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THE
LONDON, EDINBURGH, and DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

CONDUCTED BY

SIR DAVID BREWSTER, K.H. LL.D. F.R.S.L.&E. &c.
RICHARD TAYLOR, F.L.S. G.S. Astr.S. Nat.H.Mosc.&c.
RICHARD PHILLIPS, F.R.S.L.&E. F.G.S. &c.
SIR ROBERT KANE, M.D. M.R.LA.

" Nee aranearum sane textus ideo melior quia ex se fila gignunt, nee noster
vilior quia ex alienis libamus ut apes. Just. Lifs. Polit. lib. i. cap. 1. Not

VOL. XXXIV.

NEW AND UNITED SERIES OF THE PHILOSOPHICAL MAGAZINE,
ANNALS OF PHILOSOPHY, AND JOURNAL OF SCIENCE.

JANUARY— JUNE, 1849.

LONDON:

RICHARD AND JOHN E. TAYLOR, RED LION COURT, FLEET STREET,

Printers and Publishers to the University of London;

SOLD BY LONGMAN, BROWN, GREEN, AND LONGMANS ; SIMPKIN, MARSHALL
AND CO.; S. HIGHLEY ; WHITTAKEU AND CO. ; AND SHERWOOD,

GILBERT, AND PIPER, LONDON ! BY ADAM AND CHARLES

BLACK, AND THOMAS CLARK, EDINBURGH; SMITH AND SON,

GLASGOW ; HODGES AND SMITH, DUBLIN ; AND

WILEY AND PUTNAM, NEW YORK.

" Meditationis est perscrutari occulta; contemplationis est admirari

perspicua Admiratio generat quaestionem, quaestio investigationem,

investigatio inventionem." — Htigo de S. Victore,

CONTENTS OF VOL. XXXIV.

(THIRD SERIES.)

NUMBER CCXXVL—JANUARY 1849.

Page
Mr. J. H. Alexander on a new Empirical Formula for ascertain-
ing the Tension of Vapour of Water at any Temperature . . 1

Mr. J. Brown on the Products of the Soda Manufacture 15

Mr. J. Cockle on a new Imaginary in Algebra 37

Mr. W. Pringle on the Continuance of a Solar Spot 48

The Rev. J. M. Heath on a simple Rule for converting intervals
of Sidereal into intervals of Mean Solar Time, and vice versa,

without the help of Tables 51

Mr. G. G. Stokes on some Points in the received Theory of

Sound 52

The Rev. C. Graves on the Calculus of Operations 60

Proceedings of the Royal Astronomical Society 63

Royal Society 73

On a new Modification of Phosphorus, by M. Schroetter .... 78

Meteorological Observations for November 1848 79

Meteorological Observations made by Mr. Thompson at the
Garden of the Horticultural Society at Chiswick, near
London ; by Mr. Veall at Boston ; by the Rev. W. Dunbar at
Applegarth Manse, Dumfries-shire; and by the Rev. C.
Clouston at Sandwick Manse^ Orkney 80

NUMBER CCXXVII.— FEBRUARY.

H. C. CErsted's Experiments on Diamagnetism 81

The Rev. J. Challis's Continuation of Researches in the Ma-
thematical Theory of Aerial Vibrations 88

Mr. J. H. Alexander on a new Empirical Formula for ascertain-
ing the Tension of Vapour of Water at any Temperature
{concluded) 98

Prof. J. R. Young on the Remainder of the Series in the deve-
lopment of (1-f a;)-'*, and on a Theorem respecting the pro-
ducts of Squares 113

Dr. R. A. Smith on a Mode of rendering Substances incom-
bustible 116

a2

If

iv CONTENTS OF VOL. XXXIV. — THIRD SERIES.

Page
The Rev. C. Graves on a system of Triple Algebra, and its

application to the Geometry of three Dimensions 119

Prof. Reich on the Repulsive Action of the Pole of a Magnet

upon Non-magnetic Bodies 127

Prof. Marcet on the Action of Chloroform on the Sensitive Plant

{Mimosa pudica) 130

Mr. J. Cockle's Solution of two Geometrical Problems 132

Proceedings of the Cambridge Philosophical Society 135

. Royal Astronomical Society 138

On the Equivalent of Fluorine, by M. Louyet 150

Preparation of Iodide of Arsenic 152

Composition of the Black Yttro-columbite of Ytterby 152

On Liquid Protoxide of Nitrogen, by M. Dumas 153

On the Urates, by MM. Allan and Bensch 154

On the Presence of Copper in the Human Blood, by M. Deschamps 155

Solubility of Chloride of Silver in Hydrochloric Acid 156

Formation of Carbonate of Lime from the Neutral Malate of Lime 156

Journey to Discover the Sources of the Nile 157

Post-office Regulations 158

Meteorological Observations for December 1848 159

Table 160

NUMBER CCXXVIIL— MARCH.

Prof. J. D. Forbes on the Classification of Colours 161

Mr. A. Smith on the Calculation of the Distance of a Shooting
Star eclipsed in the Earth's Shadow 179

Mr. J. Glaisher's Remarks on the Weather during the Quarter
ending December 31, 1848 182

Prof. J. R. Young on the Expression for the remaining roots
of a complete Cubic, when one root is found 193

Mr. J. Locke on Single and Double Vision produced by viewing
objects with both eyes ; and on an Optical Illusion with regard
to the distance of objects 195

Mr. T. H. Pratt's Analytical Proof of the Parallelogram of Forces 201

Mr. N, S. Heineken's Suggestions for rendering a Meridian
mark visible at Night 202

Mr. G. G. Stokes on the ITieory of Sound, in reply to Professor
Challis 203

Mr. T. H. Henry on the Composition of the Gold from Cali-
fornia 205

Sir R. I. Murchison on the Geological Structure of the Alps,
Carpathians and Apennines, more especially on the transition
from Secondary to Tertiary Types and the existence of vast
Eocene Deposits in Southern Europe 207

Proceedings of the Royal Astronomical Society 216

■ Cambridge Philosophical Society 225

CONTENTS OF VOL. XXXIV. THIRD SERIES, V

Page

On the Rationale of the Explosion causing the great Fire of 1 845

at New York, by Dr. Hare 227

Preparation of Iodide of Lead, by M. T. Huraut 231

On the Protogine of the Alps, by M. Delesse 233

Examination of Madder, by M. Debus 1,236

Analyses of Felsite, Oligoclase and Muromontite 237

On the Ferrocyanides of Strychnia and Brucia, by M. D. Brandes 238

Meteorological Observations for January 1849 239

Table 240

NUMBER CCXXIX.— APRIL.

Dr. J. W. Draper on the Existence and Effects of Allotropism

in the constituent elements of Living Beings 241

Prof. Potter on the discovery of the Chilling Process in the cast-
ing of the Specula for Reflecting Telescopes, &c 246

Mr. J. Phillips's Thoughts on Ancient Metallurgy and Mining
in Brigantia and other parts of Britain, suggested by a page

of Pliny's Natural History 24 7

The Rev. B. Bronwin on the Determination of the Coefficients
in any series of Sines and Cosines of Multiples of a variable

angle from particular values of that series 260

MM. L. Foucault and J. Regnault on some phsenomena of Bi-
nocular Vision 269

Mr. J. Glaisher on the Meteorology of England in the year 1847 271
Prof. J. R. Young on the decomposition of Functions into Con-
jugate Factors ; with some consequences deducible therefrom 278
The Rev. J. Challis on the Theoretical Value of the Velocity of

Sound, in Reply to Mr. Stokes 284

M. A. de la Rive on the Diurnal Variations of the Magnetic

Needle, and on Auroras Boreales 286

Sir W. R. Hamilton on Quaternions ; or on a New System of

Imaginaries in Algebra {continued) 294

Notices respecting New Books : — M. A. Quetelet's Letters ad-
dressed to H.R.H. the Grand Duke of Saxe Coburg andGotha,
on the Theory of Probabilities ; Mr. R. Patterson's First

Steps to Zoology 297-298

Proceedings of the Royal Society 299

On Anhydrous Nitric Acid, by M. Deville 314

On a Series of Organic Alkalies homologous with Ammonia, by

A.Wurtz 316

On the Existence of Mercury in the Tyrol, by M. H. Rose . . 318

Rectification of Spirit of Nitrous JEther, by M. Klauer 319

Meteorological Observations for February 1849 319

Table 320

VI CONTENTS OF VOL. XXXIV. — THIRD SERIES.

NUMBER CCXXX.— MAY.

Prof. H. Rose on the Isomeric Modifications of Phosphoric

Acid , 321

Sir W. R. Hamilton on Quaternions ; or on a New System of .

Iraaginaries in Algebra (continued) 340

Mr. W. H. Barlow on the Cause of the Diurnal Variations of

the Magnetic Needle 344

Mr. T. S. Davies's Note on Numerical Transformation 347

Mr. G. G. Stokes on the Theory of Sound, in reply to Pro-
fessor Challis 348

Dr. R. D. Thomson and Mr. E. T. Wood's Note on the Com-
position of Shea Butter and Chinese Vegetable Tallow .... 350
The Rev. J. Challis's Determination of the Velocity of Sound

on the principles of Hydrodynamics 353

Mr, J. Glaisher's Remarks on the Weather during the Quarter

ending March 31, 1849 366

The Rev. B. Bronwin on the Determination of the Coefficients
in any series of Sines and Cosines of Multiples of a variable

angle from particular values of that series 374

Mr. T. J. Herapath on some Combinations of Boracic Acid with

Oxide of Lead 375

Notices respecting New Books : — Mr. R. C. Taylor's Statistics

of Coal 380

Snowy Mountain in Eastern Africa 388

On Mr. Struve's Mine Ventilator, by J. Richardson, C.E 389

Analysis of Faujasite, by M. A. Damour 394

Analysis of California Gold. 394

On the Vanadiate of Lead and the Double Vanadiate of Lead

and Copper, by M. Ignace Domeiko 395

New Mineral from Brazil , 397

Analysis of the Water of the Mediterranean on the Coast of

France 398

Impurity of Commercial Bromine 399

Meteorological Observations for March 1849 399

Table 400

NUMBER CCXXXL— JUNE.

The Astronomer Royal on a difficulty in the Problem of Sound 401

Mr. J. Cockle on the Symbols of Algebra, and on the Theory of
Tessarines 406

Mr. E. J. Lowe on Remarkable Solar Halos and Mock Suns
seen at the Observatory of Henry Lawson, Esq., F.R.S. &c.,
Lansdown Crescent, Bath. (With a Plate.) 410

Prof. J. R. Young on an Improvement in the Analysis of Equa-
tions , 413

CONTENTS OF VOL. XXXIV.— THIRD SERIES. Vll

Page

M. Duhamel on the Multiple Sounds of Bodies 415

Sir W. R. Hamilton on Quaternions ; or on a New System of

Imaginaries in Algebra (continued) 425

M. Ch. Matteucci's Further Researches on Electro- Physiology 440
Mr. S. M. Drach's easy Rule for Formulizing all Epicyclical

Curves with one moving circle by the Binomial Theorem. . . . 444
The Rev. J. Challis on Spherical Waves in an Elastic Fluid,

in reply to Mr. Stokes 449

Prof. Pliicker on the Magnetic Relations of the Positive and

Negative Optic Axes of Crystals 450

Notices respecting New Books : — Sir J. F. W. Herschel's Out-
lines of Astronomy 452

Proceedings of the Cambridge Philosophical Society 455

Royal Astronomical Society 459

Royal Society 463

On some Meteorological Phsenomena, by Prof. E. Wartmann 469
On the Reflexions of different kinds of Heat by Metals, by

MM. F. De la Provostaye and P. Desains . . . .' 471

On Chloroniceic Acid, by M. E. Saint-Evre 473

On the Nature and Composition of various Chloroniceates, by

M. E. Saint-Evre 474

On the Reaction of Sulphate of Potash and Sulphate of Copper,

by M. J. Persoz 475

On Octohedral and Cubic Alum, by M. J. Persoz 476

On Anisol and its Derivatives, by M. A. Cahours 476

Compounds of Hydrochlorate of Strychnia and Cyanide of Mer-
cury 479

Meteorological Observations for April 1849 479

Table 480

NUMBER CCXXXIL— SUPPLEMENT TO VOL. XXXIV.

M. A. Bertin on Circular Magnetic Polarization 481

Mr. G. G. Stokes on the Theory of Sound , 501

Mr. Reuben Phillips on the Magnetism of Steam 502

Prof. Challis on some Points relating to the Theory of Fluid

Motion 512

Appendix to Mr. Drach's Paper on Epicyclic Curves in the last

June Number 520

Sir R. I. Muichison on the Distribution of the Superficial De-
tritus of the Alps, as compared with that of Northern Europe 523

Mr. A. Cayley's Note on the Theory of Permutations 527

Proceedings of the Royal Society 529

■ Royal Astronomical Society 532

Deflection of the Magnetic Needle by the Act of Volition .... 543
On the Artificial Formation of Minerals in the Humid Way, by

M. de Senarmont 545

Index '. 547

PLATE,

Illustrative of Mr. E. J. Lowe's Remarkable Solar Halos and Mock

Suns.

Erratum in Mr. J. Phillips's paper.

Page 250. In the note read for tiie shield of Achilles "H^AISTOS throws
into his crucibles brass unconquered, Kaa-a-iTepos, &c., not — brass, un-
conquered Kaaairepos, &c.

Errata in Mr. J. Cockle's paper.

Page 43, line 12, /or must read may.
— 43, — \2~\Z,for for otherwise the rectangle would obviously be
greater than the square, read and then ascertain the con-
sequences of such a supposition.

Supra, p. 42, note *, line 6, for " and then n " read when the index.

Erratum in Prof. Plucker's paper.
Page 451, line 30,ybr obtain read command over, M. Plucker's meaning is
that "yfyu may give to it any declination you like, from about 25° to
the east to 65° to the west."

THE
LONDON, EDINBURGH and DUBLIN

PHILOSOPHICAL MAGAZINE .

AND

JOURNAL OF SCIENCE.

[THIRD SERIES.]
JANUARY 1849.

I. On a 7}eto Empirical Formula for ascertaining the Tension
of Vapoiir of Water at any Temperature. By J. H.
Alexander, Esq.*

THE formula which the following memoir is intended to
expose, is called ne'tx)\ because, to the best of my know-
ledge, it has never been used or suggested hitherto. It is
also rightly termed empirical^ in so far as it is not susceptible
of geometrical demonstration, but thus far only ; since in point
of fact, it was derived entirely from considerations d priori
and independent of any experiment or interpolation. Of
course, it was compared as soon as possible with the tempe-
rature corresponding to the ordinary atmospheric pressure ;
and after a satisfactory agreement had been found at this
point, the accord of the formula with observations at other
points, above and below,^ was regarded as neither accidental nor
surprising. The extent and nearness of this accord through
a range of experiment more extensive than has hitherto been
included in one and the same table, it is +he principal aim of
the present paper to exhibit, after having shown in few words
the reasonableness of the formula and its limits (or rather want
of limits) in application ; a comparison, then, of the errors ex-
isting and admitted in several of the experimental series of the
highest authority, with the differences developed at the same
epochs by the formula, will indicate the probabilities in favour
of the latter, and the nature and amount of its reliability.

It is obvious that the pressure of vapour or steam must be
always in proportion to the absolute temperature at which it
is produced. But as this temperature is only observable re-
latively and upon an arbitrary scale, it is necessary, in order
to obtain anything like a measure of the quantity of heat ex-
isting, to use the ratio of the whole extent of the scale assumed
between the two epochs where the liquid changes its state
respectively, to that portion of it {i. e. the number of its de-
* From Silliman's Journal for Sept. 1848.

PJiil. Mag. S. 3. Vol. 34. No. 226. Jan. 1849, B

2 Mr. J. H. Alexander on the Tension of Vapour of Water.

grees) which expresses the existing temperature. Or, what
amounts to the same thing, the pressure of steam whose tem-
perature is observed on any scale, is directly as the number
of degrees read for the temperature; and inversely, as the
whole number of degrees on the same scale between the
melting of ice and the boiling of water. With Fahrenheit's
scale, calling t the number of degrees at any temperature, the
pressure of steam at that temperature must be proportionate to

. A<yain, the pressure of steam must be always directly as

180 & » f J J

the absolute heat of conversion, or, as it is otherwise termed,
the latent heat ; expressed, of course, in degrees of the scale
assumed. For, the greater the number of degrees for such
latent heat, the greater also will be the repulsive force of the
heat existing in the steam ; which repulsive force may be as-
sumed to be at least a function of the elasticity of the vapour.
And as such repulsive force takes effect in part by expanding
the medium vaporized, and the greater such expansion, the
less will be the remaining elasticity, it follows that the pres-
sure of steam in the ordinary state of an atmosphere must be
also inversely as its increased volume. This increased volume
may be taken, from the experiments of Guy-Lussac, at 1695
times that of water at its greatest density. And the latent
heat of steam is generally admitted as 990° F. ; which number
results from the experiments of Clement and Desormes, is
not far from the mean of several other observers, and will pro-
bably require a very small modification only to be identical with
the deduction from a strict theory of volumes applied to vapours
generally, in the mechanical relation of the observed effects of
heat upon such vapours and the liquids producing them.
So far as we have gone, then, the pressure of steam must be

/ t 990 \ T • . 11 . • •

I 7777:+ ,^^1 )• It IS here that the empu'icism comes m,

\loO io95/

and dictates the numerical power to which this ratio must be
involved, in order to harmonize the progression of the elas-
ticities with the series of the temperature. That the simple
ratio will not present such harmony is manifest ; for that would
be the measurement, by a lineal scale of equal parts, of central
forces, which, acting upon volumes, might rather be supposed
to be in the duplicate ratio of the cube. Such a ratip, equi-
valent to the sixth power, is in fact what has been taken ; and
the complete equation becomes then, calling p the pressure,

\180 1695/
This index, however plausible upon reflection the reason to
justify its adoption might have appeared, was no doubt sug-

as

Mr. J. H. Alexander on the Tension of Vapour of Water. 3

gested too, even unconsciously, from the recollection that it
had before (though with different factors) served both Young
and Tredgold, and perhaps others, in approximating the re-
sults of experiments. Moreover, it was apparent that the
numerical results in English inches from the formula were
altogether an accidental coincidence, which, dependent upon
the properties of numbers, could not be expected to occur
upon the use of any other scale than that of Fahrenheit; un-
less, indeed, one should adopt the fancy of Mr. Woolf, who
supposed that he had discovered an immediate numerical re-
lation between the atmospheric pressure and the pound avoir-
dupois, in which case the English inch might also claim to
be among natural dimensions. It is, of course, quite possible,
by a little artifice among the terms of a formula of this shape
and retaining the same index, to produce a series of numbers
corresponding to any linear scale. Thus, for instance, sub-
stituting Centigrade degrees for those of Fahrenheit, but car-
rying the denominator of the first term down to the degree at
which the pressure becomes by the formula zero (which may
be presumed to correspond to an absolute negation of heat,
and which in fact has to be used with the present formula
when it is intended to give the pressure in atmospheres), we
obtain pressures expressed in French metres, and through a
range of several atmospheres, but little discordant with the
results of experiment. In using Centigrade degrees, how-
ever, and transforming the equation so as to express atmo-
spheres, the index (6*) gradually diverges from regular mul-
tiples ; serving to show what we might otherwise conjecture,
that such index is not based upon any general relation in
nature. It was, therefore, of less interest for me to weary
myself with comparisons of other thermometric and linear
scales; it is enough that the formula affords a remarkable
coincidence in its own terms with the measures recognized
among ourselves.

It is readily seen that the first term is positive as far as 0°
of Fahrenheit ; for temperatures lower than that it becomes
negative; and there is a point, of course, where the negative
value of the first term equals the constant positive value of the
second, and the pressure, therefore, as was said just now, be-
comes itself zero. This point occurs at —105°'l 3; of which
it is enough to say, that it is not very far from the lowest de-
gree of heat yet produced, and that long before it is reached
the mercurial thermometer becomes useless. Whether there
is in the theory of nature (for it is admitted that there is not
in fact) such a point as that of the absolute privation of heat,
and if so, how it should be reckoned and where placed, — are
questions which, although kindred to the matter in hand, are

B2

4 Mr. J. H. Alexander on the Tension of Vapour of Water.

not necessary to the elucidation purposed. Whatever the
answers might be, they would not affect the working of a rule
which is intended for practicable temperatures; and if it
should be objected to the present formula (as it was to the
method of interpolation of Dr. Dalton) that it determines a
limit which does not exist in nature, or places it where it
should not stand, it may be very well replied, that the objec-
tion may hold good against the factors without prejudice to
the form. It is quite likely that, below 32° F., it is theoreti-
cally no longer proper to use in behalf of chilled and freezing
vapour, the number (990°) which belongs to boiling steam. But
as even here and for nearly 60^ below the melting-point of
ice, the actual formula was not so very discordant from the
results of experiment, 1 had the less motive for modifying or
transforming it into a nearer agreement at this unusual tem-
perature. The determination of a limit of this sort, whether
real or assumed, is necessary in converting easily a formula
like the present, irsto one which will show the pressure in at-
mospheres;— a method wliich, as it extricates the results from
any dependence on a particular system of weights and mea-
sures and thus makes them generally applicable, is of course
in any purely scientific investigation much to be preferred.
It is obvious that such a scale of atmospheres or volumes must
start from the degree where the expansions and the tendency
to expand (which is elasticity), so far as they are due to tem-
perature, are null. The limit that we have found, then, of
105°-13 below 0° Fahr., is such a term ; and the distance be-
tween that and the other extremity of the scale, or 317°*13
(which is the measure on the thermometer of one atmosphere),
becomes the new denominator to replace the 180° actually
used for the pressure in inches of mercury.

In fact, I had expected, in advance, to find the present ma-
nometric formula (as it may be termed) becoming a barometric
one, by putting it into this shape;

^ \317-13^Vl695/ /

But this did not hold good. Applying it numerically, it re-
sults in giving,

For 212°, a pressure of 1-059 atmospheres, equal to 31 "68 inches.
And for 322°-38, ... 6263 ... ... 187-37 ...

In both these instances, to agree with the original formula,
the numbers representing the atmospheres should have been
without fractions. The excess, however (as is visible), goes
on in a converging series, and by and by disappears altogether,
the difference then changing its sign. Even then it is not
much; and at high temperatures the equation corresponds very

Mr, J. H. Alexander on the Tensian of Vapour of Water. 5

nearly with the actual observations. For instance, comparing
it with the experiments of Dniong and Arago,
Temp. 335°-3 F. gave a pressure of 7'391 atmos. ; by formula 7-478 atin.

... 371°3F 11-660 ... ... 11-938 ...

Nevertheless, as the object I had in view was not to find an
equation that merely fits any particular series of observations,
or is exact only for the higher ranges of temperature, I aban-
doned this theoretical expression and preferred to deduce a
formula for pressure in atmospheres from the original one, in
the ordinary analytic way. This results in the alternative
expressions —

^"VsiT'Ts'^ 1695 / ~\317-1 3 2986 33./ '
either of which may be adopted, according as we prefer to re-
tain in view the factor of the latent heat, or that of the expan-
sion at the unitary atmospheric pressure. In practice, the
constant fraction may be substituted by the number 0-33151.
For the original formula, the similar constant was carried no
further xhvinfour places of decimals ; in this, where the unit ol'
pressure is thirty times larger, both the attainment of equal
precision requires, and the facility of calculation allows, an-
other decimal place to be taken.

These decimal constants might, indeed, have been given in
both formulae at once, instead of the fractional expression
from which they originate, were it not that I thought it de-
sirable to preserve those factors which, besides solving the
equation, indicate also, in part accidentally and in part essen-
tially, certain elementary relations between pressure and tem-
perature (or rather certain epochs in those relations), which
are important in the future complete theory of the subject.
For example, the denominator (1695) which expresses the
number of volumes of steam under the atmospheric pressure
and at a temperature of 212°, developed from an unitary vo-
lume of water at its maximum density, shows also the number
of atmospheres, the equivalent of whose pressure will, below a
certain temperature, prohibit the development of steam beyond
the sphere of said unitary volume. On the other hand, the nu-
merator (990°) gives this limiting temperature ; and shows the
degree on Fahrenheit's scale, where the force of steam becomes
equivalent to 1695 atmospheres. Its density, therefore, would
be equal to that of water ; if its behaviour in other regards were
like that of water too, this temperature would be the limit to its
useful effect. But as there would most probably still remain the
elasticity due to its expansion at that temperature, it does not
appear that we are warranted in supposing any such limit.

At this point, 990°, the ratio of the latent and sensible heats
has just inverted itself from what it was under the unitary at-

6 Mr. J. H. Alexander on the Tetisioji of Vapour of Water.

mospheric pressure. Then the former, which went altogether
towards maintaining the state of vapour, without increasing its
apparent temperature, was b'S times the latter; now the latter is
h'b times the former. And from this consideration flows an
easy manner of connecting proportionate volumes and pressure
with simple multiples of the increments of heat. But to deve-
lope this here, would be to wander from the present aim.

If we go on to suppose the temperature increased until the
whole of the latent heat is utilized and becomes sensible
(which occurs at 1170° F.), we should then have a condition
in which steam under a pressure of 4225 atmospheres (by the
formula) is reduced to very nearly y'^ths of the unitary volume
of water to produce it, and has a density (without regard to
the expansion from temperature) 2*5 times that of water at its
maximum. The expansion due to the temperature is (with
Gay-Lussac's factor) 0"975 of the unitary volume; the ex-
pansion of water due to the same temperature (taking its
maximum density as occurring at 39°'4 F., its actual expansion
as 0'04<333 at 212°* F., and its rational expansion as the cube-
root of the fifth power of the temperature above the maximum)
is 0'99944 of the same unitary volume — an accord which,
considering the possible errors of the experiments, appears to
me sufficiently satisfactory.

The equations already given serve to find the pressures in
inches of mercury and in atmospheres respectively, when the
temperatures are given : to find the temperatures correspond-
ing to given pressure, they become as follows :

/=180 Vp — 105°-13;
where/? represents the number of inches of mercury; and

/ = 307-13 Vy'-lOS^-lS;
where/)' stands for the number of atmospheres.

After these preliminaries, may now be presented the com-
parison of results by the formula and by the experiments of
various observers ; as is done in the following table. This
table is quite extensive; and, for my own sake, rather more so
than I desired. But it will be considered that it comprehends
the assemblage of the observations of many persons through
many years ; and it was, besides, to me of an interest that I
think will be partaken of by others, to have in one view and
without interpolation nearly all the determinations that have
been made by actual experiment. In limiting it to statements
of this character, I confess I thought at first to restrict its extent,
though it appears that there are in fact more experimental
data than is generally supposed ; and I desired besides to in-
crease its utility beyond the mere comparison with the present
formula, by fitting it as a depository for general reference. In
this last regard, it may be considered as authentic and reliable.

Mr. J. H. Alexander on the Tension of Vapour of Wate7\ 7

And yet, after all, it does not include the whole of our ex-
perimental data. Those of John Henry Ziegler, of Winter-
thiir in the Canton of Zurich (whose name I give at length
because he may be regarded as having led the way in this
research), of Achard, of Schmidt, of Magnus, and of some
others whose names escape me at the moment or have failed
to come to my knowledge at any time, — either were not ac-
cessible to me at all, or only in statements at second or third
hand, whose accuracy I had not the means of verifying.

Nor does it contain quite all the results of those observers
to whose experiments I had access. To have given every
one of each, would have rendered the table, in respect to the
present aim, quite enormous. For instance, the several series
of M. Regnault, the latest experimentalist, exhibit such a
luxury of repetition, at temperatures varying very slightly —
sometimes only by small fractions of a degree — as almost to
outnumber, in themselves alone, all the quotations 1 have
made from others. I exercised, therefore, the discretion of
using only those instances which, at reasonable degrees apart,
rested upon a number of concurrent observations. In general,
such concurrences of the same mean temperature give the
mean of three observations of pressure ; the result for 32*^ F,
is the mean oi forty seven recorded pressures. For the ex-
periments earlier than Mr. Southern's, I have usually taken
only those whose recorded temperatures were already other-
wise in the table. I shall have occasion, however, to speak of
each one more particularly, presently.

The difference of apparatus and manipulations necessary
for experimenting upon the elasticity of vapour above and
below the boiling temperature, has led several of the observers,
for convenience or other motives, to confine themselves to one
or the other side of this limit; and would render proper, in
any discussion of the value and reliability of such observa-
tions, an arrangement of them in two distinct tables. But as
I have no such aim at present, and as the exemplification of
the formula belongs equally to the lowest and highest tempera-
tures, there is no reason for breaking the continuity of the com-
parison or for presenting the results in more than one table.

The order of the different columns is chronological, ac-
cording to the dates of execution ; or, when that was not
known, of publication of the experiments. As in fact each
successive observer had, or might have had, the benefit of the
experience and precautions of his predecessor, it may be pre-
sumed that this order represents too, in a measure, the
respective reliabilities of the results. Only the two French
series, while they are evidently unrivalled in the extent to
which, in opposite directions, they have been carried, are
similarly distinguished by the refined elaboration which cha-
racterizes every part of the research.

8 Mr. J. H. Alexander on the Tension of Vapour of Water.

Table of the Pressure of Steam at different Temperatures, calculated and variously
observed through a Range of 462 degrees of Fahrenheit's thermometer.

Temp, in dec. of
Fahrenheit.

ill

Pressure In inches of Mercury observed by

"3 j:
a 2

r-

S

.to

r

1

<

00

00

£*

Dalton,

if

u

•s

pq

i
1

1

i

-27-112
-13-
- 4-504

0-
4- 1-706
9-41

17-402

23-702

24-

27-626

32-

36-

40-

42-

43-25

49-514

50-

52-

54-5

55-

60-

62-

64-

64-976

65-

65-75

70-

72-

75-

77--

80-

80-762

82-

85-

88-25

90-

92-

95-234

96-

99-5
100-
102-
105-
110-
110-75
112-
115-
120-
121-244
122-
125-
130-
132-
133-25

0-0066

0-0180

0-0305

0-0400

0 0437

0-0664

0-0911

01345

0-1363

0-1611

0-1956

0-232

0 275

0-298

0-314

0-402

0-410

0-443

0-487

0-496

0-596

0-641

0-689

0-713

0-0106
0-0205
0-0284

1
if

-

ii

g°
is
is

ii
Ii

0-16

i

il

as-

0-15

0-0457
0-0638
0-0941
0-1256

0170

0-1500
0-1811

0-200

0-200
*0-229

'6"29

0-0

0-250

0-1

0-2

0-23

0-1856

0-297

0-366

0-360

0-41

__

0-435

0-416
0-516

0-35
0-55

0-52

*0-597
0-630

0-75

0-616

0-630

0-732
0-849
0-908
1-001
1-074
1-184
1-214
1-263
1-389
1-538
1-623
1-726
1-903
1-947
2-159
2-192
2-323
2-531
2-915
2-977
3 081
3-346
3-829
3-958
4-038
4-367
4-969
5-228
5-397

0-726

0-73

0-7326

0-860

0-910

1-010

0-82

1-055

1-02

1-18

1-6
2-25

3-0

3-95

1-170

1-290

1-360

1 .

1-42

1-673

*l-63
1-820

1-95

1-860

1-95

2-100
2-456

2-540

2-65

2-820
3-300

3-562

3-50

3-57 3-17

3-830
4-366

*4-60
4-76

5-07

4-fi8l

1

Mr. J. H. Alexander on the Tension of Vapour of Water. 9
Tahle of the Pressure of Steam, Sfc. {continued).

— <=- ■-

o

Si 4"

— oa

III

Pressure in inches of Mercury observed by (

t

r

1
S

m C

00

s

I

Dalton.

tf

h

i

eaoo>

<

1

i

135-

140-

142-

144-5

145-

150-

151-124

152-

155-75

160-

162-

164-

165-

167-

170-

172-

173-

175-

176-416

178-25

180-

182-

185-

189-5

190-

195-

198-05

200-

200-75

202-

205-

210-

211-27

212-

213-

216-6

220-

221-6

222-44

225-

226-3

226-5

230-

230-5

232-

233-132

234-

234-5

235-

238-5

240-

242-

245-

245-25

245-8

248-25

248-5

249-

5-638
6-381
6-699
7-117
7-203
8-109
8-327
8-497
9-271
10-214
10-685
11-174
11-427
11-944
12-752
13-323
13-612
14-209
14-647
15-230
15-801
16-477
17-540
19-237
19-435
21-491
22-839
23-731
24-087
24-679
26-164
28-802
29-500
29-915
30-48
32-61
34-73
35-77
36-33
38-07
38-97

*41-67"
42-04
43-17
44-05

5-070
5-770

"6-45
8-55

"6-66

7-85
"9"9"9

12-64
i'5'91

29-8

5-T6

10-27

15-88

29-87
44-38

bib
6-72
8-65

11-65

14-05

17-85
22-62

28-65
35-8

44-5
54-9

4-53
5-46

10-10
11-07
11-95

12-88

14-73
16-58

30-

37-

38-

44-

47-
49-

56-

** 7-698

6-600
7-530

"9-606

i6-"8o6

12-656

11-25

*1302
14-60

13-18

'li'mi

13-550

15-160

16-900

' 18-86

""2'4"-6

30-
*34-99

34-20

••*

'22-'538
2"9-'62i

29*922

34-95

38-0

41-51
43-6

45-50

56-0

51-75

54-4C

44-40

i 59-i"c

190
21-100

23-600

25-900

28-880

30-

33-40
35-54
36-70

39-11
40-10

""43-16
43-50

46-80
47-22
50-30
51-70
53-60
56-34

57-10

60-4(1

'35-78"

44-75

43-05

45-13
45-54
48-41
49-69
51-44
54-17

52-12

54-93

59-08

58-20

57-51
57-99

10 Mr. J. H, Alexander on the Tension of Vapour of Water,
Table of the Pressure of Steam, 8fc. {continued).

■s

•Ox:

U n 3

lis.

Pressure in inches of Mercury observed by |

t

c

1

J= 00

r

&5-

<

00

s

I

Dalton.

n

5g

ii

1

1-4

250-

250-25

250-3

250-5

251-6

252-662

254-003

254-5

255-

257-5

260-

260-4

262-5

262-75

262-8

263-3

264-9

265-

267-

269-

269-5

270-

270-86

271-

271-2

271-346

272-

272-5

273-7

274-

275-

275-5

275-7

276-224

277-9

278-

278-4

279-5

280-

281-8

283-8

284-5

285-2

287-2

289-

289-5

290-

292-3

292-5

293-4

294-

294-5

295-

295-6

297-1

297-464

298-5

58-99

59-12

61-90

59-6

66-8

58-

62-
70-

78-
80-

82-

58-14

59-29

60-60
61-69
63-08
63-62
64-15
66-88
69-67
70-15

63-50

66-70
67-25
69-80
72-30
72-80

80-3

63-33

64-14

63-20

64-40

70-12

72-88

73-27

75-9

77-90
78-04
81-90
84-90

se-'so

"88-""

79*94

88-90

94-1

73-55
75-47
76-44
78-08
80-63

76-15

76

78-50

81-14

"82'50

81-51

81-94
83-08
83-27
83-54
83-73
84-60
85-28
86-92
87-32
88-72

85-70

82-56

85-89

85-45

91-20
93-48
94-60
97-80

101-60
101-90
104-40
107-70

112-20
114-80
118-20

120-15
123-10

119-2

100-07
104-91

105-9

88-50
90-

88-8

92-42

89-71
90-45
92-86

93-86

93-81

99-95

93-60
95-22
95-97
98-69
101-79
102-75
103-98
107-25
110-23
111-05
111-90
115-88
116-22
117-81
118-87
119-77
120-68
121-76
124-54
125-21

97-75

110-30

112-65
114-50

119-02

120-60

122-20
124-15

126-70

129-

130-40

133-90

129-11

130-79

135-8

Mr. J. H. Alexander on the Tension of Vapour of Water. 1 1
Table of the Pressure of Steam, ^c. (continued).

Temp, in deg. of
Fahrenheit.

W Id 9

&

Pressure in inches of Mercury observed by |

1

f

^2

L

IS
<

00

So

Dalton.

H

1

i

298-8

299-5

300-

300-6

301-316

302-

303-8

304-5

305-

305-393

305-5

306-8

308-

308-606

310-

310-25

311-4

312-

314-75

319-75

320-

322-

325-76

334-50

335-21

337073

338-75

340-

341-798

343-6

345-

346-

348-

350-

352-

357-269

362-66

368-51

371-12

372-

380-66

389-345

395-375

398-813

403-043

403-88

405-041

407-61f

408-40;

410-87:

419-33:

423-25

425-03

429-08

432-

435-22

127-72
129-06
130-02
131-17
132-57
133-91
137-51
138-93
139-92
140-75

137-40

139-70
140-90

144-30
147-70

150-56

154-40
157-70

161-30

164-80
165-50
or 167-

231-

2384

142-62
136-36

133-75.

136-85

137-55 .

144-33

151-92

144-05

145-15

154-28

143-65
146-19
147-50
150-49
151-03
153-57

154-88

161-12

172-99

173-61

178-56

188-21

212-28

214-37

219-84

224-88

228-74

234-35

240-07

244-61

247-86

254-51

261-32

268-35

287-40

308-13

331-99

343-12

346-95

386-47

429-7E

462-2(

481-5f

506-3J

511-4:

518-4]

) 534-3(

7 539-2f

\ 557-5

$ 611-8

7 639-8

652-9

683-4

706-1

7 731-9

150-65
155-

152-80

163-51

159-45
179-40

176-

181-23
197-13

194-42

220-69
227-32

240-48

248-92

242-17

267-62

274-

278'33

290-35

297-36

295-28

316-35

t342-51

348-04

325-
.620

393-66

433-83

467-02

483-88

511-31

514-22

P516-84

538-76

. 540-85

. 553-69

. 610-23

. 635-95

. 644-96

. 676-49

)

!

)

i

) . ...

i

i

7

3

3

2

2

. 716-13

12 Mr. J. H. Alexander on the Tension of Vapour of Water.

Of the earlier observations in the preceding table I need
not say much. Those of Mr. Watt, which he suffered to lie
by him for forty years, and, in the caustic phrase of Tredgold,
only produced when they had become unnecessary, he was
himself dissatisfied with, but, as appears upon comparison,
with more modesty than reason. I have specially calculated
but two or three of his temperatures ; and of the whole sixty-
two experiments, have inserted but twenty-two; among which,
however, both the limits are to be found. Of his friend
Robison's, I have had to calculate none specially; but all
happened to find a place in the table. Of M. Betancourt's
numerous observations, which were reported originally in de-
grees of Reaumur and French inches, I have inserted only
those which have been reduced to English scales by Sir 1).
Brewster for the Edinburgh Encyclopaedia.

The experiments of Mr. Southern are, in fact, the supple-
ment of those of Mr. Watt; having been made and reported
at the desire of the latter. The numbers will be found to
differ somewhat from those generally found in professed
treatises on the steam-engine ; they are in fact the mean of
the actual observations; while those usually given have been
selected now from one and now from the other set, and re-
duced (by himself) to what they might have been, had the
pressure at 212° been thirty inches. For the present purpose
it seemed to me proper to state the real, not the possible
result.

Mr. Dalton's experiments were distinct, and are therefore
given in distinct columns. The numbers in the earlier column,
marked with an asterisk, were not from actual experiment,
but by interpolation, according to the method he has him-
self explained. I have inserted them opposite to experimental
numbers in the adjoining -column, for the sake of comparison
and the benefit of the inference which may flow from the va-
riations. The numbers in the later column were not in
every case given by his own experiments ; but they were ac-
cepted by him as authentic, and the most reliable he knew.
It is more complimentary' to his reputation than to their own
research, that compilers of chemical manuals, even down to
the present time, retain among their tables his ancient results
whose inaccuracy he himself has recognised. All of his expe-
riments, of Southern's, and of Dr. Ure's, are in the table.
To the originals of M. Arzberger I have not had access; but
I have found these quoted in so many authorities and so uni-
formly accordant that I have not hesitated in recording them.
Of the extensive table of Mr, Philip Taylor, whose remark-
able accord with the results from the formula I may be allowed

Mr. J. H. Alexander on the Tensioji of Vapour of Water. 13

to call attention to, I have taken only those epochs of tempe-
rature which were already in my table.

Tlie experiments of the French Academy have been already
signalized. It is enough to establish their claim to distinction,
to say that they were executed by Dulongand Arago; names
that have been long since inscribed in the very highest rank
of physical philosophers. The numbers found in the appro-
priate column, are, agreeably to what I have already men-
tioned as governing me throughout, quantities actually ob-
served. The temperatures and pressures generally quoted in
the text-books on steam, as of the French Academy, are not,
in fact, what they observed, but what they deduced (in part,
by a formula of their own, and in part by Tredgold's) from
the present experimental series. The pressure 29*92 inches
corresponding to the temperature 212°, is marked with an
asterisk, because it is not expressly declared to have been
observed. It is the height which is constantly taken in France
for the barometric standard, as thirty inches are in England:
in the latter assumption, the temperature is rated at 60° F., —
in the former, at 32° F. ; and the difference of heights is
nearly identical with the difference of expansion at the re-
spective temperatures.

The pressure in this series corresponding to the tempera-
ture 368°'51, is also noted with a dagger ; it may be presumed
to be erroneous, not only because it differs so much from the
result by my formula, but because it varies so much and so
suddenly from the rate accused by the pressure on either side
of it. Nor does it correspond at all with their own formula;
calculated by that, the pressure will be 335'87 inches. The
error is not, in this instance, of the press ; since it makes its
appearance in both ways of reckoning, by atmospheres and by
metres.

I do not know how to account for another discrepant pres-
sure, corresponding to the temperature 405°*04<; which has
been indicated by a note of interrogation. On both sides,
above and below, the observed pressures are higher than the
calculated one ; in this instance it is suddenly lower. It
agrees, to be sure, with an independent calculation by the
formula of Dulong and Arago, at the temperature ; but very
manifestly breaks the uniformity or any regular progression
of the series. What adds to the difficulty, is that the same
observation is given again in another part of the Memoir of
the Academicians ; but the ciphers do not agree. I have
neither altered nor omitted either of these instances ; it is
obvious that they are not to be used in comparison with the
present formula.

14 Mr. J. H. Alexander on the Tensioji of Vapour of Water.

The temperatures of the Franklin Institute, which were
taken for the composition of the table, come from the second
series of their reported experiments. Pressures have been
also taken from both the other series, when their temperatures
were already in the table ; and, adopting this method as a
uniform system, I did not allow myself to exclude the ano-
maly which shows itself between the different series at the
temperature of 300°.

Of the experiments of M. Regnault, I have already spoken
sufficiently.

It is apparent, upon a slight examination of the table, that
the calculated pressures do not differ more from the average
of the experimental ones, than these experimental ones do
among themselves ; which is about as much as could be de-
sired to show the validity of the formula, and the reasonable-
ness of its application, instead of others which are in general
merely means of interpolating a particular experimental series.
But in order to establish this more clearly, it will be necessary
to ascertain more distinctly what the difference is between the
results of the formula and those of observation. This differ-
ence is, of course, best expressed in the arithmetical scale of
temperatures ; as I have tabulated it here, upon the maximum
deviation in each instance.

Temperature.

Greatest Differences.

Observed.

Calculated.

Southern's experiments

343-6

340-

322-

320-

362-66

297-46

343-09
340-74
320-98
322-34
364-73
300-41

+

0°~51
1 -02

Dalton's of 1820

0°-74

Arzberger's experiments

Taylor's experiments

2 -34
2 -07
2-95

French Academy's experiments . . .
Regnault's experiments

The mean of the sum of these differences is +1°'09 Fahr-
enheit ; which is the maximum error of the formula, com-
pared with these six series.

It will be observed that I have left out of this comparison
the last observation of the Academy ; because it was the very
utmost point which the apparatus could carry, and because it
might therefore be expected to be affected by the untrustwor-
thiness which forbade the series from being extended further.
I have also neglected the last observation of Arzberger, which,
compared with the Academy's, is in error more than 10°, —
a deviation sufficient to discredit it entirely. Ure's experi-
ments I have not compared at all ; because, if we admit the
series just now tabulated, his results are altogether too high.
He may, however, be compared with himself, in the two

Mr. J. Brown on the Products of the Soda Manufacture. 15

results he has recorded for his last observation. These two
different pressures accuse a corresponding difference of tem-
perature of 0°*63 F. ; a possible error, not so materially less
than what we have found as the maximum that can attach to
the formula. The Franklin Institute experiments, which cor-
respond closely with Ure's, I have omitted for a similar reason;
they do not profess even to read nearer than 0°*25 F. They
may, however, for illustration be compared with those of the
Academy, as under : —

Academy pressure, 145" 15 inches; temp, observed, 305°'39 ; temp, calculated, 307°*51
Institute ... 154*28 ... ... ... 305 50 ; ... ... 311*73

Diffbrences g*l3 0°*ll 4°*22

Discounting the observed difference from the calculated one,
we have left 4°' 11 F. as the error of one or the other series;
an amount nearly four times that of the formula.

It is manifest that the comparative error of the formula is
only approximate; because it is based in each case upon
only one observation instead of upon the combined mean of
all the observations, or, rather, the mean of the diff*erences at
every epoch observed. Also, it can only be called an error,
upon the assumption of the mean of all the experiments re-
sulting in absolute accuracy; an assumption by no means to
be made ; for in general the utmost that can be done for any
experimental series, is to determine the limits of its necessary
or accidental errors. Such a research and determination I
have thought the present formula of sufficient interest to war-
rant. The account, which is in fact the promised and proper
conclusion of the present paper, will appear in a future num-
ber of this journal.

[On this subject, see the following papers in the Philosophical Magazine:
first series, Philip Taylor, vol. Ix. for 1822, p. 452, with engraved scale for
each degree of temperature from 212° to 320° Fahr. Mr. Ivory, vol. i.
second series, for 1827. Mr, Farey, S. 3. vol. xxx. for 1847. — Also the
papers by Holtzmann, Magnus,andRegnault, in Taylor's Scientific Memoirs,
vol. iv.]

II. On the Products of the Soda Mamifacture.
By John Brown, Esq.*

GLAUBER first showed in 1658 that common salt could
be decomposed by sulphuric acid. In the year 1736, Du
Hamel proved the base of common salt to be soda. Previous
to this, however, Cohausen, in 1717} had mentioned that salt
might possibly be decomposed by means of lime ; but as this
observation was associated with numerous errors, it was en-
tirely overlooked. In 1 737 Du Hamel succeeded in obtaining

* Communicated to the Philosophical Society of Glasgow by Dr. R.
D. Thomson, and read April 12, 1848.

16 Mr. J. Brown oJi the Products of the Soda Manufacture.

the alkali from sulphate of soda by fusing with charcoal, and
digesting the fused mass in acetic acid, evaporating the acetate
of soda thus formed to dryness, and calcining the residue.

Margraff" endeavoured to decompose sulphate of soda by
limestone, but without success. In 1768 Hagen showed that
salt might be decomposed by means of potash, chloride of
potassium and caustic soda being formed.

Bergmann succeeded in decomposing salt by caustic barytes.

In 1775 it was shown by Scheele that salt was partially
decomposed by oxide of lead.

In 1782 Guy ton and Carny decomposed salt by fusion with
felspar.

In 1781 Constantini succeeded in decomposing salt by
means o^ alum.

The sulphates of lime^ magnesia, ammonia, potash, S^c. de-
compose salt, as also iron pyrites.

To convert the sulphate of soda into caustic or carbonated
alkali was, however, the process of greatest importance. The
first step, viz. the conversion of soda into sulphuret of sodium,
was known to Glauber, Stahl, Du Hamel, Margraff and
others. The difficulty was to get rid of the sulphur. Du
Hamel effected this by means of acetic acid. But in the year
1784 the present process was discovered by Le Blanc and
Dize, and in the beginning of 1791 it was patented by Le
Blanc. He used carbonate of lime to convei't the sulphuret of
sodium into carbonate of soda.

The proportions used by him were —

2 parts dry sulphate of soda.
2 ... carbonate of lime.
1 ... ground charcoal.

These were intimately mixed and introduced into a rever-
beratory furnace, where a strong heat was applied to it. After
this had been continued for about an hour, the fused mass was
raked out of the furnace and allowed to solidify. When this
cooled, it was broken up and exposed to the action of moist
air, which caused it to crumble down. In this way the caustic
soda was converted into carbonate of soda, the carbonic acid
being derived from the atmosphere. After being ground it
was ready for use.

The soda process, as at present carried on, will be best
considered under the four following heads : —

1st. The production of sulphate of soda from salt and sul-
phuric acid.

2nd. The conversion of sulphate of soda into crude carbo-
nate of soda or British barilla.

Mr. J. Brown on the Products of the Soda Manufacture, 17

3rd. The soda-ash process.

-ith. The carbonate of soda process.

I. The first stage which thus comes under our consideration
is — The decomposition of common salt by sulphuric acid, causing
the formation of sulphate of soda and muriatic acid.

The salt used in this process is obtained from the brine-
springs of Cheshire, which exist abundantly in tiie new red
sandstone of that county. The sohition is evaporated^ till it
reaches a certain strength, when all the salt precipitates ; it is
then raked out into wicker baskets and allowed to drain. The
mother-liquor is used for the manufacture of the salts of mag-
nesia. The salt thus obtained contains, as might be expected,
numerous impurities, the principal of which ai'e lime, sulphuric
acid and magnesia.

To estimate the lime, a portion of the salt was dissolved in
water, and after separating the insoluble matter by filtration,
the lime was precipitated by ammonia and oxalic acid, a large
quantity of muriate of ammonia being added to retain the
magnesia in solution.

CaO, COj. CaO. CaO per 1 000 gis.
2000 grains of salt gave 15-10 8-456 4*228

14-60 8-176 4-088

Average 4 158

The sulphuric acid was precipitated by the addition of nitric
acid and nitrate of barytes.

BaO, SO3. SO3. SO3 per 1 000 grs.
2000 grains of salt gave 39-85 13-738 6-869

39-50 13-620 6-810

Average 6-839
The quantity of magnesia was ascertained by precipitation
by ammonia and phosphate of soda, the lime having been pre-
viously separated.

2MgO,P2 0A. MgO. MgO per 1000 grs.

2000 grains of salt gave 465 1-660 0-830

The carbonate of lime remained as insoluble matter when
the salt. was digested in water, and was separated by filtration.

CaO, CO.. CaO, CO2 per 1000 grs.
2000 grains of salt gave 3-000 1-50

By estimating the amount lost by drying the salt at 212°
the quantity of water was ascertained.

Water per 1000 grs.
330-2 grains of salt lost 17-96 54-373

In order to estimate the quantity of iodide of potassium and
bromide of magnesium, li lb. of salt was put into a funnel,
the lower end of which was closed with filtering-paper. The

Phil. Mag. S. 3. Vol. 34. No. 226. Jan. 1849. C

18 Mr. J. Brown on the Products of the Soda Manufacture.

salt was then repeatedly washed with boiling water. The
iodide and bromide were thus taken up by the water along
with a large quantity of common salt. This solution was eva-
porated to dryness and the residue digested in alcohol, which
dissolved the iodide and bromide along with a little of the salt,
leaving, however, the greater part of it, which was afterwards
separated by filtration. The filtered solution was again eva-
porated to dryness and the residue digested in water. Chlo-
ride of palladium was then added, but no precipitation of
iodide of palladium took place. The palladium was preci-
pitated by sulphuretted hydrogen, and the sulphuret of palla-
dium thus formed separated by filtration. Upon testing the
filtered solution with ammonia and nitrate of silver, no preci-
pitate was obtained. Had bromine been present, it would
have been precipitated in combination with the silver, bromide
of silver being insoluble in caustic ammonia. It is therefore
evident that the common salt manufactured, as previously men-
tioned, does not contain iodine or bromine, although it is
highly probable that these bodies are present in small quantity
in rock salt; and we might therefore be able to detect them
in the brine from which the magnesia salts are manufactured.
Upon treating the salt with bichloride of platinum a slight
precipitate of potassio-chloride of platinum was obtained.

Composition of Commercial Salt.

Magnesia.

Lime.

Sulphuric
acid.

Chloride of sodium

931-615
trace.

1066

10098

1-348

1-500

54-373

0-381

4-158

5-940
0-899

Chloride of ma^'nesium

S ulphate of magnesia

0-449

Carbonate of lime

Water

1000-000

0-830

4-158

6-839

About 6 cwt. of this salt is introduced into an iron pot, and
upon this is run, by a siphon, about 5| cwt. of sulphuric acid
of about 1*7.50 specific gravity (150° Twaddell). A violent
action immediately takes place, and large quantities of muriatic
acid gas are evolved, which pass off by a chimney. If, how-
ever, the muriatic acid can be made use of, the gas is absorbed
either by passing it tnrough water contained in large cylin-
drical vessels or through a column of coke, which retains the
gas until a considerable quantity of it is collected; a stream
of water is then allowed to trickle through the coke, and in

Mr. J. Bi-own on the Products of the Soda Manufacture, 19

this manner all the gas is absorbed. At the expiration of
about two hours the evolution of gas ceases; and the sulphate,
which is in a semifluid state, is removed to another chamber,
where it is strongly heated in order to drive off the whole of
the acid. The whole operation takes about four hours.

The foreign matters contained in the sulphate of soda thus
obtained are sand, peroxide of iron, magnesia and undecom-
posed salt.

To estimate the sand. This remained as insoluble matter
when the sulphate was digested in water containing muriatic
acid, and was separated by filtration.

1000 grains of sulphate of soda gave 2-82 grains of sand.

3-38
Average 3*10

From the solution filtered from the sand the peroxide of
iron was precipitated by ammonia, muriate of ammonia having
been previously added to retain the magnesia in solution.

1000 grs. of sulphate of soda gave 2-15 grs. peroxide of iron.

Average 2-30
After separating the sand and peroxide of iron, as men-
tioned above, the lime was precipitated by oxalic acid and
caustic ammonia.

CaO.COj. CaO.SOg,
1000 grains of sulphate of soda gave 7'000 9"656

7-367 10-019
... * ... 7-100 9-520

Average 9*731

The solution thus freed from lime, &c. was treated with
ammonia and phosphate of soda. The magnesia was thus
separated as ammonio-phosphate.

2MgO, P2 0,. MgO.SO
1000 grains of sulphate of soda gave 2*70 2-893

The quantity of chloride of sodium was ascertained by pre-
cipitating the chlorine by nitrate of silver and nitric acid.

Ag CI. Na CI per 1000 grs.
200 grains of sulphate of soda gave 4-30 8-995

1000 ... ... 29-70 12-373

500 ... ... 13-80 11-500

Average 10-956

The sulphate of soda always contains a small quantity of
free acid, the amount of which was ascertained by determining
the weight lost by heating to redness.

C 2

20 Mr. J. Brown on the Products of the Soda Manufacture.

Per 1000 grs.
200 grs. of sulphate of soda lost 1*70 8*50 grs. free acid.

1-84. 9-20

Average 8*85

Composition of Crude Sulphate of Soda.

Sulphate of soda .
Sulphate of lime
Sulphate of magnesia
Chloride of sodium
Iron peroxide . .

Sand

Free acid . . .

962-170
9-731
2-893
10-956
2-300
3-100
8-850

1000000

II. This brings us to the consideration of the second part of
the process, namely, — the conversion of sulphate of soda into
crude carbonate of soda or British barilla.

This is effected by the combined action of coal and carbo-
nate of lime. The following table shows the proportions
commonly used : —

Sulphate of soda .
Ground limestone .

cwt. qrs.
. 2 2
. 2 2i

Per cent,
lbs.
100
1029

Theoretical quantity,
lbs.
100
105-3

Coal dross . , .

. 1 3

6T7

33-6

These, after being intimately mixed, are introduced into a
reverberatory furnace and strongly heated. The mass soon
becomes soft, when care must be taken to stir it frequently in
order to expose a fresh surface to the heat. When it becomes
of the consistence of dough the chemical action commences, and
jets of inflamed carbonic oxide begin to issue from it. The
evolution of gas soon becomes very rapid, so much so, that
the whole mass appears to be in a state of ebullition. When
this ceases the operation is completed, and the fused mass is
raked out of the furnace and allowed to solidify. The cake
thus obtained is the crude carbonate of soda, or, as it is tech-
nically called, " soda ball " or " black ash."

This process consists of two subprocesses, which might be
carried on in separate furnaces. 1. The coal is consumed at
the expense of the oxygen of the sulphate of soda, causing the
formation of sulphuret of sodium and carbonic oxide —

NaO, S03 + 4C=NaS + 4CO.

2. The sulphuret of sodium thus formed is decomposed by

Mr. J. Brown on the Products of the Soda Manufacture. 21

the carbonate of lime, with the formation of sulphuret of cal-
cium and carbonate of soda —

NaS + CaO, CO^sNaO, COa+CaS.

But if this compound were digested in water, a reverse
action would immediately take place, sulphuret of sodium and
carbonate of lime being again formed. To obviate this diffi-
culty, a large excess of lime is used in the process, nearl}'
twice as much as would otherwise be absolutely necessary.
This excess of lime causes the formation of a compound inso-
luble in water, the composition of which is SCaS, CaO. This
substance has no effect on a solution of carbonate or caustic
soda.

Analy&is of Soda Ball, or Crude Carbonate of Soda.

An average sample was obtained by pounding a large quan-
tity of the soda ball, and from this the specimens analysed
were taken.

1. To estimate the amount of soluble and insoluble salts. —
A portion of the substance was thrown on a weighed filter and
washed with water at about 120° F., until a portion of the
filtered liquor left no residue on evaporation ; the filter and
insoluble matter were then dried in a water-bath and weighed.

Soda ball. Insoluble matter. Soluble matter.

100 gave . . 59-87 4.0-13

58-92 41-08

' ... 59-90 40-10

Average 59*56 40-43

2. Sulphate of soda. — After saturating the soda ball with
pure muriatic acid, and separating the insoluble matter by
filtration, the sulphuric acid was precipitated by chloride of
barium.

Soda ball. BaO, SO3. BaO, SO3 per cent. NaO, SO3 per cent.

245-20 gave 8*50 3*466 2-147

110-00 ... 1-30 1-181 0-733

78*36 ... 0*76 0*969 0*601

Average 1*872 1*160

3. Chloride of sodium. — The soda ball was digested with
nitric acid and filtered, and from the filtered solution the chlo-
rine was precipitated by nitrate of silver.

Soda ball. AgCI. CI. Na CI. Na CI percent

98 gave 5*400 1-350 2*250 2*295

100 ... 3-679 0-912 1-532 1*532

Average 1*913

22 Mr. J. Brown on the Products of the Soda Manufacture.

4. Soda. — The total quantity of available soda, that is, soda
existing as carbonate, sulphuret and hydrate, was determined in
the following manner : — A portion of the soda ball was thrown
on a filler and washed with warm water until all the soluble mat-
ter was taken up ; the filtered solution was then exactly neu-
tralized by dilute sulphuric acid, which was afterwards preci-
pitated by chloride of barium. From the quantity of sulphate
of barytes thus obtained, the amount formerly got from the
sulphate of soda was deducted, and from the remainder the
per-centage of alkali was calculated.

Soda ball. BaO, SO3. BaO, SO3. BaO, SO3 p. c. Soda p. c.

44-60 gave 40-60 91-031 — l-872 = 89-159 24*593

100 ... 88-96 88-960— 1-872 = 87-088 24-024

48-50 ... 42-76 88-164 — 1-872 = 86-292 23-800

Average 24-138

5. Sulphur. — The amount of sulphur was determined in two
different ways:--lst. The soda ball, after being very carefully
pulverized, was intimately mixed with about four times its
weight of nitrate of potash and heated in a covered platinum
crucible. The nitrate of potash was thus decomposed, and
the sulphur converted into sulphuric acid by the oxygen of
the nitric acid.

KO,N02 03+S=S03 + KO + N02.

The fused mass was dissolved by muriatic acid, and after fil-
tering the solution, the sulphuric acid was precipitated by
chloride of barium. 2. The soda ball, moistened with a small
quantity of water, was intimately mixed with a quantity of
finely pulverized chlorate of potash, and to this muriatic acid
was added, drop by drop, until upon a fresh addition of acid
no more gas was evolved. The flask containing the substance
was then gently heated by means of a water-bath, care being
taken to keep the temperature below 180° F., as chlorous acid
explodes with great violence at about 200° F. When all action
had ceased, the solution was filtered, and the sulphuric acid
precipitated by chloride of barium. From the weight of the
sulphate of barytes thus obtained, the former quantity, 1-872,
was deducted, and from the number thus found the amount
of sulphur was calculated.

Soda ball. BaO, SO3. BaO, SO3 per cent. Sulphur per cent.

By 1st / 19-34 gave 17-90 92-554- 1-872=90-628 12-507

method. I 19-53 ... 18-20 93-189— 1-872=:91-3] 7 12-595

By2nd?28-90 ... 27-00 93-425-1 •872=91-553 12-627

method. 129-60 ... 2720 91-891-1-872=90019 12-416

Average 12536

Mr. J. Brown on the Products of the Soda Manufacture. 23

6. Magnesia. — This was precipitated by ammonia and phos-
phate of soda.

Ball soda. 2MgO, P^ O5. MgO per cent.

100 gave . 0-980 0-350

7. Silica and sand. — The soda ball was dissolved in muriatic
acid and the solution evaporated to dryness. The residue
was then digested with strong muriatic acid, and the insoluble
matter separated by filtration.

Ball soda. Silica and sand. Silica and sand per cent.
56-00 gave . 4*30 7-679

The silica was separated from the sand by strong caustic
potash.

Ball soda. Sand. Sand per cent. Silica per cent.

56-00 gave 2-40 4-285 3-394

8. Iron and alumina. — A portion of the soda ball was dis-
solved in muriatic acid, and after separating the insoluble
matter, the iron and alumina were precipitated by caustic am-
monia.

Ball soda. AI2O3 & Fcj O3. AI2O3 & Fcj O3 per cent.
61-20 gave 3-45 5*637

19-53 ... 1-15 5-888

29-10 ... 1-45 4-982

Average 5*502

The peroxide of iron was separated from the alumina by
caustic potash.

Ball soda. Fe2 O3. Fe2 O3 per cent. Fe per cent. AI2O3 per cent.

61-20 gave 2-94 4-804 3-363 0-833

29-10 ... 1-20 4-123 2-886 0-859

Average 3-129 0-846

9. Lime. — From the solution filtered from the alumina and
iron, the lime was precipitated by oxalate of ammonia.

Bali soda. CaO, COj. CaO. CaO per cent.

61-20 gave 33-00 18-480 30-194

29-10 ... 15-50 8-680 29-828

21-80 ... 12-05 6-748 30-954

Average 30-325

10. Carbonic acid. — By the addition of muriatic acid to
the ball soda, sulphuretted hydrogen and carbonic acid gases
were evolved, which were passed through a strong solution of
caustic barytes. The precipitated carbonate of barytes was
filtered as rapidly as possible, care being taken to keep it
covered with a plate of glass during the process.

24 Mr. J. Brown on the Products of the Soda Manufacture.

Ball soda. BaO, CO2. COj. CO2 per cent.

4.5-35 gave 28-90 6*487 14-304

90-18 ... 59-20 13-289 14-736

Average 14*620

11. Carbon. — To determine the amount of carbon, a por-
tion of the ball was treated with muriatic acid and the solution
evaporated to dryness ; dilute acid was then added, and the
insoluble matter thrown on a filter which had been previously
dried at 212^ and weighed. The total amount of carbon,
silica and sand, was thus ascertained. The whole was then
ignited and weighed, and from the loss the per-centage of car-
bon was calculated.

Ball soda. Insoluble matter. Carbon per cent.

100 gave 15-941, which lost on ignition 7"998

12. Water. — The soda ball was dried at 212°, and the
amount lost estimated.

Ball soda. Water. Water per cent.

50-00 lost . . 0-35 0-700

Whilst washing out the soluble salts, it was observed that
the filtered solution was of a greenish colour; and upon boil-
ing it a green-coloured substance was deposited, after which
the supernatant liquor became perfectly colourless. Upon
examining this precipitate, it was found to consist principally
of silica and alumina with a little lime. From this it was
concluded to be artificial ultramarine, which is frequently
found in the crevices of the ball furnaces, and which, when
dissolved in caustic soda, yields a green-coloured solution,
precisely the same as that mentioned above.

Ball soda. Ultramarine. Ultramarine per cent.

200 gave . . 0-46 0-23

100 ... 0-36 0-36

Average 0*295

Sulphate of soda . . . 1-160

Chloride of sodium . . 1913

Soda 24-138

Lime 30-325

Sulphur 12-536

Carbonic acid .... 14 520

Sand 4-285

Silica 3*394

Magnesia 0*350

Alumina 0*846

Iron 3-129

Water 0*700

Carbon 7*998

Ultramarine .... 0*295

Mr. J. Brown on the Products of the Soda Manufacture. 25

Soda.

Lime.

Carbonic
acid.

Sulphur.

Carbonate of soda

35-640
0-609
2-350
1-160
1-130
1-913
0295

29-172
6-301
4-285
4-917
3-744
7-998
0-700

21-120
0-609
1-504

0-905

14-520

0-454

10-296

1786

Aluminate of soda

Sulphate of soda

24-024
6-301

Ultramarine

3CaS+CaO

Sand

Silicate of magnesia

Carbon

Water (hygroscopic)

100-214

24-138

30-325

14-520

12-536

It will be seen that in the above analysis I consider almost
all the soda to be united with carbonic acid, there being very
little caustic soda. Unger and others who have examined the
soda balls, fall into the error of supposing a large quantity of
the alkali to exist as hydrate, and also of always finding car-
bonate of lime; but if a portion of the ball soda be digested
in alcohol, and the alcoholic liquor carefully examined, it will
be found that it holds in solution a very small quantity of
alkali, which I consider to be as sulphuret. If, on the con-
trary, the soda balls contained caustic soda, it would be im-
mediately dissolved by the alcohol, and we should obtain a
strongly alkaline solution. This, however, is not the case.
But if the ball soda be digested in water, the liquid will be
found to contain a large quantity of caustic soda, which, how-
ever, can easily be accounted for in the following way. There
exists in the ball soda a large quantity of caustic lime ; and
whenever water is added to it a decomposition takes place,
carbonate of soda and caustic lime becoming carbonate of lime
and caustic soda, —

NaO, C02+CaO = CaO, CO^+NaO.

Some analysts have also found water of combination in ball
soda, that is, water united to soda or lime. But this is im-
possible, for where does the water come from ? The materials
contain none. A small quantity of water is certainly formed
in the combustion of coal, but this is not sufficient to account
for it. The method of analysis pursued in the determination
of the amount of water combined with soda or lime was, I
think, very incorrect : it was to burn the ball soda with chro-
mate of lead, and determine the weight of the water given off.

26 Mr. J. Brown on the Products of the Soda Manufacture.

Had any undecom posed coal existed in the waste, it would
have contained hydrogen, and water would consequently have
been formed, the oxygen being derived from the chromic acid
of the chromate of lead.

As might be expected, I found upon trying samples taken
from different furnaces, that the constituents were subject to
great variations. Thus the lime varied from 27 per cent, to
34 per cent. ; the soda from 22 per cent, to 26'5 per cent. ; the
sulphur from 10 per cent, to 16 per cent. But they always
stood in a certain fixed relation to one another ; for when the
quantity of lime was large the amount of sulphur was propor-
tionally increased, and the per-centage of soda consequently
diminished. The following table will suffice to show this : —

I. II. III.

Soda . . . 26-480 22-000 24-138

Lime . . . 26-959 33-807 30-324

Sulphur . . 10-527 13-820 12-436

I insert here two analyses of soda balls ; the one from Cassel
by Unger, the other from Newcastle by Richardson. They
both get hydrate of soda and carbonate of lime, and are, I
think, wrong in both of these, although the other parts of the
analysis are probably quite correct.

The manufacture in Cassel and Newcastle is carried on
almost exactly in the same way as here.

From Cassel. From Newcastle.

Sulphate of soda .

. . 1-99

3-64

Chloride of sodium

. 2-54

0-60

Carbonate of soda

. 23-57

9-89

Hydrate of soda .

. 11-12

25-64

Carbonate of lime

. . 12-90

15-67

3CaS, CaO . . .

. . 34-76

35-57

Sulphuret of iron

. 2-45

1-22

Silicate of magnesia .

. 4-74

0-88

Charcoal . . .

. 1-59

4-28

Sand

. 2-02

0-44

Water (hygroscopic)

. 2-10

2-17

99-78 100-00

III. This brings us to the consideration of the third division
of the soda process, viz. the manufacture of soda-ash from
hall soda.

The first point is to extract all the soluble matter from the
balls. This is done by digestion in warm water. The vessels
used for this purpose are large square iron pans, five or six of

Mr. J. Brown on the Products of the Soda Manufacture. 27

which are usually worked together. They are so contrived
that the water which runs into the first pan passes through
the whole six in succession. In this way a very saturated
solution is obtained. From the last digester the liquor is run
into a large iron vessel, where it is allowed to settle : the in-
soluble matter which remains in the pans is of no use and is
therefore thrown away. It is a source of great annoyance to
the manufacturer, as also to the whole neighbourhood of the
place where it is deposited, large quantities of sulphuretted
hydrogen being evolved from it. Numerous attempts have
been made to recover the sulphur from it, but without success.

Atialysis of Soda Waste.

The following analysis of fresh soda waste was made in the
same way as that of the ball soda.

1. Sulphuric acid. — The waste was digested in pure mu-
riatic acid, and after separating the insoluble matter by filtra-
tion, the sulphuric acid was precipitated by chloride of barium,

BaO, SO3 per cent. CaO, SO3 per cent.
7-500 4.-396

7-108 4-166

Waste.

BaO, SO3,

28-00

2-10

30-95

2-20

Average 4-281

2. Sulphur. — The sulphur was oxidized by chlorate of

potash and muriatic acid, and the sulphuric acid thus formed
precipitated by chloride of barium.

Waste. BaO, SO . BaO, SO3 p. c. Sulphur p. c.

27-75 gave 27-56 99-315 — 7-304 = 92-011 12-689

30-90 ... 32-40 104-854 — 7-304 = 97-550 13-455

26-95 ... 27-80 103-154 — 7-304 = 95-850 13*220

13-182

3. Silica and sand. — By dissolving the waste in strong mu-
riatic acid, evaporating to dryness, and dissolving the residue,
the silica, sand and carbon remained as insoluble matter, the
last of which was destroyed by ignition. The silica and sand
were then separated by caustic potash.

Waste. SiO&sand. SiO. Sand. SiOp. c. Sandp.c.

50-00 gave 5-513 containing 2-640 and 2-873 5-280 5*746

4. Peroxide of iron. — After separating the silica and sand,
the iron was precipitated by caustic ammonia. It contained
a very small quantity of alumina.

28 Mr. J. Brown on the Products of the Soda Manufacture.

Waste.

Fe2 O3.

Fca O3 per cent.

20-00 gave

. 1-10

5-500

50-00 ...

2-46

4-920

21-40 ...

1-44

6-729

Average 5*716

5. Lime. — After the iron had been precipitated by am-
monia, the lime was thrown down by oxalic acid.

Waste. CaO, CO2. CaO. CaO per cent.

21-40 gave 17-10 9-576 44-747

48-90 ... 39-10 21-896 44-777

Average 44-762

6. Magnesia. — After separating the lime the magnesia was
precipitated by phosphate of soda and ammonia.

Waste. 2MgO, P^ O5. MgO. MgO per cent.

48-90 gave 0 970 0*346 0707

7. Carbonic acid. — A quantity of the waste was put into a
flask and dilute acid slowly added to it. The carbonic acid
thus disengaged was passed through a solution of caustic ba-
rytes, and from the quantity of carbonate of barytes thus pre-
cipitated the amount of carbonic acid was calculated.

Waste. BaO,C02. CO2. CO2 per cent.

30-80 gave 15*65 3*513 11*406

27*20 ... 13*30 2*985 10*974

Average 11*190

8. Soluble and insoluble salts. — The whole of the soluble
matter was extracted by water, and the residue dried at 212°
and weighed.

Waste. Insol. matter. Insol. matter p. c. Sol. matter p. c.

.71*2 gave . 52*50 73*736 26*264

9. Carbon. — The amount of carbon was determined in the
same way as in the ball soda.

Waste. SiO sand and carbon. Carbon. Carbon per cent

50 gave . 11*552, lost on ignition 6*039 12*078
Insoluble salts per cent. 61-658

10. Carbonic acid in insoluble salts.

Waste. BaO.COa.

20-30 gave 15-70

11. Lime in insoluble salts.

Waste. CaOjCOj.

23-80 gave 20-90

C02.

CO2 in insol. salts.

3-525

10-657

CaO.

CaO in insol. salts.

11-704

30-448

Mr. J. Brown o?i the Products of the Soda Manufacture. 29

12. Bisulphuret of calcium. — A quantity of the waste was
digested with muriatic acid and a large quantity of water, and
heated till the whole of the sulphuretted hydrogen was dissi-
pated. The sulphur which remained was then oxidized by
chlorate of potash and muriatic acid, and the sulphuric acid
thus formed precipitated by chloride of barium. But as this
method does not yield very accurate results, the amount of
bisulphuret of calcium given below can only be considered as
an approximation.

Waste. BaO, SO3. Sulphur. Sulphur per cent. CaSj per cent.

35-8 gave 11-45 1-579 2-205 3'583

13. Hyposulphite of lime. — About 100 grains of the waste
were digested for twenty-four hours with a solution of oxalate
of potash ; a salt of the oxide of copper was then added, by
which all the sulphur was thrown down. The precipitated
sulphuret of copper was then separated by filtration, and to
the filtered solution sulphuric acid was added. At first no
precipitation took place; but after standing for one or two
hours, the solution became slightly turbid. The quantity of
sulphur was, however, too small for estimation.

14. Water.

Waste. Water per cent.

100 grains lost by drying at 212° 2*10

Soluble salts . . 26-264.
Insoluble salts . 73*736

100-000

Sulphate of lime ...... 4-281

Sulphur 13-182

Silica 5-280

Sand 5-746

Peroxide of iron 5-716

Lime 44-762

Magnesia 0-707

Carbonic acid 11-190

Carbon 12-078

Carbonic acid in insoluble salts . 10-657

Lime in insoluble salts . . . 30*448

Bisulphuret of calcium . . . 3-583

Hyposulphite of lime .... trace

Water 2-10

30 Mr. J. Brown on the Products of the Soda Manufacture.

Lime. Sulphur. ""^P^}'

Carbonate of lime

3CaSCaO

Carbon

Silicate of magnesia . .

Sand

Peroxide of iron

Sulphate of lime ,

Hyposulphite of lime. .
Bisulphuret of calcium
Sulphuret of calcium..,

Hydrate of lime

Carbonate of soda

Water (hygroscopic) ..,

21-220
20-363
12-709
5-987
5-746
5-716
4-281
trace
3-583
8-527
5-583
1-309
2-100

13-563
16-769

1645

1-929
6-631
4-225

10-657

7-187

2-205
3-790

0-533

100-124

44-762

13-182

11 190

As might be expected, the quantities of lime, sulphur and
carbonic acid, are subject to great variations, every sample
varying to a considerable extent.

Upon examining a sample of waste three or four weeks old,
I found the quantity of hyposulphite of lime to be much greater
than in perfectly fresh waste. Another specimen, which had
been partially exposed to the action of the atmosphere for
three years, was entirely converted into sulphate of lime,
sulphite of lime and carbonate of lime, and hyposulphate of
lime. Some specimens were obtained which consisted entirely
of sulphate of lime, carbonate of lime and causticlime. These
experiments are very interesting from their showing the gra-
dual oxidation of the sulphur which the waste contains.

The waste in the soda ball consists entirely of oxysulphuret
of lime (SCaS, CaO) and caustic lime. The SCaS, CaO soon,
however, decomposes, giving rise to sulphuret and bisulphuret
of calcium and caustic lime. The bisulphuret of calcium
being very efflorescent, forms on the waste heap a yellow coat-
ing of small prismatic crystals. The sulphur is then further
oxidized, the first products being hyposulphite and sulphite
of lime: the process still continuing, hyposulphate and sul-
phate of lime are formed ; and this oxidation goes on till
sulphate of lime remains. The caustic lime is also for the
most part converted into carbonate.

It would be very interesting to ascertain the exact amount
of each of these substances present in waste in different stages
of decomposition ; but there are as yet no methods known
by which sulphurous, hyposulphurous, and hyposulphuric
acid can be accurately determined, especially when existing
along with sulphuric acid and sulphurets, as in soda waste.
Under these circumstances, it would be impossible to make a

Mr. J. Brown on the Products of the Soda Manufacture. SI

series of analyses of the waste in its different stages of decom-
position, upon which perfect dependence could be placed ;
but it is to be hoped that as the science advances these at pre-
sent insuperable obstacles may be entirely removed.

The following is an analysis by Unger of a sample of waste
from Cassel.

Carbonate of lime

19-56

3CaS + CaO . . . .

32-80

Carbon

2-60

Silicate of magnesia .

. 6-91

Sand

3-09

Iron peroxide . . .

3-70

Sulphate of lime . .

3-69

Hyposulphite of lime

4-12

Hydrate of lime . .

11-79

Bisulphuret of calcium

4-67

Sulphuret of calcium

3-25

Sulphuret of sodium

1-78

Water

3-45

100-31
The soda waste thus affords ample room for further re-
searches, which if carefully prosecuted might yield very inter-
esting results. But without dwelling any longer on this sub-
ject, I pass on to the consideration of the remaining part of
this division of the process, viz. the manufacture of soda-ash
from the liquor containing the soluble matter extracted from
the ball soda.

This liquor contains carbonate of soda, caustic soda, sul-
phuret of sodium, sulphate of soda, and chloride of sodium,
with a little aluminate of soda, the greater part of which is,
however, soon decomposed by the action of the carbonic acid
of the atmosphere, carbonate of soda being formed whilst the
alumina precipitates. This solution is boiled down in an iron
pan until it is nearly dry.

Analysis of Soda-ash.
The analysis of this and the remaining salts were made in
the following way : —

1 . Carbonate of soda. — The amount of carbonate of soda
was determined by ascertaining the weight of the carbonic acid
which was evolved on the addition of muriatic or sulphuric
acid to the salt.

2. Sulphuret of sodium. — The amount of sulphuret of so-
dium was ascertained by passing the gases, evolved on the
addition of muriatic acid to the salt, through a solution of ar-
seniousacid in caustic potash. The sulphuret of arsenic thus
formed was precipitated by neutralizing the potash with nitric

32 Mr. J. Brown 07i the Products of the Soda Manufacture.

acid; it was then thrown on a filter, dried at 212° and weighed.
From its weight the quantity of sulphuret of sodium was cal-
culated.

3. Hydrate of soda. — To ascertain the quantity of hydrate
of soda, a portion of the substance was heated strongly with
carbonate of ammonia in order to convert the hydrate and
sulphuret into carbonate. The amount of carbonic acid was
then determined as formerly, and the difference between the
results of the two experiments gave the amount of carbonic
acid equivalent to the quantity of soda existing as hydrate and
sulphuret in the sample. The amount united to sulphur was
then deducted, and the remainder gave the per-centage of hy-
drate.

4. Sulphate of soda. — A portion of the salt was dissolved
in a pretty large quantity of water, and nitric acid added to
expel the carbonic acid. The sulphuric acid was then preci-
pitated by chloride of barium.

5. Sulphite of soda. — The salt was boiled with strong nitric
acid in order to oxidize the whole of the sulphite of soda and
sulphuret of sodium. Water was then added, and the sul-
phuric acid precipitated by a salt of barytes. From the quantity
of sulphate of barytes thus obtained, the amount got by the
former experiment was deducted, and the remainder showed
the quantity of sulphate of barytes equivalent to the amount
of sulphite of soda and sulphuret of sodium. The per-centage
of sulphuret of sodium being known, the sulphite of soda was
easily determined.

6. Chloride of sodium. — After expelling the carbonic acid
by nitric acid, the chlorine was precipitated by nitrate of silver.

7. Aluminate of soda and insoluble matter.' — A solution of
the salt was acidified by muriatic acid, and the insoluble matter
(principally sand) separated by filtration. From the filtered
solution the alumina was precipitated by caustic ammonia.

The salt obtained by evaporation from the liquor from the
keaves, after drying at 212°, yielded on analysis,-

Carbonate of soda . .

Hydrate of soda . . .

Sulphate of soda . . .

Sulphite of soda . . .
Hyposulphite of soda

Sulphuret of sodium . .
Chloride of sodium .

Aluminate of soda . .

Silicate of soda . . .

Insoluble matter . . .

100-735 99-94.1

I.

II.

6S-907

65-513

14-433

16-072

7-018

7-812

2-231

2-134

trace

trace

1-314

1-542

3-972

3-862

1-016

1-232

1-030

0800

0-814

0974

Mr. J. Brown on the Products of the Soda Manufacture. S3

This salt is then introduced into a reverberatory or car-
bonating furnace, where it is strongly heated. In this process
the sulphuret of sodium is converted into sulphate of soda,
and part of the hydrate of soda into carbonate. The salt
when removed from the furnace is ready for the market. In
Newcastle and some other places it is dissolved and carbo-
nated again, and when thus manufactured it contains less caustic
soda.

Soda-ash thus prepared contains from 48 to 53 per cent, of
available alkali, that is, alkali combined with carbonic acid
and water, and yielded on analysis, —

Analysis of ash

from Germany

I.

11.

by Utiger.

Carbonate of soda .

71-614

70-461

62-13

Hydrate of soda . .

11-231

13-132

17-20

Sulphate of soda . .

10-202

9-149

8-66

Chloride of sodium .

3-051

4-279

3-41

Sulphite of soda . .

1-117

1-136

0-35

Aluminate of soda .

0-923

0-734

1-11

Silicate of soda . .

1-042

0-986

2-56

Sand

0-316

0-464

0-62

Water

...

3-96

99-496 100-341 100-00

IV. Formation and Analysis of Carbonate of Soda.

The next stage of the process which comes under our con-
sideration is the carbonate of soda process.

The carbonate of soda balls are lixiviated with water in the
same way as in the manufacture of soda-ash. The liquor from
the settler is pumped up into a pan, where it is evaporated till
it becomes nearly dry ; it is then taken out of the pan in co-
landers, thrown up in a heap and allowed to drain. The
sulphuret of sodium and caustic soda soon deliquesce and
drain out from the salt.

This salt, after drying at 212°, gave when analysed, —

Phil. Mas. S. 3. Vol. 34. No. 226. Jan. 1849. D

34 Mr. J. Brown on the Products of the Soda Manufacture.

Carbonate of soda
Hydrate of soda . .
Sulphate of soda . .
Sulphite of soda . .
Sulphuret of sodium
Hyposulphite of soda
Chloride of sodium .
Aluminate of soda .
Silicate of soda . .
Insoluble matter . .

I.

II.

79-641

80-918

2-712

3-924

8-641

7-431

1-238

1-110

trace

0-230

trace

trace

4-128

3-142

1-176

1-014

1-234

1-317

0-972

0-768

99-742

99-854

This salt is then introduced into a reverberatory furnace and
carbonated. The last traces of sulphur are thus oxidized,
and almost the whole of the hydrate is converted into carbo-
nate.

This salt yielded on analysis, —

Carbonate of soda
Hydrate of soda .
Sulphate of soda .
Sulphite of soda .
Chloride of sodium
Aluminate of soda
Silicate of soda
Insoluble matter .

I.

84-002
1-060
8-560
trace
3-222
1-013
0-984
0-716

99-557

II.

83-761
0-734
9-495
0-386
3-287
0-620
0-780
0-846

99-909

A finer kind of soda-ash is frequently made from this salt by
dissolving it in water, evaporating to dryness, and carbonating.
It contains very little caustic soda, and should average about
50 per cent, of alkali.

It yielded on analysis, —

Carbonate of soda
Hydrate of soda .
Sulphate of soda .
Sulphite of soda .
Chloride of sodium
Aluminate of soda
Silicate of soda
Insoluble matter .

99-350 99-437

It is from this salt that the crystallized carbonate of soda is

I.

II.

84-314

84-721

trace

0-280

10-260

9-764

trace

3-480

3*140

0-632

0-716

0-414

0-318

0-250

0-498

Mr. J. Brown on the Products of the Soda Manufacture. 35

manufactured. The ash is dissolved in boiling water until
the solution attains a specific gravity of 1*250 (50° Twaddell);
it is then run into a cistern, where it is mixed with sufficient
cold water to reduce the specific gravity to 1*21 (42° Twad-
dell). This occasions the deposition of a quantity of earthy
matter. A little bleach ing-powder is then added to the liquid,
which causes another deposition. After this has been allowed
to settle, the solution is carefully decanted into another pan,
and evaporated till it attains a specific gravity of r27 (54<° T.).
From this it is run into another cistern, from which it passes
into the crystallizing pans. The average time taken in cry-
stallization is eight days; but it of course varies very much
with the season of the year and the state of the atmosphere.
It is greatly assisted by placing a few bars of wood, two or
three inches broad, on the top of the liquor.

The crystallized carbonate of soda thus obtained yielded on
analysis, —

I. II.

Carbonate of soda . 36*4.76 36*931

Sulphate of soda . . 0*94.3 0*542

Chloride of sodium . 0*424 0*314

Water . . . . ^62*157 62*213

100*000 100*000

As it contains ten atoms of water rf crystallization, its for-
mula is NaO, CO2+ lOHO, and the per-centage composition
calculated from this formula is —

Carbonate of soda , . 37*500
Water 62*500

100-000

By driving off the water from these crystals by heat, a very
pure carbonate of soda is obtained, which is used in the ma-
nufacture of glass.

It yielded on analysis —

Carbonate of soda
Sulphate of soda . .
Chloride of sodium .

99*938 99*671

98*120

97-984

1-076

1*124

0*742

0-563

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[ 37 ]

III. On a new Imaginary in Algebra, By James Cockle,
Esq.^ M.A., of Trinity College^ Cambridge, and Barrister-
at-Latio of the Middle Temple*.

ALGEBRA may be regarded under the triple aspect pre-
sented by the words Identity, Equivalence, and Impossi-
bility f; but the latter view will fall more particularly within
the scope of the present observations. The ordinary algebra,
it is true, takes cognizance, not only of negative and unreal
quantities, but sometimes of questions involving impossibility.
This impossibility is indicated either by contradictory arith-
metical results or, occasionally, by the symbol infinity %. But,
neither in the one case nor the other, does the indication of
impossibility furnish us with the elements of a calculus. Un-
real results, on the contrary, although not subjects of concep-
tion, like number, nor directly interprelable, like negative
quantities, are yet not only indirectly interpretable, but also
important instruments of investigation. Why is this? My
answer is, — because impossibility has never yet been symbol-
ized. And I would add that, before this is done, we ought to

* Communicated by T. S. Davies, Esq., F.R.S.L. & Ed., &c., who requests
us to annex the following note.

[It will of course be understood that I do not pledge myself to an agree-
ment with Mr. Cockle's views on the geo/nefnca/ signification of his i,^', i.
With respect to them as algebraical symbols, I would not here offer an
opinion : but with respect to the geometrical interpretation, I take a totally
different view, as will be inferred from a short paper of mine printed in
vol. xxix. (pp. 1 71-175) of the Philosophical Magazine, under the signature
" Shadow."

Mr. Cockle has undoubtedly the weight of cotemporary scientific autho-
rity on his side of the question; and, indeed, I believe I stand nearly alone
in the view I take of these questions. It would, however, be as unphilo-
sophical a mode of searching for the truth as it would be disingenuous in
the discussion, to suppress the expression of all views which differ from
those which 1 may happen to entertain. It is always a matter of far less
moment who is right than what is true.

Mr. Cockle very kindly put his paper into my hands for perusal before
he printed it; and J have much pleasure in fulfilling his request by forward-
ing it to you.

I may add, that it is much to be desired that the history of the attempts
that have been made to give an explanation of the symbols of incongruity
should be published. A strict discrimination between the views of differ-
ent algebraists and geometers might prevent the waste of much valuable
time and power; for there can be no doubt that a large portion of the spe-
culations which have been put forward in recent times are essentially iden-
tical with much earlier ones.

Little Heath, Charlton, T. S. D.]

Nov. 25, 1848.

t Phil. Mag. S. 3, vol. xxxii. p. 352.

X Peacock, Report on Analysis (Third Report of British Association),
pp. 237-238.

38 Mr. J. Cockle 07i a new Imaginary in Algebra.

hesitate in saying, either that impossibility is incapable of being
rendered subservient to the purposes of algebra, or even that
it is absolutely uninterpretable. A symbol for impossibility
is not only desirable, but actually necessary, provided that we
wish to classify with accuracy the various subjects of algebraic
research, and to distinguish those which are unreal from those
which are impossible. We might adopt an arbitrary symbol
to denote impossibility; but a deduced symbol is preferable,
for it gives to our investigations the character of true develop-
ments of ordinary algebra. Such a deduced symbol I have
obtained by means of a surd equation *. The next step is, to
ascertain the fundamental properties of the new symbol, its
origin and nature being duly considered. As we might ex-
pect, a priori, anomalous results offer themselves in the course
of our progress with these inquiries; such results require at
least an attempt at explanation. The geometrical interpreta-
tion (or lather capability of interpretation) of the new symbol
is another point not unworthy of consideration, and the same
may be said of the employment of that symbol in analytical
discussions. I purpose, then, in this paper, to treat, — 1, of the
Utility of the new Symbol; 2, of the Value of its Square;
3, of a certain Anomalous Result ; 4, of the Interpretation of
the Symbol in Geometry ; and, lastly, of its proposed Employ-
ment in Analysis.

1. Of the Utility of the new Symbol.

I shall show that there are i-elations, which cannot be ex-
pressed by means of the ordinary algebraic symbols. Hence,
if it be of any importance to express such relations, a new
notation must be adopted. That it is of importance, those
who value logical precision and accurate classification will, if
I am not mistaken, be disposed to admit, on such grounds
alone, and quite irrespectively of any ulterior applications of
which the new symbol may be susceptible. It will be observed,
in what follows, that I have in certain cases employed an ac-
cented zero (O'). I have done this in order to distinguish
what is, in fact, an absolute negation of existence, from the

* This symbol possesses the character which we might, almost, have an-
ticipated for it a priori. In Universal Arithmetic, a negative quantity is
impossible (Peacock, Report, p. 189), but its square is possible. So, even
when negative quantities are recognized, a negative square is contradictory,
and unreal quantities are impossidle, but their squares are possible. In like
manner, the square of a Pure Impossible — of a quantity taken as simulta-
neously positive and negative — is to be treated as possible. The contra-
diction vanishes on squaring, and there is a striking analogy with the otlier
cases.

Mr. J. Cockle on a new Imaginary in Algebra. 39

ordinary (unaccented) zero which represents a certain state of
quantity*.
Let

0'=l + <;-; (1.)

multiply both sides of this equation by 1 — V7, and we have

0'x(l->/;')=(l + y>-)(l-v'i); . . . (2.)
but

l--/;'=2-(l + Vi) = 2-0';
hence, substituting on the left-hand side of (2.), and multiply-
ing together the factors which compose its right, we obtain

0'x(2-0') = l-J. (3.)t

In (1.) substitute unity for J; the result is
0'=14-1;
hence J is not equal to unity, and consequently 1 —j is not
equal to zero. Acting upon this, let us proceed to obtain,
from the symbols of ordinary algebra, the most general ex-
pression for a quantity different from zero. We may attain
our end as follows. Let a, w, and oe be any real quantities
whatever, positive, negative or zero, and let /= ^ — 1 ; also
suppose that

W = MJ-a+( )( ),

yw — a/ \x—a}

and

V /a—a\/a—a\

X = ^-a+(, )( I;

Xw—a/ \x—a'

then

W + /X
is the most general expression of ordinary algebra for a quan-
tity different from zero, and may be made to take any given
value whatever, excepting zero. Now, as we have seen,

1 —j= some quantity other than zero ,-

hence, those who would maintain the adequacy of ihe ordinary
notation to express any relation whatever, possible or impos-
sible, must sustain the equation

l-j=W + iX;

* Peacock (Third Report of British Association), pp. 232,233, &c., and
also p. 268. The accented zero is discontinuous; and a remark of Pro-
fessor J. R. Young, on impossible equations, in the Mechanics' Magazine
(vol. xlix. p. 463) suggests the characteristic that, to 0', the right-hand
side of (1.) can make no approximation in terms of the ordinary symbols of
algebra.

t The product \/J x Vj has the double form J and v^'^ ; I here take

the former value, on the principle that ^—1 x v^— I is equal to —1.

40 Mr. J. Cockle on a 7ie'w Imaginary in Algebra.

but this equation cannot be sustained; for, if we deduce from
it the value of^, and substitute that value in (1.), we arrive at

0'=l+-v/l-W-zX,
in place of which we may write

0' = l+a+f/3;
this last equation gives /3 = 0, whence we infer that X = 0;
consequently, a being real, and the positive square root being
taken, we arrive at

0'= 1 +«= unity together with a positive number',
which cannot be. Hence j is not of the form W + fX, and
yet it is different from zero, as will be seen on substituting
zero for J in (1.). In short, J is a i\\XQ.x\\.\\.y sui generis — an
impossible quantity — a quantity the very conception of the
existence of which involves the equation (3.). And, of this last-
mentioned equation, it may be added, that it is to be regarded
as one of the principal symbolic decompositions which the
theory ofj involves, and is not to be confounded with the ap-
parently similar equation

Ox('^— 0) = W+2X.

Thus there are impossibilities not capable of being expressed
either by zero, or by quantities of the form W -f /X, unlimited
as are the values which may be given to W and X. And,
although infinity be among such values, it becomes necessary
to have recourse to the new symbol j to indicate the impossi-
bility implied in (1.). And^" would be useful, were it only to
indicate such impossibilities.

2. Of the Value of its Square.

We are now arrived at a topic, the discussion of which will,
perhaps, assist in showing that the apparent difficulties, con-
nected with the theory ofj, are not such as to justify us in re-
jecting it as unsuited to the purposes of analysis, or as inca-
pable of becoming the symbol of a calculus. In the fact, that
j has a real square, we have something like a key to the me-
thod of rendering available this anomalous symbol. Starting
from our fundamental equation, slightly changed in its form,
we have

+ -/;■= -1;

on cubing both sides of this equation, we obtain
whence* we infer that

* The present case is distinguishable from that alhuled to in the last
note.

Mr. J. Cockle on a new Imaginary in Algebra. 41

Is J then identical with either of the ordinary square roots
of unity? No. In what respect does it differ from those
roots ? This I proceed to show, and as follows. Unity may
be regarded as the square, either of positive, or of negative
unity : and, as we regard it from one or the other point of
view, we must write its square roots thus: —

+ ^(+T?, or - V{^^,
both of which are included in the expression

Let us now reverse the signs under the radical ; then the last
expression becomes

This last is a contradictory and, consequently, impossible ex-
pression, which takes the following two impossible forms,

+ v/(_i)2 and - \/(TT?
which I shall represent by/ and/' respectively. Now, if we
square + V'(+l)2, it becomes (+1)S the contradiction being
eliminated by involution. Hence we infer that

and it is not difficult to see that^',/ and/' are values of im-
possible square roots of unity*. To indicate, however, that
the discrepancy between the signs without and within the
radical cannot be eliminated by merely changing the sign pre-
fixed to the radical, I shall use additional brackets, and sup-
pose that the following equations hold; viz.

±y- ± [+ ^^r^, and +/'= ± [- -/(+!?].

Seeing thus the contradictory nature of the symbols//,/',
we must not be surprised at finding ourselves, very early in
our inquiries on the subject, face to face with such a result as
the equation (3.) given above. We see, however, that con-
tradictions may vanish and available results follow.

S.Ofa certain Anomalous Result.
In the case of the equation (3.), we have seen that there is an
anomaly, inherent in the very supposition of impossible quan-
tity, which does not occur in treating of real or unreal quan-

* I think that the following relations hold, viz, —

I shall not here attempt to discuss the relation of/ and/' to j. The
former quantities are only introduced here to illustrate what is meant by
an impossible square-root of unity.

4<2 Mr. J. Cockle on a new Imaginary in Algebra.

tity. That equation is to be considered, rather as an evidence
of the nature of the quantity which I am discussing, than as a
guide (or impediment) to us in its symbolic application ; and
although it merits further consideration, yet I do not feel
called upon to bestow that consideration here*, inasmuch as
in the theory of tessarines J is not affected with a radical sign,
and it consequently becomes unnecessary, for the purpose which
I have in view, to enter upon the subject of equations expressed
by means of radicals. But I am about to point out another
anomaly, the reverse of that which occurs in (3.) : it is that
on the supposition that J^ is equal to unity,
(H-i)(l-i) = 1-1=0;
that is to say, the (unaccented) zero may be considered as the
product of two impossible factors, neitherf of which vanishes.
It can, however, be at once shown that this anomaly cannot
lead us into error; for assuming the equation

^ a^-b^=[a^jh){a-jb), .... (4.)
and bearing in mind that a tessarine cannot vanish unless all
its constituents are zero, we see that neither a+jb nor a—jb
can vanish, unless a~0 and b = 0. Suppose that a=0 = 6,
then the equation (4.) becomes an identical one, and no error
is introduced. On the other hand, imagine that a^— 6^ should
vanish from a becoming equal to b (both a and b being differ-
ent from zero), then the right-hand side of (4.) would become

(a +»(«-»;
but, bearing in mind the fundamental property of tessarines,
we should be in no danger of inferring that one of these fac-
tors must be zero, and consequently we should introduce no
error into our investigations. It may be said, Is zero, then,
decomposable into non-vanishing factors? Impossible. I reply,
true, the factors are impossible : they are so by their origin
and nature.

4. Of the Interpretation of' the Symbol in Geometry.
In this field, I am about to indicate what (I hope) will be my

* If for no other purpose, the accent on the zero is useful for the pur-
pose of denoting an impossible equation. If the accented zero is different
from the arithmetical zero (and there are indications of a difference), what
is the square (0'^) ? a negation of a negation? I think that we must regard
0'" as identical with 0', when n is positive, and, consequently, 0'~" as iden-
tical with 0'~\ and then n is negative. Zero is not supposed to be in-
cluded in these values of n.

t If one such factor be zero, the other must be infinite. But this is in-
consistent with the forms of the factors. It is on a consideration of this
kind that my theory of congeneric surd equations is based. (See Mecha-
nics' Magazine, vols, xlvii., xlviii. and xlix.)

Mr. J. Cockle on a nem Imaginary in Algebra. 43

future course, rather than actually to commence it. But I
will not disguise the end which I am desirous of attaining —
that of vindicating, for ordinary algebra, a claim to the power
of representing space of three dimensions, as completely as, by
the aid of unreal quantities, it can denote any conditions
whatever in piano, and any modification of such conditions.
By way of illustration, let there be given two points, A and
B, and let it be required to find a third point, C, such, that the
rectangle AC x CB may be equal to half the square described
upon AB. Now, if we proceed on the supposition that C is
somewhere in the straight line which passes through A and
B, we must suppose that C lies between A and B, for other-
wise the rectangle would obviously be greater than the square.
Bearing this in mind, let AB = 2a, and ACs=:r; then the
qucBsitum of the problem gives the equation

x{2a—x)=2a% (5.)

or

x^—2ax=—'^a^,
whence

x-=a + a V' — 1.
Hence, the problem is an impossible one, if we regard C as
lying in AB. But, if we interpret the symbol v' — 1 as mean-
ing perpendicularity* — in which case we must regard (5.) as
the representation of the problem in its most general form —
we have the following construction. Along AB take AD = iAB;
from D draw DC perpendicular to AB, and equal to DA;
then, C is the point required. The point C, so obtained, evi-
dently fulfils the condition of the question. In fact, any line
in a plane being given, as an axis, we may represent any point
whatever, in the plane, by the formula jo + Q'V' — 1, p and q
being real. It is to be remarked, however, that, a line being
given in space, when the symbol \^ —\ occurs in our researches,
we may (as in the above problem) draw our perpendicular in
whatever direction we please, provided only it be in a plane
perpendicular to the primitive axis. But, the perpendicular
once drawn, we have a determined plane to which, I apprehend,
all our interpretations of V'— 1 are to be confined ; for I con-
ceive that, consistently with ordinary algebra, we cannot, on
V — I occurring a second time in our investigations, take that
symbol to denote a line perpendicular to, or making any angle
with, the former perpendicular. Perhaps I have said enough
to show what my own views are, as to the applications of the
symbol W — 1, and the limitations to be imposed on its inter-

* Tills has been done by Hamilton, Warren, and others.

ii Mr. J. Cockle on a new Imaginary in Algebra.

pretation. Supposing, then, / and its interpretation to be ad-
mitted into geometrical inquiries, the question comes, how can
j enter into such inquiries when it can never enter into the
rational equations in which such inquiries usually result?
The answer is, that geometrical conditions are not necessarily
reducible to the form of rational equations. Consider, by way
of example, the equation (5.). This equation, after reduction
to the usual form, may be resolved into congeneric surd fac-
tors ; and, the geometrical meanings of a and x being lines,
we may express the evanescence of those factors in geometrical
language. Such evanescence may be possible or impossible;
if the latter, then^* forces itself into our investigations, and the
next point is, how to interpret its occurrence. I think that the
following are the considerations which ought to guide us in
this interpretation. Imagine three points A, B, and C. If
these points are in a straight line, their relations may be re-
presented by real quantities, and the only determined space
before us is a line. But suppose that these points are required
to fulfill conditions inconsistent with the hypothesis of their
being in the same straight line; then (as will be clearly seen
on referring to the above problem) a. plane is determined, and
unreal quantities introduced ; and, supposing that a problem
respecting three points admits of solution, the most general
geometrical entity that can be determined by it is a plane. If
then we arrive at an impossible quantity, as the result of our
geometrical inquiries respecting the possibility of a supposed
relation between three points, we may be sure that the relation
cannot exist. Were the relation possible, it would be possible
in a plane ; and i is quite adequate to express (in combination
with real quantities) any possible relative position of three
points. Let us take a step further, and suppose that we are
discussing the position of four points A, B, C, D. Take AB
as the primitive axis, and let AB = 2a; then the expression
p + ig may be made to represent either C or D (or both, pro-
vided that all the four points are in the same plane). Thus
we shall have A represented by zero, B by 2a, C by c + z>,
and D by d + if. But, suppose that D is out of the plane of
A, B and C, then, from what has been before observed, d+if
will cease to represent D, for i means perpendicularity to AB
in the plane of A, B and C. Hence, in any inquiry respecting
four points, the occurrence of j would not be conclusive as to
the impossibility of the problem, but only as to the fact that
the points cannot be in the same plane. (I assume, of course,
that, in forming the condition or conditions of the problem, all
the points are represented by different values of the expression
p-\-iq.) In such a case, then, j would (provided the problem

Mr. J. Cockle Ofi a new Imaginary in Algebra. 45

were possible) be the sign of perpendicularity to the plane
ABC, and the transition, from regarding j as the sign of im-
possibility, to viewing it as the symbol of perpendicularity, is
by no means difficult*. As may be inferred, from what 1 said
in opening the question of interpretation, I am not prepared
to complete this view of the question in the present paper ;
but I cannot refrain from remarking, that j is not an unreal
root of unity, and that, although it may indicate perpendicu-
larity, yet that we must envisage it in a manner different from
that in which we regard i. In fact j indicates perpendicularity
to a plane as / does to a line; J^=l and 2^= — 1. We may
realize the distinction, geometrically, as follows. On the semi-
diameter of a sphere conceive another sphere described. Let
the point of contact of the spheres be considered as the pole
of both. Conceive two points, one at the centre of the larger
sphere, and the other situate anywhere on its equator ; let the
first point revolve in a meridian of the smaller sphere, and
the second in the equator of the greater, and let the angular
velocity of the first point be double that of the second. We
need only consider the relative positions of the points when
the first point is either at die centre of the greater sphere, or
at the common pole of both the spheres. It will be seen that
the phases (so to say) of the points correspond to those of the
above quantities ^'^ and i'^; and also that the first point repre-
sents direction perpendicular to a plane, and the second, direc-
tion perpendicular to a linef. I mention this because it might
be supposed thatj is only a second perpendicular to the primi-
tive axis, and, consequently, that it is only a second unreal root
of unity J.

5. Of its proposed Employment in Analysis.
Should the admissibility of the new symbol j be established

• The transition from unreal to impossible quantities will, perhaps, be
best exemplified by the two following problems: — (1.) Find three equi-
distant points, and, (2.), Find four equidistant points. The first may be
expressed by means of a quadratic with unreal roots ; the unreal quantities
arising from one of the points being out of the /me joining the other two.
So, unless I am mistaken, the solution of the last may be exhibited by
means of impossible quantities, which take their origin from the fourth
point being out oi the plane of the other three.

\ This remark will perhaps be better expressed when we have substituted,
for the lesser sphere, a prolate spheroid, and then diminished indefinitely the
minor axis of the spheroid. Its major axis is, of course, to remain unal-
tered, and equal to, and coincident with, the axial semidiameter of the
larger sphere.

X The symbol _/ will enter into geometrical inquiries in the following
manner. Suppose that we arrive at such an equation as

then, s z=jr is the solution.

4-6 Mr. J. Cockle on a new Imaginary in Algebra.

its use would not be confined to the discussion of the functions
which I have proposed to call tessarines. It would, as it seems
to me, be capable of other applications, and would tend to

generalize all processes to which the imaginary symbol \^ — \
has been yet applied. Thus, by way of illustration, consider
the expression

^w ±ix ±jy ikg

If we vary, in every possible way, the order of the signs in
the above; add the different values, so obtained, together; and
expand the sum ; it will be seen that the result is free from
imaginaries. Hence, that sum may, in all cases, be used instead
of the resulting series. The finite expression for the series, so
obtained, would in some instances be found useful.

It now only remains to make one remark respecting the

notation which I have adopted. In taking i to represent V' — 1,
I think that I acted under an impression that it had been so
used anterior to the quaternion theory*. The use of^" and k
followed that of /, and seemed to offer an easier and better
mode of con^paring results with that theory than I should
otherwise have had. In order, however, in future to avoid
confusion, and the misapprehensions which may arise from
employing like symbols for unlike purposes, I shall use a, /3,
and y in place of f, j and k respectively. Under this notation,
a tessarine {t) will be written

w + ax + ^y + yz,
where

a2=-l, «^ = y, -^^ = 72, ya=-/3;

and also, if the view taken in this paper be correct,

(3^=1, and /3y = «.

In my endeavours to bring space of three dimensions under
the dominion of a new species of ordinary algebra, I may per-
haps be permitted to disclaim anything like dogmatism. I
should wish all my views respecting the new symbol to be re-
garded in the light of suggestions. And if, regarded as sug-
gestions, they should have the effect of directing attention to
the theory of congeneric surd equations, they will not be with-
out their utility.

2 Church-Yard Court, Temple,
November 23, 1848.

Postscript, 7th December 1848. Perhaps I shall be per-
mitted to add the following few lines on the subject : —

* Sir W. R. Hamilton has noticed this in the Phil. Mag. S. 3, vol. xxv.

Mr. J. Cockle on a new Imaginary in Algebra. 47

6. Of the Modular Relations of Tessarines.

Let 9 be the amplitude, f the colatitude, vj/ the longitude,
and IX, the modulus of a tessarine (t). To these quantities (which
are identical with the corresponding ones in the quaternion
theory, and which may, without confusion, be adopted from
that theory) must be added another, which I propose to call
the submodulus, and to denote by v. The submoduhis is de-
fined by the equation

and the submodulus of the product (^") of two tessarine fac-
tors {t and t') is determined by the relation

and the modulus of that product by

We have, further,

WTO)" + xa^^ + yy" + zz^' = ttj'jtt^ + 2y'v\
and

The equation for the submodulus may also be expressed as
follows: —

v2=j«-2 sin 6 sin <p (cos d cos \l/ + sin 3 sin if/ cos <p).
The construction of this last equation, and of the preceding
ones, 1 hope to discuss on another occasion. And there is a
surface — that defined by the equation

jM,?/ cos ^-\-x% = 0
(where % is supposed constant), — which appears to merit at-
tention in connexion with the theory of tessarines.

Second Postscript, I'Mh December 1848. I have the satisfac-
tion of adding, that the hope above expressed has not been
disappointed, and that I have solved the problem of the four
equidistant points in the manner I proposed. I have deter-
mined the position of the fourth point by an expression of the
form

A + m+iC;

where C is to be measured in a direction perpendicular to the
plane in which the first, second, and third points are situate.
The four points are, of course, at the angles of a regular
tetrahedron. I hope to obtain for the solution a place in this
Journal.

C ^8 ]

IV. On the Continuance of a Solar Spot.
By W. Pringle, Esq.

To the Editors of the Philosophical Magazine and Journal.

Gentlemen,

IN the notice last communicated respecting the duration of
a solar spot from August to November (Phil. Mag. Dec.
1848), I stated that I should not perhaps have it in my power
to observe its re-appearance in December, should it then re-
turn. I have had, however, the opportunity of making two
observations; one on the 8th, and the other on the 13th of
December.

On the 8th there was a succession of spots, or clusters, follow-
ing each other at intervals in a straight line, and cutting down
in an oblique direction, as usual at this period. They were all
in thenorthern hemisphere,except one dispersed group of small
spots which appeared in the south-western portion of the sun.

The first spot or group next the eastern verge, which con-
sisted of two rather large spots with some minor ones attached,
appeared too little advanced to have come on upon the 4th or
Sth — the time that the spot of November might be expected
— being only about two digits from the circumference. The
second spot, which was about three digits and a lialf advanced,
seemed to me to correspond to the place the November spot
ought to have occupied ; and upon examining it attentively, it
presented such features as might result from the further decay
of the spot since it was last seen and sketched by the writer
on the 19th of November. At that time it was nearly of a tri-
angular form, the longest side measuring about 45,000 miles ;
the interior space exhibiting a mottled ground, or shallow
speckled with small dark spots. It was then large enough to
be very plainly perceptible to the naked eye, as it had been
at each period of its appearance.

If I have rightly recognised it in the spot of the Sth of De-
cember, it has dwindled into a comparatively small and un-
noticeable cluster, of an irregular outline, still preserving the
mottled penumbral interior studded with minute nuclei; and
this was one of the distinguishing characteristics of its past
appearance, that it uniformly presented a dusky ground inlaid
with lateral nuclei, never once resolving itself into anything
like a great central nucleus.

The nucleus of September 21 st, which Mr. Weld estimated
at 1' 7""2 in its longest diameter (Phil. Mag. Dec. 1848.
p. 480), lay imbedded in the western portion of the spot, and
might be considered as the nucleal remains of one of the two
separate spots of August, and which I conceive had been con-

Mr. W. Pringle on the cofilinuance of a Solar Spot. 49

joined. On the eastern margin of this nucleus, or rather on
the interior edge of the penumbra trenching on it, I may
mention, a singular patch of a half-burnished colour which
attracted my attention after the spot had passed the sun's
centre, and for several succeeding days. It was not like
those white or bright stellar appearances sometimes seen, and
of which there were also at the same time several scattered
over the general shallow or shaded space ; it rather resembled
some dim, distant fire, seen through a foggy atmosphere at
night; and I thought at times that I could detect a gradation
of tints from reddish-orange to yellowish-green. It partook
so much of a certain lustre, that I imagined at first that it was
in the glass. But it was not. Could the electric principle
have had any part there? or was the lurid reflexion merely
the result of a particular fusion of the luminous and nebulous
matter; different in degree though not in kind to that which
is thought to occasion the stellate specks ? The large nucleus
alluded to became latterly enveloped in a partial penumbra of
its own, of the usual smooth and comparatively clear surface,
though still involved in and connected with tl»e darker dis-
coloured base, or ground of the entire spot.

Another large round nucleus became formed by the combi-
nation of the small ones on the eastern side of the spot before
its disappearance at the end of September, which circumstance
at the time gave some reason to suspect the possibility of the
united group breaking up into two distinct spots, as in August.
Nor wassuch a metamorphosis even improbable; and from some
observations made in July I am much disposed to consider that
it had previously undergone a similar one. Among some
memoranda made during the latter month, I find a notice of
a spot or cluster which was seen by the naked eye on the 29th
and 31st of July, consisting of one large spot, with well-defined
nucleus and umbra, closely connected with an extensive com-
pact group of minute nuclei imbedded in a dusky penumbra.
A rough detached draft represents this spot somewhat ad-
vanced past the centre of the sun, going westwards, though
the exact date of the drawing is not given underneath. As it
was evidently somewhat past the centre, it could not have been
more than six days from its disappearing. Taking the time
from the 30th of July, the intermediate day, this would give
the 6th of August for its passing the western edge of the sun,
and the 19th of August for its arrival at the eastern limb, or
reappearance. The supplied data must, of course, render this
hypothetical, though it is far from being improbable.

But I am retrograding to the detail of former appearances,
which would have been more appropriate in the previous

Phil. Mag. S, 3. Vol. 34. No, 226. Jan. 1849. E

50 Mr. W. Pringle on the continuance of a Solar Spot.

notice, had not I feared intruding on space, as I may even
now.

The second view I had of the spot of this month was on the
13th, when it was within little more than four days of its dis-
appearance. Its general aspect was still in favour of its iden-
tity with the preceding.

From what has been observed, I am inclined to conclude
that the solar spots may last much longer than we are yet
aware of; and that the want of sufficient observation alone
has restricted our knowledge of the true extent of their dura-
tion. No doubt it would occupy almost the entire time and
undivided attention of any individual to follow out their phases
and developments so as to satisfy the objects required : but
an association, I am informed, has recently been instituted for
the express purpose of observing and studying the spots in a
more systematic manner than has hitherto been attempted,
and to whose united labours, therefore, if published, we may
look for a mass of new matter and interesting information on
the subject, such as could not be expected from mere individual
and isolated observation.

The spots seem almost the only inlets whereby to penetrate,
if possible, into the sun's physical character, with the excep-
tion of the phaenomena connected with eclipses ; and however
volatile and changeable they may appear, there is no good
reason to despair of yet reducing their evolutions to a system,
and even thus detecting laws which may bear upon the solu-
tion of the general physical organization of the sun.

Submitting these somewhat desultory notices to your con-
sideration,

I remain. Gentlemen,

Your most obedient Servant,
Edinburgh, Dec. 15, 1848. W. Pr INGLE.

P.S. Permit me to take this opportunity to state, what it
may perhaps interest Mr. Glaisher or others to know, that the
same distorted image of Mercury on entering the sun's disc
which one out of eight telescopes presented at the Greenwich
Observatory, was also observable here. The planet appeared
to me to enter like a black wedge, with undulating edges like
a Malay Krees or dagger, an appearance which 1 ascribed at
the time to the tremors of intervening smoke and vapours.
This could scarcely have been the cause at Greenwich, as the
seven other telescopes showed the planet round and clear. Dr.
Dick, of Broughty Ferry near Dundee, also described the ap-
pearance as an " indentation " on the sun's limb. — W. P.

[ 51 ]

V. A simple Rule for co?ivertitig intervals of Sidereal into
intervals of Mean Solar Tirne^ and vice versa, isoithout the
help of Tables. By the Rev. J. M. Heath*.

I. "^"^TRITE down the given time.

▼ ▼ II. Underneath its minutes, seconds, and decimals
of second, write its hours, minutes, and decimals of minute,
respectively, retaining three decimal places.

III. Subtract II. from I.

IV. Subtract II. from III., retaining the first decimals of
hours only.

V. Write the liours, minutes, and decimals of minute in III.
as minutes, seconds, and decimals of second respectively.

VI. Divide V. by VI., and place ihefrst decimal of hour
in IV. under the third decimal place of the quotient, writing
the integral number of hours to the left, and add.

VII. To I. add its third part, and the hundredth part of
their sum, retaining only the first decimal place of the hours.

VIII. As in VI., place the decimal of hour of VII. under
the tliird decimal of second in I., writing the entire hours to
the left, and add.

Then, the sum or difference of VIII. and VI. gives the si-
dereal, or the mean solar time, of which I. is solar or the
sidereal time respectively, accurately to the third decimal of
seconds. h m s

Example. I. 17 15 4.7-2'l.

II. 17 15-787

= 16 '^•7 nearly.

III.

16 58 31-453

IV.

16 41 \5-66Q

V.

16 58-524

VI.

2 49-754

•167

2 49-921

VII. 17 15

47-24

17-

5-

0
25
75
22

I.

of I.

1
j^-Q of sum.

VIII. 17

Then adding or sub
tracting VI.

Sidereal time for I. . =
Mean solar time for I. =

}

15

2

47-472
49-921

17
17

12

37-393
57-551
Communicated by the Author.
E2

[ 52 ]

VI. On some Points in the received Theory of Sound. By G.
G. Stokes, M.A., Fellow of Pembroke College, Cambridge*.

BEFORE entering on the main subject of this communi-
cation, I will make a few remarks with reference to Pro-
fessor Chaliis's last paper in this Magazine. (Vol.xxxiii,p.4)G2.)

I have no intention of rendering my reasoning liable to the
epithet " illogical," by attempting to explain away admitted
contradictions. It was my endeavour in a former communi-
cation, and will be my endeavour in this, to show that the
contradictions in question have no real existence. Since it
would seem from Professor Chaliis's words, near the bottom
of page 4:63, that he does not perceive that any step of the
mathematical reasoning by which the contradiction in the
case of plane waves was arrived at has been called in question,
I beg to state that I do not admit the validity of any mathe-
matical results obtained by a process which is equivalent to
integrating over an infinite ordinate, without inquiring whether
the passage be legitimate or not. Such a proceeding would
lead to contradictions in other subjects as well as in hydrody-
namics; for example, in central forces. It was not until after
I had pointed out, at page 352, what I conceived to be the
flaw in Professor Chaliis's reasoning, that I entered on the
physical considerations alluded to by Professor Challis at
page 462 : and when I did so, it was not in the slightest de-
gree with any intention of explaining away an admitted con-
tradiction (for I have already stated that I do not admit the
validity of the contradiction), but simply because those con-
siderations seemed to be of some interest on their own account.
I was particularly careful (see page 353) to keep the purely
mathematical question quite distinct from the physical consi-
derations which followed.

It will be necessary, in order to prevent confusion, that I
should say a few words with reference to the admission that
" plane waves are wholly incompatible with the transmission
of articulate and musical sounds." What signifies it if the
ideal elastic fluid which forms the basis of our mathematical
reasoning be wholly incompetent to transmit such sounds un-
changed ? The purely mathematical question of contradiction
or no contradiction is not in the slightest degree affected, al-
though it forms an interesting subject of physical inquiry how
far air, as agreeing approximately with our elastic fluid, may be
incapable of transmitting musical sounds without modification.

To turn now for a moment to the physical question, I would
observe that a fluid in which pap would be capable of trans-
* Communicated by the Author.

On some Points in the received Theory of Sound. 53

mitting such waves as would produce musical sounds to a
considerable distance without important alteration, even if the
waves were plane ; although sooner or later a wave of this
kind would be converted into what Professor Challis calls a
breaker^ but which I think is more nearly analogous to a bore.
An integral of the exact equation which applies to s])herical
waves has never been obtained ; but it is evident that the di-
vergence of such a wave must tend to counteract the formation
of a bore. In fact, if we suppose the velocity v represented, as

it would be approximately, by- / (r — at)^ we have, neglecting

the term involving— 2> 7-; =— /' [r— at)'i so that for a given

phase of the wave, that is for a given value ofr — at, the tangent
to the velocity curve (vol. xxxiii. p. 350) would vary inversely
as r; and therefore, as far as depended only on divergence,
the inclination of the curve would become more and more
gentle in the anterior, as well as the posterior portion. I feel
however almost certain, in consequence of an investigation in
which the effect of divergence was very approximately taken
account of, that the formation of what I have called a bore,
although much retarded by divergence, is not ultimately pre-
vented. I speak of course with reference to the ideal fluid in
which potp. I see no reason for supposing that the develop-
ment of heat and cold by sudden condensation and rarefaction
would have any tendency to prevent the formation of a bore.
I have already alluded to one cause which would have such a
tendency (vol. xxxiii. p. 356), namely the internal friction of
the fluid. If, during the rapid condensation and rarefaction
of the fluid, there should be time for any sensible quantity of
heat to pass off" or be received by way of radiation, that would
apparently have much the same effect as internal friction.
The effect of distance upon the quality of sound, and the
causes why sounds are mellowed by distance in air, whereas
under water the sound of a distant bell is heard as a crash,
would form an interesting field of inquiry.

I proceed now to notice the apparent contradiction at which
Professor Challis has arrived by considering spherical waves,
a contradiction which it is the chief object of this communi-
cation to consider. The only reason why I took no notice of
it in a former communication was, that it was expressed with
such brevity by Professor Challis (vol. xxxii. p. 497), that I
did not perceive how the conclusion that the condensation
varies inversely as the square of the distance was arrived at.
On mentioning this circumstance to Professor Challis, he

54- Mr. G. G. Stokes 07i some Points in the

kindly explained to me his reasoning, which he has since
stated in detail (vol. xxxiii. p. 463).

The whole force of the reasoning rests on the tacit suppo-
sition that when a wave is propagated from the centre out-
wards., any arbitrary portion of the wave, bounded by spherical
surfaces concentric with the bounding surfaces of the wave,
may be isolated, the rest of the wave being replaced by qui-
escent fluid; and that beingsoisolated, it will continue tobepro-
pagated outwards as before, all the fluid except the successive
portions which form the wave in its successive positions being
at rest. At first sight it might seem as if this assumption were
merely an application of the principle of the coexistence of
small motions, but it is in reality extremely different. The
equations are competent to decide whether the isolation be
possible or not. The subject may be considered in different
ways; they will all be found to lead to the same result.

1. We may evidently without absurdity conceive an out-
ward travelling wave to exist already, without entering into
the question of its original generation ; and we may suppose
the condensation to be given arbitrarily throughout this wave.
By an outward travelling wave, I mean one for which the
quantity usually denoted by <p contains a function of r — at,
unaccompanied by a function of r + «/, in which case the ex-
pressions for V and s will likewise contain functions of 7 — at
only. Let

flr^— •£ ' (1.)

r

We are at liberty to supposey'(2;) = 0, except from z = b to

2 = c, where b and c are supposed positive; and we may take

,f'{z) to denote any arbitrary function for which the portion

from z = b to z=c has been isolated, the rest having been

suppressed. Equation (1.) gives

(p=-a^J"sdt='&^-=^+^r), . . . (2.)

\I/(r) being an arbitrary function of r, to determine which we
must substitute the value of <p given by (2.) in the equation
which <p has to satisfy, namely

^^^...^t. ..... (3.)

This equation gives \|/(r) = C H , C and D being arbitrary

constants, whence

^^^f^f'{r-at) Ar-at) _D ^ ^ ,^,
dr r r^ ^2* • • v ;

received Theory of Sound. 55

Now the function y( .2) is merely defined as an integral of
f\z)dzi and we may suppose the integral so chosen as to
vanish when z=.b, and therefore when z has any smaller value.
Consequently we get from (4.), for every point within the
sphere which forms the inner boundary of the wave of con-
densation,

^=-^ (5.)

Again, if we put /(c) = A, so thaty(2r) = A when 2>-c, we
have for any point outside the wave of condensation,

A + D .^,

v= ^:^-. ..... (6.)

The velocities expressed by (5.) and (6.) are evidently such
as could take place in an incompressible fluid. Now Professor
Challis's reasoning requires that the fluid be at rest beyond
the limits of the wave of condensation, since otherwise the
conclusion cannot be drawn that the matter increases with the
time. Consequently we must have D = 0, A = 0 ; butifA = 0
the reasoning at p. ^es evidently falls to the ground.

2. We may if we please consider an outward travelling
wave which arose from a disturbance originally confined to a
sphere of radius 5. At p. 463 Professor Challis has referred
to Poisson's expressions relating to this case. It should be
observed that Poisson's expressions at page 706 of the Traits
de Mecanique (second edition) do not apply to the whole wave
from r = ai — s to r=ai + e, but only to the portion from
r=:ai — s to r=:ai; the expressions which apply to the re-
mainder are those given near the bottom of page 705. We
may of course represent the condensation s by a single function

— x(f—(^i)i where

X{-Z)=f{z\ %(^) = F(.),
z being positive ; and we shall have

K=f\{z)dz=f{s) -/(O) + F(s)- F(0).

Now Poisson has proved, and moreover expressly stated at
page 706, that the functions F,/ vanish at the limits of the
wave; so that/(s) = 0, F(6)=:0. Also Poisson's equations (6.)
give in the limiting case for which 3 = 0, /(0)+F(0) = 0, so
that A = 0 as before.

3. We may evidently without absurdity conceive the velo-
city and condensation to be both given arbitrarily for the in-
stant at which we begin to consider the motion ; but then we

56 Mr. G. G. Stokes on &ome Points in the

must take the complete integral of (3.), and determine the two
arbitrary functions which it contains. We are at liberty, for
example, to suppose the condensation and velocity when / = 0
given by the equations

r r r^

from r=b io r=c, and to suppose them equal to zero for all
other values of r; but we are not therefore at liberty to sup-
press the second arbitrary function in the integral of (3.) The
problem is only a particular case of that considered by Poisson,
and the arbitrary functions are determined by his equations (6.)
and (8.), where, however,it must be observed, that thearbitrary
functions which Poisson denotes by^, F must not be confounded
with the given function here denoted by^, which latter will ap-
pear at the right-hand side of equations (8.). The solution pre-
sents no difficulty in principle, but it is tedious from the great
number of cases to be considered, since the form of one of the
functions which enter into the result changes whenever the
value of r + a^ or of r — at passes through either b or c, or
when that of r — at passes through zero. It would be found
that unless /(6) = 0, a backward wave sets out from the inner
surface of the spherical shell containing the disturbed portion
of the fluid; and unless /(c) = 0, a similar wave starts from
the outer surface. Hence, whenever the distui'bance can be
propagated in the positive direction only, we must have A, or
f{c)—f{h), equal to zero. When a backward wave is formed,
it first approaches the centre, which in due time it reaches,
and then begins to diverge outwards, so that after the time

- there is nothing left but an outward travellinfj wave, of
a ° n »

breadth 2c, in which the fluid is partly rarefied and partly
condensed, in such a manner ihaX Trsdr taken throughout the
wave, or A, is equal to zero.

It appears, then, that for any outward travelling wave, or
for any portion of such a wave which can be isolated, the
quantity A is necessarily equal to zero. Consequently the
conclusion arrived at, that the mean condensation in such a
wave or portion of a wave varies ultimately inversely as the
distance from the centre, proves not to be true. It is true, as
commonly stated, that the condensation at corresponding points
in such a wave in its successive positions varies ultimately in-
versely as the distance from the centre ; it is likewise true, as
-Professor Challis has argued, that the mean condensation in
^'iiny portion of the wave which may be isolated varies ultimately

received Theory of Sound. 57

inversely as the square of the distance ; but these conclusions
do not in the slightest degree militate against each other.

If we suppose b to increase indefinitely, the condensation or
rarefaction in the wave which travels towards the centre will
be a small quantity, of the order 6~', compared with that in
the shell. In the'limiting case, in which b= oo , the conden-
sation or rarefaction in the backward travelHng wave vanishes.
If in the equations of paragraph 3 we write b + x for r, b(s{x)
for/'(r), and then suppose b to become infinite, we shall get
as=<T(a;), v = <t[x). Consequently a plane wave in which the
relation v=as is satisfied will be propagated in the positive
direction only, no matter vf\ie\herj'cr{x)dx taken from the

beginning to the end of the wave be or be not equal to zero ;
and therefore any arbitrary portion of such a wave may be
conceived to be isolated, and being isolated, will continue to
travel in the positive direction only, without sending back any
wave which will be propagated in the negative direction. This
result follows at once from the equations which apply directly
to plane waves ; I mean, of course, the approximate equations
obtained by neglecting the squai'es of small quantities. It may
be observed, however, that it appears from what has been
proved, that it is a property of every plane wave which is the
limit of a spherical wave, to have its mean condensation equal
to zero; although there is no absurdity in conceiving a plane
wave in which that is not the case as already existing, and in-
quiring in what manner such a wave will be propagated.

There is another way of putting the apparent contradiction
arrived at in the case of spherical waves, which Professor
Challis has mentioned to me, and has given me permission to
publish. Conceive an elastic spherical envelope to exist in
an infinite mass of air which is at rest, and conceive it to ex-
pand for a certain time, and then to come to rest again, pre-
serving its spherical form and the position of its centre during
expansion. We should apparently have a wave consisting of
condensation only, without rarefaction, travelling outwards, in
which case the conclusion would follow, that the quantity of
matter altered with the time.

Now in this or any similar case we have a perfectly definite
problem, and our equations are competent to lead to the
complete solution, and so make known whether or not a wave
will be propagated outwards leaving the fluid about the enve-
lope at rest, and if such a wave be formed, whether it will
consist of condensation only, or of condensation accompanied
by rarefaction : that condensation will on the whole prevail is
evident beforehand, because a certain portion of space which
was occupied by the fluid is now occupied by the envelope.

58 Mr. G. G. Stokes on some Points in the

In order to simplify as much as possible the analysis,
instead of an expanding envelope, suppose that we have a
sphere, of a constant radius b, at the surface of which fluid is
supplied in such a manner as to produce a constant velocity V
from the centre outwards, the supply lasting from the time 0
to the time t, and then ceasing. This problem is evidently
just as good as the former for the purpose intended, and it
has the advantage of leading to a result which may be more
easily worked out. On account of the length to which the
present article has already run, I am unwilling to go into the
detail of the solution ; I will merely indicate the process, and
state the nature of the result.

Since we have no reason to suspect the existence of a
function of the form F (r + at) in the value of f which belongs
to the present case, we need not burden our equations with
this function, but we may assume as the expression for <p

f{'>'-at) ,^ .

^ = -^. — (70

For we can always, if need be, fall back on the complete in-
tegral of (3) ; and if we find that the particular integral (7.)
enables us to satisfy all the conditions of the problem, we are
certain that we should have arrived at the same result had we
used the complete integral all along. These conditions are

(p = 0 when ^ = 0, from r=6 to r= 00 ; . . (8.)
for p must be equal to a constant, since there is neither con-
densation nor velocity, and that constant we are at liberty to
suppose equal to zero;

-^ = V when r=6, from /=0 to ; = t; . . (9.)

-r = 0 when r=:b, from t — r to t= oo . . . (10.)
dr ^

(8.) determinesy(2) from z = Z> to 2" = oo ; (9.) determines /(z)
from s = 6 to is=^b—ar; and (10.) determines y(r) from
z = b — ar to z= — 00, and thus the motion is completely
determined.

Jt appears from the result that if we consider any particular
value of r there is no condensation till at =.r—b, when it
suddenly commences. The condensation lasts during the time
r, when it is suddenly exchanged for rarefaction, which de-
creases indefinitely, tending to 0 as its limit as t tends to x .
The sudden commencement of the condensation, and its
sudden change into rarefaction, depend of course on the
sudden commencement and cessation of the supply of fluid at
the surface of the sphere, and have nothing to do with the

received Theory of Sound. 59

object for which the problem was investigated. Since there
is no isolated wave of condensation travelling outwards, the
complete solution of the problem leads to no contradiction,
as might have been confidently anticipated.

How then stands the theory of sound as usually received?
So long as we confine our attention to the first order of small
quantitiep, which is a perfectly legitimate mode of approxima-
tion, there is neither contradiction nor difficulty ; for Professor
Challis's difficulty with respect to the effect of the develop-
ment of heat by sudden compression, in the altered form in
which he has now put it, has nothing to do with the first order
of small quantities. On employing exact equations, it is true
that a very remarkable kind of motion has been brought to
light in the course of the discussion, and shown to be possible,
if not in air, at least in an ideal fluid in which the pressure is
equal in all directions, and varies as the density. The pre-
cise nature of this motion I do not pretend to describe. I
have already stated (vol. xxxiii. p. 352) that I had grounds
for believing that a sort of reflexion would take place; though
whether this reflexion would or would not prevent the forma-
tion of what I have called a surface of discontinuity I am un-
able to say, although I am inclined to think that it would.
, To prevent misapprehension, I will observe that it is the mo-
tion, whatever be the nature of it, which takes place after the
quantity denoted at page 351 by A becomes infinite, that I
have referred to in using the word bore : I did not mean, in
using that term, to assert that a surface of discontinuity was
certainly formed. An interesting field of inquiry lies open
with reference to the possibility of an actual approximation to
a bore in the case of fluids such as they exist in nature, and
generally with reference to the modification of sound by di-
stance. It is to such an inquiry that the consideration of the
effect of different functional relations between p and /?, when
the changes of p are too great to be considered proportional
to those of ^, properly belongs, and in particular the functional
relation which connects p and p in the case of air, in conse-
quence of the heat developed by sudden compression. But
as regards the purely mathematical question of the treatment
and interpretation of our equations, no contradiction arises
when the restrictions which the occurrence of infinite quanti-
ties imposes on all mathematical reasoning are attended to.

In conformity with an intention which I have already ex-
pressed (vol. xxxiii. p. 349), and with the title which I have
chosen for this article, I have confined myself to the defence
of the theory of sound as usually received. I have refrained
from calling in question the new and startling conclusions

60 The Rev. C. Graves on the Calculus of Operaiions.

at which Professor Challis has arrived with reference to ray
vibrations. I have done so partly because the subject has been
taken up by the Astronomer Royal, partly because I have not
leisure for the discussion at present, partly because the points
which would have to be noticed, and which would be likely
to arise in the course of the discussion, are so numerous, that
I think it hardly fair to take up the pages of a Magazine like
the present with the controversy. I cannot however conclude
without recording my protest, first against equation (8.)
(vol. xxxii. p. 282), and secondly against equations (B.) and
(C.) (vol. xxxiii. pp. 99 and 100), by which an attempt is made
to satisfy equation (A.).

Pembroke College, Cambridge,
Dec. 23rd, 1848.

VII. On the Calculus of Operations. By the Rev. Charles
Graves, M.A., Professor of Mathematics in Trinity College^
Dublin *.

PROFESSOR YOUNG, objecting to the method by which
the theorem of Leibnitz is usually extended to successive
integration, has lately proved its applicability in that case by
means of repeated " integrations by parts : " and he has shown
how to obtain in this way a series of supplementary integrals,
without the addition of which the theorem is not generally
true, though they are commonly suppressed in the statement
of it. Professor Young seems to impute this omission to the
nature of the Calculus of Operations, by means of which the
theorem is usually treated ; as though that method necessarily
gave the theorem in the imperfect form, and made no provision
for the correction which he suggests.

It is my purpose here to show that the omission of the sup-
plementary integrals has been caused by the use of an incom-
plete form of the binomial theorem, rather than by any inhe-
rent deficiency in the Calculus of Operations, which, if applied
to this problem with proper caution, will furnish the desired
result in a direct and elegant manner.

If we take the identity

{l+x)-'^=:l—x + x'^ — 8ic +(-!)'»- 1^;'"-'

+ ( — 1 )'"(!+ J^)"'^"*,

and differentiate it (n—l) times; we shall obtain a develop-
ment, which coincides for its first m terms with that obtained
by the use of the binomial theorem ; and furnishes moreover

* Communicated by the Author,

The Rev. C. Graves on the Calculus of Operations. 61

the supplemental terms necessary to make the equation iden-
tically true, viz.

(1 +^)-"= 1 -na!-\- ''^^Y^ -^c + (- 1)'"-'A„ x^-'

+ (-l)'"{A„(l+a;')-' + A„_.(l+^)-2 + &c

+ ^i^±il (1 4.a^)-»+2 4-m(l 4-^)-"+' + (H-a?)-«}a?»'.

An being used to stand for

«(w+l). . .. (« + OT — 2)

1.2 {»»— 1) '

or its equal

m{m-\-\) . . . . (w + w — 2)^

1.2 («-l) '

and so on for A„_i, A„_2, &c.

D'

And, if we write ^yj i" place of ^, we have the identity

+ (_l)m-iA„D"-«-'»+»D'"»-» + (-l)'"{A„(D" + D')-'
D»-n-m+i ^. A„_,(D" + D')-2D"-»-'»+2_^ &c. . . .

D"-"*-' + (D" + D')-"D"-'"}D''";

which continues to hold good when D' and D" are any two
symbols of commutative and distributive operation, just as
much as if they denoted quantities.

Using D to denote the operation of taking the diflFereritial
coefficient with respect to Xy we have

D [vu) = vDu + uDv ;
from which we derive the symbolical equation

D = D" + D':
understanding that D" shall operate exclusively on m, and D
exclusively on v. The symbol D being thus absolutely equi-
valent to D" + D', we are entitled to operate on uv with the
symbol (D"4-D')~", or its expansion as given above, for the
purpose of effecting an w-fold integration. Thus we obtain
the complete formula of which we are in search :

/" , „ dv . n(n+l)d^v .

62 The Rev. C. Graves on the Calculus of Operations.

/"d'^v , 1

M^ being used for brevity instead of / udjc^.

The series of supplemental integrals here given differs from
that exhibited by Professor Young only as regards the signs
of its terms, which by an oversight he has made alternately
positive and negative.

The reasoning remaining the same, it is enough merely to
indicate the corresponding mode of obtaining the general
formula for the finite integration of a product. Sir John
Herschel has given this formula in his excellent article on
differences and series, appended to the Cambridge translation
of Lacroix's treatise on the Differential and Integral Calculus :
and it is to be observed that he has taken care to supply those
supplementary integrals which are necessary to its correctness.

Since

^("A) = K^^. + ^.+ 1^^ = %^^. + ^''^.' ^«.'
we have the symbolical equation

it being understood that A" and e^ operate only on t?^, and A'
only upon u^. And if we put A" and e^A' in place of D" and
D' in the formula given above for (D' + D")~", we shall at once
obtain Sir John Herschel's formula.

The examples here discussed by the Calculus of Operations
are certainly instructive. Whilst they manifest the danger,
noticed by Professor Young, of substituting symbols of opera-
tion for those of quantity in divergent infinite series, they in-
dicate that, wherever we know how to express in a finite form
the value of the remainder after any given number of terms
of an infinite series, there is a safe way of effecting such a sub-
stitution. It must be made in the expression for the remai7i~
derj as well as in the terms of the series.

Dublin, November 30, 1848.

C 63 ]
VIII. Proceedings of Learned Societies.

ROYAL ASTRONOMICAL SOCIETY.

[Continued from vol. xxxiii. p. 480.]

June 9,"V[ OTICE of the principal English Observatories. Ex-
1848. -^^ tracted from official or direct sources.

Two British observatories only are, properly speaking, public, —
those of Greenwich and Edinburgh. The Observatory of Oxford
was built and is supported by a bequest under Dr. RadclifFe's will :
it is under the control of private trustees. The Cambridge Obser-
vatory was erected principally by private subscription, and is sup-
ported in part by the funds of a professorship founded by Dr. Plume,
but mainly out of the University chest. The RadclifFe observer
makes an annual report to his trustees ; and Visitors appointed by
the Cambridge Senate draw up yearly a statement to be laid before
that body. Since the appointment of Mr. Airy to the Royal Obser-
vatory, a minute report of all proceedings and changes in that esta-
blishment is read to the Board of Visitors at the Annual Visitation in
June.

Greenwich.

It has been mentioned in former numbers that the meridional in-
struments at Greenwich are deficient in optical power, and that the
Astronomer Royal proposed to replace the mural circle and transit
by a single instrument, viz. a transit circle, which is to be erected
on the site of the present circle-room. In his report to the Visitors,
Mr. Airy says, —

" An object-glass of eight inches clear aperture and eleven feet six
inches focal length having been placed in my hands by Mr. Simms,
I carefully examined it. I found that it showed some objects not of
the closest class (as e Bootis and ^ Cancri) better, I think, than I
had seen them before ; that it separated ij Corona; ; that it did not
separate y Coronae (which, having witnessed the difficulty of that
star in the great Pulkowa refractor, I was prejDared to expect) ; and
that it dispersed light no more than the best Object-glasses usually
do. At my recommendation, therefore, this object-glass was pur-
chased by the Lords Commissioners of the Admiralty, at the price of
275/. I have now to explain the form in which I propose to mount
it. No verbal description, probably, can dispense with reference to
the model*, and I will therefore confine myself to the leading points.
I propose to mount it as a transit circle, its Ys bearing solidly on
the piers far from their edges, and having no adjustments ; the axis
carrying two nearly similar circles on the east and west arms, one
for clamping, the other for the divisions. I propose that the clamps

* A small model of the proposed transit circle was exhibited at the Visi-
tation in June ; and the Astronomer Royal, having obtained the consent
of the government, is proceeding with the construction of the instrument.
A full-sized model has been made and approved. The circle-room ia
rebuilding.

64 Royal Astronomical Society.

have no tangent-screw, the bisection being in all cases effected by
the micrometer in the field of view of the telescope. I propose that
the divisions be illuminated by a single lamp in the prolongation of
the axis, without reflectors ; and that the microscopes be in a conical
surface, passing through one pier, the eye-ends being in a circle of
two feet diameter ; and that the divisions be cut upon a limb of metal
which is so bevelled on the circle that the light of the lamp will be
reflected up the microscopes. Several microscopes to be permanently
mounted, in positions proper for ascertaining with the utmost ex-
actness the errors of division. Microscopes to be mounted for ascer-
taining the laws of movement of points on the ends of the pivots.
The instrument never to be reversed ; but an apparatus to be pro-
vided for raising it so far that a collimating telescope firmly fixed on
a solid pier on the north side, and one on the south side, can be ad-
justed on each other ; then when the instrument is dropped into its
usual place, the error of coUimation and the flexure will be deter-
mined without reversion, by observation of the two collimators. No
spirit-level or equivalent instrument to be used, but the error of level
to be determined by observation of the image of the wires by reflexion
in a trough of mercury. A parallel- motion apparatus to be used for
carrying the trough, and a peculiar arrangement for facilitating the
process of cleaning the mercury. In regard to the material, I pro-
pose that the whole be made of cast-iron, the axis being in two parts
(which enables the founder to make the pivots of hard chilled iron,
while the rest is of soft iron), each end of the telescope being in one
part, and each of the two circles being cast in one piece. An instru-
ment thus constructed would probably be more accurate for right
ascensions than the present transit, in so far as the frequent obser-
vation of the well-mounted collimators would add to the knowledge
of its azimuthal error ; and perhaps more accurate for zenith distances
than Troughton's circle, in so far as the circle is in a state of less
strain, while its construction possesses greater firmness. But the
reasons for recommending it, as is known to the Visitors, are the
power of carrying a larger object-glass, and the enabling one observer
to complete the observation of the two elements."

The observations in polar distance were made with Jones's Cape
circle, until Troughton's circle was erected in another apartment,
where it is and will be used until the circle-room is rebuilt and fur-
nished. The transit will not be disturbed till the new instrument
is at work.

The zenith tube has been taken down. It was proposed by some
of the Visitors that it should be erected in another and less objection-
able position than that which it formerly occupied, and a new site
was pointed out by the Astronomer Royal. Mr. Airy, however,
greatly prefers a different construction, if a continuous series of ob-
servations of y Draconis be required. The principle of this con-
struction, which is singularly simple, is thus described by Mr. Airy : —

" Let the micrometer be placed close to the object-glass, the frame
of the micrometer being firmly connected with the object-glass cell,
and a reflecting eye-piece being used with no material tube passing

Royal Astrofiomical Society. 65

over the object-glass ; and let a basin of quicksilver be placed below
the object-glass, but in no mechanical connexion with It, at a di-
stance rather greater than half the focal length of the object-glass,
so that an image ot the star is formed on the wires after the rays are
reflected from the mercury. Such an instrument would at least be
free from all uncertainties of twist of plumb-line, viscosity of water,
attachment of upper plumb-line microscope, attachment of lower
plumb-line microscope, and the observations connected with them ;
and might be expected, as a result of this extreme simplicity, to give
accurate results."

The Astronomer Royal was recommended by the Board of Visitors
to take the necessary steps for procuring a zenith instrument on the
principle described, and he has already printed and distributed an
account of the construction which he proposes to adopt, with expla-
natory drawings. There seems scarcely any limit to the power and
probable accuracy of such a zenith tube ; and as the mounting is
exceedingly cheap and simple, it will most likely come into general
use, especially for nice determinations of latitude.

It will be remembered that an altitude and azimuth instrument,
made after Mr. Airy's designs by Messrs. Ransome and May, and
Messrs. Troughton and Simms, has recently been added to the Royal
Observatory, for the express purpose of observing the moon. Mr.
Airy says : —

" The altitude and azimuth instrument having now been in use
for an entire year, I am able to give some account of its success or,
failure as a mechanical arrangement. The first subject for remark
is the steadiness of support of the upper pivot, which is held in its
place, as the Visitors will remember, by a frame of bars whose arrange-
ment in every part is triangular. The steadiness is perfect. In the
first observations, the levels were read before and after the telescopic
observation, but it was very soon found that this caution was entirely
unnecessary. The next point is the steadiness in the position of the
horizontal axis of the vertical circle relatively to the vertical revolving-:
fran:e, and generally the steadiness of the constants of instrumental
errors. For some time the constants were so unsteady as to give
me great trouble. Several observations of stars were absolutely re-,
jected. In the month of July, after careful consideration of the dis-
cordances, I came to the conclusion that there was a wandering of
the horizontal pivots in their Ys, caused probably by the counter-
poises : the counterpoises were therefore diminished by one-third
part, and since that time the constants of instrumental error have
been steady, and not a single observation has been rejected. The
next circumstance which gave me trouble was the uncertainty in the
scale-values of 3ome of the long levels. The singular good fortune
of having four parallel levels upon the instrument, which are always
read, enabled me to compare the proportion of the scale-values in
actual use to the scale-values determined before mounting. These
were very discordant. I became at length persuaded that this was
caused by the very defective construction usually adopted in the
mounting of English levels ; and in the autumn I applied to the two
Phil, Mag. S. 3. Vol. S4. No, 226. Jan. 1849. F

66 Royal Astronomical Society.

longest levels the construction with which I had become familiar In
Germany and Russia, in which the glass tube of the level is supported
in Ys : since that time the levels have been fairly accordant. Another
contrivance extensively applied to German levels, namely the cover-
ing by glass shades, has also been applied here. A difficulty which
can be surmounted only by constant care has sometimes presented
itself, namely, that when the dome is opened very shortly before
observation, the changes of readings of the upper and lower levels do
not exactly correspond. Lastly, when the best values of instru-
mental errors of every kind are applied, the accuracy of every part
of observation, of calculation and application of instrumental errors,
and of tabular calculation, is checked by the determinations of the
zero of azimuth, lliese determinations are sufficiently steady in any
one evening, or perhaps in groups of several evenings ; but they are
not steady from time to time, the variation amounting to three or four
seconds of arc. Whether this arises from a twist of the brick pier,
or from a twist of the piers of the transit instrument (the times being
obtained from the transit- clock), or from a change in the observer's
personal equation, I cannot tell. The substitution of improved me-
ridional instruments for those now in use will enable me to remove
one of these conjectural causes.

" I have spoken hitherto solely with respect to the azimuthal ob-
servations, in which alone, from the first, I anticipated any difficulty.
The zenith distance observations have never given the smallest
trouble.

" The accuracy of the results, as estimated by the observation of
stars, is somewhat less than that of the mural circle, perhaps nearly
in the degree which might be expected in circles whose diameter is
half that of the mural circle.

" For observations with the altitude and azimuth instrument, the
following rule is laid down. The moon is to be observed if visible,
and the observer is bound to watch if necessary while the moon is
above the horizon, and the sun is not more than an hour above the
horizon. One azimuth and one altitude are to be observed, and if
possible, two azimuths and two altitudes in reversed positions of the
instrument : and if the night is fine, a low star and a high star are
to be observed in azimuth, both in reversed positions of the in-
strument, and one star in altitude, in reversed positions. Thus a
complete set includes ten observations. These rules have been fol-
lowed carefully during the thirteen lunations intervening between
1847, May 15, and 1848, May 30; and I am now able to give a
comparison of the number of days of observation of the moon with
the altitude and azimuth instrument and with the meridional instru-
ments. With the altitude and azimuth instrument no days are in-
cluded except both altitudes and azimuths are observed ; and with
the meridional instruments, no days except both right ascensions and
polar distances are observed."

The number of days during the last year in which the moon has
been observed at Greenwich are —

With the altitude and azimuth circle =203
meridional instruments ... =111

Royal Astro7i07nical Society. 67^

This statement gives only an imperfect idea of the value of the
instrument. When the moon passes from one to four hours before
or after the sun, there are thirty-four observations with the altitude
and azimuth, and not one with the transit and circle*. It is not
necessary to point out the immense utility of these results in the
lunar theory, or to geography and navigation, which dejjcnd on lunar
observations for their fundamental determinations!. The results of
the observations, as reduced to the state of apparent errors of tables
in R.A. and N.P.D., appear very good; perhaps a little, and but a
little, inferior to those of the meridional instruments.

Throughout the year 1847, the new form of star- reduction pro-
posed by Mr. Airy as a substitute for Bessel's (see M. Notices,
vol. vii. p. 189) has been used, and it has been found convenient.
At present the assistant! are employed in collecting all the star-ob-
servations in 1842-47, for the purpose of forming another grand
catalogue reduced to the epoch 1845. The Astronomer Royal pro-
poses to give in this catalogue the star-constants e,f,g, h, I; e\f',
g', h', I', and also, for a few years, the day-constants E, F, G, H, L,
which are required by his method.

The reduction of Fallow's Cape Observations was commenced
some time ago under the direction of the Astronomer Royal. This
was interrupted by the work incident to the completion of the lunar
reductions, but it will be resumed in a short time.

The ledgers of star- observations and occasional star- catalogues
found in Maskelyne's observations have been fairly written out.
Mr. Airy submitted to the Visitors the propriety of printing these
reductions, and also suggested whether it might not be advisable to
take some steps of calculation with respect to Bradley's observations
anterior to 1750.

In June last, the printing of the volume for 1846 was nearly
finished, and the volume for 1847 was commenced.

Edinburgh.

Professor C. P. Smyth has been hitherto engaged in reducing and
editing the observations of his lamented predecessor, and in exami-
ning and repairing the defects of his instruments and observatory.
The meridian buildings are reported to be now in perfect order, and
the instruments in a satisfactory state. It will be remembered that
Professor Smyth has undertaken to determine the places of stars
compared with the small planets and comets in extra-meridian ob-
servations, and when he is fully prepared (of which due notice will be
given), it will be desirable that he should have early information of

• The working of this instrument is considered to absorb one assistant
and one additional computer.

t When the limar tables are made to satisfy the places of the moon
given in the 'Reduction of the Greenwich Lunar Observations,' and are
further corrected by observations made with the azimuth and altitude in-
strument, well-observed moon-culminations will not require corresponding
observations, and occultations will yield trustworthy results wherever they
may be observed.

F2

68 Royal Astronomical Society.

the approximate places of those stars, on which accurate and import-
ant comparisons depend.

The Astronomer Royal has communicated, and doubtless will con-
tinue to communicate, the mean right ascensions of the stars em-
ployed at Greenwich, so that the Edinburgh places will harmonize
with those of Greenwich ; or, as Professor Smyth remarks, " it will
not work against but co-operate with Greenwich."

There seems reason to complain of the smoke of the city, but pro-
bably this will not very materially injure the great mass of obser-
vations. I'he transit has a noble object-glass 6| inches aperture,
and the Professor proposes to use the circle with illuminated wires on
a dark field.

The instability of the Edinburgh transit was suspected by Profes-
sor Henderson to arise from the effect of temperature on the founda-
tion. Professor Smyth has traced it to a much simpler cause, a de-
fective original construction of the Ys. He thus describes the con-
struction of the new Ys : —

" They are large slabs of cast-iron, covering the whole area of
the top of the pier, and weighing several hundredweight ; there are
no adjusting-screws, but the sides of the angles in which the pivots
rest have been filed away, until the instrument is made to move as
nearly in the plane of the meridian as could, perhaps, have been
managed with screws. One good result has been certainly proved
to have followed, viz. that the reversing of the instrument to obtain
the error of coUimation does not now sensibly throw it out in azi-
muth, which Professor Henderson used to complain of with the old
Ys. Touching the fears that the new Ys might split the stone piers,
and the hopes that they might correct the temperature fluctuations,
there has not been sufficient time yet to settle that question through
the medium of the large transit ; it may, however, be considered to
be pretty well set at rest by the experiments on the 30-inch transit.
This was mounted on a similar huge block of cast-iron screwed and
cemented down to its pier ; on this it has now been sticking for a
year, as firmly but as innocently as could be desired.

" The following is a list of azimuth errors of the 30-inch transit
in its cast-iron block, as determined by all the transits of a Ursae
Minoris, observed on all five wires, during the period which elapsed
from the final fihng of the Ys to a snapping of two of tlie wires,
which took place, it was supposed, from moisture : the clock-star
used on each occasion was 6' Ceti : —

s

1847. Nov. 1 -0-091

2 -0003

15 -0058

16 -0040
30 +0043

Dec. 14 -i-0006

1848. Jan. 7 +0020

13 —0024
" Now these apparent fluctuations of the Ys in azimuth, which
are very small, include the probable error with which each observation

Royal Astrojiomical Society. 60

may be affected, by reason of the small optical power of the tele-
scope and other matters inseparable from an inferior instrument.
Hence they may be considered to be quite insignificant : taking them,
however, as they are, and comjiaring them with the azimuth errors
of the large transit in the corresponding period of the years 1841-42,
the fluctuations of the large instrument turn out to be five times as
great as those of the small one ; a convincing proof that the cause of
the changes hitherto remarked is not in the ' hill on which the ob-
servatory is built.' The above list of errors in azimuth may also
convince observers that they may themselves rub down unadjustable
Ys to limits which will be abundantly within easy calculation (and
with transits, too, without micrometer wires).

*' The Ys of the large transit having been erected, every screw
about the instrument was tried, to make sure that it was doing its
duty : a number of the smaller ones (which seemed to be made of
brass wire — drawn brass, not cast brass) were found quite rotten ;
these were replaced, and a good many new ones introduced about
the sliding tubes at the eye end ; handles for moving the instru-
ment, and acting only in the plane of the meridian, were added ; and
then, as the line of soldering of the telescope was beginning to
show symptoms of oxidation, the instrument was painted. A nadir-
pier and mercury-trough have been established, and a collimating
eye-piece of peculiar construction, which for perfect vision seems to
leave little to be desired, and reveals almost every affection of the
instrument. On account of some of its revelations, the fixed wires
have been removed, and five in their place mounted on the micro-
meter-frame. I propose to examine the errors of collimation and
level every night, before and after the observations, as shall be found
necessary, and am now engaged in trying to cure the reversing-
carriage of a trick it has got of throwing the instrument to the west
during reversal. Collimating lenses, of the full aperture of the
object-glass, for marks on the boundary wall, are also being put up,
as the old semi-collimating semi- meridian m.arks are now seldom
seen, on account of the increased smoke of the city ; and when they
are, the 6* 5-inch aperture of the transit must be reduced to 2 inches,
and the eye-piece pulled out so far as to make the wires very indi-
stinct and unsteady."

Oxford.

The Radcliffe observer has lately published his seventh annual
volume. This consists, like the preceding volumes, chiefiy of ob-
servations of circumpolar stars contained in Groombridge's Ca-
talogue.

It is not necessary to dwell upon the merits of Groombridge's
Catalogue, one of the most laborious tasks ever undertaken by an
amateur, as well as one of the most useful. His transit circle,
though i^erhaps rather weak as a right ascension instrument, was at
the time of its construction, and for many years after, the most per-
fect instrument in existence for determinations in north polar di-
stance. On this account, and considering that the time elapsed

70 Royal Astronomical Society.

since Groombridge made his observations is sufficient to detect and
exhibit proper motions, Mr. Johnson undertook to re-observe the
Catalogue with great care, and has now nearly completed the task.

The north polar distances have almost all been redetermined, and
a large portion of the right ascensions ; there are, however, several
gaps, occasioned by the necessity of observing certain circumpolar
stars for meridian error, and fundamental stars for clock error. The
increased number of well-determined stars will now allow the ob-
server more liberty in this respect, and the blanks are rapidly filling up.

Besides the general advantage of a full standard catalogue of stars
within 50° of the north pole (which by the aid of Groombridge's de-
terminations may be carried forwards for some years), and the ma-
terials thus afforded for investigating precession, proper motion, &c.,
a special advantage will be found in geodesical operations from this
large supply of accurate places for the zenith sector*.

Mr. Johnson expects very soon to receive the heliometer by Rep-
sold, when his attention will be directed to another department of
practical astronomy. Thus limited in time, and having the aid of
only one assistant, he has been induced to confine himself in most
cases to Uvo observations of a star in the same year, and occasionally
to one. The star has, however, been observed in different years, so
that there is a considerable check on errors of computation and on
casual fluctuations in the instruments.

Care has generally been taken to note circumstances of interest
connected with stars, which have come under observation. Among
others, their magnitudes have been watched with much attention.
The method adopted has been simply to estimate the apparent mag-
nitudes in reference to ideal standards ; and pending the discovery of
some more accurate photometric measure, Mr. Johnson has instituted
an inquiry as to the degree of reliance which may be placed on the
method he has pursued. This inquiry has not yet been fully followed
out ; but the results, as far as they go, are given in the preface to
the present volume. From what is there said, it may be inferred
that Mr. Johnson would recommend that at every observation of a
star, not distinctly visible to the naked eye, an estimate of its mag-
nitude should be noted, unless there is some obvious impediment to
a correct determination ; and that a mean of such estimates should
be taken as the magnitude of any given star, just as the mean of a
number of observations in right ascension or north polar distance is
considered as the correct right ascension or north polar distance.

Mr. Johnson acknowledges the great services which he has hi-

* It is proper to remind observers who possess instruments not of the
highest class, or who cannot afford the time for deducing fundamental
places, that the partial catalogues in the Radcliffe Observations will supply
them abundantly with zone stars, from the pole to 50° of north polar di-
stance, and that the volumes of the Edinburgh Observations will afford a
sufficient number of similar stars for a complete zodiacal catalogue. With
these and the Greenwich catalogues there can be no want of a sound base
of operations. Professor Argelander has made excellent use of the Rad-
cliffe Observations in his admirable Zone Observations.

Royal Astronomical Society. 71

therto received in the voluntary revision of his work, first from Mr.
Harris, our late assistant- secretary, and latterly from Mr. William
LufF, of Oxford. Mr. LufF has most kindly undertaken to read the
proof-sheets and to revise the additions* ; and from the great care
employed, it is hoped few typographical errors escape. When any
errors are detected, Mr. Johnson hopes that they will be communi-
cated to him.

The entire expense of printing these observations is borne by the
RadclifFe trustees. The beautiful typography, and the convenient
size of the volumes, enhance their value, and it is gratefully acknow-
ledged that the trustees distribute them liberally and judiciously.

Cambridge,

The Syndicate appointed to visit the Cambridge Observatory made
a report to the Senate, of which the following is the substance : —

The total number of observations in 1847 were, —

With the transit 2540

circle 2285

Northumberland equatoreal 1400

The observations with the meridional instruments are chiefly of
the sun (of which there is a very extensive series), moon, Jupiter,
Saturn, Uranus and Neptune, with a good series of Astraea, Flora,
and Iris. About 300 stars have been also observed.

The equatoreal observations are for the most part of the minor
planets and comets, which could not be seen on the meridian. These
are Neptune, Astraea, Hebe, Iris, Flora, and the following comets : —
Hind's, Feb. 6 ; Mauvais', 3rd ; Miss Mitchell's ; Colla's.

Professor Challis finds himself so much oppressed with unreduced
and unpublished observations, that he has discontinued observations
of the sun, moon, and the older planets since the beginning of this
yearf. The recently- discovered planets are observed on the meri-
dian and with the Northumberland equatoreal, and the results com-

* ** The process of revision is as follows : — Mr. Luff receives the proof
sheet as soon as it comes from the printer. He goes over all the additions,
without having the copy by him ; he notes all the mistakes he finds ; then
the proof is collated with the copy, and it is seen which are the mistakes of
the printer and which of the copy. All being corrected, the proof is
returned to the printer. The revise is carefully read over again, and no
sheet is marked for press till it is clear of mistakes."

t It is perhaps proper to inform those who are not acquainted with the
University of Cambridge, that Professor Challis gives lectures during one
term on physics, and that he is largely engaged in the university examina-
tions. His duties as lecturer and examiner vimt be attended to in the^rs^
place, whatever the observatory business may be. The university cannot
afford to give such a salary as will secure persons competent to carry on
the computations without constant superintendence ; and when an assist-
ant has obtained the necessary acquirements, he is naturally and properly
on the look-out for a better place. It is not generally known how much
mere heavy labour has been actually performed by the late and present
professor.

72 Royal Astronomical Society.

municatfcd to the Royal Astronomical Society and to foreign astro-
nomers.

The meridional observations of 1847 are completely reduced. The
equatoreal observations are less forward.

The volume for 1843 is nearly ready for publication. It does not
contain the equatoreal observations, which are reserved for separate
publication. Two appendices are added; one containing so many
of the observations made in search of the planet Neptune as are re-
quired to substantiate the statements given in the special report of
Dec. 12, 1846 ; and the other a description of the Northumberland
telescope and dome, drawn up by the Astronomer Royal.

Liverpool.

A very fine equatoreal has recently been erected at the Liverpool
observatory. The general form of the instrument has been mentioned
in former Notices ; and it promises, so far as we have heard, to be
the most accurate and most convenient instrument of its size now
existing. The object-glass is by Merz of Munich, of eight inches
aperture ; and as it has been approved of by Messrs. Dawes and
Lassell, most capable and somewhat fastidious judges, there can be
no doubt of its superior excellence. In firmness and steadiness the
equatoreal is reported to resemble a meridian instrument. The hour-
circle is carried, as in the Northumberland telescope, by clock-work,
and the right ascension is read off at once by the verniers. The
Astronomer Royal, under whose direction the instrument was con-
structed, has given a perpetual motion to this hour-circle by clock-
work moved by a water-wheel, to which a regulator is applied. The
variation of the clock does not exceed P per hour. The declination
and hour-circles are sufficiently good to give excellent results when
objects are compared beyond the limits of the micrometer, an im-
mense advantage when time is wanting and the weather is uncer-
tain, and in all cases a great comfort, as it secures perfect identifi-
cation.

We do not know certainly what line of astronomical research Mr.
Hartnup will take up. He will do most wisely to follow his own
inclination ; but such an instrument would be very well employed in
observing the planets, for instance, especially the smaller planets,
when they cannot be observed on the meridian at Greenwich. This
would not only complete the series of the Greenwich observations,
but would greatly relieve the Cambridge observatory, on which this
branch of observing has of late pressed heavily*.

* It is desirable that a semi- public observatory like Liverpool should
take a determinate line. We have every reason to admire the zeal and
steadiness of our amateur observers, many of whom might be cited as mo-
dels in these respects ; but they ought not to be tied down to a strictness
and continuity of research which must often be inconvenient and some-
times impossible.

Rm/al Society. 73

ROYAL SOCIETY.
[Continued from vol. xxxiii. p. 551.]
Anniversary Meeting, November 30, 1 848.
The Marquis of Northampton, President, in the Chair.
The President, after returning thanks to the Royal Society for
the honour conferred on him for ten yeare, delivered the Medals
with the following words : —

Mr. Galloway,

I deliver this Royal Medal to you with great satisfaction, for your
communication on one of the most interesting and difficult problems
in Astronomy, the proper motion in space of our system ; specula-
tions which may almost seem too mighty and daring for the human
intellect.

One who, like yourself, has entered on such a path of discovery,
is not likely to turn from it. In further pursuing it, I feel assured
that your zeal for the prosperity of the Royal Society will induce
you to enrich our Transactions with other communications. Should
my hopes prove well-founded, though my successor will, from his
own pursuits, be much better able than myself to appreciate your
labours, he will not be able to hail them with greater pleasure than
myself.

Mr. Hargreave,

I am glad to deliver into your hands this Royal Medal for the
mathematical paper with which you have enabled the Council to
adorn the Philosophical Transactions.

It is a paper, from its nature indeed, more suited for the attentive
study of the closet, than for reading before an audience, however
scientific, but it is not on that account less valuable.

Mathematical analysis is doubly important : important in itself,
and important as one of the great instruments of philosophical in-
vestigation. Every extension of it must then be at all times most
highly welcome to a Society founded for the advancement of natural
knowledge, and I, therefore, in its name, tender its thanks and an
expression of the hope that it will not be the last communication that
we shall receive at your hands.
Mr. Adams,

It is a great pleasure to me to be the channel by which the
Council of the Royal Society gives you this Copley Medal.

In their award, I am sure that they have not done more than
justice to the scientific zeal, industry, and skill exerted by you in
the search of the great and distant body that caused the perturba-
tions of the planet Uranus, a search crowned with success, both in
your case and in that of your illustrious friend Le Verrier.

If he be an honour to his nation, not the less so are you to En-
gland ; if he is a worthy follower of La Place, not less so are you of
Newton. His name and yours will remain iraperishably united in
the annals of the glorious science which you both cultivate with so
much zeal and so much success.

74- Royal Society.

Lieut.-Col. Sabine,
1 have to request of you, when transmitting to M. Regnault this
Rumford Medal, to state to him the importance which the Royal
Society attaches to his researches, determining with a degree of
accuracy hitherto unobtained, the laws which govern the connexion
between the temperature and elasticity of saturated steam, and the
quantity of heat absorbed by a given weight of water under different
densities and pressures.

The laws which govern the expansion of atmospheric air, under
diiferent pressures, and the expansion and densities of different gases
and mercury, and the measurement of temperatures by these means,
form in a series of memoirs altogether the most important investi-
gations hitherto made on this subject.

Had the philosophical and philanthropical founder of this
Medal been now living, I am sure that he would have cordially
approved of the award of it to inquiries connected with the most
important power that Providence has, as yet, given to man for
lightening and assisting his industry, and for giving him speed for
crossing sea and land, compared with which, the fabled wings of
Daedalus would have been comparatively useless. My only regret
on the present occasion is, that M. Regnault is not here himself to
receive this Medal.

The Statutes relating to the election of Council and Officers
having been read by the Secretary, and Dr. Royle and Mr. Bennett
having, with the consent of the Society, been nominated Scrutators
to assist the Secretaries in examining the lists, the votes of the
Fellows present were collected.

Mr. Bennett reported the following Noblemen and Gentlemen as
being duly elected Officers and Council for the ensuing year: —

President. — The Earl of Rosse.
Treasurer. — George Rennie, Esq.
o , . f S. Hunter Christie, Esq.
Secretaries. | Thomas Bell, Esq.

Foreign Secretary. — Lieut.-Col. Edward Sabine, R.A.

Other Members of the Council. — George Biddell Airy, Esq., M.A. ;
Sir James Clark, Bart., M.D.; John P. Gassiot, Esq.; Thomas
Graham, Esq., M.A. ; William Robert Grove, Esq., M.A. ; Leonard
Horner, Esq. ; Sir Robert H. Inglis, Bart., LL.D. ; John George
Shaw Lefevre, Esq., M.A. ; Sir Charles Lyell, M.A. ; William Allen
Miller, M.D. ; The Marquis of Northampton ; Richard Owen, Esq.;
John Phillips, Esq. ; Peter Mark lloget, M.D. ; the Dean of West-
minster; Charles Wheatstone, Esq.

It was moved by Sir Robert Harry Inglis, Bart., seconded by
Mr. Broughton, and resolved unanimously : —

That on this the last occasion of the Marquis of Northampton
occupying the Chair of the Royal Society as its President, the
special thanks of the Society be cordially tendered to his Lordship,
for his able, zealous, and efficient discharge of the duties of that
office for ten years.

Royal Society. 75

On the motion of Dr. Paris, seconded by Professor Baden Powell,
it was resolved unanimously : —

That the best thanks of the Royal Society be, and they are hereby
given, to Dr. Roget for his continued and valuable services during
a period of twenty-one years, in the office of Secretary to the Society.

Dec. 7. — " Experimental Researches in Electricity." By Michael
Faraday, Esq., F.R.S. Tvpenty- second Series. § 28. On the Cry-
stalline Polarity of Bismuth and other bodies, and on its relation to
the magnetic form of force.

The author states that in preparing small cylinders of bismuth by
casting them in glass tubes, he had often been embarrassed by the
anomalous magnetic results which they gave, and that having de-
termined to investigate the matter closely, it ended in a reference of
the effects to the crystalline condition of the bismuth, which may be
thus briefly stated. If bismuth be crystallized in the ordinary way,
and then a crystal, or a group of symmetric crystals, be selected and
suspended in the magnetic field between horizontal poles, it im-
mediately either points in a given direction, or vibrates about a given
position, as a small magnetic needle would do, and if disturbed from
this position it returns to it. On resuspending the crystal so that
the horizontal line which is transverse to the magnetic axis shall be-
come the vertical line, the crystal then points with its maximum
degree of force. If it be again resuspended so that the line parallel
to the magnetic axis be rendered vertical, the crystal loses all direc-
tive force. This line of direction therefore, which tends to place
itself parallel to the magnetic axis, the author calls the Magne-
crystallic axis of the crystal. It is perpendicular, or nearly so, to the
brightest and most perfect of the four cleavage planes of the crystal.
It is the same for all crystals of bismuth. Whether this magne-
crystallic axis is parallel or transverse to the magnetic axis, the bis-
muth is in both cases repelled from a single, or the stronger, pole ;
its diamagnetlc relations being in no way affected. If the crystal
be broken up, or if it be fused and resolidified, and the metal then
subjected to the action of the magnet, the diamagnetic phenomena
remain, but the magnecrystallic results disappear, because of the
confused and opposing crystalline condition of the various parts. If an
ingot of bismuth be broken up and fragmentary plates selected which
are crystallized uniformly throughout, these also point ; the magne-
crystallic axis being, as before, perpendicular to the chief plane of
cleavage, and the external form, in this respect, of no consequence.

The effect takes place when the crystal is surrounded by masses of
bismuth, or when it is immersed in water, or solution of sulphate of
iron, and with as much force, apparently, as if nothing intervened.

The position of the crystal in the magnetic field is affected by the
approximation of extra magnets or of soft iron ; but the author does
not believe that this results from any attractive or repulsive force
exerted on the bismuth, but only from the disturbance of the lines of
force or resultants of magnetic action, by which they acquire as it
were new directions ; and, as the law of action which he gives, is, that
the line or axis of magnecrystallic force tends to place itself parallel,

76 Royal Society.

or as a tangent, to the magnetic curve or line of magnetic force, passing
through the place where the crystal is situated, so the crystal changes
its position with any change of direction in these lines.

A common horse-shoe magnet exhibits these phenomena very well :
the author worked much with one lifting 30lbs. by the keeper ; but
one that can raise a pound or two only, is sufficient for many of the
actions. When using the electro-magnet, the advantage of employ-
ing poles with large plane opposed faces is mentioned as being con-
siderable, for then diamagnetic phenomena are almost or entirely
avoided and the peculiar magnecrystallic relations then appear.

The peculiar force exerted in these phenomena is not either at-
tractive or repulsive, but has for its distinctive character the ten-
dency to place the crystal in a definite position or direction. The
author further distinguishes it from that described by M. Pliicker in,
his interesting memoir upon the repulsion of the optic axes of cry-
stals by the poles of a magnet*, in that, that is an equatorial force,
whereas this is an axial force.

Crystals of antimony were then submitted to a similar magnetic
examination, and with the same results. But there were also certain
other effects produced of arrest and revulsion, the same in kind as
those described in a former series of the ' Experimental Researches ''
(par. 2-309, &c.) ; these are wrought out and eliminated, and the re-
sults described.

Arsenic also proved to be a body capable of pointing in the mag-
netic field, like bismuth and antimony.

The paper describing the foregoing results is dated 23rd of Sep-
tember, 1848. In a later paper of the date of 20th October, 1848,
the author continues his researches. Native crystals of iridium and
osmium, and also crystallized titanium and tellurium, appeared to be
magnecrystallic : crystals of zinc, copper, tin, lead, gold, gave no
signs of this condition. Crystals of sulphate of iron are very strongly
affected by the magnet according to this new condition, and the
magnecrystallic axis is perpendicular to two of the planes of the
rhomboidal prism ; so that when a long crystal is employed, it will
not, as a mass, point between the poles, but across the line joining
them. On the other hand, the sulphate of nickel has its magne-
crystallic axis parallel, or nearly so, to the length of the ordinary
prism. Hence bodies, both magnetic and diamagnetic, are, by their
crystalline condition, subject to the magnetic force, according to the
law already laid down. Diamond, rock-salt, fluor spar, boracite, red
oxide of copper, oxide of tin, cinnabar, galena, and many other
bodies, presented no evidence of the magnecrystallic condition.

The author then enters upon a consideration of the nature of the
magnecrystallic force. In the first place he examines closely whe-
ther a crystal of bismuth has exactly the same amount of repulsion,
diamagnetic or other, when presenting its magnecrystallic axis
parallel or transverse to the lines of magnetic force acting on it. For
this purpose the crystal was suspended either from a torsion ba-

* Poggendorff's Annalen, Ixxii. Oct. 1847; or Taylor's Scientific Me-
moirs, vol. V. p. 353.

Royal Society. 77

lance, or as a pendulum thirty fevjt in length ; but whatever the
position of the magnecrystallic axis, the amount of repulsion was
the same.

In other experiments a vertical axis was constructed of cocoon
silk, and the body to be examined was attached at right angles to it
as radius ; a prismatic crystal of sulphate of iron, for instance, whose
length was four times its breadth, was fixed on the axis with its
length as radius and its magnecrystallic axis horizontal, and there-
fore as tangent ; then, when this crystal was at rest under the torsion
force of the silken axis, an electro-magnetic pole was so placed,
that the axial line of magnetic force should be, when exerted, ob-
lique to both the length and the magnecrystallic axis of the crystal ;
and the consequence was, that, when the electric current circulated
round the magnet, the crystal actually receded from the magnet
under the influence of the force, which tended to place the magne-
crystallic axis and the magnetic axis parallel. Employing a crystal
or plate of bismuth, that body could be made to approach the mag-
netic pole under the influence of the magnecrystallic force ; and this
force is so strong as to counteract either the tendency of the magnetic
body to approach or of the diamagnetic body to retreat, when it is
exerted in the contrary direction. Hence the author concludes that
it is neither attraction nor repulsion which causes the set or deter-
mines the final position of a magnecrystallic body.

He next considers it as a force dependent upon the crystalline con-
dition of the body, and therefore associated with the original mole-
cular forces of the matter. He shows experimentally, that, as the
magnet can move a crystal, so also a crystal can move a magnet.
Also, that heat takes away this power just before the crystal fuses,
and that cooling restores it in its original direction. He next con-
siders whether the effects are due to a force altogether original and
inherent in the crystal, or whether that which appears in it, is not
partly induced by the magnetic and electric forces ; and he concludes,
that the force manifested in the magnetic field, which appears by
external actions and causes the motion of the mass, is chiefly, and
almost entirely induced, in a manner subject indeed to the crystalhne
force and additive to it ; but at the same time exalting the force and
the effects to a degree which they could not have approached without
the induction. To this part of the force he applies the word mag-
neto-cry stallic, in contradistinction to the word magnecrystallic,
which is employed to express the condition, or quality, or power,
which belongs essentially to the crystal.

The author then remarks upon the extraordinary character of the
power, which he cannot refer to polarity ; and gives expression to
certain considerations and views which will be best learned from the
paper itself. After this, he resumes the consideration of Pliicker's
results "upon the repulsion of the optic axes of crystals" already re-
ferred to, and arrives at the conclusion that his results and those
now described have one common origin and cause. He then con-
siders Pliicker's results in relation to those which he formerly ob-
tained with heavy optical glass and many other bodies. In con-

78 Intelligence and Miscellaneous Articles.

elusion he remarks, " How rapidly the knowledge of molecular forces
grows upon us, and how strikingly every investigation tends to de-
velope more and more their importance and their extreme attraction
as an object of study ! A few years ago magnetism was to us an
occult power affecting only a few bodies ; now it is found to influ-
ence all bodies, and to possess the most intimate relations with elec-
tricity, heat, chemical action, light, crystallization, and, through it,
with the forces concerned in cohesion ; and we may, in the present
state of things, ■w^ell feel urged to continue in our labours, encouraged
by the hope of bringing it into a bond of union with gravity itself."

IX. Intelligence and Miscellaneous Articles.

ON A NEW MODIFICATION OF PHOSPHORUS.

MSCHROETTER has stated, that the red substance which
• forms on the surface of phosphorus exposed to the light, is
entirely an isomeric formation of phosphorus. It takes place in va-
rious gases, such as hydrogen, azote and carbonic acid, when the
phosphorus is perfectly dry ; it is therefore impossible to attribute this
effect to the oxidizement of the phosphorus.

The transformation is rapid in direct light, but it is observable
even in diffused feeble light. Heat effects the same change. "When
phosphorus which has been thoroughly dried is exposed for forty or
sixty hours to a temperature of 464° to 482° F., a great part of it
becomes of a carmine-red. A red opake powder is first detached,
which is soon uniformly generated in every portion of the mass, and
it eventually falls to the bottom of the vessel.

By operating on small quantities in close vessels, and by continuing
the action in the mode described, M. Schroetter has succeeded in
converting the whole of the phosphorus into the red modification.

In order to isolate the amorphous phosphorus prepared in rather
large quantity, M. Schroetter employed sulphuret of carbon, which is
an excellent solvent of common phosphorus, but dissolves amorphous
phosphorus with difficulty ; filtration is performed with peculiar pre-
cautions ; the residue is afterwards boiled in a solution of potash of
rS density; it is then to be washed with pure water, afterwards
with water acidified with nitric acid, and again with pure water ;
the phosphorus thus obtained is a powder varying in colour from
scarlet to deep carmine-red.

Under peculiar circumstances a blackish-brown modification of
phosphorus may be obtained. The density of amorphous phosphorus
at50°F. is 1-964.

Amorphous phosphorus is unalterable by exposure to the air, in-
soluble in aether, alcohol, naphtha, or chloride of phosphorus ; oil of
turpentine dissolves a little at a high temperature ; it is much less
combustible than common phosphorus, it gives out no light in the
dark, and does not bum till exposed to 500° F. This is the tempe-
rature at which amorphous phosphorus begins to return to the state
of common phosphorus when heated in an inert gas.

Meteorological Observations. 79

Amorphous phosphorus does not combine with sulphur at 233° F.,
it must be heated to 446"^. Chlorine combines with it without
the disengagement of light, a boiling solution of potash acts upon
it, evolving non-inflammable phosphuretted hydrogen, and the
phosphorus becomes the black modification, described by M. The-
nard; and according to M, Schroetter, phosphorus never becomes
the black modification without having previously assumed the red
one. — Comptes Rendus, Octobre 1848.

METEOROLOGICAL OBSERVATIONS FOR NOV. 1848.
Chiswick. — November 1. Rain, with fog. 2, Fine : cloudy. 3. Overcast :
cloudy and fine. 4. Overcast. 5. Clear and frosty : overcast. 6. Overcast.
7. Clear and cold : sharp frost at night. 8. Frosty : bright sun : clear and frosty.
9, 10. Clear: slight frost at nights. 11. Overcast. 12. Slight rain. 13, 14. Very
fine. 15. Clear : severe frost at night. 16. Frosty : clear and fine. 17. Densely
clouded: rain: peculiar luminosity in the evening: overcast. 18. Densely
clouded. 19. Very fine. 20. Densely clouded : rain : boisterous. 21. Clear
and fine : peculiar aurora borealis half-past seven p.m. in N.W. 22. Overcast.
23. Rain. 24. Cloudy: clear and frosty. 25. Frosty: overcast: slight rain.
26. Cloudy. 27. Very fine. 28. Cloudy. 29. Densely overcast : boisterous.
SO. Clear : cloudy : partially overcast.

JVlean temperature of the month 41°'01

Mean temperature of Nov. 1847 44 '61

Mean temperature of Nov. for the last twenty years 43 "00

Average amount of rain in Nov 2*56 inches.

Boston. — Nov. 1. Foggy. 2. Fine. 3. Rain : rain a.m. and p.m. 4. Fine:
rain early a.m. and snow p.m. 5. Fine : rain p.m. 6. Cloudy. 7, 8. Fine.
9. Cloudy : snow A.M. 10 — 14. Fine. 15. Cloudy. 16. Fine. 17. Cloudy:
rain A.M. 18. Cloudy. 19. Fine. 20 Cloudy : rain in evening. 21. Fine.
22. Cloudy : rain a.m. 23. Cloudy. 24. Fine. 25. Fine : rain p.m. 26 —

28. Fine. 29. Fine : rain p.m. 30. Fine.

Applegarth Manse, Dumfries-shire. — Nov. 1. Dull a.m. : soft rain p.m. 2. Fine
generally : flying showers. 3. Rain a.m. : cleared : looking frosty. 4. Frost
hard : hills covered with snow. 5. Frost hard : sprinkling of snow. 6. Thaw:
showers : stormy. 7. Frost : fine clear day. 8. Frost : clear : snow p.m. 9. Fine
winter day : fro£t : snow inch deep. 10. Frost : clear: snow melting. II. Frost :
dull and cloudy : snow gone. 12. Fine: no frost a.m. : gentle frost p.m. 13.
Frost a.m. : a change of weather. 14. Frost a.m.: thaw: frost again. 15. FrostA.M.:
thaw P.M. 16. Drops of rain occasionally. 17. Rain during night : aurora very
splendid. 18. Heavy rain during night : ditto day. 19. Frost a.m. : thaw p.m.
20. Storm of rain and wind : flood. 21. Bleak and dull all day. 2A Rain,
greater part of day. 23. Fair a.m. : rain p.m. 24. Frost again. 25,26. Thaw:
rain and high wind. 27. Fair and fine. 28. Wet nearly all day : high wind.

29. Frequent showers. 30. Fair, but cloudy.

Mean temperature of the month 39°'8

Mean temperature of Nov. 1847 47 -7

Mean temperature of Nov. for the last twenty five years . 40 '4

Rain in Nov. 1847 S*79inches.

Average amount of rain in Nov. for twenty years 3'60 „

Sandvdck Manse, Orkney. — Nov. 1. Drops : rain : aurora. 2. Showers: hail-
showers. 3. Snow : hail-showers. 4. Snow,: clear. 5. Showers. 6. Cloudy :
showers. 7. Showers : sleet. 8. Snow-showers : clear : frost. 9. Cloudy : rain.
10. Cloudy: clear: aurora. 11. Bright: drizzle: showers. 12. Fine: clear.
IS. Cloudy : showers. 14. Cloudy : hail-showers. 15. Cloudy: showers. i6.
Bright : cloudy. 1 7. Showers : aurora. 18. Damp : showers : aurora. 19. Hoar-
frost: showers. 20. Rain. 21. Rain : cloudy : aurora. 22. Rain. 23. Showers:
aurora. 24. Cloudy : clear. 25, 26. Cloudy : rain. 27, 28. Showers. 29.
Cloudy : showers. 30. Showers.

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THE
LONDON, EDINBURGH and DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[THIRD SERIES.]

FEBRUAR Y 1849.

X. Experiments on Diamagnetism. By H. C. QErsted*.

A T a meeting of the Royal Society of Sciences at Copen-
■^*- hagen on the 30th of June, I communicated the results
of some researches I had mnde upon diamagnetism, an abs-
tract of which appeared in the proceedings of the Society.
During the vacation I have continued my experiments, and
have obtained several new results. As the memoir will not
appear for many months, I have decided on giving an abstract
which may be communicated to my distant friends.

My researches relate to the celebrated diamagnetic disco-
veries of Mr. Faraday, and to the extensions they have re-
ceived from some learned Germans.

Mr. Faraday, in the experiments with his great electro-
magnet, met with a class of bodies which are repelled by the
two poles of the magnet. One or two examples of this repul-
sion had been known for some time ; but the researches of
the illustrious Englishman has rendered this fact general, and
of such importance as to become the object of attention of all
philosophers. Brugmanns had already discovered in 1778
that bismuth is repelled by the two poles of a magnet. Bec-
querel, sen., again met with this repulsion both in bismuth
and in antimony. Mr. Faraday found that his great electro-
magnet produced this repulsion in almost all the bodies it did
not attract. He discovered at the same time that long pieces
thus repelled, assume, under the influence of the electro-
magnet, a position perpendicular to that which an attracted
body would take under the same circumstances. This pro-
perty he called diamagnetism.

M. Reich of Freiburg fj so well known from his beautiful
experiments on the deviation in the fall of bodies from a great

* From the Annates de CMmie et de P/it/sique for December 1848.
t See p. 127 of the present Number.
P/iil. Mag. S. 3. Vol. 34.. No. 227. Feb. 1849. G

82 Prof. CErsted's Experiments on Diamagnetism,

height, applied to the discovery of diama^netism the observa-
tion, overlooked by other philosophers, that the two poles of
the magnet employed together do not produce on these bodies
a repulsion equal to the sum of the repulsions produced by
each of them, but equal to their difference; so that their joint
effect is null when their forces are equal. At the same time
he made some experiments which seemed to indicate that the
pole which repulsed the diamagnetic body produced in the
parts adjacent to it a magnetic force similar to its own, but
not an opposite force, as occurs in attracted bodies. Prof.
Weber* confirmed the opinion of M. Reich by some very
elaborate researches, and showed that diamagnetic bodies ac-
quire, under the influence of the electro-magnet, a transversal
magnetism having two poles, but so placed that each of them
has the same kind of magnetism as the pole nearest to the
electro-magnet.

M. Poggendorff conceived some very decisive experiments,
which have the advantage of proving the new view in an easy
manner; and M. Pliicker contributed a further experiment,
which, if it did not tend to establish the view, rendered the
possibility of proving it more easy.

In my experiments I made use of the large U-shaped elec-
tro-magnet of the Polytechnic School of Copenhagen, which
is capable of carrying 1400 kilogrammes f. It must, how-
ever, be remarked, that it was not necessary to put all its force
in activity for these experiments; but there was rarely less

* See Taylor's Scientific Memoirs, vol. v. Part 19, p. 477-
f I have here conformed to the usual way of indicating the power of the
magnet, although there is much uncertainty, as is proved by my experi-
ments on this electro-magnet, conmiunicated to the Society in December
1847. The object of these experiments was to ascertain the weight the
electro-magnet was capable of carrying when its poles were furnished with
different pieces of iron. Within certain limits the carrying force increased
nearly in proportion to the mass of the keeper ; but what merits most
attention is, that the force of the electro-magnet expressed in weight does
not follow the same relation to the electro-motive power of the galvanic
apparatus, when the keeper is in contact with the electro-magnet, and when
it is at a certain distance. In contact, the mean effect of each galvanic
element was 712'5 kilogrammes. But two elements combined gave but
0'72 of the sum of the individual effects of the elements ; three elements
combined gave but 0-48; eight, 0-26 ; sixteen, 0-125; so that the effect of
sixteen elements was only twice that of one. At the distance TSS millim.
the effect of one element was only 0-1 78 of that of the same element in the
case of contact, but the effect increased in a very different ratio with the
number of the elements; in this case sixteen elements gave four times the
effect of a single one. At 2-225 millims. distance the effect of one element
was only 0-051 of that produced in case of contact; but sixteen elements
gave 9-4 times the effect which a single one gave. These experiments,
which require much time, I intend to continue as soon as my other engage-
ments will allow.

Prof. QErsted's Experiments on Diamagnetism. 83

than half of this power made use of, although the greater part
of them might have been executed with a much weaker force,
even with a single element. Each extremity of the electro-
magnet supports a horizontal piece of iron, which we shall
call a polar piece. These polar pieces serve to give a hori-
zontal direction to the action of the electro-magnet. It is be-
tween the two perpendicular surfaces, which face one another,
that the diamagnetic tody is made to oscillate. These are
called polar faces. In every case, excepting those indicated,
I have made use of rectangular pieces. At the commence-
ment of my experiments I employed cylindrical pieces; but
this form is less suited for discovering all the circumstances
which should be taken into consideration in these investiga-
tions.

A diamagnetic needle suspended horizontally between the
polar faces, assumes, as is well known, an equatorial \)OS\\\ox\^
which is parallel to the polar fiices ; but if it is raised a little
above the edges of the polar faces, it takes a perpendicular
direction to the prolonged polar faces. This position is at
the same time axial; but it will subsequently be seen that the
question here is as to its perpendicularity to the polar faces.
This phaenomenon is exhibited with remarkable quickness,
which renders the experiment very convenient for many dia-
magnetic investigations. When the needle is turned from its
perpendicular position, it reacquires it by oscillation. Its di-
recting force diminishes with its elevation above the polar
pieces. The experiment was made with several diamagnetic
bodies, with bismuth, amber, mother-of-pearl, tortoise-shell,
alabaster, quills of feathers, sulphur, coal, &c.

The change of direction observed in these experiments
vanishes as the polar faces are separated. At the distance of
17 millims. the effect was still well-marked; but it is very
much stronger at short distances. When the distance was
diminished so that the diamagnetic body could not be in-
serted between the polar faces, that is to say, perpendicular to
them, the part of the effect which takes place above the polar
faces was exhibited with considerable energy. When the dia-
magnetic needle was suspended above the upper edge of one
of the polar faces, it equally assumed the position called axial,
perpendicular to that edge, but with less force than under the
influence of the two faces. On examining the position which
the needle takes above the other edges of the polar piece, it
is found that it everywhere assumes the position perpendicular
to the edge to the influence of which it is exposed. In those
cases where it is exposed to the action of two edges at once,
it takes the intermediary position. Above the edge of a wedge-

G2

84 Prof. CErsted's Experiments on Diamagnetism.

shaped piece of iron, placed with its base on one of the poles
of the electro-magnet, the needle likewise assumes a position
perpendicular to this edge. On a cylindrical polar piece, the
needle placed with its centre above the edge of the polar face,
arranges itself perpendicular to it; but at some distance from
the edge it turns and takes a perpendicular position to the line
which may be drawn parallel to the axis in the more elevated
part of the cylindrical surfece. When a perforated cylinder is
employed as a polar piece, and the diamagnetic needle is made
to descend and rise alternately, and parallel to the polar face,
the needle is found to leave the position parallel to the polar
faces, and to assume the position called axial as soon as it is
opposite the perforations. I employed for this experiment a
needle of bismuth 16 millims. in length. "With two similar
polar pieces the same effect is obtained, but it is much greater.

When the diamagnetic needle is suspended between the
polar faces, it has, in accordance with the experiments of the
German philosophers above-mentioned, magnetic poles in the
transversal direction, arranged in such a manner that the mag-
netism of each side is of the same nature as that of the pole
nearest to the electro-magnet. The easiest method of obser-
ving this is that described by M. Pliicker, who introduces
between the polar faces, and parallel to them, a small bar of
iron separated from the faces by some non-magnetic substance.
As the sides of this bar acquire by the influence magnetism
the reverse of that of the nearest face, but each side of the
needle has the same magnetism as the nearest face, the needle
now retained by two forces oscillates with much greater velo-
city than when under the influence of the polar faces only.
When the diamagnetic needle is raised above a polar piece,
and its direction is changed, its magnetic poles change their
places at the same time.

I had been led into error at the commencement by several
phaenomena, which in the novelty of the investigation seemed
very complicated, but which nevertheless appeared very sim-
ple when the law had been found for them. At first I thought
that the diamagnetic needle above the polar pieces hatl in
each extremity the opposite magnetism to that of the adjoining
polar piece ; for the lower part of a bar of iron, influenced by
the piece, repelled the extremity of the needle which was above
that piece. This effect takes place not only on placing the
repelling pole of iron near to each side, but equally above and
below. Nevertheless later experiments have led me to reject
the conclusion which I had previously drawn. I found that
a piece of iron of moderate size receives from the polar piece
acting upon it, a magnetic force sufficient to repel the diar

Prof. CErsted's Experivients on Diamag7ietism. 85

magnetic matter of the needle, notwithstanding the poles
which it had received under the influence of the electro-
magnet. To discover the diamagnetic poles in this case, it is
necessary to employ very small pieces or blades of iron ;
they should not weigh more than two or three grammes. To
experiment with them the easier, I attached slips of zinc or
pieces of wood to them. By this means 1 at last arrived at
the conviction, that the lower part of the diamagnetic needle
suspended above a polar piece has the same magnetism, and
that its upper portion has the opposite. In experimenting on
this subject, 1 finally made use of a thin blade of iron, shaped
thus ([ , and fixed to a piece of wood. When this blade is
placed on the polar piece, it has in its upper part the same
magnetism as the polar piece, ami its lower part the opposite.
When the opening of the curve faces the needle, it attracts it;
but when the upper part is beneath, or its lower part above
the needle, it repels it.

When the needle is suspended over one of the polar pieces,
so that the prolongation of one of the perpendicular faces of the
piece divides the needle into two parts, it is found that the dia-
magnetic poles produced by the electro- magnet extend beyond
the part which is above or correspondent to the upper surface
of the piece. In some experiments made with a needle of bis-
muth of 56 millims., this effect extended to nearly 14 millims.

When the needle was divided into two equal parts by the
prolonged perpendicular faces, the extremity of the needle the
more distant from the polar piece was not polarized.

When the electro-magnet was furnished with two polar
pieces placed at a distance of 48 millims., I found that the
same needle had diamagnetic poles in all its parts. The
half of the needle which was turned towards the north pole
had north magnetism at its lower edge, and south mag-
netism at its upper edge ; the other portion of the needle
had, under the influence of the south pole, the magnetism of
this pole at the lower edge, and north magnetism at its
upper edge. There is, therefore^ opposite magnetism in the
two halves of each edge, taken separately, and in each half
between the two edges, the upper and lower one.

When the diamagnetic body is made to oscillate between the
polar faces, it does so with greater velocity the nearer it is
to one of the edges of this face. In one experiment in which
the electro-magnet was excited by sixteen of Bunsen's galvanic
elements, and where the distance of the polar faces was 6
millims., a needle of bismuth at an equal distance from the
upper and lower edges of these faces made twenty-five oscil-
lations m 30 seconds; but level with the edges, it made

86 Prof. CErsted's Experiments on Diamagnetism.

100 oscillations in the same time. Above the polar pieces, in
the axial position, the needle made only nineteen oscillations
in the same time. These experiments have been sufficiently
repeated and varied to afford the most perfect certainty as to
what has been stated ; but the investigations have not yet
been carried far enough to admit of an accurate numerical
law being deduced from them.

When a horizontal needle of bismuth is suspended by a
fibre of silk to the extremity of a balance, so that the balance
can be raised or lowered, the needle is found to be repelled
with so much more force the nearer it is situated to one of the
edges of the polar faces. It will be understood that this re-
pelling power causes the needle to ascend when it is near the
upper edges, and to descend when near the lower edges ; in the
intermediate position it neither ascends nor descends. When
the needle is suspended above the polar pieces, and conse-
quently in the direction perpendicular to the edges of the
polar faces, it is again re[)elled, but much more feebly than
when in the equatorial position.

Hitherto diamagnetic effects had only been observed in
bodies which are repelled by the two poles of the magnet. My
experiments have shown that a similar effect may be produced
in most bodies which are attracted by the two magnetic poles;
so that these bodies constitute a new kind of diamagnetic sub-
stances. These two classes may be distinguished by the names
o^ repulsive and attractixie diamagnetic bodies.

A needle made of a substance that is attracted by the mag-
net, but of which the magnetism is not of the same nature as
that of iron and nickel, when suspended between the two polar
faces of the electro-magnet, acquires, as is well known, the
position called axial by Mr. Faraday ; but if it is raised above
the upper edges, or lowered beneath the lower edges of the
polar faces, it assumes the equatorial position. This property
has hitherto been found in the following substances ; viz. pla-
tiiia, palladium, iridium, titanium, an alloy of 0"S25 of tin,
0*024 of bismuth, 0*108 of ii*on, in brass, German silver, wood-
charcoal, coke (fossil coal belongs to the repulsive diamag-
netic bodies), obsidian, native carbonate of iron, attractive
glass, Prussian blue, and solutions of iron.

In the majority of these substances, the magnetic poles which
they obtain during the influence of the electro-magnet disap-
pear nearly as soon as this influence is removed ; however,
their existence is betrayed when the poles of the electro-magnet
are suddenly changed, for then many of these bodies turn half
round, as is the case with a magnetic needle ; others do not
exactly turn, but oscillate, thus indicating their tendency to a

Prof. QErsted's Experiments on Diamagnetism. 87

change of position. But we find some attractive diamagnetic
bodies, sucli as a piece of iridium in my possession, wood-
charcoul and coke, which retain the poles they have accjuired
by the influence much longer, of which it is easily to be con-
vinced by experiments on the mariner's compass. The ex-
perimental investigation of the phaenomena exhibited by these
bodies is complicated by this duration of the polarity; but
they will probably lead us to discover the connexion which
exists between magnetism and diamagnetism.

When a needle constructed of an attractive diamagnetic
substance is suspended above the upper or beneath the lower
edge of a polar piece, it assumes a position parallel to this
edge. In this parallel position, which might either be per-
pendicular to the magnetic axis of the polar piece, or parallel
to it, or have quite another position than the figure of the
polar piece requires, the disposition of the magnetic forces in
the needle is transversal, as in a repellent diamagnetic body ;
but with this difference, that its lower part has the opposite
magnetism to that of the polar piece, and the upper part that
of the same nature.

I have not succeeded in causing iron to assume the diamag-
netic state. An iron wire, of which the diameter is but y\jth
of a millimetre, still takes the axial position just as well above
the polar faces as between them, and with a force nearly suffi-
cient to break the fibre of silk. This experiment has been
varied by placing in a quill of a feather, which is repellent,
a fragment of the same iron wire, merely 2 millims. in
length ; but this arrangement still exhibited the same effects
as the iron separately. The same result was also obtained on
substituting for the bit of iron wire a very small particle of
iron filings ; but on introducing, in the place of iron, a piece
of straw which had been immersed in a solution of iron, the
diamagnetic effects of attractive bodies was obtained. Nickel
gives the same results as iron. Thus nickel and iron ought
properly to be called magnetic in the strict sense. This pro-
bably applies to some other substances, perhaps to cobalt.

There is consequently a decreasing magnetic progression
which comprises the properly so-called magnetic substances,
the attractive diamagnetic substances, and the repulsive dia-
magnetic substances. The niagnetism of these last may be
considered as negative, if tlie magnetism of iron and the at-
tractive diamagnetic substances is regarded as positive.

The eflf'ect which the polar faces exert upon the attractive
diamagnetic bodies is like that which takes place with the re-
pulsive diamagnetic bodies, — stronger when the body is placed
nearer the upper or lower edges than their intermediate

88 Prof. Challis's B£searches in the

parts. A piece of attractive glass, 27 millims. in length,
which was suspended between tlie polar faces, 29 millims.
apart, in such a manner that the extremities of this needle
were not more than 1 niillim. distant from the polar faces,
was made to oscillate each time during 30 seconds. At an
equal distance from the upper and lower edges it made only
45 oscillations in the 30 seconds, but level with the polar faces
it made nineteen.

When the polar faces are at this distance, the needle does
not assume the equatorial position when it is suspended above
their edges. At the distance of 4*5 millims., it made 5'5
oscillations; 13*5 millims. distant it made only 2"5 oscil-
lations. The polar faces were approached to within 3 millims.
The needle, which now could not assume the axial position
between the faces, nevertheless showed its tendency to take
that position ; but raised 2 millims. above their edges, it
assumed the equatorial position, and made eighteen oscilla-
tions in 30 seconds. At a distance of y^^ths of a millim., it made
thirty-five oscillations. At the least possible distance, so as just
to avoid contact with the polar pieces, it made forty-five
oscillations.

It is seen that both the repulsive as well as the attractive
diamagnetic bodies make more numerous oscillations in a
parallel position to the polar faces than in the perpendicular
position. It must however be observed, as has already been
done, that the determinations of the numbers have not yet
been carried sufficiendy far to serve for the calculation of their
laws.

I lately made some experiments on the influence which heat
exercises on diamagnetic substances. These experiments are
not yet numerous enough ; they however prove to me that
some attractive diamagnetic substances pass into the class of
repulsive diamagnetic substances by an increased temperature.
The only substance which exhibits this effect in a high degree
is brass. My analogous experiments on other bodies are not
yet sufficiently decisive to warrant their publication.

XI. Continuation of Researches in the Mathematical Theoi-y
of Aerial Vibrations. By the Rev. J. Challis, M.A.,
F.R.S.i F.R.A.S, Plnmian Professor of Astronomy and
Experimental Philosophy in the University of Cambridge'^ .

IN replying to arguments against my conclusion that plane-
waves are physically impossible, I am required by the rules
of logic to notice those only which profess to invalidate any
* Communicated by the Author.

Mathematical llieory of A'&tnal Vibrations. 89

step of the reasoning by which the conclusion was arrived at,
the proof being a reductio ad absurdum. I shall therefore
confine myself to the single objection of this nature brought
forward by Mr. Stokes in his communication to the Philoso-
phical Magazine for January, viz. that which he expresses by
saying that I "integrate through an infinite ordinate." Now
the very terms of this objection preclude the necessity of
answering it. For it admits the existence of an infinite ordi-
nate, and consequently of some valid reasoning by which it is
shown to exist. I contend for nothing more. The expres-
sion "infinite ordinate" is a symbolical term to indicate that
a finite change of velocity or density occurs in an infinitely
small space. I have expressed the same thing in plainer
terms by saying, that the same point of the same wave is at
the same time a position of maximum velocity and of no velo-
city. It is clear, therefore, that there is no question as to the
mathematical reasoning, but as to the interpretation of the
result. I consider that the infinite ordinate is condemnatory
of the hypothesis of plane-waves: Mr. Stokes says that it is
indicative of certain physical circumstances analogous to a
bore. As this difference is not likely to be removed by fur-
ther mathematical discussion, I pass on to the case of spherical
waves.

First, I propose to exhibit as distinctly as possible the rea-
soning which Mr. Stokes has undertaken to call in question.
For this purpose I shall commence with the usual hydrody-
namical equations applicable to small disturbances, and pro-
ceed through the different steps in order, numbering the pa-
ragraphs for the sake of reference.

(1.) Formation of a general differential equation applicable
to small condensations.

Let/> = a^(l +5), x> being the pressure and s the small con-
densation at the time / at a point whose co-ordinates are a;, y^ z.
Then if 11, v, w be the resolved parts of the velocity at the same
point and at the same time in the directions of the axes of co-
ordinates, we have the known approximate equations,

c,ds_ djt, „^di dv_ ^^ds_ dw_

""dx^dt^^' ""'dy^dt-^' ""'dz^dt-^'

ds du dv dw _
dt dx dy dz~ '

By differentiating the last equation with respect to /, we
obtain

dt"^ "^ dtdx ■*■ dtdy "^ dtdz ~ '

90 Prof. Challis's Researches in the

whence, by substituting for -- , — , -.— , from the tliree first
equations, we liave the equation sought, viz.

df^ ~^ ' \dx^ "^ df "^ dzV'

(2.) The hypothesis of spherical waves.
In consequence of this hypothesis 5 is a function of the di-
stance (r) from the origin of co-ordinates, and the above equa-
tion becomes

dKsr _ 2 d'^.sf
~7F~""~d^'

(3.) Integral of the last obtained equation.

The general integral contains two arbitrary functions, each
of which separately satisfies the equation. I need only write
down the solution which has been employed in the present
discussion, viz.

5=i.F(r-a/).

(4.) Interpretation of the above integral.

The meaning of this integral may be exhibited as follows.
Draw any straight line OABA'B' from O the origin of co-
ordinates, and putting z for r—at, describe a curve APB such

that any ordinate MP shall represent the value of — -^ that

is, of the condensation 5, corresponding to the value OM
of r. Because the function F is arbitrary, the form of the

curve is arbitrary. It is admitted that the function F maybe
discontinuous, and accordingly that at a given time /„ Y{r — at^
has real values from r=OA to r = OB, and that for values of
r less than OA and greater than OB, Y{r — at^ = 0. The
state of condensation which this curve represents is propa-
gated with the uniform velocity r/, so that at a subsequent
epoch ^2 it has the position of the curve A'P'B'. The change
which the curve has undergone is such that, A'B' remaining
equal to AB, any ordinate PM corresponding to an abscissa
AM has to the ordinate P'M' corresponding to an equal
abscissa A'M', the ratio of OM' to OM. Or if PM, P'M'

Mathematical Theory of A'Mal Vibrations. 91

represent respectively the condensations 5j, 53, and OM = r,,
OM'=r2, wehave ^ = ^.

(5.) The contradiction.

Draw an ordinate })m indefinitely near to PM, and />W
indefinitely near to P'M', and let the interval Mm=M'm' = a.
Then the quantity of matter contained between the spherical
surfaces of which the radii are /-j and r^ + u, beyond what would
exist in the same space in the quiescent state of the fluid, is
4Tr,^5,a; and similarly the quantity of condensed matter be-
tween the surfaces of which the radii are r^ and r^ + a is
4>Trr^\ci. The same reasoning applies to every set of corre-
sponding ordinates of the two curves. Hence by the prin-
ciple of constancy of mass employed in investigating one of
the general hydrodynamical equations, those two quantities
must be equal to each other; that is, A!yrr^s^ot. = ^%r^^s,^.

ST*

Consequently — = -\. This result, which is incontrovertible,

is at variance with the conclusion in (4?.). I infer, therefore,
that the hypothesis of (2.) is inadmissible.

I have been thus explicit for the purpose of stating distinctly
the course which this discussion must take. It will be observed
that subsequently to making the hypothesis of (2.), the velocity
of the fluid is nowhere introduced. The rules of right rea-
soning absolutely forbid the introduction of any expression
for the velocity by an opponent, simply because such expres-
sion cannot be obtained without adopting the very hypothesis
the legitimacy of which is the point in dispute. Mr. Stokes
from beginning to end has argued from a value of the velocity
derived from the disputed hypothesis. To all his argument 1
have therefore this one answer : the truth of the expression
for the velocity is not proved. It is quite necessary that the
discussion should turn on the reasons which I have alleged
for the different steps of my argument, without taking for
granted the legitimacy of the hypothesis of spherical waves. I
am discharged from the necessity of dwelling on the details
of Mr. Stokes's reasoning, because the whole of it contains a
petitio 2>rincipii. Indeed I should desert the position which
the acknowledged rules of right reasoning compel me to take,
if 1 made a single remark which implied an admission that
that reasoning required from me any answer. 1 proceed,
therefore, with the subject which more strictly accords with
the title of this communication.

I shall commence by saying that I regret Mr. Stokes does
not feel himself at liberty to give me the benefit of his stric-
tures on my mathematical theory of ray-vibrations, and to

92 Prof. Challis's Researches in the

state his reasons for objecting to equations (8.)) (B.) and(C.).
The subject, being entirely new, offers a fair field for discus-
sion ; and I am quite prepared to find that I have been mis-
taken on some points. In particular, I have recently disco-
vered that the equations just named are not generally true
beyond the first order of approximation, as I shall presently
show. From the nature of the problem the processes applied
to it must be in a great measure tentative, and can only be
tested by the results. The equation (A.), for instance (Phil.
Mag., vol. xxxiii. p. 99)? as far as I know, is the first example
of an application of analysis to a question in physics, which
presents for solution a partial differential equation containing
two principal variables and two independent sets of variables
mixed up with each other. I have proceeded on the prin-
ciple that if particular and consistent values of one set of va-
riables be substituted in the equation, the resulting equation
will be true ^oy general, if not the most general, values of the
other set. But the application of this principle is restricted
by any limitation to which the supposition which conducted
to the equation (A.) is subject, viz. the supposition by which
udx + vdy-\-'iSodz was made an exact differential. Now it is
well known that for small vibrations, the equations
du _ dv du _ dtso dv _ dw
dy dx* dz dx dz dy '
must be at least approximately verified. This will be the
case for approximate values of w, v, w, if the complete values
make udx + vdy + ii^dz an exact differential. On this account,
for the purpose of verifying those three equations approxi-
mately, it was assumed (Phil. Mag., vol. xxxiii. p. 99) that
udx + vdy + wdz was integrable for the complete values of u,
V, and w. But it is equally possible that the same three equa-
tions may be exactly verified by approximate values of u, v,
and w, in which case the condition that tidx + vdy-j-ivdz be
integrable, may be only satisfied by approximate values of
u, V, w. It does not seem possible to decide which of these
is the true state of the case but by trial. I supposed the
former to be true, and obtained equations applicable to ray-
vibrations on this presumption. It turns out on further in-
vestigation that the latter is the true theorem, when the inte-
grability depends on the supposition that {d.f(p) = udx-\-vdy
+ isodz, f being a function of s and t, andy a function of .r and y.
For this reason those equations require certain modifications
which I now proceed to develope.

The equation (B.) in page 99 of the Phil. Mag. for last
August, viz.

o_ o^ + a.^^^ ^^2 U% dzdt dz^ dz^ ' ^^'^

Mathematical Theory of Aerial Vibrations.

93

was derived from the equation (A.) in the same page, by as-
suming, since the motion is by hypothesis vibratory, thatjf
has a maximum vahie equal to unity, and that the values of jr

1-1 ■ ^ df ^ idf ^ , 9fno« im ay>lai
and y which satisfy ^ =0 and ^ =0> inake , ,, . ,, „.^,,

dx^ dip-

On the same suppositions respecting / 1 have shown in the
Phil. Mag. for last December (p. ^QB) that the equation (B.)
is satisfied by an equation of this form, > *,-. ,

''^+«,$=o CO

dt

dz

a, being a certain constant. This may be regarded as a par-
ticular integral applying to propagation in a single direction,
and is all that is required for the present investigation. An
integral of (B.) satisfying (1.) was also obtained (vol. xxxiii.
p. 363) by successive approximations. By differentiating (I.)

and substituting the resulting values of ^-|- and -j-^. in (B.),

we obtain

d^^ _
dz'^ "

b''<p

dl''

a^b^

«-G.-i

2)

dzdt

--{'^-t)'

Now substituting these values in (A.) (which equation being
long I dispense with inserting here), the result is

y

+ 2<p

d(p
dz

(-^1

) \dx^ '^ dy^

)

^ ' \dx^ dx^ doc dy dxdy dy'

dyV

(A'.)

dy dxdy dy^
It thus appears that the result is not independent of f un-

94 Prof. Challis's Researches in the

less powers of this quantity above the first be neglected. In
this case the equation becomes

and at the same time the equation (B.) becomes

It is, however, important to remark, that the equation (A'.)
is satisfied without any restriction of the value of f the more
exactly in proportion as the value of /approaches more nearly
to unity. We may hence infer that the equation (B.) is
strictly true for points on and immediately contiguous to the
axis of the ray, but not more generally. Thus, with the ex-
ception just named, J have not hitherto succeeded in obtaining
equations applicable to ray-vibrations beyond the first order of
approximation, which probably is all that will be required in
application. In equation (3.) of my communication to the
Philosophical Magazine for last November (p. 364), the terms

involvmg -7-^, -—■, and m, which arose trom relymg too im-
plicitly on the hypothesis by which udx -irvdy + 'wdz vm^ made
integrable, must be rejected. The equations (1.) and (2.) in
p. 363 hold good. It may also be stated, that the reasoning
in the December Number, by which it was shown that along
an axis of raj'-vibrations a given state of density and velocity
may be propagated uniformly and without alteration, remains
untouched.

Having stated all the modifications which, according to the
present investigation, the previous results require, I beg to add
a few general remarks. It is well known that on first apply-
ing the integrals of partial differential equations to physical
questions, a dispute arose between Euler and D.'Alernbert as
to the arbitrariness of the forms of the functions. The dis-
cussion issued in the establishment of the principle of discon-
tinuity by Lagrange. Yet where two mathematicians of so
great eminence differ, it may generally be concluded that a
share of truth is on each side. The singular contradictions
which I have pointed out as resulting from the suppositions
of plane-waves and spherical waves, must, I think, raise the
question whether, as D'Alembert appears to have supposed,
forms of solution are not discoverable which indicate motions
independent of the particular disturbance, and whether these
must not be discovered before proceeding to consider cases

Mathematical Theory of Aerial Vibrations. 95

of arbitrary disturbance. The explanation I give of those
contradictions is as follows. I regard them as proving, not
that cases of plane- waves and spherical waves cannot exist,
but that they cannot exist spontaneously. Having been led
deductively from the general equations to those which define
ray-vibrations, without meeting with any similar contradiction,
and anterior to any supposed case of motion, I conclude that
ray-vibrations are common to all instances of vibratory mo-
tion. Any instance of such motion may be conceived to be
composed of the ray-vibrations defined by the equations (2.)
and (3.), the number, magnitude, and directions of the rays
being unlimited. Accordingly if a plane-wave were generated,
it would be composed of an unlimited number of ray-vibrations
having their axes all parallel to the direction of propagation ;
and from what has been proved of these vibrations, the result-
ant wave might be propagated to any distance without under-
going any change. So if a spherical wave were generated, it
would be composed of an unlimited number of ray-vibrations
having their axes diverging from a centre, and the change of
condensation with the change of distance from the centre would
be according to the inverse square of the distance, since it
would depend only on the divergence of the rays. This re-
sult is in accordance with the principle of the constancy of
mass.

Cambridge Observatory,
January 9, 1849.

Postscript^ Jan. 11, 1849. — After despatching the foregoing
communication, I deduced from my theory of ray-vibrations
a result which ought, I think, to command the attention of
mathematicians.

I have already exhibited the course of reasoning which
finally led to the conclusion, that the transverse motion in ray-
vibrations is defined, to the first order of approximation, by
the foregoing equation (2.). lo this equation is next to be
applied the process by which, from the equation (3.) of ana-
logous form, which defines the motion along the axis of the
ray, a unique integral expressed in finite terms was obtained,
viz.

27r/ , /\ e\^\

f = 7n cos — (s-a/w H — j-l.

(See Phil. Mag. for April 1818, p. 279.) On so doing, the
result is

/=» cos {gx + hy),

96 Prof. Challis's Researches in the

the quantities g and h being subject to the condition,

g^ + h^= -o =4^.
Hence if ^=2 Vecos fl, it follows that ^=2 -/esin 9 ; so that
f=ct cos (2 -i/ ^ (.r cos 6 + 2/ sin fl)).

This result is not definite, because the angle d is indetermi-
nate. But a result which is definite may be obtained by sa-
tisfying the given equation so as to embrace all possible values
of 9. This may be done by taking account of the analytical
circumstance, that the value oi f may be expressed by an
unlimited number of terms like that above. Let, therefore,
«8fl be a given indefinitely small quantity. Then we may have

f=iXaU COS {2\/ e{x cos 9 +3/ sin fi)),

fl having all values from 9 = 0 to 9 = 27r. By performing the
summation, substituting r^ for x'^+y^i and determining a so
as to satisfy the condition that/=l where r = 0, the result is

/=i-^'''+i2:2'2-32;2^:^. +&C. . . . (4.)

It seems, therefore, that we have arrived at an expression
for f which involves no indeterminate quantity, and which
defines precisely the transverse motion. Making /= 0, the
resulting equation contains an unlimited number of possible
positive roots, and there are consequently an unlimited num-
ber of positions for which y*. -7^ =0, or the condensation va-
nishes, on any given radius. Similarly the equation resulting

from making -^ =0, contains an unlimited number of inter-
^ dr

mediate possible positive roots ; and there are consequently
an unlimited number of values of/, which are radii of cylin-
drical surfaces situated in positions where there is no transverse
velocity. Since no fluid passes these surfaces, there is no
propagation of motion in directions transverse to the axis of
the ray. The intervals between the surfaces approach to a
certain limiting value in proportion as we recede from the
axis, and the maximum values of /" go on continually de-
creasing. The limiting value of the interval between two
consecutive surfaces of no condensation may be obtained as
follows.

Let n be an indeterminate infinite whole number. Then

Mathematical Theory of Aerial Vibrations. 97

makingy*=0, the equation (4.) may be expressed in the fol-
lowing form :

Or thus :

whence it will be seen that the terms at an infinite distance
either way will become insignificant on account of the factors

(l+-) ^&c., (l--y, &c.

Hence retaining only terms of the highest order of infinity,
we have,

».. + (er2— n2)e'^-»r2«-24.(er2_w2)nV-V2«-6 + &c. =0.

This equation is satisfied if ^r^— w^=0; or

As no condition has been imposed on the quantity w, except
that it be an infinite whole number, the radius rg of the next
surface of no condensation will be given by the equation

In fact, by the same process as that by which the equation
(tr^— n^)Q=0 was obtained, we might obtain

(^r2-(»+l)2)Q' = o, (^r2-(« + 2)2)Q" = 0, &c.,

all the terms of Q, Q', Q", &c. being essentially positive.
Hence the infinite roots of the equation

are contained in the equation

{er'^ ^n^)[er'^-{n-\-\Y){er^-[n-\-^Y)^c. =0.

Consequently if A be the interval between two consecutive
surfaces of no condensation, we have

It is now to be remarked, that the transverse motion be-
PhiL Mag. S. 3. Vol. 34. No. 227. Feb. 1849. H

98 Mr. J. H. Alexander on the Tension of Vapour of Water,

tween the two consecutive surfaces either of no condensation
or of no velocity, is exactly the same as that which would take
place along the axis of the ray between two consecutive posi-
tions of no condensation or of no velocity, if two series of
waves for which X is the same were propagated along that
axis in opposite directions. The time in which each particle
executes a vibration is the same in the two cases. We must
consequently have

X

H

ence

^=2-

xVe 1 ^ e>? 4

= 1, and — 2" ^ -g.

2 71"

Recurring now to the expression for the velocity (a') o^
propagation of a ray, obtained in the Phil. Mag. vol. xxxiii.
p. 363, and neglecting the term involving 7»\ it appears that

Hence if we take for a the value in art. 66 of Sir John Her-
schel's Treatise on Sound in the Encyclopcedia Metropolitana,
we shall obtain

ft. / r

a' = 916,322Y/ 1+ -^

= 1086,25 feet.

The value of a' obtained by experiment is 1089,42 feet, as
given in the same work. The slight excess may be owing to
the neglect of the term involving m"^.

I have thus obtained a value of the velocity of sound, closely
agreeing with experiment, on purely hydrodynamical principles.
As this result is not in accordance with received ideas on this
subject, I shall at a future opportunity give a careful resume
of the course of reasoning by which it has been arrived at.

J. Challis.

XII. On a neiv Empii-ical Formula for ascertaining the Tension
of Vapour of Water at any Temperature. By J. H.
Alexander, Esq.^

[Concluded from p. 15.]
N the last number of this Journal, I gave the formula itself,
the principles from which it was deduced, and a compa-
rison of results by it, with those by experiment at numerous
identical temperatures. Want of room excluded then what
* From Silliman's Journal for Nov. 1848.

1

Mr. J, H. Alexander on the Tension of Vapour of Water. 99

remained to complete this memoir, in showing the probable
errors of the formula as compared with the principal experi-
ments, and with the probable errors affecting too those differ-
ent series of experiments themselves. Such a discussion is
the object of the present paper.

It was already said in the preceding part, that the most
proper mode of expressing these errors is by the linear scale
of temperature; which both in theory is the most important,
and in practice is the most accessible and usual. In this last
aspect, it is on this scale, too, where errors of observation are
the most easy to be made, and likely to occur. With this
view the formula need be repeated here only in its converse
form {i. e. for ascertaining temperatures from given pressures),
as under: —

^^ Fahr. = 1 80 -1^^- 1 05°- 1 3 ;

p being in inches of mercury ; and

^°Fahr. = 317-13 Vp'-105°'13;

y being in atmospheres at 32°.

As this will have to be frequently applied for interpolation
throughout the following discussion, it may be as well to re-
mark here, once for all, in justification of such application,
that there need be no apprehension of its affecting the results;
for it is easy to see, by inspecting a few instances taken at ran-
dom from the table, that the rational deviation of the formula
{i. e. the difference between calculated and observed pressures)
is, for small differences of temperature, either null, or so re-
mote a fraction as to be inappreciable in the calculation.

In applying this formula, I shall take up the principal series
of experiments separately, beginning with the most recent,
and shall then make assemblage of the mean results.

1. Experime7its of M. Regnault, — To deduce the absolute
mean error of the numerous quantities of this observer, it
would be obviously requisite to take up each experiment ; a
labour of which I am by no means ambitious, and which would
be disproportionate at once to what is admissible in the other
series presently to be noticed, and to the present aim. I shall,
therefore, in all only make use of short general methods, which,
without laying claim to the accuracy of geometrical refine-
ments, will yet be recognized as having foundation in the
theory of mathematical probabilities ; and will, by their po-
pular form, recommend themselves the more readily to the
convictions of those who are chiefly conversant with steam in

H2

100 Mr. J. H. Alexander on the Tension of Vapour of Water,

practice, and for whose benefit the whole of the present dis-
cussion is mainly intended.

It is obvious, then, in the first place, that the idea of free-
dom from error is as.sociated with symmetry in the results.
Such symmetry will always be observable in quantities that
progress (as natural quantities may be assumed to do) accord-
ing to some constant law ; and as, in our ignorance of what the
true law is in this case, all that we can deal with is relative
symmetry, it is of no importance what law or formula we take
as the other term of comparison, provided there be no material
difference between the origin and termination of the two.
I shall therefore compare a few of M. Regnault's observations
at the lower temperatures with the results of the present for-
mula, as under: —

Temp. (Fahr.).

Pressure in inches of mercury.

Differences,

Observed means.

Calculated.

-27112
-13-
- 4-504
+ 1-706
+ 9-41
17-402
23-702
27-626
32-

in.

001063

0-02047

0-02835

004567

0-06378

0-09410

0-12559

0:15000

0-18111

in.

000664
001799
003055
0-04375
0-06643
0-09111
0-13451
0-16106
019561

+0-00399
+0-00248
-0-00220
+0-00192
-000265
+0-00299
-000892
-0-01106
-0-01450

It is apparent, then, that so far these observations do not
follow any uniform or symmetrical yirogression ; and without
pretending to criticise the experiments themselves, which
doubtless have as much accuracy as the nature of the research
admitted, it follows that, in spite of all the extraordinary tact and
skill of the observer, there is yet prima J'acie evidence against
the absolute accuracy of the results. It is to be remarked upon
the column of temperatures, both here and hereafter, that the
remote decimals result from the reduction of Centigrade de-
grees to those of Fahrenheit, and are preserved because they
added to the accuracy, while they did not increase the labour
of the calculation. Nevertheless the thermometer of M. Reg-
nault could be read directly to the j^th of a degree Centigrade,
corresponding very nearly to ^^\h of a degree Fahrenheit;
and by estimation, to the next decimal place.

The temperatures of this table under 32° F. are lower than
pressures have ever been observed at before, and rest upon
single observations. They do not admit, therefore, of a com-

Mr. J. H. Alexander on the Temion of Vapour of Water, lOl

parison other than has been instituted. But the observations
at32'' F., a temperature especially disengaged from instru-
mental errors, are, as has been already said, very numerous,
and allow of being compared among themselves. Of the forty-
seven observations whose arithmetical mean pressure is given
in the table, the

ill. o

maximum was 0-18485 ; corresponding to a temp, by formula of 30-72 F.
and mmjjwMOT was 0-17717 29-77...

and the difference 0-00768

corresponding difference of temp. 0-95

This difference shows a mean error in temperature, unac-
counted for, of 0°*425 F. ; and a limiting error in pressure
rather more than half the difference between the formula and
the mean of all the observations.

In the various series of M. Regnault, the temperature is
given sometimes by one thermometer only, and sometimes by
two, and even four. Of these latter classes, I have taken out
of each series the observation where the difference of reading
of the several thermometers is the greatest, to serve for another
comparison, as follows : —

Series.

Thermometer.

Differences.

Maximum.

Minimum.

A.
B.

N.

0.

'i P.

\u Q.
R.

S.
T.

33-61 C.

42-63

43-64

47-84

47-87

91-25
12272
11072
137-75

33-49 C.

42-56

42-84

4714

47-

91-06
122-50
110-64
137-52

612 c.

007
0 81
0-70
0-87
0-19
022
0-08
0-23

Mean temp. 75°-154 C; mean difference 0°-366; corresponding with!
167°-277F. ... 0°-659F.

This difference is that of the extremes ; and as the mean error
of any number of observations is as likely to he plus as mivjis^
it is equivalent in this example to an absolute error of 0°'33 F.
This error manifests itself in a series where the thermometric
variations are the greatest. I shall now present another where
these same differences, although not perhaps the lowest of all,
are yet very much less than in the last. At least, this series
(which in fact forms part of the comparative table in the pre-
ceding memoir) was selected without any reference to the
present investigation, and with a view to the introduction of
the greatest number of accordant observations, and may be

102 Mr. J. H. Alexander on the Tension of Vapour of TVater,

considered, therefore, as offering an impartial, if not favour-
able, term of comparison.

The mean temperatures are given here, as in our former
table, in degrees of Fahrenheit; the individual differences
between the thermometer readings are, to save calculation,
retained in Centigrade degrees.

Mean temperatures.

Differences of thermometer readings

Fahrenheit.

Centigrade.

151-124.

0-52

176-4.16

0-30

198-05

013

211-27

0-06

222-44

0-08

2.S3-132

0-16

252-662

0-22

• 263-3

0-14

276-224

0-20

297-464

0-19

Mean difference 020

corresponding with mean temperature 228°-2 F. ; mean differ-
ence 0°-36 F., which is equivalent to an absolute mean error
of0°-18F.

o

We have, then, for the mean error at 32° F. 0*42

167 0-33

228 0-18

the average of which, or 0-31

is the probable amount of error, plus or minuSf with which the
various series of M. Regnault are still to be considered as
affected.

Such being the error of the experiments, I shall now show,
by the following table, the comparative error of the formula.
The quantities in the column of differences are considered as
on the same side of the equation with the results from the for-
mula; those marked + indicate, therefore, the default, while
the sign — indicates the excess of the calculated temperatures.

Mr. J. H. Alexander on the Tension of Vapour of Water. 103

Pressures in inches.

Temperatures.

Differences.

Observed.

Calculated.

in.

001083

002047

0-02835

004567

006738

009410

012559

015000

018111
7-6977
14-081
22 538
29-620

35-779
44-752
63-333
76152
93-859
130-790

-27-112 F.
-13-
- 4-504
+ 1-706
+ 9-410
+ 17-402
+23-702
+27-626

32-

151124
176-416
198050
211-27

224-44

2S3-132

252-662

263-3

276-224

297-464

-20-73 F.
-10-98
- 5-74
+ 2-49
+ 8-65
+ 16-27
+22-25
+26-08

30-26
147-80
174-58
197-38
211-49

221-62
234-03
254-24
265-45

278-59
300-42

+7-38 F.

+2-02

-1-24

-0-78

+0-76

+ 1-13

+1-45

+ 1-55.. .+1-53

+ 1-74
+3-32
+ 1-84
+0-67
-0-22... + 1-47

+2-82
-0-90
-1-58
-215
-2-37
2-96... -1-19

Mean difference +0605

This mean difference (say 0°'6i F.) is the error of the for-
mula as compared with the ob.servations above, which are
themselves the mean of more than twice as many experiments,
and which may be taken as representing impartially the whole
ranjre of M. Reffnault's results. In arriving at this mean

o CD , . .

difference, I have arranged the several instances into groups,
whose individual means furnish the definitive general (me.
This is proper, in view of the different methods of experiment
which the different relations of temperature in the respective
groups rendered necessary. The indiscriminate mean of all,
however (0°'657 F.), is not materially variant. It will be seen
that, up to the point of boiling water, the formula-temperatures
are generally /otU(?r than experiment; above that point, they
are in general higher. I believe that such a change of sign
accords with what might be anticipated, and in so far does
not diminish the reliability of the formula.

The difference ( + 0°-61 F.) would be the absolute error of
the formula, were we to assume the experiments as perfectly
accurate. But they have been already siiown to be themselves
affected by an error of +0°-31 F. ; and the absolute error of
the formula, then, may be either 0°-30 or 0°-92 F., according
as the equation is made of the sum or the difference. Either
of these quantities may in the theory represent the true error;
and we have, therefore, in fine, the case of an even chance for,
accuracy with the formula or with the observation. . .;

104 Mr. J. H. Alexander 07i the Tensionof Vapour of Wate)\

Sucli are the conclusions that arise from the comparison
with M. Regnault's experiments.

2. Experiments of the Franklin Institute. — The temperatures
were read by these observers to only the nearest quarter of a
dej^ree of Fahrenheit ; they are therefore not comparable in
precision, whatever they may be in accuracy, to those that
have just been considered. And as but one reading either of
temperature or pressure is given in each instance, they do not
allow of being treated in the method that has just now been
applied. I can only then compare them as in the following
table : —

Pressures in
atmospheres.

Temperatures,

Diffeiences of
Franklin Institution
in formula and ex-
periment.

Calculated by
my formula.

Observed by

French
Academy.

Observed by
Franklin
Institute.

Calculated by

formula of
Frank. Instit.

1

2
3
4
5

6

7

8

9

10

212 F.

250-84

275-73

294-43

309-57

322-36

333-49

343-36

352-25

360-36

212 F.

250-52

275-18

293-72

307-54

320-36

331-70

341-78

350-78

358-88

212 F.

250

275

291-5

304-5

315-5

326

336

345

352-5

212 F.

248-8

2723

2901

304-4

316-5

327-3

336-4

344-8

352-5

0

+ 1-2
+2-7
+ 1-4
+01
-10
-1-3
-0-4
+0-2

Mean differences -l°-20 -SMS +0°-36

The temperatures of the Academy in this table were not,
as has been said already, from experiments at the precise
epochs of pressure, but were interpolated from experimental
terms not remote. Under a general principle, 1 excluded
them from the comparative table in the preceding memoir ;
but they satisfied even the fastidiousness of M. Dulong, as
representing accurately the results of observation, and are
therefore fit to be compared as they are here. The last line
o{ mecm differences shows the excess of the formula-tempera-
tures above those of the Academy to be not much more than
one-third of the excess of the latter above those of the Franklin
Institute ; the probability of accuracy of these last, then, at
most cannot be more than in the same ratio. It also shows a
mean error between the formula adopted by these observers
and their observations, of 0^'36; by which deviation the
former, with the advantage of having the two extremes arbi-
trarily to coincide, yet fails to adjust itself to the latter.

3. Experiments of the French Academy of Sciences. — Out of
the whole of this series M. Dulong has himself selected ^/^w»
as the most unexceptionable, and has used them for a standard

Mr. J. H. Alexander 0)i the Tension of Vapour of Water. 105

whereby to compare the merits of divers proposed formulae as
well as his own. It is true this group contains, among others,
one of the very experiments which I have in the preceding
memoir noted as faulty, and as being differently recorded in
the two tables ; but I have not allowed myself to exclude it
here any more than before. Also for his comparison, M.
Dulong has taken the reading of only the maximum thermo-
meter, which represented the actual temperature of the water
in the boiler. For the present purpose, however, as in the
case of M. Regnault's experiments, it is necessary to take the
other thermometer-reading also and register the variation of
the extremes, as under. I have entered the pressures here in
English inches, since they have already been reduced for the
comparative table ; but to save unnecessary calculations, I re-
tain the temperatures in Centigrade degrees.

Number of
experiment.

Prcsswes in
inches.

Temperatures.

Difference in
degrees Cent.

Maximum.

Minimum.

1

in.
64-14

123-7 C.

122-97 C.

d-73

3

8570

133-3

132-64

0-66

5

136-85

149-7

149-54

016

8

194-42

163-4

163-00

0-40

9

220-69

168-5

168-40

010

15

34804

188-5

188-30

0-20

21

514-22

206-8

206-40

0-40

22

516-84

207-4

207-09

0-31

25

553-69

210-5

210-47

0-03

28

644-96

218-4

218-30

010

30

716-13

224-15

223-88

0-27

Mean d

iference

0°-3055 C

. =0''-55F.

equivalent to a mean error of +0°'28 F., arising from the
uncertainties of temperature.

To contrast this with the formula, we have from the same
experiments as under: —

Number of

Temperatures.

Differences.

experiment.

Observed.

By the formula.

1

123-7 C.

123-9 C.

-6-2 c.

3

133-3

133-8

-0-5

5

149-7

150-8

-1-1

8

163-4

164-5

-1-1

9

168-5

169-5

-1

15

188-5

189

-0-5

21

206-8

206-8

0

22

207-4

207-2

+0-2

25

210 5

210-4

+0-1

28

218-4

217-7

+0-7

30

22415

222-9

+1-2

Mean difference — 0°-5

JC. =-0°-36F.

106 Mr. J. H. Alexander on the Tension of Vapour of Water,

a deviation between the formula and the experiment but little
more than the admitted error of the thermometric readings.
The mean error of observation from this last source was found
just now to be +0°'28 F., and the mean error of the formula
then may be either 0°'08 or 0^"64 of Fahrenheit. These
quantities equally satisfy the equation, and the probabilities in
favour of each are even.

It is observable that errors in this series come out with a
different sign from those of M. Regnault, though the errors
of observation in the two experimental series are nearly iden-
tical, as might be expected in advance from the great skill and
probably equal tact of the two observers. Such a difference
of sign is favourable to the character of the formula, which
will be seen by combining the two results, as under: —

Error of formula.

Maximum.

Minimum.

From experiments of Academy of Science...
From experiments of M. Regnault

-6-64 F.
+0-92

-0-08 F.
+0-30

Mean error by both

+014

+011

The nearness of these limits, and the smallness of the num-
ber inclosed by them, warrant, I think, a sole and entire reli-
ance upon the formula in the present state of experimental
knowledge on the subject. I do not introduce into combina-
tion any of the other and earlier series of observations; be-
cause, from the way in which they have been reported by their
respective authors, they do not admit the application of the
same methods of comparison ; and because it may justly be
supposed that the apparatus, intellectual and mechanical, re-
sorted to in 1829 and since, is paramount in accuracy to what
had been at disposal in preceding researches.

I shall only, therefore, in further illustration of the present
formula, compare its results with those of expressions that
have been proposed by other mathematicians; only extending,
in point of fact, for this purpose, a similar comparison which
MM. Dulong and Arago have already instituted ; and using,
except for the last column, quantities from the calculation of
these philosophers. Their table is founded upon the same
eleven observations of their own, just now quoted ; and they
have given for each instance the individual deviations of the
several formulae from the result of experiment. Not to em-
ploy so much room, I have thought it equally satisfactory to
give the general results and inferences, as under. The devi-

Mr. J. H. Alexander on the Tension of Vapour of Water. 107

ations are given in Centigrade degrees, and belong to the same
side of the equation with the temperatures given by the re-
spective formulse.

FormuliB proposed by
Tredgold.i Roche. ] Coriolis. I Dulong. J. H. A,

0-69 I 0-63 080

Maximum deviation in excess

MaxiiBum deviation in defect 2-11 [ 075

Mean deviation without regard to signs: 0-79 0'37
Mean deviation with regard to signs... l4-0'338 —0001

0-25

0-40
073

0-44 i 0-24
-0-363! -0 007

flO

1-20

0-60

-0-20

The last formula, as is seen, although the sum of its devia-
tions is greater than two or three of the others, lies yet more
symmetrically with the curve of the experiments than any.

The three first are given by M. Dulong, after a copious
enumeration of different formulae, as agreeing the best with
observation. Of these, in that of Tredgold and of Coriolis,
the elasticity is a function of the temperature; but M. Coriolis
uses, instead of an integral, a mixed fractional index. His
exponent, instead of 7 as Dr. Young took, or 6 as Creighton
and Tredgold preferred, or 5*13 as Southern chose, or 5 as
Dulong adopted, is 5'335 ; whose coincidence with the natural
law is only empirical, and can be but accidental. In the for-
mula of M. Roche (which he offers, not as a means of inter-
polation, but as the expression of a general physical law), the
temperature is itself an element of the index by which certain
constant quantities are to be involved. The principles, how-
ever, upon which he has founded the expression, are disap-
proved both by M. Dulong and M. Regnault. The formula
of M. Dulong presents a smaller aggregate deviation than any
of the others ; and it would be singular if it did not, seeing
that it was derived from a constant furnished by his own ex-
periments. But as might also be anticipated, this constant,
taken (to four places of' decimals) from the result of thehigh-
.est experimental temperature, fails to apply in the lower ones.
The maximum deviation under his formula, given in the last
table, occurs at the lowest experimental temperature ; and in
fact in his final table of atmospheric pressures and correspond-
ing temperatures, he has preferred, below the limit of four
atmospheres (l4.5°-4 C. or 293°'72 F.), to abandon his own
formula and use that of Tredgold. Below the ordinary atmo-
spheric pressure his quantities are utterly inapplicable, as will
be seen by the following statement: — '

108 Mr. J. H. Alexander on the Tension of Vapour qfWat€7\

Pressures in
atmospheres.

0047368

0006

0000684

Temperatures (Centigrade).

Observed by
Regnault.

32-38

0-00

-25-00

Calculated by formulae of
Dulong. Franklin Instit. Alexander.

36-16

1045

- 7-25

3352

4-28

-17-31

29-80
- 108
-23-89

The last two columns are added here for illustration ; and
show, among other things, that the formula of the Franklin
Institute is, like that of the French Academy, inapplicable to
low temperatures and pressures.

Later than these, M. Biot, in 1839, propo.sed another for-
mula, and in IB^l published a table calculated by it, in which
the pressures are given in metres and for every degree Centi-
grade from —20° to 220° C, corresponding to the limits of
— 4° and 428° F. The patient labour requisite for this task
has not been overrated by its distinguished performer ; as can
be readily appreciated, when it is known that part of the cal-
culations actually were, and it was apprehended that even the
whole might require to be, executed with logarithms of eleven
decimals, and that the constants reach even the twelfth deci-
mal place. These constants were derived, for the higher
temperatures, from the already quoted experiments of Dulong
and Arago and of Taylor ; and for the lower, from unpub-
lished experiments of M. Gay-Lussac. The temperatures
are throughout given in terms of an azV-thermometer instead
of a mercurial one, a modification which undoubtedly im-
presses a more systematic accuracy upon the method ; but yet,
in spite of the aid afforded by tabular corrections for reduc-
tion, appears to diminish materially the chances of practical
resort to the table itself. These temperatures M. Biot, in the
form first proposed by Prony, (the same which Dr. Young,
with more emphasis than reflection, has called "ridiculously
complicated,") employs as the exponents of a series; the pe-
culiarity of the method, however, is in that the direct nume-
rical result of the equation gives, instead of the pressure itself,
the tabular logarithm of the pressure. It is therefore essen-
tially a logarithmic formula.

1 present the following comparison between it and the pre-
sent formula, applied to the same instances of experiment,
which have been already signalised by M. Dulong himself,
and already quoted here. To save both the tedium and ha-
zard of a reduction to English measures, 1 leave the quantities
under their original denominations ; and, in so far varying
from the preceding instances, I give the deviation of the for-

Mr. J. H. Alexander on the Tension of Vapour of Water. 1 09

mulse in terms of the pressure instead of the temperature.
This method enables me, by an easy and safe interpolation,
to extract the proper quantities from M. Biot's table, and
thus to avoid the portentous labour of working out the nume-
rical transformation of his theorem.

Number of
experiment.

Temperatures (Centigrade).

Pressures in metres.

Mercurial

Air-

By Biot's

Observed liy Du-

By present

thermometer.

thermometer.

formula.

long and Arago.

form.

1

1237

123-13

1-65020

1-62916

1-62022

3

133-3

132-47

2-26396

21816

2-14687

5

149-7

148-41

3-47146

3-4759

3-37449

8

163-4

161-69

4-94220

4-9883

4-80439

9

168-5

166-63

5-60263

5-6054

5-45121

15

188-5

185-96

8-89046

8-840

8-73476

21

206-8

203 60

13-05578

13061

1304525

22

207-4

20418

13-21410

13137

13-21202

25

210-5

20716

14-05179

14-0634

1410266

28

218-4

214-75

16-36717

16-3816

16-60100

30

22415

220-28

1818048

18-1894

18-64254

Not to embarrass this table with so many columns, I omit
the individual deviations of the two formulae, and present the
general result as under.

Biot. Alexander.

Mean deviation from experiment'! o-02'ififi O-ISIQI
without regard to signs . • J M

Mean deviation from experiment I _o*01704< —0-09114
with signs j

It is hardly necessary to repeat that the first of these for-
mulae is founded in part upon the very experiments with which
it accords so well, and that the other was not.

The table of M. Biot goes up as far as 220*^ C. ; but he
supposes that his formula is applicable much further; and in
fact he has given results, in a small supplementary table, as
high as 300^ C. or 572° F., at which temperature it makes the
pressure equal to almost exactly eighty- five atmospheres.
The present formula would make, corresponding to this pres-
sure, a temperature of 559°-92 F. or 293°-3 C, differing from
the other within the correction between a mercurial and an
air-thermometer.

It is at the other extremity, where we still have opportu-
nity of referring to experiment, that the difference between
the two formulae becomes more marked; and where that of
M. Biot, neither in its terms nor its progression, can be con-
sidered applicable. This may be seen as under: —

1 10 Mr. J. H. Alexander on the Tension of Vapour of Water.

Pressures in inches
of mercury.

Temperatures (Centigrade).

Observed by Regnault.

B.

A.

in.

0024

0-033

-23-83 -20 -22-46
-20 -17 -19-52

About the same time with M. Biot, other formulae claiming
(like that of M. Roche) a foundation on abstract theoretical
principles were proposed by Mr. Russell, who has also ap-
plied their somewhat extensive logarithmic apparatus to the
calculation of a table of pressures for each degree from 32° to
2.50° F., and then for intervals of one or more atmospheres up
to fifty. This does not properly come into this discussion,
because the author has found it necessary to employ different
terms above and below the point of boiling water, and in point
ol fact to liave two formulaj; an inconvenience, the same in
kind though not in degree, with what the object of the very
research is to avoid. Nor do they counterbalance this by a
proportionate accuracy which would warrant their results to
be substituted for those of experiment. On the contrary,
starting from their common zero, 212°, they both deviate in
their respective directions from the curve given by observation;
the pressures calculated by them are, at the two extremities,
very much above any experimental ones. Not to trouble
ourselves with the part of the scale below the boiling tempe-
rature, where the errors are not of so much practical import-
ance, I give a few instances in the higher degrees, contrasted
with the results of the French Academy.

Pressures in
atmospheies.

Temperatures

(Fah-enheit).

Differences.

French Academy.

Russell's table.

1

212

212

5

307-5*

306-8

6-7

10

358-9

355-6

3-3

20

41C-5

410

8-5

30

457-2

4446

12-6

60

510-6

491-4

19-2

The formulae of M. Regnault, to whose experimental re-
searches such resort has been had, are in one respect in the
same category as those of Mr. Russell: they are three\ one
adapted to pressures below the melting of ice, the second

* Mr. Russell, in his comparison, as well as the Franklin Institute in
theirs, give this temperature at 30>5°-8 ; an error which has arisen from
hastily reducing the actual Centigrade temperature of 153°-08, as if it were
153°-80.

Mr. J. H. Alexander on the Tension of Vapour of Water. 1 1 1

reaching from that point to the ordinary atmospheric pressure,
and the hist proper tor high temperatures only. He promises,
when his experiments in this upper part of the thermometric
scale shall have been sufficiently extended and accumulated,
to apply himself to the grouping of all three divisions in one
comprehensive expression; and from l)is well-known character
much may be expected, original and appropriate. In the
mean time it would be premature to enter here upon any dis-
cussion of what is only provisional.

To resume, then, in conclusion of this rather protracted
memoir ; it seems to me that in the various combinations and
comparisons that have been given, the claim of the formula I
propose is reasonably well-established, not to be an expres-
sion of a law of nature, for to this much it makes no preten-
sion, but to represent the pha^nomena of reciprocal pressures
and temperatures more exactly and with a more extensive
scope, than any that has yet been offered ; and that in so far
it is worthy of being taken as paramount to all that have pre-
ceded it. How far, in view of the discord yet existing between
experimental results of the most recent and reliable observers,
it is fit to come in as a substitute for any and all of those results
themselves, is not of course for me to determine. I shall only
allow myself to notice, then, its remarkable simplicity, and the
consequent facility with which it adapts itself to calculation,
either with or without logarithms ; as well as the readiness
with which, from its elements and form, it suggestsitself at all
times to the memory. One important use of a formula, it is
to be observed, is in enabling an inquirer in any emergent
case, away from books and tables, to extemporise an accurate
result ; in proportion to its complexity and arbitrariness, then,
it becomes a (juestion of individual strength of memory, and
its resort more and more limited. In the present instance all
its terms are either given in the very case to be solved, or are
physical constants at the foundation of the theory of heat,
which I may even say it is impossible for one ordinarily
well-informed to forget. And the composition of these terms,
thus susceptible of instant recall to the mind, is so plain and
necessary even, that it is equally" impossible, with a moment's
reflection, for one to go wrong. I believe I am only stating
the simple fact when I say that, in these respects, the present
formula stands alone.

Finally, then, I offer for general practical use and reference
the following table of temperatures corresponding to pressures
in atmospheres and parts through the whole range of experi-
ment hitherto. If my labour in so far shall be fortunate enough
to meet with the approval of the learned, it will be but an in-

112 Mr. J. H. Alexander on the Tension of Vapour of Water.

considerable task hereafter to complete its scope by furnishing
a table of pressures at all useful temperatures for each degree
of Fahrenheit's thermometer, to whose arbitrary and otherwise
inconvenient scale the present investigation has served not a
little to reconcile me.

Table of Temperatures corresponding to the Pressures of
Steam in Atmospheres.

Pressures in atmo-

Temperatures in de-

Pressures in inches

Pressures in lbs. avdp.

spheres and parts.

grees of Fahrenheit

of mercury at 32° F

per square inch, mer-
cury sp. gr. 13-6.

o

in.

lbs.

000025

-25 53

00075

00037

00005

-15-78

0-015

00074

0001

- 4-84

0-030

0-0147

0005

-f26-01

0-15

00735

001

42-41

0-30

0-1470

005

87-36

1-50

0-7352

010

110-93

2-99

1-4704

0-20

137-39

5-98

2-9407

0-25

14658

7-48

3-6759

0-50

177-40

14-96

7-3518

1-

212-00

29-92

14-7036

2-

250-84

59-84

29-41

3-

275-73

89-76

44-11

4-

294-43

119-68

58-81

5-

309-57

149-60

73-52

6-

322-36

179-52

88-22

7-

333-49

209-44

102-93

8-

34336

239-36

117-63

9-

352-25

269-28

132-33

lo-

360-36

299-20

147-04

ll-

367-81

329-12

161-74

12-

3;4-72

35904

176-44

13-

381-16

388-96

191-15

14-

387-20

4)8-88

205-85

15-

392-90

448-80

220-55

16-

398-28

478-72

235-26

17-

403-40

50S-64

249-;i6

18-

408-27

538-56

2UQ6

19-

412-91

568-48

279-37

20-

417-36

598-40

294-07

21-

421-62

628-32

308-78

22-

425-73

658-24

323-48

23-

429-67

688-16

338-18

24-

433-48

718-08

352-89

[ 113 ] '■'•■■''*

XIII. Ofi the Remainder of the Series in the development of
(l+ar)"", and on a Theorem respecting the products of
Squares. By J. R. Young, Professor of Mathematics^ Bel-
fast^.

IN the last Number of the Philosophical Magazine, there is
a very interesting paper, by Professor Graves, On the
Calculus of Operations, in which he has communicated a valu-
able theorem in that important department of analysis, which
I believe has not hitherto appeared in a complete form.

Professor Graves has been enabled to deduce this theorem
from the previous development of (1 +^)~"; which, by means
of the differential calculus, he has exhibited in connexion with
the remainder of the series.

This completed form of the expansion may be readily ob-
tained by a process imitative of that employed in my paper
published in the November Number of this Journal, and with-
out involving any operation of a more advanced character than
that of common algebraical division. It is as follows: —

As ip Professor Graves's notation, let

. _ w(n+l)....(n + ?^ — 2)
" 1.2 (m-1)

_ m(?M+ 1) . . . .m + n— 2

~ 1.2 (w-1) '

Put also

(H-j;)-'(-a;)'» = R„ (H-^)-2(-a^)- = R2,

(l+:r)-3(-^)'« = R3,&c.;
then, since

( 1 +^)-» = 1 -.r + a^2_gjc ( _.r)'»-i + R„

we shall have, by dividing the terms on the right severally by
(1 •\-x\ the following rows of results, namely,

(1 +:r)-2= 1 —xJf.x^-a^-\- (-a?)'«-' + Ri

— :r + x2— a?a + (_^)«»-» + Rj

■^x^—a?-\- {-xY-^ + Ri

-0^ + (-a;)'»-»+R,

&c. &c. +R2;

that is,

(1 +.r)-2= 1 -2a?4- 3a?2_4.a^ + . . . . A2(-a?)'"-» + A2R1 + Rg.

Similarly,

* Communicated by the Author.
Vhil. Mag. S. 3. Vol. 34-. No. 227. Feb. 1 84-9. I

11 4 Prof. J. R. Young on the Remainder of the Series

(1 +;r)-3= 1 -3^ + 6.r2-10x3+ ...A3(-j:)'"-» + A3R1
: +A2R2+R3

{l+a!)-=l-nx+'^^^K^-8ic...A„{-xy'^-' + A^'R,

1 • ^

+ A„_iR2 + A„_2R3+... + AiR„,
the A, being introduced before R„ for the sake of uniformity
of notation: its value is evidently unit. When R,. is replaced
by {l+x)~''{—xy"'f this result becomes the same as that in
Professor Graves's paper ; and it must certainly strike a reader
as a circumstance worthy of notice, that an expression thus
obtained by aid of only the first principles of algebra, should
virtually involve a theorem of such interest in the higher re-
searches of analysis as that given in the paper alluded to.

I fear, from the remark in the first paragraph of that paper,
that I must have expressed myself somewhat obscurely in re-
ference to the Calculus of Operations making " no provision
for the correction " I had adverted to as necessary. I think
I ought to have added, that this provision should always be
furnished by the theorem for quantity whence that for opera-
tions is derived.

I take this opportunity of mentioning that the general form
of the theorem respecting squares, namely,

which the Rev. Mr. Kirkman has done me the favour to insert
at page 500 of the last volume of this Journal, I should prefer
to have appeared in the following more comprehensive shape :

to which may be added the analogous theorems

where m and n are any positive whole numbers whatever.
Belfast, Jan. 13, 1849.

Note. I submitted the substance of the foregoing investiga-
tion to Professor Graves, who, in reply, did me the favour to
communicate to me a sketch of two other methods of arriving,
algebraically, at the same result : this I here give in his own
words : —

* * * « I indicated the method of obtaining the re-
mainder by differentiation, because that process admits of
being described in the fewest words, though it is far from
being the simplest. I know of two algebraical methods by
which the result is obtained more easily.

in the development qf{l-\-x)~". 115

" One follows the ordinary track ; showing that, if the theo-
rem holds for {l+x)~"'^\ it will hold likewise for {l+x)-\
And this is readily proved by means of the fundamental pro-
perty of the binomial coefficients; viz. that the algebraical
sum of the coefficients of a?'"^ and .r*" in the development of
(1 +x)P is equal to the coefficient of a^ in the development of
{l+x)P+K

" My other method having something peculiar in it, I shall
give it in full. Using S to denote the sum of the first m terms
in the development of (1 — <r)~", and R the remainder after S,
we shall have

^^l-(l-.r)».S

Now, when we come to examine the numerator in this value
of R, we find that it contains only powers of x, from a?™ up to
^m+n-\^ R may therefore be put into the form a;'"./(l — a)~^
/{x) being used to denote a series of n terms proceeding ac-
cording to positive integer powers of jr, from x up to a;".

" We have now ascertained the form of the remainder about
which we are inquiring, and it will be easy to determine the
coefficients inf. For this purpose let us take the equation

{l-x)-"=l + na;+ ^^^^x^-i- ,.. + AnX"'-'-hx'^.J{l-x)-\

and in it interchange x with 1— a?, and m with n. Then we
shall have

x-m=z\-^m{l-x)+ ^^^+ ^) (1 -xy+... + A„{l-xy-'

where/' ^~' stands for a series of powers of x~\ from a;~^ up
to .a?""*. Multiply the last equation by a?'" ( 1 —a?)"'*, and it will
become

(1 -x)-" = a:'» -{"(1 ---^)-»+ »?( 1 -3?)-''+' + ^^^^(l-a?)-"+2

+ .., + An{l-:t)-'^ +x^f'x-K

On comparing the several terms of the two finite developments
thus given for (1— a^)~", it is obvious that we must have

and

Rssa;"' <{l -a?)-« + w(l -a;)-»+' + ^^^^^^ (1 -x) «+2
l_ 1 « J

12

C 116 ]

XIV. On a Mode of rendering Substances incombustible.
By Robert Angus Smith, Ph.D.^ Manchester*.

I HAVE often been surprised that, considering the number
of materials which will not burn and the small number
which do burn, we should be compelled to build houses so liable
without constant watchfulness to instantaneous destruction ;
that we should go also to sea in vessels made of a most com-
bustible substance filled with enormous fires, frequently under
the care of ignorant men. I think, therefore, I may be ex-
cused when 1 endeavour to add to a knowledge of the mode
of rendering substances incombustible, or the theory of the
mode to be sought after, even if the addition which I make be
but a very small one.

Silicate of potash has been considered good. It is a soluble
glass which was expected to cover the fibre of cloth or wood,
and so protect it from heat. This does act to some extent, pro-
bably in the same manner as stones do when put into a fire of
wood or coal ; they take heat but give none, and are also bad
conductors. If silicate of potash remained as a glass, it would
act also by keeping out the air ; but this does not seem to be
the case, as it falls after a time to a powder.

It struck me that the mode of preventing combustion was
not by protecting the wood from the fire merely, as heat must
cause combustible gaj^es to rise from wood, whether there be
incombustible substances mixed with it or not, and these gases
will force their way to the surface where there is no longer any
preventive to burning. My object then was to find a substance
which would render the wood unfit to burn, and would cause
it to give out gases which would not burn; so that whilst the
wood itself was being preserved, except where in contact with
the fire, the gases would assist in extinguishing the fire.

I first tried phosphate of magnesia and ammonia, thinking
the ammonia given out would be of use in extinguishing the
fire; but this was of no value, as apiece of calico required to
be made quite stiff with it before it was rendered incombustible.
The calico was prepared by dipping in a solution of phosphate
of magnesia in muriatic acid and then in ammonia. It seemed
to me that the earthy salts are of little use for the purpose
required, and that the amount of solid matter incapable of
evaporation left on the cloth, assists in a very small degree.

Sulphuric acid, however, seemed to present the most pro-
mising characteristics of a substance incapable of burning, and
of acting so strongly on vegetable substances as to make them

• Comnuinicated by the Autlior,having been read before the Manchester
Literary and Philosophical Society, January 9, 1849.

On a Mode qfrendermg Substances incombustible. 117

incapable of burning. Sulphuric acid itself is a body perfectly
burnt, or we may say overburnt, having an atom of oxygen
given to it by artificial means, so to speak, which atom is dif-
ficult to separate, and therefore not resembling the oxygen of
many highly oxidized bodies. It requires a high degree of
heat to raise it to vapour; and the vapour formed is sluggish
and heavy, remaining long where formed, and quenching flame
wherever it is. It destroys the texture of wood also and other
vegetable substances, causing them to give out after a time
gases which do not burn, mixed with some which do burn;
but if there be enough of acid, forming a mixture which does
not burn. The wood also cannot be again induced to become
combustible until it be heated to redness, so as to remove all
the sulphuric acid, leaving only charcoal.

If sulphuric acid then could be introduced into wood just
at the time that the fire was going to take place, the fire would
cease to take place ; and this we can do easily by saturating
the wood with sulphate of ammonia. When there is no fire
present there is no sulphuric acid present, as such ; but as
soon as the heat rises, ammonia goes off, and sulphuric acid
is instantly presented to the wood. The ammonia does not
come off quite pure, it is mixed with nitrogen and sulphurous
acid ; and this disengagement of gases is of advantage in ex-
tinguishing fire; when the heat rises to 536°, the sulphuric
acid is then left to act on the wood in part and to volatilize in
part, and that which I have mentioned takes place. The out-
side of course would first undergo the change, and the inside
would be protected by the incombustible outer part ; if the fire
continued to act long, the inner layer would undergo a similar
change. I imagine, then, the acid acts in a double manner ;
it makes the wood refuse to burn and it puts out fire. As
sulphurous acid is given off in this process, the action is also
similar in one point of view to that of sulphur, which has
long been used for putting out fire in chimneys.

I have no doubt that a house built of wood prepared in this
manner might have a fire lighted on the wooden floor without
danger, burning only on the spot to which the fire was limited.
A ship also would be safe, even if the cinders did fall from
the grate in stormy weather.

1 know that muriate of ammonia has been used, and that
it acts very well ; but I think the sulphuric acid is superior,
the ammonia being merely to keep it innocent ; any other vo-
latile base might do. I am sorry, however, that this is not
perfect; its solubility in water is a great disadvantage, as it
cannot be applied to clothes to be frequently washed. True,
it is so cheap that it might be applied every washing where

118 On a Mode q/' rendering Substances incombustible.

there are peculiar dangers; but if a person were standing very
near the fire, the ammonia would in part be evaporated, and
the acid remaining would be enough to injure the fabric.
There are however cases, such as curtains, to which this could
not apply, and where it would be valuable.

Sir William Burnet's liquid is chloride of zinc: he uses it
for preserving wood and canvas, and also for preventing fire.
I am certainly surprised that more use has not been made of
it, being, as far as I have seen it, so efficient. I believe the
manner in which the chloride of zinc acts is very similar to
that of the sulphuric acid, destroying the organic matter on
the approach of heat, and rendering it incombustible. It can
he introduced into wood at a specific gravity of 2000, I be-
lieve; sulphate of ammonia cannot easily be used above 1200.
By heating the solution more may be attained. Sulphate of
ammonia is cheap and easily procured and used, not hurting
anything with which it may come in contact, and therefore
more easily managed in households.

The chloride of zinc is said to unite with the fibre. This
cannot be said for the sulphate of ammonia. It would not,
however, come from the centre of a beam of wood, even if
immersed in water, as the water enters with great difficulty
into wood ; and the solution itself cannot be introduced with-
out forming a vacuum in the saturating vessel, and so remo-
ving all the air from the wood.

The first time I used this solution I found a large quantity
of mould formed, and indeed it contains all the elements to
increase its growth. The second time the solution was boiled
in an iron vessel, and no mould formed on it ; on the contrary,
mould was destroyed by it. The sulphate of ammonia dis-
solves iron rapidly, and forms a double salt which is delete-
rious to such growths. I imagined any other metallic salt
would do, and used ordinary chloride of manganese prepared
in the laboratory, which killed all such fungi rapidly, and no
more have grown after standing eleven months in contact with
organic matter.

I believe there are many ways in which this may be used.
My wish was to find a substance suited for building fire-proof
ships, and 1 believe this would do ; at any rate the ships would
be fire-proof, experience could alone tell if any other objection
followed. It does not render the wood hard, heavy or brittle.

I believe it would be of the greatest advantage in mills, which
now suffer so much from fire, diminishing or rather entirely
removing the expense of insurance. It does not hurt colours ;
so that even coloured goods might be dipped when kept long
in one place, or when sent in vessels abroad. Possibly some

The Rev. C. Graves oti a system of Triple Algebra. 119

delicate colours may be attacked, but this must be a rare
case.

I am more desirous of seeing ships built of an incombustible
material, the means of escape at sea being few, and confined
to few ; and whilst there is any hope of doing it easily, I
scarcely think it proper for any one to neglect what informa-
tion may exist on the subject.

XV. On a sgsfem of Triple Algebra, and its apjdicatio?! to
the Geometry of three Dimensions, By the Rev. Charles
Graves, A.M., Professor of Mathematics in Trinity College,
Dublin^.

WHETHER we can directly evolve out of the ordinary
double algebra such a more general algebra as will
answer all the requisitions of the geometry of three dimensions,
is a question into which I am not about to enter at present.
But Mr. Cockle's discussion of it in the last Number of the
Philosophical Magazine, and his use of an imaginary whose
square is equal to positive unity, have reminded me of a system
of triple algebra which I devised some time ago, and which,
I am inclined to believe, will find morefavour with others than
it has (lone with myself. The outline of it was stated in a
letter addressed by me to the late Professor MacCullagh in
April 1845, since which time I have suffered it to lie without
further development or application. For my own part, I com-
mend the superior power, symmetry, and flexibility of Sir
William Hamilton's quaternion theory, of the excellence of
which use has only more and more convinced me. I find,
however, that there are mathematicians who continue to object
to it on the ground that they cannot conceive the nature of
the operations by which his symbols i,j, k transform extra-
spatial, scalar quantity into directed linear magnitude; and
who protest against the sacrifice of the commutative character
of multiplication. To those who cannot bring themselves to
waive these objections, I submit the following system. It has
the advantage of being easily and completely interpreted ; and
I may add that, in the treatment of some geometrical questions,
it has proved itself not inferior even to the quaternions.

1. Imagine an operation denoted by the symbol (5), and
of such a nature that —

1. 5(1) is a quantity not homogeneous with the real unit,
so that no equation of the form 5(a) + 6 = 0, where a and b are
real, can subsist, unless a and b be separately equal to zero.

* Communicated by the Author. .

120 The Rev. C. Graves ow a system of Triple Algebra^

2. It is distributive ; therefore

s[a) + s{b)=s{a + h)=i{a + b)s{\).

3. It is a periodic operation of the second^order ; that is to
say, s^(a) = a.

2. The nature of the operation being so far 'defined, let us
take two mixed binomials, x-\-s{y) and x' + s{y'), and operate
with either upon the other ; for it matters not which is ope-
rator and which operand; the result of the operation will be

{.V + «(j/)) (.r' + s{i/))=xa^ -\ryi/ + s{xy' +ya^) ;

and putting

a/' = xoi^ + yy\ and y = xi/ -hyx',

it is easy to see that we shall have

{x-1/){x'-7/') = x"-7/>J' ' ' ' ^'^

These are the modular equations of multiplication in the sy-
stem of double algebra, with which we are at present concerned.

3. There is no difficulty in assigning a geometrical inter-
pretation to the symbol s.

Draw in a plane the two rect-
angular axes of co-ordinates OX
and OY, and bisect the angle be-
tween them by the right line OA.
Then the symbol 5 may be taken
to express rotationfrom left toright
through an angle of ] 80° round the
axis OA.

Supposing that the real unit
be placed upon the axis of ^,
it is evident that this geometrical
representation fulfils the condi-
tions imposed upon s. For, in
the first place, 5(1) lies upon the
axis of y ; and being at right angles with the real unit, is
as much distinct from it as V — I is. Next, this rotation is
plainly a distributive operation. Lastly, it complies with the
third condition; seeing that a repetition of the rotation through
180° brings the real unit back again into its original position.
4-. We may now agree to represent the line drawn from the
origin to the point whose rectangular co-ordinates are x and
y, by the binomial x + s[y). The first consequence of this will
be, that the sum of two lines will be represented by the dia-
gonal of the parallelogram whose sides are the lines to be
added. In fact, analogy so strongly demands this, that it is a

and its appUcatioji to the Geometry of three Dimensions. 121

part of almost all modern systems of symbolic geometry. If
we next proceed to inquire how we may represent the product
of two lines {xy) and {x^i/)i we shall find the following rule for
constructing it : " The projections of the real unit line, the fac-
tor lines, and the product line, either upon the axis OA, or upon
OB, a right line perpendicular to it, foi-m an algebraic pro-
portion.^* This is the geometrical interpretation of the equa-
tions (1.).

5. So far we have been geometrizing only in piano ; but we
can pass readily into the geometry of three dimensions, since
in the course of its rotation round OA the real unit quits the
plane of jry.

As s denotes the rotation of 180° round OA, 5* may be taken
to denote half that rotation.

Again, the unit of length in the direction of OA is

l+s{l),

and the unit of length in the perpendicular direction is evi-
dently

i-sjl)

± v/2'

Now if this latter unit be operated upon by s4, it will be brought
into the position of the axis of z, perpendicular to the axes
both of x and y; that is to say,

i/2

is the positive x-unit, which we shall henceforth denote by w(l).
For the square of this ^-unit we shall find a simpler expres-
sion. Squaring the equation

we have

5(l)-l=n2(l);

and operating upon the preceding equation with s, we find

5n(l)=-«(l).

6. Having now ascertained the laws of the combination of
5 and n, we may proceed to deal with trinomials of the form
x + s{y) + n{z), which we shall take to represent the line drawn
from the origin of rectangular co-ordinates to the point (03/2).

As before, the sum of two lines is their resultant. If

a" + s{y") +»(»") = (a? + s{y) + «(z)) (a;' + 5(2^) + n{z% (2.)

122 The Rev. C. Graves oti a system of Triple Algebra,
we shall have

i/':=xi/-\-i/a^-{-zz' V (3.)

:::"={a;—i/)z' + :2{x'—y')J

From these relations it would not be difficult to determine
directly the mode of constructing the product line ; but the
result will be more readily arrived at by means of the follow-
ing method.

Let ^, tji ^ be the projections of the line (-rj/z) upon OA, OB,
and OZ the axis of Z. Then I, m, n, the unit lines upon these
axes, are respectively expressed in terms of.? and s* by means
of the equations

7_ 1+^(1) „,_l-^(t) ^^_5i(l-6(l)).

and we may wiite
instead of

l^ + mri-\- n^

'}. (*.)

And for the laws of combination of the imaginaries /, ?n, n, we
have the equations

Im — O, /« = 0, ?ww=v'^2.«

Hence the product of two lines (^jj^) and (^V^'} is

Now the part of this which is on the axis OA, viz. \^2.l^^',
is in length a fourth proportional to the projections of the real
a?-unit and the two factor lines upon that axis.
So again the part

which lies in the plane perpendicular to OA, is, in Mr. War-
ren's sense of the word, a fourth proportional to the projec-
tions upon that pla?ie of the same three lines.

7. From the geometrical interpretation which has been as-
signed to the symbol s, we can derive what seems to me to be
a satisfactory explanation of the vanishing of the product of
the factors 1—5(1) and l+s(l), although these factors are
each different from zero.

In consequence of s being distributive, the expression
(1_5(1))(1 -|-.s(l)) means the difference between 1+5(1) and
5(1+5(1)). Now, as the line l+s(l) coincides with the axis
OA, round which rotation takes place, it is unaffected by any

and its application to the Geometry of three Dimensions. VIS

amount of such rotation, and so may be considered equal to
5^(1+5), where jo is any real quantity whatsoever. There is
therefore no difference in either magnitude or direction be-
tween the lines 1 +5(1) and ^(1 +5(1)) ; and we are entitled to
put

(I-5(l))(l+5(l))=0.

8. I think the following will be found to be the true theory
of the vanishing of factors and products.

In a system of algebra in which there is but one real mo-
dulus of multiplication, a factor and its modulusvanish together.
The vanishing of a factor will cause the vanishing of a product
into which it enters along with other finite factors; and con-
versely, the vanishing of a product indicates the evanescence
of one of the factors. This rule applies to the ordinary double
algebra, whose units are 1 and (—1)^, and also to Sir William
Hamilton's quaternion algebra of four units, 1, 2,7, k.

But systems of algebra may be constructed in which the
case is different. The process of operating with one mixed
quantity upon another — that process, in fact, which, on the
score of analogy, seems to claim the title of multiplication —
may lead us to regard a mixed quantity as having more than
one real modulus: and these moduli, suppose n in numberj
need not all vanish simultaneously. When they do, the
quantity to which they belong vanishes likewise ; but we can-
not say that it does so unless its different moduli of multipli-
cation are separately equal to zero. Such a factor entering,
along with other finite factors, into a product, annihilates it
by annihilating all the « real moduli of the product. On the
other hand, the product is not reduced to zero unless all its
real moduli vanish ; and this cannot take place unless, amongst
all the moduli of all the factors, there be w, of different Icinds^
separately equal to zero. These n vanishing moduli may in
general be distributed amongst the factors in various ways
without annihilating any one of them. What has been last
said applies to the systems of double and triple algebra dis-
cussed in the present paper.

Operating with x + sy upon x' -{-s^, we found that the mixed
quantity x + s{y) had two real moduli of multiplication, <r+j/
and J^— ^. It vanishes if both these are equal to 0, but not
otherwise. And x" + s{y")f the product of these two factors,
will not vanish unless both its real moduli likewise vanish.
Now it appears from equations (I.) that this may happen in
four different ways.

1. We may have x=0 and y — 0; 2, x' = 0 and y=0;
3, a?+^=0 and x'—i/=Q\ 4?, x—y—O and «'+y=0. In

124? The Rev. C. Graves on a system of Triple Algebra,

the first and second cases, a factor actually vanishes ; but not
so in the third and fourth. In these, two real moduli, of dif-
ferent kinds and belon<Ting to different factors, becoming equal
to zero, annihilate the two real moduli of the product, and so
cause itself to vanish.

So again in the case of the trinomial x -\- s{y) -\- n[z). It
appears from the equations (3.) that it has two real moduli of
multiplication, x+y^ and {x—y)^ + 2z^. That is to say, if
{x + s{y) + n{z))[x' + s{y')-^n[z')) = x''^s{y")+n{z")',

we shall have

{x-Yy){pi^+i/)=x"+y\
and

(J^x-yf + 25;2) ( [x^ -y'f + 2z'^) = {x" -y"f + 22"\

The factor x-{-s{y) + fi{z) vanishes if both its real moduli vanish,
but not otherwise. And the product x" + s{y") + n{z") will not
vanish unless both its real moduli likewise vanish. But this
may happen in different ways.

1. We may have x+y = 0, and {x—i/)^-\- 22^ = 0; condi-
tions equivalent to x=y = 2 = 0, and therefore involving the
annihilation of a factor.

2. x'+y' = 0, and {x' —y'f + 22'^=i0; conditions which, in
like manner, annihilate the other factor.

3. x + y = 0, and {x^—y'f + 2z'^=0; equivalent to a'+^=0,
x'—y'=0, and 2;' = 0.

4. {x— y)'^ -{-22^=0, and x'+y' = 0; equivalentto .r— 3/=0,
2=0, and x'-\-y'=0.

Here, as before, we see that the complete annihilation of
the product may be brought about either by the complete
annihilation of either factor, or by the vanishing of two mo-
duli, of different kinds, occurring one in each of the two
factors.

Similar observations apply to the system of triple algebra
discussed by Professor De Morgan in the Transactions of
the Cambridge Philosophical Society, vol. viii. ; and to the
closely related system, of which 1 have given a brief account
in the Proceedings of the Royal Irish Academy, vol. iii. pp. 5i
and 57.

In both the systems just referred to there are two real mo-
duli of multiplication ; and in both a product of two factors
may vanish without the vanishing of either of the factors them-
selves.

9. It is not only when the two factors 1 +s(l) and 1— s(l)
come together that we meet with a singular result. The
occurrence of either of them in a product affects it in a re-
markable manner.

and its application to the Geometry of three Dimetisions. 125

If we consider the two right lines which represent the quan-
tities X -\- s{y) -\- n{z) and 5(a? + s(^) + ;?(«)), we shall see that
either of them is obtained from the other by causing it to turn
round the axis OA through an angle of 180°. The sum of
the two, therefore, compounds a line in the direction of that
axis ; and their difference is a right line in the plane perpen-
dicular to OA. Thus it appears that the product (1+5(1))
{x-{-s{y)->tn{z)) denotes a right line, which, as well as the fac-
tor 1 + 5(1), coincides with the axis of rotation; whilst the
line denoted by the product {l—s{\)){x + s{y) + n{z)) lies, like
the factor 1—5(1), in the plane perpendicular to that axis.
Here we have a geometrical interpretation of the strange-
looking results,

(l+5(l))(^ + 5(.t/) + «(-2r)) = (l+5(l))(.r+j,)

{\--s{l)){x+s{y)) = {\-^s{\)){x-y)

10. Lest it should be supposed, however, that the explana-
tions just given of these seemingly anomalous results have arisen
accidentally out of the geometrical interpretation assigned to
the symbol 5, I proceed to show that these results admit of
being explained by reference to the analytical conditions im-
posed upon 5 at the outset. I do so, because in the course of
the inquiry we shall be led to notice some general principles
useful in other systems of algebra as well as in the one be-
fore us.

On reviewing the operation which conducted to the result

{x + s{y)){a^ + s[y'))=a^' + s{f),

we observe that this equation holds good, whatever distributive
operation 5 may be, provided it satisfies the equation 5^(a) = a.
Now this equation has two purely algebraic solutions 5= + 1
and 5= —1. It follows, then, that we may write successively
+ 1 and — 1 instead of 5 in it, and so we obtain at once the
two modular equations (1.). And further, whatever equation
we find subsisting, in which s appears along with real quanti-
ties, it must continue to hold good for 5= + I and 5= — 1.
Thus, for instance, from the equation

c*W = Hyp. cos x-\-s Hyp. sin x

we derive both

fr'=Hyp. cosa:+ Hyp. sin a:
and

e~*= Hyp. cos x~ Hyp. sin x.

Suppose now that we had before us some expression of the
form (1 +5(l))!p(*, a-, 3/, z) ; we might expect to find in all trans-
formations of it the operator (1+5(1)) still remaining, or ca-

26 The Rev. C. Graves on a system of Triple Algebra.

pable of being readily put, in evidence, because the expression
vanishes for 5= + 1. In other words, as 5 partakes of the cha-
racter of + 1, these two factors are of such a nature that either
of them, like zero, assimilates the product into which it enters
to itself. If both enter into the same product, they must make
it like their own product 1—5^(1), which, by the definition of
5, is equal to zero.

11. The same kind of reasoning admits of a more interest-
ing application to the problem of finding the moduli of multi-
plication in the triple algebra whose units are 1, s, and «; or
in that other whose units are I, m, and w.

As the operation n is defined by the equation

^1-5(1))
''" a/2 '
it appears that, when s degenerates into + 1, « becomes 0 ; and
when s is equal to — I, n is equal to + -/ — 2. In the former
case the equation (2.) is reduced to

(.r +3/) (.2'' -f- y ) = x" +y' •

In the latter we shall have
{x-y± \/^^.z){a,'-i/'±\/^^.z') = x"-y"± x^'^.z".

Thus we obtain the two real moduli of multiplication of the
system whose units are 1, 5, and n.

12. Let us next pass to the consideration of the triple sy-
stem, whose three units are /, m, and «, as defined by the equa-
tions (4<.).

According as 5= +1, or —1, these latter become

/= v/2, 7rt = 0, n = 0;

or

/=0, m= s/% n—± a/— 2.

And, if we introduce these two systems of vahies successively
into the equation

(/? + mij + «^)(/r + mii + vXl) = /r' + mil' + «$",
we shall derive from it the following :

__ >/2.fr=^",

a/2(>)± v/^.?)(»)'± ^-1.^') = V'± 'vZ-U",
which furnish the modular relations belonging to this system
of triple algebra.
DiibIin,Jan. 12, 1849.

[To be continued.]

[ 127 ]

XVI. On the Repulsive Action of the Pole of a Magnet upon
Non-magnetic Bodies. By Professor Reich of Freiberg^.

THE repulsion which takes place, according to Faraday's re-
cent observations, between the pole of a magnet and every
diamagnetic substance, apparently with the exception of the
atmosphere, is to my mind so new and surprising an exhibi-
tion of force, that probably some observations on the subject
will be considered worthy of attention, even though they
merely confirm what Faraday has found, if they exhibit this
repulsion in a more easy and direct manner.

For these observations I employed a torsion- balance which
had been arranged to determine the mean density of the earth.
A horizontal wooden rod two metres in length is suspended by
means of a copper wire to a strong iron beam fastened into a
massive wall, and at each of its extremities a metallic ball is
suspended by a fine wire. The whole is inclosed in a wooden
case, which, however, is nowhere in contact with the torsion-
balance. The torsion-rod carries a mirror, in which the posi-
tion of the rod is observed with a telescope upon a distant scale.
The force required to deflect the rod a certain quantity from
its position of rest results from the following expressions, which
will likewise show the very great sensitiveness of the apparatus.
The mass of the whole moveable portion of the torsion-balance
reduced to the central point of one of the two balls is
ss^'s 1031 560 milligrammes. The distance of the central
point of either ball from the axis of rotation is = r = 10005
millimetres. The horizontal distance of the mirror from the
scale which is divided into millimetres is =]«. = 42827 milli-
metres ; if, therefore, we suppose the deflection of the ball from
its position of rest to be = A millimetres, and the number of
divisions of the scale corresponding to this deflection to be
= B millimetres, then

A — ' R — ^^Q^^ R

~ 2/Z ~ 85654- '

and the force which deflects the ball A millimetres from the
position of rest, with a time of vibration = N seconds,

g-.A __ rg.B

Wl ~2fi.W.l'

I being the length of the seconds' pendulum in millimetres.
When the torsion-rod vibrates without any external action upon
the balls, N is very nearly = 350 seconds, which gives a de-
flecting force of 0*00098956 B milligrammes. But B may

* From Poggendorff's Anruden, vol. Ixxiii. p. 60,

128 Prof. Reich on the Repulsive Action of a

be estimated to a tenth, and the force therefore to 0*0001 of
a milligramme.

I first tried the effect of magnets upon one of the balls
which had been employed in the determinations of density,
and which consisted of tin mixed with 10 percent, of bismuth
and about 2 per cent, of lead. Magnet bars on being brought
up in a horizontal direction to the case near a ball produced
a very distinct repulsion, both when the north and the south
pole were brought near. But when several similar bars were
brought near, half with their north and the other half with
their south poles, there was no effect perceptible, or merely a
slight one, arising from the inequality of the magnets em-
ployed. A horse-shoe magnet with its two poles was just as
ineffective. A four-pound magnet bar belonging to a mag-
netometer was brought as close to the south ball as the wooden
case would permit, and in a direction perpendicular to the
rod. The rod previously stood at 41*50 of the scale, the ap-
proach of the north pole re. laved it 5314'; the south pole
then raised it to 55*45, and the north pole again brought it to
54*05. After removing the magnets, the position of rest found
was 42*80. If we take the mean from the first and last posi-
tion of rest with the magnets removed, and also with the north
pole brought near, we obtain —

Repulsion by the north pole 11*445 divisions of the scale,
south pole 13*300

The diflference may be owing to unsymmetrical distribution
of the magnetism in the bars.

As is well-known, the repulsive action of a magnet upon
bismuth had been observed ; I therefore had a ball made of
pure bismuth of the same weight, and hung it in the place of
the one hitherto used upon the south extremity of the rod of
the torsion-balance.

The position of rest of the rod with the magnet at a distance
was observed to be —

Divisions of scale.
previously . . 11*200
subsequently . _9*775 Time of vibration.
mean . , . 10*488 350*5 seconds.

North pole close to the case . . 69*250 280*8

at 10 millims. distance 43*670 307*4

at 30 millims. distance 21*205 333-7

This gives —

Pole of a Magnet upon t^ on-magnetic Bodies. 129

Distance of the

position of rest

from 0.

Observed repul-
sion.

Divisions
of scale.

millims.

Divisions

of scale.

B.

millims
A.

Distance of the pole of

the magnet from the

central point of action

in the ball.

Millims.

Time of
vibra-
tion.

Seconds.

Repulsive force,
milligram.

Observed. Calculated.

10-488
69-250
43-670
21-205

1-2223

8-0889
5-1010
2-4769

0
58-762
33-182
10-717

0
6-8666
3-8787
1-2546

x-\- 8-0889
.r+151010
.r+32-4769

350-5
280-8
307-4
3337

0
009038
0-04260
0-01169

0
009038
0-04084
001042

The last column i.s calculated upon the assumption that x,
i. e. the distance of the pole of the magnet when close to the
case from the central point of action in the ball, is 1504' mil-
limetres for the position of the torsion-rod at 0, and that the
repulsive force acts in an inverse ratio to the third power of
the distance. The differences do not exceed the possible
error of observation.

At the distance of 15 millimetres from the surface of the
case, and with the position of the rod at 0 of the scale, scarcely
the surface of the ball is attained, which would show that the
principal action is upon the nearest surface of the ball.

A second observation gave the following measures of re-
pulsion : — Millims.
For the magnet close to the case . . . 7*388
At a distance of 10 millims. from the case 4*365
20 ... 2-64.1
30 ... 1*628
40 ... 0*856

Since, however, the corresponding periods of vibration were
not observed, the repulsive force exerted in each instance can-
not be calculated from them. That the effect was found to be
greater on this occasion, is explained by the circumstance of
the position of rest of the rod, with the magnet removed, being
on an average at 0*994 of the scale instead of, as in the first
series of experiments, at 10*488 divisions, so that consequently
the distances vvere less than in the first observation.

A third series of experiments were undertaken after the ball
of bismuth had been suspended from the north end of the rod.
In this instance it approached less close to the case than pre-
viously on the south side, as will be seen in the followuig tiom
the determination of :r.

The direct observations gave —

Position of rest in
divisions of the scale

With the magnet removed . 72*6229

Magnet close to the case . 33*2583

At 10 millims. distance . . 50*1167

At 20 ... distance . . 58*46875

At 30 ... distance . . 63*73125
PhiL Mag. S. 3. Vol. 34. No. 227. Feb. 1849.

Duration of vibra-
tion in seconds.
349*46
301*59
323*75
336-56
342*44
K

ISO Prof. Marcet on the Action of

Whence we obtain —

Distance of the

position of rest from

100.

Divisions
of scale.

millims.

Observed repulsion.

Divisions of

scale.

B.

niLLlims,
A.

Distance of the pole Time of
of the magnet from vibra-
the central point of
action in the ball.

millims. _ ,

Seconds,

tion.

Repulsive force,
milligrammes.

Observed. Calculated.

27-3771

66-7417

49-8833

41-53125

36-26875

31978
7-7959
5 8267
4-8511
4 '2364

0

39-5646

22-5062

1415415

8-89165

0
4-5981
2-6289
1-6533
1-0386

x^ 7-7959
•^+15-8267
.:r-|-24-85U
.r+34-2364

349-46
301-59
323-75
336-56
342-44

0
005246
002603
0-01515
000919

0
0-05246
002719
001494
000899

The three last values of the last columns are calculated from
the second, upon the assumption that x — 25 millims., and that
the action is inversely as the third power ot'lhe distance. With
this assumption we again constantly arrive nearly at the surface
of the ball of bismuth.

Although the calculated values agree sufficiently with ob-
servation, I by no means regard the experiments sufficient to
deduce from them the two following positions, that —

I. The repulsive action acts principally upon the nearest
surface of the diamagnetic body.

II. That this repulsion decreases as the third power of the
distance of the pole of the magnet increases.

In the first place, the experiments are not sufficiently nu-
merous, and require to be repeated with modifications; and
secondly, it should be observed that the ball was contained in
a cylindrical wooden case, the inside and outside of which was
coated with tinfoil. Now if the cause of the repulsive action
is owing to an induction, perhaps, of electric currents which
the pole of the magnet excites in or upon the ball, it is highly
probable that it would excite similar induction upon the coat-
ing of tinfoil, or even upon the wood of the case, which
would react upon the ball and so complicate the total effect.

XVII. On the Action of Chloroform ow the Sensitive Plant
(Mimosa pudica). By Professor Marcet of Geneva*.

WHEN one or two drops of pure chloroform are placed
on the top of the common petiole of a leaf of the sen-
sitive plant, this petiole is seen almost immediately to droop,
and an instant after the folioles close successively pair by pair,
beginning with those which are situated at the extremity of

* Read before the Societe de Physique et d'Histoire NatureUe, Oct. ] 9,
1848, and communicated by the Author.

Chloroform on the Sensitive Plant, 131

each branch*. At the end of one or two minutes, sometimes
more, according as the plant is more or less sensitive, most of
the leaves next to the chloroformed leaf and situated beneath it
on the same stalk, droop one after another, and their folioles
contract, although generally in a less complete manner than
those of the leaf placed in immediate contact with the chloro-
form. After a rather long time, varying according to the
vigour of the plant, the leaves open again by degrees; but on
trying to irritate them by the touch, it is seen that they have
become nearly insensible to this kind of excitement, and no
longer close as before. They thus remain as if torpid for
some time, and generally do not recover their primitive sen-
sitiveness till after some hours. If, however, when they are
in this state of apparent torpidity, they are subjected again to
the action of the chloroform, they close as they did the first
time. It is not till after they have been chloroformed several
times, that they lose all kind of sensitiveness, at least until
the next day ; sometimes they even fade completely at the end
of too frequent repetitions of the experiment. In all cases the
effects observed are the more marked in proportion to the
purity of the chloroform employed and the degree of sensitive-
ness in the plant.

An analogous phasnomenon is produced if, instead of placing
the drop of chloroform on the base of the petiole, it is laid on
the folioles situated at the extremity of a branch. The folioles
of this branch immediately begin to close pair by pair, the
common petiole droops, lastly the folioles of the other branches
close in turn. At the end of two or three minutes, the nearest
opposite leaf, and if the plant is vigorous, most of th6 other
leaves situated below on the same stalk follow their example.
When, after some time, the leaves open again, the saii^e want
of sensitiveness is manifested as in the preceding case.?

A singular feature in this phaenomenon is the manner in
which the action of the chloroform is propagated from one
branch to another, then from one leaf to another, even when the
liquid disappears by evaporation almost as soon as it is deposited.
This action, as we have just seen, appears to be communicated
from the leaf to the stalk, following in the latter a descending
direction; generally the leaves situated beneath the chloro-
formed leaf are not at all affected. DeCandolle, in making
an analogous experiment on a sensitive plant with a drop of
nitric or sulphuric acid, remarked, on the contrary, that it
was the leaves above the leaf touched which closed, without

* I previously convinced myself by experiment that a drop of water,
placed delicately on a leaf of the sensitive plant, caused no movement.

K 2

1 S2 Mr. J. Cockle's Solution of two Geometrical Problems.

those situated beneath participating in this motion*. The
observation of our learned countryman is quite naturally ex-
plained by attributing to the ascending sap the transport of
the corrosive poison, a transport which, in this case, would
take place in the direction from below upwards. But how to
account for the apparent transmission of the effects of the
chloroform in the contrary direction, from above downwards?
Might the descending sap more peculiarly have the property
of transmitting the narcotic effects of this singular compound
from one part of the sensitive plant to the other ; or might
there exist in this plant some special organ susceptible of being
affected by certain vegetable poisons in a manner analogous
to the nervous system of animals ? Notwithstanding the in-
teresting investigations of Dutrochet and other physiologists,
there still prevails too much obscurity on this subject to ha-
zard an opinion. But in any case the fact is singular, and
appears to me to merit the attention of persons accustomed to
engage in questions of this nature.

Experiments of the same kind, made on the contractility of
the sensitive plant with rectified aether, have furnished me with
results nearly similar to the preceding ; with this difference,
however, that whilst one drop of chloroform placed on the
common petiole of a leaf situated at the extremity of a branch
of a sensitive plant suffices to cause most of the other leaves
situated beneath on the same branch to close, aether in general
produces an effect only on the leaf itself with which it is put
in contact. The next leaves have generally appeared to me
not affected. I must however add, that my experiments with
aether having been made after the others, and at a time of
year when the sensitiveness of the plant already began to di-
minish, it is possible that the intensity of the effects produced
may have thereby been affected.

XVIII. Solution of two Geometrical Problems. Py James
Cockle, Esq.y M.A., of Trinity College, Cambridge , and
Barrister-at-Law of the Middle Temple f.

THE following solutions are effected by what may be termed
a Uniaxal or an Imaginary Geometry. The equations of
the problems are formed and treated as if the points which
constitute the data and quaesita were in the same straight line.
The sketch here given of such a geometry is necessarily short
and confessedly imperfect. And yet, perhaps, it will be found

* DeCandolIe, Physiologie VegHale, vol. ii. p. 866.
f Conrinuinicated by the Author.

Mr. J. Cockle's Solution of two Geometrical Problems. 133

sufficient for my purpose, — which is, to show the iiiterpreta-
bility of impossible quantity.
Definition. In the equation

t;=A + ?B+iC (1.)

let

A2 + B2+C2=a2;

then I propose to call (1.) ^ virtual solution of the ecjuation

v=a.

Rule. To solve a problem by the imaginary geometry, let
its conditions be expressed by the independent relations U = 0,
and V = 0 J form the equation

U + wV=0 (2.)

where m is a disposable multiplier: then, if a solution of (2.)
is a virtual solution of

V = 0,

the thing required is done*.

Problem 1. Find three points equidistant from each other.

Let A be one of the points. Draw AB equnl to any quan-
tity «, and in any direction, and let B be another of the points.
Let C be the third point. Then, since C is equidistant from
A and B, we have ACxCB=AC^; and also, since A is
equidistant from B and C, we obtain AC^=AB^. These
equations will, on putting AC=,r, be expressed algebraically
as follows : —

.v{a—x)=x\ ...... (3.)

^2=a2. (4.^)

add (3.) and (4.) and we obtain

x{a—x)=a\ (5.)

and hence

a , a \/~^
2 — 2
a .aVS

and, the solution of (5.) being a virtual solution of (4.), the
problem is solved. ABC is, of course, an equilateral triangle.

Problem 1L Find four points equidistant from each other.

Complete the rhombus ACDB. Then D is equidistant
from B and C. And, by symbolical geometry, we have
AD = AB+BD. But, we must also have, since A is equi-

* Observations on this Rule and its grounds are reserved for another
opportunity.

1 34- Mr. J. Cockle's Solution of two Geometrical Problems.

distant from B and D, AD = AB. Let AD =2;; then, we
may express the two last conditions by

z = a-\-Xi ....... (6.)

and

z=ai (7.)

add (6.) and (7.) and we obtain, on reducing,

z=«+|, ....... (8.)

or

^ T ^ " ^ -^

Now, in its present form, the last equation does not furnish us
with any solution of the problem. But it may be rendered
available by decomposing it into two congeneric surd equa-
tions and selecting the impossible congener. For, the left-hand
side of (9.) may be resolved into two factors, both of which
are included in the expression

±^Z + ^^---^-^+Z. . . (10.)
Assume that

""-^-'-^ +Z==;Z, . . . (1.1.)
then one of the values of (10.) takes the form

and, consequently, vanishes. This, then, is the solution of
(9.) which we are in quest of, and, further, this is a solution
which must not be neglected supposing that we admit impos-
sible quantities into algebra. We must now consider D as
situate out of the plane of ABC.

But, the question occurs, which of the infinite number of
values of Z are we to select ? The answer is, that which sa-
tisfies the condition* that the orthographical projection of D
on the plane of ABC shall be the centre of the circle inscribed
in ABC. But, this condition gives,

whence,

r, Sa

* The condition in question is, perhaps, the one best adapted to the
problem before us ; the determination of Z under its most general aspect
will be discussed on a fitting occasion.

Cambridge Philosophical Society. 135

Let

A=-, B=:-^-, C = (Z=)-,

then we have

2;=A + ?B+iC; (12.)

and, since (12.) is a virtual solution of (7.), the problem is
solved. A BCD is a regular tetrahedron.

Scholium. These solutions may be readily verified. The
form, which imaginary geometry will finally take, may possibly
be very different from that exhibited here, but I have endea-
voured to show a j)riori that, under certain limitations,,/ indi-
cates perpendicularity to a plane. As to those limitations the
reader is referred to pp. 44, 45 of this volume. The geome-
trical illustration given at the latter of those pages will be more
correct if we suppose the small sphere to be moved, parallel
to itself and perpendicular to its axis, until its pole is at a di-
stance unity from its former position. The reader who is in-
terested in the subject of the impossible quantity j is referred
to my papers at pp. 435-439 of the last, and pp. 37-47 of the
present volume of this Journal. For distinctness of reference
I have in this paper used / andj instead of a and /3. I have
not thought it necessary to refer to the case where k (or y)
enters into a geometrical problem, as it was beyond my pre-
sent object.

Nb/^. The value of Wjix.'^ + 23/1/^ {supra^ p. 47) should be

tt) W -f y.r" + t/y" + 2/2;".

The omission of accents has occasioned the error.

2 Church -Yard Court, Temple,
January 16, 1849.

Correction. Supra, p. 42, note*, line 6,

for " and then n " read when the index.

XIX. Proceedings of Learned Societies.

CAMBRIDGE PHILOSOPHICAL SOCIETY.
[Continued from vol. xxxiii. p. .394.]

Nov. 13, QECOND Memoir on the Fundamental Antithesis of
1848. »^ Philosophy. By W. Whewell, D.D.
This memoir is a continuation of a former one in which the anti-
thesis of thoughts and things, of ideas and facts, of subjective and
objective, were shown to be at bottom the same antithesis, and to be
a fundamental antithesis, the union of the two elements entering
into all knowledge, and their separation being the test of all philo-

1 36 Cambridge Philosophical Socictjj,

eophy. The present memoir is employed in illustrating the proposi-
tion that the progress of science consists in the transfer of some truth
from the factorial to the ideal side of the antithesis, or as it may be
termed, in the idealization of facts. This is exemplified in mecha-
nics, astronomy, botany and chemistry.

In a note, the author remarks on certain German systems of phi-
losophy with reference to this antithesis. The Sensatorial school
having reduced all knowledge to facts, Kant re-established the neces-
sity of Ideas, which Fichte made almost the exclusive element. Schel-
ling founded his philosophy upon the absolute, from which he derives
both facts and ideas, but which a wiser philosophy shows us that we
can never reach ; and Hegel took the same foundation, but in a cer-
tain degree rightly pointed out that the progress towards the identity
of fact and idea is to be traced in the history of science ; which view,
however, he has carried into detail by rash and blind conjecture.

Nov. 27. — On a Difficulty suggested by Professor Challis in the
Theory of Sound. By Robert Moon.

In a paper by Professor Challis contained in the Supplementary
Number of the 32nd Volume of the Philosophical Magazine, I find
the following : —

" The difficulty respecting the augmentation of the velocity of
sound by the development of heat, cannot be so summarily disposed
of as Mr. Airy appears to imagine. I shall perhaps succeed better
in conveying my meaning by using symbols. If S be the tempera-
ture where the pressure is p and density p, and fli the temperature in
the quiescent state of the fluid, we have, by a known equation,

;j=a2^(H-a.9-9i).

Hence

d'^z dp a-dp c ft t\^P 0 d& ., v

— = — _£- = J:_a2a(9— 9j— L — a^a—-. . (1.)

dt"^ pdz pdz paz dz

" The usual theory explains how the third term of the right-hand
side of this equation may be in a given ratio to the first ; but my
difficulty is to conceive how the same can be the case also with the
second term, since it changes sign with the change of sign of 0 — fl,."

I conceive that the explanation, according to the usual theory to
which Professor Challis here alludes, depends upon the principle,
" that for very small condensations of air, the rise of temperature will
be proportional to the increase of density." (Vide Herschel On
Sound, Encyc. Met., art. 72.) Thus we may put

where A is a constant, and 1 is put for the density of equilibrium :
on which hypothesis it is obvious that the third term of equation (1.)
will be a multiple of the first, as described by Prof. Challis. It also
follows that the second term vanishes, since it has (1 — p) for a fac-
tor, and in reducing (1.) to the ordinary form of the differential
equation of sound the difference between p and 1 is neglected. It
thus, I think, appears that the difficulty suggested by Prof. Challis
has no real existence.

Cambridge Philosophical Society. 137

Dec. 11. — On the Formation of the Central Spot of Newton's
Rings beyond the Critical Angle. By G. G. Stokes, M.A., Fellow
of Pembroke College, Cambridge.

It has long been known that when Newton's rings are formed be-
tween the under surface of a prism and the upper surface of a lens,
or of a second prism, so as to allow of increasing the angle of inci-
dence at pleasure, the rings disappear when the critical angle is
passed, but the central spot remains. The existence of the spot
under these circumstances has even been attributed to the disturbance
in the second medium, which, when the angle of incidence exceeds
the critical angle, takes the place of that disturbance which at a
smaller incidence constitutes the refracted light ; but the expression
for the intensity has not hitherto been given, so far as the author is
aware. The object of the author in the present paper is to supply
this deficiency.

The author has not adopted any particular dynamical theory, but
has deduced his results from Fresnel's expressions for the intensities
of reflected and refracted polarized light. When the angle of inci-
dence becomes greater than the critical angle these expressions be-
come imaginary. "When the imaginary expressions are interpreted
in the way in which physical considerations show that they must be
interpreted, it becomes easy to obtain the expression for the intensity
of the light, whether reflected or transmitted, in the neighbourhood
of the spot. When the first and third media are of the same nature,
the following expression is obtained for the intensity (I) of the re-
flected light, the incident light being polarized in the plane of inci-
dence, and its intensity being taken for unity.

In this expression jw, is the refractive index of the fii*st medium, i the
angle of incidence on the surface of the second medium, or inter-
posed plate of air, D the thickness of that plate at the point con-
sidered, A the length of a wave in air, 2 9 the acceleration of phase
due to total internal reflexion. When the light is polarized perpen-
dicularly to the plane of incidence, it is only necessary to replace 2 fl
by 2 f , the angles 9, ^ being those so denoted in Airy's Tract. The
intensity of the transmitted light is obtained by subtracting that of
the reflected light from unity.

From the expression for the intensity, the author has deduced the
following results, all of which he has verified by observation.

The spot is comparatively large near the critical angle, and becomes
smaller and smaller as the angle of incidence increases. Near the
critical angle the fainter portion, or ragged edge, of the bright spot
seen by transmission is broad ; at considerable angles of incidence
the light decreases with comparative abruptness. Towards the edge
of the spot there is a predominance of the colours at the red end of
the spectrum, causing the ragged edge to appear brown. Near the
critical angle the spot is larger for light polarized perpendicularly to
the plane of incidence than for light polarized in that plane : at con-

1 38 Royal Astronomical Society.

siderable angles of incidence the order of magnitude is reversed.
The difference is far more conspicuous in the former case than in the
latter, and in that case consists principally in the greater extent of
the ragged edge. When the incident light is polarized at an azimuth
of 45°, or thereabouts, and the transmitted light is analysed so as to
extinguish the light transmitted near the point of contact, there is
seen a central dark patch surrounded by a luminous ring.

ROYAL ASTRONOMICAL SOCIETY.

[Continued from p. 72.]

Nov. 10, 1848. — The Method in Use at the Cambridge Observa-
tory of Measuring Diiferences of Right Ascension and North Polar
Distance by an Equatoreal provided w^ith Clock-movement, and of
Correcting the Observations for Refraction. By Professor Challis.

" Diiferences of north polar distance are usually measured by the
Northumberland equatoreal, by means of a small sector of a large
circle, on the limb of which are inscribed equidistant divisions, se-
parated by an arbitrary but ascertained interval. A similar sector
can be clamped to any part of the hour-circle, and differences of
right ascension measured in an analogous manner. This is effected
by an arrangement contrived by Mr. Airy (who contemplated the
kind of observation here described), by which the instrument may
be moved about its polar axis independently of the hour-circle, while
the latter is carried nearly at the rate of sidereal time by a clock.
The hour-circle sector has been substituted for the hour-circle itself,
because the divisions of the latter are on brass, and not so well-
adapted for accurate bisection as those of the sector, which are on
white metal ; and because the equidistance of the divisions, which is
the essential condition, is more likely to be secured in a small por-
tion of a circle than in a complete circle. The intervals of both sec-
tors are subdivided by microscope-micrometers. The following is
the method of taking the observations.

" It is generally required, and always desirable, to measure simul-
taneously differences of right ascension and north polar distance.
Accordingly the object is bisected by the equatoreally adjusted wire,
very near the transverse wire, so that the rate of the clock, gaining
or losing as the case may be, soon brings it upon the latter wire, the
observer taking care in the meantime that it remains bisected by the
other. The instant of simultaneous bisection by the two wires is noted,
and the microscope-micrometers of the two sectors are then read off
in integral intervals and revolutions, and parts of a revolution. This
process is commenced with the star, or point of reference ; the object
referred is next observed in the same manner, and so on alternately,
the series concluding with the reference star. In case the compared
object be too faint for observation with micrometer wires, the prac-
tice with the Cambridge equatoreal is to use a diaphragm bounded
by straight edges at right angles to each other, and the object being
placed near the angular point in the prolongation of the edge which
is equatoreally adjusted, the instant at which its centre is brought

Rm^al Astronomical Society. 1 39

into coincidence with the angular point by the clock's rate is noted.
In other respects the operation is the same as that just described.
The chronometer is compared with the transit-clock at the end of
the series (sometimes, also, before its commencement), and finally
the barometer and thermometer are read off.

" With respect to the reduction of the observations, the chief
things to remark upon are the corrections for the clock's rate, and
for refraction. The differences of the hour-circle sector-readings for
the star are entirely due to these two causes, if the instrument be
supposed to be in good adjustment. The star being known, and the
times of bisection known, the effects of refraction on the hour-angles
are calculated for each observation of the star, by a process which
will be presently stated. Corrections for refraction being applied to
the hour-circle sector-readings for the star, the remaining differences
are due to the clock's rate, and by comparison with the times of bi-
section, determine the rate. The correction for rate of hour-circle
is a part of the loss or gain in the interval between consecutive bisec-
tions of the star, which bears the same ratio to the whole, as the
interval from either bisection of the star to the bisection of the planet
or comet bears to the interval between the two bisections of the star.
ITie following is the formula for this correction, the sidereal times
of the three bisections, in the order of their occurrence, being s,, tr,
5,2 ; H being the excess of the hour-circle sector-reading for the star
at 5, above the reading at s^ converted into time, and R the excess
of the correction for refraction in hour-angle for the star at s, above
that at Sjj : —

Correction for rate of hour-circle = ^ (H-f-R).

Sg — Sx

This formula gives the quantity to be added to the algebraic excess
of the sector-reading for the comet or planet, above that sector-read-
ing for the star which was taken at the time s,, and is sufficient for
all cases.

" It is to be remarked, that if the difference of the sector-readings
be affected by any other source of error acting proportionally to the
time, as, for instance, want of adjustment of the instrument, such
error is eliminated by the above calculation. For this reason, to
ensure greater accuracy, the excess of the reading of the declination
sector for the compared object, above that for the star at the time Sj,
is also corrected by the process just indicated, although that excess
is unaffected by the clock's rate. The formula for this purpose is
precisely the same as that given above ; H, in this case, representing
the excess of the declination sector- reading for the star at s^ above
the reading at 5.^, converted into arc ; and R the excess of the correc-
tion for refraction in north polar distance for the star at s^ above
that at ^2.

" After applying the corrections now considered, it is presumed
that the instrumental measures of differences of apparent right ascen-
sion and north polar distance are affected only by refraction. The
total refractions for the star in R.A. and N.P.D. have been already
required, and therefore the obvious course is to calculate also the

140 Royal Astronomical Society,

total refractions for the planet or comet, and thence deduce the dif-
ferences of refraction corresponding to the measured differences of
11. A. and N.P.D. It may be questioned whether any approximate
formulae, requiring only the calculation of differences of refraction,
would lead to a less amount of calculation in this kind of observation.
If P be the pole of the heavens, Z the zenith of the observer, S the
place of the object, and ZQ be dravvn a perpendicular on PS, the
formula used for the total corrections for refraction in R.A. and
N.P.D. are the following : —
Correction for refraction in N.P.D. = A. tan (PS— PQ)

Correction for refraction in R.A. = A. — tf^* cosec PS. sec (PS — PQ).

15

The factor A is given by the tables in Bessel's Astronomische Unter-
suchungen, vol, i. pp. 198, 199, the argument in. the case of the star
being the true zenith distance, which is obtained by the formula
sec ZS = sec Q,Z sec (PS— PQ). The argument in the case of the
compared object is the apparent zenith distance, which is deduced
from the same formula, the apparent N.P.D. and hour-angle being
first obtained by applying the corrections for refraction in N.P.D.
and R.A. of the star (with signs changed) to its true N.P.D. and R.A.,
together with the measured differences of N.P.D. and R.A. affected
only by refraction.

" The above calculations will be much facilitated by two tables,

one containing the values of PQ, log sec QZ, and log — — — (to

five figures) for every minute of hour-angle from 0^ to 6^, which will
be found to require interpolations only to first differences, and which
is, in fact, merely an expansion of the table mentioned in the Monthly
Notices, vol. viii. No. 9, p. 210. The other is a table for obtaining
the factor A. It will save much trouble, and be sufficiently accurate
to take account of the barometer and thermometer by the empirical
formula given in the Monthly Notice above cited, viz.
logA = log/c+0-015B4-0-001(100°-T),
in which log k is log a or log aJ of Bessel, according as the argu-
ment is the true or the apparent zenith distance, diminished by the
constant 0-49572. Any error which the use of this formula induces,
will very nearly disappear in the differences of the refractions. Thus
the second table need merely consist of values of log a— 0*49572,
and log a'— 0*49572 ; and the most convenient argument is log sec
ZS, the consecutive logs differing by 0"01. This table would, there-
fore, very well range with the table of values of log a" — 0*4957, re-
quired in the computation of differential refractions."

The Astronomer Royal gave a description of the gigantic tele-
scope erected by the Earl of Rosse, at Birr Castle, which he visited
and carefully examined this autumn. The mode of grinding and
polishing the speculum, the mounting, &c. were fully described and
illustrated by models, and the residual difficulties stated. He also
exhibited models of Mr. Lassell's grinding and polishing machine,
and of the mounted instrument, dome, &c. It was clearly shown

Royal Astronomical Society. 1 4- 1

that, though pursuing different courses, the Earl of Rosse and Mr.
Lassell had each attained almost absolute perfection in figuring and
polishing their specula, and that the difficulties in mounting, &c.
were gradually being overcome by Lord Rosse, while they were
already nearly got rid of by Mr. Lassell in his comparatively small
instrument.

Mr. Drew, who has lately built and furnished a very convenient
observatory at Southampton, adopts a collimating telescope for get-
ting rid of his error of collimation. To this latter telescope he has
attached a wire micrometer, which supplies the object to be viewed
by the transit. He also uses the wire-micrometer to measure the
intervals of his wires. The results are more readily obtained than
by slow moving stars, and he conceives with at least equal accuracy.
Specimens of the determination of the intervals by both methods are
given, which agree very nearly.

Dec. 8, 1848.— Transit of Mercury, Nov. 8-9, 1848. By the
Rev, W. R. Dawes, at Cranbrook.

" My attention was directed principally to the appearance of the
planet at its ingress, and to measurements of its diameter during the
transit.

" The ingress was observed with my 8^-foot achromatic, the aper-
ture being limited to 4 inches, the eye-piece magnifying eighty-
seven times. So extremely undulating was the edge of the sun in
general, that no advantage seemed to arise from an increase of power.
Nothing remarkable was noticed till Mercury had advanced on the
sun's disc to about three-quarters of its jjwn diameter, when the
cusps appeared much rounded off, giving a pear-shaped appearance to
the jjlanet. The degree of this deformity, however, varied with the
steadiness and definition of the sun's edge, being least when the de-
finition was best. A few seconds before the complete entrance of
the planet, the sun's edge became much more steady, and the cusps
sharper, though still occasionally a little broken towards their points
by the undulations. At the instant of their junction the definition
was pretty good, and they formed the finest conceivable line. Mercury
appearing at the same time perfectly round.

" The impression upon my mind was, that the distortion of the
planet arose entirely from the rounding off of the points of the cusps
by the tremor and diffusion of the image. I have repeatedly ob-
served precisely the same appearance at the ingress and egress of
the shadow of a satellite of Jupiter, when the edge of the planet has
been rather undulating and diffused.

" For the measurement of the diameter of Mercury I had prepared
several different instruments. The filar micrometer was applied to
the 8^ foot equatorially mounted achromatic, the clock motion being
in use. A 5-foot achromatic by DoUond was furnished with one
of his spherical crystal double-image micrometers, and mounted on
a very stout floor-stand with an equatorial socket. An excellent
Gregorian reflector, of 5 inches aperture and 20 inches focus (the
large metal figured by Cuthbert), and furnished with its own di-
vided object-glass heliometer, was also employed. And lastly, j^ij..

14fi Royal Astronomical Society.

spherico- prismatic crystal double-image micrometer was applied to
the 8^-foot equatoreal. Measurements were obtained with each of
these instruments ; but from the excessive tremor which usually
affected the image, the results were not very satisfactory.

With the 8i-foot equatoreal and filar micrometer, power 163, aperture

reduced to 2"84 inches,

Polar diameter of Mercury =:9"'3694, six observations.
Same instrument and power, aperture 4'02 inches,

Polar diameter =:9"'38!)0, six observations.

The mean of the two sets =9"-393.
Same telescope, and spherico-prismatic micrometer, power 184,

Polar diameter =8"'89, three observations.
With the 5-foot achromatic and the spherical micrometer, power 117,

Polar diameter, by four observations =:9""02 "1 yen nf/.Qi

Equatoreal diameter, by two do. =9"'36j * '

With the heliometer on the 20-inch Gregorian, power 115,

Polar diameter, by four observations =8""89 \ ,•«. A'^Q1

Equatoreal diameter, by ten do. =9'''20j '

" No difference is recognised in the Nautical Almanac between the
polar and equatoreal diameters of this planet ; yet my observations,
both with the 5-foot achromatic and the Gregorian, show a percep-
tible difference, and nearly to the same amount. And it was noticed
with each of the double-image micrometers that a satisfactory mea-
sure of the equatoreal diameter was always perceptibly too large for
the polar diameter, the images appearing slightly separated ; and that,
on the contrary, with a gQpd measure of the polar diameter, the images
overlapped when placed in the direction of the equator. The change
was repeatedly made from one to the other, and always with the
same result. The compression would thus appear to be about -^.

" It will be remarked that no sensible difference was produced in
the apparent diameter by varying the aperture from 4"02 inches to
2'84 inches. The same darkening glass was employed with both
apertures ; and therefore, though the telescopic irradiation would be
least with the larger aperture, yet, the image being brighter with
that aperture, the ocular irradiation would be greater. Probably,
therefore, the two effects might counteract each other.

" The measurements, though few, were taken with extreme care,
each of them having been repeatedly examined under the best views
before it was read off."

By Mr. T. Dell, at Dr. Lee's Observatory, Hartwell.

" The time was taken from the transit-clock, the error of which
was well known from observations on the 7th and 8th. The first
contact was not noted with any degree of certainty ; the interior
contact was well- observed.
Interior contact 14''18'"55'-3sid. time, or 23*" 3" 57' mean time at Hartwell.

" My attention was directed by the Rev, Mr. Reade to a phseno-
menon described by the late Professor Moll (Mem. Ast. Soc. vol. vi.
p. 116), a recurrence of which we all observed, — Mr. Reade and his
assistant, with a Gregorian telescope, at Stone, and again with me

Royal Astronomical Society. l-l-S

here. This is a considerable grayish spot on the disc of Mercury,
very indefinite, but gradually shading off from the brightest point in
the centre to the blackness of the rest of the planet. I have at-
tempted to give some idea of this appearance in the drawing annexed,
as seen with a power of 240 ; with a less power we could not di-
stinguish it."

By the Rev. Mr. Reade, at his observatory, Stone.

Mr. Reade has sent a drawing of the gray spot observed in Mer-
cury, which agrees with Mr. Dell's. The observations consist of a
numerous series of angles measured from Mercury to spots on the sun,
from which M. Fazell has made an elaborate chart of the path of
Mercury over the sun's disc.

By Mr. Hartnup, at the Observatory, Liverpool.

Equatoreal, 8^ inch achromatic ; power, 134.

Internal contact 23** 6™ 54*-4 Greenwich mean time.

"The instant is noted at which the sun's light was first seen to
surround the planet completely."

Description of a Machine for Polishing Specula. By Mr. Lassell.

" The twelfth volume of the Memoirs of the Royal Astronomical
Society contains a description of a Newtonian Reflecting Telescope,
of 9 inches aperture and 112 inches focus, equatoreally mounted in
a revolving dome of 14^ feet diameter.

" Several years' experience in the use of this instrument so well
convinced me of its general efficiency, and especially of the con-
venience and stability of its mounting, that I determined, two or
three years ago, to carry out precisely the same principle on a much
larger scale.

" With a view of informing myself what degree of perfection is
attainable in figuring surfaces of larger mirrors than can be wrought
by hand, and also of ascertaining the proportion of aperture to focus
which it would be most desirable to adopt, I visited Birr Castle ;
and, by the kindness of the Earl of Rosse, enjoyed the opportunity
of two nights' observations with the 3-foot telescope erected by his
lordship.

" I was also favoured with an examination of the whole of the
machinery employed in grinding and polishing the great speculum :
and I returned so well satisfied with all I had seen, that I very
shortly resolved to cast a speculum of 2 feet diameter and 20 feet
focus.

"The mode of casting the large speculum which I employed
involved the principle, discovered, I believe, and first published, by
Lord Rosse, of casting the speculum on what is technically called a
chill, i. e. an iron base, slightly warmed, which causes the speculum
to cool upwards in horizontal strata.

" Principally, however, from the difficulty of forming it, I did not
employ a base constructed with iron hoops placed edgewise, and
turned to the gauge, as Lord Rosse recommends, but, instead of it,
a disc of cast iron, with its upper surface convex, according to the
required radius of curvature, and a rebate formed on the edge of its

144> Royal Astronomical Societr/.

upper surface, which, receiving a stout iron hoop equal in hreadth
to the thickness of the speculum, formed an iron mould, and dispensed
altogether with the use of sand in the casting. The disc does not
require to be turned, but if cast from a well-made wooden pattern
will be sufficiently true ; neither do I think turning the hoop essential,
though it might be well to turn the inside surface and the edges, if
the means of doing so were at hand.

"As it is necessary that the pouring should be pretty quick, in
order that there may not be time for the base to solidify any ])ortion
of the metal before it is completely covered, I inclined the base a
little, pouring on the lowest side, in order that the fluid might rise
in one compact wave ; and when the disc was nearly covered, it was
restored to a truly horizontal position, and the pouring continued,
until the mould was sufficiently filled, namely, to the depth of about
two inches and three quarters. The hoop was about three inches
broad, and having been turned parallel, the mould was in the first
instance placed horizontal, by a spirit-level being placed upon its
edge. The inclination was produced by the application of a lever,
which, when withdrawn, restored the base to its horizontal position,
and ensured the equable thickness of the speculum at every part of
its circumference."

Mr. Lassell then describes the very ingenious method which he
adopted to procure the requisite quantity of metal in the proper state,
and his mode of ascertaining that the dose of tin was sufficient. The
final proportion which he used is 32 lbs. of copper to 15*09 lbs. of
grain tin, and 18 lbs. of white arsenic were stirred up with 438 lbs.
of the melted mixed metal.

" The speculum was ground and polished on a machine almost
precisely the same as that described by Lord Rosso in his lordship's
very interesting paper, published in the second part of the Philoso-
phical Transactions for 1840.

" I found, however, the grinding process much facilitated by in-
terposing a piece of sheet-lead, about a tenth or twelfth of an inch
thick, between the speculum and the iron grinding-tool. This saved
the rapid wearing down of the tool and also cut the metal much
faster, as the softness of the lead suffered the particles of emery to
imbed themselves into it, and thus to form a very keen grinding
surface. When the lead, fully charged with the emery, had become
smooth, it was exchanged for a fresh piece. When an entire sur-
face had been obtained upon the speculum, the smoothing and per-
fecting of the surface previous to polishing was produced by the iron
tool and the finest washed emery.

" The speculum was polished many times on the same machine,
following as nearly as practicable the directions given by Lord Rosse ;
but, after several months' trial, I did not succeed in obtaining a
figure which satisfied me, the best I got being very inferior to the
surfaces I had obtained by hand on specula of various sizes, up to
nine inches diameter. In despair of success by this process, I
ultimately contrived a machine, in which I endeavoured to represent
as closely as possible the evolutions of the hand, by which I had

Royal Astronomical Society. liS

been accustomed to produce very satisfactory surfaces on smaller
specula."

The machine invented by Mr. Lassell, and constructed by Mr.
Nasmyth, for figuring and polishing specula, cannot be made intel-
ligible without figures*. The speculum rests with its face uppermost
in a horizontal position, and is carried slowly round by a vertical
axis. The polisher rests, with its grinding and polishing surface,
upon the speculum, and is moved by a pin which fits loosely into a
hole in the centre, at the back of the polisher. The motion of the
polisher is that of the driving-pin.

Now this, by a very ingenious and very compact mechanism, re-
ceives a compound motion which may be thus imagined. Conceive
a circular motion given to a point round the centre of the speculum,
and then conceive that the driving-pin has a circular motion round
this point. The curve is an epitrochoid, and the adjustments of the
mechanism enable the workman to give any radius to either circular
motion, from 0 up to a certain number of inches. The proportions
of these radii, in order to give a parabolic figure, are determined ex-
perimentally, in which the relation of aperture to focal length must
be considered. The size of the polisher, and even the hardness of
the pitch, must also be proportioned to the figure and aperture re-
quired. Mr. Lassell finds no difficulty in getting a true parabolic
figure when the aperture is one-eighth of the focal length f. The
speculum, while grinding and polishing, is supported in the same way
as it is in the tube when in use. The principle of this mode of sup-
port is mentioned by Lord Rosse, Philosophical Transactions, 1840,
p. 524.

"The polisher should possess as much stifl'ness as is compatible
with the requisite lightness, and I have found these qualities best
combined by making it of white American deal, in two strata, well-
united by glue and a few screws, with the direction of the grain at
right angles, the wood well-seasoned, and, if possible, cut out of
the same board. The polisher for the 2-foot speculum is made out
of \\ inch board, and has, for symmetry, both the upper and under
surfaces convex, to fit the speculum. It is about 2 inches thick at
the circumference, '2Q\ inches diameter, and weighs about 12 lbs.
with the pitch surface upon it."

Mr. Lassell then enters very minutely into the mode of coating
the polisher with pitch evenly and to a proper thickness, of dividing

* A model to half the true size, and tiie drawing by Mr. Nasmyth, may
be seen at the Society's apartments. A model of Lord Rosse's engine, and
of the mounting, &c. of the B-foot reflector at Birr Castle, may also be
seen. These were made by Mr. Airy, and |)resented by him to tlie Societj'.
Mr. Williams will explain the action and details of all the models to any
fellow who wishes for information.

\ Mr. Lassell has given so full an account of all his processes that we
conceive any person of ordinary intelligence would be able to execute
them ; but they do not admit of compression, and extend beyond the limits
of a Monthly Notice. It ought to be mentioned that the Eail of Rosse
and Mr. Lassell have at all times freely conmiunicated the steps of progress
as soon as these became evident to themselves.

Phil. Mas, S. 3. Vol. 34. No. 227. Feb, 1849. L

146 Royal Astronomical Society.

the surface into equal squares, and the various manipulations which
are required to produce a perfect result. The grinding powder is
known as rouge, and the best quality may be had from Mr. Fox of
Saffron Hill.

" The whole time occupied in obtaining the requisite lustre varies
from about one hour to three, and it ought to be steadily advancing
throughout.

" A good idea may be formed of the quality of the operation as it
proceeds by watching the motion of the tool. It should be regular
and uniform, without any apparent labouring or inequality of speed,
and the spontaneous motion which the tool has upon the pin as a
centre should be slow and regular. No firm adhesion is ever to be
allowed between the tool and speculum : this will take place if a due
and regular supply of water be not afforded.

" A second application of powder will rarely be required, and never
in any quantity, but many applications of water probably will, and
the more rapidly the polish is advancing the more frequently will
water be required. It is best applied through a hole in the back of
the polisher as near the centre as is convenient, which may perhaps
be at about the distance of one-third of the radius. But care should
be taken not to give the water in excess. The speculum must never
be dry, but there must be no superfluous water. It is very conve-
niently applied with a flat camel's-hair brush, half or three-quarters
of an inch broad ; but as much as the brush would take up would
generally be too much for one application. Towards the end, the
water should be added more sparingly, and if needful more fre-
quently, going as near to dryness as may be but never reaching it.

" The lustre in this state of the process advances most rapidly.
If the process has gone on well the powder will have become almost
black at the close. The machine having been stopped, the tool is
to be carefully taken off by a sliding motion, and the speculum may
then be cleaned with a soft linen cloth or leather ; or it may be
washed with a soft sponge and water, and then dried, and ultimately
rubbed lightly with some very soft wash-leather. If the polishing
has apparently wrought smoothly, and the aspect of the tool when
taken off, both during the process and at its close, is everywhere of
even texture when viewed by an oblique light, the speculum will
most likely have a uniform curve of some description, whether para-
bolic or not, for it is a characteristic quality of this machine gene-
rally to produce a uniform curve. The quality of the curve is best
examined by placing the mirror in its tube, and, by means of dia-
phragms, exposing separate portions of the mirror of equal area
from the centre to the circumference.

" I have been accustomed to produce by hand surfaces of, I believe,
great excellence, on various sized specula up to nine inches diameter,
of which I may instance my 9-foot equatoreal, which enabled me to
discover independently (for I did not previously know of its existence)
the sixth star in the trapezium of Orion, and with which also the
observations of a second divison of the ring of Saturn were made, as
described in the Astronomical Notices, vol. vi. p. 11. Such sur-

Royal Astronomical Society. 147

faces as these were, however, produced with some degree of anxiety,
much manual labour, and perhaps some admixture of accident, espe-
cially in the union of a perfectly parabolic curve with regularity of
surface. The superiority of the machine in these respects is so
striking as almost to put comparison out of the question.

" If driven by a steam-engine the manual labour is of course an-
nihilated. The control over the machine, by the setting of the
cranks, is such, at least with all foci not less than eight diameters
of the speculum, that the curve can be changed almost at pleasure
from the spherical side to the hyperbolic side of the parabola, and
vice versd ; the alterations of the curve being, ceteris paribus, almost
exactly commensurate with the adjustments of the cranks. In fact,
one of the most anxious and laborious operations is, by this machine,
converted into an intensely interesting amusement. With moderate
care and a little experience a bad figure never need to be feared,
though it may require two or three successive trials to satisfy the
feistidiousness of a cultivated and long-practised eye. The lustre of
polish transcends even my best efforts by hand, and is the easiest
quality of all to obtain ; and however erroneous the figure may be
after any unsuccessful effort, the proper curve may be recovered
without resorting to the grinder,* or indeed materially impairing the
polish — at least, I have not found it needful, even when the differ-
ence of foci of the central and exterior portions of a mirror has
amounted to fifteen hundredths of an inch. In 3 or 3^ hours by the
polisher alone, it is possible to annihilate an error even as enormous
as this, I have a strong persuasion that this machine might prove
eminently serviceable in working the curves of object-glasses of large
dimensions, though of this I have no experience."

Mr. Lassell then briefly describes the mounting of the telescope,
the form, weight, and dimensions of its component parts, and the
covering dome. They are in principle almost the same as were used
on a smaller scale in his 9 -foot Newtonian. There is a very good
model of the dome and mounting, presented by Mr, Lassell, at the
apartments of the Society.

" To afford some notion of the degree of facility attained in the
management of so large a dome and telescope, I may mention, that
with an assistant I can, without hurry, place an object, invisible to
the naked eye, within the field of the telescope in nine or ten
minutes from leaving my house. This includes opening the dome,
uncovering the large speculum, attaching the eye-piece, setting
from the catalogue for the object, and turning the dome to the
required azimuth. Without an assistant, I should require three or
four minutes longer, which would be principally occupied in open-
ing the shutters of the dome.

" One of the greatest difficulties I have encountered in support-
ing the speculum in its various positions equably, is to avoid the
effects of the friction of its edge under considerable changes of alti-
tude of the telescope.

" It is obvious, that when the altitude is low, the principal part
of the weight of the speculum must be borne upon its edge, and the

L 2

148 Royal Astronomical Society.

supporting plates being thus in a great measure relieved from tlie
pressure of the speculum, must, by their elasticity, tend to distort
the metal by pressure at its back ; and when the telescope is moved
towards the zenith, the plates yield again by the weight of the spe-
culum, while the lower edge, still in hard contact at the points of
support, is unduly borne up there, and the equilibrium is destroyed.
To remedy this evil I have slung the speculum in a hoop of thin
iron, equal in length to half its circumference, the ends of the hoop
being attached to swivels fixed in each of the two horizontal brackets,
aiid the lower part of the hoop being thus quite at liberty to rise and
fall with the plates.

" This has nearly, if not entirely, removed all perceptible distor-
tion ; yet in some positions, and under some circumstances, vestiges
of it are to be perceived. I have devised a plan of supporting the
metal laterally by an equal tension on the several points of support,
and think it may probably be useful ; but I have not yet had leisure
to carry it into effect,

" Instead of a plane speculum I usually employ a prism, which
transmits a pencil of two inches in diameter, made for me by Messrs.
Merz and son, of Munich. I am persuaded, from repeated experi-
ments, that the prism has an obvious advantage in light over a spe-
culum, and the material is so fine, and the surfaces so exquisitely
wrought, that no perceptible injury of the image exists. The only
care necessary in the use of the prism is to preserve it from dew,
which it is extremely liable to collect ; this I have remedied by ha-
ving a chamber made in the mounting of the prism, which receives
a cube of cast iron enveloped in thick felt : this, being moderately
warmed and placed in the chamber, effectually prevents the deposi-
tion of dew for at least some hours, while the extremely slow radia-
tion through the felt does not produce any sensible disturbance in
the formation of the image. The prism is rather small ; for though
it transmits the entire pencil, there is scarcely anything to spare ;
and had it been easy to obtain a sufficiently good one half an inch
larger, I should have procured it."

A short notice of the Equatoreal of the Liverpool Observatory.
By Mr. Hartnup.

As the Astronomer Royal will probably give some account of this
instrument, which has been constructed on his recommendation and
entirely under his superintendence, Mr. Hartnup states, in a few
■words, that it is of the English construction ; that is, the telescope
is a transit supported at each end, between two long supports
which form the polar axis. The telescope is by Merz of Munich,
8| inches in aperture, and 12 feet focal length. The circle and
declination-circle are each 4 feet in diameter, divided by Mr.
Simms upon his "self-acting circular dividing engine*." The
hour-circle revolves independently of the instrument, and is carried

* Described in vol. xv. of the Memoirs. The new altitude and azimuth
instrument at Greenwich, which was divided on the same engine, is consi
dered by Mr. A'ry to he exceedingly well divided.

Royal Astronomical Society. 1 4-9

by clock-work, the moving power of which is a water-mill, regu-
lated by " Siemen's Chronometric Governor." This is so success-
fully applied, that the rate of the hour-circle is not sensibly altered
by clamping the polar axis to it. When the hour-circle is properly
adjusted, the instrument reads off right ascensions at once *. The
polar axis, which is of wrought iron-plate, is very massive and stiff.
The weight of the whole instrument is between 70 and 80 cwt.
This keeps all steady, even in very hard gales. The instrument is
abundantly supplied with eye-pieces and micrometers. The stiff
frame and large circles were evidently designed by Mr. Airy to
supply a peculiar power to the instrument. In ordinary mountings,
great accuracy is not to be expected when the star of reference is
more than a few minutes distant from the object compared. The
screw of the micrometer is not to be relied u])on for larger spaces,
and the circles, though sufficient for finding and identifying, are
seldom intended for accurate measures. Stars of comparison can,
indeed, generally be found which are contained in some of the special
and extended catalogues, but such stars can only be considered to be
roughly known, and in many cases fail altogether. The Liverpool
equatoreal is intended to measure hy its circles intervals of a few
degrees, with as much accuracy as the average stars of our extensive
catalogues possess, and thus to give excellent places by reference to
well-known stars.

Mr. Hartnup has made some observations to test the powers of his
equatoreal in this respect. The observations of y, a, (3 Aquilse, of
a and /3 Lyrse, of Castor and Polluxf, show satisfactorily, that within
such limits as these the instrument will measure differences of right
ascension and north polar distance almost, if not altogether, as well
as can be expected from the best meridian instruments.

Mr. Hartnup further remarks, that the instrument keeps its ad-
justments steadily, which seems to show that it is not only firm in
itself, but, also, that it rests on a sound foundation. The observa-
tions by Mr. Hartnup, of standard stars in all parts of the heavens,
are not sufficiently numerous to yield a safe estimate of the probable
error of a single independent determination, but it is evidently very
small, even for stars at 6^ from the meridian.

Mr. Bishop's Ecliptic Charts, from Observations at the South
Villa Observatory.

Our treasurer, Mr. Bishop, has lately published the first hour of
an ecliptic chart for the epoch 1825. This contains all the stars to
the 10 mag. inclusive in a zone of 6° of latitude, 3° on each side the
ecliptic. The scale is 1'2 inch to 1°, which gives a clear and open
map. The execution is very good.

In the notice which accompanies the chart Mr. Bishop says, " It
is the first of a series of twenty-four charts, which I hope to pub-
lish. . . . The discovery of planets is materially facilitated by mapping
down the stars within a few degrees on each side the ecliptic ;

* This contrivance is peculiar to the equatoreals of Cambridge and
Liverpool, and in some researches is of great convenience.

t These observations are given in detail in the accompanying memoir.

150 Royal Astronomical Society,

and it is for this purpose I have undertaken the present series of
charts. . . . The stars included in Weisse's Catalogue from Bessel's
Zones were first laid down for 1825 as points of reference. All
other stars, to the tenth magnitude inclusive, were then entered by
estimation of their positions with respect to the neighbouring mem-
bers of Weisse's Catalogue. . . . The charts for the hours of right
ascension in which the ecliptic falls beyond the declination limits of
the Berlin maps ( — 15°) are in a state of forwardness, and will be
published as soon as they are completed. They are regularly com-
pared in their present state with the heavens, so that the search for
planets and the formation of the charts are going on at the same
time. ... I take this opportunity of expressing my warmest thanks
to Mr. J. R. Hind, for the great care and indefatigable zeal he has
displayed in the formation of this chart, which, to my knowledge,
he has examined with the heavens from fifty to sixty times ; but the
success of his research, as shown by the discovery of two planets,
speaks for itself, and will, I am sure, dispose astronomers to receive
these charts with confidence."

Extract of a Letter from Lieut. Giliiss*, U.S.N.

" The computations for the longitude of Washington, from cor-
responding moon-culminations observed by me between 1838-1842,
are nearly completed. The results for 1839 and 1840 give the fol-
lowing corrections of the (hitherto received) longitude : —

8

1st Limb —5-39 by 182 comparisons.

2nd Limb — 4'41 by 74 comparisons.

Mean . . — 4'84 by 256 comparisons according to weight.
The European observatories with which the comparisons are made,
are Edinburgh, Oxford, Greenwich, Cambridge, and Hamburg : the
individual results very accordant ; those from Cambridge strikingly so.
Comparisons have also been made with the observations of Copen-
hagen, Kremsmunster, Cracow, and Wilna, which seem to show
considerable errors in the longitudes assigned to those observatories."

XX. Intelligence and Miscellaneous Articles.

ON THE EQUIVALENT OF FLUORINE. BY M. LOUYET.

TN some previous experiments the author had deduced the equiva-
■■- lent of fluorine from the quantity of sulphate of lime yielded by a
certain weight of the purest natural fluoride of calcium, and also by
artificial fluoride. As the two series of experiments agreed perfectly,
M. Louyet had presumed that the results to which they led were
sufficiently correct. Nevertheless he decided with some reserve ;
for having demonstrated that sulphuric acid did not completely de-
compose fluoride of lead, it occurred to him that this acid might act
in an analogous manner on fluoride of calcium. His doubts were
* Lieut. Giliiss became very favourably known to many members of this
Society on his visit to England a few years ago. The observations made
at Washington were published in 1846 by the order of the Senate, and
have been very freely distributed here and on the continent. They are a
proof of what may be done with moderate means by a skilful and conscien-
tious observer.

Intellwence a?id Miscellaneous Articles. 151

"b

the stronger, because he had found the equivalent of fluorine, de-
duced from the analysis of fluoride of lead, a higher number than
that obtained by the fluoride of calcium. M. Louyet had also an-
nounced his intention of studying and consequently of analysing all
the fluorides, in order to attempt a discovery of the cause of these dif-
ferences; the present notice gives the additional researches on this sub-
ject, and the author states his belief that he has decided the question.

In a previous memoir M. Louyet had fixed the equivalent of fluo-
rine at 239"81 ; but this number ought to be raised to 240 in making
calculations with 250 for the equivalent of calcium and 200 for that
of sulphur. Calculation then indicated, if this calculation was cor-
rect, that 1 gramme of fluoride of sodium should yield 1'680 grm.
of anhydrous sulphate of soda. In decomposing this fluoride by
sulphuric acid, it is extremely difficult to avoid loss. The vapours
of sulphuric acid appear to carry off very small quantities of sulphate
of soda ; besides which it is requisite to expose the crucible for a
long time to a strong red heat, in order to completely decompose
the alkaline bisulphate formed. It is not useless to dwell on this
point, in order to show that all the sources of error tend to produce
loss ; that is to say, to lessen the weight of the sulphate obtained.
In three experiments, 1 gramme of fluoride of calcium gave succes-
sively 1*686, 1*685. 1683 of anhydrous sulphate of soda. All these
figures are higher than that which calculation indicates the equiva-*
lent of fluorine to be 240.

These observations induced M. Louyet to repeat the analysis of
fluoride of calcium. A fresh series of experiments, which he con-
siders as exact as possible, gave him the following results : — In his
experiments 1 gramme of fluoride gave of sul])hate of lime, 1*742,
1-744, 1-745, 1-744, 1-7435, 1-7435, the mean being 1-7436. If
the equivalent of fluorine be taken as 237-50, that is to say, 19,
calculation shows that 1 gramme of fluoride of calcium should yield
1-74358 of sulphate of lime ; it results, therefore, from these experi-
ments, that the equivalent of fluorine is 237-50.

M. Louyet has analysed other fluorides to verify this result. Fluo-
ride of barium converted into sulphate in a manner to ensure its per-
fect decomposition, gave the following results : — 1 gramme of the
fluoride of barium gave of sulphate of barytes 1-332, 1-331, 1-330;
the equivalent of fluorine being 237-50, the calculated amount would
be 1-33090.

Lastly, the author repeated the examination of fluoride of lead,
and he discovered the cause of the diff^erences that had formerly re-
sulted between the equivalent deduced from the analysis, and that
obtained from the fluoride of calcium. He had not observed, that
on account of tjie great difference existing between the equivalents
of fluorine and lead, that the slightest error in the analysis would
lead to great diflferences in the calculated results. This being stated,
the following are the figures obtained in his last experiments : — 5
grammes of fluoride of lead gave of sulphate, 6-179, 6-178, 6-178 ;
the theoretical number is 6*1828. All these amounts are too small,
a circumstance which might readily arise from manipulation or other
causes which the author enumerates in his memoir.

152 Intelligence and Miscellaneous Articles,

In his first memoir on fluorine and the fluorides, M. Louyet, guided
by various considerations, rejected the hypothesis of Ampfere, in
which fluorine is ranged with chlorine, bromine and iodine. The
equivalent of chlorine, indicated by the later researches, strengthen
this ojjinion ; for whilst the equivalents of chlorine, bromine and
iodine, are not exactly divisible by the equivalent of hydrogen, that
of fluorine is, on the contrary, a multiple of this equivalent, which
associates it with the series of oxygen, sulphur, nitrogen, phosphorus,
arsenic and carbon. — L'Institut, Janvier 4, 1849.

SOLUBILITY OF CHLORIDE OF SILVER IN HYDROCHLORIC ACID.

M. J. Pierre states hydrochloric acid is capable of dissolving at
least l-200dth of its weight of chloride of silver ; and when diluted
with twice its weight of water, it is capable of holding l-600dth of
its weight of the chloride in solution. — Journ. de Ch. Med. ,Ja.n. 1849.

PREPARATION OF IODIDE OF ARSENIC.

M. Meurer proposes the following method of obtaining this com-
pound : — :Pass arseniuretted hydrogen gas into a solution of 4 parts
of iodine in 120 parts of alcohol until the liquid is decolorized; a
fresh quantity of iodine is then to be added, and the current of arse-
"niuretted hydrogen is to be again passed through the solution to the
same point. The liquid ought not then to become turbid ; but if a
brown turbidness should be produced, it mu.st be made to disappear
by an addition of iodine.

By spontaneous evaporation the liquor deposits microscopic hexa-
gonal tables, which, according to M. Kuhn's analysis, are iodide of
arsenic. — Journ. de Ph. et de Ch., Decembre 1848.

COMPOSITION OF THE BLACK YTTUO-COLUMBITE OF YTTERBY.

According to the analysis of M. Peretz, this mineral consists of —

Columbic acid 58'65

Yttria 21-25

Tungstic acid 0*60

Lime 7*55

Magnesia 1*40

Protoxide of uranium .... 3 "94

Protoxide of iron 6*29

Oxide of copper 0*40

100-08

The density of this mineral at an average temperature is 5*67 ; it
becomes 6'40 by calcination.

M. Rose states, on the occasion of this analysis, that an orthite
occurs at Ytterby so much resembling yttro-columbite, that it is
impossible to distinguish these minerals from each other by appear-
ance.

According to M. Rose, the columbite of Finland possesses the
same composition, density, and metallic acids as the yttro-columbite
of YXXQxby.—Ibid.

Intelligence and Miscellaneous Articles. 153

ON LIQUID PROTOXIDE OF NITIIOGEN. BY M. DUMAS.

M. Natterer of Vienna has constructed a forcing-purap for the
liquefaction of gases, by means of which carbonic acid and protoxide
of nitrogen can readily be obtained in the liquid state. Having
procured one of these instruments, and employed it more especially
for the liquefaction of the protoxide of nitrogen, I soon perceived
the necessity of using a series of indispensable precautions, but
which, once adopted, have enabled me to effect with promptitude
and security, as well as oeconomy, the liquefaction of large quantities
of protoxide of nitrogen.

As this liquid furnishes a means of producing an excessively low
temperature, and is very easily operated with, I will here briefly
point out the observations I have made. The first relates to the
principal piece of the apparatus, that is to say the reservoir. In
my opinion the Viennese manufacturer has not given it sufficient
strength. I have had it surrounded with a belt of forged iron,
capable of resisting 800 atmospheres, and very nicely made by
M. Bianchi. Moreover, I arranged things so that the reservoir
being surrounded by ice, the body of the pump was cooled un-
interruptedly by a circulation of water around it, and that even
the stem of the piston was always moistened by cold water; in
this manner there is no danger of the valve of the piston being
injured by the heat proceeding from the compressed gas, and by
its special action as a combustible gas. With these precautions,
we may compress into the reservoir in the course of two hours 200
litres of gas, of which 20 suffice to produce a pressure of 30 atmo-
spheres, about which liquefaction commences. The remainder of
the gas furnishes a liquid ; 100 litres yield 200 grms., or very nearly.
The gas should be absolutely dry in order to succeed, and likewise
as pure as possible. I prepare it from the nitrate of ammonia as
usual, and after having dried it, pass it into Macintosh bags ; a
couple of pounds of nitrate of ammonia suffices.

Once compressed, the liquid gas may be preserved for one or two
days at least in the reservoir ; the valve however is slightly injured
by it. When the stopcock of the reservoir is opened, the gas
escapes ; a portion freezes at first, but it then flows liquid ; the solid
portion resembles a mass of snow ; it melts upon the hand, and
rapidly evaporates, leaving a severe burn. The liquid portion, which
is by far the most abundant, and of which it is easy to obtain in one
operation 40 to 50 grras., being received in a glass, keeps for half
an hour, or even more, in the air.

In order to observe more readily its properties, I collected it in
open tubes, contained in vessels at the bottom of which was placed
some pumice-stone moistened with sulphuric acid. It then retains
its transparency for a very long time.

The protoxide of nitrogen is liquid, colourless, very mobile and
perfectly transparent ; each drop that falls upon the skin produces
a very painful burn. The gas, which is incessantly liberated by a
slow ebullition, possesses all the properties of the protoxide of ni-

154; Intelligence and Miscellaneous Articles.

trogen. When metals are dropped into this liquid, they produce a
noise like that of red-hot iron immersed in water. Quicksilver
causes the same noise, instantly freezes, and affords a hard brittle
mass, white like silver, which it perfectly resembles in appearance.
Potassium floats upon the liquid, and experiences no change ; the
same is the case with charcoal, sulphur, phosphorus and iodine.
Ignited charcoal floats upon the surface of the liquid, and burns
with considerable brilliancy, and frequently until the whole is con-
sumed. Ordinary sulphuric acid and concentrated nitric acid freeze
immediately, i^ther and alcohol mix with the liquid without
freezing. Water is instantly converted into ice; but it produces
such a sudden evaporation of a portion of the liquid, that it causes
suddenly a kind of explosion, which would be dangerous if merely
a few grammes of water were poured at once into the liquid. —
Comptes Hendus, Nov. 6, 1848.

ON THE URATES.

MM. Allan and Bensch have examined several of these salts.

Neutral Urate of Potash is obtained with greater facility than has
been supposed. It is prepared by saturating a cold dilute solution
of potash, free from carbonate, with uric acid diffused in water, and
then concentrating the solution by ebullition in a retort. At a cer-
tain point of concentration the salt separates in fine needles ; the
matter is allowed to remain a few minutes, the liquid is poured off
and the crystals are washed, first with weak and afterwards with
stronger alcohol.

The salt thus obtained is very soluble in water, has a strong caustic
taste, attracts carbonic acid from the air quickly, and is gradually
decomposed by boiling in water.

The crystals are anhydrous, and gave by analysis a composition
corresponding to C^ N* H^ O^, KO.

This salt is soluble in 44 parts of cold and 35 of boiling water.

Urate of Soda. — The preparation of this salt succeeded by adojjting
a process corresponding to the preceding. One part of it dissolved
in 77 parts of cold and 75 parts of boiling water.

Neutral Urate of Ammonia and Neutral Urate of Magnesia. —
Neither of these salts could be obtained. Attempts were made, but
also in vain, to prepare double salts of magnesia and ammonia, potash
or soda.

Neutral Urate of Lime. — ITiis is readily obtained by adding,
drop by drop, a neutral solution of urate of potash to a boiling solu-
tion of chloride of calcium, \mtil the precipitate, which at first re-
dissolves, begins to be permanent ; the limpid liquid is then to be
boiled for an hour ; the neutral urate is then deposited in the state
of anhydrous grains at 212° ; they contain C^ N'' H^ O^, CaO. One
part of this salt dissolves in 1500 of cold water and 1400 of boiling
water.

The acid salt of lime is more soluble than the neutral salt ; it re-
quires only 603 parts of cold and 276 of boiling water for solution.

Intelligence and Miscellaneous Articles. 1 55

Urate of Strontia. — Uric acid diffused through water is to be added
to a boiling and saturated solution of strontia, taking care that the
acid is greatly in excess. The first portions of acid are entirely dis-
solved ; but by the addition of successive portions a salt separated,
■which, examined by the microscope, appeared to be acicular and
grouped in stars. This urate of strontia contains C-^ N* H- O'^, Sr O^
+ 2aq.

The two equivalents of water are expelled at 329° F. The salt
attracts moisture rapidly from the air, and decomposes at 338°. One
part of it is soluble in 4300 parts of cold, and 1790 parts of boiling
water.

The acid salt is much more soluble : one part dissolves in 603
parts of cold, and 276 parts of boiling water.

Neutral Urate of Barytes is obtained similarly to the neutral salt
of strontia. It contains C^ N^ H'^ 0-, BaO. One part of this salt
requires 7900 parts of cold, and 1790 parts of boiling water for so-
lution.

Urate of Lead. — When a dilute solution of neutral urate of potash
is added, drop by drop, to a solution of nitrate of lead also dilute and
boiling, a yellow precipitate is at first obtained ; this is to be sepa-
rated by filtration, and a fresh portion of urate of soda [potash ?]
is to be added to the liquid. A heavy precipitate is thus obtained,
which is perfectly white and is easily washed. It is quite inso-
luble in water and alcohol. It may be heated to 320° F. without
decomposing; it appears to be an anhydrous salt composed of
C^N«H2 02PbO.

The authors did not succeed in preparing any other neutral urates.
—Ibid.

ON THE PRESENCE OF COPPER IN THE HUMAN BLOOD.
BY M. DESCHAMPS.

The author observes, that when the numerous examinations of
the question of the existence of copper in human blood which have
been published are considered, it will be found that these experi-
ments cannot be adduced either to oppose or support the existence
of copper in organized beings, because many authors forget to de-
scribe their processes of analysis, neglect to examine the precipitate
which is formed in a liquid by hydrosulphuric acid, either gaseous
or liquid, do not state the length of time which the liquid, treated
with sulphuretted hydrogen, is allowed to deposit the precipitate,
and they do not state whether they have prepared their hydrochloric
acid, whether they have analysed their distilled water and acids, and
particularly the hydrochloric acid, for the pure hydrochloric acid of
commerce almost always contains copper.

After considering the different processes which have been proposed
for the detection of metallic substances in the blood, M. Deschamps
followed a method analogous to that which he employed to extract
copper from vegetables.

The acids and distilled water which be employed contained no

156 Intelligence and Miscellaneous Articles.

metallic substance whatever. The hydrochloric acid was prepared
expressly for the purpose ; nitric acid only was sometimes employed ;
the filters were made of paper which was analysed and found to con-
tain no copper, and they were washed with concentrated nitric acid
diluted with an equal volume of distilled water. The capsules, cru-
cibles, glass rods, bottles, funnels and glasses, were washed with
aqua regia, with nitric acid, and in some cases with boiling nitric
acid.

The blood employed in these experiments weighed 162 grs.,
200 grs., 300 grs., 315 grs., 380 grs., 472 grs.; it was cautiously
evaporated to dryness in a porcelain capsule, and burnt in a porcelain
crucible ; the ash was treated with aqua regia or nitric acid ; the
solution was evaporated to get rid of the greater part of the acid,
then treated with water, filtered into a bottle, subjected to the action
of hydrosulphuric acid, and allowed to stand at least eighteen hours
that the precipitate might subside ; the liquid was filtered to separate
this : the filter after being washed with water containing a little
hydrosulphuric acid, in a small porcelain capsule, treated with a
few drops of aqua regia of nitric acid, allowed to stand, or slightly
heated till the colour of the precipitate was so modified as to possess
the colour of sulphur. The filter was washed, the liquid evaporated,
and the residue calcined and treated, after cooling, with two drops
of nitric acid ; it had all the properties of a solution of a salt of cop-
per, for ammonia rendered it blue, and the ferrocyanide of potassium
gave a reddish-brown precipitate, and lastly it deposited copper
on metallic iron.

From the facts above detailed, the author considers that the exist-
ence of copper in the blood cannot be questioned ; and he is of opi-
nion, as stated in a memoir presented to the Academy in 1848, that
vegetables take from the soil part of the copper which they contain ;
that herbivorous animals receive it from plants, and man from plants
and animals which serve him for food. — Journ. de Ph. et de Ch.,
Decembre 1848.

FORMATION OF CARBONATE OF LIME FROM THE NEUTRAL
MALATE OF LIME. BY M. DESSAIGNES.

The researches of M. Piria have proved that asparagine may be
regarded as the amide of malic acid. When it is impure and dis-
solved in water, it soon ferments, and is converted into succinate of
ammonia. It occurred to M. Dessaignes that if malic acid, or one
of its salts, was susceptible of undergoing the same kind of fermen-
tation, the relation discovered by M. Piria would receive from it a
more complete demonstration.

Neutral malate of lime, such as obtained by M. Liebig's pro-
cess from the berries of the mountain ash, was exposed to a some-
what deep stratum of water, in a vessel covered merely with paper.
This was in the autumn of 1847 ; after three months, the superna-
tant water was partly filled with a mucilaginous and unquestionably

Intelligence and Miscellatieous Articles. 157

organized product; on this and on the sides of the vessel there
were formed abundance of fine crystals of hydrated carbonate of lime.
The filtered water slightly precipitated acetate of lead. The forma-
tion of carbonate of lime and mucilage ceased as the spring advanced
and in the summer. M. Dessaignes observed beneath the malate of
lime, which diminished insensibly, the formation of a stratum of very
fine and compact prismatic crystals. This stratum was raised by
some large bubbles of gas which were given out by the malate of lime.
This mass of crystals was dissolved in hot water, precipitated by car-
bonate of soda and filtered. By this there was obtained a slightly
coloured solution, which, with the addition of alcohol and ammonia,
precipitated nitrate of lead, nitrate of silver, neutral perchlorideof iron,
and chloride of barium. The liquor was concentrated, treated with
a slight excess of hydrochloric acid, and evaporated to dryness, the
residue being repeatedly treated with boiling aether. The aethereal
solution gave by spontaneous evaporation fine crystals of an acid
which was volatilized without decomposing, burnt without residue
on platina foil, and was in fact succinic acid.
It appeared to be composed of —

Carbon 40-68

Hydrogen 5*08

Oxygen 54*24

100-00

At the sitting of the Academy of Sciences on the 2nd inst. Sir
David Brewster was elected one of the eight foreign associate mem-
bers of the National Institute of France, vacant by the death of the
celebrated chemist, M. Berzelius.

JOURNEY TO DISCOVER THE SOURCES OF THE NILE.

In our number for December last* we announced the arrival of
Dr. Bialloblotzky at Alexandria. According to letters since received
from him by Dr. Beke, he left Suez for Aden on the 22nd of Novem-
ber, by the East India Company's steam-packet " Adjdaha," by which
a free passage had been granted him by the Court of Directors ; and
at the latter place he was awaiting (Dec. 11) the arrival from Djid-
dah of a small steam-vessel with pilgrims returning from Mecca to
the Persian Gulf, by which he intended to proceed to MakuUa, on
the south coast of Arabia. He there expected to meet with an Arab
vessel to take him to Mombas, on the east coast of Africa, from which
place he would commence his journey into the interior.

Dr. Beke informs us that he has just received a letter from Cap-
tain Haines, I.N., Political Agent at Aden, dated Dec. 24, informing
him that Dr. Bialloblotzky had left that place for Makulla by the
steamer " Sir Charles Forbes," Capt. Lichfield, and it was expected
that by the end of the month he would be able to sail from Makulla
for Mombas.

• Vol. xxxiii, p. 481.

158

Intelligence and Miscellaneous Articles.

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Meteorological Observations. 159

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M

ETEOROLOGICAL OBSERVATIONS FOR DEC. 1848.

Chiswick. — December 1. Foggy and drizzly : cloudy: rain, and boisterous at
night. 2. Fine. 3. Clear: overcast: boisterous, with rain at night. 4.
Boisterous, with heavy rain : clear at night. 5. Overcast : clear : slight rain.
6. Clear : heavy clouds. 7. Rain. 8. Slight rain. 9. Very tine. 10. Clear
and very fine. 11. Foggy : cloudy. 12. Foggy ; uniformly overcast. 13. Ex-
ceedingly fine. 14. Fine. 15. Hazy: rain. 16. Drizzly : constant heavy
rain. 17. Cloudy: foggy. 18. Hazy: fine: densely overcast. 19. Foggy.

20. Hazy : clear and frosty at night. 21. Clear and frosty. 22. Frosty : clear :
frosty. 23. Foggy : hazy : sharp frost. 24. Frosty : slight haze : overcast. 25.
Hazy : cloudy. 26. Densely clouded. 27, 28. Fine. 29. Overcast. 30. Foggy :
fine : foggy. 31. Foggy : hazy : foggy at night.

Mean temperature of the month 41°*75

Mean temperature of Dec. 1 847 41 -09

Mean temperature of Dec. for the last twenty years 39 '66

Average amount of rain in Dec 1*58 inch.

Boston. — Dec. 1. Cloudy : rain a.m. and p.m. 2. Fine. 3. Fine : rain p.m.
4. Cloudy : rain p.m. 5. Cloudy : rain a.m. 6. Fine : rain a.m. 7. Rain :
rain a.m. and p.m. 8. Fine: rain p.m. 9 — 11. Fine. 12. Cloudy. 13. Fine.
14. Cloudy : rain p.m. 15. Cloudy : stormy, with rain from s.w. p.m. 16.
Cloudy: rain p.m. 17,18. Fine. 19. Rain: rain early a.m. 20. Cloudy.

21. Fine : plenty of ice this morning. 22 — 24. Fine. 25,26. Rain. 27,28.
Cloudy. 29. Fine. SO. Cloudy. 31. Cloudy : a remarkable dark day.

Apjilegarth Manse, Dumfries-shire. — Dec. 1. Frost a.m. : rain and high wind p.m.
2. Rain: sleet: high wind: lightning. 3. Snow inch deep : heavy rain p.m.
4. Very high flood : heavy rain and high wind, 5. Fair, after very wet night :
flood again. 6. Dull : drizzling : frost a.m. 7. Frost : damp and drizzly p.m.
8. Soft, moist and foggy. 9. Rain ail day : high wind p.m. 10. Fair : high wind.
11. Fairand fine. 12. Dull and foggy a.m. : rain p.m. 13. Rain a.m. : showery
all day. 14. Fair a.m. : rain and high wind p.m. 15. Fair a.m. : rain p.m., with
storm of wind. 16. Fair and fine. 17. Frost a.m.: slight showers p.m. 18,
Fair A.M. : cloudy: showery p.m. 19. Fair: fog: cleared p.m. 20. Frost:
thaw P.M. 21. Frost, hard : clear and bracing. 22. Frost very hard : clear.
23. Frost keen : clear : wind rising. 24. Frost ; high wind p.m. 25. Frost, slijjht :
thaw P.M. 26. Rain very heavy : high wind. 27. Fair and clear : threatening
frost. 28. Hard frost all day.' 29. Hard frost. 30. Frost moderate: dull.
31. Frost moderate: cloudy.

Mean temperature of the month 39®*8

Mean temperature of Dec. 1847 40 '2

Mean temperature of Dec. for the last twenty- five years . 38 '2

Average amount of rain in Dec. for twenty years 2*94 inches.

Sandwick Manse, Orkney. — Dec. 1. Hoar-frost: rain. 2. Cloudy. 3. Rain:
cloudy. 4. Showers: thunder: hail-showers. 5. Hoar-frost : showers. 6. Bright:
showers. 7. Bright : clear. 8. Showers : rain. 9. Cloudy : rain. 10. Hazy :
rain: clear. 11. Cloudy: clear. 12. Cloudy: rain. 13. Bright: showers.
14. Cloudy. 15. Bright : rain. 16. Showers: clear. 17. Showers : cloudy.
18. Cloudy. 19. Bright: clear. 20. Cloudy. 21. Bright: clear: aurora.

22. Clear : frost : clear : aurora. 23. Clear : frost : clear. 24, 25. Cloudy. 26.
Rain : cloudy. 27. Showers : clear. 28, 29. Clear : frost : clear. 30, 31. Cloudy,

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[THIRD SERIES.]
MARCH 1849.

XXI. Hints toiscards a Classification of Colours.
By Professor J. D. Forbes*.

TO classify and describe colours is not so easy a matter as
it might at first sight appear to be; and yet it is of very
considerable interest, as well scientific as artistical. It is im-
possible that this shall be completely done until we understand
far better than at present the cause of the colours of bodies,
natural or artificial. Had Newton's explanation of these
colours been as satisfactory as it was for the phagnomena of
the solar spectrum, the classification of colours would be more
simple and obvious than it is. My present object is to treat
the subject not at all as a matter of art, neither as to the effect
which colours produce in painting as an imitative art, nor as
to the chemical art of producing and combining pigments ; but
simply as a matter of description and nomenclature, so that
the objective effects of varied lints and hues may be referred
to some standard classification of colours and their modifica-
tions. I shall then state what progress I have made in form-
ing such a stantlard suite of colours.

I am unable, and have not the intention, to give a complete
history of the principles and methods adopted at different
times for classifying and compounding colours for the use of
the painter, and in imitation of natural hues which probably
exhaust all which art can succeed in producing. For what I
shall now state I am in part indebted to the ingenious essay
of the profound Lambert, called Beschreibung einer Ausge-
mahlten Farbenpyramide-\.

Pliny, in describing the pigments used by the most famous
painters in their pictures, mentions four, which, according to

* Communicated by the Author having been read before the Royal
Society of Edinburgh, December 4, 1848, and January 15, 1849.

t 4to. BerUn, 1772.
Phil, Mag, S. 3. Vol. 3*. No. 228. March 1 849. M

162 Prof. J. D. Forbes on the Classification of Colours.

Lambert, were white, ochre-yellow*, red and black; and he
supposes that a bluish tint may have been obtained by diluting
the black with white. Leonardo da Vinci, himself a painter
of the first order, appears to have had clear ideas on the sub-
ject of the formation of compound colours from their simplest
elements; and although he reckons six, namely white, yellow,
green, blue, red and black, yet it is scarcely to be supposed
that he was not aware that green could be compounded of
yellow and blue, and therefore we may probably admit that
he regarded blue, red and yellow as the primary colours, to
be mixed with white or black according to the degree of sha-
dow in which they are to be represented.

Waller, in the Philosophical Transactions for 1686, attempts
a classification of colours and pigments, proceeding upon the
basis (which ajipears to be there assumed as generally ad-
mitted) that red, yellow and blue are the sole primary colours.
This paper was originally accompanied by a diagram of hues,
which is, however, wanting in the copies which I have ex-
amined.

Newton's discovery of the composition of white light was,
of course, an important step in the theory of the composition
of colours generally ; although that apparent paradox did not
fail to introduce some difficulties into the explanation of the
action of pigments, which have not unnaturally affected the
views of persons accustomed to regard the subject solely as one
of art, and at the same time to complicate it somewhat by the
introduction of seven colours in the complete spectrum, red,
orange, yellow, green, blue, indigo and violet. Perhaps it is
not too presumptuous to say, that but for some peculiar re-
spect for the number seven, and more particularly from a
fancied analogy between the spaces occupied by the colours
and the musical intervals, Newton would not have classed blue
and indigo as distinct colours; in which case we may consider
that the Newtonian spectrum consists of the three primary
colours, red, yellow and blue, and the three secondary, orange,
green and purple. Newton was perfectly aware that by com-
bining the primary colours, such as yellow and blue together,
a green, not distinguishable from that of the spectrum except
hy its refrangibilityf, will be formed ; and he also observed
the effect of combining three or more coloured rays, which
generally tend to a more or less perfect whiteness, though it
does not appear that Newton ever actually formed white light
by the partial combination of certain rays of the spectrum.

* SUaceus, the word translated ochre-yellow, is of very doubtful signifi-
cation.

t Optics, Book I. part 2, prop. iv.

Prof. J. D. Forbes on the Classification of Colours. 163

For he slates (Prop. VI. p. 136, edit. 1730) expressly, "I
could never yet by mixing only two primary colours produce
a perfect white. Whetlier it may be compounded of a mixture
of three taken at equal distances in the circumference [of the
figure of a circular spectrum which he is describing] I do not
know; but of four or five I do not much question but it may.
But these are curiosities of little or no moment to the understand-
ing of the phcenomcna of Jiatiire. For in all whites produced
by nature there uses to be a mixture of all sorts of rays and
by consequence a composition of all colours'!'-."

Every optician knows Newton's empirical rule for the esti-
mation of the colour produced by the mixture of any number
of the elementary colours of the spectrum and in any propor-
tions. But it is necessary here to repeat it, because it would
appear to simplify the scale of colours much indeed, were
colours only such as the composition of the coloured rays of
light present. Imagine a circle drawn whose centre is pure
white, and its circumference presents in order all the colours
of the spectrum in succession, and occupying arcs proportioned
to their lengths in the true spectrum, the two ends of the spec-
trum or the extreme violet and red coalescing at one placef.
Let these colours pass gradual!}', by the mixture of white light,
from their intensest development at the circumference to per-
fect whiteness at the centre of the circle. Suppose it were
required to find the effect of mixing two parts of green, one
of red and one of violet. At the circumference of the circle and
at the centre of the green space, describe a small circle whose
area is 2; at the middle points of the red and violet describe
circles whose areas are I ; find the centre of gravity of the
three circles : it will be found on the diagram over the tint
which the mixture of rays, if actually formed, will present.

Since the quantity of colour" in any mixture is to be regarded
in this arbitrary construction as applied at the centre of the
arc appropriated to each colour, and since no two such points
are directly opposed to one another in the circle, it follows by
construction (as M. Biot has remarked^) that a perfect white

* Optics, Book I. part 2, prop. iv.

t The limits of the several colours will occur, according to the New-
tonian proportions, at the following angles in a continued graduation : —

o /

Red begins at 0*0

Orange 60-46

Yellow 94-57

Green 149-38

Blue 210-23

Indigo 265-4

Violet 299-15

X Traite de Physique, vol. iii. p. 449. 'i'" '• J«> "i* ■'

M 2

164 Prof. J. D. Forbes on the Classificatioji of Colours.

cannot be compounded out of only txw colours in the spec-
trum. This corresponds with Newton's experience, that such
a colour (the mixture of two opposites) " shall not be perfectly
white, but some faint anonymous colour*." But these expe-
riments merit well a careful repetition, which I am not indeed
aware that they have ever received ; and it is very probable
that Newton never made them with a pure, or even an ap-
proximately pure spectrum f.

But Newton's celebrated experiment of mixing together
coloured powders until he obtained a perfectly indefinite gray
is most to our present purpose. He describes in the fifteenth
experiment of the second part of his first book of Optics, the
various dry pigments which he employed, the most effective
of which was a mixture of orange, purple, green and blue,
which " became of such a gray or pale white as verged to no
one of the colours more than to another" (p. 131), which
when powerfully illuminated by the sun was an exact match
for a pure white paper less perfectly illuminated. The reason
why it does not appear absolutely white under ordinary cir-
cumstances Newton thus explains: — "All coloured powders
do suppress or stop in them a very considerable part of the
light by which they are illuminated. For they become co-
loured by reflecting the light of their own colours more co-
piously and that of all other colours more sparingly, and yet
they do not reflect the light of their own colours so copiously
as white bodies do." [This he illustrates by illuminating red
lead and white paper with the red ray ; the white paper ap-
pears the more brilliantly red of the two.] " Therefore by
mixing such powders [powders, namely, of various colours,]
we are not to expect a strong and full white, such as is that
of paper, but some dusky obscure one such as might arise
from a mixture of light and darkness, or from white and black,
that is, a gray, or dun, or russet-brown, such as are the co-
lours of a man's nail, of a mouse, of ashes, of ordinary stones,
of mortar, of dust and dirt in the highways and the like." —
P. 130.

Whatever may be thought of Newton's theory of the colours

• Optics, ed. cit. p. 136.

t If Newton's circular ti<jfiire was intended (which, however, M. Biot
questions) to be divided according to the accurate proportions of the colours
in the spectrum, it would be a matter of great difficulty to assign the due
proportions to the extreme red and violet ; any variation in this respect
would alter the character of the diametrically opposed tints, and make a
perfectly white compound of two tints possible, or the reverse. It is hardly
necessary to add here, that the perfect white produced by the comple-
mentary tints which occur in experiments of depolarization, arise:) from the
mixture of colours of a very complex constitution.

Prof. J. D. Forbes on the Classification of Colours. \ 6.5

ofnr.tural bodies (which refers them to the colours of thin
plates), the reasoning of the above paragraph will hardly be
questioned. There will therefore be always this essential
difference between compounding rays of the spectrum and
compounding pigments ; that in the former case, by throwing
light of two or more colours upon a white screen, each of
these colours being reflected with equal vividness, the bright-
ness of the screen will be the sum of the brightnesses due to
the several rays (and if a sufficient number of rays be com-
bined, the result will be a dazzling white) ; but, on the other
hand, by combining pigments we do not add together lights,
but merely construct a ground or screen capable of scattering
a greater number of the constituents of a beam of white light
which falls upon it. Thus there will be an inevitable quantity
of darkness or absorbent faculty in the constitution of every
artificial colour, whatever be its predominant reflecting hue;
and the mixture of pigments will not tend to increase the
brightness, as the mixture of lights would do, but only to mix
with the fundamental darkness of the surface a portion of light
which shall be of a mixed instead of a simple hue.

Let us suppose for a moment a simple case. Let us admit
that a paper thickly coated with ultramarine can reflect none
but blue rays, and that a paper coated with chrome-yellow
reflects only yellow rays. But further than this, a share of
the blue and of the yellow light falling on each is absorbed;
suppose one-half to be thus lost. If a compound pigment of
blue and yellow be formed and exposed to the same white ray
as before, we cannot expect that it should have more brilliancy
than either one or other of the primitive colours, whilst it is
evident that the union of rays of yellow and blue light upon a
white screen would have a twofold splendour*. For we must
admit the reflecting particles in each of the separate pigments
to be so densely spread that a ray of light can fall nowhere,
on the ultramarine for instance, but it finds a particle of
colour ready to decompose and reflect it, and the same of the
pure yellow pigment; in a mixture, therefore, of the two, the
surface may be regarded as equally divided amongst an infi-
nite number of the blue and the yellow reflecting points, so
that the reflected light is half yellow, half blue, but altogether
is no more than the amount which either pigment covering
the whole surface would have reflected. We must not there-
fore suppose that by mixing pigments we render the surface
on the whole more reflective, that is to say, more luminous,
than before. Experience confirms this anticipation. On the

• What is meant here by speaking of rays of different colours having
"equal" or " two-fold" brilliancy will be explained by and by.

166 Prof. J. D. Forbes on the Classification of Colours.

whole, then, in this experiment, half the light is still absorbed
and half of the remainder is yellow, the rest blue, constituting
a green colour.

Similar reasoning will hold true of any number of separate
colours combined into one; and as perhaps no pigment reflects
only one pure colour of the spectrum, the mixtures will always
be more compound than they are assumed to be, and give
hues of always increasing impurity.

The process of mixture cannot in any case be expected to
improve the power of reflecting the pure colours residing in
the constituent pigments. It is much more likely to deterio-
rate it, which will tend to give a tone of always less absolute
brightness to more complex colours.

The process of mixing pigments may often affect their rela-
tive strength, especially if moisture be used. There may be
a mutual action, which will cause an undue preponderance of
one or other constituent. These difficulties have constantly
been felt by those who have endeavoured to compound colours
from their elements.

We have assumed above that the reflective power of pig-
ments of different colours is the same. But this does not
appear to be the case. The ingenious experiments of Lam-
bert*, which though probably but rough approximations are
yet valuable as such, inform us that the whitest surface re-
flects Y%ths of the white incident rays, and king's yellow almost
as much of rays of its own colour : but the brightest red
(cinnabar) reflects but |rd of its own coloured light, and a
blue surface (mountain Berlin blue) but yth of the blue i*ays.

On these principles we may expect that if the circumference
of a wheel be painted with stripes of red, yellow and blue, alter-
nating with one another, so tliat the extent occupied by these
colours shall be in certain determinate proportions, the mix-
ture shall appear white, or rather tieufral gj-ay, the wheel being
put in rapid rotation. We can estimate the illumination of the
surface compared to that of white paper in the following
manner.

The proportional extent of the surfaces may be found by
direct experiment, or otherwise thus. By Newton's rule of
compounding colours (see p. 163), we may deduce that a white
compounded of red, yellow and blue, must consist of 38*6,
19*6, and 41 '8 rays out of 100 of these colours respectively;
for the centres of gravity of the red, yellow and blue sectors
make angles of 91° 54>'-5 between the red and yellow, 1 15° 26'
between the yellow and blue, and 152° 39'*5 between the blue
and red. In order that the centre of gravity of the whole shall
* Photometria, § 747; and Farbenpyr amide, § 5.

Prof. J. D. Forbes on the Classification of Colours. 167

coincide with the centre of the circle, the primary colours
must be in proportion to the sines of those angles, which are
•9785, -9031 and •4'593; the first being blue, the second red,
and the third yellow, which give the proportions above stated.
And there can be little doubt that this rule is sufficiently cor-
rect, though we restrict the colours of the spectrum to three
only; for the centre of gravity of the blue (for example) may
be regarded as the centre of gravity likewise of the blue con-
tained in the green, and that in the indigo and violet ; and so
of the other colours.

Let then the proportions 0'386 = R, 0'196 = Y, 0-41 8 = B
represent the constituents of white light in the spectrum, their
sum being =1. But by what has been said of Laml)ert's expe-
riments, it appears that red, yellow, and blue pigments reflect
but ^rd, y'^^lhs, and ^th of the rays ot those respective colours
which fell upon them. Therefore to \vA\e reflected Wghioiihe
same composition with the white of our tricoloured spectrum,
we must have the surfaces of the colours larger in proportion
as their reflecting power is less. Hence the spaces in our
coloured wheel must be

Red . . 3R or .

. l-157=n

Yellow . — - Y or .
4

. 0'490=j^.

Blue . . 7B or .

. 2-927 = &.

The sum of these .

. 4*574 = w.

Consequently, of all the red rays which fall upon our tri-

l r
coloured surface only the fraction - •- are reflected (for of

^ S n ^

those which fall on the yellow and blue spaces, none are re-
flected, and but one-third of those falling on the red), that is,

R

— or '0843

71

Y

Of the whole yellow rays are reflected — or "0429

Of the whole blue rays are reflected . — or •0914
•' n

In short, the reflected light is white light) whose intensity is
attenuated by reflexion m or 4*57 times ; whereas had it been
incident on white paper, it would have still had y^oths of its first
brightness, or been attenuated only 2*50 times. This abun-
dantly explains why the result is a grai/ colour, not a bright

168 Prof. J. D. Forbes on the Classification of Colours.

white*. The proportions of the surfaces of bright colours
whose mixture produces white (r, i/, b in the preceding nota-
tion) is 5, 3 and 8, as given by Field f.

It is to Mayer, the mathematician, that we owe a complete
and perfect diagram of mixed colours, staiting from red, yel-
low and blue, as constituents. Let the extreme corners of a
triangle be painted of these colours, and let the periphery of
the triangle be composed of graduating colours between each
pair of these respectively ; then the centres of the sides of the
triangle will be occupied by perfect orange, perfect green and
perfect purple, each of which will pass in each direction towards
the predominating primary colour. The periphery of Mayer's
triangle includes, therefore, all the colours of the spectrum, or
primary colours mixed two and two. But combinations of
three colours may be represented by selecting points in the
interior of the triangle which shall be the centre of gravity of
the constituent colours. Thus if the three colours, red, yel-
low and blue, be mixed in equal proportions, the resulting
colour, which will be neutral gray, will be found at the centre

of gravity of the
triangleatW. But
this would also re-
sult from the mix-
ture of one portion
of red and one of
blue united at P to
form twoofpurple,
which then being
compounded with
one of yellow, Y,
will give the centre
of gravity at one-
third of the di-
stance from P to-
wards Y. The

Fig. 1.

* Goethe, in his Theory of Colours, seems to think that he has over-
turned Newton's experimental demonstrations by calling the opinion that
" all the colours mixed together produce white," " an absurdity which
people have credulously been accustomed to repeat for a century in oppo-
sition to theevidenceof their senses." (Eastlake's translation, p. 225.) 'Ihe
truth is, that " gray " is not an affection of Light at ail, but of Surface
merely. All Light combining the coloured elements in due proportion is
essentially white, though more or less intense ; but no Surface can be said
to be 'perfectly/ white rather than graj/, except by comparison with another.
A surface of white paper illuminated by common daylight is gray relatively
to a similar one placed in full sunshine.

t Field's Chromatography, p. 247.

Prof. J. D. Forbes oti the Classification of Colours. 169

same result will evidently flow from mixing two parts of orange
with one of blue, or two of green with one of red.

It also clearly follows from this construction, that a point in
the triangle may always be found which shall represent any
2)ossible proportional mixture of the three colours, because the
centre of gravity of the three elements, however unequal, must
of necessity be found within the triangle.

Also, a complex colour of three elements may be regarded
as composed of primary colours and their binary compounds
in an infinite variety of ways. Thus, the colour called citrine
by some authors, and which is described as a compound of
equal parts of orange and green, has its place in the triangle
at a, which shows that it is intermediate between pure yellow
and neutral gray in the proportion of 1 of the first to 3 of the
second ; or it is a mixture of pure yellow and pure purple in
equal proportions.

The annexed diagram shows the principle of Mayer's mix-
ture of colours, the subscribed figures denoting the relative
portions of each colour in any compound, the sum of the units
making up 8 in every case. The same principle of numerical
ratios may be extended to any degree of nicety ; but it is soon
found that the power of the eye in distinguishing hues is over-
passed.

Fig. 2.

.^4 u lyA^'sKy^r^Uy^rM y^ h

1.% ^5 \y^^^fi^\y^^'A\y?r^Ky^^\K\y^ h

\y-i ^6 lj/2^-5^i \y'i^'Myi^Myz^Myi^\h\yi h I
\y\ ^7 \y\>A \yxrA\y\rA\yx^sh\yi^'A\yirA\y^ ^7 1

I r^ I ^7 ^1 I ^6 h \rsh\ f'4 f>4 I ^3 h I ^'2 ^6 kl ^7 I ^8 |

The unit of mass for any primary colour or pigment is the
proportion which, mixed with the other two primaries, forms
a perfectly neutral gray. This must be found by experiment,
and resembles the atomic weight or equivalent of the simple
bodies of chemistry. Lambert found it by uniting carmine
and gamboge until a perfect orange was formed, which (judging
by the eye) inclined neither to red nor yellow ; so with yellow
and blue forming green, and with blue and red forming purple.
The quantities being weighed in each case, two such expe-
riments were sufficient to determine the relative powers of the

170 Prof. J. D. Forbes 07i the Classification of Colours.

colours, but the third was used to confirm them. He thus
found the combining proportions by weight of carmine, gam-
boge and Berlin blue, to be 1, 10 and 3*.

We are now to consider how far this triangle carries us
towards a complete scale of colours. It is manifest that the
««/f»5iV?/ of the colours depends upon the reflective power of
the pigments used, and that this essentially varies for the dif-
ferent primary colours. In no sense, tiien, can we be said to
have red, yellow and blue, of equal brightness at the corners
of our triangle; for even if we assume as merely convenient
definition, that by equal brightness of different colours we
mean the proportions in which, when combined, white light
results, we have already seen that the yellow pigment, being
far most reflective, will be brightest, then the red, and after all
the blue. But in fact we have no scale at all for comparative
brightness of heterogeneous colours. We must take the Tp\g-
rvxQnls purest in quality, and most lucid or reflective as regards
the quantity of light which they scatter, and consider these
as the primary colours. The mixed colours also vary in their
lucidity, according to the prevalence of a more or less lucid
component; the yellow hues will be most lucid, the blue least.

When the triangle is exposed to a brighter light, the pro-
portions of the colours remain unchanged, and the whole will
be more lucid.

It is probable, however, that the decomposing action of all
pigments upon light is limited; and that a coloured surface
may be so drowned in white light, that much of the light is
returned undecomposed, and the colour is thus diluted.

If less light tall on the triangle, a different kind of dilution
occurs, only piu'e coloured light will be reflected, but so little
of it as to affect the eye but slightly, or not at all with the
sense of colour.

In the latter case all colours pass into indistinguishable
blackness, in the former case into indistinguishable whiteness.

If we mix black and white pigments with coloured pigments,
we may have both these variations exhibited at once under a
common external illumination.

If we have a series of triangles thus constructed, they will
embrace under one common illuminating influence (as ordinary
daylight) all possible varieties of hue and shade under that
illumination. Every conceivable natural or artificial object,
such as a piece of stuff", a feather, or a flower, ought to be
capable of being matched with one or other of the spaces in
these triangles. This is all that we propose to accomplish.
If we choose the most lucid known bodies for our primary
* Farbenpyramide, § 63.

Prof. J. D. Forbes on the Clasiification of Colours. 171

colours, we shall be sure to have none to match which are
not included in our suite.

The question now arises of the number of intermediate
mixed colours which can be interpohited between any two
primary or simpler colours, so that each may be distinguish-
able by the eye upon a close comparison. The number is
much smaller than might be supposed. Lambert states, that
from perfect black to perfect tichite he could only trace thirty
intermediate shades distinguishable by the eye under themosi
favourable circumstances possible*. The number of gradations
of even the most positive colours is probably considerably less,
and of the more neutral colours much less again; at least if
we do not repeat those semi-neutral compounds which are
indistinguishable from one another.

As to the transition from one quality of colour to another
without regard to its dilution with light or shade, as, for in-
stance, from red to blue, the sensible intermediate stages are
also probably much fewer than might be suspected. Mayer
affirms t that the distinction of mixed colours is evident so
long as the sum of the component parts denoted, as in the dia-
gram, fig. 2, does not exceed 12. Thus a bright yellow,
such as king's yellow, being denoted by z/jg, yellow ochre is
^10 ^'s' umber is j/g r^ b^ ivory black j/g r^ b^, in all which cases
the sum of the parts is equal to 12. Upon this scale it is easy
to show that the fundamental triangle, whose side is 13, will
contain 91 coloured spaces. These embrace all possible com-
binations of colour, of the fundamentally greatest intensity
which the imperfection of our pigments enable us to procure.

Allowing 4 gradations of each colour into blackness and
4 into whiteness, Mayer reckoned*819 colours in all; a num-
ber which will certainly appear small considering the appa-
rent infinity of hues and shades. It is probably sufficient,
however, for matching any colour by reference to two others,
one above and the other below it, in any of the scales ; and
such subdivision may probably be carried by the eye to greater
accuracy than one intermediate step. The gradation of nine
steps from perfect black to perfect white through any colour
is perhaps too small ; but on the other hand, the neutral co-
lours, as already observed, some of them at least, lose their
distinguishable characters compared with one another, when
diluted either with black or with white, but especially the
former f. Taking advantage of this consideration, Lambert
modified Mayer's triangles by reducing them continually in

* Lambert, Farbenpyramide, § 10, 11. f Ibid. § 29.

X I suspect, indeed, that in some instances the dilution of the semi-
neutrals with white renders them more easily distinguished, but only down
to a certain point.

172 Prof. J. D. Forbes ow the Classification qfColojirs.

size and in the number oF elements, as the standard colours
approached white on the one hand and black on the other ;
forming thus a double pyramid, whose common base was
Mayer's triangle, the colours vanishing into white at one apex
and into bhick at the other*. A triangular pyramid or tetra-
hedron of 13 elements in the side would contain 4-55 elements,
and the double pyramid or hexahedron 910 elements.

Other writers have attempted to adopt primary colours dif-
ferent from red, yellow and blue; and with this subject has
been mixed up the inquiry into the actual composition of the
solar spectrum, which though not immediately connected with
it, may be mentioned in passing. Mayer maintained, not merely
that all colours whatever may be formed by the combination of
red, yellow and blue, but that in reality these colours alone
exist in the solar lightf) which he inferred by looking through
a prism at a black spot on a white ground, an essentially
faulty mode of operating. But later. Dr. Wollaston came to
the conclusion that the solar spectrum is composed of four
colours only, namely red, green, blue and violet, without any
gradations in the quality of the colour. Dr. Young, finding ex-
perimentally that " the perfect sensations of yellow and of blue
are produced respectively by mixtures of red and green, and
of green and violet light J," assumes that the primary colours
are red, green and violet ; a singular opinion, which appears
to rest on no particular evidence further than the disjunction
of the red and violet rays at the two ends of the spectrum, and
which has met with no support any more than that of Abbe
Nollet, who maintained the primary qualities of orange, green
and purple. In truth no synthetical experiment can give any
sure countenance to one or other of these views ; for the fact
that red and green combined in certain proportions produce
yellow, admits of equally sound interpretation by supposing
that the green, being a compound of yellow and blue, the
whole of the blue and a part of the yellow combine with the
red to produce a perfect white, which then dilutes the out-
standing portion of the yellow; and in like manner a perfect
purple mixed with perfect green must make a perfect blue
diluted with a perfect white. Analysis, however, where pos-
sible, must lead to more conclusive results; and Sir David
Brewster considers that the orange, green, and purple of the
spectrum are really composed of two, if not three colours

* In the coloured plate accompanying Lambert's work we find only the
pyramid of colour dihited with white, which he seems to have considered
sufficient in practice, § 39. In reality, however, the shades or mixtures
with black are indispensable components of such a system.

+ Mayer, Gottinmschen Anzeigen, quoted by Lambert, p. 30.

t Lecture XXXVII.

Prof. J. D. Forbes on the Classificalion of Colours. 178

each *. The analysis he employs is the absorbing power of
media, for these colours are (as is well known) undecomposable
by refraction.

Various attempts have been made to carry out Mayer's
principle of compounding colours from red, yellow and blue,
and some elaborate attempts have been made to obtain model
suites of colour. I shall at present only refer to Mr. D. R.
Hay's ingenious work called Nomenclature of Colours, which
he has illustrated by a very large number of selected hues and
shades all compounded from lied. Yellow and Blue, variously
diluted with Black and White, which, from Mr. Hay's skill
in the choice and use of colours, are probably as pure and
vivid as we can expect to produce in the present state of art.
It is unnecessary here to speak of the taste and skill with which
the harmony and contrasts of colour are used and illustrated
in the plates to iiis work.

As a mere classification of colours, Mr. Hay's work does
not adopt the simplest form ; nor is the nomenclature, I con-
ceive, by any means free from objection. It would be difficult,
for instance, to refer any required colour to its place in a com-
plete system of hues and shades by merely looking over Mr.
Hay's plates. The specimens which have the closest affinity
are often widely separated ; but then the object, a purely
artistic one, was different from ours. The mixtures used by
Mr. Hay in his gradations of colour were made, I understand,
by the eye, and not by weight; but an experienced eye will
perhaps make a gradation at least as good as a quantitative
one. The dilutions with white (or tints, as Mr. Hay calls
them,) appear to be less perfect in this respect.

The primary colours in Mr. Hay's work are red, yellow
and blue, or those which occupy the angles of Mayer's triangle,
fig. 1. They are composed of carmine, chrome-yellow, and
French ultramarine.

The secondary colours, or orange, green and purple, with
their gradations into their component primaries, exhaust all
the combinations two and two of the primaries, embracing all
the colours of the spectrum, and are represented by the ex-
terior row of colours in Mayer's triangle.

The combination of colours by three at a time leads to more
complexity, and the advantage of Mayer's system is here most
evident. Mr. Hay, following Field, calls tertiary colours
those produced by a union of the secondaries : thus —
Orange and green form Citrine,
Orange ... purple ... Russet,
Purple ... green ... Olive.
* Edinburgh Transactions, vol. xii.

174 Prof. J. D. Forbes on the Classification of Colours.

** Their distinctions arise from a double occurrence of yel-
low in the first, of red in the second, and of blue in the third*."
In like manner, by the combination of the tertiary colours (or
primary hues, as Mr. Hay calls them), he produced a system
of quaternaries (or secondary hues), and so on until the small
predominance of any one or two primary colours in the com-
pound reduces the whole to a neutral gray. Now a simple
inspection of Mayer's triangle, p. 168, shows that the confusion
of the colours, by drawing a series of perpetually inscribed
triangles, increases with great rapidity, and that consequently
the gradations of shades will not be such as affect the eye most
sensibly, but will be deficient in the brighter and redundant
in the grayer colours.

The law of composition of the secondary and tertiary colours
is however worthy of notice, and may be represented in the
following diagram (employing the notation of page 169).

Orange,,^^

= ^43/4

"^Citrine .^^

Green <^

=yA'-<,

"~^ Green hue

=i^4^4

> Olive <^

'^ 3/3^3^2

Purple<C^

= V2i/2

"^ Purple hue

= ^4^4

^^Russet <^

h'-sV-i

Orange \^

=r^A

"^^Orangehue

= ^43/4

^^~^ Citrine --^

^'zVA

Hence citrine^ for instance, is a mixture of four parts of yellow,
with two of blue and two of red, which is equivalent to two
parts of yellow and six of neutral gray (since gray ^ybr) ;
citrine is therefore yellow verging into gray, one-fourth of the
mixture being yellow and three-fourths gray ; its place in
Mayer's triangle, fig. 1, will therefore be at Qi dividing the line
YW in the ratio of 3 to 1 ; and so it does by construction,
taking the centre of gravity of the orange and green, O, G.

Let us take an instance of a still more indefinite colour, a
colour of the fourth order, which Mr. Hay calls a secondary
hue, and to which he gives the distinguishing names of orange
hue, green hue^ and purple hue; we find these to be thus
composed :

Hy?,K yA'>\-> h^^y^^

or thus:
2 orange + 6 gray, 2 green + 6 gray, 2 purple + 6 gray.
* Hay, p. 18.

Prof. J. D. Forbes oti the Classification of Colours. 175

We therefore refer them at once to their places in Mayer's
triangle as intermediate between tlie secondaries and neutrality,
and dividing the interspace in the ratio of 3 to 1, as in the
former case; only here the compound is more neutral, because
the secondary colours are themselves one stage on the way to
neutrality.

Thus we arrive at this conclusion, that all combinations of
three primary colours (as far as difference of quality is con-
cerned) may be represented by transitions from the primary and
secondary colours into gray ; and thence, though it may appear
at first sight paradoxical, though the quality of a primary or
secondary colour (such as red or green) is not changed by
diluting it with white, it ^5 changed by mixing it with gray,
or by first mixing it with white and then diminishing the in-
tensity of light in the mixture.

Hence a classification of colours may be made, which,
although redundant in some parts, has the advantage of point-
ing out clearly the composition of each in this point, and also
of suggesting a convenient nomenclature, which I propose to
adopt in preference to Mr. Hay's (where they differ), as more
expressive of the composition of each. This diagram, like
Mayer's triangle, includes colours varying in quality, but of
standard ititensity and of the highest attainable purity. This
diagram was obligingly arranged for me by Mr. Hay out of
the coloured specimens in his work.

The places marked by asterisks will supply a sufficient rela-
tive number of intermediate hues, as these evidently approach
the absolute uniforniity of neutral gray in the last column,
whilst the first contains the graduated colours of the spectrum.

All these colours may be varied by mixing them with white
or black, forming what Mr. Hay judiciously calls tints or
shades of any colour.

It is sometimes convenient to have these tints and shades
arranged in immediate apposition for the purpose of compa-
rison. This may be conveniently done for the principal co-
lours by having two diagrams. In one, the colours of the
spectrum form a circular ring, the colours passing through
tints into perfect white at the centre, and the shades continued
in outward radiating lines till they coalesce in a perfectly
black circumference. In the other, the principal intermediates
between the principal colours and gray may be exhibited with
like transitions. These intermediates may conveniently be
denominated by the following terms, sufficiently expressive
and in common use.

176

Prof. J. D. Forbes on the Classification of Colours.

rray.

Russet intermediate between red and g
Brown ... ... orange

... yellow

green
blue
... purple

Citrine
Drab
Olive
Slate

IRed.t
Orangish-red.

Grayish-red.
*

Gray-red.

[Rus
*

Red- gray.

,et.]

Reddish-gray.

Gray.

Red-orange.

*

*

*

Reddish- orange.

*

*

Orange.
YeUowish.orange

Grayish-orange.
*

Gray-orange.

[Bro
*

Orange-gray.

wn.]

Orangish-gray.

Gray.

Yellow-orange.

»

*

*

Orangish-yellow.

*

*

Gray.

|Yellow.|
Greenish-yellow.

Grayish-yellow.
*

Gray-yellow.

[Citr
*

Yellow-gray,
ine.]

Yellowish-gray.

Yellow-green.

*

*

»

Yellowish-green.

*

*

Gray.

Green,
Bluish-green.

Grayish-green.
*

Gray-green.

[Dr
*

Green-gray,
ab.]

Greenish-gray.

Blue-green.

*

*

*

Greenish-hlue.

*

*

Gray.

|Blae.|

Grayish-blue.
*

Gray-blue.

[01
*

Blue-gray.

ve.]

Bluish-gray.

Purplish-blue.

Blue-purple.

»

*

*

Bluish-purple.

»

K

Purple.
Reddish-purple.

Grayish-purple.

*

Gray-purple.

[Sis
*

Purple-gray,
ite.]

Purplish-gray.

Gray.

Red-purple.

*

»

»

Purplish-red.

*

»

Prof. J. D. Forbes on tlie Classification of Colours. 177

Mr. Hay has been kind enough to arrange for me his ex-
tensive suite of artificial colours according to these diagrams.

But it must be owned to be highly desirable to possess such
a suite of colours in more perfect and durable materials than
any pigment as usually applied presents. Painted porcelain
and coloured enamels alone appear to possess this valuable pro-
perty. I'he immense collection of artificial enamels employed
in the Vatican fabric of mosaic pictures seenis to offer an unri-
valled opportunity of forming such a classification.

This gigantic establishment was founded about two centu-
ries ago tor the express purpose of adorning the interior of St.
Peter's with the elaborate mosaic pictures and ceilings which
astonish every visitor. The whole interior of the stupendous
dome is incrusted with mosaic patterns and pictures, of coarse
execution indeed, but such as suits best the vast distance from
which alone they can be properly viewed ; whilst the finished
mosaic works which adorn the altars reproduce in unfading
colours and with consummate skill in shading the chefs-
c?'a?«3;r^ of Raphael, Domenichino, and other artists preserved
in the Vatican gallery. The material is a soft and fusible
enamel, and the formation of 1 8,000 tints was effected by an
ingenious artist named Matteoli, at the time I have mentioned.
The rough cakes of enamel are preserved in separate cup-
boards or pigeon-holes, surrounding a hall of great length ap-
propriated to this purpose by Pope Pius VI. But the main
intention of the work being completed with St. Peter's, it has
not been thought worth while to preserve the integrity of the
collection (which, indeed would be no easy matter) ; and it is
certain that though still reputed to contain 18,000 modified
colours, the effective number is vastly smaller.

Having been fortunate enough in 1844< to make the valu-
able acquaintance of Monsignore de' Medici Spada, an en-
lightened and influential prelate residing at Rome, I entreated
his influence to procure a selection of specimens of the leading
colours of the Vatican mosaics. For a long time official slug-
gishness rendered the application fruitless ; at length the im-
portunity of my friend overcame all difficulties, but not until
I had long left Rome, and was therefore quite unable to
superintend the selection. My instructions were therefore
general, to prefer the most varied tints which the collection
presented. At last an assortment of no less than 94 1 pieces
of mosaic, classified in separate packets, arrived. A close
examination rather disappointed me. They presented a great
preponderance of indefinite colours, and a great deficiency of
many of the livelier and brighterprimary and secondary colours.
But particularly whole packets were composed of specimens

Phil. Mag. S. S. Vol. 34. No. 228. March 1849. N

178 Prof. J. D. Forbes on the Classification of Colours.

scarcely sensibly differing from each other. This last circum-
stance was probably occasioned by the carelessness and indo-
lence of the workmen who selected them. The former cir-
cumstances might naturally be expected in a collection con-
structed for the purpose of imitating paintings, in which, as is
well-known, optically pure colours are almost never used ; but
their effect is invariably produced by skilful contrasts. Many
of the suites of indefinite colours are exquisitely beautiful.
With Mr. Hay's assistance, I selected a sufficient number of
distinct hues to represent tolerably Mayer's triangle of colours;
but the great mass of colours being only detached suites, it was
impossible to combine them into a connected whole. As I have
no doubt, however, that the collection is one which faithfully
represents the colours chiefly used by artists, it may not be
uninteresting to copy the catalogue forwarded to me by Mon-
signore Spada, with the local names and the principal denomi-
nations on the scale of nomenclature proposed in this paper,
which they include.

Local Names. Technical Names.

100 specimens. Bigi. Tints of yellow-gray, and tints of gray.

100 ... Cabnagioni. Tints of orange-gray [brown], reddish-

yellow gray, reddish-purple gray, purple-
gray.
60 ... GiALLi. Tints of yellow-gray and reddish-yellow

gray.
Orange passing into red and yellow.
Grayish-red, reddish-gray, yellowish-gray,

purplish-gray, shades of piuplish-gray.
Yellow-gray passing into purple-gray,

shades of purple-gray.
Gray-purple and tints of ditto.
Red and grayish-red.
Tints of yellow, tints of orange, tints of

yellow-gray and of red-gray.
Tints of blue, tints of purplish-blue, gray-

ish-greenish blue.
Tints of green-gray, tints of blue-green.

This number of specimens would have been sufficient to
make a complete series of colours ; but, as has been said, they
were very deficient in the more positive hues. I have still
hopes, however, of being able to obtain a series of perfect
matches for the whole series of Mr. Hay's pigments, speci-
mens of which have indeed been already sent to Rome for the
purpose.
Edinburgh, January 1849.

20
60

GlUGIOLINI.

Lacche.

60

Leonati.

60

76 ...
172

Pavonazzi.

porporini.
scorzetti,

91

TURCHINI.

142

Veroli,

941 Total of s]

pecimens recei'

[ 179 ]

XXII. On the Calculation of the Distance of a Shooting Star
eclipsed in the Earth* s Shadow, By Archibald Smith,
Esq., of Lincoln* s- Inn, Barrister-at-Law, late Fellow of
Tritiitij College, Cambridge*.

1, TN a paper in the Philosophical Magazine for February
J. 1848, Sir John Lubbock has suggested that shooting
stars may be small planetary bodies which shine by reflected
light, and that their sudden disappearance may be occasioned
by their immersion in the earth's shadow ; and he has given
a formula for calculating, on this hypothesis, the distance of
a shooting star, at the moment of its disappearance, from a
spectator on the earth's surface. The formula, however, as it
is given in the paper in question, even when simplified by
supposing the earth's shadow to be cylindrical instead of coni-
cal, is not well adapted for numerical calculation, and may
repel some who would be inclined to pursue the investigation
if the necessary calculations were less laborious.

The data assumed by Sir J. Lubbock are the zenith di-
stances, and the difference of the azimuths of the sun and of
the star at the moment of its disappearance. In repeating his
calculations, I find that by introducing the angular distance
between the shooting star and the sun, or rather the point
opposite the sun, the formula is very much simplified, and
thus the time requisite for calculating a distance does not ex-
ceed a very few minutes. As this is a point of some import-
ance in furnishing a test for the theory, this communication may
not be considered inappropriate to the Philosophical Magazine.
2. Let the centre of the earth be the origin of co-ordinates.
Let the axis of Z be directed to the zenith, of ^ to the north,
and of j/ to the east.

Let x,y,x be the rectangular co-ordinates of the shooting
star at the moment when it enters the earth's
shadow,
p its distance from the spectator.
5, « its zenith distance and azimuth.
a,h,c the co-ordinates of the vertex of earth's shadow.

S the length of the shadow = V'a^ + 6^ -f- c^.
Z, A the zenith distance and azimuth of the point in the
heavens which is diametrically opposite to the
sun.
^ the angular distance of the shooting star from the

point opposite the sun.
R semidiameter of the earth.

* Communicated by the Author.
N2

180 Mr. A. Smith on the Calculatmi of the Distance
3. The equation to the cone of the earth's shadow is

{a?2+y+^^-R^}{s^-R'} = {«^+2'j/+c^-R2}2 . (1.)

Also, since

a:=f) sin ?cosa «=SsinZcosA

y=psin5sina J=SsinZsinA

2;=fCos^+R' f=ScosZ,

it follows that

^^2_|.j^2^^2=p2^2RpC0S^+R^ . . . (2.)

ax 4- i?/ + c^ = Sp { sin ?sin Z cos ( A — «) + cos ?cos Z } + SR cos Z

= (by spherical trigonometry) Sp cos <p + SR cos Z. . (3.)

It is the introduction of (p at this part of the process which
so much simplifies the result.

Substituting in equation (1.) the values of x^ + y'^ ■\- z'^ and

o^ ax-\-hy + c?^i given by equations (2.) and (3.), and putting gr

■p
for ^T^, q being a small quantity, the average value of which is

about 'OOie, we have
f)2{sin«(p-y2}4.2Rp{(l-g2)cos?-cos^(cosZ~y)}

-R2{cosZ-g}2=0

} (4.)

In this formula sin (p may be considered as always greater
than (7, or <^ greater than 16'; since, if smaller, it would be
impossible to make any approximation to the distance of the
shooting star. This equation, therefore, will have two roots ;
one positive, the other negative. The positive root is of course
the only admissible solution. Solving the equation and ma-
king R = l, we obtain

-n/{^

-g^)cos^- cosf(cosZ-<7)'\^ {cosZ-qf

^ (1—9^)008 ^— cos <p (cos Z — g) . - V

sin2<p-<72 ^^'>

The calculation ofp from this formula will be facilitated by
the use of a subsidiary angle 4', such that

CP,tifr- (l-g')cOS^-COSf(cOsZ-g), ^ ^ ^gj

-/ sin^ <p -- g^ ( COS Z — g')
we then have

cosZ— » ^ ^ ,^v

vsin^(p — ^-^ ^

4. In almost all cases a sufficiently approximate result will

/ fcOS ^— COS (p COS Z ]

1 2 cos'^ Z

COS^— COS (p COS Z

sin^ f

q/'flr Shooting Star eclipsed in the Earth* s Shadow. '181

be obtained by supposing the earth's shadow to be cylindrical
instead of conical ; the formulae then become extremely simple.
This is the same thing as supposing q = 0; we then have

p2sin2(p + 2Rp(cos?-cosf cosZ)-R2cos2Z=0 . (8.)

(9.)

and, as before, using a subsidiary angle ^, but which is now
determined by the equation

cot \I/ = cos 5 sec Z cosec f — cot <p, . . (10.)
we have

p= cos Zcosec ftan— (11.)

These two formulae furnish the means of calculating p in a
very few minutes with the aid of a table of logarithms and of
natural tangents. Such tables (Hutton's, for instance,) re-
quire to be opened seven times only.

5. The formulae may be still further simplified by introdu-
cing, as one of the data, the angle contained between the great
circles which pass through the shooting star and the sun, and
the star and the zenith. As this angle may be directly ob-
tained from a celestial globe without calculation, it may be
worth while to exhibit the formulae with this substitution. I
believe, however, that although the formulae thus become more
simple in appearance, the calculation will be very little, if at
all, facilitated thereby.

If we call this angle B, then by a well-known formula in
spherical trigonometry we have

cos ?— cos <p cos Z= sin (p sin Z cos B ;

and substituting this value in equation (8.), it becomes

p2sin2^-t-2Rpsin(psinZcosB— cos2Z = 0. . (12.)

Solving this equation, we have

sin f

P gnrz ^ ^^^^^ ^ "^ cot2 Z- cos B ; . (13.)
and using, as before, a subsidiary angle 4'j such that

cotv|/=cosB.tan Z, (14.)

we have

/J = cos Z cosec (p tan — . . . » . (15.)

182 • Mr. J. Glaisher's Remarks on the Weather

The calculation by means of equations (14.) and (15.) does
not require a table of natural tangents. It requires the lo-
garithmic tables to be opened six times.

6. For the use of persons who may wish to make the cal-
culations without being able to follow the steps of this investi-
gation, it may be desirable to give the result separately ; it is
this.

Let ^ be the zenith distance of the shooting star at the mo-
ment of disappearance, Z the zenith distance of the point of
the heavens diametrically opposite the sun, and ip the distance
of this point from the snooting star. Find an angle vp such
that

cotan 4'= cos ^ sec Z cosec (p — cotan <p.

Then the distance p of the shooting star from the spectator in
terms of the earth's semidiameter as the unit is

p = cos Z cosec <p tan - \I/.
' 2

, 3 Stone Buildings, Lincoln's-Inn,
February 14, 1849.

XXIII. Remarks on the Weather during the Quarter ending
December 31, 1848. Bi/ James Glaisher, Esq., of the
Royal Observatory, Greenwich^.

THE meteorological returns for the past quarter furnished
to the Registrar-General have been received from sta-
tions spread over the country. The observations have been
made, for the most part by experienced observers, upon an
uniform plan. The following remarks are based upon obser-
vations which have been furnished either to myself or to the
Registrar-General, and drawn up to accompany the meteoro-
logical tables published by the Registrar- General, all of which
have been examined by myself, and reduced under my direc-
tion.

The weather during the period has been variable. 'J'he
changes of temperature have beon frequent and great, there
has been an unusually large number of exhibitions of the
aurora borealis, and the magnetic instruments have been
greatly disturbed. The amount of electricity in the atmo-
sphere has been small, many days together having passed
without the instruments at Greenwich being affected.

From the 1st of October to the 10th the excess of tempe-
rature above the average of seven years was 6°*6 ; the greatest
daily excess was 12°*3 on the 7th. Between the 11th and
* Communicated by the Author.

during the Quarter ending December SI, IS-iS. 183

21st the temperature was 4°*5 below the average ; on the 18th
it was 10° in defect. From October 22 to October 30 it was
5°*3 in excess; the greatest was 7''7 on the 27th. From
October 31 to November 16 the temperature was mostly be-
low the average, its mean defect was 4<°'2, its greatest within
the period was 10°*2 on the 4th. From November 17 to
December 19 the temperature exceeded the average by 4°'8,
On December 7 the excess was 12°'4 ; on the 8th was 15°*7;
on the 9th was l^^-'i; and on the 10th was 10°'l. From
December 20 to December 24 the defect was 6°*2 ; from
December 25 to December 29 the excess was 5°*8 ; and it
was 2° '3 below the average on December 30 and 31. The
following are the particulars of each subject of investigation
arranged as in the preceding quarters.

The mean temperature of the air at Greenwich —

For the month of October was 51°-6, which is 2°-5, 6°'2,
3^-6, 2°-l, 1°'4, and l°'l aioue those of the years 1841 to 1846
respectively, and 1°*3 beloiio ihat in the year 1847; oritis2°-3
above the average of these seven years ;

For the month of November was 43°'8, which is l°'l and
1°*0 above those of the years 1 841 and 1842, of the same value
as that of 1843, 0''-2, 2°-0, 2°-2, and 3°-l below those of the
years 1844 to 1 847 respectively ; or it is 0°'7 below the average
of these seven years;

For the month of December was 44°'0, which is 3*^*5, 0°'l,
ll°-0, 2°-3, 11°-1 and l°-2 «Z»ot;e those of the years 1841, 1843,
1844, 1845, 1846, and 1847 respectively; and \°'Obelow that
of the year 1842, or it is 4°"1 above the average of these seven
years.

The mean value for the quarter was 46°"5; that for 1841
was 44°-0; for 1842 was 44°-4 ; for 1843 was 45°-2 ; for 1844
was 42°-2; for 1845 was 45°-9 ; for 1846 was 43°'l ; and for
1847 was 47°*5 ; so that the excess of temperature this quarter
above the corresponding quarter in the years 1841 to 1846
was 2°'5, 2°'l, l°-3, 4°-3, 0°-6, and 3°-4 respectively ; the only
year between 1841 and 1847 whose mean temperature for this
period was ^r^a^£?r than that for the present year was 1847,
and the difference is 1°'0. The average value for this quarter
from the seven preceding years was 44°*6 ; so that the mean
temperature of the air for the quarter ending December 31,
1848, was above that of the corresponding quarter in the pre-
ceding seven years by 1°'9.

The mean temperature of evaporation at Greenwich —

For the month of October was 49^*3, which is 1°'5 above
that for the preceding seven years ;

For the month of November was 41°*7, which is 1°'7 below
that for the preceding seven years;

1 84 Mr. J. Glaislier's Remarks on the Weather

For the month of December was 4-2°*3, which is 3'^*5 above
that for the preceding seven years.

The mean value lor the quarter was 44'°'4', whicii is l"vl
above that for the preceding seven years.

The mean temperature of' the dew-point at Greenwich —

For the month of October was 47°'4-, which is 2^-3, 5°-0,
2°-7, 1°'4, 0°-9, and 0°-2 above those of the years 1841 to 1846
respectively, and l""*? below that of the year 1847; or it is 1^-6
above the average of tliese seven years ;

For the month of November was 38°'8, which is 1°*0, 1^*6,
2--1, S""-!, 4'''0, 4°-3, and 5°-3 below those of the years 1841
to 1 847 ; or it is 3°'0 below the average for these seven years ;

For the month of December was 40°- 1, which is 4°*9, 10°'l,
2^-4, 10°-7, and 0°-3 above those of the years 1841, 1844,
1845, 1846, and 1847 respectively, 3°'l and l°-9 beloxo those
of the years 1842 and 1843 ; or it is 3°'3 above the average of
these seven years.

The mean value for the quarter was 42°'l, which is 0°*7
above the average for the corresponding period of the preceding
seven years.

The mean weight of water in a cubic foot of air for the
quarter was 3*3 grains, which is O'l grain greater than the
average of the preceding seven years.

The additional weight of water required to saturate a cubic
foot of air was 0*54 grain. This value from the preceding seven
years was 0*38 grain.

The mean degree of hurnidity o^ i\\e atmosphere for October
was 0*853, for November was 0*848, and for December was
0*873. The averages for the seven preceding years were 0*888,
0*909, and 0*900 respectively. The value for the quarter was
0*858, which is 0*041 less than the average for these years.

The mean elastic force of vapour for the quarter was 0*285
inch, which is 0*008 less than the average for the preceding
seven years.

The mean reading of the barometer at Greenwich for October
was 29*646 inches, for November was 29*785 inches, and for
December was 29807 inches; these values are 0*014 inch
below, 0*075 inch above, and 0*028 inch below the average for
the same months from the preceding seven years. The mean
value for the quarter was 29*746 inches, which is 0*011 above
the average for these years.

The average weight of a cubic foot of air under the average
temperature, humidity, and pressure, was 540*3 grains; the
average for the seven preceding years was 542 grains.

The rain fallen at Greenwich in October was 3*50 inches ;
in November was 1*20 inch; and in December was 2*55 inches.
In October, in the years 1841 to 1847, were 5*95, 1*41, 4*25,

during the Qtiarter ending December 31,1 848. 185

4*03, 1*38, 5*13, and 2*00 inches respectively; the mean of
these values is 3*45 inches. In November, in the years 1841
to 1847, were 3-70, 428, 230, 4*32, 2*40, 1-52, and 200
inches respectively ; the mean of these values is 2*92 inches.
In December, in the years 1841 to 1847, were 2-40, 0*74, 0*40,
0*42, 2*00, 1 "13, and 200 respectively; and the mean of these
values is 1*29 inches. The depth of rain in October this year
was nearly the same as the average from the seven preceding
years, the fall in three instances being less, and in four exceed-
ing that of this year. In November the fall in this year was
less than that in any corresponding period since the year 1828,
its amount being 1*72 inch less than the average from the
seven preceding years. In December the fall exceeded that
in every December since 1833, the mean excess being 1*26
inch above the average since 1841. In October rain fell on
twenty -four days, on fourteen of which the amount was less
than 0*1 inch ; on six it was between 0*1 inch and 0"2 inch;
on three it was greater than 0"2 inch and less than 0*3 inch ;
there was one instance exceeding 0*3 inch, one exceeding
0*4 inch, and one between 0*5 inch and 06 inch. In Novem-
ber there were only two instances of the fall in one day ex-
ceeding 0*1 inch ; on one of these it amounted to 0*390 inch.
In December there were three instances exceeding 0*1 inch,
five exceeding 0*2 inch, and one amounting to 0*685 inch ; on
all other days the fall was less than 0*1 inch. The amount
for the quarter is 7*25 inches, and the average from the seven
preceding years is 7*66 inches.

The fall of rain during the year 1848 at Greennsoich was 31*9
inches; in 1841 it was 33*3 inches; in 1842 it was 22*6
inches ; in 1843 it was 24*5 inches; in 1844 it was 25 inches ;
in 1845 it was 22*3 inches ; in 1846 it was 25*3 inches; and
in 1847 it was 17*6 inches. The mean of their values is 24*4
inches; so that the excess of the fall of rain this year over
the average from the seven preceding years is 7*5 inches. At
Beckington it was 43*16 inches; in 1845 it was 24*94 inches;
in 1846 it was 32*30 inches; and in 1847 it was 28*74 inches.
In 1845 it fell on 134 days; in 1846 on 168 days; in 1847 on
151 days; and in 1848 on 219 days, as registered by the Rev,
Charles Blathwayt.

At St. John's Wood, London, the fall exceeded the average
from ten years by 5*05 inches, as observed by George Leech,
Esq.

At Aylesbury it fell on 174 days, amounting to 34*68 inches,
exceeding the average from the preceding six years by 9*5
inches, as observed by Thomas Dell, Esq.

At Empingham it amounted to 30*36 inches, which is the

186 Mr. J. Glaisher's Remarks on the Weather

largest fall since 1830, as observed by William Fancourt,
Esq.

At Derby was 'tO'O? inches, exceeding the average from
the preceding four years by more than 10 inches, and by 12
inches the average from twenty years, as observed by John
Davis, Esq.

At Leeds was 37*86 inches, it having fallen on 244 days.
In the year 1846 it fell on 218 days, and in 1847 it fell on
174 days ; and the amount was 28*442 inches, as observed by
Charles Charnock, Esq.

At Hereford was 46*41 inches ; the average fall from a long
series of years is rather more than 30 inches, as observed by
James Pendergrass, Esq.

The fall of rain during the year 1848 all over the country
was about one-third larger than the average fall, and this ex-
cess fell during the first three quarters. The fall in the last
quarter was about its average value at most places.

The temperature of the isoater of the Thames was 47°'5 by
day, and 45°'7 by night. The water, on an average, was of
the same temperature as that of the air. During the quarter
the temperature of the water has changed more than usual ;
the decrease of temperature from November 4 was rapid.

The direction of the wind at Greenwich —

From October 1 to 11 was chiefly S.W. ; between October
11 and 20 was chiefly N. ; and from October 20 to 31 was
mostly S.;

From November 1 to 7 was variable, but was chiefly S.W.
and N.W. ; from the 7th to 15th was N.; from the 16th to
the 21st was S.W.;' from the 21st to the 23rd was S.E. ; and
was chiefly S.W. to the end of the month;

From December 1 to 9 was S.W. ; from the 9th to the 15th
was mostly S. by E., and was then N. and N.E. to the end of
the month.

Jlie daily horizontal movement of the air —

From October 1 to 11 was about 160 miles; the greatest
value during the period was 300 miles, and the least was 80
miles; from October 11 to the 20th was 130 miles ; the great-
est was 270 miles, and the least was 30 miles; and from October
20 to the end of the month was 150 miles; the greatest being
240 miles, and the least 40 miles. The average for the month
was 150 miles daily;

From November 1 to 7 was 150 miles, the greatest and least
being 245 miles and 10 miles; from November 7 to 15 was
110, the extremes being 200 miles and 80 miles; from the
16th to the 2 1 St was 250 miles, the extremes being 495 miles
and 185 miles; from the 21st to the 23rd was 190 miles; and

during the Quarter ending December 31, 1848. 187

from the 24th to the end of the month was 230 miles, the ex-
tremes being 300 miles and 70 miles; the average for the
month was 165 miles;

From December 1 to 9 was 290 miles daily ; from the 9th
to the 15th was 170 miles ; and it was 94- miles from the 15th
to the end of the quarter. The extremes in December were
320 miles and 10 miles. The average for the month was 170
miles, and that for the quarter was 160 miles daily.

In October the readings of the thermometer on grass were at
and below 32° on four nights ; between 32° and 40 ' on fourteen
and above 40° on thirteen nights. In November the lowest
reading was 21°*5 ; the readings were below 32° on eighteen
nights, and above 32° on thirteen nights. In December the
lowest reading was 18°, and the readings were below 32° on
twelve nights, between 32° and 40° on fifteen nights, and above
40° on four nights.

The mean amount of clouds was 7*3 in October, and 6*7 both
in November and December. The averages for the seven pre-
ceding years were 6'9, 7*2, and 7*2 respectively.

There were no less than twenty-four exhibitions of the
aurora borealis during the quarter ending December 31, 1848,
which occurred on October 18, 19, 20, 22, 24, 25, 27 and 80,
both in the morning and in the evening of the 30th ; Novem-
ber 13, 14, 17, 18, 21, 23, 24, 25, 26, 30; December 13, 17,
22, 27 and 29. At all these times the magnets were more or
less disturbed. In the weekly reports it was stated that from
October 1 7 to 30 the magnetic instruments were almost always
under some cause of disturbance, and particularly on the 17th,
18th, 19th, 23rd and 24th, slightly on the 21st and 22nd, and
moderate on the remaining days. The finest aurora was that
on the 17th of November ; this was best observed by Professor
Challis, and described by him in the Cambridge Chronicle.
The most important part of his communication was that rela-
tive to the varying position of the corona. Professor Challis
says, " I took twenty-four observations of the position of the
corona, partly by reference to stars, and partly by a small
altitude and azimuth instrument expressly constructed for this
kind of observation, which I call a meteoroscope. A compa-
rison of the results of the several observations seemed to show
that the central point has not a fixed altitude and azimuth,
but oscillates in a capricious manner about a medium posi-
tion, more especially in the azimuthal direction." Observa-
tions of this kind are of the highest importance for com-
parison with the varying positions of the corona with the
simultaneous variations of the magnetic dip and positions of
the magnets. "'

188 Mr. J. Glaisher's Remarks on the Weather

Thunder-storms occurred at Whitehaven on October 9, 23,
28, 29, November 22, December 1 ; at Preston on Dec. 1 ;
at Stonyhurst on December 9, distant thunder and lightning
were noticed. Thunder was heard at Exeter on October 22
and on December 1. Lightning was seen at Truro on Oc-
tober 16, at Stone on October 28, at Saffron Walden on De-
cember J and 6, at Durham on October 18 and 28, at White-
haven on December 2, at Greenwich on October 25, and at
Stone on October 6 and November 3.

Hail fell at Truro on October 18, November 4, 7 and 8,
at Greenwich on December 1, at Exeter on December 23, at
Whitehaven on October 23, 28, 29, December 1 and 4.

Sno'w fell at Exeter, Empingham, and Saffron Walden on
October 18, at Truro, Southampton, Greenwich, and Em-
pingham on November 4, at Truro on November 7 and 8, at
Hartwell on November 23 and December 2, and at Exeter
on December 23.

Solar halos were seen at Maidenstone Hill, Greenwich, on
October 5, 24, 29, and November 25 ; at Stone on Nov. 30 ;
at Greenwich on October 24 ; at Highfield House on Oct. 1,
4, 18, 29, and December 2.

On November 8 a mock sun was seen at Highfield House.

Limar halos were seen on October 8, December 2, 4, 10
and 12.

Large and continuous falls of rain. — On October 23, at
Latimer Rectory, rain to the depth of 1*7 inch fell in twenty-
four hours following 9 a.m.

At Falmouth, on December 27, there was a heavy fall of
rain ; in a few hours 1*5 inch fell. At Truro, on December 27,
rain fell to the depth of 2*1 inches. In some parts of the
county of Cornwall the fall of rain on December 27 exceeded
2 inches; at Penzance more than 2 inches fell. Great da-
mage was done by the consequent floods.

The mean monthly values of the several subjects of research
for the times of observations are appended to the report of
the Registrar- General.

The monthly mean temperatures in the counties of Cornwall
and Devonshire exceeded those at other places; but there
seems to have been a good deal of bad weather in these coun-
ties, and more snow, hail and sleet seems to have fallen in
these counties than elsewhere.

The readings of the barometer till October 4 were between
29-5 inches and 29*7 inches ; after October 4 it steadily in-
creased, and passed the point 30 before noon on the 5th, and
remained above this point until the 7th ; the highest reading
was 30-062, and took place at 9^ a.m. on the 6th. Between

during the Quarter ending December SI, 1848. 189

the 8th and the 20th the fluctuations were very frequent, with
generally larger decreasing than increasing readings. On the
25th the reading was 29*1 11, and was the lowest in the month.
On the 26th, at 6^ p.m., it had increased to 29-749, and after
this the readings were low, and with slight variation to the
end of the month. The extreme difference of the readings
during the month was 0953 inch.

From November 1 to 6 the readings were between 29*6 and
29-4; it then increased from the latter reading to 30*248 on
the 10th at midnight. On the 15th the reading was 30*348,
which was the highest during the month. On the 18th, at
midnight, the reading was 29'417; on the 19th the increase
was 0*520 inch, and on the 20th the decrease was 0*454. On
the 23rd, at midnight, the reading was 29*048, which was the
lowest in the month. On the 25th, at noon, the reading was
29*984 ; after this the changes were small till the end of the
month. The range during the month was 1'300 inch.

On December 1 the reading decreased 0*436, and was
29284 at midnight; on the 2nd it increased 0*253, and on the
3rd, at 10^^ a.m., it was 29*730; it then decreased rapidly,
and the lowest reading during the quarter took place on the
5th at 6^ A.M. ; it increased slowly till the 7th, and then quickly
from the 7th to the 10th. The reading was above 30 from
the 10th to the 13th ; it was between 29*5 and 30 from the
13th to the 18th. On this day, at 6^ p.m., it was 29*677, and
on the 22nd, at midnight, the reading was 30*266, the highest
during the month. The reading was generally high till the
end of the month. The range during the month was 1*432
inches.

At Stonyhurst, from Nov. 1 to 6, the readings were between
29-098 and 29*355, it then increased to 30*150 at 1 1 p.m. ; on
the I2lh it remained above 29*8 till November 17, when it
decreased suddenly to 29*518, and gradually to 28*923 on the
20th ; it increased to 29*110 on November 21, but decreased
to 28*624 by 3^ p.m. on the 22nd ; it then increased steadily
till November 25 at 9 a.m., when it was 29*615, and the varia-
tions afterwards were small.

On December 5, at 9^ a.m., the reading was 28*421, the
wind at the time blowing strongly from the west.

Charles Charnock, Esq., of Stourton Lodge, Leeds, has
kindly furnished me with the following agricultural repprt for
the North Riding of Yorkshire.

" The continued rain from the 20th of September to No-
vember 1 prevented any large quantity of wheat being sown,
even on dry lands; and that which was sown was finished in
a very unsatisfactory manner. The comparative dry weather

1 90 Mr. J. Glaisher's Remarks on the Weather

from November 1 to 12 enabled the farmers to sow a great
portion of their wheat. On strong wheat soils a large breadth
remains for spring sowing with wheat or oats. The seed time
upon an average was nearly a month later than usual, and the
seed since has vegetated very slowly, owing to the wetness and
coldness of the soil.

" The continued fall of rain in September completely de-
stroyed the crops of corn in the backward situations, and large
quantities of barley, oats and beans, in the straw have been
carried into the yards for the cattle and pigs, as not worth the
expense of thrashing.

" The disease among potatoes has not been found so destruc-
tive as was anticipated, and will be more injurious to the grower
than to the consumer. In some situations the crops were
totally, and in others partially destroyed; yet from the great
extra breadth planted with this vegetable last spring, there
will perhaps be no great scarcity felt. I was most surprised
by seeing field potatoes taken up as late as the 18th of De-
cember.

" The crops of corn now thrashing are very deficient both in
quantity and quality. Turnips are an indifferent crop, and
do not bear much eating ; the sheep folded upon them have
been prevented from doing well by the wetness of the weather.
Symptoms of rot are apparent among many flocks of sheep.

"From the open weather the grass land has been full of meat,
and has kept cattle out of the straw yards longer than usual.
The disease on the lungs of beasts and milch-cows has been
prevalent and exceedingly fatal; the mortality is calculated to
have been 95 per cent, of those attacked.

" Within the last few weeks the epidemic prevalent in the
years 1839 and 184-0 has appeared among lean stock; its
symptoms are blisters on the tongue and lameness. It is not
often fatal, but reduces the catde attacked by it very much.

" Employment for agricultural labourers is scarce, and its ill
effects are much augmented by the great number of men who
have been discharged from the railways, whose intemperate
and vicious habits tend greatly to demoralise the agricultural
districts.

" Many of the low grounds have been flooded, and forming
operations prevented in consequence."

Themean of the numbers in the first column is 29"608 inches,
and this value may be considered as that of the pressure of dry
air for England during the quarter ending December 31, 1848.
The differences between this number and the separate results
contained in the first column show the probable sums of the
errors of observation and reduction; the latter arising partly

during the Quarter ending December 31, 1848. 191

from erroneously assumed altitudes, and partly from the index
error of the instruments not having been determined. In most
cases the sums of their errors are small.

The mean of the numbers in the second column, for those
places situated in Cornwall and Devonshire, is 47°*9 ; for those
places situated south of latitude 52°, including Chichester
and Hartwell, is 44'°'6 ; for those places situated between the
latitudes of 52° and 53°, including Saffron Walden and High-
field House, is 44-°'2 ; for those places situated between the
latitudes of 53° and 54°, including Liverpool and Whitehaven,
is 43°-3 ; and for Durham and Newcastle is 43°*0. These
values may be considered as those of the mean temperature of
the air for their parallels of latitude during the quarter ending
December 31, 1848.

The average daily range of the temperature of the air in
Cornwall and Devonshire was 9°*6 ; at Liverpool and White-
haven was 6°*9; south of latitude 52° was ll°'6; between the
latitude of 52° and 54° was 9°*6; and at Durham and Newcastle
was 8°-9.

The greatest mean daily ranges of the temperature of the air
took place at Greenwich, Hartwell, Latimer Rectory, and
Aylesbury respectively; and the least occurred at Whitehaven,
Guernsey, Torquay, Liverpool, and Truro respectively.

The highest thermometer readings during the quarter were
76° at Hartwell and Leicester, 74° at Greenwich and Aylesbury.
The lowest thermometer reading was 20°'5 at Stonyhurst,
and readings about 21°occurred at several places. The extreme
range of temperature of the air during the quarter was there-
fore about 55°.

The average quarterly range of the reading of the thermo-
meters in air in Cornwall and Devonshire was 37^*0 ; at Li-
verpool and Whitehaven was 36°'9 ; and the mean of the
numbers at all the remaining places is 48°*7.

The mean temperature of the dew-point in Cornwall and
Devonshire was 43°-5 ; at all places south of 53° was 41°*6 ;
and it was 39°'6 at places north of 53°.

The great mass of air has passed from the south-west in
all places except Exeter and Stonyhurst, at both of which
places it seems to have passed from the north.

From the numbers in the tenth column the distribution of
clouds has been the same at all places, and such as to have
covered somewhat more than three-fifths of the whole sky.

Rain has fallen on the greatest number of days during the
quarter at Highfield House, Stonyhurst, Derby, Leeds, Hel-
ston and Latimer, and the average number at those places
was 63. It fell on the least number of days at Aylesbury,

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On the Expression for the roots of a complete Cubic. 19S

Hereford, Newcastle and Saffron Walden, and the average
number at these places was 41. The places at which the
largest falls have taken place were Guernsey, Truro, Wake-
field, Whitehaven, Falmouth and Helston. The falls were
smallest in amount at Saffron Walden, Stone, Hartwell and
Liverpool. The average fall in the counties of Cornwall
and Devonshire was \2°'S ; and at all other places except
Southampton, Beckington, Hereford, Stonyhurst and York,
was 8°-5.

The numbers in column 13 to 17 show the mean values of
the hygrometrical results at every station ; from which we find
that—

The mean weight of vapour in a cubic foot of air for all
places (excepting Cornwall and Devonshire) in the quarter
ending December 31, IMS, was 3*2 grains.

The mean additional weight required to saturate a cubic
foot of air in the quarter ending December 31, ISIS, was 0*49
grain.

The mean degree of humidity (complete saturation = 1) in
the quarter ending December 31, 184-8, was 0'884.

The mean amount of vapour mixed with the air would have
produced water, if all had been precipitated at one time on
the surface of the earth to the depth of 3*7 inches.

The mean weight of a cubic foot of air at the level of the
sea, under the mean pressure, temperature and humidity, at
the mean height of 160 feet, was 541 grains.

And these values for Cornwall and Devonshire were 3*5
grains; 0*6 grain ; 0'839 ; 4*1 inches, and 540 grains respect-
ively.

XXIV. On the Expression for the remaining roots of a cotn-
plete Cuhicy Xfihen one root is found. By J. R. Young, Pro-
fessor of Mathematics, Belfast^.

THE following expression for two roots of an incomplete
cubic equation, in terms of the third root, is given at
page 216 of the Analyse Algebrique of Gamier, namely,

where q represents the coefficient of x, the term in x^ being
absent, and where x^ is a root already found.

This formula I obtained independently, and gave it at
page 322 of the second edition of my book on Equations. 1

* Communicated by the Author,

Phil, Mag, S. 3. Vol. 34. No, 228. March 1849. O

194 On the Expression for the roots of a complete Cubic,

here propose to deduce from it the suitable expression when

the equation is complete, as this is as likely to be useful as the

other.

Let the equation be

x^ +pa,^ + gx + r = 0,

then it is plain that we may accommodate the former expression

to this case, provided we increase each of the roots of this by

1 p^

- Pf and employ here S'— "^ for g' above; so that, still repre-

senting a root by x^, the foregoing expression under the ra-
dical will now become

{3x,^ + 2px,)+p^

A. -t

and consequently the expression for the remaining roots is

which may be stated in a rule, as follows : —

Add «", to the coefficient of <2?'^, and call half the sum, with
changed sign, a.

Subtract « from So^j, multiply the remainder by a.

Subtract the coefficient of a; from the quotient, and call the
result /3.

Then a+ \//3 will be the two remaining roots sought.

Dr. Rutherford of Woolwich has recently published a neat
and ingenious method of attaining the above object when the
root ^, is developed by Horner's method, without contraction
of the decimals ; in which case it is remarkably easy when the
coefficient of a?'^ does not consist of many figures. The pre-
ceding rule involves nearly the same numerical woi'k whether
the coefficients be large or small, as x^ will usually have several
decimals ; and its application does not preclude a free abridge-
ment of Horner's columns.

It is plain that, by a similar process, rules for the higher
equations, when all the roots but two are found, might be
contrived : but they would, in general, be complicated when
the equations are complete : when the second term of the
equation is absent, formulas for two roots, in terms of the
others, are given in the work before referred to.

Belfast, Feb. 16, 1849.

[ 195 ]

XXV. On Single and Double Vision produced hy viewing
objects iioith both eyes ; and on an Optical Illusion mth regard
to the distance of objects. By Joum Locke*.

I HAD commenced the investigation of this subject so early
as 1816, while I was a student of medicine in Yale Col-
lege, and I have occasionally turned my attention to it up to
the present time.

Although I have been fairly anticipated in the publication
.of some of my results, perhaps most of them, by the late in-
vestigations of Prof. Wheatstonef and Sir David Brewster J,
yet 1 deem it not useless to give you an account of the history
of my own experiments and conclusions, especially as some of
them are not, so far as I know, contained in the publications
of either of the distinguished philosophers who have just
written upon the subject. It is a well-known phaenomenon,
that with both eyes open we can see a single object either
single or double, according as the axes of the eye^ are made
to converge and meet either at the object, or at a point nearer
than that object. Having acquired the power of voluntary con-
vergence of the optical axes to an extreme degree without the
aid of viewing near objects, such as the nose, or a finger held
near to the eyes, I commenced my experiments as follows : —

Experiment I. — I viewed a burning candle at the distance
of about eight feet, the axes of the eyes being " crossed " or
extremely converged. Two images were of course seen, the
distance between which could be varied at pleasure by the
amount of that convergence. The two images of the candle
being thus seen, I suddenly closed one of my eyes, when the
image on the same side of the closed eye vanished. Thus on
closing the right eye, the right image disappeared ; and on
closing the left eye, the left image became extinct.

Inferences, — 1st. As the axis of the right eye was directed
to the left of the object, and the image which disappeared on
closing that eye was to the right of it, that image must have
been an oblique one, seen as v/e see lateral objects to which
the eyes are not directed. It appears, too, that while the axes
were converged upon vacancy, the oblique image in the right
eye took the place of an image formed directly in the axis of
the left eye, and the same relatively of the left eye; thus each
eye appeared to have an image in its axis, which image was
really in the opposite eye§.

* From Silliman's Journal for January 1849.
t I have not seen bis paper. % Phil. Mag., May 1847.

§ This is not always the condition of strabismus, for one eye may be so
directed that the axis shall be on the object while the other is oblique.

02

196 Mr. J. Locke on Single and Double Vision^

2nd. The two oblique images on the retina must have been
formed on points nearer to the nose, or nearer to the medial
line of the body than the principal axis of perfect vision.

3rd. The images appeared in such position as objects
should have been to produce pictures on the same parts of the
retina, the axes being at the same time parallel, or nearly so.

4th. As with both eyes we see a lateral object ordinarily
single, especially when at the same distance as the principal
object viewed, it is inferred that the two pictures, one in each
eye, must fall on parts of" the retina not correspondent to the
medial line of the body in order to produce single vision. For
example, on looking at a person standing ten yards in front,
the image of a person standing two yards or more to the right
will appear siiigle though not well-defined. The picture of
this second person must in such case be formed to the left side
of the retina of both eyes, towards the nose in the right eye,
Six\(\from it in the left eye. In these two situations on the
retina, and in no other, will the two oblique pictures present
a single image to the mind.

5th. All this establishes the principle, that certain parts of
the retina of one eye correspond to certain specific parts of the
retina of the other eye, in such a manner that "when identical
pictures fall on those correspond itig parts, single visioti is the
7'esult. Those corresponding parts lie inward in one eye and
outward in the other, viz. both to the right or both to the left.
From each of those corresponding parts of the retina it is pro-
bable that the fibres of the optic nerve proceed, and severally
unite at the point of anatomical communication where the
optic nerves cross before entering the brain ; hence the single
impression or single image.

It may be added to this experiment (I.), that if the finger
be pushed against the under part of the eyeball so as to roll it
upward, the image in that eye will appear to descend, and
double vision will thus be the result. Here the eye being
rolled upward, the image falls on the upper part of the retina;
hence the impression of a lower object would form a picture
on that part, were the eye not distorted*.

Experiment II. — Two candles of equal size and height were
placed side by side on the table, and by converging the axes
of the eyes, four images were produced. As the convergence
progressed, each pair of images receded gradually from the
original place of the single image until the two contiguous
ones, the second and third, approached, and finally coalesced

* In all cases the mind seems to make no allowance for distortion of the
eye, but refers the image to its true place were the eye in its natural po-
sition. Uij

and an Optical IllusiofU' ' • 197

into one, when three images only were in view. The same
experiment may be made with two letters, or any other figures
or objects which are equal in size and form, as follows: —
J* A A Natural single vision.

'2. AA A A View with axes slightly converged.

f View with greater convergence of the op-

3. A A A< tical axes and the two intermeUJatp

[_ images coalesced into one. j-ya^K/

On suddenly closing either eye, this middle or superimposed
image did not disappear, and it was evidently made of two
images from two objects formed on corresponding parts of the
retinaj. Hence we have the converse of the case of double
vision of a single object, for two objects are made to produce
a single impression. Thus far I had proceeded in 1816, when
I read a paper on this subject to a club of my fellow- students
at Yale. *^;

Experiment III. — In 1843 I made the experiment of cori-
verging the optical axes upon two contiguous figures on the
wall-paper of my office, in the same manner as I had done
with the two images of the two candles in Exp. II. When I
had thus succeeded in taking up optically the two figures and
superimposing them one upon the other, suddenly the whole
wail appeared to leap out from a distance of ten feet to within
half a yard of my eyes, where it remained in miniature beauty
as palpable to vision as it had been in its original place. To
this image, suspended as it were between the observer and the
object, 1 shall in the subsequent part of my paper apply the
term illusive image.

It then appeared that the right eye was directed to the left
one of two contiguous figures, and the left eye to the right
figure, which, being identical in form and size, gave the im-
pression of a single object at the point of intersection of the
optical axes. Here we have two triangles formed by the two
optical axes intersecting each other and joined at their ex-
tremes; on one part by a line from one eye ''>q"«' ^"'J ^■"'-"
to the other, and on the other part by a line ^ ' '

from one figure or object to the other. These
last lines being parallel (see figure), where A
and B represent the eyes, C and D the ob- \^^ rrtt ^c\i U
jects, or two figures on the wall, AD the axis
of the eye A, BC the axis of the eye B, and
E the point of intersection of the axes at the
place of the illusive image. As these trian-
gles are equiangular and similar, we can de-
duce from them all of the equations of such
triangles and apply them to the optical phse-

198 Mr. J. Locke oft Single and Double Vision^

nomena. Thus the distance from the eye to the illusive image
( AE) will be to the distance from the object to the same image
(DE) as the distance between the eyes (AB) is to the distance
between the objects (CD) the figures or pannels on the paper,
&c.

It is not merely the two objects directly in the axes of the
eyes which coincide, but every contiguous pair of objects seen
obliquely will also coincide, and form the illusive picture in
extenso. Indeed the optical operation of convergence seems
like taking up a duplicate copy of the figures lyingin the first
place exactly over them, and slipping it gradually to the ex-
tent of one figure, until again the figures coincide in a new
place.

Some of the phenomena of the Illusive Image. — It is quite
perfect, and can be viewed deliberately and critically as if it
were a real substance in place as it appears; the figures are
smaller than the originals in proportion as they are nearer;
as the outlines are a little blended by double pictures not ex-
actly coincident, an elegant softening and a playful beauty
exalts their effect above that of the original ; as the head
moves sideways, upward or downward, the illusive image
moves, but with a diminished motion ; as the head is inclined
to the right or left, the superimposed pictures slide out from
each other, the one ascending and the other descending to the
extent of the inclination.

Optical Equivalency. — The illusive image and the erroneous
distance at which it appears, show evidently that philosophi-
cally we do not see an object, but we contemplate an image on
the retina. If this image can by any means be formed without
the object, we still contemplate the substance such as would
produce that image. Thus in Exp. III., and the figure illus-
trating that experiment, the two objects C and D produce
each picture in the eyes at A and B, exactly as would be pro-
duced by a single object of smaller size at E. Thus the two
objects, one at C and the other at D, " fulfill the conditions
of the problem" of the images on the retina, exactly as it
would be fulfilled by a single smaller object at E. In both
cases identical pictures are formed on "corresponding" parts
of the two retinae. Hence the two objects produce the im-
pression of a single image.

Directions how to make the experiment of the Illusive Image.
— With two identical objects only, although it is easy to su-
perimpose them as in Exp. II., yet the illusion of distance
can scarcely be attained. But with a papered wall having a
repetition of the same figure at equal distances, a person who
has voluntary command of the optical axes will soon move the
double images to coincide, when presently the illusion will be

and an optical Illusion, 199

perfect. Persons who have not this command of their eyes
may succeed in obtaining the proper convergence by looking
at a finger held about fifteen inches from the face, while stand-
ing ten feet from a wall with figures twenty inches apart.

Apparent distances of objects. — It seems that we judge of
moderate distances by a kind of triangulation, the distance
between the eyes being a constant base-line. In order to put
this to the test, I have several times made the actual measure-
ments as in the following cases : —

Having measured the distance between my eyes, 2*6 inches,
the distance between the figures on the wall 21 inches, the
distance from the wall 10 feet, the distance of the illusive
image was calculated to be 14"7 inches, when it had been mea-
sured as near as may be 14<'5 inches. In a second experiment
we endeavoured to ascertain the distance of the observer from
the wall. The other data were —

Distance between the eyes . . 2*6 inches.

Distance between the figures . 21 inches.

Distance of illusive image . . 16*75 inches.

Calculated distance of the wall . 12*5 feet.

Measured distance of wall . . 13*15 feet.

When it is recollected that the observer is obliged to range
lengthwise on his measure while he determines the distance
of the aerial image, and that the base-line is only 2' 6 inches,
the above results appear quite as accurate as we ought to an-
ticipate.

There is peculiar beauty and accuracy in some of the results
of these experiments ; and it had occurred both to Sir David
Brewster and myself, that when a strip of wall-paper was placed
at a greater or less distance from its fellow than others, the
illusive image would not appear in the same place, some strips
would advance a little and others would recede, so as to fulfill
the conditions of the triangles above named ; even the six-
teenth of an inch would be appreciable.

In the history of my examination of this subject, I would
observe that my friend Dr. D, S. C. H. Smith, of Sutton, was
present when my paper was read at New Haven. In 1845,
my assistant, Thomas K. Beecher, A.M., witnessed and re-
peated most of the experiments above named- Among other
things we made the equations dependent upon the above tri-
angles, and verified our calculations.by actual admeasurement
of the distances between the eyes, between the objects, and
to the illusive image. I attempted a popular lecture on this
topic, but found it difficult to interest an audience in a matter
requiring so much previous optical knowledge. In the spring

200 On Single a?id Double Vision, and an optical Illusion.

of 1846 I communicated the leading principles of what I
thought then questionable discoveries, either to Prof. Bache
or to Prof. Henry, and consulted him as to their originality.
He gave his opinion that they were new. Without the least
disposition to contest the point of originality, which I have
failed to establish by neglecting to publish my results, I wish
merely to inform my friends of what I have in fact done, and
thus appear as a collateral witness to the trutii and interest of
Sir David Brewster's paper. He has brought forward some
points which had never presented themselves to me. That
figures less distant than the two eyes may be so viewed as to
form an illusive image at a greater distance than the object
itself, is evidently true, yet I had never made or anticipated
the experiment. Two such small figures might occupy such
a situation as to form the pictures on the retina due to a single
larger object placed at a greater distance, and thus become an
optical equivalent to that object. I am now experimenting on
the subject of single vision produced by two identical figures
of different colours. So far the results have not excited any
very surprising interest. The illusive image, as would be
anticipated, usually exhibits the effect of a commingling of the
colours ; but by directing the attention to one or the other
eye, one or the other colour may be made to predominate.
Thus a cameleon picture is formed, changing colour at the
will of the inspector.

Sir David Brewster alludes in his paper to some discoveries
made by Prof. Wheatstone, in reference to "binocular" vision
of objects of three dimensions. I have not seen the paper on
that subject, nor had I turned my attention in the least to its
consideration ; yet so intimately is it connected with the prin-
ciples just laid down, that upon its being named certain im-
portant conclusions at once present themselves. Thus when
the hand is held edgewise, within three inches of the nose,
one eye will receive an image of the palm and the other
of the opposite side ; and the two pictures, being dissimilar,
cannot fall on corresponding parts of the retina and produce
a single perfect image. Let any one make the experiment,
and he will perceive that Hogarth's caricature of bad perspec-
tive, in the figure of a barrel with both ends visible at the same
time, was not altogether absurd ; for if the barrel be shorter
than the distance between the eyes, it is practicable. The
same thing will occur with regard to any solid, as a cube,
which has several aspects, and the imperfection will be evi-
dently greater as the object is smaller and nearer the eye.

The experiments on this interesting subject can be extended
and varied in many ways highly interesting and instructive ;

Analytical Proof of the Parallelogram ofFoites, 201

and as no other apparatus is required than our eyes and the
objects of our inspection, it would seem that they were easily
made. But it requires rather an acquired power over the
organs of vision to be readily successful. Sir Ddvid Brewster
applied "binocular" convergence upon two figures, drawn
side by side to superimpose one upon the other, and compare
their exactness in point of size and form. I have extended
the same operation to figures of unequal size, though of the
same form. My son had just completed a half-size copy of a
drawing representing an Arab on horseback, the correctness
of which had been questioned. It was evident that, being
placed at distances proportionate to their size, the images
of the original and copy on the retina would be equal when a
consistent illusive image might be obtained by convergence.
The original was hung on the wall, and the half-size copy sus-
pended at about half the distance from the observer, at such
an angle that one could be fully seen beside the other. I
converged or superimposed the images, and found them so
nearly to coincide, that the common outline was merely ele-
gantly softened by the inequalities. In this experiment it ap-
peared as if the eye, when the figures did not exactly coincide,
had some power to complete the work or conceal the imper-
fections. "^''^
I have just succeeded in substituting a blank tablet for one
of the pictures, and in tracing upon it with a pencil the illusive
image converged from the other tablet. But this is not a
very practicable method of copying pictures, requiring unusual
command and steadiness of the optical axes for even the most
moderate success in the operation. '•• <--^-^\y'^^ -^^'i'

: ao!?gfobjdnor>

XXVI. Analytical Proof of the Parallelogram of Forces^ oq
By T. H. Pratt. - Tifinii adi

7b the Editors of the Philosophical Magazine and Journa^f .

Gentlemen,

IN my work on Mechanical Philosophy, I have given a proof,
of the Parallelogram of Forces which depends on the sonnj
hition of the following functional equation, _>, ,,,,, ,., ^^tm

C fit \1 2 -B ioti aew fSiotJt

T 1 111- . iOi' -[.'Ti "P'^i'^* em«K

1 there solved this equation indirectly, ihe following ih^^^

rect solution may be acceptable to some of your readers, .jg}^

202 Suggestiotisfor rendering a Meridian marie visible at NighU
and the equation (1.) becomes after reduction

9(S)-'p(|-^)=o,

which is rendered identical by putting (p(d) equal to any arbi-
trary function of sin 2d, as F(sin 2fl).

Hence the most general solution of equation (1.) is

(/(6)}2=:cos2fi+(cos9-sina)F(sin29). . . (2.)

In the application of this formula to the Parallelogram of
Forces, we must make use of some condition that the function
F may be determined ; as mechanical experience shows it
cannot remain arbitrary. Such a condition is the following :
that if fi be increased by tt, the resultant will obviously be the
same as before in magnitude, but opposite in sign : this will
therefore be the Ciise with/(fl).

Put, therefore, 7r + 6 for 9 in (2.), and equate the two values
of the square of the function (as they must be the same), and
reduce, and we have

cos2d+(cos5-sind)F(sin2d) = cos2d-(co5 9-sinfl)F(sin29);

.-. F(sin25)=0

for all values of fl.

Hence in the case of the Parallelogram of Forces,

/(6) = cos5 or — cos 9 J

the latter is excluded, because when 9 = 0, the resultant is the
force itself, or/(0)=l.

Hence/'(9) = cos d is the complete solution.
I am. Gentlemen,

Your obedient Servant,
Bombay, Dec. 14, 1848. T. H. PllATT.

XXVII. Suggestions for rendering a Meridian mark visible at
Night. By N. S. Heineken, Esq.

To the Editors of the Philosophical Magazine and Journal.
Gentlemen, Sidmouth, Feb. 14,1849.

SHOULD you deem the following suggestions likely to be
of service, 1 shall be glad if you will give them a place in
the Philosophical Magazine.

I am. Gentlemen,

Respectfully yours,

N. S. Heineken.

It occurred to me eleven years since, that platinum wire,
rendered incandescent by the galvanic battery, might be ap-

Mr. G. G. Stokes on the Theory of Sound. 203

plied to the purpose of rendering a meridian mark visible at
night; and that the same means might also be used for illu-
minating the wires of the transit instrument. I proposed that
the meridian mark should consist of a hole in a plate of brass,
adjustible in a vertical and horizontal direction by screws,
and that behind this hole should be placed the incandescent
wire in a glass tube for the purpose of illumination, the bat-
tery being of course at any required distance. For illumina-
ting the wires of the telescope, I proposed that the platinum
wire, protected by a glass tube, should be placed either within
or at the side of the eye-piece, and thus obviate the necessity
of piercing one of the transit-arms, as is usual. I was only
able to tr}' the above plans upon a small scale for want of a
more powerful battery ; but the experiments of Mr. Staite
lead me to think that there may be cases in which his method
of illumination by galvanism might be used with the greatest
advantage for rendering a veyy distant meridian mark visible.
I have found that even platinum wire, rendered incandescent
by alcohol^ may be distinguished by the telescope at a con-
siderable distance; as may also the hydrogen and platinum
lamp. By any of the above plans, the necessity of attention
to the lamp itself by an assistant is done away.

While upon the subject of micrometer-wires, may 1 also be
allowed to state, that so far back as 1831 I invented a substi-
tute for them by lines drawn upon glass with a diamond, which
lines were illuminated through the edge of the glass? but I
was led to abandon the plan after trial, fearing to introduce
the errors of the two surfaces of the glass, though I found that
in other respects the plan fully answered. I am induced to
mention this, having lately seen in the public prints that the
same method has been since independently discovered, and I
rejoice to find satisfactorily employed, by the Earl Rosse.

XX VIII. On the Theory of Sound, in reply to Professor
Challis. By G. G. Stokes, M.A., Fellow of Pembroke
College, Ca m bridge *.

AS the subject of plane waves does not seem likely to be
elucidated by further discussion, I pass on to spherical
waves.

Professor Challis has divided his demonstration of the
"contradiction" arrived at in this case into five heads. I
entirely agree with the first four ; the fifth I beg leave to dis-
pute. The part to which I object is contained in the sentence,

* Communicated by the Author. iKti^iiiiJi

204 Mr. G. G. Stokes om the Theory of Sound.

" Hence by the principle of tlie constancy of mass employed
in investigating one of the hydrodynamical equations, those
two quantities must be equal to each other." I deny that the
equality follows in any manner from the principle in question.
As the onus ■probandi evidently rests with Professor Challis,
I might here stop; but to render everything as definite as
possible, I will give a precise enunciation of the principle of
the constancy of mass, at least as I myself regard it, and I
have no reason to suppose that Professor Challis differs from
me in this respect.

Let S be any closed surface, finite or infinitesimal, drawn
in the fluid at the time /, r any finite or infinitesimal interval
of time ; and at the time ^ + t let the surface S' be the locus
of the particles which at the time t were situated in the sur-
face S : then the mass contained within the surface S' at the
time ^ + T is equal to the mass contained within the surface S
at the time /.

The principle of the constancy of mass might have been
enunciated somewhat differently, as follows ; and it will be
easily seen that the two enunciations come to the same thing.

Let S be any closed surface, finite or infinitesimal, drawn
within the fluid and remaining fixed in space; and let M be
the whole mass of fluid which flows across the surface S during
the time t, those portions being reckoned positive which flow
from without to within S, and those negative which flow from
within to without: then the mass contained within the surface
S at the time t-{-r exceeds the mass contained within the same
surface at the time t by the quantity M.

I will not at present say anything about the paragraph
Vi^hich follows (5.) ; because if Professor Challis and 1 can
agree about (5.), we shall probably have little difficulty in
agreeing about the paragraph in question.

Neither will I pursue the discussion further in the present
communication ; because, in addition to the motives which I
have already mentioned for declining to do so, it seems to me
that too great discursiveness is an evil in controversy, espe-
cially in mathematics, where one false step may invalidate all
that follows. I think it best to discuss only one fundamental
point at a time.

Pembroke College,
Feb. 3, 1849.

[ 205 ] ,-.^

XXIX. On the Composition of the Gold from California.
^,S; By T. H. Henry, Esq., F.R.S*

/^OLD as found in nature is never chemically pure, being
^^ combined with variable proportions of silver and traces
of iron and copper, and occasionally it occurs with palladium
and also with tellurium. '

The amount of silver was found by Boussingault in a series
of analyses of the native gold of Columbia to vary between 2
and 35 per cent., from which he drew the conclusion that the
gold and silver were combined in atomic proportions, 1 atom
of silver being constantly combined with more than 1 atom of
gold. 'J.'he specimen containing 35 per cent, of silver he con-
sidered to be a compound of 1 atom of silver and 2 atoms of
gold, Ag Au^, and that containing 2 per cent, of silver 1 atom
of silver and 12 gold, Ag Au^^.

This view of Boussingault was controverted by Gustav
Rose on his return from his celebrated journey to' the Ural
Mountains theoretically, on the ground that gold and silver
were isomorphous bodies, and such substances are not gene-
rally met with combined in atomic proportion. " It would be
as remarkable as if antimony, arsenic and tellurium were found
combined in atomic proportion," he remarks f ; " but as iso-
morphous substances do sometimes occur combined in atomic
proportion, as in bitterspar, diopside, &c., the only remarkable
result of the analyses of Boussingault is, that the gold and
silver should be constantly so combined ; " and experimentally
by the analyses of several specimens of native gold from the
Ural Mountains, in the greater number of which no such defi-
nite composition existed.

The purest specimen analysed by Rose contained 98*96 per"
cent, of gold and 0*16 per cent, of silver; the others contained
from 60 to 94< per cent, of gold.

The gold from California, a small quantity of which I ob-^
tained from Mr. Tennant of the Strand, was taken from a
quantity of about 60 lbs. weight, and was considered a fair
average sample of the whole : the greater part of it was in the
form of flattened grains or spangles, varying from j\jth of a
grain to 2 or 3 grains in weight ; one piece however weighed
upwards of 30 grains : the surface was rough and irregular,
with minute portions of siliceous matter imbedded in it. The
specific gravity of a number of the smaller grains taken toge-
ther by the bottle was 15*96; an analysis was made of these

• Communicated by the Author.

t PoggendorfTs Annalen, vol. xxiii. p, 164.

206 On the Composilion of the Gold from California,

by treating them with aqua-regia, separating the chloride of
silver, after dilution, by decantation of the solution of gold,
and the chloride of silver, after having been well washed, dried
and weighed, was dissolved in ammonia, leaving a white sili-
ceous residue, but no gold. The solution of the gold was,
after the destruction of the nitric acid by heat and hydrochlo-
ric acid, digested with oxalic acid until all the gold was pre-
cipitated ; the acid solution from the precipitated gold was
treated with sulphuretted liydrogen ; the precipitate of sul-
phuret of copper produced was ignited strongly, the metal
estimated from the oxide and a minute button of metallic
copper was procured from this by the blowpipe. After the
precipitation of the copper, the solution was evaporated to
dryness, the oxalic acid was expelled by heat, leaving a minute
quantity of chloride of iron*, which was dissolved in water aci-
dulated by hydrochloric acid and precipitated by ammonia.
The gold precipitated by the oxalic acid was entirely dissolved
by aqua-regia. In this manner these grains were found to be
composed of per cent. —

Or after abstraction
of siliceous matter.

Gold 88*75 . . . 90-01

Silver 8-88 . . . 9-01

Copper with trace of iron 0*85 . . . 0*86

Siliceous residue . . . 1*40

99-88 99-88

The larger piece or "pepite" weighed 30*92 grains, and its
specific gravity was 15-63. After being flattened out on a
polished steel anvil until it appeared free from foreign matter
and gently ignited, it weighed 30-24 grains, and the specific
gravity was now found to be 16-48.

10-96 grains, mostly of this larger piece, were analysed in
the manner described above, and were found to consist of in
100 parts, —

Gold 86-57

Silver 12*33

Copper 00-29

Iron 00*54

99-73

0-688 grain of this larger mass, assayed by the blowpipe by
the method described by Plattnerf, gave 86-33 per cent, gold ;
and a very thin spangle, which weighed 0*483 gr., and after

* Unless the oxalic acid has been prepared by sublimation, a small quan-
tity of carbonate of lime will be left after dissipation of the acid.
•j- Probirkunst mit dem Lothrohre. Leipzig, 1847.

On the Geological Structure of the AlpS) Carpathians^ Sfc. 207

fusion and separation of the siliceous matter '^Gl, gave 85'03
per cent.

I could detect no platinum, palladium, or any of the metals
usually combined with them, such as osmium, iridium, &c., in
this gold ; but the small amount at my disposal would not
allow me to employ a quantity sufficient to enable me to pro-
nounce absolutely on the absence of any traces of them.

The remark of Dumas ( Traite de Chimie appliquee auxArts,
tome iv, p. 434-), that the proportions of gold and silver are so
nearly constant in the mineral from the same locality {gisement)
that the assayers know the composition when they have ascer-
tained the precise locality which furnished it, is not confirmed
by the above analyses, in which the gold varies from 8.5 to 90
per cent., nor indeed by those of G. Rose, of four specimens of
gold from the same spot (Boruschka), which contained re-
spectively 5*23, 8-35, 9*02 and 16*15 percent, of silver.

This gold has very nearly the colour of the pure metal;
after fusion however it becomes of a brass-yellow colour. This
fact, together with the appearance of the grains under the mi-
croscope, would almost induce one to believe that the surface
of the grains was purer or ^^Jiner" than the interior, and that
a portion of the silver had been removed from the surface by
some chemical agent in nature. Prof. G. Rose, at the end of
his memoir already quoted, refers to the opinion prevalent
both in the Ural and at St. Petersburg, that the gold from the
washings is purer than that from the mines, and appears suc-
cessfully to combat both this opinion and the speculations of
Ferussac, who would account for it by the action of sea-water,
&c.; but I must refer the reader to his memoir for his argu-
ments, which are of great interest.

XXX. On the Geological Structure of the Alps, Carpathians
and Apennifies, more especially/ on the transition fro/n SecoU'
daryto Tertiary Types and the existence of vast Eocene De-
posits in Southern Europe. By Sir Roderick Impey Mur-
CHisoN, F.E.S., V.P.G.S.i <^c.; Mem. Imp. Ac. Sciences of
St. Petersburgh, Corresp. Member of the Academies of Paris,
Berlin, Turin, Sj-c*

nPHIS memoir, chiefly the result of the author's last excursion on
the Continent, consists of three parts, the first of, which is an en-
deavour to bring up to the present standard of knowledge the work
on the Eastern Alps, published long ago by Prof. Sedgwick and him-

* Abstract of a Memoir read before the Geological Society Dec. 13. 1848
and Jan. 17, 1849. '

208 Sir R. I. Murchison on the Geological Structure

self*, and to extend the sun'ey from that jDortion of the chain to
Switzerland and Savo)'. The second part is a brief explanation of
his present views respecting the northern flank of the Carpathian
mountains, and the third relates to Italy and the Apennines.

The Alps. — The central masses of the Eastern Alps, though in
parts highly crystalline, contain recognizable remnants of Upper Si-
lurian, Devonian, and carboniferous deposits, as proved by organic
remains; but no traces of the Permian system f of the author, so
abundant in Russia, Germany and England, have been found in
them or in any part of Southern Europe. In the same regions, viz.
in the South Tyrol and the Salzburg Alps, the above-mentioned palae-
ozoic formations are succeeded by trias, vi'ith true " muschelkalk "
fossils, as recently expounded by Von Buch, Emmerich, Von Hauer,
and other geologists. But in following the central parts of the chain
from Austria into Switzerland and Savoy, all fossil evidences of these
paleozoic and triassic deposits cease ; which, if ever they existed,
have been obliterated by the very powerful action of metamorphism
■which has affected the Western Alps, llie presence, however, of
undoubted species of old coal plants in Savoy has led some geolo-
gists to believe that the carboniferous system had some representa-
tive there ; whilst M. E. de Beaumont and M. Sismonda contend,
that the association of such plants with belemnites proves that they
occur in the lias of this part of the chain (Mont Blanc, Taren-
taise, and Maurienne), so clearly recognized by its numerous animal
organic remains. Sir R. Murchison allows, after personal inspection,
that in the much-disputed locality of Petit Coeur, the coal-plants
and anthracite really appear to lie in the same formation with the
belemnites as described by M. E. de Beaumont.

After a notice of the better acquaintance of geologists at this day
with the fossils of the secondary rocks of the Alps than when Prof.
Sedgwick and himself described them, and after showing the great
value of the Oxfordian group of Von Buch as the clear uppermost zone
of the Jurassic limestones, the author goes to his chief point, and
proves by a number of natural sections, that the opinion for which his
colleague and himself formerly contended, and which met with so
much opposition, is at length completely established ; — that the flanks
of the Alps exhibit a true transition from the younger secondary into the
older tertiary strata. But whilst this principle was correct, the author
allows that his friend and himself were in error in applying it to the
Gosau deposits ; all the lower and fossiliferous parts of which he now

• Trans. Geo). Soc. Lond. N. Ser. vol. iii. p. 301, and Phil. Mag. N.S.
vol. viii. Aug. 1830.

f The term " Permian," derived from the vast region of Russia, where
this uppermost Palaeozoic system is more largely developed than in any
part of the world hitherto examined (see Russia in Europe and the Ural
Mountains), embraces in its meaning the Rothe Todt-liegende, Kupfer
Schiefer and Zechstein of the German geologists, and among the latter.
Professor Naumann, Dr. Geinitz and Capt. Gutbier have recently adopted
the new term. In England the term includes the Lower Red sandstone,
and the magnesian limestone. As far as researches have gone, it would
appear that the Permian system is omitted in Southern Europe.

qfi/ie Alps, Carpathians and Apennines, * ' 209

aamits to be cretaceous. In common with all the geologists of their
day, they also formed an erroneous opinion of the age of the " flysch "
in viewing it as secondary greensand.

He now specially refers, as the base of all his subsequent results,
to a memoir of his own, read before the Geological Society in 1829
(Annals of Phil, and Phil. Mag. June 1830), which proved that on
the edge of the Venetian Alps, near Bassano and Asolo, the white
and red scaglia, or chalk, is there conformably succeeded by the
nummulitic and shelly deposits of the Vicentine, which are unques-
tionably of lower tertiary age, and graduate upwards through other
shelly strata and sandstones into marls and conglomerates with sub-
Apennine fossils. It has since been ascertained that deposits with
the same shells, Echinidae and nummulites of older tertiary age,
enter far into the higher Alps of the South Tyrol, and are there ele-
vated to great heights on the surface of limestones v/hich represent
the chalk. Natural sections are then described in Savoy, Switzer-
land, and Bavaria, which show a clear ascending order from the
Neocomian limestone (a formation unknown when he formerly
visited the Alps), or equivalent of the lowest greensand of Eng-
land, through a zone charged with fossils characteristic of the gault
and upper greensand into a limestone containing Inocerami and Anan-
chytes ovata, which, whether of white, gray or red colour, unques-
tionably stands in the exact place of the white chalk of Northern
Europe. Certain conformable transitions from this inoceramus lime-
stone up into shelly and nummulitic strata, like those of the Vicen-
tine, are pointed out, particularly at Thones in Savoy, at the Hoher-
Sentis in Appenzell and near Sonthofen in Bavaria, where these
intermediate beds, partaking of all the mineral characters of the great
eupracretaceous groups, or "flysch," are still characterized by a
Gryphaea, which is not to be distinguished from the G. vesicularis
of the upper chalk. Above this zone (?'. e. in tracts free from
dislocation and inversion) no traces have been discovered of any
one fossil referable to the cretaceous system ; the overlying strata
being unequivocally nummulitic and shelly rocks, which ai-e linked
together by position and fossils, and which on the north flank of the
Alps (especially at Sonthofen and Kressenberg), as well as on the
high summits of the Diableretz and Dent du Midi, represent the
lower tertiary of the Vicentine. The upper portion of this group, so
vastly expanded on the north flank of the Alps, is a collection of
shale, impure limestone and sandstone, the " flysch " of the Swiss,
to a great extent the " Wiener Sandstein," or fucoid grit of the
Austrians*, and the " Macigno " of the Italians. The whole group
of nummulite rocks and " flysch," much loaded with chlorite, pre-
eminently a " greensand," and often assuming a very ancient litho-
logical aspect, is not, as many geologists (including himself) sup-

* In an able map of the Northern Alps of Bavaria and Austria, M. Mor-
lot had placed the nummulite and flysch rocks above the chalk. Now,
however, great confusion prevails among the Austrian geologists respecting
the position of the " Wiener Sandstein," which has recently been mapped
as " Keuper."

Phil. Mag. S. 3 . Vol. 84. No. ^28. March 184-9. P

210 Sir R. I. Murchison on the Geological Structure

posed, an upper member of the cretaceous rocks, but really represents
the true eocene. The adoption of this view, which it is supposed all
palaeontologists must adhere to, seems already to be also in great
part taken by M. Boue, in opposition to his former opinion. In
reviewing the physical relations of the upper secondary and lower
tertiary rocks of the Alps, it is made manifest, that the independence
of any one member of this succession cannot be assumed from its
unconformability to others in certain localities ; inasmuch as such
appearances are proved to be local pheenomena only, by a more
general survey which detects the order to be unbroken and conti-
nuous. In the Alps therefore, as in Russia, where deposits of several
ages are conformable, the limits of formations can alone be defined
by their imbedded organic remains.

The author next developed the true age of the " Molasse and Na-
gelflue" of the northern portion of the Alps. Citing the researches
of Prof. Studer, M. Escher and others, he showed that the axis or
older part of these tertiary deposits was usually removed to some
distance from the higher ridges of cretaceous and eocene rocks, and
consisted of freshwater strata ; that the central or marine accumula-
tions are from their fossils (as collected in the Cantons St. Gallen
and Berne) of sub-Apennine or pliocene age, and that the great over-
lying portion of molasse and nagelflue, which frequently (owing to
enormous dislocations) seems to dip under the older rocks, out of
which it has been formed, is again, as far as can be ascertained, of
terrestrial and freshwater origin. Following these deposits in as-
cending order to their outermost and superior zone, they are found
to be surmounted by the well-known lacustrine formation of CEnin-
gen, formerly described in some detail by the author*. The remark-
able feature of this deposit is, that although it has unquestionably
been formed long after pliocene marine deposits (in which shells
exist undistinguishable from those now living), its fauna and flora
consist entirely of lost species. The examinations of its quadrupeds,
chelonia and reptiles by Herman von Meyer and Owen, of its fishes
by Agassiz, and of its plants by Goppert, all lead to this conclusion.
Even in respect to the insects of Qilningen, Prof. Heer of Zurich
has recently satisfied himself that in a multitude of species which
he is about to describe not one is identifiable with a living form.
Hence Sir Roderick maintains, that the terms miocene and pliocene
cannot be correlatively deduced from submarine and terrestrial for-
mations ; since if this be done in Switzerland, types of lost terrestrial
species overlie existing marine forms.

In concluding his observations on the Alps, attention was called
to the extraordinary contortions and convulsions they had undergone.
By diagrams of various transverse natural sections it was shown that
the Oxfordian, cretaceous, and eocene or nummulitic groups had
conjointly undergone such great flexures as in many instances to
produce absolute inversions, and in others great ruptures, both longi-
tudinal and transverse. Whilst the direction of the sedimentary
rocks is shown to conform to the axes of certain great ellipsoids
* See Trans. Geol. Soc. Lend. vol. iii. N. Ser. p. 277-

of the Alps, Carpathians and Apennines, 211

of crystalline rock, whether eruptive or purely metamorphic, the
deviations from such conformity are very numerous, particularly
where the strata wrap round the ends of each separate crystaUine
mass ; in illustration of which a geological map of the Canton of
Glarus, by M. Escher, was appealed to. Seeing that the forms of
the anticlinal and synclinal folds exhibited in his sections coincided
with the illustrations of the Appalachian mountains and other chains
recently produced by Prof. H. Rogers, the author — without offering
any opinion on the theory of that able geologist — pointed out that
in the Alps, as in the United States, the long and slightly inclined
slopes of each anticlinal face the great centre of disturbance, whilst
the short and steep sides of the same dip away from the chain. In
reference to the very frequent phaenomenon of the younger strata
(including the molasse) dipping under the older, particularly along
the line of great longitudinal faults. Prof. Rogers presented diagrams
explanatory of such overlaps in accordance with his theory.

Carpathians. — A brief sketch, the result of a survey in 1843, is
then given of the northern flanks of the Carpathian mountains. Indi-
cating the general succession northwards from the Tatra chain, the
author points out how a mass of nummulitic limestone, overlying
secondary rocks, dips under shale and sandstone like the flysch of the
Alps, such deposits representing, as in those mountains, the eocene
of geologists. An outer ridge (Zafflary and Rugosnik) of Oxfordian
Jura and chiefly Lower Neocomian according to Zeuschner and Key-
serling, rises abruptly through these superior deposits, and between
it and Cracow are undulating hills, much broken up and dislocated,
consisting of sandstones, shale, &c., in parts of which Prof. Zeusch-
ner has discovered many secondary greensand or Neocomian fossils.
These sandstones have a wide range, extending into Moravia, and
doubtless constitute a large portion of what has been termed Car-
pathian grit. But the author observes that in tracts like this, where
the cretaceous system assumes an arenaceous and earthy form, and
particularly in those districts where the nummulitic limestones no
longer exist, it is exceedingly difficult to draw any clearly defined
line of separation between sandstones of secondary and tertiary age.
He therefore believes that under the name of " Gres des Carpathes "
rocks both of eocene and cretaceous age have hitherto been con-
founded, and that arguments concerning the age of any given portion
of these sandstones in a country so constituted and so full of dislo-
cations are valueless without the test of organic remains.

The Apennines. — A general view of the structure of Italy is then
offered ; and whilst on the authority of General della Marmora the
existence of Silurian rocks in Sardinia is cited, it is shown that
the lowest fossiliferous deposits of the Peninsula are liasso-jurassic,
followed by limestones, often of red colours, of Oxfordian age (am-
monitico -rosso). These constitute a number of parallel ridges of
various altitudes, overlaid by or forming troughs with younger accu-
mulations, and thus forming numerous backbones, of which the
Apuan Alps and their crystalline marbles, the hills of La Spezia
and Pisa, are the most prominent examples in the North.

P2

212 Sir R. I. Miirchison on the Geological Slmcture

Admirably exposed on the flanks of the Venetian Alps, and
scarcely less so at Nice, the cretaceous system in all its members
(from the Neocomian limestones of foreign geologists or equivalents
of the English lowest greensand up to the white chalk inclusive) is
surmounted by nummulitic eocene deposits, which near Asolo and
Bassano are followed by miocene and pliocene shelly strata. After
showing how they occupy a trough between such Alps and the Eu-
ganaeans, the author explains how the latter hills have recently been
described by M. de Zigno as composed of Oxfordian Jura and a
full cretaceous system up to the white chalk inclusive, overlaid by
the nummulitic group. In Liguria, Modena, Lucca and Tuscany,
such clear evidences do not exist ; for there the formations above
the Oxfordian Jura are singularly devoid of fossils, and the series
between that horizon and the deposits of miocene age, with the ex-
ception of certain flaggy limestones (Alberese), assumes an arena-
ceous type. At very rare intervals only, and chiefly to the south of
Florence, are any bands of nuramulites observable ; but where they
occur the author refers all the " macigno " sandstone which is asso-
ciated with or overlies them to the eocene epoch ; such rocks being
perfectly undistinguishable from the " macigno Alpin," or flysch of
the Alps. As, however, these rocks repose upon others, including
vast thicknesses of the Alberese limestone, so largely seen in the
Apennines between Bologna and Florence and in the northern part
of the Tuscan Maremma, it is presumed that much of the latter mat/
represent the chalk. For although these rocks contain fucoids, one
or more of them being said to be similar in species to those which
overlie the nummulite strata of the Alps, no sort of reliance can be
placed on the presence of such marine vegetables, which in the Alps
range from the lower chalk high into the eocene ; whilst in Tuscany
an ammonite and a hamite have actually been found in these infra-
nummulitic masses. The inference of the author is, that Prof. Savi,
though correct in viewing a portion of this series as cretaceous, has
erred in including it in the nummulitic rocks.

In paying a just tribute to the talents, labours and character of
the lamented Prof. Pilla, the author avows the impossibility of ad-
mitting his term of " Systema Etruriano '"' as an equivalent for any
true geological division, as in it are comprehended strata which that
■writer had admitted to be cretaceous, with others which it is the
chief object of this memoir to establish as lower tertiary.

In passing into the Papal States and Naples, the superposition
of the nummulitic limestones with their usually associated fossils
to hippuritic limestones, the equivalents of the chalk, is seen to be
resumed ; and thus the same general succession as in the Alps and
Carpathians is maintained. Cases illustrative of this order, with
much overlying macigno, are pointed out in the Sabine Hills and
in the kingdom of Naples.

A transverse section of the Monferrato Hills (Superga) near Turin
exposes a most instructive tertiary succession. A coralline concre-
tionary limestone with small nummulites (Gassino), though de-
scribed and mapped as cretaceous by CoUegno and others, is shown

of the Alps, Carpathians and Apennines. 213

to lie at the top of the eocene or bottom of the miocene, and to
pass up through conglomerates, marls and sandstones replete with
the well-known miocene types of the Superga into the blue marls
and yellow sands of the Astesan, which are of sub-Apennine age.
The great interest of this section lies in its exposure of a vast
thickness of intermediate beds, in which the per-centage of recent
and fossil species is of so mixed a character, that for a league across
the inclined strata the able palaeontologists, E. Sismonda and Bellardi,
who made the section with the author, found it impossible to draw a
defined line between miocene and pliocene accumulation, so com-
pletely do they inosculate.

After describing the relations of the miocene and pliocene for-
mations near Bologna and in the Tuscan Maremraa, including the
great coal beds in the latter, which are believed to be of the older
miocene date, the relations of all these marine tertiary deposits to
younger ten-estrial and freshwater travertines and limestones is
traced; and reference is made to the more recent changes in the
configuration of the Campagna di Roma and valley of the Tiber,
with allusions to the labours of Monsignore Medici Spada and Prof.
Ponzi, from whom he announced future communications ; the one
on the igneous rocks of Latium, the other on the sedimentary de-
posits of the Papal States.

After briefly recapitulating the principal phsenomena in the Alps,
Apennines and Carpathians, the author dwells in conclusion on the
chief aim of his present communication, viz. the establishment of a
true equivalent of the eocene in Southern Europe. He analyses the
writings of the geologists who have recently described the nummu-
litic formations in the south of France,viz.Leymerie,Pratt,D'Archiac,
Delbos, RauUin, Tallavignes, Rouant, &c., and indicates how their
facts and his own are in harmony in showing the superposition of
such deposits to the true cretaceous system, no characteristic fossil
of which has been continued into the nummulitic group. Two or
three species of Gryphsese are alone common to the upper beds of
the one and the lower beds of the other. All the other fossils as-
sociated with the nummulites, whether from the Vicentine on the
south or from Sonthofen and Kressenberg on the north of the Alps,
are of tertiary forms, a certain number of them being absolutely
identical with species of the London and Paris basins. Looking to
the very great thicknesses and fine lamination of these accumula-
tions, including the shale, sandstone and lime^stone above the num-
mulites in the Alps, it is contended that as all these surmount the
white chalk, they must be an equivalent in time of what is legiti-
mately eocene, and that they do not merely represent, as suggested
by that eminent geologist M. E. de Beaumont, the interval which in
the North of Europe has occurred between the termination of the
chalk and the commencement of the plastic claj%

Extending the application of his view to still more southern and
eastern regions. Sir Roderick Murchison is of opinion that the great
masses of the nummulitic limestone of the Crimsea, Africa, Egypt
and Hindostan are also of eocene age ; or in other words, that from

214 Sir R. I. Murchison on the Geological Structure

the Carpathians to Cutch at the mouth of the Indus, a space of not
less than 25° lat. has been occupied by sea-basins in which creatures
of this sera lived. In reference to Ej^ypt, he cites copious collections
of shells and nummulites, chiefly those at the Royal Museum of
Turin, examined by M. Bellardi and himself ; and in regard to Hin-
dostan (after reverting to the Cutch fossils collected by Grant and
described by Sowerby), he pointedly dwelt on the rich and instructive
supplies of them recently sent home to him by Capt. Vicary from
Scinde and Sabathoo, and examined by Mr. Morris, which not only
demonstrate the existence of this same group in the Hala range, ex-
tending northwards towards Caubul, but also along the southern
edge of the Himalaya mountains.

The inference then is, that it is necessary to separate the vast
nummulitic formation, which the author believes to be eocene, from
the cretaceous system with which it has hitherto been merged, and
hence that a great change must be made in geological maps and
in the classification of the rocks of this age in South Europe and
other parts of the world. The union of the nummulitic and creta-
ceous groups in one system has been almost exclusively based upon
the prevailing phsenomenon of both having undergone the same move-
ments, and having been often elevated into the same peaks and
ridges. But such agreement in physical outline cannot be admitted
as invalidating the clear testimony borne by organic remains, and
from the study of which Brongniart, Deshayes, Agassiz, D'Orbigny
and Bronn have all placed the nummulitic group as lower tertiary.
Patient geological researches therefore at length prove that, when
clear from obscurities and unbroken, the order of superposition is in
harmony with the distribution of animal remains.

[P.S. In the course of the memoir, of which it is difficult to ex-
plain even the chief points in an abstract, the author particularly
cites Professors Studer and Brunner, jun., and M. Arnold Escher
von der Linth, as having rendered him very great services in his
examination of the Swiss Alps. In reference to Savoy, he mentions
the Canon Chamousset and M. Pillet; and respecting the East-
ern Alps, he points out the assistance he received, first from the co-
operation of his old associate M. de Verneuil in his re-examination of
Styria, Gosau, &c., and afterwards from M. Leopold de Buch, to
travel with whom through any part of that chain is to ensure good
results. It was when with M. de Buch and M. de Verneuil that he
explored the Triassic deposits of the South Tyrol. In attending the
Venetian Meeting of the Italian •' Scienziati" in the autumn of 1847,
the author further necessarily acquired much additional knowledge
there from intercourse with the geologists who have worked out the
details of that region, including Pasini, Catullo and De Zigno, and
he was then led to institute comparisons between some of the re-
sults of the Marquis Pareto in the western shores of the Southern
Alps, and with those of the Austrian geologists, V. Hauer, Morlot,
&c. in theEast.as well as from that excellent palaeontologist M.Ewald
of Berlin, But as at that time Sir Roderick had not examined either
the Swiss and Savoy Alps, the Monferrato, Apennines or Southern

of the Alps, Carpathians and Apennines. 215

Italy, any words he may be cited as having spoken at that Meeting
are not to be taken as affecting his ultimate conclusions expressed in
this memoir. Since it was read he has received a letter from M. Al-
cide d'Orbigny, which he willingly cites both as confirming his ge-
neral conclusions and as bringing these deposits into close comparison
with the lower tertiaries of Northern Europe. " For three years,"
M. d'Orbigny writes, " I have made the most extensive and most ge-
neral researches on the strata containing nummulites ; and in com-
paring all the stratigraphical and palaeontological results, it is im-
possible not to recognize therein two distinct epochs superposed the
one to the other, and having each its proper fauna. One of these
epochs, which 1 have recognized in the French Alps, the Pyrenees
and the Gironde, corresponds to the plastic clay of Paris and Lon-
don, and which, belonging to the lower sands of Soissons, I have
named ' ttage Suessonien' ; the other, equally common in the Alps
and the basins of the Gironde, and which includes the ' Calcaire
Grossier' of Paris up to^the gypsum of Montmartre and the London
clay, &c., I designate ' Etage Parisien.' These divisions, based upon
a considerable number of facts, are detailed in the work I am now
printing, and the entire composition of their characteristic faunas is
given in my ' Prodromus of Universal Palaeontology.' The habit I
have acquired of determining these fossils makes me regret that I
cannot go to inspect your collections in London ; but the portions
of them I have seen in the hands of our friend M. de Verneuil has
led me to recognize at once what I was already acquainted with in
the Pyrenees and the French Alps. Again, the fossils I have exa-
mined in the collection of M. TchichatchefF, confirm me in my opi-
nion, and would lead me to extend the limits of these tertiary stages,
as you have suggested, through Asia Minor and other tracts even to
Hindostan."

It may be added, that in citing the able memoir of M. Coquand*,
Sir Roderick has expressed his opinion, that the data, though con-
strued differently by that author, may be so interpreted as to lead to
the conclusion that the mass of the rocks containing nummulites in
the Barbary states and the shores of the Mediterranean are, like
those of the Alps and Apennines, supra- cretaceous ; his own limited
observations in the Neapolitan territories being confirmed by the
local knowledge of Professor Savi. Similar conclusions are, he thinks,
inevitable respecting the nummulitic rocks of Egypt, on the part of
any one who has read Russegger's work on that country. At the
same time, though well-assured of his own facts, he would not con-
tend against the possible existence in certain southern regions, not
examined by him, of some one species of nummulite in strata of the
age of the uppermost chalk, as insisted upon for the Crimsea by M.
Dubois de Montpereux, and for Cape Passaro in Sicily by M. Con-
stant Prevost. AH that he contends for is that the great nummulitic
group, as characterized by a multitude of species of shells, Echino-

* Description geologique de la partie septentrionale de I'empire de Ma-
roc, par H. Coqaand.—Bull. de la Soc. Geol. de France, 2nd ser. vol. iv.
p. 1188.

91V6 Royal Astronomical Society/.

derms, nutnmuUtes, &c., is a formation superior to and distinct from
the chalk ; and if there be situations (which however he has never
seen) in which a species of nummulite be common to the uppermost
chalk and lowermost tertiary, they would only the more confirm his
view of transition from the one epoch to the other in some regions
of the surface of the globe. In the memoir about to be published,
the author will give the result of the comparison of the species of the
nummulites, whether collected in the Alps or Hindostan, with those
of the south of France by M. le Vicomte d'Archiac, who has obli-
gingly compared them.]

XXXI. Proceedings of Learned Societies.

ROYAL ASTRONOMICAL SOCIETY.
[Continued from p. 150.]

Dec. 8, T^XTRACTS of a Letter from Dr. Forster, of Bruges.

1848. -L-^ " I have long wished to call the attention of the So-
ciety to a very curious fact in the chronology of lunations, if I may
so express myself ; but I have always been deterred by an appre-
hension that it had so much the air of superstition about it, that it
might, in many minds, rather excite ridicule than interest. Still,
however, facts are not to be despised ; and I have resolved to point
out to you, that whenever the new moon has fallen on a Saturday,
the following twenty days have been wet and windy. This must
depend on some cycle of lunations whose influence on our atmo-
sphere has hitherto escaped the notice of meteorologists. I first
perceived the coincidence to which I allude in Sussex, in the years
1817-27, and at that time thought it accidental ; but on accurately
examining a journal of the weather kept in my family by my grand-
father, my father, and myself, in succession, I find that in every
twenty Saturday's new moons, nineteen have been actually stormy
and the rest doubtful ; and this has been the case ever since our
journal began, a.d. 1767, up to the present time. I find, too, that
the greatest storms of wind on record have been during the month
following a Saturday's moon. It would be interesting to know
whether this observation applies to other latitudes ; and with a view
of ascertaining the same, it is that I have thought it worth while to
call the attention of the Society to the subject. For, during the
last twenty-nine years, I have been enabled, in some measure, to
predict the sort of weather that we should have for a long period, by
examining the calculated times of new moon. It may here be ob-
served, that stormy months, thus indicated, are characterised by the
prevalence of S.W. and W. winds.

"Periodical and other meteors. — On the night of the 13th of No-
vember last, a clear interval occurring between 10** and 13*^ 50^, I
observed the sky to be marked by numerous small meteors shooting,
in general, towards some point in the heavens, as nearly as I could
judge N.N.W. ; but unfortunately I was not in a position to make

Royal Astronomical Society. 217

any accurate observations. Several hundreds of meteors must have
occurred during the three hours and a half to which I allude ; the
clouds then closing the sky, I gave up observation. The meteors
were small and very white, and generally left long trains behind
them : one meteor had a contrary direction, it was larger than the
rest, and moved slowly across the zenith towards the S.E. I am
most decidedly of opinion that this phaenomenon is altogether atmo-
spherical and connected with electrical changes ; nor does their mo-
tion, in the apparent direction of the magnetic poles, at all militate
against this hypothesis of their electrical origin. A few occurred
last 10th of August, during a disturbed state of the atmospheric
electricity ; and I saw three on the 20th of December."

On the Variability of A Tauri, By Mr. Baxendell.

" On the night of the 6th instant I observed that the star A Tauri
was decidedly less bright than usual, being barely equal to v, a little
less bright than y, and decidedly below o and ^ ; whilst on the pre-
vious night I had noted it down as being a little brighter than o and
^, and decidedly above y and v, and in all my former observations I
had invariably placed it above y. On the following night (the 7th)
it had nearly recovered its usual lustre, being decidedly brighter
than y, above y, and equal to o and ^. A short time previous to the
6th instant I had remarked that my former observations of the stars
0, 0, and A Tauri exhibited discordances which rendered it impossible
to fix with certainty the order in which these stars ought to be
placed. After the observations on the nights of the 6th and 7th
instant, there could be no doubt that these discordances were mainly,
if not wholly, due to the variability of A ; and on carefully re-ex-
amining all my observations of this star, I was led to infer that its
changes were accomplished in a period of only about four days. I
therefore continued to watch it very closely, and on the night of the
10th instant had the satisfaction of again observing it reduced to an
equality with v. As, however, the presence of the moon on that
night might be supposed to have interfered with the estimations, L
have continued ray observations regularly since ; and having observed
A decidedly reduced in brightness on the nights of the 14th, 18th,
and 22nd instant, I can no longer have any hesitation in concluding
that this star belongs to the list of variable stars of short period,
being, in fact, the next in order after /3 Persei, the period of which
is the shortest yet known."

Dr. Gerling, of Marburg, published (Astron. Nach. 502) an ac-
count of a method for determining the parallax of the sun by obser-
vations on Venus and Mars M'hen nearest the earth, and requested
the co-operation of American astronomers. Lieut. Gilliss, having
satisfied himself that the method was feasible, volunteered his ser-
vices to the American government to carry Dr. Gerling's proposal
into effect, and the expedition is now preparing.

Lieut. Gilliss is to place himself in the most suitable station he can
find on the coast of Chili, where he is to make meridian and extra-
meridian observations of both planets, at the proper times, in cor-,

318 Royal Astronomical Society.

respondence with other observers at home. He also proposes to
observe an extensive catalogue of southern stars, and make various
astronomical and magnetical observations. His instruments are a
3-foot meridian circle, with a telescope of 52 lines aperture, made
by Pistor and Martins, of Berlin, under Professor Encke's direction ;
a 5-foot equatoreal, with clock-motion, by Fraunhofer ; clock, chro-
nometers, &c. Lieut. Gilliss expects to leave home in about six
months, and to be absent two or three years.

At the close of the evening the chairman informed the meeting
that the Astronomer Royal had presented the models of Lord Rosse's
telescope and polishing machine to the Society. Thanks were re-
turned to the Astronomer Royal for his present. They are now in
the meeting-room for examination.

Feb. 9, 1849 The Annual General Meeting of the Society, Sir

John Frederick William Herschel, Bart., President, in the Chair.

Before commencing the usual business of the meeting, the Presi-
dent rose and said : —

Gentlemen, — Before the proper and formal business of this meet-
ing begins, I must call your attention to the bust which you have
seen in our entrance-hall ; — it is that of our late beloved and re-
spected president, Francis Baily, a name which will never be men-
tioned in this Society without calling up a lively recollection of all
that is excellent in public, and amiable in private, character. When
you trace, as you cannot but do, in that marble the faithful and.
charming reproduction of features we have so often seen in the place
I now occupy, animated with the pure love of science, and with deep
interest in the welfare of this Society, you will be surprised to learn
that it is the production of an artist by whom these features had
never been seen but in the faint reflection of an engraving from his
portrait, and in that painful memento which preserves the impress
of a form from which the spirit has departed. When I name the
eminent artist, however, who has wrought this triumph over time
and oblivion (our celebrated sculptor, Edward Hodges Baily), your
surprise will cease. It is an achievement familiar to his chisel.

'J'he bust is presented to this Society by Miss Baily, the surviving
sister of our late president. She has judged riglitly in supposing
we shall value it. No possession we have will be more precious in
our eyes. Nowhere could a memorial of the kind be more appro-
priately placed than in the meeting-place of a public body with
which his name and his fame are so largely identified, and of which
he was so distinguished an ornament. We have now his picture
and his bust — both excellent. What art can do to keep his memory
fresh is done. It remains for us to show that his spirit is not ex-
tinct among us.

I am sure you will enable me to respond as I ought to do to this
touching and munificent gift of Miss Baily, vvho has requested me
to be her spokesman on this occasion ; and as there can be but one
feeling on the subject, I shall call on the Astronomer Royal to em-
body that feeling in a motion of thanks.

Royal Astronomical Society. 219

Proposed by G. B. Airy, Esq., seconded by A. De Morgan, Esq. :
— That the cordial thanks of the Society be given to Miss Baily for
this vaUiable present.

Address delivered by the President (Sir J. F. W. Herschel, Bart.)
on presenting the Honorary Medal of the Society to William Lassell,
Esq., of Liverpool.

Gentlemen, — The Report of the Council having been read, in
whicli the astronomical discoveries of the year, and especially that
of the planet Metis, have been clearly and eloquently commemo-
rated, it is now my pleasing duty to state to you the grounds on
which it has been agreed by us to award the gold medal of the So-
ciety for this year to Mr. Lassell. And this duty, pleasing in itself,
I execute with the greater satisfaction, because I have a sort of he-
reditary fellow-feeling with Mr. Lassell, seeing that he belongs to
that class of observers who have created their own instrumental
means — who have felt their own wants, and supplied them in their
own way. I believe that this greatly enhances the pleasure of
observing, especially when accompanied by discovery, and gives a
double interest in the observer's eyes, and perhaps, too, in some
degree, an increased one in those of the public, to every accession
to the stock of our knowledge which his instruments have been the
means of revealing : upon the same principle that the fruit which a
man grows in his own garden, cultivated with his own hands, is en-
joyed with a far higher zest than what he purchases in the market.
Nor is this feeling by any means a selfish one. It arises from the
natural and healthy excitement of successful exertion, and is part of
that happy system of compensation by which Providence sweetens
effort, and honours well-directed labour. If this be true of the
labour of a m;m's hands in the mere production of material and
perishable object^, it is so in a far superior sense, when the faculties
of the intellect are called into exercise, and works elaborated with
rare skill, and wrought to an extraordinary pitch of perfection, have
yet a higher, ulterior, intellectual object, to which their existence is
subordinate, as means to an end.

Mr. Lassell has long been advantageously known to us as an
ardent lover of astronomy, and as a diligent and exact observer, in
which capacity he has appeared before us, as a reference to our
Memoirs and Notices will testify, on numerous other occasions
besides those to which I shall more particularly call your attention
presently. In the year 1840 he erected an observatory at his
residence near Liverpool, bearing the appropriate name of Starfield,
which has ever since been the scene of his astronomical labours.
Even at its first erection this observatory presented features of
novelty and interest. In addition to a good transit, it was furnished,
instead of a meridian instrument or an ordinary equatorial achro-
matic, with a Newtonian reflecting telescope of nine inches aperture,
and rather more than nine feet in focal length, equatorially mounted,
the specula of which were of his own construction, and the mode
of mounting devised by himself. This was already a considerable

220 Hoi^al Astronomical Society.

step, and forms an epoch in tlie history of the astronomical use of
the reflecting telescope. Those only who liave had experience of
the annoyance of liaving to keep an object long in view, especially
under high magnifying powers, and in micrometrical measurements,
with a reflector mounted in the usual manner, having merely an alti-
tude and azimuth motion, can duly feel and appreciate the advantage
thus gained. But the difficulties to be surmounted in the execution
of such a mode of mounting were very considerable — much more so
than in the case of an achromatic, — owing partly to the non-coin-
cidence of the centre of gravity of tlie telescope and mirror with the
middle of the length of the tube, and partly to the necessity of
supporting the mirror itself within the tube in a uniform bearing
free from lateral constraint, and guaranteed against flexure and
disturbance of its adjustment by alteration of its bearings. These
difficulties, however, Mr. Lassell overcame : the latter, which is the
most formidable, by an ingenious adaptation of the balancing
principle first devised, if I am not mistaken, by Fraunhofer and
Reichenbach for the prevention of flexure in the tubes of telescopes
— a principle which has not received half the applications of which it
is susceptible, and which, by throwing the whole strain of the weight
of instruments on axes which may be made of unlimited strength,
may be employed to destroy the distorting force of gravity on every
other part*.

The success of this experiment was such, and the instrument was
found to work so well, that Mr. Lassell cortceived the bold idea of
constructing a reflector of two feet in aperture and twenty feet in
focal length, and mounting it upon the same principle. The cir-
cumstances of his local situation, in the centre of manufacturing
industry and mechanical construction, were eminently favourable to
the success of this undertaking ; and in Mr. Nasmyth he was fortu-
nate enough to find a mechanist capable of executing in the highest
perfection all his conceptions, and prepared, by his own love of as-
tronomy and practical acquaintance with astronomical observation
and with the construction of specula, to give them their full effect.
It was of course, however, the construction and polishing of the
large reflector which constituted the chief difficulty of this enterprise.
To ensure success, Mr. Lassell spared neither pains nor cost. As
a preliminary step, he informs us that he visited the Earl of Rosse,
at Birr Castle, and besides being favoured with more than one op-
portunity of satisfying himself of the excellent performance of that
nobleman's three-loot telescope, enjoyed the high privilege of exa-
mining the whole machinery for grinding and polishing the large
speculum, and returned so well satisfied as to resolve on the imme-
diate execution of his own ideas.

* As, for example, the divided limbs of circles, and the spokes connecting
them with their centres; an easy and simple mechanism, which, devised
some time ago, and approved by the late M. Bessel, I may, perhaps, take
some future opportunity to submit to the Society.— (A^o/e added in the
Printing.)

Royal Astronomical Societi/, 221

The mode of casting and grinding the mirror, differing in some
of the details, though proceeding generally on tlie same principle as
Lord Rosse's (t. e. by a chilled casting), has been described in a
communication read to this Society on the 8th of December last.
The polishing was performed on a machine almost precisely similar
to that of his lordship. But finding after many months' trial that
he could not succeed in obtaining a satisfactory figure, he was led to
contrive a machine for imitating as closely as possible those evolu-
tions of the hand by which he had been accustomed to produce per-
fect surfaces on smaller specula. This machine has been described
(and a model of it, as well as Mr. Nasmyth's finished working
drawings of it, exhibited) in a paper of great interest read at the
la/.t meeting of tliis Society, of which also an abstract has been
printed in our Notices, and must by this time be in the hands of
every fellow here present, so that it cannot be necessary for me to
recapitulate its contents. Suffice it to say, that I have carefully
examined both the drawings and tlie model, and having myself had
some experience in the working and polishing of reflecting specula,
approaching (though inferior) in magnitude to Mr. Lassell's, 1 am
enabled to say that it seems to unite every requisite for obtaining
a perfect command over the figure ; and when executed with that
finish which belongs to every work of Mr. Nasmyth, from the
steam-hammer down to the most delicate product of engineering
and mechanical skill, cannot fail to secure, by the oily smoothness
and equability of its movements, the ultimate perfection of polish,
and the most complete absence of local irregularities of surface.
The only part which 1 do not quite like about it, or perhaps I should
rather say which seems open to an a priori objection, refutable, and,
in point of fact, refuted by the practical results of its operation, is
the wooden polisher, owing to the possibility of warping should
moisture penetrate the coating of pitch with which it is (I presume)
enveloped on every side. Some unhygrometric, non-metallic sub-
stance, such as for instance earthenware, porcelain biscuit, or slate,
would be free from this objection, though possibly open to others of
more importance.

Both Mr. Lassell and Lord Rosse appear to be fully aware of
the vital importance of supporting the metal, not only while in use,
but also while in process of polishing, in a perfectly free and
equable manner ; but the former has adopted a mode of securing a
free bearing on the supports, by suspending the mirror, which is a
great and manifest improvement on the old practice of allowing it
to rest on its lower edge, by which not only is the figure necessarily
injured by direct pressure, but the metal is prevented from playing
freely to and fro, and taking a fair bearing on its bed. As I have,
however, on another occasion enlarged on the necessity of making
provision against these evils, by a mechanism almost identical in
principle, I need not dwell upon this point further than to recom-
mend it to the particular attention of all who may engage in similar
undertakings.

222 Royal Astronomical Society*

It is right that I should now say soir.elliing of the performance
of the nine-inch and two-feet reflectors. And first, as regards the
success of the system of mounting adopted in securing the peenHar
advantages of the equatorial movement. This appears to have been
very complete. The measurements, both differential and micro-
metrical, made with them, and recorded in our Notices, show that
in this respect they may be considered on a par with refractors, and
in ficility of setting and handling they appear nowise inferior. Of
the optical power of the former, two facts will enable the meeting
to form a sufficient judgement. With this instrument Mr. Lassell,
independently and without previous knowledge of its existence, de-
tected the sixth star of the trapezium of 9 Orionis, And with this,
under a magnifying power of 450, and in very unfavourable circum-
stances of altitude, both himself and Mr. Dawes became satisfied of
the division of the exterior ring of Saturn into two distinct annuli, a
perfectly clear and satisfactory view of the division being obtained.

The ieats performed by tlie larger instrument have been much
more remarkable and important. It has established the existence
of at least one of the four satellites of Uranus, which since their an-
nouncement by Sir W. Herschel had been seen by no other observer,
viz. the innermost of all the series, and afforded strong presumptive
evidence of the reality of another, intermediate between the most
conspicuous ones. The observations of M. Otto Struve, if they
really refer to the same satellite, are of nearly a month later date.

'i'o Mr. Lnssell's observations with this telescope we also owe the
discovery of a satellite of Neptune. The first occasion on which
this body was seen was on the 10th of October, 1846, but owing to
the then rapid- approacli of the planet to the end of its visibility for
the season, it could not be satisfactorily followed until tlie next year,
when, on the 8th and 9th of July, observations decisive as to its
reality as a satellite were made, and in August and September full
confirmation was obtained. This important discovery has since
been verified both in Russia and in America. I call it so, because,
in fact, the mass of Neptune is a point of such moment, that it is
difficult to overrate the value of any means of definitively settling
it. Unfortunately, the exact measurement of the satellite's distance
from the planet is of such extreme difficulty, that up to the present
time astronomers are still considerably at issue as to the result.

I come now to the most remarkable of Mr. Lassell's discoveries,
one of the most remarkable, indeed, as an insulated fact, which has
occurred in modern astronomy : though, indeed, it can hardly be
regarded as an insulated fact, when considered in all its relations.
I need hardly say that I allude to the discovery of an eighth satel-
lite of Saturn, a discovery the history of which is in the highest de-
gree creditable, not only to the increased power of the instruments
with which observatories are furnished in these latter days of astro-
nomy, but also to the vigilance of observers. IfT am right in the
principle that discovery consists in the certain knowledge of a new
fact or a new truth, a knowledge grounded on positive and tangible

Royal Astronomical Society. 223

evidence, as distinct from bare suspicion or surmise that such a fact
exists, or that such a proposition is true — if I am right in assigning
as the moment of discovery, that moment when the discoverer is
first enabled to say to himself, or to a bystander, " I am sure that
such is the fact, — and I am sure of it, for such and such reasons,"
reasons subsequently acquiesced in as valid ones when the discovery
comes to be known and acknowledged — if, I say, I am right in this
principle (and I really can find no better), then I think the discovery
of this satellite must be considered to date from the 19th of Septem-
ber last, and to have been made simultaneously, putting difference
of longitude out of the question, on both sides of the Atlantic. In
speaking thus, I desire, of course, to be understood as expressing
only my own private opinion, and in no way as backing that opinion
by the authority of the Society whose chair I for the moment occupy.
The Astronomical Society receives with equal joy the intelligence of
advances made in that science from whatever quarter emanating,
and accords the meed of its approbation to diligence, devotion, and
talent, with equal readiness wherever it finds them — but declines
entering into nice questions of personal or national priority, and
would, 1 am sure, emphatically disavow the assumption of any title
to lay down authoritative rules for the guidance of men's judgements
in such matters. The medal of this day is awarded to Mr. Lassell,
not on account of this discovery alone, and as such, but as taken in
conjunction with the many other striking proofs he has afforded of
successful devotion to our science — both in the improvement and
in tlie use of instruments. And among the motives which have
induced your Council to place Professor Bond on the list of our
Associates (1 trust not long to be the only one of his countrymen
by whom that honour is enjoyed), though this discovery has had its
due and just weight, we liave not been unheedful of his general
merits, both as an observer and as a theoretical astronomer — merits
of which the Memoirs which have recently reached us convey the
most abundant evidence in both departments.

I have observed that, when taken in all its relations, the discovery
of an eighth satellite of Saturn cannot be regarded as quite an insu-
lated fact. Between lapetus and Titan there existed a great gap
unfilled, in which (as formerly between Mars and Jupiter) it was
not in itself unlikely that some additional member of the Saturniaii
system might exist. The extreme minuteness of Hyperion forcibly
recals the analogous features of the asteroids, and it would be very
far from surprising if a further application of the same instrumental
powers should carry out this analogy in a plurality of such minute
attendants.

Mr. Lassell, as you are all well aware, is bound to astronomy by
no other tie than the enjoyment he receives in its pursuit. But in
our estimation of his position as an amateur astronomer it must not
be left out of consideration, that his worldly avocations are such as
most men consider of an engrossing nature, and which entitle them
in their moments of relaxation, as they conceive, to enjoyments of
a very different kind from those which call into fresh and energetic

224 Royal Astronomical Society.

exertion all their faculties, intellectual and corporeal. It is no slight
and desultory exercise of those faculties which will enable any man
to carry into effect so much thoughtful combination, and to avail
himself with so much consecutiveness of their results when produced.
And however we may and must acknowledge that such a com-se of
action is really calculated to confer a very high degree of enjoyment
and happiness, we ought not to feel the less gratefully towards those
who, by their personal example, press forward the advent of that
higher phase of civilization which some fancy they see not indistinctly
dawning around them ; a civilization founded on the general and
practical recognition of the superiority of the pleasures of mind over
those of sense ; a civilization which may dispense with luxury and
splendour, but not with the continual and rapid progress of know-
ledge in science and excellence in art.

1 think I should hradly be doing full justice to my subject or to
the grounds taken by the Council in the award, if I were to conclude
what I have to say otherwise than in the pointed and em[)hatic words
of a report officially embodying the prominent features of the case.
♦' 7'he simple facts," says that document, " are, that Mr. Lassell cast
his own mirror, polished it by machinery of his own contrivance,
mounted it equatorially in his own fiishion, and placed it in an
observatory of his own engineering : that with this instrument he
discovered the satellite of Neptune, the eighth satellite of Saturn,
and re-observed the satellites of Uranus. A private man, of no
large means, in a bad climate" (nothing, 1 understand, can be much
worse), "and with little leisure, he has anticipated, or rivalled, by
the work of his own hands, the contrivance of his own brain and the
outlay of his own pocket, the magnificent refractors with which the
Emperor of Russia and the citizens of Boston have endowed the
observatories of Pulkowa and the Western Cambridge."

The President then, delivering the medal to Mr. Lassell, addressed
him in the following terms : —

And now, Mr. Lassell, all that remains for me is to place the
medal in your hands, aD,tl to congratulate you on your success and
on the noble prospect of future discovery which lies before you,
now that, free from the preliminary labour of construction, your
whole attention can be devoted to using the powerful means you
have created. In the examination of the nebulae, in the measure-
ment of the closest double stars, and the discovery of others which
have hitherto defied separation — in the physical examination of the
planets and comets of our own system, there is a wide field open and
the sure promise of an ample harvest ; and I can only add that we
all heartily wish you health and long life to reap it.

The Meeting then proceeded to the election of the Council for the
ensuing year, when the following Fellows were elected, viz. —

President,— G. B. Airy, Esq., M.A., F.R.S., Ast. Koy.

Vice-Presidents. — J. C. Adams, Esq.,M.A.; Edward Riddle, Esq.;
Rev. Richard Sheepshanks, M.A., F.R.S. ; Lieut. William Stratford,
R.N., F.R.S.

Treasurer, — George Bishop, Esq.

Cambridge Philosophical Sociclj/, ^225

Secretaries. — Augustus De Morgan, Esq. ; Captain R. H. Man-
ners, R.N.

, ,,. Foreign Secretary. — John Russell Hind, Esq.

.1, , Council. — George Dollond, Esq., F.R.S. ; Rev. George Fisher,
M.A., F.R.S. ; Sir John V. W. Herschel, Dart., K.H., M.A., F.R.S. ;
John Lee, Esq., LL.D., F.R.S. ; Rev. Robert Main, M.A. ; Charles
May, Esq.; Lieut. Henry Raper, R.N. ; William Rutherford, Esq.,
LL.D. ; Captain W. H. Smyth, R.N., K.S.F., D.C.L., F.R.S. ; J.
W. WooUgar, Esq.

CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Continued from p. 1.38.]

May 17, 1817. — A Theory of the Transmission of Light through
Transparent Media, and of Double Refraction, on tlie Hypothesis
of Undulations. By Professor Challis.

The object of the author in this, as in two preceding communica-
tions on Luminous Rays and on the Polarization of Light*, is, to esta-
blisli the undulatory theory of light on hydrodynamical principles,
by means of a system of ray-vibrations, the motions in which are
mathematically deduced from hydrodynamical equations. In ap-
plying these views to the transmission of light through transparent
media, it is assumed that the aether is of the same uniform density
and elasticity within any transparent medium as without ; and that
the diminished rate of propagation in the medium is owing to the
obstacle which its atoms oppose to the free motion of the aethereal
particle.s. Considering the proximity of the atoms to each other,
and that the retardinjj effect of each atom at a siven instant extends
through many multiples of its linear dimensions, it is presumed that
the mean retardation, though resulting from the presence of discrete
atoms, may be regarded as continuous. It is also supposed that
the mean effect of the presence of the atoms is to produce an appa-
rent diminution of the elasticity of the aether, the motion in all other
respects being the same as in free space. By the application of
these principles, it is first shown that the surface of elasticity , that is,
the surface whose radius vector drawn in any given direction repre-
sents the elasticity in that direction, is in general an ellipsoid. This
being ascertained, the velocity of a ray in any given direction is inves-
tigated ; and the result is, that the surface whose radius vectors
drawn in any given direction represent the velocities of propagation
of two oppositely polarized rays in that direction, is precisely the
wave-surface in Fresnel's theory of double refraction.

March 6, 18i8. — A Mathematical Theory of Luminous Vibra-
tions. By Professor Challis.

This paper is intended to be supplementary to three former com-
munications in which the undulatory theory is treated on hydrody-
namical principles, and to elucidate or confirm results previously
arrived at. In particular the author enters more at length into the
mathematical theory of ray-vibrations, which, according to his views,
* Phil. Mag. vol. xxx. p. 3G5.

Phil. Mag. S. 3. Vol. 34-. No. 228. March 184-9. Q

S26 Cambridge Philosophical Society.

correspond to rays of light. The principal theoretical deductions
are, — (1.) that the longitudinal vibrations of a ray are defined by a

function of the form sin t:^ — Iz—a <^ /i+iA.\ X being the
breadth of the undulation, and a, e certain constants ; (2.) that light
from any source is in general composed of rays for which a and —

are the same and X different ; (3.) that light coming immediately
from its origin is common light, whatever be the nature of the cause
producing it, and that to become polarised light, it must be acted
upon by reflexion, refraction, &c. ; (4.) that light coming imme-
diately from its origin is seen in all directions.

Nov. 27, 1848. — Observations of the Aurora Borealis of Nov. 17,
1848, made at the Cambridge Observatory. By Professor Challis.
These observations relate principally to the corona, or point of
apparent convergence of the streamers, the remarkable display of
Nov. 17 being peculiarly favourable for noting the position of this
critical point. They were taken partly by estimation of distances
from stars, and partly by a small altitude and azimuth instrument
(called by the author a meteoroscope), which is furnished with a bar,
eighteen inches long, carrying at one end a rectangular piece whose
edges are horizontal and vertical, by looking at which through an
eyelet-hole, about the size of the pupil of the eye, at the other end,
the collimation is performed. Each observed position is compared
with the point of the heavens to which the south end of the dipping-
needle was directed at the time of observation. Tiiis point was
ascertained by means of observations of declinntion,- horizontal
force, and vertical force, taken at tlie Greenwich Observatory during
the prevalence of the aurora by Mr. Brooke's photographic pro-
cess, the results of which were communicated to the author by tlie
Astronomer Royal. It is assumed that the magnetic declination and
dip at Cambridge differ from those at Greenwich at any given time
by certain constant quantities, whether the magnet be disturbed or
not. These constant differences were derived from the following
formula : —

V-Vo = 0-142518\ -I- 0-159548;

D-Do= 0027713\ + 0-5135231,
in which V and D are the declination and dip at a place not very
distant from Greenwich, Vq and Do' the contemporaneous declina-
tion and dip at Greenwich, X the longitude of the place 7vest, in
seconds of time, and I the excess of its latitude in minutes above that
of Greenwich. These are merely formulae of interpolation by sim-
ple differences derived from the following data : —

T ^ T tur i. Declination Dip

Lat. Long. West. -^^i^^^,. in 1843.

o / m. s. o / o I

Greenwich 51 28-6 0 0-0 23 17-59 69 1-9

Makerstoun 55 34-7 10 3-5 25 22-85 71 25-0

Dublin '53 21-0 25 4-0 27 9-87 70 41*3

Inlelligence and Miscellaneous Articles. 227

The above are very accurate contemporaneous values of the decli-
nation and dip at the three places, and tlie formulae derived from
them will probably apply with considerable accuracy to any place
in the United Kingdom at any date not very remote from 1843.
For the Cambridge observatory V— Vo= -f 3''7 and D— Do =

The mean result from 24 observations of the position of the
corona is, that it was situated 5' further from the astronomical
zenith, and 1° 14' nearer to the meridian than the point of the
heavens to which the south end of the dipping-needle was directed.

The places of the corona given by the different observations ex-
hibit considerable discrepancies, which are accounted for by saying,
that as the formation of the corona is merely an effect of perspective,
its position varies, since the streamers are not exactly parallel, with
the locality from which they rise ; also with any variation of their
direction at a given locality ; and, supposing the course of the
streamers to be somewhat curved in their ascent, it will vary with
the height to which they rise. Accordingly, as appeared to be the
fact, the corona would be continually shifting its position within
certain limits.

Prof. Challis has made a similar comparison with observations of
the position of the corona of the same aurora made at Haverhill, at
Darlington, and at Bath ; also with observations at Whitehaven of
the aurora of Oct. 18, 1848, and of that of Oct. 24, 1847, at Cam-
bridge. From a consideration of all the results derived from the
discussion of observations made on different occasions and at differ-
ent places, the following conclusions seem to be established : —

First, that the corona of an aurora borealis is formed near the
point of the heavens to which the south end of the dipping-needle at
the place of observation is directed.

Secondly, that the observations, while they indicate no decided
difference of altitude between the two points, show with great pro-
bability that the corona is the more westerly by about l^° measured
on an arc perpendicular to the meridian.

The paper concludes with a particular description of the aurora
borealis of Nov. 17 as observed at the Cambridge Observatory, and
iVith three tables of the observations of declination, horizontal force,
and vertical force, made at Greenwich, and used in the calculations.
These observations present so striking an instance of great magnetic
disturbances occurring simultaneously with an extraordinary display
of the aurora borealis, that the connexion in some way of the two
kinds of phaenomena must be regarded as a physical fact.

XXXII. Intelligence and Miscellafieous Articles.

ON THE RATIONALE OF THE EXPLOSION CAUSING THE GREAT
FIRE OF 1845 AT NEW YORK. BY DR. HARE*.

T^R. HARE communicated to the meeting some inferences and
■^-^ facts, tending to explain the contradictory impressions which
* Communicated by the Author.
Q2

228 Intelligence and Miscellaneous Articles.

have existed respecting the competency of fused nitre to explode with
water, or with aqueous, hydrogenous, and carbonaceous combustibles.
This subject was treated of in reference to a series of detonations ter-
minating in an explosion of tremendous force, by which, in July 1845,
the intensely ignited contents of a store in Broad Street, New York,
were thrown over an extensive district, involving the destruction of
about 200 houses and property estimated at two millions of dollars.
As far as the oaths of highly competent witnesses could avail, no
gunpowder was present, so that tlie result could only be attributed
to the reaction between an enormous quantity of nitre and combus-
tible merchandize with which the store was promiscuously occupied.
In all there were 300,000 lbs. of nitre in parcels of 180 lbs. (each
secured by two bags, an additional bag having been put over that
originally employed). About 30,000 lbs. was situated upon the first
floor, 180,000 on the second floor, and 80,000 on the third floor.

Of the merchandize, the aggregate was more than double the
weight of the nitre.

It was however the general opinion of those best acquainted with
the subject, that when ignited with combustibles, nitre produces only
that species of combustion which is called deflagration by chemists,
without being capable of the more violent and instantaneous reaction
designated by the word explosion. This impression was strengthened
by the failure of every effort (made by several eminent chemists em-
ployed by the Corporation of New York) to explode nitre by ignition
with combustibles.

Nevertheless, agreeably to Hays, of Massachusetts, an explosion
was effected in his laboratory, by bringing water into contact with
about 100 lbs. of incandescent nitre ; also the accidental falling of a
jet of melted nitre on some water in the laboratory of the University
of Pennsylvania had been productive of a similar result.

The explosion of a vessel laden with nitre, which, while lying in
Boston harbour, was burnt to the water's edge, and of others simi-
larly laden and burnt, could only be explained by supposing that
nitre, when sufficiently heated, will explode with water on due con-
tact. Consistently, it might be inferred that this salt (well-known
to be a compound of nitric acid and oxide of potassium or potash)
would explode with any substance capable of yielding either or both
of the elements of water or hydrogen. The presence of the latter
would be equivalent to water, since it would, with the oxygen of the
acid, form water.

In a letter addressed to the distinguished chemist above-mentioned,
in July 1845, Dr. Hare had adverted to the explosion which succeeds
the combustion of potassium upon water, as arising from the combi-
nation of one portion of the water with the resulting incandescent
globule of oxide, while the heat of this globule uniting with another
portion of the liquid, converts it into high steam. Moreover, it was
suggested that in this instance chemical affinity between the water
and the oxide, in causing the water and heated globule to coalesce,
is equivalent in efl&cacy to the momentum of the hammer when a bar
of iron, at a welding heat, is forced into contact with some moisture
situated upon an anvil.

Intelligence and Miscellaneous Articles. 229

'■Dr. Hare presumes that no explosion can take place unless the re-
agents for producing it are held or brought together at the moment
of reaction, by a certain force, either chemical or mechanical.

Some chemical compounds, such as are formed with fulminic acid
or with ammonia, by metallic oxides ; also the chloride of nitrogen
and perchloric aether, explode violently without confinement, so as
to fracture a plate or saucer, upon which a small quantity may be
detonated ; but pulverulent mixtures, such as gunpowder, however
powerfully explosive when employed in gunnery or rock- blasting, in
open vessels flash without fracturing them, or producing any report.
In an exhausted receiver gunpowder is far less explosive than when
subjected to atmospheric pressure in an open vessel. Nevertheless,
when gunpowder is restrained until the temperature requisite for the
appropriate reaction of its ingredients is attained, it exerts a force
far exceeding that which the chamber confining it is capable of re-
sisting. In this respect it differs from steam, of which, when the
temperature of the fire applied is sufficiently high, the explosive force
is directly as the pressure immediately before bursting, and this of
course is commensurate with the strength of the confining boiler.

The ingredients of gunpowder, sulphur, charcoal and nitre, to pro-
duce the greatest effect, require extreme comminution and intimate
intermixture by trituration, and to be so granulated that the flame of
the portion first ignited may convey inflammation to the rest through
the interstices between the grains. Its superiority over any other
mixture of nitre with combustible matter destitute of sulphur, is con-
ceived to be due not only to the pre-eminent susceptibility of this
substance, of vaporization and inflammation, but likewise to its well-
known ability to decompose metallic oxides by attracting botli the
metal and oxygen. Since an opinion was expressed in 1845, in the
letter above-mentioned to Hays, that the formation of sulphide of
potassium is the first step in the process of the explosive reaction of
gunpowder, Faraday has alleged the flame of this compound to be,
in the case in point, an important instrument in the propagation of
fire throughout the mass.

The hepatic odour of the fumes consequent to the firing of cannon,
and likewise of the washings of a gun after the customary service,
demonstrate the production of a sulphide. It has been found that a
filtered solution of the residue displays, when tested by iron, the red
hue which indicates the presence of a sulphocyanide.

Agreeably however to a qualitative examination, the solid residue
of exploded gunpowder consists mainly of nearly equal parts of car-
bonate and sulphate of potash, while the gaseous residue is consti-
tuted nearly of equal volumes of carbonic acid and nitrogen. Of
course the sulphate may arise from the oxidation of sulphide formed
at the outset. Notwithstanding that the ingredients of gunpowder
are prepared as above stated, confinement is necessary to prevent
the grains from being thrown apart and chilled, so as to prevent the
propagation of the ignition, through the congeries forming a charge,
by means of the flame of the first portions fired. This was fully
demonstrated by the exposure of a pile of gunpowder, comprising

230 Intelligence and Miscellaneous Articles.

enough for the charge of a musket, within an exhausted receiver, to
a wire intensely ignited by a galvanic discharge. The grains did
not take fire instantly, probably because the vapour evolved pre-
vented actual contact ; and when ignition did ensue, it extended only
to the production of a feeble flash. On examination, it was found
that a portion of the powder had escaped inflammation.

In the next place, a like weight of gunpowder was consolidated
into a cylinder by intense pressure. Thus prepared and ignited, by
contact with an incandescent wire in the exhausted receiver, more
than half of the cylinder remained unconsumed.

A much larger cylinder of the same mixture, similarly consolidated,
placed at the bottom of an iron pot, four inches in diameter and
twelve inches in depth, on being touched by the end of an iron rod
reddened in the fire, burnt at first like a squib, but towards the last
was dissipated with an activity in some degree explosive, probably
in consequence of the pressure created by the reaction of the gaseous
current generated by its own deflagration.

The want of confinement, which is thus capable of lessening the
explosiveness of gunpowder, of which the constituents are intimately
intermingled, is still more enfeebling, where analogous reagents are
ignited together without admixture or comminution. Under these
circumstances, the reagents are made to recede from each other by
the generation of that vapour or gas, to the evolution of which, under
confinement, the capability of exploding is due. Thus sundered, they
are chilled by radiation, so that the temperature requisite to sustain
and communicate ignition is not supported. Moreover, the rapidity
of reaction being as the multiplication of the points of contact, and
these being fewer as the substances are less divided and intermingled,
the deflagration takes place in detail, instead of having that simul-
taneousness which is indispensable to render it explosive.

In addition to the ideas above-mentioned as having been conveyed
in Dr. Hare's letter to Hays, it was urged also that his inference as
to the explosion of water with incandescent nitre being attributable to
a reaction analogous to that represented as taking place when potas-
sium is burnt with the oxide of hydrogen, was supported by the fact,
that at a white heat the base of nitre spontaneously abandons its acid,
while from water it cannot be separated by any temperature. Conse-
quently, the presentation of substances, consisting of carbon, hydro-
gen, and oxygen, by yielding water to the base, could not but be
productive of a result analogous to that which results from the pre-
sentation of sulphur and carbon.

The only obstacle is as follows : — Substances containing hydrogen
and oxygen, whether in the proportion for forming water, like sugar,
starch, gum and wood, or having an excess of hydrogen, like oils and
resins ; moreover, all the constituents of nitre, even the base, are
susceptible of the aeriform state at the temperature producible by the
reaction of nitre with them. But when kept together until that point
is attained, the explosive power must be fully equivalent to that of
gunpowder. The reagents are in a state analogous to that of two
gases extremely condensed.

Intelligence and Miscellaneous Articles. 231

The explosibility of incandescent nitre with water was illustrated
in the small way, by heating a portion in a platina capsule by the
flame of a hydro-oxygen blowpipe, and sudden immersion in the
liquid. So active was the explosion, that a portion of the resulting
hydrate flew out upon the operator. Yet, when thrown in the same
state upon molasses or sugar, no explosion ensued : nevertheless,
when a capsule containing nitre heated to the point of volatilization
was struck with the face of a hammer coated with sugar melted upon
it and made to adhere by moisture, a detonation took place. A still
more powerful detonation was produced as follows : — Upon an anvil
a disc of paper of three inches in diameter was laid, covered with
pulverized sugar : over the sugar was placed another similar disc,
covered with pulverized nitre : a bar of iron rather wider than the
discs at a welding heat was then held over them, and subjected to a
blow from a sledge. An explosion, with a report like that of a
cannon, ensjied.

Instructed by the facts and considerations above stated, it is inferred
that the explosions which contributed to extend the conflagration in
New York, as above mentioned, arose from the reaction of the nitre
with the combustible merchandize with which it was surrounded. It
is presumed that as soon as the fire reached any of the gunny bags it
must have run rapidly through the whole pile, by means of the inter-
stices necessarily existing between them, the nitre with which they
were imbued causing them to deflagrate. Much of the salt being
thus brought to the temperature of fusion, it must have run about
the floor, reached the combustibles, and soon found its way to the
next story through the scuttles which were open. All the floors
must have been rapidly destroyed by the consequent deflagration, far
exceeding in activity any ordinary combustion. Meanwhile, the nitre
being all liquefied and collected in the cellar in a state of incEin-
descence, and the merchandize conglomerated by the fusion'of sugar
and shell-lac, aided by the molasses, the weight, the liquidity, and
temperature, must have produced all the conditions requisite to in-
tense detonations. The floors having been consumed, the store must
have been equivalent to an enormous crucible of twenty feet by ninety,
at the bottom of which were nearly 300,000 lbs. of nitre, superficially
heated far above the temperature producible by any furnace, so as to
convert the reagents into nascent aeriform matter under a pressure
of half a million of pounds. The intense reaction, however, would
not permit of durable contact. At each impact the whole mass must
have been thrown up explosively, and hence the successive detona-
tions. But the chemical reaction, the heat, and the height of the fall,
growing with their growth, and strengthening with their strength, the
last elevation was succeeded by the thundering report and stupen-
dous explosion of which it has been an object to afford a satisfactory
explanation. — From the Journal of the Franklin Institute.

PREPARATION OF IODIDE OF LEAD. BY M. T. HURAUT.

The author remarks that several processes are known for the pre-
paration of iodide of lead ; all of which give tolerably satisfactory re-
sults. When carefully employed they yield a pure product, and the

^Si Intelligence and Miscellaneous Articles.

quantity obtained difft-rs but little from that indicated by theory ; it is,
therefore, of little consequence which of the processes is adopted in
preparing small quantities of the iodide; the case is, however, different
when considerable quantities of the ingredients are employed, as
in this case the differences are too considerable to be neglected.

The author thinking that some experiments which he has made
on the subject would not be uninteresting, has published them ; and
in every case such a quantity of iodine or iodide was employed as
ought to yield, by theory, 18"20 grammes of iodide of lead.

Process by Iodide of Potassium. — This process is that originally
employed ; it consists in decomposing iodide of potassium by a salt
of lead. The Codex prescribes the neutral acetate, but this salt has
been generally abandoned since it was discovered by M. Depaire
and Felix Boudet, that nearly one-tenth of the iodide of lead was
dissolved by the acetate of potash formed ; 13"10 grammes of iodide
of potassium containing 10 grammes of iodine were treated with
neutral acetate of lead ; the weight of the iodide precipitated was
15-70 to 15-80 grammes.

To avoid the loss occasioned by the use of acetate of lead, M.
Boudet proposed to substitute the nitrate for it ; by this process M.
Huraut obtained with 13' 10 grammes of iodide of potassium from
17-50 to 17-55 of iodide of lead.

Iodide of lead prepared with iodide of potassium is of a fine lemon-
yellow colour, and entirely soluble in boiling water.

Process by Iodide of Sodium. — Ten grammes of iodine converted
into this salt gave with acetate of lead 15-90 to 16" 10 of iodide, and
with the nitrate 16-85 to 16-95. It resembled that obtained with
iodide of potassium perfectly.

Process by Iodide of Calcium. — A quantity of this containing 10
grammes of iodine, gave 17-60 to 17*70 of iodide of lead, of a fine
orange-yellow colour. In one experiment, so performed as to pro-
duce a crystalline iodide, the product was remarkably brilliant ; with
acetate of lead 17*25 to 17 '40 of iodide were produced, also of a fine
orange- yellow colour.

Process by Iodide of Iron. — Ten grammes of iodine converted into
iodide of iron and treated with neutral acetate of lead, gave 16-70
to 16-75 grammes of iodide of lead; with nitrate the products were
17-50 grammes; they were orange-yellow, and totally soluble in
boiling water in both cases.

Process by Iodide of Zinc. — This salt is now perhaps that most
commonly employed in preparing iodide of lead ; the preference
given to it arises from the facility with which it is prepared, its
great solubility and unalterability in the air; 10 grammes of iodine
converted into this salt gave with acetate of lead 17-05 to 17*15
grammes of product, and with the nitrate 17-40 to 17-45. The
colour is palish orange-yellow.

Process by the double Iodide of Potassium and Lead. — This is a
complicated plan proposed by M. Thevenot; the author compared
the product with that afforded by the above-described processes ;
the comparison was in favour of the latter. M. Huraut concludes
from the above-described experiments, that in the preparation of

Intelliitence and Miscellaneous Articles. 233

"to

iodide of lead the nitrate ought to be preferred to the acetate, on
account of the greater quantity of product which it yields.

The process by iodide of calcium is the most advaDtSgeous both
as to the quality and quantity of the product. jxayin^ii

The two processes by iodide of iron and iodide of zinc yielding
equally fine and abundant products, it is nearly indifferent which
is employed.

The process by iodide of sodium offers no advantage, and that by
iodide of potassium is the least economical.

There is a loss of nearly 10 per cent, in preparing iodide of lead
on using iodide of potassium and acetate of lead ; the greater part
of which loss may, however, be avoided by substituting the nitrate,
or by adding to the supernatant liquor a sufficient quantity of nitric
acid to decompose the acetate of potash. — Journ. de Pharm. et de
Chem, Janvier 1849. :;'irnfoi iHBJoq lo a/ajyoc 3uJ ^d i)3vio8?il)

ON THE PROTOGINE OF THE ALPS. BY M. DELESSE* (' T

The author observes that protogine usually contains five different
minerals, which are, quartz, orthose, oligoclase, mica with a base of"
iron, and a variety of talc : these may be seen in the protogine of
Mont Blanc. These minerals are not however equally distributed,
and one or more of them are frequently wanting ; but then the
minerals which remain have so preserved the same characters as
those which they possessed when the five elements are present in
the rock, that it is impossible not to consider them as formed under
the same circumstances ; they constitute therefore varieties of the
original rock, into which they pass insensibly, both by their mine-
ralogical characters and their geological relations. .

Quartz. — Quartz forms one of the important elements of proto,-
gine as of all granitic rocks. When the rock has a well-character-
ized granitic structure, the quartz of the paste is sometimes con-
fusedly crystallized ; generally, however, this does not occur, and it
is hyaline, gray or violet ; when it is in crystals of several centi-
metres in thickness, as seen in some veins, instead of being reddish
or violet, it generally has a deepish black colour, and is of the^ va-
riety called smoky quartz. (Vkt- Oc ~!

It may be stated generally that in fracturing pieces havmg the
usual thickness of the grains of quartz or the paste of the rock, the
difference of colour is derived rather from the greater thickness of
the quartz in the veins than from the presence of a greater quantity
of colouring matter. i-

This colour of quartz, which is observable in many granitic rocJ{%>
is derived from organic matter, which is volatile without leaving
any residue, and disappears completely by slight calcination, the
quartz losing only twelve thousandths, and becoming wiu^e^^lld
transparent. ; ^.m! -

Tliis organic matter is not volatile in vacuo at common terapera-

turesu fox it lioes-not disappear by exposing the q}iSixuiQX:^sf:ysia\
\o noiifiTjsqaiq odJ ai jfirij tsiaamiwqzs badiioaab-avoda adi moil

284 Intelligence and Miscellaneous Articles.

days over sulphuric acid in the exhausted receiver ; nor is it de-
stroyed when the quartz is digested, either hot or cold, in hydro-
chloric acid or ammonia : this resistance to chemical and physical
agents is probably derived, in part, from the intimate admixture of
the organic matter in the pores of the quartz.

Orthose. — The colour of this is generally white or grayish-white,
sometimes, however, it is fawn- or rose-coloured, or pale scarlet.
When it has a tendency to a brownish-yellow or red colour, the
mineral is altered by incipient decomposition : it has a brilliant
pearly lustre.

According to Saussure, its density is 2'6i5. Its crystals are often
several centimetres long, well-formed, and generally possess the
characteristic made which is common to them in granitic rocks.

The crystals of orthose, oligoclase and mica analysed by M.
Delesse, were taken from an enormous block, well-known at Cha-
mouni, and which has fallen from the needles above the Mer de
Glace.

The following was found by the author to be the composition of
the grayish-white orthose, with a tint of fawn colour: —

Silica 66-48

Alumina 19'06

Lime 0-63

Peroxide of iron traces

Magnesia traces

Potash 10-52

Soda 2-30

98-99

Oligoclase. — In protogine, as in the greater number of granites,
there is besides orthose a second felspar, which in this case is oligo-
clase. It is somewhat difficult to distinguish on account of its
white colour, which is nearly the same as that of orthose : this
however is translucent, whilst oligoclase is dull, or very slightly
greenish ; it is, moreover, characterized by parallel microscopic
striae, and the crystals are often complex and macled, like those of
the albite of Carlsbad. Its density is 2-633.

The analysis of very pure crystals from the needles of the Mer
de Glace, made by carbonate of soda and hydrofluoric acid, gave —

Silica 63-25

Alumina 23-92

Peroxide of iron traces

Oxide of manganese .... traces

Lime 3*23

Magnesia 0-32

Soda 6-88

Potash 2-31

99-91
This composition is almost identical with that of the oligoclase of

Intelligence and Miscellaneous Articles. ^2$5

Warmbrum in Silesia, analysed by MM. G. Rose and Rammels-
berg.

Mica. — M. Beudant has already observed that protogine contains
mica : it is of a more or less deep green colour, and has little or no
lustre ; by calcination in an open crucible it becomes of a reddish-
bronze colour, with brilliant reflexions ; in a close crucible it becomes
blackish green. When it is in very thin laminae the action of the
air is sufficient to give it a bronze colour, which is a character that
may serve to recognize it. Its density is 3*1£7, which ii much
greater than that of the micas of granites ; this is unquestionably
owing to the large quantity of oxide of iron which it contains.

It is not crystallized in small transparent scales ; on the contrary,
it has the form of small irregular hexagonal prisms, the edges of
which are not perpendicular to the bases.

Before the blowpipe the edges are rounded with difficulty when
in small laminae ; with fluxes it indicates iron and manganese, and
it dissolves entirely in phosphate of soda.

It is perfectly acted upon even by hydrochloric acid, and the silica
separates in the form of flocculi : the facility with which it is acted
upon is probably owing to the great quantity of iron.

The analysis was performed on the mineral taken from the granite
block already mentioned ; it gave —

Silica 41-22

Alumina 13-92

Peroxide of iron 21-31

Protoxide of iron 5-03

Protoxide of manganese... 1-09

Lime' 2-58

Magnesia 'l-'TO

Potash 6-05

Soda 1-40

Fluorine 1*58

Water and loss by heat . . 0*90

99-78

Talc. — Protogine also contains a substance forming very small
contorted laminae inserted among its various minerals, and which is
to be regarded as a variety of talc. It has a pearly lustre ; its colour
varies from celadon to emerald and pale grayish-green. By calci-
nation it acquires sometimes a brownish tint and sometimes a bright
wood-brown tint, with golden reflexions ; pure talc becomes very
slightly yellowish silver-white. It is not elastic ; its hardness is
rather greater than that of talc, even when unmixed with foreign
matters ; like talc, it scratches glass after calcination.

Very thin laminae of this talc, extracted from various specimens
of protogine, were tried by the blowpipe; at a very high tempera-
ture, as already remarked by Saussure, their edges were rounded
without exfoliation, and the fused portion was coloured by iron.
Talc, on the contrary, exfoliates without fusing ; this fusibility of

236 Intelligence and Miscellaneous Articles.

the substance and the brown colour which it acquires by calcination
indicate that it is richer in iron than is the case with talc.

Independently of the minerals which have been described, proto-
gine, as observed by MM. Dufrenoy and E. de Beaumont, may
accidentally contain liornblende, sphene, iron pyrites, garnets and
serpentine.

Some veins contain fluor spar, oligiste iron and sulpliuret of
molybdenum, &c. In I'Oisan there are albite, rutile,anatase,brookite,
&c. ; lastly, there are found in veins which appear to be contem-
poraneous with the rock, and usually formed of quartz, epidote
and the variety of chlorite, to which M. G. Rose has given the
name of ripidolite ; it is also found in the paste of protogine. — Ann.
de Chtm.et de Phys., Janvier 1849.

^•^' EXAMINATION OF MADDER. BY M. DEBUS.

til 21V

In order to isolate the different colourmg matters of Zealand
madder, the author employs the following process : the root is ex-
hausted by boiling water, and the decoction is boiled with excess of
hydrate of lead. The colouring matters form with this oxide in-
soluble compounds of a reddish brown colour. The deposit is to
be separated, washed, and decomposed with dilute sulphuric acid
and heat. The colouring matters, which are slightly soluble in
water, separate with the sulphate of lead. The precipitate is to be
boiled in alcohol, which dissolves the greater part of the colouring
matters. They may be separated into two groups by agitating the
alcoholic solution with calcined oxide of zinc. Some of them pre-
cipitate in combination with the oxide of zinc, while others remain
in solution.

Tiie author has hitherto examined only the first group, that is to
say, the colouring matters combined with the oxide of zinc. They
are purified by decomposing them with weak sulphuric acid, and
dissolving the precipitated colouring matter in aether ; the solution
obtained is again heated with oxide of zinc. The zinc compound,
heated with dilute sulphuric acid, leaves as a residue a mixture of
two colouring matters, both soluble in a boiling solution of alum,
one of which precipitates on cooling and the other remains in solu-
tion ; the first constitutes what the author calls lizaric acid; this
substance is obtained in a state of purity by boiling it with a little
dilute hydrochloric acid, to free it from the alumina which it retains,
and by repeated solution in boiling alcohol.

The second colouring matter, which remains in solution in the
aluminous liquors, may be precipitated by sulphuric acid. The
separation is not completed in less than twenty-four hours ; the pre-
cipitate, exhausted by hot dilute hydrochloric acid, which removes
a little alumina, is afterwards dissolved in 150 to 200 times its
weight of boiling alcohol. In two or three hours long red needles
separate, which constitute what M. Debus calls alizaric acid.

Lizarie Acid. — It crystallizes from its alcoholic solution in long

Intelligence and Miscellaneous Articles. Hal

orange-red coloured needles : it is soluble in aether, in alcohol and
in hot water, but dissolves with difficulty in a boiling solution of
alum. Sulphuric acid dissolves it, and becomes of im intense red
colour ; on diluting the solution with water the colouring matter is
precipitated unaltered. The salts formed with lizaric acid are of a
red or violet colour, and, with the exception of the alkaline salts,
are insoluble in water or in alcohol. The composition of the free
acid is expressed by the formula C^o H'^C. The salt of lead, ob-
tained by adding lizaric acid, dissolved in alcohol acidulated with
acetic acid to an alcoholic solution of acetate of lead, is formed of
C^ W 07, 2PbO.

Oxylizaric Acid is distinguished from lizaric acid by the facility
with which it dissolves in a solution of alum. It is slightly soluble
in cold water, but dissolves more readily in boiling water, in alcohol,
iEther and the alkalies. Fuming sulphuric acid dissolves, and may be
heated with it without altering it ; the author gives C'^ H^ O^ as Us
composition ; the salt of lead would have C^^ H* O*, PbO for its
formula. Hence it is evident that by adding one equivalent of
oxygen to one equivalent of lizaric acid, two equivalents of oxyli-
zaric acid will be obtained. It is this relation between the two
acids which is expressed by the name of the latter. — Journ'.'de
Pharm. et de Chim., Janvier 1849. ^ : mo-

ANALYSES OF FELSITE, OLIGOCLASE AND MUROMON^'i'l'I^E.'

M. Kerndt has analysed the above-named minerals, with the an-
nexed results : —

Felsite. — Crystallized green felsite, density 2'54^5,,;^9ni„j^9^ein-
mais gave, taking the mean of two analyses, — idoioa m QinAmx-J

Silica 63-657 '

Alumina 17-271

Potash .^„ . ^; Jta:§59., .:..,.,, ^ , , ..,

^Oda ■■,i .•»iu(«<;iD(fftfii«3#> >'cj f»o.fi»ri«f '^-i '

Lime '. .^ 0-394

Magnesia 2-281

Protoxide of iron 0'451 f^jj,^ i)9jB9rl

Protoxide of manganese. . 0-153 niiuofoo o/^j
'03 no «100*00 fiotdw lo ano

OUgoclase. — This mineral from Boden liear Marien'6erg,^iA*^pi'e

Erzgebirge, of density 2-66-2-68, gave — fin-^.n?

Silica 61-958 --^V^ '^f^

Alumina 22-658 . ^^'^ ^1,,^"'^

Potash 3-079 'T'''',

Soda 9-432'''''»''^. ^«^"""""''«

Lime ;;.; 202^'^"7''

Magnesia 0-104*''"'''^ '

Peroxide of iron 0-348

.Biiimwifi olijii

Peroxide of manganeVeV. 0-396-'"?''f "^^ ^'^S'^'"
-3 i\u\i\'ft .eJijifiqaa

100-000- \)jrK VnosiA

238 Intelligence and Miscellaneous Articles.

"b

Muromontite. — By this name the author designates a ceriferous
mineral met with in the environs of Mauersberg near Marienberg,
in the Erzgebirge. It has the form of black grains, with a greenish
reflexion. Density, 4'263-4-265.
This mineral contains —

Silica 31-089

Yttria 37-140

Glucina 5-510

Alumina 2-230

Oxide of lanthanium. ... 3-530

Oxide of cerium 5-540

Protoxide of iron 11 -230

Protoxide of manganese . . 0- 900

Lime 0*710

Magnesia 0-420

Soda 0-650

Potash 0-170

Water 0-820

99-939
Journ. de Ph. et de Ch., Novembre 1848.

ON THE FERROCYANIDES OF STRYCHNIA AND BRUCIA.
BY M. D. BRANDES.

The author states tliat when a solution of ferrocyanide of potas-
sium is added to one of a neutral salt of strychnia, an abundant preci-
pitate is obtained, consisting of small and nearly colourless needles.

In operating on dilute solutions deprived of free acid, crystals of
two to three centimetres in length and of a very bright yellow colour
are obtained ; they are four-sided prisms, terminated by dihedral
summits : these crystals are ferrocyanuret of strychnia, represented
by the formula 2(Str, H Cy) + Fe Cy + 8H0. At 212° the salt loses
6-1 per cent., or six equivalents of water. If it be dissolved in hot
water, or if the cold saturated solution he boiled, crystals of strych-
nia separate, and the liquor, which is of a deep yellow colour, holds
ferrocyanide of strychnia in solution,

3(Str, HCy,)-fFe2Cy3+12HO.

This salt, which forms crystals of a golden-yellow colour, corre-
sponds to the red prussiate of potash, and may also be obtained by
mixing the cold saturated solutions of sulphate of strychnia and red
ferrocyanide of potassium : according to the author, this salt loses
three equivalents of water in a dry vacuum, six equivalents at 212°,
and eight equivalents at 277° F. Above this temperature it decom-
poses. When an alcoholic solution of strychnia is mixed with a
solution of hydroferrocyanic acid in alcohol, a white amorphous pre-
cipitate is obtained. This is nearly insoluble in water or alcohol,
and has a distinct acid reaction. M. Brandes assigns to it the for-
mula (Str, 2ACy + 2FeCy) + 5HO. He considers it as an acid ana-
logous to the hydroferrocyanic, and expresses its constitution as fol-
lows, deducting the five equivalents of water it contains (Str, HCy

Meteorological Observations, 239

4-2Fe Cy) + H Cy ; the author has not, however, as yet succeeded
in preparing salts directly with this acid.

The ferrocyanurets of brucia are prepared by processes analo-
gous to those above described ; they resemble, both in their pro-
perties and composition, the corresponding salts of strychnia. — Ibid.
Janvier 1849.

METEOROLOGICAL OBSERVATIONS FOR JAN. 1849.

Chiswick. — January 1. Overcast : hazy. 2. Clear and frosty. 3. Frosty : dry
haze : overcast : frosty. 4. Uniformly densely overcast : rain. 5. Drizzly and
foggy. 6. Overcast. 7. Overcast : rain at night. 8. Rain. 9. Very fine :
slight rain. 10. Cloudy: boisterous: rain. 11. Rain: densely clouded. 12.
Frosty: overcast: rain. 13. Densely clouded : rain. 14. Rain. 15. Clear.
16. Fine : rain. 17. Rain : densely overcast : clear. 18. Fine: boisterous at
night. 19, 20. Very fine. 21. Very fine : overcast : boisterous. 22. Boisterous:
fine : clear and boisterous. 23. Densely clouded: fine. 24. Cloudy : boisterous
at night. 25. Densely clouded : boisterous. 26. Rain : exceedingly fine. 27.
Slight frost : overcast : rain. 28. Cloudy : fine. 29. Rain : cloudy and cold :
frosty at night. 30. Slight fog: drizzly. 31. Fine : clear and frosty at night.

IVlean temperature of the nionth 39°'56

Mean temperature of Jan. 1848 33-62

Mean temperature of Jan. for the last twenty years 36 '40

Average amount of rain in Jan 1*59 inch.

Boston. — Jan. 1. Cloudy. 2 — 4. Fine. 5,6. Cloudy. 7. Fine: rain early a.m.
8. Rain. 9. Fine: rain p.m. 10. Cloudy : stormy all day. 11. Cloudy: rain
early A.M. 12. Fine. 13. Rain: rain early a.m. 14. Cloudy : rain early a.m.
15. Fine : rain A.M. and p.m. 16. Foggy. 17 — 20. Fine. 21. Cloudy. 22 —
24. Fine. 25. Cloudy. 26. Fine: rain early a.m. 27. Fine: rain p.m. 28. Fine.
29. Rain : rain a.m. 30. Cloudy : rain a.m. and p.m. 31. Fine.

Applegarth Manse, Dumfries-shire. — Jan. 1. Frost moderate. 2. Frost very
hard : barometer falling. 3. Frost clear : fine. 4. Frost, but cloudy. 5. Frost :
cloudy. 6. Frost : still cloudy. 7. Frost : still more overcast. 8. Thaw : rain :
fo<^ : rain again. 9. Frost again: clear a.m. : rain p.m. 10. Heavy rain during
night : rivers flooded. 11. Frost a.m. : thaw at noon : rain. 12, Soft rain all
day. 13. Soft rain : cleared : rain p.m. 14. Gentle frost: cloudy: wind rose.

15. Soft: cloudy. 16. Mild and clear after rain a.m. 17. Moist a.m. : rain and
high wind p.m. 18. Very fine till noon: rained again. 19. Frost: getting
cloudy P.M. 20. Heavy rain and high wind p.m. : thunder. 21. Storm of wind
and rain. 22. Fair, but a storm of wind. 23. Fair a.m. : came on storm, wind
and rain. 24. Rain nearly all day : wind high. 25. Fair and keen a.m. : wet
P.M. : high wind. 26. Fair a.m. : rain p.m. 27. Snow : rain : wind high. 28.
Frost : clear : dull p.m. 29. Frost and snow : thaw and rain. 30. Frost mode-
rate. 31. Thaw and showery.

Mean temperature of the month 86°'35

Mean temperature of Jan. 1848 33 "80

Mean temperature of Jan. for the last twenty-five years . 34 '90

Rain 3*70 inches.

Rain in January 1848 2*34 „

Average amount of rain in Jan. for the last twenty years 2*60 „

Sandwick Manse, Orkney. — Jan. 1 . Cloudy. 2. Bright : cloudy. 3. Cloudy.

4. Cloudy : frost : snow-showers. 5. Bright : cloudy. 6. Snow. 7. Thaw :

clear. 8. Rain : showers. 9. Showers : cloudy. 10. Rain : snow. 11. Snow.

12. Rain : showers. 13. Showers. 14. Showers : sleet-showers. 15. Showers.

16. Showers: cloudy. 17, 18, Showers. 19. Showers: clear, 20. Cloudy.
21. Rain : showers. 22, Sleet-showers. 23. Sleet-showers : rain. 24. Rain*:
sleet-showers : cloudy. 25. Sleet-showers : aurora. 26. Sleet-showers : cloudy.
27. Bright : sleet-showers. 28. Sleet-showers : clear. 29. Frost : cloudy. 30.
Snow: sleet: showers. 31. Sleet-showers: showers.

* From 9 p.m. on 23rd till 2 p.m. on 24th (about 17 hours) 2-08 inches of rain
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THE
LONDON, EDINBURGH and DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

♦

[THIRD SERIES.]

APRIL 184y.

XXXIII. On the Existence and Effects of Allotro^nsm in the
constituent elements of Living Beings. By John William
Draper, M.D.y Professor of Chemisti-y in the University of
New York"^.

IT has been completely established for the majority of ele-
mentary substances, that there are several forms under
which each may occur ; forms which differ entirely both in
their physical and chemical relations.

Thus, in the case of carbon, many such forms are known.
To three of them M. Berzelius has directed attention: — 1st,
ordinary charcoal; 2nd, plumbago; 3rd, diamond. They
are three distinct modifications of the same element. They
dilfer in specific gravity, in specific heat, and in conducting
power, both for electricity and caloric. In their relations
to light, the first perfectly absorbs it and is black ; the
second reflects it like a metal; the third is transparent like
glass. When crystallized, plumbago and diamond do not
belong to the same system : their chemical relations are also
strikingly different. Charcoal takes fire with facility, and
some varieties of it are even spontaneously combustible in the
air ; but crucibles and furnaces are made of plumbago because
of its incombustibility ; and the diamond with difficulty is set
on fire in pure oxygen gas.

It seems immaterial to what class elementary bodies belong,
whether electro-negative or positive : they present analogous
results. Silicon, sulphur, selenium, phosphorus, titanium,
chromium, uranium, tin, iridium, osmium, copper, nickel,
cobalt, iron, oxygen, chlorine, are cases in point ; and the
instances which appear as exceptions are rapidly diminishing
in number.

* Communicated by the Author.
Phil. Mag. S. 3. Vol. 34. No. 229. April 1 849. R

242 Prof. Draper on the Existence and Effects ofAllotropism

Asiswell known, to these singular modifications M.Berzelius
gave the designation of allotropic forms, and the whole phaeno-
menon passes conveniently under the designation of allotropism.
He shows that the peculiarity assumed is often of such a per-
sistent nature that it is not lost, even though the substance
affected should go into combination with others. Thus there
are two forms of silicon ; one combustible, and the other re-
markably incombustible. Each, by uniting with oxygen, gives
rise to a silicic acid ; the acid in one case being soluble in water
and in hydrochloric acid, and in the other the reverse. And in
like manner, metallic arsenic, which exhibits the same duality
of condition, gives rise to two different arsenious acids. Of
phosphorus there are at least two modifications; and accord-
ingly we have two compounds of that body with hydrogen,
one of which is spontaneously inflammable, and the other not;
and at least two oxygen acids, the monobasic and tribasic, in
which the essential difference rests in the state of the phos-
phorus they contain.

It is to be remarked, that, so far as observation extends, the
most common cause of producing these singular differences is
the action of that class of agents which we term imponderable
substances. In very many cases change of temperature brings
about allotropic change; in others it is the agency of light, as
in chlorine and phosphorus ; and again, in others, association
with foreign bodies, which apparently establish new voltaic
relations. Heat, light and electricity seem to be the general
modifying agents.

M. Berzelius, following the suggestion of M. Frankenheim,
proposes a nomenclature for pointing out the peculiar form
referred to in any special case. It de[)ends on the use of
Greek letters. Thus we have the three forms of carbon just
alluded to, designated on these principles by Ca, C/3, Cy.
But in a paper which I published in this Journal on the allo-
tropism of chlorine (Nov. 1845, p. 327), it is remarked that
we may often with greater convenience use the simple expres-
sions "active" and "passive." Thus active chlorine is that
which will decompose water in the dark, passive chlorine fail-
ing to do so. In this paper the same expressions will be em-
ployed.

Hitherto allotropism has only been considered as affecting
inorganic states of matter, but its influence can be plainly
traced in the far more interesting case of organic beings; and,
when placed in a proper point of view, yields a remarkable
explanation of some of the most obscure but important facts
in physiology and pathology. These explanations I propose
now to point out.

in the constituent elements of Living Beings. 243

In the Philosophical Magazine (March 1846, p. 178) there
is a paper by me explanatory of the causes of the circulation
of the blood in the capillary vessels. It is merely an abridge-
ment of a lecture which for eight years past has been delivered
in this university. The doctrine there set forth has been ge-
nerally received in America, and introduced into some of the
standard works on physiology published in England. The
principle on which it essentially depends, and which has been
abundantly confirmed by direct experiment, is briefly this—
that if there be two fluids occupying a capillary tube, or a
porous structure of any kind, under the condition that one of
them has a stronger chemical affinity for the substance of that
tube or structure than the other, a movement of the liquids
will at once ensue, that which has the stronger affinity driving
the other before it. On this principle a clear account of the
systemic circulation of animals may be given ; for the arterial
blood, an oxidizing liquid, having a stronger affinity for the
soft tissues with which it is in contact than the venous blood, the
affinities of which have been satisfied and therefore no longer
exist, necessarily exerts such a pressure that motion must
ensue, the arterial blood forcing the venous before it.

An application of the same principles shows that in the
pulmonary circulation the motions must necessarily be in the
opposite direction, or from the venous to the arterial side, as
is actually the case. It also explains clearly the conditions
of the portal circulation, in which the direct action of the
heart could hardly be expected to be felt. With the gene-
rality which ought to belong to a true theory, it meets all the
cases which occur in the lower orders of animal life, such
as the greater circulation in fishes, in which there is no
systemic heart; the movements which take place in the vas-
cular system of insects; and even the extreme case of the rise
and descent of sap in plants.

In this doctrine everything depends on the relationship
between the nutritive fluid, or blood, and the solid parts with
which it is brought in contact; and whatever changes that
relationship must impress a corresponding change on the cir-
culation itself.

From experiments which I made some time ago, I have
been led to suppose that the arterial ization of the blood, as it
takes place on the cell- walls of the lungs, bears a strong ana-
logy to the oxidation of white indigo. The loose hold which
the colouring matter of the blood retains on the oxygen, cou-
pled but not combined with it, is not unlike what is witnessed
in other nitrogenized colouring matters, such as indigo, which
oxidizes and deoxidizes with the utmost facility. Charged

R 2

244' Prof. Draper on the Existence and Effects of Alloiropism

with the oxygen it has thus obtained, the arterial blood passes
to all parts of the system ; and now arises that striking but
all-important physiological fact, that it does not attack indis-
criminately all those parts of the soft solids which it first en-
counters, but proceeding in a measured way, exerts its action
on such particles alone as have become effete, and accomplish-
ing the great process of interstitial death, resolves those par-
ticles into other forms, so that they can be eliminated from
the system by the lungs, the kidneys, and the skin.

Now why is it that things proceed in this way ? What is it
that regulates this interstitial death ? Why is one atom pre-
served and another surrendered ?

It is upon these obscure points that the recent discoveries
in allotropism shed a flood of light.

The three leading neutral nitrogenized bodies, fibrine,
albumen and caseine, are characterised by exhibiting allotro-
pism in a most remarkable degree, and that in a double sense.
1st. Though so different from one another in their physical
and chemical relations, it is admitted on all hands that they
are mutually convertible; the albumen of the egg, during in-
cubation, gives rise to fibrine and other allied bodies ; from
caseine, in the milk with which the young mammalia are
nourished, the albuminous and fibrinous constituents of their
systems arise; the nurse fed on fibrine and albumen secretes
caseine from the mammary gland. Indeed there is no more
reason to regard these three bodies as essentially distinct sub-
stances, than there is to apply the same conclusion to char-
coal, plumbago and diamond. Between the two cases there
is the most complete analogy; and if charcoal, plumbago and
diamond, are merely allotropic forms of one substance, the
same holds good for fibrine, albumen and caseine. But 2nd,
each of these three compounds betrays a disposition under
trivial causes to assume new forms; as with silicic acid so with
fibrine, there are two varieties, one soluble in water, the other
not. A difference of a few degrees of heat turns transparent
albumen into the porcellanous variety, and analogous obser-
vations might be made respecting caseine.

It may therefore be asserted that these proteine bodies ex-
hibit a propensity to allotropism in a far more remarkable
manner than any other substances known ; not only passing
indiscriminately into one another, but also exhibiting special
variations under the influence of the most trivial causes.

And now we may recall the fact, that of the agents which
in the inorganic kingdom bring about these changes, the im-
ponderable principles are pre-eminent. I transfer this obser-
vation to the case of organized beings, and infer that the ner-

in the constituent elements of Living Beings. 245

vous system has the power of throwing organized atoms into
the active or passive state ; that this is the fundamental fact
on which all the laws of interstitial death depend ; and that
upon this principle — its existing allotropic condition — an
organized molecule either submits to the oxidizing influence
of arterial blood, or successfully resists that action.

But it has been stated that there are certain pathological
conditions, which, upon these views, meet with a clear expla-
nation ; conditions, which, though long and laboriously studied
by physicians, remain involved in contradictions and obscu-
rity. The conditions to which I refer are those known as
inflammation and congestion.

It is agreed among chemists, that during the prevalence of
these conditions the urine assumes a peculiar constitution. In
inflammatory actions the relative quantity of urea and sul-
phuric acid is much above the normal standard, whilst in
congestive cases the reverse holds good, and the urea and
sulphuric acid are below the standard. What is the inter-
pretation of these remarkable facts ? We shall find they are
very significant.

The quantity of urea and sulphuric acid in the urine un-
doubtedly expresses the quantity of proteine matter that has
undergone oxidation in the system. In all cases where that
quantity is above the normal standard, the destruction of
proteine matter has been correspondingly accelerated ; and
w^here it is deficient, the destruction has been reduced. The
result of inflammations corresponds to the first of these cases,
and of congestions to the second.

Recalling now what has been said respecting the cause of
the capillary circulation, we see how all these apparently dis-
connected facts group themselves together in the attitude of
dependent effects. In inflammation there has been that allo-
tropic change in the soft solids involved, that they have as-
sumed a disposition for rapid oxidation — they are active.
Their relations with arterial blood have become highly exalted;
and the blood flows, on the principles I have set forth, to the
affected part with energy. Redness of that part and a higher
temperature are the result. Oxidation goes on with promp-
titude, and urea and sulphuric acid begin to accumulate in
the urine.

But in congestive cases it is the reverse ; the parts affected
are thrown into a more passive state. Oxidation goes on in
a reluctant way, the amount of tissue metamorphosed dimi-
nishes, the urea and sulphuric acid diminish in the urine; and
on the principles which I have endeavoured to explain respect-
ing the capillary circulation, we perceive that an immediate

246 On the casting of the Specula Jbr Reflecting Telescopes,

action must be exerted on the flow of the blood, the passive
condition of the tissues and diminished capacity for oxidation
restrain the flow from the arteries, and there being now less
pressure on the contents of the veins, engorgement of those
vessels is the result, and this condition of things is what a
physician designates as congestion.

In this manner, if we admit the existence of allotropism in
organic atoms, we can give a very clear explanation of the
condition of the circulation in the pathological states of in-
flammation and congestion, and also of the peculiarities which
in those states belong to the constitution of the urine.

University of New York,
Feb. 17, 1849.

XXXIV. On the discovery of the Chilling Process in the casting
of the Specula for Reflecting Telescopes, Sj-c. By Professor
Potter, A.M.^ F.C.P.S., late Fellow of Queen's College,
Cambridge.

To the Editors of the Philosophical Magazine and Journal.
Gentlemen,

I PERCEIVE at page 143 of the February Number, that
in an abstract of a paper by Mr. Lassqll read at the As-
tronomical Society on the 8th of December IS^S, there is the
following passage : — " The mode of casting the large speculum
which I employed involved the principle, discovered, I believe,
and first published, by Lord Rosse, of casting the speculum
on what is technically called a chill, i. e. an iron base, slightly
warmed, which causes the speculum to cool upwards in hori-
zontal strata."

Your readers will find in the fourth volume of the new
series of the Edinburgh Journal of Science, namely for 1831,
that in a paper on improvements in the casting, working, &c.
of Specula for Reflecting Telescopes, I had discovered and
there published {page 18) the improvement in speculum metal
by casting upon a chilling surface.

All known substances which were affected in different man-
ners by rapid and slow cooling, after being heated, had up to
that time indicated that it was a general law, that sudden
cooling induced the property of brittleness or loss of tenacity,
cracking, and frequent failing into fragments of the substance
suddenly cooled ; and that the opposite procedure of very
slow and gradual cooling, which was generally called anneal-
ing, induced toughness and tenacity in the substance. Glass,
many minerals, and steel, were known to be subject to this
law.

Mr. J. Phillips on Ancient Metallurgy and Mining, 247

All writers on the casting of specula, including Lord Ox-
mantown in the second volume of the same series of the
Edinburgh Journal of Science, had received or prescribed
annealing as the course to be used in the casting of the metal ;
but in spite of all precautions, when the metal was of the best
proportions of copper and tin for colour and polish, the cast-
ingswere continually found to be cracked, and not unfrequently
broken into fragments at the termination of the annealing
process, whilst the metal was often so brittle as not to bear
grinding with emery without tearing up.

In this state of the subject I undertook an experiment in the
contrary direction (see page 18 of the paper referred to), and
cast a metal upon a chilling surface, when I found that I had
obtained by it the great desideratum, a hard compact metal,
which bore grinding and polishing admirably. I repeated the
experiment several times with small mirrors, and obtained
always the same result. Having obtained full confidence in
the method, I recommended it to Lord Oxmantown's notice
in the before-named paper.

This chilling process in casting, and my method of making
the polishing powder, detailed in the same paper, by precipi-
tating with ammonia an oxide of iron from a solution of the
sulphate and then calcining, will, 1 am sure, in future time be
accounted amongst the most important discoveries connected
with the construction of the reflecting telescope, and I claim
the discovery and first publication of them.
I am, Gentlemen,

Your obedient Servant,

University College, March 13, 1849. RiCHARU PoTTEB, A.M.

XXXV. Thoughts on Anciejit Metallurgy and Mining in Bri-
gantia, and other parts of Britain, suggested hy a page of
Pliny's Natural History, By John Phillips, Esq., F.R.S.f
F.G.S*

TO one who meditates on the progress of natural know-
ledge, the difficulty of penetrating to a true estimate of
its condition in past ages often appears unconquerable, except
in cases which admit of the interpretation of ancient results
by modern laws and theories. Once in firm possession of
such laws, we enclose the old phaenomena, so to speak, in a
field to which are only such and such possible avenues, and

* From the Proceedings of the Yorkshire Philosophical Society for
March 1848.

^48 Mr. J. Phillips on Ancient Metallurgy and Mining

thus can sometimes declare the very mode by which thealchy-
mist was led to his golden error, and the Chaldaean shepherds
to brighter truths. Without this principle of interpretation,
many almost modern writers, nay authors of this very century,
can sometimes not be understood. The laws of modern geo-
logy and zoology, for such there are, and well-founded too,
are as much required to put a true construction on some of
the writings of Lister and Linnaeus, as the methods of Ray,
Linnaeus, and Cuvier are required for the just estimation of
Aristotle. We shall probably find the darkest pages of an-
tiquity to be precisely those which refer to subjects where our
own knowledge is least clear, least collected into lawsof phae-
nomena, and most removed from laws of causation. Ought
we not, before declaiming on the ignorance of the ancients, to
be careful to make allowance for the differences of form in
which knowledge presents itself at different periods, as well
as for the incompleteness of their records, and the imperfection
of our interpretations ?

Pliny's Natural History appears to me to be precisely in
the position of difficulty which has been already alluded to.
Its vastness, variety, and seeming disorder, may well deter
the most comprehensive master of modern science from duly
weighing its mass, or even measuring its surface ; and the
evident incompleteness and almost hap-hazard character of its
chapters are apt to disgust the student of special branches of
science and art. Yet, probably, if for each important branch
of human knowledge handled by Pliny, a special editor were
set to work, well- versed in the philosophy of his subject, Pliny
would take a higher degree on examination, and the history
of human knowledge be amended.

From the thirty-seven books of diffuse and erudite learning,
the genuine work of Pliny the elder, let us fix on the part
which treats of the nature of metals ; and passing over his
lamentations on the useless excess of gold and silver — which
may be recommended to the Chancellor of the Exchequer — his
accounts of the uses and properties of gold, electrum*, chryso-
colla, silver, quicksilver, stibium, scoria argenti, spuma argenti,
minium, cinnabar, brass, cadmium, iron, and many compounds
of metals, let us pause at the 16th chapter of the 34th book,
which treats of the metals of lead, white and black.

" The most precious of these, the white, is called by the
Greeks Kaaairepo^, and fabulously declared to be sought for
in isles of the Atlantic, to which it is brought in wicker vessels,
covered with leather (vitilibus navigiis corio circumsutis). But

* Gold with one-fifth of silver.

in Brigantia and other parts of Britain. 249

now it is ascertained to be indigenous in Lusitania and Gal-
licia, in sandy surface soil, of a black colour, and only distin-
guished by its weight. Small pebbles [of the ore] also occur
principally in dried beds of streams. The miners [metallici]
wash these sands, and what subsides they melt in furnaces.

" It is also found with the gold ores (aurariis metallis) which
are called stream works (elutia), the stream of water washing
out (eluente) black pebbles a Httle varied with white, and of
the same weight as the gold. On this account, in the vessels
in which the gold is collected, these pebbles remain with it;
afterwards they are separated in the chimneys* (caminis se-
parantur), and being melted are resolved into plumbum album.

" In Gallicia plumbum nigrum is not made, because the
adjoining Cantabria [Asturias] so much abounds in that
metal.

" Not out of white plumbum as out of the black can silver
be extracted.

" To solder together [pieces of] plumbum nigrum is im-
practicable without [the use of] white plumbum, nor the white
to the black without the addition of oil. Nor can [pieces of]
white plumbum be soldered together without the aid of the
black metal.

" That [plumbum] album was in esteem during the Trojan
time Homer is witness, who calls it Kaao-LTepo^.

" Of plumbum nigrum the source is double : either it comes
from its own vein, without admixture, or grows with silver,
and is melted while mixed with that metal. The part which
is first liquid is called stannumf, that which flows next is
silver, that which remains in the furnace galena:]:, which is
the third portion of the vein (or ore). This being again
melted § yields plumbum nigrum, [the other] two parts [of
the ore] being deducted."

This chapter is a text on which a 38th book of Natural
History might be written, embracing the history or fable of
the Kaaai,TepcBe<i, the ancient artsof metallurgy, and the eager
trade in metals which allured the Phoenician sailors on the
Atlantic, and led the Roman armies to Britain.

What is Koaatrepo^, for which plumbum album is the equi-

* What distinctive meaning should be attached to furnaces and camini
is uncertain. It seems that the canu'ni may indicate, if not what we call
chimneys, at least cavities in or above the furnace.

f Analogous to this is the process of separating silvery lead from mere
lead, invented by H. L. Pattison, Esq.

I Lib. xxxiv. cap. 18. Est et molybdaena, quam alibi galenam vocavimus,
plumbi et argenti vena communis.

§ At the present day we should perform this melting of the residual ' ga-
lena' in the slag-hearth, with a flux.

250 Mr. J. Phillips on Ancient Metallurgy and Mining

valent? what is stannum, obtained from mixed ores of silver
and lead ? what is galena, elsewhere called rnolybdajna ?
(cap. 18.) We need not ask what is plumbum nigrum, for
by that is clearly designated lead.

That KaaaLTepo<i or KarTCTepo^ was tin, appears to be ge-
nerally allowed. The mineralogist and miner who know the
mode of occurrence and character of tin ore, will have no
doubt that plumbum album of Pliny is tin, and that author
twice positively and expressly identifies this with Kaaalrepo^.

The uses to which Homer puts Kaa-acrepo^ in the thoraca
and shields of Agamemnon, Achilles, and Asteropasus, and in
the greaves of Achilles, are such as imply easy fusibility and
ductility, and indicate that the metal was highly valued and
almost precious*.

Virgil puis no tin into the arms of ^neas — perhaps the
metal was then of too vulgar use — employed too much by tin-
kers— to be fit for a heroic shield. Electrum is substituted,
and iron is the staple article in the Vulcanian workshop, as
brass was in that of "H<f>AIST02, 1000 years before.

The picture of the great artist — the Tubal Cain of the west,
the cunning worker in metal, who melted, alloyed, inlaid,
carved, and polished his work — whose multiplied bellows
breathed at the will of the god softly or fiercely — whose brass
was hardened to wound, or tempered to bend, — is perfect, and
might be paralleled on a small scale till a few hundred years in
the famous smiths of Wales, who made their own iron, and
were by the laws of that country, as renesved by Howell Dda,
allowed to sit next the sacred priest.

* The following are the principal passages in the Iliad where Kaaairepos
is mentioned : —

Xf. 25. In the thorax of Agamemnon were ten plates (ot/iot) neXavos
Kvdvoio, twelve of gold and twenty of Kaa-alrepos,

XI. 34. In the shield of Agamemnon were twenty white bosses (o/x^aXot)
of tin, and in the middle one of Kvavos.

XVIII. 474. For the shield of Achilles "H-I-AISTOS throws into his cru-
cibles brass, unconquered Ka(r(riTfpo<:, honoured gold, and silver.

XVIII. 564. He pours the tin round the border.

XX. 270, In this shield were five [ilates; the two exterior ones brass;
within these, two of Kaa-a-irepos ; and in the middle of all, one of gold.

XVHI. 612. The greaves of Achilles are made of soft Kaa-a-irepos.

XXII. 503. The chariot of Diomedes was adorned with gold, and Kao-crt-
repos.

XXIII. 561. In the brazen thorax of Asteropaeus the ornament was of
glittering Kaa-airepos-

What is here called Kvavos, and is apparently a much-valued substance,
is difficult to say. From its colour, lapis lazuli, turquois, and carbonate of
copper have been suggested. As it is only mentioned in connexion with
the arms of Agamemnon, which were the gift of Cinyras king of Cyprus,
the latter mineral may be thought to have the best title, especially if, as at
Chessy, it occurs blue in Cyprus.

iyi Brigantia and other parts of Britain. 251

Why Pliny treats as a fable the story of the Cassiterides
yielding tin, is somewhat difficult to say. He classes the Cas-
siterides with Hispania, book iv. cap. xxii. (ex adverso sunt
insuiaj, — Cassiterides dictse Graecis, a fertilitate plumbi), and
speaks of Mictis (on the authority of Timaeus the historian)
as six days' sail from Britain, and as yielding candidum plum-
bum, iv. cap. IG. If the Cassiterides are the Ocrynian Pro-
montory and the Scilly Isles, from which, as recorded by
Strabo, the Phoenicians drew their tin ("I/crt? of Diodorus,
Mt/cTt? of Timaeus, and Ovr)KTL<i of Ptolemy being Vectis or
Wight, from which the tin was carried through France to
Marseilles), we may suppose that in the early period the only
route for the tin of Cornwall to the Mediterranean was by sea
to the western parts of Spain ; but that in the latter period
the track by land through Gaul to Massilia was preferred,
and the old trade had become a tradition which Pliny chose
not to adopt from Strabo, who is never quoted on this subject
by the author of the Historia Naturalis, but may be obliquely
and slightingly alluded to. Whether tin occurs at all in any
part of the Spanish Peninsula can hardly be doubtful after the
assertion of Pliny. He had been procurator in Spain, and by
his intimacy with Vespasian* must be supposed in position to
learn much of Britain, from the despatches of Petilius Cerealis,
Ostorius Scapula, and Agricola. But he was suffocated by
the fumes of Vesuvius in 79, one year after the appointment
of Agricola to Britain — and for the greater part of his literary
life, Britain was a scene of never-ending war and confusion.
Besides this, the Cornish promontory appears to have been at
no time much occupied by Roman stations, or traversed by
roads ; and it may be thought to have had then, as afterwards
in Saxon and Norman times, a history and commerce quite
distinct from and little known to the Belgic settlers in Albion.
He might be mistaken respecting Britain, of which perhaps
he could know only Albion ; but his positive assurance of the
occurrence of tin in Spain is confirmed by a passage in Bowles's
Natural History of Spain, and, as I hear from Mr. Kenrick,
by a later German writer (Hopfensach); it occurs, in fact,
according to one of our best books of mineralogy, in beds in
the mica schist of Gallicia. (W. Phillips, 1823.) Oxide of
tin has been found, besides, on both sides of the Erzegebirge
in granite, at Puy de Vignes (Haute Vienne), also in granite
in Wicklow (granite), on the east coast of Sumatra, Siam and
Pegu, and in Banca and Malacca. It has been found in
Mexico, Chili and Greenland, and mixed with other matters
in Finland and Sweden.

* Accessit imp. a.d. 69.

252 Mr. J. Phillips on Ancient Metallurgy and Mining

Upon the wliole, the case is probably thus. It is the old
PhcBnician trade, destroyed with Carthage, which Strabo de-
sci'ibes, and Pub. Crassus went to explore in the /cacro-iTeptSe?.
Diodorus Siculus narrates the course of trade in the days of
Augustus from Ictis, when Gaul offered an easy route to the
Mediterranean; but 100 years of war and commotion inter-
rupted this trade of Cornwall with the East, and Pliny was
suspicious of the fables of Greece, and knew that tin was
obtained in Spain. Notwithstanding this fact, it appears that
Cornwall and the Asiatic Isles have been the principal, almost
the only sources of the tin of the ancient world, that of Zinn-
wald being quite unknown till a much later date.

Stannum is evidently an alloy of an argentine or tin-like
aspect — a variable pewter — a metal more easily melted than
copper, for the lining of which it was much used in Pliny's
days to obviate the danger of cupreous solutions. This pro-
cess we now call tinning; and stannum*, with its variable
meanings, is perhaps the common parent of the French dtain,
meaning as often pewter as tin ; and of the German zi7in,
which like tin in the English workshops, is used sometimes
for pewter when lining vessels, and solder when covering sur-
faces which are to be joined. Our German silver, Britannia
metal, &c. belong to this class. The process of illination with
stannum must have been well-executed to justify the exclama-
tion of Plinv, that it did not augment the weijjht of the vessel
to which It was applied. The Brundisian specula made of it
yielded to silver, indeed, at last ; but they are declared to have
been of admirable efficiency.

Stannum, then, is an alloy of tin with lead, tin with brass,
tin with antimony, lead with silver, or other variable mixtures
of metals often associated in nature.

Pliny mentions adulterate or alloyed kinds of stannum,
composed of one part white brass to three parts of candidum
plumbum ; of equal weights of candidum and nigrum (which
is called argentarium) ; of two parts of nigrum and one of
candidum (called tertiarium) ; with this last lead pipes are
soldered f. Fraudulent dealers add to the tertiarium equal
parts of album, call it argentarium, and with it plate or line
other metals.

He gives the prices of these compounds and those of pure

* Pliny's notices of stannum are frequent. See Hardouin, vol. ii. 429,
22; 528,7; 530, 30, 31, &c.

Stanno et aere mixtis, 627, H — illitum seneis vasis saporem gratiorem
facit, 669, 14 — discern! vix possit ab argento, 669, 26 — aeramentisjungitur,
669,11.

f Hoc fistulae solidantur. This is the solder of our tinmen.

in Brigantia and other parts of Britain. 253

album and nigrum ; the former twenty, the latter seven de-
narii for 100 lbs.

Plumbum album, he says, is rather of an arid nature; the
nigrum is entirely humid ; " therefore the white is of no use
unless it be mixed with another metal. Silver cannot be leaded
(lined) with it, it will be melted first."...." It is affirmed that if
there be too little nigrum mixed with the album, the silver
will be corroded by it. Album is melted into brass-work
(inlaid, an invention of Gaul), so that it can hardly be known
from silver — these works are called Incoctilia " (silvered). He
then speaks of the application of this invention to the trap-
pings of horses and carriages, and other curious productions
of Alesia and the Bituriges, a subject which our esteemed
Kenrick has lately handled with his usual felicity. One of
Pliny's sentences is remarkable as narrating a class experiment
fit for a chemical school : "Plumbi albi experimentum in charta
est, ut liquefactum pondere videatur, non calore, rupisse."

The meaning seems to be, that the metal is fluid at so mo-
derate a heat as when fused to break by its weight, not burn
by its heat, the charta on which it is poured. Tin melts at
440° to 442°; lead at 612°.

What follows is a very important passage: "India neque aes
neque plumbum habet, gemmisque suis ac margaritis hoc per-
mutat."

May we be justified by this sentence in refusing to credit
the supposition that tin (plumbum album) was brought over-
land or by other routes from the Asiatic Isles and shores
towards Western Europe ? If so, Cornwall chiefly, if not
wholly, supplied the tin which entered so many ways into the
comforts and necessities during peace and war of all the na-
tions surrounding the Mediterranean and Euxine, Baltic and
German Ocean ; in fact, the world, as distinctly known to the
Roman geographers.

Let us now inquire into the means whereby the ancient
people reduced the metals which they were so earnest in seek-
ing across mountains and oceans at the point of the sword.
To confine the inquiry within reasonable limits, we shall
speak chiefly of tin and lead, the only metallic products, as it
appears, which were regarded by the ancients as abundant in
Britain. [Iron is mentioned by Caesar as of limited occur-
rence.]

Gold, the most widely if not most abundantly distributed
metal — found near the surface of the earth, in a pure and mal-
leable state, easily fused, uninjured by fusion — was probably
the metallic substance on which the earliest processes of fire
were tried, and they could not be tried unsuccessfully.

254» Mr. J. Phillips on Ancient Metallurgy and Mifiing

Tin, the ore of which has been found at the surface in many
situations with auriferous sand and gravel, cannot have been
long unknown to the gold-finders of the East and the West.
Some one of the many accidents which may or rather must
have accompanied the melting of gold would disclose the na-
ture of the accompanying white metal, whose brilliance, duc-
tility, and very easy fusibility, would soon give it value.

The melting o^ tin ore is, however, a step in advance of the
fusion of native gold. The gold was fused in a crucible
(xxxiii. p, 617, Hard.) made of white clay*, which only could
stand the heat and the chemical actions which that generated :
but tin ore would in this way of operation prove totally infu-
sible. It must be exposed at once to heat and a free carbo-
naceous element. The easiest way of managing this is to try
it on the open hearth. Perhaps some accidental fire in the
half-buried bivouacs of the Damnonii may have yielded the
precious secret. As to the fuel, we are told that pine-woods
were best for brass and iron (Hard, xxxiii. p. 621) ; but the
Egyptian papyrus was also used, and straw was the approved
fuel for gold. In the metalliferous country of Cornwall and
Devon, peat is plentiful; and an order of King John (1201)
allows the miners to dig tin, and turves to melt the tin, any-
where in the moors, and in the fees of bishops, abbots and
earls, as they had been used and accustomed. (Confirmed by
Edward I., Richard II. and Henry IV.-j-)

These and other singular privileges, extending as far as the
lands on which the crown claimed rights, are long anterior to
the other rights of property in Cornwall, Mendip, Derbyshire
and the Forest of Dean, and go far to justify the supposition
of our modern mining laws being a relic of Roman, or per-
haps of earlier than Roman times.

As the bellows was known at least 1000 years before Pliny,
we have here all the materials for a successful tin smelter's
hearth. If the smelting work was on waste land, and a little
sunk in the ground, we recognize the old 'bole' or 'bloomery '
of Derbyshire, now only a traditional furnace, but anciently
the only one for the lead and iron of that country.

Pure tin once obtained, there must intervene a long series
of trials and errors before its effect in combination with lead,
brass, silver, &c. could be known ; before the mode of con-
quering the tendency to rust in the act of soldering could be
discovered ; oil being in this respect as valuable to the tinner
as artificial chrysocolla was to the jeweller and goldsmith
(xxxiii. p. 621. Hard.). From all this it follows that the

* Such as now is called Cornish chiy, for example,
t De la Beche, in Report on Geology of Cornwall.

ill Brigantia and other parts of Britain. 255

smelting of tin might be, and probably was, performed by the
inhabitants of the Cornish peninsula. This art they may
have brought from the far east ; Phoenicians may have taught
it them ; but all the accounts of the ancient tin trade repre-
sent the metal, and not the ore, as being carried away from
the Cassiterides. Diodorus mentions the weight and cubical
form of the tin in blocks, carried from Ictis to Marseilles
and Narbonne ; and Pliny says of the Gallician tin, that it
was melted on the spot.

Did the Cornish or Gallician miners make bronze ? For
this is generally the compound indicated by the Roman ceris
metalla^ though it is undoubted that they also knew of, and
distinguished zinc brass. There is, I believe, no instance of
a single bit of pure tin or pure copper being found with the
numerous 'celts,' which occur in so many parts of England;
nor is any other proof given that the direct union of tin and
copper was effected by the natives of Britain. Copper is so
abundant in Cornwall that it might tempt to the other hypo-
thesis; but this copper is a sulphuret; it is found united to
the sulphuret of iron, in deep veins, and in a matrix of quartz;
and these are things which render the production of pure
copper one of the most refined operations in smelting. Caesar
tells us the brass used by the natives of Britain was imported.
Probably Cyprus, — colonized by the Phoenicians, to which
old authors refer as the original source of brass — Cyprus with
its ancient copper mines (Tamassus), which has given its name
to the metal, might be one of the points from which bronze
radiated over the Grecian, Roman and barbarian world. It
was from Cinyras, the king of Cyprus, that Agamemnon re-
ceived his splendid breastplate with twenty plates of tin, and
its liberal additions of turquoise, lazulite, or rather malachite,
obtained perhaps from the soil of the island. (Pliny, xxxiii.
p. 633, Hard.)

The works of "H^ato-ro?, the Crawshay of antiquity, may
have been fixed on Lenmos on account of some volcanic ap-
pearances there; but the tradition shows at least that thevarious
operations of refined metallurgy were not strangers to the
islands of the Mediterranean ; and the uniformity of design
and composition in the ancient celts, chisels, fiuKcXXa, and in-
struments of war, implies a common, and that not a barbarous
origin. The perfection and variety and great proportions of
the brass work executed in the Grecian states and colonies,
may also be regarded as indicating the local seat of the early
as well as the later art of working in bronze.

Lead was obtained in Spain and Gaul from deep and labo-
rious mines (xxxiv. p. 669, Hard.), but so abundantly near

256 Mr. J, Phillips on Ancient Metallurgy and Mining

the surface in Britain as to suggest a law for preventing more
than a limited production — a Brigantian law of vend. The
Romans employed lead in pipes (fistulae) and sheets, which
were soldered with alloys, as already mentioned. This lead
was previously refined, and its silver removed ; the silver in-
deed being often the object of the enterprise. How earnestly
silver was sought — how well the mining operations were carried
on by the ' old men ' — appears from the notice of the Cartha-
ginian mines in Spain, the pits and levels driven by Hannibal
being mentioned as in wonderful preservation by Pliny. The
same may be said of at least one set of mining works of Ro-
man date, in the extreme parts of South Wales, viz. the Go-
gofau near Lampeter, where gold was extracted with much
labour from broken and pounded quartz, of which enormous
mounds remain. The adit still exists, and was lately entered
by Sir H. T. De la Beche, who found in it a specimen of
native gold. In the vicinity, tradition indicates a Roman
settlement ; and a massive chain of gold and other remains
were found, and are now possessed by the family of Johnes of
Abercothi *.

The districts in Britain, where lead veins coming to the
surface in abundance might justify the praises of Pliny, are,
in the south, Mendip; in the west, Flintshire, &c. ; in the
north, Derbyshire, Yorkshire and Cumberland, that is to say,
the Brigantian territory ; and it is to this last district that the
descriptions apply most correctly. Lead cast in Roman
moulds, pigs, in fact, of the age of Hadrian and other empe-
rors, have been found in Flintshire, Derbyshire, Yorkshire,
and some other counties. But few ancient mining instruments
have ever been found in the lead-bearing districts of Britain f ;
and I am strongly of opinion that much of the lead ore was
collected from the surface by aid of water, artificially directed.
The process, in fact, is described by Pliny in terms so exactly
applicable to the modern ' hushes ' of Svvaledale, that no doubt
can remain of this custom, which is now esteemed rude and
semi-barbarous, being of Roman or earlier date in Britain.

As thus from Roman or earlier times our lead-mining de-
rives its * hush,' its levels, and shafts, implements for washing,
and other processes of the workmen, and the forms, weights,
and marks of its melted metal, we may easily admit a similar
origin for the melting processes. Lead mostly occurs in the
sulphuret, which offers no particular difficulty in the fire. By
cautious roasting, its excess of sulphur may be removed, and

* See Sir R. I. Miirchison's remarks on Gogofau (Sil. Syst. p. 367,368).
t Sir R. I. Murchison mentions Roman mining utensils at Shelve in
Shropshire (Sil. Syst., p. 279).

in Brigantia and other parts of Britain. 257

the subsequent melting with charcoal, or a flux, be facilitated.
Indeed without roasting, and without flux in many cases, the
lead will flow out of the ore, if placed among flaming wood or
peat, and subjected to a sufficient stream of air.

But the use of fluxes could not long remain unknown in the
limestone districts of Norihumbria, or amid the fluoric veins
of Derbyshire — limestone and fluor being to this day valuable
aids in the furnace. Peat was the fuel in Cornwall, and still
is in Yorkshire; and perhaps the Roman smelters did really
erect their furnaces on waste ground and heaths at Dacre and
Matlock, far from the mines of Greenhow and Youlgreave,
even as is done at present with the cupolas of Lee and Langley
mills.

The use of crucibles ['xpavoi)^ bellows, cavities of some pe-
culiar sort {Ka/jiLvoi), perhaps chimneys, great variety of car-
bonaceous fuel, the power of purifying and alloying, and
knowledge of the properties of alloys, appear quite conspicu-
ous among the ancient arts.

The inscriptions * on these masses of lead are in the same
general form as the * marks ' of the different mines now in
work, and which, no doubt, are their literal and line.al descend-
ants. Thus the Aid or Auld Gang mine of Swaledale, old in
the days of the Saxons ; the mines of Greenhow Hill, which

* The following inscriptions have been recorded on pigs of lead obtained
from British mines during the Roman sway in Britain. It will be remarked
that they belong to early imperial times.

IMP. CAES. DOMITIANO. AVG. C. C. S-VII. Found at Hagshaw
Moor, Dacre Pasture, near Pately Bridge, Yorkshire, in 1734.

A Roman pig of lead, weighing 126 lbs., was found on Cromford Moor,
near Matlock, in the year 1777, having the following inscription in raised
letters on the top :

IMP. CAES. HADRIANI. AVG. MET. LVT.

A second was discovered near Matlock in 1783. It weighed 84 lbs,, and
was 19 inches long at the top, and 22 at the bottom. Its width at the top
was 31 inches, and at the bottom 4^. The inscription appears to contain
these letters :

L. ARVCONI. VERECVND. METAL. LVTVD.

A third, with the inscription also in raised letters on the top, was found on
Matlock Moor in the year 1787. It weighs 173 lbs., and was 17| inches
in length, and at bottom 20|.

TI. CL. TR. LVT. BR. EX. ARG.

Glover's Derbyshire, vol. i. p. 71,72.

A fourth is stated to have been found at Castleton, on which only the
letters IMP could be read distinctly. It was said by Mr. Mawe to be pre-
served in the museum of Mr. Green at Lichfield.

Sir R. I. Murchison records a Roman pig of lead (from the Shelve mines
in Shropshire probably), bearing the inscription, IMP. ADRIANI. AVG.
(Sil. Syst., p. 279.) This pig is said to be unlike the modern pig.

Phil. Mag. S. 3. Vol. 34-. No. 229. April 1849. S

258 Mr. J. Phillips on Ancient Metallurgy and Mining

supplied sheet and pipe lead for our baths and coffins at York,
as well as tribute to the imperial treasury ; the mines of Mid-
dleton and Youlgreave (Aldgroove), from which the Lutudce
sent not only lead, but ' exargentate ' (that is to say refined)
lead from which the silver had been removed, use to this day
the pig of the same weight of 1^ cwt., of similar shape and
similar mark to that of 1800 years antiquity*. And just as
at the present day, the countryman whose galloway is tired
drops the leaden load by the way side, for another day's work,
so in the days of Rome, the Brigantian lead was thrown down
from the tired caballus by the side of the ancient mining road,
on Matlock Moor in Derbyshire, and Dacre Pasture in York-
shire.

This fact of the discovery of the Roman lead, 7iot at the
mines, but at a distance of some miles from them on a track
leading towards a Roman or rather a Pre-Roman station, is of
much importance in archaeology. For thus we arrive, in the
first pJHCe, at the conviction of the existence of very ancient
mining roads not of Roman work, nor probably of Roman
but of earlier date, leading toward Cataractonium, Isurium,
Eburacum, Mancunium, Derventio, or rather to the Brigan-
tian towns or centres of trade, on which the Romans following
their wont in Africa, Spain, and Gaul, fixed their attention
and established their war camps and their colonies. The
politic lords of the world broke up no national industry, set
no legionaries to supplant the native miners, but stationing a
few cohorts on the ancient roads, in or close to the mining
district, as at Hope and Bainbridge, to control a rude popu-
lation, received regularly the fruits of the industry which they
might direct, but did not personally share. Viewed in this
light, how complete appears the grasp of the Roman treasury
on the mining fields of Britain ! The Fossway from the
Ocrynian promontory crosses the Mendip Hills — the road
from Mancunium to Bremetonacum traverses the Calamine
district of Bowland — the road from Derventio or Tutbury to
Mancunium runs along the west of the great Derbyshire field,
and the legionary path from Carlisle to York goes right across
the metalliferous country of Yorkshire and Durham.

We may even ask, with some confidence, whether the line
of the Hadrian wall, which cuts off from the north all the
richest mines of the Derwent, the Allen, and the Tyne, but
abandons the mossy dales of bleak Northumbria, was not

• The modern pig is made near to — of a fodder or 176^ lbs. Three
Roman pigs found near Matlock in 1777, 1783, 1787, weighed 173, 126,
and 84 lbs., these being as 1, %, and \ of the modern pig.

in Brigantia and other parts of Britain. 259

drawn with especial reference to the mining wealth of the
districts.

May we not regard, as a confirmation of all that has been
advanced touching the antiquity of our mining processes, the
fact of the existence to this day, though impaired by recent
acts of parliament, of peculiar rights and privileges in the
mining districts ? These rights are sometimes guaranteed by
and appear to emanate from royal charters, as in the stan-
neries of Cornwall and Devon, but they are probably of far
earlier date, and have merely been confirmed as old customs
by John and his successors. In Mendip, the Forest of Dean,
and Derbyshire, the miners' rights were preserved by royal
officers, but the rights themselves transcend all history and
tradition. To sink a pit or drive a level in any field ; to cover
the rich herbage with barren ore-stufF; to cut a way to the
public road ; to divert, employ, and waste the running waters ;
and to do all this without consent of owner, and without com-
pensation being so much as asked by lord or villein, landlord
or tenant, implies in Derbyshire a settlement of mining rights
long anterior to Domesday Book, the charters of Repton
Abbey*, the neighing of the Saxon horse, and the flight of
the Roman eagle. In connection with all that has been men-
tioned before, — the furnaces, the roads, the restricted vend,
the foreign trade — tliey seem to me to indicate a people wht)
came with many inventions from the metalliferous east to the
metalliferous west, before the Athenians drew silver from
Laurion, or the Carthaginians from Iberia.

To these ancient, these Semitic mining processes we have
added perhaps steel instruments, and certainly explosive
agents ; the ore-hearth still remains, but it is generally yield-
ing to the reverberatory furnace; silver is no longer obtained

* The mines in the neighbourhood of Wirksworth were wrought before
the year 714; at which period that district belonged to the nunnery
at Repton, over which Eadburga, the daughter of Adulph, king of the
East Angles, presided as abbess. In that year the abbess sent to Croyland,
in Lincolnshire, for the interment of St. Guthlac, who was originally a monk
of Repton, a sarcophagus of lead lined with linen (plumbum lintheumque).
This lead was obtained from the possessions of the old Saxon religious
establishments at Repton, part of which were the mines near Wirksworth.
In the year 835, Kenawara, then abbess of the same nunnery, made a grant
to Humbert, the alderman, in which she surrenders that estate of mines,
called Wircesworth, on condition that he gives annually, as a rent to Arch-
bishop Ceolnoth, lead to the value of 300 shillings, for the use of Christ's
Church, Canterbury. On the destruction of the religious houses by the
Danes in 874, it is probable that the lead mines became the property of
the Crown. As such they are mentioned in Domesday Book. — Glover's
Derbyshire, vol. i. p. 73.

S2

260 The Rev. B. Bronwin on the Coefficients of Sines

by oxidation of some thousand times its weight of" lead ; steam
blows our furnace fires, rolls and pipes our metals, and flies
with iron wings on roads more solid than the Appian Way.
The world of George Stephenson is much different from that
of Julius Agricola ; but some features of the past remain to
connect the earliest with the latest aspect of our country : and
among these the least altered, and the most instructive, appear
to be the mineral products and the mining processes. If by
these we judge the great Brigantian tribes which surrounded
Isurium, they must be placed far higher on the scale of civi-
lization than the place usually accorded by the Saxon to the
Celt.

I presume to think, indeed, that without full attention to the
mining history of Britain, as indicated by fragments in classic
authors, and illustrated by processes not yet extinct, the opinion
which may be formed of the ancient British people would be
altogether conjectural, derogatory, and erroneous.

XXXVI. On the Determination of the Coefficients in any series
of Sines and Cosines of Multiples of a variable angle from
particular valuesof that series. By the Rev. Brice Bronwin *.

MLE VERRIER'S method of determining the coeffi-
• cients of a series of sines and cosines of the multiples
of a variable angle requires a very great amount of labour.
Moreover his formulae contain large factorials of sines in their
denominators, which endangers the accuracy of the results.
The method of Mechanical Quadratures, strictly so called, is
attended with some uncertainty ; because each term of the cor-
rection may vanish, while the sum of all of them to infinity
may have a finite value, owing to the initial and terminal series
of differences being divergent. Some more suitable method is
therefore wanted. Sir J. W. Lubbock, in some tables which he
very kindly sent me a short time ago, has inserted one given by
Bessel and Hansen in the Astronomische 'Nachrichlen^ which
is perhaps as good a method as we have reason to expect. In
what way the authors arrived at it I do not know. As it is
important, and may be found in a very simple manner, per-
haps the investigation may not be unacceptable. Besides, the
formulae given by them, as printed in the tables above-men-
tioned, are only one out of several distinct sets, and are not
well adapted to all cases j and moreover they may be modified
and simplified.

* Communicated by the Author.

071(1 Cosines of Multiples of a variable angle. 261

cos fl + cos(fl + /0 + cos (5 + 2A) ■!-.... + cos (fl + («--l)/0

nh
sin —

cos(fl4.<'M_n- \ —

sin-

{^Hn-l)\)

sind+ sin(H^) + sin(fl + 2^)H-.... + sin(d+(«— 1)^)

> (a-)

= sin (, + (,_!)_)_-

sin-
2

are formulae which will be needed ; and we will begin with
(p(a?) = Bo + BiCos;r+B2Cos2a?+B3Cos3a;+ . . (A.)

(p(a) = Bo + Bj cos « + Bg cos 2a + Bg cos 3« + ......

f (a + ^) = Bo + BiCOs(a + ^) + B2COs2(a + Jc) + B3COs3(a + k) + ...

<p{a + 2^) = Bo+ Bi cos (« -f 2>?r) + Bg cos 2(a + 2k) + Bg cos 3
(« + 2/<:) + ...

^(«+(n— l)^") = Bo + BiCos(a+(r«-l)^) + B2Cos2
(« + («- 1)^) + B3 cos 3 (« + («- J)^)+...

Adding these together by summing the terms vertically by
means of the first of (a.), putting ix for 5 and ik for h, we have

. nik

i;^(« + i'^)=wBo+5;B,cos?(«+(«-l)|) ^- • (!•)

sin— •
2

Let B;„ be the last coefficient which is sensible, then i will
represent all the numbers 1, 2, 3, .... m. In order that all

the quantities B, may vanish, we must have— = w, or ^= — .

Then . nik . .

sm-rr- = sm«7r=0;
2

but

. ik . iv

sin — = sm —

2 n

is not nothing if « > w, or greater than the greatest value of /.
If/t=2«,

cos?

(, ^. k\ . . ink . ,

«+ (w — I) — 1= cos 2«a, sm — - = sm tnu;

262 The Rev. B. Bronwin on the Coefficients of Sines
and (1.) will reduce to

S9{(2^'-H)a}=nBo + SB,f^. . . (2.)

Ifac=— - and 2w > W2, sin 2ma=0; but sin ?a is not no-
2w

thing. Therefore

B«=^^4^~j'2''>"'- • • • («•)

where i' has all the values 0, 1, 2, ....w— 1. This value of
Bq requires a value of n only half as large as the supposition

k= — would require.

In order to find B„ multiply the first of the assumed equa-
tions by 2 cos /a, the second by 2 cos ?(a + ^), the third by
2 cos /(« + 2A), &c., and sum as before. Any coefficient B^
will now be multiplied by

2 cos^acosia= cos {p — i)ci+ cos (p + 2)«, 2 cos^(« + ^)

cos i{a. + k)= cos {p—i){oi + k)+ cos {p + *)(« + k\ &c. ;

which will constitute two series, the sums of which must be
separately taken. We shall thus have

. ink

2t<^{ot. + i'k) cos i(ot + i'k) = 2Bo cos « ( a + (« — 1) ^j

. , .. nk

. ik

sm -—

2

SBp cos {p - i) Lt^{n- 1) - j

k

2

+ 2B,cos(p + 0(« + (n-l)|) f . . ('i.)

sin (^^ + 2)2

But when ^=/, the coefficient of Bj will be

. , , sin ink
n + cosi{2ci+{n-\)k)-^^jj^.

We may give to^ all the values 1, 2, .... m, except /; ftnd if
n

and Cosines of Multiples of a variable angle. 263

flJc Tile

sin(p-/) — =0, sin (;? + i)— =:0;

but

k k

are not nothing if w > 2w. Thus the coefficients of Bq and
Bp will vanish. Also s\x\ink = Oi but sin e^ is not nothing,
and the coefficient of Bj reduces to n.
But make /5- = 2a, and (4.) becomes

22f {(2,'' + lM cosi(2i'+l)« = Bo?^|^ +SB,

. !^ • (5.)

sin ( j3 — i) 2na. sin ( /? + 0 2^*'=* "1
.2 sin (j9 — /)« 2 sin {p + i)a. J

When j!? = /, the coefficient of Bj is

sin 4 /wa

n-\ : .

2 sm 2ia.

It

{;

Make a=— , n:::^m^ and the coefficients of Bq and Bp will
vanish; and that of Bj reduces to n. Thus we shall have

K>/w, B,= ^2f|(2.'+l)£}cosf(2/'+l)^, . (6.)

where «' has all the values 0, 1, 2, .... w— 1.

If we had not restricted the value of w, we should have had

27r
a series of terms, as B„_,-, B«, B„+i, &c. when k— — ; and the

series B2„_<, B^^, Ba^+i when k='lci— -.

The formulae (3.) and (6".), making 72 > m in both, enable
us to determine all the coefficients of (A.) But

COS (2n-l)- =± cos— , cos {^n-3)^=^+cos—,

&c. to

+ cos ^ _ ,
2w

if n be even, which it will always be convenient to suppose.
Therefore

2,{(«'+ DfJ cos (.. . i)| = {, (fj ±.(^')}

2n

264 Tlie Rev. B. Bronwin on the Coefficients oj Sines

Hence if we put u^^ Wgj &c. for the sums of the first and last,
of the second and last but one, &c. of the particular values of
f{x)i and in like manner Vj, Wgi &c« fo^' the differences of the
same, we may replace (3.) and (6.) by

n \ 2'

■D 2/ iV Silt {n—i)iit\

if 2 be even; I ('''•)

r> 2/ /tt , S/tt , , (w--l)/7r\

^^ = d'^' '°^ 2^ +^^*^^' 2^ +•••• +^|"^' 2^>

if i be odd.

This will diminish the labour of numerical computation
considerably; and when we know the value of ?i, we may for
many values of i effect a further reduction.

Now let

\|/(a7) = Aisin A' + A2sin2a? + Ag sin3^+ (B.)

rJ/(a) = Ai sin a + Agsin 2a + Agsin 3a+....
vI/(a4-A:) = Ai sin (« + A;) + A2sin 2(a + ^") + A^sin S{a + k)-{- ....
\I/(a 4- 2^) = Ai sin (a + 2^) + Ag sin 2(a + '2k) + Ag sin 3(a + 2lc)
+ ....

• • • • •

t|/(a + («— 1)^) = Aj sin (a+ (?/— 1)^) + A2sin 2(a+(« — 1)^)

+ Ag sin 3(a+ (« - 1)^) + ....

Multiply the first of these by 2 sin /«, the second by 2 sin /
(a + /c), the third by 2 sin /(a + 2/?:), &c. and sum as before.
The coefficient of A^ will contain the terms

2 sin pa. sin za= cos (p — i)a— cos (p + i)a,
2 sin J3(« + k) sin i{u + A") = cos {p — i){u -\-k)— cos {p + i){u + ^),
&c., and therefore
22\I/(a + i'/c) sin /(a + i'i) = S Ap cos ( j9 — /) ( « + (n— 1 ) - j

--^ _SApCos(;)H-?)(« + («-l)-)

sin (i?-0 -

. , ..nk
sm{p + i)~

1 («•)

and Cosiries of Multiples of a variable angle. 265

But wheu^=/, the coefficient of A,- is

sin ink

(, , X , \ SH

2a-\-[n — \)k I— r
^ ' / SI

sin ik '

Suppose Am the last coefficient which is sensible, and let p ,
have all the values 1, 2, 3, .... m except i. As the coefficients
of Ap and Aj differ from those found in (4.) only in some of

their signs, the same conclusions result from them when we

2_
make k= — , and k=z2a. Therefore, passing by the formulse

^ 27r

derived from k= — , as we have before done, and taking only

that which results from — =«= — -, we have

2 2n

A,= I X^{{2i'+1) ^} sin i2i'+ 1) ^, n > 7«, . (9.)

where i' has the same values as before.
This may be reduced as (6.) was. For

• /^ , \ ^i" — • 2V . ,^ ^v zV _ . S/tt «
sm(2« — 1)— =+ sm— , sm(2n—3)——+ sm— — ,&c.
^ 2w 2n ^ 2n 2n

If therefore we make Wj, Wg, &c. the sums of the first and last,
of the second and last but one, &c. of the particular values of
'^{x)i and ^1, /g, &c. the differences of the same, (9.) may be
replaced by

. 2/ . i-Jt , . Silt , , . ln—\)i'n

^'■= ;rr''"^2^+^^'^"-2;r +--+^i'^"-W-

or

. 2 /, . i-TT , ^ . S/tt ^ , (n—l)iTr

^■■= n V' ''" 2S +'» ™ ^ + •••• +% "" -1^

the first when i is odd, the second when it is even.
We now proceed to the more general form,

(10.)

yi[a:) = Bo+B, cosiT + BgCOs 2x+ ,..,
+ A, sin j; + Ag sin 2a? +

;;;;} . . (c)

The solution of this form might be derived from those of

the two forms before treated, by taking that particular case in

27r
which k= — ; but I prefer treating it separately.

y(a) = Bo+ BiCosa4-B2Cos2a+....
+ Aj sin a+ Ag sin 2a + ....

/(a + ^) = B(,+ BiC0s(« + ^) + B2C0s2(« + ^) + ....
+ Aisin (« + ^) + A2sin 2(a + it) + ....

266 The Rev. B. Bronwin on the Coefficients of Sines

y(a + 2A') = Bo + Bi cos («+?/•) + B2 cos 2(a + 24) + ....
+ Ai sin (a + 2Z') + A2sin2(a + 2/^) + ....

/(« + (w- 1)^) = Bo+ Bi cos (« + (»- 1)A:) + Bg cos 2
(«+(«-- l)X-) + ....

+ Ai sin (a + (n— 1)^) + A2sin 2(a+ (« — 1)/^) + ....

Let B^ or A„ be the last of the coefficients which is sen-

27r
sible. Then if Z'= — , taking the sum of these as before, the
n °

terms containing the cosines all vanish, as we have found from

(1.), and the general term of the sines is

. ink

Aisin i\ot+{n — \)-j

sm-^

. ik

sm —

2

which vanishes for the same reason, having the same vanish-
ing factor. Therefore

Bo=is/(«+ — ),^>m. . . . (11.)
n \ n J ^

Now multiply the first by 2 cos ia, the second by 2 cos i

(« + A:), the third by 2 cos ?(a + 2yt), &c., and sum. The part

of this sum depending on the cosines is given in (4.) ; and the

27r
result, when k=. — , is the same as there found in the same
n

case. The coefficients of Ap will be
sin {p^i)oi.+ sin (p + ?)a,

sin (p — i) (« + it) + sin {p + i) (a + k\ &c. ;
and their sum

. , .. nk

(2,-/)(« + (/^-l)|)- ^ +sin(;, + /)

sm

sin {p — i)

2

nk
I'

The coefficient of Aj will be

• ^- • ^-z 7\ • -/^ / ,. J. sin ink

sm 2i«+ sm 2i{oi + k) + ..,.= sm 2(2«+(«— 1)a:) — ; — rr-.

sm ZfC

and Cosines of Multiples of a variable angle, 267

As the vanishing factors here are the same a9 in (4.), these all
vanish when n > 2m. Therefore

B,= |^«+ ^)cos.-(<.+ '^),n^2m. . (12.)

Again, multiplying the first by 2 sin i «, the second by
2 sin z(« + ^), the third by 2 sin /(« + 2/^), &c. and summing,
it is obvious that the coefficient of Bp will be had by changing
p — i into i—p in that of Ap last found, and therefore it will
vanish under the circumstances supposed. The coefficient of
Bi will be the same as that of A;, and consequently will vanish
also. It is to be remembered, that we do not give to p the
value p — iy as we always find the terms depending on this
value separately. The coefficient of Bq will be

. ink
sm —

2 sin?( «+(«— 1) — )

. ik
sm-

and will therefore vanish. And it is obvious that the coeffi-
cients of Ap and A; will be the same as those found in (8.),
and will be 0 and n respectively. Therefore

In (11.), (12.) and (13.), i' has the same values as in the other
forms, namely 0, 1,2, .... n—l; and these three formulae give
all the coefficients of (C). It will be evident that we may
always suppose n as large as we please, and therefore that
w > 2»? in (11.).

If «=0, i'—O, l,....w-l, or if «= — , i'=l, 2, ....n, the

two sets of formulae coincide. It will be better to make the
latter supposition, then we have

" n "^ \ n / n \ n / \ n / {

Ai= -2/1 — }s\m[ — ), ?'=1,2, ....
n '' \ n / \ n J

(HO

We may derive from these others similar to (7.) and (10.).

TT

Ifa=-, i' = 0, 1, 2 ....»— 1, n^2m.

268 On Sines and Cosines of Multiples of a variable angle,

cos<M±ih). A,= |s/((^^) . (...)

It will always be convenient to make n divisible by 2, or
even by 4, 6, 8, &c. And thus we shall have

cos(2w— 1) — = cos — , cos (2w — 3)

n

COS , &c. ;

n

sm (2w— 1)— = — Sin—,
n n

Make

(2»-l)
n
w — 1 )7r

^).

We may therefore replace (15.) by

ztt Sztt

sin {In •— 3) — = — sin — , &c.
n n

T, 2 f 27r , 3«9r ,

J3i = — <«, COS l-WoCOS V .... + W« COS

www 2 n

(w— I)z7r"|

. iTT . ^iit . in-

sm f-i'osm f-. ••.-!- t'ra sm ^^ —

w I ' w n 2

DiTT

(16.)

The only fault of this method of finding the coefficients is
that it requires a large value of ^^, and consequently a large
number of particular values of the functions <p{<x), &c. But
if we give it a smaller value, the quantities

sin(p-2)-, sin (/? + «)-,

sin ik, sin 2/«, &c..

would vanish for certain values of p and i ; which would in-
troduce the terms Bn-i, B„, B«+j, A„_,-, A„+,, &c. ; and we
should have more unknown quantities than one in an equation,
and could not determine them without employing some other
means. In certain cases where the form of <p(a;), &c. is given
and simple, and we can ascertain with certainty the correction,
the method of Mechanical Quadratures, properly so called,
may require less labour, and may therefore be preferable.

Gunthwaite Hall, near Barnsley, Yorkshire,
March 7, 1849.

[ 269 ] • '^

XXXVII. Ofi some phenomena of Binocular Visio7i,
By MM. L. FoucAULT and J. Regnault*.

IN a beautiful investigation on the vision of objects of three
dimensions, Mr. Wheatstonef states that when two visual
fields, or the corresponding elements of the two retinae, simul-
taneously receive impressions from rays of different refrangi-
bility, no perception of mixed colours is produced. The as-
sertion of this able philosopher being opposed to the opinion
of the majority of those who have attended to the same sub-
ject, we have thought it useful to repeat, modify, and extend
these experiments. The stereoscope of Mr. Wheatstone
offered a simple means of disentangling these delicate obser-
vations of all complication capable of injuriously affecting the
accuracy of the physiological results.

The recomposition of mixed tints by means of vibrations
produced on the retinae by different coloured rays is beyond
doubt. But the aptness varies in a remarkable manner in
different individuals; it is possible that it may be exceedingly
weak in some persons, and exceptionally null in others.

The tendency of one of the eyes to become inattentive in this
kind of experiment is very remarkable when the whole extent
of the visual fields is uniformly lighted up by different coloured
rays. If we cause an impression to be made on limited and
corresponding parts of the retinse, the power of the attention
constantly favours the recomposition. If two coloured rays
susceptible, on reaching a white screen, of producing a mixed
tint produce the same sensation when acting separately on the
corresponding portions of the retinae, it seems probable that
two complementary rays will produce the sensation of white
by affecting the corresponding elements of the sensitive mem-
brane.

To prove this recomposition with respect to a great number
of complementary tints, and present the phaenomenon in all its
clearness, we arranged the following experiment : — We affixed
to the stereoscope two plane mirrors, forming a variable dihe-
dral angle, the vertical edge of which is placed symmetrically
in relation to that of the two glasses of the stereoscope. The
vertical uprights bearing the grooves for the purpose of intro-
ducing the images are perforated by two large circular aper-
tures. In the grooves are placed two glasses, on which are
pasted two circular screens of white paper of the same size,
and of a diameter less than that of the apertures. Two large
luminous rays of complementary tints, obtained by chromatic
polarization, are directed horizontally upon the plane mirrors

* From the Comptes Rendus for Jan. 15, 1849.
t Philosophical Transactions, part 2, 1838.

270 On some yhcenomena of Binocular Vision.

*
which reflect them ; they traverse the glasses of the grooves
which remain dark ; but when reflected irregularly on the cir-
cular screens they give two coloured discs, exactly identical as
to form and extent, which become the images conveyed by the
stereoscope to the corresponding elements of the retinae. We
might, by means of an appropriate disposition of the polarizing
apparatus, successively present numerous complementary tints,
vary at the same time the intensity of the two coloured images,
and modify the intensity of one or other of the images separately.

The following are the physiological results we have ob-
served. When the corresponding elements of the retinae re-
ceive an impression at the same time, the alternations of acti-
vity or inertness of one of the eyes is generally manifested at
the commencement of the experiment, and sometimes one of
the tints is perceived, and at other times its complementary
one ; but after a duration of observation, varying considerably
according to the individuals, only a single white circle is seen.

When the eyes are in some degree accustomed to this un-
usual mode of impression, the tendency to recomposition be-
comes so energetic in some persons, that the screens might
present successively all the complementary tints which the ap-
paratus furnishes without there being any sensation corre-
sponding to the colours; only white light is seen.

On diminishing the intensity of one of the colours, the other
remaining constant, recomposition still takes place; but the
white disc appears to become covered more or less strongly
with the predominant tint.

If the intensity of the complementary rays is varied in the
same manner for the two collections of rays, the recomposi-
tion is made with greater facility at the commencement of the
observation, as their intensity is more moderated.

Of the complementary rays which we have examined, the
sensible blue and the yellow tints are best adapted for the ex-
periment, and immediately furnish the sensation of white.

We believe that this phsenomenon is owing to the circum-
stance, that the accommodation of the eyes being the same for
these groups of rays, according to the portions of the spectrum
which they occupy, the efforts necessary to produce recom-
position are on that account considerably less.

We find that, saving exceptional cases, the sensation of
white light may be produced by any two complementary chro-
matic impressions in ench of the eyes ; that the sensation solely
of white arising from two complementary rays is independent
of any mutual action of these rays externally to the visual
apparatus; that the luminous impressions produced on the
retinae retain their properties even to the innermost recesses
of the brain.

C 271 ]

XXXVIII. On the Meteorology of England in the year 1847.
J[y James Glatsher,^^^'., of the Royal Observatory, Green-
wich *.

THE meteorological returns for the year 1847 furnished to
the Registrar- General were from about thirty different
stations, situated between the latitudes of 51^° and 55° ^ and
between the longitudes of 5\° W. and 1:^° E. of Greenwich.
The elevations of the different places varied from 30 feet to
350 feet above the level of the sea.

The monthly returns in each quarter were published in
their respective quarterly reports, but only the monthly values
corresponding to the limes of the observations, and not
those showing the mean values for the month ; these were
reduced to mean values for the formation of the quarterly
tables; but the corrections used in their reduction were those
deduced from three years' observations only ; since that time
I have formed tables from the five years' Greenwich observa-
tions ending December 31, 1845f, and have reduced the ob-
servations again by these values. The true monthly values
thus found will probably appear in a future Annual Report of
the Registrar-General.

On discussing the true monthly values, it was found that
the temperature of the air decreased from the month of January
to that of February at all places south of the latitude 52°;
that at places situated near this parallel the temperatures of
these two months were nearly the same, and that north of this
parallel there was an increase of temperature from January
to February. Therefore the coldest month in the year, at
places whose latitude was less than 52°, was February, and at
places north of this parallel was January. The hottest month
at all places was July. By taking the monthly means from
the mean annual temperature, the annual variation is shown.
It was found to be a single progression, having one ascending
branch and one descending branch, being nearly identical
with that shown in the tension of vapour. This evidence is
conclusive upon the dependence of the monthly march of the
vapour upon the temperature ; each element has one ascending
branch and one descending branch, and the march is harmo-
nious.

The only other circumstance with respect to temperature
to which I need allude, is the fact, that whilst the decrease
of temperature month by month proceeded regularly in the

* Communicated by the Author.

t See Philosophical Transactions, part i. 1 848,

27*2 On the Meteorology of England in the year 184-7.

counties of Cornwall and Devonshire, and also, though at a
less rapid rate, in the northern latitudes, the temperatures of
September and October at intermediate places were nearly of
the same value.

It was found that the months from March to July were less
humid than the average for the year, and that the remaining
months were more humid than the yearly average. The
months of March to July are those distinguished by the tem-
perature of the air increasing, and those from August to
January or February by a decreasing or stationary tempera-
ture. The places situated in the counties of Cornwall and
Devonshire were less humid than elsewhere; for notwith-
standing the greater amount of vapour contained in the same
mass of air in those counties to that in other places, yet the
temperature increased more rapidly than the air received the
addition to its vapour necessary to keep an equal degree of
humidity.

The following Table contains the mean of all the monthly
values of each element : —

From the numbers in this table the following values have
been deduced for the year 1847: —

The mean pressure of the atmosphere of dry air at the level
of the sea was 29*641 inches. This value applies to all parts
of the country.

The mean pressure of the atmosphere of vapour in latitude
51° so' was 0"319 inch; and this value seems to diminish
O'OIO inch for an increase of 1° in latitude.

The mean temperature of the air at the level of the sea
in latitude 51° 30' was 50^^ O'; and this value at a uniform
level was found to vary 1° very nearly for a change of 1° of
latitude.

No certain law can be deduced from the observations of
1847, representing the excess of the temperature of the air
above those of evaporation and dew-point; depending upon
the difference of latitude, it seems however that the excess be-
comes smaller as we proceed north ; but the observations of
the wet-bulb thermometer during the early part of the year
1847 were unsatisfactory in many places, and the following is
all the information I can give in this respect: —

The mean excess of temperature of the air above that of
evaporation was 3°'0, and above that of the dew-point was
S°'Q^ in the counties of Cornwall and Devonshire;

The mean excess of temperature of the air above that of
evaporation was 2°'2, and above that of the dew-point was
4°'7, at places situated in the vicinity of the sea;

The mean excess of temperature of the air above that of

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P^/7. M«g. S. 3. Vol. 34. No. 229. ^/^nV 1 849. T

274- Mr. J. Glalsher on the Meteorology of England

evaporation was S^*?, and above that of the dew-point was
6°'9, at places situated inland.

The highest readings of thermometer occurred at Uckfield,
Helston, Beckington, Lewisham and Walworth respectively.

The mean 3'early range of temperature in the counties of
Cornwall and Devonshire was 55° 6, at places in the vicinity
of the sea was 52°"6, and in latitude 51° 30' was 78°-0. This
value diminished by about 5° for an increase of 1° in latitude.

The places of greatest yearly range were Uckfield, Beck-
ington, Lewisham, Walworth and Greenwich respectively.

The places of least yearly range were Truro, Liverpool,
Brighton, Torquay and Whitehaven.

The mean monthly range of temperature in latitude 51° 30'
was 40°; and this value seemed to diminish 3° for every in-
crease of 1° in latitude.

The places of greatest monthly ranges were Uckfield, Beck-
ington, Lewisham, Greenwich and Cambridge respectively.

The places of least monthly ranges were Truro, Liverpool,
Torquay and Whitehaven respectively.

The mean daily ranges were greatest at Uckfield, Cam-
bridge, Lewisham and Greenwich respectively.

The places of least daily ranges were Truro, Liverpool,
Whitehaven and Torquay respectively.

The mean daily ranges in latitude 51° 30' were 16°'l ; and
this value diminished by about 1° for an increase of 1" of lati-
tude.

With respect to the average strength of the wind, I can
speak with no confidence. 1 believe no two observers have
estimated its value upon the same scale.

The prevailing direction of the wind in the counties of Corn-
wall and Devonshire was north-east and south-west, at Liver-
pool it was north-west, and at all other places it was south-
west.

The amount of cloud may be considered to have been equal
at all places, and such as to cover about three-fifths of the sky.

The average number of days in which rain fell was 152;
this number was greatly exceeded at Truro and at White-
haven, and was much less between the latitudes of 51° and 52°.

The mean amount of rain fallen in the counties of Cornwall
and Devonshire was 35'8 inches, at Liverpool and White-
haven was 37*3 inches, and at all other places was 22*2
inches. This value was exceeded at Beckington, Hereford and
Derby. The places at which the smallest amounts of rain fell
are Walworih, Durham, Uckfield, Greenwich and Lewisham.

The mean weight of water mixed with a cubic foot of air
was 3*8 grains in the counties of Cornwall and Devonshire,
and 3*5 grains at other places.

in the year 1847. 275

The mean additional weight required to saturate a cubic
foot of air was 0*9 grain in the counties of Cornwall and
Devonshire, and 0'7 grain at other places.

The mean degree of humidity was 0'825 in the counties of
Cornwall and Devonshire, and 0*839 at other places.

The mean weight of a cubic foot of air at all places at the
level of the sea was 541 grains.

The preceding results are chiefly for one point only, viz.
that of the Royal Observatory at Greenwich reduced to the
level of the sea. To render the results most generally useful,
I have deduced the following formulae, applicable to any place
in England, taking into account the irregularities of the for-
mation of the surface of the soil, and the law of the decre-
ment of heat with increasing elevation.

The pressure of dry air at any place in England in the
year 1847 was
*°' height of place in feet above the level of the sea

^y 0*1 ~r~" . j •

82 mches

The pressure of water was Qii^'Sig + (51° 30'— latitude) x
0-010 inch.

The sum of the two preceding formulae gives the reading
of the barometer.

The temperature of the air at any place in England may be
calculated from the formula —

50°-0 + (51° 30'— latitude) x 1°— 0-00345° X height of place
in feet above the level of the sea.

The observations are not sufficient to deduce the term de-
pending upon longitude ; its coefficient however would be
very small, and in the present state of our knowledge may be
neglected.

The approximate yearly range of temperature may be cal-
culated from the formula —

78°+ (51° 30'-latitude of place) x 5°.

The mean monthly range of temperature may be calculated
from the formula

40° + (51° 30'- latitude of place) x 3°.

The mean daily range Of temperature may be computed
from the formula

16°-2 + (51° 30*- latitude of place) x 1°.

The approximate weight of a cubic foot of air may be cal-
culated from the formula

- . , . . height of place in feet above the level of the sea

541 grams + — ^ C ^ . , .

^ 82 mches

T2

276 Mr. J. Glaishel' 07i the Meteorology of England.

It is desirable to compare the results deduced by means of
the preceding formulae with the observed results, not only
for the purpose of testifying their accuracy, but also to examine
the accuracy of the several instruments which have been used
at the different stations.

The annexed table shows the results of this comparison.

In viewing the differences shown in the fourth and tenth co-
lumns between the observed and calculated pressures, it is evi-
dent that at most places the instruments are good. When we
view the diff'erences which some of the places present, we shall
readily see that the instruments at those places need correction.
The barometer at Exeter seems to read too low by 0"103 inch ;
that at Uckfield to read too high by 0*131 inch ; and that at
Derby to read too low by 0*193 inch. These corrections
should be applied to the readings of these instruments till the
true values are determined by a direct comparison with a
standard.

I proceed to consider the numbers in the thirteenth column,
showing the differences between the observed and calculated
annual temperatures of the air. These differences are generally
small, and quite within the probable errors of the instruments
themselves. This formula therefore may be considered to give
the true temperature ; and from it the mean temperature of
any place in England may be calculated for the year 1847.
The differences at Brighton and at Beckington are large, and
the annual temperatures as determined at those places are
either erroneous, or those places are subjected to a peculiar
local influence. Let us turn for a moment to the annual tem-
perature as determined from the corrected readings of the dry
bulb thermometer in the yearly table at both these places.
Theannual temperatures, as thus deduced, are nearer the com-
puted values than those determined from the maximum and
minimum thermometers ; and it would seem that the self-
registering instruments are in error, and that the differences
are not due to local disturbances. At Derby and at Notting-
ham the instruments require correction. At Liverpool the
difference shown would seem to be due to locality.

The differences in the thirteenth column, at those places
situated in the counties of Cornwall and Devonshire, are small ;
therefore the annual temperatures of these places are only those
due to their latitude. In turning our attention to the 16th,
19th, and 22nd columns, we at once perceive the cause of the
loveliness of the climate of those counties to be the uniformity
of their temperature. Their yearly, monthly, and daily ranges
are respectively 28°, 15°, and 7^ less than those due to their
latitude.

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278 Prof. J. R. Young on the decomposition of Functions

I have already alluded to the influence of the sea in equal-
ising the temperature of places in its vicinity in my remarks
on the quarter ending June 30, IS^?. It moderates the se-
verity of winter and the heat of summer, but does not seem to
exercise any influence over the mean annual temperature.
Hence we perceive the same annual temperature may be dis-
tributed in various ways in the different seasons of the year.

At Uckfield, the yearly, monthly and daily ranges are in
excess, which must be owing to local perturbations. By re-
ference to column 13, it will be seen that the annual tempe-
rature has been uninfluenced by these large ranges.

The only other places at which considerable differences are
show^n between the calculated and observed ranges are White-
haven and Liverpool ; and the smallness of the ranges at those
places are evidently to be attributed to the vicinity of the sea.

In conclusion, I have merely to remark, that I am persuaded
the spirit of the above method of reducing meteorological ob-
servations, and deducing from them general formulae, will
some day lead to empirical laws, showing the reciprocal de-
pendence of each subject of research. I would, however,
impress upon observers generally the absolute necessity of
using good instruments, and ascertaining their errors by com-
parison with standards ; such would save me a great deal of
labour and anxiety, which I have experienced in the past
year. These exertions, it is evident, could not be long con-
tinued ; and it must be borne in mind, that no system of cal-
culation can deduce good results from imperfect observations.
I must beg, however, to offer my sincere thanks to the gen-
tlemen who have furnished the observations, for their ready
acquiescence at all times to my wishes.

This is the first annual report upon the meteorology of En-
gland. May it be followed by many, more ably conducted
and more valuable to meteorology!

XXXIX. On the decompositioTi of Functions into Conjugate
Factors ,- isoith some co7iseguences deducible therefrom. By J,
R. Young, Professor of Mathematics, Belfast College"^.

I THINK that by decomposing a function of an even de-
gree into its constituent conjugate factors, some interest-
ing results in analysis may occasionally be deduced. It will
be seen, from the expressions below, that this decomposition
is very easy ; and as the component factors involve an arbitrary
function, common to each pair, we are at liberty to fix its
value so as to fulfill certain prescribed conditions that may
* Communicated by the Author.

into Conjugate Factors, 279

subserve the purpose of leading to general theorems. In the
present communication, I shall confine myself entirely to the
expression (A), which I propose to employ for the purpose
of obtaining, in a more direct and simple manner, the formula
investigated in the last Number of this Journal (p. 192), and
of giving to that formula a greater degree of generality.
Let (f{x) be any function of a;; then since

<Pi^)=f'<P{^h=fi'f'f{^)^> &c.

we obviously have these identities, in which F, /, &c, are any
functions whatever :

^(a;) = [F+v/{F2-^(a;)}]x[F-v'{F-<p(^)}] . . (A)

= [F+v'{F2-/.^(a:)}] X [F-V{V'-f.f{x)}]^

= [F+^{F2-/.fW}] X [F-^{F2-/.f(^)}]

X [f +^-[p2-i|] X [f'-v^J^F'2-^|]

and so on.

In the expression (A), let

then, as F is entirely arbitrary, it may be made to take such
a value as to render the sum of the two conjugate factors in

(A), that is to say 2F, equal to —p ; which value isF=— -p:

and as the product of the same factors is q, it follows that these
factors are the roots of the quadratic equation

x^+px + q = 0;
that is, the roots are

4^V{7-*} (')

Again : let there be the cubic equation
x^ + p.r^ -i-qx + rz^O,
and, in this case, put

<p{x)=x^+j}x + q=-^;

then, if a\ be one of the roots of the cubic, the expression
Xi^+pa?i + q

will be the product of the two remaining roots ; and therefore,
that these roots may be represented by the factors in (A), we

280 Prof. J. R. Young 07i the decomposition of Functions
shall merely have to determine F so as to fulfill the condition

and, consequently, the two remaining roots are

as in the paper referred to above

Let us now take a biquadratic equation

a/^ -\-px^ + qx^ -f r.ar + 5 = 0 ;
then if x^ and Xc^ be two of its roots, we shall have

x^ +pxc^ + ga^a + r =

X<2

.-. {x,^-:^i)+p{x,--xi)^.q{x,-x,)^ ^- " ^ = '"^^^^^
and, consequently, the expression

x^ H- x^x^ + x^ +p{xx + ^2) + 2' = = (^1 + -^2)

1<*2

(B)

+p{xx + "^2) — ^i-^a + !Z

is the product of the two remaining roots.

Let (p[x) represent this product: then, as before, in order
that the two factors in (A) may represent these roots, we
shall merely have to determine F so as to fulfill the condition

2F=-Gr,+.'2+i^).-. F=- ^i±|i±^;

so that the expressions for the two remaining roots are

__ Xy + x^+p
2

the irrational part of which may be put in the more convenient
form

into Conjugate Factors. 281

which we see reduces to the preceding expression for the roots
of a cubic when Xc^^-O.

If we were to take an equation of the fifth degree, and to
proceed as above, we should obtain three expressions of the
third degree, as the expression (B) of the second degree is
obtained : atid if from the double of one of these the sum of
the other two be taken, and an obvious division performed,

we shall get an expression for ;-, the product of the re-

12 3

maining two roots x^^ .r^: and, determining F as before, the
roots themselves are found to be

2

•T XyX(^-\-X]XQ-]rXc^Q Q ( >
of which the irrational part may be written

-f- ^1 [Xc^ + Xq) + Xc^Q ^ \ *

and thus the general form of the expressions for two roots of
any equation, when the others are found, is sufficiently indi-
cated. It is probable however that, beyond equations of the
fifth degree, these formulas would not be much more commo-
dious for actual numerical computation than those equivalent
ones in which minus the product of the two sought roots is
introduced under the radical, in the form in which it is imme-
diately obtained from dividing the final term of the equation
by the given roots with changed signs ; the formula in this
way being immediately suggested by the expression (1), from
which indeed what is here done might have been derived.
But my chief object has been to show how the conjugate factors
(A) may be turned to account in a particular inquiry ; as we
see that, from these, the form (1) has been itself obtained. The
reader will at once perceive how the term cofijugate factors
has been suggested ; and I would here venture an opinion that
the same term might with propriety be employed, instead of
congeneric, in certain equations related to one another in a
somewhat similar manner as these factors : and further, that
the expression conjugate roofs of equations seems to be unne-
cessarily restricted : all the roots of equations of an even de-

282 Prof. J. R. Young on the decomposition of Functions

gree may be expressed in pairs of the forms « + /3, a— /3 ;
aj + /3j, «i — /3i, &c. ; and these seem entitled to be called
conjugate pairs. This way of pairing the roots of equations
has already been distinctly noticed by Professor Davies in a
former Number of this Journal. See vol. xxxiii. p. 366.

In reference to the formulas established in this paper, it
may not be superfluous to observe that tliey will be found to
be more especially useful in those cases in which all the roots
but two are real ; as they will enable us to exhibit the imagi-
nary pair, by aid of the real roots, with comparatively little
expense of calculation ; and even when all the roots are real,
a saving of figures is still effected by them. But, in comparing
formulas of this kind with the numerical process of Horner,
it must always be remembered that Horner's method supplies
the roots in an explicit form ; whereas, in expressions for them
such as these, there yet remains an unperformed operation,
indicated by the radical ; which, however, in the case of ima-
ginary roots, is of course impracticable ; and therefore leaves
nothing further to be done. But, in all formulas for imagi-
nary roots, into which approximate values only of the real
roots enter, it is necessary, in delicate cases, that is in those
cases in which a very slight change in any of the coefficients
would convert unequal into equal roots — it is necessary, in
such cases, to push these approximations to a more than usual
extent, in order to avoid the conversion of imaginary roots
into real, and vice versa; for there is no hope of attaining the
imaginary forms accurately, when we employ approximations
only to the real quantities which enter into the expression of
them.

Although, as stated at the outset, it is not my intention at
present to enter into any discussion of the forms which follow
(A), yet I may perhaps be permitted briefly to notice here
one or two obvious deductions from them.

By putting s for <^{x), we at once see how easily the usual
formulas for the solution of equations of the fourth degree may
be obtained from those forms : we shall only have to multiply
together the quadratic factors

x'^-2Yx+fs

and to equate the resulting coefficients with those of the like
powers of .r in the proposed equation : and it may not be un-
deserving of notice, that when two roots are reciprocals, and
two only, then/=l, and F, F' may each be determined by a
simple equation.

into Conjugate Factors. 283

Again : if a pair of conjugate roots of a biquadratic equa-
tion be given in the form «+ \//3, then it is plain, from the
same expressions, that we must have

and consequently the values of the two remaining roots will be
2a-fj? ///2a+^\2 s \

which might indeed have been deduced from the form (1),
though I believe that form has not hitherto been employed for
this purpose. The same values, as furnished by the expres-
sions previously given, take the somewhat more convenient
forms

I shall merely add, in conclusion, that, as far as equations
are concerned, the conjugate factors of <^{x) do little more than
express the fact that the roots of every equation of an even
degree,

{x^-\-ax-\-b){x'^ + a-^x-\-b^ .... =0,

may be exhibited under the forms

F+i/{F2-6}

&c. &c. ;

a truth which, however obvious from the theory of common
quadratic equations, has never, I believe, been turned to any
account elsewhere. The conjugate factors here referred to
express obvious identities : they do not presuppose the solution
of a quadratic, but actually supply that solution, by aid of the
fundamental property that the coefficient of the second term is
the sum of the roots with changed signs, and the third term
the product.

P.S. In my last paper (p. 194, line iS)for "quotient" read
" product."

Belfast, March 8, 1849.

[ 284 ]

XL. On the Theoretical Falne of the Velocity of Smmd^ in
reply to Mr. Stokes. By the Rev. J. Cuallis, M.A., F.R.S.,
F.R.A.S., Pliimia?i Professor qfAstro?iomt/ and Experimental
Philosophy in the University of Cambridge^.

nnHE velocity of propagation of waves in an elastic medium
•*■ so constituted that the pressure varies in the same pro-
portion as the density, is usually deduced from the hydrody-
namical equations by assuming, either that the motion of the
vibrating particles is a function of the distance from a fixed
plane, or that it is a function of the distance from a fixed
centre. On the former assumption, an exact integral, appli-
cable to propagation in a single direction, may be obtained,
which conducts to the inference that a point of maximum ve-
locity of a given wave travels at a rate different from that of
a point of no velocity, so that, however large the maximum
velocity may be, one of these points may overtake the other,
without any indication on the part of the analysis of the phy-
sical impossibility of such an occurrence. The inevitable con-
clusion from this result is, that the integral admits of no inter-
pretation compatible with fluid motion, and that the assumption
of plane- waves is inadmissible.

The assumption of spherical waves is shown to be inad-
missible by conducting to an incompatibility of another kind,
as I have proved by an argument contained in the Number
of the Philosophical Magazine for last February. The argu-
ment is divided into five heads ; the four first of which include
the proof of incompatibilit}', and the fifth is merely an appeal
to an admitted principle in physics, viz. that of constancy of
mass, to which the result of the previous reasoning is opposed.
Mr. Stokes, in the March Number, after assenting to the four
first heads, meets the fifth by a simple denial for which he
gives no reason. But surely the weight of this denial falls
very harmlessly on a part of the argument which admits of
no dispute ; for I presume that Mr. Stokes does not intend to
maintain that in physics there is such a thing as generation or
annihilation of matter.

My argument put in syllogistic form is as follows : —

Let the waves be supposed to be spherical.

Then, as the analysis shows, the same portion of matter has
a different value (expressed, for instance, in cubic feet of the
matter in a given state of density) at onetime from that which
it has at another time.

But by the principle of constancy of mass the same portion
of matter has the same value at all times.

* Communicated by the Author.

On the Theoretical Value of the Velocity of Sound. 285

Therefore the waves cannot be supposed to be spherical.

That there may be no excuse for misapprehension as to the
result attributed to the analysis in the second member of the
syllogism, I proceed to exemplify that result by a numerical
instance. The pressure being a^(l +5), the value of the con-
densation s at any distance r from the centre, and at any time
ti is admitted to be given by the equation

Yir-at)

s— — -.

r

Since the function F is arbitrary, it may be supposed that

a . 27r , ^ ,

5=-sm — {r—at-\-c),

[Xf X, and c being certain constants. It is also admitted that
the function F may be taken discontinnously, that is, from one
zero value to another zero value; and that all other values of
s not included between those limiting values may be zero.
Let therefore the values of the circular function be taken from

r—ati + c=0 to r—at^-\-c= -. Then, the mean density of

the medium being unity, the quantity of condensed matter in
the space occupied by the wave above matter of mean density
occupying the same space, is the integral o^ ^irr^sdr taken be-
tween the limits just mentioned. Call this quantity «, and
for the sake of definiteness of conception, let the fluid under
consideration be contained at the time /j between two rigid
spherical surfaces, the radius of one of which is 1000 feet, and
that of the other 1,000,000 feet. There is nothing in the an-
tecedent investigation to exclude such a supposition, and for
the purpose of the argument these numbers will serve as well
as any others. Let the fluid of mean density which would
fill the space between these surfaces be A in cubic feet, which
of course is a constant quantity. Then a. being expressed in
cubic feet, the whole quantity of matter at the time ^1 is A + a.
To express a numerically let X=\ foot, and let the constant
/u.= 1, which amounts to supposing that the maximum conden-
sation at a distance of 10,000 feet is 0,0001. Consequently

« = 47r / (

Ir sin — {r—ati + c)dr

the exact value of which integral between the limits r—at^

+ c=0 and r—at^-\-c=- is 4Arj, 7\ being the distance of the

maximum condensation from the centre at the time/j. Hence
whenr, = 10,000feet, thewhole quantity of matter is A +40,000

286 M. A. de la Rive on the Diurnal Variations of

cubic feet. But since the wave is propagated from the centre
with a velocity «, the distance of the maximum condensation
from the centre may at certain time tc^ become 100,000 feet,
in which case the whole quantity of matter is A + 400,000
cubic feet. Thus in the interval from /j to tc^ the analysis has
generated 360,000 cubic feet of matter ! After obtaining such
a result from the part of the argument to which Mr. Stokes
has expressed his assent, I am at a loss to conceive for what
reason he asserts that any onus probatidi rests with me.

I have no doubt whatever that I have pointed out r^aZ con-
tradictions resulting from the suppositions of plane-waves and
spherical waves, of the utmost importance in hydrodynamics,
since they prove that the true theoretical value of the velocity
of sound cannot be deduced from those suppositions. By
another supposition which conducted to ray-vibrations, I ob-
tained in the Philosophical Magazine for February a value of
the velocity of sound very nearly agreeing with observation,
without meeting with any similar contradiction. To this sub-
ject, however, I hope to find time to recur on a future occasion.

Cambridge Observatory,
March 22, 1849.

XLI. On the Diurnal Variations of the Magnet Needle^ and
on Aurorce Boreales. By Auguste de la Rive, being an
Extract from a Letter to M. Arago*.

ALLOW me to communicate to you, with the request that
you will make it known to the Academic des Sciences,
an extract of a memoir recently read before our Societe de
Physique et d'Histoire Naturelle, on the cause of the diurnal
variations of the magnet needle and of auroras boreales. In
assigning successfully these two classes of phaenomena to the
same origin, I have but followed the path you have pointed
out; for more than thirty years ago you established with in-
defatigable perseverance, by your numerous observations, the
remarkable agreement which prevails between the appear-
ances of the aurora borealis and the disturbance of the mag-
net needle.

The following is my theory. You will observe that it rests
solely upon well-ascertained facts and on principles of physics
positively established.

I had already, in 1836, in a notice upon hailf, attempted
to show that the atmospheric electricity owes its origin to the

* From the Annates de Chimie et de Physique for March 1849.
f Bibliotheque Umverselle, vol. iii. p. 217, nouvelle serie.

the Magnetic Needle and on Aurora Boreales. 287

unequal distribution of temperature in the strata of the atmo-
sphere. It is well known that, in a body of any nature what-
soever heated at one of its extremities and cooled at the other,
the positive electricity proceeds from the hot part to the cold,
and the negative electricity in the contrary direction ; it thence
results that the lower extremity of an atmospheric column is
constantly negative and the upper one constantly positive.
This difference of opposite electric conditions must be so much
the greater the more considerable is the difference of tempe-
rature ; consequently more marked in our latitudes in summer
than in winter, more striking in general in the equatorial than
in the polar regions. It must be observed that the negative
state of the lower portions of the atmospheric columns must
be communicated to the surface of the earth on which they
repose, whilst the positive state of the upper portions is dif-
fused more or less, from above downwards, through nearly
the whole of each of the columns, according to the facilities
offered by the greater or less degree of humidity of the air to
the propagation of the electricity. An atmospheric column
therefore resembles a high-pressure battery on account of the
imperfect conductibility of the elements of which it is com-
posed,— a battery the negative pole of which is in constant and
direct communication with the terrestrial globe, discharges
itself upon the globe, whilst it becomes itself charged with the
electricity of its positive pole, which is distributed over it
with an intensity decreasing with the distance from this pole ;
— this explains why the positive electricity increases with the
height of the atmosphere.

The causes which determine the accumulation of negative
electricity at the surface of the earth and of positive electricity
in the upper regions of the atmosphere, act in a continuous
manner : there should thence result an unlimited tension of the
two opposite electric states, if, having attained a certain degree
of energy, they did not neutralize each other by the aid of
diflferent circumstances. In other words, having reached a
certain limit of tension which varies with the state of the atmo-
sphere and the surface of the earth, the two electricities cannot
go beyond it, and unite or neutralize each other as regards
the excess over that limit. This neutralization is effected in
two ways, in a normal or constant manner, and in an irregular
and accidental manner.

This second mode is exhibited under a variety of forms ;
sometimes it is simply the humidity of the air, and better still
the rain or snow, which re-establish the electrical equili-
brium between the earth and the atmosphere; in some cases
waterspouts manifest in an energetic form the mutual action

288 M. A. de la Rive on the Diwiial Variations of

of the two electricities, which tend to unite. Sometimes the
winds, by mixing the air in contact with the surface of the
earth, and like it negative, with the positive air of the more
elevated regions, give rise to sheet-lightning, or to storms,
when there is at the same time a formation of clouds and con-
densation of aqueous vapours, owing to the humidity and dif-
ferent temperature of the strata of air which become mixed.
The attraction of clouds by mountains, the luminous phaeno-
niena exhibited at the extremity of elevated points, are like-
wise due to the same cause. But I will not stop to discuss
further all these natural and intelligible consequences of the
theory which I expound. I shall confine myself to one single
remark, which is, that we must bear in mind that in observa-
tions of atmospheric electricity the intensity of the electric
signs perceived is not always a proof of the intensity of the
electricity itself; for the humidity of the atmosphere, by
favouring the propagation of the electricity of the upper
strata, may give rise, as is frequently seen in winter, to
very powerful electrical manifestations even when the cause
producing them is not very powerful. The contrary is fre-
quently seen in summer.

I now pass to the regular and normal mode of neutralization
of the two electricities. I had already suspected the existence
of this mode in my notice of 1836 ; but 1 did not announce it
positively, because there was then wanting a fact which science
now possesses, viz. the perfect conductibility of the terrestrial
globe with which the employment of the electric telegraph has
made us acquainted.

To make it understood how I conceive this mode of neu-
tralization, I divide the atmosphere into annular strata parallel
with the equator; the positive electricity accumulated at the
external portion of this layer cannot exceed a certain degree
of tension without traversing rarefied and more or less humid
air until it reaches the polar regions, where, finding an atmo-
sphere saturated with humidity, it will combine readily with
the negative electricity accumulated on the earth. We have
thus the circuit formed ; each annular stratum of the atmo-
sphere gives rise to a current which proceeds in the elevated
regions from the upper portion of the stratum towards the
pole, redescends to the earth through the atmosphere sur-
rounding the poles, and returns by the surface of the globe
from the pole to the lower part of the stratum from which it
started. These currents will consequently be the more nu-
merous and the more concentrated the nearer we approach
the pole; and as they all proceed in the same direction, that
is to say from south to north in the upper portion of the at-

the Magnetic Needle and on Aurora Boreales. 289

mosphere and from nortli to south on the surface of the earth,
their effect will become the more perceptible in proportion as
we leave the equator and approach the pole. But as the cur-
rents produced by the equatorial strata are individually stronger
than those proceeding from more northerly strata, the dif-
ference, although real, will notwithstanding be less than
would be believed. What passes in our northern hemisphere
must occur in exactly the same manner in the southern hemi-
sphere; the currents proceed equally from the equator to the
pole in the upper regions of the air, and from the pole to the
equator on the surface of the earth ; consequently, for an ob-
server travelling from the north pole to the south, the current
would proceed in the same direction from the northern pole
to the equator, and in a contrary direction from the equator
to the southern pole : I speak here of the current circulating
on the surface of the earth. I ought moreover to observe,
that the limit which separates the regions occupied by each of
these two great currents is not the equator properly so called,
for it must be variable; it is, according to my theory, the pa-
rallel between the tropics which has the sun at its zenith ; it
changes consequently each day.

Now it is easy to conceive the cause of the diurnal varia-
tions of the magnetic needle. In conformity with the laws esta-
blished by Ampere, the current which proceeds from the
northern pole to the equator ought to cause the north pole of
the needle to deviate to the west, which is what takes place in
our hemisphere ; and the current which proceeds from the
southern pole to the equator should cause the north pole of
the needle to deviate to the east, which is precisely what oc-
curs in the southern hemisphere. The deviation should be,
in one and the same place, the more considerable the greater
the difference of temperature, and consequently of the electric
conditions between the lower and the upper stratum of the
atmosphere ; thus the deviation increases from the morning to
jh 3om p jyj^ j[ jg more considerable in those months during
which the sun is longer above the horizon ; it is at its minimum
in the winter months. Lastly, these diurnal variations increase
in magnitude in proportion as we recede from the equator and
approach the pole, a result which again perfectly agrees with
what I have stated respecting the increase in number of the cur-
rents towards the polar regions. In these regions themselves
the variations may be very irregular, and may be entirely
absent if the magnetic needle happens to be [)lacedin those very
localities where the electric currents traverse the atmosphere
to reach the earth; in fact, a needle surrounded thus on all
sides by currents is no longer affected by them, or at least is

Phil. Mag. S. 3. .Vol. 34-. No. 229. April 1849. U

290 M. A. de la Rive on the Diurnal Variations of

no longer affected in a regular manner. This remark may
explain certain observations, especially those made at Port
Bowen, which appeared rather exceptional.

On examining carefully all the magnetic observations I was
able to consult, and in particular those of Colonel Sabine, I
was especially struck by the remarkable manner in which they
agreed with my theory. I will cite but one example — the
observations recently made at St. Helena, and just published,
by Colonel Sabine. At St. Helena the diurnal variation oc-
curs to the west as long as the sun is to the south of the island,
and to the east as soon as the sun is to the north. lu fact, in
the first case, as 1 have previously observed, St. Helena must
form part of the region in which the electric currents proceed
on the surface of the earth from the north pole to the equato-
rial regions ; and, in the second case, it forms part of the re-
gion in which these currents pass from the south pole to the
equator. The hour of the maximum of the diurnal variation
is not the same at the island of St. Flelena as in the continental
countries, which is owing to the temperature of the surface of
the ocean not following the same laws in its diurnal variations
as the temperature of the surface of the earth. Now the tem-
perature of the lower stratum of the atmospheric column is
always that of the surface of the ocean, or of the soil on which
it rests. This same circumstance explains certain apparent
anomalies exhibited by tlie diurnal variations in some parts of
the globe, as for instance at the Cape of Good Hope, which
is surrounded almost on every side by a vast extent of
ocean.

I wish it to be understood that in the preceding I have only
taken notice of the causes disturbing the direction of the mag-
netic needle, and not of the cause of this direction itself, that is
to say of terrestrial magnetism — a cause which I do not at all
believe to be of the same nature, but upon which I at present
express no opinion. I am content to consider the terrestrial
globe as a large spherical magnet, and to study the external
causes capable of modifying the direction which it tends to
impart in its quality of magnet to magnetic needles.

Now what is the aurora borealis according to the theory
which I have just expounded? It is the luminous effect of
electric currents travelling in the high regions of the atmo-
sphere towards the north pole — an effect due to the combina-
tion of certain conditions, which are not always exhibited in
the same manner, nor at all seasons of the year.

It is now well proved that the aurora borealis is an atmo-
spheric phaenomenon, as you long ago suspected. The name
of magnetic stormy by which Von Humboldt designates it in

the Magnetic Needle and on Aurorce BorealeS. 291

his Cosmos, implies the same idea, which is moreover con-
firmed by the interesting details which he gives of this meteor.
The observations of Parry, Franklin, and especially those of
MM. Bravais and Lottin, so numerous and carefully made,
are likewise quite favourable to this opinion, which followed
equally from the observations of M. Biot at the Shetland
Isles.

Admitting this point, I explain the production of the aurora
borealis in the following manner: — When the sun, having
passed into the southern hemisphere, no longer heats so much
our hemisphere, the aqueous vapours which have accumulated
during the summer in this part of the atmosphere begin to
condense, the kind of humid cap enveloping the polar regions
extends more and more, and facilitates the passage of the
electricity accumulated in the upper portions of the air. But
in these elevated regions, and especially at this period of the
year, the aqueous vapours must most frequently pass into the
state of minute particles of ice or snow floating in the air,
similar to those which give rise to the halos ; they form, as it
were, a kind of semitransparent mist. Now these half-frozen
fogs conduct the electricity to the surface of the earth near
the pole, and are at the same time illumined by these currents
or electric discharges. In fact, all observers agree in asserting
that the aurora borealis is constantly preceded by a mist
which rises from the pole, and the margins of which, less
dense than the remainder, are coloured the first; and indeed
it is very frequent near the pole in the, winter months, and
especially in those where there is abundance of vapour in the
air. For it to be visible at great distances from the pole, it is
necessary that these clouds, composed of frozen particles, ex-
tend in an almost uninterrupted manner from the polar regions
to somewhat southern latitudes, which must be of rare occur-
rence. These same clouds, when they are partial, which is
frequently the case, produce the halos.

Now the analogy pointed out by nearly all observers be-
tween the mists which accompany, the aurora borealis and
those wiiich produce the halos, is a somewhat remarkable cir-
cumstance. It is easy to verify by direct experiment the iden-
tity which exists between the light of the aurora borealis and
that obtained by passing a series of electric discharges into
rarefied air containing a large quantiiy of aqueous vapour, and
especially through a very thin layer of snow or a slight layer
of hoar frost deposited on the glass. I have ascertained that
highly rarefied but perfecdy dry air gives but a very faint light,
and that in the experiment of the vacuum-tube it is essentially
the moisture adhering to the inner sides of the tube which,

U2

292 M. A. tie la Rive on the Diurnal Variations of

by conducting the electric discharges, gives rise to the lumi-
nous effects. It will be conceived that the electric discharges
transmitted by this kind of network of ice must, on becoming
concentrated near the pole, produce there a far more brilliant
light than they develope when they are distributed over a
much greater extent.

But why does the magnetic pole, and not the terrestrial
pole, appear to be the cause of the phaenomenon ? Here is
my answer. Place the pole of a powerful electro-magnet be-
• neath a large surface of mercury ; let this surface communi-
cate with the negative pole of a powerful battery; bring near
to it the point of a piece of charcoal communicating with the
positive pole of the battery; immediately the voltaic arc is
formed, and the mercury is seen to become agitated above the
electro-magnet; and wherever this is placed, luminous cur-
rents are observed to rotate around this pole, and throw out
from time to time some very brilliant rays. There is always,
as in the case of the aurora borealis, a dark portion in the
form of a circular point over the pole of the magnet ; this pecu-
liar effect disappears without the voltaic light being inter-
rupted when the electro-magnet ceases to be magnetized.
With a continuous current of ordinary electricity arriving at
the pole of a powerful electro-magnet in rarefied and moist
air, luminous effects, still more similar in appearance to those
of the aurora borealis, are obtained.

These phaenomena result from the action of magnets on
currents; now the same should apply to the action of the
magnetic pole of the earth ; the neutralization of the two elec-
tricities probably takes place over a somewhat large extent of
the polar regions ; but the action of the magnetic pole causes
the conducting mists to rotate around it, sending forth those
brilliant rays which by an effect of perspective appear to us
to form the corona of the aurora. The sulphurous odour,
and the noise which is said sometimes to accompany the ap-
pearance of the aurora, would not be inexplicable ; for the
odour would be due, like that which accompanies lightning,
to that modification which the passage of electric discharges
produces upon the oxygen of the air which M. Schonbein has
called ozone; while, as regards the noise, it would be analogous
to that which, as I have shown, the voltaic arc produces when
it is under the influence of a very near magnet. If it seldom
occurs in the case of the aurora, it is owing to its being very
rare that the luminous arch is sufficiently near the earth, and
consequently' to the pole. However, the description which
has been given of this noise by those who have heard it, is
perfectly identical with that which I have given, without sus-

the Magnetic 'Needle and on Aurora Boreales. 29S

pecting the analogy, of the noise which the voltaic arc pro-
duces in the action of the magnetism.

The magnetic disturbances which always accompany the
appearance of an aurora borealis are now easily explained.
This accidental union of a greater proportion of the accumu-
lated electricities must derange the normal action of the regu-
lar current; with respect to the directions of the disturbance,
it will depend on the portion of the current acting upon the
needle, and consequently on circumstances impossible to fore-
see, since they depend on the extent of the phaenomenon and
the position of the needle in relation to it. In fact, according
as the horizontal plane in which the declination needle moves
comprises above or below some of the region in which the
greatest activity of the phaenomenon takes place, it will be
either the current circulating on the earth or that travelling
in the air (currents which proceed in a contrary direction)
which will act upon the needle ; even during the same aurora,
it may be sometimes one sometimes the other of these two
currents which will act. The variable directions in which the
needle is deflected during an aurora borealis agree very well
with this explanation, at least as far as I have been able to
judge from the different observations published in the Annates
de Chimie et de Physique and in several scientific voyages.
The remarkable effect observed by M. Matteucci in the appa-
ratus of the electric telegraph between Ravenna and Pisa,
during the magnificent aurora of the 17th of last November,
fully proves the existence of a current circulating on the sur-
face of the earth, and which, ascending the wire of the tele-
graph, passed in part through this better conductor. The
sounds which long iron wires strung in the direction of north
to south give out under certain meteorological circumstances,
are undoubtedly a proof that they are traversed by a current
which is probably derived from the currents circulating on
the surface of the earth from north to south in our hemisphere.

It would be highly interesting and important to profit by
those telegraphic wires, which are found to have a direction
more or less approaching to that of the declination needle, in
order to make with them, when they are not in use for ordinary
purposes, some observations which would enable us to demon-
strate and to measure the electric currents which probably
traverse them ; it would be easily accomplished by means of
a multiplying galvanometer, by completing the communication
of these wires with the earth at one of their extremities. The
comparison of the results obtained in this manner with those
furnished by the simultaneous observation of the diurnal va-
riations of the needle, would certainly present considerable

294- Sir W. Rowan Hamilton on Quaternions.

interest, and might lead to meteorological results of a remark-
able nature.

I cannot conclude this abstract without drawing attention
to the circumstance, that M. Arago had already pointed out
in 1820, short!}' after CErsted's discovery, the possibility of
acting upon the voltaic arc by this magnet, and the analogy
which might result between this phasnomenon and that of the
aurora borealis.

XLII. On Qicaternions ; or on a New System of Imaginaries
in Algebra. By Sir William Rowan Hamilton, LL.D.,
M.R.I. A. .f F.Ii.A.S., Correspotiding Member of the Insti-
tute of France, Sec., Andrews' Professor of Astronomy in the
University of Dublin, and Royal Astronomer of Ireland.

[Continued from vol. xxxiii. p. 60.]
65. TF we make

p-X=:A^; p— jtA=^^; p—yj=x',', p—fx,'=zij,l; . (115.)

and in like manner, (see (106.),)

p-0=-6S = ^,; (116.)

and if we regard these five newvectors, Apjw,pX/,|«,/,and^pas lines
which, being drawn from the centre A, terminate respectively
in five new points, l^, Mp l/, m/, and h; while the vector p,
drawn from the same centre a, still terminates in the point E,
upon the surface of the ellipsoid; then the equations (113.),
(114.), of art. 62, will give:

TK,=Tix.,= Thl = Ti^'=b; . . . (117.)

while the equations (101.) will enable us to write

■li^itLili^!hz:h=Y-'o; . . (118.)

and in like manner, (see (112.),)

-7- --7- - /_^^ -'^ ^^' • U^9-^

this symbol V~^0 denoting (as already explained) a scalar.
We shall have also, by (84.), (89.),

ri^ = A-=\-'0; tZJ!:i=:^^=V-'0; (120.)

X — »

the scalars denoted by the symbol V ^0 being not generally
obliged to be equal to each other, and being, in these last

Sir W. Rowan Hamilton on Quaternions, 295

equations (120.), respectively equal, by (86.), (91.), to those
which have been denoted above hy h and h'. In like manner,
by (110.),

^^'=,-^=V-0; ez:^ = V^,=V-'0. (121.)

And because, by (107.), / has a scalar ratio to x, and x' has
a scalar ratio to i, we may infer, from (118.), (119.), the ex-
istence of the two following other scalar ratios :

ttlh^Y-'O; htl^' =Y-'0. . . (122.)

Finally we may observe that, by (120.), (121.), there exist
scalar ratios between certain others also of the foregoing vec-
tor-differences, and especially the following :

^-^::^=v-^o; ^nV-v-'o. . . . (123.)
p-h p-h

66. Proceeding now to consider the geometrical significa-
tion of the equations in the last article, we see first, from the
equations (117.)} that the four new points, l^ m^, l/, m/, are
all situated upon the surface of that mean sphere, which is
described on the mean axis of the ellipsoid as a diameter;
because the equation of that mean sphere has been already
seen to be

p2 + /;2=o* equation (100.), article 58;

which may also be thus written, by the principles and nota-
tions of the calculus of quaternions :

Tp = b • (124.)

From the relations (122.) it follows that the two chords l^m/
and l/Mp of this mean sphere, both pass through the point h,
of which the vector ^^ is assigned by the foimula (116.) ; for

* Thh form of the equation of the sphere was published in the Philo-
sophical Magazine for July 1846 ; and it is an immediate and a very easy
consequence of that fundamental formula of the whole theory of Quater-
nions, namely

which was communicated under a slightly more developed form, to the
Royal Irish Academy, on the 13th of November 1843. (See Phil. Mag. for
July 1844.)

It may perhaps be thought not unworthy of curious notice hereafter, that
after the publication of this form of the equation of the sphere, there should
have been found in England, and in 1846, a person with any mathematical-
character to lose, who could profess publicly his inability to distinguish the
method of quaternions horn that of couples ', and who could thus confound
the system of the present writer with tliose of Argand and of Fran9ais, of
Mourey and of Warren.

296- Sir W. Rowan Hamilton <?« Qiialerm'ofis.

the first equation (122.) shows that the three vectors X^, ju,/, ^^,
which are all drawn from one common point, namely the
centre a of the ellipsoid, all terminate on one straight line ;
since otherwise the quotient of their differences, fJ-, —^i and
^i~^p would be a quaternion*, of which the vector part would
not be equal to zero : and in like manner, the second equation
(122.) expresses that the three lines A/, |«,y, ^^ all terminate
on another straight line. The four-sided figure L,M^L/My' is
therefore &. jdane quadrilateral, inscribed [genexoWy) in a small
circle of the mean sphere, and having the point h for the in-
tersection of its second and fourth sides, m^l/ and m/l^, or of
those two sides prolonged. And these two sides, having re-
spectively the directions of hm^ and hl^ or of the vector-
differences yt6y — ^y and \— ^y, are respectively parallel, by
(118.), to the two fixed vectors, » and x; or (by what was
shown in former articles), to the two cyclic normals, ac' and
AC, of the original ellipsoid. The plane of the quadrilateral
inscribed in the mean sphere is therefore constantly parallel
to the principal plane c\d of that ellipsoid, namely to the
plane of the greatest and least axes, which contains those two
cyclic normals. The first and third sides, l^m^ and l/m/, of
the same inscribed quadrilateral, being in the directions of
ja.^— Xy and jw,/— A/, are parallel, by (118.), (119.)> ^o two
other constant vectors, namely < — x and <' — x', or to the axes
AB, ab', of the two cylinders of revolution which can be cir-
cumscribed about the same ellipsoid. And the point of inter-
section of this other pair of opposite sides of the same inscribed
quadrilateral is, by (123.), the extremity of the vector p, or
the point e on the surface of the original ellipsoid; while the
point H, which has been already seen to be the intersection of
the former pair of opposite sides of the quadrilateral, since it
has, by (1 16.), its vector ^^=— i\ is the reciprocal point, on
the surface of that other and reciprocal ellipsoid, which was
considered in article 61 ; namely the point which is, on that
reciprocal ellipsoid, diametrically opposite to the point which
was named f in that article, and had its vector ■=hS.

67. Conversely it is easy to see, that the foregoiiig analysis
by quaternions conducts to the following mode o^constructing-\,
or generating, geometrically, and by a graphic rather than by

* A Quaternion, geometrically considered, is \}ci& 'product, or the quotient,
of any two directed lines in space.

-f This construction, of two reciprocal ellipsoids from one sphere, was
'communicated to the Royal Irish Academy in June 1848; together with an
extension of it to a mode of generating two reciprocal cones of the second
degree from one rectangular cone of revolution ; and also to a construction
of two reciprocal hyperboloids, whether of one sheet, or of two sheets,
from one equilateral liyperboloid oi revolution, of one or of two sheets.

Notices respeclmg New Books, 297

a metric process, a system of txm reciprocal ellipsoids, derived
from one Jixed sphere, and of determining, also graphically,
for each point on either ellipsoid, the reciprocal point on the
other.

Inscribe in the fixed sphere a plane quadrilateral (LyMyL/M/),
of which the four sides (l^M;, m^l/, l/m/, m/l^) shall be re-
spectively parallel to four fixed right lines (ab, ac', ab', ac),
diverging from the centre (a) of the sphere; and prolong
(if necessary) the first and thirtl sides of this inscribed qua-
drilateral, till they meet in a point e ; and the second and
fourth sides of the same quadrilateral, till they intersect in
another point h. Then these two points, of iniersection k and
H, thus found from two 2>airs of opposite sides of this inscribed
quadrilateral, will be two reciprocal points on two reciprocal
elli-psoids', which ellipsoids will have a common mean axis,
namely that diameter of the fixed sphere which is perpendi-
cular to the plane of the four fixed lines : and those lines, ab,
ac', ab', AC, will be related to the two ellipsoids which are
thus the loci of the two points e and h, according to the laws
enunciated in article 61, in connexion with a different con-
struction of a system of two reciprocal ellipsoids (derived there
from one common moving sphere) ; which former construction
also was obtained by the aid of the calculus of quaternions.
Thus the lines ac, ac' will be the two cyclic normals of the
ellipsoid which is the locus of e, but will be the axes of cir-
cumscribed cylinders of revolution, for that reciprocal ellipsoid
which is the locus of h; and conversely, the lines ab, ab' will
be the axes of the two cylinders of revolution circumscribed
about the ellipsoid (e), but will be the cyclic normals, or the
perpendiculars to the cyclic planes, for the reciprocal ellip-
soid (h).

[To be continued.]

XLIII. Notices respecting New Books.

Letters addressed to H.R.H. the Grand Duke of Saxe Coburg and
Gotha, on the Theory of Probabilities, as applied to the Moral and
Political Sciences. By M. A. Quetelet, Astronomer Royal of
Belgium, Corresponding Member of the Institute of France, 8(C. SfC.
Translated from the French by Olinthus Gregory Downes, of the
Economic Life Assurance Society.

OF this work, which was begun by M. Quetelet in 1837, and
published at Brussels, we believe, early in 1845, the author
thus describes the object in his preface.

" Certain circumstances, which have left me many pleasant remi-
niscences, made it necessary for me nearly ten years since to devote

298 Notices respecting New Books,

my whole attention to the application of the Theory of Probabilities
to the study of the moral and political sciences. I then felt how
desirable it was that this science should be rendered more elementary,
and that it should be brought down from the higher regions of ana-
lysis, and placed within the reach of those who have most frequently
to make use of it. It links itself to numerous questions which inter-
est the legislator and the statesman, — both are often obliged to infer
from the statistics of the past what' it may be useful to do for the
future, and they feel the want of means to enable them to judge of
the results produced by modifications of the laws which connect
events with each other, and to assign the weight to be assigned to
symptoms which announce the adversity or prosperity of a country."

The subject is discussed under several general heads — the Theory
of Probabilities — Means and Limits — the Study of Causes — and
Statistics — eacli branching out into subordinate departments. The
whole discussion, though perfectly elementary and practical, is a
masterly performance. The illustrations are selected with singular
judgement ; and, for such a subject, the work is a book of very plea-
sant reading.

"We have no doubt that the work will be eminently useful in this
country, and that it will make the name of its author known to many
who have never before heard of the astronomer of Brussels.

The translation is executed with faithfulness, and is creditable to
the taste and judgement of the translator ; and we shall rejoice if
this is only a foretaste of what we may expect from the same hand.

First Steps to Zoology. By Robert Patterson. Simms and
M'^Intyre. London, 1849.

In this little volume, Mr. Patterson has presented to the young
naturalists of this country an abridgement of his recently published
Zoology for Schools His object in so doing (as appears from the
preface) is to convey some knowledge of the natural history and
classification of the various animals which inhabit our globe, to a
younger class of readers than would easily understand his more ex-
tended work above-mentioned. In most instances, accordingly, he
has confined himself to giving short notices of the different orders of
animals, selecting as individual examples of each, when such could
be done conveniently, those which inhabit our own islands and the
seas surrounding them. On the whole, the entire range of animated
nature is very fairly represented ; the vertebrated animals, the birds,
beasts and fishes of the old natural history books, preponderating,
as indeed, from the fact of their exhibiting the largest amount of in-
telligence and the greatest number of individual traits of character,
must almost necessarily be the case in a popular book.

A large number of woodcuts are inserted, illustrating the different
subjects treated of; but the impressions are by no means so good as
one could wish ; and we think that, if Mr. Patterson would have a
little more care bestowed upon the getting up of these illustrations,
in case of the aj^pearance of a second edition, his work would be
greatly improved. On the whole, however, we can safely recommend
it to the notice of our readers.

[- 299 ]

1 t!

XUV. Proceedings of Learned Societies.

ROYAL SOCIETY.

[Continued from p. 78.]

Dec. 14-, "/^"|N the effect of surrounding Media on Voltaic Tgni-
1848. V/ tion." By W. R. Grove, Esq., M.A., F.R.S.

The author refers to some experiments of his published in the
Philosophical Magazine for December 184-5, and in the Bakerian
Lecture for 1847, relating to the difference of ignition generated in
a platinum wire heated by the voltaic current, when the wire is im-
mersed in atmospheres of different gases. In the present paper
these experiments are continued, the current being passed through
two platinum wires both in the same voltaic circuit, but immersed
in atmospheres of different gases.

It appears from these experiments that the heat generated in the
wire is less in hydrogen and its compounds than in other gases ;
and that when the wires and their atmospheres of gas are immersed
in given quantities of water, the water surrounding the hydrogenous
gases is less heated than that surrounding those which contain no
hydrogen.

Similar experiments, in which the wires ai'e immersed in different
liquids, are then given ; the heat developed appears not to depend
on the specific heat of either the gases or the liquids.

The two series of experiments are brought into relation by one wire
being immersed in hydrogen and the other in water, by which it
appears that the cooling effect of the hydrogen nearly equals that of
water.

Further experiments are then given, in order to ascertain, if possi-
ble, to what chemical or physical peculiarity these cooling effects are
due ; and from them it appears that they are not due to the specific
gravity, specific heat, or to any conducting power of the gases for
electricity ; and that they do not follow the same law as that by which
gases escape from minute apertures. They apparently depend upon
some molecular character of the gases, by which either the inter-
change of hot and cold particles is facilitated, or a superficial action
takes place, the surface of the hydrogenous gases presenting a more
ready escape to the heat, similarly to that which has been long ob-
served with regard to the different molecular constitutions of solid
bodies, such for instance as the more rapid radiation or absorption
of heat by black than by white surfaces, in the present case the
epipolie action being dependent on the surface of the aeriform me-
dium, and not on that of the solid substances.

Jan. 11, 184'9. — " Contributions to the Physiology of the Alimen-
tary Canal." By W. Brinton, Esq., M.B. Communicated by R.
Bentley Todd, M.D., F.R.S.

The paper consists of two parts, having a real relation to each
other, though apparently little connected.

I. Qn the Movements of the Stomach. — The anatomy of its nius-

300 Royal Society.

cular coat is first briefly mentioned, and the so-called oblique fibres
of some authors stated to be really transverse, i. e. at right angles to
the altered direction of the canal.

The muscular actions of the digesting stomach are then con-
sidered.

These Haller regarded as alternate contractions in two directions,
now forwards, now backwards, forcing the contained food in corre-
spondingly reversed directions, and rested this conclusion on expe-
riment and argument ; but the author believes tlie experiment to be
solitary, and not parallel with the fact sought to be established, and
the argument to be inconclusive.

Beaumont's views are cited as analogous to Haller's, but are con-
sidered as having been by no means clearly stated.

The author indicates an argument from analogy, but chiefly bases
his conclusion on the observations of Owen and others on Fishes, and
his own observations in animals immediately after death : — in the
empty or non-digesting stomach ; and in the stomach which contains
food ; first, in the early stage of digestion ; and, secondly, at a later
period.

From a contrast of these three states it is found, that in the first
there is no movement ; in the second and third a considerable one ;
that in the latter, the opening of the pylorus, and the preponderance
of the contractions of the pyloric half of the viscus, constitute its
chief distinction from the second. The two latter movements are
both peristaltic, or in one direction only — being never reversed, so far
as the author has seen.

The movement impressed on the food is next considered. Ac-
cording to the observations of Beaumont and others, the food passes
in two directions or streams, forwards and backwards. These ob-
servations the author has been unable to repeat, but regards them
as established.

Assuming the truth of these observations, and contrasting them
with the muscular actions previously stated, it appears that the latter
are uniformly in one direction, the former in two, — an apparent in-
congruity, which the author next seeks to explain.

By experiment he attempts to imitate the natural conditions, and
with the production of the like result. He next ofl^ers an exjjlana-
tion and illustration of the fact (which might almost be predicated,
d, priori^ and adduces some (possible) analogues from the animal
kingdom.

He then seeks to establish a general law — that transverse con-
tractions, occurring in a closed tube filled with fluid, and proceed-
ing in one direction only, imply two currents ; a peripheral of ad-
vance, taking the same course as the peripheral contractions ; and an
axial of return, in the opposite direction.

He next points out the modification of this law for stomachs of
human shai)e, and shows how compatible this is with the careful
observations of Beaumont, none of which are essentially opposed
to it.

The author indicates a probable modification correlative with the

Royal Society. 301

consistence of the food in some animals, and thus shows a dependence
of this physical process on a previous one.

A solitary experiment is adduced to show that, as in the healthy
movement, so also in vomiting, no backward or antiperistaltic con-
traction necessarily occurs.

A conjecture concerning regurgitation of fluid from the stomach
concludes this part of the paper.

II. On the Physiology of Intestinal Obstructions. — In the preceding
part of the paper it has been stated, that two currents probably ob-
tain in the liquid contents of the stomach. Many of the conditions
of the intestinal tube approximate to those of the stomach ; and if
disease or experiment add to these occlusion and distension, the ana-
logy of the two organs is rendered tolerably complete, and the results
will hence probably be referrible to the same general principle.

The most remarkable and constant symptom of this state of ob-
struction is the occurrence of faecal vomiting.

The author briefly states the theory of an antiperistalsis by which
this phenomenon is ordinarily explained : and from an inquiiy into its
experimental basis he deduces this general result, that an antiperi-
staltic movement has never yet been observed in any part of the ali-
mentary canal. He regards the irregular actions seen on laying
open the bellies of healthy animals recently killed, as not definedly
peristaltic or the reverse, but as dependent on the irritation pro-
duced by the admitted air. So also, in the case of the occluded in-
testine, an inverted movement likewise fails to be recognized. In
general, the vermicular actions are more energetic, and more peri-
staltic, than in the healthy bowel.

He next adduces the following arguments :—

1 . The antiperistalsis is usually attributed to irritation ; but
irritation is present in almost every disease of the tube, while faecal
vomiting is limited to cases of obstruction. This renders it pro-
bable that the latter is the cause, and that the process of causation
is, like the cause, physical.

2. The starting-point of the supposed inverted movement is the
fullest part of the bowel, while the place towards which it has set is
the emptiest. This condition is inconsistent with the supposition
of an antiperistalsis, yet perfectly consistent with a forward move-
ment, and analogous to the obstructions of other tubes conveying
fluids.

3. Intus-susception is often the cause of obstruction. But, both
from experiment and argument, it appears probable, that an anti-
peristalsis would at once remove this condition, and would there-
fore be incompatible with it.

4. The supposed inverted movement is continuous, while the
vomiting is occasional. Hence a theory which showed the essential
independency of the return of faecal matters to the stomach, and
their ejection thence, would be, so far, preferable.

5. Experiment and observation agree in showing that the ordi-
nary peristalsis obtains immediately below the strangulation. And
it is difficult to imagine how or why the same irritation should pro-
duce two opposite movements in reversed directions.

90S Royal Society.

6. The general and comparative date of accession of the vomit-
ing is scarcely compatible with the antiperistaltic theory.

The author next adduces experiments in which the intestine of
animals was artificially occluded by a ligature. In exceptional cases,
the ligature sloughed into the canal, and the obstruction was thus
destroyed. In all others, the tube was distended above the stricture
to a variable extent. Below the stricture, the intestine was usually
empty and contracted for an inch or two. The contents of the tube
varied both in quality and quantity ; uniform fluidity being asso-
ciated with a lar^e quantity of contents, while their smaller amount
was often attended with differences of consistence. The date at
which the vomiting acceded varied considerably. In one or two in-
stances this symptom did not occur at all. These difi^erences ap-
peared mainly dependent on —

1 . The amount of fluid ingesta,

2. The distance of the stricture from the stomach.

The date of death seemed to vary chiefly with the degree of dis-
tension.

He therefore deduces the theory, — That, in an obstructed intes-
tine, a movement of the ordinary (and probably peristaltic) character
propels the contents onwards to the seat of occlusion ; that a con-
tinuance of the process distends, first this part of the tube, and next,
those portions above it ; that, if the contents are fluid, the ordinary
peristalsis tends to develope an axial and reversed current, which re-
turns matter from a lower to a higher point of the intestine ; — often
from the obstruction to the stomach, whence they are ejected by
vomiting.

That in some cases, however, the action is probably much less
perfect than this ; the consistence of the contents preventing the
perfection of these currents throughout the whole course of the tube.
But still a mixture results, although a less intimate one.

The author next glances at the mode in which obstruction ap-
pears to aff'ect peristalsis, and the nature of the distending fluid.
He compares the obstructed intestine to the healthy stomach, to the
obstructed artery and duct ; referring its peculiar appearances to
the dilatable yet muscular structure of its coats.

In conclusion, he indicates the possible result of this theory on
practical medicine.

"On the Determination of the Diflference of Longitude, by means
of the Magnetic Telegraph." By Elias Looniis, Esq., in a Letter
to Lieut.-Col. Sabine, R.A., For. Sec. R.S. Communicated by
Lieut.-Col. Sabine, R.A., For. Sec. R.S.

Tlie writer first refers to a series of experiments made under the
direction of Professor Bache, for the determination of the difference
of longitude between New York, Philadelphia and Washington, by
means of the magnetic telegraph. By this series of experiments he
considers it established that, by means of Morse's telegraph, two
clocks distant from each other 200 miles, can be compared together
with the same precision as if they were placed side by side ; and
that the difference of longitude of two places can be determined
with the same precision as the relative error of the clocks. These

Royal Society, 303

results were so satisfactory that Professor Bache determined to pro-
secute them more extensively, and during the past summer conipa-
risons have been made between New York and Cambridge observa-
tory near Boston. The plan of operation this season was more
matured than during the former. The comparisons were all made
between a solar chronometer at Cambridge and a sidereal clock at
New York. At ten o'clock in the evening, the two observatories
having been put in telegraphic communication, when the seconds
hand of the solar chronometer came round to 60', a signal was given
at Cambridge, by pressing the key of the telegraph-register ; at the
same instant a click was heard at New York, and the time was re-
corded according to the sidereal clock. At the end of 10'' a second
signal was given, which was also recorded at New York ; at the end
of another 10^ a third signal was given, and so on for sixty seconds.
The Cambridge astronomer then commenced beating seconds by
striking the key of the telegraph-register in coincidence with the
beats of his chronometer. The New York astronomer compared the
signals received with the beats of his clock, and waited for a coin-
cidence. When the beats were sensibly synchronous the time was
recorded, and the astronomer waited six minutes for another coin-
cidence of beats. The Cambridge astronomer continued beating
seconds for fifteen minutes, during which time the New York ob-
server was sure of two coincidences, and might obtain three. When
these were concluded, the New York astronomer in the same man-
ner gave signals for one minute at intervals of 10% and then beat
seconds for fifteen minutes,during which time the Cambridge astro-
nomer obtained four or five coincidences upon his chronometer.
This mode of comparison was practised every night, and it is con-
sidered that the uncertainty in the comparison of the time-pieces
cannot exceed two or three hundredths of a second on any night;
and in a series of comparisons the error may be regarded as entirely
eliminated.

Another mode of comparison which was practised is that of tele-
graphing star transits. A list of stars which culminate near our
zenith at intervals of five or six minutes was prepared, and the ob-
servers, both at New York and Cambridge, were furnished with a
copy. They then proceeded as follows : Cambridge selected two stars
from the list, which we will call A and B, and struck the key of his
register at the instant when the star A passed each of the seven wires
of his transit. These signals were heard at New York, and the times
recorded. Cambridge then observed the transit of star B in the
ordinary manner without telegraphing. New York then observed
the transit of star A on his meridian in the usual manner ; and struck
his key at the instant the star B passed each of the seven wires of
his transit, which signals were heard and recorded at Cambridge.
The difference of longitude between New York and Cambridge is
nearly twelve minutes, affording ample time for all these observations.
Thus New York obtained upon his own clock the times of transit of
star A over the meridians of Cambridge and New York ; and Cam-
bridge obtained upon his chronometer the times of transit of star B

S0§ Royal Society.

over the same meridians. The difference of these times gives the dif-
ference of longitude independent of the right ascension of the stars.
Both observers then reversed the axis of their transit instruments ;
Cambridge selected a second pair of stars from the list, and the same
series of observations was repeated as with the first pair. The error
of coUimation was thus eliminated, and by confining the observations
to stars within about five degrees of the zenith, the influence of azi-
muthal error was avoided. The level being reail at every reversal,
the correction for it was applied by computation. In this manner
it is hoped to eliminate every possible source of error, except that
which arises from the personal habits of the observers. In order to
eliminate this error, a travelling observer worked for a time at Cam-
bridge and compared with the Cambridge astronomer; then came
to New York and compared with the New York astronomer ; then
returned to Cambridge again, and so on as often as was thought
necessary. Finally, at the conclusion of the campaign all the ob-
servers were to meet at Cambridge and make a general comparison
of their modes of observation.

On one or two nights the preceding programme was changed, and
each observer telegraphed both star A and star B.

"On the peculiar cooling effects of Hydrogen and its com-
pounds in cases of Voltaic Ignition." By W. F. Stevenson, Esq.,
F.R.S.

In this communication the author gives several theorems which
he considers to be established by the experiments cited in a pam-
phlet which he published, entitled " The Non-decomposition of Water
distinctly proved." He then states, that when we apply the prin-
ciple of these theorems to Mr. Grove's discovery of the cooling
properties of hydrogen, it will be found to admit of a most simple
solution : " for instance, when the coil of platinum wire is connected
with the poles of the electric battery, and the current is established,
it is evident that the electric matter thus passed through the wire
must escape at the contrary end (the air with which the wire is
surrounded not being a conductor of electricity), and as the quan-
tity of electric matter thus transmitted is considerable, and its exit
from the wire confined but rapid, that commotion before noticed
(in one of the author's theorems) necessarily ensues and causes the
ignition of the wire ; but when the coil of wire is immersed in
hydrogen, which is a conductor of electricity, it is evident that the
electric matter must be, at the same moment, abstracted or con-
ducted from every portion of the wire, and consequently the com-
motion or rush of the electric matter at the extremity of the wire,
which causes the ignition, is suspended and the comparative cool-
ness of the wire is the necessary result."

Postscript to a paper " On the Ganglia and Nerves of the
Heart," with two drawings. By Robert Lee, M.D., F.R.S.

The author states that since his former communication Avas pre-
sented to the Royal Society he has made a very minute dissection
in alcohol of the whole nervous system of the young heifer's heart.

Royal Society, 305

In this preparation the distribution of the ganglia and nerves over
the entire surface of the heart, and the relations of these structures
to the blood-vessels and muscular substance, are considered by the
author to be far more fully displayed than in any of his former dis-
sections. He states, that on the anterior surface there are distinctly
visible to the naked eye, ninety ganglia or ganglionic enlargements
on the nerves, which pass obliquely across the arteries and the mus-
cular fibres of the ventricles from their base to the apex; that these
ganglionic enlargements are observed on the nerves, not only where
they are crossing the arteries, but where they are ramifying on the
muscular substance without the blood-vessels ; that on the posterior
surface the principal branches of the coronary arteries plunge into
the muscular substance of the heart near the base, and many nerves
with ganglia accompany them throughout the walls to the lining
membrane and columnae carneae.

The author considers that, in the accompanying beautiful draw-
ings, Mr. West has depicted with the greatest accuracy and minute-
ness the whole nervous structures demonstrable in this preparation
on the surface of the heart; but that the ganglia and nerves repre-
sented in these drawings constitute only a small portion of the
nervous system of the heart, numerous ganglia being formed in the
walls of the heart which no artist can represent.

" On the Aurora Borealis which occurred on the evening of
Friday, the 17th of November, 1848." By Mr. R. Smith, Black-
ford, Perthshire. Communicated by P. M. Roget, M.D., F.R.S.

The author states that the 17th of November was a fine day with
a clear sky and bright sunshine : towards evening the sky became
cloudy and a few drops of rain fell, but it soon again became clear,
with the exception of a few fleecy clouds that here and there dimmed
its brightness. At 6^' 45"" a soft and gentle light began to illumine
the northern region of the sky ; and at 7 o'clock a considerable
portion of it was covered with dark-red streams of light towards
the east ; while streamers moving to and fro, arrayed in colours of
golden and silvery hues, overspread the south and north. About
8 o'clock there appeared near the zenith, and upon the magnetic
meridian, a ring of an elliptical form, from which proceeded in all
directions towards the horizon, beams or columns of light, giving to
the heavens the appearance of a splendid vault, with its top adorned
with a crown or wreath ; while around and within the vault were to
be seen clouds of brilliant light flashing towards and from the crown
or central circle of the aurora, sometimes tinged with prismatic rays,
at other times intensely white and lucid. About half-past nine
nearly the whole of the aerial canopy was clad with clouds of a
bright red colour, casting a curious reddened hue over the objects
on the surface of the earth. After a short period of time had
elapsed, the red colour began to diminish in intensity, and was again
replaced by the white dome. However, in various parts of the sky
the red colour still remained, principally in the north-west, south-
west, and north-east. Between the hours of twelve and one beams
of brilliant white light commenced shooting up in the south from

Phil. Mag. S. 3. Vol. 84-. No. 229. April 1849. X

306 Royal Society.

the horizon to the central ring or pole. The beams appeared to be
at nearly equal distances from each other, the entire colunm of them
stretching over a space equal to about one-fifth part of the visible
horizon, in the form of a fan. The whole figure rapidly changed
from a pure white light into a glow of brilliant colours of every tint,
variegating the undulating waves as they rolled on their way to the
pole of the aurora. In the course of three minutes these gave place
to the white flashing radiations.

During the time of the aurora there were a great number of small
meteors, the direction of whose motion was from east to west, and
which appeared to be considerably below the sphere of the aurora.

A box containing a delicately balanced needle, was exposed upon
the ground during the display of the aurora, but. did not appear to
be affected in the slightest degree till about one o'clock, when it was
observed to be considerably deflected. At the time when the needle
was disturbed, there was a dense column of radiating light in the
north-west and south-east. The reflexion from the north-west was
so clear, that when made to fall upon the polarizing plate of
M. Biot's polarizing apparatus, and a film of mica was placed upon
the stage of the instrument, the various colours produced by the
mica were beautifully clear and distinctly seen in the analysing glass.

The author considers that the phenomenon of the colours which
were noticed, was probably caused by exhalations or vapour floating
in the atmosphere, betwixt the light of the aurora and the observer,
causing a refraction of the rays transmitted to the eye, analogous to
that which produces the phenomenon of halos. The continued
undulations of the auroral light, and also the passing of the rays
through thick and thin portions of the vapour, may, he considers,
have produced the great variety of colours. During the time of
the exhibition of this phenomenon, a thin fog or vapour was observed
on both sides of the auroral fan. The author is of opinion that the
cause which produced the variety of tints, is different from that
which occasioned the red-coloured auroral clouds. At the time of
the latter phenomenon the moon's position was nearly due east, and
a cloud moved from the west towards the east, which in its course
passed between the moon and the observer; as soon as the cloud
obscured the light of the moon, the red colour to the north-west
disappeared, but became visible when an opening in the cloud allowed
the rays to pass through, and again vanished when another portion of
the cloud cut off the light; and when the cloud had finally passed
over, the red colour in the different parts of the sky resumed the
same tint that it possessed before the moonlight was obscured by
the cloud. The author states that it would thus appear, that when
the light of the moon was incident at a certain angle upon the white
light, or some kind of vapour that surrounded it, a red colour was
produced ; and hence that the moon is in some way or other con-
nected with the phenomenon. He remarks, that the red colour
was first observed in the east, and the moon being in that quarter
of the heavens, the rays proceeding from it would first come in
contact with that part of the aurora towards the east. When the

Royal Society. 307

aurora commenced, the moon was considerably below the horizon ;
but this, it is considered, does not form any serious objection to
what has been stated, since the aurora soared to so great a height,
that the rays of light proceeding from her would strike the aurora a
considerable time before she arose above the horizon.

The aurora continued for upwards of six hours, and during that
time the thermometer stood at 34°.

Jan. 18. — "On the Development and Homologies of the Cara-
pace and Plastron of the Chelonian Reptiles." By Professor Owen,
F.R.S.

The author commences by defining the several parts of which the
osseous thoracic-abdominal case of the Chelonian Reptiles is com-
posed, and briefly discusses the several opinions that have been
published with regard to their nature and homologies, dwelling
chiefly on that recently proposed by Prof. Rathke, in his work on
the Development of the Chelonia, in which it is contended that the
carapace consists exclusively of the development of parts of the
endo-skeleton, viz. the neural spines and vertebral ribs (pleurapo-
physes), agreeably with the opinion of Cuvier and Bojanus, and
that the remainder of the thoracic-abdominal case, consisting of the
"marginal pieces" and "plastron," are formed entirely from bones
of the dermal system.

Adverting to the hypotheses of Cuvier, GeoflProy and Meckel, that
the thoracic-abdominal case is a modification of parts of the endo-
skelelon exclusively, the author tests their determinations by com-
parisons with the corresponding parts of the bird and crocodile, and
infers, from the latter animal, that the hyosternal, hyposternal and
xiphisternal bones are not parts of the sternum, but are homologous
with the haemapophyses (sternal ribs and abdominal ribs) ; those in the
Plesiosaurus making the nearest approach to the peculiar develop-
ment of the parts in the Chelonia, especially as they appear in the
plastron of the immature Terrapenes and Sea-turtles.

Admitting that any hypothesis framed from the comparison of the
completed structures in the adult Vertebrata requires for confirma-
tion its agreement with the important phenomena of the develop-
ment of those structures, the author proceeds to apply that test.

He details his observations on the development of the skeleton,
and especially of the thoracic-abdominal case, in the embryos and
young of different genera of Chelonia. The chief facts that have
governed his conclusions are the following : —

With respect to the carapace. The cartilaginous basis of the
neural plates is developed in the substance of the derm ; and of
these, the 9th, 10th, 11th, and the ' nuchal' plate are ossified from
independent centres, and remain permanently free from anchylosis
with the subjacent spines of the vertebrae : they are, therefore,
" dermal bones," homologous with those that overlie the vertebrae
of the crocodile. But the first to the eighth neural plates inclusive
are serial homologues with the foregoing, and must, therefore, have
the same general homology. The objection that ossification extends
into their dermal cartilaginous basis from the neural spines is met

X2

308 Royal Society.

by the remark, that other parts, e.^. the radius and ulna of the frog,
are ossified from a common centre, without their honiological di-
stinctness being thereby masked or destroyed. The course or start-
ing-point of ossification does not determine the nature and homology
of parts, and the author refers what he believes to be an erroneous
conclusion of Prof. Rathke to undue value being given to the cha-
racter of connation.

The cartilaginous basis of the costal plates is developed in the
substance of the derm ; the subjacent ribs are previously ossified and
present the normal slender form. But ossification extends from near
the head of each of the eight pairs of dorsal ribs, from the second
to the ninth pair inclusive, into the superincumbent dermal carti-
lages. This had been described as the development of the tubercle
of the rib. But Prof. Owen observes that, in the development of
the carapace of the young of the Testudo indica, the connation of
the costal plate with the rib commences at a different point in each
rib alternately, and appears to be governed by the arrangement of the
horny scutes above. Another objection to these ossific expansions
being the tubercles of the ribs is presented by their abutment me-
sially against the neural plates, not against the vertebral diapo-
physes, as in the bird and crocodile.

In regard to the development of the plastron, the author describes
two situations in which the primitive cartilages are developed, cor-
responding with those in the embryo -carapace, viz. one belonging to
the endo-skeleton, the other in the derm. The first form under
which the endo-skeletal parts of the plastron appear agrees with the
evidence afforded by the comparison of the fully-developed parts
with those of the crocodile, and proves the hyosternals, hyposternals
and xiphisternals to be ' haemapophyses' or abdominal ribs: the
hyosternals and hyposternals are primitively long, slender, trans-
verse bars, which join the vertebral ribs in the Tortoises and Terra-
penes, without the intervention of any marginal pieces. The ossifi-
cation of the superadded dermal portions proceeds from the pre-
viously ossified endo-skeletal elements.

The author concurs with M. Rathk^ in regarding the marginal
pieces as ' dermal bones,' and concludes by a full discussion of the
facts and arguments which have led him to a different conclusion
respecting the nature and homologies of the carapace and plastron.

The memoir is illustrated by figures of the carapace and pla-
stron, and of the corresponding segments of the skeleton in the bird
and crocodile, and of the development of the thoracic-abdominal case
in land- and sea-chelonians.

Jan. 25. — Some remarks on a paper entitled " On the Depth of
Rain which falls in the same localities at different Altitudes in the
Hilly districts of Lancashire, Cheshire, &c., by S. C. Homersham,
C.E." By John Fletcher Miller, Esq. Communicated by Lieut.-
Col. Sabine, R. A., For. Sec. R.S.

The author, after alluding to the discordance between the con-
clusions at which he had arrived from a discussion of his meteoro-
logical observations in the lake district of Cumberland and West-

Royal Society, 309

moreland, described in a former paper, and those drawn from the
same facts by Mr. Homersham, in a paper read before the Society
on the 25th of May last, states that the results for the year 1848
show a precisely similar gradation to those of the two preceding
years ; and that the whole of the observations appear to warrant the
conclusion which he had ventured to draw from those detailed in
his former paper.

He remarks that, as the rain-gauges are, with one exception, si-
tuated on the high mountains surrounding tiie head of the Vale of
Wastdale, this valley is the only one which can fairly be selected as
a standard in comparing the quantities of rain obtained at the differ-
ent mountain stations. The discordance between his conclusions and
those arrived at by Mr. Homersham, he considers, has arisen from
that gentleman having selected the distant and excessively wet loca-
lity of Seathwaite at the head of the southern fork of Borrowdale,
as a representative of the quantity of water deposited in the valleys
generally.

If the receipts of the mountain gauges, he observes, be compared
with the rain-fall at Wastdale Head, or in any of the other valleys
except Seathwaite, it will be found that the quantity increases con-
siderably up to 1900 feet, where it reaches a maximum; and that
above this elevation it rapidly decreases, until at 2800 feet above the
sea the amount is very much less than in the surrounding valleys.

In conclusion, the author states that it appears to him, that much
of the discordance in the resultsobtained atvarious elevations amongst
the mountains has arisen from the circumstance of the instruments
having been placed on the slope or breast of the hill nearly in a line
with each other ; in which positions, he is convinced from experi-
ence, that when strong winds prevail, the gauges are exposed to eddies
or counter-currents, which prevent a portion of the water from enter-
ing the funnel, and thus a less depth of rain is obtained than is due
to the elevation.

The gauges under his superintendence being all stationed either
on the top or shoulder of the mountain, and exposed to the wind
from every point of the compass, are not, he observes, open to this
objection.

Supplement to a paper " On the Theory of certain Bands seen
in the Spectrum." By G. G. Stokes, Esq., M.A., Fellow of Pem-
broke College, Cambridge. Communicated by the Rev, Baden Powell,
M.A., F.R.S.

The principal object of the author in this communication is to
point out some practical applications of the interference bands re-
cently discovered by Professor Powell, the theory of which was con-
sidered by the author in the paper to which the present is a supple-
ment. The bands seem specially adapted to the determination of
the dispersion in media which cannot be procured in sufficient purity
to exhibit the fixed lines of the spectrum. The ordinary experi-
ments of interference allow of the determination of refractive indices
with great precision ; but in attempting to determine in this way the
dispersion of the retarding plate employed, there is the want of a

jllt lloyal Society.

definite object to observe in connection with the different parts of
the spectrum. In Professor Powell's experiment, the wire of the
telescope, placed in coincidence with one of the fixed lines of the
spectrum previously to the insertion of the retarding plate into the
fluid, marks tlie place of the fixed line, and so affords a definite ob-
ject to observe when the retarding plate is inserted into the fluid,
and the spectrum is consequently traversed by bands of interference.

The practical applications considered by the author are princi-
pally four. In the first, the variation of the refractive index of the
plate in passing from one fixed line to another is determined, the
absolute refractive index for some one fixed line being supposed ac-
curately known. The observation consists in counting the number
of bands seen between two fixed lines of the spectrum, the frac-
tions of a band-interval at the two extremities being measured or
estimated.

In the second application, the absolute refractive index of the plate
is determined for some one fixed line of the spectrum. The oljser-
vation consists in counting the number of bands which move across
the wire of the telescope, previously placed in coincidence with the
fixed line in question, when the plate is inclined to the incident light.

The third application is to the determination of the change in the
refractive index of the fluid, for any fixed line of the spectrum, pro-
duced by a change in the temperature. The observation consists in
counting the number of bands which move across the wire of the
telescope while the temperature sinks from one observed value to
another, the temperature being noted by means of a delicate ther-
mometer which remains in the fluid. For this observation a know-
ledge of the refractive index of the retarding plate is not required.

The fourth application is to the determination of the change of
velocity of the light corresponding to any fixed line of the spectrum,
when the direction of the refracted wave changes with reference to
certain fixed lines in the plate, which is here supposed to belong to
a doubly refracting crystal. The observation consists in counting
the bauds as they pass the wire when the plate is inclined. It re-
quires that the plate should be mounted on a graduated instrument.
It would be possible in this way to determine, by observation alone,
the wave surface belonging to each fixed line of the spectrum.

While considering the theory of Professor Powell's bands, the au-
thor was led to perceive the explanation of certain bands, described
by Professor Powell, which are seen in the secondary spectrum formed
by two prisms which produce a partial achromatism. Although the
account of these bands has been published many years, they do not
seem hitherto to have attracted attention. It is easily shown by com-
mon optics that when two colours are united by means of two prisms,
the deviation, regarded as a function of the refractive index, the
angle of incidence being given, is a maximum or minimum for some
intermediate colour. For the latter colour, two portions of light of
consecutive degrees of refrangibility come out parallel ; and there-
fore the diffraction bands belonging to different kinds of light, of
very nearly the same refrangibility with the one in question, are su-

Royal Society. 31 1

perposed in such a manner that the dark and bright bands respect-
ively coincide. Thus distinct bands are visible in the secondary
spectrum, although none would be seen in the spectrum formed by
a single prism, in consequence of the mixture of the bright and dark
bands belonging to different kinds of light of nearly the same degree
of refrangibility. The diffraction bands here spoken of are of very
sensible breadth, in consequence of the small width of the aperture
employed in the actual experiment.

When a spectrum is viewed through a narrow slit half covered by
a plate of mica, the edge of which bisects the slit longitudinally, and
is held parallel to the fixed lines of the spectrum, the bands described
by Sir David Brewster are seen, provided the mica plate lie at the
side at which the blue end of the spectrum is seen, and provided the
thickness of the plate and the breadth of the slit lie within certain
limits. When these bands are invisible in consequence of the slit
being too narrow, or the spectrum too broad, it follows from theory
that the bands ought to appear when the slit and plate are turned
round the axis of the eye, so that the edge of the plate is no longer
parallel to the fixed lines of the spectrum. The author has verified
this conclusion by experiment, employing plates adapted to obser-
vations with the naked eye, which are best suited to the purpose.

Feb. 1. — "On the Chemistry of the Urine ;" in three Parts. By H.
Bence Jones, M.D., M.A., F.Il.S.

Part I. On the variations of the Acidity of the Urine in Health.

The mode of examination adopted by the author was the following :
Two test solutions were made ; the one with carbonate of soda ; the
other with dilute sulphuric acid, of such strength that each measure
of a graduated tube, when filled with either solution, was equivalent
to one-twelfth of a grain of dry and pure carbonate of soda.

A weighed quantity of urine was neutralized by one or other of
the test solutions, and thus the degree of acidity or alkalescence was
determined.

Diurnal variations in the acidity of the urine were observed. The
acidity of the urine was found to ebb and flow ; it was greatest a
short time before food was taken, and was least about three hours
after breakfast, and five or six hours after dinner, when it reached
the minimum point ; after which it again increased, and attained the
maximum point previous to food being again taken.

If no food was taken, the acidity varied but slightly for twelve
hours.

By comparing the effect of vegetable food with animal food, it
appeared that the food which irritated the stomach most and caused
most secretion of acid in the stomach, caused the greatest oscillations
in the urine.

Dilute sulphuric acid taken in large doses produced but little
effect on the variations of the acidity of the urine ; but it was proved
to increase the acidity of the urine.

Part II. On the simultaneous variations of the amount of Uric Acid
and the Acidity of the Urine in a healthy state.

The result of these experiments is, that there is no relation be-

S 12 Royal Society.

tween the acidity of the urine and the amount of uric acid in it.
The urine that was most acid contained least uric acid; that which
contained most uric acid was not most acid. All food causes an in-
crease in the amount of uric acid in the urine ; and there is no de-
cided difference between vegetable and animal food, either as to the
increase or diminution of the amount of uric acid in the urine.

Part III. Variations oj the Sulphates in the Urine in the healthy
state, and on the influence of Sulphuric Acid, Sulphur and the Sul-
phates, on the Sulphates in the Urine.

The result of these experiments is, that the sulphates in the urine
are much increased by food, whether it be vegetable or animal. Ex-
ercise does not produce a marked increase in the sulphates. Sul-
phuric acid, when taken in large quantity, increases the sulphates in
the urine. In small quantity, even when long-continued, no effect
on the amount of sulphates is manifest.

Sulphur taken as a medicine increases the sulphates in the urine.
Sulphate of soda and sulphate of magnesia produce the most marked
increase in the sulphates in the urine.

Feb. 8. — "On the application of the Theory of Elliptic Functions
to the Rotation of a Rigid Body round a Fixed Point." By James
Booth, L.L.D.,F.R.S.

In the introduction to his investigation, the author, after noticing
the investigations of D'Alembert and Euler, and the solution of this
problem by Lagrange, refers more particularly to the memoir of
Poinsot, in which the motion of a body round a fixed point, and free
from the action of accelerating forces, is i*educed to the motion of
a certain ellipsoid whose centre is fixed, and which rolls without
sliding on a plane fixed in space ; and likewise to the researches
of MaccuUagh, in which, by adopting an ellipsoid the reciprocal of
that chosen by Poinsot, he deduced those results which long before
had been arrived at by the more operose methods of Euler and La-
grange ; observing, however, that it is to Legendre that we are in-
debted for the happy conception of substituting, as a means of inves-
tigation, an ideal ellipsoid having certain relations with the actually
revolving body. He then states, that several years ago he was led
to somewhat similar views, from remarking the identity which exists
between the formulae for finding the position of the principal axes of
a body and those for determining the symmetrical diameters of an
ellipsoid ; and further observing that the expression for the per-
pendicular from the centre on a tangent plane to an ellipsoid, in
terms of the cosines of the angles which it makes with the axes, is
precisely the same in form as that which gives the value of the mo-
ment of inertia round a line passing through the origin. Guided by
this analogy, he was led to assume an ellipsoid the squares of whose
axes should be directly proportional to the moments of inertia round
the coinciding principal axes of the body. This is also the ellipsoid
chosen by MaccuUagh. Although it may at first sight appear of
little importance which of the ellipsoids — the inverse of Poinsot, or
the direct of MaccuUagh and the author — is chosen as the geome-
trical substitute for the revolving body, it is by no means a matter

Kmjal Society. 313

of indifference wlien we come to treat of the properties of the in-
tegrals which determine the motion. Generally those integrals de-
pend on the properties of those curves of double flexure in which
cones of the second degree are generally intersected by concentric
spheres ; and it so happens that the direct ellipsoid of moments is
intersected by a concentric sphere in one of these curves. By means
of the properties of these curves a complete solution may be ob-
tained even in the most general cases, to which only an approxi-
mation has hitherto been made.

In the first section of the paper, the author establishes such pro-
perties as he has subsequently occasion to refer to, of cones of the
second degree, and of the curves of double curvature in which these
s!irfaces may be intersected by concentric spheres, some of which
he believes will not be found in any published treatise on the sub-
ject. He considers that he has been so fortunate as to be the first
to obtain the true representative curve of elliptic functions of the
first order. It is shown that any spherical conic section, the tan-
gents of whose principal semiarcs are the ordinates of an equilateral
hyperbola whose transverse semiaxis is 1, may be rectified by an
elliptic function of the first order; and the quadrature of such a
curve may be effected by a function oi" the same order, when the
cotangents of the halves of the principal arcs are the ordinates of
the same equilateral hyperbola.

This particular species of spherical ellipse the author has called
the " Parabolic Ellipse," because, as is shown in the course of the
investigation, it is the gnomonic projection, on the surface of a sphere,
of the common parabola whose plane touches the sphere at the focus.
As in this species of spherical ellipse either the focus or the centre
may be taken as the origin of the spherical radii vectores, in effect-
ing the process of rectification, we are unexpectedly presented with
Lagrange's scale of modular transformations, as also with the other
equally well-known theorem by which the successive amplitudes are
connected. Among other peculiar properties of the spherical para-
bolic ellipse established in this paper, it is shown that the portion of
a great circle touching the curve, and intercepted between the per-
pendicular arcs on it from the foci, is always equal to a quadrant.

In the second and following sections, the author proceeds to dis-
cuss the problem which is the immediate subject of the paper.
Having established the ordinary equations of motion, he shows that,
if the direct ellipsoid of moments be constructed, the motion of a
rigid body acted on solely by primitive impulses may be represented
by this ellipsoid moving round its centre, in such a manner that its
surface shall always pass through a point fixed in space. This point,
so fixed, is the extremity of the axis of the plane of the impressed
couple, or of the plane known as the invariable plane of the motion.

But a still clearer idea of the motion of such a body is presented
in the subsequent investigations, it being there shown, that the most
general motion of a body round a fixed point may be represented by
a cone rolling with a certain variable velocity on a plane whose axis
is fixed, while this plane revolves about its own axis with a certain

314 Intelligence and Miscellaneous Articles.

uniform velocity. This cone may always be determined. For the
circular sections of the invariable cone coincide with the circular
sections of the ellipsoid of moments ; whence the cyclic axes of
the ellipsoid, or the diameters perpendicular to the planes of these
sections, will be the focal lines of the supplemental cone ; and as
the invariable plane is always a tangent plane to this cone, we have
sufficient elements given to determine it.

From these considerations it appears that we may dispense alto-
gether with the ellipsoid of moments, and say that if two right lines
be drawn through the fixed point of the body in the plane of the
greatest and least moments of inertia, making angles with the axis
of greatest moment, the cosines of which shall be equal to the
square root of the expression

L(M-N)
M(L-N)'

(L, M, N being the symmetrical moments of inertia round the prin-
cipal axes) and a cone be conceived having those lines as focals, and
touching moreover the invariable plane, the motion of the body will
consist in the rotation of this cone on the invariable plane with a
variable velocity, while the plane revolves round its own axis with
an uniform velocity.

Although it is very satisfactory, the author remarks, in this way
to be enabled to place before our eyes, so to speak, the actual mo-
tion of the revolving body, yet it is not on such grounds that the
paper is presented to this Society. It is as a method of investiga-
tion that it must rest its claims to the notice of mathematicians ; as
a means of giving simple and elegant interpretations of those definite
integrals on the evaluation of which the dynamic state of a body at
any epoch can alone be ascertained.

In these applications of the theory of elliptic functions, the au-
thor has been led to the remarkable theorem, that the length of the
spiral, between two of its successive apsides, described in absolute
space on the surface of a fixed concentric sphere, by the instantane-
ous axis of rotation, is equal to a quadrant of the spherical ellipse
described on an equal sphere moving with the body, by the same
instantaneous axis of rotation.

The last section of the paper is devoted to the discussion of that
particular case in which the axis of the invariable plane is equal to
the mean semiaxis of tbj ellipsoid of moments.

XLV. Intelligence and Miscellaiieous Articles.

ON ANHYDROUS NITRIC ACID. BY M. DEVILLE.

M DUMAS presented to the Academy in the name of M. De-
• ville. Professor at the Faculty of Sciences of Besan^on, the
first results of his researches on the action of chlorine on the anhy-
drous salts which oxide of silver forms both with organic and inor-
ganic acids.

IntelUsence and Miscellaneous Articles. 315

'to

By treating nitrate of silver with perfectly dry chlorine, M. De-
ville has succeeded in isolating anhydrous nitric acid, the existence
of which was demonstrated by numerous analyses. This beautiful
substance is obtained in colourless crystals, which are perfectly bril-
liant and limpid, and may be procured of considerable size ; when
they are slowly deposited in a current of gas rendered very cold,
their edges are a centimetre in length. These crystals are prisms
of six faces, which appear to be derived from a right prism with a
rhombic base. They melt at a temperature not much exceeding
85°*5 F.; their boiling-point is about 113°; at 50° the tension of
this substance is very considerable. In contact with water it becomes
very hot, and dissolves in it without imparting colour, and without
disengaging any gas ; it then produces with barytes the nitrate of
that base. When heated its decomposition appears to commence
nearly at its boiling-point ; this circumstance is an obstacle to the
determination of the density of its vapour by the process of M.Dumas.

The process by which M . Deville obtained anhydrous nitric acid
is very simple ; but the readiness with which it penetrates tubes of
caoutchouc renders it necessary to unite all the pieces of the appa-
ratus by melting them. The following is the process : — the author
employs a U-shaped tube capable of containing 500 grs. of nitrate
of silver, dried in the apparatus at 356° F. in a current of dry car-
bonic acid gas. Another very large U-tube is connected with this,
and to its lower part is attached a small spherical reservoir ; it is in
this reservoir that a liquid is deposited which always forms during
the operation, and which is excessively volatile (nitrous acid ?). The
tube containing the nitrate of silver is immersed in water covered
with a thin stratum of oil, and heated by means of a spirit-lamp
communicating with a reservoir at a constant level. The chlorine
issues from a glass gasometer, and its displacement is effected by a
slow and cons^tant flow of concentrated sulphuric acid. The chlorine
must afterwards pass over chloride of lime, and then over pumice-
stone moistened with sulphuric acid. At common temperatures no
effect appears to be produced. The nitrate of silver must be heated
to 203° F., the temperature being then quickly reduced to 136° or
154°, but not lower. At the commencement, hyponitrous acid, di-
stinguishable by its colour and ready condensation, is produced ; and
when the temperature has reached its lowest point, the production
of crystals begins, and they soon choke the receiver cooled to 6°
below zero ; they are always deposited upon that part of the receiver
which is not immersed in the freezing mixture, and M. Deville states
that ice alone is sufficient to occasion their formation.

The gases are coloured, and the small sphere of the cooled tube
contains a small quantity of liquid, which must be taken from the
apparatus before the nitric acid is removed to another vessel ; this
latter operation is readily effected by replacing the current of chlorine
by one of carbonic acid. The condenser is then to be no longer
cooled, and the vessel for receiving the crystals is to be immersed in
a freezing mixture ; this is fastened to the producing apparatus by
means of a caoutchouc tube furnished with amianthus. The chlo-^

316 Intelligence and Miscellaneous Articles.

rine should pass very slowly, at the rate of about three or four litres
in twenty-four hours. All the gas however is not absorbed by the
nitrate of silver. Oxygen is evolved, the volume of which appears
to be equal to that of the chlorine employed. An apparatus thus
constructed operates day and night without watching, care being
however taken to renew the sulphuric acid which displaces the chlo-
rine, the spirit of the lamp, and the ingredients of the freezing mix-
ture.

. The author states that he shall forward hereafter a more complete
memoir, in which he will describe the chemical properties of the
anhydrous nitric acid, and detail the results of his researches on the
action of chlorine and hypochlorous acid on the salts of silver. —
L'Institut, Fevrier 21, 1849.

ON A SERIES OF ORGANIC ALKALIES HOMOLOGOUS WITH
AMMONIA. BY A. WURTZ.

The history of the ammoniacal compounds, so complete and so
important in a theoretical point of view, forms in some measure a
transition between inorganic and organic chemistry. Ammonia
should decidedly be regarded as the most simple and the most power-
ful of the organic bases ; and it would be for all chemists the type
of that numerous class of bodies, did it not differ in one undoubtedly
important character, but to which an exaggerated value has been
attributed. Ammonia contains no carbon. This difference how-
ever of coHjposition does not suffice, in my opinion, to separate
ammonia from the organic bases ; I have succeeded, in fact, in con-
verting this alkali into a true organic compound, by adding to it the
elements of the hydrocarbon C^ H- or C* H*, without depriving it
of its characters of a powerful base, or of its most striking properties,
for instance its odour. By adding to the elements of ammonia, NH^,
the elements of 1 equiv. of methylene, OH^ the compound C^H^N,
which may be called methylic ammonia, is obtained. By adding
to ammonia the elements of ethylene, C"*!!^, ethylic ammonia,
C* H7 N, is obtained.

The compounds C^ H* N and C* H^ N may be viewed as methylid
sether, C^ H' O, and ordinary aether, C^* H^ O, in which the equiva-
lent of oxygen is replaced by 1 equiv. of amidogen, NH'^; or as
ammonia in which 1 equiv. hydrogen is replaced by methylium,
C^ H3, or ethylium, C* H*. The following formulaB will exhibit the
relations which exist between these substances and ammonia : —

H3 N, ammonia. NH^, H, hydramide.

O^ H^ N, methylic ammonia. NH% C^ H^, methylamide.
C-* H7 N, ethylic ammonia. NHS C H*, ethylamide.

I shall employ in preference the names methylamide and ethylamide
to designate these new bases.

In the present communication I shall restrict myself merely to
communicating the circumstances under which these substances are
produced, and to communicating the results of some analyses which

Intelligence and Miscellaneous Articles. S 1 7

establish their composition. They are produced iinder three dif-
ferent circumstances, — 1st, by the action of potash on cyanic (Ethers;
2nd, by the action of potash upon cyanuric cethers ; and 3rd, by the
action of potash upon the ureas. These reactions will be best exhi-
bited by a few formulae : —

C2NO,HO + 2KO + 2HO=2C02,KO + H3N

Cyanic acid. Ammonia.

C^ NO, C2 H3 0 + 2KO + 2HO=2COSKO + C2 H* N
Cyanate of methylene. Methylamide.

C« NO, C* H* 0 + 2KO+2HO=:2COS KO + C* H? N
Cyanic ather. Ethylamide.

Cyanuric acid and the cyanuric aethers being isomeric with the
cyanic compounds, it will suffice to multiply the preceding formulae
by 3 to explain the second mode of formation. With respect to the
ureas, the following equations will show how they give rise to these
bases : —

C2H4N202 + 2KO + 2HO=:2COSKO + H3N + H3N

Urea.

C^H6N^02 + 2KO+2HO=2COSKO + H3N + C2H^N
Acetic urea.

C6 H8 N2 02+2KO+2HO=2CO^ K0 + H3 N + C* W N

Metacetic urea.
Hydrochlorale of Methylamide. — I obtained this salt by boiling
cyanurate of methylene with an excess of potash in an apparatus
arranged so that the vapours of methylamide, after having passed
through a refrigerator, were condensed in a receiver containing a
little pure water. The excessively caustic liquid thus obtained has
a strong odour of ammonia, but does not contain a trace of that
alkali; for if saturated with hydrochloric acid and evaporated to
dryness, the residue, consisting of hydrochlorate of methylamide,
dissolves very readily in hot absolute alcohol. The salt crystallizes
on cooling in beautiful laminae, which are iridescent so long as they
float in the liquid, and assume a nacreous appearance when dry.
Analysis gave —

Carbon 17'4 2 = 12 17-7

Hydrogen 8*7 6 6 8-8

Chlorine 52-2 1 35-5 52-5

Nitrogen 21'7 1 14- 21-0

Hydrochlorate of Methylamide and Chloride of Platinum. — Beau-
tiful golden scales, which are soluble in hot water, and contain
CIH, C'^ H^ N PtCR Analysis gave—

Carbon 5*3 2=12 5-0

Hydrogen 2*8 6 6 2-5

Chlorine 44''4 3 106*5 44-9

Platinum 41-4 1 98-6 415

Nitrogen 1 14

Nitrate of Methylamide — beautiful transparent prisms, which are
soluble in alcohol.

$18 Intelligence and Miscellaneous Articles.

Hydrochlorate of Ethylamide. — I have prepared this substance
both with cj'anic and with cyanuric aether. It dissolves readily in
absolute alcohol, and crystallizes in laminae; it melts below 212°,
and solidifies on cooling into a crystalline mass. When distilled
with burnt lime, it gives ofF ethylamide in the form of an excessively
caustic liquid, which diffuses a strong odour of ammonia. This
liquid precipitates all the metallic salts, and even the salts of mag-
nesia. In solutions of salts of copper it at first forms a blue preci-
pitate, which it afterwards redissolves, forming an azure-blue liquid ;
it produces a green precipitate in salts of nickel, which however is
not redissolved, as is the case with ammonia. I ascertained that
the liquid did not contain a trace of ammonia, by saturating it with
hydrochloric acid ; the residue, evaporated to dryness, dissolved en-
tirely in absolute alcohol, and formed with chloride of platinum a
double salt, the analysis of which will be found below.

The composition of the hydrochlorate of ethylamide is represented
by the formula CIH, C^ W N. Analysis gave—

Carbon 28-9 29-4 4 = 24- 29*4

Hydrogen 9-9 9-9 8 8 9-8

' Chlorine 43-7 .. 1 35-5 43-6

Nitrogen 17*5 .. 1 14 17*2

Hydrochlorate of Ethylamide and Chloride of Platinum — golden
scales, soluble in water. They gave on analysis —

Carbon 9-5 4 = 24 9'5

Hydrogen 3-2 8 8 3*2

Chlorine 42-0 3 106*5 42-4

Nitrogen 1 14

Platinum 39*0 1 98*6 39-2

I hope soon to give a complete history of these alkalies. — Comptes
Pendus, Feb. 12.

ON THE EXISTENCE OF MERCURY IN THE TYROL.
BY M. H. ROSE.

M. Weidenbusch, in analysing in the author's laboratory a speci-
men of tender gray copper ore, stated to be from Schwarz in the
Tyrol, found it to contain a notable quantity of mercury, amounting
to 15'5 per cent. This gray copper is mixed with quartz and sul-
phuret of copper. Its powder is almost black, and has a specific
gravity of 5" 1075 ; when heated in a flask, it yields a little metallic
mercury with a light reddish-brown sublimate. If it be mixed with
carbonate of soda and heated, a larger quantity of mercury is obtained.
It contains also zinc, iron, antimony and sulphur, and traces of
arsenic and silver. These substances exist in it in the same pro-
portions as in other gray copper ores. A crystallized gray copper,
also stated to be from Schwarz in the Tyrol, did not contain any
mercury. — L'/ws^iVm^ Fevrier 21, 1849.

Meteorological Observations. 319

RECTIFICATION OF SPIRIT OF NITROUS ^THER.

M. Klauer, pharraacien of Mulhouse, states it to be well known,
that when spirit of nitrous aether has become acid, it is usual to
rectify it from calcined magnesia, which, however, does not prevent
this preparation from becoming again acid in a few weeks. The
author observes that this is not the case when it is rectified from
neutral tartrate of potash instead of magnesia ; it may then be kept
for several months without exhibiting any sign of acidity. — Journ.
de Ph. et de Ch., Fevrier 1849.

[The rationale of this operation is not evident. — Edit. Phil. Mag.]

METEOROLOGICAL OBSERVATIONS FOR FEB. 1849.

Chismck. — February I. Frosty: foggy: rain. 2. Drizzly: hazy: rain. 3. Hazy
and damp : densely overcast. 4. Overcast. 5. Very fine : overcast. 6. Hazy :
densely overcast. 7. Overcast. 8. Very fine : clear. 9. Fine : overcast. 10.
Overcast : clear at night. 1 1. Clear : very fine : barometer unusually high : clear
and frosty at night. 12. Frosty and foggy : fine: clear and frosty. 13. Dense
fog : fine at noon: foggy. 14. Foggy : fine. 15. Very fine. 16. Foggy: clear
at night. 17. Frosty: exceedingly fine. 18. Overcast. 19. Overcast: fine.
20. Slightly overcast : cloudy : rain. 21. Cloudy and fine : rain. 22, 23. Fine.
24. Drizzly : rain : lightning in the evening : densely overcast. 25. Hazy :
boisterous, vfith rain and thunder : constant heavy rain at night. 26. Cloudy
and fine : frosty. 27. Frosty : cloudy and fine : clear. 28. Boisterous, with
heavy rain. — On the 11th the barometer was higher than it has ever been observed
in this locality.

Mean temperature of the month 41°*35

Mean temperature of Feb. 1848 43*06

Mean temperature o^Feb. for the last twenty years 40 "36

Average amount of rain in Feb 1*61 inch.

Boston. — Feb. 1. Fine. 2, .3. Foggy. 4,5. Fine. 6,7. Cloudy. 8. Fine:
rain P.M. 9— 12. Fine. 13. Foggy. 14 — 17. Fine. 18. Cloudy. 19. Cloudy:
rain P.M. 20. Fine: rain p.m. 21. Cloudy. 22,23. Fine. 24,25. Cloudy.
26,27. Fine. 28. Rain : snow a.m. and p.m.

Applegarth Manse, Dumfries-shire. — Feb. 1. Frost and snow; looking moist p.m.
2. Fog and drizzling all day. 3. Fog and drizzling. 4. Dull a.m. : drizzling
rain p.m. 5. Still dull, but fair : cloudy and moist p.m. 6. Mild : cloudy : high
wind p.m. 7. Rain during night : fair and clear. 8. Fair, but dull a.m. : rain p.m.
9. Fair early a.m. : rain at noon : rain p.m. 10. Fine morning : one shower:
clear P.M. 11. Frost: fog: cleared p.m. 12. Fair: slight shower: cleared.
13. Frost A.M.: clear: rain p.m. 14. Fine spring day : dry throughout. 15.
Fine : drying wind. 16. Frost : clear and fine : high wind p.m. 17. Fair and
clear : storm of wind. 18, 19. Rain and wind. 20. Dull and moist. 21. Frost:
rain and wind p.m. 22. Dull a.m. : came on storm, wind and rain. 23. Fair:
slight frost : snow on hills. 24. Frost and snow : clear : freezing. 25. Hard
frost: fine. 26. Very hard frost: hail-shower. 27. Hard frost. 28. Rain heavy :
wind high.

Mean temperature of the month 41°*2

Mean temperature of Feb. 1848 40 'I

Mean temperature of Feb. for the last twenty-five years . 37 '3

Rain in Feb. 1848 5-53inches.

Average amount of rain in Feb. for the last twenty years 2-04 „
Sandwich Manse, Orhiey. — Feb. 1. Clear: frost: cloudy. 2. Cloudy. 3. Bright:
drizzle. 4. Bright : clear : hoar-frost. 5. Drizzle : rain. 6. Clear : cloudy.
7. Drizzle: cloudy. 8. Bright: hail, thunder and lightning. 9. Cloudy:
damp. 10. Thunder and lightning : sleet-showers. 11. Bright: cloudy. 12.
Fine : clear : aurora. 13. Showers. 14. Rain. 15. Rain : drizzle. 16. Hazy :
showers. 17. Cloudy: showers. 18. Showers. 1 9. Showers : aurora. 20. Snow-
showers. 21. Frost : snow-showers. 22,23. Snow : frost : aurora. 24. Snow-
showers. 25. Snow-showers : clear : frost. 26. Snow-showers : clear : aurora,
27. Snow : fine : clear ; aurora. 28. Rain : sleet-showers.

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THE
LONDON, EDINBURGH and DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE,

[THIRD SERIES.]

MAY 1849.

XLVI. On the Isomeric Modifications of Phosphoric Acid.
By H. Rose, Professor of Chemistry in the University of
Berlin^.

IT has frequently been remarked, that no substance presents
greater difficulties in every respect to the chemist in its
examination than phosphoric acid ; whilst the longer and more
constantly we are engaged in studying the reactions of this
acid, the greater do the difficulties become. Every time the
investigation is renewed, the chemist meets with new anomalies,
and numerous new obscure phaenomena are exhibited, whilst
even those previously known are far from being satisfactorily
explained.

Some time since, in an investigation upon the isomeric mo-
difications of the peroxide of tin, I drew attention to the fact,
that such modifications are principally met with in those me-
tallic oxides which possess acid properties; they are almost
entirely absent in those which possess strong basic properties.
The latter certainly frequently alter in density to a consider-
able extent when exposed to various elevated temperatures,
becoming at the same time soluble with difficulty or insoluble
in acids ; when, however, they are once dissolved, they always
exhibit in solution the same reactions.

Phosphoric acid exhibits paradoxical isomeric properties
still more than any metallic acid. After the important disco-
very of Clark, the ingenious investigations of Graham did
much towards the elucidation of the difficulties ; but by new
investigations new facts have been discovered, part of which
cannot be satisfactorily explained by Graham's views.

Assuming, in accordance with Graham's views, the exist-
ence of three modifications of phosphoric acid, viz. metaphos-

* Translated from PoggendorfTs Amialen der Physikund Ckemie,vo\. Ixxvi.
p. 1.

Phil. Mag. S. 3. Vol. 34-. No. 230. May 1849. Y

322 Prof, H. Rose on the Isomeric Modifications

phoric, pyrophosphoric, and ordinary phosphoric acid, which
withBerzelius we may denominate a-,6-, and c-phosphoric acid,
it is the metaphosphoric acid which especially exhibits the
greatest anomalies. The pyrophosphoric acid also presents
some ; and the ordinary phosphoric acid, the c-phosphoric
acid, is that which has most analogy with other oxy-acids in
its behaviour towards reagents. This is the one which can be
best separated and determined quantitatively. Fortunately
the other modifications can be more or less easily converted
into the c-phosphoric acid ; hence in their quantitative deter-
mination this conversion must in most cases be previously
effected.

I shall now make some remarks upon each of these three
modifications of phosphoric acid.

a-Phosphoric Acid {Metaphosphoric Acid).

For the purpose of elucidating to a certain extent, if only
for the present, the chaos of anomalous phaenomena which
this acid exhibits, we are compelled to admit the existence of
several submodifications of it : at least three of these are
now distinguishable.

1. One of these submodifications consists of the acid exist-
ing in Graham's metaphosphate of soda, which is obtained by
fusing the acid phosphate of soda with the phosphate of soda
and ammonia (microcosmic salt). After fusion, the mass
must be cooled rapidly, and not slowly. The solution of this
salt, as is well known, possesses a neutral or very slightly acid
reaction, and is especially characterized by giving precipitates
with neutral solutions of several salts of the earths and me-
tallic oxides, which are generally soluble in an excess of the
salt of soda, and possess the remarkable property of forming
a heavy, thick, oleaginous mass when shaken. The solution
of the salt alone does not yield any precipitate with a dilute
filtered solution of albumen, but a copious dense precipitate
immediately appears on the addition of acetic acid.

The following are the special reactions of a solution of the
metaphosphate of soda : —

Chloride of barium produces a voluminous precipitate, the
supernatant liquid reddening litmus-paper; it is entirely so-
luble in excess of the salt of soda; ammonia does not produce
a precipitate in this solution. The precipitate is not oleagi-
nous, nor does it become so, either when set aside for some
time, or on ebullition.

Chloride of calcium produces a voluminous precipitate, which
on agitation, even in the cold, collects at the bottom of the
vessel in the form of a thick, oily or turpentine-like mass.

of Phosphoric Acid. 323

The supernatant fluid reddens litmus-paper. The mass is not
affected by ebullition, but when heated with muriatic acid, it
is dissolved. The precipitate is entirely dissolved by excess
of the soda salt; ammonia does not produce a precipitate in
the solution.

A solution of sulphate of magnesia does not produce any
precipitate, even on ebullition. If much of the soda salt has
been added, no precipitate is produced in the solution by am-
monia ; in the opposite case, ammonia produces a precipitate
which is soluble in chloride of ammonium.

Nitrate of silver produces a dense, voluminous, white pre-
cipitate, which is soluble in ammonia and nitric acid. It is
also perfectly soluble in a large excess of the salt of soda.
The supernatant liquid reddens litmus-paper. The precipi-
tate does not become oleaginous when shaken in the cold, but
when boiled it contracts, occupying a smaller volume, and
becomes perfectly resinous. Heat renders it tenacious, so
that it can be drawn into threads; and on cooling, it solidifies
to a brittle mass.

A solution of pernitrate of mercury^ which necessarily con-
tains free acid, produces a white precipitate; this, when shaken,
even in the cold, subsides to the bottom of the vessel as a
dense oleaginous mass.

Solution of bichloride of mercury produces no change.

Solution of protonitraie of mercury produces a dense white
precipitate, which is soluble in an excess of the soda salt ;
on boiling it becomes resinous, like the salt of silver.

Solution of sulphate of copper Y^todnces no change; but
chloride of copper produces a bluish-white precipitate, which
is soluble both in excess of the soda salt and of the chloride
of copper.

Solution of acetate of lead YiToduces a dense voluminous pre-
cipitate, which is soluble in excess of the soda salt, and when
shaken forms a coherent mass, but does not become oleagi-
nous ; when set aside, however, it becomes somewhat resinous.

Solution of protosulphate of manganese produces a white
precipitate, which when shaken becomes an oily mass. The
precipitate is soluble in an excess of the soda salt; sulphuret
of ammonium throws down sulphuret of manganese from this
solution.

Solution of protosulphate of iron does not produce a preci-
pitate. Nor is any precipitation caused by the addition of
ammonia to this solution, but it renders it dark green.

Solution of sidphate of zinc produces no change.

Solutions of the sulphates of cobalt and nickel also produce
no change. However, solutions of the chlorides of cobalt and

Y2

324' PiDf. H. Hose on the Isomeric Modifications

nickel produce red and (jreenish-wliite precipiiates, which on
agitation subside in the form of lieavy oily drops of a red and
greenish-white colour. The precipitates are soluble in excess
of the soda salt.

Solution of the nitrate of bismuth^ although it contains free
acid, produces a white precipitate, which on agitation becomes
somewhat resinous, but not oily. The precipitate is soluble
in the soda salt.

If the acid contained in the salt of soda be separated from
the base, its aqueous solution exhibits somewhat different pro-
perties from those of the aqueous solution of the metaphos-
phoric acid produced by combustion in oxygen gas. The
solution of the soda salt was precipitated with nitrate of silver,
and the precipitate allowed to remain in the liquid during one
night; it was then washed with cokl water, and after suspen-
sion in water, was decomposed by a current of sulphuretted
hydrogen. The sulphuret of silver formed remains suspended
in the tree acid for a long time, and is separated by filtration
with extreme difficulty.

The aqueous solution of the acid does not at first produce
any precipitate in chloride of barium ; after a considerable time
merely some flakes subside. Earytic water however produces
a precipitate in it, even when not added in excess, and whilst
the solution is still acid.

Chloride of calcium produces no precipitate. Lime-water
only produces a precipitate when added in excess.

A solution of sulphate of magnesia mixed with chloride
of ammonium, only produces a precipitate in the solution of
the acid saturated with ammonia, when the solutions are con-
centrated. This is, however, soluble in a considerable quan-
tity of water, and hence does not appear in dilute solutions.

SoXntXou o^ nitrate of silver produces a white precipitate,
which on the saturation of the solution with ammonia becomes
more considerable.

In a solution of albumen a copious white precipitate is im-
mediately formed.

The reactions of a solution of metaphosphoric acid produced
by the combustion of phosphorus in oxygen gas, differ some-
what from those of the acid as procured from the salt of soda.

Thus, a solution o'i chloride of barium immediately produces
a copious precipitate in it. A very large excess of the acid
is requisite to redissolve the precipitate ; ammonia does not
produce any precipitate in this solution. Barytic water pro-
duces a precipitate in the acid, even when the latter is in great
excess j an excess of barytic water, however, renders it more
copious.

of Phosphoric Acid. 325

Solution o^ chloride of calcium produces an extremely small
precipitate, which is perfectly soluble in excess of the acid.
Ammonia produces a dense voluminous precipitate in this
solution. Lime-water does not cause any precipitate until
added in excess. Both acids exhibit the same reactions with
sulphate of magnesia, nitrate of silver and albumen.

These two acids therefore differ principally in their reac-
tion with chloride of barium. But as solutions of Graham's
metaphosphate of soda, as stated above, when decomposed
by solutions of neutral salts, yield liquids which possess an
acid reaction, the acid separated from the precipitates must
be different from that in the soda salt used.

As the composition of the oily and resinous precipitates
produced by Graham's salt was entirely unknown, the salt of
silver was analysed by M. Weber. It was prepared in the
same manner as the salt precipitated for the purpose of ob-
taining the free acid. After having been dried for a long
time at 212^ F., 0*930 grm. lost at a red heat 0-019 grm.,
or 2"04' per cent, of water. It fused into a mass of a yellowish
colour, which, when dissolved in nitric acid, yielded on the
addition of hydrochloric acid 0'815 grm. of chloride of silver,
corresponding to 70*09 oxide of silver. The phosphoric acid
in the filtered liquid was precipitated in the form of the phos-
phate of ammonia and magnesia; 0*257 grm. of pyrophos-
phate of magnesia was obtained after heating to redness,
corresponding to 27*87 per cent, of phosphoric acid.

These results correspond to the formula 3AgO, 2P^ O^
+ HO. Berzelius examined this salt long ago, but he ob-
tained it in a different manner*. He precipitated a solu-
tion of phosphoric acid which had been recently heated to
redness, with nitrate of silver; he then obtained a precipitate,
the composition of which but little resembled that of AgO
P^ O^, the amount of oxide of silver differing nearly 3 per cent,
from that calculated. This salt was placed in boiling water;
in a few minutes it fused into a viscid mass resembling tur
pentine, and had the same composition as the compound men-
tioned above.

The composition found above explains the acid reaction of
the liquid, when the neutral solution of Graham's salt is pre-
cipitated with solution of nitrate of silver. It also renders
evident why the acid separated from the silver salt may possess
different properties from metaphosphoric acid as obtained by
the combustion of phosphorus. The question, whether the
other oleaginous salts, which a solution of Graham's salt forms
with other neutral salts, possess an analogous composition to
* PoggendorfTs Annalen, xix. p. 333.

S26 Prof. H. Rose on the Isomeric Modifications

the silver salt, as might be expected from the circumstance
that on their formation the solutions which are originally
neutral acquire an acid reaction, well deserves investigation.

A salt of silver having an analogous composition can be
prepared from Graham's salt. M. Fleitmann decomposed
Graham's salt with nitrate of silver, filtered the precipitate
immediately, washed it slightly with cold water, and then
pressed it strongly between blotting-paper. The water used
in the washing possessed a barely acid reaction. The fused
compound gave the following per-centage composition :—

Phosphoric acid . . . 37*62
Oxide of silver . . . 61-18

98-80

which closely approximates the composition calculated ac-
cording to the formula AgO, P^ O^, which requires 61-89 oxide
of silver and 38 -11 phosphoric acid. The loss probably arises
from its containing a little soda, which was not removed in
consequence of the rapidity with which the compound was
washed. It is thus evident, that when the silver salt is preci-
pitated and separated as quickly as possible from the liquid,
it corresponds in composition with Graham's salt; but that,
on prolonged contact with the liquid, even when cold, it be-
comes decomposed and loses acid.

Whether anything similar occurs in the case of the other
precipitates has not been determined ; but it is not improbable,
as at first they are precipitated in a pulverulent form, and only
acquire the oily appearance when powerfully shaken, by which
they probably lose a portion of the phosphoric acid.

2. The acid existing in those remarkable salts which Fleit-
mann and Henneberg obtained from the acid phosphate of
soda, or rather from the microcosmic salt, by fusion and very
gradual cooling*, may be regarded as a second submodifica-
tion of metaphosphoric acid. This salt has exactly the same
composition as the metaphosphates; it contains the same
number of atoms of base and acid, and has therefore the com-
position of Graham's salt ; but differs from it in its opacity and
crystalline structure, whilst Graham's salt is transparent and
amorphous. It crystallizes from its solution with 45 atoms of
water, whilst Graham's salt cannot be made to crystallize. Its
solution, like that of Graham's salt, exerts a neutral reaction.

The most remarkable property of the acid of this salt is,

that it forms with all bases compounds soluble in water, in

which respect it differs essentially from all the modifications

of phosphoric acid. The salts, even the silver salt, can be

* Chem. Gaz., vol. vi. p. 289.

of Phosphoric. Acid, 327

obtained in a crystalline state. According to Fleitmann, the
solution of the soda salt produces no precipitates with solu-
tions of nitrate of silver, nitrate of lead, chloride of barium,
chloride of calcium, chloride of strontium, sulphate of mag-
nesia, protosulphate of manganese, protosulphate of iron, the
sulphates of zinc, cobalt or nickel. In the solutions of the
proto- and pernitrates of mercury, the solution of the soda
salt at first produces no turbidity, but after a considerable
time a precipitate is formed. It also produces a precipitate
in a solution of acetate of lead.

The solution of the soda salt, as also that of Graham's salt,
produces no precipitate with albumen i but this is the case
when acetic acid is added. According to Fleitmann, the acid
may be readily isolated from the solution of the crystallized
silver salt by a current of sulphuretted hydrogen. The solu-
tion of the free acid immediately produces a copious precipi-
tate with albumen. When saturated with carbonate of soda,
the original soda salt is again obtained ; and if its solution,
after neutralization with ammonia, is treated with nitrate of
silver, the crystallized silver salt can be procured from the
solution.

3. The acid contained in those salts which were formerly
called acid phosphates and have long been known, may be
regarded as forming the third modification of metaphosphoric
acid. These salts have recently been examined by Maddreli*.
In obtaining them, Maddreli made use of salts of the most
different kinds — metallic chlorides, sulphates, nitrates, car-
bonates and chlorates, which were heated to + 600° F.
with free phosphoric acid. I shall presently show, that an
insoluble pyrophosphate, possessing similar properties to those
presented by Maddrell's metaphosphates, may be simulta-
neously formed. The heating with free phosphoric acid must
therefore be continued until a portion of the heated mass,
when removed, is found to precipitate a solution of albumen.

The insoluble metaphosphates dissolve when heated with
concentrated sulphuric acid. According to Fleitmann, the
acid cannot be isolated, or at least only very imperfectly, by
transmitting a current of sulphuretted hydrogen through the
copper salt suspended in water. The decomposition of it is
best effected by treating the above salt with sulphuret of
sodium. A soluble soda salt is then obtained, which some-
what resembles Fleitmann 's metaphosphate of soda, but differs
from it in many respects, and can only be procured with half
its water of crystallization.

* Phil. Mag., S. 3, vol. xxx. p. 322.

328 Prof. H. Rose on the Isomeric Modifications

The various sub-modifications of metaphosphoric acid all
agree in having the same capacity of saturation ; one atom of
acid saturates one atom of a powerful base. Graham supposes
that the different capacity of saturation of the various modifi-
cations of phosphoric acid is the cause of their different reac-
tions. This difference in the capacity of saturation of the
various phosphoric acids, however, is indisputably a result of
their isomeric state, and, as I have remarked on a former
occasion, cannot be the cause of them*.

A second general property of all the varieties of metaphos-
phoric acid is, that their aqueous solutions precipitate a solu-
tion of albumen. This is almost the only property by which
the various kinds of metaphosphoric acid can be recognised in
qualitative examinations, and unequivocally distinguished from
the other modifications of phosphoric acid ; for neither pyro-
phosphoric acid nor the ordinary c-acid precipitates albumen.
The soluble salts of metaphosphoric acid do not precipitate
albumen until acetic acid has been added to their solutions.

The property of metaphosphoric acid to produce a copious
precipitate in the solution of chloride of barium is especially
peculiar to the acid produced by the combustion of phosphorus
only.

When a concentrated solution of c-phosphoric acid is heated
very gently for several hours, so that none of it volatilizes, an
acid is obtained, the aqueous solution of which does not pro-
duce any precipitate with albumen; nor does it produce a
precipitate with chloride of barium, or merely an inconsider-
able troubling after a long time. Nitrate of silver however
produces a white precipitate. These are the properties of
pyrophosphoric acid. When the same acid is heated in a
platinum crucible longer and more strongly, so that it com-
mences to be copiously volatilized, its aqueous solution then
immediately produces a copious precipitate with albumen and
chloride of barium, and a white precipitate with nitrate of
silver, which when shaken becomes resinous. Thus by strongly
heating it, metaphosphoric acid is formed, and apparently the
same modification as that obtained by the combustion of phos-
phorus.

. By the rapid application of a certain degree of heat, how-
ever, an acid can be obtained, the aqueous solution of which
affords with albumen a copious piecipitate, but none with chlo-
ride of barium, andwhich, after saturation with ammonia, yields
a white precipitate with nitrate of silver, in which, after some
time, an admixture of yellow can be distinctly perceived. In
this case, the same acid as that which 1 separated from the
* Chem. Gaz. vol. vi. p. 383.

of Phosphoric Acid. 329

metaphosphate of silver appeared to have been formed, in ad-
mixture with a Uttle undecomposed f-phosphoric acid.

Some uncertainty still continues regarding the composition
of fused phosphoric acid. A very long time since I made
several experiments upon this point*, and found that the acid,
fused for a considerable time over a spirit-lamp, in three ex-
periments, contained a slightly less amount of water than is
required by the compound P^O^+HO. In another experi-
ment, probably with some acid which had been heated more
strongly, and for a longer time, the amount of water was still
less, and nearly corresponded to the compound 3P^O^ + 5
^p2Q5_j_ piO) : hence it is thus rendered probable that phos-
phoric acid vk^ould be obtained in a perfectly anhydrous statf
by a very long and continuous application of heat.
' My experiments have been recently confirmed by M.
Weber, who examined an acid which had been exposed for a
considerable time to a temperature at which it began to be
slightly volatilized; 3*127 grms. of this acid, when treated
with 16*891 grms. of oxide of lead, left, after having been
heated to redness, a residue of 19*700 grms. The per-centage
composition of the acid was therefore —

Phosphoric acid .... 89'84<

Water 10*16

100*00
In this case also the quantity of water is slightly less than
the composition P'^0^ + HO requires. The quantities of oxygen
are in the proportion of 50*34; : 9*03.

In addition to these three submodifications of metaphos-
phoric acid, there are undoubtedly others. That acid which
is formed on burning phosphorus in dry atmospheric air or
oxygen gas, may be considered as a fourth submodification, for,
as has been stated above, the reactions of its solution are dif-
ferent from those of the other modifications. The salts which
it forms with bases have not been prepared and examined. I
shall merely remark here, that anhydrous phosphoric acid does
not exhibit any affinity towards dry ammoniacal gas, nor does
it absorb it; hence it differs in this respect from anhydrous sul-
phuric acid. Probably the various submodifications of me-
taphosphoric acid should be considered as conjugate acids,
as the difference in their reactions would then be more satis-
factorily explained. Anhydrous phosphoric acid may consti-
tute the conjunct, which is capable of combining in different
proportions with pyrophosphoric acid or with c-phosphoric
* PoggendorfF's Annden, vol. viii. p. 203.

330 Prof. H. Rose on the Isomeric Modifications

acid, giving rise to the numerous modifications of metaphos-
phoric acid. This conjunct per se probably alone possesses
the property of precipitating albumen, and thus this property
is communicated to all the varieties of metaphosphoric acid.

b-Phosphoric Acid {Pp-ophosphoric Acid).

At least two submodifications of this modification of phos-
phoric acid must also be admitted ; for there are two different
kinds of pyrophosphates. One of these consists of the well-
known pyrophosphate of soda, which is obtained by heating
the c-phosphate of soda, 2NaO, P^O^ HO, to redness, and
the salts which are formed from this soda salt by decomposi-
tion. The second variety is produced in the same manner as
Maddrell's insoluble metaphosphates, that is, by heating the
salts with excess of phosphoric acid, the heat not being so
great as to cause the production of metaphosphates. Thus
by treating nitrate of copper with phosphoric acid, a salt of
copper is formed which resembles the insoluble metaphospliate
of copper, especially as regards insolubility. But the acid
existing in it may be readily isolated by a current of sulphu-
retted hydrogen, and its aqueous solution possesses the same
properties as the solution of ordinary pyrophosphoric acid.
As this modification of the pyrophosphates has not been suffi-
ciently examined, a more detailed notice of it cannot now be
given.

As is well known, the pyrophosphates are formed when the
c-phosphates, which contain two atoms of a fixed and one atom
of a volatile base (oxide of ammonium or water), are heated
to redness. The usual process is the conversion of the ordi-
nary phosphate of soda (2NaO, P^O^HO) into pyrophosphate
of soda {2NaO, P^O^).

Graham ascribes the difference in the properties of the py-
rophosphates from the phosphates, to the difference in the
capacity of saturation of the two acids contained in the two
kinds of salts. It cannot be denied that pyrophosphoric
acid especially saturates two atoms of a base, and thus differs
characteristically from the c-phosphoric acid, which requires
three for its saturation. But I have remarked above that
this is a consequence of the isomerism of the two acids; hence
it must appear as not improbable, that an atom of water may
be expelled from the ordinary phosphate of soda without con-
verting it into a pyrophosphate.

Some experiments which were made with this point in
view, have not however yielded a favourable result. The ordi-
nary phosphate of soda was exposed to a very gentle heat, so
that it still contained more than one atom of water ; 3*0635

of Phosphoric Acid. 331

grms. of it, when heated to redness, gave 2*7900 grms. of pyro-
phosphate of soda. This phosphate of soda, therefore, still
contained 0*2735 grm., or 8*92 per cent, of water. 3*126 grms.
of the same salt were exposed for a considerable time to de-
finite temperatures ; it was found that by gradually heating the
salt, almost the whole of the water it contained could be ex-
pelled at a temperature of 464'° F. The following are the de-
tails of this experiment : —

The above quantity of the salt weighed after exposure to a
heat of

320° F. during 1 1 hours 3*054' grms.

320

6

3*053

320

8

3*043

446

4

3*015

446

6

2-967

464

2

2*920

464

2

2*894

464

2

2-887

464

2

2*883

Had the salt been kept for a still longer time at a temperature
of 464° F., it would undoubtedly have lost the whole of the
water it contained. The quantity used would then have
weighed 2846 grms.

But when the salt was examined, it was found that even at
the above temperature it had become almost completely con-
verted into pyrophosphate of soda. The solution gave a white
precipitate with nitrate of silver, which was mixed with as
much of a yellow one as would have been expected from the
quantity of water still existing in the heated salt.

Solution of pyrophosphate of soda yields, with very many
neutral salts of metallic oxides, precipitates which are partly
soluble in excess of the pyrophosphate of soda. The peculia-
rity of the pyrophosphate of soda in readily formingdouble salts,
has already been very distinctly pointed out by Stromeyer.
Persoz has recently studied this point, without alluding to Stro-
meyer's memoir; he has however confirmed all the facts given
by him. Schwarzenberg has recently examined most of the
pyrophosphates quantitatively*, and Baer has made the inter-
esting discovery, that those insoluble precipitates which are
produced by a solution of pyrophosphate of soda, and are not
soluble in excess of it, are frequently insoluble double salts
of the soda salt with the pyrophosphates formed, in which
the soda replaced the other base, apparently without the
two existing in the double salt in a definite simple propor-

* Chem. Gaz., vol. vi. pp. 181, 190.

332 Prof. H. Rose on the Isomeric Modifications

tion*. Even the silver salt contains some, although a small
quantity of soda. Persoz and Fleitmann have also procured
and examined insoluble double salts of the pyrophosphate of
soda with the pyrophosphate of copper.

The following are the special reactions of a solution of
pyrophosphate of soda with salts of the metallic oxides : —

Solution o^ chloride of barium produces a precipitate, which
is insoluble in excess of the soda salt; at least the filtered
liquid either yields no precipitate, or at most a very slight
troubling with dilute sulphuric acid.

Solution o^ chloride of calcium produces a precipitate, which
is soluble in a very large quantity of the pyrophosphate of soda.
The clear liquid however becomes spontaneously turbid when
set aside, and in 24 hours a very slight precipitate only of
oxalate of lime is caused by a solution of oxalate of potash.

Solution of sulphate of magnesia produces a precipitate,
which is soluble in excess of the pyrophosphate of soda ; but
on ebullition a copious precipitate is produced in this solution,
which does not disappear as the liquid cools. Ammonia
does not cause a precipitate in a solution of the pyrophosphate
of magnesia in pyrophosphate of soda, even when set aside for
a long time. The precipitate of the pyrophosphate of mag-
nesia is also readily soluble in excess of sulphate of magnesia.
Ebullition causes a precipitate in this solution, which does not
disappear on cooling.

Solution o^ nitrate of silver produces the well-known white
precipitate. It is not wholly insoluble in a very large excess of
the pyrophosphate of soda. The supernatant liquid does not
affect litmus paper, and only renders it blue when excess of
the soda salt has been added.

Sol ution of pernitrate of mercury , although it necessarily
contains free nitric acid, produces a copious white precipitate,
which becomes basic and of a reddish-yellow colour on the
addition of excess of the pyrophosphate of soda.

Solution of the protonitrate of mercury produces a white
precipitate, which is soluble in excess of the pyrophosphate of
soda. In this solution, ammonia causes a blackish gray, sul-
phuret of ammonium a black, and hydrochloric acid a white
precipitate ; the latter consists of chloride of mercury.

Solution oi bichloride of mercury does not immediately pro-
duce any precipitate. After a considerable time, a dense red
precipitate is formed, which is still more rapidly thrown down
when heat is applied.

Solution oi sulphate of copper produces a bluish-white pre-
cipitate, which is readily soluble in excess of the pyro-
* PoggendoriTs Anncden, vol. Ixxv. p. 152.

of Phosphoric Acid. 333

phosphate of soda. The solution is of a blue colour ; on the
addition of ammonia it becomes of a darker blue, sulphuret of
ammonium immediately produces in it a brown precipitate of
sulphuret of copper. The pyrophosphate of copper is also
soluble in a very large excess of sulphate of copper. Heat
produces a precipitate in the solution which does not dis-
appear on cooling.

Solution of acetate of lead causes a white gelatinous preci-
pitate, readily soluble in the pyrophosphate of soda. Sulphuret
of lead is immediately thrown down from this solution by
sulphuretted hydrogen.

Solution of protosulphate of manganese produces a white
precipitate, which is insoluble in excess of the protosalt of
manganese, but is soluble in the pyrophosphate of soda. This
solution is not troubled by ammonia ; sulphuret of ammonium
does not produce a precipitate of sulphuret of manganese in
it (which certainly deserves special notice), even when set
aside for a considerable time.

Solution of protosulphate of iron produces a white preci-
pitate, soluble in excess of the pyrophosphate of soda. A
black precipitate of sulphuret of iron is immediately formed
in the solution on the addition of sulphuret of ammonium ; the
solution is not however rendered turbid by ammonia; it
merely renders it of a dark colour. The pyrophosphate of
iron is also soluble in excess of the solution of the proto-
sulphate of iron.

Solution of perchloride of iron causes a white precipitate,
readily soluble in excess of the pyrophosphate of soda. Sul-
phuret of ammonium immediately produces in the almost
colourless solution, a black precipitate of sulphuret of iron,
which deserves particular notice, because Persoz denies the
production of sulphuret of iron in the solution by sulphuret of
ammonium. Ammonia however does not render the solution
turbid ; it immediately turns it of a blood-red colour.

Solution of sulphate of zinc produces a white precipitate,
soluble in the pyrophosphate of soda. The solution is neither
precipitated by ammonia nor by boiling, but sulphuret of am-
monium throws down sulphuret of zinc. The precipitate is
also soluble in excess of the solution of sulphate of zinc.
On boiling, the solution becomes turbid ; the turbidity does
not disappear on cooling.

Solution of sulphate of cadmium produces a precipitate
which is soluble in pyrophosphate of soda. The solution be-
comes turbid when heated, the turbidity not disappearing as
the solution cools. Sulphuret of cadmium is immediately
precipitated from it by sulphuret of ammonium.

884 Prof. H. Rose on the Isomeric Modifications

In a solution oi sulphate of nickel a greenish- white preci-
pitate is produced, which is readily soluble in the soda salt.
The solution is not rendered turbid by heat. With chloride
of nickel the same reaction occurs, except that the solution of
the precipitate in excess of the soda salt is rendered turbid
when heated and does not become clear on cooling, Sul-
phuret of nickel is immediately thrown down from this solution
by sulphuret of ammonium.

Solution of sulphate of cobalt is precipitated of a pale red
colour; the precipitate is readily soluble in the soda salt.
The solution is red, and when heated becomes perfectly blue,
but not turbid ; on cooling it reacquires the red colour. Sul-
phuret of ammonium immediately causes the formation of sul-
phuret of cobalt in it.

Solution of alum produces a white precipitate, soluble in
the soda salt. No precipitate is caused in this solution either
by ammonia or sulphuret of ammonium. But the preci-
pitate is soluble in excess of the solution of alum.

Solution of nitrate of bismuth, although it contains free
acid, produces a white precipitate, soluble in the soda salt.
On applying heat a precipitate is formed. Sulphuret of am-
monium produces sulphuret of bismuth in it.

No precipitate is caused in a dilute filtered solution o^ albu-
men by solution of pyrophosphate of soda, even after the addi-
tion of acetic acid.

As is well known, aqueous solution of pyrophosphoric acid
is best obtained by decomposing the pyrophosphate of lead,
suspended in water, by a current of sulphuretted hydrogen.
In this manner the conversion of the pyrophosphoric acid into
the c-phosphoric acid is avoided, which, as we know, ensues
after the lapse of some time even by repose, but much more ra-
pidly when the acid is heated. If however the solution of pyro-
phosphoric acid has been saturated or supersaturated by a
strong base, it may be keptin it without undergoing any change.
Neither by ebullition nor by long repose is the ^-phosphoric
acid converted into the c, and a solution of the pyrophosphate
of soda may be preserved for many years without change.
With excess of alkali, the solution of pyrophosphoric acid is
not converted into the c-phosphoric acid, until the mass, after
evaporation to dryness, has been completely fused. But even
in this case, according to Weber, the entire conversion into
the c-phosphoric acid does not take place, until the pyro-
phosphate has been completely decomposed by fusion with
excess of alkali, especially an alkaline carbonate. As this does
not occur completely on fusing pyrophosphate of lime with
excess of an alkaline carbonate, the ^-phosphoric acid of the

of Phosphoric Acid. 335

undecomposed portion of the lime-salt does not therefore be-
come converted into the c-phosphoric acid. With the appli-
cation of a high temperature, the conversion can be better
effected in this manner in the case of the pyrophosphate of
strontia, and still better of the pyrophosphate of baryta;
pyrophosphate of magnesia may be completely decomposed,
and the pyrophosphoric acid contained in it completely con-
verted into c-phosphoric acid, even by fusion over a spirit-
lamp with a mixture of carbonate of potash and carbonate of
soda in atomic proportions.

It is well known that the conversion of pyrophosphoric acid
is perfectly effected by acids, especially when it is heated with
them. The stronger the acid, the more completely the con-
version is effected ; it succeeds best when concentrated sul-
phuric acid is used.

An aqueous solution of pyrophosphoric acid, immediately
after its preparation, exhibits the following reactions : —

Solution oi chloride of barium produces no precipitate; after
a long time a very inconsiderable troubling occurs. A pre-
cipitate is produced in the liquid by ammonia.

Solution of chloride of calcium produces no precipitate, even
after long repose. Ammonia causes a precipitate, although it
is not very copious. Lime-water immediately causes a pre-
cipitate, even when the solution is slightly acid ; if however
excess of pyrophosphoric acid is added, the precipitate is re-
dissolved ; ammonia does not then cause a precipitate in this
solution.

If chloride of ammonium be added to pyrophosphoric acid,
and it then be supersaturated with ammonia, solution of sul-
phate of magnesia produces a precipitate, which is however
soluble in a very large quantity of water. If a solution of
c-phosphate of soda be added to this liquid, a precipitate is im-
mediately thrown down.

Solution of nitrate of silver usually causes no precipitate.
On saturation with ammonia, a white precipitate is produced,
which, if the solution of pyrophosphoric acid has stood for
some time, is of a yellowish tint. Solution of acetate of silver
produces a white precipitate, which is soluble in a large quan-
tity of pyrophosphoric acid.

A dilute filtered solution of albumen causes no precipitate.
It is a remarkable fact, that even in the last edition of
'Qev'/.eXms^sLehrbiichy he ascribes to pyrophosphoric acid the
property of precipitating albumen, and thus distinguishes
it from the c-phosphoric acid. Also, according to his state-
ment, solutions of the pyrophosphates after the addition of
acetic acid precipitate albumen. It is however a very cha-

336 Prof. H. Rose on the Isomeric Modifications

racteristic property of pyrophosplioric acid, that it does not
precipitate albumen, and by this very property it differs most
essentially from metaphosphoric acid, all the submodifications
of which possess the property of precipitating albumen in a
remarkable degree.

I have already remarked above, that the aqueous solution
of pyrophosphoric acid obtained from the insoluble pyrophos-
phate of copper by sulphuretted hydrogen gas, exhibits the
same reactions as the acid obtained from the lead salt.

Pyrophosphoric acid differs from metaphosphoric acid, in
addition to its characteristic reaction with albumen, also in
that with a solution of chloride of barium, although, as I have
remarked above, all the modifications of metaphosphoric acid
are not alike in this respect ; also in the difllerent properties
of the precipitate produced by a solution of silver, regarding
which, it must be observed, that one of the modifications of
metaphosphoric acid (the submodification described at p. 326)
forms a soluble salt with oxide of silver. The difference be-
tween the reaction of metaphosphoric acid and pyrophospho-
ric acid with albumen therefore forms the most important
distinction between these two acids.

c-Phosphoric Acid [ordinart/ Phosphoric Acid).

This modification of phosphoric acid is the one most com-
monly occurring in analytical investigations, because the other
modifications are converted into it by the action of acids.

Its salts have been so frequently examined, that most of
their properties are known. One property however, which
especially characterizes the c-phosphates, appears to have been
hitherto overlooked. This consists in the solubility of a large
number of the insoluble phosphates in excess of the saline so-
lution from which they have been precipitated by means of
the phosphate of soda. The solution generally possesses
the property of producing a copious precipitate when heat
is applied, which disappears however on cooling. Hence
double salts are formed, which are decomposed by a high
temperature. Many of the precipitates thrown down by
pyrophosphate of soda, frequently dissolve, as has already
been mentioned, even in excess of the solution of the salt ;
this solution is also troubled by heat, but the precipitate is
not dissolved on cooling. As in chemical investigations the
c-phosphate of soda is so frequently used to precipitate oxides
fron\ the solutions of their salts, and as it appeared to me im-
portant to be accurately acquainted with the properties of the
precipitates, I shall offer no excuse for stating here the re-

of Phosphoric Acid. 337

actions of the most important salts of the metallic oxides with
a solution of c-phosphate of soda.

Sohition of chloride of barium produces a copious precipi-
tate, which is neither sohible in excess of the phosphate of
soda nor of the chloride of barium.

Solution of chloride of calcium reacts in the same manner.
Traces of the precipitate are soluble in excess of the chloride
of calcium, and may be precipitated from the filtered solution
by ammonia.

Its reaction with a solution o^ sulphate of magnesia is known
generally, but not perfectly in detail. The sulphate produces
a precipitate in a solution of the phosphate of soda, which is
insoluble in the latter, but soluble in excess of the solution of
sulphate of magnesia. If this clear solution be treated with
ammonia, a copious precipitate falls, part of which consists of
hydrate of magnesia and is soluble in chloride of ammonium;
another portion, which is composed of the phosphate of mag-
nesia and ammonia, is insoluble in it. The clear solution of
the phosphate of magnesia in the sulphate of magnesia, when
boiled, yields a copious precipitate, which however com-
pletely disappears as the liquid cools, reappearing if the ebul-
lition be repeated. If however this experiment be repeated
many times, the precipitate thrown down on ebullition at last
ceases to disappear entirely on cooling.

The precipitate produced by nitrate of silver is insoluble
both in excess of the phosphate of soda and in the salt of
silver.

Solution of pernitrate of mercury produces a white precipi-
tate, which is not insoluble in excess of the solution of the
mercurial salt; but as this always contains free acid, the solu-
bility of the precipitate may arise from this.

Solution of the yrotonitrate of mercury causes a white pre-
cipitate, insoluble in excess of the mercurial solution.

Solution of the bichloride of mercury at first produces no
change. After standing for a long period, a slight red de-
posit subsides, which is produced sooner and in greater abun-
dance by heat. The reaction is the same as with the pyro-
phosphate of soda.

Solution of sulphate of copper produces a bluish-white pre-
cipitate, soluble in a large quantity of the cupreous solution.
A copious precipitate is produced by heat in the clear solu-
tion, which completely disappears on cooling.

Solution of ihe protosulphate of manganese produces a white
precipitate, which is only soluble in a very large excess of the
solution of manganese. A precipitate is caused in this solu-
tion by ebullition, which completely disappears on cooling.
Phil. Mag. S. 3. Vol. 34.. No. 230. May 1849. Z

338 Prof. H. Rose on the Isomeric Modifications

Solution o^ protosulphate of iron produces a white precipi-
tate, which is readily soluble in excess of the solution of the
protosalt. A copious precipitate is thrown down by heat,
and does not completely disappear on cooling.

In a solution of the perchloride of iron a white precipitate
is formed, which is readily soluble in excess of the solution of
the perchloride.

Solution of the sulphate of zinc produces a white precipi-
tate, which is readily soluble in excess of the solution of zinc.
When the solution is heated, a troubling ensues; it is but
inconsiderable, and does not completely disappear on cooling.

Solution of sulphate of cadmium causes a white precipitate,
readily soluble in excess of the solution of cadmium. The
solution yields a copious precipitate when heated, but this
completely disappears on cooling.

Solution of chloride of nickel yields a greenish-white pre-
cipitate, soluble in excess of the solution of nickel. The solu-
tion, which yields a precipitate when boiled, becomes per-
fectly clear on cooling.

Solution o( the sidphate of cobalt produces a blue precipi-
tate, soluble in excess of the solution of cobalt. The solution
is red. On ebullition, a red precipitate is produced in it,
which completely dissolves on cooling.

Solution of alum gives a white precipitate, soluble in a con-
siderable excess of the solution of alum. When heated, the
solution yields a copious precipitate, which partly, but not
entirely, disappears on cooling.

Solution of nitrate of bismuth gives a white precipitate, in-
soluble in excess of the solution of bismuth.

Phosphate of soda does not cause a precipitate in a dilute
filtered solution of albumen ^ even when acetic acid is added.

The aqueous solution of the c-phosphoric acid differs from
pyrophosphoric and metaphosphoric acids, as is well known,
by its reaction with a solution of silver.

Solution of chloride of barium produces only an inconsider-
able turbidness; but on the addition of ammonia a copious
precipitate is immediately formed.

Barytic water causes a precipitate, even when the liquid is
acid. The phosphoric acid is not completely separated by
carbonate of baryta in the cold. The liquid, filtered at the
end of several days, still yields a precipitate on the addition
of sulphuric acid.

Solution of chloride of calcium gives no precipitate, even
after standing for a considerable time ; but ammonia imme-
diately causes a copious precipitate. Lime-water produces a
precipitate, even when the liquid is somewhat acid.

of Phosphoric Acid. 339

A dilute filtered solution of albumen, as we know, gives no
precipitate with the sokition of phosphoric acid.

A short time since we were made acquainted, by Svanberg
and Struve, with an excellent reagent for phosphoric acid* in
the molyhdate of ammonia. This is so delicate in the detec-
tion of the smallest trace of phosphoric acid, and is capable
of showing its presence even in those compounds in which the
acid is discovered with difficulty or cannot be so at all, that
an important service has been rendered to analytical che-
mistry by the recommendation of this reagent.

If a solution of the molybdate of ammonia be added to a so-
lution of any phosphate, and then so much muriatic, or what
is better nitric acid, that the precipitate which is formed at
first disappears again, the liquid immediately becomes yellow,
and deposits, even when the smallest quantity of phosphoric
acid is present, a yellow precipitate, which consists of molybdic
acid, but which is a different modification, and possesses dif-
ferent properties from the molybdic acid which is obtained
when phosphoric acid is not present. If the phosphoric
compound to be examined is insoluble in water, it is used
in solution in acids, especially nitric acid. The precipita-
tion is accelerated by heat. The yellow precipitate is soluble
in ammonia, as also in an excess of the phosphate. Hence
only very small quantities of phosphoric acid are most easily
detected in this manner; and it is quite possible that a larger
quantity might be overlooked, because in this case a very
large quantity of the molybdate is requisite to produce the
precipitate after saturation with nitric acid.

The yellow precipitate can be readily recognised, even when
it is precipitated from a coloured liquid, as from a nitric solu-
tion of phosphate of copper, or from acid solutions of other
coloured phosphates.

It must however be remarked, that c-phosphoric acid and
its salts only are able to produce this reaction. The other
modifications of phosphoric acid do not give the yellow preci-
pitate with the molybdate of ammonia, unless they have been
converted into the c-phosphoric acid by the nitric acid added.
As is well known, this often takes place slowly and incom-
pletely in the cold. Hence the pyrophosphate of soda may
be allowed to remain for a very long time in dilute solutions
with the molybdate of ammonia and free nitric acid, without
any effect being perceptible. But if the whole be made to
boil, a yellow liquid is instantly produced, and soon after-
wards a yellow precipitate.

* Phil. Mag., S. 3, vol. xxiii. p. 524.
Z2

S¥) Sir W. Rowan Hamilton on Quaternions.

Experiment to separate Phosphoric from Pijrophosphoric Acid.

The different reaction of the phosphate and pyrophosphate
of soda towards a solution of sulphate of magnesia and am-
monia, led me to hope that it might form the basis of a method
of separating these two modifications of phosphoric acid.

When pyrophosphate of soda is dissolved in a large amount
of water, and the solution is mixed with a very large quantity
of chloride of ammonium, no precipitate is produced on the
addition of sulphate of magnesia and solution of ammonia.
But at the end of a considerable period a precipitate falls, and
is deposited firmly upon the sides of the vessel. If, however,
the quantity of chloride of ammonium is very considerable, it
frequently does not appear for several days.

1 '828 grm. of hydrated c-phosphate of soda, which had lost
a small quantity of its water of crystallization by efflorescence,
was dissolved in water with 1*521 grm. of the same salt, which
had been previously heaied to redness and furnished 0"611
grm. of pyrophosphate of soda. The solution was mixed with
100 grms. of chloride of ammonium, then diluted with 1600
grms. of water, and sulphate of magnesia and solution of am-
monia added. The precipitate was filtered ofFafter an interval
of two hours, then washed, first with water containing chloride
of ammonium, afterwards with water containing ammonia.
0-814- grm. of calcined phosphate of magnesia was obtained,
which contains 0*516 grm. of phosphoric acid. But the 1*828
grm. of phosphate of soda contains only 0*391 grm. of phos-
phoric acid : hence a considerable amount of pyrophosphoric
acid was precipitated with the phosphate of anmionia and mag-
nesia. This method of separation is consequently inapplicable.

XLVII. On Qjiiaternions ; or on a New System of ImagiJiaries
in Algebra. By Sir William Rowan Hamilton, LL.D.,
M.R.I. A. y F.R.A.S.y Corresponditig Member of the Insti-
tute of France, S^c, Andrews' Professor of Astronomy in the
University of Dublin, and Royal Astronomer of Ireland.

[Continued from p. 297-]
68. ^T^HE equation of the ellipsoid (see Philosophical
JL Magazine for October 1847, or Proceedings of the
Royal Irish Academy for July 1846),

T(»p-[-px)=x2-i^, eq. (9.), art. 38,

which has so often presented itself in these researches, maybe
anew transformed as follows. Writing it thus,

^(^+H(pO^^^(,„,)^ . . . (125.)

Sir W. Rowan Hamilton on Qualernions. 341

which we are allowed to do, because the tensor of a product
is equal to the product of the tensors, we may observe that
while the denominator of the fraction in the first member is a
pure scalar, the numerator is a pure vector ; for the identity,

i/> + l?x=S.(» + x)|5 + V.(«-x)p, . . . (126.)
gives

S.(ip+p)(»-x)=0: .... (127.)

the fraction itself is therefore a pure vector, and the sign T,
of the operation of taking the tensor of a quaternion, may be
changed to the sign TV, of the generally distinct but in this
case equivalent operation, of taking the tensor of the vector
part. But, under the sign V, we may reverse the order of
any odd number of vector factors (see article 20 in the Philo-
sophical Magazine for July IS'tS); and thus may change, in
the numerator of the fraction in (125.), the partial product
<p(i — x) to (i — x)pj. Again, it is always allowed to divide
(though not, generally, in this calculus, to multiply) both the
numerator and denominator of a quaternion fraction, by any
common quaternion, or by any common vector; that is, to
multiply both numerator and denominator into the reciprocal
of such common quaternion or vector: namely by writing the
symbol of this new factor to the right (but not generally to the
left) of both the symbols of numerator and denominator, above
and below the fractional bar. Dividing therefore thus above
and below by i, or multiplying into »~^, after that permitted
transposition of factors which was just now specified, and
after the change of T to TV, we find that the equation (125.)
of the ellipsoid assumes the following form:

^y(-.V + f(x-xV')^^ . . (128.)

(< — x) + (x — x^» *) ^ '

the new denominator first presenting itself under the form
>?r^ — i, but being changed for greater symmetry to that
written in (128.), which it is allowed to do, because, under the
sign T, or under the sign TV, we may multiply by negative
unity.

69. In the last equation of the ellipsoid, since

X — x^<~*=x(» — x)«~^,
we have

T(x-x2i-0 = TxT(i-x)Ti-»; . . (129.)

and under the characteristic U, of the operation of taking the
versor of a quaternion, we may multiply by any positive
scalar, such as —x,~^ is, because x^ and x"^ are negative*

• By this, which is one of the earliest and most fundamental principles
of the whole quaternion theory (see the author's letter to John T. Graves,

342 Sir W. Rowan Hamilton on Quaternions.

scalars ; whereas to multiply by a negative scalar, under the
same sign U, is equivalent to multiplying the versor itself by
— 1 : hence,

U(x-x2*-0 = -U(jc2i-i-x)=-U(x-i-i-i). . (130.)

If then we introduce two new fixed vectors, rj and 6, defined
by the equations,

>, = TiU(.-Jt); 9 = TxU(x-»-i-»); . . (131.)
and if we remember that any quaternion is equal to the pro-
duct of its own tensor and versor (Phil, Mag. for July 1846);
we shall obtain the transformations,

.-jc=>jT-^^; x-xV»=-flT-^— ?^; . (132.)

which will change the equation of the ellipsoid (128.) to the
following :

TV^?^^=T(»-x) (133.)

70. To complete the elimination of the two old fixed vec-
tors, », X, and the introduction, in their stead, of the two new
fixed vectors, )3, 9, we may observe that the two equations
(132.) give, by addition,

,-x2,-»=(>;-9)T'-^; .... (134.,)

taking then the tensors of both members, dividing by T >

and attending to the expression (81.) in article 56, (Phil. Mag.
for May 1848,) for the mean semiaxis h of the ellipsoid, we
find this new expression for that semiaxis :

^^"^-^^^w^r^ (^^^•)

Esq., of October 17th, 1843, printed in the Supplementary Number of the
Philosophical Magazine for December 1844), namely by the principle that
the square of every vector (or directed straight line in tridimensional
space) is to to be regarded as a negative number, this theory is not merely
distinguished from, but sharply contrasted with, every other system of alge-
braic geometry of which the present writer has hitherto acquired any know-
ledge, or received any intimation. In saying this, he hopes that he will not
be supposed to desire to depreciate the labours of any other past or present
inquirer into the properties of that important and precious Symbol in Geo-
metry, V — 1. And he gladly takes occasion to repeat the expression of
his sense of the assistance which he received, in the progress of his own
speculations, from the study of Mr. Warren's work, before he was able to
examine any of those earlier essays referred to in Dr. Peacock's Report:
however distinct, and even contrasted, on several fundamental points, may
be (as was above observed) the methods of the Calculus of Quaternions
from those of what Professor De Morgan has happily named Double Al-

eSBRA.

Sir W. Rowan Hamilton on Quaternions. 34-3

But also, by (131.), or by (132.),

T>i = T.; T6=Tx; .... (136.)
and therefore,

92_>j2 = x2-l2 (137.)

Hence, by (135.), we obtain the expression,

T(.-x)=^^; .... (138.)

which may be substituted for the second member of the equa-
tion (133.), so as to complete the required elimination of i and
X. And if we then multiply on both sides by T()j— d), we
obtain this new form* of the equation of the ellipsoid:

'r^^=*'-'" • • • • ('«»•)

which will be found to include several interesting geometrical
significations.

• This form was communicated to the Royal Irish Academy, at the stated
meeting of that body on March 16th, 1849, in a note addressed by the pre-
sent writer to the Rev. Charles Graves, It was remarlced, in that note,
that the directions of the two fixed vectors x, 6, are those of the two asym-
ptotes to the focal hyperbola; while their lengths are such that the two ex-
treme semiaxes of the ellipsoid may be expressed as follows :

a=T)7+T(J; c=Tj7— T^j
the mean semiaxis being, at the same time, expressed (as in the text of
the present paper) by the formula

i=T(„-0).
It was observed, further, that >j — 6 has the direction of o«<? cyclic normal of
the ellipsoid, and that jj-i — ^-i has the direction of the other cyclic normal;
that >j4-^ is the vector of one umbilic, and that >5-i + ^~' has the direction
o^ another umbilicar vector, or umbilicar semidiameter of the ellipsoid; that
t\\e focal ellipse is represented by the system of the two equations

S.gU>j=S.eU^,
and

TV.eU>5=2S Vn6,

of which the first represents its plane, while the second, which (it was re-
marked) might also be thus written,

TV.gU^rrgSV^,
re\ive?,ent% a. cylinder of revolution (or, under the latter form, a second cy-
linder of the same kind), whereon the focal ellipse is situated ; and that
the focal hyperbola is adequately expressed or represented by the single
equation,

To which it may be added, that by changing the two fixed vectors n and 6
to others of the forms t-^m and t&, we pass to a confocal surface.

[To be continued.]

.,0

[ 344. ]

XLVIII. On the Cause of the Diurnal Variations of the Mag-
netic Needle. By W. H. Barlow, Esq., M.I.C.E.

To the Editors of the Philosophical Magazine and Journal.

Gentlemen,

IN the Number of your Journal for April, an extract from
a letter from M. de la Rive to M. Arago is published, in
which the author attributes the diurnal variations of the mag-
netic needle and the auroras boreales to the effect of electric
currents at the surface of the earth and in the atmosphere.

In confirmation of this theory, mention is made of a re-
markable effect observed by M. Matteucci in the apparatus of
the electric telegraph between Ravenna and Pisa during the
magnificent aurora on the 17th of last November; and the
author concludes by observing that " it would be highly in-
teresting and important to profit by those telegraph wires,
which are found to have a direction more or less approaching
to that of the declination needle, in order to make with them,
when they are not in use for ordinary purposes, some obser-
vations which would enable us to demonstrate and to measure
the electric currents which probably traverse them."

My object in addressing you is to state, that in the early
part of 1847 I was led to undertake extensive observations on
this subject, in consequence of the peculiar disturbances occa-
sionally visible on the telegraph instruments of the Midland
Railway (on which line the telegraph was erected under my
superintendence as the company's engineer).

These disturbances were at first attributed to atmospheric
electricity passing to the earth by means of the wires ; but
from certain effects observed, I was led to infer that they were
due to other causes; and in order to explain these effects, it
is necessary to state that the Midland system of telegraphs
consists of four principal lines centring in Derby, as follows: — ■

1st. From Derby northwards to Leeds.

2nd. From Derby north-east to Lincoln.

3rd. From Derby southwards to Rugby.

4th. From Derby south-west to Birmingham.

The disturbances on these four telegraphs were observed
to occur simultaneously, with rare exceptions; and the direc-
tion of the current in the two telegraphs proceeding northerly
and north-easterly was always contrary to those proceeding
southerly and south-westerly; that is to say, when the deflec-
tion was such as to indicate that the current was towards
Derby on the first two, it was from Derby on the last two ;
and when it changed in one, it changed in all. It was also
observed that on the 19th of March 1847 there was an un-

On the Diurnal Variations of the Magnetic Needle. 34-5

usual degree of disturbance during the presence of aurorae
boreales.

As these effects could not be attributed to the transit of
ordinary atmospheric electricity along the wires to the earth,
I determined to make a set of experiments on the subject.

Having obtained delicate galvanometers, I first ascertained
that currents are at all times perceptible in the telegraph
wires to a greater or less extent when the galvanometer is
applied on a sufficient length of wire, and between two earth
connections ; but that wires having no earth connexion, or
only one, exhibited no currents.

I also found by simultaneous observations on two galva-
nometers, applied one at each extremity of a wire forty-one
miles long, that the changes of force and direction of the cur-
rents were simultaneous at both ends; the current piissing
direct from one earth connexion to the other.

But the most interesting fact which appeared during these
observations, and that which bears immediately on the remarks
contained in the letter of M. de la Rive, is that there is a daily
movement of the galvanometer needle, similar to that of the
horizontal magnetic needle, produced by the electric currents
travelling in one direction from about 8 a.m. to 8 p.m., and
returning in the opposite direction during the remainder of
the twenty-four hours. The times of zero are not regularly
maintained, and vary from 7 to 10 o'clock both in the morning
and evening; but the greatest regularity is observable in the
morning, and the mean result of numerous observations is as
above stated.

This regular diurnal movement of the galvanometer needle
is subject to disturbances of greater or less force and duration,
which are found to be of greatest energy during magnetic
storms, and when aurora is visible; and in these cases the
currents are so strong as to affect the ordinary telegraph in-
struments, and sometimes prevent altogether the transmission
of messages.

The next experiments were made with a view to ascertain
the direction in which these currents alternate ; and the result,
as determined from numerous observations, denotes it to be
from north-east to south-west. The nearer this line is ap-
proached, the more decided is the effect on the galvanometer ;
but between east and south, and between north and west, the
effect is smaller ; and in approaching north-west and south-
east, it becomes indefinite and irregular, but never ceases
entirely.

It also appeared that the effect depended, not on the direc-
tion of the wire itself, but on the relative directions of the two

S^S On the Diurnal Variations of the Magnetic Needle.

earth connexions ; that is, the points where the wire was con-
nected with the earth. I next made simultaneous observations
with the galvanometers and a declinometer needle; from which
it appeared, taking the mean of numerous observations, that
that part of the day in which the currents flow southwards
(that is, from 8 or 9 a.m. until the evening), the variation of
the declinometer needle is westerly ; and that during the night
and early part of the morning (at which time the currents
travel northwards) the variation is easterly ; also, that the
large disturbances called magnetic storms are simultaneous on
both instruments.

But although there is this resemblance in the general fea-
tures of the movements of both needles, the paths described
are not similar. The movements of the galvanometer needle
are more frequent and rapid than the declinometer, and the
deflection frequently changes over from right to left without a
corresponding movement of the declinometer.

The observations thus briefly recorded formed the subject
of a paper which was read at the Royal Society on the 17th
of June ] 847 ; and I have thought it desirable to make this
communication to your Journal on reading M. de la Rive's
letter, because it rather curiously happens, that the unusual
delay which has arisen in the publication of my paper by
the Royal Society is attributable to the fact, that I arrived
from these experiments at the same conclusion as M. de
la Rive, as to the electric origin of the diurnal variation of
the magnetic needle, which 1 considered to be the effect of
the alternating electric currents exhibited by the telegraph
wires.

The Royal Society were unwilling to give their sanction to
this view of the case, and only consented to the publication of
the observations above described on my omitting that portion
of the paper.

The paper is, however, now in the hands of the printers,
and will, I hope, be shortly before the public.

I ought to state in conclusion, that my idea of the origin of
the currents differs in one respect from the theory of M. de
la Rive ; inasmuch as he considers them to arise in the atmo-
sphere, whei'eas I have attributed them to thermo-electric
action in the crust of the earth. I speak of course with great
deference on a subject of this kind ; but there is an important
fact tending to this conclusion which is now well-ascertained,
namely, that in the telegraphs which are laid entirely under
ground, deflections occur similar to those before described ;
while wires suspended in the air exhibit no deflections, unless
they are connected with the earth in two places, and then the

Mr. T. S, Davies's Note on Numerical Transformation. S^l

direction Jn .which the current travels depends on the relative
positions of the earth connexions, however circuitous may be
the roiite'of the" wire itself.

I am, Gentlemen,

Your obedient Servant,
Derby, April 12, 1849. W. H. Barlow, M.I.C.E.

XLIX. Note on Numerical Transformation.
By T. S. Davies, Esq., F.E.S. L. Sf E.* ,,

IN my notes on Mr. Cockle's paper (Phil. Mag., S. 3.
vol. xxxii. p. 351), it was incidentally suggested to express
the conjugate pair of roots of one of the quadratic factors of
an algebraic equation by « + /3 and a— 13, without assigning
the algebraical form of ^. It was, moreover, proposed to form
the two subordinate equations which contained relations be-
tween a and /3, as is now generally done, after the example of
Lagrange.

It has been objected by analysts with whom I have corre-
sponded or conversed — analysts who would not have raised
frivolous objections to any proposition whatever — that by
the assumption of these forms I deprived myself of the means
of forming those two equations upon any legitimate principle ;
inasmuch as their derivation is founded entirely on the appli-
cation of the principle of incongruity, by showing that in

X + Y/3^/^=0

*ixie must have simultaneously

X=0, andY=0.

I may remark in the first place, that though I do not ques-
tion the legitimacy of this argument when all the roots are
imaginary, I still think it ambiguous when some of the roots
are real, and altogether fallacious when there are no imaginary
roots at all.

And, in the second place, that any process founded on this
for the determination of the real roots of an equation, is totally
deficient of all legitimate foundation.

I proceed, however, to the object of this note, which is to
form the equations X' = 0, Y' = 0 on the unrestricted forms of
the pair of conjugate roots, « + /3 and «— 13. I use accented
letters, because the quantities represented by X' and Y' differ
in the signs of the alternate terms from X and Y.

If/(,r) = 0 be an algebraic equation of an even degree, and
a + jS, «— 13 two of its roots, we have

* Communicated by James Cockle, Esq., M.A., Barrister-at-Law.

348 Mr. G. G. Stokes on the Theory of Sound,

0=/(« + S) = X' + Y'/3 (1.)

0=/(a-/3) = X'-Y'/3 (2.)

where X' and Y' contain only even powers of + /3 and rational
functions of « ; and these powers and functions are the same
in (1.) and (2.).

When /3 is different from zero, we get, by addition and sub-
traction of (1, 2),

X'=0, and Y'=0,

precisely as in Lagrange's method, where /3 -/ — 1 is put for |3.

The separation into two subordinate equations is therefore
as legitimate in the general case as in Lagrange's restricted
one.

When ^3 = 0, then Y' is indeterminate: but this does not
seem to be of much consequence, since the original equation
is then reduced, as it should be, to

/(«) = 0,and/,M=o,

indicative of a common measure .r— a between the equation
and its derivee. This we already know to be the consequence
of the given equation having equal roots.

I shall shortly send you for insertion some extracts from
Mr. Horner's rough notes on the effect of the transformation
of/(<z') = 0 by means of the quadratic divisor

employing his synthetic method of transformation.

Royal Military Academy, Woolwich,
April 6, 1849.

L. On the Theory of Sound, in reply to Professor Challis.
By G. G. Stokes, M.A., Fellotx) of Pembroke College,
Cambridge^.

IN my last communication I contented myself with a simple
denial in the case of the fifth head of Prof. Challis's de-
monstration, having nothing, as I conceived, to meet but a
simple assertion. I am unable to perceive the slightest con-
nexion between the conclusion that 47rrj^5ia = 47rr2%aand the
principle of the constancy of mass, irrespective of some tacitly
assumed step of reasoning. Without noticing this connexion
explicitly, Prof. Challis has certainly rendered his reasoning
more definite by the introduction of the supposition of rigid
envelopes.

* Communicated by the Author.

in reply to Professor Challis. Sl-Q

Prof. Challis's argument, as it now stands, may be divided
into the following heads : —

1. The condensation s may be expressed by r~^¥{r—at),

2. The function F may be given arbitrarily from r—at^=-h
to r—at^=.h-\-\ and may be supposed to vanish beyond those
limits, so that the wave of condensation is comprised between
the spheres whose radii are at-\-h and a/ + Z> + A.

3. The fluid may be supposed at rest beyond the outer, and
within the inner boundary of the wave of condensation.

4. Therefore we may introduce two rigid envelopes, &c.
The third head I wholly deny, and have already, as I con-
ceive, disproved. (Phil. Mag., vol. xxxiv. p. 54.) I have

shown that unless / F(r)</r=0, the fluid cannot be at rest

both outside and inside the wave of condensation.

It is vain to reply (as at p. 91) that the expression for the
velocity is not proved. It is proved on the hypothesis (I do
not myself regard it as an hypothesis) of spherical waves, as
Prof. Challis does not seem to deny. It is plainly illogical
reasoning to make the hypothesis of spherical waves, obtain a
part of the results to which the analysis leads on that hypo-
thesis, refuse to attend to other results to which the analysis
equally leads, and then, oti the gratuitous assumption of the
possibility (according to the hypothesis under trial) of a state
of things which does not result from the partial solution already
obtained, and which the analysis, if wholly carried out, would
have proved to be impossible, to argue that the hypothesis is
absurd.

In conclusion, I will merely explain what I meant by saying
that I did not regard the hypothesis of spherical waves, as an
hypothesis. I consider it axiomatic that the initial conden-
sation and initial velocity may be conceived to be given arbi-
trarily in a mass of elastic fluid ; at least if no abrupt varia-
tions be supposed to exist in the initial condensation or velo-
city: such abrupt variations, in so far as they are admissible,
may be afterwards considered as limiting cases of continuous
variations. Consequently we may without absurdity conceive
the initial condensation and initial velocity to be arranged
symmetrically about a centre. But in this case the conden-
sation and velocity must be symmetrical with respect to the
centre during the whole motion ; because if we draw any two
radii vectores from the centre, whatever we can say of the one
we can say of the other ; and therefore the velocity will be
directed to or from the centre, and the condensation and ve-
locity will be functions of /• and of /. This will be equally
true whether we neglect or take account of the development

350 Messrs. Thomson and Wood on tlte Composition of

of heat, whether we suppose the pressure equal in all direc-
tions, or adopt any otlier hypothesis. Accordingly, when we
adopt the equations which are obtained on the usual theory,
and suppose the initial condensation and initial velocity given
arbitrarily as functions of r, no contradiction is arrived at,
either in the general case (see Phil. Mag., vol. xxxiv. p. 5.5,
paragraph marked 2), or in the particular case which might
seem beforehand most favourable to tlie contradiction (see
paragraph 3), as might have been confidently anticipated,
inasmuch as one truth cannot contradict another.
Pembroke College, April 12, 1849.

LI. Note on the Composition of Shea Butter and Chinese
Vegetable Tallow. By Dr. R. D. Thomson a7id Mr.
Edward T. Wood*.

^HEA Butter. — This substance is a vegetable product of
Western Africa, and was brought into notice by the cele-
brated Mungo Park during his journey in 1796. The tree
from which it is procured he describes as very much resem-
bling the American oak, and the fruit (from the kernel of
which, being first dried in the sun, the butter is prepared by
boiling the kernel in water) has somewhat the appearance of
a Spanish olive. The kernel is enveloped in a sweet pulp
under a thin green rind, and the butter produced from it,
besides the advantage of keeping the whole year without salt,
is whiter, firmer, and, according to Park, of a richer flavour
than the best butter he ever tasted made from cows' milk.
The growth and preparation of this commodity seem to be
among the first objects of African industry, and it constitutes a
main article of their inland commerce. This butter is abun-
dantly produced, not only towards the Gambia, but also in
the countries adjoining the Niger, as it is mentioned by the
Landers and other recent travellers. Mr. John Duncan, who
penetrated by Dahomey, describes the tree as resembling a
laurel, and growing to the height of eighteen or twenty feet.
The leaf is somewhat longer than the laurel and a little broader
at the point. The nut is of the size and form of a pigeon's
egg, and of a light brown colour; the substance of the shell
about that of an egg. The kernel when new is nearly all
butter. The shell is crushed from the kernel, which is also
crushed and boiled with a little water in a pot for half an hour ;
it is then strained through a mat, when it is placed in a grass

• Read before the Philosophical Society of Glasgow, April 26, 1848.

Shea Butter atid Chinese Vegetable Tallow, 351

bag and pressed. A good sized tree will yield a bushel of
nuts.

Shea butter appears to be the same as that which is called
Galam butter, and is derived from a species of Bassia ; but the
species has not yet been made out, as no specimens of the
flower and fruit have reached botanists. The oil upon which
the following experiments were made was obtained through
the kindness of Dr. Carson of Liverpool, from Mr. Jameson,
formerly of this city and now of Liverpool, whose benevolent
exertions for the improvement of Africa are so well known.
The colour of the oil is white with a shade of green. It is
solid at the usual temperature in this country; at 95° it assumes
the consistence of soft butter, and at 110° is a clear and liquid
oil. When boiled in alcohol the greater part is dissolved, and
crystallizes on cooling in needles : it dissolves in cold aether,
and separates in needles by evaporation. The oil was sapo-
nified by means of caustic potash in a silver basin ; the soap
separated from its solution by common salt and decomposed
by tartaric acid. After being crystallized out from alcohol
five or six times, and freed by pressure from adhering oleic
acid, the acid was obtained in fine pearly scales fusing at 142°:
it was united with soda, and yielded a salt in fine pearly scales.
Its atomic weight was estimated by means of the silver salt.
In the first, second and third experiments, the silver salt was
formed by precipitating an aqueous solution of nitrate of silver
by an aqueous solution of the fatty acid united to soda. In
the fourth and fifth experiments, an alcoholic solution of the
acid was precipitated by a solution of nitrate of silver in
alcohol, and hence the excess of acid.

I. 3*73 grains of silver salt gave 1'05 metallic silver = 1*126
oxide of silver = 30*19 per cent. AgO.

II. 10*65 grains of silver salt gave 301 silver =3*221 oxide
of silver = 30*23 per cent. AgO.

III. 2-85 grains gave -861 AgO = 30-21 per cent.

IV. 4*71 grains gave 1*30 silver =1*394 AgO= 29*53 per
cent.

V. 2*72 grains gave -743 silver = *797 AgO = 29*30 per
cent.

The following table will express the per-centage composition
of the silver salt by these five experiments : —

I. II. III. IV. V.

Acid . . . . 69-81 69*77 69*79 70*41 70*70
Oxide of silver . 30*19 30-23 30*21 29*59 29*30
Taking the mean of all these experiments, the constitution
of the silver salt will be —

Acid .... 70*10
Oxide of silver . 29*90

352 On Shea Butter and Chinese Vegetable Tallow.
and the atomic weight of the anhydrous salt is —

Acid .... 33*97
Oxide of silver . 14<'50

I.

ir.

III.

Mean.

Anhydrous acid,

54-73

54.-88

54-54.

54-71

77-83

8-94

8-78

9-22

8-98

12-77

6-12

6-75

6-94

6-60

9-40

30-21

29-59

29-30

29-71

48-47

or leaving out the two last determinations, we shall have as a
mean for the three higher results the atomic weight of the acid
equal to 33-82. To determine the composition of the anhy-
drous acid, the three following analyses were made by means
of oxide of copper and chlorate of potash. : —

I. 2-85 grs. of silver salt gave HO=2-30 grs. and C02=5-73 grs.
11. 3-91 ... ... =3-39 ... =7-87 ...

III. 3-667 =3-058 ... =7-334 ...

The following table gives the composition of the above salt
in 100 parts : —

C

H
O
Ago

From the facts which have been stated in reference to the
acid contained in the shea butter, it is obvious it is margaric
acid, the same substance which is found in the human fat
and in butter. There is little doubt that on examination this
acid will be found extensively distributed in the vegetable
kingdom : its presence in the shea butter may assist in ex-
plaining the statement of Park, that this substance when fresh
is equal in taste to butter.

Ckitiese Vegetable Tallow. — This is a solid oil, long known
to those who are acquainted with China, where it is exten-
sively used for making candles. It is derived from the seeds
of Siillingia sebi/era, which, according to Fortune ( Wander-
ings in China, p. 65), are pulled in November and December.
They are placed in a wooden cylinder with a perforated bottom
over an iron vessel filled with water, which is boiled and the
seeds well-steamed to soften the tallow; in ten minutes they
are thrown into a large stone mortar, and beat with stone
mallets to separate the tallow from the other parts of the seed :
the tallow is thrown on a sieve heated over the fire and sifted,
and is then squeezed out by a peculiar pro«:':oS. As imported
it is a hard, white, solid oil, with a green shade. It fuses at
about 80°. The oil was saponified, and the acid separated
and purified according to the method already noticed. A soda
salt was formed, and from this a silver salt was precipitated.
14-38 grains of this salt when burned left 4*03 grains of me-

Prof. Challis's 07i the Velocity of Sound. 353

tallic silver, which gives the following for the composition of

the salt: —

Atomic vveiglit. Per cent.
Oxide of silver . 4>'3'28 1 4<-.50 30-03

Acid .... 10-052 33*67 69-97

The acid was not quite pure ; for when heated it softened
at 143°, became very soft at 149°, of the consistence of cream
at 150°, and quite fluid at 154°; it obviously therefore retained
some stearic acid, but must have consisted principally of mar-
garic acid, as stearic acid fuses at 167°. There is no doubt
that both of these oils might be advantageously employed in
soap-making, the supply apparently, from the statements of
the traders, being unlimited.

LI I. Determination of the Velocity of Sound on the princi-
ples of Hydrodynamics. By the Rev. J. Challis, M.A.,
F.R.S.y F.R.A.S., Plumian Professor of Astronomy and Ex-
perimental Philosophy in the University of Cambridge^.

IN conformity with the intention expressed at the close of
my communication to the Number of the Philosophical
Magazine for last February, I propose now to exhibit collec-
tively the whole course of the mathematical reasoning by which
I obtain, entirely on hydrodynamical principles, a value of the
velocity of sound closely agreeing with that found by obser-
vation. The importance of the result, and the novelty of the
considerations on which it depends, will be my excuse for
going through the reasoning somewhat in detail, and for re-
peating some parts of previous communications. It may be
proper to state at once, that I do not regard as defensible, or
pertinent, all that 1 have written in the course of this difficult
investigation; for instance, I have found that the new hydro-
dynamical equation, the necessity of which I have elsewhere
insisted upon, is not, as I supposed, essential to the present
inquiry. My immediate object is to extract and put in logical
order what is really legitimate and essential.

The problem to be solved is, the numerical determination
of the velocity of sound from the equations of hydrodynamics.
As this may be considered to be a case of small vibrations,
powers of the velocities and condensations above the first will
be neglected. The pressure {p) being such that J9 = a^(l +s),
and «, 'y, W being the resolved velocities at the point ocyz and
at the time /, the equations applicable are the following: —
« Communicated by the Author.

Phil. Mag. S. 3. Vol. 34. No. 230. May 1 849. 2 A

354- Prof. Challis's Determination of the Velocity of Sound

ods , du ^ ^ds ^ dv .ds dw __

dx dt ' dy dt dz dt

ds du dv , ——o
di'^di'^'dj'^Tz'

all the differential coefficients being partial. Hence by inte-
gration,

p ^ds ,, , d.fahdt ,^

u— —I a^ — dt + c =— ♦^ +c

»/ ax fij^

r.= -ra-%dt^c^=-^jf^^d
-^ dy dy

r^ds ,, d.faHdt ,u

J dz d^

the arbitrary quantities c, c', c" being functions of ^,3/ and «. As
the motion is by supposition vibratory, it will be assumed that
c=0, c' = 0, c" = 0, which is to assume that no part of the velo-
city is independent of the time. Now substitute vf/ for —Jahdt.

Then

d-h t/4/ f/v^

rf.r Gj/ dz

It follows that udx + vdy + wdz is an exact differential.

Here it must be particularly remarked that the above result
has been arrived at prior to the consideration of any particular
case of disturbance. Consequently ndx + vdy + wdz is an exact
differential for some reason which applies equally to all cases
of small vibrations. Such a reason would be given if it were
proved, that the motion in every case is composed of motions in
plane-'waves, that is, waves in which the motion of each particle
is perpendicular to a fixed plane and a function of the distance
from the plane, the number of such waves and the directions
of motion being taken arbitrarily. The consequences of the
general supposition of plane-waves have been traced in an in-
genious paper by Mr. Earnshaw, contained in the Transac-
tions of the Cambridge Philosophical Society (vol. vi. part 2,
p. 203). It is there shown that the velocity of propagation on
that supposition is the constant «, which, as will afterwards
appear, is a first approximation to the true theoretical velocity
of sound. Mr. Earnshaw finds also that "a plane-wave can-
not be transmitted through any fluid unless it extend com-
pletely across the medium from boundary to boundary." This
result makes it impossible to conceive how the motion can in
every case be composed of motions in plane-waves. There is,

on the principles of Hydrodynamics, 355

moreover, an antecedent objection to the supposition of plane-
waves. For on making this supposition, a particular and exact
integral of the resulting differential equation may be obtained,
which, as I have recently shown in this Magazine by argu-
ments that need not now be repeated, admits of no interpreta-
tion consistent with fluid motion. Such inconsistency must
necessarily be significant. The reasoning by which it was
arrived at being good, it clearly means that tne general sup-
position of plane-waves is not legitimate.

Again, a general reason for the integrability of m^x + i;^
+ 'wdz is given, if it be proved that in every instance of small
vibrations the motion is composed of motions in spherical
waveSf that is, waves in each of which the motion is directed
to or from a fixed centre, and is a function of the distance from
the centre. But such composition of the motion does not
admit of being proved, because the hypothesis of spherical
waves is liable to an antecedent objection. For on making
this hypothesis, a result is arrived at inconsistent with the
principle of constancy of mass, on which one of the general
hydrodynamical equations rests. This I conceive that I have
shown in my communication to the Number of the Philoso-
phical Magazine for last February.

It appears, therefore, that vibratory motion is not generally
compounded of motion either in plane- waves or spherical waves,
and that the integrability of uda^ + vdy + wdz is not accounted
for on either of those suppositions. At this stage of the argu-
ment it is important to remark, that although inconsistencies
have resulted from the general suppositions of plane-waves and
spherical waves, it does not thence follow that these are not
possible cases of arbitrary disturbance. As such, however,
they must plainly be treated by a different process. Before
treating these, or any other instances of arbitrary disturbance,
it is absolutely necessary to assign a general reason for the
integrability o^udx + vdy + isodz. I proceed, therefore, to make
another supposition.

Let the function which we have called rj' be composed of
two factors, y and <p, such that /is a function of ^* and y only,
and f a function of s: and t only. Then

_d^_ ^
dx dx

_d^^ df
^~ dy ~'^ dy

2 A2

SH6 Prof. Cliallis's Determination of the Velocity of Sound
and

udx + vdy + 'wdz=<^ [£ dx+ dy^yj^^'d'z'^'''

It hence follows that the left-hand side of the last equality is an
exact differential. We have now to trace the kind of motion
resulting from this general hypothesis.

For this purpose I shall assume, for the present, that the
supposition by which uda; + vdy-\-'wdz was made integrable,
holds good for exact values of w, v and w, although the proof
of the integrability of that quantity extended only to terms of
the first order of approximation. Thus we shall have the
following seven exact equations : —

»=^l <'•>

^-4 ^'-^

»=/| («•)

:^ ^. ^ + iifl + '^'P'^ =0 . . . (4.)

dt dx dy dz

,., + ©=" («•)

"^ + (7:)-' (^•'

with an eighth deducible from the first three and the last three
by integration, viz.

By means of equations (1.), (2.), (3.) and (8.), u, v, w,

—4-, -4-, — T-» and —i-. may be eliminated from (4.), and
pdt' pdx* pdy' pdz* ^ ^ '

the resulting equation is

(A.)

on the principles of' Hydrodynamics. 357

^ \dx^ dx^ doc dy dxdy dy^ dy^ /

J dz'dzdt ^ dz^' dz^

In treating this equation, it may be assumed, since/* and <p
contain no variables in common, that if particular and con-
sistent values be substituted fory, x and y, the resulting equa-
tion is true for general values of (p, z and t. Now since, by
hypothesis, the motion is vibratory, the function /must have
a maximum value. This value may be assumed to be unity,
becauseymay be regarded as a numerical quantity, and one
value of it may be taken arbitrarily. Let therefore the values
of .r and^ given by the equations

dx ^ dy ^
satisfy the equations ^

•^~ ' dx^ "^ %2- ^2»

the negative sign in the latter being a consequence of the sup-
position of a maximum. Then by substitution in (A.) we
obtain the following equation for determining <p :

dz^ dt^ dz dzdt dz^ dz^'

(B.)

It does not appear that this equation is generally integrable:
but an integral applicable to the present inquiry may be ob-
tained by successive approximations, the terms involving the
first power of <p being first considered. The arbitrary func-
tion of the time may be supposed to be included in <p. Thus
to the first approximation we have

The integral of this equation, as I have shown in the Phi-
losophical Magazine for April 1848 (p. 278), may be obtained
in an infinite series involving two arbitrary functions. In
investigating the velocity of propagation, one of the arbitrary
functions must be made to vanish. This having been done,
and the following substitutions made, viz.

368 Prof. Challis's Determinationqf the Velocity of Sound

^■=z-\-at, v=z—aty Gi(v)==/G(v)(afv,

G2(v)=/Gi(v)rfv, &c.,
it was found that

^=G(v) + .^Gi(v)+ '^G,(v)+ £|-^G3(v) + &c.

No inference respecting the propagation of the motion can be
drawn from this result, unless <^ be expressed in exact terms.
The form of the series shows that this can be done only by
satisfying the following equations :

1 r* f \ _i_ d.Giy)

&c.= &c.

Now since Gi(v)= /'G(v)<?v, it follows that

^i^)=G(v)

dv ^

or

d^.GAv) _ „ „ , .
-^— +^'-Gi(v)=0.

The upper sign would give to the function G a logarithmic
form, which is clearly excluded by the nature of vibratory
motion. Taking, therefore, the lower sign, the integral of
the last equation gives

G,(v)= sin(Mv4-c).

Consequently

G(v)=wcos (wv + c).

This form of G satisfies all the other equations. Hence

on the principles of Hj/drodynamics. 359

=4-1)

■■mcos

= m cos < n{z—ai) {z+at)+c X

Let, now,

so that
Then

e /4>v'^ , \\

f = w cos — ( ;2-a^ y/ 1 + -^ +Cj-

Hence it appears that the velocity of propagation of the
wave, or series of waves, defined by the above form of (p, is

the propagation taking place along a straight line parallel
to the axis of ^, or, if we please, along the axis of z itself.
Here it is important to remark, that the particular expres-
sion obtained for (p, and the consequent velocity of pro-
pagation, have been arrived at by a strict induction. The
course of the investigation leads to these results and to no
others of a like kind. Hence as there is at present no case
of disturbance under consideration, these results have, with
regard to vibratory motion, a general significance. The in-
ference from them is, that whatever be the disturbance, the
motion consists of vibrations defined by a circular function of
the above form, and that the velocity of propagation exceeds
the value a by a quantity depending on the numerical value

of-2-

If the investigation be conducted on the hypothesis of plane-
waves, the solution to the first order of approximation is
rJ; = G(2f— rt/), and the velocity of propagation is «, the form
of G remaining arbitrary. I have already argued that these
results have no significance, because the exact integral, of
which this is the first approximation, conducts to results in-

360 Prof. Challis's Determination of the Velocity of Sound

compatible with fluid motion. The same objection might be
raised against the results of the hypothesis now under consi-
deration, ifj on carrying the investigation to higher degrees of
approximation, any similar incompatibility appeared. To
determine whether this be the case, it is now required to in-
tegrate equation (B.), taking account of the two last terms.

This may be done by successive approximations, beginning
with the value of ^ already obtained. The result to three
terms, as given in my communication to the Philosophical
Magazine for November 1848 (p. 363), is

(p = w cos q{z— alt + c)

"■"3k~^^"^2'(^~'^''^ + ^^

and

4^2 V W

It:

=«'+r2+^V\^-^2

q being substituted for

So far as this result indicates, <p is a function ofz—a't, and
the velocity of propagation at all points of a wave is the con-
stant a'. To ascertain whether constant propagation, the same
for all points of the same wave, accords exactly with equa-
tion (B.), let us introduce into this equation the condition
(p = F(z — a^t). For the sake of brevity write v for z — a^^t, and
F for F(v). Then, supposing «j constant while z and t vary,

^_ ^ (l^ ^_^ _^__ ^

dt"" ~^i • 'd^' dz~ rfv' dzdt ~ '^^ dv^'

Consequently

d^F^ o o ^ dF dF^\ ,01. ^ , V

-^(«,^-a^-2«,-^4--^)+6^F=0. . . («.)

It may be here observed, that if the term involving b^ be
omitted, the resulting equation, which applies exactly to the
case of plane-waves, is satisfied by either of the two equations,
d^F ^ ( dF\^ 2 ^

Neither of these equations gives a form of F compatible with
vibratory motion : whence we may infer that, on the hypo-
thesis of plane- waves, all the parts of a wave are not propa-
gated with the same velocity. This result leads to the incon-
sistency already spoken of.

on the principles of Hydrodynamics, 361

Retaining now the term involving h^ in equation (a.), this
equation gives by integration a certain form of F, which, if it
be compatible with vibratory motion, is the particular form
that satisfies the condition of constant and identical propaga-
tion of all parts of the same wave. A first integral of the
equation is readily obtained ; but the complete integration can
probably be effected only by successive approximations. To
the first approximation we have

-X2 +-T— o-F=0.

dv^

a^—c^

Whence

17 ( ^^ \

i'=OTC0sl , +c|

consequently, putting

2-K c b

— for

we obtain

and

f=:F{v) = m cos -^( IS— ai<\y 1+ ^ -\-c].

This is the first approximate value of <p already obtained by
the integration of the partial differential equation (B.). By
continuing the solution of equation («.) to the third approxi-
mation, I find precisely the same expression for <p as that
above given for the solution of equation (B.) to the same
degree of approximation. We may hence conclude that the
solution of (B.) carried to an unlimited number of terms, has
the property of satisfying exactly the condition of uniform and
identical rate of propagation of all the parts of a wave. 1 have
proved this proposition in another way in the Philosophical
Magazine for December 1848, p. 465.

The next step is to introduce into the equation (A.) the
condition which it has just been shown that the function <p
must satisfy, viz. the condition expressed analytically by the
equation

When by means of this equation the differential coefficients
of the second order are eliminated from (A.), the result is

362 Prof. Challis's Determination of the Velocity of Sound

«=«'nS» + #,) +*y^ T7 5A^

-^t{--4M-¥)

r • (A'.)

^ \dx^ ' dx^^ da;' dy' dxdy dy"^ ' dy^) '

Hence it appears that an equation altogether independent of
f is obtained by neglecting powers of this quantity above the
first. When this is done, the equation becomes

d^f dH by ,^.

At the same time equation (B.) becomes

-§-'S-*^^ • • ; ■ (-)

We have thus arrived at two equations, one of which shows
thatyis a function of ^ and y only, and the other that <p is a
function of ;s and t only. These results are in accordance
with the original suppositions respecting these quantities, by
which /^c/o^ + uc^^z + wc^z was made integrable. The condition
of integrability of that quantity has therefore now been satis-
fied in a manner consistent with the hydrodynamical equations.
It is to be remarked that the equations (^.) and (y.) contain
only the first powers of / and f. Now since the reasoning
given at the commencement of this communication, by which
It was shown that udx + vdy + wdz must be Integrable for vi-
bratory motion, extended only to the first power of the velo-
city, there was no reason to expect greater generality in the
equations which determine f and f. It may, however, be
remarked, that the equation (/3.), as equation (A'.) shows, is
satisfied without any restriction of the value of <p, the more
exactly in proportion as the value of/" approaches more nearly
to unity. We may hence conclude that for points on and imme-
diately contiguous to the axis of %, the motion is exactly de-
termined by the equations (/3.) and (B.), and that for these
points udx + vdy + wdz is integrable for exact values of u, v,
and TO. This result is important, because it enables us to infer
from the reasoning that has preceded, that along the axis of
z waves are propagated without undergoing any change what-
ever, all the parts of a given wave being propagated with pre-
cisely the same velocity.

To obtain a complete idea of the nature of the motion, it is

on the principles of Hydrodynamics. 363

now required to treat equation (/3.) by the same process as
that applied to equation (y.), for the purpose of deducing the
particular form of the integral which defines the motion trans-
verse to the axis of z independently of any arbitrary disturb-
ance. It may be presumed that such a form exists, because
the motion along the axis is already so defined. I have ex-
hibited the mathematical reasoning by which an integral of
this nature is obtained, in ihe Postscript to my communication
to the Philosophical Magazine for last February (p. 96). The
following is another process which conducts to the same re-
sult. \i (^=x-\-y V — I, SLVidiy=x—y V — 1, the general inte-
gral of (|8.) is

/=F(f.) + G(v)-.{vFi(^) + ^G,(v)}

-&c.,
where

Fi(,.)=/F(^y/., F,(,.)=/Fj(,.)rf^,&c., G,(v)=/G(vyv,

G2(v)=/Gi(v)^v,&c.

Now a specific form may be given toy* by supposing the arbi-
trary functions to be arbitrary constants. Let, therefore,
F(jx) = c, and G(v) = d. Then

F,(^)=c^, F,(^)=^, F3(^)=-j^^,&c.

G,(v)=c'v, G,(v)=^, G3(v) = -^^,&c.
Hence

/=s(c+c') {\^euv+ ^^ - ^jT^r^^ +&c.j,

by putting r^ for x^-\-y^. Buty has already been required to
satisfy the condition, y= 1 when r=0. Consequently c + c'=l,
and the arbitrary constants disappear of themselves. Thus
we obtain

/=l-^r^+ jl;^, ~jr^+&c., . . (8.)

a result independent of all that is arbitrary. This form of/*
indicates that the motion is the same in all directions trans-
verse to the axis of z.

By thus obtaining, prior to any supposed case of disturb-

364 Prof. Challis's Determination of the Velocity of Sound

ance, particular forms of <p and /, which define a particular
kind of vibrations, it is shown that in all cases of small dis-
turbances the motion is composed of such vibrations. As the
vibrations are symmetrically disposed about an axis, I have
called them ray-vibrations. The number of the rays (the
equations defining them being linear), the directions of their
axes, and the values of m and A, may be assumed so as to
satisfy the conditions of given disturbances.

This theory is not complete without obtaining a numerical
value of the velocity of propagation. It is unnecessary to re-
peat here in detail the mathematical reasoning by which I
succeeded in doing this in the Postscript to my communication
to the February Number of this Journal. It will suffice to say
that by making /=0 in the equation (8.), an equation results,
which is satisfied by an infinite number of values of r, such
that the difference between two consecutive values approaches

continually, as r increases, to the limit ~~7^' These values of

r correspond to positions of no condensation, since to the first
approximation we have

The values of r which satisfy the equation -4-=0, are also

unlimited in number, being intermediate to those which satisfy
y=0, and correspond to positions of no transverse velocity,
since the velocity transverse to the axis of the vibrations is

<p -J-. Between two consecutive cylindrical surfaces of no

velocity at an infinite distance from the axis, the transverse
vibrations are the same in kind as those which would take
place along the axis between two points of no velocity, sup-
posing two series of waves exactly equal were propagated
along that axis in opposite directions. The effect of the si-
multaneous propagation of two such series would be, to pro-
duce vibrations along the axis like the transverse vibrations
at an infinite distance ; and as the time of a vibration would
be the same in the two cases, it follows that

—p. = -, and -^——.

Hence, from what has already been shown, the velocity of
propagation is

""s/

4

1 + ^.

on the principles of Hydrodynamics. 365

The numerical value of this quantity is less by only 3*17 feet
than the mean result (lOSQ'^S feet) of a large number of de-
terminations of the velocity of sound by direct experiment.

The course of this investigation has shown that the velocity
of propagation (a) resulting from the hypothesis of plane-
waves, is not the correct value deducible from hydrodynamical
principles. The only true value given by hydrodynamics is that
I have exhibited above. The difference between the observed
velocity of sound and the value a, has been attributed, accord-
ing to a well-known theory, to the effect of the development
of heat accompanying sudden compressions of the air, and of
absorption of heat accompanying sudden dilatations. The
fact of such development and absorption is established by ex-
periments made on air in enclosed spaces. The walls of the
enclosure, by preventing the immediate escape of the deve-
loped heat, and the immediate restitution of the absorbed
heat, allow of ascertaining the fact by the thermometer. The
same development and absorption of heat must accompany the
condensations and rarefactions of aerial vibrations. But the
absence of enclosing walls makes an essential difference be-
tween this case and that of the experiment just mentioned. It
may not unreasonably be supposed that the thermometric
effect in the experiment is wholly due to reflexions at the
walls of the enclosure, by which the developed heat or cold is
made to traverse the enclosed air numberless times in an in-
appreciable interval, and thus produces a sudden change of
temperature. In a similar manner, when heat is radiating
from the earth's surface into a clear sky, as soon as a cloud
comes over, the air between the cloud and the earth, beco-
ming in some degree enclosed, and being traversed by the heat
reflected from the one to the other, has its temperature im-
mediately raised. As the heat developed or absorbed by
aerial vibrations cannot be supposed to produce a difference of
temperature by any analogous operation, a particular hypo-
thesis is required to account for an analogous effect in this
case. It is assumed that the air possesses the property of de-
taining the heat or cold set free by sudden compression or
dilatation, and that as the development or absorption is greater
the greater the condensation or rarefaction, there is always by
this detention an excess of temperature in the denser of two
contiguous elements of the vibrating air, and consequently an
effective accelerative force from the denser to the rarer por-
tion, which produces an apparent increase of the elasticity of
the medium.

Respecting this theory, it must be said that it cannot be
considered as fully established, unless the property which it

366 Mr. J. Glaisher's Remarlcs on the Weather

ascribes to the air of preventing the immediate escape of the
developed heat or cold be demonstrated. At present this pro-
perty is hypothetical. The argument contained in this com-
munication would rather point to the inference, that the de-
veloped heat or cold instantly passes off in a radiant form,
without producing any very appreciable alteration of the state
of temperature of the air where it is generated.

The mathematical investigation of the velocity of sound
which I have now expounded, originated in an attempt to ex-
plain theoretically the polarization of light exclusively on
hydrodynamical principles, the aether being treated as a con-
tinuous medium, the pressure of which, as in air of given
temperature, varies in the same proportion as the density.
My views on this subject are contained in several recent com-
munications to the Cambridge Philosophical Society. In the
course of the inquiry I found that the velocity of pi'opagation
in such a medium was not the value a, as generally supposed,
but a certain greater quantity a -v/ 1 +/", and that the expla-
nation of the phainomena of polarization essentially depended
on this result. I', was clearly, therefore, important to obtain
a numerical value of « i^l -f/r which could be tested by actual
observation of the velocity of sound. This I consider that I
have now done, and that I have thus removed an objection
which otherwise might have been urged against the proposed
theory of the polarization of light.

Cambridge Observatory,
April 18, 1849.

LIII. Kemarks on the Weather during the Quarter ending
March 31, IS^Q. ^j/ James Glaisher, Esq., of the Royal
Observatory^ Greejmich *.

THE meteorological returns for the past quarter furnished
to the Registrar-General and myself have been received
from thirty-four different places, whose returns have passed
the necessary examination. The observations generally indi-
cate a decided improvement, having been made for the most
part by experienced observers, who have generally paid more
attention to their instruments than hitherto. The results are
therefore found to be more accordant with each other than
any previously received.

Till January 7 and after March 18, the temperature of the
air was below its average value; the mean amount of the
deficiency of daily temperature in the former period was 6°'9,
and in the latter it was 3°'7.

* Communicated by the Author.

during the Quarter ending March 31, 1849. 367

The interval of time between January 8 and March 17 was
distinguished by very unusual warmth for the season. The
average daily excess of temperature within this period was
6°'l ; on four of the days this exceeded 12°, on three days it
exceeded 13°, and on two days it was greater than 14°.

The mean temperature of the three montiis ending Febru-
ary, constituting, in fact, the three winter months, was 42°*5,
being no less than 4°*7 above the average temperature of the
same time for seventy years. The warmest winters within this
period were those ending February 1796, 1822, 1834 and 1846,
and which were 43°-2, 42°-4, 43°-0 and 43°-2 respectively.

The pressure of the atmosphere during the month of Fe-
bruary was very unusual. The average reading of the baro-
meter from the 1st of February till the 18th was 30*36 inches
at the height of 160 feet: this was fully half an inch above its
average value. This denotes an increase in the volume of air
of about one- sixtieth part above the usual quantity. On the
11th day the very unusual reading of 30-715 inches took
place. The true reading for the whole day, reduced and cor-
rected to 32° Fahrenheit, was 30*695 inches, showing that
about one-thirtieth more than the usual quantity of air was
over England on this day. The reading of the barometer on
the 1 Ith day, reduced to the level of the sea, was 30'91 inches.
In December 1778 the reduced reading was 30*90 inches;
in January 1825 it was 30*92 inches ±.

The condition of the atmosphere, therefore, during the
greater part of the past quarter, both with respect to pressure
and heat, has been very unusual.

From the discussion of the observations which have been
made at the Apartments of the Royal Society since 1774, there
appears to be no foundation for the opinion that a hot summer
either precedes or follows a cold winter ; on the contrary, the
hot summers have for the most part been accompanied by
warm winters.

From the long continuance of high temperatures, it would
seem that for some time past causes have been in operation
which have raised the temperature: these causes probably
still exist, and therefore there seems to be every probability
of a fine and warm summer.

I proceed now to detail the results of the several subjects ol
research in the past quarter.

The meaji temperature of the air —

For the month of January was 40°*1, exceeding \he average
of seventy years by 4°'3. The temperatures in this month in
the years 1775, 1796, 1804, 1806, 1819 and 1834, were those
only which exceeded that of this year. In the year 1796 it
was 45°'4p being the warmest on record ;

368 Mr. J. Glaisher's Remarks on the Weather

For the month of February was 4'3°'2, exceeding the ave-
rage of the preceding seventy years by 4°'7. The temperature
of this month in the year 1779 was 45°*2, being the only in-
stance within the period of seventy years in which the tem-
perature exceeded that of this year;

P^or the month of March was 42°'5, exceeding the average
of seventy years by l"'2.

The mean for the quarter was 41°*9. The average value for
seventy years is 38°'6. In the year 1779 the mean was 4?2°*0 ;
in 1822 it was 43°-4; in 1834 it was 42°-S; and in the year 1846
it was 43^*6 : in all the remaining years it was less than 42°'0.

The excess of temperature above the average of the prece-
ding eight years was in January 3°'0; in February was 5°'4;
in March was 0°'l ; and for the quarter was 2°*9.

The mean temperature of evaporation at Gree7iwich —

For the month of January was 38°'6 ; for February was
41°*4 ; and for March was 39°'8. These values are 2°*0 above,
6°'-2 above, and 0°*3 beloio, respectively, the averages of the
preceding eight years.

The mean value for the quarter was 39°*9, which is 2°'6
above that of the average of eight years.

The mean temperature of the dew-point at Greenwich —

For the months of January, February and March, were 36°'4,
38°'8, and 36°'5. The average values for the preceding
eight years were 35°- 1, 34°-5, and 36°-6.

The mean value for the quarter was 37°'2, which is 1°*8
above the average for the preceding eight years.

The mean elastic force of vaponr for the quarter was 0*239
inch, which is 0*012 inch greater than the average for the pre-
ceding seven years.

The mean 'weight of *water in a cubic foot of air for the
quarter was 2*8 grains, which is 0*1 grain greater than the
average of the preceding seven years.

The mean additional weight of water required to saturate a
cubic foot of air was 0*6 grain. This value for the preceding
seven years was 0*38 grain.

The mean degree of humidity in January was 0*883, in Fe-
bruary was 0*863, and in March was 0*801. The averages
for the seven preceding years were 0*903, 0*888, and 0*841.
The mean value for the quarter was 0-849, which is 0*028 less
than the average for these years. These values denote a con-
siderable degree of dryness in these months.

The mean reading of the barometer at Greenwich in January
was 29*771 inches, in February was 30*106 inches, and in
March was 29*915 inches; these values are 0*005 inch above,
0*415 inch aioi'f, and 0*186 inch aiow respectively the averages
of the same months for the preceding eight years.

during the Quarter endhig March 31, 184-9. 369

The reading for the month of February, exceeding 30*1
inches at the height of 160 feet, is very remarkable. Since
the year 1774- there have been eight such instances only:
these occurred in July 1800, April 1801, November 1805,
April 1817, February 1821, January 1825, December 1834-,
and December 184'3.

The average weight of a cubic foot of air under the average
temperature, humidity, and pressure, was 54-9 grains; the
average for the seven preceding years was 546 grains.

The rain fallen at Greenwich in January was 1"6 inch; in
February was 2*2 inches; and in March was 0*5 inch. The
amount for the quarter was 4'*24 inches. The average amount
for the preceding eight years was 5* 14 inches.

The temperature of the water of the Thames was 43°'8 by
day, and 42^*1 by night. The water, on an average, was 1°
warmer than the air.

The direction of the wind at Greenwich from January 1 to 6
was N.E. ; from January 7 to 28 was S.W. ; on January 29
was N.N.W. ; from January 30 to March 7 was at times
variable, but chiefly S.W. ; from March 8 to 17 was mostly
N.W. ; from March 18 to 28 was chiefly N.E. ; and after-
wards it was mostly S.S.VV. to the end of the month.

At Leicester the direction of the wind was S.W. during
seventy-six days within the quarter.

The daily horizontal movement of the air from January 1 to
6 was about 90 miles; the greatest value was 200 miles; from
January 7 to 28 was 240 miles; the greatest was 500 miles;
and from January 30 to March 28 it was 110 miles ; the great-
est was 320 miles. The movement of the air in the month of
March was small.

The average daily ranges of the thermometer in air at the
height of four feet, were 10°*8 in January, 12^*9 in February,
and 13°*8 in March. The average ranges for these three
months, from the observations of the eight preceding years,
were 8°'l, 9°-6, and 13°-4 respectively.

2"he readings of the thermometer on grass in January were
below 20° on three nights, the lowest was 17^; at and below
32° on fourteen nights ; between 32° and 40° on six nights ;
above 40° on six nights ; and on one night it was 50°.

In February the lowest reading was 20°, and the readings
were below 32° on eleven nights; between 32° and 40° on eight
nights, and above 40° on seven nights. In March the lowest
reading was 21° ; and the readings were below 32° on sixteen
nights ; between 32 and 40° on twelve nights ; and above 40
on three nights.

The mean amount of cloud was 7'5, being the same as the
average for the preceding eight years.

Phil. Mag. S. 3. Vol. 34. No. 230. May 1849. 2 B

S70 Mr. J. Glaisher's Remarks on the Weather

There were nine exhibitions of the aurora borealis during
the quarter ending March 31, IS-iQ, which occurred on Janu-
ary 14, and were seen at Aylesbury, Whitehaven and Maiden-
stone Hill; on January 15 at Hartwell; on February 18 at
Wakefield ; on the 19th at Stone, Whitehaven and Wakefield;
on the 20th at Whitehaven, Hartwell and Greenwich ; on the
2 1st at Hartwell ; on the 22nd at Holkham, Aylesbury, Stone,
Norwich, Newcastle and Greenwich ; on the 23rd at W^hite-
haven and Hartwell ; and on March 18 at Stone.

Thunder-storms occwvvedi on January 10 at Whitehaven ;
on January 14 at Norwich; on February 25 at Truro; and
on March 31 at Uckfield.

Hail fell at Norwich and Hartwell on January 14 ; at New-
castle on February 22 ; at Wakefield on February 23 ; at Saf-
fron Walden on March 8.

Siwiio fell at Saffron Walden on January 4 ; at Leicester,
Saffron Walden, Highfield House, and Southampton on Ja-
^-nuary 5; at Saffron Walden on January 29: at Stone, Hart-
well, Norwich, and Saffron Walden on February 28 ; at Stone
and Saffron Walden on March 24 and 25 ; at Stone on the
28th; and at Norwich on the 31st.

Solar halos were seen at Greenwich on February 26 and
on March 30.

Lu7iar halos were seen at Greenwich on January 6, and at
Stone on February 27, March 7, 8 and 31.

Zodiacal light was seen at Whitehaven on February 11.

The reading of the barometer was above 30 inches on the
1st of January; it decreased to 29*58 by the 3rd, and in-
creased to 29*93 by the 6th. During the evening of the 7th
it decreased quickly, and was 29 4 on the 8th and 9th. On
the 1 0th the lowest reading in the month took place, and v/as
28*83; on the 11th it increased rapidly, and was 29*85 at
midnight, and passed the point 30 early on the morning of the
12th. On the 13th and 14th the readings decreased, and
were 29*31 on the latter day, and then it increased with slight
exceptions till the evening of the 23rd, when the reading was
30' 328, being the highest during the month ; it then decreased,
slowly at first, and rapidly during the afternoon of the 27th.
The reading was 29*26 on the evening of the 28th, after which
it increased, and was 30*16 at the end of the month. In
February the average reading from the 1st to the 18th was
30*36. On the 11th, in the evening, the very extraordinary
reading of 30*715 took place. On the 19th the reading de-
scended below 30 inches ; on the 20th the decrease was 0*4 ;
on the 21st and 22nd the reading was about 29*7; on the
24th and 25th the decrease both days was about 0*25, and the
reading was 29*21 on the evening of the 25th ; it increased 0'5

during the Quarter ending March 31, 1849. 371

on the 26th, and still further increased 0*25 on the 27th. The
reading at this time was 29*91 at midnight. On the 28th it
decreased rapidly, and was 29*20 at midnight, being the low-
est in the month. The range within this month was 2*52 in-
ches.

On the 2nd of March the reading passed the point 30 inches,
and on the 6th the highest reading in the month took place,
viz. SO'^tS ; after this the changes were small till the 26th,
the reading being above its average value. On the 27th the
reading decreased half an inch, and on the 28th the lowest
reading took place in the month, 29*18, and it remained low
till the end of the month.

The reading of the barometer on February 1 1 at Aylesbury
was 30*369 ; at Leicester was 30*800; at Durham was 30*440 ;
at Whitehaven 30*62 ; at Newcastle was 30*764 ; at Exeter
was 30*838; at Liverpool was 30*861 ; at Truro was 30*74 ;
at Norwich was 30*910; and at Cardington was 30*846.

The monthly mean values of the several subjects of investi- *
gationare shown in the Registrar- General's report.

The observations have been corrected for diurnal ranges,
and the results are all comparable with each other.

The observer at Southampton has kindly furnished me with
the following agricultural report for Hampshire, the particu-
lars having been supplied by John Clark, Esq., of Finsbury
Farm, near Romsey.

" The weather during the quarter has been most propitious
for cropping. The fine dry March, followed by the gentle
showers of April, have benefited to a great degree both the
soil and cattle. Sowing is in a forward state, and young
wheat looks well.

" The lambing season is over, and it is believed will prove
to be an average. Some strange anomalies have been preva-
lent. On adjacent farms no loss has been experienced in one,
whilst the loss both of ewes and lambs have been great in the
other."

Themean of the numbers in the first column of the subjoined
table is 29837 inches, and this value may be considered as
the pressure of dry air for England during the quarter ending
March 31, 1849. The differences between this number and
the separate results contained in the first column show the pro-
bable sums of the errors of observation and reduction ; the
latter arising partly from erroneously assumed altitudes, and
partly from the index errors of the instruments not having been
determined. In most cases the sums of these errors are small.

The mean of the numbers in the second column, for Guern-
sey and those places situated in the counties of Cornwall and
Devonshire, is 45°'2 ; for those places situated south of latitude

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Mr. J. Glaisher"^s Remarks on the Weather. 373

of 52°, including Chichester and Hartwell, is 41°'4 ; for those
places situated between the latitudesof 52° and 53°, including
Saffron Walden and Leicester, is 40°*7 ; for those places
situated between the latitudes of 53° and 54°, including Derby
and York, is 40^-2 ; and for Whitehaven, Durham and New-
castle is 40°'5. These values may be considered as those of
the mean temperatures of the air for those parallels of latitude
during the quarter ending March 31, 1849.

The average daily range of temperature in Cornwall and
Devonshire was 10°'4; at Liverpool and Whitehaven was
6°*7; south of latitude 52° was 12°-6; between the latitudes of
52°and 54° was 1 1°'5; and at Durham and Newcastle was 1 1°'5.
The greatest mean daily ranges of the temperature of the air
took place at Hartwell, Aylesbury, and Latimer ; in fact, in
and near the vale of Aylesbury; and the least occurred at
Whitehaven, Liverpool and Guernsey.

The highest thermometer readings during the quarter were
65° at Highfield House, 63°- 7 at Hartwell, and 63°-5 at La-
timer. The lowest thermometer readings were 10°"0 at Saf-
fron Walden, ll°-0 at Leeds, and 12°-0 at Hartwell. The
extreme range of temperature of the air during the quarter in
England was therefore about 55°, most likely somewhat less
than this value.

The average quarterly range of the reading of the thermo-
meter in Cornwall and Devonshire was 28°* 1 ; at Liverpool
and Whitehaven was 33°*5; south of latitude 52° was 40°'3;
and north of 52° was 42°- 1.

The mean temperature of the dew-point in Cornwall and
Devonshire was 41°-1 ; south of latitude 52° was 37°-6; be-
tween the latitudes of 52° and 53° was 36°-5, and north of 53°
was 37° 2.

The direction of the wind has been mostly south-west ; at
some few places it seems to have prevailed for some time from
the north-west.

From the numbers in the tenth column the distribution of
clouds has been such as to cover about three-fifths of the whole
sky.

Rain has fallen on the greatest number of days at Wakefield,
Falmouth, Truro and Helston. The average number at these
places was 53. It fell on the least number of days at Oxford,
Saffron Walden, Durham and Leicester, and the average
number at these places was 35. The stations at which the
largest falls have taken place were Stonyhurst, Falmouth,
Whitehaven and Leeds. The falls were smallest in amount
at Durham particularly, York, Holkham and Oxford. The
average fall in the counties of Cornwall and Devonshire was
7*2 inches; south of latitude 52° was 5'1 inches; between

374; On Sines and Cosines of Multiples of a variable angle,

latitudes 52° and 53° was 4 inches; and south of 53°, omitting
Stonyhurst, was 4*3 inches.

The smallness of the fall at Durham is remarkable ; between
January 31 and March 28 only 0*14 inch fell.

The numbers in column 14 to 18 show the mean values of
the hygrometrical results at every station ; from which we find
that—

The mean weight of vapour in a cubic foot of air for all
places (excepting Cornwall and Devonshire) in the quarter
ending March 31, 1849, was 2-8 grains.

The mean additional weight required to saturate a cubic
foot of air in the quarter ending March 31,1 849, was 0*4 grain.

The mean degree of humidity (complete saturation = 1) in
the quarter ending March 31, 1849, was 0'860.

The mean amount of vapour mixed with the air would have
produced water, if all had been precipitated at one time on
the surface of the earth, to the depth of 3*3 inches.

The mean weight of a cubic foot of air at the mean height
of 160 feet under the mean pressure, temperature and humi-
dity, was 547 grains.

And these values for Cornwall and Devonshire were 3*2
grains; 0'5 grain ; 0*878 ; 3*8 inches ; 547 grains, at the mean
height of 120 feet.

Errata. — In the formula for calculating the pressure of dry air, in the
last Number of the Magazine,/or + read — ; and/or 83 inches read 820 feet.

For the formula for calculating the weight of a cubic foot of air, substi-
tute the following :

541 trains /^^^^ght of place in feet above the level of the sea ,q\

^ \ 820 feet /*

LIV. On the Determination of the Coefficients in any series of
Sines and CosiJies of Multiples of a variable angle from par-
ticular values of that series. By the Rcv.Brice Bronwin*.

MY last paper in this Journal having been terminated
rather in haste, I did not observe that the step con-
tained in (16.) might be repeated. Thus

/ 1 N ^'''' , ^''^ / r.\ ^T 3/7r .

COS in— 1) — = 4- cos — , cos (n — S)— — + cos , &c. :

' n ~ n ^ ' n — n

•I ,s.i'n _ . iit i% _ , Siir „

sm (w— 1)— = + sm— , sm {71— S)— = + sm — , &c.
n n ^ ' n n

Therefore if we make

Wj + Un = w„ 2^2 ± "«._ = '^a* &c. ;
2 2 '

V{^Vn — a?!, Vcp^Vn = X^t &c. ;

2 2 ^

* ConuQunicated by the Author.

On some Combinations of Boracic Acid with Oxide of Lead. 375
we may replace the values of B, and A, in (16.) by

„ 2 f «7r Stir 1

Jd; = — <TO,cos \-WqCOS [■"">

' w [ * w ^ n j

h . . (IV.)

Af= — -^57, sin h^osm 1-.... ^

n \^ n ^ n (J

Then if the particular values of/(^ —) be given,

or

tabulated, we may with very little trouble form tables of u^,

u^y &c. ; t^i, ^25 &c. ; and from these last tables of w^, ttJg, &c. ;

a-i, ^2J &c. Thus the labour of computation required by (15.)

would be reduced to one half of the same in (16.), and to one

fourth of it in (17.).

We may reduce (14.) once in a similar manner, omitting

2
the last terms in the values of B^ and A., which will be -f{2v)

and 0 respectively ; but this can be done only once. It ap-
pears, therefore, that (15.) is prefei'able to (14.) when we have
tables of the particular values.

If we employ a smaller value of w, we shall have for certain
values of i

iJc k k

sin- =0, sin {p-i) ~ =0, sin (p + i) j =0;

and we should find

Bo + B„ +B2« +

B^ + B„_i + B„+j + B2„_i +...»..
A.— A„_i + A„+<— Aj8„_< +

in the place of Bq, B., and A. respectively. These might be
employed to determine some of the quantities B^ and A., when
a part of them has been found by the method before given.
Gunthwaite Hall, April 9, 1849.

LV. On some Combinations ofBoracic Acid toith Oxide of Lead,
By Thornton J. Herapath, Esq.*.

l^EUTBAL Borate of Lead, PbO+BO^, maybe obtained

by digesting the heavy white precipitate which is formed

when biborate of soda is added to a solution of any neutral

salt of lead, for twelve or fourteen hours, in a strong solution

* Read before the Bristol Philosophical and Literary Society, April 19,
1841, and now communicated by the Author.

376 Mr. T. J. Herapath on some Combinations of

of caustic ammonia (I., IV.). It would also appear to be pro-
duced when the basic acetate of lead is imperfectly precipitated
by biborate of soda (II., V.), or when an acid solution of either
of the subsequently described borates is supersaturated by
strong liquor a mmonice (III., VI.).

It is a heavy, white, amorphous powder, which is almost
insoluble in water, both hot and cold. It is perfectly inso-
luble in alcohol. It dissolves with great facility in dilute nitric
acid, even when cold, and likewise in boiling acetic acid ; from
these solutions it may be again precipitated unaltered, by
adding a large excess of ammonia. It is easily decomposed
by sulphuric and hydrochloric acids, and likewise by a boil-
ing solution of caustic potash or soda. Before the blowpipe,
it intumesces, gives off water, and becomes dark in colour ;
and at a low red heat fuses into a clear colourless glass, which
possesses a specific gravity of .5*5984<, at 5Q° F., and is softer
than common flint-glass. Heated to redness on charcoal, it
is partially reduced, and the fused mass contains numerous
globules of metallic lead.

The following are the results of my analysis of the hydrated
salt, after it had been exposed to a temperature of about 212°F.,
in a Liebig's drying-tube, for three or four hours,

I. 9*2 grains were heated to redness in a platina capsule ;
loss =0-598 gr. = 6-5000 per cent.

II. 10 grs., treated as before, lost 0-683 gr. in weight
= 6-8300 j)er cent.

III. H-Ol grs. lost 0-684 gr. in weight=6-2125 per cent.

IV. 10 grs. were dissolved in dilute nitric acid, and the
solution precipitated by an excess of diluted sulphuric acid;
PbO, SO^ 9-77 grs. = PbO 7-198 grs. = 71-9800 per cent.

V. 20 grs. gave of PbO, SO^ 19-80 grs. = PbO 14-578 grs.
= 72-8900 per cent.

VI. 20 grs. gave of PbO, SO^ 19-03 grs. = PbO 14-0221
grs. = 70- 11 05 per cent.

I. II. III. IV. V. VI. Mean.

HO . 65000 6-8300 62125 ... ... ... 6-5141

PbO . ... 71-9800 72-8900 70-1105 71-6601

BO^ . .. ... ... 21-8258

Now, if we consider the composition of this hydrated salt to
be represented by the formula PbO, BO'^-f HO, it ought to
contain —

Water .... 1 9 or 5-7692 per cent.

Oxide of lead. . 1 112 ... 71*7939

Boracic acid . . 1 35 ... 22*4369
This salt begins to lose water between 250° and 300° F. ; and
by long-continued desiccation at a temperature of from 450°

Boracic Acid with Oxide of Lead. 377

to 500*' F., it may be rendered perfectly anhydrous without
experiencing any perceptible change of colour.

Sesquiborate of Lead, 2PbO + 3BO^. — The salt which is
produced when a boiling solution of the nitrate, or any other
soluble salt of lead is precipitated by a great excess of bibo-
rate of soda, has been hitherto considered by chemists to be
composed of PbO + SBO^. According to my experiments,
however, it would appear to consist of 2PbO + SBO^. It is a
white powder, like the preceding, which it closely resembles
in its properties. Before the blowpipe it fuses into a colour-
less glass, the specific gravity of which is rather lower than
that of the neutral borate, being 5*2352; its hardness is very
nearly equal to that of flint-glass. The following are the re-
sults of my analysis of this salt dried at 2] 2'^ F. : —

I. 10 grs., heated to redness, lost 0'918 gr. in weight
= 9-1800 per cent.

II. 11-26 grs., treated as before, lost 1-106 gr. =9-8223
per cent.

III. 25 grs. lost 2-3 grs. in weight=9-2000 per cent.

IV. 10 grs. gave of PbO, SO^ 8-62 grs.= PbO 6-3515
= 63-5150 per cent.

V. 10 grs. gave of PbO, SO^ 8-34 grs. = PbO 6-14.52
= 61-4520 per cent.

VI. 10 grs. gave of PbO, SO^ 8-41 i grs. = PbO 6-1975
= 61-9750 percent.

I. II. in. IV. V. VI. Mean.

HO . 9- 1800 9-8223 9-2000 ... ... ... 9-4007

PbO 63-6150 61-4520 61-9750 62-3140

B03 ... ... ... 28-2853

These numbers indicate a composition very closely approxi-
mating to the formula 2PbO, 3BO^ + 4HO, as will be seen
upon comparing them with those given below : —

Water 4 36 or 9-8630 per cent.

Oxide of lead . . 2 224 ... 61-3690
Boracic acid ... 3 105 ... 28-7680
Dried between 350° and 400" F., it loses two of its atoms of
water, and its composition is now expressed by the formula
2PbO, 3B03 + 2HO.

I. 10 grs. of the salt in this state of hydration lost 0-495 gr.
upon being heated to redness, =4*9500 per cent.; calculation
requires 5-1873 per cent.

Biborate of Lead, PbO + 2BO^, may be easily obtained by
boiling either of the preceding recently-precipitated salts,
whilst still moist, in a concentrated solution of boracic acid.
It is a light amorphous powder, which at a red heat fuses with
difficulty into a vitreous mass. From the almost impossibility,

378 Mr. T. J. Herapath on some Combinations of

however, of obtaining this glass free from air-bubbles, I was
unable to ascertain its true specific gravity. It was slightly
superior to flint-glass in hardness.

The hydrated salt, dried for some time at 212^ F., yielded
upon analysis the following numbers: —

I. 10 grs., when heated to redness, lost \'519 grs. in weight
= 15*790 per cent.

II. 25 grs., treated as before, lost 4'021 grs. in weight
= 16"084< per cent.

III. 10 grs. gave of PbO, SO^ 7*07 grs. = PbO 5-2095
= 52-095 per cent.

IV. 25 grs. gave of PbO, SO' 17-784. grs. = PbO 13-1040
= 52*4' 16 per cent.

I. II. III. IV. Mean.

Water . . . 15*790 IS-OSi ... ... 15*9370

Oxideoflead. ... ... 52-095 52*4.16 52*2555

Boracic acid , ... ... ... ... 31*8075

Now, supposing its composition to be expressed by the for-
mula PbO, 2BO^ + 4HO, it ought to be composed of—

Water 4 36 or 16*513 per cent.

Oxideoflead. . . 1 112 ... 52*376
Boracic acid ... 2 70 ... 32*111

Dried between 400° and 450*^ F., it contains 4*435 per cent,
of water=one atom ; calculation requires 4*712 per cent. •

NitrO'Borate of Lead. — When either of the above-described
borates of lead are dissolved in moderately strong nitric acid
to saturation, the solution filtered and concentrated by evapo-
ration until a pellicle appears upon the surface, and then
allowed to cool, the sides of the vessel containing the solution
in a short time become covered with numerous irregular,
glistening crystals. These, when heated to somewhat above
250° F., become nearly opake, slightly decrepitate, and give
off water and traces of nitric acid vapour. Heated to redness,
they evolve large quantities of nitrous acid fumes, and the re-
sidue fuses into a transparent colourless glass. They are,
therefore, obviously a nitro-borate of lead; but from the dis-
cordant results of my analyses, I have as yet been unable to
satisfy myself with regard to their true composition. They
are most probably composed of PbO, BO^, -f PbO, NO^
+ HO.

Chloro-Borate of Lead. — This curious and interesting double
salt was formed accidentally whilst attempting to prepare a
borate of lead by precipitating a hot solution of biborate of
soda by a boiling concentrated solution of chloride of lead.
By filtering the mixed solutions whilst still warm, and wash-.

Boracic Acid with Oxide of Lead. 379

ing the white flocculent precipitate which remained upon the
filter with lukewarm water, the new salt was obtained in a
state of purity.

This, when examined under a microscope of high power,
was found to consist of exceedingly minute irregularly-acicular
crystals, which depolarized light and possessed a nacreous
lustre.

The compound thus obtained does not appear to be acted
upon by cold water ; boiling water, however, slowly but gra-
dually decomposes it into its constituent salts. It is perfectly
insoluble in alcohol. It dissolves with facility in hot dilute
nitric acid, being decomposed, and chloride of lead set free,
which, upon cooling, separates from the solution in long
needle-formed crystals. When heated to from 250° to 300"' F.,
it loses about 3*59 per cent, of water, and then becomes anhy-
drous. At a low red heat it readily fuses into a clear amber-
coloured globule ; this, upon cooling, solidifies into a trans-
parent and almost colourless glass, which is slightly opales-
cent. When heated to redness, however, on charcoal, or in
an open platina capsule, it behaves differently ; white fumes
are now given off, and the fused mass becomes gradually
darker in colour, and of a thicker consistence, until it very
much resembles melted sulphur in appearance. If it be now
allowed to cool, it will be found to have undergone a very
considerable change ; it rapidly concretes into an opake, straw-
coloured brittle mass, which is made up of a multitude of long,
radiating, acicular crystals, and bears a striking resemblance
to molybdic acid.

The crystallized hydrated salt, dried by exposure to sul-
phuric acid at the ordinary temperature, yielded upon ana-
lysis the following numbers : —

I. 4*42 grs. were taken, and heated to redness in a tube of
Bohemian glass ; the aqueous vapour having been drawn off
by suction, and the apparatus allowed to cool, it was found to
have lost 0-180 gr. in weight=4<'072 per cent.

II. 3*86 grs., treated as before, lost 0*12 gr. in weight
= 3*109 per cent.

III. 5*00 grs. were dissolved in boiling dilute nitric acid,
and the solution was precipitated by nitrate of silver; Ag, CI,
(fused) =2-60 grs. in weight =0-64.11 Cl= 12*822 per cent.

IV. 3-32 grs., treated as above, gave of Ag CI 1*82 grs.
= Cl 0-4487 gr.= 13-515 per cent.

V. 4-581 grs. gave of FbO, SO^ 4*824 grs. = Pb 3-3009
= 72-0580 per cent.

VI. 3-6 grs. gave of PbO, SO^ 3-833 grs. = Pb 2-6226 grs.
= 72-8510 per cent.

380 Notices respecting New Books.

I. II. III. IV. V. VI.

Water . . 4072 3-109 ... ... ... ... 3-5905

Chlorine 12-822 13-515 ... ... 131685

Lead 72-058 72-851 72-4545

Oxygen. . l in-ySf?-^

Boracicacid/ - 1« 7865

The formula that agrees best with these numbers is PbO,
BO^ +Pb, Cl-f HO; supposing this to be its composition,
it ought to contain —

Water 1 9 or 3 '04)0 per cent.

Chlorine .... 1 36 ... 12-162. ...

Lead 2 208 ... 70-030

Oxygen 1 8 ... 2'703

Boracic acid ... 1 35 ... 12*065
The excess of chlorine and lead shown by the analysis was
doubtlessly owing to the difficulty of entirely removing the
excess of chloride of lead, which was carried down by the
salt, without producing a decomposition of the salt itself.

All subsequent attempts to reproduce this compound having
failed, I have been unable to verify the above results by a re-
petition of my analysis.
Mansion House, Old Park,
Bristol, March 4, 1849.

LVI. Notices respecting New Books.

Statistics of Coal. The Geographical and Geological distribution of
Fossil Fuel or Mineral Combustibles employed in the Arts and Ma-
nufactures : their production, consumption, commercial distribution,
prices, duties, and international regulations, in all parts of the tvorld ;
including four hundred statistical tables and eleven hundred analyses
of mineral bituminous substances. With incidental statements of the
statistics of Iron Manufactures, S[C., derived from official reports
and accredited authorities. Illustrated ivith Coloured Maps and
Diagrams. 5y Richard Cowling Taylor, i^.G.S. London: John
Chapman, 142 Strand. Philadelphia: J. W. Moore. 1848.

COMPREHENSIVE as the title of this work appears, it does not,
yet, convey a just idea of its scope, or of the extent of its sub-
ject-matter. Did its title stand, " Coal, the civilizer ; its natural
history, production and applications," it would perhaps convey to
the casual reader a more just idea of the object and contents of the
work. We confess thait we ourselves closed the book with very
different feelings from those with which we opened it. We have no
hesitation in saying that we have long ceased to entertain that ex-
traordinary respect for mere statistics which it has been very much
the habit of late years to inculcate. We have seen too many in-
stances, and too many instances are daily occurring, in which sta-
tistics are made the mere instrument of the partizan and the theorist.

Notices respecting Neiv Books, 381

He must have been very unobservant of public events of late years
who is not aware that statistics may be made use of with equal con-
fidence to support any side of any question. Hence the cautious
inquirer who really desires to get at some actual and permanent
result, will always look with extreme suspicion upon every thing
that comes before him with an ostentatious parade of statistics,
aware that there is nothing so easily abused, nothing which is more
liable to abuse.

But a perusal of these pages has shown us that mere statistics form
but a very subordinate part of the design of the author. A long and
intimate practical acquaintance with mines and mining operations in
different parts of the world had necessarily led him to amass a great
quantity of materials ; the value of which, as a constant object of
reference for his own use, led him to feel the utility of a digested
and methodized arrangement of those materials, in a permanent
shape, for the use of others. But there is found throughout these
pages a pervading spirit beyond that merely materialistic and dry
one which the title would indicate, and which the professional en-
gagements of the author might have led us to anticipate. We per-
ceive impressed on every section the idea, not of coal the mere wealth
producer, the mere material instrument of the human animal, but of
coal as an important agent in promoting civilization. " We take it
for granted," are the first words of the introduction, *' that every one
who may chance to peruse the summary of statistics of mineral fuel
which we have embodied in the present section will be impressed
with the immense importance of those substances, particularly as
developed of late years ; how vastly enlarged the area and bulk of
their production in all countries ; how essential they now are to
the comfort of the human family ; how much they have done towards
the extension of the useful arts ; how gloriously they have aided the
progress of invention and improvement ; how mighty are the results
which have followed their increased application." (P. xiii.) And
again in p. Ixxxiv. the author justly says; "Respecting the won-
drous influence which the employment of mineral combustibles has
had, even in our own days, upon the whole world, by the acquisition
of new forces ; by the extension of mechanical powers, of manufac-
turing capabilities ; by the impulse given to the industrial arts and
the creation of new sources of wealth ; by rapid and cheap modes of
transportation and enlarged commercial facilities ; above all, by the
improved condition of the people, we will not here dilate. Abundant
evidence of all these will be found in this volume." It is in the same
spirit, and imbued with the same everywhere pervading high moral
sentiment, that the author more than once (pp. xiv. and xxxviii)
calls attention to the vastly greater importance of iron than of gold
and silver, — a truth which it is not by any means beside the mark
to touch upon in these days of California-mania. Adam Smith long
ago remarked that the adventurer in a silver mine ran every pro-
bability of being ruined, but that the adventurer in a gold mine
was certain of being ruined. It will not be amiss to put, beside
such an authority, the following passage from the work before us : —

i82 Notices respecting New Books.

" It would be no difficult task to show in figures how vastly more
profitable is the application of labour in the mining and working and
transporting of coal than in that of the precious metals. The an-
nual production of all the gold and silver mines of North and South
America was estimated by Baron Humboldt at 9,243,000/., and at
present at less than 5,000,000/. Now the value of the coal produced
annually, in Great Britain alone, is computed at near 10,000,000/,
at the pit's mouth, and at from 15,000,000/. to 20,000,000/. sterling
at the places of consumption. At the same time, the value of the
iron brought into a manufactured state through the agency of this
coal is 17,000,000/. more. We shall enter more particularly into
this subject in a future page. We cannot but mark also the supe-
rior character and condition of the inhabitants of the coal-producing
and consuming countries, such as those of the northern hemisphere,
especially since the introduction of steam-power, to that of the
people of the southern and tropical latitudes, to whom coal has either
been wholly denied or is not applied to any use. The industry, acti-
vity, moral culture, and intelligence concentrated around any of the
great depositories of coal and iron in the temperate regions, have no
parallel in the countries from which such treasures have been with-
held." (P. xiv).

And it is not only in these respects that the author departs from
a mere dry statistical detail. He justly considers every matter con-
nected with the history of the formation of coal, and with its most
important applications, to be necessary parts of the information which
will be desired by those who would thoroughly understand the
subject. Thus we find him including the iron manufacture, and the
extent and application of railroads and steam-vessels, as parts of his
subject; " so closely," he justly observes, " do all these matters seem
interwoven with each other." But, beyond this, a very large amount
of valuable and very interesting information is communicated on the
methods of working mines; on the casualties to which mines and
miners are liable ; and on the various means which have been
adopted in various countries for the mitigation of these casualties,
and for the promotion of the healthiness and security of this occu-
pation. Benefit societies and provident institutions, as they exist
in mining districts, claim a large share of the author's evidently
cordial and sympathetic attention.

A very interesting and comprehensive sketch is further given of
fossil botany and of the organic remains found among the coal
measures. The author's observations on the interest of this branch
of his subject seem to us so just and pertinent that we transcribe
some of them.

" In intimate connection with the matter of the present volume,
a knowledge of the forms, the botanical classification, the geological
arrangement, of the vegetable remains of our ancient world seems to
be almost indispensable. It embraces facts, at least, sufficiently
valuable to ensure for it, as a collateral branch of natural science, a
conspicuous section of this book. Independently of its usefulness,
there is a never-failing interest attached to such an investigation

Notices respecting New Books. 383

which enables us to trace the history, as it were, of the past con-
dition, the present adaptation of the primaeval flora ; that magnificent
vegetation which amidst the mutations of our planet yet survives for
our use ; its character changed, it is true, but only to become more
serviceable to man."

" A happy provision was it that secured for the ultimate advantage
of the human race, ages before its appearance upon the globe, the
trees of gigantic size, the densely growing shrubs, the most delicate
even of the lesser plants — that flora which covered in such pro-
fusion the islands and plateaux, and filled the humid valleys, of the
early world. A happy provision was it that, amidst the early ca-
tastrophes of the earth, those convulsions which modified its entire
surface, overwhelmed its primaeval forests, and buried them beneath
enormous accumulations of earthy debris, of sediment, and of rocky
debacle, — still perpetuated and matured, during the lapse of countless
ages, that primitive vegetation which finally, in the form of mineral
combustibles, we are now busy in exploring, mining, and appro-
priating in a thousand ways and for a thousand purposes. A happy
provision was that, — a beneficent one surely, — by which, at the mo-
ment when man is compelled to level the existing forests to make
room for the progress of agriculture and the cultivation of the present
surface, he finds nigh at hand, yet buried beneath that surface
within the shallow basins and woody islands of the antediluvian world,
those inexhaustible stores of a combustible now rendered infinitely
more precious and effective than that existing vegetable fuel whose
destruction is the inevitable consequence of advancing civilization."
(P. Ixxxiv.)

It must be very obvious that the labour of collecting the materials
for a volume so wide in its scope, so multifarious in the subject-
matters of which it treats, must have been very great, and the task
one of great difficulty. " The information required was not access-
ible in any single work, nor even in a number of works : it was no-
where to be found." In a work like this it is no slight matter to
have gathered together, in an accessible shape, certain and definite
information which may be used as a key by other inquirers in indi-
vidual regions or departments ; which may have something of the
character of a complete skeleton by which the general relations of
the various parts of the subject may be seen. That imperfections
should be found to exist in the work when any single part is sub-
mitted to a rigid test, is a necessary result from the very nature of
the subject, in which new facts are hourly arising and being re-
corded. But the value of this, as a general work of reference, will
not thereby be lessened.

There is one cause of the difficulty with which our author has had
to grapple on which it is not uninteresting to remark. This is the
circumstance that, while there are " oflScial" returns and other do-
cuments in all the continental countries of Europe, none such exist in
the two by far most important coal-producing countries in the world,
namely Great Britain and the United States of America. Our author
several times calls attention to this circumstance, as having occa-

384 Notices respecting New Books.

sioned him much difficulty. Though, however, he sometimes does
intimate something like a wish that we had, in Great Britain, a
" corps of state engineers," he does not, like many theorists, allow this
personal inconvenience to himself to blind him to the true circum-
stances out of which the want of such " official" returns arises. He
is awake to the fact that such " official" completeness is only to be
obtained at the price of the sacrifice of national liberty and individual
independence. He candidly admits that such a " process must, at
all times, be unpopular, and the results extremely uncertain. This
species of investigation savours too much of scrutiny into the private
concerns of men." (P. xl.) The volume before us supplies additional
illustrations to the numberless ones which every honest inquirer will
find, of the importance to the prosperity of any country, and of any
branch of industry, that the latter should be unshackled by the med-
dling interference of government officials. It is a heavy price to pay
for the merely superficial, but never really reliable, result of regularly
published official returns, that enterprise should be checked, indi-
vidual energy cramped, self-dependence prohibited ; and that two
or three revolutions in a century should be necessary to keep the
state from the anarchy of despotism. There is too much tendency
in England at this time to follow in that centralizing path which
has brought so much suffering on the continent. The specious pre-
tences of schemers and theorists have already succeeded too far in
their attempts at this official interference. We are quite content to
have it still said that " as there is no system of supervision adopted
in the mining regions here, as in all the other countries of Europe,
it is impossible to arrive at any exact account of the quantity of
coal which is annually raised in the mines." (P. 259.) It is far
more satisfactory to us than the most perfect returns could be to
find it stated that " what the wise direction of public authority has
established in Germany, the spirit of association, the sentiment of
individual independence, the habit of calculation and of observation,
have consecrated in Great Britain." (P. cxxvi.) And, while we have
every reason to be grateful for " the bounteous supply of mineral
wealth which nature has assigned to England," we are infinitely more
grateful for the spirit of independence which has resisted that de-
spotic and pernicious " system of supervision" which has elsewhere
prevailed ; for that " enterprising character of her people, who have
turned that supply of mineral wealth to such good account." (P. 257.)
Mr. Taylor, very properly, does not confine himself to that de-
scription of mineral which is commonly called Coal. He includes
full and valuable information on the Lignites of the geological for-t
mations above the carboniferous group, and also on the recent Peat.
It is clear that, in strictness, all these may with exactly as much
propriety be included under the term " Coal" as the substance itself
which is commonly known by that name. The use of that substance
is comparatively modern ; but the word itself is an ancient Saxon
one, and one common indeed to the dialects of the old northern lan-
guages. As our author dates from the other side of the Atlantic, it
will not be out of place to illustrate this fact by a reference to an

Notices respecting New Books. 385

ancient Icelandic Saga in which we find this word used in a way
which clearly shows that something very different was meant hy it
from that heavy mineral which we commonly now call " coal." In
the Saga which relates the explorations of Thorfinn Karlsefhi on
the American coast, a.d. 1008, we find it related that, when his
party were on that part of the coast now known as Rhode Island,
they had some encounters with the natives. It is expressly told
how these latter came, on one occasion, in their canoes in numbers
" as many as if coals had been strewn upon the bay" (svd marl sent
kolum vceri sat fyrir hopit). What the sort of coals were which,
as floating on the surface of the water, suggested this simile we will
not undertake to say, but they were certainly much more likely to
have been charcoal or peat than stone coal. The name remains
however no less apt at this day than it was at that in its application
to the substance, in each case of vegetable origin, which is used as
fuel, — whether that substance be the peat, spongy and light, of yes-
terday's gi'owth, or the prostrate giant trees, compressed and heavy,
which grew and flourished and were embedded in ages of unknown
remoteness. It is interesting to find that the following passage will
properly include coals of every age, and of every growth, and of
every shade of meaning of the word, ancient or modem : —

" Each stratum of coal is the product of a peculiar vegetation,
frequently diff'erent from that which precedes and that which follows
it, — vegetations which have given rise to the superior and inferior
layers of coal. Each stratum resulting in this manner from a distinct
vegetation, is frequently characterized by the predominance of
certain impressions of plants ; and the miners, in numerous cases,
distinguish the diff'erent strata which they remove by the practical
knowledge they possess of the accompanying fossils. Any seam of
coal and its overlying rock or slate should consequently contain
the various parts of the living plants at the period of its formation :
and, by carefully studying the association of these various fossils,
which form so many special floras, containing generally but few spe-
cies, we may hope to be able to reconstruct these anomalous forms
of the ancient world." (P. xc.)

The value of this volume is greatly enhanced by a series of maps,
in which the position and extent of all the ascertained coal-basins
throughout the world are laid down.

It is certainly a remarkable spectacle to see the extraordinary
increase of the production and consumption of mineral coal, and the
changes which have been wrought in the habits of millions of hu-
man beings thereby. Our author tells us that " in Great Britain coal,
according to some authorities, was mentioned as occurring in England
as early as the ninth century, a.d. 853 [query, stone coal, or such coal
as above alluded to in the Saga of Thorfinn Karlsefni] . It was cer-
tainly known and applied to various economical purposes in the middle
of the twelfth century. In 1239,King Henry III. granted the privilege
of digging coals to the good men of Newcastle. But it is little more
than 250 years since coal came to be in general use, as fuel, in
London. Upon its first introduction there, one or two ships were

FMl. Mag. S. 3. Vol. 34. No. 230. May 1849. 2 C

386 Notices respecting Nevo Books,

sufficient for the whole trade. At the present day there are several
thousand ships constantly engaged in the transportation of that
combustible," In 1845 upwards of thirty-four millions of tons of coal
were produced in Great Britain (p. 259) ; and in the same year
11,987 ships' cargoes of coal were entered for duty at the port of
London alone. As many as 282 cargoes, — amounting to upwards of
80,000 tons, — have been sold in the City of London Coal-market in
one day (p. 263). And the iron manufacture has followed with no
halting steps upon this amazing increase in the production of coal.
It is not seventy years since William Hutton wrote his History of
Birmingham. He alludes to an iron furnace in the neighbourhood
of that town, and calculates the age of those iron-works from the
mound of calx or cinder which lay near the refuse of the furnace.
Reckoning by the ordinary rate of the increase to that mound, even
at the accelerated ratio of his own day, and while in constant active
work, he concludes that that very furnace must have been in active
working there for at least three thousand years, — twelve centuries
prior to the invasion of the Romans. And his calculation was pro-
bably within the m£irk instead of exceeding it*. But, within the
part of one century which has passed since he wrote, many and
many a mountain has grown up, to the disfigurement, alas ! of
many a fair and fertile plain, in many parts of England, any one
of which throws that ancient mound at Aston, the growth of more
than thirty centuries, altogether into the shade ; and this, although,
from the superior manner of working the ore, a given quantity would
not leave nearly so much refuse as formerly. This curious calcu-
lation of Hutton's would have formed a striking introduction to Mr.
Taylor's account of the progress of the iron manufacture. It ap-
pears from his work that the production of iron in that same district
(the StaiFordshire district) had increased from 13,210 tons in 1796,
fifteen years after Hutton wrote, to 500,760 tons in 1846; the total
produce of Great Britain in the same year being upwards of
2,000,000 of tons.

Another striking picture of the progress in the production of
coal, with all its innumerable physical and moral consequences, may
be drawn entirely from our author's pages. He is speaking of the
anthracite beds of Pennsylvania. A quarter of a century ago, he
tells us, a few tons of an unknown combustible were brought to Phi-
ladelphia from a wild and desert tract, known by the not unapt title
of "the Wilderness of Saint Anthony."

" But the miner," proceeds our author, " has entered into this
Wilderness of Saint Anthony, and canals have penetrated it, and
railroads have traversed it ; basin after basin of this combustible has
been discovered in it ; tract after tract has supplied productive col-
lieries in it; until in a single year, 1847, it had furnished the sur-
prising amount of 3,000,000 of tons, and 11,439 vessels cleared from

* See, further, a very interesting paper in the last Number of this
Journal, by Mr. John Phillips, calling attention to the antiquity of metal
works in England.

Notices respecting New Books. 2>» /

the single port of Philadelphia in that season, loaded with a million
and a quarter of tons, for the service of the neighbouring states."
(p. 20.) Further on, the author, in the spirit which we have already
remarked as pervading the entire work, concludes another division
of the same subject by the observation : " It is beyond the scope of
human vision to contemplate, in our day, the results associated with
these millions, — the industrial facilities, the wealth, and power, and
influence at home and abroad, which they must inevitably confer
upon the future inhabitants of the country." (P. 33.)

Another illustration of a more special nature, of a means of the
consumption of coal accompanied by a vast amount of benefit and
comfort to the community, is touched upon in this work, which we
cannot omit to notice. " It was in 1803," says the author, " that
Mr. Winsor first exhibited the effect of gas-light at the Lyceum
Theatre, London." It appears that there were in London in 1845
nineteen gas companies, who produced, on an average, 10,000,000
cubic feet of gas every twenty-four hours, which is at the rate of
3,650,000,000 cubic feet a year. The whole number of lights is
calculated at 100,000. In 1838 these London gas companies alone
consumed 340,000 tons of coals annually.

It would be obviously impossible to direct attention to all the
matters, however great their interest, which are discussed in this
volume. We must content ourselves with referring to two more
points, both of them of much importance, and both of them among
those numerous ones which we have already said that the title of the
volume will hardly lead the reader to expect to find discussed
within it.

The comparative merits of anthracite and bituminous coal are very
fully discussed in all their bearings, as to the application of each,
both to domestic use and to manufacturing purposes. We are espe-
cially glad thus to see done "justice to anthracite in pointing out
the incalculable value of a species of fuel previously rejected and
despised, as amongst the most inferior and most impracticable of all
the combustibles." (P. 365.) It cannot be denied that an absurd
prejudice still exists in this country against the use of anthracite
cobJ. Our countrymen seem in love with smoke. Two centuries ago
*' the famous city of London petitioned the parliament of England
against two nuisances or offensive commodities which were likely to
come into great use and esteem," one of which was " Newcastle
coals, in regard of their stench," &c. (P. xl.) But the citizens seem
to have now become in love with that same stench ; and, while no
effectual means have been taken, as they might be, to prevent the
waste of material which takes place in the escape of smoke, —
which is the mere result of imperfect combustion — the use of the
smokeless anthracite encounters hopeless prejudices. Our author
well remarks, that "the difiiculty suggested about ignition, evea
were it found so in practice [wliich it is not in reality], is deprived
of all weight, from the consideration that, with ordinary attention,
a fire, when once kindled in the fall of the year, may be kept up
until the return of summer if needed. The supposed tendency of

2 C 2

388 Intelligence and Miscellaneous Articles,

anthracite to emit a greater amount of noxious vapours during com-
bustion than bituminous coal, is contradicted by the daily experience
of those who employ the former in their apartments, and. is much
less objectionable on that head than bituminous coal. Our tables of
analyses at the end of this volume will, if doubts remain, decide
this matter. In fact, they show that the anthracites contain less
sulphur than the blazing coal." (P. 92.)

We can speak on this matter from our own experience, having
ourselves used the anthracite coal for many years, both in open
grates and in various other ways, and having always found it pro-
ductive of far less trouble and of far greater comfort than any kind
of bituminous coal. We cannot conceive the inducement by which
we should be persuaded to return to the use of the smoke-giving
system. The matter is one of importance in other respects in this
country, besides those of present health, cleanliness, and comfort ;
for we are told by our author that '* Great Britain possesses a far
larger area of anthracite than exists in America or any other part
of the world." (P. 92.)

Another matter, not unconnected in several respects with the last,
is the incredible amount of waste which takes place annually in the
coal after it has been actually fetched to the pit's mouth. It ap-
pears that nearly one-third part of the best coals are thus wasted at
Newcastle, — amounting to more than a million of chaldrons of coal
annually (p. 368). The author directs special attention to this
subject in a section (p. 367) on " preparediuel," in which he notices
the fact of clay balls — coal dust and clay mixed — having long been
in use in Wales and elsewhere, and being actually preferred to pure
lump coal, where both are at hand. He gives a sketch of several
attempts made to introduce the use of different compounds, by which
at the same time the present enormous and quite needless annual
waste would be saved, and a fuel provided as convenient, oeconomical
and useful as the best coal.

We cannot conclude without cordially recommending this work
to the attention of our readers. While it will be an invaluable book
of reference to every future inquirer into the numerous ceconomic
questions connected with our most important industrial operations
and manufactures, and into the great social questions arising out ol
them, it will form an indispensable part of the library of every
geologist.

LVII. Intelligence and Miscellaneous Articles.

SNOWY MOUNTAIN IN EASTERN AFRICA.

THE Rev. Mr. Rebmann, of the Church Missionary Society's East
Africa Mission, has recently sent home an account of a journey
made by him into the interior. At about 100 miles due west from Mom-
bas, in 4° S. lat., he came to the foot of an elevated table-land, and
saw before him a lofty mountain named Kilimandjaro, the summit of

Intelligence and Miscellaneous Articles. 389

which is covered with perpetual snow. The elevation of this moun-
tain can scarcely be less than about 20,000 feet, and from other
sources we learn that it is crossed by the road to the country of Mono-
Moezi. In the native languages of this part of Africa " Moezi "
means " moon ; " so that it is not unreasonable to conclude, as Dr.
Beke does, that Mount Kilimandjaro forms a portion of the " Moun-
tains of the Moon," in which Ptolemy places the sources of the Nile,
and the snows of which he describes as being received into the lakes
of that river.

It is by proceeding into the interior westwards from Mombas that
Dr. Bialloblotzky, whose exploratory journey into Eastern Africa has
on several occasions been noticed in this Journal, expects to reach
the sources of the Nile ; and the discovery of this snowy mountain
and high table-land so near the coast argues favourably for the suc-
cess of his undertaking. According to the last letters received from
him, he arrived on January 3rd at Muscat, whither he had gone on
by steamer from Aden and Maculla ; and he was there looking for a
native vessel to take him across to Mombas or Zanzibar.

ON MR. STRUVE S MINE VENTILATOR. BY J. RICHARDSON, C.E.

This machine has now been three weeks in full operation at the
Eaglesbush Colliery, near Neath, and the unequivocal success which
has attended it is a matter of sincere congratulation, not only to the
talented inventor, but to all engaged in raining. Its beautiful simpli-
city of design, its easy adaptation to the peculiar circumstances of any
mine, and its certainty of effect, are its chief characteristics ; whilst the
comparatively small amount of capital required in its construction, and
the slight annual expense incurred by it, are strong recommendations
for its general adoption. It is well known, that in the best-managed
collieries recourse is had to the furnace as a means of ventilation —
not because it is perfect, but as the best system known. Without
entering into a description of this mode, with which most of your
readers are familiar, it will perhaps be suflScient to mention some of
the most serious objections to it, and see how far they are remedied
or avoided by this invention.

After describing the imperfections and evils of the modes of ven-
tilation by furnaces, Mr. Richardson continues : — By Mr. Struve's
machine all the advantages resulting from the use of the furnace are
retained and augmented, additional benefits are secured, the evils
complained of are removed, and are not replaced by others ; at least
such is the opinion of the writer, who devoted a day to the careful
examination of it and its effects, both above and underground, and
who is uninfluenced by any partiality arising from pecuniary inter-
ests or connexion with either the inventor or the proprietors of the
colliery.

By referring to the annexed plans, section, and description of the
machine now in operation at Eaglesbush, the reader will be able

390 Intelligence and Miscellajieous Articles.

readily to understand the construction of the ventilator and the mode
of its operation. The machine could have scarcely been tried under
circumstances more unfavourable to its success than in this instance ;
for independent of the additional friction caused by drawing a large
quantity of air through ways of little more than 11 feet area,
owing to the men having been watered out of that part of the mine
where the principal works have recently been carried on, it was found
needful to change the direction of the air- ways, and, in consequence,
it is at present conducted through temporary passages, which are
ill- calculated for such a purpose, and which permit an immense leak-
age of the air into the waste parts of the colliery. The enlarge-
ment of the areas of the upcast shaft, which is now only 3 feet dia-
meter, and of the air- ways, is now in progress, which, when com-
pleted, will materially add to the effective performance of the ma-
chine. The engine, too, is an old one, and has been injured by long
exposure, is less than half the power necessary to work the machine
to its full effect, and is of a defective construction. Yet under all
these disadvantages and impediments to the development of its
powers, the machine worked steadily at 1\ strokes per minute. The
diameter of the aerometers is 12 feet, and the length of stroke 4 feet.

Therefore 12 feet diameter = 113 feet area x 2 = 226 feet area
and 4 feet stroke x 2 = 8 x 7^ strokes per minute 60 feet velocity

= to 13,560 cubic feet of
air drawn out of the mine per minute. The greatest quantity of air
passing through this mine previous to the erection of this machine
was 3000 cubic feet per minute ; whilst this machine, if worked to
its full extent, is capable of drawing 40,000 cubic feet per minute.
By increasing the diameter of the aerometers to 15 feet, then
70,000; and if to 25 feet, then 125,000 cubic feet of air per
minute would be drawn out of the mine, provided the engine-power
was also increased. No sooner was the machine set to work, than its
effects were immediately felt in every part of the mine. Stalls in
which the fire-damp was so prevalent that it required the utmost
caution to be used even with the Davy lamp, the cylinders of which
were so heated as to require to be frequently taken into another part
of the mine to be cooled, were cleared of this dangerous enemy as if
by magic, and all indications of the presence of fire-damp vanished ;
indeed, so effectually has the machine removed all apprehensions of
danger, that naked lamps and candles are now substituted for the
safety-lamp. Even thewaste parts of the mine, which are at a consider-
able distance from the direct course of the air- way, and which were so
foul and fiery as to render the introduction of even a safety-lamp into
it very hazardous, were unexpectedly, and to the astonishment of the
men, completely cleared. The abandoned stalls, which have hitherto
been magazines of explosive air, can now be entered with safety with
a candle ; and the whole atmosphere of the mine is so much improved
and purified, that, according to the concurrent testimony of both
masters and men, a collier now cuts three tons of coal with less

Intelligence and Miscellaneous Articles. 39 1

fatigue than he could previously cut two tons. The effect of the
machine in clearing the colliery of noisome vapours is plainly indi-
cated by the offensive odour of the air discharged from it, and by the
fact observed yesterday, of a dense volume of powder smoke issuing
from the outlet valves of the machine, almost immediately after the
discharge of a blasting shot in the mine. When the ventilator was
first put in operation, the furnace, which is situated at a short di-
stance from it, in one of the air-tunnels on the surface, was also in
action, when, with almost the first motions of the pistons, the fire
was swept off the bars, and the red-hot cinders carried along the
inlet air-passages to the aerometers — a fact clearly illustrative of the
superior draught of this machine, as compared with that caused by
the furnace.

Other facts, confirmatory of what has been stated, might be added ;
but it is presumed sufficient has been said to prove that, although
tried under great disadvantages, yet the success of this new mine
ventilator has been unequivocally demonstrated ; and that with the
slight improvements which experience may point out as expedient,
this mode of ventilating mines will constitute a means far superior
to the furnace, high-pressure steam, or any other mode which has
hitherto been attempted.

It will be seen, on a reference to the plans, that it is perfectly un-
affected by the thermometrical and barometrical changes in the
atmosphere ; that it is capable of being so constructed, as to double
or treble the quantity of air ordinarily required, on the occurrence of
a diminished pressure of the atmosphere, or on other emergences ;
that its eflFective operation is uninfluenced by fogs or wind ; that it
constitutes an air-gauge, indicating the quantity of air at any time
passing through the colliery, and gives immediate and unequivocal
indications of the neglect of the man attending it ; that it may be
applied with facility to drawing as well as to other upcast shafts
and levels, and thus effect a great saving of the expense now incurred
by the rapid destruction of the chains, ropes, tubbing, &c., besides
diminishing the existing dangers caused thereby to the men.

These, Sir, are a few of the benefits which will result from this
valuable invention ; to which may be added its easy adaptation to
the peculiar circumstances of any mine, without requiring any alte-
ration in its internal works. The cost of erecting a machine of the
same dimensions as that at Eaglesbush, independent of the engine
power, is about 300/., and it requires the attendance of only one
man, consuming somewhat less than two tons of small coal per week ;
so that, to its other numerous advantages, cheapness of cost and a
small annual expense are to be added. In many collieries there is
sufiicient spare power, so as to render the erection of an engine for
this purpose unnecessary ; and where this is not the case, 200/. will
amply provide the requisite power.

It is highly gratifying to Messrs. Penrose and Evans, the pro-
prietors of Eaglesbush colliery, that their efforts to improve the ven-
tilation of their valuable mine have resulted in such signal success,
and converted one of the most dangerous into one of the safest col-

S92

Intelligence and Miscellaneous Articles.

lieries in the district ; and it is to be hoped that their laudable ex-
ample in thus providing for the comfort and safety of the numerous
men in their employment will have a beneficial influence on other
coal-owners, whose mines are in a dangerous state from imperfect
ventilation. — J. Richardson, C.E.

STRUVE'S PATENT VENTILATING APPARATUS.

Description of the Engraving.
Figs. 1, 2, 3 are a plan, section and elevation of the mine ventilator.
A represents the upcast pit, which may be either the coal or pumping-pit.

Intelligence and Miscellaneous Articles. 393

B culvert, 5 feet by 6 feet, connecting the upcast pit with the mine ven-
tilator ; thus an uninterrupted communication is established with the whole
of the air-passages of the colliery.

DD are two cylinders of masonry, 14 feet interior diameter, and 16 feet
lon|.

EE are interior cylinders, 9 feet 6 inches long, and 4 feet 6 inches dia-
meter : the space between the cylinders is filled with water, 7 feet deep,
and marked J.

FF are two aerometers, of 12 feet in diameter and 8 feet 6 inches long,
made to balance each other, and to move vertically in the water by means
of guides.

GG, connecting-rods, with the chains from the crank shaft, and which
also serve as guides.

HH, two cranks, placed in opposite direction on a shaft, and to which an
engine is attached to give them a rotatory motion.

II, two chains connecting the cranks with the aerometers, and giving a
vertical motion to the aerometers.

KKKK, four sets of inlec-valves to admit the air from the mine.

LIXL, four sets of outlet- valves for the discharge of the air into the at-
mosphere.

M is the framing, which supports two shelves, of 2 feet in diameter, over
which the chain moves, and which have to support the whole weight of the
aerometers.

N, embankment formed from the cuttings of the foundations.

The operation of the machine is as follows : — A steam-engine or other
power gives a rotatory motion to the shaft and cranks HH, and by means
of the chains 11, a reciprocating motion is given to the aerometers FF,
equal to twice the length of the cranks ; in this case the machine can
work a 4-feet or a 6-feei stroke : the aerometers balance each other, and
descend by their own weight — the lower inlet-valves opening at the same
moment as the upper outlet-valves,a rapid passage of air takes place through
the pumps.

The water forms the packing, or hermetical seal, which prevents air
escaping, or being admitted, except through the inlet or outlet-valves.
This machine is capable of discharging 40,000 cubic feet per minute when
moving at the rate of 200 feet per minute ; and there is no reason why it
may not be worked much faster. The machine is moved by a five-horse
power high-pressure engine.

The following testimonial has been received from Me&srs. Penrose
and Evans, relative to the patent mine ventilator, which has been
lately erected at the Eaglesbush colliery,

" Dear Sir, — Your patent mine ventilator has now been at work
at our colliery for a month, and gives us perfect satisfaction. In
our case not only is the gas and foul air dravsTi from the stalls and
general workings of the mine, hut the old goaves and abandoned
parts are likewise kept clear. Our men now all work in their stalls
with naked lamps. We work the ventilator from about five in the
morning till six in the evening, it heing unnecessary to work it at night,
as on entering the mine in the morning the overman takes a Davy
lamp with him ; and however much of gas there may be there, it is
immediately drawn off on the working of the ventilator. Our men
say that the mine is now cool, and wholesome to work in, and we
observe that they finish their labour in a much shorter time. The

391 Inlelligence and Miscellaneous Articles.

current of air underground is uniform, and quite independent of ba-
rometrical or thermometrical changes. We shall at all times be ready
to give facilities to any parties you may wish to have an opportu-
nity of viewing the working of the machine. — Penrose and Evans,
Eaglesbush Colliery, Neath, March 5." — From theMining Journal for
March 20.

ANALYSIS OF FAUJASITE. BY M. A. DAMOUR.

The author states, that in a notice inserted in the first volume of
the fourth series of the Annales des Mines, he gave a description of
a mineral belonging to the zeolite family, which, on account of its
crystalline form and composition, appeared to him to constitute a
distinct species ; to this mineral he gave the name of Faujasite. The
rarity of the mineral at the time it was discovered prevented M.
Damour from employing more than a very small quantity for ana-
lysis. Having, however, lately procured several specimens, they
were employed in repeating the analysis.

The fresh analysis gave as the composition of this mineral : —

Silica 46-12

Alumina 16*81

Lime 4-79

Soda 5-09

Water 27*02

99-83
In his first notice the author stated that faujasite retained its
transparency after heating to redness, and was acted upon by acids ;
he has since found that the mineral loses these properties when
heated to near its melting-point ; it then disengages the last traces
of water, becomes milk-white, and hydrochloric acid cold or boiling
does not act upon it. — Ann. des Mines, torn. xiv. p. &T.

ANALYSIS OF CALIFORNIA GOLD.

M. Rivot, mining engineer, has analysed a specimen of California
gold sent by Mr. Peabody to the Ecole des Mines. The specimen
contained — small flattened grains, of a fine yellow colour, and ex-
tremely small and smooth grains, attracted by the magnet, which
appeared to be titaniferous iron. A rather large, yellow and irre-
gularly rounded grain, weighing 0-628 grs., the density of which was
only 14-60, was fused on a small cupel in a muffle, and gave a but-
ton of alloy, the density of which was 17*48.

The analysis of the grains of gold, performed on one gramme,
gave the following results : —

Gold 90-70

Silver 8-80

Iron 0-38

99-88--/6frf,

Intelligence and Miscellaneous Articles. 395

ON THE VANADIATE OF LEAD AND THE DOUBLE VANADIATE
OF LEAD AND COPPER. BY M. IGNACE DOMEIKO.

The formation of secondary porphyry of Chili, which has been
already remarkable for specimens of the native amalgam of Arque-
ros, and of iodide of silver of Algodones, also contains a very rich
formation of vanadiate of lead and of copper.

The mine in which these specimens were discovered is twelve kilo-
metres to the east of the silver mines of Arqueros, and is known by
the name of the Mina Grande, or Mina de la Marqueza, and was con-
sidered as one of the richest silver mines in Chili. A miner who
was about to recommence working the mine, found a heavy yellow
mineral, which he brought to Coquimbo for analysis. It was found
by M. Domeiko to be very poor in silver, but contained vanadium.

This mineral is of a dirty yellow colour, sometimes of a sulphur-
yellow, or slightly orange or greenish ; its powder is of a yellowish
white colour ; its texture is compact, sometimes slightly earthy, and
sometimes of a weak resinous lustre. It contains numerous irregular
cavities, the interior of which is always incrusted with a brownish
matter, often consisting of globular concretions ; the mass sometimes
exhibits greenish earthy particles, coloured by carbonate of copper,
and also white carbonate of lead.

Before the blowpipe, the mineral fuses with intumescence into a
gray metallic scoria, slightly frothed, and giving a blue colour to the
flame. On charcoal, with the addition of carbonate of soda, there
are obtained a perfectly malleable button of lead and a yellowish
gray scoria ; when melted on a platina wire, with the salt of phos-
phorus, it yields a transparent bead, which assumes a fine green
colour in the interior flame, and becomes yellowish brown in the ex-
terior flame ; when heated in the matrass, it yields a little water
derived from the argillaceous gangue ; nothing sublimes in the open
tube.

Dilute nitric acid dissolves it readily, even when cold, without
producing either eff^ervescence or nitrous vapours, and leaves only
a residue of brownish or reddish gelatinous matter. Acetic acid has
no action upon it. The action of sulphuric acid determines the
absence of fluorine.

The process which succeeded best in analysing this mineral was
the following : —

The mineral reduced to an impalpable powder is treated with cold
dilute nitric acid, and digested for 24 hours ; it is then to be gently
heated and filtered to separate the ferruginous clay, unacted upon.
The chlorine is to be determined by nitrate of silver, the excess of
which is to be precipitated by a little hydrochloric acid. The greater
part of the lead is then to be precipitated by sulphuric acid ; the
filtered liquor is to be largely diluted, and sulphuretted hydrogen
is to be passed through it cold, and the operation is to cease as soon
as the lead and copper are precipitated ; filter and precipitate the
arsenic by saturating and repeatedly heating the solution. Evapo-
rate the filtered liquor to dryness ; treat the residue with hot dilute

896 Intelligence and Miscellaneous Articles.

nitric acid, dilute the solution and precipitate with excess of ammo-
nia the phosphates of lime, zirconia, iron and alumina (A). The
filtered liquor is concentrated, and a fragment of sal-ammoniac is im-
mersed in it, with the addition of a few drops of ammonia. The
vanadium is immediately precipitated in the state of vanadiate of
ammonia, which is to be collected on a filter and washed, at first with
a saturated solution of sal-ammoniac, and then with alcohol. The
solutions containing sal-ammoniac are evaporated and the residue
slightly calcined. Water is then to be added, which separates a
little silica, and the phosphoric acid is to be determined by iron ac-
cording to Berthier's process. As to the precipitate (A), if it
contain a notable quantity of vanadium, it is to be redissolved in
nitric acid, and to be again precipitated by excess of ammonia, &c.
In the opposite case, it is to be fused with one part of silica and
three parts of carbonate of soda, and treated with water ; phosphoric
acid is to be sought for in the alkaline liquor, and the insoluble re-
sidue, composed of silica, alumina, lime, oxide of iron, and zirconia,
is to be analysed.

The mean of several analyses yielded —

Chloride of lead 9-05

Oxide of lead 58-31

Oxide of copper 0'92

Arsenic acid 11*55

Phosphoric acid 5*13

Vanadic acid 1"86

Lime 7-96

Alumina, zirconia (?), traces of oxide of iron. . 1*10

Argil 2-00

Water 1-12

99-00
The presence of copper in the above-described mineral induced
the author to examine whether vanadiate of copper might not be
found among the accompanying minerals, this mineral having been
stated to exist in Siberia.

The green earthy portions were first examined, and these were
associated with traces of vanadium ; other portions examined did not
contain more, and M. Domeiko was about to give up the examina-
tion, when he found that the blackish-brown portion which he had
taken for ferruginous argillaceous gangue, was much richer in vana-
dium than the yellow mineral.

This substance is amorphous, porous, heavy, of a more or less deep
blackish-brown colour, and of a texture which is either compact or
earthy ; by the heat of the taper, it melts into a black bead, which
is somewhat frothed. By the blowpipe, it gives a green bead, with
phosphorus salt, a cupreous globule of lead upon charcoal, and in the
matrass it yields a little water. It is rather soft, and its powder is
brownish yellow. It lines the cavities of the yellow arsenic -phos-
phate mineral, and is frequently associated with the amorphous
carbonates of lead and copper. At first sight it is mistaken for hy-

Intelligence and Miscellaneous Articles. 397

drate of iron, from which it differs by its great fusibility, its ready
solubility in dilute nitric acid, and its reaction with the blowpipe,
&c.

The mean of two analyses gave as its composition —

Oxide of lead 53-43

Oxide of copper 15*78

Vanadic acid 13*41 ,

Arsenic acid 4*64

Phosphoric acid 0*64

Chloride of lead 033

Silica? 116

Lime 0*54

Oxide of iron, alumina, &c 3*46

Argillaceous residue 1*26

Water 2*70

97*35
From these results the author is induced to suppose that the above
mineral contains a double vanadiate of lead and copper, the com-
position of which approaches the formula Ph'^V + Cu'^V, or Pb^V
+ Cu*V. — Ann. des Mines, tom. xiv.

NEW MINERAL FROM BRAZIL.

M. Dufrenoy exhibited before the Academy a specimen of a mi-
neral from Brazil, which appears to be to the diamond what emery
is to corundum, as stated by M. Elie de Beaumont. Among some
specimens recently sent to the Ecole des Mines by M. Hoffman, a
dealer in minerals, were two which were stated to be hard enough
to polish the diamond ; and in fact the hardness of these specimens
was found to be superior to that of the topaz.

This substance was analysed by M. Rivot, mining engineer, who
had at his disposal one large fragment weighing 65*760 grs., and
several small pieces weighing rather less than 0*50 gr. ; the latter
only were analysed. The large fragment appeared to come from the
same alluvial formation as that in which the Brazilian diamonds
occur. Its edges are rounded by long friction ; but it has not the
appearance of a rolled flint. It is of a slightly brownish dull black
colour. Viewed with a glass, it appears riddled with small cavities
separating very small irregular laminae, which are slightly translu-
cent and iridescent. The brown colour is very unequally distributed
throughout the mass. On one of the faces the cavities are linear,
which gives it a fibrous aspect similar to obsidian. It cuts glass
readily, and scratches quartz and topaz; its density is only 3*012.
The small fragments subjected to analysis weighed 0*444 gr.,
0*410 gr. and 0*332 gr. ; their densities were respectively 3*141,
3-416 and 3*255.

These numbers indicate great difference in the porosity of the
specimens ; they lead, however, to the conclusion, that the density
of the substance is very nearly the same as that of the diamond.

598 IntelUi'ence and Miscellaneous Arliclis.

•to

By means of long calcination at a bright red heat in a covered cru-
qible, the specimens were not altered ; they retained their aspect,
hardness and weight ; they do not therefore contain any substance
volatilizable by calcination out of contact of the air. This result
certainly does not prove the igneous origin of these diamonds, but
renders improbable the idea expressed by M. Liebig, that diamonds
are derived from the transformation of organic vegetable matter.

The three specimens were successively burnt in pure oxygen gas
in the apparatus employed by M. Dumas for the combustion of the
diamond. The oxygen obtained from chlorate of potash was con-
tained in a gasometer ; it was dried and purified before it reached
the combustion- tube by passing through two tubes containing sul-
phuric acid and pumice, and one tube with potash ; employing this
method with the precautions indicated by M, Dumas, 100 of the

Carbon.

Ash.

Loss.

1st specimen gave 9 6" 84

203

1-13

2nd .. 99-73

0-24

003

3rd . . 99-37

0-27

0-36

In the combustion of the first specimen only one bulb-tube with
potash was employed, so that a portion of the carbonic acid produced
by the combustion was lost ; but in the other two experiments, in
which two bulb-tubes containing potash were used, the second in-
creased in weight some centigrammes.

The two last analyses prove perfectly that the specimens are com-
posed entirely of carbon and ash. The ash was yellowish, and in
the first specimen it had retained the form of the diamond. When
examined by the microscope, the ash appeared to be composed of
ferruginous alumina and small transparent crystals, the form of
which could not be ascertained. — Ulnstitut, Mars 2, 1849.

ANALYSIS OF THE WATEll OF THE MEDITERRANEAN ON THE
COAST OF FRANCE.

M. J. Usiglio analysed the water taken from the foot of Mount
St. Clair, about 4000 metres from the port of Cette.
100 parts gave —

Chloride of sodium 2-9424

Bromide of sodium 0-0556

Chloride of potassium 0-0505

Chloride of magnesium 0-3219

Sulphate of magnesia 0-2477

Sulphate of lime 0-1357

Carbonate of lime 001 14

Peroxide of iron 0*0003

Water 96-2345

100000
Comptes Rendus, October 1848.

Meteorological Observations. S99

IMPURITY OF COMMERCIAL BROMINE.
M. Poselger, in distilling some samples of commercial bromine,
found that the boiling-point was not 122° F., but 248° F. ; and that
the colour of the liquid became gradually lighter, till it was even-
tually quite colourless. On continuing the distillation to dryness,
he obtained a residue of charcoal. On separating the bromine from
the last portions of the distilled liquid by means of a solution of pot-
ash, an aromatic, oily, colourless liquid was obtained, which analysis
proved to be carburet of bromine ; this existed in various specimens
of bromine to the extent of 6 or 8 per cent., and there is every reason
to conclude that it was derived from the aether employed in the pre-
paration of this substance. — Journ. de Ph. et de Ch., Fevrier 1849.

METEOROLOGICAL OBSERVATIONS FOR MARCH 1849.
Chiswick. — March 1. Cloudy: clear and windy: cloudy. 2. Fine: cloudy:
clear, 3. Overcast. 4. Clear : cloudy and fine : clear. 5. Fine : frosty. 6.
Frosty : fine : overcast. 7. Cloudy : boisterous, with rain at night. 8. Fine :
hail-shower : clear and frosty at night. 9. Clear and frosty : very fine : slight
snow. 10. Clear: cloudy. 11, 12. Overcast throughout. 13. Fine. 14. Cold
dusky haze. 15, 16. Overcast. 17. Foggy: fine: clear. 18. Foggy, with heavy
dew: hazy: foggy and damp. 19. Foggy: overcast. 20. Dusky haze : over-
cast: clear. 21. Overcast : clear. 22. Foggy : cold haze : overcast. 23. Over-
cast : cold haze : densely overcast. 24. Fine, but cold : clear and frosty at night.
25. Snowing : cloudy and cold : overcast. 26. Densely clouded. 27. Dry haze.
28. Foggy : overcast throughout. 29. Hazy : rain : cloudy, SO. Heavy showers.
31 Clear: fine : cloudy.

Mean temperature of the month 41°*56

Mean temperature of March 1848 42*43

Mean temperature of March for the last twenty years 42 '62

Average amount of rain in March l*36inch.

Boston. — March 1. Cloudy: rain a.m. and stormy. 2. Cloudy. 3. Fine.
4. Cloudy. 5,6. Fine. 7. Fine: rain p.m. 8. Cloudy: snow p.m. 9, 10.
Fine. 11. Cloudy. 12. Fine, 13. Fine : rain early a.m. 14 — 16. Cloudy.
17. Fine. 18, 19. Foggy. 20 — 22. Cloudy, 23. Cloudy: rain p.m. 24. Snow.
25. Cloudy. 26. Fine. 27. Cloudy. 28. Rain : rain a.m. 29. Cloudy :
rain p.m. 30, 31. Fine: rain p.m.

Applegarth Manse, Dumfries-shire. — March I. Fair and clear a.m.: getting
cloudy : rain p.m. 2. Fair a.m. : sliowers. 3, Fair : wind rising. 4. Fair :
cloudy : fine sunset. 5. Slight shower : cleared. 6. Fair : cloudy : high
wind p.m. 7, Rain all day, but light, 8. Frost hard : clear all day. 9. Frost
keen. 10. Frost increasing in keenness. 11. Frost slight: shower. 12. Fine
spring morning : got colder. 13—17. Fine dry weather. 18. Very fine day.
19. Still finer, like May. 20. The same, beautiful we^ither. 21. The same :
fog P.M. 22. Frost during the night : fine day. 23, 24. Fine, though dull.
25. The same : brighter. 26. Frost : dull p.m. 27, The same dullness : no frost.
28, Clear and cool. 29. Shower early. 30. Showery. 31. Slight drops of rain:
slight frost.

Mean temperature of the month 41''8

Mean temperature of March 1848 41 '2

Mean temperature of March for twenty-five years 39 '1

Rain in March 1848 4*1 inchet.

Rain in March for twenty years 2*3 „

Sandwick Manse, Orkney. — March 1, Bright : hail : light showers. 2. Cloudy :
showers. 3. Cloudy : clear. 4. Clear : showers. 5. Sleet : showers. 6. Showers.
7. Sleet: snow-showers, 8, Snow-showers, 9. Drift: snow-showers, 10.
Thaw : showers. 11, Rain: drizzle. 12, Cloudy: snow, 13. Snow-showers:
thaw. 14. Drizzle. 15. Fog : damp. 16, Fog: fine, 17, Fine: fog, 18,
19. Fog. 20. Fine: clear aurora, 21, Fine : cloudy, 22, Fog : cloudy. 23.
Rain. 24, Damp : drizzle. 25, Drizzle : aurora. 26. Sleet-showers : clear
aurora. 27. Cloudy : showers. 28. Cloudy : clear aurora. 29. Cloudy. 30.
showers. 31. Damp : clear.

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THE
LONDON, EDINBURGH and DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[THIRD SERIES.]

JUNE 1849.

LVIII. The Astronomer Royal on a difficulty in the Pro-
blem of Sound.

To the Editors of the Philosophical Magazine and Journal.

Gentlemen,
A N apparent difficulty in the problem of sound has been
■^■*" discussed in successive numbers of your Journal by Pro-
fessor Challis and Mr. Stokes. Upon this subject I beg leave
to lay before you a ^evf remarks. My communication will
consist, not of strict mathematical investigation, but of ana-
logies and conjectures ; and will therefore be fairly liable to
the charge of vagueness to which all papers of this character
are exposed. At the same time I must assert that I should
not think it proper to transmit such a paper for insertion in
your respectable Journal, if I were not persuaded that the
analogies are valid and the conjectures probable.

Attention has been called by Professor Challis to the nature
of the accurate solution of the partial differential equation
applying to plane waves of air. And Mr. Stokes has shown
that a distinct meaning can be given to this solution up to a
certain point in the progress of the wave.

This solution, although complete as far as it goes, is still
but a functional differential equation of the first order with
regard to the co-ordinates of a particle of air. For my present
purpose, it will be more convenient to take the differential
equation applying immediately to the co-ordinates of the par-
ticles ; although in that case also the complete solution cannot
(so far as I am aware) be obtained.

If the original ordinate of a particle of air be w, and its
disturbed ordinate be ,2?+X, then the differential equation
applying to its motion is

"" dx''~V^ dx) ' dt^'
Phil, Mag. S. 3. Vol. 34. No. 231 . June 1 849. 2 D

402 The Astronomer Royal on a difficulty

Now, as the first subject of my communication, I propose to
compare this with the equation applying to the motion of long
waves of water in shallow canals, when the proportion of the
vertical movement of the particles to the depth of the canal
is not neglected ; a case which I have in some measure dis-
cussed in the Encyclopaedia Met7-opoUtanaf article Tides and
Waves. That equation has the form

2^_/ dxy d^X

The equations are generally similar in form, with the differ-
ence that the exponent in one case is 2 and in the other is 3.
Imperfect solutions of both (as my friend Professor De Mor-
gan has pointed out to me) may be obtained in the following
forms : —
For the first.

For the second,"

dX ^ , / dX\

dX ^ 2a

-df=^-^

x/l-f}"

These solutions give us no assistance further than this, that
they indicate a critical state of the movement of the particles

J'SJ'

in both as occurring when -t-= —1, which in both equations

makes — r— infinite.
dt

But a very distinct idea of the nature of the progressing

wave in both cases may be obtained by solving the equations

by the method of successive substitution. Thus, of the equation

rf^ _ ^ _ r^ (dxyi cpx

' ds^ dt'' ~ V ^^ \da;J J * dt^*
a first solution may be obtained by neglecting the right-hand
side (a quantity of the second order as regards the movement
of the particles), which solution, if the wave be supposed to
travel in one direction only, is

X = c.f{at-,v).

If this value be substituted in the right-hand side of the equa-
tion, terms of the second order only being retained, and if the
solution be then effected under the conditions proper for suc-
cessive substitution, namely that no new terms of the first
order be introduced, but that the terms of the second order
may have the most general form ; and if we avail ourselves

in the Problem of Sound, 403

of that generality to secure the conditions that the general
value of X shall not contain at + x, and that when a? is =0 the
value of X shall have strictly the form c./{at) ; then the value
of X to the second order will be

X=c./(^/~^)+^.;r.{A+/V-^)V},

where A is an arbitrary constant which may be used to remove
the non-periodical part oif'{at—xjf. Thus if

f{at—x)= sin n{at—x\
then

f\at—x)]^=zn^. cos^ n{at—x)-= o" V + ^^^ 2n{at—x)h

and here it will be proper to take A= — — ; and then the
value of X to the second order is

c-'n"

^ = c .sin n{at—x) + —--- .X .cos 2n{at—x).

If we had treated the equation for waves of water in the same
manner, we should have found, to the second order,

ScV

X=c.sin n{at—x) a — - . x . cos 2n{at—x)»

And in both cases, if we carried on the solution to the 3rd,
4th, &c. orders, we should introduce the second power of a?
multiplying trigonometrical functions of the simple and triple
argument, the third power of x multiplying functions of the
double and quadruple argument, and so on. The forms of
the solutions would be similar, but the coefficients would be
different.

If now for any definite value of x we construct a curve
graphically representing the motion of the particles, we find

that the value of -^ is represented by an unsymmetrical curve,

the deviation from symmetry increasing as larger and larger
values of ^ are taken. The nature of the asymmetry is this,
that the interval of time from the extreme backward motion
of a particle to the extreme forward motion is less than half
the whole period of vibration ; and that this inequality is
greater as we consider the movement of particles whose ori-
ginal position is more and more advanced. And this happens
in the same manner for the particles of air and for the particles
of water.

I might have shown this at once by merely requesting your
readers to compare the diagram drawn by Mr. Stokes, in the

2D2

404^ The Astronomer Royal on a difficulty

November Number of your Journal, with the diagrams drawn
by me in the Eiicyclopcedia Metropolitana. But I have pre-
ferred to show, step by step, that there is strong analogy in
every part of the mathematical process in the two cases.

The rapid change of velocity implies in both cases very
violent forces of compression among the particles ; and very
sudden alterations of the surface for the water, and very rapid
expansions for the air.

Now in the case of waves of water we cannot by mathema-
tical process (so far as I know) pursue the investigation of the
condition of the particles to its limit, but we can in some
measure observe it experimentally. And the fact which we
observe is this, that in a long river, where the magnitude of
the tide is considerable, the tide in the upper parts of the river
presents the phsenomenon of a bore. The surf formed upon
extensive flat sands is probably a phsenomenon of the same
class. Physically considered, the expression of this fact is,
that the continuity of the particles is interrupted.

Now I imagine that in the critical state of the waves of air
the same thing occurs, namely that the continuity of the par-
ticles is interrupted ; and that with it the laws of motion of
the air, depending essentially as they do upon the existence
of that continuity, are interrupted. I imagine that the air is
in a state exactly analogous to that of a bore or a surf. Adopt-
ing this analogy, I do not think that there is anything of the
nature of reflexion of the wave (for I have never been able to
observe the smallest trace of reflected wave from a surf),
although at the same time I am utterly unable to account for
the disposal of the vis viva.

Taking this view of the matter, it appears to me that the
expression " a plane wave of air is impossible " stands on pre-
cisely the same footing as the expression " a tide in the Severn
is impossible." In both cases, if the assumption is made
*'that the waves are to preserve the same character through
infinite space and infinite time," the wave is impossible. In
both cases, the ordinary understandings on the theory of waves
will apply for a considerable distance.

I believe also that in both cases, if by friction or other causes
the coefficient of vibration be materially diminished, the waves
may travel to a very great distance without any aquatic or
aerial bore.

The second subject of my communication is the probable
sensational indication of the physical phaenomenon, " inter-
ruption of continuity of particles of air." And it is my belief
that it produces that sound which, according as it is in a high
key, or is in a low key, or is articulately interrupted, is called

in the Problem of Sound. 405

a hiss, a buzz, or a whisper, and of which the phonetic symbol
is S or Z. Perhaps also the sound which we call a roar, of
which the symbol is R, is related to it; for philologists appear
to take for granted that these letters represent cognate sounds
(thus Niebuhr assumes as indubitable that Aurunci and Au-
sonii are the same word). My reasons for connecting the
sound of S with the interruption of continuity, or with the
broken character of the aerial wave, are the following: —

1. It has long ago been remarked that the sound of S is
not returned by an ordinary echo. In like manner, a broken-
headed sea is not reflected by a vertical pier. When a broken-
headed sea strikes a pier perpendicularly, it is thrown upwards ;
when it strikes it obliquely, it is partly thrown upwards and
partly it runs horizontally along the face of the pier. In
neither case is there any reflexion of the broken head, or any
creation of a broken wave travelling in the opposite direction,
although the swell is reflected according to the usually under-
stood laws.

2. It is well known that in whispering galleries the sound
of a whisper is carried along the surface of the dome, and never
quits that surface, and can be heard on the opposite side only
by applying the ear very near to that surface ; while an ordi-
nary sound is not transmitted along the surface, and is not
heard at the opposite side of the dome in any striking inten-
sity. In like manner, a broken-headed sea will run along the
face of a vertical pier for a considerable distance without show-
ing any disposition to quit it; while an ordinary fluctuation is
thrown ofl'it at once, in the form of a swell rolling away at an
angle given by the usual laws of refiexion.

Whether there is any well-established instance of the con-
version of a clear musical sound into a hiss or buzz by mere
distance of transmission, I am not able to say ; but I should
think that, long before the wave could have received its change
of character to the degree necessary to produce discontinuity
of particles, even supposing that its coefiicient had remained
undiminished, that coefficient would from friction or from ex-
pansion of the wave have become so small, that there would
be no perceptible tendency in the wave to change its form.
So far however as the tendency to change the form exists, it
will be greater for loud sounds than for faint sounds, and
greater for sounds in a high key than for sounds in a low key ;
or in other words, the sounds which might be expected soonest
to degenerate into hiss or buzz would be loud sounds on a
high key. I am, Gentlemen,

Royal Observatory, Greenwich, Your obedient Servant,

May 17, 1849. G. B. AlRY.

[ 406 ]

LIX. On the Sy7nhols of Algebra, and on the Theory of Tes-
sarines. By James Cockle, Esq., M.A., of Trinity Col-
lege, Cambridge, and Barrisier-at-Law of the Middle
Temple*.

AT page 436 of the last (33rd) volume of this Journal, I
proposed to include under the genus imaginary two
independent species of quantity which I distinguished by the
respective terms unreal and impossible. But, if we extend the
meaning of the word "imaginary" so as to make it compre-
hend all quantity that is not real, a third species of quantity,
for which I would suggest the name of ideal, must be added
to the two already included in the common genus of imagi-
naries. And perhaps it will be conducive to distinctness of
conception and to convenience if we admit a fourth species of
imaginaries, which I propose to call typal, from their squares,
&c. running into certain types.

The peculiar symbols of the quaternion theory of Sir W.
R. Hamilton, of the octave theory of Mr. John T. Graves
and Mr. Cayley, of the triple algebra of Professor De Morgan,
and of the pluquaternion theory of the Rev. T. P. Kirkman,
all belong to the species ideal. It may be stated as the cha-
racteristic of ideal quantities, that, in their combinations, they
do not follow the laws of that ordinary algebra to which Mr.
De Morgan applies the term Double Algebra. The imagi-
naries in some of the systems of Mr. Cayley's theory of couples
(the systems marked C, D, C, and D', Phil. Mag., S. 3,
vol. xxvii. pp. 39-40) are ideal. And so, in general, are those
of the last two systems (E and E', lb. p. 40) in Mr. Cayley's
paper. Mr. J. T. Graves would call the couples involved in
the above systems anomalous.

But in other of the systems of Mr. Cayley (those marked
A, B, A', B', lb. p. 38 et seq.) the imaginaries axe typal', and
this is also the case in Mr. J. T. Graves's theory of couples
{lb. vol. xxvi.). Borrowing a term from the learned writer
last mentioned, we may call all the couples mentioned in this
paragraph normal. The characteristic of typal quantities is,
that, although in their laws of combination they follow the
rules of ordinary algebra, yet the types or conditions by which
they are defined are not consistent with that algebra. Ktypal
differs from an impossible quantity in this : that the ordinary
algebra /orcf-s impossible quantity upon our notice, and defines
it by means as purely algebraic as those by which unreal
quantity is defined ; while on the other hand, typal (as well as

* Communicated by Dr. John Cockle, FJl.C.S.

Mr, Cockle on the Symbols of Algebra, 407

ideal) quantity is the oiFspring of arbitrary ultra-algebraic
definition*.

I should propose to apply the term hyper-algebraic to typal
and ideal quantities, and to confine it to those quantities.
Ought we to apply the word hyper-algebraic to impossible
quantity ? I think not. If I may be permitted to use the term
possible so as to include under it not only real quantities but
also the unreal quantities of ordinary algebra, I would suggest
that it is only in respect of certain anomalous resultsf (re-
sults, however, that do not defy explanation J) that impossible
differs from possible quantity, and consequently that impossible
quantity must be regarded as algebraic. It is unquestionably

* I must not be understood as desiring to underrate these symbolic
children of definition. So far from it, I think it within the limits of pos-
sibility so to define symbols as that they may have their prototypes in na-
ture, and serve to expound other of her phaenomena than those to which
symbols have been yet applied. For instance, might not arbitrary symbols
be made the representatives of chemical phaenomena ? Mr. Boole's Ma-
thematical Analysis of Logic is a step out of the beaten track which sym-
bolic science has hitherto persevered in j and although to pass from mental
to chemical phaenomena may not authorise us to hope that such sciences
as chemistry may be rendered symbolic, yet I cannot help thinking, that by
a proper notation for affinity, &c. chemical decompositions might be repre-
sented : at any rate it may be worth a trial.

t Vide supra, pp. 41, 42. It is not a little singular, that if we abandon
the principle laid down, supra, p. 39, note -j-, we have

(i+Vi)a-Vi)=i-v'/=i-i=0;

and also that if we preserve that principle, we derive from the equation (1.),
supra, p. 39, the following :

(-1)4=(+ ^jf={ /;X ^j)x(Vjx VJ)=JxJ,
or

1=/.
On the other hand, although from the relation

il+k)^=2k=z2ij
we may deduce

still, seeing, from the relation {supra, p. 40)

the anomalous nature of the evolution of impossibles, we must not attempt
to express \/J as a linear function of i,J, k.

I may add, that in previous papers (supra, pp. 45, and 135,) the radius
of the (larger) sphere is supposed to be unity.

I See paragraph 8 (and 10) of the Rev. Prof. Charles Graves's paper on
Triple Algebra (supra, p. 119-126). I propose to call Mr. C. Graves's
system a trinar, and Mr. De Morgan's a ternary algebra; the latter form
of name being given to the quadratic system, in which the square of the
imaginary is negative. And hence the respective terras trine and temion
suggest themselves as distinctive^

408 Mr. Cockle on the Symbols of Algebra^

algebraic in its origin, nor is it to be considered as other than
algebraic in its laws of combination with other symbols ; and
if in first approaching the subject of impossible quantity some
mysteries present themselves and some difficulties arise, it
may be worth considering whether our estimate of the range
of ordinary algebra has not been too limited, and whether its
own inherent powers, when sufficiently developed, may not
explain the mystery and clear away the difficulty. With these
views it will not excite surprise if I state here, that I regard
paragraph 8 of the Rev. Charles Graves's paper on Triple
Algebra {supra, pp. 123, 124) as a valuable contribution to
ordinary algebra.

I take the opportunity of adding one or two remarks on
tessarines, premising that I shall use i' f and ^ to denote the
imaginaries which enter into those expressions.

1. The product of two tessarines of the form

Yi'x -\-j'y + k'z

is of the same form, and the moduli and amplitudes of the fac-
tors and product are related in the same manner, and the latter
may be constructed as readily, as if the factors and the product
were quaternions.

2. The product of the two tessarines

2'^i +/j/i + A'zi, and i'x^-^j'yc^+Wzcif
will be of the form

provided that

But, if the two systems of values ^-j, yj, ^j, and oe^^y^, z^, re-
spectively satisfy the condition

f-a:^-z^=0, (w.)

and if, moreover,

■^1 "^a

then (w.) may be satisfied*. But (w.) represents a right-

• If we multiply the two equations that result from substituting the two
systems of values of .r i/ and z respectively, we shall have after reduction, &c.

which suggests matter for future observation. With reference to the sub-
ject of Imaginary Geometry {supj-a, p. 132-135), and indeed of analytical
geometry in general, I may remark that I propose to call the real primary
axis Cthat of jr) the axe, the unreal secondary axis (that of 5^) the^erp^, and
the impossible tertiary axis (that of 2) the norme. Thus, in the equation

and on the Theory of Tessarines. 409

angled cone, whose axis is the axis of y, and whose vertex
is the origin, the axes being rectangular. Hence, taking
i'x+j'y-\-k'z to denote a point whose rectangular co-ordinates
are .r, ?/, 2, we see that if two points be taken in the same
generatrix of the cone (w.), their tessarine product, considering
the vertex as origin, will be the point* whose co-ordinates are

(1.) (supi'h, ]). 133) A is the axe, B the perpe, and C the norme. Let me
add that the equations

a—x—0, a2_j_y— 0, //«+ '/«=0,
(of which the respective solutions are

x=:a, i/=i'a, z=j'a,)

denote, when considered separately, three points, each at a distance a from
the origin, but in axes at right angles to each other.

* My friend Professor Davies has {supra, p. 37, and vol. xxix. p. 171-
175) intimated or expressed an opinion adverse to the interpretability of
the symbol \/ — i in geometry. If eminence in geometric science can
confer a right, not only to express such an opinion, but to have that ex-
pression duly weighed, then I think that there are few, if any, English geo-
meters who possess those rights more unquestionably. I must however
confess that I do not see the force of the reasoning employed in Mr. Davies's
proposition {lb. p. 174). The inconsistency alluded to in paragraph 4 of the
proposition could never arise — at least 1 am unable to perceive how itcould.
In saying that a rectangle is equal to the product of its sides, we mean that
the numbers of linear units in the sides, when multiplied together, give a
number equal to the number of square units in the rectangle. But the
signs are not elements in the consideration when we multiply the sides.
Unless I am mistaken, the inconsistency in question must arise in some such
manner as the following : — Take A in BB'; and let it be required that the
rectangle B' A X AB shall equal half the square on BB'. We should then (as
to this vide supra, p. 43) find AC=4:a'V/- 1, and that would be a solution
of the problem. I consider, then, that my much-valued friend's argument
rests solely on the inductive ground previously assigned (vol. xxix. p. 172).
But there are strong inductive reasons on the other side of the question.
The symbol V' — 1 appears to indicate that move dimensions o? the sub-
ject-matter must be taken into consideration than are stated in the data of
a question. If we are dealing with a two-dimensioned subject, and meet
with the symbol s/ — 1, that symbol indicates impossibility or not according
to the fact of our having or not having two dimensions given in our data.
Thus, in that spherical geometry with which Mr. Davies's name must ever
be associated, and in which a point may be determined by its longitude (<p)
and its latitude (^^) (see Camb. Math. Journ., vol. i. p. 193; ii.p. 37, &c.),

such an expression as

cos (p -f cos;^V — 1

is unmeaning; while in plane analytical geometry the expression x-\-yV — 1
indicates possibility in space. I may here observe, that if we regard space
under a purely graphic aspect, and consider all lines drawn through a point
as determined by their inclination to two fixed lines, then we have a strictly
two-dimensioned science. With reference to the subject of space, I would
add that " Symbolical Geometry " may be made to take an almost infinite

410 Mr. E. J. Lowe on Remarkable

This will readily be verified by multiplication, or by substitu-
tion in the formulae of page 437 of the last volume of this
Journal. So, we might have considered the conditions requi-
site in order that the product

may have the same form as its factors ; and the same of other
forms. These concluding remarks are not offered as conti-
nuations of the general development of the tessarine theory,
but as individual instances not perhaps altogether unworthy of
notice.

2 Church Yard Court, Temple,
May 8, 1849.

LX. Remarkable Solar Halos and Mock Suns seen at the
Observatory of Henry Lawson, Esq., F.R.S. S/'c., Lans-
doivti Crescent, Bath. By Edward Joseph Lowe, Esq.,
F.R.A.S.

[With a Plate.]

1849. T^EBRUARY 12<i and 13^. From 12^ 22^'^ a halo,
-i- A, Plate L fig. 1, was formed encircling the sun,
S, apparently in a very thin vapour, for the sky was quite
blue. All the morning there had been a dense ground-fog of
some extent.

23^ 30m. The horizontal diameter of the halo A 45° and
its vertical diameter 47°. The portion of the heavens en-
closed within this circle was many shades darker than that
without, especially so near the circle, for it gradually became
lighter near the sun. The sun was also sun-ounded with a
burr, and had his rays carried out considerably, at times 20°.
The inner edge of the halo was well-defined, but the outer
edge very confused ; the breadth about 1°.

23h 4om, Xwo mock suns, B and C, fig. 2, faintly visible,
partaking of the same tint as the halo, viz. pale yellow, were
formed in the edge of the halo A on the sun's horizontal level,

variety of forms. Thus, let A, B, C, D be four points in space, then the
equation

D— C=B— A ........ (a.)

may be used to indicate either that the lines DC and BA are equal and
parallel, as is done in the system of Sir W. R. Hamilton ; or we may use
(a.) to denote that those lines are equal and directed to some one point in
space; and, in this latter system, (when the last-mentioned point is at an
infinite distance,) we may be reconducted to the former one. And other
systems might be devised, each geometrical system having probably a corre-
sponding one in analysis, and vice versa.

Solar Halos and Mock Suns. 411

the one on the right and the other on the left side of the true
sun.

23^4ini. Mock sun C had disappeared.

23^ 44°'. Mock sun B disappeared.

23^ 45'^. From the part of the halo A where the mock
sun C had appeared, there proceeded outwards an arc of a
circle, D, which was apparently a segment of a circle whose
diameter was about 90°. There were 30° of this arc visible.

23^ 46"". Another segment of a circle, E, of seemingly the
same diameter as D, cut the circle A at 12° on either side the
base. D had disappeared.

23^^ 55^. E had vanished, and the circle A was indistinct ;
much linear cirri now appearing.

13doiiiom Circle A again brilliant, and the upper half
of another circle, G, fig. 3, appeared, of 46° radius, which
had the sun for its centre; it was colourless and 1° in breadth.
There was about 140° of this circle visible. Clouds much
thinner, the sky being now clear, with the exception of a very
thin uniform haze.

0^ 15°^. Mock sun B again just visible, and another, H,
at the apex of the circle A, colourless.

Qh 20™. The phaenomenon had vanished excepting the
circle A.

0^ 35°^. A indistinct. From this time until 1 o'clock no
change, the halo being feeble for a time, and then brightening
up again.

1*". A curious feature now showed itself; within the circle
A an ellipse, I, fig. 4, was formed, and immediately became
brilliant; its horizontal diameter (measured in the centre of
the ellipse) was 20° and its vertical diameter 30° ; within this
ellipse the sky was much brighter than that without; its lower
edge blended with the solar burr; at its apex was a mock
sun, K, and also the mock sun H again formed.

jh 5m^ 'pijg appearance had changed ; mock sun K and
the ellipse I gone; but a segment of a circle, L, fig. 5, of
about 140° in diameter, rose from the sun S,- and cut the
circle A at its summit, and extended towards the north-west j
90° in length of this segment was plainly visible. Where it
cut the circle A at H was a mock sun ; also two other mock
suns, M and N, were formed in the circle A at the distance of
8° on either side the vertex of this circle. All colourless.
Sky becoming loaded with colourless cirri, but were less abun-
dant near the phaenomenon. Prospect, foggy; wind, west;
temperature in shade, 43° ; Franklin's hygrometer, 95 ; baro-
meter, 30*15 inches.

I*' 9°^. Another portion of a circle, O, fig. 6, of like dimen-

4; 12 On Remarkable Solar Halos and Mock Suns.

sions with L, rose from the true sun, passed through the ver-
tex of the circle A, where it cut the segment L, and stretched
out towards the north-east. This was inverted with respect
to L. 10° of a circle of large dimensions also cut the seg-
ment L at P. On the west horizon a few long muddy cirro-
strati were just visible above the fog.

1^ 15"^. The circle A alone remained, the other portion of
the phaenomenon having disappeared soon after 1*^ 10°*.

1^ 30™. More clouds ; the halo A fainter ; but another
feature was at this hour traceable — a ring, Q, fig. 7, whose
centre was a few degrees above the true sun, cut the circle A
at R and the circle G at T; the vertical diameter was 70°
and the horizontal diameter 55°. The circle A had become
more elliptical ; its vertical diameter was 48° and its horizontal
diameter 4:4!^°.

jh 37111^ Another change took place; the circle Q vanished,
but there were two other segments of circles, U and W, fig. 8,
of apparently 140° in diameter; these crossed each other at
the point X, which was distant 70° from the sun. These
segments both went eastward ; there were again three mock
suns visible, viz. M, N and C; the latter was bright but
colourless, and had a tail of 8° in length, diminishing to a
point, and proceeding diametrically opposite to the true sun.

Ih 40™. All had vanished but the circle A.

1^ 50™. Circle A disappeared. Sky becoming clear.
Temperature, 44° ; hygrometer, 94 ; wind, west and calm.

1^ 55^. Faint portion of the circle A again formed above
the true sun, and at this time prismatic. It finally disap-
peared at 2^^, and the sky became thinly and partially scat-
tered over with clouds of cirrocumuli.

The width of all the circles was 1°.

A fine night, succeeded by a foggy morning and slight
solar halo. On the 16th all day a prismatic solar halo, and
at 3^ 25"* there were two prismatic mock suns formed on the
horizontal level of the true sun in the halo of 45° diameter;
these lasted 7 minutes.

It is well to add, that all the measurements were made
with an admirable and at the same time simple instrument
invented by Mr. Lawson for this purpose. It is hung in the
observatory, and is ready for use at a moment's notice.

Observatory, Lansdown Crescent, Bath,
February 17th, 1849.

[ 413 ]

LXI. On an Improvement in the Analysis of Equations. By
J. R. Young, Professor of Mathematics in Belfast College"^.

IN the analysis of numerical equations our chief difficulties
are with those of an even degree, into which equations of
an odd degree may always be changed. The following brief
sketch of an improved method of discussing such equations
will, I think, be acceptable to the readers of the Philosophical
Magazine, although 1 reserve minute details, as well as some
important extensions, for a future communication.
Let

a;2«+flA-2«-i + i^«-2 + c^2«-3^,,^^_,_;5._0 . . (A)

be any equation of an even degree with numerical coefficients j
then the left-hand member may be decomposed into the fol-
lowing pair of conjugate factors, namely,

and

«+ia^«-' + ,Y/^Q«^-M^'"-'-f^^'''-^-....-^T»

and, consequently, if these be separately equated to zero, and
either of the roots of the proposed equation substituted for x,
one of these two new equations will always be satisfied. It
follows, therefore, that when a real root is substituted, the
expression under the radical, namely,

Qa2_^,V,2«-2_c^2n-3_...._yt, . , . . (B)

must always be positive ; otherwise a real expression, viz. that
which precedes the radical, would neutralize an imaginary
one.

Hence, if we perform the partial analysis of the equation
(A) by the method of Budan, commonly called the method
of Fourier, we may discuss the doubtful intervals, which that
analysis leaves for further examination, by an appeal to the
inferior polynomial (B). Applying the before- mentioned
analysis to this, in reference to the doubtful intervals, we
should infer that whenever, throughout any such interval, (B)
was plus, the sought roots may be real ; and that when,
throughout, the sign was minus, they must be imaginary.
When the sign of (B) passed from plus to minus, or from
minus to plus, in the interval, the doubt would remain; and

* Communicated by the Author.

414 On an Improvement in the Analysis of Equations,

our business would then be to take a portion of only the
positive region of this interval, and to employ its limits in the
corresponding interval of the original analysis : if two roots
were still indicated, within these limits, we may suspect
them to be real : if no roots be indicated, we must widen the
positive interval in the second analysis : if the roots are real,
we shall thus at length inclose them; but if no roots become
indicated till we trench upon the negative portion of the in-
terval in (B), we may be sure that the roots are imaginary.

At present I shall give but a single example in illustration.
Let the equation be

^'^ — 4.r3 — 3a? + 23 = 0,
the complete analysis of which, by the method of Fourier, is
attendecl with a good deal of trouble. (See Analyse des Equa-
tions, p. 137.)

As in this example b is zero, the expression (B) is

4a;'2 + 3^ — 23;

and as the only doubtful interval, as ascertained by the partial
analysis of Budan (ibid. p. 138), is the interval [1, 10], it is in
reference to this alone that we have to examine the signs of
the quadratic expression just written. Of this interval, the
portion [1,2], being negative, we reject it: taking therefore
the interval [3, 10], which is wholly positive, and employing
it, instead of the wider interval [1, 10], in Budan's analysis,
a root is detected ; and consequently, since no root can lie in
the rejected region, the other root must lie in the interval
[2, 3] . And thus the character and places of the roots are
ascertained.

It is proper to add, that if the second term be absent from
the proposed equation (A), then the rational part of each con-
jugate factor will be

and the expression under the radical will be

to which all the preceding remarks apply.

1 need scarcely observe, after what has been shown in my
paper in the April Number o!" this Journal, that the conjugate
factors here employed form but one pair, out of several that
might be used for a similar purpose ; although the forms here
given will, as far as they go, be those most eligible; and
1 venture to think that, by our availing ourselves of them in

M. Duhamel on the Multiple Sounds of Bodies. 415

the analysis of equations, considerable facilities will often be
introduced into the numerical process. I shall only observe,
in conclusion, that, by using other conjugate forms, it is easy
to see how equations of an odd degree may be brought, in a
direct manner, under the operation of the above method ; or,
without using any change of form at all, the method becomes
immediately applicable upon introducing, into an equation of
an odd degree, the new root .r = 0; or by supposing this to be
done after the partial analysis of the equation in its original
state.

But the general principle here sketched, admits of import-
ant modification. I shall hereafter show that, for an equation
of the degree 2w, the polynomial under the radical may always
be reduced to the degree « — 1 ; and that for an equation of
the degree 2«+ 1, it may be reduced to the degree n+lx so
that the analysis of a biquadratic equation will depend on the
examination of an expression of the first degree; that of an
equation of the fifth or sixth degree, upon the examination of
an expression of the second degree ; and so on.

In what is stated above, I have alluded to certain circum-
stances of doubt, of which no mention is made in the general
announcement in this last paragraph. I have done so because,
in this brief sketch of the initial steps of the investigation in
which I am engaged, no explicit guidance in these circum-
stances is actually furnished. My present purpose is simply
to make known the path I am pursuing in an important in-
quiry; and to point to the results which I confidently expect
to arrive at.

Belfast, May 12, 1849.

LXII. On the Multiple Sounds of Bodies.
By M. Duhamel, Member of the Institute*.

THE question under consideration is far from being new,
and still it may be said to be as yet unsolved ; that is to
say, scientific men have not yet arrived at a common and un-
contested opinion on this point. The object of the present
memoir is to establish such an opinion.

What follows will perhaps appear so evident, and so little
different from what is already known, that persons who have
for a long time been too easily satisfied on this subject, may,
whilst they accept my explanation, remain persuaded that
they have never viewed the matter otherwise. I expect this,
and consent, if they will have it so, that all my ideas and ex-

♦ From the Annates de Chimie et de Physiqtie for January 1849.

416 M. Duhamel on the Mtdtiple Soutids of Bodies.

perimenls are to be found in previous memoirs of physicists
and geometers. All I ask is, that they should acknowledge
that there can no longer be two opinions on this matter: 1
shall but have learnt one thing, — that all are agreed ; that is
enough for me.

The coexistence of several sounds emanating from one and
the same vibrating body is undoubtedly one of the most re-
markable phaenomena of acoustics. Experience shows that
sounds which may be produced singly by the same body may
often be produced in it simultaneously. This phaenomenon
has given rise to very various explanations, none of which has
obtained the complete assent of geometers and physicists. I
propounded, in 1840, some new views on this subject; and the
experiments which I made to confirm them appeared to throw
some light on the question. I was however not entirely satis-
fied, and announced that my researches on this point were not
terminated. The problem now appears to me completely
solved. I have for several years been in possession of its so-
lution. It seemed to me so natural, that I thought it would
present itself to others besides myself; and hence, no doubt,
the little eagerness I felt to call the attention of men of science
to it. Perhaps I might have remained silent still longer, had
not recent publications proved to me that there was still some-
thing to learn on this point.

And we shall remark first, that there can be no occasion to
explain how we perceive several sounds at a time, any more
than to explain how we experience at once several sensations
of any other kind. Our aim should be to include the phaeno-
menon in question in a more general class of recognized phae-
nomena; but this is precisely what has not been sought by
those who have studied the subject ; it is what I have attempted
for several years, and in this 1 think I have now succeeded.

I shall begin by recalling in a few words what had been
said before me on this subject.

Father Mersenne, in his Harmonie Universellef after refuting
certain explanations that had been given of this phaenomenon,
sought to prove that it is produced by different reflexions of
the air on the surface of the body which emits several sounds
at a time. His theory not having been adopted, we shall
abstain from stating it in detail, and shall pass to that which
has obtained the assent of the greatest number of physicists.

This theory, first suggested by Sauveur, has been so much
developed and extended by Daniel Bernouilli, that he is in
some measure considered as its author. This illustrious phi-
losopher, in his solution of the problem of vibrating strings,
considers the initial figure of the string as formed by the su-

M. Duhamel on the Multiple Sounds of Bodies. 417

perposition of an indefinite number of curves, having for their
bases, some the entire length of the string, and others the half,
the third, the quarter, &c. If these different curves were taken
separately for the figure of the string, the fundamental note
would be heard for the first, the octave for the second, the
twelfth for the third, &c. Now analysis shows that if the
ordinates of the initial figure were the sums of those which
correspond to any number of these curves, the ordinates vari-
able with the time would always be the sums of those which
should correspond to them, at the same instant, in the partial
motions answering to each of the curves, taken as initial figure.
Thence Daniel Bernouilli concludes that the movement of
each point being capable of being decomposed into a succes-
sion of others, which, if they existed singly, would give the
sounds 1, 2, 3, 4, &c., all these sounds must necessarily be
heard at once. He expresses himself as follows, in the par-
ticular case in which the octave alone is heard along with its
fundamental sound: — "This absolute movement of the point
D comprises really two periodical movements, one with relation
to the point C, and the other with relation to the point B.
The number of the first periodical returns will always be
double that of the second. The mind perceives each species of
these periodical returns^ and thence remarks two sounds, one of
•which is the octave of the other"

This rather subtile explanation, however, did not satisfy
all geometers ; and in reality it is not an explanation, since
the fact in question is not referred to other admitted facts.

Lagrange well observed, that this decomposition of the
movement was a purely geometrical conception, but one which
proved nothing relative to the sound produced ; he added,
that this sound could alone result from the absolute movement,
which was single. Daniel Bernouilli was much surprised at
these objections, which in no degree altered his views. The
following passage occurs in his memoir entitled Recherches
Physiques, Mecanigues, et Analytiques, sur le son, et sur les
tons des tuyaux d'orguesx — "I have given the explanation of
this phaenomenon with relation to strings in the Memoires
de Berlin o{ n 53. This explanation is so luminous and so
self-evident, that the celebrated M. de Lagrange cannot have
examined it with sufficient attention. Not content with reject-
ing it, he blames the learned M. Euler for having approved
it. Let I be the length of the string, tt the semicircle in the
radius = 1 ; can it be doubted that the equation

. ntx o . Swa^
^=«sin-y- +bsin— T-

PhiL Mag, S. 3. Vol. 84. No. 231. June 1849. 2 E

418 M. Duhamel on the Multiple Sounds of Bodies.

defines the state of a string which makes at the same time
vibrations of the first and the third order, and which conse-
quently gives at once the fundamental note and its twelfth ?
Why should the string rather give the fundamental sound
than its twelfth; since on making a = 0 only the said twelfth
will absolutely be heard, and making §=0, only the funda-
mental note will be heard ?"

It must be acknowledged that this reasoning is little con-
clusive ; for might it not be that a single sound should be
heard, and be determined by the relation of a. to §? and if
several were heard, why should they be only those obtained
when a or § are null ?

The obscurity of all this reasoning was inevitable; it par-
takes of the faulty direction which this great philosopher follow-
ed : he tried to demonstrate directly that two sensations must
be experienced at once, instead of seeking simply to refer the
phsenomenon to another. This reduction is the object which
ought always to be proposed ; and even when one class of
phaenomena is referred to another class as yet little known,
science has always made progress, new relations being disco-
vered, and instead of two difficulties, but one remaining. In
the present case, the object cannot be, as we have already said,
to explain the double sensation ; we must only attempt to
bring the phaenomenon investigated into another class, in
which this double sensation is admitted.

Struck by the want of solidity in the reasons alleged in
favour of these various theories upon a question in itself so
interesting, I endeavoured, as Lagrange required, to ascer-
tain the absolute movement of the diflf'erent points of the vi-
brating body. Reasonings applicable to every kind of body,
followed by precise calculations relative to the simple case of
strings, led me to a proposition which may be stated thus : —

When a body is made to vibrate by several causes which would
produce separately the simple sounds which it can give, its sur-
face generally divides itself into a certainjinite number of parts,
in each of which the vibrations have unequal durations. These
different durations have relation to sounds corresponding to the
different causes; and we are in the same position as if we had
several separate surfaces, each having a peculiar movement of
vibration.

In order to ascertain the truth of this proposition in the
simple case of two coexistent sounds, I must first observe that
if each of them existed singly, it would correspond to a differ-,
ent system of nodal lines. Now if the two causes which would
produce each of these movements are made to act at the same
time, or successively, a movement would result composed of

M. Duhamel on the Multiple Sounds of Bodies. 419

those two movements, according to the general principle of
superposition of small movements. Whence it follows that,
in all the points of the nodal line of any one of these two move-
ments, the other movement alone will be produced. But it is
easy to ascertain, by calculation, that the neighbouring points of
one of these lines will make an equal number of vibrations ;
and in passing, for example, from that in which the number of
vibrations is greatest towards the other, there will be a certain
number of vibrations the amplitude of which will successively
diminish, and will become null when we pass the line of sepa-
ration of the two regions related to each of the sounds : the
number of the vibrations will then remain the same until we
reach the limit of the region entered.

This reasoning shows clearly that the surface will divide
generally into two or more parts, in each of which the num-
ber of vibrations will correspond to one of the two sounds ;
and it was very natural to conclude from thence that these
two sounds would be heard as proceeding from different sur-
faces.

This proposition appeared entirely new when I stated it in
1840. And, in fact, in the works published up to that time,
none but vague suppositions are to be found, such as are met
with in the work of Father Mersenne, and to which he did
not himself adhere ; or again, some accidental result of calcu-
lation offered in particular examples, and from which more-
over no induction was drawn similar to mine. I may even
say that I did not remark these remote relations until a long
time after I had demonstrated my general proposition, and
upon a minute investigation of all that might have any analogy
with it. 1 shall add lastly, that it was so little expected as to
be at first disputed, principally by M. Savart, when I an-
nounced it as a theoretical consequence of the laws of motion ;
he even predicted to me that the experiments which I proposed
to make to confirm it would not yield the result I expected.
For my part, I had no doubt on the point ; and I shall in a
few words describe these experiments, confining myself to
those which relate to plates or to bells, as the most easily
performed, and more conclusive than those which relate to
strings. I first took a square plate about two decimetres
wide by four millimetres thick, fixed at the centre and free at
all its other points. On moving the bow perpendicularly to
the plane of the plate and at any one of its corners, the deepest
sound was obtained, and the nodal lines were the two parallel
to the sides, drawn through the centre. On the contrary,
when the bow was placed in the middle of one of the sides,
the plate quitting its state of rest, a sound was obtained rather

2E2

420 M. Duhamel on the Multiple Sounds of Bodies.

higher than a fifth, and the nodal Hnes were the two diagonals
of tlie plate.

If now, after producing one of these sounds, which con-
tinues for a considerable time, the bow is employed in the
same manner as when it is used to produce the other sound
when the plate is at rest, the two sounds are heard at once :
the nodal lines disappear, as M. Savart had remarked, and
as Daniel Bernouilli had already observed in the case of
strings. There only remains to determine the relation of the
number of vibrations executed in the same time by two points
situated near each of the nodal lines, which only exist when
each sound takes place separately. This 1 have done by means
of the apparatus already mentioned; and I constantly found
the relation of 31 or 32 to 45, as I ought to find for the two
sounds, whose distance was a little less than a fifth.

I should add that another verification, less susceptible of
accuracy, might be made. On bringing the ear close to one
of the corners, scarcely any but the low note was heard ; and,
on the contrary, on bringing it near to the middle of one side,
only the high note was heard ; whence it appeared to follow,
that the two sounds were produced by different parts of the
plate.

I sought on the outline the point of separation of the parts
related to each number of vibrations; but this point varies
with the relation of the intensities of the two causes of vibra-
tion, and this inquiry is only of secondary interest.

I communicated these results to M. Savart, and he wished
immediately to repeat the experiment with me, employing
resonant tubes, which he put in unison, first with one sound
and then with the other. On bringing one of the tubes in
succession close to the different parts of the plate, the sound
was considerably strengthened in some, and was not percep-
tibly in others : the first appeared therefore only to produce
the sound of this tube; the second produced a similar effect
with the second tube. These results perfectly agree with
mine.

We finally considered the simultaneous sounds produced
by a large bell, and thought that we perceived in the case of
two, and even of three sounds, that each of them existed singly
in the same part of the surface. We did not perform this
experiment with the same degree of precision as the first,
because the result presented itself naturally as it was expected.

Both the reasonings, therefore, and the experiments in
question, led to the admission of this general law of the co-
existence of sounds, that —

Wheii ofie and the same vibratijig surface simidtaneously emits

M. Duhamel on the Multiple Sounds of Bodies. 4-21

several sounds, each of them exists in one or morejinite farts of
the surface, and appears to be perceptible there only ,- so that
the ear is affected as it would be by several separate surfaces,
ixhich should each give one only of the sounds in question.

In this memoir, resting on new facts, which 1 first arrived
at theoretically and then demonstrated by experiment, I
attempted to bring the phaenomenon in question under another
class of phsenomena admitted without dispute, and which con-
sists in our perceiving simultaneously the sounds produced by
the vibrations of different points. These inductions were not
disputed by any physicist. M. Poisson himself, who was much
occupied with acoustics, raised no objection to them. It is to
this same class of phaenomena that I shall proceed now to refer
those under consideration ; but it will be by means of a more
simple and at the same time more general theory, which will
supply the voids and uncertainties which still existed, and
which induced me to engage anew in this subject.

General explanation of the simultaneous sounds produced by
a single body.

We admit that when several points of a medium have dif-
ferent vibratory movements, we hear, in general, the different
sounds which each of them would give if it alone were in
motion ; and we propose to refer to this phaenomenon that of
the perception of several simultaneous sounds produced by a
single point in motion. In other words, it has to be proved
that our organs are sensibly affected in the same manner by
several movements existing in distinct points of the surround-
ing medium, as by a single movement resulting from the com-
position of the several movements in a single point of this me-
dium.

In the first place, when a point of the medium is not at a
very small distance from the ear, its movement produces in
all parts of our organ movements which do not differ percep-
tibly from those which would take place, if for the first point
of the medium any other were substituted not very distant
from it, and which was affected by the same movement. This
is easily demonstrated by calculation and experiment.

This being settled, we know, by the principle of the super-
position of small movements, that in any system of material
points, homogeneous or not, but whose mutual actions depend
only on their distances ; if one or more of these points have
movements resulting from the composition of several others,
the displacement and velocity of every point of the system
may be considered at every instant as the resultant of the
composition of those which would be observed in them at the

422 M. Duhamel on the Multiple Sounds of Bodies.

end of the same time, in the movements of the system corre-
sponding to the different movements composing the first points.

But, according to the preceding remark, our organs will
be affected in the same manner by the movement of a point of
the medium, or by an identical movement attributed to another
point near the first. Hence results the following propo-
sition : —

When any point of the medium 'vohich surrounds us is affected
by a movement resulting from the composition of several other Sy
all parts of our organs are sensibly affected in the same man-
ner as they would be if these different component movements^
instead of being united in the same point, existed separately at
different points of the first.

And reciprocally : —

If several points of a medium affected by different vibratory
movements cause us to hear several sounds at once, it nsoill suffce^
in order that a single point of the medium should cause us to
hear all these same sounds at a time, to give to this point the
movement resulting from the composition of the former ernes.

It is seen, then, as we stated, that the phaenomenon of the
multiplicity of the sounds which the same body gives, enters
into another class of phaenomena, that of the coexistence of
the sounds produced by distinct bodies which simultaneously
agitate the medium. It suffices, in fact, that the initial state
of a sonorous body, with respect to the displacement of its
molecules and the impressed velocities, be considered as re-
sulting from the composition of several initial states corre-
sponding to different simple sounds which it can produce, for
all these sounds to be produced in us, by each of the points
of the surface of this body.

It is possible, moreover, that one of these sounds may be
produced more strongly than another in certain parts of the
body, and even that there may be points in which it entirely
predominates. These different circumstances will depend on
the velocity in the different vibrations which are compounded
at each point. In fact, since we receive the same impressions
as if distinct points of the medium were respectively excited
by these elementary movements, the simultaneous sounds
which will proceed from the same point will have very differ-
ent intensities, if the magnitude of the velocities is itself very
different in the component vibrations. Experiment confirms
this proposition ; for, as I have already had occasion to say,
when a body emits several sounds at a time, there are portions
of its surface which seem to give only one sound, although we
may convince ourselves, by particular processes, that they
emit several others.

M. Duhamel 07i the Multiple Sounds of Bodies. 423

Experiments tvhich confirm the preceding theory.

The theoretical considerations on which I have founded
the explanation of the multiple sounds of bodies, seem to me
incapable of giving rise to any difficulty; and the assump-
tion cannot be denied, that the movement of a single point
may produce the sensation of several sounds, as soon as we
admit that this effect may result from the movement of several.
Nevertheless I have thought it not uninteresting to demon-
strate this fact experimentally. The first thing required was to
find an accurate method for determining the sound given by
each point of the surface of the vibrating body. I was at first
obliged to give up that which I had already employed, to depict
upon a disturbed plane the movement of the vibrating point,
since the object was to verify a sensation ; and it could not
even have been affirmed that a movement which might have
appeared to depict the same vibrations as when only one sound
was heard, would not have contained some difference imper-
ceptible to the eye, but which would have produced an effect
sensible to the ear : it was therefore to be referred solely to
this last sense. I tried different processes, the results of which
were always attended with uncertainty, and I at last stopped
at that which I shall proceed to describe and which is free
from all error.

I shall first call to mind, that when a rod or an elastic wire,
indefinitely extended in one direction, has its extremity sub-
jected to any small movement, each of its points is affected
successively with this same movement, which is propagated
with a constant velocity. If the wire is of a definite length,
this first movement is complicated with a second, which de-
pends on the length of the wire, but is insensible with relation
to the other ; and experiment shows, in fact, that the only
sound which the wire or the rod transmits, is that which
corresponds to the vibrations communicated to its extremity.

Hence it results that an effectual mode of studying the
proper movement of any point of the surface in a vibrating
body, will be to fix to it one of the extremities of an elastic
wire, to put the other extremity in communication with one
ear, closing the other carefully, and preventing the sound
reaching the first except by the medium of the wire. This
is very easily accomplished, and it is easy also to verify its
success. In fact, it is to be remarked, that the wire must be
stretched for the sound to be perceptible: the entire wire can
at pleasure be stretched, or that part can be left unstretched
which is near the vibrating surface ; and we find that in the
first case a very distinct sound is heard, but none in the second.

424 M. Duhamel on the Multiple Sounds of Bodies.

This proves two important things ; namely, first, that the sound
which is heard is transmitted by the wire alone ; and in the
second place, that it only proceeds from the point where it is
joined to the surface, and that the other parts of this surface
do not act perceptibly upon it by the medium of the air; for
if that were the case, we should still hear a sound when this
wire was stretched throughout its whole length, except near
the point where it is attached.

Once in possession of so simple and sure a process for ascer-
taining the sound emitted by any point of the surface of a vi-
brating body, I applied it to the investigation for which 1 had
devised it, and the following are the results at which I have
arrived.

I caused a square plate to vibrate so as to produce two
sounds ; I fixed the end of a thread of caoutchouc succes-
sively at different points of the surface, and I always heard the
two sounds, satisfying myself that they were transmitted only
by the wire ; this took place even at the points where the geo-
metrical influence of one of the movements was imperceptible.
Whence it follows that each point of the plate emits the double
sound, as the theory which I have explained had rigorously
established ; and they are distinguished by this method, even
when one of them has become so feeble that it would no longer
be heard through the medium of the air.

When the plate emitted three sounds, the wire still gave
the sensation. Instead of a plate, bells, strings, or bodies of
any form may be chosen, and the same fact will generally be
observed. Nevertheless we may imagine such forms as that
this law would be liable to exceptions, and not to hold good
over the whole extent of the surface. It might be that the move-
ment relative to one of the sounds would be weak in certain
parts of the surface, that even though it existed there singly,
it would make no sound heard ; in this case, still more, it
would not be heard if this movement were combined with
another ; and it will always be easily ascertained that the par-
ticular cases which, at first sight, would seem to constitute
exceptions, are explained naturally by meansof our principles.

I shall sum up the whole of this memoir by saying that I
have established theoretically and experimentally the following
proposition : —

If the vibratory motion of a -point be decomposed into several
others, the ear is perceptibly affected in the same manner by the
movement of this point, as it would be by so many distinct points,
each under the influence of one of these component motions.

The phaenomenon of the multiplicity of sounds emitted by
a single point is thus referred to that of the simultaneous

Sir W. Rowan Hamilton on Quaternions* 425

audition of the sounds emitted by separate points. Being
referred to an admitted phaenomenon, it is explained ; and 1
think I may say that it had not been completely explained
before.

The conclusion of these researches is, therefore, that the
phaenomena of simultaneous perception of several sounds pro-
ceeding from the movement, whether of several points or of a
single one, are only modifications of one general phaenomenon,
which may be stated in the following manner : —

" When our organ of hearing is affected by a movement
that may be geometrically decomposed into several others,
which, if they existed separately, would yield different sounds,
we generally perceive all these sounds at the same time."

LXIII. On Quaternions; or on a New System of Imaginaries
in Algebra. By Sir William Rowan Hamilton, LL.D.,
M.R.I. A. ^ F.R.A.S.i Corresponding Member of the Insti-
tute of France, ^c., Andrews* Professor of Astronomy in the
University of Dublin , and Royal Astronomer of Ireland.
[Continued from p. 343.]

71. T>EFORE entering on any discussion of this new form
XJ of the equation of the ellipsoid, namely the form

TV|j^j =fi'->l% eq. (139.), art. 70,

it may be useful to point out another manner of arriving at
the same equation of the ellipsoid, by a different process of
calculation, from that construction or generation of the surface,
as the locus of the circle which is the mutual intersection of a
pair of equal spheres, sliding within two fixed cylinders of
revolution whose axes intersect each other; while the right
line, connecting the centres of the two sliding spheres, moves
parallel to itself, or remains constantly parallel to a fixed right
line in the plane of the fixed axes of the cylinders : which
mode of generating the ellipsoid was published in the Philo-
sophical Magazine for July 1848 (having also been communi-
cated to the Royal Irish Academy in the preceding May),
as a deduction from the Calculus of Quaternions. And
whereas the fixed right line, through the centre of the ellip-
soid, to which the line connecting the centres of the two sli-
ding spheres is parallel, may have either of two positions,
since it may coincide with either of the two cyclic normals,
we shall here suppose it to have the direction of the cyclic
normal », or shall consider the second pair of sliding spheres

4<26 Sir W. Rowan Hamilton on Quaternions.

mentioned in article 64, of which the quaternion equations
are, by article 62 (Phil. Mag. for July 1848),

T(p-,.)=T(p-X') = i. (114.)

72. Here (see Phil. Mag. for May 1848), we have for jtx,
the value,

l«,=^'(x— »)i eq. (91.), art. 57 j
and

a'(x'- »') = x'p + px', eq. ( 1 1 0.), art. 60 ;
also

»x'=»'x = T.»x, eq. (107), same article;

whence we derive for >! the expression,

>!='^^^^ = ^:l^. . . . (HO.)
» '— X ' * — rx *

But

{i-^^K-')-'={i{it-i)K-'}-^ = )i{}i-i)-h-'', . (141.)

and by (104.),

,p + p, = -A'(x-02; .... (142.)
therefore

X'=-^'x(x-.)»-i=A'(x-x2,-i). . . (143.)

If then we make, for abridgement,

g=-/i'T^, (144.)

and employ the two new fixed vectors >j and 9, defined by the
equations (see Phil. Mag. for May 1849),

,] = T.U(i-x), 5 = T:<U(x->-»-i), (131.)
which have been found to give

i~x=>,T'-^, x-xVi=-9T*-:=^, (132.)

we shall have the values,

(i—gri; A'=g9; (146.)

and the lately cited equations (114.) of the two sliding spheres
will become,

T(p^gri)=b', T{p-gQ)^b; . . . (146.)

between which it remains to eliminate the scalar coefficient gf
in order to find the equation of the ellipsoid, regarded as the
locus of the circle in which the two spheres intersect each
other.

73. Squaring the equations (146.), we find (by the general

Sir W. Rowan Hamilton on Quaternions, 4t9,*J

rules of this Calculus) for the two sliding spheres the two fol-
lowing more developed equations :

Taking then the difference, and dividing by g, we find the
equation

^(92->,2) = 2S.(fl-„)p; .... (148.)

which, relatively to p, is linear, and may be considered as the
equation of the plane of the varying circle of intersection of
the two sliding spheres ; any one position of that plane being
distinguished from any other by the value of the coefficient g.
Eliminating therefore that coefficient g^ by substituting in
(146.) its value as given by (148.), we find that the equation
of the ellipsoid, regarded as the locus of the varying circle,
may be presented under either of the two following new forms :

T(f-?%^0=*' • • ■ ("«•'

t(p-?^^0=^=- • • C^"-)

respecting which two forms it deserves to be noticed, that
either may be obtained from the other, by interchanging *j and
6. And we may verify that these two last equations of the
ellipsoid are consistent with each other, by observing that the
semisum of the two vectors under the sign T is perpendicular
to their semidifference (as it ought to be, in order to allow of
those two vectors themselves having any common length, such
as b) ; or that the condition of rectangularity,

, (9 + »i)S.(a-~y,)p
P pIT^a -L9 — >Jj . • • (151.)

is satisfied : which may be proved by showing (see Phil. Mag.
for July 1846) that the scalar of the product of these two last
vectors vanishes, as in fact it does, since the identity

(9-,,)(9 + >,) = 6Hfl')->i9->}%

resolves itself into the two following formulae:

S.(d-„)(d+„)=fl^->,^1

V.(d-,,)(fl + i,)=fl,-„d;J • • • V . •;

of which the first is sufficient for our purpose. We may also
verify the recent equations (149.) (150.) of the eUipsoid, by
observing that they concur in giving the mean semiaxis b as
the length Tp of the radius of that diametral and circular sec-

428 Sir W. Rowan Hamilton on Quaternions.

tion, which is made by the cyclic plane having for equation

S.{6~ri)p = 0; (153.)

this plane being found by the consideration that vj — Q has the
direction of the cyclic normal i, or by making the coefficient
^=0, in the formula (148.).

74. The equation (149.) of the ellipsoid may be successively
transformed as follows :

= T{&^p-ri{6p-{-p^)+r,pri}

= Ty{{Q-r,)&P^np{d-ri)}

= TV.{pd-np){Q-ri)

= TV.(>,p-p9)(,,-9); (154.)

and by a similar series of transformations, performed on the
equation (150.), we find also (remembering that fl^— >3^, being
equal to x^—t,% is positive),

b{l^^n^)^Ty.{pyi-Qp){yi-6). . . . (155.)

The same result (155.) may also be obtained by interchanging
vj and 9 in either of the two last transformed expressions (154.),
for the positive product b{Q'^—ry^) ; and we may otherwise
establish the agreement of these recent results, by observing
that, in general, if Q and Q' be any two conjugate quaternions
(see Phil. Mag. for July 1846), such as are here rip—pQ and
pYi — &p, and if a be any vector, then

TV.Qa=TV.Q'a; (156.)

for

and because

V.Qa=:«SQ-V.«VQ, n

V.Q'«=aSQ + V.«VQ;/ • • • ^ '•-'

0 = S.aV.aVQ, (158.)

the common value of the two members of the formula (156.) is

TV.Q«=V'{(TV.«VQ)2 + (T«.SQ)2}. . (159.)

If then we substitute for b its value,

b=T{r}-$), eq. (135.), art. 70,

and divide on both sides by this value of b, we see, from (154.),
(155.), that the equation of the ellipsoid may be put under
either of these two other forms :

TV.()jp-pd)U()j-5)=a2-,,2, . . . (160.)

TV.(p)j-9p)U(*3-fl) = fl2-)j2. . . . (161.)

Sir W. Rowan Hamilton on Quaternions. 429

But the versor of everi/ vector is, in this calculus, a square
root of negative unity ; we have therefore in particular,

(U(n-9))2=-l; (162.)

and under the sign TV, as under the sign T, it is allowed to
divide by —1, without affecting the value of the tensor: it is
therefore permitted to write the equation (160.) under the
form

which form is thus demonstrated anew.

75. A few connected transformations may conveniently be
noticed here. Since, for any quaternion Q,

(TVa)2= -(VQ)2 = (TQ)2-(SQ)S . (163.)

while the tensor of a product is the product of the tensors,
and the tensor of a versor is unity ; and since

S.(p>j-6/j)(ij-fl) = S(p)j2-p»j6-ep>} + flp5) = ~2S.>jdp, (164.)

because

0 = S.pr=:S.QpQ,andS.pyiQ = SJpri=sS,ri$p; . (165.)

we have therefore, generally,

T.(/,,-5p)U(,,-fl) = T(p,,-fl|.); 1 .

S.(p>}-fl^)U(»-9)=-2T(.j-d)-'S.)j9p;J * ''

and there results the equation,

TV.{pn-^p)\3{r,-^)=V{T{pn-^Y-^T{yi-^)-^{S.n^pf},{\61,)

as a general formula of transformation, valid for any three
vectors, >j, $, p. We may also, by the general rules of the
present calculus, write the last result as follows,

TV.(/pij-flp)U(>3-9)=i/{(p.j-9p)(>jp-p^)

-^{n-^y^yfip-phf}', ..... (168.)

the signs S and T thus disappearing from the expression of
the radical. For the ellipsoid, this radical, being thus equal
to the left-hand member of the formula (167.), or to that of
(168.), must, by (161.), receive the constant value &^ — r^; so
that, by squaring on both sides, we find as a new form of the
equation (161.) of the ellipsoid, the following:

Or, by a partial reintroduction of the signs S and T, we find
this somewhat shorter form :

T(f'J-flf)H4(>j-fl)-^(S.>)flp)2=(fl2-,,2)2. . (170.)

4S0 Sir W. Rowan Hamilton on Quaternions.

And instead of the square of the tensor of the quaternion
prj—Qp, we may write any one of several general expressions
for that square, which will easily suggest themselves to those
who have studied the transformations (already printed in this
Magazine), of the earlier and in some respects simpler equa-
tion of the ellipsoid, proposed by the present writer, namely
the equation

T(<^ + ^x) = x2_,9. eq. (9.), art. 38.

For instance, we may employ any of the following general
equalities, which all flow with little difficulty from the princi-
ples of the present calculus:

-{pri-6p)(r)p-pQ) = (rip-p&){pri-$p)
= {r}^ + Q^)p^-py}pQ-6prip

= {n + Q)Y-{rip+pri){dp+pQ)

= (>j' + flV-2S.r]^flp

z={yi + Q)y-4^S.rip.SJp

= (l-fl)y + 4S(V.,,p.V.p5); .... (171.)

and which all hold good, independently of any relation between
the three vectors v}, $, p.

76. As bearing on the last of these transformations it seems
not useless to remark, that a general formula published in the
Philosophical Magazine of August 1846, for any three vectors
a, a', «", namely the formula

aS.«'a"-a'S.a"« = V(V.aa'.a"), eq. (12.) of art 22,
which is found to be extensively useful, and indeed of constant
recurrence in the applications of the calculus of quaternions,
may be proved symbolically in the following way, which is
shorter than that employed in the 23rd article:

V(V.a«'.a") = |(V.a«'.a"-a"V.«a') = l(««'.a"-«".aa')

= !«(«'«" + «V)-i(aa" + «"«)«' = aS.a'a"-a'S.«"«. (172.)
The formula may be also written thus:

V.a"V.a'a = «S.a'a"-«'S.««"; . . . (l73.)
whence easily flows this other general and useful transforma-
tion, for the vector part of the product of any three vectors,

a, a', a":

V.a"a'a = aS.a'a"-«'S.a"« + «"S.a«'. . (174.)

Operating on this by S.a'", we find, for the scalar part of the
product of any four vectors, the expression :

S .«"'«"«'« = S'. «'"«. S . u'x" - S.«"'a'. S.«"« + S. a"'«".S.««'.( 1 75.)

Sir W. Rowan Hamilton 071 Quaternions, 4«31

But a quaternion, such as is u'a or «'"«", is always equal to the
sum of its own scalar and vector parts ; and the product of a
scalar and a vector is a vector, while the scalar of a vector is
zero : therefore

a'«=S.a'« + V.a'a, «'"«"= S.a"'«" + V. «'"«", . (176.)
and

S . ec"'a"u'ct = S . a"'u". S.u'u + S(V. «"'a". V. a'«). (177.)

Comparing then (175.) and (177.), and observing that

S.«a'=+S,«'«, V.aa'=-V.a'a, . . (178.)

we obtain the following general expression for the scalar part
of the product of the vectors of any two binary products of
vectors :

S(V. «'"«".¥. a'«)= S . «'"«. S . «'a"-S . u"W. S.a"« ; (179.)
while the vector part of the same product of vectors is easily
found, by similar processes, to admit of being expressed in
either of the two following ways (compare equation (3.) of
article 24) :
V(V. a"'a".V.«'a) =a"'S . a"a'a-a"S . u"'u'a

= aS.a"'a"a'-a'S.a"'a"«; . . . (180.)

of which the combination conducts to the following general
expression for any fourth vector a'", or p, in terms of any
three given vectors a, a', a", which are not parallel to any one
common plane (compare equation {4t.) of article 26):

pS.cc"u'u = uS.u"u'p + u'S.u"fcc + u"S.pa'a, . (181.)

If we further suppose that

«"=:V.«'«, (182.)

we shall have

S.a"«'a=(V.«'«)2=«"2; . . . (183.)

and after dividing by a"% the recent equation (181.) will be-
come

whereby an arbitrary vector p may be expressed, in terms of
any two given vectors a, a', which are not parallel to any com-
mon line, and of a third vector u", perpendicular to both of
them. And if, on the other hand, we change «, a', «", «'" to
e, p, p, >), in the general formula (179.), we find that generally,
for any three vectors »j, 9, p, the following identity holds good:
S(V.)jp.V.p9)=p2S.»ifl-S.)}p.S.p9; . (185.)

which serves to connect the two last of the expressions (171.)»
and enables us to transform either into the other.

4-32 Sir W. Rowan Hamilton on Quaternions.

77. To show the geometrical meaning of the equation (185.),
let us divide it on both sides by T.p^)}^ ; it then becomes, after
transposing,

-SU.)39=SU.*3p.SU.pfl + S(VU.))p.VU.p9). (186.)

Here, by the general principles of the geometrical interpreta-
tion of the symbols employed in this calculus (see the remarks
in the Philosophical Magazine for July 184-6), the symbol
SU . ijS is an expression for the cosine of the supplement of
the angle between the two arbitrary vectors >j and fl; and
therefore the symbol — SU . >3fl is an expression for the cosine
of that angle itself. In like manner, — SU .r^p and — SU .pfi
are expressions for the cosines of the respective inclinations of
those two vectors ij and 6 to a third arbitrary vector p ; and at
the same time VU.ijp and VU . p9 are vectors, of which the
lengths represent the sines of the same two inclinations last
mentioned, while they are directed towards the poles of the
two positive rotations corresponding; namely the rotations
from >j to p, and from p to 9, respectively. The vectors VU.jjp
and VU.p9 are therefore inclined to each other at an angle
which is the supplement of the dihedral or spherical angle,
subtended at the unit-vector Up, or at its extremity on the
unit-sphere, by the two other unit-vectors U>] and U5,or by the
arc between their extremities : so that the scalar part of their
product, in the formula (186.), represents the cosine of this sphe-
rical angle itself (and not of its supplement), multiplied into
the productofthe sines of the two sides or arcs upon the sphere,
between which that angle is included. If then we denote the
three sides of the spherical triangle, formed by the extremities

of the three unit-vectors Urj, U9, Up, by the symbols, »)d, vjp, p9,
and the spherical angle opposite to the first of them by the

symbol rjpfl, the equation (186.) will take the form

cos >39= cos>)pcos pS-f sinijp sin pfl cos >]p9; . (187.)

which obviously coincides with the well-known and funda-
mental formula of spherical trigonometry, and is brought for-
ward here merely as a verification of the consistency of the
results of this calculus, and as an example of their geometrical
interpretability.

A more interesting example of the same kind is furnished
by the general formula (179.) iov four vectors, which, when
divided by the tensor of their product, becomes

S(VU . a'V'.VU . u'u) = SU . a"'« . SU . a'a"

-SU.«'V.SU.«"«; (188.)

Sir W. Rowan Hamilton on Qitaierniorts. 433

and signifies, when interpreted on the same principles, that

sin ««'. sin «"«'", cos (««' «"«'")= cos«a". cos «'«'"

— cos aa'". cos «'«" ; (189.)

where the spherical angle between the two arcs from a to a'
and from a" to a!" may be replaced by the interval between the
poles of the two positive rotations corresponding. The same
result may be otherwise stated as follows: If Z/, L', L", V",
denote any four points upon the surface of an unit-sphere,
and A the angle which the arcs LL', L"L"' form where they
meet each other, (the arcs which include this angle being mea-
sured in the directions of the progressions from L to Z/', and
from Z/" to Z<"' respectively,) then the following equation will
hold good :

cos LZ". cos L'L"'- cos LUK cos LIL"

=z sin LL'. sin L"L"'. cos A. . . . (190.)

Accordingly this last equation has been incidentally given, as
an auxiliary theorem or lemma, at the commencement of those
profound and beautiful researches, entitled Disquisitio7ies
Generates circa Superficies Curvas^ which were published by
Gauss at Gottingen in 1828. That great mathematician and
philosopher was content to prove the last written equation by
the usual formulae of spherical and plane trigonometry ; but,
however simple and elegant may be the demonstration thereby
afforded, it appears to the present writer that something is
gained by our being able to present the result (190.) or (189.),
under the form (188.) or (179.), as an identity in the quater-
nion calculus. In general, all the results of plane and sphe-
rical trigonometry take the form of identities in this calculus ;
and their expressions, when so obtained, are associated with
a reference to vectors, which is usually suggestive of graphic
as well as metric relations.
78. Since

p>j-ep = S.f(>j-d) + V.p(>!+fl), . ... (191.)

the quaternion prj—Qp gives a pure vector as a product, or as
a quotient, if it be multiplied or divided by the vector ri + Q
(compare article 68) ; we may therefore write

pn-Qp=:\,{ri + $), ..... (192.)

Ai being a new vector-symbol, of which the value may be thus
expressed :

A,=p-2(>j + 0-iS.fip (193.)

Phil. Mag. S. 3. Vol. 34. No. 231. June 1849. 2 F

494 Sir W. Rowan Hamilton on Qiiaternions,

The equation (192.) will then give,

T(p>j-6p)2=V()3 + 5)2. / • • • ^
We have also the identity,

(92_^2)2^(^__5)2(^4.e)S+(^9_5^)2. . . (195.)

which may be shown to be such, by observing that

(,,_e)2(^ + fi)2=(,2 4.e2_2S.>j9)(,,2 + 9H2S.>)5)

= (^2 + fi2)2_4(S.^9)2^(^2_52)2^4.(X.,,fl)2_4(S.>)fl)2

= (,,2_fl2)2_4.(V.>j9)2=(62_»,2)2_(^9_e^)2. , . (igg.)

or by remarking that (see equations (152.) (163.)),

,^-feS.(,,-fl)(>, + e), >]9-fl>,=V.(),-S)(,, + 9),-|

and(>)-S)2(n + e)2=(T.(^-fl)(l + 5))^; -J ^

or in several other ways. Introducing then a new vector e,
such tliat

,,9-fi>j = sT(»i4-S), or, £=2V.>i3.T(*) + 5)-'; . (198.)

and that therefore

(,,9-e,,)2=_s2(^ + 5)2, .... (199.)
and

2S.o36p = S.£p.T(*)+9), 4(S.r)9^)2=-(S.e^)2(») + 6)2; (200.)

while, by (135.),

T(*j-fi)=5, ()j-6)^=-&^; . . . (201.)
we find that the equation (170.) of the ellipsoid, after being
divided by (>) + 5)^ assumes the following form :

Ai2^&-2(S.ep)2 + 6^ + 82=o. . . . (202.)
But also, by (193.), (198.),

S.eXi = S.g^; (203.)

the equation (202.) may therefore be also written thus :

0=(Xi-e)2+ (6 + 6-^8. sp)2; . . . (204.)

and the scalar b + b~^^.sp is positive, even at an extremity of
the mean axis of the ellipsoid, because, by (195.) (199.) (201.),
we have

(^2-»,2)2=-.(^,2 + s2)(^^g)2=(^,2_X62)T()3 + e)2, . (205.)

and therefore

Ts<6 (206.)

We have then this new form of the equation of the ellip-
soid, deduced by transposition and extraction of square roots

Sir W. Rowan Hamilton on Quaternions. 435

(according to the rules of the present calculus), from the form
(204.) :

T(Ai-6)=^+&~'S.6^ (207.)

By a process exactly similar to the foregoing, we find also the
form

T(Al + ^) = &-i-^S.s/>; . . . (208.)

which differs from the equation last found, only by a change
of sign of the auxiliary and constant vector s : and hence, by
addition of the two last equations, we find still another form,
namely,

T(Ai-6)+T(Ai + 6) = 26; . . . (209.)

or substituting for Xj, e, and b their values, in terms of »), 9,
and p, and multiplying into T(jj4-9),

= 2T.(»,-9)(,, + 6) (210.)

79. The locus of the termination of the auxiliary and vari-
able vector Aj, which is derived from the vector p of the original
ellipsoid by the linear formula (193.), is expressed or repre-
sented by the equation (209.) ; it is therefore evidently a cer-
tain new ellipsoid, namely an ellipsoid of revolution, which has
the mean axis 2b of the old or given ellipsoid for its major
axis, or for its axis of revolution, while the vectors of its two
foci are denoted by ihe symbols +e and — e. If « denote the
greatest, and c the least semiaxis, of the original ellipsoid,
while b still denotes its mean semiaxis, then, by what has
been shown in former articles, we have the values,

T,,=Ti=i(a + c); T9 = Tx=i(a-c); . (211.)

and consequently (compare the note to art. 70),

a=Tij + Tfl; c=Tti-TQ; . . . (212.)

therefore

«c=T)j2-Td2=62_;,2. .... (213.)

also

= 2T>j2 + 2Tfl2 = (T>j + Te)2 + (T)} - T&)% (214.)
and

T(>) + d)2 = fl2_j2 + c2; .... (215.)

whence, by (205.),

T.W.-^-^ = (-^^^. . (...)
2F2

45^ Sir W. Rowan Hamilton on Quaternions.

Such, then, is the expression for the square of the distance
of either focus of the new or derived ellipsoid of revolution,
which has A^ for its varying vector, from the common centre
of the new and old ellipsoids, which centre is also the common
origin of the vectors X, and p : while these two foci of the new
ellipsoid are situated upon the mean axis of the old one.
There exist also other remarkable relations, between the
original ellipsoid with three unequal semiaxes a, b, c, and the
new ellipsoid of revolution, of which some will be brought
into view, by pursuing the quaternion analysis in a way which
we shall proceed to point out.

80. The geometrical construction already mentioned (in
articles 64, 71, &c.), of the original ellipsoid as the locus of
tlie circle in which two sliding spheres intersect, shows easily
(see art. 72) that the scalar coefficient^, in the equations (146.)
of that pair of sliding spheres, becomes equal to the number 2,
at one of those limiting positions of the pair, for which, after
cutting, they touch, before they cease to meet each other. In
fact, if we thus make

5=2, (217.)

the values (145.) of the vectors of the centres will give, for
the interval between those two centres of the two sliding
spheres, the expression

T{ix,-X<)=gT{n-Q) = 2b; .... (218.)

this interval will therefore be in this case equal to the diameter
of either sliding sphere, because it will be equal to the mean
axis of the ellipsoid : and the two spheres will touch one an-
other. Had we assumed a value for g, less by a very little
than the number 2, the two spheres would have cut each other
in a very small circle, of which the circumference would have
been (by the construction) entirely contained upon the surface
of the ellipsoid; and the plane of this little circle would have
been parallel and very near to that other plane, which was
the common tangent plane of the two spheres, and also of the
ellipsoid, when g received the value 2 itself. It is clear, then,
that this value 2 of ^ corresponds to an umbilicar point on the
ellipsoid ; and that the equation

S.(9->,)p=S2_,,2, .... (219.)

which is obtained from the more general equation (148.) of
the plane of a circle on the ellipsoid, by changing^ to 2, re-
presents an umbilicar tangent plane, at which the normal has
the direction of the vector *j— 5. Accordingly it has been
seen that this last vector has the direction of the cyclic normal
«; in fact, the expressions (131.), for yj and 5 in terms of » and

Sir W. Rowan Hamilton on Quaternions. 437

X, give conversely these other expressions for the Jatter vectors
in terms of the former,

. = T)jU(.j-fl); x = T9U(S-'->)-»): . (220.)
whence (it may here be noted) follow the two parallelisms,

U,-Ux = U(>j-5) + U(.)-»-fl-') II U»j + U5; . (221.)
U* + Ux = U(,j-S)-U()j-»-fl-') II U>3-U3; . (222.)

the members of (221.) having each the direction of the great-
est axis of the ellipsoid, and the members of (222.) having
each the direction of the least axis ; as may easily be proved,
for the first members of these formnlae, by the construction
with the diacentric sphere^ which was communicated by the
writer to the Royal Irish Academy in 1846, and was published
in the present Magazine in the course of the following year.
The equation (219.) may be verified by observing that it gives,
for the length of the perpendicular let fall from the centre of
the ellipsoid on an umbilicar tangent plane, the expression

j9 = (d2_,,2)T(,j-S)->=:flC^»-'; . . . (223.)

agreeing with known results. And the vector to of the um-
bilicar point itself must be the semisum of the vectors of
the centres of the two equal and sliding spheres, in that limit-
ing position of the pair in which (as above) they touch each
other ; this umbilicar vector oo is therefore expressed as follows:

co = Yi + Q; (224,.)

because this is the semisum of ft and x' in (145.), or of g>) and
gd when ^=2. (Compare the note to article 70.) As a veri-
fication, we may observe that this expression (224.) gives, by
(215.), the following known value for the length of an umbi-
licar semidiameter of the ellipsoid,

u=Tco=T{ri + d)= Via^-b^+c^ . . (225.)
By similar reasonings it may be shown that the expression

«;' = T>,Ufi + TflU)j, .... (226.)
which may also be thus written, (see same note to art. 70,)

a,' = -T.»)fi.(»j-' + 9-0, .... (227.)
represents another umbilicar vector ; in fact, we have, by (224.)
and (226.),

Tco'=T«>, (228.)

and

co-c«'=(Trj-Tfl)(U>]-U9)U * • • ^ •'
so that the vectors w co' are equally long, and the angle between

(230.)

438 Sir W. Rowan Hamilton on Quaternions.

them is bisected by Ur +Ud, or (see (221.)) by the axis major
of the ellipsoid j while the supplementary angle between on
and — ctt' is bisected by U>j — Ud, or (as is shown by (222.)) by
the axis minor. It is evident that —m and —aJ are also um-
bilicar vectors ; and it is clear, from what has been shown in
former articles, that the vectors )j and Q have the directions of
the axes of the two cylinders of revolution, which can be cir-
cumscribed about that given or original ellipsoid, to which all
the remarks of the present article relate.

81. These remarks being premised, let us now resume the
consideration of the variable vector Aj, of art. 78, which has
been seen to terminate on the surface of a certain derived
ellipsoid of revolution. Writing, under a slightly altered form,
the expression (193.) for that vector Aj, and combining with
it three other analogous expressions, for three other vectors,
Ag, A3, A4, as follows,

it is easy to prove that

TAi = TA2 = TA3=:TA4; .... (231.)
and that

S.)39Ai = S.>)5a2=S.>)9a3=S.>)9A4=S.»i9p; . (232.)

i^hence it follows that the four vectors Aj, Ag, A3, A^, being sup-
posed to be all drawn from the centre a of the original ellip-
soid, terminate in four points, Lj, Lg, Lg, L4, which are the
corners of a quadrilateral inscribed in a circle of the derived
ellipsoid of revolution ; the plane of this circle being parallel
to the plane of the greatest and least axes of the original ellip-
soid, and passing through the point e of that ellipsoid, which
is the termination of the vector p. We shall have also the
equations,

iC:£ = |:f=V-0; tsZl=^=V-'o; (233.)

Aj— p b.dp ^4 — P 0.9 'p

which show that the two opposite sides L^Lg, L3L4, of thisin-
scribed quadrilateral, being prolonged if necessary, intersect
in the lately- mentioned point e of the original ellipsoid. And
because the expressions (230.) give also

V^=«. V;;^.=0, . . (234.)
these opposite sides LjLg, L3L4, of the plane quadrilateral thus

Sir W. Rowan Hamilton on Quaternions. 439

inscribed in a circle of the derived ellipsoid of revolution, are
parallel respectively to the vectors >j+fl, >)~* + ^"^ or to the
two umbilicar vectors co, «', of the original ellipsoid, with the
semiaxes abc. At the same time, the equations

wbzh^Oy V^i~^=0, . . . (235.)

hold good, and show that the two other mutually opposite
sides of the same inscribed quadrilateral, namely the sides L2L3,
L4L1, are respectively parallel to the two vectors >), $, or to the
axes of the two cylinders of revolution which can be circum-
scribed about the same original ellipsoid. Hence it is easy to
infer the following theorem, which the author supposes to be
new : — If on the mean axis 2b of a given ellipsoid^ abc, as the
major axis^ and isoith ixm foci Fi, Fg, qf>which the common di-
stance Jrom the centre a is

- - K/{a^-h^s/{h^-c^) .

we construct an ellipsoid of revolution ; and if in any circular
section of this new ellipsoid, we inscribe a quMdrilateral^iu^Li^^^i
of which the two opposite sides LjLg, L3L4 are respectively parallel
CO the two umbilicar diameters of the given ellipsoid ; while the
two other and mutually opposite sides, i-gLg, L4L1, of the same
inscribed quadrilateral, are respectively farallel to the axes of
the two cylinders of revolution which can be circumscribed about
the same given ellipsoid; then the point of intersection E of the
Jirst pair of opposite sides (namely of those parallel to the um-
bilicar diameters), will be a point upon that given ellipsoid. It
seems to the present writer that, in consequence of this re-
markable relation between these two ellipsoids, the two foci
Fj, Fg, of the above described ellipsoid of revolution, which
have been seen to be situated upon the mean axis of the ori-
ginal ellipsoid, of which the three unequal semiaxes are de-
noted by a, b, c, may be not inconveniently called the two
MEDIAL FOCI of that original ellipsoid : but he gladly submits
the question of the propriety of such a designation, to the
judgement of other and better geometers. Meanwhile it may
be noticed that the two ellipsoids intersect each other in a
system of two ellipses, of which the planes are perpendicular
to the axes of the two cylinders of revolution above mentioned;
and that those four common tangent planes of the two ellip-
soids, which are parallel to their common axis, that is to the
mean axis of the original ellipsoid abc^ are parallel also to its
two umbilicar diameters.

[To be continued.]

[ 440 ]

LXIV. Further Researches on Electro-Physiology.
By M. Ch. Matteucci*.

I HOPE that the Academy, which has always been pleased
to encourage me in my researches upon electro-physio-
logy, will permit me to communicate some new investigations
upon this subject. I cannot commence the exposition of these
researches without very briefly recapitulating the four principal
points from which I started, and which, to a certain extent,
form a summary of the whole of my former labours.

1. In each cell of the electric organ of fishes, the two elec-
tricities become separated under the influence of the nervous
action propagated from the brain towards the extremities of
the nerves. A relation exists between the direction and the
intensity of the nervous current, and the position and the
quantity of the two electricities developed in the cell. In acr
cordance with this relation which has been established expe-
rimentally, if, as was done by Ampere in the case of electro-
magnetic action, we represent the nervous current by a man
lying extended upon the nerve, and with his face turned
towards the caudal extremity of the Torpedo or the dorsal
surface of the Gymnotus, the positive electricity of the cell
always exists on the left of the man: since each cell of the
organ forms a temporary electi'ic apparatus, this explains the
position of the poles at the extremities of the prisms, and the
intensity of the discharge being proportional to the length of
the prisms, as established by experiment.

2. It has been shown by experiment, that the greatest ana-
logy exists between the discharge of electric fishes and mus-
cular contraction. There is no circumstance which modifies
one of these phaenomena which does not act equally upon the
other.

3. The contraction of a muscle developes in a nerve which
is in contact with this muscle, the cause by which the nerve
excites contractions in the muscles through which it ramifies.
Although experiment has not yet enabled us to decide whether
this phaenomenon is an instance of nervous induction, or a
proof of an electric discharge developed by muscular contrac-
tion, we are led by all analogy to admit the second hypothesis.

4. The electric current modifies the excitability of the nerve
according to its direction. The electric current, when pro-
pagated in the direction of the ramification of the nerve, de-
stroys its excitability ; when propagated in a contrary direc-
tion to the ramification, it augments the excitability of the
nerve. The phaenomena brought into play by the cessation

* From the Comptes Rendus for April 30, 1849.

M. Matteucci's Researches on Electro-Physiology. ^^l

of an electric current traversing the nerves of an animal, de-
pend upon the modification which the excitability of the nerve
has experienced by the passage of the current, according to
its direction. The same cause explains the voltaic alterna-
tions, /. e. the muscular contractions excited by a current,
which is made to traverse a nerve in a contrary direction to
that in which it had ceased to produce any effect.

In this first extract I shall confine myself to communicating
to the Academy a result which I regard as fundamental to the
theory of electro- physiological phaenomena. By a very simple
experiment, and one which is easily repeated, J have shown
that an electric current which traverses a muscular mass in
the direction of its fibres, and consequently in a direction
which is normal or oblique to that of the ultimate nervous
ramifications which are distributed through it, developes in
these filaments a nervous current, the direction of which varies
according to that of the electric current, relatively to the ra-
mification of the nerve. This law is the same as that which
establishes the relation between the direction of the nervous
current and the position of the contrary electric conditions in
the organ of electric fishes ; in other words, it is the reaction
of electricity upon the nervous force. In discovering a new
analogy, and that the most intimate possible, between the
electric discharge of fishes and muscular contraction, I have
shown that, just as in the electric apparatus of the torpedo,
the nervous current developes the two electricities in a deter-
minate direction, according to its own direction. In a mus-
cular mass the two electric states, diffused through the elements
of its fibres, produce a current, the direction of which, varying
with that of the electric current, is established, like the direc-
tion of the discharge in the torpedo, by that of the nervous
current which excites it. I have taken every pains to establish
by experiment this result, which I shall henceforth consider
as the foundation of the theory of electro- physiological phae-
nomena. Whatever may be the nature of the nervous force,
of which we are ignorant, as of that of the other great natural
agents, it is a fact that this force is propagated in the nerves
sometimes from the brain to the extremities, sometimes in a
contrary direction. It is entirely independent of hypothesis,
and, in fact, in accordance with experiment to admit, that in
the act of muscular contraction excited by the action of the
will or by the stimulation of the nerve, a nervous current is
propagated in the direction of the ramification of the nerve :
on the other hand, the nervous current follows an opposite
direction, when sensation is experienced by the stimulation of
the extremities of the nerve.

i'4-2 M. Matteucci's Researches on Electro-Physiology.

I have already shown in my former researches, and by
direct experiments, the great difference between the nervous
and the muscular substance as regards the conduction of the
electric current. Regarding these experiments, which it would
be impossible for me to describe here in detail, I shall confine
myself to the account of one, the evidence afforded by which
is perfect, and which may be applied to the case in point. This
experiment consists in introducing the nerve of a very sensitive
galvanoscopic frog into the interior of a muscular mass, cut
with a knife in the direction of its fibres. On passing a tole-
rably strong electric current through the muscular mass, con-
tractions are never excited in the prepared frog. In this case,
besides the better conductibility of the muscular substances,
we have for the production of the effect observed the great
difference between the relative mass of the muscle and of the
nerve. It is unnecessary to state, that the contraction of the
prepared frog occurs if the poles of the battery are closely
approximated to its nerve, or if the muscular mass, by its con-
tractions, produces the phsenomenon called induced contrac-
tion. The experiment succeeds perfectly on taking the
muscles of one of the mammalia or a bird, after their irrita-
bility has ceased ; so that the passage of an electric current
through these muscles does not excite any sensible contraction.

It is then proved by experiment, that when a muscular mass
is traversed by an electric current, the nervous filaments dif-
fused through the mass do not produce any sensible part of
this current, so that the effects obtained can only be due to
the direct action of the electric current upon the muscular
fibre, and to the indirect action or the injluence oi ihe electric
current upon the nervous force.

The following are these effects : — If, in a living rabbit, dog
or frog, we expose the muscles of the legs, by entirely re-
moving the integuments, and pass an electric current from a
pile oF thirty or forty elements through these muscles, apply-
ing one of the poles to the upper and the other to the lower
part of the leg — if the positive pole is placed above and
the negative pole below, so that the electric current traverses
the muscular substance in the direction of the ramification of
the nerves, a very powerful contraction is produced, not only
in the muscles of the leg, but also in those of the foot. If the
current is passed in the contrary direction, the animal cries
out from pain, the contraction is much less, and only occurs
in that muscle which is traversed by the current.

On repeating these experiments many times and upon dif-
ferent animals, which I have taken care to do, we readily
discriminate the principal results which I have described

M. Matteucci's Researches on Electro-Physiology, 443

from those slight modifications which sometimes occur, espe-
cially at the commencement of the experiment.

These results can only be explained in one way. The very
powerful contraction excited in the muscles of the leg and in
the foot by the passage of the electric current, proves the
existence of a nervous current propagated from the extren)i-
ties towards the centre, and developed under the influence of
an electric current which traverses the muscular mass in the
contrary direction to that of the ramification of the nerve.

As an electric current, when propagated through a muscle,
never leaves the muscular fibre to follow the nervous fila-
ments, we have perfect evidence that the nervous currents of
which we have spoken are due to the influence of the electric
states which are propagated in the muscle.

To demonstrate the entire importance of these conclusions,
we only require to be made acquainted with their connexion
with the law of the electric discharge in fishes ; this connexion
is as intimate as is possible. In fishes, the electric discharge
arises from the production of a nervous current by the stimu-
lation of the nerve which is distributed in the organ. In the
experiments which we have described, a nervous current is
produced by the electric discharge which is passed through
the muscle. When this discharge is passed through the
muscle in such a manner that the positive and negative elec-
tric states are disposed with regard to the nerves in the same
manner as in the discharge of the electric fishes, a nervous
current is produced by the influence of the electric current.
This nervous current has the same direction in both cases;
but in the discharge of the torpedo the electric states are pro-
duced by the animal, 'whilst in the experiment of the muscidar
contractioti the nervous current is produced by the influence of
the electric cur-rent.

When the electric current traverses a muscular mass in a
contrary direction to that of the ramification of the nerve, it fol-
lows, from the facts which we have established, that the electric
current developes a nervous current, the direction of which is
opposed to that which it developes on traversing a muscle in
the opposite direction. This is shown experimentally by the
phaenomena of sensation or of pain which are produced by an
electric current traversing a muscle in the contrary direction
to the ramification of these nerves.

[ 4.44 ]

LXV. An easy Rule for FormuUzing all Epicyclical Curves
with one moving circle by the Binomial Theorem. By S. M.
Drach, Esq., RR.A.S.'^

I REFER to the monography " Trochoidal Curves " in the
Penny Cyclopaedia for the various forms, but which recent
article does not mention the following generalization, extend-
ing the use of Newton's theorem to these curves as well as to
the interpolation series.

Origin is at the deferent's center, x positive towards one
apo-center.

;r=rcos 9=a cos q<p + b cos p<p
y=r sin 9 = a s'm qip -\- b sin p<p
i^szx^+y^—a^-\-b'^ + 2ab cos {pf — q<p:=^)

The resulting function (r% x) shows the general symmetry
as regards y ; when w = -, interchange a and b.

Case 1. a=bi p positive.

,: a;=:2a cos^—-^ <p X cos^--^<p, r=2acos^—-^<p,

fl--^i— 2^, _=2cos(p-g').— — , -=cos(;? + g')

qn=:p

<pn—(p of Pen. Cyc.

2 ^ a '^ ^' P + T '' P + q

Hence 2 cos {p—q)Q developed as

22'A.(cosS = ^y

is to be put equal to 2cos{p + q)d' developed as

2B.(2cos«'=^y

for the general equation of the curve.

^*^- n=L p^q:p + q::^:l0::2'.5,

*i2 f^ ^f& f

2 cos 26 = 4.-^-2=2 cos 56'= —^ g- +5-

?-2 a^ a-* a

is the equation.

Case 2. a=6,;? negative; change sign of 5' in first case,
and

2 cos (;)-|-2')d=22'AYcos d= ^j

* Communicated by the Author.

An easy "Rule for Formulizing all Epicyclical Curves. 44-5
is to be put equal to

2 cos {p - q)d' = SB A cos fl' = ^ ) .

Ex. 7

»«=— 3> p—q'P + gi''^:lO:2:5,

2 cos 5Q = —r ^ + — =2 cos 25' = -5 — 2.

Case 3. a and 5 unequal, p positive. For brevity, put

cos«+sin«. -/— l = (c + s)«=(c + s)*,
.'. rcosfl + rsin^. ^^^=a(c±s)q^ + b{c±s)p^z=a{c±sy^

+ %±5)J=r(c±5/,
.*. 2rP-9cos {p-'q)^ = rP-'t{c + s)^"^ + rP-i{c—sYf''

-{a{c+s)l + b{c + s)l}P-i+ {fl(c-5)'+6(c-s)5}^-»

h'aP~'i
= 2aP-? cos ( j05'(p — q^<^ = y^|/) -+• S r— .

l.2..i V + ((;— 5)?^+'^ = 2cosg\I/ + i\I//

=2^>p-^cos«4.+S2a'^-^-'ig::ig-^i-P::ig~^^--(^"g~^'+^)

1 . 2 .. 2

COS {p—i)'^,
agreeably to the binomial development. Hence

2 cos (j3— g)a=22*A/cos 5= -j
put equal to the sum of

2cosxf/= — -^ =^j

is the equation of the curve, and easily expansible by the for-
mula for expressing the cosine of a multiple angle in powers
of the cosine of the simple angle. Thus, for

p—q=i ; 2^=:2ar cosj94' + 2arcos {p — l)4f

p—q^^; ^x^ — 2r^ = 2b'^r^ cosp^ + ^abr^ cos' [p—l]'^

-f2aVcos(^ — 2)^^

^=5, (7=3; a*J3(4^2_2r2)=Q5-5a2^,2Q3_,_5^4^4Q^2a2Q4

Case 4, a and i still unequal, but p negative ;

446 Mr. S. M. Drach's easy Rule for

'^=p<^ + qfr{c±s)i-a{c±s)\-\-h{c':\isYp
rP+?{(c+s)r' + (c-s)r*} =2rP+*cos (i? + y)^ =2aP+«cos q^^

1 . 2 . . 2

[_ + (c — 5)p^+9^-''' (c + s)!* = 2 cos r^'iZ — ^V J *
As before, de^^elope

2 cos ( ^ + g) 9 = 22' A . (cos 9 = ^ j \

and put it equal to the sum of

2 cos (^-i)rl;' = 2B,(2 cos 4;'= '^'f'^' = ^)\

each having its binomial multiplier for the general equation
of the curve, in both cases.

Ex, w=-3, p + g'=4, q=l', .'. 16^'^-16^V2 + 2^-4=^

+ 8ba^ + 6baQ+ ^ (Q2-2a2Z>2^+ ^ (Q^-Sa^i^Q).

All these examples verify a previous individual and trouble-
some method of eliminating cos <p from the original equa-
tions.

Generally

^=rcos9, r^— «^— 5^=2a5cos\I/;

and for p positive,

rP-^. 2 cos {p-q)Q = {bK° + aK)P-'iy K'=2 cos (p-i)^;
p negative,

rP+1. 2 cos {p + q)Q={aK^ + bK)P+9^ K»=2 cos(5'-2>,

an easily memory-retained formula, to be developed by the
binomial theorem, akin to the finite difference series

i/.= (l+A2/o)^.

The final equation with p negative, p + </ = ^, being

r^.2cos^& = {aK^-i-bKf^

the left-hand member is

F{F--j-l){F-J-2)..{V-2j+l)
- 1,2.3, .J ^ '

Formulizing all Epicyclical Curves. 447

and the right-hand member is

,P„P-.(K..i£rM=2.^,K3.c.).

divided into odd and even angles as cos Pd= funct. (cosfi)
decreases by ex)en exponential differences. Hence the first
{ } is represented by

1 ,, y(!y--i-i)(g-i-2)..(g-2;+i)'

1.2.. 3.. 7
._.^^ P(P-l) ,, (g-2)(g-i+i)(y-jf)..(g-2;-+3)

o^* 2 ^ 1.2..(j-l)

&VP(P-l).(P-3) (g-4)(<y-i+3) . (g-2; + 5)
a* 1.2.3.4 1.2.. 0-2)

_^ P(P-l)..(P-5) (g-6)(g-y+5) . . (g-2;-+7)
a^' 1.2. .6 1.2.(j-3)

&c. &c.

and the second { } is replaceable by

(y-l)(!7-i)(2'-i-l) . . (?-2/)

1 X

1.2.3..^
- ^ (P-^) y (g-3)(9-i+2)(g-i+l)..(g-2;+2)

fl^* 2 1.2. .(7-1)

^ (P-l).(P-3) (g-5)(y-i+4) . . (g-2;-+4) l>
"^ a^- 1.2.3.4 1.2. .(7-3)

__66 (P-l)..(P-5) (g-7)(g-i+6) . . (g-2i+6)
a^* 1.2. .6 1.2. .(7-5)

&c. &c.

the sum of these two 2 is the right-hand member developed
as SQ^.

When a=bf w=— 3, there results a three-looped curve,

t^ = 4a V — 1 Gr^a^w^ + 1 Qx'^a^ ;
hence

^=4a2(/r2-j/2)2, r4=4aV^-j/^), r2=4flV-2/^)^>
rO= 4a2(a?2 -/)- \ r'^ = 4^2(^2 _y )-2,
are this three-looped curve, two-looped lemniscate, one-looped
circle, equilateral hyperbola, quadrilateral equal hyperbola,

448 An easy Rule for Formalizing all Epicyclical Curves,

&c. The f oral appearance of many of these curves induces
me to suggest the name of peialoids. They may possibly one
day lead to geometrical disclosures on the structure oi'Jlo'wers,
as Naumann and Moseley (Phil. Trans. 1838) have success-
fully shown in shells; each individual shell having its own
numerical parameter, which a verbal nomenclature would
vainly follow, as every additional digit increases the number
of varieties tenfold, three already denoting 999 varieties.
Mr. Perigal's finite spiroeids are very curious, especially the

cos
retrogressive syphonoids(a?=acosg'<p, ?/ = 6 ^^^ p<p), the two-
branched syphonoid being the common conical parabola. {Vide
Mr. Sang's paper On the Vibration of Wires, Edin. Phil.
Journ. 1832, p. 317.) All these which terminate in points
Sive finite portions of generally infinite curves ; for in the latter
y'^ = ax, \f X be proportional to the periodic quantity cos^ A,
and ?/ proportional to cos X, 3/^ -r a; is still constant: the pa-
rabola is described, but only between the limits of + 1 = cos A,
towards which points the motion gradually slackens to zero,
as publicly shown by Mr. Perigal at Lord Northampton's
scientific smVe'^ in March 1846'. The Royal Society, Astro-
nomical Society and Royal Institution, possess three volumes
of various singular epicyclical curves executed by Mr. Perigal's
machinery, some of which are highly ornamental, and I think
might be useful for the arts, e. g. the drawing of volutes to
Ionic columns, &c.

Kinematic Parabola. A retrogressive Syphonoid produced by compound
circular motion.

7 Bedford Place, Hampstead Road,
September 1848.

[ 449 ]

LXVI. On Spherical Waves in an Elastic Fluids in reply to
Mr. Stokes. By the Rev. J. Challis, M.A.^ F.R.S.,
F.R. A.S., Plumian Professor of Astronomy and Experimental
Philosophy in the University of Cambridge* .

THE question which has been discussed by Mr. Stokes and
myself in several recent Numbers of this Magazine, is of
the following nature. On supposing the waves in a compress-
ible fluid to be spherical, I deduced from that supposition,
by reasoning given at length in the Philosophical Magazine
for last February, a result inconsistent with one of the funda-
mental principles of hydrodynamics, viz. that of constancy of
mass. Hence I concluded that the supposition of spherical
waves is inadmissible. Mr. Stokes undertook to dispute this
inference. Now although by an acknowledged rule of logic,
the inference could not be set aside except by showing some
fallacy in the reasoning which conducted to the absurdity,
Mr. Stokes, in three attempts to set it aside, has not once
alluded to any step in the reasoning. In the first attempt he
produced an argument which took for granted the very point
in dispute; in the next he denied, without giving any reason,
what was altogether undeniable; in the third attempt (Phil.
Mag. for May) he admits what he before denied, and denies,
again without assigning a reason, what in the second attempt
he admitted. The denial in this instance refers to the possi-
bility of the propagation of a solitary wave of arbitrary' con-
densation and constant type. I infer the possibility from the
principle of discontinuity. Mr. Stokes calls this inference a
gratuitous assumption, without making the slightest allusion
to the principle on which it rests ; and yet he has drawn a like
inference from the same principle in the same way. ( Phil.
Mag., vol. xxxiv. p. 54. 1.27-31.)

With respect to my having deduced only a part of the re-
sults to which the supposition of spherical waves leads, I am
able to give a very good reason for not proceeding further.
I obtained, as already stated, from that supposition, by rea-
soning of which Mr. Stokes has not shown, and I am unable
to perceive, the fallacy, a result inconsistent with one of the
fundamental principles of hydrodynamics. The supposition
I consider to be thereby condemned. If I allowed myself to
qualify this course of reasoning by another from the same
supposition, I should proceed in direct opposition to an in-
controvertible rule of logic, and neutralize a highly important
and significant result.

Mr. Stokes appeals with great confidence to the results he
* Communicated by the Author.

Phil, Mag. S. 3. Vol. 34. No. 231. June 1849. 2 G

450 Prof. Pliicker on the Magnetic Relations of the

has obtained by two courses of reasoning in the Phil. Mag.,
vol. xxxiv. pp. 54-57 and 57-59. I cannot concede that these
results have any weight against my position, because the rea-
soning from which they are derived takes for granted the
question in dispute; but I may adduce them in favour of it so
far as they exhibit inconsistencies. The first argument, which
professes to be a general consideration of a solitary wave of
arbitrary condensation, conducts to the result that the sum of
the condensations is exactly equal to the sum of the rarefac-
tions. Now if the reasoning be restricted to the case in which
the sum of the condensations is equal to the sum of the rare-
factions by the original disturbance, it ceases to be general,
and the result is a mere truism without meaning ; and if it be
not so restricted, it is impossible that the result can be true.
Mr. Stokes's argument cannot escape from this dilemma. The
second argument, which applies to a case in which condensa-
tion prevails over rarefaction, is included in the general argu-
ment above mentioned, if that be of any value, and its leading
by a different process to a different result is only another phase
of contradiction.

As I consider that this hydrodynamical question has now
been so fully discussed that it is not likely to receive any ad-
ditional elucidation, as far as I am concerned the discussion is
closed.

Cambridge Observatory,
May 23, 1849.

LXVII. On the Magnetic Relations of the Positive and Nega-
tive Optic Axes of Crystals. By Professor Plucker of
Bo7i?if in a letter to, and communicated by, Dr. Faraday.

ALLOW me, Sir, to communicate to you several new facts
which, I hope, will spread some light over the action of
the magnet upon the optic and magnecrystallic axes.

L The first and general law I deduced from my last expe-
riments is the following one : — " There will be either repulsion
or attraction of the optic axes by the poles of a magnet, ac-
cording to the crystalline structure of the crystal. If the
crystal is a negative one, there will be repulsion ; if it is a po-
sitive one, there will be attraction."

The crystals most fitted to give the evidence of this law are
diopside (a positive crystal), cyanite, topaz (both negative),
and other ones, crystallizing in a similar way. In these cry-
stals the line (A) bisecting the acute angles made by the two
optic axes, is neither perpendicular nor parallel to the axis (B)
of the prism. Such a crystal, suspended horizontally like a

Positive and Negative Optic Axes of Crystals. 451

prism of tourmaline, staurolite, or " red ferridcyanide of po-
tassium," in my former experiments, will point neither axially
nor equatorially, but will take always a fixed intermediate
direction. This direction will continually change if the prism
be turned round its own axis B. It may be proved by a
simple geometrical construction, which shows that during one
revolution of the prism round its axis (B), this axis, without
passing out of two fixed limits C and D, will go through
all intermediate positions. The directions C and D, where
the crystal returns, make, either with the line joining the two
poles, or with the line perpendicular to it, on both sides of
these lines, angles equal to the angle included by A and B; the
first being the case if the crystal is a positive one, the last if a
negative one. Thence it follows, that if the crystal by any kind
of horizontal suspension should point to the poles of a magnet,
it is a positive one ; if it should point equatorially, it is a negative
one. This last reasoning conducted me at first to the law
mentioned above.

The magnecrystallic axis, I think, is, optically speaking,
the line bisecting the (acute) angles made by the two optic
axes ; or in the case of one single axis, this axis itself. The
crystals of bismuth and arsenic are positive crystals ; antimony,
according to my experiments, is a negative one : all are uni-
axal.

II. The cyanite is by far the most interesting crystal I
have examined. If suspended horizontally, it points very
well to the north, by the magnetic power of the earth only.
It is a true compass-needle, and more than that, you may
obtain its declination. If, for instance, you suspend it in
such a way that the line A bisecting the two optic axes of
the crystal be in the vertical plane passing through the axis
B of the prism, the crystal will point exactly as a compass-
needle does. By turning the crystal round the line B you
may make it point exactly to the north of the earth, &c. The
crystal does not point according to the magnetism of its sub-
stance, but only in obedience to the magnetic action upon its
optical axes. This is in full accordance with the different law of
diminution by distance of the pure magnetic and the opto-mag-
netic action. If you approach to the north end of the suspended
crystal the south pole of a permanent magnetic bar, strong
enough to overpower the magnetism of the earth, the axis
B of the prism will make with the axis of the bar (this bar
having any direction whatever in the horizontal plane) an angle
exactly the same it made before with the meridian plane, the
crystal being directed either more towards the east or more
towards the west,

2G2

452 Notices respecting New Books.

The crystal showed, resembling in that also a magnetic
needle, strong polarity; the same end being always directed
to the north. I think this may be b. polarity of the opto-mag-
netic 2)ower. Two questions too may easily be answered : —
1st. Is the north pole indicated by the forms of crystalliza-
tion ? 2nd. Did the crystal obtain, when formed, its polarity
by the magnetism of the earth ? Between the poles of the
strong electro-magnet the permanent polarity disappeared as
long as the magnetism was excited.

I am obliged, by the new facts mentioned above, to take up
my former memoir ; I must reproduce it under a quite new
shape. I will examine again the rock-crystal, which, being
acted upon weakly by a magnet, induced me to deny in that
memoir, what I ascertain now and what I thought most pro-
bable, as soon as I received the first notice of your recent re-
Searches. [That you will find in the memoir given to M. Pog-
gendorff two or three months ago.] Perhaps the exceptional
molecular condition of rock-crystal, as indicated by the passage
of light through it, will produce a particular magnetic action.

I should be very much obliged to you, if you would give notice
of the contents of my present letter to M. De la Rive, when he
calls on you, as he intended to do. I showed him several of
my experiments when he passed through Bonn the 12th of
May. The following day I obtained the different results men-
tioned above.

My best wishes for your health.

Very truly yours,

Plucker.
Bonn, the 20th of May 1849.

LXVIII. Notices respecting New Books.

Outlines of Astronomy. By Sir John F. W. Herschel, Bart., K.H.,
8(C. 8vo. London : Longman and Co., 1849.

THE treatise on Astronomy by Sir John Herschel, published in
1833 as a Part of the Cabinet Cyclopaedia, is familiar to every
student of that science, and justly prized as containing one of the
most lucid and eloquent expositions of its facts and principles ever
given to the world. A new edition of this popular and standard
"work, brought down to the present day, had become necessary by
the progress of discovery. In the course of the sixteen years which
have elapsed since the original publication, astronomy in all its
branches has been assiduously and successfully cultivated. Six new
planets have been added to the solar system (and w^hile we write
a seventh is announced) ; a satellite to Saturn, and one or two to
the recently discovered large planet ; a multitude of comets have

Notices respecting New Books. 4>63

been observed and their orbits computed ; and our knowledge of the
sidereal heavens has been greatly extended, not merely in conse-
quence of continued observation, but by the more general use of large
instruments. Hence the work of 1833, however complete at that
time, necessarily leaves the student much behind the point to which
this branch of knowledge has actually attained. To supply the de-
fects, record the recent discoveries, state the modifications of former
views to which they lead, and represent the science as it exists at
the present moment, are the objects of the work now before us. It
appears under an enlarged form, and with a different title; and though
the substance and arrangement of the original treatise have been
preserved, the additions and alterations are so extensive and import-
ant as to render it rather a new book than a new edition of the
former one.

The present work is divided into four parts. The first embraces
what is sometimes called Descriptive Astronomy, beginning with the
general notions of the science, and including all the topics usually
treated of under this branch of the subject ; — the .shape and size of
the earth, the atmosphere, refraction, the theory and use of instru-
ments, the apparent and true motions, as well as the appearances
and physical nature of the bodies of the solar system. On many
of these heads there was little to alter in the original work ; much
new matter has however been introduced, and we may instance in
particular the chapter on comets, which will be found to be replete
with interest. In this we have a condensed account of the remarkable
phsenomena exhibited by some comets which have recently appeared ;
by Halley's at its return in 1835, when it was observed under very
favourable circumstances by the author himself at the Cape ; by the
comet of 1843, which approached so near to the surface of the sun,
that the intensity of light and radiant heat must have been 47,000
times greater than at the surface of the earth, and whose tail at the
perihelion passage was whirled round, unbroken, through an angle
of 180° in little more than two hours; by Biela's comet, which at
its last apparition in 1846 was seen to separate itself into two di-
stinct bodies, which, " after thus parting company, continued to
journey along, amicably, through an arc of upwards of 70° of their
apparent orbit, keeping all the while within the same field of view of
the telescope pointed towards them." Some other comets recently
observed, which seem to describe elliptic orbits in short periods,
are also taken notice of ; and the description of the phaenomena is
followed by some most ingenious and highly interesting speculations
on the physical nature of those enigmatical bodies.

The second Part treats of the planetary perturbations ; a subject
which from its nature can never be rendered very pojmlar, but which,
nevertheless, as is proved in the present instance, may be explained
in such a manner that the principal effects on the motion of a system
of bodies produced by their reciprocal attraction may be clearly and
readily apprehended by a reader having no more than an elementary
knowledge of geometry and mechanics. The accurate and minute
computation of these effects is quite another thing, and must be left

454 Notices respecting New Books.

to the few who possess the requisite technical knowledge. Many
attempts have been made to give elementary explanations of the
inequalities of the celestial motions, but seldom with much success :
indeed, if we except Maclaurin's Account of Newton's Discoveries,
Mr. Airy's Treatise on Gravitation, and the work now under
review, it would be difficult to point out one from which a student
about to enter on the works of Newton and Laplace would derive
any considerable aid ; and even in respect of these master-pieces it
may perhaps be said, that their merit will hardly be appreciated ex-
cepting by those who have proceeded to some extent in the technical
examination. But Sir John Herschel has not confined himself merely
to the illustration of methods already known. In discussing certain
effects of perturbation he has struck out an entirely new path, and
presented the subject in a light which certainly renders it much
easier of comprehension ; and on this account the work must be re-
garded as an important contribution to physical astronomy. To these
new views he thus alludes in his preface : —

" In delivering a rational as distinguished from a technical expo-
sition of this subject, the course pursued by Newton in the section
of the Principia alluded to has by no means been servilely followed.
As regards the perturbations of the nodes and inclinations, indeed,
nothing equally luminous can be substituted for his explanation ;
but as respects the other disturbances, the point of view chosen by
Newton has been abandoned for another which it is somewhat diffi-
cult to perceive why he did not, himself, select. By a diflferent re-
solution of the disturbing forces from that adopted by him, and by
the aid of a few obvious conclusions from the laws of elliptic motion,
which would have found their place, naturally and consecutively, as
corollaries of the seventeenth proposition of his first book (a propo-
sition which seems almost to have been prepared with a special view
to this application), the momentary change of place of the upper
focus of the disturbed ellipse is brought distinctly under inspection ;
and a clearness of conception introduced into the perturbations of
the eccentricities, perihelia and epochs which the author does not
think it presumptuous to believe can be obtained by no other method,
and which certainly is not obtained by that from which it is a de-
parture The reader will find one class of the lunar and

planetary perturbations handled in a very difi^erent manner from that
in which their explanation is usually presented. It comprehends
those which are characterized as incident on the epoch, the principal
among them being the annual and secular equations of the moon,
and that very delicate and obscure part of the perturbational theory
(so little satisfactory in the manner in which it emerges from the
analytical treatment of the subject), the constant or permanent efl^ect
of the disturbing force in altering the disturbed orbit. I will venture
to hope that what is here stated will tend to remove some rather
generally diflfused misapprehensions as to the true bearings of New-
ton's explanation of the annual equation."

The third Part relates to Sidereal Astronomy. If the former is
that which is likely to have the fewest readers, this undoubtedly is

Cambridge Philosophical Society. 455

the one whicli will have the greatest number. The extreme interest
inherent in the subject would naturally secure this result ; but in
the present case the interest is greatly heightened by the circum-
stance, that the author himself stands in the foremost rank among
those to whose labours we are indebted for the progress recently
made in this department of astronomy. The subject, indeed, may
be said to belong to him of hereditary right. He has devoted to
it a large portion of the labours of his liffi ; and after scrutinising
the heavens from pole to pole, he has here become the expositor of
the great discoveries in which he has taken so large a part, and in
language alike remarkable for its eloquence and perspicuity, has pre-
sented us with an epitome of all that is known about the fixed stars,
— their parallaxes, distances and distribution — their motions, relative
and systematic — about variable and periodic stars — double stars and
binary systems — clusters and groups of stars — the classification, dis-
tribution and resolvability of nebulae, the zodiacal light, &c. It may
be said without any exaggeration, that in the whole range of natural
philosophy there is nothing more interesting in respect of subject-
matter, or more admirable as regards the mode of treatment, than
the three chapters forming this part of the work.

Part IV. contains a single chapter which treats of the account of
time, or the calendar.

We had marked some passages for the purpose of extracting them ;
but on consideration this appeared to be unnecessary, as the book
itself will be in the hands of every one who takes an interest in the
science of astronomy.

IjXIX. Proceedings of Learned Societies.

CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Continued from p. 227.]

Nov. 13, I^N the Elements of Plane Geometrical Trigonometry,
1848. v^ applicable to Trigonometrical Formulae. By the Rev.
F. Calvert.

The object of this paper is to define as distinctly as possible the
elementary terms of trigonometry, and to explain the conventional
use of the negative sign in expressing such simple functions of
angles as the sine, cosine, tangent, &c.

Nov. 27.— On Clock Escapements. By E. B. Denison, Esq., of
Trinity College.

The object of this paper is, first to point out the real cause of the
general excellence of tiie dead beat escapement ; and secondly, to
show that in a gravity, or remontoir escapement, in which the pen-
dulum raises an arm carrying a small weight, from an angle y up to
its extreme semiarc a, which follows the pendulum down again to an
angle /3 (either + and less than y, or = —y), there is a certain pro-
portion between a, /3, and y, which will cause the errors of the clock

456 Cambridge Philosophical Society,

for small variations of a to be much smaller than in the dead escape-
ment, and in fact inappreciable.

The author adopts the equations obtained by Mr. Airy in his paper
on this subject in vol. iii. of the Transactions of the Society, and
shows that the increase of the time of an oscillation

\9 cc J
where A is the diiFerence between the time of oscillation of a free
pendulum and one affected by this escapement (which in clocks of
the best construction he shows will amount to about 1 second a day) ;
(p is the angular accelerating force of the escapement on the pen-
dulum ; d<p the variation in this force due to the variation of the
friction of the train and of the state of the oil on the acting part of
the pallets ; da the variation of the arc from the same causes, and
also from the state of the oil on the dead or circular part of the
pallets. It appears therefore that the two causes of error have a
tendency to correct each other ; and in practice it is found that

is generally not far short of — , which is the reason of these

clocks going so well.

In a gravity escapement there is no variation of the force ; and

rfA
the author shows from Mr. Airy's equations that -^sOif a=:y ^2

in that escapement where the remontoir weight is taken up at y and
follows the pendulum again to — y ; and in the other kind of gravity

</A «?A _ „ ,
escapement -rr -3 " wnen

a^ = 2 Va'^ -y* Va?' — /3«.

This last construction however is barely practicable, if this con-
dition is to be satisfied, on account of the small difference between
/3 and y which is allowed by the deduction of the value necessary
for a — y, the angle in which the unlocking of the escapement is
eflFected ; although this is the construction which has been used in
nearly all the gravity escapements that have been tried ; and of course
the proper condition has been very far from satisfied, and the clocks
nave failed.

In a supplement to this paper the author proposes, chiefly for
turret clocks, a new construction of a spring remontoir on the axis
of the escape-wheel. The object of such remontoirs is to remove
from the escapement (of any ordinary kind) the great inequalities of
force caused by the varying friction of the heavy train and dial- work,
and by the action of the wind on the hands ; and also to cause the
minute-hand to move only at visible intervals, such as i a minute, and
the striking to take place exactly at the right second. The Royal Ex-
change clock, made under the superintendence of the Astronomer
Royal, has a gravity remontoir in the train introduced for these
purposes ; but it is too complicated and expensive for ordinary use.

Cambridge Philosophical Society. 4*57

and has a good deal of friction, from which the proposed remontoir
is free. Spring remontoirs winding up at similar intervals have been
tried in France, but without success, from defects in their con-
struction.

March 12, 1849. — On the Intrinsic Equation to a Curve, and its
application. By the Master of Trinity.

The author remarked that the expressions for the lengths of
curves, their involutes and evolutes, in the ordinary methods, are
complex and untractable, which arises in a great measure from the
properties of extrinsic lines being introduced, namely, coordinates.
But a curve may be represented without any such additions, by an
equation between the length and the angle of flexure, which is
therefore called the intrinsic equation. This equation gives, with
remarkable facility, the radii of curvature ; involutes and evolutes
of most curves. It also expresses very simply what may be called
running curves ; namely, curves which run like a pattern along a
strip of ornamented work. A very simple equation expresses, for
instance, the inclined scroll pattern so common in the antique, and
by altering the constants, gives to this pattern an endless variety of
forms. If s be the length of the curve and f the angle, the intrinsic
equation to the circle is s^=a(p ; to the cycloid s = a sin (p. The
equation to an epicycloid or hypocycloid is s = « sin m<p, according
as m is less or greater than unity. The equation to an undulating
pattern is ^ = m sin s, which assumes very various shapes by varying
m. The method was also used in proving that if we take the suc-
cessive involutes of a curve an indefinite number of times, the re-
sulting curve (with certain limitations) bends to become the equi-
angular spiral if the unwrapping be always in the same direction,
and tends to become the cycloid if the unwrapping be alternately in
opposite directions. The latter proposition had already been dis-
covered by Bernouilli and proved by Euler.

Feb. 26. — On a New Method of finding the Rational Roots of Nu-
merical Equations. By Robert Moon, Esq.

The author proposes to found a new experimental method of find-
ing the integral roots of numerical equations upon the following
theorem.

If the equation

x"+piX''-'^+p<^"-^+ &C +PiX''-'+ &C.+J9„=0
has a positive and integral root m, we shall have —p^ equal to m
terms of the following series : —

A„_i + 1. A„_2-h 1.2. A„_3 -1-1.2.3 A„.„+ &c. -|- 1.2 ... (n-i+ 1) A^

-h&c. +1.2...(w— l)Ao
-|-A,.,-f2.A„.2-f-2.3A„_3-f2.3.4A„_„+ &c. +2.3 ...(n-i+2) A^

+ &c. -|-2.3...nAo
-t-A»_i + 3.A,.2-f-3.4 A,_3+3.4.5 A„_„+ &c. +3A...{n-i+3) A^

-j- &c. +3.4...(«+l)Ao
-hA„.,-H4.A„_2+4.5A„_3+4.5.6 A^_.+ &c. +4.5...(«-i-j-4) A^

+ &c. -H4.5...(« + 2)Ao

&c. &c.

45ft Cambridge Philosophical Society.

where

and

A,=the sum of the natural numbers 1, 2, 3 {n—i).

^=the sum of the homogeneous products of the same quantities of

two dimensions.
Agssthe sum of the homogeneous products of the same quantities of

three dimensions, and so forth.
From the above formula for A; may be determined all the coeffi-
cients A except the first, which is determined from the equation

K-\=Pn-l—Pn-2+Pn-i — ^^' ±Pv^^-

Having determined the quantities A in any particular case, let them
be substituted in the first line of the series. If the sum of that line
be equal to — /)„ unity is a root of the equation. Let the second line
be then written down and added to the first. If the sum of the two
equals— j9„, 2 is a root of the equation, and so by adding successive
lines we shall ascertain whether the successive integers 3.4., &c. are
or are not roots of the equation.

The quantities h in the expression for Aj depend upon the number
of the coefficient and the number of the dimensions of the equation.
The author proposes that these should be calculated and tabulated
for equations of all dimensions up to a certain limit, by which means
we should be in possession of so many skeletons of equations, ready
for ap])lication in any particular case, and the calculation in particular
instances would be thus greatly facilitated.

It will be observed that each successive line is derived from that
preceding by a simple division and multiplication of the separate
terms of the latter, and thus each succeeding trial facilitates those
which follow ; contrary to what obtains in the ordinary method by
successive substitutions, in which each attempt proceeds de novo.

If the addition of a term makes the series from being greater than
/»„ less than it, or vice versa, a fractional or surd root will lie between
the number expressing the number of the term so added and the
number next below it.

If all the roots are impossible, the series will be either always
greater or always less than p„, whatever be the number of terms
taken.

For an example take the cubic

Here — r=3 x 1.2 + 2(;j— 3)1+2-— ;?+l

+ 3x2.3 + 2(;?-3)2 + 5'-;)+l

+ 3x3.4 + 2(;) — 3)3 + y— ;>+l

+ &c. to X terms,
if a; is a positive integer.

The method in common with other experimental methods applies
to the discovery of all roots, possible or impossible, which do not
involve surds.

Royal Astronomical Society. 459

ROYAL ASTRONOMICAL SOCIETY.
[Continued from p. 225.]

March 9, 1849.— On Irradiation. By Professor Powell.

After adverting to the history of researches on this subject, the
author dwells particularly on the method of exhibiting the phseno-
menon adopted by M. Plateau, which forms the basis of all his own
experiments, and which consists of a card or lamina, cut so that one
half of a long parallelogram is cut out whilst the other remains,
having the portions at the sides cut away. Viewed against the
light, the enlargement of the bright half, in breadth, is seen con-
trasted with the opake, and might be subjected to measurement.

The first question on the subject refers to the supposition of a
ij}QC\x\\BX physiological cause affecting the eye to produce the apparent
enlargement of the bright image. After fully allowing for some
portion of such phaenomena being fairly attributable to ocular causes,
such as dazzling, contrast, &c., experiments are adduced to show
that precisely similar phcenomena are produced in an artificial eye, or
camera obscura ; whence the hypothesis of any peculiar affection of
the retina is rendered unnecessary. The same conclusion is further
confirmed by photographic impressions of the image of the card cut
as before, which exhibit the same enlargement. Specimens of these
impressions, taken by Mr. N. S. Maskelyne, were exhibited.

These results, clearly pointing to an optical cause, agree with the
conclusions of the undulatory theory, relative to the " diffraction of
a lens," as investigated by Mr. Airy, which apply to the eye con-
sidered as an optical instrument, as well as to the object-glasses of
telescopes ; in either case the image of a point being an extended disc,
which, if the light be bright enough, will be surrounded by rings. A
luminous surface will exhibit a like enlargement.

Without reference to any theory, it is an ascertained law that the
enlargement increases with the intensity of the light. The enlargement
also is formed with a rapid decrease in brightness towards the edge.
On these grounds it is easy to explain the fact of the great diminu-
tion or total destruction of irradiation by the interposition of lenses,
which would follow immediately from the weakening of the intensity
in proportion to the square of the linear magnification. The author
has examined particularly into the extent to which this effect takes
place, and announces that low powers (from 5 to 20) are sufficient to
obliterate all irradiation even in the most intense light which the eye
can bear.

Various results of M. Plateau and others as to the effects of con-
trast in making a narrow bar or wire continue visible, though the
irradiations ought to overlap, have been examined, and found only
to hold good with low intensities.

The author next considers the effect in telescopes. Here that
portion of the effect which regards the ocular image being placed out
of consideration from the influence of the magnifying power (already
referred to), we have only to consider that part which affects the
focal image of the object-glass. The diminution of the aperture in-

460 Royal Astronomical Society.

creases the irradiation, but at the same time it diminishes the light.
At a certain point, then, these two causes counterbalance each other,
and no further enlargement takes place. This limit will vary with
each instrument, and we have no certain grounds on which to de-
termine it. Various observations are referred to in which its influ-
ence is evinced.

The astronomical facts connected with these causes are then ex-
amined from the testimony of various observers. In particular the
application of these principles to some of those singular phaenomena
occasionally noticed in eclipses, transits, occultations, &c. seems easy
in theory abstractedly considered. The difficulty lies in explaining
why they are observed only in some cases and not in others. The
author dwells particularly on the desirableness of a closer attention
to stating all the conditions of the telescopes employed, especially the
apertures.

In particular the phsenomenon " the necic," in the transits of Mer-
cury and Venus, would be an obvious consequence of irradiation,
which would diminish the planet's disc and enlarge that of the sun
except at the small portion of the circumferences in contact, when
the absence of both irradiations would produce a " neck."

Both theory and experiment show that a small dark disc would
have for its image a diminished disc with a bright internal concentric
ring, which, if the disc be very small, will be contracted to a central
bright point. This seems to agree with the appearance noticed by
several observers in the transit of a white spot on the centre of the
planet. On a former occasion, however, Professor Moll and others
saw such a spot excentrical. The projection of a star on the bright
limb of the moon would also be an effect of irradiation, which would
cause the disc of the moon simply to overlap the star.

Lastly, the author suggests a method for obtaining measures of the
amount of irradiation under any given light, by placing a card, cut as
before, at the focus of a lens, opposite to the object-glass of a tele-
scope, and attached to it by a short tube ; when the enlargement of
the image of the card, illumined by the light from any source, can
be subjected to the exact measurement of the micrometer of the
telescope.

On a New Method of Observing Transits. By A. D. Bache, Esq.,
Director of the American Coast Survey.

" Permit me to invite your attention, and that of the members of
the Royal Astronomical Society, to a brief abstract of an official
report made to me on the 15th inst. by Mr. Sears C. Walker, one
of the assistants of the United States Coast Survey, It relates to
the printing, by the use of an electro-magnetic clock, in connexion
with Morse's telegraph register, of the actual dates of any celestial
phsenomena, which are ordinarily made the subject of observation
by astronomers.

"The electro-magnetic clock of Mr. Wheatstone is described in the
Proceedings of the Royal Astronomical Society for Nov. 19, 1841.
Mr. Steinheil has described his in Schumacher's Astronomical Jahr-
buch for 1844.

Royal Astrofiomical Society. 461

" Recently Prof. Bond and Dr. Locke have invented different pro-
cesses, which are described in Mr. Walker's Report.

" Prof. Bond proposes to make circuit by the metallic contact of
insulated portions of the pallet and escapement-wheel. Dr. Locke,
like Mr. Wheatstone, uses a metallic wheel on the arbour of the
seconds' hand. This wheel has sixty teeth, each of which when
horizontal strikes against a platinum lever or tilt-hammer, weighing
two grains. The rising and falling of the hammer from a bed of pla-
tinum breaks and makes the galvanic circuit. The fulcrum of the tilt-
hammer and the platinum bed rest severally on a small block of wood.

" The object of all these methods is to cause a delicate astrono-
mical clock to make and break the galvanic circuit every second,
without injury to the machinery or rate of the clock. The mode of
action of such alternations on Morse's electro-magnetic telegraph
register, as now in daily use in the United States, is the same for
each of these methods.

" llie automatic clock register thus formed consists of a graduated
fillet of paper delivered pretty uniformly at the rate of an inch per
second. The beginnings of minutes, and fives and tens of minutes,
and of seconds, and fives and tens of seconds, are distinguished from
each other by the lengths of the corresponding imprinted blank spaces.
The printed second consists of an indented line of about nine-tenths
of a second or less, and of a blank space for the remainder. The rate
of the delivery of the paper is regulated by a centrifugal clock like
those of the Munich equatoreals. An error of two seconds per
minute in the rate of delivery causes only an average error of one-
hundredth of a second in the register of a date.

" The printing of the date of any event not susceptible of auto-
matic register, but dependent for our knowledge of its occurrence
upon human sensations, is effected by tapping gently at this date on
a break circuit telegraph key, so as to insert in the line of the auto-
matic clock register a short blank space, whose beginning marks the
instant of the tap. Should this blank space occur near that of the
automatic clock register, the fact would identify its date. For iso-
lated events the finger dwells long enough on the key to be sure of
cutting off some portion of one of the indented lines. The dates
susceptible of impression with advantage on the automatic clock re-
gister are such as the phases of an eclipse or occultation, or the
bisections of a star or comet, or of a planet's centre or limb, by the
wires of a transit instrument. The association of the nerves and
sensations of sight and touch is known to be far more intimate than
that of those of the eye and ear. The art of tapping at the proper
dates requires far less practice and experience than that of counting
beats and estimating fractions of a second. The labour of counting
beats and of writing down the dates being here dispensed with, the
equatoreal intervals of the transit wires may be reduced to two
seconds of time, or even to less, and fifty bisections may now be
registered in the same time as seven are in the ordinary way. The
three advantages of Mr. Walker's method are respectively, —
- " 1st. The facility of acquirement of the practical skill for observing.
5 " 2nd, The twofold precision nearly of a single observation.

482 ' Royal Astronomical Society.

" 3rd. The sevenfold multiplication of observations in the same
interval of time, or in the single transit of one, or the relative trans-
its of two or more heavenly bodies.

" From all these sources it will be apparent that Mr. Walker's
method of printing dates has nearly a tenfold advantage over the
ordinary mode of using the transit instrument.

" A single transit of a star, or a night's or even a year's work by
this method of printing, may take the place of some ten times those
quantities by the method now in use.

" The experiment of printing the dates of bisections of transit
wires by a star, on the ordinary registering fillet of Morse's telegraph,
was made by Mr. Walker in 1846. It was repeated this last summer
for some twenty or more stars, in connexion with Prof. Bond and
Prof. Loomis, for a distance of some three hundred miles from Cam-
bridge to New York. In October last it was repeated for a like
number of stars between Philadelphia and Cincinnati, in connexion
with Prof. Kendall and Prof. Mitchell, through a distance of seven
hundred and fifty miles. The taps made on the telegraph key at
the time of bisection at each place were registered at both. In these
operations, however, the year was used to estimate fractions of a
second by the audible beats of the telegraph and observing clocks,
and no use was made of the visible register.

" Dr. Locke's electro-magnetic clock of his own invention and
construction (Wheatstone's method not being known to him at the
time) was used for some two hours or more, on the 17th of November
last, to make the automatic clock register such as is described above.
The distance tried was about four hundred miles from Cincinnati to
Pittsburg.

" The experiment was completely successful. The interruption
of the line from Pittsburg to Philadelphia that night prevented the
actual continuation of the two operations on the same fillet of paper,
namely, the graduation of the paper by the automatic clock, and the
reciprocal imprinting of the dates of transits of stars at the two ob-
servatories. Each process, however, has been tried by itself to a
sufficient extent by Mr. Walker and his associates, to warrant his
conclusions with respect to their combination, for a more full trial of
which he now waits for the construction of the most approved ap-
paratus.

" In order to make the precision of the other ajjpendages of a
transit instrument commensurate with the tenfold increase of that of
the art of imprinting the dates of bisections for a single culmination,
Mr. Walker recommends the use of a cast-iron box for the frame.

" Each side should carry three or more levels.

" The number read on each occasion should depend upon the
degree of precision aimed at. The instrument should admit of rapid
reversal, even on equatoreal stars. For use at the station of the
Coast Survey, Mr. Walker prefers to retain the micrometer adjust-
ment of the azimuth, like that of the new Simms's transit instru-
ments recently made for the survey.

" In the telegraph operations for longitude, two such transit in-
struments of moderate size are to be mounted, at any two stations.

Royal Society, 463

distant one or more thousand miles. All the levels are to be read
with the instrument pointing to the zenith, then twenty bisections
of a circum-zenith star are to be imprinted on the automatic clock
register previous to reversal. The like number for the same star on
the same wires are to be imprinjted after reversal, and the levels are
again to be read.

" A similar operation is performed for the transit of the same star
at the western station.

"The primitive astronomical clock may be located and rated at
the central station of the coast survey. The automatic clock register
may be made and kept there, even if the distance be a thousand
miles from either station.

" Clock registers in any number may be made at the separate
stations. The transits of two fundamental stars at remote dates, at
either of the three stations, may give the rate of the primitive or
central clock.

" One such transit of the same star over each station with twenty
printed registers of normal bisections, and six normal levelings, with
independent levels, at or near the position of actual observation,
with the increased precision of the instrumental adjustments, will
give in the form of a permanent printed record (with multiplied
copies) the relative longitude of the two stations.

" The uncertainty of such a result need be only a few hundredths
of a second, and may be such only as attends our present knowledge
of the relative longitudes of Greenwich and Paris, the two oldest
observatories extant.

"I avail myself of the occasion to remark, that the Coast Survey-
operations were completely successful this autumn between Phila-
delphia and Cincinnati, while actually working on the line from
Philadelphia to Louisville. The distance of the line in the air is
nine hundred miles, that of the circuit is eighteen hundred. 1 learn
from an authentic source that the same success attends the use of
the line from Philadelphia to the Mississippi river opposite St. Louis.
The length of this circuit is one-tenth of the circumference of the
earth. The inference from this trial is clear, that a line round the
earth, if such could be constructed, might be worked with facility at
one stroke. The expense of acids to supply the thousand Grove's
pint cups, required for the motive power, would be about one pound
sterling (five dollars) per day."

ROYAL SOCIETY.

[Continued from p. 314.]

March 29, 1849. — " Examination of the Proximate principles of
some of the Lichens." — Part II *. By John Stenhouse, Esq., F.R.S.

Gyrophora pustulata.

The author states that this lichen, which is the " Tripe de Roche"
of the Canadian hunters, has been long employed by the manufac-

* [An abstract of Part L will be found at p. 300 of the 32nd volume of
this Journal. — Ed-]

464« Royal Society.

turers of archil, though the quantity of colouring matter contained
in it is by no means considerable, being little more than a twelfth
of that in the Roccella Montagnei. The Gyroplwra pustulata, on
which the author operated, was brought from Norway, where it is
annually collected in considerable quantity for the manufacture of
archil. The colouring principle was extracted by maceration with
milk of lime, and was precipitated in a gelatinous state by neutral-
izing the lime solution by muriatic acid precisely in the way so fre-
quently described in the author's former paper (Phil. Trans. 18i8).
The precipitate was gently dried, and then dissolved in hot spirits
of wine. On the cooling of the liquid, the colouring principle was
deposited in small soft crystals, which by digestion with animal
charcoal and repeated crystallizations were rendered quite colour-
less. This principle, to which the author has given the name of
Gyropfioric acid, is almost insoluble in either hot or cold water, and
is also much less soluble in hot spirits of wine than either orsellic,
erythric, or any of the analogous colouring principles. It is neutral
to test-paper, and possesses no saturating power, as the smallest
quantity of an alkali gives its solutions an alkaline reaction. Gyro-
phoric acid strikes a bright red fugitive colour with hypochlorite of
lime; and when macerated with a solution of ammonia, it is slowly
converted into a purplish-red colouring matter, similar to that
yielded by the analogous acids under the same circumstances. When
subjected to analysis, the formula of gyrophoric acid was found to
be C36 His O15.

Gyrophoric acid when boiled for some hours in alcohol yields an
ether similar in appearance and properties to the erythric and leca-
noric ethers ; its formula is C4 H5 O -|- C.,6 H,8 O15.

Gyrophoric acid unites with the alkalies and metallic oxides, but
the compounds which it forms possess little stability and cannot be
procured of an uniform composition.

Lecanora tartarea.
This lichen, like the Gyrophora pustulata, has been employed
from an early period in the manufacture of archil. It is found in
considerable abundance in the hilly districts of the northern parts
both of Scotland and Ireland. The lichen on which the author
operated came from Norway. He found it also to contain gyro-
phoric acid, in much about the same quantity as the Gyrophora
pustulata. This fact was established by the analysis of the acid
itself and of its ether compound.

Brom-orcine.
In the author's former paper on the proximate principles of the
lichens, read before the Royal Society on the 3rd of February 1848,
he described a crystalline body obtained by cautiously adding bro-
mine to an aqueous solution of orcine. In this second part he
states that, in the ' Comptes Rendus' for August of the same year,
Messrs. Laurent and Gerhardt describe the very same compound
obtained in precisely the same way, without even hinting that it had
been previously discovered. These gentlemen however give a dif-
ferent formula for the compound, viz. CnHaBrjO^, or orcine ia

U(n/al Society, ^Q5

which three equivalents of hydrogen are replaced by three equiva-
lents of bromine ; and the author is disposed to adopt this formula,
as on repeating the analysis of the compound he found that he had
somewhat over-estimated the amount of bromine contained in it,
while its other constituents were determined correctly enough.

Beta-orcine.

This substance, described by the author in the Philosophical Ma-
gazine for July 1848, may be obtained from nsnic acid, either by
destructively distilling it, or by acting on it with alkalies.

Beta-orcine crystallizes very beautifully in four-sided prisms sur-
mounted at either end by four-sided pyramids. These crystals have
a brilliant lustre, and are from three quarters of an inch to an inch
long. Their solution strikes a fugitive bright-red colour with hypo-
chlorite of lime, and with a solution of ammonia it yields a perma-
nent blood-red colouring matter which becomes darker on standing.
The formula of beta-orcine, which however is merely empirical, is

^IG "10 '^4'

Quintonitrated-erythromannite.
In his former paper on the lichens, the author has described, under
the name of pseudo-orcine, a remarkably beautiful crystalline body
which is obtained by boiling either picro-erythrine, or erythric acid,
with an excess of lime or baryta. This substance he then regarded
as very analogous to mannite both in its composition and properties,
and this view having been amply verified by an experiment which
he has recently made, he has been induced to change the name of
this compound to erythro-mannite, as at once indicating its origin
and its most striking properties. After referring to the discovery by
Messrs. Flores Domonte and Menard, of "Mannite quintonitrique"
or mannite in which five equivalents of water are replaced by five
equivalents of nitric acid, and which possesses the remarkable pro-
perty of detonating so violently when struck by a hammer that M.
Sobrero has proposed employing it, instead of fulminate of mercury,
in the manufacture of percussion caps, the author states that when
erythro-mannite is treated with fuming nitric acid, in exactly the
same way as mannite, it yields a perfectly analogous compound, or
erythro-mannite in which five equivalents of water are replaced by
five equivalents of nitric acid. This compound, which he has called
quintonitrated erythromannite, is also insoluble in water, but cry-
stallizes out of hot spirits in large flat crystals resembling those of
benzoic acid, only larger and exhibiting a much more pearly lustre.
Quintonitrated-erythromannite also detonates with great violence
when it is mixed with a little dry sand, and is strongly struck with
a hammer.

In order to exhibit more distinctly the close analogy which sub-
sists between the four compounds, their rational formulae are given,
viz.

Mannite =Ci2Hi4 0io;

Erythro-mannite = CuHi4 0n;

Quintonitrated mannite = CiaHgOy-j-SNO^ ;

Quintonitrated erythromannite = Cn Hg Oe + SNO^
PhiU Mag. S. 3. Vol. 34. No. 231. June 1849. 2 H

460 Royal Society,

May 10. — " Remarks on M. De la Rive's Theory for the Physical
Explanation of the Causes which produce the Diurnal Variation of
the Magnetic Declination," in a letter to S. Hunter Christie, Esq.,
Sec. R.S., from Lieut.-Col. Sabine, For. Sec.R.S. Communicated by
S. Hunter Christie, Esq.

My dear Sir, Woolwich, April 16, 1849.

The Annates de Chimie et de Physique for March last contains a
letter from M. Dela Rive to M. Arago*, in which a theory is pro-
posed, professing to explain, on physical principles, the general
phenomena of the diurnal variation of the magnetic declination, and,
in particular, the phenomena observed at St. Helena and at the Cape
of Good Hope, described in a paper communicated by me to the
Royal Society in 1847, and which has been honoured with a place
in the Philosophical Transactions.

Although I doubt not that the inadequacy of the theory proposed
by M. De la Rive for the solution of this interesting problem will be
at once recognised by those who have carefully studied the facts
which have become known to us by means of the exact methods of
investigation, adopted in the magnetic observatories of recent esta-
blishment ; yet there is danger that the names of De la Rive and
Arago, held in high and deserved estimation as authorities on such
subjects, attached to a theory, — which moreover claims reception on
the ground of its accordance with "well-ascertained fticts" and
" with principles of physics positively established," — may operate
prejudicially in checking the inquiries which may be in progress in
other quarters into the causes which really occasion the phenomena
in question ; I have thought it desirable therefore to point out, in a
very brief communication, some of the important particulars in which
M. De la Rive's theory fails to represent correctly the facts which it
professes to explain, and others which are altogether at variance
with, and opposed to it.

1. M. De la Rive's theory, in a few words, is as follows : —
In consequence of the inequalities of temperature in the higher
and lower strata of the atmosphere, electric currents are generated,
which in the higher regions proceed from the equator to the poles,
and return at the surface of the earth from the poles to the equator ;
the return current causing in the northern hemisphere the north end
of the magnet to deviate in the one direction, and in the southern
hemisphere in the opposite direction ; the deviation being at any
given place greatest at the hour (about 1"*30 p.m.) when the differ-
ence of temperature in the upper and lower strata of the atmosphere
is greatest, and of course increasing until that hour, and subse-
quently diminishing.

That the north end of the magnet does thus deviate in the fore-
noon towards the west in the northern hemisphere, and towards the
east in the southern hemisphere, and return in both cases in the
opposite directions in the afternoon, were facts known before the
establishment of the magnetic observatories ; but M. De la Rive's

* [A translation of this letter appeared in the Phil, Mag. for April 1849,
p. 386.— Ed.]

Royal Society* 467

explanation of them appears to have been suggested, and its appro-
priateness, as he considers, shown, by its affirmed accordance with
the remarkable peculiarity in the phenomena made known to us by
the observations at the Magnetic Observatory at St. Helena, and
communicated to the Royal Society in the paper referred to. This
peculiarity is briefly as follows : the deviation which constitutes the
principal part of the diurnal variation at St. Helena is not uniform
in its direction throughout the year ; in one part of the year it is to
the west, and in the other part of the year to the east ; and conse-
quently during certain months of the year the movement of the
magnet is in the contrary direction to that which prevails at the
same hours during the other months of the year.

Now St. Helena is situated within the tropics, and M. De la Rive
infers from his theory that in all places so situated, the diurnal va-
riation should be in one direction when the sun's declination is north
of the latitude of the place, and in the contrary direction when the
sun's declination is south of the latitude of the place : and hence he
too hastily concludes that his theory accords with the characteristics
of the diurnal variation at St. Helena ; when however the facts are
more closely examined it is seen that they do by no means accord
with M. De la Rive's supposition.

That it may be quite clear that I do not misapprehend either M.
De la Rive's theory, or his supposition in regard to the facts at St.
Helena, I subjoin his own expressions, which convey his meaning,
as that gentleman's writings generally do, with most commendable
precision.

The first extract defines the limit which, according to his theory,
should separate the electric currents proceeding respectively from
each of the poles to the equator ; and should consequently separate
the parts of the globe in which the diurnal variation is in the one
direction, from the parts in which it is in the opposite direction ;
whilst the second extract describes what he believes to be the facts
of the phenomenon at St. Helena.

Extract 1.

" La limite qui s6pare les regions occupies par chacun de ces
deux grands courants n'est pas I'equateur proprement dit, car elle
doit etre variable : elle est, d'apres la theorie que je developpe, celui
des paralleles compris entre les tropiques, qui a le soleil a son zenith ;
elle change par consequent chaque jour."

Extract 2.

" A St. Hel^ne, la variation diurne a lieu k I'ouest tant que le
soleil est au sud de I'ile, a Test d6s que le soleil est au nord. En
eiFet, dans le premier cas, ainsi que j'ai remarque plus haut, St.
Helene doit faire partie de la region dans laquelle les courants elec-
triques vont sur la surface de la terre du pole boreal aux regions
6quatoriales ; et, dans le second cas, de la region dans laquelle ces
courants vont du pole austral vers I'equateur."

Whoever will be at the pains to refer to the paper printed in
the Philosophical Transactions, describing the phenomena of St.
Helena, or to the volume containing the details of the observations

2H2

468 lioyal Society.

on the diurnal variation in each month during the five years in
which hourly observations were maintained day and night at that
observatory, will perceive, — on evidence which admits of no uncer-
tainty,— that the two portionsof the year in which the diurnal variation
is in contrary directions at that island, are not determined as M. De
la Rive supposes by the declination of the sun relatively to the la-
titude of the place, but by the declination of the sun relatively to the
equinoctial line. The sun is vertical at St. Helena, passing to the
south in the first week of November ; and again when passing to the
north in the first week of February : consequently the two portions
into which the year is thus divided, are respectively the one of three,
and the other of nine months' duration ; but the actual portions in
which the contrary diurnal movements of the magnets take place at
St. Helena are of equal duration, and consist of six months and six
months ; the dividing periods coinciding unequivocally, not with the
sun's verticality at St. Helena, but with the equinoxes.

2. But if M. De la Rive's explanation be thus inconsistent in respect
to the dates of the transition periods of the phenomena at St. Helena,
it must be regarded as altogether at variance with, and opposed to,
the phenomena described in the same paper at the Cape of Good
Hope, where also they have been observed at the Magnetic Obser-
vatory at that station with an exactness which leaves no uncertainty
whatsoever as to the facts themselves. The Cape is not situated within
the tropics ; its latitude is 33° 56' south ; the sun is consequently
throughout the year well to the north of its zenith ; and therefore
according to M. De la Rive's theory, the deviations should be in one
and the same direction throughout the year. But the fact is not so ;
for the same contrariety in the direction of the diurnal variation at
different portions of the year takes place at the Cape as at St. He-
lena ; the two portions of the year in which the opposite phenomena
prevail, are also identical at the two stations ; and at both the
change in the direction of the deviation takes place when the sun
crosses the equinoctial line ; the deviation being to the west at both
stations when the sun is in the northern signs, and to the east when
he is in the southern signs.

3. But in considering a theory which comes before us, claiming
the high distinction of affording a physical explanation of facts which
are known to us by well-assured observation, we ought not to con-
fine our view to those points only for which it professes to supply
the explanation : these are certainly tests as far as they go ; — and in
the present instance the conclusion from them is not favourable to
the theory proposed ; — but we should also notice its deficiencies ; or
those points wherein it neither furnishes, nor attempts to furnish, ex-
planations of circumstances which are certainly amongst the most
remarkable facts of the case. They may be possibly amongst the
most difficult to explain, but no physical theory can be regarded as
meeting the facts which does not at least attempt an explanation of
them. I may name as the most prominent in interest amongst these
the striking fact, that the Cape of Good Hope should be one of the
stations at which this remarkable peculiarity of a contrariety of move-
ment at different periods of the year takes place.

hitelUgence mid Miscellaneous Articles. 4-69

It is known that it does not occur at places situated in correspond-
ing latitudes north of the geographical equator ; at Algiers, for ex-
ample,— which is moreover nearly in the same geographical meridian
as the Cape, and where the magnetic inclination is nearly the same
towards the north as i& the case at the Cape towards the south. It
may be quite correct perhaps to view the corresponding phenomena
at St. Helena and the Cape as those belonging to magnetically -nc^^-
torial stations ; but they are certainly not those peculiar to or cha-
racteristic of geographically -Q(\Vi9.tox'wX stations, which would be the
condition in M. De la Rive's theory. There are thus two parts in
the problem demanding physical explanation ; on the one hand,
the cause is required of the contrariety of movement, as well as of
the times at which the different movements occur, the latter having
obviously a dependence on the sun's position either in the northern
or the southern signs ; and on the other hand, the cause must be
shown why certain stations are thus aiFected and others not : a di-
stinction which obviously does not depend on situation in regard to
the geographical equator or to the tropical divisions of the globe.

I have myself been led to infer that the peculiarity in question re-
sults from and is indicative of proximity to the line of least mag-
netic force, regarded as physically the separating line on the surface
of the globe between the northern and southern magnetic hemi-
spheres ; the peculiarity would thus be strictly a magnetically- equa-
torial one.

It results from the present position of the four points of maximum
intensity at the surface of the earth, that the intermediate line of
least intensity departs considerably in the Southern Atlantic from
the middle or geographically-equatorial portion of the globe, passing
between the Cape and St. Helena, and consequently not far from
either of these stations.

As far as I have yet been able to examine, I have found that the
same remarkable peculiarity does exist at all other stations which
are near this line, and at none which are remote from it. But
however this may be, the accordance of the phenomena at the Cape
of Good Hope and St. Helena, and their dissimilarity from those at
other stations is a well-ascertained fact, of far too much bearing and
importance to be passed without notice ; and we may safely anti-
cipate that its cause must occupy a prominent place in the theory
which shall be ultimately received, as affording an adequate solution
of the problem of the diurnal variation.

Believe me, my dear Sir, sincerely yours,

Edward Sabine.
S. H. Christie, Esq., Secretary to the Royal Society.

LXX. Intelligence and Miscellaneous Articles.

ON SOME METEOROLOGICAL PHENOMENA.

BY PROF. E. WARTMANN.

f\N a mirage with a strong bise. — Most of the mirages described

by authors appear to be manifested in a tranquil state of the air.

M. Kamtz even affirms that a calm atmosphere is indispensable to their

4i*!(> Intelligence and Miscellaneous Articles.

production*. Although this may be the ordinary circumstance, it
is not always the case. More than once vessels have been carried
along by the wind, and their images shifted in a similar manner.
This phsenomenon is often remarked on the Black Sea, from Odessa.
The accounts of the observations of Woltmann near Cuxhavenf, and
of Vince at RamsgateJ, leave no doubt with respect to this fact. The
following is a similar instance, observed at Nyon in the summer of
1848 by M. Thury, formerly professor at Lausanne, and which is
still better characterized than the previous ones.

It was twenty minutes to 8 o'clock, a.m. The bise had raised
foaming waves upon the lake. In the south-east some vapours
floated on the horizon : the sky in every other part was of a clear
blue. By the aid of a good telescope of 0*068 millim. aperture, and
which magnified thirty times, M. Thury descried on the heights of
Coppet, in the direction of Geneva, the two lateen sails of a bark
the hull of which was not at all visible. A little below the lower
extremity of these sails, the commencement of their image was seen
inverted. This incomplete image terminated abruptly on the agitated
and clear surface of the water. The space which separated the sails
from their image was of a uniform greenish-blue colour. The lowest
strata of air undulated in a very perceptible manner.

This last circumstance, and the situation of the image below the
object which it represented, are proofs that the phsenomenon resulted
from a greater heating of the atmosphere in the lower strata than in
the more elevated regions. But for the hull of the vessel to be in-
visible, and for the contours of the objects, examined with a telescope
at fourteen metres above the level of the water, to be perfectly well-
defined, the warmest zone of air must have terminated under the
wind toward the base of the sails, that is to say, at three or four
metres above the surface of the lake. The existence of a zone thus
limited is therefore possible with the bise blowing sufficiently strong
during the few minutes necessary for the observation. This fact
recalls the persistence of the undulation in the strata of air which
are close to the roofs and the ground during the warm hours of a
summer's day, or above the chimneys in winter, notwithstanding the
agitation produced by intense winds.

II. On blue rays. — On the 30th of November last, a little before
sunrise, M. Thury perceived at Nyon, above the mountains which
border the lake on the east, horizontal strata of light clouds tinged
with a beautiful yellow. The sky, seen in the spaces between them,
was of a deep azure colour. Toward the point of the horizon where
the sun was about to appear, a dark blue ray rose divergingly up to
a great height, and occupied a space in which no cloud was percep-
tible. This appearance vanished after two or three minutes.

Dr. Gosse has found, among his father's papers, the account of an
analogous observation made at Lyons toward the end of the last

* Lehrbuch der Meteorologie, vol. iii. p. 87. — Cours Complet de Meteoro-
logie, translated by Ch. Martins, p. 4?2.

t Gilbert's Annalen der Physik, vol. iii. p. 397.
J Philosophical Transactions, 1799, p. 13,

Intelligence and Miscellaneous Articles. 4<71

century by this venerable founder of the Helvetic Society of Natural
Sciences. The manuscript unfortunately has not been preserved.

What has been called a ray appears to me, on the contrary, to be
a phsenomenon of shadow. The light of the sun, arrested by some
obstacle out of sight, left the transparent vapours situated near the
horizon invisible in a determinate region of the sky, whilst to the
right and left it tinged them by playing upon them. The space not
illumined should offer to the eye a blue colour, the more remarkable
as it contrasted strongly with those of the adjacent strata of a golden
yellow. The diverging form of the ray is a well-known illusion in
perspective. What apparently increased the separation of its mar-
gins toward the upper part, was the less quantity of vapour existing
in elevated regions.

It is to be regretted that M. Thury was not able to measure the
height of the ray ; this would have aided in determining the position
and perhaps the nature of the screen {dcran) which produced it. If this
was one of the principal summits of the Savoy Alps, the appearance
should recur periodically toward the 30th of November and the 13th
of January, when the atmospheric conditions are favourable. Ob-
servers placed in proper localities would easily decide the truth of
this supposition.

III. On solitary crepuscular rays. — In 1846* I described a meteor
of a character quite different from that of the blue rays, and which
consisted of a single, vertical, luminous band, 35° high, without any
trace of divergence. I afterwards found that Professor Christie had
twice seen a phaenomenon nearly analogous to this in 1834t. The
rays examined by Mr. Christie were less extended than the band
seen at Lausanne and Geneva. Moreover they had a perceptible
divergence, whilst the edges of the latter were absolutely parallel.
Lastly, on the 31st of May 1846, there was such a bise blowing that
the sky was perfectly clear. The meteorological registers of Saint-
Bernard, Lausanne and Geneva, prove this. It therefore does not
seem that these various appearances can be entirely assimilated.
The theory proposed by the English savant must be subjected to
new observations, as he himself admits. It will be useful to make,
with the polariscope, some researches on the state of the light of the
solitary rays, and of that of the atmosphere in the adjacent parts. —
Extracted from the Bibliotheque de Universelle Geneve.

ON THE REFLEXIONS OF DIFFERENT KINDS OF HEAT BY
METALS. BY MM. F. DE LA PROVOSTAYE AND P. DESAINS.

Those philosophers who have been occupied with the study of
heat, seem to admit that the rays of different natures are reflected
in the same proportion on pohshed metals.

On the other hand, the very precise experiments of M. Jamin

* Archives des Sciences Phys. et Natur., vol. ii. p. 166.
t Report of the Fourth Meeting of the British Association for the Ad-
vancement of Science : Transactions of the Sections, p. 666. London, 1835.

472 Intelligence and Miscellayieous Articles.

agree with the formulae of M. Cauchy, to prove that the intensity of
metallic luminous reflexion depends on the colour of the light em-
ployed. The numerous analogies which exist between heat and
light, scarcely admit of an essential difference on this head. The
authors, indeed, are of opinion, that they have in fact proved by ex-
periment that this difference does not exist, and that the rays of heat
of different natures are reflected in very unequal proportions on the
same metallic mirror.

The plan adopted and followed by the authors is precisely the
same as that previously followed in their Researches on Metallic
Reflectors. The source of heat was always a Locatelli's lamp ;
only they operated successively with the direct rays, and with the
same rays transmitted, sometimes through a plate of natural sal
gem, and at others with it smoked, and lastly through a lamina of
glass 5 millimetres in thickness. The incidence of the rays being
about 60°, the following results were obtained : —

Experiments made with the Metal of the Reflectors of Telescopes . —
The metal of the reflector employed reflected 0*80 or 0-84 of the
direct heat of Locatelli's lamp. It reflected only 0*74 of the heat
derived from the same source when modified by passing through
a lamina of glass of 0"*005 in thickness. Lastly, it reflected 0'82
to 0*83 of the same heat transmitted through sal gem.

Experiments with Silver. — The silver mirror reflected 0*95 to 0*9 6
of the natural heat, and 0*91 of the heat which had passed through
0-005 of glass.

Experiments with Platina. — The platina employed reflected 0'79
of the natural heat; 0*77 to 0*78 of the heat which had traversed
sal gem ; 65 to 66 of that which had traversed 0'"'005 of glass ; and
lastly, 83 of that which had passed through smoked sal gem.

Some experiments were also made with plates of gold and unpo-
lished silver, which were employed by the authors in an investiga-
tion respecting the diffusion of heat. The proportion of the incident
flow which these plates reflect back to the pole when it is placed in
the direction of the regular reflexion, is extremely different, according
as the heat has previously traversed glass or sal gem.

It results from these numbers, that the heat most transmissible
through glass is reflected in smaller proportion on the various metals
tried ; and that the heat, which is transmitted in larger proportion
through smoked sal gem, is reflected more abundantly upon the
same substances. A marked consequence of these experiments is,
that a bundle of heat rays reflected on a metallic mirror has generally
a composition entirely different from that of the incident bundle,
and that consequently it should not suffer the same loss in traversing
diathermanous substances. This has been directly verified by the
authors in the following manner : —

1. They determined the loss of intensity which the heat of a Lo-
catelli's lamp suffered in traversing a lamina of glass of O'^'OOS
in thickness.

2. The loss suffered by heat from the same source twice reflected
on parallel mirrors.

Intelligence and Miscellaneous Articles. 473

In the first case the lamina of glass employed transmitted 0*44 of
the incident heat ; in the second only 0-33 to 0'34.

These two methods afford then concordant results ; and the
authors are of opinion that they have determined that, on a great
number of metals, and probably on all, the different kinds of heat
are reflected unequally, and that the reflexion on polished metals
changes the j)roportions of the different kinds of heat which compose
the incident bundle. — Comptes Rendus, Avril 16, 1849.

ON CHLORONICEIC ACID. BY M. E. SAINT-EVEE.

For the preparation of this acid, the first operation consists in
passing a current of moist chlorine gas into a cold and strongly alka-
line solution of benzoate of potash. After many trials, the propor-
tions which succeeded best were found to be 60 grms. of benzoic
acid, 200 grms. of hydrate of potash, and 300 to 350 grms. of water,
according to the degree of hydratation of the potash of commerce,
which is far from being constant.

The potash is to be dissolved at a gentle heat, and the benzoic
acid afterwards added ; and the chlorine is not to be passed till
everything is dissolved. After some time the solution assumes suc-
cessive shades of yellow, greenish-yellow and bright green ; it then
again becomes yellow, and eventually deposits an abundant pulpy
compound, which is grayish and crystalline. An abundant evolution
of carbonic acid takes place during the operation, which is readily
detected by passing it into a vessel containing barytes water. An
operation performed with the quantities above described continues
about two days.

The precipitate is composed, — 1st, of chlorate of potash, the cry-
stallization of which is completely modified by the presence of or-
ganic matter, the salt being usually obtained in the form of four-
sided prisms ; they are long, hard and brittle ; 2nd, of a small quan-
tity of unaltered benzoate of potash ; 3rd, of a salt of potash con-
taining the new acid ; and lastly, the supernatant liquor contains
benzoate of potash and chloride of potassium.

About half its volume of water is to be added to the mass. The
solution is to be saturated, at a moderate temperature, by means of
a current of carbonic acid, the saturation being completed by the
addition of a small quantity of dilute hydrochloric acid. The solu-
tion is then to be boiled. The magma which is precipitated gra-
dually redissolves, and there soon appears an oleaginous substance
which is fusible at about 270° F. ; it is of an amber colour, and heavier
than water. It sometimes precipitates to the bottom of the capsule,
and sometimes, on the contrary, it floats, according to the degree of
concentration of the solution. The solution being poured off, an
oily matter remains, which soon concretes by cooling.

The crude acid thus prepared is hard, brittle, of a slight yellow
colour, and contains a notable quantity of benzoic acid. It is freed
from this by repeated solutions in boiling distilled water ; and the

474 Intelltge?tce a7id Miscellaneous Articles.

purification is eventually completed by repeated crystallizations in
alcohol, or in a mixture of alcohol and sether.

The benzoic acid separated in this manner is added to that which
the decanted liquor contains, which is then strained through a cloth
and washed with cold water. It is to be carefully pressed ; and by
treating it in a capsule with a small quantity of hot water, a fresh
and often considerable quantity of the new acid is separated, which
had been dissolved during the first separation.

The pure acid consists of granular crystals of a cauliflower form.
When examined by the microscope, they present the form of four-
sided acicular prisms. It melts at 302°, and when melted its den-
sity is 1-29 ; it boils at 270° F. It volatilizes without decomposing,
and is deposited during distillation on the sides of the vessel in flat
needles of a greasy lustre and grouped around a common centre.

Its odour, especially when it has been melted, is sharp and pene-
trating, like that of all chlorinated compounds in general, but en-
tirely difi^erent from that of the acid from which it is derived.

Analysis showed that it was composed of— •

C«4 72 50-00

H>o 5 3-47

CP 35 24-30

O* 32 22-23

144 100-00

"When chloroniceic acid is treated with fuming sulphuric acid, they
combine so as to form with barytes a soluble salt, probably repre-
sented by 2S03, C24 H8 Cl^ O^, BaO, H^ O, like the corresponding
sulphobenzoate.

When distilled with lime or barytes, with proper precautions, two
hydrocarburets are formed ; the first is liquid and the second solid.
It resists long-continued exposure to the action of dry chlorine, even
under the influence of heat, and also the dechlorizing action of the
amalgam of potassium. — Ann. de Chim. et de Phys., Avril 1849.

ON THE NATURE AND COMPOSITION OF VARIOUS CHLORO-

NICEATES. BY M. E. SAINT-EVRE.
Chloroniceate of Ammonia. — When freshly prepared by the direct
saturation of [chloro ?] niceic acid dissolved in alcohol, [chloro ?]
niceate of ammonia crystallizes in large micaceous laminae, which
undergo change by the action of light, becoming brown and acid to
litmus-paper. When pure, this salt is fusible and volatile, without
undergoing decomposition. By analysis it appeared to be composed
of—

C24 72 44-72

H'6 8 4-96

Cl^ 35 21-73

O* 32 19-90

Az2 14 8-69

161 100-00

Intelligence and Miscellaneous Articles. 4.75

Chloroniceate ofBarytes. — This salt is a white crystalline powder,
which is slightly soluble in water, but readily so in hot alcohol. It
is decdYnposed by heat, yielding a mixture of two hydrocarbons, one
solid, the other liquid ; and a coaly residue is formed. It appears
to be composed of —

C2* 72 34-12

H8 4 1-89

CP 35 16-58

O* 32 15-19

Ba 68 32-22

in 100-00

Chloroniceate of Silver. — When prepared in the usual manner, in
alcoholic liquors, this salt is precipitated in the form of white flocculi,
which washing and drying convert into a crystalline powder. By
analysis it yielded —

C24 72 28-68

H8 4 1-59

CP 35 13-94

Ag 108 43-02

O* 32 12-77

^51 100-00

Ann. de Chim. et de Phys., Avril 1849.

ON THE REACTION OF SULPHATE OF POTASH AND SULPHATE
OF COPPER. BY M. J. PERSOZ.

When a saturated solution of sulphate of potash containing some
sulphate of copper is made to boil, it becomes in a very short time
extremely acid, and yields a very dense precipitate, which adheres
to the bottom of the vessel, and causes so much bumping that it is
requisite to use a porcelain capsule.

In the first experiment, 210 grms. of sulphate of potash were dis-
solved in 1*5 litre of water; and 150 grms. of sulphate of copper in
1*2 litre.

These solutions, previously filtered, were mixed, and the solution
was kept boiling for an hour, and then suflFered to cool ; then having
poured off the acid liquor, and washed the triple salt formed till the
washings ceased to be acted upon by ferrocyanide of potassium,
the salt was pressed between folds of blotting-paper, and dried in a
stove at 212° F. This triple salt weighed 9-95 grms.

The second experiment, made with the same proportions of the
salts, but with less water, yielded 27-25 grms. of triple salt.

Lastly, a third experiment was performed, in which 210 grms. of
sulphate of potash were dissolved in 1-2 litre of hot water ; and when
made to boil, 155 grms. of crystallized sulphate of copper were added
in small portions to it, so as not to reduce the temperature. In a
quarter of an hour an abundant precipitate of the triple salt was
obtained, which, washed with cold water, expressed and dried by
the water-bath, weighed 56-7 grms. When the sulphate of potash

4-76 Intelligence and Miscellaneous Articles.

is added to the sulphate of copper, the same quantity of precipitate
is not obtained.

It results from the above-stated experiments, that the quantities
of the triple salt vary according to the proportions of water, and
whether the sulphate of copper is added to the sulphate of potash, or
the reverse ; when the proportion of sulphate of potash is increased,
a larger quantity of the sulphate of copper is decomposed, but with-
out in any case obtaining bisulphate of potash and the triple salt
only.

As the proportions of the triple salt thus formed are variable, it
is natural that the solutions which yield it should differ in composi-
tion ; some are found to contain much of the copper salt, whilst in
others the sulphate of potash is in excess.

On subjecting these solutions to careful evaporation, crystals of a

double salt, S Cu SK + 6H'^ O, are formed, which by two or three
successive concentrations separate completely. It is a curious phae-
nomenon to observe such very soluble salts produced so perfect by
simple crystallization. The mother- waters eventually resulting from
these crystallizations are merely bisulphate of potash. — Ann. de
Chim. et de Phys., Mars 1849.

ON OCTOHEDRAL AND CUBIC ALUM. BY M. J. PERSOZ.

It is well known, that when a solution of octohedral alum is satu-
rated with potash, or for a short time put in contact with trissulphate
of alumina, it cannot be heated to 140° F. without becoming turbid ;
and there are formed octohedral alum, soluble at all temperatures,
and trissulphate of alumina, which is precipitated. When the solu-
tion, however, instead of being subjected to so high a temperature,
is subjected to evaporation at a gentle heat, cubic alum is obtained,
which readily becomes octohedral alum by dissolving it in water
slightly acidified with sulphuric acid. It may then be evaporated
and redissolved at pleasure without undergoing any alteration.
Lastly, if a certain quantity of cubic alum be dissolved in water and
boiled, it yields basic sulphate of alumina insoluble in water ; and
the mother-water and that used in washing, when mixed and evapo-
rated, give only octohedral alum. Hence it is concluded that these
two alums are not identical, and that cubic alum contains most
alumina. — Ann. de Chim. et de Phys., Mars 1849.

ON ANISOL AND ITS DERIVATIVES. BY M. A. CAHOURS.

Anisol presenting with respect to toluol (benzoene of M. Deville)
the same relations of composition that phenol does to benzene, the
author resumed the examination of this product, in order to complete
his researches respecting the compounds of the anisic series.

It has been shown that anisol treated with fuming nitric acid ex-
changed 2 or 3 equivalents of hydrogen for 2 or 3 equivalents of
hypoazotic vapour. There was therefore wanting in this series of
the derivatives of anisol, its first term, that is to say, that which
would result from the replacement of 1 equivalent of hydrogen by

Litelligence and Miscellaneous Articles. 4-77

1 equivalent of hypoazotic vapour, and which the author calls in the
nomenclature adopted for these compounds mononitric anisol. M.
Cahours has succeeded in obtaining it by treating anisol with small por-
tions of fuming nitric acid, taking care to keep the vessel containing
the reacting substances extremely cold. Operating with these pre-
cautions, a thick liquid of a blackish blue is obtained, which is purified
by submitting it, at first, to repeated washings with slightly alkaline
water, and afterwards by distilling it, having first digested it over fused
chloride of calcium. Thus prepared, mononitric anisol is a liquid of
an amber colour and heavier than water. It boils between 503° and
507° F., and possesses an aromatic odour. Solution of potash, even
when heated, does not alter it. Concentrated sulphuric acid dissolves
it when gently heated, and water added to the solution separates the
product from the liquor unaltered. Treated with an alcoholic solu-
tion of hydrosulphate of ammonia, it is readily acted upon ; sulphur is
deposited, and the alcohol holds in solution a new organic base, which
differs from toluidine only in containing two molecules of oxygen.
Mononitric anisol submitted to analysis yielded nearly —
14 eqs. Carbon ........ 84 54-90

7 ... Hydrogen 7 4-57

) ... Nitrogen 14 9'14

6 ... Oxygen 48 31-39

153 10000

The substance thus formed differs from anisol by the substitution
of one equivalent of hypoazotic acid for one equivalent of hydrogen,
which justifies the name of mononitric anisol bestowed upon it.

The analysis of the new base, formed by the action of mononitric
anisol and hydrosulphate of ammonia, leads to the formula C'* H^NO^;
it forms a crystalline salt with hydrochloric acid. As mononitric
anisol is prepared with difficulty, M. Cahours has obtained only a
small quantity of it, and he proposes to call it anisidine.

Benzene and binitric cumene being easily acted upon by hydro-
sulphate of ammonia, and transformed into nitric alkaloids, binitric
anisol was submitted to the same reagent. By treating an alcoholic
solution of binitric anisol with hydrosulphate of ammonia, an abun-
dant deposit of sulphur is obtained, whilst the alcohol retains in so-
lution a substance which perfectly saturates acids, and forms with
them crystallizable acids [salts ?].

The new base thus formed crystallizes in long needles of a reddish
brown colour, possessing much lustre ; it is insoluble in water, but
dissolves readily in boiling alcohol, the greater portion separating on
cooling. This alkaloid yields well-formed crystalline salts with
sulphuric, nitric and hydrochloric acids ; some of them are perfectly
colourless when pure.

The analysis of this substance indicated its composition to be —
14 eqs. of Carbon 84 50*00

8 ... Hydrogen .... 8 4-76

2 ... Nitrogen 28 16-67

6 ... Oxygen 48 28-57

168 100-00

It will be observed that this substance differs from the preceding

478 Intelligence and Miscellaneous Articles,

only in one equivalent of hydrogen being replaced by one equiva-
lent of hypoazotic vapour ; for this reason the author gives it the
name of anisidine nitree. This base forms with hydrochloric acid,
a colourless salt crystallized in long needles, represented by the for-
mula CIH, C'^Hs N^Oe.

The chloroplatinate crystallizes in needles of a golden yellow
colour ; its formula is CIH, PtCl^ C'^ H^ N^Qs ; the nitrate has the
form of prisms of considerable size, which are slightly soluble in
water ; the formula is N0^ HO, C'* H^ N* O^ ; the sulphate is very
soluble in water, it crystallizes in very fine needles, grouped around
a common centre ; its formula is SO^, C* H^ N^ O^.

When toluol is treated with fuming nitric acid, it forms two com-
pounds, one of which is liquid, and is the mononitric toluol ; the
other is crystallized, and is the binitric toluol ; when the latter was
treated with an alcoholic solution of hydrosulphate of ammonia, it
yielded a very fine alkaloid corresponding to anisidine nitree, diflFering
from it only by two equivalents of oxygen. This new alkali the
author calls toluidine nitree ; its formula is C'"* H'^ N'^ O*.

The number of alkaloids increases daily ; their study affords re-
sults of great interest, and the hope may be entertained that those
presented by nature may eventually be formed bj'- art. M. Wiirtz
has described two very remarkable alkalies obtained by the action of
potash on cyanicaether, alcohol, and pyroxylic spirit; petinine, recently
discovered by M. Anderson in the products of the distillation of ani-
mal matters, is to be added to the group. The strong ammoniacal
odour, the manifest analogy of the properties of its salts, with those
of the salts formed by the alkalies of M. Wiirtz, induced M. Cahours
to suppose that petinine belongs to this series. Adopting the for-
mula C^ H^^ N, proposed by M. Gerhardt, from the analysis of the
chloroplatinate, it will be seen that petinine is merely butyrammonia
C* H2, NH^. M. Anderson has also noticed, in the oil derived from
the distillation of animal substances, some very volatile alkaline pro-
ducts, among which will probably be found the curious alkalies of
M. Wiirtz.

When fuming nitric acid is made to react upon anisic acid, or
nitranisic acid, binitric or trinitric anisol is formed, according to the
proportion of the matters reacting and the duration of the reaction ;
besides these two substances, there is formed, and often in great
abundance, an acid which crystallizes from an alcoholic solution as
it cools, in the form of rhomboidal plates of a magnificent golden
yellow colour ; this acid, which M. Cahours calls chrysanisic acid, has
a very remarkable composition : it is isomeric with trinitric anisol ;
consequently it is an homologue of picric acid (phenol trinitree).

This acid submitted to analysis gave —

14 eqs. of Carbon 84 34-57

5 ... Hydrogen .... 5 2*05

3 ... Nitrogen 42 17-29

14 ... Oxygen 112 46-09

243 loO^^

This acid, differently from all others of the same kind, forms a very
soluble salt with potash. — Comptes Rendus, March 19, 1849,

Meteorological Observations. 479

COMPOUNDS OF HYDROCHLORATE OF STRYCHNIA AND CYANIDE
OF MERCURY.

M. Brandes states that when a mixture is made of hydrochlorate
of strychnia with one of cyanide of mercury, a crystalline precipi-
tate is obtained, the composition of which has been hitherto un-
known. The author thinks that it is a combination of hydrochlorate
of strychnia and cyanide of mercury, corresponding to the formula
Str, HCl + 4. UgC\.—Journ. de Ph. et de Ch., Janvier 1849.

METEOROLOGICAL OBSERVATIONS FOR APRIL 1849.
Chiswick, — April 1. Rain : fine : showery. 2. Densely clouded, 3. Fine :
cloudy. 4. Foggy: rain. 5. Drizzly: fine. 6. Heavy dew: very fine. 7. Cloudy:
drizzly. 8. Hazy. 9. Foggy: densely overcast. 10. Hazy: heavy clouds.
11. Cloudy and cold : clear and frosty at night. 12. Cloudy. 13. Rain. 14.
Slight haze : fine : clear and frosty. 15. Rain. 16. Cloudy throughout. 17,
Cold and dry: clear and frosty at night, 18. Clear: snow; cloudy. 19. Heavy
fall of rain, sleet and hail throughout the day. 20. Snow and hail in forenoon,
stormy showers : clear and frosty at night. 21. Clear : cloudy : clear and frosty.
22. Overcast : rain at night. 23. Rain. 24. Cloudy. 25. Drizzly : fine. 26.
Rain : cloudy : clear : slight frost. 27. Foggy ; cloudy : slight rain. 28. Fine :
heavy showers : partly hail. 29. Slight haze : very fine : overcast. 30. Fine :
clear.

jNlean temperature of the month 44°"29

Mean temperature of April 1848., 47 '33

Mean temperature of April for the last twenty-three years 47 '53

Average amount of rain in April 1*46 inch.

Boston. — April 1, Rain: rain a. m, 2. Cloudy : rain p.m. 3. Cloudy. 4. Fine.
5. Rain : rain a.m. and p.m. 6. Fine. 7. Fine : rain p.m. 8, 9. Rain :
rain A.M. and P.M. 10. Rain : rain a.m. 11, 12, Cloudy. 13. Rain : rain a.m.
and rain and snow P.M. 14. Cloudy : rain a.m. 15,16. Cloudy. 17. Fine:
snow P.M.: stormy. 18. Fine: rain and snow p.m. 19, 20. Cloudy. 21. Fine.

22. 23. Fine : rain p.M, 24, 25. Fine. 26. Cloudy : rain early a.m. 27. Fine.

28. Fine : rain p.m. 29, 30. Fine,

Applegarlh Manse, Dumfries-shire. — April 1. Showers: thunder. 2. Spring
showers. 3. Showers : thunder. 4. Frost : calm and fine. 5. Showers. 6.
Fair : beautiful day. 7. Fair, but rigid and ungenial. 8. Slight shower a,m, :
parching wind. 9,10. Fair and chilly. 11. Fair and chilly : frost a.m. 12.
Cloudy A.M. : rain p.m. J 3. Frost: snow: hail: fine p.m. 14. Snow an inch
deep: rain p.m. 15. Fairandfine. 16. Frost: slight shower of snow. 17. Frost
very hard : shower of snow. 18. Frost: snow-showers: rain p.m. 19. Frost:
shower of snow : fair and keen p.m. 20. Frost : clear and cold : slight shower of
snow. 21. Frost very hard : clear and cold. 22. No frost : rain gentle : cloudy.

23. Rain : soft and warm : blessed change of weather. 24, Fine a.m. : grew
cloudy : rain p.m. 25. Fine a.m. : shower of hail : rain. 26. Fine a,m, -. shower.
27. Shower early : rain and wind p.m. 28, Growing day : one slight shower.

29. Fair and fine : cloudy p.m. 30. Most beautiful spring day.

Mean temperature of the month 42°'3

Mean temperature of April 1848 43 '2

Mean temperature of April for the last twenty-five years . 44 '4

Rain in April 1848 2*52inches.

Average amount of rain in April for the last twenty years 1*76 „
Sandwich Manse, Orkney. — April 1. Cloudy : rain. 2. Showers : drizzle. 3.
Damp : showers. 4. Drizzle, 5. Damp : showers. 6. Damp. 7. Damp :
cloudy. 8. Bright: clear. 9. Bright: cloudy. 10, 11. Cloudy. 12. Showers:
cloudy. 13. Bright : cloudy. 14. Bright : clear : aurora, 15. Cloudy. 16.
Snow-showers : snow. 17. Snow-showers: drift: snow-showers. 18. Snow-
showers : frost : aurora, 19. Sleet: hail-showers : aurora. 20. Bright : frost :
clear : aurora. 21. Bright : clear : aurora. 22. Cloudy. 23. Drops : cloudy.

24. Bright : hazy. 25. Clear. 26. Cloudy. 27. Clear. 28. Clear : very clear,
29, 30. Fine: cloudy.

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THE
LONDON, EDINBURGH and DUBLW

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

SUPPLEMENT to VOL. XXXIV. THIRD SERIES.

LXXI. On Circular Magiietic Polarization,
By M. A. Bertin*.

SINCE the discovery of Prof. Faraday, circular magnetic
polarization has formed the subject of several important
investigations ; but nevertheless there still remain some con-
ditions of the phaenomenon to discuss, some conclusions to
verify, some obscure points to clear up. I have attempted to
do this, aided by the valuable assistance of MM. Pouillet and
Edmund Becquerel, who have kindly placed at my disposal
the apparatus they made use of in their researches on this
subject.

The method of experimenting is now too well known to
require description. I shall only say that all the numbers
referred to in this paper represent the total rotation produced
in the plane of polarization by a change in the direction of the
current. It is this total rotation which I have always mea-
sured, as it is immediately presented in the experiment; be-
cause, corresponding to a more intense phaenomenon, it is for
that very reason measured with greater accuracy ; and lastly,
because it is independent of the determination, always very
uncertain, of the zero, that is to say of the position of the
analyser, for which the light was extinguished before the pas-
sage of the current.

As is seen, the measurement of the rotation results from the
determination of two planes of polarization, or of two tints of
passage ; each of these observations being liable to an error of
\ degree, it is necessary to admit the possibility of an error of
\ degree in the rotation, which moreover also is subject to the
influence of small irregularities in the transmission of the cur-
rent by the commutator. The variations which affect the cur-

* From the Annates de Chimie et de Physique, vol. xxiii. p. 5.
Phil. Mag, S. 3. No. 232. Suppl, Vol. 34. 2 I

482 M. A. Bertin on Circular Magnetic Polarization.

rent itself during a series of experiments would produce dif-
ferences far more considerable, which must be avoided by
comparing only the rotations observed at short intervals, and,
as it were, the one following the other.

It has been observed, that for the success of these experi-
ments it is indispensable that the glasses experimented with
should be annealed ; but happily this is not the case, other-
wise these researches would be impossible, since the greater
part of them are unannealed, or soon become so. When
such a glass is suitably placed between two crossed Nichol's
prisms, one or several black lines appear in it, which serve as
index points. On viewing one of these lines, which may nearly
always be isolated in the field of vision, it is seen to disappear
when the current passes ; and it is again found, on turning
the analyser, absolutely like the black image of annealed
glass. In white light it experiences the same variations of
tints as this, and it is always easy to determine the azimuth in
which it presents the tint of passage. It is true that, in con-
sequence of the movement of the plane of polarization, it is
not the same black ray which reappears, but another slightly
different. However, the rotation is not influenced by this
circumstance, for I have ascertained that it is independent of
the black ray to which the eye is directed.

The most happy modification which has been made in the
original apparatus of Prof. Faraday, has been to cause the ray
of light to pass, not merely near the line of the poles, but
through this line itself, by piercing the electro-magnet in this
direction. This condition is satisfied in M. E. Becquerel's
electro-magnet by the perforated terminations which he places
on the two poles*, and to them must be attributed the greater
part of the force of this apparatus. The following table leaves
no doubt on this point.

Rotations observed with the Electro-magnet ofM. Becquerel.

Substances placed between the two poles.

With the
terminations.

Without the
terminations.

mm.

Very thick flint-glass 55-1 thickness.

Faraday's flint-glassf 48-3

... 18-3
Distilled water 1300

21 00
25 6

18 20
5 30
3 50

1 30
6 30

2 30

3 00
0 00

... 300

* Ann. de Chim, et de Phys., third series, vol. xvii. p. 437.
f I am indebted to M. Dumas for this flint-glass.

M. A. Berlin on Circular Magnetic Polarization. 483

The same condition is still better satisfietTin Rwhmkorff's*
apparatus, where the helices themselves are pierced in the direc-
tion of their axes. The Ecole Normale has such an apparatus,
54; kilogrammes in weight, which is as powerful as that be-
longing to M. Becquerel, although it is not more than one-
third its weight.

The effect produced by an electro- magnet of the same shape
depends on the mass. Thus, an apparatus one-fifth the size
of the preceding, produced, under the same circumstances,
rotations twice as feeble.

The dimensions of the wire have also a certain influence. In
general it should be thick. That used in the Ruhmkorfi^ ap-
paratus is 2*50 millims. in diameter ; and in M. Becquerel's
electro-magnet the maximum effect is obtained by doubling
the section of the conducting wire. With respect to the mass
employed, the instrument-makers at present are in the habit
of rolling around the iron nucleus a thickness of wire equal
to its radius, so that the external diameter of the reel is twice
that of the interior cylinder.

Lastly, the intensity of the phaenomenon likewise depends
on the current, or rather on the relation existing between the
dimensions of the electro-magnet, and the force of the battery
exciting it; so that a very powerful apparatus may appear
very feeble when it is not set in action by a battery of suffi-
cient strength. With the same apparatus the intensity of the
effects produced increases with the intensity of the current;
as to this, it increases with the number of the elements of the
battery, but is far from being proportional to it. Again, when
the current has a certain force, it is more advantageous to
increase the quantity of electricity than the tension ; that is to
say, it is better to enlarge the surface than the number of the
elements of the battery. It is therefore desirable to ascertain
what is the arrangement for a given battery which will pro-
duce the maximum effect. Thus, having at my disposal 80
Bunsen's elements, I have found that the best arrangement
to be given to them in order to excite the great Ruhmkorff
apparatus, was to join, by poles of the same name, four bat-
teries of 20 elements. The results just stated follow from the
subjoined table, which contains the rotations observed in one
of Faraday's flint-glasses of 39 millimetres in length placed
between the two poles of the Ruhmkorff apparatus.

* Ann. de Chim. et de Pht/s., third series, vol. xviii. p. 318.
21 2

iS'l' M. A. Berlin 07i Circular Magnetic Polarization.

Number of elements.

Tension of
the battery.

Surface of the
battery.

notation
observed.

80

80

40

20

40

20

10

20

10

5

10

5

1

1

'4
1
2
4
1
2
4
1
1
1

23 30

25 20

26 30
20 30
22 30
18 30
17 30
16 40

13 00

14 30
10 00

9 30

80

80

40

40

40

20

20

20

10

6

1 selected element

By means of the above-mentioned apparatus I have been
able to make the experiment so as to render it visible at a
public lecture. The arrangement presents no other difficul-
ties than those resulting from the narrovi^ness of the beam of
light traversing the electro-magnet. I therefore removed the
diaphragms placed at the extremities of the helices, and was
then able to work with a ray of light of two centimetres in
diameter. This ray proceeded from a lamp of M. Soleil's
construction, placed before one of the reels. It traversed
successively a polarizer formed of one large pile of glasses,
a Faraday's flint-glass, of 48 millimetres, placed between the
two reels and in contact with them, an analyser formed of a
large doubly refracting prism, and then a convergent lens
which projected it upon a screen. One of the two images of
the prism having been extinguished while the current passed
in a certain direction, it was seen to reappear as soon as the
direction of the current was changed; and it was again ex-
tinguished, or rather it was brought to the tint of passage by
a suitable rotation of the analysing prism. But the expe-
riment is much more striking when a doubly rotating Soleil's
plate of quartz is placed behind the polarizer. The lens then
projects upon the screen two images of complementary tints,
the two halves of which, brought at first to an equality of tints
by means of the analyser, change in a contrary direction as
soon as the current is altered. On turning the analyser a
certain extent, we reobtain an uniformity of tint in each image.
This experiment is precisely the same as that of M. Pouillet*.

The direction of the rotation impressed on the plane of po-
larization was correctly recognized by Prof. Faraday, and may
be determined in a simple manner. The rotation has the same
direction as the current which produces magnetization, or rather

* Comptes Rendus des Stances de fAcadeviie des Sciences, vol. xxii. p. 1 35.

M. A. Benin on Circular Magnetic Polarization. 485

to'

it has the same direction as the current which, according to
Ampere, would be established under the influence of an
electro-magnet in a piece of soft iron inserted in the place of
the substance experimented with.

It may not be without interest to connect this general law
with the theory which M. Fresnel has given of circular pola-
rization. After having shown that a ray of light polarized at
right angles can be decomposed into two others polarized
circularly in a contrary direction, and vice versa, it sufficed
for him to suppose that a plate of quartz cut perpendicular to
the axis has the property of allowing to pass, with unequal
velocities, rays polarized circularly in one or other direction,
and all the rotatory properties of the quartz followed from
this simple supposition. Let us admit, in like manner, that
the presence of the electro-magnet, or of a circular current,
which is the same thing, communicates to transparent bodies
the property of allowing the circularly polarized rays to pass
more easily, the luminous molecules of which rotate in the
same direction as the current, and the general law which I
have announced will result from this simple hypothesis.

To be convinced of the accuracy of this law, it suffices to
observe, as I have done, the direction of the rotation for all
positions of the glass, or of the transparent substance in gene-
ral, in relation to the current.

First. If the glass is placed between the two poles of the
electro-magnet f two cases may occur.

Either these two poles are directly opposed to the glass, as is
the case in RuhmkorfF's apparatus: then there is no doubt as
to the direction of the current. For instance, on viewing the
glass from the surface which touches the south pole, we see
that it is subject to a current proceeding from left to right ;
and the rotation observed is, in effect, in this direction ; — it
changes direction with the current. Or the two branches of
the electro-magnet, instead of being in the axis of the glass,
are perpendicular to it, which takes place in the iron horse-
shoe electro- magnet, and then the currents are oblique in re-
lation to the glass, or parallel to its axis. But the effect ought
always to be the same as above; for a piece of soft iron in-
serted in the place of the glass would assume the same poles
in both cases: only the intensity would be much less; and to
increase it, it would be necessary to bring the polar axes
nearer to that of the glass. This effect is produced by M.
Becquerel's terminations.

Secondly. In a straight electro-magtietic helix traversed in
the direction of its axis by a ray of light, the current is in
the same direction the whole length; consequently the rota-

486 M. A. Bertin on Circular Magnetic Polarization.

tion observed ought to be in the same direction, whether the
glass is placed in front or behind ; and if the helix is viewed
from the south pole, a rotation to the left should occur: it
would be to the right if the helix were viewed from the other
pole, or, which comes to the same thing, the current is changed
without altering the position of the eye.

Hence it follows, that in an apparatus formed of two similar
helices, the direction of the rotation should be the same when
the flint-glass is between the two helices or reels as when it is
at the extremities ; so that in the whole length of a file of
reels thus arranged, the direction of the rotation should not
change.

If, then, several glasses are placed in the intervals separa-
ting these reels, the rotations produced by all these glasses
will add to each other, and thus give the means of multiplying
indefinitely the action of a substance, and consequently of
rendering that action visible, however feeble it may be.

All these suppositions are confirmed by experiment, as I
have found by means of two systems of straight reels. In
the one system, consisting of two reels, ^28 centimetres in
length, and inclosing an iron nucleus, 8 centimetres in diameter,
both of them in contact with the flint-glass of Faraday, of
48 millimetres in length, produced a rotation of nine degrees.
The other system, consisting of four reels 10 centimetres in
length, and inclosing a cylinder of 3 centimetres in diameter,
perforated likewise in the direction of the axis ; they are centred
one after the other in a wooden trough*. This file of reels,
including the external ones, presents five intervals in which the
substances submitted to magnetism may be placed. Some
experiments made with this apparatus are here added.

1st. Experiment made mthjive cells containing sulphur et of
carbon 1 centimetre in thickness. Rotations.

With five cells placed in the five intervals ... 8 5

With the two outer cells removed 6 25

With only the centre cell left 2 00

The five cells in contact, with two reels on either"! , „^

side j- 4 00

0 55

2nd. Experiment on water.

With one cell placed between the first and second

reels

With a second placed between the second and', , ,^
third reels / ^ *^

* This apparatus was lent to me by M. Pouillet.

M. A. Berlin on Circular Magnetic Polarization, 487

Rotations.
With a tliird placed between the third and fourth"! ^ '

reels J

With two reels placed on either side of the three\ . ^q

cells J

3rd. Experiment on Jlint-glass.

With a very thick flint-glass, of 55 millimetres,! - _

placed between two reels J

With Faraday's flint-glass, of 48 millimetres . . 6 10
With two flint-glasses placed at two different in-\ ^^ ^^

tervals J

The two flint-glasses in contact, with two reels\ ^ ^^

on either side j

The last experiment of each series clearly shows that the
increase observed in the rotation depends, not on the in-
crease in thickness of the magnetized substance, but on the
distribution of its different layers in the intervals between the
reels. I need not observe that, in employing successively all
these intervals during the experiment, the reels remained
strictly in the same position, and consequently retained their
magnetism intact.

Thirdly. In a helix perpendicular to a polarized ray, as, for
instance, in a vertical reel receiving on its upper base the glass
traversed by the light : if this is rotated around the pole,
placing it successively on all the radii of the reel, a rotation in
the same direction is found on viewing it always from the same
surface ; for instance, from that which is turned towards the
pole; and this rotation is to the right if the pole is southern,
to the left if it Is northern. The rotation again changes di-
rection when the glass is viewed from the opposite surface.

It follows, therefore, that if the glass is viewed placed in two
symmetrical positions in relation to the pole, the position of
the analysing prism being fixed, we shall observe rotations of
contrary directions. Consequently, if employing the horse-
shoe electro-magnet, we look through the flint-glass always
placed on the line of the poles, but successively at the extre-
mities and then in the middle, we should observe, as M.
Pouillet did, rotations of the same direction outside the poles,
but in the middle a rotation in the contrary direction.

These positions, where we observe change of direction in
the rotation, are separated by others where there is no effect ;
the latter are precisely at the poles. But it was seen that in
this case a rotation was perceived on looking in the axis of
the current, or of the magnet supposed to be perforated in
this direction.

488 M. A. Benin on Circular Magiietk Polarization.

Wishing to assure myself that in the electro- magnet of M.
Becquerel, the axis of which is filled up, the rotation on the
pole occurred in the same manner as in the hollow reels where
I had observed it, I endeavoured to receive the ray in the axis
of the reel by the aid of reflexion. For this purpose I fixed
Norremberg's apparatus on the pole. The tinned horizontal
mirror being placed directly on the end face of the electro-
magnet, I received on the oblique glass the light of the clouds.
This light is, as is known, reflected a first time fi'om above
downwards upon this glass ; then a second time on the hori-
zontal glass, which sends it vertically from below upwards to
the analyser ; only, from its being very imperfectly polarized,
it is difficult to determine the plane of polarization, and con-
sequently the rotation which it might experience. But this
becomes easy by placing upon the crystal bearer the double
rotating plate of M. Soleil. The position of the plane of po-
larization is then determined by that of the analysing prism,
which gives the equality of tints in the two halves of the plate.
With this arrangement, let us place upon the horizontal mirror
the flint-glass of Prof. Faraday. As long as the current does
not pass, we observe no change, except that which arises from
a slight unannealed condition of the glass ; but as soon as the
current passes, we see the double rotation plate vary its tints
in an extremely brilliant manner ; and to re-establish the iden-
tity, it will be necessary to turn the analyser ten degrees if the
flint-glass is 18 millimetres in thickness, and twenty-one de-
grees if it is 48. As to the direction of the rotation, it takes
place from right to left when the pole is southern, and from
left to right when it is the contrary.

This method allows of our observing the action of an elec-
tro-magnetic reel parallel to its axis, in another direction than
this axis, and the results thus obtained deserve attention.

Let us picture to ourselves the horizontal section of the
electro-magnet of M. Becquerel. It is composed of two equal
circles corresponding to the two vertical arms, not exactly
touching, but only 1 centimetre apart and 23 centimetres in
diameter. Each of these circles is formed by an interior
circle of 11 centimetres, which is the section of the iron nu-
cleus, surrounded by a copper ring of 6 centimetres in width,
appertaining to the reel properly so called. If the flint-glass is
moved along the line of the centres whilst the electro-magnet
is in action, the following takes place. In the middle, equi-
distant from the centres, the rotation will be null; it will in-
crease until in contact with the iron, where it will be 9 de-
grees; then quite close to it, on the iron nucleus, it will in-
crease suddenly to 21 degrees. It will remain nearly fixed

M. A. Berlin on Circular Magnetic Polarization, 489

throughout the whole extent of this circle except in the centre,
where it will be somewhat less ; then on the outside of this
circle it decreases, but less rapidly than it increased at first,
being 13 degrees at the interior portion of the copper ring,
7 degrees at the circumference, and 3 degrees at 1 centimetre
distance, which corresponds with the initial position of the
centre ; finally, it will still be perceptible at more than 1 de-
cimetre.

During this progress the rotation will not have changed di-
rection; it will always take place from right to left if the pole
is southern, and from left to right if it is northern. These
phaenomena are interesting when compared with the directly
opposite phaenomena, which are observed in the direction of the
line of the poles ; so that with the same position of the flint-
glass the rotation may be right or left, null or very powerful,
according as it is viewed parallel or perpendicular to the cur-
rent. Is it necessary to add, that in all cases the direction of
the rotation is always determined by the general law which I
have stated at the commencement?

As I have just observed, that which strikes us at first,
when Niirremberg's apparatus is employed, is the great in-
tensity of the action observed over the pole. It depends on
two causes; one portion must be attributed to the circum-
stance that the current acts in the very direction of the lumi-
nous ray, instead of being oblique to it; but this especially
depends on the reflexion of the ray, which is thus compelled
to traverse the magnetized substance twice. This double
passage through the quartz would have the effect of causing
its natural rotatory power to disappear, by producing two
equal rotations of contrary direction, because the rotation of
the quartz is independent of the direction in which it is viewed.
This is also an excellent method of proving the circular mag-
netic polarization in quartz, as it is requisite first of all to de-
stroy the atomic polarization in this substance, as remarked
by M. Becquerel. In magnetized flint-glass, on the contrary,
during the double passage of the light through its thickness,
the current acts so as to produce two rotations of the same
direction, and consequently the effect is found to be double.
I convinced myself of this by making two experiments; the
first according to the usual method, by looking directly through
the flint-glass, and the second by causing the ray of light to
traverse it twice by means of Norremberg's apparatus. The
rotation was always twice as great in the second case as in the
first. This influence of the reflexion on the intensity of the
magneto-rotatory power had already been discovered in a
different manner by Prof. Faraday*.

* Phil. Mag. S. 3. vol. xxix, p. 153.

490 M. A. Berlin ofi Circular Magnetic Polarization*

The change of rotation with the direction in which it is
observed establishes between the magnetized flint-glass and
quartz a difference which is rendered still more perceptible by
the experiments just cited. However, that is nearly the only
difference. The dispersion of the planes of polarization for the
different colours is the same in the two bodies. I have proved
it in the following manner.

With the flint-glass placed between the two poles of the
electro-magnet under the most favourable conditions to pro-
duce a great rotation (29 degrees), I counterbalanced this ro-
tation by the contrary effect of a plate of quartz of the requi-
site thickness, which is easily obtained with M. Soleil's com-
pensator. The system was then perfectly neutral, and would
remain so in every position of the analyser, if the quartz and
magnetized flint-glass acted in the same manner on the light,
which is, in effect, what I have observed in all the different
kinds of flint-glass I have experimented with.

Let us now examine the various circumstances which cause
the magnitude of the rotation to differ.

The nature of the bodies should rank first. The differ-
ences are considerable in the various kinds of glasses ; they
are less perceptible in liquids, and indeed, according to some
experimenters, all solutions have the same rotatory power.
Thus Prof. Faraday (in 2185 of his memoir) considers as
probable that, in aqueous solutions, the [ruling] rotative
matter is the water, and not the other substance. But this
opinion will no longer be entertained, when we observe in the
first place that the most energetic liquids are precisely those
which are anhydrous; and secondly, that among the dissolved
bodies there are some which increase the rotatory power of
the water, and others that diminish it. Moreover, on increasing
the proportion of water in one and the same solution, the
rotatory power is seen gradually to approach to that of pure
water, — a conclusive proof of the influence of the substance in
solution. Alcoholic solutions lead to the same result,

I will here enumerate a few of the numerous experiments
made on this subject. The concentration represented by 1 is
that of the most saturated solution ; i represents the degree of
concentration of the same solution diluted with water, and so on.

I . Rotations produced by some anhydrous liquids.

Rotation of
Name of liquid. Thickness,

Centim.
Bichloride of tin .... 1
Sulphuret of carbon ... 1
Sulphuret of carbon ... 8
ProtochJoride of phosphorus 1

Rotations.

water.

o /

7 30

o /

2 20

7 00

2 20

14- 5

4 30

5 00

2 20

M. A. Berlin on Circular Magnetic Polarization,

4-91

2. Rotations jfroduced hy some aqueous solutions.
Name of liquid. Concentration. Thickness. Rotations.

Chloride of calcium .
Chloride of calcium .
Chloride of calcium .
Chloride of calcium .

Water

Chloride of magnesium
Chloride of magnesium
Chloride of magnesium

"Water

Chloride of zinc . .

Water

Chloride of strontium

Water

Nitrate of ammonia .

Water

Sulphate of iron . .
Water . . . • .

Chloride of magnesium . ...

Chloride of strontium

Alcohol of 36° Beaume . ...

Distilled water

The rotatory power of sulphuret of carbon is remarkable ;
it is three times greater than that of water, and only twice less
than that of the flint-glass of Faraday. This liquid is conse-
quently valuable, because it may be substituted for the majo-
rity of the rare glasses required for this class of experiments.
In the same substance the rotation varies in intensity with the
thickness ; but the law which regulates this variation has been
differently expressed by the various experimenters who have
studied this subject. Some have stated that the rotation was
proportional to the thickness ; others that it was independent
of it; and others, again, that it increased with the thickness
up to a certain limit, from which it diminished, and was finally
reduced to zero. It is easily seen how much is true and how
much is erroneous in these assertions.

In the first place, it is evident that if we consider the action
of a single pole on a substance of indefinite length, this action,
variable with the distance, will decrease from the first layer
to the second, from the second to the third, to a certain di-
stance, beyond which it will be null; so that the more distant
layers will no longer be acted upon by the magnetism. The
effects upon all the successive layers being added together, it

Centim,

o /

13

6 20

• ••

4 55

• ••

4. 40

• ••

4 00

.••

3 40

• ••

6 5

•*•

5 30

»•«

4 5

• ••

3 30

8

10 00

• ••

4 30

• ••

5 30

• ••

4 15

13

8 45

*..

4 55

• ••

4 20

...

6 00

lie solutions.

13

3 20

...

3 50

»..

3 00

• •«

4 15

4-92 M. A. Berlin on Circular Magnetic Polarization.

will be seen that if we submit to the influence of a single pole
increasing thicknesses of the same substance, the rotation will
increase with the thickness up to a certain point, beyond
which it will remain constant, an increase of thickness only
producing layers which are no longer influenced.

It is likewise evident, that if the substance is submitted to
the contact action of two poles of constant equal force, the
effects will only be doubled, and that in consequence the law
will still be the same.

But on interposing fresh strata between the poles, it becomes
requisite to separate them more and more, which somewhat
diminishes their intensity by decreasing the influence which
they exert one on the other. Three cases may then occur ;
either this diminution of intensity will compensate the eff*ect
produced by the increase of thickness, or it will have a more
feeble influence, or lastly, it will preponderate.

In the first case, the rotation will be independent of the
thickness ; in the second, it will increase with the thickness
up to a certain point, beyond which it will remain constant ;
and lastly, on the third hypothesis, it will attain a maximum,
beyond which it will decrease, hut ^without returning to zero^
the two poles always producing effects which conjoin ; so that
the extent of the rotations will be twice the effect produced by
a single pole.

It will be seen from the following table that the two first
cases may occur with M. Becquerel's electro-magnet, where
the variation of the poles of the keepers must be considerable.
Substances experimented with. Thickness. Rotations.

mm. o ,

Flint-glass of Faraday ... 18*3 18 20

Flint-glass of Faraday . . . 48*3 25 5

Very thick flint-glass .... 55*1 22 30

Very thick flint-glass . . . . IIO'S 23 30

Distilled water 10*0 2 00

Distilled water 20'0 3 30

Distilled water 30*0 4 20

Distilled water 80-0 4 30

Distilled water 130'0 5 00

Distilled water 155*0 5 00

If instead of always bringing the poles in contact with the
magnetized substance, they are left at the same distance, by
placing between them a gradually increasing number of strata,
the rotation will be seen to increase in a continuous manner
until the thickness is equal to the distance of the poles. Again,
if these poles are sufficiently removed from the various strata
of the substance, so that the variations of their distance do not
produce perceptible variations in their rotations, the action

M. A. Berlin ofi Circular Magnetic Polarizatio7i. 493

will be equal upon all, and the rotation observed will be pro-
portional to the thickness of the substance. This, in fact, is
the law discovered by Prof. Faraday, on employing horse-
shoe electro-magnets not furnished with keepers.

The law of the variations with the thickness is evidently
connected with that of the variations with the distance ; but
this is not better known than the first. It is therefore to
the simultaneous examination of these two laws that my at-
tention was necessarily directed, and I made it as soon as I
had RuhmkorfPs great apparatus at my disposal.

Law of the thickness and of the distafice.

The action of the two reels or coils of the apparatus being
no more than the sum of the rotations produced by each of
them, I had at first, in order to simplify the problem, to study
the action of a single coil upon a substance of known thick-
ness, placed upon the axis at a fixed distance.

Action of a single pole. — One of the coils being removed, I
placed the flint-glass upon which I wished to experiment in con-
tact with the remaining coil, then I removed it a certain distance,
which I valued by the passage of its support over a divided
scale. Now, on increasiyig the distance of the fiint-gla%s from
the coil in arithmetical -progression^ the rotations of the plane of
polarizatio7i decrease in geometrical progression. In proof of
this I will enumerate only three series of experiments in which
I varied the distances ; at first 1 millimetre, then 5 millimetres,
and lastly 10 millimetres. The relations of the successive
rotations are —

In the first case . . 0'97587=r^
In the second case . 0*88504 = r^,
In the third case . . 0-78233 = r*'^.

1. ExperimeJits isoith the flint-glass of Mr. Faraday, thickness
38 "9 millims.

Distance from

the flint-glass to

the coil.

X.

Rotation
observed.

y-

Relation of the
rotations.

y

Calculated

rotation.

y'i=y. 0-97587.

Difference.

y{-y'-

0
1

11 12
11 00

0 (

0 00
10 56

-4

""o'982l'

2

10 26

0-9470

10 44

+19

3

10 7

0-9712

9 57

lio

4

9 50

0-9719

9 51

+ 1

5

9 30

0-9661

9 35

+ 5

6

9 20

0-9824

9 16

- 4

7

8 47

0-9417

9 4

-17

8

8 35

0-9772

8 34

- 1

9

8 20

0-9709

8 22

+ 2

10

7 55

0-9508

8 6

+ 11

0

9 50

494 M, A. Berlin on Circular Mametic Polarization.

2. Experiments mth the Jlint-glass of Mr. Faraday, thickness
38-9 millims.

. Distance from

the flint-glass to

the coil.

X.

Rotation
observed.

y-

Relation of the
rotations.

y

Calculated

rotation.

J/j'=j/. 0-88504.

DiflFerence.

y{-y'-

0
5

12 30
11 10

6 00

11 4

- 6

0-8934

10

9 35

0-8582

9 54

+19

15

8 30

0-8870

8 30

20

7 25

0-8726

8 31

+ 6

25

6 35

0-8876

6 33

- 2

30

5 45

0-8735

5 50

+ 5

35

5 5

0-8840

5 5

40

4 35

0-9016

4 31

- 4

45

4 00

0-8728

4 4

+ 4

50

3 35

0-8957

3 32

- 3

Experiments mth the Jlint-glass of Matthiessen, thickness
44 millimetres.

Distance from

the flint- glass to

the coil.

0
10
20
30

40

Rotation
observed.

7 40
6 20
5 00
3 40
2 50

Relation of the
rotations.

y'

0-8261
0-7895
0-7333
0-7727

Calculated

rotation.

2/j'=z/. 0-7823.3.

0 00
6 1
4 56
3 53
2 53

Difference.

yi'-y'-

-19
- 4

+14
+ 3

This law may be represented by a very simple formula.
On expressing by A the rotation produced by the flint-glass
in contact with the coil, if Ar is the rotation produced by the
same flint-glass at the distance of 1 millimetre, the action of
the coil at a distance x millimetres will be y — Ai-^.

This formula proving true for all thicknesses, it must be
concluded that it represents the elementary action of a pole
upon any stratum whatsoever; for instance, upon a stratum
of 1 millimetre. It may consequently lead us to the law con-
necting the rotation with thickness, if in every instance each
of the different sections of a substance receives an impression
as if it were a single one. To convince myself of this, I placed
two flint-glasses in contact between the two poles in certain
positions, and observed the rotations produced by the two
flint-glasses collectively, and by each of them singly, in the
position which it first occupied.

The following experiments show that the first rotation is
always the sum of the two others.

M, A. Bertin on Circular Magnetic Polarization. 495

Flint-glass in juxtaposition.

Rotations produced.

Difference be-
tween thethird
and the sxun
of the two
first.

By the flint-
glasses sepa-
rately.

By the sum of
the two flint-
glasses.

Flint-glass of Faraday, of 18-3

do. 38-9
do. 38-9
do, 48-3
do. 38-9
Flint-glass of M. Matthiessen of 44-0

Flint-glass of Faraday 38-9

Common flint-glass of 43-5

Experiment with a single pole.

Flint-glass of Faraday, of 18-3

do. 38-9

§101
17 5/
12 121
14 12/
12 321

11 20/

12 151
7 5t

6 351

7 10/

0 /

25 10

26 10
24 10
19 32

12 55

- 5
-14

+18
+12

+ 10

Thus the action of a pole upon any section whatever of a
substance depends solely on the distance of this section from
the pole, and according to a known law. If, then, in a thick-
ness of e millimetres, we consider e sections of 1 millimetre,
and represent by c the rotation which each of these sections
would produce if it were in contact with the pole, the rotation
produced on contact by the thickness e will be the sum of the
terms of a geometrical progression, the first term of which is
c, the cause r, and the number of the terms e; that is to say,
we shall have

1— r«
A^c- ,

whence

J/

-<\^}

This formula represents the general action of a single pole.
It may be proved by comparing the rotations observed at the
same distance a? by two thicknesses e and e' of the same flint-
glass; for if y and ?/' represent the two rotations observed, it
is evident that we ought to have

y _ l-r^

and we are able to compare this rotation with that given by
experiment. This comparison confirms the accuracy of the
formula, as will be seen by the following table :—

496 M. A. Bertin on Circular Magnetic Polarization,

Relation

Calculated

Calcu-

Di-

Thick-

Rota-

of the

relation.

lated

Differ-

Name of the flint-glass.

stance.

ness.

tion.

rotations.

1— re'

rota-

ence.

X.

e.

y-

?1.

y

l-J-e

tion.

yx-y-

Flint-glass of Mr. Faraday...

0

48-3

9 55

1951

1-916

9 51

- 4

18-3

5 5

1

1

5 9

+ 4

Flint-glass of Mr. Faraday . . .

0

18-3

4 47

1

1

4 54

- 7

.. .

48-3

8 50

1-847

1-916

9 23

+33

38-9

8 10

1-704

1-697

8 19

+ 9

■ ••

57-2

10 30

2-195

2-082

10 12

-18

87-2

11 50

2-474

2-438

11 57

+ 7

Flint-glass of Mr. Faraday...

13-3

18-3

3 25

1

1

3 19

- 6

48-3

6 10

1-823

1-916

6 16

+ 6

...

38-9

6 00

1-756

1-697

5 36

-24

...

57-2

7 20

2146

2-082

6 52

-28

87-2

8 10

2-390

2-438

8 24

+ 14

Flint-glass of M. Matthiessen

0

440

7 57

2-695

2-374

7 40

-17

13-3

2 57

1

1

3 14

+17

Flint-glass of M. Matthiessen

0

440

7 0

2-540

2-374

6 51

- 9

13-3

2 45

1

1

2 54

+ 9

Common ilint-glass

6

43-3
14-5

4 25
2 0

2-210

1

2-190
1

4 24
2 1

- 1

+ 1

Action of the two poles of the apparatus. — The formula
^ = Ar*, which represents the action of a single coil, gives us
also that of two electro-magnetic coils facing each other with
the poles of opposite names, as in the case of our apparatus.
If, in fact, these two coils are at a distance d, the flint-glass of
e thickness, placed at a distance x from the first, will be di-
stant d—e—x from the second; and as the two actions add to
each other, the total rotation will be

■ = c(^~){r--^7^-'-').

Even the form of this expression shows that if we only vary
the distance x in taking three consecutive rotations z, z\ »",
observed in the same Jlint-glass placed successively at the distances
.r, ^ + a, ^ + 2a, the sum of the two extreme rotations will be to
the intermediaiy rotation iti a fixed relation equal to ?•« + ;■-«;
that is to say, that

2 + 3''

:/•« + ;—«.

This conclusion is confirmed by experiment, as is seen in the
following tables : —

M. A. Berlin on Circular Magnetic Polarization. 497

1 . Eicper intent with sulphuret of carbon,
e=4ri d-11.

Distance.

X,

Rotation.

Z'

Relation.

Calculated

rotation.

z' -«+«"
' 206 ■

Difference.

z',-z'.

5
15
25
35
45
55
65

75

6 00

5 00
4 25
4 10
4 5
4 20
4 50

6 30

2*08
2-08
2-02
2-07
2-06
203

6 (M)
5 3
4 27
4 7
4 7
4 19
4 46

6

+3
+2
-3

+2
-1
-4

Mean 206

2. Experiment xoith the Jlint-glass of Mr. Faraday,
^ = 48-3 r/= 125.

0

10
20
30
40
50
60
50
40
30
20
10
0

9 40
8 25
7 35
6 45
6 25

6 30

7 30

6 25

e'ss

8 50

2'04
202
206
2-06
214

9 40
8 20
7 36
6 47
6 26

6 45

7 15
6 39
6 25
6 28

6 55

7 38

8 39

'o

- 3

+ 1
+ 2
+ 1
+ 15

0

0

-11

Mean 2-06

The rotations compared being here more considerable, I
have deduced the value of r from the equation

ri"+r-

;2-06,

whence

r=0'97587.

The numbers compared with the experiments in the pre-
ceding tables have all been calculated with this value of r.

The general formula also furnishes us with another series
of proofs. If we vary merely the thickness of the flint-glass
by placing it in contact with one of the coils, we shall obtain
as the relation of two rotations z and 2', produced by two
thicknesses e and ef,

z ~ Kl—r")

'1 +

.1+;^

FhiL Mag. S. 3. No. 232. Suppl. Vol. 34.

2K

498 M. A. Berlin on Circular Magnetic Polarization.

In the following table is shown the comparison of the re-
sults calculated in this manner with those furnished by expe-
riment.

Name of flint-glass.

d.

Thick-
ness,
e.

Rota-
tion.

Relation.

z'
z

Calculated
relation.

Calcu-
lated
rotation.

Differ-
ence
z^—x.

Faraday's flint-glass ...

48-3

48-3

22 12

2-537

2-587

22 19

+ 7

18-3

8 45

1

1

8 38

- 7

do.

'48'3

48-3

21 45

2-534

2-587

21 57

+ 12

18-3

8 35

1

1

8 23

-12

do.

57-2

57-2

25 10

3-073

3014

25 19

+ 9

38-9

17 5

2-086

2-002

16 49

-16

18-3

8 10

1

1

8 24

+14

do.

730

48-3

12 45

2-390

2-350

12 40

- 5

18-3

5 20

1

1

5 25

+ 5

do.

77-0

48-3

12 45

2-250

2-314

12 52

+ 7

18-3

5 40

1

1

5 33

- 7

do.

87"2

87-2

26 10

2147

2-197

26 8

- 2

48-3

14 12

1164

1-204

14 20

+ 8

38-9

12 12

1

1

11 56

-16

do.

iib-3

48-3

11 20

2-261

2113

11 10

-10

18-3

5 5

1

1

5 15

+ 10

Matthiessen's flint-glass

'440

440

17 30

3-365

3-226

17 18

-12

13-3

5 12

1

1

5 24

+ 12

do.

Ts's

440

16 20

3-322

3165

16 8

-12

13-3

4 55

1

1

5 7

+ 12

do.

770

440

10 10

2-652

2-836

10 21

+11

13-3

3 50

1

1

3 39

-11

Common flint-glass

Ts's

43-3

10 25

2-841

2-869

10 28

+ 3

14-5

3 40

1

1

3 37

- 3

do.

730

43-3

6 10

2-400

2-620

6 20

+ 10

14-5

2 35

1

1

2 25

-10

do.

lid's

43-3

5 20

2-667

2-385

5 13

- 7

14-5

2 0

1

1

2 7

+ 7

This comparison is the last verification which can be made
of our formula. We might, it is true, vary d, and then, as
a?=0, we have

z = A{l+r^-%
or

A A ,

s—A= —r^.
r"

We conclude that the quantity z— A ought to decrease in
geometrical progression as the distance of the poles increases
in arithmetical progression. But experiment no longer con-
firms this conclusion, which is owing to the coefficient A no
longer being constant, but varying with the distance of the
poles ; for these, by reacting one upon the other, change the
intensity of their magnetism. If this reaction did not take
place, the action of the two poles in contact with the flint-glass

M. A. Bertin on Circular Magnetic Polarization. 499

would be twice the action of a single one, whilst it is much
stronger. In one experiment, for example, it was 28° lO' in
the first case, and only 12° 30' when one of the coils was re-
moved.

To sum up, we may state that the rotation produced by the
two coils of our apparatus is represented by the formula

—ci^-^Yr^ + r^-"-"),

which gives the action of a single coil in making d= co .

In this formula r appears neither to depend on the intensity
of the magnetism nor on the nature of the substance. As to
c, it depends on both ; but it remains constant in all the ex-
periments compared, because they were always made at very
short intervals, and moreover upon the same substance, and
with the same distance between the poles.

It would undoubtedly be curious to ascertain why c varies
with the intensity of the magnetism ; but it may already be
said that the law is the same for all substances ; so that the
relations of the rotations produced by these substances do not
depend on the force of the magnetism, as may be seen by the
following experiment made with M. Becquerel's electro-
magnet.

Flint-glass of Mr. Faraday, Sulphuret of carbon, Relation between
of 18'3 millims. of 10 millims. the rotations.

7 42 3 18 0-43

13 48 6 00 0-43

19 00 8 18 0-43

Three other experiments made with M. RuhmkorfTs ap-
paratus gave —

For Faraday's flint-glass
For Matthiessen's flint-glass
For common flint-glass . .

And in these three series the rotation of the flint-glass of
M. Matthiessen remained nearly equal to 0*8, and that of the
common flint-glass equal to 0*5 of the rotation produced by
the flint-glass of Prof. Faraday.

For this reason I propose to call c the coefficient of magnetic
polarization. The value is calculated by comparing two ro-
tations observed at short intervals upon two substances placed
under certain circumstances, but always between two poles at
the same distance, that is to say, by deducing the value of c

2 K 2

27 30

16 25

13 40

21 40

13 40

10 30

13 45

8 50

6 45

500 M. A. Berlin on Circular Magnetic Polarization.

from the equations which give y or z. We shall have, for
example, for a:=0,

^' — ^ ^~^^
c ~ y 1— r®'*
or

d _^ l—r" 1 + r^~^
7 ~T' 1— r*'* 1+r^-*"

The following table has been formed in this manner : it
contains the coefficients of magnetic polarization of the differ-
ent substances experimented with, compared with the flint-
glass of Prof. Faraday.

Faraday's flint-glass 1*00

Guinant's flint-glass 0*87

Matthiessen's flint-glass 0-83

Very thick flint-glass (from the Conservatoire) 0*55

Common flint-glass 0*53

Bichloride of tin 0*77

Sulphuret of carbon 0*74'

Protochloride of phosphorus 0*51

Chloride of zinc dissolved 0'55

Chloride of calcium dissolved 0*4-5

Water 0*25

Ordinary alcohol of 36° Beaume 0*18

^ther 0-15

I ought not to conclude without remarking, that all the pre-
ceding experiments merely relate to the action of electro-
magnets upon exterior substances. When the flint-glass ex-
perimented with, instead of being outside the electro-magnetic
coil was placed in the interior, I did not observe the rotation.
I have noticed a very faint one when the second coil was
brought near to the first; but this action was very consider-
ably less than that which this second coil would have produced
upon the same flint-glass placed at the same distance exter-
nally. These negative experiments are not opposed to those
of Prof. Faraday, for they were not made under the same cir-
cumstances.

If Prof. Faraday has observed a rotation in flint-glasses
placed in helices, still it was very faint; again, the inter-
position of iron nuclei only increased it when they were longer
than the helix; and lastly, this interposition diminished, on
the contrary, the rotation when the interior iron cylinder,
having the same length as the helix, had at the same time a
convenient thickness (§ 2209 of his memoir). Thus in a
helix of 673 millimetres in length, of 121 millimetres in ex-

Mr. G. G. Stokes on the Theory of Sound. 501

ternal diameter, and of 63 millimetres internal diameter, the
insertion of an iron nucleus of 9 millimetres in thickness di-
minished the rotation of the substances placed in the interior.
In my experiments I have never made use of coils enclosing
such thin iron cylinders ; the thickness of these cylinders was
not less than 25 millimetres ; their length was moreover always
equal to that of the external helix.

LXXII. On the Theory of Sound. By G. G. Stokes, M.A.,
Fellow of Pembroke College^ Cambridge.

To the Editors of the Philosophical Magazine and Journal.

Gentlemen,
A S I see no advantage likely to result from further discus-
■^^ sion of the question of the possibility of the existence of
spherical waves of sound, it is not my intention to continue
the controversy. I have, as 1 conceive, shown that the " con-
tradiction " arrived at by Professor Challis has no real exist-
ence ; and I am quite content to leave the question as it now
stands to the judgement of mathematicians.

I feel it, however, to be but justice to myself to notice one
sentence in Professor Challis's last paper; for if your readers
take their views of the controversy from this sentence, they
must think I have strange notions of reasoning.

Professor Challis, in speaking of my last three papers, ob-
serves, " In the first attempt he produced an argument which
took for granted the very point in dispute ; in the next he
denied, without giving any reason, what was altogether unde-
niable; in the third attempt he admits what he before denied,
and denies, again without assigning a reason, what in the
second attempt he admitted." In the first of these papers it
was not until I had, as I conceived, overthrown Professor
Challis's a priori demonstration of the impossibility of sphe-
rical waves by pointing out that it rested on a tacit assumption
(Phil. Mag., vol. xxxiv. p. 54), that I proceeded to inquire
whether there was any foundation for this assumption in the
received equations of motion. Properly speaking, the mere
pointing out of the omission in Professor Challis's train of
reasoning was my answer to his ai'gument. As to my second
paper, what Professor Challis regards as unde?iiable I regard
as untrue. In this paper 1 admitted the possibility (as a par-
ticular case of possible motion) of a solitary wave of conden-
sation, and I continue to admit it; in my next I denied that
in this instance of motion the velocity of the fluid can be con-
fined to the wave of condensation, except when a special con-

502 Mr. R. Phillips on the Magnetism of Steam,

dition is fulfilled, and I continue to deny it. Hence Professor
Challis is mistaken in supposing that I admitted what before
I denied, and denied what before I admitted.

It is not my intention to attack Professor Challis's new
views respecting the theoretical velocity of sound ; because if
Professor Challis and I cannot agree on what to my own
mind seems so plain a matter as the theory of spherical waves,
I see little chance of our agreeing on the subject I have men-
tioned.

I am, Gentlemen,

Your obedient Servant,
Pembroke College, Cambridge, G. G. StokeS.

June 6, 1849.

LXXIII. On the Magnetism of Steam.
By Reuben Phillips, Esq.*

1 . rp^HE following investigation has resulted from an at-
-I- tempt made with a view to the better understanding
of the relation between electric and magnetic forces, by ascer-
taining whether the only form of the electric current, the na-
ture of which, principally from the researches of Dr. Faraday,
is very completely comprehended, possesses the usual magnetic
properties. In this I was baffled (but the way is now open) by
an unexpected phsenomenon, the nature of which it became of
primary consequence to develope, and which forms the subject
of the present paper.

2. A little wooden stick was laid across the mouth of a
small Bohemian beaker ; the stick was placed parallel with
the bottom of the glass, and held in its position with sealing-
wax : the beaker was 3*5 inches high. A common sewing-
needle No. 7, and another No. 8, were magnetized and stuck
through a slip of thin card, the north end of one magnet being
opposed to the south of the other, the needles being two inches
apart. This partially astatic arrangement was suspended from
the stick by a single fibre of silk, and the length of the fibre
between the points of suspension was one inch. The needles
made one vibration in about two seconds. I found if the
needles were much more astatic than this, they were very
subject to slow irregular variations of position, which appeared
to proceed from a twisting of the silk arising from changes in
its hygrometric or calorific condition. The mouth of the
glass was closed with a card cover furnished with a rim.

3. The beaker was now placed on the stage of a microscope

* Communicated by the Author.

Mr. R. Phillips on the Magnetism of Steam. 503

and fastened to the triangular bar by means of cork and thread,
and a hole about an inch across was cut in the card cover,
the centre of the hole being over the point of the uppermost
needle ; the arm for carrying the optical tube was now brought
over the hole, and then the optical tube, fitted with an object-
glass, was screwed into its place ; the arm came down nearly
close upon the card cover, and the tube of the object-glass
passed through the cover and stood over the point of the upper
needle. The fine adjustment was used to focus the instru-
ment; and to bring the point of the needle to its right position
under the object-glass, I employed the motion of the arm
which carried the optical tube, and also that imparted to
the whole instrument by laying hold on the extremity of one
of the prongs of the tripod stand and making it describe a
small portion of a circle on the table. The point of the needle
was viewed by means of a pencil of light thrown from the
mirror through the bottom of the beaker. The magnifying
power used was about 450 diameters.

4. A micrometer eye-piece was employed, the scale of which
formed an angle of about 25° with the edge of the needle to
be observed. The optical power of this micrometer corre-
sponded to the third, or deepest eye-piece, with which micro-
scopes are generally furnished ; because that for such experi-
ments a shallow object-glass is to be preferred.

5. This galvanoscope was sheltered by a rectangular zinc
plate, '1 inch thick, bent somewhat into the shape of the smaller
segment of the convex surface of a right cylinder, made by a
plane parallel to, and at a considerable distance from, its axis.
The length of a line drawn perpendicularly from one straight
side to the other was 10 inches, and the maximum length of a
perpendicular drawn to this line and extending to the nearest
point of the zinc was 2*8 inches; in consequence of this curve,
the zinc plate could easily be made to stand on end, in which
position it was 18*5 inches high. This shield was employed
throughout all the following experiments on steam.

6. In order to obtain accurate results, I found it necessary
to avoid moving any mass of iron near the galvanoscope, and
also to keep moderately still, although the hands and arms
could be freely used without affecting the galvanoscope.
During these experiments I sometimes observed a very singular
effect produced on the magnetic needles: when the steam was
turned on, the needle would begin to move across the field of
view with a peculiar slow motion ; and when the steam was
shut off, it as slowly or more slowly returned to its former
position, one of such vibrations occupying half a minute or
more. This I at length found was produced by the steam

$0^ Mr. R. Phillips on the Magnetism of Steam.

heating the zinc shield ; and thus this motion could be pro-
duced or avoided at pleasure, and was, I think, sufficiently
accounted for.

7. The steam was obtained from a small hydro-electric
machine; and the various apparatus for effecting the discharge
were screwed into the condenser at the place made to receive
the Armstrong's jet. The condenser was always dry, except
where I have noted the contrary, and the steam was discharged
horizontally towards the north.

8. A galvanic current was sent through a wire in the neigh-
bourhood of the galvanoscope ; the wire lay parallel with the
needles in the path taken by the steam (9.), and acted about
equally on both ; the wire of course lay about north and south,
and the electricity passed along it from north to south, sup-
posing the current to pass through the conducting wire of a
voltaic circle from the platinum to the zinc. When the circuit
was completed, the needle moved to one side, A, of the field
of view from the opposite side C. Throughout these experi-
ments the motion of the same end of the same needle was
always recorded, and the galvanoscope always stood to the
east of the current of steam.

9. To a brass jet, 1*8 inch long, was fastened a piece of glass
tube, 1 1'Sinches long and /^ inch diameter inside,and thejunc-
ture was made tight, or nearly so, with caoutchouc ; the aper-
ture at the end of the brass jet which projected from the con-
denser (7.) was circular, and yg inch in diameter. The nearest
point of the convex surface of the zinc shield was about j^
inch from the glass tube; the stage of the microscope in this
and the following experiments came nearly close to the shield.
The fibre which suspended the needles was about If inch
from the nearest part of a plane drawn through the end of the
brass jet, and making a right angle with its bore; the steam
was used in this experiment at about 35 lbs. on the inch.
Things being so arranged, I found when the steam was turned
on that the needle immediately began to move towards C ; and
by alternately checking the steam and letting it off", a consi-
derable swing of the needles was produced; and by reversing
the times of letting off" the steam, the swing of the needles
could be again reduced. I had ascertained by pi'evious trials,
that turning the cock of the boiler without letting off" the steam
produced no effect on the needles. The experiment was made
by screwing a stop-cock in the place of the above-mentioned
brass jet; and then the cock of the boiler could be worked
without letting off* the steam, and without affecting the galva-
noscope.

10. The galvanoscope was raised a few inches, so that the

Mr. R. Phillips o?i the Magnetism of Steam. 505

steam might act principally on the lower side of the undermost
needle, instead ot acting equally on both as before, everything
else being as in the former experiment. When the steam was
turned on, the needle began to move towards A ; and by
alternately shutting oft' the steam and letting it issue at the cor-
responding positions of the needle, a vibration through full
half of the micrometer was obtained; and then by making the
blasts of steam synchronical with the opposite vibrations of
the needle, the motion was checked.

11. The glass tube was now taken away, the galvanoscope
lowered as in (9.), and the shield and galvanoscope moved
horizontally about f inch in a perpendicular direction to the
path of the steam, which was used at about 35 pounds on the
inch. Operating as before, I could with this jet of steam
more easily obtain the swing, the motion being towards C
when the steam was turned on. Water being placed in the
Armstrong's condenser, produced no alteration in the magnetic
effects of the jet of steam.

12. The effect of this jet (11.) was much greater than that
of the current of electricity of an Armstrong's jet under the
most favourable circumstances. The comparison was made
in the following manner: — 1 found that an Armstrong's jet
could discharge more steam in a given time than the brass
jet (11.), also that the electricity produced by the Armstrong's
jet could deflect the needle of a galvanometer of the ordinary
construction 3 or 4 degrees. I then found that a small voltaic
arrangement capable of deflecting the needles of the galvano-
meter 4°, and having a conducting wire lying in the path of
the steam, acted far less on the needles of the galvanoscope
than the blast of steam from the brass jet (11.). Also when
an Armstrong's jet was substituted for the brass jet, and water
placed in the condenser, I could perceive no difference in the
swing, whether the steam passed by the galvanoscope in a
highly electrified condition, or whether the electricity was, in
a great measure, collected by means of a number of fine points
almost as soon as it left the wooden channel. The points
were supplied by two small concentric loops of wire-gauze,
placed edgeways in the steam at a distance in different expe-
riments of from ^ to ^ inch from the end of the Armstrong's
jet. The galvanoscope was as before (11.), except that it was
placed about | inch further from the jet, but at about the
same distance from the path of the steam.

13. There is a singular variation which I have sometimes
observed in the magnetic effect produced by the steam issuing
from an Armstrong's jet; namely, that the action of the steam
on the galvanoscope is much stronger when one of the needles

$06 Mr. R. Phillips oji the Magnetism of Steam.

is in the same horizontal plane as the central line of the path
of the steam, than when the galvanoscope is placed so that
the steam may pass equally near to both needles. I have
occasionally observed the swing to be five or six times greater
in one position than in the other ; but sometimes the swing is
the same in either position. I have not perceived a similar
effect with any other jet.

14-. A pewter tube 6 feet long, having a bore 2% inch dia-
meter, was coiled up after the fashion of the wire of a galva-
nometer: it made six convolutions. One end of the tube
was jambed on the brass jet (9.), and the coil stood horizon-
tally like the coil of a galvanometer; the zinc shield was
brought nearly close to the coil, and the height of the galva-
noscope was adjusted so that the coil might act on both the
upper and lower sides of the lower needle ; the steam entered
on the upper side of the coil in the same direction as before.
By this arrangement I succeeded in producing a mai'ked
swing with steam of a very low pressure, I think below 5 lbs.
on the inch ; but I did not exactly ascertain the amount of
this low pressure, because the safety-valve of the boiler is not
graduated below 4C lbs. As the pressure rose, the power
exerted on the needles became greater ; and at 40 lbs. on the
inch, a few puffs of steam caused the needle to move through
the whole length of the micrometer; the motion, when the
steam was turned on, being towards C.

15. Instead of allowing the steam to pass continuously
through the tube during the whole of each alternate vibration,
it was shut off before the vibration was completed. The swing
was now much less, showing that it was not the first gust of
steam which alone moved the needles; and this agreed with
the visible motion of the needle, which manifestly increased
in velocity after it began to move.

16. Water being placed in the condenser of the hydro-
electric machine did not sensibly alter the force exerted on
the needles by the coil.

1 7. The coil was now attached to the boiler by means of a
piece of brass, the steam-way of which was so large that the coil
only might be looked upon as opposing the exit of the steam.
Various pressures were tried, as the boiler became heated, up
to 40 lbs. per inch. A lead pipe was, in this experiment and
frequently afterwards, placed at a short distance from the coil
to catch the steam and convey it up a chimney, whereas in
former experiments it had freely escaped into the apartment ;
everything else remained as before (14.). The needles began
to be affected at a very low pressure; and as the pressure in-
creased, the swing became certain and steady, the needle

Mr. R. Phillips on the Magnetism of Steam. 507

moving towards C when the steam was turned on. The in-
tensity of the swing-producing force was, 1 think, at its maxi-
mum at 10-15 lbs. per inch; as the pressure rose, the swing
was obtained with more difficulty, and at 40 lbs. on the inch
no certain swing could be produced. On repeating this ex-
periment at a subsequent period with steam at 40 lbs. on the
mch, a feeble swing was obtained ; but in this latter case the
coil, instead of being bright, had become covered with a de-
posit of carbonate of lime.

18. The galvanoscope was lowered until the upper needle
was in the same horizontal plane as the undermost side of the
upper portion of the coil, and the lower needle was in a hori-
zontal plane which came to about ^ inch below the lower sur-
face of the lower half of the coil. When the steam was turned
on, the needle made a start towards A ; and by shutting off
the steam during the next vibration, and repeating these ope-
rations two or three times, a considerable swing was obtained ;
but on attempting to increase this swing by continuing these
intermittent operations, the swing rapidly diminished. The
order in which the jet of steam and its cessation took place
was now inverted, which soon produced a very powerful swing,
the needle moving towards C when the steam was turned on.
The steam was used at from 30 to 40 lbs. on the inch. These
rather irregular motions (17, 18.) are probably connected with
those of the Armstrong's jet (13.).

19. The coil was moved through an angle of 180°, the
angular motion being performed parallel to a plane forming a
right angle to the path of the steam, and the galvanoscope
was adjusted so that the interior surface of the coil might act
on the lower needle; by this arrangement the direction of the
steam as regards the needle was reversed. Many different
pressures were observed from a pound or two on the inch
to 40 lbs. As soon as any distinct swing was produced, it
was occasioned by the needle moving towards A when the
steam was turned on; as the pressure rose, this swing in-
creased until the steam was, I think, about 25 lbs. on the
inch, after which the increased pressure only occasioned a
somewhat diminished swing.

20. The apparatus (19.) was now rather differently disposed,
the shield and galvanoscope were moved horizontally about
f inch, the direction of the motion being perpendicular to the
path of the steam ; also a piece of ani ron gun-barrel 7 inches
long and open at both ends, was fixed to a support, so that
the gun-barrel might easily be thrust into, or removed from,
the coil without bearing upon it; the axis of the gun-barrel
when placed in the coil formed a right angle to the path of
the steam, and was horizontal, and its direction lay not far

508 Mr. R. Phillips on the MagJieiism of Steam.

from east and west; also one end came to \ inch of the shield
opposite to the lower needle of the galvanoscope. This piece
of a gun-barrel had been mnde red-hot and slowly cooled,
and its magnetism when in the above position was nearly
= 0; the diameter of its external surface at that end which
was placed in the coil, in this and the following experiments,
was about one inch, and this end was made of iron about —^
inch thick, the other end being thinner. There was a distance
of about ^ inch between the nearest upper or lower part of
the iron and the respective inside surfaces of the coil.

21. The steam being at 40 lbs. per inch, and the gun- barrel
in the coil, five puffs of steam, each puff acting during every
alternate vibration of the needles, produced a swing of about
25° of the micrometer ; the gun-barrel was now removed, when
five puffs of steam, acting as before, produced a vibration of
only 10°. This was done many times, and always with the
same result. I think better results would have been obtained
at a lower pressure than 40 lbs. ; for on examining the vibra-
tion at various pressures, the iron being in the coil, I found
that a considerable swing was produced almost as soon as the
water began to boil at the atmospheric pressure; and shortly
afterwards, as the pressure rose, the vibrations became very
strong, much stronger than at 40 lbs. The point of the needle
moved towards A when the steam was turned on.

22. During these last experiments, I ascertained that the
first puff of steam which passed through the coil when it was
cold produced a much greater effect on the galvanoscope than
any immediately succeeding puff. This was guarded against
by letting off the first puff, then checking the motion of the
needles by some inverse puffs, and then proceeding to make
the vibration which was to be compared (21.).

23. I next endeavoured to find the cause of the strong
action of the first puff.

24. The jet and pewter coil (14.), instead of being affixed
to the boiler, were attached to a copper box, which inclosed
about ninety cubic inches, and the shield and galvanoscope
adjusted as before (14.) ; air was now pumped into the box
until the pressure rose to about 40 lbs. on the inch, and then
discharged through the jet as the steam had been. The
needles of the galvanoscope were quite unaffected; conse-
quently, air of about the same temperature as the surrounding
atmosphere cannot act on a magnet like steam.

25. It now appeared very possible that the increased action
produced by the iron arose from its cooling powers ; and also
that by further cooling the coil, a more intense action would
be obtained.

26. The apparatus (17.) had the coil partly immersed in

Mr. R. Phillips on the Magnetism of Steam. 509

water, by bringing a copper pan filled with water under it, so
that the water might cover the lower portion of the coil ; the
upper portion of the coil was kept moist by having had some
loosely spun cotton twisted about it, the ends of which dipped
into the water. The gun-barrel was placed in the coil much
as before (:20.), and was supported without touching the coil ;
consequently any little alteration, produced by the heat of the
steam, in the position of the coil, could not move the iron; the
gun-barrel lay entirely under water. The steam was used at
35 to 40 lbs. on the inch, as at that pressure a difference of a
few pounds did not much affect the galvanoscope. A few puffs
of steam made the water about the coil to boil ; after which,
when the circumstances of the experiment appeared to be very
steady, I observed that one puff of steam could move the edge
of the needle across the field of view. The gun-barrel was
removed, and now two or three puffs of steam, acting through
each alternate vibration, could only produce a swing half
across the micrometer. A solid brass rod, •? inch diameter
and about 6 inches long, being now laid in the place of the
gun-barrel, produced no alteration in the swing of the mag-
netic needles ; the brass rod being removed, and the gun-
barrel replaced, the motion of the magnets at once became as
strong as before. It now follows that the action of the iron
on the galvanoscope is independent of any little change of tem-
perature it may produce. The swing was towards C when
the steam was turned on.

27. In these experiments I did not perceive any difference
between the first and following puffs of steam ; and the swing
was, I think, at least as great at 40 lbs. as at any lower pres-
sure.

28. The brass jet (9.) was screwed into the end of the coil
from which the steam escaped (26.), which apparatus in other
respects remained as before; this alteration caused an in-
creased pressure in the coil, and the condensation was con-
sequently very rapid ; so much so, that what escaped from the
brass jet looked more like water than steam. The swing,
when the gun-barrel was either in or out of the coil, remained
just as before (26.), the pressure in the boiler being about
40 lbs. on the inch.

29. A pewter tube, 9 feet long and \ inch internal diameter,
was made into a dense cylindrical coil 4 inches long, the dia-
meter of the external surface of which was 2*5 inches, and the
diameter of the interior 1'25 inch. This coil was attached to
the boiler by means of a short brass connecting piece, the
steam-way of which was cylindrical, and /^ inch diameter ;
the coil was supported horizontally, and pointed about east

510 Mr. R. Phillips on the Magnetism of Steam.

and west. The shield stood ^ inch at its nearest part from
the end of the coil, and the galvanoscope was adjusted so that
the lower needle was opposite to the coil ; the steam circulated
in the same direction as before (14, 26.). At 40 lbs. four puffs
of steam produced a swing about one-fourth across the micro-
meter, and the swing was not very much stronger at any lower
pressure. The copper pan, which had previously been placed
in position under the coil, was now filled with water, by which
the coil was about half-covered. The first motions of the
needle were not very powerful ; but alter three or four puffs,
which heated the water about the coil, one puff of steam would
move the needle from A quite out at C; the swing was in the
same direction when there was no water in the pan.

30. The cylindrical pewter coil had a turn or two opened
out, in order to give sufficient length between the brass con-
necting piece and the coil to allow the latter to be immersed
in the water of the copper pan. The gun-barrel was placed
in the coil, and was supported by wires attached to the copper
pan, and so prevented from touching the coil ; the gun-barrel
projected about ^ inch beyond the end of the coil on the side
nearest to the galvanoscope. The optical tube of the galva-
noscope was removed, and the hole in the cover was closed
with a piece of glass. The steam was used at about 40 lbs.
on the inch. After the water in the copper pan had been
heated, I could easily by successive puffs of steam produce a
vibration through an arc of 20 or 30 degrees.

31. Putting together these gun-barrel experiments, and
that with condensed air (24.), I come to the conclusion that a
difference of temperature is necessary to produce these peculiar
magnetic effects ; which accounts for the greater force of the
first blast of steam (22.), and for the superior force of a jet
(11.). Also the similarity which exists between the magnetic
effects of the steam current, and the magnetic effects of the
voltaic current, both as regards magnets and soft iron, renders
it nearly or quite certain that this force of the steam is mag-
netism.

32. Now a jet of steam, even when mixed with much water,
is an excellent non-conductorof electricity; for when discharged
from an Armstrong's jet, it is seen that electricity even of a
very high intensity cannot pass through it, and a jet of dry
steam must be at least as good a non-conductor ; hence the jet
of steam (11.) cannot be travelled by any feeble current of
electricity, a thermo-electric current for instance. Again, this
magnetic effect of the steam must be independent of any elec-
tricity carried forward by the steam, as when discharged by an
Armstrong's jet ; for the greatest amount of electricity which

Mr. R, Phillips 07i the Magnetism of Steam. 511

I have been able to obtain from an Armstong's jet was found
sufficient to charge a Leyden jar of about 34-0 square inches,
both sides taken together, twenty-eight times in a minute ; the
spark being ^ inch long, and the steam at 40 lbs. on the inch :
now whether the steam passed by the galvanoscope in this
highly excited condition, or nearly divested of its electricity,
the effect on the galvanoscope was the same, and always many
times greater than what could be produced by this largest
quantity of frictional electricity that could be obtained (12.).
Besides, the frictional electricity of steam increases much as
the pressure rises from 10 to 40 lbs. (Armstrong, Matteucci).
But I have not generally perceived that the swing of the gal-
vanoscope thus increases with the pressure, but rather the
reverse. Also Dr. Faraday has shown, that dry pure steam
(11.) cannot develope frictional electricity.

33. From these considerations, I conclude that no continu-
ous electric current passes through or by means of the steam
jet; however, many very small currents may circulate in it
For instance, if we may suppose that a particle of steam when
brought into contact with a particle of colder water deve-
lopes a momentary current of electricity in a direction bearing
some fixed relation to those particles, and then if a continual
succession of such particles ensues, the majority of which are
similarly placed, we should have something answering to an
ordinary electric current, and not very unlike those currents
imagined in Ampere's theory of magnetism. This notion
accounts for the change in the direction of the magnetism
produced by changing the direction of the steam, the effect of
the difference of temperature, and the manifest want of equi-
valency between the steam power expended and the magnetic
force obtained. But it may be well to bear in mind, that
perhaps magnetism may ultimately come to be regarded as
some function of ordinary matter and the aether. I can only
look upon the experiments (12.) as going to show that mag-
netism is not always bound up with current electricity ; I
should probably have made a decisive experiment on this
point, but that the steam apparatus at my disposal was not
sufficiently powerful.

34. It is possible instances may be found on board steamers
in which the compasses are much disturbed by the steam.
Clouds, too, in the act of formation and passing rapidly over
a magnet may somewhat affect it.

7 Prospect Place, Ball's Pond Road.

[ -512 ]

LXXIV. On sojne Points relating to the Theory of Fluid
MotioTi. % the Rev. J. Challis, M.A., F.R.S., F.R.A.S.,
Plumian Professor of Astronomy and Experimental Philo-
sophy in the University of Cambridge^.

IN a memoir on certain questions in the theory of the motion
of fluids, published (1847) by Professor P. Tardy of Flo-
rence, for a copy of which I am indebted to the author, re-
ference is made to a communication contained in torn, xxiii. of
the Comptes Rendus of the Academy of Sciences of Paris, in
which several of my investigations on fluid motion are brought
under review by M. J. Bertrand. Professor Tardy states at
the same time, that he had himself previously made remarks
on one of the points on which I am supposed to be in error.
The citation of Professor Tardy first made me aware that
M. Bertrand had taken notice of my labours. On turning to
the article in the Comptes Rendus, I perceived that the im-
portant errors [erreiirs graves) attributed to me were partly
due to misconception of my reasoning, which, I am ready to
admit, may not have been developed with sufficient clearness;
and partly to the circumstance, not unusual in the history of
science, that new truths appear to be errors so long as the
errors they replace are supposed to be truths. It will suffice
for the present to advert to one point of primary importance.
I have repeatedly contended that, to complete the analytical
theory of hydrodynamics, a new general equation is absolutely
required, M. Bertrand first calls in question the principles
on which this equation is established, and then contends that
an equation which I derived from a combination of the new
equation with that which is usually called the equation of con-
tinuity, is identical with a particular case of this latter equa-
tion. The following investigations will supply answers to
these objections.

1. I propose first to exhibit the principles on which the
new equation rests, and to deduce it accordingly. If the equa-
tion i>{x,y, 2f t) = 0 express any given relation between the
coordinates a:, y, z and the time t, any other relation be-
tween the same quantities may be expressed by the equation
^{x + da;, y + ^y, z + dz, t + df) = 0, the increments dx, 8j/, dz
being in general functions of the coordinates and the time.
Supposing the increments to be indefinitely small, we obtain

* Communicated by the Author.

On some Poinds relating to the Theory of Fluid Motion. 513

If w, Vi w be the resolved parts of the velocity of a given par-
ticle of fluid in motion, and we suppose that

la!=uU ly-i^t Iz—'w'^t,
the above equation becomes

d"!/ d^ d^ d^ ^ H

dt dx dy dz

The signification of this equation depends entirely on the na-
ture of the curve surface defined by the equation •^{x^y,z^t)^=^Qi.
If, for instance, this be the surface of a fixed or moveable
boundary with which the fluid is in contact, the equation
affirms that the same particle remains in contact with the
boundary in successive instants. If the function vj/ be the
general expression for the pressure/?, then since /> = 0 is the
equation of the free boundary, the above equation would ex-
press in this instance the condition that a given particle is
situated on the free boundary in successive instants. Let now

(«4')= r dj;+ —dy+ —dz.

AAA

Then, as is known, 4/=:0 is the equation of a surface cutting at
right angles the directions of the motions of the particles

through which it passes. The factor — is applied for the

A

sake of generality, because it may be assumed that such a sur-
face always exists, and consequently that the right-hand side
of the above equality is an exact differential, although it can-
not be affirmed that uda,- + vdy + wdz is always an exact differ-
ential. Assuming, therefore, the integrability of

—■dx+ — dy+ —dz,

AAA

it follows that

«-i"('-) -^S"(^-) »-i"(«-)-

Hence, substituting in the foregoing equation,

d^> ^^(d^'^ dV .d:\,^\

This is the new equation which it was proposed to obtain.
The course of the investigation shows that this equation ex-
presses the condition that the directions of the motion in a
given element are in successive instants normals to surfaces of
continued curvature. The fulfilment of this condition ensures
the continuity of the motion ; and the above equation may
Phil. Mag. S. 3. No. 232. Suppl. Vol. 34. 2 L

51*" Prof. Challis oti some Points relating to

consequently be called the equation of continuity, while the
equation usually so named may with more propriety be called
the equation of constancy of mass, with reference to the prin-
ciple on which it is based.

To the equations (1.), (2.), (3.), and (4.) are to be added
the two following : -

the fluid being supposed to be acted upon by no impressed
forces. When the relation between the pressure p and density
p is given, these six equations serve to determine the six un-
known quantities, \|/, X, ?/, t;, w and p.

The equation (5.) is equivalent to the following :

dp dp dx dp dy dp dz /du ,dv dwX _
dt dx' dt dy' dt ds' dt '^Xdx dy dz/~

Hence, i^u^ + v^ + ^'^-V^ and ds=\dt,

dp TT dp (du dv d'm\

But by what is proved in the Cambridge Philosophical Trans-
actions (vol. vii. part 3. p. 385, 386), where, however, it is
proper to remark, the use of equation (4.) is not absolutely
necessary, we have

du dv dw _ dV ,y{^ -4- M
dx dy dz ~ ds \r r^J*

r and r^ being the principal radii of curvature at the point
xyz, of the surface which cuts the directions of motion at
right angles. Hence, by substitution,

This equation has also been derived from elementary consi-
derations in the memoir above cited (p. 387), By whatever
process it be obtained, it involves the principle expressed ana-
lytically by the equation (4.), viz. that the directions of motion
are normals to surfaces of continued curvature. It cannot,
therefore, be identical with any equation which does not in-
volve the same principle, or does not contain exiMcitly the
radii of curvature r and r^. An application of the equation

the Theoi^ of Fluid Motion. 515

will illustrate this remark. Suppose the fluid, ta be i^com<p
pressible: then )H| aioai ihiw yiini b^riuin oa vlltuii-ij rujiii .; .

ds \r r^ )~ '

Hence, since ds=dr=dr^i we have by integrating,

V=

_ <pW

rr

1 *

This expression for the velocity is general, having been ob-
tained prior to the consideration of any particular case of mo-
tion. It establishes the general law, that the quantity of fluid
which passes in a given small time a given small element of a
surface of displacement, being proportional to Yrr^, is given
for a given value oC (p{f), and consequently that the moment-
ary trajectory of the surfaces of displacement to which a single
disturbance gives rise is a straight line.

If the motion take place in space of two dimensions, we
have

r

I formerly obtained this result for the case in which udx +
vdy is an exact differential, by a process which Professor
Tardy and M. Bertrand object to, and which I do not now
insist upon, because the above reasoning is inclusive of the
particular case, and the result is obtained in a more direct
manner.

To apply the above general value of V to a given instance:
suppose a perfectly smooth sphere to move in any manner in
an incompressible fluid of unlimited extent, its centre remain-
ing on a given straight line. The general value of V applies
to the velocity impressed on the fluid by the surface of the
sphere. If V' be the velocity of its centre, then the motion
impressed at any point of the surface the radius to which
makes an angle Q with the direction of motion, is V^ cos Q.
Hence, R being the radius of the sphere,

VI cos 9 = -^, or (p{t) = R2 V^ cos Q.

Consequently, as there is no other arbitrary quantity to satisfy,

R2

v= % VI cose,

where r^ is substituted for rr^, because as one surface of dt^-'
placement is spherical, all are spherical, their momentary tra-
jectory being rectilinear* This result does not agree with that

2 L2

SIB Prof. Challis on some Points relating to

given by the received hydrodynamical equations employed in
the usual manner, — first, because those equations do not in-
clude the principle of continuity expressed by equation (4.);
and secondly, because in treating this problem, udx + vdy + "wdz
has vi'ithout reason been assumed to be an exact differential.
2. I proceed next to trace the consequences of introducing
into the general equations the condition that udx + vdy + 'wdz
is an exact differential. Let {d<^)=udx + vdy + 'wdz. Then
{d(^) =0 is the differential of the equation of a surface cutting
at right angles the directions of motion. Hence the value of

— + — in equation (7.) may be expressed by means of the

partial differential coefficients of (p. This expression being
substituted in (7.), and p being eliminated by means of (5.) and
(6.), the result is identical with the known equation (n) in the
Mecajiique Analytique (Part 2. Section XII. p. 344). I have
indicated this process for the purpose of remarking, that it
does not thence follow that equations (7.) and (w) are identical,
or that the former is a particular case of the latter. Both
equations are equally general. The essential difference be-
tween them is, that (7.) involves an expression of the condition
that the directions of motion are normals to surfaces of con-
tinued curvature ; whereas («) involves no expression of this
condition, being usually obtained by simply supposing (p to be
a certain function, the partial differential coefficients of which
with respect to x, y and z are respectively ^<, v and w. It is
not possible to pass from (w) to (7.) without introducing this
condition of continuity, and equation (7.) consequently sig-
nifies something more than equation (w).

We have now to consider the change which equation (4.)
undergoes by supposing udx + vdy + 'wdz to be an exact dif-
ferential. It might at first sight be supposed that for this
purpose it is sufficient to put unity for A. That this would
be a false step is clear from the consideration, that there would
then be six equations and but five imknown quantities, without
any reasons for concluding that the equations would be con-
sistent with each other. In fact, it would be found on trial that
on this supposition they are inconsistent. The only legitimate
process is to trace the consequence of supposing wrfj^ + u^^ + Mxia
to be an exact differential, by reasoning generally according
to the rules of analysis.

Between the functions (p and ^ we have the relation, that
(<i\J/) = 0 and {d(p) = 0 are both differential equations of the same
curve surface. But (df) =0 being the differential equation of
a curve surface, it is clear that (^d, F(<p))=0, or F'((p)(t/(p) =0,

the Theory of Fluid Motion, 517

is a differential equation of the same surface. Consequently

(#) = F'(^)(^^).
But

(#)=iW;

therefore

A=

Also by integration,
Hence

Consequently by substituting in (4.), and having regard to the
value of A, we obtain

df.dfdfdf:^

dt "^ dx' ■** ^y "*■ dz'' "^ F(f) ""• • • ^^'^

Thus we have an equation involving the same variables as
the general equation (w) already referred to, and yet not iden-
tical with it. The interpretation of this analytical circum-
stance is, that the function <^ has not an arbitrary, but apar-
ticidar form ; and it is a matter of importance to ascertain
what that form is. The following investigation may perhaps
suffice for this purpose, but is not the most general that might
be adopted. For the sake of avoiding Umg processes I shall
confine the reasoning to the first order of approximation.

The equation (8.) may be put under the form,

dt ^ ds^ + F(^) -"' .... {9,}
where ds is the increment of a line drawn in the direction of
the motion, so tha
ving V^, we have

and by integration,

F(?)+xW=e(5).
Hence

the motion, so that -^ =V. Now neglecting the term invol-

^fS Prof. Ghallis ofi some Points relating to

But we have already found, without using equation (9.), that

F(f)+X(0=^.
Hence

A comparison between these two values of f gives

The above value of <p must satisfy the following linear equation:

0 = a<^ + ^^- + ^V ^ (10)

It may therefore be assumed that Q{s) and )({t) are linear quan-
tities, and accordingly that

6(5) =5, and ^{t)=ct.

Also since 4/=0 is the equation of a curve surface, we have
in general,

4'=5; + c' + g'(.r, y,0.
Hence

s=:z + (/ + q[x,i/,t).

It does not appear possible to satisfy this last condition
unless the line of motion be rectilinear ; that is, unless the mo-
tion be along an axis, which may be supposed to coincide with
the axis oi' z. The function q, being of given form depending
on the equation of the surface, cannot express the value of s.
This function must therefore disappear on making a; = 0 and
«/=0; and we thus obtain

S=:Z+c\

and

<p=/{z + c'-ct).

This value applies strictly to motion along the axis of ^. Be-
fore proceeding to substitute in equation (10.), it is necessary
to express the value of <p for points immediately contiguous to
the axis. For this purpose suppose

f=J{2 + c' + g{x,i/, t)-ct).

Then substituting the letters^ and q for the functions them-
selves, we shall have

d2f_,dy ^_ d^_ „(d_q__Y.f,^q

dt ~'' ' dy'^' dz" '-f ' dt^ "-^ \dt ) ^-^ df" '

the Theory of Fluid Moiion, . , MW

Also supposing that 'H '.^ iff . "^iviul -»\m tiM

grzh flo? + /3j/ +gx^ -{- hxy + ^y + &c.,
it follows that

^-ojr ^-0 and ^-0

The coefficient h may be made to disappear by changing the
direction of co-ordinates, and g and k must be supposed to be
independent of the time. Hence by substituting in (10.) we
obtain for determining the form ofy the following equation ;

Putting m for the coefficient of/", and v for the quantity of
which /is a function, the integral of this equation becomes

/=Ae'"''+B.

The form of/ is thus ascertained. Since q is of arbitrary
value, we may multiply it by v'—l, and the value of /will
then become

/= Ae~(^*^'''"y^^e'"(*+^-'''^^~^+ B.

The equation (10.) being linear, may be satisfied by the sum
of this value and that which results by changing the sign of
\/ — \. So that putting x — Q and y — % and suppressing the
constant B, the final result is,

<^—\i.QQ%m{z-\-d^ct),
By putting — for w, the resulting value of c is the following,

,=,(,+ te±S.^)*.

This value agrees with what I have previously obtained in a
different manner. The reasoning in the present method is
more direct, in consequence of the use that has been made of
the new hydrodynamical equation. I shall conclude this investi-
gation with the remark, that the results arrived at are wholly
incompatible with those deduced from the supposition of
plane-waves, although the reasoning proceeded on the general
hypothesis that udx + vdy-{-isodz was an exact differential, and
ought not, if that supposition were allowable, to have led to

520 Appendix to Mr. Drach's Paper on Epicyclic Curves.

any contradiction. I infer that the supposition of plane-waves
is not allowable.

I have been induced to make the last remark from having
seen it asserted by the Astronomer Royal in the Number of
the Philosophical Magazine for June, that I have pointed out
a difficulty in the interpretation of an equation applying to the
caseof plane- waves. Mr. Stokes asserted the same thing before;
and I then disclaimed, as I now disclaim, having pointed out
any difficulty. The equation is a very simple one, and easily in-
terpreted. A few steps of plain deduction conducts to a result
incompatible with fluid motion. It follows in due course that
the supposition of plane-waves cannot be made. This infer-
ence is in perfect accordance with the argument contained in
this communication, which I think Mr. Airy may find to be
worthy of some consideration. Any other inference would
have presented a real difficulty.

Cambridge Observatory,
June 22, 1849.

LXXV. Appejidix to Mr. Drach's Paper on Epicyclic
Curves in the last Jime Number.

A NOTHER example, with series expanded.

5

n^-^-.-.p + q^^l, g=2.

Let Q = Q'.a6. The equation is

1^7 _ 7^2^ (2a?)5H- 14r4(2^)3-7r6(2^) = a7(Q/2_2) + 7«^^Q'
+ A<2a^b^-\-S5a'^t^Q!-^S5aH\Q^-2)-^2la^b\Q!^-S(^)

+ 7«i6(Q'4_4Q^2 _^ 2) + 67(Q/5__5Q^3 ^. 5Q^).

When a = 6, the second member

= a7{(Q' + 2)5^3(Q' + 2)4=rio«-J0-3r8a-8}
= r'^[r^a~^ — 3ra~^"^f
agreeably to Case 2.

The following errors have to be corrected in the last
Number.

Ex. o ^ Q ^ Q«^

n^^Sfor ^ read -^;

Appendix to Mr. Drach's Paper on Epicyclic Curves, 521
the general or 5th term of ( -r j 's bracketed factor is

1.2. 3. ..25

. {q^^i){q^j^8^ X)(^q^j-s^2) . . (g-2; + 1) '
1.2.3..(i-5)

and of (Q-^ai)?-=y-' is

(bY. (P-l)(P-2)..(P-25)
-\a) ' 1.2.3..(2s+l)

(g->25-l).(g~7-5-g).(y-i-5-3)..(g-2;')
1.2.3.J-S

+ as stated, so that both series end always with §' — 2;+ 1 or
q—%j'i two consecutive terms being in the ratio of

b^ P— 25 P — 25— 1 <7— 25 — 2 j—s

1 to -^ X

a2'^25+l 25 + 2 j-25 q-j—s—V

and

i^ p_2s—i P-25— 2 q-2s—B J—s
^°a2 X 25 + 2 25 + 3 ' q — 2s—l' q—j—s — 2

respectively.

Mr. Perigal's finite syphonoids, strongly resembling a di-
stiller's * worm,' are expressed by

Hence

2 cosj9.g'<p=SAJ cos 5-^= -j = 2 cos 5' . j»<p

=2B/(cosj9f ="^j.

The nodal or lemnoid curves of a finite number of knots
are comprised in

x=^acosq<^i y=bs\np<p.
Hence

2cos;).(2gf)==2A,.(cos2g^=-^-lj ==2cosg(2pf)
= SB^(cos2;,^ = l-^y.

$22 Appendix to Mr. Drach's Paper on Epicyclic CurveSi^O

Retrogressive Syphonoid. x=^a cos (p, y=zh cos 2(p. .'. (^•\-y)=.'ibx^.

^^

^

-=.=.

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Oblate Lemnoid. a

'=0 COS (p

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-h sin 2<p.

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— ii

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=J

Prolate Lemnoid and retrogressive Syphonoid.
Particular cases (extremes) of the same law of compound circular motion.

On the Distribution of the Superficial Detritti^ dftTitjitps. W^

Oblique Lemnoid and retrogressive Syphonoid.

S. M. Drach.

June 20, 1849.

'\ LXXVI. On the Distribution of the Supeijicial Detritus of the
Alps, as compared with that of Northern Europe. By Sir
Roderick Impey Murchison, F.R.S., V.P.G.S. Sfc;
Mem. Imp. Ac. Sciences of St. Petersburgh, Corresp. Member
of the Academies of Paris, Berlin, Turin, Sfc*

REFERRING to his previous memoir upon the whole structure
of the Alps and the changes which those mountains underwent,
the author calls attention to the fact, that whilst during the forma-
tion of the molasse and nagelfiue a warm climate prevailed, so after
the upheaval of these rocks an entire change took place, as proved
by the uplifted edges of these tertiary accumulations being surmounted
by vast masses of horizontally-stratified alluvia, the forms of whose
materials testify that they were deposited under water. The warm
period, in short, had passed away and the pine had replaced the
palm upon the adjacent lands, before a glacier was formed in the Alps
or a single erratic block was translated.

Though awarding great praise to the labours of Venetz, Charpen-
tier and Agassiz, which have shed much light on glaciers, and par-
ticularly to the work of Forbes for so clearly expounding the laws
which regulate the movement of these bodies; Sir Roderick conceives,
that the physical phsenomena of the Alps and Jura compel the
geologist to restrict the former extension of the Alpine glaciers
within infinitely less bounds than have been assigned to them by
those authors. True old glacier moraines may, he thinks, be always /
distinguished, on the one hand, from the ancient alluvia, and on the
other from tumultuous accumulations of gravel, boulders and far

* Abstract of a Memoir read before the Geological Society May 30, 1 849, \
when His Royal Highness Prince Albert honoured the meeting with his"
presence for the first time as a Fellow of the Society.

524> Sir R. I. Murchison on the Distribution of

transported erratic blocks, as well as from all other subsequent detritus
resulting from various causes which have affected the surface. He
first shows, from the remnants of the old water-worn alluvia which
rise to considerable heights on the sides of the valleys, that in the
earliest period of the formation of the Alpine glaciers, water, whether
salt, brackish or fresh, entered far into the recesses of these moun-
tains, which were then at a considerably lower level, ^. e. not less
than 2500 or 3000 feet below their present altitude.

He next appeals to the existing evidences in the range of Mont
Blanc to show, that as each glacier is formed in a transverse upper
depression, and is separated from its neighbour by an intervening
ridge, so by their movement such glaciers have always protruded
their moraines across the adjacent longitudinal valleys into which
they descended — and were never united to form one grand stream
of ice. It is stated that there are no traces of lateral moraines
on the sides of the main valleys at considerable heights above their
present bottoms, whether on the flank of the great ridge from whence
the glaciers issued or on the opposite side of each longitudinal valley,
which must have been the case if a large mass of glacier ice had ever
descended the general valley. On the contrary, examples of the
transport of moraines and blocks across such longitudinal depressions
are cited from the valley of Chamonix on the one flank and from
the Allee Blanche and Val Ferret on the other flank of the chain of
Mont Blanc. Another proof is seen in the ancient moraine of the
Glacier Neuva, the uppermost of the valley of the Drance ; and a
still stronger case is the great chaotic pile of protogine blocks accu-
mulated on the Plan y Bceuf, 5800 French feet above the sea, which
have evidently been translated right across the present deep valley of
the Drance, from the opposite lofty glacier of Salenon.

Having thus shown that none of the upper longitudinal and flank-
ing valleys around Mont Blanc were ever filled with general ice-
streams, the author has still less difficulty in demonstrating that all
the great trunk or lower valleys of the Arve, the Doire, and the
Rhone, oflfer no vestiges of what he calls a true moraine ; all the
detritus from great heights above their present bottoms exhibiting
either water- worn pebbles or occasional large erratic blocks, more
or less angular, — the latter being for the most part irregularly and
sporadically dispersed. As Venetz and Charpentier have attached
great imjjortance to the original suggestion of an old peasant of the
Upper Vallais,that a great former glacier alone could have carried the
erratic blocks to the sides of the lower valley of the Rhone, so on the
other hand the author relies on the practised eye of his intelligent
Chamonix guide Auguste Balmat, who declares that he has never
recognized the remains of " moraines " in that detritus of the larger
valleys which has been theoretically referred to glacier action. In
descending from the higher Alps into such trunk valleys, Sir Roderick
found many examples of rocks rounded on the side which had been
exposed to the passage of boulders and pebbles, with abrupt faces on
the side removed from the agent of denudation, all of them remind-
ing him forcibly of the storm and lee sides of the Swedish rocks over
which similar water-worn materials have passed.

the Superficial Detritus of the Alps, 525

Seeing, then, that this coarse drift or water- worn detritus is dis-
tributed sometimes on the hard rocks and often on the summits of
the remnants of the old valley alluvia, he believes that the whole df
the phsenomena can be explained by supposing that the Alps, Jura,
and all the surrounding tracts have undergone great and unequal
elevations since the period of the formation of the earliest glaciers —
elevations which, dislodging vast portions of those bodies, floated
away many huge blocks down straits then occupied by water, and
hurled on vast turbid accumulations of boulders, sand and gravel.
To these operations he attributes the purging of the Alpine valleys
of the great mass of their ancient alluvia, and also the conversion of
glacier moraines into shingle and boulders. He denies that the
famous blocks of Monthey opposite Bex, can ever have been a portion
of the left lateral moraine of a glacier which occupied the whole of the
deep valley of the Rhine, — as Charpentier has endeavoured to show ;
and he contends that if such had been the case they would have been
associated with numberless smaller and larger fragments of all the
rocks which form the sides of the valley through which such glaciers
must have passed. They are, however, exclusively composed of the
granite of Mont Blanc ; and must therefore, he thinks, have been
transported by ice rafts, — which, having been forced with great
violence through the gorge of St. Maurice, served to produce many
of the striae which are there so visible on the surface of the limestone*.
Fully admitting that the stones and sand of the moraines of
modem glaciers scratch, groove, and polish rocks. Sir Roderick
Murchison still adheres to the idea he has long entertained from
surveys in Northern Europe, that other agents more or less subaque*
ous, including icebergs and great masses of drift, have produced pre-
cisely similar results. He cites examples in the Alps, where per-
fectly water-worn or rounded gravel being removed, the subjacent
rocks are found to be striated in the directions in which such gravel
has been moved ; and he quotes a case in the gorge of the Tamina,
above the Baths of Pfeffers, where this ancient stiiation, undistin-
guishable from that caused by existing glaciers, has, by a very recent
slide of a heavy mass of gravel from the upper slope of the same
rock, been crossed by fresh scorings and striae, transverse to those
of former date, from which the markings made in the preceding
year only differ in being less deeply engraved. He also adverts to
the choking up of some valleys, particularly of the Vorder Rhein
below Dissentis, by the fracture, in situ, of mountains of limestone,
which constitute masses of enormous thickness, made up of innu-
merable small fragments, all of which have been heaped together
since the dispersion of the erratic blocks ; and he further indicates the
effects of certain great slides or subsidences within the historic sera.

* Mr. Charles Darwin, in a recent letter to the author, adheres to his old
opinions derived from observations in America, and says, " I feel most en-
tirely convinced ih&t floating ice and glaciers produce effects so similar, that
at present there is, in many cases, no means of distinguishing which for-
merly was the agent in scoring and polishing rocks. This difficulty of di-
stinguishing the two actions struck me much in the lower parts of the
Welsh valleys."

526 On the Distribution of the Superficial Detritus of the Alps.

In considering the distribution of the erratic detritus of the Rhone,
Sir Roderick having denied that it can ever have been carried down
the chief valley to the Lake of Geneva in a solid glacier, still more
insists on the incredibility of such a vast body of ice having issued
from that valley, as to have occupied all the low country of the can-
tons Vaud, Friburg, Berne and Soleure, and to have extended its
erratics to the slopes of the Jura, over a region 100 miles in breadth
from north-east to south-west as laid down in the map of Charpen-
tier. He maintains that in the low and undulating region between
the Alps and the Jura, the small debris derived from the former has
everywhere been water- worn, and that there is in no place anything
resembling a true moraine ; and he therefore believes, that the great
granitic blocks of Mont Blanc were translated to the Jura by ice-
floats, when the intermediate country was under water. He further
appeals to the water- worn condition of all the detritus of the high
plateaux around Munich, 1600 and 1700 feet above the sea, to show
that a subaqueous condition of things must be assumed when the
great erratic blocks were carried to their present positions.

Prof. Guyot of Neufchatel has endeavoured to show, that the de-
tritus of the rocks of the right and left sides of the upper valley of
the Rhone have also maintained their original relative positions in
the great extra Alpine depression, and that these relations are proofs,
that nothing but a solid glacier could have arranged the blocks in
such linear directions. But the author meets this objection by
suggesting that there are notable examples to the contrary. He also
refers to the great trainees of similar blocks which preserve linear
directions in Sweden and the low countries south of the Baltic, to
show that as this phaenomenon was certainly there produced by power-
ful streams of water, so may the Alpine detritus have been arranged
by similar agency. In alluding to the drainage of the Isere he further
points to the admission of Prof. Guyot, that nearly all its erratic de-
tritus, both large and small, is rounded and has undergone great at-
trition ; and he quotes a number of cases in which such boulders and
gravel, derived from the central ridges of Mont Blanc, have been
transported across tracts now consisting of lofty ridges of limestone
with very deep intervening valleys ; and therefore he infers that the
whole configuration of these lands has been since much changed, in-
cluding the final excavations of the valleys and the translation of
enormous masses of broken materials into the low countries of
France.

In conclusion it is suggested, that the dispersion of the far-travelled
Alpine blocks is a very ancient phsenomenon in reference to the
historic sera, and must have been coeval with the spread of the
northern or Scandinuvian erratics, which it has been demonstrated
was accomplished chiefly by floating ice, at a time when large
portions of the Continent and of the British Isles were under the
sea. Viewing it therefore as a subaqueous phsenomenon. Sir
Roderick is of opinion that the transport of the Alpine blocks to
the Jura falls strictly within the dominion of the geologist, who
treats of bygone events, and cannot be exclusively reasoned upon by
the meteorologist, who invokes a long series of years of sunless and

Mr. A. Cay ley's Note on the Theory of Permutations. 527

moist summers to account for the production of gigantic glaciers
upon land. This last hypothesis is at variance even with the physical
phaenomena in and around the Alps, whilst it is in entire antagonism
to the much grander and clearly established distribution of erratics
of the North during the glacial period. The effect in each case is
commensurate with the cause. The Scandinavian chain, from
whence the blocks of central Europe radiated, is of many times larger
area than the Alps, and hence its blocks have spread over a much
greater space. All the chief difficulties of the problem vanish when
it is admitted, that enormous changes of the level of the land in
relation to the waters have taken place since the distribution of
large erratics ; the great northern glacial continent having subsided,
and the bottom of the sea further south having been elevated into
dry land, whilst the Alps and Jura, formerly at lower levels, have
been considerably and irregularly raised.

LXXVII. Note on the Theory of Pernmtations.
By A. Cay LEY*.

IT seems worth inquiring whether the distinction made use
• of in the theory of determinants, of the permutations of a
series of things all of them different, into positive and nega-
tive permutations, can be made in the case of a series of things
not all of them different. The ordinary rule is well known,
viz. permutations are considered as positive or negative ac-
cording as they are derived from the primitive arrangement
by an even or an odd number of inversions (/. e. interchanges
of two things) ; and it is obvious that this rule fails when two
or more of the series of things become identical, since in this
case any given permutation can be derived indifferently by
means of an even or an odd number of inversions. To state
the rule in a different form, it will be convenient to enter into
some preliminary explanations. Consider a series of n things,
all of them different, and let abc... be the primitive arrange-
ment; imagine a symbol such as {xyz) [u) {vw)... whereas, y, &c.
are the entire series of n things, and which symbol is to be
considered as furnishing a rule by which a permutation is to
be derived from the primitive arrangement abc... as follows,
viz. the {a^yz) of the symbol denotes that the letters x^y^z in
the primitive arrangement abc... are to be interchanged x intoj/,
y into 2;, z into x. The [u) of the symbol denotes that the
letter u in the primitive arrangement abc... is to remain unal-
tered. The {vw) of the symbol denotes that the letters <r, y in
the primitive arrangement are to be interchanged x into ?/ and
y into Xf and so on. It is easily seen that any permutation

* Communicated by the Author.

528 Mr. A. Cay ley's Note on the Theory of Permutations.

whatever can be derived (and derived in one manner only)
from the primitive arrangement by means of a rule such as is
furnished by the symbol in question*; and moreover that the
number of inversions requisite in order to obtain the permu-
tation by means of the rule in question, is always the smallest
number of inversions by which the permutation can be derived.
Let «, /3... be the number of letters in the components {scyz)^
(w) {vw) &c., K the number of these components. The num-
ber of inversions in question is evidently a— 1 +/3 — 1 +&c.,
or what comes to the same thing, this number is (n — A). It
will be convenient to term this number X the exponent of
irregularity of the permutation, and then {n — X) may be termed
the supplement of the exponent of irregularity. The rule in
the case of a series of things, all of them different, may con-
sequently be stated as follows: " a permutation is positive or
negative according as the supplement of the exponent of irre-
gularity is even or odd." Consider now a series of things,
not all of them different, and suppose that this is derived from
the system of the same number of things abc... all of them
originally different, by supposing for instance « = 6 = &C.,
f=g=:Si,c. A given permutation of the system of things not
all of them different, is of course derivable under the suppo-
sition in question from several different permutations of the
series abc... Considering the supplements of the exponents of
irregularity of these last-mentioned several permutations, we
may consider the given permutation as positive or negative ac-
cording as the least of these numbers is even or odd. Hence we
obtain the rule, " a permutation of a series of things not all of
them different, is positive or negative according as the minimum
supplement of irregularity of the permutation is even or odd,
the system being considered as a particular case of a system
of the same number of things all of them different, and the
given permutation being successively considered as derived
from the different permutations which upon this supposition
reduce themselves to the given permutation." This only
differs from the rule, "a permutation of a series of things, not
all of them different, is positive or negative according as the
minimum number of inversions by which it can be obtained is
even or odd, the system being considered, &c.," inasmuch as
the former enunciation is based upon and indicates a direct
method of determining the minimum number of inversions re-
quisite in order to obtain a given permutation; but the latter
is, in simple cases, of the easiest application. Asa very simple

* See on this subject Cauchy's Memoire sur les Arranfjemens, Sec, Exer-
cices d' Analyse et de Physique Mathermtique, t. iii, p. 151.

Royal Society. 529

example, treated by the former rule, we may consider the per-
mutation 1212 derived from the primitive arrangement 1122.
Considering this primitive arrangement as a particular case
of abcd^ there are four permutations which, on the suppositions
a — h= 1, c=rf = 2, reduce themselves to 1212, viz. acbd^ bead,
adbc, bdac, which are obtained by means of the respective
symbols {a){bc){d)', {abc){d); {a){bdc)\ (abdc), the supplements
of the exponents of irregularity being therefore 1, 2, 2, 3, or the
permutation being negative; in fact it is obviously derivable
by means of an inversion of the two mean terms.

58 Chancery Lane,
June 1849.

LXXVIII. Proceedings of Learned Societies.

ROYAL SOCIETY.

[Continued from p. 469,]

Feb. 22, T^ESCRIPTION of an Infusory Animalcule allied to
1849. -^-' the genus Notommata of Ehrenberg, hitherto unde-
scribed. By John Dalrymple, Esq., F.R.C.S.

The examination of various specimens of the animalcule described
by the author, disclosed the dioecious character of one of the more
highly organized of the rotiferous class of Infusoria, hitherto sup-
posed to be androgenous. This discovery was first made by obser-
ving the difference in the form and development of the embryo
while still enclosed in the ovisac of the parent animal. From the
extreme transparency of this form of rotifer, it is possible to trace
the progressive development of the young from the Graeffian vesicle
in the ovary to the period of mature gestation, when the embryo is
expelled, the whole machinery of whose organs has been perfected
while still within the body of the female.

Thus, although the young one observed in the ovisac, when nearly
ready to be expelled, was in the great majority of instances a mi-
niature portrait of the parent, yet occasionally an embryo was seen
of a different aspect, within whose body a vesicle was discovered
filled with actively moving spermatozoa.

A further investigation of the subject brought clear evidence of
the functions performed by this male, — its copulation with the young
females; but it also displayed the singular fact, that although the
organs of reproduction and locomotion were highly developed, there
was a total absence of those of assimilation ; in fact, that neither
mouth, nor stomach, nor other digestive cavity or glands, were pre-
sent in its curious organization.

In the early part of the paper the author describes the anatomy
of the female, which differs from the family of Notommata of Ehren-
berg, in the absence of intestine and anal orifice, and forcipated or
caudal foot. In every other respect the organization is so similar to
that class, that the author believes the proper place for this animal-
cule to be in a sub-genus of Notommata.

Phil. Mag, S. 3. No. 232. Suppl. Vol. 34. 2 M

&$0 Royal Society*

In relation to physiology, the author submits a new theory of the
mechanism of circulation and respiration in the general group of
Rotifers, a subject which is but obscurely treated of by the great
German observer, who appears to have believed in the existence of
tubular vessels or true vascular system. The author thinks, how-
ever, that these functions are performed in a manner more resem-
bling that of insects, viz. that the blood is contained in the general
cavity of the animal and circulates round the lung, which is here
represented by a contractile vesicle that receives and expels the
water in which the animalcule lives, and so comes to be in interme-
diate relation with the air mixed with the water. The difference
therefore between the aeration of the blood of insects and that of this
rotifer is rather due to the difference of the media they respectively
inhabit, than of design. In both, the blood is contained in a general
cavity and brought in contact with the air, without the intervention
of any true vascular system.

The beautiful transparency of the animal, and the facility with
which the development of the ovum may be traced through all its
stages, induces the author to believe it to be well-suited to the in-
quiries of the embryologist and of those who devote themselves to
the study of the metamorphosis of cell into tissue.

This animalcule has hitherto been discovered only in a few situa-
tions (in Norfolk near Norwich, and in Warwickshire near Coven-
try), but it is believed, from the vei'y general dispersion of Infusoria,
that it may be more extensively met with, especially in the months
of June, July, August and beginning of September.

The author concludes by expressing his belief that re-examina-
tion of the whole order of Rotifera is necessary to determine the
disposition of the sexes, and to assign them their proper situation in
the scale of animated beings.

"On the Integration of Linear Differential Equations." By the Rev.
Brice Bronwin. Communicated by C J. Hargreave, Esq., F.R.S.

The method chiefly employed in this paper, is analogous to one
which the author had previously applied (Camb. Math. Journal,
No. 4) to the integration of such equations in cases where the co-
efficients are integer functions of the independent variable. Here
they are any functions of that variable, it being however understood
that in all integrable cases there must be some relation among these
coefficients. The integration is effected by a general theorem of
the form

where D denotes any function of x, and ot a function of symbols
both of operation and quantity. By means of this theorem, and the
substitution m=ct-j ■ss^. . . zHn v, or some other similar one, the equa-
tion is either reduced to an integrable form, or to an equation of a
lower order ; or, when neither of these objects can be accomplished,
the method may be employed to effect a transformation.

The method applies most readily to equations of the second order ;
but may be applied to those of a higher order, the coefficients be-

Royal Society, 5S1

coming more restricted as the order rises. The iiitegrable cases are
very numerous and vary considerably in form ; and, as each distinct
form requires a variation in the process, they are distributed into
classes. In each class, a few particular examples, derived from the
general cases, are given.

By means of the general theorem, the equation

may be integrated in the most general case, or when the coefficients
are any functions of x, having, however, certain relations between
them.

Several theorems of the form ifnpu^pirn-iu, where p^D+Q,
*»=D'^ + AnD+B„, or similar to it, are given. They are not found
without difficulty ; are much more restricted in their application
than the general theorem ; and lead to but few results ; but they
are deserving of notice on the ground that they may possibly suc-
ceed in a particular case when all other methods fail.

A few general examples of a class of equations, the solution of
which is attended with considerable difficulty, are next given. These
are of the forms,

and others varying a little from them.

The concluding part of the paper is occupied with the transfor-
mation and application of one or two of the general theorems which
have been given by the author in the Cambridge Mathematical
Journal, New Series, vol. iii., from which a few examples, more or
less particular, have been derived.

March 1, 184-9 " Minute Examination of the Organ of Taste

in Man." By Augustus Waller, M.D. Communicated by Richard
Owen, Esq., F.R.S.

The author commences by describing his mode of observation,
which differs from that followed by previous observers. It consists
in removing from the living tongue one of the papillae, and imme-
diately subjecting it to examination. He then proceeds to describe,
— 1st, the epithelium; 2nd, the fungiform papillae; 3rd, the conical
papillae ; and ^th, the inferior surface with its mucous glands, &c.

1. The epithelium is of two kinds ; the flat plates with a central
nucleus, which are mostly found clothing the stem and other regions
of the fungiform papillae ; and the globular cells which compose
most of the external parts of the processes of the conical papillae.

2. The fungiform papillae are found to consist of numerous small
cones seated on a common stem. These secondary cones, already
described by Albinus, are completely hidden by a common invest-
ment of epithelium which fills up the irregular spaces between them.
Each of these cones contains capillary vessels, which, at the apex of
the cone, either form a simple loop or a complex coil which is covered
only by epithelian scales of the most attenuated nature. The author
states that in these capillary vessels the motion of the blood may be
observed for several seconds after the removal from the living body,

2 M2

6S2 Royal Astronomical Society.

and may be excited for a long time by the application of a slight
degree of pressure. By these means he has been enabled to watch
the passage of the red and white globules contained in the blood,
and to detect in the human papillae all the various phenomena in the
transparent membranes of the lower animals. By allowing the blood
to coagulate in the vessels, beautiful examples of injected papillae
may be obtained. The congestion of the vessels is much increased
by compressing the point of the tongue before the removal of the
papillae. The capillaries are connected together at the bases of the
secondary papillae, and arise from a common trunk immersed in the
body of the papilla. The nerves are found to subdivide in the se-
parate cones, in which they ascend to the apex and terminate in
abrupt extremities, as in the frog, toad, &c. In the foetus the fun-
giform papillae are stated to consist of a simple cone without any
secondary papillae.

3. The conical or filiform papillae of man are described to be of
a compound nature, consisting of numerous secondary cones spring-
ing from a common stem. Each of these secondary cones is clothed
with an elongated process which is fitted on the cone like a sheath.

• This process consists of elongated epithelial scales ascending towards
the summit, and resembling in general appearance the feather of an
arrow. At their summit these processes are clothed with an exter-
nal zone of granular matter, which considerably adds to their thick-
ness. This granular matter is often detached after the papilla has
been removed a short time from the tongue. The blood-vessels form
a simple loop at the summit of the papilla, and the nerves are ar-
ranged in a similar manner.

4. The inferior surface is described as very smooth, presenting
numerous follicles abundantly supplied with blood-vessels and nerves.
These follicles are generally of a conical shape and surrounded with
an arch composed of epithelial cells. The nerves may frequently be
detected and followed over the surface of the follicle, but their ex-
tremities are hidden amidst the blood-vessels.

The author has illustrated the paper by several drawings.

ROYAL ASTRONOMICAL SOCIETY.
[Continued from p. 463.]

March 9, 1849. — Substance of the Lecture delivered by the
Astronomer Royal on the large Reflecting Telescopes of the Earl
of Rosse and Mr. Lassell, at the last November Meeting.

The Astronomer Royal gave that evening an account of the large
reflecting telescopes of the Earl of Rosse and Mr. Lassell, which he
had personally examined in the course of the last summer.

Premising that the subject might be considered interesting to the
Society on these two grounds, first, that the reflecting telescope is
exclusively a British instrument in its invention and improvement,
and almost exclusively so in its use ; and secondly, that it had been
almost exclusively the instrument of amateurs — a circumstance which
seemed to prove both the difficulty of constructing it and its great

Royal Astronomical Society. 5SS

excellence when properly constructed — the Astronomer Royal re-
marked that his account would consist in some measure of a state-
ment of the differences between the processes of these two amateur
constructors. These differences, he thought, would be found well-
worthy the attention of all who were engaged in or contemplated
the construction of reflecting telescopes. It is certain that both
systems of methods are successful ; and it may be doubtful how far
the differences are connected with the difference of dimensions of the
telescopes : for in all that follows it must be borne in mind, that
Lord Rosse's largest telescopes are 6 feet in clear aperture and more
than 50 feet in focal length, while Mr. Lassell's are 2 feet in clear
aperture and about 20 feet in focal length ; that the thicknesses of
the specula are nearly in the same proportion as their diameters ;
and hence that the weights of the specula are nearly as 27 to 1 (that
of Lord Rosse's being about four tons, and that of Mr. Lassell's
being about three hundred- weight), a difference which in itself might
be expected to require some difference of construction.

L The first difference is in the constitution of the metal-mixture
used for the speculum.

In Lord Rosse's specula the metal is purely a mixture of tin and
copper, in a proportion understood to be an atomic proportion, the
weight of the copper being something more than double that of the
tin. In the process itself there appears to be strong evidence that
this proportion is truly atomic. When the metals are mixed in the
intended proportions, it is found that the addition of one or two
ounces of either metal to a mass of forty pounds of the mixture pro-
duces a difference so striking, that it is at once recognised by every
person who is employed on the works. The mixed metal, when in
fusion, possesses a remarkable union of penetrating power and of
viscidity. It is so penetrating that Lord Rosse found it impossible
to retain it in the ordinary cast-iron crucibles, which are cast with
the mouth downwards ; and he was compelled to have crucibles
made expressly for this purpose, cast with the mouth upwards. It
is at the same time so viscous, that when, in the casting of the spe-
culum, the liquid metal is poured upon what Lord Rosse calls a " bed
of hoops," that is, a broad base of the mould, formed by pressing
together the flat surfaces of a great number of iron hoops, whose
edges (trimmed into shape by turning and other mechanical opera-
tions) form the base, and through the interstices of which the air can
escape, the metal itself does not in the smallest degree enter into
these interstices. As soon as the cooling metal has acquired some
consistency, it is dragged into the annealing furnace, every part of
which has been brought to a low red heat. Great attention is given
to the form of the floor of this furnace, as the metal is still in such
a state that it receives figure from the furnace-floor. The furnace
is then closed (care being taken to prevent one part from cooling,
by radiation or convection, more rapidly than another), and after
some weeks it is opened and the metal is taken out.

In Mr. Lassell's specula, the metal is made by first mixing copper
and tin in the same proportions as those employed by Lord Rosse,

5S4 Royal Astronomical Society.

and then adding a small quantity of white arsenic. It is well known
that very great importance was attached to the admixture of arsenic
by the principal constructors of reflecting telescopes in the last cen-
tury. Lord Rosse has uniformly stated it as the result of his expe-
rience, that when the copper and tin are properly proportioned,
nothing is gained by adding arsenic ; Mr. Lassell is equally confident
that the brightness of the metal is much improved by it. It would
be idle to express an opinion on this point without comparing speci-
mens of the two kinds of speculum-metal side by side ; and the
Astronomer Royal only undertook to say that both bear a very high
polish. The Astronomer Royal believed that the operation of an-
nealing Mr. Lassell's mirrors was nearly similar to that for Lord
Rosse's, although the trouble and risk of the process are of course
materially diminished by the much smaller dimensions of the spe-
culum.

The next step in Lord Rosse's operations is to turn (by a grinding
process, with emery as the abrading powder) the edge of the mirror,
the mirror being placed with its broad surface horizontal, with its
lower surface and about one-third of its depth immersed in water,
and being turned horizontally by a vertical spindle passing through
a stufling-box in the bottom of the water vessel. The purpose of
this turning is, to make a nearly air-tight fitting for a covering which
is to be applied to the mirror when it is not in use, and with which
is connected a box of quicklime, for the desiccation of the air in
contact with the mirror ; and the object of turning it in water is, to
keep all parts of the speculum in the same temperature, a caution
which is necessary in every part of the operations. This caution is
probably requisite only for specula as large as those of Lord Rosse.
For these it was found by Lord Rosse, that if the iron grinder (to
be mentioned presently) be washed with warm water, and be then
applied to the speculum, the metal almost infallibly cracks. A sur-
face of warm pitch may, however, be applied without producing the
same bad effect.

II. The subject next worthy of attention, rather for the similarity
of the methods of the two constructors than for their dissimilarity,
is the method of mounting the mirrors. And this seems to be a
proper place for mentioning the mounting, because it is indispensable
(at least with Lord Rosse's specula) that the mirror be ground and
polished on the very same supports, applied in the very same manner,
as when the mirror is in use in the telescope.

Tlie special object of Lord Rosse's support is this : supposing the
surface of the speculum to be divided into any number of equal por-
tions, each of a form not particularly elongated in any direction, then
the supports are to be so arranged that every one of these portions
shall necessarily sustain the same upward pressure, acting at its
centre of gravity. At the same time it is necessary, for definiteness
of support, that the ultimate support upon the fixed frame be upon
three points. These objects are thus attained : the surface of the
mirror is divided into twenty-seven equal portions, and these are
grouped into nine groups, of three in each ; to each portion is at-

Royal Astrotwmical Society. SS6

tached, by felt and pitch, a small plate of cast iron ; and in a small
hole sunk in this cast iron, opposite to the centre of gravity, a pro-
jecting pin of a triangular plane lever takes its bearing. This trian-
gular plane lever is merely a triangle, having three points to sustain
pressure, and a small hole for the fulcrum at the centre of gravity of
the three points considered as equal weights ; it is, then, evident
that if that point is made really the fulcrum, the pressures at those
three points are necessarily equal. The same construction being
applied to each of the nine groups, then it is only necessary to sup-
port the nine fulcra in such a way that the pressures upon them
are necessarily equal ; and this is done by grouping three of their
fulcra as points of pressure upon the three points of another and
stronger triangle, whose fulcrum is at the centre of gravity of
these three points considered as equal weights ; and this fulcrum
is one of the points of the fixed frame. It will thus be evident
that every one of the first- mentioned pins sustains exactly one
twenty- seventh part of the whole weight of the mirror, or rather
of the whole pressure of the mirror perpendicular to its surface,
neither less nor more. [When the mirror is in the telescope, it ex-
erts also an edgewise pressure, which in Lord Rosse's construction
was at first sustained by fixed pillars of the fixed frame ; of these
more will be said hereafter.] The fixed frame has small wheels ;
while it is in the grinding-trough, it is so lifted off the wheels that
it takes a firm bearing upon the rotating frame ; when it is to be
carried to the telescope it is lowered to take bearing upon its wheels,
the side of the grinding-trough is taken off, the fixed frame carrying
the speculum is wheeled upon a proper carriage, the carriage con-
veys it to a place very near the telescope, where is a railway at
proper height for receiving the small wheels ; the telescope is placed
vertical, its lower end is opened, and continuation-rails are laid to
it, and the fixed frame is thus wheeled into the telescope, carrying
the mirror ; then by powerful screws the bearing of the fixed frame
is received upon three points, the wheels being entirely lifted off
their bearings. [Since giving this account, the Astronomer Royal
has learnt from Lord Rosse that a speculum which was raised for a
few minutes only from its lever-bearings received in that time a
permanent change of figure.]

In Mr. Lassell's operations the speculum is supported on eighteen
points, the grouping being first made by two and two, with straight
levers, and then the fulcra of the straight levers being by means of
triangular levers supported upon three ultimate points. The Astro-
nomer Royal was not able to say whether the same cautions as to
retaining the speculum at all times upon the same bearings, which
Lord Rosse found necessary, were required for Mr. Lassell's mirrors ;
but it is evident that the difliculties of support of every kind are here
very much less. The edgewise pressure, when in the telescope, is
here supported by a semicircular iron hoop, of which more will be
said hereafter.

III. The next point deserving special comparison is the apparatus
for grinding and polishing.

The apparatus used by Lord Rosse imitates very closely, but with

536 Royal Astrotiomical Society.

that superior degree of regularity which is given by machinery, the
operation of polishing by hand. Proceeding from the quick motions
to the slow ones, the order of movements is as follows : — (I.) By a
steam-engine, a rotatory motion round a vertical spindle is given to
a crank, which by a connecting rod acts upon a sliding rod that
moves the grinder or polisher backwards and forwards. This sHding
rod passes through a fixed guide at the end next the connecting rod,
and through a slowly moving guide [described under (2)] at the other
end ; and thus every stroke of the grinder is very nearly a straight
stroke. The sliding rod is interrupted in its middle, and there its
place is supplied by a hoop which loosely surrounds the grinder :
this construction is necessary, because, on account of the great size
and weight of the grinder, it is necessary that it be supported by
counterpoises in various parts, at the same time that it is necessary
that (in order to prevent the formation of striae in a definite direction
on the grinder) the grinder be left free to revolve : that motion of
revolution is given merely by the friction upon the mirror, whose
rotation will be mentioned under (3), and depends upon the weight
with which the grinder presses the mirror. (2.) A band from the
crank-spindle (1) passes round a wheel which carries a crank, in
which is the forked guide (turning by a spindle in the crank end)
that guides the distant end of the sliding rod ; and thus the strokes
of the grinder do not pass uniformly over the centre of the mirror,
but pass in their direction a certain distance to the right and left.
But as, in the ordinary crank motion, the duration of the strokes «t
the extreme right and left would be too great, the wheel on the
spindle of this grinding- crank is elliptical, the proportion of its axes
being about three to one : its angular motion is therefore unequal ;
and the strokes are thus made to dwell a shorter time near the ex-
treme right and left, and a longer time near the centre. (3.) Anothev
band from the crank spindle (1) passes round a large wheel on the
vertical axis, that passes through a stuffing-box in the bottom of the
grinding-trough, and there carries a broad frame, on which is placed
the fixed frame of the mirror, supporting the mirror itself. Thus the
mirror turns slowly round to receive the strokes of the grinder in
every direction. It is only necessary to add to this, that a large
part of the grinder's weight is equally supported at twelve points by
a lever counterpoise above it, which supports the centre of a triangle,
each point of which (by a roller over which a cord passes) supports
two points, each of which i;)oints is the middle of a straight lever
whose ends are attached to the grinder.

In Mr. Lassell's apparatus (omitting some details, which are not
now necessary, as a detailed account of this apparatus has lately
been given by Mr. Lassell himself to the Society), the steam-engine
puts in motion (1) a vertical spindle below, which turns the mirror
slowly, and (2) a vertical spindle above, which has a horizontal arm,
in which is a planet- wheel turned by its tooth-connexion with a
fixed sun -wheel. The axis of this planet- wheel carries another
wheel that works in a third wheel, whose spindle is carried by the
horizontal arm, but whose place can be fixed at pleasure near to or
far from the centre of the vertical spindle (2). An arm from the

Royal Astronomical Society. 537

spindle of the third wheel carries a spike pointing downwards (the
distance of this spike from the centre of the spindle of the third
wheel being adjustable), and this spike lodges loosely in the centre
of the grinder and moves it upon the face of the mirror. Thus it
will be seen that the centre of the grinder is always moved in an
epitrochoid (including the circle as a possible case, if the adjustment
is made for that curve), in which the proportion of velocities of the
two circles is fixed, but the radii and their proportions are adjustable.
The grinder, as in Lord Rosse's construction, is allowed to take
freely the rotatory motion which it may receive from the friction on
the mirror ; but no counterpoise is used, the weight of the grinder
being comparatively small.

The essential difference of these constructions, as regards the
movements of the grinder, is therefore this : that in Lord Rosse's
apparatus every stroke is very nearly straight, while in Mr. Lassell's
apparatus there is no resemblance to a straight movement at any
part of the stroke.

IV. The process of grinding is nearly the same (with differences
corresponding to the difference of dimensions) in the operations of
both constructors.

In Lord Rosse's grinding, the speculum having received an ap-
proximate figure from the form of the annealing furnace, the cast-
iron grinder is brought very exactly to form by turning upon the
lathe, with proper reference to a gauge, and (if not done before) its
surface is scored with cross-furrows about two inches apart and
nearly an inch deep, leaving the acting part of the surface in squares.
The grinder is then mounted in the apparatus ; and this is the most
dangerous part of the whole operation. The slightest jar of the iron
grinder upon the mirror would break the mirror ; and to avoid this
risk, a great number of thin wooden wedges is placed upon the edge
of the mirror, the grinder is slowly lowered upon them, and then by
degrees they are gently withdrawn. The grinder is then used to grind
the surface, with the intermediate powder of emery and water;
coarser and finer emery in succession. A heavy weight is allowed
to press the grinder upon the mirror ; and as the grinder itself suf-
fers much in form, it is repeatedly re-turned upon the lathe. This
operation sometimes lasts many days.

Of the grinder used by Mr. Lassell the Astronomer Royal could
give no account, but believes that it is of wood, the same which is
used for polishing. The abrading powder used is the same as Lord
Rosse's (emery, coarser and finer in succession).

V. The next point deserving attention is the important process of
polishing.

When the figure of the speculum given by grinding is supposed
by Lord Rosse to be sufficiently accurate, the projecting squares of
the cast-iron grinder are covered with a coating of resin and tur-
pentine, of such a consistence that, at a temperature of about 50°
Fahrenheit, the nail can easily make an indentation in it. This is
then covered with another coating, of a substance formed by com-
bining the mixture last-mentioned with a certain quantity of wheat-

188 Royal Astronomical Society.

flour, and of such a consistence that, at a temperature of 80° Fahr-
enheit, the nail can make an indentation in it. It is not to be un-
derstood that there is any particular virtue in the temperatures 50°
and 80°, but (for reasons to be hereafter given) it is necessary to
have two strata of different degrees of hardness (the harder being
the exterior) ; and the hardnesses defined by those two temperatures
having been used in many experiments, the other adjustments have
been determined, using these as bases ; and if these bases were now
changed, every other adjustment must be changed.

The necessity for the different strata of resin is thus explained.
It is necessary that the polisher yield a little, else the polishing sur-
face could not be in contact with the mirror at all parts of its stroke
(the mirror being supposed parabolical), and this yielding is given
by the first or soft stratum. But it is also necessary that the stra-
tum next to the metal be hard ; for if, in passing across a scratch or
furrow, it were able to accommodate its form to that of the furrow, it
would round off and polish the edges of the furrow, and this would
very much injure the image of a bright object.

The coating is heated to softness by the flame of a torch, and the
grinder is then lowered upon the mirror, and the coating takes the
proper form. The mirror is exposed to no danger of cracking from
this application.

The powder used is the red oxide of iron, prepared by precipitating
(by means of ammonia) the black oxide of iron from a solution of
sulphate of iron, and then heating the black oxide in a furnace with
access of air. Lord Rosse finds that no other method of preparing
the red oxide is successful. No polishing-putty or other powder of
any kind is employed.

The powder is moistened with water to a degree known by ex-
perience, and then, the counterpoise of the grinder being so much
increased that the remaining friction is enough to turn the grinder
only once for about sixteen revolutions of the mirror, the operation
goes on for about eight hours. It is essential, for reasons similar to
those lately mentioned, that the temperature of the air and the tem-
perature of the dew-point have nearly certain definite values (the
latter, that the water mixed with the powder may dry in the proper
degree) : if the external air is too dry, the air of the room is moist-
ened by a jet of steam ; if it is too damp, the polishing is not at-
tempted.

After about eight hours the grinder is lifted and a fresh applica-
tion is made of powder mixed with " ammonia soap," a substance
formed by treating common soap with ammonia. The metal then
dries more rapidly, and the labour to the engine becomes much
greater : the work is continued till the surface is dry, or very nearly
dry ; the grinder is taken off, and the mirror is found finished, ha-
ving a parabolic figure, and a very high pohsh.

It is to be remarked here that the smaller mirrors made by Lord
Rosse (3 feet diameter, about 25 feet focal length,) were tried before
they were used, by means of an object fixed something more than
50 feet above thepi, whose image accordingly is found at something

Royal Astronomical Society, 5SS

less than 50 feet above them. But for the larger mirror such a trial
is impracticable (the height of the object must exceed 100 feet),
and the mirrors have therefore been placed in the telescope without
any trial, and their definition has been found perfect.

Mr. Lassell uses for polisher a wooden plate formed of two thick-
nesses of pine- wood with the grain crossed ; and this apparently
yields to accommodate itself to the form of the mirror. In the arrange-
ment of its square protuberances, it is similar to Lord Rosse's ; but
it is covered with only one coating of pitch. The polishing powder
used is the same as Lord Kosse's. The Astronomer Royal believed
that the attentions to temperature, moisture, &c., which Lord Rosse
found indispensable for his large mirrors, are not found necessary by
Mr. Lassell. Mr. Lassell finishes the operation with the speculum wet.

VI. The next point to which allusion was made is the form and
mounting of the telescopes.

Lord Rosse's telescope js a wooden tube, its interior diameter
exceeding 6 feet in every part, being at the middle about 7 feet, and
nearly 50 feet in length. This is fixed to a cube of 10 feet, which
has folding-doors on that side which, when the telescope is hori-
zontal, is the upper side (at which side the fixed frame supporting
the mirror is introduced, as has already been said), and which carries
the fixed frame by three large screws in that side of the cube which
is opposite the mouth of the telescope. To this side of the cube is
attached the universal joint by which the lower end of the telescope
is connected with a fixed support, the joint being a few feet below
the general surface of the ground. On each side (east and west)
of the telescope is an enormous pier of solid masonry, about 70 feet
long, in the north and south direction, between 40 and 50 feet high,
and in its thickest part nearly 20 feet thick. [None of these dimen-
sions are taken from actual measure.] The fixed support is nearer
to the north than to the south ends of these piers. Near the top of
the piers, on the interior faces, in the east and west plane passing
through the universal joint, are two cranes with pulleys (the turning
crane being no bigger than suffices to carry a large pulley, whose
edge is in the vertical axis of the crane) ; over these cranes the
chains pass which are attached to the telescope ; and to the lower
ends of the chains, after they have passed fixed pulleys on the walls,
are attached the counterpoises, weighing about four tons each.
These counterpoises are not allowed to depend freely, but are con-
nected by bridle-chains with wooden horns that project from the
north ends of the piers ; the effect of this arrangement is, that when
the telescope tube is nearly horizontal, and the force required to
support it is very great, the weight of the counterpoises acts very
nearly vertically on the chains, and is entirely effective for the sup.-
port of the telescope ; but when the telescope is considerably ele-
vated, and less supporting force is required, the weight of the coun-
terpoises is supported in a great measure by the bridle-chains, and
very little tension is given to the supporting chains. For the sake
of supplying some slight defects in the laws of tension thus produced,
and also for the sake of constantly producing a small tendency in the
telescope towards the south horizon, other counterpoises, in a pit

540 Royal Astronomical Society.

south of the fixed support, are brought successively into action as
the telescope is raised. There is then only a comparatively small
and very manageable tendency of the telescope towards the south,
and this is supported by a light chain which passes over a pulley on
a bar connecting the horns before mentioned (the pulley being in
the direction of a polar axis passing through the lower universal
joint, and the motion of the telescope, therefore, for a given length
of the chain, being equatoreal), and this chain is shortened or
lengthened, and the telescope is thereby raised or depressed, by a
windlass a little way north of the fixed support. Upon the inner
face of the eastern pier is an iron arc of a circle, upon which slides
a runner connected with a rod that passes through a frame on the
telescope tube and near to its mouth, and is there racked for working
with a pinion : by the movement of this pinion the distance of the
telescope from the pier is altered, and thus a motion in hour- angle
is given. At the south ends of the piers there are strong ladders,
upon which (assisted by counterpoises) there slides a stage ; upon
which stage a small observing-box travels east and west : this is
used for observing, so long as the mouth of the telescope is below
the end of the pier. For greater elevations, the top of the western
pier being shaped by slopes so as to approximate to a circular arc,
there are mounted upon it curved galleries, which are carried by
beams that run above and below pulleys fixed to the top of the pier ;
and the galleries are carried out by rack-and-pinion work, to ap-
proach the side of the telescope. It is intended to give the power
of observing as far north as the pole ; but at present the galleries
extend only to the zenith. The telescope is Newtonian, the minor
axis of the small mirror being about six inches, and the observer
looks into the side of the tube.

Mr. Lassell's tube is of sheet-iron ; and this tube is not carried
immediately by the mounting, but is inserted in a long box of cast
iron, in which it can be turned round its own axis. This movement
is necessary to place the eyepiece exactly in the same side-position
in all directions of the telescope, and also to cause the edgewise
support of the mirror to act always in the same way. The long box
is mounted equatoreally, the polar axis turning in two bearings be-
low the declination axis, and carrying an hour-circle, upon which
are fixed two supports, in which turn the two pivots of the declina-
tion axis of the long box. The telescope is Newtonian, the eye-
tube being in one side ; but the smaller dimensions of the small
mirror (a diameter of two inches only being required) enable Mr.
Lassell to use the reflexion at the internal surface of a glass prism,
by which much more light is reflected than by a metallic reflector.
At first much annoyance was caused by the deposition of dew on
the glass, but this was remedied by attaching to it a case carrying a
small piece of heated lead ; and when proper attention is given to
the inclosure of the lead, no inconvenience is sustained from the
efi^ect of the hot metal in disturbing the air in the tube of the tele-
scope. The whole is covered by a revolving dome thirty feet in
diameter, and the observer is mounted for observation on a stage
which is carried by the dome.

Royal Astronomical Society, B41

The Astronomer Royal then proceeded to describe some of tht
difficulties to which instruments of this class are yet liable, founded
partly upon his own observations with Lord Rosse's telescope.

Upon directing the telescope to an object very near the zenith, it
was seen very well defined, or at least with no discoverable fault.
It must be remarked that the image of a star never assumes the neat
spherical form to which the eye of an observer with a fine refractor
is so much accustomed. This arises evidently from the circumstance
that (from the great aperture of the mirror) the diffraction image and
diffraction rings are invisibly small, and the form of the blurred
image is probably determined by the irregular sensibility of the ner-
vous membranes of the eye. The same effect exactly is produced
by a large refractor when a power is employed too low to exhibit
the rings.

But when the telescope was directed to a star as low as the equator
its image was very defective. The defect, however, followed that
simple law which the present Master of Trinity College has described
by the word astigmatism. When the eyepiece was thrust in, the
image of the star was a well-defined straight line, 20 seconds long,
in a certain direction ; when the eyepiece was drawn out a certain
distance (about half an inch from the former position) the image of
the star was a well-defined straight line, 20 seconds long, in a di-
rection at right angles to the former. Between these two positions
the image was elliptical, or, at the middle position, a circle of 10
seconds diameter. The image of Saturn (then without a ring) was,
in the two positions above-mentioned, an oval (not an ellipse) whose
length was about double its breadth ; or, in the middle position, it
was a confused circle, whose diameter was about 30 seconds instead
of 20. The position of the astigmatic lines had no distinct relation
to the vertical plane ; and this circumstance, as well as the magnitude
of the astigmatism, proved that it was not produced by a tilt of the
mirror.

Lord Rosse immediately suggested a probable cause of this defect.
The triangular levers which support the mirror are all, to a certain
degree, elastic. When the telescope is dropped from the vertical
position, the edge of the mirror begins to press the fixed pillars in
the fixed frame mentioned under No. II. ; and as the edgewise pres-
sure increases faster than the excess of the elastic force of the levers
over the pressure on the levers, the edge is firmly locked to the pil-
lars by the friction against them. When the telescope is much de-
pressed, this friction is perhaps not much less than two tons, or is
equal to the greatest strain of six horses, all exerting a force per-
pendicular to the face of the mirror at a part intended to sustain no
such force, and therefore tending to bend the mirror out of shape.
The obvious remedy was to suspend the edge of the mirror in such
a manner as to leave it free to play in a direction perpendicular to
the face of the mirror ; and for this purpose, first an iron hoop, and
secondly a chain, were used, the bearing against the pillars being
entirely destroyed. The effect was at first very satisfactory ; defi-
nition was made very good ; Saturn's ring was well seen on Sep-
tember 2 as a narrow line. But in subsequent observations the effect

542 Royal Astronomical Society.

\vas not so satisfactory. Here, again, Lord Rosse discovered the
cause. The triangular levers vphich immediately support the mirror
are necessarily not plane, inasmuch as the points which take hold
of the plates that adhere to the mirror must project higher (the tele-
scope being vertical) than the fulcra of these levers. If, then, when
the telescope is lowered, the mirror is allowed to slip edgewise, it
throws the lower points of each lever partly out of bearing, and then
the mirror is not supported in the way to which its figure is adapted.
It was found that some of the points were not in bearing ; and it
was also found that, when the telescope was directed to a star, the
image was rendered extremely good by screwing or unscrewing to
tile proper degree the supports of the hoop or chain. It was found,
moreover, that all this adjustment was affected by the bending of
the iron base of the fixed frame. On the whole, this part of the
mounting is not in a satisfactory state. The Astronomer Royal was
not able to say what is the nature of the edge-bearing adopted by
Lord Rosse at the present time ; still it is evident that, with due
attention, the mounting above-mentioned may be made perfectly
available. [The Astronomer Royal has lately been informed, that
Lord Rosse has entirely overcome these difficulties, by placing be-
tween the back of the mirror and the plates in which the points of
the triangular levers act, sheets of tin, which allow the mirror to
slip upon the plates, instead of the felt and pitch which formerly
united the mirror and the plates.] It may be proper here to men-
tion, that Lord Rosse has informed the Astronomer Royal that the
mirror which he saw under the polisher has been mounted, and that
it shows very well the third star of / Andromedae, — no small proof
both of the perfection of the figure and of the efficiency of the sup-
port of the mirror in that position.

The Astronomer Royal then explained his own ideas upon the
nature of the mounting, to which (whatever might be the practical
difficulties in the mere mechanical operation) he thought it would
be necessary to approach. He thought that it was absolutely ne-
cessary to give the edge-support by counterpoises ; and this might
be done, retaining the present levers, by making the point of each of
the small triangles a socket for a ball-and-socket joint, in which
turns a lever whose point lodges in the mirror. The extreme
hardness of the speculum metal makes it, however, difficult to drill
the holes.

In terminating now the account of the mirror, the Astronomer
Royal alluded to the impression made by the enormous light of the
telescope ; partly by the modifications produced in the appearances
of nebulae figured by Sir John Herschel, partly by the great number
of stars seen even at a distance from the Milky Way, and partly
from the prodigious brilliancy of Saturn (the only planet which he
had an opportunity of seeing). The account given by another
astronomer of the appearance of Jupiter was, that it resembled a
coach-lamp in the telescope ; and this well expresses the blaze of
light which is seen in this instrument.

The Astronomer Royal then stated that he had had no opportu-
nity of trying Mr. Lassell's telescope ; but he had understood from

Intelligence and Miscellaneous Articles. SiS

*t>

Mr. Lassell that some difficulties had been found in the arrangement
of the edge-bearing of the mirror, which had been overcome by sus-
pending it in an iron hoop. These, however, from the great differ-
ence of dimensions, would probably be trifling compared with those
of Lord Rosse's mirrors.

Adverting again to the mounting of these telescopes, the Astro-
nomer Royal suggested for consideration whether it might not be
advantageous to mount the telescope with an altitude and azimuth
movement, by an overhanging fork inserted in a vertical pillar. If
a rod were joined to the stalk of this fork, and, by means of an or-
dinary parallel motion, were compelled to move parallel to the axis
of the telescope, and if any part of this rod were connected by an-
other rod (adjustable in length according to the polar distance of
the object) to a universal joint on the ground, in such a position
that the line drawn from that universal joint to the stalk of the fork
is a polar axis, then the motion of the telescope would be equatoreal.
For the observer, it was proposed that an observing-box should be
fixed to the telescope, and that the access should be by a spiral
staircase round the pillar, by a narrow platform near its top, and
by a staircase along the side of the tube.

In conclusion, the Astronomer Royal observed that it was impos-
sible to overrate the advance that had been made in the construction
of telescopes by the two amateur constructors of whom he had spoken.
Lord Rosse had shown that it was possible, without any important
manual labour, to produce with certainty, by means of machinery,
mirrors of a size never before attained, and perhaps with a perfect-
ness of definition which had not been reached before ; and he had,
by publication and by private communication, made these methods
accessible to the world. This success was the more remarkable,
because the whole of the work was done by workmen found on the
spot ; even the steam-engine, by which (to a late time) the whole
of the machinery was driven, was made by native workmen under
Lord Rosse's personal instructions. To Mr. Lassell also much was
due, for the example which he had set of what may be done by a
man possessing less ample means, and whose time is fully occupied
in business ; and much also for the elegant and compendious and
manageable apparatus arranged by him, which promises to be of the
greatest use in the construction of large object-glasses as well as of

LXXIX. Intelligence and Miscellaneous Articles.

DEFLECri'ION OF THE MAGNETIC NEEDLE BY THE ACT OF
VOLITION*.

THIS curious and interesting experiment is due to the investigations
of M. DuBois Reymond of Berlin, and his method of performing
it is as follows : — He takes a very sensitive galvanometer, and attaches
to the terminal wires thereof two perfectly homogeneous strips of
* We are indebted to the kindness of W. G. Lettsom, Esq., for the com-
munication of this notice, by whom we are also informed that the experi-
ment has been repeated with success.

54>4i Intelligence and Miscellaneous Articles.

platina. These strips are dipped down into two vessels filled with
salt and water, into which fluid, as contained in each vessel, two
corresponding fingers of the two hands are to be plunged. On the
first immersion of the fingers there is almost always observable a
more or less decided deflection of the needle, this deflection not being
amenable to any known law, and being in the opinion of the experi-
menter due to the diflference existing to some extent and in some
way or another between the cutaneous covering of the two fingers.
Whenever there is a Avound on one of the fingers the deflection is
greater than usual ; and its direction is uniformly such that the injured
finger behaves like the zinc- side of an arc of zinc and copper, which
we may conceive to be inserted between the two vessels instead of
the human body. It need hardly be remarked, that it is not this sort
of action to which in the experiment in question it is purposed to
direct the attention. On the contrary, in order to observe the eff^ects
alluded to, it is requisite to wait either till the needle has gone back
to the zero-point of its scale, or at least until it has assumed a con-
stant deflection attributable to the residue of a current which it is
beyond us to eliminate. As soon as this state is attained, the whole
of the muscles of one of the arras must be so braced that an equili-
brium may be established between the flexors and the extensors of
all the articulations of the limb, pretty much as in a gymnastic school
is usually done when one wants to let a person feel the development
of one's muscles.

As soon as this is done the needle is thrown into movement, its
deflection being uniformly in such a sense as to indicate in the braced
arm " an inverse current," according to Nobili's nomenclature ; that
is to say, a current passing from the hand to the shoulder. The
braced arm then acts the part of the copper in the compound arc of
zinc and copper mentioned above.

With his own galvanometer, and when M. Du Bois Reymond him-
self performs the experiment, the deflection amounts to 30°. He
obtains however movements in the needle of far greater extent by
contracting alternately the muscles, first of one arm and then of the
other, in time with the oscillations of the needle. On bracing simul-
taneously the muscles of both arms, inconsiderable deviations are
observable, sometimes in one direction, sometimes in another ; and
these minute deflections are evidently attributable to the diflfer-
ence between the contractile force of the two limbs. Hence it arises
that when the experiment is repeated many times in succession, the
results diminish gradually in amount, not only in consequence of the
energy of the contractions becoming less and less, but also because
it becomes more and more difficult to restrain the act of slackening
or letting down the muscles to only one of the two arms.

The amount of deviation, cateris paribus, depends upon the amount
of the development and the exercise of the muscles. The author is
said to have an arm of considerable power, and among the number of
savans that have tried the experiment at his residence, there has not
as yet been found one who excelled or even came up to him in this
respect. There are indeed individuals who do not possess the power
of producing a sensible deflection in the jieedle of his galvanometer.

Intelligence a7td Miscellaneous Articles. 54-5

but it is readily ascertained that in these instances there is a want of
sufficient muscular tension.

I'here is one remark, to conclude, which the author has been fre-
quently led to make, namely, that the habitual superiority of the right
hand over the left in this experiment is to be interpreted by the prepon-
derance of the amount of deflection produced by the tension of the
right arm. This peculiarity was likewise observed when the experi-
ment was performed by M. von Humboldt. The impulsion impressed
on the needle by the contraction of the muscles of his right arm
was appreciably more considerable than that produced by his left
arm.

For his own part, M. von Humboldt has addressed to M. Arago a
letter of the following tenor : — He says, the fact of the experiment
of affecting a magnetic needle by the alternate tension of the mus-
cles of the two arras, an effect due to volition, is established beyond
all question or doubt. Notwithstanding my advanced years and tlie
little strength that I have in my arms, the deflections of the needle
were very considerable ; but they were naturally more so when the
experiment was performed by M. J, Muller or by M. Helmkoltz, who
are younger men. To facilitate the experiment it is advisable to
plunge the fore-fingers into the water, and to support the palms of
the hands, to enable one to brace up well the muscles of the arm
which it is purposed to bring into play.

ON THE ARTIFICIAL FORMATION OF MINERALS IN THE HUMID
WAY. BY M. DE SENARMONT.

Many mineral species very nearly approach some compounds
obtained by the usual processes of chemistry, and even supply the
deficiencies, which are still left in certain natural series. Such for
example are the carbonates of magnesia, of protoxide of iron, man-
ganese, nickel, cobalt and zinc, which are placed near rhombic car-
bonate of lime, and are met with in nature in a state of purity, or of
isomorphic union, yielding hybrid species, forming connecting links
between pure species.

These natural compounds, which have not been hitherto formed
artificially, could not evidently have arisen under the usual conditions
of laboratory experiments ; for we have no reason for supposing that
the same causes could produce different eflTects at different epochs.
It is therefore a subject of great interest to determine with precision
the circumstances necessary to the production of all minerals ; and
the solution of this question would undoubtedly be the best method
of raising the veil behind which are hidden the phaenomena which
attended the formation of a great number of rocks and of a part of
the terrestrial globe.

Some happy synthetical attempts have already afforded valuable
data on this head. MM. Mitscherlich and Berthier have obtained
several fusible minerals in the dry way, and M. Ebelmen has made
an additional step in his researches on the formation of infusible
silicates and aluminates. M. G. Rose has ably examined the con-
ditions under which carbonate of lime is precipitated in the state of
aragonite; lastly, the beautifuUexperiment of M. Haidinger has

Pkil. Mag. S. 3. No. 232. Suppl. Vol. "S*. 2 N

54:6 Intelligence and Miscellaneous Articles.

thrown great light on the controverted question of the formation of
dolomites and the general problem of metaraorphism.

The author conceived the idea of forming certain mineral species
in the humid way, and his first essays were made with the carbonates.
As it is clearly demonstrated that heat generally favours dehydra-
tion, it occurred to the author that the formation of neutral car-
bonates might be a simple question of pressure and temperature ; but
before attempting to precipitate them by the disengagement of the
excess of carbonic acid serving them as a solvent at a high tem-
perature under strong pressure, the author attempted double decom-
positions in the humid way. These first attempts set out there-
fore from the beautiful experiments of M. Haidinger.

The substances were exposed to each other in glass tubes, herme-
tically sealed, after having exhausted them. If they were of a kind
to react upon each other immediately, they were at first separated,
and were then by turning moved at the proper time. For high
temperatures the tubes were inclosed in a gun-barrel, hermetically
sealed and half-filled with water, so as to equalize as much as possi-
ble the internal and external pressures on the glass tube. The tubes
were heated in small closed chambers in the furnaces of a gas-work.

The following mineral species were by these means produced : —

Carbonate of Magnesia. — ^By the double decomposition of sulphate
of magnesia and carbonate of soda, at about 322° Fahr. It was in
the state of white crystalline sand, which was scarcely acted upon
by weak acids. Having understood, during the course of his experi-
ments, that M. Maurignac had performed analogous experiments on
the reaction of chloride of magnesium and carbonate of lime, hereafter
to be described, the author did not proceed further.

Carbonate of Protoxide of Iron. — Obtained by double decomposi-
tion : first, of sulphate of protoxide of iron and carbonate of soda,
at about 302° F. and above ; second, from protochloride of iron and
carbonate of lime, at temperatures between 266° F. and 392° F.,
kept up during twelve, twenty-four, and thirty-six hours. The
carbonate of iron was in the state of crystalline sand, of greater or less
fineness, of a grayish-black colour, nearly unalterable in dry air, very
slowly acquiring a flaxen colour in moist air, and scarcely acted upon
by diluted acids. This crystalline sand has then all the properties
of spathose iron ore. Its gray colour is more or less dark, and its
spontaneous alterability is the smaller according as it is formed at
high and long- continued temperatures. Perhaps to circumstances
of this kind may be attributed the differences which exist in this
respect among natural spathose iron ores ; differences which are not
sufficiently explained by variation in their composition.

Carbonate of Manganese. — Obtained by double decomposition :
first, of chloride of manganese and carbonate of soda, at about 320° F. ;
second, chloride of manganese and carbonate of lime, at tem-
peratures between 284° F. and 338° F., kept \yp twelve and forty-
eight hours. It is in the state of a fine white powder, slightly rose-
tinted, no crystalline appearance, and is unalterable at a moderate
heat.

Carbonate of Zinc. — Obtained under the same conditions as that
of iron ; it is a fine white powder, not crystalline, unalterable at a
moderate heat. — Comptes Rendus, Juin 4, 1849.

547

INDEX TO VOL. XXXIV.

Acids : — succinic, 157 ; lizaric, 236 ;
oxylizaric, 237; anhydrous nitric,
314; phosphoric, 321; margaiic,
352; gyrophoric,464; chloroniceic,
473; nitranisic, 478.

Aerial vibrations, on the mathematical
theory of, 88, 225.

Airy (Prof.) on a difficulty in the
problem of sound, 401 ; on the
large reflecting telescopes of the
Earl of Rosse and Mr. Lassell,
532.

Akxander (J. H.) on a new empirical
formula for ascertaining the ten-
sion of vapour of water at any tem-
perature, 1, 98.

Algebra, on a new imaginary in, 37 ;
on a system of triple, 119 ; on the
symbols of, 406 ; on a new system
of imaginaries in, 425,

Alkalies, organic, on a series of new,
316.

Allan (M.) on tke urates, 154.

AUotropism, on the existence and
effects of, in the constituent ele-
ments of living beings, 241.

Alps, on the geological structure of
the, 207 ; on the distribution of the
superficial detritus of the, 523.

Alum, on octahedral and cubic, 476.

Ammonia, on a series of organic alka-
lies homologous with, 317-

Animalcule, description of a new
genus of, 529.

Anisidine, on the production and com-
position of, 477.

Anisol and its derivatives, observa-
tions on, 476.

Arsenic, on the prepai-ation of iodide
of, 152.

Aurora borealis of Nov. 17, 1848,
observations of the, 226, 305.

Aurorse boreales, remarks on the cause
of, 286.

Bache (A> D.) on a new method of
observing transits, 460.

Barlow (W. H.) on the cause of the
diurnal variations of the magnetic
needle, 344.

2

Baxendell (Mr.) on the variability of

X Tauri, 217.
Bensch (M.) on the urates, 154.
Bertin (A.) on circular magnetic po-
larization, 481.
Bishop's (Mr.) ecliptic charts, notice

of, 149.
Bismuth, on the crystalline polarity

of, 75.
Blood, human, on the presence of
copper in, 155,

Bodies, on the multiple sounds of,
415.

Books, new: — Quetelet's Letters to
the Grand Duke of Saxe Coburg
and Gotha, on the Theory of Pro-
babilities, as applied to the Moral
and Political Sciences, 297 ; Patter-
son's First Steps to Zoology, 298 ;
Taylor's Statistics of Coal, 380; Sir
J. F. W. Herschel's Outlines of
Astronomy, 452.

Booth (Jas.) on the application of the
theory of elliptic functions to the
rotation of a rigid body round a
fixed point, 312.

Boracic acid, on some combinations
of, with oxide of lead, 375.

Brandes (M. D.) on the ferrocyanides
of strychnia and brucia, 238; on
compounds of hydrochlorate of
strychnia and cyanide of mercury,
479.

Brinton (W.) on the physiology of
the alimentary canal, 299.

Bromine, impurity of commercial,
399.

Brom-orcine, observations on, 465.

Bronwin (Rev. B.) on the determina-
tion of the sines and cosines of
multiples of a variable angle, 260,
374; on the integration of linear
differential equations, 530.

Brown (J.) on the products of the
soda manufacture, 15,

Brucia, on the ferrocyanide of, 238,

Cahours (M, A.) on anisol and its de-
rivatives, 476,

Calculus of observations, on the, 60,
N2

54-8

INDEX.

Calvert (Rev. F.) on the elements of
planegeometricaltrigononietry,455.

Cambridge Philosophical Society,
proceedings of the, 135, 225,
455.

Cayley (A.) on the theory of permu-
tations, 527.

Challis (Rev. J.) on the mathematical
theory of aerial vibrations, 8S, 225 ;
on the method in use at the Cam-
bridge Observatory of measuring
differences of right ascension and
north polar distance, 138; on the
transmission of light through trans-
parent media, 225 ; on the aurora
boreahs of Nov. 17, 1848, 226; on
the theoretical value of the velocity
of sound, 284, 353; on spherical
waves in an elastic fluid, 449 ; on
some points relating to the theory
of fluid motion, 512.

Chelonian reptiles, on the develop-
ment and homologies of the cara-
pace and plastron of the, 307.

Chloride of silver, on the solubility of,
in hydrochloric acid, 152.

Chloroform, action oiF, on the sen-
sitive plant, 130.

Chloroniceic acid and chloroniceates,
on the, 473.

Clock-escapements, observations on,
455.

Cockle (Jas.) on a new imaginary in
algebra, 37; on two geometrical
problems, 132; on the symbols of
algebra, and on the theory of tessa-
rines, 406.

Colours, hints towards a classification
of, 161.

Copper, on the presence of, in the
human blood, 155; on a new dou-
ble salt of the sulphate of, with sul-
phate of potash, 475.

Crystals, on the magnetic relations of
the positive and negative optic axes
of, 450.

Cubic, on the expression for the roots
of a complete, 193.

Curve, on the intrinsic equation to a,
457.

Curves, epicyclical, on an easy rule for
formuhzing all, 444, 520.

Dairy mple (J.) on an infusoiy ani-
malcule allied to the genus Notom-
mata, 529.

Damour (M. A.) on faujasite, 394.

Davies (T. S.) on numerical transfor-
mation, 347.

Dawes (Rev.W. R.) on the transit of
Mercury, 141.

Debus (M.) on madder, 236.

Delesse (M.) on the protogine of the
Alps, 233.

Dell (Mr. T.) on the transit of Mer-
cury, 142.

Denison (E. B.) on clock escapements,
455.

Desains (M. P.) on the reflexions of
different kinds of heat by metals,
471.

Deschamps (M.) on the presence of
copper in the human blood, 155.

Dessaignes (M.) on the formation of
carbonate of lime from the neutral
malate of lime, 156.

Deville (M.) on anhydrous nitric acid,
314.

Diamagnetism, experiments on, 81.

Domeiko (M. J.) on the vanadiate of
lead and the double vanadiate of
lead and copper, 395.

Drach (S. M.) on an easy rule for for-
mulizing all epicyclical curves with
one moving circle by the binomial
theorem, 444, 520,

Draper (Dr. J. W.) on the existence
and effects of allotropism in the
constituent elements of living be-
ings, 241.

Duhamel (M.) on the multiple sounds
of bodies, 415.

Dumas (M.) on liquid protoxide of ni-
trogen, 153.

Electricity, experimental researches
in, 75.

Electro- physiology, researches in, 440.

Equations, on improvements in the
analysis of, 413 ; on the integration
of linear differential, 530.

, numerical, on a new me-
thod of finding the rational roots of,
457.

Erythromannite, on some nitrogen
compounds of, 465.

Ethylamide, on the preparation and
composition of, 318.

Faraday (M.) on the crystalline pola-
rity of bismuth and other bodies, "Jb.

Faujasite, analysis of, 394.

Felsite, analysis of, 237.

Fluid motion, on the theory of, 512.

Fluorine, on the equivalent of, 160.

INDEX.

5«&

Forbes (J. D.) on the classification of
colours, 161.

Forces, analytical proof of the paral-
lelogram of, 201.

Forster (Dr.) on the weather after a
new moon, 21 G.

Foucault (M. L.) on some pha;nomena
of binocular vision, 269.

Functions, on the decomposition of,
into conjugate factors, 278.

• , elliptic, on the application

of the theory of, to the rotation of
a rigid body round a fixed point,
312.

Geometrical problems, solution of
two, 132.

Gilliss (Lieut.) on the longitude of
Washington, 150.

Glaisher (Jas.) on the weather during
the quarters ending December 31,
1848, 182; and March 31, 1849,
366 ; on the meteorology of Eng-
land in the year 1847, 271,

Gold from California, on the compo-
sition of the, 205, 394.

Graves (Rev. C.) on the calculus of
operations, 60 ; on a system of tri-
ple algebra, 119.

Grove (W. R.) on the effect of sur-
rounding media on voltaic ignition,
299.

Gyrophoric acid, on the preparation
and composition of, 464.

Hamilton (Sir W. R.) on quaternions ;
or on a new system of imaginaries
in algebra, 294, 340, 425.

Hare (Dr.) on the rationale of the ex-
plosion causing the great fire of
1845 at New York, 227.

Hartnup (Mr.) on the transit of Mer-
cury, 143 ; on the equatoreal of the
Liverpool Observatory, 148.

Heart, on the ganglia and nerves of
the, 304.

Heat, on the reflexion of different
kinds of, by metals, 471.

Heath (Rev. J. M.) on a simple rule
for converting intervals of sidereal
into intei-vals of mean solar time,
and vice versa, 51.

Heineken's (N. S.) suggestions for
rendering a meridian mark visible
at night, 202.

Henry (T. H.) on the composition of
the gold from California, 205.

Herapath (T. J.) on some combina-

tions of boracic acid with oxide of

lead, 374.
Herschel's (Sir J. F. W.) Outhnes of

Astronomy, noticed, 452.
Huraut (M. T.) on the preparation of

iodide of lead, 231.
Hydrogen, on the peculiar cooling

effects of, in cases of voltaic igni-
tion, 304.
Irradiation, observations on, 469.
Jones (Dr. H. B.) on the chemistry of

the urine, 311.
Kemdt (M.) on felsite, oligoclase and

muromontite, 237.
Klauer (M.) on the rectification of
. spirit of nitrous aether, 319.
Lassell (Mr.) on a machine for pohsh-

ing specula, 143.
Lead, on the preparation of the iodide

of, 231.
Lee (Dr. R.) on the ganglia and

nerves of the heart, 304.
Lichens, on the proximate principles

of some, 463.
Light, on a theory of the transmission

of, through transparent media, 225.
Lizaric acid, on the preparation and

composition of, 236.
Locke (J.) on single and double vision,

and on an optical illusion, 195.
Longitude, method of detennining

differences of, 302.
Loomis (E . ) on the determination of the

difference of longitude, by means of

the magnetic telegraph, 302.
Louyet (M.) on the equivalent of

fluorine, 150.
Lowe (E. J.) on remarkable solar

halos and mock suns, 410.
Luminous vibrations, on a mathemati-
cal theory of, 225.
Madder, examination of, 236.
Magnecrystallic force, on the nature

of the, 7^-
Magnet, on the repidsive action of a

pole of a, upon non-magnetic bo-
dies, 127.
Magnetic declination, on the causes

which produce the diurnal varia-
tion of the, 466.
Magnetic needle, on the cause of the

diurnal variations of the, 286, 344 ;

on the deflection of the, by the act

of voUtion, 543.
Magnetic polarization, observations

on, 481.

550

INDEX.

Magnetic relations of the positive and
negative optic axes of crystals, 450.

Magnetic telegraph, on the determi-
nation of the difference of longi-
tude, hy means of the, 302.

Magnetism of steam, observations on,
502.

Malic acid, conversion of, into succi-
nic acid, 156.

Man, on the organ of taste in, 531.

Marcet (Prof.) on the action of chlo-
roform on the sensitive plant, 130.

Matteucci(M. Ch.), researches in elec-
tro-physiology, 440.

Mediterranean, analysis of the water
of the, 398.

Mercury, on the existence of, in the
Tyrol, 318.

, on the transit of Nov. 8-9,

1848, 141.

Metallurgy, thoughts on ancient, 247.

Metals, on the reflexions of different
kinds of heat by, 471.

Meteorological observations, 79, 159,
239, 319, 399, 479. .

phsenomena, obsei-va-

tions on some, 469.

Meteorology of England during 1847,
on the, 271.

Methylamide, on the preparation and
composition of, 316.

Meurer (M.) on the preparation of
iodide of arsenic, 152.

Miller (J. F.) on the depth of rain
which falls in the same localities at
different altitudes, in the hilly di-
stricts of Lancashire, Cheshire, &c.,
308.

Mimosa pudica, on the action of chlo-
roform on, 130.

Mine ventilator, on Mr. Struve's, 388.

Mineralogy : — analysis of the black
yttro-columbite of Ytterby, 152;
Californian gold, 205, 394 ; proto-
gine of the Alps, 233 ; orthose, 234 ;
oligoclase, 234, 237 ; felsite, 237 ;
muromontite, 438 ; gray copper ore,
318; faujasite, 394; vanadiate of
lead, 395 ; diamond, 397.

Minerals, on the artificial formation
of some, 545.

Moon (R.) on the theory of sound,
1 36 ; on a new method of finding
the rational roots of numerical
equations, 457.

Murchison (Sir R. I.) on the geolo-

gical structure of the Alps, Cai-pa-
thians, and Apennines, 207 ; on the
distribution of the superficial detri-
tus of the Alps, 523.

Muromontite, analysis of, 237.

Muscular contraction, on the deflec-
tion of the magnetic needle by, 543.

Newton's rings, on the formation of
the central spot of, 137.

Nile, notice of Dr. Bialloblotzky's
journey to the sources of the, 157.

Nitric acid, observations on anhy-
drous, 314.

Nitroanisic acid, on the rafeparation
and composition of, 478.

Nitrogen, on liquid protoxide of, 153.

Nitrous aither, on the rectification of
spirit of, 319,

Numerical transformation, note on,
347.

Observatories, notice of the principal
English, 63.

CErsted (H. C.) on diamagnetism, 81.

Oligoclase, analysis of, 237.

Owen (Prof.) on the development and
homologies of the carapace and
plastron of the chelonian reptiles,
307.

Oxylizaric acid, on the preparation
and composition of, 237.

Patterson's (R.) First Steps to Zoo-
logy, notice of, 297.

Peretz (M.) on the composition of
the black yttro-columbite of Ytter-
by, 152.

Permutations, on the theory of, 527.

Persoz (M. J.) on the reaction of sul-
phate of potash and sulphate of
copper, 475; on octohedral and
cubic alum, 476.

Phillips (John) on ancient metallui'gy,
and mining in Brigantia and other
parts of Britain, 247-

• (Reuben) on the magnetism

of steam, 602.

Philosophy, on the fundamental anti-
thesis of, 135.

Phosphoric acid, on the isomeric mo-
difications of, 321 ; separation of,
from pyrophosphoric acid, 340.

Phosphorus, on a new modification
of, 78,

Pierre (M. J.) on the solubility of
chloride of silver in hydi-ochloric
acid, 152.

Pliicker (Prof.) on the magnetic rela*

INDEX.

551

tions of the positive and negative
optic axes of crystals, 450.

Polarization, on circular magnetic,
481.

Poselger (M.) on the impurity of com-
mercial bromine, 399.

Post-office regulations, notice respect-
ing, 158.

Potter (Prof.) on the discovery of the
chilling process in the casting of
the specula for reflecting telescopes,
246.

Powell (Prof.) on irradiation, 459.

Pratt (T. H.) on an analytical proof
of the parallelogram of forces, 201.

Pringle (W.) on the continuance of a
solar spot, 48.

Protogine of the Alps, on the, 233.

Provostaye (M. F. de la) on the re-
flexions of different kinds of heat
by metals, 471.

Quaternions, a new system of imagi-
naries in algebra, on, 294, 340, 425.

Quetelet's (M. A.) Letters to H.R.H.
the Grand Duke of Saxe Coburg and
Gotha, on the Theory of Probabili-
ties, notice of, 297.

Rain, on the depth of, which falls at
different altitudes in Lancashire,
Cheshire, &c., 308.

Reade (Rev. Mr.) on the transit of
Mercury, 143.

Rebmann (Rev. Mr.) on a snowy
mountain in Eastern Africa, 388.

Regnault (J.) on some phsenomena of
binocular vision, 269.

Reich (Prof.) on the repulsive action
of the pole of a magnet upon non-
magnetic bodies, 127.

Reymond (M. Du Bois) on the deflec-
tion of the magnetic needle by the
act of volition, 543.

Richardson (J.) on Mr. Struve's mine
ventilator, 388.

Rive (A. de la) on the diurnal varia-
tions of the magnet needle, and on
aurorae boreales, 286.

Rivot (M.) on California gold, 394.

Rose (H.) on the existence of mercury
in the Tyrol, 318 ; on the isomeric
modifications of phosphoric acid,
321.

Royal Astronomical Society, proceed-
ings of the, 63, 138, 216, 459, 532 ;
anniversaryaddi'ess of the President,
218.

Royal Society, proceedings of the, 73,
299, 463, 529.

Sabine (Lieut.-Col.) on the causes
which produce the diurnal variation
of the magnetic declination, 466.

Saint-Evre (M. E.) on chloroniceic
acid, and the chloroniceates, 474.

Schroetter (M.) on a new modification
of phosphorus, 78.

Senarmont (M.) on the artificial pro-
duction of several minerals, 545.

Shea butter, on the composition of,
350.

Shooting star, on the calculation of
the distance of a, 179.

Smith (A.) on the calculation of the
distance of a shooting star, 179.

Smith (Dr. R. A.) on a mode of ren-
dering substances incombustible,
116.

Smith (Mr. R.) on the aurora borealis
which occurred on the 1 7th of No-
vember 1848, 305.

Soda manufacture, on the products of
the, 15.

Solar halos, on some remarkable, 410.

Solar spot, on the duration of a, 48.

Sound, on some points in the re-
ceived theory of, 52, 88, 136, 203,
348, 401, 501 ; on the theoretical
value of the velocity of, 284; on
the determination of the velocity of,
on the principles of hydrodynamics.

Sounds, multiple, of bodies, on the,

415.
Spectrum, on the theory of certain

bands seen in the, 309.
Specula, description of a machine for

polishing, 143 ; on the discovery of

the chilling process in the casting

Squares, on a theorem respecting the
products of, 113.

Steam, tables of the pressure of, at
different temperatures, 8; on the
magnetism of, 502.

Stenhouse (J.) on the proximate prin-
ciples of some lichens, 463.

Stevenson (W. F.) on the peculiar
cooling effects of hydrogen and its
compounds in cases of voltaic igni-
tion, 304.

Stokes (G. G.) on some points in the
received theory of sound, 52, 203,
348, 501 ; on the formation of the

552

INDEX.

central spot of Newton's rings be-
yond the critical angle, 137 ; on the
theory of certain bands seen in the
spectrum, 309.

Struve's mine ventilator, observations
on, 389.

Strychnia, on the ferrocyanide of,
238 ; on the compounds of the
hydrochlorate of, and cyanide of
mercury, 4/9.

Succinic acid, formation of, from ma-
lic acid, 156.

Tallow, vegetable, on the composition
of, 350.

Tayloi-'s (R. C.) Statistics of Coal, no-
tice of, 380.

Telescopes, on the large reflecting, of
the Earl of Rosse and Mr. Lassell,
532.

Tessarines, on the theory of, 406.

Thomson (Dr. R. T.) on the composi-
tion of Shea butter and Chinese
vegetable tallow, 350.

Transits, on a new method of ob-
serving, 460.

Urates, on the preparation of the, 154.

Urine, on the chemistry of the, 311.

Usiglio (M. J.) on the composition of
the water of the Mediterranean, on
the coast of France, 398.

Vanadiate of lead, observations on
the, 395.

Vapour of water, on a new empiincal
formula for ascertaining the ten-
sion of, 1, 98.

Vision, on single and double, 195 ;
on some phsenomena of binocular,
269.

Voltaic ignition, on the effect of sur-
rounding media on, 299 ; on the
peculiar cooling effects of hydrogen
in cases of, 304.

Waller (Dr. A.) on the organ of taste
in man, 531.

Wartmann (Prof. E.) on some meteo-
rological phaenomena, 469.

Waves, spherical, observations on, 449.

Weather, remarks on the,182,271,366.

Whewell (Dr.) on the fundamental
antithesis of philosophy, 135 ; on
the intrinsic equation to a curve,
and its application, 457.

Wood (E.T.) on the composition of
Shea butter and Chinese vegetable
tallow, 350.

Wurtz (A.) on a series of organic al-
kalies, homologous with ammonia,
316.

Young (J. R.) on the remainder of
the series in the development of
i+x)-^, and on a theorem respect-
ing the products of squares, 113;
on the expression for the remaining
roots of a complete cubic, 193; on
the decomposition of functions into
conjugate factors, 2/8; on an im-
provement in the analysis of equa-
tions, 413.

Yttro-columbite of Ytterby, on the
composition of, 152.

END OF THE THIRTY-FOURTH VOLUME.

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