Enrico Ferri.

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^' Nee araneuum sane iextas ideo mdior quia ex se flia gignunt, nee notter
Yilior qnia ex alienis libamiu at apes." Just. Lips. P6^t» lib. i. cap. 1 . Not.



Prffif«r« and Publishers to the Universiiy of London ;






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Meditationit est perscmtari occulu; contempUtionii est adminuri

penpicua Admiratio generat qutestionem, quettio iiiTestigationem,

invesdgatio inventioiiem." — Hugo de S, Vietore,

— " Cur Spirent venti, cur terra dehiscat.
Cur mare turgescat, pelago cur tantus amaror.
Cur caput obscura Phoebus femigine coudat.
Quid toties diros cogat flagrare cometas;
Quid pariat nubes, veniant cur fulmina coelo,
Quo micet igne Iris, superos quis conciat orbes
Tam vario motu."

J. B. PinelU ad Masonimm,

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Prof. R. Oausius on different Forms of the Virial 1

Mr. F. P. Purvis on Amsler's Planimeter 11

Prof. A. W. Wright on the Polarization of the Zodiacal Light, 13
Baron N. Schilling on the Constant Currents in the Air and

in the Sea : an Attempt to refer them to a common Cause . . 21
Mr. R. Mallet on the Tidal Retardation of the Earth's Rotation. 88
Mr. E. W. Hilgard on some points in Mallet's Theory of Vul-

canidtir 41

Mr. J. W . L. Glaisher on a New Formula in Definite Litegrals. 53
Dr. J. Rae on some Physical Properties of Ice ; on the Trans-
position of Boulders from below to above the Ice; and on

Mammoth-remains ... '. 56

Mr. P. Clowes on a Glass Cell with Parallel Sides 61

Notices respecting New Books : —

Mr. T. M. Goodeve's Prindples of Mechanics 62

The Rev. S. J. Johnson on Eclipses Past and Future, with

General Hints for Observing the Heavens 64

Proceedings of the Royal Society : —

Mr. W. Crookes on the Action of Heat on Gravitating

Masses 65

Mr. G. Gore on Electrotorsion 70

Proceedings of the Geological Society : —

His Grace the Duke of Argyll on* Six Lake-basins in

ArgyUshire 72

Prof. R. Owen on the Skull of a dentigerous Bird 73

Mr. J. W. Hulke on the Anatomy of Hypsilophodon Foxii. 74
Mr. J. Geikie on the Glacial Phenomena of the '* Long

Isknd" 74

Mr. J. F. Campbell on the Glacial Phenomena of the

Hebrides 75

Prof. P. M. Duncan on Fossil Corals from the Eocene

Formation of the West Indies 76

Mr. R. Etheridge on the Lignite-deposit of Lal-Lal, Vic-
toria, Australia 76

On the Flow of Saline Solutions through Capillary Tubes, by

Theodore Hiibener 77

On Melde's Experiment, by W. Lowery 78

On Constant Electric Currents, by M. Heine, of Halle ..... 79
On the Nature of the Action of Light upon Silver Bromide,
by M. Carey Lea, Philadelphia 80

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Mr. W. Crookes on Attraction and Eepulsion accompanying

Eadiation. (With a Plate.) 81

Mr. J. O'Kinealy on Fourier's Theorem 96

Baron N. Schilling on the Constant Currents in the Air and

in the Sea : an Attempt to refer them to a common Cause 97
Prof. M'Leod on an Apparatus for the Measurement of Low

Pressures of Ghw 110

Dr. W. H. Stone on Wind-pressure in the Human Lungs du-
ring Performance on Wind Listruments 113

Dr. W. H. Stone on the Fall in Pitch of Strained Wires

through which a Ghdvanic Current is passing 115

Mr. H. G. Madan on an Improvement in the Construction

of the Spectroscope 116

Mr. L. Schwendler on the General Theory of Duplex Tele-
graphy 117

Dr. W . H. Stone on a simple Arrangement by which the Co-
loured Eings of Uniaxial and BiaxuJ Crystals may be shown

in a common Microscope 138

Prof. W. F. Barrett on the Modification of the usual Trombone
Apparatus for showing the Literference of Sound-bearing

Waves 139

Notices respecting New Books : —

M. J. Plateau's Statique Exp^rimeutale et Theorique des

Liquides sounus aux seuleis Forces Mol^culaires 140

Mr. W. B. Birt's Contributions to Selenography 141

Proceedings of the Boyal Society : —

Dr. A. C. Bamsay on the Comparative Value of certain
Geological Ages (or groups of formations) considered

as items of Geological Time 143

Prof. O. Eeynolds on the Forces caused by Evaporation

from, and Condensation at, a Sur&ce 146

Proceedings of the Geological Society : —

Prof. W. H. Flower on the Skull of a Species of Halithn'

rium from the Bed Crag of Suffolk 163

Mr. H. Woodward on Forms intermediate between Birds

and Eeptiles .,*.,.»... 154

Mr. J. W. Hulke on the Astragalus of IguanodonManUUi;
and on a very 1r^ Saurian Limb-bone from the Kim-

meridge Clay of Weymouth, Dorset 155

On a Simple Ocular-Spectroscope for Stars, by F. Zollner . . 156
Note on the Cause of Tides, by E. J. Chapman, Ph.D., Professor
of Mineralogy and Geology in University College, Toronto. 157

On the Temperature of the Sun, by J. Violle 158

On a Peculiar Phenomenon in the Path of the Electric Spark,
by Prof. Toepler, of Graz 160

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Captain Abney on the Opacity of the Developed Photographic

Image 161

Mr. C. Homer on the Behaviour of certain Fluorescent Bodies

in Castor-oil 165

Baron N. Schilling on the Constant Currents in the Air and

in the Sea : an Attempt to refer them to a common Cause. 166
Prof. Challis on the Hydrodynamical Theory of the Action

of a Ghilvanic Coil on an external small Magnet. — Part I. . . 180
Prof. A. Stoletow on the Magnetization-Eunctions of various

Iron Bodies 200

Mr. A, Tylor on Tides and Waves.— Deflection Theory. (With

Three Plates.) 204

Proceedings of the Boyal Society : —

Mr. H. E. Boscoe on a Self-recording Method of Measu-
ring the Intensity of the Chemical Action of Total Day-
light 220

Mr. J. Cottrell on the Division of a Sound- Wave by a
Layer of Flame or heated Gas into a reflected and a trans-
mitted Wave 222

Mr. A. E. Donkin on an Instrument for the Composition

of two Harmonic Curves ' 223

Proceedings of the Geological Society : —

Mr. J. W. Hulke on the Anatomy of Hypsilophodon

Foxii 227

Mr. T. Mellard Eeade on the Drift-beds of the North-
west of England 227

Mr. E. D. Darbishire on a deposit of Middle Pleistocene

Gravel near Leyland, Lancashire 228

Mr. H. G. Fordham on the Structure sometimes deve-
loped in Chalk 228

Mr. £. Pinchin on the Geology of the Eastern Province

of the Colony of the Cape of Good Hope 229

Lieut. A. W. Stiffe on the Mud-craters and geological

structure of the Mekran Coast 230

On the light reflected by Permanganate of Potassium, by

Dr. Eilhard Wiedemann 231

On the Temperature of the Sun, by M. J. Violle 233

Physics of the Internal Earth, by D. Vaughan, Esq. 237

On the Conversion of Ordinary into Amorphous Phosphorus
by the Action of Electricity 239


Dr. E. J. Mills on Gladstone's Experiments relating to Che-
mical Mass 241

Dr. E. W. Davy on a very singubr Sulphuretted Nitrogenous

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Compound, obtained by the Action of Sulphide of Ammo-
nium on the Hydrate of Chloral 247

Dr. A. Schuster on Unilateral Conductivity 251

Lord Bayleigh on the Vibrations of Approximately Simple

Systems 258

The late W. S. Davis on a simple Method of Illustrating the
chief Phenomena of Wave-motion by means of Flexible

Cords. (With a Plate.) . . , 262

Prof. A. M. Mayer's Eesearches in Acoustics. — Xo. V 266

Prof. J. J. Miiller on a Mechanical Principle resulting from

Hamilton's Theory of Motion 274

Mr. J. O'Kinealy on a New Formula in Definite Integrals . . 295

Mr. F. Guthrie on an Absolute Galvanometer 296

Notices respecting New Books : —

The Eev. J. F. Twisden's First Lessons in Theoretical

Mechanics 298

Mr. E. Butler s Supplement to the First Book of Euclid's

Elements 300

Mr. F. Cuthbertson's Euclidian Geometry 300

Proceedings of the Boyal Society : —

Mr. J. H. N. Hennessey on Displacement of the Solar Spec-
trum 303

Mr. J. H. N. Hennessey on White liines in the Sdair

Spectrum 305

Messrs. Negretti and Zambra on a New Deep-sea Ther-
mometer 306

Proceedings of the Geological Society : —

Mr. A. B. Wynne on the Physical Geology of the Outer

Himalayan region of the Upper Punj&b, India 310

Mr. E. J. Dunn on the mode of occurrence of Diamonds

in South Africa 311

Mr. J. C. Ward on the Origin of some of the Lake-basins

of Cumberland 311

Mr. D. Mackintosh on the Traces of a Great Ice-sheet in
the Southern part of the Lake-district and in North

Wales 313

Mr. A. W. Edgell on some Lamellibranchs from the Bud-

leigh-Salterton Pebbles 313

On the Action of two Elements of a Current, by J* Bertrand. 314

On Earth-currents, by L. Schwendler, Esq 315

Experiments on the Dissipation of Electricity by Flames, bv

J. W. Fewkes \ 319

On the Stratification of the Electric Light, by M. Neyreneuf . 320


Mr. H. A . Rowland on the Magnetic Permeability and Maxi-
mum of Magnetism of Nickel and Cobalt 321

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Dr. A. Schuster's Experiments on Electrical Vibrations .... 340
Prof. ChaUis on the Hydrodynamical Theory of the Action of

a Gkdvanic Coil on an External Small Magnet. — Part U. . . 350
Sir W. Thomson on the Perturbations of the Compass pro-
duced bj the rolling of the Ship 363

Br. W. M. Watts on the Spectrum of Carbon 369

Prof. A. M. Mayer's Besearches in Acoustics. — No. V 371

Mr. C. Tomlinson on the Action of Solids and of Friction in

liberating G^ &om Solution 385

Prof. O. Eeynolds on the Surfiice-Forces caused by the Com-
munication of Heat 389

Proceedings of the Boyal Society : —

Mr. W. N. Hartley on the Chemical Constitution of Saline

Solutions 391

Mr. G. Gore <hi the Attraction of Magnets and Electric

Conductors 393

On the Temperature of the Sun, by J. Violle 395

Preliminary Notice on a new Method for Measunng the Specific

Heat of Gases, by Eilhard Wiedemann 398

On a new Formula in Definite Integrals, by J. W. L. Glaisher. 400


Dr. C. B. A. Wright on the Relations between Affinity and
the Condensed Sjrmbolic Expressions of Chemical Facts and

Changes known as Dissected (Structural) FormulaB 401

Prof. Challis on the Hydrodynamical Theory of the Action of
a Gblyanic Coil on an external small Magnet. — Part III. . . 430

Prof. A. M. Mayer's Besearches in Acoustics. — No. V 445

Lord Bayleigh on a Statical Theorem 452

Dr. W. M. Watts on Carbon-Spectra 456

Mr. J. W. L. Glaisher on the Problem of the Eight Queens . . 457
Notices respecting New Books : —

The Hon. Sir W. E. Grove's Correlation of Physical

Forces 467

Mr. W. G. WiUson's Elementary Dynamics 471

Proceeduigs of the Eoyal Society : —

Dr. W. Huggins on the Motions of some of the Nebul©

towards or from the Earth 471

On the Intensity of the Light reflected from Glass, by Dr. P.

Glan 475

Pohurization of the Plates of Condensers, by A. S. Thayer . . 478
On Electrical Currents accompanying the non-simultaneous
Immersion of two Mercury Electrodes in various Liquids,
by G. Quincke 479

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M. H. Herwig : the Heat-conducting Power of Mercury in-
dependent of the Temperature 481

Prof. J. Lovering on the Mathematical and Philosophical

State of the Physical Sciences 493

Mr. B. H. M. Bosanquet on Temperament, or the Division

of the Octave 507

Mr. S, Sharpe on Comets and their Tails 512

Prof. A. M. Mayer s Researches in Acoustics. — No. V 513

Mr. F. Guthrie on an Absolute Galvanometer 526

Notices respecting New Books : —

Mr. D. D. Heath's Elementary Exposition of the Doc-
trine of Energy 527

Mr. B. A. Proctor's Transits of Venus 529

Dr. W. Huggins's Approaching Transit of Venus 529

Proceedings of the Boyal Society : —

Prof. O. Beynolds on the Befraction of Sound by the

Atmosphere 530

Mr. T. Grubb on the Improvement of the Spectroscope. 532
Drs. Stewart and Schuster's Preliminary Experiments on

a Magnetized Copper Wire 535

Proceedings of the G^ologicil Society : — '

Mr. J. W. Judd on the Secondary Bocks of Scotland . . 541
Mr. A. W. Waters on Possils from Oberburg, Styria . . 545
On the Cosmic Dust which falls on the Surface of the Earth

with the Atmospheric Precipitation, by A. E. Nordenskiold. 546
On the Passage of Gases through Liquid Films, by Dr. F.
Exner 547

Index 548

Page 203, note f", line ^A^for liniiteil r*ff<iiClo»ed.


L lUuatratiTe of Mr. W. Crookes's Paper on Attraction and Repulsion
accompanving Radiation.

II.. III., and iV. Illustrative of Mr. A. Tylor's Paper on Tides and Waves.

V. Illustrative of Mr. W. S. Davis's Paper on a simple Method of Illus-
trating the chief Phenomena of Wave-motion by means of
Flexible Cords.

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JULY 1874.

I. On different Forms of the Virial. By R. Clausius*.

MY theorem of the virial has already given rise to some
discussions on the forms which the virial can assume.
I myself, in my first memoir relative to itfi indicated that when
the movable points partly exert forces upon one another, and
partly are acted on by forces from without, the virial can be
analyzed into an internal and an external, and gave their forms
for certain frequently occurring cases. Yvon Villarceau subse-
quently [Comptes Rendus, vol. Ixxv.) effected other transforma-
tions of the equation relating to it, especially by resolving the
total motion of the system of material points into the motion of
the centre of gravity and the relative motions of the individual
points about the centre of gravity, and referring the equation to
each of these two constituents singly. Prompted by this, in a
note published in the same volume of the Comptes Rendus I
added a series of further transformations. As, however, in that
brief note results only, without demonstrations, could be com-
municated, and those but imperfectly, a more connected treat-
ment of a subject so important in itself will not be void of

1. The simplest form of the equation in question is the fol-
lowing. If m denotes the mass of a material point which is in
stationary motion together with other material points, of, y, z
its rectangular coordinates at the time /, and X, Y, Z the com-

* Tianslated from a separate impression, communicated by the Author,
from PoggendorflTs Annalen, Jubelband, p. 41 1.

t Bertchie der Niederrhein. Gesellsch. fur Natur- u. Heilkunde, June
1870; Phil Mag. S. 4. vol. xl. p. 122 ; Pogg. Ann. vol. cxli. p. 124.

PhiL Mag. S. 4. Vol. 48. No. 315. July 1874. B

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2 Prof, R. Clau8iu8 on different Forms of the ViriaL

ponents of the force acting upon it^ then

or, if (as will always be done in the following) the first differen-
tial coeflScient according to time be indicated by affixing an

-;r«=:-^X^+5-^ (1«)

From this results, indicating mean values by drawing a hori-
zontal stroke above : —

'^¥'=-\y^x (2)

If we name the quantity ^ a^^ the vis viva with respect to the

^-direction, and the quantity "" 5 ^ ^^^ virial relative to the

a?-direction, since the x- is any direction we please, the meaning
of the equation can be expressed thus : — For each freely movable
point f the mean vis viva relative to any direction is eqital the virial
relative to the same direction.

If we form for a point the equations relative to the three di-
rections of its coordinates and add them up, we get (r denotin<^
the velocity of the point, and / its distance from tlie origin of
the coordinates): —

J. - |(Xx + yy + Z.) + ^^P. ... (3)

If, further, we denote by L the component, in the direction
of /, of the force acting on the point, and i*eckon it positive fi-om
the origin of the coordinates onward, the equation (as is readily
seen) becomes: —

2*'=2^'+4-rfl* (*)

It is obvious that these equations, which are valid for each
individual point, can be extended by simple summation to the
entire system of points. We thus obtain : —

-. m « 1 -., , 1 d^XmP

2.jr»=22L/+^-^^-5- (7)

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Prof. R. Clausius on different Forms of the ViriaL 3

In the formation of the mean values, in all these equations
just as in (1), the last term on the right-hand side falls away ;
and the expression then remaining on that side represents the

2. The first method of transformation of these equations is
based on the fact that when the points are acted on by forces of
different sorts which we wish to consider singly, the force-com-
ponents can be separated into as many summanda as the kinds
of force that are to be distinguished, whereby the virial is divided
into just as many parts.

If, for instance, the above-mentioned distinction be made be-
tween the forces which the points of the system exert on each
other, and those which act upon the system from without, and
this be denoted by the indices i and e, we can put X = X, -f X^;
and the same holds for the components Y, Z, and L. It is
readily seen how the above equations are changed by the inser-
tion of these sums. Equation (6), for example, thereby changes

^is?- ^^)

When more special assumptions are made concerning the
nature of the forces, the expressions also take more special forms,
of which I will briefly cite two which are exhibited in my first
memoir. When, namely, the internal forces consist of reciprocal
attractions or repulsions^ which, according to any law, depend
on the distance, so that for two points whose distance is r the
force (which as an attraction is reckoned positive, and as a repul-
sion negative) can be represented by a function ^(r), we can put

-\l.{7.f + Y,y^Z,z) = \lr<i>{r), ... (9)

in which the sum on the right-hand side refers to all combina-
tions of two mass-points each. When the system of points is
further considered as a body on which the only external force
acting is a symmetrical pressure/? normal to the surface, we can put

-|2(X,^+Y,y + Z,r)=|pV. .... (10)

in which V denotes the volume of the body.

3. Another mode of transformation depends on the separation
of the coordinates of the points into summanda.

To this belongs the transformation effected by Yvon Villar-
ceau. If, namely, besides the fixed systems of coordinates, we


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4 Prof. R. Clausius on different Forms of the Virial.

introduce a movable system having for its origin the centre of
gravity of all the material points^ and parallel to the fixed sy stem,
and if we name the coordinates of the centre of gravity in rela-
tion to the fixed system Xc, y^ ^o ^^^ the coordinates of any one
of the material points in relation to the movable system ^, 77, ^,
then is

^=^c+f, y-Vc^Vy ^='8'r+(;
If we now form the equation

and consider that we may put

we gct^ if M denotes the total mass of all the material points^
consequently the sum Sm, the equation

2wa^=Ma:J + 2wf« (11)

In precisely the same manner we obtain

2W*=M;r'J + 27n^« (12)

Finally, the mere substitution in SXj? of a?^ + f for the coor-
dinate X, Xp denoting the sum 2X, gives

2Xa- = X,^e + 2Xf (13)

If now we form for the centre of gravity the identical equation
which for a single material point has served for the derivation
of (1), viz.

2 <//« ~\dt) '^''' dt*'

which, after multiplication by ^,canbe writtcD thu.4.

and suppose herein

we then obtain

** rf?^^'"5?''^^^^"

2^'- 2^*'+ 4^r« (^*)

With the aid of this equation in conjunction with (11), (12),
and (13), the following equation can be immediately derivtd from


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Prof. R. Clausios on different Forms of the VirtaL 5

2^r»=-^2Xf+l^. . . . (15)

All the equations above derived for the j?-direction, of course
bold good in a corresponding manner for the other two direc*
tions of coordinates ; and when each three equations thereby
arising are added together^ a new system of equations is ob-
tained. In order to write these conveniently^ let us introduce
the following symbols. We will nanie the distance of the centre
of gravity from the origin of the fixed coordinates /<.; and the
distance of a mass-point from the centre of gravity, \. Let the
velocity of the centre of gravity be called Vc, and the relative
velocity of a mass-point about the centre of gravity, consequently
the quantity \/f 4-»/*+|^*, be called w. Further, of the force
whose components in the coordinate-directions are X^, Yc, Z<.,
the component in the direction of /« may be denoted by 17^ ; and
of the force acting on a mass-point, let the component in the

Online LibraryEnrico FerriThe London, Edinburgh and Dublin philosophical magazine and journal of science → online text (page 1 of 58)