wheel) being
12*2^5 lbs., A^ is found to be about 1'24 sq. ft

A rope passing up the outside of one of the beams, over two small pulleys, and
down between the rails, serves to raise and lower the block, next to be described,
or to keep it suspended by a hook at any desired height. A cord running paj*allel
to the rope is attached to the catch of the hook at the end of the rope, so that
by pulling this cord the hook is tilted and allows the block to fall.

29—2

228

ON IMPACT.

[lxxxviii.

The block is rectangular, and formed of hard wood (plane-tree along the grain),

IlJ by 7| by 2^ inches, weighing 5^ lbs. Down the centre of each of the edges runs
a deep groove, at the ends of which pieces of iron with a polished groove of U
section are screwed on. It is on these that the guides bear while the block is falling.
The guides and Us being well oiled, the friction is reduced to a minimum.

A brass plate, 5^ inches by £| inches, is sunk into the face of the block about
J inch, and through the plate and wood a longitudinal slot, 3 inches by | inch, is
cut, the centre of the slot coinciding with the centre of the block. Another plate
of brass, 3J inches by 2\ inches, with two parallel slots 2| inches long and | inch
broad, half an inch distant troin, and on either side of the centre, lies on the fixed
plate, and can be clamped to it by means of flat- headed screws passing through the
slots. This movable plate has, therefore, a longitudinal (vertical) play of about 2 inches
when the screws are loose. It carries the tracing- point and its adjusting mechanism.

The tracing-point is at the extremity of a steel rod, one inch of whose length
is of J inch diameter, the remaining | inch being of rather less than ^ inch diameter.
The thicker part works freely, but not loosely^ in a cylindrical barrel, the thinner
part parsing through a collar at the front end* The cylinder is fixed, at right angles,
to the movable brass plate, and passes through the slot in the block. The rod is
lightly pressed forwards at the thicker end by a piece of watch-spring, so as to keep
it, when required, steadily in contact with the revolving disc. In the wall of the
cylindrical barrel is a long slot which runs backwards for ^ inch parallel to the axis,
and then, turning at right angles to its former direction, runs through a small fraction
of the circumference of the barrel. In this slot works a stout wire screwed perpen*
dicularly into the rod which carries the tracing-point. Of course when this wire is
in the transverse part of the slot the needle-point is retracted; but as soon as it
is turned into the axial part the spring makes the needle-point project through the
collar. Before the block falls, the wire is in the transverse part of the slot, and the
needle-point is retracted. But when, in its fall, the point has passed the edge of
the glass disc, a pin fixed at the proper height catches the end of the wire and
turns it into the axial slot. As soon as the tracing is complete, the wire is forced
back (by means of a system of jointed levers) into the transverse slot, and thus the
tracing- point is permanently withdrawn from the disc, so that the block can be pulled
up, and adjusted for another falL

The last part of the apparatus to be described is that for recording the time*

It consists of an electrically controlled tuning-fork, making 128 vibrations per second,
A circular bar of iron, 8 inches long, is fixed pei'pendicutarly to one of the beamsj
and in the plane of the beams. From this the tuning-fork is suspended by means
of circular bearings. It therefore has a swinging motion perpendicularly to the disc,
as well as a translatory motion parallel to it. By means of a screw it can be fixed
in any position, and to any degree of stiflFness. The bar is at such a height that
the end of the tuning-fork carrying the tracing-point is in the same horizontal plane
with the centre of the revolving glass plate. By this means it can be adjusted to
trace its record anywhere between the edge of the plate and a circle whose radius
IS 5 or 6 inches, measured from the centre of the glass.

LXXXYm,]

ON IMPACT,

229

Theory of ike Experifnents.

So far as coBcems the motion of the block between two successive impacts, the
investigation is extremely simple. For we assume (in fair accordance with the results,
ad shown above) that the friction is practically constant. Thus the motion of the

block is represented by

Mr = Mg±F,

the positive sign referring to upward motion.

We have also, taking the angular velocity, &>, of the disc as uniform throughout
the short period of the experiment,

d0 = mdt

Thus

de*'

^(si J)/«'-2fl, say;

so that

r = A + Be^,

if we agree that 8 is to be measured in each case from the particular radius which
is vertical at the moment when the block is at one of its highest positions.

If our assumptions were rigorously correct, the equations of those branches of the
curve which are traced during each successive rise of the block should differ from one
another solely in the values of the constant A. Similarly with those traced during
successive descents. The ascending and descending branches of the same free path
should differ solely by the change of value of B, according as the friction aids, or
opposes, the action of gravity. Also the two values of B should differ from their
mean by a smaller percentage the greater is the mass of the block. This, however,
will be necessarily true only if the friction be independent of the w^eight of the block »

As a test of the closeness of our approximation, to be applied to the experimental
results below, it is clear that, if we call B^ the mean of the values of B for the
parts of the curve due to any one rebound, we have

But, in the notation of the Tables as explained in the next section, we have

Taking the value of ^ as 32 2 when a foot is unit of length, it is Ml 4 to

millimetres; and the two equations above give the following simple relation between
B, and N

B,^^N\

which is sufficiently approximate to be used a» a test, the fraction being in defect
by about 0'14 per cent, only, say 1 /700th.

230 ON IMPACT. [lxxxviii.

Thus, in the first experiment of those given below for date 23/7/90, we have

i\r= 21-25,
which gives as the calculated value

5o = 12316; or, with l/700th added, = 123-33.

The actual value, as given by the equations for the two parts (/8i, ff^) of the first

rebound, is

i(125-73 + 120-81) =123-27,

the difference being less than 0-05 per cent. In this case the acceleration due to
friction bears to that of gravity the ratio

2-46:123-27;
almost exactly 2 per cent.

From the data (71, 7,) for the second rebound we find the actual value of B^ to be

i (131-31 4- 121-46) = 126-33 ;

and the percentage of acceleration due to Motion rather less than 4. As the whole
rise in this second rebound was considerably less than an inch, these results are
highly satisfactory.

It is a fairer mode of proceeding, however, to calculate the value of N from
that of Bq, by means of the above relation The values, thus calculated, are inserted
in the tables below, in the same column as the measured value of N, with the
prefixed letters 13, 7, &c., to show from which rebound, the first, second, &c., they
have been calculated. These agree in a very satisfactory manner with the value of
N given by the record of the tuning-fork.

From the facts, that the time of impact is nearly the same for all small dis-
tortions, and that it diminishes rapidly as the distortion is greater, it follows that
the equation of motion must be of the form

during the first stage of the impact; and of approximately the same form, but with
the square of the coefficient of restitution as a fiskxtor of the right, during the second
stage. In this equation x (which is confined to positive values) is measured from
the datum line, so that no term in g comes in explicitly. X is a function of x,
which is small for small values of x, but increases faster than does the first power
of X for larger values. Hence, for small relative speeds, the time of compression is

TT /M

2y C

and that of rebound 1/e times as much. The utmost distortion is

^ y c^ c

where V is the speed at the datum line. The first term is due to the fall; the
second, which is due to the weight of the block, does not appear in our Tables, as

LXXXVIIl]

IMPACT.

281

the measures are made from the datum line. Its value, however, is usually only a
small fmction of that of the first term.

To compare the distortion with the duration of impact in experiments made with
the same mass, falling from dififerent heights, the following equation was tried : —

^ = — n'x —

2ct'

where the numerical factors are introduced for convenience. This assumes X, above,
to vary as the square of the distortion measured from the datum circle, and it gives,
for the time of compression, in terms of the greatest distortion, «, the expression

J ft

to a sufficient approximation. Here p is a numerical quantity which is about 1"6
when a/a is small in comparison with n', and continuously approaches the value 1'4
as m gradually increases. It is easy to give similar expressions for other assumed
laws of relation of stress to distortion ; but, as will be seen later, this part of the
inquiry has not yet led to any result of value,

In testing the results obtained with the earlier apparatus I assumed the force
(for the more violent impacts) to be as the square of the distortion simply. This gives,
in the notation of the Tables below,

Of course any investigations, based on such simple assumptions as those made
above, can be only very rough approximations, since they ignore altogether the true
nature of the distortion of either of the impinging bodies, as well as the internal
wave disturbance which is constantly passing to and fro in the interior of each;
part of it, no doubt, becoming heat, but another part ultimately contributing to the
resilience* In such circumstances the impact may perhaps sometimes consist of a
number of successive collisions; certainly the pressure between the two bodies will
have a fluctuating value.

Meamtrefnents of the Tracings^ and their Bedmtion^

From the tracing for each separate experiment the following quantities were
carefully determined* Their values are given in the subsequent Tables, under the
corresponding letters below*

1. Number of vibrations of the fork corresponding to one-sixth of a complete
revolution of the disc • . * ^ • . * N.

Three diameters of the disc were drawn, making angles of 60° with one another,
and the number of undulations of the fork-tracing intercepted between each pair of

232 ON IMPACT. [lxxxviii.

radii was counted. This process was preferred to the simpler one, of counting the
undulations in the entire circumference, for two reasons: — it tests the uniformity of
the rotation, or a possible shrinking of the photographic paper; and it makes one
common process of measurement applicable to complete traces, and to others which
from some imperfection of adjustment presented only parts which were su£Sciently
distinct When only one measurement is given under this head, it means either that
only one was possible or that all six gave the same result When two are given,
they are chosen as the least and greatest of the six. They usually di£fer by a small
quantity only, and may indicate distortion of the paper or irregularity of the fork
(due to the bristle's being clogged with printers' ink, or to its pressing too strongly
on the plate?). In these cases the arithmetical mean is to be taken for any sub-
sequent calculation.

2. The radius of the datum circle R.

This, and the other measurements of length, are in millimetres.

3. The height of fiJl, or of rebound H.

For the first &11, this was of course measured on the rails: — for the subsequent re-
bounds it was measured on the tracing.

4. Chord of the arc of datum circle intercepted by the trace during impact ... C.
As this arc was, on the average, considerably less than one-tenth of radius, the chord
is practically equal to it (differing at most by l/1200th onlyX and it is thus a measure
of the duration of the impact. The duration is, in £BM^t,

C 6N _ 3 CN ,
2«-iil28"400- R ^^^^'

this approximation being much within the inevitable errors of experiment. It is
tabulated under T.

5. Greatest dist<Nrtion — ie., greatest distance of the trace beyond the datum circle
(of coarse not including the (small) distortion due to the weight of the block). This
datum is always, to a small but uncertain amount, increased by the distortion of the
lower part ot the fidling blocL This is jHrobably nearly proporti<Hial to that of the
elastic cylinder, so that the numbers given are all a little too large, but they are
increased nearly in a comnnm ratio D.

It was found impracticable to estimate with certainty the relative distances of
this greatest ordinate from the ends of the intercepted arc; as the radial motion
generally ranains exceedingly small during a sensible fraction of the whole time of
impact. This is true <^ all the substances examined, even when they have jHroperties
so difierent as those of vulcanite and vulcanised india-rubber. It seems as if the
elastic substance were for a moment stunned (if such an expressi^m can be permitted)
when the sudden distortion is complete.

We can easily assign limits within which the time <^ compression must lie. For.
since the elastic force resists the moti<m, and increases with the distortion, its time-

LXXXVriT.]

ON IMPACT.

233

average during the compression is greater than its space-average : — i,e.

mV mF*

t ^ 2D *

where m is the mass of the block, V its speed at the datum line, and t the time

of compre^on. Hence

D 2D

y <t< y ,

If we make the assumptioiL that the force at each stage during restitution is e times
its value during compression, this gives

D T W

and the values tabulated satisfy these conditions. Thus the somewhat precarious
assumption as to the circumstances of restitution is, so far, justified,

6, The tangents of the inclitiation of the trace to the radius of the datum cii"cle
drawn to the intersection of these curves before and after impact ., *,.Ai, Ai,

These values were determined directly by drawing tangents to the trace; and
indirectly by calculation from the equation of each part of the trace. The agreement
of the observed (o) and calculated (c?) values is satisfactory.

Attempts to farm the equation of the part of the trace made before the first
impact were not very successful, as the available range of polar angle was small, and
the radius vector increases rapidly for small changes of that angle* Hence the calculated
value of A^ was obtained simply as the ratio of the tangential and radial speeds of
the tracing-point at the moment of its first crossing the datum circle. This was
taken as

Rui R

In this numerical reduction H is taken as 4 feet, i^., 1219 mm.; and the full
value of g is employed, as we do not know the amount by which friction diminishes
it, the contact of the tracing-point with the disc coming about only during an un-
certain portion of the lower range of the fall ; while it is not possible to estimate
with any accuracy the effect of the impact on the trigger. The calculated value of
the tangent will therefore always be too small, but (since the square-root of the
acceleration is involved) rarely by more than 1 per cent. On the other hand, the
graphic method employed for the direct measurement of this tangent usually exaggerates
its value,

7. The ratios of these pairs of tangents — i£., the values of the coefficient of
restitution _., ,,,,, ,,, , ,.€,

The equation of each distinct part of the trace (alluded to in 6, above) was
found thus:— The minimum (or maximum) radius- vector was drawn approximately for
T. H. 30

234 ON IMPACT. [lxxxviii.

each separate free path, and other radii were drawn, two on either side of it, making
with it convenient angles: — usually 40°, 80°, —40°, —80°, or such like. The notation
employed below for the measured lengths of these radii-vectores is simply square
brackets enclosing the value of the angle- vector, thus: —

[80], [40], [0], [-40], [-80].

If X be the angular error introduced in the estimated position of the minimum radius,
we determine it, as well as the A and B of the equation of the corresponding half
of the branch of the curve in question, from three equations of the very simple form

[0] = il+£a:»,
[40] = il+5(40 + a;)»,
[80] = il4-5(80 + a?)»,

(which may be made even more simple for calculation by putting y for 40 + x). The
assumed initial radius was in most cases so near to the minimum that very little
difference was found between [0] and A ; x being usually very small.

We now write the equation of this part of the branch in the form

r^A^B{0-¥xf',

the numerical values of A, B, x being inserted, after x has been reduced to radians,
and B modified accordingly. The equations, in this final form, are printed below —
each with the data from which it was obtained (A fine protractor, by Gary, London,
reading to one minute over an entire circumference, belongs to the Natural Philosophy
Class collection of Apparatus; so that it was found convenient to deal with degrees
in all measurements of angle, and iu the bulk of the subsequent calculations: — the
results being finally reduced to circular measure.)

In the Tables below, after the data (enumerated above) from each experiment,
come the equations of the successive parts of each trace in order. In these, ^i, )8,
refer respectively to the rise and fall due to the first rebound; 71, 7j to the second
rebound, &c.

To test the formulae thus obtained, other radii were measured, as far as possible
from those already employed, say for instance [20], [60], [—20], [-60], &c. These
measured values, and the corresponding values calculated from the equation (before
reducing to circular measure), are also given below. The agreement is, in most cases,
surprisingly close; and shows that the assumption of nearly constant friction cannot
be far from correct.

The whole of the above statement presupposes that the adjustments have been
so exact that the line of fall of the needle-point passes accurately through the centre
of the disc. On a few occasions, only, it was not so: — but the necessary correction
was easily calculated and applied, by means of the trace preceding the first impact;
even if the trace of the first rebound did not reach to the level of the centre of
the disc. In fact, if we wish to find the curve which would have been traced on
the disc had the adjustment been perfect, it is easy to see that we must draw from

Plata IV

Mepr^Jyccd 9m » acsh ^Amtt ih9 «

LXXXVin.] ON IMPACT. ^^^^^^^^235

each point of the trace a tengent to the cu*cle described about the centre of th^
disc so as to touch the true line of fall The position of the centre of the disc,
relatively to the point of contact of this tangent^ is the same as that of the true
point, relatively to the actual point, of the trace. This applies, of course, to all parts
of the trace, including the datum circle.

In the special trace which has been selected for photolithography as an illustration
(see Plate IV) this adjustmoDt is markedly imperfect * much more so than in the worst
of the others. The path of the tracing- point passed, in fact, about 3 mm. from the
centre of the disc; while, in the worst of the other cases, the distance was not more
than half as great. But this very imperfection serves to enable the reader to follow
without any difficulty the various convolutions of the trace. The measurements and
reductions, obtained from this specially imperfect figurCj BgreB wonderfully with those
obtained from the best traces. It would only have confused the reader had we selected
one of the latter for reproduction, since each of them contains the record of four
experiments — i.e., it contains four times as much detail as does the trace reproduced.

Conclnsians from the Experiments.

It will be observed from the following Tables that the assumed initial radius-
vector was never very far from the true position of the minimum ; the correction (in
circular measure) being usually of the order 0"01, i.e., about O^G, and very often much
less. When the minimum was small, the correction was usually larger; but in few
cases did it amount to 0^05, i,€„ Z^, This correction ought, of course, to have equal
values for the two parts of each free path.

The substances experimented on were fresh specimens, not those which had beea
frequently battered by 8 and 12 foot falls in the earlier experiments. They were
limited to four. Plane-tree, Cork, Vulcanised India-rubber, and Vulcanite. The first
material was chosen the same as that of the falling block, in order that (if possible)
a correction for the compression of the block might be determined, and applied ta
the results of the experiments on other materials. I do not as yet see any simple
mode of obtaining approximately such a correction ; — and the data from diflferent
experiments with the same materials are scarcely sufficiently consistent with on©
another to warrant the application of rigorous analysis, a task which would involve
immense labour as well as difficulties of a most formidable order Hence there is
not much to be said, for the present at least, about the behaviour of a hard body
such as vulcanite, whose distortion is only of the same order as that of the block.
The time of the impact between it and the wood-block is somewhere about I /500th
of a second when the speed of the block is about 16 feet per second. For lower
speeds it is longer; while for very low speeds this substance seems to show a
peculiarity which is specially marked in cork, and will be considered below.

With vulcanised india-rubber, when the speed is 16 feet per second, the time of
impact is about l/130th of a second; it becomes longer as the relative speed is less;
until f with very low speeds, it becomes practically constant*

30— 11

236

ON IMPACT.

[lxxxviii.

With cork the period of impact for a speed of 16 feet per second is about l/70th
of a second ; it increases as the speed is reduced to about 8 feet per second ; and
again steadily diminishes as the speed is still further reduced. This seems to indicate
that (at least in circumstances of rapid distortion) the elastic force in cork increases
in a slower ratio than does the distortion, while both are small, but at a higher
ratio when they are larger.

In all the cases tested the coefficient of restitution seems steadily to diminish
as the speed of impact is increased.

In some of the experiments the mass of the block was doubled; and occasionally
the doubled mass was allowed to fall from half the previous height, so that its energy
remained unaltered. But the number of cases is as yet too small to enable us to
judge with certainty the consequences of these changes. I hope to discuss this point
in a subsequent paper.

23/7/90. Plane Tree, I.

T D

c c

• (

21-25

292-5 1219-2

3-8

0-00206 2-0

0-421

0-377 1-474 1-608

•286*

670

4-7

255 0-8

1-600

1-626 2-651 2-720

-604

fi 21-24

221

4-8

260 0-6

2-798

2-844 4-198

-667

7 21-6

9-1
4-2
21

5-0

6-8

271 0-3
314

[0]

225-0

[15]

232-8

232-9

20:

239-3

r =225 + 125-73 (^ -

-0115)*

35'

270-3

270-2

[40]

284-3

>2:

291-8

290-5

[o:
20
40:

225-0
240-8
286-0

r = 226 + 120-81 (^ -

-0128)»

[-30]
[-42:

269-7
293-0

269-9
291-7

7..

[O'

270-8

10'

274-4

r = 270-8 + 131-31 ((?

- -0087)'

[24]

293-0

292-9

20'

2860

7.. 1

[0-

270-8

274-7 1

r = 270-8 + 121-46 (^-

--0047)*

[-24]

292-7

292-6

'—

20:

286-0

T D

c 0

22-9

301

-5 1219-2

3-0

0-00170 1-6

0-388

0-36 0-924 1-028

-42

155-2

3-8

215 12

1-037

1-039 1-867 1-919

-655

i8, 22-5

42-7

225 -6

1-982

1-942 3-271

-606

7, 22-7

16-2
7-5
3-7

4-2

237 -4

* Note.— It is dear from this value of e, and from the amotmt of the first rebound, that the cylinder was
not home in the lead-block. This fall is therefore not trustworthy in some of its details.

Lxxxvni.]

ON IMPACT.

237

•>]

146-6

■20=

162-3

:*o'

212-2

146-6

164-5

—

40'

215-5

7i.

ro'

258-8

10"

262-8

275-5

7i.

258-8

262-9

—

20'

275-5

r = 146-6 + 140-50 (6 - •0143)»

r = 146-6 + 135-91 (d - •0141)«

r = 258-8 + 142 80 (6 - -0072)*

r = 258-8 + 139-52 (6 - -OOS)*

[30 ] 183-0 183-0

[60-5] 300-9 299-1

-10 ] 151-4 161-4

-50 ] 253-7 253-5

-60-5] 302-2 302-2

[32] 302-2 302-2

[- 32] 301-1 301-8

T D

c e

> (

22-33 322-4 6096

4-1

0-00212 1-9

0-575

0-56 1-281 1-356

-449

104-4

5-6

289 1-0

1-385

1-368 2-718 2-680

•509

^, 22-23

27-5

5-2

269 -6

2-592

2-634 4-705

-551

7, 22-06

9-7
4-0
2-0

7-8
8-2

403 -5
424 -3

[0]
'20'
'40-

218-3
234-9
284-7

r = 218-3 + 136-24 (Oy

[30 ]
[50-08

256-1
322-9

256-6
322-4

0
20'

40;

218-3
235-1
284-6

r =.218-3 + 133-61 (^ -

-0056)'

[-30 ]
[- 50-08;

255-6
321-6

255-7
321-7

7i.

[0"

294-6

10'

298-4

r = 294-6 + 132-95 C^-

-oosiy

[26-04]

321-6

20'

310-4

7a.

[oi

294-5

[-10]

298-8

r = 294-5 + 132-95 (<? -

•0064)*

[- 26-04]

3221

322-6

[-20.

3112

IV.

N R

T D

c c

» <

22-8 331

-8 1219-2

4-5

0-00231 2-4

0-408

0-4 1-072 1-150

•381

155-0

5-3

272 1-3

1-098

1-133 2-179 2-238

-504

A 22-5

36-7

5-6

287 -8

2-371

2-397 4-127

•575

7, 22-8

11-6
4-7
2-3

8-0
8-1

410 -6
416 -5

A. [0]

20
40'

176-6
193-3
243-6

r = 176-6 + 137-88 (^ -

•0017)'

[30 ]
[60-05;

214-3
331-7

2142
327-6

^.,

[0]
20
40"

176-6
194-5
246-2

r = 176-6 + 138-70 C^-

•OlOS)*

[-30 ]
;- 60-05]

216-0
332-8

216-0
332-0

238

7.. [ 0 ] 295-9
[10] 300-2
[20] 313-5

7.. [0

295-9

OX ncPACT. [lxxxvui.

o e
r = 295-9 + 147-73 (^--OOSgf [29-04] 332-8 3330

[-10] 300-3 r = 295-9 + 136-24 (^--OOlOf [-29-04] 3317 3310
[-20] 3130

14/6/90. CoBK, I.
N R

21-7

21-8

A 21 -79

296-4

1219-2

122-8

220

4-4

30-5
44-5
39-8
370

0O167 190

243 8-2

218 3-3

202 1-5

A. A,

0 C 0 c

0390 0-373 1-10 1124

1-250 1-230 271

•355
-461

0
40

173-5

186-5 r= 173-24 + 141-16(^-0428)"

233-8

17-45]

182-5

182-9

32-45'

2113

211-9

56-06°

296-9

296-8

A, [01 173-5

.[01

r- 20] 191-2 r = 173-29 + 11818 (0 ■

[-40] 2380

[-12-57] 1810 181-4
•0424)' [-28-57] 2076 2079
[-5606] 295-9 29645

II.

H C

T D

A. A,

224
225

22-2

306-1 1219-2 29-6

131-5 46-2

23-7 40-3

4-9 38-6

0O162 190
247 8-9
220 3-6
210 1-6

0 C 0 c
0-394 0-373 1-065 1106
1-204 1-196 2-578

•37

-467

[0]
'20'
'40'

174-8

1890 r =174-7 + 144-44 C^-

238-3

0 c
•0244)' [5635] 306-6 3065

A, [0] 174-8
[-20] 193-8 r= 174-6 + 124-75 (^-•0408)' [-5635] 305-7 305-5
[-40] 243-6

III.

22-76
22-8

3210

i8, 22-47

1219-2

128-2

23-8

302
470
42-1
39-2

0O160 18-8

249 8-7

223 36

208 16

/8..

20'

40]

A. [0]
[-20
r-40'

1920
2066
2570

1920
2110
2610

0 C 0 c

0414 0386 1107 1167
1226 1249 2-633

0 0

•0340)» [553] 321 3199

r = 1918 + 147-73 (d

r« 191-8 + 12803 (<?--0380)» [-653] 320-9 3216

-374
-465

■ LXXXVIII."

^^^^^ ON IMPACT. '

^^^^^^^^^241 ^^^^^^H

0^.

[0"

-86-7

^^^^^^^H

80"

199

r = -87 + 135-92(^-1 OSiiy

[93-5 3 300-5 299-2 ^^^H

272

^^^H

A. [o:

-86-7

^^H

[-80-

145

r = -86-7 + 118-52 (d)" [-

-103-2 ] 300-5 297-8 ^^H

[-90'

206-6

^^^1

Yi.

1425

[10 ] 144-9 144-8 ^^1

20'

155-8

r = 14225 + 14313 ($ - 0414)'

30 ] 175-3 175-5 ^^H

2040

62-97] 300-8 3023 ^^H

7.. [0]

142-5

[-30 ] 1810 1810 ^^H

[-20
t-40

160-6

r = 142-36 + 124-42 {0 - OSSS)*

-50 ] 244-5 2446 ^^M

209-0

-62-97] 301-2 302 0 ^^|

S>. [0

228-5

[10 ]

231-8 231-7 ^^H

;2o

243-7

r = 228-44 + 141 '49 {0 - 0206)*

^30 '

264-4 264-2 ^^H

40'

293-4

;42-27]

3012 301-2 ^^H

V [o;

[-20-
t-40i

228-5
245-5
292-6

r = 228-44 + 12343 (0 - -0227)'

-10 ] 233-0 233-2 ^^|
-42-27] 300-0 300-0 ^^H

N ]

Et H

C T D A.

^^^H

21-7 31

0-8 1219-2

15'5 00080 11-6 0-412 0*38 0-680 0683 -606 ^^H

216

389-3

21-3 111 8-8 0 742 0*722 1054 1058 704 ^^H

^.21'5

159-8

26-3 131 6 3 1132 1-132 1600 1-589 707 ^^M

-V, 21-49
B, 21-39

73-0

26-6 138 45 1-689 1682 2238 16& ^^M

351

26-5 138 32

^^^H

170

26-9 140 2-2

^^^H

7-9

27-7 144 1-5

^^^H

3-6

27-6 143 09

^^^^1

1-5

27'6 0-5

^^H

[0]

-78-5

60=

r = - 78-8 + 132-63 (0 + -O*?!)"

[20] -61-5 -65-9 ^^1

197-5

^^^1

A>[0]

-78-5

^^^1

[-80
[-90:

136

r = - 78-8 + 11917 ($ + -OoSe)*

[-60] 37 38-7 ^^H

195-5

^^^1

7i.

151-2

30 ] 1823 182-7 ^^|

■20

1640

r- 151-0 + 13492 (0 - •0386>>

50 ] 244-6 244-8 ^^H

:*o:

209-7

:64-68 311-2 311-3 ^^H

7.. [0]

151-2

■

-10 ] 156-6 156-36 ^^H

[-20

168-7

r = 15102 + 117-36 (^--0391)'

-50 248-3 248 6 ^^H

-40^

214-8

-64-68] 310-3 310-9 ^^H

s„

238-4
252-4
298-7

r = 238 32 + 13253 (0 - -0231)»

[10* ] 241-4 241-4 ^^1
43-56] 3103 310-3 ^^H

«„[0
-20

;-4o:

238-4
2550
300-2

r = 238 3 + 117-36 {0 - OSSiy [

-10 ] 2431 243-1 ^^H
-43-55] 3111 3111 ^^H

^^^^^ T. U.

^^1

fU2

ON DCPACT.

[LXXXVm.

21-76 323-5 1219-2 15-6 0-0078 11-5

A 22-0
21-73
21-66

392-0 22-1

162-3 25-8

76-0 27-5

36-0 27-5

17-3 27-8

8-1 28-3

3-8 30-0

16 Sli)

110
129
138
138
139
142
150
156

8-7
6-4
4-5
3-2
2-2
1-6
1-0
-6

0-419
0-765
1-160
1-698

A, e
c o c

0-407 0-670 0679 625

0-741 1087 1-091 703

1141 1-616 1-581 718

1726 2-484 -683

[0]
80'

;90

(^1

80'
100'

7i.

'20'

7».

0
20'

40;

8..

«,.

0
20

40'
0

[-20
[-40;

-69-4
202-4
276-0

-69-4
158-2
287-0
160-9
1739
220-0
160-9
178-5
225-9
248-3
263-0
311-3
248-3
264-7
310O

r = - 69-6 + 144-45 {0 - 0244)'

[60]

c
81-6

r = - 69-5 + 119-83 (^ - -OSM)* [-60] 56 541

r = 160-7 + 135*81 (^ -
r =160-8 + 122-28 (^ -
r = 248-2 + 137-88 (^ -
r = 248-2 + 118-51 (d-

•0374)«
-0316)"
•0218)'
-0237)'

[30 ] 193-0 192-8

[64-67] 323-4 322-5

[-10 ] 166-0 1660

[-64-67] 323-8 3252

[10 ] 251-5 251-4

[43-8] 323-8 3242

[-30 ] 283-7 283-7

[-43-8] 322-5 321-8

IV.

21-75 333-5 12192 160 00078 115

fi. 21-6
7, 21-92
S, 21-81

392-6

1640

74-2

35-4

17-3

8-2

3-9

1-7

-6

22-6
26-5
281
28-6
29-2
29-4
31-5
311

110
129
137
139
142
143
163
151

8-9
6-3
4-4
3-1
2-3
1-6
10
-6

0-432
0-771
1-170
1-739

c
0-42
0-767
1185
1-759

0-722
1-097
1-669
2-402

c
0-723
1-088
1-652

•598
703
706

•724

/8,.

0a,

80
90'
0

[-80

[- loo;
7.. [o;

'20

40'

-60-5
219
291

-60-5
1650
2945
1699
183-3
231-0

r = - 60-7 + 135-26 (6 - 0419)'
r = - 60-6 + 12016 {0 - •0262)«
r = 1697 + 141-16 {6 - 0384)'

[60 ] 100

c
99-7

[-20] -44 -481

10
50
64-6

172-3 172-3
267-5 2679
334-0 336-6

^^ LXXXVIII.] ^^^^^^^Ol^MPACT. ^^^^^^^^^f^ 243 ^^]

■

^^ 7.. [ 0
m [-20

H s„[o

■P 20
[40

169-9

187-4 r = 169-8 + 12140 {0 - -0318)»

234-5

259-6

274-5 r = 259-6 + 138'21 (0 - -0201)=

323-1

[-10
-30
-50
- 64-5'

[30
;4297;

0 c
174-9 174-9
2071 207-2
269-7 2691
333-3 332-5

294-6 294-6
1 333-3 3332

s., [o:

-20'

259'6

2768 ,- = 259-5 + 121-79 (.-0278)' [:J«.„j g^ ^

28/6/90. VuLCANLsED Indu-Rubbeb. I.

^^1

NRHCTD A, A, e

^^H

21-75 25)3-4 12192 13'7 00076 lie 0370 037 0-601 0629 -616
jS, 22-0 418-1 18-2 100 92 0'637 0-621 0875 0922 728
% 22-28 182-6 225 124 67 0954 0-942 1-358 1-326 702
S, 22-31 89-5 24-0 133 51 1368 1342 1-836 -745
45-7 24-1 134 3-7
23-9 24 5 135 27
12-5 25-2 139 19
6-3 25-0 138 1-4
30 25-3 139 09

A. [01

'80

[100'

-124-5
158 r = - 124-8 + 130-00 (6 - -0524)= [90] 217 217'8
293-5

■

[-80

[-90;

-124-5
123 r = - 124-7 + 133-29 {$ - -0332)* [- 30] - 863 - 90 0
190-5

■

[0]
20^
40

125-7 r = 110 55 + 139-12 (^-0191)« [^,^] ^^^ g|«

7.. 1

20'

110 6 [•
128-0 r= 110-6 + 13213(5- 0141)'
177-6

-30 ]
-50

- 66-68'

148-9 148-6
2140 214-3
2931 293-7

s..

:2o

Si -204-6 + 138 2(^-0128)' [JO..J ^JOS 240-6

5.,

^i] iH ^ = 204-6 + 133-77(^-0054). [l 3« ^,] 242;2 242-0

II.

NEHCTD A. A, •

0 c 0 e
2175 3020 1219 2 144 00077 119 0384 038 0618 0 644 "622
/9, 216 423-0 19-5 106 93 0 652 0-65 0-914 0 952 -713
«, 21-96 183-9 231 124 69 0-983 0989 1428 1383 '689
8.21-87 90-0 24-4 131 51 1402 1-411 1'954 718
461 25-9? 139 3'7
24-4 25-4 136 27
13-0 251 135 1-9
^^^B 6'8 26-2 141 1-4
^^^H 3-4 26-5 142 10

^^1

244

[90'

[-80
[-90]
0
20
40
0]

7i.

1*>

«.,

8..

[-20

r-4o;

0
20'

40;

[-20]

[-*o."

-121
146
215

-1210
114-7
178-6
1180
133-0
181-2
118-0
135-0
183«
212-6
228-2
276-2

212-6
228-9
276-5

ON DCPACT. [lXXXVTH.

0 e

r = - 121-2 + 130^)0(d--0377)i» [+20] -106 -1018

r = -121-1 + 125-41 (d- •0261)' [-1062] 302-5 2985

r = 118 + 136-24 (6 - -0168)«

r = 118-0 + 127-21 (d - ■0169)' [~ JJ^g]

r = 212-6 + 132-96 (0 - •0066)»

163-0 153-0

217-5 217-8

302-3 302-9

218-8 218-6

301-6 300-8

[10 ] 216-4
[47-35] 301-6

216-3
302-0

[-10 ] 216-7 216-8

r=212-6 + 128-36((?-0074). p-g^j -- --

in. Double Mass.

0 c

22-45 325-6 1219-2

16-7

0-0086 13?

0-401

0-4 0-712 0-744 -663

A 22-5

360-7

24-5

126 9-9

0-749

0-736 1-124 1-149 -667

t. 22-67
«,22-5

147-2

31-0

159 8-0

1164

1-131 1-648 1-641 -700

70-5

35-1

180 6-0

1-723

1-663 2-356

-781

35-6

36-6

188 4-4

18-4

37-2

191 3-2

9-7

39-2

201 2-3

4-9

89-8

204 17

40-6

209 1-0

1-2

A. [0]

[60-
80^

-25-3
147-0
271-0

r = -26-1 + 136-57 (^ -

--0787)«

[50

[87-2;

100
325-8

97-6
323-8

A. [0]

-25

[-60

102

r = -25-7 + 139-63 (^-

--0879)*

[- 96-25]

1 325-8

327-8

[-80'

214

7i.

[0]
20-
40]

178-2
194-8
245-0

r = 178*2 + 137-88 (^ -

-0021)»

[50]

:59;

1 283?
1 325-7

282-7
323-8

7»

ro]

178-2

[-30]

216-1

216-2

[-20

1950

r = 178-2 + 140-34 (^ -

-OOSl)*

;-5o:

284-5

284-2

-40]

246-0

-59-

325-9

32.5-9

«.,

[0]

255-3

'20'

272-3

r = 255-3 + 139-52 (d)«

[40-77]

325-9

40'

323-3

20'
40'

255-3
272-2
322-5

r = 255-3 + 137-06 C^-

■•0021)»

10 ]
40-77;

259-5
325-4

259-6
826-0

LXXXVIII.]

^^^^^ON IMPACT.

^^ 245 ^^^

■

IV.

^^^^1

N R H

C T D

^^^^^1

C (

J c

^^^^1

21-6 33Y-5 1219-2

17-8 00085 13?

0-434

0-428 0-774 0787 -561

^^^H

A 220 360-9

25-6 122 10-2

0772

0-759 1163 1162 664

^^^H

7, 22-3* 1550

32-0 153 8-2

1-180

1160 1-668

■707

^^^1

7*-3

36-8 176 6-2

1-741

2-376

-733

^^^H

380

38-5 184 4-6

^^^H

19-7

39-6 189 3-5

^^^H

10-5

40-3 192 2-5

^^^H

5-4

40-3 192 1-7

^^^H

40-3 192 1-2

^^^H

1-3

40-3 192 0-7

^^1

fiu

[0
=60

:8o

] -22-5
] 146-5
] 2650

r = - 24 + 127-38(5 -

•1094)»

[40 ]
902

63-5
3378

c
337

'■

0
6C
80

-225
] 101
] 209

r = - 21-71 + 1369(5

- •092iiY

[-40 ]

[-98-6 ;

30
337-8

28-4
340-6

7.. [

1827

[30

219-7

219-6

^^^^H

1990

r = 182-7 + 136-24 (5-

■ -0031)«

285-5

285-7

^^^1

'40

248-5

6r27

337-5

^^^1

1:'

20
40

182-7
1991

248-8

r = 182-7 + 136-56 (5 -

-0026)*

[-61-27;

286-3
337-9

286-1
3381

■

24/7/90. Vulcanised India-bubbeb. I.

^^1

N R H

C T D

^^^1

C 0

^^^H

216 302-9 1219-2

14-3 00076 12-0

0-387

0-384 0-617 0-628 628

^^^H

/9,21-7 451-4

19-6 104 9-6

0(139

0632 0917 0924 -697

^^^1

7.2206 199-8

23-2 123 74

0-960

0940 1-297 1-310 740

^^^H

S. 22-42 97-0

25-6 135 5-7

1-309

1-320 1-778

■736

^^^1

49-0

26-2 139 4-0

^^^1

25-5

27-0 143 31

^^^H

13-3

28-0 149 2-3

^^^^1

6-9

29-5 157 1-6

^^^H

3-5

29-5 157 1-0

^^^H

1*6

^^H

A. [0]

-148

^^^H

[80

124-5 r = - 148-4 + 129-02(5

- •0581)'

[1036 ]

1 303

300-7

^^^^H

[100

271-3

^^^H

[-80
[- 100'

7.-[0
[20

-148

91 r = - 148 + 12738 {6 -
228-3

•0262)^

[-40 1
[- 109-4 .

-86
303

-90-5
304-5

103-7
119-0 I

■ = 103-7 + 135-81 (5 -

•0132)'

[30

50 ■

00 '-

1390
2041

249-6

1391
2041
2490

167-4

'69-97'

3031

3020

^^1

7.. [ •> ]
[-20,

103-7

121-0 r = 103-7 + 1 30-0 <5~ -01 61)

170-0

-30
-50
-60 =
- 69-9r

141-8
206-7
250-8
302-3

141-6

206-4
250-7
302'7

244

[0]

-121

80^

146

215

[0]

-121-0

[-80]

114-7

;-9o:

178-6

7i.

118-0

133-0

■40]

181-2

7«.

118-0

[-20-
-40'

135-0
183-0

8..

[0]

212-6

'2a

228-2

'40'

276-2

ro]

212-6

-20]

228-9

40]

276-5

ON IMPACT.

r = - 121-2 + 130-00 {0 - •0377)'

o e

[+20] -105 -101-8

r = - 121-1 + 125-41 (^-•0251)' [-106*2] 3025 2985
r = 118 + 136-24 {6 - -0168)«
r = 118-0 + 127-21 {6 - ■0169)»
r = 212-6 + 132-96 {0 - •0065)'
r = 212-6 + 128-36 (6 - -0074)»

30 1 163-0 163-0
50 ] 217-5 217-8
67-72] 302-3 302-9

[-60 ] 218-8 218-6

[-67-72] 301-6 300-8

[10 ] 216-4 216-3

[47-36] 301-6 302-0

[-10 ] 216-7 216-8
[-47-35] 3021 301-8

m. Double Mass.

0 e

22-45 325-6 12192

16-7

0-0086 13?

0*401

0*4 0-712 0-744 -668

A 22-5

360-7

24-5

126 9-9

0-749

0-736 1-124 1149 -667

% 22-67
%, 22-5

147-2

31-0

159 8-0

1-164

1-131 1-648 1-641 -700

70-5

35-1

180 6-0

1-723

1-663 2-356

-781

36-6

188 4-4

18-4

37-2

191 3-2

9-7

39-2

201 2-3

4-9

39-8

204 1-7

2-4

40*6

209 1-0

1-2

[0]
60'
:80'

-25*3
1470
271-0

r = -26-l + 186-57(^-

-0787)'

[50
87-2]

100
325-8

97-5
323-8

-25

[-60]
[-80]

102

r = -25-7 + 139-53 (^ -

-•0879)'

[- 96-25]

325-8

327-8

214

7i.

[0]
'20'
'40'

178-2
194-8
245-0

r =178-2 + 137-88 (d-

-0021)»

[50]

:59:

1 283?
325*7

282-7
323*8

7».

178-2

[-30]

216*1

216-2

[-20J

195-0

r = 178-2 + 140-34 (^ -

-OOSl)*

-50'

284-5

284-2

-40'

2460

-59-

325-9

s.,

[0]

255*3

'20'

272-3

r= 255-3 + 139-52 (<?)•

[40-77]

326-9

325-9

'40'

323-3

"0
20
40'

255-3
272-2
322-5

r = 256-3 + 137-06 C^-

-0021)«

10 ]
40-77;

269*5
325*4

259-6
825-0

Lxxxvin.]

^^^^^ON IMPACT.

■

IV.

^^^^1

N R H

C T D

A. A.

C 0 c

^^^1

21-6 337-5 1219-2

17-8 0-0085 13?

0-434

0-428 0-774 0787 -561

^^^^H

j8. 220 360 9

25-6 122 10-2

0-772

0-759 1163 1162 -664

^^H

7, 22-34 1550

32-0 153 8-2

1-180

1160 1668

-707

^^^M

74-3

36-8 176 6-2

1-741

2-376

■788

^^H

380

38-5 184 4-6

^^H

k 19-7

39-6 189 3-5

^^^1

H 10-5

40-3 192 2-5

^^H

40-3 192 1-7

^^H

40-3 192 1-2

^^H

40-3 192 0-7

^^1

1 '"

] -22-5
] 146-5
] 2650

r^-24 + 127-38 (5-

■1094)'

[40 ] 63-5
[902 ] 337 8

c
337

-60
-80

-22 5

] 101
] 209

r = - 21-71 + 136-9(5

- '0928)?

[- 40 ] 30
(-98-6 ] 337-8

28-4
340-6

182-7

[30 219-7

219-6

^^1

■ [20

199-0

r = 182-7 + 136-24 (0 ~

• -0031)*

[50 ] 285-5

285-7

^^^1

■

248-5

[61-27] 337-5

337-5

l^^^H

H 7>> [ 0

H [- 40

182-7
1991
248-8

r = 182-7 + 136-56 i^-

■ -0026)'

- 50 ] 286-3
-61-27] 337 9

286-1
3381

■

^1 24/7/90. Vulcanised India-bubber. I.

■ N R H

C T D

A, A,

^^1

e 0 0

^^^1

^ 21-6 302-9 1219-2

14-3 0-0076 12 0

0-387

0-384 0-617 0-628 628

^^H

1 ^,21-7 451-4

19-6 104 9-6

0-639

0-632 0917 0-924 "697

^^H

1 7, 2206 199-8

23-2 123 7-4

0-960

0940 1-297 1-310 740

^^^^M

I S. 22-42 97-0

25-5 135 5-7

1-309

1-320 1-778

•736

^^^^H

H 49-0

26-2 139 4-0

^^^1

H 25-5

270 143 3-1

^^H

■ 13-3

280 149 2-3

^^^1

■ 6-9

29-5 157 1-6

^^^^1

29-5 157 1-0

^^^^1

H 1-6

^^1

1 jdi. [0]

-148

^^^M

■ [80

124 5 r=- 148-4 + 129-02(^

- -0581)*

[103-6 ] 303

300-7

^^^H

■ [loo;

271-3

^^^1

t [-80'

k [iioo'

-148

91 J
228-3

• = -148 + 127-38 (e-

■0262)'

[-40

[- 109-4 :

-86
303

-90-5
304-5

-ii

0]
20'
40

103-7

1190 r
167-4

- = 103-7 + 135-81(5-

•0132)'

[30 ]
50 ■

60 ]

;69-97;

1390
204-1
249-6
303-1

1391
2041
2490
3020

H '^1-^0^

103-7

121-0 r
170-0

•=103-7 + 130-0 (5- -0161)

[-30 ]
-50 =
-60
- 69-97

1418
206-7

250-8
302-3

141-6
206-4

2.'>0-7
302-7

■ :^^^ Hi(

246

ON IMPACfr.

[lxxxvtii.

«..

206-3

20"

222-5

r = 206-3 + 13919 (^ ■

-•0077)'

[47-98]

302-3

801-7

40'

272-6

«..

206-3

1-20]

224-0

r = 206-3 + 135-42 (d-

- •0127>'

[-47-93;

1 303-6

803*9

[-40]

274-7

II. DOTTBI.K

Mass.

R H

C 0

22-1 311-8 609-6

19-5

0-0103

11-5

0-563

0*645 0-814 0-863

692

22-2

241-0

26-6

141

9-2

0*885

0-875 1-242 1-263

713

113-0

321

170

1*294

1-286 1-831 1-798

707

;9, 22-11

56-9

35-9

190

5-9

1*782

1-791 2-482

718

7, 22-03
S, 2206

29*3

34-8

184

15-6

34-2

181

2-8

8-2

33-6

178.

2-0

4-3

33-0

176

1-3

2-3

27-5

146

0-6

1-4

70-4

[10

74*3

74*1

860

r = 70-4 + 135-81 (^ -

-oiooy

60 '

216-0

216*6

1347

76-87'

312-0

311-2

70-4

[-30
-60

108-1

108-4

[-20

87-7

r = 70-4 + 131-31 (^ -

*0142)»

218*8

218-3

[-40

137-0

- 76-87;

312*2

311-8

[0'

198-0

'20

213-5

r = 198-0 +135-42 (^ -

- -oioay

[63*18]

312*2

3120

'40

262-0

7..

198-0

-20

214-5

r =198 + 129-67 (^ -

0077)»

[- 53*18]

311-6

-40=

262-6

«..

2651

10'

259-0

r = 2561 + 132-96(5-

--0032)'

[37-62]

311-5

'20

271-0

«..

255-1

-10

;-2o:

¥TT

259-2

r = 256-1 + 182-95 (6 -

-•0001)»

[-37*52]

312-0

271-4

III.

N ]

R H

C 0

22-25 82

8-8 1219-2

16-2

0-0081

12-2

0-417

0-4 0-645 0-704

-647

22-1

894-8

24-2

122

10-4

0-711

0-710 1052 1089

676

A 22-4

169-6

30-6

154

8-4

1-099

1-089 1*563 1-510

707

7,22-26
S. 22-87

82-7

34-6

173

6-6

1-530

1-522 2-087 2-061

-783

42-2

36-3

182

4-9

2102

2-893

6,23-12

22-0

37-3

187

3-7

2-917

f. 22-93

11-6
6-1
3-1

370
38-9
38-9

186
196
195

2-6
1-8
1-3

1-5

Lxxxvin/

^^^^^^^ OW raPACT.

^^^^^^^^^47 ^^^^^H

y8>.

-669

^^^^^H

'60"

865 r=- 66-9 + 1 3821 {6 - '0063 J"

[96-7] 329 323-8 ^^H

=80'

205

^^^H

A. [0.

-66-9

^^H

[-60

74-6 r = - 67 + 135-91 {$ - Ot&^f

[-99-4] 329-0 329-6 ^^|

[-80,

188

^^^H

yu [o;

[20
40

159-0

175-1 r=159 + 136 24(^--0063)'

2240

[50 ] 261-3 861-5 ^^|
[63-87] 329-1 326-7 ^^H

7.. [o;

[-20
-40-

159*0

175-9 r= 159 + 134-59 f^-OOSS)"

226-6

[-50 ] 262-9 262-7 ^^|
-63-87] 328-2 3278 ^^H

[0]

246-4

^^^1

=20'

263- 1 r = 246-4 + 1 4405 (6 - -OOSfi)*

[43-72] 328-2 328-4 ^^1

=40=

314-9

^^^1

K [0]

240-4

^^^1

-20=

2C3 8 T = 246-4 + 141 49 (0 - 'OOUY

[-43-72] 329-2 329-1 ^^1

■-40

315-7

^^^1

«.. 1

286-2

^^^1

=10

2906 r = 286-2 + 147 73 (0 - -OOig)"

[31-08] 329-2 329-36 ^^H

'20

304-0

^^^1

f.. [0]

286-2

^^^1

[-10

290-5 r = 286-2 + 144-44 (0 - 0020)'

[-31-08] 328-2 328-4 ^^H

:-2o

303-6

^^^1

K.,

"0]

306-2

^^^1

^10'

3104 r = 306-2 + 14608 {0 - 0049)'

[22-53] 328-2 3282 ^^1

[20]

323-5

^^^1

K.. [0]

3062

^^^1

-10

310 8 r = 306-2 + 141 16 (0 - -0061)*

[-22-53] 328 9 3287 ^^H

-20;

324-0

^^H

21/8/90. VuL

CANisiiD India-rubber.

(This ia the trace reptodneed in ibe plate, and the details on; given here to Dhow that fur renilts ^^^^|

can lie obtained even when the adinHtmeot is veiy imperfeot.) ^^^^^

N I

I H C D T

A, • ^^H

^^^H

22-75 3

38 12:9-2 15-3 11-9 00077 "383

-636 -655 -602 ^^H

/8, 22 5

456 21-5 9-6 0108 660

-693 966 1066 683 ^^H

7, 22-9
h, 23-4

197-5 250 7-6 0125 "933

1-000 1-354 1-42 -689 ^^H

950 26-6 5-5 0133 1-418

1-42 1-842 -770 ^^H

48-0 28-0 4-2 0140

^^^H

24-5 30 0 30 0150

^^^H

125 31-5 2-4 -0158

^^^H

60 33-0 1-9 0165

^^1

A, [0 1

-118

[55"

18 ^^H

[80-

168-2 ) 118 + 146-09 (^)"

70=

100 101 ^^M

[100

3317

^90'

242-4 244 ^^H

A> [o;

-118

[-70]

85-5 88-6 ^^1

[-80

1 50-6 r = - 1 17-8 + 1 30-33 {0 - 0368)'

-75'

116 118-2 ^^H

[-100

300-1

-90]

217-5 219-0 ^^H

248

ON IMPACT.

[lxxxvul

7i.

[0]

131-5

30'

1710

'60'

2890

7..[0]

131-5

[-30
-60

171-0

290-0

81.

[0]

232-6

251-0

40'

306-8

«..[0]

232-6

[-20'

250-8

40]

305-4

r = 131-5 + 14314 {6 + •0017)»
r = 131-5 + 144-45 {6 - •0007)«
r = 232-6 + 149-37 (6 + -OOig)*
r = 232-6 + 149-37(tf)»

[15j
[*5j

141-5
219-8

c
141-5
220-2

[-10"

-50'

t-65i

135-9
241-8
3170

135-9
241-3
317-2

[10 ]
'30 '

:4o-5;

237
274
327-6

237-2
273-8
327-3

[-30 ] 273-5 273-5
[-40-75] 327-6 327-8

DESCRIPTION OF THE PLATE.

The chief figure is, as above stated, photo-lithographed on the scale of 0'3 from the record
of a 4-foot fall on Yulcanised India-rubber. Even in this reduced scale it shows fairly enough
the relative details of at least eight of the successive rebounds. These are numbered in order.
The original showed several more. As its lines were not only very fine, but in blue, they
had to be carefully gone over with a photographically inactive colour, so that much of the
more delicate detail is unavoidably lost. The tuning-fork was kept in contact with the disc
for a little more than a complete revolution. The consequent overlapping of the trace enables
us to see that the angular velocity had not sensibly changed during one revolution of the disc.

The three figures immediately below are (pencil) records of successive impacts on Native
India-rubber (9/1/89). Time of rotation of disc 0»-3.

Then follow records of impacts on Pine Tree (7/11/88) from heights of 8, 4, and 2 feet.
These show the "wriggles" spoken of in the text. Time 0**3.

The group of five which follows belongs to the experiment III. of 23/7/90 with Plane
Tree, whose details are given in the Table. Some of these show traces of wriggles.

The final group contains details of the first eight successive impacts of lY. of 7/6/90
on Vulcanised India-rubber. To save space, the first and third, as also the second and sixth,
which took place at the same portions of the datum circle, have been drawn together.

In each of the two later groups the time of rotation of the disc was a little more than
one second.

The disc always had positive rotation; so that the older figures (those in pencil) must
be read the opposite way to the others, which were reversed in printing from the disc: —
i.6., the compression part of the impact is to the left on the pencilled figures, to the right
on the others.

LXXXIX.]

249

LXXXIX.

ON IMPACT. IL

[Trmiaaations of the Royal Society of Edinburgh, VoL xxxvii. Bead January IS, 1892.]

[Since this second mstalment of my paper was read to the Society my attention
has been called to a remarkable investigation by Hertz* ; in wbich the circumstances
of collision of two elastic spheres are fully worked out, under the special limitations
that both are smooth, and that their deformations are exceedingly small This forms
a mere episode in the paper, which is devoted mainly to the statical form of the
problem of deformation ; as, for instance^ the case of the ordinary apparatus for the
production of Newton s rings. But it contains a definite numerical reault; giving for
the duration of impact between two iron spheres of 50 mm. diameter, which encounter
one another directly with a relative speed of 10 mm. per second » the value 0'W038.
This seems to be the earliest reckoning of the time of collision. The experimental
verification of Hertz' formulae was undertaken with success by Schneebeli'fj who obtaineii
results in close accordance with them. His mode of measuring the duration of impact
was defective, though ingenious. But the speeds employed by him, though for the
most part considerably greater than those contemplated in Hertz' work, were far inferior
to the lowest of which I have availed myself: — and thus no comparison can be
instituted between my results and the theoretical formulae; first, because I have
neoemarily dealt with deformations so large as to be directly measurable; secondly,
because the formulae, being originally obtained for the statical problem, have left
aside tbermodynamical considerations, and thus assume equal duration for compression
and for restitution, which is certainly incorrect; finally, because one of my colliding

* Jawmal f^r die reine und mtffewandu Matkematik, xeU., 1883. Uber die Beriihruiig renter elftatiBohfiir Kdrper.
t ArchiviM de» Scitnee* phy»iqu€$i &c-, Oentee^ xv., 1885* B^oherohes exp^rimeat&les sur le Ohoo dea Corpa

T, IL 32

250 ON IMPACT. [lXXXIX.

bodies was fixed, and thus virtually struck on both sides, besides being notably
deformed throughout the greater part of its substance; while, except in the case of
very hard bodies, the surface of contact was nearly equal to the whole section of
the cylinder. I regret, however, that I had not seen Hertz paper before I made
my apparatus, as a study of it might have led to improvements in my arrangements;
especially in the choice of the form of the elastic substance to be operated on. But
my results have the advantage of being applicable to many practical questions (besides
those of Golf, to which they owe their birth), such as the driving of a nail by a
hammer, or of a pile by a ram, &c. One of Hertz* results is specially interesting,
viz. that the duration of impact between two balls is infinite if the relative speed
bo indefinitely small. This may easily be seen to depend upon the fact that (in
consequence of their form) the total force between them, at any instant, varies as a
power of the deformation higher than the first.]

The experiments, whose results are tabulated at the end of the paper, were (with
the exception of the first, presently to be noticed) made with a new set of specimens
of various elastic substances, considerably larger in all their dimensions than those
previously employed. They were, as before, cylinders very slightly rounded at theii*
upper ends; but their lengths, as well as their diameters, were 56 mm. instead of
32 mm. as formerly. As I could not procure a piece of good cork of the requisite
dimensions, the cylinder of that substance employed was built up of two semi-cylinders,
gently kept together by two india-rubber bands. The glass cylinder turned out to be
somewhat difficult of manufacture, and the experiments with it are altogether defective.
But, after the third, and most considerable, impact to which it was subjected it
presented a very interesting appearance. There was formed inside it a fissure some-
what in the shape of a portion of a bell; meeting the upper surface in a nearly
circular boundary 12 mm. in diameter. This fissure showed the colours of thin plates
in a magnificent manner. It gave the impression that the portion of the glass
contained within it had, by the shock, been forced downwards relatively to the rest.
Its lower, and wider, extremity did not come within 4 mm. of the sides of the
cylinder, and this was at a depth of about 6 mm. below the upper surface.

One result of the new experiments is obvious at the first glance. The duration
of impact is notably longer than before; in consequence of the increased dimensions
of the elastic bodies operated on. But the coefficient of restitution is only slightly
afTcctod.

As the old block had been split daring some experiments in which it was allowed
to fall on vulcanite from heights of 3 m. and upwards, a new one (also of plane tree)
was obtained. The mass of this new block was 3'75 lbs., and (except where it is
otherwise specially noted in the tables of experimental results) had its lower end
shod with a flat plate of hard steel G mm. in thickness, and 1 lb. mass. The main
object of this was to prevent the "wriggles" formerly noticed. Another plate of the
same material, with a blunted wedge-shaped ridge projecting firom its lower surface,
WAS i>cca8ionally substitut^Kl for this (as noted) in some of the experiments on vul-
canised india-nibber. It was tried on cork also, but the result was disastrous.

I*XXXIX.]

ON IMPACT.

!51

The object of this ridge was to teat the effect, on the coefficient of restitution
and on the duration of impact, produced by applying a given raonitintuni of the
falling body in a more concentrated form, by restricting the surface-region of its
application to the elastic solid* The results obtained by this process, though un-
fortunately limited to one elastic substance, are very intereating. The duration of
impact is notably increased, in spite of the increased distortion; but the coefficient
of restitution is practically unaltered.

The first set of experiments given below (7/4/91) was made with the old cylinder
of Vulcanised India- Rubber. They were designed to form a link between the present
experiments (with the steel plate) and the former set (in which the impinging surface
was hard wood).

Mr Shand has again made the measurements of the tmces, and reduced the
observations, precisely in the same manner as before: — and it will be seen, from the
numbers in the columns headed N, that the new series of results is at least as
trustworthy as the old one. But I was not satisfied with the numbers in the columns
Ai, A,; nor, of coui^se, with those in e, which are their respective ratios. These data
are derived from the very difficult and uncertain process of dmwing tangents at the
ewtremities of portions of curves. I therefore calculated (to two places only) the values
of the square-root of the quotient of each pair of suocessive numbers in the column
H, If there were no friction, the results thus obtained should be the successive
values of the coefficient of restitution. And, even taking friction into account, if we
suppose the acceleration it produces to be m-fold that of gravity (m being, aa shown
in the first part of the paper, nearly constant and somewhere about 0"03) the values
in the table so formed should be those of

Vr+S^"^^"'"^"^*"'^"

This (though at a first glance it might not be suspected) is the result to which we
should be led by calculatiug from the equations of the various parts of the trace
the tangents of the inclination of the curve to the radius-vector at the points where
it meets the datum circle. For

^"**^(4f)r?7ror

(F-X)'

so that

_tan^_ /B^{Jt-A^)_ /B.M,
' "tan^ yB,{Ii-A,) \/ B,H,^

Unfortunately, it is in general difficult to get a ti-ustworthy value of B for the (first)
incomplete branch of the curve. But, by various modes of calculation and measure-
ment, I have made sure that the friction is practically the same ivhatever be the
mass of the block, so that its effects are the less sensible the gr^iter is that mass. The
numbers thus obtained fluctuated through very narrow limits, at least for such bodies
as native and vulcanised india-nibber, and therefore give for very extensive ranges of
speed of impact a thorough verification of Newton s experimental law; viz, the con-
stancy of the coefficient of restitution for any given impinging bodies. This bad,

32—2

252 ox IMPACT. [UQCXIX.

however, been l<Hig ago carefiilly tested by the ehibomie experiments of Hodgldiisoii*.
There was, it is tme, a slight fidling off for the very high speeds, and likewise fin*
the very low: as wiU be seen from the table of Approaeimate Coefficients of Restitution
which follows the experimental resolts. The first may be due in part to a defect in
the apparatus, the second wiU be accounted f<»r below.

The approximate constancy of e, for all relatiye speeds, proves merely that the
force of restitution is, at every stage, proportional to that required for compnessioii.
We must therefore look to the values of the total distortion, or to those of the doratioii
of impact, for informaticm as to the relation between the distortion and the force
producing it. The equation of motion during the compresaon is, say,

Mx^Mg^F--f{x) (IX

Hence, as F may be considered to be nil while the datum circle is being traced, we
have for the correction, d suppose, to be applied to the tabulated values of Z>, that

positive root of

Mg-f\x)^0 (2)

which vanishes with If.

Integrating the equation of motion, we have

Jfi»;2 = JfF^/2 + (%-i0x-/(x) (3),

where V is the speed at impact, and f{x) vanishes with x. Thus, at the turning point,

0 = ifF'/2 + {Mg - I'XD + d) -f{D + ?>

Now, by (2X we see that (Jfgr — F) is of the order /'(3) only, so that, when F (and
therefore D) is considerable, we may write this in the approximate form

0 = JfF^/2-/(D).

This equation enables us to get an approximate estimate of the form of the function
f, A graphical representation of D in terms of MH, based on the various data of
the experiments of 22/6/91, below, on vulcanised india-rubber, gave three nearly parallel,
but closely coinciding curves, whose common equation (when the different values of d
for the different masses were approximately taken account of) was of the form

JfffxD*;

for the subtangents were 2*5-fold the abscissae. Hence we are entitled to write (3)
in the tentative form

lfi»,2 = JfrV2 + (Jtfy-10x-ilx* (4X

Equation (2) now becomes Mg^\Ad^\

whence d may be found, A being determined from one of the larger values of D
(and the corresponding kinetic energy) by the relation

MgH^AD* {51

* Britiik Attociatiom Report^ ISM.

LXXXIX.] ON IMPACT. 253

These give the approximate value

1 - (^\^

Thus I found that the values of D, for the experiment of 22/6/91 on vulcanised
india-rubber, must be augmented by 0'75 mm., 1*2 mm., and 1*9 mm. respectively: —
according as the mass was single, double, or quadruple. These agree remarkably well
with the relative positions of the parallel curves already spoken of: and also with
direct measurements of d which have been recently made for me, by a statical process,
by Mr Shand. In what follows, I shall assume that the values of D have had this
(positive) correction applied.

By the help of (4) we now have, for the time of compression, the expression

IM [O

Except for the very small values of D, we may neglect the last term under the
radical, and the expression, slightly diminished in value, becomes

V 24V2)Jo

•Jl-z^'

The numerical value of the integral is approximately 1*5. For any one substance the
time of compression is therefore inversely as the fourth root of D; and, of course,
directly as the square root of M, But we may also write the expression, by means
of equation (5) above, in a form which applies to all substances for which the elastic
force is in the sesquiplicate ratio of the distortion, viz.

VoD

This result lies just half-way between the limits, D/F and 2D/V, assigned (from
general considerations) in the first part of this paper.

With the data for the first fall of the quadruple mass in the experiment last
referred to, this expression becomes almost exactly 0"'01. The value of e is about

0*77, so that the whole time of impact should be (I+aTt^) ^''^^' ^^ 0'023; while

the experimental value of T is 0"'0211. But, in consequence of the quantity 3, above
spoken of, all the measurements of arcs from which T is calculated are necessarily
too small. Add to 0, as measured, the product of 9 by the sum of the two tangents,
as given in the table; and diminish R by the amount 3; the observed time becomes
0»-0224; so that the formula gives a tolerably close approximation.

If we bear in mind that the values of D ought to be increased by the quantity
9, we see at once the reason, already referred to, for the apparent falling oflF of the
values of e at low speeds, when they are calculated fi-om the values of H given in
the tables.

254 ON IMPACT. [lXXXIX.

Among the practical applications of the results above, we see that when a nail
is driven, say by a J-lb. hammer moving at the rate of 10 feet per second, the time
of impact being taken as 0»*0004, the time-average force is some 300 lbs. weight- If
the head be one-tenth inch square, this corresponds to a pressure of more than 2000
atmospheres.

Finally, to finish as I began, with an application to golf, although from the nature
of the case, the experimental data are not very directly applicable: — ^we see that, as
the coefficient of restitution from wood is about 0*66, and the mass of the ball about
01 lb., the club must be moving at some 300 feet per second to produce an initial
speed of 500 feet per second: — and the time-average of the force during collision must
be reckoned in tons' weight. The experiments on hammered, and on unhammered
balls, all made at the same time and of the same material, show clearly how very
small is the gain in coefficient of restitution, and therefore in initial speed, which
is due to the hammering: — and thus force us to look in another direction for an
explanation of the unquestionable superiority of hanmiered over unhammered balls.

[It is very curious that the law of force in terms of the distortion (as given
above) is the same as that which results from Hertz' investigations. For, what is
called D above is the diminution, in length, of the whole cylinder operated on; while,
in Hertz' work, the quantity which he calls a, and to whose 3/2th power the force
is proportional, is the advance towards one another (since the first contact) made by
points chosen in the two bodies, whose distance from the (infinitesimal) surface of
contact is finite, yet very small in comparison with the dimensions of the bodies
themselves. In my experiments the vertical shorteuing extends throughout the whole
of at least the protruding part of the cylinder, and in extreme cases the distortion
is so great that the diameter at the middle becomes more than double that at the
ends; in Hertz' investigations it is assumed to be mainly confined to the immediate
neighbourhood of the surface of contact.

It is even more curious to find that the same law holds, at least in a closely
approximate manner, for the very large and unsymmetrical distortions produced by the
ridged base, as shown by the data of 7/11/91.

Some additional details connected with this investigation, including a sketch of
the apparatus and of the trace of 13/7/92, will be found in an article Sur la Duree
du Choc, which appeared in the Revue des Sciences pures et appliques, 30/11/92.]

LXXXIX.]

ON IMPACT.

255

[In the following experiments, unless some contmry statement is made, the falling
block terminated in a horizontal plate of steel.]

7/4/91. Vulcanised India-Rubber. (Old Small Cylinder.)

I. Single Mass.

21-7 3370

1219-2

15-5

11-0

-0075

•4396

-6200

•709

/S, 20-8

510-6

181

8-5

•0087

•6903

-8693

•794

7, 210

8, 21-6

283-5

19-3

6-7

-0093

-9163

11165

•821

163-6

19-7

5-3

•0095

1-1875

1-4496

■819

96-7

21-5

4-4

•0103

58-6

22-2

3-6

•0107

35-5

22-9

2-8

-0110

21-8

23-5

2-3

-0113

13-6

24-1

1-8

-0116

8-4

25-0

1-6

•0120

5-0

27-1

-0130

3-1

27-1

-0130

1-8

27-1

0-9

•0130

[ 0 '

-1740

20'

-158-4

r = -

-174 + 120-16 (^+01133)»

100'

212-5

/9.. [ 0 ]

-174

[-20'

-ir,7-6

>• = -

- 174 + 116-22 (d + -02754)'

[- loo;

207-6

7i.

[ 0 ]

.53-4

30 =

94-5

/• = 53-4 + 137-89 {0 + -02266)"

[80-75]

337

336-1

60'

211-3

7., [ 0 ]

53-4

[-30

85-3

>• = i5

3-4 + 117-20 (^--l

30174)»

[-89-15]

337

336-5

-60

181-7

a.,

[ 0 ]

173-3

30'

209-7

r =173-3 + 133-29 (^-

•00105)=

[63-17]

337-0

335-0

eo;

319-2

s=. [ 0 ]

173-3

[-30^

208-3

/• = 173-3 + 12213 (^ + -01180)'

[-65^7]

337-0

337-2

[-60

3102

II. Single 5

I ASS.

21-7 t

126-7

1219-2

15-0

10-8

•0074

•4330

-5639

•768

/3, 20-9

5480

17-6

8-7

-0087

•6403

-7766

•824

7. 21-4

303-0

19-0

7-0

-0004

•S606

10289

-837

8, 21-5

1770
lOfi-0
64-5
40-4
2.-.1
1.V7

19-1
199
21-0
22-3
22-5
230

5-6
4-5
3-7
3-0
2-5
20

-0095
-0099
-0104
•0110
•0116
-0114

1-1054

1-3581

-814

101

23-3

1-5

-0115

256

ON IMPACT.

[lxxxix.

[ 0 ]
'30

loo;

-221-8

-1841

177-3

r = - 221-8 + 128-14 (6 + -01918)»

[117]

326-7

c
322-6

A.[0
[-30

[- no;

-221-8

-188-5

22G-6

r = - 221-8 + 121-7 {6 - •0005)»

[-121-1]

326-7

322-0

7i.

:o]

[4o:

SO'

23-8

97-5

305-5

r = 23-7 + 137-89 {8 + -03365)«

[83]

326-7

326-6

7., [ 0 ]
[-40-

-so;

23-8

790

244-2

r = 23-8 + 112-94 (0 + •00105)«

[-93-4]

326-7

324-3

Si. [ 0 ]
[30
[60]

148-9
185-1
293-3

r = 148-9 + 131-32 (0 + 00140)'

[66-4 ]

326-7

325-7

8„ [ 0 ]
[-30

;-6o;

148-9
184-0
286-5

r = 148-9 + 122-78 (^+ -01116)'

[-6815]

326-7

3259

I, Quad. I

tiASS.

H C D T

21-3 i
A 21-1
7. 211
S, 21-4

)20-6

1219-2 21-6 14-6 -0107

359-5 29-8 127 -0146

197-8 34-7 10-9 0172

121-5 38-3 9-1 0190

790 40-6 7-8 0201

53-0 41-7 6-5 -0207

35-4 41-7 5-3 0207

24-1 39-7 4-1 0198

161 41-0 3-6 0203

10-8 43-2 3-1 -0214

•4383

•7669

1-0241

1-3206

•7813
10052
1-2305
1-5911

-561
-763
•832
-830

[ 0
'60'

'90-

- 38-7

78-0

241-1

r = - 39-6 + 127-38 (6 - 08595)'

[101-7 ]

320-6

c
323-7

i8., [ 0 ]
[-60

:-9o;

- 38-7
106-6
273-5

r = - 39-5 + 114-58 {0 + 08194)'

[-96-5 ]

320-6

317^9

7».

[ 0 ]
'30'
'60 =

122-8
157-6
259-9

r = 122-8 + 123-11 (0 + -00819)'

[71-83]

320-6

3188

7«. [ 0 ]
[-30

:-6o:

122-8
156-8
256-6

r = 122-8 + 119-83 (0 -t -00907)'

[- 72-75]

320-6

318-7

[ 0 ]
20'

40;

199-5
214-7
260-8

r = 199-5 + 126-72 (0 - •00279)'

[55-9 ]

320-6

319-4

[0
-20

-4o;

I 199-5
214-2
259-4

r = 199-5 + 126-08 (0 - •00610)'

[- 56-95]

320-6

321-5

CXXIX.J

OX IMPACT.

\ Quad.

Mass.

2145

3070

1219-2

20^0

15-3

•0104

•4142

•7050

•588

0. 21-2

3946

273

130

•0142

•707G

•9088

•779

7. 21-3

2128

340

11 3

•0177

•9725

11813

•823

B, 215

1280
80-3
54^0
365
247
10-4
112

369
400
408
420
420
450
450
47^9
503
.-)40
544

90
81
70
5^0
46
40
33
2^8
21
17
12

■0192
•0209
•0213
•0219
•0219
•0235
•0235
•0250
■0202
•0282
•0284

12052

1-5409

•782

257

/3..

[ 0 ]

- 88^3

80'

1500

100'

286-8

/S„

[ 0 i

- 88^3

[-80]

1570

[-100]

290-9

7..

0 ■

93-8

'30'

1270

'co;

2295

7..

938

[- 30]

1279

-60'

2290

8i,

[ 0 '

1793

20'

1949

40'

2410

«•..,

r 0 1

1793

-20]

194-0

-40]

239-5

r = - 88-4 + 126-72 (6 - 0246)=

0 c

[102-33] 307 3047

;- = -88-5 + 11917(^+0389)= [10163] 307 303

r = 93-8 + 126-07 (0 - •01046)'^

[75-1 ] 307 300-9

»• = 93-8 + 122-13 (^ + -00488 )•-• [-7515] 3070 305-4

r = 179-3 + 125-08 (^ + -00401)= [571 ] 307 3045

r = 179-3 + 126-40 (^-•00802)-' [-.5793] 307 30G-5

12/6/91. New Native India-Rubber.
I. Single Mass.

A..

22-7

265-5

750-0

28-4

213

-018

•4317

•4866

•8872

22-7

501-2

29-5

181

-019

•5441

•6017

•9042

343-8

30-6

15-9

-0195

•6358

■7167

•8871

240-7

31-7

137

-020

•7550

•8627

•8751

165-7

33-3

11-6

-021

•9004

1-0514

•8563

110-5

35-4

10-0

-023

11403

12892

•8845

74-2

360

8-3

-023

49-5

391

7-1

-025

3.S-2

40^8

-(I2(i

221

423

5-2

-027

14-7

451

4-5

-029

9 2

460

3-7

-029

5-7

486

3-0

•031

3-4

510

2-3

•033

T. II.

258

ON IMPACT.

[lxxxix.

«!.

0
30
60

99-2
1401
262-3

€„ [ 0 ] 99-2
[- 30] 135-2
[- 60] 243-5

II. Single Mass.

N
21-8
21-8

274-5

€„[0

[30

reoi

108-0
145-5
257-8

e„ [ 0 ] 1080
[- 30] 143-5
[-60] 246-2

H
2-2
1-3
10

-7

48-6
48-6
48-6
48-6

1-6

•9

-6

•031
-031
-031
•031

r = 99-2 + 148-39 (0 + •0014)»

r = 99-2 + 131-98 {d - -0014)»

7500

5080

347-8

244-6

167-3

112-2

75-3

50-2

33-5

22-1

14-6

9-0

5-7

3-7

2-7

1-7

1-2

C
29-9
31-8
330
34-2
360
37-9
39-5
41-3
43-1
45-8
48-1
49-5
50-9
50-9
50-9
509
50-9

21-6

18-3

15-8

13-5

11-8

10-1

8-5

7-5

6-4

5-4

4-6

3-7

2-9

2-3

1-9

1-3

-9

•018

-019

-0195

-020

-021

•023

•024

•025

•026

•027

-029

•0295

•030

-030

r = 108 + 136-24 (0 + 0010)'

r = 108 + 122-46 (0 - •0005)'

0 c

[40] 171-8 171-8

[- 40] 163 163-3

A,
-4581
-5860
•6873
•8064
•9833

A,
•5131
•6473
•7646
•9163
1-1086

e
-8928
-9062
-8989
-8800
-8870

0 c

[40] 175-0 175-0

[- 20] 123-7 122^9

III. Double Mass.

N
212
e, 21-8

R
2921

7500

536-8

399-3

303-8

232-9

177-3

133-7

100-0

73-7

54-7

40-2

C.
37-6
40-9
410
42-9
440
45-5
46-6
481
50-1
52-6
52-6

D
26-1
231
20-4
183
16-3
14-7
13-2
113
10-6
9-2
8-0

T
-020
•022
•022
•023
-024
-025
•025
-026
-027
-0286
-0286

A,
-4956
-5715
-6590
•7632
-8561
-9556

-5528
-6590
•7220
-8337
•9270
1-0831

•897
•867
•913
•915
•923
-882

LXXXIX.]

ON IMPACT.

259

«„ [ 0 ] 59-5

[ 30 ] 991

[ 60 ] 212-1

e„ [ 0 ] 59-5

[- 30] 94-0

L- 60] 197-5

29-3

22-2

16-2

11-5

8-2

5-7

4-0

2-9

1-6

•8

C
52-6
52C
52-6
52-6
51-7
520
511
50-2
48-3
44-7
37-6
34-2

6-9
60
5-2
4-4
3-6
30
21
1-5
11
00
-4

-0286

•028

•028 ■

-028

•027

026

•024

-020

•018

r = 59-6 + 133^95 (0 + •0209)-'

r = 59-5 + 125-74 (^)=

0 c

[744] 2921 292-5

[-77-6] 292-1 290-1

IV. Double Mass.

21-8
219

313

Cl.

[ 0 ]
30'

!6o:

780
1161
228-5

e...

[0]
-30

-6o;

780
114-5
220-9

7500

539-2

401-7

305-9

237-9

180-8

137-2

103-4

77-7

58-1

435

32-3

23-9

17-3

130

9-5

6-9

4-9

3-4

2-3

1-6

1-2

-8

C
39-5
41-7
441
46-2
47-4
49-0
51-0
52-7
53-8
56-0
58-0
59-5
61-3
61-0
610
61-0
61-8
62-6
63-1
630
62-9
60-9
50-8
48-?

26-4

23-3

20-8

18-7

16-7

15-0

13-6

120

10-6

9-5

8-6

7-5

6-5

5-8

4-8

3-9

3-5

2-8

2-2

1-7

1-2

-7

•4

•3

T
-021
-022
-023
-024
-025
•026
•027
-028
-028
-029
-030
-031
-032
-032
-032
-032
-032
-033
-033
-033
-033
-032
-020
•025

r = 78 + 135-59 {d + •0066)-

- = 78 + 127-38 (^ + -01 185)-

A.
•5165
-6120
-0997
-8012
-9244
1-0441

A,
-5879
-6681
-7879
-8795
1-000
1-1504

-879
-916
-888
-911
-924
-908

0 c

[75-2] 313 314

[-77 ] 313 312
33—2

260

ON IMPACT.

[lxxxix.

22/6/91. New Vulcanised India-Rubber.

I. Single Mass.

220 269-2

914-4

25-3

184

•0155

-4262

•5325

■800

S, 22-4

459-4

26-7

13-8

•0163

•5844

•7150

•817

252-7

27-0

10-8

•0165

•7627

•9331

•817

151-3

277

8-5

■0170

-9833

11918

•825

90-8

28-4

6-7

-0174

53-7

20-1

5-2

•0178

.311

201

•0178

18-0

300

-0184

10-4

30-9

•0189

6-1

31-1

-0190

3-6

311

1-4

•0190

2-0

31-1

1-0

•0190

Br,[0

117-6

[20

1331

r = 117-6 + 141-17 (d-

-0174)»

[60-5 ]

2692

2698

[4o;

183-0

s..[o:

117-6

[-20

-4o;

134-5

r =

117-6 + 132-63(^4

-00645)»

[- 61-15]

2692

270^9

183-7

11. SmoLB A

TAS.S.

21-2 S

!83-2

914-4

261

18-5

•0146

-4434

•5438

•815

B, 22-4

464-5

27-2

139

-0152

•5961

•7400

•805

2603

28-5

11-0

•0160

•7921

•9675

•819

1553

29-0

8-7

-0163

1-0082

12349

•816

93-4

29-9

6-9

•0168

56-6

31^

5-5

-0174

33-2

31-5

4-2

■0177

18-9

321

-0180

10-7

335

2-5

-0188

3-9

34-4

2-0

-0193

35-2

1-5

-0198

368

1-1

-0207

«,.

[ 0 ]

128-4

146-1

r =

128-4 + 142-81 (0 + -00296)'

[5955]

2832

2834

:-*o;

198-6

S..[0]

128-4

[-20

1 144-1

r =

128-4 + 131-65 (tf-

■ -00401 y»

[-622]

283^2

284^6

:-•«>:

191-9

III. Double

Mass.

22-0 :

>99-7

914-4

33-9

23-6

•0186

•4455

•5704

•781

S, 21-7

5051

37-4

18-8

-0206

•6009

•7655

•785

3059

;w-8

15-3

0212

•7790

•9523

•818

193-7

39-8

124

•0219

•9573

11875

•806

1241

40-7

10-1

•0224

79-8

422

8-0

-0232

500

42-2

6-6

•0232

LXXXIX.

ON IMP

ACT.

31-4
19-5
12-2
7-6
4-6
2-7
1-6

43-6
44-4
44-6
44-6
44-6
44-6

D
50
4-1
3-3
2-5
1-8
1-3

T
•0240
-0244
-0246
-0246
•0246
-0246

[ 0

106-4

'20
'40

1220
1700

r=]

L06-4 + 132-96 (tf-

- -00645)'

[70

] 2997

3027

B„

106-4

[^20

-4o;

122-5
169-4

r = 106-4 + 126-40 (6 + •0078o)«

[-70-4] 299-7

2997

IV. Quad. A

Iass.

22-3 ;

J17-7

914-4

402

28-2

0211

•4621

-6340

•729

8, 22-3

486-3

47-0

23-7

-0248

•6290

•8021

•784

293-3

510

19-5

-0268

•7974

10035

•795

183-4

541

160

•0285

1-0355

1-2.527

-827

117-5

55-5

12-9

•0292

1-2783

15587

-820

75-2

56-9

10-5

•0300

48-7

57-8

8-7

•0304

31-5

59-5

6-7

•0313

19-9

59-5

5-2

•0313

12-3

60-5

4-0

•0318

7-4

61-9

3-0

•0326

4-6

620

2-2

•0326

2-7

62-0

1-5

•0326

1-7

620

1-1

•0326

Si. [ 0 ]

134-8

'40'

l.-,ll
201-6

r =134-8 + 140-51 (d-

- •00837)'

[66-5 ]

3177

321-2

««. [ 0 ]

134-8

[-20
-40:

152-0
202-0

r = 134-8 + 134-60 (6 + -00854)»

[- 66-25]

3177

317-2

16/6/91. Cob

I. Single M

ASS.

22-5 2

1723

1219-2

14-4

9-9

-0089?

•3640

•6732

•541

/3, 220

250-0

71-4

24-5

8-9

14-1
14-1
14-7
15-4

5-2
30
1-7
1-2

-0087
-0087
-0091
-0095

•7664

13230

•579

/3., [ 0 ]

230

[30
'60'
/8=. [ 0 ]

78-2
214-9

r = 21-7 + 148-72 (d + •0924)*

[691 ]

2723

272-4

23-0

■30]
-60:

50-2
1411

r = 22-8 + 116-22 (tf-

■03836)'

[- 8525]

2723

267-0

261

262

ON IMPACT.

[lxxxix.

II. Single Mass.

22-15
P, 21-9

286-8

A.[0
[30
[60]

A. [ 0 :

[-30
[-60

34-3

76-9
200-7

34-3

65-0

1580

III. Double Mass.

N
21-15
/3, 21-4

296-5

30]

[60]

116-1
151-5
258-6

A, [ 0 ] 116-1
[-30] 148-9
[- 60] 247-3

IV. Quad. Mass.

N
21-45
/3, 21-3

iSx,

321-7

[0]
20]

:40]

A, [ 0 ]

[-20]
[-40]

186-4
2020
248-8

186-4
201-5
246-0

1219-2

252-9

71-5

25-1

9-4

3-5

16-0
15-6
15-7
16-3
16-5
17-6

D
10-3
6-4
3-2
1-9
1-3
-8

T
•0093
-0090
-0091
-0094
-0096
-0102

Ai Aj

•3959 -7341
•8156 1-4176

e
•539
•575

r = 343 + 148^06 {0 + •0129)'

0 c

[74-4 ] 286-8 2886

r = 34-3 + 113-59 (d- •00384)' [-8483] 2868 282-0

1219-2

179-8

44-5

14-0

4-9

291
27-4
26-1
25-6
28<3

D
15-7
6-7
3-6
2-2
1-4

T
-0156
•0147
•0140
•0137
-0161

r = 116-1 + 130-66 (6 - -00732)«

r = 116-1 + 119-50 C^)*

H
1219-2
135-1

26-4
6-8

C
52-1
54-2
46-2
46-9

24-7

100

4-4

2-5

T
•0261
•0271
-0231
-0235

r = 186-04 + 128-4 {dy

A A« e

-4032 -9884 408

-9977 1-9458 513

0 c

[67 5 ] 296^5 2963

[- 7033] 2965 2962

-4245 1-2218 347
1-2550 2-7450 '457

0 c

[58-75] 321-7 3210

r = 186-4 + 120-65 {6 - -00663)' [- 606 ] 321-7 3196

24/6/91. Cork.

I. Single Mass.

N R

22-3 2731
7, 22-3

22860

373-6

88-6

29-7

11-4

4-6

24-1
19-7
17-2
16-9
17-4
17-8

20-5
7-5
3-7
2-1
1-4
1-0

T
-0148
•0121
-0105
-0104
-0107
-0109

Ai Ai e

•2568 ^5844 -439

•6322 12009 526

LXXXIX.]

ON IMPACT.

263

I\'

7i.

[ 0
20

;4o

184-2
202-1
254-7

»• = 184-2 + 142-15 (<? +

-00610)'

[45-1]

273-1

273-9

7., [ 0 ] l«4-2
[- 20] 109-4
;- 40] 2460

r = 184-2 + 129-02 C^-

-00558)'

[-30 ]

218-7

218-8

I. Single Mass.

22-2!

7, 221

7i.

J 289-2

[ 0 ] 198-1
20 ] 214-8
40 ] 265-3

H C D

2286-0 26-1 21-1

390-0 22-3 8-0

91-5 18-8 3-8

30-3 18-5 2-3

11-5 19-1 1-6

4-7 20-4 1-0

»• = 1981 + 138-87 (^ -

T
-0151
•0129
•0108
■0107
-0110
-0118

■00209)»

-2754
-6590

[46-7]

■6200

1-3079

289-2

e
-444
-.')04

c
289-7

7., [ 0 :

[-20-
[-40]

1 198-1
214-2
262-1

/• = 198-1 + 130-66 (^ +

•00192)'

[-30 ]

234-0

234-3

[I. Double Mass.

X R

21-8 30«-0
7. 21-8

H C D

2286-0 38-6 292

312-0 41-6 12-2

64-3 33-5 4-9

18-0 31-8 2-7

5-9 31-8 1-6

1-8 36-1 ? 1-1

T
•0206
-0222
-0179
•0170
•0170
•0193?

A>
•2908
•7776

A,
-7463
1-6697

-390
-466

7..[0-

[20'

401

242-4
257.5
304-7

r = 242-4 + 131-65 (^ -

•01011)'

[30]

277-1

c
277-1

7c. [ 0 ]
[-20

[-40]

242-4
2580
304-7

;• = 242-4 + 128-04 (^y

[-10]

246-3

V. Quad. Mass.

N R
22-1 323-8
/3. 22-0

H C D

2286-0 45-5 339

232-6 72-1 17-8

42-0 62-3 6-8

8-4 57-1 3-0

T
-0233
-0369
•0319
-0292

A.
•3102
-9244

A,
-9163
2-1283

e
•3385
•4743

" 0 ] 910

30] 12G-5

; 60 ] 236-0

r = 91-0 + 134-93 (0 - -01064)'

[75-5]

323-8

c
321-6

/?=.!

0
-30'
-60

910
1280
237-6

r = 91 + 132-30 (^ + -0052)»

[-40 ]

156-1

1564

264

ON IMPACT.

[lxxxix.

26/6/91. Vulcanised India-Rubber.

I. Single Mass.

N R
22-5 310-8

H
2438-4

C
26-4

29-8

T
-0143

A.
•2836

A,
•3578

e
•793

n. Single Mass.

N R
21-9 3000

2438-4

C
26-0

29-7

T
•0142

A,
•2733

A,
•3518

e
•777

m Double Mass.

N R
2205 285-7

2438-4

C
260

D
311

T
•0151

A,
•2586

A.
■3457

e
•748

IV. Quad. Mass.

N R
2205 270-1

H
2438-4

C
25-6

D
35-4

T
•0157

A,
•2348

A,
•3275

e
•717

3/7/91. Vulcanite.
I. Single Mass.

D T

22-35

2714

1219-2

1-6

1-2 -0010

•3620

•7002

•517

^, 22-3

2593

72^7

312

16-8

9-8

5-9

19
22
2-6
2-9
3-2
2-8
3-2

11 0012
-5 0014
•5 -0016
•3 -0018
•3 0020
-2 0017
•2 ^0020

•7265

13663

•532

[ 0 ]

12^4

•30^

59-8

r = 12-1 + 148-72 {0 + •04254)'

[734]

271-4

2720

60'

188^7

)3„ [ 0 ]

12-4

[-30

440

r = 124 + 12213 (0 - 01464)=

[-83^4]

2714

265-9

[-60

142-6

II. Double

Mass.

D T

224

2791

12192

2-1

2-3 -0013

•3561

•5250

-678

7. 22-5

492-3

1517

48^2

230

115

2-4
2-8
31
3-6
4-2

1-6 -0014

10 0017

•6 0019

•4 0022

•3 ^0025

•5372

•9424

570

7i,

[ 0 ]

127-5

'20'

1440

r = 127^5 + 14150 (^ -•0075)'

[59-7]

2791

278-9

[40]

195-0

7». [ 0 ]

127-5

[-20;

144-7

r = 1275 + 136-24 {9 - 00401)'

[600]

279^1

275-8

-40]

1951

LXXXIX.]

ON IMPACT.

265

III. Double Mass.

N
23-35
7, 23-2

R
299-3

7i.[0
[30

[60;

7..[o:

-30
-60^

127-7
167-5
289-7

127-7
168-0
286-8

2438-4

542-8

172-3

57-0

25-3

11-4

1-9

2-4

3-3

3-7

1-8

1-4

1-0

-5

-4

•3

T
-0011
-0012
-0014
-0019
-0022
-0023

r = 127-7 + 150-36 (0 - -009070)=

A] As
-2586 -4986
-5486 -8601

•519
-638

0 c

[61-75] 299-3 2999

r = 127-7 + 143-14 (6 + 0136)-' [- 624 ] 2993 301-7

IV. Quad. Mass.

N
22-85
)3, 22-5

308-7

30
60

17-6

66-0

197-4

A, [ 0 ] 17-6
[- 30J 47-3
[- 60] 145-7

V, Quad. Mas-s.

N R

22-75 321-1
/3, 22-5

/8,.[0
30
'60
)3„ [ 0
[-30
[-60]

49-0
90-3

2100
490
83-6

1910

1219-2

291-0

58-0

14-7

4-8

1-7

C
3-4
3-3
5-2
9-6
9-6
9-6

2-7
11
•9
-9
-5
-3

T
•0019
-0018
-0029
-0053
-0053
-0053

r = 17-3 + 151^35 (^+0436)=

A, A, e

•3805 7673 496
•7841 17090 -459

0 c

[775] 308^7 3123

r = 17-4 + 12541 (^-•03487)= [-882] 308-7 3012

24384

2721

539

13-3

C
23
26
50

8^5
8^4

D
1-5
0-8
0-8
0-8
0-4

T
-0012
-0014
•0027
•0045
•0045

r = 49 + 14281(d+0143)»

r = 49 + 132^63 {0 - 01255)'

A, A, e

•2867 8878 323
•8466 18040 ^469

0 c

[78-7] 3211 324

[-826] 3211 319^8

10/7/91. Lead.

I. Single Mass.

N R

22^55 279^0
T. II.

H
1219-2

C
2-5?

D
1-4?

T
•0015?

A,
•3640

A, e

•8230? -442?
34

266

IMPACT

LTiXXS

Single Mass.

N R

23-35 2950

1219-2

1206

211

1-8
2-0

0-8
04

•0011
•0012

IIL

Double Mass.

N R

22-55 304-2

1219-2

168-6

170

1-7
2-6
26

1-3

-7
•4

•0009
•0014
•0014

•3899

10355

•3765

IV.

Quad. Mass.

N R

23-0 325-3

1219-2

84-7
9-4

3-0

•0016

-4265

17461

•244

Quad. Mass.

N R

230 317-8

2438-4

4-5

-0024

-2773

19375

•143

29/6/91. Plane Tree.
L Single Mass.

21-1

269-6

1219-2

1-7

-0012

-3719

•7590

-490

$, 21-0

215-3

45 0

15-4

7-3

2^8
2-6
2-6
33

11
-6
-3
-2

•0016
-0015
-0015
-0019

-8785

1^6977

•617

[ 0 ]

54-6

30'

89-3

r = 54-6 + 130-66 (^ -

•00802)'

[73-9]

2696

269-2

'60'

195-6

)3,.

[0]

54-6

[-30]

84-7

r = 54-6 +110-64 (^-

-00209)'

[-80 ]

269^6

269-6

[-60]

175-5

[. Single S

Iass.

22-3

277-6

1219-2

1-1

-0007

-3640

•7618

•478

)3, 21-7

229-9

54-1

170

7-9

2-6
23
3-0
3-1

1-3
•7
-4
-2

-0016
-0014
-00l8
-0019

-8273

15340

•439

A, [ 0 ]

47-7

30"

85-3

r = 47-7

+ 134-93 {0 + -00436)»

[74-25]

277-6

276^8

'60]

196-9

/8«. [ 0 ;

47-7

[-30]

82-0

r = 47-7 + 123-77 (^ +

-00296)»

[-78 ]

277-6

2781

-6o;

184-2

LXXXIX.]

ON IMPACT.

267

III. Double Mass.

J2o 288-7

1219-2

2-2

1-9

•0013

•3679

•8754

•420

J2-5

198-6

2-9

1-2

•0017

•9004

17251

•522

48-8

3-4

•8

•0020

15-2

•6

•0030

5-5

8-5

•5

•0050

A. [ 0 ]

90-6

1280

r = 90-6+ 141-83 (^--(

)0994)''

[68-4]

288-7

289-3

'60-

243-2

/8.. [ 0 ]

90-6

[-30

1280

r = 90-6 + 134-60 {0 + •00349)»

[-69-4]

288-7

289-2

[- 60'

239-3

IV. Single Mass.

22-2
7. 211

299-9

2438-4

408-8

90-7

25-0

11-1

5-2

2*5
21
2^9
3-5
4-3

10
14
•9
•5
-3
•2

•0014
-0012
-0016
-0019
-0024

•2830
•6720

•6494
1-2846

•436
•523

7i.

[0]
20
40'

210-0
225-2
274-1

r = 210-0 + 138-21 C^-

-00174)=

[47-1]

299-9

c
303^0

72.

[o:

-20
-40'

210-0
227-4
2761

r = 210 + 128-37 {B + -00174)=

[-30 ]

247-9

245-4

. Double ]

Mass.

22-4
$, 21-4

316-4

2438-4

255-4

58-6

15-6

6-0

3-6

4-0

5-7

11-0

1-4
1-0

-7
-6

•0019
•0021
•0030
-0058

•2943
-9131

•9067
1-7747

•324
-515

/3,.

[ 0 ]
'30'
'60'

61-2

9.V8

201-2

r = 61-2 + 12902 (<>--

00575)=

[80-75]

316-4

c
314-3

■ 0 ■

-30'

-6o;

61-2

95-2

195^0

r = 61-2 + 119-83 {0 + -00907)=

[-83 ]

316-4

315-8

I. Quad, i

Iass.

21-9
/3, 22-4

313-4

2438-4

1261

17^4

3-2

5-2?
3-2
9^0
9-5

3-3?
•9
•8
•4

•0027?
-0017
•0047
•0050

•2867
12364

11263
33122

•255
•374

34—2

268

ox DfPACT.

Tlxxxel

A.I

[0]

A.L

:o]

[-20]

40]

lH7i>
2038
2.S4-3
l%7-0
203^
253-3

r=\Vi-^\9fi-i.\(0f

o c

[54-6] 313-4 3125

/•= 1 S7 + 1 34-27 «d + -00453 >» [-55-2] 3134 312-8

6 7 '91. Steel.

L SixoLE Mjum.
N R

22-3 2750

IL SlXGLE ILlss.
X R

21-0 2786

m Double Mass.
N R

21-75 284-3

IV. Single Ma.s8.
N R

218 292-5

V. Double Mass.
X R

21-.55 297-5

VL Quad. Mass.
N R

22-25 303-5

VII. Quad. Ma.ss.
N R

21-55 324-7

1219-2

2525

50-4

14-5

1219-2

2.53-9

51-6

15-0

1219-2

286-6

68-3

19-2

2438-4

.3850

57-7

113

2438-4

309O

62-5

102

H
1219-2
118-7
49

H
2438-4
83-5
6-7

1-1

1-5

2^)

2-7

C
-9
1-6
1-9
2-2

C
17
19
24
33

C
0-9

22
3-5

C
1-8
15
25
46

C
31

D
1-6
1-0
-5

1-5
-9
3
•1

2-0
-9
-6

1-6

•8

-3

T
-0007
-0009
■0012
•0016

T
•0005
•0009
•0011
•0012

T
•0010
OOll
■0014
•0019

T
•0005

•0012
0019

OOIO
•0008
•0014
•0025

•0017

A,
•3502
•7590

A,
-3939
•8079

A.
3620
•7346

A.
•2679
•6745

A,
-2720
-7813

A.
•3799

A-

•7292

1-6709

•8012
17113

•7028
14229

A,
•5716

1-6088

•7490

1-6755

A.
11145

e
■343
•454

e
•492
•472

e
•515
•516

e
•469
•419

e
•364
•466

e
•341

LXXXIX.]

ON IMPACT.

269

8/7/91. Glass.

I. Single Mass.

N R

22-5 279-7

68-0
6-1

2-0

-5

•0012

14882

5-0504

•294

II. Single Mass.

III.

Single Mass.

R H

282-8 609-6

1219-2

73-8

124-6

11-6

19-1

7/11/91. Vulcanised India-Rubber.

(Single Mass.)

I. Flat Base.

N R

21-4 277

1066

17-3

•0127

-3899

-4791

•814

624-2

22-8

14-3

-0132

•4986

•6168

•808

371-2

11-6

•0138

•6581

-8040

-818

221-3

-0144

130-5

25-9

7-2

•0149

77-5

27-2

5-8

-0157

4.5-5

281

4-4

•0162

26-1

29-0

3-5

•0167

15 0

300

2-9

•0173

31-2

-0180

II. Flat Base.

N R

21-65 283

1066

22-2

17-7

•0127

•3939

•4942

•797

...

23-9

14-7

-0136

•4986

•6368

•784

384-6

24-7

12-0

-0141

•6519

•8391

•777

2300

26-5

9-4

-0151

136-3

271

7-3

•0155

81-5

28-2

6-0

•0161

48-0

29-0

4-9

•0166

27-7

307

4-0

-0175

16-0

320

•0183

9-0

32-5

-0186

III. Ridged Base.

N R

21-8 283

1066

34-9

26-2

0201

-4122

•5195

•793

559-6

37-9

21-2

-0218

-5430

-6681

•813

338-5

39-0

17-3

•0224

•6916

-8481

•815

207-8

40-8

14-3

•0235

128-0

41-7

11-2

•0240

78-8

417

8-9

•0240

48 0

430

7-0

•0247

290

440

5-6

-0253

170

44-3

•0255

270

ox IMPACT.

[T.v-gTrr

IV. Ridged Base.
N R

21-6 296-5

1066

604-5

363-0

221-3

135-5

81-9

50-3

30-1

17-2

35-5
30-9
41-5
43-8
44-9
46-7
4«-7
472
48-6

27-0

22-2

18-0

145

11-9

5-7

4-6

T
-0193
•0217
•0226
•023s
^244
-0254
-0254
•0257
•0264

A.
•4142
•5430
•7062

A.,

5250
-6S73
-8770

e
-789
-700
-805

13/7/92.
N

19-783

^,2005
f, 20-54
S, 20-53
e. 20-78

VULCAXISED IXDIA-RUBBEB. (SiSGLE MaSS.)

R
333-3

1000

592-8

354-6

213-2

1291

74-9

42-6

23-3

12-6

2-8

1-3

27-8
291
303
31-6
32-8
34-1
35-9
37-7
39-3
403
421
42-9

16-5
133
10-9
8-9
7-2
5-6
4-5
3-5
2-5
1-7
1-3
08

-0123
-0129
•0134
•0140
•0145
•0151
•0159
•0167
-0174
•0178
•0186
-0190

-5206

-6728

-8511

1-0756

1-4097

1-8040

2-4142

3-2540

4.3433

-5098

•6617

•8356

1^0776

•6556
•8069
1-06S0
13151
17205
2.3314
3^0656
43373
6-1066

•6486

•8138

10540

[-
[-

75.

[-
[-

[-
[■

30
60
0
30
-60

[60]

[90"

[0]
-60'
-90

[0]

[30

[60]

[0]
-30'
-60

[0]

[20

[40

[o:

-20
■ 40

- 259-7
-228-0
-134-7

- 259-7

- 2.30-5
-14-2-7

- 21-9
-HllS-O

277-8

- 21-9
+ 94-0

243-9
120-5
153-0
250-0
120-5
152-5
246-1
203-9
219-2
203-9

2039
217-8
259-7

r = - 259-7 -I- 112-33 {6 + •0077)'

r = - 259-7 + 106-88 {6 - -0009)'

r = - 21-9 -I- 118-34 {0 + 0208 )»

r = - 21 -9 -1- 1 1 1-81 (0 - -0288)*

r = 120-5 -f- 117-62 {6 + •0021)'

r = 120-5 -1- 112-33 {6 + -OlOiy"

r = 203-9 + 120-64 {0 + -0072)=

[120 ]
[130-57;

+ 2420
+ 333-3

236-7
3276

[-120 ]
- 135-05]

-1- 207-0
333-3

208-8
333-7

[70 ■
[97-95;

161-0
333-3

160-8
332-4

[-70 ]
[- 103-85;

137-6
3333

137-1
333-7

[7C-9 ]

333-3

333

[- 4-5 ]

;- 78-3 ;

191-5
333-3

191-6
333-4

[58-9 ]

333-3

333-2

[-50

;-60-5 ;

291-6
333-3

290-7
331-1

r = 203-9 -!- 114-89 {0 - -0037)=
[This was a single experiment, specially designed for the Niimberg Exhibition.]

•794
-834
•797
•818
•819
•774
•788
•750
•711

LXXXIX.]

ON IMPACT.

271

20/5 01. UXHAMMERED GoLF BaLL. (WoOD BlOCK UnSHOD.)

SlNOLK

Mass.

227

240-7

1219-2

6-6

6-0

•00465

•3272

•5902

•555

30t)-0

8-6

-00005

-0371

-9325

•683

lO.S-9

9-9

2-7

•00097

1-0110

1-5008

-648

43-0

10-1

1-9

•00711

18-3

10-1

1-4

•00711

s-1

10-5

-9

-00739

.•5-8

11-6

-8

-00817

1-7

II.

21-9

257 3

1219-2

7-3

6-3

•00464

-3504

-6050

•579

337-4

9-5

4-3

•00003

-6140

-9896

•621

llS-6

10-8

2-8

•00086

1-1039

1-6022

•089

47-3

111

1-9

•00705

205

11-2

1-3

•00711

9-5

11-4

•9

•00724

4-6

11-8

-6

•00750

2-3

11-8

-4

•00750

III.

DocHLE Mass.

A.,

21-9

273-5

1219-2

10-9

7-6

•00651

•3689

•7360

•501

272-0

14-7

5-5

•00878

-7536

12916

•583

98-8

170

•01016

1-27.53

1-9774

•645

401

177

2-6

•01058

17-9

1-8

•01070

8-4

17-9

1-2

-01070

4-0

18-6

-9

•01111

18-7

•7

•01117

IV.

Qu.\D.

Mass.

21-9

2S8-7

1219-2

15-6

101

-00?«83

•3819

•8430

•453

233-0

22-6

7-7

-01279

-8391

1-4154

•593

82-1

2G-5

5-3

-01500

1-4200

2-2198

640

32-6

28-1

3-5

-01591

14-3

28-5

2-4

•01613

G-4

28-5

1-6

•01013

2-8

29-5

-01070

2S/

5 91. Hammered Golf Ball. (Block Unshod.)

SiNULE

Mass.

21-8

245-3

1219-2

3780

7-8

4-1

•00517

•5902

-9163

-644

144-3

2-7

•00597

■9358

1-3814

•677

61-5

9-3

2-0

•00617

27-5

10-5

1-5

•00096

12-4

11-9

1-1

•00789

0-2

12-4

-9

•00822

2-9

272

11.

HI.

IV.

OlS

1 IMPACT.

[LXXXIX.

N R

21-6 254-3

1219-2

6-3

5-7

•00499

•3581

•5384

•665

386-3

8-2

4-0

•00520

•6285

•9490

•662

149-2

9-0

2-9

•00571

•9725

1-4388

•676

63-4

10-1

2-2

•00640

29-9

10-2

1-5

•00647

14-2

11-0

1-2

-00697

6-7

11-7

-8

•00742

3-0

Double Mass.

N R

21-6 272-5

1219-2

9-2

6-7

•00544

•3679

•6745

•545

321-3

12-3

5-6

•00728

•6835

10486

•652

132-6

13-3

3-9

•00787

10432

1-6085

•648

59-8

14-7

2-7

•00870

280

15-6

2-1

•00923

13-7

16-3

1-6

•00964

6-9

17*5

1-2

•01035

3-5

18-2

1-0

•01077

1-6

Quad. Mass.

N R

22-6 288-1

1219-2

13-4

9-4

•00784

•3640

•7646

•476

279-9

17-6

6-9

•01030

•7308

1-2505

•584

108-1

20-1

4-6

•01177

11771

1^9596

-601

44-0

22-0

.3-2

-01288

18-9

23-8

2-6

-0l:!93

23-0

1-6

-01346

4-3

2-0

1/3/92. Hammered Golf Ball.
I. (Steel Plate.)

(All Single Mass.)

N R

21-75 263

1219-2

5-9

5-0

-00364

•3410

•5820

•586

297-2

7-8

3-5

-0048

•6330

1-0176

•622

1050

9-0

-0056

10913

1-6865

•647

39-2

9-5

1-6

•00586

15-8

10-9

1-2

-00617

6-9

111

0-9

•00685

2-7

114

-7

•00703

II. (Steel Plate.)

N R

23-2 273-0

1219-2

6-9

5-9?

•00438

•3551

•6009

•591

354-2

8-6

3-7

•00545

•6627

1^0247

•647

123-1

9-3

2-5

•00590

11132

16842

•661

45-9

1-7

•00609

19-0

10-6

1-2

•00672

8-4

111

-9

•00704

3-7

III.

IV.

VI,

II.

XXIX.J

OX IMPACT.

(Wood.)

N R

A...

210 27!)-5

121!>2

5-S

4-3

•00325

•3462

•5774

•600

3.S30

7-7

3-5

-00432

•6208

•9691

•641

131-9

8-3

2-1

-004(55

10283

15911

•646

49-4

.S-9

1-G

•00499

201

9-5

1-1

•00533

H-6

9-8

•8

•00549

(Wood.)

N R

22-3 2l>3()

121!>2

«J-4

4-7?

-003(i3

•4040

•7028

•575

3S4-7

S-2

3-0 ?

-004(J5

•6644

10538

•630

134-0

9-5

2-4

-00539

11028

16G43

•663

50-9

10-1

1-8

•00573

21-0

10-9

•00(518

10-9

-9

•00G18

10-9

•5

•00(518

(Wood.)

N R

22-6 3()(;-5

1219-2

(\l?

4-2

•00336

•3939

•6656

•5!»3

390-7

H-^f

3-0?

•004(52

•6758

1-2685

•533

10(iO

10-u

2-2

•00550

12647

19500

•649

41-0

10-7

1-7

•00589

lC-9

11«

1-2

•00(538

7-4

12-5

•00688

3-2

14-4

•00792

(Steel Plate.)

N R

21-35 310\S

1219-2

8-0

5-5

•0041

•4204

•6681

•629

3SI-H

10-.S

4-0?

•00554

•7178

12572

•571

1027

12-3

2-5

•00(531

1-3968

21742

•642

37-0

12-7

1-7

•00652

15-0

13-7

•00703

0-()

14-8

•9

•00759

3 J)2. Uxhammered Golf

Ball.

(All Sin

OLE Ma.SS.)

(Steel Plate.)

N R

22-45 275-4

1219-2

4-7

•00335

-3581

•6009

•596

373-9

(>-8

3-0

•00414

•6334

10000

•633

12.S-0

8-G

2-1

•00523

10724

17217

•623

45-5

8-«

1-3

■00523

17-4

0-9

•00548

(i--S

100

•7

•0()60.s

N R

(.;

21-15 2N3-(i

1219-2

5-7

4-8

•00317

•3819

•5914

•645

420-2

7-3

3-5

•00406

•6249

1-0053

•622

144-3

8-7

2-3

•00484

10488

17532

•598

52-0

9-9

•00551

19-4

10-3

•00573

7-5

10-6

•8

•00590

273

T. II.

274

■J9

•juTt

r. m m m 1^

1 H

5^ 1 H

lis H)5-i

J*3

Jr-I

1*5

J-)

S-1 •••MSI

tir:^ ix, 1 r

1J3-I
±15

i*-j

4f4 ^MSK

-ear

Hal

fli

♦*r3

:5.>s

1-! #3" -Jt'sa^
:3'i :-j -Ji^m

S3w

#
^s«

- ;7cvi.

-«15

3f 1 H

y 1 H

HfT ^r^ lilt

."5 ~*'«'»T^y7 _-». !_?■ ;

A A,

•«»* inr* .^::5

II.

Ill,

IV,

VI.

XXIX.J

ON IMPACT.

Single Mass.

N R

22-2 268-6

1219-2

8-0

7-4

-00494

•3310

•5200

•636

458-2

9-7

5-4

-00598

•5362

•8332

•644

1732

11-3

4-0

-00697

•8894

13151

•676

68-5

121

2-9

-00747

28-8

13-6

2-0

-00839

12-8

14-3

1-4

•00882

5-6

15-4

1-0

-00950

17-4

-7

-01074

. Single Mass.

N R

221 2770

12192

8-9

7-5

-00530

•3696

•5543

•667

4C7-8

10-3

•00613

•5766

•8682

•664

178-0

120

3-8

-00714

•9358

13900

•673

71-3

131

2-8

•00780

30-7

14-1

2-0

-00840

137

14-6

1-4

•00869

6-1

12-8

-8

•00762

Double Mass.

N R

22-6 285-2

1219-2

11-8

9-2

•00698

•3682

•5695

•646

37.3-9

141

6-9

-00834

•6273

•8926

•702

143-3

173

-01023

10064

1-4804

•680

59-8

18-2

•01076

25-9

19-7

2-4

•01165

11-5?

20-2

1-7

•01196

5-0

21-6

1-2

•01277

Double Mass.

N R

2312 2971

1219-2

121

9-9?

•00703

•3705

-5670

•653

408-2

16-0

7-3

•00929

•6350

-9025

•704

1613

18-0

5-2

•01045

•9896

1-4770

•670

68-0

19-5

3-6

•01132

29-9

20-8

2-6

•01208

133

21-4

1-8

■01243

5-9

20-5

1-2

-01190

2-9

22-2

0-9

•01289

Quad. Mass.

N R

21-5 312-2

1219-2

19-5

12-5?

•01002

•4215

•7248

•582

286-6

26-5

9-3

•01362

•8069

11840

•682

115-3

29-0

6-6

•01490

12746

19170

-665

47-7

31-4

4-6

•01614

20-0

31-5

3-0

•01619

9-0

35-6

2-2

•01829

3-7

35-6

1-5

•01829

275

35—2

276

ox IMPACT.

[lxxxix.

24/8/92. Unhammered Golf
I. Single Mass.

Ball on Unhammered Golf Ball.

II.

VI.

N R

22-35 262-3

12192

7-1

•00528

.3424

-5206

658

419.>

9-6

-00610

-5774

8214

•703

1610

11-0

3-6

-00699

-8988

1-3556

•663

64-1

114

•00725

26-6

132

•00839

122

136

1-4

-00865

5-5

151

•00960

Single Mass.

X R

22-05 270*3

1219-2

ts-1

•00493

•34«8

•5392

■647

461-9

9-8

•00597

5693

•8243

•691

1765

11-2

i:h)682

-9099

13352

6S1

69-1

12 3

-00749

305

13-0

1-9

•00791

135

133

-00810

6-1

136

0-9

•00828

. Single Mass.

N R

214 27S-2

1219 2

S-7

-00499

3M9

•5658

675

4732

106 ?

54?

-00608

-5758

8391

•6S«

1.N46

11-8

3-9

-00677

9244

13238

•698

736

12 5

-00718

317

141

-00vS09

143

138

•00792

151?

-8?

-00i>67

DijCBLE Mass.

X R

2112 2>»}"5

1219 2

12-7

9-2

-00699

-.3819

6342

•60S

:iM-2

l.V>

6-8

D0N69

-6745

•9657

•698

l.MJl

ISl

5-2

-00996

1D538

15014

702

667

195
19 7

36
24

1)1073
-010S4

126

211)

•01155

5-7

211)

-01155

I>:.r-BLE Mass.

X R

A-

*2«> 2>T1

121:'2

130

1X)718

•3>»09

■5957

•639

4239

16-4

7 5

00!X)6

-6494

•9as3

•715

16:r5

lv9

t)1044

11X»>8

14578

692

71i>

201

-01111

314

20-9

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1.35

22 2

1)1227

."•:•

233

1-3

<»1 2^^

24<>

-nl3-26

. QVAD. Ma-j?.

X R

«;

21;* 3136

121:»2

19<»

120

i»!''90

41. i6

•7400

•562

321 i»

2.-6

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01.3:J4

-79s3

112»J3

709

13-3

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1)1501

1-1988

17321

■692

5^;>

311)

1)1615

25-9

310

HI 662

11-6

34S

DIMS

5 1

.352

Dis;i4

LXXXIX.J

ON IMPACT.

277

2/6/92. Eclipse Ball — Steel Plate.
I. SiXQLE Mass.

III.

IV.

VI.

22-25 27.3-8

1219-2

9-5

7-3

•00576

3541

•6346

•558

11-2

4-6

-00679

•0669

11504

-580

107 0

12-2

3-0

•00740

38-1

14-0

2-0

-00850

13-9

140

1-3

-00850

Single Mass.

N R

22-4 281-7

1219-2

9-8

7-3

-00582

•3696

•6656

•555

333-8

11-5

4-7

-00682

•7107

1-1648

•610

106-0

12-8

3-0

-00760

39-3

140

2-0

•00831

14-6

1-3

-00866

«!,

[0]

267-7

= 6'

2690

r —

267-7 + -04444 {0 -

■ -5625)'

[1848]

282

273-5

fj.

267-7

[-6]

269-4

r =

267-7 + -04028(^4

■ •ol7)»

[18^38]

282

2821

-12;

2740

. Double Mass.

N R

22-2 290-8

1219-2

140

9-9

-00798

•3676

-7146

•514

291-9

17-8

6-4

-01014

•7590

13143

•577

87-4

19-9

-01134

29-6

20-6

2-3

-01174

10-5

20-4

•01162

Double Mass.

X R

21-55 300

1219-2

15-1

10-2

-00809

•4061

-7391

•549

297-0

19-3

6-4

-01035

■8142

1-3865

•687

88-8

20-9

4-0

-01120

30-4

22-8

2-5

-01222

10-7

24-2

1-7

-01297

3-5

26-0

1-0

•01394

Quad. Mass.

N R

22-22 314-5

12192

19-7

12-2

•01039

•4115

-8746

•471

241-2

27-4

8-0

•01445

•9179

1-6577

•554

66-6?

30-6

4-5

•01G13

20-5

343

2-9

•01808

6-2

36-8

1-5

•01940

Quad. Mass.

N R

22-6 324-3

1219 2

20-1

12-5

•01045

•4149

-8889

•467

245-0

270

7-8

•01404

•9163

1-8094

•506

63-6

32-0

4-5

•01664

19-7

35-5

2-8

•01846

5-8

36-8?

1-3?

•01913

278

ON IMPACT.

[t.xxxix.

APPROXIMATE COEFFICIENTS OF RESTITUTION.

Successive Values of e (1 — m), calculated by the First Formula in the Paper. (The
suffix to the number of the experiment indicates the mass, and the height of the
first fall is quoted.)

7/4/91. Old V. I. R.
1219.
II. Ill, IV,

•64

•54

•74

•78 •'

•77

78-

•80 •

•78

•82 •.

•78

•82- •.

•78+

•83

•79

•83

•78

•82 •

•77

•79

■>Tft —

•57
•73

•78-

•79

•82

•82+

•82

•81

•83

12/6/91. New N.

750.

I. II. Ill,

•82 -82 84

•83 82 86

•84 85 -87

•83 82 87

•82 ^82 -87

•82 82 ^86

•81 -82 86

•82 81

•81 -81

•81 81

•79 -79

•79 80-

•77 80+

•80 85

•74 79

•88 -83
•82

86
85
85
87
85
84
84
84
84
84
84
89

I. R.

IV.

85-

87 +

22/6/91. New V. I. R.
9144.

I,
71

74
77
77

II.

•71
•75

•77
•77

75+ -77

76 -77

76 76-

76 75

77 ^75-
76+ ^77
74 ^75+

III,
•74-

•78

•79+

•80

•79

•78

•77

IV.
•73

•78

•79

•80

•80+

•80

•79

•78

•77

•79

•77

13/7/92
1000.

•77
•77
•77
•77
•76
•75
•74
•74
•70
•68
•68

16/6/91. Cork.
1219.

I. II, III, IV,

•46 45- 38 -33

•53 -53 49 44

•59- 59 56 50

•60 61 59
•61

24/6/91. Cork.
2286.

Ii
•40
•49
•58
•63
•63

II.
•41
•49
•58
•61
■64

III, IV,

•32 31+

•45 -42

•53- 45
•57 +
•55

I.
•46
•53
•65
•71
•76
•78
•78
•76

3/7/91. Vulcanite.

1219. 2438. 1219. 2438.

HI, IV, V,

•47 49

•56 -45

•58 -50

•67 57

•67+ -59

II,
•63
•55
•56
•69
•71

33
•44
•49
•58

29/6/91. Plane.

1219.

2438.

II.

III, IV.

VI,

•42

•43

•40 41

•32

•46

•49

•49 ^47

•48

•37

•58

•56

•56- 52

•51

•43

■69

•69

•60 66
•68

•56

■45
■42
•53

6/7/91. Steel.

1219.

II. Ill,

•45 49

•53 52

2438.
IV. V,

•39 36
•39- -45
•44 40

1219. 2438
VI, VII,

•31
•20

•19
•29

LXXXIX.]

ON IMPACT.

279

7/11/91. V. I. E.
1066.
Flat Base. Ridged Base.
I, II, III, IV,
•74
•77
•78
•78
•78-
•78
•78-
•75

Golf Ball, Wood Base.

•72 •

•77

•78

■77

•77

•78

•77

•78

•76

•77

•78

•74

•77

•78

•75

•76

•78- •'

•76

•77 •'

•75

26/6/91. Unhammeeed.
1219.
II, III,

•50
•59
•63
•65
•66
•68
•66

•62

•59

•63

•66

•68

•70-

•70

47
60
63
67

IV.

•44
•59
•63
•66

69- -66
69 -66
70

28/5/91. Hammered.

1219.

I, II, III, IV,

55 56 -51 48

63 62 64 60-

65 65 67 64

67 69- 68 65

67 69 -70 -69

70 69- -71 69

69 67 71 ^68

280 [xc.

xc.

QUATERNION NOTES.

[Proceedings of the Royal Society of Edinburgh, June 4, 1888.]

(a) Prof. Cayley's paper*, which was read at last meeting, reminded me of an
old investigation which I gave only in brief abstract in our Proceedings for 1870 \^Ant^,
No. XVIII.]. There is, unfortunately, a misprint f in the chief formula of transformation.
In fact, we have quite generally, as a matter of quaternion analysis,

= (Vcr)2 - S . Vi(7iV .a-Va-. SVa + S VVj . a-.a.
The hydrokinetic equation is

so that F.V2)^o- = 0;

or, by the above transformation,

F.2),V(7=F(VififcriV.cr),
which is the equation treated by Cayley.

It is worthy of note that the right-hand member may be written as
V(Vay-S,V,ay .a-Wa.S^a- [or as W<t .SVtr - S .V^a.V .a']
because S Wj . Va^a = 0 identically.

* [Collected Papers, Vol. xin., No. 890. Note on the Uydrodynamical Equations,']

t [Also an omiesiou, corrected in the Heprint. The expression itself occurred to me whUe I was mAlriTig the
translation of v. Helmholtz' paper on Vortex Motion which appeared in Phil, Mag,, u., 1867. The multifonD
transformations of the expression V^c^c furnish a very interesting and instructiTe exercise in Quaternions. 1899.1

XC] QUATERNION NOTES. 281

If we now introduce the equation of continuity

flfVo- = 0,

we have (as in the abstract referred to)

D„Va = - S . VitTiV . a = S^aCT,
with the further result

(b) The second note contains additions, of which

JJJV.VVaTd^^JJ{TS<rUv-(rSTUv)d8

may be given as a specimen, to the paper on Quaternion Integrals printed in abstract
as No. XXII. above. [One of the chief special applications, for which these formulae
were devised, was the comparison of integrals taken over the same finite closed
surface. Thus for instance, even in the simple particular case cited, we have some
remarkable equalities on the right from the mere assumption that o* and r satisfy
(ill any of the infinite variety of ways possible) the condition

V VoT = scalar,
or F<rr = Vt;. 1899.]

T. II. 36

282 [xci.

XCI

OBITUARY NOTICE OF BALFOUR STEWART.

[Proceedings of the Royal Society of London, 1889.]

Dr Balfour Stewart was bom in Edinburgh on November Ist, 1828, and died
in Ireland on December 18th, 1887, having just entered his sixtieth year. He was
educated for a mercantile profession, and in fact spent some time in Leith, and after-
wards in Australia, as a man of business. But the bent of his mind towards physical
science was so strong that he resumed his studies in Edinburgh University, and soon
became assistant to Professor J. D. Forbes, of whose class he had been a distinguished
member. This association with one of the ablest experimenters of the day seems to
have had much influence on his career; for Forbes s researches (other than his Glacier
work) were mainly in the departments of Heat, Meteorology, and Terrestrial Magnetism,
and it was to these subjects that Stewart devoted the greater part of his life. In
the classes of Professor Kelland, Stewart had a brilliant career; and gave evidence
that he might have become a mathematician, had he not confined himself almost
exclusively to experimental science.

In 1858, while he was still with Forbes, Stewart completed the first set of his
investigations on Radiant Heat, and arrived at a remarkable extension of Provost's
" Law of Exchanges." His paper (which was published in the Transactions of the RoycX
Society of Edinburgh) contained the greatest step which had been taken in the subject
since the early days of Melloni and Forbes. The fact that radiation is not a mere
surface phenomenon, but takes place like absorption throughout the interior of bodies
was seen to be an immediate consequence of the new mode in which Stewart viewed
the subject. Stewarts reasoning is, throughout, of an extremely simple chco-acter, and
is based entirely upon the assumption (taken as an experimentally ascertained fact) that
in an enclosure, impervious to heat and containing no source of heat, not only will
the contents acquire the same temperature, but the radiation at all points and in all

xcl]

OBXTUABY NOTICE OF BALFOUR STEWART.

!83

directions will ultimately become the same, in character and in intensity alike. It
follows that the radiation is» throughout, that of a blaek body at the temperature of
the enclosure. From this, by the simplest reasoning, it follows that the radiating and
absorbing jjowers of any substance must be exactly proportional to one another (equal,
in fact, if measured in proper units), not merely for the radiation as a whole, but
for every definitely specified constituent of it. In Stewart s paper (as in those of the
majority of young authors) there was a great deal of redundant matter, intended to
ihow that his new views were compatible with all that had been previously known,
and in consequence his work has been somewhat lightly spoken of, even by some com-
petent judges. These allow that he succeeded in showiBg that equality of radiation
and abi^orption is consistent with all that was known ; but they refuse to acknowledge
that he had proved it to be necessarily true. To such we would recommend a perusal
of Stewart's article in the Philosophical Magazine (Vol, xxxv,, 1863, p. 354), where they
will find his own views about the meaning of his own paper. The only well-founded
objection which has been raised to Stewart's proof applies equally to all proofs which
have since been given, viz., iu none of them is provision made for the peculiar phenomena
of fluorescence and phosphorescence.

The subject of radiation, and connected prcjperties of the luminiferous medium,
occupied Stewart*s mind at intervals to the very end of his life, and led to a number
of observations and experiments, most of which have been laid before the Royal Society*
Such are the " Observations with a Rigid Spectroscope " and those on the ** Heating
of a Disk by rapid Rotation in Vacuo/' in which the present writer took part. Other
allied speculations are on the connection between " Solar Spots aod Planetary Con-
figurations," and on *' Thermal Equilibrium in an Enclosure containing Matter in Visible
Motion/'

From 1859 to 1870 Stewart occupied, with distinguished success, the post of Director
of the Kew Observatory. Thence he was transferred to Manchester as Professor of
Physics in the Owens College, in which capacity he remained till his death. His main
subject for many yc^rs was Terrestrial Magnetism; and on it he wrote an excellent
article for the recent edition of the Encyclopedia Briiminica. A very complete summary
of his work on this subject has been giveu by Schuster in the Manchester Memoirs
(4th Series, Vol. L, 1888). In the same article will be found a complete list of Stewart*s
papers.

Among the separate works published by Stewart, his Treatise on Heat, which haa
already reached its fifth edition, must be specially mentioned* It is an excellent intro-
duction to the subject, though written much more from the experimental than from
the theoretical point of view. In the discussion of radiation, however, which is given
at considerable length, a great deal of theoretical matter of a highly original character
is introduced.

Of another work, in which Stewart took a great part, 77* e Vtiseen Universe, the
writer cannot speak at length* It has passed through many editions, and has experienced
every variety of reception^from hearty welcome and approval in some quarters to the
extremes of fierce denunciation, or of lofty scorn, in others. Whatever its merits or
demerits it has undoubtedly been successful in one of its main objects, vix., in showing

36—2

284 OBITUARY NOTICE OF BALFOUR STEWART. [XCI.

how baseless is the common statement that ''Science is incompatible with Religion." It
calls attention to the simple fact, ignored by too many professed instructors of the public,
that human science has its limits ; and that there are realities with which it is altogether
incompetent to deal.

Personally, Stewart was one of the most lovable of men, modest and unassuming,
but fiill of the most weird and grotesque ideas. His conversation could not fail to set
one a-thinking, and in that respect he was singularly like Clerk-Maxwell. In 1870 he
met with a frightful railway accident, from the efiFects of which he never folly recovered.
He passed in a few months from the vigorous activity of the prime of life to grey-headed
old age. But his characteristic patience was unruffled and his intellect unimpaired.

He became a Fellow of the Royal Society in 1862, and in 1868 he received the
Bumford Medal

His life was an active and highly usefol one; and his work, whether it took the
form of original investigation, of accurate and laborious observation, or of practical teaching,
was always heartily and conscientiously carried out. When a statement such as this can
be truthfolly made, it needs no amplification.

XCII.]

285

XCII.

THE RELATION AMONG FOUR VECTORS.

[Proceedings of the Royal Society of Edinburgh, March 4, 1889.]

A SYSTEM of five points is completely determined by the vectors joining one of
them with the other four. If a, yS, 7 be three of these, the fourth is necessarily

Hence any property characteristic of a group of five points will remain when a?, y, z
are eliminated. But we have

Sah =s xSaa + ySap + zSaiy ,

SI3S = xSfix + ySfil3 + zSfiy,

SyS = xSya + ySyl3 + zSyy,

SBB = xSBa + ySBfi + zSSy .

Hence, at once, a determinant of the 4th order.

If we note that each term, as Sfiy for instance, can be written either as

i()8» + 7»-)8-7«) or as ^T^Tycosfiy,
we see that the determinant may be written either in Dr Muir's form or as

0= 1 cosa)9 cos 07 cosa^

cosySa 1 008)87 COB/3B

cos 7a cosy 13 1 C0S7S

cosSa coscjs cos 87 1

286

THE RELATION AMONG FOUR VECTORS.

[xcn.

which is the relation among the sides and the diagonals of a spherical quadrilateral. The
method above can, of course, be extended to any number of points. One additional
point introduces three new scalars to be eliminated, and six new scalar equations for
the purpose.

{Addition— Betid March 18.)
If we operate, as above, with any other four vectors, we have

Sa^a 8ail3 Sa^y Sa,8 =0,

Sl3,a SM Sl3,y 8/3,8

8y,(x Sy,l3 Sy,y 8y,S

8S,a 88,13 8S,y 8S,B

and the tensors are again factors of rows or columns. Thus, if ABCD, abed, be any
two spherical quadrilaterals,

cos^a cos -46 cos -4c cos Ad =0.

cos 5a cos Bb cos Be cos Bd

cos Ca cos Cb cos Ce cos Cd

cos Da cos Db cos Dc cos Dd

This has many curious particular forms; one, of course, being the former result, when
the two quadrilaterals coincide. Another is when the quadrilaterals are "polar." Let
a be the pole of AB, b of BC, &c., then

cos Ab cos Be cos Cd cos Da — cos Ac cos Bd cos Ca cos Db = 0.

And numerous other relations can be obtained, with equal ease, by the same simple
process.

Cayley's form of the expression connecting the distances, two and two, among five
points in space is an immediate consequence of the identity

txi^a - ey = Ixa^ - 28eixa -f 6^tx,

where «!, a,, &c., are n given vectors, 6 any vector whatever, and x,, a?j, &c., n undeter-
mined scalars.

For, provided that n is greater than 4, we may always assume

2a? = 0, 2a:a = 0,
which are equivalent to four homogenecms linear relations among the xs.

Let, then, n = 5, and write the above identity separately for each a, put in place of 0.
Thus we have

tx (a - a,)» = lxa\

lx{a-a^y=^xa\

2^ (a — a^ = 2a:a».

XCII.] THE RELATION AMONG FOUR VECTORS. 287

Take, with these, 2a: = 0,

and we obtain six linear equations from which to eliminate the five values of x. The
result is, at once. A, B, C, D, E being the points,

AA^ BA^ CA^ DA' EA' 1 Sa?a» = 0.
AB» BB' G& D& E& 1

AE' BE' CE' DE' EE' 1
111110

As ^xa' may have any value, this is Cayley's expression*. An interesting variation of it
is supplied by taking X {xx) = 0, instead of X (x) = 0, as the sixth equation.

* [Collected Papers, No. 1. Dr Mair*s expression, mentioned above, is given in Proc, i?. 5. £., xvi., p. 86.
1899.]

ia8 son:

xcnL

atm sr ite JLi^md iictga if Jjoamrtk^ Ajr'i L licif.

ITT MZS = I ^ T^UfiL

r T^ -ae -Tnaattm^ t£ -a*- «iic& ^ -iui iri>& if me -huL sad x a unn-^nicair _ ^

-w^uisst a jt -3**^ i^gra. if UL ^^mtmr if "atf 'wnmmic mr^ if "aitf ^ecdin^ joii 3i»

-aar _

> x. 7 = - tj. if = Til

'ir ^ii:mwk«nna \t ia Tumrri* inmxKT if ii^iaisst "nrTnrr>*ij anskl nr%sai -Mffntffii»L z •3inr

IftT-* IT lOr^^

▼mm J* "r^ut nuuianiKiiEAi irni \i "3ut T?iir!acf!r ixiii Lin** r-rraeynL

-*fiiiar.i)n .r * viPinniiiry ' *"i "^k ^emnii j* Tn*rsiT ihe T«rdcTLiir 3aee sncrsspiHiiixiiic ^

XC17.]

289

XCIV.

QUATERNION NOTE ON A GEOMETRICAL PROBLEM.

[Pfvceedings of the Royal Society of Edinburgh, June 4, 1889.]

The problem referred to is that of inscribing in a sphere a closed n-sided polygon,
whose aides shall pass respectively through n given points which are not on the surface,
Hamilton evidently regarded his solution of this question as a very tough piece of mathe-
matics (see bis Life, Vol IIL pp, 88, 426). In preparing the third edition of my Quaterniojt^,
I was led to a mode of treating this question which enables us to dispense with the
brilliant feats of analysis which seem to be required in Hamilton's method.

[A sketch of his very curious analysis is given in § 250 of that work* § 250*
gives the full text of my own process. As I have since found it to be needlessly
prolix, it is considerably pruned down and concentrated in the present reprint. 1899.]

The quaternions which Hamilton employed were such as change the radius to one
comer of the polygon into that to the next by a conical rotation. La the present Note
I employ the quaternions which directly turn one side of the polygon to lie along the
next. The sides, severally, are expressed as ratios of two of these successive quaternions.

Let pit pa, &c., p„ be (unit) vectors drawn from the centre of the sphere to the
comers of the polygon; a,, Oa, ,»*«„, the points through which the successive sides are
to pass. Then (by Euclid) we have n equations of the form

(/?^i - flrn) (pm " Am) = 1 + ^m' = A^ , SUppOSe.

These equations ensure that if the tensor of any one of the p'b be unit, those of all
the others shall also be units. Thus we have merely to eliminate p^, --*> Pnl ^^^ then
remark that (for the closure of the polygon) we must have

Pn+l = /?l*

That this elimination is possible we see from the fact already mentioned, which
shows that the unknowns are virtually mere unit-vectors; while each separate equation
contains coplanar vectors only. In other words, when p,„ and a„ are given, p^+i is
determinate without ambiguity.

T. n.

290 QUATERNION NOTE ON A GEOMETRICAL PROBLEM. [XCIY.

The general equation aboTe may obviously be written as

or, if we introduce the quaternion

9«-i = (p« - «•) (pm^i - a»-i) ... Oh - Oi),
as 5» = ^«9«M + /3«?»-i-

Here ^, = a« - a,H-i

is one of the rector sides of the polygon whose comers are the assigned points.
And the statement above as to the nature of the quaternions employed is expressed as

9»-i = (Pm — ««) ?»-«.
Since we have

?#=Pi-ai, ?i = (/>i-as)(/>i-ai) = ^i + A?., ?. = ^,9# + A}i, &c.
it is clear that the values of q are all linear functions of pi, of the form

where r^ and 9^ are definite functions of Ui, a,, ... a. only.

Again, from Pm-a« = ~=^,

we have p^=M??=i±25=?=:P!?r:?, suppose.

This gives at once, by the definition of p,

fm-i = Pm^ " <lmqm-9 = (pm — ««) 9«-j ;

and, as an immediate consequence,
We now see at once that

Pm-i = i-lr-' (pm - Om) (/>,*-, " O^i) • • • (^a - «,) ;>o = (-)"^' C (1 + O^).

Thus, finally,

^^ = ^ = ^Cpr^^Tc' "^^ ^^ ''''^' =]DT^=.^:rc' ^^ ^ ^ odd ...(a).

6' and D being quaternions to be calculated (as above) from the values of a. The two
cases require to be developed separately.

Take first, the odd polygon: — then piD -{- piCpi = C — Dpi,

or Pi (c? + S) + pi (c + 7) />! = c + 7 - (d + S) pi,

if we exhibit the scalar and vector parts of the quaternions C and D. Cutting out

the parts which cancel one another, and dividing by 2, this becomes

dpi + SSpi + piSypi - c = 0,

XCIV.] QUATEBNION NOTE ON A QEOMETBICAL PBOBLEM. 291

which, as /9i is finite, divides itself at once into the two equations

Sypi + d = 0, SSpi - c = 0.

These planes intersect in a line which, by its intersections (if real) with the sphere,
gives two possible positions of the first comer of the polygon.

For the even polygon we have

PiD-piOpi = C + Dpi, or VpiS-piSypi-y^O;

which may be written V. p^ (S — Vypi) = 0, or S — Vypi = xpi.

This equation gives pi = (x + yy'^(S + SyBlx),

where a: is to be found from «■ — 7* = S^yS/o" — S*.

The two values of a^ have opposite signa Hence there are two real values of a, equal
and with opposite signs, giving two real points on the sphere. Thus iJiia case of the
problem is always possible.

[We might have arrived at equations (a), which involve the complete solution of
the problem, by the following direct and simple process: —

Let pff^i, pm be any two successive comers of the polygon, a„»»i the point through
which the corresponding side is to pass; we have at once

(pm - Om-i) (/>m-i " «m-i) = 1 + a*m-i,

or Pfn--: 1 — •

This is general, so that

^"^' pm - ««! (a«v-i - Om) pti^-i + (OmOm-i + 1) *

Note that, in these quaternion fractions, the coefficients of the linear expressions
^ Pn^u above and below, are the same pairs of quantities, in direct and inverted
order, viz.

Om-i, 1 , OmOf^-i + l, -(Ofi^-i-am) .

, and . ^ , &a

1 , - Om-i flW-i - «m , fimOm-i + 1

Their ostensible signs are, obviously, either alike above and unlike below, or unlike
above and alike below, alternately.

Hence, as p, = - (signs aiike above),

Pi — «!

, Cp,±D

we nave P^^'^ f^^"^ Dp x c*

where the upper signs belong to the case of n odd, and the lower to n even, 1899.]

37—2

292 [xcv.

xcv.

NOTE APPENDED TO CAPTAIN WEIR'S PAPER "ON A NEW

AZIMUTH DIAGRAM."

[Proceedings of the Royal Society of Edinburgh, July 15, 1889.]

[As Sir W. Thomson was unable personally to conmmnicate Capt. Weir's paper to
the Society, he asked me to add to it a Note on the principle of the new method.]

Capt Weirs singularly elegant constmction not only puts in a new and attractive
light one of the most awkward of the formuke of Spherical Trigonometry, but it
practically gives in a single-page diagram the whole contents of the two volumes of
Burdwood's Azimuth Tables. Further, it supplies a very interesting graphical plane
construction of a function of three independent variablea

In the usual notation for spherical triangles, if il be the zenith, C the pole, and
B a heavenly body (whose declination is S), C is the hour-angle (A), 6 the colatitude

:— — XJ, and A the supplement of the azimuth. Hence, fix)m the formula

cot a sin 6 = cot A sin C+ cos 6 cos C,
we have at once

4. / • *u\ sin A

tan (azimuth) = ^ — i ^ .

smXcosA — tanocosX

Oapt. Weir, in his diagram, virtually puts

j: = sin A sec X)

y = cosAtanXJ '^^

so that tan (azimuth) = ^ ,

^ ^ y - tan S '

XCV.] NOTE TO CAPT. WBIR's PAPER "ON A NEW AZIMUTH DIAGRAM.*'
X and y being found by the intersection of the confocal conies

298

and

sin'

-r— "Ta~^' *^® hour-angle hyperbola.

The Amplitude is the value of the azimuth at rising or setting, so that the
corresponding hour-angle is to be found from

cos h + tan X tan S = 0.

(2).

With this value of h, equations (1) become

a?=secXVl-tan"Xtan*S)

y = — tan"XtanS J

Elimination of h gives, of course, the latitude ellipse as before. But elimination of X
gives, instead of the confocal hyperbola, the curve

«» + [y -i(tan S -cot S)]" = \ (tanS + cot S)»,

or «■ + (y + cot 28)»= cosec" 28,

which is a circle passing through the common foci of the ellipses and hyperbolas.

The construction of the "Diagram" by means of (1) is, theoretically, a very simple
matter. Thus, take OA as unit length on the axis of x^ and draw AP parallel to y.
Make AOP^\ and yOH=h. Draw the circles whose centre is 0, and radii OP and
AP respectively. Let OH meet them in p, q. From p and q draw lines parallel to Oy,

Ox, respectively. Their point of intersection, Q, belongs obviously to the ellipse X, and to
the hyperbola h. A somewhat similar, simple, construction can easily be given for the
circle.

294 [xcvi.

XCVL

ON THE KELATIONS BETWEEN SYSTEMS OF CURVES WHICH,
TOGETHER, CUT THEIR PLANE INTO SQUARES.

[Proceedings of the Edinburgh Mathematical Society, November 9, 1889. Vol. vn.]

If p be the vector of a comer of a square in one system, o- that in a system
derived without inversion, we must obviously have

da-^ulcoB^ + k ain^ j dp (cos^ — k sin^j,

^u{(ico8 <f> +j ain <f>) dx— (i sin <f> —j COB <l>) dy] (1),

k being the unit-vector perpendicular to the common plane.
This requires that

-T- {u (ico8<l> + j sin <f>)} = -r- {m (— t sin ^ +j cos <f>)},

which gives the two equations

du , du . . f . .d<t> . dd>\

du . . du . ( .d6 . . d6\

or, in a simpler form,

1 du d<f>
udx" dy
ldu_^d4>
udy dx

•(2>

XOVI.] ON THB BELATIOKS BETWEEN SYSTEMS OF OUBVES, ETC. 295

Eliminating ^ and u separately, we have

cPlogjt ePIogtt_-

'^'^^ ^«>fi^r?} (3)

represent associated series of equipotential, and current, lines in two dimensions.

Assuming any lawful values for the members of (2) we obtain u and ^, and thence,
by integration of (1), o- is given in terms of p.

Thus o- = tf+ji;,

where { and 17 are known functions of x and y. From this x and y can be found in
terms of {, 17. Thus if

F^{x,y) = A^, F,(x,y) = A, (4)

be a pair of sets of curves possessing the required property, we obtain at once another
pair by substituting for x and y their values in terms of {, 17. These may now be written
as Xf y, and the process again applied, and so on.

Thus, let the values of the pairs of equal quantities in (2) be 1, 0, respectively
(which is obviously lawful), we have

w = €», <^ = y;
so that (1) becomes

do- = €* {(% cos y + j siny)dx'- (t sin y — j cos y) dy],

and (T = €* (i cos y -k-j sin y)

or f = €*cosy, i; = €*siny.

From these we have

a; = logVf« + i7«, y = tan-^|;

or, using polar coordinates for the derived series,

a; = logr, y^0,

[This is easily seen to be only a special case of (3) above.] Hence, by (4), another
pair of systems satisfying the condition is

F^ (log r, 0)^A,, F, (log r, 0) = A,.

This, of course, is only one of the simplest of an infinite number of solutions of the
equation (1), which may be obtained with the greatest ease fix)m (2).

296 ON THE RELATIONS BETWEEN SYSTEMS OF CURVES, ETC. [XCVI.

If there is inversion, all that is necessary is to substitute p"^ for p, or — p^^clpp"^
for dp. But the necessity for this may be avoided by substituting for any pair of
systems which satisfy the condition their electric image, which also satisfies it, and
which introduces the required inversion.

The solution of this problem without the help of quaternions is interesting. Keeping
as far as possible to the notation above, it will be seen that the conditions of the
problem require that

whatever be the ratio dxidy.
This gives at once

^^4.^^ = 0
dxdy dxdy

From these the equations (2) can be deduced by introducing ^ as an auxiliary angle.

XCTH*]

297

XCVIL

ON THE IMPORTANCE OF QUATERNIONS IN PHYSICS^

[Philosophical Magazine, January, 1890.]

Mt subject may usefully be treated uuder three heads, viz. : —

1. The importance of mathematics, in general^ to the progress of physica

2. The special characteristics required to qualiiy a calculus for physical applications.

3* How quaternions meet these requirements*

The question has often been asked, and frequently answered (one way or other)
in the most decided manner; — Whether is experiment or mathematics the more important
to the progress of physics? To any one who really knows the subject, such a question
is simply absurd. You might almost as well ask : — Whether is oxygen or hydrogen
the more necessary to the formation of water? Alone, either experiment or mathematics
is comparatively helpless: — to their combined or alternate assaults everything penetrable
must, some day, give up its secrets.

To take but one instance, stated as concisely as possible: — ^think of the succession
of chief steps by which Electroraagnetism has been developed. You had first the funda-
mental experiment of Oersted; — next, the splendid mathematical work of Ampire,
which led to the building up of a magnet of any assigned description by properly
coiling a conductmg wire* But experiment was again required, to solve the converse
problem : — ^and it was by one of Faraday*a most brilliant discoveries that we learned
how, starting with a magnet, to produce an electric current. Next came Joule and
V, Helmholtz to show (the one by experiment, the other by analysis) the source of

* Abfltmcl of an Addreae to the PhystGal BocieCj of the Univeraitj of Edinhurghf November 14^ 1389.
See the Anihor't AddreM to Section A at the BritiBh AsBooiatioiip 1871. [^ti(€, 17o, XXIIL]

T- n, 38

298: ON THE IMPORTANCE OF QUATERNIONS IN PHYSICS. [XCVIL

the energy of the current thus produced: — in the now-a-days familiar language, why
a powerful engine is required to drive a dynamo. Passing over a mass of important
contributions mathematical and experimental, due to Poisson, Green, CSauss, Weber,
Thomson, &a, which, treated from our present point of view, would furnish a nanatiTe
-of extraordinary interest, we come to Faraday's Lines of Farce. These were suggested
to him by a long and patient series of experiments, but conceived and described by
him in a form requiring only technical expression to become fully mathematical in the
most exclusive sense of the word. This technical expression was given by Clerk-Maxwell
in one of his early papers, which is still in the highest degree interesting, not only
as the first step to his Theory of the Electromagnetic Field, but as giving by an
exceedingly simple analogy the ph}^cal interpretation of his equations. Xext, the
narrative should go back to the establishment of the Wave-theory of Light: — to the
mathematics of Young and Fresnel, and the experiments of Fizeau and Foncanlt.
Maxwells theoiy had assigned the speed of electromagnetic waves in terms of electrical
quantities to be foimd by experiment The close agreement of the speed, so calculated,
with that of light rendered it certain that light is an electromagnetic phenonienon.
But it was desirable to have special proof that there can be electromagnetic waves:
and to measure the speed of propagation of such as we can produce. Here experiment
was again required, and you all know how effectively it has just been carried ont bv
Hertz. It is particularly to be noticed that the more important experimental steps
were, almost invariably, suggested by theorj' — that is, by mathematical reasoning of
some kind, whether technically expressed or not. Without such guidance experiment
can never rise above a mere groping in the dark.

I have to deal, at present, solely with the mathematical aspect of physics; but
I have led up to it by showing its inseparable connection with the experimental side,
and the consequent necessity that every formula we employ should as openly as possible
proclaim its phx'sical meaning. In presence of this necessity we must be prepared to
forego, if required, all lesser considerations, not excluding even such exceedingly desirable
qualities as con\})actness and elegance. But if we can find a language which secures

those to an unparalleled extent, and at the same time is transcendently expressive

bearing its full moaning on its face — it is surely foolish at least not to make habitual
use of it Such a language is that of Quaternions; and it is particularlj noteworthy
that it wivs invontwl by one of the most brilliant Analysts the world has yet seen,
a man who had for years revelled in floods of symbols rivalling the most formidable
combinations of l^grango, AbeK or Jacobi. For him the most complex trains of formnhe,
of the mast artificial kind, had no secjnet*: — he was one of the veiy few who could
affonl to dis|H>nso with simplifications: yet, when he had tried quaternions, he threw
over all other mothiHls in thoir favour, devoting almost exclusively to their develop-
ment, the hust (wonty years of an exooodingly active life.

Kvoryono hjvs hoaixl the somewhat [HH^ulijU". and more than doubtful, assertion

NwmwMitw jm, numma injuria. Wo may, without any hesitation, make a parallel but
mon^ o*isily admittod statomont :— rA** hitjbest art is the absence (not, as Horace would
have it, the coucealmeut) of artijiiX. This c^nnmends itself to reason as well as to

XCVII.]

ON THE IMPORTANCE OF QUATERNIONS IN PHYSICS,

29»

experience; but nowhere more forcibly than in the application of mathematics to phj^ical
Hcieoce. The difficulties of physics are sufficiently great, in themselves, to tax the highest
i-ei*ources of human intellect; to mix them up with avoidable mathematical difficulties ia
unreason little short of crime. (To be obliged to evaluate a definite integral, or to solve
a differential equation^ i^ a necessity of an unpleasant kind, akin to the enforced extraction
of a cube root : and here artifice is often requisite in our present state of ignoraQce : but
its introduction for such purposes is laudable. It does for us the same kind of service
which has been volunteered in the patient labour of the ciilculators of logarithmic tables.
It is not of inevitable, but of gratuitous, complications that we are entitled to complain.)
The intensely artificial system of Cartesian coordinates, splendidly useful as it was in its
day, is one of the wholly avoidable encumbrances which now retard the progress of
raathematical physics. Let any of you take up a treatise on the higher branches of
hydrokinetics, or of stresses and sti^ins, and then let him examine the twofold notation
in Maxwell's Electricity. He will see at a glance how much expressiveness as well as
simplicity is secured by an adoption of the mere notation, as distinguished from the
processeSj of quaternions. It is not difficult to explain the cause of this* But let us
first take an analogy from ordinary life, which will be found to illustrate fairly enough
some at least of the more obvious advantages of quaternions.

There are occasions (happily rare) on which a man is required to specify his name
in full, his age, height, weight, place of birth, family history, character, &c. He may
be an applicant for a post of some kind, or for a Life Policy, &c* But it would be
absolutely intolerable even to mention him, if we had invariably to describe him by
recapitulating all these particulars* They will be forthcoming when wanted ; but we
must have, for ordinary use, some simple, haudy, and unambiguous method of denoting
him. When we wish to deal with any of his physical or moral qualities, we can easily
do so, because the short specification which we adopt in speaking of him is sufficient
for his identificatioii. It includes all his qualities. We all recognize and practise this
in ordinary life; why should we outrage common-sense by doing something very different
when we are dealing with scientific matters, especially in a science such as mathematics,
which is purely an outcome of logic ?

In quaternions, a calculus uniquely adapted to Euclidian space, this entire freedom
from artifice and its inevitable complications is the chief feature. The position of a
point (relative of course to some assumed origin) is denoted by a single symbol, which
fully characterizes it, and depends upon length and direction alone, involving no reference
whatever to special coordinates* Thus we use p (say) in place of the Cartesian a?, y, i^
which are themselves dependent, for their numerical values, upon the particular scaffolding
which we choose to erect as a (temporary) system of axes of reference. The distance
between two points is

instead of the cumbrous Cartesian

* Note here that though ahBatnte position ia an idea too ftbaord ev^n for the nmjonty of metaphyaicmiiB^
Abeolnte direction h a perfect! j definite phyfiical Idea. It b one essential part of the first law of motion.

38—2

300 OK THB DfFORTAXCB OF QUATERSIOK8 IN FHTBIGB. [XCTH.

Bat the disiaDce in quesdoo is faU j symbolind as to diiectaon as weU as leDgth hj
the ample foim

p-p'.

If three oonteimiDoas edges of a pandlelepiped be p^ p\ p\ its Tolmne is

S' It
.pp p .

Eren whoi adTantage is taken of the lemarkable condensation aecored \rj the
intenselT ardfidal noiadon for detenninants, Cartesian methods most content tKp>wMi^|^^^
with the much moie cumbrous expression

X y X

f »f 90

X y s

As ve advance to higher maners, the Cartesian oomplexitj teDs more and mate;
while quaternions pres^re their simididtf. Thus any central sur&ee (tf the ^^f^^n^
degree is expressible by

Sp^ = -1, or r^p = l;
vlule the Ourteaian fonn develops into

The hc«DOgeneoQS strain which changes p into p is expnsBible bj « smgle
lettiar: — thus

te Outesian form TCqoires three equations.

x'=ar J-fty + cr,

r ^jpur+iv-^ it

Theti^ may bo simplitiod* but only a little, by employing the notation for a »*^trrT
1\> expiv«8 in qnatortiions the conjugate strain, a mere dash is required: thos

while with the artidcial sonflolding we must wriie our three equaskiks again, ami^;ing
the ci>e£cients as below: —

a d f

h € &

If wx^ uow ask tho qucstiv\u. \Vha5 stnun will cccTiers *Jk eLipscii abox^e into the
uuil s}J^oi\\ iho ax\s>fcxT will bo a^:uo ur.x* ir: ccciin^ rrcci lise jKoSa^xss
&\nuuU\ The ^uaioxtu^\i\ f>rti\xiU a^oi^i^ius ii a: vYkv as d^

xcvil]

ON THE IMPORTANCE OF QUATERNIONS IN PHYSICS

When Qauss gave his remarkable expression for the number of mterlinkings of
two endleBs curves in space, he had to print it as

What an immense gain in simplicity and intelligibility is secured when we are enabled
to write this in the form

or as

47rJ "^J Tp~f^^

m that we instantly recognize in the latter factor the vector force exerted by unit
current, circulating in one of the closed curves, tipon a unit pole placed anywhere on
the other; and thus see that the whole integral represents the work required to carry
the pole once round its circuitt

Without as yet defining V, I shall take, as my final example, one in which it
is involved. A very simple term, which occurs in connection with the stmin produced
by a given displacement of every point of a medium, is

Its Cartesian expression is, with the necessary specification^

<r = if + ji? + &f,
made up of three similar terms of which it is sufficient to write one only, viz.,

<«»'-^')(£|4jf)-<—-)(SS-SS)-<'^-*>(|S-SD-

Now, suppose this to be given as the ^-coordinate of a point, similar expressions (formed
by cyclical permutation) being written for the y and £ coordinates^ How long would it
take you to interpret its meaning?

Look again at the quaternion form, and yon see at a glance that it may be written

in which its physical meaning is more obvious than any mere form of words could

make it.

Or you may at once transform it to

which shows clearly why it vanishes when a and 0 are parallel.

I need not give more complex examples: — because, though their quaternion form
may be simple enough (containing, say, 8 or 10 symbols altogether), even this unusually
large blackboard would not suffice to exhibit more than a fiBction of the equivalent
Cartesian form.

y>2 f«r THE iifpoiirrAXCB or quATEB3io%^ is phtsi€& [xctil

Any nrukth^matical method, which u to be aipplied to pfty^cal probienia, mist be
capable 6f expreivting not only jipeu^-feUtiood b<zt alao the grand rharak'tiPTwtiiflH (so fiu*
M we jet know them) of the materials of the physical world. I ha^e jist faneflj shown
how exactly and tiniqaely qaatermona are ads^kted to Eaclkiian space; we most next
inqnire how they meet the other reqairementa

The grand charactenAtic^i of the physical world are: — CoDserratioQ of Matter, with
absolnte preservation of it« identity; and Conaenration of Energy, in spite of perpetnal
change of a charact^^r such a8 entirely to prevent the reoogniticxi of identity. The first
of these i.s very simple, and needs no preliminary remarks. But the methods of symbolizing
change are almost as nnmerous as are the varions kinds of change. The more impcnrtant
of them employ forms of the letter D: — viz. d, ?, D, A, i. and V.

From our present point of view little need be said of Jl, whkh is the equivalent
of (D- 1) or of (e^'^-'l), because the changes which it indicates take place by starts
and not continuously. Good examples of problems in which it is required are furnished
by the successive rebounds of a ball from a plane on which it &lls, or by the motion
of a light string, loaded at intervals with pellets.

Various modes of applying the symbol d are exemplified in the equatiiHi

In the terms dx, dy, dz, the symbol d stands for changes of value (usually small) of
quantities treated as independent. In the term dQ it stands for the whole consequent

change of a quantity which is a function of these independents. By the fisu^tors (-f^)

&c. we represent the rates of increase of Q, per unit of length, in the directions in
which X, y, z are respectively measured. The contrast between the native simplicity of
the left-hand, and the elaborate artificiality of the right-hand, member of the equation,
shows at once the need for improvement. To express the rate of change per unit of
length in any other direction, we have to adopt the cumbrous expedient of introducing
three direction-cosines, and the result is given in the form

dx dy dz'

The above equation may be read as pointing out, at any one insta/nt, how a function
of position varies from point to point. To express the change, at any one place, from one
instant to the next, we write in the usual notation

^-m

But if we have to express the changes, from instant to instant, of some property of
a point, which is itself subject to an assigned change of position with time, we have

xctil]

ON THE IMPOBTANCE OF QUATEBNIONS IN PHYSICS,

303

to combine these expreseions, and to indicate the relation of position to time. Thus we
build up the complicated expression

Here the fiymbol d in called in, to effect a slight simplification ; and we go a little
further in the same direction hy putting % v^ w for

dx di/ dz
dt' di' dt'

which are obviously the components of velocity of the point for which Q is expressed*
Thus we write

dyj

Of course you all know this quite well; and you may ask why I thus enlarge upon it.
It is to show you how completely artificial i«id unnatural are our recognized modes
of ex|}reBsion.

Freenel well said: — La nuture ne s'est pas embaiTossfy des diffi.culti& d^ancdyse. elle
n*a 4m^i que la cmnplication des 'moyem. Why should we not attempt, at least, to imitate
nature by seeking simplicity ?

The notation S^ as commonly used, is (like the d in dQ above) quite unobjectionable.
At least we cannot see how to simplify it further. Its effect is to substitute, for any
one point of a figure or group, a proximate point in space, so that the figure or group
of points undergoes slight, and generally continuous, but otherwise wholly arbitrary dis-
placement and distortion. It thus appears that d and S are entitled to take their places
in a calculus, such as quaternionSi where simplicity, naturalness, and direct intelligibility
are the chief qualities sought. We have now to inquii^e how such expressions a^

jdQ_^ dQ^ dQ
€Lm ay dz

can be put in a form in which they will bear their meaning on their ^ce.

It was for this purpose that Hamilton introduced bis symbol V«
origiually defined in the cumbrous and unnatural form

No doubt, it was

. d

dm^ dy di *

But that was in the very infancy of the new calculus, before its inventor had succeeded
in completely removing from its formulae the fi^gmenta of their Cartesian shell, which
were still persistently clinging about them. To be able to speak fi*eely about this
remarkable operator, we must have a name attached to it, and I shall speak of it

804 ON THB IMPORTANCE OF QUATERNIONS IN PHYSICS. [XCVH.

as Nabla*. We may define it in many ways, all independent of any system of co-
ordinates. Thus we may give the definition

meaning that, whatever unit-vector a may be, the resolved part of V parallel to
that line gives the rate of increase of a function, per unit of length, along it. From
this we recover, at once, Hamilton's original definition: — thus

V = - OiSfaV - 138^ - 7/S7V = ad.+ fidg + ydy,

a, )8, 7 being cmy system of mutually rectangular unit-vectors.

But, preferably, we may define Nabla once for all by the equation

- SdpV = d,

where d has the meaning already assigned. The very nature of these forms shows
at once that Nabla is an Invariant, and therefore that it ought not to be defined
with reference to any system of coordinates whatever.

Either of the above definitions, however, shows at once that the effect of applying
V to any scalar function of position is to give its vector-rate of most rapid change,
per unit of length.

Hence, when it is applied to a potential, it gives in direction and magnitude
the force on unit mass; while firom a velocity-potential it derives the vector velocity.
From the temperature, or the electric potential, in a conducting body we get (employing
the corresponding conductivity as a numerical fisujtor) the vector flux of heat or of
electricity. Finally, when applied to the left-hand member of the equation of a series
of surfisu^es

it gives the reciprocal of the shortest vector distance firom any point of one of the
surfisu5es to the next; what Hamilton called the vector of proximity.

If we form the square of Nabla directly from Hamilton's original definition, we find

simply the negative of what has been called Laplace's Operator: — that which derives
from a potential the corresponding distribution of matter, electricity, &c.

Thus Laplace's equation for spherical harmonics &c. is merely

and, as l/r(p — a) is evidently a special integral, an indefinite series of others can
be formed from it by operating with scalar fimctions of V, which are commutative

* Hamilton did not, bo far as I know, suggest any name. Clerk-MaxweU was deterred by their yemaoolar
•ignifieation, nsnally ludicrous, from employing such otherwise appropriate terms as Sloper or Orader ; but adopted
the word Nabla, suggested by Bobertson Smith from the resemblance of y to an ancient Assyrian harp of that
name.

XCVn.] ON THE IMPORTANCE OF QUATERNIONS IN PHYSICS. 305

with V, such as SfiV, e"^^, &c. In passing, we may remark that if /8 be a unit
vector, i7+jm + An, we have

-SfiV^l^ + m^ + n^.
ax ay dz

This is the answer to the question proposed a little ago.

The geometrical applications of Nabla do not belong to my subject, and they
have been very fully given by Hamilton. But, for its applications to physical problems,
certain fundamental theorems are required, of which I will take only three of the
more important; — an analytical, a kinematical, and a physical one.

I. The analytical theorem is very simple, but it has most important bearings
upon change of independent variables, and other allied questions in tridimensional
space. Few of you, without the aid of quaternions or of immediately previous pre-
paration, would promptly transform the independent variables in a partial differential
equation fix)m x, y, z to r, 0, ^: — and you would certainly require some time to recover
the expressions in generalized (orthogonal) coordinates. But Nabla does it at once.
Thus, let

-. , d , d , , d

where a = tf +jrf + *f,

f , ff, f being any assigned functions of x, y, z. Further, let

da = <l>dpf

where (/>, in consequence of the above data, is a definite linear and vector function.
Then, from the mere definition of Nabla,

SdaV^^^d^SdpV,
which gives at once

S . <l>dpV^ = S . dp(f>'V^ = Sdp^.

As dp may have absolutely any direction, this is equivalent to

where <f>' is the conjugate of <f>,

II. The fundamental kinematical theorem is easily obtained from the consideration
of the continuous displacement of the points of a fluid mass. (It is implied in the
word "continuous" that there is neither rupture nor finite sliding.)

If a be the displacement of the point originally at p, that of p + dp is

a + da^ a — SdpV , a ;

and thus the strain, in the immediate neighbourhood of the point p, is such as to
convert dp into

dp — SdpV . a-.

Thus the strain-function is yfrr = t — SrV . a.

T. II. 39

806 ON THE IMPORTANCE OF QUATERNIONS IN PHYSICS. [XCVU.

If this correspond to a linear dilatation e, and a vector-rotation e, both being quantities
whoso squares are negligible, we must have

^T = (1 + e) T + Fer.

Comparing, we have — StV . cr = er + Ver,

from which at once (by taking the sum for any three directions at right angles to
one another),

so that iSVcr represents the compression,

and ^rVcr „ ., vector-rotation,

of the element surrounding p.

By the help of these expressions we easily obtain the stress-function for a homo-
geneous isotropic solid, in terms of the displacement of each point, in the form

^o) s » Fi (iSo)V . a- + VSttxr) - (c — | w> «5Vcr ;

whore n and c are, respectively, the rigidity and the resistance to compression ; and
^ is the stre^. per unit of surface, on a plane whose unit normal is a».

III. The jfundamental ph\-sical relation is that expressing conservation of matter,
commonly called the oi^uation of cinitinuity. We have only to express symbolically
that the increaso of mi\ss in a tinito simply-connected space, due to a displacement,
is the oxot\!5S of what enters over what leaves the space. This gives at once

whon^ Cr is unit nonnal drawn outuxirds from the bounding surface. If we put for
a the oxprx^ssion uVr. when^ u and r are any two scalar functions of position, this
bovvmes OrtvnV Theorem,

If the sjviiV vvnsidoroil be imaj^^ntni as Kninded by two indefinitely close paralM
Mirfacos. and by the normals at o,'^oh jxMut of a closed curve drawn on one of them,
this is t>asily nnlucwl * to the form of the lino and surfece iniogral

The simp'iosi forms of those t\|u,^tions iuv respeotivcly

and i I r ^ TrV ^ >,o> = I jV .-.,

\^:u:v ' ;s a:\\ ?VA*;Ar :V,no::.>n ot jv\si:ion. Bu: :: i? o.tAr rVvni :he mode in which
;i c:.:<r> :hA: •, :v4A\ ly av*\ qv*A:onv.o;i. Ar,d :: :> ivisy io buijd on these an

XCVII.] ON THE IMPORTANCE OF QUATERNIONS IN PHYSICS. 307

indefinite series of more complex relations. Thus, for instance, if cr and t be any two
vector-functions of p, we have

jjY(crVV + KVfT . Vt) d9 = jja UpVrds,

which has many important transformations. You will find it laborious, but alike
impressive and instructive, to write this simple formula in Cartesian coordinates. It
consists of four separate equations, containing among them 189 terms in all !

In the three relations just given we have the means of applying quaternions to
various important branches of mathematical physics, where Nabla is indispensable. But
I must confine myself to one example, so I will take very briefly the equations of
fluid motion.

Let e be the density, and a- the vector-velocity, at the point p in a fluid.
Consider the rate at which the density of a little portion of the fluid at p increases
as it moves along. We have at once, for the equation of continuity,

s-'^^'-

which we may write, if we please, as

This is the result we should have obtained if we had considered the change of
contents of a fixed unit volume in space. Next consider the rate at which the
element gains momentum as it proceeds. We write at once, since momentum cannot
originate or be destroyed by processes inside the element,

,%.-.VP^

iUUvds,

where P is the potential energy of unit mass at p, and <f>Uv is the stress-function
due to pressure and viscosity. We have already had the form of this function; so
that the equation transforms at once into

6^ = - gVP - Vjo - n(VV + iVSVcr) ;

which contains the three ordinarily given equations. Here n is the coefficient of
viscosity, and the pressure p enters the equation in the form

cSVcr.

To obtain v. Helmholtzs result as to vortex-motion, put n = 0, and we deduce
for the rate of change of vector-rotation of an element, as it swims along,

|-FVcr=F.VF.crFV,cr,.

39—2

308 ON THE IMPORTANCE OF QUATERNIONS IN PHYSICS. [XCVII.

If the fluid be incompressible, this becomes

From either it is obvious that the rate of change of the vector-rotation vanishes
where there is no rotation. But time forbids any further discussion of formuIsB.

Hydrokinetics, as presented by Lagrange and Cauchy, was rather a triumph of
mathematical skill than an inviting or instructive subject for the student. The higher
parts of it were wrapped up in equations of great elegance, but of almost impene-
trable meaning. They were first interpreted, within the memory of some of us, by
Stokes and v. Helmholtz, after we know not what amount of intellectual toil. The
magnificent artificers of the earlier part of the century were, in many cases, blinded
by the exquisite products of their own art. To Fourier, and more especially to Poinsot,
we are indebted for the practical teaching that a mathematical formula, however brief
and elegant, is merely a step towards knowledge, and an all but useless one, until
we can thoroughly read its meaning. It may in fact be said with truth that we
are already in possession of mathematical methods, of the artificial kind, fully sufiScient
for all our present, and at least our immediately prospective, wants. What is required
for physics is that we should be enabled at every step to feel intuitively what we
are doing. Till we have banished artifice we are not entitled to hope for full success
in such an undertaking. That Lagrange and Cauchy missed the import of their
formulae, leaving them to be interpreted some half-century later, is merely a case of
retributive justice: —

" Deque enim lex aequior ulla

Qnam necis artifices arte perire sul"

Lagrange in the preface to that wonderful book, the Micanique Analytiqae, says: —

"Les m^thodes que j'y expose ne demandent ni constructions, ni raisonnemens
g^om^triques ou m^caniques, mais seulement des operations algdbriques, assuj6ties k
une marche rdgulifere et uniforme.*'

But note how different is Poinsot's view: —

" Gardons-nous de croire qu*une science soit faite quand on Ta rtWuite k des
formules analytiques. Rien ne nous dispense d'dtudier les choses en elles-mSmes, et
de nous bien rendre compte des iddes qui font Tobjet de nos speculations."

No one can doubt that, in this matter, the opinion of the less famous man is
the sound one. But Poinsot's remark must be confined to the analjrtical formulae
known to him. For it is certain that one of the chief values of quaternions is
precisely this: — that no figure, nor even model, can be more expressive or intelligible
than a quaternion equation.

xcviil] 309

XCVIII.

GLISSETTES OF AN ELLIPSE AND OF A HYPERBOLA.

[Proceedings of the Royal Society of Edinburgh, December 16, 1889.]

Last summer, while engaged with some quaternion investigations connected with
Dr Plarr*s problem (the locus-boundary of the points of contact of an ellipsoid with three
rectangular planes), I was led to construct the glissettes of an ellipse. I then showed
to the Society a series of these curious curves, drawn in my laboratory by Mr Shand,
who had constructed for the purpose a very true elliptic disc of sheet brass. I did
not, at the time, think it necessary to print my paper; but, after the close of the
session, I made the curious remark that precisely the same curves can be drawn each
as a glissette of its own special hyperbola. This double mode of sliding generation of
the same curve seems to possess interest. It is somewhat puzzling at first, since the
ellipse turns completely round, while the hyperbola can only oscillate. But a little
consideration shows the cause of the coincidence.

Let 0 be the origin, C any position of the centre of the ellipse, CA that of the
major axis, and P the corresponding position of the tracing point. This does not require
a figure.

Then it is easy to see that if ^ be the inclination of OC to one of the guides,
0 that of CA to the same, we have

Va* cos'd + 6" sin>(? ^y/a^ + b" cos<^.

But this gives Va'cos'^ — 6'8in"^ = Va" — 6'cosd,

which is the corresponding relation for the hyperbola. In fact the one equation is
changed into the other by changing the sign of 6", and interchanging the angles
6 and (f>.

310 GLISSETTES OF AN ELLIPSE AND OF A HYPERBOLA. [XCVHI.

Let the polar coordinates of the tracing point, referred to the centre of the ellipse
and the major axis, be r, a, we obtain a position of P by the broken line OC, CP\
their lengths being VoM^, r, and their inclinations to the guide <f>, d + a, respectively.

If we now turn the guides through an angle a, and use a hyperbola whose axes
are to those of the ellipse respectively as r : Va' — 6* ; and consider the curve traced
by a point Q in its plane, whose central polar coordinates are Va* + 6", — a ; the position
of the point Q is given by the broken line 0(7, C'Q, Of these 0(7 is equal and parallel
to OP, while CQ is equal and parallel to OC, Thus the points Q and P coincide.

In fisict the motion of either is the resultant of two circular motions, one of which
is complete (viz., 6, which has all values from 0 to 27r), the other reciprocating
(viz., ^, which varies between sin-*(6/Va' + 6') and sin"* (a/Va' + 6')). But, in the case
of the ellipse, the centre has the reciprocating motion; while, in the hyperbola, it
describes the complete circular path.

Mr Shand has constructed a hyperbolic disc, comprising a considerable portion of
each of the branches of the curve, and it gives very fair glissettes. It is very curious
to watch the proper point of the hyperbola gliding over the curve already traced by
the ellipse. But this apparatus is not so easily managed as is the elliptic disc, so
that the figures in Plate V were drawn by means of the latter, and reproduced on
a diminished scale by photolithography.

To exhibit, by a few forms, as completely as possible the general nature of these
glissettes, I selected a series of tracing points equidistant from the centre of the ellipse,
and situated within and on the boundaries of the various regions, to each of which
belongs a special form. For this purpose I traced the curve formed by successive positions
of the instantaneous centre of rotation on the disc. The disc, with this curve on it,
is represented in the upper central figure. The equation of the curve is

6»ar^ + ay "" ' a« - 6»

It is easily traced as follows. Draw the ellipse whose semiaxes parallel to x and y
respectively are

, and T ,- ;

diminish every radius vector in proportion to the cosine of double the angle vector;
and then diminish the ordinates in the ratio 6 : a, so that the ellipse itself becomes
a circle.

In the disc from which the glissettes were drawn, a (rather more than a foot in
length) was made double of 6.

This equation suggested, as a useful distance of the tracing point from the centre,

the quantity

a»-62

2 Va'Vi^ '

Plate V

XCVin.] GLISSETTES OF AN ELLIPSE AND OF A HYPERBOLA. 311

and accordingly the points 0, A, B, (7, D, E, F were taken on the corresponding circle.
The glissettes of B and D, of course, have cusps: — and it is interesting to study the
changes of form from one to the next of the seven just named. Two groups of figures
give the glissettes of successive points on each of the axes separately, viz., (?, 0, K, M
on the major axis, and J, F, L, N on the minor. Of these K and L have cusps. The
figures G, H, J were drawn to show how the glissettes of points near the centre
approximate to the (theoretical) four cusps which belong to the path of the centre
itself, the finite circular arc described four times over during a complete rotation of
the ellipse. The point P was chosen as close as possible to the intersection of the
ellipse and the centrode.

The locus of the instantaneous axis in the guide-plane is of no special interest. It is
easy to construct it geometrically from its polar equation, which may be written generally as

r (2 VS*T^ - r) = 4a*6»/(a« + 6^ sin» 2d,

or in the present special case r (\/5a — r) = 4a'/5sin'2d.

It is an ovoid figure, symmetrically situated between the guides, with its blunter end
turned from the origin.

The equation of the glissettes is found by eliminating 6 between the equations

X = Va» cos'd + 6" sin'd + r cos {6 + a),
y = Va»sin*d + 6»cos«d + r sin {6 + a).

This seems to lead to a relation of the 12th degree in x and y\ but it must contain
a spurious factor, as Professor Cayley informs me the final result ought to be of the
8th degree. And in fact we see at once that, if the tracing point be at a very great
distance from the centre (in comparison with the major axis of the ellipse) the glissette
will consist practically of four circles, with centres in the four quadrants between the
guide-lines.

812

[xcix.

XCIX.

NOTE ON A CURIOUS OPERATIONAL THEOREM.

[Proceedings of the Edinburgh Mathematical Society, January 10, 1890.]

The idea in the following note is evidently capable of very wide development,
but it can be made clear by a very simple example.

Whatever be the vectors a, ^, 7, B, we have always

But vector operators are to be treated in all respects like vectors, provided each be
always kept before its subject.

Let cr = if+ji7 + A;f,

where f , 17, f are functions of x, y, z\ and let

— ^ . d ' ^ _i_ z, ^
^ dx *' dy dz '

as usual Also let ctj, V^ be their values when x^, yi, Zi are put for x, y, z.

Then by the first equation, attending to the rule for the place of an operator,

V . FVcrFVjcr, = VS . aV,a^ - S (V.a^V) a.

If we suppose the operations to be completed, and then make Xi=^x, yi = y, -8^1 = 5,
the left-hand member must obviously vanish. So therefore must the right.

That is :— VS. crV,o-i = S (VjcriV) a ;

if when the operations are complete, we put Ci = cr, V^ = V.

In Cartesian coordinates this is equivalent to three equations, of the same type.
I write only one, viz.: —

da>

dx.

dyi

dzi

dx.

dyi

dzi

d
dx

d
dy

d
dz

if, after operating, we put a?, = a?, f 1 = f , &c., &c.

c] 313

NOTE ON RIPPLES IN A VISCOUS LIQUID.

[Proceedings of the Royal Society of Edinburgh, March 3, 1890.]

The foUo^ving investigation was made in consequence of certain peculiarities in the
earlier results of some recent measurements of ripples by Prof Michie Smith, in my
Laboratory, which will, I hope, soon be communicated to the Society. These seemed
to suggest that viscosity might have some influence on the results, as might also the
film of oxide, &c., which soon gathers on a free surface of mercury. I therefore took
account of the density, as well as of possible rigidity, of this surface layer, in addition
to the surface tension which was the object of Prof Smith's work. The later part of
the paper, where Cartesian coordinates are employed, runs somewhat on the lines of an
analogous investigation in Basset's Hydrodynamics. My original object, however, was
different from his, as I sought the effects of viscosity on waves steadily maintained
by means of a tuning-fork used as a current interrupter; not on waves once started
and then left to themselves. Besides obtaining his boundary conditions in a singular
manner, I think that in his § 521 Mr Basset has made an eiToneous investigation of
the eflfects of very great viscosity.

The stress-function in a viscous liquid may be obtained {Anti, No. XCVIL, pp. 306-7)
from that in an elastic solid, by substituting velocity for displacement ; in the form

<^o = - /A (/SoV . cr -h V/Socr) - (c - f/t) cdSVct (1),

where, in order to include the part of the pressure which is not due to motion, we
must write p instead of the quantity

cSVcr.

Here a is the vector velocity of an element at p, and /a is the coefficient of viscosity.
T. II. 40

314 NOTE ON RIPPLES IN A VISCOUS LIQUID. [c.

Hence, supposing the volume of the element to be unity, we have for the equation
of motion

where e is the density of the liquid, and P the potential energy of unit mass at />;
and the double integral is taken over the surface of the element. This is a perfectly
general equation, so we must proceed to the necessary limitations.

First. Let the displacements be so small that their squares may be neglected.
Then we may write d for 9.

Second. Let the liquid be incompressible; then

SVcr = 0 (2).

With these, the equation of motion becomes

e'^^-'^ip^eD-fiV^c (3).

Third. Let the motion be parallel to one plane, and we have

Ska = 0 (4).

From (2) and (4) we have at once

(T^Vw.k (5),

where w is a scalar function of Vkp.

Operate on (3) by F.V, and substitute from (5), and we have

(ej^+/iV»)v^ = 0 (6).

Fourth. Limit w to disturbances which diminish rapidly with depth. Here the
problem has so far lost its generality that it is advisable to employ Cartesian co-
ordinates, the axis of x(i) being in the direction of wave-motion, and that of y(j)
vertically upwards. Then it is clear that a particular integral of (6) is

w = (^6'y-|-56^)€('^+^«)' (7),

where t denotes V— 1. The only conditions imposed on r, «, and n, are that the real
parts of r and «, in so far as they multiply y, must be positive; and by (6)

fi(^-r^)^eni (8X

The speed of vertical displacement of the surface is

^=:-(Sjo-)o = (SiVi(;)o = -n(^+5)6<«'+^)* (9).

From this, j-^ and ^, which will be required below, are found by using the factors
— r* and r*.

C] NOTE ON RIPPLES IN A VISCOUS LIQUID. 315

The stress on the free surface (where y = 17, a quantity of the order A) is, by (1),

(<A?)o = -l>oj-/i(SjV.cr + VS;cr)o (10),

where, in jpo, we must include the effects of the tension T, and of the flexural-rigidity
Ef of the surface-film.

But, by (3) and (5), we have

V ( p + eP) = - ^e ^ + fiVA Vw.k]

so that as ^ ^9y

we have dp + egdy = eni (ndy — rdx) il6»v+ <'■«+»<)».

From this, by integrating, and introducing the surface conditions,

If we now substitute this in (10) and, for the boundary condition,* make

(omitting terms of the second degree in A and B\ we have by means of (9) the
two equations

R(A+B)- en*A + 2fimi (rA + sB) = 0,

r»(^ + 5) + r»^+«»5 = 0,

where, for shortness, R=^egr+ Tr' + Er* (11).

Thus, finally. P-en'-h Vnr»t + 4^r»(r-«) = 0 (12).

This must be treated differently according as /i is small or great.

I. Let fjL be small ; and let n be given, and real. This is the case of the sustained
waves in Prof. Smith's experiments.

The equation obtained by neglecting /a, viz.

gives one, and only one, positive value of r, whose value is diminished {i,e., the wave-
length is increased) alike by surface tension and by surface-flexural-rigidity. Call it r©,
and let

r = ro + fip,

then by (12), keeping only terms of the first order in /i,

(ge + STn^ + 5Ero*)p + ^roh = 0 (13).

Thus p is a pure imaginary, and therefore the viscosity does not affect the length of
the wavea It makes their amplitude diminish as they leave the source. (For the real

* W. Thomson, Camb, and Dublin Math. Journal, in. 89 (1848).

40—2

316 NOTE ON RIPPLES IN A VISCOUS LIQUID. [c.

part of w belongs in this case, if we take n as positive, to waves travelling in the
negative direction along x, and vice versa,) The factor for diminution of amplitude
per unit distance travelled by the wave is

This expression gives very curious information as to the relative effects of viscosity on
the amplitudes of long and short waves, when we suppose gravity, surface-tension, or
surface-flexural-rigidity, alone, to be the cause of the propagation.

If the waves be started once for all, and allowed to die out, r is given and n is
to be found. This is the first case treated by Mr Basset. If then n = n© be found from

en" = JB,

we may put n=noH-/ii/.

By (12) we have, keeping only the first power of /x,

ev = 2r26,

which coincides with the result given in § 520 of Basset's Treatise,

II. Let /i be large. Suppose r to be given, a real positive quantity. Then, by
(8), we may eliminate n from (12) and obtain

— + (5*-r»)2 + 47-»(5*-r^) + 4r»(r-«) = 0 (14).

The first term is very small, and the rest has the factor s — /*. Omit the term which

contains this factor twee, and we have

Re
«(*-^)=-4;s-. (15).

This has real positive roots if, and only if,

/A^r^ > Re,

and thus, by (8), when this condition is satisfied n is a pure imaginary, and there can
be no oscillation. Of the two roots of (15) we must, in consequence of our assumption
(that {s — r)" is negligible), choose that which is nearly equal to r. It might be fancied
that, as this assumption leads to B = — A very nearly, a new limitation would be
introduced as regards the magnitude of 17. But we have

17 = - - (^ -\-B) e<'^+»<'* = - - ^ fl - ^!l-'\ ^(rx+ntu

n^ n \ r^-^^J

= -^^g(nr+n<)»^ nearly.

The wave-pattern, in this case, does not travel but subsides in situ, its amplitude
diminishing according to the approximate factor

Thus, as was to be expected, the subsidence is slower as the friction is greater. Also,
if gravitation is the sole cause of subsidence, the longer waves subside the faster;

C] NOTE ON RIPPLES IN A VISCOUS LIQUID. 317

while if the main cause be surface-tension, or surface-flexural-rigidity, the shorter waves
subside the faster.

III. If there be a uniform film of oxide or dust, in separate particles which
adhere to and move with the surface, we must add to the expression for surface-stress
in (10) the term

= - m [jrn (^ -h -B) -h ini (rA + sB)] e'"^^**"*,
where m is the surface-density of the film.

The equations for the elimination of AjB become

{R + mm^ - en^ + 2fn^nC) A-¥{R + mrn^ + 2firmi) 5 = 0,

(2r»-^^ + (2r»+^-*-^5 = 0;

so that instead of (12) we have

{c — m (« - r)} {R + mm') = eW — ^/irhiei + 4>fi^ (« — r) - mn^es.

When fjL and m are small, this is approximately

R + 2mrn^ = en^-~ ^fir^m.

There is no other term in the first power of m, independent of /a ; so that, to
this degree of approximation (which is probably always sufficient), the dust layer has
no effect except to increase R. When there is no viscosity this increases the ripple-
length (ie,, diminishes r) for a given period of vibration.

When terms of the first degree in the viscosity are taken account of, the eflfect on
n (for a given value of r) is merely to add to it the pure imaginary

2/17^4/(6— 2mr),

whose value increases alike with m and with r.

Thus the period is not affected, but the surface layer aids viscosity in causing
waves to subside as they advance.

This investigation above may be easily extended to the case in which a thin
liquid layer is poured on mercury to keep its surface untarnished. The only difficulty
is with respect to the relative tangential motion at the common surface of the liquids.

318 [CL

CI.

NOTE OX THE ISOTHERMAUS OF ETHYL OXTOE.

[Proceedinffs of the Royal S*yci€ty of EdiMbnTyk, July 6, IS&l/

The first three pressure-colnnms of the following linle table were coDstmcteii firoci
the elaborate data given by Drs Bamsaj and Yoang in their impt^rtant paper " Oa
Evaporation and Dissociation." Part nr. \PkS. Mag^ Mav 1V>7)l Thev give, in znecres
of mercury, the pressures re«)aired to Djnfine Mie gramme of oxide of ethyl to v-ari-xxs
specified numbers of cubic centimetres, at temperatures near to that of the eritiical
point.

193 -S

...

73-

7*9

as6

...

38-55

383

34-

34-3

a* 43

3416

31-2

31^

3153

31-55

2-75

2S-

fejl

2824

2841

...

277

27 42

27 45

...

27-2-

2719

273

3-7

...

27-2

27-19

27-2

...

27^

27-20

27^

27-

271

2712

271

266

26-7

26^

26-46

25-&

25<f

26^)0

25«

22-&

2*<r

22-S9

22-86

1%^

1%4

IS -26

18^

I5<>

150

14^7

148

• '

« '

7^)1

7^)2

100

3-69

3-75

300

- . .

1-27

l^S

132

CL]

NOTE ON^ THE ISOTHEKMALS OF ETHYL OXIDE*

319

The values in the second coluirm are taken directly from the paper referred to
(Table I,), in which lOS'^'S C* is regarded by the Authora as the critical temperature*
Those in column A were calculated for temperature 194'' C. from the pressures given
in the same table for 195' C. and 200^ C. (occasionally 210' or 2i0^ C)- Those in
column B were calculated, also for 194* C, from Table II. of Drs Ramsay and Young,
which contains their ''smoothed" values of the constants. Finally, column C has been
computed from my own formula, in forms (given below) which are adapted to volumes
greater and less than the critical volume, respectively, A glance at column B shows
that, so far as the "' smoothed *' data are concerned, the critical point should be sought
slightly above 194' C. For, at that temperature, the pressure has still distinctly a
maximum and a minimum value, both corresponding to volumes between 3 and 5.
Column A, calculated from the unsmoothed data, does not show this peculiarity. Hence
I have assumed, as approximate data for the critical point.

The last of these is, I think, probably a little too large ; but we have the express
statement of Drs Ramsay and Young that the true critical volume is about 4'06.

From their Table II., above referred to, I quote the first two lines below, giving
(usually to only 3 significant figures) values of dpidt at constant volume : —

dt
Calc.

2-5

160 0-92

622

414

•319

■133

616

•426

320

•131

■633

•405

i >«

. « ■

•056 019

056 019

"' 1 1-65 0-90
The third and fourth lines are calculated respectively from the expressions

(0.85+ «)i. and (l■2 + -i:^)^

representing the coefficient of {t — i) in my general fonnula

100

300

009

■0029

•009

•0029

t-t

Approximate values of the other constants are now easily obtained ; and we have^ for
the critical isothermal, while the volume exceeds the critical value,

P - ^^'^ V " v{v+\)iv-0'5)) '

In attempting to construct a corresponding formula for volumes lower than the critical
range, I assumed 3*5 as an inferior critical volume, and obtained

p-^^H^-'T^m^

As will be seen by the numbers in column C above, which are calculated from them,
these formulae represent the experimental results very closely :— but I am not quite

320 NOTE ON THE ISOTHERMALS OF ETHYL OXIDE. [CL

satisfied with the first of them, because the value (3), which it assigns to a, seems
to be too large in comparison with v. But, on the other hand, if we much reduce
this value of a, the closeness of representation of dp/dt is much impaired. Again, the
value (—1*5) which is assigned for a in the second of these formulae is inconsistent
with the fact that at 0° C. and 1 atm. the volume of one gramme is 1*4 c.c. nearly.
But a very small change of a will entirely remove this objection, and will not per-
ceptibly impair the agreement of the formula with experiment.

The general formula is applicable to temperatures considerably imder that of the
critical point, for volumes greater than 4. In fact Drs Ramsay and Young seem to
assert that at any constant volume p is a linear function of t But I think even
their own experiments show that, for t; < 4, there is diminution of the value of dp/dt
as soon as the temperature falls below the critical value: — i.e., as soon as we begin
to deal with liquid alone. And certainly such is the result which theory would lead
us to expect.

[It is curious to note that if, in my general formulse (No. LXXX. above, p. 200),
we assume

a = 7,

we have pv^E (1 + —^ — ) ; — + t— — r^;

^ \ v + yj v+y (v + yf

and this leads to P^pU- ^7^^^) + -B fl + -^] — ;

^ ^\ v(v + yyj \ v + yj V

with the condition Sv •\'2y = Rt/p.

This formula diflfers by want of one disposable constant from (C) of the paper
referred to, but approximates much more closely to it than does either (A) or (B).]

cu.]

321

NOTE APPENDED TO Dr SANG^S PAPER, ON NICOLAS
POLARIZING EYEPIECE.

[Proceedififjs of the Royal Society of Edinburgh^ November 23, 1891.]

At the yeiy urgent request of the late Dr Sangj who regarded the abo%e paper
as one of Ms chief contributions to science, I brought before the Council of the Society
the question of its publication. From the Minute-Book of the Ordinary MeetingSj I find
that it was read on the 20th February 1837, though it is not mentioned in the
published Proceedings of that date. On 21st July 1891 the Council finally resolved that
the paper should be printed in the Proceedings " if otherwise found desirable/* The
reasons in favour of printing it seem to outweigh those which may, readily enough, be
raised against such a course.

The subject is one with which, except of course in its elementSp I have long ceased
to be familiar. But, from the imperfect examination which I have found leisure to
make^ I have come to the following conclusions.

The paper contains a very important suggeation which (one would have thought)
should have been forthwith published, whatever judgment might be passed on the rest
of the work: — viz*, the proposal to construct the polariser of two glass prisms, separated
by a thin layer, only, of Iceland spar. In view of the scarcity of this precious substance,
Buch a suggestion was obviously of great value,

I am not sufficiently acquainted with the early history of the Nicol prism to be

able to pronounce on the question of Dr Sang's claim to priority in the explanation

of its action : — but he told me that he believed himself to have been the fiiBt to

demonstrate that the separation effected was due to the total reflection of the ordinary

T. IL 41

322 NOTE APPENDED TO DB SAXO's PAPEB, [cil.

rajr*. And it is qaite certain that, long subsequent to 1837, varioos yerj singalar
attempts at explanation have been given in print. The inyentor, himself^ seems to have
thought that the effect of his instrument was merely to "increase the diTergency" of
the two rays.

The numerical error which Dr Sang has pointed out in Mains' work seems to
have been a slip of the pen only, as the minutes and seconds of the angle in question
are correctly given. He supplies no reference to the passage, but I find it in the
list of calculated angles at p. 125 of the Th/crie de la Double lUfraetion. It cannot
be a mere misprint, because the supplement is given along with the angle, and is
affected by the a^rresponding error. But I do not think that Dr Sang's further remark
is justified, as Malus not only gives the correct expression for the cosine of the angle
in question, but seems to have employed in his subsequent calculations the inclination
of the axis to a /ace, not to an edge* of the crystal : — and he gives the accurate
numerical value of this quantity, as deduced firom Wollaston's measure of the angle
between two fSsu^es.

There is an altogether unnecessarily tedious piece of analysis in Dr Sang's in-
vf5Stigation of the limits within which the prism works: — and it is so even although
he shortens it by the introduction of the terribly significant clause "after repeated
simplifications/' I will give below what I consider to be a natural and obvious mode
of dealing with the question ^one which, besides, leads to some elegant results): —
but I have reproduced Dr Sang's manuscript as it was read, for the circumstances
of the present publication seem to require literal accuracy. Dr Knott has kindly
verified for me the agreement of my final equation with that of Dr Sang.

Dr Sang's problem is equivalent to the following: —

A tangent is drawn to an ellipse from a point of a concentric circle; find when
it subtends the greatest angle at the common centre.

Let the curves be

(a:/a)» + (y//8)* = l, and ^i* + yi' = y, respectively.

Then, if a; == a cos 0, y==/9 sin 0,

a:i = 7C0Si^, yi=78inv,

the condition of tangency is obviously

cos if> cos V sin 0 sin y ^ 1

Also, since the angle at 0 is to be a maximum,

|;|tan-(ftan^)-.} = 0.

* [See, however, a Note by Fox Talbot (Pwc, R. S, E., yii. 468; 15/5/71) which appears to settle this
important matter of soientiflc history by reference to a paper published by him in 1834 {PhiL Mag., iv.
389). 1899.]

cii.] ON nicol's polabizing eyepibce, 328

DifTerentiating the first equation, and eliminating d<l>ldv between the two, we get at
once the remarkably simple relation

(tan0)« = -|tani; (1).

But we may put the first into the form

cosi; sini/^ . 1 .

(cos i/V l,2co8»'sini'. ,, /(sin v)* l\,. ... ^ /«x

or -^--y+ ^ tan^ + (^5^--j(tan^)« = 0 (2).

The elimination of tan (f> between (1) and (2) is easily effected by multiplying (2) twice
over by tan^, using (1) after each operation. We thus avoid the radicals which make
Dr Sang's work so complicated, and we have only to eliminate tan^ and (tan^)* among
three equations of the first degree. The resulting equation is of the fourth degree
in (sinvy, but it contains the irrelevant factor

(cosy)* (sini/y

(Another method of effecting the elimination, while quite as simple as that just
firiven, has the advantage of not introducing the irrelevant hctor. Write for shortness

cos V sin V

and we have pcos(f> + qeinif>^- ,

p (sin <l>y + q (cos 4>y = 0.
From the second of these, by the help of the first, we at once obtain

p sin <f> + q cos (f> = - cos <f> sin <f>,

p . q 1
or -^ + -r-^ = - .

cos 9 &in<f> y

The following are immediate consequences: — obtained, respectively, by multiplying to-
gether the first and fourth of these equations, and by squaring and adding the first
and third: —

'^ ^ sin ^ COS ^ y
p* + 5" + 4p9 sin 0 cos 0 = ~ {1 + (sin 0 cos 0)"}.
From these the final result may be written by inspection, in the form

41—2

324 NOTE APPENDED TO DR SANG'S PAPER. [CII.

or {p' + ^-^J-4>P'^{p' + ^~)=^^-.

which is obviously of the third degree in (sin v)\)

[It is particularly interesting to compare these plane results with those of the
corresponding space-problem as given by the obvious quaternion process. 1899.]

It is clear that there are other parts of Dr Sang's paper which might be greatly
simplified by the use of an auxiliary angle; but it suffices to have shown the value
of the method in the most complicated part of the investigation.

[P.S, — J\rot;. 23, 1891. — Mr R. T. Glazebrook has kindly given me a reference to
Comptes Rendus, xcix. 538 (1884), where M. E. Bertrand has suggested the employment
of glass prisms separated by a thin layer of Iceland spar.]

cm.] 325

cm.

NOTE ON Db MUIKS SOLUTION OF SYLVESTERS
ELIMINATION PROBLEM.

[Proceedings of the Royal Society of Edinburgh, May 2, 1892.]

The following method of treating the question occurred to me while Dr Muir was
reading his paper at the last meeting of the Society. It seems to throw some new
and curious light on the intrinsic nature of the problem. I have confined myself to
an exceedingly brief sketch, but it is clear that the proposed mode of treatment
opens a wide field of interesting work.

Write the equations as

or f*-2e,ff; + i;»=0, &c.

The two values of f/i;, &c., are evidently reciprocals of one another. In fact, if we
were to put

f/i; = tantf„ &c.,
the equations might be written 1 - ^ sin 2tf, = 0, &c.

Since we have - . ^ . i = 1,

^ ? f

while the values of the factors on the left are, respectively,

^ or -, ^ or -, ^ or -,

326 NOTE ON DR MUIB'S SOLUTION OF SYLYESTEB's ELIMINATION PROBLEM. [CUL

it is obvious that the fourth equation required for the elimination is

{t^ (^^^ - ^y (hu - uy {uu - t,y {uu - uy = o.

Put T^tiUU, and this is

^(r-i)»(r-f,«)»(r-tf)»(r-v)»=o.

Expanding and regrouping, the expression is easily transformed to

16 ( /I j-/_t\« /I a./_«\a /I j-f_a\*

?.{-(^T-e-a'-c-^^^^- ^"^ ^}-».

The factor in brackets is the square of the determinant

\ I e^ e,

\ et 1 ei

I e, e^ 1

and thus Dr Muir's result is reproduced when we insert the values of 6i, e^, ^ in
terms of A, B, C, A\ B, C\

One interestiug point of the transformation seems to be the breaking up of this
determinant into the four factors above specified; so that the equation

1 (sin2^)-i (sin2a)-^ =0

(sin2tf)-i 1 (sin 2/8)-^ !

(sin2a)-^ (sin2/9)-» 1

has for roots, as values of tand.

tan a tan /8,

tan a tan /8' tan/8

.tan a , tan B

, and '^

tana

But the novelty and value of the process seem to lie in the mode in which the
elimination is effected by mere general reasoning.

CUV.]

327

CIV.

NOTE ON THE THERMAL EFFECT OF PRESSURE ON WATER.

[Proceedings of th^ Royal Society of Edinburgh, July 18^ 1892,]

I HAVE just seen in the Comptes Rendiis (June 27) an account of some experiments,
on this subject, made by M. Galopin io the laboratory of Professor Pictet As the
effects obtained by him seem to be somewhat greater than my own experiments had
led me to expect, I was induced to repeat my calculations with the view of trying to
account for the difference. Unfortunately, M. Galopin's work is confined to 500 atmo-
spheres, a pressure which lies a little beyond the range of my experiments; so that
no very trustworthy comparison can be made- M» Galopin's results have one advantage
over those of the direct experiments of the same kind which I made, inasmuch as
he was able to use ordinary thermometers, while I employed thermo-electric junctions,
in measuring the rise of temperature by compression. But they have a corresponding
disadvantage, in the fact that mine were obtained instantaneously (by means of a dead-
beat galvanometer) and required no correction ; while his had to be corrected for the
heat-equivalent of his apparatus to an amount not easy to estimate with accuracy.

I had assured myself of the general accuracy of my own work by showing that
three altogether independent modes of estimating the effect of pressure on the maximum
density point of water gave closely concordant results : — viz. a lowering of that
point by about 1* C, for every 50 atmospheres. These investigations were described to
the Society in 1881 — 4, and appear (in abstract) in our Proceedings] more fully in
the Challenger Reports. [See No. LXI. above.] One mode of determination was dived
(a modification of Hopes experiment); the others were theoretical deductions, from the
compressibility of water at different temperatures, and from the rise of temperature
produced by compression, respectively. M. Amagat subsequently obtained a result very
closely agreeing with mine as given above. His method differs from any of mine, for he

328 NOTE ON THE THERMAL EFFECT OF PRESSURE ON WATER. [ciV.

seeks two temperatures, not very different, at which water has the same Tolome at the
same pressure.

So far, I had been dealing with pressures of little more than 200 atmospheres.
Higher pressures led to the result that the displacement of the maximum density point
increases very much faster than does the pressure. For the terms in higher powers of
the pressure begin to tell more and more; and another cause comes prominently into
play, depending on the fact that water has a temperature of minimum compressibility
(about 60"" C. at ordinary pressures). This affects to a very much greater extent the
lowering of the maximum density point by pressure than it affects the amount of heat
developed by the compression. Both of these causes are indicated in my formulae as
contributing to such a result, but the small numerical fisurtors of the terms which express
them are not accurately known; and the calculation of the thermal effect of large
pressures from data obtained by measuring compressibility at different temperatures is a
very severe test of their accuracy. Besides, in giving a formula which exactly represented
my determinations of the change of volume of water, under pressures from 150 to
450 atmospheres, and at temperatures 0"" to IS"" C, I expressly said that *^ it must not
be extended, in application, much beyond" these limits. If, however, we venture to
extend it to 500 atmospheres, it leads to the following expression, for the heating of
water by the sudden application of that pressure,

^ 4- 3-2 ,
26 '

where t is the original temperature (C.) of the water operated on. In obtaining this
result it is assumed, in accordance with Eopp's data, that the expansibility of water at
ordinary temperatures and at atmospheric pressure is approximately {t — 4f)/72flO0. Other
experimenters make it somewhat greater. [If the maximum density point were lowered
V for every 50 atmospheres, the heating by 500 atmospheres would be about (« + l)/22
only. Comparing this with the result above, we see how considerably the causes, alluded
to, affect the calculated amount of heating.]

Now I find that M. Galopin's results may be represented very closely (from 0** to
10° C, which are his temperature limits) by the analogous expression

t + 5
25 •

The difference between the denominators of these expressions is not serious, and may
depend upon the uncertainty of the assumed expansibility of water, or upon an over-
correction of his results by M. Galopin. [He increases his observed data by 52 per cent,
in consequence of the thermal capacity of his apparatus.] But the difference between
thu numerators seems to show once more that M. Galopin's data have been over-corrected,
or that it was scarcely warrantable to extend the application of my formula so far
AH 500 atmospheres.

CT,]

329

cv.

NOTE ON THE DIVISION OF SPACE INTO INFINITESIMAL

CUBES.

[Proceedings of the Roi/al Soctett^ of Edinburgh j December 5, 1S92-]

The propasition that '' the only series of surfaces which, together, divide space into
cubes are planes and tht^ir electric images*' presented itself to me twenty years ago,
in the course of a quaternion investigation of a class of Orihogmial Isothermal Surfaces
[No, XXY. above]. I gave a second version of my investigation in voL ix. of our Pro-
cmdings. [No. XLIV. above,] Prot Cay ley has since referred me to Note vi., appended
by liouville to his edition of Mongers Application de I' Analyse A la Giomih^ (1850),
in which the proposition occurs, probably for the first time. The proof which is there
given is very circuitous ; occupying some eight quarto pages of small type, although
the reader ia referred to a Memoir by Lam^ for the justification of some of the steps.
But Liouville concludes by saying: — "Tafialyse pr^cMente qui ^tablit ce fait important
n'est pas indigne, ce me semble, de I'attention des g^om^tres," He had previously
stated that he had obtained the result " en profitant d'une sorte de hasard/' As
Liouville attached so much importance to the theorem, and specially to hb proof of
it, it may not be uninteresting if I give other modes of investigation* The first of
them is merely an improved form of what I have already given in our Proceedings;
the second (which is the real object of this note) seems to have secured nearly all the
advantages which Quaternions can afford, in reapect alike of directness, clearness, and
conciseness. It is very curious to notice that much of this gain in brevity is due
simply to the &ct that the Conjugate of a certain quaternion is employed along with
the quaternion itself in my later work ; while I had formerly dealt with the reciprocal,
and had, in consequence, to introduce from the first the tensor explicitly. The in-
vestigation should present no difiiculties to anyone who has taken the sort of trouble
T, n. 42

330 NOTB ON THE DIVISION OF SPACE INTO INFINITESIMAL CUBES. [CV.

to remember elementary quaternion formulae which every tyro in integration has to take
to fix in his memory the values of d\AnXy d\og , or dtan'"*^;, &c.

The only peculiarities of the question seem to be due to the contrast between the
(apparently) great generality of the initial equation and the extremely restricted character
of the sole solution. This will be abundantly evident from the discussions which follow,
since it would almost appear as if the conditions arrived at were too numerous to be

simultaneously satisfied. I find it very convenient to use a sjrmbol 3 [in the sense of -7-)

to express rate of increase per unit of length. Thus

dx ^ dy dz
may be written V = a9i + y83, + 73,,

where a, )8, 7 are any rectangular unit-system.

The equation da^ruf^dpq (1)

(where u is a scalar, and q a versor, function of p) ensures that an element of space
at <T corresponds to a similar element at p; so that the transformation from p to <r,
or vice versd, is from one mode of dividing space into infinitesimal cubes to another.
[From the purely analjrtical quaternion point of view the question may be regarded
as simply that of finding u and q as functions of p, so that the right-hand member
may be a complete differential.] We have at once

Soda- = — SdpV . Sao- = u^ . dpqaq-^,

whatever constant unit vector a may be. Thus

-VSa<r = uqa^^ (2).

A part, only, of the information given by this is contained in

W .uqaq-^^O (3),

or V . qaq-^ — = FV . ja j-*

= F . (Vgrgr-igragr-i + qaq-^Vqq-^) - 25 . gag-^Vj . Vq^q^^
= 2qagr^8 . Vq^^ - 28 . jag-^Vi . Vq^q-K

From the sum of the three equations of this form (each multiplied by its jagr-i)
it appears at once that

S.Vqq-^^0 (4);

so that, as jag~* may be any unit vector,

= 2a,9g^> (5).

CV.] NOTE ON THE DIVISION OF SPACE INTO INFINITESIMAL CUBES. 331

ions of this form we have

From two of the three equations of this form we have

whence

-F.,<* + W?-^»%^".

where the V is obviously superfluous ; so that

^v^ + a.^=?»?^^ (6).

There are, of course, three equations of this form, and they give by inspection

The first and last of these equals give
whose general solution is known to be

where m and e are constants. The other members of (7) show that one term only of
this 2 is admissible; so that, as no origin was fixed,

ti* = -Sir .

From the three equations (5) we get also

V(ti5) = 0,
so that J = Up .a,

a being any constant versor. Thus we have the complete solution. It gives by ^1)

da- = mHr^p'-^dpp''^a = — m*a~^dp''^a,
so that <r is merely — p~^ multiplied by a constant and subjected to a definite rotation.

But the following process is very much simpler. For we may get rid of the
factor u, and greatly shorten the investigation, by writing the equation of condition
in the form

da^Kqdpq (1).

It gives at once 3«3i<r = 3a (-^?«9) = 3i (JSTg/Sj),

or F.7(V9-70,g)(r^ = O (a).

42—2

332 NOTE ON THE DIVISION OF SPACE INTO INFINITESIMAL CXJBBS. [CV.

Multiplying by 7, and adding the three equations of this form, we have

By the help of this we may write (a) as

Tq ~^\Tq VqJ'

» .h.. ^T,"^-^'?,'^-?,">'?,-i^P, W

Thus, as the form of the three middle terms shows that their common value most
be some constant quaternion,

for we need not add a constant vector to p, and the form of the first of the five
equal quantities above shows that no qiuUernion constant (except, of course, one of the
form 6a abready referred to) can be added to the right-hand side.

Thus, finally, as before d<r = — ap~*dpp"^ira.

Though the methods employed in these two investigations are, at least at first
sight, entirely different, it will be easily seen that the equations (7) and (6) to which
they respectively lead are identical in meaning with one another, term by term. Yet
the former shows two differentiations in every term, while the second appears to involve
one only. Thus also, two distinct integrations were required in the first solution, while
one sufficed for the second. But in the first, the tensor and versor of the quaternion
were all along separated; in the second the quaternion itself was directly sought.

• [Note that Sdqq-^ = dTqlTq. 1897.]

cvi.] 333

CVI.

NOTE ON ATTRACTION.
[Proceedings of the Edinburgh Mathematical Society, Febnuiry 10, 1893. Vol. xi.]

It is well known (see Thomson and Tait, §§ 517, 518) that a spherical shell,
whose surface-density is inversely as the cube of the distance from an external point,
as well as a solid sphere whose density is inversely as the fifth power of the distance
from an external point, are centrobaric. The centre of gravity is, in each case, the
"image" of the external point.

To show that these express the same physical truth, we may of course recur to
the method of electric images from which they were derived. But we may even
more easily prove it by a direct process, for it is obviously only necessary to show
that a thin shell, both of whose surfaces give the same image of an external point,
has everywhere its thickness proportional to the square of the distance from that
point.

Call 0 the object, and / the image, point; and draw any radius- vector /PQ,
meeting the respective surfaces of the shell in P and Q. Then, ultimately,

0(2-OP = QPco80P/,
or, in the usual notation,

sQ = Sr cos OP/,

whence (introducing the new tBciot r)

r»^« Sr (^-rcos OP/) ^hrOIcosIOP.

But lOP is equal to the angle between IP and the normal at P, so that the
thickness of the shell at P is

Srcos/0P = 7^.
O/.e"

334 [cvj

CVII.

ON THE COMPRESSIBILITY OF LIQUIDS IN CONNECTION WIT]
THEIR MOLECULAR PRESSURE.

[Proceedings of the Royal Society of Edinburgh, March 6, 1893.]

That liquids, if finitely compressible, must (at any one temperature) beoon
steadily less compressible as the pressure i^ raised, seems to be obvious without ar
attempt at proof. Yet the assertion is even now generally made, mainly in consequent
of an erroneous statement of Orsted, which has been supported by some compan
tively recent investigations of Cailletet and others, that the compressibility of wat^
(at any one temperature) is practically the same at all pressures not exceeding a fe
hundred atmospheres.

But in 1826 (Phil. Trans., cvi.), Perkins had clearly established the feet tb
the compressibility of water at 10° C. diminishes: — rapidly at first, afterwards moi
and more slowly: — as the pressure is gradually raised. Perkins' estimate of h
pressure-unit seems to have been considerably too small, so that his numerical dat
are not very trustworthy: — but this does not in the least invalidate the proof I
gives of the gradual diminution of compressibility; for that depends of coarse upo
relative, not upon absolute, value&

In the very earliest determinations which I made, some ten or twelve yeai
ago, while examining the pressure-errors of the "Challenger" thermometers, this dim
nution of the compressibility of water was prominently shown: — and in 1888 I oav
as a £Ekirly close approximation to the average compressibility for the first p atmosphere
the empirical expression

in which the constants depend on temperature only.

cvu.]

ON THE C0MPBES8IBILITY OP LIQUIDS.

335

This, it will be observed, is in complete agreement with the form of the result
of Perkins, I also found that the addition of coraraon salt^ to the water operated
on J had the effect of increasing the constant B in this formula by a quantity pro-
portional to the amount of salt added; A being practically unchanged, so long as
the temperature was kept couatant.

These considerations seemed to point to the quantity B as being at least closely
connected with the internal molecular pressure (usually named after Laplace); and,
speculative as the idea confessedly is, it seemed worthy of further development.
Another argument in its favour is furnished by a consequence of the hypothesis.
For it is easy to see that when the average compressibility of a subatance can be
represented by the expression above, the equation of its isotherm a Is must have the
form

{B-^p)(V'a)=C;

approximately that given by the kinetic theory of a gas, when it is regarded as an
assemblage of hard spherical particles.

Neitrly three years ago, wHile I was preparing for press the second edition of
my text- book Properties 9' -\**^V^^ \ ^^"^^"^^^ kindly gave me several unpublished

*Q^ ^jid column some of these results for

numerical details of his n ., ° , *^^on the compressibility of water and

ether. The following short taw.
water at O'^C:—

^^,

1
501
1001
1501
2001
2501
3001

Volume,

100000

97668

'95645
'93924
9239S

^91065
'89869

100000 0
•97664+ 4
■95662-17
•93925- 1
•92405-12
•91064+ 1
•89870- 1

b
•00000 0
•97652 + 16
•95644+ 1
■93909 + 15
•92393 0
•91058+ 7
•89873- 4

1-00000 0
•97657 + 11
■95652- 7
•93916 +
■92399 -
91062 +
•89875 -

The numbers io the columne a, b, c are volumes calculated respectively from the
following formulae for the average compressibility for p atmospheres: —

•80454 -30 3015

6019 + jo' 5887+J3' 6933+p"

The first was calculated from the data for 1, 1501, and 3001 atm.; the second
from those for 1, 1001, and 2001 atm.; the third was obtained from them by inter-
polation. After the numbers in each column the difference " observed — calculated " is
given. These are all small ; and, especially in the case of formula c, the coincidence
seems almost perfect throughout, for the differences have regular alternations of sign.
But it is to be noticed that simultaneous increase, or diminution, of A and B by
as much as 2 per cent, does not seriously affect the agreement of the formula with
the results of experiment.

336 ON THE COMPRESSIBILITY OF LIQUIDS [cvn.

I have been for some time preparing to undertake an extended series of experi-
ments on the compressibility of various aqueous solutions, with the view of finding
(although by an exceedingly indirect and possibly questionable process) how the
addition of a salt to water affects its internal pressure. But the recent publication
of the final results of Amagat's experiments on the compression of water by pressures
rising to 3000 atmospheres (more than six-fold the range attained in my own work)
has led me to make a new series of calculations with the view of testing how far
the above speculations, suggested by the results of pressures limited to some three
tons' weight per square inch, are borne out by the results of pressures of twenty
tons. The agreement, as will be seen, seems on the whole highly satisfactory; though,
for a reason abready given, and presently to be even more forcibly illustrated, the
calculations are necessarily of a somewhat precarious character.

Thus we obtain from Amagat's paper (Comptes Rendus, January 9, 1893) the
following determinations of the volume of water at 0°C., for additional pressures of
400 and 800 atmospheres: —

Fiet8Qre.

Table, No. 1

Table, No. 2.

100000

401

•98067

^"1

801

•96371

..1
Hdinbur'''

The pressures in Table 1 extend to 1000

atm. only, those in Table 2 to 3000 atm.

These give, respectively, for the average compressibility of water per atmosphere
for the first p additional atmospheres, p ranging from 0 to 800,

0-296 03057

5725 -hp' 5939 +p'

whence the compressibility at ordinary pressure may be either

00000517 or 000005147.

To enable us to choose between these formul» we have the following comparison
with the data for higher pressures in Amagat's second table : —

Fresanre.

Amagat.

First formola.

Second formnla.

1001

•95596

•95595

2001

•92367

•92337

•92299

3001

•89828

•89824

•89741

The first formula, therefore, represents with remarkable closeness the average
compressibility of water at 0°C. for any range of pressure up to 3000 atmospheres;
while the second obviously gives considerably too much compression at higher pressures.
Yet there is but one numerical difference between the sets of data from which
these two formulas were derived, and that is merely a matter of four units in the
fifth decimal place of the volume at 401 atmospheres! Thus very small inevitable
errors in the data may largely affect the values of the constants in the formula.

cvn.]

m ooHimQOOir with theih moleculab pressure.

337

The only certain method of overcoming this difficulty would be to work with prea-
surea of the same order as B.

The expreasioa which I gave in 1888 for the average compressibility per atmo-
sphere at C'C. was {Gkallengm' R&pm% Physics and Ckmiistry, Vol ii,, Part 4, p. 36;
anil No. LXL p. 34),

0-QQ1863
36 +i) *

the unit for p being 1 ton weight per square iuch. To atmospheres (1523 per ton
weight per square inch) this Lb

0-284

5483 + p'

giving 0*0000518 as the compressibility at ordinary pressures. This agrees closely with
the first, and more accurate, of the two formulae just given ; and yet it was derived
from data ranging up to 450 atmospheres only. I stated at the time that " probably
both of the constants in this formula ought to be somewhat larger" This would make
it still more closely agree with Araagat's results,

I have worked out the values of the quantities A and B for the ten special
temperatures (from 0° to 48°*95C, inclusive) in Amagat's table No. 2; taking for each
temperature the data for pressures 1, 1501, and 3001 atmospheras. The resulting
f^jmiute give results agreeing very fairly with the compressions given for 501, 1001 1
2001, and 2501 atmospheres i — the agreement being in fact almost perfect for the two
higher pressures, but the compression being (as a rule) slightly in defect for the
lower pressures, M, Amagat himself has stated that his results for lower pressures
are given more accurately in the series of e^cperinaents where the pressure was never
very great, than in those where it was pushed to 3000 atm. In fact his manometer
had to be made considerably less sensitive when very gi^eat pressure was employed.
For the reasons just pointed out I cannot wholly trust these calculations, and there-
fore I think it unnecessary to give them here* But they agree (with only one
exception, for 29^*43 C) in a very remarkable manner in showing that the values of
A and B steadily increase with rise of temperature up to about 40' C, and thence
apparently diminish. That the value of A should at first steadily increase with rise
of temperature was of course to he expected as a consequence of the known change
of molecular structure if (in accordance with the supposed analogy of the kinetic
gas formula above quoted) it represents the utmost fractional diminution of volume
which can be produced by unlimited pressure* And Cantons old discovery, that rise
of temperature involves diminution of compressibility, requires that B should at fixst
increase more rapidly than does A. [This is not necessarily inconsistent with the
commonly received statement that the surface-tension of water is, in all cases,
diminished by rise of temperature,] The turning*point seems to be connected with
the temperature of minimum compressibility^ discovered by Pagliani and VincentinL

T, IL 43

338

ON THE COMPRESSIBILITY OF LIQUIDS.

[cvii.

The behaviour of water at ordinary temperatures is of such an exceptional
character that we cannot feel certain that aqueous solutions may not show more than
mere traces of it. In my projected experiments, therefore, I intend to employ at
least three different solutions of each of the salts to be examined, one of them
being only a little below saturation strengtL The comparison of the results for
solutions of very different strength may enable me to eliminate the effects of the
peculiarities of the solvent.

As a contrast to the behaviour of water, above discussed, I give some results
for sulphuric ether; also founded on data furnished to me three years ago by M.
Amagat. These data were given to four decimal places only.

'reegnre.

Amagat.

Formnla.

W-20.
Amagat. Formula.

10000

1^0320

10320

501

■9468

■9498

•9673

•9722

1001

•9130

•9156

•9294

■9311

1501

■8884

•8885

•9018

■9018

2001

■8684

•8805

•8797

2501

■8522

•8524

•8630

•8624

3001

•8394

•8395

•8484

■8484

The agreement is not by any means so complete as in the case of water: —
but it is probable that slight changes in the values of the constants may greatly
improve it where defective, while otherwise scarcely interfering with it.

The formulae for average compressibility employed were, respectively,

•3016

•2863 . ^o ,

for 20°-2.

(Note that calculation from the data, direct, gives 0*31126 as the value of A
in the second of these, but this has to be divided by the volume at one atmosphere.)
Here, according to the previous mode of interpretation, the Laplace-pressure is
diminished, and the ultimate volume seems to be increased by rise of temperature,
as was to be expected.

cviii.] 339

CVIIL

PEELIMINAEY NOTE ON THE COMPEESSIBILITY OF AQUEOUS
SOLUTIONS, IN CONNECTION WITH MOLECULAE PEESSUEE.

[Proceedings of tiie Royal Society of Edinburgh, June 5, 1893.]

The experiments referred to in my paper of March 6th (antd, No. CVII.) have
been completed, but the results are by no means so exact as I hoped to make them.
There was great difficulty in procuring the small bore tubes for the piezometers, and
thus I had to employ them without previous calibration, as the solutions to be experi-
mented on had already been prepared, and their densities determined at definite
temperatures. Delay might have led to evaporation. When I proceeded to the
calibration, after completing a large series of experiments, I was greatly annoyed to
find that the bores of many of the tubes were by no means uniform. This accounts
for the fact that my experiments, though fairly concordant, are not sufficiently so to
afford more than a very strong probability in fiavour of the general result of the inquiry.
For this reason I have described my paper as a Preliminary Note.

The idea I sought to develop was of the following nature. I had found that the
average compressibility of water, at any one temperature, could be well represented by
the simple formula

where p is the range of pressure through which the compressibility is measured ; A and
B being functions of temperature. But I also found that for aqueous solutions of
common salt, of different strengths, and at the same temperature as the water, the
formula was altered to

A
5 + « + p'

43—2

340

NOTE ON THE COMPRESSIBILITY OF AQUEOUS SOLUTIONS,

[cvnL

where A and B were as before, and 8 was proportional to the weight of salt dissolved
in 100 of water. In particular that, when 1 ton weight per square inch (152'3 atmo-
spheres) is the pressure unit, 8 is nearly the weight of salt in 100 of water.

Theoretical speculations (given at some length in my Report an 8ome of the
Physical Propertiee of Water, ante, No. LXI.) led me to look on the B, and the jB + «,
of these formulae as being connected with the molecular pressure in the liquid, and
I developed one application of them, relating to the maximum density points of various
solutions of common salt.

The present series of experiments was conducted precisely as were the earlier ones,
but unfortunately many of the piezometers (of which a large number were required
in order that several solutions should be operated on at the same time) were new,
and (as I afterwards found) faulty. The selection of the salts was undertaken by
Dr Crum Brown, and the solutions were made and the density determinations effected
in his Laboratory by Mr A. F. Watson.

I give these at once, as they have intrinsic value altogether apart from my work
and my hypothesis.

In the following table the letters S and W stand for the masses of salt, and of
water, respectively. Mr Watson remarks that the error in the numbers of the first
column, from which the second was calculated, does not exceed 1 in 1000. The error
in the densities does not exceed unit in the fourth decimal place.

'""■stw '""w

Temp.
C.

Sp. Gr.

Temp.
C.

Sp. Gr.

Potassium Iodide —

14-538 17011

5°-5

1-1197

13°-5

1-1179

9-302 10-256

5°-6

1-0737

12°-2

1-0727

4-313 4-507

5°-4

1-0329

12°0

10323

Potassium Ferrocyanide —

14089 16-399

5°-5

1-0987

13°-5

1-0967

9-411 10-389

6°-3

10620

12°-1

1-0610

4-733 4-990

6°0

1-0328

ll°-4

1-0322

Ammonium Sulphate —

15-938 18-960

6°-8

1-0954

ll°-2

10944

9-232 10-171

6°-3

1-0559

12°-7

10547

5-301 5-597

5°-7

1-0326

12°-1

1-0317

Magnesium Sulphate —

13-836 16-058

6°-8

1-1489

ll°-2

11479

9-508 10-507

5°-8

1-1005

13°-1

10990

5-869 6-235

5°-7

10614

12°1

1-0602

CVIII.] IN CONNECTION WITH MOLECULAR PRESSURE. 341

'%!w ^«»fr

Temp.
C.

8p. Gr.

Temp.
C.

Sp. Or,

Barium Chloride —

13-798 16006

5°-8

1-1366

11°'2

11354

9096 10006

5°-8

1-0869

13°-1

10855

4-585 4-805

5°-6

1-0423

12°-2

1-0416

To these may be added the following, due to Dr Gibson, from my Challenger
Report referred to.

O^C. 6°C. 12«C.

Sodium Chloride —

17-6358

1-138467

1-136040

1133565

13-3610

1-101300

1-099341

1-097244

8-8078

1067689

1-066144

1-064485

3-8845

1-029664

1028979

1-027935

Although I made at least two observations at each of the pressures 1, 2, and
3 tons, on each solution, in each of two piezometers, I publish in this Abstract nothing
beyond some mean results at one temperature and for one pressure: — viz. 12°C. and 2 tons.
These are fairly representative of the whole work. The columns of mercury used in
calibration corresponded nearly with the parts of the tubes concerned in the measured
compression at that pressure; and, on such lengths of tube, errors of measurement due
to slight changes of temperature of the solution, &c., are comparatively insignificant.

The change of (unit) volume of water per ton at 12"* C. and 2 tons is (by my
former work)

«|^* = 0««M.

If to the 36 in this expression be added the product of the quantity 8 below given
for any one salt, multiplied by the percentage of the salt, we have the numbers in
the column headed Cede, Those headed Obs. were obtained as stated above; and the
agreement is on the whole satisfactory. The old determinations for common salt are
included in the table, though they show rather less concordance than the others.

100^ $ ObB. Calc.

Sodium Chloride —

17-6 11

13-4

8-8

3-9

Magnesium Sulphate —

1606 10

10-51
6-23

0-00428

0-00431

472

470

524

519

594

585

450

457

510

555

559

342 NOTE ON THE COMPRESSIBILITY OP AQUEOUS SOLUTIONS. [cvm.

looj

Ob«.

aue.

Ammonium Sulphate —

18-96

0-77

0-00475

0-00470

10-17

542

640

575

580

Potassium Ferrocyanide —

16-4

0-62

512

513

10-4

654

556

5-0

605

602

Barium Chloride —

16-0

052

530

534

100

573

4-8

612

611

Potassium Iodide —

1701

0-29

576

10-26

602

603

4-5

627

629

As stated in my previous note, my formula agrees extremely well with the recent
determinations of Amagat, of compression of water up to 3000 atmospheres. But the
values of A and B which I deduced firom them (especially about 12** C.) are some-
what larger than mine, though they bear to one another nearly the same ratio. If
I had used his value of B, the coincidences above would not have been sensibly impaiied,
but the values of 8 would have come out a little greater.

cix.] 343

CIX.

ON THE COMPRESSIBILITY OF FLUIDS.

[Proceedings of the Royal Society of Edinburgh, January 15, 1894.]

The recent publication of the full results of Amagat's magnificent experiments has
led me to make further comparisons with the empirical formula (originally suggested
by the graphs of my Challenger work) which I have on several occasions brought
before the Society: — viz.

Vo — v_ e

I find that Amagat's results, for a number of common liquids, from 1 to 3000 atm.
may be fairly represented by substituting the following values of e and 11 in the
above formula: —

0° 10^ 20° 80° 40° C.

p 0-291 -296 -302 -310 '319

.... 2420 2240 2100 1980 1860

Ethylic Alcohol
Methylic „
Propylic

Bisulphide of Carbon
Iodide of Ethyl

0-274

•280

•281

•287

•288

3230

3130

2970

2865

2700

0-283

-290

•295

•302

3240

3180

2990

2870

0-265

•271

•

•277

•274

3510

8390

B200

2880

0-286

•286

•291

•294

•299

3970

3720

3560

3370

3190

•

0-288
3570

•

•291
2920

344 OS THE 00MPBE88IBILITY OP FLUIDS. [CUL

(f- lO" XP atP WC.

,,. , .J , p, u . 0-278 . -293 .

Chlonde of PhoRphonw . ^^^ ^^

. , 0-284 -298

Acetone • • • • gjgo 2570

For the curiouHly exceptional case of water we have

(TC. 2'^1 4«'85 6^a5 lOP-1 14'-25 2(r'4 29^-43 40^-45 48*'-85

0-303 -303 '307 -311 '313 314 -314. -313 327 -323
5940 6030 6220 6390 6560 6680 6830 6940 7520 7440

whence compressibility for low pressures,

00000511 503 493 486 478 470 459 449 434 434

The agreement with the experimental data would be somewhat closer if II for
any one temperature were (in accordance with theory) regarded as a quantity which
increases with the compression produced.

For the present, as no definite theoretical basis has been assigned for it, the formula
must be regarded merely as an exceedingly convenient mode of summarizing the ex-
perimental results; justified by the closeness of its general agreement with them.

On these numbers remark

First, that e is nearly the same for all the liquids in the table: — its lowest value
being for propylic alcohol, and its highest for water. But the differences of these
extremes from the mean of all are less than 7 per cent. Hence it seems that ordinary
liquids, as a rule, would be reduced by infinite pressure to about 70 per cent, of their
usual volume: — provided, of course, that the formula remains applicable for pressures
immensely exceeding even the enormous ones applied by Amagat.

Second, e increases, as a rule, with rise of temperature. [But it does not appear
to increase, in any case, so much as to make the ultimate volume diminish when
temperature rises.]

Ttiird. Except in the case of water, 11 falls off rapidly with rise of temperature.
This was, of course, to be expected from the increase of volume ; and it is the chief
cause of the increase of compressibility as given by the formula. But the value of 11
does not seem to vary inversely as the square of the volume.

Fourth, In the exceptional case of water, 11 increases steadily with rise of tempera-
ture, at least up to 40"* C. This is the immediate cause of the diminution of com-
pressibility given by the formula as the temperature is raised. But, so far as the present
rough calculations go, Amagat's data would seem to make the temperature of minimum
compressibility considerably lower than that assigned by Pagliani and Vincentini [This
may be due to the great range of pressure, or to the fact that the formula treats 11
as a constant instead of taking account of its increase with compression.]

It is interesting to compare, with these, some (necessarily very rougli) results fo

CIX.]

ON THE COMPRESSIBILITY OF FLUIDS.

345

a subfltanee which requires considerable extemal pressure to keep it in the liquid state.
It is shown that if the empirical formula, above, be true generally for any substance,
it holds from any initial value of i?o*, provided that we give e aud 11 proper corre-
sponding changes of value. The new H is greater than the old by the pressure at
the new t*„. The new e must be employed with the new initial volume to give the
ultimate volume. The following data were calculated from Amagat's Tables 13 and 10
(Ann. de Ckimie, XXIX,, 1893). The first of the three volumes given for each temperature
is that of the substance when just wholly liquefied by pressure. This comparison is
by no means a fair one. for the range of volumes is very different alike in extent
and in situation, for the diflerent temperatures. And, from the extremely great com-
pressibility of the liquid when just fonned, we should expect to find the assumption
of constant 11 very far from the truth.

Carbonic Acid.

np. (f C.

si* -002145

44-4

■002338 50-4

20"
•002609

80°
70-7 003282

500 1781

500

1826 500

1876

500 1926

1000 1656

1000

1685 1000

1716

1000 1748

From these we obtain the following sets of values of e and 11 : —

0335 0-373 0*424 0527
420 276 170 48

The value of IT, calculated from the altered formula, has, in each case, been diminished
by the corresponding initial value of p. We see that e increases with great rapidity
as the temperature rises r — but the indicated ultimate volumes of carbonic acid, under
infinite pressure, are not much affected thereby, being respectively

000143 147 150 155

where the unit is the volume of the gas at 0"" C. and 1 atm.

The values of 11 are, of course, small ; and they diminish rapidly with rise of
temperature, [The critical point is about 31^*35 C, which is but little above the
highest temperature in the table.]

A fairer test than the above, from one point of view at least, might have been
based upon Amagat's important Table 17, had it given data for (say) voh = *00225
at each temperature in addition to those at *0025 and '0020. I have done the best
I could, by taking the nearest data directly given in Table 13. Here are a few of

the results obtained.

* [TliDfij from
we haye st onoe

if we write

T* n.

-V _ ep

b- ^L.

{n + 1 - epi) n ^p ' ih + ip-Pi) *

n-hl-epi

.— t a^^ ni = n+|j,» 1890.]

346

ON THE COMPRESSIBILITY OF FLUIDS.

[cix.

Carbonic Acid.

20° C.

SSO"

40"

60°

V "

64-4 -0025

109 0025

155 0025

201 -0025

150 00217S

200 0022

225 -00228

300 -002255

300 002

384 002

470-5 002

660 002

e -2833

•2936

•3136

3312

n 35-6

22-7

24-5

34-6

Ult. Vol. 001792

•001766

•001716

•001672

Other deductions from Amagat's data are given, in considerable numbers: — from regions
of the COa diagram in which 11 is respectively +, — , or even zero, the latter be-
longing of course to the conditions under which it behaves as a true gas. Thus, taking
the data for volumes

001636, 0013, and 001

we obtain the values of IT given in the first line of the table below. Here the
substance was, throughout, at density less than the critical. The second line gives
the corresponding results for a range of volumes which includes the critical volume : — viz.

0-00578, 000428, 000316.

The application of the formula to this series (where the part of the isothennal which
is treated contains a point of contrary flexure) is obviously a matter rather of curiositv
than of science.

VIZ.

Finally, the third line gives data for volumes all well under the critical volume: —

000316, 000250, and 0002.

100«

198"

- 21

- 8

-803

-80-5

-46-5

Values of n for COa (in Atmospheres).

Temp. 30° C. 36° 40° 50° 60° 70° 80° 90°

(58-5) 34-3 14-2 49 24 05 - 1-2

-73-5 -75-5 -77 -786 -805 -80 -81-1

[35-6] -38-6 -43 -42 -46*5 -47 -46-5

The single number in ( ) refers to vapour, that in [ ] to liquid; all the others
to gas. The results for volumes greater than the critical volume are very interesting.

The rest of the paper deals with (unsuccessful) attempts to apply, to Amagat's data,
the equation of Van der Waals: — viz.

(p + f,)(«-/3) = -B2'.
The arguments in consequence of which the constituent A/v^ was originally introduced

CIX.] ON THE COMPRESSIBILCTY OF FLUIDS. 347

and, as I have elsewhere* endeavoured to show, incorrectly introduced, were specially
based upon the properties of liquids, rather than of fluids in general; and it is
therefore to be expected that the formula, if valid, should be specially applicable to
liquids.

The most valuable characteristic of the equation above, in addition to its special
merit of giving in certain cases three real values of v, and therefore, in a sense,
representing the results of Andrews and the conclusions of J. Thomson, is its simplicity.
But this simplicity depends essentially upon the understanding that A, 0, and R are
genuine constants; or, at least, may be treated as such through moderate ranges of
volume: — as, for instance, in the compression of an ordinary liquid by 3000 atmo-
spheres. The equation loses its value (from this point of view) entirely if, as has been
suggested, yS is a sort of adjustable constant! For if it be so, it ought to be
expressed as a function of 7^, or of t; and t, and then the simplicity of the whole
is gone.

Selecting, as before, a set of three corresponding pairs of values of p and t; for
any one temperature, we form three equations which lead to a quadratic in A, when
0 and R are eliminated. This involves heavy numerical work, and the results are so
much modified by very slight changes in the data (quite within the limits of experi-
mental error) that I was fain to try the simpler process of assuming tentative values
for A, and determining the other constants from them: — the equations being then
linear. But I found that very wide ranges of tentative values of A seemed to suit
the conditions, to the same (extremely rough) approximation. I could get nothing
satisfactory. The reason is easily found by making a case in which the labour of
calculation shall be, to a considerable extent, avoided. It is clear, from the numbers
in the early part of this paper, that we may lawfully assume the existence of a liquid
which, for some special (ordinary) temperature, shall give

n = 2700 atm., e = 03.
With these numbers the calculation is very much simplified. For such a liquid, if
its volume were 1 at atmospheric pressure, would be reduced to 25/28 by 1500 atm.,
and to 16/19 by 3000 atm. The quadratic to which Van der Waals' formula leads,
is found to have imaginary roots!

The main cause of this totally-unexpected result seems to be the factor 1/t^ in
the term corresponding to K. Its effect is to make K increase at a rate quite in-
consistent with the experimental data, at least if the rest of the equation is to retain
its present form. This is easily seen by taking the following roughly approximate values

of -^ for ether, at constant volume, which I obtained by a graphic process from

Amagafs Table 29.

•95

•9

•85

(-^\ v const..

14-5

p at 0° C.

460

1250

2570,

* Trans, JR. 8, £., xxxvi. ii. (1S91). AnUt No. lxzz. See also Correspondence with Lord Bayleigh and
Prof. Korteweg {Nature, xliv. and xlv.) [Part of this has been given on p. 208 above. 1899.]

44—2

348 OS THE COMPRESSIBILITY OF FLUIDS. [CIX.

Since Van der Waals' equation gives, for constant Tolome,

we easily find the approximate values

fi^O^eS, fi = 3-8;
and the complete formula is something like

iP + ^ (r-0^) = 1037 + 3«,

where t is temperature centigrade.

This cannot be very &r wrong, so &r at least as 0 and R are concerned, for
it gives the following calculated values of -^ (at the four selected volomea above) which
are compart with the obeerved values: —

Obs. 10 12 14-5 17

Calc. 10^27 11-9 141 17-3.

But when we calculate the ci^nesponding pressures and compare them with those
observed, we have

ObsL 1 460 1250 2570

Calc 1 134 379 S3S.

The diffei^Aces between the numbers in each pair are due to the very mpid inoeafie
of the K term in the formula^ for moderate diminuticns of volume^ The following
ci^mpari^n is instructive. The first numbers are calculated on the hypothesis tK^x £
is inversely as r*. Thoe^ in the second line are the c^Hiesponding valaes of JT re-
quired to make an approximate agreement between Amagat s data, and the ^numerical)
formula above: —

2S04 3107 34«2 3881

2S03 2781 2591 2144.

Thus the ivquisite values of K diminish rapidly, instead of iacremaing, as die
ov^mpressiv>n pnxv^dsw In feet it would s«em as if Van der Waals' equmtion gives
imiKkssible rvx^ts in precisely that limited region where experimi^t shows »>^^t real
ones aie to be found. I intend ;^xmi to examine the cause of this strange lesah fiom
a puivly ma:hematical jx^ini of \iew.

ex.] 349

ex.

ON THE APPLICATION OF VAN DER WAALS' EQUATION TO
THE COMPRESSION OF ORDINARY LIQUIDS.

[Proceedings of the Royal Society of Edinburgh, June 4, 1894.]

In a paper, read for me to the Society in January last {ante, No. CIX.) I pointed
out the difficulties I had met with in trpng to reconcile Van der Waals* equation
with Amagat's experimental data for common liquids, and I promised to recur to the
question when the state of my health should permit. I now find that, as I had then
only surmised, the constants in Van der Waals' equation necessarily become non-real
when we try to adjust it to Amagat's data.

The proof of this assertion is very simple. Suppose the equation

to hold for any three pairs of values of p and v; say p and a, q and 6, r and c.
Eliminating BT among the three resulting equations, we have

The values of A are therefore to be found from the quadratic

-^'^°"'^^a'^e^^^'"°^ + ^s{/ff (06-60 -Hca)}-2{pg(a-6)} = 0.

Write, for brevity, P^p-^. Q = 9^' •^ = *"^;

350 ox THE APPUCATIOX OF VAX DEB WAALS' EQUATION [CX.

80 that one at least of P, Q, R is essentially negative, if p, q, r he all positiTe. The
condition that the Talaes of A shall be real is

Bot it is an obvious theorem of ordinary algebra, that, whatever be the quantities
involved, the two expressions

(/jc + my + nz>* + [ary {/ - my + yz (m - »)» + zor (n - /)«;

and (ar + y + z) (ftp + mhf + n^z)

are absolutely identical except in form.

Hence the condition for real values of ^ is simply that

(P + Q4-i?){P(a6-6c + ca)» + Q(a6 + 6c-ca)» + JJ{-a6 + 6c + ca)»}
shall be positive: — t.e. that its factors shall have the same sign.

To compare with experiment, let us take r = 1 atm., c ~ 1 ; and find the relation
between the values of p and q, the pressures when the volume is reduced to a = 0*9,
and 6 = 0*95, respectively.

The factors of the above quantity are

0O5 01 0O5

"l^/7vovS+9

(0-95)» " ^ (o-gy (o-95)«(o-9y

, _ OOoCO-SOoy O-ljO-dOoy (H)5(0'995y

P (o-dof "*■' (0'9y (o-9oy{o^y'

or, quite approximately enough for our purpose,

-/) + 2 2285 -1-234
and - p + 2-8169 - 1-886.

In the latter form each has been divided by the (essentially positive) multiplier of p:
and, as p and q are each of the order 1000 atm., the last terms may usually be
disregarded. Thus it appears that the values of A cannot be real if p/q lie between
the approximate limits 2-23 and 2*82. But from Amagat's data we easily calculate
the following sufficiently accurate values: —

Ratio of Pressures at 0° C. for Volumes 0-9 and 0-95.

TXTm*^^ Bisulphide Methylic Ethylic Chloride Propylio «.•

^^•*®^- of Carbon. Alcohol. Alcohol. of Ethyl. Alcohol. *•"»«■.

2-51 2-61 2-65 265 269 271 2*73

[The values of q range from 458 atm. in the case of ether to 1166 atm. in that of
water.] All of these ratios lie well within the limits of the region in which the

ex,]

TO THE COMPRESSION OF ORDINARY LIQUTBS,

351

constants of Van der Waals* equation are non-real ; though they are, as a rule, nearer
to the upper than to the lower limit.

But it is well to inquire what values A assumes at the limits of this region ^
when it has just become real A rough calculation shows that when pjq = 2^23 we
have A= — 18'lq (a tensian); and for pjq ^2-S2, A = %Oq, Outside these limits A has
of course two values.

It thus appears that Van der Waala' equation becomes altogether meaoingless except
for liquids in which the compressibility alters very much with increase of pressure: —
i.e. for substances which have ju^t assumed the liquid form under considerable pressure.
For, of course, under the lower limit we are dealing with substances naturally in a state
of tension. As I said in nay previous paper, this state of things is due mainly to
the factor Ijv^ with which A (if taken as coiTesponding to my 11) is affected. There
is little doubt that the n term in my formula does increase as the volume is diminished,
but much more slowly tlian in the inverse ratio of the square of the volume,

(Added 6/6/94*) It may be interesting to look at the above result fi^om a different
point of view, so as to find why it is impossible to reconcile the general equation of
Van der Waals with the experiments of Amagat-

For this purpose let us take 0 as independent variable, and (using the same data
as before) find the value of pjq. Eliminating BT and A, we obtain the equation

-{''V<-«G^c-'')}-»^

from which, at once^

In the further discussion of thia equation we may neglect the last term (which
is usually t/ery much smaller than the preceding term, and becomes infinite for the
same values of 0), Its only noticeable effect is to slightly alter the values of 0 for
which p/q vanishes. We therefore have, to a quite sufficient approximation,

£ = 21712

<«-^'(5^-»)'

where the literal factors have been retained in the more important portion. The value
of p/q in terms ot 0 'm thus seen to be a numerical multiple of the ratio of the
corresponding ordinates of two equal and similarly situated parabolas, whose vertices
do not coincide. The first cuts the axis of x at b and €a/{c + a\ the second at a and
bcl{b-^c\ so that the second lies wholly within the first while y is negative* They

352

OS THE APPLICATIOX OF TAX DEB WAALS' BQUATEOS'

[a-

imersect in the single point whose absdan is abc \ab -r be -^ cat These parabofas ue
shovn in the car belov.

The Talaes of p q are the ordinate^ of the chief curve. This has three asymptotes r^ —
two parallel :o y. and cutting x at a and hcib-^o respectively: and the third a:
a constant distance. 21712. from the axis of x. Its marimnm ordinate^ are given by
the equation

" dx ^:

I'J — X» I X

0 = ^^oi • 6c -h cu^ X* - 2aicx.

ex.] TO THE COMPRESSION OF ORDINARY LIQUIDS. 858

Thus the maximum (at A in the cut) is on the axis of y; and the minimum (at B)
corresponds to a? = 0*6321. Their values are 2*228 and 2*816 respectively; and the
ordinate of the point of intersection of the construction-parabolas lies midway between
them.

Thus, since the minimum numerically exceeds the maximum, the curve has no
ordinate intermediate to these values; and therefore no selection of real constants can
make Van der Waals* equation applicable to a liquid in which the pressure, required
to reduce its volume by 10 per cent., exceeds that required for a 5 per cent, reduction,
in any ratio between 2*228 and 2*816.

Moreover, in accordance with what has been said above about the term A/i^, it
is only while the ratio of pressures exceeds the higher of these limits that this term
represents a pressure, and not a tension. For the graph of A/q in terms of )8 is easily
seen to be a rectangular hyperbola whose asymptotes are parallel to the axes; cutting
X at bc/(b-^c), and y at l^c^ j {b^ - (f). The curve cuts x at 6, and so its ordinates are
positive from bc/ib + c) to 6, anly.

T. n. 45

354 [cxL

CXI.

NOTE OX THE COMPRESSIBILITY OF SOLUTIONS OF SUGAR

[Pn.ymf«jia» of Oe R^yal &xuN of Edimbwrak. Jn/jr IS, 18a&]

Ix eontiiiaadoQ of former inresturmuoas ot the ah^mdoa ot compKcaaifaOitj of wmset.
vhieh is prvxhiced br dLssoIriiu: T;ftnoas sihs in it^ I «» led to imgine thai some
iKtradiTie results might Ke tumished by soiatioos soch as tiuee of sogmr. vb<Ke bulk
is neariv the sum of the balks of their coostitaents : — for. in thenL. ve mi^t expect
Iii«2e chanj^ in eompnessibili^ &om that of w;ftter itself: L«. in acoov^iaace with mr
hjpocheticil formula. Ht^e change in the term regarded as repressentii^ the moiecixiir
pressureL

The fcliowing pceifnraarT resuhs hare recentiy bten obtained fe aae by Mr Sbazii
Xkh'X FixiDiiiftdcoer. vho exuplcy^ the Fraser gun azMi the ^iwiigat gmu;^ pnKuiec
fcr my '^Ckallec^r'^ wv?rk: — A&i & new 5^5 of pieat^meters -r-f the saoke * Fc-ncTs* giass
;a$ that vhose cocipresKbility I bj^i de^errt^ir.ed to be CKKKNDOdd. Tbese cat^ ^Ken
caierilly gauged, bet fcaTe !=.o5 as yet been d£r«rtly cccipttced vith thotse iscstsiT
eEipif:TOl

T!&e jii^ti-Ma? eiprriziecicd cc were pceMrec. in Rr Crim BK^*a * Ldbcca&xx. br
Mr W. W. Tayi:r. M \ . RSr^ xz«i xc^Ainec rct?pe«iT^Iy 3. !•>. 15l 30 parG& by w^^t.
of iugar to ICO ci -waiter. T^e teciperai^ir^ Tiri-sd bet slightly mn IS^^ C. dzdic
the vbrie cccrse cf the rxrerizients.

AwT'itJH C

■wrrsasftftJjJTf pur

^7.«j«»wr». z:

ir-4 C-

Soar ?« :•»! ^aatt

}

F.x frss ica . .

. «>-:i»:4<cj*>

usiy

43:5

4h»

, rv: V-cs .

*5fi>

4S:rf

4iTr»

MIS

, iar«e v:cs

4410

4il0

40C5

3»3>

*55^5

CXI.] NOTE ON THE COMPRESSIBILITY OF SOLUTIONS OF SUGAR. 855

The numbers in the first column were taken direct from the Plate in my second
Challenger Report {ante. No. LXL), 0*0000026 being (of course) added to each.

The Reciprocals of these numbers are, in order,

2151 2257 2344 2439 2522
2212 2317 2404 2492 2581
2268 2375 2460 2551 2640

Comparing with the formula, we see that these reciprocals should be, in the
first column proportional to 11, 11 + 1, 11 + 2 ; in the second to 11 + 5a7, 11 + 1 + 5a?,
n + 2 + 5j? ; etc., where a? is the increase of 11 for 1 part sugar in 100 (by weight)
of water.

The results are not very concordant, especially in the second and fifth columns
(which seem to indicate some error in the gauging of the corresponding piezometers),
but they are all fSeiirly satisfied by taking

n : 1 : a? = 2151 : 681 : 192 ;

so that the actual value of 11 appears to be 37 tons* weight per sq. inch.

Thus it appears that the effect of sugar is, weight for weight, barely one-third
of that of common salt in reducing the compressibility of water; for, with common
salt, fl? = 1 nearly.

45-2

356 [cxii.

CXII.

ON THE PATH OF A ROTATING SPHERICAL PROJECTILE.

[Transactions of the Royal Society of Edinburgh, Vol. xxxvii. June 5, and July 3, 1893.]

The curious effects of rotation upon the path of a spherical projectile have been
investigated experimentally by Robins and many others, of whom Magnus is one
of the more recent. They have also been the subject of elaborate mathematical
investigation, especially by Poisson, who has published a large treatise on the question*.
For all that, we know as yet very little more about them than Newton did in 1666,
when he made his famous experiments on what we now call dispersion. Writing to
Oldenburg an account of these experiments in 167l-2f, he says: —

"Then I began to suspect whether the rays, in their trajection through the prism,
did not move in curve lines, and according to their more or less curvity, tend to divers
parts of the wall. And it increased my suspicion, when I remembered that I had
often seen a tennis-ball, struck with an oblique racket, describe such a curve line. For,
a circular as well as a progressive motion being communicated to it by that stroke,
its parts, on that side where the motions conspire, must press and beat the contiguous
air more violently than on the other; and there excite a reluctancy and re-action of the
air proportionably greater. And for the same reason, if the rays of light should possibly
be globular bodies, and by their oblique passage out of one medium into another acquire
a circulating motion, they ought to feel the greater resistance from the ambient sether,
on that side where the motions conspire, and thence be continually bowed to the other."

From this remarkable passage it is clear that Newton was fully aware of the effect
of rotation in producing curvature in the path of a ball, also that it could be of
sufficient amount to be easily noticed in the short flight of a tennis-ball; that he

* Recherchet mr le Mouvement det ProjectiUs dan* VAir. Paris, 1839.
t Uaaci Newtoni Opera qua exstant Omnia (Horsley), vol. ly. p. 297.

CXII.]

ON TEE PATH OF A EOTATING SPHERICAL PROJECTILE.

357

correctly described the direction of the deviation, and that he ascribed the effect to
difference of air-pressure for which he assigned a cause. All that has since been done
experimentally seems merely to have given various more or less striking illustrations
of these facts, without any attempt to find how the deflecting force depends upon the
velocities of translation and rotation : and I am not aware of any successful attempt
to extend or improve Newton's suggeation of a theoretical explanation. It seems in
&M;t to have been altogether unnoticed, perhaps even ignored.

Thus Robins*, writing some seventy years later than the date of Newton's letter,
speaks of

"the hitherto unheeded effects produced by this resistance; for its action is not
solely employed in retarding the motions of projectilea^ but some part of it exerted
in deflecting them from their course, and in twisting them in all kinds of directions
firom their regular track ; this is a doctrine, which, notwithstanding its prodigious
import to the present subject, hath been hitherto entirely unknown, or unattended
to; and therefore the experiments, by which I have confirmed it, merit, I conceive,
a particular description ; as they are themselves too of a very singular kind.'*

Bobine measured accurately, by means of thin screens placed across his range, the
deviation (to right or left) of successive shots fired from a gun which could be exactly
replaced in its normal position, after each discharge ; and found that it increased
much more rapidly than in simple proportion to the distance. Then he experimented
successfully with a gun whose barrel was bent a little to the left near the muzzle,
with the view of forcing a loose-fitting bullet to rotate by making it roll on one side
of the bore. The bullet, of course, at first deviated a little to the left; but this was
soon got over, and it then persistently curved away to the right. And he showed the
effect of rotation very excellently by suspending a ball by two strings twisted together,
so as to give rotation to it when it was made to vibrate as a pendulum. The plane
of vibration rotated in the same sense as did the ball

I have not had an opportunity of consulting, in the original, Eulers remarks on
this question. The following quotations are taken from a retranslationf of his German
version of Robins* work, but the statements they contain are so definite that the
translator cannot be supposed to have misrepresented their meaning: —

" The cause which Mr Robins assigns for the uncertainty of the shot cannot be
the true one, since we have indisputably proved, that it arises from the figure of the
ball only/* p. 313.

" if the ball has a progressive motion, we may, as has been already shown, consider
it at rest, and the air flowing against it with the velocity of the balls motion; for
the force with which the particles of air act on the body will be the same in both
cases." [Then follows an investigation.] , . . » . "hence this proposition appears

* New FrincipUi vf Qunmty (new edItOi 1805, p. 206. The paper referred lo is lUted to have been read
to the Eoyal Society Id 1747.

t ** The true Pnnciplea of Guimety iitve«iigated and explained, eomprehending trauslationi of Profeseor
Eoler'i ObBerrationa, &c. Ssc" Bj Hugh Brown* London . 1377 [tic].

358 ON THE PATH OF A ROTATING SPHERICAL PROJECTIUE. [CXIL

indisputably true ; that a perfectly spherical body which, besides its progressive motion,
revolves round its centre, will suffer the same resistance as if it had no such rotation.
If, therefore, such a ball should receive two such motions in the cannon, yet its progressive
motion in the air would be the very same as if it had no rotation." pp. 315—7.

Poisson's treatment of the subject is altogether unnecessarily prolix, and in con-
sequence not very easily understood. It is sufficient to say that, like Euler, he rejects*
Robins' explanation; and that his basis of investigation of the effects of rotation on
the path of a homogeneous sphere really amounts to no more than this: — that, since
friction is greater where the density of the air is greater, the fix)nt of the ball suffers
greater friction than does the back. Thus there is a lateral force, which he shows
to be very small, tending to deflect the ball as if it were rolling upon the air in
front of it. As this is exactly the opposite of the effect described by Robins, I feared
at first that I must have misunderstood Poisson's mathematics. But this feeling gave
way to one of astonishment when I read further; for there can be no doubt of the
meaning of the following passage which occurs in his comments on the investigation:

"C'est ce que Ton pent aussi regarder comme Evident d priori, si Ton consid^re
que cette deviation est due k Texc^ de la density de Tair en avant du projectile,
sur sa density en arri^re; exc^s qui donne lieu k un plus grand frottement du fluide,
contre Th^misph^re ant^rieur, et Jt im moindre centre Th^misphfere post^rieur ....
il en r^sultera une force horizontale qui poussera ce point [the centre of inertia] dans
le sens du plus grand frottement ou en sens contraire de la rotation k laquelle il
r^pond, c'est-&-dire vers la gauche, quand les points de la partie antdrieure du projectile
toumeront de gauche k droite, et vers la droite, lorsqu'ils tourneront de droite k gauche."
Recherches, &c., p. 119.

In fact, Poisson's elaborate investigation leads to no term, in the expression for
the normal component of the force, which can have different values at corresponding
points of the two front semihemispheres of the projectile: — and it is to a force of
this nature that Newton's remarks and Robins' experiments alike point.

The paper of Magnusf commences with a historical sketch of the question, but
it contains no reference to Newton. The author obviously cannot have read Robins'
papers, for he mentions his work only once, and in the following altogether inadequate
and unappreciative fashion: —

"Robins, der zuerst eine Erklarung dieser Abweichung in seinen Principles of
Gunnery versucht hat, glaubte, dass die ablenkende Kraft durch die Umdrehung des
Geschosses erzeugt werde, und gegenwartig nimmt man dies allgemein an."

* Poisflon, in fact, says of his own results :— " N^anmoins, d'aprds la composition de la formnle qui ezprime
la deviation horizontale k la distance du canon oil le boulet retombe sur le terrain, on reoonnait facilemeot
que oette deviation ne peut jamais Stre qu'une tr^s petite fraction de la longeur de la port^; en sorte que
ce n'est pas au frottement de la surface du boulet contre la couche d'air adjacente et d'in6gale density que
sont dues principalement les deviations observes, ainsi que Bobins et Lombard Tavait pens^." M^moirt wr
le Mouvement des Prqjectilesy Ac, Comptes Rendus, 6 Mars, 1838, p. 288.

t '*Ueber die Abweichung der Geschosse," Berlin Trans,, 1862.

cxuJ]

ON THE PATH OF A ROTATING SPHERICAL PEOJECTILE*

359

Had Magnus known of the experiments with the crooked gun-barrel and the
rotating pendulum, he would surely have employed a stronger expression than '"glaubte"!
For Bobins says (p. 208) of his own pendulum experiment: —

"it was always easy to predict, before the ball was let go, which way it would
deflect, only by considering on which side the whirl would be combined with the
progressive motion ; for on that side always the deflecting power acted ; as the resistance
was greater here, than on the side where the whirl and progressiva motion were opposed
to each other"

This passage strongly resembles part of the extract already made from Newton's
letter. But Robins justly adds (two words have been italicized) —

'* This experiment is an incontesiible proofs that, if any bullet, besides its progressive
motion, hath a whirl round its axis, it will be deflected in the manner here described."

The one novelty in the experiments of Magnus (so far as spherical projectiles are
concerned) consisted in blowing a stream of air against the rotating body, instead of
giving it a progressive as well as a rotatory motion ; thus, in fact, realizing the idea
suggested by Euler in one of the quotations made above. He was thus enabled, by
means of little vanes, to trace out in a very interesting and instructive manner the
character of the relative motion of the air and the rotating body. This was a cylinder
instead of a sphere, so the effects were greater and of a simpler character, but not
so directly applicable to bullets. Otherwise, his experiments are merely corroborative
of those of Robins.

But neither Robins nor Magnus gives any hint as to the form of the expression
for the deflecting force, in terms of the magnitudes of the translatory and the rotatory
speed. That it depends upon both is obvious from the fact that it does not exist
when either of them is absent, however great the other may be.

1. For some time my attention has been directed to this subject by the singularly
inconsistent results which I obtained when endeavouring to determine the resistance
which the air offers to a golf-ball*. The eoeflScient of resistance which I calculated
from Robins* data for iron balls, by introducing the mass and diameter of a golf-ball,
was very soon found to be too small :^ — and I had grounds for belief that even the
considerably greater value, calculated in a similar way from Bashforth's data, was also
too small Hence the reason for my attempts to determine its value, however indirectly.
The roughness of the baU has probably considerable influence ; and, as will be seen
later, bo possibly has its rotation. I collected, with the efficient assistance of Mr T.
Hodge (whose authority on such matters, alike from the practical and the observational
point of view, no one in St Andrews will question), a fairly complete set of data for
the average characteristics of a reall^^ fine drive : — elevation at starting, range, time

• "The Unwritten Chapter on Golf,'* Natvre, 22/9/87; and •* Some PoinU in the Physics of Golf/' Ibid.,
2e/S/90, 24/9/9 If 29/S/9S. Also a popular article **Hamm«»ring and Driving,*' Gal/, 19/2/^2; where the importanoft
of underspin is coneldteredf malntjr from the point of new of ^lability of motion of ft projectile which ia
always somewhat imperfect aa regsjids both sphenoity and homogeaeitjp

360

ON THE PATH OF A ROTATING SPHERICAL PROJECTILE.

[cxn.

of flight, positioD of vertex, &c. Assuming, as the definite result of all sound experiment
from Robins to Bashforth*, that the resistance to a spherical projectile (whose speed
is less than that of sound) varies nearly as the square of the speed, I tried to determine
from my data the initial speed and the coefficient of resistance, treating the question
as one of ordinary Kinetics of a Particle. We easily obtain, for a low trajectoiy,
simple but sufficiently approximate expressions for the range, the time of flight, and
the position of the vertex, in terms of the data of projection and the coefficient of
resistance. If, then, we assume once for all an initial elevation of 1 in 4, the only
disposable initial element is the speed of projection. Making various more or less
probable assumptions as to its value, I found for each the corresponding coefficient
of resistance which would give the datum range. Thus I obtained the means of
calculating the time of flight and the position of the vertex of the path. The greats
the assumed initial speed (short, of course, of that of sound) the larger is the coefficient
of resistance required to give the datum range, and the more closely does the position of
the vertex agree with observation; though it seems always considerably too near the
middle of the path. But the calculated time of flight, which is greatest (for a givoi
range) when there is no resistance, is always less than two-thirds of that observed:—
while, for high speeds, and correspondingly high resistances, it is diminished to less
than half the observed value. To make certain that this discrepancy was not due to
the want of approximation in my equations, yet without the slightest hope of success
in reconciling the various conflicting data, I made several calculations by the help of
Bashforth's very complete tables, which carry the approximation as far as could be
wished; but the state of matters seemed worse rather than better. It then became
clear to me that it is impossible for a projectile to pursue, for so long a period as
six seconds, a path of only 180 yai-ds, no part of which is so much as 100 feet abo?e
the ground: — unless there be some cause at work upon it which can, at least partiallj,
counteract the effect of gravity. The only possible cause, in the circumstances* is
underspin: — and it must, therefore, necessarily characterise, to a greater or less d^^ee,
every fine drive. (And I saw at once that I had not been mistaken in the opinion,
which I had long ago formed from observation and had frequently expressed, that the
very longest drives almost invariably go ofl* at a comparatively slight elevation, and are
concave upwards for nearly half the range.) In Nature (24/9/91) I said : —

''it thus appears that the rotation of the ball must play at least

as essential a part in the grandest feature of the game, as it has long been known
to do in those most distressing peculiarities called heeling, toeing, slicing, &c"

This conclusion, obvious as it seemed to myself, was vigorously contested by nearij
all of the more prominent golfers to whom I mentioned it : — being generally re^nded as
a sort of accusation, implying that the best players were habitually guilty of something
quite as disgraceful as heeling or toeing, even though its effects might be beneficial
instead of disastrous. The physical cause of the underspin appears at once when we
consider that a good player usually tries to make the motion of the club-head as
nearly as possible horizontal when it strikes the ball from the tee, and that he stands a

* 071 the Motion of Projectiles, 2Dd edD., London, 1890.

cxil]

ON THE PATH OF A ROTATING SPHERICAL PROJECTILE.

361

little behind the tree. Thus the club-head b moviog at impact in a direction not
perpendicular to the striking face; and, unless the ball be at once perfectly spherical
and perfectly smooth, such treatment must give it underspin: — the more rapid the
rougher are the ball and the face of the club. This is, simply, Newton's ''oblique
racket"

In fact J if the ball be treated as hard, and if the friction be sufficient to prevent
elippingp there is necessarily a vm^timum elevation (about 34°) producible by a club
moving horizontally at impact, however much " spooned '^ the face may be. This
maximum is produced when the face of the club makes, with the sole, an angle of
about 28'' : — which is less than that of the most exaggerated " baffy '* I have seen.
This^ taken along with the remark above (viz. that the longest drives usually go off at
very small elevations), is another independent proof that there is considerable underspin.

Hence the practical conclusion, that the face of a spoon, if it is to do its proper
work eflBciently, ought to be as smooth as possible,

2, I next considered how to take account, in my equationSj of the effects of the
rotation ; and it appeared to me most probable that this could be done, with quite
sufficient approximation, by introducing a new force whose direction is perpendicular
at once to the line of flight and to the axis of rotation of the ball:^ — concurrent in
fact with the direction of rotatory motion of the foremost point of the surface. Various
considerations tended to show that its magnitude must be at least nearly proportional
to the speed of rotation and that of translation conjointly* Among these there is the
simple one that its direction is reversed when either of these motions is reversed
This may be generalised ; for if the vector axis, t^ be anyhow inclined to the vector
of translation, a, the direction (why not then the magnitude also^ to a constant
multiplier prh) of the deflecting force is given by Vea. Another is that, as the
resistance (t.e. the pressure) on the non-rotating ball is proportional to the square of
the speed, the pressures on the two front semihemispheres of the rotating ball must
be (on the average) proportional to (v + e^w)^ and (v — et^Y respectively : — where t; is the
speed of translation, ^ that of rotation, and e a linear constant. The resultant of
these, perpendicular to the line of flight, will obviously be perpendicular also to the
axis of rotation, and its magnitude will be m vm. But I need not enumerate more
arguments of this kind. In the absence of anything approaching to a complete theory
of the phenomenon we must make some assumption, and the true test of the assumption
is the comparison of its consequences with the results of observation or experiment.
This I have attempted, with some success, as will be seen below,

3. Another associated question, of greater scientific difficulty but of less apparent
importance to my work, was the expression for the rate of loss of energy of rotation
by the ball Is it, or is it not, seriously modified by the translation? But here I
had what seemed strong experimental evidence to go on, afforded by the fact that I
had often seen a sliced or heeled ball rotating rapidly when it reached the ground
at the end of its devious course. This isj of course, what would be expected if the
deflecting force were the only, or at least the principal, result of the rotation : — for,
being always perpendicular to the direction of translation, it does no work But, on

T, n.

362 ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. [cxn.

the other hand, if the friction on a rotating ball depends upon its rate of translation,
the ball while flying should lose its spin faster than if its centre were at rest. This
is a kind of information which might have been obtained at once from Magnus'
experiments, but unfortunately was not.

4. As I felt that there was a good deal of uncertainty about the whole of these
speculations, I resolved to consult Sir G. G. Stokes. I therefore, without stating any
arguments, asked him whether my assumptions appeared to him to be suflSciently
well-founded to warrant the expenditure of some time and labour in developing their
consequences: — and I was much encouraged by his reply. For he wrote: —

"if the linear velocity at the surface, due to the rotation, is small compared with
the velocity of translation, I think your suggestion of the law of resistance a reasonable
one, and likely to be approximately true. This would make the deflecting force vary
as Vfo. I think too that the resistance in the line of flight will vary nearly as v*,
irrespective of the velocity of rotation of the ball.

"As to the decrement of the energy of rotation, I think the second law which you
suggested is likely to be approximately true. The linear velocity due to rotation, even
at the surface where it is greatest, being supposed small, or at least tolerably small,
compared with the velocity of translation, I think you are right in saying that the
force acting laterally upon the ball will vary, at least approximately, as vto. If this
acted through the centre, it would have no moment. But I think it will not act
through the centre, though probably not far from it, so that it would have a moment
varying as vto. Hence the decrement of angular velocity would vary as i;a>, and the
decrement of energy of rotation as a> (— day/dt), or as a> . vco, or as V(o\ according to
your second formula.

"However, I think the force at any point of the surface, of the nature of that
which we have been considering, would act very approximately towards the centre, and
therefore would have little moment, so that after all the moment of the force tending
to check the rotation may depend rather on the spin directly than on its combination
with the velocity of translation. But, if this be so, I doubt whether the diminution
of rotation during the short time that the ball is flying is sufficient to make it worth
while to take it into account."

5. For a first inquiry, and one of great consequence as enabling us to get at
least general notions of the magnitude of the deflecting force, let us take the simple
case of a ball, projected in a direction perpendicular to its axis of rotation, in still
air, and not acted on by gravity. [This would be the case of a top or "pearie,"
with its axis vertical, travelling on a smooth horizontal plane.] Suppose, further, that
the rate of rotation is constant. Then, in intrinsic coordinates, the equations of tan-
gential and normal acceleration given by our assumptions are

« = — «"/a, and s^/p = «« ^ =

cxn,]

ON THE PATH OF A ROTATING SPHERICAL PROJECTILE.

363

respectively. The second may be put in either of the forma

9 = k^t

kmjL

The first shows that the direction of motion revolves uniformly; the second, that the
curvature h inversely as the speed of translation. Andp as the first equation gives

the iBtrinBic equation of the path is evidently

if 0 be measured from the initial direction of projection, and F be the initial speed*
This is an endless spiral, which has an asymptote, but no multiple points, and whose
curvature is

It therefore varies continuously frotu nil, at negative infinite values of a, to infinity
at positive infinite values. Any arc of the spiral has therefore preeisely the character
of the horizontal projection of the path of a sliced, toed, or heeled, golf-ball; for it
is obvious at once that the curvature steadily increases with the diminishing speed of
the ball, thus far justifying the assumptions made in forming the equations of motion.
We have only to trace this spiral, once for all, to get the path for any circumstances
of projection. For the asymptote is obviously parallel to

kmd

0 — — 'p. = — a suppose.

Measure ^ from this direction, and the equation becomes

a gives the length correapouding to unit in the figure; and a (which determines the
point of it from which the ball starts) depends only upon a and the ratio of the spin
to the initial speed* This, with <^/a and sja interchanged, is the equation of the
equiangular spiral, which would be the path if the resistance were directly as the speed.

6, This enables us to get an approximate idea of the possible value of km in the
flight of a golf-balL For if it be well sliced, its direction of motion when it reaches
the ground is often at right angles to the initial direction, although the whole deviation
from a straight path may not be more than 20 or 30 yards. Assume for a moment,
what will be fully justified later, that in such a case we may have (say) s = 480 feet,
a ^ 240 feet, and V = 350 foot-seconds. We see that

^ I 24 „ .
2 = ^a.K^.x6'4;

so that

k^^ -^r^ =0357, nearly,

gives a sort of average value, which may safely be used in future calculationa In
the case just considered, the acceleration (at starting) due to the rotation, is 0*357 x 350

46—2

364 ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. [CXII.

or nearly four-fold that of gravity: i.e., the initial deflecting force is four times the
weight of the ball.

7. In trying to find the positions of the asymptote, and of the pole, of the spiral
of § 5, I spent a good deal of time on integrals like

r* 8in0d<(»
Jo a + <^ '

with the hope of adapting them to easy numerical calculation by transformation to
others with finite limits, such as 0, 7r/2. Happily, I learned from Professor Chrystal
that they had been tabulated by Mr J. W. L. Glaisher : — and from his splendid paper
(Phil. Trans. 1870) I obtained at once all that I sought. In fact his 8ul> and Ct^
are simply the x, y coordinates of this spiral (each divided by a)\ the axes being
respectively the perpendicular fi-om the pole on the asymptote, and the asymptote itself.
Thus I traced at once, as shown in Plate VI. Fig. 1, the first three-quarters of a turn : —
and the transformations I had already obtained enabled me to interpolate points when
(after 0 = 5) those given in the tables were too distant from one another for sure
drawing. Another help in completing the curve graphically is given by the fact that
the tangent, at any point, makes with the asymptote the angle <f> which belongs to
the point. This spiral does not, perhaps, exhibit the courses of the two functions
so clearly as do the separate curves given by Glaisher; but it certainly shows their
mutual relation, and their maximum and minimum values, in a very striking manner.

The numbers, affixed to various points of the figured spiral, are (in circular measure)
the corresponding values of 0, or (by the equations of § 5) they may be taken as
proportional to the Umes of reaching these points by the moving ball, starting with
infinite speed from an infinite distance.

8. Even in the plane problem of § 5, the introduction of the effects of a steady
current of wind in the plane of motion complicates the equations in a formidable
manner. Suppose <f> be measured from the reversed direction of the wind, and let the
speed of the wind be W. Then if U, with direction '^, be the relative velocity of
the ball with regard to the wind (for it is upon this that the resistance, and the
deflecting force, depend), we have

CTpos -^ = TT + « cos 0,

I7'sin'^= «sin0;

and the equations of motion are

S = cos (0 — '^) -h A; CT sin {<f> — ^),

- = — sin (0 — '^) -h A; CT cos (0 — '^) ;

where, once for all, we have written k for kna.

cxu.]

ON THE PATH OF A ROTATING SPHEEICAL PROJECTILE.

365

Putting V for i, and elimmating f, these become

V J- ^ — (Tf coB^ + fi) + ifcTFfim^,

Ifsini^ +it(TFcosi^ + t;);

where, of course,

U^^W*'^^ + 2WvcoB<p.

These equations reduce themselves at once to the simpler ones above treated, when
we put W^O, and therefore U = v. As they stand they appear intractable, in general,
except by laborious proeeases of quBdrature, But while ^ is small, ie,, while the
ball is advancing nearly in the wind's eye, they may be written approximately as

as a ^

W-^v

W4^-^k{W^v).

From the first of these we see not only that the space-rate of diminution of
speed is increased in the ratio {W+vflv^ which was otherwise obvious; but also that
the rotation tends, in a feeble manner, to counteract this effect From the second we see
that the space-rate of change of direction is increased, not only by the factor (W^v)/v
in the term due to spin, but by a direct contribution from the resistance itself The
eflfect of a head-wind in producing upward curvature, even in a "skimmer," is well
known; and we now see that it is, at first, almost entirely due to the underspin
which, without being aware of it, long drivers necessarily give to the ball. As soon
as sin ^ has, by the agency of the underspin, acquired a finite value, the direct
resistance comes in to aid the underspin in further increasing it. We now see the
true nature of the important service which (in the hands of a powerful player) the
nearly vertical face of a driving putter renders against a strong wind. It enables him
to give great translatory speed, with little elevation, and with just spin enough to
neutralize, for the earlier part of the path, the effect of gravity.

9. Before I met with Robins' paper, I had tried his pendulum experiment in a
form which gives the operator much greater command over the circumstances of rotation
than does his twisting of two strings together. Some years ago, with a view to
measuring the coefficient of resistance of air, even for high speeds^ in the necessarily
moderate range afforded by a large room, I had procured a number of spherical
wooden shells, turned very thin. My object, at that time, wa^i to make the mass as
small as possible, while the diameter was considerable: — but, of course, the moment of
inertia was also very smaU. So, when I fixed in one of them the end of a thin
iron wire, the other end of which was fastened to the lower extremity of a vertical
spindle which couid be driven at any desired speed by means of multiplying gear, the
wire suffered very little torsion, except at the moments of reversal of the spin. The
pendulum vibrations of this ball showed almost perfect elliptic orbits, rotating about
the centre in the same sense as did the shell : — and with angular velocity approximately

366

ON THE PATH OF A ROTATING SPHERICAL PROJBCTILE.

[CH

proportional to that of the shell. These two experimental results are in full accordin
with the assumed law for the deflecting force due to rotation. For, the oidinaiy vect
equation of elliptic motion about the centre is

cr = — m V.

If the orbit rotate, with angular velocity fl, about the vertical unit vector o, peipei
dicular to its plane, cr becomes

Eliminate a from these equations, and we have at once

p = - (m« - n») p + 2fta/5.

The part of the acceleration which depends upon the motion of translation of tl
bob : — viz.

2na/>,

is proportional to the speed, and also to fl, that is (by the results of obserratioii
proportional to the rate of spin; and it is perpendicular alike to a and to the directki
of translation* These statements involve the complete assumption above. The otlw
part of the acceleration depends upon position alone, and must therefore be — n^p, thi
of the non-rotating ball. Hence we see that

or the period in the rotating ellipse is always shortened: — whether the ball move romn
it in the sense of the spin or not. This test cannot be applied with any certainty ii
the experiment described above, for in general fl is much less than n, so that i
exceeds n by a very small fraction only of its value.

A very beautifril modification of this experiment consists in makiug the path oi
the pendulum bob circular, before it is set in rotation. Then rotation, in the mm
sense as the revolution, makes the orbit shrink and notably dioiinishes the period
Reverse the rotation; the orbit swells out, and the period becomes longer.

10. The equations of motion of a golf-ball, which is rotating about an axis per
pendicular to its plane of flight, and moving in still air, are now easily seen to be

- ^ 'J.

« = ---^sm(^.

^ = A; — T cos ^.

The most interesting case of this motion is a "long drive," as it is called when
if) is always small, so long at least as it is positive ; its utmost average value for tk
first two-thiids of the range being somewhere about 0*25. This applies up to aul
about as much beyond, the point of contrary flexure. A little after passinsr that poini
^ begins to diminish at a considerably greater rate than that at which it had i^evioosb
increased.

OXn.] ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. 367

A first approximation gives, as above,

if we omit the term ^ sin <^ in the first equation. With this, the second equation gives
at once, on integration,

^ = a + ^(.««-l)-|^.(6^/--l).

We might substitute this for sin ^ in the first equation, and so obtain a second, and
now very close, approximation to the value of i. But the result is far too cumbrous
for convenient use in calculation. We will, therefore, be content for the present with
the rudely approximate value of i written above.

Integrating again with respect to «, we have

/>=-^'(--'-S-4^.(--'-|)-
Now, for rectangular coordinates {x horizontal) and the same origin,

a?=j'cos^ = j Vl -^' + &c.^cfo, y=j 'sin <^ = jY</)-^ + &c.jcfo;

so that, to the order of approximation we have adopted, the equation of the path is

^-«--T(«"->-S-|r.(<-"-'-|)-

The only really serious defect in this approximation is the omission of ^rsin^ in the
first equation. This renders the value of 8 too large for the greater part of the path,
and thus the value of y will be slightly too small up to the point of inflection, and
somewhat too large up to (and some way beyond) the vertex of the path.

11. When this paper was first read to the Society, it contained a considerable
number of details and sketches of the paths of golf-balls, based on three very different
estimates of the constant of resistance: — respectively much less than, nearly equal to,
and considerably greater than, that suggested by Bashforth's results. These details have
just been printed in Nature (June 29), and I therefore suppress them here, replacing
them by calculations based on experiments made between the two dates at the head of
the paper. One important remark, suggested by the appearance of these curves, must,
however, be made now. Whatever, firom 180 to 360 feet, be assumed as the value of a,
the paths required to give a range of 180 yards and a time of 6''5, have a striking
family resemblance. So much do they agree in general form, that I do not think
anything like an approximation to the true value of a could be obtained firom eye-
observations alone. We must, therefore, find a or F directly. Only the possession of
a really trustworthy value of a, found by such means, would justify the labour of
attempting a closer approximation than that given above. I have not as yet obtained
the means of making any direct determinations of a, but I have tried to find its value
indirectly; first, from experimental measures of V made some years ago by means of
a ballistic pendulum; secondly, a few days ago, by (what comes nearly to the same

368 ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. [CXH.

thing) measuring directly the speed of the club-head at impact, and thus determining
the speed from the known coefficient of restitution of the ball All of these experiments
have been imperfect, mainly in consequence of the novelty of the circumstances
and the feeling of insecurity, or even of danger, which prevented the player from
doing his best. The results, however, seem to agree in showing that V is somewhat
over 300 foot-seconds (say, for trial, 350) for a really fine drive. Taking the carry as
180 yards, and the time as 6', the value of a given by the formulae above is somewhere
about 240 feet. With these assumed data, the initial (direct) resistance to the ball's
motion is sixteen-fold its weight. Bashforth's results for iron spheres, when we take
account of the diameter and mass of a golf-ball, give about 280 feet as the value of a.
The diflference (if it really exist) may possibly arise from the roughness of the golf-ball,
which we now see to be essential to long carry and to steady flight, inasmuch as the
ball is enabled by it to take readily a great amount of spin, and to avail itself of
that spin to the utmost. One of the arguments in § 2 above would give the resistance
as proportional to t;* -h €^to\ instead of to t^ simply.

12. We have thus all the data, except values of a and of k, required for the
working out of the details of the path by means of the approximate x, y equation just
given. The best course seems to be to assume values of a from 0*24 (according to
Mr Hodge) down to zero; and to find for each the corresponding value of k which
will make y = 0 for a? = 640. This process gives the following values with a = 240,
F=350, as above: —

o k kVjg alogkVlg

0-24 0182 200 1663

012 0-246 209 2375

00 0-309 3-37 291-6

It will be seen that the values of k are of the order pointed to by the behaviour
of a sliced ball, though they are considerably less than that given in the example of
§ 6. This, of course, is a strong argument in favour of the present theory ; for, even
in the wildest of (unintentional) heeling, the face of the club is scarcely so much inclined
to its direction of motion as it is in good, ordinary, driving with a grassed club.
(Slicing is very much less susceptible of accurate quantitative estimation by means of
eye-observations.) The third column gives the ratio of the initial deflecting force to
the weight of the ball. As this is more than unit in each of the three cases, all
these paths are at first concave upwards. The numbers in the fourth column indicate
(in feet) the distance along the range from the origin to the point of inflexion.

The approximate equation of the first of these paths is

y = 57-6 ^-h 3005 (e^/^-l- I)- 3-76 (c^^-l-^V

The abscissa of the maximum ordinate is given by

0 = 57-6 + 3005 (€*/• - 1) - 7-52 (e^^^ - 1),
which leads to €*/* = 4-93, whence a? = 384 nearly.

The vertex is therefore at 0*71 of the range.

cxn,]

ON THE PATH OF A BOTATING SPHERICAL PROJECTILE,

369

13, Under exactly the same circumstances, had there been no rotatiouj the equation
of the path would have been

y = 57-6 -^ 3-76 U^f^ - 1 - -V

This gives for ^ — 0, x^lTla^ 4slO feet onlj,

The position of the vertex h given by

0 = S7*6-7 52(£«'^«-l);
m that m = 258 feet, nearly.

In this case the vertex is at U'63 of the mnge, only, and the time of flight
is 3» 1.

We have here, in consequence of a very moderate spin only, (in fact about half
of that given by a good slice), all other initial circumstances being the same, an
exceedingly well-marked difference in character between the two paths, as well as
notable differences in range, and time of flight. Thus, while a player who gives no spin
has (say) a carry of 136 yards only ; another, who gives the same initial speed and
inclination of path but also a very moderate amount of spin, accomplishes 180 yards
with ease; his ball, in fact, remaining twice as long in the air.

14. For the sake of farther illustration, let us consider the course by which the
ball| sent oif at the same inclination, but without rotation, may be forced by mere
initial speed to have a range of 540 feet. Here the condition for V is

240\

0=129'e-8(=^) 84*5,

so that the requisite speed is 648 foot-seconds ; an increase of 56 per cent., involving
about 2 5-fold energy of translation, which I take to be entirely beyond the power
of any playen And the time of flight is reduced to 3''7 only, a mpidity of execution
never witnessed in so long a carry. The initial resistance in this case rises to nearly
forty- fold the weight of the ball The equation of the path is

y = 57-6 ^-1-54 (e-/-- 1-2^).

and the vertex is at 355, or about two-thirds of the range, only.

15. Fig, 2 shows the three paths just described, which start initially in the same
direction; the uppermost is that with speed 350 and moderate spin. The lowest has
the same speed, but no spin. The intermediate course, also, has no spin, but the
initial speed is 548 to enable it to have a range of 540 feet. Thus the two upper
paths in this figure are characteristic of the two modes of achieving a long carry :^ —
viz. skill, and brute force, respectively. In fig, 3 the first of these paths is repeated,
and along with it are given the corresponding trajectories with the same initial speed
T. II. 47

370 ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. [CXII.

350, but with inclinations of 012 and 00 respectively, and with the values of k, given
above, which are required to secure the same common range. [To increase this range
from 180 to 250 yards, even in the lowest and thus least advantageous path where
there is no initial elevation, all that is required is to raise the value of JfcF (the
initial acceleration due to rotation) from 108 to 219; i.e. practically to double it.
V might, perhaps, be increased by from 25 to 30 per cent, by a greatly increased
effort in driving: — but k is much more easily increased. A carry of 250 yards, in still
air, is therefore quite compatible with our data, even if there be no initial elevation.
It can be achieved, for instance, if F is 400 foot-seconds, and k about 50 per cent,
greater than that which we have seen is given by a good slice. Of course it will
be easier of attainment if the true value of a is greater than 240 feet. When there
is no rotation there must be initial elevation; and, even if we make it as great as
1 in 4, the requisite speed of projection for a carry of 250 yards would be 1120 feet
per second, or about that of sound.] Each of the curves has its vertex marked, and
also its point of inflexion, when it happens to possess one. Fig. 4 gives a rough,
conjectural, sketch of the probable form of the path if, other things being the same, the
spin could be very greatly increased. As I do not see an easy way to a moderately
approximate solution of this problem, either by calculation or by a graphic process,
I intend to attempt it experimentally. I am encouraged to persevere in this by the
fiBM^t that in one of the few trials which I have yet made, with a very weak bow,
I managed to make a golf-ball move point blank to a mark 30 jrards off. When
the string was adjusted round the middle of the ball, instead of catching it lower,
the droop in that distance was usually about 8 feet. With a more powerful bow, and
with one of the thin wooden shells I have mentioned above, the circumstances will
be very favourable for a path with a kink in it.

PlauVI

Totoe«|i3

Ill

oxni.]

371

CXIIL

ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. II.

[Tranaaddmis of the Royal Society of Edinburgh, Vol sxxix. Part II.]

(Read 6th and 20tb January, 1896.)

The first instalment of this paper was devoted in great part to the general Eobject
involved in its title, but many of the illustrations were derived from the special case of
the flight of a golf-ball. Since it was read I have endeavoured, alike by observation
and by experiment, to improve ray numerical data for this interesting application,
particularly as regards the important question of the coefficient of resistance of the air.
As will be seen, I now find a value iutermediate to those derived (by taking average
estimates of the mass and diameter of a goU-ball) from the results of Robins and of
Baishforth. This has been obtained indirectly by means of a considerable improvement
in the apparatus by which I had attempted to measure the initial speed of a golf-balL
I have, still, little doubt that the speed may, occasionally, amount to the 300, or
perhaps even the 350> foot-seconds which I assumed provisionally in my former paper z-^
but even the first of these is a somewhat extravagant estimate; and I am now of
opinion that, even with very good driving, an initial speed of about SiO is not often
an underestimate, at least in careful play. From this, and the fact that six seconds at
least are required for a long carry (say 180 yards), I reckon the "terminal velocity" at
about 108, giving tP/360 as the resistance -acceleration.

I hope to recur to this question towards the end of the present paper: — but I
should repeat that I naturally preferred the comparatively recent determination to the
much older one, and that in formerly assuming a resistance even greater than that which

47—2

372 ON THE PATH OF A BOTATING SPHERICAL PROJECTILE. [cxm.

Bashforth's formula assigns, I was to some extent influenced by the consideration of
the important effects of roughening or hammering a golf-ball. For I fancied that this
might increase the direct resistance, as well as the effects due to rotation, by the better
grip of the air which it gives to the ball. [See last sentence of § 11. Of course the
assumption of increased coeflScient of resistance required a corresponding increase of the
estimate of initial speed.] The time of describing 180 yards horizantaUt/, t,e., when
gravity is not supposed to act, if the initial speed is 240 and the "terminal velocity"
108, is about 5'-2; and this has to be increased by at least 1', if we allow for the
curvature of the path and the effect of gravity. I have employed this improved
value of the coefficient of resistance in all the calculations which have been made
since I obtained it. But various considerations have led me to the conclusion that the
resistance, towards the end of the path, may be somewhat underrated because of the
assumption that it is, throughout, proportional to the square of the speed. This point,
also, will be referred to later, as I wish to make at once all the nece^ssary comments and
improvements on the part already published.

Though the present communication is thus specially devoted to some curious
phenomena observed in the game of golf, it contains a great deal which has more
extended application: — to which its results can easily be adapted by mere numerical
alterations in the data. Therefore I venture to consider its subject as one suitable for
discussion before a scientific Society.

In my short sketch of the history of the problem I failed to notice either of two
comparatively recent papers whose contents are at least somewhat closely connected
with it. These I will now very briefly consider.

The first is by Clerk-Maxwell* " On a particular Case of the Descent of a Heavy
Body in a Resisting Medium" The body is a fiat rectangular slip of paper, falling with
its longer edges horizontal. It is observed to rotate about an axis parallel to these
edges, and to fall in an oblique direction. The motion soon becomes approximately
regular; and the defiection of the path from the vertical is to the side towards which
the (temporarily) lower edge of the paper slip is being transferred by the rotation.
[When the rectangle is not very exact, or the longer edges not quite horizontal, or the
slip slightly curved, the appearance, especially when there is bright sunlight, is often
like a spiral stair-case.] Maxwell examines experimentally the distribution of currents,
and consequently of pressure, about a non-rotating plane upon which a fluid plan's
obliquely; and shows that when the paper is rotating the consequent modification of
this distribution of pressure tends to maintain the rotation. The reasoning throughout
is somewhat difficult to follow, and the circumstances of the slip are very different fix)m
those of a ball: — ^but the direction of the defiection from the unresisted path is alwaj's
in agreement with the statement made by Newton.

Much more intimately connected with our work is a paper by Lord Rayleigh-f*
''On the Irregular Flight of a Tennis Ball*' in which the "true explanation" of the

* Cambridge and Dublin Mathematical Journal^ ix. 145 (1854).
t Messenger of Mathematics, vii. 14 (1878).

CXin.] ON THB PATH OF A KOTATING SPHERICAL PKOJECTILB. 373

curved path is attributed to Prof. Magnus. The author points out that, in general, the
statement that the pressure is least where the speed is greatest, is true only of perfect
fluids unacted on by external forces; whereas in the present case the whirlpool motion
is directly due to friction. But he suggests the idea of short blades projecting from the
ball, the pressure on each of which is shared by the contiguous portion of the spherical
sur&ce. Here we have practically Newton's explanation — 1.6. the "pressing and beating
of the contiguous air." Lord Rayleigh's paper contains an investigation of the form of
the stream-lines when a perfect fluid circulates (without molecular rotation) round a
cylinder, its motion at an infinite distance having uniform velocity in a direction per-
pendicular to the axis of the cylinder. And it is shown that the resultant pressure,
perpendicular to the general velocity of the stream, has its magnitude proportional alike
to that velocity and to the velocity of circulation. [There are some comments on this
paper, by Pro£ Qreenhill, in the ninth volume of the journal referred to.]

In the Beihldtter zu d, Ann. d, Phys. (1895, p. 289) there appears a somewhat
sarcastic notice of my former paper. The Reviewer, evidently annoyed at my remarks
on Magnus' treatment of Robins, which he is unable directly to controvert, refers to
H^lie, Traits de Balistigtie, as containing an anticipation of my own work. I find
nothing there beyond a very small part of what was perfectly well known to Newton
and Robins; except a few of the more immediately obvious mathematical consequences,
deduced from the hypothesis (for which no basis is assigned, save that it is the simplest
possible) that the transverse deflecting force due to rotation is proportional to the first
power of the translational speed.

In the present article I give first a brief account of my recent attempts to deter-
mine the initial speed of a golf-ball, and consequently to approximate to the coefficient
of v* in the assumed expression for the resistance.

Next, instead of facing the labour of the second approximation (suggested in § 10)
to the solution of the differential equations, I have attempted by mere numerical
calculation to take account of the effect of gravity on the speed of the projectile, and
have thus been enabled to give improved, though still rough, sketches of the form
of the trajectory when it is not excessively flat. This process furnishes, incidentally,
the means of finding the time of passage through any arc of the trajectory.

Third, I treat of the effects of wind, regarded as a uniform horizontal translation of
the atmosphere parallel, or perpendicular, to the plane of the path.

Finally, recurring to the limitation of a very flat trajectory, I have treated briefly the
effects of gradual diminution of spin during the flight. This loss is shown to be in-
adequate to the explanation of the unexpectedly small inclination of the calculated
path when the projectile reaches the ground. Hence some other mode of accounting
for its nearly vertical fall is to be sought, and it is traced to the rapid diminution
of the resistance (assigned by Robins' law) when the speed has been greatly reduced.

374 ON THE PATH OF A ROTATING SPHERICAL PROJECmiiE. [CXHL

Determination of Initial Speed.

16. The bob of my new ballistic pendulum was a stout metal tube, some 3 feet
long, suspended horizontally, near the floor, by two parallel pieces of clock-spriDg about
2*5 feet apart, and 8'63 feet long. On one end of the tube was fixed transversely a
circular disc, 1 foot in diameter, covered with a thick layer of moist clay into which
the ball was driven firom a distance of 4 feet or so. The whole bob had a mass of about
33 lbs.; and, in the most favourable circumstances, its horizontal displacement was
about 3*5 to 4 inches. As the ball's mass is 0*1 lb., the average indicated speed was
thus about 200 foot-seconds*. Though I had the assistance of two long drivers, whose
habitual carry is 180 yards or upwards, the circumstances of the trials were somewhat
unfavourable, for there was great difficulty in hitting the disc of clay centrally. The
pendulum was suspended in an open door-way; and heavy matting was disposed all
about the clay so as (in Robins' quaint language) "to avoid these dangers, to the
braving of which in philosophical researches no honour is annexed"; so that the whole
surroundings were absolutely unlike those of a golf-course. I therefore make an allow-
ance of 20 per cent, and (as at present advised) regard 240 foot-seconds or something
like it as a fair average value of the initial speed of a really well-driven ball: — while
thinking it quite possible that, under exceptionally favourable circumstances, this may
be increased by 20 or 30 per cent at least. Now, it is certain that the time of flight
is usually about six seconds when the range is about 180 yards: — considerably more
for a very high trajectory, and somewhat less for a very flat one. As we have by § 5
the approximate formula

<=^(6"--l),

we may take a = 360 as a reasonable estimate. This number is possibly some 10 per
cent, in error, but it is very convenient for calculation, and golf-balls diflFer considerably
from one another in density as well as in diameter. With it the "terminal velocity" of
a golf-ball is about 108 foot-seconds; intermediate to the values deduced £jx)m the
formulae of Robins and of Bashforth, which I make out to be 114 and 95 respectively.

* If Z be the length (in feet) of the supporting straps, d the (smaU) horizontal deflection of the bobt
its vertical rise is obviously (2^/22, so that its utmost potential energy is

{M-hm) gd*l2l,

where M is its mass and m that of the ball. But, if V was the horizontal speed of the baU, that of bob
and baU was mVI(M+m). Equating the corresponding kinetic energy to the potential energy into which it is
transformed, we find at once (M+m) gd''l2l=m*V^I2(M-i'm), leading to the very simple expression

With the numerical values given in the text we easily find that this is equivalent to

r=331^ 1-93 = 63-22);

where V is, of course, in foot-seconds, but the deflection is now (for convenience) expressed in inches, and
called D, Hence the numerical result in the text.

CXni.] ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. 375

With this value of a, it is easy to see that air-resistance, alone, reduces the speed of a
golf-ball to half its initial value in a path of 83 yards only. This is the utmost gain of
range obtainable (other conditions remaining unchanged) by giving four-fold energy of
propulsion. With the value (282) of a deduced from Bashforth's formula, this gain
would have been 65 yards only! [So far for the higher speeds, but it is obvious from
all ordinary experience of pendulums (with a golf-ball as bob) that slow moving bodies
suffer greater resistance than that assigned by this law.]

In passing, I may mention that, on several occasions, I fastened firmly to the ball
a long light tape, the further end being fixed (after all twist was removed) to the
ground so that the whole was perpendicular to the direction of driving. After the
4-foot flight of the ball, the diameter at first parallel to the tape preserved its
initial direction, while the tape was found twisted (in a sense corresponding to under-
spin) and often through one or two full turns, indicating something like 60 or 120
turns per second. This is clearly a satisfiswstory verification of the present theory.

Nv/merical Approximation to Form of Path.

17. The differential equations of the trajectory were integrated approximately in
§ 10 by formally omitting the term in ^ in the first of them, that is so far as the
speed is concerned. In other words : — ^by assuming that (f) is always very small, or the
path nearly horizontal throughout. It was pointed out that if the value of (f>, thus
obtained from the second, were substituted for sin<^ in the first, equation, we should
be able to obtain a second approximation to the intrinsic equation of the path, amply
sufficient for all ordinary applications. But the process, though simple enough in all
its stages, is long and laborious: — and it is altogether inapplicable to the kinked path,
discussed in § 15, which furnishes one of the most singular illustrations of the whole
question.

The fact that one of my Laboratory students, Mr James Wood, had shown himself
to be an extremely rapid and accurate calculator led me to attempt an approximate
solution of the equations by means of differences: — treating the trajectory as an equi-
lateral polygon of 6-foot sides, and calculating numerically the inclination of each to
the horizon, as well as the average speed with which it is described. For we may write
the differential equations in the form

lrf(t)»)^t;« . ,

2-dr + a=-^^^^*'

d6 k g .
-f - — ^ cos 6,

and these involve approximately

t;'^ - v* + 2 ^^ +^sin <^^ &= 0,

f-<^=(*-fcos<^)s..

376 ON THE PATH OF A ROTATING SPHERICAL PROJEOTILB. [CXHI.

Thus we find, after a six-foot step, the new values

v'a = ^1 - ^ V - 384 sin <^,
., . 6* 192 cos A

[If we take account of terms in (Say, we find that we ought to write for 12/a
the more accurate expression 12/a. (1 — 6/a). But this does not alter the form of
the expression for v\ It merely increases by some 2 per cent, the denominator of
the coefficient of resistance, of which our estimate is, at best, a very rough one; so
that it may be disregarded. But the successive values of v^ are all on this account
too large; and thus the values of 0, in their turn, are sometimes increased, some-
times diminished, but only by trifling amounts. This is due to the fact that the
change of <}> depends upon terms having opposite signs; and involving different powers
of V, so that their rdative as well as their actual importance is continually changing.
These remarks require some modification when k is such that <f) may have large values,
as for instance in the kinked path treated below. But I do not pretend to treat the
question exhaustively, so that I merely allude to this source of imperfection of the
investigation.]

Let, now, a = 360, k = 1/3, and suppose <^ to be expressed in degrees. We have,
to a sufficient approximation,

., .^120 12000 ./ 1\

and successive substitutions in these equations, starting from any assigned values of
V and if), will give us the corresponding values for the next side of the polygon, with
the more recent estimate of the coefficient of resistance. See the two last examples
in § 19 below, which lead to the trajectories figured as 5 and 6 in Plate VII.

Unfortunately, many of Mr Wood's calculations were finished before I had arrived
at my new estimate of the value of a ; but their results are all approximately repre-
sentative of possible trajectories : — the balls being regarded as a little larger, or a little less
dense, than an ordinary golf-ball; in proportion as the coefficient of resistance assumed
is somewhat too great. And no difficulty arises from the assumption of too great an
initial speed; for we may simply omit the early sides of the polygon, until we come
to a practically producible rate of motion.

18. To discover how far this mode of approximation can be trusted, we have
only to compare its consequences with those of the eauct solution. For the intrinsic
equation can easily be obtained in finite terms when there is no rotation. In fact,

cxin.]

ON THE PATH OF A ROTATING SPHERICAL PROJECfTILB.

377

by elimination of g between the differential equations of § 10, assuming k^O, we have
at once the complete differential of the equation

€"*t;cos0= Fco8<^o= y<i suppose;

where it is to be particularly noticed that Fo is the speed of the horizontcd component
of the velocity of projection, not the total speed. By means of this the second of
the equations becomes

?-? — ^ ^/a cos* (h

ds Fo»^ ^ ^'

whence

^.(^-l>=.ec*.t«.*.-«c*.«.* + l.«^^±^

The following fragments show the nature and arrangement of the results in one of
the earlier of Mr Wood's calculated tables. Having assumed (for reasons stated in
the introductory remarks above) that a = 240, I supplied him with the following
formulse : —

r'»= (l - ^) t;«- 400 sin <^ (1-0-04),

f =<^-^-^cos<^(l-004),

and I took as initial data F = 300, <}>=15''; [whence, of course, Fo*= 84,000 nearly.
This is required for comparison with the eauct solution].

Working from these he obtained a mass of results from which I make a few
extracts : —

./6
1.
2.
3.

90,000
85,401
81,032

300

292-2

2846

•003

•00342

•00351

•003

-00675

•01026

15°

14-876

14-746

•2588
•2565
•2546

•2588
•5153
•7699

oos^
•9659
•9665
•9671

2(00.0)

•9659

19324

28995

20.
21.

33,045
31,319

181-8
177-0

•00.550
•00565

•08666
•09231

11-028
10-686

•1914
•1854

4^6102
4-7956

•9815
•9826

19-4569
20-4395

40.
41.

11,440
10,875

106-9
104-3

•00935
•00959

•23391
•24350

- 1023

- 2^030

-•0178
- ^0355

6-6163
65808

•9998
•9994

393178
40^3172

60.
61.

5453

5377

•

738
733

•01354

•01363

•

•46935
•48298

- 30-748

- 32-564

•

- 5113

- -5383

14677

•9294

•8595
•8428

58-3988

59-2416

This table gives simultaneous values of 8, v, and <^ directly, t is obviously to be.
found by multiplying by 6 feet the numbers in column fifth; while by the same process
we obtain rectangular coordinates, vertical and horizontal, from the eighth, and the last,
columns respectively. Thus for instance we have simultaneously

T. n. 48

378 ON THE PATH OF A BOTATING SPHERICAL PBOJECTILE. [cXin.

♦

120

181-8

0-52

ll°-028

27-66

116-74

240

106-9

1-404

- 1 023

39-69

235-9

(The trajectory is given as fig. 3 in the Plate, and will be further analysed in the
next section of the paper.)

From the complete table we find that, in this case, <f) is positive up to the 38th
line inclusive, and then changes sign. It vanishes for « = 233 (approximately) after the
lapse of l'-35. The rectangular coordinates of the vertex are about 230 and 40, and
the speed there is reduced to 110. From the exact equation we find 9 = 232 for <^ = 0°.
This single agreement is conclusive, since the earlier tabular values of a for a given
value of (f) ought to be somewhat in excess of the true values; while the later, and
especially those for negative values of <f> greater than 30"* or so, should be somewhat too
small: — i,e, the calculated trajectory has at first somewhat too little curvature, but
towards the end of the range it has too much. It is easy to see that this is a necessary
consequence of the mode of approximation employed : — look, for • instance, at the fact
that the initial speed is taken as constant through the first six feet. See also the
remarks in § 17. On the whole, therefore, though the carry may possibly be a little
underrated, the numerical method seems to give a very fair approximation to the truth.
This admits of easy verification by the help of the value of d<f>ld8 last written, for it
enables us to calculate the exact value of s for any assigned value of <^ by a simple
difference calculated fi:om the result obtained from an assumed value.

19. Taking the method for what it is worth, the following are a few of the results
obtained from it by Mr Wood. I give the numerical data employed, plotting the
curves from a few of the calculated values of x and y. But I insert, at the side of each
trajectory, marks indicating the spaces passed over in successive seconds. This would
have been a work of great difficulty if we had adopted a direct process, even in cases
where the intrinsic equation can be obtained exactly; — and it must be carried out when
we desire to find the effects of wind upon the path of the ball

Fig. 1 represents the path when a = 240 (properly 234), F=300, <^o = 0°, and A? = 1/3.
This will be at once recognised as having a very close resemblance to the path of a
well-driven low ball. The vertex (at 0*76 of the range) and the point of contrary
flexure are indicated. This trajectory does not differ very much from that given (for
the same initial data) by the roughly approximate formula of § 10; which rises a little
higher, and has a range of some ten yards greater. But the assumed initial speed, and
consequently the coefficient of resistance, are both considerably too great.

In fig. 2 all the initial data are the same except k, which is now increased to 1/2 : —
%.e, the spin is 50 per cent, greater than in fig. 1. We see its effect mainly in the
increased height of the vertex, and in the introduction of a second point of contrary
flexure. A further increase of k will bring these points of contrary flexure nearer to one
another, till they finally meet in the vertex, which will then be a cusp, a point of
momentary rest, and the path throughoiU wiU he concave upwards! This is one of the

oxin,]

ON THE PATH OF A ROTATTNG SPHERICAL PROJECTILE,

379

most curious results of the investigation, and I have realized it with an ordinary golf-
ball : — using a cleek whose face made an angle of about 45" with the shaft and was
furnished with parallel triangular grooves, biting domnwar<h, so as to ensure great under-
epin. [The data for this case give extravagant results when employed in the formula of
I 10, The vertex it assigns is 510 feet from the starting-point and at nearly 172 feet of
elevation : — while the range is increapsed by 60 or 70 yards. And that formula can never
give more than one point of contrary flexure. All this was, however, to be expected ;
since the formula was ba^d on the express assumption that gravity has no direct effect
on the speed of the projectile.]

Fig. 3 shows the result of dispensing altogether with initial rotation ^ while
endeavouring to compensate for its absence by giving an initial elevation of IS"". This
figure, also, will be recognised as charaeteristic of a well-known class of drives; usually
produced when too high a tee is employed, and the player stands somewhat behind hia
ball. Notice, particularly, how much the cany and the time of flight are reduced,
though the initial speed ia the same. The slight underspin makes an extraordinary
difference J producing as it were an unbending of the path throughout its whole length,
and thus greatly increasing the portion above the horizon. But of course the pace of
the ball, when it reaches the ground, is very much greater than in the preceding cases,
it usually falls more obliquely, and it has no back -spin. On all these accounts we
should expect to find that the "run** will in general be very much greater. Still, in
consequence partly of the greater coefficient of resistance at low speeds, presently to be
discussed, overspin (due to the disgraceful aet called "topping'') is indispensable for
a really long run. In such a case the carry will, of course* be still further reduced,
unless the initial elevation be very considerably increased. (Some of Mr Wood's
numerical resulta, from which fig» 3 was drawn, were given in the preceding section.)

In fig* 4j a and V are as in fig. Ij but k — 1 and ^ = 45"", Hei-e we have the kink,

of which a provisional sketch (closely resembling the truth) was given in the former
instalment of the paper* I have not yet obtained it with a golf- ball, though as already
stated I have got the length of producing the cusp above spoken of* But the kink can
be obtained in a striking manner when we use aa projectile one of the large balloons
of thin india-tubber which are now so common. We have only to '* slice*' the balloon
Bharply downwards (In a nearly vertical plane) with the flat hand. This is a most
instructive experiment, and its repetition presents no difiiculty whatever. It is to be
specially noticed that, in the particular kink sketched, there is a point of minimum
speed somewhat beyond the vertex, and a point of maximum speed, both nearly in
the same vertical with the point of projection. The first (where the speed is reduced
to 58'7) is reached in a little more than two seconds, the other (where it has risen
to 73'8) in rather more than four.

It may be interesting to give a few details of Mr Wood s calculations for this
oase : — ^selecting specially those near the points of maximum and minimum speed, and
along with them those for closely corresponding elevations on the ascending side. Also

48—2

380

ON THE PATH OF A ROTATING SPHERICAL PROJECTILE.

near the vertex. The equations were

V = t;> ^ - JL^ _ 400 sin <^ (1 - 0-04)

. .360 12000 ... ^.^..
i>i^i> + — ^- cos <^ (1-004)

[CXUL

./6

r»

i/f

2(l/»)

gin^

2(8m^)

008^

2(008*)

90000

300

•003

45°

•7071

-7071

•7071

23.

24582

156-8

•00638

•10693

78°-72

•9807

19-6186

-1956

113075

41.

5583

74-7

•01359

•27640

145°-3

•5693

35-8751

•8221

62814

•

44.

4278

65-4

•01529

•32038

166°-46

•2343

36-9422

--9722

34951

45.

3974

630

•01586

■33624

174°-68

•0944

37-03G6

--9956

24996

46.

3739

611

•01636

•35260

183°^16

-0553

36-9813

-•9981

1.5015

•

48.

3475

590

•01697

•38630

201'-3

-•3633

36-4078

-•9317

- 6921

49.

3441

58-7

•01704

•40334

210°5

- -5075

35-9003

-•8616

-1-4537

50.

3464

58-9

•01700

•42034

219°^5

-•6363

35-2640

-•7714

-2-2251

•

67.

5434

73-7

•01357

•67179

313°-1

- 7302

20-0274

•6833

- -3162

68.

5443

73-8

•01355

•68534

316°-5

-•6880

19-3394

•7258

+ 4096

69.

5435

73-7

•01357

•69891

319°-9

-•6446

18-6948

-7646

+ -1742

•

The following data belong to the last elements for which the calculations were
made: —

80.

4374

66-1

•01512

•85485

352°-9

- ^1224

14-6898

•9926

112602

81.

4202

64-8

-01542

•87027

356°-8

--0732

14-6166

•9973

12^2575

As the last five values of <f) have been increasing steadily by nearly 3° for each
element, it is clear that the direction of motion again rises above the horizontal; but
whether the path has next a point of contrary flexure, or another kink, can only be
found by carr3dng the calculation several steps further. [The second kink is very
unlikely, as the speed is so much reduced at the point where the calculations were
arrested. Mr Wood has gone to Australia, and I had unfortunately told him to stop
the numerical work in this particular example as soon as he found that 2 (cos <f>\
after becoming negative, had recovered its former maximum (positive) value.]

The trajectories represented in figs. 5 and 6 may be taken as fairly representative
of ordinary good play by the two classes of drivers. For we have in both a = 360,
F=200. These are the new data, representing (as above explained) the best information
I have yet acquired. In fig. 6 A: = 1/3, <^o=10°; but in fig. 6 A; = 0, ^ = 15^ In spite
of its 50 per cent, greater angle of initial elevation, the carry of the non-rotating
projectile is little more than half that of the other : — and it takes only one-third of the

cxin.]

ON THE PATH OF A BOTATING SPHERICAL PBOJECTILl*

S81

time spent by the other in the air. But the contrast shows how mueb more important
(so far as carry is concerned) is a moderate amount of underspin than large initial
elevation. And we can easily see that initial elevation, which is always undesirable
(unless there is a hazard close to the tee) as it exposes the ball too soon to the action
of the wind where it is strongest, may be entirely dispensed with. This point is
discussed in next section.

On account of their intimate connection with actual practice, I give a few of the
numerical results for these two closely allied yet strongly contrasted caaes, belonging
to two different classes of driving; — choosing sides of each polygon passed at intervals
of about V, as well as those near the vertices and the point of contrary flexure. The
formulae for these cases are those given at the end of § 17 above: — the second term
in the expression for if/ being omitted for the latter af the two trajectori^,

For Fig. 5.

,16

f»

Ijv

s (iM

sin0

2 (da*)

eoK*

Z (con p)

40,000

200

•00500

■00500

10°

■1736

•1736

■9848

-9848

•

25.

15,497

1245

■00803

•16549

17^552

•3015

62345

■9534

25-2200

♦

•

39.

8,216

90'6

•01103

■29869

19^789

•3388

107983

■9410

38-4544

•

42.

7,042

83-9

•01192

-33353

19-665

■3366

11-8116

•9417

412783

•

64.

3,511

59-3

•01687

■50626

13-611

•2354

15-3925

-9719

52-7246

61.

2,387

48-9

■02046

•63904

r727

•0303

16-3078

■9996

59-6508

62.

2,296

47-9

■02088

•65992

- 0-675

- -0120

16-2958

-9999

60-6507

•

70.

2,249

47-4

•02109

•83156

- 21 807

- -3714

14-5533

■9285

68-4117

•

79.

3,157

56-2

•01780

100513

- 35-890

- -5862

9-0647

-8103

76-1309

•

89.

4.338

65-9

■01519

116748

- 40-840

- -6538

3-6521

•7566

83-8830

•

94.

4,853

69-7

•01436

124081

- 41 548

- -6633

0-3507

•7484

87-6381

For Fig. 6.

40,000

200

•OO50O

■00500

15°

•2588

-9659

•9659

•

26,

16,035

126-6

■00790

'16507

3-523

•0613

4-5617

■9981

255497

•

30.

13,940

1181

•00847

•19809

0472

•0082

4-6769

•9999

29-5476

31.

13,472

116-1

•00861

•20670

- 0360

-0064

4-6705

•9999

30-5475

♦

•

44.

9,147

95-6

■01046

■33189

- 13-854

- -2393

3-0442

■9709

43-4147

52.

7.850

88-6

•01129

•41952

- 24-208

--4099

•3650

-9121

50-9412

382 ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. [CXIU.

I regret that Mr Wood was obliged to give up his calculations before he had
worked out more than about a third of the requisite rows of figures for a trajectoiy
diflfering initially from fig. 6 in the sole particular <^ = 5'' instead of 10^ This would
have been still more illustrative than fig. 5 as a contrast with fig. 6. But a feirly
approximate idea of its form is obtained by taking the earlier part of fig. 5, regarded
as having the dotted line for its base. See a remark in § 22 below, which nearly
coincides with this.

Effect of Wind.

20. So far, we have supposed that there is no wind. But with wind the con-
ditions are usually very complex, especially as the speed of the wind is generally
much greater at a little elevation than close to the ground. Hence I must restrict
myself to the case of uniform motion of the air in a horizontal direction. We have
in such a case merely to trace, by the processes already illustrated, the path of the
ball relatively to the air; and thence we easily obtain the path relatively to the
earth. Here, of course, it is absolutely necessary to calculate the time of passing
through each part of the trajectory relative to the air. If the wind be in the plane
of projection, and its speed U, the relative speed with which the ball starts has
horizontal and vertical components Fcosa— t^, and Fsina, respectively. Thus, rela-
tively to the moving air, the angle of elevation is given by

, Fsina

tana = ^^^ jf,

Fcosa— U

and the speed is

F'= ^V*^2UVcoaa+U\

The relative trajectory, traced from these data, must now have each of its
points displaced forwards by the distance, Ut, through which the air has advanced
during the time, t, required to reach that point in the relative path. Of course, for
a head-wind, U is negative; and the points of the relative trajectory must be displaced
backwards.

Figs. 7, 8, 9 illustrate in a completely satisfactory manner, though with some-
what exaggerated speeds and coefficient of resistance, the results of this process.
Mr Wood had calculated for me the path in still air, with a = 288 (or, rather, 282),
F=300, 0 = 6^ A; = 1/3. Since the time of reaching each point in this path had
been incidentally calculated, it had only to be multiplied by 25, and subtracted fix>m
the corresponding abscissa, in order to give the actual path when the speed of the
head-wind is about 17 miles an hour, and the initial speed about 275. (The exact
values of this and of the actual angle of projection must be calculated by means
of the preceding formulae: — but they are of little consequence in so rough an illus-
tration as the present, especially as <^o and U/V are both small.) The corresponding
trajectory is shown in fig. 7. If we use the same relative path for wind of 25'5
miles per hour, the actual initial speed must be about 262*5, and the true path is
fig. 8. Finally, fig. 9 gives the result with actual initial speed 250, and head-wind

CXin.] ON THE PATH OP A ROTATING SPHERICAL PROJECTILE,

383

blowing at 34 miles an hour. Here, again, a kink 1^ produced in the actual patb^
but it is due to a completely different cause from that of fig. 4. And it is specially
to be noted how much the vertex is displaced towards (and even beyond) the end
of the range.

2L It ia not necessary to figure the result of a following wind, for such a
cause merely lengthens the abscissae in a steadily increasing ratio, and makes the
carry considerably longer, while placing the vertex more nearly midway along the
path. But it is well to call attention to a singularly erroneous notion, very prevalent
among golfers, viz., that a following wind carrieB the ball rniwarcb! Such an idea
is. of course, altogether absurd^ except in the extremely improbable case of wind
moving faster than the actual initial speed of the ball. The true way of regarding
matters of this kind is to remember that there is always resistance while there is
relative motion of the ball and the air, and that it is less as that relative motion
is smaller; ao that it is reduced throughout the path when there is a following wind.

Another erroneous idea, somewhat akin to this, is that a ball rises considerably
higher when driven against the wind, and lower if with the wind, than it would
if there were no wind* The difference (whether it is in excess or in defect will
depend on the circumstances of projection, notably on the spin) is in general very
small ; the often large apparent rise or fall being due mainly to perapective, as the
vertex of the path is brought considerably nearer to, or further from, the player.

These approximations to the effect of wind are, as a rule, very rough ; because
in the open field the speed of the wind usually increases in a notable manner up
to a considerable height above the ground, so that the part of the path which is
most affected is that near the vertex. But the general character of the effect can
easily be judged from the examples juat given.

When the wind blows directly across the path, the same process is to be applied.
It is easy to see that the trajectory is no longer a plane curve; and also that, in
every case, the carry is increased. But, in general, '* allowance is made for the wind,"
i.e. the ball is struck in such a direction as to make an obtuse angle with that of
the wind, more obtuse as the wind is stronger. In this case the carry must invariably
be shortened But without calculation w© can go little beyond general statements
like these.

Effect of Gradual Diminution of Spin.

22, In my former paper I assumed, throughout, that the spin of the ball remains
practically unchanged during the whole carry* That this is not far from the truth, is
pretty obvious from the latter part of the career of a sliced or a heeled ball If,
however, in accordance with § 4, we assume it also to fall off in a geometric ratio
with the space traversed : — an assumption which is probable rather than merely plausible;
so long, at least, as we neglect the part of the loss which would occur even if the

384

ON THE PATH OF A ROTATING SPHERICAL PROJBCnL.E.

[cxm.

ball had no translatory speed: — ^the equations of § 10 require but slight inodificatiQD.
For we must now write, instead of k,

The time rate at which this falls off is proportional to itself and to v, directly, and
to 6 inversely.

If we confine ourselves to the very low trajectories which are now characteristic
of much of the best driving, we may neglect (as was provisionally done in § 10) the
effect of gravity on the speed of the ball, and write simply

Thus the approximate equation of the path becomes

$^ = a+^'(^'«'-l)-^,(.--l).

2r>

Here

and finally

1 _1_1

d'~a 6'

where a is always very small, perhaps even negative; and may, at least for our present
purpose, be neglected. Its main effect is to elevate, or depress, each point of the path
by an amount proportional to the distance from the origin; and thus (when positive)
it enables us to obtain a given range with less underspin than would otherwise be
required.

23. For calculation it is very convenient to begin by forming tables of values
of the functions

for values of p at short intervals from 0 to 3 or so. [Note that the same tabte
are adaptable to negative values o{ p, since we have, obviously,

f(-p)^e'Pf(pl and F(^p)^ e'^(/(p)^F(p))l

These we will take for granted. We may now write

y=Y,(k VF (x/a') - gF (Zx/a))

The range, and the horizontal distances of the vertex and of the point of con-
trary flexure, respectively, are given by the values of x which make the second ftctofs

CXni,] ox THE PATH OF A ROTATING SPHERICAL PROJECTILE.

985

vanish r^and it is curious to remark that (to the present rough approximation, of
course, and for given values of a and a') these depend only upon the value of kV/^,
i.e. the initial ratio of the upward to the downward acceleration Thus so far as
the range is concerned, the separate values of k and V are of no consequence* all
depends on their product. But it is quite otherwise as regards the flatness of the
trajectory, for the maximum height is inversely as the square of F, Of course we
must remember that one indispensable condition of the approximation with which
we are dealing is that the trajectory shall be verj^ flat; and thus, if the range is
to be considerable, V cannot be small, and (also of course) k cannot be very large.
We have already seen how to obtain a fairly approximate value of a (say 360), but
b presents much greater difiGcuIty, We may, therefore, assume for it two moderate,
and two extreme values, and compare the characteristics of the resulting paths. If 6
be infinite, we have the case already treated, in w^hich the spin does not alter during
the ball's flight ; while, if 6 be less than a, the spin dies out faster than does the
speed and we approximate (at least in the later part of the path) to the case of
no spin. Hence we may take for the values of 6 the following : — ^ » 900, 360, and
180 : — so that a' has the respective values 360, 600, qo , and — 360. Let the cany
(x) be, once for all, taken as 180 yards. Then, for y=0, we must have 2x/tt = 3; and
the respective values of xja are 1*5, 0 9, 0, and — 15, With these arguments the
values of F are, in order,

1-7873; 0-8807, 0*6908, O'o, and 0^3258;

so that we have the following approximate values of the ratio kVjg

2-03, 269, 3-57, 549.

The fii^st two require a moderate amount of spin, only, if we take 240 as the
initial speed.

The approximate position of the vertex (jjJ of the first of these paths is given by
/{2a?,/a) = 2'03/(^,/a), or e*»^«=^3^06, (j?Ja = M184)
whence a^o = 402"6, or about three-fourths of the c^rry.

The corresponding value of y is about 27 feet.

The point of contrary flexure is at €*'**= 203, so that ^i = 255, and the value of
-^ there has its maximum, about 007 only.

In the other three paths above, the maximum ordinate and the maximum in-
clination both increase with the necessarily increased value of i, while the vertex and
the point of inflexion both occur earlier in the path. The approximate time of flight,
in all, is a little over five seconds. The paths themselves are shown, much fore-
shortened, in figs, 10, 11, 12, 13, where the unit of the horizontal scale is 3*6 times
that of the vertical This is given with the view of comparing and contrasting them.
Fig. 14 shows the first, and flattest, of these paths in its proper form. It is clearly
a fair approximation to the actual facts; and when we compare it with the others,

T. U, 49

386 OK THE PATH OF A Rr/TATEVG SPHERICAL PBOJECTILE. [CXIIL

tm in th<; forenh^^t^med fifpiren, we iK?e that the aABmnption of oonstant spin (| 4| is
pn/bably iK/t far from the tnith. For, in the great majority of cues of drives of
thii! character, there in obiter^-ed to be very little mn: — and this can be acooanted
for only on the anKiimption that there is considerable nnderspin left at the pitch.
But it is also clear that the falling off of the spin produces comparatively little
incr^;ase of the obliquity of impact on the gronnd, even in the exaggerated f<Hin in
which these paths are drawn. Their actoal inclinations to the groand have tangents
about 0'49, 066, 0-78, and 108 respectively. The last, and greatest, of these angles
is just over 48'.

24. It is interesting to compare this Het of data, and their consequences, with
those of |§ 11, 14, 15. The latter were in fair agreement with many of the more
easily observed features of a good drive, but they gave too high a trajectory. The
new measure of initial speed, and the consequent reduction of the estimated value of
the coefficient of resistance, have led to results more closely resembling the truth.

But in all, as we have seen, there is one notable defect. The ball comes down
too obliquely, and this is the case more especially when the carry is a long one, and
the ball's speed therefore much reduced. I was at first inclined to attribute this to
my having assumed the spin to remain constant during the whole flight. This was
my main reason for carrying out the investigations described in ^ 22 sq. But these
give little help, as we have just seen, and I feel now convinced that the defect is
due chiefly to the assumption that the resistance is throughout proportional to the square
of the speed. I intend to construct an apparatus on the principle described in § 16
above, but of a much lighter t3rpe, to measure the resistance for speed of 30 feet-
seconds or so, downwards. But I shall probably content myself with verifying, if I
can, the idea just suggested; leaving to some one who has sufficient time at his
disposal the working out of the details when the resistance is proportional (towards
the end of the path) to the speed directly, or to a combination of this with the
second power. The former is considerably more troublesome than Robins' law ; and
a combination of the two may probably be so laborious as to damp the ardour of
any but a genuine enthusiast. The possibility that the law of resistance may change
its form for low speeds {i,e., towards and beyond the vertex of the path) throws
some doubt upon the accuracy of the determination of the coefficient of resistance
from the range, the time of flight, and the initial speed. But, at present, I have no
means of obtaining a more accurate approximation.

25. The whole of this inquiry has been of a somewhat vague character, but its
value is probably enhanced, rather than lessened, in consequence. For the circum-
stances can never be the same in any two drives, even if they are essentially good
ones, and made by the same player. To give only an instance or two of reasons
for this: — Two balls of equal mass may have considerably different coefficients of
renstance in consequence of an apparently trifling difference of diameters, or of the
amount or character of the hammering: — or they may have very different amounts
of resilience, due to comparatively slight differences of temperature or pressure during

Piai

Pl«. 18

Plff. O FIff. 8 FIff. 7

fOO PUT.

CXIII.] ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. 887

their treatment in the mould The pace which the player can give the club-head
at the moment of impact depends to a very considerable extent on the relative
motion of his two hands (to which is due the "nip") during the immediately pre-
ceding two-hundredth of a second, while the amount of beneficial spin is seriously
diminished by even a trifling upward concavity of the path of the head during the
ten-thousandth of a second occupied by the blow. It is mainly in apparently trivial
matters like these, which are placidly spoken of by the mass of golfers under the
general title of *' knack/' that lie the very great differences in drives efifected, under
precisely similar external conditions, by players equal in strength, agility, and (except
to an extremely well-trained and critical eye) even in style.

[Oct 5, 1898. — The printing of this paper has been postponed for nearly three
years in the hope, not as yet realised, that I might be able to determine accurately
by experiment the terminal speed of an average golf-ball, as well as the average
value of ky when (as in § 5) kayv represents the transverse acceleration, in terms of
the rates of spin and translation. Another object has been to measure the efifect
of rapid rotation upon the coefficient of resistance to translatory motion. These ex-
periments, in various forms, are still being carried out by means of various modes
of propulsion, from a cross-bow to a harpoon-gun. I hope abo to procure data, for
speed and resistance, applicable to various other projectiles such as cricket-balls, arrows,
bird-bolts, etc.]

[1899.— A popular sketch of the main results of Nos. CXII. and CXIIL, so
far as they are applicable to the game of Golf, will be found in the Badminton
Magazine for March, 1896.]

49—2

388

[CXIT.

CXIV.

NOTE ON THE ANTECEDENTS OF CLERK-MAXWELL'S ELECTRO-
DYNAMICAL- WAVE-EQUATIONS.

[Proceedings of the Royal Society of Edinburgh, April 2, 1894.]

The first obvious diflBculty which presents itself, in tr3ring to derive Clerk-Mazweirs
equations from those of the elastic-solid theory, appears in the fact that the latter, being
linear, do not impose any relations among simultaneous disturbances. Thus, for instance,
they indicate no reason for the associated disturbances which, in Maxwell's theory, con-
stitute a ray of polarised light. Hence it appears that we must look on the vectors of
electric and magnetic force, if they are to be accounted for on ordinary dynamical principles,
as being necessary concomitants, qualities, or characteristics of one and the same vector-
disturbance of the ether, and not themselves primarily disturbances. From this point
of view the disturbance, in itself, does not correspond to light, and may perhaps not
affect any of our senses. And the very form of the elastic equation at once suggests
any number of sets of two concomitants of the desired nature, which are found to be
related to one another in the way required by Maxwell's equations.

For the moment, as sufficiently illustrating the essential point of the above remarks.
I confine myself to disturbances, in the free ether, such as do not involve change of
volume. The elastic equation is

with the limiting condition fi>V^ = 0*.

[Had not this condition been imposed, the dynamical equation would have involved
on the right, the additional term

♦ Stokes, **0n the Dynamical Theory of Diffraction,*' Camh. Phil. Tram., ix. (1849).

CXI v.] NOTE ON THE ANTECEDENTS OF CLERK-MAXWELL's EQUATIONS. 389

From any vector satisfying these equations let us derive (by means of the
operators d/dt and aV, which are the only ones occurring in the equation of motion)
the concomitants

or € = d', ^ = — aV0, &c., &c.,

and we have between them Clerk-Maxwell's equations

e =s aVfjL, /i s= — aVe,
with the conditions flfVc^O, SVfi^O.

The extension to dielectrics, whether they be isotropic or not, is obtained at once : —
and it secures (in the latter case) all the simplicity which Hamilton's linear and vector
function affords. Thus the properties of double refraction, wave-surfaces, &c., follow
almost intuitively.

When we come to conducting bodies, we have to introduce further conditions.
But I do not enter on these at present, as the problem is essentially altered in
character. Nor do I, for the moment, discuss the bearing of the above notions upon
the profound question of the possible nature of electricity and of magnetism.

There is a sort of analogy to the above, in the case of sound. For it is not
the (vector) disturbance of the air which affects the sense of hearing, but the (scalar)
concomitant change, or rate of change, of density.

Thus, possibly, the widely different results obtained by observers of the alteration
of plane of polarisation in diffracted light, may ail really be in accordance with Stokes'
splendid investigation : — if we look upon light as an effect produced by the concomitants
of the ether disturbance, and not directly by the ether disturbance itself

390 [cxv.

cxv.

ON THE ELECTRO-MAGNETIC WAVE-SURFACE.

[Proceedings of the Royal Society of Edinburgh, April 2, 1894.]
We may write the electro-magnetic equations of Clerk-Maxwell as

For plane waves, running with normal velocity t;a = — /tft""S we have
01 = €f(vt + Sap\ e^ = fif{vt + Sap),
whence at once ^e = F/lm;, '^ = — Vfie,

so that Sfi<l)€=^0, /Si/A'^ = 0.

[For the moment, we assume that 0 and yjt are self-conjugate, so that a linear
function of them is also self-conjugate. And we employ the method sketched in Tait's
Quaternions, §§ 438-9.]

We have n-^^^^e = Vyfrfiy^ = — F. yjrfiVfie,

or yjrfjLSesfrfi = n(f>€ H- Sfiyjrfi . yjte = «re, say.

Thus we have, to determine fi, the single scalar equation

S . fi(f>^-'^yjrfi = Sfi (nylt-^ + Sfiyltfi . <t>'-^)-'^ fJi = 0 (a).

This is the index-surface, and the form of «r shows that it has two sheets: —
i.e., there are two values of Tfi for each value of Ufi,

The tangent plane to the wave is

S^^P — 1 (6).

CXV.] ON THE BLECTKO-MAGNETIC WAVE-SURFACE. 391

To shorten our work, introduce in place of e the auxiliary vector

80 that 'sjrfi^ n<lyr + Sfiyjrfi.y^ (c).

(a) may now be written 5/Lt0T = O (a).

Hence (c) gives, by operating with S.ft, S.r, and S.yjt'^py

Sfiyfrr^l (1),

l^^nSrffn + Sfi'ylrfjLS'nln' (2),

-l=^nSpylr'^it>T + STpSfiy^^fjL (3).

These preliminaries being settled, we must find the envelope of (6) subject to the
sole condition (a). We have at once by differentiation

SpdfjL^O, and iSfd/u, (<^t - i/r/iST^T) = 0,

80 that fl?p sa ^T — -^/tftiST^T (d).

Treat this with the three operators used before, and we have respectively

X '^ S/A^^fA Srifyr (4),

Srp^O (5),

xSpyjt-^p ^ Spylt-^ifyr -i- St<I>t (6).

By means of (5), (3) becomes - 1 «= nSp^'^^r,

so that (6) takes the form aj/Sp^-^p = - - + St^^t (6).

Substitute for -^fi in (d) its value in terms of r from (o); and x becomes, by
(4) and (6), a factor of each term ; so that

p = — n5/3^~*p .0T — -^ (d).

Eliminating r between this and (5), we have finally

• S.p{ylt + nSpylt-^p . 4>y^p = 0.

(Equation (2) above, has not been, so far, required: — but it is necessary if we
desire to find the values of Sfi'^fi and other connected quantities.)

It is obvious that, if we had originally eliminated e instead of 17, we should have
obtained the (apparently) different form

S . p (^ + mSpifr^p . y^y-^p « 0.

It is an interesting example in the treatment of linear and vector functions to
transform one of these directly into the other. (Tait's Quaternions, § 183.)

392 [cxvi.

CXVI.

ON THE INTRINSIC NATURE OF THE QUATERNION METHOD.

[Proceedings of the Royal Society of Edinburgh, July 2, 1894.]

My title is purposely ambiguous, because it has to represent two things: — I intend
to treat not only of what a quaternion really is, but also of its self-containedness, or
independence.

Professor Cayley has just stated* that "while coordinates are applicable to the
whole science of geometry, and are the natural and appropriate basis and method in
the science, Quaternions seem to me a particular and very artificial method for treating
such parts of the science of three-dimensional Geometry as are most naturally discussed
by means of the rectangular coordinates x, y, z.**

On this I would remark as follows: —

1. I have always maintained that it is not only not a reproach to, but one of
the most valuable characteristics of, Quaternions that they are uniquely adapted to
tridimensional space. In my Address to Section A, at the British Association Meeting
in 1871 (No. XXIII. above), I said:—

"It is true that, in the eyes of the pure mathematician. Quaternions have one
grand and fatal defect. They cannot be applied to space of n dimensions, they are
contented to deal with those poor three dimensions in which mere mortals are doomed
to dwell, but which cannot bound the limitless aspirations of a Cayley or a Sylvester.
From the physical point of view this, instead of a defect, is to be regarded as the
greatest possible recommendation. It shows, in fact, Quaternions to be a special in-
strument so constructed for application to the Actual as to have thrown overboard

* ''Coordinates versuB Quatermons," Proc, R.S,E,, July 2, 1894; or Collected Papers, No. 962.

CXVI.] ON THE INTRINSIC NATURE OF THE QUATERNION METHOD. 393

everything which is not absolutely necessary, without the slightest consideration whether
or no it was thereby being rendered useless for applications to the Inconceivable,"

2. Whether Quaternions are to be regarded as arti6cial, or the reverse, will obviously
depend wholly upon what is to be understood by the term Quaternions. This forms the
main object of the present paper.

3. Though the passage quoted above contains no statement as to the relative
merits of Quaternions, and Coordinates, as instruments (in the region which is
common to them), it is clear from other passages in his paper that Prof. Cayley holds
that Quaternions are, at best, superfluous: — he allows that they enable us to effect
great abbreviations, but he insists that, to be applied or even understood, they must
be reconverted into the x, y, z elements of which they are, in his view, necessarily
composed.

But their Inventor himself, who certainly devoted vastly more time and attention
to Quaternions than it can have been possible for Prof Cayley to devote, took a very
different view of the matter: —

"It is particularly noteworthy that [Quaternions were] invented by one of the most
brilliant Analysts the world has yet seen, a man who had for years revelled in floods
of symbols rivalling the most formidable combinations of Lagrange, Abel, or Jacobi.
For him the most complex trains of formulae, of the most artificial kind, had no
secrets: — he was one of the very few who could afford to dispense with simplifications:
yet, when he had tried Quaternions, he threw over all other methods in their favour,
devoting almost exclusively to their development the last twenty years of an exceedingly
active life."

It will be gathered from what precedes that, in my opinion, the term Quaternions
means one thing to Prof. Cayley and quite another thing to myself: — thus

To Prof Cayley Quaternions are mainly a Calculus, a species of Analytical Geo-
metry; and, as such, essentially made up of those coordinates which he regards as
"the natural and appropriate basis of the science.'' They artfully conceal their humble
origin, by an admirable species of packing or folding : — but, to be of any use, they

-doubly dying, must go down

To the vile dust from whence they spnmg !

To me Quaternions are primarily a mode of representation : — immensely superior to,
but of essentially the same kind of usefulness as, a diagram or a model. They are,
virtually, the thing represented: and are thus antecedent to, and independent of,
coordinates: giving, in general, all the main relations, in the problem to which they
are applied, without the necessity of appealing to coordinates at all. Coordinates
may, however, easily be read into them: — when anything (such as metrical or numerical
detail) is to be gained thereby. Quaternions, in a word, exist in space, and we have
only to recognize them : — but we have to invent or imagine coordinates of all kinds.
The grandest characteristic of Quaternions is their transparent intelligibility. They give
the spirit, as it were, leaving the mere letter aside, until or unless, it seems necessary

T. II. 50

394 ON THE INTRINSIC NATURE OF THE QUATERNION METHOD. [CXVI.

to attend to that also. In this respect they give a representation analogous to the real
image of a planet in the focus of an object-glass or mirror: — all that is obtainable is
there, and you may apply your microscopes and micrometers to it if you please. But,
theoretically at least, you may dispense with them and have recourse to your eyes
and your yard-stick alone, if you increase your focal length, and along with it the
aperture, of your object-glass suflBciently. Of course Newton's "most serene and quiet
air'' would be indispensable. For the development of this feature of my subject, and
for illustrative examples, I refer to the B. A. Address above cited ; and to the Address
to the Edinburgh University Physical Society (No. XCVIL above), alluded to by Prof.
Cayley.

To those who have read Poe's celebrated tale, The Purloined Letter, it will be
obvious that the contrast between these two views of Quaternions is even greater than
that between the Parisian Police and M. Dupin himself, though of very much the
same kind.

There was a time, in their early history, when Professor Cayley's view of Quaternions
was not merely a correct one, it was the only possible one. But, though the name
has not been altered, the thing signified has undergone a vital change. To such an
extent, in fact, that we may almost look upon the Quaternion of the latter half of
this century as having, from at least one point of view, but little relation to that
of the seven last years of the earlier half.

Hamilton's extraordinary Preface to his first great book shows how from Double
Algebras, through Triplets, Triads, and Sets, he finally reached Quaternions. This was the
genesis of the Quaternion of the forties, and the creature then produced is still essentially
the Quaternion of Professor Cayley. It is a magnificent analytical conception; but it
is nothing more than the full development of the system of imaginaries i, j, k ; defined
by the equations

iJ=j» = A;» = i;A=-.l,

with the associative, but not the commutative, law for the factors. The novel and
splendid points in it were the treatment of all directions in space as essentially alike
in character, and the recognition of the unit vector's claim to rank also as a quadrantal
versor. These were indeed inventions of the first magnitude, and of vast importance.
And here I thoroughly agree with Prof. Cayley in his admiration. Considered as an
analytical system, based throughout on pure imaginaries, the Quaternion method is
elegant in the extreme. But, unless it had been also something more, something very
diflferent and much higher in the scale of development, I should have been content
to admire it: — and to pass it by.

It has always appeared to me that, magnificent as are Hamilton's many contributions
to mathematical science: — his Fluctuating Functions, and his Varying Action, for in-
stance:— nothing that he (or indeed any other man) ever did in such matters can be
regarded oh a higher step in pure reasoning than that which he took when he raised
Quat<;mionH from the comparatively low estate of a mere system of Imaginaries to the
proud j)08ition of an Organ of Expression ; giving simple, comprehensive, and (above all)

CXVI.] ON THE INTRINSIC NATURE OF THE QUATERNION METHOD.

395

transpareDtlj mtelligible, embodiment to the most complicated of Real geometrical and
physical relations. From iiiB most intensely artificial of systems arose, as if by magic t
an absolutely natural one!

Most unfortunately, alike for himself and for his grand conception, Hamilton *s nerve
failed him in the composition of his first great Volume* Had he then renounced, for
ever, all dealings with % j\ k, his triumph would have been complete, He spared Agag^
and the best of the sheep, and did not utterly destroy them I He had a paternal
fondness for i, j\ k\ perhaps also a (not unnatural) liking for a meretricious title such
as the mysterious word Qimiennon; and, above all, he had an e^niest desire to make
the utmost return in his power for the liberality shown him by the authorities of
Trinity College, Dublin* He had fully recognized, and proved to others, that his i, j, k
were mere excrescences and blots on his improved method : — but he unfortunately
considered that their continued (if only partial) recognition was indispensable to the
reception of his method by a world steeped in Cartesian ism 1 Through the whole
compass of each of his tremendous volumes one can find traces of his desire to avoid
even an allusion to i, j, k; and, along with them, his sorrowful conviction that, should
he do so, he would he left without a single reader. There can be little doubt that,
by thus taking a course w*hich he felt to be far beneath the ideal which he had
attained, he secured for Quaternions at least the temporary attention of mathematicians.
But there seems to me to be just as little doubt that in so doing he led the vast
majority of them to take what is still Professor Cayley's point of view; and thus, to
regard Quaternions as (apparently at least) obnoxious to his criticisms* And I further
believe that, to this cause alone, Quaternions owe the scant favour with which they
have hitherto been regarded,

[I am quite aware that, in making such statements, I inferentially condemn (to
aome extent, at least) the course followed in my own book. But, since my relations

w*ith Hamilton in the matter have been alluded to more than once, and alike incompletely
and incon^ectly, by Hamilton's biographer, I may take this opportunity of making a slight
explanation, not perhaps altogether uncalled for. That Hamilton can altogether have
forgotten the permission (limited as it was) which he had given me, when, a little
later, I proposed to avail myself of it (strictly mtfiin ttie liinits imposedX seems incredible.
Mr Graves should either have let the matter alone, or have gone into much greater
detail about it. As it stands, he virtually represents Hamilton as being unaccountably
capricious. The following extract from the letter (of date July 10, 1859) in which
Hamilton gave his sanction to my writing a book on the subject, speaks for itself.
I had, of course, no rights in the matter; — and I cheerfully submitted to the restrictions
he imposed on me; especially as I understood that he expressly (and moat justly) desired
to be the first to give to the world his system in its vastly improved form*

" [2.] If 1 ah&U go on to speak of tuy views, wishes^ or feelings, on the subject of future publication,
I request you beforehand to give to any such eipreaaion of mine your most indulgent construction ; and
not to attribute to me any jealousy of you, or any wish to interfere^ in any way, with your freedom,
as Author and as Critic*

[3j If we were altogether strangera, I could have no right to address you on such a subject at

50—2

396 ON THE INTRINSIC NATURE OF THE QUATERNION B£ETHOI>. [CXVL

alL [Here follow, as an example, some allusions (which need not be quoted) to a then recent pamphlet
of Mdbius, dealing with the Associative Principle in Quaternion Multiplication.] But between you and
me, the case is perhaps not exactly similar ; as we have so freely corresponded, and as you are an
Author in the same language, and of the same country : — England, Scotland, and Ireland, being here
held to have their sons compatriots.

[4] To M5bius*s excellent Pamphlet, it is likely that I may retirni. Meanwhile I trust that it
cannot be oflfensive to you, if I confess, — what indeed your No. 38 encourages me to state, — that in
any such future publication on the Quaternions as you do me the honour to meditate, I shoukl
prefer the establishinent of * Principles' being left, for some time longer, — say even 2 or 3 yeare,—
in my own hands. Open to improvement as my treatment of them confessedly is, I wish that
improvement, at least to some extent, to be made and published by myself. Briefly, I should like
(I own it) that no book, so much more attractive to the mathematical public than any work of mine,
as a book of yo\u*s is likely to be, should have the api)earance of laying a * FonNDATiON ' : althou^
the richer the * Superstructure,' on a preWously laid foundation, may be, the better shall I be
pleased. I think, therefore, that you may be content to deduce the Associative Law, from the mte
of ?', j\ k ; leaving it to me to consider and to discuss whether it might not have been a fatal
objection to these rules^ if they had been foimd to be inconsistent with that Prtncipi«b.

[7.] For calculation, you know, the rules of t, y, k are a sufficient basis, although of course we
have continual need for transformations, such as

which may at last be reduced to conseqiiences of those rules ; and also require some NoiatUm^ sudi
as 8, F, JT, T, r, which 1 have been glad to find that you are willing, at least for the presentj to
retain and to employ. But my peculiar turn of mind makes me dissatisfied without seeking to go
deeper into the philosophy of the whole subject, although 1 am conscious that it will be imprudent
to attempt to gain any lengthened hearing for my reflections. In fact I hope to get much more
rapidly on to rtdes and operations, in the Manual than in the Lectures ; although I cannot consent
to neglect the occasion of developing more fully my conception of the Multiplication op VficiOBS.
and of seeking to establish such mult[iplication] as a much less arbitrary process, than it may seem
to most readers of my former book to be."

I do not now think that Hamilton, with the "peculiar turn of mind" of which
he speaks, could ever, in a hook, have conveyed adequately to the world his new
conception of the Quaternion. I got it from him by correspondence, and in conveisatioa
When he was pressed to answer a definite question, and could he kept to it, he replied
in ready and effective terms, and no man could express vivd voce his opinions on such
subjects more clearly and concisely than he could: — but he perpetually planed and re-
polished his printed work at the risk of attenuating the substance: and he fatigued
and often irritated his readers by constant excursions into metaphysica One of hi«
many letters to me gave, in a few dazzling lines, the whole substance of what afterwards
became a Chapter of the Elements ; and some of his shorter papers in the Proc. IL I. A,
are veritable gems. But these were dashed off" at a sitting, and were not plaiied and
repolished.

Should I be called upon, in the future, to produce a fourth edition of my book,
the Chapter which Prof. Cayley so kindly furnished for the third edition will probably
preserve by far the greater part of the allusions to i, j, k (except, of course, the
necessary introductory and historical ones) which it will contain.]

CXVI,] ON THE TNTBraSlC NATURE OF THE QUATEHNION MirraOD~ 39^

In the sense above explainedi I consider Pro£ Cay ley's remarks to be so for
warranted, hard to bear though some of them undoubtedly sure. But the Quatemion,
when it is regarded from the true point of view, is seen to be untouched, in fact
unassailable, by any criticism based upon such grounds as reference to coordinates. It
occupies a region altogether apart. To compare it to a pocket map is to regard it
as a mere artificial mode of wrapping up and concealing the *, j, k or the x, y, m
which are supposed to be its ultimate constituents. To be of any use it must be
unfolded, and its neatly hidden contents turned out. But, from my point of view^ thia
comparison is entirely misleading* The quaternion exists, as a space-neahty, altogether
independent of and antecedent to i, j, k or w^ y, z. It is the natural, they the altogether
artificial, weapon. And I venture further to assert (1) that if Descartes, or some of
his brilliant contemporaries, had recognised the quaternion, (and it is quite conceivable
that they might have done so)^ science would have then advanced with even more
tremendous strides than those which it has recently taken; and (2) that the wretch
who, under such conditions, had ventured to introduce i\ _;, t, would have been justly
regarded a« a miscreant of the very basest and most depraved character: possibly
subjected to "brave punishments,*' the peine foi'ie et dure at the very least! In a word,
Hamilton invented the Quaternion as Prof Cayley sees it; he afterwards discovered
the Quaternion as I see it.

If Quaternions are to be compared to a map, at all, they ought to be compared

to a contoured map or to a model in relief, which gives not only all the information
which can be derived from the ordinary map but something more : — something of the
veiy highest importance as regards the features of a country.

A much more natural and adequate comparison would, it seems to me, liken
Coordinate Geometry (Quadriplanar or ordinary Cartesian) to a steam-hammer, which
an expert may employ on any destructive or constructive work of one (general ktnd^
say the cracking of an egg-shell, or the welding of an anchor. But you must have
your expert to manage it, for ijiathout him it is useless. He has to toil amid the
heat, smoke, grime, grease, and perpetual din of the suffocating engine-room* The work
has to be brought to the hammer, for it cannot usually be taken to its work. And
it is not, in general, transferable; for each expert, as a rule, knows, fully and confidently,
the working details of his own weapon only. Quatemiom, on the other hand, are like
the elephant's trunks ready at any moment for anything^ be it to pick up a crumb
or a field-gun, to strangle a tiger, or to uproot a tree* Portable in the extreme, applicable
anywhere: — ^alike in the trackless jungle and in the barrack square: — directed by a little
native who requires no special skill or training, and who can be transferred from one
elephant to another without much hesitation. Surely this, which adapts itself to its work,
is the grander instrument ! But then, it is the natural, the other the artificial, one*

The naturalness of Quaternions is amply proved by what they have effected on

I their first application to well-known, long threshed-out, plane problems, such as seemed
particularly ill-adapted to treatment by an essentially space-method. Yet they gave,
at a glance, the kinematical solution (perfectly obvious, no doubt, when found) of that
problem of Fermat's which so terribly worried Viviani I And, without them, where

398 OK THE lyXBIXSIC KATTBE OF THE QUATEEXIOX XETHOD. [CXVL

would have he^iu even the Circular Hodograph, with its marvellaas power of simpliftiiig
the elementary treatment of a planet'j^ orbit ? I could give many equally strikiiig
injstancea

Ah to the neceKsity, in modem mathematical physics, for aom^ substitute for what
I muHt (with all due deference to Prof Cayley> call the cumbersome, unnatural, and
unwieldy mechanism of coord inatef?, I have elsewhere fiilly expressed my own opinion,
and need not repeat it.

Of course it will be obvious from what precedes that I adhere to every word of
the first extract which Prof Cayley has made from my original Preface.

The phrase which he afterwards extracts for comment: — "such elegant trifles as
Trilinear Coordinates*': — seems somewhat too sweeping, and I should certainly hesitate
to use it without qualification. But the context shows that, in my Preface^ it was
used to characterize the so-called *' Abridged Notation" which had then been for some
years introduced into Cambridge reading and examinations, not at all because of its
suj>eriority in completeness to the ordinary x, y system: — and therefore not on scientific
grounds : — but mainly for the purpose of " aggravating" students, whether in the lecture-
room or in the Senate House, at very small additional labour on the part of the
lecturer or the examiner. But I made no reference whatever to Quadriplanar Co-
ordinates; for which I feel all due respect, not altogether free from an admixture of
wholesome awe!

cxvii.] 399

CXVII.

SYSTEMS OF PLANE CURVES WHOSE ORTHOGONALS FORM

A SIMILAR SYSTEM.

[Proceedings of the Royal Society of Edinburgh, May 6, 1895.]

(Abstract.)

While tracing the lines of motion and the meridian sections of their orthogonal
surfaces for an infinite mass of perfect fluid disturbed by a moving sphere: — the
question occurred to me, "When are such systems similar?" In the problem alluded
to the equations of the curves are, respectively,

(r/a)« = cos ^, and (r/6)* = 8inft

It was at once obvious that any sets of curves such as

1
(r/ay^ = cos0 and (r/6)"* = 8in^

are orthogonals. But they form similar systems only when

7?l» = 1.

Hence the only sets of similar orthogonal curves, having equations of the above
form, are (a) groups of parallel lines and (6) their electric images (circles touching
each other at one point). As the electric images of these, taken from what point
we please, simply reproduce the same system, I fancied at first that the solution must
be unique: — and that it would furnish an even more remarkable example of limitation
than does the problem of dividing space into infinitesimal cubes. (See No. CV. above.)

400 SYSTEMS OF PLANE CURVES WHOSE ORTHOQONALS [CXVH.

But I found that I could not prove this proposition ; and I soon fell in with
an infinite class of orthogonals having the required property. These are all of the
type

rg = (tan^r+i (1).

which includes the straight lines and circles already specified. The next of these in
order of simplicity among this class is

1
r = ae*®^ cos 0,

1
with r=6€«^^sinft

In order to get other solutions from any one pair like this, we must take its electric
image from a point whose vector is inclined at 7r/4 or 39r/4 to the line of reference.
For such points alone make the images similar. And a peculiarity now presents itself^
in that the new systems are not directly superposable : — but each is the perversion
of the other.

If we had, from the first, contemplated the question from this point of view, an
exceedingly simple pair of solutions would have been iumished at once by the obviously
orthogonal sets of logarithmic spirals

and another by their electric images taken from any point whatever. The groups
of curves thus obtained form a curious series of spirals, all but one of each series
being a continuous line of finite length whose ends circulate in opposite senses round
two poles, and having therefore one point of inflection. The excepted member of each
series is of infinite length, having an asymptote in place of the pK>int of inflection.
This is in accordance with the facts that: — a point of inflection can occur in the image
only when the circle of curvature of the object curve passes through the reflecting
centre, and that no two circles of curvature of a logarithmic spiral can meet one
another. [See No. CXVIII. below.]

We may take the electric images of these, over and over again, provided the
reflecting centre be taken always on the line joining the poles. All such images will
be cases satisfying the modified form of the problem.

If we now introduce, as a factor of the right-hand member of (1), a function
of 0 which is changed into its own reciprocal (without change of sign) when 0 increases
by 7r/2, we may obtain an infinite number of additional classes of solutions of the
original question; and from these, by taking their electric images as above, we derive
corresponding solutions of the modified form. We may thus obtain an infinite number
of classes of solutions where the equations are expressible in ordinary algebraic, not
transcendental, forms.

cxvil]

FORM A SIMILAB SYSTEM.

401

Thus we may take, as a factor in (1), taii*(0 + a). The general integral is com-
plicated, so take the very particular case of m=l, a^irj^t. This gives the curves

tan0 sec0

r^a

g^d+iMif)^

(l+tantf)«

Again, let the factor be tan (tf — a) tan (tf + a). With m = l, and tana = l/V3, we get
the remarkably simple form

But such examples may be multiplied indefinitely.

[As the last example given above, though a specially simple one, is curious from
several points of view, I append a tracing of the four curves

for the particular cases of numerical equality between a and 6. The ±a curves are
full, the others dotted.

T. II. 51

402 SYgTEMS OF PLANE CUBYEB. [CXVII.

Whatever be the values of a and b, we have at an intersection of these carves

respectively, so that their orthogonality is obvious.

Each of them consists of a single symmetrical kink, without contraiy flexure;
having its double point at the origin, where its (infinite) branches cross its axis at
angles of ± 60^

Their form is, of course, unique, the constant determining merely the scale of
each figure ; except when it changes sign, and then the figure is simply reversed. But,
even in that case, two curves of the same series cannot intersect, except of course at the
origin ; as, at either side of the origin the parts of the two lie respectively between, and
outside, the common tangents to the series. Also it is obvious that one member of each
series can be made to pass through any other assigned point in their plane, provided
it be not taken on one of the tangents at the origin. For then the substitution
of its coordinates in either equation determines the characteristic constant without
ambiguity.

When, as in the cut, a and b are numerically equal, the curves intersect one
another at their points of maximum distance firom their respective axes of symmetry,
where they are necessarily perpendicular to one another. And the common tangents to
one series intersect members of the other series in points which separate, on each curve,
the regions in which it is met by the kinked parts, fi-om those in which it is met
by the branched parts, of those of the other series. 1899.]

cxvra.] 403

CXVIII.

NOTE ON THE CIRCLES OF CURVATURE OF A PLANE

CURVE.

[Proceedings of the Edinburgh MathenuUical Society, December 13, 1895.]

When the curvature of a plane curve continuously increases or diminishes (as is
the case with a logarithmic spiral for instance) no two of the circles of curvature can
intersect one another.

This curious remark occurred to me some time ago in connection with an accidental
feature of a totally different question. (Syeteme of Plane Curves whose Orthogonais
form a similar System. Ant^, No. CXVII.)

The proof is excessively simple. For it A, B, he any two points of the evolute,
the chord AB is the distance between the centres of two of the circles, and is necessarily
less than the arc AB, the difference of their radii. (This is true even if the evolute
be sinuous, so that the original curve has ramphoid cusps.)

When the curve has points of maximum or minimum curvature, there are corre-
sponding [keratoid] cusps on the evolute ; and pairs of circles of curvature whose centres
lie on opposite sides of the cusp, (7, may intersect: — for the chord AB may now exceed
the difference between CA and CB.

51—2

404 [cxix.

CXIX.

NOTE ON CENTROBARIC SHELLS.

[Proceedings of the Royal Society of Edinburgh, February 3, 1896.]

It is singular to observe the comparative ease with which elementary propositions
in attraction can be proved by one of the obvious methods, while the proof by the
other is tedious.

Thus nothing can be simpler than Newton's proof that a uniform spherical shell
exerts no gravitating force on an internal particle. But, so far as I know, there is
no such simple proof (of a direct character) that the potential is constant throughout
the interior.

On the other hand the direct proof that a spherical shell, whose surface-density

is inversely as the cube of the distance from an internal point, is centrobaric is neither
short nor simple. (See. for instance, Thomson and Tait's Elements of Natural Philosophy,

CXIX.] NOTE ON CENTBOBAPIC SHELLS. 405

§ 491.) But we may prove at once that its potential at external points is the same
as if its mass were condensed at the internal point.

For if an elementary double cone, with its vertex at 8, cut out areas E
and E, we have

E _ E
SE^^SE*'

Let P be any external point, and take T on PS (produced) so that

PS.ST^ES.SE^b'.

Then we have obviously, from similar triangles,

SE.EP^SP.ET.

Th„« ^^"'^ __JL___-i J_ A.

TF^'SE.SE SE.EP'^b^' SP' ET'

But the sum of the values of jpp is the (constant) potential at T for unit surfieu^

density; so that the sum of the values of the first side of the equation is inversely
as SP; and the proposition is proved.

Although no mention has been made of Electric Images, in the above investigation,
it is obvious that nearly all their chief elementary properties have been proved, almost
intuitively, in the course of these three or four lines. The others are obtained at once
by applying the same method to the case in which P is inside the spherical shell, and
T outside: — ^remembering that the potential at T is now inversely as the distance of T
from the centre, 0, of the sphere ; and referring the potential of ^ to a point 8' on 08
produced till 08.08' is the square of the radius of the shell

[This investigation has been at once further simplified and extended, in § 52 of
my little book Newton's Laws of Motion, 1899.]

406 [cxr

cxx.

ON THE LINEAR AND VECTOR FUNCTION.
[Proceedings of the Royal Society of Edinburgh, May 18 and June 1, 1896.]

In the following Abstract I refer to such Linear and Vector Functions, only, as
correspond to homogeneous strains which a piece of actual matter can undergo. There
is no diflScuIty: — though caution is often called for: — in extending the propositions to
cases which are not realizable in physics*.

The inquiry arose from a desire to ascertain the exact nature of the strain when,
though it is not pure, the roots of its cubic are all real: — i.e., when three lines of
particles, not originally at right angles to one another, are left by it unchanged in
direction.

1. The sum, and the product (or the quotient), of two linear and vector functions
are also linear and vector functions. But, while the sum is always self-conjugate if
the separate functions are so (or if they be conjugate to one another), the product
(or quotient) is in general not self-conjugate: — though the determining cubic has, in
this case, real roots. The proof can be given in many simple forms.

If fir and o) represent any two pure strains, there are three real values of g, each
with its corresponding value of p, such that

vTp = ga>p (1).

* [Thus the transformations, given below, are presumed to involve real quantities only. Dr Muir, in
making some valuable comments on one of the results (Phil, Mag. 1897, i. 220), appears to have overlooked
this important preliminary condition. 1899.]

CXX.] ON THE LINEAR AND VECTOR FUNCTION. 407

Assume (o^p s a ; and the equation becomes

But to'^mtur^ is obviously self-conjugate. Hence the three values of ^ are real, and
the vectors a form a rectangular system. Thus (1) is satisfied by three expressions
of the form

p « «*~o- = ^tDr~*o- (2);

f.e., there is one rectangular set of vectors which have their directions altered in the
same way by the square roots of the inverses of each of the given strains.

But (1) may be written in the form

where tiT^m is in general not a self-conjugate function. Thus

Two pure strains in succession give a strain which is generally rotational, but whose
cubic has three real roots.

Conversely, when a strain is such as to leave unchanged three directions in a body,
it may be regarded as the resultant of two successive pure strains.

These are to be found from (2), in which the values of g and p are now regarded as
given, so that the problem is reduced to finding eo (a pure strain), and the (rectangular)
values of a from three equations of the form

to\p as a.

When ci) is thus found, the value of « is given by (1). The solution is easily seen
to express the fact that <o and «, alike, convert the system />i, p%i pi iuto vectors
parallel to Vp^pt, Vp,pi, Vpip^, respectively.

2. Other modes of solution of (1) are detailed, of which we need here mention
only that which depends upon the formation of the cubic in

the calculation of the coefficients in Mg^ and the comparison of these forms with their
equals found from

a process which gives interesting quaternion transformations.

3. Some curious consequences can be deduced from these formulae, which have
useful bearing upon the usual matrix mode of treating the problem algebraically.

408 ON THE LINEAR AND VECTOR FUNCTION.

For, if we take

«r = ( -4 c b ) and «=»( |) 0 0 )

[cxx.

c
b

B a

a C

0 q 0
0 0 r

which involve complete generality since %, j, k are undefined, we have for the cubic
(1) in g

A'-pg c b

c -B — g^ a

b a C ^rg

The transformation of (1) given above is equivalent to dividing the successive rows,
and also the columns, of this determinant by \/p. Vji V^ respectively. It thus becomes

A/p-g cjjpq b/Jpr =0,
c/Jpq B/q-g a/Jqr
bjjpr ajjqr C/r — g

from the form of which the reality of the roots is obvious.

A somewhat similar process* shows that the roots of

A'-x b c =0

d E-x f
g h / — a?

are always all real, provided the single condition,

cdh=^bfg,
be satisfied.

* [Multiply the rows, and divide the oolumns, respeotively, by p, 9, r. It becomes

A-x bpfq cpjr
dqlp E-x fqjr
grip hrjq I-«

so that, to make it azi-symmetrical, we mast have

{plqy^djb,

(qlrY^hlf,

(rlpf^clg.
Thos finally it becomes

A-x ^bd tjcg
Jdb E-x Jfh
tjgc Jhf I-x

if the condition in the text above is satisfied. 1896.]

CXX.] ON THE LINEAR AND VECTOR FUNCTION. 409

It is easy to see that this statement may be put in the form : — ^The roots of Jlf^= 0
are real, provided a rectangular system can be found such that

S%(f3J8j<f>kSk4>% = 8k4:j8jifnSi<f>k

The quaternion form, of which this is an exceedingly particular case, expresses
simply that the roots of the cubic in ^ are all real, if a self-conjugate function eo
can be found, such that o)^ is self-conjugate. This is merely another way of stating
the chief result of § 1 above. But it may be interesting to illustrate it from this
point of view. We may write, in consequence of what has just been said,

Spifhp» . <f>p =giVpipi8pip +gtVp^pi8pip + gtypipt8p^py

and <oa = PiP\8pi<r 4- PtPt8pta' + p»p»8pta.

These give at once

w<f>p ==^pigipi8pip +Pigip28pip + ptg»p98ptp,
which is obviously self-conjugate.

4. The results above have immediate application to fluid motion. For, when there
is a velocity-potential, the motion is ''differentially irrotational": — i.e., the instantaneous
change of form of any fluid element is a pure strain; a particular cubical element
at each point becoming brick-shaped without change of direction of its edges. But
if we think of the result of two successive instantaneous changes of this character,
we see that there is in general at every point a definite elementary parallelepiped,
the lengths, only, of whose edges are changed by this complex strain. In special cases,
only, is a similar result produced by three successive pure strains.

[The remainder of this Abstract referred to the genesis and history of No. CXV.
above.]

T. II. 52

410 [CXXL

CXXI.

ON THE LINEAR AND VECTOR FUNCTION.

[Proceedings of the Royal Society of Edinburgh, March 1, 1897.]

In a paper read to the Society in May last, I treated specially the case in which
the Hamiltonian cubic has all its roots real. In that paper I employed little beyond
the well-knowD methods of Hamilton, but some of the results obtained seemed to
indicate a novel and useful classification of the various forms of the Linear and Vector
Function. This is the main object of the present communication.

1. It is known that we may always write

</)/o = 2 (oiSaip)

and that three terms of the sum on the right are sufficient, and in general more than is
required, to express any linear and vector function. In fact, all necessary ^nerality is
secured by fixing, once for all, the values of a, y8, 7, or of ai, A, 71, leavings the others
arbitrary : — subject only to the condition that neither set is coplanar. Thus as a
particular case we may write either

^p = SaSip,

or <f>p = XiSaip.

In either case we secure the nine independent scalar coefficients which are required
for the expression of the most general homogeneous strain. But forms like these are
relics of the early stage of quaternion development, and (as Hamilton expressly ur^ed)
they ought to be dispensed with as soon as possible.

2. A linear and vector function is completely determined if we know its effects on
each of any system of three non-coplanar unit-vectors, say a, /8, 7. If its cubic have

CXXI.] ON THE LINEAR AND VECTOR FUNCTION. 411

three real roots, these vectors may, if we choose, be taken as the directions which it
leaves unaltered; if but one, we may take a corresponding system in the form

a, 13 coa a ±iy sin a,

where i is >/- 1. But it is preferable to keep the simpler form a, 0, y, with the under-
standing that 13 and 7 may be bi- vectors, of the form just written.

3. In terms of the three roots thus designed, we may form, with the help of
three arbitrary scalars (two of them bi-scalars of the form y ± iZt if necessary), three
very simple but distinct varieties of linear and vector function : — viz.

(a) Strains leaving three directions, a, 13, y or Vfiy, F7a, Fa/8, unaltered, so that
their reciprocals have the same form.

Sa0y . </)p = xaSffyp + yfiSyap 4- zySafip,

with Sa^y . <^p = xVfiySap ^-yVyaSfip + zVa/3Syp.

In this case, if x, y, z are the same in each, ^ is the conjugate of <f>.

(When x = y = z, these strains leave the form and position of a body unaltered ;
but each linear dimension is increased x fold.)

(6) Pure strains: —

wp = xaSap 4- yfiSfip + zySyp,

with tDTi p = a? VfiySfiyp -f y VyaSyap + z VafiSafip,

The second of these changes the system a, /8, 7, into V0y, F7a, Va/3] while the
first effects the reverse operation.

father form of (a) repeated (with altered scalar constants), simply perpetuates the
form. In </>^ and <^</> we have new forms, which are pure when x : y : z are the
same in each of the factors.

The two forms (6), in succession, give one or other of the forms (a); and, con-
versely, either form of (a) may be regarded as the resultant of the two forms (6)
taken in the proper order. This is the main result of my former paper: — for it is
obvious that, having between them twelve disposable constants, tor and ^sri may be made
to represent any two pure strains.

But, while <f>w and «r<^ merely repeat the type w; and tDTj^, and ^tDTj the type
Wi] we have novel forms in the combinations

'BT<f>, <t>i'ST, if>vTi, and Vi<l)i.

Many of these are useful in the solution of equations among forms; such as, for
instance,

52 — 2

412 ON THE LINEAR AND VEC5T0R FUNCTION. [CXXI.

where ;^ is to be found wheu -^ is given. One simple result of the above discussion,
which is often of great use in such matters, is the obvious condition that two such
forms shall be commutative in their successive application.

4. When two roots are imaginary, all the forms above are still real; since, when
/8 and 7 take the forms 13 ±iy, y and z must be written y ± iz. In the forms (6),
the imaginary terms cancel one another; in (a) the real terms do so, and the whole
is divisible by t.

5. Of course, with a, /8, 7 (as in 2, above) and three scalar constants, we can
produce any form of linear and vector function. And the paper concludes with forms
in which these constants are merged in a new arbitrary vector.

cxxii.] 413

CXXII.

NOTE ON THE SOLUTION OF EQUATIONS IN LINEAR AND

VECTOR FUNCTIONS.

[Proceedings of the Royal Society of Edinburgh, June 7, 1897.]

In a paper read to the Society on March 1 {anUy No. CXXI.) I spoke of the
application of some of its results to the solution of equations involving an unknown
Linear and Vector Function. These results depended chiefly upon the expression of the
function in terms of its roots, scalar and directional; and I now give a few instances
of their utility, keeping in view rather variety of treatment than complexity of subject.
The matter admits of practically infinite development, even when we keep to very
simple forms of equation, and is thus specially qualified to show the richness in re-
sources which is so characteristic of quaternions. But it will be seen also to be strongly
suggestive of the extreme caution required even in the most elementary parts of this
field of inquiry.

In what follows, I employ x ^ denote the unknown function; <f>, yjr, etc., known
functions, w is specially reserved for a self-conjugate function, and eo for a pure
rotation.

1. Given </>X = X^ Wy

i.e., to find the condition that two functions shall be commutative in their successive
application. Let a be a root of <f>, real or imaginary, so that

<f>a=ga.
We have at once, by applying the members of the proposed equation to a,

414 SOLUTION OF EQUATIONS IN LINEAR AND VECTOR FUNCTIONS. [CXXIL

Thus, except in the case of equal roots of </>,

so that the required condition is merely that ;^ has the same directional roots as f
When two values of g are equal, two of the directional roots of x *"^ limited only to
lie in a definite plane: — when all three are equal, ^ becomes a mere mag^nificatioD, and
X is, of course, wholly undetermined.

[When the roots of <f> are all real, we have

/Sa/87 . XP = hioSfiyp + hi/38yap + h^ySafip.

When two are imaginary we may preserve this form; or, if we wish to express it in
terms of real quantities only, we may write it as

SalSy . XP = Ih OiSfiyp + iK0 - h,y) Syap + (h^y + A,/9) Sa/3p,

where the meanings of A,, A,, 0, 7, are entirely changed.

It is well to notice that the squares of these functions preserve the form, so that
in the first

Sal^y ' XV = hi*aS/3yp + K^fiSyap + h^^ySyap ;

and in the second we have the value

h,'aSl3yp + {(V - A3*) 13 - 2h^h,y} Syap + {(V - V) 7 + 2A,A,y8} Sa/3p.

Thus the square roots of such expressions may be obtained by inspection.]

2. Had the known factors been difierent in the two members, %,e., had the
equation been

<^ = X^ (!'),

the same process would still have been applicable, though the result would have been
very different. For a being a root of '^, we have

as before. But we can no longer conclude firom this anything further than that the
scalar roots of '^ must be the same as those of </>, in order that the given equation maj
not be self-contradictory. Thus, if ^fr have three real roots, so must ^, and oonveiseljr.
If this necessary condition be fulfilled, x ^^ ^^7 function which changes the directiawil
roots of yit into those of <f>. Its own scalar roots remain indefinite.

3. Let the equation be

<^' = xf

.(2>

The members, besides being equal, are conjugates ; so that they represent any pure strain
whatever.

Thus X = </>""^^» ^^d X ^ ^^'""^ which are of course consistent with one another.
Remark that, as a particular case, w may be a mere number. If «• be taken = ^', we
have the obvious solution x ~ ^-

CXXII.] SOLUTION OF EQUATIONS IN LINEAR AND VECTOR FUNCTIONS. 415

4. If we alter the order of the factors on one side of (2) we have an altogether
new form: —

^' = 0'X (3).

Since <f> is given, this may be written

where y^ is known. An immediate transformation by taking the conjugate gives

a type which is obviously a particular case of (1^); and, besides, will be treated later,
with the sole difference that x ^^^ ^^^^ ^^ ^^^ given function, and ^jr that to be found.
But when a solution has thus been obtained, it must be tested in the original equation;
for selective eliminations, such as that just given, often introduce irrelevant solutions.
(See § 8, below.)

5. A curious modification of (3) is produced by making in it ^ and x identical,
so that it becomes

xx!^xx (*)•

Though no longer linear, this equation is in some respects analogous to (1). It thus
imposes the condition that x ^^^ ^^^ conjugate shall have the same directional roots. If
all three be real they must therefore form a rectangular system. If two be imaginary,
the vectors of their real and imaginary parts form a rectangular system with the third.
Thus X ™*y ^^ *^y P^r^ strain, or a rotation associated with a pure strain symmetrical
about the axis of the rotation.

A simpler mode of dealing with (4) is suggested by the last remark. For we may

always assume

X = v(a,

and (4) becomes vrtaa^^w = «•' = cii~*tDr'a),

from which (coupled with the results of (1)) the above conclusions are obvious.

6. The form X^x' = <^' (5)

also admits of simple treatment. Its conjugate is

x<i>Y = f .

Now we can always write ^ = w + Fe,

with ^' = BT — Ve,

and the equations above become, by addition and subtraction,

Put the first of these in the form

where eoi and o), are, so far, arbitrary. As each side is the product of a strain and

416 SOLUTION OF EQUATIONS IN LINEAR AND VECTOR FUNCTIONS. [CXXII.

its conjugate (because the conjugate of a pure rotation is its reciprocal), we may at
once write

or X ~ «r*c«>«r~*,

where a) = a),a)i~^ is still arbitrary. To determine it, the second equation above, viz.

gives me = X€>

where m is the product of the numerical (scalar) roots of x 5 obviously unit in this case,
as there is no change of volume. This gives

so that the axis of a> is cr^e, but the angle of rotation remains undetermined.

The direct algebraic verification of this solution is troublesome, unless we refer the
strain to the axes of its pure part «r, when it becomes feirly simple. For ^ can
then be written as

( ^« -1/ ,1 )

V jB» -X

whence it is easy to see that

^ {- n/+ (1 - e) Im] ^ {m/+ (1 - e) In] )

^{n/+(l-6)Zm}

e + (1 - c)m» pi" If-t (1 - e) mn]

_ {- ^/+ (1 - e) In] -g {lf+ (1 - e) mn]

e + (l'-e)n*

where
and

I = AX/jA^X" + &fi^ + UV, etc.,

7. A similar mode of treatment can, of course, be applied to the more general form

X<^ = t (6).

After what has just been said, it is easy to see that if '^ = 'Bri+ Vci, we shall have

X = tJi^onsr-*,
with the condition for « (and for the possibility of a solution)

where m is the product of the numerical roots of x-

[In connection with the results above it may be interesting to find the relations

CXXII.] SOLUTION OF EQUATIONS IN LINEAR AND VECTOR FUNCTIONS. 417

among the various constituents of the two different modes of breaking up a linear
vector function into pure and rotational parts : — i.e.,

^ = «r + Fe = tTiO).

(See No. XXI. above, for another solution.)

The general form of a pure rotation is

o) = a^/» ( ) a-^/» = cos u4 + sin -4 7 . a - (1 - cos -4) aS . a,

where a is the unit vector axis and A the angle of rotation.

Thus, writing for shortness c^cosA and 8 = sin A,

vrp + Vep = ctJip + OTJi Vap — (1 — c) VioSap,

vrp — Vep = cvTip — sVarsFip — (1 — c) aSavr^p,

so that 2 Vep = 5 (tsri Vap + Vavip) + (1 - c) F. ( Va^i a) p.

Now Hamilton (in giving his cubic) showed that

{ni2 — wi) Vap = V'GTiap + Vavipy

so we have 2 Fe/5 = « (7^2 Fa/3 — Vv^ap) + (1 — c) F. ( Fa^rja) p ;

and, as this is true for all values of p,

26 «= « (matt — -BTia) + (1 —c) Fa^TiCr,

the second tenn disappearing when the rotation is about one of the axes of the pure
part of the strain. Again

2vrp = 2c'GTip + s ('SFiVap — FatTip) — (1 — c) {'SFiaSotp + atSa«r,p}

is obviously self-conjugate.]

8. An instantaneous, and (at first sight) apparently quite different, solution of
(5) is obtained by multiplying each side into the reciprocal of its conjugate. For we
thus have a case of (1) in the form

But this equation, which would assign to ^ any value commutative with <f><l>'^\ is
very much more general than (5) from which it is derived. [This is an excellent
example of the necessity for caution already pointed out.]

To analyse this solution, with the view of restricting it, note that by Hamilton's
method we have at once

in (^'~* - <^~M = 2 F. «r€ = 2eV. «r*a, suppose,

where m is the product of the scalar roots of ^; a a unit vector, and e a scalar
constant, both definite.

T. II. 53

418 SOLUTION OF EQUATIONS IN LINEAR AND VECTOR FUNCTIONa [CXXII.

Thus >f»f>'-'p = p + --<!> Vv^ap

\ m 1^ m ^ m ^

where rr^ is the product of the scalar roots of «r*, and therefore

m = m — Sewe = m + 6*.
[The former solution, giving

= p cos A + sin il«r*yiatT~*p — (1 — cos -4) w^OiSatT^^p,

contains this as a particular case, for it is easy to see that the two expressions agree
if we are entitled to assume simultaneously

. - 2e= . . 2em* , . 2e»

cos-d = l , sm-d= , 1— 008-4 = — .

mm m

The first and last are identical ; and the first and second require merely that
we shall have

\ m) m*
which is satisfied in consequence of the expression for m above.]

That the complete admissible value of ;f is what we have already found, and
contains only the one scalar indeterminate A, is easily verified by expressing ^ as a
linear combination of the operators 1, «r*Fa«j~*, -BrioSa^r"*, which are suggested by its
relation to ^^'~\ and are obviously commutative with one another; and independent,
in the sense of not producing any new operator by their combinations. Then the
required relations among the coefiicients are determined by comparing term by term
the expressions for ^ and x"^^.

9. Finally, we may treat (5) by a method similar to that adopted for (1). Let
a now be a directional root of x'> so that xd — gou Then we have

X<^a = -<^a.

But the cubics of x ^^^ X *^^ necessarily identical, and thus their common

numerical roots can be no others than 1, g, \jg. Also, since ^ is assumed to be real,

g is imaginary, for ^ changes the g directional root of x' ^^ ^^ ^l9 root of x> *^d
conversely.

CXXII.] SOLUTION OF EQUATIONS IN LINEAR AND VECTOR FUNCTIONS. 419

But, if we operate by the conjugate of (5) upon a, we get

Thus the directional roots of x *^ treated alike by ^' and by ^, and must there-
fore belong to ^~^^'. So those of ^ belong to ^<^'""^ Thus we are again conducted
to the previous result; but this third method gives us great additional information
as to the intrinsic nature of the strains involved, and the relations which exist among
them.

10. It is, of course, only in special cases that simple methods like these can
be applied to linear vector-function equations of a little greater complexity. But
when they are applicable they often give singularly elegant solutions. As an instance
take the equation

<^iX + X<^ = ^ (7),

or, as it may obviously be written,

X-'*i+<NX"' = X"'^^*-
Let a be a directional root of ^, then at once

<^iX« + fl'X« = '^'>
or X« = (<^ + 5')"''^«-

If the roots of ^ be unequal, the three equations of this form completely deter-
mine X'

11. Again, let <^iX + X^ = ^^^X^* + '^ (8)-

If 9\i ^i» etc., are roots of ^, this gives three equations of the form

(^ + fl'i) X«i = <^X (<^4«i) + '^"i-
If the values of a be unequal, we can of course find the coefficients in

<^4aj = aaai + 6jaa+ ...

Then, putting \i for x«i» etc., we have finally

The three equations of this form give Xi, etc., that is, x^\y etc., and thus x i® found in
terms of its effects on three known vectors.

12. The most general linear equation in x ^^^ x' '^*y ^® written as

53—2

420 HOLUTIOX OF EQUATIONS IN LIKEAB ASD VBCTOB FUXCTIOSBL [cXXIL

Take; «, ^, 7, three non-copUnar veotors, and let

^ifi^pa -I- ?')S + ^7 r etc
^,7 = p"« + /V3 + r7j

^'a = «c + ^)3 + U7 1
^'P^^a +f/3 + ...' etc.
^V =«"'« + j

Apply the memberM of the given equation to a, )3, 7 separately; and operate on
each of the renultN with H.a, 8.fi, S,y, We obtain nine scalar equations in x« = X,
X^^fi, X7"^' '^^ which two are

2 \Sfiif>(pX + y^ + ry) + ;5f^,a («'X + f/i + up)] = S^Sfd.
TheHc are nea^ssary, and ffufficient, to determine \, ^, 1/; and thence x*

cxxiil] 421

CXXIII.

ON THE DIRECTIONS WHICH ARE MOST ALTERED BY A

HOMOGENEOUS STRAIN.

[Proceedings of the Royal Society of Edinburgh, December 7, 1897.]

The cosine of the angle through which a unit vector p is turned by the homo-
geneous strain ^ is

Tp.T<t>p-
This is to be a maximum, with the sole condition

Diflferentiating, &c., as usual we have

a?p = — 2^pSp<f>'<l>p + <f><f>pSp<f>py
where 2^ = ^ + <l>.

Operate hy S . p and we have

— a; = — Sp<f>pSp<l>'<f>p ;

^'''^' f' — 's%p'-8%'

Hence the required vector, and its positions after the strains ^ and ^'^, lie in one
plane ; and the tangent of the angle between p and $p is half of the tangent of the
angle between p and <f>'<f>p^ [In the original, <f> was (by an oversight) written for ^, so
that the last statement has been modified. 1899.]

When the strain is pure, the required values of p are easily found. Let the chief

422 DIRECTIONS MOST ALTERED BY A HOMOGENEOUS STRAIN. [CXXIU.

unit vectors of (f> be a, )8, 7, and its scalars gi, g^, g^. Then the equation above gives
at once three of the form

^■{'*it- £-)-'■

Sp<l>p 8p(f>*pJ
There are two kinds of solutions of these equations.
First, Let the first factor vanish in two of them, e.g.^

fif)8p = 0, Syp = 0, or /3 = a.
Then the remaining equation is satisfied identically, because its second factor becomes

whence u* = 1.

Thus, as we might have seen at once, the lines of zero alteration (minima) are the
axes of the strain.

Second, Let the second factor vanish in two of the equations, e.g.,

Sp<f>p Sp<i>^p ' Sp4>p 8p4>^p

These give at once Sp<f>p = "7 -, Sp<^^p = — g^gt;

ffi'rgs

80 that ^'=7-V"^-

In this case it is evident that we have also

Sap = 0.

[In fact, neither the first factors, nor the second factors, in the three equations, caD
simultaneously vanish: — except in the special case when two of giy g^, g^ are equal.]

Of the three values of u^ just found, the least, which depends upon the greatest and
least of the three values of g, gives the single vector of maximum displacement: —
the other two are minimaxes, corresponding to cols where a contour line intersects
itself.

(Read February 21, 1898.)

The self-intersecting contour-lines, corresponding to 3, 2, 1 as the values of the

g's, were exhibited on a globe; whose surface was thus divided into regions in each

8
of which the amount of displacement lies between definite limits. The contour w* = 5

encloses the regions in which the maximum iu^ = -\ is contained: — and (where its separate
areas are superposed) one of the minima. This minimum is surrounded by a detached

FlateVni

FiiZ

Fis^

F,f 3

FifS

CXXIII.] DIRECTIONS MOST ALTERED BY A HOMOGENEOUS STRAIN. 423

24
part of w* = 5K> while the rest surrounds the other two minima (u*=l); and the double

points of these contours are the minimaxea

A general idea of their forms may be gathered from their orthogonal projections on
the principal planes, as shown in Figs. 1, 2, 3 of Plate VIII. These projections are

curves of the 4th order: — but w^=q (dashed) splits into two equal ellipses on the xy

24
plane, and hyperbolas on that of xz\ while i*' = s^ (dotted) gives ellipses on yz and

hyperbolas on xz. Fig. 4 gives, on a fourfold scale, the region near the z pole of the
projection on yz^ of which the details cannot be shown on the smaller figure.

The curves were traced from their equations. One example must suffice. Thus

** 9 "9aj« + 4y» + 2^
gives, eliminating z by the condition a;" + y' + ^'=l,

(2^+y«+l)» = |(8^+3y« + l),

2(.:±|)Vy« = |.

The forms of these curves depend only on the ratios gi : g^ : g^, so that I have
appended Fig. 5, in which we have 5:4:3, for comparison with Fig. 3 where we
have 3:2:1.

424 [cxxiv.

CXXIV.

ON THE LINEAR AND VECTOR FUNCTION.

[Proceedings of the Royal Society of Edinburgh, May 1, 1899.]

Three years ago I called the attention of the Society to the following theorem: —

The reaxdtant of two pure strains is a homogeneous strain which leaves three directions
unchanged; and conversely.

[It will be shown below that any strain which has three real roots can also be
looked on (in an infinite number of ways) as the resultant of two others which have
the same property.]

As I was anxious to introduce this proposition in my advanced class, where I
was not justified in employing the extremely simple quaternion proof, I gave a number
of diflferent modes of demonstration ; of which the most elementary was geometrical, and
was based upon the almost obvious fact that

If there be two concentric ellipsoids, determinate in form and position, one of which
remains of constant magnitude, while the other may swell or contract wiHiout limit; there
are three stages at which they touch one another.

[These are, of coui-se, (1) and (2), when one is just wholly inside or just wholly
outside the other (that is when their closed curves of intersection shrink into points),
and (3) when their curves of intersection intersect one another. The whole matter
may obviously be simplified by first inflicting a pure strain on the two ellipsoids, such
as to make one of them into a sphere, next considering their conditions of touching, and
finally inflicting the reciprocal strain.]

But the normal at any point of an ellipsoid is the direction into which the radius-
viicUfT of that point is turned by a pure strain; so that for any two pure strains

CXXIV.] ON THE LINEAR AND VECTOR FUNCTION. 425

there are three directions which they alter alike, (These form, of course, the system
of conjugate diameters common to the two ellipsoids.) This is the fundamental pro-
position of the paper referred to, and the theorem follows from it directly.

In the course of some recent investigations I noticed that if ^ have real roots, so
also has

whatever real strain yft may be. This is, of course, obvious, for they are ^Ira, yfrfi, ^7, if
a, )8, 7 be the roots of ^. At first sight this appeared to me to be a generalisation
of the theorem above, of a nature inconsistent with some of the steps of the proof.
But it is easy to see that it is not so. For all expressions of the form

correspond to pure strains if od is pure. Hence

and is thus, as required by the theorem, the product of two pure strains.

Of course we might have decomposed it into other pairs of factors, thus

In the former case the factors have each three real roots, in the latter they have not
generally more than one.

A great number of curious developments at once suggest themselves, of which I
mention one or two.

Thus, let there be three successive pure strains (which may obviously represent
any strain). We may alter them individually, as below, in an infinite number of ways
without altering the whole.

= a)i~^ft>~^a)i"^ . t0i(O(Oia)O)i . cu, = etc.

The expression ©tsr itself, when its three roots are given, t.e., a, ff, 7 with
9iy gty g^y givcs (o and w separately, with three scalars left arbitrary. For we may take

wp = XiaSap + W2/3S/3p + ...,
'^P = }/if^^yS^yp + y2VyoLSyap + ...,
and then obviously there are three conditions only, viz.

^lyi a?ayj ^8^3
T. II. 54

426 ON THE LINEAR AND VECTOR FUNCTION. [CXXIV.

Another portion of the paper deals with a sort of converse of the above problem : —

The relation between two strains (whether with three real roots or with one) when

their successive application gives a pure vStrain; and various questions of a similar
kind.

In these inquiries we constantly meet with a somewhat puzzling form, which
repeats itself in a remarkable manner under the usual modes of treatment, viz.: —

€oV€p+ Veeop.
A little consideration, however, shows that it can be put into the form

F(m2€ - me) p,
which is thoroughly tractable.

cxxv.] 427

cxxv.

NOTE ON CLERK-MAXWELUS LAW OF DISTRIBUTION OF
VELOCITY IN A GROUP OF EQUAL COLLIDING SPHERES.

[Proceedings of the Royal Society of Edinburgh, June 15, 1896.]

The sarcastic criticism which M. Bcrtrand {Comptes Rendus, May 4 and 18, 1896)
again bestows on Clerk-Maxwell's earliest solution of the fundamental problem in the
Kinetic Theory of Oases, together with Prof. Boltzmann's very different, but thoroughly
depreciatory, remarks (ib,, May 26), have led me to reconsider this question, already
discussed by me at some length before the Society. Both of these authorities declare
Maxwell's investigation to be erroneous: — but, while Prof. Boltzmann allows his result to
be correct, M. Bertrand goes further, and bluntly calls it absurd. He had, in his Calcul
des ProbabiliUs (1888), already given Maxwell's proof as an example of illusory methods.
I have the misfortune to agree with Maxwell, and to hold that his reasoning, though
not by any means complete, is (like his result) correct. (Trans, R,S,E., vol. xxxiii.
pp. 66 and 252.)

I have not found anjrthing in these communications of mine (so far at least as
the present question is concerned) which I should desire to retract; but they can
be considerably improved; and I think that, by the introduction of the Doppler-
(properly the Rorner-) principle, the true nature of a part of the argument can be
made somewhat more immediately obvious. Also I will venture to express the hope
that Prof. Boltzmann may at last recognise that I have, in this matter at least, not
deserved the reproach of having reasoned in a circle*.

1. The following quotation from my first paperf (in which I have italicized the
greater part of one sentence) shows the general ground of my reasoning, which was
expressly limited to a very numerous group of equal, perfectly hard, spherical particles.

♦ Phil. Mag., xxv. (1888), pp. 89, 177. t [AnU, No. lxxvii. pp. 126, 129. 1899.]

54—2

428 ON clerk-maxwell's law of distribution of velocity. [cxxv.

"Very slight consideration is required to convince us that, unless we suppose the
spheres to collide with one another, it would be impossible to apply any species of
finite reasoning to the ascertaining of their distribution at each instant, or the distri-
bution of velocity among those of them which are for the time in any particular
region of the containing vessel. But, when the idea of mutual collisions is introduced,
we have at once, in place of the hopelessly complex question of the behaviour of innu-
merable absolutely isolated individuals, the comparatively simple statistical question of
the average behaviour of the various groups of a community. This distinction is forcibly
impressed, even on the non-mathematical, by the extraordinary steadiness with which
the numbers of such totally unpredictable, though not uncommon, phenomena as suicides,
twin or triple births, dead letters, &c., in any populous country, are maintained year
after year.

"On those who are acquainted with the higher developments of the mathematical
Theory of Probabilities the impression is still more forcible. Every one, therefore, who
considers the subject from either of these points of view, must come to the conclusion
that continued collisions among our set of elastic spheres will, provided they are all
equal, produce a state of things in which the percentage of the whole which have,
at each moment, any distinctive property must (after many collisions) tend towards a
definite numerical value; from which it will never afterwards markedly depart."

**When [the final result, in which the distribution of velocity-components is the
same for all directions] is arrived at, collisions will not, in the long run, tend to
alter it. For then the uniformity of distribution of the spheres in space, and the
symmetry of distribution of velocity among them, enable us (by the principle of averages)
to dispense with the only limitation above imposed; viz., the parallelism of the lines of
centres in the collisions considered."

2. Now, considering the 3.10" absolutely equal particles in each cubic inch of a
gas, where could we hope to find a more perfect example of such a community?
Where a more apt subject for the application of the higher parts of the Theory of
Probabilities 1 If we are ever to find an approach to statistical regularity, it is surely
here, where all the most exacting demands of the mathematician are fully conceded.

Is it not obvious, at once, that such a group must present at all Hmes, and from
ail sides, precisely the same features? In other words: — that the solution of the problem
is UNIQUE. (This word practically contains the whole point of the question.) If
not, the higher part of the Theory of Probabilities (in which M. Bertrand himself is
one of the prominent authorities) is a mere useless outcome of analytical dexterity;
and even common-sense, with consistent experience to guide it, is of no value
whatever.

A first consequence of this perfect community of interests is that (on the average,
of course) the fraction of the whole particles, whose component speeds in any assigned
direction lie between x and x+ Bx is expressed by

f(x) Sx

where / is a perfectly definite (and obviously even) function.

cxxv.] ON clerk-maxwell's law of distmbution of velocity. 429

It is dear from this that the density of ends in the velocity space-diagram
depends on r only; but we require further information before we c€m find how,
(M. Bertrand seems to admit the first statement; but he insists that, otherwise, the
solution is wholly arbitrary,)

3. [But, before seeking this, we may take another mode of viewing the situa-
tion:— as follows. It is, of course, nothing more than an illustration of the argument
just given.

Suppose, merely for the purpose of examining the condition of the gas, and there-
fore without any inquiry into other physical possibilities, which have nothing to do with
the argument: —

That (a) each particle of the group is self-luminous, and all give out, with equal
intensity, light of one definite period. (To illustrate the remark just made, note that
this luminosity is not attributed to collisions, nor to any assigned physical causes.)

(6) The wave-length of light reaching the eye from a moving source is altered
by an amount proportional to the speed with which its distance from the eye alters.

(d) An ideal grating may be assumed, of any requisite regularity and fineness;
and, again for the sake of argument only, it may be supposed to act, however fine
it be, in the same manner as do ordinary gratings.

These premised, the spectrum of the gas will be a band, whose visible breadth
depends only on the fineness of the grating and the luminosity of a particle. But
this band will present, at all times and from all sides of the group, exactly the same
appearance.

Its brightness, therefore, at any given distance from its central line, will be constant.
But this means that the fraction of the whole number of particles which have any given
speed in the line of sight, depends on that speed alone. The utmost speed of a gaseous
particle is exceedingly small compared with that of light, and the alteration of wave-
length is not affected by the part of the motion of the luminous particle which is trans-
verse to the line of sight.]

4. We have not yet exhausted the consequences of absolutely perfect (average)
community. For every particle, in virtue of citizenship, has a right to, and obtains, its
due quota of whatever is shared among the group. Its tenure of any one value of
X ceases (usually in a most abrupt way) some 10" times per second, but leaves it
absolutely free to have, during each of these brief periods, any values of y and z
which may fall to it. There are, in fact, definite specifications of x, y, z speeds ;
but they are distributed among the particles with absolute independence of one
another, in a manner which is perpetually changing at an exceptionally rapid rate.
And the entire independence of x, y, and z speeds is shown by the fact that, in a

CXXV.] ON CLERK-MAXWELL*S LAW OF DISTRIBUTION OF VELOCITY. 431

6. What Maxwell did propose, and solve, was a very different problem indeed.
Here are his words (Phil. Mag. XIX. (1860), p. 22):—

"Prop. IV. To find the average number of particles whose velocities lie between
given limits, after a great number of collisions among a great number of equal
particles.*'

He had already pointed out that the particles are regarded as spherical and
perfectly elastic; and that, though collisions are perpetually altering the velocity of
each, the tendency is to some regular law of distribution of vis viva among the group.
I am far from asserting that his paper (which, epoch-making as it was, is evidently a
somewhat hasty and unmatured effort) is free from even large errors: but it certainly
does not contain such palpable absurdities as those now laid to its charge.

M. Bertrand entirely ignores the fact that Maxwell was dealing with a " community."
And his comment on Maxwell might justly be retorted on himself in a slightly altered
form. For he asserts that the a?, y, z speeds are not independent, which is virtually
the equivalent of the statement that when the latitude of a ship at sea has been any-
how determined, its longitude is no longer wholly indeterminate!

[July 6, 1896. Prof. Boltzmann, to whom I sent a proof of the above, requests me
to add, on his part, as follows: —

"I have given expression to my high respect for Maxwell in the Prefaces to the
two Parts of my Lectures on Ma^xwelVs Theory of Electricity and LigJU, and specially in
the Motto to Part II. And, besides, I regard Maxwell's discovery of the Law of Distri-
bution of Velocity as so important a service that, in comparison, the trifling mistakes
which appear to me to occur in his first proof are not worthy of consideration. The letters
which I wrote to M. Bertrand, who was good enough to communicate them to the French
Academy, had thus by no means the object of expressing my concurrence in M. Bertrand's
dissentient (abfdllig) judgment of Maxwell's work on the Velocity-distribution-law. I
wished rather to say that M. Bertrand was so much the less justified in this opinion
because the one objection he was able to make had already been made by others, who
agree in all essentials with Maxwell."]

432 [cxxvi.

CXXVI.

ON THE GENERALIZATION OF JOSEPHUS' PROBLEM.

[Proceedings of the Royal Society of Edinburgh, July 18, 1898.]

In the third Book of The Wars of the Jews, Chap. viii. § 7, we are told that
Josephus managed to save himself and a companion out of a total of 41 men, the
majority of whom had resolved on self-extermination (to avoid falling into the hands
of Vespasian) provided their leader died with them. The passage is very obscure,
and in a sense self-contradictory, but it obviously suggests deliberate fraud of some
kind on Josephus' part.

"And now," said he, "since it is resolved among you that you will die, come on,
let us commit our mutual deaths to determination by lot. He whom the lot falls
to first, let him be killed by him that hath the second lot, and thus fortune shall
make its progress through us all; nor shall any of us perish by his own right hand,
for it would be unfair if, when the rest are gone, somebody should repent and save him-
self." Whiston, Works of Flavins Josephus, iv. 39.

Bachet, in No. xxiii. of his Prohlhmes plaisants et diUctahles, makes a definite
hjrpothesis as to the possible nature of the lot here spoken of; so that the problem,
as we have it, is really his.

"Supposons qu'il ordonna que comptant de 3 en 3 on tuerait toujours le troisi^me,...
il faut que Jos^phe se mit le trente-uni^me apr^s celui par lequel on commenfait k
compter, au cas qu'il visat h, demeurer en vie lui tout seul. Mais s'il voulut sauver
un de ses compagnons, il le mit en la seizi^me place, et s'il en voulut sauver encore
un autre, il le mit en la trente-cinquifeme place."

Thus stated, the problem can be solved in a moment by the graphical process of
striking out every third, in succession, of a set of 41 dots placed round a closed curve.
When three only are left, they will be found to be the 35th, 16th, and 31st; and,
if the process were continued, they would be exterminated in the order given. And
any similar question, involving only moderate numbers, would probably be most easily

CXXVI.] ON THE GENERALIZATION OF JOSEPHUS' PROBLEM. 433

solved in a similar fashion. But, suppose the number of companions of Josephus to
have been of the order even of hundreds of thousands only, vastly more if of billions, this
graphic method would involve immense risk of error, besides being toilsome in the
extreme ; and the whole process would have to be gone over again if we wished
the solution for the case in which the total number of men is altered even by a
single unit.

It is easy, however, to see that the following general statement gives the solution
of all such problems: —

Let n men be arranged in a ring which closes up its ranks as individuals are picked
out. Beginning anywhere, go continuously round, picking out each mth man until r only
are left. Let one of these be the man who originally occupied the pth place. Then, if
we had begun with n + 1 men, one of the r left would have been the originally (p + m)th,
or (if p-|-?/i>7i + l) the (p + m — n— l)th.

In other words, provided there are always to be r left, their original positions are
each shifted forwards along the closed ring by m places for each addition of a single
man to the original group.

A third, but even more simple and suggestive, mode of statement may obviously be
based on the illustrations which follow. In these the original number of each man is
given in black tjrpe, the order in which he is struck off, if the process be carried out to
the bitter end, in ordinary type.

By threes: — for groups of 8, and of 9, men respectively: —

351742860
12346678

971462853
12346678 9.

Increase by unit every number in the first line (to which a 0 has been appended)
and write it over the corresponding number in the third. We have the scheme

46285397 1,
97146285 3.

Here the numbers, and their order, are the same, but those in the lower rank are
three places in advance.

By fives : —

11 8

6 7

13 11

6 7

13.

The numbers of the first line, increased by units, and those of the third, are

13 11 4 6 2 12 9 8 5 3 10 7 1
5 3 10 7 1 13 11 4 6 2 12 9 8,

again the same order, but now shifted forwards by five places.

T. IL 55

434 ON THE GENEBALIZATION OF JOSEPHUS' PBOBLEM. [CXXVI.

It is easy to see that the two rows thus formed are identical when m = n+l.
Thus

By tens: —

9 5 0

8 9

10 6 1

8 9 10,

and the statement above is obviously verified.

To show how rapidly the results of this proceas can be extended to higher numbers,
I confine myself to the Josephus question, as regards himself alone, the last man. For
the others, the mode of procedure is exactly the same.

Given that the final survivor in 41, told off by threes, is the 31st, we have

71 last man

41 31

The rule just given shows that succeeding numbers in these columns are formed
as follows: — taking only those which commence, as it were, a new cycle: —

41 +x, 31 + 3a? - (41 + a?) = 2a? - 10.

The value of x which makes the right-hand side one or other of 1, and 2, is there-
fore to be chosen, so we must put a? = 6, and the result is

47 2

Successive applications of this process give, in order

13,655

105

20,482

158

30,723

237

46,085

355

69,127

533

103,691

799

155,536

1,199

233,304

1,798

349,956

2.697

524,934

4,046

787,401

6,069

1,181,102

9,103

1,771,653

CXXVI.] ON THE GENERALIZATION OF JOSEPHUS' PROBLEM. 43&

provided the (merely arithmetical) work is correct. And, of course, we can at once
interpolate for any intermediate value of n.

Thus, in 799 men, or in 30,723, the first is safe:— in 1000 the 604th; in 100,000
the 92,620th, and in 1,000,000 the 637,798th.

The earlier steps of this process, which lead at once to Bachet's number for 41
(assumed above), are

11 9 1

2 2 14 2

3 2 21 2

4 1 31 1
6 1

so that the method practically deals with millions, when we reach them, more easily
than it did with tens.

Unfortunately the cycles become shorter as the radix, and with it the choice of
remainders, increases; so that a further improvement of process must, if possible, be
introduced when every hundredth man (say) is to be knocked out.

From the data above given, it appears that up to two millions the number of
cases in which the first man is safe is 19, while that in which the second is safe
is only 16. (The case of one man, only, is excluded.) As these cases should, in the
long run, be equally probable, I extended the calculation to

13,059,835,455,001 1

with the result of adding 20 and 19 to these numbers respectively. But the next
15 steps appear to give only 2 cases in favour of the first man!

55-2

436 [cxxvii.

CXXVII.

KIRCHHOFF.

[Nature, Vol. xxxvi. October 27, 1887.]

Geheimrath Gustav Robert Kirchhoff was bom at Konigsberg on the 12th
of March, 1824. He commenced his professorial career at Berlin University as Privat
Decent; became Extra-ordinary Professor in Breslau from 1850 to 1854, thereafter till
1874 Professor of Physics in Heidelberg, whence he was finally transferred (in a
somewhat similar capacity) to Berlin. His health was seriously and permanently
affected by an accident which befell him in Heidelberg many years ago, and he had
been unable to lecture for some time before his death.

It is not easy, in a brief notice, to give an adequate idea of KirchhofTs numerous
and important contributions to physical science. Fortunately all his writings are easily
accessible. Five years ago his collected papers (Gesammelte Abhandlungen von G.
Kirchhoff, Leipzig, 1882) were published in a single volume. His lectures on Dynamics
(Vorlesungen uber MatJiematische Physik, Leipzig, 1876) have reached at least a third
edition; and his greatest work (TJntersuchungen iiber das Sonnenspectrunif Berlin, 1862)
was, almost immediately after its appearance, republished in an English translation
(London, Macmillan). To these he has added, so far as we can discover, only three
or four more recent papers; among which are, however, the following, published in
the Berlin Abhandlungen: —

Uber die Formanderung die ein fester elastischer Korper erfahrt, wenn er mag-
netisch oder dielectrisch polarisirt wird. (1884.)

A subsequent paper gives applications of the results (1884).

Additions to his paper (presently to be mentioned) on the Distribution of Electricity
on Two Influencing Spheres. (1885.)

CMLVn.]

KTRCHHOFF.

While there are nowadays hundreds of men thoroughly qualified to work out, to
its details, a problem already couched in symbols, there are but few who have the
gift of putting au entirely new physical question into such a form. The names of
Stokes, Thomson, and Clerk-Maxwell will at once occur to British readers as instances
of men possessing such power in a marked degree, Kirch h off had in this respect no
superior in Germany, except his life- long friend and coUeague v. Helmholtz.

His first published paper, Ofi electric cotiduction in a thin plate, and ^pedaUp in
a eircidar one (Pogg. Ann. 1845), gives an instance. The extremely elegant results
he obtained are now well known, and have of coui'se (once the start was given, or
the key-note struck) been widely extended from the point of view of the pure
mathematicians. The simpler results of this investigation, it must be mentioned, were
fully verified by the author's experimental tracing of the equipotential lines, and by
his measurements of their differences of potential A remark appended to this
paper contains two simple but important theorems which enable us3 to solve, by a
perfectly definite process, any problem concerning the distribution of currents in a
network of wires. This application forms the subject of a paper of date 1847*

Kirchhurt* published subsequently several very valuable papers on electrical questions,
among which may be noted those on conduction in curved sheets, on Ohm's Law, on
the distribution of electricity on two influencing spheres, on the discharge of the
Leyden Jar, on the motion of electricity in submarine cables, &c. Among these is
a short, but important, paper on the Deterrninution of the constant on which depends
the Iniemity of indmed currents (Pogg- Ann, 1849), This involves the absolute measure-
ment of electric resistance in a definite vnta. Kirchhoff was also the inventor of a
valuable addition to the Wheatstone Bridge- To the above class of papers may be
added two elaborate memoirs on Induced Magnetism (Crelle, 1S53 ; Pogg. Ergdnz-
LUig^htind, 1870),

Another series of valuable investigations deals with the equilibrium and motion
of elastic solids, especially in the form of plates, and of rods. The British leader
will find part of the substance of these papers reproduced in Thomson and Tait*s
Natural Philosophy, There are among them careful experimental determinations of
the value of Poisson's Ratio (that of the lateral contraction to the axial extension
of a rod under traction) for different substances. These results fully bear out the
conclusions of Stokes, w^ho was the fii-st to point out the fallacy involved in the
statement that the ratio in question is necessarily 1/4.

Kircbhoff*s Lectures on Dy^mmics are pretty well known in this country, so that
we need not describe them in detail. Like the majority of his separate papers they
are somewhat tough reading, but the labour of following them is certainly recom-
pensed. They form rather a collection of short treatises on special branches of the
subject^ than a Bystematic digest of it. One of the most noteworthy features of the
earlier chapters is the mode in which dynamical principles (e.g. the Laws of Motion)
are intmduced* While recognizing the great simplification in processes and in verbal
expression which is made possible by the use of the term Force, Kirchhoff altogether

438 KIRCHHOFF. [CXXVII.

objects to the introduction of the notion of Cause, as a step leading only to confusion
and obscurity in many fundamental questions. In fact he roundly asserts that the
introduction of systems of Forces renders it impossible to give a complete definition of
Force. And this, he says, depends on the result of experience that in natural motions
the separate forces are always more easily specified than is their resultant. He prefers
to speak of the motions which are observed to take place, and by the help of these
(with the fundamental conceptions of Time, Space, and Matter) to form the general
dynamical equations. Once these are obtained, their application may be much facilitated
by the introduction of the Name Force; and we may thus express in simple terms
what it would otherwise be difficult to formulate in words. So long as the motion
of a single particle of matter only is concerned we can, from proper data, investigate
its velocity and its acceleration, as directed quantities of definite magnitude. Thus
we proceed from Keplers Laws to find the acceleration of a planet's motion. This
is discovered to be directed towards the sun, and to be in magnitude inversely as
the square of the distance. We may call it by the name Force if we please, but
we are not to imagine it as an active agent. Something quite analogous appears in
the equations of motion when we introduce the idea of Constraint. The mode in
which the idea of Mass is introduced by Kirchhoff is peculiar. It is really equivalent
to a proof (ultimately based on experiments) of Newton's Third Law. Once, however,
it is introduced, the same species of reasoning (which differs but slightly from what
we should call Kinematical) leads to the establishment of D'Alembert s and Hamilton's
Principles, with the definition of the Potential Function, the establishment of Lagrange's
Generalized Equations, and the proof of Conservation of Energy, &c. The observational
and experimental warrant for this mode of treatment is, according to Kirchhoff*, the
fact that the components of acceleration are in general found to be functions of
position. [Kirchhoff^s view of Force has some resemblance to, but is not identical
with either of, the views previously published by Peirce and by the writer.] This is
the chief peculiarity of the book, and very different opinions may naturally be held
as to its value, especially as regards the strange admixture of Kinematics and
Dynamics.

Of the rest, however, all who have read it must speak in the highest terms.
A great deal of very valuable and original matter, sometimes dealing with extremely
recondite subjects, is to be found in almost every chapter. Among these we may
specially mention the investigation of surface conditions in the distortion of an elastic
solid, with the treatment of capillarity, of vortex-motion, and of discontinuous fluid
motion {Flilssigkeitsstrahlen),

Besides these definite classes of papers, there is a number of noteworthy memoirs
of a more miscellaneous character: — on important propositions in the Thermodjmamics of
solution and vaporization, on crystalline reflection and refraction, on the influence of heat
conduction in a special case of propagation of sound, on the optical constants of
Aragonite, and on the Thermal Conductivity of Iron.

Finally we have the series of papers on Radiation, partly mathematical, partly
experimental, which, in 1859 and 1860, produced such a profound impression in the

CXXVII.] KIRCHHOFF. 439

world of science, and which culminated in the great work on the solar spectrum
whose title is given above. The history of Spectrum Analysis has, from that date,
been one of unbroken progress. Light from the most distant of visible bodies has
been ascertained to convey a species of telegraphic message which, when we have
learned to interpret it, gives us information alike of a chemical and of a purely
physical character. We can analyze the atmosphere of a star, comet, or nebula, and
tell (approximately at least) the temperature and pressure of the glowing gas. But,
at the present time, the fact that such information is attainable is matter of common
knowledge.

This is not an occasion on which we can speak of questions of priority, even
though we might be specially attracted to them by finding v. Helmholtz and Sir
W. Thomson publicly taking (in full knowledge of all the facts) almost absolutely
antagonistic views. However these points may ultimately be settled, it is certain that
Eirchhoif was (in 1859) entirely unaware of what Stokes and Balfour Stewart had
previously done, and that he, with the powerful assistance of Bunsen, MADE what is
now called Spectrum Analysis: Kirchhoff, by his elaborate comparison of the solar
spectrum with the spectra of various elements, and by his artificial production of a
new line whose relative darkness or brightness he could vary at pleasure; Bunsen
by his success in discovering by the aid of the prism two new metallic elements.

440 [cxxviii.

CXXVIIL

HAMILTON.

[Fi'om Encyclopcedia Britannica, 1880.]

Hamilton, Sir William Rowan, one of the really great mathematicians of the
present century, was bom in Dublin, August 4, 1805. His father, who was a solicitor,
and his uncle (curate of Trim), migrated from Scotland in youth. A branch of the
Scottish family to which they belonged had settled in the north of Ireland in the
time of James I., and this fact seems to have given rise to the common impression
that Hamilton was an Irishman.

His genius displayed itself, even in his infancy, at first in the form of a wonderful
power of acquiring languages. At the age of seven he had already made very con-
siderable progress in Hebrew, and before he was thirteen he had acquired, under the
care of his uncle, who was an extraordinary linguist, almost as many languages as
he had years of age. Among these, besides the classical and the modern European
languages, were included Persian, Arabic, Hindustani, Sanskrit, and even Malay. But
though to the very end of his life he retained much of the singular learning of his
childhood and youth, often reading Persian and Arabic in the intervals of sterner
pursuits, he had long abandoned them as a study, and employed them merely as a
relaxation.

HLs mathematical studies seem to have been undertaken and carried to their
full development without any assistance whatever, and the result is that his writings
belong to no particular "school," unless indeed we consider them to form, as they
are well entitled to do, a school by themselves. As an arithmetical calculator he was
not only wonderfully expert, but he seems to have occasionally found a positive delight
in working out to an enormous number of places of decimals the result of some

cxxvtil]

HAMILTOSf.

441

irksome calculation. At the age of twelve he engaged Colburn, the American '* cal-
culating boy/' who was then being exhibited as a curiosity in Dublin, and he had
not always the worst of the encounter But, tw^o years before* he had accidentally
fallen in with a Latin copy of Eiiclid. which he eagerly devoured; and at twelve he
attacked Newton's Arithmetica Universalis. This was his introduction to modem
analyds. He soon commenced to read the Prindpia, and at sixteen he had mastered
a great part of that work, besides some more modern works on analytical geometry
and the diflFerential calculus.

About this period he was also engaged in preparation for entrance at Trinity
College, Dublin, and had therefore to devote a portion of his time to classics* In
the summer of 1822, in his seventeenth year, he began a systematic study of Laplace's
Micanique Cdleste, Nothing could be better fitted to call forth such mathematical
powers as those of Hamilton ; for Laplace's great work, rich to profusion in analytical
processes alike novel and powerful, demands from the most gifted student careful and
ol^n laborious study, It was in the enccessful effort to open this treaaure-house that
Hamilton's mind received its final temper, **J)hs lors il commen^a k marcher seul,"
to use the words of the biographer of another great mathematician. From that time
he appears to have devoted himself almost wholly to original investigation (so far at
least as regards mathematics), though he ever kept himself well acquainted with the
progress of science both in Britain and abroad.

Having detected an important defect in one ol' Laplace's demonstrations, he was
induced by a friend to write out his remarks, that they might be shown to Dr
Brinkley, afterwards bishop of Cloyne, who was then royal astronomer for Ireland and
an accomplished mathematician. Brinkley seems at once to have perceived the vast
talents of young Hamilton, and to have encouraged him in the kindest manner.
He is said to have remarked in 1823 of this lad of eighteen, — "This young man, I do
not say mill be, but is, the first mathematician of his age,"

Hamilton's career at college was perhaps unexampled. Amongst a number of
competitors of more than ordinary merit, he was first in every subject, and at every
examination. His is said to be the only recent case in which a student obtained the
honour of an optime in more than one subject. This distinction had then become very
rare, not being given unless the candidate displayed a thorough mastery over his subject.
Hamilton received it for Greek and for physics. How many more such honours he
might have attained it is impossible to say ; but he was expected to win both the
gold medals at the degree examination, had his career as a student not been cut
short by an unprecedented event. This was his appointment to the Andrews pro-
feiiSorBhip of astronomy in the university of Dublin, vacated by Dr Brinkley in 1827.
The chair was not exactly offered to him, as has been sometimes asserted, but the
electors, having met and talked over the subject, authorized one of their number,
who was Hamilton's personal friend, to urge him to become a candidate, a step which
his modesty had prevented him from taking. Thus, when barely twenty-two, he was
established at the Dublin Observatory* He was not specially fitted tor the post, for
although he had a profound acquaintance with theoretical astronomy, he had paid but

T, IL 56

442 HAMILTON. [CXXVIII.

little attention to the regular work of the practical astronomer. And it must be
said that his time was better employed in grand original investigations than it would
have been had he spent it in meridian observations made even with the best of
instruments, — infinitely better than if he had spent it on those of the observatory,
which, however good originally, were then totally unfit for the delicate requirements
of modem astronomy. Indeed there can be little doubt that Hamilton was intended,
by the university authorities who elected him to the professorship of astronomy, to
spend his time as he best could for the advancement of science, without being tied
down to any particular branch. Had he devoted himself to practical astronomy they
would assuredly have furnished him with modem instmments and an adequate staff
of assistants.

In 1835, being secretary to the meeting of the British Association which was
held that year in Dublin, he was knighted by the lord-lieutenant. But far higher
honours rapidly succeeded, among which we may merely mention his election in 1837
to the president's chair in the Royal Irish Academy, and the rare and coveted
distinction of being made corresponding member of the academy of St Petersburg.
These are the few salient points (other, of course, than the epochs of his more
important discoveries and inventions presently to be considered) in the uneventful
life of this great man. He retained his wonderful faculties unimpaired to the very
last, and steadily continued till within a day or two of his death (September 2,
1865) the task (his Elements of Quaternions) which bad occupied the last six years of
his life.

The germ of his first great discovery was contained in one of those early papers
which in 1823 he communicated to Dr Brinkley, by whom, under the title of CavMics,
it was presented in 1824 to the Royal Irish Academy. It was referred as usual to
a committee. Their report, while acknowledging the novelty and value of its contents,
and the great mathematical skill of its author, recommended that, before being pub-
lished, it should be still further developed and simplified. During the next three
years the paper grew to an immense bulk, principally by the additional details which
had been inserted at the desire of the committee. But it also assumed a much
more intelligible form, and the grand features of the new method were now easily
to be seen. Hamilton himself seems not till this period to have fully understood
either the nature or the importance of his discovery, for it is only now that we
find him announcing his intention of applying his method to djmamics. The paper
was finally entitled "Theory of Systems of Rays," and the first part was printed in
1828 in the Transactions of the Royal Irish Academy. The second and third parts
have not yet been printed ; but it is understood that their more important contents
have appeared in the three voluminous supplements (to the first part) which have
been published in the same Transactions, and in the two papers "On a General
Method in Djoiamics," which appeared in the Philosophical Transaxiti(ms in 1834-.5.
The principle of "Varying Action" is the great feature of these papers; and it is
strange, indeed, that the one particular result of this theory which, perhaps more
than an}rthing else that Hamilton has done, has rendered his name known beyond

cxxvni,]

^ilULTON.

443

the little world of true philosophers, ehould have been easily withiji the reach of
Fresnel and others for many years before, and in no way required Hamilton's new
conceptions or methods, although it was by them that he was led to its discovery.
This singular result is still known by the name *' Conical Refraction/' which he
proposed for it when be first predicted its existence in the third supplement to his
Stfstenis of Rays, read in 1832.

The step firom optics to dynamics in the application of the method of " Varying
Action'* was made in 1827, and commumcated to the Royal Society of London, in
whose Philosc^kical Trahsactions for 1834 and 1835 there are two papers on the
subject. These display, like the "Systems of Rays,*' a mastery over symbols and a
flow of mathematical language almost unequalled. But they contain what is far more
valuable still, the giv^atest addition which dynamical science had received since the
grand strides made by Newton and Lagrange* Jaoobi aud other mathematicians have
developed to a great extent, and as a question of pure mathematics only, Hamilton's
processes, and have thus made extensive additions to our knowledge of differential
equatLons* But there can be little doubt that we have as yet obtained only a mere
glimpse of the vast physical results of which they contain the gerra, Aud though
this is of course by far the more valuable aspect in which any such contribution
to science can be looked at, the other must not be despised. It is characteristic of
most of Hamilton's, as of nearly all great discoveries, that even their indirect con-
sequences are of high value.

The other great contribution made by Hamilton to mathematical science, the
Calculus of Quaternions, is fully treated under that heading. [No. OXXIX. below J
It is not necessary to say here more than this, that quaternions form as great an
advance relatively to the Cartesian methods as the latter, when first propounded,
formed relatively to Euclidian geometry. The following characteristic extract from a
letter shows Hamilton's own opinion of his mathematical work, and also gives a hint
of the devices which he employed to render written language as expressive as actual
speech. His first great work, Lectures on Quatemiotfs (Dublin, 1852), is almost painful
to read in consequence of the frequent use of italics and capitals.

" I hope that it may not be considered as unpardonable vanity or presumption
on my part, if, as my own taste has always led me to feel a greater interest in
vwihoda than in rmults, so it is by methods, rather than by any theorems, which
can be separately quoted, that I desire and hope to be remembered. Nevertheless it
is only human nature, to derive Bome pleasure from being cited, now and then, even
about a '^Theorem'*; especially where . . . > . the quoter can enrich the subject, by
eombining it with researches of his own/'

The discoveries, papers, and treatises we have mentioned might well have formed
the whole work of a long and laborious life. But, not to sp^ik of his enormous
collection of MS, books, full to overflowing with new and original matter, which have
been handed over to Trinity College, Dublin, and of whose contents it is to be hoped
a large portion may yet be published, the works we have already called attention

56—2

444 HAMILTON. [CXXVIII.

to barely form the greater portion of what he has published. His extraordinary in-
vestigations connected with the solution of algebraic equations of the fifth degree,
and his examination of the results arrived at by Abel, Jerrard, and Badano, in their
researches on this subject, form another grand contribution to science. There is next
his great paper on Fluctiuiting Functions^ a subject which, since the time of Fourier,
has been of immense and ever increasing value in physical applications of mathe-
matics. There is also the extremely ingenious invention of the Hodograph. Of his
extensive investigations into the solution (especially by numerical approximation) of
certain classes of differential equations which constantly occur in the treatment of
physical questions, only a few items have been published, at intervals, in the Philo-
sophical Magazine. Besides all this, Hamilton was a voluminous correspondent. Often
a single letter of his occupied from fifty to a hundred or more closely written pages,
all devoted to the minute consideration of every feature of some particular problem :
for it was one of the peculiar characteristics of his mind never to be satisfied with
a general understanding of a question; he pursued it until he knew it in all its
details. He was ever courteous and kind in answering applications for assistance in
the study of his works, even when his compliance must have cost him much valuable
time. He was excessively precise and hard to please with reference to the final polish
of his own works for publication; and it was probably for this reason that he pub-
lished so little compared with the extent of his investigations.

Like most men of great originality, Hamilton generally matured his ideas before
putting pen to paper. "He used to carry on," says his elder son, "long trains of
algebraical and arithmetical calculations in his mind, during which he was unconscious
of the earthly necessity of eating; we used to bring in a 'snack' and leave it in
his study, but a brief nod of recognition of the intrusion of the chop or cutlet was
often the only result, and his thoughts went on soaring upwards."

For further details about Hamilton (his poetry and his association with poets,
for instance), the reader is referred to the Dublin University Magazine (Jan. 1842),
the Oentleman's Magazine (Jan. 1866), and the Monthly Notices of the Royal Asti-o-
nomical Society (Feb. 1866); and also to an article by the present writer in the
North British Review (Sept. 1866), from which much of the above sketch has been
taken. [See, also, especially in connection with some of the opening statements above.
Life of Sir W. R. Hamilton by the Rev. R. P. Graves (3 vols.; Dublin 1882-89).
And, in particular, Addendum to that work (Dublin 1891). This Addendum refers
particularly to the notice of Hamilton in the Dictionary of National Biography, On
this I remarked (Nature, XLiii. 608), "the patent error of that notice is the confusion
of Hamilton's Varying Action with his Quaternions, The consequence is that Hamilton
gets no credit for his absolutely invaluable contribution to Theoretical Dynamics V*
1899.]

CXXIX,]

445

CXXIX.

QUATEENIONa

[From Encyclopimiia Britannicat 1886.]

Thb word quaternion properly means *' a set of four." In employing such a word
to denote a new mathematical method. Sir W, R. Hamilton (No. CXXVIII.) was probably
influenced by the recollection of its Greek equivalent, the Pythagorean Tetractys, the
mystic source of all things.

Quaternions (as a mathematical method) is an extension, or improvement, of
Cartesian geometry, in which the artifices of coordinate axes, &c„ are got rid of, all
directions in space being treated on precisely the same terras. It is therefore, except
in some of its degraded forms, possessed of the perfect isotropy of Euclidian space*

From the purely geometrical point of view, a quaternion may be regarded as
the quotimU of two directed lines in space — or, what comes to the same thing, as ifie
factor, or operator, which changes on^ directed line into anotlwr. Its analytical definition
cannot be given for the moment; it will appear in the course of the article.

HiBt&ry of the Method. — The evolution of quaternions belongs in part to each of
two weighty branches of mathematical history^ — the interpretation of the imafjtnury (or
impossible) quantity of common algebra, and the Cartesian application of algebra to
geometry. Sir W. R, Hamilton was led to his great invention by keeping geometrical
applications constantly before him while he endeavoured to give a real significance to
V— 1. We will therefore confine ourselves, so far as bis predecessors are concerned^
to attempts at interpretation which hatl geometrical applications in view.

One geometrical interpretation of the negative sign of algebra was early seen to
be mere reversal of direction along a Une. Thus, when an image is formed by a
plane mirror, the distance of any point in it from the mirror is simply the negative

446 QUATERNIONS. [CXXIX.

of that of the corresponding point of the object. Or if motion in one direction along^
a line be treated as positive, motion in the opposite direction along the same line is
negative. In the case of time, measured from the Christian era, this distinction is
at once given by the letters a.d. or B.C., prefixed to the date. And to find the
position, in time, of one event relatively to another, we have only to subtract the
date of the second (taking account of its sign) from that of the first. Thus to find
the interval between the battles of Marathon (490 B.C.) and Waterloo (1815 A.D.) we have

+ 1815 - (- 490) = 2305 years.

And it is obvious that the same process applies in all cases in which we deal with
quantities which may be regarded as of one directed dimension only, such as distances
along a line, rotations about an axis, &c. But it is essential to notice that this is by
no means necessarily true of operators. To turn a line through a certain angle in a
given plane, a certain operator is required; but when we wish to turn it through
an equal negative angle we must not, in general, employ the negative of the former
operator. For the negative of th^ operator which turns a line through a given angle
in a given plane will in all cases produce the negative of the original result, which is
not the result of the reverse operator, unless the angle involved be an odd multiple of a
right angle. This is, of course, on the usual assumption that the sign of a product
is changed when that of any one of its factors is changed, — which merely means that
— 1 is commutative with all other quantities.

The celebrated Wallis seems to have been the first to push this idea further. In
his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of
a quadratic equation by going out of the line on which the roots, if real, would have
been constructed.

In 1804 the Abbe Bu^e*, apparently without any knowledge of Wallis's work,
developed this idea so far as to make it useful in geometrical applications. He gave, in
feet, the theory of what in Hamilton's system is called Composition of Vectors in one
plane — 1.6., the combination, by + and — , of complanar directed lines. His constructions
are based on the idea that the imaginaries ± V— 1 represent a unit line, and its reverse,
perpendicular to the line on which the real units ± 1 are measured. In this sense
the imaginary expression a+iV— 1 is constructed by measuring a length a along the
fundamental line (for real quantities), and from its extremity a line of length 6 in some
direction perpendicular to the fundamental line. But he did not attack the question of
the representation of products or quotients of directed lines. The step he took is really
nothing more than the kinematical principle of the composition of linear velocities, but
expressed in terms of the algebraic imaginary.

In 1806 (the year of publication of Buee's paper) Argand published a pamphlet i- in
which precisely the same ideas are developed, but to a considerably greater extent. For

♦ PhiL Trans., 1806.

+ Etiai 8ur ur^e manihre de reprisenter Us Quantitis Imaginaires dans Us Constructions Oiom^triques, A second
edition was published by Hoiiel (Paris, 1874). There is added an important Appendix^ consisting of the papers
from Gergonne^s AnnaUs which are referred to in the text above. Almost nothing can, it seems, be learned of
Argand*s private life, except that in all probability he was bom at Geneva in 1768.

CXXTX.]

QUATEBNIONS*

an interpretation is assigned to the product of two directed lines in one plane, when each
is expressed as the sura of a real and an iraaginary part. This product is interpreted
as another directed line, forming the fourth tenn of a proportion, of which the first term
is the real (positive) unit-line, and the other two are the factor-lines. Argand's work
remained unnoticed until the question was again raised in Qergonms Annates, 1813,
by Fran9aia. This writer stated that he had found the germ of his remarks among the
papers of his deceased brother, and that they had come from Legendre, who had himself
received them from some one xm named. This led to a letter from Argand, in which he
stated his communications with Legendre, and gave a resum^ of the contents of his
pamphlet. In a further commnnication to the AnnaleSf Argand pushed on the appli-
cations of his theory. He has given by means of it a simple proof of the existence of
n rootSi and no more, in every rational algebraic equation of the nth degree with real
coefficients. About 1828 Wan^en in England, and Mourey in France^ independently of
one another and of Argand. reinvented these modes of interpretation; and still later, in
the writings of Canehy, Gauss, and others, the properties of the expression a + frV—l
were developed into the immense and most important subject now called the theory of
complex numbers. From the more purely symbolical point of view it was developed by
Peacock, De Morgan, &c., as double algebra.

Argand's method may be putj for reference, in the following form* The directed line
whose length is a, and which makes an angle d with the real (positive) unit line, is

expressed by

a (cos d + isind),

where i is regaixled as + V— 1, The sum of two such lines (formed by adding together
the real and the imaginary parts of two such expressions) can, of course, be expressed as
a third directed line^ — the diagonal of the parallelogram of which they are conterminous
sides. The product, P, of two such lines is, as we have seen, given by

1 : a (cos e + 1 sin 0) :: o'(cos 0' + 1 sin 0") : P,

oer

P = €ui' [cm {0 + r) + i sin (0 + ff)].

Its length is, therefore, the product of the lengths of the factors, and its inclination to
the real unit is the sura of those of the factors If we write the expressions for the
two lines in the form

the product is

AA'-BB-i-i(AB-^BA');

and the fact that the length of the product line is the product of those of the factors
m seen in the form

{A' + S») (A"* + B^) = {A A' - BRf + (AB + BA%

In the modem theory of complex numbers this is expressed by saying that the Norm of
a product is equal to the product of the norms of the factors^

Argand's attempts to extend his method to space generally were firuitless. The
reasons will be obvious later; but we mention them just now because they called

448 43UATEBNION8. [CXXIX.

forth from Servois {Oergonne's AnncUes, 1813) a very remarkable comment, in which was
contained the only yet discovered trace of an anticipation of the method of Hamilton.
Argand had been led to deny that such an expression as i* could be expressed in the
form A + Bi, — although, as is well known, Euler showed that one of its values is a real
quantity, the exponential function of — 7r/2. Servois says, with reference to the general
representation of a directed line in speice : —

"L'analogie semblerait exiger que le trin&me fftt de la forme

JO cos a + 5 cos yS + r cos 7 ;
a, /3, 7 ^tant les angles d*une droite avec trois axes rectangulaires ; et qu'on eiit

(pcos a -h gcosyS + r cos 7)(p' cos a + q' cosyS + r'cos 7) = cos»a + cos"y8 + cos"7 = 1.

Les valeurs de p, q, r, p', q\ r' qui satisferaient & cette condition seraient absurdes ; mais
seraient-elles imaginaires, reductibles k la forme g^n^rale -4 + jB>/— 1? Voili une ques-
tion d'analyse fort singulifere que je soumets k vos lumiferes. La simple proposition que
je vous en fais suffit pour vous faire voir que je ne crois point que toute fonction
analytique non r^elle soit vraiment reductible k la forme A + B*/^l"

As will be seen later, the fundamental i, j, k of quaternions, with their reciprocals,
furnish a set of six quantities which satisfy the conditions imposed by Servoia And
it is quite certain that they cannot be represented by ordinary imaginaries.

Something far more closely analogous to quaternions than anything in Argand's work
ought to have been suggested by De Moivre's theorem (1730). Instead of regarding, as
Bu^e and Argand had done, the expression a (cos 0 + i sin d) as a directed line, let us
suppose it to represent the operator which, when applied to any line in the plane in
which d is measured, turns it in that plane through the angle 0, and at the same
time increases its length in the ratio a : 1. From the new point of view we see at
once, as it were, why it is true that

(cos 0 + isin 0)^ = cos mO 4- i sin mO.

For this equation merely states that m turnings of a line through successive equal
angles, in one plane, give the same result as a single turning through m times the
common angle. To make this process applicable to any plane in space, it is clear that
we must have a special value of i for each such plane. In other words, a unit line,
drawn in any direction whatever, must have — 1 for its square. In such a system there
will be no line in space specially distinguished as the real unit line: all will be alike
imaginary, or rather alike real. We may state, in passing, that every quaternion can be
represented as a (cos ^ + «r sin 0), — where a is a real number, 0 a real angle, and w a
directed unit line whose square is — 1. Hamilton took this grand step, but, as we have
already said, without any help from the previous work of De Moivre. The course of his
investigations is minutely described in the preface to his first great work* on the subject.
Hamilton, like most of the many inquirers who endeavoured to give a real interpre-
tation to the imaginary of common algebra, found that at least two kinds, orders, or

* Lecturei on QuatemiatUf Dnblin, 1858.

CXXIX,]

QUAT1RNI0N8,

racks of quantities were necessary for the purpose- But, instead of dealing with points
on a line, and then wandering out at right angles to it, as Bu^e and Argand had done,
he choae to look on algebra as the science of pure time*, and to investigate the pro-
perties of "sets" of time-steps. In its essential nature a set is a linear function of any
number of distinct units of the same specie.s. Hence the simplest form of a set is
a couple; and it was to the possible laws of combination of couples that Hamilton
fii'st dii^ected his attention. It is obvious that the way in which the two separate time-
steps are involved in the couple will determine these laws of combination. But
Hamilton's special object required that these laws should be such as to lead to certain
assumed results; and he therefore commenced by assuming these, and fi*om the
assumption determined how the separate time-steps must be involved in the couple.
If we use Roman letters for mere numbers, capitals for instants of time, Greek letters
for time-steps, and a parenthesis to denote a couple, the laws assumed by Hamilton as
the basis of a system were as follows : —

(a, b)(a, /8)-(aa-b^, ba-hajS)t.

To show how we give, by such assumptions, a real interpretation to the ordinary
algebraic imaginary, take the simple case a = 0, b = 1, and the second of the above
formulie gives

(0, l)(a, j9)=(-^, a).

Multiply once more by the number-couple (0, l)j and we have

(0, 1)(0, l)(a, ^) = (0, 1)^^, «)=(-a, -0)

Thus the number-couple (0, I), when twice applied to a step-couple, simply
changes its sign. That we have here a perfectly t^eal and intelligible interpretation
of the ordinary algebraic imaginary is easily seen by an illustration, even if it be a
somewhat extravagant one. Some Eastern potentate, possessed of absolute power, covets
the vast possessions of his vizier and of his barber. He determines to rob them both
(an operation which may be very satisfactorily expressed by —1); but, being a wag, he
chooses his own way of doing it. He degrades his vizier to the office of barber, taking
all his goods in the pmcess; and makes the barber his vizier. Next day he repeats the
operation. Each of the victims has been restored to his former i*ank, but the operator
— I has been applied to both*

Hamilton, still keeping prominently before him as his great object the invention of
a method applicable to space of three dimensions, proceeded to study the properties of
triplets of the form a*+iy + j>, by which he proposed to represent the directed line in
space whose projections on the coordinate axes are ^, y, z. The composition of two
such lines by the algebraic addition of their several projections agreed with the

* Th^Gry of €Qi\jugtit0 Funetiofu, or Algebntic CoupUtr with a PfelimtnaTy and Elementartf Enaif on Algebra
at the Stier\ce of Purt Time, read in 1833 and 1835* and pubUahed in Trans. E. L A., xvii- »i. (1835).
t Compare these with the loug-flubaa^iuejit id^aa of Graasmazm, presently to be described,

T. IL • 57

450 QUATERNIONS. [CXXIX.

assumption of Bu^e and Argand for the case of coplanar lines. But, assuming the
distributive principle, the product of two lines appeared to give the expression

^ " yy* " ^^' + * (y^' + ^y') + i (^^' + ^^') + ^iiy^' + ^y )•

For the square of j, like that of i, was assumed to be negative unity. But the inter-
pretation of i] presented a difficulty, — in fact the main difficulty of the whole investiga-
tion,— and it is specially interesting to see how Hamilton attacked it. He saw that he
€Ould get a hint from the simpler case, already thoroughly discussed, provided the two
factor lines were in one plane through the real unit line. This requires merely that

y \ z w y' \ z'\ or yz'-zy' = 0\

but then the product should be of the same form as the separate factors. Thus, in
this special case, the term in ij ought to vanish. But the numerical factor appears
to be ys! ■\- zy\ while it is the quantity yz' ^zy' which really vanishes. Hence Hamilton
was at first inclined to think that ij must be treated as niL But he soon saw that "a
less harsh supposition" would suit the simple case. For his speculations on sets had
already familiarized him with the idea that multiplication might in certain cases not be
commutative; so that, as the last term in the above product is made up of the
two separate terms ijyz and jizy', the term would vanish of itself when the factor
lines are coplanar provided ij=—jiy for it would then assume the form ij {yz' — zy'y
He had now the following expression for the product of any two directed lines

XX — yy — zz' + i {yx' + acy'^ + j {xz' + zx^ + ij {yz — zy').

But his result had to be submitted to another test, the Law of the Norms. As soon
as he found, by trial, that this law was satisfied, he took the final step. "This led
me," he says, "to conceive that perhaps, instead of seeking to confine ourselves to

tripletSy we ought to regard these as only imperfect forms of Quaternions, and

that thus my old conception of sets might receive a new and useful application." In
a very short time he settled his fundamental assumptions. He had now three distinct
space-units i, j, k ; and the following conditions regulated their combination by
multiplication : —

i2=j2=:^3 = — 1, ij^''ji=^k, jk = ^kj^i, ki = ^ik=^j*.

And now the product of two quaternions could be at once expressed as a third
quaternion, thus —

(a + ib + jc + kd) (a' + iV +jc' + kd') = A+iB +jG -h kD,

where

A = aa'^ bV —cc'— dd\

B=^ab' + ba' + cd' - dc\

C = ac' + ca + dV - bd\

D^ad' + da' -{- b& - cb'.

Hamilton at once found that the Law of the Norms holds, — not being aware that
* It wiU be easy to see that, instead of the last three of these, we may write the single one ijk~ — l.

CXXIXp]

QUATERNrONS.

451

Euler had long before decoiuposed the product of two sums of four squares into this
very set of four squai-es. And now a directed line in space came to be represented
fts iar-hj^-hi^, while the product of two lines is the quaternion

— (xx' -¥ yy -¥ zz') + i{yz' — zy) + j {zt^ — mz') ■¥ k {xy* — yx*).

To any one acquainted, even to a slight extent, with the elements of Cartesian
geometry of three diniensioos, a glance at the extremely suggestive constituents of
this expression shows how justly Hamilton was entitled to say — *' When the con-
ception had been so far unfolded and fixed in my mind, I felt that the nrnv

insh^imwnt for applying calculation to geometry , fur which I had so long sought, was
now, at least in part, attained." The date of this memorable discovery is October 16,
1843.

We can devote but a few lines to the consideration of the expression above.
Suppose, for simplicity, the factor lines to be each of unit length* Then ^, y, ^,
of, y\ z* express their direction -cosines. Also, if 6 be the angle between them, and
m'\ y, z** the direction -cosines of a line perpendicular to each of them, we have

$im* + yy ' + zz* = cos 8, yz' — z^/^sf' sin $, &c.,

so that the product of two unit lines is now expressed as

- cos 6 + (m?" -h jy" -h kz") sin &,

Thus, when the factors are parallel, or ^ ^ 0, the product, which is now the square of
any (unit) line, is —1. And when the two factor lines are at right angles to one
another, or 5 ^ 7r/2, the product is simply id* +Jy" + kz*\ the unit line perpendicular
to both. Hence, and in this lies the main element of the symmetry and simplicity
of the quaternion calculus, all sy sterns of three muttially rectangular nnit lines in space
have the sams properties as the fundamental system i, j, k. In other words, if the
system (considered as rigid) be made to turn about till the first factor coincides with
i and the second with j, the product will coincide with k. This fundamental system,
therefore, becomes unnecessary; and the quaternion method, in every case, takes its
reference lines solely from the problem to which it is applied. It has therefore, aa
it w^ere, a unique internal character of its own,

Hamilton, having gone thus far, proceeded to evolve these results from a train
of a priori or metaphysical reasonings which is so interesting in itself, and so
characteristic of the man, that we briefiy sketch its nature.

Let it be supposed that the product of two directed lines is something which
has quantity ; i>., it may be halved, or doubled, for instance. Also let us assume
(a) space to have the same properties in all directions, and make the convention
(6) that to change the sign of any one factor changes the sign of a product. Then
the product of two lines which have the same direction caimot be, even in part, a
direded quantity. For» if the directed part have the same direction as the factors,
(6) shows that it will be reversed by reversing either, and therefore will recover
its original direction when both are reversed. But this would obviously be inconsistent

57—2

452 QUATERNIONS. [CXXIX.

with (a). If it be perpendicular to the factor lines, (a) shows that it must have simul-
taneously every such direction. Hence it must be a mere number.

Again, the product of two lines at right angles to one another cannot, even in
part, be a number. For the reversal of either factor must, by (6), change its sign.
But, if we look at the two factors in their new position by the light of (a), we see
that the sign must not change. But there is nothing to prevent its being repre-
sented by a directed line if, as farther applications of (a) and (b) show we must do,
we take it perpendicular to each of the factor lines.

Hamilton seems never to have been quite satisfied with the apparent heterogeneity of
a quaternion, depending as it does on a numerical and a directed part. He indulged in
a great deal of speculation as to the existence of an eoctra-apatial unity which was to
furnish the raison cCetre of the numerical part, and render the quaternion homogeneous
as well as linear. But, for this, we must refer to his own works.

Hamilton was not the only worker at the theory of sets. The year after the
first publication of the quaternion method, there appeared a work of great originality,
by Grassmann*, in which results closely analogous to some of those of Hamilton were
given. In particular two species of multiplication ("inner" and "outer") of directed
lines in one plane were given. The results of these two kinds of multiplication
correspond respectively to the numerical and the directed parts of Hamilton's quaternion
product. But Grassmann distinctly states in his preface that he had not had leisure
to extend his method to angles in space. Hamilton and Grassmann, while their earlier
work had much in common, had very different objects in view. Hamilton, as we have
seen, had geometrical application as his main object; when he realized the quaternion
system, he felt that his object was gained, and thenceforth confined himself to the
development of his method. Grassmann's object seems to have been, all along, of
a much more ambitious character, viz., to discover, if possible, a system or systems
in which every conceivable mode of dealing with sets should be included. That he
made very great advances towards the attainment of this object all will allow; that
his method, even as completed in 1862, fully attains it is not so certain. But his
claims, however great they may be, can in no way conflict with those of Hamilton,
whose mode of multiplying couples (in which the "inner" and "outer" multiplication
are essentially involved) was produced in 1833, and whose quaternion system was
completed and published before Grassmann had elaborated for press even the rudi-
mentary portions of his own system, in which the veritable difficulty of the whole
subject, the application to angles in space, had not even been attacked. Grassmann
made in 1854 a somewhat savage onslaught on Cauchy and De St Venant, the former
of whom had invented, while the latter had exemplified in application, the system of
*' clefs algibriqueSy' which is almost precisely that of Grassmann. [See letter now
appended to this article. 1899.] But it is to be observed that Grassmann, though
he virtually accused Cauchy of plagiarism, does not appear to have preferred any such
charge against Hamilton. He does not allude to Hamilton in the second edition of

• Die Atudehnungtlehret Leipsic, 1844; 2d ed., '' vollsUindig und in strenger Form hearheitet^*^ Berlin, 1862.
See also the collected works of Mdbius, and those of Clifford, for a general explanation of Orassmann^s method.

cigaca*]

QUATERNIONS,

45S

his work. But in 1877, in the Mathematische Annalen, xiL, he gave a paper "On
the Place of Qiuiternioos in the Ausdehnuiifjfslehrey* Ln which he condemns, as far as
he can, the nomenclature and methods of Hamilton.

There are many other BjBtems, based on various principles, which have been given
for application to geometry of directed lines, but those which deal with products of
lines are all of such complexity as to be practically useless in application. Others, such
as the Barycenirimhe Catcul of Miibius, and the Mithode des £quipoltmc€s of Bellavitis,
give elegant modes of treating space problems, so long as we confine ouraelves to projective
geometry and matters of that order ; but they are limited in their field, and therefore
need not be diseussed here. More general systems, having cloae analogies to quaternions,
have been given since Hamilton s discovery was published. As instances we may take
Goodwin's and O'Brien's papers in the Cambridge Philosophical Trafisaciions for 1849.

Melations to other Branches of Science. — Even the above brief narrative shows how
close is the connexion between quaternions and the ordinary Cartesian space -geometry.
Were this all, the gain by their introduction would consist mainly in a cleai^er insight
into the mechanism of coordinate systems, rectangular or not — a very important
addition to theory, but little advance so far as practical application is concerned. But
we have now to consider that, as yet, we have not taken advantage of the perfect
symifietry of the method. When that is done, the full value of Hamilton's grand step
becomes evident, and the gain is quite as extensive from the practical as from the
theoretical point of view* Hamilton, in fact, remarks*, *' I regard it as an inelegance
and imperfection in this calculus, or rather in the state to which it has hitherto been

unfolded, whenever it becomes, or ^eemB to become, necessary to have recourse .to

the resources of ordinary algebm, for the solution of equations in quaterniomJ* This
refers to the use of the ^r, ^, £ coordinates, — associated, of course, with i, j, k But
when, instead of the highly artificial expression ir+j^+fo, to denote a finite directed
line, we employ a single letter, a (Hamilton uses the Greek alphabet for this purpose),
and find that we are permitted to deal with it exactly as we should have dealt with
the more complex expression, the immense gain is at least in j>art obvious. Any
quaternion may now be expressed in numerous simple forms^ Thus we may regard it as
the sum of a number and a line, a + «, or as the product, ^7, or the quotient, Se~^, of
two directed lines, &c., while, in many cases, we may represent it, so &r aa it is required,
by a single letter such as 5, r, &c.

Perhaps to the student there is no part of elementary mathematics so repulsive as
is spherical trigonometry. Also, everything relating to change of systems of axes, as
for instance in the kinematics of a rigid systemp where we have constantly to consider
one set of rotations with regard to axes fixed in space, and another set ^^^th regard
to axes fixed in the system, is a matter of troublesome complexity by the usual
methods. But every quaternion formula is a proposition in spherical (sometimes de^
grading to plane) trigonometry, and has the full advantage of the symmetry of the
method. And one of Hamilton's earliest advances in the study of his sj^tem (an advance
independently made, only a few months later, by Gayley) was the interpretation of the

454 QUATERNIONS. [CXXIX.

singular operator q( )5~S where 3 is a quaternion. Applied to a7iy directed line, this
operator at once turns it, conicallyy through a definite angle, about a definite axis. Thus
rotation is now expressed in symbols at least as simply as it can be exhibited by means
of a model. Had quaternions eflFected nothing more than this, they would still have
inaugurated one of the most necessary, and apparently impracticable, of reforms.

The physical properties of a heterogeneous body (provided they vary continuausly
from point to point) are known to depend, in the neighbourhood of any one point
of the body, on a qiiadric function of the coordinates with reference to that point.
The same is true of physical quantities such as potential, temperature, &c., through-
out small regions in which their variations are continuous; and also, without re-
striction of dimensions, of moments of inertia, &c. Hence, in addition to its geometrical
applications to surfaces of the second order, the theory of quadric functions of position
is of fundamental importance in physics. Here the symmetry points at once to the
selection of the three principal axes as the directions for i, j, k; and it would appear
at first sight as if quaternions could not simplify, though they might improve in
elegance, the solution of questions of this kind. But it is not so. Even in Hamilton's
earlier work it was shown that all such qxiestions were reducible to the solution of linear
equations in quaternions) and he proved that this, in turn, depended on the deter-
mination of a certain operator, which could be represented for purposes of calculation
by a single symbol. The method is essentially the same as that developed, under
the name of "matrices" by Cayley in 1858; but it has the peculiar advantage of
the simplicity which is the natural consequence of entire freedom from conventional
reference lines.

Sufficient has already been said to show the close connexion between quaternions
and the theory of numbers. But one most important connexion with modem physics
must be pointed out, as it is probably destined to be of great service in the im-
mediate future. In the theory of surfaces, in hydrokinetics, heat-conduction, potentials,
&c., we constantly meet with what is called Laplace's operator, viz.,

d? d^ d«

We know that this is an invariant; t.e., it is independent of the particular directions
chosen for the rectangular coordinate axes. Here, then, is a case specially adapted to
the isotropy of the quaternion system ; and Hamilton easily saw that the expression

dx ^ dy dz

could be, like ix+jy^-kz, effectively expressed by a single letter. He chose for this
purpose V. And we now see that the square of V is the negative of Laplace's
operator; while V itself, when applied to any numerical quantity conceived as having
a definite value at each point of space, gives the direction and the rate of most rapid
change of that quantity. Thus, applied to a potential, it gives the direction and
magnitude of the force; to a distribution of temperature in a conducting solid, it
gives (when multiplied by the conductivity) the flux of heat, &c.

CXXIX.]

QUATEEKIONa

455

No better testimony to the value of the quateniion method could be desii'ed than
the constant use made of its notation by nmthematicians like Clifford (in his Kinematic)
and by phyBicists like Clerk-Maxwell (in his Electricity and Maf^ftetwrn). Neither of
theae men professed to employ the calculus itself, but they recognized fiilly the extra-
oniitiary clearness of insight which is gained even by merely translating the unwieldy
Cartesian expressions met with in hydrokinetics and in electrod^Tiamics into the
pregnant language of quaternions.

Works on the Subject — Of course the great works on this subject are the two
immense treatises by Hamilton himself. Of these the second {EletHmita of Quaternions,
London, 18f}6; 2nd ed. 1899) was posthumous — incomplete in one short part of the
original plan only, but that a most important part, the theory and applications of V,
These two works, along with Hamilton's other papers on qnatemions (in the Dublin
Proceedings and Transactions, the Philosophical Maffasine, &c,), are storehouses of in-
formation, of which but a small portion has yet been extracted. A German translation
of Hamilton's Elements has recently been published by Glan.

Other works on the subject, in order of date, are Allegret, Bssai sur le Cal€^d des
QmxtemioTis (Paris, 1862); Tait, An Elementart/ Treatise on Quatermons (Oxford, 1867;
2nd ed., Cambridge, 1873; 3rd, 1890; German translation by v. Scherff. 1880, and French
by Plarr, 1882 — 84) ; Kelland and Tait, Introduction to Quaternions (London, 1873 ; 2nd
ed. 1882); Hoiiel, SUtnents de la Tkeorie des Qiiateimiojis (Paris, 1874); Unverzagt,
Tftmrie der Quaiemianen (Wiesbaden, 1876); Laisant, Introduction d la MSthode dea
Qmjte?'nions (Paris, 1881); Graefe, Vorlemngen itber die Theorie der Qnuternionen (Leipsic,
1884). [To these must now be added M*^Aulay, Utiliit/ of Quaternions in Physics,
London, 1893 ; as well as a number of elementary treatises. 1899,]

An excellent article on the " Principles *' of the science, by Dillner, will be found in
the Maihematische Annalen, vol XI., 1877. And a very valuable article on the general
question, Linear Associative Algebra, by the late Prof. Peirce, was ultimately printed in
vol IV, of the Avnei'ican Journal of Mathematics, Sylvester and others have recently
published extensive contributions to the subject, including quaternions under the general
class matriwt and have developed much farther than Hanulton lived to do the solution
of equations in quaternions. Several of the works named above are little more than
compilations, and some of the French ones are painfully disfigured by an attempt to
introduce an improvement of Hamilton's notation ; but the mere fact that so many
have already appeared shows the sure progress which the method is now making*

[In an article by Prof. F. Klein (Mat^K Ann. U. 1898) a claim is somewhat
obscurely made for Gauss to a share, at least, in the invention of Quaternions, Full
information on the subject is postponed till the publication of Gauss' Ncichlass, in
Vol. VI I L of his Oesammelte Werke^ From the article mentioned above, and from a
"Digression on Quaternions'* in Klein und Sommerfeld Ueber die Theorie des Kreisels
(p» 58), this claim appears to rest on some singular misapprehent^ion of the nature
of a Quaternion : — whereby it is identified with a totally different kind of concept, a
certain very restricted form of linear and vector Operator. 1809,]

456 QUATERNIONS. [CXXIX.

APPENDIX.

(Reprinted an account of the passage now marked. See p. 452 above,)

QUATERNIONS AND THE AUSDEHNUNGSLEHRE.

[Nature, June 4th. 1891.]

Prof. (iibW second long letter was evidently written before he could have read my reply to the
first. This is unfortunate, as it tends to confuse those third parties who may be interested in the
question now raised. Of course that question is naturally confined to the invention of methods, for
it would be prei)osterous to compare Grassmann with Hamilton as an analyst.

I have again read my article "Quaternions" in the Encyc, Brit, and have consulted once more
the authorities there referred to. I have not found anything which I should wish to alter. There is
much, of course, which I should have liked to extend, had the Editor permitted. An article on
Quaternions, rigorously limited to four pages, could obviously be no place for a discussion of
Qrassmann's scientific work, except in its bearings upon Hamilton's calculus. Moreover, had a similar
article on the Aiudehnungslehre been asked of me, I should certainly have declined to imdertake it
Since 1860, when I ceased to be a Professor of Mathematics, I have paid no special attention to
general systems of Sets, Matrices, or Algebras; and without much further knowledge I should not
attempt to write in any detail about such subjects. I may, however, call attention to the facts
which follow : for they api)ear to be decisive of the question now raised. Cauchy (Comptes Rendus,
10/1/53) claimed quatemia as a special case of his *'clofs algdbriques." Grassmann, in turn (Comptes
Retidus, 17/4/54 ; and Crelle, 49), declared Cauch/s methods to be precisely those of the Aus-
dehnungslehre. But Hamilton (Lectures, Pref. p. 64, foot-note), says of the clefs alg^briques (and
therefore, on Qrassmann^s ovm showing, of the methods of the Ausdehnimgslehre) that they are ^^ included
in that theory of Sets in algebra announced by me in 1835 of which Sets I have always con-
sidered the Quaternions to be merely a particular case."

But all this has nothing to do with Quaternions, regarded as a calculus ** uniquely adapted to
Euclidian space." Grassmann lived to have his fling at them, but (so far as I know) he ventured
on no claim to priority. Hamilton, on the other hand, even after reading the first Ausdehnungslehre,
did claim priority and was never answered. He quoted, and commented upon, the very passage (of
the Preface to that work) my allusion to which is censured by Prof. Gibbs. [Lectures, Pref. p. 62,
footnote.] I still think, and it would seem that Hamilton also thought, that it was solely because
Qrassmann had not realized the conception of the quaternion, whether as fia or as ^~\ that he felt
those difficulties (as to angles in space) which he says he had not had leisure to overcome. I have
not seen the original work, but I have consulted what professes to be a verbatim reprint, produced
under the author's supervision. [Die Ausdehnungslehre von 1844, oder die lineale Ausdehnungslehre, ttc.
Ziceite, iin Text unveninderte Aufiage. Leipzig, 1878.] Prof. Gibbs* citations from my article give a
very incomplete and one-sided representation of the few remarks I felt it necessary and sufficient to
make al)out Grassmann. I need not quote them here, as anyone interested in the matter can readily
consult the ai-ticle.

In regard to Matrices, I do not think I have ever claimed anything for Hamilton beyond the
separable <^ and the symbolic cubic (or biquadratic, as the case may be) with its linear factors ;
and those I still assort to he exclusively his. My own work in this direction has been confined to
Hamilton's 0, with its square root, its applications to stress and strain, &c.

As to the general history, of which (as I have said above) I claim no exact or extensive
knowledge, Cayley and Sylvester will, no doubt, defend themselves if they see fit It would be at
once ridiculous and im^tertinent on my part were I to take up the cudgels in their behalf.

P. G. Tatt.

cxxx,]

457

cxxx.

RADIATION AND CONVECTION.

[From Encydopiedia Brttannicat 1886.]

1, When a red-hot canDon ball is taken out of a furnace snd suspended in the
air it is observed to cool, i.e„ to part with heat, and it continues to do ao at a
gradually diminishing rate till it finally reaches the temperature of the room. But
the process by which this effect is produced is a very complex one* If the hand be
held at a distance of a few inches from the hot ball on either side of it or below
it, the feeling of warmth experienced is considerable ; but it becomes intolerable when
the hand is held at the same distance ab&ve the ball. Even this rude form of experi-
ment is sufficient to show that two processes of cooling are simultaneously at work,
— one which apparently leads to the loss of heat in aU directions iudifiFerently,
another which leads to a special loss in a vertical direction upwards. If the experi-
ment is made in a dark room, into which a raj of sunlight is admitted so as to
throw a shadow of the ball on a screen, we me that the column of air above the
ball also casts a distinct shadow. It is, in fact, a column of air very irregularly
heated by contact with the ball, and rising, in obedience to hydrostatic laws, in the
colder and denser air around it. This conveyance of heat by the motion of the heated
body itself is called convection ; the process by which heat is lost iudifiFerently in all
directions is called radiation. These two processes are entirely different in their
nature, laws, and mechauism ; but we have to treat of both in the present article.

2- To illustrate how the third method by which heat can be transferred, viz.
Conduction, is involved in this process, let the cannon ball (which for this purpose
should be a large ooe) be again heated and at once immersed in water until it just
ceases to be luminous in the dark, and then be immediately hung up in the air.
After a short period it again becomes red-hot all over, and the phenomenon then
T, IL 58

458 RADIATION AND CONVECTION. [CXXX.

proceeds precisely as before, except that the surface of the ball does not become so
hot as it was before being plunged in the water. This form of experiment, which
requires that the interior shall be very considerably cooled before the surface ceases
to be self-luminous, does not succeed nearly so well with a copper ball as with an
iron one, on account of the comparatively high conductivity of copper. In fact, even
when its surface is covered with lamp-black, to make the loss by radiation as great
as possible, the difference of temperature between the centre and the surface of a
very hot copper ball — which is only an inch or two in diameter — is inconsiderable.

3. In conduction there is passage of heat from hotter to colder parts of the
same body; in convection an irregularly heated fluid becomes hydrostatically unstable,
and each part carries its heat with it to its new position. In both processes heat is
conveyed from place to place. But it is quite otherwise with radiation. That a body
cools in consequence of radiation is certain ; that other bodies which absorb the radiation
are thereby heated is also certain; but it does not at all follow that what passes in the
radiant form is heat. To return for a moment to the red-hot cannon ball. If, while
the hand is held below it, a thick but dry plate of rock-salt is interposed between
the ball and the hand there is no perceptible diminution of warmth, and the tem-
perature of the salt is not perceptibly raised by the radiation which passes through
it. When a piece of clear ice is cut into the form of a large burning-glass it can
be employed to inflame tinder by concentrating the sun's rays, and the lens does the
work nearly as rapidly as if it had been made of glass. It is certainly not what we
ordinarily call ''heat" which can be transmitted under conditions like these. Radiation
is undoubtedly a transference of energy, which was in the form commonly called heat
in the radiating body, and becomes heat in a body which absorbs it; but it is trans-
formed as it leaves the first body, and retransformed when it is absorbed by the second.
Until the comparatively recent full recognition of the conservation and transformation
of energy it was almost impossible to form precise ideas on matters like this; and, con-
sequently, we find in the writings even of men like Provost and Sir J. Leslie notions of
the wildest character as to the mechanism of radiation. Leslie, strangely, regarded it
as a species of "pulsation" in the air, in some respects analogous to sound, and
propagated with the same speed as sound. Provost, on the other hand, says, " Le
calorique est un fluide discret ; chaque dl^ment de calorique suit constamment la mdme
ligne droite, tant qu'aucun obstacle ne Tarrdte. Dans un espace chaud, chaque point est
traverse sans cesse en tout sens par des filets de calorique."

4. The more intensely the cannon ball is heated the more luminous does it become,
and also the more nearly white is the light which it gives out. So well is this
kuown that in almost all forms of civilized speech there are terms corresponding to
our " red-hot," " white-hot," &c. As another instance, suppose a powerful electric current
is made to pass through a stout iron wire. The wire becomes gradually hotter, up
to a certain point, at which the loss by radiation and convection just balances the gain
of heat by electric resistance. And as it becomes hotter the amount of its radiation
increases, till at a definite temperature it becomes just visible in the dark by red
rays of low refrangibility. As it becomes still hotter the whole radiation increases;

CXXX.] HADIATION AND CONVECTION. 459

the red rays formerly given oflf become more luminous, and are joined by others of
higher refrangibility. This process goes on, the whole amount of radiation still increasing,
each kind of visible light becoming more intense, and new rays of light of higher
refrangibility coming in, until the whole becomes white, i.e., gives oflf all the more
efficient kinds of visible light in much the same relative proportion as that in which
they exist in sunlight When the circuit is broken, exactly the same phenomena occur
in the reverse order, the various kinds of light disappearing later as their refrangi-
bility is less. But the radiation continues, growing weaker every instant, even after
the whole is dark. This simple observation irresistibly points to the conclusion that
the so-called "radiant heat" is precisely the same phenomenon as "light," only the
iu visible rays are still less refrangible than the lowest red, and that our sense of sight
is confined to rays of a certain definite range of refrangibility, while the sense of touch
comes in where sight fails us. Sir W. Herschel in 1798, by placing the bulb of a
thermometer in the solar spectrum formed by a flint-glass prism, found that the highest
temperature was in the dark region outside the lowest visible red, — ^a result amply
verified at the time by others, though warmly contested by Leslie.

5. This striking conclusion is not without close analogies in connection with the
other senses, especially that of hearing. Thus it has long been known that the "range
of hearing" differs considerably in different individuals, some, for instance, being pain-
fully affected by the chirp of a cricket, which is inaudible to others whose general
hearing is quite as good. Extremely low notes, on the other hand, of whose existence we
have ample dynamical evidence, are not heard by any one; when perceived at all they
are felt

6. We may now rapidly run over the principal facts characteristic of the behaviour
of visible rays, and point out how far each has been found to characterize that of so-
called "radiant heat" under similar conditions.

(a) Rectilinear propagation: an opaque screen which is placed so as to intercept
the sun's light intercepts its heat also, whether it be close to the observer, at a few
miles from him (as a cloud or a mountain), or 240,000 miles off (as the moon in a total
eclipse), (b) Speed of propagation: this must be of the same order of magnitude, at
least, for both phenomena, i,e., 186,000 miles or so per second; for the sun's heat ceases
to be perceptible the moment an eclipse becomes total, and is perceived again the
instant the edge of the sun's disk is visible, (c) Reflexion: the law must be exactly
the same, for the heat-producing rays from a star are concentrated by Lord Rosse's great
reflector along with its light, (d) Refraction: when a lens is not achromatic its
principal focus for red rays is farther off than that for blue rays; that for dark heat
is still farther off Herschel's determination of the warmest region of the spectrum
(§4 above) is another case in point, (e) Oblique radiation: an illuminated or a self-
luminous surface appears equally bright however it is inclined to the line of sight. The
radiation of heat from a hot blackened surface (through an aperture which it appears
to fill) is sensibly the same however it be inclined (Leslie, Fourier, Melloni). (/) In-
tensity: when there is no absorption by the way the intensity of the light received
from a laminons pointHBOuree is inversely as the square of the distance. The same

58—2

458 RADIATION AND CONVECTION. [CXXX.

proceeds precisely as before, except that the surface of the ball does not become so
hot as it was before being plunged in the water. This form of experiment, which
requires that the interior shall be very considerably cooled before the surface ceases
to be self-luminous, does not succeed nearly so well with a copper ball as with an
iron one, on account of the comparatively high conductivity of copper. In &ct, even
when its surface is covered with lamp-black, to make the loss by radiation as great
as possible, the difference of temperature between the centre and the surface of a
very hot copper ball — which is only an inch or two in diameter — is inconsiderable.

3. In conduction there is passage of heat from hotter to colder parts of the
same body; in convection an irregularly heated fluid becomes hydrostatically unstable,
and each part carries its heat with it to its new position. In both processes heat is
conveyed from place to place. But it is quite otherwise with radiation. That a body
cools in consequence of radiation is certain ; that other bodies which absorb the radiation
are thereby heated is also certain; but it does not at all follow that what passes in the
radiant form is heat. To return for a moment to the red-hot cannon ball. If, while
the hand is held below it, a thick but dry plate of rock-salt is interposed between
the ball and the hand there is no perceptible diminution of warmth, and the tem-
perature of the salt is not perceptibly raised by the radiation which passes through
it. When a piece of clear ice is cut into the form of a large burning-glass it can
be employed to inflame tinder by concentrating the sun's rajrs, and the lens does the
work nearly as rapidly as if it had been made of glass. It is certainly not what we
ordinarily call " heat " which can be transmitted under conditions like these. Radiation
is undoubtedly a transference of energy, which was in the form commonly called heat
in the radiating body, and becomes heat in a body which absorbs it; but it is trans-
formed as it leaves the first body, and retransformed when it is absorbed by the second.
Until the comparatively recent full recognition of the conservation and transformation
of energy it was almost impossible to form precise ideas on matters like this; and, con-
sequently, we find in the writings even of men like Provost and Sir J. Leslie notions of
the wildest character as to the mechanism of radiation. Leslie, strangely, regarded it
as a species of "pulsation" in the air, in some respects analogous to sound, and
propagated with the same speed as sound. Provost, on the other hand, says, " Le
calorique est un fluide discret ; chaque dl^ment de calorique suit oonstamment la mdme
ligne droite, tant qu'aucun obstacle ne TarrSte. Dans un espace chaud, chaque point est
traverse sans cesse en tout sens par des filets de calorique."

CXXX.] HADIATION AND CONVECTION. 459

the red rays formerly given oflf become more luminous, and are joined by others of
higher refrangibility. This process goes on, the whole amount of radiation still increasing,
each kind of visible light becoming more intense, and new rays of light of higher
refrangibility coming in, until the whole becomes white, i,e., gives oflf all the more
efficient kinds of visible light in much the same relative proportion as that in which
they exist in sunlight. When the circuit is broken, exactly the same phenomena occur
in the reverse order, the various kinds of light disappearing later as their refrangi-
bility is less. But the radiation continues, growing weaker every instant, even after
the whole is dark. This simple observation irresistibly points to the conclusion that
the so-called "radiant heat" is precisely the same phenomenon as "light," only the
iu visible ra}rs are still less refrangible than the lowest red, and that our sense of sight
is confined to rays of a certain definite range of refrangibility, while the sense of touch
comes in where sight fails us. Sir W. Herschel in 1798, by placing the bulb of a
thermometer in the solar spectrum formed by a flint-glass prism, found that the highest
temperature was in the dark region outside the lowest visible red, — ^a result amply
verified at the time by others, though warmly contested by Leslie.

(a) Rectilinear propagation: an opaque screen which is placed so as to intercept
the sun's light intercepts its heat also, whether it be close to the observer, at a few
miles from him (as a cloud or a mountain), or 240,000 miles off (as the moon in a total
eclipse), (b) Speed of propagation: this must be of the same order of magnitude, at
least, for both phenomena, i,e., 186,000 miles or so per second; for the sun's heat ceases
to be perceptible the moment an eclipse becomes total, and is perceived again the
instant the edge of the sun's disk is visible, (c) Reflexion: the law must be exactly
the same, for the heat-producing rays from a star are concentrated by Lord Rosse's great
reflector along with its light, (d) Refraction : when a lens is not achromatic its
principal focus for red rays is farther off than that for blue rays; that for dark heat
is still farther off Herschel's determination of the warmest region of the spectrum
(§ 4 above) is another case in point, (e) Oblique radiation: an illuminated or a self-
luminous surface appears equally bright however it is inclined to the line of sight. The
radiation of heat from a hot blackened surface (through an aperture which it appears
to fill) is sensibly the same however it be inclined (Leslie, Fourier, Melloni). (/) In-
tensity: when there is no absorption by the way the intensity of the light received
from a luminous point-source is inversely as the square of the distance. The same

58—2

458 RADIATION AND CONVECTION. [CXXX.

proceeds precisely as before, except that the surface of the ball does not become so
hot as it was before being plunged in the water. This form of experiment, which
requires that the interior shall be very considerably cooled before the surface ceases
to be self-luminous, does not succeed nearly so well with a copper ball as with an
iron one, on account of the comparatively high conductivity of copper. In &ct, even
when its surface is covered with lamp-black, to make the loss by radiation as great
as possible, the diflference of temperature between the centre and the surface of a
very hot copper ball — which is only an inch or two in diameter — is inconsiderable.

3. In conduction there is passage of heat from hotter to colder parts of the
same body; in convection an irregularly heated fluid becomes hydrostatically unstable,
and each part carries its heat with it to its new position. In both processes heat is
conveyed from place to place. But it is quite otherwise with radiation. That a body
cools in consequence of radiation is certain ; that other bodies which absorb the radiation
are thereby heated is also certain; but it does not at all follow that what passes in the
radiant form is heat. To return for a moment to the red-hot cannon ball. If, while
the hand is held below it, a thick but dry plate of rock-salt is interposed between
the ball and the hand there is no perceptible diminution of warmth, and the tem-
perature of the salt is not perceptibly raised by the radiation which passes through
it. When a piece of clear ice is cut into the form of a large burning-glass it can
be employed to inflame tinder by concentrating the sun's rays, and the lens does the
work nearly as rapidly as if it had been made of glass. It is certainly not what we
ordinarily call '* heat " which can be transmitted under conditions like these. Radiation
is undoubtedly a transference of energy, which was in the form commonly called heat
in the radiating body, and becomes heat in a body which absorbs it; but it is trans-
formed as it leaves the first body, and retransformed when it is absorbed by the second.
Until the comparatively recent full recognition of the conservation and transformation
of energy it was almost impossible to form precise ideas on matters like this; and, con-
sequently, we find in the writings even of men like Provost and Sir J. Leslie notions of
the wildest character as to the mechanism of radiation. Leslie, strangely, regarded it
as a species of ''pulsation" in the air, in some respects analogous to sound, and
propagated with the same speed as sound. Provost, on the other hand, says, " Le
calorique est un fluide discret ; chaque dldment de calorique suit oonstamment la mdme
ligne droite, tant qu'aucun obstacle ne Tarrdte. Dans un espace chaud, chaque point est
traverse sans cesse en tout sens par des filets de calorique."

CXXX.] HADIATION AND CONVECTION. 459

the red rays formerly given oflf become more luminous, and are joined by others of
higher refrangibility. This process goes on, the whole amount of radiation still increasing,
each kind of visible light becoming more intense, and new rays of light of higher
refrangibility coming in, until the whole becomes white, i.e., gives oflf all the more
efficient kinds of visible light in much the same relative proportion as that in which
they exist in sunlight. When the circuit is broken, exactly the same phenomena occur
in the reverse order, the various kinds of light disappearing later as their refrangi-
bility is less. But the radiation continues, growing weaker every instant, even after
the whole is dark. This simple observation irresistibly points to the conclusion that
the so-called "radiant heat" is precisely the same phenomenon as "light," only the
iu visible rays are still less refrangible than the lowest red, and that our sense of sight
is confined to rays of a certain definite range of refrangibility, while the sense of touch
comes in where sight fails us. Sir W. Herschel in 1798, by placing the bulb of a
thermometer in the solar spectrum formed by a flint-glass prism, found that the highest
temperature was in the dark region outside the lowest visible red, — a, result amply
verified at the time by others, though warmly contested by Leslie.

(a) Rectilinear propagation: an opaque screen which is placed so as to intercept
the sun's light intercepts its heat also, whether it be close to the observer, at a few
miles from him (as a cloud or a mountain), or 240,000 miles off (as the moon in a total
eclipse). (6) Speed of propagation: this must be of the same order of magnitude, at
least, for both phenomena, i.e,, 186,000 miles or so per second; for the sun's heat ceases
to be perceptible the moment an eclipse becomes total, and is perceived again the
instant the edge of the sun's disk is visible, (c) Reflexion: the law must be exactly
the same, for the heat-producing rays from a star are concentrated by Lord Rosse's great
reflector along with its light, (d) Refraction: when a lens is not achromatic its
principal focus for red rays is farther off than that for blue rays; that for dark heat
is still farther off. Herschel's determination of the warmest region of the spectrum
(§4 above) is another case in point, (e) Oblique radiation: an illuminated or a self-
luminous surface appears equally bright however it is inclined to the line of sight. The
radiation of heat from a hot blackened surface (through an aperture which it appears
to fill) is sensibly the same however it be inclined (Leslie, Fourier, Melloni). (/) In-
tensity: when there is no absorption by the way the intensity of the light received
from a luminous point-source is inversely as the square of the distance. The same

58—2

458 RADIATION AND CONVECTION. [CXXX.

3. In conduction there is passage of heat from hotter to colder parts of the
same body; in convection an irregularly heated fluid becomes hydrostatically unstable,
and each part carries its heat with it to its new position. In both processes heat is
conveyed from place to place. But it is quite otherwise with radiation. That a body
cools in consequence of radiation is certain ; that other bodies which absorb the radiation
are thereby heated is also certain; but it does not at all follow that what passes in the
radiant form is heat. To return for a moment to the red-hot cannon ball. If, while
the hand is held below it, a thick but dry plate of rock-salt is interposed between
the ball and the hand there is no perceptible diminution of warmth, and the tem-
perature of the salt is not perceptibly raised by the radiation which passes through
it. When a piece of clear ice is cut into the form of a large burning-glass it can
be employed to inflame tinder by concentrating the sun's rays, and the lens does the
work nearly as rapidly as if it had been made of glass. It is certainly not what we
ordinarily call " heat " which can be transmitted under conditions like these. Radiation
is undoubtedly a transference of energy, which was in the form commonly called heat
in the radiating body, and becomes heat in a body which absorbs it; but it is trans-
formed as it leaves the flrst body, and retransformed when it is absorbed by the second.
Until the comparatively recent full recognition of the conservation and transformation
of energy it was almost impossible to form precise ideas on matters like this; and, con-
sequently, we find in the writings even of men like Provost and Sir J. Leslie notions of
the wildest character as to the mechanism of radiation. Leslie, strangely, regarded it
as a species of "pulsation" in the air, in some respects analogous to sound, and
propagated with the same speed as sound. Provost, on the other hand, says, " Le
calorique est un fluide discret ; chaque dl^ment de calorique suit oonstamment la mSme
ligne droite, tant qu'aucun obstacle ne larrdte. Dans un espace chaud, chaque point est
traverse sans cesse en tout sens par des filets de calorique.''

4. The more intensely the cannon ball is heated the more luminous does it become,
and also the more nearly white is the light which it gives out. So well is this
kuown that in almost all forms of civilized speech there are terms correspondinir to
our " red-hot," " white-hot," &c. As another instance, suppose a powerful electric current
is made to pass through a stout iron wire. The wire becomes gradually hotter, up
to a certain point, at which the loss by radiation and convection just balances the gain
of heat by electric resistance. And as it becomes hotter the amount of its radiation
increases, till at a definite temperature it becomes just visible in the dark by red
rays of low refrangibility. As it becomes still hotter the whole radiation increases*

BADIATION AND CONYECTION.

459

the r^ raye formerly given off become more luminons, and are joined by others of
higher refraiigibility. This process goes oE, the whole amount of radiation still increasing,
each kind of visible light becoming more intense, and new rays of light of higher
refrangibility coming in, until the whole becomes white, i.e., gives off all the more
efficient kinds of visible light in much the same relative proportion as that in which
they exist in sunlight When the circuit is broken, exactly the same phenomena occur
in the reverse order, the various kinds of light disappearing later as their refrangi-
bility is less. But the radiation continues, growing weaker every instant, even after
the whole is dark. This simple observation irresistiblj points to the conclusion that
the so-called "radiant beat'' is precisely the same phenomenon as "light," only the
tu visible rays are still less refrangible than the lowest red, and that our sense of sight
is confined to rays of a certain definite range of refrangibility, while the sense of touch
comes in where sight fails us. Sir W, Herschel in 1798| by placing the bulb of a
thermometer in the solar spectrum formed by a flint-glass prism, found that the highest
temperature was in the dark region outside the lowest visible red, — ^a result amply
verified at the time by others, though warmly contested by Leslie,

5. This striking conclusion is not without close analogies in connection with the
other senses, especially that of hearing. Thu;^ it has long been known that the " range
of hearing" differs considerably in different individuals, some, for instance, being pain*
fully affected by the chirp of a cricket, which is inaudible to others whose general
hearing is quite as good* Extremely low notes, on the other hand, of whose existence we
have ample dynamical evidence, are not heard by any one ; when perceived at all they
are felt

6* We may now rapidly run over the principal facta characteristic of the behaviour
of visible rays, and point out how far each has been found to characterize that of so-
called "radiant heat" under nimilar conditions,

{a) Rectilinear propagation : an opaque screen which is placed so as to intercept
the sun's light intercepts its heat also, whether it be close to the observer, at a few
miles from him (as a cloud or a mountain), or 240,000 miles off (as the moon in a total
eclipse )♦ (t) Speed of propagation ; this must be of the same order of magnitude, at
least, for both phenomena, t.eL, 186,000 miles or so per second; for the sun's heat ceases
to be perceptible the moment an eclipse becomes total, and is perceived again the
instant the edge of the sun*s disk is visible, (c) Reflexion : the law must be exactly
the same> for the heat-producing rays from a star are concentrated by Lord Rosse's great
reflector along with its light, (d) Refraction : when a lens is not achromatic its
principal focus for red rays is farther off than that for blue rays; that for dark heat
is still farther off. Herschel's determination of the warmest region of the spectrum
(§ 4 above) is another case in point* {b) Oblique radiation ; an illuminated or a self-
luminous surface appears equally bright however it is inclined to the line of sight. The
radiation of heat from a hot blackened surface (through an aperture which it appears
to fill) is sensibly the same however it be inclined (Leslie, Fourier, Melloni)^ (/) In-
tensity: when there is no absorption by the way the intensity of the light received
from a luminous point-soiirce is inversely as the square of the distance. The same

58—2

460 RADIATION AND CONVECTION. [CXXX.

is true of dark heat. But this is not a new analogy; it is a mere consequence of
(a) rectilinear propagation, (g) Selective absorption: light which has been sifted by
passing through one plate of blue glass passes in much greater percentage through
a second plate of the same glass, and in still greater percentage through a third. The
same is true of radiant heat, even when the experiment is made with uncoloured
glass ; for clear glass absorbs certain colours of dark heat more than others (De Laroche,
Melloni). (h) Interference bands, whether produced by two mirrors or by gratings,
characterize dark heat as well as light; only they indicate longer waves (Fizeau and
Foucault). (%) Polarization and double refraction: with special apparatus, such as plates
of mica split by heat into numerous parallel films, the polarization of dark heat is
easily established. When two of these bundles are so placed as to intercept the heat
an unsplit film of mica interposed between them allows the heat to pass, or arrests
it, as it is made to rotate in its own plane (Forbes), (j) By proper chemical ad-
justments photographs of a region of the solar spectrum beyond the visible red have
been obtained (Abney). We might mention more, but those given above, when con-
sidered together, are conclusive. In fact (b) or (t) alone would almost settle the
question.

7. But there is a superior as well as an inferior limit of visible rays. Light
whose period of vibration is too small to produce any impression on the optic nerve
can be degraded by fluorescence into visible rays, and can also be detected by its
energetic action on various photographic chemicals. In fact photographic portraits can
be taken in a room which appears absolutely dark to the keenest eyesight. By one
or other of these processes the solar spectrum with its dark lines and the electric arc
with its bright lines have been delineated to many times the length of their visible
ranges. The electric arc especially gives (in either of these ways) a spectrum of extra-
ordinary length; for we can examine it, as we can not examine sunlight, before it has
suffered any sensible absorption.

8. Thus radiation is one phenomenon, and (as we shall find) the spectrum of a
black body (a conception roughly realized in the carbon poles of an electric lamp) is
continuous from the longest possible wave-length to the shortest which it is hot enough
to emit. These various groups of rays, however, are perceived by us in very different
ways, whether by direct impressions of sense or by the different modes in which they
effect phjrsical changes or transformations. The only way as yet known to us of treating
them all alike is to convert their energy into the heat-form and measure it as such.
This we can do in a satisfactory manner by the thermo-electric pile and galvano-
meter.

9. Of the history of the gradual development of the theory of radiation we can
give only the main features. The apparent concentration of cold by a concave mirror,
which had been long before observed by Porta, was rediscovered by Pictet, and led
to the extremely important enunciation of the Law of Exchanges by Prevost in 1791.
As we have already seen, Provost's idea of the nature of radiation was a corpuscular one,
no doubt greatly influenced in this direction by the speculations of Lesage. But the

oeupl]

BADTATION AND CONTECTION.

461

value of his theory as a concise statement of facts and a mode of co-oTdiDating them
is not thereby materially lessened. We give hie own statements in the following close
paraphrase, in which the italics are retained, from sect, ix. of his Du OalofHque Rayon-
mnt (Geneva, 1809).

**L Fi^e caloric is a radiant fluid. And because caloric becomes free at the surfaces
of bodies every point of the mrface of a body is a centre, towards and from which JUa-
7nmits (filets) of caloric move in all directione,

*' 2. Heat equilibrium between two neighbouring free spaces oo^iMsta in equalittf
of exchange.

"3. When equilibrium is interfered with it is re-established by inequalities of
exchange. And, in a medium of constant temperature, a hotter or a colder body reaches
this temperature according to the law that difference of temperature diminishes in
gemnetrical progression in successive equal intervals of time,

*' 4. If into a locality at unifoT^m temperature a reflecting or refracting surface is
introduced, it has no effect in the way of changing the temperalure at any point in that
locality,

" 5, If into a locality otherwise at uniform temperature there is introduced a
warmer or a colder body, and newt a reflecting or refracting surface , the points on
which ih£ rays emanating from the body are throum by these surfaces vrill be affected,
in the sense of being warmed if the body is warmer, and cooled if it is colder.

'* 6* A reflecting body, heated or cooled in its interior, will acquire the mrronnd-
ing temperature more sloivly than would a n on ^ reflector,

**7- A reflecting body, heated or cooled in its interior, will less affect (in the
way of heating or cooling it) another body placed at a little distance than would a
non-reflecting body under the same circumstances,

'*A11 these consequences have been verified by experiment, except that which re^rds
the refraction of cold This experiment remains to be made, and I confidently predict
the result, at least if the refraction of cold can be accurately observed. This result is
indicated in the fourth and fifth consequences [above], and they might thus be subjected
to a new test. It is scarcely necessary to point out here the precautions requisite to
guard against illusory results of all kinds in this matter/'

10. There the matter rested, so far as theory is concern^, for more than half a
century. Leslie and, after him, many others added fact by fact, up to the time of
De la Provostaye and Desains, whose experiments pointed to a real improvement of
the theory in the form of specialization. But, though such experiments indicated, on
the whole, a proportionality between the radiating and absorbing powers of bodies and
a diminution of both in the case of highly reflecting surfaces, the anomalies frequently
met with (depending on the then unrecognized colour-difl^erences of various radiations)
prevented any grand generalisation. The first real step of the general theory, in
advance of what Provost had achieved, and it was one of immense import, was made
by Balfour Stewart in lS5d. Before we take it up, however, we may briefly consider

cxxx.]

BADIATION AND CX>NVECTION,

463

is the idea of iJte ah&olntB uniformity {qualitative as well as qimntitative) of the radiation
at all points, and in all directions, within an enclosure impervione to heat» when thermal
equilibrittm has once been arrived at. (So strongly does he insist on this point that
he even states that, whatever be the nature of the bodies in the enclosure, the
radiation there will, when equilibrium is established, be that of a black body at
the same temperature. He does not expressly say that the proposition will still be true
even if the bodies can radiate, and therefore absorb, one definite wave-length only;
but this is a legitimate deduction from his statements. To this we will recur.) His
desire to escape the difficulties of surface-reflexion led him to consider the radiation
inside an imperfectly transparent body in the enclosure above spoken of. He thus
arrived at an immediate proof of the existence of internal radiation, which recruits the
stream of radiant heat in any direction step by step precisely to the amount by which
it has been weakened by absorption. Thus the radiation and absorption rigorously
compensate one another, not merely in quantity but in quality also, ao that a body
which is specially absorptive of one particular ray is in the same proportion specially
radiative of the same ray, its temperature being the same in both cases. To complete
the statementp all that is necessary is to show how one ray may diifer from another,
viz., in intensity! wave-length, and polarization.

14. The illustrations which Stewart brought forward in support of his theory are
of the two following kinds. (1) He experimentally verified the exiatence of internal
radiation, to which his theory had led him. This he did by showing that a thick
plate of rock-salt (chosen on account of its comparative transparency to heat-radiations)
radiates more than a thin one at the same temperature, — surrounding bodies being
in this case of course at a lower temperature, so that the effect should not be masked
by transmission. The same was found true of mica and of glass. (2) He showed
that each of these bodies is more opaque to radiations from a portion of its own
substance than to radiation in general. Then comes his conclusion, based, it will be
observed, on his fundamental assumption as to the nature of the equilibrium radiation
in an enclosure. It is merely a detailed explanation that, once equilibrium has been
arrived at, tbe consequent uniformity of radiation throughout the interior of a body
requires the step-by-step compensation already mentioned. And thus he finally arrives
at the statement that at any temperature a body's radiation is exactly the same both
as to quality and quantity as that of its absorption fix)m the radiation of a black
body at the same temperature. In symbolical language Stewart's proposition (extended
in virtue of a principle always assumed) amounts to this : — at any one temperature
let ii be the radiation of a black body, and eR (where e is never greater than 1)
that of any other substance, both for the same definite wave-length; then the substance
willj while at that temperature, absorb the fraction e of radiation of that wave-length,
whatever be the source from which it comes. The last clause contains the plausible
assumption already referred to. Stewart proceeds to show, in a very original and
ingenious way, that his result is compatible with the known facts of reflexion, refraction,
&c., and arrives at the conclusion that for internal radiation parallel to a plane the
amount is (in isotropic bodies) proportional to the refractive index. Of coui-se, when
the restriction of parallelism to a plane is removed the internal radiation is found

464 BADIATION AND CONVECTION. [[CXXX.

to be proportional to the square of the refractive index. This obvious completion of
the statement was first given by Stewart himself at a somewhat later date.

15. So far Stewart had restricted his work to "dark heat," as it was then
called; and he says that he did so expressly in order to confine himself to rays
"which were universally acknowledged to produce heat by their absorption." But he
soon proceeded to apply himself to luminous radiations. And here he brought forward
the extremely important fact that ''coloured glasses invariably lose their colour in the
fire" when exactly at the temperature of the coals behind them, %.e., they compensate
exactly for their absorption by their radiation. But a red glass when colder than
the coals behind appears red, while if it be hotter than they are it appears green.
He also showed that a piece of china or earthenware with a dark pattern on a light
ground appears to have a light pattern on a dark ground when it lb taken out of
the fire and examined in a dark room. Hence he concluded that his extension of
Provost's theory was true for luminous rays also.

16. In this part of the subject he had been anticipated, for Fraunhofer had
long ago shown that the fiame of a candle when examined by a prism gives bright
lines (t.e., maxima of intensity of radiation) in the position of the constituents of a
remarkable double dark line (i.e., minima of radiation) in the solar spectrum, which
he called D, Hallows Miller had afterwards more rigorously verified the exact coin-
cidence of these bright and dark lines. But Foucault* went very much farther, and
proved that the electric arc, which shows these lines bright in its spectrum, not
only intensifies their blackness in the spectrum of sunlight transmitted through it,
but produces them as dark lines in the otherwise continuous spectrum of the light
from one of the carbon points, when that light is made by reflexion to pass through
the arc. Stokes about 1850 pointed out the true nature of the connection of these
phenomena, and illustrated it by a djmamical analogy drawn from sound. He stated
his conclusions to Sir W. Thomsonf, who (from 1852 at least) gave them regularly
in his public lectures, always pointing out that one constituent of the solar atmo-
sphere is certainly sodium, and that others are to be discovered by the coincidences
of solar dark lines with bright lines given by terrestrial substances rendered incan-
descent in the state of vapour. Stokes's analogy is based on the fact of synchronism
(long ago discussed by Hooke and others), viz., that a musical string is set in
vibration when the note to which it is tuned is sounded in its neighbourhood.
Hence we have only to imagine a space containing a great number of such strings,
all tuned to the same note. Such an arrangement would form, as it were, a medium
which, when agitated, would give that note, but which would be set in vibration by,
and therefore diminish the intensity of, that particular note in any mixed sound which
passed through it.

17. Late in 1859 appeared Kirchhoff's first paper on the subject^. He supplied
one important omission in Stewart's development of the theory by showing why it is

* VInttitut, 7th Febniftiy, 1S49; see Phil. Mag., 1860, i. p. 193.

t Brit, Assoc., Preaident's address, 1871. :;: Pogg, Ann., or Phil Mag., 1860.

oxxx.]

RADIATION AND CONVTICTION,

465

neceasary to use as an absorbing body one colder than the soiLrce in order to produce

reversal of spectral lines. This we will presently consider, Kirchhoff s proof of the
equality of radiating and absorbing powera is an elaborate but unnecessary piece of
mathematics, called for in consequence of his mode of attacking the question. He
chose to limit his reasoning to special wave-lengtha by introducing the complex
mechanism of the colours of thin plates aod a cooaequeot appeal to Fourier's theorem
instead of to the obviously permissible assumption of a substance imperfectly trans-
parent for one special wave-length, but perfectly transparent for all others; and he
did not, as Stewart had done, carry his reasoning into the interior of the body.
With all its elaboration, his mode of attacking the question leads us no farther than
could Stewart's. Both are ultimately based on the final equilibrium of temperature
in an enclosure, required by Camot's principle, and both are, as a consequence, equally
inapplicable to exceptional cases, such as the behaviour of fluorescent or phosphorescent
substances. In fact (see "Thermodynamics/* No. CXXXI, below)jCarnot's principle is
established only on a statistical basis of averages, and is not necessarily tnie when
we are dealing with portions of space, which, though of essentially finite dimensione,
are extremely small in comparison with the sentient part of even the tiniest instrument
for measuring temperature,

18. Kirchhotf s addition to Stewart's result may be given as follows. Let radiation
r, of the same particular wave<length as that spoken of in § 14, fall on the substance;
er of it will be absorbed, and (1— e)r transmitted. This will be recruited by the
radiation of the substance itself, so that the whole amount for that particular w^ave-
length becomes (1 — e)r + ei£, or r — €(r — Ry Thus the radiation is weakened only
when R<r, sl condition which requires that the source (even if it be a black body)
should be at a higher temperature than the absorbing substance (§ 4, above). But
the converse is, of course, not necesimrily true. This part of the subject, as well
a» the special work of Kirchhoff and of Bunsen, belongs properly to spectrum
analysis.

19. From the extension of Provost's theory, obtained in either of the ways just
explained, we see at once how the constancy of the radiation in an enclosure is
maintained. In the neighbourhood of and perpendicular to the surfaces of a black
body it is wholly due to radiation, near a transparent body wholly to transmission.
A body which reflects must to the same extent be deficient in its radiation and
transmission ; thus a perfect reflector can neither radiate nor transmit. And a body
which polarizes by reflexion must supply by radiation what is requisite to render the
whole radiation unpolarized A body, such as a plate of tourmalioe, which polarizes
transmitted light, must radiate light polarized in the same plane as that which it
absorbs, Kirch hoflf and Stewart independently gave this beautiful application*

20. Empirical formulee representing more or less closely the law of cooling of
bodies, whether by radiation alone or by simultaneous radiation and convection, have
at least an historic interest What is called Newton's Law of Cooling (see p. 462 above)
was employed by Fourier in his Theorie AnalytiquB de la Gkaleur, Here the rate of

T. n, 59

466 RADIATION AND CONVECTION. [CXXX.

surface-loss was taken as proportional to the excess of temperature over surrounding
bodies. For small differences of temperature it is accurate enough in its applications,
such as to the corrections for loss of heat in experimental determinations of specific
heat, &c., but it was soon found to give results much below the truth, even when the
excess of temperature was only 10° C.

21. Dulong and Petit, by carefully noting the rate of cooling of the bulb of a
large thermometer enclosed in a metallic vessel with blackened walls, from which the
air had been as far as possible extracted and which was maintained at a constant
temperature, were led to propound the exponential formula Aa^ + B to represent the
radiation from a black surface at temperature t As this is an exponential formula,
we may take t as representing absolute temperature, for the only result will be a
definite change of value of the constant A, Hence if ^o be the temperature of the
enclosure, the rate of loss of heat should be -4(a* — a*»), or -4a^(a^"^— 1). The quantity
A was found by them to depend on the nature of the radiating surface, but a was
found to have the constant value 10077. As the approximate accuracy of this ex-
pression was verified by the experiments of De la Provostaye and Desains for
temperature differences up to 200° C, it may be well to point out two of its con-
sequences. (1) For a given difference of temperatures the radiation is an exponential
function of the lower (or of the higher) temperature. (2) For a given temperature
of the enclosure the radiation is as (1-0077/ -1, or 5(1 + 0-00386?+ ...), where 0 is
the temperature excess of the cooling body. Thus the (so-called) Newtonian law gives
4 per cent, too little at 10° C. of difference. ^

22. Dulong and Petit have also given an empirical formula for the rate of loss
by simultaneous radiation and convection. This is of a highly artificial character, the
part due to radiation being as in the last section, while that due to convection is
independent of it, and also of the nature of the surface of the cooling body. It is
found to be proportional to a power of the pressure of the surrounding gas (the
power depending on the nature of the gas), and also to a definite power of the
temperature excess. The reader must be referred to French treatises, especially that
of Desains, for further information.

23. Our knowledge of the numerical rate of surface-emission is as yet scanty,
but the following data, due to Nicol*, may be useful in approximate calculations.
Loss in heat units (1 lb. water raised 1° C. in temperature) per square foot per minute,
from

Bright copper 109 0*51 0*42

Blackened copper 203 146 135.

The temperatures of body and enclosure were 58° C. and 8° C, and the pressure of
contained air in the three columns was about 30, 4, and 0*4 inches of mercury
respectively. The enclosure was blackened.

♦ Proc. R. S. E., VII. 1870, p. 206.

cxxx.]

RADIATION AND CONVECTION-

24. Scanty as is our knowledge of radiation, it is not at all surprising that
that of convection should be almost nil, except as regards some of its practical
applicationa Here we have to deal with a pix)blem of hydrokiuetics of a chai'acter,
even in common cases, of far higher difficulty than nmuy hydrokinetic problems of which
not even approximate solutions have been obtained.

25* What is called Doppler*s Principle has more recently* led Stewart to eome
curious apeculationSj which a simple example will easily explain. Suppose two parallel
plates of the same substance, perfectly transparent except to one definite wave-length,
to be moving towards or from one another Each, we presume, will radiate as befom,
and on that account cool; but the radiation which reaches either is no longer of
the kind which alone it can absorb, whether it come directly from the other, or is
part of its own or of the others radiation reflected from the enclosure* Hence it
would appear that relative motion is incompatible mth temperature equilibrium in
an enclosure^ and thus that there must be some effect analogous to resistance to
the motion. We may get over this difficulty if we adopt the former speculation of
Stewart, referred to iu brackets in § 13 above. For this would lead to the result
that, as soon as either of the bodies has cooled, ever so slightly, the radiation in
the enclosure should become that belonging to a black body of a slightly higher
temperature than before, and thus the plates would be furnished with radiation which
they could at once absorb, and be gradually heated to their former temperature*

26. A very recent speculation, founded by Boltzmannf upon some ideas due to
Bartoli, is closely connected in principle with that just mentioned* This speculation
is highly interesting, because it leads to an expression for the amount of the whole
radiation from a black body in terms of its absolute temperature. Boltzmann's in-
vestigation may be put, as follows, in an exceedingly simple form. It was pointed
out by Clerk-Maxwell, as a result of his electro -magnetic theory of light, that
radiation falling on the surface of a body must produce a certain pressure. It is
easy to see (most simply by the analogy of the virial equation), that the measure
of the pressure per square unit on the surface of an impervious enclosure, in which
there is thermal equilibrium, must be one- third of the whole energy of radiation per
cubic unit of the enclosed space. We may now consider a reversible engine conveying
heat from one black body to another at a different temperature, by operations
alteraately of the isothermal and the (Odiabatic character, which consist in altering the
volume of the enclosurej with or without one of the bodies present in it. For one
of the fundamental equations (p, 478 below) gives

' dE_ dp

dv ^ ^ dt ^'

where t is the absolute temperature. If / be the pressure on unit surface, 3/ is the
energy per unit of volume, and this equation becomes

^di-/=3-^

* BriL Amoc, Rtpart, 1871.

t Wiedfttiaitn't Ann.^ 1884| ini.

59—2

468 RADIATION AND CONVECTION. [CXXX.

Hence it follows at once that, if the fundamental assumptions be granted, the energy
of radiation of a black body per unit volume of the enclosure is proportional to the
fourth power of the absolute temperature. It is not a little remarkable that Stefen*
had some years previously shown that this very expression agrees more closely with
the experimental determinations of Dulong and Petit than does their own empirical
formula.

27, It would appear from this expression that, if an impervious enclosure con-
taining only one black body in thermal equilibrium is separated into two parts by
an impervious partition, any alteration of volume of the part not containing the black
body will produce a corresponding alteration of the radiation in its interior. It will
now correspond to that of a second black body, whose temperature is to that of
the first in the inverse ratio of the fourth roots of the volumes of the detached part of
the enclosure.

28. Lecherf has endeavoured to show that the distribution of energy among
the constituents of the radiation from a black body does not alter with temperature.
Such a result, though apparently inconsistent with many well-known facts, appears
to be consistent with and to harmonize many others. It accords perfectly with the
notion of the absolute uniformity (statistical) of the energy in an enclosure, and its
being exactly that of a black body, even if the contents (as in § 25) consist of a
body which can radiate one particular quality of light alone. And if this be the
case it will abo follow that the intensity of radiation of any one wave-length by any
one body in a given state depends on the temperature in exactly the same way as
does the whole radiation from a black body. Unfortunately this last deduction does
not accord with Melloni's results; at least the discrepance from them would appear
to be somewhat beyond what could fairly be set down to error of experiment. But
it is in thorough accordance with the common assumption (§ 14) that the percentage
absorption of any particular radiation does not depend on the temperature of the
source. The facts of fluorescence and phosphorescence, involving the radiation of
visible rays at temperatures where even a black body is invisible, have not yet been
dealt with under any general theory of radiation ; though Stokes has pointed out a
dynamical explanation of a thoroughly satisfactory character, they remain outside the
domain of Carnot*s principle.

♦ Sitzungtber, d, k, Ak, in Wien, 1879. t Wiedemann' t Ann,, 1882, xvii.

CXXXl.]

CXXXI

THERMODYNAMICS

[Fp&m EncijclopfFdia Bntamnca, 1888.]

In a strict interpretation, this branch of science, somebimes called the Dynamical
Theory of Heat, deals with the relations between heat and work, though it is often
extended so as to include all transformations of energy. Either term is an infelicitous
one, for there is no direct reference to force in the majority of questions dealt with
ill the subject. Even the title of Camot's work, presently to be described, is much
better chose a than is the more modem desiguatiou. On the other hand, such a
German phrase as die bewegende Kraft der Wdrnw is in all respects intolerable.

It has been shown ♦ • * that Newton's enunciation of the conservation of energy
as a general principle of nature was defective in respect of the connection between
work and heat, and that, about the beginning of the present century, this lacuna was
completely filled up by the researches of Rumford and Davy. Joule*s experimental
demonstration of the principle, and his determination of the work-equivalent of heat by
various totally independent processes, have been discussed*

But the conservation of energy, alone, gives us an altogether inadequate basis for
reasoning on the work of a heat-engine. It enables us to calculate how much work
is equivalent to an assigned amount of heat, and vice verad, provided the trans-
formation can be effected ; but it tells us nothing with respect to the percentage of
either which can, under given circumstances, be converted into the other. For this
purpose we require a special case of the law of transformation of energy. This was
first given in Carnot's extraordinary work entitled RefiexioTis swr la Puissance Motrice
du Feu, Paris, 1824*.

^ The anthotp N* L, Badi Camoi (1796 — i@S2), wae Ihe eecond aon of Napoleon 'it celebrated miniAtfix of war,
biinaelf a mattiematician of real note eveti among the wonderful galasy of whlclt France conld then boast.

470 THERMODYNAMICS. [CXXXI.

The chief novelties of Carnot's work are the introduction of the idea of a cycle
of operations, and the invaluable discovery of the special property of a reversible
cycle. It is not too much to say that, without these wonderful novelties, thermo-
dynamics as a theoretical science could not have been developed.

Carnot's work seems to have excited no attention at the time of its publication.
Ten years later (1834) Clapeyron gave some of its main features in an analytical form,
and he also employed Watt's diagram for the exhibition of others. Even this, how-
ever, failed to call attention properly to the extremely novel processes of Carnot, and
it was reserved for Sir W. Thomson (in 1848, and more at length in 1849) to point
out to scientific men their full value. His papers on Carnot's treatise, following closely
after the splendid experimental researches of Colding and Joule, secured for the
djmamical theory of heat its position as a recognized branch of science. James
Thomson, by Carnot's methods, predicted in 1849 the lowering of the freezing point
of water by pressure, which was verified experimentally in the same year by his
brother. Von Helmholtz had published, two years before, a strikingly original and
comprehensive pamphlet on the conservation of energy. The start once given, Rankine,
Clausius, and W. Thomson rapidly developed, though from very different standpoints,
the theory of thermodynamics. The methods adopted by Thomson differed in one special

The delioate constitntion of Sadl was attributed to the agitated circumstanoes of the time of his birth, which
led to the proscription and temporary exile of his parents. He was admitted in 1812 to the £cole Polytechnique,
where he was a fellow-student of the famous Chasles. Late in 1814 he left the school with a commission in
the Engineers, and with prospects of rapid advancement in his profession. But Waterloo and the Restoration
led to a second and final proscription of his father; and, though Sadl was not himself cashiered, he was
purposely told off for the merest drudgeries of his service; *M1 fut envoy6 sucoessivement dans plusieurs places
fortes pour y faire son metier d'ing^nieur, compter des briques, r6parer des pans de muraiUes, et lever des
plans destines k s'enfouir dans les cartons," as we learn from a biographical notice written by his younger
brother. Disgusted with an employment which afforded him neither leisure for original work nor opportunities for
acquiring scientific instruction, he presented himself in 1819 at the examination for admission to the staff-corps
(^tat-major), and obtained a lieutenancy. He now devoted himself with astonishing ardour to mathematics,
chemistry, natural history, technology, and even political economy. He was an enthusiast in music and other
fine arts; and he habitually practised as an amusement, while deeply studying in theory, aU sorts of athletic
sports, including swimming and fencing. He became captain in the engineers in 1827, but left the service
altogether in the following year. His naturally feeble constitution, farther weakened by excessive devotion to
study, broke down finaUy in 1832. A relapse of scarlatina led to brain fever, from which he had but partially
recovered when he feU a victim to cholera. Thus died, at the early age of thirty-six, one of the most profound
and original thinkers who have ever devoted themselves to science. The work named above was the only one
he published. Though of itself sufficient to put him in the very foremost rank, it contains only a fragment
of 8adi Carnot's discoveries. Fortunately his manuscripts have been preserved, and extracts from them have
been appended by his brother to a reprint (1878) of the Puissance Motrice. These show that he had not only
realized for himself the true nature of heat, but had noted down for trial many of the best modem methods
of finding its mechanical equivalent, such as those of Joule with the perforated piston and with the internal
friction of water and mercury. W. Thomson's experiment with a current of gas forced through a porous plug
is also given. One sentence of extract, however, must suffice, and it is astonishing to think that it was written
over sixty years ago. *' On peut done poser en th^se g^n^rale que la puissance motrice est en quantity invariable
dans la nature, qu'elle n'est jamais, k proprement parler, ni produite, ni d^truite. A. la v^rit^, elle change
de forme, c'est-^-dire qu'elle produit tantdt un genre de mouvement, tantdt un autre ; mais elle n'est jamais
an^ntie."

CXXXI.] THERMODYNAMICS. 471

characteristic from those of his concurrents, — they were based entirely on the experi-
mental facts and on necessary principles; and, when hypothesis was absolutely required,
attention was carefully directed to its nature and to the reasons which appeared to
justify it.

Three specially important additions to pure science followed almost directly from
Camot*s methods: — (1) the absolute definition of temperature; (2) the thermodynamic
function or entropy; (3) the dissipation of energy. The first (in 1848) and the third
(in 1852) were given by W. Thomson. The second, though introduced by Rankine, was
also specially treated by Clausius.

In giving a brief sketch of the science, we will not adhere strictly to any of the
separate paths pursued by its founders, but will employ for each step what appears
to be most easily intelligible to the general reader. And we will arrange the steps
in such an order that the necessity for each may be distinctly visible before we
take it.

1. Oeneral Notions. — The conversion of mechanical work into heat can always be
effected completely. In fact, Motion, without which even statical results would be all
but unrealizable in practical life, interferes to a marked extent in almost every problem
of kinetics, — and work done against friction is (as a rule) converted into heat. But
the conversion of heat into work can be effected only in part, usually in very small
part. Thus heat is regai'ded as the lower or less useful of these forms of energy,
and when part of it is elevated in rank by conversion into work the remainder sinks
still lower in the scale of usefulness than before.

There are but two processes known to us for the tionversion of heat into work,
viz., that adopted in heat-engines, where the changes of volume of the "working
substance" are employed, and that of electromagnetic engines driven by thermoelectric
currents. To the latter we will not again refer. And for simplicity we will suppose
the working substance to be fluid, so as to have the same pressure throughout, or,
if it be solid, to be isotropic, and to be subject only to hydrostatic pressure, or to
tension uniform in all directions and the same from point to point.

The state of unit mass of such a substance is known by experiment to be fully
determined when its volume and pressure are given, even if (as in the case of ice in
presence of water, or of water in presence of steam) part of it is in one molecular state
and part in another. But, the state being determinate, so must be the temperature, and
also the amount of energy which the substance contains. This consideration is insisted
on by Camot as the foundation of his investigations. In other words, before we are
entitled to reason upon the relation between the heat supplied to and the work done
by the working substance, Camot says we must bring that substance, by means of a
cycle of operations, back to precisely its primitive state as regards volume, temperature,
and molecular condition.

472

THERMODYNAMICS.

[cxxxi.

2. Watt*8 Diagram, — Watt's indicator-diagram enables us to represent our operations

graphically. For if OM (fig. 1) represent the volume,
at any instant, of the unit mass of working substance,
MP its pressure, the point P is determinate and cor-
responds to a definite temperature, definite energy, &c.
If the points of any curve, as PP\ in the diagram
represent the successive states through which the
working substance is made to pass, the work done
is represented by the area MPP*M\ Hence, a cycle
I of operations, whose essential nature is to bring the
working substance back to its primitive state, is
necessarily represented by a closed boundary, such as
PP'Q'Q, in the diagram. The area enclosed is the
excess of the work done by the working substance

over that spent on it during the cycle. [This is positive if the closed path be described

clockwise, as indicated by the arrow-heads.]

3. Carnot*8 Cycle. — For a reason which will immediately appear, Camot limited
the operations in his cycle to two kinds, employed alternately during the expansion
and during the compression of the working substance. The first of these involves
change of volume at constant temperature; the second, change of volume without direct
loss or gain of heat [In his hypothetical engine the substance was supposed to be
in contact with a body kept at constant temperature, or to be entirely surrounded
by non-conducting materiala] The corresponding curves in the diagram are called
isothermals, or lines of equal temperature, and adiabatic lines respectively. We may
consider these as having been found, for any particular working substance, by the
direct use of Watt's indicator. It is easy to see that one, and only one, of each of
these kinds of lines can be found for an assigned initial state of the working sub-
stance; also that, because in expansion at constant temperature heat must be constantly
supplied, the pressure will fall off less rapidly than it does in adiabatic expansion.
Thus in the diagram the adiabatic lines PQ, P'Q' cut the lines of equal temperature
PP\ QQ^ downwards and to the right. Thus the boundary of the area PP'Q'Q does
not cross itself To determine the behaviour of the engine we have therefore only
to find how much heat is taken in along PP' and how much is given out in Q'Q.
Their difierence is equivalent to the work expressed by the area PP'Q'Q,

4. Camot's Principle of Reversibility, — It will be observed that each operation
of this cycle is strictly reversible; for instance, to take the working substance along the
path P'P we should have to spend on it step by step as much work as it gave out
in passing along PP\ and we should thus restore to the source of heat exactly the
amount of heat which the working substance took from it during the expansion. In
the case of the adiabatics the work spent during compression is the same as that
done during the corresponding expansion, and there is no question of loss or gain of
heat directly.

CXXXI.] THERMODYNAMICS. 475

on, after exhaustive experiments, runs: — "The temperatures of two bodies are propor-
tional to the quantities of heat respectively taken in and given out in localities at
one temperature and at the other respectively, by a material system subjected to a
complete cycle of perfectly reversible thermodynamic operations, and not allowed to
part with or take in heat at any other temperature; or, the absolute values of two
temperatures are to one another in the proportion of the heat taken in to the heat
rejected in a perfect thermodynamic engine, working with a source and refrigerator
at the higher and lower of the temperatures respectively*." If we now refer again
to fig. 1, we see that, t and t' being the absolute temperatures corresponding to PP'
and QQ, and H, H' the amounts of heat taken in during the operation PP' and
given out during the operation Q'Q respectively, we have

whatever be the values of t and t\ Also, if heat be measured in terms of work,

we have

H-'H'^eLresL PP'QfQ.

Thus with a reversible engine working between temperatures t and t' the fraction of
the heat supplied which is converted into work is (t — t')/t.

It is now evident that we can construct Watt's diagram in such a way that the
lines of equal temperature and the adiabatics
may together intercept a series of equal areas.
Thus let PP' (fig. 2) be the isothermal t, and
on it so take points P', P'\ P"\ &c., that, as
the working substance passes from P to P\ P'
to P", &c., t units of heat (the unit being of
any assigned value) shall in each case be taken
in. Let QQ\ RR\ &c., be other isothermals, so
drawn that the successive areas PQ, QR\
&c., between any two selected adiabatics, may
be equal. Then, as it is clear that all the
successive areas between each one pair of iso- ) \ *v \

thermals are equal (each representing the ai*ea /^ Js* ^xS" j^

t — t'\ it follows that all the quadrilateral areas ^^""^

in the figure are equal.

It is now clear that the area included between PP' and the two adiabatics
PQR, P'QR is essentially finite, being numerically equal to t Thus the temperature
for each isothermal is represented by the corresponding area. This is indicated in the
cut by the introduction of an arbitrary line 8S\ supposed to be the isothermal of
absolute zero. The lower parts of the adiabatics also are unknown, so that we may draw
them as we please, subject to the condition that the entire areas PS\ P'S'\ P''S"\ &c.,
shall all be equal. To find, on the absolute scale, the numerical values of two definite
temperatures, such as the usually employed freezing and boiling points of water, we

• Trans. R. S. £., May 1854.

60—2

474 THERMODYNAMICS. [CXXXI.

We have now to consider that, if an engine (whether simple or compound) does
work at all by means of heat, less heat necessarily reaches the condenser than left the
boiler. Hence, if there be two engines A and B as before, and the joint system be
worked in such a way that B constantly restores to the source the heat taken firom it
by -4, we can account for the excess of work done by A over that spent on B
solely by supposing that B takes more heat from the condenser than -4 gives to %L
Such a compound engine would transform into work heat taken solely firom the con-
denser. And the work so obtained might be employed on B, so as to make it convey
heat to the source while farther cooling the condenser.

Clausius, in 1850, sought to complete the proof by the simple statement that
"this contradicts the usual behaviour of heat, which always tends to pass from warmer
bodies to colder." Some years later he employed the axiom, "it is impossible for a
self-acting machine, unaided by any external agency, to convey heat from one body
to another at a higher temperature." W. Thomson, in 1851, employed the axiom, "it is
impossible, by means of inanimate material agency, to derive mechanical effect from any
portion of matter by cooling it below the temperature of the coldest of the surrounding
objects." But he was careful to supplement this by further statements of an extremely
guarded character. And rightly so, for Clerk-Maxwell has pointed out that such
axioms are, as it were, only accidentally correct, and that the true basis of the second
law of thermodynamics lies in the extreme smallness and enormous number of the
particles of matter, and in consequence the steadiness of their average behaviour. Had
we the means of dealing with the particles individually, we could develop on the
large scale what takes place continually on a very minute scale in every mass of
gas, — the occasional, but ephemeral, aggregation of warmer particles in one small region
and of colder in another.

6. The Laws of Thermodynamics. — I. When equal quantities of mechanical effect
are produced by any means whatever from purely thermal sources, or lost in purely
thermal effects, equal quantities of heat are put out of existence, or are generated.
[To this we may add, afber Joule, that in the latitude of Manchester 772 foot-pounds
of work are capable of raising the temperature of a pound of water from 50** F. to 51*" F.
This corresponds to 1390 foot-pounds per centigrade degree, and in metrical units to
425 kilogramme-metres per calorie.]

II. If an engine be such that, when it is worked backwards, the physical and
mechanical agencies in every part of its motions are all reversed, it produces as much
mechanical effect as can be produced by any thermodynamic engine, with the same
temperatures of source and refrigerator, from a given quantity of heat.

7. Absolute Temperature. — We have seen that the fraction of the heat supplied
to it which a reversible engine can convert into work depends only on the temperatures
of the boiler and of the condenser. On this result of Camot's Sir W. Thomson based
his absolute definition of temperature. It is clear that a certain freedom of choice
is left, and Thomson endeavoured to preserve as close an agreement as possible between
the new scale and that of the air thermometer. Thus the definition ultimately fixed

CXXXI.] THERMODYNAMICS. 475

on, after exhaustive experiments, runs: — "The temperatures of two bodies are propor-
tional to the quantities of heat respectively taken in and given out in localities at
one temperature and at the other respectively, by a material system subjected to a
complete cycle of perfectly reversible thermodynamic operations, and not allowed to
part with or take in heat at any other temperature; or, the absolute values of two
temperatures are to one another in the proportion of the heat taken in to the heat
rejected in a perfect thermodjmamic engine, working with a source and refrigerator
at the higher and lower of the temperatures respectively*." If we now refer again
to fig. 1, we see that, t and t' being the absolute temperatures corresponding to PP'
and QQj and H, H' the amounts of heat taken in during the operation PP' and
given out during the operation Q'Q respectively, we have

whatever be the values of t and t'. Also, if heat be measured in terms of work,

we have

JI-fi'' = area PP'QfQ.

Thus with a reversible engine working between temperatures t and t' the fraction of
the heat supplied which is converted into work is (t — t')/t

It is now evident that we can construct Watt's diagram in such a way that the
lines of equal temperature and the adiabatics
may together intercept a series of equal areas.
Thus let PP' (fig. 2) be the isothermal t, and
on it so take points P\ P", P'", &c., that, as
the working substance passes from P to P\ P'
to P", &c., t units of heat (the unit being of
any assigned value) shall in each case be taken
in. Let QQ\ RR\ &c., be other isothermals, so
drawn that the successive areas PQ, QR\
&c., between any two selected adiabatics, may
be equal. Then, as it is clear that all the
successive areas between each one pair of iso- ) \ *\ \

thermals are equal (each representing the ai'ea /^ Js* ^xS" _^

t — t'\ it follows that all the quadrilateral areas ^^""^

in the figure are equal. *

It is now clear that the area included between PP' and the two adiabatics
PQR, P'QR is essentially finite, being numerically equal to t Thus the temperature
for each isothermal is represented by the corresponding area. This is indicated in the
cut by the introduction of an arbitrary line 8S\ supposed to be the isothermal of
absolute zero. The lower parts of the adiabatics also are unknown, so that we may draw
them as we please, subject to the condition that the entire areas PS\ P'S'\ P''S''\ &c.,
shall all be equal. To find, on the absolute scale, the numerical values of two definite
temperatures, such as the usually employed freezing and boiling points of water, we

• Trans, R. S. £., May 1854.

60—2

476 THERMODYNAMICS. [CXXXI.

must therefore find their ratio (that of the heat taken and the heat rejected by a
reversible engine working between these temperatures), and assign the number of degrees
in the interval.

Thomson and Joule experimentally showed that this ratio is about 1"365. Hence,
if we assume (as in the centigrade scale) 100 degrees as the range, the temperatures
in question are 274 and 374 nearly.

8. Entropy. — Just as the lines PP\ QQ', &c., are characterized by constant
temperature along each, so we figure to ourselves a quantity which is characteristic of
each adiabatic line, — being constant along it. The equation of last section at once
points out such a quantity. If we write (f) for its value along PQ, if> for P'Qf, we
may define thus

0' - 0 = Hit

From the statements as to the equality of the areas in fig. 2 the reader will see
at once that the area bounded by t, If, (f>, <f>' is {t — if) (<f>' — <f>). We are concerned only
with the changes of <f>, not with its actual magnitude, so that any one adiabatic may
be chosen as that for which 0 = 0.

9. I%€ Dissipation of Energy. — Sir William Thomson has recently introduced the
term thermodynamic motivity to signify " the possession the waste of which is called
dissipation.'* We speak of a distribution of heat in a body or system of bodies as
having motivity, and we may regard it from without or from within the system.

In the first case it expresses the amount of work which can be obtained by
means of perfect engines employed to reduce the whole system to some definite
temperature, that, say, of the surrounding medium. In the second case the system
is regarded as self-contained, its hotter parts acting as sources, and its colder parts as
condensers for the perfect engine.

As an instance of internal motivity we may take the case of a system consisting
of two equal portions of the same substance at different temperatures, say a pound of
boiling water and a pound of ice-cold water. If we neglect the (small) change of
specific heat with temperature, it is found that, when the internal motivity of the
system is exhausted by means of perfect engines, the temperature is about 46"* C,
being the centigrade temperature corresponding to the geometrical mean of the original
absolute temperatures of the parts. Had the parts been simply mixed so as to dissipate
the internal motivity, the resulting temperature would have been 50° C. Thus the
work gained (i.e., the original internal motivity) is the equivalent of the heat which
would raise two pounds of water from 46° C. to 50° C.

As an instance of motivity regarded from without we may take the simple case of
the working substance in § 2, on the h)rpothesis that there is an assigned lower tem-
perature limit. As there is no supply of heat, it is clear that the maximum of work
will be obtained by allowing the substance to expand adiabatically till its temperature
sinks to the assigned limit.

cxxxl]

THERMODYNAMICS.

477

Thus if P (fig. 3) be its given position on Watt's diagram, PQ the adiabatic through
P, and P'Q the isothermal of the lower temperature
limit, Q is determinate, and the motivity is the
area PQNM. If, again, we wish to find the motivity
when the initial and final states P and P' are given,
with the condition that the temperature is not to
fall below that of the state P', the problem is re-
duced to finding the course PP' for which the area
PP*M'M is greatest. As no heat is supplied, the
course cannot rise above the adiabatic PQ, and by
hypothesis it cannot fall below the isothermal P'Q, —
hence it must be the broken line PQP\ Thus,
under the circumstances stated, the motivity is represented by the area MPQP'M',
If any other lawful course, such as PP\ be taken, there is an unnecessary waste of
motivity represented by the area PQP\

10. ,Elementary Thermodynamic Relations. — From what precedes it is clear that,
when the state of unit mass of the working substance is given by a point in the
diagram, an isothermal and an adiabatic can be drawn through that point, and thus
0 and t are determinate for each particular substance when p and v are given. Thus
any two of the four quantities p, v, t, <f) may be regarded as functions of the other
two, chosen as independent variables. The change of energy from one state to
another can, of course, be expressed as in § 9, above. Thus, putting E for the energy,
we have at once

dE^td<t>-pdv (1)

if 0 and v be chosen as independent variables, and if heat be measured, as above,
in units of work. This equation expresses, in symbols, the two laws of thermodynamics.
For it states that the gain of energy is the excess of the heat supplied over the work
done, which is an expression of the first law. And it expresses the heat supplied
as the product of the absolute temperature by the gain of entropy, which is a
statement of the second law in terms of Thomson's mode of measuring absolute
temperature.

But we now have two equations in partial diflFerential coefficients : —

, , , J .
Equating them, we are led to the thermod3mamic relation

\dv) \d6j '

the differential coefficients being again partial.

478 THERM0DYNAMIC5S. [CXXXI.

This expresses a property of all "working substances/' defined as in § 1. To
state it in words, let us multiply and divide the right-hand side by t, and it then
reads : —

The rate at which the temperature falls off per unit increase of volume in adiabatic
expansion is equal to the rate at which the pressure increases per dynamical unit of
heat supplied at constant volume, multiplied by the absolute temperature.

To obtain a similar result with v and t as independent variables, we have only
to subtract from both sides of (1) the complete differential d{t<f>\ so that

d{E'-t<f>)=^-<l>dt-pdv.

Proceeding exactly as before, we find

(d^\ _ /dp\
UvJ^UtJ'

In words this result runs (when both sides are multiplied by t): —

The rate of increase of pressure with temperature at constant volume, multiplied by
the absolute temperature, is equal to the rate at which heat must be supplied per unit
increase of volume to keep the temperature constant

Very slight variations of the process just given obtain the following varieties of
expression : —

which are to be interpreted as above.

11. Increase of Total Energy under various Conditions. — The expression (1) of
§ 10 may be put in various forms, each convenient for some special purpose. We give
one example, as sufficiently showing the processes employed. Thus, suppose we wish
to find how the energy of the working substance varies with its volume when the
temperature is kept constant, we must express dE in terms of dv and dt. Thus

But we have, by § 10, under present conditions

\dvj ~ [dtj '

a result assumed in a previous article (Radiation, No. CXXX. above).

If the working substance have the property (that of the so-called "ideal" perfect gas)

pu = Rt,

CXXXI.J THERMODYNAMICS. 479

we see that, for it, (/T / ^*

The energy of (unit mass of) such a substance thus depends upon its temperature
alone.

12. Specific Heat of a Fluid. — Specific heat in its most general acceptation is the
heat required, under some given condition, to raise the temperature of unit mass by
one degree. Thus it is the heat taken in while the working substance passes, by some
assigned path, from one isothermal t to another ^ + 1 ; and this may, of course, have
as many values as there are possible paths. Usually, however, but two of these paths
are spoken of, and these are taken parallel respectively to the coordinate axes in
Watt's diagram, so that we speak of the specific heat at constant volume or at con-
stant pressure. In what follows these will be denoted by c and k respectively.

Take v and p for the independent variables, as in the diagram, and let k be
the specific heat corresponding to the condition

f(v, p) = const.
Then Kdt = td(l>^t(-^dv + ^ dp);

while 0^4- dv'\--^dp,

and dt — -T-dv + -j- dp.

dv dp ^

d^df^ ^d<f>d£
rjyy . dv dp dp dv

dv dp dp dv

This expression vanishes if / and 0 vary together, i.e., in adiabatic expansion, and
becomes infinite if / and t vary together, i.e., in isothermal expansion ; as might e€wily
have been foreseen. Otherwise it has a finite value. It is usual, however, to choose
V and t as independent variables, while we deal analytically (as distinguished from
diagrammatically) with the subject. From this point of view we have

But the last term on the right is, by definition, cdt\ so that

(/c - c) (ft = ^ ^ dv,

with the condition -r-dt -^-^dv^O.

at dv

480 THERM0DYNAMIC5S. [CXXXI.

«-«=-'gf/|.

which is a perfectly general expression. As the most important case, let f represent
the pressure, then we see, by § 10, that

dv^di'
and the formula becomes ^"^^"^Vth) / d'

13. Properties of an Ideal Substance which follows the Laws of Boyle and
Charles. — Closely approximate ideas of the thermal behaviour of a gas such as air,
at ordinary temperatures and pressures, may be obtained by assuming the relation

pv = Rt,

which expresses the laws of Boyle and Charles. Thus, by the formula of last section,
we have at once

a relation given originally by Camot.

Hence, in such a substance,

J, dt ,, .dv

or 0 — ^0 = c log ^ + (i — c) log v.

In terms of volume and pressure, this is

<f> — (f>o-c log p/R + k log V,

or J9^;*/c = i^€<*-^/^

the equation of the adiabatics on Watt's diagram.

This is (for 0 constant) the relation between p and v in the propagation of sound.
It follows from the theory of wave-motion that the speed of sound is

where t is the temperature of the undisturbed air. This expression gives, by com-
parison with the observed speed of sound, a very accurate determination of the ratio
k/c in terms of R. The value of R is easily obtained by experiment, and we have
just seen that it is equal to A; — c ; so that k and c can be found for air with great
accuracy by this process, — a most remarkable instance of the indirect measurement of
a quantity (c) whose direct determination presents very formidable diflSculties.

CXXXI.]

H, Eff&^i of Presmre on the Melting or Boiling Point of a Substance. — By the
second of the thermod^Tiamic relations in § 10, above » we have

so that

But, if the fraction e of the working substance be in one molecular state (say liquid)
in which Fp is the volume of unit mass, while the remainder 1 — e is in a state (solid)
where F| is the volume of unit mass, we have obviously

Let L be the latent heat of the liquid, then

^dv/

Also, as in a mixture of the same substance in two different states, the pressure
remains the same while the volume changes at constant temperature, we have dp/dv =* 0,
80 that finally

which shows bow the temperature is altered by a small change of pressure.

In the case of ice and water, Fj is greater than Fi, so the temperature of the
freedng-poiut is lowered by increase of pressure. When the proper numerical values
of Fo, F,, and L are introduced, it is found that the freezing-point is lowered by
about O^'OOT^ C, for each additional atmosphere.

When water and steam are in equilibrium, we have Fg much greater than Fj,
so that the boiling-point (as is well known) is raised by pressure. The same happensi,
and for the same reaeoUj with the melting point, in the case of bodies which expand
in the act of melting, such as beeswax, paraffin, cast-iron, and lava. Such bodies
may therefore be kept solid by sufficient pressure, even at temperatures far above their
ordinary melting points.

This is, in a slightly altered form, the reasoning of James Thomson, alluded to
above as one of the first striking applications of Camot*s methods made after his
work was recalled to notice^

15- Effete of Pressure on Maximum Density Poini of Wcder. — One of the most
singular properties of water at atmospheric pressure is that it has its maximum
density at 4* C. Another, first pointed out by Canton in 1764, is that its com-
pressibility (per atmosphere) is greater at low than at ordinary temperatures — ^being,
according to his measurements, 0000,049 at 34' K, and only 0 000,044 at 64' F. It
is easy to see (though it appears to have been first pointed out by Fuschl in 1875)

T, n. 61

CXXXI,]

16. Motivitff wnd Entr&py, Dii^sipation of Energy. — The motivity of the quantity
H of heat, in a body at temperature *, is

where t^, is the lowest a^'ailable tempemture.
The entropy is expressed simply as

being independent of any limit of temperature.

If the heat pass, by conduction, to a body of temperature i' (lower than t^ but
higher than ^o), the change of motivity (ia, the dissipation of energy) is

which is. of coursej losg; while the corresponding change of entropy is the gain

The numerical values of these quantities differ by the factor ^, so that, if we
could have a condenser at absolute zero, there could be no dissipation of energy.
But we see that Clausius's statement that the entropy of the universe tends to a
maximum is practically merely another way of expressing Thomson's earlier theory of
the dissipation of energy.

The whole point of the matter may be tiummarised as follows. When heat is
exchanged among a number of bodies, part of it being transformed by heat-engines
into work, the work obtainahle (i.e., the motivity) is

The work obtained, however, is simply

S(F).
Thus the waste » or amount needlessly dissipated, is

This must be essentially a positive quantity, except in the case when perfect engines
have been employed in itll the operations. In that case (unless indeed the un-
attainable condition U ^ 0 were fulfilled)

S(H/«) = 0.
which is the general expression of reversibility.

17. Works on the Subject — Ca mot's work has, as we have seen, been reprinted.
The scattered papers of Rankine, Thomson, and Ctausius have also been issued in
collected forms. So have the experimental papers of Joule. The special treatises on
Tkenjiodynamics are very numerous ; but that of Clerk-Maxivell {Theory of Heat),
though in some respects rather formidable to a beginner, is as yet far superior to
any of its rivals.

61—2

484 [cxxxii.

CXXXII.

MACQUORN RANKINE.

[From a Memoir prefixed to Rankine's Scientific Papers, 1881.]

The life of a genuine scientific man is, from the common point of view, almost
always uneventful. Engrossed with the paramount claims of inquiries raised high above
the domain of mere human passions, he is with difficulty tempted to come forward
in political discussions, even when they are of national importance; and he regards
with surprise, if not with contempt, the petty municipal squabbles in which local
notoriety is so eagerly sought. To him the discovery of a new law of nature, or
even a new experimental fact, or the invention of a novel mathematical method, no
matter who has been the first to reach it, is an event of an order altogether diflFerent
from, and higher than, those which are so profusely chronicled in the newspapers. It
is something true and good for ever, not a mere temporary outcome of craft or
expediency. With few exceptions, such men pass through life unnoticed by, almost
unknown to, the mass of even their educated countrymen. Yet it is they who, far
more than any autocrats or statesmen, are really moulding the history of the times
to come. Man has been left entirely to himself in the struggle for creature comforts,
as well as for the higher appliances which advance civilization; and it is to science,
and not to so-called statecraft, that he must look for such things. Science can and
does provide the means, statecraft can but more or less judiciously promote, regulate,
or forbid, their use or abuse. One is the lavish and utterly unselfish furnisher of
material good, the other the too often churlish and ignorant dispenser of it. In the
moral world their analogues are charity and the relieving officer ! So much it is
necessary to say for the sake of the general reader; to the world of science no
apology need be made. In it Rankine's was and is a well-known name.

It is high eulogy, but strictly correct, to say that Rankine holds a prominent
place among the chief scientific men of the last half century. He was one of the

cxxxn.]

MAGQUORN RANK FN E,

485

little group of thinkers to whom, after the wondrous Sadi Carnot^ the world is
indebted for the pure science of modem thermodynamics. Were this all, it would
be undoubtedly much. But his services to applied science were relatively even
greater. By his admirable teaching, his excellent text-books, and his original memoirs,
he has done more than any other man of recent times for the advancement of
British Scientific Engineering, He did not, indeed, himself design or construct gigantic
structures; but he taught, or was the means of teaching, that invaluable class of
men to whom the projectors of such works entrust the calculations on which their
safety, as well as their efficiency, mainly depend* For, behind the great architect or
engineer, and concealed by his portentous form* there is the real worker, without
whom failure would be certnin. The public knows but little of such men. Not every
von Moltke has his services publicly acknowledged and rewarded by his Imperial
employer I But he who makes possible the existence of such men confers lasting
benefit on his country. And it is quite certain that Ba.nkine accomplished the task.

In concluding the scientific part of this brief notice of a true man, we need
scarcely point out to the reader how much of Rankine's usefulness was due to steady
and honest work. The unscientific are prone to imagine that talent (especially when,
as in Rankine's ca^e, it rises to the level of genius) is necessarily rapid and ofi'-
hand in producing its fruits. No greater mistake could be made. The most powerful
intellects work slowly and patiently at a new subject. Such was the case with Newton,
and so it is still. Rapid they may be, and in general are, in new applications of
principles long since mastered; but it is only your pseudo-scientific man who forms
his opinions at once on a new subject. This truth was preeminently realized in
Rankine, who was prompt to reply when his knowledge was sufficient, but patient
and reticent when he felt that more knowledge was necessary. With him thought
was never divorced from work:— both were good of their kind: — the thought profound
and thorough, the work a workman-like expression of the thought. Few, if any,
practical engineers have contributed so much to abstract science, and in no case hag
scientific study been applied with more effect to practical engineering, Rankine's name
will ever hold a high place in the history of science, and will worthily be associated
with those of the great men we have recently lost. And, when we think of who
these were, how strangely does such a list i — including the names of Babbage, Boole,
Brewster, Leslie Ellis, Forbes, Herschel, Rowan Hamilton, Clerk-Maxwell, Rankine, and
othei3 ; though confined to physical or mathematical science alone : — contrast with the
astonishing utterance of the Prime Minister of Great Britain and Ireland, to the eflfect
that the present is by no means an age abounding in minds of the firat order! Ten
such men lost by this little country w^ithin the last dozen years or so — any one of
whom would have made himself an enduring name had he lived in any preceding age,
be it that of Hooke and Newton, or that of Cavendish and Watt [ Nay more, even
such losses as these have not extinguished the hopes of science amongst us. Every one
of these great men has, by some mysterious infiuence of his genius, kindled the sacred
thirst for new knowledge in younger but kindred spirits^ many of whom will certainly
rivals some even may excel, their teachers I

486 [ex XXIII.

CXXXIII.

ON THE TEACHING OF NATURAL PHILOSOPHY^.

[Contemporary Review, Janvury, 1878.]

At the very outset of our work two questions of great importance come pro-
minently forward. One of these, I have reason to conclude from long experience, is
probably a puzzling one to a great many of you: the other is of paramount con-
sequence to us all. And both are of consequence not to us alone but to the whole
country, in its present feverish state of longing for what it but vaguely understands
and calls science-teaching. These questions are, Wftat is Natural Philosophy? and, How
is it to be taught?

A few words only, on the first question, must suffice for the present The term
Natural Philosophy was employed by Newton to describe the study of the powers of
nature: the investigation of forces from the motions they produce, and the application
of the results to the explanation of other phenomena. It is thus a subject to whose
proper discussion mathematical methods are indispensable. The PHndpia commences
with a clear and simple statement of the fundamental laws of motion, proceeds to
develop their more immediate consequences by a powerful mathematical method of
the author's own creation, and extends them to the whole of what is now called
Physical Astronomy, And in the Preface, Newton obviously hints his belief that in
time a similar mode of explanation would be extended to the other phenomena of
external nature.

In many departments this has been done to a remarkable extent during the two
centuries which have elapsed since the publication of the Principia. In others,
scarcely a single step of any considerable magnitude has been taken ; and in con-
sequence, the boundary between that which is properly the subject of the natural

* Extended from Notes of the Introductory Lecture to the ordinary course of Natural Philosophy in Edinburgh
University, October Slst, 1877.

CXXXIII.] ON THE TEACHING OF NATURAL PHILOSOPHY. 487

philosopher 8 inquiries and that which is altogether beyond his province is at present
entirely indefinite. There can be no doubt that, in many important respects, even life
itself is dependent upon purely physical conditions. The physiologists have quite
recently seized, for their own inquiries, a great part of the natural philosopher's
apparatus, and with it his methods of experimenting. But to say that even the very
lowest form of life, not to speak of its higher forms, still less of volition and conscious-
ness, can be fully explained on physical principles alone — i.e., by the mere relative
motions and interactions of portions of inanimate matter, however refined and sublimated
— is simply unscientific. There is absolutely nothing known in physical science which
can lend the slightest support to such an idea. In fact, it follows at once from the Laws
of Motion that a material system, left to itself, has a perfectly deterpained future,
i.e., that upon its configuration and motion at any instant depend all its subsequent
changes; so that its whole history, past and to come, is to be gathered from one almost
instantaneous, if sufficiently comprehensive glance. In a purely material system there is
thus necessarily nothing of the nature of a free agent. To suppose that life, even in its
lowest form, is wholly material, involves therefore either a denial of the truth of
Newton's laws of motion, or an erroneous use of the term "matter." Bothl are alike
unscientific.

Though the sphere of our inquiries extends wherever matter is to be found, and
is therefore coextensive with the physical universe itself, there are other things, not
only without but within that universe, with which our science has absolutely no power
to deal. In this room we simply recognize them, and pass on.

Modem extensions of a very general statement made by Newton enable us now to
specify much more definitely than was possible in his time the range of physical
science. We may now call it the Science of Matter and Energy, These are, as the
whole work of the session will be designed to prove to you, the two real things in
the physical universe ; both unchangeable in amount, but the one consisting of parts
which preserve their identity; while the other is manifested only in the act of trans-
formation, and though measurable cannot be identified. I do not at present enter on
an exposition of the nature or laws of either; that exposition will come at the proper
time ; but the fact that so short and simple a definition is possible is extremely
instructive, showing, as it unquestionably does, what very great advances physical
science has made in recent times. The definition, in fact, is but little inferior in
simplicity to two of those with which most of you are no doubt already to a certain
extent familiar — that of Geometry as the Science of Pure Space, and of Algebra as the
Science of Pure Time.

But, for to-day at least, our second question, viz.. How is Natural Philosophy to be
taught? is of more immediate importance. The answer, in an elementary class like this,
must of course be — "popularly." But this word has many senses, even in the present
connection — one alone good, the others of variously graduated amounts of badness.

Let us begin with one or two of the bad ones. The. subject is a very serious one
for you, and therefore must be considered carefully, in- spite of the celebrated dictum of

• 488 ON THE TEACHING OF NATURAL PHIIX)SOPHY. rcXXXIII.

\ '-

! Terence, Obsequium amicoSy Veritas odium parit (In other words, Flatter your audience

J and tickle their ears, if you seek to ingratiate yourself with them ; tell them the

3 truth, if you wish to raise enemies.) But science is one form of truth. When the

ij surgeon is convinced that the knife is required, it becomes his duty to operate. And

I Shakspeare gives us the proper answer to the time-serving caution of Terence and

j Cicero in the well-known words, " Let the galled jade wince."

One of these wholly bad methods was recently very well put by a Saturday critic,
as follows: —

"The name of * Popular Science* is, in itself, a doubtful and somewhat invidious one, being
commonly taken to mean the superficial exposition of results by a speaker or writer who himself
imderstands tham imperfectly, to the intent that his hearers or readers may be able to talk about
them without understanding them at all.''

Clerk-Maxwell had previously put it in a somewhat different form: —

"The forcible language and striking illustrations by which those who are past hope of even being
beginners may be prevented from becoming conscious of intellectual exhaustion before the hour has
elapsed."

This, I need hardly say, is not in any sense science-teaching. It appears, however,
that there is a great demand for it, more especially with audiences which seek amuse-
ment rather than instruction; and this demand of course is satisfied. Such an
audience gets what it seeks, and, I may add, exactly what it deserves.

Not quite so monstrous as that just alluded to, yet far too common, is the
essentially vague and highly ornamented style of so-called science-teaching. The
objections to this method are of three kinds at least — each independently fatal.

First It gives the hearer, if he have no previous acquaintance with Physics, an
altogether erroneous impression of the intrinsic difficulty of the subject. He is exhorted,
in grandiloquent flights of laboured earnestness, to exert his utmost stretch of intellect,
that he may comprehend the great step in explanation which is next to be given; and
when, after this effort, the impression on his mind is seemingly quite inadequate, he
begins to fancy that he has not understood at all — that there must be some pro-
found mystery in the words he has heard which has entirely escaped his utmost
penetration. After a very few attempts he gives up in despair. How many a man has
been driven away altogether, whose intellect might have largely contributed to the
advance of Physics, merely by finding that he can make nothing of the pompous dicta
of his teacher or text-book, except something so simple that he fancies it cannot possibly
be what was meant !

Second, It altogether spoils the student's taste for the simple facts of true science.
And it does so just as certainly as an undiluted course of negro melodies or music-hall
comic songs is destructive of all relish for the true art of Mozart or Haydn, or as
sensation novels render Scott's highest fancies tame by contrast. And,

" as if increase of appetite had grown

By what it fed on, "

OXXXIII,]

ON THE TEACHING OP NATURAL PHILOSOPHY,

489

the action oa the listener ia made to react on the teacher, and he is called upon for
further and further outrages on the simplicity of science* Sauces and spices not only
impair the digestion, they create a craving for other stimulants of ever-increasing
pungency and deleteriousnesSt

But, thirdly. No one having a true appreciation of the admirable simplicity of
science could be guilty of these outrages. To attempt to introduce into science the
meretricious adjuncts of " word-painting/* &c., can only be the work of dabblers — not of
scientific men, just as

'*To gild refinM gold, to paint the lily^
To throw a perfume on the violet,
To smooth the icoi or add anotber hue
Unto the rainbow ; or with ta^»er light
To seek the beauteouB eje of heaven to garnish,
le waMefyl and ndicnlout ej;mMj'

None could attempt such a work who had the smallest knowledge of the true beauty of
nature. Did he know it, he would feel how utterly inadeqtiate, as well as uncalled-
for» were all his greatest efforts. For, a^^in in Shakapeare's words, such a course

" Slakas sound opinion sick, atid truth susp^ted.
For putting on so new a fashioned robe/'

"In the great majority of 'popular' scientific works the author, as a rule, has
not an exact knowledge of his subject, and does his best to avoid committing him-
selfj among difficulties which he must at least try to appear to explain. On such
occasions he usually has recourse to a flood of vague generalitieSp than which nothing
can be conceived more pernicious to the really intelligent student. In science ' fine
language ' is entirely out of place ; the stern truth, which is its only basis, requires
not merely that we should never disguise a difficulty, but, on the contrary, that we
should call special attention to it, as a probable source of valuable information. If you
meet with an author who, like the cnttle-fish, endeavours to escape from a difficult
position by darkening all around him with an inky cloud of verbiage, close the book at
once and seek information elsewhere."

But I must come back to the really important point, which is this : —

True science is in itself simple, mid should be explained in as simple and definite
language as possible.

Word-painting finds some of its most appropriate subjects when employed to deal
with human snobbery or human vice — ^where the depraved tastes and wills of mortals
are concerned— not the simple and immutable truths of science. Battles, murders,
executions ; political, legal, and sectarian st|uabbles j gossip, ostentation, toadyism, and
Buch like, are of its proper subjects* Not that the word -painter need be himself
necessarily snobbish or vicious — lar from it. But it is here, as our best poets and
satirists have shown, that his truest field is to be found. Science sits enthroned, like
the gods of Epicurus, far above the influence of mere human passions, be they
virtuous or evil, and must be treated by an entirely different code of rules. And a
T, II. 62

490 ON THE TEACHINO OF NATURAL PHILOSOPHY. [cXXXin.

great deal of the very shallowest of the pseudo-science of the present day probably
owes its origin to the habitual use, with reference to physical phenomena, of terms
or synonyms whose derivation shows them to have reasonable application to human
beings and their actions alone— not at all to matter and energy. In dealing with
such pseudo-science it is, of course, permissible to me, even after what I have said,
to use word-painting as far as may be thought necessary.

The Pygmalions of modem days do not require to beseech Aphrodite to animate
the ivory for them. Like the savage with his Totem^ they have themselves already
attributed life to it. "It comes," as v. Helmholtz says, "to the same thing as
Schopenhauer's metaphysics. The stars are to love and hate one another, feel pleasure
and displeasure, and to try to move in a way corresponding to these feelings." The
latest phase of this peculiar non-science tells us that all matter is alive; or at least
that it contains the "promise and potency" (whatever these may be) "of all terrestrial
life." All this probably originated in the very simple manner already hinted at ; viz.,
in the confusion of terms constructed for application to thinking beings only, with
others applicable only to brute matter, and a blind following of this confusion to its
necessarily preposterous consequences. So much for the attempts to introduce into
science an element altogether incompatible with the fundamental conditions of its
existence.

When simple and definite language cannot be employed, it is solely on account
of our ignorance. Ignorance may of course be either unavoidable or inexcusable.

It is unavoidable only when knowledge is not to be had. But that of which
there is no knowledge is not yet part of science. All we can do with it is simply to
confess our ignorance and seek for information.

As an excellent illustration of this we may take two very common phenomena — a
rainbow and an aurora — the one, to a certain extent at least, thoroughly understood;
the other scarcely understood in almost any particular. Yet it is possible that, in our
latitudes at least, we see the one nearly as often as the other. For, though there
are probably fewer auroras to be seen than rainbows, the one phenomenon is in general
much more widely seen than the other. A rainbow is usually a mere local pheno-
menon, depending on a rain-cloud of moderate extent; while an aurora, when it occurs,
may extend over a whole terrestrial hemisphere. Just like total eclipses, lunar and
solar. Wherever the moon can be seen, the lunar eclipse is visible, and to all alike.
But a total solar eclipse is usually visible from a mere strip of the earth — some fifty
miles or so in breadth.

The branch of natural philosophy which is called Oeometricai Optics is based upon
three experimental facts or laws, which are assumed as exactly true, and as repre-
senting the whole truth — the rectilinear propagation of light in any one uniform medium,
and the laws of its reflexion and refraction at the common surface of two such media;
and as a science it is nothing more than the developed mathematical consequences of
these three postulates.

CXXXIII.]

ON THE TEACHING OF NATURAL PHILOSOPHY*

491

Hence, if these laws were rigorously true, and represented all the truth, nothing
but mathematical investigatioti based on them would be required for the complete
development of the phenomena of the rainbow — except the additional postulate, also
derived from experiment, that falling drops of water assume an exact spherical form
— andj as data for numerical calculation, the experimentally-determined refractive index
for each ray of light at the common surface of air and water

Thus for instance we can tell why the rainbow has the form of a portion of a
circle surrounding the point opposite to the sun ; why it is red on the outer edge ;
what is the order of the other colours, and why they are much less pure than the
red ; why the whole of the background enclosed within it is brighter than that just
outside; and so on. Also why there is a second (also circular) rainbow; why it is
concentric with the first ; and why its colours are arranged in the reverse order, &c.

But, so long at least as we keep to Geometrical optics, we cannot explain the
spurious bows which are usually seen, like ripplea, within the primary and outside the
second rainbow ; nor why the light of both bows is polarized, and so forth. We must
apply to a higher branch of our science; and we fiud that Pkt/mcal Optics, which
gives the results to which those of geometrical optics are only approximations, enables us
to supply the explanation of these phenomena also.

When we turn to the aurora we find nothing so definite to explain. This may,
to some extent at least, account for our present ignorance. We remark, no doubt, a
general relation between the direction of the earth's magnetic force and that of the
streamers: but their appearance is capricious and variable in the extreme. Usually
they have a pale green colour, which the spectroscope shows to be due to homogeneous
light; but in very fine displays they are sometimes blood-red, sometimes blue. Auroral
arches give sometimes a sensibly continuous spectrum ; sometimes a single bright line.
We can imitate many of the phenomena by passing electric discharges through rarefied
gases; and we find that the streamers so produced are influenced by magnetic force. But
we do not yet know for certain the source of the discharges which produce the aurora,
nor do we even know what substance it is to whose incandescence its light is due. We
find by a statistical method that auroras, like cyclones, are most numerous when there
are most spots on the sun : but the connection between these phenomena is not yet
known. Here, in fact, we are only begiiming to understand, and can but confess our
ignorance.

But do not imagine that there is nothing about the rainbow which we cannot
explain, even of that which is seen at once by untrained observers. All the phenomena
connected with it wfiich we can explain are mathematical deductions from observed
facts which are assumed in the investigation^ But these facts are, in the main, them-
aelvea not yet explained. Just as there are many exceedingly expert calculators who
habitually and usefully employ logarithmic tables without having the least idea of what
a logarithm really is, or of the manner in which the tables themselves were originally
calculated ; so the natural philosopher uses the observed facts of refraction and reflection
without having as yet anything better than guesses as to their possible proximate
cause. And it is so throughout our whole subject: assuming one result, we can prove

62—2

492 ON THE TEACHING OF NATURAL PHILOSOPHY. [OXXXIH.

that the others miLst follow. In this direction great advances have been made, and
every extension of mathematics renders more of such deductions possible. But when we
try to reverse the process, and thus to explain our hitherto assumed results, we are met
by difficulties of a very diflFerent order.

The subject of Physical Astronomy, to which I have already alluded, gives at once
one of the most striking and one of the most easily intelligible illustrations of this point.
Given the law of gravitation, the masses of the sun and planets, and their relative
positions and motions at any one instant, — the investigation of their future motions, until
new disturbing causes come in, is entirely within the power of the mathematician. But
how shall we account for gravitation? This is a question of an entirely different
nature from the other, and but one even plausible attempt to answer it has yet been
made.

But to resume. The digression I have just made had for its object to show you
how closely full knowledge and absolute ignorance may be and are associated in many
parts of our subject — absolute command of the necessary consequences of a phenomenon,
entire ignorance of its actual nature or cause.

And in every branch of physics the student ought to be most carefully instructed
about matters of this kind. A comparatively small amount of mathematical training
will often be found sufficient to enable him to trace the consequences of a known truth
to a considerable distance; and no such training is necessary to enable him to see (pro-
vided it be properly presented to him) the boundary between our knowledge and our
ignorance — at least when that ignorance is not directly dependent upon the inadequacy
of our deductive powers.

The work of Lucretius is perhaps the only really successful attempt at scientific
poetry. And it is so because it was written before there was any true physical science.
The methods throughout employed are entirely those of d priori reasoning, and there-
fore worse than worthless, altogether misleading. Scientific poetry, using both words
in their highest sense, is now impossible. The two things are in their very nature
antagonistic. A scientific man may occasionally be a poet also; but he has then two
distinct and almost mutually incompatible natures ; and, when he writes poetry, he
puts science aside. But, on the other hand, when he writes science, he puts poetry and
all its devices aside. Mark this well ! A poet may, possibly with great effect on the
unthinking multitude, write of

" the huger orbs which wheel

In circuits vast throughout the wide abyss
Of imimagined Chaos — till they reach
-Ethereal splendour "

(The word "unimagined" may puzzle the reader, but it probably alludes to Ovid's
expression " sine imagine" For this sort of thing is nothing if not cUissiccU ! The
contempt in which "scholars" even now hold mere "physicists" is proverbial. And
they claim the right of using at will new words of this kind, in whose company
even the " tremendous empyrean " would, perhaps, not be quite out of place.)

cxxxiil]

ON THE TEACHING OF NATURAL PHILOSOPHY.

493

But, whether this sort of thing be poetry or not, it is in no aenae science. " Huge,"
and *' vast," and such-like (for which, if the rhythm permit, you may substitute their
similars, " Titanic/' " gigantic/* &c.), good honest English though they be, are utterly
unscientific words. In science we restrict ourselves to sviall and great, and these amply
suffice for all our wants. But even these terras are limited with us to a mere relative
sense; and it can only be through ignorance or forge tfiiln ess of this that more sonorous
terms are employed. The size of every finite object depends entirely upon the unit in
terms of which you measure it. There is nothing absolutely great but the Infinite,

A few moments' reflection will convince yon of the truth of what I have just said.
Let US only go by easily comprehensible stages from one (so-called) extreme to the
other. Begin with the smallest thing you can see, and compai*e it with the greatest,
I suppose you have all seen a good barometer. The vernier attached to such an
instrument is usually read to thousandths of an inch, but it sometimes leaves you in
doubt which of two such divisions to choose. This gives the limit of vision with the
unaided eye. Let us therefore begin with an object whose size is about 1 -2000th of
an inch. Let us choose as our gcalo of relative magnitude 1 to 250,000 or there-
abouts. It is nearly the proportion in which each of you individually stands to the
whole population of Edinburgh. (I am not attempting anything beyond the rudest
illustration, because that will amply suffice for my present purpose.) Well : 250,000 times
the diameter of our rmnimum vmbile gives us a length of ten feet or so— three or four
paces. Increased again in about the same ratio, it becomes more than 400 miles, some-
where about the distance from Edinburgh to London, Perform the operation again,
and you get (approximately enough fur our purpose) the sun^s distance from the earth.
Operate once more, and you have got beyond the nearest fixed star. Another such
operation would give a distance far beyond that of anything we can ever hope to
see. Yet you have reached it by repeating, at most five times, upon the smallest
thing you can see, an operation in itself not very difficult to imagine. Now as there is
absolutely nothing known to science which c^in preclude us from carrying this process
farther, so there is absohvtely no reason why we may not in thought reverse it, and thus
go back from the smallest visible thing to various successive orders of small ness. And
the first of these that we thus reach has already been pointed t-o by science as at
least a rough approximation to that coarse -grain edness which we know to exist (though
we shsAl never be able to see it) even in the most homogeneous substances, such as glass
and water. For several trains of reasoning, entirely independent of one another, but
baaed upon experimental facts, enable us to say with certainty that all matter becomes
heterogeneous (in some as yet quite unknown way) when we consider portions of it
whose dimensions are somewhere about l-500,000,000th of an inch. We have, as yet,
absolutely no information beyond this, save that, if there be ultimate atoms, they are at
least considerably more minute still.

Next comes the very important question — How far w experimental illustratioti
necessary and useful? Here we find excessively wide divergence^ alike in theory and
in practice.

In some lecture-theatres, experiment is everything ; in others, the exhibition of
gorgeous displays illustrative of nothing in particular is said occasionally to alternate

494 ON THE TEACHING OF NATURAL PHILOSOPHY. [CXXXIII.

with real or imagined (but equally sensational) danger to the audience, from which they
are preserved (or supposed to be preserved) only by the extraordinary presence of mind
of the presiding performers — a modern resuscitation of the ancient after-dinner amuse-
ment of tight-rope dancing, high above the heads of the banqueters, where each had
thus a very genuine, if selfish, interest in the nerve and steadiness of the artists.

Contrasted in the most direct manner with these, is the dictum not long ago laid
down: —

"It may be said that the fact makes a stronger impression on the boy through the medium
of his sight — that he believes it the more confidently. I say that this ought not to be the case.
If he does not believe the statements of his tutor — probably a clergyman of mature knowledge,
recognized ability, and blameless character — his suspicion is irrational, and manifests a want of the
power of appreciating evidence— a want fatal to his success in that branch of science which he is
supposed to be cultivating."

Between such extremes many courses may be traced. But it is better to dismiss
the consideration of both, simply on the ground that they are extremes, and therefore
alike absurd.

Many facts cannot be made thoroughly intelligible without experiment; many others
require no illustration whatever, except what can be best given by a few chalk-lines on
a blackboard. To teach an essentially experimental science without illustrative experi-
ments may conceivably be possible in the abstract, but certainly not with professors and
students such as are to be found on this little planet.

And, on the other hand, you musrt all remember that we meet here to discuss
science, and science alone. A University class-room is not a place of public amuse-
ment, with its pantomime displays of red and blue fire, its tricks whether of prestigiation
or of prestidigitation, or its stump-oratory. The best and greatest experimenter who
ever lived used none of these poor devices to win cheap applause. His language (except
perhaps when non-experimenting pundits pressed upon him their fearful Greek names for
his splendid discoveries) was ever the very simplest that could be used: his experiments,
whether brilliant or commonplace in the eyes of the mere sight-seer, were chosen solely
with the object of thoroughly explaining his subject; and his whole bearing was
impressed with the one paramount and solemn feeling of duty, alike to his audience and
to science. Long ages may pass before his equal, or even his rival, can appear ; but the
great example he has left should be imitated by us all as closely as possible.

Nothing is easier in extempore speaking, as I dare say many of you know by
trial, than what is happily called "piling up the agony." For, as has been well said,

" men there be that make

Parade of fluency, and deftly play
With i)oints of speech as jugglers toss their balls ;
A tinkling crew, from whose light-squandered wit
Ko seed of virtue grows."

Every one who has a little self-confidence and a little readiness can manage it without
trouble. But it is so because in such speaking there is no necessity for precision in

CXXXIII.]

ON THE TEACHING OE NATCTRAL PHILOSOPHY.

495

the use of words, and no objection to any epithet whatever, even if it be altogether
misplaced. But the ess^ence of all aitch discourae is necessarily fanc^t ^nd not fact Herep
during the serious work of the session, we are tied down almost exclusively to facts.
Fancies must appear occasionally ; but we admit them only in the carefully -guarded
form of a reference to old opinions, or to a '*good working hypothesis.'* Still, facts
are not necessarily dry ; not even if they be mere statistics. All depends on the way
in which they are put. One of the most amusing of the many clever songs, written
and sung by the late Professor Banlcuie in his moments of relaxation, was an almost
literal transcript of a prosaic statistical description of a little Iriah town, taken from
a gazetteer] He was a truly original man of science, and therefore exact in his state-
ments; but he could be at once both exact and interesting. And I believe that the
intrinsic beauty of science is such that it cannot suffer in the minds of a really
intelligent audience, however poor be the oratorical powers of its expounder, provided
only he can state its facts with clearness. Oratory is essentially art^ and therefore
essentiaUy not &ciBnce^

There is nothing false in the theori/f at least, of what are called Chinese copies.
If it could be /idly carried out, the results would be as good as the original — in
factj nndistingaishable from it. But it is solely because we cannot have the theory
carried out in perfection that true artists are forced to slur over details, and to give
" broad effects " as they call them. The members of the Pre-Raphaelite school are
thoroughly right in one part at least of theu' system : unfortunately it is completely
unrealizable in practice. But the " broad effects '* of which I have spoken are ti'ue
art, though perhaps in a somewhat modified sense of the woixi (which, not being
a scientific one, has many shades of meaning). To introduce these '* broad effects"
into science may be artful, but it is certainly unscientific. In so-called " popular
science,*' if anywhere, Ai^s est cetitre insoientiam. The '* artful dodge" is to conceal
want of knowledge. Vague explanations, however artful, no more resemble true science
than do even the highest flights of the imagination, whether in Ivanhoe or Qumtin
Darward, Knickerbocker's New York or Macau lay's England, resemble history. And when
the explanation is bombastic as well as vague, its type is the same as that of the w^ell-
known speech of Sergeant Busifuz.

One ludicrous feature of the " high-falutin " style is that if you adopt it you throw
away all your most formidable ammunition on the smaller game, and have nothing
proportionate left for the larger. It is as if you used a solid shot from an 81 -ton
gun upon a single skirmisher 1 As I have already said, you waste your grandest terms,
such as huge, vast, enormous, tremendous, i&a, on your mere millions or billions; and
then what is left for the poor trillions ? The true lesson to be learned from this is,
that such terms are altogether inadmissible in science,

But even if we could suppose a speaker to use these magnificent words as a genuine
description of the impression made on himself by certain phenomenal you must remember
that he is describing not what is known of the objective fact (which, except occasionally
from a biographic point of view, is what the listener really wants), but the more or less
inadequate subjective impression which it has produced, or which he desires you to think

496 ON THE TEACHING OF NATURAL PHIIX)60PHT. [CXXXHL

it has produced, on "what he is pleased to call his mind" Whether it be his own
mind, or that of some imaginary individoal, matters not. To do this, except peiiiaps
when lecturing on psychology, is to be unscientific. True scientific teaching, I cannot
too often repeat, requires that the facts and their necessary consequences alone should
be stated (and illustrated if required) as simply as possible. The impression they are
to produce on the mind of the reader, or hearer, is then to be left entirely to himself.
No one has any right to suppose, much less to take for granted, that his own notions,
whether they be "so-called poetic instincts" (to use the lowest term of contempt) or
half-comprehended and imperfectly expressed feelings of wonder, admiration, or awe, are
either more true to £Etct or more sound in foundation than those of the least scientific
among his readers or his audience. When he does so he resembles a mere leader of a
claque, * * * If your minds cannot relish simple food, they are not in that healthy
state which is required for the study of science. Healthy mental appetite needs only
hunger-sauce. That it always has in plenty, and repletion is impossible.

But you must remember that language cannot be simple unless it be definite;
though sometimes, from the very nature of the case, it may be very difficult to under-
stand, even when none but the simplest terms are used. Multiple meanings for technical
words are totally foreign to the spirit of true science. When an altogether new idea
has to be expressed, a new word must be coined for it. None but a blockhead could
object to a new word for a new idea. And the habitual use of non-scientific words in
the teaching of science betrays ignorance, or (at the very least) wilful indefiniteness.

Do not fiEuicy, however, that you will have very many new words to learn. A month
of Botany or of Entomology, as these are too often taught, will introduce you to a
hundredfold as many new and strange terms as you will require in the whole course of
natural philosophy; and, among them, to many words of a far more "difficult com-
plexion " than any with which, solely for the sake of definiteness, we find ourselves
constrained to deal.

But you will easily reconcile yourselves to the necessity for new terms if you bear
in mind that these not only secure to us that definiteness without which science is
impossible, but at the same time enable us to get rid of an enormous number of
wholly absurd stock-phrases which you find in almost every journal you take up,
wherever at least common physical phenomena are referred to. When we are told
that a building was "struck by the electric fluid*' we may have some difficulty in
understanding the process; but we cannot be at all surprised to learn that it was
immediately thereafter "seized upon by the devouring element which raged unchecked
till the whole was reduced to ashes.'' I have no fault to find with the penny-a-liner
who writes such things as these : it is all directly in the way of his business, and
he has been trained to it. Perhaps his graphic descriptions may occasionally rise even
to poetry. But when I meet with anything like this, — and there are but too many
works, professedly on natural philosophy, which are full of such things, — ^I know that
I am not dealincr with science.

''o

A wild and plaintive wail for definiteness often comes from those writers and

cxxxiil]

ON THE TEACHINa OP NATUEAL PHILOSOPHY,

497

lecturers who are habitually the most vague. A few crocodile teauB are shed, appear-
ances are preserved, and they plunge at once into greater mistiness of verbosity than
before.

Considering the actual state of the great majority at least of our schools and our
elementary text^books, I should prefer that you came here completely untaught iu
physical science. You would then have nothing to unlearn. This is an absolutely
incalculable gain. Unlearning is by far the hardest task that was ever imposed on
a student, or on any one else. And it is also one of those altogether avoidable tasks
which, when we have allowed them to become necessary, irritate us as much as does
a perfectly unprofitable one^snch as the prison cmnk or shot-drill. And in this lies
by far the greatest responsibility of all writers and teachera Merely to fail in giving
instruction is bad enough, but to give false information can be the work only of utter
ignorance or of carelessness, amounting, so far as its effects go, almost to diabolical
wickedness-

Every one of you who has habitually made use of his opportunities of observation
must have already seen a greal deal which it will be my duty to help him to under-
stand. But I should prefer, if possible, to have the entire guidance of him in helping
him to understand it. And I should commence by warning him in the most formal
manner against the study of books of an essentially unscientific character. By all
means let him read fiction and romance as a relaxation &om severer studies ; but let
the fiction be devoted to its legitimate object, human will and human action ; don*t
let it tamper with the truths of science. From the Arabian Nights, through Dan
Qaiwote, to Scott, the student has an ample field of really profitable reading of this
kind; but when he wishes to studf;^ let him carefully eschew the unprofitable, or
rather pernicious, species of literary fiction which is commonly called " popular
science/'

As I have already said, in this elementary class you will require very little
mathematical knowledge, but such knowledge is in itself one of those wholly good
things of which no one can ever have too much. And, moreover, it is one of the
few things which it is not very easy to teach badly, A really good student will leani
mathematics almost in spite of the badness of his teaching* No pompous generalities
can gloss over an incorrect demonstration ; at least in the eyes of any one competent
to understand a correct one* Can it be on this account that there are so many more
aspirants to the teaching of physics than to that of the higher mathematics ? If eo,
it is a very serious matter for the progress of science in this country ; as bad, at
least, as was the case in those old days when it was supposed that a man who had
notoriously failed in everything else must have been designed by nature for the vocation
of schoolmaster; a truly wonderful application of teleology.

But even this queer kind of Dominie was not so strange a monstrosity as the
modern manikins of p«jt?er scie/ice, who are always thrusting their crude notions on
the world ; the anatomists who have never dissected, the astronomers who have never
used a telescope, or the geologists who have never carried a hammer ! The old
metaphysical pretenders to science had at least some small excuse for their conduct in
T. II, 63

498

ON THE TEACHING OF NATURAL PHiLOeOPHy.

[CXXXIIL

the feet that true science was all but uiiknowii iu the days wheo they chiefly flouriBhed,
and when their A priori dogniatiam was too geneiullj looked upon as science. But
that singular race is now welUnigh extinct, and in their place have come the paper-
scientists (the barbaroug word suits them exact ly}-*those who, with a strange mixture
of half-apprehended fact and thoroughly appretiated nonsense, pour out continuoua
floods of information of the most self-contradictory character* Such writers loudly claim
the honours of discovery for any little chance remark of theirs which research may
happen ultimately to substantiate, but keep quietly in the background the mass of
imreason in Nvhich it was originally enveloped. This species may be compared to midges,
perhaps occasionally to mosquitos, continually pestering men of science to an extent
altogether disproportionate to its own importance in the scale of being. Now and then
it buzzes shrilly enough to attract the attention of the great sound-hearted, but un-
reasoning because non-scientiiic, public which, when it does interfere with scientific
matters, can hardly fail to make a mess of them.

Think, for a moment, of the late vivisection critsade or of the anH^vaceiHaiors, What
absolute fiends in human form were not the whole race of really scientific medical raea
made out to be» at least in the less cautious of these heated denunciations ? How
many camels are unconsciously swallowed while these gnats are being so carefully strained
out, is obvious to all who can take a calm, and therefore a not necessarily unredSODable^
view of the matter.

But the victims of such people are not in scientific ranks alone. Every man who
occupies a prominent position of any kind is considered as a fit subject for their attacks.
By private letters and public appeals, gratuitous advice and remonstrance are perpetually
intruded upon him. If he succeed in anything, it is of course because these unaought
hints were taken : if he fail, it is because they were contemptuously left unheeded !

Enough of this necessary but unpleasant digression. I knoio that it is at least quite
as easy to understand the most recondite mathematics as to follow the highest of
genuine physical reasoning; and therefore, when I find apparently profound physical
specuiation associated with incapacity for the higher mathematics, I feel convinced that
the profundity cannot be real. One very necessary remark, however, must be made
hei'e: not in (jualification, but in explanation, of this statement. One of the greatest
of physical reasoners, Faraday, professed, as most of you are aware, to know very little
of mathematics. But in fact he was merely unacquainted with the technical use of
symbols. His modes of regarding physical problems were of the highest order of mathe-
matics. Many of the very best things in the recent great works on Electricity by
Clerk-Maxwell and Sir William Thomson are (as the authors cheerfully acknowledge)
little more than well- ex ecu ted translatimta of Faraday's conceptions into the conventional
language of the higher analysis.

I hope that the time is not far off when no one who is not (at least in the same
sense as Faraday) a genuine mathematician, however he may be otherwise qualified, will
be looked upon as even a possible candidate for a chair of Natural Philosophy in any of
our Universities, Of course such a danger would be out of the question if we were

OXXXIU.]

ON THE TEACHING OF NATUEAL PHILOSOPHY.

499

to constantly bear in miod the sense in which NewtoD understood the tenn Natural
Philosophy, There is nothing so well fitted as mathematics *' to take the nonsense
out of a man" as it is popularly phrased. No doubt a man may be an excellent mathe-
matician, and yet have absolutely no knowledge of physics; but he cannot possibly know
physics as it is unless he be a mathematician. Much of the most vaunted laboratory
work is not nearly of so high an order of skilled labour as the every-day duty of a good
telegraph clerk, especially if he be in charge of a syphon-recorder. And many an
elaborate memoir which fills half a volume of the transactions of some learned society
is essentially as unsightly and inconvenient an object as the mounds of valueless dross
which encumber the access to a mine, and destroy what otherwise might have been an
expanse of fruitful soiL

There are many ways in which these mounds may grow. The miner may be totally

ignorant of geology, and may thus have bored and excavated in a locality which he
ought to have known would furnish nothing. Or he may have, by chance or by the
advice of knowing friends, hit upon a really good locality. Even then there are many
modes of failure, two of which are very common. He may fail to recognize the ore
when he has got it ; and so it goes at once to the refuse- heap, possibly to be worked
up again long after by somebody who has a little more raineralogical knowledge — as
in the recent case of the mines of Laurium. Here he may be useful — at second-hand.
Or, if it be fossils or crystals, for instance, for which he is seeking, his procedure may
be so rough as to smash them irreparably in the act of mining. This is dog in the
manger with a vengeance. But, anyhow, he generally manages to disgust every other
digger with the particular locality which he has turned upside down ; and thus
exercises a real^ though essentially negaiim, influence on the progress of mining.

The parallel here hinted at is a v^v^ apt one, and can be traced much farther.

For there are other peculiarities in the modes of working adopted by some miners,

which have their exact counterparts in many so-called scientific inquiries ; but, for the
present, we must leave them unnoticed.

There is but one way of being scientific: but the number of ways of being
unscientific is infinite, and the temptations alluring us to them are numerous and
strong. Indolence is the most innocent in appearance, but in fact probably the most
insidious and dangerous of all. By this I mean of course not mere idleness, but that
easily acquired and fatal habit of stopping just short of the final necessary step in
each explanation. Faraday long ago pointed this out in his discourse on Mental
Inertia. Many things which are excessively simple when thoroughly understood are
by no means easy to acquire ; and the student too often contents himself with that
half learning which, though it costs considerable pains, leaves no permanent impression
on the mind, while '*one struggle more" would have made the subject his own for
ever after.

Science, like all other learnings can be reached only by continued exertion. And,
even when we have done our utmost, we always find that the best we have managed
to achieve has been merely to avoid straying very far from the one true path.

500 ox THE TEACHING OF NATURAL PHILOSOPHY. [CXXXIH.

For, though science is in itself essentially simple, and is ever best expressed in
the simplest terms, it is my duty to warn you in the most formal manner that the
study of it is beset with difficulties, many of which cannot but constitute real obstacles
in the way even of the mere beginner. And this forms another of the fatal objections
to the school-teaching of physical science. For there is as yet absolutely no known
road to science except through or over these obstacles, and a certain amount of
maturity of mind is required to overcome them.

If any one should deny this, you may at once conclude either that his mental
powers are of a considerably higher order than those of Newton (who attributed all
his success to close and patient study) or, what is intrinsically at least somewhat more
probable, that he has not yet traversed the true path himself. But it would be a
mere exercise of unprofitable casuistry to inquire which is the less untrustworthy guide,
he who affirms that the whole road is easy, or he who is continually pointing out
£Euicied difficulties.

Here, as in everything to which the human mind or hand can be applied, nothing
of value is to be gained without effort; and all that your teacher can possibly do for
you is to endeavour, so far as in him lies, to make sure that your individual efforts
shall be properly directed, and that as little energy as possible shall be wasted by
any of you in a necessarily unprofitable direction.

END OF VOLUME II.

CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.

Stanford University LibraH
Stanford, CaUfomia

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