niiLOSoriiicAL
TRANSACTIONS

OF TIEE

ROYAL SOCIETY

OF

LONDON.

FOR THE YEAR MDCCCLXIII.

VOL. 153.

LONDON:

PRINTED Br TAYLOR .VST) FRANCIS, RED LION COURT, FLEET STREET
MDCCCLXIV.

Q
41

CONTENTS

OF VOL. 153.

I. On the Relation of Radiant Heat to Aqueous Vapour. By John Ttxdall, F.R.S.,

Member of the Academies and Societies of Holland, Geneva, Gottingen, Zurich,
Halle, Marburg, Breslaii, Upsala, la Societe Philomathique of Paris, Cam. Phil.
Soc. &c. ; Professor of Natural Philosophy in the Royal Institution . . page 1

II. On the Volumes of Pedal Surfaces. By T. A. Umsi, F.R.S. 13

III. On the Archeopteryx of vox Meyer, with a description of the Fossil Remains of

a Long-tailed species, from the Lithographic Stone of Solenhofen. By Professor
Owen, F.R.S. &c 33

IV. On the Strains in the Interior of Beams. By George Biddell Airy, F.R.S. , Astro-

nomer Royal 49

V. On the Reflexion of Polarized Light from Polished Surfaces, Transparent and Me-

tallic. By the Rev. Samuel Haughtox, M.A., F.R.S., Felloio of Trinity College,
Dublin 81

VI. On the Exact Form of Waves near the Surface of Beep Water. By Williaji Joh.v

Macquorn R.\xkixe, C.E., LL.D., F.R.SS. L. & K 127

VII. Photo-chemical Researches. — Part V. On the Direct Measurement of the Chemical

Action of Sunlight. By Robert Bunsen, For. Mem. R.S., Professor of Chemistry
in the University of Heidelberg, and Henrt E. Eoscoe, B.A., Ph.D., Professor of
Chemistry in Owens College, Manchester 139

VIII. On the Immunity enjoyed by the Stomach from being digested by its oivn Secretion
during Life. By F. W. Pavy, M.D 161

IX. On Thallium. By William Crookes, Esq. Communicated by Professor G. G.

Stokes, Sec. R.S 173

[ iv ]

X. On the Listnlution of Surfaces of the Third Order into Species, in reference to the

absence or presence of Singular Points, and the reality/ of their Lines. By
Dr. ScHLAFLi, Professor of Mathematics in the University of Berne. Com-
municated hy xIktiiur Caylet, F.Ji.S. page 193

XI. On the Tides of the Arctic Seas. By the Rev. Samuel Haughton', M.A., F.R.S.,

Felloiv of Trinity College, Dublin 243

XII. Results of the JIagnetic Observations at the Kew Observatory, from 1857 and 1858

to 18G2 inclusive. — Nos. I. and II. By Major-General Edward Sabixe, R.A.,
Pre-ndent of the Royal Society ... 273

XIII. On the Diurnal Inequalities of Terrestrial 2Iagnetism, as deduced from observa-
tions made at the Royal Observatory, Greenwich, from 1841 to 1857. By George
BiDDELL Airy, F.R.S., Astronomer Royal 309

XIV. Researches on the Refraction, Dispersion, and Sensitiveness of Liquids. By J. H.

Gladsto-\e, PLD., F.R.S., and the Rev. T. P. Dale, M.A., F.R.A.S. . . 317

XV. Researches into the Chemical Constitution of Xarcotine, and of its Products of

Decomposition. — Part I. By Augustus Mattiiiessen, F.R.S., Lecturer an
Chemistry in St. Marys Hospital, London, and G. C. Foster, B.A., Lecturer
on Natural Philosophy in Anderson's University, Glasgow 345

XVI. On the Influence of Temperature on the Electric Conducting-Power of Thallium
and Iron. By A. Matthiessen, F.R.S., Lecturer on Chemistry in St. Marys
Hospital, and C. Vogt, Ph.D 369

XVII. On the Molecular Mobility of Gases. By Thomas Grah.ui, F.R.S., Master of
the Mint 385

XVIII. On the Peroxides of the Radicals of the Organic Acids. By Sir B. C. Brodie,
Bart., F.R.S., Profes.sor of Chemistry in the University of Oaford . . . 407

XIX. An Account of Experiments on the Change of the Elastic Force of a Constant
Volume of Atmospheric Air, between 32° F. and 212° F., and also on the Tem-
perature of the Melting-point of Mercury. By Balfour Stewart, M.A.,
F.R.S 425

XX. On some Compounds and Derivatives of Glyoxylic Acid. By Henry Debus, Ph.D.,

F.R.S. 437

XXI. On Skew Surfaces, otherwise Scrolls. % Arthur Cayley, i^.i^./S. . . . 453

X .\ 1 1 . On the Differential Fjiuafions of Dynamics. A sequel to a Paper on Simultaneous
Differential Eqtmtions. By (iicuuciE Boole, F.R.S., Professor of Mathematics in
Queen's College, Cork 485

[ V ]

XXIII. On the Xature of the Suiis Magnetic Action itpmi the Earth. By Charles
Chambers. Communicated by General Sabine, P.B.S page 503

XXIV. On the Calculus of Symbols. — Third Memoir. ByW. H. L. Rvssell, Esq., A. B.
Communicated by A. Cayley, F.R.S -517

XXV. Numencal Elements of Indian Meteorology. By Hermann de Schlagintweit,
Ph.D., LL.D., Corr. Memb. Acad. Munich, Madrid, Lisbon, &c. Communicated
by Major-General Sabine, P.B.S 525

XXVI. On the Structure and formation of the so-called Aj)olar, Unipolar, and Bipolar
Nerve-cells of the Frog. By Lionel S. Beale, F.R.S., M.B., Fellow of the Royal
College of Physicians ; Professor of Physiology and of General and Morbid Ana-
tomy in King's College, London; Physician to King's College Hospital, ctr. 543

XXVII. On the Rigidity of the Earth. By W. Thomson, LLL., F.R.S., Profes.wr of
Natural Philosophy in the University of Glasgow 573

XXVIII. Dynamical Problems regarding Elastic Spheroidal Shells and Spheroids of
Incompressible Liquid. By Professor W. Thomson, LL.D., F.R.S. . . 583

XXIX. First Analysis of One Hundred and Seventy-seven Magnetic Storms, registered by
the Magnetic Instruments in the Royal Observatory, Greentvich,from 1841 /o 1857.
By George BiDD^hh AiUY, Astronomer Royal G17

XXX. Results of hourly Observations of the Magnetic Declination made by Sir Francis
Leopold M*^Clintock, and the Officers of the Yacht ' Fox,' at Port Kennedy, in
tlie Arctic Sea, in the Winter o/ 1858-59; and a Com.parison of these Results
with those obtained by Captain Rochfort Maguire, and the Officers of Her
Majesty's Ship 'Plover,' in 1852, 1853, and 1854, at Point Barroiv. By Major-
Ge7ieral 'Ed\\mib Sabine, R. A., President of the Royal Society .... 649

Index GC5

Appendix.
Presents [1]

LIST OF ILLUSTRATIONS.

Plates I. to IV. — Professor Owen on the Archeoptcryx.

Plates V. to VII. — Mr. G. B. Airy on the Strains in the Interior of Beams.

Plate VIII. — Rev. S. Haugiiton on the Reflexion of Polarized Light from Polished
Surfaces, Transparent and Metallic.

Plate IX. — Professor Bunsen and Dr. H. E. Roscoe's Photo-chemical Researches.

Plates X. to XII. — Rev. S. Hauohton on the Tides of the Arctic Seas.

Plates XIII. to XV. — Major-General Sabine on the Magnetic Observations at tlie Kew
Observatory.

Plates XVI. to XXIII. — Mr. G. B. Airy on the Diurnal Inequalities of Terrestrial
Magnetism.

Plate XXIV. — Mr. Balfodr Stewart on Experiments with an Air-Thermometer.

Plates XXV. to XXVII. — Mr. C. Chambers on the Nature of the Sun's Magnetic
Action upon the Earth.

Plates XXVIII. to XXXII. — Dr. H. de Schlagintweit on the Numerical Elements of
Indian Meteorology.

Plates XXLXIII. to XL. — Professor Beale on the Structure of the so-called Apolar,
Unipolar, and Bipolar Nerve-cells of the Frog.

Plate XLI. — Major-General Sabine on the Results of hourly Observations of the Mag-
netic Declination at Port Kennedy.

PHILOSOPHICAL TEANSACTIONS.

I. On the Relation of Radiant Beat to Aqueous Vapour. By John Tyndall, F.R.S.,
Member of the Academies and Societies of Holland, Geneva^ Gvttingen, Zurich,
Halle, Marburg, Breslau, Uiisala, la Societe Philomathique of Paris, Cam. Phil.
Soc. &c. ; Professor of Natural Philosophy in the Royal Institution.

Received November 20, — Read December 18, 1862.

I HAVE already placed before the Royal Society an account of some experiments whicli
brought to light the remarkable fact that the body of om- atmosphere, that is to say
the mixture of oxygen and nitrogen of which it is composed, is a comparative vacuum
to the calorific rays, its main absorbent constituent being the aqueous vapour which
it contains. It is very important that the minds of meteorologists should be set at rest
on this subject — that they should be able to apply, without misgiving, this newly
revealed physical property of aqueous vapom- ; for it is certain to have numerous and
important applications. I therefore thought it right to commence my investigations
this year with a fresh series of experiments upon atmospheric vapoui% and I now have
the honour to lay the results of these experiments before the Royal Society.

Rock-salt is a hygroscopic substance. If we breathe on a polished surface of rock-salt,
the affinity of the substance for the moisture of the breath causes the latter to spread
over it in a film which exhibits brilliantly the colours of thin plates. The zones of
colour shrink and finally disappear as the moisture evaporates. Visitors to the Inter-
national Exhibition may have witnessed how moist were the pieces of rock-salt exhi-
bited in the Austrian and Hungarian Courts. This property of the substance has been
referred to by Professor Magnus as a possible cause of cn-or in my researches on aqueous
vapour ; a film of brine deposited on the surface of the salt would produce the effect
which I had ascribed to the aqueous vapour. I will, in the first place, describe a method
of experiment by which even an inexperienced operator may avoid all inconvenience of
this kind.

In the Plate which accompanies my former paper, the thermo-electric pUe is figured
with two conical reflectors, both outside the experimental tube ; in my present expcri-

MDCCCLXIII. B

2 PEOFESSOE TTNDALL ON TIIE EELATION

ments the reflector wliicli faced the experimental tube is placed tvifhin the latter, its
naiTow aperture. Avhich usually embraces the i)ile, abutting against the plate of rock-
salt ^yhich stops the tube. Fig. 1 is a sketch of this end of the experimental tube. The

Fis;. 1.

=i3

edge of the inner reflector fits tiglitly against the interior surface of the tube at ah; cd
is the diameter of the wide end of the outer reflector, supposed to be turned towards
the "compensating cube" situated at C*. The naked face of the pile P is tm-ned
towards the plate of salt, being separated from the latter by an interval of about ^th
of an inch. The space between the outer surface of the interior reflector and the inner
surfece of the experimental tube is filled with fragments of freshly-fused chloride of
calcium, intended to keep the circumferential portions of the plate of salt perfectly
dry. The flux of lieat coming from the source C being converged upon the central
portion of the salt, completely chases every trace of humidity from the surface on which
it falls.

With this arrangement I repeated all my former experiments on humid and diy air.
Tlie result was the same as before. On a day of average humidify the quantity of vapotir
diffused in London air j)rodueed upwards of (JO times the absorption of the air itself.

It has been suggested to me that the air of our laboratory might be impure ; the
suspended carbon particles in a London atmosphere have also been mentioned to me as
a possible cause of the absorption which I had ascribed to aqueous vapour. With
regard to the first objection, I may say that the same results were obtained when the
apparatus was removed to a large room at a distance from the laboratory ; and with regard
to the second cause of doubt, I met it by procurhig air from the following places : —

1. Hyde Park.

2. Primrose Hill.

3. Hampstead Heath.

4. Epsom race-course.

5. A field near Newport, Isle of Wight.
G. St. Catharine's Do^^^l, Isle of Wight.
7. Tlie sea-beach near Black Gang Chine.

TIte arpieous vapour of the air from these localities exerted absorptions from 60 to 70
times that of tlie air in ivhich the vapour was diffused.

I then purposely experimented with smoke, by carrying air through a receiver in
which ignited brown ])ai)er had been permitted to smoulder for a time, and diying it

• I here assume an acfiuaiiitanco with my two last contributioas to the rhilosophioal Transactions, in whicli
the method of uompcnsatioii is described.

OF radiaot: heat to aqueous vapoue. ^

afterwards. It was easy, of coui-se, in this way to intercept the calorific rays ; but, con-
fining myself to the lengths of air actually experimented on, I con^^nced myself that,
even when the east wind blows, and pours the carbon of the city upon the west end of
London, the heat intercepted by tlie suspended carbon paiticles is but a minute fraction
of that absorbed by the aqueous vapour.

Fui-ther, I purified the air of the laboratory so well that its absorption was less than
unity ; the purified air was then conducted through two U-tubes filled with fragments
of clean glass moistened with distilled water. Its neutrality when dry proved that all
prejudicial substances had been removed from the air ; and in passing through the
U-tubes it could have contracted nothing save the piu"e vapour of water. The vapour
thus carried into the experimental tube exerted an absorption 90 times as great as that of
the air which carried it.

I have had the pleasure of shomng the experiments on atmospheric aqueous vapour
to several distinguished men, and among others to Professor Magnus. After operating
with common undried air, which showed its usual absorption, and while the uudried air
remained in the experimental tube, I removed the plates of rock-salt from the tube and
submitted them to the inspection of my friend. They were as dry as polished rock-
ci7stal, or polished glass ; their polish was undimmed by humidity ; and a dry handker-
chief placed over the finger and drawn across the plates left no trace behind it *.

I would make one additional remark on the above experiments. A reference to the
Plate which accompanies my two last papers ■nill show the thermo-electric pile stand-
ing, with its two conical reflectors, at some little distance from the end of the experi-
mental tube. Hence, to reach the pUe after it had quitted the tube, the heat had to
pass through a length of au- somewhat greater than the depth of the reflector. It has
been suggested to me that the calorific rays may be entirely sifted in this interval — that
aU rays capable of being absorbed by air may be absorbed in the space of air inter-
vening between the experimental tube and the adjacent face of the pile. If this were
the case, then the filHng of the experimental tube itself with dry air would produce no
sensible absorption. Thus, it was imagined, the neutrality of di-y air which my experi-
ments revealed might be accounted for, and the difierence between myself and Pro-
fessor ;NL\g>us, who obtained an absorption of 12 per cent, for dry air, explained. But
I think the hypothesis is disposed of by the foregoing experiments; for here the
reflector which separated the pile from the tube no longer intervenes, and it cannot be

* The present Number of the 'Monatsbericht' of the Academy of Berlin contains an account of some experi-
ments executed with plates of rock-salt by Professor Magnus. The plates which stopped the ends of a tube
were so far wetted by humid air that the moisture trickled from them in drops. As might be expected, the
plates thus wetted cut off a large amount of heat. The experiments are quite correct, but they have no bear-
ing on my results. In the earlier portions of my journal man)- similar ca-ses are described. In fact, it is by
making myself, in the first place, acquainted with the anomalies adduced by Professor Magnus, that I have
been able to render my results secure. I may add that the communication above referred to was made to the
Academy of Berlin before my friend had an opportunity of examining my rock-salt plates. I do not think he
would now urge this objection against my mode of experiment.

b2

PEOFESSOE TTXD-\XL OX THE EELATIOX

siipposed that in an inteiTal of ^th of an iucli of air an absorption of 12 per cent, has
taken place. If, however, a doubt on this point should exist, I can state that I have
purposely sent radiant heat through an intenal of 24 inches of dry aii- pre\-ious to per-
mitting it to enter my experimental tube, and found the effects to be the same as when
the beam had traversed 24 inches of a vacuum.

In confii-mation of the results obtained when my tube was stopped by plates of rock-
salt, I have recently made the following experiments with a tube in which no plates were
used. S is the source of heat, and ST the front chamber which is usually kept exhausted,
being connected with the experimental tube at T. This chamber is now left open.
A B is the experimental tube, with both its ends also open. P is the thermo-electric

Fifr. 2.

<^

d

JL

c

pile, the anterior face of which receives rays from the source S, while its posterior sur-
face is warmed by the rays from the compensating cube C. At c and d are two stop-
cocks— that at c being connected with an india-rubber bag containing air, while that at
d is connected ^^ith an air-pump.

My aim in this arrangement was to introduce at pleasure, into the portion of the tube
between c and d, diy air, the common laboratory air, or air artificially moistened. The
point c, at which the air entered, was 18 inches from the source S; the point d, at which
the air was withdrawn, was 12 niches from the face of the pile. By adopting these
dimensions, and thus isolating the central portion of the tube, one kind of air may Arith
ease and certainty be displaced by another without producing any agitation cither at
the source on the one hand, or at the pile on the other.

The tube A B being fiUed by the common air of the laboratoiy, and the needle of the
galvanometer pointing steadily to zero, dry air was forced gently from the india-rabber
bag through the cock c; the pump was gently worked at the same time, the diy air
being thus gradually dra^\Ti towards d. On the entrance of the dry air, the needle com-
menced to move in a direction which showed that a greater quantity of heat was now
passing through the tube than before. The dry air proved more transparent than the
common ah", and the final deflection thus obtained was 41 degrees. Here the needle
stopped, and beyond this point it could not be moved by the further entrance of diy air.

Shutting off the india-rubber bag and stopping the action of the pump, the appa-
ratus was abandoned to itself; the needle retui-ued ^vith great sloA\Tiess to zero, thus
indicating a correspondingly slow diffusion of the aqueous moisture through the dry air

OF RADIANT HEAT TO AQUEOUS VArOUE. 5

within the tube. By working tlie pump the descent of the needle was hastened, and it
finally came to rest at zero.

Diy air was again admitted ; the needle moved as before, and reached a final limit of
41 degrees ; common air was again substituted, and tlie needle descended to zero.

The tube being filled with the common air of the laboratory, which was not quite
saturated, and the needle pointing to zero, air from the india-rubber bag was now forced
through two U-tubes filled with fragments of glass wetted with distilled water. Tlie
common air was thus displaced by air more fully charged with vapour. The needle
moved in a direction which indicated augmented absorption ; the deflection obtained in
this way was 15 degrees.

I have repeated these expeiiments himdreds of times, and on days mdely distant fiom
each other. I have also subjected them to the criticism of various eminent men, and
altered the conditions in accordance with their suggestions. The result has been inva-
riable. The entrance of each kind of air is always accompanied by its characteristic
action. The needle is under the most complete control, its motions are steady and
unifonn. In short, no experiments hitherto made with solids and liquids are more free
from caprice, or more certain in theu- execution, than ai"e the foregoing experiments
with diy and humid au*.

The quantity of heat absorbed in the above experiments, expressed in hundredths of
the total radiation, was found by screening off one of the sources of heat, and deter-
mining the full deflection produced by the other and equal source.

By a careful calibration, repeatedly verified, this deflection was proved to correspond
to 1200 units of heat, — the unit being, as before, the quantity of heat necessary to move
the needle of the galvanometer from 0°to 1°. According to the same standard, a deflec-
tion of 41° coiTesponds to an absorption of 50 units. From these data we immediately
calculate the number of rays per hundi-ed absorbed by the aqueous vapour,

1200: 100 = 50 : 4-2.

An absorption of 4'2 per cent, was therefore efiiected by the atmospheric vapom- which
occupied the tube between the points c and d. Air j^erfedly saturated on the day in
question gave an absoi-ption of 5^ per cent.

These results were obtained in the month of September, and on the 27th of October
I determined the absorption of aqueous vapour with the above tube when stopped with
plates of rock-salt. Three successive experiments gave the deflections produced by the
aqueous vapour as 46°*6, 46°"4, 46°*8. Of this concuiTent character are aU the experi-
ments on the aqueous vapour of the air. The absorption corresponding to the mean
deflection here is 66. The total radiation through the exhausted ^tube was on this day
1085 ; hence we have

1085 : 100 = 00 : Gl;

that is to say, the absorption of the aqueous A'apom- of the aii* contained in a tube 4 feet
long, was on this day 6 per cent, of the total radiation.

6 PEOFESSOR TTNDALL ON THE EELATION

The tube with which these experiments were made was of brass, polished within ; and
it was suggested to me that the vapour of the moist air might have precipitated itself
on the interior surface of the tube, thus diminishing its reflective power, and producing
an effect apparently the same as absorption. In reply to this objection, I would remark
that the air on many of the days on which my experiments were made was at least
25 per cent, under its point of saturation. It can hardly be supposed that air in this
condition would deposit its vapour upon a polished metallic surface, against which,
moreover, the rays from our source of heat were imjiinging. More than this, the
absorption was exerted even when only a small fraction of an atmosphere was made
use of, and found to be proportional to the quantity of atmospheric vapour present in
the tube. The following Table shows the absorptions of humid air at tensions varying
from 5 to 30 inches : —

Humid

Air.

Tensions

Absorpt

ion.

in inches.

Observed.

Calculated.

5

16

16

10

32

32

15

49

48

20

64

64

25

82

80

30

98

96

The third column here is calculated on the assumption tliat the absorption, within the
limits of the experiment, is sensibly proportional to the quantity of matter in the tube.
The agreement with observation is almost perfect. It cannot be supposed that results
so regular as these, agreeing so completely with those obtained with small quantities
of other vapours, and even with small quantities of the permanent gases, can be due
to the condensation of vapour on the surface of the tube. When 5 inches were in the
tube it had less than one-sixth of the quantity of vapour necessary to saturate the space.
Condensation under these circumstances is not to be assumed, and more especially a
condensation which should produce such regular effects as those above recorded.

The subject, however, is so important that I thought it worth while to make the
following additional experiments : —

C is a cube of boiling water, intended for our source of lieat ; Y is a hollow brass
cylinder, 3-5 inches in diameter and 7 "5 inches in depth; P is the thermo-electric pile,
and C the compensating cube; S is an adjusting screen, used to regulate the amount
of heat falling on the posterior surface of the pile. The apparatus was entirely sur-
rounded by boards, the space within being divided by tin screens into compartments
which were loosely stuffed with paper or horsehair. The formation of air-currents near
the cubes or the pile was thus prevented, and irregular motions of the external air were
intercepted. A roof, moreover, was bent over the pile, and this was flanked by sheets

OF RADIA>fT HEAT TO AQUEOFS VAPOUE. 7

of tin. The action here souj^ht I know must be small, and hence tlie necessity of
excluding every disturbing intlucnce.

Fijr. 3.

The cylinder Y was first filled with fragments of quartz moistened with distilled water.
A rose burner r was placed at the bottom of the cylinder, and from it the tube Hed to a
bag containing aij-. The bag being subjected to gentle prersm-e, the air passed upwards
amid the fragments of quartz, imbibing moisture from them, and finally discharged itself
in the open space between the cube C and the pile. The needle moved and assumed a
permanent deflection of 5 degrees, indicating that the opacity of the intervenino- space
to the rays of heat was augmented by the discharge of the saturated air.

The moist quartz fragments were now removed, and the vessel Y was filled with frag-
ments of the chloride of calcium. The rose burner being, as before, connected with
the india-rubber bag, air was gently forced up among the calcium fragments and
discharged in front of the pile. The needle moved and assumed a permanent deflec-
tion of 10 degrees, indicating that the transparency of the space between the pile and
source was augmented by the presence of the diy air. By timing the discharges the
swing of the needle could be augmented to 20 degrees. Repetition showed no devia-
tion from this result — the saturated air always augmented the opacity, and the dry air
always augmented the transparency of the space between the soui-ce and the pile.

Not only, therefore, have the plates of rock-salt been abandoned, but also the experi-
mental tube itself, the displacement between dry and humid air being effected in the open
atmosphere. The experiments are all perfectly concurrent as regards the actim of the
aqueous vapour upon radiant heat.

8 PROFESSOE TTXDALL ON THE EELATIOX

The power of aqueous vapour being thus established, meteorologists may, I think,
apply the result •nithout fear. That 10 per cent, of the entii-e terrestrial radiation is
absorbed by the aqueous vapour which exists withru ten feet of the earth's surface on
a day of average humidity, is a moderate estimate. In warm weather and air approach-
ing to saturation, the absorption would probably be considerably greater. This single
fact at once suggests the importance of the established action as regards meteorology.
I am persuaded that by means of it many difficulties will be solved, and many familiar
effects, which we pass over without sufficient scrutiny because they are familiar, will
have a novel interest attached to them by their connexion with the action of aqueous
vapour on radiant heat. While leading these applications to be made in all their
fullness by meteorologists, I would refer, by way of illustration, to one or two points
on which I think the experiments bear.

And first it is to be remarked that the vapoiu- which absorbs heat thus greedily
radiates it very copiously. This fact must, I think, come powerfully into play in the
tropical region of calms, where enormous quantities of vapour are raised by the sun, and
discharged in deluges upon the earth. This has been assigned to the chilling conse-
quent on the rarefaction of the ascending air. But if we consider the amount of heat
liberated in the formation of those falling toiTents, the chilUng due to rarefection will
hardly account for the entire precipitation. The substance quits the earth as vapour,
it retui-ns to it as water ; how has the latent heat of the vapour been disposed of? It
has in great part, I think, been radiated into space. But the radiation which disposes
of such enormous quantities of heat subsequent to condensation, is competent, in some
measure at least, to dispose of the heat possessed prior to condensation, and must there-
fore hasten the act of condensation itself. Saturated air near the surface of the sea
is in circumstances totally different from those in which it finds itself in the higher
atmospheric regions. Aqueous vapoiu' is a powerful radiant, but it is an equally power-
ful absorbent, and its absorbent power is a maximum when the body which radiates
into it is vapour like itself. Hence, when the vapour first quits the equatorial ocean
and ascends, it finds, for a time, a mass of vapour above it, into which it poiu-s its heat,
and by which that heat is intercepted and in part returned. Condensation in the lower
regions of the atmosphere is thereby prevented. But as the mass ascends it passes
through successive vapoui--strata which diminish far more speedily in density than the
associated strata of aii-, until finally om- ascending body of vai)our finds itself lifted above
the screen which for a time protected it. It now radiates freely into space, and con-
densation is the necessary consequence. The heat liberated by condensation is, in its
turn, spent in space, and the mass thus deprived of its potential energy retui-ns to the
earth as water. To what precise extent this power of aqueous vapour as a radiant comes
into play as a promoter of condensation, I will not now inquire; but it must be influen-
tial in producing the torrents which are so characteristic of the tropics.

The same remarks apply to the formation of cumuli in our own latitudes. They ai-e
the heads of columnar bodies of vapour which rise from the earth's surface and are

OF RADIANT HEAT TO AQCT:0U.S YAPOTTR, • 9

condensed to cloud at a certain elevation. Tlius the ■\isible cloud forms the capital of
an innsible pillar of saturated air. Certainly the top of the column, piercing the sea of
vapour wliich Imgs the earth, and offering itself to space, must lose heat by the radiation
from its vapour, and in this act alone we should have the necessity for condensation.
The " vapour plane" must also depend, to a greater or less extent, on the chilling effects
of radiation.

The action of mountains as condensers must, I think, be connected Avith these con-
siderations. AVhen a moist wind encounters a mountain-range it is tilted upwards, and
condensation is no doubt to some extent du(> to the work performed by the expanding
air ; but the other cause cannot be neglected ; for the air not only performs work, but
it is lifted to a region where its vapour can freely lose its heat by radiation into space.
During the absence of wet winds the mountains themselves also lose their heat by radia-
tion, and are thus prepared for actual surface condensation. We must indeed take into
account the fact that this radiant quality of water is persistent throughout its three
states of aggregation. As vapour it loses its heat and promotes condensation ; as water
it loses its heat and promotes congelation ; as solid it loses its heat and renders the
surfaces on which it rests more powerful refrigerators than they would otherwise be.
The formation of a cloud before the air which contains it touches a cold mountain, and
indeed the formation of a cloud anywhere over a cold tract of land, where the cloud is
caused by the cold of the tract, is due to the radiation from the aqueous vapour. The
uniformly diffused fogs which sometimes fill the atmosphere in still Aveather may be
due to cold generated by uniform radiation throughout the mass, and not to the mixture
of currents of diflFerent temperatures. The cloud by which the track of the Nile and
Ganges (and sometimes the rivers of our own country) may be followed on a clear morn-
ing is, I believe due to the chilling of the saturated air above the river by radiation
from its vapour.

ObseiTation proves the radiation to augment as we ascend a mountain. Maktins and
Bravais, for example, found the lowering of a radiation-thermometer 5°'7 Cent, at Cha-
mouni; while on the Grand Plateau, under the same conditions, it was 13°'4 Cent.
The following remarkable passage from Hooker's Himalayan Journals, 1st edit. vol. ii.
p. 407, bears directly upon this point : — " From a multitude of desultory observations I
conclude that, at 7400 feet, 125°-7 or 67° above the temperature of the air, is the average

maximum effect of the sun's rays on a black-bulb thermometer These results,

though greatly above those obtained at Calcutta, are not much, if at all, above what may
be observed on the plains of India [because of the dryness of the air. — J. T.]. The effect
is much increased with the elevation. At 10,000 feet, in December, at 9 a.m. I saAv the
mercury mount to 132° [in the sun], with a difference [above the shaded air] of 94°,
while the temperature of shaded snow hard by was 22°. At 13,100 feet, in January, at
9 a.m. it has stood at 98°, with a difference of 68°-2, and at 10 a.m. at 114°, with a
difference of 81°-4, whilst the radiating thermometer on the snow had fallen at sunrise
to 0°"7." This enormous chilling is fully accounted for by the absence of aqueous

MDCCCLXIII. c

10 PEOFESSOE TTINDALL OX THE EELATION

vapour overhead. I never nnder any circumstances suffered so much from heat as in
descending on a sunny day from the so-called Corridor to the Grand Plateau of Mont
Blanc. The air was perfectly still, and the sun literally blazed against my companion
and myself We were hip deep in snow ; still the heat was unendurable. Immersion
in the shadow of the Dome du Goutc soon restored our powers, though the air of the
shade was not sensibly colder than that through which the sunbeams passed. Not-
withstanding the enormous daily accession of heat from the sun, terrestrial radiation at
these altitudes preserves an extremely low temperature at the earth's surface.

AVithont quitting Europe we find places where, even when the day temperatiu-e is
high, the hour before sunrise is intensely cold. I have often experienced this even in
Germany; and the Hungarian peasants, if exposed at night, t-ake care, even in hot
weather, to prepare for the nocturnal chill. The rccjuje of temperature augments with
the dryness, and an " excessive climate " is certainly in part caused by the absence of
aqueous vapour.

Regarding Central Australia, Mr. Mitchell publishes extremely valuable tables of
observations, from which we learn that, when the days are at the same time calm and
clear, the daily thermometric range is exceedingly large. The temperature at noon being
68° on the 2nd of INIarch 1835, that at sunrise next morning was 20°, showing a differ-
ence of 48°. The 7th and 8th were also clear and calm ; the difference between noon
and sunrise on the former day was 38°, while on the latter it was 41°. Indeed between
April and September a range of 40° in clear weather was quite common — or more than
double the amount which it is in London at the corresponding season of the year.

A freedom of escape similar to that from bodies at great elevations would occur- at
any other level were the vapour removed from the air above it. Hence tlie withdrawal
of the sun from any region over which the atmosphere is dry, must be followed by quick
rc'frigeration. This is simply an a j)riori conclusion from the facts established by expe-
riment ; but I believe all the experience of meteorology confirms it. The winters in
Tibet are almost unendurable from this cause. The isothermals dip deeply from the
north into Central Asia during the winter, the earth's heat being wasted without impe-
diment in space, and no sun existing sufficiently powerful to make good the loss. I
believe the fact is well established that the desert of Sahara, which during the day is
burning hot, is often extremely cold at night. This effect has been hitherto referred in
a general way to the "purity of the air;" but purity, as judged by the eye, is a very
imperfect test of radiation, for the existence of large quantities of vapour is consistent
with a transparent atmosphere. The purity really consists in the absence of aqueous
vapour from those so-called rainless districts, which, when the sun is Avithdrawn, enables
the hot surface of the earth to run speedily down to a freezing temperature.

On the most serene days the atmosphere may be charged with vapour ; in the Alps,
for example, it often happens that skies of extraordinary clearness arc the harbingers of
rain. On such da\s, no matter how pure the air may seem to the eye, terrestrial radia-
tion is arrested. And here we have the simple explanation of an interesting fact noticed

OF EADIA2fT HEAT TO AQUEOUS VAPOUE. 11

by Sir John Leslie, which has remained without explanation up to the present time.
This eminent experimenter devised a modification of his differential thermometer,
which he called an ^EfhrioscojJe. The instrument consisted of two bulbs united by a
vertical tube, of a bore small enough to retain a little liquid index by its own adhesion.
The lower bulb was protected by a metallic coating ; the upper or sentient bulb was
blackened, and was placed in the concavity of a polished metal cup, which protected
it completely from terrestrial radiation. " This instrument," says its inventor, " Avill at
all times during the day and night indicate an impression of cold shot downwards from

the higher regions But the cause of its variations does not always appear so

obvious. Under a fine blue sky the JEtkrioscope will sometimes indicate a cold of
50 millesimal degrees ; yet on other days, when the air is eqiuiUij hright, the effect is
hardly 30°." It is, I think, certain that these anomalies were due to differences in the
amount of aqueous vapour in the air, which escaped the sense of vision. Leslik him-
self connects the effect with aqueous vapour by the following remark: — "The pressuie
[apparently a misprint iox presence'\ of hygrometric moisture in the air probably affects
the indications of the instrument." In fact, the moisture opened and closed an invisible
door for the radiation of the "sentient bulb" of the instrument into space. The follow-
ing observation in reference to radiation-experiments with Pouillet's pyrheliometer,
now receives its explanation. " In making such experiments," says M. Schlagixtweit,
" deviations in the transparency are often recognized which are totally inappreciable
to the telescope or the naked eyes, but which after^\•ards announce themselves in the
presence of thin clouds," &c.

In his beautiful essay on Dew, Wells gives the true explanation of the formation of
ice in India, by ascribing the effect to radiation. I think, however, his theory needs
supplementing. Given the same day-temperatui'e here as at Benares, could we, even in
cleai" weather, obtain a sufficient fall of temperature to produce ice I I think not.
The interception of the calorific rays by our humid air would too much retard the chiU.
It is apparent, from the descriptions given of the process, that a dry still air was the
most favourable for the formation of the ice. The nights when it was formed in greatest
abundance were those during which the dew was not copious. The flat pans used in
the process were placed on dry straw, and if the straw became wetted it was necessary
to have it removed. Wells accounts for this by saying that the wetted straw is more
dense than the dry, and hence more competent to transfer heat from the earth to the
basins. This may be to some extent true; but it is also certain that the evaporation
from the moist straw, by throwing over the pans an atmosphere of aqueous vapom-,
would check the radiation and thus tend to diminish the cold.

Mellom, in his excellent paper " On the Nocturnal Radiation of Bodies," gives a
theory of the serein, or excessively fine rain which sometimes falls in a clear sky a few
moments after sunset. Several authors, he says, attribute this effect to the cold resulting
from radiation of the air during the fine season immediately on the departure of the sun.
" But," writes Melloxi, " as no fact is yet known which distinctly proves the emissive

12 ON THE EELATION OF EADIANT HEAT TO AQUEOUS VAPOUR.

power of pure transparent elastic fluids, it appears to me more conformable to the prin-
ciples of natural philosophy to attribute this species of rain to the radiation and subse-
quent condensation of a thin veil of vesicular vapour distributed through the higher
strata of the atmosphere "*. Now, however, that the power of aqueous vapour as a
radiant is known, the difficulty experienced by Mellom disappears. The former hypo-
thesis, however, though probably correct in ascribing the efi'ect to radiation, was incor-
rect in ascribing it to the radiation of '■^the air."

Dr. Hooker encourages me to hope that this newly discovered action may throw some
light on the formation of hail. The wildest and vaguest theories are afloat upon this
subject. But the same action which produces serein must, if augmented, freeze the
minute rain, and the aggregation of the small particles thus frozen would form hail. I
cannot think the hail that I have had an opportunity of examining to be due to the
freezing of drops of water, each hailstone being merely the ice of the drop. The "stones"
are granular aggregates, the components of which may, I think, be produced by the
chill of radiation. I will not, however, dwell further on this subject, but will now com-
mit the entire question to those who are more specially qualified for its investigation.

* Tatloe's Scientific Memoirs, vol. v. p. 551.

[ 13 ]

II. 0)1 the Volumes of Pedal Surfaces. By T. A. IIikst, F.Ii.S.

Eeceived August 25, — Read jS'ovember 20, 1862.

1. In accordance with the proposition recently made by Dr. Salmon in his excellent
treatise on Surfaces*, the term jaedal surface is here adopted, as tlie English equivalent
of the French surface-podaire and the German FussjJiincts-Flache, to indicate the locus
of the feet of perpendiculars, let fall fi-om one and the same point in space, upon all the
tangent planes of a given primitive surface.

The point of contact of the tangent plane, and the foot of the perpendicular upon the
latter, are said to be corresponding j)oints on the primitive and its pedal. The point
whence perpendiculars are let fall may be termed the pedal-origin. It is obvious that
the pedal surface may Ukewise be regarded as the envelope of spheres lia\ing for their
diameters the radii vectores from this origin to the several points of the primitive f.

The primitive siuface remaining unaltered, the magnitude and form of the pedal will
of course y-Axy with the position of its origin. Between the volumes of all such pedals,
however, certain very general and remarkable relations exist. The object of the present
paper is to establish these relations.

2. Twenty-four years agoj Professor Steiner, in one of his able and purely geometrical
memoirs presented to the Academy of Berlin, established analogous relations between
the areas oi pedal ctirves corresponding to different origins in the plane of the primitive.
I am not aware, however, of any attempt having been made to extend his results to
surfaces, although such an extension can scarcely ha^•e failed to suggest itself, not only
to Steixer himself, but to many of his readers §. For the sake of comparison I will here
state a few of these results.

* A Treatise on the Analytic Geometry of Three Dimensions, hy G. Salmon, D.D., 18G2, p. 369.

t The pedal origin being the same, the surface derived from the pedal, in the same manner as it was derived
from the primitive, would bo called the second pedal ; the pedal of this, again, the third pedal, and so on. It
has, further, been found convenient to apply the term positive to the pedals of this series, in order to distin-
guish them from another series of surfaces obtained by reversing the above process of derivation. Thus the
surface of which the primitive is the pedal is termed the Jirst ymjative pedal, and so on. I may also remark
that the whole series of positive and negative pedals is identical with the series of derived surfaces which forms
the subject of papers published by Messrs. Tortolini and "W. Roberts, as well as by myself, in Toetoli.vi's
' AnnaU ' and the ' Quarterly Journal of Math.' for 1859. In the present paper first positive pedals arc alone
considered, though it would no doubt be interesting to examine the volumes of pedals of higher order.

X See Crelle's Journal, vol. xxi. p. 57.

§ Dr. BoRCHASDT has quite recently (April 1863) apprized me of the existence of an Inaugural Dissertation,
entitled " De supcrficierum pedaUum theorcmatibus quibusdam," whose publication was sanctioned, in 1859,
by the University of Berlin, and in which the two fundamental theorems of art. 3 are established. To English
mathematicians, however, the theorems in question will probably be still new, since, so far as I can ascertain,
their discoverer. Dr. FiscnER, has never given fidl publicity to the results of his investigations.

JIDCCCLXIII. D

14 JIE. T. A. HlflST OX THE VOLIJIMES OF PEDAL SUEFACES.

" The primitive curve being closed, but otherwise perfectly arbitrary, the locus of the
origins of pedals of constant area is a circle. The several circular loci, corresponding to
different areas, are concentric, and theu- common centre is the origin of the pedal of
minimum area."

ST£I^'ER signalizes, as a veiy remarkable mechanical property of this common centre,
the fact that it always coincides with the Krummungs-Sch'werpunct of the primitive
curve, — that is to say, with the centre of gra\ity of that primitive, regarded as a material
ciu've whose density is ever3'where proportional to the curvature.

In 1854, sixteen years after the appearance of Stelnee's memou', Professor Raabe of
Zurich* extended Steiner's theorem so as to embrace the pedals of unclosed curves.
The general definition of the area of a pedal being the space swept by the perpendicular
as the point of contact of the tangent describes the pi-imitive arc, Raabe found that " the
origins of all pedals of the same area lie on a conic." The several quadric loci, corre-
sponding to different areas, are concentric and co-axal ; theu* common centre is again
the origin of the pedal of least area; and though it no longer coincides with the
Kr'dmmurigs-Schwerpunct of the primitive arc, it is intimately comiected therewith, as
has been more recently shown by Dr. Wetzig of Leipzigf .

3. With respect to surfaces, the volume of the pedal may be stated, in general terms,
to be that of the cone whose vertex is the pedal-origin and whose base is that portion
of the pedal surface which corresponds to the given portion of the primitive. This
definition being accepted, it will be shown in the sequel that, whatever the nature of the
jprimitive surface may he, the origins of pedals of equal volume always lie on a surface of
the third order; and further, that when the primitive surface is closed, but otherwise
perfectly arbitrary, this cubic locus degenerates to a quadric, tlie wliole of the loci,
corresponding to all pjossible volumes, then forming a system of similar, similarly placed,
and concentric quadrics whose common centre is the origin of tlte pedal of least volume.

4. For the sake of comparison it is desirable to treat, by a uniform method, the two
analogous qviestions respecting pedal curves and pedal surfaces. I commence, therefore,
witli a brief consideration of Stelxer's theorem.

Let (C) represent the primitive curve, (P) the pedal whose origin A has the coordinates
X, y, and (P,,) the pedal whose origin O coincides with that of the coordinate axes. The
curve (C) may be regarded as dividing the plane into two parts, distinguishable as
external and internal ; let a, and /3 then be the angles, each positive and less than -tt,
between the positive dirc^ctions of the coordinate axes and that of the normal at any
point M of (C), this normal being always supposed to be drawn from the curve into the
external part of tlie plane. Further, let^J and j'^ be the perpendiculars let fall respect-
ively from tlie point A, and from the origin O upon the tangent at M, so that their feet
m and m^ are tlie points on the pedals (P) and (P„) which correspond to M on the primi-
tive. The direction-angles of each perpendicular will be

«, /3, or T— a, w — /3,

• Creij.k's Journal, vol. 1. p. 19,3.

t Zeitschrift fiir Mathematik iind I'liysik, 1800, vol. v. p. 81.

ME. T. A. HIRST ON THE VOLUMES OF PEDAL SUBFACES. 16

according as its direction coincides with, or is opposed to that of the normal ; so that if
we regard p and ^;, as positive or negative according as the one or the otlier of these
circumstances occurs, we shall have, generally,

p-=Po — X cos a — y sin a.

If we, further, denote by d6 the ai"c of the unit-circle, around the origin, intercepted
between radii whose du'ections coincide with those of the normals at the extremities of
the element ds of the primitive arc at M, and agree to consider the parallel elements ds
and do as alike or unlike in sign according as their directions coincide with or are
opposed to each other, the corresponding elements (ZP and dV^ of the areas of the pedals
(P) and (P,) will be

and, by the preceding relation, we shall have

2f7P=(^o — A* cos u—y sin a)V7^ ;
whence, by integration, we deduce the equation

P=P.-A,a:-A^+i(A„a'^+2A,^rj/+A,^=), (A.)

wherein P and Po denote the areas of the two pedals, and the coefficients have the values

Aj =\2)od6.cosu, A2^=\ Pod 6. cos 13,

A„=W^cos*a, A,2=:|f7^cosacosj3, A„o=\d^ cos^ft,

dependent only on [the position of the origin O, and on the curvature of the primitive
curve. The integration in each case is, of course, to be extended to all points of the
primitive arc.

5. The above formula, by means of which the area of any pedal (P) may be found
when the area of any other (?„) is known, shows at once that the locus (A) of the origin
A of a pedal (P) of constant area is a conic, and that all such loci constitute a system of
similar, similarly placed, and concentric conies, the common centre of the loci being the
point at which the integrals A„ A^ vanish. K we suppose the origin of our coordinate
axes to coincide with this point, the equation of the locus (A) may be written thus :

P=P„+ir( ^'cos Dc+y cos^ydO,

whence we learn that the common centre of all the quadric loci (A) is the origin of the
pedal of least area.

6. This is Eaabe's theorem ; in order to deduce Steiner's from it let us consider, in
the first place, the pedals of a primitive arc containing a point of inflexion and having
parallel normals at its extremities. The normals along such an arc vriU consist of
pau-s of like-directed parallels ; but in passing from one extremity to the other the sign

d2

16 IME. T. A. HIEST OX THE VOLUMES OF PEDAL SUEFACES.

of d6 will change, so that the integrals A,„ A,„, Ajj will each consist of equal and
opposite elements and vanish in consequence*.

If, now, the primitive be a closed curve, but othervrise perfectly arbitrary, w^e may
always conceive it to consist of arcs (C) of the kind just considered, and of other arcs
(C) the directions of whose normals represent exactly all possible du-ections round a
point. But it has already been shown that for every arc (C) the integrals A,,, A, 2, Aj,
vanish, and it is easy to see that, extended over the arcs (C"), these integrals have the

values

A,, = A2,=j;7r, A,2=0,

where n represents the number of such arcs, in other words, the number of convolutions
of the primitive curve. In this case, therefore, the equation of art. 5 becomes

P = Po + f(a'^+/) = Po + ?'^

and for constant values of P represents a circle around the origin of the least pedal.

7. In order to illustrate by an example what is meant by the area of a pedal, let
us consider for a moment the case of an ellipse with the semiaxes a, b. The focal pedal,
as is well known, is a circle whose diameter is the major axis; so that putting for P, 11-, r"
the values z-a^ 1, fr — b' respectively, we find, for the area of the central pedal, the value

Y,=l(rr+b-^),

equal to the area of the semicircle whose radius is the line joining the extremities of
the axes ; and the area of any other pedal is

For the circle ff=ib, we liave

P=-(.r + i=+r^).

P=-«^+^r^

which clearly represents the sum of the areas of the two loops of which the pedal con-
sists when its origin is \Wthout the circle. When a vanishes, the pedal is well known
to be the circle on r as diameter. Our last formula shows, however, that we must con-
ceive this circle to be doubled. A glance at the expressions for jj and dP in art. 4
explains this distinctive feature of pedal areas. It will be there seen that the sign of
the increment fW do(\s not depend upon that of ^*. which latter changes according as the
pedal-origin lies on one or the other side of tlu^ tang(>nt. For pedal surfaces, to which
,we will now proceed, the case is otherwise.

8. Let a\ //. z be the coordinates of the origin A of a pedal (P) of a siu-facc (S) ; and,
as before, let (PJ denote^ the pedal of the same surface whose origin O coincides with
.that of the coordinate axes. Then if «, /3, y be the direction-angles of the external

* The locus (.-V) of equal pedal origius eoiiioidos, in this ca-se, with the right line 1' = ]',, — A,a-— .V^y, as was
•first shown bv Wktzio.

MB. T. A. HIRST OX THE VOLUMES OF PEDAL StTRFACES. 17

normal at a point M of (S), andp, ^)„ the perpendiculars from the origins of (P) aiid(Po)
upon the tangent plane at M, we shall, again, have the general relation

2>^Po — ^ cos u — 1/ cos j3— z cos y,

pro^ided the sign of ^ he understood to depend upon the side of the tangent plane upon
which the pedal origin is situated.

Further, let d<r be the sm-face-element of the unit-sphere intercepted by radii hanng
precisely the same directions as the external normals at the contour of the element ds at
M on the primitive surface. According to Gauss's definition drr will also be the total
curvature of the element ds, and will have the value kds, where k is the measure of
curvature at M, in other words, the reciprocal of the product of the principal radii of
curvature. The volume-element of the pedal P will, obviously, have the value

fZP=i/(7(r,

and will change sign with j) as well as with d<r. By means of the preceding relation,

then, we have

BdF=(2>—iV cos a—}/ cos /3 — z cos 7)V7(t,

which expression, when developed and integrated, assumes the form

P=Po-(A„ A„, A3X^^', >/, -) + (A„, A,„ A33, A„3, A3,, A,„Xx, y, zf ]

¥(AllH -^.222? -^^3335 -A-U25 -^1135 ■^■2235 -^-221? -^3315 -^3325 -^123jL'^' Vl ^) 'J

where the nineteen coefficients are independent of the position of the pedal origin A,
and represent double integrals to be extended to all points of the primitive surface. Of
these coefficients it will suffice to write the values of the following six, the remaining
thirteen being deducible therefrom by permutations of a, j3, y in accordance with those
of the suffixes i, 2, 3.

A, =yjGf/<rcosa , A,, =WJ„(Zff cos'a , Ai,i=|(7ff cos'«,

A,2= i^V?<^ cos a cos /3, A,,„= j da cos^ a cos j3, A,23= IfZ-r cos a cos /3 cos y.

. The above fonnula for the volume of the pedal (P) at any point A shows at once, as
stated in art. 3, that the origins of j)edals of equal volume are situated on a surface of
the tliird order.

9. The analogy between the cases of pedal curves and sui-faces will be evident on
observing that the above cubic locus proceeds essentially from the three dimensions of
space, just as the quadric locus, in the case of pedal curves, was due to the two dimen-
sions of a plane. It is interesting to note, however, that whilst the hypothesis of a
jclosed primitive curve had merely the effect of altering the species, not the order, of the
locus (A), the hypothesis of a closed primitive surface leads to a reduction of this locus
from a cubic to a quadric. The former effect was produced by the equalization of the
coefficients of .r* and /, and the vanishing of that of xy (art. G); the latter is due to

18 :\rR. T. A. HIEST ox THE YOLTBIES OF PEDAL SURFACES.

the vanishing of each of the ten integi'als A,,,, A, ,2, &c which, not involving p,,

have vahies dependent solely upon the curvature of the primitive siu-face.

I do not attempt any complete discussion of all possible singularities of curvature, but
merely obsene that the above-mentioned property of the ten integrals is easily recognized
when the primitive surface is not only closed, but everywhere convex ; for since all direc-
tions round a point are then exactly represented by its normals, the integrals in ques-
tion each represent a sum of pairs of equal and opposite elements. In the more general
case, where certain directions are represented more than once, and consequently an odd
number of times, by the nonnals of the primitive, the property in question may be
verified by a method similar to that employed in art. 6.

10. The primitive being a closed surface, the form to which the equation (A.) of art. 8
becomes reduced, at once shows that the several quadric loci corresponding to pedals (P)
of different , but constant volumes, constitute a system of similar, siinilarly situated, and
concentric quadrics, their common centre being the origin of the ])edal of least volume.
For if this centre, which is determined by the conditions

A, = y)^(Z<T cos «=0, A,= yjo'^^ircos/3=0, A3=i jjof?(7cos7=:0,

were chosen as origin of coordinate axes, the equation (A.) of art. 8 would assume the

form

P=P„+(A,„ A,,, A33, A,3, A3,, A„X^-,y, zf,

which may be also written thus.

P=Po-f yjo(.r cos a-\-y cos/3+2 cos y^da.

in which form it renders apparent the minimum property in question.

When the closed primitive has itself a centre, the latter mil also be the common
centre of the loci (A); for, the centre of the primitive being taken as origin of coordinate
axes, each of the integrals A,, A.^, A3 will again consist of pairs of equal and opposite
elements.

11. To illustrate the foregoing principles, as well as to facilitate future applications,
we will consider for a moment the simplest of all cases — where the primitive is a
sphere with radius a. Taking its centre for origin, sixteen of the integrals of art. 8
will be found to vanish, and the remaining ones, A,„ A22, A33, to acquire the common
value f Trt ; so that the volume of any pedal (P) becomes

P=|,rff(«^+a:H?/Hz')=-3'r«(ff'+J'').

When the origin of (P) is without the sphere, the pedal consists, of course, of two
distinct sheets, each passing through the origin and touching the primitive ; the volume
of the pedal, as above given, is the difference of the volumes enclosed by these sheets.
When the sphere diminishes to a point, the volumes of all pedals vanish according to
the formula; so that we must regard the pedal of a point as consisting of two coin-
cident spherical sheets.

MR. T. A. HIEST OX TIIE VOLUlVIEa OF PEDAL SURFACES. 19

In like manner the pedal of a tubular surface would, in general, consist of distinct
sheets which would coincide when the primitive degenerated to a line. Although the
pedal surface, therefore, still exists when two dimensions of the primitive are sui)posed
to vanish — being, in fact, still the envelope of spheres whose diameters are the radii
vectores of the cui've — its volume must be regarded as evanescent.

The case is othenvise, however, when one only of the three dimensions of the primi-
tive is supposed to vanish. Such a surface (S') would consist of two coincident sheets,
and would, therefore, enclose no space ; to the eye, in fact, it would not be distinguish-
able from some definite portion of an ordinary surface. Its pedal, however, would be
of a compound nature — consisting, Jirst, of a surface (P') of the same nature as (S'),
undistinguishable to the eye from ^a portion of its ordinary pedal, and, secondly, of the
simple pedal (P) of the curve (C) forming the contour of the primitive (S'). The
volume of the compound pedal, however, would be simply that of the pedal (P) of the
contour (C). This volume, therefore, properly interpreted, ought to be deducible from
our general formulae.

It must be observed, however, that although the form of the pedal of a curve (C) is
invariable, its volume must be diflferently estimated according to the nature of the two-
dimensional surface (S') of which the curve is supposed to form the contour. To render
this more e\ident, it wiU be convenient to regard the pedal of a curve, not only as the
envelope of a sphere, but also as the locus of a circle whose magnitude varies at the
same time that its plane rotates about a fixed point, the pedal-origin. This circle, in
fact, is the characteristic of the pedal ; its plane is perpendicular to the tangent at a
point M on the curve (C), and its chords, though the origin, are the perpendiculars upon
the several tangent planes of (S') at the point ISI of its contour.

Remembering now the convention of art. 8 with respect to the signs of these perpen-
diculars, and the relation between the same and those of the corresponding volume-
elements, we easily conclude that the volume of the pedal (P) will be the difference of
the volumes of the sm-faces generated by the segments into which the characteristic
circle is divided by the perpendicular ^' upon the ordinary tangent plane of (S') at the
point M of its contour.

The most interesting case, and the only one we shall examine further, is when the
surface (S') coincides with the developable of which (C) is the cuspidal edge. The per-
pendicular p' then coincides with that let faU on the osculating plane of the primitive
curve (C) ; through it pass the planes of two consecutive characteristics, and the locus
of its extremity is the cuspidal edge of the pedal (P), and at the same time the curve to
which, as is well known, the pedal surface of the developable (S') resolves itself.
The volume of the pedal (P) has now the simplest possible definition, and the double
integrals of art. 8, by means of which this volume may be expressed, are easily reducible
to single ones.

12. To effect this reduction, we vtall first express any perpendicular J5„ by means of^'
the perpendicular on the osculating plane of (C) — parallel therefore to the binomial — and

20 JIE. T. A. HIEST OX THE YOLUIMES OF PEDAL SUEEACES.

f the perpendicular on the rectifjdng plane ; the latter will of course be at right angles
to x^ and parallel to the principal normal. Since ^' and p are perpendicular chords of
a chcle, passing through the same point of its circumference, we have at once

^„ =^' cos <p +^J sin <p,

where ^ is the angle between p' and ^j^-

Further, the direction-cosines of ])^, that is to say cos a, cos /3, cos y, may in like
manner be expressed by means of those oi jJ and j), which we will denote respectively
by >/, \j1^ v' and "k, fji,, v. For the projections on J^o^ p\ P of the linear unit, set off on
any line through the origin, are clearly, again, chords of a circle, so that, operating
successively on the three coordinate axes, we readily deduce the relations

co« « =X' cos ^+^ sin <p,

cos j3=;u-' cos ^+ f* sin ip,

cos 7=1/' cosip+i' simp.

Lastlv, representing by (16 the angle between the planes of two consecutive character-
istics, in other words the angle of contact of the primitive curve (C), the surface-element
da of the unit-sphere will have the value

d(r^ sin (p d<p dd.

We have now merely to substitute the above values in the several integrals of art. 8,
and to effect the integration according to <p between the limits 0 and -r, regarding
thereby 'p, p\ a, j!a, v, X', jU.', v as constants. This may be readily done ; the nineteen
results are deducible by appropriate permutations of X, X' ; (Jj, i/J ; and c, f', in accordance
with the corresponding suffixes i, 2, and j, from the following six expressions : —

A. =-l^{?>}r+1}!pp'+K/^)d6.

A,„=^Ja(X'-'+X'^>/^.
Au.=iJ[(3>r+X'^)f/.+2XAy](7^.

■Ry means of the equations to the curve the nine quantities involved in thes(> int(>grals
arc readily expressible as functions of a single variable. This done, the integration in
each case is to be extended to all points of the primitive curve (C).

13. I do not enter into the several interesting questions which here suggest tliem-

and consequently

MR. T. A. HIRST OX THE VOLUMES OF PEDAL SUEFACES. 21

selves — as to the nature of the cubic locus of the origins of pedals of the present kind
which have a constant volume, the conditions under which this locus dco-encrates to a
quadric, and the position of the origin of tlie pedal of least volume — but pass at once to
the case of a plane primitive curve, every pedal of wliich will be a surface generated by a
circle, through two fixed points, whose magnitude varies at the same time that its plane
rotates around the lino joining those points. Taking the plane of the primitive as the
coordinate plane of xy, we lla^■o clearly

^;' = 0, //=j«,'=r^=0, ^' = l,

A, =^yjj-(U, Aa ~\u.]j-iU,

An =^J>^7«?^, A^, =-^Wp(U,

A,o ='-^y.i^2)d0, A33 =1 XjhJO,

A.„ = |'JaV^, A,,,=^J^W^,

An2= y J>>f7^, A20, = -^ i^X[/.'dO,

A33i=y>.f/^, A,,,='^yd0.

'When the primitive is a plane closed curve, the last six integrals, in general, vanisli,

and the locus of origins of equal pedals again degenerates to a quadric surface. The

origin of the least pedal does not generally coincide with the Krihnmungs-Schwerjmnct,

since A,, Aj have no longer the same values as in art. 4 ; it coincides with the centre of

the primitive, however, whenever the latter possesses such a point. For instance, for a

primitive circle {n) it will be found, on taking its centre for origin, tliat, with the

exception of three, all the foregoing integrals vanish, and that these three acquire the

values

A,i=A22=-g'3-"a, Aj^^-^'TTci.

The volume of the central or least pedal, (Pq), which is here a surface generated by the

rotation of a circle with radius -^ about one of its tangents, is easily found to be I^W, so

that the volume of any other pedal will, by art. 8, be

and the locus of origins of pedals of the same volume a prolate spheroid.

14. To return to the case of surfaces: I propose to consider next the pedals of the
ellipsoid, which, ever since the publication of Feesxel's researches on Light, have been

MDCCCLXIII. E

22 ism. T. A. HIKST OX THE T0Lr:ME3 OF PEDAL STJETACES.

regarded with especial interest. The application to them of the foregoing principles
will lead us to several new results.

With a view to this application, and in continuation of the subject of art. iO, I may
add that when the primitive siu-face is symmetrical ^nth respect to three rectangular
planes, the integrals A,,, A^s, A3, likewise vanish, on taking these planes of symmetry for
coordinate planes. In -sirtue of this property, which is evident from an inspection of
the values in art. 8, the expression for the volume of any pedal assumes the simple form

P=Po+A„r=+A,^^+A33C^
If, further, as, in the case of the ellipsoid, the primitive be a dosed convex surface, the
coefficients

A,, = 1 ^^(Zff cos^ u, Ar,2=-\pod<^ cos^ /3, A33=: \ po(7iT cos^ y

will manifestly be sums of elements of the same sign, so that fJie locus (A) of equal pedal
origins will he an ellipsoid whose axes coincide with the axes of symmetry of the primitive .
15. For the primitive ellipsoid

*,2 ,,2 Ji

^ + L + 1.^1

the squares of whose semiaxes, written in descending order of magnitude, we will sup-
pose to be rt„ «o, «3, we have the well-known formulae

cosa=^^^, co3 3=^^v cos7=^^^^o,

}__^,f,^ 1

pj a^""" c^ "*" Oj «! cos-' a. + a.2 cos* |3 + a^ cos* 7'

Both these equivalent expressions for the volume of the central or least pedal have
then* advantages. In the second the integi-ation is supposed to be extended to all points
of the ellipsoid ; in the fii'st, after expressing a, j3, y and thence ^0 by means of two
suitable independent variables, to all points of the unit sphere. The limits in the latter
case will not involve the axes, and by partial diflTerentiation we shall clearly have

with similar formulae for A.^ and A33 ; so that the volume of any pedal whatever will be
given by the formula

that is to say, it will be obtained by simple differentiation of the expression for P„. At
the same time it will be obseiTcd that Po, being a homogeneous function of «„ a.^, a^ of
the degi-ee |, satisfies, identically, the relation

ME. T. A. HIEST ON TUE VOLUMES OF PEDAL STIRFACES. 23

or, retaining the more convenient symbols A,„ Ajj, A33,

3P„=fl',A„+fl'„A.o+ff3A33.

IG. From this, and the general formula for P in art. 14, a very simple relation may be

at once established between the volume of the central pedal and that of any other whose

origin is on one of the diagonals of the rectangular parallclopiped circumscribed to the

ellipsoid. For the coordinates of any point on such a diagonal are given by the equations

•where r is the radius vector to the point, and «=«i+a2+a3 the square of the semi-
diagonal in question. On substituting these values the two fonnulae for P and Po give

When r'=fl', the origin of (P) coincides with a comer of the parallclopiped ; and when
3;''=Q', it is a point on the ellipsoid; so that we may say, the vohme of the x^edal whose
origin is at any comer of the rectangular parallelopiped circumscribed to the primitive
ellipsoid is four times that of the central piedal, and douhle that of the pedal ai any one
of the eight points wherein the ellipsoid is pierced by the diagonals of the parallelopiped.

17. In order to establish fui'ther relations we will represent, generally, by a-,, 2^,, Zj and
/•( the coordinates and radius ^•ector of any point (?) in space, and consider, fii-st, the
pedals (P,), (P2), (P3) whose origins are at the extremities (i), (2), (3) of any three con-
jugate diameters of a quadric (S') concentric and co-axal with the primitive ellipsoid (S).
The squared semiaxes of (S') being a\, a'n, a'3 we have, of course,

fi+yl+yl=a2,

zl+zl-}-zl=a',;
so that by substituting successively, in the general formula for (P), art. 14, the coordi-
nates of the three points under consideration, and adding together the resulting equa-
tions, we have _
P,+P,+P3=3P„+a;A„+«;A,,+«;A33=3P.

The pedal (P), whose volume is here put equal to one-third of the constaiit sum of the
other three volumes, is easily seen, by the general formula for P, art. 14, to be that whose

origin is at one of the points \\/ —■> \/ -T' "v v ) ' ""'^'^^'^ ^^^ quadric (S') is pierced by

the diagonals of its circumscribed parallelopiped. If, then, we agree to take the volume
of a pedal positively or negatively according as the diameter upon which its origin lies
meets the quadiic (S') in real or imaginaiy points, we may say that the algebraical sum
of the volumes of three ellipsoid-pedals, whose origins are at the extremities of any conjitr
gate diameters of a conce^itric and co-axal quadric, is constant, and equal to three times
the volume of the pedal at the point where this quadric is pierced by a diagonal of its
circumscribed rectangular parallelopiped.

e2

24 3IE. T. A. IIIEST ON THE Y0LU3IES OF PEDAL SURFACES.

We may add, too, that the sum of the three pedal-vohimes corresponding to origins
situated at the extremities of conjugate diameters is not only invariable for one and
the same quadric (S'), but for all concentric and co-axal quadrics which are inscribed in
rectangular parallelopipeds, themselves inscribed in one and the same locus (A) of equal
pedal origms. For the axes of all such quadrics clearly satisfy the condition

«'iA„+«lAo2+«3A33= const.

18. "\Mien the quadric (S') is not only concentric and co-axal with the primitive
ellipsoid, but also similar to it, the diagonals of their circumscribed rectangular paral-
lelopipeds coincide in direction; so that by art. IG, and putting

.3;-- = a\ -f a'., + "J = «',
the last relation becomes

r,+p,+r,=3T'=3'^'p„.

"When ((!=((, that is to say, when the quadric (S') coincides with tlic primitive ellip-
soid, we learn that the sum of the volumes of the three pedals tvhose origins are at the
extremities of any conjugate diameters of the primitive ellipsoid is constant, and egualto
six times the volume cf the central or least pedal.

The three pedals whose origins are the vertices of the primitive ellipsoid are, of course,
included in this theorem.

19. "When the quadric (S') is a sphere, the conjugate diameters are at right angles to
each other, and the diagonals of the circumscribed parallelopiped (cube) are equally
inclined to the axes of the ellipsoid ; hence the sum of the volumes of the elhpsoid-2)edals
whose origins are the three vertices of any tri-rcctangular triangle on a concentric sphere
is constant, and equal to three times the volume of the pedal at a pioint on the sphere
crpiidistant from the axes of the ellipsoid. The value of this constant sum is

SP„+r(A„+A,,+A33).

20. Lastly, when the quadric (S') is an ellipsoid confocal witli the primitive, we may

put

a\ — a,=d.,—a.,=d.,—a,=]c\

and substitute the values of a\, a'.,, a'^ in the general equation of art. 17. By so doing
we find

P,-fP,+P3=GP„+/r(A„+A,,+A33).

Comparing this, therefore, with the expression at the end of the last article, Ave learn
that the sum of the volumes of the three pedals wliose origins are at the extremities of
any conjugate diameters of an ellipsoid confocal with the primitive is eqtial to double
the svm of the volumes of the three p)('dah at the extremities of any three orthogonal
diameters of a concentric sphere the square on whose radius is half the difference of the
squares on the like-directed semiaxes of the confocals. Of this general theorem the one
at the end of art. 18 is a particular instance, corresponding to the case where the con-
focal ellipsoids coincide, and consequently /i-=0.

MR. T. A. IIIEST OX TIIE VOLt'-MES OF PEDAL SUEFACES. 25

21. From the fundumoutal lonnula, written tlius,

P— P

H=— ^=A„ cos' >.+A„., cos>+A33 cosS,

we may tlecluce further relations, as well as a construction for the volume of the pedal
at any point. In the first place we learn that the Uucnr magnitude H is constant at all

P

points of the same radius vector ; and secondly, that it is the limit to which ,:.=/t

approaches as the origin of tlic pt-dal recedes from the centre. Tliis line A, being the
altitude of a parallelepiped (of the same volume as the pedal) having for its base the
square on the radius vector, we propose, for convenience of enunciation, to call the
jjedal-altitude at the point under consideration. Thus H will be the pedal-altitude at
infinity on the line (X, [l, v); A,,, Ao^, A33, respectively, the pedal-altitudes at infinity on
the three axes, and (A,,+Aoo-j— A.33) that on the line equally inclined to the three axes.
Imagine now a central ellipsoid-pedal (P), concentric and co-axal with the primitive,
and such that the squares on its semiaxes are respectively proportional to the altitudes
A,,, A22, A33. It is plain from the last equation that the squares on its radii vectores
will be proportional to the pedal-altitudes at infinity on those vectores. The pedal-
altitude at infinity on any line being thus determined by the auxiliary pedal (P), that
at any other point on the same line is easily found, and thence also the parallelopiped,
equal in volume to the pedal which has that point for origin.

22. Between the pedal-altitudes at difierent points in space numerous relations might
be established; we shall limit ourselves to one or two. Let (1), (2)' (0 ^"^'^^ denote the
extremities of any three diameters, at right angles to each other, of the concentric and
co-axal quadric (S') before considered. Then the addition of the three formulaj (similar
to the one last written) which refer to tlicse extremities gives

7,,-f/,,-f/,3=P„(-+4+-)+A,. + A,,+A33=37i,
since by a well-known theorem

'1 '2 3 1 "a 3

The pedal-altitude h, which is here put equal to the constant arithmetic mean of tlie
other three, corresponds to the point on the quadric (S') which is equidistant from its
three axes, as may be easily seen by putting, in the formula of art. 21,

cos'' X= cos'' |«,= cos'' 1'=^,
and observing that for such a point

'■' «, «2 "3

Hence the ahjclraical sum of the three 2^edal-altitudes at the extremities of any three
orthogonal diameters of a quadric^ concentric and co-axal with the primitive ellipsoid, is
constant, and equal to three times the pedal-altitude at the extremity of a diameter of
the quadric equally inclined to its axes.

26

jME. T. a. HIEST on the VOLITMES of PEDAIi STJEFACES.

We may add, too, that this sum Is not only invariable for one and the same quadric
(S'), lut for all concentric and co-axal quadrics which pass through one and the same
point, equidistant from the xmncipal diametral planes of the primitive ellipsoid. The
quadric (S') bemg a sphere, the pedal-altitudes at its several points are, of com'se,
proportional to the pedal-volumes; so that we obtam again the theorem of art. 19.

23. Before proceeding to the actual calculation of the volume of an ellipsoid-pedal,
we may remark, lastly, that for any four origins situated on a concentric and co-axal
quadric the coiTesponding pedal-volumes satisfy the relation

= 0,

Pn

x\,

f

P.,

xl

y\

P3,

a%

y\

P.,

.11,

y

into the geometrical meaning of which, however, we will, at present, not inquii-e fm-ther.

24. I propose to show, in the next place, that the volume of any pedal may be expressed,

symmetrically, by means of the first partial differential coefficients of the definite integral

It is well known that when the coordinates x, y, z of any point of a surface are regarded
as functions of two independent variables <p and v, we have the following equivalent
expressions for three times the volume of the pyramid whose vertex is the coordinate
ori"-in, and base the siuface-clement ds enclosed between the curves (p-=comt., v=const.
and their respective consecutives :

Pi>ds=

X,

y ^

z

By

d<^

By
dv'

dv

dvdp.

25. Now the equation of the primitive ellipsoid will obviously be satisfied, identically,
by the assumptions

^■'=«i7:p^cosY.

?/^=ff,— --— sur <p,

^y + «3 '

which, when substituted in the above determinant and in the expression for^„ given in

art. 15, lead at once to the expressions

lh<is- —^— ^^^^^^^

-L-r'^cos=^-i-^i±^sin»pl4-.

ME. T. A. HERST OX THE VOLUMES OF PEDAL SURFACES. 27

Substituting these values in the second expression for P^, given in art. 15, extending
the integi-ation over the ellipsoid-octant, — whereby the limits of <f> will clearly be 0 and ^,
whilst those of v will be 0 and oo, — and taking eight times the result, wc have

___l_rp {v + a,)^dv.df

'~ v/«,.,\ \ ^'ll^ cos' ^ +^"^ sin' ^] '

whence, by differentiation, we deduce

A _ 9^_ _ _i_r r ^" ^ ''^^^'^" ^^

VO •'0

The integi-ation according to p presents no difficulty, and when effected gives the result
A _^rr 3a? , 2«,«2 3al -\{v + a^)dv

where, for brenty, we have put

26. A more convenient form can be given to the above expression for A33 by intro-
ducing the partial differential coefficients of the two symmetrical integrals

«/0 */0

V=^^' w=\^.

In fact if, for brevity, we indicate the results of the operations

rffli' da2 ■ • ■ ■ </aJ ' da^da^'' ' ' ' '

performed on any subject, by gi\ing to the sjinbol of that subject the suffixes
■ 1, J, .... 11, 12, . . ., wc shall have

- A33=fl3(aiV„-f 2a,a2V,2+f4V22)+oiW„ +2«,«,W,2+f4W22,

as may be easily verified.

27. This expression, however, may itself be resolved into a simpler one invohing
V], Vj, V3 alone. To effect this resolution we may observe that, in virtue of the identity

dv

we have

rfR -r. -r-K -r-.

■w,+W.+W.= -ij;"(R.+R.+E.)^=f;«?{-i5);

jfrom which, by partial integration, we deduce

W,+W,-fW3=-V,

28 JIE. T. A. niEST ox THE VOLOIES OF PEDAL SUEFACES.

since —7== clearly vanishes at both limits. From this expression, again, wc obtain by
differentiation the relations

W,,+W,,+W„3=-Vo,

W,3 + W,3 + W33=-V3.

By subjecting the integral Y to a precisely similar treatment, it will be found that

1

= -2r/,(V„+V„+Y,3),
= -2r.,(Y,,+Y,,+Y,3),

= -2«3(Y,3 + Y,3 + Y33).

"■ ■'l«2"3

Further, since

1/(1
V -if ^^ -^ &c

we have, on resolving the coefficients of ^= in Y,,, Y03, Y3, into partial fractions,

2(^,-r/,)Y„=Y,-Y„
2(^,-^3)Y,3=Y,-Y3,
2(rr3-«,)Y3,=Y3-Y.; •• : ' :

and in like manner we also iuul that

2(a,-cu)^^\,=a^',-a^\,

2(a,-a,)^Y.^^=a;V,-a^',, - " ' ./ _

2^— r/,)W3,=ff,Y,— «3Y3.

28. Now the last four groups of equations clearly suffice for the expression of A33
(art. 2G) in terms of Y,, Y„, Y3, and thence, by mere permutations of suffixes, we may
obtain the values of A,,, A22. The results, after due simplification, may be thus written :

A„ = -^[( r/,+ r/3>/,Y, + ( a,+2a,)a^',+{2a,+ (r^r^X^, '

A..= -l[{2'U+ r/3>^Y, + ( a,+ r/,>/X,+( f/,+2ff,y.3V3], - - . ,-

From these values of the pedal-altitudes at infinity on each of the axes (art. 21) wc

MK. T. A. llUiST OX THE VOLUMES OF PEDAL .SURFACES. 20

obtain, by addition, the following value of the i)edal-altitude at infinity on a line equally
inclined to these axes:

Again, in virtue of the relation at the end of art. 15, we at once deduce tlie foUowino-
expression for the volume of the central pedal of the ellipsoid,

if, for brevity, we put

^m, = {n,-\-a.,+a^){a._-\-a^)+al-\-al,

3?«3=(«,+«,+«3)(^/3+a,)+f4+«?,

^m^={a,+a^+a^){a,-\-a^)+a\+dl.

Lastly, for the volume of any pedal (P) whose origin A is at x, y, s, we have the
expression

where again, for brevity, we put

3M,=(3r^+«X«3+«.)+3(«32'+«ia^')+«3+«?,

3M3=(3r'+aXffl,+«,)+3(«,a;'+a22/')+«?+«2,

r' and a being, as usual, abbreviations for af-\-if'-{-z^ and a^-\-a.i-\-a.i. The volume of
the primitive ellipsoid, when expressed by means of V,, Vj, V3, is

S= — -| [^ffj«3.r/,V, + «3«7, . a^^-^a,a.,.a^^J^ ,

as is at once e^ident from one of the relations in art. 27. The integral V itself, Avhen
thus expressed, has the value

V=-2[«,V,+«,V,+«3V3];

for it may readily be shown to be a homogeneous function of «„ «,, rt, of the degree —\.

I do not dwell upon the many interesting expressions of S and V by means of pedal-
volumes, but proceed at once to the expression of the foregoing results by means of
elliptic integrals.

29. The integral V, by means of whose partial differential coefficients the volumes of

all pedals have been expressed, is at once converted into an elliptic integral of the

first kind by the substitution

. , a, — flo

^ f + «,

whereby the limits 0 and (X) of w will con-espond, respectively, to the limits 6 and 0

MDCC'CLXIII. F

30 ME. T. A. HIEST OX THE VOLUMES OF PEDAL SUEFACES.

of (p. provided

^=cos-'a A=sm-' \/'hz:^.

The result of this substitution is easily found to be

Jo

F

where

is dearly positive and less than unity.

Kepresenting also, with Leoendee. by E the elliptic integral, of the second kind,

E(^J') = ( (/?^/l-/?-"sin';p,
and differentiating the preceding value of V, it will be found that

1

1' 1

1
«, — flj

E

ff, — Oj

Va,-a.^'^

\/fl, — Og'

1 _,_ 1

F

0,-^3

E

do — ^3 Va^a^u^ ('\~"2

\/«,-ff3 (

>1

-«2)("2-

-"a)

Va^—a^

-ffo 1

+

1

G2-ff3

E

(ic, — "a ^"i^'a^s

v/ffi-ag'

By substituting these values in the formula? of art. 20, we might at once obtain tlie
values of An, A.,, A33, P„, and P expressed in elliptic integrals. Since the volume of
any pedal (P), hoAvever, may be deduced from that of the central pedal (P„) by mere
differentiation (art. 15), the following complete expression for P,, will here suffice: —

Po=^^(2^/-^^)\/^^ + 07ff3 + ^/■5 — f/,rg --== + 2(r/, — r?3)a --— _1.

This expression, I may add, agrees precisely with the one first obtained by Professor
ToKTOLiNi in 1844*.

30. If we allow ^/,, to diminish indefinitely, the amplitude 6 approaches the limit
^, and the modulus k ac(piires the value

^.=\/^'-

* Ciu;i.i-k's .Jonnial, vol. .\xxi. p. 28. At the time the present paper wa.s communicated to the Eoyal Soeiet)-
I wa.s under the impression that the central j)edal of the ellipsoid was the only one whose volume had liithcrto
bcon calculated. I have since found that Dr. M.vokneh, in a paper " On the Cubature of Ellipsoid-pedals "'
(Okunert's Journal, t. xxxiv. 18(!0), first gave the complete expression for I' in elliptic integrals. Although the
simple relation between Pg and P, above referred to, appears to have escaped Dr. Maoexeh's notice, it is due to
him to state that lie not only detennincd the loci (A) of the origins of ellipsoid-pedals of e(|ual volume, but also
succeeded in giving to P a very interesting and symmetncal form, by introducing tlu; jiartial differential coeffi-
cients of the well-known double integral to which Jacoiu, in 18;}l! (C'keli.e's Journal, vol. x.), reduced the
quadrature of the reciprocal of the primitive ellipsoid.

MR. T. A. HIRST OX THE VOLUMES OF I'EDAL SURFACES. Ml

The elliptic functions E and F tlms become transformed into the complete ones

E(q' fcA and F( -^ A, j, or more simply, E, and F,.

Representing generally by [U] the limit to which any function U apprt)aehes when rt.,
diminishes indefinitely, we deduce from the expressions in art. 29 the limiting values

• ry"i=-^-— L-^+ " -^^

[Po]=i\/«.{2(^/.+^OE,-ff,F,}.

This last is the volume of the central pedal surface of an ellijjse (art. 13). By substitu-
tion in art. 28, it will be found that the volume of any other pedal of this curve is given
by the formula

[P]-[P„]=^/^{[(2«,-«.,)E,-«,FJr^+[(r^-2r/,)E, + .*,FJ/+(r^-.,,)E,^4'

to which expression we should have been led at once had we sought, directly, the
values of A,,, A22, A33 as exhibited in art. 13. In fact when a,=a2, the above formula
may be easily reduced to the one already found in art. 13 for the volume of the pedal
surface of a circle.

31. I give, lastly, the modifications of the preceding formulne which correspond to
the special cases of ellipsoids of rotation.

For the prolate spheroid a.^^^a^, and

P.=^{(2«, + 3„V,^+^^ log [v/«^'+\/^] }.

At either focus a''^=«, — «3, j/=:z=0, and the volume of (P) becomes

P=P„+2 1^° («,-«3)=f «,x/^,

which is, of course, the volume of the sphere whose diameter is the major axis of the
generating ellipse.

For the oblate spheroid rti=«2, and hence

V=— ^-^ cos-\A,

32 ME. T. A. HIRST ON THE VOLUMES OF PEDAL SURFACES.

which last formula, when ^^3=0, is also reducible to the last formula in art. 13 for the
volume of any pedal (P) of a circle, regarded as the limit of a surface, one of whose
dimensions has been allowed to diminish indefinitely.

[ 33 ]

III. On the Archeopteryx of voN Meyer, with a description of the Fossil Eemains of
a Long-tailed species, from the Lithographic Stone of Solenhofen. By Professor
Owen, F.R.S. &c.

Ecceived November 6, — Read November 20, 1862.

The first e\a(ience of a Bird in strata of the Oxfordian or Corallian stage of the Oolitic
series was afforded by the impression of a single feather, in a slab of the lithographic
calcareous laminated stone, or slate, of Solenhofen ; it is described and figured Avith
characteristic minuteness and care by M. Hermann von Meyer, in the fifth part of the
'Jahrbuch fiir Mineral ogie*.' He applies to this fossil impression the term Arclico-
pteryx lithographica ; and although the probability is great that the class of Birds was
represented by more than one genus at the period of the deposit of the lithographic
slate, and generic identity cannot be predicated from a solitary feather, I shall assume
it in the present instance, and retain for the genus, wliich can now be established on
adequate characters, the name originally proposed by the distinguished German palae-
ontologist f.

At the Meeting of the Mathematico-Physical Class of the Eoyal Academy of Sciences
of Munich, on the 9th of November, 1861, Professor Andreas Wagner commu-
nicated the discovery, in the lithographic slate of Solenhofen, of a considerable
portion of the skeleton of an animal with impressions of feathers radiating fanwise

* 1861, p. 561.

+ A specific diagnosis deduced from the characters of a single feather presupposes that such characters arc
common to everj- feather of the bird so defined, and the impression of a second feather differing greatly in its
shape and proportions, as in Plate IV. fig. 8, would represent a distinct species in Pateontology ; otherwise the
characters aflForded by a feather cannot be held to be distinctive of a species.

From the number of species of Pterodactyhis, some having short, some having long tails, in the lithographic
slate of Bavaria, it is probable that there may have been different species of Archeopteryx so charaeteriited : the
future possible discovery of a short-tailed Archeopteryx with impressions of feathers corresponding with that of
the Archeopteryx litlwgraphicM, v. Meyer, would impose upon its dcscriber the duty of applying a new specific
name to the long-tailed Archeopteryx with the differently-shaped feathers, to which the name lithoyraphica would
thus prove to have been wrongly applied. Moreover, as winged reptiles are not peculiar to the lithographic
modification of oolitic deposits, the term litlioyraphica may prove as little distinctive of an Archeopteryx as of a
Pterodactylus.

On these grounds the author distinguished in his original communication, as in the Catalogue of the Fossils
in the British Jluseum, the species of Archeopteryx, indicated by the specimen which, for the first time, has
yielded any knowledge of the specific characters of one of the genus, by the term expressive of the best-marked
of those characters, and by which Archeopteryx macrura, Ow., difiers most conspicuously rem every other
known species of bird.

MDCCCLXIII. Q

34 PEOFESSOE OAVEN ON THE AECHEOPTEEYX.

£i"om each anterior limb, and diverging obliquely in a single series from each side of a
long tail.

These and other particulars of the fossil Professor Wagner gave oir the authority of
M. WiTTE, Law-Councillor (Oberjustiz-Rath) in Hanover, who had seen the fossil in the
possession of M. Haberleix, District Medical Officer (Landarzt) of Pappenheim.

Upon the report thus furnished to him. Professor Wagner proposed for tlie remark-
able fossil the generic name Griphomunis, conceiving it to be a long-tailed Ptero-
dactyle witli feathers. His state of health prevented his visiting Pappenheim for a
personal inspection of the fossil ; and, unfortunately for pala^ontological science, which
is indebted to him for many valuable contributions, Professor Wagner shortly after
expired.

I thereupon communicated with Dr. Haberlein, and reported on the nature and de-
sirability of the fossils in his possession to the Trustees of the British Museum : they
were accordingly inspected by my colleague Mr. Waterhouse, F.Z.S. ; and an interesting
and instructive selection, including the subject of the present paper, has been purchased
for the Museum.

The specimen is divided between the counterpart halves of a split slab of lithographic
stone : the moiety (Plate I.) containing the greater number of the petrified bones exhibits
such proportion of tlie skeleton from the inferior or ventral aspect.

The lower half of an arclied furculum (merry-thought, ss) marks, by its relative posi-
tion to the wings, the fore part of the trunk. From this portion of the furculum to the
root of the tail measures 4^ inches; the length of the caudal series of vertebrte ( 6'(l, Cdl)
is 8 inches ; but the terminal tail-feathers extend 3 inches further, making the length of
the tail 11 inches. From the end of the tail to the anterior border of the wing-feather
impressions is 1 foot 8^ inches. From the outer border of the impression of the left
wing [d) to that of the riglit wing measures 1 foot 4 inclies. The front margin of the slab
of stone has been broken away shoit of tlie anterior border of the impression of the
outspread left wing, and the head or skull of the specimen may have been included in
that part of the quarry or stone from wdiich the present slab has been detached. The
pres(>r\ed parts of the feathered creature indicate its size to have been about that of a
Rook or Peregrine Falcon. The exposed bones on one moiety of the split slab
(Plate I.) are—

The lower portion of the furculum (sh) above mentioned.

Portion of the left os innomiuatum, showing part of the ilium (62) and ischium (a), with
the acetabulum {a).

Twenty caudal vertebriK (CV/) in a consecutive and naturally articulated series.

Several slender curved ribs {pi), most of them sternal [h], irregularly scattered about
the i-egion of tlie trunk.

Left scapula (51).

Proximal half of left humerus (r,;.'), entire, and part of the distal half.

Left radius (51') and ulna (si-).

PEOFESSOE OWEN ON THE AECHEOPTEETX. 35

Left carpus (so) and portion of a metacarpal bone (57).

Right scapula (51).

Right humerus (53). wanting part of the bony wall and the proximal end.

Right radius (m) and ulna (a).

Two metacarpal bones (v).

Two unguiculate phalanges (i and 11).

Right femur (ss), tibia (ea), and bones of tlu^ foot (es, ?', ii, in, iv).

Left femur (cy) and tibia (os').

Impressions of the quill-feathers of the wings and tail. Impressions of parts of finer
feathers and do^vn at the side of the body.

The opposite moiety of the split slab contains only one claw-bone (Plate I. fig. i'),
belonging to the impression of the unguiculate digit (i) of the right wing, and a few
slender curved rib-like bones, in addition to those shown on the lower moiety ; of which
bones the counterpart displays the impressions, and in some instances, as in the femora,
the thin outer crust of the shaft.

The furculum, pelvis, and bones of the tail are in their natural undisturbed position,
as in the skeleton of the animal. The left scapula has been displaced backward, and
lies outside of, and nearly parallel with, the left os innominatum. The left humerus
extends outward and a little forward from its scapular articulation, from which it has
not been dislocated. The antibrachium is bent directly inward towards the trunk ; and
the wing-feathers, of which twelve primaries may be counted, diverge about an inch or
less in advance of the cai-pus.

The right scapula retains almost its natural relative position to the trunk, and is im-
bedded in the matrix, exposing its lower sharp margin. The right humerus extends
backward; the right antibrachium is bent forward, outside of and close upon the
humerus ; the two metacarpal or proximo-phalangeal bones (57) extend forward in the
same du-ection, but have been dislocated inward. Impressions of about fourteen long
quill-feathers, from 6 to 7 inches in length, like those of the left wing, diverge from an
extent of about 3 inches, parallel with and outside of the metacarpo-phalangeal
bones.

The right femur extends from its acetabular articulation backward and a little outward,
reaching as far as the eighth caudal vertebra. The tibia extends directly outward and
backward from the knee-joint ; the metatarse is bent upon the tibia obliquely forward
and inward ; and the toes extend in nearly the same direction, the foot being contracted.
The left femur is dislocated from the pelvis ; its head is opposite the eighth caudal
vertebra ; the shaft extends fonvard and a little outward ; the tibia extends fi-om the
knee-joint more directly outward and a little forward.

The best-preserved impressions of the quill-feathers of the wing measure 6 inches
in length, with a breadth of vane of nearly 1 inch ; the anterior series of barbs being the
shorter, or the anterior part of the vane being less broad than the posterior part ; and
the end of the vane is obtusely rounded, as at d, Plate I., and in fig. 7, Plate IV. The

q2

36 PEOFESSOE OWEN ON THE AECIIEOPTEEYX.

area covered by the diverging quill-feathers of the left wing measures 6 inches across
its widest part, near the ends of the feathers ; that of the right wing occupies a space
of 11 inches from before backward; but this difference is due to the three posterior
primaries being dislocated from the rest and directed backward. The under pai't of
the wing being exposed, a few shorter feathers, ' under-coverts,' are seen crossing
rather obliquely the ' primaries ' : one of these (Plate IV. fig. 7 a) is exquisitely
preserved.

The impressions of the tail-feathers may be discerned from the third to the last
caudal vertebra? (Plate I. CcZ), the right series being complete ; the anterior fifth of the
left series being wanting. Twenty feathers succeed each other, from before backward,
on the right side ; and the last thirteen feathers of the left side are preserved. The
principal tail-feathers correspond in number with the tail-vertebra?, and di\"erge, outward
and backward (a pair- from each vertebra), at an angle of 45° with the Ime of the tail,
becoming more acute towards the end (Plate I. Cd'), where the two feathers forming
the pair from the sides of the last caudal vertebra extend nearly parallel with each
other, and in the axis of the tail, about 3^ inches beyond the end of that vertebra.
The length of the anterior tail-feathers is about an inch, and they gradually increase to a
length of about 5 inches in the 15th, IGth, and 17th pairs (Plate IV. fig. 8) ; and gra-
dually decrease, with a more backward direction, to the last pair, which have a length of
3 inches 8 lines. Thus the tail, which is about 11 inches in length, gradually expands
to a breadth of Z^ inches opposite the last two vertebrae, and terminates by an
obtusely rounded or almost squared or truncate end.

In general shape and proportions it resembles rather the tail of a Petam-us or Squirrel
than of a modern bird ; while the wings, in their present state of preservation, agree in
form and proportion with those of the Gallinaceous or ' round-winged ' birds.

The scapula (Plate I. si , and Plate II. fig. 1, si) is 1 inch 10 lines in length, 4 lines across
the articular end, 2 J lines across the neck, and very gradually expanding, towards the
base, to a breadth of 3 lines. In its slightly bent, lamelliform or sabre-shaped figure,
and in the concavity between the glenoid articulation and the short acromial projection
on the outer side, it closely resembles the scapula of a bird.

In tlie Pfcwdarfijlus Huevicus (Plate II. fig. 3, si), a species whicli accords in general
size with the Archcojjfcrijx, the scapula is broader in proportion to its length, and
exhibits a slight double or sigmoid flexure lengthwise.

The extent of the furcular arch (Platt^ I. ss, and Plate IV. fig. 1), or connate clavicles,
which is preserved, measures from end to end, following the curve, about 2 inches ; the
breadth at the apex of the curve is 2 lines ; but this is obtuse, and the piers diverge at
a right angle, but curving from each other; so that the arch is an open or rounded one,
not contracted and pointed as in the true Gallinaceous birds ; the furcular bone, moreover,
is as thick as the slender part of the shaft (jf the humerus.

No skeleton of the Pterodactyle has shown a furculum. The best-preserved speci-

PROFESSOR OWEN ON TTIE ARCHEOPTERTX. 37

mens, such as the Pterodacfyhis siievicvs, figured by Quexstedt *, exhibit the scapula
and coracoid entire, without a trace of cla^^cle, separate or confluent.

The prominence beyond the left scapula (Plate I. si') suggested at first view the humeral
end of the coracoid, but I believe it to be part of the humerus corresponding with the
tuberosity on the ulnar side of the sessile semioval head, overarching the pneumatic
foramen in the bird. The humerus of Archeopteryx (Plate 1. 53, 53', and Plate II. fig. 1, 53)
is nearly 3 inches in length, with the same slight sigmoid flexure as in the bird.
The pectoral ridge (ib. h) has a basal extent of 1 inch : the breadth of the humerus at
this part is 6 lines, one-half of which breadth appears to be due to the pectoral ridge.
In contour it most resembles that in the Corndce (Plate II. fig. 4), the border being
continued almost straight down from the low upper angle ; but there is a better-marked
lower angle in Archeoptei-ijx, where the border of the process curves with a slight con-
cavity to subside in the shaft.

The Pterodactyle (Plate II. fig. 3, 53) presents a well-marked difference from the
bird in the greater extent to which the pectoral ridge projects from the shaft of the
humerus, and in the minor relative extent of its base. The humerus, moreover, is
straight, shorter in proportion to the antibrachium, and thicker in proportion to its
length, with a different character of the distal articulation. In Archeopteryx the
humerus closely resembles that of the bird, and presents about the same proportion,
in length, to the trunk as in the Peregrine Falcon (Plate II. fig. 2, 53), the Touraco,
and most GaUince.

The radius (Plate I. 54, and Plate II. fig. 1, 54) is slender and straight. The ulna [ih. 55)
is thicker, rather longer, and slightly bent, leading a well-marked interosseous space
between the two bones : it expands at both ends to contribute the chief share in both
the elbow- and wrist-joints. The right ulna (Plate I. 55) shows the convexity at the
part of the proximal end next the radius, as in modern birds. Both ulna and radius
closely resemble the antibrachial bones of the bird. The length of the ulna is 2 inches
8 lines — bearing nearly the same proportion to the humerus as in some Scansores and
Gallinacece.

In Pterodactyles (Plate II. fig. 3) the radius (m) and ulna (55) are of equal thickness,
are straight, leave no interosseous space, take equal shares in the formation of the elbow-
and wi-ist-joints, and the antibrachium is always much longer than the humerus.

A single carpal, of large size, wedged between the end of the radius and the base of a
metacarpal, is shown on the left side of Archeopteryx (Plate I. se), indicating a structure
of the wiist like that in the bird.

On the right side an irregular mass of spar occupies the position of a thick carpus or
metacarpus, twisted inward at right angles to the antibrachium ; but this is a doubtful
indication. An inch from the antibrachium, nearer the medial line, but, like the anti-
brachium, directed forward, are two longish bones, with expanded proximal articulations
and straight shafts, growing slender to their distal ends, which come in contact (Plate I. s-,
• Ueber PUrodactylus suevtevs, 4to, Tubingen, 18.55.

38 PEOFESSOE OWEN ON THE AECHEOPTERTX.

and Plate II. fig. 1, 5,-). The proximal articular surfaces are convex, indented by grooves ;
that of the shorter bone is 4 lines in advance of that of the longer. The latter
is 1 inch 5 lines in length, or about half the length of the ulna ; its small distal end
is obtuse, and may have been articular. The contiguous shorter bone extends beyond
the end of the other, and seems to terminate by a small convex condyle. These appear
to be metacarpals ; they bear the same relation of length to the antibrachium as do
the two terminally coalesced metacarpals in the bird (Plate II. fig. 2, 57). If they be
the homologues of these, they retain their original individuality or distinctness, and they
are more equal in thickness. If they be proximal phalanges of the two digits answering
to those which constitute the penultimate joint of the pinion of the bird (Plate II.
fig. 2, iv), they differ in being relatively longer and more equal in length and thickness.

Half an inch from the outer of the two bones of the pinion, and external to, but
on the same transverse parallel as, its distal articulation, is the impression of a slender
bone, about 11 lines long, extending forward in the same line or direction as the above
pinion-bones. At the distal end of the slender bone is the impression of part of a com-
pressed curved bone, grooved along the side, 4 lines in length, 1 line in breadth ; this
dimension slightly decreasing as the bone recedes, curving from the longer slender
supporting bone : it is most like the basal half of an ungual phalanx, supported by a
long and slender penultimate phalanx (Plate I. 11).

In advance and external to the foregoing is the bone itself, of a corresponding penul-
timate phalanx, 11 lines in length, half a line in thickness of shaft; expanded at both
ends, but most so at the distal one, which supports a beautifully perfect claw-phalanx,
preserved in the opposite slab (Plate I. i'), and indicated by its impression {ib. i) in the
moiety which retains most of the bones of Archeopteryx. The claw-phalanx is 8 lines
in a straight line, 2^ lines broad at the base, with a degree of curvature equal to that
of the claw-phalanx of a Raptorial bird ; grooved along the side ; with the base pro-
duced, at the under or concave side, for the insertion of the flexor tendon, and with a
sharp apex.

This claw resembles that of the mid-claw of the hind foot (Plate I. Hi) ; but the bone,
which plainly appears to be in penultimate phalangeal relation with it, is twice as long
and only half as thick as the penultimate phalanx in the foot, and the repetition of the
same character of penultimate phalanx in the less definite or less perfect indication of
the other claw (Plate I. 11) indicates that the hand of Archeojtteri/x, besides being
concerned in supporting the remiges or quill-feathers of a wing, also suppoi'ted two
moderately long and slender free digits, each terminated by a strong, curved, sharp-pointed
claw (as in the restoration, Plate II. tig. 1, 57, I, 11, ill, iv).

It is true that the parts of the present skeleton show a certain amount of dislocation,
and one of the claw-bearing digits might have belonged to the left \ving ; but this is less
probable than that they are on their right side. So much of the skeleton of the hand
as is exposed to view in the present specimen unquestionably accords in its proportions
with that of the bird (compare fig. 1, 1,7, with fig. 2, s?, Plate II.).

PROFESSOR OWEN ON THE ARCHEOPTEETX. 39

The anterior of the three digits Avhich are developed in the bird's pinion (ib. fig. 2, li)
remains free, and in some species supports a claw or spur *. The digit answering
to the middle one in the pinion of birds of flight, supports, in Apteryx, a terminal
curved claw. But if my interpretation of the appearances above described in the
present fossil be correct, Arc/ieopferi/.r differs markedly from all known birds in
having two free unguiculate digits in the liand ; and these digits, in the slenderness
of the penulthnato plialanx, do resemble the unguiculate digits in the hand of the
Pterodactyle (Plate II. lig. 3, ii). But the claw has not the characteristic depth or
breadth of that of the Pterodactyle ; and there is no trace of the much-lengthened
metacarpal and phalangeal bones of the fifth digit, or peculiar wing-finger, of the
flying Reptile (ib. v).

Had the manus of Archeoptenjx been constructed for the support of a membranous
wing, the extent to which the skeleton is preserved, and the ordinary condition of the
fossil Pterosmria in litliographic slate, render it almost certain that some of these
most characteristic elongated slender bones of the wing-finger (Plate II. fig. 3, v, i, 2, 3, 4)
would have been preserved if they had existed in the present specimen. But. besides
the negative evidence, tlie positive proof of the ornithic proportions of the hand or
pinion, of the existence of quill-feathers, and the manifest attachment of the principal
ones, or ' primaries/ to the carpal and metacarpal parts of a short terminal segment of
the limb, sufficiently evince the true class-aflfinity of the Archcoptcrijx.

The pehis is chiefly represented by a bone on the left side (Plate I. 62), bearing the
nearest resemblance to the iliac bone of a bird. A circular acetabulum, 3 lines in
diameter (ib. a), is defined by a sharp border backed by matrix, not by bone. An
oblong plate of bone extends in advance of the acetabulum 11 lines, with a breadth
at the acetabulum of 7 lines, diminishing to a breadth of 4 lines, and then exjjanding
to one of 5 lines. The margin of the bone next to the sacrum is nearly straight ; the
opposite or outer border is sinuous, being concave as it leaves the acetabulum, and
then convex with an obtusely rounded anterior end. The exposed surface is smooth
and polished. Transversely this surface is concave at the medial, con^ex at the lateral
half. The bone is continued backward along the medial side of the acetabulum, of a
breadth equal to that of the cavity ; and behind it for the same extent, with a breadth
of 7 lines, where it is interrupted by the well-defined curve of the anterior border of a
large oval vacuity, one boundary of wliich is broken away at 6 lines' distance from the
acetabulum.

I conclude that here is shoAvn the left os innominatum, including the anterior two-
thirds of the ilium, and the anterior half, or more, of the coalesced ischium. The
anterior iliac border of the acetabulum ends abruptly and obtusely, precisely at the part
where the acetabular end of the os pubis articulates with the ilium in the young

* E. g. Syrian Blackbird {Menda dactyhptcra), Spur-winged Goose {Anser fjamhcnsis), Jacana {Parra jacana).
The Screamer {Palamedea comuta) has two spurs ; the Megapode {Megajpodius) has a tubercular rudiment of a
pinion-claw.

40 PEOFESSOE OWEN ON THE AECHEOPTEEYX.

bird ; the ischium, however, appearing to meet that part of the ilium at a lower level
(in the exposed surface of the fossil), and sending a very short process towards the ace-
tabulum. The ischium (Plate I. 63 ), behind the acetabulum and external (as it lies)
to the oval interspace between it and the ilium, shows the anterior curved boundary
of a smaller or narrower vacuity, which I take to have intervened between the ischium
and pubis.

We have here, therefore, plain indications of a large ischio-iliac interspace, answering
to that called 'great ischiatic foramen or notch' (ib. /'), and the smaller ischio-pubic
vacuity called ' obturator foramen ' (ib. o), under conditions of size, formation, and rela-
tive position to the acetabulum, known only in the class of birds. The acetabulum
itself, moreover, instead of being a bony cup, is a direct circular perforation of the os
innominatum, as in birds.

Sufficient is known of the pelvis of the Pterodactyle to show that the ilium is rela-
tively shorter and narrower than in the present fossil ; that the pubic and ischial bones
are distinct, short, broad, subtriangular plates, and that they contribute to form, Avith
the ilium, a bony cup for the head of the femur.

Whether the pubis has retained its individuality in Archeoptcryx, or has been broken
away from the part of the ilium indicative of the place of its original attachment and
relations to the acetabulum, I cannot determine. So far as the appearance of the pelvis
can be discerned and, by me, interpreted, they give no evidence of a reptilian structure.

A confused mass of coalesced vertebra;, much shorter and broader than those of
the tail, covers the proximal end of the right femur, and extends forward between
it and the left innominatum. The sparry material which has crystallized in the vacuities
of all the widely and apparently pneumatically excavated bones of the Archcopteryx
chiefly represents the sacral portion of the spine, in which a series of six or seven short
and broad transverse processes, in close contact on the right side, can alone be distin-
guished. From this indication, the sacrum would seem to have been at least 2 inches
in length, and nearly 1 inch in breadth. The inferior or central surface, as in the case
of the slightly dislocated left innominatum, is towards the observer, but is much
mutilated.

The broad, subcpiadrate, short, compressed spines of one or two lumbar vortebra3 are
dimly discernible in front of the sacrum. No trace of the vertebral column in advance
of these is visible, nor any part of the sternum ; trunk, neck, and head arc all wanting.
The remains o^Arclteopteryx, as preserved in the present split slab of lithographic stone,
recalled to mind the condition in which I have seen the carcase of a Gull or other sea-
bu-d left on estuary sand after having been a prey to some carnivorous assailant. The
viscera and chief masses of flesh, with the cavity containing and giving attachment to
them, are gone, with the muscular neck and perhaps the head, while the indigestible
quill-feathers of the wings and tail, with more or less of the limbs, held together by
parts of the skin, and with such an amount of dislocation as the bones of the present
specimen exhibit, remain to indicate what once had been a bird.

PROFESSOE OWEN ON THE AECHEOPTERYX. 41

Perhaps the most decisive mark of the chiss-relationship of the Arclicopterrjx is
afforded by the bones of the pelvic appendage or extremity, especially of the foot.

The mark of reptilian nature on Avhich Cuvier mainly relied in his masterly analysis
of tlie Pterodactyle's sk(>leton, was the separate state of the tarsals, and of the metatar-
sals supporting tlie digits, witli the different number of joints in each digit. In the
present specimen, a single coalesced tarso-metatarsal bone (Plates I. & III. fig. 1, es)
articulates at one end Avith the tibia ; at the other, by a trifid trochlear end, with three
toes (//, ///, iv) directed forward : a shorter opposing toe (?') is connected with the meta-
tarse a little above and behind tlic inner trochlea.

The femur (Plates I. & III. fig. 1, «) is 2 inches 4 \ lines long, and 2 lines in diameter
at the middle of the shaft, which is slightly bent, witli the concavity backward. In the
Pterodactyle (Plate III. fig. 4, 65) the femur is straiglit. In some birds [Corythaix,
Plate III. fig. 2, 65) it shows the same bend as in Archeopta-yx.

The tibia of Archeojdcryx (ib. fig. 1, co) is 3 inches 2 lines long, with a shaft of 1-^ line
in diameter; it is straight. On the left side (Plate I. es), where its back sm-face
appears, it shows the division of the hinder border of the upper articular surface into
two lobes ; but these arc thicker, more rounded or convex, and with a deeper mid-cleft
than in those birds that best show this division. In the fossil, however, the sharper
contour of this part of the bone is indicated by the thin layer imbedded in the depression
on the counterpart slab.

The right tibia (ib. ee) exposes its inner or tibial side, and neither the bone nor the im-
pression exhibits a procnemial ridge. The head of the tibia is produced obtusely below
the fore part of the knee-joint. The procnemial production varies much in different birds ;
in some Baj)tores [FaJco trmrgatus, Plate III. fig. 3, ee), and in most Volitores, it would
not leave a more marked indication than in Arclieopteryx. Tlae distal end of the tibia
expands anteriorly, and the contour shown by the inner surface of the right tibia, and
the hinder and inner part of the left one, agrees with the peculiar structure of that
part in birds.

In the proportion of the tibia to the femm*, exceeding as it docs the latter bone by
rather more than one-fourth of its OAvn length, Archcoptcryx (Plate III. fig. 1) resem-
bles some birds (Grouse, Touracos [ib. fig. 2), many Insessorcs) ; but the thigh is propor-
tionally longer in Archeopteryx than in the majority of birds, especially those (e. g.
Cursorcs, GraUaforcs) which are remarkable for the length of leg. In tlie Pterodactyle
[lb. fig. 4) the tibia (ee) is more nearly of equal length with tlie femur (ss). Whatever
trace or proportion of tlie fibula may have existed in Archeopteryx, if preserved, is buried
in the matrix beneath the exposed parts of the tibia.

There is no indication, in either the fossil bones or their impressions, of a separate or
distinct tarsus. The upper end of the coalesced metatarsals (Plates I. & III. fig. I, e?)
shows the calcaneal process and the tendinal groove on its inner side. The thin bony
crust of the inner side of this single composite bone adheres to the impression on the
counterpart slab ; the cast of the medullary cavity in the usual clear, liglit-coloiu'cd spar

JIDCCCLXIII. H

42 PEOFESSOE OWEX OX THE AECHEOPTEEYX.

represents the major pail of the shaft ; but the iiiiiermost and the middle of the three
distal condyles, or trochlear joints, are well preserved. The length of the tarso-metatarsal
to the end of the mid-trochlea is 1 inch 10^ Knes, to the end of the imier trochlea 1 inch
8^ lines : this characteristic bii"d-bone in Archeojyteri/x thus resembles the same in Galli-
naccce and some other groups in which the inner trochlea is least produced, and differs
from the Sapforcs (Plate III. tig. 3) and others in which the trochlea; terminate ou
the same or nearly the same level.

The short metatareal of the innemiost or back toe (Plate III. fig. 1, ?') begins at the
lower thml of the metatarsus (•^*) ; has an extent of attachment, shown to be ligamentous
by a linear tract of matrix, of 2i lines ; and its convex articidar end is about the same
distance above the inner trochlea as that is above tlie middle trochlea of tlie connate
metatarsals. Thus the proportion of the metatarsus to the tibia resembles the average
or common proportion in birds [ib. fig. 2), havuig neither the extreme length of the
Grallatorial. the extreme shortness of the Volitorial, nor the robustness of the Eaptorial
modifications {ib. fig. 3) of this characteristic bone.

Tlie difference from the Reptilian structure, and especially from the Pterosaurian
modification thereof (Plate III. fig. 4. e-), is here most striking. The tarsus (ib. a 1) is
a distinct segment in the volant reptiles, and the metatarsals {ib. gs) equally retain their
distinctness, and con-espond in number with the toes. Tlie entire tarso-metatarsal seg-
ment of the limb in the Pterodactyle is much shorter in proportion to the tibia than ui
Archeopteryx and most birds.

The innermost or back toe of Archeopteryx (Plate III. fig. 1, i) consists of two
phalanges, each 4 lines in length : the second phalanx is curved, slender, pointed, with
an obtuse process on the under or plantar side of the articulation, closely resembling the
claw-phalanx of the bu-d: the toe is shorter and more slender than in the Maptores
{ib. fig. 3. ?", 1. 2), longer and more slender than in the Basores, more ciuTcd than in the
Grallafores, corresponding in its proportions, as in the relative length of the proximal
phalanx, with the same toe in perching birds. The second toe {ib. fig. 1, ?"/)' t^ie innermost
of the tlu'ee directed forward, consists of three phalanges (i, 2, 3) of nearly equal length,
that of the entii-e toe being 1 inch 3 lines. The third (ib. ///), or mid-toe of the three
front ones, is 1 inch 9 lines in length, and consists of four phalanges, the second ( 2) and
penultimate ( 3 ) being rather the shortest. These toes, with their claw-phalanges, equally
accord in structure and proportions with the Insessorial type of foot. The termination
of the claw-phalanx of the outermost (fourth) toe (ib. iv) projects beyond and from
beneath that of the second toe, indicatmg a length intermediate between that of the
second and third toes, but more nearly that of the second toe : traces of the other joints
of the fourth toe are sufficiently plain to detennine that it was not bent biick. but that
it accorded in position and direction with the Insessorial, not the Scansorial, type of
foot. All the claw-bones correspond in the proportions of breadth to length with the
bird-type of those bones, and not with the compressed deep form \\lii(li they present in
Pterodactj'les.

PEOMSSOR OWEN OX THE ARCIEEOPTEETX. 43

The structure of the foot, and the proportion which its metatai-sal bone bears to the
tibia, load me to restrict the account of the closer comparisons of the bones of Archeo-
pteryx mth those of other bii-ds to the species of Insessores and Raptores which best
accord with the fossil in general size. Tlie furculum oi Archcoptenijc (Plate IV. fig. 1)
presents the proportional strength, thickness, :uid span of the arch which characterize
the diurnal Raptores {ib. fig. 3) : but the piers or crura do not arcli into one another
below by so open a cuitc ; they have converged in a form more angular, more like that
in the Owls {ib. fig. 4, Xi/ctea nivea), and still more like that in some Gralla', with a strong
furculum, as, e. g., in the Spoonbill {Flatalea leucorodia) and Argala {ib. fig. 5) ; only,
as before remarked, the type of pelvic limb precludes any useful comparison with bii'ds
of the Wading order. The furcvdum in Colwn^idcc and Gracidm {ib. fig. G) is feeble in
comparison with that of Archeopten/x : in the more typical Gcdlinacea', the still more
slender piers of the furculum meet at an acute angle, and devclope a compressed plate of
bone from the apex. The furculum of ArcJieopteryx is that of a bii-d of a more powerful
flight than in the true Gallinacece. In the Corvidxe {ib. fig. 2), in which the furculum is
naiTower in proportion to its length than in Falconidce, the piers unite by a wider curve
than in Archeopteryx.

The scapula of Archeopteryx (Plate II. fig. 1, m) bears nearly the same proportion in
length to the humerus and femur as in some of the more slender-limbed Falconidw
{Falco trivirgafu^, Plate II. fig. 2, si). But the humerus seems to have been more
slender than in the Falcon {Falco trivirgatus), which comes nearest to Archeopteryx in
this respect. The form of the pectoral ridge presents the dificrence prcnously pointed
out.

In the Kites {Milvus) and Perns {Pernis) the humerus is proportionally longer than
in Arclieopteryx : in the Corvidcc it is proportionally thicker (Plate II. fig. 4, Cormis
corax). It is by the proportion of the antibrachium {ib. fig. 1, 54, 5.5) to the humerus
that Archeopteryx departs furthest from the Raptorial and Insessorial types, whilst it
closely resembles the true GaUinacea?, the antibrachium being rather shorter than the
humerus ; and this condition of the wing-bones accords with the indication of the
pi'oportions of the primary quill-feathers, as in the short rounded wing of Grouse
and Pheasants. The bones of the segment of the hand giAing attachment to the pri-
maries are not preserved in the left wing of Archeopteryx; two of those on the
right side are preserved, and the manus shows, apparently, in the two distinct sets
of phalanges, terminated each by a compressed, curved, sharp-pointed claw, the de-
parture, next in importance after the tail, from the structures of modem and kno-mi
tertiary birds.

Few of the bones, even the best-preserved ones of Archeopteryx, permit a close or
minute comparison of superficial features and markings with their recent homologues in
birds or reptiles.

The osseous remains of Archeopteryx being included between the halves of a split
slab, it might be supposed that the configuration of the outer surface of the fossilized

h2

44 PEOFESSOE OWEN OX THE AECHEOPTERYX.

bone must be demonstrable on one or other of the moieties : it is not so. The long
contact of the phosphate with the carbonate of lime has resulted in a certain degree of
dismtegration or partial decomposition of the former, Avhich has baffled every attempt
to detach the matrix from the bone, or the bone from the matrix, Avhcre they have
come to hand in their original contact. Only in the instances of the bones with the
thickest osseous walls, as those of the feet, and especially the claw-bones, is the surface
entire ; and this has been exposed by the splitting of the slab, and needs no working
out by tool.

Were it not for the large proportional size of their canties, the general configuration
of the long bones of the limbs could not have been so well preserved and presented for
the requisite comparison. AVhen these bones sank in the soft fine calcareous mud
which has hardened into the peculiar stone which the progress of lithographic art has
rendered so valuable, the sparry matter in solution, percolating the matrix and entering
the canities of the bones, has slowly crystallized there, and ultimately filled them by a
compact body of spar. The degree to whicli this represents the original bone gives the
measure of the pneumatic canities and cancelli in the skeleton of Archeopteryx, and
shows that the proportion of the original osseous matter must \\\\\e been tliat wliich we
observe in the present day in birds of flight.

The gi'eat and striking cUfFerence, and that which gives its enigmatical character to
this fossil bird's skeleton, is the number, or rather the proportions and distinctness, of
the caudal vertebi'se ; their under surface is exposed, or rather the sparry casts of the
canities of their bodies, the thin crust of the bone adhering to the impressions of the
counterpart. The best aipw of the under sm-face of the caudal centrum, thus obtainable,
shows a slight expansicm of the two articular ends, wliich join those of the contiguous
vertebras by simple flattened surfaces, having tlie margin obtuse. The mid-line of the
under surface is slightly canaliculate, the impression probably of tlie caudal artery
(Plate IV. fig. 8). There is no trace of hicmal arch, or spine, or articular surface for
such, in any part of the caudal series ; nor is there any appearance of the ossified thread-
like ligaments which are so conspicuous in tlie tail of tlic Pterodactyle. The first five
of these vertebrne show transverse processes progressively diminishing in breadth and
length to the fifth caudal : no trace of such processes is visible in the succeeding
vertebra?. Tlie length of tlie first caudal vertebra is 3^ lines ; this dimension gradually
increases to the eighth caudal, the centrum or body of which is 6 lines in length, and
that dimension is retained to the sixteenth caudal, when it gradually cUminishes to the
last caudal, which is 5 lines in length, and terminates in a point.

The impressions of the quills of the antcn'ior shorter tail-feathers show that they
were attached, ligamentously, to the end of tlie transverse processes in the anterior ones,
and in the succeeding caudals to the sides of the vertebnr, each of these vertebnc sup-
porting a pair of plumes. The under surface of the tail-feathers being exposed, the
median groove of the shaft f)f the vane is clearly shown. The barbs of tlie vane are
as distinctly and inimitably preserved in this delicate and fine-grained litliograpliic matrix

PROFESSOR OWE!^ ON THE ARCHEOPTERTX. 45

(Plate IV. fig. 8), as in the impression of tlic single shorter and broader feather from
the same formation described by M. IIkkmanx vox Mkyer*. The narrower series at
the fore part of one feather o\erla])s the margin of the broader series of barbs of the
preceding feather.'

With the exception of the caudal vcrtebroc, and possibly of the bi-unguiculatc and
less confluent condition of the manus, the parts of the skeleton preserved in this rare
fossil feathered animal accord with the strictly ornithic modifications of the vertebrate
skeleton.

The main departure therefrom is in a part of that skeleton most subject to variety.
In Bats tlierc are short-tailed and long-tailed species, as in Rodents, Pterodactyles, and
many other natural groups of air-breathing vertebrates ; and it now is manifest that, at
the period of the deposition of the lithographic slate, a like variety obtained in the
feathered class. Its unexpected and almost startling character is due to the constancy
with which all birds of the neozoic and modern periods present the short bony tail, ac-
companied in most of them with that fvu'ther departure from type exemplified by the
coalescence and special modification of the terminal vertebra?, to form the peculiar
'ploughshare bone' supporting the coccygeal glands, and giving attachment to the
limited number of fanwise radiating rectnces, constituting the outward and visible tail
in existing birds. All birds, however, in their embryonic state exhibit the caudal verte-
brae distinct, and, in part of the series, gradiially decreasing in size to the pointed
tenninal one.

In the embryo Rook (PL III. fig. 6), the proper extent of the caudal vertebra; is shown
by the divergence of the parts of the ilia (b2) to form the acetabula (a) ; and as many as
ten free, but sliort, vertebra? are indicated beyond this part {Cd). Five or six of the
anterior of these subsequently coalesce with each other and with the hinder halves of the
ilia, lengthening out the sacrum to that extent. The tail is further shortened by the
welding together of three terminal vertebrae to form the ploughshare bone.

In the young Ostrich from eighteen to twenty such vertebrae may be counted, freely
exposed, between the parts of the ihac bones behind the acetabula ; of which vertebra;
seven or eight are afterwards annexed to the enormously prolonged sacrum, by coalescing
with the backwardly produced ilia ; while two or three vertebra; are welded together to
form the terminal slender styliform bone of the tail, without undergoing the ' plough-
share' modification. In Archeopteryx the embryonal separation persists with such a
continued growth of the individual vertebrae as is commonly seen in tailed Vertebrates,
whether reptilian or mammalian.

The modification and specialization of the terminal bones of the sjiinal column in

modem birds is closely analogous to that which converts the long, slender, symmetrical,

many-jointed tail of the modern embryo-fish into that short and deep symmetrical sliapc,

with coalescence of terminal vertebra? into a compressed lamelliform bone, to wliich

* Jabrbuch fiir Miiieralogie, &c., 1861, p. -SGI.

46 PROFESSOR OWEN ON THE ARCHEOPTERTX.

the term ' homocercal' applies ; sucli extreme development or transformation passing
through the protocercal and usually the heterocercal stages, at which latter stage, in
pala:ozoic and many mesozoic fishes, it was in different degrees arrested.

Thus we discern, in the main differential character of the by-fossil-remains-oldest.
kno%Mi feathered Vertebrate, a retention of a structure embryonal and transitory in the
modern representatives of the class, and a closer adhesion to the general vertebrate type.
The same evidence is afforded by the minor extent to which the anchylosing process
has been carried on in tlie pinion, and by the apparent retention of two unguiculate
dio-its on the radial side of tire metacarpo-phalangeal bones, modified for the attachment
of the primary quill-featliers. But wdien we recall the single unguiculate digit in the
wing of Ptei'ojms, and the number of such digits, equalling that in Pterodactijlus, in the
fore foot of the Flying Lemur (Galcojnthecus), the tendency to see only a reptilian
character in what may have been the structure of the manus in Archeojyteryx receives
a due check.

The best-determinable parts of its preserved structure declare it unequivocally to be a
Bird, with rare peculiarities indicative of a distinct order in that class. By the law of
correlation we infer that the mouth was devoid of lips, and was a beak-like instrument
fitted for preening tlie plumage of Afchcoptenjx. A broad and keeled breast-bone was
doubtless associated in the living bii'd with the great pectoral ridge of the humerus, with
the furculum, and with the otlier evidences of feathered instruments of flight.

ESPLANATIOX OF THE PlATES.

PLATE I.

The moiety of the split slab of Lithographic Slate, containing, with the impressions of
the feathers, the major part of the fossilized skeleton of Arclieopteri/x : — nat. size.

n. Concretionary nodules : the larger one consists of matrix, which filled a cavity, 7i',
formed by a thin layer of brownish and crystalline matter ; which may be, as
suggested by Mr. John Evans, F.G.S., part of the cranium with the cast of
the brain of the Arclicopteryx.
n'. Cavity with a layer of brown matter, in the counterpart slab, which was applied
to the nodule, n.
Fig. 2. Fore part of the brain of a Magpie {Covvus pica, 1^.).

Fig. 3, y . Premaxillary bone and, fig. 1 , ])■, its impression, resembling that of a fossil
fish. Tlie other letters and figures are explained in the text.

PLATE II.

Wing-bones — of the Anhcoplery.v (restored, fig. I), of a Bird (Fr/Ico fn'r/'ri/rffus, fig. 2),
and of a Pterodactyle {Pterodactylus snevicus, Qnenst., fig. 3), and tlie humerus
of a Haven (Corvns corax, fig. 4).

PEOFESSOK O^VEN OX THE AECHEOPTEETX. 47

PLATE III.

Fig. 1. Bones of the leg of Archeoj^feri/.r.

Fig. 2. Bones of the leg of a Touraco {Corythaix).

Fig. 3. Bones of the leg of a Falcon (Faico trivirgatus).

Fig. 4. Bones of the leg of Pterodadylus suevicus, Qucnst.

Fig. 5. Pehis and caudal vertebnie of a nc\vly-hatched Ostrich : — nat. size.

Fig. 6. Pehis and caudal vertebife of an emhiyo Rook (magnified 6 diameters).

In both figures, 62 ilium ; 63 ischium ; ci pubis ; a, acetabulum ; Ccl, caudal vertebi-a;.

PL.\TE IV.

Fig. 1. Portion of the furculum oi Archeopteryx.

Fig. 2. Fui-culum of a Raven (Corvus corax).

Fig. 3. Furculum of a Falcon {FaIco ])ere(jrimis).

Fig. 4. Furculum of an Owl (Ki/cfca nhca).

Fig. 5. Furculum of a Stork {Ciconia argala).

Fig. 6. Furculum of a Cui-assow {(h'ax alector).

Fig. 7. Impressions of the basal part of two ' primaries ' and of four entiri^ ' under-
coverts ' of the left wing of Archcopteryx.

Fig. 8. Impressions of the caudal plumes of the 15th and 16th caudal vertebrse of
Archeopteryx.

Fig. 9. Two bone-cells or lacunae, femur of Dinornis.

Fig. 10. Two bone-cells or lacunae, wing-bone of Pterodactyhis. (From Quekett's
' Catalogue of the Histological Series, Museum of the Royal College of Sur-
geons,' 4to, vol. ii. plate 9. fig. 29, and plate 10. fig. IG, showing identity of
character.)

[ 49 ]

IV. On the Strains in the Interior of Beams.
By George BiDDELii Aiky, F.R.S., Astronomer Eoyal.

Received November 0, — Kead December 11,1 SG2.

I HAVE long- ck'sired to possess a theory which should euahle nie to express and to com-
pute numerically the actual strain or strains upon every point in tlie interior of a beam
or girder, under circumstances analogous to those which occur in <n-dinary engineering
applications, — partly for information on the amount of force actually sustained by the
different particles of the cast or wrought iron in a solid beam, partly as a guide in the
construction of lattice-bridges. The memoirs and treatises on the theories of elasticity
and strains, to which I have referred, have given me no assistance*. I have therefore
constructed a theory, in a form which (I believe) is new, which solves completely the
problems that I had proposed to myself, and which, as I think, may, Avitli due attention
to details, be applied to all the cases that are likely to present themselves as interesting.
This theory, with some of its first applications, I ask leave to place before the Eoyal
Society.

1. It is supposed, in the following investigations, that the beam consists of one lamina
in a vertical plane, — the idea of a solid beam being supplied by the conception of a mid-
titude of such laminte side by side, all subject to similar strains, and therefore exertuig
no force one upon another. It is also supposed that tlie thickness of the lamina is
uniform, and that its form is rectangular, the depth of the beam being equal through-
out : these suppositions are made only for the sake of simplicity, as there does not
appear to be any difficulty of principle in applying the theory to cases not restricted by
these conditions, altlxough the complexity would be much increased. It also appears
necessary to suppose that the material of the beam yields equally, with equal forccvs, in
different directions. Another physical supposition, Avliich appears to be necessary for
complete solution of tlie problem, will be stated Avhen we reach the discussion of the
first instance.

2. It is to be remarked that our theory is not intended to take account of aU the
strains possible in a beam, but only of those which are introduced by the weight of the
beam or its load in the position in which it is used. A beam, whether of cast ii'on or of
wrought iron, is, by the process of its manufacture, in most instances affected by perma-
nent strains ; so that, while the lamina is lying on its flat side, some parts are ready to

* Since completing this essay, I have found that considerable progress had been made in the case of figure 5,
by Professor W. .1. 3[. IIa.vkine.

SIDCCCLXIII. I

50 ME. G. B. AIRY OX THE STRAINS IN THE INTERIOR OF BEAMS.

burst asunder, while others are severely compressed. When the lamma is placed in ;i
vertical plane, these accidental strains will be combined -with the strains which are
jiroduced by the weight of the beam, &c. ; nevertheless our attention ^y\^l be confined
strictly to the latter. The algebraical expression of this idea is, that we do not want
complete solutions of oiu- differential equations ; we only want solutions which will
satisfy those equations ; and among solutions which possess this property, we may have
respect to the laws of pressiu-e antecedently known from simpler investigations.

3. For the imit of force we shall use the weight of a unit of surface of the lamina ;
but ill writing the expressions, we shall omit the word " weight," as no ambiguity can
be produced by its absence. For the unit of the force of compression, or of tension
(which is merely compression ^^^th changed sign, or negative compression), we must
refer to such considerations as the following. A force of tension is not a force acting
in a single line ; it is a force acting in parallel or nearly parallel lines, with nearly con-
stant magnitude over a considerable extent of surface. In a large structiu-e, like the
Britannia Bridge for instance, on any space one inch broad there is a certain force of
tension ; but on the neighbouring space of one inch broad there is the same force of
tension, and so for each inch in a long succession there is sensibly tlie same force of
tension. The force of tension, acting on a ccrtam breadth measured perpendicularly to
the direction of tension, will therefore be proportional to that breadth, or will be equal
to the weight of a surface or ribbon whose breadth is the breadth which sustains the
action, and whose length varies with the magnitude of the tension. Tliat lengtli is the
proper measure of tension. AMien the breadtli subject to the action =1 (the unit of
linear measure), the amount of action is expressed simply by that length ; when the
breadth has another value, the amount of action is the product of the value of breadth
by the length which measures the tension. The same remarks apply to the measure of
compression.

4. We must now consider the effect of tension estimated in a direction inclined at an
angle p to the direction of tension. Suppose that a cut is made through the lamina, at
right angles to the direction of tension, and that the effect of tension is to separate the
sides of the cut. And suppose* the direction of tension to rotate in the plane of the
lamina. As the rotation proceeds, the tendency to open the cut diminislies, till, when
f r=90\ tlie tendency vanishes entirt^ly. But when p becomes greater than 90', the ten-
dency to open the cut is restored, and when (p = 180', it is exactly as great as when
ip=0. As (p is further increased, the tendency diminishes by the same degrees, and
vani.shes for ^=270"; then increases till 2)= 360°. It is never convert(>d into a force of
compression, and its changes are the same for positive and for negative changes of p.
These considerations sliow that the effect must be represented by a formula containing
only even powers of cos <p. And the following consideration will show that there \v\\\
be only one term, multiphing cos'^ <p. AVhen the tension acts at right angles to the cut, if
t be the length which measuiTs the tension, and if I be the length of a portion of the ctit,
the force wliicli acts is the weight of tlie ribbon whose length is f and breadth / ; and is

MR. G. 15. AlKV ON THE STRAINS IN THE ENTERIOE OF BEAMS. 51

therefore = It. But when the direction has rotated through <p, the force acting obliquely
on / is the weight of the ribbon whose length is t and breadth / . cos <p, and is therefore
=lf. cos p. And this force is not normal to the cut, but makes the angle p witli the
normal ; and therefore the force which is normal to the cut, acting on the length /, is
It . cos <p X cos <p=lt . cos^ (p=lX t . cos" (p. Consequently the measure of the tension, at
the angle <p to the original tension, is t . cos'' (p. The same theorem applies to com-
pression.

5. AVe must now proceed to consider the coexistence of two or more forces of com-
pression or extension. There is no difficulty in conceiving that a plate of metal may at
the same time be extended in one direction and compressed in another dii-ection trans-
versal to the former. But on consideration it ■nill be f()und eqiially easy to conceive
that a plate of metal may sustain at the same time several forces of compression, or of
extension, or of both. It is easy to devise an apparatus which will produce these effects.
Such forces may exist in the strains of a beam ; and it is imjxntant to show that tliey
can be included in a simple investigation. The following theorem is now to be proved.
" Whatever be the number and directions of the forces of com])ression and extension,
theii' combination may in all cases be represented by the combination of two forces at
right angles, — these forces being sometimes both of compression, sometimes both of
extension, sometimes one a force of extension and the other a force of compression, and
generally unequal in magnitude." The follomng is the demonstration. Suppose that
there are forces of compression (forces of extension being represented as negative
forces of compression) of magnitudes A„ Aj, &c., acting in directions which make angles
a„ «„, &c. with a fixed line. Let us estimate the effect of their combination in a direc-
tion making any angle i^ with the same line. The angles between tlie directions of the
several forces and this dii-ection are respectively a,— -4/, cc.^—^'i &c-; and therefore, by the
last article, theii" effects in the direction i^ are Ai . cos^ (aj — \|/), A2 . cos^faj— i//), &c. ; or

A, . cos' a, . cos' -v// +2A, . cos a, . sin «, . cos -4/ . sin -^z + A, . sin' a, . sin' -ip,
A3 . cos' u-i . cos' 4' +2 A, . cos U2 . smu.2. cos -v^ . sia t^ + Aj . sin' u.^ . sin' -i^,

&c.;
the sum of which may be represented by

2(A . cos' a) . cos' •4/+ 2(2 A . cos a . sin a) . cos -4/ . sm -4/+ 2(A . sin' u) . sin' •v//,
^^ «.cos'-4/ -f i.cos-v^.sin-v^ +c.sin'-4';

where a, b, c may have any magnitude and either sign. And it is to be shown that we
can find a force B acting at the angle (3, and a force C acting at the angle ^-f-90°, whose
combination will produce the same effect.

Now the effect of these forces, by the theorem of last article, is

B.cos'(^-^^) -^-c.cos'(^+9o°-^^),

or

B . cos' /3 . cos' ■i^ -f 2B . cos (3 . sin /3 . cos \|/ . sin \|/ +B . sin' /3 . sm' ^

4-C . sin' /3 . cos' vj/ — 2C . sin /3 . cos /3 . cos \{/ . sin \J/ +C . cos' /3 . sin' xj/.

i2

52 "Sm. G. B. AIRY ON THE STRAINS IN THE INTERIOR OF BEAMS.

Comparing this, term by term, ■\vitli the former,

B.cos'-^/3 + C.sin'/3=«;

B.sm=/3 + C.cos'/3=c;

(B-C).sm2/3 =b.

The clifFercncc of the first and second equations gives

(T,_C).cos2/3=«-c;

and the quotient of the third by this gives

^ ^ Z-*
tan 2/3= ;

riien B — C= - „. or = - ,„, wliicli is always possible. And, by adding the first

sm 2/3 cus 2(3' •' ^ jo

■which always gives a possible value for p.
Then B-C=r — ^

sm 2/;

and second equations,

B + C'=ff+e.

By the combination of B+C and B— C, B and C are found. Thus all the elements
may be found, for representing the effect of any number of forces of compression or
extension, by the effc'ct of two forces of compression or extension acthig at right angles
to each other. Our succeeding investigations therefore will be confined to the consi-
deration of two such forces acting at each point.

"We are now in a state to proceed with the consideration of the strains in a beam.

C. In fig. 1, Plate V., let the parallelogram represent a beam, supported in any way,
as for instance by having one end fixed into a wall, and subject to any force, as for
instance the vertical reaction R of a support at distance h. If II is negative, it will
represent a weight hanging on the beam. Conceive a line to pass in any curved or
crooked direction, from the lower to the upper edge, dividing the beam into two parts, a
near part and a distant part. This division is to be understood merely as a line visible to
the eye ; it is not to be contemplated as a mechanical separation ; for if it were such,
the metal on one side could be considered as acting upon the metal on the other side
only in tlie direction perpendicular to the separating line; which action, in many cases
(as when tlie separating line is vertical), would obviously be incompetent to support the
distant part of tlie beam. The compressions and tensions, which we can sup]iose to
(>xist while the continuity is mechanically uninterrupted, will suffice (with or without
other forces) to sujjjiort tlie distant part. Now if the upjier end (if the curve t(>rminates
in the upper edge of the beam, conceive the curve to continue along that e<lg(^ till it
meets the upper angle at the end of the beam ; if it terminates in the \ertical end of
the beam, conceive it carried upwards till it meets the upper angle; thus the sjiecial
actions which sometimes operate in the limiting lines will be separated from those in
the dixidiug curve. Let r and s be the length and de])th of the beam ; x the horizontal
abscissa (measui'ed from o), and y the vertical oidinate (measured from tlie lower edg(>)

MR. G. B. AIRY OX THE STRAINS IN THE INTERIOR OF BEAMS. 53

of any point of tlio curve. At the first limit of the curve, the coordinates are :, 0 ; at
the hist, tlie coordinates are r, s.

7. The distant part of the beam is sup])orted by tlie forces of compression (this term,
■\A-ith negative values, including tensions) across every part of the curve, combined ^^'ith
the reaction E. At the point whose coordinates are w, y, conceive that there is one
force of compression B whose dii'ection makes the angle /3 to the left side of i/ produced,
and another force of compression C whose direction makes the angle ft-\-00° to the left
side of y produced. And, in figure 2, consider the actions of these on the small element
Is of the curve, or ratlier the actions on a portion of the lamina, including h. Let (5 be
the angle made by Ss with y. The direction of the action of B makes with Ss the angle
|3+S; and therefore the breadth of the ribbon representing its action is hx sm (/3 + 6),
and its whole force is B.SsX sin ((3-\-0). Resolving this in the directions of x and y,
we have for the effects of B on the distant part of the beam.

In the direction ,r, B . Ss X sin (/3 + ^) X sin /3,
In the direction y, — B.SsX siu(|3 + ^)x cos/3.
In like manner, the effects of C on the distant part of the beam are,

In the direction ,r, C . h X sin (|3 + 90'+^) X sin (|3 + 90°),
In the direction y, — C.S.sX sin ((3 + 90'+^) X cos(/3 + 90°).
Expanding the sine, we have, for the whole force in the direction x,

{B. sin-/3+ C . C0S-/3} . cos ^ . os+{B. cos/3 . sin/3 — C. sin/3 . cosj3} . sin^. os,
and for the whole force in the direction ?/,

{ — B . sin /3 . cos /3+C . cos (3 . sin fi} . cos d . S.s+ { — B . cos' /3— C . sin' /3} . sin d . h.
But cos 0 . ls = oi/, sin 0 . h=ox. And using for convenience the following letters,
L=B.sin'/3+C.cos=f3,
M = (B-C).sin/3.cos/3,
Q = -B.cos'/3 — C.sin-/3,
we have for the whole forces on the element Is,

In the direction a\ L . 5_y+M . h;,
In the direction _y, — ^I . S_y-1-Q . Sa*.

It must he borne in mind that the force in direction x acts in a line whose vertical
ordinate is y, and that the force in direction i/ acts in a line whose horizontrJ ordinate
is X.

8. There is another force acting on this- portion of the distant part, nanielj-, the
weight of the lamina included between the ordinatcs corresponding to or and a'+5a';
which, estimated in the direction y, is — >/ . Ix, acting in a line whose horizontal ordinate
is X.

54 ME. G. B. AERY ON TILE STILYINS IX TilE INTEKIOE OF BEAMS.

And, besides these forces which act at every point of the ciuTe, there is the reaction
+E in the dii-ection i/. acting in a Hne whose horizontal ordinate is Ji.

9. We have now collected all the elements for the eqnations of equilibrium of the
distant part of the beam, and we proceed to form those equations. For h>/ we shall put
p . hx. The equations are as follows :

First, equation for forces in x :

^dx.(l4}+U)=0 (1.)

Second, equation for forces in y :

J^.r.(-Mp+Q-i/)+R=0 (2.)

Thii'd, equation of moments :

^dx.(L>/j)+Mi/-\-Uap-Qa'+a:>/)-'Rh=0 (3.)

It will be convenient at once to make )/ — Q = 0; and the equations become

^dx.iLp +M) = 0 (4.)

J'dr.(M^+O)-E=0 (5.)

j'<7a-.(%;+M^+M.^y+Oa-)-R/; = 0 (G.)

10. We shall now introduce a consideration which will pro^c singularly advantageous
for the solution of these equations. Eeferrmg to figiu'e 3, tlie equations which we have
obtained apply to the curve ahcdef. The same equations, mutatis mutandis, apply to
the curve a h(j d ef. Hence the variations in those equations produced by passing from
one of these ciirves to the other mil =0. Now these variations are clearly such as
are treated in the Calculus of Variations. We may therefore form the variations of
the equations according to the rules of the Calculus of Variations, and equate those
variations to zero. R and RA will disappear.

11. The left side of equations (4.), (5.), (6.), is in each case a function of x, y, ])
(L, M. and O de2iending on the position of the point in the lamina, and therefore being
fimctions of x and //), and of no other differential coefficients. Therefore the equation
of variations in each case, in the usual language of the Calculus of Variations, will have

the form N ^^^ =0. Ap])lying this in each instance we have; —

For S.j'f7.r.(T4;+M):

therefore

dh dM dL dh .

di/J^^ dy -1^-^!^ = ^^
or

dU dh ^

-~dy-lx=^^ (7.)

For I .j'(7,r(M;>»4-0): in the same manner,
dO dU ^
^-«^=0 (8.)

MH. G. B. AIRY OX THE STSAINS IN THE INTEEIOE OF BEAMS. 55

For h . Sds(Li/p + M// + M.iyj + Ox) :

d(P) dL dh . ^ . rfM , dM , -.
therefore

or

This equation, by virtue of equations (7.) and (8.), is identically true, and therefore
adds notliing to our knowledge. Tlie information, then, that we have obtained from our
process is comprised in the two equations

rfM

dy'-

dL

-dj; '

dO

dM

- dx-

(7-)
(8.)

From this it follows that L, M, O are the three partial differential equations of the
second order of a function F of x and y, such that

T.'^^F ,r_i!I O-^-
^-rf^' ^^-dxdy' ^-d.v^ '

and we may substitute these symbols for L, M, O, in the equations of equilibrium of

the distant part of the beam.

12. If it had been necessary to use expressions of the utmost possible generality, we

must have said

-. </*F , , , -- d^V „ rf^F , ,, .
L=^+<p(yX M=^, 0=^,+-4^(x},

where the forms of the functions (p and -^p are arbitrary. Suppose now that F is so
determined that the substitution of ^-^ , ^^, and -^ for L, M, O will satisfy the equa-
tions (4.), (5.), (6.) m then- entirety. Then the substitution of <p(i/) and ■4^{x) alone
must satisfy those equations deprived of tlieii- constant terms ; and therefore (p(>/) and
■>p[x) may be multiplied to any degree, or different functions of the same character may
be added to them. These remarks clearly indicate that these functions represent
accidental strains such as we have spoken of in article 2, and they are therefore to be
neglected. We confine oiu'selvcs therefore to the terms

T—^ Af_ ^ O— —

dy'^ ' * d-xdy'' dx*

56 jVIE. G. B. AIET ox the STRiUXS IX THE IXTEEIOE OF BEAMS.

13. Making these substitutions, and restoring for ])(lv its original expression di/, the
equations become the following :

jI'^-f+*--S)='>^ («■)

((''^ -H +''•'■ •S)-K=»^ (1"-)

the integrals being taken from .r. ?/=r, 0, to .r, i/:=>\ s. Now

and the same symbols appear in the bracket of equation (11.). Hence the equations
become

I-<f)-^ • • ■ ■ (-)

and

Integrating the quantities under the bracket by parts, the bracket becomes

^F .(IF C/ J (IF . 7 f/F\

But '^\7y+'^ ,lr=d(¥). The value of the bracket, therefore, is ?y^+.r'^-F; and
equation (11.) becomes

f '{/i+--S-r}-r.,=o (14.)

Attaching the subscripts r, 0 and r, s to tlu' symbols or brackets, to denote the values
which the expressions assume when z, 0 or r, s are substituted for ,r, >/, the equations
finally become

(f ).„-(?).. =» (i^-)

m.rm,.r^=" ■ • ■ ■ • (>«■)

From these, by very simple treatment, tlie form of F may be found ; and ficjm tliat form
every required expression will be deduced witli great focility.

14. A sliglit familiarity witli tlie expressions for strains, as given by sim])le tiieory
in some ordinary cases, is sufficient to connnce us that F Avill contain only integer

MK. O. B. AIRY OX TlIE STEAINS IN TILE INTEEIOK OF BEAMS. 57

powers of X uiul I/. Assume, therefore,

F=S+T^+U/+\y+W/4-&c.,
M'here S, T, U, V, W, &c. are functions of x ; then

~= T+2U^+3Vy^+4A\y+&c.

For r, s, the vahie of this is

T,+2U, . 5+3V,. .s- + -nv, . s'-{-&c.
For z, 0, its value is

The expression f — j —I'j will then^fore contain the function T., where z is abso-
lutely arbitrary. It is imjiossible that equation (15.) can subsist, except by making
T,=0, and therefore T,=0, and generally '1=0.

Again, omitting T, we find (using tlie accents to indicate differential coefficients)

g'=:S'+u'/+v'/+wy+&c.

For )', 5, the value of this is

S:+U,.s'+V,.s^+W,.«^+&c.
For z, 0, its value is

s;.

For the same reason as before, S' generally =0. Therefore if S have any value, it is a
mere niimerical constant; and this A^ill disappear in each of the equations (15.), (10.),
(17.); and therefore it may be entirely omitted. The expression for F will therefore be
reduced to U^^+V^^+W^ +&c. We shall hereafter show that ordinary investigations
entitle us to assume that the expression for F mil really be limited to the first two
terms of this series, and that the powers of x will not be higher than the second ; and
therefore Ave shall suppose

F={ax-^+Lv+ c),f + (..r +/(•+ r/)/.

We can now proceed with instances.

15. Example 1. Suppose the beam to project from a wall, and to sustain no load
except its own weight.

Here E=0; and the three equations (15.), (16.), (17.), with the last assumption for

F, become

{2av'-\-1lr-\-1c)s-\-{?ier+?>fr+?>fj)s"=[),

{2ar+hy +(2^>-+/V =0,

{?,ar"'-\-2br+c)s' +{Aer"-+?>fr+1g)s^=Q.

Determining from these the values of b, c, ff, we change the expression for V to tlie

following :

^^l{ay-^i-2ar-2ers-fs)x+ar-\-2er's+fr.s)f\

MDCCCLXIII. K

58 MR. G. B. AIRT ON THE STRAIN-S IN THE INTERIOR OF BEAMS.

To determine the constants «, e,f, which remain, we must have recourse to other con-
siderations.

16. If we suppose the beam cut through in a vertical line corresponding to abscissa x,
and if we make the usual assumptions in regard to the horizontal forces acting between
the two parts and thus sustaining the moment of tlie distant part, namely, that
there is a neutral point in the centre of the depth — that on the upper side of this
neutral point the forces are forces of tension, and on the lower side are forces of com-
pression— and that these forces are proportional to the distances from the neutral
point, ^ith equal coefficients on both sides, — then we can ascertain the horizontal force
at every point. But I remark that it appears to me that these suppositions involve a
distinct hj^j^othesis as to the physical structure of the material. They seem to imply
that the actual extensions or compressions correspond exactly to the curvature of the
edge of the lamina, and that the forces of elasticity so put into play correspond to the
amount of extension or compression. The experiments of Mr. W. H. Barlow appear
to modify this theory ; and it seems probable that, when duly followed into their
mathematical consequences, they may require the introduction into the formula for
F of other powers of y. Leading this question open, I shall now proceed, on the
usual assumptions, to compute the horizontal force at every point of the vertical
dinsion.

17. Let the horizontal force at elevation ij, estimated as compression, be represented

by (■.(^— j; the force on the element ly is the ribbon lyxc . \o—y) '> its moment is
yX^'yXc.i -^—y\ =■(■{—— y'-^ly; and the entire moment is c)^(Jyi-^—y-\-=c{~^ — -^\;

which, from //^O to y=^s, is — pj. The moment produced by the weight of the
distant part of the bar is the product of its weight by the horizontal distance of its
centi'e of gl•a^ity, or is {r — x) X s X -7r~ = <? • The equation of moments is there-

fore, — p^-l -^ — =:0. From this, f=- 3^ — ; and the horizontal compression-force

at elevation //=-2.(r—.r)lf^—//j ; or the horizontal compression-force on the element

18. But we have the means of expressing the same liorizontal force in terms of F.
For. in the last expressions of art. 7, conceive the dividing line to be vertical ; that is,
conceive S.r=0, and Ss^Sy; then we have for the compression- force on the element 5y
in 'dnection x, the expression L ,Sy ; which, giving to L its value from the end of art. 11,

becomes t-j hj.

Comparing these two expressions, j-t-2=-2(/'— a')". f2"~y)- And, using the last for-

ME. G. B. AIRY ON THE STKAINS IN THE INTEEIOB OF BE.UIS. 59

mula of art. 14,

'{2a.v'+(-iar-iers-2fs}x-{-2ar'+ier's+2frs}]G /s \

Compaiiiig the coefficients of ^, ea-'+/a,-+(— f ;•'—//•) = — ^ a- +^' A'— jj. The first

term gives e= —^ ; the second gives f=-^ ; the third gives ^,— ^= — ^5, which is iden-
tical.

Then substituting these in the term independent of y, and comparing,

3
The first term gi^es «=^ ; and this makes the second and third comparisons to become

identical equations. The circumstance, that the determination of the constants from
some terms causes the other terms to agree, gives endence of the agreement of the two
lines of theory, inasmuch as those remaining terms arc obtained in the two theories by
totally dififerent operations, each peculiar to its own theory.

We may now therefore use -XT=-:2(>"— .f)^.(o— ^)-
19. From this we find

F=i.(,.-.)-.('|-^).
from which

L =f =|.(.-,,,.,.-2,) = .,.|!.(l-5)'.(l_|);

M=S=^.(.--).(f-|>-.?.(l-9 ■!■ (i-f);

N=-2M = s.l^.(l~^yi.(l-i);

Q=,-0=.f-0=,(f-3S+2S)=;,f.(l_f).(l-|).

Put v=~, 10=-, and omit the general multiplier s. And as the succeeding operations,

while kept in the symbolical form, become rather cumbrous, assume for - a numerical

value, as 5. Then

L=75.(l-?;)^(l-2w);

N=60.(l-«).w.(l-w);

Q=w.(l-w).(l-2w).
k2

60 >£R. G. B. AIET ON THE STEAINS IX THE INTEEIOR OF BEAMS.

From these (see art. 7),

tan2/3 = L^;

p T, N L + Q.

^~^ —sin 2/3~'cos 2/3 '

C+B=L-Q;

Avhich give the numerical values of the three elements B, C, (5 of the strains at every
point.

By means of these formula?, the numbers of Table I. (end of the Memoir) have been
computed and the lines of pressure traced in Plate V. fig. 4. They give complete infor-
mation on the nature and magnitude of the strains to which such a beam is subject.

20. Ea-ample 2. A beam of length 2r and depth s rests, at its two ends, freely on piers,

and sustains no load except its own weight.

Assume, as before,

F= (ra-^+i.r + r)/+(ra-'+/r+r/)/,

and remark that the distant pier exerts a reaction vertically upwards, of magnitude rs

at distance 2r. The three equations (15.), (16.), (17.),Jaking the integrals from z, 0 to

2/'. s, become

( 8«r=+45r+2r)s+(12e^--+6/'r+3<7)s^ =0;

( 4:ar+h)s' +( 4f/- +/)s' -rs =0;

(12«/-^ + 4i;-+f)s= +(lGfy^+C/;-+2y)s'-2/-\^=0.
"NVhen from these we determine the values of h, c, g, and substitute them in the
expression for F, it becomes

^^l|,a-'^+(_4.,;.-4frs-/s+y^+(4«r^+8frs+2yb-^')}/j

21. The horizontal pressure at any point of any vertical line across the beam at
distance .v will be found on the usual theory as foUows. The compression at any eleva-
tion y being represented, as in article 17, by cJ-^—i/j, the entire moment is, as in that
article, — t^. The moment produced by the weight of the distant part of tlie beam,
whose length is 2r—x, is --^ — - ; and the moment produced by the reaction at the
distant pier is —rsx{2r—x). The equation of moment is therefore

-^%^--^--x(2r-:r)=0,
or

~12+ 2 ~ 2 "■"'
or

cs* a:{2r—x) ^

— 12~" 2 ~^"

MR. G. B. AIRY ON THE STRAINS IN TUE INTERIOR OF BEAMS. Gl

From tliis, c= j-^^-^ ; and the horizontal comprcs.sion-forcc at elevation y

_ 6x^—12rj;

Therefore, as in article 18,

d^¥_6x^-l2rx/s
'di/ — ~s^

And, using the last formula of article 20,

6 1 ^r

Comparing the coefficients of 3/, 6f.r^4-Q/'i'+( — -4fr-— 12/>-)= — 7^0,"+-^- j-. The

1 ~ Ir "ir^ '>4r*
first term gives e=. — -^ ; the second gives/ =-2 ; the third gives — , ^^ =0, which

is identical.

Substituting these in the term independent of y, and comparing,

3

The first term gives «=5- ; and on substituting this, the second and third comparisons

become identical equations. The evidence of correctness of theory is therefore satis-

factory; and we may use ^= ^^ U"^)'

22. From this we find

„ 6x'^—12rx /sT/^ ]/^\

from which

V 4 g)'

^ dn^ 6x''—12rx /s \ 6r^ x /^ x\ /^ 2i/\

^='dT^ = —V- (2-y) =-'--^-r\^-2^)-[^—i)'

d^F_12x-l2r (sy y^\ _ Gr / x\ y I y\

N= -2M = 5.^. (l_^).f.(l_^^;

As before, put v=~, w=-; and suppose - = 5. Then, omitting 5,

L=z-7d . V . (2-v) . (l-2iv);
N= 60.(l-v).w.(l—w);
Q= w.(1-m;).(1-2w);

62 ME. G. B. AIET ON THE STEAINS IN THE INTEEIOE OF BEAMS,

after which we may use the same formulse as before, namely,

tan 2/3=1;:^,

P_P. _ N _L + Q

^' -^ ~ sin 2/3 ~ cos 2/3'

C + B=L-Q;

by means of which the numbers have been computed for Table II. (end of the Memoir),
and the lines have been traced that are exhibited in Plate VI. fig. 5.

23. There is one part of the ^jressures whicli it is matter of great interest to compute,
namely, the pressures exerted on different parts of the end portion of the beam which
rests on the pier. It will be seen m figure 7 that this part is not free from pressui-e ;
there are at every point a large force of compression in one direction, and a large force
of tension in another dh-ection. And the circumstances of this part differ from those
of any other vertical section of the beam in this respect, that there is no opposing force.
In all other sections, a thrust of compression on one side is met by a thrust of compres-
sion on the other side, and so for tension ; and though there may be a tendency to
crush or to disrupt the particles of the metal, yet there is no great tendency to force a
small sectional portion horizontally or vertically. But on the end portion, where the
forces of compression and tension are not so met, there are or may be great tendencies
to force that end portion horizontally or vertically. We proceed now to investigate
these tendencies.

24. First, for the horizontal pressui-e. The force B (which is estimated as a com-
pression), acting in a dii'ection which makes the angle j3 with the vertical, upon the
element ly (as measured in the vertical direction) or sin j3 . §</ (as measured in the direc-
tion transverse to B), does really exert the pressure B sin /3 . Itj in the direction of B, or
the pressure B . sin /3 . Sj/ X sm 8, or B . sin^ /3 . ly, in the horizontal direction. Similarly,
the force C exerts the pressui-e C. sin=(/3 + 90°) . Sy, or C . cos"^ j3 . Sy, in the horizontal
du-ection. The entire horizontal force upon the element ly is therefore

(B . sur f3-f C . cos= /3) . S^=L . ly=^ly.

In the instance before us, of a beam resting on two piers,

d,/ — s" '\2 ~y) '
and at the end of the beam, where 3i:=2r, this quantity =0 whatever be the value oi y.
The same applies where x=0. There is no tendency therefore to bend or distort the
end portion.

25. Secondly, for the vertical pressure. The pressure B.sin(3.S_y m the direction
of B, found in last article, will produce the pressure B . sin /3 . cos /3 . hy in the direction
vertically downwards. Similarly, the force C will produce the pressure
C . sin (/3-f 90°) . cos (i3 + 90") . hy

ME, G. B. AIEY ON TUE STEAINS IN THE INTEEIOE OF BEAMS. 63

vertically downwards. The whole downwards pressure therefore on the element hy is

(B . sin /3 . cos /3+C . sin 90°+/3 . cos 90°+/3) . li/, or (B— C) . sin /3 . cos (3 . Sy, or M . J^ ;

12.r \2r /si/ y^\

which in the present instance = " g " • ( ^— 2 ) " ^!/- -^^ the end of the beam, where

6r
x=2r, this =^ . (si/—j/") . I//. Let >J=zs—y (that is, let the ordinate be measured from

the upper edge downwards) ; then the downwaixls pressui-e on the element ^y of the end

portion =^ • (•^'/—y'') ■ ^y'- Integrating this from the top downwards, \>c find for the
pressure whic li a horizontal section of the end portion must sustain,

At the middle of the depth this = - ; at the base it =rs. It appears therefore that

every part of the end portion which rests upon the pier is subject to a very heavy pres-
sure (such as affects no other part of the beam), increasing from the top to the bottom,
where it is equal to the weight of half the beam.

It was undoubtedly from a clear perception of the magnitude of this pressure (though
not reduced to the formula; of mathematical investigation) that Mr. Robert SiEPnEXSON,
in the construction of the Britannia Bridge, was induced to insert the strong end-frames
in each of the tubes, at the places where they rest on their piers.

26. Example 3. A beam of length 2r and depth s rests, at its two ends, freely on
piers, and carries a weight "W at the distance a from the left-hand extremity.

For convenience, we ^^^ll suppose a to be not greater than r. This will include every
case, as the supposition a' greater than r is the same as the supposition a less than r
measured from the right-hand extremity, if a^a'=2r.

In examples 1 and 2, we have selected a form for F which satisfied the equations (15.),
(16.), (17.), applying to F, and we have then sho\A-n that this form represents properly
the hoiizontal pressure determined fi-om the ordinary theoiy. In the present example,
which is \msymmetrical and complicated, we shall find the form for F (a discontinuous
form) which represents the horizontal pressui-c as determined fi-om the ordinary theory,
and shall show that this form satisfies in all parts the equations (15.), (16.), (17.).

27. The pressui-e upon the left-hand pier is rs+ W • ''~"; and that upon the right-hand
pier is rs+AV • ^. The reactions of the piers have the same values, but in the opposite

direction. For a transverse section at the ordinate x, where x is less than a, the forces
which produce moments are the following: the weight sx(2r — .r) acting at distance

''~^: the weight W at distance a—x; and the reaction )-s-\-W ■ ^ at distance 2r—x.
2 ° ' 2r

The sum of their moments, estimated as compressing the upper part, is

(2,.-a-) . (,.+W . I--S . '-r^) -AV(«-.), = (..+W . ?^)^-|^.

64 ME. G. B. AIEY OX THE STRAINS IN TIIE INTEEIOE OF BEAMS,

(The same value will be found if we consider tlie moment as produced by the weight
of bar and the reaction on the left side of .r.) Treating this as in article 17, we find
the liorizontal compression-force at elevation ij

=,M(2"+^v^")-'-'i(^-0'

This, as in preceding instances, ought to equal -r^ ; and therefore F ought to equal

This formula applies to any point of the part of the bar included between >r=0 and
a'=a, which we shall call the "fii'st part." For any point of the "second part," or the
part included between .T=ff and .r=2r, there is no weight W on the right hand; the

forces producing moments are the weight sx{^)'—.t) acting at distance ^' ~^, and the

reaction r5+ W^ at distance 2r — x ; the sum of their moments, estimated as compressing
the upper part, is

whence, as in article 17, the horizontal compression-force at elevation^
= |/?^+(2,.-Wl),,.-.,}.(,-|),

which ought to equal -p.- ; and therefore F ought to equal

This formula applies to any point of the part of the bar included between x=a and
x=2r, or to any point of the "second part." The function changes its form, or is
discontinuous, when x passes the value a, — the two formulae, however, giving the same
value for F when x=a. We have now to ascertain whether the discontinuous function
does in all parts .satisfy the equations (15.), (10.), (17.).

28. First, suppose the integrals to begin from a point z in the ''first part." It is
unnecessary to make an elaborate trial of equation (15.), because, as our assumed value

for F contains the multiplier ~ — ■—, and '^y therefore contains the multiplier \—^-k, -~r

will necessarily vanish at botli the limits for // (namely // = 0, y=><) which enter into the
formula) of (15.). In regard to tlic other ecpiations, the integrals must be taken by tlie
formula) (jf the " first part " from c, 0, to a, s ; and by the formula) of the " second part "

from a, s, to 2r, s; and the constant forces are +W at abscissa a and — (rs-\-^Y^j at

abscissa 2r.

MR. G. B. AIRY ON THE STRAINS IN THE INTERIOR OF BEAMS. G5

For equation (10.), '/^' in tlic - first part" =V-2r+yV .--^^^-2AC4~'-T-) I wlikli

for.-, 0, =0, and for a, s, = r5J-J,-_-J,, + W . r£r^l . i!. And ^? in the '■ second part"
s- ( rs ) 12 dx '•

which for a, s,

— ^ |-' -(' >* ,.^j • 12-
and for 2/", s,

The sum of the two portions of the integral will therefore be
=,,_W.?^«.

2r

r.s+'N^';^. 1, or — r.s+AV" „ ; the sum is 0.
Equation (10.) therefore is satisfied when z is in tlie "first part."

For equation (17.): omitting y ^ (because, as is explained above, it cannot produce
any term), it will be found that in the '• first part "

'^ dx ^—s^ { '^ ^ \G A )'

which for z, 0, =0, and for a. s, =-, • a" ■—. And in the "second part,"

i- 12

dx s-\ s j \ '' ^ /

•which for «, s,

-?)^+'''|r2'
and for 2)\ s,

The sum of the two portions of the integral will tliercforc bo
o- 0 + rr a--\- + 4>- • = 2>->'.

To this are to be added +^^ '' and— ny+W;^. j2;-, or — 2r,v; the sum is 0. Equation

(IT.) therefore is satisfied when z is in the '-first part."

29. Second, suppose the integrals to begin from a point z in the "second part." As
before, it is unnecessary to make a trial of equation (1-5.), which is necessarily satisfied.
In regard to equations (10.) and (17.), the integrals are only to be taken by the formulse

MDCCCLXIII. L

66

3IE. G. B. AIET ON THE STRAINS IN THE INTEEIOE OF BEAMS.

of the '-second part" from z, 0, to 2>', s; and the only constant force is "(''•5+^2^1

at abscissa 2;'.

/7F G ( (I 1 Zip si/^ \

For equation (16.), j^ in the "second part" =^-^2r— W— — 2.rV • (g" — ■^j, which

for c, 0. =0, and for 2r. .<;, =-^f-2/'-W-}, or rs+W;^. To this is to be added

— (ri+W— ) ; the sum is 0. Equation (16.) therefore is satisfied wlien z is in the

" second part."

. ,,„, f^F ,,. ^, , ... 6 1 2Wfl J /?/ s,/2x
For equation (1 / .), >^'^ — -t m the " second part =-ji\ ^ -i'" \ • I "'e ~"4" ) ' '^viiich

for r. 0, =0, and for 2r, s, = -j— - +4rH = +Wa+2r's. To this is to be added

— (;•*■+ '\V^.j2r, or — 2r"5— Wa; the sum is 0. Equation (17.) therefore is satisfied

when z is in the " second part."

30. It appears therefore that our equations (15.), (16.), (17.) are in all parts of this
loaded bar satisfied by the discontinuous formula which we found for F ; and therefore
that formula is to be adopted in the further calculations. But ditferent calculations
must be made for the " fii-st part " and the " second part."

First Part, from x^O to x:=a.

M=5.{3,.+AV.5^«-2,,|.(^-|);
N =1 . |a-+W . ~--2.r\ . {>}-,/) ;

Q = .- (1-35+2$).

Second Part, from x=.a to x=.2r.

^■=|-{™"+(='-^v.ii).-4.(.|-|).
L = t {5^+ (2,-w. ;!)„.-.»}. (,-|);

M=^.{2'-^V.^-24.(-f-f);
N=?{2'-wi-2.r}.(..3,-/);
0 = ..(8^2$);

To diminish tlie number of symbols, we will at once assume tliat ^^'=: weight of half
the bar =:rs. Then we have

L = ,|.|(4,-,>.-,r-l.(,-|);
N=|./4,-«-2.rJ.(,sv-,/);

«=«-(!-3S+2S)-

L = p|2ra+(2<— «>t-ij. (,/-|) ;

N=p|2,-«-2.,j.(,9-,/');

«=-'(!-'S+2S)-

And we will now select the cases which it appears desirable to compute numeiically.

ME. G. B. AIET OX THE STRAINS IN THE INTERIOE OF BEAilS. 67

31. The strains upon the beam are not at all affected by placing a weight upon its
end (supposed strong enough to resist distortion of form). It appears probable, there-
fore, that the extreme changes of opposite character wU be given, on the one hand, by
placing the weight upon the centre of the beam's length, or making r(=:r; on the other

hand by placing the weight upon the centre of one half of the beam, or making «=-.

We will proceed first with the formula} for the case when the weight is upon the
centre, or a=r. It is unnecessary here to make calculations for the two segments of
the beam, as the strains will be symmetrical -nith respect to the two extremities. As

before, - is taken =5.

Weight rs placed on the centre of the beam's length.

Q = s.io.{l—io).{l--lw);

from all wliich, as before, the general factor s may be omitted.
Proceeding now with the other case, or

Weight rs placed on the centre of the first half of the beam's length,

the formulae for the "fii'st part," from 0^ = 0 to x-=a=.-, or from «=0 to 'y=0'5,
will be

n4.{|-2.},.(.-,) = ,.if^(Z_f}.f.(l-|) = ,.00.{|-,}.,o.(l-,.);

Q = s.w.{l-w).{\-2io);

and those for the "second part," from x=-^ to x=.2r, or from ?; = 0'5 to i) = 2'0,
will be

L = -^..(.^+|.-.'}.{|-,}=-..^^.(2-^).(l+-:).(l-^) = -..75.(2-,).(i+,.).(l-2»);

N= J.{|-2.}.(»,-,') = ..!l'.{5_f}.|.(l_0 = ..60.(J-,).,„.(l-,„,;

Q = s.w.{l—w).{l—2w);

from all which the factor s may be omitted.

32. For all these cases, the same formula? as before are to be used in the ixltimate

L 2

08 ^IR. G. B. ATET OX THE STEAINS IN THE INTEKTOE OF BEAMS.

calculations of the magnitudes and directions of the strains, namely,

N

tan 2/3:

L+LV
N L + Q

*- -^ ~sia2/3~cosi.'/3'

C+B =L-Q.
By means of these, the numbers have been computed for Table III., and Table IV.
parts 1 and 2 (end of the Memoir), and the lines of tigs. G and 7, Plate VI. have been
traced.

33. It is worthv of remark that, in figures 4 and 5, the lines representing the direc-
tion of thrust, and also those representing the direction of pull, are continuous ; but in
figures G and 7 they are discontinuous, the two segments of each curve, at their meeting
in the ordinate vertically below the weight, having different tangential directions. This
follows as an inevitable consequence of the assumption in art. IG; I think it probable
that a hypothesis like that of Mr. "W. H. Barlow would remove the discontinuity. An
investigation similar to that of art. 25 would show that, at these points, the transverse
section of the beam must be sufficiently strong to support the weight by tlirust (if the
weight is on the top of the beam), or by tension (if the Aveight is carried by or attached
to the bottom of the beam).

34. There are cases somewhat different from those already considered, whose import-
ance and singularity of principle are such as to make them worthy of special notice.
In Mr. Egbert Stephenson's construction of the Britannia Bridge, the strength of the
tubes was nearly doubled by the following admirable arrangement. The junction of
the ends of successive tubes, at their meeting on the piers, was effected, not while the
two successive tubes rested on the bearings which they were finally to take, but while
the distant end of one of the tubes was considerably elevated. It is a problem of no
great difficulty to ascertain what elevation ought to be given in order to reduce the
maximum strains on the bridge to their smallest value; when the best arrangement is
made, the strains are reduced to one-half of their original value. The singularity of the
mathematical principle consists in this, that there is impressed on the end-frame of
the tube or beam a strain of the nature of a couple, or (as it is called in the preceding
articles) a moment. Where there arc three or more connected tubes, the middle tube,
or each of the middle tubes, has such a moment-strain at each end ; but each of the
external tubes has a moment-stram at one end only (inasmuch as, at the land termi-
naticiu of the bridge, there are no means of applying such a strain). There are there-
fore two different cases, requiring different investigations.

35. Take, first, the case of a middle tube in which a moment-strain is impressed
on each end, the directions of the two strains (sup])osed equal) being opposed, so that
both tend to raise the middle of the tube. Tlie i)ressures upon the two piers will not
be disturbed, because the effects of tlu' two strains u])on tlie entire beam balance. If
now we consider the forces which act on the distant part of the beam (using the Ian-

ME. G. 13. AIRY OX TIIE STRAINS IX THE IXTERIOR OF BEAMS. G9

xiage of art. 7), avc sliall lia\o to combine, with foiTcs formerly recognized, the mo-
ment wliich acts on the distant end. By the kno^^^l hxws of translation of the place
of application of a moment, we may suppose this moment applied at the imaginary
division of the bar. Thus, at every vertical section of the bar, there is combined \\ith
the ordinary moment of strains a moment equal to that impressed on each end. The
most advantageous magnitude for this moment is evidently lialf the magnitude of mo-
ment at the beam's centre, with opposite sign ; for if we use a smaller value we leave
too much moment at the centre, and if we use a larger value we impress too great a
straining moment at the junction above the pier.

3G. Now in ai't. 21 we found, for the horizontal thrust in a point of any vertical

section, ''^ ~^ ''^^ •(r,~l/j- -'^^ regards the variations of a-, this is greatest when .r=;',

and its value is then ^f^— y). One half of this with changed sign, or +"^(5— i' j^

is now to be applied to the expression for horizontal thrust in every part of the beam's
length. Hence the expression to be used for horizontal thrust or compression is

j^ •\2 yp

and therefore

It will be seen immediately that this quantity satisfies the equations (15.) and (10.), the
integrals being taken from z, 0 to 2r, s. But with regard to equation (17.), we must
consider that in the instance before us a moment is to be introduced which lias not
presented itself before, namely, the moment impressed on the distant end. The value
of that moment, which (with the sign contemplated in forming equation (17.)) is

— y^y-~T\Q — y"), becomes +-7-- Hence equation (17.) becomes in this case

dV , dV ,J f dY d¥ 1 ^^ „ r'^s

J dy ^ dx J „,. , [_■■' dij I dx J . „ '4

And, on making the sixbstitutions, this equation is satisfied.
37. Therefore we are to adopt

6a*-12ra:+3r«

£fom which

<f-C).

M=^=^^-(f-f) =-..f.(l-^).f(l-f);

d-F I2x — I2r
'dxdy'

X=-2M

~{'-y'{'-'^--

70 ME. G. B. AIEY ON THE STKAIXS IN THE INTEEIOE OF BEAMS.

Q=,-0=.f-0=,(f-35+2S)=..f.(l-f).(l-?).
Or, if «z=-, 10=.-, -=5, and the miiltiplier s be omitted,

L=75 .|(l-i'f -U. (l-2w) ;

Then

^ = m .{l-v).'io .{\-w);
Q=w.{l-w).{\-2iv).

tan2/3=r^; C-B=-^=^^; C+B=L-Q,
"^ L + Gl sin 2/3 cos 2/3 ! '

by which the numbers for Table V. have been computed, and the ciuves of figiu'e 8,
Plate VII. have been dra^ni.

38. Take, secondly, the case of an end tube, on which a moment is impressed only
at one end. In this case, the effect of that moment is not balanced by a moment
impressed at the other end, and must be balanced by an increase of pressm-e on the near
pier (at which the moment is impressed), and a decrease of pressure on the distant

pier. The value -j of moment Avill be balanced by an increase of pressure ,r on the
near pier, and a decrease of pressure -g on the distant pier. Hence the pressure on the
distant pier will be rs— -=-g-. From this (as in art. 21) the moment produced by

the weight of the distant part of the beam =-^^^-5 , and the moment produced by

the reaction of the distant pier is =— -g- x(2;'— a"). The equation of moments is now

-g+^-^-^i^X(2>-.0=O;

-g+.(2.-.).{.-|-^}=:0;
-g-.(2>-.-).(H)=0

From this,

{Gx-\2r){x-^
C'= 3 ;

and the liorizontal compression-force at elevation y

{Gx-\2r){x-^

(^-V)

ME. G. B. AIRY ON THE STRAINS IN THE INTERIOR OF BEAMS. 71

Therefore we are to take for trial

F=

(?-^)-

7rs
Remai'king that the reaction of the distant pier =-5-, and that its moment upwards

=-g-X2/', it will be found that this function satisfies equations (15.), (16.), (17.).

39. Adopting therefore

F=

(6.-12r).(.-l)

•[4 6J'

we have

L=- =■ — ^^-^•(H= -¥-('-^)-(H)-(i-?)^

27
12a^ — r

,,_^ 2_ /sy 7r\ _ ^ (^ A V l\ y\

^^—cLxdy— s« \2~2J —~^'s'\^~r)'s'y-~s)'

s y8 rj s \ s/

Q=^-o = .'-f-li-fXi-l)-

And, with ^'=-, w=-, -=5, s=l,

L=-75.(2-iO.(r-i).(l-2w);

N= 6oYg~A?o.(l-M0;

Q= w.(1-w).(1-2m>);

tan2i3=f^; C-B=:^=^§; C+B=L-Q;
'^ L< + ti sin2p cos2p' ' ^'

by which the numbers for Table VI. have been computed, and the curves of figure 9
have been drawn.

40. These instances wiU probably suffice as applications of the theory to the most
important cases of practice, and as examples of the modifications on subordinate points
which may be requu'ed in investigating strains where the foims or other cii'cumstances
ai"e dififerent from those considered here.

41. Perhaps useful information may be derived from the diagi-ams and tables of
numbers for guiding the construction of Latticed Bridges. Thus, in such cases as those
of figiu'es 5, 6, 7, the upper and lower edges require great longitudinal strength in the
middle of the beam's length, but very little near the ends ; on the contrary, powerful
lattice-work is requii-ed near the ends, but very little neai' the middle. In the case of
figure 8 these remarks require veiy considerable modification.

2 o >-. ■

? ^ w 9^

+3 '5 cl!5

o ^ c 5^

rJ^ i;

c_H ^ ^

I c g S

r:* > 1^ tj^

c- -^ -a ^ i''-

'^ _ "^ '-^ -S

0

0

0

^

0

1

0

0

^

0

0

0

0

0

0

9

0

=

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0

0

o

6

0

6

0

0

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0

i

6

6

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6

6

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6

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t^

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6

6

6

6

6

6

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6

6

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1

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6

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ix

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0

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0

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1

6

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6

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6

i

5

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M

tb

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■^

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(M

^o

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0 *^

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to

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% "^

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8"

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0

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6

6

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[ 81 ]

V. On the R(]/lcxion of Polarized Liijhtfrom Polislied Surfaces, Transparent and Metallic.
Bij the liev. Samuel Hauguton, M.A., F.E.S., Fellow of Trinity College, Luhlin.

Eeccivcd June 9, — Read Juno 19, 1862.

Introduction.
Amoxg tlie experimenters who have made the reflexion of polarized light the object of
their researches, there is no one to whom science is more indebted than to M. Jamix,
whose accurate observations are a model for subsequent observers. His first paper on
this subject was published (1847) in the 19th vohime of tlie ' Annalcs de Chimie et de
Physique,' 3rd series, p. 29G, on Metallic Reflexion.

In this remarkable paper M. Jajmin verified many of the prcAious observations of
Brewstee, and added many of liis own. lie employed two distinct methods in these
experiments, —

1st. The method of Comjyarative Intensities — by observing tlie relative intensities of
the same beam of light reflected from a polished surface, composed partly of glass and
partly of the substance to be examined.

2nd. lyic method of Multij^le Reflexions, pre\iously known from tlie researches of
Bkewstek.

The optical constants used by Jamix in this paper are —

(ff) The angle (/,) of maximum polarization.

{!)) The angle (A) whose tangent is the ratio of I to J, the square roots of the inten-
sities reflected in the plane of incidence, and in the perpendicular plane.

[c) The coefficient (e) used by Cauciiy, which is connected with the other two constants
by means of theoretical equations.

By the first method of observation, M. Jamix determines the constants /, and i for the
following substances : —

1. Steel,

2. Speculum metal ;

and by the second method of observation, he determines ?, and A for

3. Silver,
and ?', for

4. Zinc,

and gives the details of experiments on

5. Copper,
from which the optical constants may be found.

MDCCCLXIII. X

82 EEV. S. HATJGHTON ON THE EEFLEXION

M. Jamin's next paper on Metallic Reflexion appeared in 1848, in the Annales de
Chim. et de Phys. ord series, vol. xxii. p. 311. In this paper he makes use of the second
method of observation, by multiple reflexion, and gives valuable tables of the results of
his experiments with the various colours of the spectrum on the seven following metallic

substances : —

1. Steel.

2. Speculum metal.

3. Silver.

4. Zinc.

5. Copper.

6. Brass. -

7. Bell metal. '■ ' ■

From these Tables the constants ^'i and A may be inferred.

In 1850 M. Jmiix published his well known paper " On the Reflexion of Light at the
Surface of Transparent Bodies," in the Ann. de Chim. et de Phys. 3rd series, vol. xxix.
p. 263. In this series of experiments he used a new method of observation, founded on
the Quartz Compensator of Babinet. In this elaborate and important paper he publishes
the details of his experiments on the following substances : —

1. Fire Opal, ■.

2. Hyalite, • ,

3. Realgar, • ;

4. Blende,

5. Diamond, ... , ,
G. Fluor-spar,

7, 8. Two kinds of glass,

and, in addition, gives in a Table at the end of the paper the constants of many other
transparent bodies.

M. Jamin has also published, in 1851, in the Ann. de Chim. et de Phys. vol. xxxi.
p. 165, a memoir " On the Reflexion of Light at the Surface of Liquids," in which he
determines the optical constants of many liquids.

It occurred to me that the method of observation employed by Jamin for transparent
bodies might be advantageously used in the case of metals ; and I was thus led to
commence the series of experiments the results of which are recorded in the following
pages.

In these experiments I have added many metallic substances to Jamin's list, and have
re-examined the metals observed by him by a different method.

In transparent bodies I have examined a few not experimented on by Jajiix, and
investigated in detail the form of the reflected ellipse, under varying conditions of
incidence and azimuth.

In the course of my paper I liave employed for the second optical constant one more

OF POLAEIZED LIGHT FEOM POLISHED SURFACES. 88

readily determined than those usually employed, but which is readily deduced from the
constants A and k of Jaiiix.

At the close of the paper I shall give a Table containing a comparison of the constants
found by Jamix and myself for all the bodies which we have both examined.

Some years ago, in making observations on polaiized light, I found that by adjust-
ing properly the azimuth of the incident polaiized beam, and allowing it to fall at
-the angle of principal incidence, I could obtain a reflected beam of circularly polarized
light.

On repeating the experiment with different polished surfaces, I found that the coeffi-
cient of reflexion, or whatever property it is that gives a surface a metallic reflexion,
might be conveniently expressed by the cotangent of the azimuth at which an incident
beam of plane-polarized light should be placed so as to give, on reflexion at the prin-
cipal incidence, a reflected beam of circularly polarized light.

The following paper contains an account of my experiments on many substances, and
a Table of their Coefficients of Reflexion and Kcfraction, determined with as much
accui'acy as I was able to attain with the instruments at my disposal.

The apparatus used by me consisted of a large graduated circle (horizontal), provided
with two moveahle arms, each fmiiished with graduated circles (vertical) ; and the large
horizontal circle was capable of being hung vertically, so as to allow of experiments
being made on liquids as well as solids. The substance to be examined was placed on
• a stage provided with adjusting screws, so as to bring the surface exactly into the
centre, or intersection of the axes of the polarizing and analysing arms. These arms
were mounted with Nicol prisms, made for me by Duboscq of Paris, and without sensible
deriation. The light employed was generally sunlight, but I sometimes used a mode-
rator lamp with colza oil.

I employed the quartz compensator described by M. Jamix*, for the pui-pose of con-
verting the elliptically polarized reflected light into plane-polarized light, before allow-
ing it to pass through the analyser.

The instrument used by me in making my obseiTations on the reflexion of polarized
light, was made by Mi*. Grttbb of Dublin for the late Professor M'^Cullagii, and was
presented to me, shortly after M'^Cullagh's death, by his brother. It is substantially
the same as that described by M. Jamin in vol. xxix. Ann. de Chim. et de Phys. ser. 3.
I procured from M. Duboscq Soleil, of Paris, a compensator of Jamin's pattern, and had
■ it adapted to my own apparatus.

In making my observations I used the follomng precautions : —

1. The zero of both polarizer and analyser was determined by direct observation with
red sunlight, reflected at the angle of polarization of several glasses found to give a
reflected beam capable of being completely cut off by the Nicol prism.

2. The Nicol prisms themselves were carefully tested and found to have no deviation.

3. Each of my recorded observations is the mean of four or five ; and when these
differed from each other by more than 20', I took the precaution of repeating them

* Annalcs dc Chimic ct tic Physique, ser. 3. vol. xxix. p. 2G3 et s^q.

n2

84 • KEY. S. HAUGHTOX OX THE EEFLEXIOX

nsain. on another day, with my eye fresh and unfatigued, before I finally adopted my
mean.

4. I frequently repeated the observations, with the incident light polarized at an equal
angle, at the opposite side of the plane of incidence ; and also reversing the polarizer
and analyser, so as to read the opposite sides of their scales.

The foUomng definitions will explain the sense in which I use certain terms.

The Azimuth of a beam of plane-polarized light is the angle which its plane of
polarization makes with the plane of incidence.

The Index of Refraction is the ratio which the sine of the angle of incidence bears
to the sine of the angle of refraction.

The Coefficient of Bef faction is the tangent of the FrincijJal Incidence.

The Prineii^al Incidence is that angle of incidence at which rays polarized in any
azimuth have the major axis of the reflected elliptic light in the plane of incidence ; or
at which the components of the reflected beam, in and perpendicular to the plane of
incidence, tlificr by 90° in phase.

This angle is nearly the same as Brewster's Angle of Polarization or Maximimi
Polarization.

The Coefficient of Befiexion is the Cotangent of the Azimuth of an incident beam of
plane-polarized light, which after reflexion at the principal incidence becomes ciixularly
polarized.

The Princijxd Comjionents of the incident and reflected light are the components in
and perpendicular to the plane of incidence.

The following preliminary investigation will seiTe to show the principles on which I
have tabulated the results of my experiments : —

Let the elliptically polarized reflected beam be represented, as in the annexed figure,
inscribed in a rectangle, whose sides are parallel to O I and
O P, the plane of incidence and perpendicular plane.

Let 0 .1' be the diagonal of the circumscribed rectangle,
and O >/ the axis of the ellipse ; it is required, from the
diffc'rence of phase of the light in the planes O I and O P,
and knowing the direction of the line Ox, to find the
direction of O y and the ratio of the axes of the ellipse.

The angle a* O I = «' is the azimuth of the reflected beam,
measured by the analyser, after it has lost its elliptic polarization in the compensator;
and the diflference of i)hasc of O I and O P is measured in the compensator itself, by
the displacement necessaiy to reduce the elliptically-polarized to plane-polarized light.

We may imagine, to aid our conception, but without hypothesis, that a material
])oint traverses the ellipse, and tliat its coordinates are

;=A sin (Z'i -}-<')>

;; = B sin (/.•!■ + <?'), - ■ '

where e^—e is the diffc'rence of phase between the beams O I and O P, and A, 13 are the
Incs 0 I and O P.

OF POLAEIZED LIGHT FEOM POLISHED SUEFACES. 85

Eliniiiiating" f, wo find

g+|-2cos(.'-.)|53 = shr(.'-.) (1.)

In this clUpso, the an<;lc p madc^ by tlic axis with tlic plane of incidence is found from

the well-known expression

2E

tan2? = j3;jp,

bcloniring: to the ellipse

D.r + '2 Ya-y + Vif = const.

Substitutini:^ for D, E, F their valiu>s from (1.), we find

tan2f=tan2«'cos(f'— r>); (2.)

p denoting the angle l/Ol, and a! the angle .r O I. But if a and l> denote the axes of

the ellipse, it can be proved that

i'^ _ (D + F) + (D - F) sec 2p
(i^ "■ (D + F) - (D -F) sec 2p '

or substituting from (1.) and (2.),

^ = V' — cot ((p + a') cot (<p — a')

(3.)

-=\/ — tan (©+«') tan (ip — a')

From equations (2.) and (3.), I calculate the position of the elliptic axes and their
ratio.

The angle a! is obtained by direct measurement with the analysing prism; and e' — c
may be found, as follows, from the compensator.

In the compensator made for me by M. Duboscq, I find tliat 39--43 represents the zero,
?. e. the position in which the compensator affects equally the light in and perpendicular
to the plane of incidence ; the number of graduations corresponding to a difference of
half a wave (180°) I found to be

Red lampliglit (colza oil) .... 15-43

Red sunlight • 15-37

White lamplight (colza oil) . . . 13-29

If, therefore, C denote the reading of the compensator in any experiment, the difference
of phase of the two principal beams will be expressed for red sunlight, in degrees, by
the expression

(C-39-43)x^,

and by a corresponding formida for the other kinds of light.

The angle thus measured by the compensator is not the difference of ])]iase between
the principal components of the reflected light until it is increased by 180°, because
experiment shows that in the act of reflexion there is this constant difference between

86

EEV. S. HAUGHTON ON THE EEFLEXION

the two components, in addition to the varjing difference of phase, depending on inci-
dence, azimuth, and nature of polished siu'face. We therefore use the formula

e' - f =180'+ (C- 39-43) x^-^, (4.)

where i denotes the interval corresponding to 180' for the light used.

In tabulating my experiments, I give the original measurements of the analyser and
compensator, and use the equations (2.), (3.), and (4.) to calculate the other columns.

I. Munich Glass («).
The fii-st experiments I shall record were made with glass procured from ^Munich by
the late Professor M^Cullagii. I have four rhombs made of it, whose index of refrac-
tion I determined by the following experiments : —

Table I. — Munich Glass {a).

Ehomb.

Angle.

Minimum cleriation*
of red liglit.

Eefractive index.

No. 1.,

44 56 0

31 43 30

1-6229

No. 2.

54 28 30

41 28 0

1-623 0

No. 3.

39 50 0

27 17 0

1-6327

No. 4.

59 58 30

48 22 0

1-6221

Mean

1*6227

Calculated Angle of Polarization =58' 21'.

I also found the refractive indices of No. 2 for the extreme red and violet rays to be
1-6190 and 1-6555, which indicates a dispersive power in the glass of 0*0573.

This glass was found to contain the following constituents : — • -

Silica 42-25

Oxide of Lead . . . 46-35

Lime 0*45

Alkalies (by diff.) . . 10-95 - - -

imHK)

The following Tables contain my observations on this glass : —

* III all my experiments the rod light iwcd was pas.9ed through tht^ same piece of red glass, which waa very
homogeneous.

or POLAEIZED LIGHT FEOM POLISHED SURFACES.

Table II.— Munich Glass (a). (September 20, 1854.)
Azimuth of Polai-izer=20°. Eed Sunlight.

87

Incddenoe.

Compensator.

Analyser.

<r'-e-180°.

«•

a
6'

Taa-.(J).

34 30

39-69

11 35

2 41

+ l! 34

89-66

29 23

62 30

42-65

2 5

36 30

+ 1 40

45-80

5 42

63 30

44-09

2 0

64 31

+ 1 10

35-25

5 29

54 30

45-81

2 0

74 38

+ 0 32

29-71

5 29

65 30

48-68

2 6

108 13

- 0 39

28-68

5 45

66 30

49-86

2 37

122 2

- 1 24

25-89

7 9

73 30

63-88

11 45

169 4

-11 34

26-93

29 45

The principal incidence is therefore 54° 57'.

The last column of this Table is thus found : —

Let the principal components of the incident polarized beam, in and perpendicular to
the plane of incidence, be cos a and sin « (vmity denoting the mcidcnt beam) ; and let
I and J denote what a unit of light becomes after reflexion, in and perpendicular to the
plane of incidence respectively ; then the principal components of the reflected beam are
I cos a and J sin a, and therefore

j=tana'cota; (5.)

and tlie angle tan"'( y) may be found from this equation without any difficulty.

According to the tlieory of Fresnel,

J cos {i + r)
I cos [i—ry

an expression which vanishes at the polarizing angle (^+>'=90°), and therefore

tan"'(j) ought at this angle of incidence to vanish also; but we find, not only in this

experiment, but in those which follow, that it does not vanish, but only reaches a mini-
mum, the tangent of which is sensibly equal to wliat I have called the Coefficient of
Reflexion*.

In fact, let X denote the angle whose cotangent is this coefficient. Then I cos X, J sin X
are the principal components of the reflected light, which by definition is circularly
polarized, and therefore I cos K=J sin X, and

J

cotX= J-

The coefficients of Refraction and Reflexion, as determined by this experiment, are

Coefficient of Refraction=tan 54° 57'=l-4255.
Coefficient of Reflexion =cot 84° 31'=0-0960.

* Strictly speaking the angle of incidence at which the maximum is reached is found to be somewhat less than
the Principal Incidence,

88

EEV. S. HAUGIITOX OX THE EEFLEXIOX

Table III.— Munich Glass (a). (June 26, 1854.)
Azimuth of Pohirizer =45°. "White lamphght (Colza oil).

Incidence.

Compensator.

Aualyscr.

c'_f_lSO'.

f-

a

lan-.g).

43 37

39-54

18 30

3 11

+ 18 28

47-79

18 30

48 37

40-07

10 55

10 17

+ 10 45

29-42

10 55

50 45

40-61

8 10

17 36

+ 7 48

23-37

8 10

51 45

41-11

6 45

24 22

+ 6 10

20-70

6 45

52 45

42-67

6 10

45 29

+ 4 21

13-03

6 10

53 52

43-15

5 22

50 21

+ 3 27

13-86

5 22

54 20

44-46

5 1

69 44

+ 1 45

12-15

5 1

55 20

' 46-30

5 36

94 38

- 0 27

10-23

5 36

56 20

48-18

6 15

120 6

— 3 10

10-58

6 15

57 40

50-07

7 35

145 41

- 6 19

13-52

7 35

58 40

51-00

9 39

158 16

- 9 0

16-16

9 39

CO 35

51-60

11 10

166 24

-10 53

22-34

11 10

65 40

1 51-98

18 11

171 33

-18 2

22-84

18 11

75 35

! 52-50

30 25

178 40

— 30 25

00

30 25

The principal incidence is therefore 55" 8', and the minimum value of tan"'/ jj is
5' 1', or Circular limit = S4' o'J'. Therefore the

Coefficient of llefraction=l'4352.
Coefficient of Reflexion =0-0877.

Table IV.— Munich Glass (a).
Azimuth of Polarizer =: 80^.

(July 28, 1854.)
Eecl Sunlight.

Incidence.

Compensator.

Analyser.

e'-e-lSir.

<f-

a

Tan-. (J).

34 30

39-72

74 6

3 23

+ 74 i

88-20

o -

31 35

43 30

40-13

61 45

8 11

+ 61 53

16-90

18 10

48 30

4076

49 15

15 34

+ 49 24

7-62

11 34

50 30

42-15

38 0

31 49

+ 36 49

3-62

7 51

51 30

42-80

33 30

39 26

+ 30 36

3-09

6 40

52 30

43-85

28 0

51 42

+ 21 17

2-70

5 22

53 30

45-41

25 45

69 57

+ 11 39

2-28

4 52

54 30

47-20

26 34

90 54

- 0 36

1-99

5 2

55 30

48-80

28 45

109 37

-13 53

2-02

5 31

56 30

50-30

34 0

127 10

-28 7

2-26

6 47

57 30

51-20

40 0

137 42

-38 18

2-64

8 25

58 30

51-21

42 30

137 49

-41 38

2-82

9 10

Co 30

52-14

53 45

148 42

— 55 8

3-76

13 31

65 30

52-77

66 30

156 4

-G7 47

G-59

22 5

70 45

53-18

75 15

160 52

— 75 56

12-30

33 49

Prnicipal Incidence :=54^ 27'.
Coeff. of Kefraction =1-3993.

Tan M'j j=4' 57', or Circular limit =85^ 3'.
Coeff. of Reflexion =0-08GG.

OF POLARIZED LIGHT FROM POLISHED StTEFACES.

89

Table V. — Munich Glass (a). (Au<,nist 7, 1854.)
Azimuth of Polarizer =85°. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'-e-lSO".

9-

A"

-"-(?)■

34 30

39-68

80 12

2 55

+ 80 13

98-30

zS 52

52 30

43-37

44 30

46 5

+ 44 17

2-37

4 55

53 30

44-33

39 54

57 19

+ .35 47

1-86

4 11

54 30

46-12

38 24

78 16

+ 20 28

1-36

3 58

55 30

48-13

41 50

101 47

-30 44

1-26

4 28

56 30

49-45

44 0

117 13

— 42 49

1-64

4 50

57 30

50-42

51 0

128 34

-54 24

2-15

6 10

73 30

53-62

81 0

166 1

-81 15

26-76

28 55

Principal Incidence =54° 59'. Tan-Yj^ =3° 58', or Circular limit =86^ 2'.

Coeff. of Refraction =1-4272.

CoefF. of Reflexion =0-0693.

Table VI.— Munich Glass (a). (Sept(>mbcr 27, 1854.)
Azimuth of Polarizer = 85° 45'. Red Sunlight.

Incidence.

Compensator.

Analyser.

c'-c-lSO'.

f-

a
T

Tan-.(^).

54 30

46-05

43 26

77 23

+ 37 56

1-25

4 2

54 45

46-75

43 20

85 38

+ 26 18

1-09

4 1

55 0

46-90

43 8

87 24

+ 17 24

1-08

3 59

55 15

47-53

43 15

94 46

-26 49

1-11

4 0

55 30

48-05

45 30

100 51

-47 39

1-21

4 20

Principal Incidence =55° 6'.
Coeff. of Refraction =1-4334.

Tan-Y^)=3° 59', or Circular limit =86° 1'.
Coeff. of Reflexion =0-0696.

Table VII.— Munich Glass (a). (September 27, 1854.)
Azimuth of Polarizer =85° 55'. Red Sunlisrht.

Incidence.

Compensator.

Analyser.

e'-e-lS<r.

<f-

a
T

Tan-,(^).

54 30

54 45

55 0
55 15
55 30

45-96 45 30 7§ 24
46-65 45 12 84 28
47-00 j 45 5 88 34
47-72 i 45 40 96 59
48-00 46 30 ! 100 16

+ 47 7
+ 47 4
+ 48 19
-50 25
-53 12

1-27
1-10
1-02
1-13
1-20

4 10
4 7
4 6
4 11
4 18

Principal Incidence =55° 7'.
Coeff. of Refraction =1-4343.

Tan-'('^W4° 6', or Cii-cular limit =85° 54'.
Coeff. of Reflexion =0-0717.

MDCCCLXin.

90

EEY. S. HAUGHTOX OX THE EEFLEXIOX

Table YIII.— Munich Glass (a). (September 26, 1854.)
Azimuth of Polarizer := 86°. Red Sunlight.

Incidence.

Compensator.

Analyser.

f'-e-lSO'.

1>-

a
T

Tan-.(_5.

34 30

! 39-77

83 20

3 58

+ 83 21

120-50

30 53

52 30

43-03

53 30

42 7

+ 56 16

2-73

5 24

53 30

44-67

47 0

61 18

+ 49 8

1-70

4 17

54 0

; 45-25

46 20

68 5

+ 48 34

1-48

4 11

54 30

46-13

46 11

78 23

+ 50 48

1-23

4 10

54 45

46-48

46 0

82 28

+ 52 28

1-14

4 8

55 0

46-91

45 45

87 30

+ 60 29

1-06

4 6

55 15

47-33

47 0

92 25

-74 27

1-08

4 17

55 30

48-07

48 20

101 5

-60 39

1-25

4 30

56 0

1 48-86

50 15

111 30

— 58 25

1-53

4 48

56 30

i 49-40

51 30

116 38

-58 37

1-71

5 2

5" 30

i 50-20

57 0

126 0

— 63 34

2-26

6 9

73 30

' 53-69

83 15

166 50

-83 25

37-96

30 35

Principal Incidence =5-5^ 8'.
Coeff. of Refraction =1-4352.

Tan-'/ j") = 4° 6', or CirciUar limit =85' 54'.
Coeff". of Eeffexion =0-0717,

Table IX.— Munich Glass {a). (September 21, 1854.)
Azimuth of Polarizer = 87'. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'-c-mr.

f-

a
V

Tan-@.

1

34 30

39-48

84 50

0 27

+ 84 60

00

30 6

52 30

42-87

60 0

40 14

+ 63 33

3-27

5 11

53 30

44-06

55 0

54 10

+ 60 56

2-16

4 17

54 0

44-93

54 20

64 21

+ 63 59

1-78

4 11

54 30

45-82

54 0

74 45

+ 70 30

1-52

4 8

54 45

46-34

53 54

80 50

+ 76 48

1-43

4 7

55 0

46-60

53 55

83 53

+ 80 50

1-40

4 7

55 15

47-23

53 34

91 15

-87 59

1-35

4 4

55 30

47-92

55 30

99 20

— 78 33

1-48

4 22

56 0

48-40

56 30

104 57

-74 21

1-64

4 32

66 30

49-08

59 30

112 54

-72 28

1-97

5 5

57 30

50-00

64 30

123 40

— 72 48

2-72

6 16

73 30

53-34

85 0

165 5

-85 10

44-50

30 55

Principal Incidence =55' 13'. Tan-'(T)=4' 4', or Cu'cular limit =85"" 56'.

Coeff. of Refraction =1-431)7. Coeff. of Reflexion =0-0711.

Collecting together the ])receding results, and denoting hy}^ the azimuth of the plane
of polarization of the incident light, wliicli on reflexion at the principal incidence wiU
produce, on reflexion, circuhuly polarized light, and calling it the Circular Limit, we
obtain

OF POLAEIZED LIGHT FROM POLISHED SURFACES.
TAiiLt: X. — Constants of >[uni("n Glass (a).

91

Azimuth of
Polarizer.

Principal
Inoidonce.

Circular Limit.

Coodicient of
Refraction.

Cocflicient of
K«ilcxioii.

20 6

54 57

84 31

1-4255

0-0960

45 0

55 8

84 59

1-4352

0-0877

80 0

54 27

85 3

1-3993

0-0866

85 0

54 59

86 2

1-4272

0-0693

85 45

55 6

86 1

1-4334

O-O696

85 55

55 7

85 54

1-4343

0-0717

86 0

55 8

85 54

1-4352

0-0717

87 0

55 13

85 56

1-4397

0-0711

Means

55° o' 37"

85° 32' 30"

1-4287

0-0780

The movement of the axis of the reflected ellipse differs according as the azimuth of
the incident light is less or greater than the circular limit. This is sho^vn in Plate VIII.
fig. A, on which the values of <p are laid down for different angles of incidence in the two
cases in which the azimuth of the incident light is 80° and 8T^

When the azimuth of the incident light is less than X, the circular
limit, the axis of the ellipse moves as in the annexed figure. Let
P O A be the azimuth of the incident light, and Q O A equal to P O A ;
P O is the position of the axis major corresponding to 0° incidence ;
O A is the position of the axis major in the plane of incidence, corre-
sponding to the principal incidence ; and O Q is the position of the
axis corresponding to 90° incidence.

When, however, the azimuth of the incident light is greater than
the circular limit, the axis major moves from P to w, as in
the annexed figure, then back from a: to y, passing through
B at the principal incidence, and finally from 1/ to Q. Let
P O A be the azimuth of the uicident light, and Q O B
equal to P O B. At the incidence 0°, O P is the position of
the axis major; as the incidence increases ffom 0° to the
principal incidence, the axis major moves from O P to O a;
and turns back, attaining the position O B at the inincipal incidence ; and as the incident
angle increases from the principal incidence to 90°, the axis major moves from O B to
O y, and back again to O Q.

Having ascertained the truth of the preceding laws of the movement of the axis major
of the elliptically polarized light, I made the follomng experiments. Having removed

o2

92

EEY. S. HAUGIITOX OX THE EEFLEXION

the compensator, I set the polarizer at 88" and 89°, and found that the analyser gave a
minimum of light at 90°, showing that the axis major of the ellipse was perpendicular
to the plane of incidence.

All the experiments already given were made with the Mimich glass («) No. 1. I
made the follo-ndng experiment mth (a) No. 2, in order to cstabhsli fully the identity of
the pieces of glass with regard to reflexion, as they are certainly identical in their refrac-
tive mdices.

Table XI.— Munich Glass («). (October 11, 1854.)

Azimuth of Polarizer =86'. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'-e-mr.

?■

/,'

— ©■

34 30

39-70

83 20

3 8

+ 83 20

CO

30 53

52 30

43-33

53 30

45 37

+ 56 48

2-53

5 24

53 30

44-70

50 0

61 39

+ 55 11

1-73

4 46

54 0

45-60

40 0

72 11

+ 57 20

1-41

4 36

54 30

46-30

47 30

80 22

+ 58 48

1-21

4 22

54 45

46-79

46 45

85 58

+ 65 30

1-10

4 15

55 0

47-31

46 0

92 11

-66 15

1-05

4 9

65 15

47-70

47 30

96 45

-63 20

1-16

4 22

55 30

48-00

48 30

100 16

-62 17

1-24

4 31

56 0

48-83

50 30

109 58

-59 50

1-50

4 51

66 30

49-10

52 30

113 8

-62 9

1-64

5 12

57 30

49-95

57 0

123 5

-64 32

2-15

6 9

73 30

53-40

82 30

163 27

— 82 48

26-98

27 59

Principal Incidence =54° 53'.
Cu-cular Limit =85° 51'.

Coeff. of Refraction =1-4220.
Coeff. of Reflexion =0-0725.

The agreement of these values with those given for No. (1) in Table X. is sufficiently
satisfactory.

II. Munich Glass (b).

The glass now to be described is a rhomb which gave me the following values : —

o (

Angle of rhomb = 54 30

Minimum deviation of standard red ... 34 2

Hence the refractive index of this red is 1-5244, and the angle of polarization

50° 44'.

Table XII.— Munich Glass (b). (October 10, 1854.)

Azimuth of Polarizer ^45°. Red Sunlight.

Incidence.
54 15

Couipensiitor.

Analyser.

c'-e-l80".

?•

r

Tan-CJ).

1
39-43

i 0

6 6

+ i 6

00

i 6

54 30

39-43

0 43

0 0

+ 0 43

00

0 43

54 45

39-43

0 30

0 0

+ 0 30

oo

0 30

55 0

39-43

0 10

0 0

+ 0 10

00

0 10

55 15

54-13

2 15

172 0

— 2 14

209-4

2 15

Principal Incidence ^55° 1'.
Circular Linut =89' 50'.

Coeff. of Refraction =1-4290.
Coeff. of Reflexicm =0-0029.

OF POLAEIZED LIGHT FKOM POLISHED SUKFACES.

93

Table XIII.— Munich Glass (/>). (September 29, 1854.)
Azimuth of Polarizer = 80°. Eed Sunlight.

Incidence.

Compensator.

Anulyser.

f'_f_180°.

f-

a

■— «)■

34 30

39-43

72 30'

6 6

+ 72 30

oo

29° 13

52 30

39-57

20 15

1 38

+ 20 15

00

3 38

53 30

39-65

11 30

2 34

+ 11 29

127-3

2 3

53 45

39-65

10 15

2 34

+ 10 14

135-6

1 49

54 0

39-65

7 30

2 34

+ 7 29

160-2

1 20

54 15

39-65

6 0

2 34

+ 5 59

179-9

1 4

54 30

39-73

1 20

•3 30

+ 1 20

171-8

0 13

54 45

54-25

4 15

173 23

- 4 13

114-9

0 45

55 0

54-25

6 0

173 23

- 5 57

80-6

1 4

55 15

54-25

8 30

173 23

- 8 27

54-8

1 30

55 30

54-35

9 37

174 33

- 9 34

62-8

1 43

56 30

54-35

17 30

174 33

-17 26

37-5

3 11

57 30

54-36

25 25

174 41

-25 21

28-3

4 47

58 30

54-45

31 0

175 44

-30 58

30-3

6 3

73 30

54-68

70 0

178 25

— 70 0

90-5

25 51

83 30

54-68

76 30

178 25

-76 30

164-3

36 18

Piiucipal Incidence =54° 35'.
Cii-cular Limit =89° 47',

Coeff. of Refraction =1-4063.
Coeff. of Reflexion =0-0037.

Table XIV.— Munich Glass (fj). (October 10, 1854.)
Azimuth of Polarizer =: 87°. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'-c-lSO°.

f-

a

Tan-.g).

54 15

39-43

15 40

6 6

+ 15 40

00

0 51

54 30

39-43

2 15

0 0

+ 2 15

00

0 7

54 45

54-13

3 45

172 0

— 3 43

114-5

0 12

55 0

54-13

7 40

172 0

- 7 36

56-1

0 24

55 15

54-13

14 20

172 0

— 14 13

30-0

0 46

Principal Incidence =54° 36'.
Circular Limit =89° 53'.

Coeff". of Refraction =1-4071.
Coeff". of Reflexion =0-0020.

Table XV.— Munich Glass (b). (October 1, 1855.)
Azimuth of Polarizer =:88°. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'_c_180°.

<?■

r

Tan-.(^).

33 37

39-43

8§ 45

0 0

+ 8^ 45

00

31 36'

43 37

39-43

83 45

0 0

+ 83 45

00

17 41

53 37

39-43

37 30

0 0

+ 37 30

oo

1 32

54 37

39-43

359 45

180 0

- 0 15

oo

0 1

55 37

39-43

322 0

180 0

— 38 0

oo

1 34

56 37

39-43

301 30

180 0

-58 30

oo

3 16

63 37

39-43

278 0

180 0

— 82 0

oo

13 57

73 37

39-43

274 30

180 0

-85 30

oo

23 55

Principal Incidence =54° 37'.
Circular Limit =89° 59'.

Cceff. of Refi-acti&n =1-4080.
Coeff: of Reflexion s=0-0001.

94

REV. S. HAUGHTON OX THE EEFLEXION

Table XVI.— Munich Glass (b). (October 5, 1855.)
Azimuth of Polarizer =89^. Red Sunlis^ht.

Incidence.

Compensator.

Analyser.

(^-e-

ISO".

f-

b

Tan-g).

33 37

39-43

88 6

0

6

+ 88 6

oo

2^ 34

43 37

39-43

85 25

0

0

+ 85 25

00

12 17

53 37

39-43

55 0

0

0

+ 55 0

00

1 26

54 37

39-43

358 0

180

0

- 2 0

00

0 2

55 37

39-43

307 30

180

0

-52 30

00

1 2

56 37

39-43

292 0

180

0

-68 0

00

2 28

63 37

39-43

275 0

180

0

— 85 0

00

11 17

73 37

39-43

272 15

180

0

— 87 45

00

23 57

Principal Incidence =54° 35'.
Circular Limit = 0° 2'.

Coeff. of Refraction =1-4063.
Coeff. of Reflexion =0-0006.

From this and the foiu' preceding Tables the following results may be collected.

Table XVIL— Constants of Munich Glass (b).

Azimuth of
Polarizer.

Principal
Incidence.

Circulor Limit.

Coefficient of
Eefi-action.

Coefficient of
Reflexion.

45

55 i

89 50

1-4290

0-0029

80

54 35

89 47

1-4063

0-0037

87

54 36

89 53

1-4071

0-0020

88

54 37

89 59

1-4080

0-0001

89

54 35

89 58

1-4063

0-0006

Means

54° 4°' 48"

89° 53' 24"

1-4113

0-0019

III. Paris Glass.

This glass was supplied to me by M. Duboscq of Paris. Its refractive constants were
found to be as follows : —

o »

Angle of prism 59 55

Minimum deviation of extreme red 37 35

Minimum deviation of extreme violet .... 39 13

Index of refraction of extreme red . . . =1-5059

Index of refraction of extreme violet . . . =1-5246 ;

OF POLAEIZED LIGHT FROM POLISHED SUEFACES.

95

Table X\111.— Paris Glass. (October 1, 1855.)
Azimuth of Polarizer = 88°. Eed Sunlight.

Incidence.

Compensator.

Analyser.

e'_e_180°.

r

a

T

Tan- (9.

33 37

39-43

SS 20

6 6

+ 8^ 20

00

■s ■' ■

43 37

39-43

84 30

0 0

+ 84 30

00

ly ou

53 37

39-43

63 45

0 0

+ 63 45

oo

4 3

54 37

40-19

50 20

8 53

+ 50 24

12-76

2 25

65 37

42-43

25 30

35 6

+ 22 39

4-24

0 57

65 52

44-44

19 30

58 36

+ 11 26

3-41

0 43

66 7

46-75

17 30

85 38

+ 1 32

3-18

0 38

56 22

48-39

18 45

104 41

— 5 30

3-07

0 41

66 37

50-22

27 30

126 14

-20 5

2-65

1 2

67 37

52-70

48 0

155 15

-48 18

4-59

2 13

58 37

5:5-27

60 0

161 55

-60 38

7-32

3 28

63 37

54-00

80 0

170 28

-80 7

36-85

11 12

73 37

54-40

87 0

175 8

-87 1

180-90

33 41

Princijial Licideiice =56° 10'.
Cii-culai- Limit =89° 22'.

Coeff. of Refraction =1-4919.
Coeff. of Eeflexion =0-0110.

The compensator was then s(>t at 47' 12, which corresponds with a difference of phase of
90°, between the principal components of the reflected light ; and the compensator being
thus set, the angle of incidence was determined by trial, for which the dark band was
centrally placed. The incidence so found is the principal incidence. Having thus
found the principal incidence, I changed the azimuth of the polarizer, and read the
analyser, obtaining the following results.

T.UJLE XIX.— Paris Glass. (October 1, 1855.)
Compensator =47-12 = 90°. Eed Sunlight.

Polarizer.

Analyser.

a
T

Tan- (5).

89 30

48 6

1-110

6 33

89 0

32 0

1-600

0 37

88 0

18 0

3-077

0 39

87 0

13 20

4-219

0 43

86 0

10 0

5-G71

0 43

85 0

9 0

6-314

0 48

80 0

4 30

12-706

0 48

70 0

2 45

20-819

1 0

60 0

1 40

34-367

0 58

50 0

1 10

49-103

0 59

40 0

0 54

63-656

1 4

30 0

0 47

73-139

1 21

20 0

0 37

92-908

1 42

10 0

0 24

143-237

2 16

Principal Incidence =56° 7'.
Cii-cular Limit =89° 24'.

Coeff. of Refraction =1-4891.
Coeff. of Reflexion =00104.

96

EEY. S. HArGHTOX ON THE EEFLEXIOISr

The last column of this Table shows that the value of fjj increases slightly as the
azimuth of the polarizer diminishes.

Combinmg the preceding- results, we find

Table XX. — Constants of Paris Glass.

No.

Principal
Incidence.

Circular
Limit.

Coefficient of
Refraction.

Coefficient of
Keflcxion.

XVIII.

56 10

89 22

1-4919

0-0110

XIX.

56 7

89 24

1-4891

0-0104

INIeaiis... ,

56=" 8' 30"

89° 23'

1-4905

0-9107

IV. Fluor-Spar.

The specimen of fluor-spar on which I made my experiments was transparent and
blue. The following' are the results I obtained.

T.able XXI.— Fluor-Spar. (September 11, 1855.)
Azimuth of Polarizer = 80°. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'-e-lSO".

f-

a
T

Tan-.g).

3.3 37

39-43

73 30

0 6

+ 73 30

00

30 46

43 37

39-43

60 45

0 0

+ 60 45

00

17 29

53 37

39-43

10 30

0 0

+ 10 30

00

1 53

54 37

39-43

0 30

0 0

+ 0 30

CO

0 5

55 37

39-43

3.51 45

180 0

- 8 15

00

1 28

58 37

39-43

327 0

180 0

— 33 0

00

6 32

63 37

39-43

305 0

180 0

-55 0

00

14 8

73 37

39-43

288 15

180 0

-71 45

00

28 8

Principal Incidence =54° 40'.
Circular Limit =89° 55'.

Coeff of Refraction =1-4106.
Coeff. of Reflexion =0-0014.

OF POLARIZED LIGHT FROM POLISILED SUEFACES.

97

Table XXll.— Fluoi-Spar. (September 20, 1855.)
A/imutli of Polarizer =88°. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'-c-lgO'.

t-

53 37

54 7

54 37

55 7
55 37

39-43
39-43
39-43
39-43
39-43

43 30

31 0

15 0

344 30

331 0

d 6

0 0
0 0

180 0
180 0

+ 43 30
+ 31 0

+ 15 0
-15 30
-29 0

■-fi)-

^

,

1

54

1

12

0

32

0

33

1

7

Principal Incidence =54° 52'.
Cii-cular Limit =89° 28'.

Coeff. of Refraction =1-4211.
CoeflF. of Reflexion =0-0093.

From the preceding results combined, we obtain the following constants of fluor-spar.
T.-vBLE XXIIl. — Constants of Fluor-Spar.

So.

Principal.
Incidence.

Circular

Limit.

Coefficient of
Refraction.

Coefficient of
Ectleiion.

XXL

54 40

89 55

1-4106

0-0014

XXIL

54 52

89 28

1-4211

0-0093

Means

54 46

89° 41' 30"

1-4158

0-0053

V. Glass of AxTiMoxr.
The specimen of this glass witli which I experimented was given to inc by Professor
Apjohx.

Table XXIV. — Glass of Antimony. (October 5, 1855.)
Azimuth of Polarizer =80°. Red Sunli^rht.

Incidence.

Compensator.

Analyser.

e-_P_180'.

'<f-

a
A"

Tan-.g).

3°3 37

39-43

75 6

5 6

+75 6

OS

33° 21'

43 37

39-43

66 20

0 0

+ 66 20

00

21 55

53 37

39-82

38 50

4 33

+ 38 49

27-43

8 5

55 .^7

40-16

25 10

8 32

+ 25 1

17-84

4 44

57 37

41-90

10 30

28 54

+ 8 50

9-89

1 52

58 7

43-14

8 30

43 24

+ 6 16

9-86

I 31

58 37

46-19

G 20

79 5

+ 1 13

9-18

1 7

59 7

49-50

7 30

117 48

— 3 34

8-62

1 20

59 37

51-18

10 15

137 22

- 7 41

8-30

1 50

61 .37

53-21

26 30 ,

161 13

—25 46

7-77

5 2

63 37

53-70

39 30 1

166 57

-39 21

13-92

8 16

73 37

54-25

68 20

173 33

-63 25

27-04

23 56

Principal Incidence =58° 44'.
Circular Limit =88° 53'.

MDCCCLXIII.

Coeff. of Refraction =1-6468.
Coeff. of Reflexion =0-0195.

98

EEV. S. HAUGHTON OX THE EEFLEXION

Table XXY. — Glass of Antimony. (October 5, 1855.)
Azimuth of Polarizer =89". Eed Sunlifjht.

Incidence.

Compensator.

Ajialvsor.

<'_f_l80°.

1

b'

Xan-.({).

33 37'

39-43

89 6

6 6

+ 89° 6

<x

45 6

43 37

39-43

87 45

0 0

+ 87 45

CO

23 57

53 37

39-65

83 20

2 34

+ 83 20

oc

8 30

55 37

39-91

80 0

5 36

+ 80 3

56-19

5 39

57 37

41-25

64 30

21 17

+ 65 30

6-93

2 6

58 7

42-43

55 40

35 6

+ 57 46

3-44

1 28

58 37

45-72

48 0

73 35

+ 55 12

1-36

1 7

58 47

47-00

47 0

88 34

+ 72 11

1-09

1 4

59 7

48-76

49 30

109 9

-67 26

1-24

1 10

59 37

50-35

60 0

127 45

-60 23

9-36

1 44

61 37

52-80

78 30

156 26

-79 22 ;

12-75

6 9

63 37

53-68

82 15

166 43

-82 27

32-36

7 19

73 37

54-10

88 45

171 38

-88 46

1

281-50

38 40

Principal Incidence =58° 50'.
Circular Limit =88" 56'.

Cocff. of Refi-action =1-6533.

Coeff. of Pveflexion =0-0186.

T.UJLE XXVI. — Glass of Antimony. (October 5, 1855.)
Compensator =47-12 = 00°. E(-d Sunlight.

Polarizer.

-bialysor.

V

--©■

89

50° 0

1-192

1° n

88

28 45

1-823

1 6

87

21 15

2-571

I 10

85

1-2 40

4-449

1 9

80

6 -25

8-892

1 8

70

3 10

18-075

1 9

50

1 35

36-177

1 20

30

0 52

66-105

1 30

1 ^^

0 37

92-9O8

3 29

:Me

Ml — 1° 28' 0"

Principal Incidence =08'' 52'.
Cii-cular Limit =88° 46'.

Coeff. of Refraction ^1-6555.
Coeff. of Reflexion =00215.

It is to be remarked, that in Table XXV., in whicli the azimutli of tht> polarizer is
gi-eater than the circular limit, the movement of the axis of the elUpse follows the same
law as that of the Munich glass already described.

From the foregoing Tables, the optical constants of Glass of Antimony may be thus
inferred : —

or POLAEIZED LIGHT FEOM POLISHED SUBTACES.
Table XXVII. — Constants of Glass of Antimony.

99

-J Prindpal
■"°- 1 Incidence.

Circular
Limit.

Coefficient of
Refraction.

Coefficient of
Eefleiion.

XXIV. 58 44

88 53

1-6468

0-0195

XXV. 58 50

88 56

1-6533

0-0186

XXVI. ; 58 52 88 46

1-6555

0-0215

Means ' 58° 48' 40" 88° 51' 40" 1-6519

0-0199

VI. Quartz (a). Xatural surface. Plane of incidence perpendicular to optical cuds.

Table XXVIII.— Quartz {a). (October 13, 1855.)
Azimuth of Polarizer =88°. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'_e_180°.

<P-

a

T

Tan-(^).

33 37

39-43

87 16

6 6

+ 87 16

00

35 12

43 37

39-43

84 20

0 0

+ 84 20

00

19 23

53 37

39-90

67 45

3 9

+ 67 46

59-16

4 52

55 37

41-02 '

42 0

18 36

+ 41 50

6-09

1 48

56 7

43-40 '

32 0

46 27

+ 27 21

2-70

1 15

56 37

46-82

24 30

86 27

+ 44

2-20

0 55

57 7

50-53

35 0

129 52

-30 12

2-34

1 16

57 37

51-28

48 0

138 38

-49 0

2-65

2 13

58 37

52-10

63 15

148 14

—65 31

4-50

3 58

63 37

53-30

80 45

162 16

-81 10

20-52

12 6

73 37

53-50

86 30

164 36

-86 37

63-78

29 43

Principal Incidence =56° 40'.
Cii-cular Limit =89° 5'.

Coeff. of Refiaction= 1-5204.
CoefF. of Reflexion =0-0160,

Table XXIX.— Quartz {a). (October 15, 1855.)
Compensator =47-12 = 90°. Red Sunlight.

Polarizer.

Analyser.

a
6'

Tan-'(f).

89° 30
89 0
88 0
85 0

64° d
48 0
28 0
11 10

2-050
1-110
1-881
5-066

1 i

1 7

1 4
1 0

Mea

n = 1° 3' 0"

Principal Incidence =56° 40'.
Circular Limit =88° 51'.

CoefF. of Refraction =1-5204.
Coeff. of Reflexion =0-0200.

p2

100 EEY. S. HAUGHTOX OX THE EEFLEXIOX

Hence we obtain

Table XXX. — Constants of Quartz {a).

Xo.

Principal
lucideuce.

Circular
Limit.

Coefficient of
Refraction.

Coefficient of
Reflexion.

XXVIII.

5^ 40

89 5

1-5204

0-0160

XXIX. 1 56 40 88 51

1-5204

0-0200

Means 56 40 88 58

1-5204 i o-oi8o

VII. Quartz {h). yafnml surfacr. Plane of incidence contains optical axis.

Table XXXI.— Quartz {h). (October 16, 1S55.)
Azimuth of Polarizer =: 88°. Reel Sunlitiht.

Incidence.

Compensator.

Analyser.

f'_f_i8i.r.

f-

//

Tan-.g).

33 37

39-43

86 30

6 6

+ 86 30

00

29 43

43 37

39-43

84 30

0 0

+ 84 30

oo

19 56

53 37

39-90

66 0

5 30

+ 66 4

27-85

4 29

55 37

40-72

45 15

15 5

+ 45 16

5-57

2 1

56 7

41-82

36 30

27 57

+ 35 27

4-22

1 29

56 37

43-87

23 30

51 57

+ 16 44

3-15

0 52

57 7

48-14

27 30

101 54

— 8 12

1-99

1 2

57 37

60-97

39 0

135 0

-36 39

2-47

1 37

63 37

53-85

79 30

168 42

-82 34

7-17

10 40

73 37

54-27

86 50

173 37

-86 51

176-20

32 15

Principal Incidence =56° -57'.
Circular Limit ^89' 8'.

Coeff. of Refraction =1-5369.
Coeff. of Reflexion =0-0151.

Table XXXIL— Quartz {h). (October 10, 1855.)
Compensator =47-12 = 90°. Red Sunliijht.

Polarizer.

Analyser.

b'

T»-.(J).

89 30
89 0
88 0
85 0

44° 30
41 10
26 20
13 20

1-017

1-143
2-020
4-219

6 30
0 52

0 59

1 11

Mean

= 0° 53' o"

Principal Incidence =56' 52'.
Circular Limit =89° 34'.

CoefF. of Refraction =1-5320.
CocfT. of Reflexion =0-0070.

OF POLARIZED LIGHT I-ROM TOLISIIED SUEFACES.

101

From wliich wc obtain

Tablk XXXIII. — Constants of Quartz (b).

No.

Principal
Incidence.

Circular
Limit.

Coelliciciil of
Refraction.

CoolTicient of
Kcflcxion.

XXXL

5^ 57

89 8

1-5369

0-0151

XXXFL

56 52

89 34

1-5320

0-0076

Cleans

56° 54' 30"

89 21

i*S344

o-oio8

The following experiments were made on the metallic bodies.

VIII. SrKCULijr Metal.

Table XXXIV.— Speculum Metal. (July 29, 1854.)

Azimuth of Polarizer = 45°. lied Lamplight.

Incidence.

Compensator.

Analyser.

e'-e-180°.

P-

n

--'(D-

35 7

40-25

43 30

9 42

+ 42 27

10-06

42° 30

44 7

40-7.'?

42 45

15 12

+ 42 40

7-42

42 45

49 7

41-19

41 30

20 35

+ 41 16

5-58

41 30

51 7

41-46

40 50

23 45

+ 40 27

4-78

40 50

53 7

41-62

40 20

25 36

+ 39 50

4-45

40 20

55 7

41-89

40 3

28 46

+ 39 22

3-96

40 3

57 7

42-10

39 20

31 14

+ 38 24

3-05

39 20

59 7

42-63

39 6

37 26

+ 37 38

3-03

39 6

61 7

43-06

39 20

42 28

+ 37 24

2-64

39 20

64 7

43-68

38 50

49 42

+ 35 40

2-24

38 50

69 7

44-96

37 45

64 42

+ 29 25

1-69

37 45

74 7

46-90

37 45

87 24

+ 4 58

1-29

37 45

Pi-incipal Incidence =75° 27' (]). Circular Limit =52° 15' (?).

In this experiment the angle of incidence was not made at any time equal to tlie
principal incidence.

Table XXXV.— Speculum Metal. (August 30, 1855.)
Azimuth of Polaiizer :=50°. Eed Sunlight.

Incidence.

Compensator.

Analyser.

(.'-f-lSO''.

f-

/,'

--(•1)^

33 37

40-00

47 15

S 40

+ 47 16

16-48

42 63

43 37

40-95

45 40

17 47

+ 45 42

6-40

40 39

53 37

41-80

43 0

27 44

+ 42 45

4-13

38 3

63 37

43-34

40 15

4.5 44

+ 38 16

2-42

35 24

68 37

44-60

39 40

60 29

+ 34 .32

1-77

34 50

73 37

46-26

39 0

79 59

+ 19 38

1-32

34 12

74 37

46-40

39 32

81 33

+ 18 38

1-27

34 42

75 37

46-56

40 15

83 18

+ 17 26

1-23

35 24

76 37

47-03

39 20

87 30

+ 68

1-23

34 40

77 37

47-50

39 43

94 25

-11 13

1-22

34 53

78 37

48-00

39 45

lUO 16

-21 06

1-29

34 55

83 37

50-85

43 0

133 36

— 42 6

2-34

38 'i

88 37

53-50

49 45

164 37

— 49 55

7-65

44 45

Principal Incidence =75° 51'.
Circular Limit =55° 48'.

CoefF. of Refraction =3-9665.
CocfF. of Ecflexion =0-6796.

102

EEV. S. HAUGHTOIS ON THE EEFLEXION

Table XXXVI.— Speculum Metal. (August 28, 1855.)
Azimuth of Polarizer =:60°. Red Simli^lit.

Incideuce.

Compensator.

Analyser.

,■-(.-180°.

f.

I'

Tan-(5).

33 37

40-17

57 30

8 39

+ 57 38

14-20

42 11

43 37

40-76

55 15

15 33

+ 55 36

7-89

39 46

53 37

41-76

63 30

27 15

+ 54 30

4-31

37 58

63 37

43-40

52 10

46 27

+ 55 10

2-44

36 38

68 37

44-60

51 0

60 29

+ 56 40

1-79

35 29

73 37

46-13

49 30

78 23

+ 64 6

1-29

34 3

74 37

46-30

49 30

80 22

+ 66 43

1-26

34 3

75 37

46-55

48 45

83 18

+ 69 14

1-19

33 22

76 37

46-91

49 15

87 30

+ 81 52

M7

33 49

77 37

47-36

48 55

92 47

-80 17

1-16

33 31

78 37

48-25

51 0

103 11

-66 30

1-37

35 29

83 37

50-86

54 30

133 43

-58 14

2-53

38 59

88 37

54-00

60 0

170 28

-60 10

14-13

45 0

Principal Incidence =75° 57'.
Circular Limit =5G° 38'.

Coeff. of Eefraction =3-9959.
Coeff. of Reflexion =0-6585.

In this Table, the polarizer having been set at an angle exceeding the circular limit,
the axis major of the ellips(> passes through 90° at the i)rincipal incidence, and behaves
exactly as in the transparent bodies.

Table XXXVIL— Speculum Metal. (June 29, 1855.
Azimuth of Polarizer =80". Red Suidight.

Incidence.

Compensator.

1
AiL-dyser. 1

(.•_f_l80°.

f-

//

Tan-©.

! 33 37

40-14

80 6

8 18

+ 80 6

39-78

45 6

43 37

40-97

79 15

18 1

+ 79 44

17-57

42 53

53 37

41-81

77 30

27 50

+ 78 48

10-02

38 30

63 37

43-28

76 30

45 2

+ 80 6

6-06

36 18

68 37

44-44

76 0

58 36

+ 82 16

4-78

35 16

73 37

46-00

76 0

76 52

+ 86 33

4-13

35 16

74 37

46-32

75 47

80 36

+ 87 28

4-01

34 50

75 37

46-57

75 50

83 32

+ 88 16

3-99

34 56

76 37

46-94

76 0

87 52

+ 89 26

4-01

35 16

77 37

47-21

75 55

91 1

-89 44

3-99

35 6

78 37

48-15

76 30

102 1

— 86 58

4-27

36 18

81 37

49-38

77 30

116 24

-84 9

5-45

38 30

83 37

50-20

77 45

126 0

-82 30

5-89

39 5

85 37

51-30

79 20

128 53

— 81 48

8-24

43 7

88 37

54-19

80 0

172 41

-80 4

47-68

45 0

Pi-incipal Incidence =76° 7'
Circidiir T,imit =55"^ 10',

CoeflP. of Refraction =4-0458.
Coeff. of Reflexion =0-6959.

OF POLARIZED LIGHT FROM POLISHED SURFACES.

103

Table XXXVIII.— Speculum ^letal (fresh polished with rouge). (Sept. 11, 1855.)
Compensator =4712 = 90°. Red Sunlight.

Polarizer.

Analyser.

a

T

Tan-.(f).

80

75 40

3-7.32

34 37

70

G'3 45

1-942

35 15

60

49 45

1-181

34 18

50

38 0

1-279

33 15

40

29 30

1-767

33 57

30

21 45

2-506

34 39

20

14 10

3-961

34 43

10

7 30

7-596

36 45

Mean = 34° 41' 7"

Pi-incipal Incidence =78° 7'.
Circular Limit =55° 19'.

Coeff. of Refraction =4-7522.
Coeff. of Reflexion =0-6920.

This experiment shows tliat the fresh pohshing of the surface affected the coefficient
of refraction more than the coefficient of reflexion, on which the elHptic polarization
altogether depends.

The angle tan-'f y) i^ "ot constant, but attains a minimum at thc^ circular limit.

Additional direct experiments with speculum metal, such as setting the compensator
at 90°, making the incidence 76°, setting the analyser at 45°, and then determining
the azimuth of the polarizer, gave for the circular limit 54° 45'.

Combining all together, we find

Table XXXIX.-

— Constants of

Speculum Metal.

No.

Principal
Incidence.

Circular
Limit.

CoefEcient of
Refraction.

Coefficient of
Reflexion.

XXXV.

75 51

55 48

3-9665

0-6796

XXXVL

75 57

56 38

3-9959

0-6585

XXXVII.

76 7

55 10

4-0458

0-6959

xxxvin.

78 7

55 19

4-7522

0-6920

Direct Ex.

54 45

0-7067

Means

76 33

55° 32' 0"

4-1901

0-6865

104

EEY. 8. IIAUGHTOX OX THE EEFLEXIOX

IX. Silver.
I examined tlirce descriptions of silver. —

(a) Fine siher, rolled.

(b) Fine silver, cast.

(c) Standard silver, rolled.

Table XL. — Silver (a). (September 3, 1855.)
Compensator =47-12=:90°. Red Sunlight.

Pularuer.

-iiialyser.

o

F

— (t)-

30

79 30

5-395

43 34

70

69 0

2-605

43 29

60

56 40

1-520

41 17

50

46 15

1-044

41 14

40

36 10

1-368

41 3

30

27 15

1-941

41 44

20

18 0

3-077

41 45

10

9 40

5-870

44 1

Mean

= 42° 15' 52" 1

Principal Incidence ^72° 37'.
Circular Limit =18' 4G'.

Coeflf. of Refraction =3-1942.
Coeff. of Reflexion =0-8765.

Table XLI. — Fine Silver (<•/) (newly polislied). (September 7, 1855.)
Compensator =47-12 = 00^ Red Sunlight.

Polarizer.

Analyser.

a
T

x.„-.(5).

^

n /

0

80

79 40

5-484

44 2

70

68 15

2-506

42 23

Go

54 45

1-415

39 15

50

46 15

1-045

41 14

45

42 45

1-082

42 45 1

40

38 0

1-280

42 57

30

27 40

1-907

42 15

20

19 0

2-904

43 25

10

9 20

6-084

42 59

Mean:

= 43° 28' 20" {

Principal Incidence =: 71° 37'.
= 48' 13'.

Circular Limit

Coeff. of Refraction =3-0090.
Coeff. of Reflexion =08930.

IIa\-ing set tlie angle of incidence at 72" 37', the comix'usator at 47-12 = 90°, and the
analyser at 45°, I found, by trial, the polarizer or circular limit to be 48° 0'.

OF POLAEIZED LIGHT FEOII POLISHED SLTIFACES.

T.vULE XLII. — Silver (L). (September G, 1855.)
Compensator =47-12 = 90°. Red Sunlight.

105

Polarizer.

Analyser.

a
b'

Tan-.g).

80

79 35

5-439

43 48

70

69 5

2-616

43 36

60

58 40

1-642

43 29

50

47 50

1-104

42 49

45

43 45

1-083

42 45

40

37 30

1-303

42 27

30

28 10

1-867

42 51

20

18 40

2-960

42 52

10

9 50

5-769

44 31

iSIean

=43° 14' 13"

Principal Incidence =78° 7'.
Cir-cular Limit =47° 13'.

Coeff. of Refraction =4-7522.
CocfF. of Reflexion =0-9255.

Table XLIII.— Silver [c). (September 7, 1855.)
Compensator =47-12 = 90°. Red Sunlight.

Polarizer.

Analyser.

a
b'

x„-.(5.

80

79 30

5-395

43 34

70

68 30

2-538

42 44

60

57 30

1-570

42 11

50

47 15

1-082

42 14

45

42 30

1-091

42 30

40

37 0

1-327

41 55

30

28 0

1-881

42 39

20

19 25

2-837

44 5

10

9 50

5-769

44 30

Mean

=42° S5' 47"

Principal Incidence =78° 22'.
Circular Limit =47° 38'.

Coeff. of Refraction =4-8573.
Coeff. of Reflexion =0-9120.

By direct experiment, as before described, I found the circular limit to be 4G° 45'.
On the day preceding that on which the experiments were made on Silver (r), I
examined it before polishing, when evidently tarnished with sulphm-et, and found
Principal Incidence =67° 37'. Coeff. of Refraction =2-4282.

Circular Limit =52° 30'. Coeff of Reflexion =0-7673.

Combining the preceding results into one Table, we find,

MDCCCLXIII.

106

-REV. S. HATJQHTOISr ON THE REFLEXION
Table XLIV. — Constants of Silver.

SiLVEE («).

Principal
lucidence.

Circular
Limit.

Coefficient of
Refraction.

Coefficient of
Keflerion.

XL.

72 37

48 46

3-1942

0-8765

XLL

71 37

48 13

3-0090

0-8936

Direct exp.

48 0

0-9004

Means

72 7

48° 19' 40"

3-1016

0-8901

Silver (h).

XLII.

78 7

if 13

4-7522

0-9255

Silver (c).

XLIII.

78 23

47 38

4-8573

0-9120

Direct exp.

46 45

0-9407

Means

78 22

47° n' 30"

4*8573

0-9263

Note. — In all the experiments on silver, the miaimura value of tan~' (^)> corresponding to the circular limit,

is apparent, although, if the surfaue were mathematically smooth, it ought to ho constant, being a function of
the incidence only.

X. Gold (Standard).

Table XLV.— (September 20, 1855.)
Compensator = 47-12 = 90°. Red Sunlight.

Polarizer.

Analyser.

a

T

Tan-.(£).

80

79 45

5-530

44 17

70

68 45

2-571

43 6

60

58 15

1-616

43 0

50

47 0

1-072

42 38

45

42 30

1-091

42 30

40

37 45

1-291

42 42

30

28 0

1-881

42 39

20

19 10

2-876

43 41

10

9 40

5-870

44 1

Mean

= 43° 10' 26"

Principal Incidence =75° 37'.
Circular Limit =47° 47'.

The minimum value of tan"

■©

^Coeff. of Refraction =3-8994.
Coetf. of Reflexion =0-9073.

is here also evident.

OF POLAEIZED LIGHT FROM POLISHED SIJEFACES.

XI. Mercury (Distilled).

Table XLVI.— (November 1, 1860.)

Compensator = 47-12 = 90°. Red Lamplight.

107

Polarizer.

Analyser.

a
T

T„-,(i).

80

7§ 6

4-011

35 16

70

62 42

1-937

35 11

60

51 35

1-260

36 3

50

41 0

1-150

36 6

40

32 2

1-598

36 43

30

23 25

2-309

36 53

20

14 49

3-780

36 1

10

7 19

7-788

36 4

0

0 0

00

Mean = 36° z' 7"

Principal Incidence = 81° 4'.
Circular Limit =53° 4G'.

Coeff. of Refraction =6-3616.
Coeff. of Reflexion =0-7328.

By a direct experiment, I obtained, as before described,

arcular Limit =53° 52'. Coeff. of Reflexion =0-7301.

The value of tan"' ( j j appeai-s to be constant in mercury : can this be due to its being
a liquid ■?

XII. Platinum.

Table XLVIL— (September 21, 1855.)
Compensator = 47-12 = 90°. Red Sunlight.

Polai-izer.

Analyser.

a
T

Tan-. (5).

80

7^ 10

4-061

35 36

70

63 0

1-962

35 32

60

52 15

1-291

36 43

50

40 10

1-185

35 19

40

32 0

1-600

36 41

30

22 16

2-444

35 19

20

14 45

2-798

35 53

10

8 0

7-115

38 33

Mea

1 = 36° iz< 0"

Principal Incidence =76° 37'.
Circular Limit =54° 0'.

Coefi". of Refraction =4-2030.
Coefi". of Reflexion =0-7265.

<l2

lOS

EEV. S. HAUGHTON OX THE EEFLEXION

XIII. Pall.\dium.

Table XLYIII.— (September 21, 1855.)

Compensator =47-12 = 90°. Ked Sunlight.

Polarizer.

Analyser.

a

r

Xa„-.g).

80

75 15

3-798

33 49

70

G5 40

2-211

38 50

60

50 10

1-199

34 41

50

40 15

1-181

35 23

40

29 30

1-631

33 59

30

23 50

2-375

36 6

20

15 0

3-732

36 21

10

8 0

7-115

38 33

Mean

= 35° 57' 45"

Principal Incidence =:77° 37'
Circular Limit =51° 47'

CoefF. of Refraction =4-5546.
CoefF. of Reflexion =0-7058.

XIV. CorPER.

Table XLIX. — Copper. (October 6, 1857.)

Azimuth of Polarizer =46° 15'. Red Sunlight.

Incidence.

Compensator.

Analyser.

c'-c-im".

f-

b'

Xan-.g).

63 30

44-96

42 40

64 42

+ 39 35

1-589

41 25

68 30

46-19

42 30

79 0

+ 32 41

1-236

41 15

69 30

46-49

42 45

82 36

+ 29 17

1-164

41 30

70 30

46-77

42 25

85 52

+ 19 17

1-123

41 10

71 30

47-17

42 16

90 34

- 2 57

1-101

41 5

72 30

47-25

42 20

91 30

- 7 50

1-102

41 5

73 30

47-63

42 15

95 57

-23 33

1-152

41 4

74 30

47-81

42 32

98 4

-29 12

1-180

41 17

75 30

48-54

43 50

106 35

-37 34

1-174

42 35

76 30

48-76

43 1

109 9

-39 2

1-416

41 46

78 30

49-84

43 20

121 48

-41 51

1-804

42 5

83 30

51-74

45 24

144 2

-45 30

3-000

44 9

Principal Incidence := 71° 21'.
Circular Limit =48° 55'.

Coeff. of Refraction =2-9629.
Cocff. of Reflexion =0-8718.

OF POLAEIZED LIGHT FROM TOLISHED SURFACES.

109

Tablk L. — Copper. (October C, 1857.)
Azimuth of Polarizer =47° 45'. Red Sunlight.

Incidence.

Compensator.

Analyser.

c'-f-lSO".

f-

*■

Tan-.g).

63 30

44-93

45 10

64 24

+ 45 23

1-593

42 25

68 30

46-17

44 20

78 55

+ 41 34

1-218

41 35

69 30

46-38

44 25

81 18

+ 41 10

1-166

41 40

70 30

46-67

42 45

84 42

+ 24 47

1-129

40 1

71 30

47-15

43 50

90 20

— 7 58

1-043

41 6

72 30

47-68

43 20

96 32

-31 27

1-137

40 36

73 30

47-85

43 20

98 31

-34 16

1-174

40 36

74 30

48-23

43 45

102 58

-39 29

1-261

41 1

75 30

48-45

43 45

105 32

— 40 22

1-320

41 1

76 30

48-84

43 45

110 6

-41 23

1-435

41 1

78 30

49-71

44 45

120 17

-44 30

1-732

42 0

83 30

51-85

45 50

145 19

-46 1

3-177

43 5

Principal Incidence

Cuxular Limit

= 71° 27'.
= 49° 59'.

C'.xff. of Refraction =2-9800.
Coetf. of Reflexion =0-8390.

Table LI.— Copper. (October 6, 1857.)
Azimuth of Polarizer =55°. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'-e-lSOf.

p. 1

a

x..-.(i).

63 30

1 44-72

50 15

61 39

+ 55 40

1-734

40 7

68 30

45-93

50 15

76 3

+ 63 46

1-362

40 7

69 30

46-17

50 0

78 51

+ 66 11

1-301

39 51

70 30

46-68

50 20

84 49

+ 77 12

1-231

40 11

71 30

46-90

50 7

87 24

+ 82 57

1-204

39 58

72 30

47-24

50 26

91 22

-86 27

1-226

40 16

73 30

47-59

49 30

95 28

-74 29

1-203

39 21

74 30

47-77

49 28

! 97 34

-70 1

1-228

39 19

75 30

48-23

49 30

102 57

-62 38

1-320

39 21

76 30

48-67

49 47

108 6

-59 15

1-282

39 38

78 30

49-33

51 20

115 49

-58 39

1-683

41 11

83 30

51-37

53 45

139 42

-56 14

2-896

43 41

Principal Incidence =71° 6'.
Cii-cular Limit =50° 40'.

Coeff. of Refraction =2-9207.
Coeff. of Reflexion =0-8194.

110

EEV. S. HAUGHTON ON THE EEFLEXION

'JLABLE LII. — Copper. (September 21, 1855.)
Compensator = 47-12 = 90°. Bed Sunlight.

Polarizer.

An.alyser.

a
*•

Ta.-.g).

80

79 50

5-576

44 31

70

69 15

2-639

43 51

60

59 30

1-697

44 25

50

48 0

1-110

42 59

45

43 0

1-072

43 0

40

37 30

1-303

42 27

30

28 40

1-829

42 47

20

19 30

2-824

44 13

10

9 50

5-769

44 31

Meal

= 43° 38' 13"

Principal Incidence =73° 37'.
Cii-cular Limit =47° 0'.

Combining the preceding results, we obtain

Coeff. of Kefraction =3-4013.
Coeff. of Reflexion =0-9325.

Table LIII. — Constants of Copper.

No.

Principal
Incidence.

Circular
Limit.

Coefficient of
Kefraction.

Coefficient of
Beflexion.

XLIX.

71 2i

48 55

2-9629

0-8718

L.

71 27

49 59

2-9800

0-8396

LI.

71 6

50 40

2-9207

0-8194

LII.

73 37

47 0

3-4013

0-9325

Means

71° 52' 45"

49° 8' 30"

3-0662

0-8656

OF POLAEIZED LIGHT FROM POLISHED SUEFACES.

Ill

XV. Zinc.

Table LIV.— Zinc (April 22, 1858.)
Azimuth of Polarizer =53°. Red Sunligrht.

Incidence.

Compensator.

Analyser.

1 «'_e_180°.

1

f-

a
T

'-- ©•

63 30

43-71

47 6

50 4

48 7

2-143

38 57

68 30

44-58

46 0

60 15

47 1

1-723

37 58

75 30

46-93

45 45

87 45

61 51

1-048

37 43

76 30

46-98

45 0

88 20

+ 45 0

1-000

37 0

77 30

47-19

45 30

90 47

-70 56

1-022

37 29

78 30

47-84

45 45

98 23

-50 5

1-161

37 43

79 30

48-49

45 30

106 0

-46 49

1-327

37 29

80 30

48'65

46 0

107 52

-48 15

1-375

37 68

81 30

49-20

46 0

, 114 12

-47 26

1*548

37 68

83 30

50-47

46 30

129 10

-47 22

2-114

38 27

88 30

54-15

52 0

[ 172 13

-52 4

13-078

43 58

Principal Incidence =:77° 11',
Cii-culai- Limit =53° 0'.

Coeff. of Refraction =4-3956.
Coeff. of Reflexion =0-7535.

T.VBLK LV.— Zinc. (May 7, 1858.)
Azimuth of Polarizer =57°. Red Sunlight.

Incidence.

Compensator.

Analyser.

e'_e_180».

«>■

a
T

40 46

63 30

43-37

53 6

46 6

+ 5§ 14

2-485

68 30

44-43

50 30

58 30

+ 55 12

1-849

38 14

73 30

45-94

48 30

76 9

+ 58 35

1-315

36 17

75 30

46-49

49 30

82 36

+ 70 26

1-227

37 16

76 30

46-95

49 0

87 58

+ 82 55

1-169

36 46

77 30

47-49

50 0

94 18

-78 29

1-224

37 44

78 30

47-74

50 0

97 13

-72 16

1-244

37 44

79 30

48-34

49 30

104 14

— 61 5

1-354

37 16

80 30

48-79

50 30

109 30

-60 6

1-491

38 14

81 30

49-32

50 30

115 42

-57 4

1-658

38 14

83 30

50-17

52 0

125 39

-56 35

2-048

39 44

88 30

64-06

54 0

171 10

-54 5

14-983

41 48

Principal Incidence =76° 49'.
Circular Limit =53° 29',

Coeff. of Refraction =4-2691.
Coeflf. of Reflexion =0-7404.

112

EEY. S. IIAUGHTOX OX TlIE EEFLEXIOX

Table LVI.— Ziuc. (Septembci- 20, 1855.)
Compensator =47-12 = 90°. Eed Sunlight.

Polarizer.

Aualvser.

A"

Ta„-.g).

80

75 6

3-732

33 21

70

62 0

1-881

34 23

60

49 30

1-171

34 3

50

39 45

1-202

34 54

40

29 15

1-785

33 43

30

22 30

2-414

35 39

20

15 0

3-732

36 21

10

7 30

7-59G

36 45

Mean

= 3+° 53' 37"

Principal Incidence =78° 7'.
Circular Limit =55° 23'.

Coeff. of Refraction =4-7522.
CocfF. of Reflexion =0-6903.

Combining the preceding results, we obtain the following Table for zinc.
Table LVII. — Constants of Zinc.

No.

Principal
lucidence.

Circular
Limit.

Coefficient of
Kefraetion.

CoefBeieiit of
Reflexion.

LIV.

77 11

53 6

4-3956

0-7535

LV.

76 49

53 29

4-2691 0-7404

LVI.

78 7

55 23

4-7522

0-6903

Means

77° 22' 20"

53° 57' 20"

4-4723

0-7281

XVI. Lead (polished).
Table LVIII. (September 20, 1855.)

Compensator =47-12

= 90'. Red Sunlight.

Polarizer.

Analyser.

T

T„-,g)^

80

64 6

2-050

19 52

70

40 30

1-171

17 16

60

29 45

1-750

18 16

50

22 30

2-414

19 10

40

15 0

3-732

17 43

30

10 0

5-671

16 59

20

7 15

7-861

19 16

10

2 30

22-903

13 54

Mean=i7° 48' 15"

Principal Incidence =09° 37'
Circular Limit =71° 55',

Coeff". of Refraction =2-0913.
Coeff". of Reflexion =0-3205.

OF POLAEIZED LIGHT FEOM POLISHED SURFACES.

XVII. Bisiiurn.

Table LIX. (September 25, 1855.)
Compensator =47-r2 = 90. Ecd Sunlight.

113

Polarizer.

Analyser.

a
V

T.,.,(I)^

80

7^ 15

4-08G

35 43'

70

60 45

1-785

33 1

60

50 25

1-209

34 56

50

39 30

1-213

34 40

40

30 30

1-697

35 4

30

21 0

2-605

33 37

20

14 25

3-890

35 14

10

6 30

8-777

32 52

Mean

= 34° 23' 22"

Principal Incidence =73° 37'.

Cii-cular Limit =55° 2'.

Coeff. of Refraction =3-4013.
Coeff. of Reflexion =0-6993.

XVIII. Ti.v.

Table LX. (September 25, 1855.)
Compensator =47-12 = 90°. Red Sunlight.

Polarizer.

Analvser.

b

^-(l)-

80

76 30'

4-165

3S 18

70

64 10

2-065

36 56

60

52 35

1-307

37 2

50

40 30

1-171

35 38

40

32 0

1-600

36 40

30

22 30

2-414

35 40

20

15 20

3-647

36 59

10

8 10

6-968

39 8

Mean

= 36^ 47' 37"

Principal Incidence =75° 7'
Circular Limit =53° 43'

Coeff. of Refraction =3-7027.
Coeff. of Reflexion =0-7341.

MDCCCLXIII.

114

KEY. S. HAUGHTOX OX THE EEFLEXION

XIX. Ikon.

Table LXI.— Hard Steel. (September 29, 1855.)
Compensator =47-12 = 90°. lied Sunlight.

Polarizer.

Au;iljser.

h'

Tan-.g),

80

72 15

3-124

28° 51

70

55 30

l-4o5

27 54

60

42 35

1-088

27 57

50

33 0

]-.'i40

28 36

40

24 35

2-186

28 36

30

17 30

3-171

28 39

20

11 50

4-773

29 57

10

6 15

9-131

31 51

Mean = 29° 2' 37"

Principal Incidence =78^ 7'.
Circular Limit =G1° 52'.

CoefF. of Refraction =4-7522.
CoefF. of Reflexion =0-5347.

Table LXII.— Soft Steel. (September 29, 1855.)
Compensator =47-12 = 90"'. Red Sunlight.

Polarizer.

Aiialy.ser.

a
T

Tan-.(^.).

80

70 45

2-863

2(5 48

70

55 50

1-473

28 12

60

41 45

1-120

27 16

50

31 45

1-616

27 27

40

23 30

2-300

27 24

30

17 50

3-108

29 7

20

11 20

4-989

28 50

10

6 0

9-514

30 48

Meai

1 = 28^ 14' 0"

Principal Incidence =77° 7'.
Circular Limit =03° 13'.

CoeflP. of Refraction =4-3721.
Coeff. of Reflexion =0-5048.

OF POLAEIZED LIGHT FEOM POLISHED SUEFACES.

115

Swedish Iron {cut perpendiada)' to the (jrain).

T.\BEE LXin. (September 29, 1855.)
Compensator = 47 -12 = 90°. Reel Sunlight.

Polarizer.

Analyser.

a
T

Tan-.g).

80

71 35

3-003

27 54

70

55 10

1-437

27 37

60

40 45

1-160

26 27

50

32 0

1-600

27 40

40

22 45

2-385

26 33

30

17 0

3-271

27 54

20

11 0

5-144

28 6

10

5 30

10-385

28 38

Mean=27° 36' 7"

Principal Incidence =76° 7'.
arculai- Limit =62° 57'.

CoefF. of Refraction =4-0458.
CoefF. of Reflexion =0-5106.

Sioedish Iron [cut parallel to the grain).
Table LXIV. (September 29, 1855.)

Polarizer.

Analyser.

a
T

T..-.g).

80

71 45

3-032

28 8

70

55 20

1-446

27 46

60

41 40

1-124

27 12

50

32 0

1-600

27 40

40

24 0

2-246

27 57

30

17 30

3-171

28 39

20

11 0

5-144

28 6

10

5 35

10-229

29 0

Mea

n=28° 3' 30"

Principal Incidence =76'' 7',
Circular Limit =62° 26',

CoeflF. of Refraction =4-0458.
Coeff. of Reflexion =0-5220.

Combining the preceding results, we flnd

Table LXV. — Constants of Steel and Iron.

No.

Principal
Incidence.

Circular
Limit.

Coefficient of
Refractiou.

Coefficient of
Reflexion.

LXI. Hard steel.

78 7

61 52

4-7522

0-5347

LXIL Soft steel.

77 7

63 13

4-3721

0-5048

LXIII. Iron (a).

76 7

62 57

4-0458

0-5106

LXIV. Iron (h).

76 7

62 26

4'0458

0'5220

k2

116

EEY. S. HAUCtHTOX OX THE EEFLEXIOX

XX. Alumixium.

Table JJ^VI. (May 10, 1856.)
Compensator =4712 = 00°. Red Sunlight.

Polarizer.

Analrser.

a

T

x.,.-,(^).

80

75 45

3-937

34 46

70

60 30

1-767

32 45

60

48 0

1-110

32 40

50

37 30

1-303

33 47

40

28 15

1-861

32 38

30

20 30

2-674

32 56

20

13 45

4-086

33 55

10

7 10

7-953

35 30

-Mean

= 33" 29' 37"

Principal Incidence :=7T° 7'.
C'ii-cular Limit =57° 9'.

Coeff. of Refraction =4-3721.
CoefF. of Reflexion =0-6457.

By a direct experiment I found tlie circular limit to be 57° 15'.

XXI. Alloys of Coiter and Zinc.

The following experiments were made on fourteen alloys of copper and zinc prepared
by Mr. Robert Mallet, in atomic proportions, as follow : —

10Cu+ Zn
9Cu+ Zn
8Cu+ Zn
7Cu+ Zn
0Cu+ Zn
5Cu+ Zn
4Cu+ Zn
3Cu+ Zn
2Cu+ Zn

Cu+ Zn

Cu+2Zn

C'u+3Zn

Cu+4Zn

Cu+5Zn

The chemical and physical properties of these alloys are fully described by Mr. Mallet
in his "Report on the Action of Air and "Water upon Iron" to the British Association
for the Advancement of Science for the year 1840, p. 306.

In all the experiments red sunlight was used, and the compensator was placed at
47-12 = 00°.

No.

1

No.

2

No.

3

No.

4

No.

5

No.

6

No.

7

No.

8

No.

0

No.

10

No.

11

No.

12

No.

13

No.

14

OF POLAEIZED LICaiT FEO^f POLISHED SURFACES. 117

Table LXVII.— Alloys of Copper ami Zinc, Xo. 1. (September IG, I80G.)

Polarizer.

Analvser.

/,'

—'(?)■

80

79 6

5-144

42 13

70

G6 45

3-237

40 16

GO

5G 10

1-492

40 45

JO

45 30 '

1-017

40 30

40

35 30 !

1-402

40 22

30

27 0

1-962

41 26

20

18 15

3-032

42 11

10

9 25

6-029

43 15

3Iean

= 41° 22' 15"

Principal Incidence = 72° 5'.
Circular Limit =iO° ?>2'.

Coeff of Ecfraction =3-0930.
CoefT. of Eeflexion =0-8.531.

Table LXVIII.— Alloys of Copper and Zinc, Xo. 2. (September 16, 18-50.)

Principal Incidence =72° 15',
Circular Limit =49° 32'.

Polarizer.

AualrscT.

6'

'-'&■

80

79 40

5-484

44 3

70
60

67 35
58 0

2-424
1-GOO

41 25

42 44

50

45 35

1-020

40 35

40

35 25

1-406

40 17

30
20

27 30
17 40

1-921
3-140

42 2
41 11

10

9 30

5-976

43 30

Mean

= 41° 58' 22"

Coeff. of Eefraction =3-1240.
CoefF. of Eeflexion =0-8531.

118 EEV. S. HAUGHTON ON TIIE EEFLEXIOX

Table LXIX. — Alloys of Copper and Zinc, Xo. 3. (September 18, 1856.)

Polarizer.

.inalyser.

a
7j'

Tan-.g).

^

0 ,

0 /

80

79 10

5-225

44 20

70

67 20

2-394

41 4

60

54 0

1-376

38 28

50

46 0

1-035

40 59

40

34 50

1-437

39 40

30

27 15

1-941

41 44

20

17 35

3-155

41 2

10

9 15

6-140

42 43

Mean=4i° z' 30"

Principal Incidence =73^ 10',
Circular Limit =49^ C.

Coeff. of Eefraction =3-3052.
Coeff. of Reflexion =0-8662.

Table LXX.— Alloys of Copper and Zinc, No. 4. (September 18, 1856.)

Polarizer.

Analyser.

a

Xa„-.g).

80

78 30

4-915

40 55

70

67 45

2-444

41 40

60

57 0

1-540

41 38

50

46 0

1-035

40 59

40

35 30

1-402

40 22

30

27 0

1-962

41 26

20

18 0

3-077

41 45

10

8 50

6-435

41 23

Mean

= 41° 16' 0"

Principal Incidence ^73° 8',
Circular Limit =49° 3'

CoefF. of Refraction =3-2983.
Coeff. of Reflexion =0-8677.

Table LXXL— Alloys of Copper and Zinc, No. 5. (September 18, 1856.)

Polarizer.

Aualyscr.

//

Xa„-g).

80

79 16

5-225

42 40

70

67 0

2-356

40 37

60

5G 15

1-496

40 50

50

45 55

1-032

40 55

40

35 52

1-383

40 45

30

26 50

1-977

41 13

20

18 15

3-0.32

42 n

10

9 45

5-819

44 16

Mean

= 41° 40' 52"

Principal Incidence =74° 5'.
Circular Limit =49° 5'.

Coeff. of Refraction =3-5066.
Coeff. of Reflexion =0-8667.

OF POLAEIZED LIGHT FKOM POLISHED SURFACES.

119

Table LXXIL— Alloys of Copper and Zinc, No. 6. (September 19, 1856.)

Polarizer.

Analyser.

a

Xan-.(f).

80

79 25

5-352

4.'i 20

70

68 30

2-538

42 44

60

57 55

1-595

42 .39

50

47 40

1-098

42 39

40

36 30

1-351

41 24

30

28 15

1-861

42 57

20

18 12

3-041

42 5

10

9 45

5-819

44 16

Mean

= 42° 4S' 30"

Principal Incidence = 74°
Circular Limit =47°

37',

Coeff. of Refraction =3-5183.
CoefF. of Reflexion =0-9126.

Table LXXIII.— Alloys of Copper and Zinc, No. 7. (September 19, 1856.)

Polarizer.

Analyser.

a

Tan-©.

80

79 10

5-225

42 39

70

67 30

2-414

41 18

60

57 0

1-540

41 39

50

45 40

1-023

40 .39

40

35 0

1-428

39 51

30

26 10

2-035

40 24

20

17 45

3-124

41 20

10

9 25

6-029

43 15

Meai

= 4.° 48' 7"

Principal Incidence =73° 16'.
Cu-cular Limit =49° 23'.

Coeff. of Refraction =3-3261.
Coeff. of Reflexion =0-8576.

Table LXXIV.— Alloys of Copper and Zinc, No. 8. (July 13, 1857.)

Polarizer.

Analyser.

a

*•

Tan-.©.

80

77 30

4-511

38 30

70

65 10

2-161

38 11

60

55 40

1-464

40 13

50

43 50

1-041

38 51

40

33 5

1-535

37 49

30

24 40

2-177

38 30

20

15 40

3-565

37 37

10

8 20

6-827

39 43

Mean

= 38° 40' 30"

Principal Incidence =73° 12'.
Circular Limit =50° 53'.

Coeff. of Refraction =3-3121.
Coeff. of Reflexion =0-8132.

120

EEY. S. lIArGHTOX OX TIIE EEFLEXIOX

Table LXXV.— Alloys of Copper and Zinc, Xo. 9. (October 5, 1857.)

IVlarizor.

^ViialvstT.

--(?)•

80

79 6

5-144

42 13

70

67 30

2-414

41 18

60

54 55

1-424

39 25

50

46 20

1-047

41 19

40

35 35

1-397

40 27

30

26 0

2-050

40 11

20

17 0

3-271

40 2

10

8 40

6-560

40 50

Mean

= 4°° 43' 7"

Principal Incidence =72' 18'.
Circular Limit =48'' 4G'.

CoefF. of Refraction =3'1334.
Coeff. of Reflexion =0-8764.

Table LXXVL— Alloys of Copper and Zinc, No. 10. (October 5, 1857.)

Tolarizcr.

Analyser.

b'

T.,.-'{?).

80

79 6

5-144

42 13

70

65 0

2-144

37 58

60

54 20

1-.393

38 49

50

43 45

1-044

38 46

40

34 35

1-450

39 35

30

25 15

2-120

39 15

20

16 30

3-3/5

39 9

10

8 45

C-497

41 7

Mean

= 39° 36' 30"

Principal Incidence =72' 15'.
Circular Limit =51' 11'.

CoefF. of Refraction =3-1240.
Coeff of Reflexion =0-8045.

Table LXXVIL— All.iys of Copper and Zinc, No. 11. (October 5, 1857.)

Polarizer.

Analyser.

/,'

-.-'6)-

80

78 30

4-915

0 .

40 55

70

65 30

2-194

38 37

CO

54 0

1-376

38 28

50

43 0

1-072

38 3

40

33 30

1-511

38 16

30

24 30

2-194

38 17

20

16 15

.3-431

38 41

10

8 35

G-625

40 34

Mean :

= 38° 58' 52"

Principal Incidence =72° 15'.
Circular Limit =51° 49'.

Coeff. of Refraction = 3-1240.
Coeff. of Reflexion =0-7864.

OF POLAHIZED LIGHT FRO.M P(^LIS1IED .SURFACES.

121

Table LXXVIII. — Alloys of Copper and Zinc, No. 12. (October 5, 1857.)

Polariier.

Analyser.

a
T

Tan-©.

80

74 40

3-G47

32 45

70

58 45

I -048

;!0 57

60

48 0

1-110

32 40

50

37 45

1-291

33 1

40

28 10

1-867

31 57

30

20 0

2-747

32 14

20

13 20

4-219

.■53 4

10

6 30

8-776

32 5G

Mean

= 32° =6' 4S"

Principal Incidence =76° 7'.
Circular Limit =-57° 5'.

CoefF. of Kefraction =4-0458.
CoefF. of Keflexion =0-6473.

Taclk LXXIX. — Alloys of Copper and Zinc, No. 13. (October 6, 1857.)

Polarizer.

Analyser.

a

T..-(i).

80

76 15

4-086

35 46

70

60 50

1-792

33 7

60

49 45

1-181

34 18

50

39 10

1-227

34 21

40

29 15

1-785

33 43

30

20 40

2-651

.33 10

20

13 30

4-165

33 25

10

6 40

8-555

33 32

Mean

= 33° 55' >5"

Principal Incidence =73° 52'.
Circular Limit =55° 31'.

Coeff. of Eefraction =3-4570.
CoefF. of Reflexion =0-6868.

Table LXXX.— Alloys of Copper and Zinc, No. 14. (October 6, 1857.)

Polarizer.

Analyser.

a
V

Tan-©.

80

75 45

3-937

34 46

70

61 45

1-861

34 7

60

48 40

1-136

33 17

50

39 0

1-235

34 12

40

28 40

1-829

33 5

30

20 50

2-628

33 23

20

13 55

4-036

34 15

10

7 10

7-953

35 30

Mean

= 34° 4' 2 2"

Principal Incidence ^76° 0'.
Cii-cular Limit =56° 12'.

MDCCCLXIII.

Coeff. of Refraction =40108.
Coeff. of Reflexion =0-6694.

122

EEV. S. HAUGHTOX OX THE EEFLEXIOX

The alloys from 1 to 11 are all yellowish, and from 12 to 14 are whitish.

The following Table shows that the Coefficients of Eefraction from 1 to 11 increase
gradually, reaching a maximum at No. G (5Cu+Zn), and then diminish to No. 11, in
passing from which to No. 12 the coefficient suddenly increases. The Coefficient of
Reflexion follows an order somewhat similar, but suddenly decreases in passing from
11 to 12, which is the limit at which the zinc begins to preponderate over the copper,
in producing the optical properties of the alloy.

In Plate VIII. fig. B, I have tabulated the coefficients of refraction and reflexion of
the alloys of copper and zinc, showing the progression of these constants, as just
described.

Tabl?: I_XXXI. — Optical Constants of all the Substances examined.

Principal
Incidence.

Circular
Limit.

I.

II.

III.

IV.

V.

VI.

VII.

VIII.
IX.

XI.

XII.

XIII.

XIV.

XV.

XVI.

XVII.

XVIII.

XIX.

XX.

XXI.

(A.) Transparent.
Munich (ilass («)...
Munich Glasf* (6) ...

Paris Glass

Fluor-Spar

Glass of .\utiuiony

Quartz (a)

Quartz(i)

(B.) Metal.i.

Speculum

Silver (a)

Silver (i)

Silver (c)

Gold

Mercury

Platinum

Palladium

Cop[)er

Zinc

Lead

Bismuth

Tin

55 0 .37
54 40 48

56 S 30
54 46 0
58 48 40
56 40 0
56 54 30

76 33 0

Iron

Steel

.'Muminium

Alloys ol Copper and Zinc:

No

No.

No.

No.

No.

No.

No.

No. 8

No. 9

No. 10

No. 1 1

No. 12
No. 13

/« / u

78 7 0

78 22 0

75 37 0
81 4 0

76 37 0

77 37 0
1 52 45
7 22 20

69 37 0

3 37 0

5 7 0

6 7 0

7 37 0
7 7 0

i.' o 0

2 15 0

3 10 0

3 8 0

4 5
4 8
3 16
3 12
2 18
2 15
2 15

6 7
3 52

85 32 30
89 53 24
89 23 0
89 41
88 51

88 58

89 21

30

40

0

0

55 32 0
48 19 40
47 13 0
47 11 30

47 47

53 49

54 0

54 47
49 8
53
71 55

55 2
53 43
62 41 30
62 32 30
57 9 0

0
0
0
0
30
20
0
0
0

49 32 0

49 32 0

49 6 0

49 3 0

49 5 0

47 37 0

49 23 0

50 53 0

48 46 0

51 II 0
51 49 0

No. 14 } 76 0 0

57 5

55 31

56 12

I Coefficient of
Eefraction.

Coetficient of j Kefractive
Reflexion. Index.

1-4287
1-4113
1-4905
1-4158
1-6519
1-5204
1-5344

4-1901
3-1016
4-7522
4-8573
3-8994
6-3616
4-2030
4-5546
3-0G62
4-4723
2-6913
3-4013
3-7627
4-0458
4-5621
4-3721

3-0930
3-1240
3-3052
3-2983
3-5066
3-5183
3-.S261
3-3121
3-1334
3-1240
3-1240

4-0458
.3-4570
4-0108

0-0780
0-0019
0.0107
0-0053
0-0199
0-0180
0-0108

0-6865
0-8901
0-9255
0-9263
0-9073
0-7315
0-7265
0-7058
0-8656
0-7281
0-3265
0-6993
0-7341
0-5163
0-5197
0-6457

0-8531
0-8531
0-8662
0-8677
0-8667
0-9126
0-8576
0-8132
0-8764
0-8045
0-7864

0-6473
0-6868
0-6694

1-6227
1-5244

OF POLARIZED LIGHT FROM POLI«ILED SURFACES.

123

111 the preceding Table there are twelve pure metals ; if wc arrange these in two
Tables, according to the magnitude of the Coefficients of Refraction and Reflexion, we
obtain the following.

T.\BLE LXXXII. — Coefficient of Refi-action of pure Metals.

Metal.

CoefEciont of Refraction.

I. Mercury

6-3616
4-8047
4-5546
4-4723
4-3721
4-3039
4-2030
3-8994
3-7C27
3-4013
3-0662
2-6913

II. Silver

IIL Palladium

IV. Zinc

V. Aluniiniuui

VI. Iron

VII. Platinum

VIII. Gold

I.\. Tin

XI. Copper

XIL Lead

Table LXXXIII. — Coefficient of Reflexion of pure Metals.

Metal.

CoefBcient of Reflexion.

I. Silver

0-9259
0-9073
0-8656

IL Gold

IV. Tin

0-7341

0-7315
0-7281
0-7265
0-7058
0-6993
0-6457
0-5180
0-3265

VI. Zinc

VII. Platinum

VIII. Palladium

XII. Lead

The brilliancy of a metallic surface depends on the coefficient of refraction, and there
is, doubtless, some sensible quality, not yet clearly defined, which corresponds to the
coefficient of reflexion, which indicates the power of the surface to form elliptically
polarized light from incident plane-polarized light. This quality might be pronsionally
named lustre.

It is very remarkable that gold, silver, and copper, which from time immemorial
have pleased the eye of man, and been used as coins, should head the list of bodies
possessing a high coefficient of reflexion. Mercury, which has so brilliant a surface,
and therefore heads the list in Table LXXXII., occupies a comparatively low place in
Table LXXXIII., probably owing to its being a liquid, and its surface, therefore, in a
less favourable condition than that of a solid for imparting elliptic polarization to an
incident beam.

M. Jamin has examined optically several of the substances mentioned in the preceding

S2

124

EEV. S. HAUGHTOX ON THE EEFLEXIOX

Tables — the metallic bodies by the methods of equal intensities and multiple reflexions,
and the transparent bodies by the method employed in this paper, and originally used
by him.

I have deduced from his (niginal observations, the optical constants of the substances
comuKni to him and myself, and have recorded them for the purpose of comparison, in
the two following Tables. LXXXIV. and LXXXV.*

Table LXXX1^^ — Optical Constants of Metals, deduced from Jamin's experiments.

Substance.

rrincipal
Incidence.

Circular
Limit.

Coefficient of
Refraction.

C'oeflicient of
Eeflciion.

! Steel (1)

7^ 6

59 6

4-0108

0-5985

1 Steel (2)

77 4

61 27

4-3546

0-5441

I. Means

76° 32' 0"

60° 16' 30''

4-1827

0-5713

Silver (3)

71 40

54 0

3-0178

0-7265

Silver (4)

75 0

47 1

3-7320

0-9320

II. Means

73' 20' 0"

50' 30' 30"

3'3747

0*8292

Zinc (5)

77 0

4-3314

Zinc (5)

79 13

5-2505

Zinc (6)

75 11

60 57

3-7804

0-5554

III. Means

77' 8' 0"

60' 57' 0"

4"+5+'

0-555 +

Copper (7)

70 9

50 36

2-7700

0-8214

CopjifP (h)

71.1

53 41

2-9629

0-7350

I V. Means

70" 45' 0"

52" 8' 30"

2-8664

0-7782

Speculum metal (9)

75 50

56 45

3-9616

0-6556

Speculum metal (10)

56 40

0-6577

Speculum metal (11)

76 14

53 33

4-0815

0-7386

V. Means

76° z' 0"

55° 39' zo"

4-0215

0-6839

VI. Brass (12)

71 31

52 57

2-99 16

0-7549

Tlie.se Tables were added during the printing of the paper.

OF POLARIZED LIOIIT VROM POLISHED SURFACES.

125

Tauuc LXXXV. — Optical Constants of Transparent Bex

ies. from Ja.\ii.\

's experiments.

Substance.

Principal
Incidence.

Circular

Limit.

Coerticient of
HelVaclion.

CoelBcienl of
RcQexion.

I. Glass of Antimony (13)

63 34

88 20 j

2-0115

0-0290

H. Quartz (13)

oG 50

89 25

1-5301

0-0102

in. Fluor-Spar (13)

55 15

s!) :;i

1-4415

0-0084

(1) Anil, de Chim. et do Phys. (ser. 3) vol. xix. p. 304.

From the two Tables in tlii.s pa2;e, I find at 75° incidence, I=0-94G, and J = (>-r,C>(i. from which it follows that

■■(i)-

=30^ 54'.

(2) Ann. dc Chim. et de Phy.s. (ser. .3) vol. xxii. p. 310 (mean red).
The azimuths given in thi.s and the foUowiiig page are arcs such that

tan (azimuth) =1,--

k= tan-

From this consideration the coefficient of rctlexion is deduced.

(3) Ann. de Chim. et de Phys. (sdr. 3) vol. xix. p. 315.

(4) Ann. de Chim. et de Phys. (sdr. 3) vol. xxii. p. 310 (mean red).

(5) Ann. dc Chim. et de Phys. (ser. 3) vol. xis. p. 320.

(6) Ann. dc Chim. et de Phys. (ser. 3) vol. xxii. p. 31(5 (mean red).

(7) Ann. de Chim. et de Phys. (ser. 3) vol. xix. p. 337. I have calculated the value of the circular limit
and coefficient of refraction from the experiment recorded as made with Him reflexions.

(8) Ann. de Chim. et de Phys. (ser. 3) vol. xxii. p. 31 7 (red light).

(9) Ann. de Chim. et de Phys. (.ser. 3) vol. xix. pp. 305, 30(5. The ratio of J to 1 at the principal incidence
is found to be, from the Tables of these two pages, as 023 to 950, from which the c ircular limit is deduced.

(10) Ann. de Chim. et de Phys. (sdr. 3) vol. xix. p. 330.

(11) ^Vnn. dc Chim. et de Phys. (sdr. 3) vol. xxii. p. 310 (red light).

(12) Ann. de Chim. et de Phys. (.sdr. 3) vol. xxii. p. 317 (red light).

(13) Ann. de Chim. et de Phys. (ser. 3) vol. xxix. p. 303.

[ 127 ]

VI. On the Exact Form of Waves near the Surface of Beep Wafer.
Jill William John Macquorn Rankixe, C.E.. LJj.D., F.R.SS. L. & E.

lloceived September 21 , — Head November '11, ] S(J2.

(1.) Thk investigations of the Astronomer Royal and of some other mathematicians on
straight-crested parallel waves in a liquid, are based on the supposition that the dis-
placements of the particles of the liquid are small compared with the length of a wave.
Hence it has been very generally inferred that the results of those investigations are
approximate only, when applied to waves in wliich tlie displacements, as compared with
the length of a wave, are considerable.

(2.) In the present paper I propose to prove tliat one of those results (viz.. that in
very deep water the particles move mth a uniform angular velocity in vertical circles
whose radii diminish in geometrical progression with increased depth, and consequently
that surfaces of equal pressure, including the upper surface, are trochoidal) is exact for
all displacements, how great soever.

(3.) I believe the trochoidal form of waves to have been tirst explicitly stated by
Mr. Scott Russell ; but no demonstration of its exactly fultilling the conditions of the
question has yet been published, so far as I know.

(4.) In 'A Manual of Applied Mechanics' (first published in 1858), page 579. 1
stated that the theory of rolling waves might be deduced from that of the positions
assumed by the surface of a mass of water revolving in a vertical plane about a hori-
zontal axis ; as the theoi-y of such waves, however, was foreign to the subject of the
book, I did not then publish the investigation (m which that statement was founded.

(5.) Having communicated some of the leading principles of that investigation to
Mr. WiLLL\M Froude in April 1862, I learned from him that he had already arrived
independently at similar results by a similar process, although he had not published
them.

(6.) Proposition I. — In a mass of gravitating liquid whose particles revolve uniformly
in vertical circles., a wavy surface of trochoidal profile fulfils the conditions of uniformity
of pressicre, — such trochoidal profile being generated by rolling, on the tmderside of a
straight line, a circle tvhose radius is equal to the height of a conical pendulum that
revolves in the same period ivith the particles of liquid.

In fig. 1 (p. 128) let B be a particle of liquid revolving uniformly in a vertical circle oi'
the radius C B, in the direction indicated by the arrow N ; and let it make n revolutions
in a second. Then the centrifugal force of B (taking its mass as unity) will be

4^n' . CB.

12S

DK. W. J. .MAC\)UOE>: BAXKIXE OX THE EXACT FOKM

DrawCA vertically ii]nvards,and of such a Icniith that centrifugal force : granty : : C'B: AC;
that is to sav. make

9_

which is the well-known expression for the height of a revolving pendulum making n
revolutions in a second.

Then AC being in the direction of and proportional to granty, and CB in tlie direc-
tion of and proportional to centrifugal force, AB mil be in the direction of and pro-
portional to the resultant of gravity and centrifugal force ; and the surface of equal
pressure tra\ersing B will be normal to AB.

The profile of such a surface is ob\iously a trochoid LBM, traced by the point B,
which is carried by a circle of the radius CA rolling along the underside of the hori-
zontal straight line HAK. Q.E.D.

(7.) CoroIIanes. — The length of the wave whose period is one-?ith of a second is equal
to the circumference of the rolling circle ; that is to say (denoting that length by I.),

/.=2^.CA=/-,:

the jn'riod of a wave of a gi\en length X is given in seconds, or fractions of a second, by
the equation

n V (J
and the velocity of propagation of such a wave is

results agreeing with those of tlu; known theory.

OF WAA-ES NEAR THE SURFACE OF DEEP WATER.

129

(8.) Proposition II. — Let (oiot/ier surf ace of mllform pressure be conceived to exist
indefinitclij near to the Jirst surface ; then, ifthejirst surface is a surface of contimiity,
so also is the second.

By a sui'face of continuity is hew meant one which always j)ass('s tlirou<j;li the same
sot of particles of liquid, so that a pair of such surfaces contain between them a layer
of particles which are always the same.

The perpendicular distance between a paii- of surfaces of uniform pressure is in this
case inversely proportional to the resultant of gravity and centrifugal force ; that is to
say, to the normal AB. Hence if a curve Ibfn be drawn indefinitely near to the curve
LBM, so that tlie perpendicular distance between them, I^, shall everywhere be
inversely proportional to the normal AB, the second curve will also be tlie pi'ofile of a
surface of uniform pressure.

Conceive now that the whole mass of liquid has, combined with its wave-motion, a
uniform motion of translation, with a velocity equal and opposite to that of the propa-
gation of the waves. The dpiamical conditions of the mass are not in the least altered
by this ; but the forms of the waves are rendered stationary (as we sometimes see in a
rapid stream), and, instead of a series of waves propagated in the direction shown by the
arrow P, we have an undulating current running the reverse w^ay, in the direction shown
by the arrow Q. (This is further illustrated by fig. 2.) According to a well-known

Fig. 2.

property of curves described by rolling, the velocity of the particle B in that current is
proportional to the normal AB, and is given by the expression

2^rt.AB.

Consider the layer of the current contained betwx^en the surfaces LBM and Ihn. In
order that the latter of those surfaces, as well as the former, may be a surface of con-
tinuity, it is necessary and sufficient that the thickness of the layer B/" at each point
should be inversely as the velocity; and that condition is already fulfilled; for B/' varies
inversely as AB, and AB varies as the velocity of the current at B ; therefore LBM and
Ibm are not only a pair of surfaces of uniform pressure, but a pair of surfaces of con-
tinuity also. Q.E.D.

(9.) Corollary. — The surfaces of uniform pressui-e are identical with surfaces of con-
tinuity throughout the whole mass of liquid.

(10.) Corollary. — Inasmuch as the resultant of gravity and centrifugal force at B is
represented by

•^ AC
MDC'CCLXni. T

130 DE. W. J. MACQUORN EA^'KINE OX THE EXACT FOEM

the excess of the miifonn pressure at the surface Ihm above that at the surface LBM is
given by the expression

^ AC -^

in which iv is the heaviness of the liquid, in units of weight per unit of volume. By
omitting the factor u\ the pressure is expressed in units of height of a column of the
liquid.

(11.) Proposition' III. — Tlie i^rqfiJe of the lower surface of the layer referred to in the
2)receding proposition is a trochoid generated by a rolling circle of the same radius xvith
that which generates the first trochoid; and the tracing-arm of the second trochoid is
shorter than that of the first trochoid by a quantity hearing the same proportion to the
depth of the centre of the second rolling circle below the centre of the frst rolling circle,
which the tracing-arm of the first rolling circle bears to the radius of that circle.

At an indefinitely small depth Aa below the horizontal line HAK, draw a second
jiorizontal line hak, on the under side of which let a circle roll with a radius ca=CA,
the radius of the first rolling circle ; so that the indefinitely small depths Cc=Aa. To
find the tracing-arm of the second rolling circle, draw cd parallel to CB, the tracing-arm
of the first circle; in cd take ce=CB, and cut off eb=ed; b will be the tracing-point,
and cb the tracing-arm required ; for, according to the principle laid down in the enun-
ciation, we are to have

CB-7b = eb = Cc.^.

Let the second circle roll; then b will trace a trochoid Ibm. From b let fall 5/ per-
pendicular to AB produced ; B/' will be the indefinitely small thickness at B of the layer
between the two trochoidal surfaces.

The proposition enunciated amounts to stating that Bf is everywhere inversely pro-
portional to the normal AB ; so that lb77i is the profile of a surface of uniform pressure
and of continuity.

To pro\e this, join Be and ef Then Be is parallel to ACr-, and equal to C'r, and dcf
is evidently an isosceles triangle, ^" being =ed. Let AB (produced if necessary) cut the
circle of the radius CB in G ; then CG is parallel to ef and the indefinitely small triangle
Bef ifi similar to the triangle ACG ; consequently AC : AG : : Bf — Cc : Bf; or

t)ut, by a well-known property of the circle,

-r— AC'-CB'
-^^ = — AB— •

and therefore

— AC'-CB\
^^=^^- -AC-IS-'

OF WAVES NEAR THE SUEFACE OF DEEP WATER. 131

that is to say, the thickness of the layer varies inversely as the normal AB ; and the second
trochoid, Ihn, i^ therefore the profile of a surface of uniform jn-essure and of continuity.
Q. E. D.

(12.) Corollaries. — The proiiles of the surfaces of uniform pressure and of continuity
form an indefinite series of trochoids, described by equal rolling circles, rolling with the
same speed below an indefinite series of horizontal straight lines.

The tracing-arms of those circles (each of which arms is the radius of the circular
orbit of the particles contained in the trochoidal surface which it traces) diminish in
geometrical progression with increase of depth, according to the following laws: —

For convenience, let Cr be denoted by dk, CB by r, and cb by r—dr; then

dr=dk.-^=dk.^^.

and the integration of this equation gives the following result : —

Let k denote the vertical depth of the centre of the generating circle of a given surface
below the centre of the generating circle of the free upper surface of the liquid ;

/"o the tracing-arm of the free upper surface (= half the amplitude of disturbance) ;

r, the tracing-arm of the surface whose middle depth is k ; then

k 1-nk

r^r=.r^e"^-=rffi ~,

a formula exactly agreeing with that found for indefinitely small disturbances by
previous investigators.

(13.) Propositiox IV. — The centres of the orbits of the j)articles in a given surface of
equal pressure stand at a higher level than the same particles do when the liquid is still,
by a height which is a third i^roportional to the diameter of the rolling circle and the
tracing-arm or radius of the orbits of the particles, and which is equal to the height due to
the velocity of revolution of the particles.

If the liquid were still, the given surface of equal pressure would become horizontal.
To find the level at which it would stand, we must first find what relation the mean
vertical depth of a given layer of particles bears to the depth Cc=^dk, between the
centres of the rolling circles that generate its boundaries.

The length of the arc of the cuine LBM described in an indefinitely short interval of
time dt is

2'xn.AB.dt,

and the thickness of the layer being

AC'-CB'

Bf=dk.'-

AC.AB

let the product of those quantities be divided by the distance through which the centre
of the rolling circle moves in the same time, viz.

2Tn.AC.dt,
t2

132 DE. W. J. MACQUOEN E.AJS'KINE ON THE EXACT FOEM

and the result \Till be the mean vertical depth of the layer, which being denoted by dk\
we liave

dh

The difference by which the mean vertical thickness of tlie layer falls short of the
difference of level of the rolling circles of its upper and lower surfaces is given by the
following expression.

AC'

and this being integrated from od to k, gives the depth of the position of a given particle,
when the liquid is still, below the level of the centre of the orbit of the same particle
when disturbed, viz.

" 2.\C'^ ~2XQ~ x'

or a third i^roportional to the d/a/iutcr uf the rolling circle and the radius of the orbit of

2AC~ ^ff

the particle ; also -^rf,= ^n' '*' ^^x" height due to the reJociti/ of revolution of tlte parti

rles. Q.E.D.

(13. \.) Corollary. — The mechanical energy of a wave is lialf actual and half poten-
tial,— half being due to motion, and half to elevation. In other words, the mechanical
energy of a wave is double of that due to the motion of its particles only, there being
an equal amount due to the mean elevation of the particles abo^e tlieir position when
the water is still.

(14.) Corollary. — The crests of tlie waves rise higher above the level of still water than
their hollows fall below it ; and the difference between the elevation of the crest and
the depression of the hollow is double of the quantity mentioned in Proposition IV.,
that is to sav, it is

aC~~^~'

(15.) Corollary as to Pressures. — An expression has already been given in art. 10 for
the difference of pressure at the upper and under surfaces of a given layer. Substituting
in that expression the value of the thickness of the layer, we find

r/w=U' . ^.^ . dk . -^^=.- = w . dA{ 1 — = — - ]=w . al,„

^ AC AC.AB V ACV

(as the prec(!ding corollary shows), being precisely tlie same as if the li([uid wer(> still ;
and hence it follows that the hydrostatic j^ressure at each indiridual j'article diirivy wave-
motion is the .iaiue as if the liquid were still.

(16.) In Proposition III. it has been sliown, by geometrical reasoning fnnn the mecha-
nical construction of the trochoid, that a wave consisting of trochoicial layers .satisfies
the condition of continuity. It may be satisfactory also to show the same tiling by the
use of algebraic symbols. For that purpose the following notation will be used.

OF AVAVES NEAR THE SURFACE OF DEEP WATER. 138

Let the origin of coordinates be assumed to be in the horizontal line containing tlie
centre of the circle which is rolled to trace the profile of cydoidal waves, having cusps,
and being (as Mr. Scott Kussell long ago pointed out) the highest waves that can exist
without breaking. In such waves, the tracing-arm, or radius vector, of the uppermost
particles is equal to the radius of the rolling circle ; and that arm diminishes for each
successive layer proceeding downwards.

Let X and y be the coordinates of any particle, x being measured horizontally tKjainst
the direction of propagation, and y vertically downwards.

Let k (as before) be the vertical coordinate of tlie centre of the given particle's orbit ;
h the horizontal coordinate of the same centre.

Let K be the radius of the roUing circle, a the angular velocity of the tracing-arm
( = 2«-«). so that

is the length of a wa\ o, and

is the velocity of propagation.

I^t 6 denote the jihasc of the wave at a given particle, being the angle which its
radius vector, or tracing-arm, makes with the direction of +y, that is, with a line point-
ing vertically downwards.

Let t denote time, reckoned from the instant at which all the pai'ticles for wliich
A=0 arc in the axis of//; then

6=^^+i (!•)

Then the following equations give the coordmates of a given particle at a given instant:

_k

3:=]i-\-lxe Rsin^; (2.)

j/=A-+Re"«costf (3.)

Let u and v denote the vertical and horizontal components of the velocity of the
particle at the given instant ; then

dx -*
u=-^= all.e ^cosO= a{ji—k); (4.)

du _*
v=-^ = —aR.e ^sm6=—a{x—}i) (5.)

The well-knoAvn equation of continuity in a liquid in two dimensions is

1+1=0^ (6-)

and from equations (4.) and (5.) it appears that we have in the present case

, dv ( dk , dh\ ( dk , Rr/e\ ,_ ,

134 DK. W. J. MACQUOR^" RANKIXE OX THE EXACT FORM

In the original formnJse, k and 0 are the independent variables. "When ,r and // are
made the independent variables instead, we have, by well-known tbrmulfe,

(Ik -■ . I f/'^' (i^ dk^l e R sin 3 |

and i, (8.)

( dx\ _* .

M \ J J Jn e K Sin d

dy~ ~)~M~dk"dx(~R{y—e''n)

I ^1 J

so that the equation of continuity (6.) is exactly verified.

(17.) Another mode of testing algebraically the fulfilment of tlxe condition of con-
tinuity is the following. It is analogous to that employed by Mr. Airy ; but inasmuch
as the disturbances in the present paper are regarded as considerable compared with
the length of a wave, it takes into account quantities which, in INIr. Airy's investigation,
are treated as inappreciable.

Consider an indefinitely small rhomboidal particle, bounded by surfaces for which the
values of A and k are respectively h, h-\-(JJi, /■, k+dk. Then the area of tliat rhomboid is

/dx dy dx dy\ .. ,,
[dh-dk-dk-M)'^^'-'^^'

and the condition of continuity is that this area shall be at all times the same ; that is

to say, that

d /dx dy dx dy\ ,„ .

(it\dh'dk~dk'dh)~^ ^ ''

Upon performing the operations here indicated upon the values of the coordinates in
equations (2.) and (3.), the value of tlie quantity in brackets is found to be

l-e'^; (10.)

which is ob^'iously independent of the time, and therefore fulfils the condition of con-
tinuity.

Appendix.

Received October 1, — Read November 27, 1802.

On the Friction betiveen a Wave and a Wave-shaped Solid.

Conceive that the trougli between two consecutive crests of the trochoidal surface of
a series of waves is occupied, for a breadtli which may be denoted by z, by a solid body
with a trochoidal surface, exactly fitting the wave-surface ; that the solid boily moves
forward with a uniform velocity equal to that of the propagation of the waves, so as to

OF WAVES NEAK THE SURFACE OF DEEP 'WATEH. 135

continue always to fit the wa\e-surf;ic(\ and that there is fnctif)n between the solid
snrfoce and the contiguous liquid particles, according to the law which experiment has
shown to be at least approximately true, \iz. varying as the surface of contact, and as
the square of the velocity of sliding.

Conceive, further, that each particle of the liquid has that pressure apjilied to it
which is required in order to keep its motion sensibly the same as if there were no
friction ; the solid body must of course be urged forwards by a pressure equal and opjxi-
site to the resultant of all the before-mentioned pressures.

The action, amongst the liquid particles, of pressures sufficient to overcome tlie fric-
tion, -vnll disturb to a certain extent the motions of the liquid particles, and the figures
of the surfaces of uniform pressure ; but it will be assumed that those disturbances are
small enough to be neglected, for the purposes of the present inquiry. The smallness
of the pressures producing such disturbances, and consequently the smallness of those
disturbances themselves, may be inferred fi'om the fact, that the friction of a current
of water over a sui'face of painted iron of a given area is equal to the weight of a layer
of water covering the same area, and of a thickness which is only about '0030 of the
height due to the velocity of the current.

Those conditions having been assumed, let it now be proposed, to find approxiniately
the amount of resultant pressure required to overcome the friction between the wave and
the wave-shaped solid.

This problem is to be solved by finding the mechanical work expended in OAercoming
friction in an indefinitely small time dt, and dividing that work by the distance through
which the solid moves in that time.

Taking, as before, as an independent variable the phase 0, being the angle wliich the
tracing-arm CB=;' (fig. 1) makes with a line pointing vertically dowTiwards, the length
of the elementary arc corresponding to an indefinitely small increment of phase dd is

qdO.

where q is taken, for brevity's sake, to denote the normal AB.
The area of the corresponding element of the solid surface is

The velocity of sliding of the liquid particles over that elementary surface is

aq,

in which a, as before, denotes y, the angular velocity of the tracing-arm. Hence let ^

denote the heaviness (or weight of unity of volume) of the liquid, and f its coefficient
of friction when sliding over the given solid surface ; the intensity of the friction per
unit of area is

2ff '

136 DE. W. J. MACQL'OEX EAXKINE OX THE EXACT FOEM

That friction has to be overcome, during the time df, through the distance

aqdt=^qd6.
Multiplying now together the elementary area, the intensity of the friction, and the
distance through which it is overcome in the time dt, we find the following value for the
work performed in that time in overcoming the friction at the given elementary surface,

.id^y,^-^xqd6=^-^.<j-d0K

Xow during the time dt, the solid advances through the distance

aJ{dt=V^d&
(R. as before, being the radius of the rolling circle); and dividing the elementary portion
of work expressed above by that distance, we find the following \aluc for an elementary
portion of the pressure required to overcome the friction,

''P=f 4-'"' (!■)

The total pressure reqirired to overcome the friction is found by integrating the
preceding expression throughout an entire revolution, that is to say.

i'4f .ir*"''" (-■'

To obtain tliis integral the following value of the square of the normal q or AB is to
be substituted,

2==R=+r + 2Rr.cos^,
whence

and

VfeI!-j'"(l+|;+£+4(l+j;)jj.cos<+4,^,cos-<))<W=2.E'(l+4|;+,J.),

I'=^F'(l+4+£) (8-)

The following modification of this expression is sometimes convenient : —
Let V=«Il denote the velocity of advance of the solid ;
A = 2';rR, as before, its length, being the length of a wave;

sin jS^ jT the sin(^ of the greatest angle mad(? by a tangent to the trochoidal surface

\ntli the direction of advance ; then

l'^'->.r(l + 4sin=,3 + sin',^*) (4.)

* This formula (neglecting sin'' /3 as unimportant in piaotice) has been used to L'alculatc approximately the
renistancc of steam-vessels, and its results have been found to agree very closely with those of experiment, and
have, also been used since IS.jS by Mr. Jajiks R. Napiku and the aiithor with complete success iu practice, to
calculate beforehand the engine-power required to propel proposed vessels at given sjjeeds. The fonnula has
been found to answer approximately, even when the lines of the vessel are not trochoidal. by ]mtting for p

OF WAVES ^"EAK THE SUEFACE OF DEEP WATER. 137

It is to be observed that tlie resistuiice F, as deterniined by the preceding investiga-
tion, being deduced iVoni tlie amount of work performed against friction, includes not
only the longitudinal components of tlie direct action of friction on each element of the
surface of the solid, but the longitudinal components of the excess of the hydrostatic
pressure against the front of the solid above that against its rear, which is the indirect
effect of friction. The only quantities neglected are those arising from the disturbances
of the figiu-es of the sm'faces of equal pressure, which quantities are assumed to be
unimportant, for reasons already stated. The consideration of such quantities would intro-
duce terms into the resistance varying as the fourth and higher powers of the velocity.

Received October 22, — Head November 27, 1SG2.

Note, added in October 1862.

The investigation of Mr. Stokes (Camb. Trans, vol. \iii.) proceeds to the second degree
of approximation in shallow water, and to the thu'd degree in water indefinitely deep.
In the latter case he arrives at the result, that the crests of the Avaves rise higher above
the level of still water than the troughs sink below that level, by a height agreeing
with tliat stated in art. 14 of this paper, and that the profile of the waves is cq)2>roxi niatel//
trochoidal.

Mr. Stokj:s also arrives at the conclusion, that, when the disturbance is considerable
compared with the length of a wave, there is combined with the orbital motion of each
particle a translation which diminishes rapidly as the depth increases. No such trans-
lation has been found amongst the results of the investigation in the present paper ; and
hence it would appear that Mr. Stokes's results and mine represent two different possible
modes of w^ave-motion*.

the mean of the values of the greatest angle of obliquity for a series of water-lines. The method of using the
formula in practice, and a Table showing comparisons of its results with those of experiment, were communi-
cated to the British Association in 18G1, and printed in the Civil Engmcer and Arcliitcct's Journal for October
of that year, and in part also in the ' Mechanics' Magazine,' ' The Artisan,' and ' The Engineer.' The ordinary
value of the coefficient of friction / appears to be about -0030 for water gliding over painted ii-on. The
quantity Xr(l + 4 sin- /5 -|- sin''/3) con-esponds to what is called, La the paper referred to, the awjmenled surface.

* Note added in June 1803.
The difference between the cases considered by Mr. Stokes and by me is the following : — In Mr. Stokes's
investigation, the molendar rotation is null; that is to say,

j^fdv dti\_„ .
''[di-di,)-^'

while in my investigation it is constant in each layer, being the following function of 1-,

Jdv_du\ ary-R

^\d.v d<i)~_ _2* ^ '

From this last equation it follows that

d fdv dxi\_,,
dt\dx dy)
and therefore that the condition of continuity of pressure is verified.
MDCCCLXIII. U

138 ON THE EXACT FORM OF WAVES ISFIAE THE SUEFACE OF DEEP WATER.

The simplicity with which an exact result is obtained in the present paper, is entirely
due to the following peculiaiity : — Instead of taking for independent variables (besides
the time) the undisturbed coordinates of a paiticle of liquid, there are taken two quan-
tities, h and k, which are functions of those coordinates, of forms which are left inde-
terminate until the end of the investigation. Ji then proves to be identical with the
undistiu-bed horizontal coordinate ; but k proves to be a function of the undisturbed
vertical coordinate for which there is no sj'mbol in our present notation, bemg the root
of the transcendental equation

" 2R

in which k,, is the undisturbed vertical coordinate (see art. 13). Hence it is e\ident
that, had ^'o instead of k been taken as the independent variable, the question of wave-
motion considered in this paper could not have been solved except by a complex and
tedious process of approximation.

139 ]

VII. Photo-chemical Besearches. — Part V. On the Direct Measurement of the Chemical
Action of Sunlight. By Kobert Bu^'S]i^, For. Mem. B.S., Professor of Chemistnjin
the University of Heidelberg, and Henry E. Eoscoe, B.A., Ph.D., Professor of
Chemistry in Chcois College, Manchester.

Keceived November 11, — Read December 11, 1862.

The photo-chemical action exerted by dii-ect sunlight and by diffuse daylight upon a
horizontal portion of the earth's siuface, varies with the time of year and with the lati-
tude of the place, and constitutes an important link in the chain of physical relations
which connects the organic with the inorganic world.

In former communications* made to the Royal Society we have endeavoured experi-
mentally to determine the distribution of these chemical actions on the earth's surface,
as varying with the time of day and year, and with the geographical position of the
place, when the sky is perfectly unclouded. The methods of measiu'ement there adopted
are, vmfortunately, not applicable to the determination of the variations in photo-chemical
intensity when, as is most frequently the case, the transparency of the atmosphere is
more or less obscured by clouds, mist, or rain. To enable us to estimate the alterations
which occur- in the amount of the chemicaDy active rays falling on the earth's surface,
we must, therefore, have recoui'se to a mode of measurement totally different from that
employed in oiu- fonner investigations.

In spite of the numerous futile attempts which have from time to time been made for
the purpose of establishing a standard of measurement of the chemical action of light
by means of photographic tints, it appeared to us not impossible in this way to gain the
end we desii'ed.

The proposals which have been made to register and measure the chemical action of
light by help of photographic tints have been very numerous ; amongst others we may
mention those of Jokd.\n, Hunt, HEESCHELf, and ClaudetJ. All instruments based
upon such principles must, however, afford totally imreliable results unless we know the
conditions under which photographic surfaces of a constant degree of sensitiveness can
be prepared, and unless the relations are determined which exist between the degree of
tint produced and the time and the intensity of the light acting to produce such a tint.

Hence one of the first points of inquiry is, to determine whether the tints produced by
the photographic action vary in shade in the direct ratios of the intensities of the incident
light.

* Philosophical Transactions, 1857, pp. 35.5, 381, 601 ; 1859, p. 879. t Ibid. 1840, p. 46.

X London, Dublin, and Edinburgh Phiioeophical Magazine, Ser. 3. vol. xxxiii, p. 329.

u2

140 PROFESSOE BUySEX AXD DE. H. E. ROSCOE\S PHOTO-CHEMICAL EESEAECKES.

For the purpose of measuring the degree of tint which the paper assumed, we
employed a ciixular disc vnih black and white sectors whose relation to one another
could be altered at pleasure. By allowing the black sector to coyer j^j, -ni, -nj, &c. of
the surface of the disc, we obtained, on rotation, a disc haying the tints xo, -^, -^, Sec.
The central portions of tlie disk were filled with the papers which had been tinted by
the action of the light. It was soon found that yery slight differences in the degree of
shade could be detected so long as the tints are light-coloured, but that when deeper
tints are employed the eye loses the power of estimating such differences. Experiments
thus made proyed that the shade produced upon photographic paper is not jiroportional
to the intensity of the incident light ; thus the intensities 5 and 1 were found to cor-
respond to the shades 0'5 and 0-22. Hence we haye altogether relinquished the idea of
employing any mode of measurement foiuided upon a comparison of photographic papers
of different shades.

"We next had to examine whether equal shades of blackness produced by light of
different intensities acting for different times can be used as the basis of tlie mode of
measurement, under the supposition that equal shades of blackness always correspond
to equal products of the intensity of the incident light into the times of insolation. The
truth of this proposition, which was assumed some years ago by Mal.\guti *, has recently
been experimentally Aerified by HAXKELf withm the slight yariation of intensity from
1 to 2-J. In order to proye the truth of this proposition for the wider limits needed in
our measurements, it was necessary to determine the times of insolation yery exactly to
within small fractions of a second, and to be able to estimate with accuracy the points of
equal shade. These ends were gained by employing the following arrangement.

The iron stand (Plate IX. fig. 1) carries the metal plate (A), which can be placed hori-
zontally by three set-screws, and in which a straight slit, 15 millimetres broad and 190
millimetres long, is cut. Oyer this slit, which is shaded black in the cb'awing, is placed
a yery thin and elastic sheet of mica h c d, blackened at one end from b to c, and
fastened at d to the curyed drum (E) attached to the pendulum (F). When the pen-
dulum is allowed to yibrate, the sheet of mica as it rolls on and oft' the curyed drum (E)
at each \ibration, uncoyers and again coyers the .slit, so that each point throughout the
whole length of the .slit is exposed for a different period. If we wish to use this instru-
ment for the purpose of exposing a photographic surface to the action of the liglit for
different times, the paper is gummed upon the white sui-face of the metallic slide
(G, fig. 1); this is then coyered by a metallic lid, which does not touch the paper, and
the whole arrangement pushed into the dark groove h, placed dii-ectly under the slit,
and protected from the entrance of light by a lappet of cloth, which liangs in front.
The metallic lid is then withdrawn, the screw k turned, and tlius the paper slightly
pressed against the slit, so that no light can enter sideways between the paper and the

* Ann. do Chim. ct tie I'hys. torn. Ixii. p. 5.

t " Mcssungcn iibcr die Absori)tion dcr chcmischcn Strahkn des Sonncnlichts,"Aljli;iiidJ. d. Kiin. SiicLs.
GcsclLjchaft der "Wissenschaften. Leipsig, 1862. Bd. ix. p. 55.

PROFESSOR BIINSEN AXD DR. U. E. ROSCOE'S PUOTO-CIEE^inCAL RESK\RC11ES. 141

thin metallic edges of the slit. By raising the lever « m I at /, the pendulum is
released from the catch at m, and, after completing a vibration, it is held fast by a lower
catch at n. If it be required to double or to mirltiply the time of insolation, it is only
necessary to repeat the vibration, once or several times, care being taken before each
vibration to raise the rod of the pendulum so as to allow the end to fall into the upper
catch. In order thus to set the pendulum in motion, and to stop it with certainty and
ease, the lever is once for all balanced by a small weight at /, so that the arm « m is but
slightly heavier than the ami l.

The time which the sensitized paper is insolated at any point on the slit can be
obtained when we know the duration and the amplitude of vibration of the pendulum.
Let a u (fig. 2) represent the end of the sheet of mica in the position in which it is found
when the end of the pendulum is held in the upper catch m ; let /3 /3 (fig. 2) represent
the position of the end of the sheet of mica when the pendulum is frcelv hanging and in
cqmlibrium ; and let y y (fig. 2) be the position which the same end occupies during
the vibration when the time t has elapsed since the start of the pendulum. If we
call M the distance y (3, and r the time dming which the sensitized surface at y y
is insolated, the relation betw een u and r is found as follows : — Let a represent the
distance a (5. and let T represent the diu-ation of a simple vibration, we have

({,.).

If f, signify the time at which the end of the sheet of mica returns to the position
y y. we have

and also

and therefore

r=2{T-f);
or, if we express t in terms of n,

T:= — I TT— COS ■- I?

or

u= — aco»(^,7rj (1.)

In our instrument a=105 milhmetres, and T=f second. From formula (1) a Table
can easily be calculated, in which the time of insolation is given for every point on the
slit ; that is, the time for which, during one vibration of the pendulum, this point is
uncovered by the blackened sheet of mica. For this purpose we have applied a milli-
metre scale, commencing at S, to the slit, which in oiu- instrument, reckonmg from
|3j3, extended 85 millims. in the direction of S, and 105 millims. in the direction of s
(fig. 2). Column I. of the following Table contains the numbers of the millimetre scale,
and column II. the con^esponding times of insolation for one vibration expressed in
seconds.

142 PEOFESSOE BUXSEN AKD DE. H. E. EOSCOE'S PHOTO-CHE^nCAL EESEAECHES.

Table I.

I.

II.

I.

II.

I.

II.

I.

n.

I.

II.

I.

II.

Millims.

Seconds.

Millims.

Seconds.

MilUms.

Seconds.

Millims.

Seconds.

Millims.

Seconds.

Millims.

Seconds.

0

1-200

32

1-003

64

0-846

1 96

0-700

128

0-549

160

0-369

1

1-193

33

0-998

65

0-841

97

0-695

129

0-544

161

0-363

2

1-186

34

0-993

66

0-837

98

0-691

130

0-539

162

0-357

3

1-179

35

0-988

67

0-832

, 99

0-686

131

0-534

163

0-350

4

M73

36

0-983

68

0-828

100

0-682

132

0-528

164

0-343

5

1-165

37

0-977

69

0-823

101

0-677

133

0-523

165

0-336

6

1-158

38

0-972

70

0-819

1 102

0-672

134

0-518

166

0-329

7

1-151

39

0-967

71

0-814

i 103

0-668

135

0-513

167

0-321

8

1-144

40

0-962

72

0-809

104

0-663

136

0-508

! 168

0-314

9

1-137

41

0-957

73

0-805

105

0-659

137

0-502

169

0-309

10

1-131

42

0-952

74

0-800

106

0-654

138

0-497

170

0-300

11

1-125

43

0-947

75

0-796

107

0-650

139

0-492

171

0-291

12

1-119

44

0-942

76

0-791

108

0-645

140

0-487

172

0-283

13

1-113

45

0-937

77

0-786

i 109

0-640

141

0-482

173

0-274

14

1-106

46

0-932

78

0-782

110

0-635

142

0-476

174

0-266

15

1-100

47

0-927

79

0-777

111

0-631

143

0-470

! 175

0-257

16

1-094

48

0-922

80

0-773

112

0-626

144

0-465

i 176

0-249

17

1-087

49

0-917

81

0-768

113

0-621

145

0-459

1 177

0-240

18

1-081

50

0-913

82

0-764

114

0-617

146

0-453

178

0-229

19

1-076 ,

51

0-907

83

0-759

115

0-612

147

0-448

179

0-219

20

1-070

52

0-903

84

0-755

116

0-607

148

0-442

180

0-208

21

1-064

53

0-898

85

0-750

117

0-603

149

0-436

' 181

0-198

22

1-058

54

0-893

86

0-745

118

0-598

150

0-431

182

0-187

23

1-053

55

0-888

87

0-741

i 119

0-593

151

0-425

] 183

0'176

24

1-047

56

0-884

88

0-736

120

0-588

152

0-419

1 184

0-161

25

1-041

57

0-879

89

0-732

121

0-583

153

0-413

185

0-146

26

1-036

58

0-874

90

0-727

122

0-578

154

0-407

186

0-131

27

1-030

59

0-8/0

91

0-723

123

0-573

155

0-401

187

0-116

28

1-025

60

0-865

92

0-718

; 124

0-568

156

0-394

29

1-019

61

0-860

93

0-714

' 125

0-563

157

0-388

30

1-014

62

0-856

94

0-709

126

0-558

158

0-382

31

1-009

63

0-851

95

0-704

127

0-553

159

0-376

The paper insolated in tlie slit whilst the pendulum is vibrating, exhibits throughout
its whole length a regularly diminishing shade. In Table I. the time of insolation for
each one of these different shades is to be found.

If we wish to determine which of these shades on the strip of paper corresponds to
another tint produced by a separate insolation, we cannot make the comparison of the
two shades either by daylight or by lamp- or candle-Ught, as the weakest light of this
kind, which must be used in order to make such a comparison accurately, produces a
sensible alteration in the tmt of tlie paper during the process of reading. Still less does
it appear advisable to fix the paper with hyposulphite of soda, or other material, before
comparison, as sucli a treatment frequently causes irregular alteration of tone. We have
overcome this difficulty by employing an intense soda-Hame for illuminating the surfaces
to be compared. This light is chemically so inactive that the rays proceeding from the
flame can be concentrated by a convex lens and allowed to fall for several hours upon
the sensitized paper without producing the least change of colour. This mode of illu-
mination possesses another important advantage, inasmuch as small differences in colour.

PEOFESSOE BUNSEX AXD DE. H. E. EOSCOE'S PHOTO-CHEMICAIi EESEAECHES. 143

which leuder the comparison of shades by the eye with ordinary white light so difficult,
entirely disappear when the monochromatic soda-flame is employed.

In order to avoid the necessity of carrying the instrument into the dark after each
insolation, a millimetre scale, similar to the one upon the slit, is fastened upon a wooden
board (fig. 3, a) coAered with paper, and moveable in a groove across a fixed wooden
stand. The strip of photographically tinted paper is then cut off from the slide G and
gummed upon the board {a, fig. 3), so that it has the same position relati\ e to the scale
on the board as it had to the scale on the slit. A, fig. 4 represents a small square Avoodcn
block having a circular hole in the middle 5 to 6 millims. in diameter, the lower half
being covered by the paper of wliich the degree of shade has to be determined. This
block is pressed by means of a spring, as is seen in fig. 3, in a fixed position against the
strip of paper. On thromng the image of the soda-flame C, by help of the convex
lens D, upon the cu-cular opening in the block, it is easy, by drawing the slide backwards
and forwards, to determine the exact point at which the upper and lower halves of the
ciiTular hole appear equally dark. It is then only necessary to read off on the scale
the number representing the time which the paper at that point has been ingolated in
order to determine the degree of shade which the paper hi that time has attained. To
ensure accuracy in the obser\-ations, it is necessary that the oje should always be placed
in one and the same position; most advantageously in a direction nearly perpendicular
to the siu-face of the strip of paper.

Having proved by experiment with the above instrument that we were able to measure
the length of time required to effect equal degrees of shade within hundi-edths of a
second, it became necessary to obtain a series of lights of accurately known degi'ees of
intensity, and varying as much as possible from each other, in order to produce various
shades on photographic paper. We employed for this purpose direct sunlight, and in
the following manner we avoided any eiTors arising from the varyuig intensity of the
light caused by the alteration of the sun's zenith-distance, and the changes of transpa-
rency in the atmosphere. In the roof of the darkened loft of the laboratory at
Heidelberg, a brass plate was inserted, in which were bored round holes varying in size,
and countersunk fi-om the outside. The diameters of these holes were accurately mea-
sured with a micrometer, and through the holes sunlight was allowed to fall upon the
sensitized paper, the surface of which was placed perpendicularly to the incident rays,
and at such a distance from the brass plate that, seen from this position, the holes pre-
sented a smaller apparent diameter than the sun. The several intensities of the small
pictures of the sun thrown upon the paper, which were so well defined that the larger
solar spots were distinctly seen, are thus obtained quite independently of any alteration
which change in the height of the sun, or variation in the transparency of the au- may
produce, and are directly proportional to the areas of the openings tluough which the
light passes.

By allowing these pictures of the sun having the intensities \, Ij, \- ■ ■, to act upon
the paper for the times Iq, f^, t^, the products If^t^, l^t^, l^t,^ were obtained for the various

lU PKOFESSOE BUXSEX AND DE. H. E. EOSCOES PHOTO-CHEMICAL EESEAECHES.

tints thus iJi-oduced. A second piece of the same photographic paper was then exposed
to diffuse daylight, of the intensity ?, in the penduhun-instrument, and thus a tinted
strip was produced containing a successive gradation of shade. For each particular
shade the time of insolation r^. ti. r., could be read off on the scale, and hence the pro-
ducts ?Vo, ?>!, ?V.,, &c. could be formed. The points upon the strip were next determined
(by reading off ^Tith the soda-flame) which corresponded in shade to the several tints
produced by the sun-pictures. On the supposition that equal products of intensity into
time of insolation correspond to equal shades of blackness, we have the following equa-
tions : —

and

or, if we take a special case,

-Jlo

rj = fili . const

The following experiments prove that this equation is satisfied.

Experiment I.
8th August ISGO, at 12" 30" p.m. Cloudless sky,

(2.)

I.

11.

III.

IV.

V.

I.

f.

r.
ObserTcd.

Calcuiatcd.

Difference.

1-00

20

2-65

2-47

-0-08

1-69

20

•' 4-06

4-17

+ 0-11

2-78

20

! 7-01

6-8G

-0-15

4-00

20

9-92

9-87

, -0-05

0-44

20

13-46

13-43

-0-03

7-47

20

18-26

18-44

+ 0-18

Experiment II.
2nd August 1862, at 12" ll"" p.m. Cloudless sky.

I.

II.

III.

IV.

V.

r.

/.

Observed.

r.
Calculated.

Difference.

1-00

ISO

4-53

4-75

+ 0-22

1-71

150

8-71

8-15

-0-56

3G-81

10

n-G2

11-66

+ 0-04

45-04

10

14-08

14-28

+ 0-20

The first column contains the intensity of the sun-pictures employed to darken the
{)aper ; the second column, the length of time which the paper was thus exposed ; the
third, the times, measured with the pendulum-instrument, which the diffuse light of
day took to produce the same degree of shade ; and lastly, the fourth column gives the
same times calculated from equation (2.), in which the constant is equal to 0'12339.

PEOFESSOR BUNSEN AND DR. 11. E. KOSCOE'S PIIOTO-CIIEMICAL EESE^UICHES. 145

As the intensity of the light in these experiments \aiicil from 1 to nearly 50 without
a greater tleviation from the calculated results occurring than that \\hicli may fairly be
ascribed to the unavoidable experimental errors, we ccmclude

That equal products of the intensity of the liglit into the time of insolation
correspond, within very wide limits, to equal shades of darkness produced on
chloride-of-silver paper of uniform sensitiveness.

Upon this important proposition a method may be founded for measuring the
chemical action of light by means of simple observations. For, if we assume as the
unit of photo-chemical action that intensity of light which produces in the unit of time
a given degree of shade, we have only to determine on a strip of paper, blackened in the
pendulum-apparatus, the point where the shade of the strip coincides with the given
unalterable tint. The reciprocals of the times which correspond to these points of equal
shade give the intensity of the light expressed in terms of the above unit.

It is clear that this method is available only under the suppcjsitions,

(I.) That the phenomena of induction, accompanying the light of the intensities
employed in the measurement of the total daylight, are of so short a duration that the
variations thus produced fall within the limits of the necessary experimental errors ;

(II.) That it is possible to prepare a photographic surface possessing a p(>rfectly con-
stant degree of sensitiveness ;

(III.) That an unchangeable shade of blackness is obtainable which can be easily
prepared, and can be exactly compared to a photographically tinted paper.

In order to investigate the influence of photo-chemical induction uj)on the black-
ening of the chloride-of-silver paper, we have employed the following method. By means
of the pendulum-apparatus we exposed strips of the same sensitive paper quickly one after
the other to the light of a cloudless sky, insolating the iirst strip during «„ vibrations
of the pendulum, the second strip during m, vibrations, and determining on each of
these strips the points of equal shade. The times of insolation, ^„, ^„ t.^ • ■ ., corresponding
to each of these points are obtained from Table I. If no appreciable induction occurs,
the pi'oducts Wo^o, «,^,, njt.^, &c. must, in accordance with the former proposition, be equal.
If, on the contrary, the chemical action continued for a certain length of time after each
vibration, as is the case with photo-chemical induction, the products «„ 4 must regularly
alter with increasing n. The following experiments prove that this is not the case.

Experiment III. — Intensity No. 1.

,

Deviation

from Mean.

4

1-024

4-096

-0-067

\ 4

1-041

4-164

+ 0-001

4

1-063

4-252

+ 0-089

8

0-.132

4-256

+ 0-093

8

0-.i25

4-200

+ 0-037

12

0-341

4-092

-0-071

12

0-340

4-080

— 0-083

IIXDCCCLXIII.

146 PEOFESSOR BUySEX AXD DE. H. E. EOSCOE'S PHOTO-CHEinCAL RESEARCHES.

The same readings by a second obsen'er.

nf.

Deviation

from Mean.

4

1-048

4-192

— 0-002

4

1-Oo4

4-216

+ 0-026

4

1-054

4-216

+ 0-026

8

0-515

4-120

-0-070

8

0-520

4-160

— 0-030

12

0-342

4-104

-0-086

12

0360

4-320

+ 0-130

The same obscn'ations, £»i\ing the mean of seven readings.

Deviation

from Jloan.

4

1-028

4-112

-0-028

4

1-036

4-144

+ 0-004

4

1-036

4-144

+ 0-004

8

0-513

4-104

— 0-036

8

0-501

4-008

-0-132

12

0-354

4-248

+ 0-108

12

0-352

4-224

+ 0-084

Intensity No. 2.

Deviation

from Mean.

12

1-022

12-264

+ 0-505

12

0-982

11-784

+ 0-025

18

0-654

11-772

+ 0-013

18

0-655

11-790

+ 0-031

24

0-479

11-496

-0-263

24

0-477

11-448

-0-311

Intensity No. 3.

1

Deviation

from Mean.

3

0-975 1

2-925

— 0-011

3

0-975

2-925

-0-011

4

0-739

2-956

+ 0-020

4

0-735

2-940

+ 0-004

6

0-487

2-922

— 0-014

6

0-492

2-952

+ 0-016

Intensity

No. 4.

Deviation

from Mean.

2

1-053

2-106

-0-004

2

1-057

2-114

+ 0-004

4

0-523

2-092

— 0-018

4

0-532

2-128

+ 0-018

PROFESSOR BUNSEN AND DR. U. E. EOSCOES PUOTO-CHEMICAL RESEARCHES. 147

lutc'usity No. 5.

n.

i.

nf.

Deviation
from Mean.

9
9

12
12

0-810
0-793
0-603
0-600

7-290
7-137
7-236
7-200

+ 0-074
-0079
+ 0-020
-0-016

Intensity No. 6.

n.

t.

ni.

Deviation
from Mean.

1

1-061

1-061

+ 0-032

1

1-050

1-050

+ 0-021

2

0-iJ02

1-004

-0-025

2

0-502

1-004

-0-025

Intensity No. 7.

n.

!■.

?if.

Deviation
from Mean.

2
6

M29
0-379

2-258
2-274

— 0-008
+ 0-008

Consideiing the importance of this question, we deem it advisable to record another
series of experiments, made for the purpose of investigating the influence of plioto-
chemical induction on the results of our measurements. They were made by allowing
a circular disc of metal, having a sector cut out, to revolve for the same length of time,
but at different rates, over two papers of the same degree of sensibility. The disc
revolved at the rate of 30 revolutions per minute over one paper, and at the rate of 366
revolutions per miaute over the other paper; the object being to determine whether the
shade produced by the same intensity of the light and the same length of iusolation
remained constant, and was independent of the rate of rotation of the disk. Inasmuch
as the results of these experiments, which were made on August 1, 1859, at 12'' noon,
with the light of a cloudless sky, coincide with those of the former series, we do not
think it necessary to enter into a full description of the experiments.

We may therefore conclude

That photo-chemical induction does not exert any ])rejudical efiect with inten-
sities of light such as are employed in the measurements under consideration.

(II.)
The next question upon which the successful solution of our problem materially
depends, concerns the possibility of preparing a photographic paper which shall always
possess the same degree of sensitiveness- In describing this portion of our investigation,
we have thought it necessary to enter more minutely into the experimental details than
perhaps may be consistent with the reader's patience, because we felt that, imless a com-

x2

148 PEOFE.S.SOE BUXSEX AXD DE. II. E. EOSCOE'.S PHOTO-CHEMICAL EESEAECHES.

plete description of the experiments were given, it would be impossible to remove the
most weighty objection which can be urged against photometric measurements based
upon a comparison of photographic shades.

It appeared most rational to avoid all complicated photographic receipts for the
preparation of our sensitive surface, and we therefore limited our investigation to
the case of a simple paper co\cred ■\\dth a film of pure chloride of silver.

For the purpose of compaiing the sensitiveness of papers prepared in various ways
and under varying conditions, we employed a strip of paper which had been photogra-
phically shaded in the pendidum-photometer, and afterwards fixed in a bath of hyposul-
phite of soda. The strip exhibited a gradually increasing shade from its white to its
dark end, and was furnished with an arbitrary scale, so that the particular shade corre-
sponding to a given number could easily at any time be read off by the soda-flame.

In order to test whether any given papers possessed an equal degree of sensitiveness,
they were exposed for equal lengths of time to the same light, and then, by means of the
arrangement represented in fig. 3, they were examined to see whether they exhibited the
same degree of shade ; that is, whether they corresponded to the same number on the scale
adapted to the fixed stri[).

We always employed a solution of chemically pure crystallized nitrate of silver as the
silvering liquid. The pure chloride of sodium required for obtaining a film of chloride
of silver was prepared by passing gaseous hydrochloric acid into a concentrated solu-
tion of common salt, washing the pi'ccipituted chloride of sodium with water, and
heating it strongly in a platinum basin.

We have made the following series of experiments for the purpose of determining the
influence exerted on the sensitiveness of the paper by the concentration of the solution
of .salt, the quantity of silver contained in the silvering solution, the Cj[uality of paper
used, and the changes of atmospheric temperature and moistui'e.

1. Silvering the Paper.
Pieces of the same perfectly homogeneous salted paper, of a quality such as is usually
employed by photographers, })repared according to a method hereafter described, were
allowed to lie for two minutes upon the surface of silver solutions of different strengths,
as follows : —

ra])er (i on a solution containing 12 AgNO^ to 100 of water.
„ h „ „ 10

„ c „ „ 8

„ d „ „ G

The papers were then air-dried in the dark, exjiosed for one and tlie same time to the
daylight, and their shade (Ictcrniined. Tlie following numbers were obtained : the
readings A were made by one ind(>pendent observer, the readings B by another ; and each
number is the mean of several readings. Etjuality in the numbers denotes equality in
the shade, that is, equality in the sensitiveness of the paper.

PEOFESSOE BUNSEK AXD DE. H. E. EOSCOE'S PHOTO-CHEMICAL EESEAECHES. 149

Experiment IV.

Intensity No.

I.

Parts of nitrate
of silver to
100 of wat«r.

Observations.

A. 1 B.

12 128-6
10 128-7
8 128-7
6 129-7

129-7
127-0
128-0
130-0

Intensity No. 3.

Parts of nitrate
of silver to
100 of water.

Observations.

A. 1 B.

1

12
10

8
6

110-0 ! 110-0

109-5 109 3
109-6 109-3
119-0 120-0

Intensity No. 2.

Parts of nitrate
of silver to
100 of water.

Observations.

A.

B.

12

10

8

6

125-5
125-5
125-4
161-5

125-0
125-5
124-2
160-2

Intensity No. 4.

Parts of nitrate
of silver to
100 of water.

Observations.

A.

B.

12

10

8

6

90-6
88-0
90-7
89-6

90-0
88-3
89-4
89-0

From these numbers it is seen that the sensitiveness of the paper remains unaltered
when the concentration of the silver solution varies from 8 to 10 or 12 parts of nitrate
of silver to 100 of \vater, but that when a solution containing 6 parts of this salt to 100
of water is employed, the point at which alteration occurs is approached.

The influence of the concentration of the silver solution ha\'ing thus been determined,
it was next necessary to examine the dependence of the sensitiveness of the paper upon
the length of time during which it remained upon the silver solution. For this purpose
pieces of the same homogeneous salted paper were laid for vai-ious times upon the surface
of a silver solution containing 12 parts of nitrate of silver to 100 of water: —
Paper a silvered for ^ of a minute.
„ b „ 1 minute.

,, c „ 8 minutes.

On determining the shades of paper thus prepared and insolated for an equal time,
the following results were obtained : —

Experiment V.

Intensity No. 1.

Intensity No. 2.

Duration

of the
silvering.

Observations.

A.

B.

6 15 140-6
1 0 1 139-0
8 0 ' 139-6

140-5
1400
139-0

Duration

of the
silvering.

Observations.

A.

B.

6 15
1 0
8 0

91-0
91-5
91-5

91-0
90-5
920

150 PEOFESSOE BUXSEN AKD DE. H. E. EOSCOE'S PUOTO-CHEMICAL EESEAECHES,

Experiment V. [continued).
Intensity No. 3. Intensity No. 4.

Duration

of the
silvering.

Observations.

A.

6 15
1 0

8 0

45-9
47-1
45-0

Duration | Observations.

of the I

silvering. [ A.

0 15

1 0
8 0

89-9
90-0
89-2

Hence we may conclude that the time during which the paper lies on the surface of
the silver-bath can vary from 15 seconds to 8 minutes without any difference in the
sensitiveness of the paper being noticeable.

If the duration of the silvering be shortened below the 15 seconds, a film of chloride
of silver is obtamed which is much less sensitive than that obtained by a longer
silvering.

It appeared to be of special impoi'tance to determine by experiment how long a silver-
bath can be used without the quantity of nitrate of silver being reduced below 8 parts
to 100 of water, at whicli point the sensitiveness of the paper may begin to alter. We
found that when a paper was silvered, rather more nitrate of silver than water was
removed from the silver-bath ; that, however, two-thirds of a solution containing 12 of
nitrate of silver to 100 of water may be used up before the quantity of silver salt sinks
fi'om 12 to 8. One square decimetre of paper docs not absorb more than O'Ol grm. of
nitrate of silver from a solution of the above strength.

Not only a diminution in the silver occurs on using the silver-bath, but likewise a
formation of nitrate of soda takes place, which might, by its presence, affect the sensi-
tiveness of the paper. "NYe liave tlierefore compared a freshly prepared silver-bath with
one which had been long in use ; and the results of this examination are seen in the fol-
lowing Tables, and show that the occurrence of the nitrate of soda produces no effect
upon the sensitiveness of the paper.

Experiment VI.

Silver solution.

Intcnsit

y Xo. 1.

Intensity No. 2.

A.

B.

A.

Long used

rrt'>lily iJicpured .
Eroshiy prepared .
Long used

. 130.2
. 130-0
. 130-8
. 130-0

130-8
131-5
130-!)
130-3

73-0
73-4
73-2
74-0

The next series of observations sho^^ the length of time whicli the silvered paper may
be preserved in the dark before insolation without alteration of its sensitiveness. The
paper employed was silvered in a solution contaming 12 parts of nitrate of silver to 100
of water.

PEOFESSOR BUXSEN AXD DE. 11. E. ROSCOE'S PIIOTO-CIIEMICAL EESEAECHES. 151

Experiment VII.

Kept in the
dark for

Intensity No. 1.
A. B.

Kept in the
dark for

Intensity No. 2.
A.

Kept in the
dark for

Intensity No. 3.

1 A

1 hour

5 hours ...
9 hours ...

100-0
98-9

lon-n

101-0
99-0
101-0

5 hours ...

6 hours ...

7 hours ...

8 hours ...

99-3
98-6
98-8
98-4

5 hours ...

6 hours ...

7 hours ...

8 hours ...

111-8

109-8

1 109-4

[ 109-8

Kept in the
dark for

Intensity No. 4.

Kept in the
dark for

Intensity No. 5.

A.

^•

A.

4 hours ...
15 hours ...

99-8
100-8

99-7
101-0

4 hours ... 99-2
15 hours... 100-0

2. Salting the Taj)er.
If the paper be allowed to float upon the surface of the solution of chloride of sodium
as it is allowed to do upon the nitrate-of-silver solution, a paper is obtained which, after
drying and silvering as already described, exhibits a sensitive surface of great irregularity,
as is seen from the following experiments. In these, different parts of the same sheet
of paper lying 1 decimetre from each other were examined, by two observers, A and B.
The readings differ ^videly among themselves, a circumstance which could not occur
if the sensitiveness of the film had been equal throughout the sheet.

Experiment \J\1.

Part of Paper.

Intensity No. 1. | Intensity No. 2.

Intensity No. 3. Intensity No. 4.

Solution containing Solution containing
2 per cent. NaCl. | 4 per cent. NaCl.

Solution containing
7 per cent. NaCl.

Solution containing
8 per cent. NaCL

A. 1 B. 1 A. j B.

A. B. 1 A.

B.

Upper part of sheet ...
Middle part of sheet ...
Lower part of sheet ...

100-0 100-0 96-3
116-5 117-5 100-0
1 i 122-2

114-4 115-0 1 940 1 93-0

122-6 122-5 99-0 996

141-0 1 140-8 109-6 1 109-6

From the above experiments it appears that the most sensitive portions of the sheet
of paper were those which were lowest when the sheet was hung to dry vertically — that
is, those parts by which the salt solution had been most thoroughly imbibed. We
therefore endeavoured to obtain a homogeneously sensitive paper by immersing the
paper in the solution of salt, and allowing it to soak for five minutes. The salt solution
employed contained 4 per cent, of chloride of sodium, the silver solution contained 12
parts of nitrate of silver to 100 parts of water. The following experiments give the
results obtained by this mode of treatment : —

152 PEOFESSOE BUNSEX AXT) DE. H. E. EOSCOES PIIOTO-CIIE.MICAL EESEAECHES.

Experiment IX.

Single Sheet

of

P.iper.

1 Intensity

Xo. 1.

j Intensity No. 2.

Intensity

No. ;?.

1 -ity No. 4.

A.

B.

A. 1 B.

A. i

B.

X. \ B.

Upper part...
.Mid.lK- part
Lower part...

96-9
97-0 ,
97-5

98-0
9o-2
98-0

1 121-6
121-6
122-5

120-2
1200 ,

72-0 1
72-6

720 1
72 0

87'5 i 87-8

87 0 87-8

88 0 87*5

. Intensity Xo. 1. Intensity Xo. 2. : Intensity Xo. 3.

A.

B. 1 A. 1 A. 1 B.

No. 1. Upper part 83 6

No. 2. .Muiille part 84-2

No. 3. Lower part 85-5

83-8 1 69-7 87-0 86-5
84-0 69-7 87-3 87-8
85-6 1 69-2 88-0 88-0

These observations show that, in order to obtain a homogeneous sensitive film of
chloride of silver, the paper must not be laid iqmn but immersed in the chloride-of-
sodium solution.

From the following experiments we learn the influence which the concentration of
the salt solution exerts upon the sensitiveness of the paper. The papers salted in
different solutions were all silvered in a bath containing 12 parts of nitrate of silver
to 100 parts of water.

Experiment X.

Intensity No. 1.

Intensity No. 2

Intensity No. 3

Na CI to
100 of water.

A.

B.

Na CI to

100 of water.

A.

B.

Na CI to ,
lOOofwater. I

B.

1
o

3
4

62-6
95-7
132 6
167-0

60-4

94-7

129-6

168-0

4
5

6

93-2

9-2-9

111-5

93 0

93-3

113-2

6

8

10

12

67-6
83-4
94-7
97-0

68-6
83-7
93-7
95-0

Intensity N

0.4.

Intensity No. 5.

Na CI to 1

lOOof water. '

A.

NaClto ^
100 of water.

B.

13

14-5

16

154-5
159-6
161-6

12
15
18
21

69-0
75-0
95-0
94-5

70-0
78-5
95-0
95-0

An examination of the above Table shows

That the sensitiveness of the paper increases rapidly with increasing strength of
the chloride-of-sodium solution, and that, as far as the observations extend, no

PEOFESSOR BUXSEV AXU DR. H. E. ROSCOE'S riTOTO-ClIEMICAL RESEARCHES. 153

limit exists beyond which an increase or a diminution of the percentage of salt in
solution ceases to affect the sensitiveness of the film.

In order to obtain constant results, it is therefore necessary to employ a solution of
chloride of sodium of unvarying strength. We have decided upon using a solution
which contains 3 per cent, of chloride of sodium. Such a solution is especially conve-
nient, because the paper dipped into it removes salt and Avatcr almost exactly in the
proportions in which they are contained in solution ; thus 225 cubic centimetres of
a 3 per cent, salt-bath was altered from 2-948 per cent. NaCl to 2-935 per cent, by
impregnating 0-72 square metre of paper. In another experiment the strength of
10 litres of salt solution containing 297 per cent. NaCl, was only increased to 308 per
cent, by impregnating 4^ square metres of paper. It is therefore possible to impregnate
6 square metres of paper with a solution containing 60 grammes of chloride of sodium,
without any danger of reducing the strength of the salt solution below the point at
which differences begin to appear.

3. Infiuence of the dcscrijption of Pajter employed.

In the examination of the effect of change of quality in the paper used, we have
confined our experiments to three kinds of paper, diffei-ing extremely in thickness, from
the thickest to the thinnest commonly in use among photographers.

One square decimetre of the first of these, called paper a, weighed 0-354 grm. ; the
same area of the second, called I, w^eighed 0-732 grm. ; and the same quantity of the
third sort, called c, weighed 0-876 grm. From the first series of experiments made
\nih these papers, we thought that the varying thickness of the paper was of the greatest
moment in detennining the sensibility of the film ; thus, for instance, the three sorts of
papers, sensitized in exactly the same way, gave the following unequal readings : —

Papers.

Intensity No. 1.

a
b
c

90-0
75-3
72-5

We soon cominced om-selves, however, that this want of agreement was not caused by
any difference in the sensitiveness of the film, but solely by the difference in the partial
opacity of the papers. If the transparency was got rid of by placing a piece of thick
white paper behind the tinted papers whilst reading off, the following numbers were
obtained instead of the forenroinor : —

Papers. Intensity >'o. 1.

MDCCCLilll.

73-6
73-6
72-0

154 PEOFESSOK BUNSEN AND DE. H. E. EOSCOE'S PHOTO-CHEMICAL EESEAECHES.

The following series of readings show still move clearly that no difference in the shade

of these papers can be observed when a white background is placed behind the tinted

papers : —

Experiment XT.

Paper.

NaClto
100 of water.

1 Intensity No. 1.
A?NO«to 1

Intensity

No. 2.

109-5
112-0
109-5

100 of water. ^

B.

a
b
c

2
2
2

12 73-0
12 73-0
12 G9o

71-0
73-0
73-6

Paper.

NaClto
100 of water.

AgNCto
100 of water.

Intensify
No. 1.

Intensity
No. 2.

Intensity
No. 3.

a
b
c

16
16
16

12

8
10

89-3
91-0
90-0

1 20-0
120-0
120-0

142-4
142-9
141-9

Hence we may conclude

That variation in the thickness of white paper, such as is usually employed for
photographic purposes, is without influence upon the sensitiveness of the him of
chloride of silver.

4. Influence of the Changes of Atmospheric Temperature and Moisture.
In order to become acquainted witli the mfluence exerted by change of temperature
and moisture upon the sensitiveness of the paper, wc gummed portions of the same sheet
of sensitized air-dried paper upon two tin boxes, filled with water of different tem-
peratures, and exposed these two papers for the same length of time to the same inten-
sity of liglit. No greater chfferences in shade were observed in the papers thus tinted
than such as arose from the unavoidable expeiimental errors ; this is seen from the
following numbers : —

Experiment XII.

Intensity No. 1.

Intensity No. 2.

Temperature.

Temperature.

A. B.

+50' C.

+3''C.

-\-L<r c.

+ 4"C.

+30' C.

A. B.

A. 1 B.

A.

B.

A. B.

A.

B.

89-2 88-3

88-0 89-0

88-2 88-5

88-0

88-5

81-6

80-6

81-6

80-6

IIenc(! there can be no doubt

That tlic ordinary cliangcs of atmosplicric temperature and moisture do not affect
the sensitiveness of the paper.
From the n^sults of tin; foregoing series of e\p(!riments it is easy to select th(^ condi-
tions under which a paper of constant sensibility can be obtained.

PEOFESSOE BUNSEN AND DE. U. E. ROSCOE'S PHOTO-CIIEMICAL EESEAECIIES. 155

Preparation of the Standard Paper.

Wc give the following receipt for preparing a constant standard paper, by help of
which comparative measurements can be carried out at any time or at any place : —

300 grammes of pure chloride of sodium are dissolved in 10 litres of water, and the
solution poured into a shallow zinc vessel large enough to float the paper. The sheets
of paper, about 0-3 metre in area, are one by one laid flat in the solution, and the
vessel slightly agitated in order to remove adhering bubbles of air. AVhen the sheet
has been immersed for 5 minutes, it is taken out of the solution and dried by hanging in
the air in a vertical position. The solution thus prepared will serve to salt seventy
sheets of paper, each having an area of 0-3 metre. This salted paper can be preserved
for months without being in any way injured.

The silvering is managed, with the precautions ordinarily in use amongst photographers,
by placing the paper, after each sheet has been cut into four pieces, upon the surface of a
solution containing 120 grammes of nitrate of silver to 1 litre of water, placed in a flat
glass dish. Each piece of paper is allowed to lie upon the surface of the silver-bath for
2 minutes. This litre of silver solution serves to sensitize 500 such pieces of paper,
after which its volume will be reduced to one-half. The standard paper thus prepared
can be preserved in the dark after drying in the air for from 15 to 24 hours without
undergoing any appreciable change in its sensitiveness.

It is scarcely necessary to add that, by adhering to the proportions here given, smaller
quantities of the standard paper may likewise be prepared.

We next give a series of experiments to show that such a standard paper, prepared at
diflFerent times and under various conditions, can always be obtained homogeneous
throughout its surface, and of a perfectly constant sensitiveness.

Exp

eriment XIII.

Size of paper 03

square metre.

Intensity No. 1.

Inliii-

\ .N.i L'.

A. B.

A.

\i.

Upper part ...

115-0 1 115-0
115-4 116-4
116-0 115-4

12'J-4
l!29-0
129-5

126-6
127-2
U'H-O

Middle part
Lower part

The standard paper used for the above experiment consisted of a sheet O'o of a square
metre in size ; and the portions which were selected for trial lay at a distance of about
25 centimetres from each other. In order to show that no irregularities occurred in a
large number of sheets prepared in the same salt solution, wc add a comparison of the
sensitiveness of eighteen sheets of standard paper, each of an area of 0-075 square metre,
salted in a solution containing "2-95 per cent, of chloride of sodium.

t2

156 PEOFESSOE BUXSEX AXD DE. 11. E. EOSCOE'S PHOTO-CHE]\nCAL EESEAECHES.

Paper.

IntonsitT Xo. I.

Paper.

Int^nsii

T No. 2.

A.

E.

A.

B.

.Miiicile part of Sheet No. 1 ...
Middle part of Sheet No. 9 ...
Middle part of Sheet No. 18...

98-0
98-3
98-0

98-3
98-9
99-0

Upper part of Sheet No. 1 ...

" Middle part of Sheet No. 8 ...

Lower part of Sheet No. 17 ...

88-0
89-5
89-5

88-0
89-6
89-5

Intensity No. 3. 1

Intensity No. 4.

A. B. 1

A.

B.

70-0
70-0
71-0
70-5

Upppr part of Sheet No. 2 ... 69*0
Middle part of Sheet No. 4 ... 69-5
Middle part of Sheet No. 11...! 69-3

69-5 ' Lower part of Sheet No. 4 ...

70-8 Lower part of Sheet No. 12 ...

70-0 Upper part of Sheet No. 16 ...

Upper part of Sheet No. 18 ...

70-5
70-2
70-0
70-5

Intensity No. 5.

Paper.

Intensity No. 6.

Paper. 1

A.

B.

A. 1 B.

Upper part of Sheet No. 4 ... 99-5
Middle part of Sheet No. 6 ...[ 100-5
Lower part of Sheet No. 9 ... 101-0

99-9 Lower part of Sheet No. 8 ...
100-0 Middle part of Sheet No. 16...
101-0 Upper part of Sheet No. 17 ...

63-9
62-5
61-8

62-5
62-5
61-5

The following Tables contain the results obtained from three salt solutions of the
same approximate strength but prepared at different times, the exact composition of
each being determined by silver analysis. In these solutions three sheets of paper, each
0-07-5 sq. metre in area, were prepared, the sheets being afterwards sensitized as described.
Here, likewise, the same uniformity is strikingly seen.

Paper.

Na CI to
100 parts
of water.

Intensity
No. 1.

Intensity
No. 2.

Upper part of Sheet No. 2 ... 3-026
Middle part of Sheet No. 3 ... 2-950
Middle part of Sheet No. 2 ... 3-028
Lower part of Sheet No. 2 ...j 3-000

87-0
8C-3
86-0
85-9

75-4

74-4
74-9
74-4

Paper.

NaCl to
100 parts
of water.

■ Intensity No. 1.

Intensity No. 2.

A. B.

A.

B.

Upper part of Sheet No. 2 ...
Middle part of Sheet No. 1 ...
Middle part of Sheet No. 3 ...
Lower part of Sheet No. 2 ...

2-950
.•5-026
3-000
3-026

77"5 76-5
77-0 77-5
78-2 78-0
78-9 78-5

89-0
89-0
90-1
89-9

87-0
88-0
93-0
90-9

PEOFESSOE BUNSEN AXD DR. H. E. ROSCOE'S PTIOTO-CHE:yrTCAX RESEARCHES. 157

Paper.

Na CI to
100 parts
. of water.

Intensity
No. 1.

Upper part of Sheet No. 3
Middle part of Sheet No. 3
Middle part of Sheet No. 2
Lower part of Sheet No. 2

2-950
3-028
3-000
3-028

86-2
87-0
86-8
87-.5

Paper.

NaClto
100 parts
of water.

Intensity No. 1.

Intensity No. 2.

A.

B.

A.

B.

Upper part of Sheet No. 2 ...
Lower part of Sheet No. 2 ...
Middle part of Sheet No. 1 ...
Middle part of Sheet No. 3 ...

2-950
3-026
3-026
3-000

70-2
70-6
70-0
70-0

70-0
69-3
69-0
70-4

101-3
101-5
100-9
101-0

101-5
101-7
100-9
]00-0

We may, therefore, conclude that the standard photographic paper jjrepared
according to the above receipt possesses a degree of sensitiveness sufRciently con-
stant for all the purposes of oui- measurements.

(III.)

For the purpose of obtaining a unit of measurement, we need a perfectly fixed and unal-
terable shade of colour, which can be at any time easily prepared ; this is made by mixing
oxide of zinc and lamp-black in certain proportions, and grinding them together until no
change of tint is produced by further rubbing. The oxide of zinc is prepared in the
wet way chemically pure, and then ignited at a low red heat for 5 minutes in a closed
platinum crucible. The hinip-black is prepared chemically pure by allowing a turpen-
tine lamp to burn under a large porcelain dish filled with cold Avater, and igniting the
soot, whicli is deposited on the outside of the dish, for 5 minutes in a covered platinum
crucible. In this way a fine impalpable powder is obtained which burns without leaving
the slightest trace of ash.

Experiment showed that the tint obtained by mixing 1 part of the lamp-black with
1000 parts of oxide of zinc is that about which the eye can distinguish between very
slight alterations of shade, but that these slight alterations cannot be observed when tlic
tint is either darker or lighter. Hence we have adopted the mixture containing 1000
parts of oxide of zinc to 1 part of lamp-black as the standard tint. In order that the
colour may adhere to the paper, it is mixed with water containing 1 per cent, of isinglass.
During the preparation of the colour it was noticed that the shade of the mixture became
gradually darker when it was well ground on a glass plate and afterwards dried, but that
after a time a point was reached at which no alteration in shade was produced by repeating
the operations of grinding and drying. The following observations of points of equal
shade made upon a fixed strip illustrate this gradual change.

158 PROFESSOR BUNSEN AND DR. H. E. ROSCOE'S PIIOTO-CHEMICAL RESEARCHES.

First Preparation.

A. B.

Mean.

Alter llie Hr~t tTijiidiiif;

AftiT the seoiinil ^ririiling ...

Al'ter Ihi- tliiid rrriudiri".'

-Alter ilie fmirtli <;riniliii^ ...

66-0
72-9
72-4

72-8

G6-2
72-.5
72-6
73-0

G6-1
72-7
72-5
7ii-9

In order to obtain a perfectly constant colonr, the mixture must be aycII gronnd on a
glass plate with water for an hour, then dried in the water-bath, and this operation
repeated until no difference in shade can be detected between the mixture in various
stages on examination in the usual manner upon a fixed strip.

Four separate mixtures, made at different times and with differently prepared consti-
tuents, gave the observations contained in the following Table, in which the constant
nature of the tint produced is seen.

1st Reading. 2nd Reading.

Mean.

A. ' B. 1 A.

B.

1. First preparation

2. Spconil preparaticin ...
'.i. Tliinl preparation ...
4. Fourtli jireparation ...

72-0 j 71-8
72-5 1 72-0
72-9 i 730
72-0 72-0

71-5
72-0

72-2

72-5
72-0

73'2

71-95
72-12
72-95
72-35

We may therefore consider it proved

That the colour used as the measure of the standai'd tint can at any time be
prepared of a constant and unalterable shade.

Having, in the foregoing, described the mode in which a standard photogi-aphic paper
of constant sensitiveness, and a standard tint of unvarying shade can be prepared, we
need only apply the proposition tliat equal products of the inten.sities into the times of
insolation produce equal shades of blackness, in order to found a method of measurement
and comparison of the chemical action of the total daylight. As the unit of measurement,
we propose to adopt

That intensity of the light which in one second of time produces the standard
tint of blackn(>ss upon the standard paper.

"WIkmi the standtird juiper is insolated in the pendulum-apparatus, a strip is obtained
which is tinted with c\cy\ gradation of shade from dark to white. If the pomt on this
strip which coincides in shade with a piece of paper covered with the standard tint
be determined by m(>ans of the arrangement (fig. 3), we have only to look for the corre-
sponding reading of the millimetre scale in Table I. to obtain the time of insolation t in
seconds which was necessary in order to produce this shade. If this time of insolation
were found to be one second, th(> intensity of the liglit then acting would, according to
definition, be 1=1. For any other time of insolation, t for example, the intensity of the

PEOFESSOR BUNSEN AND DR. 11. E. ROSCOE'S PHOTO-CHEMICAL RESEARCHES. 159

chemical rays would bo j. The following Table (II.) gives in column II. tlio intensities,

for one vibration of tlie pendulum, corresponding to the points of equal shade of the
standard paper and standard tint, as read off on the millimetre scale in cohuun I.
The intensities corresponding to n vibrations of the pendulum arc obtained by dividin"
the numbers iii column II. by n.

Table II.

I.

n.

I.

II.

I.

II.

I. 1

II.

I.

II. 1

I.

II.

Millims.

Intensity.

Millims.

Intensity.

Millims.

Intensity.

Millims.'

Intensity.

Millims.

Intensity. '

Mi.lims.

Intensity.

0

0-834

33

0-997

64

1-183

96

1-429

128

1-824 ,

160

2-710

1

0-839

33

1-002

65

1-190

97 1

1-439

129

1-840

161

2-763

o

0844

34

1-007

66

1-197

98

1-448

130

1-856

162

2-816

3

0-8-J9

35

1-012

67

1-203

99

1-4.^.8

131

1-874

163

2-869

4

0-853

36

1-018

68

1-209

100

1-467

132

1-892

164

2-9-'3

5

0-858

37

1-023

69

1-215

101

1-477

133

1-911

165

2-977

6

0-864

38

1-029

70

1-221

102

1-487

134

1-930

166

3-048

7

0-869

39

1-0.S4

71

1-228

103

1-497

135

1-949

167

3-119

8

0-874

40

1040

72

1-235

104

1-507

136

1-969

168

3-190

9

0-879

1 ^1

1-046

73

1-242

105

1-517

137

1-990

169

3-262

10

0-884

42

1-051

74

1-249

106

1-528

138

2-011

170

3-334

11

0-889

43

1-0.-7

75

1-256

107

1-539

139

2-032

171

3-437

12

0-894

44

1-062

76

1-263

.108

1-551

140

2-053

172

3-534

13

0-899

45

1-068

77

1-270

109

1-563

141

2078

173

3-650

14

0-904

. 46

1-074

78

1-277

110

1-575

142

2-103

174

3-759

15

0-909

47

1-079

79

1-285

111

1-586

143

2-128

175

3-891

16

0-914

48

1-085

80

1-293

112

1-598

144

2-153

176

4-016

17

0-919

49

1-090

81

1-301

113

1-610

145

2-179

177

4-167

18

0-924

50

1-096

82

1-309

114

1-622

146

2-207

178

4-3f)7

19

0-929

51

1-102

83

1-317

115

l-6.i4

147

2-235

179

4-5(;6

20

0-935

i ^2

1-108

84

1-325

116

1-647

148

2-263

ISO

4-807

21

0-940

53

1-114

85

1-333

117

1-660

149

2-291

181

5-051

22

0-945

54

1-120

86

1-.J42

118

1-673

150

2-320

1S2

5-348

23

0-950

55

1-127

87

1-350

' 119

l-6s6

151

2-354

183

5-682

24

0-955

56

1-133

88

1-3.^9

, 120

1-700

152

2-3«9

184

6-212

25

0-961

57

1 139

89

1-367

121

1-715

153

2-424

185

6-848

26

0-966

58

1-145

90

1-376

122

1-730

154

2-459

186

7-633

27

0-971

59

1-151

91

1-385

123

1-745

155

2-494

187

8-620

28

0-976

60

Mo6

92

1-394

124

1-760

156

2-537

29

0 981

61

1-163

93

1-402

125

1-776

157

2-580

30

0-986

62

1-170

94

1-411

126

l-79i

158

2-623

31

0-992

63

1-176

95

1-420

127

1-808

159

2-666

The observations are carried out in the manner fully described in the commencement
of the present communication. In order to make several observations quickly after each
other, the screw (k, fig. 1) is loosened, and, after each observation, the slide (G) drawn
out rather more than the width of the slit. The readings are also made in the way
described, with the arrangement fig. 3, half the hole in the block (fig. 4) being occupied
with paper covered with a thick layer of the standard tint. Care must be taken that the
white background upon which the strip is placed is free from spots or dirt, which by
appearing through the paper may alter the readings. Tlie pa[)er on which the standard
tint is painted must not be too thin, and must be thoroughly au-diied before use. Each

100 PEOFESSOE BUNSEX AXD DE. H. E. EOSCOE'S PHOTO-CHEMICAL EESEAECH ES

comparison of shade is made five or six separate times, the scale being covered up dm-mg

every reading, and the mean of these observations recorded.

As an example of such measurements, we append several observations representing the

chemical action exerted upon a horizontal surface by the whole sunlight and diflPuse

daylight duruig the various hours of the day. These observations, Avhich are contained

m Table III., were carried on in Manchester, on the roof of the Laboratory of Owens

CoUege, and were made on days in which the sun sometimes shone, and sometimes w^as

obscured by clouds. The observations are represented by the curves (fig. 5), and the

maxima and minima correspond exactly with the appearance and disappearance of the sun.

From these few obser\ations an idea may l)e formed of the -\ast difierences exhibited by

the chemical activity of sun- and day-light about the periods of the longest and the

shortest davs.

Table III.

Wednesday

, Dceember 18,

18C1.

Thursday,

Deceml

er 19, 1861.

Wednesday, .July 30, 1862.

f.

. 1 ..

^=1.

/.

i.

i=i.

f.

i.

71.

-=I.

11

h m

h m

h

m

10 6 a.m.

1-05

124

0-00847

9 .39 a.m.

1-79

120

0-0149

7

0 A.M.

0-88

60

0-0147

10 16

2-49

170

0-0147

9 49

2-10

150

0-0140

7

20

0-85

32

0-0266

10 26

1-60

100

0-0160

10 1

1-89

120

0-0157

7

35

1-07

25

0-0428

10 36

l-4<)

90

0-0166

10 21

1-93

100

0-0193

7

50

0-89

16

0-0556

10 47

1-47

100

0-0147

10 31

1-72

80

0-0215

8

0

0-83

10

0-0830

10 56

1-34

80

0-0168

10 41

2-05

80

0-0256

8

35

0-92

12

0-0767

11 6

1-47

80

0-0184

10 51

1-66

80

0-0208

9

0

1-33

15

0-0887

11 16

1-59

100

0-0159

11 1

1-93

90

0-0215

9

5

1-20

10

0-120

11 26

1-41

80

0-0176

1! 11

1-91

80

0-0239

9

30

1-22

7

0-174

11 36

1-39

75

0-0185

11 21

1-91

80

0-02;i9

10

10

1-12

5

0-224

11 46

1-25

80

0-0156

11 31

1-91

SO

0-(i2:!9

10

20

0-91

5

0-182

11 .56 a.m.

1-46

66

0-0221

11 41

1-73

80

0-0216

10

30

0-83

10

0-0830

12 6 i-.M.

1-52

60

0-0253

11 51 .\.M.

1-69

Gl

0-0277

11

0

0-86

11

0-0782

12 16

1-42

50

0 0284

12 1 ,...M.

1-66

60

0-0277

11

30

0-86

4

0-215

12 26

1-42

45

0-0316

12 11

1-54

50

0-0308

12

0

0-86

3

0-287

12 36

1 20

40

0-0300

12 21

1-49

50

0-029S

12

30 P.M.

0-86

3

0-287

12 46

0-92

80

0-0115

12 41

1-10

50

0-0519

1

30

0-88

6

0-147

12 57

1-02

120

0-0085

12 51

1-37

65

0-0211

2

0

1-11

8

0-139

1 6

1-19

90

0-0132

1 1

1-02

50

0-0204

2

30

1-33

13

0-110

1 16

1-38

75

0-0184

1 11

1-12

65

0-0172

3

0

1-22

9

0-1.36

1 26

1-22

65

0-0188

1 21

1-56

90

0-0173

4

0

1-27

15

0-0846

1 36

1-05

50

0'0-210

1 36

1-69

86

0-0197

4

35

1-22

18

0-0678

1 47

0-84

60

0-0140

1 46

1-75

100

0-0175

5

0

1-49

20

0-0745

1 56

1-2G

100

0-0126

1 56

1-54

100

0-0154

5

30

1-34

25

0-0536

2 10

1-36

150

0-00906

2 6

1-22

100

0-0 1 22

6

0

1-24

40

0-0310

2 22

1-34

150

0-00893

2 16

1-40

120

0-0117

2 32

1-41

160

0-00881

2 27

1-59

160

0-00994

2 42

l"5o

200

0-00775

2 45

1-50

180

0-OOS33

2 52

1-36

225

0-00529

2 53

1-25

160

0-00781

3 5

1-56

400

0-00390

3 8

1-45

250

0-00580

3 25 P.M.

1-53

450

0-00340

3 21 P.M.

1-72

500

0-00344

At the close of this communication we may remark tliat, by help of the pendulum-
apparatus described, we have constructed a portable instrument by which a large
nvuiiber of measurements can b(> made on a few square inches of paper. "We reserve the
description of this instrument for a future occasion.

[ 161 1

VIII. On the Immunity eiijoycd hj the Stomach from being digested hj its otvn Secretion
during Life. By F. W. Pavy, M.B.

Received April 29,— Read May 7, 1863.

In a communication, bearing the above title, that was read before the Royal Society,
January 8, 1863, I brought forward experimental evidence which had conducted me to
view the immunity enjoyed by the stomach from being digested by its own secretion
during life, as resulting from the neutralizing influence on the acidity of the gastric
juice exerted by the stream of alkaline blood flowing through its parietes. The oppo-
sition that this view received on the evening of its announcement induced me to extend
my experiments, and as from the additional results obtained some important confirmatory
e^'idence can be adduced, I have deemed it desirable to present this further communi-
cation, in which the whole subject is concisely re^"iewed with the aid of the new matter
that has been brought to light.

John Huntkr directed attention to the point under consideration in a paper entitled
" On the Digestion of the Stomach after death," which is contained in the Philosophical
Transactions for 1772. After adverting to the fact that in occasional instances, espe-
cially in persons who have died of sudden and violent deaths, the stomach is found so
dissolved at its greater extremity as to have allowed of the escape of its contents into
the abdominal ca%ity, and, without an actual perforation occurring, that there are very
few dead bodies in which some degree of digestion of the coats of the organ may not be
observed, Hunter gives reasons for concluding that the condition described must be
owing to the action of the digestive fluid after the occurrence of death, and not the
result of disease in the living subject. The stomach being thus afiirmed to be suscept-
ible of digestion by its own secretion after death, it became necessary to account for
its not undergoing a similar process of digestion during life. According to Hunter's
view it was the "li%ing principle" that afforded the required protection to the living
organ.

Post-mortem examinations of the human body supply constantly recm-ring examples
of the gastric solution that Hunter has described. Experimentally, however, the effect
may be rendered much more strikingly manifest. If, for instance, an animal, as a rabbit,
be killed at a period of digestion, and afterwards exposed to artificial warmth to prevent
its temperature from falling, not only the stomach but many of the surroimding parts
will be found to have been dissolved. With a rabbit killed in the evening and placed in
a warm situation (100° to llOTahr.) during the night, I have seen in the morning the
stomach, diaphragm, part of the liver and lungs, and the intercostal muscles of the side

MDCCCLXIII. z

162 DE. PAVT ON THE IMMUNITY ENJOYED BY THE STOMACH

upon which the animal was laid, all digested away ; with the muscles and skin of the
neck and upper extremity on the same side also in a semi-digested state.

Submitted to examination, Hunter's idea about the protecting influence of the " living
principle" does not stand the test of experiment. To Claude Bernard, of Paris,
science is indebted for suggesting an ingenious mode of experimenting with reference to
this point. Through an artificial opening into the stomach of a dog, Bernard intro-
duced the hind legs of a living frog whilst digestion was going on. As the result, the
parts were digested and dissolved away notwithstanding the frog continued alive. My
own experience enables me to offer corroborative evidence as regards this experiment
upon the frog ; but further, I have found that tissues belonging to a warm-blooded
mammal have likewise shown themselves susceptible of attack under subjection to the
influence of the gastric digestive menstruum.

Taking for experiment a vigorous rabbit, I carefully introduced one of its ears throTigh
a flstulous cpening into the stomach of a dog at a period of full digestion. Precautions
were used to avoid inflicting mechanical injury upon the ear in placing and retaining it
in position ; and, at the same time, to avoid, as far as possible, obstructing the flow of
blood through its vessels. At the end of two hours, the ear was withdrawn, and several
spots of erosion, some as large as a sixpenny piece, were observed on its sm-face ; but
nowhere was it eaten completely through. On being replaced for another two hours
and a half, the tip, to the extent of rather more than half an inch, was almost com-
pletely removed, a small fragment only being left attached by a narrow shred to the
remainder of the ear ; a considerable escape of blood took place, especially towards the
latter part of the experiment.

To replace the refuted influence of the " living principle," it has been suggested,
that it is the epithelial lining which gives to the stomach the immunity from destruc-
tion it enjoys during life. The stomach, it has been said, is lined with an epithelial
layer, and this, with the mucus secreted, acts as a kind of varnish in protecting the
deeper parts. Whilst digestion is proceeding, the epithelium and mucus are constantly
being dissolved like the food contained in the stomach ; but, a fresh supply being as
constantly produced, the organ is thereby maintained intact. Death taking place, and
the epithelial layer being no longer produced, the gastric juice, after acting upon and
dissolving it, reaches the deeper coats, and then, continuing to exert its influence, may
ultimately, the temperature being maintained sufficiently favourable for the purpose,
occasion a ])erforation of the organ.

Such is tlie view that has been propounded, but, like the " living principle," it fails to
stand the test of experiment.

As regards tlie mucus, an exaggerated notion may be formed, respecting its amount and
importance, if an examination of the stomach be made when even a short time only has
been allowed to elapse after death. With the rabbit, for example, under such circum-
stances, on opening the stomach a more or less thick, pulpy, white pellicle is found to
adhere to the mass of food which the organ contains. This, however, consists of

FROM BEING DIGESTED BY ITS OWN SECRETION DURING LIFE. 163

digested mucous membrane ; for, examined immediately after deatli, the stomach lifts
off from tlie food, leaving the latter uncovered by auytliing that is visible. The mucous
membrane itself, also, is firm in structure througliout.

That the stomach cannot derive its protection from the epithelial layer, as suggested,
is proved by the fact, that a patch of mucous membrane may be rtunoved, and food will
afterwards be digested without tlie sliglitest sign of attack being made upon the deeper
coats of the organ. I have several times performed the experiment, to enable me to speak
safely on the point, and never have I had the slightest evidence, that depriving the
stomach of a portion of its mucous membrane has left the denuded part in a position of
greater insecurity than the rest, on the score of liability to digestive attack. It is upon
the dog that these experiments have been made ; and, upon one occasion, after removing
the mucous membrane, and exposing the muscular fibres over a space of about an inch
and a half in diameter, the animal was allowed to live for ten days. It ate food every
day, and seemed scarcely affected by tlie operation. Life was destroyed whilst digestion
was being carried on, and the lesion in the stomach was found very nearly repaired :
new matter had been deposited in the place of what had been removed, and the denuded
spot had contracted to much less than its original dimensions. In other experiments, I
have examined the stomach at earlier periods after the operation. Life has always been
destroyed whilst digestion has been going on. The day after the operation, I have found
the denuded spot irregular and raw. Then lymph is deposited upon its surface, and,
apparently through the organization of this, the Avails arc gradually thickened, and the
process of reparation carried out.

In addition to the evidence afforded by experiment, it may be assumed, upon reflection,
that something more constant — some condition presenting less exposure to the chance of
being influenced by external circumstances than that supplied by the existence of an
epithelial layer, would be required to account for that unfailing security from ante-
mortem solution which the stomach appears to enjoy. From the articles swallowed,
abrasion of the mucous membrane may be presumed to have been not unfrequently
produced, and ulceration is not so uncommon an occurrence ; yet, perforation has not
been observed as the necessary result. Perforation, it is true, does sometimes occur as
a consequence of ulceration, but the same is the case in other parts of the alimentary
tract, and there is reason to regard it here, as elsewhere, as resulting from a gradual
advance of the ulcerative process, and not from a special digestive action exerted by the
gastric juice.

The notion, then, that the stomach is prevented from being digested during life, because
it is a liiing stnicture, is disproved by the consideration that the parts of living animals
that have been introduced into the digesting stomach have not shown themselves capable
of resisting its digestive influence. That the epithelial layer, also, Avitli its capacity for
constant renewal, does not afford the explanation needed, is proved by the absence of any
solvent action being exerted by the digestive fluid upon the deeper coats when the part
has been completely denuded of its mucous membrane. The question, therefore (and

164 DE. PAYT OX THE IMMUNITY ENJOYED BY THE STOMACH

an exceedingly imi^ortant one it must be admitted by all to be), still remains open for
solution, Why does the stomach, composed as it is of digestible materials, escape being
digested itself whilst digestion is being carried on in its interior 1 It is evident, whatever
explanation, with any pretence to sufficiency, is given, must comprise some broad prin-
ciple of action capable of providing against all contingencies — capable of affording, in
fact, that uninterru2)ted security during life which upon looking around us we observe
the stomach to enjoy.

The view that 1 have to offer refers the immunity observed to the circulation within
the walls of the organ of an alkaline current ; and this agrees with the principle I have
laid down as indispensable, for the cii'culation of blood forms with us an essential
condition of life. It will not be disputed, that the presence of acidity is one of the
necessary circumstances for the accomplishment of gastric digestion. Now, alkalinity is a
constant character of the blood, and as diiring life the walls of the stomach are every-
where permeated by a current of this alkaline blood, we have here an opposing influence,
the effect of which would be to destroy, by neutraUzing its acidity, the solvent properties
of the digestive fluid tending to penetrate and act upon the texture of the organ.

The following point is also worthy of note in passing. In the arrangement of the
vascular supply, a doubly effective barrier is, as it were, provided. The vessels pass
from below upwards towards the surface : capillaries having this direction ramify between
the tubules by which the acid of the gastric juice is secreted. Acid being separated by
secretion below must leave the blood that is proceeding upwards correspondingly
increased in alkalinity ; and thus, at the period when the largest amount of acid is
flowing into the stomach, and the gi'eate.st protection is required, then is the provision
afforded in its highest state of efficiency. Looking to nature's secretion alone, the act
creating a demand for protection enhances the character of the protection provided.

The blood being stagnant after death, the opposing influence is lost that is offered by
the circulating current. Should life happen to be cut short at a period of digestion,
there is only the neutralizing power of the blood actually contained in the vessels of
the stomach, to impede the progress of attack upon the organ itself; and the consequence
is, that digestion of its parietes proceeds, as long as the temperature remains favourable
for the process, and the solvent power of the digestive liquid is unexhausted. There is,
therefore, no want of harmony between the effect that occurs after death, and the expla-
nation that refers the protection afforded during life to tlie neutralizing influence of the
circulation.

Having thus stated the nature of tlie view propounded, I next proceed to show in what
manner it answers to the test of experiment.

It occurred to me, that, if the circulation really fulfilled the office I have alleged, the
act of arresting the flow of blood through the walls of the stomach during life ought to
lead to the same, or about the same, effect on the organ, other circumstances being
equal, as would occur after dcnvth. The experiment being performed upon dogs and
rabbits, I observed, as the result, digestion proceed to the extent of perforation in the

FEOM BEIXG DIGESTED BY ITS OWX SECRETION DTJKING LIFE. 165

rabbit, whilst in the dog 1 did not witness a greater effect than some amount of solution
of the mucous layer.

Having before me the effect I have described as ensuing when a rabbit is killed, and
its temperature is afterwards maintained artificially, and taking this as an index of the
effect to be looked for in these experiments, I had to account for the absence of perfo-
ration occurring in the dog. I conceived, at first, that the circulation in the surround-
ing parts which would exist during life, and not after death, might produce a modifying
influence on the result. To what extent this is true is shown by the following experi-
ments.

A couple of rabbits that had been f(>d alike were killed at a period of digestion. 'J'he
stomachs were immediately removed, and the one immersed in some freshly-drawn,
defibrinated sheep's blood ; the other, in a solution of gum and sugar made to correspond
to the blood in density. The gum was introduced to take the place of the albumen, and
the sugar, the salines, so as to have a fluid that would behave about like blood as regards
osmosis. The liquids were placed side by side in an oven, and the temperature main-
tained at about 100^ Fahr. At the end of 4| hours, the stomach immersed in the solu-
tion of gum and sugar had undergone perforation, and allowed of the escape of its
contents. The other Avas still entire, but digested in its interior so as to be reduced
to only a thin layer. In another experiment the effect was not allowed to proceed so
far. Both stomachs remained externally entire, but that immersed in the solution of
gum and sugar ])rcscnted, in a distinctly marked mannin-, evidence of more extensive
attack than the other.

Through much subsequent experience I l(>arned that I had in reality been labouring
under an exaggerated notion, and that the standard I had taken from the rabbit was
unjust in its application to the dog. The result of actual experiment on tliis animal
shows a marked difference in degree of effect produced by the digestive action of the
contents of the stomach after death to that wliich occurs in the rabbit. In the experi-
ments thus performed, the animals have been killed about four or five hoius after a
meal of animal food. The temperature of the body has then been maintained for five
and six hours closely to that belonging to life. Now, at the end of this time, the stomach
has only shown signs of more or less digestion of its mvicous membrane, a condition
that has been about equalled in some of my experiments, where the flow of blood
through the stomach has been arrested at a period of digestion, and the animal allowed
to live for about five or six hours afterwards. By means of ligatures applied around tlie
pylorus and the end of the oesophagus, and also around the vessels passing between the
spleen and greater curvature of the organ, its circulation is with security and facility
arrested, and its contents at the same time prevented from escaping. It is in this way
that the experiments during life have been all conducted.

In the case of the rabbit, as I have said, I have witnessed digestion of tlie stomach
proceed to the extent of perforation, as the result of stopping the flow of blood tlirough
the vessels. The process of digestion, however, being so much influenced by the tempe-

166 DR. PA\"r ON THE IMMUNITY ENJOYED BY THE STOMACH

rature, it is necessarj' to observe, with a small animal like the rabbit, that it is not placed
in a cold situation, for the heat to decline after the operation has been performed.
Without artificial warmth and during cool weather I have seen the cardiac extremity
of the stomach digested away in less than eight hours. In an experiment, however,
where artificial warmth was employed, perforation was observed at the end of four
hours. The operation was performed four hours after food had been given. The
animal was then placed in an atmosphere with the thermometer standing at 92° Fahr.
In four hours' time it was killed, and the parts were examined immediately. The stomach
throughout was in an advanced state of digestion, and was perforated in one spot of
about the size of a shilling.

The contents of the stomach in the rabbit are always observed most powerfully acid,
much more so, according to what I have seen, than in the case of the dog. From the
nature of the food some acid may be generated in addition to that derived from the
blood by secretion. Xow, upon the quantity of acid, amongst other circumstances, the
energy of the digestive menstruum depends, and, in harmony with this, it can be shown,
that if an acid (an acid that is not of a nature to exercise of itself any direct erosive
effect) be introduced into the stomach of a dog at a period either of digestion or
fasting, and the circulation through the walls of the organ be afterwards stopped, the
efiect W'hich occurs is even considerably stronger than what has been hitherto referred to
in the rabbit. I may mention three experiments in proof of this assertion. The acids
employed were purposely selected on account of their non-corrosive properties. In the
first, the animal was taken six hours after a full meal of animal food. One fluid ounce
of the dilute phosphoric acid of the London Pharmacopana, mixed with an equal quan-
tity of water, was introduced into the stomach, and the circulation through the organ
afterwards, in the usual way, arrested. Death took place during the night, and a large
perforation was found in the cardiac extremity of the stomach. In the second, six
drachms of the same acid, diluted with an equal quantity of water, were employed, and
this time upon an empty stomach. Perforation took place in 2^ hours' time. In the
third, GO grains of citric acid, dissolved in two ounces of water, constituted the acid used,
and tliis time also it was at a period of fasting that the experiment was performed. In
four hours death occurred from perforation.

It is thus rendered evident, that all that is Avantcnl in tiie dog to produce digestive
destruction of the stomach when its circulation is arrested, is the presence of a sufficient
amount of acid in its interior. With a limited amount of acid the power of the gastric
juice soon becomes exhausted, and tliere being food as well as the stomach to act upon,
this exhaustion may occur before any marked attack upon the organ has taken jdace.
With a larger amount of acid, however, tlic exliaustion does not at this early pcniod
arrive, and the stomach continues to be acted upon until a perforation of its coats may
be effected.

In striking contrast to the (effect a1)ove narrated, of introducing a mild acid into the
stomach and ligaturing the vessels, are the results I have obtained from introducing the

FEOM BEING DIGESTED BY ITS OWN SECRETION DURING LIFE. 167

acid without the operation on tlie m'sscIs. Three experiments were performed, using the
same acids, ivnd the same quantities of them, that had been employed before. Ligatures
were placed around the end of the oesophagus and the pylorus to secure the retention
of the acid in the stomach, care being taken, however, to avoid including the vessels.
The circulation was thus left free to exercise its neutralizing influence, whilst, in other
respects, the circumstances of the experiment were the same as befoi-e. Wliere one
ounce of the dilute phosphoric acid, mixed with an equal quantity of water, was
employed, the animal was alive on the following day, and when killed, the stomach was
found free from unnatural appearance, with the exception of a number of small ulcerated
spots strewed over the internal surface. These did not extend through the mucous
membrane, and looked like what might be supposed to result from the action of an
irritant. Where the six drachms of dilute phosphoric acid, and the same of water,
were used, the animal was also alive on the following day. The mucous membrane of
the stomach presented here and there an appearance of congestion, and a few small spots
of superficial ulceration towards the pyloric end. The organ was otiierwise ibund in a
natural condition. With the 60 grains of citric acid, dissolved in two ounces of water,
the animal, from some cause, died during the night. The stomach was found everywhere
perfect ; and, in this case, was without the slightest appearance of ulceration of its surface.
The last dog had been taken at a period of digestion, the other two of fasting.

It will be seen how strongly the above results stand in support of tlie view I have
brought fonvard. The stomach yields to the digestive influence of its contents in one
set of experiments, and not in the other, the only difference in the experiments being,
that the flow of blood through the vessels of the organ is arrested in the former and
allowed to continue in the latter. The circulation being allowed to continue, a check
is offered to the penetration of the Avails of the stomach by its contents in an acid state,
and thus the freedom from attack that occurred. The circulation, on the other hand,
being arrested, there is no such neutralizing influence in operation ; the acid menstruum,
therefore, is able to attack the stomach just as it does the food in its interior.

A mode of experimenting which I am indebted to Dr. SnARPEY for suggesting, like-
wise gives confirmation to the view I have propounded. If an incision be made into the
stomach and a portion of the opposite wall be drawn forward ; and then, if a ligature be
placed around this so as to stop the circulation of blood through the part, the constricted
portion, on being returned and left projecting in the interior of the stomach, will
undergo digestion just as if it consisted of a morsel of food. In one experiment the
operation was performed on a dog at a period of fasting. Food was given on the
following morning, and 7^ hours afterwards the animal was killed. In tlie act of
remo\ang the stomach the parts surrounded by the ligature fell asunder, leaAing a large
circular opening from 1|^ to 2 inches in diameter. There was not a vestige of the con-
stricted mass to be discovered ; it had all been digested au ay. In another experiment
some food was in the stomach when the ligature was applied. Although vomiting
twice occurred soon after the operation was completed, and death took place in twenty

168 DE. pa"\t: on the immtjnitt enjoyed by the stomach

hours' time without any more food being given, more than half the projecting mass was
di2:ested away, as though it had been cleanly sliced off transversely ; so that, when the
liijature was removed and the stomach spread out, a hole fully an inch in diameter
presented itself As such an effect could not have taken place without the presence of
gastric juice, it is to be inferred, that the whole of the contents of the stomach had not
been ejected by the vomiting, and that the appearance observed was produced during
the first few hours after the operation. At death there was nothing whatever con-
tained in the organ. The special attack upon the most projecting part of the ligatured
mass is probably to be explained by the contracted state of the stomach allowing only
this portion to fairly present itself as a part of the surface in contact with the contents,
as lonjj as anv remained. I might bring forward more evidence than the above from
mv laboratory experience ; but these experiments I think suffice to show, that a portion
of the stomach, to the exclusion of the rest, may be rendered susceptible of digestion by
the removal of the protecting influence I conceive to be afforded by the circulation.

It will naturally be recjuired of me to reconcile the view I have advanced vnih the
effect that was noticed in an early part of this communication as occurring w^here the
living frog's legs and rabbit's ear were introduced through a fistulous opening into the
stomach, and submitted to the influence of the digestive menstruum. If the circula-
tion, throuijh its neutralizing power, protect the stomach, why should it uot afford equal
protection to the tissues of living animals, introduced through a fistulous opening into
the digesting organ ? I thus state the question openly, because it is one that requires to
be openly met.

According to the proposition I have offered the stomach is protected, because the
neutralizing power of its circulation is sufficient to overcome the acidity of the gastric
juice which is tending to penetrate and attack its texture. Now, this consideration, it
will be seen, involves the result in a question of degree of power between two opposing
influences. Diminish the neutralizing power of the circulation beyond a certain point,
and allow the strength of the digestive liquid to remain the same; theoretically, the
result should be in fa\our of digestion instead of protection ; practically, this may be
regarded as what ha))pens in the experiments with the frog's legs and rabbit's ear.
Allow, on the other hand, the neutralizing power belonging to the circulation of the
stomach to remain tlie same, but increase beyond a certain ])oint the strength of acidity
of the digestive li(iuid ; theoretically, digestion of the stomach's parietcs would be
looked for as the result ; practically, it can be shown that this is really what occurs, as
will be seen by an experiment to which I shall pn^sently refer.

"With the living frog's legs introduced into the digesting dog's stomach, it may be
fairly taken that the amount of blood possessed by the frog would be totally inade-
quate to furnish the required means of resistance to the influence of the acidity of the
do"-'s •mstric juice. With the rabbit's ear the vascularity is so much less than that
of the parietes of the dog's stomach, that there is nothing, to my own mind, incom-
prehensible in the fact of the one yielding to, and the other resisting attack. No com-

FEOM BEING DIGESTED BY ITS OWN SECRETION DUEING LIFE. 1G9

parison can be drawn between the position of the stomach, and that of the rabbit's
ear. The stomach is not only in itself exceedingly vascular, but is entirely surrounded
by equally vascular parts. The rabbit's ear is only supplied with blood that reaches it
at its base, and, immersed in the stomacli, it would lie completely bathed all around by
gastric juice.

From the experiments I have mentioned it has been seen, that the introduction of a
moderately strong acid liquid into the stomach leads to the production of a solvent
effect on tlie organ when its circulation is stopped, which does not occur when the
circulation is allowed to remain free. Now, if the strength of the acid be increased, the
stomach shows itself to be susceptible of attack, although the circulation may have been
left undisturbed. In an experiment upon a dog whilst fasting, I introduced 3 ounces
of a liquid, consisting of 3 drachms of muriatic acid and the remainder of water, into
the stomach, and afterwards ligatured the end of the oesophagus and pylorus without
including the vessels. In one hour and forty minutes death took place, and, on the
parts being examined immediately, perforation was found, with an escape of the con-
tents of the stomach into the peritoneal cavity. The interior of the stomach throughout
had undergone an extensive dissolution ; and in the neighbourhood of the perforation,
which was at the cardiac extremity, the textui'e presented quite a gelatinized appear-
ance. This result was evidently the effect of digestion ; for the acid, at the strength it
was employed (one part in eight), does not possess such physical corrosive properties.
A considerable escape of blood had taken place from the stomacli during its attack ; and
it may be reverted to, in comiexion with this, that there was also a considerable amount
of hemorrhage observed from the rabbit's ear, whilst being attacked in the stomach of
the dog. I take it, in the above experiment, that the height of acidity in the stomach
was very much too great for the neutralizing capacity of the circulation : and thus, the
rapid progress of digestive solution.

There is one more point that remains to receive consideration. It would be incom-
patible with my ^ievr, that a living organism could exist in a free state in the stomach,
whilst digestion is going on, without being attacked ; unless this organism should consist
of, or be protected by, an indigestible material. It is well known, however, that larva? of
the (Estrus inhabit the stomach of the horse, but it will be found that they live with
their heads firmly attached to, and buried in, the mucous membrane ; indeed, there is
sometimes quite a honeycomb arrangement in which the greater portion of the animal
can be lodged. Living upon the juices of the animal these larvae infest, they become
more or less, as it were, a part and parcel of the stomach's parietes. It is further to be
remarked, that the principle (chitine), which forms the basis of the tunic of insects, is
of an exceedingly indigestible character. By Professor Simonds I have been informed
of an entozoon (a species of Filaria) which he has found in the last stomach of the
sheep ; but this parasite also lives firmly attached to the mucous membrane, and in
connexion with the juices of the animal it infests. I have not been able to learn, that
any example can be brought foi-Avard of life being carried on, under isolated circum-

MDCCCLXIII. 2 A

170 DE. PA"VTr ON THE ODIUOTTY ENJOYED BY THE STOMACH

stances, in a digestible organism placed in the interior of an actively digesting stomach.
The older physiologists found, in their early experiments on digestion, that such animals
as leeches and earthworms, placed in perforated metal spheres, and introduced iii a
living state into the digesting stomach, were attacked, just as if they had consisted of
ordinary food.

The following resume may be taken as representating the main points of what has
been adduced in this communication : —

That Hunter's suggestion of the "living principle" forming the source of protection
to the stomach from being digested by its own secretion during life, is negatived by
CL.iUDE Ber.vard's discovery, that parts of living animals introduced tlirougli a fistulous
opening into the digesting stomach, are observed to undergo digestion like its other
contents.

That the epithelial layer, also, with its capacity for constant renewal, does not furnish
the explanation required, is proved by the experimental evidence brought forward,
showing, that a patch of mucous membrane may be removed, and digestion still be carried
on, without the denuded part being digested.

That in default of the sufficiency of these suggestions, the view I have been led to
entertain refers the immunity enjoyed by the stomach from being digested during life,
to the influence of an alkaline circulation. Acidity is necessary for digestion, and alka-
linity is a constant character of the blood. Whilst the walls of the stomach, therefore,
are permeated by a current of blood, an opposing influence is offered to digestive attack.
Death taking plaee, there is no longer a circulation of alkaline fluid to exert a neutrali-
zing effect on the acidity of the gastric juice tending to penetrate and attack the organ:
the consequence is, that digestion now proceeds, according to the nature of the circum-
stances that prevail.

That this view is supported by experimental evidence of the following description : —
By ligaturing the vessels of the stomach so as to arrest the flow of blood through
the organ, it is rendered susceptible of attack by its contents during life in like
manner as after death.

In the rabbit, digestion of the stomach has been thus observed to proceed to
the extent of perforation.

In the dog, the action has not been witnessed to proceed beyond a solution of
the mucous layer.

Upon introducing, however, into the stomach of the dog, previous to ligaturing
the vessels, a moderate quantity of a dilute acid — mineral or vegetable — a
perforation of the organ in each of the three experiments performed has ensued.
The introduction of the same acids, similarly diluted, and in like amounts,
without the; operation of ligaturing the vessels, so that the cuxulation has been
left free, has not occasioned any digestive attack.

By pinching up and ligaturing a portion of the walls of the stomach so as to

FROM BEING DIGESTED BY ITS OWN SECRETION DURING LIFE. 171

leave a constricted mass projecting into the cavity of tlie organ, this has been

found to undergo digestion like a morsel of food.
That the attack upon the li\ing frog's legs and rabbit's ear introduced into the
digesting stomach of a dog need not be looked upon as forming any valid objection to
the view propounded. The explanation is one that involves the result in a question of
degree of power between two opposing influences. Because, through degree of vascu-
larity, the neutralizing power of the circulation is sufficient to hold in check the solvent
action of the gastric juice in the case of the walls of the stomach, it does not follow
that it should similarly be sufficient to do so in the case of the legs of a frog or the ear
of a rabbit. The circumstances are far from identical in the two cases ; and, in support
of what has been stated, it can be shown by experiment that even with the stomach
itself, by increasing the acidity of its contents beyond a certain point, its circulation is
no longer adequate to enable it to resist digestion.

That the capacity of a li^'ing and digestible organism to exist in an isolated state in
the interior of the digesting stomach would be incompatible with the view that has
been annoimced. Instances can be brought forward of animals inhabiting the stomach,
but they do not foiTU examples of the above description.

[ 173 ]

IX. On ThalUum. J?yAViLLiAM Crookes, Esq.
Communicated by Professor G. G. Stokes, Sec. R.S.

Received February 5, — Read February H), 1803.

Occutrence, Distribution, and extraction from the Ore.
1. Since the date of the last paper on Thallium which I had the hcnour of communi-
cating to the Royal Society*, I have been unremittingly engaged in attempting to find
a source from which this metal could be extracted in quantity. Having first, discovered
thallium in the deposit from the chambers of a sulphuric-acid manufactory, I naturally
turned my attention towards similar deposits from English oil-of-vitriol works where
pyrites was burnt. Applications were accordingly made to several large manufacturers
for specimens of the pyrites whicli tlicy used, and also for some of the deposit from
their leaden chambers. These requests, with scarcely an exception, were readily
responded to, and in a short time I was in possession of specimens from nearly thirty
different establishments. In many instances thallium was detected in the pyrites, but I
was disappointed to find that the deposits of sulphate of lead from the chambers con-
tained no thallium whatever. I then applied to manufacturers who I had ascertained
were constantly burning thalliferous pyrites, and obtained from them specimens of the
products in different stages of tlieir manufacture, but in no instance did I find an
accumulation of thallium in any part of th(^ operations.

2. In the operation of burning the pyrites, the thallium oxidizes with the sulphur
and volatilizes into the leaden chambers; it there meets with aqueous vapour, sulpliu-
rous and sulphuric acids, and becomes converted into sulphate of the protoxide of
thallium. This being readily soluble both in water and dilute sulphuric acid, and not
being reduced by contact with the leaden sides, remains in solution and accompanies
the sulphuric acid in its subsequent stages of concentration, &c. It is not probable
therefore that any thallium can accumulate in the insoluble deposit, but it will remain
dissolved in the liquid, where indeed I have found it — not liowever in quantities suffi-
cient to be worth extracting, as it is present in scarcely a largcu' proportion tlian in the
original pyrites. That this view of the path followed by thallium is correct, I am
satisfied both from careful analyses of products from various manufactories, and also by
experiments tried on a small scale in my own laboratory. M. Lamy states that he
extracts thallium from similar deposits to those which I have examined ; but as I have
experimented on residues from English manufactories in which they burn pyrites
almost, if not quite, as rich in thallium as that used in M. Kuhlmann's works, there

* Proceedings of the Koyal Society, June 19, 1862.
MDCCCLXIII. 2 B

174 ME. W. CROOKES ON TKALLIUM.

must be some cause or local arrangement in their manufactory, different from what is
usually adopted in this country, to occasion so large an accumulation of thallium at one
particular stage of the operations*.

3. Having failed in my endeavours to find a residue from a manufactory which would
yield thallium, I turned my attention to the ores in which it was likely to occur.

The iinique collection of minerals brought together at the International Exhibition
of 1862 enabled me, through the kindness of the various commissioners and class-super-
intendents, to verify the opinion which I had formerly expressed as to thallium being a
very widely distributed element. It most frequently, indeed almost invariably, occurs
in iron pyrites containing more or less copper ; but I have also detected it in native
sulphur, in zinc, cadmium, bismuth, mercury, and antimony ores, as well as in the manu-
factured products from these minerals. Thallium is confined to no particular country,
but is at the same time by no means uniformly distributed in mineral veins from the
same locality, or even in adjacent rocks in the same mine. Hitherto I have detected
no approach to law in its distribution. In every country mineralogically represented in
the late Exhibition I have detected the presence of thallium, when the minerals examined
were at all numerous and could be regarded as fair average samples of the different
deposits. Doubtless much of this apparent abundance is to be attributed to the extreme
delicacy of the test employed, as ores in which thallium is present, only in the proportion
of 1 to 100,000, give e\ddent traces of it in the spectroscope. Many pyrites, however,
contain more than a mere trace of thallium : it is present, in sufficient quantity to be
readily extracted by direct treatment, in pyrites from various parts of North and South
America, France, Belgium, Spain, as well as Cornwall, Cumberland, and many parts of
Ireland.

4. The optical process of detecting thallium in a mineral is very simple. A few
grains only of the ore have to be crushed to a fine powder in an agate mortar, and a
portion taken up on a moistened loop of platinum wire. Upon gradually introducing
this into the outer edge of the flame of a Bunscn's gas-burner, the characteristic green
line will appear as a continuous glow, lasting from a few seconds to half a minute or more,
according to the richness of the specimen. By employing an opake screen in the eye-
piece of the spectroscope to protect the eye from the glare of the sodium line, I have
in half a grain of mineral detected thallium when it was only j)resent in tlie proportion
of 1 to 500,000. After a few experiments of this kind, and having a thalliferous
pyi'ites of known richness for comparison, it i.s easy to give a rough estimate as to tlie
quantity of thallium prcscnit.

5. One of the rich(>st thalliferous minerals in the Exhibition was the prominent block

* This anomaly has just been cleiircd uji by a iiujicr communicated to the French Academy, on the 2()th ultimo,
by M. Fitfeu. Xuiii.MANN, in which he t-xjilains that, in order to prevent the passage of arsenic from the pyrites
into the sulphuric acid, he intei-poscs, betwcon the pyrites-kilns and the ordinary leaden chambers, a largo
supplementary chamber, in wliich the jiroducts of combustion arc lowered in temperature and deposit the more
easUy coudensible volatile matters. It Ls in this deposit that thallium is found.

MR. W. CROOKES ON THALLIUM. 175

of pyrites, weighing nearly two tons, wliicli formed so conspicuous an object in the
Belgian department ; it was exhibited by the Societe Anonyme de Rocheux et d'Oiieux
of Theiix, near Spa (Belgium, Class I., No. 18 in the Catalogue). Accompanying the
principal block were several smaller specimens from different parts of the same locality.
Tliallium was detected in all, but in very varying proportions. By the great kindness
of Professor Chandelon of Liege, one of the jurors of Class II., I was enabled to
examine a series of examples from four separate chambers, and eight different rocks
from the mine whence these specimens were brought, he ha\ing most courteously taken
the trouble to visit the locality and select the specimens for me. In some, as in rocks
C, D, and G, no appreciable quantity of thallium was present; in chamber No. 1 there
was a trace only, Avhilst in chamber No. 2, and in rock II, it was present in comparative
abundance. Professor Ch.\ndelon, to whom I desire to offer my hearty thanks for his
uniform courtesy and valuable assistance, informs me that those jJortions of the mineral
vein which I have found most rich in thallium are close to a vein of blende and
calamine, Avhich is worked in the neighbourhood for zinc. I am promised some of this
blende and calamine for analysis ; and it will be of some interest to know the relative
quantity of thallium they contain in comparison witli that in the adjacent pyrites. I
have already found considerable quantities of thallium in metallic zinc and cadmium
manufactured from this ore.

Professor Chandelon not only troubled himself to make several journeys from Liege
to the mines on my account, but likewise induced M. RSnard (the manager) and the
proprietors to present me with upwards of two tons of the mineral carefully selected
from those parts of the mine which I had found to be richest. This arrived in Sep-
tember last.

6. Numerous experiments have led me to adopt the following process for extracting
the new metal from its ore. The pyrites is first broken up into pieces about the size
of a walnut, and placed in cast-iron retorts, capable of holding 20 lbs. each ; five of
these ai'e an-angcd in a reverberatory furnace, so that the flame may lick round and
heat them uniformly ; iron condensers being luted on, the temperature is raised until
the retorts become of a bright red heat, at which they are kept for about four hours.
They are then allowed to cool, and the product removed from the condenser; 100 lbs.
of pjTites usually give from 13 to 17 lbs. of sulphur. The sulphur is of various
colours, according to the temperature attained, sometimes being orange, sometimes
pui'ple-brown, sometimes bright grass-green, and is always highly crystalline. Experi-
ments tried at various times during the distillation, show that thallium is present in the
first portions of sulphur which distil over, as well as in the last, although the proportion
somewhat increases towards the end. When the heat has been sufficient, and the yield
of sulphur above 15 per cent., scarcely a trace of thallium remains behind in the retorts ;
but with a low heat, the sulphtir is almost pure yellow in colour, and contains very
little thallium. Each operation lasts about five hours; the iron retorts frequently
serve for two, and sometimes for as many as five or six heatings. They gradually,

2b2

176 ME. W. CEOOKES ON THALLIUM.

however, become converted into sulpliide, and then melt avv'ay. This sulphide of iron
contains no thallium. Within the last three months I have distilled upwards of
16cwt. of iron pyrites, yielding me 2^ cwt. of thalliferous sulphur. The results are
appended : —

November 8,1862, 14 lbs. of pyrites yielded

!5

14, ,

, 24

„

29, ,

, 120

December

11, 55 70

55

13, „ 100

n

15, „ 100

n

16, ,

, 100

Januaiy

3,1863, 100

!5

5,

100

5»

7,

100

55

8,

100

,,

9,

100

55

10,

100

55

12,

100

,,

14,

100

55

24,

100

55

26,

100

55

27,

100

55

29,

100

55

30,

100

1828

lbs. ozs.

2 6 of

sulphur.

3 8

12 7

8 12

12 5

12 8

14 5

13 5

12 9

11 4

15 4

14 7

14 13

16 3

15 2

16 4

13 13

17 2

14 5

11 9

252 3

It may be of some interest to state that, although the mineral is put in in its ordinary
air-diy state, I invariably get from cacli lOOllis. of pyrites about half a pint of liquid
distilled over before^ th(> sulphur comes. Upon evapoiating this down, water, having a
peculiar empyreumatic odour, goes off, oxide of iron is precipitated, and sulphite of the
protoxide of iron separates in large, nearly colourless c.ystals: no thallium is present.

7. I have met with gn-at difficulty in extracting economically, and without loss, the
whole of the thalliiun from this sulpluir. On the small scale, nothing is easier than to
boil it, finely powdered, in fuming nitric acid or aqua ref/ia, until the residuary sulpluu'
is of a pure yellow colour, and then to extract the thallium from the solution by pro-
cesses to be hereafter described ; but such operations are impracticable when working
by the hundredweight.

An attempt was made to separate the thallium by converting the thalliferous sulphur
into chloride. Half a pound was trc^ated in this manner; but upon rectifying the pro-
duct, thallium was found in the distillat(> as well as in the residue. An immense
number of experiments were tried to extract the thallium from tlie chloride of sulphur.

ME. W. CROOKES ON THALLIUM. 177

but with no success, unless intlood moans were employed wliicli would have hecn equally
easy and economical upon tlu^ original snlpliur. The simplest plan would doubtless be
to dissolve out the sulphur with bisulpliide of carbon, in which sulphide of thallium is
insoluble ; and this is a plan that I hope before long to have in full work ; hitherto,
however, I have found the following the readiest method : — In a large cast-iron caldron
dissolve 12 lbs. of good caustic soda in 1.} gallon of water, heat it to the boiling-pcint,
and then add 18 lbs. of the sulphur in large lumps, just as it comes from the receivers.
A few pounds should be added at a time, as dissolved, and the mixture kept gently
boiling, water being added from time to time to replace that lost by evaporation.
When no more sulphur can be dissolved, dilute the mixture with four or five times its
bulk of water, and allow it to cool. A voluminous black precipitate will separate,
which must be collected on a calico filter. The greater portion of the thallium remains
in this precipitate in the form of sulpliide, together with iron, copper, &c. ; but some
passes through dissolved in the alkaliuL' liquid. I have hitherto been unable to recover
the thallium from this solution without an incommensurate expenditure of both acids
and time. After a slight washing, the black precipitate is transferred to a laige dish,
and boiled in sulphuric acid until sulphuretted hydrogen ceases to be evolved ; nitric
acid is then added in small quantities, and the mixture is boiled until all solvent action
has ceased ; it is then diluted with ^vater, and filtered. Evaporate down until all excess
of nitric acid is removed, and then add liyihochloric acid and sulphite of soda to the
liquid ; this produces an immediate precipitation of the thallium in the state of proto-
chloride as a white crystalline precipitate, only slightly soluble in water; as, however, a
certain quantity of this chloride remains dissolved, it is advisable to add, after the
sulphite of soda, iodide of potassium, which precipitates the whole of the thallium in
the form of an insoluble yellow iodide. If the solution turns almost black upon adding
the iodide of potassium, or the precipitate comes down of a dirty grey colour, it is a sign
that an insufficient quantity of sulphite of soda has been added ; a further addition will
remedy this. A little copper, which is generally present in the sulphur, will likewise be
precipitated in the form of subiodide. The iodides are to be filtered off, and washed
until they are free from iron. They may now be decomposed by heating with oil of vitriol,
which converts them into sulphates ; but tlie temperature required for this being very
high, and the decomposition difficult to effect perfectly without loss of thallium by vola-
tilization, I prefer the following, though somewhat longer process : — Boil the iodides in
excess of sulphide of ammonium until they are entirely converted into sulphides ; filter,
and wash with weak sulphuretted water until no iodine can be detected in tlie washings;
then boil the precipitate with strong sulphuric acid, adding a little nitric acid from time
to time ; evaporate until the sulphuric acid begins to go off in white vapours, and then,
after dilution with water, add an excess of ammonia. Now add cyanide of potassium
until the blue ammoniacal solution of copper is decolorized, and then a slight excess of
sulphide of ammonium, and gently warm. Filter and wash with dilute sulphuretted
water untU all the cyanide of potassium, &c. is removed, and boil the precipitated

178 ME. W. CEOOKES ON THALLIUM.

sulphides with strong sulpliiiric and nitric acids until the mass becomes perfectly white.
Drive off the whole of the nitric acid and the greater portion of excess of sulphuric acid
by heat, and then boil the residue for some time in water. Filter, and wash well. A
white insoluble residue will generally be left, containing a little thallium ; but the greater
portion of this metal will be in the solution in the form of sulphate.

8. From this solution the metal is readily obtained in the metallic state by voltaic
precipitation. Two or three cells of a Grove's battery, with platinum terminals dipping
into the aqueous solution of the sulphate (either acid or rendered alkaline with
ammonia), produce an immediate reduction, oxygen being evolved at one pole, and the
metal coming down at the other. The appearance presented when a tolerably strong
solution of tliallium is undergoing reduction is very beautiful. If the energy of the
current bears a proper proportion to the strength and acidity of the liquid, no hydrogen
is evolved at the negative electrode, but the metal grows from it in large crystalline fern-
like branches spreading out into brilliant metallic plates, and darting long needle-shaped
crystals, sometimes upwards of an inch in length, towards the positive pole, the appear-
ance strikingly resembling that known as the tin tree. Some of the tabular crystals, as
seen in the liquid, are beautifully sharp and well defined, their angles being temptingly
measurable ; considerable difficulty is, however, met with in disengaging them from the
electrode, and removing them in a perfect state from the liquid. So long as thallium
is present in the solution, no hydrogen is evolved with a moderate current ; as soon as
bubbles of gas begin to form, the reduction may be considered complete. The crystal-
line metallic sponge may now be squeezed into a compact mass round the platinum
terminal, and, being disconnected from the battery, quickly removed from the acid liquid,
rinsed with a jet from a wash bottle, and transferred to a basin of pure water. The
metal is then carefully removed from the platinum, and kneaded with the fingers into
as solid a lump as possible. It will be found to retain its metallic lustre perfectly
under water, and coheres together readily by pressure. The lump may now, after
having been dried with blotting-paper, be put into a steel-crushing mortar and strongly
hammered until it assumes the form of a solid ingot. To obtain this in a fused mass,
the best plan is to break it up into small pieces, and drop them one at a time into a
crucible containing fused cyanide of potassium, at a low red heat ; they melt at once
and run together at the bottom of the crucible into a brilliant metallic button : allow
the crucible to cool, and dissolve out the flux with water, when the thallium wiU be left
in the form of an irregular lump, owing to its remaining liquid and contracting after the
cyanide has solidified. As long as the surface is wet, either with the solution of cyanide
or witli pure water, th(> metal presents a highly crystalline appearance, resembling tin
when washed with acids ; this disappears upon exposure to the air, owing to the forma-
tion of a pellicle of oxide.

I'/n/sical CJiaraderistics of Tliallium.

9. Thalliuia has a distinct colour of its own, not being absolutely identical with any

MR. W. CROOKES ON THALLIUM. 179

other metal. Tlie true colour can be seen by scrapininj a piece of the metal under water,
or fusing it in hydrogen and allowing the melted globule to flow away from the dross.
In appearance it most resembles tin and cadmium, not being so brilliantly white as silver,
but without the blue tinge belonging to lead. It is susceptible of taking a veiy high
polish : by rubbing it under water with a fine polishing stone, the surface can be made
smooth and briglit, reflecting as perfectly as a mirror.

It oxidizes in the air with almost the rapidity of an alkali-metal. When fre.shly cut
with a knife, if the eye follows the blade, the proper colour of the metal will be seen
to assume, in a few seconds, a yellowish cast, caused by a thin coating of the protoxide,
which continues to increase until the metallic lustre is obscured by a grey film, scarcely
distinguishable from the superficial tarnish of metallic cadmium. At this stage the
oxidation appears to be almost arrested, and the metal may be freely handled and
exposed to the air, with scarcely any further change. After having remained in the air
for some weeks, the surface becomes covered with a white powder, which easily rubs off,
and has a strong biting taste.

When rubbed between the fingers, a faint peculiar smell may be observed, unlike that
produced by any other metal under the same circumstances. If a perfectly bright sur-
face of thallium is applied to the tongue, no taste whatever is observed ; but a tarnished
surface tastes strongly alkaline, and somewhat sweet like oxide of lead ; whilst if the
surface is more oxidized than usual, from the metal having been long exposed to the
air, or prenously raised to a high temperature, the taste is very caustic and biting,
remaining on the tongue for some hours, and resembling that observed when the tongue
is applied to the terminals of a voltaic pile.

10. Thallium is the softest known metal admitting of free exposure to the atmo-
sphere : it can be cut, pressed out, and moulded with the finger-nail with the utmost
ease ; and whilst it is incapable of abrading the surface of a piece of lead, this latter
metal scratches thallium with great facility. ThaUium marks paper as easily as lead :
the streak is blue at first, but almost instantly turns yellowish, and in the course of a
few hours nearly fades out, from oxidation. The writing can, however, be restored at
any time to more than its original blackness by exposure to sulphuretted hydrogen or
sulphide of ammonium. Thallium is too soft to file well or be cut with a saw, as it clogs
up the teeth directly, and it does not become brittle when exposed to a low temperature.
It has very little tenacity, being inferior to lead in this respect. It is very malleable,
and may be hammered out into leaves as thin as paper; it may also be moulded and
pressed in a die, taking a very sharp impression. When repeatedly hammered, it does not
appear to get sensibly harder or require annealing. I have succeeded in drawing it into
wire ; but, owing to its want of tenacity, this is a matter of some difficulty ; the wire may,
however, be obtained in a very easy way by pressure. Dr. Matthiessen has been good
enough to prepare for me several specimens of Avire by pressing the metal with a power-
ful vice through a fine hole in a steel box, receiving the wire as it issued forth in tubes
filled with dry carbonic acid, and various liquids : I have also prepared it since in similar

180 am. w. crookes on thallium.

apparatus of my own. It is squeezed into wire more readily than lead, and when received
into dry carbonic acid or petroleum, without contact with air, and instantly sealed up,
its true metallic lustre and colour are very apparent ; if, however, a very minute trace of
air obtains access, it assumes a deep blue appearance. This is remarkable, as the first
superticial coat of oxide which forms on a freshly cut surface, freely exposed to the air,
is distinctly yellow.

11. I have carefully determined the specific gravity of thallium. It varies according to
the treatment the metal has previously undeigone. A lump melted and allowed to cool
slowly under cyanide of potassium, Avas found to have a specific gravity of 11 'SI. The
same lump, after being strongly hammered in a steel mortar, had its density increased
to 11-88. Another portion, experimented on in the form of thick wire, had a density
of 11 '91 immediately it had been squeezed through the die. I believe that thallium is
capable of undergoing still greater condensation. If in the process of squeezing it into
wire the vice is screwed up until the thallium is just making its appearance through the
fine aperture, and the pressure is then kept stationary, the issue of metal proceeds for a
few seconds and then stops. Upon now applying the fiame of a spirit-lamp to the die,
a piece of wire from an eighth to half an inch long quickly shoots out. This cannot be
due to ordinary expansion by heat, as it commences and terminates abruptly at an
apparently definite temperature. Were it expansion by heat, the formation of wire
would proceed nearly uniformly as the temperature increased. The most probable
explanation is, that the application of heat gives the metal power suddenly to release
itself from an abnormal state of condensation into which the enormous pressure had
forced it.

Thallium wire is almost devoid of elasticity ; it retains any form into which it is bent
with scarcely a tendency to sjjring towards its former position.

When freshly prepared, thallium wire is perfectly amorphous, and remains so if kept
at the ordinary temperature in petroleum or carbonic acid ; in water, however, it gradu-
allv assumes a superficial ci'vstalline appearance, resembling the moire mftallique of tin
plate: this efi'ect is immediately produced when thallium in wire, ingot, or plate,
tarnished or clean, is boiled in water.

12. In fusibility thallivun stands between bismutli and lead, its melting-point being
;j5()^ 1-alir. : it does not ajjpear to become soft and ])asty before undergoing complete
fusion. Two pieces of the metal, when perfectly clean, are capable of welding together
in the cold by strong pressure. I have repeatedly tilled the steel die with small scraps
and cuttings of thallium, and squeezed them out into one continuous length. Wire so
made is a])parently as tenacious as that obtained (i'oni one lunii).

Thallium volatilizes easily: when heat(>d out of contact with air, it evolves vapours at
a red heat, and Ixjils below a white lieat. in a ciu'rent of hydrogen gas it may be
easily distilled at a red licat ; it does not, however, condense very pcn-fectly ; for if the
hydrog(>n be ignited, e\en after traversing four or fi\e feet of cold glass tube, it burns
witli a bri":ht green flame.

MK. AV. CROOKES OX THALLIUM. ISl

13. Thallium is a pretty good conductor of heat and electricity. My friend Dr. ]\Iat-
TiiiESSE.v is at present engaged in the quantitative determination of these constants of
the metal, upon specimens with which 1 have supplied him. Its electro-chemical
position is very neai" cadmium, being precipitated from the sulphate by zinc and ii'on,
but not by cadmium or copper. Professor "Wiieatstoxe is examining in this respect
specimens of the metal specially purilied for the purpose.

Thallium is strongly diamagnetic. A small permanent horseshoe magnet was sus-
pended vertically, poles downwards, to one arm of a very delicate balance, capable of
turning to 0-0005 of a grain. After being accurately counterpoised, a lump of metallic
thallium, weighing about 200 grains, was placed beneath the magnet, very close, but not
touching. Upon now observing the weight of the magnet, it was seen to be decidedly
repelled by the metal, losing in weight 0-003 of a grain.

By the kindness of Professor Faraday, who himself tried most of the experiments, the
above observation was verified w ith the large electro-magnet belonging to the Royal Insti-
tution. Upon suspending a long cylindi-ical helix of thallium wu-e horizontally in the mag-
netic field, and making contact with the batter)-, it was strongly driven to an equatorial
position. A lump of the metal, suspended almost in contact with one of the magnetic
poles, was repelled from it, being permanently deflected nearly a tenth of an incli from the
perpendicular. From a comparison with bismuth under somewhat similar circumstances,
I am inclined to believe that thallium is one of the most diamagnetic bodies known,

14. Thallium readily alloys with other metals. AVith 95 per cent, of copper it forms
a hard button, flattening somewhat under the hammer, but soon cracking at the edges ;
the further addition of thallium produces a very hard and brittle gold-coloui'ed alloy ;
and when the proportion of thallium is further increased, the colour of the copper is
entu-ely lost. A very minute quantity of thallium, less than half a per cent., melted
with copper, greatly diminishes its malleability and ductility, acting in this respect like
arsenic. 1 believe that the variation in the physical properties of different specimens of
commercial copper (a variation which has never yet received satisfactory explanation)
is to be attributed to the presence of more or less minute traces of thallium, as I have
found it present in many samples of bad copper, as well as in some specimens of crystal-
lized sulphate (29). This subject is still under investigation.

Five per cent, of thallium alloy readily with tin, when they are melted together under
cyanide of potassium ; the resulting compound is perfectly malleable.

With mercury, thallium forms a solid crystalline amalgam.

Thallium melts readily Avith platinum : if a portion of the metal is placed on the end
of a platinum wire and heated to redness, a fusible alloy is obtained, which is crystalline,
very hard, and almost as brittle as glass under the hammer. If this alloy is heated
before the blowpipe, the characteristic green colour- is vividly communicated to the outer
flame ; before the oxyhydrogen blowpipe the green light is of extraordinary splendour.
A similar alloy is left when the platinochloride of thallium is heated to redness in a
crucible, chlorine going off together with a little thallium.

JIDCCCLXIII. 2 c

182 ME. W. CEOOKES OX THALLIUM.

15. Tlie remarkable simplicity of the spectrum of thallium has given rise to repeated
experiments with a 'iiew to ascertain whether, and under what circumstances, it could
be obtained compound. I have already stated* that "a flame of sufficient temperature
to bring the orange line of lithium into view produces no addition to the one thallium-
line ; and an application of telescopic power strong enough to separate the two sodium
lines a considerable distance apart still shows the thallium-line single." The former
observation has lately been verified and extended by Dr. W. Allen MiLLERf , who has
also noticed that at the high temperature of the electric spark se\eral new lines, especially
two green and a blue line, make theii- appearance. The latter observation, as to the
unresolubility of the green line, was recently put to a crucial experiment. My friend
Mr. Bkowxixg. the well-known ^philosophical-instrument maker, kindly allowed me to
examine the thallium-spectrum in a spectroscope which he is making for Mr. Gassiot.
This instrument is furnished with nine flint-glass prisms of large size ; and although as
yet unfltted \^^tli the elaborate adjustments with which it will be ultimately furnished,
it is even now capable of producing, with a moderate magnifying power, an apparent
separation of the eighth of an inch in the two sodium-lines, at the standard microscopic
distance of ten inches. Under this enormous amplifymg power the thallium-line was
still seen single, being as fine and sharply defined upon the black ground as either of
the constituents of the double sodium-line. I have stated that in a spectroscope of the
ordinary size the line appears to be identical in refrangibility with the line S in the
barium-spectrum ; a comparison of the two spectra in this large instrument showed me
that these lines do not coincide in position.

The delicacy of this optical test for thallium is very great. I prepared a standard
solution of sulphate of thallium, and diluted it until it was in the proportion of one grain
dissolved in fifteen gallons (or about 1,000,000 grains) of water. Upon dipping a plati-
num •«-ire loop into this solution, and holding the moistened end in the flame of the
spectroscope, the green line was distinctly visible. The quantity of liquid taken up by
th(> ])latinuni wire was about the fifth of a grain, containing, tlierefore, no more than
570 0 0,0 0 0*^1 of a grain of sulphate of thallium.

A flame strongly coloured with thallium can be obtained by passmg hydrogen over
chloride of thallium at a liigh temperature and then igniting it. The absolute mono-
chi'omatic character of the light renders everything illuminated by it either green or jet
black : coloured sealing-wax, ribbons, a bouquet of flowers, as well as coloured precipi-
tates, are entirely altered in appearance, whilst tlie luiman face assumes a horrible,
corpse-like green hue.

The green thallium-line can be reversed in the spectrum ; but tliis is an (experiment of
some difficulty. The effect can, however, be well seen by adopting the plan wliich I
have u.sed to show the same phenomenon in the case of sodium — by holding a small
thallium-flame in front of a lai'ger one coloured with the same metal, the mantle of the
front flame being projected as a black line on the hinder flame. If a trace of lithia is

• rrocccdings of the Royal Society, June 19, 1862. t I^id- January lo, 1863.

MR. W. CROOKES ON THALLIUM. 183

added to the larger flame, insufficient to destroy the green colour, the effect of contrast
is very striking ; tlie border of the front flame being opake to thallium, whilst it is trans-
parent to lithium, the flame appears of a beautiful green colour with a crimson edge.

IC. The atomic weight of thallium has been a subject to which my attention has been
directed for some months, in fact ever since I commenced to obtain the metal in suffi-
cient quantities to enable me to purify it to the requisite degree without too much
diminishing my stock. The investigation is far from concluded as yet, and the numbers
which I have obtained must be regarded as only approximate. M. I^amy* gives the
equivalent as 204. As, however, he gives no statement respecting the processes adopted
to arrive at this figure, and we are not even informed of the number of experiments of
which this is the mean result, of the quantities of material operated upon, or the
divergence of each result from the mean 204, it is impossible to know what value is to
be attached to it. Below I give the results of five separate determinations by difFcn-ent
methods: I admit they do not agree so closely as one would like in experiments of this
sort ; indeed the discrepancies are beyond the probable error of analysis, and seem to
point to some other disturbing cause not yet ascertained. I give, howe^•cr, all the
necessary figures, and the results may be taken for what they are worth.

About 200 grains of thallium, prepared as already described, were specially tested for,
and purified from, metals with which it was likely to be contaminated, by processes
appended in the analytical notes, and were obtained in the form of sulphate. This
was recrystallized twice, and the metal precipitated from its aqueous solution by two
Grove's batteries, platinum electrodes being employed. The metal was then fused
under cyanide of potassium, and, after being cleaned in dilute acid, preserved for use in
a dry bottle filled with coal-gas. Some of this purified metal was then dissolved in
dilute sulphuric acid, the solution was evaporated down, and the residue heated until
sulphuric acid ceased to come off; it was then redissolved in water, and the sulphate of
thallium allowed to crystallize. The salt was then considered to be sufficiently pure for
analysis.

I. Some of the crystals were heated to incipient fusion and weiglied, they were then
dissolved in water, and iodide of potassium was added until no further precipitation of
iodide of thallium took place. The precipitate was then warmed and allowed to settle,
collected on a tared filter, washed with Avatcr, dried in a water-bath and Avcighcd.

II. Another portion of sulphate of thallium was heated and Aveighed as above
described, and the aqueous solution precipitated with nitrate of baryta. The precipi-
tated sulphate of baryta was then collected as usual, well washed, and weighed.

III. A thii'd portion of svdphate of thallium was weighed as before, dissolved in a
small quantity of warm water, and mixed with a slight excess of pure hydrochloric acid ;
alcohol Avas then added, and the precipitated chloride of thallium collected on a tared
filter, washed with alcohol, and weighed.

IV. A piece of metallic thallium was weighed and converted into sulphate. The

* Comptcs -Eendus, December 8, 1862.

2c2

184 IME. W. CEOOKES ON THALLIUM.

excess of sulphimc acid being driven off by heat, the remaining sulphate of thallium
was heated to its fusing-point and then weighed.

V. The sulphate of thallium obtained in experiment IV. was dissolved in water and
mixed with an excess of bichloride of platinum. The precipitated platinochloride of
thallium, which is more insoluble than the potassium salt, was then collected on a tared
filter, washed, and weighed.

Tlie following Table shows the experimental results obtained : —

Experiment I.
7".342 grains of sulphate of thallium yielded 9-G55 grains of iodide of thallium. Call-
ing X the equivalent of thallium, we have the following proportion,
7-312 : .r+48 : : 9-COG : x+V21, .: .r=202-73.

Experiment II.
D'SSS grains of sulphate of tliallium gave 4-577 grains of sulphate of baryta.
9-883 : .r+48 : : 4-577 : .r+llC-5, .-. .r=203-55.

Experiment III.

8-555 grains of sulphate of thallium yielded 8-127 grains of chloride of thallium.

8-555 : .r+48 : : 8-127 : .r+35-5, .-. .r=r201-85.

Experiment IV.
lO'llo grains of thallium yielded 12-503 grains of sulphate of thallium.
10-113 : w:: 12-503 : .r+48, .-. .r:=203-l.

Experiment V.

12-503 grains of sulphate of thallium yielded 20-312 grains of platinochloride of

thallium.

12-503 : ^(•+48 : : 20-312 : .r+205-2, .-. .r=203-5G.

I have therefore adopted the mean result 202-96 or 203 as the equivalent of thal-
lium, wi-iting the protoxide TIO and the sulphate TIO.SO,, unless, indeed, as appears
probable from theoretical considerations, tliese compounds have to be expressed Tl^O and
TI2O.SO3, in which case the atomic weight would be half this number.

Clteiiiical I'roj)crties of T/iaUiiim.
17. Tliallium do(>s not decompose pure water, either at the common temperature or
when boiling. If, however, steam be passed over the metal at a red heat, it is decom-
posed, with formation of oxide of thallium and separation of hydrogen, the gas evolved
burning with a decided green flame. The oxide which forms superficially when thallium

:mk. w. cPxOokes ox TnAi.LTor. 185

is exposed to tlie air is the ])rotoxicle. a powerful base, soluble in water, forming a liquid
which is strongly alkaline to test-paper. If a lump of thallium, weighing 00 or 100
grains, is placed, after exposure to the air for a few days, in an ounce of water and
boiled for a few seconds, the solution will be found to possess alkaline characters.
It turns litmus pajier strongly blue, browns turmeric paper, has a metallic alkaline
taste, and perfectly neutralizes acids. It also precipitates alumina from a solution of
alum, evolves ammonia from chloride of ammonium, and reacts with hydrochloric acid,
iodide of potassium, sulphide of ammonium, &c., in the characteristic manner of a thal-
lium-salt. As might be imagined, thallium is readily acted on by air and water jointly ;
and by shaking up pure thallium wire in a bottle with an insufficient quantity of water
to cover it, allowing fresh air to have access from time to time, a strong solution of
oxide of thallium can be obtained. When thallium is melted in the air, it behaves very
similarly to lead, rapidly oxidizing and becoming coated with a fusible oxide resembling
litharge. Upon continuing the heat, this increases, whilst the bright globule in its
centre diminishes in size. The fused oxide is absorbed by bone-ash, and I have ascer-
tained that a silver-thallium alloy can be cupelled like silver-lead. When the metal is
strongly heated on charcoal before the blowpipe, it volatilizes in brownish fumes, which
are without odour. Upon removing the heat, the red-hot globule of metal continues to
bum and give off vapours for some time afterwards, like pure antimony under similar
circumstances. On cooling, the adjacent parts of the charcoal are coated with globules
of sublimed metal. The oxide resembling litharge is the same as that formed by the
superficial action of air on the metal, or steam at a high temperature ; it may also be
prepared in strong solution by decomposing sulphate of thallium with baryta water and
filtering. The oxide may be obtained in the crystalline and aiihydrous state by evapo-
rating this solution to dryness in vacuo. Its physical characters hanng been fully
described by M. Lamt, I have not fui'ther experimented with it.

Alcohol exerts a curious action upon thallium. A coU of pure thallium wire was
placed in a tube with some absolute alcohol, just sufficient to cover it. At first no
action was apparent, except a slight opalescence of the spirit. In the course of a few
hours this had disappeared, and upon close examination needle-shaped crj'stals, together
with a few drops of a colourless heaAy liquid, were observed adhering to the sides of the
tube and sinking in the alcohol. In three days the wire Avas nearly eaten away, whilst
the oily drops had considerably increased in bulk. The alcoholic liquid was carefully
decanted from the hea\7 oil and tested : dilution with water produced no change in it ;
it was strongly alkaline to test-paper, and reacted in other respects like a strong solution
of protoxide of thallium. Upon the addition of a drop of hydrochloric acid, a thick
curdy precipitate of protochloride of thallium was produced.

The oily liquid was decomposed upon the addition of water, solidifying to a yellow
crystalline mass of ])rotoxide of thallium, which dissolved on further addition of water
and heating.

186 ME. W. CEOOEES ON THALLIUM.

Upon evaporating the alcoholic liquid over a water-bath, a fiu-ther formation of the
hea'Ny oil took place. The production of this oily liquid, by dissolving oxide of thallium
in alcohol and evaporating, has been pre^iously obsened by M. L.\iiY, who calls it
ThaUic Alcohol.

18. Thallium dissolves in sulphuric acid with ease, evolving hydrogen. The gas
given off burns with a flame in which thallium can frequently be detected with the spec-
troscope, although I have hitherto failed in proving the existence of a gaseous com-
pound of hydrogen and thalUum. Upon evaporating the solution, sulphate of thallium
crj'stallizes out. I have little to add to M. Lamy's description of this salt. It forms
large, w^ell-detined colourless crystals ; when heated to a little above the boiling-point of
sulphuric acid, they fuse to a clear liquid, which on cooling appears glassy and slightly
crystalline. The salt is soluble in twenty or thirty times its weight of cold water, and
in much less when boiling, crystalHzing out y\ith. facility upon cooling.

Thallium dissolves with the utmost rapidity m nitric acid. A piece of the metal
thrown into this acid mixed with half its bulk of water, runs about on the surface like
sodium on water, rapidly dissolving, and evolving nitric oxide mixed with nitrous oxide.
I have not found any ammonia produced in this reaction. When the liquid cools, it
becomes almost solid, fi'om the crystallization of nitrate of thallium, which is nearly
insoluble in nitric acid, although it is very soluble in water.

Hydi-ochloric acid attacks thallium but slowly, the action soon ceasing, owing to the
formation of a layer of difficultly soluble chloride of thallium. When hydrochloric
acid or a soluble chloride is added to a solution of the protoxide of thallium or one of
its soluble salts, a w-hite curdy precipitate of protochloride of thallium, Tl C'l, is thrown
down, scarcely diftering at first sight from chloride of silver. It has, however, a crystal-
line appearance, is slightly soluble in cold water, moderately so in boiling water, from
which it crystallizes out on cooling like chloride of lead, and is insoluble in alcohol.
When boiled in nitric acid or aqua regia it is converted into a higher chloride, crystal-
lizing out in large spangles of a yellow colour. The same chloride is formed by the
action of nitrohydrochloric acid upon the metal or its sulphide. It is more soluble in
water and acids than the protochloride, and is precipitated in the latter form u])on the
addition of sulphite of soda to its solution.

I have already described* the properties of the sul2)hide of thallium and some other
of its insoluble salts. Having since worked upon purer as well as larger quantities of
the metal, I have an addition or two to make to my previous descriptions. Thus the
protiodide of thallium is of a bright yellow coloiu', the red tinge which 1 formerly
noticed in it Ijeing due to the presence of a persalt of thallium. It is insoluble
in excess of dilute solution of iodide of potassium, being soluble only when the latter
is concentrated. The protocarbonate is soluble in water.

Concentrated acetic acid dissolves thallium slowly Avhen heated, forming a soluble

* Proceedings of the Koy:il Society, June 11), 1802.

MK. AV. CROOKES ON THALLIUM. 187

acetate; very dilute cold acid has no action upon thallium. Owinij to the solul)ilitv of
the oxide of thallium, no precipitate is produced in the jnotosalts of this metal by
potash, soda, or ammonia.

Neutral or slightly acid protosalts of thallium are incompletely precipitated bv sul-
phuretted hydrogen, and not at all when a large excess of acid is present. Sulpliide of
ammonium, as I have already stated, precipitates them perfectly, reducing the metal
to the state of protosulphide when in a higher state of oxidation.

19. The compounds of thallium are not only volatile when heated in the drj' state, but
many of its salts volatilize when their aqueous solutions are boiled. The chlorides are
especially volatile, insomuch that loss is experienced in e^•aporating them down. Ten
grains of pure metallic thallium were dissolved in a considerable excess of nitrohydro-
chloric acid, and the solution was gently boiled down in a retort. Upon testing the
acid distillate by supersaturation with ammonia and addition of sulphide of ammonium,
a considerable precipitate of sulphide of thallium was formed. Kitrohydrochloric acid
was then added to the residue of sesquichloride of thallium remaining in the retort,
and the distillation was repeated over a water-bath, care being taken that the evapora-
tion in this case was conducted below the boiling-point of the liquid. Upon now test-
ing the distillate, traces of thallium were still found in it : the metal in this case could
not have been earned over mechanically, as the liquid in the retort had not once entered
into ebullition.

Ha\ing for upwards of a year had considerable quantities of liquids containing thal-
lium evaporated in open dishes in my laboratory, it was natural to anticipate, after the
above experiment, that some quantities of the metal had been thus volatilized along with
the aqueous vapour, and would be found adheiing to the walls and deposited with the
dust on the upper shelves of the room ; a small portion of dust was accordingly removed
from a shelf at a height of above 10 feet from the ground, and tested for thallium.
A brilliant gi-een line in the spectroscope showed me that this metal was present in
more than minute traces.

20. Thallium may be determined quantitatively by precipitation, either as proto-
chloride, iodide, or double chloride of platinum and thallium. The chloride must be
washed with alcohol, as it is slightly soluble in water. The iodide and platinochloi-ide
are practically insoluble.

Position of Thallium amongst elementary bodies.

21. When I discovered thallium two yeai's ago, owing to the excessively minute por-
tion of substance which I had under examination, misled by its constant occuiTence
with sulphur and selenium, and basing my conjectui'es upon some of tlie properties first
noticed — namely, its complete volatility below a red heat, its precipitation in the
elementaiy form by zinc, its non-precipitation from an acid solution by alkalies, and
its solubUity in water when fused with nitre and carbonate of soda — reasoning upon
these observations, I ventured to suggest that it was probably a metalloid belonging to

188 ME. AV. CKOOKES OX THALLIUM.

the sulphur group, although, I added, •• I hesitate to assert this very positively"*. In
speaking of a metalloid of the sulphur group, I should explain that I had in view, not
a decidedly non-metallic body like sulphur, but one of the connecting links between
metals and non-metals — a metalloid in the strict meaning of the word, like tellurium or
arsenic. If I had formed any particular view upon the matter, knowing so little of the
properties of the new body, it was that it might possibly prove to be a higher link in
the sulphur, selenium, and tellurium chain. It was not long before further research
showed me that the body under examination had, in addition to the characters already
mentioned, others which gave it strictly metallic characters ; and although no formal
publication of this isolated fact was immediately made, the element was commonly
spoken of in scientific circles as a new metal, and was so described by me at the Exhi-
bition on the 1st of May last. I enter thus into details on so trifling a subject because
French chemists are inclined to attacli undue importance to the term, and misinterpret
the meaning of metalloid.

Even with our present knowledge of the chemical and physical characteristics of
thallium, it is not easy to assign its true position in the scale of elements. I cannot
admit, with the French chemists, that it is an alkali-metal. Almost the only property
which thallium possesses in common with the alkali-metals is the solubility of its oxide,
and perhaps its forming an insoluble platinum-salt. But oxides of lead, silver, and
mercury are also soluble in water, reacting in many respects like alkaline solutions;
and oxide of thallium is far more analogous to these than to potash and soda, inasmuch
as it has scarcely any affinity for water, becoming anhydrous, in a vacuum, even in the
cold. In opposition to these reasons for classing it with the alkalies, we have numerous
facts to prove that its true position is by the side of mercury, lead, or silver. The ready
dehydration of its basic oxide — the insolubility of its sulphide, iodide, chloride, bromide,
chroraate, phosphate, sulphocyanide, and ferrocyanide — its great atomic weight — its
ready reduction by zinc to the metallic state — and, according to Dr. Miller, the com-
plexity of its photographic spectrum — all prove that thallium cannot consistently be
classed anywhere but amongst the heavy metals, mercury, silver, lead, &c.

22. Those who remember how readily figures can be moTdded to suit any theory,
will attach slight importance to the argument adduced by M. Dujr.vs in fiivour of thal-
lium bcnng related to potassium and sodium because its ecpiivalent is rather near a
figure obtained by adding twice the atomic weight of one metal to four times the
atomic weight of the other. By similar processes of addition, multiplication, or sub-
traction it would not be difficult to pro^(> a relationship between thallium and any
desired group. Thus twice; tlie ((luivaltiit of tellurium added to tluit of arsenic would
make one equivalent of thallium, an argument in favour of its being a metalloid ; one
equivalent of mercury and one e(jui\alent of lead added together make one e([uivalcnt
of thallium, as also do one e(iui^aleut of silver and two (>quivalents of molybdenum —
fdch proving it to be a heavy metal of the silver and lead group. Were it worth while
* Chfiuital Xows, Muixh :J0, 1S(;1, p. I'.r.',. and Phil. Mag.. April ISGL

ME. W. CEOOKES 0^'^ THALLIUM. 189

to pursue these relationships further, it would not he difficult to find many coincidences
less strained than the one hrought forA\ard by M. Dumas.

Analytical Notes on Thallium.

23. The chemical identification of thallium when associated with other metals is not
difficult. I have already discovered very exact methods of detecting the presence of
thallium in, and separating it from, most of its associated metals, and further experi-
ment will doubtless still more increase the accuracy of its analytical detection and esti-
mation.

In starting with the analysis of a thalliferous mineral by the ordinary analytical
tables, in which Group I. is precipitated by hydrochloric acid. Group II. by hydro-
sulphuric acid, Group III. by ammonia, and Group IV. by sulphide of ammonium, a
slight analytical difficulty will be at first met with, as, unless special precautions are
taken, thallium will appear in all four groups. Thus if the thallium be in the state of a
sesquisalt, no precipitate will be produced in a moderately dilute solution upon addition
of hydrochloric acid ; if, on the contrary, it be as a protosalt, the great bulk \\ ill come
down in this group. In either case it will be ad%dsable to reduce the remainder of the
metal to the state of a protosalt, by passing sulphurous acid through the filtrate and
heating. If, upon allowing the solution to cool after this treatment, a white crystal-
line precipitate of protochloride of thallium is produced, it will show that the metal
originally existed in the state of a sesquisalt. This precipitate may be filtered off and
examined separately. In the filtrate from this, even were there sufficient acid present
to prevent the sulphide of thallium by itself from being precipitated by sulphuretted
hydrogen, it will be partially carried down by other metals of this group which may be
present. What escapes this precipitant will in a similar manner be partially carried
down with the oxides of the third group, whilst the remaining thallium escaping the
first three group-tests will be precipitated by sulphide of ammonium.

Thallium may be very accurately separated from most metals. Some of the analy-
tical methods which I have employed for many months are very delicate ; others, on the
contrary, still require elucidation.

24. Thallium from Zinc. — (I will assume that a piece of commercial zinc has to be
tested for thallium.) Dissolve the metal in sulphuric acid, adding a little nitric acid
towards the end to effect the perfect solution of the black residue. Evaporate to drive
off nitric acid ; dissolve in a small quantity of water ; filter from sulphate of lead, if
any be present, and heat the moderately acid solution with excess of sulphite of soda.
Allow the liquid to cool, and add a few drops of solution of iodide of potassium. A
yellow precipitate of iodide of thallium will be thrown do^vn. Many specimens of
commercial zinc, tested in this manner, will be found to contain thallium.

25. Thallium from Iron (thalliferous iron pyrites). — Dissolve 30 or 40 grains of the
finely powdered mineral in nitrohydrochloric acid ; evaporate with excess of hydro-
chloric acid to diive off the nitric acid; redissohe in water; add sulphite of soda in

MDCCCLXIII. 2 D

190 ME. W. CEOOKES OX TII-\LLIXJM.

excess, and heat until all the iron is reduced to the proto-state ; and then add iodide of
potassium. Iodide of thallium will be precipitated. This test is sufficiently delicate to
show thallium in a pyrites which does not contain more than 1 in 10,000.

Tlialliiun from Manganese may be separated as thallium from iron.

2G. ThalUiun from Mercunj. — I have not yet ascertained a delicate and reliable
method of separating thallium from salts of the suboxide of mercury. It is, however,
verv readily separated from persalts of mercury ; and therefore the best plan is to per-
oxidize both metals by boiling witli nitrohydrochloric acid ; then reduce the thallium
with sulpliite of soda, and add iodide of potassium to the almost neutral solution. If
much mercury be present, the precipitate will be almost pure scarlet; but on further
addition of iodide of potassium, drop by drop, the iodide of mercury will dissolve, and
will lea\e the iodide of thallium as an insoluble yellow powder. Upon warming the
liquid the precipitate collects together and readily settles to the bottom. This is a very
delicate test.

27. Thallium from Lead. — Evaporate the solutions to dryness with excess of sulphuric
acid, and extract ^^•ith hot water. Sulphate of lead will be left behind, whilst sulphate
of thallium will be dissolved. This is a very ready process, but is not quite so acciu-ate
as the succeeding one.

28. Tliallium from Bismuth or Lead. — Dilute the solution and add a .•iliyht excess of
carbonate of soda ; add solution of cyanide of potassium (free from sulphide), and allow
the mixture to stand for an hour at the temperature of about 100° F. ; then filter and
wash : the residue will contain all the bisnuith or lead. To the clear filtrate add
sulphide of ammonium, and warm gently for some time ; the deep-bro-wn sulphide of
thallium will gradually collect together in fiakes at the bottom of the vessel. It must
be washed with water containing a little sulphide of ammonium, as it readily oxidizes
when moist, and might pass through the filter as sulphate of thallium.

This is an exceedingly delicate test for thallium in bismuth ; and by its means it can
be detected in most specimens of commercial bismuth and its salts, even when sold as
])ure. The presence of thallium in some samples of bismuth has been suspected by
J)r. W. IhiU) Hek.\patii*. The analytical method which he gives is not calculated to
detect it except pcrhajjs when present in comparatively large quantities. By the above
j)rocess it will be found to be a ^c'ry frequent constituent of bismuth compounds, even
when working upon no nu)r(' than oO grains of material.

20. Tliallium from Copjier.—'Vo the acid solution add ammonia in excess, and then
cyanide of pota.ssivmi until the blue colour has entirely disappeared ; then add sulphide
of ammonium, and gently warm foi- some time. Sulphide of thallium will gradually
collect together in tlie liquid. By tliis test 1 hav(> detected the presence of thallium
ill many specimens of co])per as met with in commerce, as well as in crystallized
sulphate of copper. It is extremely delicate.

Through the kindness of Dr. Matthii:.sskn I have been I'uabled to examhie for thal-
* PLarmaceutical Jouiual, Jan. ], 18G3.

t.ni. W. CKOOKES ON THALLIUM. 1 91

lium a specimen of copper prepared in Spain by a process called " cementation." This
consists in allowing copper ]iyritcs to oxidize slowly, washing out the resulting sulphate
of copper, and precipitating tlic solution with metallic iron. The pulverulent copper is
then heated until it coheres, and the metal sent into the market in the form of pigs, no
fmther purification being attempted.

This metal was found by Dr. Matthiessex to have a conducting-power for electricity
of about 15, pure copper being 100. The exact metallic impurity which rendered it so
preeminently bad had not been ascertained. It was tested for thallium in the manner
above described, and found to contain a large quantity. It is evident, from the way the
copper was extracted, that any thallium which might have been present in the pyrites
would accompany the copper.

30. Thallium from Cadmium. — To the acid solution of these metals add bichromate of
potash, then excess of ammonia, and boil; insoluble cliromate of thallium will be pre-
cipitated. This is not so delicate a test as some of the above, although by its means
I have frequently detected thallium in metallic cadmium and its salts. Commercial
sulphide of cadmium, as sold for artists' use, varies considerably in tint. Dark-
coloured samples frequently contain thallium. I may especially instance, as being
highly thallii'erous, a beautiful specimen of this sulphide from Xouvelle Montague,
near Liege, which formed a prominent object in the Belgian department of the late
Exhibition.

31. Thallium Jrom Gold. — ^The gold may be separated by the usual process of reduction
to the metallic state with oxalic acid, all the free nitric acid ha\ing been previously
removed by evaporation with hydrochloric acid.

32. Thallium from Antimony, Tin, and Arsenic. — A very good method of separating
these metals is to add excess of sulphide of ammonium to the alkaline solution. Sulphide
of thallium will be precipitated, whilst the other sulphides will remain dissolved.

Most of the above processes have been tried with weighed quantities of the different
metals, seldom taking more than 1 of thallium to 1000 parts of the other metal.
They can therefore be relied upon to that extent ; whilst some of them are much more
delicate, as, for instance, the separation of thallium from iron, copper, bismuth, and
lead.

It is adnsable, in testing for small quantities of thallium, to appeal to the spectro-
scope for confii'matory evidence of the presence of this element in any precipitate
suspected to contain it.

33. Thallium from Sulphuric and Hydrochloric Acids. — I have frequently met with
specimens of commercial hydrochloric and sulphuric acids which contained almost enough
thallium to be worth extracting. I may especially mention a very crude yellow hydro-
chloric acid now to be met with at about 8 shillings the hundredweight. Two ounces
of this was neutralized with soda, and a few drops of sulphide of ammonium were added.
A black precipitate was obtained, which in the spectroscope showed endent presence of
thallium. At my request, Messrs. Hopkj.v and Williams, the well-known manufac-
turing chemists, treated 112 lbs. of this acid in the above manner, and forwarded me

192 ME. W. CEOOKES ON THALLIUM.

the black precipitate obtained. It was worked up in the manner described in the first
part of this paper (7.), and yielded a little over four grains of metallic thallium.

From sulphuric acid it may be separated in a similar way.

It is not difficult to understand how thallium gets into these acids. Messrs. Chance
Brothers and Co., of Birmingham, makers of the thalliferous hydrochloric acid, have
obligingly informed me that the process by which it is produced is the ordinaiy one
of decomposing common salt in cast-iron pans and fire-brick furnaces. The acid is
condensed in high stone towers or chambers filled with coke, and is afterwards collected
in gutta-percha cisterns, and bottled or drawn off. The sulphuric acid used in the
manufacture is obtained from iron pyrites burnt in kilns. Upon examining specimens
of pyrites and lead-chamber-deposit forwarded by Messrs. Chance and Co., I find that
the former contains thallium, but scarcely any appreciable traces are in the deposit,
thus agreeing with the results of my previous investigations on this subject.

[ 193 ]

X. On the Distribution of Surfaces of the Third Order into Sj)ecies, in reference to the
absence or presence of Singular Points, and the reality of their Lines. By
Dr. ScHLAFLi, Professor of Mathematics in the University of Berne. Com-
municated by Aktuur Cayley, F.E.S.*

Received December 18, — Read December 18, 1862.

The theory of the 27 lines on a surface of the third order is due to Mr. Cayley and
Dr. Salmon; and the eflect, as regards the 27 lines, of a singular point or points on the
surface was first considered by Dr. Sal.mox in the paper '• On the triple tangent planes
of a surface of the third order," Camb. and Dub. Math. Journ. vol. iv. pp. 252-260 (1849).
The theory as regards the reality or non-reality of the lines on a general surface of the
third order, is discussed in Dr. Sciilafli's paper, '' An attempt to determine the
27 lines &c.," Quart. Math. Journ. vol. ii. pp. 5(3-65, and 110-120. This theory is
reproduced and developed in the present memoir under the heading, I. General cubic
surface of the third order and twelfth class ; but the greater part of the memoir relates
to the singular forms which are here first completely enumerated, and are considered
under the headings II., III. &.c. to XXII., viz. II. Cubic surface with a proper node,
and therefore of the tenth class, &c., down to XXII. Ruled surface of the third order.
Each of these families is discussed generally (that is, Avithout regard to reality or
non-reality), by means of a properly selected canonical form of equation ; and for the
most part, or in many instances, the reciprocal equation (or equation of the sui-face in
plane-coordinates) is given, as also the equation of the Hessian sui'face and those of
the Spinode curve ; and it is further discussed and divided into species accordmg to
the reality or non-reahty of its lines and planes. The following synopsis may be con-
venient : —

I. General cubic surface, or sui-facc of the third order and twelfth class. Species I.

1, 2, 3, 4, 5.
II. Cubic surface with a proper node, and therefore of the tenth class. Species II.

1, 2, 3, 4, 5.

III. Cubic surface of the ninth class with a biplanar node. Species III. 1, 2, 3, 4.

IV. Cubic surface of the eighth class with two proper nodes. Species IV.

1, 2, 3, 4, 5, 6.
V. Cubic surface of the eighth class with a biplanar node. Species V. 1, 2, 3, 4.

* Dr. ScHLAFLi authorized me to make any alterations in the phraseology of his memoir, and to add remarks
which might appear to me desirable. Passages in [ J, or distinguished by my initials, are by me, but I have
not thought it necessary to distinguish alterations which are merely verbal or of trifling importance. — A. C.

MDCCCIiXIlI. 2 E

194 DE. SCHLAFLI 0^" SUEFACES OF THE THIED OEDEE.

\I. Cubic surface of the seventh class with a biplanar and a proper node.
Species VI. 1, 2.
VII. Cubic surface of the seventh class with a biplanar node. Species VII. 1,2.
VIII. Cubic surface of the sixth class \Aith three proper nodes. Species VIII.
1, 2, 3, 4.
IX. Cubic surface of the sixth class with two biplanar nodes. Species IX. I, 2, 3, 4.
X. Cubic surface of the sixth class with a biplanar and a proper node. Species X.
1, 2.
XI. Cubic surface of the sixth class with a biplanar node. Species XI. 1, 2.
XII. Cubic siu-face of the sixth class with a uniplanar node. Species XII. 1, 2.

XIII. Cubic sm-face of the fifth class -with a biplanar and two proper nodes.

Species XIII. 1, 2.

XIV. Cubic surface of tlie fifth class with a biplanar node and a proper node.

Species XIV. 1.
XV. Cubic surface of the fifth class with a uniplanar node. Species XV. 1.
XVI. Cubic sm-face of the fourth class ^nth four proper nodes. Species XVI. 1, 2, 3.
X^'II. Cubic surface of the fourth class with two biplanar and one proper node.
Species XVII. 1, 2, 3.
XVIII. Cubic surface of the fourth class ynth one biplanar and two proper nodes.
Species XVIII. 1.
XIX. Cubic surface of the fourth class with a biplanar and a proper node.
Species XIX. 1.
XX. Cubic surface of the fourth class wath a uniplanar node. Species XX. 1.
XXI. Cubic surface of the third class with three biplanar nodes. Species XXI. 1, 2.
XXII. Ruled surface of the third order and the third class. Species XXII.
1, 2, 3.— A.C.

I. General cubic surface, or surface of the third order and twelfth class.
Alt. 1. As the system of coordinates undergoes various transformations (sometimes
imaginary ones), it becomes necessary to adhere to an invariable system of a real mean-
ing, for instance the usual one of three rectangular coordinates. AVe shall call this the
system of fundamental coordinates, and define it by the condition that the coordmates
of every real point (or the ratios of them, if they be four in number) shall be real.
Consequently any system of rational and integral equations, expressed in variables of a
real meaning, and w-here all the coefficients arc real, will be termed a real system (of
equations), whether there be real solutions or none, proA-ided that the number of equa-
tions do not exceed that of the variables, or of the quantities to be determined. The
degree of the system will be the number of solutions of it wlien augmented by a suffi-
cient number of arbitrary linear equations ; and such degrcK^ will generally be the pro-
duct of the degrees of the single equations. It is olnious that the system, whenever
its degree is odd, represents a ?'eal continuum of as many dimensions as there are

DR. SCHLAFLI ON SUEFACES OF THE THIED ORDER, 195

independent variables; for instance, every real quaternary cubic represents a reid
surface.

It is know-n* that on the surface of the third order there are 27 lines which form 45
triangles in such manner that through each line there pass five planes meetin" the sur-
face in this line and two other lines, or say five triangle-planes. Lines not iiit(>rsecting
each other may be termed independent lines, as far as a surface of the third order is
capable of containing all of them ; the greatest number of such lines is four ; that is to
say, in whatever manner we may choose two, three, or four not intersecting lines on the
sm-face, the system has always the same properties. Let two independent lines I. and
II. on the surface be given, and imagine any one of the five triangle-planes passing
through I. ; then II. must intersect one of the two other sides of this triangle ; in other
words, this triangle aflfords a line cutting both I. and XL, and a line cutting I. alone.
Hence it appears that there are five lines cutting both I. and II., five lines cutting I.
only, five lines cutting II. only, and ten Unes cutting neitlier I. nor II.

[The theory of the 27 lines depends on the expression of the equation of the surface
in the form P— Q = 0, where P and Q are real or imaginary cubics breaking up into
linear factors ; in fact, if the equation be so expressed, it is at once seen that each of the
planes P=0 meets each of the planes Q=0 in a line on the surface, so that the form
gives at once 9 out of the 27 lines. The three planes represented by the equation P=0
(or Q=0) are termed a Trihedral of the surface.]

Art. 2. Peop. It is always possible to find a trihedral represented hy a real quaternary
cubic.

The truth of this proposition is evident when all the 27 lines are real. But when
some of them are imaginary, these are conjugate by pairs. As the case when two con-
jugate lines intersect one another is fitter for our purpose, we begin with the other case
when two conjugate lines do not intersect each other.

The problem, then, of finding the five lines intersecting such pair* of conjugate lines
depends on a real system. Hence among the five lines there wiU be an odd number of
real ones; and imaginary ones, when existing, will be conjugate by pairs. Call the
given two independent and conjugate lines I. and II., and the five lines intersecting each
of them a, b, c, d, e. If d and e be imaginary and conjugate, the plane containing I.
and d will be conjugate to that containing II. and e, and these two planes wiU not
intersect in a line of the surface (for if they did, a line of the surface would unite the
intersection of II. and d with that of I. and e ; and it is obviously a great loss of generality
if three lines of the surface meet in a point). But if all the five lines a, b, c, d, e be real,
then — because they can be intersected simultaneously only by the liiies I. and II., and
because through each of the five lines there passes at least one real triangle-plane —
it must be possible to choose among all the real planes each passing through any one of the
real lines a, b, c, d, e, two real triangle-planes not intersecting in a line of the surface.

* See Cambridge and Dublin ilath. Journ. vol. iv. p. 118, the original memoirs of Messrs. Catucy and Salmon
on the triple tangent-planes of the cubic surface.

2e2

196 DE. SCHLAFLI ON SURFACES OF THE THIRD ORDER.

As to the easier case first mentioned, when there are on the surface two conjugate
lines intersecting each other, it is plain at first sight that they afford us four pairs of
conjugate triangle-planes not intersecting in a line of the surface.

Now whether we have two conjugate planes, or two real planes not intersecting in a
line of the surface, the third plane completing tliem to a triliedral is singly determined
by a real system and is therefore real ; and hence the trihedral is represented by a real
cubic.

Ai-t. 3. Prop. A real cubic surface of the tn-elfth class (or, what is the same thing,
without nodes) can always he represented hy uvw+xyz = 0, where both uvw and xyz are
real cuhics breaking ujj into linear f actors.

Let /.A+B=0 be a cubic equation expressed in the fundamental coordinates with real
coefficients, X a numerical factor imaginary if possible, A. B cubics each decomposible
into linear factors, but A real and B imaginary if possible, and let X', B' be respectively
conjugate to \, B. Then (X — X')A+B — B' = 0 must be an identical equation, and each
solution satisfying the system A = 0, B = 0 will therefore also satisfy B'=0. But it
would be a loss of generality if. througli tlie nine lines in which the two trihedrals A
and B intersect each other, there should pass a third trihedral B'. Therefore we must
have X=X', B=B'; in other words, if <me trihe(hal of a pair is represented by a real
cubic, its associate is also so, and the trihedral-pair equation does not imply any
imaginaiy numerical factor. We are therefore justified in asserting that a real cubic
surface (without nodes) can always be exhibited in one of these three trihedral-pair
forms itvw-\-xi/z=.0 ; 1. u, v, w, a; y, z are all real ; 2. u, v, w, x are real, y is conjugate
to z ; 3. M, or are real, v is conjugate to w, and y to z.

Art. 4. To save the reader the trouble of consulting my paper in vol. ii. of the
Quarterly INIathematical Journal, I will gi\e here a scheme which serves to determine
and denote the twenty-seven lines. In space, only four linear functions can be inde-
pendent ; any fifth one will be a linear and homogeneous function of these four linear
functions. Hence it is plain that in the identical equation
Au-\-B V + Civ -\-T)a-\-Ey + Fz = 0
the coefficients arc linear and homogeneous functions of two arbitrary constants; and of
course only tlicir ratio is here of importance. Tlie identical equation

Au{Bv + 'Dx){CAV + 'Dj) + 'D.r{Au + Ey}{An + Fz)=ABCuino+J)EFxyz

then suggests the propriety of making the six coefficients subject to the condition
ABC=DEF, because we have then a transformation of the original trihedral-pair form
into another like form. But tlie condition (being a cubic equation) has three roots,
according to which we may put

1au:=au-\-bi^-^C7V-\-d.v-\-ey-\-fz = 0, abc=idef;
la'u = 0, a'b'c' = d'ef'; 2a"u = Q, a"b"c"=d"e'f".

We denote the line {n=zO, .7'=0) by t(X, and so on for all the nine lines arising from

DR. SCHLAFLI ON SURFACES OF THE THIRD ORDER. 197

the intersection of the two trihedrals itvw, xijz. Again, since there arc twenty-seven
forms of the equation of the surface such as

au{bv-\-dx){cxo-\-dx)-\-dx{au-\-ey){au-\-fz)=0,

the equations «!« + dr=0, iv-\-ey=^, cw-\-fz=0 belong to a line of the surfoce which
we denote by I, while (tis), (ux)', (ux)" respectively represent the triangle-planes

au+dx=0, a'H + dx=0, n"H + d"x=0,

and so on. Now in the scheme of the nine initial lines

X

y

z

.

u

ux

uy

uz

V

vx

vy

vz

10

wx

wy

wz

through ux,

vy,

wz pass i ,

I' ,

I",

uy.

vz,

wx „ m,

m'.

m"

„ uz,

vx.

wy „ n ,

n' ,

n".

we may first perform all the positive permutations of the columns, and then deduce from
these the negative ones by permuting only y and z. In eacli permutation we keep
in view only the lines placed in the principal diagonal. AVe thus obtain the following
easily intelligible scheme

through ux, vz, wy ■pass p, p', p",
„ ttz, vy, wx „ q, (J, q",
„ uy, vx, wz „ r, r' , r".

The plane (ux) contains the lines ux, I, p, and so on ; and the plane containing /, ?n', n"
may be represented by (Imn), and so on.

I do not think it worth while to show that the equation ABC=DEF, when explicitly
written, always has real coefficients, and that each of the cases hereafter coming into
consideration can be constructed, and that it therefore aists.

Art. 5. First case. — u, v, w, x, y, z are all of them real.

A. The cubic condition (ABC=DEF) has three real roots. It is then at once plain
that all the twenty-seven lines and all the forty-five triangle-planes are real. First
species, I., 1.

B. The cubic condition has but one real root, to which let belong the coefficients
a, b, . . . Each geometincal form then changes into its conjugate by merely permuting
the two accents ' and ". So the nine initial lines and the six lines I, m, n, p, q, r
(together fifteen) ai'e real, and the remaining lines are imaginary and form a double-six

'I', m', rt! , p", 9", r"\
^l", m", n", p', q', /•'^

where any two corresponding lines are also conjugate. Fifteen lines and fifteen planes
are real. Second species, I., 2.

198

DR. SCHLAFLI ON SrEFACES OF THE THIRD ORDER.

Art. 6. Second case. — 1/ and z only are imaginary, and therefore conjugate.

A. The cubic condition has three real roots. Each form changes into its conjugate
by merely permuting 1/ and z. Therefore, in the trihedral-paii- scheme, only the first
column contains real lines, the two other columns are conjugate; and as to the eighteen
remaining lines, tlieir two schemes are conjugate in the above-mentioned order. Three
lines and thirteen planes are real ; for there is one real triangle through each side of
whicli there pass, besides the plane of the triangle, fovu- other real planes. Fourth
sjjecies, 1., 4.

B. The cubic condition has but one real root to which let belong the coefficients a, b,
. . . Each form changes into its conjugate one by permuting at once 1/, z and the two
accents ' and ". Three lines and seven planes are real. The real lines form a triangle,
through each side of which there pass, besides the plane of the triangle, two other real
planes. Fifth sj)ccies, same as third case B, infra.

T/iird case. — v is conjugate to w; y to z; and u, x are real.

A. The cubic condition has three real roots. Each form changes into its conjugate
by permuting at once «, w, and y, z. The three above-mentioned schemes (each of nine
lines) change hereby respectively into

nx

uz

uy

I

V

I"

P

P

P

wx

wz

wy

n

n'

n"

r

/

r"

vx

vz

VI

m

m'

m"

S

i'

2'

The comparison shows that only ux, I, I', I", p, p', p" keep their places, and are therefore
real. Of the planes, only «, x, (nx), (ux)', (ux)" are real. Seven lines and five planes are
real ; namely, through a real line there pass five real planes, three of which, (ux), (ux)',
(ux)", contain real triangles. Third species, I., 3.

B. The cubic condition has but one real root. To find the form conjugate to a given
one, we must at once permute v, w, also y, z, and lastly the two accents ' and ". The
three schemes of lines by this process become

I I" U

vx
wx

vx

mj
wy
vy

n
m

P

q q" ([

Only ux, I, p keep their places, and therefore are real. Besides the planes ti, x (ux), also
tlio planes (linn), (Inm), (p>qr), (prq) are real. The three real lines form a triangle,
througli eacli side of which there pass two more real planes. Fifth species, I., 5.

Art. 7. Ifow many kinds of nodes can exist on a cubic surface?

Considering in the fir>t instance the theory of an ordinary node or conical point, let
us imagine a surface of the wlh order with a node, at which we are allowed to place

the pcjiut of reference^*. Let tlien an arl)itrary line be given, through whicli tangent

* Aji to this mode of expression, see foot-note to art. 8. — A. C.

DR. SCIILAFLI ON SUEFACES OF THE THIRD OIWER. 199

planes to the surf\ice are to pass, and throngli tliis line draw tlic planes of reference r=:0
(through the node) and ii'=0 (not passing through the node). The equation of the
surface will then take the form

F=Pw"-'+Q?<)-'+R!o"-*+&c. =0,
where

P=(.r, y, zy, Q=(.r, y, z)\ R=(.r, y, z)\ Sec,

and the points of contact of tangent planes passing through the given line (r = n, ?/'=n)

dF BF

must satisfy the conditions ^ = 0, -^7=0- In the proximity of the node the system of

the three equations reduces itself to P=0, -^=0, ^ =0 (or, what is the .same thing,

^=0, ^ =0, i ^=0 j, if none of these equations be a necessary consequence of the

other two. The node ^ then represents two solutions, because the equations are

respectively of the degrees 2, 1, 1 [or, w^hat is the same thing, among the tangent planes

through the line the plane passing through the node counts for two tangent planes ; that

/pi p p^ p \

is, the class of the sui'face is diminished by 2]. The exception (-v =0, ^=0, ;r=:0)

is inadmissible; for should the plane z = 0 touch the cone P^O, the line (- = 0, w>=0)
would not be ai'bitrarily chosen. The only possible exception is when the three equations

|P = 0, 1^ = 0, |? = 0

ox ' dy o-

can be simultaneously satisfied. Consequently so long as the nodal cone P=0 does not
break up into a pair of planes, there are two solutions, or the class is diminished by 2.
In the excepted case, w^here the nodal cone P = 0 breaks up into a pair of planes, we
may assume V^xy (or V^x^, to be discussed in the sequel) ; and since now the (>qua-
tions xy=^, a:=0, y=0, are no longer independent, we must go on to consider also

where

L=(.r,y), M={x,yy, N=(.r, ^)^

For the sake of shortness, let w^l. We then have

a;y+az='+Lz*+Mz+N+&c.=0,

and unless the constant a vanish, the system (in the proximity of the node) reduces itself
to x=0, y=0, z^=0; that is to say, a biplanar node, in general, counts for three
solutions, or diminishes the cla.ss by 3.

Next it remains to put a = 0, lj=:ax-\-by, when the system becomes

x^bz'+.... = 0, y+ff.-=+.... = 0, xy-\-{ax+by)z' + ... + Kz'+... = 0,

200 DK. SCHLAFLI ON SURFACES OF THE THIRD ORDER.

where Kr* is borrowed from R ; and the hist equation of the system reduces itself by

means of the others to (K — ab)z*-\- = 0. The node here unites four solutions, unless

K—ab should vanish; that is to say, if the nodal edge (;r=0, y=0) lie on the cone
Q = 0, the biplanar node lowers the class of the surface by -4, unless the portion of the
surface surrounding the node be, in the first approximation, represented by the form
{x+h^){i/+az'')-{- terms of the fifth order in regard to r, =0.

The further supposition would be K—ab = 0 ; but let us now assume a cubic surface,
that we may have K=0, and therefore ab = 0. Selecting the case ^=0, Ave put

Q^axz'+(bx'-\-C3:i/-^df)z-\-'^,
whence

o:+(ca-+2di/)z + ... = 0, y + «z^+....=:0,

or neglecting higher orders tlian here come into consideration, y= — az'^, w=2adz^,

whereby F:=0 becomes «\/:;^ + .. . = 0, so that the system is reduced to a'=0, y:=0,

aV/c'=0. That is to say, if one of the nodal planes touch the surface along the nodal

edge, the biplanar node lowers the class of tlie surface by 5, unless the cone Q=0 have

that line of contact either for a double line (if rt=0), or for a line of inflexion (if (1=0).

The exceptional supposition then to be made sepaiates itself into a = 0 and d=0.

But a=0 would cause all the terms of F to be of the second degree, at least in respect

to X, y, so that the surface would have (.f=0, i/ = 0) for a double line. Assumuig then

d=0, we may put

Q=xz'+iax' + bxi/)z-]-cx'-\-dxy+exf+f!f,

when the system reduces itself to .i'=0, j/=0, — /i^=0. That is to say, if one of the
nodal planes osculate the surface along the nodal edge, the biplanar node lowers the
class by 6. Here we must stop; for if we supposey=0, the cubic F becomes divisible
by X.

We go on to the case where the nodal cone becomes a pair of coincident planes, or
say where we have a uniplanar node. The equation of the surface is

¥=x'w+af+Ufz + 5cyz"- + dz^+x{ef+f>/z+gz')+x%/i>/-\-jz)+Kx'==0.

For indefinitely small values of x, y, z, the equation -, =0 causes x to be of the second

order in respect to y, z. The system of conditions for the point of contact (in the
proximity of the node) of a tangent-plane passing through the line (^=0, w=0) reduces
itself therefore to

x=0, ay' + 2byz-{-cz'=0, ay'-{-oby"z+?jcyz^-{-dz^=0,
unless the discriminant of the last-mentioned cubic should vanish. Except in this case,
the system shows that the nod ■ counts for six solutions of

(i-=». »=«. !-:=«).

or, what is the same thing, that a uniplanar node lowers in general the class by 6.
But if the binary cubic ay''' -\-oby''z-{-'icyz- -{- dy' contain a squared factor, we may denote

1)]{. Si'ULAFLl ON SUEFACES OF THE THIRD OHUEU. 201

this bj' if, and then write

Y=a"W-\-aif-\-hfz-\-{nr-\-djiz-\-iz'-)x=^0
for the equation of the surface ; for it is phiin that \\c are allowed to disregard the
subsequent terms divisible by x'\ On forming- the equation in plane-coordinates, it is
immediately seen that this surface is of the iiftli class, unless i = 0; that is, in the
general case, the class is diminished by 7.
Lastly, if J^O, then we have

F = ,i-- w -\-(if-\-{ cf + dijz -\-cz-)x=^;
and by forming the equation in plane-coordinates, the surface would be found to be of
the fourth class, that is, the class of the surface is diminished by 8.

A closer discussion of the last two cases is reserved for a fit occasion.

In the whole we are to distinguish eight kinds of nodes on the cubic surface : 1, the
proper node, which lowers the class by two ; 2, the biplanar node, where the nodal edge
does not belong to the surface and which loAvers the class by three ; 3, the biplanar node,
where a plane different from both nodal planes touches the surface along the nodal edge
and whicli lowers the class hy four; 4, tlie biplanar node, where one of the two nodal
planes touches the surface along the nodal edge and whicli lowers the class hy Jive ; 5, the
biplanar node, where one nodal plane osculates the surface along the nodal edge and
which lowers the class by six ; G, the uniplanar node, where the nodal plane intersects
the surface in three distinct lines and which lowers the class by six ; 7, the uniplanar
node, where the nodal plane touches the surface along a line and which lowers tlie class
hy seven; 8, the uniplanar node, where the nodal plane osculates the surface along a
line and which lowers the class by eight.

Art. 8. On the case of two nodes on the culic siuface.

Let y be the quaternary cubic of the surface, P, Q the symbols* of two different nodes
on it ; then Vf Qf will identically vanish. If now^ R be the symbol of any third point,
the symbol aP+/3Q+yll, w-here a, /3, y denote arbitrary multipliers, will belong to a
point in the same plane with the points P, Q, R, and the equation

(aP+/3Q+7R)y= Ga/3yPQR/-f 2>y\a?+^Q)Wf+fWf= 0
will represent the section of the surface made by the plane. Then if the point R satisfy
the condition PQR/"=:0, the equation will become divisible by y", that is to say, the
equation PQR/'=0, in respect to the elements of R, represents a plane touch.ing the
surface along the line joining the nodes P and Q, and besides intersecting it in a lino
represented by

3«PR=/-f 3/3QRy+yRy=0,

if here a, /3, y are regarded as planimetrical coordinates, and the point R as fixed. In
the sequel I shall sometimes term the former line axis and the latter transversal.

* If x', y', :', w' are the coordinates of a node, ,r', y', z', w' current coordinates, then the symbol P of the node
is =,r'5j.+ )/'9j, + :'9- + M''9,„aud 1*-/Ls =:P/'=(.f5^+?/dy + rd.. + !t'9„')/', which vanishes identically, that is
independently of .r, y, z, iv, in ^-irtue of the equations 'dyf'=0, &c. satisfied at the node. — A, C.
MDCCCLXIII. 2 P

202 DE. SCHLAFLI ON SUEFACES OF THE THIRD ORDER.

If P(t'+Q=0, where P=(.r, y^ zf, Q=(.r, y, c)', be the equation of a cubic siu-face
with a node, I shall call the six lines represented by the system P=0, Q=0 nodal rays.
They belong to the sm-face, and it is plain that two of them at least must coincide in
order that the surface may have another node, and this will Ue on the line luiiting two
or more rays of the first node.

II. Cubic surface toith a proper node, and therefore of the tenth class.

Art. 9. The equation of this surface can always be thrown into the form Pw4-Q=0,
where P=(.r, y, z)\ Q=(.r, y, zf.

Let ? be a linear and homogeneous function of .r, y, z, then

P(w_;)4-Q+^P=:0

is the same equation. But we may in fifteen different ways dispose of the three coeffi-
cients in I, so that Q+/P breaks up into three linear factors, and are therefore allowed
to write

p = {as" + ^/ + f .- -^1dyz-\- lezx + 1fvy)w + Ixyz = 0
as an equation of tlie surface, where, for the sake of shortness, the ternary quadric
«a'*+&c. of the nodal cone may be denoted by %, and the derivatives of hx by X, Y, Z.
Again, let

A=al)c—ad- — le- — cf--\-2def, A=bc—d-, B = m— r, C = ab—f^,
■D=ef-ad, 'E=fd-be, ¥=de-cf
and determine the constants by the quadratic equation (aX — D)^ — BC=0, then %+2Xy2
\vi\l break up into two linear factors, and (p=:('j(^-\-2Kyz)w-\-2{x—Xw)yz will be a trihedral-
pair form of the surface. Its particularity is sufficiently determined by the condition
that an edge of one trihedral intersects an edge of the other trihedral, the point of
intersection being the node. I wished only to notice the connexion of such form with
the presence of a proper node, yet will no longer dwell upon it, because I prefer to select
hereafter one of those ten trihedral-pairs in which no plane passes through tlie node,
for investigating by its help the position of the 27 lines.

Letp, q, r, s denote plane-coordinates such that ^^1"^+^?/'+ ;•.:'+ a'w'=0 shall be the
equation m point-coordinates x', y', z', w' of a tangent plane to the surface (p=0. To
find then the reciprocal equation of the surface, we are concerned with the system

r, S<P Sip Qip B?

where the first equation may also be replaced by l-\-sw=^px-\-qy-\-rz-{'S'W=^Q. The
equations

ig=X«,-fy., \f=Yv.^xz, \f=Z^^xy, '%=x

lead to the system

px+2DL-2syz=0, qx-^2lY-2szx=^, rx+2lZ-2sxy=0,

DE. SCHLAPLI ON SURFACES OF TUE TUIEI) ORDER. 203

the equations whereof are the derivatives of

Ix-2s3->/:{ = {px+qi/+rz)x-2sXi/:) = 0

with respect to x, y, z. The reciprocal equation of the sui-face therefore is of the form
Q=0, where Q is a decimic function of (/», y, r, s), which multiplied by s^ is the discri-
minant of the ternary cubic

(Sapy + 3.(a2+2fi>yy+3.{ar+2e2))x-z-\-3.{bj)+2fq)xf

+6 . (dj)+eq+fr-s)xj/z + S. {e2)-{-2er)xz'+(Uq)f+ 3 . (/j)'+2dq}ifz

+ 3.icq+2dr)yz'-\-{5cr)z\

Hence to work out the decimic in question we may use the invariants of the fourth
and sixth order which Dr. Salmon * denotes by S and T, only that we replace the latter
notation by — 8T. Puttmg, then,

<S?=Ap'+ Bq'' + Cr' + 2 Dqr + 2'Er2} + 2 Yjjq, \d<P= Tdj) + Qdq + 'Rdr,
t-=zdp-\-eq -\-fr, U = adqr + herp + cfpq, V=: 2 Ajjgy— aqrV —h)pQ— cpqR,
W= a'Aj';-" + h^Bry + c^C])-q--\- 2pqr{hcT)p + cdEq + a JFr),
L=s=-2^s-(I>, M=U5+V, l^=2ahcpqrs+Vs\ S='-P=108s^Q,
we find

S=L^-125M, T=L'-18sLM-54s%
Q=UN+L=M^-18sLMN-16sM^-27s^N^
=-2abcpqrs' + {abc^aq-r--\-2pqr^bc{2ef— 1ad)]))s^
+2{lbc%ef-Sad)pY-\-2yqrlbc{-oah-+21ad'--{-be'+cf'-12def)p'
+2pqrla{-Sahcd-\-lQbcef-6d{be^+(f')+2d'cf)qr}s'

+( K+( y+( y+( >•

— AO-(cq^—2dq)'+br)(a)"—2e)p + c2f){b2)''—2fpq+aq'),

and Q=0 is the reciprocal equation of the surface.
If — 16H denote the Hessian of the cubic p, then

n=Axw^+2{x,lBx-3Axyz)w-lcrw*+21bcf'z'+ia'i/z'2.{ad+ef)x;

the spinode curve therefore is represented by the system

(p=0, 8Axi/zw-8xyzladx+la\T*-21bcfz^=0 ;

hence it is a complete curve of the twelfth degree, and has the node of the cubic surface
for a sixfold point, where the six nodal rays are tangents to the ciu've.

Art. 10. Starting from a trihedral-pair form umo+xi/z=0, where no four of the six

* Higher Piano Curves, pp. 184 and 18G.

2f2

204 DE. SCHLAFLI ON SUEFACES OF THE TIIIED OEDEE.

planes have a point in common, and letting |3 be a linear ciifferentiation symbol signify-
ing that the differentials of the fonr fundamental coordinates may be replaced by arbitrary
quantities (I3=aB,+/3dy+yS.+Sd,„, if for the moment {x,y, z, w) are the fundamental
coordinates), Ave see that at the node the difterential equation ^{umo-\-x>iz)=.Q ought
not to be different from tlie general identical equation

AB H + B J3 V + C33 w + DMx + EQy + FSr = 0 ;
hence the coefficients of the differentials in both equations must be proportional. But
since in the former the coefficients vw, uw, nv, //z, a'z, a'y satisfy the equation

r w .uw.nv=//z. a'z . ay.
or, which is the same thing,

( u vw + a'//z){n cw — xyz) = 0,

the coefficients in the latter differential equation belong to one of the roots of the well-
known cubic condition. Let them, for instance, be «', V, c\ d\ c',f' ; then in consequence
of the equation of the surface the proportions in question become

a'i( = b'v= c'w = — d'x = — e'l/ = —fz ;
or, because without any loss of generality (since the linear functions n, r, ... imply
arbitrary numerical factors) we may replace a'. h\ c', d\ e'.f by 1, 1, 1, 1, 1, 1, more simply

at the node. Hence, and from the first and tliird identical relations, we get

a + fj + c^d + c+f, ahr=zdef\ a" -\-/j"-{-c"=d" + €"+/', a''b"c"=d"e'f'.

But we may put

a"=zXa-\-iy., h" = }Jj-\-iM, 6cc.
and we then obtain

{he + ca -\-ah- (.f-fd-de)-k-i^ = 0.

The factor within the brackets, if vanishing, would require one of the six cases such as
((z^d, h=^e, ('=/, and leave K, (m indeterminate. Avoiding so great a restriction, and
keeping to the proper meaning of the auxiliary cubic condition, we find that it has two
equal roots ?v=0, and a single root ///=:0. Consequently the constants corresponding
to the single root are a, b, c, d, e,f, and satisfy the equations

a-\-b-\-c^d-\-c-\-f, ahc=-dcf;
the constants in the accented sets are all of them equal to unity. Hence the line I' coin-
cides with l'\ m' with ?n", and so on, and all these six pairs of coincident lines pass
through the node. It may also be observed that tliey formed in the general case a
double-six, and that now the corresponding lines (in both sixes) also coincide. Moreover,
since the tlirec independent lines /', ?«', n' (in the general case) arc intersected by each
of the three independent lines ji'\ q", )•", all these six lines lie (in the general case)
iipon a quadratic surface ; and now that all the six lines meet in a point, the (piadratic
surface must degenerate into a cone. Let

P=(r;-f.r)(«'+.r)-(»+//)(»-f--), Q = („+.r)(« + //)(« + r),

DR. SCirL'VTLI ON SUEFACES OF Till: TIUKD ORDEE. 205

then

itV^Q=tuv(H-^v-{-w-\-x+i/+z)-\-(uvw+.v>/z);

and because u-\-v-{-io-\-j;-\-i/-\-z=0 is the second (or third) identical relation, and
2lviv-\-xi/z=0 the equation of the surface, the latter is changed into mP+Q=0, which
form shows the nodal cone P=0, the equation of which may also be exhibited mider
the symmetrical form

VlV-\-WU + l(V—l/Z — Z.V — .V!/=^0.

Art. 11. Bistnbvtion into species. — It is plain that a single node of a real cubic surface
cannot but be a real point. "We may therefore draw through it three (real) fundamental
planes (which call ,r, y, z) and take the fourth plane at pleasure (call it w) ; the equa-
tion of the surface then is wP+Q=:0, where the functions P, Q contain only x, y, z, and
therefore represent cones respectively of the second and third orders ; and it is obvious
that as well in P as in Q all tlie coefficients will be real. Hence as to the six nodal rays
(P=:0, Q=0), all of them may be real, or four, or two, or none. So we might distin-
guish four species of the cubic surface Avith a single proper node ; but in the last of the
mentioned cases (when the node is an isolated point of the surface) the cone P=:0 may
be real or imaginary. Let us therefore distinguish five species.

First species, II. 1. All six nodal rays are real. — The surface is constructed, when we
assume six constants and six linear functions of the fundamental coordinates, all of them
real, and satisfy tlie equations

aJf-h-\-c=(l+e-\-f, ahc=def, w + r+H--|-a'+^+r=0, an-{-hv+cio-^dx-\-e>j-^fz=(),
where bc-\-ca-\-ab—ef—fd -de must not vanish. Then uviv-\-xyz=0 is the equation
of the surface. Not passing through the node, there are fifteen simple real lines, wliicli
form fifteen triangles, each line being common to three simple triangle-planes. Of tlie
fifteen planes to be twice counted, each contains one of the simple lines and two ncxlal
rays. This species constitutes the transition from the first to the second species of the
general surface *.

Second species, II. 2. Only four nodal rays are real. — While we keep to the same
system of equations as before, it is possible to dispose of the constants and linear func-
tions in such manner that a, h, c are respectively conjugate to d, e,f, and t(, v, w to x, y, z.
Then by permuting / and — /, the three schemes

ux tiy uz I

change into

(/'

I" )

P

(I>'

P")

(»i'

»i")

1

(!?'

2")

(n'

n")

r

{''

^')

(^'

r )

T

ip'

P")

(«'

«" )

i

{'J

2")

{ni

m!')

r

0'

n

vx vy vz
wx tvy wz

ux vx xox
uy vy wy

uz vz wz

* Viz. from I. 1 to I. 2, and so in other cases where the species of the general siu-fiice are referred to — A. C.

206 DE. SCHLAFLI ON SUEFACES OF THE THIED OEDEE.

Hence the four nodal rays (/', ^"), (/, p"), {q\ q"), (r', )■") and the remaining ones
(m', m"), (71', n") are conjugate. Of the simple lines seven only, \\z. ux, vy, wz, I,]), q, r
are real and form three real triangles whicli have the line I in common. Besides these
three simple planes there are seven real planes to be twice counted, each of which
passes through the node and one of the seven real simple lines. "Wlien the two equal
roots of the cubic condition separate themselves into real roots, the four real nodal rays
become eight real lines, and the surface changes into the general one of the second
species. In the other case, only the plane passing through the two conjugate nodal rays
resolves itself into two real planes (in the former case into two conjugate planes), so
that there arises a general surface of the third .species.

Third sjjecies, II. 3. Only two nodal nojs are real. — It is possible to satisfy the
above system in such manner that the constants «, d are real, h conjugate to c, and e to/;
again, that the planes «., x are real, v conjugate to w, and y to z. By the change of i
into — i the three orisrinal schemes then change into

"^ i

I

il'

F)

P

ip'

in

mj

n

(«'

n")

r

(,•'

/•")

¥ '

hi

(»i'

m")

'1

{'/

'/)

The two nodal rays (/', I"), [p', p") alone are real ; and (not passing through the node)
only the lines ux, I, p, forming a triangle, are real. Besides the three simple planes v,
X, (mo,-) the only'real planes are the three planes (to be twice counted), which pass through
the node and through one of the real simple lines. This case forms the transition from
the third to the fifth species of the general surface.

Fourth and fifth species, II. 4, and II. 5. Three pairs of conjugate nodal rai/s. — The
above system^ is compatible with the condition that e shall be conjugate to /, and the
plane y to ~, while all the other constants and planes are real. Then in the first of the
three original schemes of lines the second and thii'd columns interchange, and the second
and third schemes interchange. Hence the nodal rays (?, I"), (m!, W), (n', u") are
respectively conjugate to {p\p"), (?', (f)- (/, r"), and of the simple lines only iix, vx, wx,
forming a triangle, are real. Of simple planes only the seven x ; ?«, (iix) ; v, {vx) ;
w, {wx) are real, and of planes to be twice counted only those joining two conjugate
nodal rays, therefore three in number. The case is intermediate between the fourth
and fifth species of the general surface.

To decide the question, when is tlic nodal cone real or not I We throw its quadric
V = 'v-\-x){w-\-x)—{;u-{-y){u-\-z) into the form

_(^d_j,)^a-c)V={{d-c){v+x)+{^^-f){u^!j)\\{d-c){:v+x)+{a-e){u + z)\

+ W-W-c)-{a-c){a-f)Jn+y){u+z).
On the right-liand side the first term is positive as a product of two conjugate factors,
and in the second term ("+.'/)(« + ") is positive for the same reason. Ilcnce the cone is
real when {d—h){d—c) — {a—e){a—f) is negative; in the opposite case it is imaginary.

DR. SCHLAFLI ON SURFACES OF THE THIED ORDER, 207

But if we elimiuatc a and d by the help of the equations

a-{-h-\-c=^d-\-c-\-f. abc=def,
the expression becomes

{i-e){h-f){c-e){c-f):{hc-cf),
where the numerator is positive, since its factors are conjugate by pairs. The nodal cone
is therefore real when the denominator hc—ef is neyative {fourth species, II. 4), but
imaginary when Oc — ef is jfositive {fifth species, II. 5).

III. Cubic surface of the ninth class with a hijyJanar node.
Art. 12. The equation .ryw+c'=0, where, in the proximity of the node, only w remains
finite, when discussed under both suppositions of x, y being real or conjugate, gives a

preliminary ^•iew of the biplanai" node at the point ^. A plane turning about its edge

(x=0, ^ = 0) cuts the surface in a curve with a cusp, which changes its direction into
the opposite one whenever the turning plane has passed one of the two real nodal
planes; or always keeps its dir-ection if the nodal planes be conjugate, so thai in the
latter case the siu-face here terminates in the form of a thorn [\'iz. in such a form as is
generated by the revolution of a somicubical parabola about the cuspidal tangent].

The equation of the surface is «i'W+Q=0, where «i, v are linear functions and Q a
cubic one of.r, y, z. Denote the three nodal rays(M^O, Q = 0 by 1, 2, 3, and the thi-ee
(i':=0, Q^O) by 4, 5, 6. Then each combination such as (14, 25, 36) gives a deter-
minate position of the plane w=0, in virtue of which the cone Q breaks up into three
planes. Keeping to the order of 1, 2, 3 and permuting only 4, 5, 6, we see there are
six such transformations. But whenever Q,=^xyz, the surface contains a simple triangle
(■M)=:0, .i''^z=0); and it is also easy to see that the three positive permutations give
one trihedral, and that three negative ones give the other trihedral of a trihedral-pair
where no four of the six planes meet in a point, the only possible trihedral-pair of sucli
kind.

If in art. 9 we put y=1(lx-\-my-\-nz'){f x-\-'n'iy-\-nlz),

I, m, n \ =Xp-\-[ji,q-\-vr=(r, 2U'{mn' -\-m'n)qr=v, 2lI!Xqr=yp,

H, m', n'

Pr q. r
then we have

K=—-k\ B = -//,», C=—v\ D=-^^, E =— vX, F = — Xjti,

A=0, t =X{mn!-\-m'H)p, V=2v, \=2a4^, W=-4^/.^ l.=s'-2ts+i^,

M=2(us+(T'4/), 'N =4:{Umnl'm'n'pqrs—->p'),

— = ilmn^m'n';pqr{U— 36sL(ys4-(T'4/) + 216s°'^'^ — 4:32lmnl'm'n'j)qrs^}

-^V{s{v■'-^')+2<Tv■4.-\-2t^'} + 3QL■4,'{vs-\-ff^P)-^2{vs-\-<T^y-l08s^\

208 DK. SCIILAFLI OX SUEFACES OF THE THIED OEDEE.

The first term of the expression according to tlie descending powers of s is

and the last is

— il;M(T^(uq—mr)(Ir—u2i)(in2) — I(j){/t'(j^ — m'r)(ri-—)i'2j){iii'2' — lq).

The system

(Lv-\-)ii>/-{-ii:){I'x-\-m'i/ + ii':)w-\-a://z=0,

l!T\v'-22mm'))>i'fz"--ix//:'ZU'{mn'+m')i)d'=0

represents the spinode curve, \Yhich is therefore a complete curve of the twelfth degree

and has the node for an eiglitfold point, where the tangents arc determined by the

system

since the cone drawn from the node through the spinode curve may also be tll^o^^^l into

the form

1I.V . 2/'j' . llU\r-l{mn'+m'n)i/:}-{-TAyz'-2a'//zl[j.KV=0.

Art. 13. Let us represent by uvw-\-.ri/z^O the only possible trihedral-pair no plane
of whicli passes through the node, and considering this as a particular case of art. 10, let

„ + , + „,+.,.+^+c = 0 •
be that identical relation which answers to the two ccjual roots which we know must
exist of the cubic condition, and

A« + Bi' + CH' + D,c+E//+Fr = 0
any other identical relation. Then the coefficients in the relation corresponding to the
single root of the cubic condition will be

r;=:XA+/A, Z' = /.B+|7-, &c. ;
and since this condition

(X A + ij.)(XB+!^){W+im) - (/.D +!j.)(xE+!m){-aV+!m) = 0

must bo divisible by a', it follows

A+B+C=D + E+F, r, = (A-DXA-EXA-F), &c., r/=:(A-DXB-D)(C-D), &c.

Again, at the end of art. 11 we had a form of the nodal cone P containing only the three

variables v-\-j; u-{-ih "+~5 "^ respect to which the discriminant of P is

{h-d){c-(l)-[u-e}\a-j ) _SQC--'2.'E,'P
(h-d)^ — (B-D)'^ '

Now in order that the nodal cone may break up into two planes, wc must have

BC+CA+AB=EF+FU + DE,

wliicli reduces the cubic condition to

(ABC-DEF)a'=0.
llLyectiug the solution

ABC=DEF
as giving rise to

A=D, B=E, C=F

DR. SCHLAFLI OX SUELACES OF THE TUIRD ORDER. 209

for iiibtance, and tlius bringing

into one and the same plane, wc infer that if a trihedral-pair form, explicitly not
singnlar, belong to a cubic surface of the ninth class, the cubic condition inherent to
such a trihedral-pair must have three equal roots.

Eeciprocally, let uviv-{-3:i/z=0 be the equation of tlie surface, and

u-\-v+io-{-x+//+z=0, Au+Bv+Cw+'Dx+Ei/+Yz=0
identical relations, where

A+B+C=DH-E+F, BC+CA-fAB=EF+FD+DE,

biit where ABC — DEF is different from zero, then wc have a set of proportions such as

A-E_C-D.
B-D— A-F'

and since at the node u = v=:w=—x= — fj:=—z, the nodal cone is represented by

But because the equation

{(B-E)(«+^-)-(A-r>)(.+//)i-.!(B-F)(«+.r)-(A-D)(.+r),^
=[(B-E)(B-F)-(A-D)(C-D)](«+.r)'^

-(A+B+C-D-E-FXA-DX"+.r-)(''+,'/+~)
+(A-B)[(A-T>)(a + v)+C(u+a-)-]{ic+vi-io+w+>/+z)
— {A-T>)(u+x){Au+Bv+Civ+'Dx+E>/+Fz)
— (A—T>)-(viv-\-ino-\- uv —y~ — xz — xy)
is cxjjiicifl)/ identical, therefore the equation

— (A—'D)-(vw + wn-\-iiv—}/z—zx—xi/)

= \(B-E)iu+x)-{A-B)(v+y)\ \(B-Y){u+x)~(A-T))(v+z)}

is identical in respect to the fundamental coordinates ; in other words, the nodal cone
breaks up into a pair of planes. The nodal edge may be represented by

uz=s-{-Af, r=s+B^, tv=zs-\-Cf, x=—s—Df, y=—s—Yjt, z=—s—Ff,

whei'e s, t denote independent variables.

Now it is plain that the equation n-{-x = 0, for instance, represents at once the three
planes previously denoted by (ux), (nx)', (nx)", wherefore now the three lines I, I', I"
coincide, and so on. Each of the sLx nodal rays thus unites three (independent)
lines of the surface; only the lines uvw=^0, xi/z=0 are simple lines. AVe have in all
6'3+9-l = 27 lines. One nodal plane unites all the six planes sucli as (ban), and the
other all the six planes such as (j^Qi')- Of the nine planes joining any ray of tlie one
nodal plane with any ray of the other, each unites three planes such as (ux), (nx)', (vx)" ;

MDCCCLXIir. 2 G

210 DE. SCHLAFLI ON SURFACES OF THE THIED OEDEE.

only the six planes of the trihedral-pair here chosen are simple triangle-planes. There
ai-e in all 2-6 + 9-3 + 6-1 = 45 triangle-planes.

Ai't. 14. There are four species.

First sjjecies, III. 1. — u, v, w, .r, //, z are real. Everything is then real.

Second sjjecies, III. 2. — u is conjugate to x, v to ?/, and w to z. Both nodal planes
are real; one of them contains the real ray I and the two conjugate raysni, n ; the other
nodal plane contains the three real rays p, q, r. Of the nine simple lines three only,
ux, vy, 'Wz, are real.

Third sj^ecies, III. 3. — u, x are real, v is conjugate to w, // to z. Both nodal planes
are real, and each of them contains a real and two conjugate rays; for I and^> are real,
and m is conjugate to n, q to r. Of the nine simple lines one only, ux, is real.

Fourth s])ecies. 111. 4. — u, v, «', x are real, // is conjugate to z. The two nodal planes
are conjugate; for I,vi, n are respectively conjugate to |;, q, r. Of the nine simple lines
thi'ee only, forming the triangle (.^=^0, i(vw=^0), are real.

The enumeration is complete, because all cases that can happen in respect to the
nodal rays are exhausted.

IV. Cubic surface of the eighth class ivith tivo proper nodes.
Art. 15. From art. 8 we akeady loiow that the line joining the two nodes, or axis,
unites t«o and the same rays of each node, and that there is a singular tangent plane
which touches the surface, and therefore also each nodal cone along the axis, and besides
intersects the surface in a single Ime which we have termed the transversal. Since then,
besides the axis, each nodal cone has four rays not passing through the other node, there
are in all ten nodal rays which represent twenty lines of the surface (considered as
though it \\'ere general), so that there remain only seven simple lines, one of which is
the transversal above mentioned. Because this transversal is not intersected by the
eight disengaged nodal rays, but only by the axis, that is by four lines, it must meet all
the six other lines, and will therefore form with them three triangles. Besides such
triangle, there pass through each of the six lines four other planes, which are of course
those passing through one or the other node, each of them counting for two triangle-
planes. Again, a plane through the axis and a disengaged ray of one node must intersect
the surface in a third line, which cannot but be a disengaged ray of the other node.
Such plane counts for four triangle-planes ; for any one of the four disengaged rays of one
node, since it determines with each of the three remaining rays three triangle-planes,
must determines with the axis two such planes ; and because it is made up of two inde-
pendent lines of the surface, the two planes must be twice counted. As to the singular
tangent plane, it counts twice, because through the transversal there already pass three
simple triangle-planes. The surface thus has a line representing four lines, viz. the
axis ; eight other nodal rays, each representing two lines ; and seven simple lines, viz.
the transversal and the remaining sides of the three simple triangles standing upon it;
in all l-4 + S-2 + 7-l=:27 lines. Again, the surface has four planes each representing

DE. SCHLlPLI ON SURFACES OF THE THIRD ORDER. 211

foui- triangle-planes] of the surface, aiz. tliose passing through the axis and one ray of
either node; thirteen planes each representing two triangle-planes, viz. the singular
tangent plane and the twice six other planes each of them through two diseno-a^ed rays
of the same node ; lastly, the three simple triangle-planes passing through the trans-
versal; in all 4-44-13'2-f-3-l = 45 triangle-planes.

We proceed now to reduce the equation of the surface in question to its simplest
form. Let .r=0 be the equation of the smgular tangent-plane, and let the plane y=0
pass through the axis, while the planes z=0 and iv=0 touch respectively the nodal cones
in lines belonging to the plane y=0, then the term i/zw and those divisible by s', w^, xz,
XIV will disappear, and we may therefore write

xziv-\-f{z-\-w)-\-a3f + iki~>j-\-%cxf+ 4:df=0.

But this cubic if multiplied by x becomes

{xz+f){x^+f)-(y-dxy+Q(c+(r-yf+i{b-dy-^>/+{a-{-d*)x\
while

.rz-{-f=xiz + 2di/-d\v)-\-(i/-dxy, xw+f^x{w+2di/-d\r)+(>;- dxy.

Now it will be readily seen that the equation of the smface can in but one way be
reduced to the form

xzw-}-i/%z+'w)-^ax^-\-bx'y+cxf=0,

where we might also put unity instead of one of the tkree constants a, b, c. In respect
to the fundamental coordinates, the equation implies seventeen constant elements, as
it should do, since the two nodes take away two disposable constants from the full
number 19.

Let us attempt to form the equation reciprocal to this. We have

6j)=zu'-\-oax^-\-2bx!/-{-c!/-, 0(i=2i/{z-\-tc)-\-bx--\-2cxi/, 6r=xw-\-y-, 6s=xz-\-if.

Putting then

(p=px'+gxy-{r-\-s)y\ y^=ax'-\-bx'y-\-cxy--f,

regarding p, q, r, s in respect to ^, ^ as constants, and eliminating z, w by the help
of the original equation of the surface, we find

f-rs=-^, 6^=^, 6^=P^
^ 0^ ox oy oy

whence 6ip=2x, 26rs=—/p; and lastly, on eliminating 6,

that is to say, the discriminant of the binary- quartic

{^px'+qxt/—(r+s)fy+irs{ax*+bx'y+cjr'f-—y*)

must vanish, and divided by ?V it will give the reciprocal equation required.

2 g2

212 DE. SCHLAFLI OX SrEFACES OF THE THIED OEDEE.

Denoting the Hessian of tlie primitive cubic by 4H, we have

+{b'-Sacy+hex'f/+(c'+12ayy+4:krf.
Hence arises for the spinode cur^e the system

a-zw +//-(c + «') + ax' + bx'// + cxf =0,]

where the axis (.r=0, ^^0) counts for two solutions; therefore the spinode curve is a
partial curve of the tenth degree, and each node of the original surface is a quadruple
point of tlie curve, the nodal rays at such point being tangents to the curve.

Art. 16. We proceed to determine the lines and triangle-planes of the surface. The
transversal is (.r=0, z-\-w=:0). The nodal cones are xz-\-tf = 0. xw-\-//-=^0; besides
touching one another along the axis, they intersect in a conic the plane of which is
z — w^O. This plane and the transversal therefore cut the axis harmonically in regard
to the two nodes.

Cutting the surface by the plane _;/ — A.r=0, and omitting the solution x=0, we obtain
the equation

(z + y.-x){ w + K\t) + (a +//>.+ TA^ - K'}x' = 0 ;

and in order tliat this break up into factors, the condition a^ — rX^ — l/\—(f^=0 must be
fultilled, and the equation of the section then becomes {z-^yJ'x)(^o -\-'k^x)=0. Now, as is
well known, the solution of the quartic condition depends upon that of the cubic
equation

X^-2rX^ + (r^ + 4rOX-/r=n.

Accordingly, in order to avoid irrationalities, we put

2c=u'+iy-\-y-, ,•^ + 4r^=/3y-l-7V + aV3^ b = afty,
and, for tlie sake of shortness, (7:=^(a+/3 + y), whence

« = <r(^_«)(ff_/3)(ff-y), r = J-(«' + /3= + y'),

and K has the four values c, u—a, (o — (x, y—n. Hence the four triangle-planes passing
through tlie axis are

ffo.-— y=0, (ff— «).(.' 4- ?/=0, (ff_/3),r+//=0, (<r— y).r+//=0.
The plane

passing tlirough the transversal, cuts tlie surface in the trilateral

or

0.- { (<r — /3 )((r - y)x + « y -w]{ n{a—a )x + a_y + w } = 0,

in each of whicli representations the consecution of the three sides is the same, while in
tlie first all the planes (or factf)rs) pass through the node W, and in the second througli
the node Z. Denoting the sides of the triangle corresponding to the constant k in the

BE. SCIILAFLT OX SUHFACES OF THE TIIIUD ORDEK. 213

same order by a.vi^, a, a. and tlie i)lanes passiiin^ tlu()ui;li tlicm and the nodes by (Wa),

(W^?), (Za), (Za), we see tliat tlnontcli each no(hd ray there pass tliree planes, as

follows : —

(Wa), (W'b). (AVc) tlirough tlic ray o-.r— y=(), a\v+z=0,

(Wa), (WJ), (We) through the ray (ff ~a).r+/y = 0, (ff-a)\v+z=0,
(Wb), (Wr), (W«) through the ray («r -/3).c+//=0, (^_/3)-=^-+r=0,
(Wc), (Wa), (W5) through the ray (ff _y).i-+y=0, (^_y)^r+z=0.

If we permute the nodes W and Z, we must in this sclieme also permute a with «, b

with b, c with c, and z with u'.

Art. 17. In order to get a trihi-dral-pair form, let

V = {T-(i)(>T-y)x + a>/-Z, Q = (^-y)(^-u)x+(3>/-Z,

K = ((r— a)(ff— /3),r + 7_y— r, S = w—z,
and

(,3-y)(y-«)(«-/3) = (1;

then it will be found that

2a(Q-E)P(P-S)+2<7(Q-rx)(R-P)(P-Q) = (J{.rr«'+y-'(c+«')+«.v'+i.f'^+c.i/};
but the left-hand side of this identical equation is equal to

-2QR{(,3-y)S+(y+«)Q--(«+/3)R},
where

(/3-7)S + (7+a)Q-(«+/3)K = (/3-7).f^(^-4r+«^+w}.
Put therefore

p = (/3-7){ff((r-a).r + «^+«'}, q = {y-c^)'A<T-i3):V-^fi>/-\-w},

r=(a-/3){ff(<7-7>r+7^+w},
and then

jtQn+qllF+r'PQ^O

will be the equation of the surface, where the six linear functions fulfil tlic identical
relations

2>+q+r=0, «j,+/32 + yy+(/3=-y=)P+(/-«=>)Q+(«^-/3^)R=0.

If h denote a number which is ultimately made to yanisli, this equation may be exhibited
under the form

{V-\-I>2j){Q+h(i)(ll-^Io-)-(F-h_i)){Q-hqXR-hr)=0.
Let^>=XP, q=i/jQ„ r=vR', then X+p.+j' = 0 in virtue of the equation of the surface.
Again, if for shortness we put

/=«X+/3^+y!', ^=a'^>,+/3=j(/.-fy'v, h=uf—g,
from the foregomg relations it will next be found

S=^+/./^v, P=r/ + (/3-y)^v, &c., ^.r=-2(/3-y)P,
^3^=2((3-y)(<r-«)P, ^r=2(/3-y)(<r-a)^P,

and then

0x=/^, Ou=fh, 6z=—dij—]i'', Ow—OXf/jn—h';

214 DE. SCHLAFLI OX SURFACES OF THE THIED OEDER.

the coordinates of a point of the sm-face are thns expressed m terms of two independent
variables; only the values X=/3 — y, (oo=y— a, i'=a— /3 are inadmissible. To verify the
equation of the surface vs'e have

6{(yx-y)=frj, 0-y){(ff-a).r+y}=>/, &c.,
whence

and on the other side

S(xz+f)=-fff\ 0{a-w+f) = l[j,vf\
This gives indeed

but the values of the nodal cone quadratics show that three rays of the node W and one
ray of the node Z cannot be expressed.

We have still to divide this sort of surface into species. Whether both nodes be real
(when z, lo are real) or conjugate (:, w conjugate), there are but three cases to be
distinguished.

1. a, |S, 7 are real. Then the four trionp-le-planes passing through the axis and the
three passing through the transversal are real. IV. 1, and IV. 4.

2. a is real, /3, 7 are complex and conjugate. Then of the planes passing through the
axis only two are real, the two others are conjugate ; and of those passing through the
transversal but one is real and two are conjugate. IV. 2, and IV. 5.

3. a is real, /3, y are lateral (according to the denomination proposed by Gauss, that
is to say, /3- and y^ are negative). Then the planes passing through the axis are conju-
gate by pairs ; and those passing through the transversal are all of them real. IV. 3, and
IV. 6.

Hence arise six species.

V. Cuhic surface of the eighth class tv/th a hiplanar node.
Art. 18. From art. 7 it appears that the equation of this surface can be written
xyw + [x -\-y)z' + 2{ax''-\-hy'')z + ex"" -\-df=^,
since all the terms divisible by xy may be joined to the first term. But givmg this
equation the form

xy\_w-2{a-\-h)z-{2ah-\-c)x-{2ah-\-d)y'\

we see that more briefly it may also be thus written,

2xyw + {x->ry){z''-ax''-hf-)=^.
The equation reciprocal to this is contained in the discriminant of the binary qiiartic

r\TY + 2s(;w + qy)xy{x-\-y) -f s^aa"" + by'') {x-\-y)\
If we put

']^=[a-\-b)s[-\-2{2y-\-q)s-\-r\ '\\={hp-\-aq)s-\-2yq, 'N=abr''-—bp'' — aq^,

DE. SCHLAFLl ON SITJFACES OF THE THIED OEDEE. 216

and denote by S, T the same invariants as are found in Dr. Salmon's ' Higher Algebra,'
p. 100, then we have

12S=L'-12s^M, 216T=-L'+lSs^LM+54s*N, lC(S'-27P) = s*0,
and ultimately

Q=ab(a+by{ia+b)r-(p-qy}s''

-\-{3ab(a'-7ab+b'y+[b{9a''+26ab-b-y--2Gab(a+b)j)q+a{-a'+2Qab
+ db')q'y+(p-2y[b{-12a+b)p'-+22abj>q+a(a-12b)q'']}s'

+2{5ab[{2a-b)2)+{-a + 2b)qy + [b{-2a+6b)p'+b(5a-2b)p'q

-\-a{-2a+ob}pq-+a{5a—2b)q'y+2(p—qy[—2bf+bp-q-\-aj)q'-2ag^']}s'

+ {Zab(a+by + [b{0a-2by+8ab2)q+a{-2a-^%)q-y*

+2[—6bp*+bfq—(a+b)pY+a2)q'-Gaq'y--\-4:j)Y{j)—qy}s''

+2{3ab{p+qy-{Up'+2bp'q+2apq'- + 5aqy*-\-ijjYip+qy}s

is the reciprocal equation required.

Let 16H be the Hessian of the primitive function, then

B.=2{x+y)xyio+{x-i/y:"-+(d+>/)(3ax'-cu">/-bxf-\-?ibfy

whence the system

,r!/ ,ax--\-bif- — z"-, aj.^-\-bt)'^ ^0

x-\-y , 2w , 2^
will represent the spinode curve, which is therefore a partial cuiTe of the tentli degree,
and in which there pass through the node six branches, in lowest approximation repre-
sented by the systems {2xio+z-=Q, 2byho-\-z*=0), {2yw+z' = Q, 2(1X^10 -\-z*=()y and
having the axis for a common tangent with a singular kind of contact. Any plane passing
through the node intersects here the curve in six coincident points, any plane passing
through the axis in eight, and each nodal plane in ten coincident points,

Ai-t. 19. Let «=«-, b=^-, U=2a|3(,i'+^)— W, V=r+a.r+/3y, ^\=z — cl.x—^y,
X=2a/3(a-+y)+W, Y=—z — ouc-\-^y, Z=—z—aa;-{-(3y; then the original equation
takes the form UVW+XYZ, and the six new functions fulfil the two identical relations
V+W+Y+Z = 0, U-(«-f-/3)V+(«+f3)W+X-(a-/3)Y+(a-/3)Z=0. Imagine
instead of these the relations

;iU+V+W+/,X+Y+Z=0. A/(U+BV+CW+D/(X+EY+FZ=0,
where

A=^-(«+/3), B=-(<.+/3)+/<«+/3)% C=a+/3-^(«-/3)%
D^}-(«+/3) + 4a/3/;, E=-(«-/3){l-;i(a-i3)}, F=-E.

21G DE. SCIILAFLI OX SUEFACES OF THE THIED OEDEE.

Because the six constants fulfil the equations

A+B + C=D+E+F, BC+CA+AB=EF+FD+DE,

the cubic condition inherent to the trihedral pair /(UVW-j-/;XYZ — 0 has three equal
roots. Let then li vanish, and the former system will be reproduced. At the same time
such equations of triangle-planes as in art. 13 were ?;+'"<'^=0, u-\-y = Q, 6'+.r^0 will
now become respectively U+X = 0, y=0, V=0, and so on; but we shall continue to
denote them bv ('«'), ("j/), (r.r) as before, yet omit accents, since all tliree accents coin-
cide. So we get the following survey of the twenty-seven lines on the surfoce, showing
in what manner tliey coincide : —

The nodal edge (or here axis, since the surface is along it touched by a plane) (a,'=0,
y=:0) unites the six lines I, ]). The four other nodal rays unite each of them four lines
such as follow, [vy, r), {to:, q) ; [vz, n), (w//, m). The transversal iix, and the other sides
of the two simple triangles standing upon it, iiy, u:, i\v, uw, are the only five simple
lines. In all l-G + 4--±-i- 5-1=27 lines.

Each nodal plane unites twelve triangle-planes, viz. A'=0 unites (>•:), («'//), (bn/i);
and // = 0 unites (vy), («'"), {pi'')- The four planes, joining a ray of one nodal plane
with a ray of the other nodal plane, unite each of them four triangle-planes, viz.
V=0 {r, (c.r)}, W=0 {«-, («\r)}, Y = 0 {//, (w/)}, Z = 0 {z, (»-)}.

The singular tangent plane x-\-y = 0 unites the tlirce planes (".*). Lastly, there are but
two simple triangle planes, those passing through the transversal U=0, and X=:0. In
the whole 2-12 + 4--l + l-3-f 2T = 45 triangle-planes.

Since the functions z, w, x-\-y, xy must always be real, there are four species.

1. All is real, and a, c are positive. V. 1.

2. X, y are real, a is positive, and b negative. The only real lines are the axis, two
rays in only one nodal plane, and the transversal. V. 2.

3. X, y are real, a and b are negative. The axis and transversal are the only real lines.
Each nodal plane contains two conjugate rays. V. 3.

4. X, y are conjugate, and so also arc a, h. The axis and transversal are real ; of the
two real planes jiassing through the transversal, one only contains a real triangle ; these
four lines only arc real. V 4.

VI. Cubic .surface of the seventh class with a biplanar and a proper node.

Art. 20. If a cubic surface have two nodes, chosen for points of reference ^-, ^, its

equation necessarily takes the form hiv-\-niz-\-nw-\-p-^^^, where I=(x, y) ; m, n^=[x, y)\

p=^{x, yY ; and if ^ be a biplanar node, h-\-n must break up into factors, whence I

must di\'idc n, so that Iz-\-n may then be replaced by xz. And joining the terms in
VIZ, whicli are di\isible by xz^ to the term xzw, we may write

xzw + xy-z + ax'^ + okry + '^)cxy- + (/// ' = 0
as the equation of tlie surface.

DE. SCHLAFLI OX SURFACES OF THE THIRD ORDER. 217

The equation reciprocal to this is found by dividing the discriuunant of the binary
quartic {j)x- + qxi/—s//-y + irs.r(a.v^ + o0.v'i/ + oc.r//'' + il//^) by >-V and equating the quo-
tient to zero. Let

L = ^- + 4(^; + 3cr)s, ]M = ~ djxj + 3( — 2q/; + ^-^ — 2b(Ji')s + 2as\

N = (ly + 2d{ohp — 2f/y + 2adr)s + 3( 3*= — iac)s\

12S=P + 24r5M, -216T=L'+ 36>-sLM + 216rVX,

M=-LN=46P, S^-27P=;-V0,

then S, T are the two invariants of the quartic in question, and
0=L=r + 8/-:\P- 9;-L:\IN -27r6N^=0
is the equation of the surface in phmc-coordinates. If

S=3K^+24+fZ|-+j|— 26-#,

thenSN=-2<?M, hMz=-dL, lL=12drs, whence SS=0, ST=0, he=0.
The quartic function, the Hessian of the original cubic, is
{z + 3(c.v + dy) } {.vzio -\rifz -f ax^ + okv^ij + ocnf + df) — 4r(«.i'^ + oki">j + ocxf + df)
— 3 { (4ff c - W-yv* + i^adx^y + Ud.f-if- + 4.cdxif + d-y ' } .

The spinode curve is therefore a partial curve of the ninth degree, which has the
biplanar node for a quintuple and the proper node for a triple point. The tangents at
the latter node are the three disengaged nodal rays ; but of those at the former node
one tangent is (.('=: 0, 3dy-\-^z=0), and the four remaining tangents are

c= 0, {iac- Wy + 4f«7.r> + QhdxY + icdaf + d-y'=^.
Art. 21. If d vanish, the edge of the biplanar node would belong to the surface, and
its class would therefore sink to six, contrary to the supposition. AVe are therefore
allowed to change z into dz and write

xzw +f-z + (y + ux){y + \3x){y + yx) = 0
as the equation of the surface. Again, let

Q = It; — yux — (7 + a,)y,
E=w — a/3.i,' — (K + /3)y,
;; = (,3-7)(«.r+y + c),
!?=(7-«)(f5-'-+.y+~X
7-= («-/3X7.r +// + -);
the six new linear functions will satisfy the identical relations

p+<^+r=0, (f3-7)P+(7-a)Q+(«-,3)R + «i7+/32+7r=0,

pQR+5EP+rrQ==-03-yX7-«)(«-/3){a"w-f/c+(//+«.r)(y+/3.i'Xy+7>r)};

MDCCCLXIII. 2 H

218 DE. SCHLAFLI OiSi SFEFACES OF THE THIED OEDEE.

and the equation of the surface is now changed into

2jQR-\-qTiF+rYQ=0.

Introducing then a number h which is ultimately made to vanish, we put

U=P+/iiJ, Y=Q-\-I>q, \Y=n+h; X=-P+7^^>, Y=-Q+% Z=-R+/;r,

whereby the equation of the surface becomes UVW+XYZ; and if, for the sake of

shortness, we put

A = u + {^-y)h, B=i3 + (y-«)/;. C = y+(a-(i)k
D = a-()3-y)/?, E=(3-{y-u)h. F^y-{cc-^)h,

the above-mentioned two identical relations become

U+V+W+X + Y + Z=0, AU+BV+CW+DX+EY+FZ=0,

where the six constants satisfy the relations

A+B+C=rD+E+F, BC+CA+AB=EF+FD+DE

strictly, while ABC — DEF is different from zero. All three roots, therefore, of the
cubic condition inherent to this trihedral-pair form coincide, and the corresponding
relation U+V+W+X+Y+Z = 0 counts for three such intersections.

Hence the axis (>r=0, //=:0) unites six lines, viz. the lines m, n. Each of the remaining
four rays of the biplanar node unites tlirec lines; viz. (a'=0, _?/+r = 0) unites the three
lines I, (~=0, aor+^^O) unites the three lines ^j, (c=0, (ix-\-y=^^) unites the three lines q,
(s=0, y.r+_?/=0) unites the three lines n Each of the remaining three rays of the
proper node unites two lines, viz. (a.i'+_y = 0, w — a^=:0) unites vz, wi/ ; (/3,r+//^0,
20— /3_y=0) unites ivx, uz; (yx-\-i/=0, w — y>/=0) unites iri/, vx. Three lines are simple
-liz. (P=0, iJ = 0) or ?«-, (Q=0, q=0) or v//, (Tl^O, r=0) or ivz. In the whole
l-6 + 4-3+3-2 + 3-l = 27 lines.

Of the following five triangle-planes each counts for six. Tlic singular tangent plane
x=zO unites all the six planes (Imn); the other plane r = 0 of the biplanar node unites
the six planes {])(ir); of the three further planes which (besides ;i'=0) pass through the
axis, the plane; «*■+//= 0 unites the two triads {rz), (w//), the plane (5x-\-i/=:0 unites the
two triads (?ar), («r), and the plane yx-\->j^i) unites the two triads (ui/), (vx). The
three planes which combine the single ray I of the biplanar node with any one of the
rays p, q, r in the opposite nodal plane count each of them for three triangle-planes,
viz. p=:0 unites the three planes (ux), q=0 unites the three planes (vi/), r=0 unites
the three planes (wz). I^astly, the three planes which combine any two of the three
disengaged rays of the proper node count each of them for two triangle-planes, viz. P^O
unites the planes u, x; Q^O unites ;;, >/ ; K^O unites w, z; they are the jdanes of the
two coinciding trihedrals. In all 5-G + 3-3 + 3-2 = 45 triangle-planes.

Art. 22. As to the reality of the linear functions in the original equation, it appears
that botli X and z must be real, since the two planes of the bi[)lauar node j)lay different

DR. SCULAFLI ON SUEFACES OF T]IE THIRD ORDER. 219

parts, and that it is always allowed to assume y as real, since the corresponding plane
may be turned about the real axis ; but w will then also be real, and of the three
constants a, /3, y one at least must be real. There are therefore but two species.

1. AU is real. VI. 1.

2. a is real, /3 and y arc conjugate. Only the axis (a.':=0, 2/=0), two rays of the
biplanar node (.r=0, ^+r=:0) and (^=0, a.r+y=0), one ray of the proper node
(aa'+3/ = 0, ?i'— a^=0), and the simple line (P=0, p=0) are real. VI. 2.

VII. Cubic surface of the seventh class vnth a Uplanar node.

Art. 23. According to art. 7 wo put

xyw-\-a:z'-\-{2a.x''+htf)z-\-cx'+dy^=0
as an equation of the surface in question, since all the terms divisible by a'y can be
carried into the single term xyw. The mark of this sort of biplanar node is that one
of its planes (here ^•:=0) touches the surface along the nodal edge; if b were to vanish
it would osculate the surface, and then the class would sink to six. Since therefore b is
not allowed to vanish, we may put the above equation ^^nder the form

xy{w—2-^{z-\-ax)-(^ab-\-'j^yj\+c(-iz + ax+'-i^y\ +%-Yr+«4-+^yj +(c-«>'=0,

or more simply

xyw-{-xz^-\-y'^z—ax^=0.

We shall in the sequel retain the constant a, because its being positive or negative
decides as to reality or non-reality. But now that we are concerned with the reciprocal
equation of the surface, we may, on putting a=:f*, change y, z, w respectively into
Xy, X'z, ?.'w, and we get

xyw -\-xz^-\-y^z — .r'= 0.

The reciprocal septic 0, when multiplied by 6•^ is the discriminant S^ — 27T^ of the binary
quartic

hence
where
and

12S=LH24s=M, -216T=L^+36.s^LM+216s%

L=r'-'-f4js, M=j;?--f2s% N=p'— 42's;

M*— LN
0=L=. — 7-^^+8sM^-9sLMN-27s^N

4s

=Q4:s'+2>2(?>jpr-4:q')s'-[-lQq(5r--^9f)s*
-f(r^+30/r=+lG0jY'--27^/+64(^^)s'
+ iq{ 1 Ij))-' + 1 2(/r' — 9fr — ifcfy
+r^(^>-"+ 12jV-^V— 8^y )s + /?r'(r^ -f) = 0

is the equation of the sui'face in plane-coordinates.

2 H 2

220 DE. SCHLAFLI OX SUEFACES OF THE THIED OEDEE.

The qi;aitic function, the Hessian of the cubic y=.r^?o+•^'~"+//~—•^'^ is

,rf-Aafz + 4x'+f;
and since the system

(/=0, 4.i-^+y_4.r/r = 0)

contains the axis (a'=0, i/=0) four times, the spinocle cur\e of the original surface is a
partial curve of the eighth degree. An arbitrary plane passing through the node inter-
sects the quartic cone in four lines, each of which also cuts the cubic surface in a point
distinct from the node. This arbitrary plane thus intersects the spinode curve in four
points distinct from tlie node, so that this must be a quadruple point of the curve, since
it unites the remaining points of intersection. One of the four branches passing through
the node is (at the lowest approximation) represented by

and therefore osculates the nodal plane which is a singular tangent plane to the surface,
and merely touches the other nodal plane. If f denote a very small variable number,
the three other branches may be represented by

z=f'W, a-=fui, y=—f'w.

Art. 24. The nodal plane .r=0, which touches the surface along the nodal edge or
axis, contains only a single disengaged ray (call it/"), the otlier nodal plane ^=0 con-
tains two rays (call them ^, h); and the planes combining the former ray mth any one
of the two latter rays are (_/r/) or c+a'=0, and (fh) or c— a.'=0. It is manifest that,
besides the nodal planes and these two planes, there pass no other triangle-planes
through the node. The planes {f<j) and {fh) intersect tlic surface respectively in the
simple lines (r+a;=0, w— y=0) or ^ and (c— .r=0, io-\-ij=^{)) or X'. Now as a plane
containing the node and any distinct and therefore simple line of the surfoce must be a
triangle-plane and therefore combine two nodal rays, there cannot, on all such planes
being exhausted, be found any other simple line of the surface. Hence these distinct
lines y and Jc are tlie only simple lines of the surface, and it is obvious tliat they do not
intersect each other. Again, since two independent lines are cut by five lines, and these
lines {j and Ic) are simultaneously cut only by the ray/^ this ray y unites five lines of
the general surface; and tlien, because each of the simple lines ^ and /(• must besides be
cut by five more lines, each of the nodal rays g and h also unites five lines. But all the
lines thus far mentioned count as 3*5 + iiT=:17 lines. Therefore the nodal edge (or
axis) unites ten lines of the general surface, precisely those ten (as we know from art. 1)
lines disengaged from the two independent lines y and 1,\

It is already pro^•ed that the two planes passing through tlie simple lines and tlie node
count each of them as five. No one of the five lines united in the ray jf intersects any
oth(;r of them ; wherefore no two of them can lie in the same triangle-plane. But two
triangle-planes passing through any one of them have been already spoken of, viz. {fg)
and {fh)\ the three remaining ones must therefore coincide with the nodal plane .r=0;

DE. SCIILAFLI OX SURFACES OF THE THIRD ORDER. 221

hence tliis plane counts as fifteen. The ray rj unites the five lines intersecting^ but not k,
the ray h unites the five lines intersecting A' but not,^', and each of the former five lines is
(as may be inferred from the consideration of a simple triangle) cut by four of the latter
five, which determine with it four different triangles. Therefore twenty triangle-planes
coincide in the nodal plane y=0. In all l-20+l-15+2-5 = 45 triangle-planes.
The same consequences may be derived from a trihedral-pair form. Let
U=^.r+r-//(«'-y), V=-_y-/,(,r+r), W=a--r,

X=.)— r+/'(;r+y), y= y-h{.v-z), Z =,,■+-,
then

^j\\\+^XZ=lh[xijw-\-xz"- + ifz-x'+hy{z'--x"-)}, V4-/AV + Y4-//Z = 0,
U+//V-(l+7r)W+X-//Y-(l-/r)Z=0.
If the number/; vanish, the equation UV\V+XYZ=0 will, at the limit, exhibit the
present surface, and the former of the linear relations, by reason of the latter, counts
for three relations answering to the cubic condition. Omitting the three accents as in
{i(x) and the permutations as in {Imn), we then get the following survey of the manner
of coincidence of the 27 lines and 45 triangle-planes of the general surface.

The axis (.r=0, ^=0) unites the ten lines {vy, l,_p, r). The nodal rays unite each of
them five lines; viz. the ray (a.'=0, ^=0) unites (ux, ivz, q), the ray (i/=0, a--\-z=0)
unites {ill/, re, n), the ray (y=0, a-—z — 0) unites {vx, wy, m); and there remain but
two simple lines (.r+"=0, w— y=0) or uz, and (o-'— 2=0, w-}-y=0) oi' t-O-v.

The nodal plane y=0 unites the twenty triangle-planes r, y, (ny), (*',r), (i'~), (wy),
(bmi); tlie nodal plane x=0 unites the fifteen planes (nx), (vy), («'~), (2'^Jf)', the two
remaining planes unite each of them five triangle-planes, viz. x-\-z=0 unites u, z, (tiz);
and X — 2=0 unites iv, x, (wx).

Art. 25. In the equation xyiv-\-xz'^-{-y'Z — aar'=0 the vanishing of the constant a

would give rise to a second node ^,' Therefore wc have hero only the two cases when

a is positive and when it is negative. Since no two of the linear functions .r, y, z, w
play a like part in the equation, wc are obliged to suppose tliem all real. So there are
only two species.

1. a is positive; all is real. VII. 1.

2. a is negative; the two simple lines are conjugate, and so also the two rays in the
nodal plane y=0. VII. 2.

VIII. Cubic surface of the sixth class tvith three proper nodes.
Art. 2G. If we place the points of reference =^i =^, ^ at the three nodes, tlie equa-
tion of the surface will contain the terms a^, x^y, x% xr'to, xzw, X2vy, xyz, yzw ; but tlie
last term is capable of taking up the three next preceding terms; in other words, the
three singular tangent planes which touch the surface along the axes (or lines joining two
nodes) may be chosen for the planes y=0, z:=0, ?(;=:0 ; then we arc at liberty to write

^+{y+z-\- t<').r^ + ayzw = 0

222 DE. SCHLiFLI ON SURFACES OF THE THIED OEDEE.

as an equation of the surface. The term x', if disappearing, would not alter the class,
but would merely form a particular case of the sort of surface here to be considered,
which case might readily be restored from the more general form by changing a, x
respectively into /i\ ha\ dividing by /i' and letting h vanish. But the term a"// cannot

disappear without bringing the class down to Jive; for the point ^ would then become

a biplanar node. Nor is the constant a allowed to be zero or —4; for in the former case
the cubic would be divisible by a", in the latter it would be half the expression

xX{.v+2z){a-+2w)-{,-+2>/){.v+2z){a-+2w),
which shows a fourth proper node at the point

If we denote the surface by/=0, and let
g=j;», l{=qn, |{=m. ^=su, x^=t, ^ = Xaf(t-2my+(t-rju)it-ru){f-su).
where f, v. are to be regarded as the independent variables, then we have not only

but also

whence

1^=0. 1^=0,
dt 0"

whenever /=0. That is to say, the equation reciprocal to/=0 is the discriminant 0 of
the binary cubic )^, when equated to zero. Putting, for shortness,

a=2+/-+s, (i=7'S-]rsq-\-qi\ 'y=qrs,
we find

2'lQ=-^6^y{p-'j){p-r)i2>-s).

+-EVaM(12/3-«V-4(2«/3 + 97);r'+2(15ay+4/3=)/-36j3yi;+27/}
+ i«{(6/3'^-«'[3-9«7)/+2(fi«V-«/3=-9/3y)i^+2/3^+27y^-9a/3y}
_ ^r-sris-qnq-rT=Q,
as the equation of the surface in plane-coordinates.
The TTossian of the primitive cubic is

'id'{a7jztv{3x-]-'//+z-\-iv)-\-x%f+;:'^+w''—2zw—2wij—29/z)}.
Ilencc the spinode curve is a complete curve of the sixth degree represented by the

system

a^-{-x-(7/-\-z-{-w)-\-ra/zw=f^, 1

^x^+x{?/+z+2v)+zw+Wf/+>/z = 0,l

wliich shows that the nodes arc^ doul)]e points of the curve, and that at these points

the (disengaged) nodal rays of the surface are tangents to the curve.

Art. 27. By what has been said in art. 15 we can at once judge of the disposition of

DE. SCHLiFLI ON SUEFACES OF THE TIllKD ORDER.

223

the lines and tiianglc-plancs. The three transversals arc the only simple lines, and form
a triangle (.v-\-i/-\-z-\-w=0, i/zw = 0), the plane whereof is the only simple triangle-plane.
The planes determined by a transversal and the opposite node intersect the surface in
thrice two (disengaged) nodal rays, each of which unites two lines. Each of the three axes
unites fom- lines. Together 3'l+6-2 + 3-4:=27 lines. The singular tangent planesy=0,
z=0, w:=0 count each of them twice, and so also does each of the three planes passing
through a trans^■crsal and the opposite node. Through each axis and two nodal rays there
pass two planes, together six planes, each of which coimts four times. Lastly, the plane
.r=0, containing the three nodes, counts eight times. Together l-l + (r2 + G'4; + l'8 = 45
triangle-planes.

If we assume the trihedral-pair form UVW+XYZ=0, where on putting «=

we have

U=— (a— lX.r+y+~+w), V=— a.r— (a— l)y, W=.r— («— l)y,
X= («-l)y, Y={c.-l)(x+>/+w), Z=(a-l)(x+//+c),

then the constants in the auxiliary relations
aV-\-...=0, rtTT + ... = 0, a"U + ...=:0 are«=0, 1=1. c=a, (7=«+l. ^=0, /=0

(therefore when e^fsxe imagined to be indefinitely small, a is of the order ef, whence,
for instance, aV-\-fY=.^ reduces itself to Y=0),

a! = V=d=cl!=^^f, a!'=b"=c" = il!'=e'=f,
and at length we get the following survey: viz., the lines are

(a,'=0, y=0) \yx, ivx; I, p],

{x=0, 2=0) lm\ m", /, r"],

(x^O, ^=0) [n', 7i", q', q"],

{x: z:w = a-l:-x:l) [l', l"l

(x: z:w=u— 1:1: —u) [_p',p"'\;

(x: w : y=u — l : —a. : 1) [_wy, r],

(x:w: y=K~l : 1 : —a) [iv/, ?jt] ;

{x: y: z = a — l:—a:l) \yz, q],

{x: y: z = k — 1:1:—u) [^wz, ?<] ;

(x-\-z-\-w=0, y=0) [war],

(x+y+w = 0, z=0) \uy'],

{x+y+ z = 0,w = 0) [^];

and the planes are

(x=0) [(vx)', (vx)", (wx)', (wx)", (Imn), (Inm), (pgr), (prq)'],
( ax + (cc—l )y =0) [y, (vy), {vz), {wx)'],
(— x-{-{a-l)y=Q) \io, {vx\ (wy), (wzj],
( a.r + («-l)j=0) [{wz)',(tvz)",(nlm),(nml)'],

224 DE. SCHL.lTLI OX SUEFACES OF THE TIIIED OEDEE.

( «>r + («-l)"' = 0) [i>n/)\ (in,)", ()j>q), (r</p)],
(- .r + («-l)«' = 0) [{>■>/)'. (r>/y\{mlnl{mnl)l

(,,+,+«, = ()) [(^^0'. (»■'•)"]'

Art. 28. One node at least must be real, for instance ^^, and tln^n the two others may

be real or conjugate. Accordhigly ,r is always real, and wliile we keep i/ real, r and w
may be either real or conjugate. On the other hand the constant a may be between
— 4 and 0, or beyond these limits. From these two reasons of partition there arise four
species of the surface witli three proper nodes. But we prefer to distinguish five species.
For if r, ir be conjugate, tlie nodal cone .r"+fc«'=0 becomes imaginary or real, accord-
ing as a > 0 or « < — 4.

1. :, w are real; a(a-\- ■i)>0, and therefore a real. All is real. VIII. 1.

2. ZAV real; — 4<«<0. Lct«= — 4 sin'- 7,, then « = ;'". The real lines are the three

axes and the three transversals. The real planes are the plane of the three nodes, the
three singular tangent planes, the plane passing through a transversal and the opposite
node, and the transversal plane. VIII. 2.

3. z, %o conjugate, «>0, and therefore a positive. The two nodes s, and -.— ^ are con-
jugate, the nodal cone at the real node ^ is imaginary. Tlie real lines are tlie axis

joining the conjugate nodes, and its transversal. The real planes are the plane of the
three nodes, the singular tangent plane through the real axis, two other planes which
pass through the real axis, the plane passing tluough tlie real transversal and tlie real
node, and the transversal plane. VIII. 3.

4. :. ?(' conjugate, a< — 4, and tlierefore a negative. The nodal cone at tlio real
node is real, but its two (disengaged) rays are imaginary and conjugate. The rest as
before. VIII. 4.

5. .:, w conjugate, — 4<«<0. The nodal cone at the real node is real. The real
lines are the axis joining tlie conjugate nodes, its transversal, and the two (disengaged)
rays of the real node. The real planes are that of the three nodes, the singular tangent
plane through the real axis, the plane passing through the real transversal and the
real node, and the transversal plane. ATI I. 5.

IX. Cahic surface af the slxih class wifli iwo h'l planar nodes.
Art. 29. From art. 20 it appears that the reduced equation of this sort of surface is
a-:w + (/y + ax)(y + ftx){>j + yx) = 0,
where ^ , " are tlie bi planar nodes. These have in common the nodal plane x=0.

1)K. SCIILAFLI OX SURFACES OF TllK TIirRl) ORDER. 225

which osculates the surface ahmg the axis (.r=:0, .y=0). Tlie other nodal pUxnes are
2^0, «'=0, eacli of whicli intersects the surface in three nodal rays. In order however
to find the reciprocal equation it is more convenient to write

1 2xz w + a.v^ + oba"^ + oc.ri/" + dif = 0
as the equation in point-coordinates. Then the discriminant of the binary cubic

rs{ax'' + Zki-y + ocxf + dy^) + 3.r( px ^-qtj)\

divided by rs and equated to zero, will furnish the equation in plane-coordinates as
follows ; viz. this is

[ci"d' — Gabcd - obx" + 4ac^ -\- Wdys'

4-G[(fffZ-— 3Jc(Z+2c')^^ + (4JV— 2«rrZ-2ir)^^(? + (2«c--«i(?-*-f)^=]?-V
-f- ?,\ZdY—\1cdp\ + (1 Old + ^r)pY-{^<t<-^ + ^hc)pq' + {Aac-Uyiys
— ^(f{dp^ — ^cp-q-\-ohpq- — ((q^)^=0.
The Hessian of the cubic \2xziv-{-ax^-irZhx"i/-\-ocxy'^-\-dy^ is

6 -l-SLr { Azio{cx + dy) - (ac - ^»=).l■= -(ad- lc)x-y - {hd - c"-)xy- } .

The system of the two expressions equated to zero breaks up into four times the axis
(,r=0, y=^) and four conies which lie in tlie planes

{\ac - U-)x' + Aculfy + Udxnf + icdxy^ + d-y' = 0,
and touch the nodal planes ^=0, w=0 at the corresponding nodes. Two of these four
planes are always imaginary and conjugate, the two remaining ones are real. For let

K= a=(Z' - Gahcd - Wc" + 4ae' + ih'd, A;'= JrZ^K,
and take for k the single real value ; again, let P^k + c'^-bd, which is positive, since
^•3 4- (^.= _ My = 1 ^ad' - Ucd + 2c J,

and determine the value of I by the condition that I{ad- — 2>bcd-\-2c'^) shall become
positive ; then the constants m", if determined by

l{m" + n') =2(ad' - Zbcd + 2(f) , n- - wr = 2/c + i{bd - c")
will be positive, because this system implies ?h«= Vo .k; and the equation of the four
planes breaks up into

{{dy + {c + l)xf-vvx-] {{dy-^{c-l)xf + n\f)=0.
Tlie section made by the real plane dy-\-{c-\-l-\-m)x^=^ is represented by

2id'zw-m({2l+m)'' + ')f)x'' = Q.

In the case therefore when both z and w are real, the two real planes contain also real
conies; but when z and lo are conjugate, one only of the two real planes intersects the
surface also in a real conic, the other real plane has, besides the axis, no real point in
common with the surface.

Art. 30. We now suppose xzw-\-{y-\-ux){y-\-(ix){y-{-yx) to be the equation of the
surface. As this form results from that of art. 21, by changing z, iv respectively into kz,

MDCCCLXIII. 2 I

226 DR. SCIILAPLI ON SUEFACES OF THE THIRD ORDER.

T-, and letting Jc vanish, we may readily tlience get a knowledge of the disposition of the

twenty-seven lines and forty-five triangle-planes, and we shall in partictdar see that the
axis here unites all the nine lines immediately afforded by a trihedral-pair. Changing
then the notation for the sake of greater symmetry, we can regard the surfoce as though
the six planes of uvw-\-.vyz^O coincided with the singular osculating plane (a'=0), while
the nine lines ux, &c. coincided with the axis. One of the two remaining nodal planes
will then unite all the six planes such as (Inm), and contain the three nodal rays (/, m, n),
(I', 7n', «'), (l", m", n") ; the other nodal plane will unite all the six planes such as (j^qr),
and contain the three nodal rays (j), q, r), QV, q', /), (^j", q", r"). One of the remaining
triangle-planes passing through the axis, for instance the plane which combines the
nodal rays {I, m, n) and (j>, q, r), would then unite the nine triangle-planes

{ux), {uijl {uz), {vx), {vy), {vz), {wx), {toy), {loz),
and the other two like planes would answer to the two remaining accents. In the whole
l-9 + 6-3 = 27 lines and 3-9 + 3-6 = 45 triangle-planes.

The singular osculating plane .r:=0, and one at least of the three other planes passing
through the axis y-\-dx—0, for instance, must be real. But z, lo can be either real or
conjugate, and so also the constants |3, y. From this double reason of partition we get
four species, IX. 1 ; IX. 2 ; IX. 3 ; IX. 4.

X. Cubic surface of the sixth class with a hiplaiiar node and a proper node.
Art. 31. The cubic surface with a biplanar node which lowers the class by four can
only in this way have a second node distinct from the first, when the two (disengaged)
rays of one nodal plane unite themselves together apart from the nodal edge. The
equation then takes the form

xyw + {x ^-yXz^ — aa.'') = 0 .
Changing z, w respectively into ^/a . z, aw, we might reduce this equation to

xyw + {x+y)(z--x') = 0.
But since a may be either positive or negative, in the latter case we should get z as
the product of the numerical factor i {=^\/ — 1) by a real function; and to avoid this
we shall retain the constant a.

If we let «=1 and denote the discriminant of the binary cubic

s''x{x +3^)' + sy{x +y)(2)X + qy) + ^r'xy*
by YJ *■"©, then
Q = ]j'-(p-qyy + [i2p-5qy-2{j)-2q)(p-qyy +

[ir' + hip' -pq + ^'Y -f{ V - ifY + \:4.'^-P + "^lY - \r)\p + qY^s + i^Ar'' -p') = 0
is the reciprocal equation of the surface.

The quartic function, the Hessian of .ryw + (x+y)(~'— ar), is

x{^ + y) {yw + 3^= - x^) + z'{x ~y)\

DE. SCILLAFLI ON SUEFACES OF THE THIRD OEDEE. ' 227

Hence the spinode cui've breaks up into four times the axis (;r=0, 3=0) joining both
nodes, twice the uodal edge, or also axis, (a"=0, y=0), and the complete curve

of the sLxth degree. It has passing through the biplanar node three branches, repre-
sented in the lowest approximation by

where only w is finite, and through the proper node two branches, the tangents whereof
are the two disengaged rays of this proper node, represented by

Art. 32. Let

U = -w+2/i(j-+y), V= z+x+%, W= z-x-ht/,
X= w+2//(.i-+?/), Y=-z-x-hy, Z=-z->rx-hy,
where h denotes a constant which ultimately vanishes ; then

U VW + XYZ = Ah { xi/to + {.V + i/){z"- - a- - hy) } = 0
represents the surface in question, and

v+w+Y+z=o, u-(i+/ov+(i+;ow+x-(i-/i)Y+(i-/oz=o

are identical relations, the former of which, in virtue of the latter, stands for the three
equations which correspond to the roots of the condition ABC=DEF. Hence we get
the following survey of the manner of coincidence of some of the twenty-seven lines and
forty-five triangle-planes (accents and permutations are omitted).

The axis joining both nodes (^^"=0, z=0) unites vz, wy, m, n, eight lines. The nodal
edge, also an axis, (.i'=0, y=0) unites ^, j), six lines. The two disengaged rays of the
biplanar node count each of them four times, viz. (^=0, x+z = 0) unites I'y, r, and
{y — 0, a.'— 2 = 0) unites wz, q. The two disengaged rays of the proper node count each
of them twice, viz. (w = 0, x + z = 0) unites ny, vx, and (w = 0, .r— 2 = 0) unites nz, tox.
Lastly, the transversal of the nodal edge (w = 0, x+y=0) is the only simple line ux.
Together l-8 + l-6 + 2-4 + 2-2 + l-l = 27 lines.

The planes of the biplanar node count twelve times, viz. a;=0 (a singular tangent
plane) unites (vz), (wy), (hnn), and j/=0 unites (vy), (wz), (i>2>'). The two planes com-
bining the double ray of the biplanar node with each of its two simple rays count
eight times, viz. z+x—O unites v, (vx), y, (uy); and a*— 2=0 unites w, {ivx), z, (uz).
The plane >r+y=0 touching the surface along the nodal edge unites [ux) three triangle-
planes, and the plane w = 0 combining the two simple rays of the proper node, unites u, x,
two triangle-planes of the general surface. In all 2-12 + 2-8 + l*3 + l'2 = 45 planes.

Because no two of the four linear functions x, y, z enter in a similar manner into the
form xyw-\-(x-\-y)(z^—ax^)^^^, all of them must be real. Only the constant a, accord-
ing as it is positive or negative, gives rise to a distinction between tvoo species. X. 1 ; X. 2.

2 i2

228 DE. SCHLAFLI OX SrEFACES OF THE THIED OEDEE.

XI. Calic surface of the sixih class icifh a bijilanar node.

Ai-t. 33. From nrt. 7 (sec art. 23) wc know that one of the nodal phxnes must osculate

the surface alonsr the nodal edge, in order that the node ^- mav knver tlie class by six,

and since in the first term .vijio of the equation of the surface all other terms divisible by
•ly/ may be included, we ■mite the equation immediately in the form

A-yw + xz" -Y 2aa": + b.i^ + elf = 0,
or, what is the same thing,

a' . (Iff . (Iw + .r{il: + adx)- + d-{h — a-)x^ + {df/y = 0,
or, to save constants,

xi/w + xz- + ax^ +f = 0,

which is the assumed form for the equation of the cubic. It is well to observe that here
all the letters are necessarily real, provided that the surface be real. Putting a = — |u<'
and changing ?/, z, w respectively into //,''//, f//r, f//«', we might get

x>/w + xz-—x^+f = 0,
where no explicit constant remains ; but then ~ would cease to be necessarily real.
If we denote the discriminant of the binary cubic

{oas-, —ps, —(qs + l^r), os%i\ yf
by iij-5^0, then

0 = _ C4s'j/- (4<2s + r'^Yj) ~ 12as\\qs + r)p — (({Aqs + rj + 432ft V= 0

is the reciprocal equation. It is obvious that

27«'e = {8/>^ + 9ff(42s + >-')i^-10SrtV}=-{4j/ + 3ft(4;^,s + r)}\

The Hessian of the cubic is

4:x{xyw -{- xz" —Zax'^ —oy'^).

The spinode curve then breaks up into six times the axis (.*'=0, ^ = 0), and the three
distinct conies

(«a-^+y = 0, yw^z'=Q).

Axt. 34. The trihedral-pair form can only be obtained by the help of two constants
which ultimately vanish. Let them be h and &;, the finite constant a be = —f, and
\] = {l-Yu + uh''^)y-\-h{l-\-2co)z + hx—uh-io,
V = ( 1 + M)y ■\-uhz-\- uh^x,
V,^=.y^hz-h^j',

X= —{l+aj-ojh'^)y — h{\+2c^)z^hcX + u>hhL\
Y= — {\ -\- u)y — uliz -\r ultPX,

'L = —y — hz~]i^x;
then

U V W + XYZ = 2a;/;\e { xy^V + {x + <^h'y){z' - p\f) + (1 + co)f \

DR. SCIILAFLI OX SUEFACES OF THE TIIIKD OllDER. 229

becomes tlie cubic of the surface as soon as h and u vanish. Of tlie two identical

relations

V + 4,W + Y + (yZ = 0,

U - s;A^g V + (1 + coVi^ofsY + X + coie^Y + (1 - *Vt=f )Z = 0,
the former, in virtue of the latter, stands for the three equations -which correspond to
the condition ABC = DEF. We ma)- therefore, in the following survey of lines and
triangle-planes, omit accents and permutations.

The nodal edge (.r=0, j/ = 0) unites ux, vy, wz, I, j), q, r, fifteen lines. The two nodal
mys count six times, viz.

(j/ = 0, r-|-fa'=0) unites uy, vz, wx, n,
(^=0, z — §x=0) unites i\v, wi/, uz, m;
in the same order as they here arc written, they form a double six. Together 1 •1-5 + 2 '6
=27 lines.

Each line of tlie one six, combined successively with the five not corresponding lines of
the other six, gives rise to five triangle-planes ; all the thirty planes so obtained coincide
with the nodal plane y=0, viz. n, v, w, a; y, z, {xiij), (iiz), (vx), (vz), (wx), (loi/), [hnn).
Again, as to the fifteen lines first mentioned, which, as we know, form fifteen triangles,
all their planes here coincide with the osculating nodal plane a*=0, viz. {tix), (vy), (wz),
(j)qr). Together 1"30 + 1-15 = 45 triangle-planes.

We can distinguish only two species, according as the constant a is negative or positive
(1, the two disengaged nodal rays are real ; 2, they are conjugate). XI. 1 ; XI. 2. The
case where a=0 is not considered, because it would imply a proper node at the point

^ with the cone y7u + z-=0.

XII. Ctibic surface of the sixth class tvith a nniplanar node.
Art. 35. The simplest form of the equation is

(x+y+zyio^'xyz = 0.
If all four letters are real, the three nodal rays (.;■ -f-y + c = 0, xyz = 0) are all of them real,
and imply the applanislicd proximity of the node into six angular spaces alternatively
full and empty, so that there appear three flat thorns having the node for their common
point*. (The sui-face here considered arises from III. 4, if there all the conjugate
values be allowed to coincide by pairs.)

Let sw=t, x+y+z=u, and -^7-9 be the discriminant of the binary cubic

(t—j)u)(f—qu)(f—ni) + lst'ic

in respect to f, u ; then 9 = 0 will be the equation reciprocal to u'w + xyz — O. Putting

a =i' + !? + ?•, /3 = (jrr + rp +2)q, y —l^ir,

* A notion of the form of the surface may be most readily acquired by taking the equation to be

r- + .r»/(r— )ii.r— j!>/)=0. — A. C.

230 DR. SCHLAFLI ON STJEFACES OF THE THIED OEDEE.

we have

The Hessian of the original cubic is

4(.r +y + -')-^(a-' +/ + :' - 2j/z - 2zx-1xy).
The spinode curve therefore breaks up into twice the nodal rays (or axes)

and a complete curve of the sixth degree, arising from the intersection of a quadi-atic
cone, which cone is inscribed in the trihedral {xijz = 0) of the singular tangent planes in
such manner that the lines of contact are harmonical with the nodal rays in respect to
the edges of the trihedral. The nodal plane does not really intersect this cone when all
three planes of the trihedral are real ; but it does so when one of them is real and the
two others are conjugate. The node is a quadruple point on the curve of the sixth
degree, and the two intersection-lines last mentioned are here a kind of cuspidal
tangents.

Ai"t. 30. lu order to get a trihedral-pair form, let «, b, c be finite numbers, h a
number which ultimately vanishes, and put {h — c){c — a){a~h)^^m, and moreover

U —{h-c){\-\- ah)x + mh-to,

V ={c—a)(l-\-'bli)y-\-mhho,

\N={a-h){l +ch)z+mJew,

X. —— nihho,

AY= -mh'w + {l + ah)(l +cli}x + (l+bh){l+ah)/j + (l+ch)(l + bli)z,
-hZ= mh'w + {l+aJi){'i-+bI))x + (l+bh}{l+ch)>/ + {l+cI>)(l + ah)z;
then the equation

U\^Y + XYZ=m(l+«70(l + ^'/0(l + ^/0{"t''+.!/ + ~i[-''+.y + - + /'(«>^' + ^^i/ + f-)]+-'-i/2}
is identically true, and the six functions U, V, W, X, Y, Z satisfy the identical relations

U + V-fW+X-f-Y + Z = 0, AU + BV + CW + DX + EY + FZ=:0,
where the numbers

A={c-a){a-b){l+ch), B = {a-b){b-c)(l + aJ>), C = (b-c){c-a){l + bh),
D=A-(i-c/(l + «//), E= -?«/(, F=0
satisfy the conditions

A + B + C = D + E + F, BC + CA + AB = EF + FD + DE,
witliout ABC — DEF vanishing. As long therefore as h is finite, the surface
UVW + XYZ = 0 has a biplanar node at the point ^, and this becomes uniplanar

when h vanishes. Omitting then accents and permutations, because the three roots of
th(! auxiliary cubic condition are equal, we get the following survey.

DB. SCHLAFLI ON SURFACES OF THE THIRD ORDER.

231

The three nodal rays count eight times ; for

(u = 0, x=0) unites u>/, uz , I , p,
(u=0, y = 0) unites vi/ , vz , n, r,
(ii=0, 2=0) unites wy, wz, m, q.
The sides of the triangle (to = 0, xi/z = 0) are simple, because they do not pass througli
the node ; they are ux, vx, wx of tlie old notation. Together 3'8 + 3'1 =27 lines.

The nodal plane ((,=0 unites y, z, (ui/), (nz), (vi/), (rz), (?('y), (ivz), (hmi), {prp'), thirty-
two triangle-planes. The three singular tangent planes count four times; for .r=0
unites u, {ux), and so on. The transversal plane w = 0 is the only simple triangle-plane
.T of the old notation. In the whole 1-32 + 3-4 + 1-1 = 45 triangle-planes.

All this might have been foreseen by the help of easy geometrical considerations.
As to reality, the function w must be real, and so must also one at least of the three
functions .r, y, z, for instance x. We then have only two species, according as y, z are
real or conjugate. XII. 1 ; and XII. 2.

XIII. Cubic surface of the fifth class ivith a hiplanar and two jproper nodes.
Art. 37. Such surface arises from art. 21, when there the binary cubic

(y + ux)(y+^x)(!/ + 7x)

has two equal roots. We are then at liberty to put /3 = y = 0, a = l, and permuting x

and y we get

7/zw+xXx+y + z)^0

as the equation of the surface, where ^ is the biplanar and ^, ~ arc the proper nodes.
And the survey given in the same article changes into the following : —

Lines unite.

(^.=0,3/=0)
(a,-=0, z =0)
(x=0, w=0)
(y =0, ^-1-2 = 0)
{z =0,x+y^O)
(x =—z=w)
(x^-y^w)
(w = 0,x+y+z=0)

m, n.

tiy, uz, vx, wx, 4

I,

vy, wz,
vz, wy,
ux,

3
3
2
2
1
27

Planes imite.

X

=0

{uz), (uy), (vx), (wx), 12

y

=0

(Imn), 6

z

= 0

(pqr), 6

x^y

= 0

(vz), (ivy), 6

or+s

= 0

(vy), (wz), 6

X —w

= 0

V, IV, y, z, 4

^+y+:

= 0

(ux), 3

w

= 0

ic, X, 2
45

The discriminant of the ternary cubic s,v'(x+y + z)—yz(px-\-qy + vz) divided by qVW
and then equated to zero is the reciprocal equation of the surface. But this may also

232 DE. SCIIL:^^!! ox SUEFACES OF TITE THIED OEDEE.

be derived from art. 20, and ^A■ill be found to be

= lQ(q-rys' + 8{2fiq + r) + 22iq--4q>-+r"-)-i>2 + r){2q-r)(q-2r)}s'-
+ {p' + S2J%q + r) - 22r( iq"- + 2Zqr + 4:r) + Z%2W{i + r) - 21 (fr } s
+f{l>-<2){p-r) = Q-
The Hessian of yzw-\-x-{d--\-ii + z) is ^{)jziv{ox+u-\-z)-\-x''{ij — z)-}. Hence the
spinodc curve breaks up into three times the axes joining the bipLinar to the two proper
nodes, twice the third axis, and a complete curve of the fourth degree formed by the
intersection of the cones

(3x + 4y + izf - (G.r + 5.y + 5.-)'^ + 9( // - zf = 0,

the latter of which passes through the vertex of the former, i. e. through the biplanar
node. Tliis is therefore a double point of the curve, and the tangents are
(3,r + 4j/ + 4r = 0, >jz = ()).
There are but two species; for x, w must be real, and only y, z can either be real or
conjugate.

1. All is real. XIH. 1.

2. !/ and z are conjugate. The two proper nodes are conjugate, and so arc also the
two planes of the biplanar node. The axis joining the two proper nodes, and the
transversal of this axis are the only real lines. XIII. 2.

XIV. Cubic surface of fJtcffth class xoifh a hijihvmr node and a 2))'02}er node.

Art. 38. As we have seen above (art. 23), the presence of a biplanar node such as
lowers the class hxfive reduces the equation of the surface to the form

x>/w + xz--\-fz — ax^ = 0.
Because the nodal plane .r=0 contains brit one disengaged ray (>r=0, ~ = 0), only the
union of the tico disengaged rays (^ = 0, -- — «.«■" = 0) in the other nodal plane can give
rise to a proper node. Hence the constant a must ^•anish. The surface in question is
therefore represented by

xi/w + xz- +fz = 0
in point-coordinates, and consequently by

0 = 27j^V + (oQ2)q)' + 1 Oq')s"- + {pr' + Sq-r)s + qr' = 0
in plane-coordinates; 48s'0 is the discriminant of tlic binary cubic

1 2sx"-{2>'r + q>j) + 3y(y.r - .s^)-' ;
and

1 O8^^0 = {h-\ps' + o^rs + r'f + {\2qs - r^.

The Hessian of tlic original cubic is 'i{x'')jio -{-x-z^ — 2)xfz -\-y*} . Tlie spinode curve

DE. SCIIL.iFLI ON SUEFACES OF THE TIIIED OEDEE. 233

therefore breaks up into five times the axis (^=0, 3 = 0) joinini;- the two nodes, four
times the nodal edge (.r=0, ?/=0), and a partial curve of the tliird degree, wliich nun-
be represented by

^, z, to ' =0,
4.r, y, —52 \\

or, vhich is the same thing, by jj = 2K.v, z=X\r, w=—^jX\v, where K denotes a variable
number. Since the plane touching the original surface at this current point has tlie
equation

- Sk\v + 3?.7/ + l2K: + iiv = 0,

the spinodc dcvelope is represented by the vanishing of the discriminant of the binary
cubic ( — 8.r, y, 4z, 4«»X^, 1)^ that is to say, by

V=6ix'w' + (4:8xi/z+f)w-128xz'-Sfz'=0 ;

and we have in feet

a/V = (GiA-po - C U"z"- - 1 Qafz +^%rpv +a'z"- +fz) + z(ixz -ff,

which shows that the curve is contained three times in the intersection of the original
surface and the developable V=0. The cuspidal line of this developable is repre-
sented by

y=8x.r, z=-2K\r, w=2x\v,

and is therefore a partial curve of the tliird degree. The equation in plane-coordi-
nates of the spinodc curve is

0^5jjV- + lG2>r' + S(J02)q}'s-o202's-lQ2-i"~0.

In point-coordinates the developable formed by the tangents of the spinodc curve is
represented by the vanishing of the discriminant of the binary cubic

(10.r, -5^, 10.^ 4«.Xa, 1)'.
Art. 39. On putting

U= z-1m+Ji{x-\-hy), Y=—y—hz — hl\\\ '\Y=—z + hr,
'K.= -z-\-]m-\-Jc{x-\-hj), Y= ij + liz — lt/uv, Z = z + tr,

whence arise the identical relations

V + /(W + Y + /;Z = 0 (holding three times),

U + Id'V - (1 + Irk)\\ + X - hkY - (1 - I>'l-)Z = 0 (accidental),

the identical equation

U^'W + XYZ = 2hk{xyiv + {x + Jnjy +y-'r - JAf - Ii/rxy} ,

when the constants Ii, k are made to vanish, enables us to perceive what arrangement is
here undergone by the 27 lines and 45 triangle-planes of the general surface.

MDCCCLXIII. 2 K

234 DE. SCHLiFLI ON SUHFACES OF THE THIRD OEDER.

The edge (.r=0, y =0) unites vi/ , I , p , r , 10

the axis (^ = 0, z =Q) unites 7(i/, vx , vz, loi/, m., n, 10
the ray (a'=0, :; =0) unites «.r, wz, q , 5

the ray {z — 0, w — 0) unites uz, icx, 2

27 lines.

The axis joining the two nodes thus unites five rays of the proper node.

Of the nodal planes, ?/=0 unites v, i/, {ay), (v.v), (vz), {unj), (Imn), 20 triangle-planes;
,r=0 unites (?a-), (vi/), (ivz), {pqr), 15 triangle-planes; and the only plane containing an
actual triangle, r = 0, unites u,iv, x, z, (kz), (wx), 10 triangle-planes; 20 + 15 + 10 = 45.

The only disengaged ray of the proper node unites two independent lines of the
surface. The five lines intersecting both of these coincide in the disengaged ray of the
biplanar node. The ten lines meeting but one of the two original lines coincide in the
axis joining both nodes. And the ten remaining lines coincide in the edge of the
biplanar node.

There is but one species, because all four linear functions x, i/, z, w must be real.
XIV. 1.

XV. Cubic surface of tlic Jifth class with a uniplanar node.

Art. 40. We have seen above that the cubic surface with a uniplanar node can
always be represented by an equation of the form .r''«' + P + Q.r, where P = (//, r)\

Q=(y, z)-, and that, whenever P has no two equal factors, the uniplanar node ^ for

itself lowers the class by six ; but upon considering the case where P has two equal
factors, it appears that there is a further reduction of one, making the whole reduction
of class to be equal seven. We are here allowed to write P^y-^; and tlie equation of
the surface accordingly is

xhv + //-Z + x{af + 2i//- + cz') = 0

or, what is the same thing,

x''[cw — ac{ac + h-)x — lahcy — c{1ac + h')z'] + (// + hx)-{cz + ncx) + x{cz + acx)" = 0

or simply

x-w + i/-z + xz'' = 0,

where all the variables are necessarily real. The equation in plane-coordinates arises
when the discriminant of ( — fg'^ qr, 2ps, Sqs'X-i\ '/f is cleared of the factor ^q''s;
hence it is

- G ^''s' - 1 6^>=;-^s + 'J22Jrrs + 2 7q's + 1 Gfr' = 0.

The Hessian of x^w-\-y'z-\-xz'^ is \C)x'^{xz—y'^). Hence the spinodc curve breaks up
into six times the double nodal ray {x—(i, y=0), twice the simple nodal i"ay (.r=0)
3 = 0), and once the complete curve (j,r— y^=0, aw + 22;'- = 0), which has the node for a

DE. SCHLAFLI O'S SURFACES OF THE THIED ORDER. 235

double poiut, where the double nodal lay is a tangent common to both branches of the
curve.

Art. 41. Denoting by h a number which ultimately vanishes, the surface in question
may also be represented by the equation

af'(z + h^w) - z(x + hij){x - hy - Ii'z) = 0.

Hence wc can see that there is but one simple line (z=0, w=0}, and that all the ten
lines intersecting it coincide^in the simple nodal ray"(.r=0, 2=0), while the double nodal
ray (jr=0, i/=0) unites all the sixteen remaining lines. Again, the plane 2 = 0 unites
all the five triangle-planes that pass through the only simple line (2 = 0, w=0), and the
nodal plane .r=0 unites alone all the forty remaining triangle-planes.

There is but one species. XV. 1.

XVI. Cubic surface of the fourth class ivith four proper nodes.
Ai"t. 42. If we choose the four nodes as points of reference, the equation of the surface
necessarily takes the form ayzw-\-hxzw-\-cxy'W-\-dxi/z^=^(i. None of the four constants
can vanish, unless the surface break up into a plane and a quadratic sui-face. We are
therefore at liberty to change x, y, z, w respectively into ax, by, cz, dw, when the equa-
tion of the surface becomes

yzw -\- xzw + xyio -\-xyz=Q.
Since

1111

p:q:r:s=-^:j^:j^:-^,

we have

/P+ ^2+ vA-+ ^s=0,
or in a rational form

(Sp" — 2Spj)' — 6 ipqrs = %p* — iXj^'q + QXj^Y + i%p-qr - 4 O^^''^ = ^^
as an equation in plane-coordinates.
The Hessian of Xyztv is

— iX(xz +yw)(xw -}- yz) = — 4:'Xa^yz = 4 { 4.r_y2W — %x . Xyzw } .

The spinode curve consequently breaks up into twice each axis (or edge of the tetra-
hedron of reference).

Art. 43. Trihedral-pair fonns are, for instance,

(x +y)zio + iz + io)xy = 0,

{x+y){x + z)(x+w)-a"(x+y+z+w) = 0.

The latter shows a transversal triangle-plane .r+^ + 2 + w=:0, which is simple as con-
taining none of the four nodes. Its sides are the transversals of the axes ; each of them
belongs to two opposite axes, as for instance (x+y=0, z+V) = 0), being the transversal
common to both singular tangent planes x+y=0 and 2+w=0. The six singular
tangent planes lie harmonically in regard to the point x^y=z=w. Let any plane
px + qy + rz + su'=^0 pass through tliis point, whence yj + (2 + '' + ^=0 ; then to this plane

2 k2

230 DE. SCHLAFLI OX SUEFACES OF THE TIIIED OEDEE.

will harmonically answer the Yioint2kr=q>/=rz=su\ and this will describe the surface

1111a

-+-+-+-=0,

while the plane turns about that fixed point.

The six axes count four times ; the three transversals are simple ; G-4 + 3-l = 27 lines.

The four planes each of which contains three nodes count eight times, the six singular
tangent planes coimt twice, the transversal plane is simple ; together 4-S4-G-2 + l-l = 45
triangle-planes.

There are three species ; the transversal plane is always real.

1. All is real. XVI. 1.

2. a; y are real, c, %o conjugate. Two nodes arc real, and two are conjugate. Two
axes and but one transversal arc real. XVI. 2.

o. X, y are conjugate, and so also c, w. All four nodes are imaginary and conjugate
by pairs. Two axes and the three transversals are real. XVI. 3.

XVII. Cuhic surface of the fourth class with two Injjlanar nodes and one propter node.

Art. 44. Such surface arises from the kind IX. when there /3 = y. With a change
of letters

a-yz-\-xiv- + 10^ = 0

(implying only fourteen constants) is a form to which the equation of such a surface can
always be reduced.
Let

then

]) : (2s-R) : (R-s) = (yc + w=) : 3w' : 2.rw,
whence

0XR-s) + 2(R- 25)^=0,

or in a rational form

{s- -{-■^<ir)- —ps^ — 3 (jj)qrs + 2 Ijfqr = 0

is the reciprocal equation of the surfoce.

The Hessian of the original cubic is Ax{.vyz-\-oy):xo-\-xiv''-); consequently the spinode
curve breaks up into four times the axis joining both biplanar nodes, three times the
two other axes, and once the conic (4a,' + 3w=0, 3_ys— W" = 0). Along this last conic
the cone {%x -{-^wf — ll yz^Q osculates the surface.

The axis (a'=0, ?o = 0) joining both biplanar nodes counts nine times, tlie two other
axes (w = 0, ?/- = 0) count six times, and the two disengaged rays of the biplanar nodes
(y = 0, x-\-xo — Q) and (c=0, x-\-iv = ^) count three times; I-9 + 2-G + 2-3=:27 lines.

The plane w=0 passing through the three nodes counts eighteen times, the plane
a.--|-w = 0 counts nine times, and the three nodal planes count six times; in the whole
1T8-1-1-9 + 3-6 = 45 triangle-planes.

Three species may be distinguished.

DE. SCIILaFLI ox SURFACES OF THE THIRD ORDER. 237

1. All is real. XVII. 1.

2. y, z are conjugate. The two biplanar nodes are conjugate, and the cone of the
proper node is imaginary (it has but one real point). XVII. 2.

3. y and — z are conjugate. The two bijjlanar nodes are conjugate, and the cone of
the proper node is real. XVII. o.

XVIII. Cubic surface of the fourth class with a hiplanar node and two proj^er nodes.
Art. 45. The equation of the surface in point-coordinates is
.r-yw + (.v+y>-=0,
and in plane-coordinates it is

(/' - V^' + \{P+ !?)'•>■ + TO '■ - 0.
this last equation arising from the discriminant of the binary quadric

The Hessian of the original cubic is

The spinode curve therefore breaks up into four times each of the lines joining the
biplanar node to the two proper nodes, twice the line joining both the proper nodes,
and tmce the nodal edge.

Let h, k be numbers which idtimately vanish, and write

U = -w + 27M(.r+y), V= s+/u- + /?v/, W= z-Lv-ky,

X= w + 2hJc{x+y), Y=-z-hj;-^J>y, Z = —z + kv-ky;

then

VYW+X.YZ = 4h/c{a'yw + {x+y)(z'-h\r-kY)},

V+W+Y + Z=0 (holding three times),

U- (/« + k)V + {h + ^•)W + X- (/i - >{-)Y + {h -k)Z = 0 (accidental).

Then the lines are as follows, aIz.

The axis (.r=0, z=0) unites vz, wy, m, n, 8

the axis (y=0, 2=0) unites vy, loz, q, r, 8

the edge (.r=0, y=0) unites Z, p, 6

the axis (^=0, io = 0) unites uy, iiz, i\v, w.r, 4

the line (.(■ + ?/ = 0, tt' = 0) is v.v, 1

27

the last-mentioned line iix being the transversal common to the nodal edge and the axis
joining the two proper nodes.

The plane of the three nodes r=0 unites v, w, y, z, (uy), (i(z), (w), (wx), sixteen
triangle-planes; the nodal planes count each of them twelve times, since .r=0 unites
(vz), (tvy), (linii), and y=0 unites (vy), (wz), (pqr). Of the singular tangent planes, that

238 DK. SCHLAFLI OX SUEFACES Of THE THIED OEDEE.

aloiig the nodal edge, d' + i/ — 0, unites (kx), three triangle-planes, and that through the
two proper nodes. w = 0, unites if, x. two triangle-planes. In all 16 + 12 + 12 + 3+2 = 45
triangle planes.

There are two species, according as .r, y are real or conjugate. As an example of the
latter species, I may notice the surface generated by a variable circle the diameter
whereof is parallel to the axis of a fixed parabola and intercepted between this curve
and its tangent at the vertex, while the plane of the circle is perpendicular to that of
the parabola.

XIX. Ci'/i/c surface of the fourth class with n hiplanar and a froj)er node.

Art. -16. Such a surface is represented xtjw -\- xz" ^ if = Q in point-coordinates, and by
(^\ps^j^(^Aqs-\-ff — () in plane-coordinates. The Hessian of the original cubic is
4a-(a-yw+.('r — 3/), whence the spinode curve breaks up into six times the edge
(a-=0, «/=0) of the biplanar node and six times the axis (y = 0, z = i)) joining the two
nodes.

From art. 34 it appears that the axis (j/ = 0, c = 0) joining the two nodes unites the
twelve lines of a double six, and that the edge (.(•=0, j/ — 0) unites the fifteen remaining
lines. 12 + 15 = 27 lines. INIoreover it is plain that the axis ruiites all six rays of the
proper node. The nodal plane j/ = 0 containing the proper node unites the thirty
triangle-planes immediately arising from the double six, and the osculating nodal plane
.r=0 imites all the fifteen remaining triangle-planes, 30+15=45 triangle-planes.

The plane :; = 0 is not fixed, for we may also Avrite

xyixv - 2XZ - l?y) + x{z + -k,jf+f=0.
The equation of the surface therefore implies but thirteen disposable constants.

There is but one species, because everything must be real. XIX. 1.

XX. Cubic surface of the fourth class with a nni2)Janar node.

Art. 47. When in the form .('w + P + Q.!— 0 of art. 40, P is a perfect cube, which we

may denote by /, this equation can be reduced to x^io-\-y^ -\-xz- = Q. The equation

reciprocal to this is 27(4^;,s' + r-)- — G42^s=0. Smce we may also write the equation in

the form

.r'-'(w + 2}.z-}?x) -^f+x{z--kxy = 0,

there is nothing tc) fix the positions of the planes ;: = 0 and w = 0 ; and the equation of
the surface implies only thii-teen disposable constants.

The Hessian is 48,r'_y, and the spinode curve breaks up into ten times the line (.r=0,
y = 0) and once the conic section (</=0, xiv+z''—0), along which the cone ;no-{-z' = 0
osculates the surface.

By the help of a constant //, which ultimately vanishes, we may represent the surface
here considered in the form of art. 40,

.(.»( _ 2,r- h'y- Wz + hhv) + (.r- V'y )"-(,r + % + h'z) + x{x + h'y + h'zf = 0,

DE, SCHLAFLI ON SUEFACES OF TILE THIRD ORDER. 239

whence wc see tluxt here all the twenty-seven lines of the general surface coincide in the
line (.r=0, y=0), and all the forty-five triangle-planes in the plane ,r=0.
There is but one species. XX. 1.

XXI. Cubic surface of the third clans with three hiplanar nodes.

Art. 48. The equation is xy.s+w'=0 in point-coordinates, and 27jJ<7r— .s^ = 0 in plane-
coordinates. The Hessian is 12xyzw ; hence the spinode curve breaks up into four times
the three axes.

The three axes, as counting each for nine lines, unite ;ill tlie twenty-seven lines of
the surface, and this distribution of them into three groups of nine lines answers to a
triad of triliech'al-parrs*. The plane w = 0 of the three nodes counts for twenty-seven
triangle-planes, and each of the singular osculating planes .i'=0, y=0, x: = 0 counts for
six triangle-planes, 27 + 6-|-6 + 6:=45 triangle-planes.

There are two species (if x, w be supposed to be real), according as y, z are real or
conjugate. XXI. 1 ; XXI. 2.

XXII. Ruled surface of the third order and third class.

Art. 49. Let us imagine a continuous system of straight lines forming a surface of
the nth. order, and take at pleasure any one of these lines as an axis about which avc
turn an intersecting plane. The section will then consist of the axis itself and a plane
curve of the (n — l)th order, which, of course, intersects the axis in 7i—l points. But
of these one alone can move, while the n — 2 remaining intersections must be fixed. For
the piano cuts an indefinitely near (or consecutive) straight line of the system in only
one pomt, and this alone moves. Should any one of the other intersections also move,
the axis would be a double line of the surface, whereas it was taken at hazard. Because
then the n — 2 remaining intersections on the axis ai-e fixed, they must arise from a
double line of the surface, such double line being met by every generating line in n — 2
points. Again, to investigate the class of this surface we take an arbitrary line in space ;
it will intersect the surface in n points, and therefore meet the same number of generating
Unas. Each plane passing through the arbitrary line and one of these n generating
lines will be a tangent plane to the surface. And since there are no other tangent
planes than such as pass through a generating line, therefore the class of the surface is
equal to its order.

Art. 50. For «=3 the double line cannot be a curve; for else an arbitrary plane
section of the surface would have two double points at least, and would therefore consist
of a straight line and a conic section ; but this cannot be the case, unless the surface
break up into a plane and a quadi-atic surface. The double line must therefore be a
straight line. Again, since througli any point of it tliere pass (in general) two distinct
generating lines, the plane of these two lines must besides cut the surface in a third
line (not belonging to the system of generating lines), and this mil meet all the gene-
* See Quart. Math. Joum.'il, vo]. ii. p. 114.

240 DE. SCHLAFLI OX SUEFACES OF TILE TIIIED OEDEE.

rating lines. For if we turn a plane about it, the section Avill always break up into

this line itself and a quadratic curve having a double point on the double line of the

surface ; in other words, the section will be a triangle whereof the vertex moves along

the double line, while the two sides are current generating lines, and the base rests on

a straight line fixed in position*, which we shall term the transversal. It plays the

part of the node-couple-dcvelope, since every plane passing through it touches the

surface in two points away from the double line, whereas every plane passing tlirough a

generating line touches the surface in only one point away from the double line.

Suppose now that there pass through the double line the planes .r=0, >/—0, and

through the transversal the planes ^; = 0, w = 0. Then the equation of the surftxce will

assume the form Mz + Nw=0, and for indefinitely small -salucs of ,r, // this cubic must

become of the second order. Therefore M, X cannot contain 0, w, but must be of the

form (,r, >/)-, whence the equation may also be presented in the form A.!"-f-2B.iy/ + CV/"=0,

where A, B, C mean homogeneous linear functions 6( z, w. If then we inquire for

what value of the ratio s : w this equation gives two equal values to the ratio .r : 1/, the

corresponding condition AC— B" is of the second degree in respect to the ratio required.

Hence there lie on the double line only two uniplanar nodes f. We are allowed to let

pass through them respectively the planes r=0, w = 0. But then M, N are perfect

squares, and we are also at liberty to represent them by —i/', .j", so that now the equation

of the surface becomes

a"iv — >fz = 0.

Since it obviously implies only thirteen constants, the existence of a double line counts
in the cubic surfiice for six conditions. The system i/=Xa; w^^K'z, where X is an arbitrary
parameter, shows the generating line in mo^•ement, and affords an easy geometrical con-
struction of the surface, which I think it is not necessary to explain.

The equation reciprocal to a"w — //"c = 0 is ^r5-|-(^-y=0 ; hence the surface keeps its
properties, though point and plane be interchanged.

The Hessian is — \Q)d"ij'. The spinode cur^e therefore breaks up into eight times
the double line and twice the generating lines which pass through the uniplanar nodes
and along which the surface is touched by the two singular tangent planes ;:^0 and

There are two species, according as the two uniplanar nodes are real or conjugate. In
the first species .?■, y, z, w are real, and whenever the ratio ;:: ; w is negative, the ratio .r : 1/
becomes lateral. In other words, Avhen the double line between the two uniplanar

* 'I'lio same tiling might also lie Ibus pi'ovrd. Take any I'uur ilistiiiel g(>iuTatiiig linos; tliry will in general
not lie on a quadratic surface, and, because they are already intersected by the straight doid)le lino of the surface,
there will bo a second straight line intersecting all of tlieni. ]iut since this now has four points in common witli
the cubic sm-face, it must lie wholly in the siu-facc.

The problem of drawing through a given generating line a triangle-])Line is of the fifth degree, and it may bo
foreseen that the plane passing through it and the transversal is a single solution ; tliefour remaining solutions
must all coincide in the plane passing througli the given line and the double line.

t In the language of Dr. Salmon and Tiiyself, cuspidal i)oints. — A. C.

DB. SCHLiFLI ON SUEFACES OF TlIE TUIRD OEDEE. 241

nodes is contiguous with the rest of the surface, then it is isolated without them ; and
when isolated within, then it is contiguous without. The two planes througli the trans-
versal and one of the uniplanar nodes are singular tangent planes, and both real. XXII. 1.
In the second species we may assume w conjugate to — z and y to .r. and write

(.1- + '»'(-- + '■«') + G^-- '»'(--'■«') = 0,

or, what is the same thing,

(.1- —If): — 2xi/iv = 0,
whence arises the system

{to=Xz, :r~2}a->/—f—0),

which for all real values of X gives also real values to the ratio .r : >/. The double line is
therefore throughout contiguous to the rest of the surface, and the two singular tangent
planes are conjugate. XXII. 2.

p3r. ScHLAFLi has' omitted to notice a special form of the ruled surface of the third
order which presented itself to me, and which I communicated to M. Ceemon.v and
Dr. Sauiox, and which is in fact that in which the transversal coincides with the
double line. For this species, say XXII. 3, the equation may be taken to be

see Salmon's ' Geometry of Three Dimensions,' pp. 378, 379, where however in the
construction of the surface a necessary condition was (by an oversight of mine) omitted.
The correct construction is as follows, viz., Given a cubic ciu've hanng a double point,
and a line meeting the curve in this point (the double line of the surface) ; if on
the line we have a series of points, and through the line a series of planes, corresponding
ardiarmonically to each other, and such that to the double jioint considered as a point of
the line, there corresponds the plane through one of the tangents at the double point, then
the line drawn through a point (of the double line), and in the corresponding plane, to
meet the cubic, generates the surface. The special form in question must, however, have
been familiar to M. Chasles, as I find it alluded to in the foot-note, p. 188, to a paper
by him, "Description des Courbcs, &c.,"' Coraptes Rendus, 18 November 1861. — A. C]

MDCCCLXm. 2 L

[ 243 ]

XI. On the Tides of the Arctic Seas.

By the Rev. Samuel Haugetox, M.A., F.R.S., Fellow of Trinity. College, Dublin.

Pait I. On the Diurnal Tides of Port Leopold, North Somerset.

Received November 7, 1861, — Head January 9, 1862.

I .oi indebted to the courtesy of Captain Washington, R.N., Hydrographer to the Navy,
for the opportunity I have had of investigating the tides of Port Leopold. Having
heai-d that I was engaged in the discussion of the Arctic Tides, he kindly placed at my
disposal the observations made on board Her Majesty's Ship ' Investigator,' during the
expedition of 1848-49, under the orders of Sii- J.arES C. Koss, R.N., in search of Sir
John Fkanklin.

The ' Investigator' was anchored, or rather fast in the ice, during the winter of 1848,
in Port Leopold, North Somerset, lat. 74° N., long. 90° W., in three fathoms water; and
the observations on the tides were made by Lieut. Feedeeick Robinson, whose care and
skill m observing are highly to be commended.

By cai-efully laymg down the daUy high and low waters, 1 have succeeded in com-
pletely separating the Diui-nal from the Semidiurnal Tide, and iu resolving each tide
into the portions due respectively to the action of the Sun and of the Moon.

In the following discussion of the Diiu-nal Tide, I shall first give the results of the
actual observations, when graphically laid down, and afterwards draw the inferences
which appear to foUow from them, when compared with theory. The mode of
reduction used by me wU be evident from an inspection of the MS. diagrams which
accompany this paper.

The following Table contains the Range of Diurnal Tide at High and at Low Water,
and the Times of Vanishing of the Diui-nal Tide at High and at Low Water.

Table I. — Range and Time of Vanishing of Diurnal Tide at Port Leopold,
Prince Regent's Inlet, 1848-49.

Eange of Tide.

Time of Vanishing.

High Water.

Low Water.

High Water.

Low Water.

ft.

2-16
2-41

ft.
1-55

1-20

November 1848.
2id gh 15m

November 1848.
17 20 0

2l2

244

EEY. S. HAUGHTON OX THE TIDES OF THE
Table I. (continued).

Kange

of Tide.

Time of Vanishing.

High Water.

Low Water.

High Water.

Low Water.

ft.

ft.

2-17

1-22

December 1848.

December 1848.

6" IS^ 10"

gd oih 45"

2-40

157

19 18 50

15 18 30

2-31

1-19

Janu:ny 1849.

December 1848.

4 2 15

31 16 0

2-40

1-57

January 1849.

17 9 30

12 23 30

2-05

1-33

February 1849.

1 16 55

28 13 0

2-23

1-38

February 1849.

14 17 0

11 8 40

1-84

0-95

27 6 45

27 3 30

1-46*

0-87*

March 1849.

March 1849.

13 8 30

11 17 15

1-53*

1-07*

2:, 13 15

24 10 20

1-31*

0-84*

April 1849.

April 1849.

6 2 20

5 17 15

1-55*

0-63*

21 11 15

18 3 30

1-96

0-86

May 1849.

3 18 15

30 12 50

2-26

0-87

May 1849.

i^Iav 1849.

19 3 15

14 'lO 19

2-34

1"22

31 14 15

27 23 20

2-03

0-97

June 1849.

June 1849.

16 6 0

11 22 20

2-44

1-35

29 5 20

24 17 30

2-14

1-23

Julv 1849.

Julv 1849.

14 'l2 45

10 '18 50

2-27

1-28

27 19 30

23 23 20

To render more evident the law of range of Diurnal Tide, I here give in Plate X.
figs. I. and II., a graphical representation of the first two columns of the preceding
Table, by means of which the relation of the range of Diurnal Tide, at High and Low
Water, to the Solstices and Equinoxes is made apjjarent.

There is no difficulty in understanding, as will be presently shown, wliy the Diurnal
Tide should reach a maximum at the Solstic(>s, and a minimum at the E(]uinoxes, as is
shown by the curves for High and Low AVater, because the Solar Diurnal Tide vanislies at
the Equinoxt^s, and consequently the equinoctial Diurnal Tide is duo solely to the Moon,
wliile the Solstitial Diurnal Tide is due to the united action of both Sun and Moon.

The following Tables II. and III. show the interval between the vanishing of the
Diurnal Tide and the time of tlie vanishing of the Moon's Declination.

* Transactions of the Royal Irish Academy, vol. xxiii. pp. i;i;5, 134, 137.

AECTIC SEAS. — PAKTS 1. AND II.— POJ{T LEOPOLD.

245

Table II. — Relation of the Times of Vanishing of the Uiuinal Tide at High Water,
to the Vanishing of the Moon's Declination.

Sato.

Vanishing of
Tide.

Interval from Moon's
Declination Vanishing.

1848. November

„ December

1849. January

d h ra
21 9 15

6 18 10
19 18 50

4 2 15
17 9 30

1 16 55

d h
+ 1 6
+ 2 3
+ 2 6
+ 2 20
+ 3 8
+ 4 8
+ 4 12
+ 2 18
+ 3 18t
+ 1 I6t
+ 0 6t
+ 1 4
+ 0 12
+ 1 10
+ 1 6
+ 1 22
+ 2 12
+ 2 22
+ 3 14

„ March

14 17 0
27 6 45
13 8 30
25 13 15

6 2 20
21 11 15

3 18 15

„ May

19 3 15
31 14 15
16 6 0
29 5 20
14 12 45
27 19 30

," Jiiiy '.'.'.'.'..

Mean ...

+ 2^ 9"

In this Table, the positive sign denotes that the Vanishing of the Diurnal Tide
followed the Vanishing of the Moon's Declination.

Table III.— Relation of the Times of Vanishing of the Diurnal Tide at Low Water,
to the Vanishing of the Moon's Declination.

Date.

Vanishing of
Tide.

Interval from Moon's
Declination Vanisliing.

1848. November

„ December

1849. January

d h m

4 12 30

17 20 0
2 21 45

15 18 30
31 16 0
12 23 30
28 13 0
11 8 40
27 3 30
11 17 15
24 10 20

5 17 15

18 3 30
30 12 50
14 10 19
27 23 20
11 22 20
24 17 30
10 18 50
23 23 20

d h
-3 3
-1 21
-1 14
-1 12

— 0 21
-0 18
+ 0 9
+ 0 16
+ 2 15
+ 2 4t
+ 0 14t
-0 4t
-2 3
-2 22

— 2 15
-2 11

— 2 4
-2 2
-0 18
-0 4

„ March

" May .".."'.

„ June

„ July

Mean ...

— O" 22" 30"

246

EEV. S. HAUGHTON OJf THE TIDES OF THE

In this Table, the positive sign denotes that the Vanishing of the Diurnal Tide follows
the Vanishing of the Moon's Declination, and the negative sign denotes that it precedes
it. From the mean result of the two Tables, it would seem that the Vanishing of the
Diurnal Tide at Low Water precedes the Vanishing of the Diurnal Tide at High Water,
by a mean amount of 3'^ 7'' 30"".

The intervals at which the vanishing of the Diurnal Tide at High Water follows the
Vanishing at Low Water are showoi in detail in the following Table and in fig. IIL,
Plate XL

Table IV. — Intervals from the Vanishing of the Diurnal Tide at Low Water to
Vanishing of Diurnal Tide at Hia;h Water.

Date.

Intervals.

1848. November

„ December

1849. January

(1 h m
3 13 15

3 20 25

4 0 20

3 10 15

4 10 0
4 3 55
3 8 20

0 3 15

1 15 15
1 2 55
0 9 5
3 7 45

3 5 25

4 16 56

3 14 55

4 7 40
4 11 50
3 17 55
3 20 10

," March

" July

Mean ...

3^ s" ^r

From the foregoing Table it is evident that the interval between the vanishing of the
Diui-nalTide at the time of High and of Low Water, increases from the Equinoxes to the
Solstices — an effect which is in a great degree due to the Solar Tide, wliich disappears
at the Equinoxes and reaches a maximum at the Solstices. The regularity with which
this increase of interval takes place is still better sho^vn by the figm-e, wliich represents
the Table, the abscissa) denoting time, and the ordinates the interval from the vanishing
of the Diurnal Tide at Low Water to its vani.shing at High Water. The minimum
interval, 12 hours, occiu's at the time of the Equinoxes, and the maximum interval,
4 days to 4} days, occurs at the time of the Solstices.

I am not aw^are that this feature of the Diurnal Tide has been before noticed ; it is
perfectly in accordance with what might be expected from Tidal Theory.

According to the best theories of the Tides, the Diurnal Tide may be represented by
the expression

D=Ssin2ff cos (s—?',)+M sin 2/* cos (hi— /„) (1.)

ARCTIC SEAS. — PAETS I. A^s^D II. — POBT LEOPOLD. 247

In this equation

D is the height of the Diurnal Tide, in feet.

S trnd M are the coefficients, in feet, of the Solar and Lunar Diurnal Tides.

a and (x are the Declinations of the Sun and Moon, at a period preceding the moment
of observation, by an unkno^\Ti inten-al to be determined for each luminan-, and cidled
the Age of the Solar and Lunar Diurnal Tide.

*• and m are the hour-angles of the Sun and Moon, west of the meridian, at the time
of observation.

i, and ?„, are the Diurnal Solitidal and Lunitidal intervals, or the times which elapse
between the Sim and Moon's southing, and the time of Solar and Lunar Diurnal Higli
Water.

At any time near the Equinox, the declination <t of the Sun is either zero or vei-}'
small, and therefore D will -vanish when fju, the Moon's declination, vanishes, and this
will happen at both High and Low Water, or at any other time of the day ; therefore
at the equinoxes the vanishing of the divu-nal tide at the time of High or of Low Water
ought to be sensibly the same ; but at the time of the Solstices, both members of the
right-hand side of equation (1>) wiU have sensible values, and the Diurnal Tide wU
vanish when these members are equal and of opposite signs ; therefore, to find the time
of vanishing of the Diurnal Tide, we have

, ., M sin2acos (m— i^) ,^ ,

cos(s— ?J = — -TT-- . '- (2.)

At the time of High Water, 9?i, the moon's hour-angle is sensibly constant, or at least
varies within narrow limits ; also, since the vanishing of the Diurnal Tide at High Water
occurs at intei-vals of about a semilunation, the moon's declination, ///, at each vanishing
of the Diui-nal Tide will also varj' within small limits; hence in passing from the
equinox to the solstice, the right-hand side of (2.) will have its change of value depend-
ing chiefly on the change of a ; and it will therefore diminish as <r increases ; therefore
cos(5— ?,) will diminish, and (s— ?',) increase; but s is the houi'-angle of the Sun at the
time of High Water, and increases day by day (48" mean) ; therefore as we approach
the Solstice, the day on vA'hich we ai'e to expect the Diurnal Tide to vanish at the time
of High Water will occur later and later.

But at the time of Low Water the angles s and m must be increased by 90° or
6 hours; and therefore (2.) becomes

• / •\ M sin2i*sin (tw— i„) .„ ,

sm(s— ?,)=— "ij-. ■ '- (3.)

By reasoning similar to that used with respect to equation (2.), we can show that
sin (s— /,) diminishes in passing from the Equinox to the Solstice, and therefore that
(s—i,) also diminishes; therefore the time of vanishing of the Diumal Tide at the time
of Low Water occurs earlier and earlier as we approach the Solstice. We thus see
that the times of vanishing at High and Low Water move in opposite directions, and

248 EEV. S. HArGHTON ON THE TIDES OF THE

become most widely separated at the time of the Solstice. This result agrees perfectly
^^^th the facts of observation at Port Leopold recorded in Table IV. and the accom-
paming fig. III. Plate XI.

I shall now endeaAour to separate, in the Diumal Tide, the effects of the Sun and
Moon. In equation (1.), the effect of the Sun, represented by the first member of the
light-hand side of the equation, when observed at High Water, may be considered to
owe its periodical change almost altogether to the change in cos(s — /J, the angles'
increasing day by day as the tide becomes later and later ; for the angle a may be
regarded as sensibly constant during the semilunation. On the other hand, the Lunar
portion of the Diurnal Tide owes its change to the change of ^w, the moon's declination,
for the angle (»« — ?,„) is sensibly constant. Tlie Solar Diurnal Tide disappears at the
equinox, because then o-=0 ; hence we may find the Lunar Diurnal Tide, at that period
of the year, uncomplicated by the coexistence of the Solar Tide.

Taking the means of the Diurnal Tide Eanges at High and Low "Water for March
and April *, I fhid

At High Water

D = 1-4G2 ft. = M sin (2/-o) . cos (»i-/„) ;
at Low Water

D = 0'852 ft. = M sin (2^) . cos (?h-?„ + 90°),

tn denoting the moon's hour-angle at High Water, and ^l denoting the moon's maximum
declination. Dividing one of these equations by the other, we find

cot(m-?,„) = -^^:^; (4.)

from which we deduce

,„_;; -149^ 4G'=10'' 19'".

'/??, the moon's hour-angle at High AVater, is shown by the observations made in March
and April to have at New and Full Moon a mean value of 12'' 0". Substituting this
value in the preceding equation, I obtain,

(1) The TAinitidal Diurnal Interval = i„=V' 41™.

The coefficient M of the liunar Diurnal Tide may be found as follows : —

Let H=: Range of Diurnal Tide, at High Water, at the equinox.
Let L=Range of Diurnal Tide, at Low Water, at the equinox.

Then

2M sin 2 (max. declination of moon)=:\/ir-|-L^ (5.)

Substituting in this equation the values of H, L, and jt*, we find

2M sin ,S7'=x/(l-4G2f-}-(0-852)^=l-C92 feet ;

* The Tide Ranges used in obtaining these means and marked (*) in Table I.

ARCTIC SEAS.— PARTS I. AXD II.— PORT LEOPOLD,
and therefore, tinally.

249

(2) Coefficient of Lunar Diurnal Tide = 11 = 1- ^S)'^ feet.

The Age of the Lunar Diurnal Tide is found by examining the interval, at the Equi-
nox, from the vanishing of the Moon's declination to the vanishing of the Diurnal Tide,
at High and at Low Water. The interval from the Moon's declination vanishing to the
Tide vanishing is given in Tables IL and IIL for High and Low "Water ; and the figs. IV.
and V. of Plate XI. represent tlie results of those Tables.

From these figures, or from the Tables which they represent, it is evident that tlie
difference in the time of vanishing of the Diurnal Tide at tlie times of High and Lcnv
Water is not altogether due to the Solar Tide ; for at the Equinox, when the Solar Tide
has disappeared, the Age of the Lunar Diurnal Tide at High Water is 1'' 21'', while tlie
Age of the same Tide at Low Water is only 0'' 21''; showing a permanent difference of
one whole day in the times of disappearance of the Tide at High and Low Water quite
independent of the Solar Tide, which, as I have already shown, tends to increase this
difference as we approach the Solstices.

The Means of the Ages of the Diurnal Tide, taken from the Tides of the 13th and 25th
of March and the 6th of April, 1819, at High Water, and from the Tides of the 11th
and 24th of March and the 5th of April, 1849, at Low Water, are V2\^ and 0'^21''*.

I am unable to explain why the Age of the Lunar Diurnal Tide at High Water should
be greater than its Age at Low Water ; but there is good reason to believe that it is an
established fact, as I found the same kind of difference of Age in the Tidal Observa-
tions made in 1850-51, on the coasts of Ireland, by the Royal Irish Academy. The
following Table shows the difference in the Irish stations.

Age of Lunar Diurnal Tide, deduced from Observations at High and Low Water, on
the Coasts of Ireland, 1850-51 (from Transactions of the Eoyal Irish Academy,
vol. xxiii. p. 137).

Station.

Ago at High Water.

Age at Low Water.

Difference.

Caherciveen

(1 h
5 4

4 9

5 10

5 9

6 19
6 5
6 17
6 23
5 19

d h
4 17
4 9
4 20

4 19

5 3
5 2
4 11
3 12
.I 14

d ll
0 11
0 0
0 14

0 14

1 16

1 3

2 6

3 10
0 5

Portrusli

Donaghadee

Courtown

Duimiore East

Means

c*" 2o'' a.o"' /i'' ^-!^ T 7'"

i" 3" 27™

" ' •' 1

Bringing together the Constants of the Lunar Diurnal Tid(> just determined, tliey are
as follow : —

MDCCCLXIU.

* Tlicsc tides are marked (+_) in Tables II. and III.

2m

250

EEV. S. HAITGHTOX OX TlIE TIDES OF THE

1. Diurnal Luni tidal Interval 1'" 41"

2. Age of Lunar Diurnal Tide at High Water . l"* 21*'

Low Water . 0" 21''

3. Coefficient of Lunar Diurnal Tide .... 1-409 ft.

It remains now to determine, if possible, the corresponding Constants of the Solar
Diurnal Tide. In order to effect this object, I laid down the Lmiar Tide, both at High
and Low Water, from tlie preceding constants, on the observed Diurnal Tide at the time
of the Solstices, and thus obtained the constants of the Solar Tide, which at those
periods of the year is a maximum.

Ha\ing thus constructed the Lunar Tide, I found, by tlie difference between it and
the Observed Diurnal Tide, that the maximum Solar Diurnal Tide was as follows : —

Range of Solidiurnal Tide at High Water : — ft.

Summer Solstice, 1849 0'82

Winter Solstice, 1848 0-91

Range of Solidiiuiial Tide at Low Water: —

Summer Solstice, 1849 0-86

Winter Solstice, 1848 0-88

Mean . . . 0-867 ft.

The following Table shows the time at which the Solar Diurnal Tide vanished at the
Solstices.

Vanishing of Solar Diurnal Tide at High Water.

Summer Solstice, 1849.

Tide passing from — to +. h m

May 31 8 34

June 29 8 20

Winter Solstice, 1848.
Tide pas>ing from + to — .

Dec. 19 8 1,1

Jan. 17 8 35

Mean 8'^ 26"'

June 16
July 14

Summer Solstice, 1849-
Tide passing from + to —

20 10
18 50

Dec. 6
Jan. 4

Winter Solstice, 1848.
Tide passing from — to +.

20 50

20 10

Mean

Vanishing: of Solar Diurnal Tide at Low Water.

Summer Solstice, 1849. Summer Solstice, 1849.

Tide pa-sing from — to -f . h m Tide passing from + to — .

June 23 7 30 I June 19

July 23 8 19 } July 8

Winter Solstice, 1848.
Tide passing from + to — .

Dec. 28 730 Dec. 13

Jan. 27 8 30 Jan. 12

Mean

7 57

Winter Solstice, 1848.
Tide passing from — to +.

b m
19 45
19 48

19 40

20 0

Mean ig** 48"

AECTIC SEAS. — PAKTS I. AND II. — POET LEOPOLD. 251

It will be obsened in the preceding Table, that the time of the Diurnal Solar Tide
vanishing may be referred to one or other of two hours, which differ by 12'', and that
the times of passing from -|- to — at the two solstices are reversed. These changes
are evident from the consideration of the expression for the Solar Diurnal Tide,

S sin 2(7 . cos(>'? — ?',),
which changes sign, from Solstice to Solstice, by the cliange of sign of (a), the sun's
declination, and also changes sign at the two high waters or low waters of the same
day by the increment of 180° which the sun's hour-angle s imdergoos. Combining all
the results together, I find that the Solar Diurnal Tide vanishes at High Water \\'lien

8h 2G'"-/,=18\
and

20'' — /,=6'',
or

?,= — 9'' 34"', and +14''.

Mean value of i,= 14'' 13"° ;

and that the Solar Diurnal Tide vanishes at Low Water when

7'' 57"'-/,=18'',
and

19h 48'"-?;= 6'',
or

/.= -10'' S-", and +13'' 48'".

Mean value of /,= I3'' 52"' 30'.

Hie Mean of the values of the Solitidal Interval, at High and Low Water, is

/j=i4'' 2'" 45^

From the preceding data we can readily find the coefficient of the Solar Diurnal Tide ;

for

Sx sin (max. declination of Sun)^0'867 feet,

or

The Age of the Solar Diurnal Tide cannot be deduced from observations such as those
under discussion, because the Sun's declination changes so slowly at the Solstices, that
it may be ccmsidered constant durmg a fortnight, and therefore the Coefficient Ssin(2(r)
is also constant during that period. The Constants of the Solar Diurnal Tide, as just
found, are as follow : —

1. Diurnal Solitidal Interval . . . 14'' 2"^ 45^

2. Age of Solar Diumal Tide . . . UnknowTi.

3. Coefficient of Solar Diurnal Tide . 1-18G feet.

The ratio of the Solar to the Lunar Coefficient is

2 Jl 2

252 EEV. S. HArGHTOX OX THE TIDES OF THE

This result differs widely from the ratios of S to M found by me at the Irish Stations,
which were as follow : —

Ratio of the Solar to the Lunar Coefficient of the Diiirnal Tide, on the Coasts of
Ireland, 1850-51 (from Trans. Eoy. Irish Acad. vol. xxiii. p. 128).

station.

S

Calierciveen

0-698
0-529
0-498
0-659
0-427
0-441
0-504
0-570
0-436

Cusheiulall

Mean

0-5305

I shall now deduce, according to received theories, the mean depth of the channel of
the Atlantic Sea, which conveys the tide from the South Atlantic Ocean to Port Leopold.
The theoiy which I select for this purpose is that given by Mr. Airy in his ' Tides and
Waves,' which is considerably in advance of that given by Laplace and the earlier mathe-
maticians, and, as it is directly founded on the motion of water in canals, seems particu-
larly well adapted to the discussion of a tide like that of Port Leopold, which is situated
at the extreme northern end of the Atlantic Ocean, which may be regarded as a Canal
occupying a meridian circle, and nearly 10,000 miles in length. From the discussion of
the Dimnal Tide, in a meridian canal, given by Mr. Airy (Tides and Waves, p. 356), it
may be deduced that the following equation is true, and that it contains the means of
finding the mean depth of the Atlantic Canal :

7i"b
S mass of Sun d^ g
M- ^- — -

mass of Sun d^
"mass of Moon-^iP-^

A

K~b

(G)

In this equation,
S, M are the coefficients of tlie Solar and Lunar Diurnal Tide, found, as at Port Leopold,

by observation.
D, d arc the mean distances of the Sun and Moon from the Earth.
N, ti are the angular velocities of the Sun and Moon about the Earth.
h is the mean radius of the Earth.
7l is the mean depth of the Atlantic C/'anal.
<j is the velocity acquired in a second by a falling body.

The left-hand side of eqriation (0.) is known by observation, and all the quantities on
the right-hand side are known, except k.

AECTIC SEAS. — PAllTS I. AXD IT. — PORT LEOPOLD. 263

Substituting, therefore, {ov the symbols the ibllowiiiii: vabies,

s

mass of Sun o-nr-i or
mass of Moon ^ '

^ 59-964

D~2x 12032'

mass of Sun ^_n (-900
mass of Moon X jjj - U'-l / ^bb ;

5_.__j; niiles;
^^=0-00345;

we fiiid

from which I deduce

—=0-00323;

0 00323-4^

0-842=0-473 . ;

0-00345-4 5-
u

6=To72' -^-3 "69 miles.

In discussing the Solar and Lvinar Diurnal Tides of nine stations on the Irish Coasts,
I found the following results*: —

VT (mean of nine Stations) = 0-5305;

1=7^' ^'=5-^2 miles.

The mean depth of the Atlantic Canal may be also deduced, by means of Mr. Amy's
Theory of Tides with Friction, from a comparison of the Solitidal and Lunitidal Intervals,
and from the Lunitidal Interval compared with the Age of the Lunar Tide.

According to the Theory of Tidal Waves without Friction, Low Water should occur
at the time of the meridian passage of the luminary ; in consequence, however, of fric-
tion, the phase of High Water is accelerated by an interval equal to the difference
between the Tidal Interval and lialf the period of a Tidal Oscillation. According to
Mr. Amy's Theory, taking account of friction (supposed proportional to the horizontal
velocity of the tidal current), the acceleration of High Water is represented byf

/ .
* Trans. Eoyal Irish Acadoniy, vol. xxiii. pp. 128, 131. t Tides and Waves, p. 332.

254 EET. S. HAUGHTOX OX THE TIDES OF THE

where

f-= coefficient of friction ;

?? = angular velocity of Luminary ;

^=32 feet ;

A-= depth of the sea;

m = -;

>.= length of the tide-wave.
Therefore

Lunitidal Acceleration 'N^—ghn'^
Solitidal Acceleration ri^—gkm- '

and, substituting the following values,

o_ n_

(7.)

S9-'S0 b6400

(8.)

25000x5280'

we find, k being expressed in miles,

Acceleration of Lunar Diurnal Tide 13-815— A

Acceleration of Solar Diurnal Tide 12-938 — /:

To find the Lunitidal and Solitidal Accelerations, we must subtract the Lunitidal and
Solitidal Intervals, /„ and ?'„ from 12'' 24"" and 12'', respectively ; but

/„= I'' 41",
/. =13'- 52'"-5;
therefore

Acceleration of Lunar Diunial Tide =+10'' 43"",

Acceleration (Set m'dafioii) of Solar Diumal Tide =— 1'' 52'"-5.
Substituting these values in equation (8.), I find

^=13-07 miles.
Again, according to Mr. Airt's Theory of Tidal Waves with friction*, the greatest
tide follows the greatest force by an intei-val (Age of Tide),

f{n^+yhrr) .

but the acceleration of the Tide is

/ .

Therefore

Age of Lunidiurnal Tide n^ + c/km^ .

Acceleration of Lunidiurnal Tide n^—gkm^ '
or

Age of Lunidiurnal Tide _ 12-938 + A
Acceleration of Lunidiurnal Tide 12-938 — A

* Tides and Waves, p. 333.

(9.)
(10.)

ARCTIC SEAS. — PARTS I. AND IT. — PORT LEOPOLD. 255

In applying this equation to determine the depth of the sea, the difficulty ah'eady
noticed, as to the Age of the Lunar Tide, deduced from High "Water and Low Water
observations, meets us again. The Age at High Water is 1'^ 21'', and at Low Water
C 21»'.
Substituting these values respectively, I find

miles.
Depth of sea (k) deduced from Age of Lunar Diurnal Tide at High Water=7-96
Depth of sea deduced from Age of Lunar Diurnal Tide at Low Water =4"19

Mean .... =6-07
Bringing together all the preceding results, we find the follomng mean depths of the

Atlantic Canal, as deduced by the various methods described : —

miles.

1. Depth deduced from Heights of Solar and Lunar Diurnal Tides . 3-69

2. Depth deduced fi-om Accelerations of Solar and Lunar Diurnal

Tides, caused by friction 1307

3. Depth deduced fi-om Acceleration and Age of Lunar Diurnal Tide,

caused by friction 6'07

Of the three methods just given for finding the mean depth of the sea, the fu'st is the
most trustworthy, for the folloAving reasons : —

1st. The determination of Heights of the Solar and Lunar Diurnal Tide by observa-
tion is more accurate than the determination of Acceleration and Age.

2iid. The theory by which the depth of the sea is deduced from Heights is independent
of friction, the introduction of which requires additional hj'potheses, which are, at best,
of a doubtful character.

At the same time it should be remarked that the depth of the sea deduced from
Acceleration and Age, at eight stations on the coasts of Ireland, exceeded the depth
deduced from Heights, in a manner similar to that which is found to occur at Port
Leopold.

The Irish depths are —

1. Depth of sea deduced from Heights of Solar and Lunar Dim-nal miles.

Tides 5-12

2. Depth deduced from Accelerations of Solar and Lunar Diumal

Tides, caused by friction 11-98

3. Depth deduced from Acceleration and Age of the Lunar Diui'nal

Tide 11-32

In the present state of our knowledge of the Theory of the Tides, I think it is safer
to adopt the results deduced from Heights as the most reliable, and to wait until mathe-
matical researches shall have further perfected the Theory of friction in Tidal Waves,
before we draw conclusions from it as to the depth of the sea, especially when we con-
sider that this Theory has not yet explained the anomaly discovered by observation as
to the difiierence in Age of the Dimnal Tides deduced from High and Low Waters.

256 REV. S. HAUGHTOX OX THE TIDES OF THE

TART II.— The Semidiurnal Tides of Port Leo])oId, JVorth Somerset.

Picccivcd Oct.ibcr 8,— Read Xovember 27, 1862.

When the daily heiglitof High and Low "Water has been cleared of the Diurnal Tide,
as explained in Part I., and as is shown in the MS. diagrams that accompany this
paper, it is easy to estimate the successive Heights of Spring and Neap Tides, cleared of
the Diurnal Tide.

Bringing together the Spring Tides and the Neap Tides, the following Tables I. and
II. are constructed; and from the second column of these Tables the diagram No. 1
Plate XII. is prepared, of which the following explanation may be useful.

Tlie interval in the abscissae corresponds to live Lunar weeks, or intervals between the
greatest Spring Tide and least Neap Tide. The ordinates are divided, as usual, into feet.
The Curve a, drawn through alternate Spring Tide Heights, is the curve of New
Moon Springs.

Tlie Curve «' is the Curve of Full Moon Springs.

The Curve b is the Curve of First Quarter Neap Tides.

The Curve I' is the Curve of Third Quarter Neaps.

These Curves are constructed from Tables I. and II., which are themselves formed
from the Curves of the MS. diagram.

The diagram No. 2, Plate XII. is formed from the tirst column of Table I. Its
abscissaj are the same as those of diagram No. 1, and its ordinates are the Solar Hours
at which the Maximum Spring Tide occurred.

Curve a represents the Time of New Moon Springs.

Curve ft' the Time of Full Moon Springs.

T.UBLE I. — Semidiurnal Maximum Spring Tide Kangcs, 1848-49.

Time.

Eaiige.

Moon's Hour-Angle.

1848. October

d h in

28 0 30
12 1 0
27 0 45
12 1 0
27 1 15
10 0 50
2G 1 15

'.) 1 0
25 1 45
10 1 0
2G 1 :!0

7 0 30
25 1 20

7 0 30
24 1 15

H 1 30

ft.

5-42

6-67

5-01

C-42

5-03

6-56

5'G5

6-28

6-50

G-11

6-C3

5-7G

6-GO

5-50

G-47

4 -GO

6-2S

5-OG

G-4G

li ni
1 0
1 48

1 8

2 29

1 31

2 4

1 54

2 21
2 17
1 45

1 53

2 24
2 32
2 35
2 17
2 28

1 0

2 48
2 S3

„ Nuvcmbtr

„ December

1849. January

'', :\Iarc!i

„ April

',[ May '.'.'.'..'.

„ Jii'l'v

21 0 30
8 1 30

22 1 15

Mean ... i'' 4"

2'- 2'"

AECTIC SEAS. — PAETS I. .VXD II. — PORT LEOPOLD.

257

Table II. — Semidiurual Minimum Xcap Ranges, 1848-49.

Sun's Hour- Angle. Range. Moon's Hour-Angle.

184S,

1849,

(1 li m

November 4 5 30

„ 18 5 45

December 4 6 0

, i 13 6 20

Januarj' 2 6 0

17 6 30

February 2 7 0

„ ■ 17 7 30

March 3 6 45

18 6 45

April ; 1 7 0

IG 6 30

Mav j 1 7 30

i 16 7 30

30 7 45

June • 15 6 30

I 29 8 0

July 14 7 0

29 8 10

' I

Mean...! 6" 50"

ft.

2-33

2-83

2-95

2-89

3-55

2-54

3-65

1-98

3-25

2-00

3-42

0.07

3-18
2-76
3-.i8
3-14
315
3-27

8 40

7 58

8 8
7 46

7 37

8 17

7 54

8 33
8 16
7 59

7 53

8 11

A. — Parallactic Inequality of Semidiurnal Tide.
The general expression for the Semidiurnal Tides is, as is well kuonTi,

T=s(^^ycos^7cos2(.s-?,)+M(|-ycos',;:^cos2 (w-2„), .

(1-)

where

S and M are the Solar and Lunar Coefficients ;

P and ]) the Solar and Lunar parallax ; and r„., ^;„, the mean values of the same.

<r, (ij the declinations of the Sun and Moon, at periods preceding that of observation
by unknown intervals called the Solar and Lunar Age of the Semidiurnal Tide.

s, in the hour-angles of the Sun and Moon, west of the meridian at the time of
observation.

?„ i„ the Solar and Lunar Tidal intervals, or time after southing of the luminary, at
which its higli water is found to occur.

From an inspection of diagrams 1 and 2, Plate XII., it is plain tliat tlie conditions
of the tide are different at New and Full Moon, and in the 1st and 3rd Quarter. But
the Solar conditions may be supposed constant for a fortnight, and as the moon's decli-
nation only enters equation (1.) by the square of the co.sine, it must be q. p. the same at
the beginning and end of a Lunar fortnight. The difference, therefore, sho^^^l in the
diagrams must depend on the Moon's Parallax, which takes four weeks to complete its
changes.

From diagram Xo. 1, it appears that the greatest difference in the Parallactic Tide,
between the opposite quarters of the Moon, both at Spring and Neap Tides, amounts to
I '47 ft. in the range of the Tide.

MDCCCLXIII. 2 N

258 REV. S. HAUGHTON ON THE TIDES OF THE

I shall presently prove that the Lunar Tide Range at the time of this maximum

Parallactic inequality is 4-62 ft. ; adding, to this, half the Parallactic inequality, and

subtracting it from it, we find

5-35_/l+A^

3-89~\^l-e^ (^.)

that is, the cube of the ratio of the Apogee to the Perigee. Solving this equation for e,
the eccenti-icity of the moon's orbit, we obtain

e= ^=0-05303 (3.)

This value of the eccentricity of the moon's orbit is very near the true value 0-05484*,
as near, indeed, as could be expected from any Tidal Observations.

B. — Solar Semidiurnal Tide.
In diagram No. 1 the Curves marked A and B are drawn so as to eliminate the
Parallactic inequality, and they represent the Curves of Spring and Neap Tide Ranges ;
but from the expression (1) it appeal's that if we could find two sets of tides, for which
the moon's declination and houi'-angle should be the same, we could eliminate the
Lunar Tide and calculate the Solar Tide separately.

These conditions are fulfilled by the Solstitial Springs and Equinoctial Neaps — as
may be thus sho^vn.
Solstitial Springs : —

New Moon June 20'' 2'' 19'" (Greenwich).

Declination 18° 49' N. j". ^;. maximum.

Equinoctial Neaps : —

First Quart <r March 30-' IS'' 58'" (Greenwich).

Declination 18" 27' N. y. p. maximum.

And the Curve of Spring Tide Range, cleared of Parallax, attains its Solstitial Minimum,
June 21'' 12'' 40"^n.W., and its Equinoctial Maximum, April 1'' 12'- 20">L.W.

We may therefore safely assume the part of equation (1.) which depends on the Moon
to be the same at both these times.
We theref(jre have

d = 2S . cos^ ff' cos 2(.>,-' - /,,) + 2M cos= ]j cos 2(wi' -/„).! ,..
_ , . . . . (4.^

<7j^=2S . cos^ff^cos 2(.s^| — ?,) + 2Mcos-|a,^, cos2(nt;,— /„),J

where _

«', T*, 6'', (jJ , m' refer to Solstitial Springs,

a' being the Solstitial Spring Range ; and

^in "■//' *//' 1^111 ^11 I'efer to Equinoctial Neaps,
a,, being the Efjuinoctial Neap Range.

But

a' =5-56 ft.

a,,=2-67ft.

* IIkksciiki/s Astronomy. J^oiulun, 1850, p. (!4W.

AECTIC SEAS. — PARTS I. A\D 11.— POET LEOPOLD. 259

Subtracting from each other the two equations (4.), we tind

(rt'-r/J=2S[cosV cos2(i''-/,)-cosXcos2(s,^-/,)], .... (5.)

and differentiating equations (4.). so as to express that the tide in question is a max.

maximoriun or min. minimorum, and subtracting, we tind

I u _

0= cosV sin2(.s' — ?^) — cos''ff^^sin2(s^^ — /,) (6.)

In these equations (5.) and (G.), </ and o-^,, are practically the declinations of the Sun at
the Equinox and Solstice, i.e. (r'=0 and (r,^=23° 28'; and s' and s^, are the mean values
of the first column of Tables I. and II. : or .s'=li' 4"'=16°, and s„=6'' 50'»=102° 30'.
Hence equation (5.) becomes

2-89 ft. = 2S[0-84 cos 2(lG°-g-cos 2(102° 30'-g], (7.)

and equation (6.) becomes

0 = 0-84 sin 2(lC°-g-sin 2(102° 30'-g, (8.)

or

9 • _6in25°-fO-84 sin 32°
tan -i*.— jjog 25° + 0-84 00^32°'
or finally,

f,=14° 5'=.5G"' 20^ (9.)

Substituting this value in equation (7.), we find

2-89=2S[0-84 cos 17° 55'+cos 10° 55'],
and, finally,

2S=f||=l-624 ft (10.)

Similar reasoning may be used with reference to the Solstitial Neaps and Equinoctial

Springs, fi'om which we derive the equations

rt"=2S cos^ ?' cos 2(s"— g+2M cos^^^" cos 2(m"-?;„),] ^ x

a^ =2S cos'-* <r, cos 2(s^ — ?j)-|-2M cos" f*, cos ^^{m^ —?„■),]
but

«" = 3-15 ft.

«^=6-33 ft.]
Substituting these numerical values, and proceeding as above, we find

2S~=l-786 ft., (12.)

and the mean of (10.) and (12) will be

2S=i705ft (13.)

C. — Lunar Semidiurnal Tide.
The constants of the Solar Tide being found, (9.) and (13.), nothing was easier than
to calculate its amount at each Spring and Neap, and to subtract it from the former and
add it to the latter. In this way the curves (a) and (/3), diagi"am No. 1, were con-
structed, and represent the Lunar Semidiurnal Tide Range at Springs and Neaps
respectively.

2 N 2

260 EEY. S. HA^GHTO^' OX THE TIDES OF THE

Its maximum and minimum amounts are

Maximum Lunar Range =4-G2 ft.
Minimum Lunar Range =4-23 ft.
If a and a' denote the maximum and minimum Lunar Ranges which correspond with
the Spring and Neap Tides of either the Solstice or Equinox, and if in and ??i' denote
the hour-angles of the moon at these tides, we have, if 1=18^ to 19° be the inclina-
tion of moon's orbit,

a =2M cos- 1 . cos 2(//« — ?„),"|

a'=2Mcos=(0) cos 2(»i'- /„,).]'

From whicli we deduce

a. cos^ I COS 2 (m — «'„,)

a' COS 2 (/«' — ?„)

and

_. a cos 2m' — a' cos'' I cos 2n» ,, ,. .

tan2?„.= = — 7r~r, — ( o , • „ (lo.)

"' —a sin 2m + a' COS'' 1 sm 2m ^ '

But, from Tables I. and II..

yl■^ -^2^ 2""= 29° 31'

m'=7'^ 48«> = 113° 14'.

Substituting these values in (15.), we find

^. _463 cos (46° 28') +42.3 x "9 cos (59° 2')
— tan -^m— 462 sin (46° 2b') +423 x '9 sin (59° 2')'

or

5141

2/„=142° 7'; [or -37° 53'];

?;,= 71° 3'; [or -18° 5G'];

or ?m = 4'' 54"M [or —I'' IS-"].

Substituting the value of ?,„ in the second of equations (14.), we find

4.93
-^^=c5r(8i°^'l=4-309ft (16.)

I have not been able to deduce from the observations any close approximation to the
Age of the Lunar Tide, but think it is probably rather over than under five days. It
would require more observations than the heights of liigh and low water to determine
this important constant witli accuracy.

D. — Mass of the Moon and Depth of the Sea deduciblc from the Semidiurnal Tide.
According to the Statical Theory of the Tides, excluding the terms depending on
Parallax, Declination, Hour of Tide, &c., the ratio of the Solar to the Lunar coefficient
is as follows : —

M'^mass of Moon ^ VD j ' ^^ ' '}

AKCTIC SEAS. — PAHTS I. AyT> II.— PORT LEOPOLD. 2G1

where D and d denote the mean distances of the Sun and Moon from tlic Earth.

Hence we find

mass of Sun S /1^\^

mass of Moon Al \d J

S /2xl2032\3

S^ /2xl2032y

But since
and
we have

and, finally,

S = l-705,
M = 4-3U9,

M

mass of Sun
mass of Moon

il=0-395G, (18.)

=359551x71-11 (19.)

From which we deduce tlie mass of the Moon to be . j!^ ^th cf that of the Earth.

From the Dynamical Theory of the Tides given by Mr. Airy (Tides and Waves, p. 357),
the ratio of S to M, for the Semidiurnal Tide, is the following : —

S mass of Sun / d \^ a b

fl\\J-JL . . (20.)

M mass of Moon '

where N and n are the angular velocities of the Sun and Moon, I:, h the mean depth of
sea and radius of earth, and r/ the force of gra%ity.

Substituting for these quantities theu" usual values, we find

«5 0-00323-^
1=0-47288 X ^;

0-00345 -T

0

or

^ =0-47288 xJSIS: (21.)

M~ ^'-"' •^13-64S-/:

1 f o Tolno I

Substituting for tt its value 0-3956, we obtain by sohing for k,

k = depth of sea = 8-32 miles (22.)

A\Tiatever doubt may attach to this high value of the mean depth of the Atlantic
Canal, and to the depths 13-07 miles and 6-07 miles obtained from the Diumal Tide in
Part I., should properly be considered as belonging to the imperfect condition of the
Theoi7 of Canals of equal depth, as apphed to such a body of water as the Atlantic
Ocean, and not to the obsen'ations.

It is, however, well Avorthy of remark, that we can obtain from the Solitidal and Luni-
tidal Intervals a value for the mean depth of the sea that is much more probable, and
is also very close to the depth derived from the ratio of S to M in the Diurnal Tide.

If we call the Acceleration of the Tide the difference between the Tidal Interval and

J62

EEY. S. HAUGHTON ON THE TIDES OF THE

the period of half a Tide Oscillation, we find, from Mr. Airy's Theory, the following
equation for the Semidiurnal Tide : —

Lunitidal Acceleration

Solitidal Acceleration

.'her(

_^'--ffkm^ ,23.)

■"^ — ghn^'' v." V

Lunitidal Acceleration 1.3'815— 4^

Solitidal Acceleration 12-938— 4F
Substituting in this equation.

h m h m m

Lunitidal Acceleration = 6 12 — 4 54= 78,
Solitidal Acceleration =G 0 — 0 56=304,

(24.)

we find

and, finally.

78 _
304'

13-81 5 -4A-
■12-938 — 4/:'

, 3190-6 „ .,^ .,
A':=-q— r-=:o-0J'J miles.

From what we Iviiow, independently of the Tides, of the depth of the Atlantic Ocean,
this value, and that found from the ratio of S to M in the Diurnal Tide, will seem
nearer the truth than the higher values found from other considerations. From this it
may be inferred tliat the theory of Tides, with friction, in these two cases, comes nearer
to the truth than it does in the other cases from which the higher values are derived.
I do not know why this is so, and would recommend the fact to the notice of those
mathematicians who are conversant with the Theorv of the Tides.

Her Majesty's Ship ' Investigator' was secured in her Winter position by Noon of the
18th day of October, the Moon's Age being 21 days. The Eegister of the Tides is com-
menced with the A.M. High Water on the 25th, four Tides before the change of the
Moon.— Lat. 74° N., Long. 90° W.

Register

of Tides observed at Leopold Harbour in the Mon

h of October 1848.

1

High Water. low Water.

Wind.

Dar.

1

1

Time.

Height. } Time.

Height.

Direction.

Force.

li m ft. in. li m

ft. in.

25. A.M.

10 45 22 92 5 0

17 4i

S.E.

6

V.M.

11 0

23 0

5 0

18 3

26. A.M.

11 30

23 6

5 30

17 6i

Easterly.

7-8

P.M.

10 0

22 6

5 30

18 6

27. A.M.

Noon

23 9

5 0

17 lOi

S.Easterly.

3

P.M.

6 45

IS 4

28. A.M.

12 20

23 1

6 10

17 6.',

N. Easterly.

2

P.M.

12 30

24 4

G 45

18 10

29- A.M.

12 30

23 3, J

6 30

18 3

S.Easterly.

3-4

P.M.

1 0

24 6

7 30

19 1

30. A.M.

1 30

22 11

7 15

17 9

Variable.

1

P.M.

1 30

24 0

7 30

18 10

Northerly.

1

.■51. A.M.

2 0 1 21 0

7 30

17 8

3

P.M.

2 0 1 23 7

8 0

18 10

N.VVcsterly.

3

ARCTIC SEAS. — PARTS I. AND n. — PORT LEOPOLD.

263

Register of Tides observed at Lei))

old Harbour

in tlie Mont

1 of November 1848.

nigh Water.

low Water.

Wind.

D»T

•*'*.'•

Time.

Height.

Time.

Height

Direction.

Force.

h m

ft. in.

h ni ft. in.

1. A.Nf.

2 30

21 4 7 45 17 3

N.W.

1

P.M.

2 40

22 8 9 0 1 18 4

»

2.' A.M.

3 0

20 9 9 30 18 0

,,

P.M. 3 30

23 0

9 30 19 10

Nortlu'ily.

5-6

3. A.M.

3 40

21 6

9 40 19 3

,,

„

P.M.

3 40

22 8

10 30

19 6

„

„

4. A.M.

4 30

20 6

11 15

19 0

Northerl}-.

6-7

P.M.

5 10

22 7

12 30

19 0

N.Westerly.

3

5. A.M.

6 0

20 0

Noon I 19 0

Northerly.

3

P.M.

6 10

22 0

10 ! 18 6

N.Westerly.

2

6. A.M. 1 7 30

20 G

10 1 18 6

„

„

P.M. ! 7 45

20 1

1 10 1 18 9

Northerly.

5-6

7. A.M. j 8 40

21 2

2 30 1 18 8

N.Westerly.

5-6

P.M. , 9 30

22 4

3 0 17 9

N.Westerly.

4-5

8. A.M. 9 10

22 3

3 30 I 18 7

N.Westerly.

6-7

P.M.

JO 20

22 8

4 0

17 5

N.Westerly.

6

9. A.M.

10 15

22 9

4 20

18 2

N.Westerly.

2-3

P.M.

11 0

22 9

5 15

16 10

„

„

10. A.M. 10 45

22 10

5 30 18 1

jj

2

p.^r.

11 30

22 11

5 0 1 IG 6

9>

1-2

11. A.M.

12 20

22 11

5 45

17 6

S.Easterly.

4-5

P.M.

12 30

22 9

5 45

16 3

jj

6-7

12. A.M.

12 30

22 11

6 45

17 4

S.S.E.

4

P.M.

1 0

23 1

7 0

16 7

1.3. A.M.

12 30

22 3*

7 15

15 7

P.M.

12 50

22 11

7 30

17 3

14. A.M.

1 40

22 4

7 45

16 5

P.M.

1 45

24 2

8 30

17 7

15. A.M.

2 30

22 2

8 15

16 9

P.M.

2 30

23 10

9 0

17 6

16. A.M.

2 45

21 6

9 0 17 6

P.M.

3 15

23 7

10 30 i 18 3

17. A.M.

4 0

21 0

10 0

18 3

P..M.

5 0

23 3

11 45

18 9

18. A.M.

5 0

20 11

11 0

18 9

P.M.

5 45

22 7

19. A.M.

7 0

20 10

12 50

18 5

P.M.

7 0

22 5

10 i 19 3

20. A.M.

8 20

21 5

1 50

18 6

P.M.

8 15

22 2

2 0

19 2

21. A.M.

9 15

21 9

2 45

18 0

P.M.

9 15

2) 11 3 30

18 8

22. A.M.

10 0

22 3

3 30

17 8

P.M.

10 15

22 0

4 0

18 9

23. A.M.

11 30

22 11

4 20

17 9

P.M.

11 0

22 2

4 45

18 8

24. A.M. 11 10

23 5 5 0

17 9

P.M. j 11 25

22 4 1 5 30

18 8

25. A.M. 11 20

23 10 5 45 17 7

P.M. Midnight

22 2 6 20 18 7

26. A.M.

G 0 ; 17 7

P.M. 12 15

24 0 6 30

18 6

27. A.M. 12 30

22 9

6 20

17 6

P.M. 12 45 i 24 0

7 00

18 7

28. A.M. 1 0

22 0

7 00

17 6

P.M. 1 30

24 0

8 15

19 0

29. A.M. 1 30

21 11

7 30 i 17 10

P.M. 1 45

24 0

8 20 ■ t 18 9

30. A.M. 2 15

21 9

8 0 17 11

P.M. 2 30

1

23 9

8 45 18 9

First quarter at 3' 3" 18", Greenwich. Full Moon at 10"

13' 35"", Greenwicl

.

Last quarter at 17'' 6'' 46"', Greenwich. New Moon at 25''

9'' 29"', Greenwich

On the 12th found the lines attached to moorings of Pole frozen in, thereby vitiating the late semilunar
tidal observations.

264

EEV. S. IIAUGHTOX OX THE TIDES OF THE

Register of Tides obsen-ecl at Leopold Harbom- in the Mouth of December 1848.

Day.

High Water.

Low Water.

Wind.

Time.

Height.

Time.

Height.

Direction.

Force.

h ni

ft. in.

h m

ft. in.

1.

A.M. 2 40

21 6

8 15

18 1

P.M. 3 00

23 7

9 45

18 1

N.Westerly.

4-5

2.

A.M. 3 00

21 5

9 30

18 5

P.M. 4 00

23 3

10 45

19 1

N.Easterly.

2-3

3.

A.M. i 4 00

21 6

9 45

18 10

P.M. 1 4 45

22 11

11 45

18 11

Northerly.

2-3

4.

A.M. ; 5 30

21 4

11 0

19 5

P.M. j 6 0

22 8

Northerly.

3-4

5.

A.M.

6 30

21 4

0 30

18 7

P.M.

6 50

22 3

0 45

19 2

N.W.

4-5

6,

A.M.

8 10

21 6

1 30

17 11

P.M.

8 0

22 1

2 0

18 10

„

4-5

7.

A.M. 1 9 10

22 2

2 45

17 7

P.M.

8 30

22 2

3 0

18 7

3-4

8.

A.M.

10 0

23 4

3 30

17 4

P.M.

10 0

22 10

3 50

18 10

2-3

9.

A.M.

10 45

24 4

4 15

17 5

Northerly.

2-3

P.M.

10 50

22 9

5 30

18 8

10.

A.M.

Noon.

24 6

5 15

16 11

N.N.E.

3-4

P.M.

IMidnight.

22 7

6 0

17 11

11.

A.M.

5 45

16 6

South.

2

P.M.

12 io

24 10

6 30

17 11

12.

A.M.

1 0

22 7

5 20

16 7

S.S.W.

1-2

P.M.

1 0

24 10

7 30

17 8

13.

A..M.

1 20

22 5

7 15

16 8

Calm.

0

P.Nf.

2 45

24 10

8 15

17 11

,,

»

14.

A.M.

2 20

22 5

8 0

17 5

P.M.

2 45

24 8

9 15

18 2

Northerly.

1

15.

A.M.

3 20

22 3

90 18 0

P.M.

3 30

24 3

9 45 18 5

16.

A.M.

3 45

21 11

9 30 18 5

N.N.W.

2-3

P.M.

4 15

23 5

10 45 18 4

S.W.

1

17.

A.M.

4 15

21 9

11 0 i 18 9

P.M.

5 40

22 7

S.Easterly.

1

18.

A.M.

6 20

21 2

I2T0

18 3

P.M.

6 20

22 3

12 20

19 3

South.

1-2

19.

A.M.

7 30

21 7

1 10

18 8

„

I'.M.

7 30

22 4

2 0

18 9

„

20.

A.M.

7 50

22 2

2 0

18 10

Northerly.

1

P..M.

8 30

22 2

2 30

19 10

,,

21.

A.M.

9 15

23 1

3 0

18 11

"

1-2

P.M.

9 15

22 0

3 20 ! 19 10

22.

A.M.

10 20

23 1

3 30 18 4

Northcily.

2-3

I'.M.

10 30

22 5

4 40 1 19 9

23.

A.M.

10 45

24 0

4 30

18 11

N.We.stcrly.

4-5

P..M.

11 0

22 0

5 15

19 5

24.

A.M. 11 30

24 11

5 0

18 6

P.M. 11 30

22 6

G 0

19 11

N.Westerly.

4-5

25.

A.M.

5 30

18 S

„

P.M. i 12 30

24 "7

6 20

19 3

,,

26.

A.M. ; lii 20

22 11

5 45

18 4

.Southerly.

1-2

P.M.

12 30

24 9

7 0

19 2

27.

A.M.

1 0

22 7

6 30

18 4

Calm.

0

P.M.

1 15

24 9

7 40

18 11

„

28.

A.M.

1 00

22 6

7 10

17 4

ss'f.

4

P.M.

1 40

24 7

7 30

18 10

„

„

29.

A..M.

1 30

22 7

7 15

18 2

South.

4-5

P.M.

2 .'iO

24 6

8 30

18 6

„

„

30.

A..M.

2 30

22 3

8 30

18 2

S.S.E.

8

P..M. 2 45

24 2

9 20

18 4

S.E.

4

31.

A.M. I 3 15

21 9

North.

1

First quarter at 3' 8", Grrcnvvicli.

Full Moon at 9'' '

!.3'' 45'", Greenwich

Last quarter at 16" 23" 13'", Grccnwi

eh. New Moon at 25'

4'' 21'", GreenwicI

1.

ARCTIC SEAS. — PAItTS I. AND 11. — POliT LEOl'OLD.

2C5

Register of Tides observed at Leopold Harbour in tlie Month of January 1849.

High

Water.

Low Water.

WiImI,

Day.

Time.

Height.

Time.

Height.

Direction.

Force.

h m

ft. in.

h m

ft. in.

1

A.M.

4 0

21 5

9 40

17 11

S.S.E.

I'.M.

3 40

22 7

10 .-^o

17 11

„

6-7

2

A.M.

4 30

21 2

9 30

18 9

S. Easterly.

5-6

I'.M.

6 0

22 5

S.S.E.

5-6

3.

A.M.

G 10

21 6

12 10

1/ 10

S.E;islcily.

5-6

P.M.

G 20

22 5

12 30

19 0

S.S.E.

5-6

4.

A.M.

! 7 0

22 0

12 30

18 1

„

4-5

PM.

7 30

22 0

1 20

19 1

jj

3-4

5.

A.M.

8 30

22 G

2 0

17 G

N.\Ve.<tcrly.

3-4

P.M.

8 20

21 11

3 10

18 10

J,

2-0

C.

A.M.

9 40

22 11

2 45

17 1

N.W.

3-4

I'.M.

10 0

21 10

4 30

18 6

J,

6-7

7.

A.M.

10 30

24 1

4 30

17 0

Viirialilc.

P.M.

11 20

22 4

5 0

18 9

N.N.Westcrly.

2'3

8.

A.M.

11 15

24 5

5 0

17 0

A'aiialilc.

J,

P.M.

11 40

22 5

5 50

18 4

S. Easterly.

3-4

9.

A.M.

11 30

25 2

5 45

IG 10

„

P.M.

6 30

18 5

N.Wcstc rly.

2-3

10.

A.M.

12 20

23 2

€ 20

17 2

,,

P.M.

25 2

7 30

18 1

J,

l'2

11.

A.M.

i To

22 7

7 15

17 0

Variable.

1

P.M.

1 30

25 1

8 15

17 9

Westerly.

1 .

12.

A.M.

2 0

22 6

8 0

17 0

N.Wei-ti'ily.

o

P.M.

2 15

24 4

8 30

17 6

Northerly.

34

13.

A.M.

2 40

22 6

8 30

17 11

^j

4

P.M.

2 30

24 6

9 40

18 8

N.We.sterly.

4-5

14.

A.M.

3 15

22 7

9 30

18 G

,,

4

P.M.

3 30

23 9

9 15

18 5

,^

4

IJ.

A.M.

4 30

22 3

10 30

18 10

Nortlierlv.

4-5

p.>r.

4 30

22 11

10 30

18 6

J,

4-5

IC.

A.M.

5 0

22 2

10 50

19 8

N.Westerly.

C-7

I'.M.

5 30

22 9

11 30

18 10

N. by i:.

7

17.

A.M.

6 0

21 11

1 30

19 U

N.N.W.

4-5

P.M.

6 30

21 10

,,

4

18.

A.M.

7 15

21 9

12 50

18 8

N.N.E.

4-5

P.M.

7 15

21 6

1 30

19 9

N.Wrstirly.

5-G

19.

A.M.

8 40

22 1

2 15

18 G

N.N.W.

5-6

P.^^.

8 50

21 0

3 0

19 9

Northerly.

5-6

20.

A.M.

10 0

22 7

3 15

IS 4

1-2

P M.

9 30

21 10

4 15

19 6

N.W. to S.E.

1-2

21.

A.M.

10 50

23 4

3 30

18 4

S.Easteily.

4-2

P..M.

10 30

22 2

5 0

19 10

J

3

2~.

A.M.

11 15

24 1

4 30

18 9

J,

3

P.M.

11 0

22 5

5 15

19 4

,,

4-5

23.

A.M.

11 30

24 5

5 20

18 7

Northerly.

4

P.M.

11 30

22 2

6 30

19 4

Calm.

0

24.

A.M.

5 45

17 9

N.N.W.

1-2

P.M.

12 30

24 2

C 40

18 8

1-2

25.

A.M.

12 20

22 3

6 15

17 4

j^

2

P.M.

1 0

24 1

7 25

IS 3

1-2

26.

A.M.

1 0

22 2

7 0

17 5

Soutlicrlv.

4-5

P.M.

1 15

24 3

7 45

17 11

'

3

27.

A.M.

1 30

22 3

7 15

17 4

,,

3-4

P.M.

2 15

24 1

8 30

17 8

Northerly.

2-3

28.

A.M.

2 0

22 3

8 0

17 2

N.N.W.

5

P..M.

3 0

2.1 8

9 0

17 8

,,

G

29.

A.M.

2 15

22 5

8 45

17 11

N. Westerly.

8-9

P.M.

3 20

23 9

9 50

18 0

1

7'8

30.

A.M.

3 45

22 8

9 30

18 6

jj

3

P.M.

3 50

2.'? 5

10 0

IS J

N.N.W.

2

31.

A.M.

4 20

22 4

10 50

18 10

1

P..M.

5 0

22 10

11 0

18 0

N.Westerly.

3

First quarter at 1'' 19" 38". Greenwich.
Last quarter at 15'' IS"" 54", Greenwich.
First quarter at 31^ 4'' 42", Greenwich.

Full -Moon at S"" 10" 50", Greenwich.
New .Moon at 23' 22", Greenwich.

266 EEV. S. HArGHTON ON THE TIDES OF THE

Eegister of Tides observed at Leopold Harbour in the Month of February 1849.

High Water.

Low Water.

Wind.

P^T-

Time.

Height.

Time. j Height.

Direction.

Force.

h m i ft. in. j

h m 1 ft. in.

1.

A.M.

5 40

23 2

11 35 1 19 5

N.N.W.

6

P.M.

5 30

23 1

11 20 ; ][) 0

N.Westerl}-.

8

2.

A.M.

6 55

23 4

,,

8

P.M. 7 0

21 11

1 30 20 3

,,

5

3.

A.M. 8 0

22 5

1 30 17 10

S.S.E.

3

P.M. 8 20

21 4

2 30 19 1

N.Westerly.

2

4.

AM. 1 9 15

22 9

2 30 1/6

Northerly.

1

P.M. 9 30

21 4

4 20

18 8

„

3

5.

A.M. 10 40

23 5

3 30

17 3

N.N.W.

6

P.M.

10 30

22 0

5 0

18 5

„

6

6.

A.M.

11 15

24 5

4 30

17 5

„

6

P.M.

11 30

23 5

5 45

18 4

„

6

7-

A.M.

11 45

24 5

5 30

17 3

„

8

P.M.

6 30

17 8

„

7

8.

A.M.

12 30

22 3

6 15

16 6

„

7

P.M.

12 40

24 2

7 0

17 6

N.Westerly.

7

9.

A.M.

1 0

22 6

7 0

IG 6

„

8

P.M.

1 0

23 10

7 40

17 4

N.N.W.

7

10,

A.M.

1 30

22 9

7 45

17 4

„

7

P.M.

1 50

24 2

8 30

17 8

N.Westerly.

8

11.

A.M.

2 0

22 10

8 15

17 6

„

6

P.M.

2 30

23 6

8 40

17 9

,,

7

12.

A.M.

3 10

22 10

9 15

18 6

„

9

P.M.

2 45

23 10

9 40

18 3

N.W.

6

13.

A.M.

3 15

22 3

9 30

18 2

N.Westerly.

4

P.M.

3 30

22 6

9 45

17 8

„

2

14.

A.M.

4 30

21 5

10 30

18 5

S.S.E.

1 to6

P.M.

4 30

21 8

10 30

18 4

,,

7

15.

A.M.

5 0

21 10

11 20

19 6

Southerly.

7

P.M.

5 0

21 4

11 30

18 2

S. by E.

5

16.

A.M.

6 30

21 3

12 30

19 6

S.S.E.

5

P.M.

6 10

21 0

„

7

17.

A.M.

7 30

32 7

12 30 19 1

„

7

P.M.

7 30

21 5

2 15

20 6

,,

7

18.

A.M.

8 30

22 11

2 0

19 6

„

7

P.M.

9 0

21 6

3 30

20 3

„

9

19.

A.M.

10 0

23 1

3 0

19 3

„

9

P.M.

10 0 21 2 1

4 30

19 6

S.Easterly

6

20.

A.M.

10 40

22 11

4 0

18 1

Southerly.

3-1

P.M.

11 0

21 1

5 15

18 4

Calm.

0

21.

A.M.

11 30

23 0

5 0

17 7

S.Wfstcrly.

1

P.M.

5 30

17 11

Calm.

0

22.

A.M.

,12 20

21 4

5 45

17 5

Southerly.

1

P.M.

12 15

23 2

6 30

17 6

Northrrlv.

1

23.

A.M.

12 15 21 8

6

IG 9

N.N.w:

2.

P.M.

12 30 23 3

7 0

IG 8

Calm.

0

24.

A.M.

12 45 21 6

6 30

]5 11

Northerl}'.

1

P.M.

10 22 10

7 15

IG 1

Calm.

0

2').

A.M.

12 45

21 9

7 20

15 8

„

0

P.M.

1 45

22 8

8 0

15 10

N.N.W.

3

26.

A.M.

1 45

21 0

7 40

IG 3

N. Easterly.

6.

P.M.

2 0

23 7

8 15

17 3

Easterly.

7

27.

A.M.

2 28

22 11

8 30

17 G

N.Westerly.

5.

P.M.

1 45

23 0

9 0

17 9

„

5

28

A.M.

3 0

22 9

9 15

17 10

„

5

P.M.

1 3 15

22 0

9 30

17 5

N.W. to S.E.

4-2

Full Moon at 6" 23" IS"". Greenwich

Last quarter at 1

4" 6" 2", Greenwich

.

New Moon at 22'^ 13" 29", Greenwic

h.

AECTIC SEAS. — PAETS I. AND II. — POET LEOPOLD.

267

Register of Tides observed at Leopold Harbour in the Month of March 1849.

High Water.

Low Water.

Wind.

Day.

Time.

Height.

Time.

Height.

Direction.

Force.

h m

ft. in.

h m

ft. in.

1. A.M.

4 30

22 4

10 30

18 0

Calm— N.W.

0-4

P.M.

4 30

21 10

10 30

17 6

N.Wcsteriy.

2

2. A.M.

5 30

22 3

11 20

18 10

S.S.E.

2

P.M.

5 30

21 8

11 30

18 3

S.

2

3. A.M.

6 40

22 8

Calm and 1

2-3

P.M.

6 45

21 6

1 30

19 0

Variable. /

4. A.M.

7 30

22 9

1 10

18 6

S.Easterly.

4

P..M.

8 20

21 7

2 45

19 5

jj

4-7

5. A.M.

9 0

23 5

2 30

18 8

ji

5

P..M.

9 45

21 11

4 20

19 1

jj

7-3

6. A.M.

10 0

23 4

4 0

17 9

jj

3-2

P.M.

11 15

22 2

4 45

18 4

,,

3-7

7. A.M.

11 10

24 4

4 30

18 4

E.S.E.

6-7

P.M.

11 20

23 0

5 45

18 8

Easterly.

8-9

8. A.M.

Noon

24 7

5 30

18 1

S.E.

7-4

P.M.

6 25

18 4

S.S.E.

2

9. A.M.

12 15

23 1

6 15

17 4

N.N.W.

1

P.M.

12 30

23 10

6 20

17 7

N.N.E.

0

10. A.M.

12 45

22 11

7 0

17 0

N.Westerlj'.

2

P.M.

1 0

23 8

7 15

17 3

^

2

11. A.M.

1 30

22 11

7 45

16 0

^,

3

P.M.

1 40

23 4

8 0

17 5

jj

6

12. A.M.

1 50

23 1

8 15

17 8

^j

5

P.M.

2 30

23 2

8 30

17 6

W.N.W.

6

13. A.M.

2 0

23 0

8 30

18 5

,^

6

P.M.

2 30

23 1

8 20

17 10

,,

7

14. A.M.

3 0

22 10

9 30

18 6

ji

9

P.M.

3 0

22 4 1 9 20

18 0

„

6

15. A.M.

3 30

22 8 9 45

19 2

N.Westerly.

5

P.M.

3 30

22 1

10 10

18 7

^j

2

16. A.M.

4 30

22 5

10 30

19 10

N.N.W.— S.S.E.

1-1

P.M.

4 0

21 7

11 0 : 18 11

S.E.-lv.— S.W.'ly.

1-3

17. A.M.

5 30

22 6

U 45 ; 20 5

"Calm.

„

P.M.

4 45

21 8

12 0

19 8

Easterly.

6

18. A.M.

6 40

22 6

1 30

20 3

,,

4

P.M.

6 45

21 3

N.Easterly.

4

19. A.M.

7 45

22 5

1 0

19 "9

jj

4

P.M.

8 30

21 0

3 15

19 8

J,

5

20. A.M.

9 30

22 8

2 30

19 5

j^

6

P.M.

9 45

21 3

4 20

19 4

^,

5

21. A.M.

10 0

23 2

3 30

19 1

Northerly.

3

P.M.

10 31

21 11

4 30 19 0

,j

1

22. A.M.

10 45

23 9

4 30

19 0

^j

3

P.M.

11 0

22 7

5 30

18 10

]^

4

23. A.M.

11 10

23 11

5 0

18 7

J,

2

P..M.

11 40

22 5

5 45

17 11

Calm.

J,

24. A.M.

5 30

17 6

J,

jj

P.M.

12 5

23 9

6 20

17 4

Southerly.

2

25. A.M.

• 12 25

22 9

6 15

16 11

S.S.Easterly.

3

P.M.

12 40

23 1

6 45

16 0

j^

3

26. A.M.

1 15

22 11

7 0

16 4

^1

5

P.M.

1 30

22 9

7 10

16 2

]^

5

27. A.M.

1 30

23 5

7 50

16 10

J,

5

P.M.

2 20

23 0

8 0

16 6

Southerly.

4

28. A.M.

2 30

23 7

8 30

17 2

Easterly.

4

P.M.

2 20

22 8

8 45

17 0

Northerly.

1

29- A.M.

3 15

23 8

9 10

18 1

Easterly.

7

P.M.

3 0

22 7

9 15

17 10

J,

7

30. A.M.

3 45

23 6

10 30

18 9

N.Easterlj'.

5

P.M.

3 50

21 9

10 15

17 11

N.VVesterlj'.

5

31. A.M.

5 0

22 11

11 30

19 0

Northerly.

2

P.M.

5 15

21 1

11 30

18 5

Southerly.

3

First quarter at 1" 12\ Greeinvicb. Full Moon at S"" IS"", Greenwich. Last q

uarter at 16'' 12'', Greenwich.

New Moou at 24'' 2\ Greenwich. First quarter at 30'' 1

9*", Greenwich.

2o2

2G8

EEV. S. HAUGHTOIN' OX TIIE TIDES OF THE

Kcgister of Tides obseiTed at Leopold Harbour in the Month of April 1849.

8.
9.
10.

11.

12.
13.
14.
15.
IG.

17-

A.M.
P.M.
A.M.

p..\r.

A.M.
P..M.
A.M.
P..M.
A .M.
P.M.
A.M.
P.M.
A.M.
F..\I.
A.M.
P.M.
A.M.
P.M.
A.M.
P.M.
A .M.
P..M.
A.M.
P.M.
A.M.
P.M.
A.M.
P M.

A.^f.

P.M.
A.M.

I'.M.

A.M.

r.M.
18. A.St.

P.M.
la- AM.

P.M.

20. AM.
P.Nf.

21. A.M.

P.M.

22. A.M.

P.M.

23. A.M.
P.M.

24. A.M.
P.M.

25. A.M.
P..M.

26. A M.
P.M.

27. AM.
P..M.

28. A.M.
P.M.

29. A.M.

P.M.

30. A.M.
P.M.

h m

G 20

7 20

7 20

8 30

8 40

9 50

10 0

10 30

10 30

11 0

11 20

11 50

12 0

0 20

0 30

0 45

1 0

1 0

1 30

1 50

2 30

2 30

2 30

3 15

3 0

3 35

3 30

5 0

4 45

5 30

6 30

7 0

8 30

8 30

9 30

9 l.>

9 45

10 0

11 »

10 50

11 15

11 40

12 0

12 0

12 40

12 40

1 20

2 0

2 15

3 0

3 30

4 0

4 15

4 45

5 10

5 45

6 40

ft.

in.

23

0

21

6

23

0

21

G

23

4

22

4

23

11

22

7

23

3

22

6

23

3

23

10

23

6

23

9

23

7

23

8

00

11

23

8

22

7

23

7

00

3

24

0

22

7

23

0

21

3

22

6

20

11

20 9

20 7

22

3

21

0

23

0

21

9

23

»

21

6

2:t

1

23

9

23

9

23

9

24

2

23

9

24

7

23

;i

24

7

22

11

24

G

22

G

24

0

21
23

7

23
21

1

.5
5

23

4

21

6

1

0

12

20

2

15

3

20

4

0

3

30

4

30

4

15

5

20

5

15

5

30

6

0

G

15

6

30

6

30

n

45

7

10

7

30

7

30

8

30

8

10

8

45

8

45

9

30

9

0

10

35

9

45

11

30

10

45

12

30

11

30

2

30

1

30

3

30

3

15

4

20

4

0

4

15

4

30

5

15

5

30

5

45

5

45

6

20

7

0

7

0

7

30

7

30

8

20

8

20

9

0

9

10

10

30

10

0

11

30

11

20

19

5

19

0

19

3

19

1

19

1

19

4

IS

11

18

5

17

10

17

't

17

8

17

10

17

11

18

2

17

9

17

10

17

18

18

0

18

3

18

1

19

3

18

3

18

8

18

0

18

10

18

5

19

1

IH

9

19

8

19

G

19

5

19

0

19

0

18

G

18

11

18

2

18

5

17

8

IS

0

1 7

10

IG

11

17

9

17

0

18

0

17

0

1 7

11

1 7

3

IS

0

1 7

11

18

9

18

11

19 4 1
19 3 /
19 4

19

E.S.E.
N.Ea^tclly.
N.We-tcrly.

Culm.

Northerly.

Ciilni.
N.\Veslerly.

N.N.W.

N.N.E.

N.Ea,tcrly.

Nortlicrly.
Soutlifrly.
Nortlierly.
N.N.VV!
Variable.
S.S.E.

Soutlierlv.
N.Westerly.

S.S E.
N.N.W.

Northerly.

N.N.E.

N.Easterlv.
N.N.W."
N. by E.
N.N.E.

N.Westerly.
S.S.E.

Full Moon at 7'' 3'' 49'", Gre-pnwich.
New Moon at 22^ 11'' 54'", Greenwich.

Last qiiaiter at 15'' 7'' 7"'. Greenwich.
First f|iiarler at 29'' 2^ 17", Greenwicli.

AECTIC SEAS. — PARTS I. AND TT. — PORT LEOPOLD.

2G9

Register of Tides observed at Leopold Harbour in the Month of May 1849.

High Water.

Low Water.

Wind.

Day.

Time.

Height.

Time.

Height.

Direction.

Force.

h m

ft. in.

h m

ft. in.

I. AM.

7 15

23 2

12 35

19 5

N.N.W.

1

P.M.

8 30

21 9

2 30

18 11

N. Westerly.
N.N.E.

3

2. A.M.

8 30

23 1

2 0

19 5

3

1"..M.

9 15

22 3

3 30

18 8

„

4

3. A.M.

2 45

19 2

Northerly.

3

I'.M.

10 15

22" "9

4 20

18 5

J,

3

4. A.St.

10 15

23 1

4 15

19 0

^,

5

P.M.

10 20

23 9

4 30

IS 5

N.N.E.

4

5. A.M.

11 0

23 4

5 0

18 9

,,

3

I'.M.

11 45

23 10

5 0

18 5

,1

3

C. A.M.

11 30

23 0

5 50

18 11

Northerly.

3

I'.M.

1-2 10

23 9

5 30

17 9

N.N.W.

5

7. A.M.

12 15

23 7

G 15

18 G

S. Masterly.

2

P.M.

12 30

23 10

6 0

17 G

Southerly.

0

8. .A.M.

12 30

22 4

6 45

18 4

Variable.

1-3

P .M.

6 30

17 9

Easterly.

7

9. A.M.

0 45

24 "2

7 0

18 U

„

7

r.M.

0 50

22 10

7 0

18 3

Southerly.

10. A.M.

1 30

24 2

8 0

18 7

Variable.

2-4

P.M.

1 40

21 9

8 0

17 9

^,

3

11. A.M.

2 20

23 7

8 40

18 7

S.S.E.

5

P..M.

2 30

21 8

8 30

18 1

Southerly.

5

12. AM.

3 0

23 8

S 15

19 2

Variable.

2

P.M.

2 40

21 9

9 15

18 6

S.S.E.

2

13. A.M.

3 20

23 5

10 15

19 2

,j

2

P .M.

3 20

21 4

9 30

18 10

,,

2

14. A.M.

4 15

23 1

11 10

19 2

^^

4

P.M.

4 10

21 0

10 19

19 3

Easterly.

5

1 5. A..M.

5 0

23 11

12 0

19 4

S.Ea>terly.

4

P..M.

6 0

21 2

11 0

19 7

J,

4

16. A..M.

6 0

22 11

,,

t,

r.M.

7 30

21 7

i 20

19 5

„

4

17. A.M.

7 15

22 11

0 45

19 11

Easterly.

3

P..M.

8 30

21 11

2 30

19 1

S.H.

3

18. A.M.

8 30

22 8

2 0

19 8

N.Easlcily.

5

V.M.

9 30

22 0

3 0

18 4

Nonljeily.

1

19. A.M.

9 15

22 5

3 20

18 8

N.Westeriy.

4

P.Nf.

10 15

22 7

4 0

17 7

ji

4

20. A.M.

10 15

22 7

4 15

18 6

N.N.W.

5

P.M.

10 40

23 2

4 20

17 3

N.We-lerly.

4

21. A.M.

11 0

22 5

5 0

18 2

N.N.W.

2

I'.M.

12 0

24 4

5 20

17 0

S.S.E.

0

22. A.M.

11 30

23 2

G 0

18 4

Northei-ly.

0

P..\f.

12 15

24 :i

G 5

17 1

S.S.E.

2

23. A.M.

12 20

23 3

6 45

IS 5

„

0

P.M.

G 30

17 3

N.N.W.

3

24. A.M.

1 0

2.V' 3

7 30

18 4

,,

2

P.M.

1 15

23 4

7 0

17 7

N.Westeriy.

3

25. A.M.

2 0

25 5

8 15

18 1

Calui.

0

)>.M.

1 30

22 10

8 15

17 8

N.\Ve>terly.

G

26. A M.

2 45

25 0

9 30

18 7

J,

7

P..M.

2 45

22 6

9 0

18 1

J,

5

27. A.M.

3 30

24 7

10 30

18 9

J,

5

P.M.

4 0

22 0

9 45

1« 5

jj

6

28. A.M.

4 40

23 11

11 20

18 8

jt

5

p..^r.

4 0

21 4

10 45

IK 11

3

29. A.M.

5 30

23 5

11 45

18 n

.,

2

P.M.

6 40

21 10

11 40

19 8

Varial.lo.

I

30. A.M.

C 30

23 5

Easterly.

2

P..M.

8 9

22 3

1 30

19 1

ij

1

31. A.M.

8 15

23 1

1 30

19 11

Northerly.

2

P.M.

8 15

22 8

2 20

19 1

>'

2

Full Moon at G' 19\ Grepiuvich.

Last qiiartrr at 14''

22\ Greenwich.

New Moon ;it 21'' 19\ Greenwich.

Fii-st (juartiT at 28''

1 1*", (iretnwieh.

270

EEV. S. HATJGHTOIS' ON THE TmES OF THE

Register of Tides observed at Leopold Harbour in the Month of June 1849.

High Water.

Low Water.

Wind.

Day.

Time.

Height.

Time.

Height.

Direction.

Force.

h m

ft. in.

li m

ft. in.

1. A.M.

8 45

22 9

2 30

19 9

Northerly.

1

P.M.

9 30

23 6

3 0

18 8

N.N.W.

2

2. A.M.

9 20

22 11

3 30

19 9

J,

4

P..M.

10 30

23 7

3 45

18 9

,,

3

3. A.M.

10 20

22 9

4 30

19 9

jj

4

P.M.

11 0

24 2

4 30

18 10

,j

4

4. A.M.

11 0

22 9

5 15

19 9

,,

5

P.M.

11 40

24 4

5 0

18 10

jj

7

5. A.M.

11 30

22 9

5 50

19 7

„

6

P.M.

12 10

24 6

5 30

18 8

„

7

6. A.M.

6 45

19 7

N.Westerly.

7

P.M.

12 15

22 8

6 10

18 8

jj

7

7. A.M.

12 45

24 6

7 20

19 2

ji

7

P..M.

1 10

23 4

6 45

18 5

jj

5

8. A.M.

1 30

24 5

7 45

19 2

Northerly.

2

P.M.

1 30

22 5

7 30

18 4

,j

1

9. A.M.

1 45

24 2

8 30

19 4

N.Easterly.

3

P.M.

2 0

22 3

7 30

18 11

E.N.E.

6

10. A.M.

2 20

24 4

9 0

19 4

Easterly.

5

P.M.

2 40

22 2

8 30

19 1

N.N.E.

2

11. A.M.

3 15

24 3

9 15

IS 10

E.N.E.

6

P.M.

3 0

21 4

9 15

18 7

6

12. A.M.

3 20

23 5

10 20

IS 11

Variable.

3

P.M.

3 40

21 7

10 0

19 4

N.Westerly.

1

13. A.M.

4 30

23 7

11 20

19 4

N.N.E.

3

P.M.

5 0

21 10

11 0

19 8

jj

1

14. A.M.

5 15

23 5

12 0

19 5

Variable.

1

P.M.

5 30

22 0

12 0

20 0

Easterly.

2

15. A.M.

6 30

23 1

jj

6

P.M.

7 30

22 4

1 15

18 11

„

6

16. A.M.

7 50

22 9

1 30

19 9

S.Easterly.

2

P.M.

9 0

22 10

2 15

IS 6

i,Variable.

1

17. A.M.

8 30

22 9

1 45

19 7

S.Easterly.

4

P.M.

10 45

23 6

3 0

18 1

„

2

18. A.M.

9 30

22 8

3 40

19 3

E.S.E.

3

P.M.

10 30

24 2

3 45

17 9

,,

5

19. A.M.

10 30

22 11

5 0

19 0

Easterly.

2

P.M.

11 30

24 9

4 40

17 7

Northerly.

0

20. A..M.

11 15

22 10

G 0

18 8

Ea>terly.

2

P.M.

12 0

25 0

5 30

17 1

J,

4

21. A.M.

6 30

18 4

„

4

P..M.

12 30

23 ""0

6 10

17 3

Variable.

2

22. A.M.

12 40

25 5

7 15

18 3

N.N.E.

2

P.M.

1 10

22 0

7 15

17 3

N.Easterly.

3

23. A.M.

1 30

25 4

8 30

18 4

J,

2

P.M.

1 50

23 6

7 40

17 10

,,

2

24. A.M.

2 15

25 3

8 50

18 5

Northerly.

1

P.M.

2 30

22 9

8 30

18 3

Southerly.

2

25. A.M.

3 15

24 11

9 30

18 5

Variable.

1

P.M.

3 20

22 5

9 30

18 9

S.S.E.

2

26. A.M.

3 50

24 4

10 30

18 5

J,

5

P.M.

4 30

22 3

10 30

19 2

S.Westerly.

5

27. A.M.

4 45

23 7

10 30

18 9

S.S.E.

5

P.M.

5 30

22 1

11 30

19 5

jj

5

28. A.M.

5 40

23 2

ij

4

P.M.

7 0

22 5

12 30

18 11

,j

5

29. A.M.

7 0

22 10

12 50

20 2

„

3

P.M.

8 20

22 7

1 30

19 0

„

1

30. A.M.

8 0

22 7

2 30

20 0

„

1

P.M.

9 0

23 3

2 30

19 0

"

2

Full .Moon at 5'' 1G\ (Jreernvich.

Last quarter at

1.3" 10'', Greenwich.

New Moon at 20" 2", Gieonwich.

First quarter at

26" 22'', Greenwich

AECTIC SEAS. — PJlRTS I. A^'D IT. — rORT LEOPOLD.

271

Register of Tides observed at Leopold Harboui- in the Month of July 1849.

; High Water.

Low Water.

Wind.

Dav.

Time.

Height

Time.

Height

Direction.

Force.

h m

ft in.

h m

ft. in.

1. A.^r.

9 30

22 7

3 15

20 4

S.S.E.

4

P.M.

9 45

23 10

3 15

19 0

„

4

S. A.M. 1 10 0

22 6

4 15

20 3

Variable.

1

P.M.

10 45

24 0

4 30

18 11

Northerlv.

2

3. A.M.

10 40

22 5

5 15

19 2

N.Westoriy.

4

P.M.

11 15

24 2

5 0

18 7

„

5

4. A.M.

11 30

22 5

5 50

19 8

„

4

P.M.

11 50

24 6

6 15

18 6

„

4

5. A.M.

11 50

22 8

6 0

19 6

„

6

P.M.

Midnight.

24 9

5 45

18 7

„

2

6. A.M.

6 45

19 4

„

3

P.M.

12 30 22 10

6 20

18 4

j>

2

7. A.M.

12 50 I 24 9

7 10

19 4

„

2

P.M.

12 50 22 9

7 0

18 6

„

3

8. A.M.

1 20

24 10

8 0

18 11

„

5-6

P.M.

1 30

22 6

7 20

18 2

»

6

9. A.M.

2 0

24 6

8 45

18 10

Variable.

2

P.M.

2 0

22 6

8 0

18 7

S.S.E.

3

10. A.M.

3 0

24 6

9 0

19 1

S.Easterly.

4

P.M.

3 0

22 7

8 50

19 0

Northerly.

2

11. A.M.

3 20

24 5

9 50

19 3

„

3

P.M.

3 20

23 0

9 20

19 8

„

2

12. A.M.

3 45

24 8

10 40

19 10

Variable.

2

P.M.

4 0

23 2

10 15

20 2

N.Westerly.

4

13. A.M.

5 0

24 0

11 30

19 5

„

4

P.M.

5 40

22 11

11 0

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Variable.

2

14. A.M.

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12 20

19 7

N.Westerly.

2

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16. A.M.

8 0

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2 10

20 4

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9 15

23 10

2 30

18 7

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2

17. A.M.

9 0

22 9

3 20

19 10

S.S.E.

2

P.M.

10 10

24 8

3 10

18 8

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3

18. A.M.

10 0

23 0

4 15

19 7

,,

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P.M.

10 45

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N.Westerly.

3

19. A.M.

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19 1

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P.M.

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25 5

5 0

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5

20. A.M.

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23 3

6 30

18 8

„

3

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17 8

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4

21. A.M.

12 30

25 8

7 15

18 7

Northerly

4

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23 6

6 40

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3

22. A.M.

1 15

25 5

7 50

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N.N.E.

2

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17 11

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Northerly.

5

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N.Westerly.

4

26. A.M.

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N.W.

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10 20

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27. A.Af.

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22 10

11 20

18 10

„

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5 40

22 4

11 30

20 0

„

6

28. A.M.

5 30

22 6

N.Westerly.

5

P.M.

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22 9

12 30

19 3

29. A.M.

6 30

22 5

1 0

20 8

P.M.

8 10

23 0

1 40

19 5

30. A.M.

8 10

22 2

2 30

20 7

N.Easterly.

3

P.M.

9 0

23 6

2 30

19 5

E.N.E.

4

31. A.M.

9 15

22 3

3 45

20 6

„

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P.M.

10 10

23 4

3 50

19 4

East.

6

Full Moon at 5'' 1", Greenwich.

Last quarter at 12'' 19" Greenwich

New Moon at 19'' 9" Cireenwich.

First quarter at 26'' 12" Greenwich.

ox THE TIDES OF THE ARCTIC SEAS.— PAETS I. AND II. — PORT LEOPOLD.

Register of Tides observed at Leopold Harbour in the Month of August 1819.

Hieh W.ifer.

Low Water.

Wind.

Day.

Time.

Height.

Time.

Height.

Direction.

Force.

h m

ft. in.

h m

ft. in.

1. A.M.

10 15

22 5

4 45

20 3

E.S.E.

7

P.M.

10 45

24 8

4 0

19 4

N.Eastctlv.

6

-'. A.M.

10 50

22 9

5 20

20 3

E.S.E.

5-6

...M.

11 45

23 0

6 0

19 10

,,

6

3. A.M.

11 55

25 0

5 15

18 10

Ea^tcfly.

4

P.M.

\-2 0

23 2

G 30

19 5

S.Eastfilv.

5

4. A.M.

Noon.

23 2

6 30

19 5

„

6

P.M.

,,

6 0

18 9

Vanaltle.

3-7

1 T). A.M.

1-2 30

25 3

7 30

19 6

S.Eastcrlv.

7

P.M.

12 0

23 5

6 30

18 4

,,

6

6. A.M.

1 0

25 1

7 20

18 11

Southeily.

4

P.M.

i

1 15

23 3

i

Fu

U Moon at 3" 13\

Edphinution of fin- MS. Liai/ram accomjmniiinrj this Paper, referred to in pcuje 243.

This diagram was formed by laying down on ruled paper the observations of heiglit
recorded in the Tables of Observation.

The alternate high waters were then joined by lines, forming two curves, the distance
between whicli is double the Diurnal Tide at the time of High Watei'.

The same construction was made for the Low Waters. As tlie Diurnal Tide vanishes
witli the declination, the equinoctial Diurnal Tide is altogether Lunar; accordingly
the Lunar Diurnal Tide was found from the equinoctial portion of the diagram.

When found, it was superposed on the Solstitial poi'tion of the diagrams, — tlie points
of intersection being found by means of the interval from |U,^0 to D = 0. or age of tide
already obtained from tlie equinoctial observations.

The part of the diurnal curves remaining, after the superposition of the Liniar Tide,
was the Solar Diurnal Tide.

The Lunar and Solar Tides at Higli and Low ^^ ater luiving been tlius found, tlu'ir
remaining constants, viz. C'oefficients and Tidal Intervals, were worked out. as shown in
the paper itself.

A Curve, drawn bisecting the range of the Diurnal Tide at High Water, and anothei-
curve in like manner at Low Water, give the High and Low "^^"ater Curves of the
Semidiurnal Tides that I have used in the second part of my paper on the Arctic Tides.
This part of the subjt'ct is given in detail in the paper its(>li'.

[ 273 ]

XII. Hesults of the Magnetic Observations at the Kew Observatory, from 1857 a7id 1858

to 1862 inclusive. — No. I.
By Major-General Edward Sabixe, ^..i., President of the Royal Society.

Ecceived ilay 21,— Head June 18, 1863.

^ 1. A tabular synopsis of ninety-five of the principal disturbances of the inagnetic declina-
tion recorded by the Kew Photograms between January 1858 and December 1862
inclusive ; and a comparison of the Laws of the Disturbances derived therefrom, with
the Laws derived by the more usual method.
It seems difficult to understand how any one having the opportunity of examining the
daily photographic records of a magnetic observatory, and viewing them with an intelli-
gent eye, can fail to discern in the magnetic disturbances the systematic operation of
laws depending upon the solar hoiu's ; and to perceive that these laws are different from
those which govern the regular solar-diurnal variation (upon which the distui-bances,
whensoever occurring, are superposed).

There are, however, many persons who have not the opportunity of examining for
themselves these full and complete records, but who may, nevertheless, be desirous of
obtaining a clearer and more distmct understanding of the true character of these
remarkable phenomena, in the belief that such knowledge is indispensable as the first
step of an inductive inquu-y which may ultimately reveal to us their causes, and the
nature of the causation by which they are produced ; and also from the prominency
which is given to such an investigation in the Report of the Royal Society in 1840,
wherein it is asserted that " the progressive and periodical variations are so mixed up
with the casual and transitoiy changes, that it is impossible to separate them so as to
obtain a correct knowledge and analysis of the progressive and periodical variations, with-
out taking exjiress account of and eliminating the casual and transitory changes.'' The
elimination of the disturbances was thus early foreseen to be an essential preliminary
step in the systematic investigation of the periodical magnetic variations generally. The
whole course of subsequent research has manifested the sagacity and importance of this
early precept, and the necessity of placing this fundamental point of our investigations
on a secure basis. I have thought, therefore, that it might be desirable to place before
the Royal Society a sjTiopsis of the deflections from the normal positions of the declino-
meter, tabulated from the photograms of the Kew^ Obsen'atory, in a large portion of the
most notable distuibances which occurred between January 1858 and December 1862,
showing the direction and the amount of disturbance at twenty-four equidistant epochs
in each of the disturbed days — in the belief that those who may desii-e to do so will
MDCCCLXIII. 2 p

274 MA.TOE-GEXEEAL SABtN'E OX THE RESULTS OF THE

obtain, by a careful examination of such a tabular new, and of the appended com-
ments, a more distinct and definite perception of the character of the magnetic
disturbances than appears to be usually possessed.

In forming the Table which occupies pages 276 and 277, the principle of selection
adopted, and invariably adhered to, has been to take all those days in which twelve at
the least of the twenty-four equidistant epochs have been disturbed to an amount equal-
ling or exceeding 0-15 inch of the photographic scale, or o'-3 of arc, on either side of
the normal of the month and hour to which the recorded position corresponds, the
normal itself having been obtained by recomputation after the omission of all disturb-
ances amounting to o'-o. The figures in the Table are the difierences of the disturbed
positions from the normals as above defined. By the process thus described the solar-
diurnal and other minor variations are eliminated. There have been ninety-five such
days in the five years. The Summary at the close of Table I. shows the resulting
aggregate values, both of Easterly and of Westerly deflection, at each of the twenty-four
equidistant epochs in each of the five years, as well as in the whole period. The hours
of astronomical time at the Kew Observatory have been taken for the twenty-four equi-
distant epochs.

It is obvious, on the most cursory view of the Summary at the close of the Table
(page 277), that the Easterly and Westerly deflections are both subject to systematic laws,
and that these laws are distinct and cUssimilar in the two cases. Thus the easterly deflec-
tions prevail during the hours of the night, and the westerly dming the hours of the day.
In the day-hours the easterly are small, and vary but slightly ; they begin to increase
about 5 or G p.m., and augment progressively until 11 or 12 p.m., when they attain a
value (speaking always of aggregate values) nine or ten times as great as on the average
of the day-hours. This great development of easterly disturbance continues until one or
two hours after midnight, when it as steadily and progressively subsides until 5 or 6 a.m.
The westerly deflections, on tlie other hand, are distinguished not only by their great
prevalence at the hours when the easterly deflections are small, viz. 5 a.m. to 6 p.m.,
but also by havmg two distinct epochs of maximum about eight or nine hours apart,
viz. one about 6 or 7 a.m., and the other about 3 p.m. This last-named distinction
between the two classes of deflection, viz. a single maximum in the one, and a double
maximum in the other, is the more worthy of notice, because, as will be shown here-
after, a similar distinction prevails at the greater part of the stations wliere the laws of
the disturbances have been investigated, although, whilst in certain localities of the
globe it is, as at Kew, the easterly disturbances which have the single maximum, and
the westerly the double maximum, in other localities the converse is found to take
place. Tlie increased prevalence of each of the two classes of deflection for about half
the twenty-four hours, and diminished prevalence during the other half, appears also
to be a tisual characteristic, — but with the reservation, that the hours of the prevalence
of each class are not the same in diflerent localities, and tliat they vary indei^endently
of >fidi (jfJier — so mucli so that at some stations the two classes of disturbance, instead

MAGNETIC OBSEEYATIOXS AT THE KEW OBSERVATOET. 210

of affecting opposite parts of the twenty-four hours as at Kew, may even have their
greatest prevalence at the same hours.

If we now take the pains to compare tlie summaries of the easterly and westerly
deflections in each year with those of the means of the Jive years, the accordance is too
manifest to admit of a douht remaining as to the general and systematic chai-acter of the
laws which have been thus placed in endence. And if we fmther proceed to examine
seriatim the general progression of disturbance in each of the ninety-fi\e days, we shall
see reason to conclude that by far the greater part of the disturbances are in conformity
with these laws (which are of course more fully and clearly shown by the annual and
quinquemiial summaries) — thus manifesting the general prevalence of a common type in
the disturbing action, even when the days ai-e regarded individually.

In the greater part of the ninety-five days it is easy to trace the presence of both
the features which may be regarded as the leading characteristics of a disturbance :
viz., 1, a deflection (of very considerable amount at certain hours) from the mean or
normal position of the magnet ; and 2, rapid fluctuations on cither side of the
deflected position. AU days of disturbance arc marked by one or the other of these
two features, and frequently by both. The deflections from the normal are variable in
amount, but in direction they are generally conformable to the systematic laws which
have been already adverted to, and which will be more fully discussed in the sequel.
The fluctuations ai-e extremely ii-rcgular both in du-ection and amount, conveying the
impression that the magnet at such times is under the action of two opposing forces, of
which sometimes the one and sometimes the other preponderates. A tremulous motion
of the magnet is occasionally shown by the photographic traces unaccompanied by
changes of direction, as if both the opposing forces were at such times in a state of
agitation, but without more than a merely momentary preponderance of either, ^^^len
large and rapid fluctuations present themselves, we sometimes find considerable and
apparently in*egular differences in the successive tabulated directions of the magnet
(taken, as must be remembered, at the precise instants of the equidistant epochs) ; but
the more regular and systematic prevalence of easterly deflection at particular hours, and
of westerly deflection at other hours, usually oven-ides, even in the indi^idual cases as it
does altogether in the means, the partial influence of the fluctuations.

The excess of easterly over westerly, or of westerly over easterly deflection at the
several houi"s in the ninety-five days is a measure of the influence which the disturbances
would necessarily exercise on the "diurnal inequality" derived fi-om the hoiu'ly means
of the ninety-five days, if the elimination of the disturbances were unattended to : the
excess thus referred to constitutes, in fact, the disturhance-diurnal variation due to that
portion of the disturbances occurring in the five years which is included in the ninety-
five days contained in the Table. This part of the subject will be resumed in the third
section of this paper.

2 p2

276

MAJOE-GEXEEAL SABI>T: ON THE EESULTS OF THE

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MAGXETIC OBSERVATIONS AT THE KEAV OBSERVATORY.

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278 3IA.J0E-GEXEEAL SABEsE OX THE RESULTS OF THE

^ 2. Coiiqiarison of the Imvs of the disturbance-diurnal variation derived from the ninety-
five days of disturbance tahulated in the first section of this pajier, with the con-
clusions derived, at the same i)lace and for the same period, from the wider basis of
investigation supplied by the process first introduced and published hy myself eighteen
years ago.
The process here referred to consists, as is well known, in separating from the whole
body of observations employed, all, without exception and whensoever occurring, which
differ from their respective normals of the same month and hour by a certain value,
constant for the same element at the same station — the amount of this arbitrary
standard, or minimum value of a disturbance, being regulated by one condition only, viz.
that it shall not be so small as to endanger the inclusion amongst the separated observa-
tions of any in which the cause of the irregularity may with probability be ascribed to
anv other source than that of the class of phenomena whose laws we desire to study.
In the case of the homiy positions tabulated from the Kew Photograms from January
1858 to December 18G2, 0-15 inch of the photographic scale, or 3-3 minutes of arc
measured from the normal of the same month and hour after the omission of the
disturbed observations, has been taken as the standard or minimum value of a distm-b-
ance. There are altogether in the photograms of the five years at Kew the effective
records of 43,456 hourly positions; the number of failures in the photographic regis-
tration from all causes being only 368. Of these 43,456 recorded positions, 5941, being
about 1 in 7 of the whole body, differed by an amount equalling or exceeding 3'-3 fi-om
their respective normals. The aggregate value of the differences of the disturbed posi-
tions, measured fi-om the normals, was 36,580-8 minutes of arc, of which 19, 748'-7 were
easterly, and 16,832'-1 were westerly deflections.

Table II. exliibits the aggregate values of the disturbances distributed into easterly
and westerly deflections, and into the several hours of their occurrence. The easterly
deflections derived from tlie ninety-iive days are in column 2, and those derived from
the 5941 disturbed positions {i. e. from all disturbances equalling or exceeding 3'-3)
in column 3 ; the westerly deflections derived from the ninety-five days occupy column 4,
and those obtained from the 5941 distm-bed positions column 5. The Satios which the
aggregate values of easterly and westerly deflection at the different hours bear to their
respective mean hourly values are shown in the same Table (II.), the easterly in columns
6 and 7 ; the westerly in columns 8 and 9. By comparing the values in columns 6
and 7 mth each other, it will be seen that the Ratios of the easterly deflections exhibit
approximately the same law, whether obtained from the ninety-five days, or fi-om all
disturbances equalling or exceeding 3'-3 ; and by comparing the ratios in columns 8 and
9, it will be seen, in like manner, tliat there is a similar general accordance in the
ratios of the westerly deflections, whether obtained fi-om the ninety-five days, or from
the more extensive induction : the latvs, when examined by tlie ratios, arc s(>en to be
approximately tlie same when derived by either process, although the aggregate values
are very dissimilar — bemg more than tliree times as great when the method of inves-
tigation is such as to comprehend all disturbances equalling or exceeding 3'- 3, as when
it is limited to the disturbances in ninety-five days of |)rinci[)al note.

MAGNETIC OBSERVATIONS AT THE Ki:\V OBSERVATOIJY.

279

Table II.

Kcw
Astrono-
mical

-iggrcgat* Values.

Eat

i09.

Kew
Astrono-
mical

Easterly Deflections.

Westerly Deflections.

Easterly Deflections.

Westerly Deflections.

Hours.

From
'.);') Days.

From

all Distiu-b-

aiioes.

From
9.') Days.

Prom

all DisUirb-

auces.

From
95 Days.

From
all Disturb-
ances.

From
95 Days.

From
all Disturb-
ances.

Hours.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

0

59

247

276

1074

0-25

0-30

1-15

1-53

0

1

48

269

311

1157

0-21

0-33

1-57

1-G5

1

2

50

308

379

1232

0-21

0-37

1-56

1-76

2

3

53

245

454

1175

0-23

0-30

1-88

1-G7

3

4

9<>

371

443

1039

0-41

0-45

1-86

1-48

4

5

105

478

254

718

0-46

0-58

1-07

1-02

5

6

140

647

216

506

0-60

0-79

0-89

0-72

6

7

249

999

108

314

1-07

1-21

0-44

0-45

7

8

315

1322

78

233

1-35

1-61

0-33

0-33

8

9

420

1562

110

257

1-81

1-90

0-45

0-37

9

10

524

1983

55

160

2-26

2-41

0-23

0-23

10

11

576

1978

65

156

2-47

2-40

0-27

0-22

11

12

572

1946

95

301

2-46

2-36

0-40

0-43

12

13

553

1706

86

371

2-38

2-07

0-35

0-53

13

14

576

1558

90

441

2-48

1-89

. 0-37

0-63

14

15

396

1245

57

376

1-71

1-51

0-24

0-54

15

IG

262

862

164

488

1-12

1-05

0-68

0-70

16

17

136

419

340

839

0-58

0-51

1-40

1-20

17

18

65

243

464

1058

0-28

0-30

1-91

1-51

18

19

95

262

473

1175

0-41

0-32

1-95

1-67

19

20

71

289

406

1039

0-31

0-35

1-67

1-48

20

21

84

274

390

981

0-36

0-33

1-Gl

1-40

21

22

84

287

285

838

0-36

0-35

M8

1-19

22

23

63

250

216

910

0-27

0-30

0-89

1-30

23

Slims...

5592

19750

5815

16838

r Mean
J. hourly
Rvalues.

oT? — i-nn aoi — i.fin

243=1-00 'TfiiTTzi.nn

Means..

233

823

243

701

AOO 1 uu

1

For the convenience of those who prefer graphical to tabular representation, the
diurnal course of the easterly deflections, corresponding to the Katios in columns 6
and 7 of Table II., is exhibited in Plate XIII. figure 1, where the broken line sliows
the diui-nal march indicated by the ratios obtained from the ninety-five days, and tlie
unbroken line the diurnal march obtained from all the disturbances equalling or exceed-
ing 3'- 3. Figure 2 is a similar representation of the diurnal march of westerly disturb-
ance-deflection, obtained, as shown by the broken line, from the ninety-five days, and
by the unbroken line from the more comprehensive investigation. The general aspect
of the two figures seems to establish in the most conclusive manner —

1. That the disturbances have systematic laws:

2. That the easterly and westerly deflections liave each their own systematic laws,
distinct and different from each other :

3. That these laws are approximately the same, whetlier derived from the more
limited or from the more comprehensive basis, although in the latter case tlie aggre-
gate values of disturbance are more than three times as great as wlien the disturbances
of the ninety-five days only ai-e taken into account.

>so

MAJOE-GEXEEAL SABIXE ON THE EESULTS OF THE

Hence it follows that by taking only the most notable days of disturbance in five years
(averaging nineteen in each year), we may gain an approximately correct view of the
character of the disturbance-diurnal variation ; but if we desire not only to learn its
character, but also to elimiitate its injlnencc. in compliance with the prescribed condition
of '• eliminatmg the casual and transitory changes as a first and essential step towards a
correct knowledge of the more regular periodical variations," then we see that the mere
omission of those ninetij-five days is altogether inadequate for the desired object, as it
would scarcely eliminate a third part of the systematically distui'bing element, sho^A^l to
admit of elimination by a more suitable process.

§ 3. Disturhance-diurnal Variation.
Table III. exhibits the excess of easterly over westerly, or of westerly over easterly
deriection at twenty-four equidistant epochs of the solar day, derived, in column 2, from
the disturbances in the ninety-five days, and in column 3, from all disturbances equal-
ling or exceeding 3''3 from their respective normals. These columns consequently show
the distm'bance-diurnal variation corresponding to the more complete, and to the less
complete, process of elimination. The character of the progression is seen to be sub-
stantially the same in both cases, but the amount of disturbance is between three and
four times as great in column 3 as in column 2.

T.iBLE III. — Disturbance-diurnal Variation ; or Excess of Easterly over "Westerly, or of
Westerly over Easterly Deflection, at twenty-four equidistant epochs in the twenty-
foiu- hours.

Derived from the ninetv-five

Derived from all disturb-

Kew

Astronomicai Ilours.

days of most notable dis-
tiu'bance.

ances equalling or exceeding

3'-y.

Kew

Astronomical Ilours.

a)

(-')

(3)

(4)

0

217 w.

826 w.

0

1

263 w.

889 w.

1

2

329 w.

924 w.

2

3

401 w.

930 w.

3

4

347 «'.

607 w.

4

5

149 w.

240 w.

5

6

76 w.

141 E.

6

7

141 E.

686 E.

7

8

237 E.

1089 E.

8

9

310 E.

1305 E.

9

10

469 E.

1823 E.

10

11

.ill E.

1822 E.

11

12

477 E.

1644 E.

12

13

467 E.

l.-i36E.

13

14

486 E.

1117 E.

14

15

340 E.

869 K.

15

16

97 E.

374 E.

16

17

204 w.

420 w.

17

18

399 w.

815 w.

18

-19

378 w.

913 w.

19

20

334 w.

750 w.

20

21

206 w.

706 w.

21

22

201 vv.

551 vv.

22

23

153 w.

660 w.

23

.AIAOXETIC OBSERVATIONS AT THE KEW OBSERVATORY. 281

For those who prefer fjrapliical representation, tlie cnrvcd line in Plate XIII. fig. 3
exhibits the excess of easterly over westerly, or of westerly over easterly deflection, i. e. the
disturbance-diurnal variation, obtained from tlie 5941 disturbance's equalling or exceed-
ing 3'-3 fi-om the respective normals, as shown in column 3 of Table III. The straight
horizontal line in figure 3 represents the mean or normal position of the magnet at the
several hours, after the omission of tlie disturbances. It is figured for convenience as a
straight line, though in reality it is itself a curve following the progression of the solar-
diurnal variation. The lengths of the ordinates which are above the normal line indi-
cate the excess of the easterly over the westerly deflections at the hours when the
easterly preponderate, and those which are below the normal line the excess of the
westerly over the easterly at the hours when the westerly deflections predominate.

The easterly portion of the disturbance-diumal variation is seen to be continuous for
about ten hours, or from about 6 p.ir. to 4 a.m. The westerly portion is also continuous,
extending o^■er the remaining fourteen hom-s, or from about 4 a.m. to 6 p.m. The
easterly has a single maximum occurring about midway between its commencement and
its termination. The westerly is more complex, having two maxima separated by an
interval of about 8 or 9 hours. But whilst the westerly excess extends over more hours
than the easterly, the areas of the two portions have neai'ly the same dimensions ; or,
in other words, the sums of the hourly deflections in opposite du'ections are at Kew
nearly equal, and any small difference between them is not a persistent one, the easterly
exceeding in some years and in others the westerly. The equality or otherwise of the
sum of the deflections in opposite directions is apparently a point of some theoretical
significance, as will be further noticed when the analogous phenomena in other localities
come to be discussed.

As we find the same general forms of the two portions of the disturbance-diurnal
variation, which have been thus derived from the Kew photograms, reproduced in other
localities in tlie separated portions of the easterly and westerly deflections (\\'ith only
such slight variations as may well be supposed to be due to accidental or subordinate
causes), it may be desirable to examine somewhat more closely what may be Adewed as
the characteristic differences of the deflections in the two directions. The easterly
deflection is represented, as we have already seen, in Plate XIII. fig. 1: it is distinguished
by its approximately conical form and single maximum, and by the small and nearly
equable amount of variation during the ten or eleven hours when the ratios are least-
Its general form thus bears a striking resemblance to the diunial curve of the solar-
diumal variation (as obtained after the careful separation and omission of the casual and
transitory changes) ; but the two phenomena differ from each other in the important
circumstance, that in the solar-diurnal variation the solar hours corresponding to its
different featui'es are the same in all meridians in the extra-tropical parts of the same
hemisphere, whilst in the portion of the ilisturbance-diurnal variation which is now
under notice, the solar hours corresponding to its different features vary, apparently
without limit, in different meridians. This is a distinction which may well be supposed

MDCCCLXIII. 2 Q

282 MAJOE-GEXERAL SABINE OX THE EESULTS OF THE

to indicate a difference in the mode of causation, altliough it would not justify an
inference that the sun may not be the originating cause in botli cases.

The westerly deflections at Kew, represented in Plate XIII. fig. 2, have a decided
double maximum, with an intervening interval of about eight or nine hours. The
analogous form in other localities has the double maximum sometimes more and some-
times less decidedly marked. The interval intervening between the maxima is usv;ally
of about the same duration at stations in the northern hemisphere ; at some stations in
the southern hemisphere it is apparently somewhat longer.

The conical form and single maximum which characterize the easterly deflections at
Kew belong also to the easterly deflections in all localities in North America where the
laws of the disturbances have been investigated. But when we view the phenomena at
Nertschinsk and Pekin, which are the only two localities in Northern Asia for which
the investigation has yet been made, we find, on the contrary, that the conical form and
single maximum characterize the wester! y deflections, whilst the easterly have the double
maximum. Further, we find that at the two Asiatic stations the aggregate values of
the westerly deflections decidedly predominate, whilst in America the easterly deflections
are no less decidedly predominant ; and at Kew, which we may regard as an interme-
diate locality, the amount of deflection in the two dkections may be said to be balanced,
there being in some years a slight preponderance of westerly, and in other years of
easterly deflection.

There is another circumstance which seems to connect, in what may prove even a
more mstructive relation, the westerly deflections in Northern Asia with the easterly in
other parts of the northern hemisphere. I refer here to an approximate accordance in
ahaolate time which appears in the most marked features of the diurnal cvu've at the
widely separated localities of Pekin, Nertschinsk, Kew, and Toronto, at each and all of
which the curves as they are presented in Plate XIII. figs. 1, 4, 5, and 6 are the mean
result of several years of hourly observation*. These localities appear to be particularly
well suited for a comparison of this nature, being not very dissimilar in geographical
latitude, whilst they include a difterence in longitude of no less than 105"\ If we select
the epoch of the maximum deflection (or the apex of the curve) as the most marked
feature, the comparison would stand nearly as follows; commencing with the most
easterly, and proceeding in succession from east to west : —

* The figures 1, 4, 5, and 6 in Plate XIII., representing respectively the Easterlj' deflections at Kew and
Toronto and the Westcrl}- at Nertschinsk and Pekin, are delineated from the following formulae, in which o,
expressed in degrees 1.3 to the hour, is reckoned from the mean noon at the station: —

Kew . . . ]+0-yS (sinrt + 280 22)— 0-417 (sin 2« + 28(; 29):

Toronto . . 1 +1-05 (sin a+285 58)— 0-332 (sin 2a + 334 07) :

Nertschinsk . 1 —0-94 (sin o + 309 02)— 0-238 (sin 2a+ 13 11):

Pekin . . . l-o-TG (sina + 289 12)-0-200 (sin 2«+ 142).

Assuming that the formulas represent correctly the ratios at the several hours, the observed values are in very

tolerable accord with them ; at Kew and Nertschinsk they are the most so ; at Kew the probable error of a .single

hourly ratio is +0-0.jf;, at Nertschinsk +0-002.

MAGNETIC OJJSEin'ATlON.s AT THE KEW OB.SEKVATOin'.

283

Deflections. Localities.

Latitudes.

Longitudes.

Approximate

Local solar Absolute Hour
Hour. at Kew.

1

39 54 N.
51 19 N.
51 29 N.
43 40 N.

116 6 E.=7-8

114 9 E.=7-7

0 =0-0

79 0 w.= 5-3

22 14
21 i:i
11 11
10 15

^^''^"'^y l;Nc.>t.cl,insk

y, ^ , f Kew

^''^'"'y (Toronto

It must be remembered that the time of the occurrence of the apex (or maximum of
deflection) scaixely admits of very precise determination ; and further, that assuming
for the disturbing impulse a common origin at any other point of the terrestrial surface
than at the geographical pole, and an equable but appreciable velocity of propagation,
the difference of the geographical meridians would not be the sole consideration in
deducing the absolute epoch from the local hours at different stations.

Could we thus identify the westerly deflections in Asia with the easterly in Europe
and Ameiica, we should have a confirmation on a very extended scale of M. Gau.ss's con-
clusion derived from the comparison of synchronous disturbances at stations remote
from each other, viz. that " the synchronous disturbances of the same element not only
differ widely in amount, but occasionally appear to be even reversed in du'cction."

It may be that this may prove the first step in the inductive inquiry which may lead
eventually to a complete understanding of the systematic distinction which we find in
comparing the solar-diurnal with the disturbance-diurnal variations, — by referring the
first to causes which, within the sphere of theu- operation, produce the same phenomena
at the same solar hours ; and the second to effects originating (as far as the terrestrial sur-
face is concerned) in special localities from whence they are propagated, and admitting
of classification by means of the absolute hours to which they approximately correspond.
For a conclusion of such moment, however, much preliminary investigation is still
required, for which materials either do not yet exist, or have not yet been submitted to
the necessary processes of examination. It seems especially important that the laws of
the disturbances, and of their respective easterly and westerly deflections, should be
known at a station or stations intermediate between Nertschinsk and Kew.

The propriety of making the easterly and the westerly deflections the subjects of
distinct investigation will be still more apparent by reverting to Plate XIII. fig. 3, and
remembering that the areas containing respectively the ordinates above and below the
normal line are subject at diflerent stations to horizontal displacements, each indepen-
dent of the other ; and thus that at some stations the opposite deflections may have a
tendency to mask each other's influence in the resultant mean deflection {i. e. in the
excess of easterly over westerly, or of westerly over easterly deflection). It happens at
Kew that the large disturbances in opposite directions take place at opposite hours of
the twenty-four, and that they thus record themselves in gi-eat measure independently
of each other ; but experience has already sho^vn that there are stations where large
distiu-bances show themselves in both dii-ections, on different days, at the same hours ;
and such deflections would of coui'se tend to neutralize each other in the resultant
mean, thus masking the operation of the general causes whose laws we desire to learn.
This inconvenience is in gi-eat measure remedied by the method of analysis which has
been adopted, wherel)y the deflections are exhibited separately as well as in their com-
bination.

2q2

284

MAJOR-GEXEEAL SABIKE ON THE EESULTS OF THE

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MAGNETIC OBSERVATIONS AT TILE KEW OBSEKVATOEY. 285

§ 4. Munial Inegiiality and Solar-diurnal Variation. Tables IV. and V. p. 284.

The dim-nal inequality (which is by no means identical with the solar-diurnal Muiation
as has been sometimes assumed) has in fact two principal constituents, viz. the solar-
diurnal variation itself, and the disturbance-tUui'nal variation. It is obtained for each
month by taking the differences between the mean positions of the magnet at each of
the twenty-four hours and the mean position in the month (the latter being the mean of
all the days and aU the houi-s in the month). It is the first step in the process of
obtaining in a separate form the several periodical variations from the combination in
which they appear in the photographic records, and includes all the positions tabulated
from the records, without the exception of any. Table IV. (page 284) exhibits the
diurnal inequality in each month, on the average of the five years, from January 1858 U>
December 18G2 inclusive.

The solar-diurnal variation, shown in Table V. (page 284), is obtained by a similar
process from the hourly positions in the same period, exclusive of those which differed
3'' 3 or more fi-om theii" respective normals of the same month and hoiu- — the normals
being the hoiu-ly means m each month after the exclusion of all the disturbed positions.
By this process the effects of the "casual and transitory changes" become in a very
great degree " eliminated ; " and we obtain a measure of the solar-diurnal variation which
is only very slightly affected by the small portion of the disturbance-diurnal variation
which remains after the separation and omission of the disturbances equalling oi-
exceeding 3'* 3 from theii* respective normals.

Plate XIV. exhibits the solar-diurnal variation at Kew (fig. 2), in comparison with the
same at Toronto (fig. 1), Xertschinsk (fig. 3), Peldn (fig. 4), St. Helena (fig. 5), Cape of
Good Hope (fig. 6), and Hobartou (fig. 7). Figs. 1, 2, 3, & 4 show the march of the
solar-diui-nal variation at stations in the middle latitudes of the northern hemisphere,
fig. 5 in the equatorial region, and figs. 6 »& 7 in the middle latitudes of the southern
hemisphere.

In figs. 3 »& 4, compared with 1 & 2, we see the gradual flattening of the curve as
the magnetic equatorial region is approached. In fig. 5 (geographical latitude of
St. Helena, 15° 55' S.) we perceive the incipient reversal of the diurnal march; whilst
in figs. 6 & 7, and particularly in fig. 7 (Hobarton, where the dip is more than 70°,
and the total force 13-6 in British units), we see the reversal completed, and the full
development of the characteristic featiu'es appertaining to the southern hemisphere.

It is seen in the Plate that at the stations in the northern hemisphere yenerally, the
north end of the magnet passes rapidly from its extreme eastern limit (about 8 a.m., or
nearer 9 a.m. at Pekin) to its extreme western Kmit (about 1 p.m., or rather later at
Pekin), the motion being more rapid between 10 and 11 a.m. than at any other hour of
the twenty-four- ; and that during the remaining nineteen hours the north end retiu-ns
to its eastern limit by a progression tolerably rapid from about 2 to 7 p.m., scarcely
sensible from 7 p.m. to 3 or 4 a.m., and again more rapid until 8 a.m. The turning hours

286 MAJOE-GEXEEAL SABIXE OX THE EESULTS OF THE

are approximately the same at the fom- stations, having apparently no relation whatsoever
to the varied circumstances of sea or land in the vicinity. Taking Hobarton as the best
representative which we possess of the southern hemisphere, we see in the Plate the
analogy of its phenomena to those of the northern stations. We have the same rapid
movement from one extreme to the other, occupving the same portion of time, viz. live
hours, and the return occupying the remaining nineteen hours ; but the directions of the
two movements are inverted, the 5-hour movement bemg in the southern hemisphere
from West to East, and the slower, or 19-hour return, being from East to West. The
epochs are nearly but not quite the same, being apparently about an hour later in the
southern hemisphere.

In the curve of the Cape of Good Hope (fig. 6) we have the same general features as
at Hobarton, but with a more flattened curve, indicating- a nearer proximity to the equa-
torial region.

If, now, we permit ourselves to depart from the general custom of expressing the
variations of the Declination in the northern as well as in the southern hemisphere by
the directions of the nort/i end of the magnet, and to speak of the solar-diurnal varia-
tion in the southern portion of the magnetic sphere as a movement of the south end of
the magnet (using the same phraseology as before for the northern hemisphere), we
appear to gain a greater simplicity in describing the general characteristics of the
phenomena in the two hemispheres. In such case the Hobarton curve is reversed, —
the westerly deflections of the north end becoming easterly deflections of the south end,
and vice versa, — the inflections of the curve of the annual solar-diurnal variation at
Hobarton then appear altogether as tlie counterparts of those at Kew and Toronto
(excepting in the one feature peculiar to the southern hemisphere, of the turning hours
being about an hour later than in the northern hemisphere). This correspondence is
shown in plate 1 of the first volume of the Hobarton Observations, published in 1850,
and in fig. 7 of Plate XIII. accompanying this paper. In like manner the annual curve
at the Cape of Good Hope, when reversed, becomes the counterpart of those at Pekin
and Ncrtschuisk ; whilst at St. Helena, so near the dividing Ime between tlie hemi-
spheres, the annual solar-diurnal variation has almost entirely disappeared, the small
remaining inflections, seen in Plate XIV. fig. 5, being due, for the most part at least, to
the semiannual ineqiialitij, which is the subject of the next section {^ 5).

§ 5. Semiannual Inerj^ualifi/ of the Solar-diurnal Variation.
The solar-diurnal variation exhibited in Table V. is seen by the semiannual means,
April to September, and October to ISIarch, to be subject to a systematic dificrencc
in the two halves of the year, coinciding, or nearly so, with the sun's position on ()])posite
sides of the equator. In Plate XV. fig. 1, the curve corresponding to the mean solar-
diurnal variation at Toronto, from April to September, is represented by the black
line, and the cun'e corresponding to October to March by the red line. The systematic
character of tliis half-yearly variation is shown by the corresponding curves similarly

MAGNETIC OBSERVATIONS AT THE KEW OBSERVATORY. 287

represeuted by black and by red lines for Kew (fig. 2), Nertschinsk (fig. 3), Pekin
(fig. 4), St. Helena (fig. 5), Cape of Good Hope (fig. 6), and Hobarton (fig. 7). The scale
is the same in all the figures. It will be seen that at all these stations the cur\e from
April to September is on the upper or East side of the October to March curve, from
about midnight, or a little later, to about 9 or 10 a.m. in the northern hemisphere, and
about 10 or 11 a.m. in the southern herhisphere; and on the lower or West side of the
October to March curve during the remainder of the twenty-four hours. On successively
considering the figures in Plates XIV. and XV., we perceive that tlie annual curves progres-
sively lessen as the equatorial region is approached, reappearing in a reversed direction in
the southern hemisphere, and gradually increasing in magnitude so as to have at Hobar-
ton, in the middle latitudes of the southern hemisphere, nearly the same magnitude as at
Toronto and Kew in the northern hemisphere ; but that through all tliese changes both
of magnitude and du-ection in the annual curves, the semiannual variation (or the differ-
ence between the two semiannual curves in each case) remains persistent throughout;
the same in direction at the same hours, and the amount approximately the same in
all parts of the globe. Thus in the equatorial region, where the annual inflection almost
or entirely disappears, the semiannual portion still subsists, and presents in each of
the half years, separately viewed, the phenomenon of a solar-diurnal variation. This
is approximately exemplified at St. Helena, which, however, is a little on the southern
side of the magnetically dividing line between the hemispheres. As the southern mag-
netic latitude increases, the annual solar-diurnal variation, as shown in Plate XIV., pro-
gresively increases in magnitude, but in a reversed direction from those of the analogous
phenomena in the north, as has aU-eady been noticed. Thus it will be seen that the two
portions, viz. the annual and the semiannual, both of which we recognize to be due to
the sun's action, inasmuch as they foUow the order of the solar hours, e\ince apparently
a dissimilai'ity in the mode of operation of the producing cause ; in the one class of
efiects, ^^z. in the annual, the north end of the magnet is deflected in opposite directions
in the two hemispheres, the deflection disappearing altogether at the magnetic equator ;
whilst in the other class, \iz. the semiannual difference, no such mversion takes place,
and the deflections are approximately the same in amount and dii-ection in the equa-
torial as in all other parts of the terrestrial surface*.

* The general custom of si)eakiug always of the north end of the Decimation magnet is here followed : if
this were modified as suggested in page 286, the reasoning upon the characteristic distinction between tlic two
phenomena would, it is obvious, remain essentially the same.

288

:\rA.JOE-GEXEEAL SABIXE OX THE EESULTS OF TIIE

^ G. Lunar-diurnal Variation.
Table VI. contains the lunar-diurnal variation on the mean of each year, from 1858 to
1862 inclusive, and a "eneral average taken for the five years.

Table VI. — Lunar-diurnal Variation in Seconds of Arc.

Lunar
Hom-s.

Years endiu

« December

31,

Means.

Lunar
Hours.

_^

S5S.

1

"^.'i',).

1800.

ISUl.

180:2,

0

w.

6-0

E.

0-6

w.

12-6

w.

5-4

w. 7-8

w.

6-2

0

1

^v.

14-4

W

7-2

w.

12'6

w.

6-0

w. 6-6

vv.

9-6

1

2

w.

10-8

w

9-6

w.

5-4

w.

7-2

w. 9-0

vv.

8-4

2

3

w.

7-8

w

4-2

w.

3-0

E.

2-4

E. 2-4

w.

2-0

3

4

\v.

30

w

4-2

w.

2-4

E.

4-2

E. 2-4

w.

0-6

4

5

E.

h-i

w

6-6

E.

5-4

E.

6-6

E. 9-0

E.

4-0

5

6

E.

12-0

E.

1-2

E.

3-0

E.

14-4

E. 14-4

E.

9-0

6

7

E.

9-0

E.

4-2

E.

9-6

E.

16-2

E. 17-4

E.

11-3

7

8

E.

9-6

E.

8-4

E.

7-8

E.

6-6

E. 14-4

E.

9-6

8

9

E.

7-2

E.

6-6

W.

0-9

W.

1-2

E. 12-0

E.

4-7

9

10

E.

3-0

E.

7-2

w.

1-8

W.

6-6

w. 2-4

VV.

0-1

10

11

W.

3-6

W

1-2

w.

4-2

w.

10-8

w. 7-8

w.

5-5

11

12

W.

4-8

\v

9-0

w.

18-0

w.

8-4

w. 7-8

w.

9-6

12

13

w.

3-0

w

13-2

w.

15-0

w.

13-2

w. 12-0

vv.

11-3

13

14

w.

3-0

w

8-4

vv.

9-6

w.

10-8

w. 15*6

w.

9-5

14

15

w.

7-2

w

3-6

E.

4-2

w.

8-4

w. 12-0

w.

5-4

15

16

E.

3-0

E.

3-6

E.

7-8

vv.

6-6

w. 10-8

vv.

0-6

16

17

E.

7-8

E.

9-6

E.

13-8

w.

2-4

w. 4-2

E.

5-1

17

18

E.

7-8

E.

14-4

E.

17-4

E.

3-0

0-0

E.

8-5

18

19

E.

4-8

E.

18-0

E.

15-0

E.

9-0

E. 2-4

E.

9-8

19

20

E.

3-0

E.

12-6

E.

6-0

E.

10-2

E. 12-0

E.

8-8

20

21

W.

2-4

E.

18-6

E.

2-4

E.

10-8

E. 7-8

E.

7-4

21

22

W.

7-8

i:.

9-6

\V.

3-0

E.

7-8

E. 5-4

E.

2-4

22

23

W.

6-0

E.

5-4

W.

3-6

E.

0-6

w. 4-2

W.

1-6

23

We see in this Table a form of diurnal Aariation systematically and essentially diiferent
from that of the solar-diurnal variation. This characteristic form, which is sho\ATi alike
bv each of the three magnetic elements in all parts of the globe for which the investigation
lias l)een made, consists in a double fluctuation taking place in every twenty-four hours,
with two extreme deflections in each direction, — the zero-line, or line in which the moon's
action produces no deflection, being passed through four times at nearly equal intervals
of .six lunar houi's. At Kew the extreme westerly deflections occur at 1 and 13 hours, and
the extreme easterly at 7 and 19 hours. The extremes at 7 and 13 hours appear to be
.somewhat larger than those at 1 and 19 hours (the two greater elongations being each
ll'^*3, and the lesser 9"-8 and 9"-6, on the average of the five years). This diflerence
may have an important theoretical bearing if confirmed by the results in future years,
and in other parts of tlic globe.

In considering the lunar-diurnal variation of the three elements in diflerent parts of
tlie globe, the division of the lunar day into four alternate and nearly equal deflections
in opposite directions appears, as already stated, to be a general feature ; but the
amount of deflection (speaking of the declination) appears to diminish as the equator is

:magketic obseevations at the KEW OBSEEYATOET. 289

approached, reincreasing in the southern hemisphere, and attaining at Hobarton nearly
the same vahic as at Kew. The hours of extreme deflection are not the same at all
stations: the north end of the magnet has its extreme westerly deflections at Kew (in
the northern hemisphere), and its extreme easterly deflections at Hobarton (in the
southern hemisphere) at the same hours, and vice versd ; there is a similar correspondence
of hours in the opposite deflections of the same end of the magnet at Pckin in the
northern, and at the Cape of Good Hope in the southern hemisphere ; but the hours
at Kew and Hobarton are different from those at Pekin and the Cape of Good Hope :
however, results at more stations must be obtained before we can draw any certain
inferences as to the systematic character and theoretical bearing of such diflercnces.
There are six stations where the lunar-diumal variation of the declination has been
computed by myself, viz. Toronto, Kew, Pekin, St. Helena, the Cape of Good Hope, and
Hobarton : the results at these stations are published in the second volume of the
Magnetical and Meteorological Observations at St. Helena, pp. cxlvi- cxl\iii.

JIDCCCLXIII. 2 E

290 MAJOE-GEXEEAL SABIXE ON THE EESULTS OF THE

Besidfs of the Magnetic Observations at the Kew Ohsen-atory,froin 1857 to 1862 inclusive.

No. II.

Eeceived June IS, — Ketid June IS, 1S63.

§ 7. Secular Change and Annual Variation of the Declination.
It is desirable to axb'ert brictly to the process by which these results are elaborated
from the photographic records. The twenty-foiu- eqiiicUstant hoiuly positions ha^ing
been tabulated from the photograms, are written in monthly tables, having the days of
the month arranged vertically, and the twenty-four hourly positions in each day hori-
zontally. The hourly positions in each vertical line are then examined, and those in
which the difference fi'om the iiormal of the same horn- equals or exceeds O'lo in. in
the photographic scale, or 3'-3 in arc, are marked as distiu-bed positions, and are put
aside for separate consideration. This process is repeated until the fnal normals are
the means of the positions in each vertical line after the omission of all those which
diflPer from them by an amount equal to 3''3 or upwards. A mean is then taken of the
positions which remam in each horizontal line after the exclusion of the disturbed posi-
tions, omittmg only days on which the disturbed hours equalled or exceeded six in number,
or one-foiu"th of the whole number of the tabiUated positions. The means thus obtained
are considered to show the mean declination at the observatory for each day. The
daily values are then collected in weekly groiips, of which there are consequently fifty-
two in each year, and mean weekli/ values are taken, such as are exhibited in columns 2
to 6 of Table VII. (page 292), for the five years commencing in January 1858 and end-
ing in December 18G2. The mean of the weekly values in each year corresponds to
the mean dechnation on the 1st of July of that year ; and these mean values are placed
at the foot of each annual column in Table VII., whilst the means of the values in the
several horizontal lines, seen in column 7, show the weekly values in a mean or typical
year, derived fi-om the hourly positions in the five years, and corresponding chrono-
logically, in the case of Table VII., to the successive weeks in the year 1800. The
mean declination of the whole Table, corresponding to July 1, 1860, is seen at the foot
of column 7 ; it is 21° 39' 18"-1 W., and is based upon 260 weekly values, or upon
6240 hoiuly positions (diminished by the positions omitted, as above stated, on account
of disturbance). The differences from tliis mean value seen in the several weekly means
in the typical year (column 7) are ascribable (partly, of course, to casual errors, but)
chiefly, as will be seen, to the effects of systematic variations. The presence of one of
these, known commonly by the name oi' secular change (inasmuch as its period is of long
and yet undetermuied duration), is conspicuous, and its mean amount during the five
years embraced by Table VII. becom(>s knovATi by comparing with each oth<>r the mean
declination in each successive year, placed at the foot of the respective colunuis. Here
we find that

MAGNETIC OBSEEYATIOXS AT THE KEW OBSEEVATOKT. 291

From July 1858 to July 1859, the West Decimation decreased 6 45-9

1859 „ ' 1860, „ „ „ 7 31-0

1860 „ 1861, „ „ „ 8 15-7

1861 „ 1862, „ „ „ 8 03-6
whence we have 7' 39" as the mean annual amount of decrease in tlie West Declination
at Kew in the five years, corresponding, (as a precise deduction,) to July 1, 18G0, the
middle epoch of the mean or tj^pical year.

It is obnous that if we apply a proportional part of this secular change to the several
weekly values in the mean or t)-pical year, we obtain fifty-two corrected values of the
declination, each of which, if there were no other systematic variation than that of the
secular change, should agree with 21° 39' 18"-1; or should show only such small and
unsystematic differences as might reasonably be ascribed to casual errors. The character
of the differences actually pi-esented sufficed to show that something more was involved,
not explicable by the small variation in the rate of secular change itself which appeared
to be pointed out by the Table. Small, however, as was this last-named variation, it
seemed proper that it should be taken into account before we should be prepared to
take a final new of the results.

It is well kno^A^l that a few years ago the secular change in London was a small annual
increase of west declination, and that from causes yet but imperfectly understood, this
increase first diminished and then ceased, giving place to a change in the opposite direc-
tion, at fii-st slow, but becoming progressively more rapid ; so that at present the rate
of decrease is very nearly if not quite equal to the rate of increase which existed at the
time first spoken of. Thus the secular change at Kew (which we may regard as the
same as at London) appears to have been somewhat less in 1858 and 1859 than in 1861
and 1862, and therefore, inferentially, less in the earlier than in the later portions of
each year ; so that we may possibly obtain more exact values of the corrections to be
applied for secular change in the different parts of the mean or typical year by substi-

7' 39"
tuting for a mean value of — ^2- = 8"'83, weekly corrections commencing with 8"-5 and

progressively increasing to 9""1. These corrections are shown in column 8, and pro-
duce the corrected values in column 9. The differences of the values in column 9, from
21° 39' 18"-1, have been placed in column 10, to which I desire to direct attention. The
mere aspect of the -j- and — signs in this column appears to point to a semiannual
inequality coinciding very nearly with the sun's position in respect to the equator. If
we arrange the differences in two categories, one includmg the twenty-six weeks from
March 26 to September 23, and the other the twenty-six weeks from September 24 to
March 25 (which is the di\-ision of weeks most nearly according with the equinoxes),
the almost constant prevalence of the — sign in the first, and of the -\- sign in the second
categoiy, indicates vrith a very high degree of probability an annual variation, whereby
the north end of the magnet points more towards the east when the sun is north, and
towards the west when the sun is south of the equator ; and we obtain in the first

2 e2

292

3IAJ0K-GEXEEAL SABIXE OX THE RESULTS OF THE

categoiy (correspouding to the interval between March 2G and September 23) an average
weekly diminution of 2S"-95 ofWest Declination, and in the second category (coiTe-
sponding to the interval between September 24 and March 25) an average weekly
augmentation of 29"-90 of West Declination, — making together an annual variation
amounting to 5S"-S5.

Table VII. — "Weekly Means of West Declination at Kew, from January 1, 1858 to

December 31, 1862.

Mean weekly

Diiferences

Correc-

values

of the several

p v/1

1508.

1809.

1S60.

1861.

1SG2.

Means.

tions for

corrected

weekly cor-

en

secular

for secular

rected values

change.

change.

from

21"+

■2r+

21°+

21°+

■2V+

21"+

21°+

21° 39' 18"-1.

(1)

(2)

(3)

W

(5)

(fi)

(7)

(8)

(9)

(10)

Jan.

4.

56 32-6

52 30-6

43 lo-2

35 00-6

28 47-7

43 13-3

-3 41

•1

39 32-2

+ 141

11.

54 49-5

52 46-5

42 26-3

34 24-9

28 50-3

42 39-5

-3 32

•6

39 06-9

-11-2

18.

55 23-8

52 42-5

42 44-8

33 43-9

28 37-1

42 38-4

-3 24

■1

39 14-3

- 3-8

25.

55 50-9

52 35-9

42 17 0

34 22-3

28 33-1

42 45-0

-3 15

•5

39 29-5

+ 11-4

Feb.

1.

57 11-0

52 30-6

42 14-4

34 51-4

27 24-4

42 50-4

-3 06

•9

39 43-5

+25-4

8.

57 53-3

52 39-8

41 57-2

34 48-7

27 04-6

42 52-7

-2 5!:

■4

39 54-3

+36-2

15.

58 34-3

51 54-9

41 46-6

35 00-6

26 56-6

42 50-6

-2 49

•8

40 00-8

+42-7

22.

57 00-4

51 41-7

41 22-8

35 20-5

27 151

42 32- 1

-2 41

•2

39 50-9

+32-8

March 1.

od 32-6

51 08-6

41 04-3

35 32 3

26 44-7

42 12-5

-2 32

•6

39 39-9

+21-8

8.

56 23-3

51 16-6

40 4!l-7

35 04-6

26 500

42 04 8

-2 23

9

39 40-9

+22-8

15.

56 53-8

51 41-7

41 54-5

34 55-3

27 07-2

42 30 5

-2 15

■3

40 15-2

+57-1

22.

56 20-7

51 16-6

40 36-5

34 27-5

26 20-9

41 48-4

-2 oe

•7

39 41-7

+23-6

2U.

56 19-4

51 25-8

40 511

33 360

26 07-7 1

41 400

-1 58

■0

39 42 0

+23-9

April

5.

56 06-2

50 38-2

41 00-3

33 08-2

24 58-9 '

41 10-4

-1 49

•3

39 211

+ 3-0

12.

54 40-2

49 58-6

41 17-5

33 45-2

24 470

40 53-7

-1 40

•6

39 131

- 50

19.

55 11-9

49 54-5

40 44-5

34 03-8

25 109

41 Oil

-1 33

■0

39 29- 1

+110

26.

54 17-7

50 02-5

40 28-6

33 10-9

25 122

40 38-4

-1 23

•3

39 151

- 3 0

May

3.

53 52-6

48 51-2

40 02- 1

32 23-3

24 12-7

39 52-4

-1 14

•6

38 378

-40-3

10.

54 03-2

47 11-9

40 35 2

32 23-3

24 06- 1

39 39-9

-1 05

•8

38 341

-44-0

17.

53 24-8

47 09-2

40 193

32 07-4

24 32-5

39 30-6

-0 57

•1

38 33-5

-44-6

24.

53 57-9

47 17-2

40 511

31 595

23 23-7

39 29-9

-0 48

•3

38 41-6

-36-5

31.

54 111

46 33-5

39 54-2

32 25-9

23 05-2

39 14 0

-0 39

■6

38 34-4

-43-7

June

7.

53 55-3

46 071

39 48-9

31 370

22 57-3

38 53-1

-0 30

•8

38 22 3

-55-8

14.

54 03-2

45 47-3

39 58-2

30 37-4

23 01-3

38 41-5

-0 22

■0

38 19-5

-58-6

21.

54 521

45 22- 1

40 03-5

31 105

22 08-3

38 43-3

-0 13

•2

38 30- 1

-48-0

28.

53 27 5

46 01-8

40 511

31 15-8

21 551 ,

38 42-3

-0 04

•4

38 37-9

-40-2

July

5.

53 08-9

44 46-4

38 111

30 29-5

21 525

37 41-7

+0 04

•4

37 46-1

-92 0

12.

53 34- 1

45 06-3

39 07-9

30 46-7

21 485

38 04-7

+0 13

•2

38 17-9

-60-2

19.

53 15-6

44 570

39 05-3

30 30-8

21 208

37 49-9

+0 22

•1

38 120

-661

26.

53 12-9

44 491

38 01 8

30 52-0

21 45-8

37 44-3

+0 31

•0

38 15-3

-62-8

August 2.

52 39-8

45 01 0

37 27 4

31 06-5

22 00-4

37 39-0

+0 3!)

■8

38 18-8

-59-3

9.

53 23 5

45 06-3

38 23 0

31 26-4

21 51-2

38 02- 1

+0 48

■7

38 50-8

-273

16.

54 09-8

46 08-5

38 20-3

31 51-5

21 34 0

38 24-8

+0 57

■6

39 22-4

+47-3

23.

53 16 9

45 34- 1

38 49-4

31 14-5

21 03-5

37 59-7

+ 1 06

■6

39 06-3

-11-8

30.

52 18-7

45 30- 1

39 22 5

30 34-8

21 39-2

37 531

+ 1 15

•5

39 08-6

_ 9-5

Sept.

6.

52 01-5

44 21-3

40 33-9

30 28-2

21 141

37 43-8

+ 1 24

•4

39 08-2

- 9-9

13.

51 39-1

45 27-4

40 33-9

30 01-7

21 35-3

37 51 -5

+ 1 33

•4

39 24-9

+ 6-S

20.

52 43-9

44 280

40 20-7

30 28-2

21 221

37 52-6

+ 1 42

■3

39 34-9

+ 16-8

27.

52 10-8

44 55-7

41 13-5

29 08 9

21 41-9

37 50-2

+ 1 51

•3

39 41-5

+23-4

Oct.

4.

52 45-2

45 10-3

40 27-3

28 25-2

22 30-8

37 51-8

+2 OC

■3

39 52-1

+34-0

11.

53 14-3

44 530

40 15-4

28 51-7

22 14-9

37 53-9

+2 09

•3

40 03-2

+ 45-1

18.

52 51-8

44 54-4

40 220

29 02 3

22 30-8

37 56-3

+2 l!i

3

40 14-6

+56-5

25.

52 30-6

44 21-3

40 35-2

29 07-6

22 44 0

37 51-7

+2 2/

■3

40 190

+62-9

Nov.

1.

52 41-2

44 47-8

39 15-9

28 35-8

21 47-2

37 25-6

+2 36

•3

40 01-9

+43-8

8.

52 08-2

44 18-7

38 190

28 51-7

20 590

36 55-4

+2 45

•4

39 40-8

+22-7

15.

52 46 5

44 18-7

36 45- 1

28 23 9

21 340

36 45-6

+2 54

•4

39 400

+21-9

22.

52 161

44 451

36 47-8

28 43-7

21 48-5

36 52-2

+3 OS

•5

39 55-7

+376

29.

52 26-6

45 53 0

36 59-6

28 451

21 23-4

36 53-5

+3 12

■5

40 060

+47-9

Dec.

6.

52 571

45 03-7

36 26-6

28 5«-9

20 39-8

36 48-8

+3 21

•6

40 10-4

+52-3

13.

53 22 2

44 54-4

36 13-4

28 31-9

20 040

36 37-2

+3 3C

•7

40 07-9

+49-8

20.

52 491

44 06-8

35 50-9

28 371

19 37-6

36 12-3

+3 3S

•8

39 52- 1

+340

27.

52 46-5

43 21-8

35 12-5

28 25-2

19 310

35 51-4

+3 490

39 40-4

+22-3

Annual l

Mcaus J

54 08-0

47 22- 1

39 51-1

31 36-4

23 32-8

39 181

MAGXETIC OBSERVATIONS AT THE KEAV OBSEEVATOEY. 293

We may compare with Table VII., and the conclusions derived from it, a corre-
sponding Table (VIII. page 294) of the weekly means of the hourly observations of the
Declination at the Hobai-ton Observatory, between October 1843 and September 1848,
made by Captain Kayk, R.N., and his assistants in that establishment. The observations
themselves are published in the second and third volumes of the Hobarton Observa-
tions, and have been treated for the present purpose precisely in the same way as those
of the Kew Observatoiy, 2'-13 haAing been taken as the standard of a distmbauce,
instead of 3'-3 as at Kew, a somewhat lower standard being required at Hobarton to
sepai'ate the same proportion of disturbed observations for the investigation of their
laws, and being otherwise unobjectionable. The mean declination in the successive
years is placed at the foot of columns 2, 3, 4, 5, and 6 of Table VIII., and from these
we obtain the secular change in those years as follows : —

From April 1844 to March 1845, the East Declination increased 1 27"6

1845 „ 1846, „ „ „ 2 02-2

1846 „ 1847, „ „ „ 1 07-3

1847 „ 1848, „ „ „ 0 55-8

Whence we have 1' 23"-2 as the mean annual increase of East Declination at Hobarton
in the five years, corresponding precisely to the middle epoch of the mean or typical
year, i. e. the beginning of April 1846, and which has for its mean decimation
9° 56' 13"*9 E. Column 7 of Table VIII. contains the weekly means in the typical
year, each on the average of the five years. Column 8 shows the corrections for secular
change, being proportional parts of an annual change of 1' 23"-2. Column 9 contains
the weekly means in column 7 corrected for secular change to the beginning of April
1846 ; and column 10 the difiierences in the values in column 9 fii'om the mean declina-
tion 9° 56' 13"'9, derived dii'ectly from all the weekly means in the five years.

The aspect of the + and — signs in column 10 appears conclusive in respect to the
existence at Hobarton of a semiannual inequality analogous to that which has been
shown to exist at Kew. The du'ection of the inequality in the two semiannual periods
is also the same in the two hemispheres, the north end of the magnet pointing more
towards the east both at Kew and at Hobarton when the sun is north of the equator
and to the west when the sun is south of the equator. If we regard the equmoxes as
the approximate epochs of the semiannual change, we find in the weeks from April to
September an average increase of east declination of 19""1, and in the weeks from
October to March an average decrease of east declination of 19"'0, making together an
annual variation of 38""1.

294

MAJOE-GEXERAL SABIXE OX THE EESFLTS OF THE

Table VIII. — Weekly Means of East Declination at Hobarton, from October 1, 1843

to September oO, 1848.

Years ending September 30,

Correc-
tions for

Mean weekly
values cor-
rected for

Difi'erence of

the several

■n-eekly

Period.

JVf^QUS.

1844.

1845.

1846.

1847.

1848.

secular
change.

secular
change.

corrected
values from

9' +

9'+

9'+

9°+

9'+

9°+

9»+

9°56'13"-9.

(1)

(2)

(3)

W

(5)

(6)

(7)

1

(S)

(9)

(10)

Oct. 4.

51 570

54 39-6

55 2o-8

57 06-6

58 09 6

55 277

+40-8

56 08-5

-05-4

11.

51 57-6

53 56 4

55 480

57 16-8

58 090

55 25-6

+392

56 04-8

-091

18.

51 45-6

53 18-4

55 39-6

57 31-4

58 40-8

55 23-2

+37-6

56 00-8

-131

25.

51 462

53 37-6

55 420

57 15'2

57 34-2

55 110

+360

55 470

-26-9

Not. 1.

52 05 4

53 23-4

55 390

57 22-8

58 222

55 22-6

+34-4

55 570

-16 9

8.

51 51-6

53 34 8

55 57 6

57 26-4

57 40-8

55 18-2

+32-8

55 51 0

-22-9

15.

51 46-8

53 00 0

55 51-6

57 17 4

57 40-2

55 07-2

+31-2

55 38-4

-355

22.

51 408

53 22-2

55 34-2

57 29-4

57 168

55 04 7

+29-6

55 33-8

-401

29.

51 522

53 08-4

55 01-2

57 06-3

58 240

55 06 4

+280

55 34 4

-395

Dec. 6.

52 06 8

53 114

56 02 4

57 01 2

57 46-8

55 137

+264

55 40- 1

-338

13.

52 026

53 18-6

55 27-9

56 55-2

57 57-6

55 08-4

+24-8

55 33-2

-40 7

20.

52 04-8

53 46-8

55 22-2

56 52 8

57 35-7

55 08-5

+23-2

55 31 7

-42 2

27.

52 05 6

53 49-2

55 30-9

57 108

57 39-0

55 151

+21-6

55 36-7

-37-2

1844.

1845.

1846.

1847.

1848.

Jan. 3.

52 37-8

53 22-8

55 27-6

57 06 3

57 40-2

55 14-9

+200

55 34 9

-390

10.

52 55-2

54 01-2

55 32 4

56 480

57 48-0

55 25 0

+ 18-4

55 43-4

-30 5

17.

52 270

53 55-8

55 40-2

56 57-6

57 47-4

55 21-6

+ 16-8

55 38-4

-355

24.

52 34 2

53 42 6

55 450

57 25-5

57 51 6

55 27-9

+ 15-2

55 43 1

-30-8

31.

52 58-8

53 58-2

55 25-8

57 036

58 10-8

55 31-4

+ 13-6

55 45 0

-289

Feb. 7.

52 59-4

54 270

55 58-8

57 26-4

58 40-2

55 54-4

+ 120

56 06-4

-075

14.

52 55-8

54 19-8

56 120

57 132

58 05-4

55 45-2

+ 10-4

56 55-6

-18 3

21.

53 08 4

54 43-8

56 26- 1

57 28-4

58 243

56 02-3

+ 8-8

56 111

-02-8

28.

52 59-4

54 39-6

56 25-2

57 45-6

58 27-6

56 03 5

+ 7-2

56 107

-03 2

Mar. 7.

53 55-8

54 31-8

56 21 0

58 030

59 030

56 22 9

+ 56

56 28-5

+ 14-6

14.

53 420

54 43-8

56 53-4

58 07-2

58 47-7

56 26 8

+ 4 0

56 30-8

+ 16-9

21.

53 420

55 07-2

56 420

57 57-8

59 30 3

56 35 9

+ 2-4

56 38-3

+24-4

28.

53 31 2

54 46-8

56 58-2

57 522

58 52 2

56 24 1

+ 0-8

56 24-9

+ 110

April 4.

53 57-6

55 07-8

57 15-6

58 16 8

58 33 0

56 38-2

- 0-8

56 37-4

+23-5

11.

53 49-8

55 10 2

57 12-6

58 183

59 06-9

56 43-6

- 24

56 41-2

+273

18.

53 40-8

55 07 2

57 20-4

58 Ofi-6

59 132

56 41-6

- 40

56 37-6

+237

25.

53 39-6

55 08-4

57 090

58 08-1

58 44-7

56 340

- 5-6

56 28-4

+ 145

May 2.

54 03-6

55 06-6

57 23-4

58 33-6

58 58'2

56 49 1

- 7-2

56 41-9

+28-0

9.

53 50-4

55 132

57 24-6

58 09-6

59 27 0

56 490

- 8-8

56 402

+26-3

16.

54 00 0

55 28-8

57 12-3

58 27-6

59 150

56 52-7

-104

56 42-3

+28-4

23.

54 01 2

55 24-6

57 17-4

58 31-2

59 15-6

56 540

-120

56 42 0

+28-1

30.

54 10-2

55 35-4

57 30-6

58 270

59 186

57 00-4

-13-6

56 46-8

+32-9

June 6.

53 570

55 01 2

57 240

58 27-6

59 17-4

56 49-4

-15 2

56 34-2

+20-3

13.

54 02 4

55 30 0

57 28-2

58 39-6

59 31 8

57 02-4

-16 8

56 45-6

+317

20.

53 39 0

55 39 0

57 306

58 24 0

59 30-6

56 56-6

1 -18-4

56 38-2

' +24-3

27.

53 450

55 33-6

57 318

58 22-8

59 210

56 54-8

-20 0

56 34 0

+20 1

July 4.

53 37-2

55 240

57 59-4

58 16-8

59 27 6

1 56 570

-21-6

56 35-4

1 +21-5

11.

53 47 4

55 26-4

57 53 4

58 36-6

59 31-8

57 03- 1

-23 2

56 39-9

1 +260

18.

53 51-6

55 24 6

57 36-6

58 28-2

59 303

; 56 58-3

-24-8

56 33-5

+ 196

25.

53 52 2

55 24 6

57 40 8

58 25-8

59 14-4

56 55-6

-26-4

56 29 2

! +15-3

Aiig. 1.

53 594

55 26-4

57 49-2

58 10-8

59 36 0

! 57 004

-28-0

56 320

; +18-1

8.

53 55-4

55 25 2

57 44-4

58 24-6

59 240

1 56 58-7

-29-6

56 291

+ 152

15.

53 54-2

55 29-4

57 27-6

58 07-2

59 201

56 51-7

-31-2

56 20-5

+06-6

22.

54 06 1

55 30 0

57 40-2

57 54 9

59 40 8

56 58-4

-32-8

56 25 6

+ 117

29.

54 01-9

! 55 35-4

57 29-4

58 114

59 31 2

56 57-9

-344

56 23 5

+09-6

Sept. 5.

54 04-2

55 30-6

57 40-2

58 126

59 14-4

56 56-4

\ -360

56 20-4

+06-5

12.

54 07-2

55 29 4

57 270

58 07-2

59 240

56 550

1 -376

56 17-4

+ 03-5

19.

53 40 8

55 33-6

58 00-6

57 420

59 390

56 55-2

-39-2

56 160

+ 021

26.

53 41-4

55 27-6

57 34-2

58 18-3

60 02-4

57 00-8

' -40-8

56 200

+06-1

Annual \
means J

9 53 12-4

1

j 9 54 400

9 56 42-2

9 57 49-5

9 58 45-3

9 56 13-9

Secular "I
change J

1' 2

,

, ,

J

r^ 2' (

>2"-2 r c

7"-2 0' .

)5"-8

MAGNETIC OBSERVATIONS AT THE IvEW OBSERVATORY. 295

In volumo II. of the Magnetic Observations at St. Ilelona, p. v, an examination is
made of the monthly valnes of the declination obtained from eif^ht years of observation,
corrected for sccnlar change, and collected in a Table. These also indicate the exist-
ence of a semiannual inequality having epochs coincident, or nearly so, Avith the equi-
noxes— the north end of the magnet pointing, as at Kew and Hobarton, more to tlie east
in the montlis from April to September, and to the west from October to March. The
amount of the inequality is less than at Kew or Hobarton, " the semiannual difference
being about 14 seconds of arc."

The first volume of the Magnetical Observations at the Cape of Good Hope, published
in 1851, contains the fortnightly means of the hourly observations of the declination
fi'om July 1842 to July 184G ; these are corrected for secular change in Table 111. of
that volume, and the differences of the declination in each fortnight (so corrected) from
the mean declination of the whole period, are shoAvn in the final column. The mean
of the thirteen fortnights (in the four years) between March 20 and September 23 is
0'"40 more easterly^ and of the thirteen fortnights between September 24 and March 25
0''40 more loesterly than the mean of the year, — thus showing an annual variation of
0''80 or (48"-0), or a semiannual inequality averaging 24" to the East in the thirteen
fortnights from March 26 to September 23, and 24" to the West in the thirteen fort-
nights from September 24 to March 25. This is in accordance with the other stations
previously discussed.

The fact of the existence of an annual variation with analogous phenomena at the
four widely separated stations of Hobarton, St. Helena, the Cape of Good Hope, and
Kew appears to be thus substantiated ; its amount is least at St. Helena, intermediate
at the Cape and Hobarton, and greatest at Kew ; the difference in amount is doubtless
to be ascribed, in part at least, to the difference in the amount of the antagonistic force
of the earth's magnetism, tending to retain the magnet in its mean place in opposition
to all disturbing causes. This force (the horizontal component of the earth's magnetic
force) is, in British units, approximately 5-6 at St. Helena, 4'5 at the Cape and Hobarton,
and 3-8 at Kew.

§ 8. Annual Variation, or semiannual inequality, of the Dip, and of the Horizontal

and Total Force.

In the year 1850 I communicated to the Royal Society a paper entitled "On the
means adopted in the British Colonial Magnetic Observatories for determining the abso-
lute valvies, secular changes, and annual variations of the Magnetic Force." This paper
is published in the Philosophical Transactions for the same year, No. IX.

In this communication I endeavoured to show the importance of introducing into such
determinations greater accuracy than had previously been customary ; and by making
known the success which had attended the improvements adopted in the instruments
and methods employed in the Colonial Magnetic Observatories, I hoped to be the means
of promoting the adoption of similar instruments and processes (or the devisal and

296 ]\L\JOE-GEXEEAL SABIXE ON THE EESULTS OF THE

emplo}Taent of others which might sevre the purpose as well, or still more effectually)
in other observatories wliich had been instituted for the purpose of cooperating with
or aiding in the plan of magnetic research proposed by the Royal Society.

Amongst the results referred to in that paper, obtained by means of the mstruments
and processes therein described, there was one which appeared to myself to be highly
deserving the conftnnation (or otherM-ise) wliich it might receive from similar researches.
By a comparison of the monthly determinations of the Dip and of the Horizontal Force at
Toronto and Hobarton, between the years 1843 and 1848, there was shown a high pro-
bability of the existence of an "annual variation" in the direction and intensity of the
magnetic force, common to both hemispheres, the mean values being passed through
about the equinoxes, and the intensity of the force being greater, and the inclination
more nearly vertical, in the months when the sim is south of the equator than in the
months in which the sun is north of the equator. The facts thus made kno-\™ appeared
to indicate the existence of a general affection of the globe ha\ing an annual period, and
conducting us natm-ally to the position of the earth in its orbit as the first consideration
towards an explanation of the periodic change. The importance of follo-«ing up with-
out delay, and in the most effective manner, a branch of research which gave so fair a
promise of establishing a conclusion of so much theoretical moment upon the basis of
competent experiment was earnestly pointed out, and specially so with reference to those
national observatories in which magnetical researches were professed objects, and from
which exact determinations might most reasonably be expected.

In 1856 the Committee of the Kew Observatory, impressed with the importance of
prosecuting an investigation which appeared to lead to the establishment of a preriously
unsuspected cosmical relation in the minor variations of terrestrial magnetism, and
perceiring that no adequate provision had been made for this purpose in any establish-
ment in the British Islands, took the matter in hand, and having obtained permission
fi-om the tenant under the Crowni, caused a suitable wooden building, copper fastened,
to be erected in Richmond Old Deer Park, at a distance of 300 feet from the observatory
itself, and ha\ing no other buildings in its ricinity. A series of monthly determinations
of the dip and of the horizontal force was commenced in this building in April 1857,
with inclinometers made by Mr. He\rt Barrow, and with a unifilar magnetometer
made by the late Mr. Willlui Joxes. These instruments were the property of Her
Majesty's Government, having been originally made (under my own direction) for the
Arctic Expedition under Sir J.uiES CLu\rk Ross in 1846-1847, and replaced in my
charge, on the return of the expedition, for repair and subsequent use. Several minor
modifications, which experience had suggested since the publication of the memoir in
1850 ab'eady adverted to, were introduced in the instruments prerious to April 1857,
and in this improved state they have been described and practical directions given for
their use in the " Instructions for Magnetic Surveys by Land and Sea," published in 1859
in the third edition of the Admiralty Manual of Scientific Inquiry. The series of
determinations with these instruments has been steadily maintained from Ajjril 1857 to

MAGNETIC OBSEEYATIONS AT THE KEW OBSERVATOET. 297

the present time, and still continues. The unifilar magnetometer employed has been
the same througliout, no change whatsoever having been made either in the instrument
itself, or in its collimator magnet. In respect to the dip observations, from April 1857
to September 1860 inclusive, twelve dip circles and twenty-four needles, all by Barrow
and all of the same size and pattern, were employed, the mean of all the observations
made in a month vnih. anij of Barrow's 6-inch circles furnished with microscopes and
verniers having been taken as the mean dip of that month. A detailed statement of the
results of these observations, specifying in each case the name of the observer and the
distmguishing marks of the circle and needle, has been published in the ' Proceedings
of the Royal Society,' vol. xi. p. 144-162. In the discussion accompanying that com-
munication it was shown that the probable error of a single determination of the dip
with instruments of this pattern does not exceed +1'"5, this being the conclusion
derived from 282 determinations on 121 different days, chiefly by four observers, employing
twelve different circles and twenty-four needles all of the same sizx' and pattern.
Between October 1860 and March 1863, the mean monthly dip has been obtained with
one of the twelve circles alone, \vi. Barrow's circle No. 33 (one of the twelve previously
adverted to), and was generally the mean of a single determination in each month with
each of the two needles of that circle. This department of the Kew observations has
been placed by the Du-ector, Mr, Stewart, in the charge of Mr. Charles Chambers, one
of the assistants in the establishment, and to that gentleman I am indebted for the
results which are embodied in Tables IX. and XI., and which afford most satisfactory
evidence of Mr. Chajibers's skill and devotion to the duties with which he is charged.

"With reference to the values of the Horizontal Force in Table IX. Mr. Chambers
remarks, "The constants for the reduction of observations with collimator magnet
' K C 1 ' are as follows : —

" K the moment of inertia, being the mean of independent determinations with six
different inertia-cylinders by the late Mr. Welsh, F.R.S., =4-4696 (log K= 0-65027 at
60°Fahr.).

" HcncG

log^r'K at 30°=1 -64439, at 70°=l-64463
40°=l-64445, at 80°=l-64469
50°=1-64451, at 90°=l-64475
60°=l-64457.
" The correction for the decrease of the magnetic moment of the collimator magnet
produced by an increase of l°Fahr. =(y)=0•000119(^„-^) + -000000213(^o-#)^ t, bemg
the observed temperature, and f=35°. The induction coefficient (f*)=: -000194. These
were both determined by Mr. Welsh. The angular value of one division of the colli-
mator scale=2'-50. Comparisons of the deflection-bar with the verified standard measure
of the Kew Observatory gave the errors of graduation as follows : —
At 1-0 foot distance =--000075 of a foot at 62°Fahr.
At 1-3 foot distance =--000097.

MDCCCLXIII, 2 S

298

3IAJ0E-GEN'ERAL SAEENTl OX THE EESFLTS OF THE

" The arc of vibration ^yas always too small to require any correction ; and none has
been applied on account of the rate of the chronometer when the rate was less than
five seconds, as was generally the case. The constant P was determined from twenty-
foiu" repetitions of experiments of deflection made nearly simultaneously at each of the
two distances 1-0 and 1-3 feet, gi™ig P= — •00192.

" Generally there have been three or four observations of deflection and two of
vibration made in each month."

Table IX. — Monthly Values of the Horizontal Component of the Magnetic Force at
Kew, computed from the Experiments of Deflection and Vibration with the
Collimator Masnet '• K C 1".

April to Sep-
tember.

1857.

1858.

1859.

1860.

1861.

1862.

Means of the
six yeai-s.

April 3-7887

May 37920

June 3-7901

JuIt 1 3-7950

August 3-7871

September 3- 7883

37932
3-7984
3-7889
3-7980
3-7942
37920

3-7897
3-8008
3-8053
[3-8052]
3-8052
3-7995

3-8038
3-S022
3-8142
3-8065
3-7979
3-8056

3-8078
3-8157
38189
3-8115
3-8113
3-8115

3-8162
3-8209
3-8150
3-8179
3-8162
3-8158

3-7999
3-8050
3 8054
3-8057
3-8020
3-8021

Means, April to "1 o.^qa.-j
September...!/ ^ '^^-^

3-7941

3-8010

3-8050

3-8128

3-8170

3-8033

October to
March.

1857 and 1858.

1858 and 1859.

1859 and 1860.

1860 and 1861.

1861 and 1862.

1862 and 1863.

Means of the
six years.

3-7925
[37906]
[3-7887]
3-7868
3-7917
3-7873

3-7962
3-7964
3-7919
3-7951

[37967]
3-7983

37914
3-7963
3-8056
3-8038
3-8016
3-8036

3-8066
3-8074
3-8075
3-8101
3-8071
38075

3-8081
3-8085
3-8113
3-8144
3-8136
3-8125

3-8144
3-8161
3-8124
3-8127
3-8188
3-8212

3-8015
3-8025
3-8029
38038
3-8052
3-8051

November

December

January

February

Means, October
to March ...

} 37896

3-7958

3-8004

3-8077

3-8114

3-8159

3-8035

Yearly means

3-7899

3-7950

3-8007

3-8063

3-8121

3-8165

3-8034

The values within brackets [ ] are inteqiolated.

The absolute values of the horizontal force, corresponding to the beginning of October
in each of the years comprehended in Table IX., and the secular change in each year,
were therefore as follows : —

From April 1857 to March 1858
From April 1858 to March 1859
From April 1859 to March 18G0
From April 18C0 to March 18G1
From April 18G1 to March 18G2
From April 18G2 to March 18G3

Mean of the six years, corresponding to
the middle epoch, April 1860

3.gO,7}-c.ch. +-005/.

3-80G3
38121
3-8165

}sec. ch. +-0056.
}sec. ch. +-0058.
}sec. ch. -(-•0044.

' ^°}3-8034[

with a mean annual secular
increase of '0053.

MAGNETIC OBSERVATIONS AT THE KEW OBSERVATORY.

299

The "Annual Variation" or "Semiannual Inequality" (April to September, and
October to Marcli) may be sliown from the monthly values in Table IX. to have been
as follows: —

Table X.

Date.

Corrections for
Secular Change.

3-8034
+ Secular Cliange.

Observed Values.

Observed — Calculated.

April to September.

October to March.

July 1, 1857

Jan. 1, 1858

Julv 1, 1858

Jan. 1, 1859

July 1, 1859

Jan. 1, 18G0

July 1, I860

Jan. 1, 1861

July 1, 1861

Jan. 1, 1862

July 1, 1862

Jan. 1, 1863

-•0146

— •0119

— •0093
-•0066
-•0040
— -0013
+ -00 13
+ -0040
+ -0066
+ -0093
+ -0119
+ -0146

3-7888
3-7915
3-7941
3-7968
3-7994
3-8021
.•5-8047
3-8074
3-8100
3-8127
3-8153
3-8180

3-7902
3-7896
3-7941
3-7958
3-8010
3-8004
3^8050
3-8077
3-8128
3-8114
3-8170
3-8159

+ •0014
-0000

—0019
— -0010

+ •0016
+ •0003
+ •0028
+ •0017

--0017
+ •0003
—0013

—0021

Mean differences between the observed and calculated values in 1
the respective semiannual periods J

+ •0013

—0013

It is seen then by Table X. that there exists a variation in the amount of the hori-
zontal force having an annual period ; that the value of this variation is on the average
of the six years approximately -0020 ; and that it consists of a semiannual inequality,
the horizontal force being on the average -0013 higher in the six months from April to
September, and •0013 lower in the six montlis from October to March than would be
due to its mean value.

I pass to the contemporaneous determinations of the Dip.

2s2

300 :\IA.TOE-GEXEE_\L SAEENTl ON THE EESULTS OF THE

Table XI. — Monthly Values of the Magnetic Dip at Kew.

April to t j^-._ : ig5g_
September. j

1859.

1860. 1861. 1862.

1

Means of the
SLS years.

1 68°+

April 27-2

May 24-9

June 240

July ... 26-1

68 +

225
23-0
22-7
23-7
21-5
21-4

6S~' +

211
19-4

20-6
220

68 +

20-5
193
19-1
18-4
166
19-4

68=+

17-6
15-7
17-7
16-8
18-7
17-1

68'+

181
14 1
14-0
14-0

68°+

21-17
19-40
19-67
20-10

August 24-1

September 24-9

15-1 19-43
13-8 19-77

Means. Aprilto 1 25-20 1 22-47
September ... j

20-87

18-88

17-27

14-85

19-92

October to ig57 ^^ jg^g
Mareb.

1858 and 1859.

1859 and 1860.

1860 and 1861.

1861 and 1862.

1862 and 1863.

Means of the
six years.

1 68"^+

October 243

November 25 -6

December [248]

Januarv 240

Februan- 24 0

March ■ 24-6

68°+

238
237
21-2
22-3

[22-4]
22-5

68'+

240
224
208
224
21 1
21-0

68 +

196
20-8
18 5
195
194
20 4

68'+

18-4
179
17 9
190
151
17-1

68°+

16-0
15-8
15-6
145
142
135

68°+

2102
21 03
19-80
20-28
1937
19-85

Means October 1 2455
to March ... J

22-65

21-95

19 70

17-57

14 93

20-22

Yearly means... 24-87

22-56

2141 ! 19 29

17 42

1489

20-07

The values -oithin brackets [ ] are mtcrpolated.

The absolute values of the dip corresponding to the beginning of October in each of
the years comprehended in Table XI., and the secular change in each year, ai-e as
follows : —

From April 1857 to March 1858
From April 1858 to March 1859
From April 1859 to March 1860
From April 1860 to March 18G1
From April 1861 to March 1862
From April 1862 to March 1863

Mean of th(> six years, corresponding ] ^..
to middle epoch, April 1, 1860 . j

^^ -^'^Ssec ch -9-31
68 22-50^'^'^- , : ^,
^g^^.^^}sec.ch. -1-15

68 i9-29^''^'^-'^--2"^2
T f. ,,,}sec. ch. —1-87
17-42

}sec. ch. — 2'53

68

68 14-89

-{

with a mean annual secular
decrease of 2'-00.

The "Annual Variation" or "Semiannual Inequality" (April to September, and
October to March) may be shown from the monthly values in Table XI. to have been
as follows : —

MAGNETIC OBSEEVATIONS AT THE KEW OBSEEVATOET.
Table XII.

301

Date.

Corrections for
Secular Change.

88° 20'-07
+ Secular Change.

Observed Values.

Obscrred — Calculated.

April to September.

October to March.

July 1, 1857

Jan. 1, 1858

July 1, 1858

Jan. 1, 1859

July 1, 1859

Jan. 1, I860

July 1, I860

Jan. 1, 1861

July 1, 1861

Jan. 1, 1862

July 1, 1862

Jan. 1, 1863

+ 5-50
+ 4-50
+ 3-50
+ 2-50
+ 1-50
+ 0-50
-0-50

— 1-50
-2-50
-3-50

— 4-50

— 5-50

68 25-57
68 24-57
68 23-57
68 22-57
68 21-57
68 20-57
68 19-57
68 18-57
68 17-57
68 16-57
68 15-57
68 14-57

68° 25-20
68 24-55
68 22-47
68 22-65
68 20-87
68 21-95
68 18-88
68 19-70
68 17-27
68 17-57
68 14-85
68 14-93

-0-37
-MO
-0-70
-0-69
-0-30
-0-72

-0-02
+ 0-08
+ 1-38
+ 1-13
+ 1-00
+ 0-36

Mean differences between the observed and calculated values in the "1
respective seaiiannual periods J

-0-65

+ 0-66

It is seen therefore by Table XII. that there exists a variation in the amount of the
Dip having an annual period ; that the value of this variation is on the average of the
six years approximately l'-31 ; and that it consists of a semianmial inequality, the dip
being on the average 0'-G5 lower in the six months from April to September, and O'-GG
higher in the six months from October to March than would be due to its mean value.

Total Force. — We find in Table IX. that the mean of the April to September values
of the horizontal component of the force in the six years is 3-8033, corresponding in
epoch to January 1, 1860; and in Table XI. that the mean of the April to September
values of the dip in the same six years is 68° 19''92, corresponding to the same epoch.

We find also in Table IX. that the mean in the six years of all the October to March
values of the horizontal comi)oncnt is 3-8035, and of the di]> (Table XI.) 68° 20'-22,
corresponding to the epoch (six months later) of July 1, 1860.

We may reduce these values to a common epoch by applying to cither (with the
proper signs) a proportional part of the mean secular change derived from the observa-
tions of the six years. The mean secular change of the horizontal force is an annual
increase of -0053 (page 298), and of the dip an annual decrease of 2'-00 (page 300).
Hence we have the corrections for the secular change (in six months), of the horizontal
force = + -00265, and of the dip =— I'-OO, to be applied to the mean values of April
to September (coiTespondmg in epoch to January 1, 1860) in order to bring them into
strict compaiison with the mean values, October to March, corresponding to the later
epoch of July 1, 1860. The values then become as folloAVs: —

302 3IAJ0E-GEXEEAL SABI^'E OX THE EESULTS OF THE

From the April to September observations, ^

Values of the Horizontal Force, January 1, 18G0 3-80o3; and of the Dip 68 19-92

Corrections for Secular Change +0-00265; - 01-00

Con-esponding values July 1, 1860 .... 3-80595 68 18-92

And from the October to March observations) „_„ j, rrii v r^oonoo

T 1 -, -.nnr. >3-80350 and trom Tablc X. 68 iO-Ji
(Table IX.), also corrcspondmg to July 1, 1860 )

whence 3-80595 sec. 68° 18'-92 = 10-30032 from the April to September observations,
and 3-80350 sec. 68° 20'-22 = 10-30349 from the October to March observations, are
the values of the total force derived respectively for the same epoch (July 1, 1860)
from the determinations of the dip and horizontal force in the two semiannual periods ;
these show a difference of 0-00317 in British units, as the measure of the greater inten-
sity of the terrestrial magnetic force in the October to March period, than in the April
to September period.

For the satisfaction of tho'^e who are accustomed to be guided by the theory of pro-
babilities in their estimate of the dependence to be placed on the results of physical
investigations, it may be desii-able to state the "probable errors" of the mean results of
the seveutj-two monthly determinations of the Horizontal Force and of the Dip in
Tables IX. and XI., as well as the probable error of a single monthly determination of
each of these values.

The mean result of the seventy-two monthly determinations of the Horizontal Force,
shown in Table IX., is 3-8031 in British units : this has a " probable error " of +-00027.
The mean result of the seventy-two monthly determinations of the Dip (Table XL) is
68' 20'-07: this has a probable error of +0'-083.

The probable error of a single monthly determination of the Horizontal Force, derived
from the seventy-two monthly determinations, and after the application of the correc-
tions for secular change and annual variation have been made, is + -00233; and of a
single monthly determination of the Dip, after the application of the corrections for
secular change and annual variation have been made, is +0'-71.

It has been already stated that for rather more than half the whole period, viz. from
April 1857 to September 1860 inclusive, twelve dip circles and twenty-foiu- needles were
employed in the monthly determinations of the Dip, the circles and needles being all
made by the same artist (Mr. Hexry Barrow), and of the same size and pattern ; there
were also several observers, but chiefly four, viz. the late Mr. John Welsh, Mr. Stewart,
Dr. Bergsma, Director of the Netherlands Magnetic Observatory at Batana, and
Mr. CiiAJiBERS. Tlic means of all the obsei-vations thus made at the Kew Observatory
in the same month, and recorded in the books of the Kew Observatory, have been taken
as the mean Dip in that month. From October 1860 to April 1863 there has been
only a single observer, Mr. Chambers, with one circle, viz. No. 33, one of the twelve in
pre\-ious use, with its two needles. Some relative advantages or disadvantages may be

MAGNETIC OBSEEVATIONS AT THE KEW OBSERVATORY.

303

supposed to attend observations made by one or by more observers, and with one or
with several instruments; and it may therefore be useful to see how far these circum-
stances have modified the probable error in the two periods. The forty-two monthly
determinations from April 1857 to September 1860, give a probable error of +0'-70 for
a single determination; and the thirty from October 1860 to March 1863, give a pro-
bable error of 4:0'-73 ; whence we may infer that the greater number of partial results
which contributed to produce the monthly mean in the earlier period rather more than
counterbalanced the diversities which may be supposed to have been occasioned by the
peculiarities of the different observers, and of the different instruments employed. But
the small amoimt of probable error in either case is well worthy of the notice of those
who have been engaged, or who are likely to be engaged, in similar investigations.

In Tables XIII. and XIV. are placed the residual errors of the observed mtmthly
determinations of the Horizontal Force and of the Dip, after the application of the
corrections for secular' change and annual variation.

Tablk XIII. — Ilesidual Errors in the Monthly Determinations of the Horizontal Force.

1857.

1858.

1859.

1860.

1861.

1862.

1863.

Means.

April

-•0003

-•0011

-•0099

-•0011

-•0024

+ •0007

— •00231

Mav

+ •0026

+ •0037

+ •0007

— •0031

+ •0051

+ •0049

+ •0034

•0000

June

+ •0003

— •0062

+ •0048

+ •0085

+ •0079

-•0013

+ •0023

Sun nortli

July

+ •0047

+ •0024

+ •0043

+ •0004

0000

+ •0011

+ •0021 f of the
— •0020 equator.
-■0023

August

-•0036

— •0018

+ •0038

-•0088

— •0006

— •0010

September ...

— •0028

-•0044

-•0023

— •0015

— •0008

— 0018

October

+ •0035

+ •0020

— •0082

+ •0017

-•0021

-•0011

-•00071

— •0001 1 +-0001

November ...

+ •0012

+ •0017

— •0038

+ •0031

-•0021

+ •0001

December ...

— •00)1

-•0032

+ •0051

+ •0018

+ •0002

— •0040

— •0002 1 Suti .south

.January

-•0034

-•0004

+ •0031

+ •0039

+ •0029

— •0041

+ •0003 r of the

February ...

+ •0010

+ •0006

+ •0^)05

+ •0004

+ •0017

+ •0015

+ •0009 1 equator.

March

-•0039

+ •0019

+ •0021

+ •0004

+ •0001

+ •0035

+ ^0006J

Table XIV.-

-Residual Errors

in the Monthly

Determinations

of the Dip.

1857.

1858.

1859.

I860.

1861.

1862.

1803.

Means.

April +i'8

— 6-8

— 6-2

+ 1-2

+ 6-3

+ 2^8

'

+ 0^85^

May -0^3

-0^2

-1^8

+ 0^1

-1^5

-1-1

— 0-80 j +0'^01

June

-\-0

-0-3

-0^5

+ 0-1

+ 0-7

-1-0

— 0-33 [ Sun north

July

+ 1-3

+ 0^9

+ 0^8

— 0-4

0-0

— 0^8

+ 0^30

r of the

August

-0-6

— 1-3

-0-1

— 2-1

+ 2-0

+ 0-4

-0-25

equator.

September ...

+ 0^4

-M

+ 1-5

+ 0^9

+ 0-6

-0-7

+ 0^37

October

-1-4

+ 0^1

+ 2^3

-0^1

+ 0^7

+ 0-3

+ 0^32

November ...

+ 0-1

+ 0-2

+ 0^9

+ 1-3

+ 0^4

+ 0-3

+ 0^53 — 0'^02

December ...

-0^6

— 2^2

-0-5

— M

+ 0-6

+ 0^2

-0^47 1 Sun south

January

-1-2

-0^9

+ 1^2

+ 0^3

+ 1^8

-0-7

+ 0-09 [ of the

February ...

-1^0

-0^6

+ 0^1

+ 0^4

-1-9

-0^8

— 0*63 equator.

March

-0^2

-0^3

+ 0-2

+ 1-5

+ 0-2

-1-3

+ 0-03J

The eiTors have no systematic appearance ; and thus the Tables are thoroughly con-
firmatory of a semiannual inequality having its epochs coincident, or nearly so, with the
sun's passage of the equator.

304 MAJOE-GE^^:EAL SABIXE OX THE EESULTS OF TlIE

The second volume of the Hobarton Magnetic Observations, published in 1852, con-
tains the particulars of the monthly determinations of the absolute values of the hori-
zontal force from January 1846 to December 1850 inclusive, all made with the same
imifilar magnetometer, and presening throughout the same experimental process. The
mean value, corresponding to July 1, 1848, is 4'o0427. The secular change obtained
by least squares from the sixty equations of condition is correctly stated m the publica-
tion referred to, as an annual diminution of 0-0006. Treatmg these results in the
same manner that the Kew results have been treated in this paper, we obtain 4-5036 in
the months from April to September, and 4-5048 in the months from October to March ;
or a diminution in the horizontal component of the force of 0-0007 in the months when
the sun is north of the equator, and an increase of 0-0005 in the months when the sim
is south of the equator; constituting a semiannual inequality of 0-0012. When the
corrections for secular change and annual variation are applied, the probable error of a
single monthly determmation is found to be +0-00125; and the probable error of the
mean result of the sLxty months is less than 0-0002.

The first volume of the Hobarton Observations, published in 1850, contained the
details of a series of monthly determinations of the Inclination, commencing in Januai-y
1841 and ending in December 1847. The second volume, published in 1852, contained
a similarly detailed account of the continuation of the series to December 1850; com-
prising, with the observations stated in the preceding volume, an uninterrupted series of
monthly determinations during ten years. The mean secular change derived from sixty-
eiglit monthly results obtained with the same circle and needle throughout, was found
to be a decrease of 0'-0G7 in each year — an amount so small as to be practically insigni-
ficant in the consideration of the questions at present under notice. The mean value of
the Inclination in the ten years, taking all the months into account, was — 70° 36'-01 ;
the mean of the months from April to September inclusive was — 70° 35'-42, and from
October to March inclusive — 70° 36'-6. The difference between these half-yearly values
is l'-18, the (south) dip being 0'"59 less in the months from April to September, and
0'-59 greater in the months fi-om October to March, than on the mean of the whole year.
We have therefore for the values of the total force at Hobarton in the two semi-
annual periods, 4-5048 sec. 70° 36'-6 = 13-5688 (in British units) from October to
March, and 4-5036 sec. 70° 35'-42 = 13-5520 from April to September. The difference,
viz. 0-0168, expresses the greater intensity of the terrestrial magnetic force in the
semiannual period from October to March than in the semiannual period from April
to September. This value may undergo a slight alteration, when the results of the
continuation of the series of monthly determinations of the horizontal force and of the
inclination until the final close of the Hobarton Observatory are added to those already
stated ; but it will be substantially tlie same. The later results will be published in the
fourth Hobarton volume, now preparing for tlie press.

MAGNETIC OBSEEVATIOXS AT THE KEW OBSEKYATORY.

305

In the second and third volumes of the Toronto Observations are published the details
of the monthly determinations of the Horizontal Force and of the Dip during eight
years, viz. 1845 to 1852 inclusive. From these we may form the following Tables,
similar to Tables IX. and XI. of the Kcw Observations.

Table XV. — Monthly determinations of the Horizontal Component of the Magnetic
Force at Toronto, 1845 to 1852 inclusive.

April to
September.

1845.

1846.

1847.

1848.

1840.

IS-W.

1851.

1852.

Means
of the

8 years.

April

3-5446
3-5481
3-5514
3-5508
3-5473
3-5466

3-5414

3-5414
3-5458
3-5446
3-5397
3-5390

3-5348
3-5386
3-5399
3-5366
3-5424
3-5338

3-5361
3-5386
3-5366
3-5376
3-53C0
3-5332

3-5378
3-5413
3-5389
3-5428
3-5394
3-5382

3-5373
3-5366
3-5380
3-5284
3-5199
3-5217

3 5311
3-5328
3-5311
3-5317
3-5318
3-5286

3-5054
3-5142
3-5083
3-5139
3-5138
3-5119

3-5336
3-5365
3-5363
3-5358
3-5338
3-5319

May

June

July

August

September

, ■^/!''"c . !• 3-5481 3-5420
April to Sept. J j

3-5385

3-5363

3-5397

3-5303

3-5312

3-5113

3-53465

October to
March.

1S45,

1840.

1*47.

1S4,S.

1849. ISoO.

1801.

1852.

Means
of the
8 years.

January

3-5472
3-5471
3-5471
3-5466
3-5471
3-5479

3-5475
3-5413
3-5441
3-5386
3-5360
3-5433

3-5435
3-5426
3-5386
3-5345
3-5366
3-5347

3-5329
3-5352
3-5372
3-5263
3-5249
3-5318

3-5319
3-5312
3-5339
3-5343
3-5366
3-5351

3-5344
3-5354
3-5387
3-5320
3-5361
3-5283

3-5249
3-5243
3-5321
3-5311
3-5304
3-5286

3-5305
3-5231
3-5237
3-5110
3-5140
3-5149

3-5366
3-5350
3-5369
3-5318
3-5327
3-5331

February

March

October

November

December

Means, 1
Oct. to Mar. J

3-5472

3-5418

3-5384

3-5314

3-5338

3-5341

3-5286

3-5195

3-53435

Yearly means ...

3-5476

3-5419

3-5384

3-5339

3-5367 3-5322 j 3-5299 3-5154

3-53451

The two half-yearly results are intercomparable, requiring no correction for secular
change, as they have both the same mean epoch, \iz. January 1, 1849.

MDCC^LXIII.

2t

30f3

MAJOE-GEXEEAL SABIXE OX THE EESULTS OF THE

Table XVI. — Monthly Values of the jNIagnetic Inclination at Toronto.
1845 to 18-52 inclusive.

April to
September.

1845.

1*4<1. i

1

1847.

1,84.8.

1840.

18.-.0.

1851.

1852. 1

!

Means of
the ^ years.

75° +

11-5
15-4
15-2
14-2
14-4
16-6

75° +

14-3
14-4
14-8
14-0
14-4
15-7

75° +

15-9
16-1
13-1
11-6
12-6
1 .5-4

75° +

18-0
17-2
16-8
16-4
19-0
17-3

75° +

18-4
18-4
18-5
18-0
19-3
21-6

75° +

19-7
19-5
19-1
19-9
18-4
21-0

75° +

21-9
20-0
20-7
19-0

19-8
20-8

75°+ 1

20'-0
20-8 i
20-8
19-9
20-0
21-6 I

1
75° 17-46 '
75 17-73 1
75 17-37
75 16-63
75 17-24
75 18-75 1

May

June

July

, J

September

Means,
April to Sept

}

75 14-55 75 14-60

75 14-12

75 17-45

75 19-03 '[75 19-60 75 20-37 75 20-52

75 17-53

October to March.

1845.

184G.

1847.

1R48.

1840.

1850. 1851.

1852.

Means.

75° +

11-4
19-5
14-5
14-3
16-8
15-2

75° +

13-9
14-2
13-8
15-4
15-0
15-1

75° +

15-0
15-2
16-3
17-6
17-7
17-0

75° +

20-3
18-7
17-2
19-0
19-4
20-6

75° +

19-5
18-1
16-7
20-6
20-1
18-1

75° +

19-9
18-7
18-0
21-8
21-3
22-5

75° +

21-6
20-0
21-5
20-0
20-4
19-4

75° +

19-3
19-6
10-6
22-2
21-3
21-2

75 18-49
75 18-00
75 17-20
75 18-86
75 19-00
75 18-64

February

Novemltcr

December

Means,
Oct. to Marcl

}

75 16-45

75 14-57

75 16-47

75 19-20

75 18-85

75 20-37

75 20-48

75 20-53

75 18-36

These two half-yearly results are also intercomparable, requiring no correction for
secular change, as they have both the same mean epoch, viz. January 1, 1849.

We have then for the Total Force corresponding to the semiannual period April to
September, 3-53465 sec. 75° 17'-53 = 13-9220 (in British units), and for the Total Force
corresponding to the semiannual period October to March, 3-53485 sec. 75° 18'-3G
= 13-9336; the difference, 0-0116, is the measure of the greater intensity of the
terrestrial magnetic force in the October to March period than in the April to September
period: or, applying to the values of the horizontal force tlie induction-correction of
— -0040 (Toronto Observations, vol. iii. pp. cxv, cxvi), we have the total force in the
April to September period 3-53065 sec. 75° 17'-53=13-D0G2, and in the October to
March period 3-53035 sec. 75° 18'-36 = 13-9178 ; and the corresponding difference,
-0116, as the excess of the total force in the October to March period over the April to
September period.

The observations of the Inclination at Toronto were carried on previous to 1845 and
continued sub.scquent to 1852, completing a series of fifteen years, for which period,
therefore, a corresponding inference, in regard to the; annual variation of the Inclination,
may be drawn, resting on a still Avider basis. The second volume of the Toronto Obser-
vations, published in 1853, and the third volume, publi.shed in 1867, contain the details
of 1020 determinations of tlic dif) nearly ('(jually distribiitod in the different months of

MAGNETIC OBSERVATIONS AT THE KEW OBSERVATORY.

301

tlie liftc'cn years, 1841 to 1855 inclusive, of which the following is a summary, arranged
in the two categoi'ies, April to September, and Octolicr to March : —

April to Scpfombor. OctoluT to Miirrli.

April 75 18-33

May 75 18-08

June 75 17*38

. . 75 17-13

. . 75 17-33

. . 75 19-09

July . .

August

September

75 17-90

January .
February .
March . .

October .
November
December

75 18-78

75 18-43

75 18-17

75 19-09

75 19-53

75 19-15
75~18-86

The scmiaiuiual results require no correction for secular cliange, as they have both
the same mean epoch. They show a semiannual inequality in the Dip at Toronto,
causing its value to be, on the average, 0'-9r) higher in the months from October to
March than in those from April to September. Table XVI., resting on a smaller
number of years, gave a semianiaual inequality of 0'-83.

We have therefore the concurrent endenco of tlie three observatories of Toronto,
Hobarton, and Kew for the existence of an amiual vaiiation in the dip, and in the inten-
sity of the total magnetic force, referable apparently to the earth's position in its orbit,
with epochs of maxima and minima coincident, or nearly so, with the solstices. The
conclusion terminating the previous section of this paper (§ 7) has shown the probabi-
lity, resting also on the concui-rent evidence obtained at four observatories, Hobarton,
the Cape of Good Hope, St. Helena, and Kew, of the existence of a corresponding
semiannual inequality in the Declination.

The phenomena thus submitted to the consideration of the Royal Society may be
briefly stated to be an increase of the Dip and of the Total Force, and a deflection of the
north end of the Declination magnet towards the West, in both hemispheres, in the
months from October to March, as compared with those from April to September. It
seems difficult to assign to such effects any other than a cosmical cause. The greater
proximity of the earth to the sun in the December c(>ra})ared with the June solstice
most natui-ally presents itself as a not improbable cause ; but we are as yet too little
acquainted with the mode of the sun's action on the magnetism of the earth to enter
more deeply into the question at present. The inequalities may in themselves seem to
be small, but judged oi scientifically, i. e. by the proportions they bear to their i-espective
probable errors, they are not so.

The tabulation from the Photograms, and the calculation of the values contained
in the Tables, liave been performed by the Non-commissioned Officers of the Royal
Artillery, imder the superintendence of the principal clerk, Mr. John Mageath, in the
Government Establishment at Woolwich for the reduction and publication of magnetic
observations.

[ 309 ]

XIII. On the Diurnal Inequalities of Terrestrial Magnetism, as deduced from observa-
tions made at the Royal Observatory, Greenwich, from 1841 to 1857.
By George Biddell Airy, F.R.S., Astrcmomer Royal.

Keceived April 8,— Read April 23, 1863..

It has been usual for the Koyal Society to receive among their communications and to
publish in theu* 'Transactions' the epitomized results of long series of voUiminous
observations and laborious calculations, of which the fundamental details have been
printed in works specially devoted to those subjects. The paper which I have the
honour now to submit to the Society consists principally of results of this class. It
exhibits m cuives the Diurnal Inequalities of Terrestrial Magnetism, as obtained by the
use of instriunents essentially the same through the whole period of the seventeen
years ; diuing the last ten years of which the magnetic indications have been automa-
tically recorded by photographic self-i-egistration, on a system which has been continued
to the present time (18G3) and is still to be continued. I offer these results to the
Eoyal Society in the hope that they will prove no luiimportant contribution to a record
of the state of Terrestrial Magnetism at Greenwich, through a period which is likely to
be esteemed a very important one in the general history of the science.

The magnets of the three magnetometers (Declination, Horizontal Force, Vertical
Force), fi-om which these indications are obtained, are 2-foot magnets, such as were
introduced by Gauss about the time of commencing this series of observations ; two of
them were prepared at Gottingen. If I had now to establish a magnetical apparatus, I
should probably adopt magnets of smaller dimensions. Yet there are advantages in the
use of large magnets, as the power of carrying large mu-rors, &c., which I would not
lightly forego. And, judging from the completeness and delicacy of the registers of
magnetic storms made by all three instruments, I have reason to believe that the general
accuracy of the records is almost as great as it will be possible to obtain with any instru-
ments. I have therefore not thought it necessary to make any change in the instru-
mental system.

From the beginning of 1841 for the Declination and Horizontal Force, and from
the beginning of 1842 for the Vertical Force, to the end of 1847, the obseiTations
were made by eye, every two hours. From the beginning nearly of 1848 (with the
exception of the Vertical Force Magnet, of which the auxiliary apparatus was com-
pleted so late in the year that it has been thought best to suppress the few observations
of 1848 entirely) the positions of the magnetometers are registered by the photographic
apparatus planned and established at the Royal Observatory by CHAEXiES Bkooke, Esq.

The details of the observations, as far as 1847, are printed in the 'Greenwich Mag-

MDCCCLXIU. 2 U

310

IME. G. B. AIEY ON THE DIUENAL

netical and Meteorological Observations' for each year. The means, however, printed
in those volumes are not, in every instance, adopted here. This arises fi'om the circum-
stance that, in order to give unity to the plan of reduction for this memoir, the days in
which there prevailed a certain amoimt of magnetic disturbance (not defined numeri-
cally, but estimated by the judgment of the Supermtendent of the Reductions) have
now been separated from the rest, in the same manner as had previously been done for
the reductions 1848-1857; and the means have been taken without these separated
days. The days thus excluded are the following : —

1841: September 24, 25, 26, 27; October 25; November 18, 19; December 3, 14.

1842: January 1; February 24; AprU 14, 15; July 1, 2, 3; November 10, 21;
December 9.

1843 : January 2; February 6, 16, 24; May 6 ; July 24, 25.

1844: March 29, 30; October 1 ; November 16, 22.

1845: January 9; February 24; March 26; August 29; October 3.

1846: May 12; August 6, 7, 24, 25, 28; September 4, 5, 10, 11, 21, 22; October
2, 7, 8 ; November 26 ; December 23.

1847: February 24; March 1, 19; April 3, 7, 21; May 7; June 24; September
24, 26, 27 ; October 22, 23, 24, 25 ; November 22 ; December 17, 18, 19, 20.

The differences between the means at different hours and the mean of the twelve two-
hourly means, which have been actually used in the fonnation of the curves in Plates
XVI. and XX., are the followmg; : —

Declination.

Horizontal
Force.

Vertical
Force.

Declination.

Horizontal
Force.

Vertical
Force.

1841.

h
0

+ 3-7

-0-00123

li
1843. 4

+ 3-4

+ 0-00042

+ 0-00043

2

+ 5-9

- 53

6

+ 0-7

+ 53

+ 38

4

+ 3-7

+ 10

8

-0-7

+ 52

+ 29

6

+ 0-7

+ 32

10

-2-0

+ 36

+ 8

8

— 1-3

+ 52

12

-2-3

+ 25

- 9

10

-2-5

+ 42

14

— 1-8

+ 01

- 22

12

— 2-8

+ 39

16

-1-9

- 03

- 26

14

0.0

+ 33

18

-2-1

+ 01

- 26

IG

— 1-5

+ 28

20

-2-3

- 29

— 25

18

-1-5

+ 43

22

-0-8

- 110

- 17

20

-1-9

- 03

1844. 0

+ 4-2

-0-00092

-0-00022

22

-0-4

- 105

2

+ 5-5

+ 06

+ 18

1842.

0

+ 3-9

— 0-00102

+ 0-00013

4

+ 2-9

+ 53

+ 55

2

+ 5-6

— 23

+ 33

6

+ 0-5

+ 80

+ 55

4

+ 3-4

+ 33

+ 40

8

-M

+ 78

+ 37

6

4-0-8

+ 45

+ 24

10

-2-1

+ 49

+ 15

8

-0-9

+ 58

+ 11

12

-2-3

+ 27

- 5

10

-2-3

+ 36

7

14

-1-8

— 03

- 19

12

-2-6

+ 38

- 27

16

-1-5

— 13

26

14

-2-1

+ 12

- 34

18

-1-9

— 13

— 30

16

-1-9

- 04

— 28

20

— 2-1

- 51

36

18

— 1-8

+ 10

— 20

22

-0-3

— 108

- 33

20

-2-0

- 02

- 10

1845. 0

+ 3-9

-0-00103

-0-00008

22

0-0

92

+ 3

2

+ 5-8

+ 10

+ 23

1843.

0

+ 4-1

— 0-00087

-0-00008

4

+ 3-3

+ 55

+ 46

2

+ 5-9

+ 02

+ 20

6

+ 0-6

+ 62

+ 43

INEQTJALITrES OF TEEEESTRIAL MAGNETISM.
T-VBLE (continued).

311

Declination.

Horizontal
Forc-e.

Vertical
Force.

Declination.

Horizontal
Force.

Vertical
Force.

1846.

h
8

-0-7

+ 0-00061

+ 0-00026

h
1846. 16

-i'9

+ 0-0O002

-0-00038

10

-1-6

+ 43

+ 5

18

—2-3

+ 08

- 45

IS

-1-7

+ 31

— 13

20

-2-4

39

39

14

-1-9

+ 01

- 25

22

-1-1

— 133

- 32

16

— 1-8

- 03

27

1847. 0

+ 4-4

-0-00119

+ 0-00015

18

-2-1

+ 01

- 25

2

+ 6-9

- 16

+ 35

20

-2-6

- 40

- 20

4

+ 3-8

+ 41

+ 37

22

-1-1

- 118

- 19

6

+ 1-0

+ €7

+ 27

1846.

0

+ 4-0

-0-00114

-0-00014

8

-0-4

+ 73

+ 9

2

+ 6-2

- 12

+ 26

10

-1-6

+ 62

— 3

4

+ 3-7

+ 53

+ 58

12

-2-7

+ 34

33

6

+ 0-9

+ 83

+ 63

14

-2-5

+ 03

- 33

8

— 1-1

+ 72

+ 49

16

-2-5

+ 08

- 32

10

-20

+ 48

+ 21

18

-2-9

+ 08

17

12

-2-1

+ 29

- 8

20

-2-9

- 24

- 3

14

-20

+ 13

27

22

-0-9

— 120

+ 13

The differences of the means for the separate hom-s from the mean of the twelve
two-hourly means, in the aggregates of the numbers for the same nominal month in
diflFerent years, through the periods 1841-1847 for Declination and Horizontal Force,
and 1842-1847 for Vertical Force, are the following: —

Declination.

Horizontal
Force.

Vertical
Force.

Declination.

Horizontal
Force.

Vertical
Force.

January.

h
0

+ 2-9

-0-00047

+ 0-00006

h
March. 12

-2-8

+ 0-00030

-0-00017

2

+ 3-8

+ 5

+ 18

14

-1-9

- 1

- 27

4

+ 1-7

+ 17

+ 34

16

-2-0

- 11

- 30

6

0-0

+ 8

+ 22

18

-1-9

+ 7

- 30

8

-0-9

+ 2

+ 14

20

-1-9

— 4

- 23

10

-2-5

— 2

+ 6

22

-0-6

- 101

- 10

12

-2-8

17

- 4

April. 0

+ 4-3

-0-00160

0-00000

14

-1-4

- 20

— 14

2

+ 7-9

- 39

+ 33

16

-1-0

- 13

16

4

+ 4-6

+ 56

+ 67

18

-0-6

+ 35

- 22

6

+ 1-1

+ 84

+ 58

20

0-0

+ 43

- 18

8

-1-2

+ 77

+ 37

22

+ 0-8

- 2

- 10

10

-1-9

+ 50

+ 3

February.

0

-4-3-6

-0-00046

+ 0-00003

12

-2-7

+ 44

_ 23

2

+ 4-9

+ 7

+ 27

14

-2-5

+ 30

_ 38

4

+ 2-7

-1- 26

+ 40

16

-2-6

+ 10

- 47

6

4-0-6

+ 29

+ 27

18

-2-3

+ 21

_ 40

8

-1-2

+ 29

+ 13

20

-3-3

- 19

_ 28

10

— 2-3

+ 4

0

22

-1-3

164

_ 12

12

-2-8

+ 1

- 13

May. 0

+ 4-4

-0-00134

-0-00005

14

-2-3

26

- 18

2

+ 6-7

9

+ 27

16

-1-6

- 17

- 22

4

+ 4-2

+ 66

+ 42

18

-1-4

+ 17

- 18

6

+ 1-2

+ 113

+ 50

20

-0-7

+ 27

- 20

8

-0-1

+ 119

+ 30

22

+ 0-3

— 30

- 10

10

-1-1

+ 70

■+ 2

March.

0

-f-4-0

-0-00109

-0-00002

12

-1-6

+ 43

_ 20

2

+ 6-7

- 3

+ 27

14

-2-1

+ 14

_ 33

4

+ 3-9

+ 56

-1- 50

16

— 2-1

_ 6

_ 30

6

-fO-5

+ 47

+ 40

18

_3-3

_ 23

_ 22

8

-1-2

+ 47

-f- 20

20

-4-6

86

_ 8

10

-2-8

-f 30

0

22

-1-7

- 167

_ 8

2u2

312

JIE. G. B. AIET OX THE DIUEXAL

Table (continued).

Decimation.

Horizontal
Force.

Vertic-al
Force.

Declination.

Horizontal
Force.

Vertical
Force.

h

h

June. 0

+ 4-0

— 0'00130

-0-00005

September. 12

-2-5

+ 0-00054

-0-00018

2

+ 6-4

- 10

+ 28

14

— 2-2

+ 37

- 32

4

+ 4-8

+ 57

+ 48

16

-2-6

+ 23

- 35

6

+ 1-8

+ 101

+ 53

18

-2-2

+ 20

- 32

8

+ 0-3

+ 110

+ 42

20

— 2-4

- 49

- 27

10

-0-7

+ 71

+ 5

22

+ 0-1

170

- 27

12

-1-9

+ 01

- 18

October. 0

+ 4-4

-0-00120

-0-00007

14

-2-2

+ 26

— 30

2

+ 5-9

- 27

+ 23

16

— 2-4

+ 11

37

4

+ 3-2

+ 17

+ 48

18

-3-3

— 11

- 33

6

+ 0-1

+ 37

+ 33

20

— 4-4

— 106

- 28

8

-1-4

+ 50

+ 18

22

-2-0

)73

17

10

— 2-5

+ 53

+ 2

July. 0

+ 4-1

— 0-00146

-0-00008

12

-2-5

+ 41

- 13

2

+ 6-7

- 20

+ 25

14

-1-8

+ 21

- 25

4

+ 4-8

+ 73

+ 52

16

-1-4

+ 27

- 27

6

+ 2-0

+ 109

+ 62

18

-1-2

+ 26

- 23

8

+ 0-3

+ 120

+ 47

20

-1-5

0

- 18

10

-1-2

+ 77

+ 13

22

-1-0

- 121

- 12

12

-2-0

+ 54

- 13

November. 0

+ 3-5

-0-00063

-0-00002

14

— 2-5

+ 14

- 35

2

+ 4-4

- 14

+ 23

16

-2-7

- 3

- 30

4

+ 2-2

+ 11

+ 42

18

-3-7

- 11

- 38

6

+ 0-3

+ 33

+ 28

20

-4-1

90

- 30

8

-1-2

+ 26

+ 17

22

-1-6

170

- 25

10

-2-5

+ 13

+ 5

August. 0

+ 5-1

— 0-00141

-0-00015

12

— 2-2

+ 10

- 12

2

+ 7-7

- 11

+ 27

14

-1-8

0

- 20

4

+ 4-6

+ 66

+ 52

16

-0-9

+ 1

- 23

6

+ 0-9

+ 89

+ 60

18

-0-9

+ 19

- 20

8

-0-7

+ 99

+ 40

20

-0-7

+ 19

- 15

10

-1-9

+ 83

+ 13

22

-0-1

- 51

- 10

12

-2-4

+ 59

- 20

December. 0

+ 2-5

-0-00043

-0-00002

14

— 2-5

+ 33

- 32

2

+ 3-7

— 10

+ 22

16

— 2-5

+ 11

- 42

4

+ 1-6

+ 13

+ 25

18

-3-6

— 14

- 33

6

+ 0-1

+ 21

+ 17

20

-3-9

- 83

- 28

8

-M

+ 14

+ 15

22

-1-0

- 190

27

10

-2-1

+ 9

+ 7

September. 0

+ 5-6

-0-00141

-0-00013

12

-2-1

+ 4

- 5

2

+ 6-9

16

+ 28

14

-1-2

- 29

— 12

4

+ 3-5

+ 39

+ 57

16

-0-6

- 14

- 13

6

+ 0-4

+ 53

+ 50

18

-0-6

+ 19

- 15

8

-2-0

+ 71

+ 30

20

-0-2

+ 26

17

10

-2-7

+ 63

+ 7

22

+ 0-2

- 6

- 8

These means are used in forming the curves of Plates XVIII. and XXII.

For the observations from 1848 to 1857, the details of the record (in the form of
measvu'es of the ordinates of every salient point of the photographic curve) mil be
found in the ' Greenwich Observations ' for each year — a few being omitted in the
earlier portion of the period. These numbers, however, have not actually been used in
forming the means. For that purpose (as is explained in the Eeductions printed in the
'Greenwich ObseiTations, 1859') curves have been traced by hand upon the photo-
graphic sheets, smoothing down their most rapid inequalities ; and the hourly ordinates
of these curves have been measured upon the sheets. Tlie means of these are given in
the 'Greenwich Observations, 1859 ;' they are used without alteration here.

USTEQUALITIES OF TEREESTELy;. MAGISTSTISM.

313

The list of days omitted in the period 1848-1857 will be louiid in the volume for
1859. It may be interesting to collect here the numbers of omitted days for the several
years of the entire period from 1841 to 1857.

1841 .

. 9

1842 .

. 10

1843 .

7

1844 .

. 5

1845 .

. 5

1846 .

. 17

1847 .

. 20

1853

. . 18

1848 .

. 20

1854

. . 13

1849 .

2

1855

. . 4

1850 .

. 6

1856

. . 0

1851 .

. 13

1857

. . 10

1852 . .

. 17

These numbers, as I believe, give a very fair measure of great magnetic disturbances in
each year. There is no appeai-ance of decennial cycle in their recurrence. Nor does
the number of distui-bed days appear- to have any distinct relation to the magnitude of
diumal change, as will be seen on comparing the list of omitted days with the curves at
the end of this memoir.

I trust to have another opportunity of explaining" more fully the reasons which have
induced me to separate entire days of distui-bed observations from the general mass,
instead of separating special observations on every day when their departure from the
mean exceeds a previously-defined limit, as has usually been done in late years. For
the present, I will only remark that every digest may be considered in some measure
satisfactory which actually renders account of the influence of evei-y observation, but
that the method which I have followed, and which puts it in my power completely to
dissect the whole storm occui-i-ing on each disturbed day, appears to me much more
satisfactory than any other.

Reverting now to the reductions which form the special subject of this memoir,
I mil fii-st state that the curves which occupy the four Plates XVI.-XIX. are formed
from the means to which I have referred, by comparing the mean for each hour witli
the mean for the twenty-four hours, and using their difference to form one of the
coordinates, — the horizontal ordinate to the left being the measure of hourly westerly
declination (as compared with the mean for the twenty-four hours) of the needle's north
end, expressed in terms of the whole horizontal force for the year by dividing its
measure in minutes of arc by 3438 ; and the vertical ordinate upwards being the
measure of hourly horizontal force (as compared wdth the mean for the twenty-four
hours) acting in the magnetic northerly direction on the needle's north end, expressed
in terms of the same horizontal force. The origin of coordinates (the intersection of
the straight lines in each diagram), fi-om the nature of the process, necessarily repre-
sents the mean declination and mean horizontal force in each month.

Now the means for each month are themselves svibject to an annual inequality,
which, it seems probable, does not depend on the same causes that produce the secular
changes. From 1841 to 1847 the mean secular change of western decimation appears
to proceed at the rate of — 4'-2 nearly per annum; and from 1848 to 1857 the rate is
about — 7'-9 per annum. Applying the proportional parts of these, with changed sign,
to the mean of the determinations for months of the same name through their proper

314

JIE. G. B. AIEY ON THE DIUENAL

periods, comparing each so corrected result with the mean of all, and converting the
difference into parts of horizontal force, the following excess for each month is found : —

Annual Inequality of Western DecUnation.

Period 1841-1847.

Period 1848-1857.

Januaiy . .

-O'OOOT

-0-0002

February

-0-0004

-0-0003

March . .

-0-0006

-0-0001

April . . .

-0-0007

+0-0001

May . . .

-0-0004

+ 0-0001

Jime . . .

+ 0-0001

0-0000

July . . .

+0-0002

+ 0-0003

August . .

+0-0004

+ 0-0002

September .

+ 0-0010

0-0000

October .

+ 0-0005

-0-0001

November .

+0-0006

-0-0001

December .

0-0000

+ 0-0001

Treating the Horizontal Force m the same way, it is necessary to observe that, for the
first period, the secular change can be derived only from the monthly means of Hori-
zontal Force (as the Deflection Apparatus was not used in the earlier years), and that
for this purpose several corrections must be made to the printed numbers, either for
changes in the position of the scale or mirror, or for the omission of constants in the
scale reading (as \vill be fully explained in the Greenwich Observations, 1862). The
annual rate is +0-0012. For the second period (Greenwich Observations, 1859), the
annual rate is +0-0022. The year 1843 is omitted because adjustments were changed
in the middle of the year, and 1847 because one month is defective. Thus we obtain —

Annual Inequality of Northern Horizontal Force.

Period 1841-1846.

Period 1848-1857

January .

. +0-0004

+ 0-0004

February

. -0-0006

+0-0003

March .

. -0-0006

0-0000

April . .

. +0-0005

+0-0002

May . .

. -0-0013

-0-0004

June . .

. +0-0002

0-0000

July . .

. -0-0001

— 0-0004

August .

. -0-0001

-0-0006

September

. -0-0002

-0-0010

October .

. . +0-00()1

+ 0-0002

November

. +0-00(I5

+0-0005

December

. . +0-0008

+0-0008

Althougli there arc irregularities, the general law of these numbers is sufficiently
distinct. There is nothing surprising in the sliglit diminution of tlie luunbers in the

INEQUALITIES OF TEREESTEIAL MAGNETISM. 315

second period, as compared with those of the first ; for, as we shall see, every inequality
of Declination and Horizontal Force is much larger in the period 1841-1847 than in
the period 1848-1857. Some great cosmical change seems to have come upon the
earth, affecting in a remarkable degree all the phenomena of terrestrial magnetism.

If now we desuvd to refer the hourly state of magnetism to the state corresponding
to a uniform secular progression through the course of each year, we must apply the
ntimbers just found (theu- irregularities being first smoothed do^\^l), with changed signs,
as ordinates froiu the intersections of lines in the diagrams ; and we should so obtain the
new point of reference for all the hourly points in each month-diagram. No change is
produced in the yeai'-diagrams. It does not appear, so far as I can see, that anything
is gained by this. I should have been glad to find that my new point of reference was
so related to some one of the hourly points that I could be justified in fijjdiig on that
hourly point as a magnetic state which is independent of the periodical daily disturb-
ances. For instance, if the new point of reference bore a constant relation to the point
corresponding to 12'', I should have concluded that there is no diumal disti;rbance at
12''. I haA'e not, however, succeeded in finding a point which possesses this property.

I have now to call attention to the remarkable change in the magnitude and form of
the diurnal curves representing the hourly magnetic forces in the horizontal plane.
From 1841 to 1848 (see Plates XVI. and XVII.) their magnitude very slowly increases,
with a small change of form. From 1848 to 1857 (see Plate XVII.) their magnitude
very rapidly diminishes, with a great change of form. Possibly one step in the physical
explanation of the change may be made by comparing the change from 1848 to 1857
(in Plate XVII.) with the change from the summer months to the winter months (in
Plate XIX.). It would seem that the later years have become entirely winter years;
and this seems to imply that the magnetic action of the sun on the earth's southern
hemisphere has remained nearly unaltered, while that on the northern hemisphere has
undergone a great diminution.

I will now allude to the curves representing the hourly state of Vertical Force, as
referred to the mean on each day. The force in these is represented by a simple ordi-
nate, the numerical value of which will be found, either in the preceding pages of this
paper, or in the printed books to which I have already referred. On examining the
curves in the separate months. Plates XXII. and XXIII., it will be seen that there is
considerable difference between those of the first period and those of the second period,
both in the place of " node " (or intersection of the curve vnth. the mean line) and in
the magnitude of ordinates ; also that in the fii'st period there is a sensible difference of
magnitude of ordinates between summer and winter, and in the second period a sensible
difference in the place of the " node " between summer and winter. On referring to the
cur\'es for the different years, a very great change will be found. From 1847 to 1849
the magnitude of the ordinates has somewhat increased; from 1849 to 1850 it has
increased still more; and no diminution follows. And on observing the place of the
node, a still more remarkable change will be seen. In 1846 the descenduig node is at

816 ON THE DIUENAL INEQUALITIES OF TEEEESTEIAL MAGNETISM.

llf' nearly; in 1847 it is at 9'' nearly; in 1849 at 7^ nearly; in 1850 at b^; in 1851
at i^ ; and there it continues \nth little alteration. (The loss of the observations of
1848 is here unfortunate.) It is important to obsei-ve that, though the instrument
was changed in 1848, the change in the place of the node did not then occur suddenly;
it had begun with the old instrument, and continued to advance gradually for several
years \A-ith the new instrument.

I have sought for collateral evidence of this remarkable change, but hitherto without
success. I have received observations which support the determinations for the earlier
period, but I have not yet found any corresponding in date with those of the later period.
I have no reason, however, to believe in the possibility of any error. And the change
in magnitude is not greater (though in reverse order) than that for the forces in the
horizontal plane ; and the change of law is not more striking.

These are the principal results that I have yet obtained from discussion of the obser-
vations on the less disturbed days. A reduction of the observations on the more
disturbed days is far advanced, and may be the subject of another communication.

The following are the subjects of Plates XVI.-XXIII.

Diurnal Curves of Combination of Declination and Horizontal Force.
Plate XVI. Mean of every day in each year, 1841 to 1847.
Plate XVII. Mean of every day in each year, 1848 to 1857.
Plate XVIII. Mean of every day in each nominal month through the period 1841

to 1847.
Plate XIX. Mean of every day in each nominal month through the period 1848
to 1857.
And

Diurnal Curves of Vertical Force.

Plate XX. Mean of every day in each year, 1841 to 1847.
Plate XXI. Mean of every day in each year, 1849 to 1857.
Plate XXII. Mean of every day in each nominal month through the period 1841

to 1847.
Plate XXIII. Mean of every day in each nominal mouth through the period 1849

to 1857.

[ 317 ]

XIV. Researches on the Refraction, IHspei'sion, and Sensitiveness of Liquids.
By J. H. Gladstone, Ph.D., F.R.S., and the Rev. T. P. Dale, 31. A, F.R.A.S.

Ecccivcd February 5, — Eead March 5, 18G3.

Ix a previous paper "On the Influence of Temperature on the Refraction of Light*,"
we started some inquiries which have been since pursued, and we now lay before the
Royal Society some of the later results.

The same apparatus has been employed, with a hollow prism of 61° 0' angle, and the
method of observation has been essentially the same. But experience has led to some
modifications, the most important of which is this : instead of attempting to take the
angular measurements at certain foredetermined temperatures, aslO°C., 20° C, they
were taken first at the temperature of the room, whate^"er that might be, and then at
such other temperatures as seemed to offer the most trustworthy results. This involved
more calculation, but it still saved time, and secured greater accuracy. The plan of
measuring to 10" was abandoned as a useless nicety; but, as a rule, two or more
observations of each fixed line at each temperatui'e were taken, and if they differed
slightly the mean was adopted, but if the discrepancy amounted to 2' or 3' the observation
was repeated. The average of these observations of the lines A, D, and H at different
temperatures gave the refractive indices which are placed together in the Table that
constitutes Appendix I., and they afford the data for nearly all the comparisons about
to be instituted. Appendix II. contains the mean determinations made of the refractive
indices of some of these liquids for a larger number of the lines at the temperature of
the room. To it have been added some observations on other liquids, and deter-
minations published in our former papers, so as to render it as complete as possible for
any w^ho may desii'e to investigate the irrationality of the spectrum, or the truth of the
formulae of Caucht.

An attempt has been made to determine the amount of probable error, not so much
absolutely as with reference to the different purposes for which the observations have
been made. The conclusions arrived at are as follows :—

Where the refraction of diff"erent fixed lines at the same temperature is compared,
the probable error is very small. The measurements may be easily obtained accurate
to +1', corresponding to about +0*0002 in the refractive index, and thus the relative
refraction of A, D, and H in Appendix I., or of all the lines in Appendix II. for any
one substance will rarely differ from the truth by more than that amount.

When the refraction of a substance at one temperature is compared with its refraction

* Philosophical Transactions, 1858, p. 887.
MDCGCLXIII. 2 X

318 DE. .T. H. GL-VDSTONE -\^T) THE EEY. T. P. DALE OX THE EEFRACTIOX.

at another temperature, there exists a source of error iii the determmation of the precise
temperature of that part of the liquid through which the solar beam is passing at the
time Avhen the measurement is taken. It is diiRcult to avoid this error, or to estimate
its amount. It is, as may be supposed, generally greatest at the temperatures fiu'thest
removed fi'om that of the surrounding objects, and in these cases there is reason to fear
that it not unfrcquently amounts to 1 or 2 degrees Centigrade. Even at the ordinary
temperature an error may arise from the heating power of the sunbeam that passes
through the liquid, and which may not affect the thermometer equally with the substance
whose refraction is measured. In some of our more exact and our later determinations
a strong solution of alum in a flat-sided glass was interposed in the path of the ray to
reduce its heating power.

"Wliere the refi-action of oue substance is compared with that of another, error may
also arise from inaccuracy in obtaining the minimum deviation. Though several adjust-
ments have to be made, the error from this source is practically confined -within very
narrow limits, and rarely if ever passes beyond the fourth place of decimals even with
very dispersive substances. This error was not so well guarded against in the observa-
tions recorded in oui- pre\ious paper ; and it may also affect the determination of the
sensitiveness of a few substances, namely those where a different adjustment of the
prism was made at different temperatures ; but these are easily known, as that was only
done for low temperatures such as 8" C, and they are all marked in Appendix I. with an
asterisk. In order to be rigidly correct, the hollow prism ought to have been adjusted
afresh for minimum deviation in the case of each line and at each temperature, but the
movement of the apparatus necessitated by this would practically have introduced greater
errors than resulted from the neglect of it. Yet this has an appreciable effect on the
length of the spectmm in highly dispersive substances ; and in order to obviate the en-or
as much as possible in the later measurements of such substances, care was taken to fix
the minimmn de%'iation not for either of the extremities, but for the middle of the
spectrum. It would not have been difiicult to make a coiTection by the usual formula
for a small deviation fi-om the minimum angle, but we doubted whether practically any-
thing would be gained, considering the greater complexity of the calculation.

If the indices of refraction were to be considered not relatively, but absolutely, other
sources of error would have to be taken into account ; for instance, inaccuracy in the deter-
mination of the prism-angle, faults of workmanship in the apparatus. For these it is
more difficult to assign a limit : they may even affect the thii-d place of decimals, whereas
the combined errors from all the other sources are probably confined to the fourth place.
But the ahmlute accuracy of an index is of minor importance in the present research.

The purity of the liquids experimented on is of course a matter of tlie utmost con-
sequence. ^^^len commercial specimens were employed they were always purified, or
their purity ascertained. Many of the liquids were prepared in Dr. Gladstoxe's labo-
ratory with special reference to this inquiry, and many others were kindly placed at our
disposal by those chemists who had paid special attention to them, and we have

DISPEESIOX, AND SENSITIVENESS OF LIQUIDS.

319

generally taken their word for the purity of the specimen. In this way we are under
obligations to Professor AVilliamsox, Professor Hofm.van, Professor Fkankland, Dr.
Wakkex De la Rue and Dr. Hugo Muller, Mr. BucKTOJf, Dr. Odlixg, Mr. A. H. Church,
Mr. Greville "Williams, and Mr. Piesse, to whom wo return our best thanks.
The present paper takes up five points.
T. The relation between sensitiveness and the change of volume by heat.
II. The refraction and dispersion of mixed liquids.

III. The refraction, dispersion, and sensitiveness of different members of homologous
series.

IV. The refraction, dispersion, and sensitiveness of isomeric liquids.
V. The effect of chemical substitution on these optical properties.

Section I. — The relation bettoeen Sensitiveness and the Change of Volume hy Heat.

Having examined now about ninety different liquids, we have uniformly found that
the refraction diminishes as the temperature increases. This property we have already
named " sensitiveness."

We have uniformly found also that the spectrum diminishes in length as the tempe-
ratui'e increases. In a very few instances this diminution is lost within the limits of
errors of obseiTation, but we believe it always occurs.

This diminution in length is progressive, the different rays being more sensitive in
the order of their refrangibility. The following observations on a most dispersive and
sensitive substance exhibit this : —

Substance.

Temp.

Eefractive indices.

A.

B.

D.

E.

G.

H.

oC.
11
36-5

1-6142

1-6207

1 -6.'i.S.3

1-6465
1-6248

1-6584
1-6.S6''

1-6836

1-70Q0

1-5945 1-6004 1-6)20

i-nnnnl i-6s27

Difference

0-0197 0-02031 0-0213

0-0217 0-0222! 0-0236

0-0263

That there is some intimate connexion between the sensitiveness of a liquid and its
change of volume by heat was pointed out in our former paper ; and our subsequent
expeiiments only confirmed this opinion.

It became therefore a matter of interest to determine, if possible, what this relation is.
The determinations of the sensitiveness of bisulphide of carbon, water, benzole, alcohol,
wood-spirit, fousel-oil, ether, acetone, acetic acid, formic, acetic, and butyric ethers,
and the iodides of methyl and ethyl afforded an opportunity of examining the matter,
since the alteration of their volume by heat has been very accurately determined by
KoPP and others ; cumole, xylole, nitrobenzole, hydi-ate of phenyl, oil of turpentine,
rectified oil of Portugal, eugenic acid, bromofonn, and saUcylate of methyl also answered
the same purpose, since we determined the expansibility of the specimens employed for
measuring the refr-active mdices at different temperatiues.

2x2

S20 DE. J. H. GLADSTOXE A^'D THE EET. T. P. DALE ON THE EEFEACTION,

In the case of every one of these liquids the refractive index of any ray alters less
rapidly than the volume ; but it was found that the refractive index minus unity, multi-
plied bv the volume, gives nearly a constant.

It is othermse with the contraction of the spectnim itself. In some cases, as bisul-
phide of carbon, it contracts much more rapidly than the volume increases, and in other
cases, as ether, much less rapidly.

Here it must be borne in mind that every refi-active index contains at least two
coefficients. Whatever may be the physical reason, and to whatever extent we may
accept such theoretical explanations as those given by Caucht, Lubbock, Sir- William

B C

Ha^iiltox, B. Powell, and others, the formula ^=:A + ^ + ^+ . .. does certainly give

results ven- near the truth, ^ being the refractive index, a the length of an undulation,
and A, B, C coefficients depending on the nature of the medium. As we must employ
A, B, C for the fixed lines of the spectrum so designated by Fraunhofer, we shall write

the above formula for the future f,(,=^+ ^+ -^4 + . . . and shall suppose «' and all sub-
sequent coefficients too small to be sensible within the limits of error. Hence we have
V the coefficient of refraction, and y. the coefficient of dispersion ; and v may e\idently be
considered the refractive index of any substance freed from the influence of dispersion.
As it appears that the function f* — 1 is of peculiar interest in these investigations, we
propose giving it a distinct name, that of " refractive energy" this number really repre-
senting the mfluence of the substance itself on the rays of light, (f* — 1) x vol., or, which

is the same thing, ||^~^ ', we propose calling the ''specific refractive energij."

As the value of jk- for any particular luminous ray is affected by the dispersion, it was
clearly desii-able to calculates in certain cases, and see whether (c — l)x vol. would give
a constant. Some doubt rests on the position of this theoretical limit; but its value
was calculated by the formula given on pages 82 and 132 of Badex Powell's treatise
' On the Undulatory Theory as applied to Dispersion.' It will easily be seen by refer-
ring to the example on p. 132, that, in consequence of an accidental relation between
the coefficients, v=fj,a — Z{(^2^l^n) to very considerable exactness. This formula has
been used by us, but in all cases given below the results have been verified by the accu-
rate one.

Bisulphide of carbon and water were the liquids chosen, being very definite sub-
stances and extremely different in their degree of expansibility, water also having the
advantage of a v(n-y irregular rate of change of volume. The refractive indices of the
fixed lines B, F, and H (on which the calculation of c depends) were determined at
different temperatures with every precaution*.

* Tlie determinatious for water in the accompanying Tabic were substituted during the printing for less
accurate numbers.

DISPERSIOX, iV^'D SENSITIVENESS OF LIQUIDS.

321

Substance.

Temp.

Eefractivc indices.

B.

ff.

H.

Bisulphide of Carbon...
Bisiiiphidi? of Carbon...
Bisulpiiide of Carbon...

11

22-5

36-5

1-6207
1-6116
1-6004

1-6584
1-6484
1-6362

1-7C90
1-6972
1-6827

Water

1
15-5
27-5

48

1-33005 1-33685
1-3298 1-3364
1-3289 1-3355
1-32595 1-33245

1-3431
1-3426
1-3416
1-3387

Water

Water

The subjoined Table contains the calculations founded on these numbers. Column I.
gives the re&active index of the theoretical limit, or v. Column II. the specific rcfracti"\'e
energy for this limit, or (v— l)vol. Column III. the specific refractive energy for the
line B, or (f*!, — l)vol. Column IV. the same for H, or (/aj£ — l)vol. Column V. gives
what Newton called the "absolute refractive power" reckoned for the limit, or
(^^-I)vol.

Substance.

Temp.

Volume.

I.

II.

III.

IV.

V.

Bisulphide of Carbon...
Bisulphide of Carbon...
Bisulphide of Carbon...

11

22-5

36-5

0-9554
0-9685
0-9854

1-5960
1-5865
1-5753

0-5694
0-5680
0-5669

0-5930
0-5923
0-5916

0-6773
0-6752
0-6727

1-4782
1-4714
1-4599

Extreme difference

25-5

0-0300

0-0207

0-0025

0-0014

0-0046

0-0183

Water

1
15-5

27-5
48

0-9999
1-0007
1-0034
1-0109

1-3227
1-3228
1-3216
1-3193

0-3227
0-3230
0-3227
0-3227

0-3300
0-3300
0-3300
0-3295

0-3431
0-3429
0-3428
0-3429

0-7495
0-7497
0-7492
0-7486

Water

Water

Water

Extreme difference

47

-0110

0-0035

0-0003

0-0005

0-0003

0-0011

It thus appears that the specific refractive energy is nearly a constant, whether we
take the limit v or the line B as the basis of calculation. The " absolute refractive
power" is evidently not a constant.

The following Table exhibits the specific refractive energy at various temperatures
for some of the other liquids mentioned above, the selection being made not of those
which give the most accordant results, but of those which may be considered repre-
sentative bodies, or of which we happen to possess observ^ations at the longest range of
temperature. The columns are numbered as before, the only difference being that in
Column III. the Hne A is taken instead of B. The refi-active indices observed will be
found in Appendix I., or in our previous paper.

322 DE. J. 11. GL.ADSTOXE AJST) THE EEV. T. P. DALE ON THE EEFEACTION,

Substance.

Temp.

Volume.

I.

II.

III.

IV.

6

20
40
60

0-9132
0-9326
0-9534
0-9762

1-3598
1-3518
1-3435
1-3347

0-3286
0-3280
0-3275
0-3260

0-3340
0-3337
0-3332
0-3326

0-3480
0-3478
0-3473
0-3473

Alcohol

+ 0-0630

— 0-0251

-0-0018

-0-0014

— 0-0007

22
31
40

1-0305
1-0436
1-0573

1-3476
1-3434
1-3390

0-3582
0-3584
0-3584

0-3650
0-3653
0-3654

0-3807
0-3811
0-3815

+ 0-0268

-0-0086

+ 0-0002

+ 0-0004

+ 0-0008

Iodide of Ethvl

23-5
36

48

0-9440
0-9583
0-9730

1-4878
1-4795
1-4718

0-4604
0-4595
0-4590

0-4720
0-4712
0-4710

0-5116
0-5103
0-5108

Iodide of Ethyl

Iodide of Ethvl

+ 0-0290

— 0-0160

-0-0014

-0-0010

— 0-0008

20-5

2.S-5

40

47-5

1-0228
1-0305
1-0432
1-0517

1-3G56
1-3624
1-3579
1-3543

0-3739
0-3734
0-3733
0-3726

0-3794
0-3792
0-3791

0-3786

0-3969
0-3967
0-3964
0-3963

Acetic Acid

Acetic Acid

+ 0-0289

— 0-0113

-0-0013

-0-0008

-0-0006

10-5

23

39

1-0125

1-0278
1-0481

1-4777
1-4704
1-4601

0-4836
0-4834
0-4822

0-4940
0-4939
0-4929

0-5371
0-5370
0-5353

Benzole

+ 0-03.56

-0-0176

-0-0014

-0-0011

-0-0018

Oil of Turpentine

Oil of Turpentine

Oil of Turpentine

24
41
47

1-1621

1-1778
1-1831

1-4521
1-4449
1-4414

0-5253
0-5240
0-5322

0-5341
0-5323
0-5308

0-5630
0-5611
0-5594

+ 0-0210

-0-0107

— 0-0031

-0-0033

-0-0036

18
27-5

0-9349
0-9412

1-5159
1-5119

0-4818
0-4817

0-4942
0-4934

0-5403
0-5383

+ 0-00G3

-0-0040

-0-0001

— 0-0008

-0-0020

These results suffice to show that uiiy refractive index minus luiity, niultiphed into the
vohuue or divided Ijy tlie density, gives nearly a constant. Indeed the numbers gene-
rally fall within the limits of experimental error. It is worthy of notice, too, that in the
majority of cases, as bisulphide of carbon or alcohol, the products show a tendency to
diminish as the temperature rises ; but there are other case.s, as formic ether, where
the tendency seems to be to increase. Again, in some cases (v — l)vol. gives the most
accordant results; in other cases (ja-H — l)voI.

Supposing tliis true of the coefficient of refraction, does the law equally hold good

of the coefficient of disiK-rsion ? It is evident from the formula jm^ii-{-j^ that in the
difference of any /y. and c, or of the refractive indices of any two rays, we have a measure

DISPEESION, AND SEN-SITIYENESS OF LIQUIDS. 323

of the cooffioient of dispersion x.. For convenience sake we adopt (ji'h—U'a as this mca-
sui'e; ami this is what is headed "Dispersion" in many subseqnent tabk'S. It is the
same as " Length of Spectrum " in our former paper. This, multiplied by the Aolume,
or (|£Au— |U,J vol., we call "Specific Dispersion." But, as already stated, there is no
simple relation holding good for different liquids between the increase of volume and
the decrease of dispersion by heat. The phenomena seem independent.

We therefore arrive at the empirical law, that the refractive energy of a liquid varies
directly with its density %mder the influence of change of temperature, or, in other words,
that the specific refractive energy of a liquid is a constant not affected hy temperature.
But in concluding thus, we wish it to be borne in mind that there is some influence,
arising wholly or partially from dispersion, which we have not been able to take into
account, but Avhich gives rise to the slight progression of most of the calculated products,
and perhaps to the non-inversion of the sensitiveness of water at 4° C, remarked on
already by J.oiix and ourselves.

Section II. — Tlie Refraction and Dispersion of Mixtures of Liquids.

This subject engaged the attention of M. Deville as far back as 1842*; and of late
years Messrs. H.vndl and A. and E. WEissf have published elaborate papers on it, but
without arriving at a solution of the question. M. HoEKlf, however, proceeding on the
assumption of Fresnel, that the density of the ether enclosed in a medium is jM-^ — 1 if
the density of the ether in space is 1, found that the formula deduced from it gave
numbers closely agreeing with those found experimentally by Deville for mixtures of
alcohol and water, or wood-spirit and water. Yet it happens that these results can
equally well be explained on the supposition that the specific refractive power of a mix-
ture is the mean of the specific refractive power of its components. And this sujiposi-
tion seemed also warranted by most of the results of Messrs. Weiss, and by several that
we oiirselves obtained.

It was clearly desirable to test these two, or any other suppositions, in a case where
the refi'active indices of the liquids mixed were very wide apart. Fortunately bisul-
phide of carbon and ether, substances almost at the opposite limits of the scale, were
foimd to mix, and that without perceptible condensation, not indeed in equal volumes,
but in the proportion of three volumes of ether to one of the bisulphide at low tempe-
ratui-es, and in the proportion of two to one at 20° C.

Two experiments were made at diifei'ent seasons on mixtures of commercially pure
specimens of these substances. The greatest care was taken to prevent evaporation as
far as possible during the progress of the experiments.

It will be seen that in a case such as this, where there is no condensation on mixture,
the calculation is much simplified, since for the specific refractive powers we may sub-

* Ann. de Chim. et de Phys. (ser. 3) tome v. p. 129.

t Wien. Ber. xxv. xxx. xsxi. and xxxiii. 589-656. % Poggcndorff's Aimalcn, cxii.

324 DE. J. H. GLADSTONE AND THE EEY. T. P. DALE ON THE EEFEACTION,

stitute the refractive indices themselves, and the supposition will stand thus : the refrac-
tive index of a mixture is the mean of the refractive indices of its components. And in
such a case Hoek's formula resolves itself into the mean of ,(/-- — 1.

Liquid.

Temperature.

Specific
gravity.

Refractive index.

A.

D. j H.

8°C.
8

8

8
8

1-2790

1-6184

1-6366
1-3575

1-4235

1-4272
1-4323

1-7093
1-3692

1-4480

1-4542
1-4619

Ether

0-7374 1 1-3542

Mixture of 1 vol. Bisulph.l
and 3 vols. Ether J

0-8710

1-4165

1-4202
1-4247

■ ' 1

20
20

20

1-26S5
0-7246

0-9059

1-6121
1-3487
1-4305

1-4365
1-4417

1-6299
1-3525

1-4390

1-4450
1-4509

I-7OO8
1-3636

1-4686

1-4760
1-4845

Ether

Mixture of 1 vol. Bisulph.l
and 2 vols. Ether J

These two experiments confirm one another, but they fail to support either hypothesis.
The calculation founded on ^-—1 gives numbers which are far too high ; and though the
mean of the indices is certainly much nearer to the calculated numbers, the discrepancy
in each case is beyond the limits of probable error. The calculation for A is certainly
nearer than that for H, but evidently not much would be gained by assuming the
theoi'etical limit as the basis of calculation.

Similar experiments were made by mixing aniline and alcohol of 90 per cent, together
in equal volumes, but in this case a slight condensation ensues.

Liquid.

Temperature.

Specific
gravity.

Refractive indices.

A.

D.

H.

23-5° C.
23-5

23-5
23-5

1-0073
0-8154

0-9167

1-5642
1-3576

1-4621

1-4636
1-4668

1-5772
1-3614

1-4707

1'4721
1-4764

1-6263
1-3729

1-5018

1-5025
1-5070

Mixture of equal vols., 1
mean of tw o experiments J

Mean deduced from spe- 1
cific refractive powers... J

This shows precisely the same thing as the previous mixture ; and, as m that case, the
experimental numbers are slightly below those deduced from the mean of the specific
refractive powers. This is also the case in other mixtures examined ; yet no other simple
formula gives numbers so closely approaching those obtained by cxpci-iment. The hypo-
thesis that tlie Hfec'ific refractive power of a mixture of liquids is the mean of the specific
refractive poioers of its constituents must therefore stand as the nearest approximation
to the truth.

In one or two cases, as in the mixtures of sulphuric acid and water examined by
Messrs. Weiss, the refi-action is not at all in accordance with the above theory. This

DISPERSION, AXD SENSITIVENESS OF LIQUIDS,

325

probably arises from some chemical combination between the two substanceSj different
hydrates being formed.

We hope to revert to this subject more fully on some future occasion, when we
propose extending our inquiry to solutions of solids.

Section III. — The Befraction, Dispersion, and Sensitiveness of different member's

of Homologous Series.
In our paper on the influence of temperature we remarked an advance in refraction
and dispersion with each increment of Cg Hj in the alcohol series. This has been
examined more carefully, and the investigation has been carried much further in the
same dii-ection. The new data for the comparisons are given in Appendix I., from
which the subsequent Tables are calculated, a reduction of the indices to 20° C. of tem-
perature being always made, and the sensitiveness being calculated for 10 degrees rising
from that temperature. The length of the spectrum, or the dispersion, is also reckoned
at 20° C. The refractive index for only one line is given, in order to save space ; and
A is the line chosen, as it is least affected by dispersion. Where two specimens of the
same substance have been examined, the mean of the observations has usually been

adopted.

The Alcohol Series.

Liquid.

Formula.

Refractive
index of A
at 20' C.

Length of
spectrum or
dispersion.

Sensitiveness Specifio g j,.^ Specifio
for 10= C. refractive dispersion, sensitive-
energy, nt-ss.

Methylic Alcohol

Etbylic Alcohol

C, H, 0^
C, H, O,
C,„H„0.,

1-3268 0-0128
1-3578 t 0-0151
1-4005 0-0174

00036 ' 0-4105 ' 0-0163 1 0-0045
0-0041 0-4482 0-0190 0-0052
0-0039 0-4»<)5 0-0-212 0-0047

Amylic Alcohol

Caprylic Alcohol

Cl6 His ^2

1-4186

0-019.')

0-0042 0-5096 0-0237 0-0051

1 i

From this it is evident that on ascending the series the refraction increases, the
dispersion more rapidly stiU, while the sensitiveness remains nearly the same.

It shoidd be borne in mind that on account of the small numbers by which the
sensitiveness is expressed, and the serious source of error arising from the difficulty of
determining the temperatiu-e with accuracy, comparisons of the sensitiveness of different
liquids camiot be so satisfactory as comparisons of their refractive indices, or the length
of the spectrum. As all the degrees of sensitiveness at 20° C. known to us lie between
0-0007 and 0-0074, we propose in future omitting the zeros, and simply stating that the
sensitiveness of methylic alcohol for instance is 36. We shall omit the zeros also in the
last three columns.

As we have already learnt the importance of comparisons of specific refractive energy,
we have added in the three last columns the refi-active energy ([/j^ — 1), the dispersion
{(ti^—^itf), and the sensitiveness ((^o^ at 20° C.—^j^ at 30° C), all dirided by the density.
It will be seen that the progression is maintained.

But some might prefer that the different alcohols should be compared, not at the
same absolute temperature, but at the same distance from theii- boUing-points. This is

MDCCCLXIII. 2 T

326 DE. J. II. GL.VDSTOXE AND THE EEV. T. P. DALE OX THE REFKACTION,

attempted in the follo^^^ng Table for 82° below the boiling-points, but as the observa-
tions do not extend nearly to that in the case of methylic or caprylic alcohols, there is
more left for calculation than is desirable.

Liquid.

Temper-iture. ' j„^l^i5; ^ Dispersion.

Sensi- Specific
tiveness. refraction.

1

Specific
dispersion.

Specific
sensi-
tiveness.

Methylic Alcohol

— 22 1-3410 i 0-0135

— 4 1-3674 0-0154
+ 50 1-38S8 0-0167

100 ' 1-3807 0-0180

33
40
40
51

4079
4515
4914
5123

161

189
211

242

40
49
50

68

Caprylic Alcohol

Here the advance of the refraction and dispersion with each addition of C^ H^ appears
again (with one exception), and the sensitiveness advances likewise ; and this is still more
evident when the numbers are di\ided by the density.

It was a matter of interest to compare with these results the refractive indices of other
homologous series belonging to the same group.

Iodide of Methyl Series.

Substance.

Formula.

Eefractivo
index A.

Dispersion.

Sensitirc-
ness.

Specific
refracrive
energy.

Specific
dispersion.

Specific
sensi-
tiveness.

Iodide of Methyl

Iodide of Ethyl

C, H3 I
C4 H, I
C« H, I
CioHijI

1-5171
1-5026
1-4934
1-4804

0-0460
0-0420
0-0408
0-0335

73

66
63
50

2359
2614

209
218

33
34
36
33

Iodide of Propyl

2882 235
3213 224

In this case the refraction, dispersion, and sensitiveness are also progi'essivc, but in the
opposite direction, for they all decrease as we ascend the series, mstead of increasing, as
was the case with the alcohols. This may be attributed to the larger proportion of
iodine which the earlier members of the series contain, for iodine has a very great influence
on the rays of light. If the numbers be divided by the specific gravity, the progression
becomes in the direction of increase as with the alcohols, both in regard to refraction
and dispersion, while in regard to sensitiveness the four members give nearly the same
number, as was also the case in the series of alcohols.

Formic Ether Series.

Substance.

Formula.

Kcfractive
index A.

Dispersion.

Sensi-
tiveness.

Specific
refractive
energy.

Specific Specific

dispersion. ,.««"«'-
1 tiveness.

Formic Ether

C,H,,0, C, II 0,

c,ii;,o,c^ n,o;

1-3549
1-3659
1-3707

0-0154
0-0157
0-0164
0-0168
0-0172

44
48
44
48
42

3905
4152
4333
4402
4502

168
178
191
191
198

48
55
51
54

48

Riityric Ijther

C4H, 0, ('^ 11- O3 ' 1-3864

Valerianic Ether

C,II,0,C.,H,03

1-3908

DISPERSIOX, AXD SEXSITR-EXESS OF LIQUIDS.

327

Here, as in the case of the alcohols, there is a progressive increase of refraction, dis-
persion, and specific energy. The numbers representing the sensitiveness appear I'ather
irregular, but it is difficult to say how far this may be due either to impurity of speci-
mens or to errors of observation.

During the progress of these experiments we found Professor Delffs has preceded
us in examining the refi'action of members of the formic ether series*. He gives as the
indices of the red ray —

Formic Ether .... 1-3570

Acetic Ether . . . . 1-3672

Butyric Ether . . . . 1-3778

Valerianic Ether . . . 1-3904

CEnanthylic Ether . . 1-4144

Laurostearic Ether . . 1-4240

He does not note the temperature. His conclusion is, that " the indices of refraction
of the compound ethers increase -with theii- equivalents." His experiments afforded him
no means of drawing a conclusion in regard to the dispersion ; and the sensitiveness was
a property not fully recognized at that time.

Acetate of Ethyl Series.

Substance.

Formula.

RefractiTO „.
index A. Dispersion.

Sensi-
tiveness.

Specific
refractive
energy.

Specific
dispersion.

Specific
sensi-
tiveness.

Acetate of Ethyl C^ H^ O, CjH.^Og

Acetate of Amyl €,„ Hn O, C^ H^ O3

Acetate of Capryl Cig H,, 0, C^ H3 O3

1-3659
1-3911
1-4088

0-0157
0-0172
0-0211

48
43
58

4152
4506

178
198

55

49

This resembles the preceding series, or that of the alcohols, as might be anticipated.
Professor Delffs in his second paper gives

Acetate of Methyl .... 1-3576
Acetate of Ethyl .... 1-3672
Acetate of Amyl .... 1-3904

He also gives the following indices, which bear similar witness : —

Butyrate of Methyl
Butyrate of Ethyl
Butyrate of -Imyl
Oxalate of Ethyl .

1-3752

1-3778
1-4024
1-3803

Oxalate of Amyl .
Formiate of Ethyl
Formiate of Amyl

1-4108
1-3570
1-3928

* Poggendorff's Annalen, Lsxxi. 470.

2t2

328 DE. J. H. GLADSTONE A]S'D THE EEY. T. P. DALE OX THE EEFEACTION,

Hydride Series.

Liquid.

_ , 1 Eefractive i _:
Formula. i injex A. Dispersion.

Sensi-
tiveness.

Specific
refractive
energy.

Specific Specific
dispersion, s'^nsi-
'^ tiveness.

Hydride of CEnanthyl

Ci^HjsH ' 1-3898 0-0172 55
C,„H,,H i 1-3971 1 0-0170 47

5499
5522

242 77
236 1 65

This also bears similar e%idence.

Mercuric and Stannic Series.
Through the kindness of Mr. BrcKTOX and Dr. Fkaxkland, we have been able to
examine some of the combinations of the metals with the compound radicals. Unfor-
tunately the specimens had all suffered a partial decomposition on standing, and thus
the results are not so trastworthy as might be desired.

Substance.

Formula.

Eefractive
index A.

Dispersion.

Sensi-
tiveness.

Specific
retractive
energy.

Specific
dispersion.

Specific
sensi-
tiveness.

C,H,Hg
C^H.Hg

1-5241
1-5162

0-0431
0-0416

43

?

1707
2112

140
170

14

?

Stannic Ethyl-methyl ... <
Stannic Ethyl

C.,H,"1 c
(C4H5),Sn

1-4550
1-462)

0-0313
0-0301

50
50

3727
3876

256
268

41

42

The specific index here, as in every preceding case, increases mth the addition of
C2 H2 ; the great absolute influence of mercury on the rays of light makes itself manifest,
as iodine did, in the inversion of the order of progress in regard to actual refraction and
dispersion ; it should be remembered that mercuric methyl contains close upon 87 per
cent, of mercury.

It is worthy of notice that in the two series last given there occur the heaviest and
about the lightest known liquid in the whole range of organic chemistry ; and the light
hydiide of ccnanthyl has a veiy high, and the hea'S'y mercuric methyl a verj' low specific
refractive energy.

All these series containing the compound radicals methyl and its congeners, agree in
exhihiting a progressive change in refraction and dispersion with the advancing members
of the series; hut in which direction and to what extent depend on the other sxibstances
with which the radical is combined. Yet, if we regard not the actual indices, hit these
minus unity, divided by the specific gravity, we find an invariable increase as the scries
advances. The following Tables exhibit this : —

DISPERSION, AXD SENSITIVENESS OF LIQUIDS.
Specific Eefractive Energy.

329

BadicaL

Formula.

Alcohol.

Iodide.

f^Z Formiate.
of acid.

Acetate.

S^Jy- Oxalate,
rate.

Mercuric
compound.

Stannic
compound.

Hydride.

Methyl

Ethyl

C2H3
C' H,

4105

4482

4895
5096

2359
2614

2882

3213

.3905

4152 3905

4333

4402

4502 4432

4750

4890 '.'.'.

4152
4506

4402
4724

3502
4306

1707
2112

3727*
3876

5499
5522

Propyl

Butyl

Amvl

CEnanthyl ...
Capryl

Laurostearvl .

•

C,oH„

CieH,:
C24H2J

Specific Dispersion.

BadicaL

Alcohol.

Iodide.

Ether of
acid.

Acetate. Mercuric
compound.

Stannic
compound.

Hydride.

Methyl

163
190

212
237

209
218
235

224

168

178

191
191
198

178
198

140
170

256»
268

242
236

Ethyl

Propyl

Butyi

CEnanthyl

Other Homologous Series.
It seemed desirable to examine other groups of homologous bodies in order to see whe-
ther there existed in them the same progressive change in the optical properties answer-
ing to the progressive additions of the increment C^ H^. Through the kindness of
Mr. Chitrch and others we were able so to test the benzole, the pyridine, and the
chinoline series.

Benzole Group.

Substance.

Formula,

EefractiTe
index A.

Dispersion.

Sensitive-
ness.

Specific

refractive

energy.

Specific Specific
dispersion, sensitive-
ness.

clll

1-4823
1-4835

0-0419
0-0402
0-0408
0-0377
0-0312

60
55
58
52
53

5564
5584
5583
5547
5454

483
464
472
425
362

69
63
67
60
61

Toluole

C,6H,o 1-4835
CigH,, 1-4819
Co„H" l-46q6

The first four members of this series, all of which were derived from coal-tar, bear a
close resemblance to one another, instead of showing that progression in refractive and
dispersive properties which marks all the series of the preceding gi-oup. Cymolc gives
lower numbers ; but the difficulties arising from isomerism, which we shall shortly advert
to, render any deduction fi-om this gi-oup very doubtful.

This compound contains both methyl and ethyl.

330 DE. J. II. GLADSTOXE AXD THE EEV. T. P. DALE ON THE REFEACTION,

Hydrate of Phenyl Series.
Allied to benzole and toluole are the two main constituents of creasote.

Substance.

Formula.

Eefractire
iudex A.

_,. . Sensitire-
Dispersion. ^g^^^

Specific
refractive
energy.

Specific Specific

dispersion, sensitive-

^ ness.

1

Hydrate of Phenyl C;, H^O, HO 1-5344

Hydrate of Cresyl C^ H, O, HO 1-5319

0-0503 46
0-0467 33

1
5034 475 43
5122 450 32

Pyridine Group.

Substance.

Formula.

Refractive
index A.

Dispersion.

Sensitive-
ness.

Specific

refractive

energy.

Specific
dispersion.

Specific
sensitive-

C,„ H, N

c,:h, N

C„H„N

1-4948
1-4902
1-4909
1-4946

0-0447
0-0427
0-0416
.0-0404

55
56

51

5081

458

56

57

53

5132 1 446
5244 448
5370 444

In this series the actual refractive indices are nearly the same, but somewhat u-re-
gular ; yet the density is progressive, and in such a manner that when the refractive
power is di-\ided by it, a series of increasing numbers is obtained. The dispersion
decreases regularly and more rapidly than the density does, so that an addition of C2 H.,
yields a lower number in regard to specific dispersion, though a higher one ui regard to
specific refractive energy.

Chinoline Group.

Substance.

Formula.

Refractive _. Sensitive-
index A. T1>n^"-^'i«"- ness.

Specific
refractive
energy.

Specific
dispersion.

Specific
sensitive-
ness.

C.,0 H„ N

1-5590 0-0631 55
16045 0-0783 58

5170

5639

583
730

50
54

In this case, unlike the pyridine group, which it so closely resembles (chemically
speaking), the refraction and dispersion increase rapidly, whether we consider the abso-
lute numbers or these divided by the specific gravity.

Lepidine, kindly given by its discoverer ]\Ir. C. Greville William.?, proves to be the
most refractive organic liquid known, very nearly equalling bisulphide of carbon.

This examination of other homologous groups sliows that the wfiucnce of each addition
of Qi-i Hj, which was observable thjrn({ih.out the series of the methyl (jrouj), docs not ncces-
sarihj hold (jood ivhen we jjciss to substances of (j^tiite another type. . .. . ■

PosTSCRii'T TO Sectiox III., February 2G, 18G3. — A few days after the above was pre-
sented to the Royal Society, we observed, on taking up the last number of Poggendorff 's

DISPEESIOX, AND SENSITIVENESS OF LIQUIDS.

331

Annalcn (cxvii. 353), a paper by M. Landolt "On the Refractive Indices of Fluid Ho-
mologous Compounds." He has examined, evidently with great care, the acids of the
C„H„04 type, and finds that on ascending the series the refraction and dispersion increase,
and the sensitiveness \ery slightly diminishes, with the exception of formic acid, which
appeai-s unconformable. This, however, is clearly due to the high density of that acid ;
and if we divide the numbers of Landolt by the densities, the anomaly disappears, and
we obtain a series of valuations confirmatory in every way of those drawn out in the
preceding Tables. Landolt measured, not A and H, but a and y of the hydrogen light,
which are nearly coincident with C and G of the solar spectrum.

Liquid.

Formula.

Specific refractiTO
energy ( jz k — 1 ) H- density .

Specific dispersion
(;ty-f<af)-=- density.

Formic Acid

C2 H2 O4
C4 H, 0,
Ce "r. O4
C« H, 0,

CioH,o04
C,oH,,,0,

C,4H.4 04

3024
3517
3860
4115
4318
4449
4569

91
98
105
114
121
125
129

Acetic Acid

Butyric Acid

Q^nanthic Acid

This also shows, what is apparent both in our Tables given above and in some in
Section V., that the amount of optical change is less between the higher than between
the lower members of the series.

Section IV. — The Befraction, Dispersion, and Sensitiveness of Isomeric Liquids.

There are some isomeric bodies which we know differ from one another in their
chemical constitution, while there are others to which we cannot yet assign any different
arrangement of their elements. We have exammed instances of both tliese classes.

Benzole Group. — This group offers a remarkable number of isomeric bodies differing
slightly in their physical and chemical characters : —

Substance.

Boiling-
point.

Density.

Refractive
index A.

Dispersion.

Sensi-
tiveness.

Specific
retractive
energy.

Specific
dispersion.

Specific
sensi-
tiveness.

80-8
97-5

•8667
•8469

1^4823
1^4814

0-0419
0-0402

60

5564
5684

483

474

69

103-7
119^5
113

•8650
•8333
•8658

1^4739
1-4715
1-4835

0-0377
0-0363
0-0402 ?

58
59
55

5478
5658
5584

435
435
464?

67
70
63

Toluole (tind specimen)

Cumole (frum Cuminic Acid)...
Cumole (from Wood-spirit) ...
Pseudo-curaole (from Coal-tar)

148-4
149-5
140-5

•8710
•8580
•8692

1^4825
1-4631
1-4819

0-0372
0^031 1
0^0370

56
51
52

5547
5400
5544

427
363
425

65
59
60

Cymole ^from Oil of Cumin)...
Cymole (from Camphor)

171
170

•8600
•8565

1-4696
1^4693

0-316
0-317

63

48

5460

5478

367
370

61
56

332 DE. J. 11. GLADSTONE AND THE REV. T. P. DALE ON THE EEFRACTION,

Here we have a variety of results : — isomeric bodies probably identical in refractive
index, specific energy, and dispersion (cumole from cuminic acid, and Dr. H. Muller's
pseudocumole, and the two cjTnoles); isomeric bodies nearly identical in theu' actual
optical properties, but, on account of a difference in theii- densities, differing in their
specific refractive energy (benzole and parabenzole, toluole and paratoluole) ; isomeric
bodies identical in density, but differing in optical properties (two toluoles) ; isomeric
bodies difiiering in density, and in each of the optical properties (two cumoles).

Essential Oil Groiij). — There are a large number of essential oils which consist of
carbon and hydrogen in the proportion of 5 equivs. of the first to 4 equivs. of the
second, and which chffer from one another slightly in physical characters. Through
Mr. PiESSE we obtained piu'e specimens of the crude oils, from which many of these
hydrocarbons were prepared, carefully purified, and examined. Tlaey are an-angcd in
the foUowmg Table according to their boiling-points. When two or more from different
plants appeared to be identical, or nearly so, in all their knowai physical properties
except odour, only one is given : —

Substance.

Boiling-
poiiit.

Sj:iecific
gravity.

Refractive ' .
index A. Di^P'^'-siou.

Sensi-
tiveness.

Specific

refractive

energy.

Specific
dispersion.

Specific
sensi-
tiveness.

Hydrocarbon from Turpentine

„ „ Anise

„ „ Tliyme ...
„ „ Carraway ..
„ „ Berganiot..

Bay

„ „ Cloves ...
„ „ Cubebs ...

160
160
160
160
173
173
252
259

•8644
-8580
-8635
•8530
-8467
-8510
•9041
•9260

1-4612
1-4608
1^4617
1^4610
1-4619
1-4543
1-4898
1-4950

0-0250
0-0269
0-0262
0-0261
0-0297
0-0257
0-0284
0-0302

47
48
48
48
49
47
45
41

-5319
-5370
-5346
•5391
•5456
•5337
•5417
•5345

289
313
326
305
350
302
314
326

55
56
56
56
67
55
50
44

Here, as in the case of parabenzole and paratoluole, we have five isomeric bodies with
sensibly the same refraction 1-4G1, although there are slight differences in the density
and other properties, differences that seem to be real. The thspersion varies consider-
ably ; and the difference between turpentine, the least, and bergamot, the most dispersive,
is only increased when the difference of density enters into the calculation. The sensi-
tiveness seems the same in each of the five. The hydi-ocarbon from bay seems slightly
lower in refraction ; while those from cloves and cubebs, with much higher boiling-
points and densities, are much higher in refraction and dispersion, and lower in sensitive-
ness. The specific refi-active energies of the whole group do not differ mdcly.

Surjars. — We do not propose entering now on the subject of solutions, but we may
state that solutions of cane-, gi-ape- and honey-sugar, and gum, of the same strength,
gave the same, or very nearly the same, amount of refraction and dispersion.

Compound Ethers. — It is well known that among the compound ethers pairs exist
which have the same ultimate composition and similar density, but which are broken
up by alkalies into different acids and alcoliols. If it does not matter, as to its action
on light, whether the increment C2II2 be in the electro-positive or electro-negative

DISPEBSIOX, AND SENSITIVENESS OF LIQUIDS.

333

element, these pairs will present an identity in refraction, dispersion, and sensitiveness.
Such indeed seems to be the case.

Liquid.

Formula.

Refractive
index A.

Dispersion.

Sensi-
tiveness.

Specific
relraclive
energy.

SpeciGo
dispersion.

Specific
sensi-
tiveness.

Valerianic Etlicr ...
Acetate of Amyl ...

C, H, 0, C,.,H„03
C,„H„0, C, H3O,

I-39O8
1-3911

0-0173
0-0172

42
43

4502 199
4506 198

48
49

In this also we find that Professor Delffs has preceded us, as far as refraction is

concerned. He gives

Formic Ether .... 1-3570

Acetate of Methyl . . 1-357G

and

Valerianic Ether . . . 1-3904
Acetate of Amyl . . . 1-3904

Aniline and Picoline. — These two bodies have each the ultimate composition Cj.^H^N,
but the action of chemical reagents proves tliat they are constructed very differently.
The following comparison \\\\\ show that they differ widely in refraction, dispersion, and
sensitiveness, even when the difference in specific gravity is taken into account.

Liquid.

Rational
formula.

Refractive
index A.

_. . Sensi-
Dispersion, jj^^^^^^

1

Specific
retractive
energy.

Specific
dispersion.

Specific
sensi-
tiveness.

Aniline

1-5650 0-0653 47
1-4902 0-0428 56

650
513

635

448

47
59

1

It thus appears that isomeric bodies are sometimes widely different in these ojitical
properties ; hut in nxany cases, especially when there is close chemical relationship, there
is identity also in this respect.

Sectiox V. — The Effect of Chemical Suhstitution.

The doctiine of types and substitution is fully recognized, at least by all the students
of organic chemistry. It becomes a matter of interest to determine the amount of
change in the optical properties which results from a replacement of one clement by
another, the type remaining the same. By this means we may attain to a knowledge of
the influence of the individual elements on the rays of light transmitted by them. The
data for such an inquii-y ought to be very numerous, but those which Ave have already
collected point to some conclusions.

MDCCCLXIII.

it L

334 DE. J. H. GLADSTOXE AND THE EEV. T. P. DALE OX THE EEEEACTION.

Tlic Siihsfitiifion of Jlydrogen hy an Organic Radical.

In the following instance the substitution of amyl has not produced much change
when the refractive power is divided by the density.

Liquid. Formula.

E«fractiTe t^-
index A. Dispersion

Sensi-
tiveness.

Specific Specific 1 Specific
refractive dispersion. .^™^^-
energy. "^ tiveness.

Aniline

C,oH„U
H J

1-5650
1-5130

0-0653
0-0508

47
46

550
559

635 45
554 49

Amyl-aniline ...

But a more interesting case is that of ether, alcohol, and water, which, according to
WiLLiAiisox, are of the same tj-pe, — a theory which the subjoined numbers favom*, as
the optical properties of the thi-ee are analogous, and those of alcohol are intermediate
between those of water and ether when the proper allowance is made for the diflFerence
of specific gi-a^ity. The comparison is made, not at the same temperature, but at 30°
below their respective boiling-points.

Liquid.

Williamson's
formula.

Refractive
index A.

Dispersion.

Sensi-
tiveness.

Specific

retractive

energy.

Sjiecific
dispersion.

Specific
sensi-
tiveness.

Water

1-3203
1-3460
1-3575

143

148
154

24
44

-3272
•44QQ

145
192
210

25
57
70

Alcohol

Ether

52 •4S8fi

Tlie Suhsfifufion of Hydrogen hy Oxygen.
Of the effect of this replacement we have the following instances : —

Liquid.

Formula.

Refractive
index A.

^. . Sensi- Specific

Dispersion. Hveness. refractive

energy.

Specific
dispersion.

Specific
sensi-
tiveness.

Alcohol

^■"JJs jo, 1-3578
CjHsOjJQ^ 1-3690

0-0151 41 -4482

190
162

52
35

Acetic Acid

0-0172 37

-3483

Ether

C^H, Iq
C4H5 i -
C4H, \o

1-3487
1-3659

0-0149
0-0157

51 -4868

208

71

Acetic Ether

48 -4152

178

55

Carvcne

Co Up

1-4610 0-02fil i 48 ' -5301

305
362
463

56

48
39

Carvole

C.U H,.0„ 1-48S6 0-0345 46 I -5126

Eugenic Acid ...

c;hi,o.

1-5277

0-0495

42

-4945

In all these cases the replacement of hydrogen by oxygen has increased tlie actual
refraction and dispersion, but decreased the specific refractive energy.

DISPEESIOX, AND SENSITIYEXESS OF LIQUIDS.

335

The two pairs, alcohol and acetic acid, ether and acetic ether, are interesting for com-
parison, since the chemical change is the same in the two, and it will bo observed that
the optical change is nearly the same also.

The three last bodies occur together, or under similar circumstances in nature, and
form a series of which the second (carvole) is precisely intermediate in composition. It
is also intermediate in optical properties.

ITis Substitution of Hydrogen by Peroxide of Nitrogen.
Of this the following instances have been examined : —

Substance.

Formula.

Befractive
index A.

Dispersion.

Senai-
tiveness.

Specific
refractive
energy.

Specific SP'^'?"
dis^ion. ^^°«-
tiveness.

C„H, \
(NO,)/

C,., H, 1
(NO,) J

1-4823
1-5356

1-5486

0-0323*
0-0501*

0-0570*

60
50

48

5671
4610

3880

380* 70
423* 43

404* 34

1

Nitrobenzole

Dinitrobenzole f ..

CfiH.Oe
QH,(N0,)3 0e

1-4659
1-4654

0-0191
0-0264

25
45

3690
2909

151
165

19

28

Kitroglj-cerine ...

Amvlic Alcohol... C,„ Hj., 0„
Nitrate of Amyl... C,o Hii(NO,) 0„

1-4005
1-4065

0-0174
0-0210

39
45

4895
4061

212
202

47
44

Here we obsene in the case of the benzole compoiuids a considerable increase of
actual refraction, but in those of glycerine and amylic alcohol little change, whUe in
each case there is a verj' marked increase of dispersion ; yet when the numbers are
divided by the density the refraction at least shovps a decrease. The sensitiveness is
greatly diminished by the substitution in the benzole compounds.

The Substitution of Hydrogen by Chlorine.

Substance.

Formula.

Refractire
index A.

Sensi Specific

Dispersion. tJTene^. "-efractive

energy.

Specific
dispersion.

Specific
sensi-
tiveness.

C„ H3 CI

C,oH3Cl3

1-4823
1-5135
1-5563

0-0419 60
0-0437 53
0-0502 46

5564
4634
3836

483
394
346

69

48
31

Chlorobeijzole

Trichlorobenzole ...

In this case there is also an increase both of actual refi'action and dispersion, and a
decrease of sensitiveness; but when the density of the chlorinated products is taken
into account, the result is a great diminution in each of the optical properties.

* In these cases 1x0—1^.^ L> taken as the measure of disj)ersion, since H was in\-isible thi'ough nitro-
benzole.

t Calculated on the assumption that the specific refractive energj- of a mixture is the mean of the specific
refiractive energies of its constituents.

2 Z 2

336 DE. J. H. GLADSTONE AjST) THE EEV. T. P. DALE ON THE EEFEACTIOX,

The Siihstitidion of Cldorine hij Bromine.
Of this we have the foUowini? instances : —

Substance. Formula.

Eefracfire
index A.

Dispersion.

Sensi- Specific

tiveness. refraetive

energy.

Specific
dispersion.

Specific
sensi-
tiveness.

Teichloride of Pho.^phorus ...i PClj 1-5062 394
Terbroiuide of Piiusphorus ..., PBi'. 1-6730 808

58 3489
64 2338

271
280

40

22

C, HCI,
ClHBrj

1-4400
1-5554

220
418

54 2949

55 2107

148
158

37

21

Bichloride of Chloretliylene...
Bibroinide of Cidoretliyiene...
Bibromide of Bromethylene...

CHjCl, Cl„
C, H, CI, Br',
C^H^Br, Br,

1-4619
1-5430
1-5809

228
354
430

59 3259
56 2415
50 i 2220

160
157
164

42
25
19

Here in eacli case the bromine has greatly increased the refraction ; but that this is
o\A-ing to its great weight is evident from the fact that the specific refractive energy is
mucli diminished. The dispersion is increased, but this is very nearly counterbalanced
by the increase of weight. The sensitiveness is diminished, at least in the ethylene
grouy..

It mil be observed that, in each of the five cases mentioned in this section where
there are two substitution products, the lower one is intermediate between the original
substance and the higher product.

These observations put us in a position to consider the question. Does an element
retain its special influence on the rays of light with whatever other elements it may be
combined \

As the specific refractive energy of a mixture, or a feeble combination such as alcohol
and water, is approximately the mean of the specific refractive energies of its constituents,
we are prepared to find the rule holding good in more distinct chemical compounds.
In the only case in which we have been able to try it among liquid elements, namely
terbromide of phosphorus, the re-sidt was pretty near ; but there is no doubt that
chemical combination often greatly changes the optical as it does the other properties
of elementary bodies.

Yet it is quite conceivable that an element may retain a specific influence on the
rays of light through many if not all its compounds ; and this ^iew certainly finds some
support in our experiments. Witness the fact of the great increase both of refraction
and dispersion caused by the addition of nitrogen, whether combined with oxygen or
not, to compounds of carbon and hydrogen (see Appendix I., Nos. 32, 52-55, 57-G2,
75, 70). But when we look more narrowly at the numbers, we find this general perma-
nence of special optical properties subject to much modification. Thus the difference in
the optical properties of some isomeric bodies shows that such a generalization cannot
be strictly true. Again, we may examine the diff'erent cases of replacement of hydrogen
by oxygen mentioned above ; and as in each case the atomic volume of the substitution-

DISPERSION, AXD SENSITIVENESS OF LIQUIDS.

337

product is the same, or nearly so, as the atomic volume of the primarj-^ compound, the
comparison is pecuharly legitimate. A^'e infer at once that oxygen in combination is
actuaUy more refractive and dispersive than hydrogen, but that, if we take into account
its much higher density, its specific refractive energy is less. But when we come to com-
pare the different cases quantitatively, we see that a good deal depends on the peculiar
natiu'e of the compound. In the follomng Table the effect of the replacement of two
equivalents of hydrogen by two of oxygen is given both with respect to refraction and
dispersion. The specific refractive energy of c is taken as the best exponent of the
influence of refraction, and ju-,,— iU-a? di^^ded by density, is assumed, as before, for the
specific dispersion.

Substance.

Atomic
volmne.

Specific
refractive
enerpri,- (»).

Effect of
substitution.

Specific
dispersion.

Effect of
Bubstitu^on.

58
57

444
344

— 100

190
162

- 28

Acetic Acid

Carvene

62
63
64

530
501

474

-"29

- 27

305
362
463

+"57
+ 101

Eugenic Acid

Hence we find that the substitution of two equivalents of oxygen for two of hydro-
gen has produced a far greater reduction in specific refractive energy in the case of
alcohol than in that of the essential oil ; while in specific dispersion it has produced a
reduction in the one case, and an augmentation in the other.

As the main conclusions have been marked by italics under each head as they were
arrived at, they are not recapitulated ; but the following may be taken as a larger
generalization deduced from them, and approximately if not absolutely tnie. Every
liquid has a specific refractive energy composed of the specific refractive energies of its
component elements, modified by the manner of combination, and which is unaffected by
change of temperature, and accompanies it zvlien mixed toith other liquids. The product
of this specific refractive energy, and the density, at any given temperature, is, when
added to unity, the refractive index.

338 DE. J. IL (;iL.U)STOXE AXD THE EEV. T. P. DALE ON THE EEFEACTION,

Appendix I.
Table of refractive indices of the lines A, D, H at different temperatures.

The initials in the column headed '• From whose laboratory," are those of Messrs.
G. B. BucKTox, A. H. Church, Wakrex De i^v Rue, E. Frankl^^nd, J. H. Gladstoxe
A. "W. HoFiLv:s.\, W. Odlixg, C. Greville Williams, and A. W. Williamson.

The sign ? attached to a liquid denotes that the purity of the specimen is doubted.

An asterisk * attached to a degree of temperature signifies that the observations at
that temperature were made on a different occasion to the observations at other tempe-
ratures.

Specific gravities not determined fi'om the specimens examined are included in
brackets.

Liquid.

Metlij'lic Alcohol

Ditto from oxalate

Amylic Alcohol

Caprylic Alcohol

Iodide of iNIethvi

Iodide of Ethyl ,

Iodide of Propyl.
Iodide of Amyl .

Formic Etiier ....

I

Acetic Ether

Acetic Ether ...

12. Pro[)ioiiic Etiicr

Butyric I'^tiier

Valerianic Ether

Acetate of .\myl

:Ditto, second specimen
Acetate of Capryl?

From

whose

laboratory.

E. F.
J. H. G.
J. H. G.
C. G. W.
A. W. W.

A. W. W.

A. VV. H.
J. H. G.
A. W. W.
A. VV. W.
J. H. G.

A. W. W.

A. W. W.
.1. H. G.

J. II. G.

J. H. G.
C. G. W.

Specific grarity.

0-797^2 at 20
0-7S6 at 20
0'8I79 at I5-0
0-82U at 15-5
2-I9I2 at 20

1-9328 at 20

1-7117 at 20
1-4950 at 20
0-9088 at 20
0-8648 at 20
0-8972 at 20

0-8555 at 20

0-8778 at 20
0-8G8 at 20

0-SG80 at 20

Tempe-
rature of
observa-
tion.

J 20

137

20
37

20

29-5

24-5

41

27

47

23-5

29-5

23-5

36

48

8-5
20
30
17-5
37
22
31
40
20
28
23-5
33
41

32

42

23

40

18

32-5

24-5

34-5

44

8-5
21-5
35
27-5
40

Refractive indices.

A.

1-3264
1-3205
1-3268
1-3230
1-3988
1-3924
1-4157
1-4073
1-5203
1-5104
1-5003
1-4918
1-4841
1-5001
1-4934
1-4871
1-4816
1-4720
1-3540
1-3500
1-3456
1-3C45
1-3606
1-3653
1-3606
1-3563
1-3696
1-3657
1-3610
1-3850
1-3768
1-3916
1-3856
1-3910
1-3867
1-3817
1-3944
1-3886
1-3820
1-4045
1-3972

1-3299
1-3238
1-3297
1-3262
1-4030
1-3966
1-4202
1-4118
1-5307
1-5202
1-5095
1-5006
1-4934
1-5095
1-5024
1-4963
1-4892
1-4797
1-.3582
1-3540
1-3494
1-3685
1-3644
1-3692
1-3643
1-3602
1-3736
1-3698
1-3651
1-3888
1-3808
1-3958
1-3898
1-3950
1-3905
1-3859
1-3988
1-3928
1-3866
1-4092
1-4020

1-3395
1-3330
1-3396
1-3359
1-4161
1-4093
1-4351
1-4266
1-5670
1-5549
1-5420
1-5326
1-5250
1-5418
1-5342
1-5272
1-5149
1-5046
1-3694
1-3652
1-3608
1-3798
1-3755
1-3S09
1-3757
1-3711
1-3860
1-3819
1-3771
1-4018
1-3933
1-4089
1-4024
1-4081
1-4037
1-3985
1-4113
1-4058
1-3990
1-4255
1-4181

DISPEESION, AND SENSITIVENESS OF LIQUIDS.
Table (continued).

339

No.

18.
19.

20.

21.
22.

23.

24.

25.

26.
27.
28.

29.
30.
31.

32.
33.

34.

35.
36.
37.
38.

39.
40.
41.

Liquid.

Hydride of CEiiaiithyl

Hydride of Capryl

Mercuric Methyl?

Mercuric Ethyl ?

Stannic Ethyl-methyl ?

Stannic Ethyl?

Triethylarsine

Acetic Acid

Acetone
Amvlene

Carbonic Ether

Boracic Ether

Silicic Ether

Salicylate of Methyl

Nitrate of Amyl

Chloroform ,

Bromoforra

Dutch Liquid

Bibromide of Bromethylene
Bibromide of Chlorethylene
Bichloride of Chlorethylene

jBenzole

Parabenzole

Toluole

From

wlioso

laboratory.

Specific grarity.

J. H. G.
J. H. G.

G. B. B.

G. B. B.
E. F.

G. B. B.

A. W. H.

A. H. C.

J. H. G.
A. W. W.

E. F.

E. F.

E. F.

J. H. G.

VV. D. L. R
J. H. G.

J. H. G.

J. H. G.

A. W. H.
W. D. L. R.
W. D. L. R.

A. H. C.
A. H. C.
A. H. C.

0-7090 at 20
0-7191 at 20

(3-069)

(2-444)
1-222 at 20

1-19^ at 20

1-0592 at 20

O-8II7 at 15-5
0-7151 at 20

0-972 at 20

0-876 at 20
0-932 at 20
1-176 at 20

1-0008 at 20
1-498 at 20
2-636 at 12

2-616 at 20
2-2477 at 20
1-4177 at 20

0-8667 at 20
0-8469 at 20
0-865 at 20

Tempe-
rature of
observa-
tion.

Ecfractive indices

9-5
22
36

9
28-5
41

8-5"
15*
26-5

8-5«
24-5
19
34-5
23
35
48
19-5
26-5
24
34-5
45
25*5
40
23
35
22
31
40
22-5
40-5
20
33-5
21
37
10
22-5
36-5
18
30
44
]5-5
29
39
21
38
18
39-5
13
24
13
29-5
10-5
23
39
20
25-5
32-5
39

D.

1-3956
1-3888
1-3811
1-4022
l-39.-!l
1-3870
1-5274
1-5262
1-5197
1-5300
1-5124
1-4555
1-4479
1-4606
1-4551
1-4481
1-4598
1-4588
1-36/4
1-36.35
1-3596
1-3540
1-3469
1-3832
1-3786
1-3773
1-3746
1-3692
1-3664
1-3578
1-3781
1-3724
1-5206
1-5140
1-4109
1-4053
1-.3988
1-4411
1-4346
1-4253
1-5579
1-5505
1-5437
1-4175
1-4082
1-5819
1-5701
1-5477
1-5413
1-4661
1-4563
1-4879
1-4806
1-4703
1-4814
1-4709
1-4672
1-4629

1-3996
1-3931
1-3854
1-4065
I-.3972
1-3911
1-5378
1-5355
1-5296
1-5397
1-,W17
1-4625
1-4747
1-4673
1-4621
1-4549
1-4669
1-4657
I-37I8
1-3680
1-.3634
1-3580
1-3512
1-3878
1-3834
1-3810
1-3787
1-3734
1-.3698
1-.3604
1-3821
1-3768
1-5319
1-5253
1-4157
1-4097
1-4035
1-4463
1-4397
1-4308
1-5674
1-5598
1-5531
1-4221
1-4126
1-5915
1-5787
1-5559
1-5495
1-4714
1-4619
1-4975
1-4900
1-4793
1-4903
1-4794
1-4755
1-4710

H.

1-41.35
1-4059
1-.3976
1-4197
1-4097
1-4032
1-5726
1-5694
1-5626
1-5729
1-5538
1-4868
1-4783
1-4905
1-4844
1-4769
1-4919
1-4906
1-3846
1-.3803
1-3757
1-3706
1-3631
1-4028
1-3982
1-3936
1-3898
1-3846
1-3815
1-3724
1-3940
1-3881
1-5810
1-5735
1-4320
1-4256
1-4191
1-4630
1-4561
1-4471
1-5998
1-5921
1-5846
1-4371
1-4276
1-6249
1-6112
1-58.39
1-5770
1-4892
1-4789
1-5305
1-5225
1-5108
1-5216
1-5090
1-5048
1-5001

340 DE. J. H. GL.\J3ST0^-E .\M) THE EEY. T. P. DALE OX THE EEFKACTIOX,

Table (continued).

Liquid.

From whose
laboratory.

Specific gravity.

Paratoluole
Toluole . ..

A. H. C.
W. O.

W. O.

Xylole

Cumole (from Cuminic Acid) ... A. H. C
Cumole (from impureWood-spirit) J- H. G

Pseudocumole j\V. D. L. K.

Cymole j A- H- C

Cymole (from Camphor) .

!W. D. L. R

Chlorobenzole ....
Trichlorobenzole
Nitrobenzole —

Dinitrobenzole in 2 equivs. o(\
nitrobenzole i

Aniline

Amyl-aniline

Hydrate of Cresyl

57- ; Pyridine

68. Picoline

59. Lutidinc

60. Icollldine

61. IChinoline

62. Lepidiiic

63. Hydrocarbon from Anise

64. „ Turpentine ..

65. „ Carraway ..

\V. D.L.K.
A. W. H.
A. H. C.

A. H. C.

A. H. C.

A. W. H.

W. D.L.K.

A. W. H.

A. H. C.

A. H. C.
A. H. C.

C. G. W.

C. G. W.
A. H. C.

J. H. G.

J. H. G.

Tempe-
rature
of obser-
vation.

0-8333 at 20
0-8658 at 20

0-866 at 20

0-871 at 20

0-858 at 20

0-8692 at 20
0-861 at 20

0-8565 at 20

1-108 at 20
1-450 at 20
1-159 at 20

1-267 at 20

1-027 at 16

0-9177 at 20
1-0364 at 20
0-9738 at 20

(O955)

(0-936)
(0-921)

(1-081 at 10)

1-072 at 15
0-858 at 20

0-8644 at 20

0-8529 at 20

28
40
14
33
11

Refractive indices.

42

7

27-5

8-5

24

34

f 12-5

\35-5

( '
129

ni

9

27-5

20

37

38

23-5

35

56

21-5

37

42

47

23-5

42

11-5

32

21-5

36

22-5

37-5

52
8-5"

22-5

23-5

45

24

35

37

21

47

11

30

10*
4
L47
rl4
137

1-4667
1-4590

1-4S69

r23

rio

1-4888

1-4788

1-4716

1-4898

1-4783

1-4687

1-4608

1-4555

1-4843

1-4728

1-4760

1-4648

1-4731

1-4659

1-4614

1-5194

1-5095

1-5563

1-5495

1-5331

1-5266

1-5460

1-5404

1-5296

1-5644

1-5567

1-5537

1-5520

1-5114

1-5035

1-5341

1-5281

1-4940

1-4860

1-4888

1-4803

I-47I8

1-4932

1-4894

1-4927

1-4820

I-55G7

1-5466

1-5496

1-6039

1-5909

1-4653

1-4669
1-4596
1-4487
1-4640
1-4529

1-4-51

1-4671

1-4957

1-4856

1-4982

1-4879

1-4805

1-4983

1-4864

1-4759

1-4680

1-4634

1-4932

1-4812

1-4834

1-4717

1-4803

1-4729

1-4684

1-5290

1-5189

1-5671

1-5600

1-5465

1-5399

1-5600

1-5542

1-5425

1-5784

1-5701

1-5676

1-5647

1-5222

1-5138

1-5454

1-5377

1-5030

1-4951

1-4980

1-4890

1-4807

1-5028

1-4987

1-5013

1-4907

1-5687

1-5587

1-5617

1-6189

1-6054

I-47I8

1-4625

1-4734

1-4653

1-4545

1-4701

1-4589

1-5030

1-4944
1-5271

1-5300

1-5192

1-5116

1-5280

1-5148

1-5008

1-4919

1-4848

1-5236

1-5093

1-5076

1-4957

1-5050

1-4975

1-4927

1-5636

1-5528

1-6065

1-5983

1-5832 G

1-5766 G

1-5994 G.

I-5932G

1-5S16G

1-6297

1-6183

1-6145

1-5622

1-5532

1-5824

1-5733

1-5387

1-5301

1-5314

1-5213

1-5122

1-5353

1-5308

1-5329

1-5210

1-6198

1-6084

1-6124

1-6822

1-6473 G.

1-4921

1-4934
1-4S45
1-4730
1-4901
1-4783

DISPERSION, AND SENSITI^TINESS OF LIQUIDS.
Table (continued).

341

1

Tempe-

Refractive indii

No.

Liquid.

Prom whose

Specific gravity.

rature

laboratory.

of obser-

1

vation.

A.

D.

H.

66.

Hydrocarbon from Thyme

J. H. G.

oC.

0-8635 at 20

(25
135-5

1-4594
1-4545

1-4652
1-4606

1-4856
1-4805

f 23
143

1-4545

1-4610

1-4818

67.

Bay

J. II. G.

0-851 at 20

1-4468

1-4528

f26-5
138

1-4574

1-4640

1-4865

68.

„ Hergamot ...

J. II. G.

0-8467 at 20

1-4517

1-4578

1-4800

fl7

I-49I8

1-4985

1-5209

69.

„ Cloves

J. 11. G.

0-9041 at 20

< 28-5

1-4870

1-4936

1-5157

L39

1-4828

1-4892

1-5110

rio-5

1-4988

1-5055

1-5294

70.

Cubebs

J. H. G.

0-927 .It 20

< 20

1-4950

1-5014

1-5252

|31

1-4905

1-4977

1-5209

ri2-5

1-4913

1-4992

1-5270

71.

Carvole

.1. H. G.

0-9530 at 20

< 24-5

1-4862

1-4935

1-5196

134

1-4812

1-4884

1-5145

fl8
127-5

1-5285

1-5394

1-5780

7S5.

J. H. G.

1-064 at 20

1-5244

1-5347

1-5722

/30
\43

1-4503

1-4553

1-4703

73.

0-8786 at 43

1-4451

1-4505

1-4653

r2o

1-4659

1-4705

1-4850

74.

Glycerine

J. H. G.

1-261 at 17

<^ 30

1-4634

1-4680

1-4823

148

1-4586

1-4631

1-4773

ri3-5

\32-5

1-4683

1-4749

1-4947

'Ji>-

Nitroglycerine?

J. H. G.

(1-60)

1-4596

1-4662

76.

Nicotine

J. H. G.

1-026 at 18

fl8
\32

1-5149
1-5107

1-5234
1-5194

1-5542
1-5493

125
\36

1-6698

1-6866

1-7506

77.

Terbromide of Phosphorus

J. li. G.

2-88 at 20

1-6627

1-6792

1-7422

125-5
\38

1-5030

1-5118

1-5418

78.

Terchloride of Phosphorus

J. H. G.

1-453 at 20

1-4957

1-5042

1-5334

fl7
\26

1-4810

1-4882

1-5118

79.

Oxy chloride of Phosphorus

J. H. G.

1-680 at 20

1-4756

1-4832

1-5067

The determinations of iodide of propyl were added, and those of acetic acid and
terchloride of phosphorus were altered during the printing of the paper.

Appendix II. — Table of Refractive Indices.
The liquids in this Table are arranged according to their power of refracting the line
A at 20° C.

Liquid.

Temp.

Refractive indices.

A. B.

C.

D.

E.

F.

G.

H.

oC.

35

?
25
11
21
IS
28
21-5
24
20
15-5
23-5
12-5

2-0389

1-9209 1-9314
1-6698 1-6752
1-6142 1-6207
1-6039 1-6094
1-5819 1-5851
1-5649 1-5699
1-5644 1-5684
1-5567 1-5617
1-5563 1-5602
1-5579 1-5610
1-5460 1-5506
1-5472 1-5500

2-0746
1-9527
1-6866

1-9744

2-1201

2-1710

2-2267?
2-0746
1-7506
1-7090

1-6822
1-6249

1-6297
1-6198
1-6065
1-5998

1-5830

Phosphorus in Bisulphide of Carbon. .

1-9941! 2-0361
1-7083| 1-7300
1-6584 1-6836
1-6403 1-6615
1-6037 1-6149
1-6014 1-6244
1-5951 1-6125
1-5879 1-6030
1-5809 1-5945
1-5790 1-5901
1-5791 1-5994
1-5659 1-5748

1-6240 1-6333! 1-6465

Lepidine

1-5727

1-6189
1-5915
1-5801
1-5774
1-5687
1-5671

1-5909
1-5737

Rectified Oil of Cassia

1-5628 1-5674

1-5600

1-5554

. Dinitrobenzole in nitrobenzole

MDCCCLXIII.

3a

342 DE. J. H. GLADSTONE A^^ THE KEY. T. P. DALE ON THE EEFEACTION,

Table (continued).

Liquid.

Temp.

C.

Nitrobenzole 25

Hydrate of Phenyl 13

Hydrate of Cresyl 11 '5

Eugenic Acid 18

Mercuric Methyl 26-5

Salicylate of Methj-l j 21

i Iodide of Methyl I l6

Mercuric Ethyl I 8-5

Nicotine ' 18

9
23-5
23-5
23-5
25-5
10-5
21*5
22-5
23-5
17
12-5
17-5
17
10-5
14
29
13-5

Chlorobenzole

Amyl-aniline

Terchloride of Pliosphorus ....

Iodide of Ethyl

Eectified Oil of Santal-wood .

Hydrocarbon from Cubebs

Pyridine

Lutidine

CoUidine

Hydrocarbon from Cloves

Pseudocumole

Iodide of Amyl

O.xycliloride of Phosphorus

Benzole

Toluole

Cymole

Nitroglycerine

Hydrocarbon from Portugal

Cumole (2t)d specimen)

Stan nic Etliy 1

Bichloride of Chlorethylene

Hydrocarbon from Turpentine .
Hydrocarbon from Carraway ,
Hydrocarbon from Bergamot

Rectified Oil of Citronella

Hydrocarbon from Bay

Stannic Ethyl-methyl

Chloroform

Caprylic Alcohol

Nitrate of Amyl

Amylic Alcohol

Hydride of Capryl

Hydride of CF.nanthyl

Acetate of Amyl

Butyric Ether

Amylene

Carbonic Ether

Propionic Ether

Boracic Ether

Acetic Ether

Alcohol

Acetone

Formic Ether

Ether

Water

Methylic Alcohol

8-5
23
13
24
24
26-5
19
23
19
10

9-5
10
25

9

9-5

8-5
23

22-5

22-5

20

15

25-5

RcfVactiTe indices.

1-5331

1-5377

1-5341

1-5285

1-5197

1-5206

1-5203:

l-5300l

1-5149!

1-5194

1-5114

1-5052

1-5003

1-4954

1-498S

1-4940

1-4894

1-4927

1-4918

1-4843

1-481G

1-4810

1-4879

1-4869

1-4648

1-4683

1-4617

1-4687

1-4606

1-4661

1-4596

1-4594

1-4574

1-4598

1-4545

1-4555

1-4438

1-4230|

1-4109

1-3981|

1-4022)

1-3956

1-3944

1-3850

1-3850

1-3773]

1-.3696

1-3664

1-3645

1-3600J

1-3540,

1-3540

1-3529

1-3284

1-3264

1-5374
1-5416
1-5377
1-5321
1-5232
1-5241
1-5234
1-5333
1-5174
1-5223
1-5150
1-5088
1-5034
1-4977
1-5012
1-4967
1-4924
1-4958
1-4944
1-4872
1-4S43
1-4840
1-4913
1-4898
1-1671
1-4706
1-4640
1-4709
1-4629
1-4680
1-4616
1-4615
1-4598
1-4619
1-4567
1-4578
1-4457
1-4246
1-4127
1-3999
1-4037
I-39G8

i-:{958

1-3864
1-3S66
1-3785
1-3713

1-5398
1-5433

1-5341
1-5263

1-3658
1-3612
1-3554
1-3553
1-3545
1-3300
1-3277

1-5168

1-493!

1-4590
1-4466
1-4255

1-3621

1-3554
1-3307

1-5465
1-5488^
1-5445'
1-5394
1-5296
1-5319
1-5307)
1-5397
1-5234
1-5290
1-5222
1-5148
1-5095
1-5015
1-5055
1-5030
1-4987
1-5013
1-4985
1-4932
1-4892
1-4882
1-4975
1-4957
1-4717
1-4749
1-4684
1-4759
1-4673
1-4714
1-4653
1-4652
1-4640
1-4655
1-4610
1-4625
1-4490
1-4279
1-4157
1-4024
1-4065
1-3996
1-3998
1-3888
1-3896
1-3810
1-3736
1-3698
1-.3685
1-3638
1-3582
1-3582
1-3566
1-3324
1-3299

1-5554
1-5564

1-5464

1-5402
1-5377

1-5292
1-5156

1-4941

!

1-5036|

1-4766

1-4691

1-4674
1-4526
1-4309

1-3661

1-3590
1-3347

1-5643

1-56.39

1-5573

1-5528^

1-5368'

1-5478J

1-5440;

1-5518

1-5346

1-5418

1-5361

1-5252

1-5214

1-5093

1-5145

1-5155

1-5100

1-5127

1-5064

1-5040

1-4987

1-4967

1-5089

1-5072

1-4808

1-4824

1-4758

1-4853

I-475S

1-4784

1-4724

1-4724

1-4721

1-4730

1-4690

1-4716

1-4555

1-4338

1-4219

1-4078

1-4076

1-4045

1-4035

l-:?938

1-3944

1-3856

1-3785

1-3742

1-3728

1-368

1-3629

1-362

1-.3606

1-3366

1-3330

1-5832;
1-5763;
1-5699

1-5526
1-5640
1-5558'
1-5634
1-5449!
1-5530
1-5491
1-5357
1-5321
1-5161
1-5227
1-5278
1-5204J
1-5232;
1-5140
1-5146:
1-5074
1-5047
1-5202
1-5174;
1-48661
1-4899!
1-4826!
1-4936,
1-4838
1-4841
1-4790
1-4789
1-4798
1-4795
1-4756

1-4795

1-4614

1-4386

1-4274

1-4122

1-4141

1-4087

1-4077

1-3981

1-3992

1-3896

1-38

1-3785

1-3766

1-3720

1-3670

1-3666

1-3646

1-3402

1-3369

1-5886
1-5813
1-5780
1-5626
1-5810
1-5670
1-5729
1-5542
1-5636
1-5622
1-5446
1-5420
1-5223
1-5294
1-5387
1-5308
1-5329
1-5209
1-5236
1-5149
1-5118
1-5305
1-5271
1-4957
1-4947
1-4894
1-5008
1-4905
1-4892
1-4845
1-4844
1-4865
1-4860
1-4818
1-4868
1-4661
1-4429
1-4320
1-4161
1-4197
1-4135
1-4113
1-4018
1-4033
1-3936
1-3860
1-3815
1-3798
1-3751
1-3706
1-3694
1-3683
1-3431
1-3395

DISPERSION, AJ^D SE-VSITIVEXESS OF LIQUIDS. 343

P.S. [^Received May 28.] — It was not till after this paper was read that we became
aware of the existence of an elaborate treatise by Dr. Sciirauf, " On the Dependence of
the Velocity of Liglit on the Density of Bodies," in Poggendorff's Annalen, cxvi. 193,
in which he investigates the question mathematically, taking as the basis of his calcula-
tions our former experiments, and those of Deville, Weiss, and others. Our own line of
tliought has many points of analogy with that pursued by him. but there is this differ-
ence in the conclusion: he believes that — g— and ^^ (or in our notation -ry- and g^,

D being the density J are the constants at all temperatures, and are the functions on

wliich depend tlic optical properties of mixtures ; while we are led by our new exjieri-

ments to accord that quality rather to -^, and to doubt any such simple formula as gj

for the changes of dispersion. To this point we propose to recur at some future period
if we have the opportimity.

There is one point in reference to our method of observation which seems to call for
a remark. Schrauf thinks that there is a slight change in the refringent angle of our
prism on its being heated. Now our hollow prism has glass ends as well as glass sides ;
but supposing such a change actually occurs, it is evident it will produce a uniform
error running through all our observations in Section I. This may be the reason why
at high temperatures the observed is almost always less than the calculated index ; but
as bisulphide of carbon and water agree so closely with either his or our theory, this
source of error must be extremely mmute.

We await with curiosity the publication of the experiments referred to in Dr. Sciirauf's
short note, " On the Velocity of Light and Chemical Composition," in the April Number

[ 345 ]

XV. Researches into the Chemical Constitution of Karcotine, and of its Products of
Decomposition. — Part I. By Augustus Matthiessen, F.R.S., Lecturer on Cliemistry
in St. Marys Hosjirital, London, and G. C. Foster, B.A., Jjccturer on Natural
Philosophy in Anderson's University, Glasgoio.

lleceived February 26,— Ecad March 26, 1803.

§ I.— COiirOSITION OF NARCOTINE AND COTAENIKE.

The existence of Narcotine was indicated by Derosne as early as the year 1803, but its
chemical nature remained almost entirely unknown until Robiquet *, in 1817, showed
that it belonged to the class of vegetable alkaloids. Numerous analyses of narco-
tine were subsequently published by Dumas and PELLETiERf, TelletierJ, Liebig §,
Regxault II , and others ; but its composition was first determined to the general satisfac-
tion of chemists by Bltth % who, in 1844, proposed the formula C^g H25 N'0i4, support-
ing it by numerous analyses of the double hydrochlorate of narcotine and platinum, and
sho^ving, at the same time, that it accorded well with the most trustworthy results of
previous investigators, and also accounted satisfactorily for the formation of the remark-
able decomposition-products of narcotine discovered by himself and by Wohler **.
Since the publication of Blyth's investigation, the formula which he proposed has been
generally adopted as expressing correctly the composition of this base. More recently,
however, Wertueim f f , founding his ojiinion chiefly on tlic composition of the volatile
bases obtained by distilling narcotine with potash, has maintained the existence of two
additional varieties of narcotine, homologous with that examined by Blyth, and repre-
sented respectively by the formula: C44 H23 NOj, and C48 H2; N0]4 ; while Hinter-
berger %X has analysed a compound of chloride of mercury with, what he considers as a
fourth variety, still homologous with the preceding, and represented by the formula
C42H21NOJ4.

Such being the results of previous investigations, it was plainly necessary to begin
any new research into the chemical nature of narcotine by endeavouring to ascertain, by
dii'ect analysis, whether there existed in reality more than one kind of narcotine, and,

* Ann. dc Chim. et de Phys. vol. v. p. 275. t Ibid. vol. xxiv. pp. 186 and 191. J Iltid. vol. 1. p. 271.

§ Jahresboricht iiber die Fortschritte der physischen Wissenseliaften, von Jacob Berzclius, vol. xi. (1832)
p. 231.

II Ann. dc Cliim. et de Phys. vol. Ix^-iii. p. 138.

% Memoii's and Proceedings of the Chemical Society, vol. ii. p. 1C7.

** Annalen der Cheniie und Pharmacie, vol. 1. p. 1. ft Journal fiir praktisoho Chcmie, vol. liii. p. 131.

Xt Annalen dor Chemie und Pharmacie, vol. Ixxsii. p. 312.
MDCCCLXIII. 3 B

346

DE. A. MATTHIESSEN AjSTD JIE. G. C. FOSTER ON

if so, which of these kinds was being operated upon. The following are the results
obtained on analysing specimens of narcotine procured from several distinct sources.

A. Narcotine prepared hj Mr. Moeson _/}"o??i a mixture of various kinds of opium*.
I.
II.
III.

rv'.

Y.

VI.

YII.

VIII.

0-631I grm. substance gave I'-iTS!) grm. carbonic acid and 0'3438 grm. water.
•2372 grm. substance gave •5552 grm. carbonic acid and •1234 grm. water.
•2244 grm. substance gave •52GG grm. carbonic acid and

•7576 grm. carbonic acid and

•1724 grm. platinum.

■1338 grm. platinum.

1304 grm. platinum.

•3258 grm. substance gave
•7470 grm. substance gave
•5708 grm. substance gave
•5555 si'iii- substance gave

1148 grm. water.
1682 arm. water.

•6153 grm. substance gave '1482 grm. platinum.

B. Karcotine p>repared from Turkish ojjium hy Dr. G. INIerck of Darmstadt.
0^3584 grm. substance gave 0-8344 grm. carbonic acid and 0-1847 grm. water.

C. Xarcotine prepared from Efinptian opium hy Dr. G. Merck.

0-3172 grm. substance gave 0-7451 grm. carbonic acid and 0-1630 grm. water.

D. Narcotine prepared from Persian opium by Dr. G. Merck.

0-3192 grm. substance gave 0-7460 grm. cai-bonic acid and 0-1660 grm. water.

E. Narcotine prepared from Efjyptian ojmim ly Mr. Morson.

©•3460 grm. substance gave 0-8094 gi-m. carbonic acid and 0^1790 gi-m. water.

F. Narcotine from Turkish opium ohtaincd from Messrs. Hopkin and Williams. —
(The whole of this sample of narcotine was di^ided into three portions by fractional
crystallization from alcohol : the following analyses were made with crystals of the first
and last crops respectively, each of which was considerably smaller in quantity than the
intermediate crop.)

I. 0-4046 grm. substance gave 0-9503 grm. carbonic acid and 0-2085 grm. water.
II. -3366 grm. substance gave -7843 grm. carbonic acid and •1730 grm. water.

These numbers correspond to the following percentages :-

A.

D.

F.

I. & V.

II.&VI.

iii.&vii.Tv.iviri.

r
I.

II.

Carbon .

63^91

G3^83

64-00 6.3-42

6.3-49

64-01

63-74

63-80

64-05

63-55

Hydi'ogen

6^05

5-77

569 5-74

5-73

5-71

5-77

5-75

5-72

5-71

Nitrogen .

3^2G

3-31

3-.32 3-40

Oxygen .

• Tfds was the material that served for the greater number of the experiments which follow.

THE CHEMICAL COXSTITUTION OF NARCOTINE. 347

The percentages required by the formula C^g H25 NOj^ are —

Carbon 64-63

Hydrogen .... 5-85

Nitrogen .... 3-28

Oxygen 26-24

from wliich the above results differ very considerably : we are therefore led to regard the
formula C^^ H23 NOj^, or preferably Coo Ho , NO-, which accords mucli better with our
analyses, as more exact: it corresponds to the following proportions in 100 parts: —

Carbon 63-92

Hydrogen . . . . 5-57

Nitrogen .... 3-39

Oxygen 27-12

And since all the different samples of narcotine which we were able to procure gave on
analysis identical results *, we conclude that there is no sufficient evidence of the exist-
ence of more than one kind of narcotine, especially as we believe that the observations
which formerly gave rise to the contrary opinion are explained by an experiment to be
hereinafter described.

If the formula G02 H23 NO7 be adopted for narcotine, it is impossible to account in

* Tho diiferenco between the highest quantity of c<arbon (F. I.) and the lowest (A. IV.) given by tho whole
Beries of analyses amoimts to 0-63 per cent.; but this extreme difference among tho analyses of seven distinct
specimens is scarcely perceptibly greater than the difference (0-58 per cent.) between the highest and lowest
(A. III. & IV.) numbers resulting from the analysis of the same specimen. Although these differences are
rather high, we believe that they are due entirely to accidental errors of experiment, and chiefly to the diffi-
culty of ensuring the complete combustion of narcotine. No analysis of that substance over gave us more than
64-05 per cent, of carbon, but several gave smaller quantities than the lowest quoted above (for instance, 62-7,
62-8, 63-1 per cent.) ; but in these cases traces of unosddized carbon could always be detected in the combus-
tion-tube after tho experiment. The combustions were made with oxide of copper, and, like all those given in
this paper, were finished in a stream of oxygen.

The difference between the percentage of nitrogen required by the old formula of narcotine (Bi-TTn's) and by
that which we adopt being no greater than the probable errors of analysis, it was not thought worth while to
determine that element in more than one sample.

For the sake of comparison, we append hero the results obtained by other chemists who have analysed
narcotine or its salts.

Analyses of Naecotini;.

Dumas and
Pelletier (1823).

Pelletier

(1832).

65-16

Liebig
(1832).
64-09

Regnault (1838).

Hofmann (1844).

Caxbon . . 68-88

64-01

64-51

64-09

64-53

Hydrogen . 5-91

5-45

5-50

5-96

5-99

5-73

6-21

Nitrogen . . 7-21

4-31

2-51

3-46

3-52

3-30 2-92

Muldei

■ (1838).

A

Varrentrapp

and Will (1841).

Nitrogen .... 303 2-98 2-73 2-44 3-77 3-72

Note. — The analyses of Dumas and PELLEirER are calculated with the old equivalent of carbon (6-11);
Ltebig's analysis is given as recalculated by Bltth (Mem. Chem. Soc. vol. ii. p. 167) with the corrected equi-
valent ; Regnatjlt's analyses are similarly recalculated from the original weighings. For all the foregoing,

3b2

348 DE. A. MATTHIESSEN AKD ME. G. C. FOSTEE 02^

any simple manner for the well-known decomposition of that substance, under the
influence of oxidizing agents, into opianic acid and cotarnine, unless the received formula
of either one or other of these bodies be also modified.

It seemed impossible to doubt the accuracy of the formula Cjo Hk, O5 for opianic acid,
but nevertheless, for the sake of greater certainty, we have repeated the analysis of this
acid, though with no other result than a fresh confirmation of the admitted formula.
I. 0-2494: grm. acid gave 0-5198 grm. carbonic acid and 0-1066 grm. water.
II. -1921 grm. acid gave -4030 grm. carbonic acid and -0855 grm. water.
III. -2492 grm. acid gave -5260 grm. carbonic acid and -1117 grm. water.
Calculated. Found.

^ ^I. II. III. Mean.

Cio 120 57-14 56-84 57-21 57-57 57-21

Hio 10 4-76 4-75 4-95 4-98 4-89

O3 80 38-10 • 37-90

C,„ Hio 0, 210 100-00 100-00

On the other hand, our analyses of cotarnme lead us to adopt the formula G^g Hjj NO3

see references +, i, §, m, page 1 ; for Hofiia>-:n-'s analysis see Bltth's memoii- {loc. cit.); Mttldee's results are
quoted from Yakeestbapp and Will (Ann. Chem. Phai'm. vol. sxsix. p. 282).

A)ml>/ses 0/ CirLOEorLATiNATE OF Nabcotixe.
Liebig (1S.38). Regnault (1839). Hofmann (1844). Blytli (1844). How (1854).

Platinnm . . 14-.51 14-64 15-81 15-97 15-85 15-80 15-65 15-73 15-88
Blvth (1844). Wertheim (1856). Calculated.

■/■^ A A .

C„,. €

Carbon . . 43-72 43-56 42-92 42-27 42-44 43-17 42-62 43^55

Hydrogen . 4-17 4-30 3-94 4-12 4-14 4-15 3-87 4-14

Platinum . 1600 15-95 15-05 15-72 15-98 15-62

Note. — Liebig, Ann. Pharm. vol. xxvi. p. .52; Eegxault, Ibid. vol. xxix. p. 00 ; Hofmajtn' and Bltth, Mem.
Chem. Soe. vol. ii. pp. 166, 167 ; How, Trans. Eoy. Soe. Edinb. vol. xxi. p. 31 ; WEETiiEnr, Traite de Chimie
Organique, par M. Charles GERnABnT, vol. iv. p. 67.

AnaJi/scs 0/ Htdkociexoeate or Naecotine.
Eobiciuet Eegnault (1838). Eeiinault Calculated.
(1832). , ^ , (1839). , ^ ,

Hydrochloric acid . . 8-22 8-14 8-17 7-96 7-43 8-12 7-87

Note. — EoBiaiTET, Ann. do Cliim. et de Phys. vol. i. p. 231 ; Eegnault, Ibid. vol. ls\-iii. p. 138, and Ann.

Pharm. vol. xxix. p. GO.

Aimhjsis of CnxoROMEBCUEATE OF Narcotine.

Hinterbcrger (1852). Calculated.

Ann. Chem. Phanu. Ixxxii. 312. ^ -^ ^

C,, H,, NO,, H CI, Hg CI. G,3 H,, NO,, H CI, Hg CI.

Carbon 43-64 45-13 46-03

Hydrogen 3-90 4-10 4-34

Mercury 18-02 17-09 16-69

THE CHEMICAL CONSTITUTIOX OP N.VECOTINE. 349

in preference to the usual formula, G,., Hj.; NO3, which represents that substance as
containing one atom of carbon more. The following analyses were made with crystal-
lized cotamine prepared by the action of dilute nitric acid on narcotine, the method
recommended by ^■LvDKRSo.v for the preparation of pure cotarnine : it had a slight yellow
or buff colour" which it was found impossible to remove.

I. 0'3473grm. substance gave 0'7746 grm. carbonic acid and 0'2043grm. water.
II. •2261 grm. substance gave '5020 grm. carbonic acid and •1324 grm. water.

III. •5633 grm. substance gave -2320 grm. platiniun.

IV. -5057 grm. substance gave •2113 gnn. platinum.

Calculated. Found. Calcukted.
A A

I. n. III. IV. Mean. CijHi^NO^.

C12 144 60-76 60-83 60-55 60-70 62^65

Hi5 15 6-33 6-54 6-50 6-52 602

N 14 5-90 5-82 5-91 5-87 5-62

O4 64 27-01 26-86 25-71

Gi2Hi.,N03, HO2 237 100-00 100-00 100-00

V. 0-6619 gi-m. substance lost 0-0493 grm. water at 110° C.

VI. -5450 grm. substance lost -0406 grm. water at 110° C.

Calculated. Found. Calculated.

, A . . A ,

r \ ! \

V. VI. c,3.

GiaHiaNOg 219 92-41 92-77

H2O 18 7-59 7-45 7-45 7-23

G12H13NO3, H2O 237 100-00 100-00

These results are confirmed by the following determinations of the proportion of

platinum in chloroplatinate of cotarnine, di-ied in vacuo over sulphuric acid * :

I. 0-4312 grm. salt gave 0-1005 grm. platinum.

II. -3161 grm. salt gave -0732 grm. platinum.

III. -4000 grm. salt gave -0924 grm. platinum.

Calculated. Found.
, ^ -, , ,

€12. Gjg. I. II. III.

Platinum per cent .... 23-3 22-6 23-3 23-2 23-1

The adoption of the formula C12 Hjj N03f for cotarnine enables us to represent its
• Five platinum-determinations, made with cUoroplatinate of cotamine that had been dried in the water-
bath, gave quantities of platinum varying from 23-5 to 23-9 per cent., nimibers which differ even more than
those quoted in the text from that rcquii-cd by the hitherto admitted formula. The substances used for analyses
n. and III. were prepared from two distinct specimens of cotamine : the salt II. was, moreover, precipitated
in presence of excess of bichloride of platinum, salt III. in presence of excess of hydroclilorate of cotamine.

t The formula originally proposed for cotamine by 'Wohxee, but which he himself only regarded as provi-
sional, was C^g Hj, If O5 ; Bltth's formula was C25 Hjj NOg ; the formula G^ H13 NO3 was proposed by Geehaedi
(Precis de Chimie Organique, 1845, vol. ii. p. 298).

350 DE. A. MATTHIESSEN AJSD ME. G. C. FOSTEE ON

formation by the action of oxidizing agents on narcotine, by means of a very simple
equation, —

€,2H,3XO- + 0=Gi,,H,,,N03 + GioHjoO,,

Xarcotine. Cotaruine. Opianic Acid.

and, as \Aill be sho^Ti hereafter, it is further supported by the manner in which cotar-
nine is decomposed by nitric and hydi-ochloric acids.

Assuming the accuracy of the formulse here proposed as siifRciently established, we
next endeavoured to ascertain the chemical constitution of narcotine by studying the
action of various reagents on cotarnine and on its other principal derivative, opianic
acid. Hitherto we have made but few experiments on the action of reagents on
narcotme itself, from the conviction that their results would hardly be intelligible
mthout a previous knowledge of the transformations of the bodies into which it splits
up with so much ease.

It appears, as the result of a good many trials, that the following is perhaps the
most advantageous method of transforming narcotine into opianic acid and cotarnine.
100 grms. of narcotine are dissolved in a considerable excess of dilute sulpliuric acid
(loOgrms. acid and 1500 grms. w'ater), the solution is heated to boiling, and 150 grms.
finely powdered peroxide of manganese * is then added as quickly as possible, care being
taken that it does not cause the liquid to froth over; when the whole quantity of
peroxide of manganese has been added, the mLxtm-e is quickly filtered througli a funnel
suiTounded by boiling water. The filtrate, on cooling, becomes half-solid, from the
separation of crystals of opianic acid, and by twice recrystallizing this product from
boiling water it is obtained sufticiently pure for most purposes, though still retaining a
slight brownish-yellow colour, which can be got rid of, when needful, by boiling the acid
■\^ath a dilute solution of hypochlorite of sodium, in the manner indicated by Wohlee.

The cotaniine contained as sulphate, together with a large quantity of sulphate of
manganese in the original mother-liquor of the opianic acid, may be conveniently
extracted therefi-om as follows. The mother-liquor is mixed with a quantity of milk of
lime sufficient to neutralize the free sulphuric acid and to precipitate part of the man-
ganese, then, without filtering, an excess of carbonate of sodium is added and the whole
is heated to boiling for a few minutes, in order to precipitate completely the manganese
and lime (or the neutralization and precipitation may be effected entirx-ly by means of
carbonate of sodium, without using lime). Tlie mixture is then filtered, fir-st through
calico and afterwards, if ne(>dful, through paper ; the filtrate is neutralized with dilute
sulphuric acid, evaporated rapidly to a small bulk, allowed to cool completely, poured
off" from any sulphate of sodium that may have crystallized out, and finally mixed with
an excess of strong potash- or soda-lye, Avhereby the cotarnine is precipitated.

It is a matter of some importance for the successful preparation of cotarnine in this
mannei-, that its extraction from the original mother-liquors containing it should be

* 100 parts of our oxide of manganese corresponded to 00 parts of pure peroxide.

THE CHEMICAL CONSTITUTION OF NAECOTINE. 351

proceeded with without much delay, otherwise the quantity obtained is small and its
colour dark. Under all circmnstances the odour of ammonia, or metliylamine, is
perceptible on adding potash for the final jirecipitation of the cotarnine ; and tliis odour
is strongest Avhen a long time has (dapsed during the pre^'ious parts of the process. The
decomposition which is thus indicated appears to lessen the amount of product obtained
in two ways : not only is the quantity of cotarnine existing in the solution diminished
by the amount that has suffered decomposition, but the resulting volatile alkali seems
to prevent the precipitation of the unaltered base.

Cotarnine that is very dark-coloured may be decolorized, to a considerable extent, by
dissolving it in hydrochhn-ic acid, digesting the solution upon bon(>-charcoal, and re-pre-
cipitating with potash ; but we have never, by any process, succeeded in obtaining
cotarnine quite without colour.

§ II.— DECOMPOSITIONS AND DERIVATIVES OF OPIANIC ACID.

1. Action of Hydrochloric Acid. — "NMicn opianic acid is heated with three or four times
its weight of ordinary strong hydrochloric acid, either to 100° C. in a sealed tube, or
to the boiling-point of the acid in an open vessel, it is decomposed with evolution of
chloride of methyl and carbonic acid, and on evaporating the remaining solution, lirst
on the water-bath, then at the common temperature over lime and sulphuric acid, a
residue is obtained which appears to consist of three distinct crystallizable substances ;
but the ease with which at least one of them undergoes alteration by heat or exposure
to air, and the difficulty of completely separating them from each other, have hitherto
prevented us from making a thorough examination of these products, and we therefore
postpone to a future communication any further description of them *.

Opianic acid seems to undergo a similar decomposition when boiled with fuming
hydriodic acid ; in this case iodide of methyl is given off, without separation of iodine,
but we have not found it possible to remove the excess of hydriodic acid from the residue
without destroying the organic constituents.

2. Action of Potash. — "When opianic acid is mixed with a large excess of potash-lye,
and the liquid is evaporated nearly to dryness, the acid splits up into meconin and hemi-
pinic acid. No blackening occurs unless too small a quantity of potash is employed ;
but when a certain degree of concentration is reached, the decomposition apjicars to
take place almost instantaneously : the mixture, which up to that point is a thick fluid,

• According to Wohier, opianate of ethyl is not formed when a solution of opianic acid in :dcohol is saturated
with hj-drochloric acid. The experiments described in the text made it seem probable that the non-formation
of opianic ether under these circumstances might be owing to the decomposition of the opianic acid ; but, on
sealing up opianic acid with an alcoholic solution of hydi-ocliloric acid and heating it in the water-bath for an
hour, it was found that opianate of ethyl was formed in abundance. The ether was precipitated by pouring the
contents of the tube into water, and purified by crj-staUization from alcohol. Thus obtained, it formed hemi-
spherical masses of brilliantly white radiating needles, insoluble in water or dilute alkalis, melting to a colour-
less oil under hot water, and easUy soluble in alcohol and ether. In the dry state it melts at 88° C.

352 DE. A. MATTHIESSEN AXD ]\IR. G. C. FOSTEE ON

suddenly becomes nearly solid, and as soon as this change has occuiTed, no more opianic
acid can be detected in the mass. The decomposition may even be effected, without the
application of external heat, by simply mixing opianic acid with a very strong and warm
solution of potash.

In an experiment made in order to ascertain the proportions in which meconin and
hemipinic acid are formed in this reaction, 42 grms. of opianic acid yielded
13-5 grms. pure meconin,

18'5 grms. pure hemipinic acid (weighed as hemipinate of ammonium),
4"7 grms. meconin and hemipinic acid mixed,
5"0 grms. uncrystalhzable residue.
Total . . 41- 7 grms.
In another experiment, a small quantity of the mixture of opianic acid and potash was
boiled down in a small flask, fitted mth a glass tube about a yard long, bent vertically
do^^^lwards just outside the cork, and dipping by its lower end into mercury, ^\^len the
reaction had taken place, the flask was allowed to cool, and the mercmy then rose in
the vertical tube to within about an inch of the height of the barometer at the time ;
thus pronng that no permanent gas had been evolved. Hence the following equation —
according to which 42 grammes opianic acid should yield 19-4 grammes meconin and
22"6 grammes hemipinic acid — may be taken to represent the decomposition : —
2CioHioO, = C\oH,„G, + C\,U,,0,.

Opianic acid. Meconin. Hemipinic acid.

The following process was adopted for the purification of the meconin and hemipinic
acid thus formed. The alkaline mass obtained by heating opianic acid with potash, was
dissolved in a moderate quantity of warm water and mixed with an excess of hydro-
chloric acid : in this way the meconin was caused to separate out as an oil and to carry
doA\ai most of the hemipinic acid. After the acidified liqiiid had been allowed to cool
completely, it was poured off" from the solid cake of meconin and hemipinic acid which
had formed at the bottom, and evaporated to a small bulk, so as to cause the separation
of the greater part of the chloride of potassium ; this was washed with alcohol ; the
alcoholic washings were mixed with the concentrated mother-liquor, and the fresh quan-
tity of chloride of potassium which was thus precipitated was removed by filtration or
decantation, and the clear liquid was evaporated on the water-bath nearly to dryness ;
the residue thus obtained was again treated with alcohol, in order to separate the last
portions of chloride of potassium, and the alcoholic solution filtered and evaporated.
The product of these operations, together with the original precipitate of meconin and
hemipinic acid, was next dissolved in boiling water and the solution made slightly
alkaline with ammonia. Nearly the whole of the meconin then crystallized out as
the solution cooled, and was obtained quite pure by recrystallization from water ; the
hemipinic acid, on the other hand, remained in solution as hemipinate of ammonium,
together with a small quantity of meconin ; for although this substance dissolves only

THE CHEMICAL CONSTITUTIOX OF NAECOTINE. 353

very slightly in cold water, it is perceptibly more soluble in a solution of hemij^inate of
ammonium. In order therefore to complete the purification of the hemipinic acid, the
solution of its ammonium-salt was precipitated with acetate of lead, and the hemipinate
of lead, after being thoroughly washed, was decomposed under water by liydrosulphuric
acid.

Meconin and hrmipinic acid thus prepared were found to have all the properties
ascribed to tliem by previous observers. Their identity was further established by
analysis.

uinaJyses of Meconin.

I, 0-2G40 grm. substance, dried at 100", gave 0-5938 grm, carbonic acid and 0-12G4 grm.
water.

II. 0-3078grm. substance, dried in vacuo, gave 0-6928grm. carbonic acid and 0-14G4 grm.
water.

Calculated. Found.

Gjo . . .

. 120

Gl-85

I.
Gl-34

II.
Gl-39

Hjo . . .

. 10

515

5-32

5-28

o

. G4

33-00

V4 . . .

^10 Hifl O4

. 194

100-00

Chloromeconin was prepared from this product by treating it in aqueous solution with
hypochlorite of sodium and hydrochloric acid.

I. 0-1940 grm. chloromeconin gave 0-1213 grm. chloride of silver.
II. -2920 srrm. chloromeconin gave '1801 sxm. chloride of silver.

Calculated. Found.

C,„H„C1G,. I. II.

•io-"^a

Chlorine per cent 15-54 15-46 15-7G

Nifromeconin was also prepared, but not analysed.

Analyses of Ilemipinic Acid.

I. 0-3234 grm. acid, dried at 100", gave 0-G28G grm. carbonic acid and O-loGl grm.
water.

II. 0-3980 grm. acid gave 0-7748 grm. carbonic acid and 0-1654 grm. water.

Calculated. Found.

C,o .... 120 53-10
H,o . . . . 10 4-42

Og 96 42-48

OjoHioOg . . 226 100-00
1-3447 grm. air-dry acid lost 0-1862 grm. at 100'.
MDCCCLXIII. 3 C

I.
53-01

II.
53-09

4-68

4-62

354 DE. A. MATTHIESSEN AXD MK. G. C. FOSTEE ON

Calculated. Found. ' ■. ' "

C,oH,oO,2H,0. • • ■ \

Water per cent 13-50 13-85

The sUvcr-salt of this acid was found to have the composition and properties of hemi-
pinate of silver.

I. 0-3855 gnu. salt gave 0-1885 grm. silver.
II. -4225 grm. salt gave -2060 grm. silver.

Calculated. Found.

C,„H«Ag:,0,. I. II.

Silver per cent. . . . 49-09 48-90 48-76

3. Action ofXascent Hydrogen. — AMicn an aqueous solution of opianic acid is -warmed'
for several hoiu-s with sodium-amalgam, the subsequent addition of hj'drochloric acid
causes a precipitate of meconin. The formation of meconin is not due, in this case, to
the decomposition of the opianic acid by the soda formed from the sodium-amalgam ; for
it takes place in a dilute solution, and at a temperature very much below that at whicli
opianic acid is decomposed under the influence of alkali ; the quantity of meconin
formed from a given weight of acid is also considerably greater than that produced
under the latter circumstances ; for instance, 5 grms. opianic acid gave 3-65 grms. pure
meconin, whereas 5 grms. opianic acid decomposed by alkali would yield 2-3 grms.
meconin and 2-7 grms. hemipinic acid. The barely possible supposition that the addi-
tional quantity of meconin is owing to the reduction of hemipinic acid formed in a
previous stage of the reaction is excluded by the fact that hemipinic acid is not acted on
by sodium-amalgam in presence of water. Moreover opianic acid is similarly converted
into meconin by the action of zinc and dilute sulphuric acid. Hence the transforma-
tion consists in a direct reduction or deoxidation of the acid under the influence of
nascent hydrogen.

C.oHioO, + n, = C.oHioO, + H.,0.

Opianic acid. Meconin.

According to this equation, 5 parts of opianic acid correspond to 4-6 parts of meconin.

Of the two following analyses, the first was made \\\i\\ the product of the action of
sodium-amalgam, the second with that of the action of zinc and dilute sulphuric acid on
opianic acid ; this latter product retained a sliglit colour, even after being several times
recrystallized, and was obviously not quite pure.

I. 0-3906 grm. substance gave 0-8845 grm. carbonic acid and 0-1834 grm. water.

II. -3700 grm. substance gave -8348 grm. carbonic acid and -1094 grm. water.

Calculated. Found.

G,Q - . .

r
... , 120

1
61-85

Hio • •

- . 10

5-15

0, . . .

. . 64

33-00

I. II.

61-76 60-55

5-22 5-01

CjoHjoO, . . 194 100-00

fHB CHEMICAL CONSTITUTION OF NAECOTINE. 85S

Chemists have long been awaie of the simple relation subsisting between the formula;
of

Meconin CioHjuO^,

Opianic acid C,o Ilm O5,

and

Hemipinic acid . . . . CjoHjo O,;;

the foregoing experiments prove that this relation is not confined to tlie formula-, but
that the bodies themselves are intimately connected. Hence it was evident that, in
order to arrive at a knowledge of the constitution of opianic acid, the chemical nature
of the other two members of the group must also be examined. The experiments
which we have made in this direction are still very far from complete ; they relate prin-
cipally to the action of acids on meconin and hemipinic acid, and the following are the
most important results yet obtained.

Actiou of Ilj/driodic and Hydrochloric Acids on 2Ieconin. — "When meconin is boiled
with concentrated hydriodic acid, a considerable quantity of iodide of methyl is formed,
but the other products of the reaction aa-e so unstable that it has not been found possible
to remove the excess of hydriodic acid without completely destroying them. It was not
till after many attempts to purify the products thus formed that it was found that
meconin undergoes a similar decomposition when heated in a sealed tube, to a little
above 100°, with thi-ee times its weight of strong hydrochloric acid. The chloride of
methyl, formed under these circumstances, quickly volatilizes when the tube is opened,
and by careful evaporation, 'over lime and sulphuric acid, a- crystalline residue may be
obtained, whence it is possible to remove completely the excess of hydrochloric acid;
but it has not yet been further investigated. . .

Action of Hydriodic Acid on Hemipinic Acid. — Hemipinic acid is decomposed, when
Boiled with concentrated hydriodic acid, into carbonic acid, iodide of methyl (boiling-
point 42°-8C., vapour-density 5"05; calculated vapour-density 4-92), and an acid of the
formula Cj Hg O4. It will be seen that the composition of this acid is intermediate
between that of salicylic acid, €7 Hg O3, and that of gallic acid, G>, Hg ©5, and, as will be
seen by the description Avhich follows, it is analogous to these acids in some of its pro-
perties. Accordingly, in order to recall the fact of its containing one atom of oxygen
less than gallic acid, we propose to name it provisionally hypogallic acid, reserving to
o«rselves to suggest, if possible, a more appropriate name when its chemical relations shall
have been more thoroughly investigated. The reaction, by which these products are
formed, takes place according to the equation

CioHioOg + 2 HI = CO, + 2GH,I + CVHgO,.

Hemipinic acid. . Iodide of mclhyl. Hypogallic acid.

Hypogallic acid, when pure, is, only slightly soluble m cold water, but dissolves easily
ia hot water, alcohol, and ether ; its solution reacts strongly acid with litmus-paper. It
separates from hot water in small prismatic crystals, united into stellate groups, and

3 c 2

356

DE. A. MATTIIIESSEN A^D im. G. C. FOSTEK ON

containing 1^ atom water of crystallization, which they lose at 100". The acid melts at
about 180°, but, as it begins to decompose even at a lower temperature, its melting-point
could not be accurately ascertained. Dried at 100" C, it gave the following results on
analysis : —

I. 0'34G5 grm. substance gave 0-G904 grm. carbonic acid and 0-1238 grm. water.

II. -4670 grm. substance gave -9326 gi-m. carbonic acid and -1662 grm. water.

III. -4968 grm. substance gave -9900 grm. carbonic acid and -1710 grm. water.

Calculated.

Fouiul.

€- . .

. . 84

54-55

I.
54-34

II.

54-40

III.'
54-35

He. .

. . 6

3-89

3-97

3-95

3-82

0, . .

. . 64

41-56

^VH,;0,

154 100-00

I. 1-486 grm. crystallized acid lost 0-2200 grm. at 100° C.
II. 2-132 grms. crystallized acid lost "3120 grm. at 100°.
III. 1-138 grm. crystallized acid lost -1716 grm. at 100°.

Calculated.

CjHeO^ 154 85-08

H3 0,j 27 14-92

Found.
II.

14-80 14-63

III.
15-08

€,ILO,, liH.,0

181 100-00

Hypogallic acid gradually turns browni when heated in the air to a little above 100° C. ;
the same change occurs more quickly when a solution of it, especially if neutral or alka-
line, is evaporated. Added to solution of nitrate or ammonio-nitrate of silver, it causes
an immediate precipitation of metallic silver, even in the cold; with sulphate of copper
and a slight excess of potash it gives a yellowish-green solution from which an orange-
yellow precipitate is thrown down on warming; in a mixture of sesquichloride of iron
and red prussiatc of potash, it immediately produces a blue i^rccipitate ; when boiled
with solution of corrosive snhlimatc, it reduces it to calomel, ^^'ith sesquichloride of
iron, it gives an intense indigo-blue coloration, Avhich is changed to violet by a very
small quantity of ammonia, and to blood-red by excess of ammonia, no precipitate being
produced, unless too much chloride of iron has been used ; the colour is destroyed by
strong acids, but restored by neutralization with alkali, and partially by addition of
water. A solution of the acid immediately becomes brown on addition of alkali, the
colour quickly becoming darker by exposure to the aii". With ammonia and chloride of
barium or calcium, it gives a dirty brown flocculcnt precipitate ; with acetate of lead, a
pale yellow precipitate.

THE CHEMICAL CONSTITUTION OF NAECOTINE. 357

ITypogallic acid is decomposed by heat into carbonic acid and a substance which
soHdifics in the neck of the retort to a colourless crystalline mass. Tlie decomposition
begins at about 170° C, and goes on rapidly at 200°. The crystalline product melts, in
the crude state, at about 90° C. ; it dissolves easily in water and crystallizes in needles
when the solution is evaporated. It is rapidly attacked by nitric acid, even when
diluted, giAing a red-brown solution. AVith sesquichloride of iron it gives a bluish-black
amorphous precipitate ; with acetate of lead it gives a white or yellowish-white precipi-
tate, soluble in an excess of acetic acid. It slowly assumes a darker colour by exposure
to air in contact Avith alkali. This substajice has not yet been prepared in sufficient
quantity for complete investigation.

In addition to hypogallic acid, no less than three other acids are known, having the
same composition, and resembling it to a remarkable degree in some of their most
characteristic properties. They are carbohydrochinonic acid* (obtained by Otto He.sse
by the action of bromine in presence of water on chinic acid), protocatcchuic acidf
(obtained by Strecker as a product of the action of fused potash on piperic acid), and
oxysalicylic acidj (obtained by Lautemanx by decomposing iodosalicylic acid with pot-
ash). All these acids are described as having about the same solubility in water, alco-
hol, and ether as hypogallic acid ; like it, they all give a dark coloration with the
smallest trace of sesquichloride of iron, they all reduce nitrate of silver, they all become
dark broAvn when mixed with alkali and exposed to the aii", all give a yelloAvish-white
precipitate with acetate of lead, and at a high temperature they are all decomposed into
carbonic acid and oxyphenic acid or hydrochinone^. Nevertheless no two of these acids
appear to have quite the same properties. The following are the most important points
in which diiferences have been observed. Hypogallic acid crystallizes with 1^ mole-
cule of water (14'9 per cent.), carbohydrochinonic and protocatcchuic acids with 1 mole-
cule (10-4 per cent.), and oxysalicylic acid without water. Hypogallic and oxysalicylic
acids give a dark blue colour with sesquichloride of iron, the other two acids a dark
green colour. Hypogallic acid reduces nitrate of silver immediately in the cold ; carbo-
hydrochinonic acid reduces it slowly in the cold, rapidly when boiled ; oxysalicylic acid
has no action on nitrate of silver in the cold, but reduces it completely when boiled.
Carbohydrochinonic acid reduces cuprous oxide from a mixture of cupric acetate,
tartaric acid, and excess of potash ; protocatcchuic acid causes no reduction of the same
solution. Hypogallic acid causes a precipitate in a mixture of chloride of barium and
ammonia ; protocatcchuic acid only on addition of alcohol.

Whether some of these differences may not be due to accidental causes, depending on
the different sources and modes of preparation of the several acids, is a question that
readily suggests itself, but it can be answered only by further investigation.

* Annalen der Chcmie und Pharmacie, vol. cxii. p. 52 (1859); vol. cxiv. p. 292; vol. cxxii. p. 221,
t Ibid. vol. exviii. p. 2S0. J Ibid. vol. cxviLi. p. 372; more fully vol. cxx. p. 311.

§ The product obtained by the action of heat on hypogallic acid docs not fully agree in its reactions with
either of these bodies, so far as yet examined.

35S DR. A. MATTIUESSEX AND JIE. G. C. FOSTEE Olf

Action of Uydrochloric Acid on Hemipinic Add. — Hemipinic acid is rapidly decom-
posed when heated with two or three times its weight of strong hydrochloric acid, either
in a sealed tube to about 110° or in an open vessel connected with a condenser so
arranged that the condensed vapour flows back into the mixture, and with an apparatus
for evohing gaseous hydrochloric acid, whereby the liquid can be kept constantly satu-
rated with that acid. The products of the reaction are chloride of methyl, carbonic
acid, and an acid crystallizing in beautiful long transparent prisms.

This acid is almost insoluble in cold water, and not much more soluble in boiling
water; alcohol and ether dissolve it more easily. When heated it begins to subhme,
without decomposition, at about 200' C, and supports a temperature of more than 245°
without any further change, though at a still higher temperature it melts, and solidifies
again on cooling to a crystalline mass. It dissolves in strong sulphiiric acid, and is
precipitated unchanged on addition of water. It gives no coloration with sesquichloride
of u"on ; with nitrate of silver it gives a white precipitate, which blackens on boiling.
It gave, on analysis, numbers agreeing nearly with the formula t-g Hg O^.

0'2747 grm. substance gave 0-5702 grm. carbonic acid and 0-1277 grm. water.

' . / ' ■ ' " • Calculated. Found. •: , ■ - . :

■■€„...

96

57-14

56-64

S

4-76

5-17

64

38-10

. -GgHgO^ . . 168 100-00

The formation of this body therefore probably takes place in accordance with the

equation

GioHioG)^ + 11 CI = CO, + CII3CI + CgHgO,.

Hemipiuic acid.

We hope to give a more complete description of it in a future communication.

By the prolonged action of hydrochloric acid on the mother-liquor from which this
acid has crystallized, hypogallic acid appears to be formed. The product tlius obtained
has not yet been analysed, but it is identical in all its qualitative reactions with that
formed by the action of hydriodic acid on hemipinic acid.

Nascent Ilydroyen, resulting from the action of sodium-amalgam on an aqueous solu-
tion of hemipinic acid, leaves that acid unacted upon, as has been already stated. A
portion of hemipiuic acid that had been subjected for a long time to the action of
sodium-amalgam was converted into silver-salt, the salt being precipitated in two frac-
tions. Both portions, as shown by the following analyses, consisted of pui-e hemi-
pinate of silver, and no other organic substance than hemipinic acid could be detected
in the solution.
- I. 0-7050 grm. salt (fir-st precipitate) gave 0-346G grm. silver.

II. -2655 grm. salt (second precipitate) gave -1302 grm. silver. ■. ' ■ ■ ■ r y.y

THE CIIEJnCAL CONSTITirTION OF NARCOTINE, 359

Calculated. Found.

C.„H,Ag,G,.
Silver per cent. . . . 49-10

We may here mention a peculiar property of hemipinatc of barium which does not
seem to have been before observed, but which has often been of service to us as afford-
ing the means of recognizing hemipiuic acid when present only in small quantity.

Wlicn a solution of hemipiuic acid is neutralized with baryta-water, or when solutions
of hemipinatc of ammonium and chloride of barium are mixed together, the liquid
remaiirs clear for a long while if left to itself; but if it is boiled for a short time, small,
shining, crystalline plates of hemipinate of barium are precipitated and soon fill the
whole liquid, if the solutions used were not too dilute. On allowing the liquid to cool,
it redissolves this precipitate, and becomes almost or quite clear ; but after standing for
a few hours, or for a day or two, according as more or less of the salt is contained in it,
it again deposits hemipinate of barium, but this time in the form of feathery tufts of
very small silky needles ; if the liquid be now again heated, these feathery crj'stals
dissolve, and the crystalline plates already mentioned make their appearance once more.

§ III.— DECOMPOSITIONS AND DEllIYATIVES OF COTARNINE.

I. Action of Nitric Acid. — By gently heating cotarnine with dilute nitric acid, we
have obtained nitrate of methylamine and a new acid, cotarnic acid, but have not
hitherto found out the conditions necessary for the certain production of the latter
substance, many attempts to obtain it having been completely unsuccessful.

Cotarnic acid dissohes easily in water, giving a solution which reacts strongly acid
with litmus-paper: it dissolves to a less extent in alcohol, and is precipitated from its
alcoholic solution by ether. It yields no trace of cyanide when heated with metallic
sodium in excess, and is thus proved to contain no nitrogen. It gives a white precipitate
with acetate of lead, and is not affected by sesquichloride of iron. Nitrate of silver
throws down a very stable silver-salt, which may be crystallized from boiling water, in
which it is slightly soluble, without alteration. This salt contains Gjj H,o Ago Oj.

I. 0'2513 gi-m. salt gave 0-2693 grm. carbonic acid, 0-049 grm. water, and 0-1235 grm.
silver.

II. -2065 grm. salt gave 0-2279 grm. carbonic acid, 0-0403 grm. water, and 0-1019grm.
silver.

Calculated. Found.

Oil . . . . . . 132"

Hjo 10

Ag, 216

Q5 . ■ ■ . . , 80

ei^IVAg^Os '. . 438

I.

II.

3014

29-22

.30-10

2-27

2-16

2-17

49 -.32

49-14

49-35

18-27

100-00

360

DE. A. MATTHIESSEX AND ME. G. C. FOSTEE ON

Hence cotarnic acid must contain C'n IIjo O^, and its formation from cotarnlne must
take place according to the equation

CJ2H13NO., + 211.0 = C,iH,,0, + CH5N.

Cotarnine. Cotarnic acid. Mcthylamine.

Tlie formation of methylamine in this reaction was proved by the analysis of its chloro-
platinatc.

0-3408 grm. salt gave O-l-iOT grm. platinum.

Calculate,!.
CH5 N, H C:i, rt (.%.

. . 41-7

Found.

Platinum per cent. . . 41-7 41 '3

Cotarnine heated with undiluted nitric acid was found to yield oxalic and apophyllic
acids, in accordance with the statements of Wuiilkr and Andkeson.

2. Action of Ilijdrochloric Acid. — Cotarnine heated with three times its weight of
strong hydrochloric acid to about 140' C, in a sealed tube, is decomposed into chloride
of methyl — identified, inter alia, by its formation of the solid hydrate described by
Bayer * — and a substance which crystallizes in very small, pale yellow, silky needles.
This body we designate, provisionally, lujdrochlomte of cotcmiamic acid, assigning to it
the formula CnHigNO^, HCl, which, though not agreeing perfectly with the analyses
hitherto obtained, expresses their results more closely than any other formula that seems
equally probable. The substance was purified for analysis by several crystallizations
from water slightly acidulated with hydrochloric acid, and was dried in vacuo over lime
and sulphuric acid.

I.

II.

III.

IV.

V.

VI.

VII.

VIII.

IX.

X.

3295 crm. substance sravc 0-G020 grm. carbonic acid and 0'1729 grm. water.

404G grm. substance gave
3378 grm. substance gave
3649 grm. substance gave
4826 grm. substance gave
5074 grm. substance gave
4590 grm. substance gave
4654 grm. substance gave
3712 grm. substance gave
5200 grm. substance gave

Calculated.

■7429 grm. carbonic acid and
•6182 grm. carbonic acid and
•1480 grm. platinum.
•2760 grm. chloride of silver.
•3220 grm. cliloride of silver.
•2626 grm. chloride of silver.
•2593 grm. chloride of silver.
•2128 grm. chloride of silver.
•2905 grm. chloride of silver.
Found.

•2017 grm. water.
•1745 grm. water.

Gn 132

Hh 14

N 14

O, 64

CI 35-5

Cu Hi3 NO4, HCl . 259-5 100-00

50-87

5-40

5-40

24^65

13^68

I.,IV.itV.

49-83
5-83
5-73

II.&VI.

50-08
5-54

]I1.&V1I.

49-91
5-74

Ylll.

IX.

14-15 1404 14-15 13-78 14-18 13-82

Mean.

49-94

5-70

5-73

24-01

14-02

100-00

» Aunalou der C'liomic uud Pharmacio, vol. elii. p. 183.

THE CHEMICAL CONSTITUTION OF NAECOTINE. 361

Assuming for the present the accuracy of the proposed formula, the action of hydro-
chloric acid on cotamine will be represented by the following equation : —

C,2 H,3 NO3+H2 G+2H C1=GH3 Q+Gu H,3 NO^, H CI.

Cotarnine. Hydrochlorate of

cotarnamic acid.

Hydrochlorate of cotarnamic acid is only slightly soluble in cold water, but is very
soluble in hot water ; it is less soluble in alcohol than in water, and almost insoluble in
ether. When dissolved in hot water, it always undergoes partial decomposition, as is
indicated by the separation of an orange-coloured granular precipitate if the quantity of
water used is only moderate ; when more water is employed, this precipitate remains
dissolved, the smallest trace imparting a bright orange-colour to the solution. A similar
precipitate (cotarnamic acid, CjjHjjNO^^) is thrown down on cautiously adding an
alkali, or alkaline carbonate or sulphite, to an aqueous or slightly acid solution of hydro-
chlorate of cotarnamic acid ; this precipitate regenerates the original compound when
treated ■n-ith hydrochloric acid ; it dissolves with orange-colour in excess of alkali, giving
a solution that rapidly becomes brown in the air ; by washing with water, out of con-
tact with air, it may be obtained quite free from chlorine. The hydrochlorate dissolves
without alteration in water containing a small quantity of free hydrochloric acid ; the
solution, which is but slightly coloured at first, gradually acquires a beautiful dark-green
colour by exposure to the air. If nitxic acid is slowly added to a solution of the com-
pound in boiling water, the portions of the liquid with which the acid comes in contact
assume a fine opaque crimson colour when seen by reflected light, but appear of a trans-
parent orange tint when seen by transmitted light. After a few minutes a slight efier-
vescence takes place, and this effect disappears. Hydrochlorate of cotarnamic acid
mixed with a slight excess of dilute sulphuric acid, and evaporated nearly to dryness on
a water-bath, acquires a fine crimson colour- rivalling that of acetate of rosaniline ; this
colour is destroyed by addition of water, but it appears again when the water is evapo-
rated. Nitrate of silver added in excess to a hot solution of the hydrochlorate is rapidly
reduced.

Hydriodic and sulphuric acids appear to act upon cotarnine in the same way as
hydrochloric acid. AVith hydriodic acid iodide of methyl is foi-med in unmistakable
quantity: the formation of a methyl-compound by the action of sulphuric acid was
not proved, but in both cases appropriate treatment of the resulting solutions yielded
hydrochlorate of cotarnamic acid with all its characteristic properties above described.
Analyses of specimens prepared in this way are Lucluded among those already given
(page 360).

We have not yet obtained cotarnamic acid itself, nor any of its compounds except the
hydrochlorate, in a state that invited analysis ; but we hope to be able to do so on con-
tinuing our experiments, and thus to remove any uncertainty that may at present exist
as to the true formula of this substance.

3. Action of Potash. — Cotai-nine distilled -n-ith caustic potash yields ammonia and

MDCCCLXIII. 3 D

362 DE. A. MATTHIESSEX AXD JME. G. C. FOSTEE ON

methylamine, biit apparently no di- or tri-methylamine or other similar compounds.
The ammonia and methylamine were separated by treating their hydrochlorates with
absolute alcohol in the usual way. The chloroplatinate of the latter was analysed.
0-.3423 grm. salt gave 0-141.3 grm. platinum.

Calculated. Found.

GH.X, HCl, PtCl,.

Platinum per cent. . . 41'7 41'3

§ lY.— COXCLUSIOX.

The conclusions which the foregoing experiments enable us to draw, relatively to
the constitution of narcotine and its derivatives, are far from being sufficiently com-
prehensive and precise to admit of expression by a series of rational formulfe ; never-
theless it may be allowable to recapitulate briefly the chief points which we think have
been established, and to offer some suggestions towards the interpretation of the results
that have been obtained.

1. Our analyses indicate the existence of only one kind of narcotine — that, namely,
which contains G.,., H23NO;. It has already been stated (page -345) that this formula
was assigned by Wertheim to " methyl-narcotine," which he believed to be only one of
several varieties of the base ; in addition he recognized, on the authority of Bltth's
investigation, the existence of " ethyl-narcotine," G.,^ ^25 NO^ ; and in consequence of
having obtained a volatile base, of the composition of propylamine, G3 Hy N, by distil-
ling narcotine with potash, he also admitted the existence of " propyl-narcotine,"
C24 H27 NO;.

In a note near the beginning of this paper, the results of all the published analyses
we have been able to find are put together. These results do not appear to us to
afford any strong evidence that other chemists have operated upon a kind of narcotine
different from that represented by the formula C22 H23 NO; ; and we think that the
following experiment shows that such a supposition is not required to explain the
observation which caused the existence of propyl-narcotine to be admitted.

By distilling 20 grammes of narcotine with an excess of concentrated hydriodic acid,
we obtained 19 grammes of pm'e iodide of methyl, a quantity which is nearly in the
proportion of three molecules of iodide of methyl to one molecule of narcotine

(€22^23 NO; : 3CH3I : : 413 : 436 : : 20 : 21-1).
Narcotine tlierefore contains three atoms of methyl so combined as to be easily sepa-
rated, and hence we think it likely that Wertheim's supposed propylamine was the
isomeric trimethylamine. Tliis experiment makes it also seem probable that the distil-
lation of narcotine with potash woidd yield trimethylamine, dimethylamtne, methyla-
mine, and ammonia in proportions varying with the conditions of the experiment, in
which case the nature of the product could afford no trustworthy evidence as to the
composition of the material employed.

THE CHEXnCAL CONSTITFTION OF NAECOTINE. 363

On the other hand, the existence of narcotine containmg three atoms of methyl may,
at tii'st sight, seem to render probable the existence of other varieties of the base in
which ethyl or a similar radicle takes the place of the whole or part of the methyl.
Analogy, however, does not support such an inference. Many natural products are
known which contain, in some form or other, methyl as one of their constituents,
although corresjionding compounds containing ethyl are either entirely miknown or
have been obtained only by artificial means. We may mention as exami)les methyl-
salicylic acid, methyl-coniine (von Plaijta and KekulS), brucine (Stkecker), morphine
(WERTnEiJi), codeine (Anderson), caffeine (Wurtz, Kochleder, Strecker), theobro-
mine (RocHLEDER and Hlasiwetz, Strecker), creatine (Strecker), sarcosme (Strecker,
Volhard).

2. Accordmg to our analyses, narcotine contains the elements of meconin and cotar-

nine: —

G,, H23 NO;=C\o H,o 0,+C,, H,3 NO3.

Narcotine. Meconin. Cotamine.

After having discussed the probable nature of these two bodies, we shall return to the
consideration of their possible functions as constituents of narcotine.

3. We have been unable to find a second reaction altogether analogous to the trans-
formation of opianic acid into meconin and hemipinic acid, regarded as a mere trans-
ference of an atom of oxygen from one molecule of opianic acid to another, according to
the equation

^10 Hio Og+Gjo Hjo 05=€io Hjo 04 + Gio Hjo Gg.

Opianic acid. Opianic acid. Meconin. Hemipinic acid.

Possibly, however, this equation, though correctly expressing the final result, does not
represent the actual decomposition which occurs in the first instance. Of course, instead
of hemipinic acid, hemipinate of potassium is produced at first, but perhaps also, instead
of meconin, a compoixnd of that body with potash, GjoHjoO.,, KHO( = C,qHjj KO^),
may be the coiTclative product. The reaction would then be (substituting for potash,
in the equation, its equivalent of water)

2 Gio Hio Os-f H2 0=G,o H^^ O^-f C,,, H,o G«,

strictly analogous to the transformation of oil of bitter almonds into benzylic alcohol and
benzoic acid :

2G-HeO +H2O = G-HgO -f G-HcOa-

Oil of bitter almonda. Benzylic alcohol. Benzoic acid.

According to this supposition, meconin would be an anhydride of a less stable com-
pound Gjo H12 O5, probably a body possessing more or less the characters of a polyatomic
alcohol. If, \vith Bertiielot*, we compare meconin to ethylene, opianic acid to
aldehyde, and hemipinic acid to acetic acid, the compound Gjo H12 O5 might be com-
pared to alcohol:

* Ann ales de Chimie et de Physique, 3rd series, vol. Ivi. p. 77.

3d2

364 DE. A. MATTHIESSEX AXD IME. G. C. FOSTER OX

Mcconin Cjo Hjj G4 Cj H4 . . . Ethylene.

Hypothetical hydrate of mecouin . . . Cjq Hjq O3 G2 Hg O . . Alcohol.

Opianic acid Cjq Hjq O5 Gg H^ O . . Aldehyde.

Hemipinic acid Gjg Hjo Og Cj H4 O2 . . Acetic acid.

The supposed existence of a hydi-ate of meconin derives some slight support from the
fact that, when the product obtained by heating opianic acid with potash is dissolved
in water and the solution made acid v^-ith hydrochloric acid, the meconin remains for a
long time dissolved in the acid liquid unless a great excess of hydrochloric acid is added —
apparently as though it did not exist as such in the solution, but required a certain
time, or else the aid of a great excess of acid, to enable it to separate from a pre\ious
state of combination *.

4. The conversion of opianic acid into meconin by means of nascent hydrogen is a
transformation as anomalous as that last considered, if viewed as a direct deoxidation of
the acid ; but if it be supposed that a compound G^q Hj., O5 is first formed and is con-
verted by subsequent dehydration into meconin, the reaction appears (not as a removal
of oxygen, but) as a fixation of hydrogen, and takes its place among a large number of
similar transformations which have been observed withm the last few years to occur
under like conditions : —

G10H10G5 + H2 = G10HJ2O5.
Opianic acid. Hydrate of meconin.

Among the many analogous reactions we may mention especially the conversion of
glucose into maunite, lately effected by LixxEMANN'f',

^6 H12 Og -f H2 = Gg Hi4 Og,

Glucose. Mannite.

— a reaction which perhaps justifies the following comparison of formulfe: —

Meconin Gjo Hjq O4 Gg Hjj G5 . . Mannitan.

Hydrate of Meconin . . . G^ Hjg Oj Gg H^^ Og . . Mannite.

Opianic acid G^ Hjq O5 Gg H12 Og . . Glucose.

Hemipinic acid C^q Hjo Og Gg Hj2 O^ . . Mannitic acid.

If this \iew of the action of nascent hydrogen on opianic acid be adopted, and if the
decomposition of that acid by potash be interpreted in the manner suggested in the last
paragraph, both reactions may be regarded as giving additional weight to the views of
Bertiielot, who has already pointed out that opianic acid possesses many properties

* Among the products which Andkuson obtained by the action of nitric acid on narcotine, was one which
he sui)posed to bo a hydrate of meconin. Anderson did not obtain a .sufficient quantity of this substance to
enable him to ascertain its chemical relations ; and we have already shown elsewhere (Proc. Eoy. Soc. vol. xi.
p. CO) that his analyses agree precisely with the foi-mula of cotamic acid, C^^ H,.^05: they also agree, though
not quite as well, with the composition of a hydrate of meconin =2(G||,Hm04) HjO, but not at all with that
of the hydrate whose existence is sujiposcd in the text.

t Annalon der Chemie und Pharmacio, vol. cxxiii. p. 136,

THE CHEMICAL CONSTITUTION OF NAECOTINE. 365

usually considered characteristic of the aldehydes. In any case, however, the fact that
nascent hydrogen converts opianic acid into meconin, but has no action on hemipinic
acid, excludes the possibility of regarding the first of these bodies as a hemipinate of
meconin.

5. The action of hydi'ochloric and hydriodic acids on opianic acid and its congeners
proves that methyl is a constituent of each of them ; and since one nioU'cule of hemi-
pinic acid yields two molecules of iodide of methyl when distilled with hydriodic acid,
we must assume the existence of two atoms of methyl in each molecule of meconin,
opianic acid, or hemipinic acid. It is useless to discuss the function of the methyl in
the fLfst two of these substances before possessing more definite knowledge of theii-
chemical nature and of the bodies formed from them when the methyl is eliminated.
In the case of hemipinic acid, our knowledge on these points is already sufficient to
enable us to form a tolerably clear idea of the state *of combination in which its two
atoms of metliyl may exist.

Hemipinic acid was shown by Anderson to be a well-characterized bibasic acid, and
to contain, in one molecule, six atoms (0=16) of oxygen. Among the better-known
acids, one which resembles it in these particulars is tartaric acid, G4 Hg Og. Now the
experiments of Perkin and Duppa, and of Kekuue, as well as those of Schmitt and
Dessaignes, clearly show that, although tartaric acid is only bibasic in the strict sense,
yet it is tetratomic, or that, in the language of the modern theory of types, it contains
four atoms of hydrogen outside the radicle. If we assume the existence of an acid
Gg Hg Og strictly comparable to tartaric acid, that is, containing outside the radicle four
atoms of hydrogen whereof two only are replaceable by ordinaiy processes of saline
double decomposition, and if we further suppose the other two atoms to be replaced by
methyl, the resulting compound would have the composition of hemipinic acid

(C3H,(CH3)2 0g = GioHioOg),
and might be expected to resemble it in being a bibasic acid, yielding two molecules of
iodide of methyl when boiled with hydriodic acid, but no methylic alcohol when boiled
with potash. The two atoms of methyl in such a compound would be combined in the
same manner as the one atom of ethyl in ethyl-lactic acid, which, as proved by the
experiments of Wdrtz and Butlerow, cannot be eliminated by the action of alkalis, but,
as found by the latter chemist *, is easily eliminated by the action of hydi'iodic acid.

If hemipinic acid be allowed to have the constitution here suggested, the decomposi-
tion which gives rise to hypogallic acid must be supposed to consist of two stages : — fii'st,
the replacement of (GH3)2 by Hg, resulting in the formation of a hypothetical normal
hemipinic acid, Gg Hg Og,

Gg H, (GH3)2 Og + 2HI=G8 Hg Og + 2GH3 1 ;

* On the dissymmetrical constitution of the radicle of lactic acid as the probable cause of the dissimilar
functions of its two replaceable hydrogen-atoms, see KEKUxfi, Lehrbuch der Organischen Chemie (Erlangen,
1861), vol. i. pp. 174 and 730. The same considerations apply, mutatis mutandis, to the case of a tetratomic
acid such as tartaric acid.

366 DE. A. MATTHIESSEN AJS'D ]\IE. G. C. FOSTEE ON

secondly, the decomposition of this product, at the tempcratiu-e of boiling hydriodic acid
(125C.), into carbonic and hypogallic acids: —

6. The action of acids on cotarnine proves that one molecule of that substance con-
tains one atom of methyl. When the methyl is eliminated, as by the action of hydro-
chloric acid, the first phase of the reaction probably consists in the simple replacement
of methyl by hydi-ogen, gi™ig rise to a non-metliylized or normal cotarnine, —

€11 Hio (CH3) NO3 + HCa = Cii Hu NO3 + CH.,, CI ;

Cotai'umc. Xormal coturniiie.

but this compound, in presence of aqueous hydrochloric acid, fixes the elements of
water and is converted into cotarnamic acid : —

G„ Hji NO3 + U,0 = G,, H,, NO,.

Normal cotarnine. Cotarnamic acid.

Cotamic acid, Gjj Hjj O5, which may be regarded as the central member of the
cotarnine group, is a bibasic acid containing five atoms of oxygen, and is therefore
analogous to malic acid, C, Hg O5. Accordingly^, we find, among the derivatives of malic
acid, compounds which coiTespond to all the derivatives of cotarnic acid.

Cotamic acid .... Cjj H,., O5 €4 H^ O5 Malic acid.

Cotarnamic acid . . . ( \i Hjj NO4 G4 H- NO4 Aspartic acid.

Hydr-ochlorate of cotar-|^ ^^^^ ^^^ rHych-ochlorate of

namic acid ... .J ■* / i ^ aspartic acid.

Cotarnimide (hypothetical] ^^

normal cotarnine)
Methyl-cotarnimide (co-]

tarnine)

The substance here called malimide is produced by the dry distillation of acid malate
of ammonium (Pasteur), or of a mLxture of equivalent quantities of aspartate of barium
and ethyl-sulphate of potassium (Dessaignes): aspartic acid is produced when this com-
pound is boiled with hydrochloric acid (Dessaignes), exactly as we have supposed
cotarnamic acid to be formed from cotarnimide.

The close analogy existing between the derivatives of cotarnic acid and those of malic
acid indicates that cotamic acid is probably triatomic ; hence the typical formulse by
which we represented this acid and cotarnine in our preliminary communication*,
require to be somewhat modified, and would probably be more correctly written thus : —

" }o H }0

(G„n,(),r (GaH,0,r .

H, Vh CIT, }N

Cotarnic acid. Cotarnine.

* Proceedings of the Eoyal Society, vol. xi. p. GO.

|c,iHi3N04, HCl C4H;N04, HCl ^
}CiiH„NG3 G4H5NO3 I

}Cii Hio (CH3) NO3 C4 II4 (C, H.) NO3 Phenyl-malimide.

THE CHEMICAL CONSTITUTION OF NAECOTINE. 367

This foi'mula for cotarnino, whicli represents it as derived from the double type

TT^^l, is still in accordance with the experiments of How* (who found it impossible to
H3NJ

replace hydrogen in cotarnine by means of iodide of methyl), for it will be seen that

the wliole of the hydrogen of the ammonia is represented as already replaced.

7. !Meconin being regarded as a polyatomic anhydride, and cotarnine as an imide, it
is allowable to suppose that the constitution of narcotine may be similar in some degree
to that of the oxygenated alkaloids obtained by Wurtz by the union of oxide of ethylene
with ammonia.

Meconin digested with aqueous ammonia in a sealed tube forms a solution from which
it is not deposited on cooling, nor even when the excess of ammonia is expelled by
cautious evaporation. Possibly the investigation of the compound thus formed may
throw further light on the constitution of narcotine.

* Transactions of the Roj-al Society of Edinburgli, vol. xxi. p. 31.

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PlauJ^l.

tiuvfx itprt.sentinff the Dninial i'hangt in
MaqnUiidt luid Diirdivn li' tht Mupnelic
Tones in thf Nori^mial I'Uuie admg in Iht
Nerlh Bid ul ' the Netdit.. at the- lioyaJ
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liirii tlif Vuu.s IS II h< ISn. iifliri
iht Mean of all (hi DthservaUcn.s.
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PM, 7ra/w MDCCCI JtlH . PUUe^ STE.

turves irpresftttifUf thr Duinuil Change ui Ma/jniOidt and Dirfrlcon of the Alaxfrielia J'orves I'lv
the Honjontdl Hane nrtiriff on the North Hnd c/" thf. NuAlt. at th^ ltoyal()f)Sfr\'<i«iir. (hwnwirh. lor
the Years hS4S to 1iSi>7, t'rrm the Means oP the Ohservalwns lU cvrrespcndinq Hours thivii/flv fach'Year.

^

Scati III/ terms vt' Wliott /foiiyjnUiJ/ three,.

/'M 7hww MDCCCIJOH. PlauJmK.

Fhxl. Tiuns. M DCCCl .XI 1 1. PLoU XIX.

Fhxl. Tmns MDCCCLXIIl . Hale-yCL

iitr\i.s fipruftnluKj fJie DuiiritK ifi<tii<n iti Magrntudt and Sign or thf- l^rticaJ/ Mag rvelw Force/ actirig onj
'' " ■• '• ■ ■! '»■ /' ■ t/if /,(HYfJ Oh,st'r\(tlon; (Ircmwtrk, for iJw ]'ears IS42 ti

t/if Aorth lirui of the Nadle al
of all (ha. Obgt.t'valwna '// coiiws-poruiuuf flours Uirouipi each Year

to 1847, from IhfMean

Yecrr^v.

Scale, m

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F}al.T:

Uiryes rtpre.sni'ni'i leu /hnmal Otariqe in Mia/ufude
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aU ihf/ Ob.fi-rmtit'n,'; at rvrrY^-pcnJ/ni/ tlciirs through

niid Sif/n
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vrHars m2-1&l7.

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XVI. On the Influence of Temperature on the Electric Conducting-Poiocr of Thallium and
Iron. Bij A. Mattiiiessex, F.R.S., Lecturer on Chemistry in St. Mary's Hospital,
and C. VoGT, Ph.D.

Received Febnwiy 12,— Ecad March 12, 1863.

It has been shown* that the conducting-power of several of the pure metals decreased
between 0° and 100° to the same extent, namelj' 29-307 per cent. On continuing the
research ^vith other metals and alloys, w'e have found that thallium and iron form an
exception to the above ; and in the foUoA^ng we will describe the experiments made
with these metals.

The thallium was kindly lent to us by Mr. Crookes, the discoverer of this new metal,
who with great readiness placed a small quantity of it at our disposal.

On account of its extreme softness (for it is much softer than pure lead), it was easily
cut with a knife to fit one of the small presses described in the ' Philosophical Magazine '
(Februai-y 1857), and pressed without the application of heat. The wire, as soon as it came
out of the small hole of the press, passed into a tube filled with water ; for although the
metal readily oxidizes in air, yet it may be kept under water, free from air, for some time
without oxidation. When, however, the wire is exposed to the air for a short time, it
loses its lustre and is soon co\ered with a coating of oxide, which, as in the cases of lead
and zinc, protects in a great measure the metal fi-om further oxidation. The error caused
by the slight oxidation during the short time we w^ere obliged to manipulate with the
■wire in aii- may therefore be overlooked, more especially as wires of the same piece of
metal showed the same conducting-power when pressed at difiierent times. Now, if the
slight oxidation had any marked effect, we ought to have found great differences in the
conducting-pow"er of different wires, for it can scarcely be supposed that in all cases the
same amount of oxidation had taken place. Again, although thallium appears to be
attacked by alcohol, yet we found we might varnish the wire with a solution of shell-lac
in alcohol ; for the small quantity of alcohol contained in the first coating of varnish vola-
tiUzes so quickly that it has very little time to act on the wdi'e ; in fact the resistance of
the wire was fovmd to be the same when determined before being varnished, and after
three or four coatings of varnish had dried on it. The reason of varnishing the wire was
to protect the metal fi'om the action of the hot oil.

The apparatus and precautions taken whilst determining the conducting-powcrs at
different temperatures have already been fully described f. The normal wires used were

* '• On the Influence of Temperature on the Electric Conducting-Powcr of iletals. By A. ilATTHrEssES and
M. VON BosE," Philosophical Transactions, lb»;2, p. 1. t Philosopliical Transactions, 1862, p. 1.

MDCCCLXIII. 3 E

370 DRS. A. MATTHIESSEN A2vD C. VOGT OX THE INFLUENCE OF TEMPEEATUEE

made of German silver ; these were compared -nith the gold-silver aUoy *, and the a alues
so obtained for their resistances reduced, for the sake of comparison ^^'ith former obser-
vations, to tliat of a hard-dra\^ai siber wire whose conducting-power at 0°is eqnal to 100,
that of a hard-drawn gold-silver wire being then 15-03 at that temperature. As in the
case of wii-es of most other metals, those of thallium were found to alter in conducting-
power after having been kept at 100° for some time. It was therefore necessary to
heat them for several days imtil their conducting-power showed no longer any alteration
after cooling to the original temperature.

The length of the wire experimented with was 158 millims., aud its diameter 0-50'2
miUim.

Conducting-power found before heating the wire .

„ „ after being kept at 100° for 1 day

2 days

„ „ „ ,, 3 days

„ ,, „ „ 4 days

The means of the conducting-powers for each of the following temperatures were-

o Eeduced to O^t.

8-808 at 13-2 9-290

8-939 „ 10-8

9-338

8-941 „ 11-S

9-378

8-949 „ 11-6

9-378

8-990 „ 10-2

9-368

Conducting-power.

T.

1 Observed.

Calculated.

l6-40

8-987

8-979

+ 0-008

25-07

8-460

8-466

-0-006

39-22

7-996

8-006

— 0-010

54-43

7-551

7-550

+ 0-001

69-68

7-138

7-132

+ 0-006

85-95

6-742

6-729

+ 0-013

100-13

6-404

6-414

-0-010

The formula deduced from the ob.sei-vations, from which the conducting-powers were
calculated, was

X = 9-364-0-037936i-f0-00U08467f,
or

Xi = 100-0-40513)'-f0-0009042f,
corresponding to a percentage decrement of 31-471 per cent.

To check the above value another wire was made, and its conducting-power determined.
The length of the wire was 187 millims., its diameter U-421 millim., and the con-
ducting-power found was

8-610 at 15°-6, or, reduced to 0° by the above formula, 9-169.
Now the first observation made with the first wire was 9-29 at 0°; or, as mean, we
find the conducting-power of pure thallium at 0° equal to 9-23.

* Pliilosophical Magazino, Febraary 1861.

t The manner in which these valuer were reduced is ftilly described in the paper already alluded to (Philo-
sophical Transactions, 180^, p. 10).

ON THE ELECTRIC COXDUCTIXG-POWER OF TILILLIUAI AXD IRON. 371

Although we had already found, as will be shown in the following, that the con-
ducting-power of iron decreases between 0°and 100° more than 29-307 per cent., the
mean value deduced from the percentage decrements of eleven metals, yet we thought
it would be interesting to check the above results, and we therefore applied to Pro-
fessor L.\MT. of Lille, for a specimen of the metal, who with great kindness lent us two
small bars prepared at diiFerent times. The results obtained «-ith these specimens fullv
confirm those obtained ^\^th Mr. Crookes's metal, both in respect to the conductin"-
power and to the percentage decrement.

The results obtained with these specimens were as follows : —

Isf Bar.

Length .
Diameter

205 millims.
0'553 millim.

0 Reduced to 0".
Conducting-power found before heating the wii-e . . . 8-881 at 11-8 9-330

after being kept at 100° for 1 day . 8-98C „ 10-0 9-370

2 days . 8-953 „ 11-9 9-410

3 days . 8-901 „ 13-4 9-413
The means of the conducting-powers for each of the follomng temperatures were —

Conducting-power.

T.

Difference.

Observed.

Calculated.

13-75

8-903

8-894

+ 0-009

25-23

8-477

8-483

-0-006

38-80

8-021

8-031

-0-010

54-68

7-544

7-547

-0-003

70-56

7-122

7-111

+ 0-011

82-71

6-819

6-811

+ 0-008

99-57

6-433

6-441

-0-008

The fonnula deduced from the observations, from which the conducting-powers were

calculated, was

X = 9-419-0-039520^+0-00009656!5»,
or

Xi=100-0-41958^+0-001025f,

con-esponding to a percentage decrement of 31-706 per cent.

Another wire from the same piece of metal, the length of which was 136 millims. and
the diameter 0-532, conducted

8-433 at 17°-8, or, reduced to 0°, 9-082.
A third wire, the length of which was 141 millims., the diameter 0-449 millim., con-
ducted

8-758 at 12°-4, or, reduced to 0°, 9-223.

The first observation made mth the first mre reduced to 0° was 9-33; and if we now

3 E 2

372 DES. A. MATTHIESSEX A2sD C. VOGT ON THE INFLUENCE OF TEMPEKATUEE

take the mean of these three values, we find the conductmg-power of the first bar to be
at 0° equal to

9-212.

2nd Bar.
Length . . . . 155 millims.

Diameter . . . 0-502 millim.

g Eeduced to 0°,

Conducting-power found before heating the wire . . . 8-377 at 14-8 8-8G6

„ ^ „ after being kept at 100° for 1 day . S-G92 „ 10-0 !)-032

,. " „ 2 days . S-764 „ 8-G 9-058

The means of the conducting-powers observed at the following temperatures were —

T. Conducting-power.

9-0 8-747

54-5 7-258

100-0 6-240

from which numbers the following formula was deduced,

X =9-054 — 0-034C97/ + 000006554f,
or

Ai= 100-0-38322!^ +0-00(i"239f-,

corresponding to a percentage decrement of, 31-083 per cent.

A second wire, 135-5 millims. long and having a diameter of 0-542 millim.., was found

to conduct

8-507 at 21°-2, or, reduced to 0°, l'-22ri.

Now, as before, taking the mean of this value and that found for the first determina-
tion, we find the conducting-power of the second bar equal to, at 0",

9-04G.

From the foregoing it will be seen that the values obtained for the conducting-powers

of the different specimens agree very closely with each other ; for that found

Percentage decrement for
Conducting- the conducting-power

power at 0°. between 0° and 100°-

For Mr. Crookes's metal . . 9-230 31-471

For M. Lamt's 1st bar. . . 9-212 31-700

2nd bar . . 9-04G 31-083

Mean . . . 9-1G3 31-420

and calculating, from the mean of the above, the fin-mula for the correction of the con-
ducting-power for temperature of thallium, we find it to be

x=9-lG3-0-03G894^+0-OOOOS104f,
where A represents the conducting-power at f°.

The mean of the three formula; reduceil to the same unit at 0° was used for calcu-
lating the above, viz.

x, = lO0-0-402G4i-f0-000S844!:'-'.

ox THE ELECTRIC COXDUCTING-POAVER OF TllAl^LIU.M AND 11U)N. oi')

As thallium in many respects resembles lead, it seemed to us of i)eculiar interest to
see whether in its electric behaviour it would resemble that metal, namely, if when
alloyed with tin, cadmium, or zinc, the conducting-power of the alloy would be equal to
the mean conducting-power of the volumes of the component metals. AVe are indebted
to Mr. Ceookes for an alloy of thallium and tin containing only traces of the latter
metal, and on testing its conducting-power we found it lower than that of pure
thallium, shomng that the additi(m of a better conductor (the conducting-power of tin
being at 0° 12-3GG) causes a decrement in the conducting-power of the metul. Again,
we alloyed a portion of M. Lamy's thallium Avith traces of cadmium ; and here we also
found the above observation confirmed (the conducting-power of cadmium being at 0"
23-725). Thallium, therefore, appears to belong to that class of metals* which, when
alloyed with lead, tin, cadmium, or zinc, or with one another, do not conduct electricity
in the ratio of their relative volumes, but always in a lower degree than the mean of their
volumes, and not to that class of metals to which lead belongs, namely, those which when
alloyed \^ith one another conduct electricity in the ratio of their relative volumes.

Respecting the conducting-power of the alloys of thallium, we shall discuss them in
our paper " On the Influence of Temperatiu-o on the Electric Couducting-Power of
Alloys," which will shortly be ready for publication.

We are greatly indebted to the kindness of Professor Percy, who placed at our disposal
the specimens of ii-ou used for the following experiments ; in fact, with the exception of
the two last, they are all from his collection. As several of them have been analysed by
Mr. TooKEY in his laboratory, the results we have obtained will be the more interesting,
as they show how traces of foreign matter influence the conducting-power of iron.

We will first give the numerical results, and then make some remarks on them.

1. Electroti)i>e Iron, deposited from a soliition of pure sulphate of iron. The strips
were very thin and porous ; we could not, therefore, obtain concordant values for the con-
ducting-power, but were able to determine the percentage decrement in the conducting-
power between 0° and 100°. We have_ therefore taken the first observed conducting-
power equal 100.

^ Iteduccd to 0°.

Conducting-power found before heating the strip . . 100 000 at 18-1 109'698
„ after being kept at 100° for 1 day. 100-520 at IG-S 109-539

2 days 100-894 at 15-9 109-443

The means of the conducting-powers observed at the following temperatures were —

T. Conducting-power.

10-0 103-92G

55-0 82-866

100-0 67-528

from which numbers the following formula was deduced,

7. =109•38-0■55983if-f0•001413if^
or

Xi = 100-0-51182^-f00012915f,
corresponding to a percentage decrement of 38-262 per cent.

* Philosophical Transactions, 1860, p. 162.

374 DES. A. MATTHIESSEX AND C. VOaT OX THE INFLUENCE OF TEMPERATUEE

2. No. 1, annealed and cooled in hydrogen.

(, Reduced to 0°.

C'onducting-power found before heating the strip . . lOOOOO at 2U-S lir?)75
„ after being kept at 100" for 1 day . 100-518 at 19-5 111-202

2 days 100-400 at 19-8 111-234

3 days 101-243 at 18-8 111-560
The means of the conducting-powers observed at the following temperatures were —

T. Conducting-power.

12-0 104-504

50 -0 83-622

100-0 68-460

from which numbers the following formula was deduced,

X =lll-275-0•57745^+0-001492S^^
or

Xi = 100-0-51894f+0-0013415f,
corresponding to a percentage decrement of 38-479 per cent.

3. Electrotype iron, a strip cut from the same foil as No. 1.

,, Reduced to 0=.

Conducting-power found before heating the strip . . 100-000 at lO'S 108-997

after being kept at 100° for 1 day 99-867 at 17-6 109-299

The means of the conducting-powers observed at the following temperatures were —

T. Conductiug-poTvcr.

li-0 102-958

55-5 82-324

100-0 67-396

from which numbers the following formula was deduced,

■K =10S-943-0-55947i+0-0014404f,
or

XI = 100-0-51355^'+0-0013221^•^
corresponding to a percentage decrement of 38-134 per cent.

4. No. 3, annealed in air.

o Reduced to 0°.

Conducting-power found before heating the strip . . 100-000 at 21-3 111-436

„ after being kept at 100° for 1 day. 102-944 at 17-2 112-356

2 days 102-705 at 17-0 112-323

The means of the conducting-powers observed at the following temperatures were —

T. Conducting-power.

ld-0 107-025

55-0 85-427

100-0 69-636

from which numl)ers the following formula was deduced,

X =112-615-0-57315^;+0-00^4341<^
or

x, = 100-0•50895^+0-0012735^^
forrosponding to a percentage decrement of 38-10 per cent.

5.

In 100 parts.

0-190

6.

In 100 parts.

0-121

7.

In 100 parts.

0-104

8.

In 100 parts.
0-118

0-020

0-173

0-106

0-228

0-014

0-lGO

0-122

0-174

0-230

0-(l4U

0-020

0-020

ON THE ELECTEIC C0XDUCTING-P0\V:ER OF THALLIUM AND lEON. 375

5. Tliis, and likewise Nos. 0, 7, and 8, were cut and drawn into wire from pieces of
metal which had been analysed. The analyses were as follows : —

Sulphur .

Phosphorus .

Silicon

Carbon .

Manganese]

Cobalt i .. 0-110 0-029 0-280 0-250

Nickel J

The length of the wire used from No. 5 was 752 millims. and its diameter 0-058

miUim.

^ Reduced to 0°.

Conducting-powor found before heating the wire . . 17-887 at 11*4 15*712

„ after being kept at 100° for 1 day . 15-004 at 9-8 15-710

The means of the conducting-powers observed at the following temperatures were —

T. Conducting-power.

ld-0 14-993

55-0 12-162

100-0 10-045

from which numbers the following formula was deduced,

A =15-719-0-()7437#+0-0001763f,
or

Xi=100-0-47312^+0-0011242f,

con-esponding to a percentage decrement of 36*07 per cent.

6. Length .... 1047 millims.

Diameter . . . 0*778 millim.

J, Reduced to 0".

Conducting-power found before heating the wire . . 14*543 at 15*4 15*640

„ after being kept at 100" for 1 day . 15*002 at 9*4 15*682

The means of the conducting-powers observed at the following temperatures were —

T. Conducting-power.

12-0 14*809

56*0 12-078

100-0 10-029

from which numbers the following formula was deduced,

x= 15*672-0*074045^+0-0001761^,
or

x^ = 100-0*47247^+0•0011237i^

coiTesponding to a percentage decrement of 36*01 per cent.

376 DES. A. .MATTHIESSEX A^D C. VOGT ON T]IE INFLUENCE OF TEMPERATURE

7. Length . . . . 5SU millims.

Diameter . . . 0-622 millim.

^ Reduced to 0°.

Conducting-power found before heating the wire . . 13'351 at 13*8 14-204

„ after being kept at 100= for 1 day . 13-469 at 12-8 14-266

2 days . 13-435 at 13-4 14-208
The means of the conducting-powers observed at the following temperatures were —
T. CoudiK-tiiig-power.

15-0 13-340

57-5 11-063

100-0 0-312

from which numbers the following formula was deduced,

X = 14•269-0■(l64133^'+0-0001456^'^
or

XI = 100-0-44946^+0•0010204^;^

corresponding to a percentage decrement of 34-742 per cent.

8. This piece of metal was drawn into wire with great difficulty, owing to its being
very brittle. The wu-e used was somewhat faulty, and this may account for the low
conducting-power found.

Length .... 100 millims.

Diameter . . . 0-479 millim.

p Reduced to 0°.

Conducting-power found before heating the wire . . 11-242 at 10-9 12-132

„ after being kept at 100° for 1 day . 11-275 at 17-9 12-222

2 days . 11-287 at 18-0 12-241

The means of the conducting- powers observed at the follomng temperatures were —

T. Conducting-power.

9-07 11-814

54-77 9-694

99-80 8-137

from which numbers the following formula was deduced,

A = 12-342 — 0-055894^'+0-0001379^^
or

Ai = 100-0-4529U+0-0011174^^

corresponding to a percentage decrement of 34-117 per cent.

9. This specimen formed the basis for some experiments made by Dr. Percy on the
absorption of carbon by iron, and was cut from the same piece of foil as Nos. 10,
1 1, 12. The strips of foil were first annealed in a current of dry hydrogen at a red
heat for about two hours. No. 9 was therefore annealed in hydrogen, Nos. 10, 11, 12
treated first with No. 9, and then separately under a layer of sugar charcoal in a current
of hydrogen for different Icngtlis of time. They were all hardened. Dr. Percy's expe-
riments have not yet been pid)lis]ied, but he informs us they will be given in his work
on Metallurgy.

ox THE ELECTEIC COXDUCTIXG-PO\VEE OF T]r.\XLIU.M AXD IKOX. 377

Length . . . 171 millims.

DiametiT . . . 0--102 millim.

o Reduced to 0°.

Conducting-poAver found before heating the strip . . 14096 at 'J"-i 14-723

„ after being kept at 100° for 1 day . l-i-123 at 9-0 14-724

The means of the couductiug-powers observed at the following temperatures were —

T. Conducting-power.

l6-0 14-009

55-0 11-416

100-0 9-471

from which numbers the following formula was deduced,

X = 14-G73-0-0G7999^+0-0001597f,
or

Xi=100-0-46343i+0-0010884f,

coiTCsponding to a percentage decrement of 35-459 per cent.

10. Heated for three hours under sugar charcoal in a current of hydi-ogcn; the
carbon taken up was 0-99 per eent.

Length . . . . 105 millims.

Diameter . . . 0-281 millim.

o Reduced to 0'.

Conducting-power found before heating the strip . . 10-376 at G-6 10-066

„ after being kept at 100' for 1 day . 10-282 at 9-0 10-076

The means of the conducting-powers observed at the following temperatures were —

T. Conducting-power.

10-0 10-218

55-0 8-499

100-0 7-177

from which numbers the folloA-s-ing formula was deduced,

X = 10G54-0-0445GO^+0-00009789f,
or

Xj=100-0-41825!5+0-0009188f,

corresponding to a percentage decrement of 32-637 per cent.

11. Heated for four hours under sugar charcoal in a cui-rent of hydrogen; the
carbon taken up was 0-933 per cent.

Length . . . . 177 millims.
Diameter. . . . 0-435 millim.

0 Reduced to 0^.

Conducting-power found before heating the strip . . 9-568 at 9-0 9-921

„ after being kept at 100' for 1 day . 9-668 at 6-4 9-021

The means of the conducting-powers obsen-ed at the followmg temperatures were —
T. Conducting-power.

8-0 9-610

54-0 8-027

100-0 6-832

MDCCCLXIII. 3 F

378 DBS. A. lyiATTIUESSEX AXD C. YOGT OX TIIE IXFLUEXCE OF TEMPEEATURE

from which uumbers the follomng formula was deduced,

X = 9-925-0-04009T^+0-00009168f,
or

Xi=100-0-404i'+0-0009237f,
coiTCsponding to a percentage decrement of 31-1G3 per cent.

12. Heated for three hours under sugar charcoal in a current of hydrogen ; the

carbon taken up was 1"06 per cent.

Length . . . . 101 millims.

Diameter . . . U-43G millim.

e Reduced to 0°.

Conducting-power found before heating the strip . . 9-032 at 11"4 9-449

„ „ after being kept at 100" for 1 day . 9-157 at 8-8 9-482

2 days . 9-162 at 8-6 9-480

The means of the conducting-powers observed at the following temperatui-es were —

T. Conducting-power.

10-0 9-090

55-0 7-652

100-0 6-564

fi-om which numbers the following formula was deduced,

A = 9-457-0-037573i5+0-00008642f,
or

Xi=100-0-3973^+0-0009138f,
coiTesponding to a percentage decrement of 30-592 per cent.

13. Thin music wu'e melted with a quarter of its weight of peroxide of iron under a
flux of plate glass.

Length . . . . 145-2 millims.
Diameter . . . 0-455 millim.

Conducting-power found before heating the wu-e . . 12-537 at 13-8
„ after being kept at 100^ for 1 day . 12-727 at 11-2

2 days . 12-731 at 11-6
„ „ „ „ 3 days . 12-639 at 13*6

The means of the conducting-powers observed at the following temperatures were —
T. Conducting-power.

14-0 12-610

57-0 10-542

100-0 8-929

from which numbers the following formula was deduced,

7. = 13-381-0•05G829^ + 0•000123<^
or

Xj = 100-0-4247^+0-0009192^,

coiTesponding to a percentage decrement of 33-278 per cent.

Eeduced to 0°.

13-293

13-346

13-374

13-389

ON THE ELECTEIC CONDFCTING-POWER OF THALLIUM ANB mON. 379

14. A piece of narrow watch-spring.

Leugtli .... 270 millims.

Diameter . . . OGlo millim.

0 Reduced to 0°.

Conducting-power found before heating the wire . . 8-254 at 11-0 8-568

„ after being kept at 100' for 1 day . 8-297 at 9-4 8-56G

The means of the conducting-powers observed at the following temperatures were —

T. Conducting-power.

10-0 8-279

55-0 7-127

100-0 6-193

from which numbers the following foi-mula was deduced,

X = 8-565-0-029099^+0-00005383^,
or

Xj=100-0-33974f+0-0006285f,

corresponding to a percentage decrement of 27-689 per cent.

15. Commercial iron wire.

Length .... 1150 millims.

Diameter . . . 0-971 millim.

g Reduced to 0°.

Conducting-power found before heating the wire . . 13-163 at 10-6 13-774
„ after being kept at 100" for 1 day . 13-157 at 10-8 13-779
The means of the conducting-powers obsencd at the following temperatures were —

T. ConductLng-po-wer.

12-0 13-082

56-0 10-859

100-0 9-117

from which numbers the following foi-mula was deduced,

X = 13-772-0-05897^+0-0001242f ,
or

Xi=100-0-43514!f+0-0009018f,

corresponding to a percentage decrement of 33*801 per cent.

We may here mention the reason of our having taken only obsenations at thi-ee
intervals between 0° and 100°. It was found that almost the same formula may be
deduced from thi-ee obseiTations as from seven or more, if the temperature of the second
obser^•ation is exactly the mean of the other two. Now as at each interval we always
made three observations, it was easy with a little practice to regulate the temperature
so as to obtain the wished-for temperatiu-e as mean. Of course sometimes we had to
make fom-, five, or more obsei-vations in order to bring out the desii-ed temperature. By
only taking observations at three intervals, the labour of the reseai-ch, especially of the
calculations, was materially diminished.

In the following Table we have placed together the results obtained with the different
sorts of ii-on : —

3f2

380 DBS. A. MATTinESSEX AXD C. TOGT OX TIIE IXTLUEXCE OF TEMPEEATURE

Percentage decrement
in the eonducting-
C'onducting-poM"cr power between
at U\ 0°.and 100=".

Electrotype iron — 38'262

Electrotype iron (annealed in hydrogen) — 38-479

Electrotype iron — 38-184

Electrotype ii-on (annealed in air) . . — 38-160

No. 5 15-712 3G-070

No. G 15-040 30-010

No. 7 14-204 34-742

No. 8 12-132 34-117

No. 0 14-723 35-459

No. 10 10-000 32-037

No. 11 9-921 31-103

No. 12 9-449 30-592

No. 13 13-293 33-278

No. 14 . 8-508 27-089

No. 15 13-774 33-801

If we look at the above Table, the following important fact will be obvious, namely,
the higher the conducting-power the higher the percentage decrement in the conducting-
power between 0° and 100^ ; in fact we have always found this to be the case ; and from
experiments made \nth about 100 alloys in this direction we have not found a single
case where the percentage decrement in the conducting-power between 0° and 100^ is
greater than that of the pure metals ; further, we have found that we may deduce the
conducting-power of the pure metal from that of the impure one when the impurity in it
does not reduce it more than, say, 10 to 20 per cent. ; for we have proved experimentally
that within those limits the percentage decrement between 0° and 100° in the conducting-
power of an impure metal varies in the same ratio as the conducting-power of the
impure metal at 100°, compared with tliat of the ptu-e metal at 100°. Thus the per-
centage decrement in the conducting-power of pure iron between 0° and 100° is 38-201
per cent., that of No. 5, 30-07 per cent. Now, if the above statement be correct, by

38-261
multiplying the conducting-power of No. 5 at 100° by ^^.[j,^^, we shall obtain the con-
ducting-power of the pure metal at 100°, and the conducting-power at 0° by dividing
that number by 0-01734.

In the following Table wc give the results of such a calculation with those specimens
of iron the conducting-powers of whicli do not vary more than 20 per cent, from that
deduced for the pure metal.

ox THE ELECTEIC COXDUCTIXG-POWER OF TIULLITIM AND lEON. 381

No. 5 .

Observed eonducting-
power at 0°.

. . 15-712

Calculated conducting-
power of pure
iron at 0°.

17-257

No. G .

. . 15-640

17-223

No. 7 .

. . 14-204

16-533

No. 9 .

. . 14-723

16-606

No. 13 .

. . 13-293

16-516

No. 15 .

. . 13-774
Mean .

l(i-717

. . 10-809

The reason for making the above deduction will be fully explained in our paper " On
the Influence of Temperature on the Electric Conducting-power of Alloys." One glance,
however, will show that the deduced values agree as well together as can be expected,
considering that the results may be modified by the presence of carbon, sulphur, &c.
In cases where we may assume that we have only solutions of one metal in another, the
concordance in the deduced values is very great ; in fact, when we experiment with
metals whose conducting-power in a pure state is known, and when the impm-ity is
only dissolved in it, then the deduced conducting-power agrees almost exactly with
that found experimentally.

For resistance-thermometers, as described by Siemens*, the use of an iron wire would
give much greater difiercnces for the same increment of temperature than copper ; for
the resistance of pure copper wu-c increases for each degree about 0-4 per cent., whereas
that of pure u-on increases about 0-6 per cent, for each degree.

When we foimd that iron decreased in conducting-power between 0° and lOO'' more
than the pm-e metals (and here again we will call attention to the fact that we have
as yet found no alloy to decrease in conducting-power between 0° and 100° to a greater
extent than that which the pui-e metals composmg it would do), we thought that its
being a magnetic body might possibly be the reason of it ; but after having tested
thallium, and found that the conducting-power of that metal also decreases more than
that of the pure metals, we knew that this could not well be the case, for thalUum is
strongly diamagnetic f , Hearing, however, that Professor Wohler possessed specimens
of pui-e cobalt and nickel wire prepared by M. Deyille, we wrote and asked him to lend
them to us ; this he immediately did, and on testing them we obtained the folloAving
results : —

Cobalt tvire.
Length .... 270 millims.
Diameter . . . 0-468 millim.

Conducting-power foimd before heating the wire

„ „ after being kept at 100° for 1 day

2 days

* Eeport of Govemment Submarine Cable Committee, p. 454.

t Lurr, Compt. Eend. 1S02, vol. Iv. p. ^36. llr. Ceooees informs us he has also found thallium strongly
diamagnetic.

12-495 at li-6

Reduced to 0°.

12-899

12-466 at 12-6

12-905

12-428 at 13-4

12-894

382 DES. A. MATTHIESSEN Ai^D C. VOGT ON THE INFLUENCE OF TEMPEEATUEE
The means of the condiicting-powers observed at the following temperatures were —

Conducting-power.

T.

Difference.

Observed.

Calculated.

S-65

12-623

12-626

-0-003

24-97

12-080

12-073

+ 0-007

39-95

11-586

11-589

-0-003

54-88

11-127

11-128

-0001

70-44

10-671

10-670

+ 0-001

84-00

10-289

10-291

— 0-002

99-78

9-873

9-872

+ 0-001

The formula deduced from the observations, from which the conducting-powers were
calculated, was

X = 12-930-0-035521^+0-00004887f,
or

Xj=100-0-27472^+0-000378f,

con-esponding to a percentage decrement of 23'C92 per cent.

Nickel wire.

Length .
Diameter

240 millims.
0-408 millim.

p Ecduced to 0°.

Conductmg-power foiuid before heating the wire . . 11-594 at 11-2 12-035
„ after being kept at 100° for 1 day . 11-030 at 14-0 12-185

2 days . 11-739 at 12-4 12-235

3 days . 11-735 at 12-5 12-235
The means of the conducting-powers observed at the following temperatiu-es were —

Con ducting-power.

Difference.

T.

Observed.

Calculated.

12-38

11-735

11-728

+ 0-007

24-20

11-2G0

11-276

-0-016

40-06

10-708

10-701

+ 0-007

53-86

10-239

10-230

+ 0-009

70-14

9-700

9-710

-0-010

83-93

9-302

9-298

+ 0-004

100-03

8-850

8-851

-0-001

The formula deduced from the observations, from which the conducting-powers were

calculated, was

x= 12-222 -0-040787i^+0-0000708SA
or

Ai=100-0-33o72#+0-0005799f,

con-esponding to a percentage decrement of 27-573 per cent.

From our experiments with alloys we should deduce that the cobalt and nickel wires

ON THE ELECTRIC COXDUCTES'G-POWER OF TUALLItHM AND IRON. 383

were not pure, aud we are justified in makinf;- this statement by the fact that wc have
not yet found any metal in a pure and solid state to decrease in conducting-power
between 0° and 100° less than 29"307 per cent., and, further, when we consider that,
although these metals may have been pure when in the state of powder, yet very little is
knowni about theu- behaviour to the crucibles at the high temperatures at which they fuse.
It is well known how difficult it is to procure chemically pure iron in a fused state, on
account of its decomposing the crucibles in which it is melted and taking up some im-
purities.

Assuming, therefore, that cobalt and nickel behave like most other pure metals,
namely, decrease in conducting-power betwedn O'" and 100", 29*307 per cent., we may
deduce fi.-om the above data the couducting-power of the pure metals.

The conducting-power of pure cobalt would then be 17-223 at 0\ aud that of pui-e
nickel 13-lOG at 0°.

Wo hope shortly to be able to prepare some pure cobalt and nickel by depositing
galvanoplastically those metals in the form of foil from solutions of their pure salts, and
so to check the above deduced values for the conducting-power of the pure metals.

In conclusion, we give, in the following Table, the conducting-power of some of the

pui"e metals, in order to show the places which the metals treated of in this paper take.

Conductixig-pQWcr at 0°.
Silver (hard drawn) 100-00

Copper „ 99-95

Gold „ 77-96

Zinc 29-02

Cadmium 23-72

Cobalt* 17-22

Iron* 16-81

Nickel* 13-11

Tin 12-36

ThaUium 9-16

Lead 8-32

Arsenic 4-76

Antimony 4-62

Bismuth 1-245

* Probable value for the pm-e metal deduced from the observations 'svith the impure one.

[ 385 ]

XVII. On the Moleadar MohUity of Gases.
By Thomas Graham, F.B.S., Master of the Mint.

Ecceived May 7,— Read June 18, 1863.

The molecular mobility of gases will be considered at present chiefly in reference to the
passage of gases, under pressure, through a thin porous plate or septum, and to the
partial separation of mixed gases which can be effected, as will be shown, by such means.
The investigation arose out of a renewed and somewhat protracted inquiry regarding the
diffusion of gases (wliich depends upon the same molecular mobility), and has afforded
certain new results which may prove to be of interest in a theoretical as well as in a
practical point of \iew.

In tlie Diffusiometer, as first constructed, a plain cylindrical glass tube, about 10 inches
in length and rather less tlian an inch in diameter, was simply closed at one end by
a porous plate of plaster of Paris, about one-third 'of an inch in thickness, and was
thus converted into a gas-receiver*. A superior material for the
porous plate has since been found in the artificially compressed
graphite of Mr. Brockedon, of the quality used for making writing-
pencils. This material is sold in London in small cubic masses
about 2 inches square. A cube may easily be cut into slices of
a millimetre or two in thickness by means of a saw of steel
spiing. By rubbing the surface of the slice without wetting it
upon a fiat sand-stone, the thickness may be fui'ther reduced to
about one-half of a millimetre. A circular disc of this graphite,
which is like a wafer in thickness but possesses considerable
tenacity, is attached by resinous cement to one end of the glass
tube above described, so as to close it and form a diffusiometer
(fig. 1). The tube is filled with hydrogen gas over a mercurial
trough, the porosity of the graphite plate being counteracted for
the time by covering it tightly with a thin sheet of gutta percha
(fig. 2). On afterwards removing the latter, gaseous diffusion im-
mediately takes place through the pores of the graphite. The
whole hydi'ogen will leave the tube in forty minutes or an horn-, and is replaced by a much
smaller proportion of atmospheric air (about one-fourth), as is to be expected from the
law of the diffusion of gases. During the process, the mercury will rise in the tube, if

* " On the Law of the Diffusion of Gases," Transactions of the Royal Society of Edinburgh, vol. .\ii. p. 222 ;
or Philosophical Magazine, 1834, vol. ii. pp. 175, 209, 351.
MDCCCIAIII. 3 G

Fig.]

L.
1

Fig.

2.

1 3

1

^1

41

iiif'tll

"|1

'4

1 ' 4'

\j_

', T ■.

i:4

i 4'

1

j 1

|4.

1

'"i

ii

[

J|

386 ME. T. GE.\HAM ON THE MOLECULAE MOBILITY OF GASES.

allowed, formiug a column of several inches in height — a fact which illustrates strikingly
the intensity of the force with which the mtei-penetration of different gases is effected.
Native graphite is of a lamellar structure, and appears to have little or no porosity. It
cannot be substituted for the artificial graphite as a diffusion-septum. Unglazed
earthenware comes next in value to graphite for that purpose.

The pores of artificial graphite appear to be really so minute, that a gas in mass
cannot penetrate the plate at all. It seems that molecules only can pass ; and they may
be supposed to pass wholly unimpeded by friction, for the smallest pores than can be
imagined to exist in the graphite must be tunnels in magnitude to the ultimate atoms
of a gaseous body. The sole motive agency appears to be that intestine movement of
molecules which is now generally recognized as an essential property of the gaseous
conchtion of matter.

According to the physical hypothesis now generally received*, a gas is represented
as consisting of solid and perfectly elastic spherical particles or atoms, which move in
all dii'ections, and are animated with different degrees of velocity in different gases.
Confined in a vessel the moving particles are constantly impurging against its sides and
occasionally against each other, and this contact takes place ■\\-ithout any loss of motion,
owing to the perfect elasticity of the particles. If the containing vessel be porous,
like a dirlusiometer, then gas is projected through the open channels, by the atomic
motion described, and escapes. Simultaneously the external air is carried mwards in
the same manner, and takes the place of the gas which leaves the vessel. To this
atomic or molecular movement is due the elastic force, >vith the power to resist compres-
sion, possessed by gases. The molecular movement is accelerated by heat and retarded
by cold, the tension of the gas being increased m the first instance and diminished in
the second. Even when the same gas is present both Avithin and without the vessel, or
is m contact with both sides of ovu- porous plate, the movement is sustained without
abatement — molecules continuing to enter and to leave the vessel in equal number,
although nothing of the kind is incUcated by change of volume or otherwise. If the
gases in communication be different but possess sensibly the same specific gra\ity and
molecular velocity, as nitrogen and carbonic oxide do, an interchange of molecules also
takes place without any change in volume. With gases opposed of unequal density and
molecular velocity, the permeation ceases of course to be equal in both directions.

These observations are preliminary to the consideration of the passage through a
graphite plate, in one dii'ection only, of gas under pressure, or under the influence of its
own elastic force. We are to suppose a vacuum to be maintained on one side of the
porous septum, and air or any other gas, under a constant pressure, to be in contact with
the other side. Now a gas may pass into a vacuum in tliree different modes, or in two
other modes besides that immediately before us.

* D. Bernoulli, J. Hf.uapath, .TouLr,, Kro.vig, C'Lvrsnis, Clerk Maxavell, and Cazix. The niorit of re\-iring
this hypothesis in recent times and first apjjlj-ing it to tlie facts of gaseous diffusion, is faiily due to Mi-. Hi:iiAi'ATu.
See ' Mathematical Physics,' in two volumes, by Jons Herai'atu, Esq. (1S47).

jNIE. T. GEAHAM ox the :\rOLECULAR MOBILITY OF GASES. 387

1. The gas may enter tlic vacuum by passing througli a minute aperture in a thin
plate, such as a puncture in platinum foil made by a fine steel point. The rate of
passage of different gases is then regulated by their specific gravities, according to a
pneumatic law which was deduced by Professor John Robison from Tokricelli's well-
known theorem of the velocity of efflux of fluids. A gas rushes into a vacuum ^^-ith the
velocity which a hea\7 body would acquire by falling from the height of an atmosphere
composed of the gas in question, and supposed to be of uniform density throughout.
The height of the uniform atmosphere would be inversely as the density of the gas,
the atmosphere of hydi-ogen, for instance, sixteen times higher than that of oxygen.
But as the velocity acquii-ed by a hea\7 body in falling is not directly as the height,
but as the square root of the height, the rate of flow of different gases into a vacuum
will be inversely as the square root of their respective densities. The velocity of oxygen
being 1, that of hydrogen will be 4, the square root of 16. This law has been experi-
mentally verified*. The relative times of the effusion of gases, as I have spoken of
it, are similar to those of molecular diffusion ; but it is important to observe that the
phenomena of effusion and diffusion are distinct and essentially different in their nature.
The effusion movement affects masses of gas, the diffusion movement affects molecules ;
and a gas is usually carried by the former kind of impulse with a velocity many thousand
times as great as is demonstrable by the latter.

2. If the aperture of efflux be in a plate of increased thickness, and so becomes a
tube, the effusion-rates ax'e distmbed. The rates of flow of different gases, however,
assume again a constant ratio to each other when the capUlarj' tube is considerably
elongated, when the length exceeds the diameter by at least 4000 times. These new
proportions of eflJux are the rates of the " Capillary Transpiration" of gases f. The
rates are found to be the same in a capillary tube composed of copper as they are in
glass, and appear to be independent of the material of the capillary. A film of gas
no doubt adheres to the surface of the tube, and the fiiction is really that of gas upon
gas, and is consequently unaffected by the tube-substance. The rates of transpiration
are not governed by specific gi'a^ity, and are indeed singularly unlike the rates of
effusion.

The transpiration-velocity of oxygen being 1, that of chlorine isl"5, that of hydi'Ogen
2-26, of ether vapoui- the same or nearly the same as that of hydi-ogen, of nitrogen
and carbonic oxide half that of hydrogen, of olefiant gas, ammonia, and cyanogen 2
(double or nearly double that of oxygen), of carbonic acid 1-376, and of the gas of
marshes 1"815. In the same gas the velocity of transpiration increases with increased
density, whether occasioned by cold or pressure.

The transpiration-ratios of gases appear to be in direct relation with no other kno'mi
property of the same gases, and they form a class of phenomena remarkably isolated
from all else at present known of gases.

* " On the Motion of Gases," Philosophical Transactions, 1846, p. 573.
t Ibidem, p. .591 ; and Philosophical Transactions, 1849, p. 349.

3g2

388 ME. T. GRAILVM ON THE MOLECULAE MOBILITY OF GASES.

There is one property of transpii-ation immediately bearing upon permeation of the
graphite plate by gases. The capillary offers to the passage of gas a resistance analogous
to that of fiiction, proportional to the surface, and consequently increasing as the
tube or tubes are multiplied in number and diminished in diameter, with the area of
dischai-ge preserved constant. The resistance to the passage of liquid through a capil-
lary was observed by Poiseuille to be nearly as the fourth power of the diameter of the
tube. In gases the resistance also rapidly increases ; but in what ratio, has not been
observed. The consequence, however, is certain, that as the diameter of the capillaries
may be diminished beyond any assignable limit, so the flow may be retarded indefinitely,
and caused at last to become too small to be sensible. We may then have a mass of
capillaries of which tlic passages form a large aggregate, but are individually too small
to allow a sensible flow of gas xmder pressiu-e. A porous solid mass may possess the
same reduced permeability as the congeries of capillary tubes. Indeed the state of
porosity described appears to be more or less closely approached by all loosely aggre-
gated mineral masses, such as lime-plaster, stucco, chalk, baked clay, non-crystalline
earthy powders like hydrate of lime or magnesia compacted by pressure, and in the
highest degree perhaps by artificial graphite.

3. A plate of artificial graphite, although it appears to be practically impermeable to
gas by either of the two modes of passage previously described, is readily penetrated by
the agency of the molecular or diffusive movement of gases. This appears on com-
paring the time required for the passage through the plate of equal volumes of different
gases under a constant pressure. Of the three gases, oxygen, hydrogen, and carbonic
acid, the time required for the passage of an equal volume of each tlirough a capillary
glass tube, in similar cu'cumstances as to pressure and temperature, was formerly
observed to be as follows : —

Time of capillary transpiration
of equal volumes.

Oxygen 1

Hydrogen 0-44

Carbonic acid .... 0'72

Now through a plate of graphite, half a millimetre in thickness, the same gases were
obsei-ved to pass, under a constant pressure of a column of mercui-y of 100 millimetres
in height, in times which are as follows : —

Time of molecular passage. Square root of density (oxygen 1).
Oxygen .... 1 1

Hydi-ogen . . . 0-2472 0-2502

Carbonic acid . . 1-1886 1-1760

It appears that the times of passage through the graphite plate have no relation to
the capillary transpiration-times of the same gases as first quoted. The new times in
question, however, show a close relation to the square roots of the densities of the

ME. T. GRAHAM ON THE MOLECULAE MOBILITY OF GASES.

389

respective gases, as is seen in tlie last Table ; and they so far agree with theoretical times
of diffusion usually ascribed to the same gases.

These results were obtained by means of the graphite difFusiometer ah-eady referred to,
which was a plain glass tube about 22 millimetres in diameter, closed at one end by the
graphite plate. In order to conduct gas to the upper surface of the graphite plate, a
little chamber Avas formed above the plate, to which the gas was conveyed in a moderate
stream by the entrance-tube e (fig. 3); while the Fig. 3.

gas brought in excess was constantly escaping
into the air by the open issue-tube i. The
chamber was formed of a short piece of glass
tube, about 2 inches in length, cemented over
the upper end of the difFusiometer. The upper
opening of this short tube was closed by a cork
perforated for the entrance- and exit-tubes.
It will be observed that by this arrangement
the upper surface of the graphite plate was
constantly swept by a stream of gas, which was
under no additional pressure beyond that of
the atmosphere, a free escape being allowed by
the exit-tube. The gas also was always dried
before reaching the chamber. The diffusi-
ometer stood over mercuiy, and was raised or
lowered by the lever movement introduced by
Professor Bunsen in his very exact experiments
upon gaseous diffusion*. To obtain the pres-
sui"e of 100 millimetres of mercury, the diffa-
siometer was first entirely filled with mercury
and then raised in the trough. Gas gradually
entered till the column of mercury in the tube
fell to 100 millimetres. The mercury was then
maintained at this height, by gi-adually raising the tube in proportion as gas continued
to enter and the mercuiy to fall, so as to maintain a constant difference of level of 100
millimetres, as observed by the graduation inscribed upon the tube itself, between the
level of the mercury in the tube and trough. The experiment consisted in observing
the time in seconds which the mercury took to fall 10 millimetre diAisions with each
gas. The constant volume of gas which entered was 2-2 cubic centimetres (0"1342 cubic
inch). Two experiments were made with each gas.

Oxygen entered in 898 and 894 seconds ; mean 896 seconds.

Hydrogen in 222 and 221 seconds; mean 221-5 seconds.

Carbonic acid in 1070 and 1060 seconds; mean 1065 seconds.
* Bitksen's ' Gasometry ' by Roscoe.

390

IHR. T. GEAHAjM ox THE MOLECULAE MOBILITY OF GASES.

In such experiments the same gas exists on both sides, and also occupies the pores
of the diaphragm. But the molecular movement mthin the pores in a downward dii-ection
is not fully balanced by the molecular movement in an upward dii-ection, o-^\ing to the
less tension, by 100 millimetres, of the gas below the diaphragm and within the tube
than the gas above and without. The influx of gas indicates the difference of mole-
cular movement in opposite directions. Taking the full tension of the gas above the
diaphragm at 760 millimetres, that below would be 660 millimetres, and the movement
downwards and that upwards are represented by these numbers respectively.

To increase the inequality of tension and favour the passage of gas through the
graphite plate, a diffusion-tube was now used, 48 inches in length, or of the dimensions
of a barometer-tube, by which a Torricellian vacuum could be commanded. The pneu-
matic trough in which this gas-tube was suspended consisted of a pipe of gutta percha of
equal length, closed at the bottom by a cork, and widening into a funnel-form at the top.
In one modification of the instrument it was found convenient to cement a capillary
glass tube to the side of the glass diffusiometer, within about 15 millimetres of the upper
end of the tube. An opening into the upper part of the glass tube was thus obtained,
bv means of which the gas contained in the ditfiisiometer could escape when the latter
was depressed in the mercurial trough. A flexible tube %vith cUp was attached to the
capillary tube referred to, so that the latter could be closed. From the same opening
a specimen of the gas contained in the diffusiometer could be drawn when required for
examination.

In another and more ser\iceable modification of this barometrical diffusiometer a lai'ge
space was obtained above the mercurial column, by surmounting the long glass tube,
unprovided with a graphite plate, by a glass jar about half a litre in capacity. This jar
was more correctly a small bell jar (fig. 4) open at top. It was fitted in an inverted posi-
tion, as in fig. 5, to the open end of the long glass tube (L by means of a cork and
Fig. 4. Fig. 5. Fig. 6.

cement. The large upper opening was closed by a circular plate of gutta percha (fig. 5),
about 1 0 millimetres, or nearly half an inch, in thickness. This disc of gutta pcrclia had
two perforations at/and y (fig. 0), the fonner of which was fitted above mth a wide glass

ilB. T. GRA1L\M OX THE MOLECULAR MOBILITY OF GASES.

391

tube. The tube / was closed below by the plate of graphite, and above with a perfo-
rated cork can-j-ing a quill tube, e. This quill tube was the entrance-tube for gas, and
was accompanied by the usual issue-tube, i. The other aperture in the gutta-percha cover
was fitted with a plain quill tube /;, which did not descend below the level of the gutta
percha, and formed a tube of exit. No difficulty was
found in making all these junctions air-tight, by applying
the heated blade of a knife to fuse the gutta percha in
contact with the glass. Gutta percha is indeed of no
ordinaiy value in the construction of pneiunatic appa-
ratus. The graphite plate itself required to be not less
than 1 millimetre in thickness, in order to support the
pressure of a whole atmosphere, to which it is exposed
in the present apparatus. This barometrical difFusio-
meter is supported from above by a cord passing over a
pulley, and is duly counterpoised by a hanging weight.

In operating, the fii-st point is to expel the air from
the baiometer-tube and upper chamber. The instru-
ment (fig. 7) is sunk completely in the mercurial trough
pre\iously described, till the whole is filled, and mercury
enters the quill tube of exit, h. The caoutchouc exten-
sion of this tube is then closed by a pinch. The difirisi-
ometer is now elevated 30 or 40 inches, when the mercury
sinks in the glass tube till it comes to stand at the baro-
metric height for the time, leading the upper chamber
entii-ely vacuous. The gas to be tried has in the mean
time been made to stream over the upper sui'face of the
graphite plate, exactly as in the experiment >vith the
fomier diffusiometer. The graphite is permeated by the
gas, and the mercuiy in the diff'usiometer-tube begins to
fall, but it now falls slowly, owing to the considerable
vacuous space to be filled. It is allowed to fall about
half an inch, and the exact time is then noted, by a
watch, when the mercury passes a certain point in the
graduation of the tube, and again when the mercury
descends to another fixed point an inch or two below the
former. The time of permeation of a certain volume of
gas is thus ascertained in seconds. The experiment is
immediately repeated with two or more gases in succes-
sion, in similar circumstances as to pressure, and with
great care taken to ensure uniformity of temperature during the whole period.

In a series of four experiments made with hydrogen, the mercury fell from 758 to

392 aiE. T. GEAHAM ON THE MOLECULAR MOBILITY OF GASES.

685millims. (29-9 inches to 27 inches) in 252, 256, 254, and 256 seconds; mean 254-5
seconds.

In three experiments with oxygen the mercury fell through the same space in 1019,
1025, and 1024 seconds; mean 1022-7 seconds:

1^ = 4-018.
254-5

The times of these gases appear therefore to be as 1 to 4-018, while the times
calculated as being inversely as the square root of the densities of the same gases are
as 1 to 4.

On another day, with a cUiferent height of the barometer, four gases were passed
through the graphite plate in succession through a somewhat shorter range, namely,
from 754 to 685 miUims. (29-7 to 27 inches).

The time of permeation of air was 884 and 885 seconds ; mean 884-5 seconds.

The time of carbonic acid was 1100 and 1106 seconds; mean 1103 seconds.

The time of oxygen was 936, 924, and 930 seconds ; mean 930 seconds.

The time of hydrogen was 229, 235, and 335 seconds; mean 233 seconds.

These times of permeation are in the following proportion : —

Times of the permeation of equal
volumes of gas through graphite.

Oxygen 1

Air 0-9501

Carbonic acid . . . 1-1860

Hydrogen .... 0-2505

These numbers approach so closely to the square roots of the density, or the theoretical
diffusion-times of the same gases, namely, oxygen 1, air 0-9507, carbonic acid 1-176,
and hydi-ogen 0-2502, that they may be held to indicate the prevalence of a common
law. They exclude the idea of capillary transpiration, which gives to the same gases
entirely different numbers.

The movement of gases through the graphite plate appears to be solely due to
their own proper molecular motion, quite unaided by transpiration. It seems to be the
simplest possible exhibition of the molecular or diffusive movement of gases. This pure
result is to be ascribed to the wonderfully fine (minute) porosity of the graphite. The
interstitial spaces appear to be sufficiently small to extinguish capillary transpu-ation
entirely. The graphite plate is a pneumatic sieve which stops all gaseous matter in
mass, and permits molecules only to pass.

It is worth observing what result a plate of more open structui-e, such as stucco, will give
in comparison with graphite. For the graphite plate, a cylinder of stucco, 12 millims.
in thickness, was accordingly substituted, and gas allowed to percolate at both low and
high pressures, as in the former experiments with graphite.

1. Under a constant pressure of 100 millims. of mercury, gas was allowed to enter
through 100 millim. divisions of the diffusiometer.

MR. T. GRAHAM ON THE :MOLECTrLAR MOBILITY OF GASES. 393

With air, the time iu two experiments was 515, and again 515 seconds.
With hydrogen 178 seconds, and again 178 seconds:

SS =2-894.

178

2. Under a pressure beginning with 710 millims. (28 inches) and ending with 660
millims. (26 inches), the time with air was 374 and 375 seconds; mean 374-5 seconds.
The time with hydrogen was 129 and 130 seconds; mean 129-5 seconds:

^I^ =2-891.
129-5

The stucco cylinder of the preceding experiments liad been dried over sulphuric acid,
without the apphcation of heat. It was further desiccated at G0° C. for twenty-four
hours, in order to find whether the porosity woukl he altered. The ratio of the time of
hydrogen to that of air now became 1 to 2-788 at the lower degree of pressure, and
1 to 2-744 at the higher degree of pressure.

It will be observed that the theoretical difiusion-ratio of hydrogen to air, which is
1 to 3-80, is greatly departed fi-om in these experiments Avith stucco. The ratio appears
to be tending to the proportion of the transpiration-times of the same gases, namely
1 to 2-04. In an experiment recorded by Bunsen, the ratio observed between the times
of hydrogen and oxygen in passing, under a small degree of pressui-e, through stucco
dried by heat was so low as 1 to 2-73, the stucco being probably less dense than in the
experiments before us.

With stucco the permeation of gases imder pressure appears to be a mixed pheno-
menon— to some extent molecular difhision into a vacuum, such as holds with the plate
of gi-aphite, but principally capillary transpiration of gas in mass.

The diflFusiometer was now closed by a plate of white biscuitware, 2-2 millims. in
thickness. The time of fall at the constant pressure of 100 millims., through a range
of forty divisions of the diffusiometer, was, for air 1210 seconds, for hydi-ogen 321 seconds.

A^- :^=3-769.

Hydrogen. . . 321

The time, again, from 736 to 685 millims. (29 to 27 inches) was, for air 685 and
684 seconds; mean 684*5 seconds; and for hydrogen 183, 183, and 184 seconds; mean
183'3 seconds.

Air 684-5_ _

Hydrogen . . . 183-5

The stoneware was evidently of a much closer tcxtme than stucco, and the ratio
appears again less influenced by capillary transpiration. In fact the moh^cular ratio of
1 to 3-80 is approached Avithin 1 per cent. Biscuitware therefore appears to be but
little inferior to graphite for such experiments, a circumstance which is important, as

MDCCCLXIII. 3 H

394 JIE. T. GEAHAJM ON THE MOLECULAE MOBILITY OF GASES.

the latter is not easily procured and cannot be converted into tubes and other convenient
forms like plastic clay.

Further, the rate of passage of gas through the plate of graphite appears to be closely
proportional to the pressure. The resistance was mcreased by augmenting the thick-
ness of the plate to 2 milliras. ; and A^dth air and hydi-ogen at a pressm'e maintained
constant at 50 and 100 millims., the time was observed that the gas took to enter 10

linear millimetre di\isions of the tube.

Seconds. Ratio.

Ail- under pressure of 100 millims. . . 1925 1

Air under pressure of 50 millims. . . . 3880 2"015

Hydrogen under pressure of 100 millims. 497 1

Hydrogen under pressure of 50 millims. 1022 2*056

By halving the pressure the time of passage is doubled, or increased somewhat more.

Greater pressui"es might probably give a rate of passage corresponding more exactly

with the pressure.

The ratio between the comparative times of the two gases in the last experiments

may also be noticed, the observations having been made in similar circumstances as to

pressure and temperatm-e.

Barom. 760 niillims. ; At pressure of 50 millims. Barom. 760 millims. ; At pressure of 100 millims.

Therm. 12°-9 C. Therm. 12°-9 C.

Air 3880 Aii- 1925

Hydrogen. . . . 1022 Hydrogen. . . . 497

The observation was repeated at the pressure of 100 millims. with barometer at 754
millims. and thermometer at 10° C.

Air 1920_

Hydi-ogen. . . 498 -

The velocity of hydi-ogen appears, as usual, to be nearly 3*8 times that of aii-;

An experiment was made at the same time as the former series upon a mixture of 95
hydrogen and 5 air, which gave an imlooked for result that led to a great deal of inquiry.
It is knowTi that such a mixture is effused through an aperture in a fine plate in a time
which is as the square root of the density of the mixtui-c, and therefore nearly the arith-
metical mean of the two gases effused separately. But in transpiration by a capillary, a
mixture of 95 hydrogen and 5 air requires a considerably longer time than the gases
transpired separately. In fact 5 per cent, of air retards the transpiration of hydi-ogen
nearly as much as 20 per cent, of air would retard the effusion of hydrogen*. Now the
mixture in question permeates the graphite plate in 527*5 seconds, while the calculated
mean of the times of the two gases is 562-1 seconds.

The mixture has therefore passed neither in the effusion time, nor in a longer time
* Philosophical Transactions, 1846, p. 028.

ME. T. GR.\HAM OX TilE MOLECULAE MOBILITY OF GASES. 395

as it would do by capillary transpiration, but, singulai- to say, in a time considerably
shorter than either. The gas that came through was found by analysis to be altered in
composition. It contained more liydrogcn and less air than the original mixture. Hence
it passed through with increased rapidity. On consideration it appeared that such a
separation of the mixed gases must follow as a consequence of the movement being
molecular. Each gas is impelled by its own peculiar molecular force, which, as has
been seen, is capable of causing hydrogen to permeate tlie graphite plate about 3*8
times as rapidly as air.

Each gas may permeate a graphite plate into a vacuum with the same relative velocity
as it diffuses into another gaseous atmosphere, but it remains a question whether the
velocities of permeation and diffusion are absolutely as well as relatively the same. To
illustrate this point, hydrogen and air were lii'st allowed to permeate into a vacuum, and
then to diffuse into each other, through the same graphite plate, wliich was 1 millim.
in thickness. The plate was a circular disc of 22 niillims. in diameter.

The mercurial column in the barometrical diffusiometer fell from 762 to 685 mUlims.
(30 inches to 27) with air in 878 seconds, and with hydrogen in 233 seconds.

^1- 878 _„

Hydrogen . . 233~

The volume of gas Avhich produced this effect was found by the calibration of the tube to
be 8"85 cub. centims. Hence 1"22 cub. centim. of the hydrogen entered the diffusiometer
in 60 seconds, or one minute. But the pressure under which the hydrogen gas entered
was the mean of 762 to 685 millims., or 723-5 mUlims. ; while a whole atmosphere (the
height of the barometer at the time) was 765 millims. The volume of the gas has
therefore to be increased as 723'5 to 765 to give the full action of a vacuum. The
volume becomes 1"289 cub. centim. in one minute.

"V\Tien the diffvisiometer was filled vatli hydrogen and the gas allowed to diffuse into air,
the rise of the mercury was pretty uniform for the first five minutes, being 15 '5 millim.
divisions in the first two miautes, 7 in the third minute, 7'5 in the fourth minute, and
7 in the fifth minute, making 37 divisions in five minutes. But as in diffusion 1 air
may be supposed to enter the tube for 3-8 hydrogen which escape, the hydrogen which

diffused was more than 37 divisions, by — , that is, by about 10 di\'isions. Hence

3*8

47 divisions of hydrogen have diffused into air in five minutes. These divisions mea-
sured, by the calibration of the tube, 6-215 cub. centims. One-fifth of this amount, that
is, 1"243 cub. centim., diffused in one minute. The result of the whole is that in one
minute there passed of hydrogen through the graphite plate,

1-289 cub. centim. by permeation into a vacuum,

1-243 cub. centim. by diffusion into air.
The numbers indicate a close approach to equality in the velocities of permeation into
a vacuum and of diffusion into another gas, through the same porous diaphragm. The
diffusion appears the slower of the two by a small amount ; but this is as it should be,

3h2

396 ME. T. GEAHAM ON THE MOLECFLAE MOBILITY OF GASES.

our estimate of the diffusion-velocity being certainly undeiTated ; for the initial diffu-
sion, or even the diffusion in the first minute, must obviously be somewhat greater than
the average of the first five minutes, which we liave taken to represent it — the hydrogen
necessarily diffusing out in a diminishing progression, or more slowly in proportion as
air has entered the diffusiometer. It is strictly the initial velocity of diffusion (that of
the first second if it could be obtained) that ought to be compared with the percolation
into a vacuum.

In fine, there can be little doubt left on the mind that the penneation through the
graphite plate into a vacuum and the diffusion into a gaseous atmosphere, through
the same plate, are due to the same inherent mobility of the gaseous molecule. They
are the exhibition of this movement in different circumstances. In interdiffiision we
have two gases mo^ed simultaneously through the passages in opposite directions, each
gas under the influence of its o^sm inherent force ; Avhile with gas on one side of the
plate and a vacuum on the other side, we have a single gas moving in one direction
only. The latter case may be assimilated to the former if the vacuum be supposed to
represent an infinitely light gas. It will not involve any error, therefore, to speak of
both movements as gaseous diffusion, — the diffusion of gas into gas (double diffusion)
in one case, and the diffusion of gas into a vacuum (single diffusion) in the other. The
inherent molecular mobility may also be justly spoken of as the diffusibility or diffusive
force of gases.

The diffusive mobility of the gaseous molecule is a property of matter fundamental
in its natiu'e, and the source of many others. The rate of diffusibility of any gas has
been said to be regulated by its specific gravity, the velocity of diffusion having been
observed to vary inversely as the square root of the density of the gas. This is tnie,
but not in the sense of the diffusibility being determined or caused by specific gravity.
The physical basis is the molecular mobility. The degree of motion which the molecule
possesses regulates the volume which the gas assumes, and is obviously one, if not the
only, determining cause of the peculiar specific gravity which the gas enjoys. If it were
possible to increase in a permanent manner the molecular motion of a gas, its specific
gravity would be altered, and it would become a lighter gas. AVith the density is also
associated the equivalent weight of a gaseous element, according to the doctrine of
equal combinmg volumes.

I^iffusion of mixed gases into a vacuum, with ])artial separation — Atmolysis.
Oxijgen and hydrogen. — A dift'usiometer of the same construction as that described
(fig. .3, p. 389), with a graphite plate of 1 milhm. in thickness, was now employed.
The upper surface of the plate was swept by a cun*ent of the mixed gas proceeding fiom
a gas-holder, the excess of gas being allowed to escape into the atmosphere, as usual, by
an open exit-tube. The gas was di-awn through the graphite by elevating the diffusio-
meter containing a column of mercui-y, from its well, so as to command a })artial vacuum
in the upper part of the tube. Care is taken that any gas, left in the upper pai't of the

MR. T. GRAHAM ON THE MOLECULAR MOBILITY OF GASES. 397

diffusiometer-tube before the experiment begins, should be of the same composition as
the gixs to be allowed afterwards to enter, so that, on starting, the gas may be uniform
in composition on both sides of the graphite plate. The height of the mercurial column,
which measures the aspii-ating force of the difliisiometer, is preserved uniform by
gradually raising the tube in the mercurial trough in proportion as gas enters and the
mercury falls. The cliiFusiometer is suspended from the roof of the apartment by a cord
passmg over a pulley and properly weighted, as in former experiments.

The mixture to be diffused consisted of nearly equal volumes of oxygen and hydrogen.
The effect of different degrees of pressure on the amount of separation produced was
first obsei'ved. It will be seen that as the pressure or aspirating force is increased the
amount of separation becomes greater. Barom. 0"759 mUlim. ; therm. 18°'3 C.

Diffusion into a partial vacuum.

Oxj-gen. Hydrogen.

Composition of original mixtui'e in 100 parts 49-3 50'7

Diffused by pressure of 100 millims 47"0 53

Diffused by pressure of 400 millims 37'5 62-5

Diffused by pressure of 673 millims. (mean of 635-710) . . . 26*4 73-6

Diffused by pressm-e of 747 millims. (mean of 736-759) . . . 22-8 77-2

In the last observation, or that with the greatest pressm-e (747 millims.), the oxygen
is reduced to 22-8 per cent, and the hydi-ogen increased to 77-2 per cent, of the diffused
mixtui-e, showing a considerable separation. The mixed gases appear to make their way
through the graphite plate independently, each follomng its owTi peculiar rate of diffusion.

But it is only under the aspiration of a complete vacuum that the separation can
attain its maximum, and reach the full difference that may exist between the s})ecial
difFusibilities of the two gases. The reason is that while we have the original mixture
on both sides of the plate, and of equal tension, the gases are not at rest, but difiusion
is proceeding as actively through the plate in opposite directions, as if the gases were
different or the tension unequal on the two sides. This is a condition of the molecular
mobility of gases (p. 38G). The tension therefore being supposed to differ by 100 millims.
only, as when the gas above the plate was of 759 millims. tension, and below of
659 millims. (in the first experiment of the last series), then 100 volumes only out of
759 of the mixture are subject to separation. But while these 100 volumes press
through they are accompanied by G59 volumes of unchanged mixture. The latter 659
volumes are replaced by an equal bulk of unchanged mixture diffused from below, so
that the volumes are not distui-bed by this portion of the molecular interchange.

The amount of separation, then, attainable by transmitting a mixed gas thi-ough a
porous diaphragm by pressure will be in proportion to the pressure — that is, to the
inequality of tension on different sides of the diaphragm.

Oxygen and nitrogen. — The separation of the gases of the atmosphere by transmission
through the gi-aphite plate ha.s a peculiar interest.

In an experiment resembling those last described, atmospheric air was swept over the

398 jmr. t. graham ox the molectjlae mobility of gases.

upper surface of a graphite plate ha^ing a thickness of 2 millims. The gas that pene-
trated into the vacuum contained, as was to be expected, the hghter and more diffusible
constituent in excess. It gave by the pyrogallic acid and potash process of Liebig,

Oxygen 20

Nitrogen 80

This -was an increase in the nitrogen of quite 1 per cent. ; for aii', analysed for comparison
at the same time and m the same manner, gave oxygen 21-03 and nitrogen 78-97.

It may be legitimately inferred from the last experiment, that if pure hydrogen in a
diffiisiometer were allowed to diffuse into the atmosphere through a porous plate, the
portion of ail' Avhich then enters the diffusiometer should also have its composition dis-
turbed. A diffusion of hydrogen through a graphite plate was interrupted before com-
pletion. The air which had entered was found to consist of

Oxygen .... 19-77
Nitrogen .... 80-23

100-00
The increase of nitrogen is 1-23 per cent.

While the nitrogen is increased and the oxygen diminished in the air which makes
its way under pressure through the graphite, the converse effect must be produced on
the air left behind. But the latter result of atmolysis cannot be made apparent Avithout
a change ui the mode of experimeutmg.

AVith the view of effecting an increase in the proportion of oxygen, a volume of air,
confined in a jar suspended over mercury, was allowed to commmiicate through a
graphite plate of 2 millims. in thickness, with a vacuum sustained by means of an air-
pump, the gauge being about 1 inch only below the height of the barometer dui-ing the
whole time of experimenting.

The jar containing the air to be atmolysed was formed of a plain glass cylinder, open
at both ends, and about 400 millims. in height (15-75 inches). The upper end was
closed by a thick \Aatv of gutta percha cemented on. This plate was itself penetrated
by a wide glass tube, descending about an inch into the jar. The last tube carried the
graphite disc, which was 27 millims. (1-04 inch) in diameter, sufficient to close the
lower end of the tube upon which it was cemented. The other or upper end of the same
tube was fitted Avith a cork and quill tube, and was put into communication mth a large
bell jar upon the plate of the air-pump.

The permeation was slow, owing to the unusual thickness of the graphite plate,
occupying three hours to drain away one-half of the original volume of air in the jar.
The air remaining behind in the jar was examined in a series of experiments, in w'hich
the original volume was reduced to one-half, one-fourth, one-eighth, and one-sixteenth.

The residual air, reduced to one-half, gave in two experiments 21-4 and 21-57 per cent,
of oxygen, the air of the atmosphere being by the same analytical process 21 per cent.

Reduced to one-fourth of its volume, the residual air gave, in two experiments, 21-95
and 22-01 per cent, of oxygen.

:MR. T. GEAHA^r ox THE MOLECTLAE ISrOBTLTTT OF GASES.

399

Reduced to one-eighth of its volume, the air gave 22"5 t per cent, of oxygen.

Reduced to one-sixteenth of its volume, the air gave 23-02 per cent, of oxygen. The
proportion of oxygen had therefore increased about one-tenth in the last experiment,
where the effect is greatest.

"MMien the numbers are compared, it appears that by a reduction to half its volume
the air gains about one-half per cent, of oxygen ; vehen this last air is reduced to one-
half again, another half per cent, of oxygen is gained, and so on — the gain in the
proportion of oxygen increasing in an arithmetical ratio, while the volume of air is
diminished in a geometrical ratio, or as the powers of the number 2.

Reduction of 1 Toluiue of air

Proportion of
oxygen per cent.

Increase of
oxygen.

21

21-48
21-98
22-54
23-02

0

0-48

0-98

1-54

2-02

To 0-25 volume

To 0-125 volume

To 0-0625 volume

The densities of oxygen and nitrogen approach too nearly to admit of any consider-
able separation being effected by this method. The density of oxygen being taken as 1,
that of nitrogen is 0-8785. The square roots of these numbers are 1 and 0-9373, which
are inversely as the diffusive velocity of the two gases.

Oxygen .

Nitrogen

Diffusive velocity.
. 1

. 1-0669

The velocity of nitrogen therefore exceeds that of oxygen by about 6-7 per cent.
Hence by a simple diffusion of a whole volume of air the oxygen could only be increased
6-7 per cent., according to theor)% In experiments such as the preceding only one-half
of the volume of the air is diffused, and consequently only one-half of the stated amount
of concentration of oxygen could possibly be produced at each step. About three-fourths
of the theoretical separation is actually obtained, although the apparatus works at an
obvious disadvantage from the air within the jar being at rest.

This diffusive method of separation recalls the original observation of Doberein:er on
the escape of hydrogen gas from a fissm-ed jar standing over water, which will always
hold its place in scientific history as the starting-point of the experimental study of
gaseous diffusion. That observation proved to be an instance of double diffusion, air
entering the jar by the fissure at the same time that hydrogen escaped by it — although,
as DoBEREiNER looked upon the phenomenon, it was more akin to single diffusion or the
passage of gas in one direction only *.

The atmolytic power of other diffusing plates was tested, besides the artificial graphite.

The barometrical difiusiometer already described was closed by a plate of red unglazed

• Annales de Chimie, 1825.

400 ME. T. GEAHAM ON THE MOLECUIAE MOBILITY OF GASES.

earthenware, 4 millims. in thickness, which was attached to the glass by resinous
cement.

Dry air was swept over the upper surface, as in operating with the graphite plate.
"With a mercurial column of 340 millims. falling to 200 millims., the air which entered
was found to contain 79-45 per cent, of nitrogen, instead of 79. With a column of mer-
curv. maintained at 508 millims. in the tube, the air entering contained 79-72 nitrogen,
and with a column beginning at 761 millims., the full barometrical height, and falling
to 679 millims. in seven minutes, the air entering contained 80-21 nitrogen. This is a
fuU degree of separation, exceeding 1 per cent., while the time was greatly shorter
than with graphite. Thermometer 19°-5 C.

With a diffusing plate of gypsum (stucco) 10 millims. in thickness, the proportion
of nitrogen was also increased, although less considerably than with biscuitware. The
standard proportion of nitrogen observed in atmospheric air being 78-99 per cent, the

air drawn into the diifusiometer was as follows : —

Proportion of nitrogen per cent.

In air entering over column of 330 — 200 millims. mercury . 79-26

In au- entering o^er column 508 millims 79-32 -

In air entering over column 761 — 685 millims 79-53

In air entering over column 761 — 685 millims 79-69

The separation is sufficiently decided, and is certainly remarkable considering the
comparatively loose texture of the stucco plate. The gas entered in the two last expe-
riments in about one minute, which appears too. rapid a passage, and not to be attended
with increased separation, compared with the immediately preceding experiment, in
which the pressure was less and the passage of the gas proportionally slower. In all
such highly porous plates, we have always to apprehend the passage of a large pro-
portion of the gas in the manner of capillary transpiration, where no separation takes
place.

It may be concluded that all porous masses, however loose their texture, will have
some effect in separating mixed gases moving through them under pressure. The air
entering a room by percolating through a wall of brick or a coat of plaster will thus
become richer in nitrogen, in a certain small measure, than the external atmosphere.

TJie Tube Afniolyser.

In the application of diffusion through a porous septum to separate mixed gases, as a
practical analytical method, it is desirable that the process should be more rapid than
it can be made with the use of graphite and other diffusing-plates of small size, and also
that the process should if possible be a continuous one. Both objects are attained in a
considerable degree by adapting a tube of porous earthenware to the purpose. Nothing
has been found to answer better than the long stalk of a Dutch tobacco-pipe used as
the porous tube. A tube of this description, about 2 feet long and ha\'ing an internal
diameter of 2-5 millims., is fixed by means of perforated corks within a glass or metallic

:\rR. T. GEAHAM OX THE MOLECULAR MOBILITY OF GASES.

401

tube, a few inches less in length and about 1^ inch in diameter (e, i fig. 8), as in the
construction of a Liebig condenser. A second quill tube {v) is inserted in one of the

end corks, and affords the means of communication between the annular' space and the
vacuum of an air-pump. The external sui-face of the corks, and of those portions of
the pipe-stalk which project beyond the enclosing tube, should be coated with a resinous
varnish, to render them impermeable to air. Now, a good Aacuum being obtained mthin
the outer tube, and sustained by the action of an air-pump, the mixed gas is made to
enter and traverse the clay tube. More or less of gas is drained off through the porous
walls and pumped away, while a portion courses on and escapes by the other extremity
of the clay tube, where it may be collected. The stream of gas diminishes as it pro-
ceeds, like a river flowing over a per\dous bed. The lighter and more diffusixe consti-
tuent of the mixed gases is di'awn most largely into the vacuum, leaAong the denser
constituent, in a more concentrated condition, to escape by the exit end of the clay tube.
The more slowly the mixed gas is moved through that tube, the larger the proportion of
light gas that is drained off into the vacuum, and the more concentrated does the heavj-
gas become. The rate of flow of the mixed gas can be commanded by either discharging
it from a gas-holder, or drawing it into a gas-receiver, in either case by a regi^latcd
pressure.

To observe the effect of a more or less rapid passage through the tube atmolyser, the
impelling pressure was varied so as to allow a constant volume of half a litre of atmo-
spheric air to pass through and be collected in different periods of time. The clay tube
used in these particular experiments was not a tobacco-pipe, but a wide unglazed tube,
about 431 millims. (17 inches) long and 19 millims. (0'75 inch) in internal diameter.
It was required to place so vnde a tube in a vertical position, and to admit the air by
the upper and di"aw it off" by the lower extremity of the tube. Tlie proportion of
oxygen in the half-litre of au- collected was as follows : —

Oxygen per cent.

Experiment 1.

Experiment 2. Mean.

When collected in 1 minute

When collected in 13 minutes

When collected in 75 minutes

When collected in 120 minutes

When collected in 304 minutes

21-00
22-33

22-77
23-25
23-54

22-25 22-29
23-02 22-89
23-22 23-23
23-51 2.3-63

MDCCCLXIII.

3i

402 ME. T. ft-RAHAM OX THE MOLECULAR MOBILITY OF GASES.

The proportion of oxygen in the aii- cii-culated appears thus to increase with the slow-
ness of its passage through the tube atmolyser. The proportion of air drawn into the
aii--pump vacuum must be very large when the time is protracted ; but the additional
concentration of oxygen appears small.

The preceding observations being made by means of a porous tube which may be
considered vnde and of considerable capacity with reference to its internal surface, the
experiment was varied by substituting a porous tube about eight times as long, very
narrow, and therefore of small internal capacity. This second atmolyser was composed
of twelve ordinary tobacco-pipe stems, each about 10 inches in length and of 1-9 millim.
internal diameter, connected together by vulcanized caoutchouc adapters so as to form
a siiagle tube. Having flexible joints, the tube was folded up and placed within a glass
cylinder that could be exliausted. Air was then circulated through this atmolyser by
the pressure of several inches of water. The instrument appeared to work with most
advantage when the air delivered at the exit-tube amounted to about one-fourth of a
litre per hour. A volume of 268 cubic centimetres, which had ciixulated in one hour,
was found to contain 24-.37 per cent, of oxygen. The current was then made slower,
so that only 108 cub. centims. of gas passed and were collected in one hour, but with
little further concentration of the oxygen. The result, however, is interesting as being
the highest concentration of oxygen yet obtained by an instrument of this kind. The
air collected was composed of

Oxygen .... 24-52

Nitrogen . . . . 75- 48
100-00

The increase of oxygen is 3-5 per cent. ; that is, an increase of 16-7 upon 100 oxygen
originally present in the air.

A^'ith the single pipe-stalk, 24 inches long, fii-st described, the oxygen of atmospheric
air was concentrated about 2 per cent, when one litre was transmitted in one hour.
Of 450 cub. centims. of air collected in that time, the composition proved to be

Oxygen .... 23-12

Nitrogen .... 76-88

100-00

About 9 litres were drawn into the vacuum at the same time.

The separation of the gases of atmospheric air is a severe trial of the powers of the
atmolyser, owing to the small diiference in the specific gravities of these gases. But
where a great disparity in density exists, the extent of the separation may become very
considerable.

Several exp(>rimcnts were made upon a mixture of etpial volumes of oxygen and
hydi-ogen carried through th(> single tube atmolyser, 24 indues in length.

1. Of the mixture described, 7-5 litres entered the tube and 0-45 litre was collected
in one experiment. The mixture was composed as follows :

^re. T. OnAIT.\'\r ON THE :\rOLECULAR JIOBTLITY OF GASES. 403

Oxygon. Hydrogen.

Before traversing the atmolyser . . 50 +50

After traversing the atmolyser . . 92-78 -|- 7-22

2. In another simihir experiment, 14 Htres of the mixed gas entered the tnbe and
0-45 Utre was delivered in a period of two hours. The result was —

Ox3-gL'n. Hydrogen.
Before traversing the atmolyser .50 + 50
After traversing the atmolyser . 95 +- 5
Here the proportion of hydrogen is reduced from 50 to 5 per cent.

3. Of the explosive mixture, consisting of 1 volume oxygen and 2 volumes hydrogen,
9 litres were transmitted and 0-45 litre collected in one hour. The change effected was

found to be as follows : —

Oxygen. Hydrogen.
Before traversing the atmolyser . 33-33 + 66-66
After traversing the atmolyser . . 90-7 + 9-3
The result in such experiments is striking, as the gas ceases to be explosive after
traversing the porous tube, and a lighted taper burns in it as in pure oxygen. A
mixtm-e of oxygen and hydrogen is not explosive till the hydrogen rises to 11 per cent.
To illustrate the analogy of diffusion into a vacuum with diffusion into air, the outer
glass tube of the diffuser was now withdrawn, and the porous tube of the instrument was
exposed directly to the air of the atmosphere. A mixture of equal volumes of oxygen
and hydrogen was again transmitted at the same rate of velocity as in experiment 1.
The gas atmolysed and collected was found to consist of

Oxygen 5T75

Hydrogen .... 5-47
Nitrogen . . . . 42-78

100-00
And may be represented as containing

Oxygen 40-38

Hydrogen .... 5*47

Air 54-15

100-00
A nearly similar concentration of the oxygen of the mixed gas is here obser\'ed as
appeared in experiment 1 ; but the gas collected is now diluted with air which has
entered by diffusion. The external air manifestly discharges the same function in the
latter experiment which the air-pump vacuum discharged in the former experiment.

Interdiffusion of Gases — double diffusion.
The diffusiometer was much improved in construction by Professor Bunsen, from the
application of a lever an-angement to. raise and depress the tube in the mercurial

3i2

404 MR. T. GEAHAM ON THE MOLECULAE MOBILITY OF GASES.

trough ; but the mass of stucco formhig the porous plate in his instrument appears too
voluminous, and, from being dried by heat, is liable to detach itself from the walls of
the glass tube. The result obtained of 3-4 for hydrogen, which diverges so far from the
theoretical number, is, however, no longer insisted upon by that illustrious physicist.
It is indeed curious that my old experiments generally rather exceeded than fell short
of the theoretical number for hydrogen; >/0-06926 = 3-7994. With stucco as the
material, the carities existing in the porous plate form about one-fourth of its whole
bulk, and affect sensibly the ratio in question according as they are or are not included
in the capacity of the instrument. Beginning the diffusion always with these cavities,
as well as the tube, filled mth hydrogen, the numbers now obtained with a stucco plate
of 12 millims. in thickness and dried -n-ithout heat, were 3-783, 3-8, and 3-739 when
the volume of the carities of the stucco is added to both the aii- and hydrogen volumes
diffused; and 3-931, 3-949, and 3-883 when such addition is not made to these volumes.
The graphite plate, on the other hand, being very thin, and the volume of its pores too
minute to requii-e to be taken into account, its action is not attended vrith the same
uncertainty. With a graphite plate of 2 millims. in thickness, the number for hydro-
gen into air was 3-876, instead of 3-8 ; and for hydrogen into oxygen 4-124, instead of 4.
With a gi-aphite plate of 1 miUim. in thickness, hydi-ogen gave 3-993 to air 1. With
a plate of the same material 0-5 millim. in thickness, the proportional number for
hydrogen to air rose to 3-984, 4-0G8, and 4-067. An equally considerable departure
from the theoretical number was observed when hydrogen was diffused into oxygen or
into carbonic acid, instead of air. All these experiments were made with di-y gases and
over mercury. It appears that the numbers are most in accordance with theory when
the graphite plate is thick, and the diffusion slow in consequence. If the diffusion be
very rapid, as it is with the thin plates, somethhig like a current is possibly formed
within the channels of the graphite, taking the direction of the lij'drogen and carrying
back in masses a little air, or the slower gas, whatever it may be. I cannot account
othen\-ise for the slight predominance which the lighter and faster gas appears always
to acquire in diffusing through the porous septum.

Interdiffusion of Gases without an intervening septum.

The relative velocity witli which different gases diffuse is sho\^^l by the diffiisiomcter,
but the absolute velocity of the molecular movement cannot be ascertained by the same
instrument. For that purpose it appears requisite that a gas should be allowed to
diffuse into aii- through a wide opening.

In certain recent expei-iments, a heavy gas, such as carbonic acid, was allowed to rise
by diffusion into a cylindrical column of air, pretty much as the saline solution is allowed
to rise into a column of water in ray late experiments upon the diffusion of liquids.
This method of gaseous diffusion appears to admit of considerable precision, and deserves
to be pursued further. A glass cylinder of 0-57 metre (22-44 inches) in height had
the lower tenth part of its volume occupied with carbonic acid, and the upper nine-

MR. T. GlLULiM ON TILE MOLECULAR MOBILITY OF GASES.

405

tenths with air, in a succession of experiments : theiinometer 16° Cent. After the lapse
of a certain number of minutes, the upper tenth part of the volume was drawn off from
the top of the jar and examined for carbonic acid. Before the carbonic acid appeared
above, it had ascended, that is, it had diffused a distance of 0-513 metre, or rather more
than half a metre. After the lapse of 5 minutes, the carbonic acid so found in two
experiments amoimted to 0-4 and 0'32 per cent, respectively. In 7 minutes, the carbonic
acid observed was 1-02 and 090 per cent. ; mean 0-96 per cent. The effect of diffusion
is now quite sensible, and it may be said that about 1 per cent, of carbonic acid has
diffused to a distance of half a metre in seven minutes.

A portion of carbonic acid has therefore travelled by diffusion at an average rate of
73 millims. per minute. It may be added that hydrogen was found to diffuse do-miwards,
in air contained in the same cylindrical jar, at the rate of 350 milUms. per muiute, or
about five times as rapidly as the carbonic acid ascended. In these experiments the glass
cylinder was loosely packed with cotton wool, to impede the action of currents in the
column of air ; but this precaution was found to be unnecessary, as similar results were
afterwards obtained in the absence of the cotton. To illustrate the regularity of the
results, I may complete this statement by exhibiting the proportion of carbonic acid
found in the upper stratum already referred to, after the lapse of different periods of
time.

Carbonic acid per cent.

Experiment 1.

Experiment 2.

Mean.

After 5 minutes

After 7 minutes

After 10 minutes

After 15 minutes

After 20 minutes

After 40 minutes

After SO minutes

0-4

1-02

1-47

1-70

2-41

5-60

8-68

0-32
0-90
1-56
1-68
2-69
5-15
8-82

0-.36
0-96
1-51
1-69
2-55
5-37
8-75

In eighty minutes the proportion of carbonic acid had risen to 8-75 per cent., 10 per
cent, being the proportion which would indicate the completion of the process of
diffusion.

The same intestine movement must always prevail in the air of the atmosphere, and
with even greater velocity, in the proportion of 1 to 1-176, the relative diffusion-ratios
of carbonic acid and air. It is certainly remarkable that in perfectly still air its molecules
should spontaneously alter their position, and move to a distance of half a metre, in any
direction, in the course of five or six minutes. The molecules of hydrogen gas disperse
themselves to the distance of a thhd of a metre in a single minute. Such a molecular
movement may become an agency of considerable power in distributing heat through
a volume of gas. It appears to account for the high convective power observed in
hydrogen, the most diffusive of gases.

[ 407 ]

XVlll. On the Peroxides of the Radicals of the Organic Acids.
By Sir B. C. Brodie, Bart., F.R.S., Professor of Chemistry in the University of Oxford.

Received June 18,— Read June 18, 1863.

L\ a former communication* I announced to the Royal Society the discovery of a new
group of chemical substances, the peroxides of the radicals of the organic acids — bodies
which, in the systems of the combmations of these radicals, occupy the same relative
position as is held by the peroxides of hydrogen, barium, or manganese in the systems of
the combinations of those elements.

The investigation of these peroxides is attended with peculiar difficulties. It is by no
means easy to prepare in any considerable quantity the anhydrous acids and chlorides
themselves, which is only the fh'st step in the preparation of the peroxides. The greater
number also of these substances is excessively unstable ; they are decomposed in the
very reactions by which they are produced, and the quantity actually obtained is very
far from corresponding to that which is indicated by theory. There can be little hope
of a complete and successful investigation of the decompositions of these bodies, until
methods are discovered by which the substances themselves can be more readily procured.
I have for these reasons not yet been able to submit their transformations to the
profound study which the subject merits, and which will doubtless be some day followed
by a rich harvest of discovery.

One exception should be made to the above remarks, the peroxide of benzoyl. This
beautiful substance can be procured with comparative facility, and I hope to pursue the
investigation of its metamorphoses. It appeared to me, however, of primary importance
to establish the perfect generality of the fundamental reaction by which these bodies are
prepared. This I have effected by forming several members of the group ; and I have
also, in one instance at least, succeeded in ascertaining the constitution of the peroxide
of a bibasic acid, a member of a new class of chemical substances, fundamentally different
(as the chemist would perhaps anticipate) from the peroxides of the monobasic acids,
and characterized by well-marked reactions.

The peroxide of barium, in respect to the definite and universal character of its

reactions, may be placed by the side of the alkalies themselves. Every anhydrous

organic acid with which I have made the experiment, without any exception, has been

found to be converted by its agency into an organic peroxide. It is a new instrument

* Proceedings of the Roy<al Society, vol. ix. p. 301.

408 SIR B. C. BEODIE ON THE PEEOXIDES OF THE

of chemical research, admh-able for the power and the simplicity of its action, and
which will certainly find in the future of chemistry many applications besides those
which arc here recorded. Its preparation is a matter of importance.

Prejxt ration of Peroxide of Barium.

The peroxide of barium as prepared by leading oxygen over heated baryta is useless
for the purposes of the following experiments, for the reason that the oxidation of the
baryta is never complete, and that the peroxide is mixed with large quantities of the
oxide of barium. However, the first step towards the preparation of a pure peroxide is
the preparation of a crude material.

AVhcn oxygen gas is passed over fragments of baryta heated in a porcelain tube, the
absorption of the gas proceeds at first with great rapidity ; and if the heat be properly
regulated, not a trace of oxygen will pass through the apparatus. It is nevertheless
extremely difiicult to prepare in this manner a peroxide which shall contain more than
about G parts of oxygen to 100 of baryta, however long the action of the oxygen be
continued, — the theoretical amount of oxygen required for the formation of the peroxide
being 10-4G parts of oxj'gen to 100 of baryta. By far the simplest and most practical
process for the oxidation of baryta is that de\ised by Liebig, which consists in exposing
to a gentle heat an intimate mixture of powdered baryta and chlorate of potassium.
The mixture is thrown by degrees into a crucible heated to low redness ; an ignition is
perceived when the chlorate of potassium melts. The fused mass is powdered and
extracted with water, which leaves an insoluble residue containing large quantities of
peroxide of barium.

On determining the amount of oxygen combined with the baryta in this experiment,
1 found that the peroxide did not contain above half the theoretical amount of oxygen.
The following experiments leave little doubt that in this reaction the baryta is oxidized
to the condition, not of peroxide, but of sesquioxide of barium. When oxygen is
passed over baryta, the action does not appear to be absolutely arrested at this point,
but nevertheless only small quantities of peroxide are formed.

Expt. I. 4-726 grms. of baryta were powdered, and mixed with one-third that weight of
chlorate of potassium. The mixture was thrown by degrees into a platinum crucible,
heated to low redness. After the first ignition, which proceeded without any external
increase of temperature, the crucible was heated somewhat more strongly, so as to ensure
tlie decomposition of the chlorate of potassium. The mass was powdered, mixed with
water, and dissolved in very dilute hydrochloric acid, to which the hydrate was added. The
solution was rendered feebly alkaline by the addition of baiyta water, which precipitated
the traces of iron and alumina invariably contained in the baryta. The solution was
then rapidly filtered, and immediately rendered acid ; and by the addition of water, it
wa.s brought to the bulk of 500 cub. ccntims. 10 cub. centims. of this solution, titred
with a standard solution of permanganate, of which 1 part was equivalent to 0-000516
grm. of oxygen, required in two experiments 8-G and 8-7 parts of that solution. Since

fiADICALS OF THE OKGAXIC ACIDS. 409

8-65 parts of permanganate were equivalent to 0-00044G5 grm. of oxygen, 0'22325 grm.
of oxygen were contained in the 500 cub. centims. of solution of peroxide of barium ;
100 parts of baryta had therefore absorbed 4-718 parts of oxygon.

In three other experiments conducted in a similar mamier, in wliich respectively
5 gnns. of baryta were mixed mth ^, \vith f, and with an ecpial weight of clilorate of
potassium, precisely the same result was obtained.

Expt. II. 5'084 grms. of baryta were mixed Avith ^ that weight of perchlorate of potas-
sium: the experiment was otherwise conducted precisely as before. 9'8cub. centims. of
the resulting solution of peroxide required 10 parts of permanganate. 1 part of the per-
manganate employed was equivalent to 0*000548 grm. of oxygen. Hence 100 parts
of baryta had combined A\atli 5"375 parts of oxygen. In the case of the formation of
the sesquioxide of barium, 100 parts of baryta would combine with 5*228 parts of
oxygen.

"When lime or strontia were substituted for baryta in the preceding experiment, not
a trace of peroxide of hydrogen could be detected in the resulting solution.

To prepare piu-e peroxide of barium, the crude peroxide, as prepared by either of the
above processes, is finely pulverized and rubbed with water in a mortar, so as entirely to
convert it into hychate. It is then mixed gradually with a very dilute solution of
hydi-ochloric acid, care being taken to keep the solution constantly acid. This solution
is filtered, and rendered alkaline with a slight excess of baryta-water. The addition of
the latter effects the precipitation of the alumina and ii-on. The alkaline solution,
which immediately begins to decompose, is rapidly filtered through Unen filters, and to
the clear filtrate is added an excess of baryta-water. The hydrated peroxide of barium
is precipitated in brilliant plates, which are insoluble in Avater, and may be washed by
decantation. In order to ascertain whether the whole of the peroxide is precipitated, a
small portion of the solution may be filtered, rendered acid, and tested Avith a dilute
solution of bichromate of potassium.

The washed precipitate is to be collected on a filter, pressed out between blotting-
paper, and dried under the aii"-pump, by which means the whole of the water of cry-
stallization may be dii\en off. The di'y peroxide appears in the form of a fine white
powder, resembling magnesia, I have analysed this substance, and found it to consist
of anhydrous peroxide of barium, Ba., O2, the only impurity being a trace of carbonate.
It is in this condition a perfectly stable substance.

The absolute amount of peroxide of barium contained in the different preparations
employed in the following experiments was ascertained either by a direct determination
of the oxygen evolved by the action of platinum-black and a dilute acid, or by means of
a standard solution of permanganate of potassium, according to the method which I have
given in a former paper.

Peroxide of Benzoyl, Ci^H^qO^.
The peroxide of benzoyl is prepared by the action either of the chloride of benzoyl,

MDCCCLXIII. 3 K

410 SIE B. C. BEODIE ON THE PEEOXIDES OF THE

or of the benzoic anhydride on hydi-ated peroxide of barium. When the following pre-
cautions are taken, the reaction is perfectly definite.

Equivalent quantities are to be weighed out of chloride of benzoyl and pure peroxide
of barium. The peroxide of barium is converted into hydrate, and pressed between
blotting-paper, to remove any great excess of water ; it is then added by degrees to the
chloride of benzoyl in a small mortar, and the two substances well mixed by means of a
pestle. The mixture is allowed to remain for some hours ; and the resulting substance
having been mixed with water is thrown on a filter, and washed until the chloride of
barium is removed. It is then treated with a weak solution of carbonate of sodium,
so as to render the solution decidedly alkaline. After thus removing the benzoic acid,
of whicli a certain portion is always fonned in the reaction, the substance is pressed out
between blotting-paper, and dried under the air-pump. When perfectly dry, it is to
be dissolved in bisulphide of carbon at a temperatiu'e not exceeding 35° C, and three
or four times crystallized from that fluid.

Of several slightly diiferent methods of preparation this gave by far the most satis-
factoi-y results. In one experiment, for example, from 20 grms. of chloride of benzoyl,
15-2 grms. of the crude peroxide, as dried under the air-pump, were obtained, corre-
sponding to 88-26 per cent, of the theoretical yield. If an excess of water be present,
the amount of peroxide of benzoyl obtained is greatly reduced. The hyckation of the
peroxide of barium appears to be essential to the reaction. An equivalent quantity
of peroxide of barium, mLxed witli anhydi-ous benzoic acid dissolved in ether and heated
for five or six hours in a water-bath, was entirely without action. The same appeared
to be the case with equivalent quantities of cliloride of benzoyl and peroxide of barium
under the same circumstances.

If, in the preparation of this substance, the peroxide of barium be taken in excess,
that is, more than one equivalent of that peroxide, Ba^ O.,, to two of chloride of benzoyl,
2Bz CI, the amount of peroxide of benzoyl formed is reduced ; and if a great excess of
peroxide of barium be employed, as for example one equivalent of that substance, Bag Og,
to one equivalent of chloride of benzoyl, Bz CI, oxygen gas is evolved, and hardly a trace
of peroxide will be f(jrmed. This arises from the circumstance that the reaction in which
the peroxide of benzoyl is formed is immediately succeeded by a second reaction in which
that substance is destroyed, according to the equations

2Bz Cl+Ba^ 02=2Ba CI+BZ2 O2,
Bz2 Og+Ba^ 02=2Ba Bz O+O2.

I have ascertained by direct experiment that the peroxide of barium, when mixed in
water with the peroxide of benzoyl, is decomposed with evolution of oxygen gas.

This affords a striking example of a class of decompositions which I recently brought
before the Society, — in which one ecjuivalent of the peroxide of barium acts as an agent
of oxidation, while a second equivalent acts as a reducing agent, destroying the substance
formed in the first reaction.

RADICALS OF TUE ORGAiS'IC ACIDS. 411

The peroxide of benzoyl thus prepared is in splcnflid crystals. From large quantities
of solution I have occasionally obtained these crystals as much as three-fourths of an
inch in chameter. They are referable to the right prismatic system, and their form, as
crjstallized from ether, has been examined by Professor W. H. Miller, of Cambridge*.
It is difficult to ascertain with absolute accuracy the melting-point of this substance.
The point of decomposition lies close u]>on the melting-point ; and it is only in very small
quantities that it can be melted without being decomposed. My experiments, however,
place the melting-point at 103°-5 C. At 15° C. 100 parts of bisulphide of carbon dis-
solve 2'53 parts of the peroxide of benzoyl. It is also soluble in ether and benzole.

This substance gave on analysis the following results : —

I. 0'3975 grm. of substance gave

Carbonic acid 10103

Water 0-1513

II. 0'4412 grm. of the same substance, twice recrystallized, gave

Carbonic acid 1-1213

Water 0-1661

These numbers give, as the percentage composition,

I. II.

Carbon . . . 69-31 69-31

Hydrogen . . 4-23 4-18

Oxygen . . . 2646 26-51

100-00 100-00

The numbers required by theory are

Ci^ =168 69-42

Hio= 10 4-13

O4 = 64 26-45

242 100-00

I have repeatedly prepared and analysed this substance with the same results.
When the peroxide of benzoyl is boiled with a solution of hydrate of potassium,
oxygen gas is evolved and benzoate of potassium is formed.

Bz2 02+2KHO=2Bz KO+Hg 0-f O.

If the peroxide of benzoyl be heated, it is decomposed with a slight explosion. By

mixing the finely powdered peroxide with sand the action may be moderated ; under

these circumstances carbonic anhydride is evolved. The decomposition commences at

about 85° C. I have estimated the loss of weight which the substance undergoes in

this decomposition : in two experiments 100 parts of peroxide lost 18-6 and 1818 parts ;

in three other experiments somewhat lower numbers were obtained, 17-78, 16-56, and

16*7 per cent. The theoretical loss, if one equivalent of carbonic anhydi-ide, COj, were

* See Proceedings of the Royal Society, January 15, 1862.

3k2

412 SIE B. C. BRODIE ON THE PEROXIDES OF THE

evolved from one equivalent of the peroxide, Cj4 Hjo O^, would be 18-18 per cent. The
substance formed by the removal of one equivalent of carbonic anhydride from the per-
oxide of benzoyl, Cjg Hjq Oj, would be isomeric vnih the benzoate of phenyl.

I have not, however, yet succeeded in so moderating the action as to form only one
substance. During the decomposition a small quantity of benzoic acid sublimes, and
on extracting the sand with ether, filtering, and evaporating the ethereal solution, a soft
glutinous residue is obtained, of which a portion dissolves on prolonged boiling in
Avater. Benzoic acid passes over with the vapoiu- of the water, and ultimately a hard
and perfectly transparent resin remains, which is soluble in potash, and in all respects
resembles a natural resin. I hope again to recur to this substance.

If peroxide of benzoyl be treated with a large excess of concentrated nitric acid, it is
dissolved by the acid. When this solution is poured into water, a slightly yellow floc-
culent substance separates, which, dried under the aii--pump, is soluble in bisulphide of
carbon.

3 grms. of peroxide of benzoyl were thro'nni into about 3 fluid-oimces of fuming nitric
acid, specific gravity 1'505. There was no perceptible increase of temperature or evo-
lution of gas. The peroxide was rapidly dissolved, the mixture became deeper in colour,
and after some time the vessel was filled vdth fumes of hyponitric acid. After standing
about twenty-four hours, the solution in nitric acid was mixed with ten times its bulk of
water. The precipitate formed was brought on a filter and washed free from acid. It was
then diied under the air-pump, and dissolved m bisulphide of carbon. On the cooling of
the bisulphide, a slightly yellow flocculent body separated. This was again dried under
the air-pump and analysed.

I. 0-4167 grm. of the substance gave

Carbonic acid . . . 0-7735 grm.
Water 0-0969 grm.

II. 0-433 grm. of the same substance, at a temperature of 18° C, and a barometric
pressure of 758-5 millims., gave 32-5 cub. centims. of nitrogen gas. This corresponds to
a weight of 0-0374 grm. of nitrogen.

We have as the results of these determinations,

Carbon 50-60

Hydrogen .... 2*58
Nitrogen .... 8-49

Oxygen 38-33

100-00
The formula of the substance derived from the peroxide of benzoyl by the substi-
tution of two atoms of peroxide of nitrogen, NO2, for two atoms of hydrogen, H, is
C,4 Hg (N02)2 0, = Ci, Hg N2 Og, and requires

RADICALS OF THE OEGAjSTIC ACIDS. 413

Ci, = 168

50-60

H«= 8

2-41

N2= 28

8-43

08=128

38-56

332

100-00

This body, when heated, decomposes ^vnth a slight explosion, leading a resinous matter
similar in appearance to that fonncd by the decomposition of the peroxide of benzoyl.

Gerhaedt did not succeed in procuring in a state of purity the anhydrous nitro-
benzoic acid ♦, on account of the facility mth which it decomposes water. Tlie nitro-
benzoic peroxide stands to the acid in the same relation as does the peroxide of benzoyl
to the anhydrous benzoic acid.

Peroxide of Cumenyl.

The peroxide of barium is decomposed by the chloride of cumenyl precisely as by the
chloride of benzoyl. The resulting substance crystallizes from ether in long and beau-
tiful needles ; when heated it ex2)l()des, leaving a resinous residue.

I have only once prepared this substance, and did not succeed in procuring it in a
state of absolute purity.

0-3798 gi-m. of the substance gave 1-020 gi-m. of carbonic acid and 0-2392 grm. of
water. These nvunbers give per cent.,

Carbon 73-24

Hydrogen .... 7-00
Oxygen 19-76

100-00

The formula Cgo H22 0^ requires

C20 =240

74-23

H22= 22

6-75

O4 = 64

19-02

326

100-00

The peroxide of benzoyl itself is mixed with traces of some substance which it is veiy
difficult to remove by crystallization, and which lowers the percentage of carbon in the
analyses ; and, notwithstanding the difference of 1 per cent, in the carbon from that
required by theory, we may assume the substance to be the peroxide of cumenyl.

Peroxide of Acetyl.

In the preparation of the peroxides of the acetic series, the use of the anhydrous acid

has great advantages over the use of the corresponding chloride. By the action of the

anhydi'ide of the acid on peroxide of barium, I have succeeded in preparing three of

these peroxides, the peroxides of acetyl, butyl, and valeryl.

* Annales de Chiinie, iii. p. 37 & 321.

414 SIR B. C. BEODIE ON THE PEEOXIDES OF THE

The peroxide of acetyl is prepared by dissolving anhydrous acetic acid in pure ether,
and adding gradually to this solution an equivalent quantity of peroxide of barium.
The decomposition takes place according to the equation

2C, H, 03+Ba,, 0,=-2a H., Ba 0.,+C, Hg O,.

The reaction is attended with an elevation of temperature which causes the ether to
boil ; the temperature is not to be allowed to reach this point. After standing some
time, the solution is filtered from the gelatinous residue, which does not contain a trace
of peroxide of barium, and the ether is distilled off at a very low temperature, great
care being taken not to allow the temperature to rise towards the end of the operation.
The residue, washed first with water and then with a veiy weak solution of carbonate
of sodium, appears as a thick and \iscid fluid. I have in this manner experimented on
as much as 20 grms. of anhydrous acetic acid, dissolved in about four times its bulk of
pure ether. The addition of the equivalent quantity of peroxide of barium occupied
two hoiu-s. From these 20 grms. of anhydrous acetic acid, only as much peroxide of
acetyl was procured as to be sufficient for the two following determinations. The
analysis was thus effected.

An undetermined quantity of the peroxide of acetyl was placed in a little water at
the bottom of the bulb-apparatus used for the estimation of oxygen in peroxide of
barium, which I have described in a former paper. The bulb was filled with baryta-
water, a small tube containing platinum-black introduced into the apparatus, and the
whole was weighed. The peroxide was now decomposed by allomng the baryta-water
to flow into the flask fi-om the bulb. Acetate of barium and peroxide of barium are
formed. The peroxide of barium was decomposed by bringing the platinum-black con-
tained in the small tube in contact with it. After the completion of the reaction, the
apparatus was again weighed, and thus the loss of oxygen determined.

A current of carbonic acid was now passed through the solution, which was boiled and
filtered, and the barium estimated as sulphate. The sulphate of barium thus formed is
the measiu-e of the acetate produced by the decomposition of the peroxide of acetyl.

Experiment I. The weight of the apparatus before and after the experiment gave a
loss of oxygen of 0T225 grm.

The solution precipitated by sulphuric acid gave 1"776 ga-m. of sulphate of barium.

Experiment II. The loss of oxygen estimated as before was 0T37 grm.

The solution precipitated by sulphuric acid gave 1-944 grm. of sulphate.

In Experiment I., 100 parts of sulphate being formed, G-89 parts of oxygen were
evolved.

In Experiment II., 100 parts of sulphate were obtained and 7*04 of oxygen evolved.

Theory requires that for every 100 parts of sulphate of barium formed C-80 parts of
oxygen should be evolved.

When a small drop of the peroxide of acetyl is heated in a watch-glass, it is decom-
posed with an explosive violence, only to be paralleled by the decomposition of chloride

EADICALS OF THE OEQANIC ACIDS. 415

of nitrogen. Hence the greatest care is necessary in its preparation, especially during
the distillation of the ether in which it is dissolved. I had frequently effected this ope-
ration without accident ; but on one occasion my assistant was engaged in distilling off
the ether from a rather considerable quantity of the substance, which was contained
in a flask placed in warm water on a small copper water-bath ; the temperature was
probably allowed to rise too high, and towards the close of the operation a %iolent
explosion took place with a report as of a cannon. A large hole was made in the copper
water-bath, through which the hand might be passed, the copper being folded back
upon the sides of the bath. The explosion, though of excessive violence, was local, and
nothing in the laboratory in which the explosion took place was injured.

The peroxide of acetji is readily decomposed under the influence of sunlight. A
measured quantity of the substance was kept unaltered in bulk for above eigliteen
hours in the dark, but when placed in water in the bright sunlight, the same substance
rapidly disappeared.

This peroxide is a most powei-ful agent of oxidation ; like chlorine it rapidly bleaches
indigo, it separates iodine from hydriodic acid and from iodide of potassium, it converts
a solution of ferrocyanide of potassium into fcrricyanide, and immediately oxidizes the
hydrated protoxide of manganese.

These properties it has in common with the peroxide of hydrogen ; but it is readily
distinguished fr-om that substance by not producing the peculiar effects of reduction, by
which the peroxide of hydrogen is characterized. It does not reduce an acid solution
of chromic or peiTuanganic acids. The addition of baryta-water to the peroxide of
acetyl suspended in water, causes an immediate precipitate of crystals of the hydrated
peroxide of barium.

We can have no more convincing proof, if such proof were needed, than that fur-
nished by this experiment, that the difference of properties which oxygen manifests in
its different combinations is due. not, as has been imagined, to the existence of certain
distinct varieties of that element, but to the circumstance that the combining properties
of oxygen, as of other elements, vary with the nature of the chemical substances Avith
which it is combined or associated.

Peroxide of Butyl.

The butylic peroxide is readily prepared by mixing hydi'ated peroxide of barium with
anhydrous butyric acid. Experiments made with the riew of preparing this substance
by the action, in ether, of the dry peroxide of barium on anhydrous acid, were imsuc-
cessful.

The result of this reaction is given in the equation

2C8 H,, 03+Ba^ 0,,=2C, H; Ba O^+C^ U,, O,.

The experiment may be advantageously conducted as follows : —

The anhydi-ous acid is placed in a small mortar, and an equivalent quantity of hydrated
peroxide of barium, fi-om which any great excess of water has been removed, is

416 SIE B. C. BEODIE ON THE PEEOXIDES OF THE

gi-adually added to it, the whole being well mixed after each addition of the peroxide.
An excess of peroxide of barium is to be carefully avoided, as it again decomposes the
peroxide of butyl. To this end it is desirable towards the close of the operation, to
examine from time to time the contents of the mortar, by placing a drop on a watch-
glass, acidifying with hydrochloric acid, and testing with a dilute solution of bichromate
of potassium. The appearance of a feeble blue colom- indicates that sufficient peroxide
has been added. The substance is mixed with a small quantity of water, and the solution
agitated repeatedly with ether, which dissolves the peroxide of butyl. This operation
is readily effected in a biuette, provided with a glass stopcock. The ethereal solution is
then repeatedly washed, fii-st with dilute hydrochloric acid, then with a weak solution of
carbonate of sodium, until the solution has a strong alkaline reaction, and then again
vrith water imtil the alkaline reaction disappears. The solution is filtered and allowed
to evaporate in a current of air at a low temperature. An oily residue is left, which is
to be washed once or twice mth a small qiiantity of water, in which it is only slightly
soluble. It is then removed with a pipette, and allowed to stand for some time in
contact vnth. a few fi-agments of chloride of calcium. The substance thus prepared is
pure peroxide of butyl.

This peroxide was analysed with oxide of copper in the usual manner.

I. 0-3562 grm. of the substance gave 0-721 grm. of carbonic acid and 0-2668 grm.
of water.

II. 0-3144 grm. of the substance gave 0-6355 grm. of carbonic acid and 0-2344 grm.
of water.

These analyses give

per cent.,

I.

n.

Carbon . . .

55-21

55-11

Hydrogen . .

8-29

8-28

Oxygen . . .

36-50
100-00

36-61

100-00

Theory requires

Cg =96

0, =64
174

i

55-172

8-046

36-782

LOO-000

A drop of the peroxide of butyl on a watch-glass decomposes with a slight explosion.
Suspended in water, it possesses the oxidizing properties of the acetic peroxide.

Peroxide of Valeryl.

The peroxide of valerj'l is prepared by the action of anhydrous valerianic acid on
hydi-ated peroxide of barium, the result of the reaction being expressed by the
equation

2Cio Hi8 03+Ba2 0,=2C, H^ Ba O^-f C^o l\,, O,.

IJAUR'ALS OF TUE OEGAXIC ACIDS. 417

The method of preparation is in all respects the same as that by which the pero.xidc
of butyl is prepared.

The peroxide of valeryl is a dense oily fluid, heavier than water. It gives a slight
explosion when heated, and possesses tlie oxidizing properties of the other analogous
peroxides.

The substance, ch-ied by means of chloride of calcium, gave to analysis the following
numbers : —

I. 0-3055 grm. of the substance gave 0-G615 grm. of carbonic acid and 0-2523 grm.
of water.

II. 0*4005 grm. of the substance gave 0-873 grm. of carbonic acid and 0-3310 grm.
of water.

These numbers give as the percentage constitution of the substance,

I.

Carbon . . . 59-05

II.
59-39

Hydrogen . . 9-17

9-17

Oxygen . . . 31-78
100-00

31-44

100-00

Theory requires

Cio=120

59-40

H,s= 18

8-91

0^ = G4
202

31-69

100-00

Peroxide of Camjyhonjl.

The action of the anhydrides of the bibasic acids on the alkaline peroxides affords a
remarkable illustration of the profound differences by wliich this group is distinguished
from the anhydrides of the monobasic acids. In the latter case we have seen that
the monobasic anhydride decomposes with the alkaline peroxide, forming the peroxide
of the radical and the baryta salt of the corresponding acid. In the case of the bibasic
anhydride, a combination takes place of the anhydride with the peroxide, with the
formation of a new and peculiar compoimd, which Ave may regard as the baryta salt of
the peroxide of the bibasic radical. The compounds thus formed have but little
permanence ; and although in several cases we ha^'e e\idcnce of their formation, in only
one example, namely that of camphoric acid, have I been able to eflect the analysis of
the compound.

If hydrated peroxide of barium be gradually added to anhydrous succinic aCid, and
carefully mixed with it in a small quantity of water, the mixture becomes fluid ; but
long before the addition of the equivalent quantity of peroxide of barium, oxygen gas is
evolved. If the fluid be filtered when this effervescence commences, it will be found to
have the following properties : —

MDCCCLXIir. 3 L

418 SIR B. C. BEODIE OX THE PEROXIDES OF THE

1. The solution is alkaline. It maybe assumed therefore to contaui but little^ if any,
succinate of barium, which is insoluble in water.

2. The solution rendered acid gives no blue colour ^^ith bichromate of potassium, and
does not discolour permanganic acid. It therefore contains no peroxide of hydrogen.
If peroxide of barium be mixed with hydrated succmic acid, a solution is obtained con-
tammg peroxide of hydrogen ^^'ith the above characteristic reactions.

3. The solution bleaches indigo, gives a precipitate of peroxide of manganese with a
■solution of acetate of manganese, oxidizes ferrocyanide of potassium, and boiled with
hydi-ochloric acid evolves chlorine.

4. When the solution is boiled, oxygen gas is evolved and a crystalline precipitate
formed of succinate of barium.

Similar results are obtained if hydrated peroxide of barium be mixed with lactide,
the lactic anhydride. The peroxide is rapidly dissolved, and a powerfully bleaching
solution is obtained, possessing the same oxidizing properties as that procui'ed fi'om the
succinic anhydride. This solution is, however, excessively unstable ; even when cooled
by ice, it is in a constant state of decomposition ; and although it doubtless contains
the lactic peroxide, I have been unable to effect its analysis.

With the camphoric anhydride I have been somewhat more successful. The anhy-
di'ous camphoric acid used in the following experiments was prepared by the oxidation
of camphor by means of nitric acid. It is better not to attempt the prior preparation
of a piu'e camphoric acid, which is attended with much difficulty, but after the product
of the oxidation of camphor has been once or twice crystallized, to distil the crude acid.
After two distillations and two or three crystallizations from alcohol of the distilled
product, the camphoric anhjxlride is obtained quite pure. The substance was analysed
with the following results : —

Calculated.
, ^ ^ Found.

Cjo . . . 120 65-93 65-51

Hi, . . . 14 7-69 7-87

O, ... 48 26-38 26-62

CioHi.O, . 182 100-00 100-00

A portion (about 3 grms.) of anhydrous camphoric acid thus prepared was tritui-ated
in a mortar with ice-cold water, and the equivalent quantity of hyth-ated peroxide of
barium was gradually added to the same, fragments of ice being mixed with the solution.
No evolution of gas was observed dui-ing the experiment. The filtered solution was
slightly alkaline, doubtless from the trace of baryta present in the peroxide. The solu-
tion, rendered acid, had the foUo^-ing properties. It gave no blue reaction ^^-ith chromic
acid, nor did it discolom- permanganic acid. It bleached indigo, oxidized ferrocyanide
of potassium, and decomposed hydriodic acid. The residue from which the solution was
filtered was small in amount, and ccmtaincd a little peroxide of barium. The solution
when boiled evolves oxygen. Evaporated to dryness, it leaves a residue, ^^■hich, dissolved

EADICALS or THE ORGANIC ACIDS. 419

in water, gives a precipitate with a solution of acetate of lead. This precipitate was sus-
pended in water, and decomposed by sulphide of hydrogen. The acid thus separated
was after one crystallization analysed. It was pure camphoric acid.

Calculated.

Cjo . .

. 120

60-00

60-37

H,, . ,

. . 16

8-00

8-13

0, ...

. 64

32-00

31-50

CioHigOj . 200 100-00 100-00

A solution of the camphoric peroxide thus prepared was analysed in the following
manner : —

1. A measured quantity of the solution was rendered acid, and titrcd by means of a
standard solution of iodine.

2. To another measured quantity of the same, a solution of carbonic acid in water
was added. The liquid was raised to the boiling-point, filtered, and precipitated by sul-
phui-ic acid. The addition of the carbonic acid effects the removal of a small quantity
of bai-yta, invariably present through the decomposition of the peroxide.

3. Another portion of the solution, similarly treated, was precipitated by acetate of
lead, and the precipitate collected and weighed. The precipitate thus obtained is pure
camphorate of lead, as is shown by the following determmation : —

0-5901 grm. of the precipitate, ignited in a porcelain crucible, gave 0-3257 grm. of
oxide of lead. Hence 100 parts gave 55-19 parts of oxide of lead. 100 parts of neutral
camphorate of lead, Cjo H14 O4 Pb.^, contain 55 parts of oxide.

Experiment I. — (1) One part of this solution requii'ed for its titration in two concord-
ant experiments, 12-98 cub. centims. of a standard solution of iodine, which contamcd
in 1 cub. centim. 0-002531 grm. of iodine. Hence 1000 parts of the solution of cam-
phoric peroxide contained 2-07 grms. of oxygen.

(2) Six parts of the same solution, treated in the manner described, gave 0-1949 grm.
of sulphate of barium. Hence 1000 parts of the solution contained 21-511 grms. of
baryta, Ba2 0.

(3) Six parts of the same solution, tr(>ated as described, gave 0-3354 grm. of cam-
phorate of lead. Hence 1000 parts of the solution contained 25-12 grms. of anhydrous
camphoric acid, Cjo H,^ O3.

Experiment II. — (1) In a similar- experiment made with another solution, one part
titred with the same standard solution of iodine was equivalent to 12-26 cub. centims. of
the iodine solution. In a second experiment, one part of tlie same was equivalent
to 12-317 cub. centims. of the same iodine solution. Hence, on the mean of the two
experiments, 1000 parts of the solution contained 1-97 grm. of oxygen.

(2) Six pai-ts of the solution gave 0*1508 grm. of sulphate of barium. Ten parts of
the solution gave 0-2927 grm. of sulphate.

3l2

420 SIE B. C. BRODIE OX TUB PEEOXIDES OF TlIE

From the first determination, 1000 parts of the fluid contained lG-5 grms. of baryta;
from the second, 1000 parts of the fluid contained 19-22 grms. of baryta. The mean of
these U\o determinations (in which doubtless there is some error) gives 17-88 grms. of
baryta in 1000 parts of fluid.

(3) Six parts of the sohition gave 0-3354 grm. of camphorate of lead. Hence 1000
parts of the solution contained 21-43 grms. of anhydrous cami^horic acid.

These numbers agree with the hypothesis that the solution contains the substance
Cio Hji O5 Ba,, — the reaction taking place according to the equation

C'lo Hi4 O^+Ba, 0, = C\, Hj, O, Ba,.
For we should have, assuming the camphoric acid to be correct as determined by pre-
cipitation with acetate of lead in 1000 parts of the solution,

Experiment I.

Atomic weiglit.

Eatio calculated.

Found.

C,oHu03 .

. . 182

25-12

25-12

0 .

. . 16

2-20

2-07

Ba. 0 .

. . 153

21-12

21-51

Experiment II.

C,oHh03 .

. . 182

21-43

21-43

0 .

. . 16

1-88

1-9G

Ba, 0 .

. . 153

18-00

17-88*

The oxygen-determinations show that even in this case there is a gradual, although
but slight, decomposition of the substance taking place during the time which the
determinations occupy. But this peroxide is far more stable than the corresponding
succinic and lactic peroxides. I have made several unsuccessful attempts to analyse
these .substances by methods similar to the above ; but, from the excessive instability
of the solutions, I have been obliged to abandon the attempt. In the case, for example,
of the lactic peroxide, three successive determinations required CO -8, 54-3, and 48-G
cub. centims. of the standard iodine solution, showing so rapid a change as to render
hopeless the accurate determination of the oxygen. These substances stand, as it were,
upon the very v(>rge of chemical possibility, and have only a momentary and fugitive
existence.

That in the above reaction the oxygen is transferred from the peroxide of barium to
the anhydrous camphoric acid — in other words, that the compound formed is to be
regarded as the barium salt of tlie peroxide of camphoryl, and not as the camplioi-ate
of the peroxide of barium, is sh()^\^l by the reactions of the solution. The action of
acids upon it does not form peroxide of hydrogen, and the action of alkalies does not

' !Mcan of the two determinations lG-5 and 10-22.

EADICALS OF THE OEG.\]S'IC ACIDS. 421

reproduce the peroxide of barium. These reactions must take place if the solution con-
tained the salt of the peroxide*.

The analog)- of the bisulphide of carbon to the anhydrous acids induced me to try
its action on the alkaline peroxide. When bisulphide of carbon suspended in water is
agitated with hydrated peroxide of barium, the peroxide is dissolved with the formation
of a yellow solution. The solution wlien filtered is at first clear; but on standing, and
more rapidly on boiling, a precipitate is formed of carbonate of barium. The solution
contains a sulphide of barium. If sulphide of hydrogen be led through water in which
peroxide of barium is suspended, a clear yellow solution is formed similar in appearance
to the preceding.

* I have not fully investigated the reaction ; but the experiments point to the conclusion
that in the fiist instance we have formed the combmation of bisulphide of carbon and
peroxide of barium, which subsequently decomposes into carbonate of barium and
bisulphide of barium, according to the equations

CS,+Ba^ 0,=Ba,, C S^ O^,
Bag C S2 02+2Ba H0=Ba2 COa+Ba,, S^+II, O.
The reaction is undoubtedly complicated by the action of the bisulphide of barium

on the peroxide. This solution in presence of an excess of peroxide becomes coloiu'less,

hyposulphite of barium being probably formed,

Ba, S.-f 3Ba. O^+SHg 0=Ba2 83 O^+GBa IIO.
In the case of the action of carbonic acid on peroxide of barium, I could detect no
indication of the formation of a higher oxide.

The previous investigation has placed beyond doubt the existence of a new and
extensive group of chemical substances, the peroxides of the radicals of the organic
acids, a group in all probability as numerous as the anhydrides of the acids, and cha-
racterized by singular properties, which have never hitherto been discovered in any

* The question regarded as one of notation maybe considered as,-\vbetlicr wo are to write the formula of the com-

Ba' ] "Ba" ]

pound(C,„H,,OJ' lO.„ or (C,„H,,0,)" [ 0,.or on the dualisticmethodasC,(,II„(),.Ba,0,"orC',„II,,,0„Ba,,0,.
Ba' J " Ba" J

If we are to write the formula of tlie scsquioxide and its hydrate as y'^nO^, and2,, \,, I (\, we must in

consistency write the formula of the peroxide and its hydrate thus, -m-, " O2 ^^^ ^^^ " ^r (Ctcm. Soc. Q. J. vol.

xiv. p. 280.) Tho normal salt of the peroxide would be the body deriyed from the peroxide by the substitution
of the equivalent quantitj- of the acid railical for the equivalent of sodium. The compound of one equivalent
of camphoric aiiliydride with one equivalent of the peroxide of barium, would bo an oxide intermediate
between the protoxide and peroxide, just as the magneHc oxide, and its derivatives, is intermediate between the
protoxide and sesquioxide.

422 SIE B. C. BEODIE ON THE PEEOXIDES OF THE

combination of carbon, and which greatly enlarge our Aiew of the system of analogies
by which the organic and inorganic Avorlds of chemistry are connected. These
bodies are, so to say, the organic representatives of chlorine. Indeed no compound
substance, unless perhaps the peroxide of hydrogen, can be compared with them in
chemical similarity to that element. The solution of these peroxides in water can
hardly be discriminated fi-om a solution of chlorine ; the solution bleaches indigo,
oxicUzes the protosalts of iron and manganese, decomposes the alkaline peroxides, is
decomposed by the action of sunlight, and breaks up with water into the hydrated acid
and oxygen. We have

Cl2+2HKO+2Mn H0=2K Cl + H^ 0 + Mn. H,> O3,
and

also

and

also

and

Ac, 0,+2HKO+2Mn H0=2KAc 0+H,, 0+Mn. H. O3 ;

Cl,>+Ba. 0,>=2Ba Cl + 0,,

Bz2 0, + Ba. 02=2Ba Bz 0+0., ;

Cl2+H,,0=2HCl+0,

Ac2 O2+ H2 0=2H Ac 0+0.
No parallel can be more complete.

In wilting the fomiula of water and of hydrochloric acid, HO and HCl, it was
imphcitly assumed that the atom of oxygen stood in the same relation in regard to the
molecule of water as does the atom of chlorine in regard to the molecule of hydrochloric
acid. This ^iew can no longer be maintained. Peroxide of hydrogen, and not oxygen,
is the analogue of chlorine. We have

H2+Cl2=2HCl,
H2+H2 02=2H2 0,
H2+Ba2 02=211 BaO,
H2+(C2 H. 0)2 02=2H(C2 H. 0)0.
The remarkable point in the case of the organic peroxides is not only that this analogy
is an analogy of the forms of decomposition, but that the same similarity of properties
which exists in hydi'ochloric acid and acetic acid is found also in chlorine and the per-
oxide of acetyl; and precisely the same reasons, derived from similarity of chemical pro-
perties, which lead us to place in the same class of " acids " hydrochloric acid and acetic
acid, compel us also to group together chlorine and the peroxide of acetyl. These
bodies are the analogues of chlorine in the same sense in which the nitrogen base is the
analogue of potash, and in a closer sense than that in which ethyl is the analogue of
hydrogen.

The transition is obvious from the peroxide of the acid radical to the peroxide of the

RADICALS OF THE OKOAXIC ACIDS. 423

basic radical. The question is immediately suggested whether by corresponding pro-
cesses we may not be able to procure the peroxides of ethyl, of ethylene, of the com-
pound ammoniums. I am yet occupied with this subject, and ^^'ill now only remark
that the peroxide of the glycol series appears undoubtedly to be formed. The bromide
of ethylene does not, indeed, decompose the hydrated peroxide of barium ; but this per-
oxide is immediately acted on by the diacetate, and a solution is formed of a most
pungent odoui*, containing no peroxide of hydrogen, but possessing the usual charac-
teristics of the organjc peroxides. I have not made many experiments with the compound
ammonias. A solution of hydrated oxide of tetramylammonium, evaporated in vacuo
\y\i\\ a solution of pure peroxide of hydrogen, gave a residue which did not a-f.pear to
contain even a trace of a substance resembling an alkaline peroxide. But this by no
means renders it impossible that the same experiment in other cases may be more suc-
cessful, for gi'eat differences are found in the stability of the peroxides of veiy analogous
metals: the peroxide of potassium is quite decomposed on evaporation in vacuo.,
whereas the hydrated peroxide of sodium can be readily thus obtained.

[ 425 ]

XIX. An Account of Experiments on the Change of the Elastic Force of a Constant
Volume of Atmospheric Air, between 32° F. and 212° Y., and also on the Temperature
of the Melting-point of Mercury. By Balfouk Stewart, M.A., F.R.S.

Received June 18, — Eead June 18, 1863.

It was some time since proposed by the Kew Committee of the British Association to
determine the temperature of the melting-point of mercury, in order if possible to add
a third to the two familial- points which have been so long exclusively used in graduating
thermometers; and afterwards the sum of £150 was voted for this pui'pose by the
Government Grant Committee of the Royal Society.

In prosecuting this research, the final arrangement of apparatus has cost much labour-
and time ; but the results at length obtained have exhibited a precision rthich has
induced me to present them to the Society in the follovping communication.

I shall in the fii-st place endeavour to describe the appaa-atus used, and shall then give
an account of the experiments made and deduce results.

Beseription of Apparatus.

The apparatus employed was very similar in principle and construction to that used
by Regxault in his fom-th set of experiments on the dilatation of elastic iiuids, — the
coefficient sought being that which denotes the increase for 1° Fahr. of the elastic force
of a gas the volume of which is constant.

This apparatus was constructed by Mr. Beckley, mechanical assistant at Kew, to whose
skill in device and execution I am on this occasion very much indebted ; and I would
likewise desii-e to acknowledge the promptness and skill with which the requisite glass-
blovving has been executed by Mr. Casella.

The atmospheric air upon which it is desired to operate is contained in the glass
bulb B (Plate XXIV. fig. 2), and this is connected by means of a capillary tube t with
another tube T of larger bore, which is cemented into an iron fitting, D. This fitting is
tightly screwed, by means of an india-rubber washer, upon a reservoir R filled with
mercui-y, and there is thus a communication between the mercury of the reservoir and
the bore of the tube T. Another tube, T', similar- to T, is attached in the same manner
to an iron fitting D', and by means of it to the reservou- R ; but the fitting D' is fiuTiished
with a stopcock, by shutting which the commimication between the reservoir and the
bore of T' may be interrupted at pleasure. The upper extremity of T' communicates
with the atmospheric air. The reservoir R, which is made of cast iron, is fitted accu-
rately into a strong slate slab, which is in its turn supported by a solid block of masonrj'.

MDCCCLXIII. 3 M

426 ME. B. STEWAET ON EXPERIMEXTS WITH AN ATE-THERMO:\IETER.

S (figs. 1 & 2) is a screw which drives a phmger P up and down, by means of which the
capacity of the resenoir R may be enlarged or contracted at pleasure. The consequence
is that when the capacity of the reservoir is diminished the mercury will ascend in the
tube T, and also in T' if the stopcock be open, while if the capacity be increased it will
descend in these tubes. The mside of the reservoir- is so shaped as to push up any small
bubble of air that might othenvise have remained in during the process of filling with
mercury. There is also a fine screw, S', with a graduated head, which drives a fine
plrmger P'; and the change of capacity of the reservoir due to one revolution of this
screw requires to be accurately ascertained.

A little above h' there are two side tubes, terminating in bulbs b, h', which are
attached at an angle to the tube T.

One of these bulbs contains a desiccating substance, such as sulphuric acid, baryta, or
anhydrous phosphoric acid, while the other contains a little caustic potash, a substance
which has a strong attraction for carbonic acid.

Let us begin by supposing these two bulbs to be attached to the tube T, as in fig. 1,
and let us also suppose that they are open to the atmosphere at their extremities.
Suppose in fact that the tube T with its appendages has just been screwed on to the
reservoir' K. Now by means of the sci'ew S drive the mercury up T imtil it reaches
the level Ji! a little below the opening where the side tubes branch off; and when it has
reached this level, seal off" the extremities of the two bulbs. All communication between
the air in the bulbs and the atmosphere is thus intercepted.

The bulb B must now be heated as often as possible, and each time a portion of its
air will be driven into the bulbs h, b', and there deprived of moisture and carbonic acid.
The tube t should also be heated occasionally, and also the tube T above //', including
the appendage tubes, but not the bulbs, the object being to drive away any moisture
which may cling to the glass ; but at the same time care must be taken not to heat the
mercury, in case by any possibility some of its vapour may enter into the bulb B.

It is e\ident that, if this process be continued long enough, the aii- of the bulb B will
be completely deprived of aqueous vapour and of carbonic acid ; also by heating B
occasionally above 300° F., any ozone which it may at first have contained will be
destroyed.

When satisfied that the air of the bulb B has been thoroughly deprived of all these
substances, seal off the appendage tubes, thereby detaching the bulbs b, V ; and then by
means of the screw S drive up the mercury to about the level h. If the branch tubes are
properly shaped, the mercury will now have run down and filled them. We liave thus
procured a quantity of unexceptionable au', which fills the bulb B and that portion of its
attached tube above li.

It is well to remark that the tubes T and T' are supposed to be well cleaned, and the
mercury used to b(! quite pure.

In making an observation, the mercury is driven first by means of the screw S, and
afterwards by the more delicate motion of S' to a fixed level A, which is chosen near

ME. B. STEAYAET OX EXPEEIMENTs WITH AX ATR-THERMOMETEB. 427

the top of the tube T, and the reading of the cathetometer C for this point is noted.
It is clear that the height of the cohunn of mercury in the long tube T' (the stopcock
at D' being open) wall depend ujion the elastic force of the air in the bulb B as com-
pared with that of the atmosphere. If this elasticity be altered by increasing or dimi-
nishing the temperature of B, the height of the mercury in T', as read by the catheto-
meter, will be altered also ; and hence the difference given by the scale of the catheto-
meter between the fixed level h, at which the mercury in T is always set, and the
surface of the coluuiu of mercury in T' will afford an indication of the temperature of
the bulb B*. It is on this principle that the instrument is used as a thermometer.

The mercury used in these experiments was purified in the following manner. It
was first heated for some time with dilute nitric acid and then allowed to remain in con-
tact with strong sulphuric acid, being frequently stirred in both cases. It was afterwards
well washed, first with a little caustic potash and afterwards with pure water, and was
finally mixed with pounded sugar, and well filtered through paper before use.

It was thouglit desirable to ascertain whether different specimens of mercury so
treated were of precisely the same specific gravity ; and Avith this purpose the following
expeiiment was made. The mercury used in the construction of the Kew Standard
Barometer, described by John Welsh in the Philosophical Transactions, 1856, page 507,
was compared with that used in tlie construction of another standard barometer, since
erected at Kew, and also with the mercuiy used in the experiments now described ; and
the following was tlie result ; —

Specific-granty bottle, filled with mercury fi-om the
cistern of old Kew standard weighed at 62° F. 13975'8grs.

The same, filled with mercury from the cistern of the
new Kew standard weighed at 62° F. 13976-1 grs.

The same, filled with mercury used in the experiments
with air-thermometer weighed at 62° F. 13976"4grs.

It ^vill be seen from this how small is the observed difference in specific gravity between
these various specimens of mercury, and that even if this were not due to error of
observation, yet would the difference between the readings of standard barometers con-
structed from these different specimens scarcely exceed one thousandth of an inch.

But while these specimens of mercury are sufficiently pure, if the fluid be used in
measuring pressure, it might still be doubted whether they would all have the same
melting-point. The following experiment will decide this question.

An old Kew standard thermometer (No. 45) was thrust into a beaker which contained
eight or ten pounds of mercury, half frozen, half melted. The mercury was not the same
as that used in the experiments with the air-thermometer.

* It was ascertained that the strength of the bulb B was sufficient to prevent any sensible change of volume
due to increase of pressure within the bulb.

3 m2

428 MR. B. STEWAET ON EXPEEIMEXTS WITH AN AIR-THERMOMETEE.

The reading (observed by Dr. W. A. Miller and myself) was . . — 37"75

In an experiment with the mercury used in the air-tliermometer in a vessel which
contained fifty pounds half frozen, half melted —

The same thermometer read on one occasion — 37"80

The same thermometer read on another occasion — o7"70

The reading of this thermometer at the melting-point of ice

throughout all these observations was 32-45

It may therefore be concluded that the melting-point of well-purified mercuiy for
different specimens of the fluid, and for different masses, is practically a point of con-
stant temperature, and that this temperature, as indicated by a Kew standard thermo-
meter (graduated throughout according to the diameter of the bore), is 70°'2 below the
freezing-point of water, or is equal to — o8°-2 F.

The boiling-point apparatus used in these experiments was that recommended by
Eegxault. It is represented in fig. 2 ; and I need only remark that the steam, after
passing roimd the bulb B, flows down by the channel indicated by arrow-heads, and
finally escapes into the atmosphere by an orifice near the bottom of the apparatus. The
bulb is thus entirely surrounded by steam in a state of motion. This piece of apparatus
was compared with two others of the same description but of very different dimensions
(one being the small apparatus used by travellers) ; and the agreement of the three, as
tested by a thermometer, was very exact. Distilled water was always used during these
experiments. A box with a few small holes bored in its bottom was that used to con-
tam the melting ice, and care was taken that the ice was really in a melting state.

The arrangement for the melting mercury is represented in figs. 3, 4, 5, 6. The
vessel for holduig it consists of two wooden boxes, the one within the other, with a
lining of felt between. When the box was filled with mercury, the bulb, in order to
counteract the upward pressure of the fluid, was bound by a string to the bottom of the
box. An agitator (figs. 4, 6, 6), made of wire gauze, was made to surround the bulb
when the fi-eezing-experiment was in progress, the compartments of which were easily
penetrated by fluid mercury. The solid lumps of mercury were introduced outside of
the wire gauze ; and the agitator served the double purpose of keeping these from con-
tact with the bulb and of promoting currents, by means of which the whole mass was
kept at a uniform temperature. For the success of these experiments I am much
indebted to Mr. Robert Aud.^ms of London ; indeed without his ready cooperation it
would have been impossible for me to freeze mercury in sufficient quantity.

This gentleman took the trouble to bring cylinders containing liquid carbonic acid to
Kew whenever the freezing-experiment was to be performed. I need not here describe
how solid carbonic acid is procured from these cylinders, nor how by mixing this with
ether a very intense cold is produced ; it is suflficient to state that by this process a very
large quantity of mercury may be kept frozen for a length of time with great facility.

ME. B. STEAVAET ON EXPERIMENTS WITH AN AIH-THEKMOMETER. 429

Experiments and lie.-iolta.

The following are the formulae used in the reduction of these experiments : —

Let P denote the elastic force of the air in the bulb B when it is surrounded with
melting ice, and let the atmosphere ai'ound, including the mercury in the two tubes
T, T', be at the temperature 32°+ ^ Also let P' denote the elastic force of the same
air when the bulb is at the temperature 32°+T, the atmosphere being supposed to
remain at 32°+^.

Fui-ther, let V denote the internal volume at 32° of the bulb B, and of that portion
of the capillary tube which is subjected to the heating and cooling agents, and let v
denote the internal volume at 32^ of that portion of the tube T above the mercury which
is not subject to the influence of these agents, but which contains air which may be
supposed to retain the constant temperature 32°+ # throughout the experiment.

Also let k denote the coefficient of expansion for 1° F. of the glass, and let a denote
the corresponding coefficient of increase of elastic force of dry air the volume of which
remains constant ; and, finally, let us denote by unit of mass the air which occupies unit
of volume under unit of pressare at the temperature of 32° F.

Then PV denotes the mass of the enclosed air which exists at the temperature of

\ + kt
32° F. when the bulb is surrounded by melting ice; also Fv yq^^ is that which exists,

at the same time, at the temperature of the atmosphere (32°+^).
Hence the whole mass of air operated upon will be denoted by

p{v+.-'r±^;} (!■)

Now let the bulb be subjected in like manner to an agent of which the temperature is
32° +T. Hence the mass of air existing at temperatm-e 32° +T will be P'V. j-x^-j,, while

that at the temperature of the atmosphere will be P'w.-i-, and the whole mass will be

Since the mass of air remains unchanged, we have (1):=(2), or

Hence, if we wish to determine u, we shall have

l+aT=^P:il±^II-r. (3-)

Here it may be remarked that y . . is a small quantity ; so that we may in it quite

well assume as the value of « that which was previously determined by Kegnault, even
although these experiments should give a slightly different value. Now according to
this observer l+180a=l-3665. Hence a=-002036 nearly.

430 ISTR. B. STEAVAET OX EXPEEIMEXTS WITH AX AIE-THEEMOMETEE.

It must also be noted that in the boiling-water experiment T is detennined in con-
formity with the report presented to the British Government by the Commissioners
appointed to construct standard weights and measures, according to which 212° F. is
taken to represent at London the temperature of steam at the pressure of 29-905 inches
of mercury reduced to 32° F. This is also the value of 212° F., which has been adopted
by the Kew Committee of the British Association *.

When it is the freezing-point of mercury which we wish to determine, the formula (3.)
must be altered as follows : —

l+aT p"

(4.)

Here it is T which we wish to determine, and which will of course appear as a negative
quantity. The value of a, is in this case supposed to have been previously determined.

Ea-periments made in order to determine a,. — First Series.
In the first set of experiments made for this pui-pose a flint-glass bulb was used. Its
volume and coefficient of expansion were determined by cleaning it, fii-st, with nitric acid,
secondly ^\-ith sulphuric acid, afterwards with water, and, lastly, drying it with alcohol,
after which process it admits of being well filled with mercury without any specks. It
was then ascertained what weight of this fluid it held at 32°, and also at 212°.

grs.

The weight of mercury at 32° was ascertained to be 10169*3,
That of mercury at 212° was ascertained to be . . 10011-4,
shoA\ing a loss of 157-9 grs. Had the glass not dilated, the loss of weight would have
been =181-4 grs. (if we suppose the expansion of mercury between these two points
to be =0-018153, which is Regnault's determination). Hence the dilatation of the
glass envelope of the bulb between 32° and 212° w^as -00235, or the coefficient of
expansion of the glass for 1° F. =-0000131=^-.

Also, assuming 252-5 grs. to be the weight of 1 cubic inch of water at 62°, and
having found by experiment 13-584 to be the specific gravity of the mercury used

compared with water at 62°, we obtain ^ .

'■ cub. m.

Capacity of the bulb at 62° = 2-957
Capacity of the bulb at 32° = 2-956 nearly.
The bulb haAing been thus calibrated was sealed on to the capillaiy tube t, and, along
with its appendage bulbs h, b' and tube T, was attached to the reservoir at D in the
manner already described. In this experiment the bulb h contained anhydrous phos-
phoric acid, and h' fused caustic potash ; while these tubes were attached, the bulb B
was heated and cooled very many times. The potash bulb b' was detached in about a
week, but the phosphoric acid bulb was kept attached for at least three weeks.

* lU'port of the Kew Committee of the British Association for 1853-54.

ism. B. STEWART ON EXPERIMENTS WITH AN AIR-THERMOMETER. 431

It has beea already remarked that in these experiments the mercury in the tube T
is always brought to a fixed ])oint, determined by its cathetometer-reading, this being
as near the top as is conveniently possible. Also, when the boiling-point apparatus is
attached, it is arranged so as always to embrace, along with the bulb, the same
portion of the capillary tube. If this position be marked, on one side of the mark we
shall have air of the temperature of steam, and on the other side air of nearly the same
temperature as the atmosphere. In order to estimate the volume of air existing in the
tube at the atmospheric temperature when the merciuy has been set to its fixed point
by means of the cathctometer, shut the stopcock at D', and estimate, by means of the
graduated head, the number of revolutions and paits of a revolution of the fine screw
S' requisite to bring the mercury to that point in the capillary tube t which has been
marked as that where the temperature of the boiling-water apparatus commences.

It may perhaps be objected to this method of measui'ing — , that should there be a

small bubble of air lui'king in the reservoir" R, or at the points of the appendage tubes,
after the bulbs b, V have been detached, this air will contract under the additional
pressure caused by raising the mercury in T, and will consequently make v appear to
be greater than it really is. It has, however, been ascertained, by means of pushing
the mercury to a fixed point of the capillary tube t., Avith the stopcock at D' shut, and
then increasing the pressure by heating the bulb, that the error arising from this soiu-ce
is inappreciable.

It has also been ascertained, by means of inserting a small thermometer, that the
temperature of the air in the tube immediately above the mercury at li remains nearly
the same as that of the atmosphere without, even when the boiling-water apparatus is
in operation.

It requires two observers to work the instrument. When the ice or boUing-water
appai"atus has been sufficiently long attached to make the observations constant, one
obseiTer is stationed at the cathetometer, which is set to the fixed point lu while another,
by means of the fine screw S', pushes the mercury (always a very little up) to the j)roper
height for the cathetometer-setting. By making the mercury always rise, a uniform
capillary action is secured. The cathetometer-observer then records the height of the
mercury in T'. Suppose this to be higher than A, the difierence between the level of
the mercury in the two tubes, added to the barometric pressure, will give us the pressui'e
of the air in the bulb B.

Mr. George Whipple, meteorological assistant at Kew, an exceedingly accurate and
delicate observer, took most of the cathetometer- and barometer-readings in these expe-
riments. It has been found that for a length of 30 inches the cathetometer-measure-
ment requires a correction of + '003 inch, or for a difierence of 11 inches +'001 nearly.
This has been attended to in reducing the experiments. The bore of the tube T' is
about -25 inch, that of the tube T at A is generally smaller ; the capillary con-ection is
ascertained after the bulb is detached, and the same atmospheric pressure acts on the

4,32

IME. B. STEWART ON EXPEEIINIEXTS WITH AX AIR-THERMOMETEE.

siuiace of mercury in both tubes, bj' setting the mercury in T to the height /^ and then
reading by means of the cathetometer the height of the fluid in T. In the first set of
experiments the capillary correction has been found to be iiisensible. The follomng
Table exhibits the results of this series of experiments.

Table I. — Results from first Bulb.

Date.

Xumber of readings
taken

Elasticity of air in inches
of mercury, liaving

Values of

Resulting
value of
1 + 180 a.

1862.

At 32=.

At21-.

P. 1 P'- temp.

X-.

T. L^ld+AV).

\+af.

Sept. 9.

Oct. 22.

29.

8
6

7

4
6
6

30-811 41-965 65
30-791 41-868 56
30-777 41-903 52

-0000131
-0000131
•0000131

180-25 -00555 i
179-10 -005646 i
180-04 -005646

1-0672
1-0489 '
1-0407

1-36729
1-36741
1-36731

Mean value of 1 + 180

« ;

1-36733

After the experiments recorded in Table I. were made, the bulb was agam carefully
examined, and found to be free from mercury, and then a very small portion of this fluid
was pushed into the bulb, and the experiments recorded in Table II. were made in
order to ascertam the influence of mercurial vapour upon the result obtained.

Table II.—

Results

from first Bulb with

a little mercury

in it.

Date.

Number of readings
taken

Elasticity of air in inches
of mercury, having

Values of

Resulting
value of
1+180«.

1862.

At 32°. 1 At 212°.

P.

the
P'- temp. ^'■

T. \^{l + i-t).

i+at.

Nov. 21.

Dec. 2,

8.

13,

6 6
6 6
8 6
6 6

30-757
30-768
30-769
30-768

' „ 1
41-914 44-0p0000131
41-857 45-0 -0000131
41-923 50-0' -0000131
41-901 47-0 -0000131

i
180-45 -00555
179-39 -00555
180-25 1 -00575
180-10 -00565

1-0244
1-0265
1-0366
1-0305

1-36774
1-36749
1-36786
1-36755

IV

lean value of l + 180a

1-36766

Hence we see that, by forcing a little mercury into the bulb, the value of l + 180a is
apparently increased by the amount '00033.

Second Series of Experiments.

On December 22, 1862, the bulb used in the first series of experiments was detached,
and another, containing a little anhydrous barytes, was put in its place. In a couple of
days the air in this bulb seemed to have become sufficiently dried ; while there could be
no suspicion of vapour of mercury having in this time distilled over into the bulb through
a capillary tube of the length of 9 inches, and while the bulb remained at the same tem-
pcratui-e as the other parts of the apparatus.

The bulb also was not calibrated before use, nor did any mercury come in contact

MR. B. STEWAET OX EXPERIMENTS WITH AX AIR-THERMOMETER.

433

with it until after the experiments about to be described. It was finally calibrated in
the usual way. The tubes T, T' were the same as those used in the last experiment,
and the capillai-y correction was inappreciable. The result obtained by this bulb is
recorded in the following Table : —

Table III. — Results from second Bulb.

Date.

Number of readings
taken

Elasticity of air in inches
of mcreurv, having

Values of

Besulting
Tahie of
1 + 180«.

1862.

At 32°.

At 212=".

P.

1 the
P'. jtcmp.

/[-.

T.

Y(l+it)-

1+.'.

Dec. 24.
27.
29.

6
4
4

4
4
4

28-454
28-457
28-460

38-747 44
38-766 46
38-684 48

•0000130
•0000130
-0000130

180-60
180-84
179-28

•00753
-00753
-00753

1-0244
1-0285
1-0326

1-36730
1-36737
1-36745

Mean value of l + lSOa

1-36737

After the experiments recorded ui Table III. a little mercury was pushed into the
bulb ; and the results obtained are recorded in the following Table : —

Table IV. — Results obtained from second Bulb with a little mercury in it.

Date.

Number of readings
taken

Elasticity of air in inches .^ , .
of mercury, haring ^^"^^^ °'

EesiUting
value of
I + 1S0«.

1863.

At 32'.

At 21-2°.

P.

P'.

the 1
temp. i:

T.

|(l+i-0-

l+.t.

Jan. 12.
14.

6
4

6
4

28-444
28-450

38-729
38-749

41 -0000130 180-40
44 -0000130 180-60

•00753
-00753

1-0183
1-0244

1-36762
1-36762

Mean value of 14-18

Oa.

1-36762

It appeal's from a comparison of Tables III. and IV., that tlie value of l + 180a is
apparently increased -00025 by forcing in the mercury. For the first bulb this increase
was -00033 ; and the difierence between these numbers is probably owning to errors of
observation : but the agreement is sufiiciently close to show that the first bulb cannot
have at first contained any vapour of mercury ; for, if it had, the difference caused by
forcing in mercury, instead of being gi-eater than that for the second bulb, should have
been much less.

Third Series of Experiments.

The bai-ytes bulb was now removed, and another bulb of fiint glass, along with its
own drjing-arrangement and tube T, was attached to the apparatus. The bulb was
only calibrated after use, and had not come in contact with mercury until the experi-
ments were finished.

Sulphuric acid was used instead of anhydrous phosphoric acid in the bulb b, and
potash, as before, in h'.

Unfortunately the capillary correction due to the setting at h was not determined in

MDCCCLXIII. 3 N

434 ]ME. B. STEWAET OX EXPEKIMENTS WITH AJV AIE-THEEMOTIETEE.

an unexceptionable manner ; but as the third series of experiments is precisely similar

( as regards the value of — j to the fourth, to be hereafter described, we shall apply

to the former the capillary correction for the fourth series as probably near the truth.
Accordingly, for equal pressui'es on both tubes, we shall suppose that T would have
read -018 in. lower than T.

Table V. — Residts from third Bulb.

_ ^ ' Number of readings
Date. taken ^

Elasticity of air in inches
of mercury, iiariiig

Values of

Eesulting 1
value of
1+180«.

1-36729
1-36719
1-36732

18t;3. At 32'. At 21-2'.

P.

1 the
!"■ temp.

I:

T. l^(l+.0. H<.^.

Marcii 3. 6 1 6
16. s 6
31. 14 1 6

31-887
31-866
31-879

43-424 i 56
43-413 47
43-465 54

•0000141 179-75 ; -00323
•0000141 180-09 -00323
-0000141 180-55 -00323

1^0489
1-0305
1-0448

Mpan vali.P nf 1-1- 180a

1-36727

Fourth Scries of Experiments.
The third bulb was removed, and another bulb of crown glass put in its place. This
bulb had not come in contact with mercury before the experiments were made ; it was
afterwards calibrated in the usual manner. The capillary correction has been already
sriven. The results with this bulb are embodied in the following Table: —

Table VI. — Results from fourth Bulb.

Date.

Number of readings
taken

Elasticity of air in incites
of mercury, having

Values of

Resulting
value of
1 + 180».

1S63.

At 32'.

At 212'.

P.

P'.

tlie
temp.

k.

T.

^(1+A-o.

\+,tt.

June 1.
2.
5.

6.
8.

6

8

8

8
("20
1 doub

6

8

6
6

12 -1
le set. J

30-681
30-687
30-681
30-685
30-679

41-817
41^804
41-792
41-744
41-757

62
66
64
64
63

•0000141
•0000141
-0000141
-0000141
-0000141

180-42
180-18
180-01
179-18
179-37

-00335
-00335
-00335
•00335
•00335

1-0611
1-0692
1-0651
1-0651
1-0631

1-36713
1-36691
1-36715
1-36705
1-36736

Mpan valiifi of 1 + 18

)a

1-36716

We have thus, by moans of these four series of experiments, four mean values of
1 -|- 1 80a, as under : —

First bulb, flint glass, dried by anhydrous phosphoric acid, gives . 1-3G733

Second bulb, flint glass, dried by anhydrous barytes in bulb, gives 1-36737

Third bulb, flint glass, dried by sulphuric acid, gives . . . . 1-36727

Fourlli bulb, crown glass, dried by sulphuric acid, gives .... 1-36710

Mean value of 1 + 1 80a . . 1-36728

ME. B. STEWAHT ON EXPEEIMENTS WITH AN AIR-THEHMOJIETEE. 435

This, therefore, is to be regarded as the coefficient obtained by these experiments. It
differs sliglitly from that found by Regnault, who makes it to be 1-3665 ; and although
the difference is not great, I should have preferred to have agreed stiU more closely with
this eminent authority, but I am unable to tliink of any source of error in these experi-
ments.

Two sets of experiments were made in order to determine the freezing-point of mer-
cury. In these, in order to ensiu'e the greatest possible amount of precision, it was
estimated that the air in a portion of the capillary tube near the mercury had a tem-
perature lower than that of the atmosphere. The first set of experiments were made
■with the first bulb, and the second set with the thiid bulb.

Here the formula to be employed is

1-f-aT f;

and the freezing-point of water now becomes the higher temperature,
these experiments is embodied in the following Table : —

The result of

Table VII. — Experiments made in order to determine the Temperature of
Melting Mercury.

Date.

Number of readings
taken

Elasticity of air in inches
of mercury, having

Values of

Resulting

value of

T.

At 32°.

At melting-
point of
mercury.

P.

P".

the
temp.

.*.

X-.

|(i+*0.

(l+at).

1802.
Oct. 22.

1863.
Mar. 13.

6
14

12
15

30-791
31-879

26'446
27-368

6^
54

•0020404
•0020404

-0000131
-0000141

•00652 1-0489
•00329 j 1-0448

-69-91
-69-95

Mean value of T

-69-93

Hence the value on Fahrenheit's scale of the melting-point of mercury, as deter-
mined by these experiments, is — 37°-93 F., while on a standard Kew mercurial ther-
mometer the reading was — 38°"2 F. It may perhaps be gathered from this that mer-
cury, before beginning to freeze, slightly increases the rate at which it contracts ; but
further experiments must be made with other mercurial thermometers before this can
be accurately determined.

* As determined by the foregoing experiments.

[ 437 ]

XX. On some Compounds and Derivatives of Glyoxylic Acid.
By Henky Debus, Ph.D., F.R.S.

Received December 31, 1862,— Head February 12, 1863.

OKGAJfic substances of simple composition, like marsh-gas, ethylene, alcohol, and acetic
acid, are deserving of most careful study, not merely on account of their being repre-
sentative members of numerous and important classes of bodies, but also because they
form connecting links between the compounds of inorganic chemistry and the more
complicated forms of organic natui'e.

Glyoxylic acid belongs to this class of bodies, because it bears the same relation to
oxalic acid that sulphurous acid does to sidphuric acid, and because it stands to glycolic
acid as common aldehyde, C2 H4 O, does to alcohol, C2 Hg O. These relations suggested
the experiments which will be described in the following pages.

ComMnations of Glyoxylates atid Sulphites,
a. Glyoxylic Acid and Bisulphite of Soda.

K a concentrated solution of bisulphite of soda be mixed with one-fourth of its volume
of nearly anhydi'ous glyoxylic acid, a white crystalline precipitate vflU separate from the
mixture in the course of a day or two. This precipitate is to be collected on a filter,
washed with cold water, and recrystallized from its solution in the smallest possible
quantity of hot water.

The substance thus prepared presents itself in small colourless crystals, which dissolve
easily in water, and evolve sulphiu'ous acid yni\\ sulphui-ic acid. The aqueous solution
is not altered by the addition of potash or ammonia, but jdelds a copious white precipi-
tate with acetate of lead. The solid substance, heated on a piece of platinum foil, burns
without any unusual appearance. Analysis furnished the following results : —

I. 0*361 grm., heated with concentrated sulphuric acid, gave 0-254 grm. of sulphate
of soda. Another quantity of sulphuric acid, of the same quality as that used in this
experiment, was evaporated in a platinum crucible ; the weight of the latter was found
to be unchanged.

II. 0'4365 grm., dissolved in hydrochloric acid and evaporated, left, after ignition,
0"2515 grm. of chloride of sodium. This residue furnished, after treatment ^ith sul-
phuric acid, 0*306 grm. of sulphate of soda.

0'573 grm., oxidized Avith a mixtui'e of chlorate of potash and hydrochloric acid, and
the sulphuric acid precipitated with chloride of barium, gave 0"C9 grm. of sulphate of
baryta,

MDCCCLXIII. 3 0

438 DE. DEBUS OX SO^IE CO^MPOUXDS AND DEEIVATIVES OF GLTOXTLIC ACID.

0-565 grin, of another preparation, burnt mth chromate of lead, gave 0-248 grm. of
carbonic acid and 0-051 grm. of water.

The substance therefore contains in 100 parts —

Carbon .

Hycb-ogen

Sodium

Oxygen

Sulphur

22-8

II.

11-97

1-00

22-67

16-55

2

24

12

2

2

1

2

46

23

6

96

48

1

32
200

16
100

The formula C.H Na O^+S H Na O3 requires—
Carbon .
Hydrogen
Sodium .
Oxygen .
Sulphur

The substance is consequently a compound of glyoxylate of soda with bisulphite of
soda, and the above numbers confirm the formula C2H2O3.

b. Glyoxylate of Lime with Bisulphite of Lime.

Glyoxylate of lime rapidly dissolves in a satui-ated solution of sulphm-ous acid. If
this solution be concentrated on the water-bath and afterwards allowed to stand in the
exsiccator, a fine crop of colourless crystals separates. These crystals have to be puri-
fied by recrystaUization from water, and dried over sulphruric acid in vacuo.

This substance, like the soda compoimd, easily dissolves in water, and its concentrated
solution comports itself with reagents as follows: — Sulphuric acid produces a white
precipitate of sulphate of lime and liberates sulphui-ous acid ; lime-water, chloride of
barium, and ammonia respectively cause the formation of a white precipitate, whilst
carbonate of lime has no decomposing influence. The form of the crystals could not be
determined.

Analysis gave the following results : —

0-351 grm. fui-nished 0-145 grm. carbonate of lime.

0-56 grm. of another preparation, oxidized with a mixture of chlorate of potash and
hydrochloric acid, and the sulphuric acid precipitated by chloride of barium, gave
0-555 grm. of sulphate of baryta.

The substance therefore contains in 100 parts —

Calcium 16-52

Sulphur 13-62

These numbers correspond to a double salt of bisulphite and glyoxylate of lime.

DR. DEBUS ON SOME COMPOUNDS AOT) DEEIVATIVES OF GLTOXTLIC ACID. 439

The formula 2(C,, H Ca Oj+S H Ca OJ-I-SH, O roquiirs—

Caldiim lG-73

Sulphur 13-38

Considering the mode of preparation, the mother-liquor of the salt ought to contain
glyoxylic acid. In order to ascertain its presence hy experiment, the mother-liquor was
evaporated to dryness on the water-bath and the residue exhausted with alcohol. The
latter had dissolved an acid which yielded, when treated with carbonate of lime, a salt
that contained no sidphur and would not crystalhze, but which in other respects com-
ported itself like glyoxylate of lime.

If a compound be the derivative of a substance of an easily changeable natiu-e, its
composition, as represented by its rational formula, should always be checked by special
experiments. It therefore appeared desirable to regenerate the glyoxylate of lime assumed
to form a constituent part of the salt in question. For this purpose an aqueous solution
of the latter was mixed with sufficient oxalic acid to precipitate all the lime ; the clear
filtrate, in order to expel the sulphui'ous acid, was evaporated on the water-bath till it
assumed the consistency of a thin syrup ; this residue, which proved to be glyoxylic
acid, was dissolved in water and converted, by treatment ^vith chalk, into a crystalline
salt that both in form and other properties agreed with glyoxylate of lime.

Glyoxylic acid is a strong acid ; it dissolves zinc and expels carbonic acid from car-
bonates. The experiments described hereafter show that it shares some distinguishing
properties with the aldehydes. It may therefore be assumed that, if the acid properties
of glyoxylic acid were less marked, it would combine with the bisulphites like hydride
of salicyl.

Compound of Glyoxylate and Lactate of Lime.

Like as glyoxylic acid is formed from ethylic alcohol, so will the homologous acid

Cj H4 O3 probably be fonned from the alcohol C^ Hg O. But since propylic alcoliol is

only prociu'able with great difficulty, I attempted the preparation of C^ H, O, from

lactic acid.

C3H6O3 - H, = C.5H4O3.

Lactic acid. Pyroraeemic acid.

Two experiments made with niti'ic acid of 1"2 sp. gr. and lactic acid did not give the
desu-ed result. At a low temperature no action appeared to take place, and at a higher
temperature, or with more concentrated nitric acid, only the formation of oxalic acid
could be expected.

It is well kno^-n that platmum when alloyed with silver is dissolved by nitric acid.
It therefore appeared not unlikely that lactic acid, when mixed with alcohol, would be
oxidized at a moderate temperature by nitric acid. The experiment was made with a
mixtui'e of equal weights of lactic acid and alcohol and a suitable quantity of nitric acid,
on the plan which I employed for the preparation of glyoxylic acid. An acid was thus
obtained which, when treated with lime, fui-nished neither glyoxylate nor lactate of

3o2

440 DE. DEBUS OX SOME CO^NrPOUMJS AJSTD DEEIVATmiS OF GLTOXTLIC ACID.

lime. The salt obtained was white and crystalline, required more boiling water for its
solution than either C^ H Ca O3 or C3 H^ Ca O3, and, when the hot solution cooled,
separated therefrom in white crystalline crusts.
Analysis gave the following results : —
0'241 grm. furnished 0-11 grm. of carbonate of lime.

0-425 grm., biuut vi-ith chromate of lead, gave 0-423 grm. of carbonic acid and
0-144 grm. of water.

The substance therefore contains in 100 parts —

Carbon 27-14

Hydrogen . . . . 3-76
Calcium .... 18-25
Oxygen ....

The formula C5 Hg Ca2 O- requires —

Carbon. ... 5 60 27-27

Hydi-ogen ... 8 8 3-63

Calcium ... 2 40 18-18

Oxygen ... 7 112 50-92

220 100-00

According to its formula this siibstance might be composed as follows : —
Glyoxylate of lime . . . C^ H Ca O3
Lactate of lime . . . . C3 H5 Ca O3

Water H2 O

CsH^Ca^O;

The following experiments confirm this composition : — The solution of the compound
C5 Hg Ca2 O7 yields vritli lime-water a white precipitate, which immediately after its
formation is found to be soluble in acetic acid ; this precipitate, however, after the lapse
of some time, or by exposui-e to a temperatiu-e of 212°, becomes insoluble in this acid.
The same property distinguishes the soluble glyoxylates.

2(C2HCa03) + CaHO = C.^ Ca, O^ + C2 H3 Ca O3.
Glyoxylate of lime. Oxalate of lime. Glycolato of lime.

If the formula C2 H Ca O3+C3 H5 Ca O3-I-H2 O be the correct expression of the compo-
sition of the salt in question, then a quantity represented by the above formula ought to
be decomposed by boiling lime-water into one atom of lactate, half an atom of glycolate,
and half an atom of oxalate of lime ; the quantity of the latter may be easily determined.
0-76 grm. of the substance was dissolved in water and boiled with an excess of clear
lime-water until the decomposition was complete; in order to prevent carbonate of lime
from mixing with the precipitate, a slight excess of acetic acid was added ; the preci-
pitated oxalate of lime was collected on a filter and converted in the usual manner into

DE. DEBUS OX SOME COMPOUNDS AKD DERR'ATITES OF GLTOXTLIC ACID. 441

carbonate of lime; the weight of the latter was found to be 0108 <i;i-m., which corre-
sponds to 28-3 per cent, of oxalate of lime. The formula C5 Hg Ca^ O- requires 29-0 per
cent, of oxalate. In order to confirm the conclusion to be drawn from this experiment,
the substance C5 Hg Ca,2 0; was prepared from lactate and glyoxylate of lime. For this
purpose equivalent quantities of C, H5 Ca O3 and C, H Ca O3 were respectively dissolved
in the least possible quantity of boiling water, and the solutions were then mixed.
From the cooling liquid a large quantity of crystals separated, which were identical in
properties with the body C5 Hg Ca^ O7. Two detenninations of the solubility in water
of I., the substance prepared as just described, and of II., the body obtained by the oxida-
tion of a mixtui'e of lactic acid and alcohol by means of nitric acid, placed the identity
of the two substances beyond a doubt.

I. 5-003 grms. of the solution, saturated at 18° C, left after evaporation at 100° C.
0'0265 grm. of residue.

II. 3151 gi-ms. of the solution, treated in the same manner as I., left 0-017 grm. of
residue. Therefore at 18° C.

One part of I. dissolves in 187 parts of water.
One part of II. dissolves in 184 parts of water.
The formula C5 Hg Ca^ 07=(C2 H Ca O3+C3 H5 Ca O3+H2 O) may therefore be con-
sidered the true expression of the composition of the compound. The last member of
this formula may be eliminated by heat ; it appears, however, that the remainder,
C2 H Ca O3+C3 H5 Ca O3, suffers some change by this process.

The tendency to form this double salt is perhaps the reason why the oxidating
influence of the nitric acid only affects the alcohol and does not extend to the lactic acid.

Ammoniacal Compounds of some Glyoxylates.
a. Glyoxylate of Lime and Ammonia.

Pui-e and well-cr)stallized glyoxylate of ammonia was dissolved in the least possible
quantity of hot water, and the solution thus obtained was divided into two equal parts.
One part, on being mixed with chloride of calcium, assumed the appearance of a transpa-
rent jelly, which resembled silicic acid when precipitated by hydrochloric acid from a con-
centrated solution of silicate of potash ; the vessel, in fact, could be inverted without any
portion of its contents being lost ; after the lapse of a few houi's white opaque points
were obseiTed in the jelly, which gradually increased in number and magnitude, until
at last the whole of the jelly-like substance was converted into a fine crop of prismatic
crystals ; these were found to agree in form and other properties with glyoxylate of lime.

To the other part of the solution of glyoxylate of ammonia a mixtiu-e of chloride of
calcium and acetate of ammonia was added. In this instance no immediate change
took place, but in the course of twenty-four hours a white precipitate, a compound of
glyoxylate of lime and ammonia, made its appearance ; the precipitate was collected on
a filter, washed with cold water, and dried over sulphuric acid.

442 DR. DEBUS OX SOME COMPOrXDS AXD DERIVATIVES OF GLTOXTLIC ACID.

A similar result was obtained by precipitating a solution of glyoxylate of ammonia
■with acetate of lime, just sufficient ammonia being at the same time added to keep the
liquid neutral ; the compound thus prepared is colom-less or slightly yellow, dissolves
spaiiugly in water, but is easily soluble in dilute acetic acid ; it becomes highly elec-
trical when rubbed in a mortar, and bums like tinder on a piece of heated platinum foil,
leaving carbonate of lime. The aqueous solution possesses an alkaline reaction, and
shows no change on the addition of lime-water.

The follo\ving experimental results determined its composition : —

0-139 grm. furnished 0-061 grm. of carbonate of lime.

0-236 grm., burnt v^-ith soda-lime, gave 0-293 grm. of ammonio-chloride of platinum.
0-26 grm., burnt with chromate of lead, gave 0-206 grm. of carbonic acid and 0-088 gi-m.
of water.

The substance therefore contams in 100 parts —

Carbon 21-6

Hydrogen .... 3-76

Calcium 17-7

Nitrogen 7-8

Oxygen

According to the formida 3(C.^H CaO.,), 2NH3, H2O, we obtain

Carbon
Hydrogen
Calcium .
Nitrogen .
Oxygen .

6 72 21-75

11 11 3-32

3 60 18-12

2 28 8-45

10 160

331

The calculated numbers do not agree very well with those found by experiment ; the
differences, however, are not greater than those which might be expected from the
circumstance of the analysed substance having been obtauaed in the form of a preci-
pitate.

If, therefore, glyoxylate of ammonia be decomposed by acetate of lime, acetate of
ammonia and glyoxylate of lime are formed ; the latter, however, withdraws a part of
the ammonia fi-om the acetate of ammonia, and thus produces the compound which has
just been mentioned. Some grammes of this substance were mixed with oxalic acid
sf)lution sufficient to convert two-thirds of its lime into oxalate. If the composition
ascribed to the compound be con-ect, the filtrate from the oxalate of lime ought to
contain glyoxylates of lime and ammonia ; the filtered liquid was evaporated over
sulphuric acid in vacuo, when a crust, consisting of small prisms arranged round a
common centre, remained. This substance was repeatedly recrystalliz(>d from water ;
and in each process the ci-ystals which form first, were separated from those which made

DE. DEBUS ON SOME COMPOUNDS AJW DEEIVATIVES OF GLTOXTLIC ACID. 443

their appearance after a half or three-quarters of the solution had been evaporated. In
this manner two kinds of crystals were obtained : those which sepai-ated first, were found
to be identical with glyoxylate of lime ; and those which formed last, possessed the
properties of glyoxylate of ammonia. This experiment, therefore, confirms the for-
mula 3(C2H CaOj), 2NH3-}-H20, which was deduced from analytical results.

A compound of a similar natiue may be directly obtained from glyoxylate of lime
and ammonia. For this purpose a hot and concentrated solution of C2H CaOa+HgO
is to be mixed with a few di-ops of ammonia, and filtered ; ammonia is then to be added
to the filti-ate so long as a precipitate forms ; the latter is to be collected and washed
with cold water. The compoimd, thus prepared, possesses the same properties as the
one obtained from glyoxylate of ammonia and acetate of lime.

The analytical results were as follow : —

0*161 grm. gave 0*079 grm. of carbonate of lime.

0*227 grm., burnt A^'ith soda-lime, gave 0*316 grm. of ammonio-chloi-ide of platinum.
0*488 grm., burnt with chromate of lead, furnished 0*402 grm. of carbonic acid and
0*126 grm. of water.

Therefore 100 parts of the substance contain —

Carbon . . * • * 22*47

Hydi-ogen .... 2*86

Calcium .... 19*62

Nitrogen . . . . 8*74

Oxygen

and the formula 3(C2 H Ca O3), 2N H3 requires —

Carbon ... 6 72

23*00

Hydrogen ... 9 9

2*87

Calcium ... 3 60

19*16

Nitrogen ... 2 28

8*94

Oxvgen ... 9 144

This compound, therefore, differs from the preceding one by containing one atom of
water less.

If a quantity of this substance be treated with oxalic acid solution, sufficient to
convert its lime into oxalate, the supernatant liquid comports itself like a solution of
glyoxylic acid and glyoxylate of ammonia.

b. Glyoxylate of Silver and Ammonia.

A concentrated solution of glyoxylate of ammonia yields with nitrate of silver a
crystalline precipitate of glyoxylate of silver. But if the solution of glyoxylate of
ammonia contain other ammoniacal compounds, such as nitrate of ammonia, the preci-
pitate is found to contain ammonia besides the other components abeady mentioned.

444 DE. DEBUS ON SOZME CO:\IPOUXDS AM) DEEIYATITES OF GLTOXTLIC ACED.

A specimen prepared iinder the last-mentioned circumstances and afterwards dried
in vacuo, was found to contain in 100 parts —

Nitrogen 5-8

SUver 59-01

These numbers were calculated from the following determinations : —
0*442 gi'm., burnt ■v^'ith soda-lime, gave 0-408 grm. of ammonio-chloride of platmum.
0-238 grm., after treatment with hydi-ochloric acid, fiu-nished 0-172 grm. of chloride
of silver and 0-011 grm. of metallic silver.

The formula 4(C2 H Ag O^), 3X H^— SHg O requii-es—

Nitrogen .... 5-82
SHver 59-9

The absence of nitric acid in this salt was proved by a special experiment -nith proto-
sulphate of iron and sulphuric acid. The silver could not be determined by simple
ignition, because heat decomposes the compound with explosive violence.

c. Glyoxylate of Lead and Ammonia.

Acetate of lead produced, in a liquid containing glyoxylate and acetate of ammonia,
a hea\'y white precipitate.

0-409 grm. of this precipitate, dried over sulphuric acid, furnished 0-243 gi-m. of
metallic lead.

0-589 gi-m., biu-nt with soda-lime, gave 0-209 grm. of ammonio-chloride of platmum.

The compound contains, therefore, in 100 parts —

Nitrogen .... 2-22
Lead 59-4

The formula 7(0, H Pb OJ, 2N TL^-IIL.^O requires—

Nitrogen .... 2-27
Lead 58-7

This substance, on being pounded in a mortar, became highly electrical.

A solution of crystallized glyoxylate of ammonia in ammonia, when raised to tempera-
tui-es below 212 , turns brown, and forms derivatives of an acid character; neither these
substances, however, nor their salts could be obtauied in crystals.

The compounds of ammonia -with the glyoxylates are easily decomposed by heat, and
by nitric acid and other reagents ; the products of decomposition could not be exa-
mined, because their- physical proi)erties precluded their preparation in a piu-e state.
An experiment, which showed the attraction between glyoxylates and ammonia, may be
mentioned here. If a quantity of 3(C2HCa03), 2NH3 be boiled with caustic potash,
a part of the ammonia is very slowly expelled ; if the liquid be evaporated to dryness,
and the residue be raised to a higher temperature, it assumes a beautiful purple colour,

DE. DEBUS ON SOME COMPOUNDS ^VND DERIYATn-ES OF GLYOXTLIC ACID. 445

and at the same time emits streams of ammonia. Tliis red substance is very changeable,
and is formed, even under the most favourable circumstances, in but very small quan-
tities.

Action of Ilijdriodic Acid on Glyoxylatcs.

Ilydriodic acid and glyoxylate of lime were heated together for se^•eral days in sealed
glass tubes, the temperature varying from 100° C. to 110° C. In order to decompose a
part or the whole of the hydriodic acid, the contents of tlie tubes were exposed for
some time to the inlluence of the atmosphere, and finally saturated with carbonate of
lime. The whole was then boiled and filtered, and the filtrate mixed with alcohol. A
precipitate was formed which proved to be a quantity of glyoxylate of lime, little infe-
rior to that which was originally taken for the experiment.

Action of Suli)hurctted Hydrogen on Glyoxylatcs.

Through a concentrated solution of glyoxylic acid a current of sulphuretted hydro-
gen was passed untU. the liquid appeared to be completely saturated; no perceptible
action took place, and even after twenty-four houi-s' contact the liquid seemed to be
unchanged. The solution was now evaporated, at first over pieces of hydrate of potash,
and afterwards over sulphuric acid in vacuo. As soon as most of the water was gone,
small needles, radiating from a common centre, began to form, and at last the whole of
the contents of the evaporating-basin appeared one mass of crystals, which were found
to be soaked with a syrupy mother-liquor, and to be so easily soluble in the ordinary
means of solution, that all attempts at further purification were abandoned. A solution
of glyoxylate of lime comports itself like'glyoxylic acid when acted upon by sulphuretted
hydrogen ; it apparently remains unchanged ; but on allowing a part of it to evaporate
at ordinary temperatures in vacuo, a brittle, transparent and amorphous compound is
obtained.

In order to avoid the inconvenience of evaporation in vacuo, the rest of the glyoxylate-
of-lime solution, after having been treated with sulphuretted hydrogen, was mixed with
a little more than its bulk of alcohol. Nearly the whole of the new compound sepa-
rated as a precipitate, which was collected on a filter and washed with spirit of wine.

Thus prepared, the substance easily dissolved in water, forming a solution of a pale
pink colour. The liquid, after the evaporation of the water m vacuo, left a transparent
amorphous and nearly colourless substance. This compound is the lime-salt of an acid
which bears a similar relation to glyoxylic acid that thiacctic acid does to acetic acid.
In the state of powder it exhibits a striking property wlien brought in contact with
water ; it becomes as viscous as glass when rendered red-hot by means of a Bunsen's
burner, and may be drawn out in long threads. By degrees the water dissolves the
\iscous mass, and forms a solution which shows the following properties with reagents :
— A white precipitate is formed by the addition of acetate of zinc or corrosive sublimate ;
acetate of lead or nitrate of silver throws down a yellow precipitate ; and sulphate

MDCCCLXIII. 3 r

•i-46 DE. DEBrS OX SO:\IE CO:MPOrXDS AXD DEEIYATmiS or GLYOXTLIC ACID.

of copper causes the immediate separation of a black substance, which is probably
siilphui-et of copper. Both the silver and the lead precipitates turn black, the silver-
after the lapse of some time at ordinary temperatures, and the lead at once on exposure
to a temperature of 100° C. Hyckochloric acid produces no perceptible change in the
aqueous solution of the compoimd ; ammonia causes the formation of a white precipi-
tate ; and lime-water the same result as it does with glyoxylate of lime. The brown
colour of iodine immediately disappears, as it does in solutions of other sulphur com-
pounds, such as xanthate of potash. Sesquichloride of iron acts like iodine. The solu-
tion of this sulphur compound, when boiled, decomposes ; it turns yellow ; a crystal-
line powder of oxalate of lime separates ; and a lime-salt, which could not be obtained
in crystals, remains in solution. The compound burns on a hot piece of platinum foil
like tinder, and evolves, when heated in a glass tube which is sealed at one end, an odour
like that of mercaptan.

In order to obtaui some guarantee for the homogeneous nati;re of the substance, a
powdered quantity of it was well mixed and digested for a long time with such a quan-
tity of very dilute spirit of wine as was required to dissolve about half of it. The
undissolved portion will be called S, the dissolved part S'. S' was obtained by evapo-
rating the solution wherein it was contained ; both S and S' were prepared for analysis
bv being dried over sulphuric acid in vacuo.

The analysis of S gave the foUomng results : —

0-731 grm., burnt with chromate of lead, gave 0-508 grm. of carbonic acid and
0-186 grm. of water.

0-42o grm., boiled with nitric acid and precipitated with chloride of barium, gave
0-39 grm. of sulphate of baryta.

The analysis of S' gave the following results : —

0-274 gi-m., burnt Avith chromate of lead, jielded 0-192 grm. of carbonic acid and
0-07-1 grm. of water. ■

The results of the analysis of S+S' were as follows : — ^

0-557 grm. from another preparation furnished 0-381 grm. of carbonic acid.

0-4 grm., oxidized with chlorate of potash and hydrochloric acid, gave 0-357 grm. of
sulphate of baiyta.

0-49 grm., precipitated with oxalate of ammonia, gave 0-192 gi-m. of carbonate
of lime.

According to these determinations, 100 parts contain —

Carbon .

Hydrogen .

Calcium . . 15-67

Oxvgcn . . .

s.

S'.

S + S'.

18-95

19-11

18-65

2-82

.3-00

Sulphur . . 12-67 12-26

DB. DEBrS OX S03IE COMPOUNDS AjS'D DERIVATm:S OF GLTOXYLIC ACID. 447

The formula Cj HoCa._,^4^3jx^O requires—
o J

Carbon . .

. 4

48

18-75

Hydrogen

. 8

8

3-12

Calcium

2

40

15-62

Oxygen . .

. 8 •

128

Sulphur

. 1

32

12-5

The homogeneit)- of the substance may be considered as proved by the identity in com-
position of S and S' ; and its formation would be represented by the folloA^ang equation: —

2(C2 H Ca O3+H2 0)+Ho S=C, H., Ca.^ ^^l + SHg O.
Glyoxylatc of lime.
The crystalline dci-ivative from sulphuretted hydrogen and glyoxjlic acid just men-
tioned is probably represented by the formula C^ H, '"

S
Neither the salts prepared fi-om this acid, nor the compounds obtained by exchanging

the calcium in C; H, Ca., ^^l+oHjO for other metals could be obtained in crystals;
and therefore I did not pursue the investigation of these bodies.

Action of Zinc on Glyoxylic Acid.

A concentrated solution of this acid dissolves pure zinc without evolution of hydi'ogen
gas, and the liquid at the same time becomes perceptibly warm ; the reaction yery soon
ceases. Even if metal and acid be left in contact for a tiny or two at ordinai-y tempera-
tures, the Uquid will still contain a considerable portion of unchanged glyoxylic acid.
The process may be accelerated, and may in fact be completed in the course of eight or
ten hoiu's, by exposing the reacting substances to a temperature of about 80° C. The
previously colourless liquid will then have assumed a yellow colour, and does no longer
contain glyoxylic acid or glyoxylate of zinc. The pieces of imdissolved zinc are found
to be covered with a small quantity of a white crystalline powder. Glyoxylic acid
thus saturated with zinc was mixed with pure carbonate of Ume and treated with
sulphuretted hydrogen: the zinc precipitated as sidphuret of zinc, and the liberated
acid dissolved carbonate of lime and formed a lime-salt. The filtrate from the Zuj S
and the excess of Caj O, COg furnished, on evaporation, only one kind of crystals, which
possessed the form of glycolate of lime. After purification by recrystallization from
boiling water, they were diied over sulphm-ic acid and analysed, with the following
results : —

I. 0*371 gim. lost at 125" C. 0-078 grm. of water, and gave 0-154 grm. of carbonate
of lime.

II. 0-194 grm. of another preparation lost at 108° C. 0-041 grm. of water, and yielded
by the usual treatment 0-08 grm. of carbonate of lime.

3p2

448 DE. DEBrS OX S03IE C03IP0TJXDS AND DEEIVATITES OF GLTOXTLIC ACID.

The compound therefore contained in 100 parts —

I. II.

Calcium lG-5 16-49

Water 21-02 21-13

The formula of glycolate of lime, 2 (C, H, Ca O .) + .3H2 O, requires —

Calcium 16-3

Water 22-1

The percentage of water as found bj- experiment dift'ers to the amount of nearly 1 per
cent, from the theoretical number. I always found the amount of water contained in
glycolate of lime which has been dried over sulphuric acid to be a little below the
calculated quantity. If, however, from the quantity of carbonate of lime found in the
above analysis the amount of calcium contained in 100 parts of the anhydrous substance
be calculated, the number thus obtained agrees with the theoretical percentage of
calcium in anhydrous glycolate of lime.

I. n. Theory.

Calcium 21-03 20-9 21-05

The substance is therefore glycolate of lime. In order to prove the identity of the
glycolic acid obtained by the oxidation of alcohol by means of nitric acid with the acid
formed by reducing glyoxylic acid with zinc, the solubility in water of their respective
lime-salts Avas determined.

A = lime-salt of the glycolic acid prepared from alcohol.

B = lime-salt prepared froiQ the acid obtained by the action of zinc on glyoxylic
acid.

A. 3-073 grm. of solution, saturated at 12° C, evaporated at 100° C, left 0-027 gi-m.
of residue.

B. 1-7G7 grm. of solution, saturated at the same temperature, left 0-016 grm. of
residue.

3-20Ggrm. of solution, prepared at 7° C, gave 0-028 grm. of residue.
100 parts of water therefore dissolve —

12' C. 7=C.

A 0-SSG . . .

B 0-913 . . . 0-SSl

The lime-salts A and B, and consequently the acids contained therein, may therefore
be considered to be identical — a conclusion whicli is supported by the other properties
of A and B.

Glycolic acid can be obtained from glyoxylic acid by at least two different ways.

1. By direct addition of hydrogen : —

Glyo.xylic iieid. Glj-colle acid.

DE. DEBUS ON so:\rE coMPOUisDs AXD deriyatrt:s op glyoxylic acid. 449

2. By the decomposition represented in the following equation : —

2(C2H2 0.,) + ir,0 = aH2 0^ + C,,H,0, (1.)

Ox;ilic acid. Gl3"cohc acid.
Under the influence of zinc tlie folldwint;: reactions could take place: —

3(C2 H, O3) + Zn. ^ 2(a H3 Zn O,) + fi Zn^ O3, .... (2.)
Glj-colate of zinc. Glj'oxj-lato of zinc*.

or

a H., O3 + Zn^ = C, Zn, O, + H^-

Glyoxylate of ziue.

Glyoxylates which contain more than one atom of metal decompose easily at the
temperatures at which the experiments were made, and form compounds of glycolic
and oxalic acids.

2(0, Zn^ O,) +2 H2 O = C2 H3 Zn O, + C, Zn20,, + ^Ho.

Glyoxylate of zinc. Glycolate of ziuc. Oxalate of zinc.

By combining this and the preceding equation, the action of zinc on glyoxylic acid may
at once be represented by the following equation : —

2(C2H2 03) + 2H2O + Zn, = C2H3Zn03 + CgZn^, + H, + |'J0. . . (3.)

Glycolate of zinc. Oxalate of zinc.

If the reaction takes place according to (2.), then by the side of five atoms of glycolate
only one atom of oxalate of zinc can be produced ; but if according to (3.), for every five
atoms of glycolate five atoms of oxalate of zinc ought to be formed, and twenty atoms
of hydrogen be liberated.

The relative quantities of oxalic acid and glycolic acid formed by the action of zinc
on glyoxylic acid were therefore first determined. An unknoA^Ti quantity of the acid
was treated with zinc until the action was complete, and the filtrate from the undis-
solved metal was divided into two parts. In one part the oxalic acid was determined by
the usual method, and the glycolic acid in the other by converting the glycolate of zinc
into glycolate of lime, and by evaporating and drying the latter at 100° C. ; this could
be done because no other substance was present in the liquid. In this way 0-98 grm.
of glycolate of limef, and 0'005grm. of carbonate of lime from the oxalate were
obtained.

A white crystalline powder has been mentioned which settles on the zinc during its
action on glyoxylic acid ; this powder would contain oxalate of zinc. From its solution
in ammonia a precipitate of oxalate of lime was obtained, which gave after ignition
0"095 grm. of carbonate of lime. The whole of the Ca2 O, COg obtained from the

* Proceedings of the Royal Society, vol. ix. p. 711. Lir.Bio, Ann. vol. ex. p. 320.

t This glycolate of lime contained 21-4 per cent, of calcium, whereas theory requires 21-05 per cent.

450 DE. DEBUS OX SOjSIE COIMPOUKDS AND DEEIYATIYES OF GLTOXTLIC ACID.

oxalate weighed therefore 0-1 grm., which corresponds to 0-09 grm. of oxalic acid,
CgHgO^; the above-mentioned 0-98 grm. of glycolate of liine contain 0-784 grm. of
glycolic acid, and therefore for every atom of oxalic acid we obtain 10-3 of glycolic acid.
Another experiment conducted on the same plan gave 0-122 grm. of carbonate of lime
and 0-876 grm. of anhydi-ous glycolate of lime, or

1 atom of oxalic acid : 7-5 atoms of glycolic acid.

A thu-d experiment pelded nearly the same result as the second.

The action of zinc on glyoxylic acid takes place therefore according to equation (2.).

0-906 grm. of glyoxylic acid gave 0-876 grm. of glycolate of lime, corresponding to
0-7 grm. of glycolic acid, and 0-122 grm. of carbonate of lime, corresponding to 0-109 grm.
of C, H., O4. If we suppose none of the glyoxylate of zinc, formed according to (2.), to
be decomposed, then no oxalic acid and only 0-62 grm. of glycolic acid ought to have
been obtained ; if, however, all the glyoxylate of zinc had been decomposed into glyco-
late and oxalate of zinc, 0-775 grm. of glycolic acid and 0-183 grm. of oxalic acid ought
to have been found. The actual numbers are intermediate between these two extremes,
and consequently half of the glyoxylate of zinc formed from 0 -900 grm. of glyoxylic
acid, according to equation (2.). underwent the decomposition represented m equa-
tion (1.). The action of zinc on glyoxylic acid may therefore be explained as follows : —
zinc dissolves and forms glyoxylate of zinc, C'2 Zuo O-, ; the hydrogen instead of being
liberated combines with other glyoxylic acid and zinc and produces glycolate of zinc ; a
quantity of C, Zn., O,, dependent on time and temperature, decomposes into glycolate
and oxalate of zinc.

The following experiment proves that no hydrogen is liberated in this reaction.
0-906 grm. of glyoxylic acid and a few pieces of zinc were placed in a flask and the
vessel nearly filled with water ; a perforated cork holding a bent glass tube was then
attached to the mouth of the flask, the other end of the tube being placed under a
graduated receiver filled with mercury. The flask was then warmed to nearly 100° C,
and kept at this temperature for ten hours, after which time no more gas was given oflp.
The apparatus was found to be air-tight both before and after the experiment. The
quantity of the gas u'liich was collected, when measured at 11° C, was 10 cub. centims. ;
it possessed the properties of atmospheric air. If one atom of hydrogen had been set free
for each atom of glyoxylic acid wliich was taken, the 0-906 grm. of acid ought to have
liberated 0-0122 grm. of hydrogen; this quantity measures 135 cub. centims. at 0°C.
and 0-76 in. pressure.

Some years ago I directed attf>ntion to the following considerations*.
Among the products foi-med by the action of nitric acid on alcohol, which remain
after the more volatile substances have been evaporated on the water-bath, are three
acids, glycolic acid, glyoxylic acid, and oxalic acid. These bodies show the same differ-
ences in composition as bcnzylic alcoliol, oil of bitter almonds, and benzoic acid.

* Quarterly .Toiinial of tlie Clicmical >Soeiety. vol. xii. p. '2'M.

DE. DESrS OX SOJIE COMPOUXDS AXT) DEEIVATIVES OF GLTOXTLIC ACID. 451

CyH.O

0*2 H, 0;,

Benzylic alcohol.

Glycolic acid.

C;H,0

C,H,0.5

Oil of bitter almonds.

Glyoxylic acid.

C^Ie^

CJI,0,

Benzoic acid.

Oxalic acid.

They comijort themselves Avith different reagents Hke the members of the benzoyl
series. Hyckate of potash decomposes glyoxylic acid into glycolic and oxalic acids, and
oil of bitter almonds into benzylic alcohol and benzoic acid ; dilute nitric acid oxidizes
glyoxylic acid to oxalic acid, and oil of bitter almonds to benzoic acid.

a. Decomposition AAnth potash : —

2(C;H6 0) +H.O=CVHsO + C;HoO.,

Oil of bitter almonds. Benzylic alcohol. Benzoic acid.

2 (C2 H2 O;,) + H2 O = C2 H4 O3 + C2 H2 O,,

Glyoxylic acid. GlycoUc acid. Oxalic acid.

b. Oxidation with nitric acid : —

C-HfiO + O = C-H^O.

Oil of bitter almonds. Benzoic acid.

C2 11,03 + O = C2H2O4

Glyoxylic acid. Oxalic acid.

According to Friedel, oil of bitter almonds unites with hydrogen and forms benzylic
alcohol ; and the experiments described in this paper show glyoxylic acid to enter into
combination with this element, forming glycolic acid.

CyHgO + H2 = C^HgO

Oil of bitter almonds. Benzyhc alcohol.

C2H20,+ H.3 = CJI^,

Glyoxylic acid. Glycolic acid.

Glycolic acid may therefore be termed the alcohol of oxalic acid, and glyoxylic acid
the aldehyde of both ; ui fact glyoxylic acid possesses other properties which are gene-
rally only found in connexion with aldehydes. Amongst these are the great affinity of
glyoxylates for sulphites (whereby well-defined and beautifully crystallizing compounds
are fonned), the exchange of the oxygen of the acid for the sulphur of sulphuretted
hydrogen, and, finally, the production of compounds with ammonia.

Glyoxylic acid may likemse be compared with various other bodies ; it has, for
instance, in many respects a resemblance to svdphurous acid.

Kekule's interesting experiments with fumaric and maleic acids induced me to

452 DE. DEBUS OX S0:ME COMPOUXTDS AND DEEIYATmiS OF GLTOXTLIC ACID.

examine the action of bromine on a solution of gljoxylic acid. If the two bodies be
left in contact at ordinary temperatures in a closed vessel, the colour of the bromine
disappears in the course of a few days. Only hydrobromic and oxalic acids, however,
resulted from this reaction.

If a rational formula be required for glyoxylic acid, the following expression may be
adopted : —

It indicates the diatomic nature of the acid as well as its intermediate position between
glycolic and oxalic acids. Two atoms of hydrogen added to the radical C, O produce
the radical of glycolic acid, and one atom of oxygen the corresponding radical of oxalic
acid.

[ 453 ]
XXI. Oil Skew Surfaces, otherwise Scrolls. Bij Arthur Cayley, F.Ii.S.

Eoccived February 3, — Read March 5, 1S03.

It may be convenient to mention at the outset that, in the paper " On the Theory of
Skew Surfaces"*, I pointed out that upon any skew surface of the order n there is a
singular (or nodal) curve meeting each generating line in (n—2) points, and that the
class of the cii'cumscribed cone (or, what is the same thing, the class of the surface) is
equal to the order n of the surface. In the paper "On a Class of Ruled Surfaces "f,
Dr. Salmon considered the surface generated by a line which meets three curves of the
orders m, n, p respectively: such surface is there showir to be of the order ^=2mnp; and
it is noticed that there are upon it a certain number of double right lines (nodal gene-
rators) ; to determine the number of these, it was necessary to consider the skew surface
generated by a line meeting a given right line and a given curve of the order m twice ;
and the order of such surface is found to be =\m{m — 1)+/(, where h is the number
of apparent double points of the curve. The theory is somewhat further developed in
Dr. Salmon's memoir " On the Degree of a Surface reciprocal to a given one";j;, where
certain minor limits are gi\cn for the orders of the nodal curves on the skew surface
generated by a line meeting a given riglat lino and two curves of the orders m and n
respectively, and on that generated by a line meeting a given right lino and a curve of
the order m twice. And in the same memoir the author considers the skew surface
generated by a line the equations whereof are (a, • .\t., 1)"'=0 {a\ • -^.i, 1)"=0,
where «,..«',.. are any linear functions of the coordinates, and t is an arbitrary para-
meter. And the same theories are reproduced in the ' Treatise on the Analytic Geo-
metry of Three Dimensions '§. I will also, though it is less closely connected with the
subject of the present memoir, refer to a paper by M. Chasles, " Description des Courbes
a double courbure dc tons les ordrcs sur Ics surfaces reglccs du troisieme et du
quatrieme ordre"||.

The present memoir (in the composition of which I have been assisted by a corre-
spondence with Dr. Salmon) contains a further development of the theory of the skew
surfaces generated by a line which meets a given cuiTe or curves: "\iz. I consider, 1st,
the surface generated by a line which meets each of three given curves of the orders
m, n, p respectively ; 2nd, the surface generated by a line which meets a gi\ en curAc of
the order m twice, and a given curve of the order n once ; 3rd, the surface which meets

* Cambridge and Dublin Math. Joum. vol. vi. pp. 171-173 (1852). t Ibid, vol. viii. pp. 45, 4G (1S53).

i Trans. Royal Irish Acad. vol. xxiii. pp. 461-488 (read 1855). § Dublin, 1SG2.

|l Comptes Rendus, t. liii. (1861, 2<= Sein.), pp. SS4-889.
MDCCCLXIIL 3 Q

454: ME. A. CATLET OX SKEW SIJEFACES, OTIIEEWISE SCEOLLS.

a given c\u\e of the order m three times ; or, as it is very convenient to express it, I
consider the skew surfaces, or say the " Scrolls," S(?;i, n, jj), S(m-, n), S(??i^). The chief
results are embodied in the Table given after this introduction, at the commencement of
the memoii". It is to be noticed that I attend throughout to the general theory, not
considering otherwise than incidentally the effect of any singularity m the system of the
given ciu'ves, or m the given curves separately: the memoir contains however some
remarks as to what are the singularities material to a complete theory ; and, in particu^lar
as regards tlie surface S(//(^), I am thus led to mention an entirely new kind of
singularity of a cune in space — viz. such a curve has in general a determinate number
of '"lines through foiu- points" (lines which meet the curve m four points); it may
happen that, of the hues through three points which can be drawTi through any point
whatever of the curve, a certam number will unite together and form a line through
four (or more) points, the number of the lines through four points (or through a greater
number of points) so becoming infinite.

Xotation and Tahle of Sesulfs, Articles 1 to 10.

1. In the present memoir a letter such as m denotes the order of a curve in space.
It is for the most part assumed that the curve has no actual double points or stationary
points, and the corresponding letter M denotes the class of the curve taken nega-
tively and di^ided by 2 ; that is, if h be the number of apparent double points, then
M = —i\_,ii']--\-Jr. here and elsewhere [m]', &c. denote factorials, viz. [_m'\-=in{m—l),
[■inf=m{m—l){in—2), &c. It is to be noticed that for the system of two curves
m,m', if //, A' represent the number of apparent double points of the two curves respect-
ively, then for the system the number of apparent double points is :=mm'-\-h-\-h',
and the corresponding value of M is therefore — llm-^-m'y-^niDi' -\-h-{-h', which is
= --Ll.my^h-ilmJ+h', which is =M+M'.

2. The use of the combinations (m, n, 2>, g), {in^, n,ji), &c. hardly requires explana-
tion ; it may however be noticed that G(m, », j>, q) denoting the Imes which meet the
curves r?!, n, j), q (that is, curves of these orders) each of them once, G(w^^ u,^)) will
denote the lines which meet the curve 7n tmce and the curves n and p each of them
once ; and so in all similar cases.

3. The letters G, S, ND, NG, NR, NT (read Generators, ScroU, Nodal Du-ector,
Nodal Generator, Nodal Eesidue, and Nodal Total) are in the nature of functional
symbols, used (according to the context) to denote geometrical forms, or else the orders
of these forms. Thus G{m,n,2),q) denotes either the lines meeting the curves m, n,2^, 2
each of them once, or else it denotes the order of such system of lines, that is, the
number of lines. And so S{m, n, p) denotes the Skew Surface or Scroll generated by
a line which meets the curves m, n, p each once, or else it denotes the order of such
surface.

4. G(»«, 11,2), ?)• ^^^^ signification is explained abo\e.

5. S(?w, n,2>) '■ the signification has just been explained ; but as the surlaces S{m, ]i,2>)->

JVIE. A. CAYLEY OX SKEW SURFACES, OTirERWISE SCROLLS. 455

S(»j', 7i), S(m^) are in fact the subject of the present memoir, I give the explanation in
full for each of them, viz. S()n, n, p) is the surfxce generated by a Ime which mec^ts the
curves m, n, p each once,•'S(m^ n) is the surfoce generated by a line which meets the
cui-ve m twice and the curve n once ; S(»t^) the surface generated by the line which
meets the cui-ve in thrice. As already mentioned, these sui-faces and their orders are
represented by the same symbols respectively.

6. ND(w(, n,])). The directrix curves ?», n,p of the scroll S(»i, n, p) arc nodal (mul-
tiple) cm-ves on the svu-face, viz. m is an n^>-tuple cur\c, and so for n and p. Reckoning
each cuiTe according to its multiplicity, nz. the curve m being reckoned Wj^pf times, or
as of the order m.Vijip'Y-, and so for the curves n and^, the aggregate, or sum of the
orders, gives the Nodal Director ND(?«, n,p).

7. NG(»i, n, p). The scroll S(»i, n, p) has the nodal generating lines G{m-, n, p),
G{m, n-, p), G(m, n,p^). Each of these is a mere double line, to be reckoned once only,
and we have thus the Nodal Generator

NG(/», 7i,2})=G(m^, n,2))+G()n, n-,2}) + G{m, n,p-).
But to take another example, the scroll S(»i% n) has the nodal generating lines G(;?i^ n),
each of which is a triple line to be reckoned ^[o]-, that is, three times, and also the nodal
generating lines G{m-,if), each of them a mere double line to be reckoned once only;
whence here NG(??i% 7^)=3G(m^ w)+G(m-, if). And so for the scroll S(w^), this has the
nodal generating lines G{m% each of them a quadruple line to be reckoned |[4]'-, that
is, six times; or we have NG(»i^)=CG(»i').

8. NIl(m, n,p). The scroll S(w, n,p) has besides the directrix curves m, n,p or Nodal
Director, and the nodal generating lines or Nodal Generator, a remaining nodal cui've or
Nodal Residue, the locus of the intersections of two non-coincident generating lines meet-
ing in a point not situate on any one of the directrix curves. This Nodal Residue, as
well for the scroll S(?>i, n, p) as for the scrolls S(m^, n) and S(nv') respectively, is a mere
double cuiTe to be reckoned once only ; and such curve or its order is denoted by NE,
viz. for the scroll S(??i, n,2>), the Nodal Residue is NR(»J, h,^^).

9. NT(?«, n,2i)- The Nodal Director, Nodal Generator, and Nodal Residue of the scroll
S(m, n,p) form together the Nodal Total NT(»«, n,p), that is, we have

NT(w, «,j;)=:ND(?H, n,p)-Y'^G{m, ?z, j^)+NR(m, n,p) ;
and similarly for the scrolls S(»r, n) and S(»t').

10. I remark that the formula? are best exhibited in an order different from that in
which they are in the sequel obtained, viz. I collect them in the following

Table.
G(?n, n, 2>', q)^-mnpq,

G{m\ n, p) =np{[pif+M),

G(»r, if) =i[»i]=[w]^+M . i[n]-^+N. i[«iJ+MN,

G(m^ n) =?i(^[»i]^+M(m-2)),

G{m*) =lL[»^]^+w-fM(^[m]^-2H^-^-^)+M^i,

3 Q 2

456 ME. A. CATLET ON SKEW STRFACES, OTHEEWISE SCEOLLS.

S{m, n, 2) )=2vmj),
ND(??z, n,}} )=hnn2){mn-\-m2)-\-np—o),
jS'G(ot, 71, p )=mn]){m-\-n-\-i:)—"j)-\-'!sliip-{Smp-\-Vinn,
NR^???, n, p )=:hniiplim/)2) — {mn-\-mp-\-n2)) — 2(m-{-n-\-2))-\-5\
*XT(?7i, n, 2} j^iS'-— S+M«j;+X;hjj + Pwk

included in which we have

S(l, 1, m) =2m,
ND(1, 1, m) =[my,
NG(1, 1, m) =[/»]-+.M,
NE(1, 1, vi) =0,
XT(1, 1, )«) =iS-^-S+M

and

S(l, m, n) =2mn,

ND(1, ???, 7i) =i;mn{mn-{-in-\-;i — o),
IsG(l, ?», «) =mJi(in+n—2)+'S[u + :Sm,
NR(l,?n,«) =4[m]t«]%
NT(1,5»,«) =iS^-S+M« + Xm

Moreover

S(//r,«) = ji ( [mJ+U),

NG(;/i=,«) = ?« ( [m]^+M.3(«i-2))
SR(m\n) = n {l[my+'M{i[m]'-2m + o))

* In the first of the two expressions for XT(m, 9i,;<), S stands for S(-,i, n, p); and so in the first of the two
expressions for XT(»i-, n), &c., S stands for S()u", «), ic.

MR. A. CATLET ON SKEW SURFACES, OTIIERAVISE SCROLLS. 457

included in which we have

S(l, 7«=)=[»;]^+M,
ND(l,w^)=^W+[;«]'+-Ni(.j[''»?-^)+M%-;i,
NG(1, nf)=[mJ-\-U.3{m-2),
NR(1, wr)=|[ni]^+M(-^[mJ-2»i + 3),
NT(1, m^)=iS=-S+M(m-:])

=5[»']'+-D"]'+^^(["0'+'"-^)+^l'-i'
and finally

ND(m')=i[m]^+i[m]^+iW^+M(i[m]='+i[m])+]\P.im,
NG(??i')=i[»0'+6m+M(3[w]^-12»i+33)4-]\P.3,

+M(i[m]^+i[m]^+i[;«,]^-|Hi+13)

The fonnuliie arc investigated in the following order, ND, G, NG, S, NR, and NT.

The 'SD fonmtke. Articles 11 to 13.

11. ND(?», n,2))- — Taking any point on the curve m, this is the vertex of two cones
passing through the curves n,2) respectively ; the cones are of the orders w,^^ respectively,
and they intersect therefore in nj) lines, which are the generating lines through the point
on the curve m ; hence this curve is an j^^^-tuple line on the scroll S(»?, «,p), and we have
thus the term ??j.-^[hj;]'^ of ND. AVheucc

ND(»i, «,^))r=?u.|[»p]^+?i.i[???2;]^+^j. i[»m]'^
= hn)q){inn-\-mj}-^ii2) — S).

12. ND(Hr, n). — Taking first a point on the curve m, this is the vertex of a cone of
the order m — 1 through the curve vi, and of a cone of the order n through the curve n ;
the two cones intersect in (m — l)u lines, which are the generating lines through the
point on the curve m ; that is, the curve m is a (??i— l)?«-tuple line on the scroll S(?/i', 7i);
and we have thus the term m.|[(??i— 1)»]- of ND. Taking next a point on the curve n,
this is the vertex of a cone of the order m through the curve «i; such cone has
(A=)|[??2]^+M double lines, which are the generating lines through the point on the
curve n; hence this curve is a (-o-[»j]''+M)tuplc line on the surface, and we have thus
the term ?^.|r|[m]^+MT- in ND. And therefore

ND(m^ n)^m4[(m-l)nJ+n.l\Jlm']'+Mj

= n (^[m]'+W+M(i^»^?-i)+M^i)

458 ME. A. CATLET ON SKETT SUEFACES, OTHERWISE SCEOLLS.

13. XD(//i^). — Taking a point on the curve m, this is the vertex of a cone of the order
??i— 1 through the curve m; such cone has (A— ;«+2 = )J[hi]-— //< + 2 + M double lines,
or the curve ??i is a (^[m]^— »i + 2+M)tuple Ime on the scroll S(//()')- Hence we have

Preparaforif rcmnrJiS in regard io the, Gformulce, the hypertriadic suirjnlaritics of a
curve in simce, Articles 14 to 22.

14. It is to be remarked that the generating line of any one of the scrolls S(h;, n, p),
S(Hr, n), S{m'^) satisfies three conditions ; and that it cannot in anywise happen that one
of these conditions is implied in the other two. Thus, for instance, as regards the scroll
S(;/;, )i.2^)i if the ciu-ves m, n are given, and we take the entire series of lines meetmg
each of these curves, these lines form a double series of lines, all of them passing of
course through the curves m, n, but not all of them passing through any other curve what-
ever ; that is, there is no curve ^; such that every line passing through the curves in and n
passes also through the curve ^j. And the like as regards the scrolls Sfwr, n) and S(nf).

15. But (in contrast to this) if the three conditions are satisfied, it may very well
happen that a fourth condition is satisfied ?}jso facto. To see how this is, imagine a
curve q on the scroll S(m, n, p\ or, to meet an objection which might be raised, say a
cui-ve 5- the complete intersection of the scroll S(;», n,p) by a plane or any other siuface.
Every line whatever which meets the cui'ves m, n,p> is a generatmg line of the scroll
S(7«, n., p), and as such will meet the c\\y\q ff, that is, in the case in question, G{in, n,p^ q),
the lines which meet the curves m, n,p, q, are the entire series of generating lines of the
scroll S(??i, n,])), and they are thus infinite in number; so that in such case the question
does not arise of finding the number of the Imes G(»i, n,p, q). The like remarks apply
to the lines G(»t^ n,p), G(;?r, if), G{m\ n), and G{m*); but I will develope them some-
what more particularly as regards the lines G(m*).

IG. Given a curve m, then [as in fact mentioned in the investigation for 'S'D(nffj
through any point whatever of the curve there can be drawn

(A-»t + 2 = )[wJ+»i-2+M
lines meeting the curve in two other points, or say [wi]-+Hi— 2 + M lines through three
points. But in general no one of these lines meets the curve in a fourth point; that
is, we cannot through every point of the cuiTe m draw a line through four points ; there
are, however, on the curve m a certain number ( = 4G(;m')) of points through wliich can
be drawn a line through four points, or line G(m').

17. But the curve ni may be such that through every point of the ciine th(>re passes
a line through four points. In fact, assume any skew surface or scroll whatever, and
upon this surface a curve meeting each generating line in four points {e. g. the intersec-
tion of the scroll by a quartic surface). Taking the curve in question for the curve m,
then it is clear that through every point of this curve there passes a line (the generating
line of the assumed scroll) which is a line through four points, or line G(m*).

. ME. A. CATLET ON SKEW SUEFACES, OTHERWISE SCROLLS. 459

18. It is to be noticed, moreover, that if we take on the curve m any point whatever,
then of the [h«]^+"' — 2+M lines through three points which can be drawn through
this point, three will unite together in the generating line of the assumed scroll (for if
0 be the point on the ciu-ve ?«, and 1, 2, 3 the other points in which the generating line
of the assumed scroll meets the curve ?», then such generating line unites the three
lines 012, 013, 023, each of them a line through three points) ; and there will be besides
^[/hJ^'+jh— 5+M mere lines through three points. The line through four- points gene-
rates the assumed scroll taken (i[3]'=) 3 times, or considered as three coincident
scrolls ; the remaining lines generate a scroll S'(»t^), which is such that the curve m is
on this scroll a (?;[;»]-+/» — 5+M)tuple line; the assumed scroll three times and the
scroll S'(;u') make up the entire scroll S(??i^) derived from the curve m, or say ^{nf)=.
3 (assumed scroll) 4-S'(w*).

19. The case just considered is that of a curve m such that through every point of it
there passes a line through four points counting as (i[3]'^=) 3 lines through three
points, and that all the other lines through three pouits are mere lines through three
points. But it is clear that we may in like manner have a line through p points counting
as-i[^j— 1]' lines through three points; and more generally if ^j, q, . . . are numbers all
different and not < 3, and if

i[„i]=_;«+2+M=a. i[^.-l]^+/3. -i[<2-lT+ . . .,
then we may have a curve m such that through every point of it there pass a lines each
through ]} points and counting as \\^p — 1]'" lines, /3 lines each through q points and

counting as 2[2'— 1]^ lines, &c : the casepi=3 gives of course a lines each through three

points and counting as a single line. It is to be added that, in the case just referred to,
the a lines will generate a scroll S'('/?i^) taken \'\_pf times, the /3 Unes will generate a scroll
S"(?ft') taken ^[g]' times, &c., which scrolls together make up the scroll S(//t''), or say

^vf)=l[pj . S'(m')+i[2]^ . S>i,^)+ &c. ;
it may however happen that, e. g. of the a lines, any set or sets or even each line will
generate a distinct scroll or scrolls — that is, that the scroll S'(m') will itself break up
into scrolls of inferior orders.

20. A good illustration is afforded by taking for the curve m a curve on the hyper-
boloid or quadric scroll*; such curves divide themselves into species; nz. we have say
the {p, q) ciuTC on the hyperboloid, a curve of the order p+g' meeting each generating
line of the one kind in p> points, and each generating line of the other kind in q points ;
here

m=p+q, {h=\[xff+l[([\\ and .•.)M = -j^^.

Assuming for the moment that^, q are each of them not less than 3, it is clear that
the lines through three points which can be drawn through any point of the curve
are the generating line of the one kind counting as W_p — '^f lines through three points,

* It is hardly necessary to remark that (rrnVdy being disregarded) any quadric surface whatever is a hyper-
boloid or quadric scroll.

460 ]ME. A. CATLEY OX SKEW SURFACES, OTHERWISE SCEOLLS.

and the generating line of the other kind counting as ^[^ — 1]" lines through three
points, so that

The complete scroll S{vf) is made up of the hyperboloid considered as generated by the
generating lines of the one kind taken -^[pf times, and tlie hyperboloid considered as
generated by the generating lines of tlie other kind taken -^[fif times (so that there is
in this case the speciality that the surfaces S'(m^), S"(y»^) are in fact the same surface).
And hence we have

21. I notice also the case of a system of i)i lines. Taking here a point on one of the
lines, the (h — m-\-2=:)[m]- — m-{-2 lines through three points which can be drawn
through this point are the -l[}n — 1]- lines which can be dra\'STi meeting a pair of the other
()H — 1) lines, and besides this the line itself counting as one line through three points
(^\_m—l]--\-l = [m^'—vi-\-2); the line itself, thus counting as a single line through three
points, is not to be reckoned as a line through four or more points drawn through the
point in question, that is, the system is not to be regarded as a curve through every
point of which there passes a line through four points : each of the lines is nevertheless
to be counted as a single line through four points, and (since there are besides two lines
which may be di-awn meeting each four of the vi lines) the total number of lines through
four points is =Y-.^\_)iiy-{-m.

22. In the following investigations for G{m, n, p, q), &c., the foregoing special cases
are excluded from consideration ; it may liowcver be right to notice how it is that the
formulfB obtained are inapplicable to these special cases ; for instance, as will immedi-
ately be seen, the number of the lines G(;h, n^p, q) is obtained as the number of inter-
sections of the surface S(;», «, p) by tlie curve q, =:2mii2) X (l=2i)W2)q ; but if the curve
q lie on the surface S(in, n, p), then G(/», n, p, q) is no longer ^=2miii)q.

The G formula', Articles 23 to 3-1.

23. G(hi, n, 2>, </)■ — Considering the scroll S(m, «, p) generated by a line which meets
each of the cui-ves m, n, p, this meets the curve qmq S{m, n,2>) points through each of
which there passes a line G(/», it, p, q) ; that is, we have

G(?H, n, 2h '2)=(l S('/«, «,jO-

But from this equation we have

S{m, '}i,2>)=G{l, m, n, 2>)=p S(l, m, ii);

thence also

S(l, m, n) = G{l, 1, m, n)=n S(l, 1, »),
and

S(l, 1, 7,l) = G{^, \, 1, m)=m ^:{l, 1, 1); S(l, 1. 1)=G(1, 1, 1, 1)=2,

ME. A. CATLET OX SKEW SURFACES, OTIIERAFISE SCROLLS. 461

since 2 is the number of lines which can be drawn meeting eacli of four given right lines.
Hence iiltimately

G(m, n,p, q)=mn])qG{l, 1, 1, \)='lmnpq^.

24. G(??i', n, p). — In a precisely similar manner we find

G(«i', n, p)=npG{\, 1, ■)ii')=np^{\, m'),

and it is the same question to find G(l, I, m") and to find S(l, m^). I investigate
G(l, 1, m") by considering the particular case where tlie curve ?n is a plane curve
having n double points. The plane of the curve meets the two lines 1, 1 in two points,
and the line through these two points meets each of the lines 1, 1, and meets the cune
in m points ; combining the last-mentioned m points two and two together, the line in
question is to be considered as ^[w]' coincident lines, each of them meeting the lines
1, 1, and also meeting the curve m twice. But we may also through any double point
of the curve draw a line meeting each of the lines 1,1; such line, inasmuch as it passes
through a double point, meets the curve twice ; and we have h such lines. This gives
for the case in question G(l, 1, ??i*)=A+^[m]*; or, introducing in the place of k the
quantity M(=h—^[ni]'), so that 7i=|[m]^+M, we have

G(l, 1, 7)f)=[m']''+U.

And, to the double points of the plane curve, there correspond in the general case the
apparent double points of the curve m. Admitting the correctness of the result just
obtained, we then have

G(??l^ n, p)=np([m']'-\-'M.).

25. G(??i% «^). — I investigate the value by a process similar to that employed for
G(l, 1, m"). Suppose that the curves m and n are plane cun'es having respectively h
and k double points ; then the line of intersection of the two planes meets the curve m
in m points, and the curve n in w points ; or, combining in every manner the m points
two and two together, and the n points two and two together, the line in question is to
be considered as i[»i]''-i[w]' coincident lines, each meeting the curves m, n, each curve
twice. There are besides the hk lines joining each double pomt of the curve 7n with
each double point of the curve n. This gives in all ^[/^^[m J +M' lines; or, writing
A=J[m]'+M, ^=^[«]'+N, the number is

=5 WW+M . i W+N . i[m]'+MN ;

which is the value of G(m^, n') given by the investigation.

26. G(»^^ n).—We have

G(m\ n)=nG{l, m')=nS{nf),

and it is in fact the same question to find G(l, m') and to find S(w'). I assume for the
present that the value is =^[??i]^+M(w— 2); and we then have

MDCCCLXIII. 3 B

462 INIE. A. CATLET ON SKEW SUEFACES, OTHEEWISE SCEOLLS.

27. Before going further, I observe that there are certain functional conditions which
must be satisfied by the G formulfe. Thus if the curve m be replaced by the system of
the two cm-ves m, m', instead of M we have M+M'. Let G{in) denote any one of the
functions G{m, n,j), q), G(m, M^_p), G{m, n% we must have

G{m-\-m') =G{m) +G(m').
Similarly, if G(??r) denote either of the functions G(in-,n,p), G(»r, ?«^), we must have

G{m+m'y=G{m')+G{m, vi') +G{m");

and so if G(»i^) stand for G{m^, n), then

G{m+m'y=G{m')+G{m\ m') + G(m, vi") -\-G{m");
and finally

G{77i+m'y=G(m')+G{7n\ m')-\-G{tif, m")+G{m, vi'')+G(m").

28. The fii'st three equations may be at once verified by means of the above given
values of the G functions. But conversely, at least on the assumption that G(m),
G(7rf), &c., in so far as they respectively depend on the curve m, are functions of m and
M only, we may, by the solution of the functional equations, obtain the values of the
G functions. It is to be obsened that the first equation is of the form

the general solution whereof is

the second equation, supposing that G (m, m') is known — the third equation, supposing
that G(??t^ ?n') and G(wi, ??i'^) are known — and the fourth equation, supposing that
G(??t\ ?>i'), G(Ht% ??i'^), G{m, m'^) are known, are respectively of the form

<p(m-\-m') = <p7n-\-<pm' -\-fmict (m, m') ;

and hence if a particular solution be given, the general solution is

<p(?rt) = Particular Solution+a??i+/3M.

The values of the constants must in each case be determined by special considerations.

29. The value of G(wt, n,2}, 'i) was obtamed strictly ; that of G(wi^ n, ])) was reduced
to depend on G(l, 1, m^), and that of G(wt^ n) on G(l, rif). I apply therefore the
functional equations to the contu-mation of the values of G(l, 1, iif), G(m*, if), and
G(l, vf), and to the determination of the value of G{m*).

30. First, if Ginf) denote G(l, 1, nv), then G(»i, rii!) denotes G(l, 1, m, wt'), which is

=2mm'; hence

G(/«+«i7-G(y/r)-G(m'=')=2TOm', . ■

which is satisfied by G(»^^) = [w]^ This gives

G(l, 1, »i=) = [HiJ=+a»t+^M.
But if the cun-e m be a system of //< lines {m=m, M=0), then G(l, 1, m'')=[iJiY; and

SIE. A. CAYLEY ON SKEW STJEFACES, OTHERWISE SCE0LL8. 463

a^in, if the curve m be a conic {m=2, M = — 1), then G(l, 1, nf)=l. This gives
a=:0, /3:=1, and therefore

Q(l, 1, Hi')=[»i]»+M.

31. Next, if G(??i') denote G(m*, w*), then G(m, m') denotes G{m, m', n*), which is
=m?7^'([?^]*^-N), The functional equation is

G{m-\-m'y-G{nf)-G{m")=mm'{[nY+^),
which is satisfied by

GK)=|[m]»(M*+N).
Hence we have

G(m', w')=iMX[«]'+N)+a»i+/3M,

where a, j3 are functions of w, N ; and observing that G(»i^, n'') must be symmetrical in
r^ard to the curses 7n and n, it is easy to see that we may write

where a, /3, y are absolute constants. To determine them, if the curve m be a pair of
lines (m=2, M=0), then

G(»r, n') = G(l, 1, 7j=)=[«]'+?^';
and if each of the curves ?«, n be a conic (m=2, M= — 1, n=2, N= — 1), then

G(m\n')=l.
These cases give a=/3=0, y=l, and therefore

G{m\ n')=l[mJ{nJ+'M. .i[»]'+N .i[m]^+MN.

32. Again, G{m^) standing for G(l, m^), then G{nv', m') and G(m, m'-) will stand for
G(l, m\ m!) and G(l, m, m'^), the values whereof are m'([mJ+M) and »i([»t7+M')
respectively. We thus have

G{m+m'y-G(nf)-G(m")='n}!{[mJ+U)+m{[mJ+W),

a solution of which is G(m')=^[m]'+wiM. Hence we have

G(l, m')=^[m]^+7nM+a»i+/3M.

Suppose first that the cun-e m is a system of lines {m=7n, M=0), then G(l, m')= J[ot]';

and next that the curve w is a cubic in space or skew cubic (??i=3, M=— 2), then

G(l, 7?i^)=0, since a line can meet the curve in two points only. We thus find a=0,

6= — 2, and thence

^ ' G(l, 7?i')=i[m]'+M(m-2).

33. Hence, substituting for G{m\ m'), G{m', m"), G{m, m") their values
m'(i[m]»+M(w-2)),|[Jn]'[m7+M.^[»lG'+M^i[n^]'+MM',andm(^[m7+M'(w^'-2))

3e2

464 :me. a. catlet ox skew suefaces, otherwise sceolls.

respectively, we find ■•

G(m+vi'y-G(m*)-G{m")= m'(^[7»]'+M(«i-2))

+i[»i]^[»i']=+M . ![»/]= +M'. |[m]^+MM'

and thence, obtaining first a particular solution, the general solution is •• •

34. To determine the constants, suppose first that the curve Hi is a system of lines
{m=m, M=0), we must have G(»i') = -iii-[wi]'4-;», and thence u=0. Next, if the curve
m be a conic {m=2, M= — 1), we must have G(m')=:0; and this gives ^=^, and
consequently .. • .,

The 'isGfonnidce, Ai-ticle 35.

35. The NG formulte are now at once obtained, viz. we have

NG(m, «, 2))=G(m\ n, i>) + G(/», < 2))-{-G(m, n, f),
NG(»i^w) =SG(m\n) +G{m\ n%
NG(m^) =6G{m'),
which give the values in the Table.

TJie S formula', particular cases, Articles 36 to 40.

36. The S formultE have m fact been obtained in the investigation of the G fonnulEe :
we have

S(»«, n, j))=2mn2i,

S(//l^ n) =n{[my +M),

37. In confirmation of the formula S(l, ?>t^)=[?»]^+M, it is to be remarked that if
we take through the line 1 an arbitrary plane, this meets the curve m in ))i points, and
joining these two and two together we have ^\_viY lines, each of them meetmg the curve
7)1 twice and also meeting the line 1 ; that is, the lines in question are generating lines
of the scroll S(] , ni'). The line 1 is, as already mentioned, an (/i=)(i [»(,]'' +M)tuple
line on the scroll ; the section by the arbitrary plane is therefore the line 1 taken
(i[j/t]''+J^I) times, together with the before-mentioned ^\_mf lines; that is, the order of
the surface is [mJ'^+M, as it should be. This is in fact the mode in which the order
of the scroll S(l, m^) was originally obtained by Dr. Salmon.

38. As regards the formula S(»i')=^[??i]^+M(?/i— 2), suppose that the curve Hi is a
(j), q) curve on the hyperboloid, we have as before m=p-\-q, M=— ^^i^, and the formula
becomes

which is

MR. A. CATLEY OX SKEW SURFACES, OTHERWISE SCROLLS. 4G5

viz. as already remarked, the surface is in this case the hypeiboloid taken i[p]'+^[?]'
times.

39. It is to be noticed also that if the curve m be a system of lines (m=m, M^O),

then the formula gives

SK)=^[m]N

which is riglit, since in this case the scroll is made up of the ^[nij hyperboloids, gene-
rated each of them by a line which meets three out of the m lines.

In the case of a curve m, which is such that the coordinates of any point of the curve
are proportional to rational and integral functions of the order m of an arbitrai-y parameter
6, or say the case of a unicursal curve of the order ???, we have

{h=l[m-\J and .•.)M=-(m-l),
and the formula gives

S(»^')=^[m-l]^

for a direct investigation of which see post. Annex No. 1.

40. In the case of a curve m, which is the complete intersection of two surfaces of the
orders^ and q respectively, or say a complete {x^Xl) intersection, we have

m=pq, (h=ipq(p—l)(q-l) and .■.)U= — ^)q{2}+q—2);
and we find

Si»f) = hps(M-n^M-^P-H + 4)
=ii3(i3-2)(2/3-3a+4)

if u=pq, (i=2)+q. The mode of obtaining this result by a direct investigation was
pointed out to me by Dr. Salmon ; see jjost, Annex No. 2.

Particular cases of the formula for G(m*), Articles 41 <&; 42.

41. In the case of a (p, q) curve on the hyperboloid, putting as before m=p-\-q,
M=— ^^g-, we find

GK)=J^[jj+j]^+i.+?-i5j(i[j;+j]^-2(^+!Z)+ii) + i^.Y,
which is

=-iV(M^+[!?]0-2?[>-lT-2i^[>-lT,

vanishing \ip, q are neither of them greater than 3 : this is as it should be, since there
is then no line which meets the curve four times. The curves for which the condition
is satisfied ai-e (1, 1) the conic, (1, 2) the cubic, (2, 2) the quacWquadric, (1, 3) the
excubo-quartic, (2, 3) the excubo-quintic (viz. the quintic curve, which is the partial
intersection of a quadiic surface and a cubic surface having a line in common), and
(3, 3) the quadri-cubic, or complete intersection of a quadiic surface and a cubic sur-
face. If either p ox q exceeds 3, we have the case of a curve through every point
whereof there can be di-awn a line or lines through four or more points, and the
formula is inapplicable.

466

ME. A. CATLEY ON SKEW SUEFACES, OTHEEWISE SCE0LL8.

42. In the case of a complete (pxq) intersection, we have as before 7n:=2)q,
M= — ipj(i>+? — 2), and the formula for G(m*) becomes

G(wi*)=Jji3 [ _G6a+144

+i3(3«^+18a-26)
+(i\ —Qa
+/3^ 2,
a formula the direct verification whereof is due to Dr. Salmon ; see post, Annex No. 3.

TJw formula for NE(1, m, n) and NR(1, m-), Articles 4-3 to 46.

43. NR(1, m, n). — Through the line 1 take any plane meeting the curve m in m
points and the curve n in ?i points ; then if ??i,, m^ be any two of the m points, and »i,, n^
any two of the n points, the lines m^iii and m^n^ are generating lines of the scroll
S(l, m, «), and these lines intersect in a point which belongs to the Nodal Residue NR;
and in like manner the lines miii^ and 111.^11^ are generating lines of the scroll, and they
intersect on a point of NR ; we have thus

(2.i[m]^i[«]^=)i[m]^M^

points on NR, that is, the arbitrary plane through the line 1 cuts NR in ^[«^]°[w]^ points.

But the plane also cuts NR in certain points lying on the line 1, and if the number of

these be (a), then

NR(1, OT, n)=\\_mf[tiJ-\-?i.

44. The points (a) are included among the cuspidal points on the line 1. Taking for
a moment A'=0, ?/=0 for the equations of the line 1 (which, as we have seen, is a
wm-tuple line on the scroll), the equation of the scroll is of the form (A, ...\x,yy'"'=^^,
where A, . . . are functions of the coordinates of the degree mn. The entire number of
cuspidal points on the line 1 is thus =2\innf; but these include different kinds of
cuspidal points, viz. we have

2[Hi?i]^=2a+2a+2a'+R,

if (a) be the number of points in which the line 1 meets NR,
„ a „ „ „ „ S(wi', n),

„ a' „ „ „ „ S(m, w'),

„ R „ „ „ „ Torse(7?j, n),

where by Torse(/rt, n) I denote tlie developable surface or " Torse " generated by a line
which meets each of the curves m and n. The order of the Torse in question is

R=(7?([TO]^-2^)+w([«]^-2A.-)=)-2(wM+mN),

see post. Annex No. 4. And then obsei-ving that we have

a =S(w^ n)=n ([m]'+M),
a'=S(wi, %=)=w(W +N ),

MB. A. CATLEY ON SKEW SUEFACES, OTUEKAVISE SCEOLLS. 467

these values give

2a+2a' + li=24m]'+2m[ft]^
and we have

a=-i(2[mw]'-2a-2«'-K)

= [?«?j]^ — «[»«]^ — m [m]"

and thence

NR(1, m, «)=f[»^]'[«]^

45. Nll(l, vv). — Through the Hne 1 take any arbitrary plane meeting the curve min

m points ; if m„ m^, m^, ?», be any four of these, then the lines miWa and msm^ are

generating lines of the scroll S(l, nf), and their intersection is a point of the nodal

residue NR ; but in like manner the lines ??i,??i3 and ?«2?«4 are generating lines of the

scroU, and theii- intersection is a point of NE. ; and so the lines ??i,m4 and maWis are

generating hues of the scroll, and their intersection is a point of NR. We have thus

i^'X-ri[»i]*=)i\jny points of NR on the arbitrary plane through the line 1. But there

are besides the points of NR which lie on the line 1 ; and if the number of these be (a),

then

NR(1, m=)=i[m]^+a.

46. The points (a) are mcluded among the cuspidal points of the scroll lying on the
line 1. Supposing for a moment that x=0, y=0 are the equations of the line 1, then
this line bemg a (^[??i]^+M)tuple Ime on the scroll, the equation of the scroll is of
the form (A, . . .^jv, ^)*f'"'''^'"=0, where A, . . . ai'e functions of the coordinates of the
degree -^{jny : the number of cuspidal points on the line 1 is thus

(2 . ■^[m]XiW-l+M)=)[m]Xi[m]'-l+M).

Bnt these include cuspidal points of several kinds, viz., we have

[m]Xi[?/i]^-l+M)=2a+3/3+R'

if (a) be the number of points in which the Hne 1 meets NR,

» ^ » » „ » S(m'),

„ R' „ ,-, » „ Torse (m%

where Torse (nf) denotes the developable surface or Torse generated by a line which

meets the curve m twice. The order of the Torse in question is

R'=-2(m-3)M,

(see post. Annex No. 5) ; and then since

/3=S(w^)=i[m]'+M(m-2),
we find

2a=[m]ti[wi]'-l + M)-3(i[m]'+M(w-2))+2M(wi-3)

=iW+W+M([m]"-«t],
and thence

NR(1, m")=fW+i[m]'+M(i[w]'— im).

468 I\rR. A. CATLET ON SKEW SUEFACES, OTHEEWISE SCEOLLS.

But I have not succeeded iii finding by a like direct investigation the vakies of
NE(;h, «,p), NR(m^«). ^'K(»0-

Formulce for NT(1, ?«, n), NT(1, m'}, Articles 47 & 48.

47. We have

NT(1, m, 70= ^'G(l, w, «)= w2«(»i+n-2)+»iN«M
+ND(1, m, n) -\-\mn{mn-\-m+n—'i)

H-NR(1, m, n) +IWW'
which is

where S=S(1, m, ii)=2mn.

48. And moreover

NT(1, m')= ND(1, m=)= i[»0* + ['»T+^I(iW- 2)+M^ . \

+NG(l,m^) +[m]^+M(3»i-6)

+ NR(l,m^) +l["0^ +M(iW-2Hi+3),

which is

=iS^-S+M(m-|)
ifS=S(l, m')=[m]=+M.

T/(« NT ««(? NR/o™»Za', .Articles 49 to 58.

49. I proceed to find NT(?7i, n, p), &c. by a functional investigation, such as was
employed for finding G(l, 1, 7ii^), &c. Writing S(w) to denote either of the scrolls
S(yrt, n,2)), S(m, ?i^), and supposing that in place of the curve m we have the aggregate
of the two cui-ves m, vi' ; then the scroll S{»i-\-m') breaks up into the scrolls Sm, Sm',
and the intersection of these is part of the nodal total NT(?/i+?/i') ; that is, we have

NT(wi+m')=NT(7;i)+NT(TO')+S(m) . S(wi') ;
and in like manner, if S(?/i^) stands for S(??^.^ n), then

NT(m+»i')'=^'T(»i') + NT(m, ni')+'^T(m")+C,(S(m% S(m, m'), S{m")),
where Cj denotes the sum of the combinations two and two together ; and so also
NT(»i+m')'=NT(wi^)+NT(m^ m')+NT(/rt, m'^)+NT(»i'^)

+C,{S(m% S(m^ m'), S(w, m"), S(m"). ,

60. Instead of assuming

it is the same thing, and it is rather more convenient, to assume

NT=iS»-S+?>, . ■ . -

MR. A. CAYLEY OX SKEW SUEFACES, OTIIEinVISE SCROLLS. 469

viz. 'ST(m)=ii(S{m)y —S(m)-{- ^(m), Sec. Then observing that
S{m+m') = S{m)+S(yn'), Sec,

the foregoing equations for NT gi\'c

<p(m+m'y=<p{)7f)+(p(m\ m')+(p(m, m")+<p(^m");

and if in the second equation <p(m, m') and in the third equation |p(??^^ m') and f (?», m'')
are regarded as known, these are all of them of the form

Jlm+m')-f(m)-f(m')=Funct. (m, m') ;

so that, a particular solutic^i being obtained, the general solution hf(m)= Particular
Solution +a/;i+(3M, at least on the assumption that _/(««), in so fur as it depends on the
curve 7n, is a function of vi and M only.

51. First, if (p(?H) stands for p()n, n, j)), wc obtain ^(?h, ??, jj)=a?}t+/3M, or obscr\ing
that <p{)n, 71, p) must be symmetrical in regard to the curves 77i, n, and jJ, we may -vviite

and then

NT(m, «,i>)=^S^-S+?(m, «,iO

But forj;=l this should reduce itself to the known value of NT(1, 7n, n); this gives
a=:0, /3=1, 7=0; we in fact have, as will be shown, ^;osi^, Art. 55, 5=0 ; and hence
NT(ot, «,^^)=iS--S +(M«^j+N??i2J+Pwm)
= 2 [riinjpY + (Mhjj + N??y; + Vmn).

52. Next, if ^{711-) stand for (p{77i^, n), then <p()7i, 7)i') stands for (p(m, 7)i', 7i), which is
=Nm??i'+?i(?«M'+?/i'M), and the equation is

^(hi+«i')'=-?)(/«=)— ?(Hi'-)=N;HWi' + H(«iM'+wi'M).

A particular solution is 9(m-)=^[m]^N+?»»M, and we have therefore

(p{77i\ «)=|[??2.]-N+?«mM+a??i+/3M ;
or obsernng that ^(??i", 7i) considered as a function of 7i, satisfies the equation

<P(7l + 7l') = f(7l) + ^()i'),

and is therefore a linear function of )i and N, wc may Avrite
we then have

NT(W=, 7l) = lS'-S + ^{7}f, 7l),

where

S=S(?n^ w)=«([wi]+M).

MDCCCLXIII. 3 S

470 ME. A. CAYLET OX fiKE\Y SURFACES, OTHERWISE SCROLLS.

And then putting /i=:l, and comparing mth the known vakie of NT(1, 7?i"), we find
u = 0, /3= — f . It will be shown, 2J0sf, Ait. 55, that y=0, B = 0 ; and we have therefore

and thence

+N(iW-+M).

53. Xext for ?(wr'), substituting for ^{m-, m') and f(m, ml'-) their values, we have

^{m+m'y-f{nf)-(p{m") = m'M(m-^) + M'{^[mJ+'yi)
+ /HM'(W-t)+M(^[Hi'}^+M'),
which is satisfied by

and the general value then is

f{r>f) = M{Yim:]'-Ui) + W + um+ftM,
and we have

where

S=Sim')=^[_mY + U{m-2).

54. Taking for the curve ?m the (j^^u) curve on the hyperboloid (Hi=j) + 5', 1^1=— pq),
S(»i^) becomes the hyperboloid taken Z; times, if /i'=G[i']^+i[j/]' ; that is, S(««^) = 2/',
and XT(»i^) = 4 . \\_^']'-\-(p(7if) ; ^(hi'') must vanish if ^j and <7 are each not greater than 3,
this implies a =3, /3=11, for with these values the formula gives

55. I assume the correctness of the value

(p(m') = 3m + M(i[«?]-^ - fm + 11 ) + M^

so obtained, as being in fact verified by means of the six several curves (1, 1), (1, 2),
(1, 3), (2, 2), (2, 3), (3, 3); and I remark that if the forcgomg value of <p(m, n, ])) had
been increased by CaMNP, then it would have been necessary to increase the value of
ip{m-, n) by 3a^I^N, and that of <p(»i'') by ciW ; and moreover that if the foregoing value
of (p(9?i*, n) had been increased by y»iN + SMN, then it would have been necessary to
increase the vali;c of ^(m'')by ymM-j-SM^ ; this is easily seen by writing down the values
<p(»i^) =7»iM +IW + aW ,

<p(»;, Hi'^) = yWi'M4-SMM'+3aMM^

MR. A. CAYLET ON SKEW STJEFACES, OTHERWISE SCKOLLS. 471

the sum of ■which is

= 7(;«+j«')(M+M')+S(M+M')^+a(M+M')%
the coiTCsponding temi of <?(»t^) ; hence the vakie of (p{nf) being correct without the
foregoing addition, we must have 7=0, S = 0, a=rO ; which confirms the foregoing
values of (?(?», 7},2>), <P{»f- «)•

56. The equation

gives

NT(m^) = ^S=-S+3/«+M(i[m]^-|m + ll)+M=

-fM(^[»i]^+i[»H]'+^W-|"« + 13)

57. We have

NE(??i-, 7i)= NT(/«-, «)— ND(//r, n)—'^G(m-, n)

+ ["T(i["'J^+l[»rjH["'?+M(["0^-i)+M^i).

58. And moreover,

+ ^^^[my-i[mJ-i[my+8m-^2())+Wii[my-2m):
and the investigation of the series of results given in the Table is thus concluded.

Intersectims of a generating line with the Nodal Total, Articles 59 to G3.

59. We may for the scrolls S(l, in, n) and S(l, «r) verify the theorem that each
genei'ating Une meets the Nodal Total in a number of points=S — 2.

In fact for the scroll S(l, m, n), the directrix cuitcs are respectively multiple cuiTes
of the orders mn, n, m, and a generating line meets each of these in a single point,
counting for the tlu'ee curves respectively as mn — 1, n — 1, and m — 1 points respectively.
Moreover the constniction (ante, Ai-t. 43) for the Xodal Eesidue NR(1, m, n) shows that
a generating line meets this curve in (m — !)(« — 1) points; and since the curve is
merely a double curve, these count each as a single point ; and the generating line
does not meet the Xodal Generator XG(1, m, n). The number of intersections there-

?«n-l+(m-l)+(«-l) + (»<-lX«-l),

which is

=2/H«-2, =8-2.

60. Similarly for the scroll S(l, »r) ; the directrix cmTes are multiple curves, Xiz.
the line 1 is a (-|[?H]^+M)tuple curve, and the cui-ve m a (wi— l)tuple cune; the

3s2

472 3IE. A. CATLET OX SIOIW SUKFACES, OTHERWISE SCEOLLS.

generating line meets the former in a single point, counting as ^[?»]''+M — 1 points,
and the latter in two points, each counting as (m — 2) points. The construction (ante.
Art. 45) for the Xoclal Residue XE(1, nr) shows that the generating line meets this
curve in i[jn — 2y points; and since the curve is merely a double curve, these count
each as a single point. Finally, the generating line does not meet the Xodal Generator
NG(1, //(-). The number of intersections thus is

which is

= [i»]-^-2+M, =S-2.

In the remaining cases we may use tlie theorem to find the number of points in which
the generating line meets the X'odal llesidue. Using IT as the symbol for the points in
question {U.(m, n, j^) for the scroll S(y;i, «, j>), tS:c.), we tind
Gl. For the scroU S(/h, n, j^),

{mn-l)-\-{i)2)-l)+{,i>p-l) + U{,n, n. j)) = 8— 2=:2;/<«j)-2,
whicli gives

U(m, n, ij):=2innp—mn — ini) — vi)-\-'\..

This includes the before-mentioned case

n(l, m, v) = {in-l){n-l\

and the more particular one

n(I, 1, m) = i).

G2. For the scroll S()h-, v),

i[„,j^_l-}.M+2((/y^-l)»-l) + n(;«S«) = S-2 = »([/>0HM)-2,

vdiich gives

U{m-, n)=n{\mJ-2m^-2 + M)

_^[;»J^_|-1_M.

This includes the before-mentioned particular case

n(l, »r)^^[/H-2]^
63. And lastly for the scroll S()y;^),

whicli gives

The f(U'egoing expressions for FT might with propriety have been inserted in tlic Table.

Annex No. 1. — Invcdigation of the formula for S(/»'') in the case of the unicursal

curve (referred to, Art. 30).
Consider the unicursal ?«-thic curve the equations whereof arc x:y:z: 7t;= A : B : C : D,
where A, 13, C, D are rational and integral functions of a parameter 6. And let it be

:MK. a. CATLEi" OX SKEW SUEFACES, OTIIERAVISE SCEOLLS.

473

required to find the equation of a plane meeting the curve iii such manner tlxat three of
the points of intersection ai-e in lined. Taking for the equation of the plane

we find between (|, ;;, ^, u) an equation of a certain degree in (;, >?, ^, <y), which is the
equation in plane-coordinates of the scroll ^{m^), the degree of the equation is therefore
equal to the class of the scroll ; but as the class of a scroll is equal to its order, the degree
of the equation is equal to the order of the scroll, or say =S( ;«'').

Proceeding with the investigation, if ^ be determined by the equation

tlicn the roots ^,, 6„, ... 6^ of this equation belong to the points of intersection of the
plane and curve ; and the corresponding coordinates of these points are (A,, 13,, C,, D,), »&c.
Suppose that the points 1, 2, 3 are in lined, and let X, /ot, v, o be tlie coordinates of
an arbitrary point, then the four points are in jflfoio, that is, we have

= 0;

and if we form the equation

n

>- ,

H'^

" ,

f

A.,

B„

c„

r>.

Aj,

B„

C2,

D.

A3,

B3,

C3,

D3

>^ ,

/^ .

" J

0

Ao

B„

Cp

D.

XI05

B.„

Co,

D.

A3,

B3,

C3,

Ds

:0,

where 11 denotes the product of the terms belonging to all the triads of the m roots,
the result will be symmetrical in regard to all the roots ; and replacing the symmetrical
functions of the roots by their values in terms of the coefficients, we have the required
relation between (?, ;;, ^, a).

n contains ^[mf terms, whereof ^["i— 1]" contain thc»i-t]iic functions (A„ B,, C,, D,)
of the root 0^; that is, the form of n is

or, when the symmetrical functions are expressed in terms of the coefficients, the form is

Now the above-mentioned determinant is divisible by (^, — ^2)(^i — ^3)(^2— ^i)) or H is
divisible by n(^i — ^^X^i— ^3X^2— ^3); and since this product contains i^Xi[niy=)h[m'y
linear factors, and the product ^((^i, ^2, . . . d,„) of the squared difierences of the roots con-
tains (2 X i[.»i']'=)[>'i']' linear factors, so that we have

n(^,-a^ -"31^2-^3)= {C(^n ^2, ..L)}'-'"-'',

474 3IE. A. CAYLEY OX SICEW SUEFACES, OTHEEAVISE SCEOLLS.

■where

m^ ^„..(l„)=:Disct.=(;. ;;, ^, c^"'-'\
and cousequeutly

so that, omitting this factor, the remaining factor of 11 is of the form

(/, p, ., .)*-U ;;, ^, ^)-"'^'---'^
but the determinant vanishes if

A, V. K i={A,, B., C,. D,), (A„ B„ C. D,), (A3, B„ C,. D,),
or sav if

(a, i^, V, f) = (A. B, C, D). ^=^., ^,. or ^3 ;

it follows that the product IT contains the factor

or omitting this factor, and obser-\-ing that

i[m]^-[_m-lJ-i[m-f=i[_yny-[m-lJ=Km-l-\\
the remaining factor is of the form

or we have finally

which is the required expression.

I give the following investigation of the expression ^[;» — 1]- for the number of
apparent double points. Imagine through the point (.i"=0, ^=0, r=0) a line cutting
the cune in the two points corresponding to the values S^, b.^ of the parameter. "We
have

A,"~B,— Cg'

which equations determine 5; and 5^.
"SYritmg the equations under the form

and treating 5, and 60 as coordinates, each of these equations belongs to a curve of the
order 2(»i — 1), having a (/?« — l)thic pomt at infinity on each of the axes. The
number of intersections thus is "

= 40H-l)=-(»^-l)'-(»^-l)^ =2(m-l)^

But among these arc included points not belonging to the original system, viz. the
points for which (Ai = 0, A2=:0) other than those for which b^z=b^; the points so
included arc in number =?«- — m; and omitting them, the number is

(2(OT-l)'^->»(w-l)) = [m-l]=,

MR. A. CAYLET OX SKEW SURFACES, OTirERAVISE SCROLLS.

475

which is the number of points 5, lying iit Uncd with the origin and another point 5^;
the number of apparent double points is the half of this, or h=iWjn — 1]-. And thence

M=(-JU]-"+/' = )-(m-1).
I investigate also the number of lines through two pomts which meet two arbitrary
lines; this is m fact ^S(l, »r), which for the curve in question is

Let the equations of the two lines be (.i'=0, _y=0) and (c = 0, wz=0); then the con-
ditions to be satisfied are

A,_B, C,_D|

A.,— B.,' C.,— D,'
or writing these under the form

AiBg-AoB, ^ CiDa-CoDi_^

and treating 5„ 5„ as coordinates, the number of intersections of these two curves is
=2(;u— 1)", the same as in the two cui'ves last above considered. And the number of
the lines in question is one half of this, or =.{in—iy.

Lemma employed in the following Annexes 2 and 3. Fonnukc for the order and

weUjld of certain systems of equations.
Let a^- denote a function of the degree a in the order variables (.r, y, . .), and of the
degree a! in the loeiglit variables (.r', ij, . .), and so in other cases; and consider first the

equation

a«, (a+A),M-A', ••• =0,

/3,, (/3+AV^^.,

where the matrix is a square ; then

Order =2a+SA,

Weight=2«'+SA'.
Consider next the system

u^., (a+A),.,^., (a + B),,+B, ••• =0,

(3,., ((3+A),.,^., (/3+B),.+B',

where the matrix is a square +1, that is, the number of columns exceeds by 1 the

number of lines ; then

Order =SAB-Sai3+S«(SA+2a),
Weight= (5;A+Sa)(SA' +Xcc')-^.'^'-\-Xccu'.

476 ME. A. CAYLET OX SKEW SUEFACES, OTKEEWISE SCKOLLS.

And again, the system

where the matrix is a square +2, that is, the number of columns exceeds by 2 the
number of lines ; then

Order = SABC+i;«,37+S«(2AB-2«i3) + ((2«)-^-S«/3)(SA+S«),
AYeight={SAB-:S«/3+2«(SA+2«)}(SA'+2a')-(SA+S«)(SAA'-S««')
+2A-^A'+S«V.
The last formula, for the weight of the square +2 system, was communicated to me by
Dr. Salmon, the others are all in effect given in the Appendix, " On the Order of
Systems of Equations," to his Treatise on the Analytic Geometry of Three Dimensions ;
and in the in\estigation in the following Annexes 2 and 3, the route Avhich I have
followed was completely traced out for me by him, so that I have only supplied the
details of the work.

Annex Xo. 2. — Investigation of the formula for ^{nf), when the curve m is the pq

com])lete intersection, viz. ivhcn it is the intersection of two surfaces of the orders

J) and q respectivehj (referred to. Art. 40).

Let U = 0, V=0 be the equations of the two surfaces of the orders p and q respectively.

Take {a\ >/, :, w) the coordinates of a point on the curve, so that for these coordinates

we have U=0, V^O ; and in the equations of the two curves respectively, write for the

coordinates .r+^a"', y-V^f, ~+?~', w-\-^w' ; then putting for shortness

A =.t-'a^+yS^ -I- r'B.. + eo'B,,,
the resulting equations may be represented by

(AU, A'-^U, ... A'-UXl, f)""' = 0,
(AY, A% . . A'VXl, .^)'''' = 0,
where it is to be noticed that besides the expressed literal coefficients there are nume-
rical coefficients (not as the notation usually denotes, the binomial coefficients, but)

1' 1.2» 1.2.3' ^^'^*

Supposing that (.(•', //, z\ ■?(('') are the current coordinates of a point on the line drawn
through the point (.*•, y, z, w) to meet the cur^■c in two otlier points, the equations in g
must have two common roots, and this gives a system equivalent to two equations, or say
a plexus of two equations. If from the plexus and the two equations U = 0, V=:0 wc
eliminate (.;■, y, z, w), wc obtain an ecpmtion S' = 0 in (a^ y', c', iv'), which is in fact the
equation of the scroll S(m''), taken (as is easily seen to be the case) thrice ; that is, S(«i'') =
^Degree of S'. But observing that the coordinates (a/, y', z', tv') enter into the plexus only

SIR. A. CATLEY OX SKEW SURFACES, OTIIERAVISE SCROLLS.

477

and not into the functions U, Y, and treating (,(•', i/\ z', zc') as U'ci<//if variables, Degree
of S'=AVeight of System (U = 0, V=0, Plexus) =Dcg. U X Dcg. V X Weight of Plexus,
=i'l?X Weight of Plexus; or, writing ^;2=/3,

S(hi')=^/3 X Weight of Plexus.
The plexus in question is the square +1 system,

= 0,

AU, A=U, .
AU,.

AV, AW, .
AV,.

_^+2— 3 columns, (j- 2)+(jj — 2)=Q>+(^— 4) lines; or representing tlie terms accord-
ing to their order and weight, that is, degree in (.v, y, z, lo) and (.i-', y\ z', io') respectively
(the order and weight of the evanescent terms being fixed so as that they may form a
regular series with the other terms), the system is

p + ^— 3 columns.

g f(i^);ro.-2)„ .

Po ,(i^-l)n

^[(2-1)., (2-2),,
2o ,(!?-?)i,

= 0,

so that

cc,,3,.

• =1^-1, 1>, ..

.2) + fJ-4:, q-l,q, ..

i> + !?--i,

«' , /3',

.- 1,0,

■ -q + i, 1,0, .

-i> + 4.

A, B, .

• = -1,-2,

• -(i^ + !Z-4),

A', B', .

. = 1, 2,

P+q~i,

or, as regards the first two lines,

«',/3',..= 2-d, 2-^\ 1 ■ ^

We then find

2«= {q-2){p-2)+^q-2){q-l)
+ (^_2)(j_2)H-Kp-2)(j;-1),
Xa'= 2{q-2)-\{q-2){q-l)
+2(i^-2)-Ki^-2Xi>-l),

JIDCCCLSIII. 3 T

478 :ME. a. CATLET ox skew SUEFACES, OlilEEWISE SCEOLLS.

2A =-VA' = _iO,+2-^Xi^+2-3),

SurJ = 2{p-2){(i-2)-{p-i).\{i--2)[q-l)-l{r^--2){q-l){^-?,)
+ 2{q--2){p-2)-{a-A).\{p-2){p-\)-l{p-2){p-\){2p--ol

which putting t\ie\e\i\ p)-\-q^=ot., pq = (p, give
2« = /3 + ^a^-^a +10,

S«' = ,3— 'U'+ !« — 10,

SA =-2A'= -V+|«— 6,

2-^A= -t«'+ i «'- -^^ a + l-i,

and thence

2A +2a = ^_2a + l,

2A' +2«' = /3 -4,
2AA'-2aa^ = -ia/3 + Oa-12,
and therefore

^Yeight = 03 _2a + 4)((3-4)+ia8-5a + 12
= /3^_f«(3 + ea-8
=-K/3-2X2/3-3« + 4),
and consequently
S(?H^) = -|j3x weight

= -i/303-2)(2,3-3«+4),
which is right.

Annex Xo. •". — Investigation of G{m^) in the case lohere the curve m is a pq complete

intersection (referred to, Ai-t. 42).
Suppose, as before, that U=0, V=0 are the equations of the two surfaces of the
orders p and q respectively ; taking also [x, y, z, w) as the coordinates of a point on the
cune, and substituting in the equations x+gj/, y+§>/, 2-\-p', w+gio' in place of the
coordinates, then if A=x''d,. + >/''d,,+z'd,+w'b^., we have as before

(AU, A^U, . . A'-UXl, f)"-' = 0,

(AV, A=V, . . A"VX1, f)"-' = 0,
where the numerical coefficients y, j.-2, itIts, &c. are to be understood as before.

Suppose now that (.r, y, z, iv) are the coordinates of a point on the curve, through
which point there passes a lino through three other points, or line G{m*) ; and that
(a.-', ij, z\ vj') are the current coordinates of a point on such line : the two equations in §
must ha\ e three equal roots ; or we must have a system equivalent to three equations,
or say a plexus of tlirec equations. The coordinates (a-', >/', z', id), althougli four in

ME. A. CATLEY OX SKEW STRFACES, OTHER"WISE SCEOLLS.

479

number, are in fact elimiuablo from this plexus ; or what is the same thing, combining
with the plexus the equation

of an arbitrary plane, and then eliminating (o^, y, z', w'), the result is of the form

(a.r+/3y+7r+S»fn=0,

where D is a function of (.r, )/, r, ?<•) only ; and considering (.r, y, z. w) as weight variables,
^=Ordcr of Plexus. But degree in (.r, )/, z, w) of {ux+fty-^-yz+hvyn is ="NVeight of
Plexus, and therefore Degree of D is = Weight of Plexus— ^, = ("Weight— Order) of
Plexus.

The equations U = 0, V = 0, D =0 then give the coordinates (.r, y, z, iv) of the points
through which may be drawn a line G(wj*); ^iz. they give (as it is easy to see) these
points four times over. And we therefore have

G{m')=l Order of (U=0, V=0, D =0)
=1 Beg. U. Deg. V. Deg. D
= 1/3 X (Weight— Order) of Plexus.

The Plexus is here the square +2 system

AU, A=U, .
. AU,

AV, A=V,
AV,

=0,

(j) + Q — 4: columns, (g— 3)+(^j — 3)=j>/+^ — C lines). Or representing the terms by
their order and weight (the weight variables being in the present case {a\ y, z, w), and
the order variables (a^, y', z', w'), and attributing as before an order and weight to the
evanescent temis, the system is

2) + q—3 columns. — f)

o

u
•/a .

CO

I

a

CO
I

a,

3t 2

480 :\rR. A. CATLET OX SKEW SUEFACES, OTHEEWISE SCEOLLS.

SO tliat we have

cc,i3,..= 1,0,-1,.. -(2-5), 1, 0, -1, ..-(p-o),

«', /3', .. =2^— 1,2), J) + 1, ..j)+2— 5, J— 1, q, y+1, .. (2+2^-5,

A,B,..= 1, 2,... P+q-5,

A', B', .. =-1, -2, ... _(^,-f^_5),

or, as regards the fii'st two lines,

;('=! to 0 — o, (t: = l to p — .j.
c',(3\..=2)-2 + (i,2'-- + ^\ '

"NVc tlicn find
lu = 2(y-3)-Kj-3)(y-2) ^ :

+ 20,-3)-Ki^-3)0)-2),
2«' = (^,_2X^-3) + i(y-3X?-2)

+ (^-2)0>-3)+i(i'-3Xi^-2),
2«^ = 4(^_3)-4.My-3X?-2) + i(.-3X?-2X2y-o)

+ 4(^.-.3)-4.i(i^-3Xi*-2) + ^0,-3Xi.-2X2i)-5),
2«^ = 8(^-3)-12.^(y-3X?-2) + 0.i(y-3X?-2)(2:/-5)-i(y-3X(y-2X

+ 8Q>-3)-12.i(i>-3)0;-2) + G.Ki^-3X7v-2X2/.-5)-Ki>-3XO;-2X,
2««'= 20.-2X^-3)-0>-4).M?-3X^-2)-i(!/-3X'? -2X2^-5)

+ 2(^-2Xiv-3)-(y-4).i(i'-3Xi>-2)-iO>-3Xi*-2X2j;-5),
S«V= 4O.-2X?-3)-4O.-3).*(^-3Xj-2) + O;-0).-i(?-3X?-2X2!Z-5)+K!Z-3n^-2r,

+ 4(y-2Xi.-3)-4(y-3).Ki^-3Xi^-2)+(-7-C).^(i^-3)0.-2)(2j.-5)+Ki^-3X0^-2)^,

vA^=-VAA'=^0.+y-5)0>+y-4X2^-+22-9),
SA^ =-i;AvV=iO>+!2-5X(i^+!Z-4)>

which, putting therein j)4-2' = a, ^)«^=/3, and from tlic reduced expressions ohtaining the
vahies of 2a/3, &c., give

iici' =^3+?,a=-J:/a+18,

ME. A. CATLEY ON SKEW SITIFACES, OTIIER^VISE SCEOLLS. 481

SA = ia^-|«+10,

S.V =-SAA' = J a'- f a-+-F« -30,

SA^ =-SA-A'=i«*- |a='+lfla--90a+100,

SAB = Aa'-|S«'+J-|aa=~^a+G5,

V ATip -L/y" Al/yS I 29 7„4 2 6 8 9„3 I 508 1„3 J_2 70„ i 1050

Avo then fiud

2A +S« =/3-S,

SA' +2a' =/3-3a + 8,

SAB -Sap =:/3^(-?;)+/3(la--5«+-l^)-4a=+36a-12G,

SAA' -Sa«' =/3(-ia)+9a-28,

2ABC+S«,37=/3U)+/3=(-i«^+J4^a-4i)+i3(ia'-|«^+W«=-^i!^« + H^-')

-«'+^3V-162«;+i^5a-1210,

SA=A'+SaV=/3X-i)+/3(-§«='+9a-iF)-29a + 98;

and then also

S«(SA+Sa) =/3- +/3(_^a=+|a- 2C)+ 4a=-3Ga+144,

(SAB-Sa,3)+Sc.(SA+Sa) =,3=(i) +/3(-ia - 1) +18 ,

{(SAB-Sa,3)+Sa(SA+Sa)}(SA'+Sa') =

p^(i)+ /3-(-2a+i) + /3( %c^-Y^oc-lO)~Uu +144,

-(SA + Sa)(SAA'-Saa')=

(3'( ia )+/3( -13a+28) + 72a -224,

and

SA-A'+Sa"«' = ("^ siqjra)

in -i)+/3(-f«^+9«-i|i-)-29« +98;

whence, adding the last three expressions, we find

AYeight=,3Xi)+/3=(-|a+f)+/3(t«=+f«-Jo^)-ll«+18;

and for the order we have

(S«)— S«/3= (y{h) +,3(-i«=+4«-^/)

+ ia'-H«'+-F«'-Wa+133;
and then

SABC+S«/37=(h^ sKjmt)

m i)+/3=(-l«^+J;^«-41)+/3( Ia^_|«3^_£ai.<,._7|9„+ja53)

- a' +435a^-162a=+i^a-1210,
(SAB-Sa/3)Sa=

/3X-1)+/3X |«^-¥«+¥)+/3(-ia^+^«^-iF«'+^F« - 531 )
+2a''-36a^+297a'-1215a+2268,
((S«)— S«/3)(SA+S«)=

/3X 4)+/3X-A«=+4a-^^)+/3( i«^_f|«34. i|7„._i|^5^+ 241 )

482 3tR. A. CAYLET OX SKEW SUEFACES, OTHEEWISE SCEOLLS.

whence, adding these three expressious,

Order=/3Xi)+/3X-A«+i)+/3(K-i«+¥)-6;
and by means of the foregoing expression for the weight, we then have

Weight - Order = /3 '(i) + /3'( - « ) + i3( i«^ + 3« - ^) - 1 1 a + 2 4 ;
and therefore

G(Hi^) = 1/3 X (AVeight- Order)

=-/,-/3{2;3^+/3X-6«)+/3(3a^+18«-2G)-C6« + 144},
which is right.

Annex No. 4. — Order o/ Torse ()«, n) (referred to, Art. 44).

^^'e have to find the order of the developable or Torse generated by a line meeting
two curves of the orders la. n respectively; Aiz. representing by /x, v the classes of the two
curves respectively, it is to be shown that the expression for the Order is

Torse {ra, 7i)=zmi'-{-ri[j..

I remark, in the first place, that, given two surfaces of the orders j) and q respectively,
the curve of intersection is of the order 2)Q ai^fl c^fiss j^iij^+q — -), or as this may be
written, class =qi^{2^ — ^)-\-P(li!I~^)- Keciprocally for two surfaces of the classes jj and q
respectively, the Torse en\eloped by their common tangent planes is of the class j^l and
order ^i^(|>— l)+i'i(2 — !)• Now, in the same way that a sui-face of the order j; may
degenerate into a Torse of the order j), so a surface of the class 2^ may degenerate into a
curve of the class p ; and the class of a curve being j', then (disregarding singularities)
its order is — i^(i>— !)• Hence replacing ^^ and j^( p — 1 ) by /«, and m respectively, and in
like manner q and *?(!? — 1) by v and 7i respectively, we have ?hi'+«|«, as the order of the
Torse generated by the tangent planes of the curves of the orders m and n respectively ;
where by tangent plane of a ciu-ve is to be understood a plane passing through a tangent
line of the cuiwe. The intersection of two consecutive tangent planes is a line meeting
the two curves, which line is the generating line of the Torse, and such Torse is there-
fore the Torse (m, n) in question.

The foregoing investigation is not very satisfactory, but I confirm it by considermg
the case of two plane curves, orders ra and n, and classes ^ and v, respectively. The
tangents of the two curves can, it is clear, only meet on the line of intersection of the
planes of the cunos ; and the construction of the Torse is in fact as follows : from any
point of the line of intersection draw a tangent to m and a tangent to n, then the line
joining the points of contact of these tangents is a generating line of the Torse. The
order of the Torse is equal to the number of generating lines which meet an arbitrary
line; and taking for tlie arbitrary line the line of intersection of the two planes, it is
easy to see that the only generating lines which meet the line of intersection are those
for which one of the points of contact lies on the line of intersection ; that is, they are

MK. A. CAYLEY OX SKEW SURFACES, OTHERWISE SCROLLS. 483

the generating lines derived from the points m which the line of intersection meets one
or other of the two cui-vcs ; they are therefore in fact the tangents drawn to the curve n
from the points in which the line of intersection meets the curve m, together ^vith the
tangents drawn to the curve m from the points in which the line of intersection meets
the cune n. Now the line meets the curve n in n points, and from each of these there
ai-e ,a. tangents to the curve m ; and it meets the curve m in in points, and from each of
these there are c tangents to the curve n ; hence the entire number of the tangents in
question is =;?f(- +'»>', which confii'ms the theorem.

Annex No. 5. — Order of Torse (nr) (referred to. Art. 46).
We have here to find the order of the de^•elopable or Torse generated by a line meet-
ing a curve of the order m twice, viz., the class of the curve being ,«,, it is to be shown
that we have

Torse (;«-)=(»« — 3)ju,.

I deduce the expression from the formula given p. 424 of Dr. Salmon's ' Geometry of
Three Dimensions;' ^•iz. putting in his formula (5 = 0, and i^ for his r, we have

Ovdex=m(f/^ — 4i) — ^u = i)i[^ — (-ii)i-{-hoc),
where (see p. 234 et seq.)

^a, = (n~m) = om(in — 2) — Gh—m,
and thence

Sf/, — Ja=4?/«, or 4«i+ia = 3^,
so that we have

Order =(m — S)w.

A more complete discussion of the Torses (m, n) and [m") is obviously desirable ; but
as they are only incidentally connected with the subject of the present memoir, I have
contented myself with obtaining the required results in the way which most readily
presented itself.

[ 485 ]

XXII. On the Differential Ufjuctfions of Dynamics. A sequel to a FaiJcr on Simultaneous

Differential Equations.
By Geokqe Boole, F.B.S., Professor of Mathematics in Queen's College, Cork.

Kecoived December 22, 1862,— EeaJ Jaiiuaiy 22, 18G3.

Jacobi, in a posthumous memoir* which has only this year appeared, has developed two
remarkable metliods (agreeing in their general character, but differing in details) of
solving non-linear pai'tial differential equations of the first order, and has applied them
in connexion with that theory of the Differential Equations of Dynamics which was esta-
blished by Sir W. R. IIamiltox in the Philosophical Transactions for 1834-35. The
knowledge, indeed, that the solution of the equations of a dynamical problem is involved
in the discovery of a single central function, defined by a single partial differential
equation of the first order, does not appear to have been hitherto (perhaps it will never
be) A^ery fruitful in practical results. But in the order of those speculative truths which
enable us to perceive unity where it was unperceived before, its place is a high and
enduring one.

Given a system of dynamical equations, it is possible, as Jacobi had shown, to con-
struct a partial differential equation such that from any complete primitive of that
equation, i. e. from any solution of it involving a number of constants equal to the
number of the independent variables, all the integrals of the dynamical equations can be
deduced by processes of differentiation. Hitherto, however, the discovery of the com-
plete primitive of a partial differential equation has been supposed to require a previous
knowledge of the integrals of a certain auxiliary system of ordinary difterential equa-
tions; and in the case under consideration that auxiliary system consisted of the
dynamical equations themselves. Jacobi's new methods do not require the preliminary
integration of the auxiliary system. They require, instead of this, the solution of
certain systems of simultaneous linear partial differential equations. To this object
therefore the method developed in my recent paper on Simultaneous Differential Equa-
tionsf might be applied. But the systems of equations in question are of a peculiar
form. They admit, in consequence of this, of a peculiar analysis. And Jacobi's metliods
of sohdng them are in fact different from the one given by me, though connected with
it by remarkable relations. He does indeed refer to the general problem of the solution
of simultaneous partial differential equations, and this in language which does not e\en

* Nova mcthodus ODquationes differcntiales partiales primi ordinis iuter numerum variabilium qnemcunque
propositas integrandi (Crellc's Journal, Band Ix. p. 1).

t Philosophical Transactions for 1862.
MDCCCLXIII. 3 U

486 PEOFESSOE BOOLE ON THE DliTEEEXTLiL EQUATIONS OF DYNAMICS.

suppose the condition of linearity. He says, " Non ego hie immorabor qusestioni
generali quando et quomodo duabus compluribusve fequationibus differentialibus parti-
ahbus una eademque functione satisfieri possit, sed ad casum propositum investigationem
restringam. Quippe quo prceclaris uti licet artificiis ad integrationem expediendam
commodis." But he does not. as far as I have been able to discover, discuss any systems
of equations more general than those which arise in the immediate problem before him.

It is only very lately that I have come to understand the natui-e^ of the relation
between the general method of solving simultaneous partial differential equations,
published in my recent memou-, and the particular methods of Jacobi. But in arrinng
at this knowledge I have been led to perceive how by a combination of my own method
with one of those of Jacobi, the problem may be solved in a new and perhaps better,
certainly a remarkable way. This new way forms the subject of the present paper*.
Before proceeding to explain it, it 'uill be necessary to describe Jacobi's methods, to
refer to my own already published, and to point out the nature of the connexion between
them.

The system of linear partial differential equations being given, and it being requii-ed
to find a simultaneous solution of them, Jacobi, according to his first method, transforms
these equations by a change of variables ; he directs that an integral of the first equation
of the system be found ; he shows that, in virtue of the form of the equations and the
relation which connects the first and second of them, other integrals of the first equation
may be derived by mere processes of differentiation from the integral already found ;
and he shows how, by means of such integrals of the first equation, a common integral
of the first and second equations of the system may be found. This common integral
is a function of the above integrals of the first equation, and of certain variables, and
its form is obtained by the solution of a differential equation between two variables — a
differential equation whicli is in general non-linear, and of an order equal to the total
number of integrals pre\iously found.

An integi-al of the first two equations of the given system having been obtained,
Jacobi shows that by a second process of derivation, followed by the solution of a second
differential equation, an integi-al which will satisfy simultaneously the first three equa-
tions of the system may be found ; and thus he proceeds by alternate processes of deri-
vation and integration till an integral satisfying all the equations of the given system
together is obtained. In these alternations, it is the function of the processes of deriva-
tion to give new integrals of the equations already satisfied ; it is the function of the
processes of integration to determine the functional forms by which tlie remaining equa-
tions may in their turn be satisfied.

Jacobi's second method does not require a preliminary transformation of the equa-
tions ; but the process of derivation, by which from an integral of the first equation
other integrals are derived by vktue of the relation connecting the first and second

* It was stated by mc, but without (Ifmonstralion, at the Meeting of the Britisli Association in Cambridge-
in October of the present year (1862).

PEOFESSOH BOOLE 0\ TIIE DirrEREN"TUL EQUATIONS OF DYNAMICS. 487

equations, is carried furthor tlian in his first method. It is indeed carried on until no
new integrals arise. The difference of result is, that the common integral of the fii-st
and second partial differential equations is determined as a function solely of tlie inte-
grals known, and not as a mixed function of integrals and variables. But its form is
determmed, as before, by the solution of a differential equation. All the subsequent
processes of derivation and integration are of a similar nature.

On the other hand, the method of my former paper applied to the same problem
leads, by a certain process of derivation, to a system of ordmary differential equations
equal in number to the number of possible integrals, and, without being individually
exact, susceptible of combination into exact differential equations. The integration of
these would gi^e all the common integrals of the given system.

All these methods possess, with reference to the requirements of the actual case, a
superfluous generality. A single common integral of the system is all that is required.

Now the chief residt to be established in this paper is the following : —

If, with Jacobi, according to his second method, we suppose one integral of tlie
untransformed first partial differential equation to bo found, if by means of this we
construct according to a certain type a new partial differential equation, if to the
system thus increased we apply the process of my former paper, continually deriving
new partial differential equations until, no more arising, the system is complete, then,
under a certain condition hereafter to be explained, a common integral of all the equa-
tions of the complete system, and therefore of the original system which is contained in
it, may be found by the integration of a single differential equation susceptible of being
made integrable by means of a factor.

When the condition referred to is not satisfied, the results obtained may be applied
to the transforming of the original system of equations into an equivalent system of the
same character, but containing one equation less than before. To tliis system we may
apply the same process as to the former, and shall arrive at the same final alternation,
■viz. either the satisfying of the system by a function determined by the solution of a
single differential equation susceptible of being made exact by a factor, or the power of
reducing it to an equivalent system containing still one equation less. In the most
•unfavourable case the common integi-al sought wUl be ultimately given by the solution
of a single final partial differential equation.

The condition in question is gi'ounded on the theoretical connexion which exists
between the process of derivation of partial differential equations developed in my
former paper, and the process of derivation of integrals involved in Jacobi's methods.
In the actual problem, and in vii-tue of the peculiar form of the partial differential
equations employed, these two processes are coordinate, and it may even be said equi-
valent. The equations of that problem, if expressed in the s}-mbolical form

A,P=0, A,P=0, ..A„.P=0,
satisfy identically the condition

(AA-A,A,)P=0.
3 u2

4SS PEOFESSOE BOOLE OX THE DIFFEEENTIAL EQUATIONS OF DTXA]\nCS.
Each of the given equations is moreover of the form

^\dxi dp~dpi dxj—^'

H.being a given fimction of the independent variables .r,, x,, . . .r„, j),, jjo, .. j;„.

It is usual to represent the first member of the above equation in the form [H, P].
If we adopt this notation, the entii'e system of equations may be expressed in the form

[H,, P] = 0, [H„ P] = 0, .. [H„„ P] = 0.

Lastly, though this is not a new condition, being already implied in the former ones,
the functions H,, H.,, .. H,„ are all common integrals of the system. It is the object of
the problem to fuid a new common integral. With reference to such a system the con-
nexion above referred to is as follows : — If we obtain a new integral K of the first equa-
tion of the system, and, associating this \vith the functions H, form with it a new equation
of the same type as the former ones, so that, corresponding to the series of integrals of
the first equation

H„H„..H„„K,

we have the series of partial differential equations

[II„P] = 0, [fL.P] = 0, ..[H„„P]=0, [K. P]=0,

and tlien if to the former scries we apply Jacobi's process for the derivation of integrals,
to the latter the process of derivation of partial differential equations of my last paper,
carrying each to its fullest extent, the result will be that to each new partial differential
equation arising from the one will correspond a new integral (of the first partial differ-
ential equation) arising from the other. The theory now to be developed is founded
upon the inquiry whether it is possible to satisfy the completed system of partial differ-
ential equations by a function of the completed system of the Jacobian integrals, ?. e. to
determine a common integral of the completed series of equations as a function of the
completed series of integrals of tlie first equation. Tlic reader is reminded that by the
completed series of integrals is meant, not all the integrals of the first partial differential
equations that exist, but all that arise from a certain root integral by a certain process
of derivation, together with the root integral itself. Now the answer here to be esta-
blished to this inquiry is the following. The first of the partial differential equations
necessarily will, and others mai/, be satisfied by the proposed function irrespectively of its
form. If the number of (equations of the completed system which is not thus satisfied
be odd (this is the condition in question), the form of the function which will satisfy all
is determinable by the solution of a single differential equation of the first order, capable
of being made integrable by means of a factor.

I have entered into some details upon the history of the problem, partly because I
believe the theory of simultaneous partial differential equations to be an important one,
but partly also in order that I might render that just tribute to the great German

PEOFESSOR BOOLE ON TIIE DIITEREXTIAL EQUATIONS OF DYNAMICS. 489

mathematician which I was unable to pay before*. Jacobi certainly originated the
theorj- of systems in which the condition

is satisfied. I leani from his distinguished pupil Dr. Borciiardt, that this subject was
fully discussed by him in lectures delivered at Konigsberg in 1842-43, my infoiTnant
having been one of the auditors. The present memoir- is but a contribution to that theory.
And though it does not appear that Jacobi has discussed the theory of systems not satis,
fying the above conditions, it is just to observe that the more general theory is at least
to this extent contained in the particular one, that the recognition of the above equation
as the condition of a mode of integration natiu-ally suggests the inquiry how that equa-
tion is to be] interpreted when not satisfied as a condition ; and in the answer to this
question the general tlieory is contained.

PKorosiTiox I.

The solution of any non-linear partial differential equation of the first order may he
made to depend upon that of certain linear systems of partial differential equations of the
first order.

Althougli what is contained in this proposition is already known, it is presented here
for the sake of unity and to avoid inconvenient references.

First, the solution of any partial differential equation is reducible to that of one in
which the dependent variable does not explicitly appear.

For let z be the dependent variable, a\,x\, ..x„ the independent ones, and^J,,^^,, ..p^
the diiferential coefficients of z with respect to these. Represent the given equation by

f{a\,(c.„..x„,z,p,,2h,--P,) = ^, (!•)

and let

fGr.,.r„..a-„,c)=0 (2.)

be any relation between the primitive variables which satisfies the given equation.

Differentiating with respect to i\, x.^, . . x^ respectively, and representing the first
member of (2.) by <p, we have

Hence determining p,, ..p,,, and substituting in (1.), we have

/•U-„A'„..a-„, s, -^, ..-^1 = 0, (3.)

^ dz d= '

a partial differential of the fii"st order in which (p is the dependent variable, and
a:,, a",, . . a:„, 2 the independent ones. Here (p does not explicitly appear.

• The results of my former researches were communicated to Dr. Sixiioif on February 4, 1S62. At that
time I had not seen Jacobi's researches, which indeed could only just have been published. A note on the con-
nexion of the two which accompanied my paper was cancelled in the proof sheet in the prospect of that fuller
explanation which I then hoped to be able to give, and now give.

490 PEOFESSOE BOOLE 02y' THE DITFEEEXTIAL EQUATIONS OF DYlS'AinCS.

It vrill suffice, therefore, to develope the theory of the solution of partial diflferential
equations not explicitly invohing the dependent vaiiable. The general form of such an
equation is

/(j-„.r„...r,.,jA,iJ=, ••!>«) = 0.

Now, to solve this equation, we must find values of j^i, pr,^ ..])„ as fimctions of x^, a'^, • . it^
which, while satisfying it, will make the equation

^;— ^9,f/,r,— p.ff.ro..— j),/7>r„=0 (4.)

admit of a single integral containing 7i arbitrary constants. This integral will constitute
a complete primitive, a form of solution from which all other forms can be derived.
Let Hi = 0 represent the given equation, and

H, = 0, H,=c,, .. Il„=c„

the system of equations fi-om which j)^, |)o, ..j),, f^i"*? to be thus found. Their values will
contain the n — 1 arbitraiy constants c.,, c^, ..c„; and the remaining constant will be
introduced by the mtegration of (4.).

Let U and V represent any two of the functions H,. H,,, .. H„. Then differentiating
the corresponding equations with respect to any one of the independent variables .r,,
explicitly as it appears, and implicitly as involved inpi, jjj, .. p,„ we have

S+^'^l=«. (5.)

£-;+s.gi^=o. (6.)

the summations extending fromJ=:l to j=zii.

Mr
Then

Multiply the first of these equations by -— , and sum the result from ? = 1 to i=n.

or, since -f~ = -j-.
' dxi dXj

■^' dxi dp^ "r-"i-'; dp. dp dxi '

'^' da.\ dpi ' ' J dpj dp. dxj

Interchange in the second term i and j, since the limits of summation are the same with
respect to both, then

■^'' dx^ djji +^;^i dpi dpj dxi — "'
or

whence, reducing the second term by (C),
'd[] (N dU dV

V ^1^ iU. IV 1^ 5 m ivi>
'^^ dxi dpi ~^' dpi ^i dpj dx

:ond term by (C),
^'\dj:i dpi~dpi dxi)—^

PKOFESSOR BOOLE OX THE DIFFERENTI-iL EQUATIONS OP DYNAMICS. 491
Eeprcsenting the first member of this equation in the form [U, V], it appears that the
functions PI„ H,, . . H„ must satisfy mutually the — ^ — equations of which the type is

[H.,H,] = 0;
and by these conditions Hj, H3, . . H„ must be deduced from II„ which is kno'SN'n*.

Now all tliese conditions will be satisfied if we determine IIj to satisfy the single
linear partial difi'erential equation

[H„ H,]=0,
then H3 to satisfy the binary system

[H„ H3]=0, [H„ Il3]=0,
then II< to satisfy the ternary systerd

[H„ H,]=0, [H„ HJ=0, [H3, H,]=0,
and finally, H„ to satisfy the system of n — 1 equations

[H„ H„]=0, [H„ H„]=0, . . [H„_„ H„]=0.

All these are cases of the general problem of determining a function P to satisfy the
simultaneous equations

[H„P] = 0, [IL,P]=0,..[H,„,P] = 0, (7.)

H,, Hj, . . H,„ being known functions mutually satisfying the conditions

[H, H,]=0,

for each P thus determined gives the succeeding function H„+,, and so on till all are
found.

This is the problem with which we are concerned. But before proceeding to its
solution we must notice certain properties of the symbolic combination [U, V].

Projjerfics of [U, Y].
1st. It is evident from the definition that

[U,V]=-[V,U], (8.)

[U,U]=0 (9.)

2nd. The case sometimes arises in which one of the functions under the symbol [ ] is
itself a function of several other functions of the independent variables. Suppose V to
be a function of Vj, v^, . . v^, then it may be shown that

[U,V]=[U,.,]g+[U,.jg^..+[U,./^, (10.)

a theorem of especial use in transformations.

* On tte sufScicncy of tliese conditions see Professor Donkin's excellent mcrnoii' " On a Class of Differential
Equations, including those of Dynamics,'' Philosophical Transactions, 1854.

492 PEOFESSOK BOOLE ON THE DIFFEEENTIAL EQUATIONS OE DTNAIVIICS.
For

— ^\ dx, \dvi dp, + rfr.j dp. ' " ' + dv, dp, ) dp, \dv^ dx.'^dv^ dx, "^ dv^ dx, ) j

dY „, -, , dY TTT 1 J '^^ riT „ 1

In like manner, if U be a function of u„ u.,, ... u^, which are themselves functions of
the independent variables, then

[u,v]=KV]^+[»„v]'^^...+K,v]'j^ (11.)

3rd. The important theorem

[U,[V,W]] + [V,[W,U]] + [W,[U,\n] = 0 (12.)

is implicitly contained in the results of the following Proposition. All these are known
relations.

Pkopositiox II.

To determine the result of the application of the (jencraJ theorem of derivation to ani/
s>/stem of partial differential equations of the form

[»„p]=o, [»„P]=o, ...[»„„ r]=o, (1.)

u^, n.,, ... u„, leinij (jiven functions of the independent rariahles a\, a\, ... .(■„, Px,p-i^ ••■ p,,-
We adopt in the expression of this proposition ?f„ u,, . . ?/,„ in the place of H„ Hj, .. II,„,

because we suppose the functions given to be unrestrained by connecting conditions.
If we represent any two equations of the system by

[U, P] = 0, [V, P] = 0,

and then give to these the symbolical forms

A.P=0, A,P=0,
we shall have

' '"''KdXr dpr dpr '/*>/

A— ^ /'^ A_'^Y A\

■' ^^Kdxs dps dps dxgj

the summations with respect to r and s extending in each case from 1 to n inclusive.

Hence collecting into separate groups the terms which contain differential coefficients
of P with respect to p„ p,.^ and with respect to a'j, x„ we have

(A.A^— lijAJi 'r^s^^Xr dj),dxs dp, dpr dXfdx, dp, dxs dp,dxr dpr'dp^ dx,dxr dprj

_v V i'l^ <^'v dP djj dw (ip_(rv (pu dP dv d^u dP]

~ ^''"^•^i/xr dprdp, dx~ dpr dXrdp, dx~ dx, dpgdpr dxr ~^ dp, dx,dpr dxrj'

PEOFESSOE BOOLE ON TUE DIFFEEEXTIAL EQUATIONS OF DYNAMICS. 493

Now this expression will not bo aftected if in any of its terms we intercliange s and ;■. If

dV
we do this in the terms involving jy, the first aggregate will become

^^ rrfu tpv ^£_'_(u d^v ^_^ <fu dP dv d^v dP]

^•^'^dx, dp^a-f dp, dp, d<c4x, dpr dx, dp^r dpr~^ dp, dx^dxr dp J

dP_d^^(dV dV_dU dy\ _^ dV d[U, V]
'' dpr dxr '^'Kdx, dp, dp, dxgj '^'' dp^ dx^

In a similar way the second aggregate of terms will reduce to

Hence
Therefore

^ dP d[V, V]

■■ dXf dpr

,AA-AA)P=x(^S-^-^^?>

(AA-AA)r=[[u,^n,P]* (2.)

Thus the theorem of derivation applied to the two equations

[U, P]=0, [V, P]=0,
gi^es

[[U,V],P] = 0,

an equation of tlie same general form as the equations from wliicli it was deri\cd.
Apply tliis to the separate pairs of equations in the given system

[»„P] = 0,[»„P] = 0, ..[»,„, P]=0,

and to the equations thus generated, and so on in succession till no new equations arise,
and the result will be a system of the form

[«„P] = 0, [«„P] = 0, ..[;^,, P] = 0, ...J7;» (3.)

This constitutes the completed system, and it possesses, in accordance with the doctrine
of my former paper, the property that, if we form the equation

d^, ^^' • • + ^. ^^'^•'■+ df, '^1'^ • • +,^, '^1'"=^^

eliminate thence q of the differential coefficients of P, and equate to 0 the coefficients of
the 2n — q which remain, we shall obtain a system of In — q ordinary differential equa-
tions susceptible of reduction to the exact form, and yielding by integration the common
integrals of the system given.

The completed system is one of independent equations in which P is brought into
successive relations with a scries of functions w„ ?«,? • • 'W,- It is important to show that
the independence of the equations involves the independence of the functions, as also
that if the equations were dependent the functions would be so too.

1st. The independence of the equations involves the independence of tlie functions.

* This is not a new theorem. It Ls but another form of the theorem (12.) Prop. I. It has also been
explicitly given by Jacobi and Clebsch.

MDCCCIiXIII, 3 X

i9i .FEOFESSOE BOOLE ON THE DIEFEEEXTIAL EQUATIONS OF DTNAlVnCS.

For suppose the equations iudepeudeut and the functions not independent, so that
one of them, ?/,, could be expressed as a function of the others, u^, Un_, . . «f,-i. Then by
(11.), Prop. 1,

[,,, P] = [». P] ^ + [U.,, P] 4 . . +[.Vn P]

du,

5-1

which would imply that the equation

was not independent of the other equations of the system, as by hypothesis it is. Hence
the functions are independent.

2ndly. If the equations of a system of the form (3.) are algebraically dependent,
then are the functions «i, ii^, ■ . ",; dependent.

The algebraic dependence of the equations implies, in consequence of the linearity of
their developed forms, the existence of at least one relation of the form

K,P] = ?,[?«,.Pj+?,[?f,,P]..+^-,K_nP], (4.)

?.„ Xj, ..>.,_, bemg, on the most general supposition, functions of the independent
variables.

Xow the functions ?f„ u,, . . ;f,,_i are either dependent or independent. If dependent,

the proposition is granted ; if independent, then equating the coefficients of -^ in the
developed members of (4.), we have

</«,_. '^i , , cj}^ I , '■'"./- 1
djii—^^'dpi'^'-dpi--'^^''-' dpi '

(I? . .

and equating the coefficients of -r- in the same equation, we have

du,j . dji^ du^ du^_i

^. — ''' 55; +''2 ^^. • • +\-i -j^---

Hence if we represent the scries of variables t,, . . .r„, ^),, . .jh, taken in any order by
7/„ ^2, . . >/.,„, both the above equations wiU be included in the general one,

Now, ?<,, ti.j,, . . ?«,_! being by hypothesis independent, u^ will be expressible as, at most,
a function of W;, 2(.^, . . ?f,_, and 2ii—q-\-l of the origmal variables. Regard then m^ as a
function of Zf„ u.2, . . ?f,_i, i/,,, ^,+i, . .//2„. Of com-sc ?/„ il^, . . u,^_t mil be fmictionally
independent mth respect to the quantities ^1, ^2? • -^g-i which they replace.
Then for all values of ^, fi-om 1 to £—1 inclusive, the last equation becomes

du^ dui du du^^i . du^ du^~\

du^ di/i' ' ' du,j^i diji ^^di/i ' ' ~' ''"' rfy; '

or

du,/du„ \ du^/du^ \ .du^-,( du, \ :-: ■ '■ _

PH0FES90E BOOLE OX THE DTPFEHEJnnijVL EQUATTOlSrS OF FnfAlUCS. 495
while for all values of i greater than y— 1 it becomes

rfy. \du, ^') ■»■ rfy. \du^ '-' y • • + dyC \du,^, ~ '-'-' j + ^. " "• ' '
From the system of y — 1 linear equations of the first type (5.), we find

dUa

du.

rf«,'~^'-^' du,
unless the determinant

(G.)
(7.)

du^

dii^

du,,_t
d'Jx

du^

du,
d'Ji

dy.

du^

du.2

du^-i

dy,-i'

d'Jg-i'

dy,-i

vanish identically. But this Avould, by the kno^^^^ property of determinants, imply that
7fi, «o, ... ?/,_, are not as functions of y„ ^2, ... ?/,_, independent, which we have seen that
they are. Hence the system (7.) is true. Reducing by it the system represented by
(G.), we have

dyj «y?+i ' di/.^,.

It results therefore that n,^ will be simply a function of ii^, ti.,, ... ?^,^_,.

From these conclusions united we see that, if from any system of equations of the

[Mn P] = 0, [lU, P] = 0, ... [m,„, P] = 0

we sepai'ate the functions ?<,, u^, ... «,„, derive from these all possible independent
functions, of the form [tf,-, «J, and representing these by «„+„ Wm+2» "^c. continue ■ndth
the aid of these the process of derivation until no new functions can be formed, then if
we represent the completed series of functions by Ui, u^, . . . ti,,, the corresponding system

of equations

[m., P]=0, [«2, P] = 0,...[vP]=0

will be precisely that system to which the theorem of derivation of my former paper,
applied to the given system of equations, would lead.

Pkopositiox III.
To integrate the system of simultaneous partial differential eqiuitions
[H„ P] = 0, [H2, P] = 0, ... [H,„, P]=0,
it being given as a condition that H„ Hj, ... H„ satisfy mutually all relations of the form

[h;, h,]=o.

Tliis we hare seen to be the general problem upon the solution of which the inte-
gration of any non-linear partial differential equation depends.

3x2

496 PEOFESSOR BOOLE OX THE DIFFEEEXTIAL EQUATIONS OF DYNAMICS.

We learn by the last proposition that the system of equations given is already a com-
plete system. For, apphing the theorem of derivation to any two equations contained

in it. we have a result of the form

[[H.,H,]P] = 0;

but this is identically satisfied by virtue of the connected condition.

Instead, however, of solving the equation as a complete system, let us, -nith Jacobi,
deduce a new integral of the first partial diiferential equation of the system, i. e. an
integral distinct from 11,, II,, ... H,„, which, in virtue of the condition given, are already
intesi-als of that equation — in fact, common integrals of the system. For if we make
P=H, in any of the equations, that equation will be identically satisfied.

Eepresent by

this new integral. It may happen that it proves on trial to satisfy all the other equa-
tions. In that case a common mtegral is foiind and the problem is solved. Suppose,
however, the new integral of the first equation not to be a common integral, and, con-
structing the equation \_u, P] = 0, incorporate it with the given system of equations so
as to form the larger system

[II„P] = 0, [H„ P] = 0, ...[II„„ r] = 0, [», P] = 0 (1.)

Any common integral of this system will be also a common integral of the given system
which is contained in this one.

Such common integral, if it exist, we propose to seek.

First let us complete the system just formed by the last proposition. The completed
system will be of the form

[?^,p]=o, K,p]=o, ...K,P]=o, ...... (2.)

in which ?;,, ti.,, ... <<,„+, are for symmetry employed to represent H,, H,, ... H„„ u, and
2(,^^o, ^f,„+35 ... w., fire new functions.

Now all the functions «,, U2, ... ii,^ are integrals of the first partial differential equa-
tion of the system ; for, the system (2.) being a formal consequence of the system (I.),
if we substitute H, for P, tlie system

[«„ii,]=o, [«,, ii,]=o, ...K, n,]=o

will be seen to be a consequence of the system

[H„II,] = 0, [IL,II,] = 0, ...[», IT,] = 0.

But the latter system is true, tliercforc the former; therefore, since [»;, H,] = — [H,, ?«,],
the system

^ [ii„7^]=n, [ii,.«,]=o, [ii„»,]=o (3.)

is true; therefore ?/,, n^, ... ?r, are integrals of the first equation of the system. They
are independent, Prop. II. And as the process by which such of them as are new

PEOFESSOE BOOLE OX THE DIITEEEXTLVL EQUATIONS OF DTX.OIICS. 497

have been foi-med is identical in character with that which Jacobi makes use of, differ-
ing only in the extent to which it is here apjilied, I sliall speak of them as Jacobian
integrals, or Jacobian functions.

Let us inquii-e whether it is possible to satisfy tlie completed system of equations (2.)
by a function of the completed series of Jacobian integrals of the first equation of the
system.

Suppose, then, P a function of n„ u.,, ... «,. The equation [?/,, P] = 0, which is the

type of the equations of the system to be integrated, assumes, by (10.) Prop. I., the

forms

r ndP , r -^<^P , r i^^ n
["■'"■lrfi^+[""^'J.^,--- + K'''v]sr,=0 (4-)

Hence since by (3.)

[H„ w,]=[m„ «J=0,

the fii-st equation of the system is identically satisfied, as it ought to be. Let us, to
make the problem more definite, suppose for the present that none of the other equa-
tions are identically satisfied. Then, giving to i the successive values 2, 3, ... j, and
observing that always ^ ^ „ ^

we see that the system of equations represented by (4.) will be

rfP (IF (IP "I

* +[".' "3] ,7;73+["- ^'0 ^, • • • +[«- «v] ^ = 0'
[«3, «d ;^, * + ["3, " J rfiT, • • * + ["3, WvJ ,7^ = 0,

r 1 '^P . r n '^P . r 1 '^P n ^ " " ' * ^^'^

^ n '^P , r -1 f'P , r n '^P n

Now the coefficients [?<,-, m,] arc on the most general supposition functions of
?«i, Mo, ... «,, since the system (2.) is complete. Hence, if from any two equations
^u„ P]=0, [_i(j, P] = 0, we derive an equation

[[»,,«,]P] = 0,

that equation will be algebraically dependent on the equations of the system, and there-
fore, by Prop. II., [i*„ «^] will be functionally dependent on ii„ u^, ... «,;•

The svstcm (5.) is then one of partial differential equations, in which m,, 11.;^, ... 21,, are
the sole independent variables. If we express that system symbolically in the form

A,P=0, A3P=0, ... A,P=0,
all equations of the form

(A,A^.-AA)P=0

498 PEOFESSOE BOOLE OX THE DIFFEEEXTIAL EQUATIONS OF D"0"A]NnCS.

derived from it will be simply what the corresponding equations derived from (2.),
when uT, . . . a',„ p^ . . .])„ are the independent A'aiiables, would become under the limita-
tion that P is a function of ?;,, u.,, ... v,j. They are therefore not new equations, but
combinations of the old ones.

It follows, therefore, that if from the equation . . '

(lu,-\--!:-ai(.

, '^P 7 O

(6.)

we eliminate, by means of the system (5.), as many as possible of the differential coeflE-
cients of P, and equate to 0 the coefficients of the remaining ones, we shall obtain a
system of differential equations of the first order, reducible to the exact form, and
giving on integration all the common integrals of the original system which are expres-
sible as functions of ??„ v.,- .. . k,^.

First let us suppose the number of equations in the system (5.) to be odd. Then are
those equations not independent. For the determinant of the system is

[»3, U.J^ * [u,, ?? J . . [u^, » J

[?/j, ?fj [»-4, "J * •• ["j, ?',]

[ii,.n.;] [u,^,'ir;] [if,.n:] .. *

and this, since [??,, v~\ = — [_Uj, >i^, belongs to the class of symmetrical skew^ determinants,
of which it is a known property that when the number of rows or columns is odd the
determinant vanishes; and this, by another known theorem, indicates that one of the
corresponding linear eqiaations is dependent. The system (5.) is therefore in general
equivalent to a system of q—2 independent equations determining the ratios of the Q—l

differential coefficients

(IP dP^

dua du^

in the form

(rp_

dii,2

(hi.^

du,i

Uj, TJj, . . U,^ being known functions of Mi, iin. . . u,^. Eliminating by these q—1 equations

dV dV dV

the ri — 1 differential coefficients -j^-, j-, .. -j- from (6.), we have

dP dP

whence, equating to 0 the coefficients of -t- and 7^5 we have

du.

dUo

The first of these gives

U2<Z'Wo+U3r7?/3

J^\],^du, = ().

PEOFESSOR BOOLE OX THE DIFFERENTIAL EQUATIONS OF DTN.OHCS. 499

a common intcgi'al alreadj- kno-mi. The second equation being redncibic by means of
the first to the exact form, it ft)llows that, if in that equation we regard n^ as constant, it
vnll admit of an integral of the form

/(«„ u.,, .. Mj=const.;

and this is the new common integral sought.

We supposed that only the first of the differential equations of the system (2.) was
satisfied independently' of the form of P as a function of u^, it.,, . . u,^. Suppose, however,
that any number of equations

[?,,„p]=o, KP]=o, ..

are identically satisfied, independently 6f the fortn of P. Then

are common integrals of the system ; and if any one of these be not contained in the
series of Imown common integrals,

it A^Till be a new common integi'al, and the problem is solved. But whether such is or
is not the case, if the number of equations

KP]=0, [«,P] = 0,...
not identically satisfied be odd, we shall, proceeding as before, arrive at an equation

resolvable therefore into

du^=0, (]u^:=0, ...

TJJu„+Vp(lUp,...=0. ■;

Then obtaining from the first line of this system the already known integrals

we shall be able to reduce by these the equation of the second line to an integrable form,
and thence obtain the common integral sought.

Generally, then, when the number of equations of the completed system

[«,,P] = 0, [«„P]=:G,..[7VP] = 0

which, on the supposition that P is a function of «„ Uo, .. u^. Is not satisfied independ-
ently of the form of that function is odd, the form which will satisfy all is determinable
by the solution of a single differential equation of the first order*.

June 27, 18G3. Secondly, suppose the nuinber of equations of the above system not
satisfied independently of the form of P as a function of «„ u^, .. «■, to be even.

In this case no form can generally be assigned to that function that will cause all the
equations of the system to be satisfied.

* Communicated to the British Associatiou ia October 1§62, — -without demonstration.

500 PEOFESSOE BOOLE OX THE DIFFEEEXTIAL EQUATIONS OF DTXA:\nCS.

But neither is it required that all should be satisfied. It is only necessary to satisfy
the original system,

[»,,P]=0, [«„P] = 0,..[;^„P] = 0,

which is also a complete system.

If in this system wc make P a function of u^, u„, .. u^, we get

_ T «T , r -1 '^P , r 1 '^P n

* +[".' "J ^3+[«- "J ^, • • +[''- «J ^ = ^'

(«3, ?0^, * +K "J^-- +[''3, ",],7;;;=0,

in which the coefficients [;?„ ??J are aU functions of »,, J^, . . ii.,^, and in the integration of
which ?/, is to be regarded as a constant. I shall prove that to this system of «i— 1
partial differential equations, the same treatment may be appUcd as to the original
system of m partial diiferential equations.

Let V be any new integral of the first equation of the above system. As such, v will
be a function of u^, ii.,, . . u,^, and will therefore be also an integral of the prior equation

[»„ P] = 0.
Forming the equation [r,, P]=0, incorporate it with the original system, so as to form
the larger system

[H„ P] = 0, [IL, P] = 0, ... [n,„, P] = 0, [r„ P] = 0,

and, expressing II,, IIo, . . II„„ i\ as functions of the original variables, complete the system
by the theorem of derivation. Let the result be

[.„P]=0, [fv, P]=0,...[.„P]=0,
in which i\, i\, . . . r^+j are identical with H,, H3, . . H,,,, i\, and let it be sought to satisfy
this system by regarding P as a function i\, v^, . . iV- The two fii'st equations will be
satisfied identically ; the others will take the form

r idP * , r -1 '^P n

t''"''J^3 * . . . +[r„ rj ;^=0,

r -i'^ , r Y^P * o

[v,,iQ^ + [v,,v,];^^ * =0;

and if the number of these equations, or more generally the number of these equations
not identically satisfied, be odd, a common integral v>ill be found by the solution of a
single differential equation of the first order, of the form

PROFESSOR BOOLE 0\ THE DIFFERENTIAL EQUATIONS OF DYNAMICS. 501

In solving that equation r,, r.j are to be regarded as constant. If the number be even,
we must proceed as before, and we shall thus reduce the original system to a system ol
the same character, but possessing only m — 2 equations. In tlie most unfavourabl(> case,
the emerging systems being always odd, the common integral will ultimately be found
by the solution of a single final partial differential equation.

I have supposed that, of the symmetrical equations which arise from the introduction
of the Jacobian integrals as independent variables, only one is dependent when tlie num-
ber of equations is odd. and none when even. But exceptions may conceivably arise
from the splitting of the determinant into component factors each of the skew-symme-
trical form, and the corresponding resolution of the system of equations into partial
systems each complete in itself. To such partial systems, and not to the general system,
the law is to be applied. The conne.xion of the common integral mth tlie odd skew
determinant does not suffer exception even in those cases in whicli the integral is
obtained without a final integration, or is primarily given. Thus for each of the common
integrals ?/„ Mj, .. u„ the determinant reduces to a single vanishing term on the diagonal.
The possibility of cases of real exception seems to be a subject well worthy of infjuiry.

Postscript. — September 24, 1863.
Since communicating the above I have discovered that the number of independent
equations of the final symmetrical system (5.) is necessarily even. This confirms the
foregoing observations. It follows that whether we take all or some of the Jacobian
functions «,, ttj, . . ?*,, if there exist one common integral of the system expressible by
means of those functions, the determinant of the system will either be or will contain as
a component factor an odd symmetrical skew determinant.

MDCCCLXIII.

[ 503 ]

XXI 11. On the Nature of the Sun's Magnetic Action upon the Earth.
By Charles Chambers. Communicated by General Sabine, P.R.S.

Received April 30,— Read May 21, 1863.

1. I\ attempting to frame a theory which shall account for the relations which have
been shown to exist between the variations of terrestrial magnetism and the position of
the sun with respect to the place of observation on the earth's surface, the following
question presents itself for consideration at an early stage of the inquiry, " Are the mag-
netic effects produced on the earth such as could be explained by tiie simple supposition
that the sun is a great magnet, or notT' The solution of this question will, to a certain
extent, limit the range of probable sources from which to seek the true cause of mag-
netic variations, and is therefore worthy of attention.

2. In the first place, let us endeavour to find the law of the diurnal variations of the
Declination, Horizontal Force, and Vertical Force at a given place on the earth's surface,
on the supposition that these variations arise from the varying relations, as to position,
of the sun acting as a magnet upon the earth.

3. The sun would afiect the magnets used for showing the earth's changes of force in
two ways — fii-st directly, and secondly by inducing magnetism in the soft iron and other
inductive matter forming part of the body of the earth, this induced magnetism reacting
upon the observed magnets.

4. Now this subject has been discussed mathematically by PoissON in the case of
masses of soft iron having any possible arrangement, and he has given expressions for
the combined effect of the direct and induced forces upon the magnets, making only this
restriction, which is allowable in the case under consideration, that the length of the
magnets must be infinitesimaUy small in comparison with their distance from the nearest
particle of iron. The expressions are as follows : —

X'=(l + a)X + JY-}-C'Z, («•)

Y'=<ZX+(l+e)Y+/Z, (*•)

Z'=^X-1-AY + (1+^)Z, (c.)

where, assuming the direction and intensity of the magnetic force exerted du-ectly by
the sun upon the centre of the earth to be approximate values of the same elements
throughout the body of the earth, X represents the resolved part of that force along the
perpendicular to the earth's axis from the place of observation, being reckoned positive
when the resolved force acting alone would make the north end of a magnet at the
station point towards the earth's axis ; Y is the resolved part of the same force perpen-
dicular to the meridian plane, the positive direction being to eastward ; Z represents

3 y2

504 :^IE. C. CHAMBERS OX THE XATUEE OY THE

the part of the force resolved along the earth's axis, plus to northward, and a, b, c, d, e,
f\ (/. h, and ^ are constants which depend on the amount and arrangement of the soft
iron of the earth. X', Y', and Z' are the combined effects of the direct force of the sun,
and of the reaction of the magnetism which it induces in the earth, upon the magnets
at the place of observation, in the directions of X, Y, and Z respectively.

PoiSSO>''s equations * may be derived from the following considerations. The inducing
force X produces a certain magnetic state in the mass of iron, and the magnetism thus
induced (including that whicli is caused by the inducing action of the particles of iron
upon each other) will exert an attraction at the place of observation in the direction of
each of the coordinate axes, which attraction is assumed to be proportional to the
inducing force X ; the forces due to X in tlie directions of X, Y, and Z respectively will
therefore be «X, rfX, and yX, and those due to Y, iY, (\\ JiY ; those due to Z, cZ, /Z,
kZ. Now, adding the forces in the same directions and including the forces X, Y, and Z.
which act directly upon the magnets at the place of observation, we get

X' = (l+r/)X + iY + c-Z,

Y'=(JX + il+e)Y+fZ,

Z' =ffX + hY + il+k)Z.

5. In order that Poissox's formuLne should be strictly applicable to the case under
consideration, we must assume that the whole effect of an inducing force acting upon
the earth requires only a few minutes to be acquired, and that it is lost with equal
facility on the removal of the inducing force. It is not necessary that the effects of gain
and loss should be momentary.

6. We must now transform the coordinates of Poisson's equations to those directions
in which the variations of the earth's magnetism are always observed, viz. towards the
magnetic east (x), towards the magnetic north (i/), and vertically downwards (r), the forces
being reckoned positive when, acting alone, they would make the north end of a magnet
point in the directions named, and we get a new set of equations,

.r=AX + BY+CZ, (d.)

?/=DX + EY+FZ, {e.)

c=GX + HY + KZ, (/.)

in which A, B, C, . . . K are constant numbers, being functions of a, b, c, . . . /?', and of

the magnetic declination and latitude of the place of observation.

7. Let us observe that, during the time in which the earth is making a single revolu-
tion upon its axis, the variations in direction and amount of Z and of R (the resultant of
X and Y) may be disregarded, while tlie direction and value of X and Y vary with the
hour-angle (A) of tlie sun, and may be expressed as follows: —

X=:Ilcos(/i4-a), (g.)

Y = Rsin(/i + «), (h.)

where tan a = ^, . . . (/.) and ir = X^-f Y^, . . . ,. (k)
* This modi; nf Dbtaiiiinj,' Poisson's forrauhc was suggested by Mr. AucniBALn Smiiii, F.R.S.

SUN'S MAGNETIC ACTIOX tiPOX THE EARTH. 505

Xo and Ya being the values of X and Y at noon. It is evident that, Z being considered
constant, the last terms of {</.), (e.), (/.) form no part of the diurnal variations, and may
be neglected ; omitting them therefore and substituting the values of X and "\' given by
{(J.) and (/(.), tlie equations (d.), (e.), and (/.) become

a\ = 11 { A cos (/* + «) + B sin (/t + a) } 1

= lVA^ + B^sin{(A + a)+/3}, ^ (/.)

where tanp = TT

y,. = R{D cos (/(+«)+£ sin (/* + «)} ^

where tanp,=g

^;^ = R{ G COS (/i+«) + H sin (h + a)}^
= Kx/GM=ffsin{(A + «) + /3„}, I ^^n ^

where tan /3,^= 77 •

As the angles a, /3, ,3,, and /3,, may be regarded as constant for a single day, the equa-
tions (/.), {I'.), (I".) show that the deviations of the magnets from their normal positions
at any two houi-s whose interval is half a day are equal in amount but opposite in direc-
tion for each element of the earth's force ; but this is quite at variance with the real
character of the diurnal variations, the excursions of the magnets being invariably
extensive on both sides of theii- normal positions daring about eight hours of the day,
which include the sun's upper transit, while deviations of comparatively small extent
occur during the remaining hours.

This discordance will be more conclusively shown if we express the observed variations
in the form of the series (y) described hereafter (17), when the coefficients of the second
and follo\A-ing terms should be zero, while we find that those of the second and third
tei-ms have magnitudes not greatly inferior to the coefficient of the first term.

Again, it will be shown hereafter (19) that there should be no mean diiu-nal varia-
tion for the whole year due to direct action of the sun, while in fact there is such a
variation of very considerable range, in middle and high magnetic latitudes, which is
opposite in character for the north and south magnetic hemispheres.

With respect to the dii-ect action of the sun upon the observed magnets, the first of
the above-mentioned reasons was adduced by Dr. Lloyd as showing the mconsistency
with observation in the law of the diurnal variations as derived from the theory in ques-
tion, in a paper published in the ' Proceedings ' of the Royal Irish Academy, February
22, 1858. The same conclusion is here extended by the aid of Poisson's formula to
the inducing action of the sun upon the soft iron of the earth. The hypothesis of the

50(5 ME. C. CHAMBEES ON THE NATURE OF THE

magnetic nature of the sun cannot, therefore, by itself be accepted as an explanation of
the diurnal variations of the earth's magnetism.

9. It may be argued, however, that the diurnal variations are the combined effects of
direct solar action and of other forces whose origin and nature are unknown ; but if this
be the case, I think we have reason for believing that the portion of the variations
which proceeds from the former source is small in comparison with the part which is
due to the other forces that are in operation.

10. Retui-ning now to the equation (l.), if R(, and «„ be any particular values of R and
K. and any other values be R.^ and (wo+y), we shall have in the latter case

a-, = R,v/AH-B"^ sm (/;+«„+7+/3), 1

andif/3' = «„+/3, (p-)

=R.VA'+B'si"(^'+7+/3')- .1
Hence it appears that when R„ and a„ are changed to R.^ and (ao+y), the range of
variation is altered in the proportion of R^ to Ro, and the hour- at which the maximum
deviation occurs is advanced by the equivalent in time to the angle y.

As we are only seeking the law of variation and not absolute quantities, we may
reject the constant factor ^Z A" +B-, when (jj.) becomes

.r; = R-sin(/i+y+/3') (t.)

12. Now let us observe that the period embraced by a large mass of hourly observa-
tions, extending over several years, comprises many revolutions of the sun on its axis, the
sun's period of rotation being about twenty-five days, and that it is fi-om such extended
series of observations that the determinations of the regular diurnal variations, which we
shall use for comparison with the results of our hypothesis, are derived.

13. As the distance of the sun from the earth is great in relation to the diameter of
the sun, we may assume, in the calculation of the magnetic force exerted by the sun
upon the earth, that the sun's free magnetism is accumulated in the extremities of a
straight line whose middle point passes through the sun's centre ; and this supposed
magnet may be resolved into two others, one having its poles in the sun's axis of rotation,
the other with its poles in the plane of the sun's equator. Now as regards the latter,
if we notice its effect upon the eai'th when its north end points towards the earth, it is
evident that when the sun has made somewhat more than half a revolution upon its
axis, the south end will point towards the earth and the opposite effect be produced,
and similarly, for every position of this magnet there is another position, when the sun
has advanced about 180' in rotation, in which the effect neutralizes that of the former
position.

14. It Is nccessaiy therefore to consider only the magnetism of the sun parallel to its
axis of rotation as affecting the mean diumal variations on the earth's surface. Now we
have no means of finding the absolute magnetic force of the sun, neither do we require
it; what we want is the lata of change of direction and intensity of the force exerted by
the sun upon the centre of the earth as the earth revolves in its orbit ; and this we
can easily calculate from the kuowTi elements of position of the sun's axis, which by

SUN'S MAGNETIC ACTION UPON TILE EARTH.

507

Mr. C.VRRiXQTOX's latest determinations are —

Inclination of sun's axis to plane of ecliptic (z)^82^ 45',
Longitude of projection of northern half of sun's axis on plane of ecliptic
at epoch 1850-0 (X')=343° 40'.
at epoch 1845-0 (X') = 343° 35'.
(See Monthly Notices of the Royal Astronomical Society, vol. xxii. page 300.)
15. Let I be the magnetic moment of the sun in the direction of its axis of rotation,
supposed positive if the north end of the sun's axis is a north pole ;
D the radius vector of the earth ;
u the obliquity of the ecliptic ;
6 the right ascension of the sun ;
X the longitude of the sun ;

S the declination of the sun, positive when north ;
and S' and & the declination and right ascension which the sun had when his longi-
tude was (X— 90").
Now, if we suppose the magnetism of the sun along its axis to be resolved perpendi-
culai- to the plane of the ecliptic (+ to northward) = I sin ?, and in that plane, and the
latter part to be again resolved in the directions of the line joining the centres of the
sun and earth (+ towards the eart]i)= — I cos/cos(X — ?.'), and at right angles to that
line (+ to the right hand of an observer who looks towards the earth, his head being
supposed at the north pole and feet at south pole of the sun)=: — I cos? sin(X— X'), then
the forces which these exert on the centre of the earth in the same directions respect-
ively, are

L =

W

sm ?,

M = — 2 g3 cos i cos (X — X'),
N = + g3 cos i sin (X — X').

Now let us change the coordinates to those of Xj,, Y„, and Z ; this Avill be facilitated by

a reference to the accompanying figure,

where C represents the centre of the

earth, ArB the equator, and P its north

pole, S'rS the ecliptic, and K its pole,

and S the sun ; then

rCS =X,

rCs =6,

SCS' =90\

«Cs' =(^-^),

sCS =S,

s'cs' =y,

PCK=^.

•508

im. C. CHAMBEES OX THE XATURE OF THE

The resolved part of L along Z = — tts sin i cos en.

The resolved part of M along Z=-\-2 jys cos / cos (?, — >/) sin 5,

The resolved part of X along Z^ — jp cos i sin (>, — /v') sin S',
and the whole force

Z=— TTsIsin i C0S&/ — 2 cos ?' cos(>v— a') sinS4- cos/sin(?. — ?.')sinS'|--

The resolved part of L along Xo= — Tyt sin i sin co sin d,

The resolved part of ^NI along X„^ — 2 j^ cos i cos (X — K') cos S,

The resolved part of N along Xo= + jyi cos i sin (a — >.') cos V cos (^ — ^'),
and the whole force

X„=-

jT3<sin? sin4< sin^+2 cos? cos(>. — X')cosS — cos/sin(X — >.') cosS' cos(fl — ^)> (t(.)

The resolved part of L along Yu= -4- jp sin / sin u cos S,

The resolved part of M along Y„= 0,

The resolved part of N along ¥„=+ T^cos?'siu (a — //) cosS' sin(^ — ^').

and the whole force ¥„ = + ^J sin / sin a/ cos d-\- cos / sin (X— /.') cos I' sin (^ — ^)|- . (v.)

16. Taking 184-5 as a convenient year, being about the middle year of the observa-
tions which we shall make use of, and calculating by means of the formuhr (».), (v.), and
data taken from the Nautical Almanac, the values of X^ and Y^ for the middle day of
each month of the year, we obtain the following results, which it may be remarked
would be altered only in a trifling degree if any year between 1835 and 1855 were
taken instead of 1845.

Table I.

x„.

Yo-

January

+ -20618

I X

+ -08202
+ -29425
+ -41678
+ -44028
+ -35721
+ -16582

- -06992

- -28116

- -41070

- -44304

- -36388
— -17132

- -01627

- -21637

- -33852

- -33349

- -34793

- -21580

- -00104
+ -20327
+ -32934
+ -37919
+ -34916

March

April

May

.June

July

August

September

October

November

December

SUN'S MAGNETIC ACTION UPON THE EARTH. 509

The constant I multiplies all these numbers, and therefore has no influence upon the
law of change; hence it may be omitted, and we shall then call the numbers X«( =u5]

and Y,/ =-^j, the resultant of X^ and Y|, being called Ry

By means of the above Table and equations (g.), (h.), and (i.), we are enabled to con-
struct a table of values of R^ and a, when, calling a„ the January value of a. y is found
by subtracting «„ successively from the a of each month.

Table II.

January

February .

March

April

May

June

July

August

September.

October

November .
December .

r;.

-■

y

•22190

21 42

6 d

•29479

93 10

71 28

•46960

117 26

95 44

•55537

127 33

105 51

•48869

133 2

111 20

•38543

154 31

132 49

•22686

197 58

176 16

•28116

269 47

248 5

•45824

296 20

274 38

•55204

306 38

284 56

•52555

316 11

294 29

•38892

333 52

312 10

(x.)

If in equation (t.) xl become a\ when E.^ becomes R^ , we obtain, by inserting the
above values of R!^ and y in

a-:=R;sin{(A+y)+/3'}, (w.)

the following equations showing the diumal variations in the different months :
January . . . .rr=-22190 sin (/i+jS'), |

February . . . a-;'=-29479 sin {(^ + 71° 28')+|3'},
March. . . . a\ = -46960sin {(A + 95° 44')-)-|3'},
&c. &c.

It will be noticed that the angle /3' is undetermined, depending as it does upon the
distribution of soft iron in the earth as well as upon the angle a„; but it is not required
in the application that we are to make of the formulae.

17. Let us now turn our attention to the results of observation as to the diurnal
variations. General Sabot; has discussed this subject veiy fully for the obser\ations
made at the British Colonial Observatories of Toronto, St. Helena, Hobarton, &c.
After a careful separation of disturbed observations, he has given hourly mean values of
the deviations of the observed magnets fi'om their normal positions, or then- equivalents
expressed in teims of the earth's force, for every month of the year, the means being
deduced fi:om several years' observations.

Now, if from the mean hourly deviations for a given month we determine by the
MDCCCLXIII. 3 z

510 ME. C. CHL\MBEES OX THE NATlTiE OF THE

method of least squares the constants m a series of the form

A,=Bsin(/(+S) + Csm2(/; + £) + Dsm3(/«+i)+&c., .... {y.)
where A^ is the deviation at any horn- (//) of the day, it is easy to show that the values of
the constants in any one term are independent of the values of those in any other term,
if the data are complete for all the twenty-four hours, and if the series be not carried
beyond a term Wsin 2.3(/< + Q

18. We have proved (16) that the variation due to direct and inducing action of the
sun is of tlie form

.;v:=.R;sin{(/^ + y)+/3'}, (a..)

and therefore we see that the tirst term of tlie series (y.) includes the whole of this
action, the remaining terms being unaffected by it. Moreover the equations (.r.) show us
that the part of the term B sin (/< + §) which is due to direct solar action varies as K^ ,
and has its maximum advanced as y increases. Thus we are led to expect changes in
B and S from month to month which shall accord with the corresponding variations of
R! and y, and the degree of accordance which is found to exist will serve to indicate the
extent to which the du-ect and inducing action of the sun bears a part in the production
of the regular diurnal variations.

19. We see from Table II. that the values of K^ for January and July are nearly
equal, and that the angle y for the former month differs by about 180° from its value
for the latter month : consequently there should be no mean diurnal variation (or a very
small one) for these two months, inasmuch as the hour of maximum deviation in the
one is the same as the hour of minimum in the other, the magnitude of the deviations
being nearly alike iu the two months : the same observation holds with regard to any
two months separated by an interval of half a year, and thus we perceive that the mean
diurnal variation /or the whole yea)\ due to direct action of tlie sun, should be extremely
small, and that, without sensible error, we may omit the consideration of it.

We may therefore separate from the quantity Bsin (A+S) for the different months,
its normal value for the year, calling the remaining quantities B' sin (/* + §') ; and we shall
then have to consider only the latter numbers for comparison with R.^ and y, for the
reason just stated.

Now it is probable (from the simphcity of the instrument used and its independence
of temperature coriections) that no periodical magnetic variations are so well deter-
mined as the diurnal variations of declination. We shall therefore confine ourselves
to the examination of the variations of that element, for which the following Table gives
the monthly values of B' and S' at Toronto and St. Helena, these being derived from
tables of variations given by General Sabine in his discussions of the observations made
at the observatories of Toronto and St. Helena, published for the British Government
by Longman and Co., London. Positive values of B' sin (/i + S') indicate easterly devia-
tions of the north end of the decluiation magnet.

SUN'S MAGNETIC ACTION UPON THE EARTH.

Table III.

511

ll

Toronto.

St. Helena.

B'. ' >'.

B'. 1 }-.

I

January

i-208 24 20
•883 ' 35 44
•467 I 188 22

•627
•916
•570
•203
•640
•858
•922
•803
•372
•642
•688
•603

349 48
347 8
353 46

March

•664

•884
1^053

i^ooe

•963
•569

228 21
204 14
179 44
177 6
207 23
261 12

196 1 ]

May

192 0

174 35 I

175 24
183 21
174 8

50 11

4 40

348 8

•J'-iy

August

October

November

December

•444 17 49
1-09.T 14 49

1-47-2 , 17 26

li. i.

B. 1 3. j

Year

2-G25 32 37 1 •I 26 1 292 2<) 1

'

20. As a diagram conveys to the mind a more distinct conception of a variation than a
table of numbers, the curve (Plate XXV. fig. 1) is constructed to reinesent the successive
values of K^ and y: in this figure the lines rl, r2, . . . rl2 are proportional to R^,
and the angles Arl, Ar2, . . . Arl2 are equal to the angle y, for January, February,
. . . December respectively. The curves of figs. 2 & 3, Avhich are formed in a similar
manner from the numbers in the preceding Table, are intended to show the variations
of B' and 5'. Drawing Imes )' G', r H' at right angles to r G and r H respectively, it is
easy to see, from the form of the expression B'sin(/i-4-S'), that the angles G'rl, G'r2,
. . . G'rl2 (reckoned by a right-handed revolution from G'r) represent the hour-angles
of the sun at the time of the occurrence of the maximum de\iation in the respective
months from Januaiy to December, and that the extent of tliat deviation is represented
by the lines rl, r2, . . . rl2 respectively; and similarly for fig. 3.

21. Now we see that direct action of the sun is not the sole cause of the variations
of the term B sin (A+5), because in that case fig. 1 (which represents the variations of
the cause) would be similar to figs. 2 & 3 (which represent the variations of the effect),
but would not be similarly situated unless the angle /3' happened to be equal to G /• 1 or
H r 1 ; and we find but httle appearance of similarity displayed by the curves ; the
extent of the likeness will be exhibited more distinctly in- what follows. We may here
state that if the quantity I were negative, it would only have the effect of increasing
/3' by 180^

22. The lines CD, E F in figs. 2 & 3 have that direction which gives the sum of the
squai'es of the perpendiculars let fall upon them from the points 1, 2, ... 12 a mini-
mum, and it is very noticeable that the points are arranged in closer proximity to these
lines than to lines at right angles to C D and EF, the points 10, 11, 12, 1, 2 being

3z2

512

ME. C. CHAMBEES ON THE NATUEE OF THE

generally towards one extremity of the lines, while the remaining points are towards
the opposite extremity.

The expression B'sin(/«+5') may be ^^Titten

B' cos (S'-<7) sin (/i+o-)+B' sin (S'— ff) cos (A+ff),
where, a being the angle G r D or H /• F, the former term represents that part of the
variation of the term B sin (/«+§) which gives a maximum of easterly declination when
A=:90^— (7, its coefficient being the resolved part of the lines rl, r2. . . . /■ 12 along
CD or EF; and the latter term represents the part of the variation of Bsin(/; + S)
which gives a maximum of declination when /* = — <7, and its coefficient is the resolved
part of r 1, ;• 2, . . . r 12 at right angles to C D or E F. The following Table shows the
values of B' cos (S' — a) and B' sin (S' — o-) for the different months; —

Table IV.

Toronto. ,1 St. Helena. i

B'cos(5'-ff).

B'sm(S'-ff). ;B'cos(S'-(t). B'.^in^S'-a).

January

+ 1-196
+ -832

- -463

- ^562

- ^875
-1^010

- -951

- -944

+ -i72

+ •296
+ •063

— •354
-•124
+ •298
+ •328

— •188
-•516
+ •013

— •045
+ •033

+ •620 --094
+ •898 ' -^179
+ •569 , -^046
-•193 -^061
—622 ! --150
— •856 +-057
—•921 +^048
-•800 i -^069
-•371 ' +^028
+ •397 +-504
+ •684 +^075
+ •593 -^107

iMarch

April

May

July

September

— -240

+ ^444
+ 1^094
+ 1-471

November

December

In figs. 5, 6, 7, & 8 (Plate XXVI.) the monthly variation of these numbers is indi-
cated by curves whose vertical ordinates are proportional to the numbers, positi^■e values
being reckoned upwards ; and the abscissae are divided into twelve equal parts to represent
the successive months of the year.

23. Remembering what was said in art. 19, we see that if a straight line be drawn in
any direction througli the point ;• of fig. 1, it divides the curve into two sets of six con-
secutive months; and that if perpendiculars be dra-mi from the points 1, 2, .3, ... 12
to that line, their lengths are least in the extreme months on each side of the line, and
increase to maxima and then diminish as the numbers increase, tlu>re being only one
maximum on each side. Now as this is true of any lino, these conditions must be
fulfilled (whatever be the value of the angle /3') by the ordinates of figs. 5, 0, 7, & 8 if
the variations which the curves represent are due to direct action of the sun : and if we
confine our attention to figs, h & 7, these conditions are satisfied in a remarkable
manner ; but as we have found that they are to be demanded alike from figs. 6 & 8

SUX'S MAGNETIC ACTION UPON THE EARTH. 513

before wc can admit the truth of the hypothesis, let us see wliether these curves also
exhibit the requiied form.

24. In the first place, we observe that the range of these curves is much smaller
than that of figs. 5 & 7, and still smaller than the whole diurnal range of declination
at the two stations, and therefore, even if the curves had the required form, we should
still have reason to believe that the direct effect of the sun was less than that of other
forces in operation. For the ratio of the mean ordinate (disregarding signs) of fig. 6 is
to that of fig. 5 as 1 to 4-2, and the corresponding ratio for figs. 8 & 7 is as 1 to 5-3,
while the least possible ratio of the mean ordinates upon rectangular diameters of fig. 1
is as 1 to 2'7 ; and therefore, under the most favourable supposition as to the value of
the angle /3', the mean ordinates of figs. 5 & 7 should be considerably loss, or those of
figs. 6 & 8 greater, in order that the results of observation should accord with the hypo-
thesis. But figs. 6 »& 8 are f;ir from fulfilling the conditions named as regards the form
of the curves ; for we see that neither of them has but a single maximum or a single
minimum ordinate, nor are six consecutive months above and six below the horizcmtal
line deviating most from that line in the middle months of each group of six.

The general appearance of figs. 6 & 8 conveys the impression that the numbers which
they represent are possibly eiTors in the determinations of the respective values of
B' and S', arising perhaps from the uneliminated part of the distm-bances ; but if it were
possible to decompose them into two parts, one obeying the required law of variation
and another followng some other law, it is probable that the mean ordinate corre-
sponding to the former part would be much less than the mean of the combined ordi-
nates : hence it is probable that there is but little trace of direct solar action to be found
in the regular diurnal variations of declination at Toronto and St. Helena.

25. It may be objected, however, that the agreement of figs. 5 & 7 with the require-
ments of the hypothesis does not appear as well in figs. 6 & 8, because the other vaiiable
forces in operation act in partial opposition to the sun, and so mask the character which
figs. 6 & 8 would have if those opposing forces could be separated. To this objection
it will be sufficient to reply that the variations of the quantities corresponding to
B'cos(S' — a) for the second and third terms of the series (t/) have a similar character to
those of the first term, and it appears reasonable therefore to infer that the same
variable force is the cause of the variations of each of the three terms ; but the second
and third terms have been proved (18) to be independent of direct action of the sun ;
hence it is most probable that the variations of B'sin(S'. — a), though they accord with
the hypothesis of direct solar action, are due to some different cause. The similarity of
the variations of B'cos(S'— c) is shown by the curves of figs. 9 & 11 (Plate XXVll.) hi
continuous'^ black, red, and interrupted black lines, which have reference to the first,
second, and third teims of (y) respectively. Figs. 10 & 12 are similar representations
of the variations of B'sin(J' — a) for the first three terms of (y); and it is observable
that these are quite different one from another. In Table V. are given the numbers
from which the curves of figs. 9, 10, 11, & 12 ai'e constructed.

•514

ME. C. CHAMBEES ON THE NATUEE OF THE

Table V. — Showing the variations of the quantities corresponding to B' cos (5' — ff) and
B'sin(S' — (t) for the second and third terms of the series {y).

Toronto.

St. Helena.

For second term.

For third term.

For second term.

For third term.

E'cos(5'-(t)

B'sin(c'-ff).

B'cos(5'-ff).

B'sin(("-(j).

B'cos(c'-CTl.

B'sin(o'--T).

B'cos(o' — <t).

B'sin(^'-<T).

January ...

+ '-giD

-•020

1

, +^497

+ •026

+ •409

+ •196

+ •345

+ •103

Februar}'...

+ -868

-•096

! +^4 29

— 195

+ •503

-•604

+ •235

— •461

March

+ -399

+ •446

; +-120

+ •371

+ •417

— •488

+ •315

-•340

April

— -184

+ •145

— •014

+ •138

+ •050

-•069

+ •232

-•041

May

— -856

—•126

-•371

-•134

-•560

—•081

-•360

+ •089

June

— -875

+ •281

-•295

— •070

— •543

+ •098

-•533

+ •135

Julv

- -574

+ •375

— •370

+ •012

—602

+ •0-4

-•614

+ •125

August ...

-1-212

+ •061

-•751

+ •018

—825

+ •012

-•690

+ •191

September

- -697

-•714

— •459

— •049

—479

-•045

— •307

-•010

October ...

+ -499

-•314

+ •139

-•007

+ •624

+ •207

+ •506

-•082

November

+ •643

-•029

+ •374

—092

+ •568

+ •298

+ •442

+ •094

December

+ 1-075

-•006

+ •694

-•017

+ •440

+ ■398

+ •457

+ •116

An important inference that may be drawn from the curves of figs. 9, 10, 11, & 12, is
that, as those of figs. 9 & 11 exhibit a remarkable likeness of character, while those of
figs. 10 & 12 present little appearance of similarity, the former are probably representa-
tives of true changes, while the latter may ])erhaps be regarded as errors in the deter-
minations, when we obtain the following approximate expression for the variable part
of the diurnal variations in any month of the year,

A{Bsin(/;+a) + Csin2(/;+i3)+Dsin3(/i + y)+&c.},
where A alone varies from month to month, being sometimes positive and sometimes
negative. If we might regard this as a true statement, it would follow, as the law of
variation remains constant throughout the year, the range only altering, that these
variations do not result from those of different forces, but from those of a single force,
or of a gi'oup of connected forces.

26. Having seen that it is probable that if the sun be a magnetic body its direct
magnetic influence upon the earth, as evinced by the regular diurnal variations, is very
small, I think we may safely infer that terrestrial magnetic disturbances are not caused
by variations in the magnetism of the sun ; for if the whole direct action of the sun (at
ordinary times) be insensible, much more will the changes which take place in the sun's
magnetic state be insensible in their effects upon the earth, as it would be difficult to
conceive that the changes in question were ever of such magnitude as to approach the
effect of reversing the sun's ordinary magnetic condition.

27. I conclude, therefore, that the mode in which forces originating in the sun
influence the magnetic condition of the earth is not analogous to the action of a magnet
upon a mass of soft iron placed at a great distance from it, but that these forces pro-
ceed from the sun in a form different from that of magnetic force, and are converted
into the latter form of force probably b)' their action upon the matter of the earth or its
atmosphere.

SITN'S MAGNETIC ACTION LTPON THE EARTH.

515

On the existence of a variation of diurnal range of hecUnation at St. Helena
depending on the Suns altitude.
Wlicn we exjiress the diurnal a ariations of declination at St. Helena for the different
months of the year in the form (y),

^,='Qs\l\{h-\-l)-\-Csil\2{h-\-i)-\-l)ii\n^h-\-l),
the values of the coefRcients B, C, D are seen to be larger about the time when at noon
the sun is overhead at St. Helena, which occurs in the first weeks of February and
November, than they are immediately before or after. This fact indicates the existence
of a secondary variation of the diurnal range of declination which has maximum values
at the times mentioned, and which is additional to the vaiiation of range caused by the
semiannual inequality discovered by General S.ubixe. The monthly values of B, C,
and D are shown in the followin"; Table : —

B. 1 C.

D.

January

'•703 '-726
•994 1-043
-640 j -904
•226 1 -375
•629 1 -292
-808 -243
•872 1 -299

'•622
•968
•904
•619
•352
•428
•499
-534
•427
•857
•713
•663

April

May

July

•771
•331
•594
•737

•682

-524
-204
•939
•903

•817

Had this been a feature of only one or even two of the three coefficients, we might
have supposed it to be an accidental circumstance which would perhaps be reversed on
a repetition of the observations ; but as the three coefficients agree in giving the same
testimony, I think we must allow it to be not merely an apparent but a real variation.

Before concluding, I must acknowledge the important help that I have derived from
Mr. Archibald Smith's ' Mathematical Theory of the Errors of Ship's Compasses,' of
which what has preceded is little more than an application to another purpose. I am
also indebted to Mr. Thomas W. B.aker, of the Kew Observatory, for the very efficient
assistance which he has afforded me in the rather laborious computations involved in the
discussion.

Note on §§ 26 and 27. By Professor W. Thomson, M.A., LL.D., F.R.S.

Received October 28, 18G3.

If the sun were a magnet as intense on the average as the earth, the magnetic force
it would exert at a distance equal to the earth's, or, let us suppose, 200 radii, would be

516 ox THE NATUEE OF THE SUX'S MAGNETIC ACTION UPON THE EARTH.

only 8,000,000 of the earth's surface magnetic force in the corresponding position rela-
tively to its magnetic axis. Considering, therefore (according to the principles explained
by Mr. Chambers), the sun as a magnet haring its magnetic axis nearly perpendicular
to the ecliptic, we see that, with an a^■erage intensity of magnetization equal to the
earth's, the effect of reversing the sun"s magnetization would be to introduce, in a
direction perpendicular to the ecliptic, a disturbing force equal to about 8,000,000 of the
earth's average polar force, and would therefore be absolutely insensible to the most
delicate terrestrial magnetic observation yet practised. A disturbing force of this amount,
acting perpendicularly to the du-ection of the terrestrial magnetic force about the equator,
would produce a disturbance in declination of only half a second ; and the sun's mag-
netization would therefore need to be 120 times as intense as the earth's to produce a
disturbance of 1' in declination even by a complete reversal in the most favourable
circumstances. These estimates appear to me to give strong e-\ideuce in support of the
conclusion at which Mr. Chambers has arrived by a careful examination of the disturb-
ances actually observed, that no effect of the sun's action as a magnet is sensible at the
earth.

The same estimates are applicable to the moon, her apparent diameter being the same
as the sun's. It is of course most probable that the moon is a magnet; but she must
be a magnet thousands or millions of times more intense than the earth to produce any
sensible effect of the character of any of the observed teiTestrial magnetic disturbances.

[ 517 ]

XX1\'. On the Calculus of Symbols. — Third Memoir. By ^^'. H. L. Russell, Esq., A.li.
Communicated by A. Caylet, F.E.S.

Received May 15,— Read May 21, 1863.

In my second Memoir " On the Calculus of Symbols," I worked out the general case of
multiplication according to one of the two systems of combination of non-commutative
symbols previously given. In the present paper I propose to investigate the general
case of multiplication according to the other system. I commence with the Binomial
Theorem, to which the second system gives rise. In my previous researches I obtained
the general term of the binomial theorem when the symbols combine according to the
first system by equating symbolical coefficients ; here, on the other hand, I consider the
nature of the combinations which arise from the sjinbolical multiplication, and obtain
the general term by summation. I next proceed to the multiplication of binomial
factors. Here the general term is obtained by considering the alteration of weight
undergone by certain symbols in the process of multiplication. The multinomial theo-
rem according to the second system is next considered and its general term calculated.
I conclude the memoir with some applications of the calculus of symbols to successive
differentiation. This paper completes the investigation of symbolical multiplication and
division according to the two systems of combination, the general case of division having
been worked out by Mr. Spottiswoode in a very beautiful memoir recently published in
the Transactions of this Society.

I shall commence with expanding the binomial (T + ^(g'))" in terms of t.

I shall put S in place of the symbol p — ■, and write a„ for «, ^^^ ■, ?^^ , ''~°'"^ •

^ * •' ^ </f "' ' 2 .3 a.

Now {'!r-\-6^y='7r'-\re§.T-\--7r6^-\-0{^)\

(a- + %)' = ^ + (or'^f + ^^^ + 6^) + (%V + 6^6§ + ■TrOf) + 6^.

Hence we evidently have

(T+%)'-=^+S(^-'^^)-f-5;(^-=^f )+S(t"-^%^)+ ,

where X{'7r'"6f) denotes the sum of all the factors in which the symbol (t) occurs (m)
times and the symbol {6§) (r) times ii-respective of position.
Now if a-\-b^=m.

Again, if a + b + c= m,

+c^l6f-7r-' -\-b,cMg^0i^~^ + Kc,h6^l'6g^'"-'+ . .
+ c, S'^f V"-= + b,c,h''00i)g'^-'+ . .
+ c^^ef-jT-^ + . . . ;

MDCCCLXIII, 4 A

518 ME. W. H. L. ELTSSELL ON THE CALCULUS OF SYBIBOLS.

and \i a-\-h-\-c-\-e=m, we have

+ cJ^l'Sf-Tr"'-" +

Hence we see that if a-\-b-\-c + e. . . . -\-I=n, the general term of 'Tr'ilg. . . . ■tt'' 0^ir'' d^ir'' O^tt"
will be

Ib^c,^^ . . . 1,V"6p . . . h'-Sgh'-dg . 7r"'-\

where the sum 2 extends to all values of a„b„Cf, .../,, included in the equation

bo+f„+ — +/»=«;

and if (r) be the number of the factors 0§, we shall have

t/Op ^%9r"^fcr''^g'T" = (%)V"+ {i,(^f)'-'S^f +C-,(^f)'-'S(%)' + ^,(%)'-'S(%)'+ . • . }^"'^'

+ \b.idpy-'h''0g + c.J^6gy- 'h%Ogy + e,{Ogy~'h%0§f + . .. +b,c,{0gy-^hOg'b0i

+ b,e,{0gy-%0gyh6g -\- . . ..}'7r'"-^ + Scc. + lbi,c\ . . . /^h'-Og.. . h'"0§h''"0g'r"'-' + ...,

and hence we shall liave

i^"'{Ooy={dgy^"' + { (/., + /;; + . . )( c^y-^id^ + (c\ + < + . .)(%)'-'S(^^)= + {e\ +e\+.. )(%)'- '5(%)^' + . . } ^"
+ {U,+f/' + . .){flsy-'h'0g+{c,+d+ . .)iOsy-ridgy+ . . +{i/,c,+f;;c:+ . .)5%s%+ ..}7r""-'

+ . . +S,„S/V',- • /iP"^S ■ .S'"^fS''"%^"'-"+ ,

\\ here S, extends to all values of /y„C!„ . . . included in the equation

f>„ + e„ + f'„+... +/„=.s,
and Xm to all values of abc included in the equation

n+b+c+. ...+l=ni.

Now the general tcim of the binomial {'7r-\-0oy will be the term involving t'' in tlie
^um of

Xi^'-'O^y Xi^"-Hjf), &c.

MR. W. IT. L. RUSSELL OX THE CALCULUS OF SYMBOLS. 519

Hence we easily see that the coefficient of" t^ in tlic binomial is

+s„_,s„_^_,V,„ .... /,/"% . . . I' Sol''' 60+ . . .,

where

a-\-b-^-r-\-e-\- . . . . I =n — v

^i+c„+.... +^„ = ??— /!//— f.
I shall next investigate the general term of the symbolical product,

(7r + ^,.)(7r + l5)(,r + 5,.) (^ + fl„§).

Now

(7r + 5,§)(7r + 62§) = ?r- + 7r9,§ + l§7r + 5,§a.,§;

(T+6,§)(«-+a,§)(x+53§)=^'+(9,§,rHTl§T4T-63g)

+ (6,§5,§7r+9,e7rfl,§+7rl§63g) + fl,§lg93§;
+(5,fl§T-^+S,t7rS3gTr+7r5,g53§7r+9,§7r^9,g+7r9,g7r9,g+7r^63g9,§)

Hence we deduce the following law of multiplication in this case.
Consider the symbolical expression

^%.hy%Myu?) 7r%x^y^

where

«+J+c+ -\-s=n—r, (1.)

a, = «+l,

b, =z « + /> + 2,

c, = a-\-b-{-c-\-S,
&c. = &c.,

l^ = a+b-\-c+ . . . . +l+r,

and let a,b,c,...s have all the values which can be assigned consistently with equation
(1.); the sum of the symbolical terms thus formed wiU be that group of the symbolical
product in which the factor ir occurs (n—r) times. And by giving (r) the successive
values 0, 1, 2, 3 . . . «, we obtain the entire product, and may Avrite

(7r+9,g)(7r + 5,g)(7r+93?). . . . (7r+5„o) = 7r''+S9,?7r-'+S9,t9,eT"-^ + S6,§6,§63§rr"-'+ . . .
+ %6,^B,oS^o 5,o?r"--+ . . .,

where 29,f7r"-', &c. are the groups we have just considered.

4 a2

520 ME. W. H. L. EUSSELL OX THE CALCULUS OE SYMBOLS.

Now if \!/, (p, 5 are any functions of (s). and a-^Ij-^c + e=m, we have

7r''-4/7r''<p7r'67r"=:'4'(}>^3-'" + 5,\}/<pS5x"'-' -\-b„'^^l-(nr"'-^ +■■■

+ C,\!/5<pS7r"'-' +5,r,v{/S<pS57r"'-'4- • • •

+ C,^^S-(p37^"'-■- + . . .

+ C,r,S4/MT'"-'+ ...
+ f„S--J/^57r"'-'' + . . .
where J,, e„ t&c. have their original significations.

Hence we see that the general term of Tr'^.g 7r'9^_.j(§)7r''5,_i^7r*5,g . tt" is

sVc„ ^'„^ -^.f • • • 5pL,i^,(§)S'°i(§K'"-%

where

a + J +r +f + ...+/ =«J,

^^+''o+fo+ • •• +^0=5;

hence the general term of X^^^So^^^^ . . . S^ott"' is

and we find the symbolical coefficient of •a^ in (•s-+5,c)(T+S-.g)('''+536). . . (tt+S^o) to be

'''^''''' a+b+c-^...+l=n-K

Let us now expand the binomial (x'- + 5(c)t)". We might of course do this by putting

6,o = S,^ = ^,o= . . .

lc= 5,0 = ^60 = 0

in the previous investigation, but we shall proceed as follows : —

(7r-^+% . 7r)''=7r^''+S(7r7'-'(% . 7r) + %(7r')"-'{^o . xf + &C. +S(7r^)'-'-(% . 7rY+ ...,

and the general term of S(5r'^)"'(5§ . tt)"" ^^'ill be

SA(2*+l)..(2e+l),, .... (2/),S''% . . . 5<'6^S''9§7r^'"+'-',
where

« + /> +r + ....+/ =»i,

and the symbolical coefficient of tt" in the binomial will be

+.S„-.S.„-.-.(2i + lM2c+lX .... (20,S'»9g . . . S^'9?,

-MR. W. H. L. RUSSELL OX THE CALCULUS OF SYMBOLS. 521

where

a + b -\-c + . . . +/ r=n — y.

f>o+t'o+ ■ ■ ■ +''0=-"— /^ — "•
To determine the general term in the expansion of the multinomial exi)ressioii
(7r"+fi,§7r''-'+fl,,cT"-- + 9,§7r"-'+ . . . )'".
This reduces itself to finding the symbolical coefficient of x" in the expression

where a.-\-^-\-y-\- . . . . =i)i, and w", fl,§7r"~', &c. are combined in every possible way so
that tt" should occur- (a) times, fl,07r""' (j3) times, ^.^gTr""^ (y) times, &c. irrespective of
position, — or, in other words, to determining the symbolical coeflScient of tt'' in the
expression

r-.(fl,fT"-r'(6,5,r"-)v.(53gx'-)?' . . 7r''^(6,g7r'-')^(S.,f7r''-7^<63?T"-)?^" . . 7r"-(6.§;r'-')^(S.f T'-'I'^S^grr"- ')^- . .,
where

i3,+/3,+/33+....=/3,

71+72 + 73+ =7. &c.

Now the symbolical coefficient of

nm-{(3 + 2y+S>;+ ....)-«
in the preceding expression will be as follows : —

i . . (h — 3)5^(« — 3)5; . . (^3 factors)(?i — 2)y;(n— 2)^;' . . (y^ factors)(7J— l)^;(w— l)^; . . {^^ factors)(?ja3)„; . .
(n — B)j^(n—S)^ . . (^2 factors)(« — 2)y;(?i — 2)^; . . (y, factors)(?z — 1)0;(« — l)^;' . . ((S^ factorsX^a^),; . .
(«-3)j;(n-3)t; . • (?, factorsXM-2)y;(«-2)y; . . (y, factorsX«-l)fl;(«-l)3" . . (/3, factorsX//a,).;

h< 6,gs^:. (i.^iK 5,g . . . J'-; s.gS'-^ i§ . . . s^; 6,^...

5-3 fl,gS^3 6,gS^3 fl,g . . . S'-, S,eS''3 6,§ . . . 5^3 635, . . . &c.,

where the sum of the indices of S must equal (*•).
From this expression, by putting

mft-(|3 + 2y+3?+ . ...)-s=i^,

the symbolical coefficient of tt** in the expression of the given multinomial may be
immediately deduced.

There are certain methods of expressing the differential coefficients of implicit
fiinctions by means of symbolical notation which I notice in this place, as the method
of summation employed for that pm-pose is similar to some of the symbolical summa-
tions we have already considered in this paper.

522 MR. W. H. L. EUSSELL ON THE CALCULFS OF SYMBOLS.

Let F{x, y) be a function of x and y, y being a function of (.r), and //,. y^, y-, the

successive differential coefficients of (y). Then

^ _rfF dF
dx dx "' dy y^''

dx- — ;^ + -^ rfW^ ^' + di/-^ y'' + (/;/ ^- ■

d^Y d^F „ d^F „ jPF „,f^ 3 1 o ^^Z. 1"!^ 1^

rf?^=^+^flS%-^'+'' fto//.^'+</i/« ■^'+'^ dxdijy^--"^ dy^ y^y^+dy y^ '

dx-* — dx-* '^'^dx^'dyy'^^ dxdi/y^'^'^dxyy'y''^ dy^y^

d^F „ d^F d^F] , d^F d-F . , .., „, , dF

Now let Ul „ be the coefficient of -; — ;— F in the expansion of -j— ■ Then

dx"'dy'' dx'-

d^- - d^- ^' dT^y^^ 2 dx'-hb/y--^' 2 3 dx-'dy^y'

r-\ /./-F .,/^_9) ^"-'F „ , ('-2)(r-3) ^-F ^. ^ ^A

dx--^dy "^ '• ' '

We easily obtain this law of coefficients,

u™.„=| uc:+y.u::„'i,+u;-_\, „ ,

whence

u;,„=^ur,:+y,ur,'_,

MR. W. II. L. RUSSELL ON THE CALCULUS OF SYMBOLS. 523

Let )i — v=l, then Ui,„_,=Ui.,=//,, and U^,„ = Ui,„_,=^:c. = 0. Hence

where S( ■ .,-« j/"~') denotes the sum of all the symbolical products in wliicli r occurs

r — n times, and y, (>' — 1) times.
We have

u:,„=|u:,-:+y.ui:-„^\+ur„',

from which, substituting for U^„' the value just obtained, U' „ may be found in a similar
manner, and so we may proceed. I shall not, however, enter upon these higher coeffi-
cients, my object being principally to call attention to the use of symbolical expressions
in expansions of this nature.

The subject of implicit differentiation has been treated by Mr. George Scott of
Trinity College, Dublin, in a very elegant paper in the Quarterly Journal of Mathe-
matics, vol. iv. p. 77. His results have great generality, but do not appear to include
the abo^e.

[ 525 ]

XXV. Numerical Elements of Indian Meteorology. By IIermaxx de Schlagixtweit,
Ph.D., LL.B., Corr. Memb. Acad. Munich, Madrid, Lisbon, &c. Communicated by
Majok-General Sabixe, P.R.S.

Received May 4,— Read May 21, 1863.

First Series. — Temperatures of the Atmosphere, and Isothermal Lines of India.

I. Materials collected : calculation of the Daily Mean.
II. Tables of 207 Stations of Mean Temperature — Months, Seasons, and Years.

III. Decrease of Temperature \\'ith Height in the Tropics.

IV. Thermal T)-pes of the Year and the Seasons.

I. Materials collected : calculation of the Daily Mean.
The numerical elements of the mean temperatiu-e * of the atmosphere for India and thf
Indian Archipelago here presented, I had occasion to collect during the years 1864-58.
For judging of the value of the data I had obtained, and for working out the general
results, it was very favom-able that, for most of the stations, I had occasion personally
to see the instruments employed and the mode of theii' being put up.

Ab-eady some ycai-s ago a considerable number of these stations had been pubhshed
for the year 1851, by Dr. Lambe in the Journal of the Asiatic Society of Bengal, as
well as by Colonel Sykes in the Report of the British Association for 1852 ; but as the
materials sent in consisted, nearly exclusively, of results presented as means, which
however were but the plain arithmetical mean of the respective hours of observation
without any further modification, it was particularly welcome to me that the Indian
Government, by the mediation of Dr. Macphersox, handed me over the oaiginal manu-
scripts, now forming tliirty-nine volumes in folio.

A new calculation of the mean temperatures showed for many of these stations,
particularly for the warmer period of the year, results lower by many degrees than the
values formerly adopted ; the difference would have been greater still and more frequent.
if for many of the Indian stations the daily variation of temperature had not been
included altogether %\ithin comparatively narrow limits.

The pubUcation of Colonel SYKEsf in 1850, the observations communicated being
his own, or those of contemporaneous residents, contains throughout means based upon
houi'S carefully selected.

* All temperatures are Fahrenheit.

t " Disciission of Meteorological Observations taken in India," by Colonel W. H. Stkes, F.R.S., Philosophic-al
Ti-ansactions, Part II. 1850.

MDCCCLXm. 4 B

526 DR. H. DE SCHLAGIXTWEIT OX THE XOIERICAL

Also the meteorological publications of Dove and Schmidt* contained important
contributions for completing the number of the Indian stations, and for comparing
them ^\ith the surrounding regionsf.

The hours of observation at the various stations had been in general selected so as to
include the minimum at the time of sunrise and the hours 10 a.m., 4 p.m., these two
nearly coinciding mth the barometrical extremes ; also the maximum of the day a little
after 2 p.m., and an evening observation is very frequently contained in these Tables ;
but. with few exceptions, the latest period was 6 p.m., or sunset. This ciixumstance
excluded therefore the introduction of an evening hour- more distant from the maximum,
such as 9 p.m. or 10 p.m., mto the calculation of the mean. A very favourable modifica-
tion it was, however, that hourly observations existed for several stations, very accurately
made, though .situated in regions where the daily variation of temperature is not a very
great one. These stations are Bombay, Calcutta, Madras, Trevandrum. Already Dove,
so very careful in completing his collections of meteorological materials, has published
several years for each of these stations J. For calculating such Indian stations as show
a more continental character in their variation of temperature, I could take advan-
tage of the observations which we had occasion to make ourselves during our travels,
a material which, I think, presented sufficient data for defining the mode of calculation,
by their number as well as by their geographical distribution.

A combination of sunrise and sunset with either the maximum of the day or
the observation at 4 p.m. showed very unfavoiu"able results, even if variable coefficients
were introduced for the different months, since, for the various geographical regions, the
changes in the daily variation of temperature during the year are very great. Also the
( ombination of the extremes with one morning hour, as I formerly had applied them to
Alpine stations)^, gave no satisfactory results, since in India the morning hours 9 a.m.
or 10 A.M. had risen already considerably more above the mean of the day than is the case
in the temperate zone.

* Dove, " Tafel der mittleren Temperaturcn verschicdener Orte in Reaumur'sehen Graden," and " Ueber
die nieht periodischen Aenderungcn der Temperatiirvertheilnng," 6 parts.

t Among.st the 207 .stations of the numerical Tables, pages 532-537, the following stations had to be taken
over without recakidation, or without the addition of new material: from the publications of Colonel Stkes,
Atare Malle, Ahraednagar, ilahabaleshvar, Mahu, Phaltan, Tiina, Satara ; from the series of the Medical
Board Observations only Gughcra remained without the addition of new material ; and from Dove's Tables I
took over, with their values unchanged, Alor Gaja, Ava, Bangkok, Chanderaagiir, Chusan, KSlsi, Krindi,
K.-inton, Makilo, ManUla, Mozufarj)ur, Pondieheri, Trivandrum. Dove's Scring.apatam is the year 1810 for the
neighbouring fort, French Rocks, for which I was able to add 1814, 1853, and 1854. In the " Lehrbuch der
Meteorologie," von Schmidt, 1800, 1 found in addition, for the Archipelago, Banjuvangi, Palanbiing, Lahiit. For
want of details about the decrease of temperature with height in these regions I excluded thoni, their lieight
being 2138, 2110, and 2104 feet.

i On the Daily Variations of tlie Temperature of the Atmosphere, Abhandl. Beri. Akad. for 1840, pp. 104-0.

§ ScHLAOiNTWEiT, " Xcuc UutcTS. phys. Gcogr. d. Alpen," page 325 ; I had obtained there the following
coefficients for deducing the mean temperature from the extremes and 0 a.m. : for the minimum 0-5 ; for the
maximum 0-4; for Oa.m. 0-11.

ELEMENTS OF IXDLAJN: I\IETEOEOT,OOY. 527

The arithmetical mean of the extremes, wlicre rei,'i.st('iing-iiistiumcnts had been used,
showed temperatures in general too warni throughout the year ; but this Aery circum-
stance induced me to try the combination of 4 p.ii. (wliicli I had for all stations) with the
obseiTations at suni'ise ; the latter is nearly always identical (except at stations in very
great heights) with the minimum obtained by registering-instruments, and four o'clock
is cooler, though but little, than the true maximum ; the result was a mucli more satis-
factory one tlum I had expected.

The coincidence of the minimum of temperature with sunrise is particularly general
in the tropics. It materially depends upon the rapid ascent of the sun above the
horizon, whilst with us, especially in summer, the effect of insolation upon clouds and
the higher strata of tlie atmosphere is partly felt already on the surface of the earth
before the sun himself is visible above the horizon. In very great heights, again, chiefly
if it be a peak in a very isolated position, the tropics show also modifications similar to
those of the temperate zones. There I found, just as I formerly had seen, too, on the
Vincent Hiitte (southern slope of Monte liosa), tliat the temperature frequently began to
rise several hours before sunrise*.

As another characteristic modification of the morning period in the tropics, I may add
here that very frequently the absolute minimum is followed by a second, though minor
depression. This becomes best marked in the tropical seas ; I found it greatest, when
the sky was clear, five to ten minutes after sunrise, and it amounted not unfrequcntly to
a full degree, but it tlqwx went lower than the absolute minimum preceding. I con-
sidered the cause of it to be the change in the relative humidity, which has attained its
maximum nearly at the moment of sunrise. The appearance of the sun aboAC the
horizon coincides, too, -with the heariest precipitation of dew, and from this moment the
relative humidity is rapidly decreasing whilst the temperature begins to rise. Not onlj-
is radiation now increased with the transparency of the atmosphere, but also the amount
of heat becoming latent in consequence of the dissolring of vesicular vapours miglit par-
ticipate in producing the second depression of temperature.

For presenting an immediate comparison of the value '--„ — '—' with tlie mean of

the 24 hours, I have given the corrections to be applied in the following Tables (with
" — " if the calculated value is too large, with " -|- " if it is too small), and have added
the corresponding corrections for three other combinations. At Bombay and Calcutta,
hourly observations are made everyday, Sundays excepted; I took 1855 as the year the
least distant from any other observations. For Tonglo, Faliit, Islamabad, and Leh, the
periods of observation are only months. For Ambala I had no quite regular series com-
pletely including the daily period, but the considerable number of observations from
morning to night, combined (by tlie particular kindness of the observer, Dr. Tritton) with
very good extremes and isolated nocturnal observations, allowed me to define with suffi-
cient precision the form of the monthly curves, and to deduce from these the hours still
wanted.

* Neue Unters. Geogr. Alpen, pp. 278-80.

4b2

528

DK. H. DE SfHLAGINTWElT ON THE NUMEEICAL

I thought it not uninteresting to complete the comparison of my mode of calculating
with those generally used, by adding the value of ™l!^il!l — ^~ also for some other

.stations, situated beyond India, and greatly differing in reference to their climatological
character.

A. From India, the lUmdlaya, and Tilet.

Bombay in the Konkan, lat. N. 18° 53' 30", long. E. Green. 72° 49' 5", height L.a.L.S.*

ia>5.

Mean.

S. R.+IV.

Max.+Min.

VI.+II.+X.
3

ATii.+ii.+o.IX.
4

January

February . . .

March

April

74-7
76-9
79-3
820
860
83-8
820
821
81 0
82-6
80-6
77-7

-0-6
-0-5

0
+0-3
-0-3
+01
+0-1
-0 5
-0-2

0
-07
-07

-0 9
-0-8
-0-5
-0-4
-0-7
-0 5
-0-7
-0-7
-0-7
-0-7
-12
-1-2

+0-1

+01
+05
+0-6
+0-4
+ 0-2
+01
+ 0'4
+01
+0-2
-01
+0-1

0

0
+0-4
+0-3
+0-2
+0-3
+03
+0-1
+0-3

0
-01
-01

Mav . .

July . ..

August

September ...

October

November ...
December ...

Mean ...

-012

-0-38

+0-11 +0-08 1

Calcutta in Bengal, lat. N. 22° 33' 1", long. E. Green. 88° 20' 34', height L.a.L.S.

S. R.+IT.
2

Max.+Min.

VT.+II.+X.
3

vri.+ii.+2.ix.

4

2

1 January

February . . .

March

April

May

June

July

August

September . . .

October

November ...
December ...

665
72 1
793
823
85-9
85-6
82-3
837
82-3
81-2
74-4
669

0

-08
-0-6

0-
-06
+01
+04
+02
+0-3
+0 2
+02
+01

-0 9
-11
-08
-0 3
-11
-0-6
-0 5
-0 5
-0-6
-0-4
-0 9
-12

0
-0 3

+0-4
+ 11
+03
+04
+01
+0-3

0
+03

0

0

0
-03
+0-5
+ 13
+07
+0 3

0
+03
+01
+ 0-2
+03

0

Mean ...1 j -0-02 j -0-73

+011

+0-14

Ambala in tlie Punjab, lat. N. 30° 21' 25", long. E. Green. 76° 48' 49", height 1026 feet.

' 1855.

Mean.

S.R.+IV.

2

Max. + Min.
2

VI.+II.+X.
3

VII.+II.+2.IX.
4

January

February . . .
March

50- 1
59-5
565
760
921
95-4

-01
-01

-0-2
+07
+ 1-7
+ 1-2
+ 0-3
+ 11
+ 11
+03
-1-9
+08

-06
-0 7
-0-3

+0-2
+ 11
+ 0-9
+0-2
+ 0-5
+0-9
+01
-22
-0-2

+ 05
+ 1-3
+ 1-4
+ 2-3
+ 0-5
+ 0-2
+01
-0-6
+ 11
+ 1-8
-07
-0 9

+0-5
+ 10
+0-4
+0-2
-11
-10
+ 1-3
+20
+0-4
+07
-17
+0-8

July

1 August

September . . .

i October

1 Novemtrer ...
1 December ...
f

83-8
87-9
82-4
73-4
602
559

1 Mean

+0-41

-0-01

+0 58 +0-22 1

* Thi.s abbreviation i.s placed for " a little above the level of the sea."' The feet arc EuglLsh.

ELEMENTS OF INDIAN METEOROLOGY. 52'J

Tonglo Peak in Sikkim, lat. N. 27" 1' 50", long. E. Green. 28° 3' 65", height 10,080 feet.

1855.

Mean.

S.B.+IV.
2 ■

Max.-(-Min.
2

VI.+II.+X.
3

VII.+II.+2.IX.
4

M«y j 481

+0-5 -1-5

-0-2 1 0

Faliit Peak in Sikkim, lat. N. 27' 6' 20", long. E. Green. 87° 59' 0", height 12,042 feet.

1855.

Mean.

S.R.+IV.
2

Mai.+Min.

VI.+U. + X.
3

VII.+II.+2.IX.

1

May

46-9

-01

-0 5

0 0

1

Islamabad in Kashmir, lat. N. 33° 44', long. E. Green. 75° 8', height 5160 feet.

1856.

Mean.

S. R.+IV
2

Max.-|-Min.
2

vi.+n.+x.

3

vii.+n.+2.ix

October ' 51-3

+07

+0-3

+ 1-3 -07

Leh in Ladak, lat. N. 24° 8' 2", long. E. Green. 77° 14' 36", height 11,527 feet.

1856.

Mean.

S.R.+IV.

Max.+Min.
2

vi.+n.+x.

3

VII.+II.+2.IX.
4

September ...

601

-0-1 -0-2 +07

-0-2

B.* F^-om the temjjerate zone in loio elevations.
Rome, lat. N. 41° 54', long. E. Green. 12° 25', height 170 feet.

Mean.

S. B.+IV.
2

Mai.+Min.

vi.+n.+x.

3

VIT.+II.+2.IX.
4

Januarr

July

49-95
75-47

-0 07
+0-36

-1-15 -0-22
+0 20 4-1-62

+0-09
+0-97

Greenwich, lat. X. .51° 29', long. E. Green. 0° 0', height 156 feet.

Mean.

S. R.+IV.

Mai.+Min.
2

VI.+IL+X.
3

yn.+ii.-i-2.ix.

4

Janoary

July

35-45
59-65

-002
+0-40

-0-40
-0-34

-0 31
+0-45

-0-22
-0-13

St. Petersburgh, lat. N. 59° 36', long. E. Green. 30° 18', height L.a.L.S.

i S.E.+IV.
Jlean. tj

Max.+Min.

VI.+II.+X.
3

VII.+II.+2.IX.
4

January

July .

l.'!-57 +016
62-37 -0-12

-0-11
-0-13

-029
+0-47

-0-25
-Oil

* The date is not added, the means being taken from various series, all of several years' duration.

530 DR. H. DE SCHLAGES'TTTEIT ON THE NUMERICAL

Toronto, lat. N. 43"" 40', lon^. W. Green. 79° 22', height 340 feet.

Mean.

S.R. + IT.

Mai. + Min.

TI.+II.+X.
3

yiI.+IT.+2.IX.
4

January 26-37

July 65-60

+0-22
-0-06

-0.36
-0 07

-0-18
+0-94

-0-40
+0-20

C. From the Alps.
Geneva, lat. N. 40° 12', long. E. Green. 6° 10', height 1334 feet.

Mean.

S. K.+IT.

MM.+Min. VI.+II.-I-X.

YII.+II.+2.IX.
4

January

July

30-81
64-16

-013

+ 0-5!l

-0-54 -0-18
+043 0

-0-16
-081

St. Bernard Hospital, lat. N. 45° 50', long. E. Green. 6° 6', height 8108 feet.

Mean.

S. R.+IT.
o

Max. + Min. VT.+II.+X.

TII.+II.+2.IX.
4

January

July

13-41

4284

+014
+ 0 61

-0 31
-018

+002
0

-0 02
-0-31

II. Tables of Mean Temperature for the Month, Seasons, and the Year (207 Stations).

Ten geogrctpihical groups are formed of the meteorological materials, and within these
the stations are arranged alphabetically.

The number of stations is 207, and they arc distributed as follows: —

1. Eastern India : 1, Assam; 2, Khassia Hills 12

2. Bengal and Bahar, and Delta of the Ganges and Brahmaputra ... 36

3. Hindostan, the upper Gangetic plain = 27

4. Panjab, including the stations west of the Indus 24

5. Western India : Rajvara, Guzrat, Kach, Sindh 10

G. Central India : Berar, Orissa, Malva, Bandelkhfind 15

7. Southern India, hilly districts: 1, Dekhan and ^laissiir; 2, Nilgiris . . 29

8. Southern India, coasts : Konkan, Malabar, Kama tik 24

9. Ceylon 10

10. Indo-Chinese rcninsnla, Archipelago, and China 20

The transcription of the geographical names is the same used and detailed by me in
our 'Results'*; the vowels are written as in Italian and German, the consonants as in
English, with very few modifications, such as " th " being an aspirated " t," (S:c. Nasal
modifications of the vowels are indicated by a circumftex. Every word has its principal
accent marked by the usual sign. The sign ^ above a vowel shows its iinjiert'cct plio-
netic formation, such as " e " in herd.

* The full dftuil is contained in vol. iii. pp. i;5'J-G0.

ELEMENTS OF INDIAN METEOROLOGY. 531

The latitude is north, unless an S is written before the respective numbers.

The loiujitude, east of Greenwich, is referred to tlie Madras observatory, its vahie being
adopted =80° 13' 56". The sign « before the stations indicates tliat the latitude and
longitude have been determined by the great Trigonometrical Survey of India; oui-
own determinations are marked by the sign f . For the remaining stations the coordi-
nates are taken from the most detailed maps.

The lie'ujht is given in Englisli feet ; I took it from our " Hypsometry," vol. ii. of the
' Results.' Heights in round numbers, for which I had no detailed data, are put in
brackets. To places veiy little elevated above the level of the sea L.a.L.S. is added.

The seasons are formed as it is usually done also for the stations of other latitudes;
these groups coincide besides, for Central and Northern India, with the character of the
climate in general. For the stations in lower latitudes, however, the type of the climate
only allows of distinguishing a hot season, a rainy one, and a cool one.

The numerical values* arc corrected only for instrumental errors, or combinations of
hours not sufficiently careful ; but the influence of height, and in consequence the differ-
ence from the isothermal lines next to the respective stations, had to follow separately.

* The stations where no dceira.iLs arc seen (but fractions or full numbers onlyj are stations of somewhat
minor aceuracj-.

DR. H. DE SCHLAGIXTWEIT ON THE 1STTMEEICAL

K
pi

cz:

tS

75-5

73-2
751

75-8

74

75-2
75-3

73

74
73-4

H
<

W

i

Cl
CO

i

2

cc

t*

to

to

lO

?>
^*

to
<^

CO

«

O

"c*

fc

X

Q

|E5

d

807

747
76-7
77-5

76-1

77-2
77-2

76-9

76-9
75 6

<
*^ 1

817

82-1
80-9
82-6

83-7

8i-3
83-9

821

83-2

82

<

75-1

73-7
773
77-4

75-8
73-8
75-7
765

72-8

737
74-3

1-5

64-5

62-2
65-3
65-6

60-3

62

66-7

63-7

62

622
61-6

1

65-2

61

64-9

65-5

60
625
64-5
63-5

60-4

62-4
61-3

Not.

71

67-4

70-6

71-1

66

66

715

69-8

67-4

69-4

68-3

O

88-5

75-6

78-5
79-2

79-2

78-5
786

77-1

78-3

77

t

82-5

81
81-1

82-2

83

81-5
83-2

817

831
81-3

<

83

81-8
81-6

82-9

85

82
84-4

82

83-5
81-2

►?

82

83-7
81-4
83

84-5

82-5
84

82-7
83-6
82-7

a;
%

June.

80

80-7
79-8
81-8

81-5

82

79-5

83-4

81-5

82-4
82

s-

s

78-5

771

80-2
80-4

79-5
79-5
80
81-1

78-4

7805
78-9

4*

757

727

78
77-4

76-5
73-5
77
76-4

72

73-8
74-5

s

71
713

737
74-5

71-5
68-5
70
72

68

69-3
69-5

4

66-2

63-4

67-8
67-6

61-5
62-5
69
66-4

64-2

641
63-8

►^

62-2

622
63 2
63-6

595
61
66-5
61-3

61-4

60
59-7

Height.

ft.
(100)

396

(120)
134

(350)
410
155

(250)

(400)

(370)
278

t- -.C -C CO

Ml ^o 1^ i rs « m — a IN 0-. -i
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^ ci Ci ci =-. c^ c OJ 3; c; c> ii

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2i> 18

27 32
26 II

26 5-8

26 33

27 31
26 24
26 21

26 52

27 2
26 34-6

"5
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DibnigSrh t

Goal para »

GoiiAtti t

Golaghat

Lakliimpur

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<

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78

80-4

76-4

78-7

78-9

79-8

75 5

78-34

78-2

79-7

76-6

76-0

77-4

77-8

79

78-4

77-1

79-4

73-4

77-3

78-4

S.O.N.

756
80-1

78-7

79

77

79-2

77-3

79-70

76-9

787

75-5

77-7

77-2

77-8

79-6
79-8
78-7
79-2
719
78-2
81-4

'i

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63-2

69-3

65- 1

69-7

63- 1

68-5

64-3

6770

65

63-7

60

63i
66

68-7

67-7

67-2

65-5

62

64-6

64-85

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79-7

81-2

72-2

79 5

78-6

81

74

77-99

79-6

77-3

75-4

76

80

79

80-1

78-9

78-2

735

80

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ELE^IE^"TS OF IXDIAN JVIETEOROLOGY.

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DE. H. DE SCHLAGIXTWEIT ON THE XUMEEICAL

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538

DE. H. DE SCHLAGIXTWEIT OX THE NUMEEICAX

III. Decrease of Temperature 7iu'fh Hei<jht in the Tropics.

The decrease of temperature with height had to be taken into consideration, not
only on account of its practical importance for the selection of stations and sanitaria,
but also for comparing the different parts of India, independently of the accidental
height of the observer's residency, and for drawing finally the general isothermal lines.

For the Dekhan and Central India, Puna, Purandar, and French Eocks could be com-
pared mth the coasts of the Konkan and the Kariiatik; for the South I had three
stations in the Nilgiris and one in Ceylon, which could be referred to the shores of the
Indian Ocean.

The follo\^'ing Table shows the results I had obtained for the year and the seasons : —

A. Dekhan and Central India.

Places of observation.

ITeight
above the
level of
tlie sea.

Hoij;ht in ftct = decrease of T F.

Tear.

Dec. to Feb. i March to May.

June to Aiii;. | Sept. to Nov.

1781

3974

2r,2(i

410

435

750

370 360

450 GfiO
900 1201)

310 595

230 390
310 600

B. Nilgiris and Cevlon.

Places of observation.

Ileight

above the

level of

the sea.

Hciglit in fect = decrease of V F. j

Year.

Dec. to Feb.

JIarch to Mav.

.Jime to Aug. t-ept. to Nov.

NiLGIELS.

Atare Malle .

4500

7490
8640

6218

270

2.S0
310

280

310

300
350

290

200

270
310

280

220

2t;o

2t;5

270

290

290
300

290

C'ETLOV'.

For the Dekhan and Central India we see that the decrease is very slow ; for the
-\lps, for mstance, I formerly obtained 320 Englisli feet for 1° Fahr.* As the principal
cause of the decrease being not more rapid, we may consider, I think, the circumstance
that the elevation, though not very considerable, extends itself with great uniformity
over a large surface.

In the second group the values are less different from tliosc in the Alps and in High
Asia ; for botli groups of the Indian stations it is characteristic that the rainy season
shows by far the most rapid decrease.

For showing simultaneoushj the variations of the decrease with the locality and the
seasons, I have drawn tliree topographical profiles (see Plates XXIX.-XXXII.), and
have indicated for each of the single seasons the diflerence of the respective decrease

* None Uuters. pliys. Gcogr. d. Alpcn, p. 564. The numbers I liavi
PC.

'ivcii there arc 54U Frciicli feet for

ELEMENTS OF INDIAN METEOEOLOGY. 539

from its annual mean value by drawing a dotted line in connexion with the topogra-
phical outline. The dotted Ime shows tlie contour which the topographical section
ought to have for the actual temperature of the season, supposing the value of the
decrease would have remained the same througliout the year; if, therefore, the decrease
in the season is too slow, the new ideal position of the station will be below the real
topographical outline, on account of the station having a temperature as if it were in a
less elevated situation ; if, vice versd, the decrease is more rapid than the annual mean,
as we see it particularly to be the case in the rainy season, the dotted line will sliow,
from the same reason, a profile which is higher than tlie topographical contour. For
Ceylon I further added the point of its highest peak, Peduru tiilla galle, 8305 feet, for
the sake of completing the general topographical profile of the island, though I had no
higher station for its mountainous regions than Nurclia. The decrease of temperature
with height in the Himalaya, the Karakonim and Kuenliin had not to be calculated
in connexion with the construction of these maps. As I liad direct data for the begin-
ning of the isothermal lines along the western and the eastern margin of these moun-
tainous regions, the form of the dotted lines which I now have drawn across thorn could
be obtained directly by uniting the terminal points. This circumstance is very valuable,
too, when I come later to examine the influence which is exercised by the topographical
formation (including vast plateaux, ridges, and isolated lofty peaks), and by the extent
of the snowy regions, upon the alterations of the decrease of temperature ^ith height.

IV. Thermal Types of the Year and the Seasons.

The considerations about the distribution of temperatiu'e over the surface of India in
general may best be combined with the analysis of the isothermal curves on the maps
annexed (Plates XXVIII.-XXXII.).

In reference to the geographical details, I have limited myself to the principal river-
systems ; and to avoid interfering with the distinctness of the isothermal lines, the names
of the stations, as well as tlie mountain-systems, are left out ; for the means of the year, the
Indian Archipelago and countries to the north-east of it are also added on a smaller scale.

In drawing the lines, I made these distinctions : besides the lines being dotted where
they pass the regions of High Asia, also the thermal equator is distinguished, its line
being a broken one. In consequence of the great difference in latitude between the
western and the eastern end of the Himalaya, the curves extend along the western
margin of the Map from 5° to 35° of latitude ; along its eastern margin only from 5° to
30°. For the period corresponding to our summer, the isothermal lines could be con-
tinued for Central Asia somewhat further to the north, in connexion with our personal
stay in these regions.

The eastern part in the higher latitudes of the Map is throughout cooler than the
western part, as sho\^Ti in the following Table, where the numerical values are nearly the
same, though the difference in latitude amounts to 5°.

540

DE. H. DE SCHLAG-INTWEIT ON THE NTTMEEICAL

Wannest
isothermal line.

Slinimum in Minimum in
the X.W. : tlie >".E.

Year 84

73
57

89
75

73

60
73

81

74

i 1 .March, April, May 90

= 1 June, .lulj-, August ■ 92

<Z3 (_ September, October, November 82

27/^' isothermal lines of the year very decidedly show the influence of the topogra-
phical form of the Indian peninsula on the increase of the mean temperature : in the
southern parts they follow the contours of the shores, or obtain forms emlently in con-
nexion with them ; in the northern part these lines are raised to the extent of a difference
of five degrees of latitude where they pass over the central axis of India. At the
same time, southern India presents one of those insular regions of greatest heat which
aj'e connected with each other by the thermal equator ; the Indian archipelago shows
us the next of these regions which follows to the east.

^Mien comparing the seasons, we are particularly surprised by the unusually great
variety of the foiu- types, whilst in many of the more western regions of the tropics we
see that it is more the numerical value of the lines which is changed than the type of
theii' forms. In India and the Indian archipelago the thennal equator runs still to the
south of the geographical one for all the three months of the cool season ; but in the
season corresponding to our summer, from July to August, we see it has been raised up
to the latitude of 32° N. This part of the year is, for the greater part of the map, the
rainy season, though for the region in the north-west it is the ^■ery season of an absolute
maximum of heat. These variations have the more importance, as the territory here
represented has a surface considerably larger than might be expected, perhaps, from the
extent of European empires. The distance from the Bay of Biscay to the Caspian Sea
can be considered as about equal to the difference in longitude of the borders of this
ISIap ; whilst 30^ of latitude, referred to European regions, might be compared with the
distance from the southern shores of the Mediterranean to St. Petersburgh.

The cool season. — This period already shows traces of the increase of temperatui'e in
the interior of the land when compared to surrounding seas ; but, as it must be expected,
the influence of insolation is, comparatively speaking, but little felt during tliis season in
the pronnces at some distance to the north of the equator, on account of the southern
position of the sim. In the regions beyond the tropics the hibernal influence of conti-
nents, compared to that of the seas, causes depression of temperatiu'e. In reference to
the Panjnb, it must be further added that we have here, comparatively speaking, a
greater number of stations for wliicli the actual temperature is still lower than the
values represented by the isothermal lines, as the latter had to be reduced to the level
of the sea. The general elevation of the groiuid, and, tliroughout tlie season, a sky
unusually (-h^ar, so favourable to nocturnal radiation, may be mentioned as the prin-

ELE^rENTS OF rXDIAI^f JIETEOEOLOGT. 541

cipal causes. Tho decrease of temperature mtli latitude is by far the most rapid in the
cool season.

TJte second period of the year (March, April, and May), whicli is generally called the
hot season all over India, also in its north-westeni parts, shows a remarkable difference
in the tji^e of the cun-es when compared to the cool season ; the influence of the topo-
gi-aphical fonns of the peninsula has become now considerably more appai-ent. The
thermal equator enters the western border of the Map akeady at an elevation of 24° of
latitude, passes through a central region of maximum temperature exceeding 90°, and
descends from thence directly to the soiith, to the very southern end of India. Great
dryness is combined in this period with the high temperature, and is an important
element for making its difference from the other seasons still more apparent ; but it
would be erroneous to expect, as it might appear rather probable, that in consequence
the heat is felt the heavier by the human organism. Though the central parts, com-
pared to the shores of the sea, show a rapid increase of temperature with the progi-ess
towards the interior, I must add that, on account of the moisture being greater along
the shores, not only the heat is felt there more close and more oppressive, but also its
influence on the health, particularly of the Europeans, is decidedly still more unfavour-
able. For the coasts, and for the interior of India up to latitude 25° N., these months
remain the period of the year which includes the highest means, and also the gi-oatest
heat of single days.

The third jieriod (June, July, and August) is, for the greatest part of India, the rainy
season; its setting in is connected, particularly in Central India, with the most rapid
sinking of temperature. Neai'er to the shores the difference is felt less beneficial ;
the humidity has increased, too, and makes, in the shade at least, the heat the more
oppressive. The power of the insolation now being broken by a sky nearly perma-
nently clouded, must be named as the particular cause why the beginning of this
season in general is considered as a welcome period. For the state of health, how-
ever, it is less favourable ; dyspeptic complaints and fevers are particularly frequent
in the latter part of this season. In the Panjab, and partly already in the north-west
provinces of Hindostan, this period has no more the character of a rainy season. The
precipitation takes the form of our summer rains with thunder-storms, and also the
amount of precipitation most rapidly decreases towards the north-west.

At the same time the meteorological observations showed for these very regions a
maximum of temperatui'e which was unexpected to me, not only on account of the
number of stations fonnerly existing being not very gi-eat, but also since I heard fi-om
the inhabitants, the Europeans as well as the natives, no unusual complaints about the
heat being much greater than in other parts of India. Nevertheless these prorinces
include a region for which the mean temperatures during the three months exceed 92°,
which therefore must be considered as one of the very hottest regions of oui- globe ;
besides, we must take further into consideration that clear days are not unfrequent,
during which the purity of the sky is not even limited, as it was in the period pre-

MDCCCLXIII. 4 D

542 ON THE NOIERICAL ELE:\rENTS OF ESODIAN METEOEOLOGT.

ceding, by dust suspended in the atmosphere. Therefore also the absolute maxima
in the shade as well as in the sun are higher here than in any other region of India.

I may further di-aw attention to the fact that for this region also the non-periodic
variations of temperature, the variations between different years, have become much
greater than we find them to be in the more southern tropical part of the territory
examined. The thermal equator enters the west of the iSIap at the latitude of 32°, and
only leaves the Indian peninsula near Ceylon in an easterly direction.

The influence of height in the Panjab is not very considerable in this season, and the
curves I have drawn remain for some stations even still a little below the respective
means ; but in the other regions, where the character of the " rainy season " prevails,
the decrease of temperature with height is more rapid than during any other part of
the year.

Autumn (September. October, November) is the only one of the tropical seasons
which shows here a very regular form of its curves, and a very slow decrease of tem-
perature with latitude ; it is not less characteristic for this season that in most regions,
particularly in those along the banks of the larger rivers, the drying up of vast surfaces
formerly inundated is the cause of most deleterious miasmatic vapoui's ; but in the
Panjab, and in tlie hilly regions along the Brahmaputra and in Central India, where
these dangerous modifications of the atmosphere are not to be feared, this season fre-
quently approaches the mild and refreshing character of the regions of southern Europe.

A more descriptive detail, together with the personal data in reference to the
observers' names and the diu-ation of the different series, will be given by me, later,
in the 4th volume of our ' Results ' *. In the present memoir I considered it my parti-
cular object to lay down the materials officially entrusted to me, and to give them at
the same time as critically worked over as my travels allowed me to attempt.

The temperature in shade, however, can but insufficiently define the climate as it is
particularly seen in these parts of the tropics, where the power of solar radiation, rains,
and storms differ in no less proportions from those in temperate zones. I will consider
it a particular pleasure to be able to forward, in not too distant a time also for these
elements, my numerical data, together with some remarks about the principal general
results.

* It also will contain some additional stations from the materials I obtained, with their usual liberality, from
the Indian authorities during my recent visit to England. Amongst the official publications, those of Glaisheb
and Macpiierson have particularly to be quoted. These materials could no more be added to the present Tables,
aa most of these too -will have to be recalculated for being reduced to tnie daily means.

[ 543 ]

XXVI. On the Structure and formation of the so-called Apolar, Unipolar, and Bipolar
Nerve-cells of the Frog. By I-ionel S. Beale, F.R.S., M.B., Fellow of the Royal
College of Physicians ; Professor of Physiology and of General and Morbid Anatomy
in King's College, London; Physician to Kings College Hospital, &c.

Received May 7,— Read May 21, 18G3.

It is an opinion now very generally entertained by anatomists that in vertebrate and in
certain invertebrate animals, there are in connexion with tlie nervous system apolar,
unipolar, and multipolar, including bipolar cells.

It is easy to demonstrate the presence of multipolar cells, but it is another matter
altogether to prove that certain cells which seem to be apolar or unipolar are really
of this nature. An observer is justified in asserting very positively the existence of
that which he has himself distinctly seen and has shown to otliers ; but it does not
follow that he is correct in concluding that, because a fibre or other structure has not
been seen by him in certain specimens, it therefore does not exist, for the actual exist-
ence of a structure and its demonstration are two very different questions in minuto
anatomical inquiry. The one is a question of fact, no matter how it may be explained,
or how many different interpretations may be offered of it. The other is but an infer-
ence arrived at in the absence of evidence, and may result from imperfect means of
observation, or want of due care on the part of the observer in preparing the specimen.

A new fact, if capable of being demonstrated to others, will be received at once as
true, but negative assertions, although supported by the authority of the most expe-
rienced, ought in matters of observation to be received with the greatest caution.

It is one thing to assert that a cell has no fibre proceeding from it, or that a cell has
but one fibre, and another to state that no fibre or but one fibre has been demonstrated.
Although many general facts are opposed to the notion of the presence of apolar and
unipolar cells, the existence of such cells is generally admitted and taught. If apolar
and unipolar cells exist at all, they are certainly less numerous than the multipolar cells ;
and it is clear that the arrangement and action of the three classes of cells, apolar,
unipolar, and multipolar, must be very different. Indeed it almost follows, if the three
classes of cells do exist, that there must be as many distinct kinds of principles of action.
Nerve-cells, which bear such different relations to nerve-fibres as must exist in the case
of the cell unconnected with any fibre at all, the cell connected with one, and that
connected with more than one fibre, cannot possibly influence the fibres in precisely the
same manner. The su^jposition of such an anatomical arrangement involves the exist-
ence of different principles of action.

MDCCCLXIII. 4 E

544 PEOFESSOE BEALE OX THE STEUCTUEE OF THE SO-CALLED

There are, however, authorities who consider it consistent with many facts they have
themselves observed, or accepted as ha^ing been demonstrated by others, that the
arrangement of different parts of the nervous system may be upon a totally different
type — that, for example, in some central parts nerves may be in bodily connexion with
the cells that influence them, while in other cases a fibre may be influenced by a cell
with which it is not structurally connected, and that in some tissues nei"ves may termi-
nate in distinct ends, while in others they form networks, and in others again terminate
in "cells." A liigh authority. Professor Kollieer, only last year suggested to me that,
although it might be true that nerve-fibres formed plexuses or networks in the muscles
of the mouse, it did not flwvcfore follow that they did not terminate in free ends in the
muscles of the frog, although no essential difference in minute structure, or different
piinciple of action, is known to exist in the muscular tissue of these animals.

Xow I submit that such arguments are utterly imtenable unless the supposed differ-
ence in the arrangement of the particular structure be associated with a fundamental
difference in the tissues under its immediate influence or holding a fixed relation to it.
I venture to assert that if the nerve-fibres terminate in ends upon the sarcolemma of the
muscles of the frog, they terminate in ends upon the sarcolemma of the muscles of the
mouse. That in one case tliere may be more ends than in another, that there may be
all sorts of modes of branching, that fibres may pass off from the trunks at different
angles, and that an infinite number of variations in detail may exist, is consistent with
all that has been observed in nature ; but I submit that there is no reason for supposing
that any typical or fundamental difference exists with regard to the ai'rangement of
ner\'e-fibres in corresponding tissues in difierent animals. So, admitting as we must
that by far the majority of ganglion-cells in different nervous centres have more than
one fibre in connexion %vith them, it seems more reasonable to conclude that those cells
which appear to have but one, and those in connexion with which no fibres whatever
have been demonstrated, have really two or more fibres cormected with them — but too
fine to be demonstrated by ordinary means, — than to accept the necessary inference, that
there are in the very same part or organ three distinct classes of nerve-cells exhibiting
as many fundamental differences of relation to the fibres tvhich they are supposed to
influence.

I have been compelled to conclude, from anatomical observation, that although many
ganglion-cells exist which appear to be destitute of fibres, none do really exist witliout
them. I have studied tliis question in various parts of the nervous system and in dif-
ferent animals, from mammalia down to the annelida, and, with regard to the arrange-
ment of the nerve-fibres in nerve-centres, I feel justified in drawing the following general
conclusions : —

1. That in all cases tlie fibres are in bodily connexion with the cell or cells which
influence tliem, and this from the earliest period of their formation.

2. That there are no apolar cells, and no unipolar cells, in any part of any nervous
system.

APOLAE, rNlPOLAE, AXD BIPOLAK KHRVE-CELLS OP THE FROG. 545

3. That eveiy nerve-cell, central or peripheral, has at least two fibres in connexion
N^-ith it.

Now, in the ganglia connected with many of the nerves distributed to the internal
organs of the frog, there are many apparently unipolar and apolar cells ; but one object
of this memou' is to prove that all these nerve-cells have at least two fibres proceeding
from them.

Although the present inquiry will be limited to the particular cells connected with
the ganglia in difierent parts of the body of the fr'og, it necessarily involves the
consideration of many fundamental principles of the utmost interest with regard to
the intimate structm-e of the highest tissue existing in li\ing animals. Nor must it
be supposed, because I only describe the structure of certain special cells, that my
observations have been limited to these cells alone. I have studied the arrangement of
neri'e-cells and fibres, in the centres and also in peripheral parts, in many different
animals.

Upon examining some specimens of the ganglia near the branches of the large arte-
ries distributed to the "\iscera of the frog, more than two years ago, I was surprised to
find that from many of the pear-shaped cells two fibres proceeded — not from either
pole, as delineated by Wagnee in the nerves of the pike, but the fibres emerged appa-
rently pretty close together, fr-om the narrowest extremity of the cell, although they did
not appear to originate from the same pail of the cell.

My specimens have all been prepared in precisely the same manner, and my observa-
tions have been made with the aid of powers magnifying more highly than those gene-
rally employed in such researches. All the drawings illustrating the subject are magni-
fied from 700 to 1800 diameters, and a few are magnified to the extent of 2800. Many
of the preparations have been presen'ed for two years, and will probably retain their
characters for a considerable time longer.

I have arranged the questions considered in this communication under the following
heads : —

1. General description of the ganglion-cells connected with the sympathetic and other
nerves of the frog.

2. On the formation of ganglion-cells.

a. Ganglion-cells developed from a nucleated granular mass like that w^hich

forms the early condition of all tissues.

b. Formation of ganglion-cells by the division or splitting up of a mass like a

single ganglion-cell.

c. Fonnation of ganglion-cells by changes occurring in what appears to be a

nucleus of a nerve-fibre.

3. Further changes in the ganglion-cell after its formation.

4. Of the spii'al fibres of the fully formed ganglion-cell, and of ganglion-cells at
diflFerent ages.

4e2

546 PROFESSOR BEALE ON THE STRTJCTUEE OF TIIE SO-CALLED

5. Of the essential nature of the changes occurring during the formation of all nerve-
cells, and of the formation of the spiral fibres.

6. The sense in which the term "nucleus" is employed in this paper.

7. Of the fibres in the nerve-trunks continuous with the straight and spiral fibres of
the ganglion-cells.

8. Of the ganglion-cells of the heart.

9. Of the ganglion-cells and nerve-fibres of the arteries.

10. Of the connexion of the ganglion-cells with each other.

11. Of the capsule of the ganglion-cell, and of the connective tissue and its cor-
puscles.

1. General description of the ganglion-cells connected with the Sympathetic and other

nerves of the Frog.

All the ganglion-cells from which nerve-fibres proceed to the vessels and other parts
in the submucous areolar tissue of the palate, to the tongue, lungs, and neighbouring
organs, those from which nerves distributed to the viscera are derived, as well as the
cells connected with the pneumogastric, those forming the ganglia upon the posterior
roots of the spinal nerves, and some others, exhibit the same general structure —
although there are many special peculiarities, and great differences in size, observed in
the cells of which some of these ganglia are composed.

The general form of these cells is oval or spherical ; but upon closer examination it is
found that the most perfectly formed ganglion-cell is more or less pear- or balloon-
ehaped in its general outline, and by its narrow extremity it is continuous with nerve-
fibres which may be followed into nerve-trunks.

In Plate XXXIII. fig. 1, a well-formed ganglion-cell from a ganglion close to one of
the large lumbar nerves of the little green tree-frog {Hyla arborea) is represented. The
substance of the cell consists of a more or less granular material, which by the slow and
prolonged action of acetic acid becomes decomposed, oil-globules being gradually set
free. Near the fundus or rounded end is seen the \ery large circular nucleus with its
nucleolus. In some of these cells, at about the central part or a little higher, are
observed a number of small oval nuclei, fig. 20, c. These oval nuclei are arranged
transversely to the long axis of the cell, and several follow each other in lines which
pass in a direction more or less obliquely downwards towards the lower narrow part of
the cell. In fig. 1 very few of these nuclei are seen, but in figs. 20 & 26 many exist.
The matter of which the mass of the cell consists gradually diminishes in diameter, and
contracts so as to form a fibre, in which a nucleus is often seen. At the circumference
of the coll, about its centr(>, the material gradually seems to assume the form of fibres,
which contain numerous nuclei, and tliese pass around the first fibre in a spiral manner.
Thus a fibre comes from the centre of the cell [straight Jibre)^ and one or many fibres
(spiral fibres, figs. I, 3, 4, 8c 22 to 28, 31 to 38, & fig. 42) proceed from its surface.
Neither of the fibres can be traced to the large " nucleus" of the cell.

APOLAR, UNIPOLAR, AND BIPOLAR NERVE-CELLS 0¥ THE FROO. 547

2. On the formation of the ganglim-cell.
I now proceed to describe the changes which I beHeve take phice in the structure
of these ganglion-cells during their development, growth, and decay. The drawings
which illustrate these observations, although arranged so as to form as far as possible
a connected series, are actual copies from nature, made from very many different
specimens at intervals during tlie last two years and a half. The clearest as well as the
least uninteresting manner of describing the points of anatomical detail will probably
be to arrange the facts of observation so as to construct, as far as may be possible, a
connected history of the structural alterations occurring during the life of these elaborate
organs. But it is not pretended that the sketch can amount to more than a rough and
imperfect outline of the process which actually occurs ; for the nature of the inquiry is
such that it is absolutely impossible to observe the sixme cell at different periods of its
life, so that the history is made up of fragmentaiy details obtained at intervals during
a long course of investigation.

The development of these, as of most other structures, may be studied in the fully
formed animal as well as in the embryo, and the various changes occurring may be
obseiTed with greater precision and distinctness in the former than in the latter. I shall
not enter into the question of the origin of the " nucleated blastema " or of the " granular
matter " observed in such quantity in all developing structures. The word " cell " I shall
use in a general sense, because it is shorter tlian " elementary part."

The nerve-cell does not at all ages possess the well-defined structure (cell-wall, cell-
contents, and nucleus) usually accorded to it by anatomists, but always consists of mat-
ter that is living, and ^natter that has lived. The first is coloured by an ammoniacal
solution of carmine, the last is not so coloured. A reference to any of my drawings
will enable the reader to see at a glance what I understand by a cell or elementary part,
and will render further description needless*. In the very young animals these gan-
glion-cells gradually form from " nuclei " which appear to be imbedded in very soft gra-
nular matter. The fibres extend from the collection in at least two directions, and exist as
granular nucleated bands, the course of which cannot be followed for any great distance,
partly in consequence of their extreme transparency and tenuity, and partly because
they ramify amongst the " nuclei," of which all the different tissues at this early period
are in great part composed. The fibres do not grow out from the cells, but are formed
as two masses of germinal matter gradually separate from each other. But new gan-
glion-cells, nerve-fibres, and nuclei are being constantly produced, not only in fully deve-
loped young animals, and in the adult, but certainly for a considerable time after the
animal has arrived at maturity, and I believe almost to the close of the ordinary period
of life. A young cell, a fully formed cell, and an old cell are represented in figs. 2, 3, & 4,
magnified 700 diameters.

• This question has been fully considered in my lecture " On the Structure and Growth of the Simple
Tissues," delivered at the Royal College of Physicians in 1860 ; and the general conclusions then arrived at, and
since confirmed by other observations, will be found to be strongly supported by facts recounted in this paper.

548 PEOFESSOE BEALE OX THE STEUCTUEE OF THE SO-CALLED

In the fully developed frog, ganglia are formed,

1. From a nucleated granular mass like that seen in the embryo, but continuous with
nerve-fibres.

2. By the division and splitting up of a mass like a ganglion-cell.

3. By changes occiu-ring in what appears to be an ordinary(?) nucleus of a nerve-
fibre.

a. Ganglion-cells developed from a nucleated granular mass like that which forms

the early condition of all structures.
At first an elongated delicately granular mass is observed in connexion with nerve-
fibres, more or less of an oval form, tapering into a tolerably broad granular band or
fibre at either extremity. This band often appears to consist of but one broad fibre,
but it may generally be resolved into several finer fibres. It cannot be said that nuclei
as distinct and separate and isolahle bodies are imbedded in this ; but by the action of
carmine numerous rounded portions of matter are darkly coloured, and these may be
seen to pass gradually into the surrounding substance, which is only slightly tinted by
the red solution. In the small ganglion represented in Plate XXXIV. fig. 5, several
such nuclei gradually passing into the surrounding uncoloured matter may be observed.
These nuclei divide and subdivide ; and at a more advanced period of development, the
whole mass having increased in size, indications of its division into several lobes or
portions may be seen {b, c, d). These divisions become more and more distinct, until a
number of masses, all exhibiting the same structure, but still intimately connected with
each other, result. At b several of the cells distinctly separated from each other, and
beginning to assume a pear shape, are seen. As the cells approach their fully developed
form they move away fi'om the point where they were first developed, as represented in
fig. 6. In this figiu'e the cells that project furthest and have the longest stems are the
oldest. From the mode of formation, it is obvious that, at least for a certain time, the
cells are continuous with each other (fig. 5, a, b, c, d, e), while it is also clear that the
very matter of which at first the substance of a cell is formed may subsequently be
drawn out so as to form a fibre. It appears, however, that as the ganglion-cells grow
and separate from each other they change their relative positions, so that it is impos-
sible to trace any one fibre for more than a comparatively short distance. The con-
nexions of the ganglion-cells will be considered separately.

b. Formation of ganglion-cells by the division or splitting np of cc mass like a
single ganglion-cell.
One of the masses (cells) just described (b, d, fig. 5) having reached a considerable
size may divide and subdivide, and thus new cells may be developed from it. The stem
by which the cell is connected with the rest of the ganglion divides and subdivides into
numerous fibres, and in this manner it seems that a number of separate ganglion-cells
may be formed by the division of one. The fibres proceeding from each of the new cells

APOLAE, TJNIPOLAE, AXD BIPOLAR XEEVE-CELLS OF THE FROG. 549

are arranged in a bundle which corresponds to the stem of the original cell. The
changes just described are well seen in tigs. 7 & 8 from a sympathetic ganglion, and in
figs. 11 & 12 from the intervertebral ganglia*. This mode of multiplication generally
occurs before the cell has assumed its complete and perfect form (figs. 1, 3, & 4).
Although I have examined hundreds of ganglia, 1 have only seen on very few occasions
such a cliange occuning in a cell which exhibits a distinct spiral fibre, and I have not
been able to study it. The greater part of the old spiral fibres would probably waste
and disappear, and new ones would be developed from a new germinal mass, resulting
from the growth and multiplication of the " nuclei" of tlie old fibres. The fibres
connected with a mass undergoing development into several ganglion-cells often exist in
considerable number, but sometimes are so fine as only to be just visible under a power
of 3000, and I believe others exist which are too fine and delicate to be seen by means
at present at our disposal. In fig. 6 several fully foraied ganglion-cells are observed ;
and here and there are oval masses of about the same size, but containing numerous
nuclei, and connected Avith the ganglion by very fine fibres (fig. G, at a & b).

In fig. 7 one of these is represented much more highly magnified ; and in fig. 8 another
which has increased greatly in size, and contains numerous separate ganglion-masses is
magnified 1800 linear. These cells are dividing and subdividing, and from the whole
mass would ha^•e been formed a new (janglion composed of u})wards of thirty cells.

c. Formation of ganglion-cells by changes occurring in vohat aj}i)ears to be a
nucleus of a nerve-fibre.

I have several specimens which exliibit a single ganglion-cell in connexion with a very
fine nerve-fibre. Not that the fibre is actually single, for it consists of at least three or
four very fine fibres ; but it runs by itself at a distance from any other fibres or ganglia,
and is imbedded in transparent cord-lUic connective tissue.

The changes occurring during the formation of a single ganglion-cell, in connexion
with what is imdoubtedly at first a single granular fibre, are of the utmost interest. I
cannot be perfectly confident that the history I give of these changes is absolutely true
in aU points of detail ; for, as may be easily imagined, to isolate fine fibres with nuclei
exhibiting the diifcrent stages of change is a difficulty of no ordinary character, Avhile

* My friend Mr. Lockhabt Clarke has not observed this division of the cells which enter into the formation
of the ganglia on the posterior roots of the nerves of mammalia. He describes the graduiil increase in size and
the alteration in structure and form of the cells as development advances, but says nothing about their increase
in number. It is true that Mr. Clarke's obser^-ations were made upon mammalia, while the statements in ray
paper refer only to the frog ; but it is almost certain — I think indeed that it is quite certain — that if the cells do
not multiply by division in mammalia they do not do so in the frog. Upon this question of fact, therefore, I
regret to say that Mr. Clarke and myself are at issue. The specimens from which my figures 11 & 12, which
show this division most distinctly, have been copied, may be examined by any one who desires to see them.
This question shall be farther investigated by me. Mr. Lockhabt Clarke's observations will be found on
page 927 of the Philosophical Transactions for 1862, Part II.

550 PROFESSOE BEALE ON THE STEUCTURE OP THE SO-CALLED

hundreds of specimens may be examined without finding one in which such fibres are
observed.

The drawings represented in figs. 9, 10, 13, 14, 15, 16, 17, & 18 show the points I have
observed in connexion witli this question.

A nucleus which cannot at first be distinguished from the ordinary nuclei in connexion
with the nerve-fibres, grows somewhat larger than the rest (figs. 9 &14). 1 am not pre-
pared to say that any nucleus connected with the fibres could or could not undergo the
changes about to be described ; but the nuclei from which the ganglion-cells are formed
exhibit at first nothing peculiar. Sometimes several in different parts of a fibre may
enlarge to some extent ; but for the most part only one in the course of a long distance
will be developed into a ganglion-cell. Of course the opinion that a ganglion- cell may
be developed from (uiy nucleus up to a certain period of its life is quite tenable, and it may
be that the actual change is determined by the presence or absence of certain conditions,
but this speculation I will not pursue. The enlarged nucleus which is about to become
developed into a ganglion-cell soon exhibits a transparent portion at its circumference.

From what has been stated, it is clear that every ganglion-cell developed in the course
of a nerve-fibre must have a fibre proceeding, as it were, from either pole (bipolar cells) ;
but the difficulty of defining the changes with the utmost certainty arises from the rarity
with which single fibres emanating from either end of the cell, like those represented by
authors, occur. The only cells exhibiting these characters in the frog that I have seen
are, in my opinion, young and imperfectly formed cells. Such an appearance is not
uncommon in fully formed cells in ganglia; but in all cases in which I have observed it
in the frog, more careful investigation has satisfied me that it was fallacious, and
depended upon a fibre really passing close to the cell, without being connected with it,
although it appeared at first to emanate from the end opposite to that which certainly
terminated in a straight fibre. In the frog 1 feel almost certain that fully formed
cells, with a nerve-fibre coming from the opposite poles, as delineated by Wagxer and
KOLi.iKER in the pike, do not exist ; and I have failed to find examples of the cells
represented by Kolliker in three specimens of the pike which 1 have examined. My
want of success, however, probably depends upon imperfect investigation, as such fibres
undoubtedly exist in the skate. In the posterior roots of the spinal nerves of the
common skate such cells are easily found ; but there is a difference in the arrangement
of the two fibres, and their relation to the mass of the cell corresponding to th.at already
described in the frog, which deserves more attentive examination. I shall describe the
peculiar structure of these cells in another communication.

The cells which are not fully developed, delineated in figs. 9, 14, 15, & 10, have a fibre
coming from either extremity, but, as will presently be shown, there is reason to believe
that as the cell grows tlie arrangement of these fibres soon becomes altered. The
appearances represented in fig. 18, when examined by a low power, would be considered
to be due to the presence of fibres emanating from eitlier end of the cell, but under a
higher power the true arrangement, which is very different, can be cleai'ly demonstrated.

APOLAE, UXirOL.Vrv, AND EIPOLAE XEliVK-CELLS OF THE FEOG. 551

3. Further changes in the gang/ion-celt after its formation.

The ganglioii-cell having assumed such forms as represented in figs. !J & 11, now
undergoes furtlier changes. It becomes separated more and more from tlic point
where its formation commenced (figs. 32 & 33, Plate XXXIX.). The two opposite
extremities of the cell are drawn down (fig. 31). The fibres increase in length and lie
parallel to each other, and the form of the cell becomes much altered. If formed in
connexion with one of a bundle composed of numerous fibres, the cell seems to grow
away, as it were, from the bundle, and sometimes is found at some distance* from it, as
represented in fig. 41 d ; but more commonly its long axis corresponds to the direction
in which the fibres of the bundle run (figs. 17 & 21). Tliis is constantly observed in
the cells in connexion with bundles of fibres running close to the large arteries in tlie
abdomen ; and tliey may easily be mistaken for cells having a fibre springing from each
end. But in this case also the ganglion-cell is some distance from the spot wliere its
first formation commenced, and the two fibres whicli no^v extend from it may lie parallel
to each other for the distance of four or five thousandths of an inch from the cell, and
then take opposite courses in the bundle of nerve-fibres.

The ganglion-cell, when fully formed, may lie on the outside of a bundl(> of nerve-
fibres, while the fibre or fibres with which it is connected may run in the central part of
the bundle. Tlie two fibres passing from the cell run amongst the bundle of nerve-
fibres, to whicli their course is at first more or less at right angles. Sometimes the fibres
from a ganglion-cell pass partly round the circumference of a bundle of nerve-fibres,
and tlien run amongst them. Often the fibres appear to pass quite to the centre of the
bundle. At this point, in fortunate specimens, tlie two fibres may be seen to alter their
course and run with other fibres of the bundle, but in opposite directions (see figs. 32
to 37 ; also figs. 1, 20 a & c, and fig. 25). As tliesc fibres are often less than the o^jTolToth
of an inch in diameter, it is very difficult to follow them in the bundle for any great
distance ; and in ordinary specimens, in endeavouring to unra^ el tlie bundle of nerve-
fibres, they are inentably broken. Often they break close to the ganglion-cell ; in whicli
case the cell itself, especially if examined in water, appears as a round apolar cell, while
the fibres which Avere continuous Avith it might be easily mistaken for fibres of con-
nective tissue running transversely around tlie bundle of nerxe-fibres imbedded in the
nemllemma. Indeed it is probable that many authorities will still maintain not owXy
that the spiral fibre, but that many fine fibres I have described as nerve-fibres really
consist of modified connective tissue. If I had only specimens from the common frog,
I might have experienced some difficulty in demonstrating that the spiral fibre was a
true nerve-fibre to the satisfaction of every one ; but many specimens Avhich I have from
the green tree-frog settle the question beyond dispute. The spiral fibre is as large
and thick as tlie straight fibre, and, like it, has been traced into a dark-bordered fibre
(figs. 1 & 19). Figs. 17 to 28, 31, 38, and figs. 41 & 42 represent difi'erent forms of
ganglion-cells.

MDCCCLXiir. 4 F

552 PEOFESSOE BEALE OX THE STEUCTITRE OF THE SO-CALLED

4. Of the s]^)iral fibres of the fully formed ganglion-cell, and of the ganglion-cells of

different ages.

In all but very young ganglion-cells the remai-kable spiral fibres already alluded to
exist. In fig. 1, which represents a fully formed cell, the beautiful spiral arrangement
of one of the fibres is observed, and in figs. 21, 22, 27, & 28 several fibres are seen
to proceed from one ganglion-cell. Close to the cell in which there is a considerable
extent of spiral, the course of the fibres is almost transverse. The fibres seem to be
coiled around the lower thinner portion of the ganglion-cell (figs. 1, 3, 4, 20, 25, 31).
Then the fibres pass spirally round the straight fibre away from the cell, and each turn
becomes more oblique than the one above it, until at last the fibre (or fibres) lies parallel
with the other fibre which leaves the cell. The spiral fibres are necessarily longer than
the sti-aight fibre. The spiral fibre eventually takes a course in the compound nerve-
trunk to which the fibres may be traced, the very opposite of that taken by the straight
fibre ; so that, although the two fibres run parallel to each other for some distance from
the ganglion-cell, as St.\x>"IUS observed was the case in the bipolar cells of the calf, it
is certain that, at least in many instances, they at length pass in opposite directions
(fig. 1). KoLLiKER thinks that " by far the greater part, perhaps all," of the fibres from
the ganglion-cells of the spinal nerves pass in a '■'■ perijiheral direct ion f but I believe this
statement is eiToneous, both as regards the existence of but one fibre, and its coru'se.
I have never been able to find any ganglion, large or small, the fibres of which passed
in but one direction. Even when a ganglion consists of only three or foiu' cells, compound
fibres pass from it in different directions. I doubt if a nerve-cell anywhere exists whose
fibres pass to their destination in the same direction. In connexion with this question
I may revert to the fact I have already stated *, that whenever any compound nerve-
trunk, large or small, composed of any fonn or number of nerve-fibres in any part of
the organism of man or animals, passes into another trunk running at, or nearly at,
right angles to it, its component fibres divide into branches which pass in opposite direc-
tions. This general fact is most important ; and I have never seen anything to make
me believe that the disposition is not miiversal, and therefore essential even to the
simplest nervous apparatus. As the disposition exists in bundles of fibres which there
is reason to believe are purely sensitive, as well as in the case of those which are purely
motor, it looks as if fibres invariably passed in two directions, whether they be traced
from theu" peripheral distribution or from their central origin. The arrangement spoken
of is represented in fig. 40, which is from the submucous areolar tissue of the palate of
the frogf.

The spiral fibres can be shown to be continuous with the material of which the body
of the cell is composed as well as the straight fibre ; but the former are connected with
its surface, wliile the latter proceeds from the more central part ; so that, in the most

• Philosophical Transactions for 18G2, page 894.

t Sec a paper in my 'Archives,' " On the Branching of Xervc-tnmks, and of the subdivision of the individu;il
fibres composing them," vol. iv. p. 127.

APOLAE, HXIPOLAE, A^TD BIPOLAE NERVE-CELLS OF THE FROG. 553

perfect of these cells, the straight fibre forms a stem around which the spiral fibres are
coUed (figs. 1, 3, 19, 20, 25, 38, &c.).

The nuclei in connexion with the spiral fibres are well represented in figs. 1, 19, 20, 26.
Sometimes they are still more numerous, and I have seen as many as twelve nuclei in the
lower part of one of these cells. Several are imbedded in the verv substance of the
matter of which the cell is composed, but they lie for the most part near the surface.
They vary a little iu size, and dinde longitudinally and transversely. By the dinsion
of a nucleus, and the subsequent formation of a fibre from each of the two resulting
nuclei, a single spiral fibre is continued onwards from a certain point as two fibres ; but
it is, I think, probable that in some cases the fibre itself may di\idc into two, quite irre-
spective of the nuclei in connexion with it. I have seen instances in which the straight
fibre passes through a fissm-e in the spiral fibre. The connexion of the spiral fibres
with the sm-face of the body of the cell is well seen in figs. 1, 3, 19, 22, 2G, 31, & 42.

If figs. 3, 20 a, b, 24, 25, & 31 be compared with figs. 1, 4, 20 c, 23, & 25, a great
difference in the extent of the spiral portion of the fibre will be noticed ; and it has been
already shown that at an early period there is no spiral fibre at all, but that the part of
the fibre which is to become spiral pursues a course at first opposite, then perhaps at
right angles to, and afterwards parallel with the other fibre (figs. 10. 13, 10, 19, 20, & 24).
It has also been shown that, as a general rule, those ganglion-cells which have the longest
stems, or ai-e separated by the greatest distance from the general mass of the ganglion
(fig. 6), are the oldest cells. Now these are the very cells in which the spiral exhibits
the greatest number of coils ; and from numerous observations 1 am con\inced that the
number of coils increases as the fibre advances in age (figs. 2, 3, & 4).

Moreover numerous observations prove that the quantity of matter constituting the
body of the cell varies greatly in different cases ; and it is almost certain that as these
cells advance in age they diminish in bulk, while the so-called nucleus is absolutely, as
well as relatively, smaller in the oldest cells. The fundus of the cell, with its large
nucleus, exliibits the same characters for a long period ; but at the lower, narrow portion
of the cell great alterations are obsersed. In some cells scarcely anything is left except
the large nucleus, external to which is a little " formed matter," from which several fibres
proceed. Old ganglion-cells are represented in figs. 4 & 25.

Now, as has been shown (figs. 9, 10, 11, 12, 13, 14, & 17), there is only one lai-ge
nucleus at an early period of development of the cell ; while in a fully formed cell there
may be from ten to twenty smaller oval nuclei at the lower part of the cell, some of
which are connected with the spu"al fibres, as well as the large ciixular one at the
fundus of the cell. There is also very frequently a nucleus in the straight fibre near
its origin (figs. 1 & 19). It seems pretty certain that all these smaller nuclei are
developed from the larger one ; but this question vrill receive fiuther consideration in
the sequel.

There are then, in the fully developed ganglion -cell, "nuclei" connected with the
straight fibre as well as with the s])iral fibre or Jibres ; and it must be borne in mind

4f 2

554 PROFESSOR BEALE ON THE STRUCTURE OF THE SO-CALLED

that there are nuclei connected with the dark-bordered fibres, near their origin in some
instances, and near their distribution in all tissues, as well as in connexion with the pale
fibres distributed with them. And there is also the very large nucleus with nucleoli at
the fundus of the cell.

5. Of the essential nature of the changes occurring during the formation of all nerve-
cells, and of tlie formation of the spiral fibres.

Although o-anglion-cells are formed according to the three different processes just
described, it will be found, upon careful consideration, that the changes which occur
are, in their essential nature, the same in all. In every case it has been shown that
what is comraonlv called the "nucleus" takes an important part in the process; and in
the drawings (figs. 5, 9, 15, & 38) the large size of the " nucleus " in the young cells
cannot fail to be noticed.

In considering the actual changes which occur, it will be better to call the matter
which is coloured red by an ammoniacal solution of carmine ''germinal matter" and
the colourless matter around it and continuous with it ''formed material." In young
specimens the germinal matter is sometimes seen to gradually pass into the formed
material ; but in fully formed cells of all kinds there appears a line of demarcation.

Neither ganglion-cell, nor nerve-fibre, nor indeed any living tissue exists without there
being living germinal matter in connexion with it. On the other hand, masses of ger-
minal matter exist before either ganglion-C('^/.s or distinct fibres are formed. If figs. 5,
9, 14, 15, 16, & 38, which represent young ganglion-cells, be contrasted with figs. 1, 3,
4, 20, & 27, which are taken from fully formed or old cells, the different proportion of
" germinal matter " to the "formed material" at different ages will be observed ; and it is
to be noticed tliat the youngest cells (fig. 9) consist almost entirely of germinal matter,
while in the fully formed cell there is at least from ten to twenty times as much "formed
material" as there is of "germinal matter" (fig. 1).

In the fully formed cell the germinal matter (nucleus) exhibits a line around it, but
there is no cell-wall or other stritcture between it and the granular material which
sun-ounds it. In fact, by the use of high powers, an actual contimiity of structure
may usually be demonstrated. The smaller centres (nucleoli) also seem to be distinct
from the germinal matter in which they lie. Even in these so-called " nucleoli " still
smaller spherical bodies, to the number of three or four (nucleoluli), are sometimes to
be seen distinctly (figs. 26 & 27). These centres are evidently formed one within the
other.

The last or smallest centres are most darkly coloured. In the "nucleoli" the colour is
not so intense. Tlie "nuclei" again arc still paler, but nevertheless the colour is very
decided indeed. The matter more external is very fixintly coloured, or it remains per-
fectly colourless. So tliat in this, as in many other instances elsewhere, it is to be
noticed that the outer part of each cell, or that in actual contact with the colouring
solution, is not coloured, Avhile the intensity of the colour gradually increases as we

APOLAE, UXIPOLAE, AND EIPOLAE NERVE-CELLS OF TJIE FROG. 555

pass towards the iiinennosf part of the germinal matter, althougli this may be situated
at the greatest distance from tlie colouring solution. To reach i\\c nucleus and nucleolus,
it is obvious tlie solution must pass f/iroui/h a consideralle thickness of tissue. The colour
is, however, deposited here in greatest quantiti/.

Again, it is to be noticed that in the younger cells (figs. 9 & 38), which are more or
less darkly coloured over their whole extent, the one or two distinct nuclei seen in tlic
fully formed cells are not to be demonstrated, although it often happens tliat u vast
number of very darkly coloured spots are to be discerned, each being imbedded in
matter more faintly coloun^d than itself. In the young cell every part of the germinal
matter, or, in ordinary phraseology, tlic " entire cell," possesses equal power, and, as we
have seen, may divide and give rise to tlie production of several separate cells ; but when
the formed matter is produced on the surface ; the cell, as a whole, no longer possesses
this power, which is restricted to the so-called '■'nucleus" or "■nucleolus," which may
di\ide and give rise to the formation of new cells.

Suppose new cells are to be developed from a fully formed cell : the outer colourless
or formed material, the cell-wall, if present, and the surrounding connective tissue take no
2yart in the process; but the active changes are effected by the " livinrj germinal matter"
(" nucleus ") alone.

How, then, is the formed material of the cell produced \ The observations just made
seem to me to lead to but one conclusion — that iho formed material results from changes
occurring in the germinal matter. I hold that all the formed material was once in the
state of germinal matter, and that whenever the ganglion-cell increases in size, or the
fibres in connexion with it increase in length, except of course when artificially stretched,
a certain amount of germinal matter undergoes conversion into formed material. The
changes which take place in the formation of nerve-fibres occur in a similar order ; but
as the relations of structure produced are more simple, the alterations may be studied
more readily and described more clearly.

Figs. 29 & 30 represent a portion of a dark-bordered fibre in course of development. It
consists of nuclei in connexion with fibres. The fibre is seen to be thinnest about mid-
way between the respective nuclei [h). The fibre grows at the points marked a, and at
these points only. The oldest parts of the fibre are the narrowest portions, marked h.
These are narrow because at the time of their formation the masses of germinal matter
were so much smaller than they are now (fig. 30). The nuclei in connexion with these
fibres may divide, and other new fibres may be produced; and a similar process occurs
in the nuclei of the ganglion-cell which are connected with the formation of the spiral
filwes.

Fig. 29 gives the appearance which this fibre would have presented if examined at
an earlier period. Now, although this figure literally represents but my oavu view of
the matter, it is only just that I should state that fibres presenting every degree of
change have been actually observed in the same specimen, so that there can be little
doubt as to the general truth of the facts brought forward, although differences of

556 PROFESSOR BEALE ON THE STRUCTURE OF THE SO-CALLED

opiuion may be entertained ^yitll reference to some of the explanations I have vcntiu-ed
to offer.

My observations upon various tissues in different stages of development have convinced
me that the growth of the cells or elementary parts is a much more simple process than
is generally supposed, and consists merely in a certain jiroportion of germinal matter
undergoing conversion into formed material, while at the same time pabulum passes into
the germinal matter, and the wonderful properties or powers possessed by this substance
are communicated to it. By this formation of new germinal matter the proportion of
the latter converted into formed material, and the formed material which is destroyed
and removed, may be completely compensated for.

There is a certain relation between the proportion of germinal matter and formed
material of the cell, which varies at different ages and under different circumstances, as
I have shown. The rate at which pabulum undergoes conversion into germinal matter
varies according to tlie facility with which it comes into contact with the living matter.
The formed material offers a greater impediment to its passage in old than in young
cells, so that under normal conditions, the process of growth occurs more and more
sloAvly as the cell advances in age. In the young cell, more inanimate pabulum becomes
living matter, and more liviiKj matter becomes formed material than in the adult, and
in the latter, more than in the old cell.

Next, then, for ct)nsidcration is the question of the mode of formation of the spii-al
fibre. Now it must, I think, be admitted that there is a great accumulation of evidence
in favour of the general conclusion that all livin// matter j^ossesses a poiocr of move-
me)it. It seems to me that not one step in growth can be explained unless the particles
of li^ing matter move by virtue of some inherent force or power which acts independently
of, and is capable of overcoming, the force of gravitation. The movements of liraig
matter have been observed in many of the lower forms of lining structures. I have
described the phenomenon as it may be seen in the mucous corpuscles and young
epithelial cells of the nasal and bronchial mucous membranes ; and although I have not
seen the movements in the living matter of the tissues generally, there seems to me the
strongest evidence that such movements actually occur*. In these peculiar ganglion-cells
we have, I think, very convincing evidence that movements have taken place uninter-
ruptedly since the earliest changes occurring in their formation. I have endeavoui'ed
to show that the cell, when fully formed, does not occupy the same spot as it did when
its development commenced ; and upon consideration it will appear that it is not possible
that many of the ganglion-cells could have been developed in the position in Avhich they
are found in thi; fully fornu'd state.

The spiral seems to result partly fi-om a sort of splitting and subsequent condensation
of the lower portion of the cell itself, and partly by growing from the nuclei connected
with the fibres, whih^ at the same time the fundus moves away, and spiral after spiral is

* Sec a paper in my ' Arcliivcs.' vol. iv. p. 150, "New Observations upon tlio Movements of the hvlwj or
germinal matter of the tissues of Man and the higher animals."

APOLAR. UNIPOLAR, AND BIPOLAR NERVE-CELLS OF THE FROG. 557

left around the central fibre, which is of course gradually incn^asinf;: in lens^th also. I
think it doubtful if the entire cell rotates, because the central fibre does not appear to
be twisted ; but it is obviously possible that the outer portion of the cell might rotate
slowly round the inner portion without causing any twisting of the fibre, the mass of
which the cell is composed being in the natural state very soft and plastic.

It must be borne in mind that at first the two fibres of the ganglion-cell are parallel
to each otlier, and that the cell wliile altering its position continues to grow. As the
cell moves away, its fundus or large extremity preceding, the fibres projecting from it
increase in length — are drawn out, as it were.

There is a fact in favoiu- of rotation which I have observed so often that it may be
regarded as constant — that, ui peripheral parts where a dark-bordered fibre is being
developed, a fine fibre passes spirally around it ; and this may be accounted for in pre-
cisely the same way as I have attempted to explain the production of the spiral fibre of
the ganglion-cells. The arrangement described is represented in figs. 29 & oO. It is
constant, but can only be demonstrated positively at an early period of development of the
dark-bordered fibre. The frequent crossing and twisting of fibres around one another
amongst ganglion-cells, and tlie strange crossing over and under obser^■ed in the case of
all fibres in the trunks of nerves, must also be due to a corresponding change of position
between contiguous fibres after they have been formed, but at an early period of their
life's history. The arrangement of the fine fibre, represented in figs. 29 & 30, is very
remarkable ; and I have seen very many specimens exhibiting the same points. It must
also be noticed that the nuclei of the dark-bordered fibre are much nearer together than
the nuclei of the fine fibre. This fact is also constant in the case of such nerves near
their ultimate distribution. I am not yet able to give a satisfactory explanation of the
fact, but it would seem to] show either that the fine fibre has grown very much faster
than the dark-bordered fibre, or that the fine fibre Avas developed, and perhaps in an
active state, at a period anterior to the development of the dark-bordered fibre. These
points are of the utmost interest, and well deserve the most searching and minute inves-
tigation ; for it is certain that the settlement of many of the questions raised, and but
very imperfectly considered here, must lead to the establishment of new general prin-
ciples of wide application.

6. The sense in which the term '■^nucleus " is employed in this paper.
Although I have for convenience made use of the ordinary word " nucleus" it must
be understood that it is used only in a general sense ; for I maintain that the matter
around the nucleus differs from that of the nucleus itself only in having reached a further
stage of existence. My meaning will be readily understood by the following statement,
which is supported by evidence already adduced (pages 548, 549). The whole of the
germinal matter, of which the young cell is almost entirely composed, may divide and
subdivide, and from it any number of new cells may be produced. Nor is it necessary
that a " nucleus " should be present in the detached portion. The " nucleus " is often

558 PKOFESSOR BEALE OX THE STEUCTUEE OF THE SO-CALLED

not to be distinguished until some time afterwards. This fact may be observed in the
germination of pus and mucus. Neither " nucleus " nor " nucleolus," therefore, are
bodies T[>os^essmg 2)ecuIiar2>owe7's or actions upon matter around them, nor is the " nucleus"
essential to the being or to the mxdtiplication of a " cell " or elementart/ fart. Nuclei
are but new centres wliicli appear in preexisting germinal matter ; and in these again
new centres may arise, and so on, centre within centre. In some of the nerve-cells there
is but one such centre, in others more than one. In some the " nucleus " is dividing
(figs. 11 & 12). Tlie terms nucleus, nucleolus, nucleolulus are arbitrary, and indicate
germinal centres, wliicli have appeared one after and one within the other. These
consist of living matter in different phases of existence.

After a time the germinal matter of which a young "cell" is composed, at its outer part
undergoes conversion into formed material. This formed material cannot jtroduce new
formed material. It may undergo p]i)/sical and chemical changes, but it is no longer
the seat of vital changes. The germinal matter which remains (nucleus) may still, up
to a certain period, give rise to the production of new cells. The more the formed
material around it increases the greater is the impediment to the passage of nutrient
matter, and the more slowly it lives ; so that, instead of new cells being produced, the
germinal matter that remains gradually undergoes conversion into formed material ; and
it is doubtful if the germinal matter, at its outer part, where this conversion is actually
occurring, could under any circumstances give rise to the production of new cells. It
has reached a later stage of being, and has lost this power. Sucli vital j^ower, however,
undoubtedly still exists in that part of the germinal matter which in these nerve-cells is
known as the '• nucleolus." In passing from without inwards, in the case of a fully formed
cell (fig. 1) we meet with matter in different stages of existence, which exhibits a differ-
ence in power. Most externally is the formed matter, ^^-hich possesses no power of
formation or reproduction whatever ; next we come to matter in wliich the vital powers
of reproduction still exist, but to a limited degree. It may increase ; but from it no
defined complex structure like a ganglion-cell could be produced. It is gradually under-
going conversion into formed matter. Within this, again, is germinal matter, which
possesses the power of increase, and of giving rise, under certain conditions, to the pro-
duction of perfect ganglion-cells. This matter (nucleus, nucleolus) still retains tlie poiver
possessed hy the entire mass, of which the embryo cell was composed, before it exliibited
the wonderful structure evident in its fully developed state ; and from it new cells might
be developed.

7. Of the fibres in the nrrvr-trunks continuous with the strai<jht and spiral fb res

of the [/(nitjl ion-cells.
The ganglion-cells 1 have described are not connected witli one peculiar kind of nerve-
fibre only. In the frog it is probable that tlie bundle of Acry fine fibres correspond to
the grey or gelatinous fibres of mammalia. I luue shown that bundles of very fine
fibres, many of which are destitute of dark-bordered fibres, arc connected with ganglion-

APOLAE, UNIPOLAE, AND BIPOLAR NEEYE-CELLS OF THE FROG. 559

cells*. Some of these very fine fibres with the ganglion-cell are represented in figs. 22
& 27, magnified 18U0 diameters; and these remain as very fine fibres throughout every
part of their course.

There is, however, no difficulty in proving that many of the pear-shaped ganglion-cells
in the frog are connected ^\■ith dark-bordered fibres. I have traced a dark-bordered fibre
into the ganglion-cell, at the lower part of which it is sometimes convoluted in such a
way that its actual connexion ^vith the substance of the cell is not demonstrable. An
example of this is represented in fig. 26. It will be asked in this case if the dark-
bordered fibre exliibits an axis-cylinder quite up to the ganglion-cell. I have seen a
distinct and very fine axis-cylinder, in the case of some old cells in the ganglia on the
posterior roots of the nerves in the frog, within three thousandths of an inch of the gan-
glion-cell ; but nearer to the cell the so-called axis-cylinder seems not to be distinct from
the material of which the white substance is composed, as I have shown to be the case
in all dark-bordered fibres near then- distribution. In fact the distinction of a fibre into
white substance and axis-cylinder is only to be demonstrated in nerve-fibres which have
been developed for a very considerable period of time ; and wherever this distinction is
observed, we may be sure that the nerve-fibre is of considerable age.

As a general rule the straight fibre is thicker than the spiral fibres, and there is no
difficulty in obtaining specimens in which the dark-bordered fibre can be traced on as
the straight fibre of the ganglion-cell. The spiral fibres are often very fine, and are con-
nected together here and there. Sometimes several unite to form one very fine fibre,
which can be followed for some distance upon, and very close to, the outer part of a
wide dark-bordered fibre. But I have many specimens in which both straight and spual
fibres are exceedingly thin and of equal diameter, others in which the straight is much
thicker than the spmil fibre, and a few in which the diameter of the spiral fibre or fibres
is even greater than that of the straight fibre.

I have traced the spiral fibres continuous v.\i\\ a dark-bordered fibre in several instances.
A very conclusive specimen showing this fact is represented in Plate XXXIII. fig. 1, in
which both straight and sjjiral fibre were continuous with dark-bordered fibres ; also in
fig. 19, in which the connexion between the spiral fibre and the dark-bordered fibre is
represented.

So that, with regard to the fibres connected with these ganglion-cells in the frog, it
may be remarked —

1. That in some instances very fine fibres, not more than the en.oooth of ^^ i"ch in
diameter, are alone continuous with both the straight and spiral fibres of the ganglion-
ceU.

2. That a dark-bordered fibre may be traced to the ganglion-cell as the straight fibre,
while the spiral fibres are continued on as very fine fibres.

3. That the spu-al fibres may be continued onwards as a dark-bordered fibre, which
may be wider, at least for some distance, than the fibre continued from the straight fibre.

* ^Vjchives of Medicine, No. XII.
MDCCCLXIII. 4 G

560 PROFESSOR BEALE OX THE STRUCTURE OF THE SO-CALLED

4. That both straight and spiral fibres may be continuous A\'ith dark-bordered
fibres*.

There is no reason for assuming that the fine fibres in all cases are but an early stage
of development of the dark-bordered fibres ; for the two classes of fibres undoubtedly
exist at every period of life of the animal, even in old age. I conclude, therefore, that
these peculiar ganglion-cells in the frog are connected with both classes of fibres ; and
in mammalia the apparently spherical ganglion-cells, which answer to these pear-shaped
cells of the frog, are connected with both gi-ey and dark-bordered fibres.

It is exceedingly difficult to follow an individual fibre for any distance in the trunk of
a nene. I have seen nerve-fibres diA'iding in the trunks near to ganglia, and on two or
three occasions I have almost convinced myself that one of the fibres resulting from the
subdi\ision passed to a ganglion-cell, while the other passed on in the trunk of the nerve ;
but I have not succeeded in iDreser-v-ing an undoubted specimen of such an arrangement.
The difficulty of following one individual fibre is much increased by the fact that the
nerves vary so much in thickness within a very short distance. Dark-bordered fibres
may be ■very distinct, and within a distance of two thousandths of an inch may be so thin
as to be scarcely demonstrable, swelling out again a little further on. This renders it
necessary to isolate an individual fibre from its neighbours before its course can be
traced with positive certainty for any great distance. A good example of the varying
diameter of dark-bordered fibres within a short distance is represented in fig. 39.

The observations recorded in page 549 seem to show that a ganglion-cell may be
developed upon a nen'e-fibre already formed. Hence these ganglion-cells cannot be
regarded as centres from which two nerve-fibres proceed direct to their peripheral
distribution, but as centres jilaced at a 2>art of a m'cuit which eaisted as a complete
circuit before the ganglion-cell was developed in connexion with it. It is impossible to
discuss this most interesting and important question without entering into the consi-
deration of the connexion of nerve-fibres with other centres, especially with the nerve-
cells in the spinal cord and parts above ; so I will not pursue it further. But I would
state that I have not succeeded in finding ganglia from whicli fibres proceed in one
direction only : and, that I may not be misunderstood upon this point, let me say that
I have never seen a ganglion, in connexion with the nervous system of any creatui-e, the
fibres of which proceed in but one direction only, as is now believed to be the case by
many observers. From every ganglion I have ever seen, fibres proceed to their destina-
tions in at least two different dirc-ctions ; and from the majority of ganglia, even in the
case of those very small ones which consist only of three or four cells, fibres often pass
away in three or four different directions (fig. 5). And in every case in which I have
been able to obtain a separate ganglion-cell well prepared, I have seen at least two

* T do not fcol quite confident that both the fibres proecedinp; from one t/niiijl!(m-ceJK nlthoufjh broad, arc truo
dark-liordcred fibres ; but it would seem pretty cei-tain that in some cases a dark-bordered fibre is connected with
the cell as a spiral, and in others as a straight fibre — a fact which may hereafter bo of some importance, as it
may afford us a positive index of the direction in which the ncrve-curreut circulates in these elaborate organs.

APOLAR, rXIPOLAE, AND BIPOL.VE NERVE-CELLS OF TIIE FROG. 561

fibres ; and although these may run parallel to each other for some distance, they have

been so often observed to pass in opposite directions when they reach the nerve-trunk,

that I belie\e myself justified in expressing a very positive opinion that such is always

the case.

8. Of the ganglionrcells of tlie lieart.

Although the description given in this paper \v\\\ apply to the ganglion-cells of ganglia
in different parts of the frog, including those distributed to the heart and lungs, I feel it
necessary to refer particularly to the cardiac ganglia of the frog, because Kolliker has
recently made some very confident statements with regai-d to the structure, arrangement,
and action of these gangUa, which my observations fail to confirm. Nor is the differ-
ence between us one of interpretation ; we are at variance as to actual facts. With
reference to the ganglia of the heart this observer says, " It may be particularly men-
tioned that the origin of neiTe-fibrcs fiom imipolar cells, and the rarity of the double
origin of fibres, may be especially well seen in one place, namely, in the septum of the

heart of the frog, where also R. Waoxer admits the fact Here also we may most

readily convince ourselves that there are many ajtolar cells without jjrocesses, as is most
plainly shown in the heart-ganglia, and in the small ganglia upon the urinary bladder of
the Bomhinator, in which, as also in similar ganglia of the frog, tlie matter is as clear as
possible " *.

But in his Croonian Lecture last year, his opiaion as to apolar cells has become
curiously modified. Although the existence of such cells was as "clear as possible" in
1860, great doubts are expressed upon the matter only two years afterwards : — " These
cells, that is to say, all of them which are connected with nerve-fibres and whose con-
nexions can be clearly made out, are unij)olar, or send out but a single fibre, and that in
a peripheral direction, ioifhout having any connexion with the transcurrent fibres of tJie
vagus. Bipolar or multijwlar cells are not to be seen : some apparently apolar cells pre-
sent themselves, but concerning these it may be doubted whetlwr they are unipolar cells
whose issuing fibre is in some tvay hidden from view"f.

These positive statements are not illustrated by a single di-awing, nor does Professor
Kolliker give any reasons for modifjing his views as to the presence of apolai- cells.

As the result of very numerous obsen ations, I have to state —

1. That there are no apolar cells, either in the ganglia of the heart or in those of the
bladder of the frog, although Kolliker asserted that their existence was as " clear as
possible."

2. The unipolar cells of Kolliker really have two or more fibres proceeding from
them ; so that his statement, that " all the cells connected with nerve-fibres send out but
a single fibre," is not a fact.

3. Some fibres certainly pass in a central direction ; so that Kollikee's assertion, that
all the fibres pass in a peripheral dii-ection, is not true.

If these ganglion-cells be examined, it will be found that the fibres proceeding from
* Manual of Jlicroscopic Anatomy, April 1S60. t Croonian Lecture, May 1st, 1862.

4g2

562 PEOFESSOE BEALE ON THE STRUCTURE OF THE SO-CALLED

some pass for a certain distance in a peripheral direction, while others pursue the very
opposite (Plates XXXIX., XL. figs. 41 & 41^). Very many lying at the side of a nerve-
trunk pass transversely towards the central part of the bundle of fibres. The arrange-
ment and structure of the ganglion-cells of the heart differ in no essential particulars
from those I have described in other ganglia. I have succeeded in demonstrating in
several instances the straight fibre passing in one direction in the trunk of the nerve,
and the continuation of the spiral fibre pursuing an opposite course. In some of these
ganglion-cells the spiral is reduced to two or three coils (fig. 42), as is observed else-
where, but I cannot but conclude that every cell has at least two fibres.

Nor can I agree with Professor Kolliker in the statement that the ganglion-cells
have no connexion with the " transcurrent " fibres of the vagus. Although I have not
been able to demonstrate how many fibres of the vagus are connected with the ganglion-
cells, nevertheless, looking generally at the course of the fibres, and at the number of
the cells, and considering the facts observed in other ganglia, I regard it as very probable
that many of them are connected with the cells.

9. Of the gnmjUon-celh and nerve-fihres of the arteries.

In the nerve-trunks running near the branches of the arteries of the palate of the
frog are numerous ganglion-cells. These ganglion-cells are often situated at the angle
of di^ision of the nerve-trunks. Some of the fibres from the small ganglia lying near
arteries may be traced to the coats of the arteries, and some fine nerve-fibres resulting
from their subdivision may even be followed amongst the muscular fibre-cells of arteries
not more than the yij^ooth of an inch in diameter.

Ganglia and ganglion-cells are found in considerable number in connexion with the
arteries distributed to the different viscera of the abdomen, heart, and lungs, and very
many are found close to the small arteries which supply the bladder of the frog. In
many cases small ganglia and separate ganglion-cells are imbedded in the external or
areolar coat of the artery.

In fig. 40 a small ganglion in course of development upon one of tlic iliac arteries of
the frog is represented ; and several fine branches of nerve-fibres can be followed amongst
the muscular fibre-cells. I have seen very fine neiTC-fibres beneath the circular muscular
fibre-cells, and apparently lying just external to their lining membrane, composed of
longitudinal fibres with elongated nuclei — an observation which confirms a statement
of Lusciika'.s. I have not succeeded in satisfying myself that nerve-fibres are ever distri-
buted to the lining membrane of an artery, although, from the appearances I have
observed, I cannot assert tliat this is not the case. In tlie auricle of the heart and at
tlie commencement of the larg(> cavic very fine nerve-fibres are certainly distributed
very near indeed to the internal surface, being separated from the blood only by a veiy
tliiu layc'i- of transparent tissue (connective tissue).

The distribution of ncrv(>-fibres to the coats of a small artery about the y^jotli of an
inch in diameter is represented in fig. 45. In all cases (and I have examined vessels in

APOLAK, UNIPOLAR, AND BIPOLAR NERVE-CELLS OF THE FROG. 563

almost all the tissues of the frog) not only are nerve-fibres distributed in considerable
number upon the external surface of the artery, ramifjing in tlie connective tissue, but
I have also followed the fibres amongst the cu'cular fibres of the arterial coat. The
nerves can be as readily followed in the external coat as in the fibrous tissues generally ;
and the appearance of the finest nucleated nerve-fibres, already alluded to, enables one to
distinguish them most positively from the fibres of connective tissue in which they ramify.

These nerves iuAariably form networks witli wide meshes. I have demonstrated such
an arrangement over and over again. A similar disposition may be seen in the auricle
of the frog, in the coats of the venae cavae near their origin from the auricle, among the
striped muscular fibres of the lymphatic hearts of the posterior extremities of the frog,
and in other situations. Kolliker confesses that he has not succeeded in observing
distinct terminations to the nerves distributed to the vessels of muscles. This obseiTcr
has made the very positive assertion that some arteries are completely destitute of
nerves, and, appai'ently without having given much attention to the subject, says,
" hence it is e\ident that the walls of the arteries are not in such essential need of
nerves as is usually supposed." Professor Kolliker seems to conclude, in too many
cases, that what he has failed to see does not exist. It is easy to demonstrate nerves
in considerable number on the arteries of the frog generally, though these nerves, and
more especially those ramifjing in the coats of the vessels of mammalia and birds, are
still considered by many authorities in Germany to be fibres of connective tissue.

The ner\es which supply the small arterial branches in the voluntary muscles of the
frog come from the very same fibres which are distributed to the muscles. I have seen
a dark-bordered fibre divide into two branches, one of which ramified upon an adjacent
vessel, while the other w'as distributed to the elementary fibres of the muscle.

10. Of the connexion of the ganglion-cells ivith each other.

In figs. 5, 15, & 16, which rei)resent ganglion-cells at an early period of development,
several are seen connected together; in fact, the matter of which the several cells are
composed is continuous. This must be the case, at least for a certain time, because a
number of cells may be formed by the division of one (figs. 5, 11, 12, & 15).

After a time, as the new cells separate further and further from each other, the inter-
vening matter which connects them becomes thinner and thinner, and forms what would
be properly termed a fibre, figs. 15 & 16; and as the cells move away from the line
where their formation commenced, these connecting fibres become finer and finer, and
at last could not be distinguished from fibres of connective tissue. It is probable, in
many instances, that all continuity of structure between some of the cells ceases ; but it
is to be remarked in all cases, that the nerve-fibres in the substance of a ganglion cross
each other in various directions, and it is certain that fibres from several different cells
run in the same bundle which leaves the ganglion. From w hat I have observed, I think
it almost certain that, in many cases, ganglion-cells of one ganglion are connected by
fibres with cells of another ganglion.

564 PROFESSOE BEALE OX THE STEUCTURE OF THE SO-CALLED

11. Of the " ccqmile " of the ganglion-ceU, and of the connective tissue and its

corpuscles.

From what has akeady been stated, it will be inferred that there is no actual cell-
Mall or special capsule at an early period of development of the ganglion-cell. The
cell may form protrusions at various parts of its surface, like a young epithelial cell, a
mucus- or a pus-corpuscle ; and each offset may give rise to the formation of a new cell.
After a time the cell is seen to be surrounded by, or imbedded in, a transparent sub-
stance, which in some cases exhibits a definite outline and might be termed a cell-waU
(fig. 25), while in others it would be more correctly described as a matrix. In old
ganglia there is a quantity of this tissue, which accumulates and becomes condensed as
age advances ; and it exhibits a fibrous character with nuclei imbedded in it. In conse-
quence of the condensation of this tissue, it is often very difiiculc to demonstrate the
anatomy of the cells in animals of matiu'e and advanced age.

In order to understand the formation of this textui-e, it is necessary to examine gan-
glia and ganglion-cells at different ages.

Figs. 9, 14, & 15 show that at an early period the ganglion-cell consists simply of
one oval mass of germinal matter, surrounded with a little formed material, ffom each
extremity of which a fibre proceeds. The whole is imbedded in a little transparent tissue ;
and in similar tissue run the very fine nerve-fibres which proceed from the young
ganglion-cell (figs. 10, 17, 18, 22, & 23).

Figs. 4, 27, & 28, on the other hand, show fully formed ganglion-cells which are
imbedded in a tissue exhibiting striations, and a few fine fibres which resist the action
of acetic acid. Around and at a short distance from the gangUon-cells there are several
oval nuclei (connective-tissue corpuscles).

Figs. 4, 27, & 28 exhibit the characters of old ganglion-cells. The fibrous appearance
of the matrix is more distinct, the fibres w^hich resist the action of acetic acid are more
numerous, and there are more nuclei around the ganghon-cell and around the fibres
proceeding from it, but these nuclei are not connected with either.

And the important fact that the so-called connective-tissue corpuscles outside the
ganglion-cells and outside the nerve-fibres are but faintly coloured with cannine, whUe
those nuclei in connexion with tlie ganglion-cell and nerve-fibres, although separated
from the solution by a gi-eatcr distance, are more intensely coloured, must not be lost
sight of. This was very distinct in the specimen represented in figs. 27 & 28.

Before any attempt is made to explain these facts, it is necessary to consider more
particularly the relation of nerve-fibres and nerve-ceUs to the connective tissue and con-
nective-tissue corpuscles. Although it is undoubtedly true that in preparations mounted
in certain fluids it is not possible to distinguish the finest nerve-fibres from connective
tissue, this distinction can be most clearly made out in some of my specimens ; for ex-
ample, in figs. 17 & 18 the nerve-fibre can be very readily distinguished as it runs amongst
the connective tissue, and the true nature of the fibre is placed beyond question by the
presence of the ganglion-cell. All fibres whicli can be followed for a considerable

APOLAE, TJNIPOLAE, AXD ETPOLAR XETIATI-CELLS OF THE FROG. 565

distance, which refract like true nerve-fibres and exhibit an appearance more or less
granular, especially if they can be followed into ganglion-cells, must clearly be pro-
nounced nen'es. The finest nerve-fibres may often be followed amongst the connective
tissue for a long distance, and their relation to other structures most positivelv deter-
mined. Fig. 43 represents a portion of a very fine nerve-fibre running amongst con-
nective-tissue corpuscles, and crossing one of the processes of a pigment-corpuscle ; and
it is unquestionably distinct from the last two structures. In the cornea, as I have
before stated, the nerves may be followed in theu* numerous ramifications amongst the
corneal corpuscles and their processes, and it can be seen that the latter are not con-
nected with the nenes as KtJHNE supposes. The nerve-fibres and the corneal tissue
grow together, but, although closely related, they remain structurally distinct from one
another.

But although there is no structural connexion between the nuclei of a tnie fibrous
tissue (pericai'dium, tendon, cornea, sclerotic, &c.) and the nerve-fibres and nerve-
nuclei ramifying in it, there is some difficulty in deciding upon this question in the case
of certain forms of indefinite connective tissue immediately surrounding nei've-fibres and
ganglia, and it is often not very easy to decide whether a given nucleus really belongs to
a nerve-fibre or should be considered as a connective-tissue coi-puscle. For example :
what is the nature of the nuclei near the nerve-fibre and ganglion-cell in fig. 18 ] These
are undoubtedly, as they now appear, connective-tissue corpuscles ; but how were they
formed ] AVe know that during the growth of nerves and ganglion-cells new nuclei are
formed, and some of these, which lie on the surface of the fibre or cell, produce connective
tissue. The nuclei under consideration are, I beUeve, of this nature ; and I consider it
probable that they belonged to a nerve-fibre at an earlier period, or at any rate resulted
from the division of nuclei which were concerned in the formation of nerve-fibre. So I
believe that those close to the ganglion-cell were formed by it. At the lower part of
the cell may be seen three small nuclei, which are probably of the same nature. From
them, up to a certain period, new nuclei might have been developed and true nerve-fibres
might have proceeded ; but the nuclei can now only produce a low form of connective
tissue, which accumulates around the more important structures. I consider that the
ganglion-cell delineated in fig. 17, from the same specimen and not very far from the
part represented in fig. 18, represents an earlier stage of development than the last
ganglion-cell.

But I have already sho^\•n that in many cases delicate fibres from true nene-fibres,
after gradually becoming very fine, are lost in the connective tissue. I have also
sho^vn that fibres of connective tissue result from the degeneration of nerve-fibres ; and
it has been proved conclusively that connective tissue results from the wasting of nerve-
fibres in disease. Nor is the neiTous the only tissue which by normal wasting or
abnormal degeneration leaves what is termed connective tissue. A structure so special
as a uriniferous tube, or a portion of the ceU-containing network of the liver may waste,
and all that represents it will be w^hat is termed connective tissue and connective-tissue
corpuscles.

566 PEOFESSOE BEALE ON THE STEUCTUEE OF THE SO-CALLED

It seems to me that the almost structureless or delicately fibrous matrix in which both
nerve-fibres and nerve-cells are imbedded is the result of changes which have taken
place before the nerve-fibres and cells there present have made their appearance, and if
these very nerve-fibres and cells had been allowed to remain for a longer period in the
living animal, they would have become surrounded with more connective tissue. Both
fibres and cells might become altered and waste : all the fatty and other constituents
hanng been absorbed, what we term " connective tissue " alone would remain. Connective
tissue and connective-tissue corpuscles are produced from the very same masses of
germinal matter as tliose from which nerve-cells and nei've-fibres are developed ; and I
think it must be admitted that many fibres which resist the action of acetic acid, and
which are generally regarded as consisting of yellow elastic tissue, were once nerves.

Nevertheless true nerve-fibres in which the nerve-current passes do not lose themselves
in the connective tissue or blend with it, nor are they connected with its corpuscles, but
they form networks, as already described. A normal nerve-fibre can always be distin-
guished from a fibre of connective tissue.

All the structiu'es existing in the adult ganglion Avere at an early period represented
only by masses of germinal matter (nuclei), surrounded or separated from each other by,
or imbedded in, a little soft formed material. At a very early period of development
the so-called nuclei of the nerve-fibres are very close together. Nerve-centres at an early
period of development bear little resemblance to the perfectly developed structure, — a
remark which is illustrated in the most striking manner in the case of the particular
ganglion-cells which have been described in this memoir. Nor would it be possible to
prove the real nature of such a structure as that represented in figs. 5, 11, 12, 8c 15, if
seen amongst the tissues of an embryo, imbedded as it would be in embryonic tissue
almost as ricli in nuclei as the structure itself. Even when nerve-tissues have reached
the period of development when their essential anatomical characters are well marked,
and when they perform their characteristic actions, it is often very difiicult, and if the
ordinary processes of preparation be employed, impossible, to demonstrate positively the
arrangement of the nerve-fibres, although we may be quite positive, as for example in
the case of voluntary muscles, that nerve-fibres are there. Nuclei can be seen which
certainly do not belong to capillaries, and these nuclei lie transversely or obliquely across
the muscular fibres, and often several may be seen following each other in lines ; but
only in very favourable cases can any fibre at all be made out, and with the greatest
care and the highest powers a very faint and slightly granular band only can be seen.
Nor can the fibres be traced to undoubted nerve-fibres : and it is even diflScult to be
certain of the nature of what will eventually become the large trunks consisting of
dark-bordered fibres ; so closely do they resemble vessels, and so numerous are the nuclei.
But by using transparent injection the vessels may be made out positively, and by adopt-
ing certain precautions in j) reparation, which it would be tedious to refer to here, many
such difficult points have been definitely settled ; and I have traced the changes wliich
occur in tlie minute structure of many tissues from tlie earliest period at which they
could be recognized up to their fully developed state. In the case of ganglia and nei-ve-

APOL.VE, UNTPOLAR, AND BIPOLAE NEEVE-CELLS OF THE FROG. 567

fibres, wc have at an early period what would be termed nuclei and granular matter
around and between them ; then wc have the fully developed structure (cell-contents,
spii-al fibre. &c.), still beai'ing the same relation to tlie germinal matter whidi produced
it ; and lastly, when this structure has wasted, we have its remains repr(>sented by con-
nective tissue and masses of germinal or linng matter, no longer caj)al)le of producing
special tissue, but only giving rise to the simple, transparent, more or less fibrous mate-
rial, or connective tissue. These masses of li-ving matter are usually known as the con-
nective corpuscles ; but in indefinite connective tissue, neither fibrous material nor cor-
puscles are developed as a special tissue destined for a special purpose in the economy :
it is merely the remains of higher tissues, which have been in great part removed ; or it
is formed by germinal matter which is not capable of gi^dng rise to any special struc-
ture. On the other hand, the so-called connective tissue of the cornea, of tendon, &c.,
is developed as a special tissue, and it may be said to fulfil a special purpose.

The structure of th(> cells in mammalia corresponding to the pear-shaped cells of the
frog is a subject worthy of separate consideration ; but I may mention that in several
instances I have seen a fibre prolonged from the cell, corresponding to the straight fibre
of the pear-shaped ganglion-cells of the fi'og — that the "nucleated fibres" which seem
almost to encircle many of the cells correspond to the nucleated spiral fibre or fibres
described in this memoir. That these nucleated fibres are true nerve-fibres, and not, as
generally supposed in Germany, " nucleated connective tissue," is rendered evident by
careful obsenation of the changes occui-ring during the development of the ganglia, and,
I think, clearly demonstrated by the observations recorded in this paper. I regard it as
certain that if these nucleated fibres surrounding the mammalian ganglion-cells are
connective tissue, both the fibres I have described in the frog's ganglion-cell are of the
same natui'e.

It is possible that, for many years to come, some observers Mill persist in terming
eveiything in which they fail to demonstrate distinct structure connective tissue, and all
nuclei which are not seen in their specimens to be in connexion with positive vessels,
positive nerve-fibres, or other well-defined tissues besides fibrous tissues, connective-tissue
corpuscles ; but there is little doubt that when the changes occurring during the deve-
lopment of special tissues shall have been patiently worked out by the use of high
powers and better means of preparation, opinions on the connective-tissue question will
be completely changed. The idea of the necessity for a supporting tissue or framework
will be given up, and many structures now included in " connective tissue " will be
isolated, just as new chemical substances year after year are being discovered in tlie
indefinite " extractive matters."

It is remarkable how positively many authorities deny the existence of structures
which they have failed to demonstrate. Such a course is only justifiable on the pre-
sumption that the art of demonstrating structure has arrived at perfection ; but we know,
on the contrary, that it is but in its infancy. Surely it is premature to maintain tliat the
vessels of the umbilical cord are destitute of nerve-fibres because we may have failed to

MDCCCLXIII. 4 u

568 PEOFESSOE BE ALE OX THE STEUCXrEE OF THE SO-CALLED

demonstrate them — that the fibres of voluntary muscle only receive neiTous supply at
one point, because authorities vrill not admit that neiTe-fibres may exist wliich are too
delicate or too fine to be demonstrated by the means they may have employed —
that the spindle-shaped fibres of organic muscle generally are not supplied with neiTes,
because they cannot find them — that the fibres prolonged from the large cells in the
cord and in the brain are not continued into fibres, because they have failed to trace
them for any considerable distance.

I think I can con^ince any one, by positive demonstration, that the three last positions
are iittcrly untenable ; while there is every reason to believe that certain elongated nuclei
and fibres, to be seen amongst the muscular fibre-cells of the umbilical arteries and on the
smaller vessels of the placenta, really belong to the nervous tissue.

Moreover many obseners seem to have determined in their own minds what appear-
ance a fibre should present to be entitled to be regarded as a nene (that it must exhibit
the double contour), and then they arbitrarily assert that a fibre which does not present
these characters cannot be neiTous ; and even if it be continuous with an undoubted
ner\e-fibre, it is put do-mi as connective tissue. The alterations which are produced
in rmdoubted neiTe-fibres by stretching, pressure, and the influence of water must not
be forgotten.

It is ti'ue that during the last few years pale fibres have been admitted to exist in
some situations besides the Pacinian coi-puscle ; but few observers will be prepared to
admit the existence of a very extensive distribution of delicate pale nucleated nerve-
fibres in every part of the peripheral nervous system, or that the active portion of all
nerve-fibres exhibits the same essential anatomical characters, and that in all cases com-
plete circuits exist, while free ends are nowhere to be foimd: yet these general con-
clusions are justified by facts which have been demonstrated*.

Conclusions.

The following are some of the most important general conclusions 1 have amved at
in the course of this inquiry : —

1. That in all cases nerve-cells are connected with neiTe-fibres, and that a cell pro-
bably influences only the fibres with which it is structurally continuous.

2. That apolar and unijmlar nen'e-cells do not exist, but that all neiTe-cells have at
least two fibres in connexion >vith them.

3. That in certain ganglia of the frog there are large pear-shaped nerve-cells, from
the lower part of which two fibres proceed: — a, vl straight Jihre conihmovis \\\ih. the

* I have seen numerous xory fine nuclei connected together with exceedingly miniito fibres in the tissues of
many of the lower animals, especially insects. In this class, I am certain, nerve-fibres exist far too minute to
be seen by any power yet made.

I regard very fine fibres and nuclei amongst the contractile tissue of the common Actinia as nervous, and I
have seen a texture presenting similar characters ramifjing in the miiscidar tissue of the Starfish and Sea-
Urchin.

APOLAE, UXirOLAE, A^D BIPOL.YE XEEVE-CELLS OF THE FEOG. 569

central part of the body of the cell; and b, ajibre or fibres continuous with the circum-
ferential part of the cell, which is coiled spirally round the straight fibre.

4. These two fibres, after lying very near to, and in some cases, when the spiral is
very lax, nearly parallel ^\■ith each other, at length pass towards the periphery in oppo-
site directions.

5. Ganglion-cells exhibit different characters according to their age. In the youngest
cells neither of the fibres exhibits a spii-al arrangement ; in fully formed cells there is a
considerable extent of spiral fibre ; but in old cells the number of coils is much greater.

6. These ganglion-cells may be formed in three ways : —

a. From a gi-anular mass like that which forms the early condition of all structures.

b. By the division or splitting up of a mass like a single ganglion-cell, but before

the mass has assumed the complete and perfect form.

c. By changes occurruig in what appears to be the nucleus of a nerve-fibre.

7. During the development of a ganglion-ceU, there is reason to believe that the
entire cell moves away from the point Avhere its formation commenced, so that the fibres
connected \\-ith it will become elongated.

8. There are " nuclei " in the body of the cell ; and there are " nuclei " connected with
the spii-al, and also with the straight fibre. The nuclei in the cell are found upon its
surface, and also in its substance.

9. The matter of which the " nucleus " is composed has been termed by me " germinal
matter." From it alone growth takes place ; and in all cases the matter (formed mate-
rial) of which the nerve-fibre consists was once in the state of germinal matter.

10. The " nucleolus " consists of germinal matter. It may be regarded as a new centre
which originates in a preexisting centre.

11. The ganglion-cells of the frog aa-e connected with dark-bordered fibres, and also
with fine fibres.

12. Contrary to the statement of Kollikeb, that apolar cells and unipolar cells are to
be demonstrated in the cardiac ganglia, all the cells in these ganglia have two or more
fibres emanating fiom them.

13. The muscular coat of all arteries of the frog, and probably of other animals, is
supplied with nene-fibres,

14. Nerve-fibres ai'e not connected with the connective-tissue corpuscles.

15. The so-called nucleated capsule of the ganglion-cells in the ganglia of mammalia
usually consists of nerve-fibres, many of which are connected with the cell.

15. As nerve-fibres grow old the soluble matters are absorbed, leaving a fibrous
material which is kno^\^l as connective tissue. A corresponding change is obsened in
other textures, both in health and disease.

4h2

570 PEOFESSOE BK\LE OX THE STErCTIJEE OF THE SO-CALLED

ExrLA>'ATIOX OF THE PlaTES.

The dimensions of each object delineated can be ascertained by reference to the scales
at the bottom of each page, magnified by the same iwiver as the object itself. These
scales, however, have not been measured quite correctly ; the upper one beuig a little
too long, while the lower is a little too short.

The drawings, with the exception of figs. 32 to 37, are accm-ate copies of nature.

PLATE XXXIII.

Contains figui'es illustrating the structure of the ganglion-cells connected with the sym-
pathetic of the frog.

PLATES XXXIV. cS: XXXV.
Illustrate the changes occurring during the development of the ganglion-cells in the
fully-formed frog, as observed in various ganglia of the sympathetic near the arteries
supplying various internal organs of the frog, in the ganglia on the posterior roots of
the nerves, the ganglia from which the heart, bladder, palate, and other organs receive
their supply of nerve-fibres.

The changes occurring during the development of a granular mass of germinal matter
into perfect tissues (nervous, vascular, muscular, fibrous, cartilaginous, osseous, glan-
dular, &c.) can be studied more satisfactorily in the adult fi-og than in the embiyo; for
a complete history of the changes may be deduced from careful observations upon a
small 2>ortion of an organ or structure in the same animal. For example, nerve-cells in
eveiy stage of development, from a small mass of germinal matter (nucleus) to the fully
developed complex nerve-cell with its straight and spiral Jibre, can be demonstrated
even in one single microscopic ganglion (figs. 5, 6, 7, 8).

PLATE XXXVI.

Shows the relation of ganglion-cells and nerve-fibres to connective tissue and its cor-
puscles (figs. 17 »& 18), and illustrates the connexion between the matter of which the
body of the ganglion-cell consists and that Avhich enters into the composition of the
nene-fibre.

PL.\TE XXXVII.
Exhibits the structure of several different forms of ganglion-cells, all of which possess
two or more fibres.

PLATE XXXVIII.
Illustrates the relation of numerous very fine fibres to a single ganglion-cell, the con-
nexion between some of the fibres of which the bundle is formed, and the relation of
the compound bundle to the cord of connective tissue in wliich it is imbedded. In figs.
29 & 30, the manner in which a fine nerve-fibre is coiled spirally round a dark-bordered
fibre at an early period of formation of the latter is represented.

APOLAE, UXIPOLAE, AXD BIPOLAE ^'EEVE-CELLS OF THE FEOG. 571

PL.\TE XXXIX.

Contains —

1. Figures of ganglion-cells exhibiting paiticiilar characters (figs. 31 & 38).

2. A series of dl•a\^'ings showing the changes which, the author believes, take place in
the production of the ganglion-cell with the spii-al fibre (figs. 32 to 37).

3. A copy of a bundle of nerve-fibres in which the diameter of each fibre is greatly
reduced at the point where the bundle passes through constricted apertui-es (fig. 39).

4. A di-awing of a small compound nerve-trunk mth a finer trunk coming off from it
at right angles (fig. 40). It will be observed that the fine trunk is composed of fibres
which pui'sue opposite directions in the large trunk, passing as it were towards the
centre and towards the periphery.

5. A drawing of one of the pneumogastric nerves of the frog near the auiicle of the
heart. Niunerous ganglion-cells are connected with the trunk of the nerve by very fine
fibres, which are soon lost, but some pui'sue a dii-ection towards the heart, while others
pass towards the brain. The trunk of the nerve is at the lower part of the figure. The
arrow points towards the heai't. The bundle of fibres marked h connects the trunk of
the nerve with that on the opposite side.

PLATE XL.

Fig. 42 represents two of the ganglion-cells, and the fibres connected with them, from
the pneumogastric neiTC (fig. 41). The course of some of the fibres can be traced in
this drawing.

The relation of a very fine nerve-fibre to the connective-tissue corpuscles, and to a
portion of one of the processes of a pigment-cell, is shown in fig. 43, which is magnified
nearly three thousand diameters. The distribution of nen^e-fibres in the tissue external
to capillary vessels is illustrated in figs. 44 & 47. The ramification of fine nerve-fibres
upon the muscular coat of a small artery from the bladder of the Hyla is seen in fig. 45,
and in fig. 46 a portion of a branch of the iliac artery, with some small gangUon-ceUs
and nene-fibres connected with them.

[ 573 ]

XXVII. On the Rigidity of the Earth. Bij W. Thomson, LL.D., F.R.S., Professor of
Natural Philosophy in the University of Glasgow.

Received April 14,— Road May lo, 18G2.

1. That the earth cannot, as many geologists suppose, be a liquid mass enclosed in only
a thin shell of solidified matter, is demonstrated by the phenomena of precession and
nutation. Mr. Hopkins*, to whom is due the grand idea of thus learning the physical
condition of the interior from phenomena of rotatory motion presented by the surface,
applied mathematical analysis to investigate the rotation of rigid ellipsoidal shells
enclosing liquids, and arrived at the conclusion that the solid crust of the earth must be
not less than 800 or 1000 miles thick. Although the mathematical part of the inves-
tigation might be objected to, I have not been able to perceive any force in the argu-
ments by which this conclusion has been controverted, and I am happy to find my
opinion in this respect confirmed by so eminent an authority as Archdeacon Pratt-}-.

2. It has always appeared to me, indeed, that Mr. Hopkins might have pressed his
argument further, and have concluded that no continuous liquid vesicle at all approaching
to the dimensions of a spheroid GOOO miles m diameter can possibly exist in the earth's
interior without rendering the phenomena of precession and nutation very sensibly
different fi-om what they are.

3. Considerations regarding the velocities of long waves in deep sea, of tidal waves
and of earthquake waves, and the harmonic vibrations of a liquid globe, having recently
led me to think of the relative values of gravitation and elasticity in giring rigidity to
the earth's figure, I was surprised to find that the former would have a larger sliare in
this effect than the latter, unless the average substance of the earth had a very high
degree of rigidity. For instance, I found that a homogeneous incompressible liquid
globe of the same density as the mean density of the earth, if changed to a spheroidal
form and then left fr(H>, influenced only by mutual gravitation of its parts, would perform
simple harmonic vibrations in 47"' 12' half-period J. A steel globe of the same dimensions,
without mutual gravitation of its parts, could scarcely oscillate so rapidly, since the velo-
city of plane waves of distortion in steel is only about 10,140 feet per second, at which
rate a space equal to the earth's diameter would not be travelled in less than l"" 8™ 40^

4. Hence it is obvious that, unless the average substance of the earth is more rigid
than steel, its figure must jield to the distorting forces of the moon and sun, not incom-
parably less than it would if it were fluid. To illustrate this conclusion, I have investi-

* Philosophical Transactions, years 1839, 1840, 1842. t Figure of the Earth, edit. imO, § 85.

i This will bo demonstrated in a mathematical paper which the author hopes soon to communicate to the
Royal Society. See §§ 55-58 of " Dynamical Problems, ic." following the present paper in the Traasactions.
MDCCCLXIII. 4 I

574 PEOFESSOE AV. THOMSON OX THE EIGIDITY OF THE EAETH.

gated the deformation experienced by a homogeneous elastic spheroid under the influ-
ence of any arbitrarily given disturbing forces*. I thus find that if 21t! denote the
difference between the longest and shortest diameters of the tidal spheroid, calculated
on the supposition that the substance is of homogeneous (and therefore incompressible)
fluid, and 2h the difference between the longest and shortest diameters of the spheroid
into which the same mass, if of homogeneous incompressible solid matter, would be
defoi-med from a naturally splierical figure when exposed to the same lunar or solar
disturbing influence, we have (see § 34, Appendix to this paper)

, 19 n

2 gwr

where vj denotes the mass of unit volume, and n the "rigidity" of the substance (see

§ 71 of the paper following the present in the Transactions) ; and g denotes the force

of gra\-ity on a unit of mass at the surface, and r the radius of the globe.

5. The density of iron or steel (7*8 times that of water) does not differ very much

from the mean density of the earth (5*6 times that of water according to Cavendish's

experiment, or G'6 according to the Astronomer Royal's). The rigidity of iron, according

to experiments of my brother. Professor James TiiOMSOxf. is 10,800,000 lbs. per square

inch. Since the weight of 1 lb. at Glasgow, where the experiment was made, is 32 "2

British absolute units of force, we must multiply by 32"2 to reduce to kinetic measure

as to force ; and we must multiply by 144 to make the unit of area a square foot instead

of a sqviare inch. We thus find, in consistent absolute measure,

«=501xl0^—

the unit of mass being 1 lb., the unit of space 1 foot, and the unit offeree that force which,

acting on one pound of matter during a mean solar second of time, generates a velocity

of 1 foot per second. In terms of the same units we have r=20,887,700; y=32-14,

being about the average over all the earth ; and for iron or steel w = 487. Hence

, h' h< _.,^i,

'^"19 501 xlU" ~2-44~" '•
2 * 3308 X lO**

Of glass, the rigidity is, according to Wertheim, about one-fifth f)f the value we have
just used as that of iron ; and therefore if the earth were homogeneous of its actual
mean density, and had throughout the same rigidity as that of glass, the result would
be/^ = -7S//.

G. Hence it appears that if the rigidity of the earth, on the whole, were only as much
as that of steel or iron, the earth as a whole would yield about two-fifths as much to the
tide-producing influences of the sun and moon as it would if it Jiad no rigidity at all ;
and it would yield ])y mor(! than three-fourths of the fluid yielding, if its rigidity were
no more than that of glass.

* The solution of llii.s problem vaVi. bo found in llio paper referred to above (see §§ 47, 48).
t Cambridge and Duliliu iratlieniulical Journal, 1S4S.

PEOFESSOR W. TIIOMSOX ON THE RIGIDITY OP THE EAETH. 575

7. Such a defi)rmation as this would be quite uudiscovenible by any direct i^eodetical
or astronomical observations ; but if it existed, it would largely influence the actual
phenomena of the tides and of precession and nutation.

§§ 8-20. Effect of the EartKs Elastic Yieldimj on the Tides.

8. To find the effect of the earth's elastic yielding on the tides, let 2H denote the
difference between the greatest and least diameters of the spheroidal surface perpendicular
to the resultant of the lunar or solar disturbing force*, and terrestrial gravitation

supposed perfectly symmetrical about the centre, then — will be the eUipticity of that

spheroid ; and we shall caU it the elUpticity of level produced ly the lunar or solar
influence on a rigid earth. It may be remarked that H is the height of high above low
water in the "equilibriimi tide" of an ocean of infinitely small density covering a rigid
earth.

9. Let H' denote the height of the equilibrium tide for an ocean of density =m of the

earth's mean density, the eartli being still supposed licrfecthj rigid and covered by the
ocean. Then the terrestrial gravitation level will be tlistui-bed (as is proved in the
theory of the attraction of ellipsoids) fi-om the spherical surface to the spheroidal sui-face

3 1 H'
of eUipticity 7 • ^ • — , by the attraction of the ocean in its altered figure. The eUipti-
city of level induced by lunar or solar influence must be added to this to give the eUip-
ticity of actuid. level, which is of course the eUipticity of the free equiUbrium sui'face of

H'
the ocean, or according to oiu- notation — • Hence

r

H'_H 3 2 ^
r r '5 * N " r '
by which we find

H'=

5 N

For sea-water the value of N is about ^ ; and therefore

H'=||h=112H,

or only 12 per cent, more than for an ocean of infinitely small density.

10. What we have denoted above by li is the value of H' for N=l ; and therefore

* This " disturbing force" is of course the resultaut of tho actual attraction of either body on a unit of mass
in any position, and a force equal and opposite to its attraction on a unit of mass at the earth's centre.

4i2

576 PEOFESSOE W. THOMSON OX THE EIGIDITT OF THE EAETH.

and

, 5 H

, 19 n
2 gwr

11. Now, according to a proposition regarding the attraction of ellipsoids already used,
we have 7 • - for the ellipticity in the terrestrial gravitation level produced by the ellip-

ticity of deformation - experienced in consequence of want of perfect rigidity. Hence
the ellipticity of the terrestrial gi-antation level, as distui-bed by lunar or solar influence,
is -•- +—. This will be the absolute tidal equilibrium ellipticity of an ocean of infi-
nitely small density covering the elastic globe ; but since there is a tidal ellipticity
- induced in the solid itself, the height fi'om low tide to high tide of fluid relatively to
solid (that is to say, the diflerence of depth between high water and low water) mil be

or o

5

or, according to the value of A just found (§ 10),

2 gwr

, 19 n
2 gicr

12. This result expresses strictly the height of the equilibrium tide of a liquid of
infinitely small density covering an elastic solid globe. It may be regarded as a better
expression of the true tidal tendency on the actual ocean than the slightly different
result calculated with allowance for the efiect of the attraction of the altered watery
figure constituting the equilibrium spheroid, and its influence on the figui'e of the
elastic solid ; since the impediments of land and the influence of the sea-bottom render
the actual ocean surface altogether diflferent from that of the equilibrium spheroid.

13. Hence the actual tidal tendency, which would be H if the earth were perfectly
rigid, is in reality

^•— H

2 gv<r

, 19 n
2 gwr

where n denotes what we may call the eartlis tidal effective rigidity, being the " rigi-
dity" of a homogeneous incompressible solid globe of equal mass which, with an ocean
equal and similar to the earth's, would exliibit the same tides.

14. If, for example, we give n the value for ii-on or steel above indicated, the for-
mula becomes "59 xH. The comparison between theory and observation, owing to the

PEOFESSOE W. TUOMSON ON THE RIGIDITY OF THE EAETH. 577

extreme complexity of the ciicumstances, has been hitherto so imperfect that we canuot
say it disproves this result ; and therefore, from tidal phenomena hitherto obsers ed, we
cannot infer that the earth is more effectively rigid tlian steel.

15. The value of ?i for glass, according to "Wertheim, is 2160000 xl-l-lx 32-2, in

British absolute units; and it reduces the formula to jtH. Now, imperfect as the

comparison between theory and observation as to the absolute height of the tides has
been hitherto, it is scarcely possible to believe that the height is in reality only two-
ninths of what it would be if, as has hitherto been universally assumed in tidal investi-
gations, the earth were perfectly rigid. It seems therefore nearly certain, with no other
evidence than is afforded by the tides, that the tidal effective rigidity of the earth must
be greater than that of glass.

16. Any approach to a close testing of the absolute amount of the tidal influence can
scarcely be expected of either of the two great Kuietic* theories — the Oceanic theory
of Lapi^vce, or the Channel theory of Airy, — as applied to diumal or semidiurnal
tides ; but notwithstanding the strong contempt which has been expressed by the last-
mentioned naturalist f (no doubt justly as regards false applications of it) for the Equi-
librium theory:}:, we may look to it confidently for good information when it is applied
to test the difference between mean fortnightly variations of level at two well-chosen
stations, one in a low latitude, and the other in a high latitude. (Sec Note at the end
of this paper.)

17. The fortnightly tide** at each pole gives high water when the moon's declina-
tion (A), whether north or south, is greatest, and low water when she crosses the
equator ; and the whole difference in level produced by it would be

H'sin^A

if the earth were all covered vnih M'ater. The mean daily level at the equator, on the
same supposition, would vary by half that amount, being low w^ater when the moon is
furthest from the equator, and high when she crosses the equator. But, o-ning to the
actual distribution of land and water, either of those variations may be diminished by

• Dyiuunicis meaning properly the science of force, and there being precedents of the very highest kind,
for instance, inDELAUXAT's ' ile'eanique Rationncllc ' of 1S61, and Eobisos's ' Mechanical Philosophy ' of 1804,
in favour of using the term according to its proper meaning — and the modem corrupt usage, which has confined
it to the branch of dynamical science in which relative motion is considered, being excessively inconvenient and
vexatious, — it has been proposed to introduce the term "kinetics" to express this branch; so that dynamics
may be defined simply as the " science of force," and di\'ided into the two branches. Statics and Kinetics. The
introduction of this new term, derived from Kivijais, motion, or act of moving, does not interfere ^^'ith AiiPEEE's
term, now universally accepted, "kinematics" (from vi'irj^a), the science of movements.

t " Naturalist. J^ person well versed in Natural Philosophy." — Johnsos's Dictionary. Armed with this
authority, chemists, electricians, astronomers, and mathematicians may surely claim to be admitted along with
merely descriptive investigators of nature to the honourable and convenient title of Naturalist, and refuse to
accept so un-EnglLsh, unpleasing, and meaningless a variation from old usage as " physicist."

X Encyclopoedia MetropoUtana, " Tides and Waves," §§ 64, 539, &c. *• Aiby's ' Tides and "Waves,' § 45.

578 PROFESSOR W. THOMSOX OX THE RIGIDITY OF THE EAETH.

an amoiint which it is impossible to estimate theoretically ; but then the other must be
increased by nearly the same amount. And if a denote the mean height of the sea-
level above a fixed mark at .the earth's north pole, about times when the moon's decli-
nation is greatest, h the corresponding mean of observations about times when she is
crossing the equator, d and V con-esponding means derived from observation at an equa-
torial station, and ^H something intermediate between H and H', we must have

«_5 + i'_rt'=f^HsurA,
whatever be the distribution of land and water over the earth, only provided the fort-
nightly tide follows sensibly the equilibrium law, which, for moderately well-chosen
stations, we may suppose it must do.

IS. If, instead of being at a pole and at the equator, the stations are in latitudes
respectively I and 1! , we should have

rt _ i 4. ^' _ «' - 1 ^H sin- A (sin- ;- sin- V).
Now if we suppose the moon's mass to be -^ of the earth's, we have H = l-92 foot. As
H'=:1-12H, and as there is more area of water than of land over the earth, we cannot
be far wrong in taking ,H=l-08H=2-04 feet.

The greatest value of A is 28" 37'; and lience, in the most favourable lunations,
« — i + i'— «'=-713 foot X (sin-/— sin'-/').

19. Iceland and Teneriffe, in nearly the same longitude, and in latitudes 63^ 20' and
28° 30', would probably be very favom-able stations. For them sin- /—sin^/'= -571 ; and

therefore

,,_i_|_i'_«'=0-407foot,
or about 4'9 inches.

It is probable that carefully made and reduced observations, with proper allowance

for barometric disturbances, at two such stations, would not only detect this tide, but

would give a tolerably accurate determination of its amoimt.

20. It would be, for Iceland and Teneriffe, as found above, 4-9 inches if the earth
were perfectly rigid ; or 3 inches if the tidal effective rigidity is only that of steel ; or
about an inch if the tidal effective rigidity is only that of ghiss.

There seems no more hopeful way to ascertain how rigid the earth really is, than to
make careful obser\ations with a view to determuiing the fortnightly tide -with all
possible accuracy. It is possible also that very acciu-ate observations on the semi-
dim-nal tides in a deep inland lake of great extent, or at distant points of the Mediter-
ranean sea-board with only deep water intervening, might help to solve this question.

§§ 21-32. Effects of Elastic YichJing on Precession and Nutation.

21. If we suppose the sun, the moon, and the earth's centre to be reduced to rest at
any moment, the two former to be held fast m theii' places, and their attractions on the
latter to be balanced by an infinitely great mass held at the proper infinitely great
distance in the proper direction ; the tide-generating distribution of force, and the couple

PEOFESSOE W. THOMSON ON TIIE ETGIDITY OF THE EAETH. 579

tending to turn the earth round an equatorial diameter, by wliicli precession and nuta-
tion are produced, would be left precisely as they are.

22. If the sun and moon be carried round the earth at rest, according to their tme
relative orbital motions, and if the infinitely distant mass be shifted continually so as
always to balance their attractions on the earth's centre of gravity, all the phenomena of
tides, of nutation, and of precession, will take place precisely as they do in reality.

23. If, now, merely as an artifice to avoid mathematical calculations, we suppose the
earth's rotation to be stopped, and instead a repulsive force, from an infinite fixed line;
coinciding Avith the earth's axis in the first place, to be introduced — the force to vary
directly as the distance fi-om this fixed line, and to amount to -^r; of gra\ity at the
equator, — the figure of the earth will remain unchanged.

24. Let us first suppose the earth to be either perfectly fluid, or to be a perfectly elastic
homogeneous and isotropic solid, of spherical figui-e when undisturbed, and the sun and
moon to be held fast in any positions. It will clearly remain in ])erfect equilibrium in
the circumstances defined in (^^ 21, 23, whether the sun and moon are both in tlic plane
of its equator or not ; because if it is not in equilibrium it must commence rotating with
a continually accelerated motion about some axis ; the fixed repelling line not being sup-
posed to impede its motion, but merely to continue repelling according to the stated law.

25. But either sun or moon, if not in the plane of the equator, and the corresponding
part of the infinitely distant balancing attractor, will exercise a couple upon the earth,
by attracting the near protuberant equatorial parts more, and the remote less, than the
centre. How then is this couple balanced 1 Clearly it is by the repulsion of the fixed
repelling line on the tidal deformation produced by the sun or moon, as the case may be.
Considering for simplicity only one, the sun for instance, we perceive that the equili-
brium tide will produce a small elliptic deriation, superimposed on the great polar and
equatorial cllipticity, the longer axis of this smaller superimposed ellipticity being in
the line through the sun. Now without this the spheroid of revolution would expe-
rience no resultant action from the repelling line ; but with it the actual resultant sphe-
roid will experience, from the repelling line, a couple tending to carry away from this
line the longer axis of the siiperimposed ellipticity. This is exactly opposite to the
couple produced by the attraction of the sun ; and as there can be no resultant action,
we see that the equilibrium is maintained by the balancing of these two couples.

26. Hence in reality we conclude tliat the couple due to a disturbing body in any
position attracting a rotating fluid spheroid, or elastic isotropic body naturally spherical
and rendered oblate by rotation, is balanced by the couple of centrifugal force on the
crowns of the tidal elongation produced by tlie disturbing body, provided the rotation
is not so fast as to render the tidal deformation sensibly different from what it would
be if the disturbing body rotated with the same angular velocity as the spheroid. This
condition Avill, it is intended, be shown, in a subsequent communication to the Royal
Society, to be essentially fulfilled when the rotation is slow enough to allow the first ap-
proximation to be used as in the ordinary investigations regarding the figure of the earth.

27. Let us now suppose the earth to be an elastic spheroid of nearly on the whole the

580 PROFESSOR W. THOMSON ON THE EIGIDITY OF THE EARTH.

same figure in all its surfaces of equal density as a rotating fluid, and therefore on the
whole nearly free from distorting stress in its interior. If under the lunar and solar
influences it were to yield and experience nearly as much tidal elongation as it would if
quite fluid, the couple to which precession and nutation are due would be very nearly
balanced by the centrifugal force on the cro^vns of the tidal elongation, and consequently
precession and nutation would be very much less than if the earth were perfectly rigid.
For instance, if the earth were a homogeneous incompressible elastic solid of the
same rigidity as glass, it would, as stated above, experience seven-ninths of the tidal
deformation of a fluid globe of the same density. Hence seven-ninths of the couple
would be balanced, and precession and nutation would be reduced to two-ninths by the
elastic yielding. Even if the rigidity were as much as that of steel, the precession and
nutation would not be more than three-fifths of their full amount for a perfectly rigid
spheroid.

28. The close agreement between the observed amounts of precession and nutation,
and the results of theory on the hypothesis of perfect rigidity, renders it impossible to
believe that there is enough of clastic yielding to influence the phenomena to any con-
siderable extent. It is worthy of remai'k, however, that in general the theoretical esti-
mates of the amount of precession have been somewhat above the true amount demon-
strated by observation. It seems not altogether improbable that this discrepance is
genuine, and is to be explained by some small amount of deformation experienced by
the solid parts of the eartli, under lunar and solar influence.

29. But the only possible ground on which it could be maintained that the earth as
a whole is less rigid than a solid steel globe of the same dimensions, is to assume that
there is an enonnous liquid vesicle, or a solid nucleus separated by a fluid layer from
the outer crust, in the interior, and that the loss of precessional eff"ective moment of
inertia, owng to this portion not being carried round in the precessional movement, is
almost exactly compensated by a diminution of the generating couple in very nearly the
same proportion by elastic yielding. Although, considering IIalley's theory of the
secular variation of terrestrial magnetism, and the general accordance of its results with
the actual phenomena as demonstrated by the best observations made up to the present
time*, it would be most rash to say that it is very improbable there is a solid iron nucleus
sunk to the centre of a hollow central vesicle of fluid in the earth, yet it seems to me
excessively improbable that the defect of moment of inertia due to fluidity in the earth's
interior bears approximately the same ratio to the whole moment of inertia, as the actual
elastic yielding bears to the yielding which would take place if the earth were perfectly
fluid. Avoiding conjectural assumptions, however, I conclude that either this proportion
is approximately fulfilled, or l)oth the following propositions are true : —

I. The defect of moment of inertia, owing to fluidity in the interior, is small in com-
parison -with the whole moment of inertia of the earth.

II. The deformation experienced by the earth, owing to lunar and solar influence, is
small in comparison with what it would experience if it were perfectly fluid.

* General iSaiiinb, Proceedings of the Roj-al Society, 1802.

PROFESSOR W. THOMSON OX THE RIGIDITY OF THE EARTH. 581

30. It is easily seen tluit the first of these piopositious is not opposed to IIallky's
theory. For instance, if there were a spheroidal iron core* 2000 miles diameter, cool
and magnetic to within IdO miles of its surface, sunk to the centre of a spheroidal space
of lighter fluid 3000 miles diameter, enclosed within a solid crust 2-300 miles thick, the
moment of inertia would he only about one-half per cent, less than it would he if the
whole were rigid. Far less magnetism than such a nucleus could retain would be
sufficient in amount to account for either the whole of " terrestrial magnetism " as
manifested at the surface, or for the secular variations of it which have been observed.
Whatever may be thought of the probability of this hypothesis, the barest possibility
that it may be true renders it an interesting problem for mathematicians to find the
precessional movement of a rigid spheroid sunk in a lighter liquid enclosed witliin a rigid
spheroidal shell. A solution, founded on the supposition that all the bounding surfaces
are truly elliptic and of small ellipticity, and that tlie fluid portion is of uniform density,
is to be obtained with ease in an extremely simple form, and might be i;seful as a guide
for speculation.

31. That the '-tidal effective rigidity" (^ 13), and wliat we may similarly call the
"precessional effccti\e rigidity "' of the earth, may be both several times as much as that
of iron (which would make the phenomena both of tides and of precession and nutation
sensibly the same as if the earth were perfectly rigid), it is enough that the actual
rigidity should be several times as great as the actual rigidity of iron, throughout
2000 or more miles' thickness of crust.

32. At the surface, and for many miles below the surface, the rigidity is certainly
very much less than that of iron (how much less might be estimated if we had trust-
worthy data as to the velocity of natural or artificial earthquake w'aves through short
distances) ; and therefore at great depths the rigidity must l)e enormously greater than
at the surface. That both the rigidity and the resistance to compression should be much
greater several hundred miles do\\ni than at the surface, seems a natural consctpience of
the enormous pressure experienced at those great depths by the matter of the earth.

Xofe on the Forfniyhfli/ Tide.

33. In water 10,000 feet deep (which is considerably less than the general depth of tlu;
Atlantic, as demonstrated by the many soundings taken within the last few years, especially
those along the whole line of the Atlantic telegraph cable, from Valencia to Newfound-
land) the velocity of long free Avaves is 567 feet per second f. At this rate the time of
advancing through 57° (or a distance equal to the earth's radius) would be only ten hours.
Hence it may be presumed that, at least at all islands of the Atlantic, the fortiiightly
tide should follow sensibly the equilibrium law.

* All ancient cold iron meteorite which may have entered a nebula of smaller boJiis and furraed the nucleus
of our present earth, which under such circumstances coull not but be built up and heated by attracting them
to itself.

t AiRT, § 170.
MDCCCLXIII. 4 K

582 PEOPESSOR W. THOMSON ON THE EIGIDITT OF THE EARTH.

"In the Philosophical Transactions, 1839, p. 157, Mr. Wiie'\\'ELL shows that the
" observations of high and low water at Plymouth give a mean height of water increasing
" as the moon's declination increases, and amounting to three inches when the moon's
" decimation is 25°. This is the same direction as that corresponding in the expression
" above to a high latitude. The effect of the sun's declination is not investigated from
" the observations. In the Philosophical Transactions, 1840, p. 163, Mr. Whewell has
" given the observations of some most extraordinary tides at Petropaulofsk in Kam-
" schatka, and at Novo-Ai'khangelsk in the island of Sitkhi on the west coast of North
" America. From the curves in the Philosophical Transactions, as well as from the
" remaining cui-ves relating to the same places (which, by Mr. Whewell's kindness,
" we have inspected), there appears to be no doubt that the mean level of the water at
" Petropaulofsk and Novo-Arkhangelsk rises as the moon's declination increases. We
"have no further information on this point."' — Aiey's 'Tides and Waves,' i^ 533.

Appendix, added January 2, 1864.

34. Let the difference of longest and shortest radii, which would be produced by
lunar and solar mfluence in the two cases — of the earth supposed a homogeneous
incompressible fluid tending to the spherical shape by gravitation alone, and supposed
a homogeneous incompressible elastic solid without mutual gravitation but tending in
vii'tue of its elasticity to the spherical figure — be denoted by h' and h" respectively ; and
let h be the difference of greatest and least radii when both gravity and elasticity act
jointly to maintain the spherical figure. We shall have obviously

For the distorting force, being balanced by elasticity and by gravity jointly, may be
divided into two parts, one jn of the whole, balanced by elasticity alone, and the other
Yf, balanced by gra-\ity alone: and therefore 4> + 4^1-

But, by § 53 of the following mathematical investigation regarding elastic spheroids,
we have h''^-^ — Y9^?'\ where m denotes the mass of the disturbing body, and c its

distance from the earth's centre. With the same notation we have, by the aid of § 51
3 m ?-^

y

of the same paper, 'H = - ^ — ^ where H has the meaning defined above in § 8 of the

5 3 m ?•«

present paper ; and therefore, § 10, /t' = 73 ~ ' l^'om this and the value above for

// 19« . h'

h'', we have ,,,=?; — ? and, as we have just seen that /«=- r,^ we have the result used

' /," 2gwr ' •> IJ

in § 4,

[ 583 ]

XXVIII. Dynamical Problems regarding Elastic Sjtheroidal Shells and Spheroids of
Incompressible Liquid. By Professor W. Thomson, LL.L., F.R.S.

Received Augiiat 22, — Eead November 27. 18G2.

1. The theory of elastic solids in equilibrium presents the following general problem : —
A solid of any shape being given, and displacements being arbitrarily produced or
forces arbitrarily applied over its whole boimding surface, it is required to find the dis-
placement of every point of its substance. The cliief object of the present communica-
tion is to show the solution of this problem for the case of a shell consisting of isotropic
elastic material, and bounded by two concentric spherical surfaces, with the natural
restriction that the whole alteration of figure is very small.

2. Let the centre of the spherical surfaces be taken as origin, and let x, y, z be the
rectangular coordinates of any particle of the solid, in its undisturbed position, and x-\-a,,
y+f3, z-\-y the coordinates of the same particle when the whole is in equilibrium under
the given superficial disturbing action. Then, by the known equations of equilibrium
of elastic solids, we have

fd'P ,P^ d^l3\ d /da d^ dy\

^\da:^^d,/^dzy+^^d2\da;^di/^dz)—^' J

m— ^ and n denoting the two coefficients of elasticity, which may be called respectively
the elasticity of volume, and the rigidity. A demonstration of these equations, with defi-
nitions of the coefficients, will be found in § 71 of an Appendix to the present commu-
nication.

3. For brevity let

dx'dy'ds '

(2)

so that 5 shall denote the cubic dilatation at the point (x, y, z) of the solid. Also, for

J 2-r^-2 "t. M^uv^L^ix ^j Y . Then the preceding equa-

d^ d^ d^
brevity, let the operation t-2+ j-a+xa be denoted by y''

tions become

4x2

(3)

584 PEOFESSOE v. THOMSON OX DYNAMICAL PEOBLEMS EEGAEDING

4. In certain cases, especially the ideal one of an incompressible elastic solid, tlie
following notation is more convenient : —

p the mean normal pressnre per unit of area on all sides of any small portion of the
solid, round the point .r, y, z. Then (below, § 21)

p=-(,„-J„)(J+|+S); (4)

and the equations of equilibrium become

__o 7)1 dp A

m — i a ax

nvV '^f^OA ' (5)

m — \ a a:

5. If the solid were incompressible, we should have ?n=oo and

t/.r+r/y + rf.- — "'

which must be taken in^^tead of (4). and. along with (5), would constitute the four diffe-
rential equations required for the four unknown functions «, /3, y, p *.

G. To solve the general equations (3) or (o), take -7-. of the fii-st, -,j of the second,

and V cif the third ; and add. AVe have thus

(«+w)V=S=0, (6)

or, which is in general sufficient,

V'o=0 (7)

If, now, an appropriate solution of this equation for h is found, the three equations (3)
may be solved by known methods, the first of them for a, the second for /3, and the third
for y, — the arbitrary part of the solution in each case being merely a solution of the
equation V^" = "- These arbitrary parts must be determined so as to fulfil equation (2)
and the prescribed surface conditions.

The complete particular determination of I cannot, howcM-r, in most cases be effected
without regard to a, ft, y ; and tlie order of procedure which has been indicated is only
convenient for determining the proper forms for general solutions of the equations.

7. First, tlien, to solve tbe ('(juation in h generally, we may use a theorem belonging
to the foundation of liAi'LACK's remarkable analysis of the attraction of spheroids, which
may l)e enunciated as follows.

If the equation V'o=0 is satisfied for every point between two concentric spheres of

• See Professor Stokes's paper '• On the Frietiuii of Fluids in Moticjii, ntul the Fiiuililiriiini and Jtotion of
Elastic Solids," Cambridge Philosophical Society's Transactions, April, IS-l.').

ELASTIC SrilEROTDAL SHELLS AXn SPIIEKOIDS OF INCOMniESSIBLE LIQUID. 585

radii a (c^reatcr) and a (less), the valiu- of S for any point of this spaco, at distance ?*from
the centre, may be expressed by the double series

Vo +V, +V, +&C.
+V>- + V>-^+\>-'+ &c.,

of which the first part converges at least as rapidly as the geometrical progression
and the second at least as rapidly as

°'0)'("')"-'

— if V,-, V'i denote homogeneous functions of .r, y, z of the order /, each satisfying, con-
tinuously, for all values of .r, y, z, the equation

A proof of this proposition is given in Thomson andTAix's 'Xatiual Diilosophy.' chap. i.
Appendix B. It is also there shown, what I believe has been hitherto overlooked, that
V(, V;, as above defined, cannot but be rational and integral, if / is any positive integer.

8. To avoid circumlocution, we shall call any homogeneous function of (.c, //, r) whicli
satisfies the equation r72Y_f)

a " spherical harmonic function," or, more shortly, a " spherical harmonic." 'Thus V^
and \'i, as defined in ^ 7, arc spherical harmonics of degree or order /; and V,r~^'~',
being also a solution of ^¥=0, is a spherical harmonic of degree —(/+!). We shall
sometimes call the latter a spherical harmonic of inverse order i. Thus Ui l)eing any
spherical harmonic of integral degree /, and therefore necessarily a rational integral
function of this degree, Wjr"'''-' is a spherical harmonic of degree — (/+! ). or of inverse
order /.

If we put— (/+])=/, and denote tliis last function by J^, then we liave

and thus it appears that the relation between a spherical harmonic of positive degr(-e /
and of negative degree J is reciprocal. The general (well known) jn-oposition on which
this depends is that if V,- is any homogeneous function of (,r, //, z) of degree /, positive
or negative, integral or fractional, Y^r"^'"' is also a solution of the equation V^V=0
(see Thomsox and Tait's 'Natural Philosophy,' chap. i. Appendix B.).

A spherical harmonic of integral whether positive or negative degree, satisfying the
differential equation continuously for all values of the variables, will be called an " entire
spherical harmonic," because such functions are suited for the solution of acoustical and
other physical problems regarding entire splieres or entire spherical sliells.

A spherical harmonic function of (,r, y, z) will be called a '• spherical surfacc-har-

586 PEOFESSOE W. THOMSON ON DYNAMICAL PEOBLEMS EEGAEDING

moiiic " when the point (x, y, z) lies anywhere on a spherical surface having its centre
at the origin of coordinates. A spherical surface-harmonic is therefore a function of
two variables, angular coordinates of a point on a spherical surface. If Y; denote such
a fimctiou of order /, positive and integral, then Y^r'' and Yj;'"'"' are what we now call
simply sj^herical harmonics ; but sometimes we shall call them, by way of distinction,
" spherical solid harmonics." Functions Y,, or spherical surface-hannonics of integral
orders, have been generaUy called " Laplace's coefficients" by English writers.

9. From the theorem envmciated in (^ 7, we see that the general solution of our
problem, so far as S is concerned, is this : —

S=S;i„"(V,+V;r-"->) (8)

10. Now because the equation V"» = 0 is linear, it follows that differential coeffi-
cients of any solution, with reference to a\ y, z, or linear functions of such differential
coefficients, are also solutions. Hence the terms V; and V-r~^'~', of S, give haiTuonics of

the degrees i—\ and — (/4-2), in — , ^, V- To solve equations (3) we have therefore
" ^ ^ (Ix di/ (h X K /

only to solve

where <p„ denotes an entire spherical harmonic of any positive or negative degree, n.
Tryiner

which is obriously the right form, we have

But, because <p„ is a homogeneous function of x, y, z of degree «,

and because it is a spherical harmonic,

V'<fi„=0.
We have also

by diflercntiation. Hence

V-«=A.2(2?i+3)<p„,

and therefore the complete solution of the equation
is ^s

where V denotes any solution of the equation.

V^V=0.

ELASTIC SPHEEOEDAL SHELLS A^D SPHEEOIDS OP INCOMPRESSIBLE LIQTHD. 587

11. llenco, by taking for <p„ the terms of ^, ^, ^ above referred to (§ 10), and

giving n its proper value, ?'— 1, or — (i+2), for each tei-m as the ctvsc may be, we find,
for the complete solution of (3), the foUoAving: —

a=:S| »i+?/jr~^~' — -

-(V.-V/— ) ,

y=S]w. +«;;>•-"■-' —

:(V.-V>-- )

(9)

n. 2(21+1) dz^

where «„ ?<■, ?;„ v\, Wj, w- denote six harmonics, each of degree i.

12. But in order that these formulae may express the solution of the original equations
(1), the functions u, i\ &c. must be related to the functions V so as to satisfy equations
(2) and (3). Now, taking account of the following formula,

which becomes simply 2i(p.

if <Pj is a spherical hai-monic of any degree i (whether positive or negative, integral or
fractional), we derive from (9) by differentiation, and selection of terms of order ^, and
of order inverse i (or degi'ec — i — 1),

+|+l=2{4.^+^>-^■■-"-.Wl)[^•V. + (^•+l)V..--■]},

da dj3 dy
dx

where, for brevity, we put

and

^* dx ^ dy ^ dz '

Hence, to satisfy (2) and (8),

and

from which we find

dx

V.=^^.-

dy

(10)

rt(2?4-l) '*

_ m((+l)

^•— '♦'■ n(2j4-l)^"

V. "^^'+^) ^

'~(2n + 7n)t + n^"

^.^ «(2£+l)_.,_
' (2a+m)«+n + j7»^'

(11)

PROFESSOE W. THOMSON ON DYNAMICAL PEOBLEMS EEGAEDING

13. Using tliesc in (9), we conclude

n ",

/3=-;=?

y=^;-,r

v' d r ^, ■ii.r "' ' ~] 1

2 dx\-{'ln-'riii)i + n [2n-\-m)i-\- ii-{-mJ y

• +(.'_ r--'''-—-\ ^ '^'■'•""~' ll .

'^ ' ' 2 </yL (2/; + »«)' + " (2;( + m)z + ?i + ?;(J J i

^^^^^::^::^lj !

n {'2/1 + »i) i +n + iiiJ j J

(12)

1 '; -o.-j-i ""

2 (/cL(2rt + 7/i); +

(13)

for a complete solution of the general equations (1), the equations of equilibrium of an
isotropic elastic solid. The circumstances for which this solution is a])propriate will
be understood when tlu> general proposition of (^ 7 is duly considered.

14. It remains to show how the harmonics ii^, r, , n\, ii\, v\, ii\ are to be deter-
mined so as to satisfy the superficial conditions. Let us first suppose these to be
that the displacement of every point of the bounding surface is given arbitrarily.
Let 2A, , 2B, , i'C, be the harmonic series f , expressing the three components of the
displacement at any point of the outer svuface of the shell, and 2Ai , «B- , SC- the
corresponding expressions for the given condition of the inner surfoce. Thus the
surface-equations of condition to be fulfilled are

a = 2A,,
/S = 2B, ,

a=2A;,

where a and a' denote tlie radii of the outer and inner surfaces respectively, and
Aj , B, , C; , A ■ , Bj , C| spherical surface-harmonics of the order /.

15. Now collecting from the series (12) of § 13, which constitute the general expres-
sions for a, /3, 7, those terms which, being either solid spherical harmonics of degrees
i and — / — 1, or sucli functions multiplied by ?•', give, at the boundary, surface-harmonics
of the order /, and equating tlie terms of this order on the two sides of equations (13),
we have

* F(ir tlic ease ( = 0, tlir tiTm^i '/_,, ''[_,, '",_, nuiy be oinittrd : but tbi'ir full intrrpn-tution would he to
express a clisplawmcnt witbout dofurniution. Tbiis ii'_^, bciii^' of degree —1. cannot but be—, wlicrc A is
a constant ; and therefore !(;_,j—'-' + ' becomes A wlicn (=<).

t That is, series of terms each of wliirli is a spherical surfacc-hannonic of integral order /. That any
function; arbitrarily given over an entire sjihirical surface, maybe so expressed, is a well-known theorem.
A demonstration of it is given in Thomson and Tait"s • Natural Philosophy,' chap. i. A])i)endi.\ 1!, § s.

ELASTIC SPHEHOIDAL SHELLS AXD SPHEEOTDS OF IXCOMPRESSTBLE LIQUID. 589

= A, when >•=«,

' 2 dx[.(2n + m)i + 3n + m {2n + 7n)i—nj\=A'i when )' = a',

V +r'r-«->-!?^-^r '^■>' ^l^-i2L!i!ll| = B, when r=a,

' ' '■ 2 di/L(2ii + m)i + 3n + m {2n + m)i—nj\ = B\w\ionr=a',

dsl(2

Wi-^-tOi)''

mr^ d

^l/,+, vj/j-ir"""^' "|f=C( when r=ff,

(2« + TO)J + 3« + m (2n + »!)8-nJ l. = C; when r=a'.

(15)

16. These six equations would suffice to determine the six harmonics (/„ r„ w„ ?;-, vl, ?<;■,
if %!.(+, and -v^l., were known. For, since each of those six functions is a homogeneous
function of x, y, z of order /, each of them divided by y' is a function of ant>uUir coordi-
nates relative to the centre, and independent of r ; and therefore if, for instance, we denote
Ui by T*n! and u\ by yW', we have two unknown quantities m and m to be determined by
the two equations of condition relative to a for the outer and the inner surface. These

equations may be written as follows, if we further denote -—^ by r'^, and --^-^

by r~'~'^', because these are homogeneous functions of the orders ^ and — ^— 1 respect-
ively :

ti7fl«+'+tir'=A./?'^' +

a-;

2[{2n-hm)i + 3n + m] 2[{2n + m)i—n]

•a',

i_l_„' W/'' + '_L

^-:

y.

'2l{2n + 7n)i + 3n + 7n] '^ 2[{2n + m.)i—n]

Resolving these equations for vy and tu\ and returning to the original notation instead
of C7, -or', a, ^',

dx

„2i+l_„'2i+l

where, for bre\'ity,

]yj "« 1

m: ,=.

2[(2« + ffi)i — h]
Introducing, also for brevity, the following notation,
^ _«>+'Ai-a''+'A- ■>

a' — ("«')''""(a'A'i-fl"A,)

Q2i + 3 a'2i + 3

(10)

(17)

fH';.

,2t+l „'2i+I -'^^ i-2 5

^•+2 Q2i+l_g/K+I ■"■'■1+2 ? 3*i-2 g2i+l_glii+l ^^i-i

MDCCCLXIII. 4 L

• (18)

590 PEOFESSOE W. THOMSOX OX DYXAMICAL PEOBLEMS EEGAEDIXG

we have the expressions for Ui and i([ given below. Dealing mth the equations of
condition relative to j3 and y, and introducing an abbreviated notation B; , B- , C; , Cj
corresponding to (17), we find similar expressions for i',, I'j, iVi, w'i, as follows: —

«.= a/+iB...%'-iH.

(19)

dx

' ' ^ '^^ dy "2 c?^

(20)

17. It only remains to determine the functions ^^ and -4^', which we can do by com-
bining these last equations with (10) of § 12. Thus, changing i into?-|-l in (17) and
into i—1 in (18), applying equations (10) of § 12, and taking advantage of the following
properties,

v^,^,=o, vx4'r''-')=o, &c.,

dx "^ du " ds

and

we find

d'^ii diii diii

' dx

, _d{%^,-^^^) , rf(S,>.r-^') , d{€i^r^

dx ~^ dy
^' \ dx

_^<n:^,^_^^2.■+3)(^•+l)iH;_,^^i.,

'-+'-

i-ir-']

r^'+'+(2?--l)?^,^,v|/,.

(21)

dy ds

These equations, used to determine the two unkno\vii functions xj/^ and \|/(, give

._ e,+ (2i+3)(z-+i)ffl;^ie;

"^^ l-(2J + 3){2i-l)(i+l)iffi,>ifi.+i'

^;'*=

where, for bre\ity,

(2i-l);^,>ie, + e;- j

l-(2j- + 3)(2J-l)(i+l)ffHi+,JJi+,' J

0,

^^/lai+.r--^') <^(gi+i?-'+') . </(€.+ 1?-'^')

dx

'^ +

dy

+ -

(/z

' \ c^ "^ rfy "^ rfr J ■ j

(22)

(23)

ELASTIC SPHEEOIDAL SHELLS AXD SPHEROIDS OF mCOMPRESSFBLE LIQUID. 501

18. The functions -v^ and 4'' lieing expressed in terms of the data of the probU'm by
equations (22), (23), (17), (18), (IG), we have only to use (19) and (20) in (12) to tind
the following expression of the complete solution : —

a = 2(!a,>-'+9;r-'-'+(fH,-i2/-''^^-M,>-^)%'-(ffi>-^'^'+f2;-M>-=)^(^

dx

_(iH'.,^'+'+^;._M;r»)

d{^i^ir-

'^

(24)

y=2|€.r'+<r-'-' + (ltt,-J2/--^-^'-M.v-^)%^-(^;/-^'^^+f2;-M;7^)^y:^^

19. This solution leads immediately, through an extreme case of its application, to
the solution of the general problem for a plate of elastic substance between two infinite
parallel planes: — Given the displacement of every point of its surface, requii-ed the
displacement of any interior point. For if we give infinite values to a and «', and keep
a—d finite, the spherical shell becomes an infinite plane plate.

20. It is, however, less easy to deduce the result in this way from the solution for
the spherical shell, than to apply directly the general method of § G to the case of the
infinite plane plate. We shall return to this subject {) 31, below), when the details of
the investigation will be sufficiently indicated.

21. A very important part of the general problem proposed in § 1 remains to be
considered,^that in which not the displacement, but the arbitrarily applied force, is
given aU over the surface. To express the smface-equations of condition for such data,
we must use the formulae expressing the stress (or force of elasticity) in any part of an
elastic soUd in terms of the strain (or deformation) of the substance. These are

Ja.

/d^ dy

■ "•" 7/.-

(25)

Q=(.+,0|+(-^0(|+^:);

„ /d^ , dy\ ^ /dy , dcc\ ^. /du d&\

where P, Q, R are the normal tractions (which when negative are pressures) on the
faces of a unit cube respectively perpendicular to the lines of reference OX, OY, OZ,
and S, T, U the tangential forces along the faces respectively parallel, and in the direc-
tions in these planes respectively perpendicular, to OX, OY, OZ (see Appendix, § 70).
22. In teiTns of these we have the following expressions for the components F, G, H
of the force on a unit ai-ea perpendicular to any Une whose dii-ection cosines are /, y, h : —

F=P/+U^+T7^-

G=U/+Qy+S/^l (26)

H=T/+S^+E/i

4l2

592 PEOFESSOE W. THOMSON ON DYNAMICAL PEOBLEMS EEGAEDING

(see " Elements of a Mathematical Theory of Elasticity," Philosophical Transactions for
1856, p. 481).

23. Using the expressions (25) in (26), we find

and symmetrical expressions for G and H.

24. If now we suppose f, g, h to denote the direction-cosines of the normal at any
point X, 7/, z of the surface of an elastic solid, the surface condition, when force, not
displacement, is given, will be expressed by equating F, G, H respectively to three
functions of the coordinates of a point in the surface, quite arbitrary except in so far
as they must balance one another in order that equilibrium in the body may be possible ;
and therefore they must fulfil the follomng integi-al equations : —

jyFJQ=0, j}GfZQ=0, j}HcLQ=0, (28)

J[f(H^-G2)fZO=0, Jj(Fz-Hx)dQ=0, JJ(Ga,--Fy)=0, . . . (29)

where dQ, denotes an element of the surface at the point (x, t/, z), and the double
integrals include the whole surface of application of the forces F, G, H.

25. For our case of the spherical shell, with origin of coordinates at its centre, we have

/=-, f/=^, h=-; (30)

and the last triple tenn in the expression (27) for F may be conveniently written thus: —

n djux + fiij + yz) na ,^,

r di 7' ^""^f

ax^-^y+yz^r,, (32)

Then, for brevity, putting
and

'i+4y+'i='-v (33,

where — prefixed to any function of x, y, z will denote its rate of variation per unit of

length in the radial direction; and using (2) of § 3, we have, by (30) and the symme-
trical equations for G and H,

F.=(._.)a..+.[(.|-i).+-f},'

Gr=(m-,05..y + »|(r|-l)3+--J^-},
Hr=(>u-;05.c+»[(.|-l)y.f§}.

(34)

26. It is to be remarked tliat tJiese equations express such functions of [x, y, z), the
coordinates of any point P of the solid, that ¥.co, G.u, H.4; are the three components of

ELASTIC SPUEfiOID.U. SHELLS AisD Si'UEliOlDS Oi' INCOMPRESSIBLE LIQUID. 593

the force transmitted across an infinitely small area u perpendicular to OP, while, for any
point of either the outer or the inner bounding spherical surface, Fco, Gna, H» are the
three components of the force api)lied to an infinitely small element u of this surface.

27. To reduce the surface-equations of condition derived from tlicsc expressions to
harmonic equations, let us consider homogeneous terms of degree i of the complete
solution, which we shall denote by a,, /Sj , y, , and let ^,_i*, ^+1 denote the correspond-
ing terms of the other functions. Thus we have

Gr=2|(»^-,05.-,y+"(^-l)/3,+^^ ^;' },
Ilr=xl{m-n%_,z-\-nii-l)y,+n^\.

(35)

28. The second of the three terms of order ^ in these equations, when the general
solution of § 13 is used, become at the boundary each explicitly the sum of two surface
harmonics of orders i and i — 2 respectively. To bring the other parts of the cxpi-essions
to similar forms, it is convenient that we should first express ^+1 in terms of the general
solution (12) of i§ 13, by selecting the terms of algebraic degree i. Thus we have

#.-

(36)

where

' ' 2l(2n + m)i—n—m] dx
and symmetrical expressions for /3, and y^, from which we find

Hence, by the proper formulae [see (42) below] for reduction to harmonics,

'='+>— 2«+l\2[(2a+/B)i-n-m] ^^(-i-t-^^+i |' • • • • (^^

and (as before assumed in § 12)

dui , dvi dwi /qq\

Also, by (11) of § 12, or directly from (36) by differentiation, we have

S,..:=^^ "(-;-^) .4.,., (40)

{2n-\-m)i—n—m ^ ' ^ '

Substituting these expressions for 5i_,, a„ and ^+, in (35), we find

* When i— 1 is positive, J,_i will express the same function as V,_i of § 9 above. The suffixes now intro-
duced have reference solely to the algebraic degree, positive or negative, of the functions, whether harmonic or
not, to the symbols for which they are applied.

594

PEOFESSOR W. THOMSON ON DYNAMICAL PEOBLEMS EEGARDING

p _-?r r -i\ ,n{2i-l)[(7n-27i)i + 2m + n] n[2i{i-l)m-{2i—l)n'] _^d4n-i _

tr—^^n[l ljHi+ (2,;+i)[(,„ + 2»>-m-«] '^'■^'-'~(2i+l)[(m+2«)i-7«-w]'" dx

This is reduced to the required harmonic form by the ob\dously proper formula

^}

2i+l (L

^-Sr-(4

^^•->— 22-ir <^-i'

Thi;s, and dealing similarly with the expressions for Gr and Hr, we have, finally,
F,=„.{(,-l,„,-2(,-2)M,,..i^-E,^".*^^-^'^},l

where [as above, (IG) of § 16.]

(42)

(43)

M,=

and now further

E,=

2 [m + 2n)i — m — n
{m — 2n)i+ 2m + n

(44)

[2i+\)[_{m-\-2n)i—m — n\

29. To express the surface conditions by harmonic equations, let us suppose the
superficial values of F, G, H to be given as follows :

F=2A,,1

G= SB,, [when r=a,

H=SC,

and

(45)

G = 2B- , when r=a',

H=sc;J

where A,- , B^ , C; , A- , B- , Cj denote surface harmonics of order i. Now the terms of
algebraic degree ^, exhibited in the preceding expressions (43) for Fr, G?', Hr, become,
at either of the concentric spherical surfaces, sums of surface harmonics of orders i and
i — 2, when i is positive, and of orders —i — 1 and — ?'— 3 when i is negative. Hence,
selecting all the terms which lead to surface harmonics of order ?, and equating to the
proper terms of the data (45), we have

'(«•-!)» -(.•+2).._.._,-2^M,,,r='^^-i+2(/+l)M_.,,r%'

' ax

-2,_l<?(^--2>-''

dx

_ 1 /(/<p.+i_t?^-A
2i + 1 \ dx dx J

[A, when r=ff,
[A'l when r=n',

(4G)

and symmetrical equations relative to y and z.

30 These equations might be dealt with exactly as formerly with the equations (15)

ELASTIC SPHEROIDAL SHELLS AIST> SPHEEOIDS OF INCOMPRESSIBLE LIQUID. 595

of § 15. But the following order of proceeding is more convenient. Commencing with

the fii-st of the surface equations (46), multiplying it by ( - j , attending to the degree of

each term, and taking advantage of the principle that, if •v// be any homogeneous function
of X, y, z, of degree t, the function of angular coordinates, or of the ratios x:y:z,

which it becomes at the spherical surface r=a, is the same svs ( - J 4' ^o^" ^iiy value of r.

we have

dx

=A.(0', (47)

f(i-l)«,-(/+2)0y"V,._.-2tM,,,a'%^+2(?'+l)M_...,«^Q

^^ dx — -^-i--^ dx ~2i+\\_ dx ~\a) dx \

where the second member, and each term of the fii'st member, is now a homogeneous
function of degree ?', of x, y, z (being in fact a solid spherical harmonic of degree and

order i). Taking -i- of this, and -r and x of the two symmetrical equations, adding,

taking into account equations (38) and (39), and taking advantage of the equation
V'V=0 for the solid harmonic functions concerned, we have

^|[^-l+(2^>l>•E,]^/.,.,-2(^>l)«-'-V'-'<p_,-2(^>l)(2^>l)^M_,,Y^)""^J._J)

^ (48)
a^y dx '^ dy ' dz \ J

Again, multiplying (47) by a~^r~'^'~\ and taking r'^'^^i- of the result, dealing similarly

with the two symmetrical equations, and adding, we have

^|2m-^,,,-[^+2-(2^•+l)(^+l)E_._,]Q'"V-.-.+2^•(^•+l)(2^+l)M.,,^/,..,, ||

7'"'+^ rrf(A.r--') rf(B,r-'-') rf(C.r-'-')'l |

—fl'+^l dx '^ dy ~^ ds l J

Changing i into z — 2 in this equation, we have

^|2(^-2)a-?5.._,-[^-(2/-3)(^-l)E_,„]('-y'"'^^_,+2(^•-2)(^•-l)(2.-3)M,^^,_,|

?•'■-' frf(Ai_,r-'+')rf(B._,r-'-^') rf(C._ir--^') )
«' y dx ' dy ' ds j

Precisely similar equations, derived from the inner surface condition of the shell, are
obtained by changing a. A, B, C into a!, A', B', C. We thus have (48), (50), and the
two corresponding equations for the inner surface, in all four equations, to determine
the four unknown functions •4',_i, V'-i? Pi-ii 'P-o "^ terms of the data which appear in
the second members. The equations being simple algebraic equations, we may regard
these four functions as explicitly determined. In other words, we may suppose <Pi and
■^i known for every positive or negative integral value of i. Then equation (47), the

(49)

(50)

596 PEOEESSOE W. THOMSON ON DTNAIHCAL PEOBLEMS EEGAEDING

two equations symmetrical with it, and the others got by changing A, a, &c. into A', a',
&c., give ;?; , v, , x^i explicitly in terms of known functions, and the expressions (30)
for a, , /3, , 7, complete the solution of the problem.

31. The solution for the infinite plane plate is of course included in the general solu-
tion for the spherical shell, as remarked above for the case in which surface displace-
ments, not surface forces, were given ; but, as in that case, it will be simpler and prac-
tically easier to work out the problem ah initio, taking advantage of the appropriate
Fourier forms. Tlic relative ease of the independent investigation is indeed still greater
in the case in which the surface forces are given than in the other case, since the general
expressions for the surface forces assume simple forms when the surface is plane, and
require no such transformation as that which we have found necessary, and which has
constituted the special difficulty of the problem, when the siu'face was spherical. The
problem of the plane plate presents many questions of remarkable interest and practical
importance ; and although the object and limits of the present paper preclude any
detailed investigation of special cases, we may make a short digression to work out the
general solution.

32. Let the origin of coordinates be taken in one side of the plate and the axis OX
perpendicular to it. Then, according to the general expressions (25) of § 21, the three
components of the force per unit of area, in or parallel to either side of the plate, are
respectively

parallel to OX, P=(.,+„) J+(..-./) (|+ J), j

parallel to OY,U=«(|+|,),
parallel to OZ, T=»(^-f '^V

:5i)

The surface condition to be fulfilled is that each of these functions shall have an arbi-
trarily given value at every point of each infinite plane side of the plate.

33. From the indications of i^ G above, it is easily seen that the following assump-
tions are correct for a general solution of the equations of internal equilibrium, and con-
venient for the application at present proposed,

' dx

(i=v +.r

^/'

where u, v, w, and <p denote functions of (.r, //, z) which each fulfil the equation V V=:0.
From these, by differentiation, and by taking y'^ip=:0 into account, we have

la. d^ dy du dr dw f/f

'x "*" c/y "*" dz dx ' dij ' dz "^ dx '

ELASTIC SPHEROIDAL SHELLS JCSD SPHEROIDS OP INCOMPEESSIBLE LIQLHD. 597
or

if

dx

. du dv dw

(52)

and I be used mth the same signification as above (§ 2). Also, by differentiation and
application of the equations V-h=:0, V- j|=:0, we find

V.«=2^ V>3 = 2^ VV=2-^.

Hence, to satisfy the general equations of internal equilibrium (3) of § 3, we must
have

dx m-\-2n

^^•

Hence the general solution becomes

ffi +

2«V.

(i=v —

y=.W —

d\^da:

m + 2n di/

mx dSj^dx
m + 2n dz

(53)

where u, v, w are any functions whatever which satisfy the general equation V-V=0,
and >}/ is given by (52) ; and where, further, it must be understood that ^-^dx must be
so assigned as to satisfy the equation V^V=0, which \|/ itself satisfies by virtue of (52).
34. The general form of the solution of V^V=0, convenient for the present apphca-
tion, is clearly

where ^, s, t ai'e three constants subject to the equation

If now we suppose, as a particular case, the surf;xce condition to be that
P= A sin {sy) sin (tz), 1

and

MBCCCLXIII.

U=B cos {sy) sin {tz), Uvhen a:=0,
T=C sin {sy) cos {tz),\

P=A'sin {sy) sin {tz)A
1J=:B'cos {sy) sin {tz), Iwhen x=.a,
T= C sin {sy) cos {tz),]
4m

(54)

598 PEOFESSOE W. THOMSOX ON DYNAMICAL PEOBLEBIS KEGAEDJNG

where A, B, C, A', B', C are six given constants, we must clearly have

v={(ii-*"-\-(f^')coii{sy)sm{tz), \ (55)

%v={]ii-f'-{-]t'^') sin [sij) cos {tz), J

where f\ g. h, f. (j\ h' are six constants to be determined by six linear equations
obtained directly from (54), (51), (53), (52), (55). But, by proper interchanges of
smes and cosines, we have in (54) a representation of the general terms of the series
or of the definite hitegrals, representing, according to Foukier's principles, the six
arbitrary functions, whether periodic or non-periodic, by which P, U, T are given over
eacli of the two infinite plane sides. Hence the solution thus indicated is complete.

35. To complete the theory of the equilibrium of an elastic spheroidal sliell, we
must now suppose every point of the solid substance to be urged by a given force.
The problem thus presented will be reduced to that already solved, by the following
simple investigation.

36. Let X, Y, Z be the components of the force per unit of volume on the substance
at any point .r, //, z. (That is to say, let qX., qY, qZ be the three components of the
actual force on a volume q, infinitely small in all its dimensions, enclosing the point
(x, y, z). Not to unnecessarily Unlit the problem, we must suppose X, Y, Z to be each
an absolutely arbitrary function of a.', y, z.

37. AVhen we remember that a; y, z are the coordinates of the undisturbed position
of any point of the substance, and differ by tlie infinitely small quantities a, /3, y
from the actual coordinates of the same point of the substance in the body disturbed
by the applied forces, we perceive that X.(h'-\-Y dy -\-Z(lz need not be the differential
of a function of three independent variables. It actually will not be a complete
differential if tire case be that of the interior kinetic equilibrium of a rigid body
starting .from rest under the influence of given constant forces applied to its surface,
and having for their resultant a couple in a ])lanc perpendicular to a principal axis.
Nor will X(h--\-Y(ly-{-Zdz be a complete difterential in the interior of a steel bar-
magnet held at rest under the influence of an electric current directed through one
half of its length, as we perceive when we consider Faraday's beautiful experiment
showing rotation to supervene in this case when the magnet is freed from all mecha-
nical constraint.

38. The equations of elastic ('(]uilil)rium ai"e of course now

VI llO -V

■«+?«-;-= A,

du:

nV'ii+m'^=-Y,
''y

(50)

ELASTIC SPHEHOIDAL SHELLS AND SPHEROIDS OF INCOMPRESSIBLE LIQUID. 599

Let sr, f, <r denote some three particular solutions of the etjuations

VV=-X, j
V^f = -Y, (57)

vv=-z. I

These, m, o, a, we may regard as kno^Mi functions, being derivable from X, Y, Z by knowTi
methods (Thomson and Tait's 'Natural Pliilosophy,' chap. vi.). Then, if we assume

and

n '

the equations (56) of interior equilibrium become

' ' ax n ax

"'^^ dt/ n dy

where ? is a known function given by the equation

„ dijr dg dr
' dx ' dy~' dz

(58)

(59)

(60)

(61)

Now, as we verify in a moment by differentiation, equations (59) and (60) are satis-
fied by

— m rfd

if 3^ is some particular solution of

' n[7n + n) dx
_ _ —m rfd
"' 7i(m + n) dy

_ —m d^
'' n(m + n) dz

(62)

(63)

Hence (58), (57), (62), (63), (61) express a particular solution of (56).

39. We conclude that the general solution of (56) may be expressed thus:-

1/ m d^\ \

a^-( 7s — — ; — :r- I + a,
n\ m-\-n ax j ' '

(64)

4m 2

600

PEOFESSOE W. THOMSON ON DYNAMICAL PEOBLEMS EEGAEDING

where

^

=v-

'X,

=v-

'Y,

=v-

=z,

=v-

+

dg di
dy'dz

)'

(65)

according to an abbre^iated notation, which explains itself sufficiently ; and 'a, '/3, 'y
denote a general solution of the equations

„„,-,, d /d'cc d>^ , d'y\ -

„,, , d /d'u , d'^ , d'y\ .

«V-y+m;^(^^+-5^+-^j=0.

(66)

40. This solution is applicable of course to an elastic body of any shape. It enables
us to determine the displacement of every point of it when any given force is applied
to every point of its interior, and either displacements or forces are given over the
whole surface, if we can solve the general problem for the same shape of body with
arbitrary superficial data, but no force on the interior parts. For 'a, '/3, 'y are deter-
mined by the solution of this problem, to be worked out Avith the given arbitrary super-
ficial functions modified by the subtraction from them of terms due to the parts of a, j3, y
which are explicitly showai in terms of data by equations (64) and (65).

41. Hence the problem of § 35 is completely solved, — whether we have displacements
given over each of the two concentric spherical bounding surfaces, when the solution of
§§ 14-18 determines 'a, '/3, 'y; or forces given over the boundary, when the solution of
§§ 26-30 is available. In the former case the superficial values of the functions

1/ w dS)\

n\ m + n dx ) '

1 / m ^\

n\^ m + n dyj '

n\ m-\-n dz I

known from equations (65), must be subtracted from the arbitraiy functions given as
the superficial values of a, /3, y, and the residues, expressed in surface-harmonic series
by the known method, will be the harmonic expressions for the superficial values
of 'a, '/3, 'y III the latter case, we must first substitute those known functions

-(■m ^ -tA, Sic, instead of a, /3, y respectively in (34), and the values of Fr, Gr, Hr

thus found must be subtracted from the given arbitrary functions representing the true

ELASTIC SPHEROIDAL SHELLS A^^) SPHEROIDS OF IXCOlSfPRESSIBLE LIQUID. 601

superficial values of Fr, Gr, Hr. The remainders, which we may denote by 'Fr, 'Gr, 'Hr,
must then be reduced to harmonic series, as in (45), and used according to the investi-
gation of § 30, to determine 'a, '/3, 'y.

42. The general solution (G4) and the expression just indicated for the terms to be
subtracted from the data so as to find 'Fr, 'Gr, 'Hr, becomes much simplified when, as
in some of the most important practical applications, li.dx-\-Y(li/-{-Zdz is a complete

differential. Thus let

dW

dx dy

-z=

(67)

W denoting any function of w, y, z. Then, assuming, as we may do according to (65),

we have by differentiating, &c.,

and therefore

Hence the solution (64) becomes

dx^dy^d:—^^'

a=v-=w.

From this we find
and (§ 25)
if

and

Hence, by (34),

But

1 rfs

m+n da: ' '

■^ m+n dy ' ^^

_ 1 dd
' m + n dz'' ''

m + n ' '

i-l-n' dri V

dx dy dz

(68)

(69)

(70)

X='axViiyVyz.
dx^ dr—\ dr^^jdx

602

PEOFESSOR W. THOMSOX ON DYNAMICAL PROBLEMS REGARDING

Thus for Fr, and the symmetrical expressions, we have

(71)

43. These expressions become further simpHficd if W is a homogeneous function of
any positive or negative integral or fractional order ?"+!, in which case we shall denote

it by W.+, . For ^ will be a homogeneous function of order ?'+3, and ^ of order i-\-2.
Hence

d d

d^

(72)

Hence the preceding become

F'-=;;^.{('«-'OW,., .■+2»(/+2) Jj+'Fr, ^

G'--;;^.{('«-«)W,.., ^+2«(.:+2)|}+'Gr,
H'- = ^{("^-")W,., . +2«(/+2) g}+'Hr. ^

44. These expressions are the more readily reduced to the harmonic forms proper for
working out the solution, if the interior force potential, W;+, , is itself a harmonic
function. We then have (§ 10)

a L_^.w '^ ^—fm +ir^^^^^V

"^~2(2z + 5)' '■+" ffo— 2i + 5V '+'^2' dx J

and

which give
Fr=

w -_!— I •'^±' •"■+5

<^a;

'}•

(73)

1 f?n + (i+l)» ^</Wi+i ?w(2?: + 5)-w ^^^.^ rf(Wi+,r^')

7n + n

2e + 3

rfa:

}+'Fr,

'(2i + 3)(22 + 5) ' n?.r f i -'' • • • ('*)

and symmetrical expressions for Gr and Hr. Here the terms to be subtracted from tlic
arbitrary functions given to represent the superficial values of Fr, Gr, and Hr are each
explicitly expressed in sums of two surface harmonics of orders i or — i — 1, and 2+2 or
— i — ?) respectively, viz., in each cas(\ that one of the two numbers wliich is not negative.
45. Wlien the sliell is in equilibrium under the influence of the forces acting on it
through its interior, without any application of force to its surface, we must have
Fr=0,1 ■ ., ,■ ■

Gr = 0, when ;•=« and when /•=«' (75)

H>-=0,

ELASTIC SPHEEOIDAL SHELLS AIS'D SPHEEOIDS OF INCOMPRESSIBLE LIQUID. 603

Hence, for the case in which W is a spherical harmonic, the preceding equations
give the proper harmonic expressions for '¥r, 'Gr, 'llr at the outer and inner bound-
ing surfaces, for determining 'a, '/3, 'y by the method of §§ 28-30. Thus, using all
the same notations, with the exception of 'a, '/3, 'y, T, 'G, 'H, instead of a, /3, y,
F, G, H, and, for the present, supposing i+1 to be positive*, we have the complete
harmonic expressions of 'F, 'G, 'H, each in two terms, of orders i and i-\-2 respect-
ively. Hence the A, A', &c. of (45) are given by tlie following equations : —

A. _ A. _

m + (J+l)n^_irf\Vi+j

{2i + 3){7n+n)

dx

B. _ B. _

a'+' a''+'

»n+(i-fl)re irfWi+i

{2i + i){m + n)

dy

Ci _ C'i _ _ ?w + (t + l)ra i dWi+i
a'+'~a'''+'~ (2j' + 3)(m + n) rfs '

A,^2_A;+3_ (2i>5)w-w ^,-^3 rf(W.-+,r-"-^) ,
a''+' a''+' (2J+3){2f + 5)(7M + w) dx

Bi^2_B'i+^_ (2i + 5)m-n ^.^a f?(W.+ ,r-'--^) ^
a'+i" a''+' (2J + 3)(2t + 5)(»H+n) «^y

C.+._C;+2_ {2i + 5)m-n ^in-a <^(W,^,>-'-^)

a' + ' 0"+' (2f + 3)(2f+5)(OT + n)

(76)

46. The functions derived from A,, B,, C,, »&c., which are required for formulae (48)
and (49), are therefore as follows : —

rf(A.r')
dx

rf(A,r---')
dx

d{Ai^,r'^'-)
dx

+

+

rf(B^
dy

d(B,r-

■t" ds

= 0,

dy

rf(C,r--2') _{i+\){2i+\)[m + {i + \)n-] a'^.y
" dz ~ (2f + 3)(w + n) ',.2.+3>V,

+

rf(B.>ar-+°) rf(C.^,r'-^') _ (i + 2)[(2z- + 5)w-?0

rfy

+

rfy

(2i + 3)(7?j + ?i)

a-+'W,

[• (77)

d{Ai^,r-'-») //(B.^.r---") rf(C.-+,r--3) ^ ^
dx ~^ dy "* dz '

with the corresponding expressions relative to A-, B^, Q, &c., obtained simply by
changing a into a!.

Hence by (48) and (50), and the two corresponding equations for the inner surface,
we infer that each of the four functions %?/(_,, %{/_,, <p,_,, <p_, vanishes. By the same
equations, with i changed into i-\-2, Ave obtain expressions, all of one harmonic form,
direct or reciprocal, as follows, for the four functions of order ? + l : —

* As we stall not in the present paper consider particularly any case of a shell influenced by centres of
force in the hollow space within it, which alone could give a potential W,>i of negative degree, we need not
■mite any of the expressions in forms convenient for making i + 1 negative.

604

PEOFESSOK W. THOMSON ON DYNAIMICAL PK0BLEM8 EEGAEDING

\!/ =K r--'-^W-

(78)

K,+,, Kj+i, L,- 1, L|+„ which need not be here explicitly expressed, being four constants
obtained fi"om the solution of four simple algebraic equations. Lastly, by the four
equations with (^+4) instead of i, we find that ^J'i+3, 4'-i-4 5 <Pi+3? <P-i-4 ^U vanish. Using
these results for -^p and <p in (47), we see that each of the functions to must be a harmonic

congnient wath either j_ or

')

find

Ui =

-L,+,

dx

dx

Hence, by using (78) in (38) and (39) we

(2j+l)(i4-l) dx

U-i-,=

-k;.+i

_ -K,+, ,,+,rf(Wi+,r-^'-3)

{2i + 5)(j + 2)

(2f+l)(t+l)' dx '

<fa;

^'-'-' (2i + 5)(f + 2)

<ia;

(79)

and symmetrical expressions for v and w. Finally, using these expressions, (79) and
(78), in (86), and the result in (69) with (73), we arrive at an explicit solution of the
problem in the following remarkably simple form : —

i = e.

dx "*"""' d^

(80)

where

■+' (i+l)(2f+l) "'"l2(2i + 3)(m + «) '^'

K,+ ,

(7« + 2re)z + ?rt4-3

1^ j.

\-2m + 3n]

(81)

Jj + 3)(2f + 5)(m + «) 2 (TO + 2n)z + !

47. In conclusion, let us consider the case of a solid sphere. For this we have

\{/_i_2=:0, and (p_i_2=0,

as we see at once from the character of the problem, or as we find by putting a'=:0 in
the four equations by which in § 40 we have seen that Kf+i, K^+i, Lj+i, Li+, are to be
determined. Then, by (48), with i changed into i-{-2, and by (49), we find

_ (i + 2)[(7«+2n); + 7w + 3«][w(2i + 5)-n]
'^*+'-- (2i + 3)n{/w[2(z + 2f+l]-n(2i + 3)} (?« + «) '+"

_ , {i+\n2i + l)\_m(i + 3)-n-]
****+' — ^'2«{/«[2(» + 2)*+l]-«(2i + 3)} *+'•

(82)

ELASTIC SPHEEOIDAL SHELLS AND SPHEEOIDS OF EXCOMPEESSIBLE LIQUID. 005

The coefficients of AV,+, iu these expressions are the values which we must take for
K,+, and L,+, respectively in (81); and therefore, after reductions which show {m-\-n)
as a factor of the numerator of each fraclion in which it appears at first as a factor of
the denominator, we have

(<4-l)[w(« + 3)-»]a^

n{m[2(i + 2)«+l]-n(2« + 3)}

+ o

[(i + 2)(2tH-5)m-(2» + 3)n]r^
n(2^' + 3){wr[2(j + 2)*+l]-n{2i + 3)}

(83)

"n(2i + 3){m[2(i-f-2)-+l]-«(2« + 3)} j

These, substituted in (80), give expressions for a, /3, y which constitute a complete and
explicit solution of the problem.

It is easy to verify this result, by testing that (5G) (with — X='--^ '*-' , &c.) is satis-
fied for eveiy point of the solid, and that equations (.34) give F=0, G=0, 11 = 0 at the
bounding surface, r=a.

48. The case of /=1 is, as we shall immediately see, of high importance. For it the
preceding expressions, (83) and (80), become

_ — 10(4?H — w)a^ + (2lOT— 5«)?-^ >
'^~ 10n{l9m—5n)

e.,=

4mr^

10n(19m— 5?t)

ax ' dx

^ = ^'^^' + ^''^

,.,rf(W,r-^)

y=€-.

d\X.

dy

^ ^' dr"

(84)

49. As an example of the application of §i^ 45-48, let us suppose a spherical shell or
solid sphere to be equilibrated under the influence of masses collected in two fixed
external points*, and each attracting according to the inverse square of its distance.
Let the two masses M, M' be in the
axis OX; and, P being the point w^hosc m'
coordinates are x, y, z, let PM=D,
PM'=r)'. Let also OM=r, OM'=e'.
Then, if m, m' denote the two masses, for equilibrium we must have

* U our limits permitted, a highly LnterestLng example might be made of the case of a shell under the
influence of a single attracting point in the hollow space within it. The effect will clearly be to keep the
whole shell sensibly in equilibrium even if the attracting point is execntric; and under stress even if the
attracting point is m the centre.

MDCCCLXin. 4 N

606 PROFESSOE W. THOMSON ON DYNAMICAL PROBLEMS REGARDING-

The potential at P, due to the two masses, will be jj+jy. or, according to tlie notation
of § 4.2, with, besides, iv taken to denote the mass of unit volume of the elastic solid,

The known forms in the elementary theory of spherical harmonics give immediately
the development of tliis in a converging infinite series of solid harmonic terms. We
have only then to apply the solution of §§ 45, 46 to each term, to obtain a series
expressing the required solution.

50. We may work out this result explicitly for the case in which both masses are
very distant ; and for simplicity we shall suppose one of them infinitely more distant
than the other ; that is to say, we shall suppose it to exercise merely a constant balancing
force on the substance of the shell. We shall then have precisely the same bodily dis-
turbing force as that wliich the earth experiences from the moon alone, or from the sun
alone.

51. Referring to the diagram and notation of § 49, we have

if we neglect higher powers of "-, -, - than the square ; and

neglecting all higher ])owers of - , - , - . Hence, taking account of the relation ^=^
reipiired for equilibrium, we have, for the disturbance potential,

-W=5(^=-iy^-i.>,

an irrelevant constant being omitted from the expression which § 49 would give. This
being a harmonic of the second degree, we may use it for W,^.,, putting /=1 m the
formulae of § 47, and thus solve the problem of finding the deformation of a homo-
geneous spherical shell under the influence of a distant attracting mass and a uniform
balancing force. I hope, in a futiu'e communication to the Royal Society, to show the
application of this result to the case of the lunar and solar influence on a body such as
the earth is assumed to be by many geologists — that is to say, a solid crust, constituting
a spheroidal shell, of some thickness less than 100 miles, with its interior filled with
liquid. The untenability of this hypothesis is, however, sufficiently demonstrated by the
considerations adduced in a previous communication ("On the Rigidity of the Earth,"
read May 8, 1862), in which the following explicit solution of the problem for a homo-
geneous solid sphere only is used.

ELASTIC SPHEROIDAL SHELLS AND SPHEEOIDS OF INCOMPEESSIBLE LIQUID. 607

52. Using the expression of ^51 for Wj, we have
rfVV, „m

dx

rfWo , m d\\\ ni

^_^3m(^-|^-|^^^

'/y

(85)

These formulae being substituted for the diiierential coefficients which appear in (84),
we have algebraic expressions for the displacement of any point of the solid.

The condition of the body being symmetrical about the axis of a:, we may conveniently

assume

9/=zycos<p, 5^ysin<p,

so that we shall have (as we see by the preceding expressions)

/3^f/yC0S^,

y=j(;(, sin (p,

and [X, will denote the component displacement perpendicular to OX. If, further, we

assume .

a'=rcos^,

yrrrsin^,
the expressions (84) for the component displacements, with (85) used in them, give

9^e.+§ § (5 cos^ ^-3) [cos ^,

2 7^

(u,=tt;^| re,+ ?^(5cos=^-l)jsin^.

(86)

The values given in (84) for (t:.^ and (£'., are to be used for any internal point, at a distance
r from the centre, in these equations (86), and thus we have the simplest possible ex-
pression for tlie required displacement of any point of the solid.

53. If we resolve the displacement along and perpendicular to the radius, and con-
sider only the radial component, we see that the scries of concentric spherical surfaces
of the undisturbed globe become spheroids of revolution in the distorted body. The
elongation of the axial radius, obtained by putting ^=:0 and taking the value of u, is
double the shortening of the equatorial radius, obtained by putting 6=^^ and taking
the value of [ju ; which we might have infeiTcd from the fact, shown by the general
equations (80) above, that there can be no alteration of volume on the whole -Ruthin any
one of these surfaces. The expression for the excess of the axial above the equatorial
radius is q /a-

4n2

008 PEOFESSOE W. THOMSON ON DYNA]\IICAL PBOBLEMS EEGAEDING

Avhicli, if we substitute for (B., and (fa their values by (84), becomes

p m 2{Am — n)a^ — {3m — 7i)i-^
'^ 7* Wn{\i)m-5^i) '"•

2 m 5
If in this we take r=(7, and m =oo , it becomes -y- -^ :r^r- r\ which is the result used in
' 3 c'* 19« '

§ 34 of the paper on the Rigidity of the Earth, preceding the present in tlie Transactions.
54. In the case of «'=0, the result of § IS takes the extremely simple form

where

'( \"/ 2«"[«(2!-l) + m(i-l)]

P \ '[" / ^ 2a'[n(2i-l)+m{i-\)]

^ '[ '\fi ) ^ 2«'[«(2!-l)+m(z-l)]

p, _d(Kr') d(\\r) , rf((V

^'-' ikT'^ "If '^ dz

d&i_

.) ^

dx

/'

de,^

.1

d>J

/'

dP),_

,)

dz

f

(87)

This expresses the displacement at any point within a solid sphere of radius a, when its
surface is displaced hi a given manner (ISA;, SB^, SC',). And merely by making i
negative we have, in the same formula, the solution of the same problem for an mfinite
solid with a hollow spherical space every point of the surface of which is displaced to a
given distance in a given direction. These solutions are obtained directly, with great
ease, by the method of §i^ G-15, or are easily proved by direct verification, Avithout any
of the intricacy of analysis ine\itable when, as in the general investigations with which
we commenced, a shell bounded by two concentric spherical siirfaces is the subject.

[Added since the reading of the Taper.]

§§ 55 to 58. Oscillations of a Liquid Sphere.
55. Let V be the gravitation potential at any point r(.r, ij, z), and li the height of the
surface (or radial component of its displacement) from the mean spherical sui-face at a
point E in the radius through P. Then, if

/,=S,+S,+ (88)

be the expression f(jr li in terms of spherical surface harmonic functions of the position
of E, and if ^jj be the attraction on the unit of mass exercised by a particle equal in mass
to the unit bulk of the liquid, we have, by the known methods for finding the attractions
of bodies infinitely nearly spherical (Tiioiisox and Tait's ' Natural Philosophy,' chap, vi.),

V=4^4»-i ^*+ j(^)'5j|^} when r<u\
and i

ELASTIC SriIEll01i)Ai> JSIIELLS AND .srilEKOLDS OF 1NC0MPEESS1J3LE LIQUID. GOy

In these

i^lj.a=?,g, (90)

if q denote the force of gravity at tlie surface, due to the mean spliere, of radius a.

56. Now for infinitely small motions the ordinary kinetic equations give

dp (du d\\ d]> (dv d\\ dp (dw dY\ ,^_.

-z='^[dt—dx)' -dr'\jt-ii,)' -d,=\-dt-Tz)' ■ • (^1)

where f is the mass per unit of volume ; u, i\ w the component Aclocities through tlie fixed
point Pat time if; and^> the fluid pressure. Hence, possible non-ijeriodie motions being
omitted. i(iLi-^vdi/-\-W(lz is a complete differential ; and, denoting it by f, d we have

C-i.=f(|-v) (02)

57. To find the surface conditions, — first, since the pressure has a constant value, n,

at the free surface,

2)=:(joh-\-U when ?•={/, (93)

the variations of gravity depending on the variations of figure being of course neglected
in the infuiitely small term g^h. And, since ^ is the radial component of the velocity

at E, we luive, when r^a,

£ ^\L ^_i_l ^_ffj!_ (94)

r dx'r di/'r d: dl

Now since, the fluid being incompressible, V'(p = 0, (p may be expanded in a scries of
solid harmonic functions ; let

^=20,(il)\ (95)

where O, , O. , . . . are surface harmonics. Hence, as the successive terms are homoge-
neous functions of the coordinates (.c, y, z), of degrees 1, 2, &c.,

£ ^+Z ^+- ij=^L^i^(L\ , (96)

r dx ^ r ay ^ r dz r '\a J "^ ■'

and therefore, by (88) and (94),

f = >. (9^)

58. Eliminating p between (92) with r=a and (93), substituting for V by (89) and

lO), diflferentiat
order /, we have

(90), diflferentiating, substituting for -j^' by (97), and comparing harmonic terms of

^1

dt^

of which the integral is

.=Acos|Vf'(l-^)-E}. (99)

Here A is a surfece spherical harmonic function of the coordinates of E exi)ressing the
maximum value of O^, and E is the epoch (Thomson and Tait, § 53) of the simple

O

610

PEOFESSOE W. THOMSON ON THE GENEEAL THEOET

harmonic fuuctiou of the time which we find to represent O, . Using this solution in
(97) and (88), we see that if the surface be normally displaced according to a spherical
harmonic of order /, and left to itself, the resulting motion gives rise to a simple
harmonic variation of the normal displacement, having for period

V g 2z(2 — ])

that is, the period of a common pendulum of length ^^^^r^ — j^- It is worthy of remark

that the period of vibration thus calculated is the same for the same density of liquid,
whatever be the dimensions of the globe.

For the case of ^'=2, or an ellipsoidal deformation, the length of the isochronous
pendulum becomes fa, or one and a quarter times the earth's radius, for a homogeneous
liquid globe of the same mass and diameter as the earth ; and therefore for this case, or
for any homogeneous liquid globe of about 5 J times the density of water, the half-period
is 47"" 12% which is the result stated in the paper "On the Rigidity of the Earth" (§ 3),
preceding the present in the Transactions.

Appendix, §§ 59-71. — General Theory of the Eqidlihrixim of an Elastic Solid.
59. Let a soUd composed of matter fulfillmg no condition of isoti-opy in any pait, and
not homogeneous from part to part, be given of any shape, unstrained, and let every
point of its surface be altered in position to a given distance in a given dii-ection. It is
required to find the displacement of every point of its substance in equilibrium. Let
X, y, z be the coordinates of any particle, P, of the substance in its tindistvu:bed position,
and O'+a. jz+jS, 2+7 its coordmates when displaced in the manner specified ; that is to say,
let a, /3, 7 be the components of the required displacement. Then, if for bre-\ity we put

A=(l+l)+(l)+(r.
B = (|)+(|+1)+(^J)-

/(let _,

(100)

d^ d^ /dy yy
+Tzdi+{T^ + ^)d:v'

di/ ' dx \ f/y / dx dy

these six quantities A, B, C, a, b, <?, as is kno^vn*, thoroughly determine the strain expe-
rienced by the substance infinitely near the particle P (irrespectively of any rotation it
may experience), in tlie following manner : —

• Thomson and Tait's 'Natural Philosophy,' § 190 («) and § 181(5). ■ .;

OF TlIE EQFILIBRnJM OF AN ELASTIC SOLID. 611

60. Let ?, f], ^ be the undisturbed coordinates of a particle infinitely near P, rela-
tively to axes through P parallel to those of w, y, z respectively ; and let ?^ , >7, , ^^ be
the coordinates, relative still to axes through P, when the solitl is in its strained con-
dition. Then

i;+r;^-\-^-=K^-\-^rf+Cl^^+2artZ,+2hl^+1i%n; .... (101)
and therefore all particles which in the stramed state lie on a spherical sm-face

are, in the unstrained state, on the ellipsoidal sui'face.

This, as is well known *, completely defines the homogeneous strain of the matter in
the neighbouihood of P.

61. Hence the thermo-dynamic principles by which, in a paper on the Thermo-
elastic Properties of Matter in the first Number of the 'Quarterly Mathematical Journal'
(April 1855), Green's dynamical theory of elastic solids was demcmstrated as part of the
modem dynamical theory of heat, show that if io.dxd//dz denote the work required to
alter an infinitely small undisturbed volume, dxdydz, of the solid, into its disturbed con-
dition, when its temperature is kept constant, we must have

w=/(A, B, C, a, J, c), (102)

where/ denotes a positive function of the six elements, which vanishes when A— 1,
B— 1, C — 1, a,h,c each vanish. And if W denote the whole work required to jiroduce
the change actually experienced by the whole solid, we have

W=JJJwrfa%flfs, (103)

where the triple integral is extended through the space occupied by the undisturbed
solid.

62. The position assumed by every particle in the interior of the solid vn\l be such as
to make this a minimum, subject to the condition that every particle of the surface takes
the position given to it, this being the elementary condition of stable equilibrium.
Hence, by the method of variation,

SW=JJJSw(?aY7y(?r = 0 (104)

But, exhibiting only terms depending on la,, we have

V \^dwfd» ^\ dw da. dw doAdZx

(„dw da dw da dw /da , -■ \ ] dia
+ [-'ZB di^ + 5a ^ + ^ (^^+ V J ^

.jc^dwdadwdadw /da \"|(f8a
'^YdCd^ + ~d^d^ + 'db\d^+^)jlb

+ &C.
* Thomson and Tait's ' Natural Philosopliy,' §§ 155-165.

612 PEOFESSOE W. THOMSON OX THE GEXEEAL THEOEY

Hence, integrating by parts, and obsening that occ, Ip, ly vanish at the limiting siu-facc,
we have

^W = -Ji)V.T7^fL-[('^-ff+^)s« + &c.}, .... (105)

where lor brevity i . Q. R denote the lactors ot -^ ' -^ ' ■^- respectively, in the pre-
ceding expression. In order that SW may vanish, the factors of Sos, Bp, ly in the expres-
sion now found for it must each vanish ; and hence we have, as the equations of

equilibrium,

ylw /da. , -| N , dw da. div doTi
^MX'di^^j'^db'd^'^dcdi/]

d j ,^dw /da.
dx]

d {^dw da dw da. dw /da.

(106)

'^d=YdC dz'^da di/^db[d.v'T'^Jj—^'
(Sec. Sec,

of which the second and third, not exhibited, may be written down merely by attending
to the sjTumetry.

63. From the property of w that it is necessarily positive when there is any strain, it
follows that there must be some distribution of strain through the interior which shall
make ^\^wdxd//dz the least possible, subject to the prescribed surface condition, and there-
fore that the solution of equations (106), subject to this condition, is possible. If, what-
ever be the nature of the solid as to difference of elasticity in different chrectious, in any
part, and as to heterogeneousness from part to part, and whatever be the extent of the
change of form and dimensions to which it is subjected, there cannot be any internal
configuration of unstable equilibrium, or consequently any but one of stable equilibrium,
with the prescribed surface displacement and no disturbing force on the interior, then,
besides being always positive, ?r must be such a function of A, B, &c. that there can be
only one solution of the equations. This is ob\iously the case when the unstrained solid
is homogeneous.

64. It is easy to include, in a general investigation similar to the preceding, the
effects of any force on the interior substance, such as we have considered particularly
ior a sjiherical shell, of homogeneous isotropic matter, in §^ 35-40 above. It is also
easy to adapt the general investigation to sn])erticial data of force, instead of displace-
ment.

65. "Whatever be the general form of the function f for any part of the substance,
.since it is always positive it cannot change in signi when A — 1, B — 1, C — 1, a, b, c have
their signs changed ; and tlierefore for infinitely small values of these quantities it must
be a homogeneous quadratic function of them with constant coefficients. (And it may
be useful to observe that for all values of the variables A, B, &c., it must therefore be
expressible in the same form, with varying coefl[icients, each of whicli is always finite, for

OF THE KQI'lLTBRIUM OF A\ ELASTIC SOLID.

613

all values of the variables.) Ihus, tor infinitely small strains, we have Green's theory
of elastic solids, founded on a homogeneous quadratic function of the components of
strain, expressing the work required to produce it. Putting

A-1='_V. B-l=2/, C-l = 2<7, (107)

and denoting by i(e, e), \(f,f), ■ ■ ■ {c,f), . . ■ (e, o), . . . the cocfRcients, we have
w = h{{e, e)e' + (f,f)f+{!j, g)f +(«, a)a' +(/>, h)l>^ +(r, c)c^]
+ {('-/ y.f + (c^ 9)^9 +(^> o,)ea +{e, b)eb +(«?, c)ec
. +(/ 9)f9Hf^ o)faJr{f, b)p+(f, c)fc
+(y, a)(ja +(«7, h)gb -\-{g, c)gc
+ («, h)ah +(«, c)ac
+ (/>, c)hc J

The twenty-one coefficients in this expression constitute the twenty-one coefficients of
elasticity, which Green first showed to be proper and essential for a complete theory of
the dynamics of an elastic solid subjected to infinitely small strains.

66. \Vhen the strains are infinitely small, the products ~ -^ , - "' -^, &c. are each

'^ a A ax db dz

(108)

infinitely small, of the second order
(107), we reduce (106) to

d dw
dx de

We therefore omit them ; and then, attending to

d dw ^^ d dw

d= db

d dw d diu d dw .
iff ///^ ~r j^y df >^ ^-- ilji '

da: dc
d dw

dz da

d dw d dw

^0,

(109)

dx db ~^ dy da '^ dz dg

which are the equations of interior equilibrium. Attending to (108) we see that

. y.' . .. —■•• are linear functions of e. /", q. «, i, c the components of strain, ^^'ritill<r out
de da •'J ^' i ^

one of them as an example, we have

dw

-,={^, ey+(ej)f+{e, ;/)!J-\-(e, a)a-\-(c, b)b+(e, c)c.

(110)

And a. /3, 7 denoting, as before, the component displacements of any interior particle, P.
from its undisturbed position (xy, 2), we have, by (107) and (100),

^ da /• dB dy

'=dx' /=!' 3=1

dz^dy dx^dz

(in)

dy "^ (li-

lt h to be observed that the coefficients (e, e) {e,f), &c. will be in general functions of
(.r, y, 2), but will be each constants when the unstrained solid is homogeneous.

MDCCCLXin. 4 o

GU PEOFESSOR W. THOMSON ON THE GENERAL THEORY

67. It is now easy to prove directly, for the case of infinitely small strains, that the
solution of the equations of mterior eciuilibriura. whether for a heterogeneous or a homo-
geneous solid, subject to the prescribed surface condition, is unique. For let a, /3, y be
components of displacement fulfilling the equations, and let a', /3', y' denote any other
functions of (.r, //, z) ha\-iug the same surface values as u. /3, y, and let e', /',•••, iv'
denote functions depending on them in the same way as c.f, . . ., w depend on a, /3, y.
Thus, by Taylors theorem,

, (Iw, , . , dw. .,, ,,, , dw , , dw , , die , ., div , s , jr

where H denotes the same homogeneous quadratic function of (''— ^', &c. that w is off,
tS:r. If for e' — e, &c. we substitute their values by (111), this becomes

dwdla.' — a) , divd{a' — u) , dtvd{a.' — a) ^

'"-"-'=d^^h^+db^i^+Tc—d^+''''-+^-

Multiplying by dxdi/(h, integrating by parts, observing that a' — «. /3' — f3, y' — y vanish
at the bounding surface, and taking account of (109), we find simply

^\^(w'-w)da-d>/dz=^]^Ildxdi/dz (112)

But H is essentially positive. Therefore every other interior condition than that speci-
fied by (a, /3, y), provided only it has the same bounding surface, requires a greater
amount of work than w to produce it : and the excess is equal to the work that would
he required to produce, from a state of no displacement, such a displacement as super-
imposed on (a, /3, y) would produce the other. And inasmuch as («, /3, y) fulfil only the
conditions of satisfying (110) and having the given surface values, it follows that no
other than one solution can fulfil these conditions.

68. But (as has been remarked by Professor Stokes to the author) when the surface
data are of force, not of displacement, or when force acts from without, on the interior
substance of the body, the solution is not in general unique, and there may be con-
fiijuratious of unstable equilibrium, even with infinitely small displacement. For in-
stan((\ let part of the body be composed of a steel bar magnet ; and let a magnet be
held outside in the same line, and with a pole of the same name in its end nearest to
one end of the inner magnet. The equilibrium will be unstable, and there will be posi-
tions of stable ecpiilibrium with the inner bar slightly inclined to the lin(> of the outer
bar. unless the rigidity of the rest of the body exceed a certain limit.

69. Recurring to the general problem, in which the strains are not supposed infinitely
small, we see that, if the solid is isotropic in every part, the function of A, B, C, a, b, c
which expresses w must be m(>rely a function of tlie roots of the (Mpiation*

(A-r)(B-r)(c-r)-«iA-r)-i-^(B-r)-^'-'(c-r)+2«k'-^, ■ (n-')

wliich (that is the positive values of ^) are the ratios of elongation along the principal
* Thomson and Taii's ' Nuturul I'hilosophy,' § ISl (11).

OF THE KQI'ILIBKIUM OF AX ELASTIC SOLID. 615

axes of the strain-eUipsoid. It is unnecessary here to enter on the analytical expression
of this condition. For the case of A — 1, B— 1, C — 1, a, b, c, each infinitely small, it
ohviously requires that

{e,e) = {f,f)={g,g); {f,g)={g,e)={e,f); {a,a)={b,b) = {c,c);\

ind {e,a)=(f,b)={g,c)=Q;{b,c)=:{c,a)={a.b)=0; . (114)

(., b) = {e, c) = {f, .)=(/; a)={g, a)={g, b) = 0. ]

Thus the twenty-one coefficients are reduced to three —

{e, e), which we may denote by the single letter 91,

(f,9h ,, ,, ,, ,, B-

{a, a), „ „ „ „ «.

It is clear that this is necessary and sufficient for ensuring cubic iaofroj)// — that is to say,
perfect equality of elastic properties with reference to the three rectangular directions
OX, OY, OZ. But for spherical isotro])y, or complete isotropy with reference to all
directions through the substance, it is further necessaiy that

g[-$ = 2w, (llo)

as is easily proved analytically by turning two of the axes of coordinates in their own
plane through 45" ; or geometrically by examining the nature of the strain represented
bv any one of the elements «, b, c (a "simple shear") and comparing it with the resultant
of c. and /'= — e (which is also a simple shear). It is convenient now to put

9+iB=2?w; so that 9[ = 7?J+«, ^ — m—n; (IIG)

and thus the expression for the potential energy per unit of volume becomes

2w=m{e+f+gf+n{e'+f+f—2fg-2ge-2ef+a:^+b-^+(f). . . . (117)

Using this in (108), and substituting for e, f, g, a, b, c theu- values by (HI), we find
immediately, for the equations of internal equilibrium, equations the same as (1) of

70. To find the mutual force exerted across any surface within the solid, as expressed
by (26) of § 22, we have clearly, by considering the works done respectively by P, Q,
R. S, T, U (^ 21) on any infinitely small change of figui'e or dimensions in the solid,

p dw Q div -n die o ''w "y d'" tt <^>^ (US)

de df dg da db dc

Hence, for an isotropic solid, (117) gives the expression (25) of § 21, which we have
used above.

71. To interpret the coefficients m and n in connexion with elementarj' ideas as to the
elasticity of the solid; first let a=:i=c=0, and e=f=g=\l; in other words, let the
substance experience a uniform dilatation, in all directions, producing an expansion of
volume from 1 to 1+S. In this case (117) becomes

w = i(m — ^)5*;

fil6 OX THE GEXEEAL THEOET OF THE EQUILIBEIUM OF AN ELASTIC SOLID,
and we have

Hence (m — ln)i is the normal force per unit area of its surface required to keep any
portion of the solid expanded to the amount specified by S. Thus m — ^i measures the
elastic force called out by, or the elastic resistance against, change of volume : and
viewed as a coejfficienf of eJasticity, it may be called the elasticity of volume. What is

commonlv called the •• comprcssibilitv" is measured bv t--

And let next c—f=g=b = c=^\ which ijives

w = h»a^; and, by (118), S=nff.
This shows that the tangential force per unit area required to produce an infinitely
small shear*, amoimting to a, is va. Hence n measures the innate power with which
the body resists change of shape, and returns to its original shape when force has been
applied to change it; that is to say, it measures the rigidity of the substance.

[ Note added, December 1863].

Since this paper was communicated to the Royal Society, the author has found that
the solution of the most difficult of tlie problems dealt with in it, which is the determi-
nation of the effect produced on a spherical shell by a prescribed application of force to
its outer and inner surfaces, had previously been given by IvMIE in a paper published
in Liouville's Journal for 1854, under the title " Meraoire sur I'Equilibre I'elasticite
des enveloppes spheriques." In the same paper IjAJME shows how to take into account
the effect of internal force, but does not solve the problem thus presented except for
the simple cases of uniform gravity and of centrifugal force. The form in which the
analysis has been apjilied in the present paper is very different from that chosen by
TiAiiB (who uses throughout polar cooi'dinates) ; but the ]n-inciples are essentially the
same, being merely those of spherical harmonic analysis, applied to problems presenting
peculiar and novel difficulties.

* Thomson and Tait's ' Natural Philosophy," § 171.

[ *^1' ]

XXIX. First Anal 11 his of One Hundred and Seventy-seven Magnetic Storms, registered hy
the Magnetic Instrnmcnts in tJw Boijal Observatory, Greemoich, from 1841 to 1857.
By George Biddkll Airy, Astronomer Boyal.

Eeccived Xovcmbcr 28, — Eead Decemljcr 17, 1863.

1. I>' a paper which the Royal Society have printed iu their Philosophical Transactions
for 1862, I gave a series of curves exhibiting to the eye the diurnal inequalities of Ter-
restrial ^lagnctism in the three du-ections of "Westerly Force, Northerly Force, and Nadir
Force, as inferred from eye-observations and pliotographic registers at the Royal Obser-
vatory from 1841 to 1857. The paper, or the works to which it refers, exhibits also the
secular change and the annual inequality tlirough that ])criod, and tlie lunar inequali-
ties as inferred from the period 1848 to 1857. These results were obtained by excludmg
the observations of certain days (of whch a list was given) on which the motions of the
magnetometers were so violent that it was difficult to draw a mean curve through the
magnetic curve of the day. In the present paper I propose to give the principal results
dcduciblc from the days omitted in the former paper. But before entering into the
details of the numerical investigations, I think it desirable to explain the principles upon
which both parts of the investigations have been conducted.

2. The methods commonly employed in late years for measuring and classifying the
effects of magnetic disturbance have been, in my judgment, very valuable to the science,
especially in its earlier stages. But familiarity through many past years with magnetic
photograms has strongly impressed me with the feeling that a different method ought
now to be employed, taking account of relations of disturbances which perhaps could
not be kno\\ii at tlie introduction of the ancient method. I may thus describe the
general ideas which have guided me: — First, that there is no such thing as a day
really free from disturbance, and no reason in the nature of things for separating one or
more days from the general series. There is abundant reason for such sei)aration on the
ground of convenience of reduction ; but when the reduction has been effected by suit-
able process, the results of the separated days ought to be combined yni\\ those of the
unseparated days in the formation of general means (the numerical necessity for which
I propose to consider in the close of this paper), — the reduction of the separated days
serving also to throw great light \\\m\\ the nature of the acting forces on those days,
which forces in all probability are acting, though in different degrees, on other days.
Second, that, with our present knowledge of the character of magnetic disturbances, I
cannot think myself justified in separating any single magnetic indication, or any series
of indications defined only by their magnitude ; nor do I entertam the belief that any

MDCCCLXIII. 4 P

618 ME. AIKY— A2v'ALTSIS OF MAGNETIC STOEMS

special value could attach to the results which I might derive from observations from
which such indications have been removed. The study of the photograms shows clearly
that the successive indications at successive moments of the same day are a connected
series; there is no such thing as a sudden display of force in any element; the sharpest
salience which is exliibited on a generally smooth curve occupies at least an hour in
its development (I believe, never less, although the individual saliences in a continued
storm are of shorter duration), and diuing this time the force has been gradually
increasino- and gradually diminishing. Under these circumstances, I cannot think it
rio-ht that I should cut off a part of that salience, vnth. the behef of obtainmg results,
that can possess any philosophical value, from the part which is left. And I come to
the conclusion that each disturbed day must be considered in its entirety, and that our
attention ought to be given in the first instance to the devising of methods by which
the compHcated registers of each of those days, separately considered, can be rendered
manageable, and in the next place to the discussion of the laws of disturbance which
they may aid to reveal to us, and to the ascertaining of their effects on the general
means in which they ought to be included.

3. The discrimination of the classes of days which (on the one hand) are treated by
the general process in the " Results of Magnetical Observations, 1859," and of those which
(on the other hand) are to be treated by the methods of this Memoir, has been effected
entirely by the judgment of the Superintendent of Computations as to the certainty
and accui-acy with which he could di-aw a mean Hue through the distiu'bed ciu-ves.
I do however entu-ely recognize the propriety of defining the " disturbed days" by some
numerical limit, when it can be conveniently done : but, the day being defined, I then
thmk that the entire disturbed day or storm ought to be treated as a coherent whole ;
and that the laws of disturbance and the amalgamation with general means ought to be
deduced from it, as already mentioned, without reference to any numerical limit.

4, The records of disturbances from 1848 to 1857 are taken from the photograms;
and the value of these, I believe, is unimpeachable. The instruments appear to have
been in the highest state of efficiency ; I do not think that there is the least doubt on
the indications of any disturbed day. And (as the effect of adjustments made expressly
for that purpose) the traces of the most violent motions are in general perfectly pre-
served— an advantage which is possessed, I believe in a peculiar degree, by the photo-
grams of the Royal Observatory. Some sheets may be lost from defects in the paper,
defects in the chemical process, &c.; but none, I beUeve, from rapidity and violence of
motion of the magnets. The indications for every salient point of the curves have been
translated into numbers which are piinted in the " Results of Magnetical Observations "
for each year ; and those numbers are used as the basis of the followmg calculations.
For the years 1841-1847, in which observations were made by eye, it will be seen in
the printed Observations that no opportunity was lost, on the slightest appearance of
distui-bance, of following most carefully the indications of all the magnetometers : and
in fact, as regards both the number of days of such observations and the number of

OBSERVED AT TILE EOYiU, OBSEllVATOKY, GREEXAVICII. 619

observations on each day, the obseiTations taken are far more numerous than was neces-
sary. The judgment of the Superintendent has been exercised in making such a selec-
tion of days and such a limitation of records for each day as should make the adopted
register for the period 1841-1847 harmonize well with that for the period 1848-1857.

In the following investigations, whenever one instrument has exhibited such signs of
distui'bance that its indications were thought unfit for treatment m the former Keduc-
tions and are therefore included in this Analysis, tlie indications of the two other
instruments are also included in this Analysis.

5. In deciding on the method of making the disturbed curves more manageable, the
following was my train of ideas. As the photographic cuive usually consists of a series
of lines (very little curved) highly inclined to the time-abscissa and leading alternately
upwards and downwards, if each of these lines be bisected and the bisecting points bo
joined, the joining lines will form a polygon of much less violent character than the
original. If these joining lines be bisected and the bisecting points joined, we shall
have a polygon of still smoother character, mth angles sensibly corresponding to the
original times, excepting only the first and the last. If the double process be repeated,
the polygon will be still smoother, but wanting points corresponding to the two first and
two last observations. And thus we shall have a mean curve containing all the long
waves of the original curve, and freed from the irregularities of short period, whose
values, however, can be measured. Numerically, each step of the process is represented
by taking, for the numerical value of a new ordinate, the arithmetical mean of the
numerical values of adjacent ordinates, or, stUl more easily, by adding the adjacent
ordinates, adding the adjacent sums thus formed, and dividing by 4, and repeating
this operation. An instance mil make this process cleai'.

4p2

620

ME. AIEY— ANALYSIS OF MAGNETIC STORMS

Eeadings for Northerly Force (corrected for temperature) in the Magnetic Stonn

of 1854, March 6.

G-ottingen

Reading

Tmie.

li m

0 0

•1153

1 8

1153

1 32

1169

1 oO

1139

~ 7

1156

2 30

1150

2 44

1159

2 58

1153

3 30

1157

4 5

1157

4 12

1163

4 45

1160

5 23

1165

6 15

1155

6 39

1131

7 6

1168

7 15

1161

7 24

1163

7 32

1146

7 45

1153

8 25

1131

9 17

1156

9 45

1152

10 40

1164

11 23

1154

n 50

1187

12 S

1171

12 20

1172

12 39

1159

13 8

1166

13 17

1162

13 45

1158

20 0

1177

21 0

1168

22 3

1167

22 25

1161

22 46

1160

22 55

1148

23 4

1148

23 30

1117

23 59

1144

•2306
2322
2308
2295
2306
2309
2312
2310
2314
2320
2323
2325
2320
2286
2299
2329
2324
2309
2299
2284
2287
2308
2316
2318
2341
2358
2343
2331
2325
2328
2320
2335
2345
2335
2328
2321
2308
2296
2265
2261

4628

•1157

4630

1157

4603

1151

4601

1150

4615

1154

4621

1155

4622

1155

4624

1156

4634

1159

4643

1161

4648

1162

4645

1161

4606

1152

4585

1146

4628

1157

4653

1163

4633

1158

4608

1152

4583

1146

4571

1143

4595

1149

4624

1156

4634

1159

4659

1165

4699

1175

4/01

1175

4674

1169

4656

1164

4653

1163

4648

1162

4655

1164

4G80

1170

4680

1170

4663

1166

4649

1162

4629

1157

4604

1151

4561

1140

4526

1132

•2314
2308
2301
2304
2309
2310
2311
2315
2320
2323
2323
2313
2298
2303
2320
2321
2310
2298
2289
2292
2305
2315
2324
2340
2350
2344
2333
2327
2325
232G
2334
2340
2336
2328
2319
2308
2291
2272

•4622
4609
4605
4613
4619
4621
4626
4635
4643
4646
4636
4611
4601
4623
4641
4631
4608
4587
4581
4597
4620
4639
4664
4690
4694
4677
4660
4652
4651
4660
4674
4676
4664
4647
4627
4699
4563

^(h or
Adopted.

•1155
1152
1151
1153
1155
1155
1157
1159
1161
1161
1159
1153
1150
1156
1160
1158
1152
1147
1145
1149
1155
1160
1166
1172
1174
1169
1165
1163
1163
1165
1169
1169
1166
1162
1157
1150
1141

The Adopted Numbers are those to be compared with the Original Reading, in order
to ascertain what portion of the Original Heading is to be ascribed to Irregularities : and
the Adopted Numbers are also to be compared with the Monthly Means deduced from
the days of easy reduction, in order to ascertain what portion is to be considered as
Wave-Disturbance. Thus we finally obtain the following separation of numbers, whose
aggregate represents the Original Eeading : —

OBSER'V'EI) AT TIIK KOVAL OliSF.KVATORY, nUEEXAVICH. G21

Component pai'ts of Northerly Force in the Magnetic Storm of 1854, March C.

Gottingen Time.

Monthly Mean.

Wavc-Di

'turbance.

Irrof»<ilnritics.

h m

1 32

•1158

-•0003

+ •

0014

1 50

1158

- 6

-•0013

2 7

1158

- 7

+

5

2 30

1160

7

— 3

2 44

llGO

— 5

+

4

2 58

llCl

6

- 2

3 30

1162

— 5

0

4 5

1162

- 3

- 2

4 12

1162

— 1

+

2

4 45

1162

- 1

— 1

5 23

1162

— 3

+

6

C 15

1162

- 9

+

2

6 39

1163

- 13

- 19

7 6

1163

— 7

+

12

7 15

1163

— 3

+

1

7 24

1163

- 5

+

5

7 32

1163

- 11

- G

7 45

1163

— 16

+

6

8 25

1163

- 18

- 14

9 17

1163

- 14

+

7

9 45

1164

- 9

— 3

10 40

1164

- 4

+

4

11 23

1164

+ •0002

— 12

11 50

11 Go

+ 7

+

15

12 8

1165

+ 9

- 3

12 20

11 65

+ 4

+

3

12 39

1164

+ 1

- 6

13 8

1164

- 1

+

3

13 17

1164

— 1

- 1

13 45

1164

+ 1

— 7

20 0

1168

+ 1

+

8

21 0

ll6l

+ 8

- 1

22 3

1156

+ 10

+

1

22 25

1156

+ 6

- 1

22 46

1155

+ 2

+

3

22 55

1155

— 5

— 2

23 4

1155

- 14

+

7

The disturbance of Horizontal Force is thus separated into two well-distinguished
parts. One part consists of five long waves, alternately — and +. The other part
consists of ii-regularities of short period, which do not show the least symptom of dis-
appearing at the disappearance of the waves, and appear to have nothing in common
with them except the connexion of both with the same general INIagnetic Storm.

6. For fully understanding the import of these numbers, it will perhaps be necessary
to study the succession of numbers in each indi"vidual instance. In this First Analysis,
I have proceeded, as the first step, to take the means that appear to be most valuable.
As regards the AVaves, I have taken separately the mean of the wave-disturbances through
each wave. But as this quantity gives little information unless taken in conjunction
with the time through which it acts, I have multiplied it by the length of tlie wave in
hours ; and this product 1 have distinguished by the technical term Fluctuation. The

622

ME. AIEY — AJN'ALTSIS OF MAGNETIC STOEMS

following is now an Epitome of the Magnetic Storm which we have had under con-
sideration.

Epitome of Disturbances of Northerly Force in the Magnetic Storm of 1854, March G.

Times of beginning
and end of wave.

Length
of ware
in hours.

Mean
Wave-dis-
turhance.

Fluctuation.

Aggregate
Fluctu-
ation.

Sum of
Hours.

Mean

Distiu-b-

anco.

Number of
Irregu-
larities.

Mean Pe-
riod of Irre-
gularity.

Mean value
of Irregu-
larity.

Ii m h ra
0 0 11 1

11 1 12 54

12 54 13 31

13 31 22 51
22 51 23 59

h
11-02
1-88
0-62
9-33
1-13

-•0007
+ 5

- 1
+ 5

- 10

-•0077 1
+ 9

- 1

+ 47

— 11

-•0033

li
23-98

-•0001

22
5

2
6

2

h

0-50

0-38

0-31

1-56

0-57

+ -0006
8
2
4
4

The disturbances of Westerly Force and Nadir Force are treated in the same way —
the values of distui-bance, &c. being converted, at convenient stages, into values expressed
in terms of whole Northerly Force.

The numbers contained in these Epitomes serve as bases for the investigations which
follow. The Epitomes themselves, though greatly reduced from the voluminous calcu-
lations on which they are founded, are far too extensive to be included in this Memoir :
they will probably be printed in the Greenwich Observations.

7. Treating the Waves as the first subject, I take in the first instance the algebraical
aggregate of the Fluctuations for each separate Magnetic Storm. In Table I., the first
or longest of the three Tables which follow, every recorded storm is included ; and in the
second, or Table II., these are all collected to form annual aggregates. But as the days
of record do not strictly coincide for the three instruments, partly from accidents in the
chemical preparation of the photographic paper, &c., but more particularly from the
experimental state of the Vertical-Force Instrument dm-ing a part of the year 1848, I
have thought it desirable to form Table III. from the observations which are strictly
comparable. In regard to the last columns of each department of Table I., and the last
lines of Tables II. and III., it will be remarked that the " Fluctuation "' is a product of
number of hours by Magnetic Disturbance, and therefore, for the Mean Disturbance,
the Aggregate of Fluctuations must be divided by the Sum of Hours.

OBSERVED AT THE ROYAL OBSERVATORY, GREENWICH.

623

Table I. — Algebraic Sums of Magnetic Fluctuations (in terms of Horizontal Force)
on Days of Great Magnetic Disturbance.

Year, Month,

Westerly Force.

Northerly Force.

Nadir Force.

Algebraic

Algebraic

Algebraic

Algebraic

Algebraic

Algebraic

and Bay.

of

A"Kregato
of Fluctua-

Mean
of Disturb-

of

Aggregate
of Fluctua-

Mean
of Disturb-

of

Aggregate
of Fluctua-

Mean
of Disturb-

tions.

ance.

tions.

ance.

tions.

ance.

1841.

Sept. 24

13-9

— 0-0022

- 2

12-0

-0-0456

-38

14-0

-0-0392

- 28

25

22-0

- -0200

- 9

12-9

- -0054

- 4

11-3

+ -2580

+ 229

27

8-2

- -0270

-33

8-2

- -0097

-12

8-2

+ -0670

+ 82

Oct. 25

22-0

- -0417

-19

22-0

— -0484

-22

20-2

+ -0226

+ 11

Nov. 18

17-9

- -0735

-41

17-9

- -0125

- 7

18-0

- -0323

- 18

19

22-8

+ -0016

+ 1

24-0

- -0276

— 12

23-7

- -0379

- 16

Dec. 3

12-7

+ -0088

+ 7

12-7

- -0205

-16

10-9

+ -0424

+ 39

14 '

10-0

- -0296

-30

10-0

— -0130

-13

10-0

+ -0621

+ 62

1842. i

Jan. 1

6-7

-0-0240

-36

6-7

+ 0-0387

4-58

6-1

+ 0-0061

+ 10

Feb. 24

8-0

- -0090

-11

8-0

— -0400

-50

8-0

+ -0014

+ 2

April 14 1

7-6

+ -0214

-1-28

7-4

- -0423

-57

8-0

- -0784

- 98

15 '

23-1

+ -0087

+ 4

24-0

- -1416

-59

22-2

- -0061

- 3

July 1 1

7-7

+ -0008

+ 1

7-7

- -0178

-23

8-0

+ -0135

+ 17

2 i

13-6

— -0523

-39

13-4

— -0138

-10

13-2

+ -0608

+ 46

3 1

9-7

- -0003

0

10-0

- -0650

-65

10-0

- -0289

- 29

Nov. 10 '

14-2

- -0340

-24

14-3

- -0710

-50

14-2

+ -0185

+ 13

21

12-0

— -0054

- 5

12-0

— -0132

— 11

12-0

— -0312

- 26

Dec. 9

10-0

- -0220

-22

10-0

- -0187

-19

10-0

+ -0311

+ 31

1843.

Jan. 2 1

10-0

+ 0-0180

+ 18

10-0

-0-0180

-18

10-0

-0-0261

- 26

Feb. 6 1

16 1

24 1

6-0

4-0

11-6

+ -0002

- -0044

- -0129

0
— 11
-11

"4-0
11-6

-12
-16

i"i-6

+ ""3

- -0048

— -0189

+ -bosi

May 6 i

4-4

- -0216

-49

4-1

— -0226

-55

4-2

- -0064

- 16

July 24 1

13-7

+ -0145

+ 11

13-7

- -0227

-17

14-0

+ -0140

+ 10

25

6-0

+ -0210

+ 35

6-0

+ -0002

0

5-6

+ -0329

+ 59

1844.

Mar. 29

15-7

-0-0140

- 9

15-7

-0-0305

-19

16-0

-0-0448

- 28

30 1

12-0

- -0097

- 8

12-0

- -0126

-11

11-6

+ -0017

+ 2

Oct. 1

6-0

- -0156

-26

6-0

- -0198

-33

6-0

+ -0018

+ 3

20

Nov. 16

io-0

+ i'i

8-0
10-0

- -0224

- -0280

-28
-28

8-0

9-7

- -0904
+ -0398

-113

+ 41

+ -0112

22

8-0

+ -0248

+ 31

8-0

- -0196

-25

8-0

- -0062

- 7

1845.

Jan. 9

10-0

-0-0290

-29

10-0

-0-0440

-44

10-0

+ 0-080

+ 8

Feb. 24

15-7

- -0198

-13

16-2

- -0177

— 11

16-2

- -0211

- 13

Mar. 26

14-0

- -0210

-15

14-0

- -0090

- 6

14-0

- -0070

- 5

Aug. 29

6-2

- -0037

- 6

6-1

- -0024

- 4

6-2

- -0062

- 10

Dec 3

14-1

- -0022

- 2

14-2

- -0667

-47

14-2

+ -0439

+ 31

1846.

May 12

10-0

-0-0009

- 1

10-0

-0-0044

- 4

10-0

-0-0040

- 4

July 11

Aug. 6

10-0

- -0092

- 9

3-4

- -0044

- 13

11-9

+ -0099

+ 8

11-9

- -0037

- 3

12-0

- -0015

- 1

7

22-0

+ -0286

+ 13

22-0

— -0013

- 1

21-9

+ -0061

+ 2

24

14-0

- -0107

- 8

12-0

- -0036

- 3

16-0

- -0160

- 10

25

16-0

- -0096

- 6

16-0

+ -0050

+ 3

14-2

- -0071

- 5

62i

ME. AIEY — AXALTSLS OF MAGXETIC STOEMS

Table I. (coiitinued).

Year, Month,
aud Day.

1846 {confi).

Aug. 28

Sept. 4

Westerly Force.

Oct.

Nov.
Dec.

10
11
21

26
23

Number

of
Hours.

1847.
Feb. 24 .
Mar. 1 .

19 .

April 3 .

7 .

21 .
May 7 .
June 24 .
July 9 .
Sept. 24 .

26 .

Oct. 22

23*(lst)
23 (2nd)
24 '.[.

Nov. 22
Dec. 17

18
19
20

1848.
Jan. 16
28
20
21
22
23
24
17
20

Feb.

Mar.

April
Mav
July
Oct.

15-9
13-0
13-9
23-8
19-9
14-0
6-0
17-7
12-0
16-2
10-0

10-0
8-0

20-0
8-0

16-0
6-0
8-0
4-0

18-0

9-8

10-0

5-8

12-0

2-0

23-3

10-0

14-0

22-0

12-0

10-0

18-0

14-2
14-0
22*5
16-9

4-0
18-0
20-8

3-3
14-2
11-4

9-1
16-4
11-5

Algebraic
Aggregate
of Fluctua-
tions.

— 0-0058

•0091
•0056
•0005
•0030
•0286
•0158
•0156
•0073
•0059
•0069
•0160

-0

0167

+

0082

—

0121

—

0240

—

0046

+

0004

+

0344

+

0109

J-

0082

—

0159

+

0008

+

0033

+

0091

—

0025

+

0137

—

0093

+

0120

+

0157

—

0120

+

0175

—

0132

— 0'

0047
0210
0192
0047
0043
0113
0335
■0077
•0126
■0045
■0016
•0054
•0024

Algebraic

Mean
of Disturb-

+ 6

+ 4

0

— 1

— 14

— 11
-26

— 4
+ 5

— 4
-16

-17
+ 10

- 6
-30

— 3
+ 1
+ 43
+ 27

+ 5
-16
+ 1
+ 6
+ 8
-13
+ 6

- 9
+ 9
+ 7

- 10
+ 17

- 7

— 3
-15

— 9
+ 3

— 10

— 6
+ 16
+ 23

— 9
+ 4

— 2

— 3
+ 2

JCortherlv Force.

Number

of
Hours.

8-7
15-8
12^9
13-8
23-8
19-8
14-0

6-0
17-7
11-8
14-6
10-0

10-0
8-0

20-0
8-0

16-0
5^5
8-0

4-0
18-0

9-8
10-0

5-8
12-0

1-9
23-3
10-0
14-0
22-0
12-0
10-0
18-0

10-3

19-1
9-1

22-8
4-0
8-6

22"8
5-2

11-5
4-1
8-4

19-4

10-6

Algebraic
Aggregate
of Fluctua-
tions.

— 0-0075

— -0116

— -0062
+ -0029

— -0148

— -0183

— -0305

— -0102

— -0509

— -0227

— -0279
+ -0170

-0-0110

- -0047

- -0846

- -0061

- -0493

- -0120

- -0032

- -0352

- -0332

- -0401

- -0300

- -0403

- -0132

- -0030

- -2038

- -0150

- -0304

- -0208

- -0193

- -0910

- -0581

-0-0340

+ -0217

— -0335

— -0742

— -0125

— -0024

— -0503

— -0067

— -0309

— -0074
+ -0105

— -0415

— -0271

Algebraic

Mean
"of Disturb-
ance,

(Number

of
Hours.

- 9

- 7

- 5
+ 2

- 6

- 9

OO

-17
-29
-19
-19

+ 17

-11

— 6

— 42

— 8

— 31

— 4

-88
+ 18
-41
-30
-70

— 11
-16
-90

— 15

oo

— 12
-16
-91
—32

-33
+ 11
-37
-33

— 31

— 3
22

— 13

— 27

— 18
+ 12

— 21
-26

8-8
16-0
12-3
14-0
23-7
20-0
13-9

6-0
18-0
11-8
16-2
10-0

9-9

8-0
18-2

8-0
16-0

G-0
10^0

4-0
17-0
10-0

9-7

6-0
11-6

2-0
23-7

9-5
15-2
14-0

Algebraic
Ag^egate
of Fluctua-
tions.

-0

+ •

+ •

+ '

+ ■

+ ■

+ ■

0114
0208
0274
0140
0292
0100
0225
0060
0378
0905
0002
0074

— 0-0030
+ -0616

— -0783
-f -0264

— -0171
+ -0156

— -0100

— -0464
+ -0435

— -0260
+ -0603

— -0108
-f -0988
+ -0016
-1- -0538
+ -0654

— -0421
+ -1260

7-1 ; -0-0970

Algebraic
Mean

of Disturb-
ance.

- 13
+ 13

4- 22
+ 10
+ 12

- 5
+ 16
+ 10

- 21
+ 77

0
+ 7

— 3

+ 78

— 43
-f 33

— 11
+ 26

— 10

-116
+ 26

— 26
+ 62

— 18
+ 85
+ 8
+ 23
+ 69

— 28
+ 90

-136

* On October 2.3, 1847, all tlic obscirations were interrupted during 10 hours.

OBSERVED AT TILE ROYAL OH^El{\AT01{Y, GKEENWICK.

025

Table I. (continued).

Year, Month,
and Day.

MDCCCLXIII.

4Q

G26

:me. aiet — a:\'altsis of :magxetic storms

T.\BLE I. (continued).

Westerly Force.

Year. Month,
and DaT.

1S53.

Jan. 10

Mar. 7

8

11

Mav 2

3

24

June 22

July 12

Aug. 21

Sept 1

2

Oct. 1

Nov. 9
Dec. 6

21

1854,
Jan. 8
20^*

resumed 1 20

Feb. 16

24

:\Iar.

April
May

1855.
Mar. 12
April 4
July 19
Oct. 18

t

1857,
Feb. 2G
Mar.
May

13

Sept.
Nov.

Dec.

!'jf.m,berj fgebraie
fof ' Aggregate
Hours, i ofFliK^tua-
nous.

XortheriT Force.

Algebraic i ifu^iber

Jlean 1 ^f
of Disturb-: I 2o^y.j

23-8

23-0

22-0

23-7

23-3

1 23-8

23-8

23-5

23-6

23-5

23-5

23-5

23-4

23-8

; 23-9

2.3-8

23-9

23-9

23-9

23-7

" 23-5

i 23-8

23-9

23-6

23-6

24-0

23-6

23-7

22-6

23-2

24-0

23-8

24-0

23-3

I 21-3

22-8

24-0

, 22-8

— 0-0063

— -0171

— -0073

-0101
•0074
•0092
-0157
•0019

-0220
•0037

- -0119

+ ^0037

+ •0134

+ ^0044

+ 0^0029

— ^0043

— •0209

— •CHS
+ ^0049

— ^0084

— ^0099

— -0034

— •OIH

— -0076
+ •OlOS
+ -0004

-0-0117
— ^0028

•0052

+ 0-0014
+ -0052
+ -0207
+ -0056

— -0124
+ -0163

— -0073

— -0049

— -0021

— -0086

— /

- 3

— 5

+ 2

+ 6

+ 2

+ 1

— 5

— 1

+ 1
+ 2

22-6
23-9
23-9
23-9
2.3-5
23-7
23-6
23-7
24-0

22^8
23-9
24-0
24-0
24-0
23-7
24-0
23-0

23-4
23-5

24-0
23-3

23-9
24-0
24-0
24-0
24-0
23-9
24-0
24-0

23-4
23-6

22-8
24-0

22-6

24-0
22^1
24^0
23^3
23-3
22^5
24^0
22-6

Algebraic
Aggregate
of Fluctua-
tions.

— 0-0200

— -0224

- -0072

— -0253

- -0657

— -0552

+ -orm

- -0029

- -0090

— -0032

- -0616

— -0312

- ^0336

— -0092

— ^0474

- -1079

- ^0071

+ 0^0246

- •ooge

•0337
•0020
•0119
•0033
•0261
•0408
-1271
-0123
-0196
-0176

-0-0506

- -0108

- -0263

- -0477

-0-0086

— -0418
+ -0270

— •0259

— -0006

— ^0036

— ^0277

— -0304

— ^0881

Algebraic
Mean

of Disturb-
ance.

Nadir Force.

— 9

— 9

— 3

— 11
-28

— 23
+ 13

— 1

— 4

— 1
-26

— 13
-14

— 4

— 20
-45

— 3

+ 11
- 4

— 14
+ 1
+ 5

— 1

— 11
-17

— 53

— 5

— 8
+ 7

— 5

— 12

— 20

— 4

-17
+ 12

— 11

0

— 2

— 12
-13
-39

Number

of
Hours.

Algebraic Algebraic

Aggregate Mean

of Fluctua- of Disturb-

tions. ance.

23-8
22-5
23-2
22'9
23-0
23-7
23-2
23-4
23-7
23-5
23-7

24-0
23-5
23^3
23^3

23-7
7-0
14^1
23-8
23-7
23-7
23-9
23-3
23-7
22*7
22-6
23-7
23-9

23^5
20-3
23-5

23-8

23-2

22-6
24-0
24-0

24-0
24-0

+ 0-3353

+ -5298

+ -3529

+ -3595

+ -3424

+ ^2269

— ^1487
+ -0097

— ^0617

— •OoSO
+ -0331

+ •sogs

— ^0578
+ -0183

- -1790

-0^1089

— -0104

— •OSSO

— ^1165

— •UU
+ -0812

— ^1049

— -0030
+ -0498
+ -1451

— '0851
+ ^0207

— •0653

— 0-2111

— •OOIS

— -0101
+ ^1049

—0-1368

- -3191

- -0147

- -4177

— -2230
+ -0427

+ 141
+ 236
+ 152
+ 157
+ 149
+ 96

- 64
4

26

- 23
+ 14

+

+ 129

— 25
+ 8

- 77

- 46

- 15

- 39

- 49

- 60
+ 34

- 44

- 1
+ 21
+ 64

- 38

— 90

— 1

— 4
+ 44

- 59

— 141

- 6
-174

- 93

+ 18

* On Jan. 20, 18.54, the obscn'ations of the Vertical-Force Instrument were interrupted duiing 'A hours.
t In 18.50 there were no daj-s of Great Magnetic Disturbance throughout the year.

The last figure in the " Algebraic Mean of Disturbance " is in the fourth decimal place of Horizontal Force.

OBSEEVED AT THE KOYAL OBSEKVATOKY, GKKENWICII.

G27

Table II. — Algebraic Sums of Magnetic Fluctuations (in terms of Horizontal Force) for
each Year from 1811 to 1857, including all days of llecord of Great Magnetical
Disturbance.

Westerly Forca

Northerly For

■.,

Nadir Force.

Algebraic

Algebraic

Algebraic

Algebraic |

1 Algebraic

Algebraic

roar.

Number

Aggregate
of Fluctua-

Mean of

Number

Aggregate

Mean of 1

Number Aggregate

Mean of

of Hours.

Distiu-b-

of Hours. ofFIucfua-

Disturb- '

of Hours, lof Fluctua-

Disturb-

tions.

anee.

tions.

anco.

tions.

ance.

1841

129-47

--1836

-14

119-63

- ^1827

-15 1

116-19

+ ^3427

+ 29

1842 .

1 112-57

-•1161

-10

113-34

- ^3847

-34 I

111-74

- -0132

- 1

1843

55-73

+ -0148

+ 3 1

49-39

- -0868

-18

45-40

+ -017.-.

+ 4

1844

51-74

--0033

- 1 '

59-70

- •1329

-22

59-29 - -0971

- 16

1845

60-00

-•0757

-13

60-41

- ^1398

-23

60-52

+ -01 7G

+ 3

1846

244-86

--0606

- 2

250-89

- •1979

- 8

247-99

+ -1305

+ 5

1847

246-75

+ -0239

+ 1

246-29

- ^7489

-30

198-75

+ -3193

+ 16

1848

264-18

-•0666

- 3

223-83

- -5274

-24

40-65

- -0905

- 22

1849

46-00

+ -OI62

+ 4

45-25

- -0418

- 9

22-92

- -3484 '

-152

1850

141-79

+ •0493

+ 3

163-80

- -1890

-12

138-34

- -34171

- 25

1851

, 294-04

-•0830

- 3

305-70

- -4764

-16

299-17

- -6190

— 21

1852

364-65

-•0938

- 3

395-76

- -2739

- 7

353^07

-1-2159'

- 34

1853

1 327-14

-•0213

— 1

402-06

- -4789

-12

350^67

+ 2-0150

+ 57

1854

285-10

-•O6I9

- 2

285-82

- -2164

- 8

279^75

- -3937

- 14

1855

1 71-37

—0197

- 3

93-75

- -1354

-14

91-03

- -1181

- 13

1856

0-00

-0000

0-00

•0000

0^00

-0000

1857

231-53

+ •0139

+ 1

208-37

- ^1997

-10

141^73 ;-1^0686

-75

Sum

^ 2926-91

-•6675

3023-99

—4-4126

2557-21 - 1^4636 1

Mean Dis-1
turbance J

-•00023

- -00146

- -00057

Table III. — Algebraic Sums of Magnetic Fluctuations (in terms of Horizontal Force)
for each Year from 1841 to 1857, including only those days of Great Magnetic
Disturbance in which Records were made by the three Instruments.

Year.

"Westerly Force.

i Northerly Force.

Nadir Forca

Algebraic

Algebraic

Algebraic

Algebraic

Algebraic

Algebraic

Number

Aggregate

Mean of

Number

Aggregate

Mean of

Number

Aggregate

Mean of

of Hours.

of Fluc-

Disturb-

of Hours.

of Fluc-

Disturb-

of Hours.

of Fluc-

Disturb-

tuations.

ance.

tuations.

ance.

tuations.

ance.

1841

129-47

— 1836

-14

119-63

- -1827

-15

116^19

+ ^3427

+ 29

1842

112-57

— •1161

-10

113-34

— -3847

-34

111^74

- -0132

- 1

1843

45-72

+ -0190

+ 4

1 45-39

- -0820

-18

45-40

+ -0175

+ 4

1844

51-74

— -0033

- 1

1 51-70

- -1105

-22

51-29

- -0067

- 1

1845

60-00

—0757

-13

60-41

- -1398

-23

60-52

+ -0176

+ 3

1846

244-86

--0606

- 2

240-89

- -1887

- 8

1 244-61

+ -1349

+ 5

1847

202-69

+ -0207

+ 1

202-19

- -5453

-27

1 194-75

+ -3657

+ 19

1848

55-17

— -0234

- 4

45-74

- -2621

-58

40-65

— -0905

22

1849

22-92

--01 29

- 6

22-84

- -0160

- 7

1 22-92

— -3484

-152

1850

141-79

+ -0493

+ 3

! 139-88

- -1869

— 13

1 138-34

- -3417

- 25

1851

294-04

--0830

— 3

305-70

- -4764

-16

299-17 1- -6190

- 21

1852

341-34

--0812

2

' 372-27

- -2612

- 7

1 353-07

-1-2159

- 34

1853

304^41 -•OlSO

- 1

1 308-40

— -3688

— 12

303-72

+ 1-7238

+ 57

1854

285-10 --0619

- 2

285-82

- -2164

- 8

279-75

- -3937

- 14

1855

71-37 —0197

- 3

1 70-97

- -1091

-14

67-58

— -1080

- 16

1856

0-00 -0000

0

1 0-00

-0000

0

0-00

-0000

0

1857

141^04 +-0046

0

139-26

- -1678

-12

141-73

-1-0686

- 75

Sum

2504-23

--6428

2524-42

-3-6984

2471-43

-1-6035

Mean Dis-l
turbance /

--00026

— -00147

- •oooes

4q2

628

MK. AIKY — ANALYSIS OF jNIAGXETIC STOEMS

8. The most remarkable of the results of these Tables is. not only that upon the whole
the Algebraic Aggregate of Fluctuations for the Northerly Force is negative (which has
been previously recognized), but that it is negative in every separate year. It vvill be
seen in Table I. that on some separate days the Aggregate of Fluctuations is positive,
but the number of days is only 22, in opposition to 155 with negative Aggregates.

The Aggregate for the "Westerly Force is also negative: and though the different
years do not consent in the same way as for the Northerly Force, yet their discordance
is not so great as to justify us in setting aside this indication, although there may be
gi'eater doubt upon the accuracy of its value. This Aggregate (taken in <;omparison
with that for the Nortlierly Force) appears to show that, on the whole, the direction of
Disturbing Force is 10' to the East of South.

The Aggregate for the Nadir Force appears greater, but it is very uncertain ; it might
be nearly destroyed by the omission of a single year.

9. These characteristics of the directions of the disturbing forces will appear also in
the following enumeration of the instances in which the first and last waves of each
Magnetic Storm are affected in different ways. In comparing the numbers it must be
borne in mind that, when there is only one wave, that wave is considered, in different
places, both as the first and the last.

Westerly
, Force.'

Northerly
Force.

Nadir
Force.

177
277

58
114

15
157

118
120

81
64

63

82

of negative fluctuations i 302

Number of instances in which the first wave is -|- 106

in which the last wave is + 100

ending with
ending with
ending with

35

68
21
40

Number of Storms beginning with Westerly Force + and Northerly Force +
beginning with Westerly Force + and Northerly Force —
beginning with Westerly Force — and Northerly Force +
beginning with Westerly Force — and Northerly Force —
ending with Westerly Forced and Northerly Force -|-
ending with Westerly Force + and Northerly Force —
ending with Westerly Force — and Northerly Force +
ending with Westerly Force — and Northerly Force —
Number of Storms beginning with Northerly Force+and Nadir Force +
beginning with Northerly Force+and Nadir Force —
beginning with Northerly Force — and Nadir Force +
beginning vntli Northerly Force — and Nadir Force —
ending with Northerly Force+and Nadir Force +
Northerly Force+and Nadir- Force —
Northerly Force— and Nadir Force +
Northerly Force — and Nadir Force —

10. The following Tables, Tables IV., Y., and "\'l., exhibit the Aggregates of Fluctua-
tions without r(>gard of sign. They are required in order to give information on the
Mean Value of Disturbance by Wave in each of the three directions.

90

8

58

26

21

55

42

6

7

57

74

OBSERVED AT THE ROYAL OBSEUVATORY, GREENWICH.

629

Table IV. — Absolute Sums, uithout regard of sign, of Magnetic Fluctuations (in terms
of Horizontal Force) on Days of Great Magnetic Disturbance.

Te.ir, Month,

Westerly Force.

Northerly Force. 1

Nadir Force.

Absolute

Absolute

Absolute

Absolute

Absolute

Absolute

and Day.

Number

Aggregate

Moan

Number

Apgrcgate

Mean '

Number

Aggregate

Mean

of Waves.

ofKliKliia-

of Disturb-

of Waves.

af Fluctua-

of Disturb-:

of Waves.

a( Fluctua-

3f Disturb-

;

tions.

ance.

tions.

ance.

tion.".

ance.

1841.

Sept. 24

25

2
6

0-0292
•0608

21

28

1

0-0456
•0846

38

66 1

1

1

0^0392
-2580

28
228

27

1

•0270

33

3

-0101

12 1

1

•0670

82

Oct 25

4

•0427

19

1

-0484

22

2

•0434

21

Nov. 18

2

•0961

54

1

•0125

7

3

•0517

29

19

5

•0214

9

3

•0292

12

Dec. 3

3

•0152

12

3

•0207

16

1

-0424
•0621

39
62

14

2

•0312

31 ;

1 1

-0130

13

1

1842.

Jan. 1

1

0^0240

36

1

0-0387

58

1

0^006l

10

Feb. 24

3

•0148

19

1

-0400

50

2

•0044

5

April 14 i

15 1

1
3

•0214
•0311

28
13

1
1

-0423
•1416

57

59

1

2

-0784
-0465

98
21

July 1 1

o 1

4
1

•0100
•0523

13
31

1
5

-0178
•0292

23

22

3

1

•0137
-0608

17
46

3 '•

o

•0283

29

1

•06.50

65

2

•0545

55

Nov. 10

1

•0340

24

1

•0710

50

1

•0185
•0312

13
26

21

2

•0320

27

3

•0248

21

Dec. 9

1

•0220

22

3

•0189

19

1

•0311

31

1843.
Jan. 2

1

0^0180

18

1

0-0180

18

1

0-0261

26

Feb. 6

2

•0060

10

16

1

•0044

11

1

•0048

12

24

3

•0131

11

3

-0201

17

2

-0093

8

May 6

July 24

1

•0216
•0149

49

11

1

-0226
-0247

55

18

2
1

-0110
•0140

26
10

25

1

•0210

35

5

-0026

4

1

•0329

59

1844.

Mar. 29

o

0^0314

20

3

0-0309

20

1

0^0448
•0161

28
14

30

4

•0169

14

3

•0126

10

~

Oct. 1

1

•0156

26

1

-0198

33

1

•0018

3

20

1

-0224

28

1

•0904

113

Nov. 16

o

•0200

20

1

•0280

28

1

•0398

41

22

1

•0248

31

3

-0220

28

•0092

11

1845.

Jan. 9

1

0^0^90

29

1

0-0440

44

1
1

0^0080
-0211

8
13

5
10
31

Feb. 24

o

•0200

13

3

•0185

11

Mar. 26

1

•0210

15

3

•0104

7

1

1
1

-0070
-0062
-0439

Aug. 29

Dec. 3

3

4

•0053
•0310

9
22

1
1

•0024
•0667

4
47

1846.

May 12

July 11

Aug. 6

7

24

3

3
1
3

0-0073

-0209
•0286
•0109

7

18
13

8

2
2

2

! 7

; 1

0^0100
•0118
•0133
•0089
•0036

10

12

11

4

3

1
2

3

1

0-0118
-0044
-0147
-0123
•0160

12
13
12
6
10

25

28

1
3

•0096
•0122

6
14

:l 3

' 3

•0070
•0075

4
9

1

11 1
ii

•0071
-0114

5
13

630

ME. AIKT— ANALYSIS OF ]\IAG>T;TIC STOEMS

TiVBLE IV. (coutiuued).

i'car. Month,
and Day.

Westerly Force.

Number
of Waves.

1846 (cont').
Sept. 4

Oct.

o

7

8

Nov.

26

Dec.

23

1847

Feb.

24

Mar.

1

19

April

3

7

21

May

7

June

24

July

9

Sept.

24

26

27
Oct. 22

23*(]st)
23 (2nd)
24 :

25
Nov. 22
Dec. 17
18
19
20

1848.

Jan. l6

28

Feb. 20

21

22

23

24

Mar. 17

20

Apr. 7

May 18

July 11

Oct.

18
23
25

29

2
4
9
1
2
3
1
1

16
3
3
4

10
1
7
5
6

12
4
2

10

ofFluctua'
tions.

0-0201
•0148
•0187
•0342
•0474
•0352
•0156
•0295
•0185
•0237
•0160

0-0223
•0152
•0339
•0240
-0224
•0028
•0344
•0109

Absolute

Mean
of Disturb-

•0550
•0163
•0056
0-0043
•0235
•0025
•0845
•0103
•0226
•0315
•0162
•0433
•0434

0-0179
•0306
•0380
•0267
•0045
•0235
•0375
•0077
•0208
•0109
•0096
•0184
•0288
•0200
•0158
•0129

13
12
14
14
24
25
26
17
15
14
16

19
17
30
14
5
43
27

31
17
6
7
20
13
36
10
16
14
13
43
24

17
16
11
13
18
23
15
10
11
11
25
19
9

Northerly Force.

Number
of Waves.

Absolute
Aggregate
of Fluctua-
tions.

•0156
•0226
■0047
■0226
•0201
0425
■0102
0523
0235
0281
0170

0-0110
•0167
•0960
•0067
•0615
•0120
•0114

•0352
•1912
•0401
•0300
•0409
•0564
•0030
-2554
•0150
•0572
•0552
•0193
-0910
•1277

0-0340
•0375
•0335
•0962
•0125
•0028
•0525
•0093
•0309
•0084
•0123
-0689
•0323
•02;f4
•0157
•0000

Absolute

Mean
of Distiu:b-

10

17

3
10
10
30

17
29
20
19
17

32

22
14

106
41
30
71
47
16

no

15
41
25
16
91
70

33
20
37
42
31
3
23
18
27
21
14
36
30
24
9
0

Nadir Force.

Number
of Waves.

Absolute

Aggregate

ofFluctua

tions.

0^0208
•0304
•0140
•0316
-0100
•0645
•0060
•0378
•0905
•0210
•0112

0-0030
•0616
•1269
•0264
•0205
•0156
•0100

•0464
•1277
•0338
-0603
•0108
•1158
•0016
•1144
•0654
•1057
•1260

0-0970

Absolute

Mean

of Distiu-b-

ance.

13
23
10
13
5
46
10
21
77
13
11

3

77
70
33
13
26
10

Tie
75
34
62

18
100
8
48
69
70
90

136

* On October 2'3, 1847, all the observations wore interrupted during 10 hours.

OBSEEYED AT THE ROYAL OBSERVATORY, GEEEJTWICH.

631

Table IV. (coutinued).

Westerly Force.

Year, Month, Abeoluto

and Day. j Number Aggregate

I of Waves. | of Fluo-

|l tuationg.

1848 (cont'').

Nov. 17

18

Dec. 17

1849.
Oct. 30 .

Nov. 27 .

1850.
Feb. 22 ..
23 ..
Mar. 31 ..
May 7 ..
June 13 ..
Oct. 1 ..

1851.
Jan. l6
19
Feb. 18
Sept 3
4
6
7

29
Oct. 2
28
6
28
29

Dec.

Jan.

1852.

4 ,

19 .

Feb. 14 .

15 .

17 .

18 .

19 .

20 .

21 .
April 20 .
May 19 ,

20 .

June 11 .

16 ,
July 10 ,
Nov. 11 .

13 .

0-0683
•0276
•0161

0^0209
•0295

0^0172
•0186
•0214

Absolute
Mean of
Disturb-
ance.

•0285
•0487
•0421

0^0432
•0293
•0252
•0473
•0218
•0595
•0462
•0720
•0343
•0772
•0465
•0247
•0420

©•0357
•0183
•0217
•0412
•0283
•0277
■0919
•0186
•0370
•0570
•0487
•0068
•0514
•0310
•0490
•0393
•0386

34
19
17

7
8
9

12
21
18

19

12
11
25
9
25
20
32
14
33
20
11
23

15
8
10
17
12
12
43
8
18
24
30
10
23
13
23
17
16

Northerly Force.

Number
of Waree.

Absolute
Aggregate
of Fluc-
tuations.

0^1961
•0271
•0194

0-0228
•0280

0^0244
•0333
•0375
•0249
•0534
•0522
•0495

0-0517
•0468
•0505
-0522
-0444
-0378
-1108
•0972
-0636
-0506
-1264
•0243
•0627

0-0968
•0462
-0823
-0712
-0475
•1041
•1042
•0441
•0660
•0796
•0289
•0140
•0606
-0275
-0336
-0367
-0352

Absolute
Mean of
Disturb-

103
26
35

10

12

10
14
16
10
23
23
21

19

19
15
46
41

27
22
54
10

44
20
37
30
20
43
44
18
28
33
12
6
26
12
15
16
15

Nadir Force.

Number
of Waves.

Absolute
Aggregate
of Fluc-
tuations.

0-0769

-0208
•0744

0-3484

0-2030
-0303
•0000

•3882
•1188
•0212

0-1009
•1367
•1382
•1114
-1769
-0648
-1805
•3864
-1370
•1834
-0815
-0950
-0233

0-0523
-1206
•0585
•1879
•2525
-4422
-2596
•1594
•1735
•1508
•0595
•0605
-4354

•0660
•3236
-1638

41
50

87

13

0

164

54

9

44
59
59
48
77
28
76
172
58
80
36
41
11

22

54

26

80

110

185

113

69

77

67

28

44

195

"29
141

77

632

ME. AlEY — ANALYSIS OF MAGNETIC STOEMS

Table IV. (continued).

Year, Month,

■Westerly Force.

Northerly Force. i

Nadir Force.

Absolute

Absolute

Absolute

Absolute 1

Absolute

Absolute

aud Day.

Number

Aggregate

Mean of

Number

Aggregate

Mean of

Number

Aggregate

Mean of

of Waves.

of Fluc-

Disturb-

of 'WaTes.

of Fluc-

Distui-b-

of Waves.

of Fluc-

Disturb-

tuations.

ance.

tuations.

ance.

tuations.

ance.

1853.

Jan. 10

3

0-0229

10

3

0^0208

9

Mar. 7

8

•0269

11

6

•0302

13

0-3353

141

8

6

•0293

13

5

•0242

10

•5298

236

11

5

•0279

12

•3529

152

May 2

3

•0359

16

1

•0657

28

•3595

157

3

6

•0210

9

3

•0552

23

•3424

149

24

7

•0280

12

5

•0886

38

-2273

96

June 22

3

•0253

11

3

•0259

11

•1487

64

July 12

8

•0299

13

3

•0646

27

•0777

33

Aug. 21

•0617

26

Sept. 1

7

•0264

11

4

•0838

37

2

•0648

28

2

5

•0299

13

5

•0738

31

2

•0717

30

Oct. 1

1

•0312

13

2

1

•0336

14

25

4

•0153

6

3

•0160

7

1

•3093

129

Nov. 9

5

•0503

21

1

•0474

20

2

•0766

33

Dec. 6

5

•0354

15

1

•1079

45

2

•0633

27

21

5

•0176

8

3

•0097

42

1

•1790

77

1854.

Jan. 8

3

0^0249

10

3

0^0374

16

1

0^1089

46

20*

8

•0245

10

3

•0108

4

1

•0104

15

20

1

-0550

39

Feb. 16

5

•0219

9

5

•0383

16

1 1

•1165

49

24

6

•0335

14

4

•0230

10

3

•1460

62

25

3

•0135

5

6

•0235

10

2

-1310

55

Mar. 6

4

•0228

10

5

•0145

6

•1049

44

15

9

•0265

11

2

•0407

17

•0514

22

16

7

•0294

12

1

•0408

17

•0498

21

28

4

•0284

12

1

•1271

53

•1451

64

April 10

6

•0458

19

4

•06s7

29

3

•0855

38

23

3

•0233

10

4

•0334

14

o

•0897

38

May 25

6

•0112

5

3

•0188

8

'

•0759

32

1855.

Mar. 12

5

0^0395

16

o

0^0574

25

1

0-2111

90

April 4

3

•0248

10

7

•0154

7

2

•0282

14

Julv 19

o

•0653

29

2

•0719

31

Oct. 18

3

•0234

10

2

•0499

21

1

•1049

44

1857.

Feb. 26

3

0^0204

9

2

0^0180

8

1

0^1368

59

Mar. 13

3

•0194

8

-May 7

3

•0689

29

0726

30

1

•3191

141

10

9

•0118

5

0388

18

2

•0893

37

-Sept. 3

6

•0372

15

0515

21

2

-4199

175

Nov. 12

3

•0353

15

0162

7

16

o

•0271

12

0216

9

17

3

•0605

26

3

03.39

15

Dec. 16

5

•0405

17

3

0768

32

1

-2230

93

17

11

•0134

6

1

•0881

39

2

-0543

23

* ]8.')4, Jan. 20. Tlic Vertical- Force observations were interniptcd during 3 hours.
t In 18.56 there wore no days of Great llagnctic Disturbance throughout the year.

The la,st figure iu the ''Absolute Mean of Distiu-bauuc " is in the fourth decimal place of Horizontal Force.

OBSEEVED .\T THE EOTAL OBSERVATORY, GEEENAVU II. G33

Table V. — Sum.s, witliout rcijavd of sign, of Magnetic Fluctuations (in terms of Hori-
zontal I'orce) for each Year from 1841 to 1837, including all days of Eccord of
Great Magnetical Disturbance.

Year.

Number

of
Storms.

Westerly Force.

Northerly Force.

Nadir Force.

Number ..^ ,
gf Mumljer

Waves. ofH"""-

Absolute
Sum of
Fluctua-
tions.

Number

of
Waves.

Number
of Hours.

Absolute
Sum of
Fluctua-
tions.

Number

of
Waves.

Number
of Hours.

Absolute
Sum of
Fluctua-
tions.

1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857

8
10

7

6

5

18

21

19

2

7

13

17

18

12

i

25
19
11
10
11
46
100
64

6
20
59
73
75
64
11

0
48

129-47

112-57

55-72

51-74

60-00

244-86

246-75

264-18

46-00

141-79

294-04

364^65

327-14

285-10

71-37

0-00

23 1^53

•3236
•2699
•0990
•1087
•1063
•3632
•5249
•4356
•0504
•1765
•5692
•6422
•3941
-3057
. -0877
•0000
•3345

15
18
13
12

9
50
46
43

5
14
41
55
53
40
13

0
29

119-63

113-34

49-39

59-70

60-41

250-89

246-29

223-83

45-25

163-80

305-70

395-76

402-06

285-82

93-75

0-00

208-37

•2641
•4893
•0928
•1357
•1420
-3213
1-2229
•7128
•0508
-2752
•8I9O
-9785
•8065
•4770
•1880
0000
•4175

12
15

I

5

28
30
10

1

8

24

27

24

25

6

0

9

llG-1!)

111-74

45-40

59-29

60-52

247-99

1 98-75

40-65

22^92

138-34

299-17

353-07

350-67

279-75

91-03

0-00

141-73

-6143

-3452

•0933

-2021

•0862

•4155

1-0719

•2691

•3484

•7615

1^8l60

2-9661

3^2000

1-1701

•4161

•0000

1-2424

177

642 2926-91

4^7915

455

3023-99 1 7-3934

239

2557-21 15-0182

Means of Absolute!
Disturbances ... /

•00164

-00244

•00587

Table VI. — Sums, without regard of sign, of Magnetic Fluctuations (in terms of Hori-
zontal Force) for each Year from 1841 to 1857, including only those days of Great
Magnetic Disturbance in which Records were made by the three Instruments.

Year.

Number

of
Storms.

Westerly Force.

Northerly Force.

Nadir Force.

Number

of
Waves.

Number
of Hours.

Absolute
Sum of
Fluctua-
tions.

Number

of
Waves.

Number
of Hours.

Absolute
Sum of
Fluctua-
tions.

Number -. ,
f Number

Waves, of »<»"«■

Absolute
Sum of
Fluctua-
tions.

1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857

8

10

5

5

5

17

16

4

1

6

13

16

13

12

3

0

6

25
19

8
10
11
46
83
16

3
20
59
69
72
64
11

0
37

129-47

112-57

45-72

51-74

60-00

244-86

202-69

55-17

22-92

141-79

294-04

341-34

304-41

285-10

71-37

0-00

141-04

•3236
-2699
•0886
•1087
•1063
•3632
•4111
•1408
•0209
•1765
•5692
•6112
•3712
•3057
•0877
•0000
•1922

15
18
12
11

9
48
36
10

2
11
41
50
43
40
11

0
18

119-63

113-34

45-39

51-70

60-41

240-89

202-19

45-74

22-84

i;}9-88

305-70
372-27
308-40
285-82
70-97
0-00
139-25

•2641
•4893
•0880
•1133
•1420
•3095
•9497
•2749
•0228
•2503
•8I9O
•9510
•6930
•4770
•1227
•0000
•3458

12
15

7

7

5

27

29

10

1

8

24

27

22

25

4

0

9

116^19

111-74

45^40

51-29

60-52

244^61

194^75

40-65

22-92

1 38^34

299-17

353-07

303-72

279-75

67-58

0-00

141^73

•6143

•3452

•0933

•1117

•0862

•4111

1^0255

•2691

•3484

•7615

1-8160

2-9661

2-7854

1-1701

•3442

•0000

1-2424

140

553

2504-23 4^1468

375

2524-42

6^3124

232 2471-43

14^3905

Means of Absolute "1
Disturbances ... j

•00166

•00250

•00582

MDCCCLSIIl.

4r

634 3IR. AIKT — ANALYSIS OF MAGNETIC STOEMS

11. In examining the last line of these Tables, it must be borne in mind that the
numbers are affected by the constant part of the Disturbance which appears as " Mean
Disturbance " at the end of Table III. The value of mean disturbance for Nadir Force
(as has been remarked) is uncertain, and that for Westerly Force is small ; but that for
Northerly Force is important. A constant term —"00147, combined with variable
quantities whose mean value is +'00250, and whose actual value even at the maximum
of its wave will very frequently be far less, will destroy some waves entirely. It will
also increase the apparent Mean of Absolute Disturbances, even when the number of
waves is not diminished. Thus : suppose, as a simple case, that the pure disturbance
is represented by «sin^, but that, when affected with a constant term, it is asin^—b.
(As has been stated, when a is smaller than b, the addition of — b will make every
value — , and will destroy the alternation of -}■ waves and — waves, and thus the just
number of waves will be apparently diminished.) When a is greater than b, if 0 be
the fii-st value of 6 which makes a sin 0 — b=0, the positive Fluctuation wiU be found
by integrating from ^=0 to ^=■3- — 0, and the negative Fluctuation by integrating
from ^=:'r— 0 to ^=2^+0. The general value of the integral is — «cos^ — b0; the
first limited integral is 2« cos 0 — ^(t— 20) : the second is —2« cos 0—5(^+20), or
(with sign changed, to make it positive) -\-2a cosQ — b( — 7r—2Q) ; and the sum of
these, or aggregate of absolute fluctuations, is 4a cos 0 + 46 . 0. Now 0 is deter-
mined bv the condition « sin 0 — 6=0, or sin 0=-- If b be small, 0= - nearly,

a a

cos 0=1— -^ nearly, and the aggregate of absolute fluctuations =.ia-{- '^— ■ The

second term is the increase of the aggregate arising from the introduction of the term b.

If then we conceive the numbers in the last line of Table VI. to be affected with the
correction which ought to be mtroduced in order to neutralize the effect of the large
constant term in Northerly Force, it is certain that the number 375 would be consider-
ably increased, and that the number 6*3124 would be considerably diminished. A very
extensive examination of details would be necessary to enable us to say what would be
the exact proportion of the changes : but it appears to me extremely probable (though
at present far from certain) that the corrected Numbers of AVaves are sensibly equal,
the corrected Absolute Sums of Fluctuations are sensibly equal, and the corrected Means
of Absolute Disturbances are sensibly equal, for Westerly Force and for Northerly
Force.

The Number of Waves for Nadir Force is less than half that for the other forces ;
and the Absolute Sum of Fluctuations is about three times as great as that for the others.

12. It would be very important to ascertain any correspondence in the times of the
waves in the different directions. I have not yet succeeded in discovering any satis-
factory or certain relation.

First, in comparison of the Waves of Westerly and Northerly Forces, the coin-
cidences of times of wave are so rare that it seems evident that notliing can be inferred
from the few w^hich can be found. From 1849 to 1857, when the photographic appa-
ratus recorded equally the disturbances at all hours, I do not find one. In a less rigo-
rous examination of the storms from 1841 to 1847, 1 find that on Nov. 19, 1841, there

0BSERAT:D at the royal observatory, GREENWICH.

635

were contemporaneous waves from 12'' 17™ to IS** 17"', "W. F. -{-, No. F. + ; and on
Jan. 1, 1842, when the storm consisted of a single wave, G*" O" to 12'' 41", the forces
were W. F. — , No. F. + . And the second W. F.— on Jan. 16, 1848, corresponds nearly
with the sole No. F. — . Sometimes two waves in one direction correspond nearly with
one in the other direction: thus in the beginning of the storm 1854, April 10, the
W. F.+ from 0'' 7"° to 5'' 21'" and — from 5'' 21"' to 13'' 16'" occupy the same time as
No. F. + from 0'' 5"" to 13'" 9": but this relation is not supported in the remainder of
the same storm. A more frequent relation appears to be, that the evanescence of one
wave corresponds with the maximum of the other: thus on February 21, 1852, and
March 7, 1853, the waves stand in this order:

Westerly Force.

Northerly Force.

Limits of

Character

Limits of

Character

Wave*.

of Waves.

Waves.

of W aves.

1852. Feb. 21

0-27 ,

0-12 1
3-14 [

_

■

+

4-9^

I
5-16 J

+

15-15 1

\

1853. Mar. 7

0-10

23-59 J

4-5,
6-25 {
12-20 J

3-13^

_

+

5-32 1
7-19 J

+

which relation, however, in the latter instance, is not maintained through the storm.
And, generally, this relation does not appear to hold through the whole of any one
storm consisting of numerous waves.

13. As the number of Nadir Waves approximates to half the number of "Westerly
Waves, it might seem worthy of inquiry whether the maximum of Nadir Wave corre-
sponds to a change of Westerly Wave. The following instances have been remarked.

Time of Maximam of
Nadir Wave.

Sigiiof
Nadir
Wave.

Change of

Westerly

Wave.

Tira

e of Marimum of
Nadir Wave.

Sign of
Nadir
Wave.

Change of

Westerly

Wave.'

h m

h m

1841, Sept. 25.

3 35

+

+ to -

1852

Feb.

18. 4 37

+

+ to —

4 17

+

- to +

June

11. 14 28

—

- to +

6 19

+

+ to -

Nov.

11. 8 18

+

+ to -

1847. Sept. 24.

5 51

+

+ to -

1853.

Mar.

8. 6 28

+

+ to -

10 21

—

- to +

14 24

+

+ to -

Oct. 23.

5 27

+

+ to -

May

2. 17 35

+

- to +

7 1

+

+ to -

3. 3 33

+

+ to -

Oct. 24.

13 4

- to +

;

24. 10 10

+

- to +

Dec. 17.

6 15

+

- to +

July

12. 11 37

—

+ to -

8 13

+

- to +

15 57

+

- to +

1851. Sept. 4.

7 19

+

- to +

Sept.

1. 15 37

—

+ to -

7.

4 14

+

+ to -

i

2. 5 18

+

+ to -

6 30

+

- to +

Oct.

25. 13 47

+

+ to -

7 34

+ to -

: 1854.

Apr.

10. 17 56

—

- to +

10 19

_

+ to -

1857.

Dec.

17. 6 10

+

+ to -

1852. Feb. 18.

2 56

+

+ to -

1

I am unable to di-aw any inference from these.

4r2

636 ME. AlKY— ANALYSIS OF MAGNETIC ST0E3IS

14. The classification in Article 9 appears to lead to no result as to the effect of con-
nexion of special signs of the first or last waves of the different forces. The inequalities
shown in the fii-st Table of Ai-ticle 9 (of which the difference of numbers of last wave
+ and numbers of last wave — for the Northerly Force is the most remarkable) are
quite sufficient to explain the inequalities in the combinations exhibited in the latter part
of Article 9. And, on the whole, the principal conclusions which can be deduced from
the examination of the Waves appear to me to be the following : —

That, while on the whole the Westerly Force is — , yet the number of + waves is the
greater ; and at the beginnings and ends of storms the number of + waves is greater
than the number of — waves in a proportion exceeding 3:2.

That, the Northerly Force being on the whole — , in two instances out of three the first
Northerly wave is — , and in ten instances out of eleven the last Northerly wave is — .

That, due regard being had to the effect of the constant — Northerly Force, it
appears probable that the number of waves and the mean value of wave-disturbance are
nearly the same for Westerly Force and for Northerly Force ; but

That for the Nadir Force the number of waves is less than one-half the number for
the other forces, while the mean value of disturbance is more than double that for the
other forces.

15. I now proceed with the Irregularities. The following Tables (VII., VIII., IX.)
exhibit their aggregates under the same divisions as those for the Waves. It will be
remarked that, from the nature of the process by which the Irregularities are found,
their algebraic sum in each storm is sensibly =0; and therefore they are treated here
only as numbers without sign.

OBSER^TED AT THE ROYAL OBSERVATORY, GREENWICH.

C37

Table VII. — Absolute Sums, without regard of sign, of Coefficients of Magnetic Irre-
gularity (in terms of Horizontal Force), on Days of Great Magnetic Disturbance.

Tear, Month,

Westerly Force.

Northerly Force.

Nadir Force.

Number

Absolute

Mean

Number

Absolute

Mean

Number

Absolute

Mean

and Day.

of

Sum of Coem-

Coefficient

of

Siun of Coeffi-

Coefficient

of

Sum of Coeffi-

Coefficient

Irregu-

cients of

of Irre-

Irregu-

cients of

of Irre-

Irregu-

cients of

of Irre-

larities.

Irregularity.

gularity.

larities.

Irregularity.

gularity.

larities.

Irregularity.

gularity.

1841. 1

Sept. 24 1

10

0-0133

13

6

0-0060

10

2

0-0031

15

25

70

•1417

20

73

-1226

17

61

-1760

29

27

6

•0086

14

12

-0090

8

3

-0021

7

Oct. 25

33

■0437

13

36

-0354

10

14

•0157

11

Nov. 18

25

•0329

13

28

•0325

12

18

-0208

12

19

19

•0252

13

26

-0213

8

13

-01.39

11

Dec. 3

7

•0134

19

13

•0127

10

3

-0018

6

14

8

•0145

18

9

-0146

16

6

-0072

12

1842.

Jan. 1

6

0-0068

11

8

0-0038

5

5

0-0021

4

Feb. 24

7

•0132

19

9

-0162

18

3

-0013

4

April 14

12

•0152

13

11

-0168

15

6

-0090

15

15

20

•0291

15

35

-0373

11

15

-0134

9

July 1

9

-0137

15

15

-0198

13

10

-0113

11

o

23

-0349

15

35

•0502

14

10

-0134

13

3

29

•0437

15

42

•0502

12

20

•0236

12

Nov. 10

11

-0197

18

14

•0139

10

4

•0021

5

21

14

-0132

9

15

•0204

14

1

-0008

8

Dec. 9

19

•0209

11

36

•0176

5

6

-0036

6

1843.

Jan. 2

5

0-0059

12

6

0-0056

9

2

0-0005

3

Feb. 6

3

•0024

8

16

7

-0008

1

6

-0015

3

24

12

-0118

10

37

-0166

4

6

•0041

7

May 6

17

-0206

12

22

-0196

9

9

-0105

12

July 24

4

-0047

12

6

-0058

10

5

•0013

3

25

14

-0151

11

13

-0141

11

5

-0015

3

1844.

Mar. 29

21

0-0230

11

24

0-0159

7

9

0-0046

5

30

18

•0246

14

29

-0335

12

7

•0041

6

Oct. 1

9

•0056

6

9

-0070

8

1

•0005

5

20

11

•0113

10

3

-0046

15

Nov. 16

28

•0290

10

19

•0190

10

9

-0049

5

22

22

•0234

11

31

•0300

10

9

-0072

8

1845.

Jan. 9

15

0-0167

11

9

0^0105

12

4

0-0033

8

Feb. 24

16

•0163

10

26

•0123

5

13

•0072

6

Mar. 26

12

•0125

10

16

•0124

8

4

-0028

7

Aug. 29

19

•0065

3

11

•0087

8

5

-0015

3

Dec. 3

57

•0698

12

61

•O7O8

12

27

-0242

9

1846.

May 12

13

0-0161

12

15

00130

9

4

0-0044

11

July 11

14

•0178

13

7

-0057

8

Aug. 6

26

•0172

7

35

•0172

5

7

-0036

5

7

64

•0207

3

55

•0308

6

15

-0090

6

24

9

•0075

8

9

•0055

6

5

-0015

3

25

5

•0033

7

5

•0059

12

2

-0015

8

28

28

•0150

5

24

•0178

7

3

-0023

8

638

ME. AIET— ANALYSIS OF MAGXETIC STORMS

Table VII. (continued).

Year, Month,

Westerlj Force.

Northerly Force.

Nadir Force.

Number

Absolute

Mean

Number

Absolute

Mean

Number

Absolute

Mean

and Day.

i ^^

Sum of CoeiB-

Coefficient

! of

Sum of Coeffi-

Coefficient

of

Sum of Coeffi-

Coefficient

! Irregu-

cients of

of Irre-

Irregu-

cients of

of Irre-

Irregu-

cients of

of Irre-

larities.

Irregularity.

gularity.

larities.

Irragularity.

gularity.

larities.

Irregularity.

gularity.

1846 (conf).

Sept. 4

26

0-0178

7

29

0^0156

5

5

0-0028

6

5

32

•0255

8

36

•0285

8

7

•0093

13

10

6

•0049

8

6

-0056

9

3

-0008

3

11

28

•0311

11

31

•0378

12

12

•0123

10

21

23

•0162

7

18

•0158

9

7

-0041

6

22

68

•0771

11

59

•0692

12

28

-0244

9

Oct. 2

8

•0089

11

11

•0100

9

3

-0018

6

7

25

•0343

14

28

•0295

11

1 3

•0049

16

8

29

•0213

8

29

•0245

9

5

•0031

6

Nov. 26

28

•0253

9

29

•0235

9

7

•0080

11

Dec. 23

12

•0163

14

9

•0133

17

7

-0039

6

1847.

Feb. 24

20

0-0132

7

15

0^0107

7

4

0-0026

7

Mar. 1

42

•0416

10

43

•0384

9

16

•0126

8

19

49

•0835

17

36

-0518

14

24

•0283

12

April 3

15

•0214

14

18

-0232

13

3

•0039

13

7

19

-0225

12

22

-0306

14

4

•0044

11

21

12

•0142

12

8

•0095

12

o

•0018

9

May 7

6

•0088

15

4

•0047

12

2

•0010

5

June 24

3

•0046

15

July 9

8

•0134

17

5

•0075

15

Sept. 24

148

•2666

18

128

-3262

26

119

•2192

18

26

12

•0128

11

15

-0142

9

9

•0087

10

27

16

•0167

10

12

-0124

10

10

•0201

20

Oct. 22

29

•0232

8

30

•0406

14

24

•0157

6

23*(lst)

86

•1132

13

73

•1332

18

58

•0882

15

23(2nd)

3

•0016

5

1

•0021

21

2

•0088

44

24

113

•2034

18

128

•3134

24

94

•1722

18

25

20

•0225

11

17

•0184

11

7

-0121

17

Kov. 22

34

•0428

13

46

■0462

10

15

-0375

25

Dec. 17

86

•1400

16

39

•0577

15

33

-0540

16

18

29

•0297

10

21

-0236

11

19

66

•0937

14

44

•0963

22

20

97

•2546

26

64

•2191

34

1848.

.Ian. 16

21

0^0570

27

21

0-0381

18

28

18

•0361

20

19

•0422

22

Feb. 20

35

-0573

16

16

•0329

21

21

35

-1182

34

49

•1857

38

22

4

•0099

25

5

•0087

17

23

16

•0283

18

12

•0248

21

24

24

•0431

18

21

-0407

19

Mar. 17

4

•0036

9

7

-0141

20

20

28

•0553

20

20

•0470

23

April 7

21

•0390

19

9

•0241

27

May 18

20

•0233

12

12

•0252

21

July 11 '

33

•0544

16

25

•0G08

24

Oct. 18

21

•0675

32

18

•0666

37

14

0-0524

37

23

23

•0518

23

19

•0.396

21

25

20

•0284

14

22

•0300

14

29

11

•0185

17

1

•0018

18

* On Oct. 23, 1847, all the observations were intomipted during 10 hours.

OBSERVED AT THE llOYAL OBSEUVATORY, GKEENWICU.

b3y

Table VII. (contiuued).

Year. Month,
aiid Day.

1848 (conf).

Nov. 17

18

Dec. 17

1849.
Oct. 30 .
Nov. 27 .

1850.
Feb. 22 .
23 .
Mar. 31 .
May 7 .
June 13 .
Oct. 1 .

1851.
Jan. 16
19

Feb. 18
Sept. 3
4
6
7
29
2
28
6
28
29

Oct.
Dec.

Jan.

1852.

4 .

19

Feb. 14

15

17

18

19

20

21

April 20

May 19

20

11

16

10

11

13

June

July
Nov.

Westerly Force.

Number

of
Irri'gti-
Inritios.

Abeoluto

Sum of C'oefli-

oieiita of

IrrcgulurilT.

1853.
Jan. 10
Mar. 7
8

38
17
19

27
35
29

13
34
25

43
37

22
19
29
18
89
63
33
24
40
36
47

38
31
20
101
90
73
73
45
50
52
25
3
31
41
29
37
43

19
66

72

0-1225
•0272
•0396

00232

•0158

0-0219
-0506
•0249

-0180
•0384
-0390

0-0544
•0341
•0297
•0311
•0512
•0320
•1659
•1426
•0489
•0448
•0697
•0381
•0463

•0343
•0358
•0255
•0987
•1440
•0965
•1630
•0457
•0739
•0690
•0207
•0031
•0573
•0373
•0352
•0483
•0506

0^0 195
•0423
•0621

Mean

Coeflieiont

of Irro-

Northerly Force.

Number

of
Irrcgu

gularity. larities.

32

16
21

15
9

14
11
16

13
9
13
16

18
18
19
23
15
19
17
n

10

9
12
13
10
16
13
22
10
15
13

8
10
18

9
12
13
12

Absolute

Sum of CoolTi-

cients of

Irregularity.

77
17
12

26
28
17
13
14
30
25

36
35
39
28
63
40
106
122
43
46
51
37
52

59
19
62
92
66
71
60
70
52
36
37
37
39
25
38
25

16
63
57

0-2394
•0306
•0213

0^0192
•0166

0-0356
•0612
•0249
•0174
•0202
•0405
•0400

0-0429
•0420
•0410
•0231
•0843
•0558
•1899
•1828
•0602
•0509
•0615
•0313
•0452

0^0208
•0540
•0562
•0888
•1924
•1295
•1397
•0641
•0785
•1515
•0322
•0466
•0586
•0464
•0411
•0435
•0301

0^0 146
•0423
•0415

Mean
Coedicient
of Irre-
gularity.

31

18
18

14
22
15
13
14
14
16

12

12

11

8

13

14

18

15

14

II

12

9

9

9
9
30
14
21
20
20
11
11
29
9
13
16
12
16
11
12

Nadir Force.

Number Absolute

of Sum of Coefli-
Irregu- cients of
larities. Irregularity

41

1

14

4
6
20
40
42
47
86
67
29
20
30
15
12

18
31
17
53

124
54

100
17
23
41
12
14
32

15
20
12

0-2362
•0008
•0167

0-0046

0-0113

-0129
•0072

•0123
-0123
•0087

0-0090
-0077
•0165
•0355
•0460
•0388
•1367
•1115
-0414
•0180
•0404
•0144
•0098

0^0087
•0177
•0195
•0398
•1354
•0576
•1789
•0198
-0226
-0440
•0121
•0077
•0352

•0111
•0224
•0080

Mean

Coefiieient
of Irre-
gularity.

0-0201
•0147

58

8

12

12

23
43

72

31
15

12

23
13

8

9
11

8
16
17
14

9
13
10

8

5

6
11

7

11
11
18
12
10
11
10

6
II

7
11

7

640

ME. AIET — AXALTSIS OF MAGNETIC STORMS

Table

VII. (concluded

)-

Tear. Month,

Westerly Force.

Northerly For

ce.

Nadir Force.

Number

Absolute Sum

Number

Absolute Sum

Mean

Number

Absolute Sum

Mean

and Day.

of

of Coeffi-

Coefficient

of

of Coeffi-

Coefficient

of

of Coeffi-

Coefficient

Irregu-

cients of

of Irregu-

Irregu-

cients of

of Irregu-

Irregu-

cients of

of Irregu-

larities.

Irregularity.

larity.

larities.

Irregularity.

lai-ity.

larities.

Irregularity.

larity.

1853 (cont'').

Mar. 11

May 2

"59"

"e"

54

80

0^0411
•0528

8

7

11
15

0-0175
•0165

16
11

0-0367

3

63

•0391

6

61

•0556

9

21

•0157

7

24

77

•0646

8

97

•1206

12

37

•0555

15

June 22

50

•0361

7

51

-0454

9

17

•0170

10

July 12

123

•1097

9

129

-1231

10

34

•0524

15

.Aug. 21

Sept. 1

42

...„.

"46"

...„.

8
13

•0118
•0190

15
15

•0260

•0418

n

70

•0665

9

90

-0959

11

36

•0391

11

Oct. 1

25 ......

22

...„.

12

9

27

-0036
•0037
•0156

3
4
6

10

'"10"

•0187

•0105

Nov. 9

49

•0407

9

49

•0376

8

19

•0118

6

Dec. 6

60

•0489

8

41

•0461

11

26

-0321

12

21

35

•0298

8

28

•0221

8

8

•0067

8

1854.

Jan. 8

33

0-0207

6

24

0^0218

9

13

0-0090

7

20*

49

•0279

6

35

•0206

6

4

•0023

6

20

Feb. 16

"se"

s"

"%{

...„.

4
26

•0059
•0170

15

7

•0460

•0527

24

53

•0460

9

67

•0481

7

21

•0175

8

25

56

•0405

7

63

•0487

8

22

•0208

9

Mar. G

33

•0178

5

37

•0204

6

16

-0216

14

15

59

•0463

8

65

•0425

7

28

•0229

8

16

58

•0556

10

69

•0513

7

24

•0188

8

28

62

•0591

9

77

•0549

7

49

•0249

5

April 10

49

•0527

11

79

-0688

9

52

•0357

7

23

38

•0206

6

49

•0322

7

21

•0108

5

May 25

38

•0229

6

52

•0342

6

32

•0301

9

1855.

Mar. 12

55

0^036l

6

59

0^0320

5

23

0^0157

7

April 4

55

-0355

6

53

-0390

7

19

•0111

6

July 19

Oct. 18

t

1857.

80

•0451

6

21

•0152

7

40

•0267

7

60

•0311

5

13

•0111

8

Feb. 26

41

0-0128

3

21

0-0119

6

10

0-0126

13

Mar 1 3

37
90

•0155

4

Mav 7

•0778

9

102

•0883

9

58

-0504

9

10

60

-0196

3

65

•0309

5

13

•0129

10

Sept. 3

55

-0501

9

92

•0629

7

37

•0296

8

Nov 12

47
41

•0256

5

58

•0292

5

16 ...

•0265

6

56

•0191

3

17

Dec. 16

42

•0329

8

68

•0307

4

66

•0847

13

82

•1496

18

19

•0147

8

17

78

•0626

«

93

•0771

8

30

•0221

7

In the column

• Mean Coefficient of Irregularity," the last figures correspond to the fourth decimal place
of Horizontal Force.

* In 18.54, Jan. 20, the Vertical Force observations were interrupted during 3 hours,
t In 1856 there were no days of Great Magnetic Disturbance throughout the year.

OBSEETED AT THE ROYAL OBSERYATOKY, GREENWICH.

641

Table VIII. — Sums, without re^rard of sign, of Coefficients of Magnetic Irregularity
(in terms of Horizontal Force), for each Year from 1841 to 1857, including all days
of Record of Great Magnetical Disturbance.

Year.

Westerly Force.

Northerly Force.

Nadir Force.

Number
of Storms.

Number
of Irregu-
larities.

Sum of
Coeffi-
cients.

Number
of Storms.

Number
of Irregu-
larities.

Sum of
Coeffi-
cients.

Number
of Stonns.

Number
of Irregu-
larities.

Sum of
Coeffi-
cients.

1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857

8

10

7

5

5

17

20

19

2

6

13

17

14

12

3

0

10

178

150

62

98

119

430

905

408

30

163

500

782

807

584

150

0

557

•2933

•2104 ,

•0613

•1056

•1218

•3585

1^4306
•8810
•0390
•1928
•7888

1-0389
•6407
•4561
•0983
•0000
■4081

8

10

6

6

5

18

20

19

2

7

13

17

17

12

4

0

9

203
220

90
123
123
442
772
382

15
153
698
810
910
684
252
0
637

•2541
•2462 j
•0632 '
•1167
•1147
•3813

1^4857
•9736
•0358
•2398
•9109

^2740
•8034
•4962
•1472
•0000
•4997

8

10

5

6

5

18

17

4

1

6

13

16

15

12

4

0

6

120

80

27

38

53

130

431

70

4

28

418

583

277

312

76

0

167

•2406
•0806

•0179
•0259
•0390
•1034
•69S6
•3061
•0046
•0647
•5257
•6405
•3404
•2373
•0531
•0000
•1423

168

5923

7-1252

173

6514

8-0425

146

2814

3^5207

1

MeanCoef-1
ficient ...J

•00120

•00123

•00125

Table IX. — Sums, Avithout regard of sign, of Coefficients of Magnetic Irregularity(in terms
of Horizontal Force), for each Year from 1841 to 1857, including only those days of
Great Magnetic Disturbance in which Records were made by the three Instruments.

Westerly Force.

Northerly Force.

Nadir Force.

Year.

Number

Sum of

Number

Sum of

Number

Sum of

of Irregu-

CoeiB-

of Irregu-

Coeffi-

of Irregu-

Coeffi-

larities.

cients.

larities.

cients.

larities.

cients.

1841

8

178

-2933

203

•2541 1

120

-2406

1842

10

150

•2104

220

■2462

80

•0806

1843

5

52

•0581

84

•0617

27

•0179

1844

5

98

•1056

112

•1054

35

•0213

1845

5

119

•1218

123

•1147 i

53

•0390

1846

17

430

•3585

428

•3635

123

•0977

1847

16

710

1^0480

635

M333

426

•6911

1848

4

95

•2568

124

•3579 ;

70

•3061

1849

1

19

•0232

8

•0192 !

4

•0046

1850

6

163

•1928

140

•2224

28

-0647

1851

13

500

-7888

698

•9109

418

-5257

1852

16

741

1-0016

771

1-2276

583

•6405

1853

13

788

•6212

819

•7404

258

-3111

1854

12

584

-4561

684

-4962

312

-2373

1855

3

150

-0983

172

-1021

55

•0379

1856

0

0

-0000

0

-0000

0

-0000

1857

6

390

-3076

455

•4207

167

-1423

Sums ...

140

5167

5-9421

5676

6-7763-

2759

3-4584

Mean Co

efficient...

-00115

•00119

•00125

16. The most striking particulars in the last line of these Tables are the following :
First, the almost exact equality of the Mean Coefficients of Irregularity in the three
MDCCCLXIII. 4 s

642

MR. AIRY— A^'ALTSIS OF MAGKETIC STOEMS

elements. Aud this remarkable agreement proves that the Irregularities as measm-ed
here are real objective facts. For they are measured fi-om photographic sheets in Avhich
the scales are very different : on the Westerly and Northerly records, O'Ol of Horizontal
Force is represented by 2-87 inches 'and 2-55 inches, while on the Nadir record O'Ol of
Horizontal Force is represented by 0-88 inch. Yet the eye of the Reader of the Photo-
graphs has caught the IiTegularities when shown on this small scale as certainly as
when shown on the larger scale. "With reference to their physical import, I think it
likely that the equality of Coefficients of Irregularity may hereafter prove to be one of
the most important of the facts of observation.

Second, the near agreement in the number of Irregulai-ities for Westerly Force and
for Northerly Force.

Thii'd, the near agreement in the number of Irregularities for Nadir Force with half
the number of Irregularities for Westerly or for Northerly Force.

17. I have not succeeded in discovering any clear relation between the times of occur-
rence of IiTegularities of Westerly Force and of Northerly Force. They certamly do
not coincide. In their intermixtui'e, I cannot assert that an Irregularity of one element
always occurs between two of the other element, though there is a general appearance
of that law.

18. It appeared to me possible that an Irregularity of Nadir Force might occiu' at the
change between + and — Irregularities of Westerly Force ; and the following exami-
nation seems to show a certain degree of plausibility in the supposition : —

Day.

Total Number of
Nadir Irregularities.

Number of Nadir Irregu-
larities corresponding to

changes of sign for
Westerly Irrcgidarities.

1841. Sept. 25

61
119
60
94
36
42
86

52
76
36
66
20
26
68
50
42
101
42
68
22
14

8
12
13
25
25

9
25

9
23
16
35
16
39
31
21

1*^47. -Sept. 24

Oct. 23

24

Dec. 17

1851. Sept. 4

7

29

67
53

124
54

100
32
20
11
15
21
37
34
13
36
10
26
21
52
23
58
37
30

1852. Feb. 15

17

18

19

June 11

Dec. 11

1853. Mar. 8

May 2

3

24

Julv 12

Sept. 1

Oct. 25

Dec. 6

1854. Feb. 24

April 10

185.5. IMar. 12

1857. May 7

Sept. 3

Dec. 17

Total

1372

990

OBSERTED AT THE ROTAL OBSERVATORY, GREENTTTCH,

643

19. The investigations which I had proposed to myself as more peculiarly the object
of this paper are now terminated, in so far as their results can be comprehended in
tables of numerical values and remarks on the relations between the numbers. But I
think it desirable to subjoin Tables tending to exliibit the laws of frequency of the great
wave-disturbances and the irregiUai-ities, with respect to the mouths of the year and
^^•ith respect to the hours of the day.

20. First, for the months of the year. The following numbers are formed by simply
collecting from Tables I., IV., and VII. all the numbers arranged in groups under each
nominal mouth. It will be seen at once that the distribution of magnetic storms
through the year is so UTCgular that, even in the long period of seventeen years, no
inference can be drawn connecting the Magnetic Storms with the Seasons.

Table X. — Aggregates of Fluctuations and Inequalities, arranged by Months,
in terms of the Horizontal Force.

Month.

Westerly Force.

Northerly Force.

Nadir Force.

Algebraical

Absolute

Sum of
Irregulari-

Algebraical

Absolute

Sum of
Irre^ari-

Algebraical

Absolute

Sum of
Irregulari-

AMregate
of Fluctua-

Amregate
of Fluctua-

Aggregate
of Fluctua-

Aggregate
of Fluctua-

Aggregate
of Fluctua-

Aggregate
of Fluctua-

tions.

tions.

ties.

tions.

tions.

tions.

tions.

January

- -0435

•3183

•3492

+ -0679

•4827

•3169

- •5582

•6250

•0662

February ...

- -1425

•6275

1^2093

— •5521

1^0223

r3985

- •1176

2-4732

•5974

March

- -1279

•3905

•6038

- ^5193

•6071

•5640

-^ 1^0271

2^0367

•2158

April

+ -0289

•2635

•3192

- ^4074

•4596

•4330

- •2766

-5416

•1341

May

- -0266

•3052

•3533

— ^0554

•4638

•5411

+ -6293

1-9545

•2291

June

— -0453

•1471

•1533

— -0224

•1674

•1706

- -9723

•5841

•0522

July

— -0598

•2238

•3114

- -2361

•4187

•4414

+ -0109

•4423

•1430

August

+ -0087

•0875

•0702

- •OUo

•0427

•0859

- ^0988

•1294

•0312

September...

- -1198

•704G

M977

— -4614

1^0812

1-3994

- -1785

2^2337

•9391

October

+ -OOGG

•5864

•8836

— •8129

•9881

1-0282

— •2781

1^8979

•4866

November ...

— -0431

•6511

•6016

- ^6096

•7150

•6836

- ^5549

•9893

•3744

December ...

- -1032

•4860

1^0726

- -7603

•9448

•9799

- •ogeg

M105

•2516

The disproportion of Irregularities to Fluctuations in the Nadir Force, as compared to
those in the other Forces, is very remarkable.

21. Secondly, for the hours of the day. For each hour, on a day of storm, the nearest
value of wave-disturbance (not of fluctuation) and the nearest value of irregularity were
taken from the sheets in which the reductions described in Article 5 were made ; and all
the numbers thus found were collected for each hour, the + and — values of wave-
disturbance being placed iu separate columns. Thus the following Table is formed.

4s2

644

ME. AIRY— ANALYSIS OF MAGNETIC STORMS

T.iBLE XI. — Sums of Wa^•e-distl^l•bauces aud of Irregularities, arranged by hours of
Gottiugen Solar Time, in terms of Horizontal Force.

Westerly Fore.

Northerly Force.

Nadir Force.

Hour

of Got-

Sums of Wave-

Sums of Wave-

Sums of Wave-

tingen

Number

disturbance.

Simis of

Number

disturbance.

Sums of

Number

distxirbauce.

Jiuns of

Time.

of Mea-
sures.

Irregu-
larities.

of Mea-
sures.

Irregu-
larities.

of Mea-
sures.

Irregu-
larities.

4-

-

+

-

+

—

0

25

■0201

•0103

•0213

29

•0136

•0717

•0323

5

•0285

•0000

•0090

1

56

•0558

•0106

•0416

57

•0339

•0726

•0674

19

•0681

•0306

•0236

2

77

•0658

•0203

•0658

82

•0617

•0900

•0954

33

•1434

•0455

•0370

3

76

•0881

•0224

•0725

92

•1060

■0807

•1060

40

■1773

•1131

•0563

4

98

•1051

•0334

•1144

108

•1201

•0823

•1462

63

■3094

•1187

•0774

5

95

•0831

•0437

•1179

103

•1407

•1019

•1233

60

•2832

•1113

•0681

6

105

•0752

•0713

•1327

114

•1276

•1291

•1290

74

■3701

•0856

•0794

7

104

•0593

•1079

•1353

108

•0806

•1422

•1344

77

•3976

•0974

■0915

8

122

•0331

•1759

•1746

136

•0570

•2171

•1754

79

•3092

•1280

•0853

9

126

•0276

•1848

•1743

119

•0479

•2393

•1439

80

•2866

•1575

•1169

10

123

•0165

•2191

•1976

130

•0553

•2612

•1750

86

•2529

•2061

•1241

11

116

•0267

•1841

•1531

111

•0544

•2747

•1524

77

•2110

•2837

•0889

12

121

•0278

•2070

•1429

122

•0449

•2917

•1422

74

•1629

•2716

•1007

13

111

•0277

•2036

•1606

108

•0307

•2470

•1429

63

•1097

•2830

•0799

14

112

•0442

•1574

•1442

109

•0308

•2S97

•1260

74

•1768

•3133

•0941

15

99

•0601

•1324

•1604

100

•0362

•2194

•1443

59

•1329

•2598

•0717

16

102

•0537

•0951

•1359

97

•0160

•2428

•1287

59

•0966

•2881

•0825

17

84

•0695

•0508

•0926

86

•0120

•2137

•1117

54

•0910

•2963

•0619

18

87

•1016

•0315

•0970

93

•0101

•2043

•1169

46

•1010

•2038

•0532

19

76

•lOOS

•0193

•0793

85

■0112

•2531

•0990

44

•0830

•1889

•0470

20

75

•1170

•0107

•0826

81

•0076

•2646

■0713

39

•0614

•1295

•0427

21

58

•0613

•0083

•0527

65

•0087

•1919

■0694

29

•0619

•0740

•0306

22

59

•0647

•0179

•0520

69

•0038

•2241

■0694

26

•0355

•0460

■0270

23

51

•0460

•0214

•0346

57

•0052

•1463

•0441

24

•0491

•0396

•0177

It must be remarked here that the number of measures at O"" is made in this Table
unfairly small. This arises partly from the interruptions which are almost unavoidable
in the operation of changing the photographic sheets at O"", and partly from the manner
in which the measiu'ed quantities have been treated in the discussion of Storms. When
a storm has evidently occupied a part of a day, it has been usual to treat by rule the
measures of the entire sheet of that day, from 0'' to 24'' ; and in that process, as is
described in the begimiing of Article 5, the two first and two last measures are lost ; and
some of these ought, in a great number of cases, to be referred to O"". The best value
that can be taken for O*" will be the mean of the values for 23'' and for 1''.

22. It will be seen that, at the same hour, the mean value of Irregularity is nearly
the same for the three Forces, but that, from hour to liour, the mean IrregiUarities are
largest where the number of measures is greatest, that is, where storms are most
frequent. In regard to the Wave-disturbance; for Westerly Force, the aggregate is
-f from 17'' to 6'', — from 7'' to 16'' ; for Northerly Force, the aggregate is + from S""
to 5'', — from 6'' to 2'' ; and for Nadir Force, the aggregate is + from 23'' to 10\ — from
11'' to 22\ In legard to the modification which these Wave-distui'bances might be
supposed to produce on the laws of Diurnal Inequality, when it is remarked that each

OBSERVED AT THE EOIAL OBSEEVATORY, GREEISTVICU.

C45

of the hours 0\ 1'', 2'', &:c. has been repeated 17x365 times, it will be seen that the
introduction of these Storm Days into the general mass of obser\'ations will in no
instance alter the mean Diunial Inequality by a unit in the fourth decimal place. In
a year of veiy great disturbance, as 1853, they may possibly introduce a correction of
one unit, or perliaps two units, in the fourth decimal of some of the Diurnal numbers.

23. The import of the numbers of the last Table will be best seen by the following
treatment. If for either of the three directions of force, at any one hour, we form the
Algebraic sum of the + and — sums of wave-disturbances, and divide by the number
of measui'es, we obtain the mean wave-disturbance whenever a storm occui's at that
hour. If we form the Absolute sum, and diAide it similarly, we obtain the double
average departure from that mean whenever a storm occurs at that houi\ The mean
Irregidarity is obtamed by simple division.

Table XII. — Frequency of Storms, mean Wave-disturbance, average departure from
the mean, and mean Irregularity, in terms of the Horizontal Force, at each hour of
Gottingen Solar Time.

Westerly

Force.

Northerly Force.

Nadir Force.

Hour

of Got-
tingen

Fre-
quency

Mean Wave-

Average
departure

Mean
Irregu-

Fre-
quency

Mean Waye-

Average

departure

from

Mean
Irregu-

Fre-
quency
of

Mean Wave-

Average
departure

Mean
Irregu-

Time.

\f'

disturbance.

from

larity.

^ 1'
of

disturbance.

larity.

disturbance.

from

larity.

Storms.

+

±

Storms.

±

±

Storms.

±

±

0

54

+

•00039

•00061

•00085

57

- -00200

•00147

•00112

22

+

•00570

•00285

•00180

1

56

+

81

59

74

57

68

93

118

19

+

197

260

124

2

77

+

59

56

86

82

- 35

93

116

33

+

297

286

112

3

76

+

86

73

95

92

-f- 28

101

115

40

+

161

363

140

4

98

+

73

71

117

108

-t- 35

94

135

63

+

303

340

123

5

95

+

42

67

124

103

-1- 38

118

120

60

+

287

329

114

6

105

+

4

70

126

114

- 1

113

113

74

+

385

308

107

7

104

—

47

80

130

108

57

103

124

77

+

390

321

119

8

122

—

117

86

143

136

- 118

101

129

79

+

229

276

108

9

126

—

125

84

138

119

- 161

121

121

80

+

161

278

146

10

123

—

165

96

161

130

- 158

122

135

86

+

54

267

144

11

116

—

136

91

132

111

- 198

148

137

77

94

321

116

12

121

—

148

97

118

122

- 202

138

117

74

—

147

294

136

13

111

—

159

104

145

108

- 200

129

132

63

_

275

312

127

14

112

_

101

90

129

109

- 238

147

116

74

_

185

331

127

15

99

—

73

97

162

100

— 183

128

144

59

_

215

333

122

16

102

_

41

73

133

97

- 234

133

133

59

_

325

326

140

17

84

+

22

72

110

86

_ 235

131

130

54

_

380

359

115

18

87

+

81

77

112

93

- 209

115

126

46

_

224

331

116

19

76

+

107

79

104

85

_ 285

155

117

44

_

241

309

107

20

75

+

142

85

110

81

_ 317

168

88

39

_

175

245

110

21

58

+

91

60

91

65

_ 281

154

107

29

_

42

234

106

22

59

+

79

70

88

69

- 319

165

101

26

_

40

157

104

23

51

+

48

66

68

57

_ 248

133

77

24

+

40

185

74

The Soli-tidal character of the principal characteristics of the occasional Magnetic
Storms, as to frequency, magnitude, inequalities of wave-disturbance, and Irregularities,
is seen clearly in this Table.

646 JME. AIET — A^'ALTSIS OF MAGXETIC STOEMS

24. I now come to the consideration of the pliysical inference from these numerical
conclusions. And first I would remark that I do not think that they can be reconciled
with the supposition of definite galvanic currents or definite magnets, suddenly pro-
duced, in any locality whatever, as sufficient to explain the disturbances observed here.
On that hypothesis, it would seem necessary to believe that such sudden currents or
magnets woiild produce simultaneous disturbances m the three co-ordinate dii-ections,
that, if the long period of a wave permitted some deviation from this rule, yet the short
period of an inequality would admit of no such deviation, and that, on any supposition,
the number of disturbances in the three directions would be approximately equal. Yet
in fact we find that neither in Waves nor in Irregularities is there the least appearance
of simultaneity, and that, though there is close equality of numbers between the
Westerly and Northerly Forces, yet the Nadir Force (in which the In-egularities are as
strongly marked as in the Westerly and Northerly, and the Wave-disturbances much
more strongly marked) exhibits less than half the number. These considerations appear
to me quite conclusive as showing that the observed disturbances cannot be produced
by the forces of any suddenly created galvanic current or polar magnet.

25. To suggest instead of this an imperfect conjecture, based upon grounds so inade-
quate as those which we can at present use for its foundation, must be a delicate
and dangerous, I may almost say an in\-idious enterprise. Yet the impression of an
explanation of broad character, partly definite but generally indefinite, has, in the course
of this investigation, forced itself so strongly on my mind, that I should think it wi'ong
to omit to describe it. Its fundamental idea is, that there may be in proximity to the
earth something which (to avoid unnecessary words) I shall call a Magnetic Ether ; that
under circumstances generally, but not always, having reference to the solar hour, and
therefore probably depending on the sun's radiation or on its suppression, a current from
N.N.W. to S.S.E., approximately, or from S.S.E. to N.N.W. (according to the boreal or
austral nature of the ether) is formed in this Ether ; that this cun-ent is liable to inter-
ruptions or perversions of the same kind as those which we are able to observe in cur-
rents of air and water ; and that their effect is generally similar, producing eddies and
whirls, of violence sometimes far exceeding that of the general cm-rent from which they
are derived.

26. Our powers of observing the two elements to which I have referred for analogy
are somewhat different, but both imperfect. We know that in a gale of wmd, the
direction of the wind is continually changing ; the horizontal pressure and the barome-
tric pressure also are continually changing ; but the changes are so rapid that we cannot
easily determine whether there is any correspondence between them. But, in the storms
on a large scale, there is reason to think that some winds are radial, but far more are
cyclonic ; that in some instances the barometer rises in the centre, but in more it is
depressed ; and in many instances the disturbance of vertical pressure is enormous (for
1 inch of barometer corresponds to a ])rcssure of about 70 lbs. per square foot). Of
water, perhaps the best study is to be found in disturbed tidal currents, as those of the

OBSERVED AT THE EOYAL OBSERVATORY, GREEXAVICII. G-i7

Western Islands of Scotland ; here, in some places, approximately circular spaces are to
be seen which are quiet, but which appeal' to the eye to be elevated above the rest ; in
some disturbed places the water is thrown upwards ; in other places the sea is whirling
round with great speed, in a good circular form, and with a funnel of considerable depth
in the centre ; in other places, boiling currents are running very fast in opposite direc-
tions, though separated by no great space ; the general impression however is that of
circularity*; great circles and small circles coexisting. Though these circular forms
may be more prevalent in one part of the sea than another, they are not fixed, but
wander irregularly, sometimes suddenly disappearing, and sometimes as suddenly
created anew. In like manner, in the coui'se of a river, tra\-elling funnels may be seen,
whose depth sometimes exceeds their breadth.

27. Now it appears to me that if a sentient and reasoning being were immersed cither
in the air or in the water through which these circles are ^^'an(lering, he would perceive
actions nearly similar to those which we have found to exist in the magnetic storms.
The large and slowly-displaced circles would produce Wave-distm-bances, slowly changing
theii- direction, and thus ha\ing different times of evanescence in the N. and S. direction
(on the one hand) and in the E. and W. dii-ection (on the other hand) ; the smaller
circles, in like manner, would produce the rapid Irregularities. And in the relation
between E. and W. disturbances and vertical disturbances, there is a point which well
deserves attention. "NMien a water-funnel passed nearly over the observer, travelling
(suppose) in a N. direction, he would fu-st experience a strong current to the E., after-
wards a strong current to the W. (or vice versa), and between these there would be a
very strong vertical pressure in one direction, not accompanied by one in the opposite
direction ; thus he would have half as many vertical as horizontal impulses. This state
of things corresponds to the proportion which we have found throughout for the mag-
netic distui-bances, and to the relation found in Article 18. I may also add that the
rule at which we have arrived, that the waves of vertical force are few, but that their
power, when they do occm-, is very great, seems to correspond to what is reported of the
whirlwinds of great atmospheric storms ; Avhich, -siolent and even frequent as they may
be, occur very rarely at any assigned place.

28. It seems to me that there is so much plausibility in these suppositions as to justify
me in expressing a ^\dsh that some effort might be made to verify them. The imme-
diate object of observations would be, to ascertain through a locality of considerable
extent the times and magnitudes of Wave-disturbances and of Irregularities on the same
days throughout, with the view of discovering whether they could be collectively repre-
sented as the effects of such travelling vortices as I have suggested. In regard to the
extent of the locality, I should think that a portion of the Continent of Eui-ope would
suffice, and that five or six magnetic observatories would decide the points under inquiry.
In regard to the mode of observation, though eye-observation is, for a limited time, the
most accurate, yet self-registering record is the only method which can insm-e the

* I have been upon these curreuts, and in close proximitj to these whirlpools.

648 ]\IE. AIET— ANALYSIS OP IMAGNETIC STOEMS.

obseiTation of all that is required ; only, I woiild specially observe, it is indispensable
that eye-observations be used to check the zeros of time and of measiu'e, and that the
photographic traces be so strong that they will not be lost in rapid motions of the
magnet. In regard to the mode of primary reduction, I imagine that the method
followed in this Memoir (mth such small alterations as experience may suggest) will be
found best.

^*^ The computations for the " Diiimal Inequalities " were performed by computers under the immediate
superintendence of Mr. Jonjf LrcAS ; some portions of them were revised and corrected by James GiAisnEB, Esq.,
F.R.S., Superintendent of the Magnefcical and Meteorological Department of the Royal Observatory. The curves
were drawn under Mr. Glaishee's superintendence by Mr. W. C. Nash, and reduced to scale by Mr. James
Caepentek, Assistant in the Astronomical Department of the Eoyal Observatory. The computations of the
present Memoir were made under the superintendence of Mr. Glaisheb, by Mr. Nash and junior computers.

[ C49 ]

XXX. Besults of hourly Ohservatiom of the MagmticDeclinatiwi made hy Sir Feancis
Leopold M'^Clixtock, and the Officers of the Yacht ' Fox,' at Port Kennedy, in the
Aj-ctic Sea, in the Winter of 1858-59; and a Conijmrison of these Sesults with
those obtained by Captain Eochfort Maguire, and the Officers of Her Majesty's
Ship 'Plover,' in 1852, 1853, and 1854, at Point Barroio. By Major-General
Edward S^ujixe, R.A., President of the Eoyal Society.

Eeceivod December 21, 1803,— Read January 7, 1864.

Is the spring of 1857 Captain Fr.vxcis Leopold M^'Clintock, of the Royal Na\7, being
about to proceed to the Aixtic Seas in the ' Fox ' Yacht in search of the ships which
had formed Sir JoHX FR-ANKLiN's^Expedition, applied to the President and Council of the
Royal Society " to afford him such information and instructions as might enable him to
make the best use of the opportunity afforded by the voyage for the prosecution of
meteorological, magnetical, and other observations."

A committee havmg been appointed lo commmiicate with Captain M'^Clixtock, I, as
one of the Members of that Committee, drew up a memorandum respecting the mag-
netical observations which he might have an opportunity of making, and supplied him
vfiih. suitable instruments belonging to the Government Establishment under my super-
intendence. With the sanction of the Committee of the Kew Observatory, Lieutenant
W, R. HoBSOX, R.N., and Captain Allen Youxg, two of the Officers who proposed to
accompany Captain M'^Clintock, were instructed in the use of these instruments at the
Kew Observatory.

As this communication is limited to a notice of the hourly observations of the Mag-
netic Declination, which Captain M'^Clintock ^and his Officers were enabled to make
in the winter of 1858-59, it will be sufficient at present to extract fi'om the memo-
randum, adverted to ia the preceding paragraph, the portion which relates to that
branch of the inquii-y, as the most suitable iatroduction to the account of the observa-
tions themselves.

" The results of the hourly observations of the declination made at Point Barrow in
1853 and 1854, by Captain Rochfort Maguire, R.N., and the Officers of Her Majesty's
Ship ' Plover,' when compared with the hourly observations at the Toronto Observatory,
have brought into view, in accompaniment mth many cii-cumstauces of a highly inter-
esting resemblance, some features, in the magnetic disturbances at Point Barrow, which
appear as if they were the converse of those of the corresponding phenomena at Toronto.
Now, Toronto in lat. 43° 40' and long. 79° 22' W., and Point Barrow in lat. 71° 21' and

UDCCCLXIII. 4 T

650 MAJOE-GEXEEAL SABIXE OX THE EESrXTS OF HOUELT OBSEEYATIOXS

long. 156' 15' W., are situated on the same continent ; and it seems probable that there
may exist some intermediate locality where the phenomena of the distiirbances may be
of a critical character. The more precise determination of this locality is full of interest,
both as respects terrestrial magnetism and geographical physics generally. It is for this
reason veiy desirable that we should learn, by similar obsers'ations to those made at Toronto
and Point Barrow, what are the corresponding periodical laws of the disturbances of the
declination at stations which either in latitude or longitude may be intermediate between
those places. It is highly probable that, if either the ' Erebus ' or ' Terror ' be still exist-
ing, there may be found in one or the other, or in both ships, the records of observations
in at least two intermediate localities, in which the Expedition may have been stationary
in different years ; because both ships were fiunished with the proper instruments, and
some of the Officers had attended at Woolwich to practise with them before the Expedi-
tion sailed : to this it may be added, that both Sii" John Fbanklin and Captain Crozier
were sti'ongly impressed with the desirability of making the observations, and letters
are extant from both, written fi'om Da^as Strait, after they had sailed from England,
expressing their full intention to set up the instruments wherever the ships should be
detamed for a sufficient period to give the observations value. The possible existence
of such records is here referred to A^dth the -view of impressing on the attention of Captain
M*^Clixtock the scientific importance of recovering these records, if possible, and of
bringing them safely home.

"The station where Captain M^Clintock's ship wall probably remain diu'ing the
months preceding the departure of the sledge-parties, as weU as during the still longer
period when they will be employed in the search which forms the object of the Expe-
dition, will not be far distant, in all probability (whether that station be in Peel's or in
Regent's Inlet), from the latitude of Point Barrow and longitude of Toronto. It is a
locality, therefore, at which observations similar to those at Point Barrow, which have
proved in many respects very important, are extremely desirable. The duration of
Captain Maguire's hourly observations was eight months in 1853, and Qune months in
1854. The accord in the conclusions drawn from the observations of either year taken
separately, -n-ith their joint results when taken together, shows that eight or nine months
is sufficient for the purposes adverted to, if a longer diu'ation be inconvenient. Even less
than eight months would suffice for a general indication, though of course the longer
the observations can be continued, under equal circumstances of care, &c., the more
precise is the information acquired. Captain M'^Clintock has stated to the wTiter of
this memorandum that he anticipates no difficulty in maintaining hourly observations
between the time w^hcn^the ship is laid up in autumn and the departure of the sledge-
parties in the spring, and that it might be possible that, when once become a routine,
they might be kept on still by those few persons who will remain mth the ship. The
observations are in themselves extremely simple, and it happens fortunately that one of
the Officers who expects to accompany Captain M'^Clintock, Mr. Grey, was also with
Captain Maguire at Point Barrow, and is therefore acquainted with magnetic observa-

OF THE MAGNTETIC DECLIXATION AT PORT KENNEDY. 651

tions, which were remarkably well conducted in Captain Maguire's Expedition *.
Anaongst the instruments at Woolwich which have been returned from the dismantled
obseiTatories, there is a Declinometer which will be suitable for the purpose, when it
has received small repairs, which in anticipation of this opportunity are already in pro-
gress f.

"At the station where the ship will be laid up, the amount of Dip may possibly
exceed 89° ; but experience has shown that until the Dip is nearer 90° than 89°, there
may still be found a sufficient horizontal directive force to give consistent results with a
Declinometer."

On the return of the Expedition from the Arctic Seas in the summer of 18-59, Captain
M'^Clintock placed in my hands the hourly observations of the Declinometer, which had
been made at Port Kennedy, in lat. 72° 0' 49" and long. 94° 19' W., from November
1858 to March 1859 inclusive, together with remarks, from which the following extracts
are made.

" The ship took up her winter position in Port Kennedy on the 27th of September, in
thirteen fathoms water, and about 500 yards from the land. The countiy is very rugged,
in many places precipitous to the sea, and is composed of gneiss and granite with masses
of trap. No low or level spot could be found sufficiently far from overlooking hills to
suit as the site of a magnetic observatory. There remained, however, the alternative of
building upon the ice when sufficiently strong. About the middle of October, the ship
being now firmly frozen in, I selected a large hummock of old ice, elevated about 2 feet
above the recently formed ice, as the best foundation to build upon. It bore magnetic
south from the ship, tUstant 220 yards, and was about 400 yards from the land. 1 con-
sidered therefore its position to be satisfactory.

"Ice was now cut, and, being from 8 to 10 inches in thickness, served well to construct
an observatory baring an interior space of 7 feet square. The roof of the house, and also
of the porch, was of loose planks, covered and cemented together by sludge (snow and
water mixed), which also served as mortar for the slabs of ice. The porch was secured
by a door, and a fearnought screen protected the entrance from the porch. A pedestal
composed of slabs of ice cemented together stood in the centre of the room. A marble
slab was placed thereon, and, after being levelled and adjusted at right angles to the

* The application made to the Admiralty for permission to Mr. Grey to accompany the Expedition was
unfortunately not successful. By the zeal of Captain M'^Cliniock and of Messrs. HoBsoir and Alleic Youtto,
the loss of Mr. Geet's services was in great measure supplied.

t The magnet of the Declinometer was of the same pattern as that of the Admiralty Standard Compass,
consisting of four bars of steel clock-spring, fixed vertically and equidistant in a light framework of brass,
carrying a very light metallic ring divided to 5'. The pair of central needles were 7-3 inches long, and the pair
of external ones .3-5 inches. The magnet was suspended by a thread of untwisted silk passing over a pulley at
the top of a suspension-tube. When not thus suspended, the magnet rested on a pivot of " native alloy" ; its
weight could be either partially or wholly relieved by means of the suspension-thread. In the hourly observa-
tions, the opposite divisions of the graduated circle were read by microscopes carried by the general frame-
work, to which the suspension-tube was also attached.

4t2

652 ma.joe-ge>:eeal sabixe on the eesults of houelt observations

magnetic meridian, was frozen upon the pedestal. A tripod table-top, wtli brass grooves
to receive the levelling-screws, having been frozen upon the marble slab, the Declino-
meter was then mounted and levelled ; and when all seemed to be in proper working
order, the feet of the levelling-screws themselves were frozen to the table, so as to prevent
all movement. The magnet caiTied a graduated circle of 6 inches diameter, divided to 5',
and rested on a pivot supported by an agate cup ; its weight could be relieved, either
partially or entirely, by a suspension-thread composed of fibres of untwisted silk ; the
dinsious of the circle corresponding to the opposite ends of the magnet were read by flsed
microscopes. "SAHien the declinometer was first set up and the hourly series commenced,
the weight of the magnet was not entirely relieved by the suspension-thread : in this
state, and after an interval of two days from the first adjustment, the torsion-force was
observed as follows : —

Torsion Circle. Reading.

At zero 218 05 •

Turned 360° to the East 216 20

At zero 218 10 . ■

Turned 360' to the West 220 10 ,

At zero 218 05 ■ - ■ ,

whence we should have 115' as the effect of 360° of torsion, or about 0'-3 as the effect
of 1' of torsion. At first, however, and as thus adjusted, the declinometer did not appear
to work in a thoroughly satisfactory manner. This may have been occasioned by the
levelling-adjustments of the magnet suited to the magnetic latitude ha\ing altered its
centre of gravity and impaired the free action of the pivot in the cup. I therefore
removed the supporting pivot altogether on the 4th of December, but in doing so I
accidentally broke the suspension-fibre; this was replaced, and the magnet finally
adjusted on the 6th of December, supported only by the silk thread ; and from this
date I consider there could have been nothing to interfere with the exactness of the
observations.

" During the first few weeks of the series, accumulations of drift snow upon one side
or other of the observatory would slightly alter the level of the ice ; upon these occa-
sions I always relevelled the instrument, if necessary, myself. It could not move in
■azimuth, as it remained frozen to its pedestal. Both ends of the needle were always
read off and recorded. Whenever the magnet was either touched, or observed to be in
a state of agitation, a note to that effect was entered on the margin of the observation
paper.

"As auroras were of frequent occurrence, I have given a Table of those observed
dui-ing the period of the hourly observations. There was nothing in or near the
obser\'ato]7 which could possibly affect the magnet. Withinside the house there were
only a wooden candlestick, a copper lamp, and a board upon which the observation
paper was fastened with copper tacks."

OF THE MAGNTHTC DECLIXATIOX AT POET KENNEDY. 653

In the discussion of the results, the means of the readings of the two ends of the
magnet have been taken throughout as the position of the magnet corresponding to the
time of the observation. The record of the hourly observations from November 1, 1858
to March 27, 1859, comprehending 3384 observations, was placed in the hands of the
Non-Commissioned Officers of the Roj-al Artillery in the "NYoolw-ich Establishment to
undergo the usual process of examination. After a careful consideration, I judged that
a difference of 1° 10' from the mean or normal position of the same month and hour
afforded a suitable value for the standard of distui-bance, as separating about a fifth
part of the whole body of the observations. There were 748 observations which differed
from their respective noiTuals by that amount or more ; and these have been accordingly
regai-ded as " disturbed observations." They form about 1 in 4-5 of the whole number ;
their aggregate values in the different months were as follows : —

1858,

Nov. 1 to 28 .

Total aggregate values
. . 486 10

Ratios to the mean
monthly aggregate value.

1-42

5i

Dec. 1 to 31 .

397 10

1-16

1859,

Jan. 1 to 31 .

189 34

0-55

»»

Feb. 1 to 28 .

284 28

0-83

■>■>

Mar. 1 to 27 .
in the 5 months

354 47

1-04

Total

1712 09

Mean monthly

value

1712° 09'_
5

342° 26'.

Separated into their easterly and westerly constituents, and into the different houi's of
their occun-ence, the Ratios of Easterly and Westerly distui-bance at Port Kennedy to
the mean hourly easterly and westerly disturbance were obtained in the manner which
has been so frequently described ; and by expanding these in sines and cosines of the
hom'-angle and its multiples, the following approximate formulaj are obtained : —

Port Kennedy, lat. 72° 01' N., long. 94° 20' W.
Easterly Disturbances.
l + -90sin(a+89°18') + -31sin(2«+86°32').

Probable En-or of a single observed hourly Ratio +011.

Westerly Disturbances.
1 + -318 sin (a+272° 56') + -637 sin (2a + 71° 58').

Probable Error of a single observed houily Ratio 4;0-17.

654 MAJOE-GENEEAL SABIXE OX THE EESULTS OF HOTJELT OESEEYATIONS

From Table III., in the discussion of the hourly observations at Point Barrow*, we
have the corresponding formulae at that Station as follows : —

Point Barrow, lat. 71° 21' N., long. 156° 15' W. ■ ^

Easterly Disturbances.
1 +1-087 (sin a + 17r 20') + -523 (sin 2a + 200° 31').

Probable Error of a single observed hourly Ratio +0-23.
Westerly Disturbances.

1 + 0-673 (sin a + 264°17') + -568 (sin 2a+94° 49').

Probable Error of a single observed hourly Ratio +0-13.

From these formula; we have the Ratios of easterly and westerly disturbance at the

several hours of local astronomical time at the two stations, as shown in the following

Table :—

Table L,

Local
Afltrou.
Hours.

Port Kennedy. 1

Point Barrow.

Local
Civil
Hours.

Easterly
Katios.

Westerly
Ratios.

Easterly
Ratios^

Westerly
Ratios.

(1)
0

(2)
2-20

(3)
1-29

(4)
0-98

(5)
0-89

Ivoon.

1

2-14

1-32

0-48

0-81

1 P.M.

2

1-95

1-20

0-08

0-63

2 P.M.

3

1-66

0-99

— 0-13

0-43

3 P.M.

4

1-32

0-73

-0-18

0-29

4 P.M.

5

0-98

0-50

— 0-08

0-26

5 P.M.

6

0-70

0-41

0-11

0-37

6 P.M.

7

0-51

0-48

0-32

0-64

7 P.M.

8

0-39

0-70

0-.51

1-04

8 P.M.

9

0-35

1-04

0-Gl

1-48

9 P.M.

10

0-37

1-41

0-65

1-87

10 P.M.

11

0-39

1-74

0-G5

2-14

1 1 P.M.

12

0-41

1-93

0-66

2-23

Midnight.

]3

0-40

1-93

0-72

2-13

1 A.M.

14

0-39

1-75

0-88

1-85

2 A.M.

15

0-38

1-41

1-15

1-47

3 A.M.

16

0-40

1-01

1-52

1-07

4 A.M.

17

0-50

0-64

1-90

0-72

5 A.M.

18

0-68

0-57

2-25

0-51

6 A.M.

19

0-95

0-28

2-48

0-42

7 A.M.

20

1-27

0-36

2-53

0-48

8 A.M.

21

1-61

0-56

2-37

0-62

9 A.M.

22

1-91

0-85

2-01

0-77

10 A.M.

23

2-13

M2

1-63

0-88

11 A.M.

The easterly and westerly deflections at Port Kennedy and Point Barrow present the
same general features as at all other staticms where the laws of tlie disturbances have
been investigated. In Plate XLI., figs. 1 & 3 represent graphically the easterly ratios
in Table I., columns 2 & 4, as do figs. 2 & 4 the westerly ratios in columns 3 & 5
of the same Table. It will be seen that figs. 1 & 3 show the conical form and single

• Pliilosophical Transactions for 18,")7, Art. XXIV. p. 502.

OF TILE MAGXETIC DECLINATIOX AT POET KENNEDY. 655

maximum, and the small and nearly equable amount of variation during the ten or
eleven hours when the ratios are least, which characterize figs. 1, 4, 5, & 6 in Plate XIII.
Phil. Trans, 1863, Ait. XII. Similarly figs. 2 & 4 show a double maximum resembling
that which is seen in fig. 2 in Plate XIII. Phil. Trans. 1863, Art. XII. In the case of
the Easterly Disturbances, the conical summit or extreme easterly deflection occui's, as
will be seen, approximately at the same absolute time at Port Kennedy and Point Barrow ;
and the principal maximum of westerly disturbance at the same local time at the two
stations. The secondary maximum of westerly disturbance is less strongly marked in
the Point Barrow than in the Port Kennedy curve, and its epoch is not so identically
the same at both stations as is the case in the principal maximum. This may be due
to the magnitude of the disturbances, and the shortness of the time during which the
observations at either station were maintained ; or there may be a real diiFerence in the
epoch and amount of the secondary maximum. The accord at the two stations of the
principal easterly maximum in ahsolute time and of the principal westerly maximum in
local time is too remarkable to be passed unnoticed, though it is certainly possible that
the accord is in both cases simply an accidental coincidence. The stations at which
the laws of the distiu'bances have been approximately investigated are as yet too few to
make an attempt at a more extensive generalization, at present, either safe or advan-
tageous. "What seems most to be desii'ed is, that stations for further research should
be selected upon a systematic plan, and with reference especially to their geographical
relations ; and that the inquiry should not be limited to the disturbances of the decli-
nation, but should include those also of the dip and total force. By the combination
of the facts which would be thus obtained, we might have a reasonable prospect of
gaining an assured knowledge of the general laws by which these phenomena are
governed in all parts of the globe. To initiate this scheme of research, which would
have been at the same time important to science and honourable to our country, was the
object of the recommendation made to Her Majesty's Government in 1858 by the two
principal scientific institutions of Great Britain. Until some such systematic proceeding
is adopted, the progress of this branch of magnetical science is likely to remain frag-
mentary.

Port Kennedy and Point Barrow have a common magnetical relation in being both
situated to the geogi-aphical North of a critical locality in the magnetic system, viz. the
locality of greatest total magnetic force in the northern hemisphere, or the centre of
the larger loop of the isodynamic lemniscates. The geographical latitude is nearly the
same, but in geographical longitude they differ 61°, or about four hours in time. Port
Kennedy is situated on the eastern and Point Barrow on the western side of the Ameri-
can Continent and its adjacent islands. Their distance apart is about 1200 geogi-aphical
mUes. The normal dii-ection of the magnet is widely different at the two stations, the
Declination at Port Kennedy being N. 135° 47' W.;(1858), and at Point Barrow N. 41° E,
(1854); the magnet therefore points in nearly opposite geographical dii-ections at the
two stations. There is also a considerable and an important difference in the amount
of the Dip, and consequently in the antagonistic force by which the horizontal compo-

656 ilAJOE-GEXEEAli SABIXE ON THE EESULTS OF HOIJELY OBSERVATIONS

nent of the earth's magnetism opposes the action of any distiu-bing force. At Point
BaiTow, where the dip was 81° 36', the intensity of the terrestrial horizontal force had
still an absolute value of about 1"88 in British imits, being about half its value in our
own islands; whilst at Port Kennedy, tlie dip being 88° 27'-4*, the horizontal magnet
was nearly astatic. It is evident that, from this great inferiority in the retaining force
at Port Kennedy, we ought to be prepared for a generally much greater apparent amount
of distm-bance at that station than at Point Barrow ; and accordingly we find that whilst
at the latter a disturbance-value of 22''87 caused the separation in the category of large
disturbances of between one-fifth and one-sixth of the whole body of hourly observations,
it requu-ed a disturbance-value of 70' to separate a nearly equal proportion of the obser-
vations at Port Kennedy. On the hypothesis of the energy of the distm-bing force being
equal at the two stations, and taking, as a sufficient approximation, the statement that
one in e-\ery five hourly observations at Point Barrow is in excess of its normal of the
same month and hour by an amount equalling or exceeding 22'- 87, a very simple calcu-
lation will show what the amoimt of the disturbance-value should be which should place
the same proportion, or one-fifth, of the whole hoiu'ly observations at Port Kennedy in
the category of large disturbances. For this purpose we may take from the most recent
maps of the isodynamic lines the total terrestrial magnetic force, approximately the
same at both stations, =:12"9 in British units; then, having the Dip at Point Barrow
81° 36', and at Port Kennedy 88° 27', we have the terrestrial horizontal force 1-88 at
Point Barrow, and 0-35 at Port Kennedy. Whence we find that, on the hypothesis of
there being an equal energy of the disturbing force at the two stations, the disturbance-
value corresponding to 22'-87 at Pomt Barrow should have been 123' at Port Kennedy
instead of 70'. Whilst, therefore, there is an increase in the effect of the disturbing
action at Port Kennedy by reason of the diminution of the antagonistic horizontal
terrestrial force, there is ob\iously also evidence of an actual and very considerable
superiority in the energy of the disturbing force itself at Point Barrow as compared
with Port Kennedy.

The inference which we thus derive from the direct comparison of the disturbances at
Port Kennedy and Point Barrow is in accordance with the fact previously made known

* Obsurvations of tlic Dij) at Port Kennedy made on the Ice, far distant either fi'om the Ship or the Land.

1858.

Needle.

Poles.

Means.

Observer.

Du-ect.

Beversed.

Oct. 9

A 1
A2
A 1
A 1
Al
A 1
A 1

88 33-1
88 30-1
88 19"3
88 25-3
88 27-3
88 20-2
88 21-3

88 38-2
88 25-1
88 32-8
88 27-0
88 29-1
88 29-2
88 26-4

88 35-5
88 27-5
88 26-0
88 26-1

88 28-2
88 24-5
88 24-0

Capt. M'Clintock.

Capt. Allen Young.

Capt. M'Clintock.

»

„ 9

„ 21

„ 28

„ 29

Nov. 2

„ 13

Mean

88 27-4 N.

OF TIIE MAGNETIC DECLINATION AT TORT KENNEDY.

657

to us by the hourly observations of the Aurora at Point IJarrow, for which we are
indebted to Captain Maguire and the officers of II. M. Ship 'Plover,' that the preva-
lence of that well-known concomitant of magnetic disturbance is far greater at Point
Barrow than at any other part of the globe where observations have been made. The
increased assurance which we now possess by direct comparison, that the maximum of
the disturbing energy is not coincident with the j) resent locality of the dip of 90", or with
that of the present maximum of the total terrestrial magnetic force, may have hereafter
an important bearing on the theory of the physical causes which combine in producing
the magnetic phenomena of the globe. The number of days on which the Aurora is
recorded to have been seen at Port Kennedy in the five months and four days from
October 28, 1858 to March 31, 1859, was 42, or little more than one day out of four;
whereas at Point Barrow the Aurora is stated to have been seen, during two successive
winters, six days out of seven*. The disparity thus shown is further (enhanced and ren-
dered more remarkable by the circumstance that, in the decennial disturbance-period,
1853 and 1854 (which were the years of observation at Point Barrow) are years of mini-
mum, and 1858 and 1859 (which were the years of observation at Port Kennedy) are
years of maximum disturbance.

Table II. — Auroras recorded at Port Kennedy in the winter months of 1858-59.

Date.

Direction of Aurora.

Date.

Direction of Aurora.

1858.

18.59.

Oct. 28

* s. to w.

Jan. 1

# W. to S.

29

* S.S.E. to W.N.W.

2

* S.W.

30

« s.w.

3

S.E.

31

N.W.

8

W.S.W. to S.E.

Nov. G

S.E. to W.S.W.

9 A.M.

# W. to N.W.

7

* S.W.

9 P.M.

N. to s. throu;;h zenith.

8

* S.W.

10 A.M.

* N.W. to s.k', s.

9

* s. to w.

10 P.M.

N. to s. through zenith.

12

N. to zenilh.

11

# S.E. to w.

14

* W.N.W. to S.W.

31 A.M.

* N w. to s e', s.

Dec. 3

# S.W.

31 P.M.

w.s.E. to zenith.

4

E. through S.W., N.W.

Feb. 1

# N W. to S.E*, s.

5

♦ N.W. to S.E.

8

# S.W.

6

# W. to S.E.

19

N. to s. thioufjli zenith.

8

S.E.

20

s. to zenilh.

12

* N.W. to s.R. through s.

23

N.E. to .S.W.

13

# W.N.W. to S.S.E.

26

N. to s. through zenith.

14

# N.W. to E.s.E. through s.

Mar. 6

N.N.w. to S.S.E. througii zenilh.

15

N.W. through s. to e.

30

* w. to S.W.

24

All over the heavens.

31

* W.

28

# Wesl^ to S.S.E.

30

8.

The following remarks by Dr. David Walker, II.N., by whom the record of the

Auroras was kept, will be read with interest. " Of the 42 Auroras observed during our

winter, 24 (marked with an asterisk) were in the direction of a space of water open

throughout the winter, or of the vapour rising from it. More than this number might

» Philosophical Transactions for 1857, Art. XXIV. p. 512."

MDCCCLXIII. 4 U

6oS MAJOE-GEXEEAL SABINE OX THE RESULTS OF HOURLY OBSERVATIONS

be traced to it; but of these 24 I am certain. On five occasions the Aurora caused an
agitation of the Declinometer: on one of these (Dec. 24, 1858) I observed a deflection
of 15° ; on the other four times the nbration was not much more than a degi-ee ; four of
the five occurred when the Aiu-ora was from nortli to south, passing through the zenith."
Table III. contains a statement, taken from the record of the hoiu'ly observations, of
the days and hours on which disturbances exceeding 5° from the normal of the same
month and hour were observed.

Table III. — Port Kennedy. Differences exceeding 5° from the normal of the same
month and hour shown by the Declinometer.

Bay.

Hour.

Disturbance.

Day.

Hour.

Disturbance.

18J8.

ISaS.

^ 59 E.

Nov. 18

21

5 23 E.

Dec. 23

21

18

22

5 35 E.

23

22

7 09 E.

18

23

5 55 E.

23

23

6 11 E.

19

22

7 25 E.

24

11

7 51 E.

19

23

6 47 E.

1859.

28

20

6 08 E.

Jan. 14

22

8 00 E.

28

21

5 47 E.

16

0

7 56 E.

28

22

6 46 E.

Feb. 9

0

9 31 E.

Dec. 4

11

5 35 w.

9

1

6 12 E.

4

15

6 00 w.

23

1

10 10 E.

4

16

7 35 w.

23

2

5 10 E.

4

17

6 27 w.

25

21

5 01 E.

4

18

5 40 w.

Mar. 25

3

5 57 E.

12

19

6 42 E.

26

0

6 51 E.

22

22

5 04 E.

26

1

5 56 E.

December 4, 22, and 2.3, February 9 and 23, were also days of excessive distui-bance
at Kew (Phil. Trans. 18G3, Art. XII. Table I.).

JJidurhance-diurnal Variation. — Table IV. exhibits the disturbance-diurnal variation
at Port Kennedy, or the average excess at the several hours of easterly over westerly,
or of westerly over easterly, deflection.

T.-UiLE IV.

Local

Dellections.

Disturbanoc-

Local

Deflections.

Disturbance-

diumal vari-

diurnal vari-

Easterly.

Westerly.

ation.

Easterly.

Westerly.

ation.

6 A.M.

ii-7

6-2

5-5 E.

6 P.M.

ib-9

3-1

7-8 E.

7 A.M.

15-4

5-3

10-1 E.

7 P.M.

9-3

2-1

7-2 E.

8 A.M.

20-8

5-2

15-6 E.

8 P.M.

6-8

4-8

2-0 E.

9 A.M.

3,7-4

6-7

28-7 E.

9 P.M.

G-0

9-9

3-9 w.

10 A.M.

37-8

5-9

31-9 E.

10 P.M.

5-9

19-1

13-2 w.

11 A.M.

31-8

9-9

21-9 E.

11 P.M.

10-2

20-4

10-2 w.

Noon.

35-6

10-7

24-9 E.

Midnight.

6-2

24-1

17-9 w.

1 P.M.

36-6

17-6

19-0 E.

1 A.M.

7-5

18-8

11-3 w.

2 P.M.

43-1

12-3

30-8 E.

2 A.M.

6-5

15-8

9-3 w.

3 P.M.

30-6

9-6

21-0 E.

3 A.M.

6-2

11-0

4-8 w.

4 P.M.

21-2

13-7

7-5 E.

4 A.M.

6-8

10-9

4-1 w.

5 I'.M.

18-3

4-0

14-3 E.

5 A.M.

10-2

7-0

3-2 E.

OF THE MAGNETIC DECLINATIOX AT POET KEXNEDT. 659

We have in this Table the opportunity t)f perceix-ing how effectually the disturbance-
diunial variation may operate in masking or disfiguring the regular progression of the
solar-diiu'nal variation, when the disturbances are not eliminated. It is well known that
the solar-diurnal variation produces generally in the extratropical parts of the northern
hemisphere a maximum deflection to the East about 8 a.m., and a maximum deflection
to the West about 2 p.m. (and this is found to be the case at Port Kennedy, as well as
elsewhere, when the disturbances are eliminated). Now the Table shows that the East-
erly extreme about 8 a.m. must be considerably more than doubled by the occurrence at
the same hour of a large easterly disturbance-deflection, and that at 2 p.m. the Easterly
disturbance-deflection has become so large as to far more than compensate the effect
of the usual amount of the solar-diurnal westerly maximum belonging to that hour,
making the joint deflection at that hour a considerable easterly one. It will also be
seen that when the two kinds of variation are left unseparatcnl, their joint effect produces
a large nocturnal maximum of westerly deflection, which disappears in the solar-diurnal
variation when the disturbances ai-e eliminated.

Solar-diurnal Variation. — The almost extreme difference in the normal direction of
the magnet, geographically considered, at Port Kennedy and Point Barrow, gives a more
than ordinary importance to the comparison of the facts of the solar-diurnal variation
at the two stations, rendering it an apt illustration, in an almost extreme case, of the
laws by which this class of phenomena is regulated. Magnetically speaking, the mean
direction of the magnet is necessarily the same at the two stations ; that is to say, the
mean pointing of the marked end of the magnet, which we usually term its north pole, is
to the magnetic north ; but in a geographical sense this direction is at Port Kennedy
about 35° to the West of South, and at Point Barrow about 41° to the East of North.
The localities afford therefore in this respect a contrast neai'ly as great as can exist in
any part of the globe ; since magnetically the directions are the same, whilst geogra-
phically they want only 6° of being 180° apart, or diametrically opposite. The value
of this contrast appears when we proceed to consider the facts of the solar-diurnal vari-
ation at the two stations, and perceive rightly their important bearing on the correct
understanding of its true natm-e and character, and of the physical relations which
must be involved in any well-grounded explanation.

To prevent, as far as may be possible, misconception in the minds of those to whom
the subject is not familiar, it may be premised that in speaking of the direction of the
magnet the eye of the observer is here supposed to be at its middle, and directed towards
the marked or north end ; a change of direction towards the magnetic cast will thus be
to the observer's right, and a change towards the west to the observer's left. Now the
most marked features of the solar-diurnal variation, and which are found to prevail
imiversally in all the extratropical parts of the northern hemisphere, are, an extreme
deflection to the observer's right (or towards the magnetic east) about 8 A.M., and an
extreme deflection to the observer's left (or towards the magnetic west) about 2 p.m.

4 u2

C60 MAJOR-GENEEAL SABINE ON THE RESULTS OF HOURLY OBSERVATIONS

Whenever the phenomena are viewed within the aforesaid limits, the facts thus referred
to are identical when ea'pressed maxjnetically.

This description of the solar-diurnal variation, in which all geographical relations are
put aside, applies with equal correctness to the phenomena at Port Kennedy and at
Point Barrow ; but when geographical relations are again introduced, the same pheno-
mena have to be described in a very different manner, and the two stations become
widely distinguished from each other*. The marked end of the magnet, when looked
at at 8 A.M., is seen at Port Kennedy to have moved from its mean position of S. 35° W.
towards the geographical Wet^t, and at Point Barrow to have moved from its mean position
of N. 41° E. towards the geographical East ; and correspondingly at 2 p.m. the marked end
is at Port Kennedy to the geographical East of its mean position, and at Point Barrow
to the geographical West. The bearing of this distinction between the magnetical and
geographical aspects of the facts upon physical explanations will be evident if we advert
to the hypothesis of currents of thermic origin, either in the earth or in the atmosphere,
generated by the rotation of the earth in presence of the sun. It may be well therefore
to take a more general view of the phenomena of which the two stations which have
been here compared present a particular case, which fortunately is a very notable and
instructive one. Let us imagine (as in the woodcut in the next page) two stations, a
and h, botli situated in the vicinity of the dip of 90° ; and (to avoid questions of abso-
lute and local time) let us assume them to be in the same geographical meridian, a being
situated to the geogra])hical north, and b to the geograpliical south of the locality where
the dip is 90°. Then at both stations the magnet, when in its mean position, will point
magnetically north ; but at a this direction will be geographically south, whilst at b it
wil) be geographically (as well as magnetically) north. Let us next consider the direc-
tion which the magnet will be found to have assumed at a and b respectively when at the
extreme points of opposite deflection due to the solar-diurnal variation. These synchro-
nize everywhere (as far as is yet known) in the extratropical parts of the northern hemi-
sphere, approximately Avith the local hours of 8 a.m. and '2 p.m. (which for convenience
we will call precisely 8 a.m. and 2 p.m.), — at 8 a.m. the north pole of the magnet being
everywhere to the observer's right, or to the magnetic east of its mean position, and at
2 p.m. to the observer's left, or to the magnetic west of its mean position. At a the mag-
netic east is geographic west, and vice versd ; while at b the geographic and magnetic
cast are the same ; tliereforc at 8 a.m. the magnet is deflected geographically to the west
at a and to the east at b, and at 2 p.m. geograpliically to the east at a and to the west
at b. I^et us next consider the direction in which tlie north end of the magnet moves
between 8 a.m. and 2 p.m. at a and at i: at both stations the movement is from the
observer's right to his left, from the magnetic east to the magnetic west. But at a this
movement, viewed geographically, is from west to east, whilst at b it is geographically

• Tliroughoiit thi.s di.scussion regarding tbo 8u!ar-diurual viiriulion, the di.stui-bauces arc, of course, assumed
to have been eliminated.

OF THE MAGNETIC DECLINATION AT PORT KENNEDY.

661

as well as magnetically from east to west. Whilst, therefore, the direction of the move-
ment is magnetically the same, geographically it is opposite.

Now let us take two other stations c and d, c to the east and d to the west of 90° of

dip, and both situated, not as in the woodcut (where, for convenience in illustration,
they are separated by a considerable meridional interval), but in its vicinity, so that the
distance between them may not be such as to make an important difference in their local
time. The mean direction of the north pole of the magnet will necessarily be at both
stations to the magnetic north ; but at c this will be to the geographic west, and at d to
the geographic east. At 8 a.m. the deflection will be to the observer's right, or magnetic
east at both stations ; but this will be at c to the geographic north, and at d to the geo-
graphic south ; whilst at 2 r.M. the deflections at both stations will be to the observer's
left or magnetic west ; but this will be at c to the geographic south, and at d to the
geographic north. As in the former case, the direction of the movement between 8
A.M. and 2 p.m. is magnetically the same, but geographically opposite.

Now, keeping these facts in view, let us imagine a circle to be drawn round the point
of 90° of dip, passing through a, b, c, and d ; at every point in the periphery of that
circle the mean direction of the marked end of the magnet will be magnetic north, but
will have every possible diversity oi geographical direction. At 8 a.m. the deflection due
to the solar-diurnal variation will be everywhere to the magnetic east, and at 2 p.m. to
the magnetic west; whilst at both hours it will have at different points in the pei'iphory
of the circle every possible diversity of geographical direction. Likewise the movement
from 8 A.M. to 2 p.m. will be, at every point in the periphery, from the magnetic east to
the magnetic west, whilst geographically it will have every possible diversity.

It is ob\ious that what is here stated of points taken in the periphery of the circle is
equally true of every point taken in the interior of the circle, until the point of 90° of
dip is so nearly approached as to render the horizontal magnet absolutely astatic.

The facts of the solar-diurnal variation at Port Kennedy and Point Barrow, after the
elimination of the larger disturbances, furnish a practical exemplification of the justice
of this description in all its details.

662 MAJOE-GEXEEAL SABIXE OX THE EESULTS OF HOUELY OBSEEVATIOXS

The magnitude of the distiubance-diurnal variation at Port Kennedy and Point Bar-
row, compared with that of the solar-diurnal variation about the hoiu's when the diumal
inequality (which is the resultant of the two variations combined) is at its extreme
eastern and western limits, affords an instructive example to those who employ the mag-
nitude of the diurnal range in different years as a means of tracing the epochs of maxi-
mum and minimum of the magnetic variation in the decennial period. Referring to
Table IV., we iind that at 8 a.m., the usual hour in Europe of the easterly extreme of
the dim-nal inequality, that extreme is augmented at Port Kennedy by a disturbance-
deflection amounting, on the average of the five months during which the observations
were maintained, to above 15' easterly; whilst at 2 p.m., the usual hour of the westerly
extreme, there is the counteracting influence of a disturbance-deflection, which is still
easterly, exceeding 30'. Now, as both these values, 15' and 30', very considerably exceed
the ordinary deflections caused by the regular solar-diurnal variation, either to the East
at 8 A.M. or the West at 2 p.m., it is obxious that, at stations where the energy of the
distiirbing force is considerable, the magnitude of the diurnal range at such stationt
must be mainly influenced by and dependent on the amount and houi's of the disturbance-'
diumal variation. Indeed, when we duly consider the extreme liability to variation in
these last-named circumstances, we shall be prepared to find that, as magnetical researches
are extended, stations present themselves where the effect of the increase of the amount
of disturbance at the epochs of maximum of the decennial period is to cause the com-
bination of the two variations to exhibit in such years a decrease instead of an increase
in the magnitude of the diurnal range — actually causing the epochs of maximum and
minimum in the cycle to apparently change places with each other ; in such cases the
miuima of the range of the diumal inequality will coincide with the maxima of the
sun's spots and of the magnetic disturbances ; whilst other stations Avill be found where
the difference between the epochs will be apparently increased in amount ; and others
where it will be obliterated, and no cycle be traceable by this method of inquiry.

The method of tracing the epochs of maximum and minimum of the decennial period
by a comparison of the aggi-egate values of the disturbing action in different years, as
shown by the separation and analysis of the disturbances themselves, is not subject to
the inconvenience which has been thus noticed : it has also the advantage that the pro-
portionate increase in the amount of disturbance between the epochs of maximum and
minimum, 2-5: 1 (St. Helena Observations, vol. ii. p. cxxxi), is much greater than the
difference in the range of the diurnal inequality, or of the solar-diurnal variation, and
forms therefore a larger basis upon which the judgment may be grounded.

The Tabic (IV.) shows the veiy large amount of the average disturbance-diumal
variation at certain hours ; the comparison of a similar Table prepared in the same way
from the obsen^ations at Nertchinsk from 1851 to 1857 inclusive will show (when the
Table shall be published) the liability at different stations to extreme variation in the
direction of the disturbance-diumal variation at the several hours of local time. At
Nertchinsk, at 8 a.m. theWesterly disturbance-diumal variation is nearly at its maximum,

OF TUE IMAGNETIC DECLINATION AT PORT KENNEDY. 663

being a deflection in the contrary sense to that at Port Kennedy, as well as to that of
the solar-diurnal variation at botli ])laces; whilst at 2 p.m. the average disturbance-deflec-
tion is almost null (the westerly being about to pass into tlie easterly). There is there-
fore no counterbalance to the diminution which has been eftected in the 8 a.m. extreme ;
and thus at Nertchinsk the diurnal inequality is lessened by the effect of the disturb-
ances, and is necessarily most lessened at the epoch when the disturbances are greatest.
In this as in many other instances, we see how liable those are to mislead themselves,
who disregard the advice contained in the Royal Society's Report of 1840, to eliminate
the distm-bances as the first and necessary step in the analysis of the complicated phe-
nomena which constitute the " dim-nal inequality."

In the observations which have supplied the subject-matter of this communication,
the Royal Society will recognize another instance, added to the many which have pre-
ceded it, of the zeal and devotion with which recommendations proceeding from the
Society are carried out by our naval officers. Even those who have not themselves
experienced an arctic climate may readily imagine that it is no slight effbrt to maintain
with the requisite regularity, for several months together, hourly observations which
have to be made at a considerable distance from the ship, exposed to the severity of an
arctic winter. I venture to think that such a service is well entitled to our thankful
recognition.

INDEX

PHILOSOPHICAL TRANSACTIONS

FOE THE YEAR 1863.

AiKY (G. B.). On the Strains in the Interior of Beams, 49.

On the Diurnal Inequalities of Terrestrial Magnetism, as deduced from observations

made at the Royal Observatory, Greenwieh, from 1841 to 1857, 309.
First Analysis of One Hundred and Seventy-seven Magnetic Storms, registered by the

Magnetic Instruments at the Royal Observatory, Greenwich, from 1811 to 1857, 617.
Archeopteryx, 33 (see Owen).
Atmolysis, 396.

B.

Beale (L. S.). On the Structure and Formation of the so-called Apolar, Unipolar, and Bipolar Nerve-
cells of the Frog, 543. — Conclusions, 568; explanation of plates, 570.

Boole (G.). On the Differential Equations of Dynamics. A sequel to a Paper on Simultaneous
Differential Equations, 485.

Brodie (SirB. C). On the Peroxides of the Radicals of the Organic Acids, 407. — Preparation of peroxide
of barium, 408; peroxide of benzoyl, 409; of cumenyl, 413; of acetyl, 413; of butyl, 415; of
camphoryl, 417.

BuNSEN (R.) and Roscoe (H.E.). Photo-chemical Researches. — Part V^ On the Direct Measurement
of the Chemical Action of Sunlight, 139.

c.

Calculus of symbols, 517.

Cayley (A.). On Skew Surfaces, otherwise Scrolls, 453.
MDCCCLSill. 4 X

666 INDEX.

Chambers (C). Ou tbe Nature of the Suu's Magnetic Action upon tbe Earth, 503. — Note by Pro-
fessor W. Thomson, 515.

Chemical constitution, relation of, to the refraction, dispersion, and sensitiveness of liquids, 325 (see
Gladstone).

Conducting-power of thallium and iron {electric), influence of temperature on, 3G9.

Cotamine, composition of, 345 ; decompositions and derivatives of, 359 ; constitution of, 366.

Crookes (W.). On Thallium, 173. — Occurrence, extraction, &c., 173; physical characteristics, 178;
chemical properties, 184; position of thallium amongst elementary bodies, 187; analytical noteSj
189.

D.

Dale (T. P.) and Gladstone (J. H.). Researches on the Refraction, Dispersion, and Sensitiveness of

Liquids, 317 (see Gladstone).
Debus (H.). On some Compounds and Derivatives of Glyoxylic Acid, 437.
Depth of sea, deduced from tidal observations, 252, 260.
Differential equations of dynamics, 485 (see Boole).
Diffusion of gases, 385.
Dispersion of liquids, 317 (see Gladstone).

E.

Elastic force of air, variation of, between 32° Fahr. and 212° Fahr., 434 (see Stewart).
Elastic solid, general theory of the equilibrium of an, 610.
Elastic spheroidal shells, problems regarding, 583.

Foster (G. C.) and Matthiessen (A). Researches into the Chemical Constitution of Narcotine, and

of its Products of Decomposition, Part 1., 345 (see Matthiessen).
Friction between a wave and a wave-shaped solid, 134,

G.

Ganglion-cells, formation, &c. of, 543.

Gases, molecular mobility of, 385.

Gladstone (J. 11.) and Dale (T. P.). Researches on the Refraction, Dispersion, and Sensitiveness of
Liquids, 317. — Relation between sensitiveness and change of volume by heat, 319; refraction and
dispersion of mixed liquids, 323 ; refraction, dispersion, and sensitiveness of different members of
homologous series, 325; refraction, dispersion, and sensitiveness of isomeric liquids, 331 ; effect of
chemical substitution on these optical properties, 333.

Glyoxylic acid, compounds and derivatives of, 437.

Graham (T.). On the Molecular Mobility of Gases, 385.

Greenwich magnetic observations, discussion of, 309 (see Airy).

Greenwich Observatory, analysis of magnetic registrations at the, 617.

INDEX. 667

H.

Haughton (S.). On the Reflexion of Polarized Light from Polished Surfaces, Transparent and Metal-
lic, 81.

On the Tides of the Arctic Seas, 243. — Part I. On the diurnal tides of Port Leopold,

North Somerset, 243; Part II. The semidiurnal tides, 256.

Hirst (T. A.). On the Volumes of Pedal Surfaces, 13.

Hypogallic acid, 355.

I.

Indian meteoroTogy , numerical elements of, 525 (see Schlaointwkit).
Iron, influence of temperature on the electric conductiag-power of, 369.

K.

Kew Observatory, discussion of magnetic observations at, 273 (see Sabink).

M.

M'^Clintock (Sir F. L.). Discussion of Magnetic Observations made by, 649.

Magnetic action of the sun upon the earth, nature of the, 503.

Magnetic observations at Greenwich, discussion of, 309 (see Airk).

Magnetic observations at the Kew Observatory, discussion of, 273 (see Sabine).

Magnetic observations at Port Kennedy, discussion of, 649.

Magnetic storms registered at the Greenwich Observatory, analysis of, 617.

Mass of the moon, deduced from tidal observations, 260.

Matthiessen (A.) and Foster (G. C). Researches into the Chemical Constitution of Narcotine, and

of its Products of Decomposition, Part I., 345. — -Composition of narcotine and cotarnine, 345 ;

decompositions and derivatives of opianic acid, 351 ; of cotarnine, 359 ; conclusion, 362.
Matthiessen (A.) and Vogt (C). On the Influence of Temperature on the Electric Conducting- Power

of Thallium and Iron, 369.
Meconin, 352 ; analysis of, 353; action of hydriodic acid, &c. on, 355.
Mercury, temperature of the melting-point of, 435.
Meteorology of India, 525 (see Sculagintweit).

Narcotine, constitution of, 345 (see Matthiessen).
Nerve-cells of the frog, structure and formation of the, 543.

0.

Opianic acid, decompositions and derivatives of, 351.

Owen (Professor). On the Archeopteryx of von Meyer, with the description of the Fossil Remains

of a Long-tailed species, from the Lithographic Stone of Solenhofen, 33. — Explanation of the

plates, 46.

4x2

6G8 INDEX.

P.

Pa.vy (F. W.). On the Immunity enjoyed by the Stomach from being digested by its own' Secretion

during Life, 161.
Peroxides of the radiails of t/ic on/auic act/Is, 407 (sec Brodie).
Photo-chemical researches, 139 (see Buxsen).
Port Kenned'j, discussion of magnetic observations at, 049.

R.

Raxki.ve (W. J. ^I.). On the Exact Forui of Waves ncartiic Surface of Deep Water, 127. — Appendix:

on the friction between a wave and a wave-shaped soHd, 13k
Rejlciion of polarized lifjht, 81 (see IIvvgiiton).
Refraction of liquids, 317 (sec Gladstone).
Tligidittj of the earth, 573.

lloscoE {{{. E.) and Bunsen' (R.). Photo-chemical Researches, 139 (see Bunsen).
Ri'ssEi.L (W. II. L.). On the Calcuhis of Symbols.— Third Memoir, 517.

Sabine (E.). Results of the Magnetical Observations at the Kew Observatory, from 1857 and 1858 to
18G2 inclusive.— No. I. 273 ; No. II. 290.

Picsults of Hourly Observations of the Magnetic Declination made by Sir Francis Leo-
told ^rCLiXTOCK, and the Officers of the Yacht ' Fox,' at Port Kennedy, in the Arctic Sea, in the
Vt inter of 1858-5!) ; and a Comparison of these Results with those obtained by Captain RocHroRT
IVIaguire, and the Officers of Her Majesty's Ship 'Plover,' in 185.2, 1853, and 1854, at Point
Barrow, G19.

ScHLAFLi (Dr.). On the Distribution of Surfaces of the Third Order into Species, in reference to the
absence or presence of Singular Points, and the reality of their Lines, 193.

ScHLAGiXTWEiT (II. de). Numerical Elements of Indian Meteorology, 525. — Calculation of the daily
mean, 525; Tables of 207 stations, 530; decrease of temperature with height in the tropics, 538;
thermal ty])es of the year and the seasons, 539.

Scrolls, or skew surfaces, 1.53.

Sensitiveness (f liquids, 317 (sec Gladstone).

Skew surfaces, 453.

Sphere, oscillations of a li(piid, G08.

Standard jmper for phot o-ehcmical observations, preparation of, 155.

Stewart (B.). An Account of Experiments on the Change of the Elastic Force of a Constant Volume
of Atmospheric Air, between 32'^ F. and 212° F., and also on the Temperature of the Melting-
point of I\Icrciiry, 125.

Stomach, immuTiity of tlic, from digestion during life. 1(11.

Strains in the interior of beams, 19.

Surfaces of the third order, distribution of, 193 (see Sciil.uli).

Surfaces, pedal, 13.

T.

Thal'ium, 173 (sec Crookes) ; 369 (sec Matthiessen).

1 X D E X. 669

TuoMSOX (W.). Note relative to the nature of tlic Sun's Magnetic Action upon tlic Kurtli, 515.

On the Rigidity of the Earth, 573. — Effect of the earth's elastic yielding on the

tides, 575 ; note on the fortnightly tide, 581 ; appendix, 582.

Dyuaniical Problems regarding Elastic Spheroidal Shells and Spheroids of Incom-
pressible Liquid, 583. — Oscillations of a liquid sphere, G08; .Vppendix, general theory of the
equilibrium of an elastic solid, 610.

Tidal obsfiralions at Port Leopold, North Somerset, 202.

Tides nf the arctic seas, 213 (see IIavghtojj).

Tides, effect of the earth's elastic yielding on the, 575.

Tyndall (J.). On the Uelation of Radiant Heat to Aqueous Vapour, 1.

V.

Vapour, aqueous, relation of radiant heat to, 1.

VoGT (C.) and M.vttiiiessen (A.). On the Influence of Temperature on the Electric Coiiducting-powcr
of Thallium and Iron, 369.

W.

Wares, exact form of, 127 (sec Rankine).

LONDON:
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Columbus : — Fifteenth Annual Report of the Ohio State Board of Agriculture, The Board.

for 1S60. 8vo. Columhus 1S61.
Dijon : — Memoires de rAcademie Impc'riale dcs Sciences, kria et BeUos-Let- The Academy.

tres. 2°"^ Serie. Tome IX. Annc'e 1801. Svo. Dijon 1862.
Dublin :—

Geological Society. Journal. Yol. IX. Part 2. Svo. DiMin 1862. The Society.

Eoyal Dublin Society. Jom-nal. Xos. 21-29. Svo. DuhVui 1862-63. The Society.

Eoyal Irish Academy. Transactions. Yol. XXIV. Part 2. 4to. 2)i(?-///^ 1862. The Academy.
Edinburgh : —

Eoyal Society. Transactions. Yol. XXIII. Part 1. 4to. S^mJio-r/Zj 1S62. The Society.

Proceedings. Session 1861-62. Yol. lY. Xo. 56. Svo.

Edinhurgh 1862.
Eoyal Scottish Society of .\j-ts. Transactions. Yol. YI. Part 2. Svo. Edhi- The Society.

lirrrjh 1862.

Falmouth : — Eoyal Corn-wall Polytechnic Society. Twenty-ninth Annual Ee- The Society.

port. 1861. Svo. Falmouth 1862.
Franldbrt :— Abhandliuigen herausgegeben von der Senckenbergischcn Xatur- The Society.

forschenden GeseUschaft. Band lY. Lief. 2. 4to. Frankfurt a. M. 1863.
Geneva : — Memoires de la Societe de Physique ct d'Histoii-e XatiireUe. The Society.

Tome XYI. Partie 2. 4to. Gtneve 1862.
Gottingen : —

Abhandlungcn der Kciniglichen GeseUschaft der \Yissenschaften. Band X. The Society.

4to. Gottingen 1862.
Xachrichten von der Georg-Augusts-rniversitiit und der Konigl. GoseU-

schaft der Wissenschaftcn. 1802, Xos. 1-27. Svo. Gottingen.
Haarlem : — Xatuurkundige Yerhandehngen van de HoUandsche llaatschappij The Society.

der Wetensehappen. Tweede YcrzameUng. Deel. XYI., XYII. i' XIX.

Stuk 1. 4to. Haarlem 1862.
HaUe: — Zeitschrift fiir die gcsammten XatuiTvisscnschaftcn, herausgegeben Tlie Tnion.

von dem Xaturw. Yereine fiir Sachsen u. Thiiringen in Halle. Band

XYIII. Hefte 7-12; Band XIX. Hefte 1-6. Svo. Berlin 1801-62.
Kazan : — Imperial Eussiau rniversity. Outehonia Zapiski (Scientific Papers) :

1834, Parts 1 & 2 ; 1835 to 1861, Parts 1 & 2 : 26 vols, and 4 Parts Svo.

Ditto: 1848, Part 4; 1850, Part 4; 1851, Part 4; 1852, Parts 1 & 4;

1854, Part 3; 1856, Parts 3 i 4: 7 Yols. 4to. Kazan 1834-61.
Kiel:— Schriften der Universitat. Biinde Yll. & YIII. 4to. Kiel 1801-02.
Klausenburg : —

Erdclyi Tcirtenelnii Adatok, Iviadja az Erdc'lji Iluzeiim-Egyesiilet. Kotet

lY. Svo. Kolozsvdrt 1862.
Az Erdelyi iluzcimi-Egj-let Evkonyvei. Kotet II. Fiucet 1. 4to. Kolozsvdrt

1862.
Erdelyi Orsz. lliizc-um Xaptara az 1863 dik Kozonseges esztendtire. Svo.

Koloz-ivdH 1863.
Kiinigsberg :■ — Scliriften der Kilniglichcn Physikalisch-Oekonomischen GeseU- The Society.

schaft. Jahrgang UI. 1862, Abth. 1 &. 2. 4to. Kdniysbcn/ 1803.

The University, by Professor
Bolzani.

The University.
The Museum.

[ ^ ]

Presexts. Dojions.

ACADEMIES and SOCIETIES (confimuil).

Lausanne : — Bulletin do la Sociote Yaudoisc des Sciences Xatiirellcs. Tomo YII. The Society.

Xos. 4S & 49. Svo. Lausanne 1SG1-G2.
Leipzig : —
Kiinigl.-Siichsischcn Gcsellschaft der Wissenschaften. Berichto. Math.- The Society.
Phys. Classe, ISGl, I. & IL ; Phil.-IIist. Classe, 1861, II., m. & ly. Svo.
Leipzig 1862.

. Dark'giing der tlieoretischcn Bcrechnung

dor in den Mondtafcln angowandtcu Storungcn, von P. A. Hansen : erste
Abhandlung. Svo. Ldpzig 1802.
. Mcssungen iiber die Absorption der chc-

mischen Strahlen dcs Sonnenlichtcs, von W. G. Hankel. Svo. Leipzig
1862.
. Prcisschiiftcn. IX. : Bohmert, Bcitriige zur

Geschichte des Zunftwesens. Svo. Leipzig 1862.
. Die Deutsche Nationalokonomik an dor

Granzscheide des Scchzehntcn und Siebzehnten Jahrhunderts, von "W.
Eoscher. Svo. Leipzig 1862.
. Locke's Lehre von der mensclilichen Er-

kenntniss, in Yergleichung mit Leibniz's Kritik dersclber, dargcstcllt

von G. Hartenstein. Svo. Leipzig 1S62.
Preisschriften gekrdnt und hcrausgcgcben von dor FiirstHch-Jablonowski'- The Society.

schen Gesellschaft. XI. Svo. Leipzig 1863.
Liege : — Mcmoircs dc la Soeiote Royalc des Sciences. Tomo XYII. Svo. Liege The Society.

1863.
Lisbon : —

Memorias da Academia Ileal das Scicncias. Classe dc Scicncias Mathcma- The Academy.

ticas, Physicas e Naturaes. Xova Serle. Tomo II. Partes 1 & 2. 4to.

Lishoa 1857-61.
Classe de Scicncias Moracs,

Politicas c Bellas Lettras. Xova Serie. Tomo II. Parte 1. 4to. Lishoa

1857.
Annaes das Scicncias e Lettras, publicados debaixo dos Auspicios da Aca-

demia Ileal das Scicncias. Scicncias Mathcmaticas, Physicas, Historico-

Natnraes e Medicas. Tomo I. July 1857 to February 1858 ; Tomo II.

March 1858 to July 1858. Scicncias Moraes, Politicas e Bellas Lettras.

Tomo L July 1857 to February 1858 ; Tomo II. March 1858 to Novem-
ber 1858. Svo. Lishoa 1857-58.
Lendas da India por Caspar Correa. Livro primeiro, Tomo I. partes 1 & 2 ;

Livro segundo, Tomo II. partes 1 & 2, 4 vols. 4to. Lishoa 1858-61.
Portugalliaj Monumenta Historica. Scriptores : Yol. I. fasc. 2 <fe 3. folio Oli-

sipone 1860-61. Leges et Consuetudines : Yol. I. fasc. 2. folio OUsipone

1858.
Liverpool : —

Historic Society of Lancashire and Cheshire. Transactions. Xew Series. The Society.

Yol. n. Svo. Liverpool 1862.
Literary and Philosophical Society. Proceedings. Xo. 16. Svo. Liverpool The Society.

1862.

[ 6 ]

Presents.
ACADEMIES and SOCIETIES {continued).
London : —

Anthropological Society. Anthropological Ecview. jS^o.-I. 8vo. London
1863.

Board of Trade. Meteorological Department, lleport, 1SG2. Svo. London
1862.

Meteorological Paper.s. No. 11. Svo. London 1862.

British Association, Eeport of the Thirty-first Meeting, held at Man-
chester in September 1861. Svo. London 1862.

British Meteorological Society. Proceedings. Vol. I. Nos. 1-7. Svo. London
1862-63.

Eleventh lleport of the Council for 1861.

Svo. Lundon 1S6].

On the Meteorologj^ of England during

1801. by J. Glaishcr, F.R.S. Svo. London 1S61.

List of the Members, 1862. Catalogue of

the Librarj', 1862. Svo. London.
Briti-sh Museum. Description of the Collection of Ancient Marbles. Part 11.

4to. London 1861.
Select Papyri iu the Hieratic Character, fi'om the Collec-
tions of the Biitish Museum. Part 2. I'lates 1-19. folio London 1860.
Chemical Society. Journtd. Vol. XV. Nos. 7-12, July to December 1862.

New Series. Vol. I. Nos. 1-6, January to June 1863. Svo. London.
Congres International de Bienfaisance. Session de 1862. 2 vols, in 1. Svo.

Londres 1S63.
Entomological Society. Transactions. New Series, Vol. V. Parts 6-11.

Third Series, Vol. I. Parts ]-4. Svo. Loiuhn 1860-62.
Geological Society. Quarterly Joui'nal. Nos. 69-74. Svo. ixiw?o/i 1862-63.

List and Charter and Bye-Laws. Svo. London 1862. Third Supplement

Catalogue of the Library, 1860-63. Svo. London 1863.
Institution of Civil Engineers. Minutes of Procccduigs. Vol. XX. Svo.

London 1861.
Linnean Societj-. Transactions. Vol. XXIII. Part 3 ; Vol. XXIV. Part 1.

4to. London 1863.
. Journal of the Proceedings. Nos. 24-26. Svo. London

1802-63. List of Fellows, 1802. Svo.
London University Calendar. Svo. London 1863.
National Association for the Promotion of Social Science. Transactions,

1862. Svo. London 1803.
Ophthalmic Hospital, lleports and Journal of the Royal London Oj)hthalmic

Hospital, edited by J. F. Streatfield. Nos. 1-13. Svo. London 1857-60.
Pathological Society. Transactions. Vol. XIII. Svo. London 1862.
Photograpliic Society. Journal. Vol. VIIT. Nos. 123-134, December to

June. Svo. London 1862-63.
Iloyal Agricultural Society. Journal. Vol. XXI II. and Vol. XXIV. Parti.

Svo. London 1802-03.
Royal Asiatic Society. Journal. Vol, XIX, Part 4 ; Vol. XX. Parts 1 & 2.

Svo. London 1862-63,

Donors.

The Society.
Board of Trade.

The Association.
The Society.

The Ti-ustees.

The Society.
Henry Roberts, Esq.
The Society.
The Society.

The Institution.
The Society.

The University.
The Association.

The Editor.

The Society.
The Society.

The Society.

The Society.

[ ' ]

Pbesents. Donors.

ACADEMIES and SOCIETIES (condniud).
London : —

Royal Astronomical Society. Memoirs. Vol. XXX. 4to. London 1S62. The Society.

Monthly Notices. Vol. XXII. Nos. 5-9; Vol. ■■ —

XXIII. Nos. 1-6. 8vo. Lowlon 1862-63.

Royal College of Physicians. List of the Fellows, Members, Extra-Licen- The College.

tiates, and Licentiates. 8vo. London 1862.
Royal Geographical Society. Journal. Vol. XXXI. 8vo. London 1861. The Society.

Proceedings. Vol. VI. Nos. 2-5 ; Vol. VII.

Nos. 1 ik 2. 8vo. London 1862-63.

lloyal Horticultural Society. Proceedings. Vol.11. Nos. 7-12; \'m1. III. Tlic Society.

Nos. 1-6. 8vo. London 1862-63.
Royal Institute of British Architects. Papers read 1861-62. 4to. London The Institute.

1862. List of Members, &c. 4to. London 1862.
Royal Institution. Notices of the Proceedings at the Meetings of the The Institution.

Members. Nos. 35-37. 8vo. London 1862-63. List of Members, 1862.

8vo.
Royal Medical and Chirurgical Society. Medieo-Chirurgical Transactions. The Society.

Vol. XLV. Svo. London 1862.

Proceedings. Vol. IV. Nos. 2 & 3.

Svo. London 1862.

Royal Society of Literatui'e. Tran.sactions. Second Series. Vol. VII. The Society.

Part 2. Svo. London 1862.
St. Bartholomew's Hospital. Descriptive Catalogue of the Anatomical The Governors.

Museum. Vol. III. Svo. London 1862.
Society of Antiquaries. Arehaeologia. Vol. XXX'VTilT. Part 2. 4to. London The Society.

1862.

Proceedings. Second Series. Vol. I. Nos. 2-7.

Svo. London 1860-61. Lists of Members. Svo. London 1861-62.

Society of Arts. Journal. Vol. X. Nos. 501-521 ; Vol. XI. Nos. 522-551 . The Society.

Index to Vols. I.-X. Svo. London 1862-63.
United Service Institution. Journal. Vol. VI. Nos. 21-25, and Appendix. Tlie Institution.

Svo. London 1862-63.
University College. Calendar for the Session 1862-63. S\o. London 1S&'2. The College.
Zoological Society. Proceedings of the Scientific Meetings. 1801, Part 3 ; The Society.

1862, Parts 1-3. Svo. London.
■ Transactions. ^Vol. IV. Part 7. Section 2. 4to. London

1862.
Luxembourg : — Societe des Sciences Naturellcs. Tome V. 1857-02. Svo. The Society.

Luxembourg 1862.
Madrid: — Anuario del Real Observatorio, cuarto afio: 1863. 12mo. Madrid The Observatory.

1862.
Manchester : —

Literary and Philosophical Society. Memoirs. Third Series. Vol. I. Svo. The Society.

London 1862.
Proceedings. Vol. I. No. 15, Title

and Index ; Vol. II. Svo. Mmchester 1860-62.
Rules. Syo. Manchester ISQl.

[ 8 ]

The Royal Society of Tic-
toria.

Peesexts. Donoes.

ACADEMIES and SOCIETIES {confinued).

Mannheim : — Astronomische Bcobachtimgen aiif der Grossherzoglichcn Stem- The Obsen"atory.

■warte, angesteUt und herausgegeben Ton Dr. E. Schonfeld. Abth. 1. 4to.

MdnnJieim 1862.
Mauritius : —

Meteorological Society. Proceedings. Vol. Y. Svo. Jlauritius ISGl-Oi. The Society.

Eoyal Society of Arts and Sciences. Transactions. New Series. Vol. II. The Society.

Part 1. (Two copies.) Svo. Mauritius 1861.
Observations on the Watcr-Supply of ■

Mauritius. By Captain J. I!. Mann. bvo. JJaurititis IMJU.
Melbourne : —

Eoyal Society of Victoria. Transactions, from January to December ISGO.

Vol. V. Svo. Melbourne 1861.
Transactions of the Pliilosoi)hical Institute of Victoria. Vol. II. Part 2.

Svo. Milbourne 1858.

Catalogue of the Victorian Exhibition, 1861. Svo. Melbourne 1861.

Essais divers servant d'Introduction au Catalogue dc I'Exposition des Pro-

duits dc la Colonic de Victoria. Svo. Melhourne 1861.
Die Colonic Victoria in Australien, Lhi- Fortscluitt, ihi'c Ililfsquellcn und ihr

physikaJischer C'harakter. Svo. Melbourne 1861.

The Victorian Government Prize Essays, 1860. Svo. Melbourne 1861.

Statistical Eegister of Victoria, from the foundation of the Colony. Svo.

Melbourne 1854.
The .Catalogue of the Melljournc Public Library ftjr 1861. Svo. Melbourne

1861.
Statistical Xotes on the Progress of Victoria from the foimdation of the

Colony (1835-1860). By AV. H. Archer. First Scries. Parts 1 & 2. 4to.

Melbourne 1861.
Second Eeport of the Board of Visitors to tlie Observatories, Victoria, fol.

Melbourne 1861-62.
MUan: —

Memorie del Rcale Istituto Lombardo di Scienze, Letterc cd Arli. Vol. VIII. The Institute.

fasc. 6 & 7; Vol. IX. fuse. 1. 4to. Milano 1861-62.

Atti. Vol. II. fasc. 15-20 ; Vol. III. fasc. 1-4, 9 & 10. 4to. Milano ]8(;2.

Atti dt'Ua Societa Gcologica residente in Milano : Vol. I., 1855-5!.». Atti The Society.

della Societa Italiana di Scienze Xaturali : Vol. II., 1850-00; Vol. III.,

1861; Vol. IV., 1862. Svo. Milano 1859-63.
Montreal : —

Natural History Society. Canadian Xaturalist and Geologist. Vol. VII. Tlic Society.

Xos. 3-6. Svo Montreal 1862.
Numismatic Socict)'. Constitution and By-Laws. Notes on Coins, by The Society.

S. C. Bagg. 12mo. Montreal 1863.
Moscow: — Bulletin de la Societe Iniperiale des Naturalistes. Annee ISGl. The Society.

Nos. 1-4. Svo. J/osco!« 1861.
Munidi : —

Abhandlungen der Matliemat.-Physikalischen Classc dcrKoniglich-Baycris- The Academy.

chen .i\Jiademic der 'Wisseaschaftcn. Band IX. Abth. 2. 4to. MUnchen

1802. . -

[ « ]

Pkesexts. Donoks.

ACADEMIES and SOCIETIES (continued).
Munich : —

Abhandlnngen der Historischcn Classc. Band IX. Abth. 1. 4to. Miinchen The Academy.
1802.

Sitzungsberiehtc : 1862, I. Hefte 1-4 ; 18(52, II. Hefte 1-4 ; 1S63, I. Hcftc

1 & 2. 8vo. Miinchen 1862-03.

Verzeichniss der Mitglicder. 4to. Miinchen 1862.

Ueber Parthenogenesis: Yortrag von C. T. E. von Sicbuld. 4to. Miinchen

1862.

Zum Gediichtniss an Jean Baptiste Biot, von C. F. P. von Martius. 4to.

Miinchen 1S02.
Annalcn der kiiniglichcn Stcmwartc. Band II. Svo. Miinchen 1802.
Naples : — Societa Reale. Eendiconto deU' Accademia delle Scienze Fisichc c

Matematiehe. Ease. 1-4. May- August 1802. 4to. NapoVi.
Neuchatel : — Bulletin de la Societe dcs Sciences Xaturelles. Tome YI. premier

cahier. Svo. Neuchatel 1862.
Newcastle-upon-Tyne : — Tynesidc Naturalists' Field Club. Transactions.

Yol. Y. Parts 3 & 4. Svo. Neiucastle 1862-03.
New York : —
Forty-fourth Annual Keport of the Trustees of the New York State Librarj-.
Svo. Albany 1862.

Seventj'-fifth Annual Keport of the Regents of the University of the State

of New York. Svo. Albany 1862.

Thirteenth and Fifteenth Annual Reports of the Regents on the Condition

of the State Cabinet of Natural History. Svo. AVxtny 1860-62.

Report of the Regents on the Longitudes of the Dudley Observatory, the

Hamilton College Observatory, the City of Buffalo, the City of Syracuse.
Svo. Albany 1862.
Paris : —

Comptcs Rendus de TAcademie des Sciences. Tome LIY. Nos. 23-24 ; The Institute.
Tome LY. Nos. 1-26 ; Tome LYI. Nos. 1-23. Table des Matieres, Tome
LIY. 4to. Paris 1862-63. Supplement, Tome H. 4to. Paris 1861.

Memoires de I'Academie des Sciences. Tome XXXIII. (Chevroul sur les •

Couleurs). Paris 1861.

Cerdcs Chromatiqucs. 4to. Paris

1862.

The Observatory.
The Society.

The Society.

The Club.

New York State Library.

Memoires pre'sentes par divers Savants a I'Academie des Sciences. Sciences

mathematiques ct physiques. Tomes XYI. & XYII. 4to. Paris 1862.
Notices ct Extraits des Manuscrits do la Bibliotheque Impe'riale ct autres

bibHotheques. Tome XY. Table ; XIX. 1^ partie ; XX. 2' partie. 4to.

Paris 1801-62.
Memoires do I'Academie des Inscriptions et BeUes-Lettres. Tome XX.

1" partie; XXIY. 1' partie. 4to. Pans 1861.
Memoires de divers Savants. Premiere Serie, Tome YI. 1*^^ partie. Dcuxieme

Serie, Tome lY. 1'' partie. 4to. Paris 1860.
Depot de la Marine: — Aunales Hydrographiques, 1-4'' Trimestre de 1801';

1-4" Trim, de 1862 ; I'' Trim, de 1863. Svo. Paris 1861-63.
Recherches Chronometriques. Cahiers Y. & YI. Svo. Paris 1861-02.
MDCCCLXIII. h

Lc Depot dc la Marine.

[ 10 ]

Presents. Donoes.

ACADEMIES and SOCIETIES (conHnufd).
Paris: —

Annuaire des Marees dcs Cotes de France, 18()2, 1803, 1S04. Svo. Paris Lc Depot de la Marine.

1861-62.

Eoutier de la Bale de Fimdv et de la Nouvelle Eeosse. 8vo. Paris 1861.

Le vrai principe de la Loi des Oiiragans. Svo. Paris 1861.

Description Hydrogi-aphique de la cote orientalc de la Coree. 8vo. Paris

1S61.
Eenseignements jS'autiqucs surles Cotes de Patagonie. (Extraits des Ann.

Hydrog.) Svo. Paris 1862.
Avis aux Xavigatcurs. Nos. ;UG, 317, 310, 320, 323 & 324. Svo. Paris

1861-62.
Instnietions Xautiques, par J. Horsbnrgb. Pai'tics I. & II. 4to. Paris

1861-62.

Eoutier de I'Australe, par A. Le Gras. Vol. II. 8vo. Paris 1861.

Des Ouragans, Tornados, Typhons et Tempetes, par F. A. E. KeUer. Svo.

Paris 1861.
Instructions !Nautiqucs sm- les Cotes dlslandc, par Barlatier de Mas. Svo.

Paris 1862.

Instnietions sur I'isle de Crete ou Candie, par Spratt. Svo. Paris 1861.

Manuel de la Navigation dans la Mer des Antilles et dans le Golfe du Mcxique,

par P. de KerliaUet. Parties I. & II. Svo. Paris 1862.
Description Hydrographique des Cotes Septentrionales de la Eussie, par M.

Eeineke. 2^ Partie. Svo. Paris 1862.
Instructions Nautiques sur le Sound de Harris et le Petit Minch. Traduit de

I'AnglaLs, par A. Le Gras. Svo. Paris 1862.
Campagne do la Cordeliere, par le V"* Fleuriot de Langle. Svo. Paris

1862.
Nouveau Manuel de la Na\-igation dans le Rio de la Plata, par E. Mouclicz.

Svo. J'arls 1S62.
Guide pour Tusage des Cartes des Vents et dos Courauts du Golfe de Guinee,

par M. de Brito CapeUo. Svo. Paris 1862.
Table Clironologique de quatre cents Cyclones, par A. Poey. Svo. Paris

18(12.
Eenseignements Hjdrographiqucs ct Statistiqucs sm- la Cote de Svi-ie, pai-

M. Desmoidins. Svo. Paris 1S62.
Guide du Marin et du Caboteur sur les Cotes Est de la Mcr du Nord. &c.

Traduit de I'ouvrago Anglais de Norie, par P. Guerj-. Svo. Paris

1863.
Piloto du (Jolfc et du Fleuve Saint Laurent, par H. W. Bayiield. Traduction

par A. Le Gras. Partie 1. Svo. Paris 1863.

Eoutier de la Cote Sud ct Sud-Est d'AMque. Svo. Paris 1803.

Forty-seven Maps and Charts.

Annales des Mines. 0*^ Sdrie, Tomes I. & 11. , 2-(;""' livr. de 1802 ; Tome III., L'Ecole des Mines.

1 & 2""' livr. de 1863. 8vo. Paris.
Institut des Provinces, des Societes Savantes et dcs Congrcs Soicntifii]ues. The Institute.

Annuaire. 2' Serie. Vol. IV. Svo. Paris 1862,
Institut Egj^ticn. Bulletin. Annee 1862. No. 7. Svo. Paris 1862. The Institute.

[ 11 ]

Presents. Donobs.

ACADEMIES and SOCIETIES (contintud).
Paris: —
Societe de Biologic. Comptes Rendus des Seances et Memoires. 3' Seric. The Society.

Tome II. 8vo. Paris 1861.
Societe d'Encouragement pour I'lndustrie Nationale. BuUetin. Tome IX. The Society.

Nos. 112-120; Tome X. Nos. 121-124. 4to. Paris 1862-03.
Societe de Geographic. Bulletin. 5" Serie. Tomes lU.&IV. 8vo. Paris 1802. The Society.
Societe Geologique de France. Bulletin. 2° Serie. Tome XVIII. feuiUes The Society.
44-52, Table ; Tome XIX. feuilles 13-68. 8vo. Paris 1862.

Notice sur la Vie et les travaux de P. L. A. Cordier, par le C"' Jaubert. 8vo.

Paris 1862.
Philadelphia : —
Academy of Natural Sciences. Journal. New Series. Vol. V. Part 3. 4to. The Academy.
Philadeljjhia 1862.

Proceedings. Nos. 7-12. 8vo. PliiladelpMa

1862.

American Philosophical Society. Transactions. Vol. XII. Parts 1-3. 4to. The Society.

Philadelphia 1862-63.
Proceedings. Vol. VTI. No. 64 ; Vol.

Vm. Nos. 65-68. Svo. Philadelphia 1860-«2.
Franklin Institute. Journal. Nos. 438-449; Vol. XLIV. Nos. 4-6; VoL The Institute.

XLV. Nos. 1 & 2. Svo. Philadelphia 1862-53.
Prague : —

Abhandlungen der Konigliehen BiJhmischen Gesellschaft der Wissenschaften. The Society.

Fiinfte Folge, zwolfter Band. 4to. Prag 1863.

Sitzungsberichte. Jahrgang 1862, Jan.-Dec, 2 Parts. Svo. Prag 1S62.

Rome: —
Atti deU' Accademia Pontiflda de' Nuovi LinceL Anno XIll. Sess. 5-7 ; The Academy.

Anno XTV. Sess. 1-2. 4to. Boma 1860-61.
Bullettino Meteorologico daU' Osservatorio del CoUegio Romano, compUato The College.

dal P. Angelo Seechi. Vol. I. Nos. 6-24, Title Page and Index; Vol. II.

Nos. 1-7. 4to. Eoma 1862-63.
St. Petersburg: —

Academic Imperiale des Sciences. Memoires. Tome IV. Nos. 1-9. 4to. The Academy.

St. PeUrsbourg 1861-62.

BuUetin. Tome IV. Nos. 3-6. 4to.

St. Petersbourg.

Stockholm : —

Kongliga Svenska Vetenskaps-Akademiens Handlingar. Ny Foljd, trcdje The Academy.
Bandet, andra Hiiftet, 1860. 4to. Stockholm 1862.

Ofversigt af K. Vetenskaps-Akademiens Forhandliagar. Argangen XVIII.

Svo. Stockholm 1862.

Meteorologiska Jakttagelser i Sverige, andra Bandet 1860. 4to. Stockholm

1862.
Sydney : — Entomological Society of New South "Wales. Transactions. Vol. I. The Society.

Part 1. Svo. Si/dne,j 1863.
Toronto : — The Canadian Journal of Indnetry, Science, and Art. New Series. The Institute.
Nos. 34, 36-44. Svo. Toronto 1861-63.

52

[ 12 ]

Presents. Donors.

AC-iDEMIES and SOCIETIES (eontimml).

Toidouse : — ilemoires de rAcademie Imperiale des Sciences, Inscriptions ct The Academy.

BeUes-Lettres. 5* Serie. Tome YI. Svo. ToitloKse 1862.
Upsala: —

Is'oTa Acta Eegia; Societatis Scicntiamm Upsaliensis. Scrici tertiai Vol. IT. The Society.

fase. 1. 4to. Q)sn7«e 1862.
rpsalaUniversitetsArsskrift, 1861. Thoologi; PliUosoplii, Spnlkvetenskap The University.

och Histoiiska Yetenskaper ; Mathcmatik oeh Naturvetenskap ; Mcchcin ;

Eatts- och Stats-Tetenskaper. 5 Parts. Svo. Upsala 18G1.
Utrecht :—

Aanteekeningen van het Ycrhandelde in de Sectie-Yergaderingen van het The Society.

Provinciaal Utrechtsch Genootschap van Kiinsten en "\Yetensehappen,

1850-61. 10 Parts. Svo. Utrecht 18.51-61.
Yerslag van het Ycrhandelde in do Algemecne Yergadering, 1860-61.

2 Parts. Svo. nm-;(C 1860-61.
Inhouds-Opgave der ^Yerken van het Provinciaal Utrechtschc Genootschap.

Svo. Utrecht.
Oironologisch llcgister op het Ycrvolg van hot Groot-Charterboek van Yan

Mieris. Svo. Utrecht 1859.

Xatmirkimdige Yerhandelingcn. Dcel I. Stuk. 1 & 2. 4to. Utrecht 1862. ■ — — •

Koninklijk Xederlandsch Meteorologisch Instituut. Storm Kaart aangevende The Institute.

do verschiUende "NYindrigtingen, -waaruit do Stormen gewocd hebben,

alsmede de Maandchjksche Stormprocentcn. Utrecht 1862.
Meteorologische A\'aarnemingen in Nederland en zijne Bezittingcn 1861.

4to. Utrecht 1862.
Vcrzameling van Kaarten iahoudende eene proccntsgewijze Opgave omtrent

Storm, Regen, Bonder en Slist, grootendeels getrokken nit do jongste

Waamemingen onzer Ncderlandische ZeeUcden als uitkomst van Weten-

sehap en En-aring aangaandc \Yinden en Zeestroomingen in sommige

gedeelten van den Oceaan. fol. Utrecht 1862.
Venice : —

Jlemorie deU" I. P. Istituto Ycneto di Scienzc, Lettcrc ed Arti. Yul. X. Tlie Institute.

parti 2 & 3. 4to. J'cnezia 1862.
Atti. Serie terza. Tomo YII. disp. 4-10; Tomo YIII. disp. 1-3. Svo.

Vinezla 1861-63.
Vienna : —

Denksehriften der Kaisorhchen Akadcmie der "Wisscnschaftcn. Phil. -Hist. The Academy.

Classc. Pand XII. 4to. WUn 1862.
Sitzungsberichte. Math.-Xatiirw. Classe, erste Abtheilung, Band XLY.

Hefte 1-5 : zweite Abth., Band XLY. Heftc 1-5 ; Band XLYI. Hette

1 & 2. Phil.-Uist. Classe, Band XXXYIII. Heft 3 ; Band XXXIX.

Hefte 1-5 ; Band XL. Hefto 1 & 2. Svo. Wien 1862.

Almanach. Zwiilfter Jahrgang. 1862. 12mo. Wien.

Central- Anstalt fiir ileteorologie uiid Erdmag-nctismus. Uobersichten der The Institute.

"Wittorung in Oesterreich und cinigen auswiirtigen Stationcn im Jahre

1860. 4to. Wien 1S61.
Jahrbuch der k.-k. Geologischen Peichsaustalt. 18(n und 1862, XII. Bund, The Institute.

Kos. 2-4; 1863, XIII. Band, Xo. 1; General-Ilegistcr der ersten zchn

Bande. Svo. Wien 1862-63.

[ 13 ]

PitKSEXTS.

ACADEMIES and SOCIETIES (continued).
Vienna : —

Mittheilungcn dor k.-k. Gcographischen GescUschaft. V. Jahrgang, ISGl.

8vo. Wien 1801.
Verhandlungcn dos Zoologisch-Botanisehcn Vcreins. Biinde TII-IX. XT. &

XII. 8vo. Wlen 1853-02.
Bericht iibcr die Oestorreicliische Literatiir dcr Zoologic, Botanik und Talii-

ontologic aus den Jahren 18.50, 1851, 1852, 1853. 8vo. Wieti 1855.
Personen- Orts- und Sach-llegisterder fiinf ersten Jahrgiinge (1851-1855)

dor Sitzungsbcrichte, und Abhandlungen des Wiener Zoologisch-Bota-

nischen Yereines (1850-1800). 8vo. Wim 1801-02.
Kachtxiige zu Maly's Enumeratio I'lantarum phanerogamiearum Imperii

Austriaci universi, von A. Xeilreich. 8vo. Wien ISGl.
Bericht iiber die zwcite allgcmeinc Versammlung von Berg- und Iliitten-

miinnorn (21 bis 28 Sept. ISGl). 8vo. Wioi 1802.
AVashington : —

Annual Keport of the Board of llegcnts of the Smithsonian Institution,

1800. 8vo. Washhigton 1861.
Catalogue of Publications of the Smithsonian Institution, corrected to Juno

1802. Svo. Washhigton 1862.
Wiirzljurg : —

Wiir/.burger Medicinischc Zeitschrift, herausgegeben von der Physikalisch-

Medicinischcn GeseUschaft. Band III. Hefte 1-6 ; Band IV. Hcftcl&2.

Svo. Wiirzhurff 1862-03.
Wiirzburgcr Xaturwisscnschaftliche Zeitschiift. Band III. Hcfte 2-4. Svo.

Wiivzhurg 1802.
Zurich : —

XeujahrsbUittcr der Xaturforschenden GeseUschaft. Stiicke 1-04. 4to. Zurich

1709-1862.
Bericht iiber die Verhandlungcn der Xaturforschenden GeseUschaft. 1825-

20, 1832-36 & 1830-;3-. 3 Parts. 12mo. Zaricli 1826-38.
Mctcorologisehc Beobachtungen angestellt auf Veranstaltung. 1837—40.

10 Parts. 4to.
MittheUungcn. Heft 10. Xos. 119-131. 8vo. Zilrkh 1856.
Vierteljahrsschrift. Jahrgang VII. Hefte 1 & 2. Svo. Zurkh 1802.
Katalog der Bibliothek. Svo. Zurich 1855.

Denkschrift zur Feier des hundertjiihrigen Stiftungfcstcs d(>r Xaturfor-
schenden GeseUschaft, Nov. 30, 1846. 4to. Zdrich 1840.
Xouvcaux ilc'moires do la Societe Helvetique des Sciences XatureUes. Band

XIX. 4to. Ziirich 1846.
Compte-Ilendu de la 45" Session, re'unic a Lausanne Ics 20, 21 et 22 Aoiit

1861. Svo. Lausanne 1861.
AilMAXUS (J.) Stirpium rariorum in Imperio Ilutheno sponte provenientium

Icones et Descriptiones. 4to. Petropoli 1 739.
AXOXTMOUS :—

Ane'roVdes (Les). Svo. Paris 1861.

Astronomical and Meteorological Observations made at the United States
Xaval Observatory dming 1861. 4to. Washimjton 1802.

Donors.

The Society.
The Union.

The Society.

The Institution.

The Society.

The Socictv.

The Society.

J. Hogg, F.E.S.

The Author.
The Observatory.

[ 14 ]

Presents.
ANOIsTTMOUS (continued).

Catalogue of the Maps and Plans and other Publications of the Ordnance

Survey of England and Wales. Svo. London 1S62.
Catalogue of the ilaps and Plans and other Publications of the Ordnance Suri-ey

of Scotland. Svo. London 1S63.
Catalogue of the Nova Scotian Department. Internation;il Exhibition, 1S62.

Svo. Halifax 1S62.
Chemical (A) Eeview, by a B. Svo. London 1863.
Correspondence between the Society of Antiquaries and the Arlmu'alty

respecting the Tides in the Dover Channel with reference to the Landing

of Caesar in Britain, b.c. 55. 4to. London 1863.
Correspondence of Scientific Men of the Seventeenth Century, printed from

the Originals in the Collection of the Earl of Macclesfield. 2 vols. Svo.

Oxford \dA^.
Cursory Thoughts on some Natui-al Phenomena. Svo. New York 1862.
EntomologLSt's Annual for 1863. 12mo. London 1863.

International Exhibition, 1862. Report on Class X., Section A. Civil Engi-
neering and Building Contrivances. Svo. Lomlon 1862.
Judicial Statistics, 1861. England and Wales. 4to. London 1S62.

Materiaux pour la Carte Geologique de la Suisse, pubUcs par la Commission
Geologique de la Societe Helve tique des Sciences NatureUes. Livr. 1. 4to.
Neuchatel 1863.

Atlas. I'^livr. Carte Geolo-
gique du Jura Balois, par A. Miiller, 1862.

Nautical Almanac and Astronomical Ephemeris for 1865 & 1S66. 2 vols. Svo.
London 1861-62.

Ephemcridcs of the Minor Planets for 1863. Svo. London 1863.

Ninth Report of the Postmaster General on the Post Office. Svo. London
1863.

Proceedings of the Commissioners of Indian Afl'airs appointed by law for the
extinguishment of Indian Titles in the State of New York, published from
the Original MS. in the Library of the Albany Institute, with an Intro-
duction and Notas, by F. B. Hough. 4to. Albany 1861.

Eeport of a sad case^ recently tried before the Lord Mayor, Owen v. Huxiet,
in which will be found fuUy given the merits of the great recent Bone Case.
Svo. London 1863.

Report of the Commissioners of Patents for 1860, Arts and Manufactures.
2 vols. Svo. Washinijton 1861.

for 1861, Agriculture. Svo. Wash-
ington 1802.

Report of the Royal Commission on the Operation of the Acts relating
to Trawling for Herring on the Coasts of Scotland, fol. Edinburgh
1863.

Report of the Superintendent of the United States Coast Survey. Appendices
Nos. 21, 22 «& 42. 4to. Washington.

Report on the Geology of Vermont, by E. Hitchcock, A. D. Hager, C. H. Hitch-
cock. 2 vols. 4to. CJarcmont, N. H. 1861.

DONOES.

Sir H. James, F.R.S.

Unknown.

W. Francis, Esq.

The -Society of Antiquaries.

The Delegates of the Uni-
versity Press.

The Author.

H. T. Stainton. Esq.

Sii- John Eennie, F.R.S.

The Secretary of State for

the Home Department.
The Societj-.

The Lords of the Admiralty.

Sii- R. HiU, F.R.S.
The Albany Institute.

John Marshall, F.R.S.

The Patent Office, Wash-
ington.

Secretary of State for the
Home Department.

Tlie Coast Survey Office,

Washington.
The State Government.

[ 15 ]

Prese-vts.
AIs'OXTMOrS («)»i<i»iu«f)-

Report upon the Colorado River of the West, explored in 1857 and 1S5S by
Lieut. J. C. Ives. 4to. WasJilnf/ton 1S61.

Report upon the Physics and Hydraulics of the Mississippi River, by Capt.
.\. A. Humphreys and Lieut. H. L. Abbot. 4to. Philadelphia 1861.

Russian Translation of the Admiralty Manual on the Deviations of the Compass.
Svo. St. PeUrsbiirff 186,S.

Statistical, Sanitaiy and Medical Reports for ISGO. Army Medical Depart-
ment. Svo. London 1862.

Statistical Reports on the Sickness, Mortality and Invaliding among the Troops
in the West Indies ; in Western Africa, St. Helena, Cape of Good Hope
and Mauritius ; in Ceylon, Tenasserim Provinces, and Burmese Empire ; in
the United Kingdom, Mediterranean and British America. 5 parts, fol.
London 1838-53.

Wissenschaftlichen (Die) und praktischen Erfolge der Novara-Expedition. Svo.

ANSTED (D. T., F.R.S.) The Correlation of the Natural History Sciences.
The "Rede" Lecture, delivered May 12, 1863. Svo. London 1863.

AVEZAC (D'.) Restitution de deux passages du texte grec de la Geographic
de Ptolemee aux ehapitres V. et VI. du septieme livre. Svo. Paris
1862.

BACHE (A. D., For. Mem. U.S.) Discussion of the Magnetic and Meteorolo-
gical Observations made at the Girard College Obser^-atory, Philadelphia,
1840-45. Part 1. 4to. Washington 1859.

— • Lecture on the Gulf Stream. Svo. Neii) Haven

1860.

Influence of the Moon on the declination of

the Magnetic Needle. Svo. New Haven 1861.
On Declinometer Observations.

(Excerpts

from the Amer. Joum.) Svo. New Haven 1861.
BATES (H. W.) The Naturalist on the River Amazons, a Record of Adventures

during eleven years of Travel. 2 vols. Svo. London 1863.
BEATSON (A.) Tracts relarive to the Island of St. Helena, written during a

residence of five years. 4to. Loiulon 1816.
BEKE (C. T.) A few Words with BLshop Colcnso on the subject of the Exodus

of the Israelites and the position of Mount Sinai. Svo. London 1862.
— Who discovered the Sources of the NUe '? a Letter to Sir Roderick

I. Murchison. Svo, London 1863.
B6R0N (P.) Meteorologie Simplifiee et Telegraphes sans fils ct sans cables.

Svo. Paris 1863.
BERTHELOT (— .) et PEAN DE SAINT GILLES. Recherches sur les

Aifinitcs ; de la Formation et de la Composition des fithers. Svo. Paris

1862.
BESSEL'S Hypsometric Tables, as corrected by Plantamour, reduced to English

Measures and re-calculated by A. J. EUis. (Excerpt from Meteor. Papers,

Board of Trade.) Svo. London 1863.
BIANCONI (G. G.) DeK Galore prodotto per I'attrito fra Fluidi e Solidi in

rapporto colic Sorgenti Termali e cogli Acroliti. Svo. Bohfjna 1862.

Donors.

The War Department
United States.

Capt. Belavcnitz, R.I.N.

Dr. T. Graham Balfour,
F.R.S.

The I. R. Nary Board,

Vienna.
The Author.

The Author.

United States Coast Survey.

The Author.

The Author.
J. Hogg, F.R.S.
The Author.

The Author.
The Author.

Admiral FitzRoy, F.R.S.

The Author.

[ 16 ]

Presents. Donors.

BLVXCOXI (J. J.) Spccimina Zoologica Mosambicana. Fasciculus XV. 4to. The Author.

Bononkc 1862.
BOECK (W.) Eecherches sur la Sr[)lulis, appuyccs dc Tableaux do Statisticjue The University, Clmstiania.

tires des ArchiTes des Hopitaux de Christiania. 4to. Christiania 1802.
BOEEHAATE (H.) Index alter Plantarum tjuoe in Horto Academico Lugdimo- J. Hogg, F.E.S.

Batavo aluntur. 4to. Lnfjd.-Bai. 1720.
BOXD (G. P.) On the Eesults of Photometric Experiments upon the Light of The Author.

the Moon and of the Planet Jupiter, made at the Observatory of Harvard

CoUegc. 4to. Camhrhhje ISGl.
BEIOSCHI (F.) Sulla risolvcute di Malfatti per le equazioui del quinto grado. The Author, by Professor

4to. MUano 1S0:J. Cayley, F.E.S.

BEOW'X (S.) On the Eate of ilortality and Marriage amongst Eiu-opcans in The Author.

India. (Exccrjit fi-om Assurance Mag.) Svo. London.
BUEDEE (W. C.) The Meteorology of Clifton. Svo. Bristol 1863. The Author.

CALLEXDEE (G. W.) Anatomy of the parts concerned in Femoral Euptiu'e. The Author.

Svo. London 1863.
C'AMPAXI (G.) Eapporto del 8cgretario Generale del Congresso degli Scienziati The Author.

Italiani per la sczione delle Scienze Fisiche, Matematiche o Naturali. 4to.

Siena 1S62.
CAETEE (H. J., F.E.S.) On Contributions to the Geology of AVestern India. Tlie Author.

Svo. ISOO.

Fiu'ther Observations on the Structure of Foraminifera

and on the larger Fossilized Forms of Sind, ifcc. Svo. 1861.

Index to Original Papers and Compilations liy H. J.

Carter. (Excerjits from Jom'n. Bombay Branch of the E. As. Soc.) Svo.

1861.
CAYAXLI (.1.) Memou'c sui' la theorie de la Eesistance Statique ct Dynamique The Autlior.

des SoUdes. 4to. Turin 1863.
CHELIXI (D.) Delia Leggc onde im EUissoido cterogenco propaga la sua Attra- The Author.

zione da Punto a Punto. 4to. Bolnr/nti 18t;2.
COOKE (E. "W., F.E.S.) Tiews of the Old and Xew London Bridges. (Proofs.) The Author.

fol. London 1S33.
COEBKJT (J. B.) Ecvue sur Ic Systeme d'lnoeulations curatives du Dr. Telephc The Author.

Desmartis. Svo. Bordeaux 1S62.
CEACE-CALYEET (Dr. F., F.E.S.) Lectures on Cod-Tar Colours and on recent The Author.

improvements and progress in Dyeing and Calico-printing. Svo. Alanchatter

1863.
CEEMOXA (L.) Introduzione ad una Teoria d'eomctrica delle Cun'e Piano. 4to. Tlie Author.

Bolofjna 1862.
DAUBEXY (C, F.E.S.) Ecmarks on tlie recent Eruption of Vesuvius in De- The Author.

cemher 1S61. Svo. 1S62.
Climate : an Inquirj- into the causes <jf its differences, ■ — •

and into its influence on Vegetable Life. F'our Lectures. Svo. Oxford

1m;:!.
DL I.A l;i\'I-: (A., For. Mem. U.S.) ^'oiiveUes Eecherches sur les Aurorcs The Author.

Boreales et Australes. 4to. Geneve 1862.
DELI'SSE ( — .) Carte Agronomique des Environs de Paris. Svo. Paris The Author.

Ib62.

[ 1' ]

PUKSENTS.

DE MORGAN (A.) On the Syllogism A'o. 5, and on various Points of the Ony-

matie System. (Excerpt from Trans. Camb. Pliil. Soc.) 4to. CambrUh/e 180:J.
DEYINCE'S'ZI (G.) On the Cultivation of Cotton in Italy, lleport to the

Minister of Agriculture of the Kingdom of Italy. 8vo. London 18(>2.
BEWALQUE (G.) Notice sur le systeme Eifelien dans le Bassin de Xiiiiiur.

Svo. BnixeUes.
DIRCKS (H.) Pei-pctuum Mobile, or History of the scardi for Kcll'-motivc

power, during the 17th, 18th, and l!)tli Centuries. Svo. London 18(il.

Contribution towards a History of Electro-Metallurgy, establish-
ing the Origin of the Art. Svo. London 1863.

DOYE (H. W., For. Mem. U.S.) The Law of Storms, considered in connexion
with the Ordinaiy Jlovements of the Atmosphere. Translated by 11. H. Scott.
Svo. London 1SU2.

DOWLE.LXS (A.M.) Official Classified and Descriptive Catalogue of the Con-
tributions from India to the London Exhibition of 1 8(52. 4to. Calcutta 1802.

DUEAXD (Andre.) La Toscanc, Album Pittorcs(]ue et Archeologicjuc. Li-
vraison 1-. lie d'Elbe. fol. Paris 18G2.

DURAXD (F. A.) Theorie Electrique du Froid, de la Chaleur et de la Lumierc
(Doctrine do I'Unite des Forces physiques). Svo. Paris 18G3.

]':MM0XS (E.) and HALL (.1.) Xatural History of New York. Part Y. Agri-
culture, by E. Emmons. Yol. Y. -ito. Afhanii 18'yi. Part YI. Palaiontology,
by James Hall. Yol. III. Part 1. Text; Part 2. Plates. 4to. Albany
1859-61.

ERXESTI (J. 0.) Initia Doctrine Solidioris. 12mo. Li^nke 1783.

EVANS (F. J., F.R.S.) and SMITH (Arch., F.R.S.) Admiralty Manual for
ascertaining and appl)-ing the Deviations of the Compass caused by the Iron
in a Ship. Svo. London 1862.

FERGUSSOX (E. F. T.) and MITCHESOX (P. W.) Magnetical and Jfetcoro-
logical Observations made at the Government Observatory, Bombay, in 1800.
4to. Bombaij 1801.

FERGUSSOX (J., F.R.S.) An Historical Inquiry into the true Principles of
Beauty in kti, more especially with reference to Architecture. Svo. London
1849.

An Essay on a proposed new System of Fortification,

with Hints for its application to our Xational Defences. Svo. London 1849.

Observations on the British Museum, Xational

Gallery, and Xational Record Office, with Suggestions for their improve-
ment. Svo. LowJon 1849.

Illustrated Handbook of Architecture. 2 vols. Svo.

Loiulon 18-55.

FITZROY (Admiral, F.R.S.) The Weather Book : a Manual of Practical Mete-
orology. Svo. London 1803.

Second edition. Svo. Lon-
don 1803.

FUCHS (J. X. Y.) Gesammelte Schriften, redigiit und mit einem Xekrologie

versehen von C. G. Kaiser. 4to. Miinchen 1850.
GARDXER (R.) Figures illustrating the Structure of Yarious Invertebrate

Animals (Mollusks and ^irticulata). Svo. London 1800.
MDCCCLXIII. C

Donors.
The .\uthor.

The Aullior.

Tlie .Vuthor.

The Author.

Admiral FitzRoy, F.R.S.

UnknoAvn.

The Prince Dcmidoff.

The Author.

The Regents of the New
York University.

J. Hogg, F.R.S.
The Authors.

The Secretary of State for
India.

The Author.

The Author.

Dr. C. G. Kaiser.
The Author.

[ 18 ]

Presents.
GETHEE (A.) Gedaiikon iiber die Nahu-kraft. Svo. OMenhun/ 1862.
GIEALDES (J. A.) Notice siir la Vie et les Travaiix do Sii- Benjamin C. Brodie.

Svo. Paris 1863.
GOrLD (B. A.) Standard Mean Right Ascensions of Circumpolar and Time

Stars. 4to. Washington 1862.
GEAHAil (J. D.) Annual Eeport on the Improvement of the Harbors of

Lakes llichigan, St. Clair, Erie, Ontario, and Champlain, for 1860. Svo.

Wasliiwjton 1860.
GROYE (W. R., F.R.S.) The Correlation of Physical Forces. Fourth edition.

Svo. London 1862.
GIDIPACH (J. von) The Figure and Dimensions of the Earth. Second edition.

Svo. London 1862.
GrTBERLETUS (H.) Chronologia. Editio tertia. 12mo. Amsldodami 1657.
HAIDINGER (W., For. Mem. R.S.) Considerations on the Phenomena attend-
ing the fall of Meteorites on the Earth. (Excerpt from Phil. Mag.) Svo.

London 1861.
HALL (J.) and WHITNEY (J. D.) Report on the Geological Sui-vey of the

State of Wisconsin. VoL I. Svo. Alhamj 1862.
HANSEN (P. A., For. Mem. R.S.) Einige Bemerkungen iiber die Siicular-

andenmg der mittleren Ltinge des Mondes. Svo. Leipzig 1863.
HARTLEY (Sir C. A.) Description of the Delta of the Danube, and of Works

recently executed at the Sulina Mouth. (Excci-pt from Proc. Inst. Civ. Eng.)

Svo. London 1862.
HEISTERUS (L.) Institutiones aiinu'gica;. 2 vols. 4to. Amstdodami 1739.
HIORTDAHL (T.) and IRGENS (M.) Geologiske Undersogelser i Bergeus

Omegn. 4to. Chrisficmia 1862.
HIRN (G. A.) Exposition Analytique et Experimentale de la Theorie Mecanique

de la Chaleur. Svo. Paris 1862.
HOEYEN (J. van der) Bijtlrage tot de Kennis van den Potto van Bosman. 4to.

Amstcrdrim 1851.
— Bijdragen tot do Ontloedkundige Kennis aangaande

Nautilus Pompilius, L. Yooral met Betrekking tot hot Mannehjke Dicr.

4to.

AniMerdam 1856.

Ober de Taal en de Yergelij kendo Taalljennis inver-

band met de Natuurlijke Geschiedonis van den Mensch. Svo.
■ Berigt omtrent het mij verleondc Ontslag als Opper-

directcur van "s Rijks Museum van Natuui-hjkc Historie te Leiden. Svo.

AmsWrdam 1860.
HOFMANN (A. W., F.R.S.) On Mauve and Magenta: a Lecture. Svo. London

1862.
HOGG (J.. F.R.S.) On the supposed Scriptiu'd Names of Baalbec or the Syrian

Heliopohs. Svo. Lowlon 1862.
On somo Inscriptions from Cj'prus, copied by Commander

Lcyccster, R.N. Svo. Lowlon 1 862.
HOOD (W. C.) Statistics of Insanity, embracing a Report of Bcthlem Hospital

from 1846 to 1860 inclusive. Svo. London 1862.
HOPE (A. J. B. Beresford) The World's Debt to Art: a Lecture. Svo. London

1863.

DONOES.

The Author.
The Author.

The Author.

The Author.

The Author.

The Author.

J. Hogg, F.E.S.
The Author.

The State Government.
The Author.
The Author.

J. Hogg, F.E.S.

The University, Christiania.

The Author.

The Author.

The Author.
The Author.

The Author.
The Author.

[ 19 ]

Presents.
HOPE (A. J. B. Berosford) Tho Condition and Prospects of Architectural Art.

8vo. London 1863.
HOSKIXS (S. E., F.K.S.) Charles 11. in the Channel Islands : a Contribution

to his Biography and to the History of his Age. 2 \oU. 8vo. London 1854.
HUNT (J.) Introductory Address on the Study of Anthropology. Svo. London

1863.
HUXLEY (T. H., F.E.S.) Evidence as to Man's Place in Nature. Svo. Loiulon

1863.
JAilES (Colonel Sir H., F.E.S.) Abstracts fi'om the Meteorological Observations

taken at the Stations of the Eoyal Engineers in 1853-59. 4to. London

1802.

Abstracts from the Meteorological Observations

taken in the years 1860-61 at the Eoyal En^ecr Office, New Westminster,
British Columbia. 4to. London 1802.

Extension of tho Triangulation of the Ord-

nance Survey into France and Belgium, with the Measiuemcnt of an Arc of

Parallel, in Latitude 52° N., from Valentia in Ireland to Mount Kemmel in

Belgium. 4to. London 1802.
JEVONS (W. S.) A serious Fall in the Value of Gold ascertained, and its social

Effects set forth. Svo. London 1863.
JOUENALS:—

AnnaU di Matematica pura ed applicata, pubblicati da B. Tortolini. Tomo III.

Anno 1860. 4to. Boma 1860.
Astronomische Nachrichten. Biindc LIY.-LVIII. 4to. Altomi 1861-62.
Bullettino Nautico e Geografico di lloma, diretto da E. Fabri-Scarpollini.

Appcndice alia Corrispondenza Scientifica di Roma. 4to. Itoma 1802.
Cosmos. TomeXX.Hvr.25&2e,; TomeXXI.; Tome XXII. Uvr. 1-24. t^vo.

PaWs 1862-63.
Giomale per I'Abolizionc della Pena di Morte : dii-ctto da Pietro Ellcro.

Nos. 4 & 6. Svo. Bologna 1802-63.
Les Mondes. Tome I. livr. 1-18. Svo. Paris 1863.
Scheikundigc Verhandelingen en Onderzoekingen uitgegeven door G. J. Midder.

Derde Deel, tweede Stuk. Svo. liotterdara 1803.
The American Journal of Science and Arts. Nos. 100-105. Svo. Newhavtn

1802-03.
The Athenocum. June to December 1862 ; January to May 1803. 4to. London.
The Atlantis, or Eegister of Literature and Science of the Catholic Univer.sity

of Ireland. Vol. IV. Nos. 7 & 8. Svo. Lo^idon 1863.
The Builder. Nos. 1011-1063. folio. London 1862-03.
The Chemical News. Nos. 135-184. 4to. Lomlon 1802-63,
The Critic. July to December 1862 ; January to June 1863. folio. London.
The Horologieal Journal. Nos. 47-53, 55, 57 & 58. Svo. London 1862-63.
The Intellectual Observer. July to December 1862 ; January to June 1863.

Svo. London.
The Journal of Mental Science, edited by J. C. BuckniU. Nos. 42 & 44. Svo.

London 1862-63.
The London Eeview. July to December 1862; Januaiy to June 1863. folio.

London.

c2

DoNons.
The Author.

The Author.

The Author.

The Author.

The Secretary of State for
War.

The Author.

Professor Tortolini.

The ObseiTatory.
The Editor.

The Editor.

Tlic Editor.

The Editor.

The Ministry of the Interior

of the Netherlands.
The Editors.

The Editor.
The Editors.

The Editor.

The Editor.

The Editor.

The Horologieal Society.

The Editor.

The Editor.

The Editor.

[ 20 ]

Pbesexts. Donors.

JouEx.yis (««?;«!(«?):—

The Mining and Smelting ilagazine, edited by H. C. Salmon. Vol. I. (Xos.l-G). The Editor.

Vol. II. (Xos. 7-12) & Vol. III. Xos. 13-17. 8vo. Lomlon 1862-G3.
The Philosophical Magazine. July to December 18(i2 ; January to June 1SG3. W. Francis, Esq.

Syo. London.
The Eeader. Januaiy to June 1863. foUo. London. The Editor.

Verhandelingen en Berigten betrekkelijk het Zeewezen en de Zeevaartkunde, General 8abiuc, P.E.S.
Terzameld en uitgegeven door Jacob Swart. Jahrgang 18.54-57. 4 vols.
8vo. Awstcnhtm.
KA^sITZ (A.) Sertum flora? territori' Xagy-Korosicnsis. 8vo. Vicnmr 1862. The Author.

Bcmerkimgen uber ciiiige ungarische botanischo Werlce. 8vo.

W!m lb>\2.
KIRKMAX (Key. T. P., F.R.S.-) On a so-caUed Theoiy of Causation. Svo. The Author.

Liverpool 1862.
KOLLIKER (A., For. Mem. R.S.) X'"eue Untersuchungen iiber die Ent^Wcklung The Author,
des Bindegewebcs. Svo. Wiirzhirg 1861.

rntersuchungen iiber die letztcn Endi-

gungen der Xerven. AbhantUimg I. Svo. Leipzig 1S62.
— ■ ■ Ueber das Vorkommen von freien Talgdrii-

sen am rotlun Lippenrande des Menschen. Svo. Leipzig 1861.

KULEXATI (F. A.) Genera et Species Trichoptcrorum. Pars altera. 4to. The Author, by ProfessorW.

J/os^ifrt- 18.39. H. Miller, For. Sec. R.S.
Monographic dor cm-opaischen Chiroptern. Svo. Bi-iinn

1>60.

■ Die forstschiidlichen Insekten. Svo. Brilnn 1860.

KOXIG (C.) and SIMS (J.) .\nnals of Botany. 2 vols. Svo. Lomlon 1805. J. Hogg, F.R.S.

KUPFFER (A. T., For. Mem. R.S.) Annales de I'Obsei-vatoire Physique Cen- The Observatory.

tral de Russie. Anne'e 1859. 2 vols. 4to. St. Petershourg 1862.
L-iJXIOXT (J., For. Mem. R.S.) Ueber die tagliche Oscillation des Barometers. The Observatory, Munich.

8%-o. Miinchen 1862.
LAMY (A.) Do FExistence dun nouveau Metal, le Tliallium. Svo. LiUe 1862. The Author.
LA WES (J. B., F.R.S.) and GILBERT (J. H.. F.R.S.) The Rothamsted Me- J. B. Lawes. F.R.S.

moirs on Agricultural Chcmi.stiy and Physiology. 2 vols. Svo. Lomlon 1863.
■ ; Drawings and Plans of

the Lawes Testimonial Laboratory, Rothamsted, Herts. 4to.
LEA (Isaac.) Description of a new Genius of the Family Melanida^, and of The Author.

Forty-five new Species, ifec. Svo. Philadelphia 1862.
LEIGHTOX (J.) On Japanese Art : a Discourse delivered at the Royal Insti- The Author.

tution. May 1, 1863. folio. London.
LEMOIXE (E. M.) Des Caases premieres de la "\'ie Animale mateiiellemcnt The Author.

deraontrees. Svo. Pecris 1863.
LE VERRIER (U. J., For. Mem. R.S.) Annales de I'Observatoire Imperial de The Observatoiy.

Paris. Tomes III.-V. (1839-44), XYI. &XVIL (1860-61). 5 vols. 4to.

Paris 1862-63.
LLOYD (Rev. H., F.R.S.) On tlie probable Causes of the Earth-Currents. Svo. The Author.

JJidjlin 1862.
LOGAN (Sir W. E., F.R.S.) Report on the Geology of Canada. Svo. Montreal The Author.

1862.

[ 21 ]

Presents.
LOWE (E. J.) Our Native Ferns. Parts V.-XIII. Svo. London.
LUTHER (E.) Astronomische Beobachtungcn nuf dcr koniglithen Universitiits-
Stemwiirte zii Konigsbcrg. folio. Kiiniijsherg 1SG2.

Dcclinationcs StcUanim funilamontaliiim novx ex ultiniis ill'.

Bcsscl observationibus derivatsc. 4to. lierjiomonti 1859.

LYELL (Sir C, F.R.S.) Tlic Geological Evidences of the Antiquity of Man,
with Remarks on Theories of the Origin of Species by Variation. Svo. Lon-
don 1S()3.

3IAACK (R. de) Travels io the Amur, by order of the Geographical Society of
St. Pctcrsburgh. (In Russian.) 4to ; and Atlas in folio. St. Petersbunjh 1859.

Travels along the Valley of the Tssiri. (In Russian.) 2 vols.

4to. St. Petersburg 1861.

M'DONNELL (R.) On the Physiology of Diabetic Sugar in the minimal Eco-
nomj-. Svo.

Expose de quelques Experiences concemant rinfluencc de,s

Agents physiques sur le Developpcment de la GrenouiUe Commune. (Ex-
cerpt from Joum. de la Phys.) Svo. Furls 1859.

Observations on the Anatomy and Physiology of the Kidney.

Svo. Did)lin 1858.

Observations on the Habits and Anatomy'of the Lcpidosiren

annectens. Svo. Dublin ISUO.
On the System of the Lateral Line in Fishes. (Excerpt

from Trans. R. I. Acad.) 4to. nnbUa 1SG2.
On the Formation of Sugar and Amyloid Substances iu the

Animal Economy. Svo.
On the Organ in the Skate which appears to be the homo-

logue of the Electrical Organ of the Torpedo. (Excerpts fi'om Nat. Hist. Rev.)

Svo. 1861.
MACLEAY (W.) Description of Twenty new Species of Australian Coleoptcra

belonging to the Families Cicindelida; and Cetoniida;. Svo. 1862.
3IAGXIN (T. M.) Equations Rectilignes du Cercle, de la Circonfercncc de la

Surface ct du Volume de la Sphere. Svo. Lyon 1855.
MAILLY (E.) Essai sur les Institutions Scientifiques de la Grande-Bretagnc et

de I'Irlande. III. 12mo. Bru.velles 1863.
M.VIN (Rev. R., F.R.S.) Astronomical and Meteorological Obsen-ations made

at the Radcliffe Observatory, Oxford, in 1859 and 1860. Vol. XX. Svo.

Oxford 1862.
MAXENAUER (E.) Erdmagnetismus und NorcUicht. Svo. Innsbrucl- 1S61.
MALAISE (C.) Note sur quclques Ossemcnts humains fossiles ct sur quelques

Silex Tallies. Svo. Briuvelks.
De I'Age des PhyUades Fossiliferes de Grand-Manil. Svo.

Briuvelles. (Excerpts from BuU. Acad. Belg.)
ArAT.T.ET (J. W.) Cotton : the Chemical, Geological, and Meteorological Con-
ditions involved in its successful Cultivation. 12mo. Loiulon 1862.
MALLET (R., F.R.S.) Great Neapolitan Earthquake of 1857. The First

Principles of Obsei-vational Seismology. 2 vols. Svo. ion<?o)t 1862.
Marsh-land Engineering. (Excerpt from Pract. Mech.

Joum.) 4to. London 1862.

DOXOES.

Tlio Author.
The Observator}-.

The Author.

The Author.

The Author.
Tlie Author.

The Author.
The Author.
The Author.
The RadcUffe Trustees.

The Author.
The Author.

Messrs. Chapman and Hall.

The Author.

[ 22 ]

Peesexts.
MAPS, EXGEA'ST:NGS, <fec. :—

Atlas Ecliptique. Cartes Xos. 2, 2 bis, 9, 15, 39 & 46. (Annates de I'Ob.ser-
vatoire Imperial: Atlas.) folio. Paris 1862.

Carte du Mexique, reprc'sentant le plateau de I'Anahuac et son vcrsant orien-
tal, par H. de Saussure. 2 sheets. 1862.

Carte Gcologique des parties de la Savoie, du Picmont, et de la Suisse voisincs
du Mont Blanc, par A. Favre. (In case.) 1862.

Exjjlication de la Carte Gcologique. Svo. Geneve 1862.

Carte Gcologique souterraine de la ville de Paris, cxccutee par M. Delesse,
1853. (Framed.)

Chart of the Curves of equal Magnetic Variation, 1858. By F. J. Evans. F.E.S.
(In case.) Svo.

Lithographic Copies from two Brass Tables contauung Fragments of the Law
relating to the Municipalitv of Malaga. 2 sheets.

Mezzotint Engraving of the Appearance of the Total Solar Eclipse fi-om
Hai-adon HOI, May 11, 1724. By Stukeley.

Three Engi-avings piinted by Photozincography : of the Photograpliic Esta-
blishment of the Ordnance SiuTey Office ; Chapel of the Southampton Inlii--
mary; and Cairo. 1862.

Two Photographs of a fossil hiunan jaw-bone found in the Drift at Abbeville.

MARTH (A.) Memoir on the Polar Distances of the Greenwich Transit-Cii-cle.

4to. AUona lS6iJ.
MILLER (Major-General.) Patrick Miller and Steam Navigation: Letter to

B. Woodcroft, vindicating the right of Patrick Miller to be regarded as the

first inventor of practical Steam Navigation. 12mo. London 18()2.
MILLER (W. H., For. Sec. R.S.) A Tract on Crystallography. Svo. Cambrklje

1863.
MOIGNO (I'Abbe.) Lerons de Calcid DiffJrentiel et de Calcul Integral. Tome IT.

fasc. 1. Svo. Paris 1S61.
MOHET (A.) Klimatographischc Tebersicht der Erde in einer Sammlung

authentischer Berichtc, mit liinzugefiigten Anmerkungen. Svo. Lcipzif)

1862.
MUELLER (F., F.R.S.) Tlie Plants indigenous to the Colony of Victoria.

Vol. I. Thalamiflora;. 4to. Melhoume 1860-62.
MtJLLEE (M.) Rig-Vcda-Sanhita, the Sacred Hymns of the Brahmans. Vol. IV.

4to. London 1862.
MUSSY (C.) NouvcUes Recherches Experimentales siu- rHJterogenie ou Gene-
ration Spontanee. 4to. Toulouse 1862.
ct JOLY (N.) Refutation de I'luie des Experiences capitalcs de

M. Pastcui'. 4to. Paris.
OLIVEIRA (B., F.R.S.) A Few Obsen-ations upon the Works of the Isthmus

of Suez Canal, made during a -v-isit in April 1863. 8vo. London 1863.
ORE (Dr.) Fonctions de Li Veine-Porte. Svo. Bordeaux 1861.
O'SULLIVAN (J. L.) Peace the Sole Chance now left for Reunion : a Letter to

Professor Morse. Svo. Lomlon 186.3.
OWEN (Professor, F.R.S.) On the Extent and .lims of a National Museum of

Natural History. 8yo. Lomlon 1862.

DoyoEs.

The Observatory.

Mons. de Saussure.

The Author, by Sir C. Lyell.
F.R.S.

Mons. Delesse.

The Author.

Senor Bcrlanga and the

Marquis Casa Loring.
The Astronomer Royal.

Colonel Sir H. James, R.E.,
F.R.S.

Mons. Quatrefages, by G.

Busk, F.R.S.
The Author.

The Author.

The Author.
The Author.
The Author.

The Author.

The Secretaiy of State for

Incha.
The Author.

The Autlior.

The Author.
The Author.

The Author.

[ 23 ]

Pkesexts.
OWEN (Professor, F.R.S.) Ostcologioal Contributions to the Natural History

of the Anthropoid Apes. No. 7. Gorilla. (Excerpt from Zool. Trans.) 4to.

London 1851.
PERETTI (P.) Dell' Azione Chimica dell' Acqua sopra i Sali e sopra gli Acidi.

8vo. Eoma 1S61.
Memoria sopra iin Lavoro chimico da C. F. Schonbein. Svo.

Eoma 1857.
PHILLIPS (J. A.) Gold-3Iining, and the Gold-Discoveries made since 1851.

Svo. Loixdon 1862.
PHILLIPS (R.) On Atmospheric Electricity. Svo. Lo)idcn 1863.
PHIPSON (T. L.) On Sombrerite. On the transformations of Citric, Butyric,

and Valerianic Acids, with reference to the artificial production of Succinic

Acid. (Excerpts from Joum. Chem. Soc.) Svo. London.
Phosphorescence, or the Emission of Light by Minerals,

Plants, and Animals. 12mo. London 1862.
. Memoire sur la Fecule et les substances qui peuvent la

remplacer dans I'lndustrie. Svo. Briuvelhs 185.5-56.
Essai sur les Animaux Domestiques des Ordres Inierieurs.

Svo. Bnuvelles 1857.
Ueber die Phosphorescenz bei den Mineralien, Pflanzen und

Thieren. Svo. Berlin 1858.
La Force Catalytique, ou Etudes .sur les Phcnomenes de

Contact. 4to. Harlem 1858.
Note sur uiic Xouvelle Roche, sur le littoral do la Flandre

Occidentale; sur quelques Phenomenes Mcteorologiques observes sur le
littoral de la Flandre Occidentale. 4to. Paris 1857. (Excerpts from Comptcs
Rendus.)
Recherehes nouveUes sur lo Phosphore. De la Phospho-

rescence. Sur les Bolets Blemssants. Protoctista, ou la Science de la Crea-
tion. Analyse de quelques Substances mineiules. Sur une Matiere colo-
rante extraite du Rhamnus Frangula. (Excerpts from Joum. Soc. Sci. Med.
et Nat. de BruxeUes.) Svo. Bru.velks 1856-61.

PLANA (J., For. Mem. R.S.) Memoire sur I'lntcgration des Equations Difife-
rentieUes relatives au mouvement des Cometes. 4to. Turin 1861.

Memoire sur I'e.xpression du rapport qui (abs-
traction faite de la chaleur solaire) existe, en vertu de la chaleur d'origine,
entre le refroidissement de la masse totale du globe Terrestre et le refroi-
dissement de sa surface. 4to. Turin 1863.

POGGEXDORFF (J. C.) Biographisch-literarisches Handworterbuch zur Ge-
schichte der exacten Wissenschaften. Lieferungen 1-5. Svo. Leipzig
1859-60.

POGGIOLI (Michelangelo) Alcuni Scritti Inediti, pubblicati per cura di G. A.
Poggioli. Svo. Soma 1862.

PONCELET (J. v.. For. Mem. R.S.) Applications d' Analyse et de Geometiiie
qui ont servi en 1822 de principal fondement au Traite des Proprietcs pro-
jectives des figures. Svo. Paris 1862.

PRATT (H. F. A.) On Eccentric and Centaic Force : a new theory of pro-
jection. Svo. London 1862.

DONOES.

The Author.

Tlie Author.

The Author.

The Author.
The Author.

The Author.

The Author.

The Editor.
The Author.

The Author.

[ 24 ]

Presents. Donors.

QUATEEFAGES ( — . de) Koto sm- la MAchoii-o Immaiuo dc'oouvcrtc par Mons. Boucher dc Perthes.

Boucher de Perthes dans le diluvium d'Abbenlle. (i! copies.) 4to. Fan's

1SG3.
Observations sur la Maclioii-e de Mouliii-Quignon.

4to. Paris ISiXi. (Excerjits from C'omptes Eendus.)
QUETELET (A., Eor. Mem. E.S.) ^Vnnales de I'Observatoii-e Eoyal de Bruxelles. The Observatorj-.

Tome XT. 4to. Bru.velles 1SG2.
.Vnnuaire de rubservatoire Eoyal, 1803.

30- AnnJe. Svo Bnixelhsi ISOi.

C'limat de Belgique. 4to. Bru.relles. The Author.

Bolide observe dans la soiree du 4 liars 18(33.

Svo. Bru.rcU,s.
■ ■ Aurore Boreale du 14 au l.j Deccml)re 18(32.

Svo. Bnu-Jhs.
Eai)port sur uu Memoire de Malunoud Bey

sur rage dcs Pyramides. Svo. Bnixellcs.
— Etoiles filautes ; Orages des mois d'Aout et

de 8eiitembre ISCl', ie. Svo. BraxJhs.
Siu- les Xebuleuscs ; sm- rHygrometrie, &c.

Svo. Bra.cJhs.
De la Variation Aunuellc de ITnclinaison et

de la Derlinaison Magneti([ue. Svo. Brnxdhs.
— et HEREICK (E.) 8ur les Etoiles FUantes. lions. Quetelet, For. Mem.

Svo. Brv.cJIts. E.S.
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I'Electrieite pendant les Orages. Svo. Brti.nlhs. (Excerpts from Bull.

Acad. Eoy. Belg.)
EEDDIE (J.) The Meeliani.sm of the Heavens, and the New Theories of the The Author.

Sun's Electro-Magnetic and Eepulsive Influence. Svo. London 18(32.
IlEGISTArLT (V., For. Mem. E.S.) Eelation des ExjK'rienccs entrcprises pour The Author.

determiner les Lois et Doiint'es Physiques necessaircs au C'alcul des Ma-
chinos a feu. Tome II. 4to. Paris 18(32.
EHEES (IV. J.) Manual of Public Libraries, Institutions, and Societies in the The Smithsonian Institu-

L'nited States and British Pronnces of North America. Svo. PJiihvhlj^Jiia tion.

18.51).
EIATTI (V.) SuUc Induzioni Elettnche e su di una Corrente magneto-lndotta- The Author.

continua. 2 parts. Svo. llerjtjio 18(30.
ROBEETS (11.) Tho Essentials of a Healthy Dwelling. Svo. London 1802. The Author.

EOTHLAUF (K.) I'eber Vertlieilung des Magnetismus in eylinch-isehen Staid- The (_)bsei-vaton-, Munich.

stiiben. Svo. MiincJim l8(il.
SABINE (General, P.E.S.) On the Gosmieal Features of Terrestrial Magnetism. The Author.

Svo. London 1802.
— Eeport on llie repetitiuu of the Magni'tie Survey of

England. Svo. London 1802.
Circular to the Visitors of the Royal (>l)Servalory.

Svo. London 1802.
SANTI (G.) Viaggi al Montauiiata e le due Pronncic Sencsi. 3 vols. Svo. Pisa Dr. Sharpey, Sec. E.S.
171*5-08.

[ 26 ]

Fbesents.
SABS (M.) Beskrivelso over Lophogaster Typicus, en maerkvterdig form af de

lavere tifoddede krcbsd}T. 4to. Christiania 1862.
SCHO^'FELD (K.) Bcobachtungen von veriinderlichen Stemen. (Excerpt from

Sitz. Akad. Wicn.) 8vo. Wien.
SCHVAKCZ (J.) A Giirogok Geologiaja Jobb Napjaikban. 4to. Pest 1861.
■ La Geologic Antique et les fragments da Clazomenien. 4to.

Peslh 1861.

Foldtani Elmelctek a Helldnsegnel nagy Sandor Koraig.

Kotet I. Fiizet 1 & 2. 8vo. Pest 1861.

Lampsacusi Strato, Adalek a Tudomilny Tdrtenetebez. Fiizet 1.

Svo. Pest 1861.

A Fajtakerdes Szinvontda hiirom ev Eldtt. Svo. Pest 1861.

On the Failure of Geological Attempts in Greece prior to the

epoch of Alexander. Part I. 4to. Ijyndon 1862.
SCIAPARELLI (G. V.) Notizie sulla Vita e sugli Scritti di Francesco Carlini.

Svo. Milam 1S62.
SELLA (Q.) Relazionc del Ministro delle I^anze presentata alia Camera dei

Deputati. Svo. Torino 1862.
SHAJRSWOOD (W.) Catalogue of the Minerals containing Cerium. Svo. Boston

1861.

Catalogue of the Mineralogical Species AUanite. Svo.

SLEIGH (F.) Diagram illustrating a Discovery in the relation of Circles to

right-lined Geometrical Figures. Svo. London 1863.
SPITZER (S.) Allgemeine Auflosung der Zahlen-GIeichungen mit einer oder
mehreren XJnbekannten. 4to. Wien 1851 .

Studien iiber die Integration linearer Differential-Gleichmigen.

Fortsetziingen 1 & 2. Svo. Wien 1S61.

STAINTON (H. T.) The Natural History of the Tineina. Vol. Vll. Svo.

London 1862.
STUDER (B.) Geschichte der physischen Geographic der Schweiz bis 1815.

Svo. Bern 1863.
SWAVING (Dr. C.) Eerste Bijdrage tot de Kennis der Schedels van Volken in

den Indischen Archipel. (Excerpt from Natuurk. Tijdsckrift.) Svo. Batavia.
TATE (G.) Antiquities of Yevering Bell and Three-Stone Bum, among the

Cheviots, in Northumberland. (Excerpt from Berw. Nat. Club Proc.) Svo.

Berwick 1S62.
TORCIA (M.) Relazionc dell' ultima cruzione del Vesuvio accaduta nel mese di

Agosto di questo anno 1779. Svo. Napoli 1779.
TORTOLINI (B.) SuUa Divisione degli Archi di una curva del quart' ordine.

4to. Modena 1857.

Sulla Riduzione di un Integrale alle Funzioni EUittiche.

Ricerche analytiche sopra le Attrazioni esercitate da una linea plana, &c.
Sulla Curva Logociclica. Sopra alcune Curve derivate dall'EUisse e dal
Circolo. Quadratura deUa Doppia Ellissoide di Rivoluzione. (Excerpts from
Annali di Matematica.) 4to. Roma 1860-63.

TYNDALL (J., F.R.S.) Heat considered as a Mode of Motion ; being a Course
of Twelve Lectures delivered at the Royal Institution. Svo. London 1863.
MDCCCLXIII. d

DOKOM.

The University, Christiania.
The Author.
The Author.

The Author.
The Author.
The Author.

The Author,
nie .\uthor.

The Author.
The Author.
The Author.
The Author.

Dr. Sharpey, Sec. R.8.
The Author.

The Author.

[ 26 ]

Pkesexts. Donors.

VIGXOLES (C, F.R.S.) Model of the Passage of the Tudela and Bilbao Rail- The Author.

way in the International Exliibition of 1S62. 8vo. London 1862.
YOLPICELLI (P.) Sulla Elcttric-ita dell" Atmosfera,seoondaetcrza Note. 4to. The Author.

liomn 1S61.

lloma 1SC!2.
WALLICH (G. C.)

Sulla Polarita elcttrostatica, quiuta C'omimicazione. 4to.

The Xortli- Atlantic Sca-Iied: comprising a Diary of the

Voyage on board H.ll.S. 'Bulldog' in ISGO. 4to. London 18B2.
"WARD (A. F.) Universal System of Semaphoric Color Signals. 8vo. Phila-
delphia 1802.
"SVATTS (H.) Dictionary of Chemistry, and the Allied Branches of other

Sciences. Parts 1-4. Svo. iowrfoa 1863.
WEITEXA^'EBER (AV. R.) Zum .\ndenken an Maccslaw Hanka in Prag. 8vo.

rra.i.
■ Ein Beitrag zur Medicinisehcn Litcrargc>chichtc.

Svo. Prag.
WKEATLEY (H. B.) Of .Vnagrams : a Monograph treating of their History.

with an Introduction. 12mo. London 1802.
WHITING (G.) The Products and Resources of Tasmania, as illustrated in the

International Exhibition 1862. Svo. Hohart Town 1802.
WILLIAMS (J.) Account of a Deposit found in an Ancient Cliinese Statue of

Buddha, June 11, 1862. Svo. London 1863.
WORMS (H.) The Earth and its Mechanism ; being an Account of the various

proofs of the Rotation of the Earth. Svo. London 1862.
WOLLERSTORF-URBAIR (B. von) Rcisc dor ostcrreichischen Fregatte No-

vara um die Erde in den Jahren 18.57-59. 3 vols. Svo. Wicn 1861.
Reisc der ostcrreichischen Fregatte No-

vara; Nautiseh-physicalischer Thoil, Ahth. 1 & 2. 4to. Wien 1862.
— ludischcr Ocean. BeUagen 1-7. (7

Majis.)
YATES (James, F.R.S.) On the Excess of Water in the Region of the Earth
about New Zealand, its Causes and Effects. (Excerpt from Edinb. New Phil.
Joum.) Svo. 1862.

The Author.
The Author.
The Editor.
The Author.

The Author.

The Author.

The Author.

Messrs. Longman.

The 1. R. Navy Board,
Menna.

The Author.

'■&^0

" Bfcfcht/ (iri .

:S)Z'

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Phil 7o»,s .Ntnccri.x'ni /•/«/<■ x>a\'

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n.ii •j'lv.ii. .MDCdijaii riiurtwui

THE ISOTHERMAL LINES of INDIA

HERMANN ^r SCM LAOINTWE IT.

Kji^aTeS una pmrta hy K Wolf, .Mmurli

i-Kil. Iraiis. MUtCCUllI PlatcJUX.

>

U R K I S T » N

"^^^$^Z.

THE ISOTHERMAL LINES of INDIA
HERMANN d< SCH L ACINTW e I T .

DECEMBER, JANUARY", FEBRUARY.
fi^|.|.| hrica.sf witk iririca.sini lalidido.

Eii6r.ivpd .111(1 jrintea bv F WoU. MmH,!,

I'liii. i'i-..ns. MiMrcuxiu niiicxir

U n K I STAN

A » e « ^ tt '

ri...»?i*4. .

' »»■

T.i^-k

THE ISOTHERMAL LINES of

INDIA

■ '

HERMANN

a- scH

LAGINTWEIT

\

'mt '^um^$.%

■mi

Iri^^f

^^^■ikz

m\

!

MARCH,

0 PPI L

MAY

Kjljind ilrv. M.-i-i

Mllltfi

n I'm

■ :.l h.

lia

^i'

E.)<;i-/.vcil .LUiX |,j:li.i.-.I Uv I Wolf. Xhu.i.li

I'iixj JV..US. Mi>tTii.xinpiai<-mr

r' " T U « |}-J.>S TAN

I *" ^^^•. . .^i-

T~

THE ISOTHERMAL LINES of INDIA

HERMANN do SCHLAOINTWtlT.
JUNE, JULV, AUGUST

I'u.i ii..ii% MijfiTi.xiii I'l.Tir sau

Etreraved .md printed bvT.Wolf Tifuiiirli

V

One of the sAnRlion cells embedded in the trunk of -
diBsection and prcssur* in clyccririi*. _ A wtraigkt ^bre

lerence. niaii*.r oi ^^"^^j^Jj. ^^^^ cbntinuation of the nucleated sj.iral fibre is a true 'tiark border tH /ibre.'

Trcc-Froe (llvla arborea). The cell w.ih isolated by

. - .. ..._.•. ■-.. ^f "frrrt with its circnm-

line. Xlte brood flbre

■vea of the Grecu iruc-rrutr \ii.»ii» »ii"Jii-.y

ith the central part of Ihe ctfll.niid a tp<i*«( fb. .

the llbres. The frerminiil inaltCr was coloured with

'v-^-^

^'

Jrii

Younp gaiiK-lioii cell*

Falij-fortncd BMiflion cell.

Old sanRlion celL

Three g».,glin„ cell. Iron, a ganglion near the aorta of the Common FroB (Itanj t<mj'<>'"i.'>i J" !SS.J.°""Sr'nume™L""Tle
libre <IM( Ml coil spirally rounfthe flret. Id Pig. S, .evcral coiU are obserred; anJ li. Fig. 4 the colU are very numerouj. lue
'^ ' spiral increases in citeutaatlie cell advances in age.

lOOOtti of an inch
1000th of an inch

[Harri-son's Irapt.

/'

y' "TTivaw-

S

^V\ V

.J

\m

The eanglion cell marked a in Fiff. 6. X TOO diftTnetcra.
The mass is undereoinif division, and from it a number
'>r separate celb lilte tbe groups m Fi^r- 0 would result.

The mass of im,
dividing into cv
capillary vessel. ■
spiral fibres; it-

Fitr. 0. a. n mafld
■rvc .li-itribiited lo
itiriif fitraiithl and
the bundle of tint

JOOOtbof au inch
lOOOlhof an inv:h

L. .^. 3 id nat, del.

\

%-*="

Ganglion cell, with several Abres pitfCeeiliiip [
Near heart of Frop. x TOO.

vpilUry with iu nuclei. At o

'u\. Showine
three. 6 the
: the cells, is

Anuthtr mass from tlitf^i-iniotranclion as Fiif. 11. <1iridini; into six or »c\i'ii
ratr tflU. X 7yt. hi tht- lowor part of the figure are three oil trlol-uU-

Fig. 14.

tJaiiRlion cell, with fibres passing i

Yoiuiftipmplion cell. Plnddor. Hyla. The fibre *
very ia\c. There was no slieath or connective tU:
about cell or fibre, x TOO.

1000th of an inch i
lOOOih of an inch i_

1 connccttve tIsBue. Fig. 17. Kladder. Hyla. Fig, 18, r

TIic rotation or iho ner\'c flbrcs mid cclifl to the roniK

ncrrc flbrcs arc ttof connecifH wiih the connective tit

The spiral fibre diTidcs at the lower pnrt of the fignn

II rfflr* bordered fibre and twn fine fibres. 6b'
foiineclion with the n

' the nuclei in
rin from the cell.

stifral fibre, with n

^ .. _ of the

nuclei, arc very distinct in the large llgure.
the oil globului ID the upper part of the cellit.

1000th of an Inch
1000th of an Inch

»

/I!V''^

)iP

\

J •

A

OanirHoii cell with fibres connccteil with it, paNsinR olT in thrco iliffercnt directions. The Inrtre

tinrk-tmrdered dhro below the cell lias no conncetlon with it, and Is prolmbly not inDuenccil l>v it.

Une or the (lbre« passing Trom the cell heooniuH a (Inc darlc-bordcKd flbro (A), a is a cnpillary

vessel. From the bladder of the llyla. x 1800. December. IStt'^

Ganglion cell and fibres in the coat of an artery, x 700. Frog. 1881.

Fig. 24.

Fig. 26.

tianfflion cell from trun^Uon. Posterior
root of a spinal ncr%e. Hyla, ^ lm.

. _ _. renal artery. Frog

The fibres are seen to pursue opposite directions, x 700

1000th of an inch
lOOOtb of an inch

[Harnscns Imp*.

z/-

Sintlle iraiislioii .■••11. ctnhcildcJ ii) :i cord of c

1- til.' lowrr I'iirt of til-? ;iort;i. t'm-,'. Tlii» is the v.

\

V

•^

A

r. 30 is RuppOAeU t

1000th of an Inch
1000th of an Inch

L. S. B. ad nat. del.

[Haniaon's Impt.

Fig. 31.

y

/Mac arl«r.v. Frog.

Prnwiniw to illustnilc the »

I'art of trunk of pncumo«ru:tno. wliere it passes throu^li an oiiciii
ill the base of the :tkull. x ;iju.

Fine compound nerve trnnk, with a branch cominp off at riifht nn(>Ies, cnmno^cd

of flbi-cs which pursue optK»-.iTe lmufsch in the trunk. From the subniucou-*

tissue oi the I'aliiu.-. Frot;. x 700. IS02.

Vniinjr (ranglion cell near lumbar ncrrcD. llvla.

Ohdcrve the ifrcat size ol the ' nurleus.' The t-piral

fibre is seen lo divide into twi> brnnchr», one of

which runs with another nerve fibre, x i&oo.

I,;

' K

' '^

\ jifruB nerve near the heart. Hyln. a, trunk of the ncr*'e pavsing- towards the lieart. l>, a branch
wliich cunneclg thi» with the nerve of the oppcwite side. The connective tihhu.-, with it» cor-
puselet!, is also represented, c, Iwo jfanBrliori celltt, more highly majfnified in Mc 42. rf, two
Piinicliou cells, the fibres of which, after nieetiiig^ together, pui sue dillerent directions in the trunk
of the nerve, x •n&.

1000th of an inch '-
1000th of an inch 1 _

L. S. B. ad nat, de

:>r the Tftfni?. If r the heart llvia viri
^ Jiffifull lo folluw the fibrta lor auj dli

ki&A-^^ ^-v^*^.

'"V _.,/

A portion of a very small artciy from the Lladder of the HyU.

/

r »'

^l:s»Kel with p€CUl

•*^;^^

„.„..icu)a. rait.

tlnrk-bordLMfd fibre i
if the

e 6bte8art; r«i>re»ente(l

of Frop. Opposite a,
M-en. and all over the
ark of very fiiii, paie

A portion of the coat of a branch of the Hi
" * ' the niUKcuIar fihr'

\ muscular fibre.

]OOCtbof aninch

Ittnorh nf an tneh

/^/^^^,7>-a/w.MDCCCLXllL PlcUillX.

Easterly JJe flections -;

^ F(/]. I. Ihi Kervrw^'.

liiLtua

JajcclI . IfiifoncriLual. /Zulus,

-10 n U 7.1 » IS W I': li }9 Aj 2l 22_ 23 o / i ,3 ■>

iriZ J3_ » ys ?/> 17 W 1.0 ?/} 91 7? 2.-1 C 1 S

^n h 19 20 21 Z2, 2a h I -i a i s S- 1

fuj. J.Jh//// liorrcw.

Westerly De/lecUon.

1 i> .1 io 11 ■r?^ja^&jsj£^i'j j& I'l wy 22 23 0 i :
JjmthlrAsffiin'C'nuf.nl'MerM'S —

' Pig. Z. Port hi Kill (iy.

> Jin. I. /'in/ Hat row

J. Jiaeiri/ bjjo:

41
L8
V.153

Phyiical 9
Applied S«l

Royal Society/ of London

Philosophical
transactions

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