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CAMBRIDGE UNIVERSITY PRESS
PUBLICATIONS.
ELEMENTS OF NATURAL PHILOSOPHY. By
Professors Sir W. THOMSON and P. G. Tait. Part I. Second
Edition. 8vo. cloth, 9J.
THE ELECTRICAL RESEARCHES OF THE
HONOURABLE HENRY CAVENDISH, F.R.S. Written be-
tween 1771 and 1781, Edited from the ori^nal manuscripts in the
possession of the Duke of Devonshire, K.G., I>y J. Clerk Maxwell,
F.R.S. {Nearly ready.
MATHEMATICAL AND PHYSICAL PAPERS. By
George Gabkiel Stokes, M.A., D.C.L., LL.D., F.R.S., Fellow of
Pembroke CoUege a.nd Lucasian Professor of Mathematics in the
University of Cambridge. Reprinted from the Origin^ Journals and
Transactions, with Additional Notes by the Author. [Tn the Press.
HYDRODYNAMICS, a Treatise on the Mathematical
Theory of Fluid Motion, by HORACE LAMB, M.A., formerly Fellow
of Trinity College, Cambridge; Professor of Mathematics in the
University of Adelaide. [/» the Press.
THE ANALYTICAL THEORY OF HEAT. By
Joseph Fourier, Translated, with Notes, by A. Freeman, M.A.,
Fellow of St John's College, Cambridge, Demy Octavo. \f>s.
AN ELEMENTARY TREATISE ON QUATERNIONS.
By P. G. Tait. M.A., Professor of Natural Philosophy in the
University of Edinburgh j formerly Fellow of St Peter's College,
Cambridge. Second Edition. Demy 8vo. 141.
®ainbi&i(ff :
AT THE UNIVERSITY PRESS.
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NATURAL PHILOSOPHY.
,Goog^[e
CAMBRIDGE WAREHOUSE,
n, PATBKNOBTBK ROW,
Cimtiilillt i DEIQHTON, BULL, AND CO.
Etfpfig: V. A. BROCKBAlTfi.
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TEEATISE
NATURAL PHILOSOPHY
K.. ... >«?; .... .. ,
SIB, WILLIAM THOMSON,. LL.D, D.O.L., F.RS.,
PETER OOTHBIB TAIT, M.A.,
VOL. L PABT I.
NEW EDITION.
CanArRigt:
AT THE UNITEKSITY PRESS.
1879
[T** r^Att of triHUlation and reproduetion art raervti.]
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PBINTXD BY 0. J. OUT, H^.
AT IHZ CHITSBaiTZ PUBS,
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PREFACE.
IiSB OMiBM ptimoidiales ne noas sunt point oonnaM; mail dtea mnt wm-
jettieB k iet lois siicples et oonstantes, qne I'on pent dGconvrir pax I'obBei-
nUioD, at dont I'jtnda eit t'objat de It, philoaophie natareUe. — FooBims.
The term Nataral Philosophy was used by Newtoh, and is
still aaed in Britisli TTniTersities, to denote the investigation of
laws in the material world, and the deduction of results not
directly observed. Observation, classificatioD, and description
<rf phenomena necessarily precede Natural Philosophy in every
department of natural science. The earlier stage is, in some
branches, commonly called Natural Histoiy; and it might with
equal propriety be so called in all others.
Our object is twofold : to give a tolerably complete account
of what is now known of Natural Philosophy, in langutige
adapted to the non-mathematical reader; and to furnish, to
those who have the privilege which high mathematical acquire-
ments confer, a connected outline of the analytical processes by
which the greater part of that knowledge has been extended
into r^ons as yet unexplored by experiment.
We commence with a chapter on Motion, a subject totally
independent of the existence of Matter and Force. In this
we are naturally led to the coneideration of the curvature and
tortuosity of curves, the curvature of surfaces, distortions or
strains, and various other purely geometrical subjects.
62
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The Law8 of Motion, the Law of Oratn'toJton and of Electria
and Magnetic Attractions, Hookas Law, and other fundamental
principles derived directly from experiment, lead by mathe-
matical processea to interesting and useful results, for the full
testing of which our most delicate experimental methods are as
yet totally insuflicient. A laige part of the present volume is
devoted to these deductions; which, though not immediately
proved by experiment, are as cert^nly true as tbe elementary
laws from which mathematical analysis has evolved them.
The analytical processes which we have employed are, as a
rule, such as lead most directly to the results aimed at, and are
therefore in great part unsuited to the general reader.
We adopt the suggestion of Aup&re, and use the term
KinemaMcs for the purely geometrical science of motion in
the abstract. Keeping in view the proprieties of language, and
following tbe example of the most logical writers, we employ
the term Dyna/mios in its true sense as the science which treats
of the action oi force, whether it maintains relative rest, or pro-
duces acceleration of relative motion. Tbe two corresponding
divisions of Dynamics are thus conveniently entitled StaUca and
One object which we have constantly kept in view is tbe
grand principle of the Gomerva^ion of Energy. According to
modern experimental results, especially those of Joule, Energy
is as real and as indestructible as Matter. It is satisfactory to
find that Newton anticipated, so far as the state of experi-
mental science in his time permitted him, this magnificent
modem generalization.
We desire it to be remarked that in much of our work,
where we may appear to have rashly and needlessly interfered
with methods and systems of proof in the present day generally
accepted, we take the position of Bestorets, and not of Inno-
vators.
In our introductory chapter on Kinematics, tbe consideration
of Harmonic Motion naturally leads us to Fourier's Theorem,
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PEEFACE. Til
one of the most important of all analytical results as regards
usefulness in physical science. In the Appendices to that chapter
we have introduced an extension of Green's Theorem, and a
treatise on the remarkable functions known as Laplacf^s Co-
efficients. There can be but one opinion as to the beauty and
utility of this analysis of Laplace ; but the manner in which it
has been hitherto presented has seemed repulsive to the ablest
mathematicians, and difficult to ordinary mathematical students.
In the simplified and symmetrical form in which we give it, it
will be found quite within the reach of readers moderately
familiar with modem mathematical methods.
In the second chapter we give Newton's Laws of Motion in
bis own words, and with some of his own comments — every
attempt that has yet been made to supersede them having
ended in utter &iilur& Perhaps nothing so simple, and at
the same time so comprehensive, has ever been given as the
foundation of a system in any of the sciences. The djmamic^
use of the Generalized Coordinates of LaqbanQe, and the Vary-
it^ Actum of Hamilton, with kindred matter, complete the
chapter.
The third chapter, " Experience," treats briefly of Observa-
tion and Experiment as the basis of Natural Philosophy.
The fourth chapter deals with the fundamental Units, and
the chief Instruments used for the measurement of Time, Space,
and Force.
Thus closes the First Division of the work, which is strictly
preliminary, and to which we have limited the present issue.
This new edition has been thoroughly revised, and very
considerably extended. The more important additions are to
be found in the Appendices to the first chapter, especially that
devoted to Laplace's Coefficients; also at the end of the second
chapter, where a very full investigation of the "ci/cloidal
motion" of systems is now given; and in Appendix B', which
describes a number of continuous calculating machines invented
and constructed since the publication of our first edition. A
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great improvement has been made in the treatment of La~
grange's QeneraUzed Eqttaiions of Motion.
We believe that the mathematical reader -will especially
profit by a perusal of the large type portion of this volume ; ae
he vill thus be forced to think out for himself what he has
been too often accustomed to reach by a mere mechanical
application of analysis. Nothii^ can be more fatal to progress
than a too confident reliance on mathematical symbols ; for the
student is only too apt to take the easier course, and ccHisider the
formvla and not the /act as the physical reality.
In issuing this new edition, of a work which has been for
several years out of print, we recognise with legitimate satis-
fiictioD the very great improv^neut which has recently taken
place in the more elementary works on Dynamics published in
this country, and which we cannot but attribute, in great
part, to our having effectually recalled to its deserved pora-
^Xwi. Kewton's system of elementary definitions, and Laws of
Motion.
We are much indebted to Mr BtntKSiBE and Prof. Chbtstal
for the pains they have taken in reading proofs and verifying
formulas ; and we confidently hope that few eiratums of serious
consequence will now be found in the work.
W. THOMSON.
P. G. TAIT.
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CONTENTS.
DIVISION I.— PRELIMINARY.
Chutxb L-^EIHEMATICS. '
Olijeatii of the Chaptw
Hotum of • Point
CnrratiiTe ol m Plane Cnrre
Cnmtnia and Tortoorit; □! a Tortooiu Curve
Integnl Comtore ol a Onrve (eompAre g 1S6)
Flexible Line — Oord in Heohuiism
Evolnte and InTolate .
Beeolation of Talodty
Compomtion of Telodties
Besolntion and Oompomtion of Aooelerationa
Determinatioii ot the Motion from giTen Telod^ oi Au-
Mleration
Aaoeleration diregted to a Fixed Centre
'Boiognpb .
CniTM of Pnmiit
Angnlar Telocity and Aoeeleratiou
Belative Motion
Beanltant Motion
HamMoiia Motion .
Compoeitlon ot ffimide Harmonio Motiona in one Line
Heehaniam for componndjn^ and Oraphioal Bepreaentation ot,
Hannmii! Motlong in <Bte Line
Cttmpoaition ol S. H. M. in difFerant diraetionB, inclnding
Composition of two UnUom Cinnilar Motions
8,«
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7— B
10-18
14-16
17—19
20—21
26,96
97
87—89
40
41-44
46— 49
60, fil
62—67
68-61
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CONTENTS.
DiaplaeementB of & Fluie Figure in its Plana — Campoaition
ol Botations about Parallel Aiee— CompoBitiou ol Bota-
tiouB and TrauBlationa in One Plane — Bapeiposition of
Small Motion*— RoUing of Cotre on Cnrre— Oyoloida and
Trooboide— Properties of the Cjeloid — Spieydoid, H;-
poojcloida, eto. .....
MoUoQ of a Bigicl Bod; about a Fixed Point — Enter's Tbeoian
— Bodiiguee' Co-ordinatee — Composition ol Botationa —
Compoaition of Angnlar Telooities — Compoaition of roo-
eesaive Finite BotatioDa — Boiling Cones — Poeition of the
Body doe to pvea Botationa
Hast general Motion of a Bigid Bodj
Preceaaional Botation^Model illiutmtiiig Freoeasiou of Eqoi-
Free rotation of a Body IdneticaUy Eymmetrioal about an azia
Conunanication of Aognlat Telocity eqnaUy between Inclined
Axes — Hooke's Joint — UniTeraal Flexore Joint-^Elostia
UniTeraal Flemte Joint— -Moring Body attaolisd to a
Filed Objeot by a Uniyersal Fleinra Joint— Two Degreea
of Freedom to more enjojred by a Body thna anspeuded ,
General Motion of one Bigid Body tooohing another — Cmre
rolling ott' Carre — Plane Carres not in same Plane—
Gnrre rolling on Carve ; two degrees at freedom — Carre
rolling oQ' Snrfaee; three degreea of freedom — Traoa
presoribed and no Spinning permitted; two degrees ol
freedom — ' Angnlar Telodty of direct Boiling — Angalar
Velocity roand Tangent — Snrlaoe on Saiface — Both
tiaoea presoribed ; one degree of freedom
Twist — Estimation of Integral Twist in a Plane Cmrre; in
a Carre conaiating of plane portions in different Planea;
in a aoQtiuuoQsly Tortaoaa Carve — Djnamios of Twist
in Einks .....
Sorfaoe rolling on Snrfaee ; both traees given
Surface rolling on Sorfaoe withont apinning
Eiamples of Tortuosity and Twist
CnrTBtnre of Sorfaoe— Synclaatio and Antielaatia Surfaces —
Mennler's Theorem ^ Eoler'a Theorem — Defloition ol
Line of Ourrature — Shortest Line between two points
on a Bnrfaoe — Spherioal Exeesa — Area of Spberieal
Polygon — Beeiprocal Polars on a Sphere — Integral change
of direction in a Boifaoe — Change of direction in a Snr-
faee of any arc traced on it
Integral Carratore—Cureiituryi tnteffra — Horograph — Change
of direction roand the bonndary in the anrface, together
with area of borograph, eqnals tonr right angles; or "In-
tegral Carvatare" eqoals " Carvatura inUffra"
Analogy between Lines and Sorfaoea aa regards Carvatare —
Horograph— Area of the Horograph .
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C0XTKNT8.
Flexible and laexteiuible Boifoee— Flexnn of iuaitenHible
Developable— Edge ot BegreBsion — Fraotical Courtmotion
of a Developable bom its edge — General propertj of
Ineitetuible Barfaoe— Snrtaoe of aonstant Bpealfio Cnr-
Tatnte — Oeodetio Triangles on Eiuh a Smfaoe .
Strain — DeBnition ol Homoganeoiu Strain — Piopertiet of Ho-
mogeneoQfl Strain— Strain Ellipsoid — Change of Yolnme
— Axes of a Strain — Elongation and Changs of Direction
of any Line at the Body— Cbhnge of PUn« in the Body^
Conical Bnriaoe of eqnal elongation- Two Planes of DO
distortion, being the Circnlar Sections of the Strain Ellip-
Boid — Distortion in Parallel Planes without Change ot
Volume — Initial and altered Position ot Lines of no
Elongation — Simple Shear — Axes of a Shew — Mewnre
of a Shear— -Ellipsoidal speoification of a Shear — Analysis
Ol a Strain . . . . :
T not, one point of wfaidi is
kniijna of a Strain into Distortion and notation . .
Pore Stnun — Composilioa of Pnre Strftins .
Displacement of a Carre — Tangential Displaoement — Tan-
gentifll Diaplaeement of a Closed Cture — Botation of a
Bigid Closed Curve — Tangential Di^laeement in a Solid,
la terms of Components of Strain — Eeterogeneons Btrun
— EomogeneOQs Strain — Infinitely small Strain — Moat
general Motion of Matter— Change ot Poution of a Bigid
Body^Non-rotational Strain — Displaoemant Frmotion .
"Bqnation of Continnity" — Integral Eqnation of Continuity —
Differential Eqnation ot Continaity — "Steady Motion'*
defined ,,...■
Freedom and Constraint— Ot a Point— Of a Bigid Body— Oeo^
metrical Clamp— Oeometrioal Slide— Examples of Oeo-
metrical Slide— Examples of Oeometrieal Clamps and
BUdee— One Degree of Constraint of the most general
eharaoter- Uechanioal ninetration— On« Degree ot Con-
straint eipreeeed analytically
Generalized Co-ordinates ^ot a Point — of any system —
Oeneralized Components ot Telocity — Examples
195—301
203—301
Aipmna A«. — Expression in Oeneralized Co-ordinates for Foieson's
Extension ot Laplaoe's Equation.
Afrmroix A. — Extension of Qreen's Theorem.
ArPEsnix B.— Spherical Eaimonic Analysis.
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CaipnB n.— DTNiJSaCAL LAWS AST) PBIHCIPLEa
Ideu of Ufttter and Foroe intiodnoed — Matter — Foree — UosB
— Density — MeaBDKment of, Mmb — Momentam — Cbange
of Momentom — Bats of ohauge ol Momentnm—Einetio
Energy — Particle and Point — Inertia .
Force — Spedfiaation ol a Faroe~Flaoe at Applioation — Direa-
tion— UagnitDde — Aooelerative Efleot — MeaEnre o{ Foroe
Standarda of Weight ors Mauei, and not primarily intended
for MeBBOrement of Forc« — Clairant'B Formula (or the
Amount of GraTity — Osdbs'b abBoInte Unit of Foroe —
Uazwell'B two snggestionB for Abaolnte Unit of Time —
Third anggeation for Absolate Unit of Time — British
Abwlnte Unit — Comparisoii with Gravity
Besolniion of Forces — Efleotife Gomponect of a Foroe
Oeometriaal Theorem p'reliminary to Definition of Centre of
Inertia — Centre of Inertia . ' .
Moment — Moment of a Force about a Point — Moment of ft
Force about an Axis ....
DigrcBBion on projectioa of Areas . , ,
Con^e — its Moment, Ann, and Axis
Moment of Velocity— Moment of Momentum — Moment of a
Bectilinsa] Displacement — For two Forces, Motions, Te<
lodtiea, or Momentoma, in one Plane, the Sum of theii
Moments proved eqnal to the Moment of their Besnltant
round any point in that Plane — Any nombei of Moments
in one Flans eompoDuded by addition — Moment ronnd
an Axis — Moment of a whole Motion round an Ajdfl — Be-
snltant Axis .....
Virtual Telocity— Tirtnal Moment
Work— Practical Unit— Boientiflo Unit— Work of a Force-
Work of a Conple — Transformation of Work— Potential
2ai— S26
337,238
Newton's Laws of Motion — First Law — Best — Time — Ex.
amples of the Law— Directional Fixedness — The "In-
variable Plana" of the Solar Syatom — Second Law —
Composition of Foroes—Measnrement of Foroe and Mass
— Translationa from the Einematios of a Point — Third
Law — -D'Alembert's Principle — Mntual Forces between
Particles of a Bigid Body — Motion of Centre of Inertia
of a Bigid Body — Moment of Momentum of a Bigid
Body — Conserration of Momentmn, and of Moment of
Momentum — The "Inrariable Plane" is a Plane throng
the Centre of Inertia, perpeudionlar to the Besnltant Axis
—Terrestrial AppUeation—Bate of doing work— Hoiee-
povBi- EnersT ^ Abetraot Dynamics
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CONTENTS.
ConBerrative Byrtem — FoDud&tion of the Theory of Energy —
PbTUoal oiiom that "the Perpetaal Motion is impoaaible"
introdnoed — Potential E le^; of ConaerrAtiTe Bjrgtein
Ineritable logs of Energy of Visible Motione — Effect of Tidal
Frietlon — Ultiroale tendeooy of the 8olai SyBtem
CoiueivBtion of Energy ....
Kinetio Energy of a System — Moment ot Inertia — Moment of
Momentum of a Botnting Rigid Body— Badios of Qyr&tiou
—Flj-wheal— Moment ot Inertia abont any Axis
Momental Ellipeoid— Eqailibration of Centrifiigal Foroea —
Definition of Principal Axes of Inertia — ^Prinoip&l Axes —
Binet'B I^eorem— Central EUipBoid— Eiuetia Symmetry
lonnd a Point ; round an Axis
Energy in Abstraot Dynomios ....
EqDiUbrimn— Prinaiple of Tiitnal Telooities— Kentral Eqni-
librinm — Stable Eqnilibriom— Unstable Eqnilibiiiun —
Teat ol the natnie of EqniUbrinm
Dednotion of the Eqnstions ot Motion ol any System — Inda-
termiiuile Equation ot Motion ol any System — ot Gonsa-
Tatire System — Eqaation of Energy— Constraint intro-
dnoed into the Indeterminate Bqnataon — Determinate
Equations of Motion dedaoed — Oanss's Prinoipla of Leait
Conttraint , , . , .
Impaot— 'nme-integral^Ballistie Fsndalnm — Direct Impaot
of Spheres — Effect of Elastid^— Nenton's Eiperiments —
Distribation of Energy after Impnet — Newton's experi-
mental Law eonsfstent with per/ect Elasticity .
Moment ot an Impact about an Axis — BalllBtio Pendnlnm —
Work done by Impact — EqoationB ot ImpolBive Motion ,
Theorem of Enler, extended by Lagrange — Liqnid set in Motion
impnMTely— ImpnlsiTO Motion referred to Oencralized
Co-ordinates — Generaliied Expression tot Kinetic Energy
~^}eneralized ComponentB of Force — ot Impnlu — Im-
pnlEive Qeneration of Motion referred to Oeneralizsd
CtMirdinates — Momentnms in terms of Telooitiea — Kinetio
Energy in terms ol Momentnma and Velocities — Velo-
idtieB in terms ot Momentnms — Beciproeal relation be-
tween Momentum and Velocities in two Motions — Ap-
plication of Oennalized Co-ordinates to Ttieorenu ot
% 811 — Problems whose data inTOlva Impalsea and Velo-
cities— Qeneral Problem (compare g S13) — EJnetio Energy
A |ni¥^iTnT^m in iViiu oaBe— -Examplcs .
I«gnaige's Equations of Motion in terms of Generalized Co-
ordinates dednoad direct l^ transformation from the
Equations of Motion in terms of Cartesian Co-ordinates
— Eqnation ot Energy— Hamilton's Ftom— " Oauonioal
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CONTENTS.
form " of Haiiulton's general EqnationB of Ktotion ol a
ConservatiTe S7Btem — Biomples ol (he use of Iiigruige'B
Qeneralized EqnationB of Uotion— GTToaoopes and G;-
roBlata — OyioBCopio Pendolam — Ignoistion o{ Co-ordi-
Einetiea of a perfect fluid— ESeot of a Rigid Plane on tlie
Motion of a Ball throngh a Liquid — Seeming Attntctioii
between two ships tuoring Hide bj side in the aame
direction — Quadrantal Pendulum defined — Motion of a
Solid of Bevolntion with its axis always in one plane through
a Liqaid — Observed phenomena — Applications to Nautical
Dynamics and Ganneiy — Action — Time Average of
Energy — Space Average of Momentomi — Least Action —
Principle ol Least Action applied to find Lagrange's
Generalized Equations of Motion— ^hy called "Station-
ary Action" by Hamilton— Varying Action — ^ Action
expieesed aa a Function of Initial and Final Co-ordinates
and the Energy; its differential Coeffioients equal ro-
speotively to Initial and Final Momentnms, and to the
time from beginning to end — Same Fropoaitione foi 0«-
ueroUzed Go-ordinatea — Eamillon'a "Characteristic Eqna-
tion" of Motion in Carteaian Co-ordinates — Eamillon'a
Characteristic Equation of Motion in Qeneralised Co-or-
dinates— Proof that the Oharactenstte Equation deOnea
the Motion, for free particlea — Same PropoaltioD for a
Connected System, and Generalized Co-oidinatea— Ea-
miltonian form of Lagrange's Generalized Equations de-
dnoed from Characteristic Equation . . ■ I
Chaiaoteristio Fnnction — Characteristic Equation of Motion —
Complete Integral of Characteristic Equation — General
Solution derived from complete Integral — Practical In-
terpretation of the complete Solution of the Charaoteristio
Equation — Properties of Surfacea of Equal Action —
Examples of Varying Action — Application to common
Optics or Kinetioe of a Single Particle — Application to
System of free mutually influencing Fattioles — and to
Generalized System . . • > 1
BlJghUy disturbed Equilibrium— Simultaneous Transformation
of two Quadratic FnnotionB to Snms of Squares — Gene-
Tolized Orthogonal Transformation of Co-ordinates — Sim-
plified expressions for the Kinetic and Potential Energies
— Integrated Equations of Motion, expressing the fun-
damental modee of Vibration ; or of falling away from
Configuration of Unstable Equilibrium — Infinitely small
Disturbance from Unstable Equilibrium — Potential and
Kinetic Energies expressed as Functions of Time —
Example of Fundamental Modes
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CONTENTS. XV
umoHB
Oenaitd Theorem of Foudameutal Modes ol infinitely smtQ
MMoa aboQt a ConfigniBtion ol Eqnilibriimi — Nomul
JMspboemeiita fnmi Eqailibrinm — Theorem of Eiuetio
Energy — Of Potential Eoeig; — Inflniteeiiaal Uotiona in
neighbonrliood of Conflgimtidn of Unstable Eqailibriom 6S8
CaM of Bqnalitj among Periods — Graphie BepreBeatation —
IHiaipatiTe Bystemi — Tiews of Stokee an BesiBtance to a
Bolid moving throneh a Liqnid — Friction ol Solids —
BematanoeB vatjing aa Telooities — ESeot of Besiatauoe
Taiying aa Telooitj in a simple Motion . 839 — Ul
Infinitety small Motion of a DissipatiTe SjBtem— Cycloidal
System defined — Positional and Motional Forces — Differ-
ential Equations of Complex Qyoloidal Motion — Tbeit
Solution — Algebra ot Linear Equations—Minors of a De<
terminant — BelatloDS among the Minors of an Evanescent
Determinant— Case of Equal Boots— Case of Eqnal Boots
and Evanesoent Minors^Bontb's Theorem — Case ol no
Motional Forcea — Conservative Positional, and no Mo-
tional, Forces — Equation of Ene^y in Beallzed Oeneral
Bolation .... . &12— M3p
Artifimal or Ideal AecomulatiTe System — Criterion of Sta-
bility—Cycloidal System with Oonservative Positional
Poroes and Unrestricted Motional Foroes — Dissipativity
defined — Lord Bajlelgb's Theorem ot Dissipt^Tity — In-
tegral Equation of Energy — Seal part of every Boot of
Determioontal Equation proved negative when Potential
Bnei^ is positive for all real Co-ordinates ; positive for
some Boots when Potential Energy has negative valaes ;
bat alv^B negative for some Boots — -Non-oscillatory sub-
sidenoe to Stable Bqailibrinm, or falling away from Un-
stable — Osoillatocy snbsidence to Stable Equilibrium, or
falling away from Unstable— Falling away from wholly
Unstable Equilibrium is essentially non-oscillator; if
Motional Forces wholly viseone— Statiility of Dissipative
System — VarioaB origins of OyroKopie Terms — Equation
ot Energy — Qyrostatio Conservative System — Bimplifloo-
tion of its Equations — Determinant of Oyrostatio Conser-
vative System — Square Boots of Skew Symmetrio De-
terminants— Gyroatatic System with Two Freedoms — Gy-
rostatio Infloenoe dotoinant— Oyrostatio Stability— Ordi-
nary Gyrostats — Gyrostats, on pmbals ; on universal
flezuTe-joint in plaee of gimbals ; on stilts ; bifilarly slang in
looi ways — Gymstatla System with Three Freedoms — Be-
duoed to a mere rotating System — Qaadmply free Qyro-
static System without toroe — Bzoepled ease of failing gy.
lostatic predominance — Qoadn^ly free Cyoloidal System,
gyrostatically dominated— Four Irrotational Btabilltiea
oimflrmed, toor Irrotational Imtabilitiee rendered stable,
..Google
CONTENTS.
bj OyiMtatiD Links— Combined Dynuuia and Oyroatatio
Stsbilit; gyroatatiaallj connteraoted— Heolizfttion ol Com-
pleted Solution — Besultfuit Motion reduced to Motion ol
a ConserTBtive Systeni with four fandmnental periodi
equal two and two^-OrthogonalitieH proved between
two componenta of one ftmdamental OMillation; and
equality of theii Gneigiaa—OrthogonalitieE proved be-
tween different fondamental osoiUatious — Caae ol Equal
PeiiodB— Completed Bolntiou loi case of Equal Periodi
-^Two higher, and two lower, of the Four Fonda-
meotal Oscillttiona, dmilarly dealt with by Bolaticm
of two Bimilai Qnadiaties, provided that gyroitatio in-
flnence be tolly dominant — Limit* of amalleat and
seeond smallett of the four periods— Limits of the next
greatest and greatest of the tonr perioda — Qnadmply
free Cyoloidal System with non-dominant gTrostatio in-
flnenees — Oyroatatio Syslem with uiy nnmber of freedoms
— Case of Equal Boots with stability — Applioation of
Bouth's Theorem— Equal Boots with metabillty in traa-
mtional oases between Stability and Instability — Condi-
tions of gyrostatio domination — Oyrostatio Links ex-
plained— GyroBtatioally dominated System : its adynamic
osoillatione (very rapid) ; and preoessional ogeillationl
(very slow] — CompaiiBon between Adyntunio Freqnenoieg,
Botational Freqaenoies of the Fly-wheels, Preoeasionsl
Frequennes of the System, and Frequenoies or Bapidities
of the System with Fly-wheels deprived oE Botation —
Proof of reality of Adynamic and of PreoeHgional Periods
when system's Irrotational Ferioda are either all real or
all imaginary — Algebraic Theorem . . BU — 846*'"'''
Einetie Stability — Oonservative disturbanoe of motion — Ki-
netic Stability and Instability diaoriminated — Examples
— Cironlar Simple Pendolam — Einetic Stability in Cir-
cular Orbit— Einetio Stability of a Particle moving on a
Smooth Surfs oe — Incommensurable OsoiUations — Osdl-
IatoT7 Einetio Stability — Limited Einetio Stability —
Einetie Stability of a Projectile — General oiiterion — Ex-
amples— Motion of a Particle on an antiolastia finrface,
□nstable; — on a Bynolastio Surface, stable — DifFerential
EqaatioD of Disturbed Path . . &4S— S6C
Oeneral investigation of Disturbed Path — Differential Equa-
tion of Disturbed Path of Single Particle in a Plane— Ei-
netie Foci — Theorem of Minimam Action — Action never a
Minimum in a oooree inolnding Einetio Foci — Two or more
Courses of Minimum Action possible — Case of two mini-
mDm, and one not minimum. Geodetic Lines between two
Points — Differenoe between two sides and the third of a
Kinetic Uiangle — Actions on different eonrses infinitely
..Google
CONTENTS.
nev one Miother between two ooDjagate Emetic Fod,
proved nltimatel; eqnal— It two sides, deviating infinitely
little (rom the third, are together eqnal to it, they con-
stitate an nnbraken ustuial oonne — Nstanl oonne proved
not • course of Minimnm Action, beyond a Einetia Foena
— A course which indndea no FoonB oonjngate to either
eitranitf inolndee no pui of Mmjngate Pod — How many
Einetio Fod in aniy ease — Theorem ot MaxImDm Aotiou
Cupm m.— EXE'BBEENCB.
Observation and Experiment — Bolee tor the oondaot of Ei-
periments — Betidnal phenomena — Unexpected agreement
or disoordanoe ol reanlts ot different trials
Hypotheses — Dednotion ot most prohabte result from a num-
ber of observations — Law of Error— Probobla Error —
Probable Error ot a Sam, Difference, or Multiple — Prao-
tioal application — Method of Least Squares — Methods re-
presenting experimental results — Curves — Interpolation
and Empirioal Formnlfs ....
CoinBB IT.— MBABUBEB AND IHSTBUMENT8.
Heeessity of aecnrate MeasnrementB-^lasBee of Instnunents
— Calonlating Machines — Angular Msaanre — Measure of
Time— Neaessity for a Feremiial Standard. A Spring sng-
gested — Ueastue of Length, fonnded on attifidal Metallic
Standards — Measoiea of Length, Surface, Yolnme, Mass
and Work . . . . .
dook-^Electricalljoontrolled Clocks — Chronosoope — Diagi^ial
Scale — Vernier — Screw — Screw - Micrometer — Sphero*
meter — Gathetometer — Balanas — Totsion-balanoe— Pen.
dnlom — Bifllar Balance — Bifllar Magnetometer — Ahsolate
Measnrement ot Terrestrial Mognetie Foi
— Friction Brakes
414— 4B7
Appebdh B'.— CONTINUOUS CALCULATING MACHINES.
I. TidB-in«dicting Machine.
n. Machine for the Solution of Simoltaneone Linear Equations,
m. An Integrating Machine having a New Sinematic Principle — Disk-,
Globe-, and Cylinder-Iotegrator.
IT. An lustrmnent for calonlating /0[z) ^{x) dat, the Integral ot tho
Prodnct of two given Fonotione,
T. Heehanical Integration of Linear Differential Eqoationa ot the
Beoond Order with Variable CoefSoients.
TI. Meohanioal Integration ot the general Linear Differential Equation
of any Order with Variable Coefficients.
Vn. Earmonio Analyzer — Tidal Harmonio Analyzer— Secondary, ter-
tiary, quaternary, etc., tides, dna to inQuenca of shallow water.
..Google
jiGoogle
DIVISION I.
PRELIMINARY.
CHAPTER I.— KINEMATICS.
1. THEltE are many properties of motion, displacement, and
deformation, which may be considered altogether independently
' of such phyedctd ideas as furce, mass, elasticity, temperature,
magnetism, electricity. The preliminary consideration of sucli
properties in the abstract is of very great use for Natural Philo-
sophy, and we devote to it, accordingly, the whole of this our .
first chapter; which will form, as it were, the Geometry of our
subject, embracing what can be observed or concluded with re-
gu:d to actiial motions, as long as the cause is not sought.
2. In this category we sball take up first the free motion of
a point, then the motion of a point attached to an inexteneible
cord, then the motions and displacements of rigid systems — and
finally, the deformations of surlaces and of solid or fluid bodies.
Incidentally, we shall be led to introduce a good deal of ele-
mentary geometrical matter connected with the curvature of
lines and surfaces.
8. When a point moyes from one position to another it must HntionorB
evidently describe a continuous line, which may be curved or
stnugbt, or even made up of portions of curved and straight
lines meeting each other at any angles. If the motion be that
of a material particle, however, there cannot generally be any
such abrupt changes of direction, since (as we shall afterwards
aee) this would imply the action of an infinite force, except in
the case in which the velocity becomes zero at the angle. It
is ns^ul to coneidw at the outset various theorems connected
..Google
2 PBELIMIKABT. [3.
uotiouotft with tbe geometrical notion of the path desoribed hy a moving
point, and these we shall now take up, deferring the considera-
tion of Velocity to a future eectioo, as heing mrae closely con-
nected with physical idea&
4. The direction of motion of a moving point is at each
instant the tangent drawn to its path, if the path he a curve, or
the path itself if a straight line.
*r''™5Se ^' ^^ '^^ P**''' ^ ^^^ straight the direction of motion
•"■^ changes from point to point, and the rate of this change, per
unit of length of the curve f-j- according to the notation below) ,
is called the curvature. To exemplify this, suppose two tangents
drawn to a circle, and radii to the points of contact The angle
between the tangents is the change of direction required, and
the rate of change is to be measured by the rehition between
this angle and the length of the circular arc Let / be the
angle, c the arc, and p the radius. We see at once that (as
the angle between the radii is equal to the angle between
the tangents)
pl= 0,
and therefore - = -. Hence the corvature of a circle is in-
0 />
versely aa its radius, and, measured in terms of the proper unit
■of curvature, is simply the reciprocal of the radius.
6. Any small portion of a curve may be approximately
taken as a circular arc, the approximation being closer and
closer to the truth, as the assumed arc is smaller. The curva-
ture is then the reciprocal of the radius of this circle.
If $0 be the angle between two tangents at points of a curvo
distant by an arc 8s, the definition of curvature gives us at once
as its measure, the limit of g- when & is diminished without
limit ; or, aeoording to the notation of tlie difiereatial oalcnlus,
J- . But Yte have
if, the curve being a plane curve, we refer it to two rectangtilsr
..Google
KIKEUATICB.
ares OX, 0 T, mccording to the Cartesian metliod, and if 0 denote Cn
the inclination of ita tangent, at any point x, jf,to OX. Hence n
and, b; differentiation with reference to any independent variable
t, we have
dxtFjf-dyd'x
AIM, d« = (daf + dj/)i.
e, BO that
(1),
Heocie, if p denote tlie radioi of curvature, bo that
l_d$
p~ da
, , 1 dxd*ti-dy^x .„.
weconclnde -= ^-j— (2).
Although it is generally convenient, in kiaematical and
kinetic fbrmtilte, to regard time aa the independent variable, and
all the chan^ng geometrical elements as functions of it, there
are casee in vhich it is luefal to regard the length of the arc or
path described by a point as the independent variable. On this
aupposition we have
0='d(dt') = d{da^ + dy^ = 2(dKd*x + difd.'y),
where we denote by the suffix to the letter d, the independent
variable understood in the diflfereatiation. Hence
^_ djf _ {dt^ + d^^
<y ~~ ^*« ~ {(d,'i/y + K'sb)'}! '
and nnng thcee, with dif = d3^A- di^, to eliminate dx and dy
from (2), we hare
1 {(rf.'y)'-^(rf»'}V
p d^
or, according to the usual short, although not quite complete,
notation,
7. If all points of the curve lie in one plane, it is called a Tortnooi
fiane cvrve, and in the same way we speak of a plane polygon
or broken line. If various points of the line do not lie in one
plane, we have in one case what ia called a curve of double
1—2
.,,,.,.,., Google
4 PSELIHINASr. [7.
Tnrtiioin curvoture, in the other a gauche polygon. The term ' curve of
curre. double cuTvature' is very bad, and, though in very general use,
is, we hope, not ineradicabla The fact is, that there are not
two curvatures, but only a curvature (as above defined), of which
the plane ia continuously changing, or twisting, round the
tangent line ; thus exhibiting a torsion. The course of such
a curve is, in common language, well called 'tortuous ;' and
the measure of the corresponding property is conveniently
called Toriuaaity.
8. The nature of this will be best understood by consider-
ing the curve as a polygon whose sides are indefinitely small.
Any two consecutive sides, of course, lie in a plane — and in
that plane the curvature is measured as above, but in a curve
which is not plane the third side of the polygon will not be in
the same plane with the first two, and, therefore, the new plane
in which the curvature is to be measured is different from the
old one. The plane of the curvature on each side of any point
of a tortuous curve is sometimes called the OactdaHng Plans of
the curve at that point. As two successive positions of it con-
tain the second side of the polygon above mentioned, it is
evident that the osculating plane passes from one position to
the next by revolving about the tangent to the curve.
2™*"™ 8. Thus, as we proceed along such a curve, the curvature
'xi^- in general varies ; and, at the same time, the plane in which the
curvature lies is turning about the tangent to the curve. The
tortuosity is therefore to be measured by the rate at which the
osculating plane turns about the tangent, per unit length of the
curve.
To express the nuiius of curvature, the direction coBinea of
the oBculating plane, and the tortnoaity, of a carve not in one
plane, in terms of Cartesian triple oo-ordinatee, let, as before,
&$ be the angle between the tangents at two points at a distance
Sa from one another along the curve, and let S0 be the angle
between the osculating planes at these points. Thus, denoting
by p the radius of curvature, and t the tortuosity, we have
p d* '
_rf0
..Google
KINEUATieS. 5
St ' otltyT"
according to the regular notation for the limiting vtdnea of ^ , uid tortu-
and -s- , wlien S* is diminiahed withont limit. Let OL, OL'
, 8^
8> '
be liuea drawn througli aoy fixed point 0 parallel to any two
sncceesiTB positions of a moTing line PT, each in the directions
indicated by the order of the letteiB. Draw OS perpendicular
to their plane in the direction firom 0, such that OL, OL', OS
lie in the same relative order in space as the positive axes of
coordinates, OJ, OY, OZ. Let OQ bisect LOL', and let OR
biseot the angle between OL' and LO produced through 0.
Let the direction codaea of
OL be a, 6, e;
OL' „ a', b\ e' ;
OQ „ l,m,n;
OB „ a,P,y;
OS „ \.^v:
and lei S$ denote the angle ZOL'. We have, by the elements of
analytical geometry,
oMS6 = aa' + bb' + ee' (3);
'"coeiStf* "^--^JW' "-^^ySfl W.
- '^- ' "'-' ''-' m
Sc" — h'e ca —i^a ab' — a'l
.(e).
Not let the two successive positions of PT be tangents to a
cnrve at points separated by an arc of length hi. We haTe
l_Sg_2sinj8g_sinSfl
^ 8. 8« 8s "'
when ^ is infinitely small ; and in the same limit
, dx dy dz
as at d»
a'-^.dp, y-i.ij, J-c.dp (8);
de' dt' dt ^ "
,ds dz
'-■-"■'-WJt.-tJI''-- W^
da dt dt dt
..Google
6 PBELIHUIABT. [9.
Dirretuni and a, j8, y become the direction cosines of the normal, PC,
tBiif. drawn towardB the centre of curvature, C ; and X, ^ v those of
the perpend icalar to the oacuUting plane drawn in the direc-
tion i-eUtiTely to FT and PC, corresponding to that of OZ
relatively to OX and OY. Then, using (8) and (9), with (7),
in (5) and (6) respectively, we have
"^ "i-j- ■'^
"7^- ^■p=^' -I'-pn <">''
dy ,dz dz dy da ,dai dx ,dz
_^ J ^ J —a.s.-zr.'^l-
1(11).
a» dt dg ds da da d» ds da ds da da ,
The simplest expression for the cnrvature, with choice of inde-
pendent variable left arbitrary, is the following, taken from (10) :
p da ^ '
This, modified by differentiation, and application of the formula
dsip» = daid'x + dyd'y + dzd'z (13), ^
J{{d'xy*{d^y)'*id'zr-{d^»)'}
da'
..(U).
Another formula for — is obtained immediately from equations
(11); but these equations may be put into the following simpler
form, l^ diSerentiation, &c.,
dyd^z-dzd'y dzd'x-dxd'e dxdPy - dyiPa: ,.,.
* ^"^ -''=— ^-'rfe-— '"" p-^df— t"J'
from which we find
_, ^ {{dyd*z - dscPyy + {dtd^x - dxd^zf *■ {dxiPy- dyd'a:)^* .^
rfs* \ >■
Each of these several expressions for the curvature, and for the
directions of the relative lines, we shall find has its own special
significance in the kinetics of a particle, and the statics of a
flexible cord.
To find the tortuosity, ~ , we have only to apply the genenU
equation above, with \, fk, v substituted for I, m,7i, and — -t- ,
..Google
9.] KIKE1UTIC8.
Kdv dp.\' { d\ . AN' /, dfK rfX\')4
'■a-'a)n's-^a)n'*-''s)},
where X, /*, v, denote the direction coeiues of the OBculating
plane, given by the preceding formule.
10. The integral carvature, or whole change of direction of inteimi *
an arc of a plane curve, is the angle through which the tangent orknirvn
has turaed as we pass from one extremity to the other. The I ixtT^
average curvature of any portion ia its whole curvature divided
by its length. Suppose a line, drawn from a fixed point, to
move so as always to be parallel to the direction of motion of
a point describing the curve ; the angle through which this
turns during the motion of the point exhibits what we have
thus defined as the integral curvature. In estimating this, we
must of course take the enlarged modem meaning of an angle,
including angles greater than two right angles, and also nega-
tive angles. Thns the intend curvature of any closed curve,
whether everywhere concave to the interior or not, is four right
angles, provided it does not cut itsel£ That of a Lemniscate, or
figure of 3 > is iero. That of the Epicycloid ^ is eight right
angles ; and so on.
11. The definition in last section may evidently be extended
to a plane polygon, and the integral change of direction, or the
angle between the first and last sides, is then the sum of its
exterior angles, all the sides being produced each in the direc-
tion in which the moving point describes it while passing round
the figure. This is true whether the polygon be closed or not
If closed, then, as long as it is not crossed, this sum is four
right angles, — an extension of the result in Euclid, where all
re-ejitrant polygons are excluded. In the case of the star-shaped
figure "^j it is ten right angles, wanting the sum of the five
acute angles of the figure ; that is, eight right angles.
12. The integral curvature and the average curvature of a
curve which is not plane, may be defined as follows : — Let suc-
cessive lines be drawn from a fixed point, parallel to tangents
at successive points of the curve. These lines will form a
conical surface. Suppose this to be cut by a sphere of unit
radius having its centre at the fixed point. The length of the
..Google
'8 ]>KEL1M1MARY. [12.
curve of JDtersectioii measures the integral curvature of the
giveo curva The average curvaiure is, as in the case of a
plane curve, the integral curvature divided by the length of the
carve. For a tortuous curve approximately plane, the integral
curvature thus defined, approximates (not to the int^ral cur-
vature according to the proper definition, § 10, for a plane
ciure, but) to the sum of the integral curvatures of all the
part« of an approximately coincident plane curve, each taken as
positive. Consider, for examples, varieties of James Bemouilli's
plane elastic curve, § 611, and approximately coincident tor-
tuous curves of fine steel piano-forte wire. Take particularly
the plane lemniscate and an approximately coincident tortuoua
closed curve.
13. Two consecutive tangents lie in the osculating plane.
This plane is therefore parallel to the tangent plane to the cone
described in the preceding section. Thus the tortuosity may
be measured by the help of the spherical curve whicli we have
just uBcd for defining integral curvature. We cannot as yet
complete the explanation, as it depends on the theory of rolling,
which will be treated afterwards (§§ HO — 137). But it is enough
at present to remark, that if a plane roll on the sphere, along
the spherical curve, turning always round an instantaneouB axis
taugeutial to the sphere, the integral curvature of the curve of
contact or trace of the rolling'on the plane, is a proper measure '
nf the whole torsion, or integral of tortuosity. From this and
§ 12 it follows that the curvature of this plane curve at any
point, or, which is the same, the projection of the curvature of
the spherical curve on a tangent plane of the spherical surface,
is equal to tlie tortuosity divided by the curvature of the given
curve.
Let - be the curvature and t the tortnoBity of the given
curve, and d* an eleotent orits length. Then / — and | r(£^ each
integral extended over any stated length, ^ of the curve, are
reapeotively the integml curvature aad the integr^ tortuoBity.
The mean curvature and the mean tortuosity are respectively
jiGoogle
13^] KIKBaU.TIC3. 9
Infinite tortuout^ will be easil/ undentood, by cooaidering Intsftnl
ft beliz, of inolination a, described on a right circular cylinder of « • curve
radius r. The carvatnre in a circular section being - , that of
the helix ia, of course, — — . The tortuoeity ia , or
tan a X cnrvaturo. Hence, if a = 7 the curvature and tortuosity
are equal.
Let the curvature be denoted by - , bo that coa'a a - . Let a
P P
lemtun finite, and let r dinmuBh vithont limit. The ttep of the
helix being 2rriBiia — iir'/pr(\ — j , is, in the limits Srs/pr,
which is infinitely iimalL Thns the motion of a point in the
curve, though infinitely nearly in a Btnught line (the path being
always at the infinitely small distance r from the fixed strught
line, the axis of the cylinder), will have finite curvature - . The
1 1 / Al
torttioeity, being -tana or -^( 1 — 1 , will in the limit be a
mean proportional between the curvature of the circular section
of the cylinder and the finite curvature of the cnrve.
The acceleration (or force) required to produce such a motion
of a point (or materia particle) will be afterwards investi-
gated (S 35 d.).
14. A chain, cord, or fine wire, or a fine fibre, filament, or Finibia
hair, may surest what is not to be found among natural or
artifiraal productions, a perfectly flexible and inexteiisii^ line.
The elementary kinematics of this subject require no investiga-
tion. The mathematical condition to be expressed in any case
of it ifi eimpl; that the distance measured along the line from
any one point to any other, remans constant, howerer the line
be bent.
15. The use of a cord in mechanism presente us with many
practical applications of this theory, which are in general ex-
tremely aimple; although curious, and not always very easy,
geometrical problems occur in connexion with it. We shall
say nothing here about the theory of knot!>, knitting, weaving,
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10 PBELUUNABT. [IS.
plaiting, etc., but we intend to return to the subject, under
vortex-motion in Hydrokinetics.
16. In the mechanical tracing of curves, a flexible and
inextensible cord is often supposed. Thus, in drawing an
ellipse, the focal property of the curve shows us that by fixing
the ends of auch a cord to the foci and keeping it stretched by
a pencil, the pencil will trace the curve.
By a ruler moveable about one focus, and a string attached
to a point in the ruler and to the other focus, the hyperbola
may be described by the help of its analogous focal property ;
and so on.
17. But the consideration of evolutes is of some importance
in Natural Philosophy, especially in certain dynamical and
optical questions, and we shall therefore devote a section or
two to this application of kinematics.
Def. If a flexible and inextensible string be fixed at one
point of a plane curve, and stretched along the curve, and be
then unwound in the plane of the curve, its extremity will
describe an Involute of the curve. The original curve, con-
sidered with reference to the other, is called the Evolute.
18. It will be observed that we speak of on involute, and
of the evolute, of a curve. In fact, as will be easily seen, a curve
can have but one evolute, but it has an infinite number of
involutes. For all that we have to do to vary an involute, is
to change the point of the curve from which the tracing point
starts, or coosider the involutes described by diSerent points of
the string, and these will, in general, be different curves. The
following section shows that there is but one evolute.
19. Let AB be any curve, PQ a portion of an involute,
pP, qQ positions of the free part of the string. It will be seen
at once that these must be tangents
to the arc AB at p and q. Also (see
.§ 90), the string at any stage, as
pP, revolves about p. Hence pP is
normal to the curve PQ. And thus
the evolute of PQ is a definite curve,
viz., the envelope of the normals drawn at every point of PQ,
..Google
19.] KIHE1U.TIC8. 11
or, which is the same thing, the locua of the ceotres of curva- BTolnte.
ture of the curve PQ. And we may merely mention, as an
obTiouB result of the mode of tracing, that the arc pj is equal to
the difference otqQ and pP, or that the arc pA is equal to pP.
20, The rate of motion of a point, or its rate of change of Velodiy.
position, is called its Yelocity. It is greater or less as the space
passed over in a given time is greater or less : and it may be
uniform, i. e., the same at every instant ; or it may be variahle.
Uniform velocity is measured by the space passed over in
unit of time, and is, in general, expressed in feet per second ;
if veiy great, as in the case of light, it is sometimes popularly
reckoned ia miles per second. It is to he observed, that time
is here used in the abstract sense of a uniformly increasing
quantity — what in the differential calculus is called an inde-
pendent variable. Its physical definition is given in the next
chapter.
21. Thus a point, which raoves uniformly with velocity v,
describes a space of v feet each second, and therefore vt feet in
t seconds, t being any number whatever. Putting s for tho
Bpace described in t seconds, we have
Thus with unit velocity a point describes unit of space in unit
of time.
22l It is well to observe here, that since, by our formula,
we have generally
and since nothing has been said as to the magnitudes of 8 and t,
we may take these as small as we choose. Thus we get the
same Tesult whether we derive v from the apace described in a
million seconds, or from that described in a millionth of a second.
This idea ia very useful, as it makes our results intelligible
when a variable velocity has to be measured, and we find our-
selves obliged to approximate to its value by considering the
space described in an interval so short, that diuiog its lapse the
velocity does not sensibly alter in value.
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12 PBELIHINAUT. [23.
23. When the point does not move uniformly, the velocity
ia variable, or different at different successive instants ; but we
define the average velocity during any time as the space de-
scribed in that time, divided by the time, and, the less the
interval is^ the more nearly does the average velocity coincide
with the actual velocity at any instant of the iaterval. Or
again, we define the exact velocity at any instant as the space
which the point would have described in one second, if for one
second its velocity remained unchanged. Tbat there is at every
instant a definite value of the velocity of any moving body, is
evident to all, and is matter of everyday conversation. Thus, a
railway train, after starting, gradually increases its speed, and
every one understands what is meant by saying tbat at a par-
ticular instant it moves at the rate of ten or of fifty miles an
hour, — although, in the course of an hour, it may not have
moved a mile altogether. Indeed, we may imagine, at any
instant during the motion, the steam to be so adjusted as to
keep the train running for some time at a perfectly uniform
velocity. This would be the velocity which the train had at
the instant in question. Without supposing any such definite
adjustment of the driving power to be made, we can evidently
obtain an approximation to this instantaneous velocity by con-
sidering the motion for so short a time, that during it the actual
variation of speed may be small enough to be n^lected.
24. In fact, if v be the velocity at either b^;inning or
end, or at any instant of the interval, and 8 the space actually
described in time t, the equation v = j is more and more nearly
true, as the velocity is more nearly uniform during the interval
t; so that if wo take the interval small enough the equation
may be made as nearly exact as we choose. I'hus the set of
values —
Space described in one second.
Ten times the space described in the firat tenth of a second,
A hundred „ „ „ hundredth „
and so on, give nearer and nearer approximations to the velocity
at the beginning of the first second. The whole foundation of
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24.] KINEJUTICS. 13
the differsntial calculus is, in fact, contained in this simple vdooit;.
question, "Wliat is the rate at which the space described in-
creases 1" i,e., What is the velocity of the moving point?
Newton's notation for the velocity, i.e. the rate at which s
increases, or the Jtvxion of s, is i. This notation is very con-
venient, as it saves the introduction of a second letter.
Let a point which baa deecribed a space s in time t proceed
to deaoribe an additional space Ss in time Si, and let v, be the
greatest, and «, the least, velocity which it has during the iu-
terraJ St. Hien, evidently,
S«<e,S(, 8»>t»,5(,
. & &i
'■'■' Sf""" a**'-
But aa Si dinuDiahes, the values of v, and v, become more and
more nearly equal, and in the limit, each is equal to the velocity
at time L Hence
^ dt: ^
25. The preceding definition of velocity is equally appHca- S^"*}^
ble whether the point move in a straight or curved line ; but,
since in the latter case the direction of motion continually
changes, the mere amount of the velocity is not sufficient com-
pletely to describe the motion, and we must have in every such
case additional data to remove the uncertainty.
In such cases as this the method commonly employed,
whether we deal with velocities, or as we shall do farther on
with accelerations and forces, consists mainly in studying, not
the velocity, acceleration, or force, directly, but its components
parallel to any three assumed directions at right angles to each
other. Thus, for a train moving up an incline in a NE direc-
tion, we may have given the whole velocity and the steepness
of the incline, or we may express the same ideas thus — the train
ia moving simultaneously northward, eastward, and upward — ■
and the motion as to amount and direction will be completely
known if we know separately the northward, eastward, and up-
ward velocities — these being called the components of the whole
velocity iu the three mutually perpendicular directions N, E,
and up.
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FRELIHIHABT. [So.
Id general the velocity of a point at x, y, a, ia (as we hare
da (/dxY /dv\* /ilz\*)i
.ee.) g , or, which >. th. «me, l^[^J * [^) * [j,) ] ■
Nov denoting hy v, the nte at whicb x iuoreases, or the velo-
city parallel to the axis of x, and bo by v, u>, for the other two ;
angles which the direction of motion makes with the axes, and
we have «^ jT? "^^f ^-ji' Hence, calling a, j5, y the
ich the directic
dt
putting 9-^,t ve have
_^ ^ _u
~da da~ g'
di
Hence Vf = q coa a, and therefore
^ 26. A velocity in any direction may be resolved in, and
perpendicular to, any other direction. The first component is
found by multiplying the velocity by the cosine of the angle
between the two directions — the second by using aa factor the
sine of the same angle. Or, it may be resolved into components
in any three rectangular directions, each component being
formed by multiplying the whole velocity by the cosine of the
angle between its direction and that of the component.
It is oseful to remark that if the axes aix,y,z are not rect-
angnlar, -ni-^tjj '^'^ ^^-^ ^ ^^ velocities parallel to the
axes, bat we shall no longer have
m<tj^m*(M)-
We leave as an exercise for the student the determination of the
correat expression for the whole velocity in terms of ite com-
If we reaolve the velocity along a line whose inclinations to
the axes are \, /i, v, and which makes an angle & with the di-
rection of motion, we find the two expressions below (which
mast of course bo equal) according as we resolve 9 directly or
by ite componeatu, w, v, to.
V COB /I -f to cos V.
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SabBtitut« in this eqiiation the roluea of u, v, w already given, Bodutbo
g 29, and we have the well-known geometrical theorem for the
angle between two straight lines which make given angles with
the axe^
cos 0 = oos a cofl A -I- cos j3 cos f( + cos y ooe V,
From the above expression we Bee at once that
27. The velocity reBolred in any direction is the sum of the vompoti'
compoDents (in that direction) of the three rectangular com- taiwitiM.
poneats of the whole velocity. And, if we consider motion in
one plane, this is Btill true, only we have but too rectangular
components. These propositions are virtually equivalent to the
following obvious geometrical construction : —
To compound any two velocities aa OA, OS in the figure ;
from A draw A C parallel and equal
to OB. Join OC:— then 00 ia the
resultant velocity in magnitude and
direction.
00 ia evidently the diagonal of the
parallelogram two of whose sides are
OA. OB.
Hence the resultant of velocities represented by the sides of
any closed polygon whatever, whether in one plane or not, taken
all in the same order, is zero.
Hence also the resultant of velocities represented by all the
aides of a polygon but ono, taken in order, is represented by
that one taken in the opposite direction.
When there are two velocities or three velocities in two or
in three rectangular directions, the resultant is the square root
of the sum of their squares — and the cosines of the inclination
of its direction to the given directions are the ratios of the com-
ponents to the resultant.
It is easy to see that as 8a in the limit may be resolved into 8r
and rZ$, where r and 6 are polar oo-ordinatos of a plane curve,
d9
'-dt'
pwpemdicnlar to, the radius vector. We may obtain the si
resuhthiw, a! = rc08tf, y^rsinft
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PBELIMINABT. [27.
dx dr . . .d0 dy dr .
But by $ 26 the whole velocity along r is ^ oob ^ + -^ Bin fl,
i.e., bytbe&bove values, -^ . Similarly the tramTerse velocity u
28. The velocity of a point is said to be accelerated or re-
tarded according as it increases or diminishes, but the word
acceleration is generally used in either sense, on the undentand-
ing that we may regard its quantity ss either positive or nega-
tive. Acceleration of velocity may of course be either uniform
or variable. It is said to be uniform when the velocity receives
equal increments in equal times, and is then measured by the
actual increase of velocity per unit of time. If we choose as the
unit of acceleration that which adds a unit of velocity per unit
of time to the velocity of a point, an acceleration measured by a
will add a units of velocity in uait of time — and, therefore, at
units of velocity in t units of time. Hence if F" be the change
in the velocity during the interval t,
29. Acceleration is variable when the point's velocity does
not receive equal increments in successive equal periods of time.
It is then measured by the increment of velocity, which would
have been generated in a unit of time had the acceleration re-
mained throughout that interval the same as at its commence-
ment The average acceleration during any time is the whole
velocity gained during that time, divided by the time. In
Newton's notation v is used to express the acceleration in the
direction of motion ; and, if tf = ^, as in § 24, we have
a = v = a.
Let V be the velocity at time t, 8v its change in the interval
St, Qj and a, the greatest and least values of the acceleration
daring the interval Si, Then, evidently,
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29.] KIHE1UTIC& 17
8t> Sv Ai
Afl & is taken Hnftller and Hmaller, tlie valuea of a, and a, ap-
proximate infinitely to each other, and to that of a the repaired
MceleratioD at time L Hence
do
It is nseful to obserTe that ve may also write (by dum^g
the iniiependeot variaUe)
dv ds dv
'^'dedt'"'^-
Since v= jTi v^ hare ° = -j3> a^d i^ is evident from dmilar
reasoning that the component aocelerationa parallel to the axes
are j-j- , t^ , -js . Bnt it is to be cuefally observed that -r^
is not genemlly the renultont of the three component accelera-
tions, bat is so only when either the curvatare of the path, or
the velocity is zero; for [§ 9 (14)] we have
m-m^ahm-i
df
The direction cosines of the tangent to the path at any point
Xy y, «are
Ide 1 if y J dz
v'dt' v'di' v^'
Thtjae of the line of resoltant acceleration are
1 d^ l^rfV I ^
/A" /df' /df'
vhere, for brevity, we denote by / the resoltant acceleration,
l&nce the direction ooeined of the plane of these two lines are
dj/d't - dtd'y •
{(dydV- (fcrfV)'+ {dxd^x-dxd'!sy+ (dsed'y- rfytPa;)'}* '
These ($ 9) show that this plane is the osculating plane of the
carve. Agun, if 6 denote the angle between the two lines, we
bave
. . {{dgtPz - dsd'y)** (dtd^x - dax^z)' + {dxd^y - tfyif a)'}*
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IS PEELIMINABT. [29.
Aeoalen- or, acconliDg to the expression for the curvature ^ 9),
Bin ff = - -vj-i = 7- .
Henoe /miB = -'.
P
Hence /cos ^ = -j3, <i°<l therefore
B»intion SO. Th« nhote acceleratioD in aoy direction is the sum of
ritfan°tf »^ the compoDents (id that direction) of the accelerations parallel
to any three rectangular axes — each component acceleration
being found bj the same rule as component velocities, that
is, by multiplying by the cosine of the angle between the di-
rection of the acceleration and the line along which it is to
be resolved,
31. When a point moves in a curve the whole acceleration
may be resolved into two parts, one in the direction of the
motion and equal to the acceleration of the velocity — the other
towards the centre of curvature (perpendicular therefore to the
direction of motion), whose magnitude is proportional to the
square of the velocity and also to the curvature of the path.
The former of these changes the velocity, the other affects only
the form of the path, or the direction of motion. Hence if a
moving point be subject to an acceleration, constant or not,
whose direction is continually perpendicular to the direction of
motion, the velocity will not be altered — and the only effect
of the acceleration will be to make the point move in a curve
whose curvature is proportional to the acceleration at each
instuit
32. In other words, if a point move in a curve, whether
with a uniform or a varying velocity, its change of direction
is to be r^arded as constituting an acceleration towards the
centre of curvature, equal in amount to the square of the
velocity divided by the radius of curvature. The whole accele-
ration will, in every case, be the resultant of the acceleration,
..Google
32.] KINEMATICS.
thus measuring change of direction, and the acceleration of^
actual velocity along the curve.
We may take another mode of reaolving acceleratioD for a
plane curve, which is BOmetimefl useful ; along, and perpendicular
to, the radiua-veotor. By a method similar to that employed ia
g 27, we easily find for the component along the radius-vector
and for that perpendicular to the radius-veotor
1 d /.dff\
r dt \ dt) '
33. If for any case of motion of a point we have given the Deiermin*.
whole velocity-and its direction, or aimply the components of "oUonfimn
the velocitj in three rectangular directions, at any time, or, as ^^J?-
is most commonly the case, for any position, the determination
of the form of the path described, and of other circumstances of
the motion, is a question of pure mathematics, and in all cases
is capable. If not of an exact solution, at all events of a solution
to any d^ree of approximation that may be desired.
The same is true if the total acceleration and its direction
at every instant, or simply its rectangular components, be given,
provided the velocity and direction of motion, as well as the
position, of the point at any one instant, be given.
For we have in the first case
dx ,■■;■"''"■'■
— sus^cosa, etc, ,,
three simultaneous equations which can contain only x, y, z, and
t, uid which therefmv suffice when integrated to determine x, y,
and 2 in terms of t. By eliminAting t among these equations, we
obtain two equations among x, y, and »— each of which repre-
sents a sur&ce on which lies the path described, and whose
intersection therefore completely determines it.
In the second case we have
^x d'y _ d**
5?="- w^^' r^^y''
to which equations the same remsrks apply, except that here
each has to be twice integrated.
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20 FBELIHINART. [33.
^e arbitrarj cooBtants introduced by integratioii are deter-
mined at once if we know the co-ordimites, and the components
of the velocitj, of the jwint at a given epoch,
34. From the principlea already laid down, a great many
interestiDg results may be deduced, of whicti we enunciate a
few of the most important.
a. If the velocity of a moving point be uniform, aud if itB
direction revolve unifonuly in a plane, the path described is
a circle.
Let a be the velocitf , and a the angle through which its direo-
tion turns in unit of time ; then, by properly chooaing the axes,
whence {x-A)' + (i/-B)'=-^.
i. If a point moves in a plane, and if its component velo-
city parallel to each of two rectangular ases is proportional to
itfl distance from that azie, the path is an ellipse or hyperbola
whose principal diameters coincide with those axee; and the
acceleration is directed to or from the origin at every instanL
r,-m
dt
' dt'l^' df-
fxvy, and tike whele aooeleration is
towards or from 0.
Also j^ = - - > from which fi/ -»«• = <?, an ellipse or hyp«^
bola referred to its principal azee. (Compare § 65.)
c. When the velo^ty is uniform, but in direction revolving
uniformly in a right circular cone, the motion of the point is in
a circular helix whose axis is parallel to that of the cone.
of 85. a. When a point moves oniformly in a circle of radius
R, with velocity V, the whole acceleration is directed towards
V*
the centre, and has the constant value ■„- . See § 31.
..Google
35.] E1NE3UTIC3. 21
b. With uniform acceleration in the direction of motion, a lumpiw irf
point describes spaces proportional to tlie squares of the times tko.
elapsed since the commencement of the motion.
In this case the space described in any interval is that
-which would be described in the same time by a point moving
uniformly with a velocity equal to that at the middle of the
interraL In other words, the average velocity (when the
acceleration ia aniform) is, during any interval, the arithmeti-
cal mean of the initial and final velocities. This is the case of
a stone falling vertically.
For if the acceleration be parallel to a^ we have
^ = o, therefore -^ •• ti = ai, and x = ^oi*.
If at time t = Q the velocity was F, these equations become at
And initial velocity = F,
final „ -cF+ot;
AiiUimetical mean >= F+ ^at,
_x
~*'
whence the second part of the above statement,
c When there is nniform acceleration in a constant direc-
tion, the path described is a parabola, whose axis is parallel to
that direction. This is the case of a projectile moving in
vacanm.
For if the azia of y be parallel to the accderation a, and if the
[Jane of icy be that of motion at any time,
and iherefore the motion ia wholly in the plane of ay.
g=0, §..
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22 PRELIHUfAHT. [35. I
" uiil b; integration
a- Ut + a, ^=^0**+ Vt + h, I
vhere 17, T,^b are conatontB. |
The eliminatioQ of t gives the eqaadoa of a parabola of which the {
axis is parallel to jf, parameter „- , and vertex tiie point whoso go- .
ordioates are
d.. Ae an iUustration of acceleration in a tortuous curve, we |
take the case of § 13, or of § 34, c.
Let a point move in a circle of raditia r with uniform angular
velocity' M (about the centre), and let tliis circle move perpen-
dicular to its plane witb velocity V. The point describee a
helix on a cylinder of radius r, and the inclination a is given by
The curvature of the path is ~-^i ri or -t^^ ;— , , and the
i~ r ¥' + rut' V + rV '
tortuo«ty^^,j^P;^.= P^^..
The acceleration ia rw', directed perpendicularlj towards the
axis of the cylinder. — Call this A.
I«t A be finite, r indefinitely amall, and therefore w indefinitely
great
Curvature (in the limit) = -^ ■
Tortnoeity ( „ ) = p ■
Thus, if we have a material particle moving in tiie maimer speci-
fitid, and if we consider the force (see Chap. IL) required to pro-
duoe the Bcceleration, we find that a finite force perpendicular to
..Google
3o.] JCINEUATICS. 23
tlie line of motion, in a directioti revolving with an infinitely ^J^^^^
great angular velocity, maintainB constant infinitely amall de- f^o"'
flection {in a direction opposite to its own) from the line of un-
disturbed moaon, _finile curvature^ and infinite tortuosity.
e. When the acceleration is perpendicular to a given plane
and proportional to the distance from it, the path is a plane
curve, which is the harmonic curve if the acceleration be towards
the plane, and a more or less fore-Bhortened catenary (§ 680)
if from the plane.
As in case e, ^ = % ji~^> ^°^ « = 0, if tjie axis of 2 be
perpendicular to the acceleration and to the direction of motion
at any instant. Also, if we choose the origin in the plane,
Henoe -77 = const = a (tsa^igoee),
diir a" r
This gjvee, if ^ is negative,
if = Pooa(f- + Qj, the harmonio cture, or curve of slues.
If /I be pomtive, }/=F€^ + Qf~^ ;
and by shifting the origin along the axia of x this can be put in
the form
y -«(.' + ."'):
which is the catenary if 2R = & ; otherwise it is the catenarj
stretched or fore-shortened in the direction of y.
38. rCompare §8 233—236 below.] a. When the accele- A««)<«ti<m
ration is directed to a fixed point, the path is in a plane passmg ftiai eentn.
through thai point; and in this plane the areas traced out hy
the radius-vector are proportional to the times employed. This
includes the case of a satellite or planet revolving about its
primary.
Evidently there is no acceleration perpendicular to the
plane containing the fixed and moving points and the direction
..Google
e have -7-7 -F-
24 PBELmiKABT. {Z(i.
^E^mtknof motion of the second at an^ instant; and, tbere being no
Omiombre. relocity perpend iculEU" to this plane at starting, there is there-
fore none throughout the motion ; thus the point moves in the
plane. And had there been no acceleration, the point would
have described a straight line with uniform velocity, so that in
this case the areas described b; the radius-vector would have
been proportiooal to the times. Also, the area actually described
in any instant depends on the length of the radius-vector and
the velocity perpendicular to it, and is shown below to be
unaffected by an acceleration parallel to the radius-vector.
Hence the second part of the proposition.
de r' df~ r'
the fixed point being the origiQ, and P being some funcUon of
x,y,x; in natwt a function of r only.
_ cTv cPx
Hence " 5p ~ ^ rf? °= ^' **"■'
which gjve on intogradon
dx dy ^ dx dz j^ dy das ^
"s-'I''^" 'n-'s'"- '-i-'n'"--
Hence at onoe C,x + Cj/ + Cjs = 0, or the motion is in a plane
through the origin. Take this as the plane of «y, then we have
only the one equation
In polar co-ordinates this is
if X be the area intercepted by the curv«, a fixed ndius-vector,
and the radius-rector of the moving point. Hence the area in-
oreaaes uniformly with the time.
b. In the same case the veloci^ at any point is inversely as
the perpendicular from the fixed point upon the tangent to the
path, the momentary direction of motion.
For evidently the product of this perpendicular and the
velocity ^ves double the area described in one second about the
fixed point
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36.] KISEKLTIC8.
Or thna — if p be tbs perpendicnlar on the tcmgeot,
If WO refer tke motion to co-ordiiuites in its own plane, we
have ouly tlie equations
<ex ^ ^ _Py
d?~ r ' de~ r '
whence, as before, ^ Jt"^^
J£, hy the hdp ot this last equation, we eliminate t from
■^ = — , Bubetituting polar for rectangular oo-ordinates, we
siriTe at the polar differential equation of the path.
For Torietj, we may derive it from the fomtube of § 32,
-■ u, we have
,f_W = iU._W^_.A* ,._,^ W,
.(Tu
But -^ ■= tu* M ~ ~ ^ j3 • tl>wefi)re
Alao-f-^J = AV, the robetitution <^ which values gives OB
> s?*«-j=s= (')•
the equation required. The integral of this equation involves
two arbitrary coustante beeidea A, and the remaining constant
belonging to the two differential equ&tiona of the second order
^Mve is to be introduced on the farther integration of
a-*"- <?).
when the valne of u in terms of 0 is subatdtnted from the equa-
ticm of tlie path.
..Google
2C FfiELUUKAST. [36.
Other examples of these principles will be met with in the
chapters on Kinetics.
37. If from any fixed point, lines }>e drawn at eveiyinstant,
representing in magnitude and direction the velocity of ^ point
describing any path in any manner, the extremities of these
lines form a curve which is called the Hodograp^. The inven-
tion of this construction is due to Sir W. R. Hamilton. One of
the most beautiful of the many remarkable theorems to which
it led him is that of § 38.
Since the radius-vector c^ the bodograph represents the
velocity at each instant, it is evident (§ 27) that an elementary
arc represents the velocity which must be compounded with the
velocity at the beginning of the corresponding interval of time,
to find the velocity at its end. Hence the velocity in the hedo-
grapb is equal to the acceleration in the path ; and the tangent
to the bodograph is parallel to the direction of the acceleration
in the path.
If X, 3/, 2 be the co-ordinates of the moving point, £, ij, { ihose
of the correBpondiug point of the hodograph, then evidently
f~^ „^^y t-"'
'dt'
ftsd therefore
di dr, dt
or the tangent to the bodograph is parall^ to the acceleration in
the orbit. Also, if <r be Uie arc of the hodograph.
£'^/(f)^-@^-(i)^
or the velocity iu the hodograph ia equal to the rate of aocelera-
tion in the path,
Hodoemih 36> The hodograph for the motion of a planet or comet is
tn^^S^ always a circle, whatever be ffie form and dimensions of the orbit.
KepK^'° In the motion of a planet or comet, the acceleration is directed
"■ towards the sun's centre. Hence (§ 36, 6) the velocity w in-
..Google
38.] KINEMATICS. S7
versely as the perpendicular from that point upon the tangent Bodognpb
to the orhit. The orbit we assume to he a conic section, whose oDmn. da-
focus is the Bun's centre. But we know that the intersection Kspi^'i
of the perpendicular with the tangent lies in the circle whose
diameter is the major axis, if the orbit be an ellipse or hyper-
bola; in the tangent at the vertex if a parabola. Measure off
on the perpendicular a third proportional to its own length and
any constant line; this portion will thus represent the velocity
in magnitude and in a direction perpendicular to its own —
so that the locus of the new points in each perpendicular will be
the hodograph turned through a right angle. But we see by
geometry* that the locus of these points is always a circle.
Hence the proposition. The hodograph surrounds its origin if
the orbit be an eUipse, passes through it if a parabola, and the
origin is without the hodograph if the orbit is a hyperbola.
For a projectile unresisted by the air, it will be shewn in
Kinetics that we have the equations (assumed in § 36, c)
if the axis of y be taken vertically upwards.
Hence for the hodograph
or f=C, ^=C'-gty and the hodi^raph is a vertical straight
line along which the describing point movee unifonnly.
For the case of a planet or comet, instead of assiiming as Hodogi»ph
above that the orbit is a conic with the aon in one focus,
(Kewton'a deduction from that and the law of areas) that the HwbKT
acceleration is in the direction of the radius- vector, and varies
inversely aa the square of the distance. We have obviously
^x^fuc J'y_ iiy
dnosd rrom
Nnrtoo't
where r'=xf + t^.
Hence, as in §36, a,J-yJ=A (1),
■ See onr noaller work, 1 61.
..Google
PBELIHIHAET. [38.
T 7 •
°J"
H»" l*''-Sr »
Simili^rlr 3*^'-%" m-
Heuoe for the kodograph
the circle ms before etated.
We may merely mention that the equation of the orbit will be
foand at once by eliminating ^ and -rr among the three first
intf^nOa (1), (2), (3) above. We thus get
a oonio section of irhlch the origin is a focus.
S9. The inten^ty of heat and light emanating from a point,
• or from an uniformly radiating spherical surface diminiahes with
increasing difitance according to the same lav as graritatioo.
Hence the amount of heat and light, which a planet receives
from the aun during any interval, is proportional to the time
int^ral of the acceleration during that interval, i.e. (§37) to
the correapODdiug arc of the hodograph. From this it ib easy
to see, for example, that if a comet move in a parabola, the
amount of heat it receives &om the bud in any interval is pro-
portional to the angle through which its direction of motion
turns during that interval There is a corresponding theorem
for a planet moving in an ellipse, hut somewhat more com-
plicate.
40. If two points move, each with a definite uniform velo-
tnty, one in a given curve, the other at every instant directing
its couiBe towards the first describes a path which is called a
..Google
40.]
EIHE1UTIC8.
29
Curve of Purguit. The idea U said to have been suggested cn™i oi
by the old rule of steering a privateer always directly for the
vessel pursued. (Bouguer, 3/An. de VAcad. 1732.) It is the
carve described by a dog running to its master.
The simplest cases are of course those in which the first
point moves in a straight line, and of these there are three, for
the velocity of the first point may be greater than, equal to,
or less than, that of the second. The figures in the text below
represent the curves in these cases, the velocities of the pur-
suer being |, 1, and ^ of those of the pursued, respectively. In
the second and third cases the second point can never over-
take tlie firs^ and consequently the line of motion of the first
is an asymptote. In the first case the second point overtakes
the first, and the carve at that point touches the line of motion
of the first. The remainder of the carve satisfies a modified
form of statement of 1;he original question, and is called the
Oaroe of FlighU
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PEELIHINARr. * [40.
We will merely form the difiereutial equation of tlie dure,
and give ita integrated form, leaving the work to the student.
Suppose Ox to be tlie line of motion of the first point, whose
veloci^ is t^ AF the cnrre of pursuit, in which the velocitj Is u,
then the tang^it at P always passes through Q, the inatan-
taneons position of the first point. It will be evident, ob a
moment's consideration, that the curve AP must have a tangent
perpendicular to Ox, Take this as the
axis of y, and let OA = a. Then, if
0Q = (, AP = a, and if a;, y be the co-
ordinates of P, we have
AP^OQ
because A, 0 and P, Q are pairs of ai-
"" mnltaneoofl positions of the two points.
™.. . r d<B
Thugivea -«=«=a:-v-^.
"^ M " djf
From this we find, unless e = 1,
the only case in which we do not get an algebraic curv& The
axis of a; is e&sily seen to be an asymptote if e ^ I.
41. When a point moves in any manner, the line joining
it with a fixed point generally changes its direction. If, for
eimplidty, we oonuder the motion aa confined to a plane
paseing through the fixed point, the angle which the joining
line makes with a fixed line in the plane is continually alter-
ing, and its rate of alteration ait any instant is called the
Angular Vetocitif of the first point about the second. If
uniform, it is of course meaeured by the angle described in
unit of time; if variable, by the angle which would have
been described in unit of time if the angular velocity at the
instaat in question were maintained constant for so long. In
this respect, the jwooess is precisely similar to that which we
have already explained for the measurement of velocity and .
acceleration.
..Google
41.] KINEMATICa
Unit of aQgutar velocity is that of i
or would describe, ubU angle about a fixed poiDt in unit of
time. The luual unit angle is (as explained in treatises on
plane trigonometrj) that which subtends at the centre of a circle
ao arc whose length is equal to the radius; being an angle of
For brevity we shall call this angle a radian.
42. The rate of increase or diminution of the angular velo-
city when variable is called the angular accderaiion, and ia
measured in the same way and by the same unit.
By methods precisely similar to those employed for linear
velocity and acceleratioa we see that if 0 be the angle-vector
of a point moving in a plane — the
dt'
Angular velocity is u =
Angalar acceleration la -y- » -n = <>> -n; -
^^ dt dff d$
Since (J 27) r -^ ia the velocity perpendicular to the radios-
vector, we see that
The angular velocity of a point in a plane is found by
dividing the velocity perpendicular to the radius-vector by the
length of the radius-vector.
43. When one poiut describes uniformly a circle about ^
another, the time of describing a complete circumference being
T, we have the angle 2ir described uniformly in T; and, there-
fore, the angular velocity is -= . Even when the angular velo-
city is not uniform, as in a planet's motion, it is useful to
introduce the quantity-^, which is then called the mean
angular velocity.
Wheo a point moves uniformly in a straight line its angular
velocity evidently diminishes as it recedes from the point about
which the angles are measured.
..Google
PfiELIHINABT. [43-
The polar eqimtion of a str^ht line is
r = a sec 0.
But the length of the line between the limiting angtes 0 and 6
a a tan 6, and this increases wiUi uniform velocitf tr. Heuoa
dt~ a dt'
Heooe -n = -^% <^ >■ therefore inverBely as the square of the
radius-veotor.
Similarly for the angular aooeleration, we have by a second
differentiation,
v = ^(otan<^-asec*9
= -j~ ( 1 ~ ^ ) I ^^ nltiiiiatelj variee iiiTflneljr us
"■■3?
the third power of the radius-vector.
41. We may also talk of the angular velocity of a moving
plane with respect to a fixed one, as tbe rate of increase of the
angle contained by them — bat unless their line of intersection
remain fixed, or at all events parallel to iteelf, a somewhat
more laboured statement is required to give definite informa-
tion. This will be supplied in a subsequent section.
45. All motion that we are, or can be, acquainted witb, is
Relative merely. We can calculate from astronomical data for
any instant the direction in which, and the velocity with which
we are moving on account of the earth's diurnal rotation. We
may compound this with tbe similarly calculable velocity of the
earth in its orbit This resultant agiun we may compound
with the (roughly known) velocity of the Hun relatively to the
so-called fixed stars ; but, even if all these elements were aoca-
rately known, it could not be said that we had attained any
idea of an oAsoJute velocity; for it is only the sun's relative
motion among the stars that we can observe ; and, in all pro-
bability, eun and stars are moving on (possibly with very great
rapidity) relatively to other bodies in space. We must there-
fore consider bow, &om the actual motions of a set of points, we
may find tbeir relative motions with regard to any one of them ;
..Google
45.] EINEIUTICS. 33
and how, having given the relative motions of &U but one with &
r^;ard to the latter, uid the actual motioD of the latter, we
may find the actual motions of alL The question is very
easily answered. Consider for a moment a number of pas-
sengers walking on the deck of a steamer. Their relative
motions with regard to the deck are what we immediately
observe, but if we compound with these the velocity of the
steamer itself we get evidently their actual motion relatively
to the earth. Again, in order to get the relative motion of
all with regard to the decic, we abstract our ideas from the
motion of the steamer altogether — that is, we alter the velocity
of each by compounding it with the actual velocity of the vessel
taken in a reversed direction.
Hence to find the relative motions of any set of points with
n^ard to one of their number, imi^ine, impressed upon each in
compoation with its own velocity, a velocity equal and opposite
to the velocity of that one ; it will be reduced to rest, and the
motions of the others will be the same with regard to it as
before.
Thus, to take a very simple example, two trains are rumuDg ■
in opposite directions, say north and soitrth, one with a velocity
of fifty, the other of thirty, miks an hour. The relative velocity
of the second with r^ard to the first is to be found by im-
pressing on both a southward velocity of fifty miles an hour ;
the effect of this being to bring the first to rest, and to give the
second a southward velocity of eighty miles an hour, which is
the required relative motion.
Or, given one trun moving north at the rate of thirty miles
an hoar, and another moving west at the rate of forty miles an
hour. The motion of the second relatively to the first is at
the rate of fifty miles an hour, in a south-westerly direction
inclined to the due west direction at an angle of tan"* }. It
is needless to multiply such examples, as they must occur to
every one.
46. Exactly the same remarks apply to relative as compared
with absolute acceleration, as indeed we may see at once, since
accelerations are in all cases resolved and compounded by the
same law as velocities.
VOL. L 3
..Google
34! FEEXIHINABT. [46.
If X, y, z, and x", ^, «", be the co-ordinates of two points
referred to axes regunled as fixed; and i, ij, {[ their relative
oo-ordinateH — we have
i = K'-x, y} = y'-y, C=»'-^
and, differentiating,
rff rf^' dx
dt~ dt dt' '
which give the relative, in terms of the absolute, velixuties j and
d^ rfV rfV
de " df di" *""'
proving our assertion about relative and absolate accelerations.
The corresponding expressions in polar co-ordinates in a plane
are somewhat complicated, and by no means convenient. The
student can essilj write Ibem down for himself.
47. The following proposition in relative motion is of con-
siderable importance : —
Any two moving poiots describe similar paths relatively to
each other, or relatively to any point which divides in a con-
stant ratio the line joining them.
Let A and B he any simultaneous positions of the points.
Take ^ or &* in AB such that the ratio
^ jr~S 2 OA O'A, ... rrv.
u IK u i» ^^ Qi- ^.^ [jg^ (^ constant value. 1 hen
as the form of the relative path depends only upon the length
and direction of the line joining the two points at any instant, it
is obvious that these will be the same for A with regard to B,
as for B with regard to A, saving only the inversion of the
direction of the joining line. Hence ^'s path about A, is A'a
about B turned through two right angles. And with regard to
O and O' it b evident tiiat the directions remmn the same, while
the lengths are altered in a given ratio ; but this is the definition
of similar curves.
48. As a good example of relative motion, let us consider
that of the two points involved in our definition of the curve of
pursuit, § 40. Since, to find the relative position and motion of
the pursuer with regard to the pursued, we must impress on
both a velocity equal and opposite to that of the latter, we see
..Google
48.] ElNE:&tA.TlCS. 35
at once that the problem becomes the same as the following. A n
boat crossing a stream is impelled by the oars witb unifonn
velocity relatively to the water, and always towards a fixed
point in the opposite bank ; but it is also earned down stream
at a uniform rate ; determine the path described and the time of
crossing. Here, as in the former problem, there are three cases,
figured below. In the firat, the boat, moving faster than the
current, reaches the desired point ; in the aecond, the velocities
of boat and stream being equal, the boat gets across only after
an infinite time — describing
a parabola — but does not land
at the desired point, which is
indeed the focus of the para-
bola, while the landing point
is the vertex. In the third
case, its proper velocity being
less than that of the water, it'
never reaches the other hank,
and is carried indefinitely
down stream. The compari-
son of the figures in § 40 with those in the present section cannot
bil to be instructive They are drawn to the same scale, and
for the same relative velocities. The horizontal lines represent
the farther bank of the river, and the vertical lines the path of
the boat if there were no current.
We leave the Bolation of this question as an exercise, merely
noting that the equation of the curve is
in one or other of the throe cnsea, according as e is >, SjOri
When t — \ Uiis becomes
y* = a* — 2n»;, the parabola.
The time of crosung is
which is finite only .for e<l, because of course a negative valu*
..Google
36 PBELIUINABT. [49.
49. Another ezccllent example of the transformation of rela-
tive into absolute motion is afforded by the family of cycIoidE.
We shall in a future section consider their mechanical descrip-
tion, by the rolling of a circle on a fised straight line or circle.
In the mean time, 'we take a diEFerent form of enunciation,
which, however, leads to precisely the same result.
Pind the actual path of a point which revolves uniformly in
a circle about another point — the latter moving uniformly in a
straight Une or circle in the same plane.
Take the former case first : let o be the radius of the relative
circular orbit, and u the angular velocity ia it, v being the
velocity of ita centre along the atraight line.
The relative co-ordinates of the point in the drcle are a cos ad
and a sin mt, and the actual co-ordinates of the centre are vt
and 0. Hence for the actual path
(=vt + acoaiat, i] = aid.D.wt,
Hence f = - sin"' - + ^a' -if, an equation which, by giving
different values to v and w, may be made to represent the cycloid
itself, or either form of trochoid. See § 92.
For the epicycloids, let & be the radius of the circle which B
describes about A, u, the angular velocil^; a the radios of A'a
path, a the angular velocity.
Also at time ( = 0, let £ be in the radius
OA of A'a path. Then at time (, if A', B
be the positions, we see at once that
lAOA' = «^, LBCA = ,of.
Hence, taking OA as axis of a;,
x—aiXAui-^-h coBuijl, ^ = a 8in(u< + ^ Bin(ii,f,
which, by the elimination of t, give an algebraic equation between
X and y whenever <u and u, are conunensurable.
Thus, for (1), = 2u), suppose oiC = tf, and we have
x~aco8$+6coB20, j/ = aBintf + &Bin 2^,
or, by an easy reduction,
jiGoogle
49.] KWEMATICa 37
Fnt x — b for x, i.e., change the origin to & distance AS to the b
left of O, the equation becomes
o-(«?+jO.(>^ + Sr'-2tan
or, in polar co-ordinates,
a' = {r-2l>coBff)', r = a -t- 2b <x» $,
and when 2b = a, r = a{l-i- cob $), the cardioid. (See § 94.)
60. As au additiooal illustration of this part of our subject,
we may deSoe as follows : —
If one point A executes any motion whatever with reference
to a second point B; H B executes any other motion with refer-
ence to a third point G ; and so oa — the first is said to execute,
with reference to the last, a movement which is the resultant of
these several movements.
The relative position, velocity, and acceleration are in such a
case the geometrical resultants of the various components com-
bined according to preceding rules.
61. The following practical methods of effecting such a com-
bination in the simple case of the movements of two points are
useful in scientific illustrations and in certain mechanical arrange-
ments. Let two moving points be joined by an elastic string ;
the middle point of this string will evidently execute a move-
ment which is half the resultant of the motions of the two
points. But fur drawing, or engraving, or for other mechanical
applications, the following method is preferable : —
CF and ED are rods of equal length
moving freely round a pivot at P, which
passes through the middle point of each —
GA, AD, EB, and BF, are rods of half tlio
length of the two former, and so pivoted
to them as to form a pair of equal rhombi
CD, EF, whose angles can be altered at -^ ^
will Whatever motions, whether in a plane, or in space of three
dimcnsioDS, be given to A and B, P will evidently be subjected
to half their resultant.
62. Amongst the most important classes of motions which
we have to consider in Natural Philosophy, there is one, namely,
Harmonic Motion, which is of such immense use, not only in
..Google
38 PBKUHIHART. [52.
ordinary kinetics, but in the theories of sound, light, heat, etc,
that we make no apolt^ for entering here into considerable
detail regarding it
63. Dtf. When a point Q movefl uniformly in a circle, the
^ perpendicular QP drawn from its position
at any instant to a fixed diameter AA' of
the circle, intersects the diameter in a point
P, whose position changes by a simpls har-
mmic motion.
Thus, if a planet or satellite, or one of
the constituents of a double star, supposed
to move uniformly in a circular orbit about
its primary, be viewed from a very distant position in the plane
of its orbit, it will appear to move backwards and forwards in a
str^ght line, with a simple harmonic motion. This is nearly
the case with such bodies &s the satellites of Jupiter when seen
from the earth.
Physically, the interest of such motions consists in the fact
of their being approximately those of the simplest vibrations of
sounding bodies, such as a tuning-fork or pianoforte wire ; whence
their uame ; and of the various media in which waves of sound,
light, heat, etc., are propagated.
64. The Amplitude of a simple harmonic motion is the
range on one side or the other of the middle point of the coarse,
i.ft, OA or OA' in the figure.
An arc of the circle referred to, measured from any fixed
point to the uniformly moving point Q, is the ArgumeiU of
the harmonic motion.
The distance of a point, performingasimple harmonic motion,
from the middle of its course or range, is a simple harmonic func-
tion of the time. The argument of this function is what we have
defined as the argument of the motion.
The Epoch in a simple harmonic motion is the interval of time
which elapses from the era of reckoning till the moving point
first comes to its greatest elongation in the direction reckoned
as positive, from its mean position or the middle of its range.
Epoch in angular measure is the angle described on the circle of
reference in the period of time defined as the epoch.
..Google
54.] KINEMATICS. 39
The Period of a simple harmonic motion is the time which aimpis
elapsea from any instant until the moving point again moves in hm^'*
the same direction through the same position.
The Phase of a simple harmonic motion at any instant is the
fraction of the whole period which has elapsed since the moving
point last passed through its middle position in the positive
direction.
56. Those common kinds of mechanism, for producing recti- Bimpie
lineal from circular motion, or vice versa, in which a crank moiion in
moving in a circle works in a straight slot belonging to a body
which can only move in a straight line, fulfil strictly the definition
of a simple harmonic motion in the part of wliich the motion is
rectilineal, if the motion of the rotating part is uniform.
The motion of the treadle in a spinning-wheel approiiimates
to the same condition when the wheel moves uniformly; the
approximation being the closer, the smaller is the angular motion
of the treadle and of the connecting string. It is also approx-
imated to more or less closely in the motion of the piston of a
steam-engine connected, by any of the several methods in use,
with the crank, provided always the rotatory motion of the
crank be uniform.
56. The velocity of a point executing a simple harmonic ypiodty
motion is a simple harmonic function of the time, a quarter of motku.
a period earlier in phase than the displacement, and having its
maximum value equal to the velocity in the circular motion by
which the given function is defined.
For, in the fig. of § 53, if F be the velocity in the circle, it
may be represented by OQ in a direction perpendicular to its
own, and therefore by OP and PQ in directions perpendicular to
those lines. That is, the velocity of P in the simple harmonic
y
motion is ^r^ PQ ; which, when P is at 0, becomes V.
67. The acceleration of a point executinga simple harmonic Aoaeient-
motion- is at any time simply proportional to the displacement motioa.
from the middle point, but in opposite direction, or always
towards the middle point. Its maximum value is that with
which a velocity equal to that of the circular motion would
..Google
40 PHELIMINABY. [57.
be acquired ia the time in -which an arc equal to the radius
' is described.
yt
For, in the fig. of § 53, the acceleration of § (by § 35, a) is j-^
along Q 0. Supposing, for a moment, QO to represent the mag-
nitude of this acceleration, we may resolve it in QP, PO. The
acceleration of ^ is therefore represented on the same scale by
V* PO V
PO. Its magnitude is therefore -^ • -^ = ^^ PO, which is
V
proportional to PO, and has at A its maximum v^ue, -;yyy , an
acceleration under which the velocity V would be acquired in
.so.
Let a be the amplitude, * the q>och, and T the period, of a
dmple bannomc motion. Then if j be ilie displacement lirom
middle position at time t, we have
, = aco«(^-.).
Hence, for velocity, we have
gjtt . /2rt
'2rt \
Agein, for acceleration,
^=-^cos(^^-«j = -^ft (See§5r.)
IJistly, for the maximum value of the aixxleration,
2»
^"-ffhere, it may be remarked, =- ia the time of describing an arc
equal to radius in the relative circular motion.
cempMi- 68- Any two simple harmonic motions in one line, and of
B-^H. M. in one period, ^ve, when compounded, a sin^e simple harmonic
T motion ; of the same period ; of amplitude equal to the diagonal
of a parallelogram described on lengths equal to their amplitudes
measured on lines meeting at an angle equal to their difference
..Google
58.} KlNEMilTICS. 41
of epochs ; ancl of epoch differing from their epochs by angles J!"*"!"^-
equai to those which this diagonal makea with the two sides of ^^^li^'
the paraUel^ram. Let P and P* be
two points executing simple harmonic
motions of one period, and in one line
B'BCAA'. Let C and Q" be the uni-
formly moving pointe in the relative
circles. On CQ and C(^ describe a
pandlelc^fram SQO^ ; and through S
draw SR perpendicular to S'A' pro-
duced. We have obviously FS=OP
(being projections of the equal and
paraUel lines Q'S,CQ,oa, CB). Hence
CS='CP+CP'; and therefore the
point S executes the resultant of the motions P and P', But
CS, the diagonal of the parallelogram, is constant, and therefore
the resultant motion ia simple harmonic, of amplitude CS, and
of epoch exceeding that of the motion of P, and falling short
of that of the motion of P", by the. angles Q03 and SC(/ re-
spectively.
This geometrical construction has ^en usefully applied by the
tidal committee of the British Asaociation for a mechanical tide-
indicator (compile § 60, below). An arm CQ turning round G
carries an arm Q8 turning round Q', Toothed wheels, one of
them fixed with ita axis through C, and the others pivoted on a
framework carried by, CQ^, are so arranged that Q'S turns veiy
approximately at the rate of once round in 12 mean lunar hours,
if CQ be turned uniformly at the rate of once round in 12 mean
solar hours. Days and half-days are marked by a counter geared
to CQ. The distance of 8 from a fised line through C shows
the deviation from mean sea-level due to the sum of mean solar
and mean lunar tides for the time of day and year marked by
CQ and the counter.
An onalytim] proof of the same proposition is useful, being as
follow,;-
■"«'[Tf-)*'«'(-Y-)
.(.
+ a'ooB*')«
T*(ai
»fr-«).
jiGoogle
42 PRELIMINARY. [58.
- wtere r = {(« cos e + a' cos t*)' + (« an « + a' sin (^l*
'' =ja»+o" + 2(K»'cOfl(«-e^}*
and taji6-= ; r.
a cos ( + a ooa «
69. The conatruction described io the preceding eection ex-
hibits the Fcsultaot of two simple harmonic motions, whether of
the same period or not. Only, if they are not of the same period,
the diagonal of the parallelogram will not be constant, but will
diminish from a maximum value, the sum of the component
amplitudes, which it has at the instant when the phases of the
component motions agree ; to a minimum, the difference of thor j
amplitudes, which is its value wlieu the phases differ by half
a period. Its direction, which always must be nearer to the
greater than to the less of the two radii constituting the sides
of the parallelc^ram, will oscillate on each side of the greater
radius to a maximum deviation amounting on either side to the
angle whose sine is the less radius divided by the greater, and
reached when the less radius deviates more than this by a
quarter circumference from the greater. The full period of this
oscillation is the time in which either radius gains a full turn
en the other. The resultant motion is therefore not simple
harmonic, but is, as it were, simple harmonic with periodically
increasing and diminishiug amplitude, and with periodical ac-
celeration and retardation of phase. This view is particularly
appropriate for the case in which the periods of the two com-
ponent motions are nearly equal, but the amplitude of one of
them much greater than that of the other.
To express the resultant motion, let a be the dUplacemcnt at
time (; and let a be the greater of the two oomponent half-
amplitudes.
< = d cos («( — c) + a' cos {n't ~ «')
= a cos (ni — <) -^ a' COB (n( - e + 0)
= (rt-H o' cos ^) cos (ne - f ) - o' sin 0 sin (n( - (),
if * = (n'(-0-C«(-«);
or, finally, « = »■ cos (n( - « + $),
..Google
59.] KIXEIUTICS. 43
if r=(a' + 2tt<i'coB^ + a'^* SS'S^*
. rt a'ffln^ ouaUue.
and tan tf = , ■ ■-;■.
a + a cos^
The masimum value of taa^ in the last of these equations is
found by makinc 6 = t; ->■ ain~* — , and is equal to , ,
2 a' ^ (a'- a")*
a&d hence the inammnm value of $ itself is sin'' — . The geo-
metrical metliods indicated above (§ 58) lead to this conclusion
bj the following veiy simple construction.
To fiod the time and the amount of the maximum acceleration
or retardation of phase, let CA he the greater half-amplitude.
From A aa centre, with the less half-amplitude aa radius, de-
scribe a circle. CB touching this circle is the generating radius
of the most deviated resultant Hence CBA is a right angle ;
CA
60. A most interesting application of this case of the com- Bnmplaot
position of harmonic motions is to the lunar and solar tides; or s'^u"
which, except in tidal rivers, or long channels, or deep hays,
follow each very nearly the simple harmonic law, and produce, as
the actual result, a variation of level equal to the sum of varia-
tions that would be produced by the two causes separately.
The amount of the lunar equilibrium-tide (§ 812) is about 21
times that of the solar. Hence, if the actual tides conformed to
the equilibrium theory, the spring tides would be SI, and the
neap tides only I'l, each reckoned in terms of the solar tide ;
and at spring and neap tides the hour of high water is that of
the lunar tide alone. The greatest deviation of the actual tide
from the phases {high, low, or mean water) of the lunar tide
alone, would be about So of a lunar hour, that is, '98 of a solar
hour (being the same part of 12 lunar hours that 28* 26', or the
angle whose sine is ^tt , is of 36(V). This maximum deviation
would be in advance or in arrear according as the crown of the
solar tide precedes or follows the crown of the lunar tide ; and it
would be exactly reached when the interval of phase between
..Google
11 PHELIMIHABT. [GO.
BampiM or the two component tides is 395 lunar hours. That is to say,
o?R h.m!" there would be maximum advance of the time of high water 4^
*"" days after, and maximum retardation the same number of days
before, spring tides (compare § 811).
61. We may consider next the case of equal amplitudes in
the two given motions. If their periods are equal, their re-
sultant in a simple harmonic motion, whose phase is at every
instant the mean of their phases, and whose amplitude is equal
to twice the amplitude of either multiplied by the cosine of half
the difference of their phases. The resultant is of course nothing
when their phases differ by half the period, and is a motion of
double amplitude and of phase the same as theirs when they are
of the same phase.
When their periods are very nearly, but not quite, equal (their
amplitudes being still supposed equal), the motion passes very
slowly from the former (zero, or no motion at all) to the latter,
and hack, in a time equal to that in which the faster has gone
once oftener through ita period than the slower has.
In practice we meet with many excellent examples of this
case, which will, however, be more conveniently treated of when
we come to apply kinetic principles to various subjects in acou-
stics, physical optics, and practical mechanics ; such as the sym-
pathy of pendulums or tuning-forks, the rolling of a turret ship
at sea, the marching of troops over a suspension bridge, etc.
MachaninD 62. If any number of pulleys be so placed that a cord
pcwhUtir passing from a fixed point half round each of them has its
*"»™ free parts all in piirallel lines, and if their centres be moved
with simple harmonic motions of any ranges and any periods
in lines parallel to those lines, the unattached end of the
cord moves with a complex harmonic motion equal to twice
the sum of the given simple harmonic motions. This is the
principle of Sir W. Thomson's tide-predicting machine, con-
structed by the British Association, and ordered to be placed
in South Kensington Museum, availably for general use in
calculating beforehand for any port or other place on the sea
for which the simple harmonic constituents of the tide have
been determined by the "harmonic analysis" applied to
..Google
62.] KmEXATtca 43
previous observations*. We may exhibit, graphically, any case Or«pi>i»i
of single or compound simple harmonic motion in one line by gj^^j„
curves in vhich the abscissae represent intervals of time, and the !^'>j?l'"
* See BritiBh Assooiation Tidal Committee'B Beporta, 1B68, 1873, ISTE :
Iwture oM Tidet, lij Sit W. Thomson (Colliiu, Qlaagoir, 18TS).
..Google
46 PRELIMINABT. [G2.
onphie^ ordinates the correspondiag diatances of the moving point from
gni^ ita mean position. In the case of a aingle simple harmonic
^uSS.'" ™o*>o'i> t^6 corresponding cui-ve would be that described by the
point P in § 53, if, while Q maintained its uniform circular
motion, the circle were to move with uniform velocity in any
direction perpendicular to OA. This construction gives the
harmonic curve, or curves of sines, in which the ordinates are
proportional to the sines of the abscissEe, the straight line iu
which 0 moves being the axis of abscissse. It is the simplest
possible form assumed by a vibrating string. When the har-
monic motion is complex, but in one line, as is the case for any
point in a viohn-, harp-, or pianoforte-string (diEferiog, as these
do, from one another in their motions on account of the different
modes of excitation used), a pimllar construction may be made.
InvestigatioQ regarding complex harmonic functions has led to
results of the highest importance, having their most general
expression in Fourier's TkeoreTti, to which we will presently devote
several pagea We give, on page 45, graphic representations of
the composition of two simple harmonic motions in one line, of
equal amplitudes and of periods which are as 1 : 2 and as 2 : 3,
for differences of epoch corresponding to 0, 1, 2, etc., sixteenths
of a circumference. In each case the epoch of the component of
gi-eater period is a quarter of its own period. In the first, second,
third, etc, of each series respectively, the epoch of the component
of shorter period is less than a quarter-period by 0, 1, 2, etc.,
sixteenths of the period. The successive horizontal lines are the
axes of abscissa? of the successive curves ; the vertical line to the
left of each series being the common axis of ordinates. In each
of the first set the graver motion goes through one complete
period, in the second it goes through two periods.
1:2 2:3
(Octave) ■ (Fifth)
y = 8iux + Binf2:i; + -^ j , y ^^ sin 2a; + sin (3a; + ~j .
Both, &om a; = 0 to a;=2T; and for n^O, 1, 2 15, in succession.
These, and similar cases, when the periodic times are not com-
mensurable, will be again treated of under Acoustics.
..Google
63.] KINEUATtCS. 47
63. We hare next to consider the composition of simple bar- s. h. mo-
monic motions in different directions. In the first place, we see diirtrent
that any number of simple harmonic motions of one period, and
of the same phase, superimposed, produce a single simple har-
monic motion of the same phase. For, the diBplacement at any
instant being, according to the principle of the composition of
motions, the geometrical resultaut (see above, § 50) of the dis-
placements due to the component motions separately, these com-
ponent displacements, in the case supposed, all vary in simple
proportion to one another, and are in constant directions. Hence
the resultant displacement will vary in simple proportion to each
of them, and will be in a constant direction.
But if, while their periods are the same, the phases of the
Eeverai component motions do not agree, the resultant motion
will generally be elliptic, with equal areas described in equal
times by the radius-vector from the centre ; although in par-
ticular cases it may be uniform circular, or, on the other hand,
rectilineal and simple harmonic.
64. To prove this, we may first consider the case in which
ve have two equal simple harmonic motions given, and these in
perpendicular lines, and differing in phase by a quarter period.
Their resultant is a uniform circular motion. For, let SA, BA'
be their ranges; and from 0, their common middle point, as
centre, describe a circle through AA'Bff. The given motion of P
in BA will be (§ 53) defined by the motion
of a point Q round the circumference of
this circle ; and the same point, if moving
i o the direction indicated by the arrow, will
give a simple harmonic motion of P, in
BA', a quarter of a period behind that of
the motion of Pin BA. But, since A'OA,
QPO.aodQP'O are right angles, the figure "" .
QPOP is a parallelogram, and therefore Q is in the position of
the displacement compounded of OP and OP. Hence two equal
simple harmonic motions in perpendicular lines, of phases dif-
fering by a quarter period, are equivalent to a uniform circular
motion of radius equal to the maximum displacement of either
singly, and in the direction from the positive end of the range of
N
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48 PRELtMIKABV. [64.
the component in advance of the other towards the positive end
of the range of this latter.
66. Now, orthogonal projections of simple harmonic motions
are clearly simple harmonic with unchanged phase. Hence, if
we project the case of § Qi on any plane, we get motion in an
ellipse, of which the projections of the two component ranges
are conjugate diameters, and in which the radius-vector from the
centre descrihes equal areas (heing the projections of the areas
described by the radius of the circle) in equal timea But the
plane and position of the circle of which this projection is taken
may clearly he found 8o as to fulfil the condition of having the
projections of the ranges coincident with -any two given mutually
bisecting lines. Hence any two given simple harmonic motions,
equal or unequal in range, and oblique or at right angles to one
another in direction, provided only tiey differ by a quarter
period in phase, produce elliptic motion, having their ranges for
conjugate axes, and describing, by the radius-vector froni the
centre, equal areas in equal times (compare § 34, b).
66. Betuming to the compOMtion of any number of simple
harmonic motions of one period, in lines in all directions and of
all phases ; each component simple harmonic motion may be de-
terminatoly resolved into two in the same line, differing in phase
by a quarter period, and one of them having any given epoch.
We may therefore reduce the given motions to two seta, differing
in phase by a quarter period, those of one set agreeing in phase
with any one of the given, or with any other simple harmonic
motion we please to choose {i.e., having their epoch anything
we pleaae).
AU of each set may (§ 58) be compounded into one simple
harmonic motion of the same phase, of determinate amplitude,
in a determinate line ; and thus the whole system ia reduced to
two umple fully determined harmonic motions differing from
one another in phase by a quarter period.
Now the resultant of two simple harmonic motions, one a
quarter of a period in advance of the other, in different lines, has
been proved (§ Go) to be motion in an ellipse of which the ranges
of the component motions are conjugate axes, and in which equal
..Google
KINEMATICS.
areas are described by the radius- vector &om the centre in equal a^ j^""
times. Hence the general propoeition of § 63. dlSSSu.
a;, = 2,a, oos («rf — c,),!
y,=»«,a,cOB(»J-«,),l...
», = n,a, cos (<uj — «,),]
be the Cartonan spedfioatioa of the first of the g^ven motioDS ;
and so irith varied suffixes for the others ;
l,m,n denoting the direc^n coeineH,
a „ „ half UDplitnde,
« „ „ epocii,
the premier suffix being attached to each letter to apply it to each
caae, and u denoting the common relativs angular velocity. The
resultant motion, specified by z, y, « without suffixes, is
a: = S/,o, OOB (nrf - «,)= COB wCy.a, cos «j + sin wtS/jOj Bin €,,
y=etc,; 4=eto.;
or, as we may write for brevity,
(c = i* cos ut + ^«ia uJ,
y=Qcoaiat+Q'!da<at,
2 = £ cos <ii( + A' sin 01^ j
where P = y, i,<i,cofltj, i>' = S f^ajSint,,)
Q = lm,a, C08<„ ^ = Sm,a^Bini,,} (3)
if = 2 nfl^ cos <„ JfmS, n^a^ sin «,.]
The resultant motion thus specified, in terms of six component
simple harmonic motions, may be reduced to two by oompQuuding
P, Q, R, and P", Q', If, in the elementary way. Thus jf
P Q R
£' = (P"+^' + £">*,
.. P' . 0" . «
X=^. ^=-j., y=^,^
the required motion will be the resultant of {cosuf in the line
(A, ft f), and f sin at in the line (X', p.', »■'). It is therefore mo-
timt in an ellipse, of winch 2{ and 2{' in those directions are
VOL. I. 4
..Google
60 PBEUMWAItY. [66.
conjugate diameterB ; irith radiua-vector from centra tracing
equal areas in «qual times ; and of period — .
■ 67. We must next take the case of the compoaitioo of simple
harmonic motions of different periods and in different lines. In
general, whether these lines be in one piano or not, the line
of motion retumR into itself if the periods are commensurable;
and if not, not. This is evident without proof.
If a be the amplitude, < the epoch, and n the angular velocity
in the relative circular motion, for a component in a line whose
direction cotinea are X, /i, v — and if f, i}, { be the co.ordinat«B in
tiie resultant motion,
f=S.V,coH(7i,(-c,), ij==S.ft,a,C08(n,i-«^, f = S.v,a,«)a («,(-<,).
Kow it is evident that at time t^T the valueti of f , 17, f! will recur
aa soon as n,7, n^T, etc., are multiples of 2*', that is, when Z'is
the least common multiple of — , — , etc
R, »,
If there be such a common multiple, the trigonometrical func-
tions may be eliminated, and the equations (or equation, if the
motion is in one plane) to the path are algebraic. If not, they
are transcendental.
68. From the above we see generally that the composition
of any number of simple harmonic motions in any directions
and of any periods, may be effected by compounding, according
to previously explained methods, their resolved parts in each
of any three rectangular directions, and then compounding the
final resultants in these directions.
69. By far the most interesting case, and the simplest, is
that of two simple harmonic motions of any periods, whose di-
rections must of course be in one plane.
Mechanical methods of obtaining stich combinations will be
afterwards described, as well as cases of their occurrence in
Optics and Acoustics.
We may suppose, for simplicity, the two component motions
to take place in perpendicular directions. Also, as we can only
have a re-entering curve when their periods are commensur-
able, it will be advisable to commence with such a case.
..Google
C9.] KINEMATICS. 51
The following figures represent tlie paths produced by tbeg-H-mo.
combination of simple harmonic motions of equal amplitude in
two rectangular directions, the periods of the compouents being
as 1 : 2, and the epochs differing succesBively by 0, 1, 2, etc.,
sixteenths of a circumference
In the case of epochs equal, or differing by a multiple of ir,
the curve is a portion of a parabola, and is gone over twice
in opposite directions by the moving point in each complete
period.
For the cue figured above^
X = a COB (2nt — t), y - a COB nl.
Henoe x = a{coB2ntcoet + an2ntaiit]
■^j-
-l)a«. + 2|yi
.t^„.
which for any given value of € is the equati
tog curve. Thus for < = 0,
the correspond -
Thus for ( = 0,
-^ — 1 , or y* = - (a; + a), the parabola as above,
4—2
..Google
&2 PBELIHINABT. [69.
™ For ,^l we have ^ = 21^1-^, or aV = V(«'-y»),
the equation of the 5th and 13th of the above curves.
In general
x~a<xm{nt + t), y = aooR (h,f + «,),
frcou which < ia to be eliminated to find the Cartesian equation of
the curve.
70. Atiother very important case ia that of two groupe of
two eimple hannonic motions in one place, such that the resultant
of each group is uniform circular motion.
If their periods are equa], we have a case helonging to those
already treated (§ 63), and conclude that the resultant is, in
general, inotion in an ellipse, equal areas being described in
equal times about the centre. As particular cases we may have
simple harmonic, or uniform circular, motion. (Compare § 91.)
If the circular motions are in the same direction, the resultant
is evidently circular motion in the same direction. This is the
case of the motion of <$ in § 58, and requires no further comment,
as its amplitude, epoch, etc., are seen at once from the figure.
71. If the periods of the two are very nearly equal, the re-
sultant motion will be at any moment very nearly the circular
motion given by the preceding construction. Or we mayr^ard
it as rigorously amotion in a circle with a varying radius de-
creasing from a maximum value, the sum of the radii of the two
component motions, to' a minimum, their difference, and increas-
ing again, alternately ; the direction of the resultant radius
oscillating on each side of that of the greater component (as in
corresponding case, § 59, above). Hence the angular velocity
of the resultant motion is periodically variable. In the case of
equal radii, nest considered) it is constant.
72. When the radii of the two component motions are equal,
we have the very interesting and important case figured below.
Here the resultant radius bisects the angle between the com-
ponent radii. The resultant angular velocity is the arithmetical
mean of its components. We will explain in a future section
..Google
72.] KmEKATICS. 53
(§ 94} hov thiB epitrochoid is traced by the rolling of one circle compoii-
on another. ^The particular case above delineated is that of a
non- reentrant curve.)
73. Let the uDifonn circular motions be in opposite direc-
tions ; then, if the periods are equal, we may easily see, as
before, § 66, that the resultant is in general elliptic motion,
includiog the particular cases of uniform circular, and simple
harmonic, motion.
If the periods are very nearly equal, the resultant will be
easily found,-as in the case of § 59.
74. If the radii of the component motions are equal, we have
cases of very great importaace in modem physics, one of which
is figured below (like the preceding, a non-reSntrant curve).
..Google
64 PBELUCIHABT. [74.
This is intimately coimected with the explanation of two seta of
important pheDomeaa, — ^the rotation of tho plane of polarization
of light, by quartz and certain fluids on the one hand, and by
transparent bodies under nu^netic forces on the other. It id
a case of the hypotrochoid, and its corresponding mode of
description will be described in a future section. It will also
appear in kinetics as the path of a pendulum-bob which contains
a gyroscope in rapid rotation.
75. Before leaving for a time the subject of the composition
of harmooic motions, wo must, as promised in § 62, devote some
pages to the consideration of Fourier's Theorem, which is not
only one of the most beautiful results of modern analysis, but
may be said to funiish an indispensable insti-ument in the treat-
ment of nearly every recondite question in modern physics. To
mention only sonorous vibrations, the propagation of electric
signals along a telegraph wire, and the conduction of heat by
the earth's crust, as subjects in their generality intractable with-
out it, is to give but a feeble idea of its importance. The follow-
ing seems to be the most intelligible form in which it can be
presented to the general reader : —
Theorem. — A complea; harmoniG function, with a constant term
added, is the proper expression, in mathematical language,
/or any arhOrary periodic function ; and consequently can
express any function whatever between definite values of
the variable.
76. Any arbitrary periodic function whatever being pven,
the amplitudes and epochs of the terms of a complex harmonic
function which shall be equal to it for every value of the iade-
pendent variable, may be investigated by the " method of inde-
terminate coefficieDt&"
Assume equation (14) Wow. Multiply both members first
you find (13).
..Google
76.] EINEUITICS. 55
This iiiTestigation is safficient as a solution of the problem, ft
— to find a complex harmonic function expressing a given arbi-
trary peiiodic function, — when once we are assured that the
problem is possible ; and when ve have this assurance, it proves
that the resolution is deteiTninate ; that is to say, fhat no
other complex harmonic function than the one we have found
can satisfy the conditions.
For description of an int^rating machine by which the
coefficients Ai, Bt in the Fourier expression (14) for any given
arbitrary function may be obtained with exceedingly little
labour, and with all the accuracy practically needed for the
harmonic analysis of tidal and meteorological observations, see
Proceedings of the Royal Society, Feb. 1876, or Ch^. r. below.
77. The full theory of the expression investigated in § 76
wiU be made more intelligible by an investigation from a
different point of view.
Let F(x} be «aj periodic function, of peiiod p. That is to
say, let F{s^ be any function fulfilling the condition
F(x*ip) = F{,) (1),
where t denotes any ponUve or n^ative integer. Ctmsider the
integial
(•F{as) dx
yrixen a, e, <l denote any three given quantities. Its value is
less than J'(a) | -. — -^, and greater than F{e) I 1—3, if *
and ^ denote the values of x, Mt^er equal to c
between the limits e and e*, for which F{x) is greatest and least
respectively. But
r—i i= -(tan ' — t«n~' — )..
a' + a? a\ a aj
•F(x)adx
rF^
/'{.)(ta
F{z')L>,
jiGoogle
56 PRELDONABT. [77.
Hence if J be tike greatoat of all Uie yaXnea oS F{x), and J? the
least,
•^ „ >B(^-t>n-i).
Also, eiimlarl;,
and „ >£ | tan"' - + s 1 ■
Adding die first m^nben of (3), (i), and (6), and comparing
with die coneaponding muns <tf the aeoond memberB, we find
But, by (1),
P)
(6)
i:^-f/^M^z.{^,4^)}-
■■(')■
Now if w& d^iote ^- 1 by w,
,1 , .i-r ' ■_ ' 1
a + (a; + tp) 2sv \x + ip — av x + ip + aaj '
and therefore, taking the tomu eiSTesponding to positive and
eqnal negative valuea of t together, and the terma for » = 0 8q)»-
rately, i
^oot
'p'-{x-avy
x + au '-•»V-(« + <wn
-oot-
~ 2apv\
V . 2irait ir
2npu j> <ipu
COB* OOB — COS
Stoo
jiGoogle
77.]
J^a' + a^ ap\ J I ™
Kezi, denoting temporarily, for brevity, « '* by {, and putting
•"'— <»),
f" -2ooe — +t P
(1 +2«ooa — 4-2e*coe -tSs'coa + 6tc I
2«ar „ , ivx „ , 6»« ^ \
-= — -, 1 1 TAsvua — 4-2rcoe -t 2s^co8 + 6tc 1.
1-e'V P jp P }
Hence, acoording to (S) and (9),
Hmce, by (6), we inier that
/■(*) (tan-' ^- tan"' M + J ^T - taa-' % tan" ' -) >
and J'(/)(taii-'^-tan-'^ + £(»-tan-'^+tan-'^<
- r'j'(!e)da:(l + 2« COB — + eta) .
Now let <^ = -c, and x=i'-i,
C being a variable, and f constant, bo far aa the integration is
concerned ; and let
and tberefore F(z) = ^(f+z),
..Google
..(11)
PRKLIMINikBY.
The preceding pair of ineqnalities becomes
^(f+a).2taii-'- + j<(^w-2tan"' -^ >
and ^(f +/) . 2 tan"' - + sfir - 2 tan"' ^ <
where ^ denotes anjr periodic fonction whatever, of period p.
Now let e be a reiy small fraction of p. In the limit, where c
is infinitely small, tiie greatest and least values of ^({') for values
of f between f + candf— c will be infinitely nearly equal to one
another and to ^(^; that is to say,
Kext, let a be an infinitely small fraction of e. In the limit
= 1.
S}...(.2).
Hence the oompArison (II) becomes in the limit an eqaatitm
which, if we divide both members by w, gives
This is the celebrated theorem discovered by Fourier* for the
development of an arbitnuy periodic function in a aeries of simple
harmonic term& A formula included in it as a particular case
had been given previously by Lagrange t.
I^ for ooB-
, we take its value
3wf . 2tVf . 2»,f
fl = 4- Din — Bin . ?
and introduce the following notation :
■ (13)
jiGoogle
77.] KmiatiTica 59
we reduce (12) to this form :— F
i,{()^A. + t" A,coB^ + t'" B,<nn^ (14),
vliich is the general exprensioa of an arl^itrary function in terms
of a Beriee of cotdnes oad of sines. Or if we take
P,= {A'+£,')i, and taae.=:j' (IS),
wehave i,{i) = A, + 2'-;p,coe(^-A (16),
which is the general expression in a series of single simple har-
monic terms of the successive multiple periods.
Each of the equations and comparisoue (2), (7), (8), (10), and Conrerg-
(ll)isatmearithmetical expression, and may be veriiied by actual fouriar'i
calculation of the numbera, for any particular case ; provided only
that ^(x) has no in&nite Talue in its period. Hence, with this
exception, (12) or either of its equivalents, (14), (16), is a true
arithmetical expression ; and the series which it involves is there-
fore convergent; Hence we may witli perfect rigour conclude
that even the extreme esse in which the arbitrary function ex-
perience* an abrupt finite change in its value when the inde-
pendent variable, increasing continuously, passes through some
particular value or values, is included in the general theorem.
In such a case^ if any value be given to the independent variable
differing however little from one which corresponds to an abrupt
change in the value of the function, the series must, as we may
infer from the preceding investigation, converge and ^ve a
definite value for the function. But if exactly the critical value
is assigned to the independent variable, the series cannot oim-
Yerge to any definite value. The consideration of the limiting
values shown in the comparison (II) does away with all difBcuIty
in tmderstauding how the series (12) gives definite values having
a finite difference for two particular values of the independent
variable on the two sides of a critical value, but differing in-
finitely little from one another.
If ttie differential coefficient ^^— ' is finite for every value of
at
( within the period, it too is arithmetically expressible by a series
of harmonic t^rms, which cannot be other than the series ob^
tained by differentiating the series for <ft{i). Hence
..Google
60 PKEUMmAET. [77.
^ST -^.-^^tripM—-) (IT),
and tbifi Beriea ia convergent ; and we majr therefore conclude thftt
the series for ^(£) is more conre^ent than a harmonic aeries
with
1. i> h h otc-.
for its coefficienta If .^ has no infinite valuee within the
period, we may differentiate both members of (17) and still have
an equation arithmetically tme ; and bo on. We conclude that
if the n*^ differential coefficient of ^(^ has no infinite Talnee,
the harmonic seriee for ^(£) must eonrei^ more rapidly than &
harmonic series -with
1 1 1 1 *
1, 2--' 3- i-' ^^'
for its coefficienta.
Diipiwa- 78. We now pass to the consideration of the dispLacement
rigid bod7. of a rigid body or group of points whose relative positions are
unalterable. The simplest case we can consider is that of the
motion of a plane figure in its own plane, and this, as far as
kinematics is concerned, is entirely summed up in the result of
the next section.
Diipko*- 79. If a plane figure be displaced in any way in its own
Diua oeiat plane, there is always (with an exception treated in § 81) one
point of it common to any two positions ; that is, it may be
moved from any one position to any other by rotation in its own
plane about one pout held fixed.
To prove this, let A, B be any two point« of the plane figure
in its first position, A', B tiie positions of the same two after
^ a displacement. The lines AA, BB will
,^ not be parallel, except in one case to be
presently considered. Hence the line equi-
distant fjrom A and A' will meet that equi-
Oj^- -/- 1 distant from B and B in some point 0.
3oinOA,OB,OA',OB'. Then, evidently,
because OA' = OA, OB = OB and A'B
=AB, the triangles OA'B and OAB axe
equal and similar. Hence 0 is similarly
situated with regard to A'B and AB, and is therefore one and
..Google
79.] KINEUA.T1CS. 61
the same point of the plane figure in its two positions. If, for ^^"^
the sake of illustration, we actually trace the triangle 0^0 upon pi^ilg^
the plane, it becomes OAE in the second position of the figure.
80. If from the e<]^ual angles A'OS, AOB of these similar
triangles we take the common part A'OB, we have the remaining
angles AOA', BOS' equal, and each of them is clearly equal to
the angle through which the figure must have turned round the
point 0 to hring it from the first to the second position.
The preceding simple construction therefore enables ua not
only to demonstrate the general proposition, § 79, hut also to
determine from the two positions of one terminated line AB,
A'B' of the figure the common centre and the amount of the
angle of rotation.
61. The lines equidistant from A and A', and from B and R,
are parallel if AB is parallel to A'ff ; and therefore the con-
struction fails, the point 0 being
infinitely distant, and the theorem
becomes ni^tory. In this case the
motion is in fact a simple trans-
ition of the figure in its own
plane without rotation — since, AB being parallel and equal to
A'B", we have AA' piuallel and equal to BB ; and instead of
there being one point of the figure common to both positions,
the lines joining the two successive positions of all points in the
figure are equal and parallel.
82. It is not necessary to suppose the figure to be a mete flat
disc or plane — for the preceding statements apply to any one of
a set of parallel planes in a rigid body, moving in any way
subject to the condition that th© points of any one phme in it-
remain always in a fixed plane in space.
83. There is yet a case in which the construction in § 79 is
nugatory — that is when AA' is paral-
lel to BBT, but the lines of AB and
A'B' intersect. In this case, how-
ever, the point of interaection is the
point 0 required, although the former
method would not have enabled us to find it.
J>^
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62 PBELIHINABr. [84.
Eimpica 84. Very many interesting appIicaUons of this principle may
ment In ana be nuu]e, of which, however, few belong strictly to our subject,
and we shall therefore give only an example or two. Thus we
know that if a line of given length AB move with its extremities
always in two fixed lines OA, OB,
any point in it as P describes an
ellipse. It is required to find the
direction of motion of P at any in-
stant, i.e., to draw a tangent to the
ellipse. BA will pass to its next
position by rotating about the point
Q ; found by the method of § 79
by drawing perpendiculars to OA
and OB at A and B. Hence P for the instant revolves about Q,
and thus its direction of motion, or the tangent to the ellipse, is
perpendicular to QP, Also AB in its motion always touches a
curve (called in geometry its envelop) ; and the same principle
enables us to find the point of the envelop which lies in AB, for
the motion of that point must evidently be ultimately (that is
for a very small displacement) along AB, and the only point
which so moves is the intersection of AB with the perpen-
dicular to it from Q. Thus our construction would enable us
to trace the envelop by points. (For more on this subject
see § 91.)
85. Again, suppose AB to be the beam of a stationary engine
having a reciprocating motion about A, and by a link BD
turning a crank CD about 0. Determine the relation between
the angular velocities of AB and CD in any position. Evi-
dently the instantaneous direction of motion of B is trans-
verse to AB, and of D transverse to CD — hence if AB, CD
produced meet in 0, the motion of BD is for an instant as if
it turned about 0. From this
it may be easily seen that if
the angular velocity of AB be
,. , .„n. AB OD .
«,thatofCi)i8^^«. A
similar process is of course
applicable to any combination of machinery, and we shall find it
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S3.] KINEUATICS. 63
veiy coDveoient when we come to consider various dynamical
problems connected with virtual velocities. "ItSl^ "^
66. Since in general any movement of a plane figure in its Compotiiion
plane may be considered as a rotation about one point, it iB>boat
evident tliat two Buch rotations may in general be compounded ^■"^■
into one ; and therefore, of course, the same may be done with
any number of rotations. Thus let A and B be the points of
the figure about which in succession the rotations are to take
place. By a rotation about A, B is brought say to S", and by a
rotation about B, A is brought to A'. The construction of § 79
gives us at once the point 0 and the amount of rotation about it
which singly gives the same effect as those about A and B in
succession. But there is one case of exception, viz., when the
rotations about A and B are of equal
amount and in opposite directions. lo
this case A'B" is evidently parallel to
AB, and therefore the compound result
is a translation only. That is, if a body
revolve in succession through equal angles, but in opposite di-
rections, about two parallel axes, it finally takes a position to
which it could have been brought by a simple translation per-
pendicular to the lines of the body in its initial or final position,
which were successively made axes of rotation ; and inclined to
their plane at an angle equal to half the supplement of the
common angle of rotation.
87. Hence to compound into an equivalent rotation a rota- 1
tion and a translation, the latter being effected parallel to the mndt^MiT-
• p 1 y . 1 1 ■ - tloniinooe
plane of the former, we may decompose the translation mto two pUne-
rotations of equal amount and opposite direction, compound one
of them with the given rotation by § 86, and then compound
the other with the resultant rotation by the same process. Or
we may adopt the following far
simpler method. I*et OA be the B' i O
translation common to all points
in the plane, and let BO C be the
angle of rotation about 0, BO
\teiag drawn eo that OA bisects the exterior angle COR. Take
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64 PBEUWHABT. [87.
J" the point B in BO produced, eruch that SC, the space through
^ which the rotation canies it, is equal and oppOBite to OA. This
point retains its former position after the perfonuance of the
compound operation ; so that a rotation and a translation in
one plane can be compounded into an equal rotation ahout a
different axis.
In general, if the origin be takea as the point about which
rotation takes place in the plane of xy, and if it be through an
angle $, a point whose coordinates ware originally x, y wiU have
Uiem clianged to
i=xo(M0—ytaa$, i} = a;Bintf 4-ycoetf,
or, if the rotataon be very small,
£ = a!-ytf, i) = y + ««.
OmMonor 88. In considering the composition of angular velocities
u^ hiffbar about different axes, and other similar cases, we may deal with
uuiiiiuui- inlinitely small disjHacements only ; and it results at once from
the principles of the differential calculus, that if these displace-
ments be of the firat ord^ of smaQ quantities, any point whose
displacement is of the secfmd order of small quantities is to be
considered as rigorously at rest. Hence, for instance, if a body
revolve through an angle of the first order of small quantities
about an axis (belonging to the body) which during the revolu-
tion is displaced through an angle or space, also of the first
order, the displacement of any point of the body is rigorously
what it would have been had the axis been fixed during the
rotation about it, and its own displacement made either before
or af^r this rotation. Hence in any case of motion of a rigid
system the angular velocities about a system of axes moving unt/t
the system are the same at any instant as those about a system
fixed in space, provided only that tfae latter coincide at the
instant in question with the moveable ones.
sapmn^ 69. From similar considerations follows also the genei&l prin-
■«mmm. ciple <^ SuperpoaitioH of tmail motions. It asserts that if several
causes act timtiUaneoiultf un the same particle or rigid body, and
if the effect produced by each ia of the first order of small quan-
tities, the joint effect will be obtained if we ooosider the causes
to act wtfcgwt'rWy. each taking the point or syjitem in the posi-
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89.] KIMEUATICS. 65
tion in which the preceding one left it. It is evident at once 5"J'^"£\|
that this is an immediate deduction from the fact that the second mctiau
order of infinitely smaU quantities may be with rigorous accuracy
neglected. This principle is of very great use, as we shall find
in the sequel ; its apphcations are of constant occurrence.
A plane figure has given angular velocitieB about given axes
perpendicolor to its plane, find the reeoltant.
Let there be an angular velocify <u about an axis rmmng
thrangh the point a, b.
The consequent motion of the point a^ y in the time 8< is, as
we have just seen ^ 87),
-(y- 6) lafit parallel toa^ and (ai - a) uSt parallel to y.
Hence, by the superpositicm of sin^ motions, tihe whole motion
parallel to ails
and that parallel to y (xSu - Sa«>)8f.
Hence the point whose ixHudinatee are
, taa , , Sbu
is at rest, and the resultant axis posses throngh it. Any other
pcnnt X, y moves throagh spaces
-(y2ai-26«)8(, {x%u-3aio)St.
Bat if the whole had tamed abont !^,t/ with velocity 0, we should
have hod for the di^laoements oi k, y,
-(y-SOOSi, {x-aTiOSt.
Comparing, we find O = Su,
Hence if the sum of the angular velocities be zero, there is no
rotation, and indeed the above fbrmulie show that there is then
merely translation,
2(&»)^ parallel to a;, and - i(ao))St parallel to y.
These foimnln suffice for the consideration of any problem on
thesntject
90. Any motion whatever of a plane figure in its own plane BoUhnior
might be produced by the rolling of a curve fixed to the figure mm.'"
upon a carve fixed in the plane.
For we may consider the whole motion as made up of suo-
cessive elementary displacements, each of which corresponds, as
we have seen, to an elementary rotation about some point in
VOL. 1. 5
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66 PRELIHINABT. [90.
the plane. Let o,, o„ o„ etc., be the successive points of
the moving figure about which the rotations take place, O,,
0,, 0,, etc, the positions of these points when each is the
instantaneous centre of rotation. Then the figure rotates about
0, {or 0,, which coincides with itj till o, coincides with 0„ then
about the latter till o, coint^des with
0„ and so on. Hence, if we join o,,
0,, Of, etc., in the plane of the figure,
and 0,, 0„ 0,, etc, in the fixed plane,
the motion will be the same as if the
polygon 0,0/1^, etc,rolled upon the fixed
polygon OfijO^, etc By supposing the
succeesiTe displacements timall enough
the sides of these polygons gradually diminiah, and the polygons
finally become continuous curves. Hence the theorem.
From this It immediately follows, that any displacement of a
i^d solid, which is in directions wholly perpendicular to a fixed
line, may be produced by the rolling of a cylinder fixed in the
solid on another cylinder fixed in space, the axes of the cylinders
being parallel to the fixed lina
91. As an interesting example of this theorem, let us recur
to the esse of § 84 : — ^A circle may evidently be circumscribed
about OBQA ; and it must be of invariable magnitude, since in
it a chord of given length AB subtends a given angle 0 at the
circumference. Also OQ is a diameter of this circle, and is there-
fore constant. Hence, as Q is momentarily at rest, the motion
of the circle circumscribing OBQA is one of internal roUing on
a circle of double its diameter. Hence if a circle roll internally
on another of twice it« diameter, any point in its circumference
describes a diameter of the fixed circle, any other point in ita
plane an ellipse This is precisely the same proposition as that
of § 70, although the ways of arriving at it are very different.
As it presents us with a particular case of die Hypocydoid, it
warns us to return to the consideration of t^ese and kindred
. curves, which give good instances of kinematical theorems, but
which besides are of great use in physics generally.
92. When a circle rolls upon a straight line, a point in its
circumference describes a Cycloid; an internal point describes a
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92.]
KINEMATICS.
67
Prolate, an external one a Curtate, Cycloid. The two latter *?***
varieties are sometimes called Trochoids.
The general form of these corree vill be seen in the annexed
figures ; and to what follows we shall confine our remarks to the
cycloid itself, as of inunensely greater consequence than the
others. The next section contains a simple investigatioD of those
properties of the (peloid which are moat useful in our subject.
93. Let AS be a diameter of the generating (or rolling circle Ir^f****
BC the line on which it rolls.
The points A and B describe
fdmilar and equal (peloids, of
which AQC and BS are portions.
If PQB he any subsequent pou-
tioD of the generating circle, Q
and 3 the new positions of A and
B, ^QPS is of course a right
angle. I^ therefore, QB be drawn
parallel to PS, PB is a diameter
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G8 PBELOCIKABT. [93-
PrwcrtiM of the rolling circle. Produce QB to T, making RT^ QR=:PS.
ejticu. Evidently the curve A T, which ia the locus of T, is similar and
eqaal to B8, and is therefore a cycloid similar and equal to AC.
But QS is perpendicular to PQ, and is therefore the instanta-
neous direction of motion of Q, or is the tangent to the cycloid
AQG. Similarly, P8 ia perpendicular to the ir^cloid B8 at 8,
and 80 is therefore TQ to AT bA T. Hence {§ 19) AQC ia the
evolute of ^r. and arc Aq=qT=^qR.
gpii^eioid^ 94. When the circle rolls upon another circle, the curve
«^u^ described by a point in its circumference is called an Epicycloid,
or a Hypocycloid, as the rolling circle is without or within the
fixed circle ; and when the tracing point is not in the circum-
ference, we have Epitrochoids and Hypotrochoids. Of the latter
we have already met with examples, §§ 70,
91, and others will be presently mentioned.
Of the former, we have in Uie first of the
appended figures the case of a circle rollii^
externally on another of equal size. The
Curve in this case is called the Cardioid
(§«)•
In the second diagram, a circle
rolls externally on another of twice
its radius. The epicycloid so de-
scribed is of importance In Optics,
and will, with others, be referred
to when we consider the subject of
Caustics by reflexion.
In the third dif^ram, we have
a hypocycloid traced by the rolling
of one circle internally on another
of four times its radias.
O
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9t]
The carve figured ia § 72 is an epitrochoid described by a Biriardddi,
point ID the plaoe of a large circalar disc which rolls upon a <ffd3dMt«.
circular cylinder of small diameter, so that Uie point passes
through the auB of the cylinder.
That of § 74 is a bypotrochoid described by a point in the
plane of a circle irbich rolls iatemaUy on another of rather
more than twice its diameter, the tracing point passing through
the centre of the fixed circle. Had the diameters of the circles
been exactly as 1 : 2, § 72 or § 91 shows that this curve would .
have been reduced to a single straight line.
The general eqiiationa of this cIasb of curves are
«=((»+ 6) COB 0-eJ COS —r- 6,
where a is the ladiua of the fixed, b of the rolling circle ; and eb
is the distonoe of the trapsing point from the centre of the latter.
93. If a rigid solid body move in any way whatever, sub- J£|^
ject only to the condition that one of its points remtuns fixed, S"* p"*"*-
there is always (without exception) one line of it through this
point common to the body in any two positions. This most
important theorem is due to Kuler. To prove it, consider Baicr'a
s spherical surface within the body, with its centre at the
fixed point C. All points of this sphere attached to the
body will move on a sphere fixed in space. Hence the
construction of § 79 may be made, but. with great circles
instead of straight lines ; and the same reasoning will apply to
prove that the point 0 thus obtained is common to the body
in its two positions. Hence every point of the body in the
line OC, joining 0 with the fixed point, must be common to it
in the two positions. Hence the body may pass from any one
position to any other by rotating through a definite angle about
a definite axis. Hence any position of the body may be speci-
fied by specifying the axis, and the angle, of rotation by which
it may be brought to that ponition from a fixed position of re-
ference, an idea due to Euler, and revived by Bodrigues.
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70 . PRKLMINABr. [95.
^S^S^ Let ox, OT, OZ be aay three fixed axes throng tiie fixed
v»»t». point 0 round which the body turns. Let X, ;i, r be the
direction oosinea, referred to these axee, of the axia 01 round
which the body muat tarn, and ^ the sjigle through vhich it
must turn round Uiis axis, to bring it from some zero position to
any other position. This other position, being specified by the
four co-ordiDat«s \ ^ y, x (redacible, of coarse, to three by
the relation X* + /i*-i-v'= 1), will be called for brevity (X, /t, y,-^.
Let OA, OB, OC be three rcotangular lines moving vitli tbe
body, which in the "zero" position coincide reepectirely with
OX, OT, 02; and put
{XA), (YA), {ZA), (XB), (TB), (ZB), (XO), {TO), (ZO),
for the nine directicm cosines of OA, OB, OC, each referred to
OX, OT, OZ. These nine direction oomnes are of course reduci-
ble to three independent ccKirdinates 1^ the well-known six
relations. Let it be reqoired now to express these nine direction
codnea in terms of Bodiigues' co-ordinates X, /i, v, x-
Let the lengths OX, ..., OA, ..., 01 be equal, and call each
Tmity : and describe from 0 as centre a apherioal aarface of unit
radius ; so that X, T, Z, A, £, C, I shall be points on this sor^
fcce. Let XA, TA, ... XB, denote arcs, and XAT, AXB, ...
an^es between arcs, in the spherical diagram thns obtained.
W6iaTe/J = 7X = coa-'A,and J/ii = x. Hence by tlie isosceles
spherical triangle XIA,
ooeXA — ooa'IX+dn*IXcaB)(j
or (XJ) = A' + (l-A')coflx (1).
And hy the spherical triai^le XTS,
coaXS^^cxelXcoBlB + malXmilBooBXIB
= \(i + J(1-k*){l-lj.')0MXIS (2).
Now XIB = XIT+TIB = XIT+xi and "by tita spherical
triangle XIT we have
oosXr=0 = coB/XooBjr+sin/Xsin7rooeX/r
= V + V(i-x»)(i-/.') cos xir.
Hence J (I - k') {1- ft') coe XIT =-)^
and J{l-k'){l-p.'} an XIT= ^(1 - X'-/*') = »■;
by which we hare
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95.]
KIHEHATICS.
71
(3).
gfisd
and itiring ihia in (S),
co8£B = \fi(l— coax) — yon X
Similarly ve find
coeAT~\n(l~coBx) + vwiix (*)■
The otJier six formoltt may bo irritten out by symmetry from
(1), (3), and (4)j and tbus for the nine direction
(IJ)=X»+(l-X»)«wxi (IB)=X^{l-e«ix)-'sittx; iYA)=\n(,l-0M}d+'
(rB)=.^i+(i-^»)w»x; {r(7)=;»(i-<»flx)-X"i°x; (Zfl)-»«»<l-ooBx)+^
(ZC) = .^+(l-^c«x; (ZA) = ^\0.-oo<ixi-p<^Xi (i(7)-A(l-owx)+*'
Adding the three first equations of these three lines, and re-
membering that
X' + / + v' = l (6),
B^MtOM-
rinx;i
■ii»x;j-(fi
and then, by the three equations separately,
l-f(X^)~(7ffl-(^g)
3-{J[A)~{rB)-{ZC)'
, _ 1-(XA) + {TB)- (ZC)
..{7);
.(8)
i-(x^)-{rif)+(^G)
3-(Xi)-(r5)-(Z(7)".
These formuUe, (8) and (7), express, in terms of {XA), (YS),
(ZC), three out of the nine direction cosines {XA), ..., the
direction codnes of the axis round which the body most turn,
and the cosine of the angle through which it mnst tarn round
this axis, to bring it &om the zero pouti<m to Uie position
spedfied by those three direction cosines.
By aid of Euler's theorem above, eucceesive or simult^ieous Oompo
rotations about any number of axes through the fixed point ntatkHu,
may be compounded into a rotation about one axis. Doing this
for infinitely small rotatiooB we find the law of composition of
angular velocities.
Let OA, OS be two axes about which a body revolves with Compnai-
angalar velocities tv, p respectively. i«r veiooi-
Witb radios unity describe the arc AB, and in it take any
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72 PBEXIMINABT. [95.
^V*' point /. Draw fa, Ifi perpendicular to OA, OB respectiToly.
tr fstel^ _ Let the rotations ftbont the two axes be
aach that that aboat OB tends to roue I
above the plane of the paper, and lliat
about OA to depress it. In an ln£nitdy
Bhort interval of time t, the omonnts oS
these displacements will be pifi. r and
— v/a.r. The point /, and therefore
every point in the line 07, will be at rest
during the m^rval t i{Jbe sum of these
displacements is zero, that is i£ p . Ifi = te, I*. Hence iihe line
01 is instantaneously at rest, or the ttoo rotations about OA and
OB may be compounded into one about 01. Draw Ip, Iq,
parallel ta OB, OA reopectively. Then, expressing in two ways
the uea of the parallelogiam TpOq, we have
Oq.I^=Op.Ia,
Oq:Op::_£^:^
Hence, if along the axes OA, OB, we measnre off from 0 lines
Op, Oq, proportional respectively to the ftTignluT velocities about
these axes — the diagonal of Ate parallelogram of whidi these are
ctmtigueus sidfls is the resoltant axis.
Again, i£ Bhw drawn porpendicnlar to OA, and if O be ihe
angular velooify about 01, the whole displacement of B may
evidently be represented dther \ijv.Bb or 0 . 7/3.
Q '. ^ :: Bh : I^ :: onBOA : vnfOB :: tanlpO. : anpIO,
:: 01 I Op,,
Thus it is proved that, —
If lengths proportional to the respective angular velo<ntiea
about them be measured oS* on the component and resultant
axes, tbe lines so determined will be the sides and diagonal of
a parallelogram.
96. Hence tbe single angular velocity equivalent to three
co-ezistent angular velocities about three mutually perpen-
dicular axes, is determined in magnitude, and the direction of
its axis is found (§ 27), aa follows : — The square of the resultant
angular velodtj is tbe sum of tbe squares of its components,
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aod
96.] KmEUATICS. 73
and the ratios of the three components to the resultant are the ^^^
direction cosines of the axis. fciVbSSt
Hence dmnltaneoas rotations abont any nomber of azeBi^ina
meeting in a point may be compounded thas : — Let w be the ""^
angular velocity about one of them whose direction cosine are
I, m,n; O the «.>igiilftr velocity and A, /i, v the direction codnefl
of the resultant,
XO = S(H ftn = S(nu.), »n = S(n«),
whence ff = 2' (&b) + 5' (m^) + S' (jw.),
Ji(U) 3(tw<.») S(?Ma)
" Q ' '*" O ' *■" O ■
Hence also, an angnlar velocity about any line may be re-
solved into three about any set of rectanguliu: lines, the resolu-
tion in each case being (like that of simple velocities) effected
by multiplying by the cosine of the angle between the directions.
Hence, just as in § 31 a uniform acceleration, perpendicular
to the direction of motion of a point, produces a change in the
direction of motion, but does not influence the velocity; so, if a
body be rotating about an axis, and be subjected to an action
tendii^ to produce rotation about a perpendicular axis, the
result will be a change of direction of the axis about which the
body revolves, but no change in the ariguUir velocity. On this
kinematical principle is founded the dynamical explanation of
the Precession of the Equinoxes (§ 107} and of some of the
seemingly marvellous performances of gyroscopes and gyrostats.
The following method of treating the subject is useful in
connexion with the ordinary methods of co-ordinate geometry.
It contains also, as will be seen, an independent demonstration
of the parallelogram of angular velocities : —
Angular velocities w, ft, <r about the axes of x, y, and »
respectiTely, produce in time Sf diaplacementa of the point at
x,y,»*M 87, 89),
{pe~iiy)U\\x, {ux-vK)ht\\y, (wy - ptc) B( || a.
Henoe points for which
nre not displaced. I^eee are tfa^t^ore the eqnatirau of the a
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74 PBELIIUNABT. [96.
ompod- UTow the perpendicular from any pdnt m,y,x to this line is,
hr mdrf^ by co-ordinate geometry.
ins in
[..^..-<--±5ri'
_ whole displaoemeat of x, y, z
Vof' + p' + ff'li '
The actual displacemmt of x, y, 2 is therefore the same as vonld
have been produced in time 8f by a single angular velocity,
O = ^m' + p' + o', about the axis determined by the preceding
equations,
97. We give next a few useful theorems relating to the
composition of succesBiTe /mite rotations.
If a pyramid or cone of any form roll on a heterochirally
similar* pyramid (the image in a plane mirror of the first posi-
tion of the first) all round, it clearly comes back to its primitive
position. This (as all rolling of cooea) is conveniently exhibited
by taking the intersection of each with a spherical surface.
Thus we see that if a spherical polygon ttums about its angular
pointA in euccession, always keeping on the spherical surface,
and if the angle through which it turns about each point ia
twice the supplement of the angle of the polygon, or, which
will come to the same thing, if it be in the other direction,
but equal to twice die angle itself of the polygon, it will be
brought to its original position.
The polar theorem (compare § 134, below) to this is, that a
body, after successive rotations, represented by the doubles of
the successive sides of a spherical polygon taken in order, is
restored to its original position; which also is self-evident.
88. Another theorem is the following ; —
If a pyramid rolls over all its sides on a plane, it leaves its
track behind it as one plane angle, equal to the sum of the
plane angles at its vertex.
' The nmilBiity of a riglit-hand and ■ left-hand ia oall«d heteiodhinl : that
ol two ligfat-handa, homoohinL Any object and its imige in a plane minor
an heteroohiially atmilar {TliomMti, J>roc. R. S. EdiiJmrgh, 1878).
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98.] EINEXATICS. 75
Otherwise : — ^in a spherical surface, a ephericsl polygon having ^podtica
rolled over all its sides along a great circle, is found in the ^iPjJ*
same position as if the side first lying along that circle had
been simply shifted along it through an arc equal to the poly-
gon's periphery. The polar theorem is: — if a body be made to
take HucGessire rotations, represented by the sides of a Bpherical
polygon taken in order, it will finally he as if it had revolved
about the axis through the first angular point of the polygon
through an angle equal to the spherictJ excess (§ 134) or area
of the polygon.
99. The iovestigatlon of § 90 also applies to this case; and it Hotiaa
is thus easy to show that the most general motion of a spherical poi&t*BoU-
figure on a fiied spherical surface is obtained by the rolling of """^
a curve fixed in the figure on a curve fixed on the sphere.
Hence as at each instant the line joining C and 0 contains a
set of points of the body which are momentarily at rest, the
most general motion of a rigid hody of which one point is fixed
consists in the rolling of a oone fixed in the body upon a cone
fixed in space — the vertices of both b^ug at the fixed point.
100. Given at each instant the angular velocities of theP«J<^?f
° thsbodjdae
body about three rectangular axes attached to it, determine |°i^ ">'
its position in space at any time.
From the given angular velocities about OA, OB, 00, we
know, § 95, the position of the instantaneous axis 01 with re-
ference to the body at every instant. Hence we know the
conical surface in the hody which rolls on the cone fixed in
space. The data are sufficient also for the determination of
this other cone; and these cones being known, and the lines of
them which are in contact at any given instant being deter-
mined, the position of the moving body is completely deter-
mined.
If A, /I, r be the direction ooeines of 01 referred to OA, OB,
OC; ■m, p, a the asgnlar velocitaes, and u tiieir resultant:
by g 95. These equations, in which sr, p, cr, <■> are given functions
<tf t, flxprees ezplidtlf the podtion of 01 relatively to OA, OB,
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5 PBELIUIKART. [100.
th^ta^ ■* 00, and therefore detennine the cone fixed in the body. For
das to{^T<a the oone fixed in Bpaoe : if r be the radioB of curvature of its
intersection witii the nnit sphere, r* the same for the rolling
cone, we find from § 105 below, th&t if f be the length of the
ftTC of either spherical curve £rom a common initial point,
which, aa 8, / and o are known in terms of t, givee r in terms
of <, or of «, as we please. Hence, by a single quadrature, the
" intrinaio" equation of the fixed cone^
lOI. AnuDSymmetricalBystemof angular co-ordinates -^,0,^,
for specifying the position of a rigid body by aid of a line OB
and a plane A OB moving with it, and a line 0 Y and a plane
YOX fixed in space, which is easentially proper for many
physical problems, such as the Precession of the Equinoxes and
the spinning of a top, the motion of a gyroscope and its gimbals,
the motion of a compass-card and of its bowl and gimbals, is con-
venient for many others, and has been used by the greatest
mathematicians often even when symmetrical methods would
have been more convenient, must now be described.
ON being the intersection of the two planes, let YON^^,
and NOB = ^; and let d be the angle from the fixed plane,
produced through ON, to the portion NOB of the moveable
plane. (Example, 6 the "obliquity of the ecliptic," ■^ the
lon^tude of the autumnal equinox reckoned from OY,tk fixed
line in the plane of the earth's orbit supposed fixed ; ift the
hour-angle of the autumnal equinox ; B beii^ in the earth's
equator and in the meridian of Greenwich : thus '^, 0, <f> are
angular coK>rdinates of the earth.) To show the relation of
this to the symmetrical system, let OA be perpendicular to OB,
and draw 0 £7 perpendicular to both; OX perpendicular to OY,
and draw OZ perpendicular to OY and OX; so that OA, OB,
00 are three rectangular axes fixed relatively to the body,
and OX, OY, OZ fixed in space. The annexed diagram shows
^, 6, ^ in angles and arc, and in arcs and angles, on a spherical
sur&ce of unit radius with centre at 0.
To illustrate the meaning of these angular co-ordinates, sup-
pose A, B, G initially to coincide with X, Y, Z respectively.
..Google
101.] KINEMATICfl. 77
Then, to bring the body into the position specified by 0, <^, ^, JJ*'^**
rotate it round OZ through an angle equal to ■^ + 0, thus J^Mjoe'""
f_____^
-x\
' }:'
A \ -^
~r
y^ 1
"^
y
Letter O at cen-
tre of spbere
coDCealed by
r.
bringing A and B from X and F to ^' and S respectively ;
and, (taking YN=-if^ rotate the body round ON through an
angle equal to Q, thus bringing A, B, and C from the positions
A', B, and Z respectively, to the positions marked A,B,Gia
the diagram. Or rotate first round ON through 6, so bringing
O from Z to the position marked 0, and then rotate round
OC tbrongb V' + ^' ^> while OC is ttiming from OZ to the
poation shown on the diagram, let the body turn round OG
relatively to the plane ZCZ'O through an angle equal to ^.
It will be in the position specified by these three angles.
Let iXZC^'p, iZ0A = v-4>,aaAZ0^e, and sr, p, a mean
Ute same as in § 100. By considering in Hucoesaion instantaneotu
Diotifais of C ^ong and perpendicular to ZC, and the motion (rf
AB in its own plane, we Iiave
dt
:nBin^ + p<
and
-psiu^-arcoe^
tUf . dA
The nine direction cosines (^A), {YS), ix., according to the
notation of g 86, are given at once by tlie spherioal triaug^
..Google
PEELIMINAKY. [101.
XNA, TNB, &c ; eaeh having tf for one angnW point, with 6,
or its Bdpplemeut or its complement, for tbe angle at tliiH point.
Thus, by the solution in each case for the cosine <^ one dde in
tenoB of the cosine of the oppoeito angle, and the oosinea and
sines of tlte two other sides, we find
(XJ)= COB0COS^«
M 0 — Bin ^ dn ^,
(X5)--coBflcoa.^si
ii^-sin^oos.^.
{YA)= costfsin^c
«^ + COB^Bin^.
(FB)=-008tfaini^8i
n^ + co8^oos^,
(7(7)= Binflsin,;',
{ZB)= sintfsin^
{ZC)^ ixee,
(ZJ) = -sin(Jcos^
{X(7)= wntfcos^.
2^^^^ 102. We shall next consider the most general possible motion
ligidbody. of a rigid body of which no point is fixed — and first we must
prove the following theorem. There is one set of parallel planes
in a rigid body which are parallel to each other in any two
positions of the body. The parallel lines of the body perpea-
dicular to these planes are of course parallel to each other in
the two positions.
Let 0 and C be any point of the body in its first and second
portions. Move the body without rotation Jrom its second
poeition to a third in which the point at C in the second posi-
tion shall occupy its original position C. The preceding de-
monstration shows that there is a line CO common to the body-
in its first and third positions. Hence a line Cff of the body
in its second position is parallel to the same line GO iu the fint
positioiL This of course clearly applies to every line of the
body parallel to GO, and the planes perpendicular to these
lines idso remain parallel.
Let S denote a plane of the body, the two positions of which
are parallel. Move the body from its first position, without
rotation, in a direction perpendicular to S, till S comes into the
plane of its second position. Then to get the body into its
actual position, such a motion as is treated in § 79 is farther
..Google
102.] EINKUATICS. 79
reqaired. But by § 79 thifi may be effected by rotation about ^^^L
a oeit^ axis perpendicular to tbe plane 8, unless the motion rt8><> ^>^-
required belongs to tbe exceptional case of pure translation.
Hence [this case excepted] the body may be brought from the
first praitioD to the second by translation through a determinate
diatance perpendicular to a given plane, and rotation through a
determinate angle about a detenuioate asis perpendicular to
that plana This is precisely the motion of a screw in its nut.
108, In the excepted case the whole motion consists of two
translatioiiB, which can of course be compounded into a single
one ; and thus, in this case, there is no rotation at all, or every
plane of it fulfils the specified condition for 8 oi% 102.
101. Retumiug to the motion of a rigid body with one point
fixed, let us consider the case in which tbe guiding cones, § 99,
are both circular. The motion in this case may be called Pre-
cesaional Sotation.
The plane through the instantaneous axis and the axis of
the fixed cone passes through the axis of the rolling cone. This
plane turns round the axis of the fixed cone with an angular
velodty tl (see § 105 below), which must clearly bear a con-
stant ratio to tbe angular velocity » of tbe rigid body abont
ita instantaneous axis.
105. The motion of tbe plane containing these axes is
called the preoesnon in any snch case. What we have denoted
by fl is the angular velocity of the precession, or, as it is some-
times called, the rate of precession.
Tbe angular motions u, 11 are to one another inversely as
tbe distances of a point in the axis of the rolling cone from tbe
instantaneous axis and from tbe axis of tbe fixed cone.
For, let OA be the axis of the fixed .
cone, OS that of the rolling cone, and 01
the instantaneous axis. From any point
P in OB draw Py perpendicular to 01,
and PQ perpendicular to OA. Then we ^ ^
perceive that P moves always in the
circle whose centre is Q, radius PQ,
and plane perpendicular to OA. Hence
..Google
80 PRELmiHABT. [105.
the actual velocity of the point P is Q.QP. But, by the
prinoiplefl explained above, § 99, the velocity of P is the
same as that of a point moving in a circle vhose centre is N,
pUme perpendicular to ON, and radius NP, which, aa this radius
revolves with angular velocity «, is u>NP. Hence
n. gP = <o. JV"i> or » : n :: ©P : ifi*.
Let a be the semivertical angle of the fixed, yS of the rolling,
cone. Each of these may be supposed for simplicity to be
acute, and their sum or difference less than a right angle —
though, of course, the formuls so obtuned are (like all
trigonometrical results) applicable to every possible case. We
have the following three c
wBin^ = 08in(a + ^,
where AOI = o, lOB - /3.
Let ^ be n^ative, and let ^ = — j
then ^ is positiTe, and we bave
- (u Bin /y = O sin (a - /S^,
where ^0/=a, BOI=^.
f In tbe preceding let /S* > a.
It may tben be oonveniently
«Bin/5' = nidn(/S'-a),
whenAO!=ii,BOI=fi',
a. and ^ being stiU podtiTe.
106. If] as illustrated by the first of these diagrams, the
case is one of a convex cone rolling on a convex cone, the pre-
cesnonal motion, viewed on a hemispherical sur&ce having A
for its pole and 0 for its centra is in a similar direction to
..Google
106.] KIKEIIATICS. 81
that of the angular rotation about the mstaDtaneous axis, ^^^f^
This we shall call poeitive precesBional rotatioQ, It is the case '"'^'"'^
of a common spinning-top (peery), spinning on a veiy fine
point which remains at rent in a hollow or hole hored hy itself;
not sleeping upright, nor nodding, but sweeping its axis round
in a circular cone whose axis is vertical. In Case iiL also we
hwepostHve precession. A good example of this occurs in the case
of ft coin spinning on a table when its plane is nearly horizontal.
107. Case IL, that of a convex cone rolling inside a concave
one, gives an example of negaHve precession: for when viewed
as before on tfae hemispbetical surface tbe direction of angular
rotation of the instantaneous axis is opposite to that of tbe
rolling cone. This is the case of a symmetrical cup (or figure .
of revolution) supported on a point, and stable when balanced.
I.e., having its centre of gravity below the pivot; when in-
clined and set spinning non-nutatioually. For instance, if a
Troughton's top be placed on its pivot in any inclined position,
and then spun off with very great angular velocity about its
axis of figure, the nutation will be insensible ; but there will
be slow preces^on.
To this case also belongs the precessional motion of tfae earth's vodei
axis; tor which the
angle a = 23" 2728",
the period of the ro-
tation o» the sidereal
day; that of H is
25,868 yeais. If the
second diagram re-
present a portion of
the earth's surface
round the pole, the
arc j1/= 8,552,000
feet, and therefore
the circumference of
the circle in which
/ move8= 52,240,000
feet. Imagine this
circle to be tbe in-
VOL. I. 6
..Google
82 PBELIHINABY. [107.
ner edge of a fixed ring in apace (directionally fized, that is
maefc™™ to flay, but having the same translatioDal motion as the
earth's centre), and imagine a circular post or pivot of
radius BI to be fixed to the earth with its centre at B.
This ideal pivot rolling on the inner edge of the fized
ring travels once round the 52,24<0,000 feet-circumference in
25,868 years, and therefore its own circumference must be
6-53 feet. Hence 5/= 0-88 feet; and angle SOI, or ft
«= 0"00867.
ttono!^ 108. Very interesting examples of Cases I. and in. are fur-
S3iy w^' •i>^^ ''y projectiles of different forms rotating about any axis.
^M^ ^"^ *^® gyrations of an oval body or a rod or bar flung into
"* the air belong to Class i. (the body having one axis of less
moment of inertia than the other two, equal) ; and the
seemingly irregular evolutions of an iU-thrown quoit belong
to Class HI. (the quoit having one axis of greater moment of
inertia than the other two, which are equal). Case III. has
therefore the following very interesting and important appli-
cation.
If by a geological convulsion (or by the transference of a few
million tons of matter from one part of the world to another)
the earth's instantaneous axis 01 (diagram in., § 105) were at
any time brought to non-coincidence with its principal axis of
least moment of inertia, which (§§ 825, 285) is an axis of
approximate kinetic symmetry, the instantaneous axis will, and
the fixed axis OA will, relatively to the solid, travel round the
solid's axis of greatest moment of inertia in a period of about
306 days [this number being the reciprocal of the most probable
value of — ji — (§ 828)]; and the motion is represented by the
dii^ram of Case III. with BI= 306 x AI. Thus in a very little
less than a day (less by =r^ when BOI is a small angle)
/ revolves round A. It is OA, as has been remarked by
Maxwell, that is found as the direction of the celestial pole
by observations of the meridional zenith distances of stars, and
this line being the resultant axis of the earth's moment of
..Google
108.] KISEIUTICS. 83
momentum (§ 267), would remaia invariable in space did no ^^^'^
external influence such as that of the moon and sun disturb the J^J)^''
earth's rotation. When we neglect precession and nutation, J^"^
ttie polar distances of the stars are constant notwithetanding "'^
the ideal motion of the fixed axis which we are now consider-
ing; and the effect of this motion will be to make a periodic
variation of the latitude of ever; place on the earth's surface
having for range on each side of its mean value the angle BOA,
aud for its period 306 days or thereabouts. Maxwell* ex-
amined a four years series of Greenwich observations of Polaris
{1S31— 2-3— 4), and concluded that there was dunng those,
years no variation exceeding half a second of angle on each
side of mean range, but that the evidence did not disprove
a variation of that amount, but on the coDtrary gave a very
slight indication of a minimum latitude of Greenwich belonging
to the set of months Mar. '51, Feb. '52, Dec. '52, Nov. '53,
Sept. '54.
"This result, however, is to be regarded as very doubtful
"and more observations would be required to establish the
" existence of so small a variation at alL
"I therefore conclude that the earth has been for a long time
" revolving about an axis very near to the axis of figure, if not
" coincidiug with it. The cause of this near coincidence is
" either the original softness of the earth, or the present fluidity
" of itfl interior [or the existence of water on its surface].
"The axes of the earth are so nearly equal that a con-
"siderable elevation of a tract of country might produce a
"deviation of the principal axis within the limits of observa-
" tion, and the only cause which would restore the uniform
" motion, would be the action of a fluid which would gradually
" diminish the oscillations of latitude. The permanence of
" latitude esserftially depends on the inequality of the etuth's
" axes, for if they had all been equal, any alteration in the
" crust of the earth would have produced new principal axes,
" and the axis of rotation would travel about those axes, alter-
• On ft Dynsnucal Top, Tratu. R. S. E., 1867, p. 559.
..Google
84 PRIXIMINAnT. [108.
" ing the latitudes (^ all places, and yet not in the least altering
<- " the position of the axis of rotation among the etara,"
Perhaps hy a more extensive "search and tuialyuB of the
" obserrationa of different observatories, the natm« of the
" periodic variation of latitude, if it ezist, may be determined.
" I am not aware* of any calculations having been made to prove
" its non-exiBtence, although, on dynamical grounds, we have
" ev^y reason to look for some very small variation having the
"periodic time of 3256 days nearly" [more nearly 306 days],
" a period which is clearly distinguished from any other astro-
" nomical cycle, and therefore easily recognised^ ."
The periodic variation of the earth's instantaneotis aziR thns
antidpated by Maxwell must, if it exists, give rise to a tide
of 306 days period (g 801). The amount of this tide at the
equator would be a rise and fall amounting only to 5^ centi-
metres above and below mean for a deviation of the instan-
taneous axis amounting to 1" from its mean position OB, or
for a deviation SI on the earth's surface amounting tu
31 metres. This, although discoverable by elaborate analj^sis
of long-continued and accurate tidal observations, would be leas
easily discovered than the periodic change of latitude by astro-
nomical observations a(»X)rding to Maxwell's method^
* (Written twenty yeun Ago).
t H&xweU ; rraiuiulf (Hu of the Boyal Soeitty of Ediiiburg\ SOth April, 1867.
J Prof. Maxwell now lefers ns to Peten ilUeherehet tur la parallax dtt
itoilmfixu, Bt Petenbor^ ObBemtotj Pttpen, ToL L, 1B6S), who wemi to
have been the first to ralae thia interesting and important qneation. Be found
from the PnlkOTft obBerrationB of Polaris from March 11, 1813 till April 30,
1818 an angular radios of 0^-079 (probabla error 0°-017}, for the einde round
ita mean poaitjon deieribed by the inrtantaaeong axis, and for the time,
within that interval, when the latitude of Pnlkon waa a "'WTiiYinm Hoy. is, \m.
The period {ealoolated from the dynamioal theory wUoh Peters aasnmed wkb
BH mean solar dajrs: the rale therefore 1-301 tnms per annum, or, nearly
«non^, 13 tnrna per ten years. Thus it Petsra' result wore genoine, and
remained oonatant lor ten years, the latitude of Pnlkaro ironld be a maiimain
about Oie lUh of Nov. again in 1863, and Pnlfcora being in 30* East longitnde
from Qreenwioh, the latitude of Oreenwioh would be a maTininin ^ of the period,
or sbont 8E days earlier, that is to sa; abont Oel. 93, 1863. Bnt Maxwell's ex-
amination of obserratlons Boemed to indieate more nearly the minimum latitude
of Greenwieh about the same time. This discrepanoe is altogether in aoootdance
wUh a ecaitinnation of Peters' inTeatigation by Dr Hysen of the Pulkova Ob-
..Google
109.] KINEMATICS. 85
109. Id various iUustratiotis and arrangemeatB of apparatus o
useful in Natural Philosophy, as well m id MechaDics, it is ^^^
required to connect two bodiea, so tliat when either turns about JJ^^nf"
a certtun axis, the other shall turn with an equal angular"""^***
velocity about another axis in the same plane with the former,
but inclined to it at any angle. This is accomplished in
mechanism by means of equal and similar bevelled wheels, or
rolling cones ; when the mutual inclination of two axes is not
to be varied. It is approximately accomplished by means of
Hooke's joint, when the two axes are nearly in the same line, '^*»'»
hut are required to be free to vary in their mutual inclination.
A chain of an infinitely great number of Hooke's joints may be Tiaibie but
imagined as constituting a perfectly flexible, untwistahle cord, ootd.
which, if its end-links are rigidly attached to the two bodies,
connects them bo as to fulfil the condition rigorously without
the restriction that the two axes remain in one plane. If we S^JSn^t
imagine an infinitely short lei^h of such a chain (still, how-
ever, having an infinitely great number of links) to have its
ends attached to two bodies, it will fiilfil rigorously the con-
dition stated, and at the same time keep a definite point of one
body infinitely near a definite point of the other ; that is to say,
it will accomplish precisely for every angle of indination what
Hooke's joint does approximately for small inclinations.
The same is dynamically accomplished with perfect accuracy Bhrtonni-
fbr every angle, by a short, naturally straight, elastic wire of ""
0«nfa>t7, in wIubIi, Iiy A eazefol acratlujr ol serenl sedM ot Polkova obBerrntioiiB
between the yeui IMS. ..1872, he eonelnded Uwt there ia no ooiutaiioy of
nugnitodB or phaie in the deruttion iooght tar. A. sunilu negfttlTe eonolnsioii
wag arrired at by Protesaoi Newoomb of the United States Nayfll Obaervatory,
Waahington, who at onz reqneat kindly undertook an invegtigation of the ten-
month period ol latitnda from the Waahington Prime Vertical ObserratioiiB
from 1863 to 1867.. Hit leanlts, sa did thoaeol FetenandNyaen andUazweU,
uemed to indicate tbb] rariationa of the earth'a inaUntananna uda amounting
to poasib^ as mnoh as f or }" from its mean position, bnt altogether irr^nilu
both in amoDnt and direotion; in fast, just soeh as might be expected from
iiregnlar hotting* np of the ooeaae bj winds in different looilitiea of the
earth.
We intend to return to this Bnbjeot and to consider ot^^nats qnestions regard-
ing iiTBgDlarities of the earth as a timekeeper, and Tariations of its figure and
of the diatribation ol matter within it, ol the ocean on ita sortaee, and of the
atmosphere snTroonding it, in M 367, 376, 405, 406, 830, SS3, 846, 846.
..Google
TvadCBTAM
at IreedoBi
86 PRELIMINAKT. [109
I- truly circular section, provided the forces giving rise to any re-
t- sistanoe to equality of angular velocity between the two bodies
are infinitely email. In many practical cases this mode of con-
nexion is useful, and permits very little deviation from the con-
ditions of a true universal flexure joint. It is used, for instance,
in the suspenaoD of the gyroscopic pendulum (§ 74) with perfect
Success. The dentist's tooth-mill is an interesting illustration
of the elastic universal flexui'e joint. In it a long spiral spring
of steel wire takes the place of the naturally straight wire
suggested above.
Of two bodies connected by a universal flexure joint, let one
be held fixed. The motion of the other, as
long as the tmgle of inclination of the axes
remains constant, will be exactly that figured
in Case I., § 106, above, with the angles a and
j3 made equal. Let 0 be the joint ; A O the
axis of the fixed body; OB the axis of the
moveable body. The supplement of the angle
A OB is the mutual inclination of the axes ;
and the angle AOB itself is bisected by the
instantaneous axis of th^ moving body. Tho
diagram shows a case of this motion, in which the mutual in-
clination^ 6, of the axes is acute. According to the formulae
of Case I., § 105, we have
a> siu a = n sin 2z,
or (u = Sn cos a = 211 sin ^ ,
where (» ia the angular velocity of the moving body about its
instantaneous axis, 01, and Ci is the angular velocity of its pre-
cession ; that is to say, the angular velocity of the plane through
the fixed axis AA!, and the moving axis OB of the moving
body.
Besides this motion, the moving body may clearly have any
angular velocity whatever about an axis through 0 perpen-
dicular to the plane AOB, which, compounded with o> round
01, gives the resultant angular velocity and instantaneous axis.
Two co-ordinates, $=A'OB, and tft measured in a plane per-
peodicnlar to AO, from a fixed plane of reference to the plane
..Google
100.] EINEUATICS. 87
AOB, fully specify the position of the moveable body in this
110. Suppose a rigid body bounded by any curved surface Q
to be touched at any point by another such body. Any motion gj
of one on the other must be of one or more of the forms sliding,
ToUing, or spinning. The consideration of the first is so simple
as to require no comment.
Any motion in which there is no slipping at the point of
contact must be rolling or spinning separately, or combined.
Let one of the bodies rotate about successive instantaneous
axes, all lying in the common tangent plane at the point of
instantaneous contact, and each passing through this point-^
the other body being fixed. This motion is what we call rolling,
or simple rolling, of the moveable body on the fixed.
On the other band, let the instantaneous axis of the moving
body be t^e common normal at the point of contact. This is
pure spinning, and does not change the point of contact.
Let the moving body move, so that its instantaneous axis,
still passing through the point of contact, is neither in, nor
perpendicular to, the tangent plane. This motion is combined
rolling and spinning.
111. When a body rolls and spins on another body, the TraoM
trace of either on the other is the curved or straight Une along '"'
which it is successively touched. If the instantaneous axis is
in the normal plane perpendicular to the traces, the rolling
is called direct. If not direct, the rolling may be resolved into direct
a direct rolling, and a rotation or twisting round the tangent
line to the traces.
When there is no apinntftg the projections of the two traces
on the common tangent plane at the point of contact of the
two surfaces have equal and same-way directed curvature: or
they have "contact of the second order," When there is
tpinning, the two projections atiJl touch one another, but with
contact of the first ordec only : their curvatures differ by a
quantity equal to the angular velocity of spinning divided
by the velocity of the point of contact. This last we see by
noticing that the rate of change of direction along the pro-
.y Google
88 PBELUnNAST. [111.
jection of the 6xod trace must be equal to the rate of change
of direction along the projection of the moving trace if held
fixed plus the angular velocity of the spinning.
At any instant let 2a = Aa? + 2Cxy +£1^ (1)
and 2s' = JV + 2(7'ay + J'y» ..(2)
be the eqoationH of the fixed and moveable snr&cea S and S'
infinitely near tJie point of contact 0, referred to axes OX, OY
in their common tai^ent plane, and 02 perpendicular to it :
let or, p, (T be the tliree component of the instantaneoas angular
TfiloQity of S'; and let at, y, be co-ordinatea of P, the poiat of
contact at an infinitely small time (, later : the tliird co-ordin&te,
t, is given by {I).
Let P" be the point of i?'which at this latertnme ooincides withP.
The co-ordinatea of P* at the first instant are x + uyt, y — axt ;
and the corresponding value of »* is given by (2). This point is
infinitely near to {x, y, s"), and therefore at the first instant the
direction codnea of the normal to iS' through it differ but infinitely
little from
~{A'x+Cy), -(Cx + B-y), 1.
But at time t the normal to £' at P' ooinddee with the normal
to ^ at i*, and ther^ore its direction oodnefl change from the
preceding values, to
~(Ax+Cy), -(Cx + By), I:
that ia to say, it rotates tlirough angles
(C'-C)x+(B'-£)y round OX,
and -{{A'-A)x + {C'-C)y} „ 07.
Hence v!t = {C'-G)x*{B'-E)y \
pt = ~{{A'-A)x*{C'^C)y}] W'
OP flr= iC'-C)H-(S'-B)S \ ,,.
p = -{{A'-A)i*{0'-C)S]S ^*''
i£sb,^ denote the component velocities of the point of contact.
Put ? = V(i** + ^ (6),
and take components of w and p round the tangent to the traces
and the perpendicular to it in the common tangent plane of the
two sur&ces, thus ;
(twisting component) -m + -p
.(C'-C)^/ *l(B -S)-iA'-A)]^ (6),
jiGoogle
[Compare below, § 124 (2) and (1).]
And for ff, the angular yelodty of apiiming, the obrions pro-
podtUoi stated in the preceding large print givee
111.] KINEHATIC». 89
and
(diiect-rolUng component) -or — p
= ^[(A'-A)d^ + 2(C'-C)i&*{S--S)f\ (7).
Chooee OX.Orsothat C-(7'=0, and pnt A'-A=a,S'-B=p
(6) and (7) become
(twisting oranponont) -w+^p = (^_o)^ (8),
(direct-rolling component) -» — /»= -(«**+/3y^ (9).
[24 (2) and (1).]
Uar yelodty of spinning, the obrions pro-
preceding luge print givee
"'(r?) <•">■
if - and -^ be the cmratarea of the projections on the tangent
pluie <^ tlie fixed and moroable tracea. [Oompore below, § 124
(»)■]
From (1) and (2) it follows that
When one of the surfacea is a plane, and the trace on the
other is a line of curvature (§ 130), the rolling is direct
When the trace on each body ia a line of curvature, the
rolling is direct. Qenerally, the rolling is direct ivhen the twists
of infinitely narrow hands (§ 120) of tfae two surfaces, along the
traces, are equal and in the same direction.
112. Imagine the traces constructed of rigid matter, and all
the rest of each body removed. We may repeat the motion
with theae curves alone. The difference of the circumstancea
now sappoeed will only be experienced if we vary the direction!
of the instontaneoua axis. In the former case, we can only do
this by introducing more or less of spinning, and if we do so
we alter the trace on each body. In the latter, we have always
the same moveable curve rolling on the same fixed curve ; and
therefore a determinate line perpendicular to their conunon
tangent for one component of the rotation; but aloi^ with this
we may give arbitrarily any velocity of twistii^ round the
conunon tangent. The consideration of this case is very in-
..Google
Tsloslt; of
90 PBELIMINART. [112.
structive. It may be roughly imitated in practice by two stiff
wires bent into tbd forms of the given curves, and prevented
from crossing each other by a short piece of elastic tube clasping
them together.
First, let them be both plane curves, and kept in one plane.
We have then rolling, as of one cylinder on another.
Let p be the radius of curvature of the roUIng, p of the fixed,
cylinder ; ta the angular velocity of the former, Fthe linear velt>-
city of the point of contact. We have
For, in the figure, Buppose /* to be at any time
the point of contact, and Q and Q' the pointe which
are to be in contact after an infinitely amall
interval t; 0,0 the centree of curvature ; FOQ
= e,PO'Q'=er.
Then PQ = PQf = BpM)o described by point of
contact. In symbols p6 - p'ff = Vt,
Also, before O'Q" and OQ can coincide in direo-
tion, the former must evidently turn through an
angle 6 + 6'.
Therefore iU~6 + ff; and by eliminating $ and
ff, and dividing by (, we get the above result.
It is to be understood, that as the radii of curvature have
been considered positive here when both suifaces are convex,
the negative sign must be introduced for either radius when the
corresponding curve is concave.
Hence the angular velocity of the rolling curve is in this
case equal to the product of the linear velocity of the point of
contact by the sum or difference of the curvatures, according
aa the curves are both convex, or one concave and the other
convei.
113. When the curves are both plane, but in different
planes, the plane in which the rolling takes place divides the
angle between the plane of one of the curves, and that of the
other produced through the common tangent line, into parts
whose sines are inversely as the curvatures in them respec-
tively ; and the angular velocity is equal to the linear velocity
..Google
113.] KINEMATICS. 91
of the point of contact multiplied by the difference of the pro- ^JJ^ ^^
jectioQS of the two curvatnree on this plane. The projections of iS,^^
the cirolea of the two curraturea on the plane of the common
tangent and of the instantaneous axis coincida
For, l«t PQ, Pp be equal aiY» of the two aurea as before, and
let PR be taken in the common tangent (v&, the intei'section of
the planes of the curves) equal to each. Then QR, pR are
ultimately perpendicular to PR.
PS?
Hence pR = -„— -,
^-?^
Also, I QRp = a, the angle between the planes of the curves.
We have Q^ = ^(). + \.l^^.
Therefore if oi be the vdocity of rotation as before,
"-v^
2 cosa
Also the instrataneouB axis is evidently perpendicular, and there-
fore the plane of rotation parallel, to Qp. Whence the above;
In the case of a = «, this agrees with the result of § 1 12.
A good example of this ia the case of a coin spinning on a
tabic (mixed rolling and spinning motion), as its plane becomes
gradually horizontal. In this case the curvatures become more
and more nearly equal, and the angle between the planes of the
curves smaller and smaller. Thua the resultant angular velo-
city becomes exceedingly small, and the motion of the point
of contact very great compared with it.
114, The preceding results are, of course, applicable to tor- p"™ n>ii-
taous as well as to plane curves ; it ia merely requisite to sub- ^IZl}^
stitute the osculating plane of the former for the plane of the fw**""-
latter.
115. We come next to the case of a curve rolling, with orcarrerou-
without spinning, on a surface. J": *i^
It may, of course, roll on any curve traced on the surface. (»«inin.
When this curve is given, the moving curve may, while rolling
along it, revolve arbitrarily round the tangent. But the com-
..Google
PRELIHmlltT. [115.
Lxia peipendicular to the common tan-
> gent, that ia, the azia of the direct rolling of one curve on the
tnedom, other, ie determinate, § 113. If this axis does not lie in the
surface, there is spinning. Henc^ when the trace on the surface
is given, there are tvo independent variables in the motion ;
the space traversed by the point of contact, and the inclination
of the moving curve's OBCulatiag plane to the tangent plane of
the fixed surface.
Truapn- 116. If the trace is given, and it be prescribed as a condi-
" '""ilSlr* ^^^^ ^^^ there shall be no spinning, the angular position of the
rolling curve round the tangent at the point of contact is deter-
minate. For in this case the instantaneous axis must be in the
tangent plane to the surface. Hence, if we resolve the rotation
into components round the tangent line, and round an axis per-
pendicular to it, the latter must be in the tangent plane. Thus
the rolliug, as of curve on curve, must be in a normal plane to
the surface; and therefore (§§ 114, 113) the rolling curve must
T«d«rori be always so situated relatively to its trace on the snr&ce that
the projections of the two curves on the tangent plane may be
of coincident curvature.
The curve, as it rolls on, must continnally revolve about the
tangent line to it at the point of contact with the surface, so as
in evety position to fulfil this condition.
Let a denote the inclin&taoD of the plane of curvatare of the
trace, to the normal to the sarfoce at any point, a the same for
the plane of the rolling curve; -, -7 their carvatores. We
P fi
reckon a as obtnse, and a' acute, when the two curves lie on
opposite aides of the tangent plane. Then
1 . , 1 .
which fixes a' w tho position of the rolling curve when the point
<tf contact is given.
Anniitf *». Let tt be the angular velodty of rdling abont an axis p»pen-
not ToUiiw diculartothetangent, tff that of twisting about the tangsDt.and let
Fbetheliuearvelocityof the point of contact Then, dnce -.oosa'
..Google
116.] EINEtUTICS.
~3B a \eaca poeiuTe wuen me curves lie on oppoBiie mae
not nillias.
- COB a (each positiTe when the curves lie on opposite sides ^^V^ *^
of the tangent pl&ne) are the projections ot ihe two cnrvaturefi o:
a plane through the nwmal to the sur&oe containing tiieir txaor
Bum tangent, we have, by { 112,
1 \
— G"
a' being detennined by the preceding equation. Let r and r'
denote the tortaodties of the trace, and of tjte rolling cnrvc^ re-
spectively. Then, first, if the curves were both plane, we see
that one rolling on Ijie other about an axis alwajrs perpendicular
to their common taogeat oould never change the inclination of
their planes. Hence, seoondly, if they are both tortuous, such
rolling will alter ihe inclination of their osculating planes by an
indefinitely gmall amount (t - t') da during rolling which ahifts A»|pihr n
the point of contact over an arc (2*. Kow a is a known function tanswt.
off if the trace is given, and therefore so also is a'. But a— a'
is the inclination iif the osculating planes, hence
K{l<^).,-0} = .
117. Next, for one surface rolling aad epinuing on another. SDriaceon
First, if the trace on each is given, we have the case of § 113
or § 115, one curve rolling on another, with this farther con-
dition, that the former must revolve round the tangent to the
two curves so as to keep the tangent planes of the two surfaces
coincident
It is well to observe that when the points in contact, and the Both tnm
two traces, are given, the position of the moveable sur&ce isomdHiM'
quite determinate, heing found thus : — Place it in contact with
the fixed surface, the given points together, and spin it about
the common normal till the tangent lines to the traces coincide.
Hence when both the traces are given the condition of no
spinning cannot be imposed. During the rolling there must in
general be spinning, such as to keep the tangents to the two
traces coincident. The rolling along the trace is due to rotation
round the hne perpendicular to it in the tangent plane. The
whole rollii^ is the resultant of this rotation and a rotation
about the tangent line required to keep the two tangent pknee
coinddent.
..Google
Qi PBELIHIKIST. [117.
Snrfnoa on III this c&se, then, there is but one independent variable — the
both tnoM space passed over by the point of contact : and when the velocity
onj^dpjree' of the point of contact ia given, the resultant angular velocity,
and the direction of the instantaneous axis of the rolling body
are determinate. We have thus a sufficiently clear view of the
general character of the motion in question, but it is right that
we consider it more closely, as it introduces us very naturally
to an important question, the measurement of the twist of a rod,
wire, or narrow plate, a quantity wholly distinct from the tor-
tuosity of its axie (§ 7).
118. Suppose all of each surface cut away except an infinitely
narrow strip, including the trace of the rolling. Then we have
■ the rolling of one of these strips upon the other, each having at
eveiy point a definite curvature, tortuosity, and twist
Twut. 119. Suppose a flat bar of small section to have been bent
(the requisite amount of stretching and contraction of its edges
being admissible) so that its axis assumes the form of any plane
or tortuous curve. If it be unbent without twisting, i.e., if the
curvature of each element of the bar be removed by bending it
through the requisite angle in the osculating plane, and it be
found untwisted when thus rendered straight, it had no trnst in
its original form. This ease is, of course, included in the general
theory of twist, which is the subject of the following sections.
120. A bent or straight rod of circular or any other form of
section being given, a line through the centres, or any other
chosen points of its sections, may be called its axis. Mark a
line on its side all along its length, such that it shall be a
straight line parallel to the axis when the rod is unbent and
untwisted. A line drawn from auy point of the axis perpen-
dicular to this side line of reference, is called the transverse of
the rod at this point.
The whole twist of any length of a straight rod is the angle
between the transverses of its ends. The average twist is the
integral twist divided by the length. The twist at any point
is the average twist in an infinitely short length through this
point ; in other words, it ia the rate of rotation of its transverse
per unit of length along it.
..Google
120.] KINEMATICS. 95
The twist of a curved, plane or tortuous, rod at any point ia Xwnt
the rate of compoDent rotation of its transverse round its tangent
line, per unit of length along it
If t be the twist at any point, JWm over any length is the
in^ral twiat in this length.
121. Integral twist in a curved rod, although readily de-
fined, as above, in the language of the integral calculus, can-
not be exhibited as the angle between any two lines readily
constnictible. The following considerations show how it is to
be reckoned, and lead to a geotnetrical construction eshibitiog
it in a spherical diagram, for a rod bent and twisted in any
manner: —
122. If the axis of the rod forms a plane curve lying in one iftimaikm
plane, the integral twist is clearly the difference between the tiiTtf'™'
inclinations of the transverse at its ends to its plane. Forin»piua
if it be simply unbent, without altering the twist in any part, """"'
the inclination of each transverse to the plane in which its
curvature lay will remain unchanged ; and as the axis of the
rod now has become a straight line in this plane, the mutual
inclination of the transverses at any two points of it has become
equal to the difference of their inclinations to the plane.
123. No simple application of this rule can be made to a
tortuous curve, in consequence of the change of the plane of
curvature from point to point along it ; but, instead, we may
proceed thus : — ■
First, Let us suppose the plane of curvature of the axis of laftcarve
the wire to remain constant through finite portions of the curve, orpfkiw^
and to change abruptly by finite angles &om one such portion Si^ma
to the next (a supposition which involves no angu-
lar points, that is to say, no infinite curvature, in j
the curve). Let planes parallel to the planes of cur-
vature of three successive portions, jPQ, Qif, B.8 (not
shown in the diagram), cut a spherical surface in the
great circles QA& , ACA', CE. The radii of the
sphere parallel to the tangents at the points Q and R
of the curve where its curvature changes will cut its j
surface in A and C, the intersections of these circles.
..Google
96 PSELIHIHABT. [123-
Let 6 be the point in which the radius of the sphere parallel to
the tangent at P cuts the surface ; and let OS, AB, CD (lines
Botimation necessarilj in tangent planes to the spherical surface), be paral-
(viatTil^ leis to the tranBverses of the bar drawn from the points P, Q, R
■tsting or of its axia Then (§ 122) the twist &om P to (? is equal to the
tjwuSn difiference of the auirles SGA and BAG; and the twist from Q
planes. to 5 is oqual to the difference between BACanA DGA'. Hence
the whole twist fromP to £ is equal to
hoa-bag'+bag-dca:
or, which is the same thing,
A'CE+(rAC-(pGE-HGA).
Continuing thus through any length of rod, made up of portions
curved in different planes, we infer, that the integral twist be-
tween any two points of it is equal to the sum of the extenor
angles in the spherical diagram, wanting the excess of the in-
clination of the transverse at the second point to the plane of
curvature at the second point above the inclination at the first
point to the plane of curvature at the first point The sum of
those exterior angles is what is defined below as the "change of
direction in the spherical surface" &om the first to the last side
of the polygon of great circlea When the polygon is closed, and
the sum includes all its exterior angles, it is f§ 134) equal to
2w wanting the area enclosed if the radius of the spherical sur-
£m» be unity. The construction we have made obviously holds
in the limiting case, when the lengths of the plane portions are
infinitely small, and is therefore applicable to a wire forming a
tortuous curve with continuously varying plane of curvature, for
which it gives the following conclusion : —
inBMM- Let a point move uniformly along the axis of the bar: and,
«!rt»Su parallel to the tangent at every instant, draw a radius of a
sphere cutting the spherical surface in a curve, the hodograph
of the moving point From points of this hodograph draw par-
allels to the transverses of the corresponding points of the bar.
The excess of the change of direction (§ 135) from any point to
another of the hodograph, above the increase of its inclination to
the transversa is equal to the twist in the corresponding part
of the bar.
..Google
123.] KINEKiTIUS. 97
The annexed diagram, Bhowing the hodograph and the B-4inati*>ii
parallelfl to the tranarerees, illustrates this rule. Thus, for iii-!^='"*
stance, tlie exoeos of the change of direction in the spherical {^"^^^
Boiiace along the hodograph from AtoC, above DCS — BAT,'^"'-
is equal to the twist in the bar between the points of it to
which A aod C correspond. Or,
again, if we consider a portion of
the bar from any point of it, to
another point at which the tangent
to its axis is parallel to the tan-
gent at its first point, we shall have
a closed curve as the spherical hodc^raph ; and if .^ be the
point of the hodograph corresponding to them, and AB and
Aff tbe parallels to the transvenes, the whole twist in the
included part of tbe bar will be equal to the change of direction
all round the hodograph, wanting tbe excess of the exterior
angle .9'^ T above the angle BAT; that is to say, the whole
twist will be equal to the excess of the angle BAff above
the area enclosed by tbe hodograph.
The principles of twist thus developed are oE vital import-
ance in the theory of rope-making, especially the construction
and the dynamics of wire ropes and submarine cables, elastic
bars, and 8{Hral springs.
For example : take a piece of steel pianoforte-wire carefully m-nunin
strmgbtened, so that when free from stress it is straight : bend kinks.'
it into a drcle aod join the ends securely so that tbere can be
no turning of oue relatively to the other. Do this first without
torsion: then twist the ring into a figure of 8, and tie the two
parts together at the crossing. The area of the spherical hodo-
graph is zero, and therefore there is one full turn (Sir) of twist;
which (§ 600 below) is nniformly distributed throughout the
length of the wire. Tbe form of the wire, (which is not in a
plane,) will be investigated in § 610, Meantime we can see
that the "torsional couples" in the normal sections farthest
from tbe crossing give rise to forces by which the tie at the
croBsiog is pulled in opposite directions perpendicular to the
plane of the crossing. Thus if the tie is out the wire springs
back into the circular form. Now do the same thing again,
VOL. r. • 7
DigilizedbyGOOgle
98 PHEUMINABY. [123.
l^nuBiM beginniiig with a straight wire, but giritig it one full turn
I'i''^ {2ir) of twist before bending it into the circla The wire will
stay in the 8 form without any pull on the tie. Whether
the circular or the 8 form is stable or unstable depends
on the relations between torsional and flezural rigidity. If
the torsional rigidity is small in comparison with the flexural
rigidity [aa (§§ 703, 704, 705, 709) would be the ca«e if,
instead of round wire, a rod of + shaped section were used],
the circular form would be stable, the 8 unstable.
Lastly, suppose any degree of twist, either more or less
than Stt, to be given before bending into the circle. The
circular form, which is always a figure of free equilibrium, may
be stable or unstable, according as the ratio of torsional to
flexural rigidity is more or less than a certMn value, depending
on the actual degree of twist. The tortuous 8 form is not (except
in the case of whole twist = 2ir, when it becomes the plane
elastic lemniscate of Fig. 4, § 610,) a continaous figure of free
equilibrium, but involves a positive pressure of the two cross-
ing parts on one another when the twist > 2ir, and a negative
pressure (or a pull on the tie) between them when twist < Stt :
and with this force it is a figure of stable equilibrium.
snrhogToii- 121. Returning to the motion of one surface rolling and
tacei both spinning on another, the trace on each being ^ven, we may
consider that, of each, the curvature (§ 6], the tortuosity (§ 7),
and the twist reckoned according to transverses in the tangent
plane of the surface, are known; and the subject is fully spe-
cified in § 117 above.
Let -, and - be the cmraturee of the traces on the toUJuk
P P
and fixed surfaces reepectiyely; a and a the mclinationB of their
planes of curvature to the normal to the tangent plane, reckoned
as in § 116; r' and r their tortuosities; t' and t their twists;
and q the velocity of the point of contact All these bnng
known, it is required to find : —
<a the angular velocity of rotation about the transverse of the
traces; that is to say, the line in the tangent plane perpendicular
to their tangent line,
w the angular velocity of rotation about the tangent line, and
o- „ „ ofapinmng.
..Google
124.] KINEMATICS. 99 ^^gJ O." )
We have
(J,0OS.'-lcO.a) (1),
—'<'-'■) -'{t^-('-''>} (^)'
'•iQ'
')■
..(3).
Theee three fonnulas are respectlTelf equivalent to (9), (8),
and (10) of gill.
125. In the same case, suppose the trace on one only of Surhreroii-
' • ' *■ uig on mr-
toe surfaces to he given. We may evidently impose the con- S^J^^J™'
dition of so spinning, and then the trace on the other is deter-
minate. This case of motion is thoroughly examined in § 137,
helow.
The condition is that the projectiona of the curvatures of the
two traces on the common tangent plane must coincide.
If -; and - be the curvatures of the rolling and stationary
sar&cee in a normal section of each through the tangent line to
the trace, and if a, a, p, p have their meanings of g 124,
p'ar'oosa', p = rco8a (Meunier's Theorem, 1 129, below).
quired.
126. If a Straight rod with a straight line marked on one JjJ^'^"'
side of it be bent along any curve on a spherical surface, go*^'*'*-
that the marked line is Itud in contact with the spherical sur-
face, it acquires no twist in the operation. For if it is laid
so along any 6nite arc of a small circle there will clearly he
no twist And no twist is produced in continuing from any
point along another small circle having a common tangent with
the first at this point.
If a rod be bent round a cylinder so that a line marked
along one side of it may lie in contact with the cylinder,
or if, what presents somewhat more readily the view now de-
7-2
..Google
1 00 PRRUHINABT. [1 26.
^^^not sired, we wind a straight ribbon epirallj on a cylinder, tlie
•ndtwiiL axis of bending is parallel to that of the cylinder, and therefore
oblique to the axis of the rod or ribbon. We may therefore
resolve the instantaneoua rotation which conatitutes the bending
at any instant into two components, one round a line perpen-
dicular to the axis of the rod, which ie pure bending, and the
other round the axis of the rod, which is pure twist
Hie twist at any point in a rod or ribbtm, bo wound on a
circular cylinder, and ooDstitiiUng a tuiifonn helix, is
COB a rina
if r be the radius of the cylinder and a the inclination of the
apiraL For if T be the velocity at which tito bend proceeds
along the previonaly straight wire or ribbon, will be the
angular velocity of the instantaiieouB rotation round the line of
bending (parallel to the axis), and tlieref<Nre
Fcoaa . , Fcoaa
Bma and cob a
r T
are the angnlar velodtiea oi twisting and of pure bending respec-
tively.
From the latter component we may infer that the curvature of
the helix is
a known result, which agrees with the expreeuon used above
(§13).
127. The hodc^raph in this case is a mnaU circle of
the sphere. If the specified condition as to the mode of
laying on of the rod on the cylinder ie fulfilled, the trans-
verses of the spiral rod will be parallel at points along it sepa-
rated by one or more whole turns. Hence the integral twist
in a ringle turn is equal to the excess of four right angles
above the spherical area enclosed by the bodc^:raph. If a be
the inclination of the spiral, ^tt — a will be the arc-radius of the
bodograph, and therefore its area is Stt (1 — sin a). Henoe the
integral twist in a turn of the spiral is Ztrsina, which agrees
with the result previoasly obtained (§ 126).
..Google
128.] KINEtUTICS. 101
128. As a preliminary to the further consideratioD of thd^''"^
rolling of one surface on another, and aa useful in various parts
of fiur subject, we may now take up a few points connected
with the curvature of surfaces.
The tangent plane at any point of a eorfaoe may or may not
cut it at that point. In the former case, the surface bends away
from the tangent plane partly towards one side of it, and partly
towards the other, and has thus, in some of its normal sections,
curvatures oppositely directed to those in others. In the latter
case, the surface on ereiy side of the point bends away from
the same side of its tangent plane, and the curvatures of all
normal sections are similarly directed. Thus we may divide
curved surfaces into Anttclattic and Syndattic. A saddle ^ves Bm«ii
a good example of the former class ; a ball of the latter. Cur- «>
vatores in opposite directions, with reference to the tangent
plane, have of course different signs. The outer portion of an
anchor-ring is synclastic, the inner anticlast)C>
129. Msunigr's I%eorem. — The curvature of an oblique sec- Cumtnn
tion of a sor&ce is equal to that of the normal section through m
the same tangent line multiplied by the secant of the inclina-
tion of the planes of the sections. This is evident from the
most elementary consideraUons regarding projections.
180. Euler's Theorem. — There are at every point of a sjna- Wpdpri
clastic surface two normal sections, in one of which the cur-
vature is a maximum, in the other a minimum ; and these are
at right angles to each other.
In an anticlastic surface there is maximum curvature (but
in opposite direcUons) in the two normal sections whose planes
bisect the angles between the lines in which the surface cuts
its tangent plane. On account <^ the difference of sign, these
may be considered as a maximum and a minimum.
Generally the sum of the curvatures at a point, in any two 525,^iS^
normal planes at right angles to each other, is independent of SSSi"*"
the position of these planes. kl^^
liking the tangent plane u that of «, y, and tlw origin at the
pdnt <rf contact, aad puttiag
..Google
I PBELIHINABT. [130.
*ehave a = 1 (Ja!* + 2Ba;y + Cy*) + etc <1)
The curmture of the normal section which passes through the
point SE^ y, « is (in the limit)
?"»• + /" ai' + y*
If the section be indiaed at an angle 0 to the plane of XZ, this
becomes
- = J COB*0 + 2£Bin0coB0-l-(7 sin'0. (2)
an^es to each other,
1 1
+ — — A +C = oonstant.
(2) may be written
1 = i { J(l + COS 2ff) + 25 ain 2fl + C(l - COB 2tf)(
= i {JTC + J^^' oca 2fl + 2-B sin 2fi} ,
orif ^{A-C) = B(xb2<i, B^EanSa,
thatia Ji= /|j(J-C^* + iflandtan2a = ^^,
we have ^=^(A+C)+ //^ {A-C)'+ B'\wa2{6- a).
The maximum and minimum curvatures are therefore those in
normal places at right angles to each other for which 6= a and
0 = a+ X, and are respectively
i(j+0'y{i(-i-c)'+4
Hence their product is AC - £*.
If this be positive we have a synclaBtic, if native an anti-
clastic, Buriace. If it be zero we have one curvature only, and the
surface ia ej/Undricat at tJie point ocmsidered It is demonstrated
..Google
130.] KINtatATICS. 103
(g 152, below) that if this condition is folfilled at every point, the Prioi^i
Bor&ce is "developable" (§ 139, below). moUodi.
By (1) a plane parallel to the tangent plane and vety near it
cuts the surface in an ellipse, hyperbola, or two parallel Btraig^t
lines, in the three cases respectively. Thia section, whose
nature informs us as to whether the curvature be eyuclastic,
anticlastic, or cylindrical, at any point, was called by Dupin
the India^rix.
A line of curvature of a surface is a line which at every point
is cotangential with normal section of maximum or minimum
curvature.
131. Let P, ^ be two points of a surface infinitely near to 8iiart««t
each other, and let r be the radius of curvature of a normal *^?m two
section passing through them. Then the radius of curvature warttea.
of an oblique section through the same points, inclined to the
former at an angle a, is (§ 129) r cos a. Also the length along
the normal section, from P to p, is less than Uiat along the
oblique section — since a given chord cuts off an arc from a
circle longer the less the radius of that circle.
If a be the length of the chord Pp, we have
Distance Pp along normal section = 2r sin"' =- = a M + ^tj) ,
,, „ oblique section i=o(l +^rr-i ;— )■
" ^ ^ 2ir cos" o/
132. Hence, if the shortest possible line be drawn &om one
point of a surface to another, its plane of curvature is every-
where perpendicular to the surface.
Such a curve is called a Geodetic line. And it is easy to see ocodetia
that it ia the line in which a flexible and inestensible string
vrould touch the surEoce if stretched between those points, the
sur&ce being supposed smooth.
133. If an infinitely narrow ribbon be laid on a surface
along a geodetic Une, its twist is equal to the tortuosity of its
axis at each point. We have seen (§ 125} that when one body
rolls on another without spinning, the projections of the traces
on the common tangent plane agree in curvature at the point
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104 PRELIMINABY. [133
sborttrt of contact. Hence, if one of tbe surfaces be a plane, and ike
(■em two trace on the other be a geodetic line, tbe trace on tiie plane ia a
ani^^ struct line. Cooverset;, if the trace on the plane be & straight
line, that on the surface ia a geodetic lino.
And, quite generally, if the given trace be a geodetic line,
the other trace is also a geodetic line.
flpherioi 134. The ivea of a spherical triangle (on a sphere of unit
radius) is known to be equal to the " spherical excess," t^, the
excess of the sum of its angles over two right angles, or the
excess of four right angles over the sum of its exterior angles.
a™oI The area of a spherical polygon whose n sides are portions
fu\itau. of great circles — !.«., geodetic lines — is to that of the hemi-
sphere as the excess of four right angles over the sum of its
exterior angles is to four right angles. (We may call this the
" spherical excess" of the polygon.)
For the area of h spherical triangle is known to be equal to
A +B+C—K.
Divide the polygon into n such triangles, with a common
vertex, t^e angles about which, of course, amount to 2r.
Area =- sum of interior angles of triangles — nr
= 2* + sum of interior angles of polygon - nw
= 2a- - gum of exterior apgle of polygon.
BKipiKki Given an open or closed spherical polygon, or line on the
Hibfin. surface of a sphere composed of consecutive arcs of great circles.
Take either pole of the first of these arcs, and the correspo&diag
poles of all the othets (all the poles to be on the right hand, or
all on the left, of a traveller advancing along the given great
circle arcs in order). Draw great circle arcs from tbe first of
these poles to the second, the second to the third, and so on in
order. Another closed or open polygon, constituting what is
called the polar diagram to the given polygon, is thus obtained.
The sides of the second polygon are evidently eqnal to the
exterior angles in the first; and the exterior angles of the
second are equal to the sides of the first. Hence the relation
between the two diagrams is reciprocal, or each is polar to the
other. The polar figure to any continuous curve on a spherical
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134.] KI:TEHATIC8. 105
surface is the locus of the ultimate Intersections of great circles B<d|irae>l
equatorial to points taken infinitely near each other along it. ipiuca.
The area of a closed spherical figure is, consequently, ac-
cording to what we have just seen, equal to the excess of 2n-
above the periphery of its polar, if the radios of the sphere be
unity.
135. If a point more on a sur£M» along a figure whose ^^ .
sides are geodetic lines, the sum o£ the exterior angles of this ^^^" *"
polygon is defined to be the integral change of the direction in
the mrface.
In great circle sailing, unless a vessel sail on the equator, or
on a meridian, her course, as indicated by points (tf the com-
pass (true, not magnetite for the latter change even on s meri-
dian), perpetiiatly changes. Yet just as we say her direction
does not change if she sail In a meridiam, or in the equator, so
we ought to say ber direction does not change if she moves in
ang great circle. Now, the great circle is the geodetic line on
the sphere, and by extending these remarks to other curved
sur&ces, we see the connexion of the above definition with that
in the case of a plane polygon (§ 10).
Note. — ^We cannot define integral change of direction here by chuw* <i
any angle directly constructible from the first and last tangents » nrikc^
to the path, as was done (§ 10} in the case of a plane curve or tnrairaitt.
polygon ; but from §§ 125 and 133 we have the following
statement : — The whole change of direction in a curved surface,
from one end to another of any arc of a curve traced on it, is
equal to the change of direction from end to end of the trace of
tlds arc on a plane by puro rolling.
136. Def. The excess of four right angles above the into- inttena
gral change of direction from one side to the same side next
time in going round a closed polygon of geodetic lines on a
curved surfiace, is the integral curvature of the enclosed portion
(^ surface. This excess is zero in the case of a polygon traced
on a plane. We shall presently see that this corresponds exactly
to what Gauss has called the curvaiura iniegra.
Def. (OausB.) The curvaiura integra of any given portion Oanatir*
of a curved surface, is the area enclosed on a spherical surface
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106 PRELIUINABT. [136.
of unit radius by a straight line drawn from its centre, parallel
to a normal to the surface, the normal being carried round the
boundary of the given portion.
The cm-ve thus traced on the sphere is called the Horograph
of the given portion of curved surface.
The average curvature of any portion of a curved snrfoce is
the int^ral curvature divided by the area. The spedfic cwrna-
ture of a curved surface at any point is the average curvature
of an infinitely small area of it round that point.
direction
tonUur
rgiBla (bur
iiirhtui|tlm
cnlCur
137. The excess of Sir above the change of direction, in a sur-
face, of a point moving round any closed curve on it, is equal to
. the area of the horograph of the enclosed portion of surface.
Let a tangent plane roll witbont spinning on the suifoce over
eveiy point of the bounding Ima (Its instantaneoae axis vill
always lie in it, and pass throng the point of contact, but will
not, as we have seen, be at right angles to the given bounding
curve, except when the twint of a narrow ribbon of the surface
along this curve is nothing.) Considering the auxiliaiy q>here
of uitit radius, used in Oauea's definition, and the moving line
through its centre, we perceive that the motion of this line is, at
each instant, in a plane perpendicular to the instantaneous axis
of tlie tangent plane to the given surface. The direction of
motion of the point which cuts out the area on the spherical
surface is therefore perpendicular to this instantaneous axis.
Hence, if we roll a tangent plane on the spherical sur&ce also,
malting it keep time with the other, the trace on this tangent
plane will be a curve always perpendicular to the instantaneons
axis of each tangent plane. The change of direction, in the
spherical surface, of the point moving round and cutting out the
area, being equal to the change of direction in its own trace on
its own tangent plane (§ 135), is therefore equal to the change
of direction of the instantaneous axis in the tangent plane to the
given sur&ce reckoned from a line fixed relatively to this plane.
But having rolled all rotmd, and being in position to roll ronnd
again, the instantaneous axis of the fresh start must be inclined
to the trace at the same angle aa in the beginning. Henoe the
change of direction of the instantaneous axis in either tangent
plane is equal to the change of direction, in the given surfaoe, of
..Google
.137.] KINEMATICS. 107
a point going all round the bounduy of the given portion of it Cunatara
(^ 135); to which, therefore, the change of direction, in the boi^trnph.
spherical anrfiice, of the point going all round the spherical area
IB equal. But, by the well-known theorem (J 134) of the
"spherical excess," this change of direction subtracted from 2-a
leaves the spherical area. Hence the spherical are&, called by
Gauss the eurvalura integra, is equal to 2a- wanting the change
of direction in going round the boundary.
It will be perceived that when the two rollings we have con-
ridered are each oomplete, each tangent plane will have come
back to be parallel to ita original position, but any fixed line in
it will have changed direction through an angle equal to the
equal changes of direction just considered.
Note. — The two rolling tangent planes are at each inytont
patallel to one another, and a fixed line relatively to one drawn
at any time parallel to a fixed line relatively to the other, re-
mains parallel to the last-mentioned line.
If, instead of the closed curve, we have a closed poison of
geodetic lines on the given surface, the trace of the rolling of
its tangent plane will be an unclosed rectilineal polygon. If
each geodetic were a plane curve (which could only be if the
given Biu^tce were spherical), the instantaneous axis would be
always perpendicular to the particular side of this polygon which
is rolled on at the instant; and, of course, the spherical area oa
the auxiliary sphere would be a similar polygon to the f^ven
on& But the given surface being oUier than spherical, there
must (except in the particular case of some of the geodetics
brang lines of curvature) be t«rtuosity in every geodetic of
the closed polygon; or, which is the same thing, twixt in
the ocaresponding ribbons of the surface. Hence the portion
of the whole trace on the second roUing tangent plane which
corresponds to any one side of the given geodetic polygon, must
in general be a curve; and as there will generally be fiuite angles
in the second rolling corresponding to (but not equal to) those in
the first, the ti«ce of tlie second on its tangent plane will be an
nnctosed polygon of curves. The trace of the same rolling on
the Bf^erical surface in which it takes place will generally be
a spherical poison, not of great circle arcs, but of other curves.
'Site mm of the exterior angles of this polygon, and of the
changes of direction &om one end to the other of each of its sides,
is the whole change of direction considered, and is, by the proper
»8 PBELUmABT. [i37<
Application of the theorem of ( 134, equal to 2* wanting the
Bpherical are* encloeed.
Or again, if, instead of a geodetic polygon as the givea curve,
we have a polygon c^ curves, eadi fulfilling the oonditioa that
the nonnal to the surface through any point of it is parallel to a
fixed plane; one plane for the first corre, another for the
second, and so on; then the figure c»i the aoxiliary spherical
sur&oe will be a polygon of arcs of great circles; its ttace on its
tangent plane will be an unclosed rectilineal polygtm ; and the
traoe <^ the given curre on tJie bmgent plane of the first rolling
will be an oncloeed polygon of curves; The sum of diangss erf
direction in these curres, and of exterior augks in paaaing from
one to another of them, is of course equal to the change of
directum in the giv^i surface in going round the given polygon
of curves on it. The chaqge of direction in the other will be
simply the mun cf the extericM' angles of Ihe spherical polygon,
or of its rectilineal trace. Bemark that in this case the in-
stantaneous axis of the fiiat rolling, being always peqtendicular
to that plane to which the normals are all parallel, remaioa
parallel to one line, fixed mth reference to the tangent plane,
during rolling along each carved side, and also remains parallel
to a fixed tine in spaca
Iiastly, remai^ that although the whole change of directioa of
the trace in one tangent plane is equal to that in the trace on
the other, when the rolling is completed round the given orciiit;
the changes of direction in the two are generally unequal in any
part of the circuit, ^ey may be equal for particular parts
of the drcuit, viz., between those pennta, if any, at which the in-
BtantaneouB axis is equally inclined to the direction of the trace
on the first tangent plane.
Any difficulty which may have been felt in reading this Seolaon
will be removed If the following exerrises on the subject be
performed.
(1) Find the barograph of an infinitely small circular area of
any continnous curved surface. It is an ellipse or a hyperbola
aoOMding as the surface is synclastic or anticlastio (§ 128). find
the axes of the ellipse or hyperbola in either case.
(8) Find the IxMogr^h of the area cut offa Byndaatic aurfiuw
by a plane parallel to the tangent plane at any given pcAnt of it,
and infinitely near this point. Find and interpret the corre-
Bpcmding result for the case in which the sar&oe is antiolastie
in the nei^bourhood of the given p(Hut
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137]
KIHE1U.TIC3.
109
(3) Let a tangent plane roll iritbout spuming over the SSHSn^ai
boundary of a given closed curve or geodetic polygon on any •" ""
dured sur&ce. Show that the pointa at the trace in the tang^it
plane which Hncoesaively touch the same point of the given
8ui£ace are at equal distances successively on the circumference
o£ a circle, the angular values of the intermediate arcs being each
2t — JT if taken in the direction in which the trace is actually
deocribed, and X if taken in the contrary direction, £ being
the "intend curvature" of the portion of the curved snrfaoe
enclosed by the given curve or geodetic polygon. Hence if K
be commensurable with a* the trace on the tangent plane, how-
ever Gomplicatedly autotomlc it may be, is a finita dosed curve
or polygon.
(4) The trace by a tangent plane rolling successively over
three principal qnadianta bounding an eighth part of the cir-
cumference of an ellipsoid is represent«d in the accompanying
disgimm, the whole of which is tnced when the tangent plane is
B" C'
rolled four times over the statad boundary, A,B,C; Jl',B,C,
Ac repreaent the pointa of the tangent plane touched in order
by ends of the mean prindpal axis (A), the greatest principal
axis (B), and least prindpal axis (C), and AB, BC, CA' are the
lengths of ^e three principal quadrants.
138. It appears from what precedes, that the same equality f;;^^^'^^
or identity subsists between " whole curvature " in a plane ^f^^S^^
arc and the excess of x above the angle between the terminal "
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110 PKEXIMINABT. [138!.
iwrm?!^ tangents, as between " whole curvature " and excess of Sir above
M^^rtf *^t'^°&6 of direction along the bounding line in the surface for
any portion of a curved surface.
Or, according to Gauss, whereas the whole curvature in a
plane arc is the angle between two lines parallel to the terminal
normals, the whole curvature of a portion of curve sur&ce is
the solid angle of a cone formed by drawing lines from a point
parallel to all normals through its boundaiy.
. . J ■ I - chanite of direction
Again, average curvature in a plane curve is ^-i — -rr ;
and specific curvature, or, as it is commonly called, curvature.
length
Thus average curvature and specific curvature are for surfaces
analogous to the corresponding terms for a plane curve.
liBstly, in a plane arc of uniform curvature, i.e., in a circular
chance of direction 1 ..■. - -i ji/i.t\
ar<^ ^ — -7j- — — - = - . And it is easily proved [as below)
that, in a surface throughout which the specific curvature is
, . 2ir — chante of direction Int^frHl curvature 1 ,
uniform, , or — ~ — ~ . = — , . where
area area pp
p and p' are the principal radii of curvature. Hence in a sur-
face, whether of uniform or non-uniform specific curvature, the
specific curvature at any point is equal to — ; ■ In geometry of
' three dimensions, pp' (an area) is clearly analogous to /> in a
curve and plane.
ConBider a portion i?, of a Burface of any curvature, bounded
by a given closed curve. Let there be a spherical surface, radiua
r, and centre 0. Draw a radius CQ, parallel to the normal at
any point P of S. If this be done for every point of the bound-
ary, the line so obtained encloses the spherical area used in
Gaua3*s definition. Now let there be an infinitely small rect-
angle on ^, at i^ having for its sides area of angles C and f, on
the normal sections of greatest tutd least curvature, and let their
radii of curvature be denoted by p and p'. The lengths of these
sides will be pC and p'C respectively. Its area will therefore be
pp'iC- The corresponding figure at Q on the spherical surface
will be bounded by arcs of angles equal to those, and therefore of
..Google
I38-] KINEMATICS. Ill
lengths rC and rf respectively, and its area will be r'tf. Hence ^t^Sj
if d<r denote this area, the area of tbe infinitely Bmall portion of
tLe given durface ■will be —,— . In a eurface for whicb pp is
constant, the area is therefore^^ lUir-pp'x integral curvatura
139. A perfectly flexible but inezteasible surface is auff- mmibieand
'^ ' . intiMnaiblB
ge^jted, altbougb not realized, by paper, thin sheet metal, or mrisoa
clotb, when the surface is plane ; and by sheaths of pods, seed
Tessels, or the like, when it is not capable of being stretched
flat without tearing. The process of changing the form of a
surface by bending is called " developing." But the term "De-
velopaile Surface" is commonly restricted to such inextenaible
surfaces as can be developed into a plane, or, in common lan-
guage, " smoothed flat."
140. The geometry or kinematics of this subject is a great
contrast to that of the flexible line (§ 14), and, ia its merest
elements, presents ideas not very easily apprehended, and sub-
jects of investigation that have exercised, and perhaps even
overtasked, the powers of some of the greatest mathematicians.
111. Some care is required to form a correct conception of
what is a perfectly flexible inextensible surface. First let us
consider a plane sheet of paper. It is very flexible, and we
can easily form the conception from it of a sheet of ideal
matter perfectly flexible. It ia very inextensible ; that ia to
say, it yields very little to any application of force tending to
pull or stretch it in any direction, up to the atrongest it can
bear without tearing. It does, of course, stretch a little. It
ia easy to test that it stretches when under the influence of
force, and that it contracts again when the force is removed,
although not always to its original dimensions, as it may and
generally does remain to some sensible extent permanently
stretched. Also, flexure stretches one side and condenses the
other temporarily ; and, to a less extent, permanently. Under
elasticity (§§ 717, 718, 719) we shall return to this. In the
meantime, in considering illustrations of our kinematical propo-
sitions, it is necesaary to anticipate such physical circumstances.
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113 FRELIMINABT. [142.
Burhm 142. Cloth woveQ in the simple common wav, very fine
Inutmdbla ,■ - . ... - ^ , . -i .
iDtndi. mualin for instance, illustrates a sumce perfectly inextcnsible
in two directions (those of the warp ami the woof), but suscept-
ible of any amount of extension from 1 up to V^ along one
difigonal, with contisction from 1 to 0 (each degree of extension
along one diagonal having a corresponding determinate degree
of contraction along the other, the relation being 0* + «'* = 2,
where 1 : 8 and 1 : «' are the ratios of elongation, which will be
contraction in the case in which 0 or «' is < 1) in the other.
"EiuUo 143. The flexure of a surface fulfilling any case of the
niiuiiu geometrical condition just stated, presents an interesting sub-
ject for investigation, which we are reluctantly obhged to
forego. The moist paper drapeiy that Albert Diirer used or
his little lay figures must hang Yexj differently &om cloth.
P^haps the stifTnesa of the drapery in his pictures may be to
some extent owing to the &ct that he used the moist paper in
preference to cloth on account of its superior flexibility, while
unaware of the great distinction between them as retards
extensibility. Fine muslin, |H-epared with starch or gum, is,
during the process of dicing, kept moving by a machine, which,
by producing a to-and-fro relative angular motion of warp and
woof, stretches and contracts the diagonals of its structure alter*
nately, and thus prevents the parallelograms from becoming
stiffened into rectanglea
nannoi 144. The flexure of an inextensible sur&ce which can be
iWreioiaUK plane, is a subject which has been well worked by geometrical
investigators and writers, and, in its elements at least, presents
little difficulty. The first elementary conception to be formed
is, that such a surface (if perfectly flexible), taken plane in
the first place, may be bent about any stnugbt line ruled on
it, so that the two plane parts may make any angle with one
another.
Such a line is called a "generating line " of the surface to be
fonned.
Next, we may bend one of these plane parts about any other
line which does not (within the limits of the sheet) intersect
the former; and so on. If these lines are infinite in number,
..Google
145.] KINEMATICS. IIS
and the angles of bending iniinitely small, but sucb that their
sum may be finite, we have our plane surface bent into a
curved surface, which is of course "developable" (§ 139).
145. Lift a square of paper, free from folds, creases, or
T^ged edges, gently by one corner, or otherwise, without
crushing or forcing it, or very gently by two points. It will
hang in a form which is very rigorously a developable surface ;
for although it is not absolutely inextensible, yet the forces
which tend to stretch or tear it, when it is treated as above
described, are small enough to produce no sensible stretching.
Indeed the greatest stretching it can experience without tear-
ing, in any direction, is not such as can affect the form of the
surface much when sharp flexures, singular points, etc., are
kept clear of.
146. Prisms and cylinders {when the lines of bending, § 144,
are parallel, and finite in number with finite angles, or infinita
in number with infinitely small angles), and pyramids and
cones (the lines of bending meeting in a point if produced), are
clearly included.
147. If the generating lines, or line-edges of the angles of
bending, are not parallel, they must meet, since they are in a
plane when the surface is plane. If they do not meet all in one
point, they must meet in several points : in general, each one
meets its predecessor and its successor in different points.
148. There is still do difficulty in understanding the form of,
say a square, or drcle, of the plane surface whea bent as explained
above, ]m>v)ded it does not include any
of these points of intersection. When the
number is infinite, and the surface finitely
curved, the developable lines will in gene-
ral be tangents to a curve (the locus of the
points of intersection when the number is
infinite). This curve is called the edge of
regressioti. The surface must clearly, when
complete (according to mathematical ideas),
consist of two sheets meeting in this edge
VOL. I.
..Google
114
FBEUUmART.
[148.
of TegresBion (just as a cone consists of two sheets meeting in
the vertex), because each tangent may be produced beyond
the point of contact, instead of stopping at it, as in the annexed
diagram.
149. To construct a complete derelopable surface in two
sheets from its edge of r^ression —
Lay one piece of perfectly flat, onwrinkled, smooth-cut
paper on the top of another. Trace any curve on the upper,
and let it have no point of inflec-
tion, but everywhere finite curvature.
Cut the two papers along the curve
and remove the convex purtiona If
the curve traced is closed, it must be
cut open (see second diagram).
Attach the two sheets together by veiy slight paper or
muslin clamps gummed to them along the common curved
, , edge. These must be so slight as not to interfere
/ /^ sensibly with the flexure of the two sheets. Take
hold of one comer of one sheet and lift the whole.
The two will open out into the two sheets of a
developable surface, of which the curve, bending
into a curve of double curvature, is the edge of
regression. The tangent to the curve drawn in
one direction from the point of contact, will
always lie in one of the sheets, and its continuation on the
other side in the other sheet. Of course a double-sheeted
developable polyhedron can be constructed by this process, by
starting from a polygon instead of a curve.
150. A flexible but perfectly inextensible surface, altered
in form in any way possible for it, must keep any line traced
on it unchanged in length ; and hence any two intersecting
lines unchanged in mutual inclination. Hence, also, geodetic
lines must remain geodetic lines. Hence "the change of
direction " in a surface, of a point going round any portion of
it, must be the same, however this portion is bent. Hence
(§ 136) the integral curvature remains the same in any and
every portion however the surface is bent. Hence (§ 138,
..Google
150.] KINEMATICS. 115
Gaaaa's TkeorenC^ the product of the principal radii of curvature '^<"J^ ^
at each point remains unchauged. ^U^""*
151. The general statement of a converse proposition, ex-
pressing the condition that two given areas of curved surfaces
may he bent one to fit the other, involves essentially some
mode of specifying corresponding points on the two. A full
investigation of the circumstances would be out of place here.
162. In one case, however, a statement in the simplest BnrfMc of
posnible terms b applicable. Any two surfaces, in each ofiixviflc
which the specific curvature is the same at all points, and
equal to that of the other, may be bent one to fit the other.
Thus any surface of uniform positive specific curvature (i.e.,
wholly convex one side, and concave the other) may be bent
to fit a sphere whose radius is a mean proportional between its
prindpal radii of curvature at any point. A surface of uniform
negative, or anticlastic, curvature would fit an imi^inary sphere,
but the interpretation of this is not understood in the present
condition of science. But practically, of any two surfaces of uni-
form anticlastic curvature, either may be bent to fit the other.
153. It is to bo remarked, that geodetic trigonometry on G«odetic
any surface of uniform positive, or synclastic, curvature, is (uohatur^
identical with spherical trigonometry.
If a = —7^^, 6=-T^-, e= -:-i-— , where i,l,uexa the lengths
Jpp' s/PP' 'JPP
of three geodetic lines joining three points on the snr&oe, and
\S A, B, C denote the angles between the tangents to the geodetic
lines a.t these points; we have six quantities which agree perfectly
with the three sides and the three angles of a certain spherical
triangle. A corresponding anticlastic trigonometry exists, al-
though we are not aware that it has hitherto been noticed, for any
sorfaoe of uniform anticlastic curvature. In a geodetic triangle
on an anticlastic sur&ce, the sum of the three angles is of course
leas than three right angles, and the difference, or " anticlastic
defect" (like the "spherical excess"), is equal to the area divided
by p ii — p', where p and — p are positive.
154. We have now to consider the very important kinema- stnUn.
tical conditions presented by the changes of volume or figure
8—2
..Google
116 PRELIHIHAttT. ' [1&4.'
stmin. experienced by a solid or liquid mass, or by a group of points'
whose positions with regard to each other are subject to known
conditions. Any such definite alteration of form or dlmensionB
is called a Strain.
Thus a rod which becomes longer or shorter is stnuhed.
Water, when compressed, is strained. A stone, beam, or mass
of metal, in a building or in a piece of framework, if condensed
or dilated in any direction, or bent, twisted, or distorted in any
way, is said to experience a strain. A ship is said to " strain "
if, in launching, or when working in a heavy sea, the different
parts of it experience relative motions.
DeSnitioti 155, If, when the matter occupying any space is strained
B^^^ in any way, all pairs of points of its substance which are initially
at equal distances from one another in parallel lines remain
equidistant, it may be at an altered distance ; and in parallel
lines, altered, it Toa,y be, from their initial direction ; the strain
is said to be homogeneous.
ij«pwtt»ii 156. Hence if any straight line be drawn through the body
8™»»« in its initial state, the portion of the body cut by it will con-
tinue to be a straight line when the body is homogeneously
strained. For, if ABC be any such line, AB and BG, being
parallel to one line in the initial, remain parallel to one line in
the altered, state ; and therefore remain in the same strugfat
line with one another. Thus it follows that a plane remains
- a plane, a parallelogram a parallelogram, and a parallelepiped
a parallelepiped.
157. Hence, also, similar figures, whether constituted by
actual portions of the substance, or mere geometrical surfaces,
or strught or curved lines passing through or joining certain
portions or points of the substance, similarly situated (». e.,
having corresponding parameters parallel) when altered ac-
cording to the altered condition of the body, remain similar
and similarly situated among one another.
168, The lengths of parallel lines of the body remain in
the same proportion to one another, and hence all are altered
in the same proportion. Hence, and from § 156, we infer that
any plane figure becomes altered to another phme figure which
..Google
•156;] KIKKMAnCS. 117
lis a diminished or mi^ifiad orthographic projection of the first Ptopotie*
on dome plane. For example, if an ellipse be altered into a ^^*
circle, its principal axes become radii at right angles to one
another.
The elongation of the body along any line is the propori;ioD
which the addition to the distance between any two points in
that line bears to their primitive distance.
159. Eveiy orthogonal projection of an ellipse ia an ellipse
(the case of a circle being included). Hence, and from § 158,
we see that an ellipse remains an ellipse ; and an ellipsoid re-
mains a suriace of which every plane section is an ellipse ;
that is, remains an ellipsoid.
A plane curve remains (§ 156) a plane curve, A system of
two or of three stmiglit lines of reference (Carteeian) remains
a rectilineal ^stem of lines of refBrence; but, in general, a
rectangular ByHt«m becomes oblique.
a b
be the equation of aa ellipse referred to any rectilineal oonjogato
axee, in tbe substance, of the body in ita initial sbite. Let a and
P be tbe proportions in which lines respectively parallel to OX
and OT are altered. Thus, if we call £ and rj the altered values
of X and y, we have
Hence 7 — rs + tsph ~ h
which also is tbe equation of an ellipse, referred to oblique axes
at, it may be, a different angle to one another from that of the
given axes, in the initial condition of the body.
Or agam, let _, 4- ^ + -, = 1
be the equation of an ellipsoid referred to three conjugate dia-
metral planes, as oblique or rectangular planes of reference, in the
initial condition of the body. Let a, ^, y be the proportion
in which lines parallel to OX, OT, OZ are altered; so that if
j, q, £ be the altered values of x, y, z, we have
f-ar, fi=py, i=yt.
jiGoogle
118 PBEUMINABY. [169.
PmpHFiis which 13 the equatioD of an ellipsoid, referred to ctoijugate dia-
geneouT metrai planes, altered it may be ia mutual indiuatioik &oin th<Me
■mm. ^j. ^1^^ given plaaes of reference in the initial condition of the
stnin 160. The ellipsoid which any surface of the body initially
*"**'^ spherical becomes in the altered condition, may, to avoid cir-
cumlocutions, be called the strain ellipsoid.
161. In any absolutely unrestricted homogeneous strain there
are three directions (the three principal axes of the strain ellip-
soid), at right angles to one another, which remain at right
angles to one another in the altered condition of the body
(§ 158). Along one of these the elongation is greater, and
aloi^ another less, than along any other direction in the body.
Along the remaining one, the elongation is less than in any
other line in the plane of itself and the first mentioned, and
greater than along any other line in the plane of itself and the
second.
Note, — Contraction is to be reckoned as a negative eloi^tlon :
the maximum elongation of the preceding enunciation may be
a minimum contraction : the minimum elongation may be a
maximum contraction.
162. The ellipsoid into which a sphere becomes altered may
be an ellipsoid of revolation, or, as it is called, a spheroid, pro-
late, or oblate. There is thus a maximum or minimum elonga-
tion along the axis, and equal minimum or maximum elongation
along all lines perpendicular to the axis.
Or it may be a sphere ; in which case the elongations are
equal in all directions. The effect is, in this case, merely an
alteration of dimensions without change of figure of any part,
chuwaor The original volume (sphere) is to the new (eltipeoid) evi-
dently as 1 : o^y.
AioioTs 163. The principal axes of a strain are the principal axes
of the ellipsoid into which it converts a sphere. The principal
elongations of a strain are the elongations in the direction of its
principal axes.
..Google
164.] KINEUA.TICa 119
164. When the position of the principal asee, and the magni' ^7^"^
tndee of the principal elongations of a strain are given, the ^^f^j^
eloi^tion of any line of the body, and the alteration of angle oi ti>ei>odr.
between any two lines, may be obviously determined by a sim-
ple geomekical construction,
Ajialytically thna : — let a— 1, /3 — 1, y-l denote the principal
elongationa, bo that a, j3, y may be now the ratios of alteration
along die three principal axee, aa we used them formerly for tlie
ratioB for any three oblique or rectangular lines. Let l,m,n
be the direction cosines of aay Une, with reference to the three
priudpal axes. Thus,
Ir, inr, nr
bung the three initial co-ordinates of a point P, at a distance
OP=r, from the origin in the direction I, m, n; the co-ordinates
of the same point of the body, with reference to the same rect-
angular axes, become, in the altered state,
air, fimr, ynr.
Hence the alitenA length of Oi' is
{aV + ^m* + y*n')^r,
and therefore the "elongation" of the body in that direction is
(a*P + ^m' + /n')*-l.
For brevity, let this be denoted by {— 1, ie.
let t={<^'C + ^m' + yn')\
The direction cosines of OP m ito altered position are
T' T' c'
and therefore the angles XOP, TOP, ZOP are altered to having
their codnes of these values respectively, from having them of
the values i, m, n.
The cosine cf the angle between any two lines OP and Of,
specified in the initial condition of the body by the direction
eoones i, ml, n', is
U + mm' + nn',
iu the initial condition of the body, and becomes
g'S' -I- ff'www' + -/raw'
{a'P + /3'm' + /«•)* (a'f + /S'w" + /»"}*
in the altered condition.
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120 PEELIMINAET. [165.
ChuigsoT 166. Witli the same data the alteration of angle between
»a£^ any two planes of the body may also be earaly determined,
eitJier geometrically or analytically.
Let ^ m, n be the coeinea of tiie angln which a plane makes
with the planes YOZ, ZOX, X07, respectively, in the initial
condition of the body. The effects of the change being the same
on all parallel planes, we may suppoae the plane in question to
pass through 0 ; and therefore its equation will be
Ix + »ny+ na = 0.
In the altered condition of the body we shall have, as before
for the altered oo-oidinatfis of any point initiaUy x,y,x. Henoe
the equation of the altered plane is
a. (i y
But the planes of reference are still rectangular, according to our
present supposition. Hence the cocdnee of the inclinations of
the plane in question, to YOZ, ZOX, SOT, in the altered con-
dition of the body, are altered &om ^, m, n to
a5' ^' y5'
respectively, where for brevity
If we have a second plane simLlarly specified by V, m', n', in the
initial condition of the body, the cosine of the angle between the
two planes, whidi ia
W + mm' ■*■ «»'
in the initial conditiou, becomes altered to
W mm' nn'
166. Returning to elongations, and considering that these are
generally different in different directions, we perceive that all
lines through any point, in which the elongations have any one
..Google
166.1 KINEHATICS. 121
value intenaediate between the greatest and least, must lie on ^i**' *^
a determinate conical snrfiace. Thia is easily proved to be in ekogatiDn.
general a cone of tbe second degree.
For, in a direotdon denoted by direction cosines I, m, n, ve
have
where t denotes the ratio of elongation, intermediate between a
tbe greatest and y the least. Tbia is tbe equation of a cons of
tbe second d^ree, I, m, n being the direction cosines of a gene-
lating line.
167. In one particular case this cone becomes two planes. Two plana
tbe planes of tbe circular sections of tbe strain ellipsoid. tonkm,
Tiet C^P- ^0 preceding equation becomes
or, tince 1 -m' = P+n*,
The first member being tbe product of two fkctora, tbe equation
is satisfied by putting ather = 0, and therefore tbe equation re-
presents the two planes whose equations are
and ^(a'-j3^i-n(^'-y')i = 0,
respectively.
This is the case in which tbe given elongation is equal betngthe
to that along the mean priucipal azis of tbe strain ellipsoid. *^^™^^'
The two planes are planes through the mean principal axis of «uiii«*<i.
the ellipsoid, equally inclined on the two sides of either of the
other axes. The lines along which the elongation is equal to
the mean principal elongation, all lie in, or parallel to, either
of these two planes. This is easily proved as follows, without
any analytical investigation,
166. Let tbe ellipse of the annexed diagram represent tbe
section of tbe straiu ellipsoid through the greatest and least
principal axes. Let S" OS, T OT he the
two diameters of this ellipse, which are
equal to tbe mean principal axis of the
ellipsoid. Every plane through 0, per-
pendicular to the plane of the diagram,
cuts tbe ellipsoid in an ellipse of which
..Google
122 PKBLIMIHABT. [168.
Tvopiuie* one principal axis is the di&meter in which it cuts the ellipse of
tortioD, the diagram, and the other, the mean principal diameter of the
^^" ellipsoid. Hence a plane through either jSiST, or TT, peqwn-
emESd." *'^'^"'^ *•* *^® plane of the diagram, cuts the ellipsoid in an
ellipse of which the two principal axes are equal, that is to say,
in a circle. Hence the elongations along all lines in either of
these planes are equal to the elongation along the mean princi-
pal axis of the str^ ellipsoid.
Distortkm 169. The consideration of the circular sections of the strain
p^Mwiih- ellipsoid is highly instnictiTe, and leads to important viewa
of Tointus. with reference to the analysis of the most general character of
a strain. First, let us suppose there to be no alteration of
volume on the whole, and neither eloi^tion nor contraction
along the mean principal axis. That is to say, let /3 = 1,
and 7 = ^ (§162).
Let OX and OZ be the directions of elongation a— 1 and
rely. Let A be any point of the
body in its primitive condition,
and A, the same point of the
altered body.so that OA^ = aOA.
Now, if we take 0G== OA,,
and if C, be the position of that
point of the body which was in
the position G initially, we shall
have 0C, = -00, and therefore
o
Z' OC^ = 0A. Hence the two tri-
angles COA and C,OA^ are equal and similar.
itMind Hence CA experiences no alteration of length, but takes
•D <rf^> the altered position 0^A_ in the altered position of the body.
Moil Similarly, if we measure on XO produced, OA' and OA', equal
respectively to OA and OA^, we find that the line G A' experi-
ences no alteration in length, but takes the altered position G^^.
Consider now a plane of the body initially through GA per-
pendicular to the plane of the diagram, which will be altered
into a plane through G^A„ also perpendicular to the plane of
..Google
169.] KINEKATICS. 123
the diagram. AJl lines initiall; perpendicular to the plane of inibi ma
the dif^ram remain bo, and remain unaltered in length. AC HMoi^m
has juat been proved to lemtun unaltered in length. Hence ^tkm.
(§ 158) all lines in the plane we have just drawn remain un-
altered iu length and in mutual inclination. Similarly we see
that all lines in a plane through CA', perpendicular to the
plane of the digram, altering to a plane through C^A,', per-
pendicular to the plane of the digram, remain unaltered in
length and in mutual inclination.
170. The precise character of the strain we hare now under
consideration will be elucidated by the following : — Produce
CO, and take OC and 00,' respectively equal to OC and 00,.
Join <7A, CA', C'A,, and C?,'-4,', by plain and dotted lines as
in the diagram. Then we see that the rhombus CA CA' (plain
lines) of the body in its initial state becomes the rbombus
C,A,GJA,' (dotted) in the altered condition. Now imagine
the body thus strained to be moved as a rigid body (i.e.,
with its state <^ stnun kept unchanged) till A, coinddes
with A, and G,' with C, keeping all the lines of the diagram
still in the same plane. A,'0, will take a
position in CA' prodaced, as shown in the
new diagram, and the original and the
altered parallelogram will be on the same
base AC, and between the same parallels
A (J and CA], and their other aides will be
equally inclined on the two sides of a per-
pendicular to these parallels. Hence, irre-
spectively of any rotation, or other absolute motion of the body
not involving change of form or dimensions, the strain under con-
sideration may be produced by holding fast and unaltered the
plane of the body through A C perpendicular to the jdane of
the diagram, and making every plane parallel to it slide, keep-
ing the same distance, through a space proportional to this
distance (t. «., different planes parallel to the fixed plane slide
through spaces proportional to their distances).
171. This kind of strain is called a simple ahear. The simpia
plane of a shear is a plane perpendicular to the undistorted
planes, and partdlel to the lines of their relative motion. It
..Google
124 POELIMINART. [171.
has (1) the property that one set of puallel planes remain
each unaltered in itself; (2) that another set of parallel planes
remain each unaltered in itself. This
other Bet is found vhen the first set and
the degree or amount of shear are giren,
thus : — Let CG, be the motion of one
point of one plane, relative to a plane
KL held fixed — the diagram being in a
plane of the shear. Bisect (7(7, in N,
Draw NA perpendicular to it. A plane
perpendicular to the plane of the dia>-
gram, initially through A C, and finally through A G^ remains
unaltered in its dimensions.
172, One set of parallel undistorted planes, and the amount
of their relative parallel shifting having been given, we have
just seen how to find the other set. The shear may be other-
wise viewed, and considered as a shifting of this second set of
parallel planes, relative to any one of them. The amount of
this relative shifting is of course equal to that of the first set,
relatively to one of them.
173. The principal axes of a shear are the lines of maxi-
mum elongation and of maximum contraction respectively.
They may be found from the pi-eceding construction (§ 171),
thus : — In the plane of the shear bisect the obtuse and
acute angles between the planes destined not to become de-
formed. The former bisecting line is the princip^ axis of
elongation, and the latter is the principal axis of contraction,
in their initial positions. The former angle (obtuse) becomes
equal to the latter, its supplement (acute), in the altered con-
dition of the body, and the lines bisecting the altered angles
are the principal axes of the strain in the altered body.
Otherwise, taking a plane of shear for the plane of the
diagram let AB be a hne in which it is cut by one of either
set of parallel planes of no distortion.
On any portion AB of this as diameter,
describe a semicircle. Through C, its
middle point, draw, by the preceding
construction, CD the initial, and GE
..Google
173.] KINEMATICS. 125
the final, position of an uoBtretcbed line. Join DA, DB, SA,^^J^*
EB. DA, DB are the initial, and EA, EB the final, positiona
of the principal axes.
174. The ratio of a ahear is the ratio of elongation or con- Urui]i«:ot
traction of its principal Hzes. Thus if one principal axis is
elongated in the ratio 1 ;«, and the other therefore (§ 169) con-
tracted in tlie ratio a : 1, a is called the ratio of the shear. It
will he convenient generally to reckon this as the ratio of
elongation ; that is to say, to make its numerical measure
greater than unity.
In the diagram of § 173, the ratio of DB to EB, or of EA to
DA, is the ratio of the shear.
176. The amount of a shear is the amount of relative
motion per unit distance between planes of no distortion.
It is easily proved that this is equal to the excess of the
ratio of the shear above its reciprocal.
2a
a'-l'
Bnt DE= ^CNim DC^= 2C2fcat DC A.
Hen« cW = ^-^=''-a-
17b. The planes of no distortion in a simple shear are Eiiii»oid»i
clearly the circular sections of the strain ellipsoid. In thet^o'»
ellipsoid of this case, he it remembered, the mean axis remans
unaltered, and is a mean proportional between the greatest and
the least axis.
177. If we now suppose all lines perpendicular to the plane 8he»r, iim-
of the shear to be elongated or contracted in any proportion, uon. mbT
without altering lengths or angles in the plane of the shear, combined.
and if, lastly, we suppose every line in the body to be elongated
or contracted in some other fixed ratio, we have clearly (§ 161)
the most general possible kind of strain. Thus if 8 be the ratio
of the simple shear, for which case «, 1, - are the three principal
ratios, and if we elongate lines perpendicular to its plane in the
..Google
126 PEELIMIHAEY. [177,
Bhew, dm- ratio 1 : m, without any other change, we have a stnuo of
tkoivMa which the principal ratios are
Bipannon, '^ ^
aombmed. 1
I^ lastly, we elongate all lines in tlie ratio I : n, we have a
strain in which the principal ratios are
where it is clear that ns, nm, and - may have any values
whatever. It is of coarse not necessary that nm he the mean
principal ratio. Whatever they are, if we call them a, y9, 7 re-
spectively, we have
/a ,„ ^
amW» Of 178. Hence any str^n (a, A 7) whatever may be viewed as
compounded of a uniform dilatation in all directions, of linear
ratio V*y, superimposed on a simple elongation ~7=^ in the
direction of the principal axis to which j3 refers, superimposed
on a simple shear, of ratio . /- (or of amount ^ — */ )
in the plane of the two other principal axes.
179. It is clear that these three elementary component
strains may be applied in any other order as well as that
stated. Thus, if the simple elongation is made first, the body
thus altered must get just the same shear in planes perpen-
dicular to the line of elongation, as the originally unaltered
body gets when the order first stated is followed. Or the
dilatation may he first, then the elongation, and finally the
shear, and so on.
DiipiaM- 180. In the preceding sections on strains, we have con-
bod7, rieid sidered the alterations of lengths of lines of the body, and of
pointof anzles between lines and planes of it; and we have, in parti-
held axed, cular cases, founded on particular suppositions (tlie pnncipal
axes of the strain remaining fixed in direction, § 169, or oue
..Google
180.] KINEICATICS. 127
of either set of nndistorted pknes in a simple shear remain* iHqiia»>
ing fixed, § 170), considered the actual displacements of parts bod^riiid
of the body from their original positions. But to complete p^[|J,^
the kinematics, of a non-rigid solid, it is necessary to take a ''^ "'^^
more general view of the relation between displacements and
stmns. It will be sufficient for us to suppose one point of
the body to remain fixed, as it is easy to eee the effect of super-
imposing upon any motion with one point fixed, a motion of
translation without strain or rotation.
181. Let us therefore suppose one point of a body to be
held fised, and any displacement whatever given to any point
or points of it, subject to the condition that the whole substance
if stnuned at all is homogeneouBly strained.
Let OX, OT, OZ be any three rectangular axes, fixed with
reference to the initial position and condition of the body. Let
X, y, £ be the initial co-ordinatee of any point of tlie body, and
fc,, jf,, X, be the co-ordinates of the same point of tlie altered body,
with reference to those axes unchanged. The condition that the
strain is homogeneooB tliroof^out is expressed by tlie following
equations : — >
y, = [ra=]ar-l-[ry]ff + [r*]«,f (1)
«.^[^:r]a: + [^y]y + [^»]^J
where [Xx], \Xy\, etc., are nine quantities, of absolutely arbi-
baiy values, the Bame for all Talnes of x, y, «:
[Xa;], [Tir], \Zx[ denote the three final co-ordinates of a point
originally at nnit distance along OX, from 0. They ore, of
coune, proportional to the direction-cosines of the altered posi-
tion of the line primitiTely coinciding with OX. Similarly for
[■Tyl \rv\ V^y\ ««■
Let it be required to find, if possible, a line of the body which
remains unaltered in direction, during the change specified by
\Xx\, etc. Let x, y, z, and x„ y„ «„ be the co-ordinates of the
primitiTe and altered position of a point in such a line. We
must have -1 = ^ =-' = l+», -where t is the elcouration of the
line in question.
..Google
8 PBGUKtHART. [181.
Th«B we have jk, = (1 + t)x, etc, and therefore if ^ = 1 + «
[Z«> +[^yly4-{[^*]-,}* = 0. '
From theee equations, by eliminating the ratios x:y:z according
to the vrell-known algebraic process, we find
(lXx]-,)([l's-]-,)(W-,)
-[Ti\[Zy](lXx-[-„)-[l!x-\lX,-\([Ty\-,)-[X,\iYxl[Z,]-,)
*[X,][r«][^]H.[Xy][l'«][^].0.
This cubic equation is neoesearily eatisfied by at least one real
value of 17, and the two others are either both real or both ima-
ginary. Each peal value of ij gives a real solution of the problem,
dnoe any two of the preceding three equations with it, in place of
17, determine real values of the ratios x-.y.z. If the body is
rigid ({.«., if the displacements are subject to the condition of
producing no stnun), we know (arde, § 95) that there is just one
line common to the body in its two positaons, the axis round
which it must turn to pass from one to the other, except in the
peculiar cases of no rotation, and of rotation through two right
angles, which are treated below. Hence, in this case, ihe cubic
equation has only one real root, and therefore it has tivo iinagi~
nary roots. The equations just formed solve the problem of finding
the axis of rotation when the data are the actual diaptacements
of the points primitively lying in three given fixed axes of
reference, OX, 07, OZ; and it is worthy of remark, that the
practical solution of this problem ia founded on the one real root
of a oubic which has two imaginary roots.
Again, on the other hand, let the given displacements be
made so as to produce a strain of the body with no angular
displacement of the principal axes of the stnun. Thus three
lines of the body remain unchanged. Hence there must be
three real roots of the equation in i}, one for each snch axis ; and
the three lines determined by them are necessarily at right angles
to one another.
But if neither of these conditions holds, we may have three
real solutions and three oblique lines of directional identity; or
we may have only one real root and only one line of directional
identity.
..Google
181.] KINEMATICS. 129
An aonlyticad proof of these coqcIuhIoob ma; easily be given; '^'^'^^
thus we may write the cubic in tJie form — ■ iMdj, Tfgid
|[X:.], [J,l [Z,]| ,|[rj,], [rrt*|[Ji], M + |[X;«], Ml Sf"
P"^1.[-%1.W +V([X»] + []'j] + [^.])-,' = 0 (3)
In the particolar case of no atrain, since [Xz], etc, are then
«gua4 not meralj proporHonal, to the direction cosines of three
mntually perpendicalor lines, we Jutve by well-known geometrical
tiieorenu
\[Xx], [Xy], [Xz]\ = 1, and |[ry], [I'a]l= [Xar], etc
m, [Ty], [y4 \[Zs,], [Zz]\
Henoe tlie cubic becomes
1 - h - fl {[J'] + [rj] * Ml - V = 0,
of which one root is evidently if = 1, Tiua leads to the above
explained rotational solution, the line determined by the value 1
of q being tiie axis of rotation. Dividing out the factor 1 —if,
•WK get for tlie two remaining roots the equation
i+(i-[Xx]-[r,]-[Ji]),+v.o,
vhoee roots are imaginary if the coefficient of ij lies between
-f 2 and - 2. Now - 2 Is evidently itA leatt value, and for that
case the roots are real, each being unity. Here there is no
n>tati<M). Also + 2 is its grwitett value, and this gives ns a pur
ei ^oes eadt = — 1, of which the interpretation is, that there is
rotatiw ttirough two right angles. In this case, as in g^ieral,
one line {the axis of rotation) is determined by the equations (2)
with the value + 1 for ij; bnt with ij — - 1 these equations are
satiafied by any line perpendicular to the former.
The limiting case of two equal roote, when there is strain, is
an interesting subject which may be lefl as an exercise. It
separatee the coses in which there is only one axis of directional
idcaitity frmn those in which there are three.
Let it next be proposed to find those lines of the body whose
eloDgationa are greatest or least For this purpose we must find
the equations expressing that x' + >/' -t-z* is a maximimi, when
a^ + j^ + ^ = T*, & constant. First, we have
!"' + y* + o* = ■i'f + S/ + C^ + 2(nyg* txsx + exif) (4),
TOU I. 8
..Google
,..(5).
130 PBEUlflKART. [181.
W"»*M^ where
bS^.Ti^ A =[XxY + [TxV + [ZxY
ornot-OM _ J i. • i. i_i.
».[Zy][X,]+[rj,][r«]*[^y][^i]
« = [x«] [xj] * [r.] [r,] + [Zi] [^,] J
Hie equation
A3f + B^+C^*2{aifZ + bzx + cxt/)^r,' (6),
where r, is any constant, repreaents dearly the ellipscnd which a
apherical sur&oe, radius »■„ of the altered body, would beoMne if
the body were restored to its primitive condition. He problem
of making r^ a mna-rimnin when r is a given constant, leads to Uie
following equations : —
»? + y'+e^r' (7),
{Ax+t!/ + bs)die + (ex + Bjf + asi}d!/ + {bx + at/*Ce)dz--0. j ^ '
On tJie other hand, tlie problem of making r a mazimnm or
m]'n'"'"'n when r, is given, that is to say, the problem of finding
mii-^iTmiiii and minimum diameters, or principal axes, of the
ellipsoid (6), leads to these same two differential equations (8),
and only differs in having equation (6) instead of (7) to complete
the deterroination of the absolute values of x, y, and s. Hence
the ratios » : y : a will be the same in one problem as in the
other; and therefore the direetums determined are those of the
principal axee of the ellipsoid (6). We know, therefore, by tiie
propertdeB of the ellipeoid, that there are three real solntioni,
and that the directions of the three radii so determined are
matnally reotangnlar. The ordinary method (Lagrange's) for
dealing with the differential equations, being to multiply one of
them by an arbitrary multiplier, then sdd, and equate the co-
efficients of the stfvarate differentials to zero, gives, if we take
—1} as the arbitrary multiplier, and the first of the two equations
tbe one multiplied by it,
{A-tUx +cy +6« = 0, ,
ea, + (B-% +a« = 0, \ (9)
hx + ay + (C - ij)a = 0. '
We may find what ij menus if we multiply the first of these by x,
..Google
181.] KIKEHATICS. 131
the second by y, and Uie tliird by s, and add ; because we thus Dlvluc-
obtAUL bodj, ricctd
^k" + ^/+ C^ + 2 (oys + fiat + ca^/) - ij(».' + /+ «^ = 0, ^^<rf
or r '— ijr* = 0 '"'^ "****
gives
'=©■ <■»)•
EUmbi&ting the ratios x:y:z from (9), by the usnal method, we
have the well-known determinant cubic
(j_,)(i_,)(C-,)-o*(J-Tj)-&*{£-l)-c'(C-7;) + 2aic = 0.,.(U),
of which the three roots are known to be all real. Any one of
the three roots if nsed for i;, in (9), harmoniEes tiiese three equa-
tions for the true ratios x:y:z] and, making the coefficients of
^ y, 3 in them all known, allows us to determine the required
ratios by any two of the equations, or symmetrically from the
three, by the proper algebraic processes. Thus w^e have only to
determine the absolute magnitudes of x, y, and z, which (7)
enables ns to do when their ratios are known.
It is to be remarked, that when [Ye] = [Zg], [Zx] = [Xz], and
[jry] = [FiB], equation (3) becomne a cubic, the squares of whose
roots are the roots of (11), and that the three lines determined
by (2) in this case are identical with those detemiined by (9).
The reader will find it a good analytical exercise to prove this
directly from the equations. It is a necessary consequence of
§ 183, below.
We have precisely the same problem to solve when the question
proposed is, to find what radii of a sphere remain perpendicular
to the sur&ce of iJie altered figure. This is obvious when viewed
geometrically. The tangent plane is perpendicular to the radius
when the radius is a maximum or minimum, Therefore, every
plane of the body parallel to such tangent plane is perpendicular
to the radius in the altered, as it was in the initial condition.
The analytical investigation of the problem, presented in the
second way, is as follows : —
Let ;,a:, + m,y, 4- n,s, = 0 (12)
be the equatioa of any plane of the altered substance, through
the origin of conirdioatea, the axes of co-ordinates being the
same fixed axes, OX, OY, OZ, which we have used of late. The
direction cosines of a perpendicular to it are, of course, propor-
tional to ?,, m,, »»,. If, now, for »,, y,, a,, we substitute their
9—2
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132 PRELTMI.VARr. [181,
Tallies, as in (1), in tenns of the co-ordinates vluch ike Bame
point of the Bubstance hod initially, we find the equation of the
same plane of the body ia its initial position, which, when the
terms are grouped properly, is this —
{l,[X<c] + m,[Tx] + n,[^x]\x + {l,[Xs] + m,[ry] + n,[^]iy
+ {lXXz] + m,[Tx]+n,[Zz]}z = 0 (13).
The direction oounea of the perpendicular to tbe plane are pro-
portional to the co-efficienta o£ x, j/, z. Now IJiese are to be the
direction cosines of the same line of the substance as was altered
into the line I, : m, : «, . Hence, iSl :m:n are quantities propor-
tional to the direction oosines of this line in its initi&l position,
we must have
l,[Xx]^mXTx]^n,[Zx]=^ 1
l,[Zy] + mS_Ts]+n^[Zy] = rra\ (14),
where ij is arbitrary. Suppose, to fix the ideas, that I,, m,, n^
are the co-ordinates of a certain point of the substance in its
altered state, and that 2, m, n are proportional to ihe initial co-
ordinates of the same point of the substance. Then we shall
have, by the fnndamentfd equations, the expreeaiona for f, , n», , n^
in terms of I, m, tt. Using these in the first members of (14),
and taking advantage of the abhrevlated notation (6), we have
precisely the same eqnataona for ^ m, n as (9) for x,y,z above.
I 182. From the preceding analysis it follows that any homo-
geneous strain whatever applied to a body generally changes a
sphere of the body into an ellipsoid, and causes the latter ta
rotate about a definite axis through a definite angle. In par-
ticular cases the sphere may remain a sphere. Also there
may he no rotation. In the general case, when there is no
rotation, there are three directions in the body (the axes of the
ellipsoid) which remain fixed ; when there is rotation, there
are generally three such directions, hut not rectangular. Some-
times, however, there is but one.
pore rtnOn. 183. When the axes of the ellipsoid are lines of the body
whose directions do not change, the strain is said to be pure,
or unaccompanied by rotation. The strains we have already
considered were more general than this, being pure strains
..Google
183.] KIHBSUncS. 133
accompanied hy rotaUon. We proceed to find the analytical Pan itnii
conditions of the existence of a piire stiwn.
Let OH, OB', 03!' be the three principal axes of the atraiiy
and let l,m,n, V, m', n', I", m", n",
be their direction coeiues. Let a, a,', a" be the principal elonga-
tiona. Then, if i, f, £" be the poaition of a point t^ the un-
altered body, ^th reference to OU, CS', 03", ite position in
the body when altered irill be of, a'(', a."(". Bat if ib, y, a; be
its initial, and x^, y^, z^ its final, portions with reference to
OX, or, OZ, we have
f=fcc + my + »a, f'«eto., i" = eto. (15),
and fc, = tif+ra'f + r'a"f", y, = etc., 8,=ietc
For f, f ', f " Hubetitnte their Tsluee (16), and we have a;,, y,, a, in
temiE of IE, y, «, expreesed by the following equations : —
«i = (oP + aT" + a" r^jxt (olw +bT«' +o"r«")y + (iih ^a'l'n' +«T'ti") x\
y, = (a«I+a'M'i'+a"m'V')a! + («»' H-b'^' + «"«™)j + {oiwi+o'mV+o"«'VTi [.fl?
», = (a«I + a'»T-i-o"«"i") *+(«"« + «'"'"«' + «"""'"'')»+(on» + «'■'* + aVJ *J
Heno^ comparing with (1) of $ 181, wo have
[J«] = J' + «TV<x"r',et«.; I
[Zj] - [ Fe] = amn + a'ni'n' + a"m"n", ete. J ' ''
In these equations, I, F, /", m, m', m", n, n', n", are deductble
from three independent element^ the three mignlar cxHsrdinates
(§ 100, above) of a rigid body, of which one point ia held fixed ;
and tlierefore, along with a, a', a", oonstituting in all six in-
dependent elemmts, may be determined so as to make the six
members of these equations have any six prescribed valnea.
Henoe the conditions necessary and sufficient to insure no rotation
"" [^]-[r4 [x»]-m, [J-jl-m (18).
16^ If a body experience a succession of Btraios, each un- Compoti-
accompanied by rotation, its resulting coodition will generally ■tnliiu.'^
be prodacible by a strain and a rotation. From this follows
the remarkable corollary that three pure strains produced one
after another, in any piece of matter, each without rotation,
may be so adjusted as to Leave the body unstrained, but rotated
through some angle about some axis. We shall have, later,
most important and interesting applications to fluid motion,
..Google
134 PBEUHiNAHT. [184-
-which (Chap, ii.) will be proved to be instantaneously, or dif-
ferentiidly, inrotational ; but which may result in leaving a
whole fluid mass merely turned round from its primitive posi-
tion, as if it had been a rigid body. The following elementary
geometrical investigation, though not bringing out a thoroughly
comprehensive view of the subject, afTords a rigorous demon-
stration of the proposition, by proving it for a particular case.
Let us consider, as above (§ 171), a simple shearing motion.
A point 0 being held fixed, suppose the matter of the body in
a plane, cutting that of the diagram perpendicularly in CD, to
move in this plane from right to left parallel to DC; and in
other planes parallel to it let there be motions proportional to
their distances from 0. Consider first a shear from P to P, ;
then from P^ on to P, ; and let 0 be taken in a line througli
_ Q V A O T> i ^f perpendicular to
C~^ ^. r-7' 7 J^ -D CD. During the shear
from P to P, a point
Q moves of course to
Q, through a distance
QQ^^VF^. Choose Q midway between P and P,, so that
P^g = QP = \P^P. Now, as we have seen above (§ 152), the
line of the body, which is the principal axis of contraction in the
shear from Q to Q,, is OA, bisecting the angle ilOE at the be-
ginning, and OA^, bisecting ^pE at the end, of the whole
motion considered. The angle between these two lines is half
the angle (ifi^, that is to say, is equal to Pfi*^. Hence, if the
plane CD is rotated through an angle equal to PfiQ, in the
plDne of the diagram, in the same way as the hands of a watch,
during the shear from Q to Q,, or, which b the same thing, the
shear from P to P,, this shear will be effected without final
rotation of its principal axes. (Imagine the diagram turned
round till OA^ lies along OA. The actual and the newly
imagined position of CD will show how this plane of the body
has moved during such non-rotational shear.)
Now, let the second step, P, to P,, be made so as to complete
the whole shear, P to P,, which we have proposed to consider.
Such second partial shear may be made by the common shear-
ing process parallel to the new position (imagined in the preced-
..Google
184.] KIHEtUTICS. 135
ing parentbeais) of CD, and to make itself also nOQ-Totatioaal, Coi
aa its predecessor has been made, we must turn farther round, **^
in the same direction, through an angle equal to QfiP^. Thus
in these two steps, each made non-rotational, we have turned
the plane CD round through an angle equal to Q^OQ. But now,
we have a whole shear PP,; and to make this as one non-rota-
tional shear, we must turn CD through an angle PfiP only,
which is less than QfiQ by the excess of PfiQ above QOP,
Hence the resultant of the two shears, PP^, PiP^t ^^^ sepa-
rately deprived of rotation, is a single shear PP„ and a rota-
tion of its prindpal axes, in the direction of the hands of a watch,
through an angle equal to QOP^•~POQ.
185. Make the two partial shears each non-rotationally. Re-
turn from their resultant in a single non-rotation^ shear : we
conclude with the body unstrained, but turned through the ai^le
QOP^ — POQ, in the same direction as the bauds of a watch,
x^ = Ax*- CI/ +hz
y^~ ex-t-B^ + aa
«, = bx + ay +Cs
is (^ 183) the most general possible expression for the displace-
ment of anj point of a body of which one point is held fixed,
strained according to any three lines at right angles to one
another, as prindpal axes, which are kept fixed in direction,
relatively to the lines of reference OX, 07, OZ,
Similarly, if the body thus strained be ag^n non-rotationally
struned, the most general possible expressions for x,, ^,, «,,
the co-ordinatea of the position to which a;,, y„ z^, will be brought,
are
y, = c,a!, + J,y,+a,s,
*i = ^1*1 + "iffi ■*■ C,Xf
Substitnting in these, for «,,?,, z^, their preceding expresdons,
in terms of the primitive co-ordinates, as, y, % we have the follow-
ing expressions for the co-ordinates of the podtion to which the
point in question is bronght by the two strains : —
a, ={A^A + e^c + 6,6) a; + (J ,c + c,5 4- ifl) y*{Afi-t-cfl+b^(f)z
y, = {e^A A- Bfi + aft) x * (c,c +B^B + a,o) y + (o,6 -i- Bfl ■¥ a,(7) z
s, = (6,j4 + «,c + C,6) IE -I- (6,c -)- a,S + C,«) y + (6,6 + «,« +C,(7) t.
..Google
5 PRELIMINAilT. [185.
Hie resultant displaoement thus reprraonted k not generaUj of
tbe non-rotatioual character, the oouditiotu (18) of § 183 not
being fulfilled, as ire see immediatelj. Thus, for iustaooe, we
eee that ^e coefficient of y in the expresuon for te^ is not
neoeMarilf equal to tlie coefficient of a; in tbe expresnon for y^
Cor. — If both strains are iufinitelj smalt, the resultant displace-
ment is a pure strain wiUiout rottttion. For A, B, C, A^, £„ C,
are each infinitely nearly unity, and a, b, etc., each infinitely
small. Hence, neglecting the products of these infinitely small
quantitiw among one another, and of any of them with the differ-
ences between the former and unity, we b&Te a resultant dia-
x,= A^Ax +(c + c,)y + (6 + 6,)a
y, = (c, + c)x+ B^Sjf 4-(a+a,)«
which represents a pure strain nnaocompanied by rotati<Hi.
166. The measurement of rotation in a strained elastic solid,
or in a moving fiuid, is much facilitated by considering sepa-
rately the displacement of any line of the substance. We are
therefore led now to a abort digression on the displacement
of a curve, which may either belong to a continuous solid or
fluid mass, or may be an elastic cord, given in any position.
The propositions at which we shall arrive are, of course, appli-
cable to a flexible but iaextenaible cord (§ 14, above) as a
particular case.
It must be remarked, that tbe displacements to be considered
do not depend merely on tbe curves occupied by tbe given line
in its successive positions, but on the corresponding points of
these curvea
What we shall call tangential displacement is to be thus
reckoned: — Divide the undisplaced curve into an infinite num-
ber of infinitely small equal parts. The sum of the tangential
components of the displacements from all the points of division,
multiplied by the length of each of the infinitely email parts,
is the entire tanffential displacement of the curve reckoned along
the undisplaced curve. The same reckoning carried out in tbe
displaced curve is the entire tangetttial displacement reckoned
on the disj^aced curve.
..Google
187.] KiNiauTics. 137
187. The whole tangential dieplacement of a curve reckoced Twornkoii-
along the displaced curve, exceeds the whole tangential dis- gP^^i^
placement reckoned along the undisplaced curve by half the oompmi.
rectangle under the sum and difference of the absolute terminal
displacements, taken as positive wb«n the displacement of the
end towards which the tangential components are if positive
exceeds that at the other. This theorem may be proved
by a geometrical demonstration which the reader may eauly
supply.
Analytically thus : — Let x, jf, s he the co-ordinates of any
pointy P, in the undisplaced curve; a;^, y^, at^, thoee of f^ the
pcrint to vhtcb the same point of the curve is displaced. Let
dx, dj/, dz \m the increments of the three oo-ordinates corre-
sponding to any infinitely small arc, dt, of the firet ; bo that
d«={dJ^^di^+dx^i,
and let corresponding notaUon apply to the corresponding
element of the displaced curve. Let $ denote the angle between
the line FF, and the tangent to the undisplaced curve through
P I BO that we have
OOB0 =
g, -a; Ac y^ — y^dy z^-xdx
I> dt* D dt* J) di'
where for brevity
being the absolute apace of displacement. Hence
i> COB Ml = («, - x)dx + (y, - y)dy + («, - »)d^
Similarly we have
Z> oofi tf ,d«, = (as, — «) (ie, + (y, — y) dy, + (a, — «) (b,,
and therefore
/) COS fl,A, - ^ COS ft* = (flj, - «)rf{«, - a:) + (y, - y) d (y, - y)
or i> oos tf,d», -DcM0dt=i d{D').
To find the difierenoe of the tangential displaoementa reckoned
the two ways, we have only to integrate this expression. Tima
we obtain
//>oo«*.A,-pooBft&-}(/>"'-2)'^ = J(i)" + i)')P"--»').
where J)" and J)' draote tbe displacementfl of the two ends.
..Google
138 PKEtmiNABT. [188.
1 188. The entire tangential displacement of a closed curve
is the same whether reckoned along the undisplaced or the
displaced curve.
189. The entire tangential displacement from one to another
of two conterminous arcs, Is the Bame reckoned along either as
along the other.
180. The entire tangential displacement of a rigid closed
'*■ curve when rotated through any angle about any axis, is equal
to twice the area of its projection on a plane perpendicular to
the axis, multiplied by the sine of the angle.
1 (a) Prop, — The entire tangential diuplacement round a closed
curve of & homogeneously strained solid, is equal to
where P, Q, S denote for ite initial position, the areas of its
projections on the planes TOZ, ZOX, XOY respectively, and
or, p, cr are as follows : —
m.Jl[Jj,]-[r»])
a.J{[r:r]-[Xy)).
To prove this, let, brther,
— !ira + [J'*
..j|[r«]*[jr,)i.
Thus we have
y, = CK + 5y + (w + wa — (KB
a, = ftic + ay + C'z + pSB — wy.
Hence, according to the previously inveatigatad expiesaion, we
have, for the tangential displacement, reckoned altHig the ondis-
placed curve,
/{(a!,-a:)dic + (y,-y)rfy +(*,-«)(&}
= /[H{^-l)«^+<5-l)y' + ((7-l)»'+2(oj« + faa: + cxy)}
+ OT (jfdz — zdy) + p {xdx - xdz) + a {tedy — yd«)].
The first part, /J(/{ }, vanishes for a closed curve.
..Google
190.] KINEUATICS. 139
Hie remamder of the expreeuon is Tnnntia
w/(y<fo - zdy) + p/(«& -xdz) + ajixdy - ydx), ™^ '> »
vhich, ftccording to the formulie for projectioQ of areas, in equal <»mpoasn
**> 2Pw + ^Qp + 2ff<r. "^'"^
For, as in g 36 (a), we have in the plane of zy
i{xdy-ydx)=j^de,
doulile ^e area of the orthogonal prDJecti«m of the cnrre on that
plane ; and similarly for the other integraK
(b) From this and g 190, it follovs that if the body is rigid,
and therefore only rotationally displaced, if at all, [^y]~[Fe]
is equal to twice the sine of the angle of rotation multiplied by
the cowine of the inclination of the axis of rotation to the line
of reference OX,
{c) And in general \Zy\ - [Fe] measui-es the entire tangential
displacement, divided by the area on ZOY, of any closed curve
given, if a plane curve, in the plane 70Z, or, if a tortuous curve,
given BO as to have zero area projections on ZOX and XOY.
The entire tangential displacement of any closed cuirve given in
a plane, A, perpendicular to a line vhose direction cosines are
proportional to m, p, tr, is equal to twice its area multiplied by
^(t^ + p' + cr'). And the entire tangential displacement of any
closed curve whatever is equal to twice the area of its jHXijectioa
on A, multiplied by ^/(nr* + p' + it').
In the transformation of co-ordinates, w, p, ir transform by the
elementary cosine law, and of course nr' + p' + tr* is an invariant ;
that is to Bay, its value is unchanged by transformation &om one
Ket of rectangular axes to another.
(d) In non-rotational homogeneous strain, the entire tangential
displacement along any curve from the fixed point to (x, y, «),
reckoned along the undisplaced curve, is equal to
i{(^-l)a:* + (j5-I)y" + ((7-l)3' + 2(ays+fiKB + ««y)}.
Beckoned along displaced curve, it is, from this and g 187,
\{{A-\)3^-i.{B-\)y' + {C-\)s?+2{<iyz+bzx+exy))
+ 1{[{A - l}x + ey + hzY+ [cx+ <S-l)y + <M]'
+ [bx + ay + (C-l)zY).
And the entire tangential displacement from one point along
any curve to another point, is independent of the curve, i.e,,
is the same along any number of oonterminons curves, this of
..Google
0 PEELIMINJUaT. [190.
coarse whether reckoned in each case along the undisplaced or
along the displaced curve.
(e) Oiren the absolute displacement of every point, to find the
strain. Let a, ^, 7, be tlie components, relative to fixed axes,
OX, OT, OZ, of the diaplacement of a particle, P, initially in
the position x,y,s. That is to say, let x + a, y + ^, a + ^ be the
co-ordinates, in the strained bod^, of the point of it vrhioh was
initially at x, y, z.
Consider the matter all round this point in its first and second
podtions. Taking this point P as movetible origin, let f, tf, {
be the initial co-ordinates of any other point near it, and f^, 7,, {^
the final co-ordinates of the sama
The initial and final co-ordinatee of the last-mentioned point,
■with reference to the fixed axes OX, OY, OZ, will be
« + £ ff + T, * + t
and fl! + a+i„ y+y5+ij„ a + y + {,,
respectively j that is to say,
are the components of the di^lacement of the point which had
initially the 00-ordinates x + £, 9 + 7) e + ^ or, which is the same
thing, are the values of a, p, y, when x, j/,zam changed into
* + f. ff+7. « + t
Hence, by Taylor's theorem,
. . da . dn da ^
rfS. dB dB^
the higher powers and products of i, ij, t being n^lected. Com-
paring these expressions with (1) of § 181, we see tliat they ex-
press the changes in the oo-ordinates of any displaced ptnnt (rf
a body relatively to three rectangular axes in fixed directions
throu^ one pcnnt of it, when all other points of it are displaced
relatively to this <me, in any manner subject only to tiie con-
dition ot giving a homogeneous strun. Hence we perceive tliat
at distances all round any point, so small that the first terms
only of the expressions by Taylor's theorem for the differences of
displacement are senable, the strain ii sensibly homogeneous,
..Google
190.] KINEBIATIC3, 141
and we oondade that the directions of the principal azea of Uie Hataro-
nbnm at any point {x, y, z), anA the unotuits of the elongatdons itiBJa.
of tLe m&tter along them, and the tangential displaioenients in
cloeed cnrves, are to be fonod according to the general methods
dcMcribed above, b; taking
[^]-2. m-2- M-i-'-
.it rjf.i.^^
If each of these nine quantities is oonstant {i.e., the same for all ^°^|^
valnes o(x,}f, z), ihe stnun is h<»nogeiieons : not unless. itrmin.
{J') Hie condition that the Kteain may be infinitely small is that tmi°iiSilii.
da da d<t
dx' dy' d»'
dp d§ d0
dx' dy' dx'
dy dy dy
dm' dy' dz'
must be each infinitely small.
(g) These formuhe apply to the most general possible motion Hort
of any substance, and they may be considered aa the fundamental ^nS
eqaations of kinematics. If we introduce time as independent
THiiable, we hare for component velocities u, v, w, parallel to
the fixed axes OX, OT, OZ, liie following expressions ; x, y,z,t
bung independent Tariables, and a, )3, y functlcuts of them :-~
^.^ „-^^ „-^y
'*-^' "-di' ""-It-
(A) If we introduce the condition that no line of the body ex-
periences any elongation, we have the general equations for the
kinematics of a rigid body, of which, however, we have had ("■^^ '^
enough already. The equations of conditdon to express this risid bodr.
wUl be six in number, among the nine quantities -j- , etc., which
(g) are, in this case, each constant relatively to x, y, iir. There
■re left three independent arbitrary elements to express any
angular motion of a rigid body.
..Google
142 PHELIHiyART. [190.
Non-roU- fif If tlie disturbed condition ia so related to the initial con-
■troin. dition that every portion of the body can paaa from ita initial to
itB disturbed position and strain, by a translation and a strain
without rotation ; i,e., if the thi-ee principal axes of the strain at
any point are lines of the substance which retain their parallelism,
we must huve, § 183 (18),
d9~ di/' dx~ dz' di/~ dx'
and if these equations are fulfilled, the strain is Don-rotataonal, as
specified. But these three equations express neither more nor
less thsn that „^ + ^^^ +r ^
is the difierential of a function of three independent variables.
Hence we have the remarkable proposition, and its converse, that
if F(x, y, z) denote any function of the co-ordinates of any point
of a body, and if every auch point be displaced from its givea
podtion (x, J/, z) to the point whose co-ordinatea are
dF dF dF
«. = «+^.y.=y+^.». = »-H^ 0).
the principal axes of the strain at every point are lines of the
substance which have retained their parallelism. The displace-
ment back from (x^, y,, e,) to (x, y, z) fulfils the same condition,
and therefore we must have
rf^, ^F ^F
where F, denotes a function of x,, v , z , and -,-— , etc, its
partial difierential coefficients with reference to this system of
vaiiables. The relation between F and F^ is clearly
F + F^ = -^D' (3),
. „ dF' dF' dF' dF' dF' dF' ,„
where i>' = -j-7+ -tj + -ri = j-V+rr-'i + rHi (■*)■
daf rfy* dxT dx' rfy,' ds," ^ '
This, of course, may be proved by ordinary analytical methods,
applied to find x,y,zin. terms of x^, y,, z^, when the Utter are
given by (1) in terms of the former.
(j ) Let a, ^, -)> be any three functions of x, y, z. Let dS be
any element of a surface \ I, iy^ n the direction cosines of its
normal.
..Google
190.] KISEMATICS. 143
-» H'(|-D-(|-|).»(f -I)} s
=/(«fa: + j8rfy + ><?«) (5),
the former integral being over any curvilinear area bonnded by a
closed curve ; and the latter, wMch may be writtea
/*("S*^S-^£).
being ronnJ the periphery of this curve line*. To demonstrate
this, be^n with the part of the first member of (9) depending on
w^(»l-"S)^
and to evaluate it divide iS* into bands by planes parallel to ZO T,
and each of these bands into rectangles. The breadth B.t x,y,z,
of the band between the planes x—t;(£c and x + ^dx 'a—. — j:, if 0
'3 2 smtf
denote the inclination of the tangent plane of ^ to the plane a;.
Henoe if da denote an element of the curve in which the plane
X cnta the soriace S, we may toko
dS=.-}-.dxd».
tsai.9-
And we have 1= coe 6, and therefore may put
m^aintfcoB^, n^sinOsin^
Henoe
The limite of the t integration being properly attended to we see
tliat Uie remaining integration, /otfor, muHt be performed round
the perijdiery of the curve bounding .S*. By this, and correspond-
ing evaluations of the parts of the first member of (5) depending
on fi and y, the equation is proved.
* Thia theorem «u given b; Stokes in his Smith's Fme paper for IB64
{Camliridge Vnivenity CaUndar, 1864). The demonetnition is ths text in on
eipeniicm of that indicated in our flnt edition. A more synUietioal pi«of !i
BiTcn in g 69 (g) of Sir W. Thonucn'a paper on " Voiiex HotiOD," rrntu. A. 5. £.
18S9. A thoroDghly analTtiaal proof Is given 1^ Piof. Clerk Maxwell in hi*
EUetrifity and Magnttim Q 3^.
..Google
lU PBELI3ITKAET. [190.
Hitero- (k) It is remarkable Uiat
■^ //^K|-f)-(s-S-(f-S)}
u tfafl same for all mrfaces haTing common curvilinear botmdary ;
and when a, P, y are the components of a displacement irom x, y, z,
it ia the entire tangential displacement round the said cnrvi-
linear bonndaiy, being a oloeed curre. It is therefore this that ia
nothing when the displacement of everj part is non-rotational.
And when it is not nothing, we see b; the above propositions and
ocnullaiies preansel; what the measure of the rotation is.
D^UW (0 i'<*^y, "We see what the meaning, for the case of no rota^
•*""■ ti<Hi, of j{adx + ^y + ydz), or, as it has been called, " the dis-
placement fiinction," is. It is, the entire tangential displacement
along any curve from the fixed point 0, to the point P {x, y, z).
And the entire tangential displacement, being in this case the
same along all different corves proceeding from one to another
of any two points, is equal to the difference of ibe values of the
displacement functions at those points.
JJi^j"™ 191. As there can be Deither annihilation nor generation
uooiur." of matter in any natural motioD or action, the whole quantity
of a fluid within any space at any time must be equal to the
quantity originally in that space, increaaed by the whole quan-
tity that has entered it and diminished by the whole quantity
that has left it. This idea when expressed in a perfectly com-
prehensive manner for every portion of a fluid in motion con-
stitutes what is called the "equaiion of continuity'' an unha|^ily
chosen expression.
in»»g»J 192. Two ways of proceeding to express this idea present
■ themselves, each affording instructive views regarding tbe pro-
perties of fluida In one we consider a definite portion of the
fluid ; follow it in its motions ; and declare that the average
density of the Bubstance varies inversely as its volume. We
thus obtain tbe equation of continuity in an integral form.
Let a, £, c be the co-ordinates of any point of a moving fluid,
at a particular eta of reckoning, and let a^ y, « be the co-ordinates
of the position it has reached at any time t from that wa. To
specify completely tbe motion, is to give each of these three vary-
ing co-ordinates as a function of a, b, c, /.
..Google
192.] KINEMATICS. 145
Let 8a, 86, 8c denote the edges, parallel to the a»e« of ooordi- l"*25*J ^
oHtBB, of a veiy small mctangular par&Uelepiped of the fluid, when eontinnitr- ■
1 = 0. Anj portion of the fluid, if only small enough in all its
dimemsionB, must {§ 190, «), in the motion, approximately fulfil
the oonditicm of a body uniformly strained throughout its volume.
Hence if 8a, 8&, Se are taken infinitely small, the ooTreaponding
portion of fluid moat (g lfi6) remain a paiallelepiped during the
motion.
If a, &, 0 be the initial co-ordinates of one angular point of this
parallelepiped : and a + 8a, fi, e ; a,b+ib,o; a,b,o + &s; those
of the other eztremitiee of the three edges that meet in it : the
oo-ordinateg of the same points t^ the fluid at time t, will be
4».»4».-S»'
'*'^<'">*l^'-1<-
Henoe tlie l«Dgtlis and direction coainefl of the edges are le-
BpeotiTelf —
\d<i.'* da'* dj)
lb
W*d^*dp)^- M i^ i^'"^
\df*db'*dl^)
dx
Ids/ M ^ d/\i. S
\d^* d/* d^)
The volume of this panjlelepiped is therefbie
fdatdydz ^dxdydx dad^dx drdydt dxdydz dxedy dx's^ sit. »
V^dbie dadadb dbdada~ dbdadc dcdadb dcdbda/
jiGoogle
[192.
•quMJonel
• (1).
or, as it is hot UBiuJlf written,
dx dy dz
de' dc' dc
How as ^ere can be neither increase nor diminatjon of the
quautitj of matter in any portion of the fluid, the density, or the
quantity ot matter per unit of volume, in the infinitely small por-
tion we have been considering;, moBt vary ioTerseljas its rcOnme
if this varies. Henoe, if p denote the daoBity of the fluid in the
neighbourhood oi (x, y, g) at time t, and p^ the initial density,
ve have
dx dy dz
da' da' da
dat dy dz
db' db' db
dx dy dx
dc' de' de
which is the intt^^al " equation of continuity. "
193. The fonn nnder which the equation of oontinnity is
SoS£«iv' most commonlj given, or the differential eqw^ion of continuity,
fts we may call it, expresses that the i-ate of dioaiuation of the
density bears to the density, at any instant, the same ratio as
the rate of increase of the volume of an infinitely small portion
bears to the volume of this portion at the same instcvnt.
To find it, let a, ft, e denote tiie co-ordinateti, not when ( = 0,
but at any time t-dt, of the point of fluid whose co^rdinatee
are x, y, z a,t t ; to that we have
according to the radinary notation for partial difiCTentaal co-
efficients ; or, if we denote by u, v, to, tiie oompcmeiits ot the
velocity of this point of the fluid, parallel to the axes of oo-
x~a = udt, y - 6 = vdt, ■.
..Google
193.]
da ila ' da da ' da da '
dx _du,
de de
, as we maet reject all terms inTDlving lugher povecB of dt
1 the fint, the detenninant becomes aimplf
7*. ^ = 1+^
, /du dv dwf\ ,,
This therefore expresses the ratio in which the volnpie is aug-
meatad in time dt. The correepouding ratio of variation of
denuty is
i3
P
if Dp denote Qie dififerentaal of p, tiie dennij of one and the Bame
porticMi of fluid as it mOTes from the position (a, b, e) to {x, y, z)
in dte interral erf time &om t — dtiat. Henoe
1 Da du dv dur . ,,.
',^*M*M*i;''' <■'■
Here p, «, v, w aro regarded aa functions of a, b, c, and t, and
the variatioD of p implied in -3- is the rate of the actnal vuiatioD
<tf the densitj' of an indefinitely small portion of the fioid as it
moves aw»7 from a fixed position (a, h, c). If we alter the
principle of the notation, and consider p as the density of what-
ever [lortion of the fluid is at time t in the neighboarhood of the
fixed p<Hnt (a, b, c), and u,v,%b the component velocities of the
fluid paadng the same point at the same time, we shall have
Dp dfi dj) Ap djt .„.
W-rfr*«d^^''d6+"'-^ <^>-
Omitting again the suffixes, according to the usnal imperfect
notation for partial differontial co-efficients, which on our- new
nnders banding can cause no embairassment, we thus have, in
virtue of the preceding equation,
dp dp dp\ da do dw -
da do del da do do
dp . d(pu) . d{pv) . d(ptt>) ^
dt da db do
a
10—8
..Google
I PREUHIHABT. [193.
vhich Ib &% dlffereuti&l equation of oontinuity, in tiie form in
which it is most commonly given.
194. The other way referred to above (§ 192) Icade im-
mediately to the differeDtial equation of continuity.
Imi^ne a space fixed in the interior of a fluid, and consider
tho fluid which flows into this space, and the fluid which flows
out of it, across different parts of its bounding surface, in any
time. If the fluid is of the same density and incompressible,
the whole quantity of matter in the space in question mast re-
main constant at all times, and therefore the quantity flowing
in must be equal to the quantity flowing out in any time. If,
on the contrary, during any period of motion, more fluid enters
than leaves the fixed space, there will be condensation of
matter in that space ; or if more fluid leaves than enters, there
will be dilatation. The mte of augmentation of the average
density of the fluid, per unit of time, in the fixed space in
question, bears to the actual density, at any instant^ the same
ratio that the rate of acqui»tion of matter into that space bears
to the whole matter in that space.
Let the space iS' be an infinitely small parallelepiped, of which
ihe edges a, j3, y are parallel to the axes <^ ooKtrdinatee, and let
at, y, s be the co-ordinates of its centre ; ho that x •*■ ^a, y tk ^^,
2^ Jy are the co-ordinates of its angular points. Let p be the
density of the fluid at {x, y, z), or the mean density through the
space S, at the time t. The denaitj at the tjioe t-\-dl will be
dp
p + ^di ; and hence the quantitieB of fluid contained in the
■pace S, at lite times t, and t + ^ are respectively pa^y and
Ip + -jA^'j o^r- Hence the quantity of fluid lost (there will of
ODune be an absolute gun if ^ be pofdtive) in Uie time dt is
-%'f-^ (•)•
Now let u, v, w be the three components of the velocity of the
fluid (or of a fluid particle) at P. These quantities will be func-
tions of a^ y, « (involving also I, except in the case of " steady
motion "), and will in general vary gradually from point to point
of the fluid ; although the analysis which foUows is not R«trict«d
..Google
194.] riNEMATica. 149
by this consideratioD, bat holds even in cases where in certaia
places of Uie fluid there are abrupt transitions in the Telocitj,
08 may be seen bj considering them as liniitiiig cases of motions
in which tJtere are vtrj sadden conlinaous transitions of velocity.
If u be a small plane area, perpendicular to the axis of x, and
having its centre of gravity at P, the volume of fluid which
fiowB across it in the time dt will be equal to vuxil, and the
niABs or quantity will be puoidL If we subatitate jSy for «,
tJie quantity which flows across either of the faces j9, y of the
parallelepiped S, will differ from this only on account of the
nuiation in the value of fm ; and therefore the quantities which
flow acnMB the two sides /3y are respectively
»d {p„^J„^)J^,
Hence a -^^r<^ or -^^apydt, is the excess of the quantity
of fluid which leaves the parallelepiped across one of the faces
fiy above that which enters it across the other. By considering
in addition the eflbct of the motion across the other fooes of the
parallelepiped, we find for the total quantity of fluid lost from the
cqpaoe S, in the time dt,
{^-V-^'V}-^ (')•
^n»ting t^is to the expreauon {a), previously found, we have
and we dednoe
t^(p"> . ^(pv) d(pu)) .dp ,
dx * dff * <lz "^dt" ^''
which is the required equation,
195. Several references have been made in preceding f
sections to the number of independent variables in a dia- k
placement, or to the degrees of freedom or constraint under
which the displacement takes place. It may be well, there-
fore, to take s general view of this part of the subject by
itself.
..Google
160 PBELIUIKART. [196.
196. A free point has three degrees nf freedom, inasmach
as the most general displacement which it can take is re-
Bolrable into three, parallel respectively to any three directions,
and independent of each other. It is generally convenient to
choose these three directions of resolution at right angles to
one another.
If the point he construned to remain always on a given
surface, one degree of constraint is introduced, or there are
left but two degrees of freedom. For we may take the
normal to the surface as one of three rectangular directions of
resolution. No displacement can be effected parallel to it:
and the other two displacements, at right angles to each other,
in the tangent plane to the sur&ce, are independent
If the point be constrained to remain on each of two buf-
faces, it loses two d^rees of freedom, and there is left but
one. In fact, it is constrained to remain on the curve which
is common to both surfaces, and along a curve there is at each
point but one direction of displacement.
197. Taking next the case of a free rigid body, we have
evidently six degrees of freedom to consider — three inde-
pendent translations in rectangular directions as a point has,
and three independent rotatious about three mutually rect-
angular axes.
If it have one point fixed, it loites three degrees of freedom ;
a in fact, it has now only the rotations just mentioned.
If a second point he fixed, the body loses two more degrees
of freedom, and keeps only one freedom to rotate about the
line joining the two fixed points.
If a third point, not in a line with the other two, be fixed,
the body is fixed.
198. If a rigid body is forced to touch a smooth surface,
one degree of freedom is lost ; there remain ^ve, two dis-
placements parallel to the tangent plane to the surface, and
three rotations. As a degree of freedom is lost by a constraint
of the body to touch a smooth surface, six such conditions
completely determine the position of the body. Thus if six
points on the barrel and stock of a rifio rest on six i
..Google
198.] KINEBtAIICS. 151
portions c^ the Burface of a fixed rigid body, the rifle may be freedom
placed, and replaced any Dumber of times, in precisely theHn^ntork
same poeition, and always left quite free to recoil when fired,
for the purpose of testing ita accuracy.
A fixed V under the barrel near the muzzle, and another
under the swell of the stock close in &ont of the trigger-guard,
give four of the contacts, bearing the weight of the rifle, A
fifth (the one to be brokea by the recoil) is supplied by a
nearly vertical fixed plane close behind the second V, to be
touched by the tri^er-guard, the rifle being pressed forward
in its V's as far as this obstruction allows it to go. This
contact may be dispensed with uid nothing sensible of accuracy
lost, by having a mark on the, second V, and a corresponding
mark on barrel or stock, and sliding the barrel backwards or
forwards in the Y's till the two marks are, as nearly as can
be judged by eye, in the same plane perpendicular to the
barrel's axis. The sixth contact may be dispensed with by
adjusting two marks on the heel and toe of the butt to be
as nearly as need be in one vertical plane judged by aid of
a plummet. This method requires less of costly apparatus,
and is no doubt more accurate and trustworthy, and more
quickly and easily executed, than the ordinary method of
clamptog the rifle in a massive metal cradle set on a heavy
mecbuuiical slide.
A geometrical clamp is a means of applying and main- Oeometnai
taining six mutual pressures between two bodies touching
one another at six points.
A "geometrical slide" is any arrangement to apply five o^>>>c*riaa
degrees of constraint, and leave one degree of freedom, to
the relative motion of twa rigid bodies by keeping them
pressed bother at just five points of their surfaces.
Ex. 1. The transit instrument would he an instance ■if?™'"!™':'
one end of one pivot, made slightly convex, were pressed """^
i^inst a fixed vertical end-plate, by a spring pushing at
the other end of the axis. The other four guiding points are
the points, or small areas, of contact of the pivots on the Y'&
Ex. 2. Let two rounded ends of legs of a three-Ie^ed
atool rest in a straight, smooth, V-shapcd canal, and the third
..Google
152 PSELmmiBT. [198.
on s smooth horizontal plane*. Gravity maintains positive
determinate pressures on the Bve bearing points; and there
ie a determinate distribution and amount o! friction to be
overcome, to produce the rectilineal translational motion thus
accurately provided for.
Ex. 3. Let only one of ihe feet rest in a V canal, and let
another rest in a trihedral hollow-f- in line with the canal, the
third still resting on a horizontal plane. There are thus six
bearing points, one on the horizontal plane, two on the sides of
the canal, and three on the sides of the trihedral hollow : and
the stool is fixed in a determinate position as long as all these
six contacts are unbroken. Substitute for gravity a spring,
or a screw and nut (of not infinitely rigid material), binding
the stool to the ripd body to which these six planes belong.
Thus we have a "geometrical clamp," which clamps two bodies
together with perfect firmness in a perfecUy definite portion,
* Thomson's rsprint of EUetnntatia and MagmUm, g SM.
t A ooDio«I hollow is more aasilj made (ai it ean be b(«ed out at onee bjr an
ordinuy drill), md talflls nearly eDongh lor moat piaetioal qiplioations tb«
geometiioal prindple. A eonioal, oi oth«rwiM rounded, hollow ia tooohed at
three point* bj knobi or ribs projeeting from a lonnd foot resting In it, and
Ihni again Ihe geometrioal prinoiple ia rigorvnsly fal&Ued. The viitoe of the
geometrical prineiple ia well illnatcated by ita poaaiblfl violatioD in this reiy
oaae. Snppoae the hollow to hate been drilled ont not quite "trae," and
instead of being a eiroolar oone to hate slightly elliptio hoiisoDtal aeotioaa: —
A hemiipherical foot will not leet Bteadil; In it, bnt will be liable to a alight
horizontal displacement in the direction parallel to the major axes of the
elliptic sections, besides the legitimate rotation roond an; axis throngh the
eentre of the hemispherical surface: in fact, on this supposition there are jnat
two points of contact ol the foot in the hollow instead of three. When the foot
and hollow are large enough in an; particular ease to allow the poasibili^ of
thie defect to be of momeDt, it is to be obiiated, not by any vain attempt to
tnm the boUow and the foot each perfectly " trae :" — even if this could be done
the desired resnlt wonld be loat \tj the amalleat paiiiole of matter snoh aa a
chip of wood, or a (ragroent of paper, or a hair, getting into the hollow when,
at any time in the use of the instrmnent, the foot is taken out and put In again.
On the eoutiary, tlie trae geometrical method, (of which the general principle
was taught to one ol ns by the late Piotessor Willis thirty yeara ago,) is to
alter one or other of the two sarfacee so aa to render it manittetly not a figure
of rerolntion, thne : — Bonghly file three ronnd notches in the hollow ao as to
render it something between a trihedral pyramid and a tirenlar eone, leading
the toot approzimately round; or else rooghly file at three plaoee of the rounded
foot BO that horizontal eeotions throng and a little above and below tlie points
of oontaat may be (nnghly) equilateral triangle* with lonnded oomen.
..Google
198.] KINEMATICS. 153
without the aid of friction (except in the screw, if a screw Bi«npi.
is tised) ; and in various practical applications gives very ctamp.
readily and conveniently a more securely firm connexion by
one screw slightly pressed, than a clamp such as those com-
monly made hitherto hy mechanicians can give with three
strong screws forced to the utmost.
Do away with the canal and let two feet (instead of only one) JSj^Sj
rest on the plane, the other still resting in the conical hollow. S'3*
The number of contacts is thus reduced to 6ve (three in the
hollow and two on the plane], and instead of a "clamp" we
have again a slide. This form of slide, — a three-legged stool
with two feet resting on a plane and one in a hollow, — will
be found very useful in a laige variety of applications, la which
motion about an axis is desired when a material axis is not
conveniently attainable. Its first application was to the
"azimuth mirror," an instrument placed on the glass cover of
a mariner's compass and used for taking azimuths of sun or
stars to correct the compass, or of landmarks or other terrestrial
objects to find the ship's position. It has also been applied to
the " Deflector," an adjustible magnet laid on the glass of the
compass bowl and used, according to a principle first we believe
given by Sir Edward Sabine, to discover the "semicircular"
error produced by the ship's iron. The movement may be
made very trictionless when the plane is horizontal, by weight-
ing the moveable body so that its centre of gravity is very nearly
over the foot that rests in the hollow. One or two guard feet,
not to touch the plane except in case of accident, ought to be
added to give a broad enough base for safety.
The geometrical slide and the geometrical clamp have both
been found very useful in electrometers, in the "siphon re-
corder," and in an instrument recently brought into use for
automatic signalling through submarine cables. An infinite
variety of forms may be given to the geometrical shde to suit
varieties of application of the general principle on which its
definition is founded.
An old form of the geometrical clamp, with the six pressures
produced by gravity, is the three V grooves on a stone slab
bearing the three legs of an astronomical or magnetic instru-
..Google
154 ruEUuiSARY. [198.
■hwragj of ment It is not generally however so "well-conditioned" as
cKiiiidft the trihedral hole, the V groove, and the horizontal plane
contact, deecribed abora
For investigation of the pressures on the contact surfaces
of a geometrical slide or a geometrical clamp, see § 551, below.
There is much room for improvement by the introduction of
geometrical elides and geometrical clamps, in the mechanism
of mathematical, optical, geodetic, and astronomical instru-
ments : wbich as made at present are remarkable for disregard
of geometrical and dynamical principles in their slides, mi-
crometer screws, and clamps. Good workmanship cannot com-
pensate for bad design, whether in the safety-valve of an iron-
clad, or the movements and adjustments of a theodolite.
199. If one point be constrained to remain in a curve, there
remain four d^reee of freedom.
If two points be constrained to remain in given curves, there
are four d^rees of constraint, and we have left two degrees of
ireedom. One of these may be regarded as being a simple
rotation about the line joining the constrained points, a motioD
which, it is clear, the body is free to receive. It may be shown
that the other possible motion is of the most general character
for one degree of freedom ; that is to say, translation and rota-
tion in any fixed proportions as of the nut of a screw.
If one line of a rigid system be constrained to remain parallel
to itself, as, for instance, if the body be a three-l^ged stool
standing on a perfectly smooth board fixed to a common window,
sliding in its frame with perfect freedom, there remiun tfirea
translations and one rotation.
But we need not further pursue this subject, as the number
of combinations that might be considered is endless; and
those already given suffice to show how simple is the determi-
nation of the degrees of freedom or constraint in any case that
may present itself.
ono dTKTM) 200. One degree of constraint, of the most general character,
■Miniot is not producible by constrdniog one point of the body to a
g™«nii curve surface ; but it consists in stopping one line of the body
from longitudinal motion, except accompanied by rotation round
this line, in fixed proportion to the longitudiuai motion, and
..Google
200.] KINEIUTICS. 15i
leaving unimpeded eveiy other motion : that la to aaj, free
rotation about an; axis peqwndicular to this line {two degrees of
freedom) ; and translation in any direction peqwndicular to the
Bame line (two degrees of freedom). These four, with the one
degree of freedom to screw, constitute the five degrees of freedom,
which, with one degree of constraint, make up the six elements.
Remark that it is only in case [b] below (§ 201) that there is
any point of the body which cannot move in every direction.
301. Let a screw be cut on one shaft, A, of a Hooke's joint, and ^*°^*n^'
let the other shaft, L, be joined to a fixed shaft, B, by a second
Hooke's joint. A nut, N, turning on A, has the most general
kind of motion admitted by one degree of constraint ; or
it is subjected to just one degree of coostraint of the most
general character. It has five degrees of freedom ; for it may
move, Ist, by screwing on A, the two Hooke's joints being
at rest; 24, it may rotate about either axis of the first Hooke's
joint, or any axis in their plane (two more degrees of freedom :
being freedom to rotate about two axes through one point) ;
3d, it may, by the two Hooke's joints, each bending, have
inrotational translation in any direction perpendicular to the
link, L, which connects the joints (two more degrees of freedom).
But it cannot have a translation parallel to the line of the
shafts and link without a definite proportion of rotation round
this line; nor can it have rotation round this line without a
definite proportion of translation parallel to it. The same
statements apply to the motion of £ if Jf is held fixed ; but it
is now a fixed axis, not as before a moveable one round which
the screwing takes place.
No simpler mechanism can bo easily imagined for producing
one degree of constraint of the most general kind.
Particular case (o).-\step of screw infinite (straight rifling),
i.e., the nut may slide freely, but cannot turn. Thus the
one degree of constraint is, that there shall be no rotation about
a certain axis, a fixed axis if we take the case of N fixed and B
moveable. This is the kind and degree of freedom enjoyed
by the outer ring of a gyroscope with its fly-wheel revolving
infinitely fast. The outer ring, supposed taken off its stand,
and held in the hand, cannot revolve about an axis perpen-
..Google
156 PBELUnNABT. [201.
dicutar to the plane of the inner ring* but it may revolve
freely about either of two axes at right angles to this, namely,
the axia of the fly-wheel, and the axis of the inner ring
relative to the outer ; and it is of course perfectly free to
translation in any directign^^
Particular case (b). — i^tep of the screw ** 0. to this case
the nut may run round freely, but cannot move along the axis
of the shaft. Hence the constraint is simply that the body
can have no translation parallel to the line of shafts, but may
have every other motion. This is the same as if any point of the
body in this line were held to a fixed surface. This constraint
may be produced less frictionally by not using a guiding sur-
face, hut the lint and second Hooke's joint of the present
arrangement, the flrst Hooke's joint being removed, and by
pivoting one point of the body in a cup on the end of the
link. Otherwise, let the end of the link be a continuous
surface, and let a continuous surface of the body press on it,
FoUing or spinning when required, but not permitted to slide.
* A single degree of constraint is expressed hj a single equation
among the six co-ordinates specifying the position of one rigid
body, relatively to another coaddered fixed. The effect of this
on the body in any particular poaition b to prevent it from getting
ont of this poution, except by means of component velocities (or
infinitely small motions) iulfilling a certain linear equation among
themselves.
Thus if W|, w^ w^ w^, w„ Wp be the six co-ordinates, and
F{in^ ) = 0 the condition; then
is tLe linear equation which guides the motion through any par-
ticular position, the special values of w^, sr,, w^ etc., for the
particular position, being used in ~ — , - — , &c.
Now, whatever may be the coordinate system adopted, we may,
if we please, reduce this eqoation to one between three relocitiea
of translation u, r, to, and three angular velocities ro, p, tr,
rata ring" is the plane oi the siii of the flj-wheel
a liag by whioh it ia piroted on the onter ring.
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201.] KINE1UTIC3. 157
Let this eqtiation be 0
Thia 18 equivalent to the foUowiog : — %
q + aia = 0,
if q denote the componest velocity along or parallel to the line
whose direction ooeines are proportional to
A, B, C,
n the component angular velocity round an axis through the
origin and in the direction whose direction cosines are propor-
tional to A', B, C,
and lastly, a = / — ,,■ —^ — >^ .
It might be supposed that by altering the origin of co-ordinates
we could do away with the angular velocities, and leave only a
linear equation among the components of tranalational velocity.
It is not BO ; for let the origin be shifbed to a point whose co-
ordinates are £, i\, ^ The angular velocities about the new axes,
parallel to the old, will be unchanged ; but the linear velocities
which, in composition with these angular velocities about the
new axe«, give tr, p, a, u, v, w, with reference to the old, are
(§89)
v-iiri+iT$=v',
w - pf + onj = UJ'.
Henoe the equation of constraint becomes
Ati' + Bvf + Cii>' + (A' ■'-Bt-Cri) a + ebo. = 0.
Now we cannot generally determine f , t; , C, so as to make v,
etc, disappear, because this would require three conditions,
whereas their coefficients, as fiinctions of (, i), Ct ^^^ "o^ ut*
dependent, since there exists the relation
A(Bt'Cr,) + B{C(~At) + O{A^~B£)='0.
The fflmplest form we can reduce to is
lu' + m^ + nu/ -v a (Jiw + mp+ no) = 0,
that is to say, every longitudinal motion of a certain axis must be
accompanied by a definite proportion of rotation about it.
202. These principles constitute in reality part of the general
theory of "co-ordinates" in geomeby. The three co-ordinates
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158 PHELIHINART. [202.
Oencmitud of either of the ordinary systems, rectangular or polar, required
loia*. to specify the poeition of a pointy coirespond to the three
Of* point, degrees of freedom enjoyed by an unconstrained point. The
most general system of co-ordinates of a point consists of
three sets of surfaces, on one of each of which it lies. When
one of these surfaces only is given, the point may be any-
where on it, or, in the language we have been using ahove, it
enjoys two degrees of freedom. If a second and a third sur-
face, on each of which also it must lie, it has, as we have seen,
no freedom left ; in other words, its position is completely
specified, heing the point in which the three surfaces meet.
The analytical ambiguities, and their interpretation, in cases in
which the specifying surfaces meet in more than one point,
need not occupy us here.
To express this analytically, let ^ = 0:, ^t^ff, B^y, where
^, <l>, 0 are functions of the position of the point, and a, /3, •/
constants, be the equations of the three sets of sur&ces, different
values of each constant giving the different surfaces of the cor-
responding set. Any one value, for instance, of a, will determine
one surface of the first set, and so for the others : and three
particular values of the three constants specify a particular
point, P, being the intersection of the three surfaces which
they determine. Thus a, 0, y are the "co-ordinates" of P;
which may be referred to as " the point (a, jS, 7)." The form
of the co-ordinate surfaces of the (^, ^, 0) system is defined
in terms of co-ordinates (a;, y, z) on any other system, plane
rectangular co-ordinates for instance, i£ ^,i^,0 are given each
as a function of (x, y, z).
OrWnoftbe 203. Component velocities of a moving point, parallel to
«>iaiiw. the three axes of co-ordinates of the ordioaTy plane rectangular
system, are, as we have seen, the rates of augmentation of
the corresponding co-ordinates. These, according to the
Kewtonian flusional notation, are written x,i/, i; or, according
to Leibnitz's notation, which we have used above, -jr , -^ i "^ •
Lagrange has combined the two notations with admirable skill and
taste in the first edition* of his M4canique Analytiqw, as we shall
■ In UtUt «clitioiia the Neivtoiiuti noteiioo U veiy onlufipilr alterad I7 the
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203.] EiHiaiATiCB. 159
Bee in Chap. It. In Bpecifying the motion of a point according to
the generalized system of co-ordinates, ^, ijt, 6 must be considered
as vaiying with the time: -^i ^, &, or -^ , -^ , -y-, will
then be the generalized components of velocity : and ^, ^, 8, or
^' « ■ -Si' " 1?' il • ^' '"'" ** ^° 8<'M'^»i
components of acceleration.
201. On precisely the same principles we may arrange sets Co-onu-
of co-ordinates for specifying the position and motion of a miem.
material system consisting of any finite number of rigid bodies,
or material points, connected together in any way. Tbns if
^, ^, 8, etc., denote any number of elements, independently
variable, which, when all given, fully specify its position and
coniignration, being of course equal in number to the d^rees
of freedom to move enjoyed by the system, these elements are
its co-ordinates. When it is actually moving, their rates of
variation per unit of time, or '^, ^, etc., express what we shall
call ita generalized component velociUea ; and the rates at which
^, ^, etc, augment per unit of time, or ^, ^, etc., its component aeiwimiijrf
accelerations. Thus, for example, if the system consists of <**««"''■
a single rigid body quite free, ^, ^, etc., in number six, may be Bnmpiai.
three common co-ordinates of one point of the body, and three
angular co-ordinates {§ 101, above) fixing its position relatively
to axes in a ^ven direction through this point Then ^, ^, etc., .
will be the three components of the velocity of this point, and
the velocities of the three angular motions explained in § 101,
as corresponding to variations in the angular co-ordinates. Or,
agun, the system may consist of one rigid body supported on
a fixed axis ; a second, on an axis fixed relatively to the first ;
a third, on an axis fixed relatively to the second, and eo on.
There will be in this case only as many co-ordinates as there
are of rigid bodies. These co-ordinates might be, for instance,
the angle between a plane of the first body and a fixed plan^
through the first axis; the angle between planes through the
d ■■ Bigniiying velooitiea und
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160 PBELIMINABT. [204.
o«naniiwd secoiid aziB, fixed relatively to tbe first and second bodiee, and
oiTeiadtf. so on ; and the component velocities, -^j ^, etc. would tbeD be
Xumpin, the angular velocity of the first body relatively to directions
fixed in space; the angular velocity of the second body re-
latively to the first ; of the third relatively to the second, and
so on. Or if the system be a set, i in number, of material
points perfectly free, one of its 3i co-ordinates may be the sum
of the squares of their distances from a certain point, either
fixed or moving in any way relatively to the system, and the
remaining 3i - 1 may be angles, or may be mere ratios of
distances between individual points of the system. But. it ia
needless to multiply examples here. We shall have illustrations
enough of the principle of generalized co-ordinates, by actual
use of it in Chap. II., tuid other parte of this book.
APPENDIX TO CHAPTER I.
A,. — ^EXBREBSION IN GeneHALIZED Co-OHDIKATE3 FOR
Poisson'3 extension of Laplace's equation.
(a) In § 491 (c) below is to be found PoisBon's extension
of litplace's equation, expressed in rectilineal rectangular oo-ordl-
nates; and in § 492 an equivalent in a form quite independent
of the particular kind of co-ordinates chosen : all with reference
to the theory of attraction according to the Kewtonion law.
The same analysis is largely applicable through a great range of
physical mathematics, including hjdro-kinematics (the "equation
of continui^" §193), the equilibrium of elastic solids (§734),
the vibrations of elastic solids and fluids (ToL li.), Fourier's
theory of beat, tec Hence detaching the analytical sulgect fr<mt
particular physical applications, consider the equation
dfU <PU d'U , ...
-^^-dP*-^"^" ('>
where f> is a given function Qix,jf,ss, (arbitrary and discontinnous
it may be). Let it be required to express in temu of generalized
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A, (a).] EINEIUTIC3. IGl
oo-ordioafes (, f , f, the property of U which thia equation ex- **^'^j
preaaes in terms of rectangular rectilinear oo-ordiiiates. This HuwimUMd
may be done of course directly [g (m) below] fay analytical tians- diintai.
fonnation, finding the expressioa in terms of f, f, $", for the
opeiatioii -i-? + ^r-i+ .-i- But it b done in the form most con-
'^ A^ dif a^
renient for physical applications much more easily as follows, by
taking advantage of the formula of § 492 which expresses the
same property of TJ independently of any particular system of
co-ordinatea. This expression is
ffSUdS=-i^}fSpdB (2),
wh^re JJdS denotee integratjim over the whole of a closed sttr&ce
S, fff dB int^p-atdon throngbout the volume B enclosed by It,
and hU the rate of variation of {7 at any point of S, per unit of
l^igth in the direction of the normal outwards.
(&) Fcnr £ take an infinitely small carrilineal parallelepiped
having its centre at (£, f, ^"), and angular points at
«*i«, f-isf, r*i«f').
Let SU, BfSt, B"S^' be the lengths of the edges of the paral-
lelepiped, and a, a', a" the angles between them in order of
symmetry, so that RS" sin a Sf Sf", &c., are the areas of its &oes.
Let DU, jyU, D"U denote the rates of. variation of U, per
unit of length, perpendicuUr to the three surfaces {-const.,
f = const, f = const., intersecting in (f, f , ^') the centre of die
parallelepiped. The value otjJSUdS for a section of the paral-
lelepiped by the sur&ce ( = const through {i, f , f") will be
Hence the values of //8 U dS for the two oorresponding sides
of the parallelepiped are
ir*" rin o Sf 8£" i) F * ^ (fl'i?" sia a Sf Sf ' 2>CQ . J 8^.
Henoe the value of fJBUdS for the pair of sides is
w ^{B'li"miaDU)SiS^&i
Dealing similarly with the two other pairs of sides of the
pandlelepiped and adding we find the first member of (2). Its
VOL. 1. 11
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L (JiS'ana"D"U)\ = -i^p ... (3).
162 FEELIHIIIABT. [A, (b).
£*piue*i seoond member is - ivp . Q . JtS'S" Si Sf !if", if Q denote the ratio
pneniii^ of the boUc cf the parallelepiped to a rectangular one of equal
dituW edges. Hence equating and dividing both sides bj the bulk of
the poiaUelepiped we find
^^{|{i?'J?'sin«i>I7)+^(JJ"j;dna'Z>-P)
(c) It remaina to express D U, D'JJ, Jy'V in terms of the co-
ordinates {, £', ^'.
Denote by K, L the two points <f , f , f") and (f + 8^, f , ^').
From Z (not shown in the diagram) draw LM perpendicular to
, the surface £=conHL through K.
ffi Taking an infinitely small portion
of this surface for the plane of our
diagram, let KS, £H" be the lines
in which it is cut reepectiTely by the
surEaoes £"- const and £' = const.
through K. Draw MN parallel to
~^ H"jr, and UQ perpendicular to JH'.
Let now p denote the angle LKM,
A' „ „ „ LQM.
We have
Jf £ - £"£ sin p = A dn p S£,
JVJf = QM coeec a = ML coeec a cot j4' = fi sin p coseo a oot J' S£.
Similarly KN= Bamp coseo a cot j1" 8^,
if A" denotes an angle corresponding to J'; ao that A' and A"
are leBpectively the angles at which the surfaces ^' - const, and
^ = const, out the plane of the diagram in ^e lines fB* and So".
Now the difTerence of values of f for X and N is -^,
"ij , „ „ r „ if „ if „ ^.
Hence VU{Kj,U(U),U(l) denote the valttM of U renpectiTeiy
St the pomts j[, M, L, Te faaTe
nttr. n,ir. ''^ '^'^ <"' ">>
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A, (c).] KINEMATICa . 163
Bat PU.m^P^,
and flo nuDg the preceding expressioiiH in. the terms involved we ^
find
„„__!_ dP 1 rfP- 1 rfp
JtNnp d4 ^sinotan^" df ^'ainatan^'rff" ""^ '"
Using this and the synunetrical ezpresaiona for I/U and Il^'U,
in (3), we have the required equation.
(d) It is to be remarked tluit a, a', a" are the three odea of
a spherical triangle of which A, A', A" are the angles, aad^ the
perpendicular &om the angle A to the opposite side.
Hence by spherical trigonometry
, COB a - COB a COB a'
• c<3eA = -. ; — -;— ;
. . lyfl-coe"** — cos'a'— co8'a" + 2coa<»co«o'ooBa") „,
BatA=:-Si-^ ; : — i..,.(6):
BinaBina
■in p = waA'a.na"
^(l-coH'n-oOB*a'-co8'CT" + 2ooaaooBot'ooBa") . .
To find Q remark that the volume of the parallelepiped is
equal to/taup .ghtdna it/, ;, A be its edges : therefore
^ = aapKiia (7)|
whence by (6)
^ = ^(l-ooB^a-co8»a'-coe*o" + 2co8acoerfco8a") (8).
Lastly hy (5) and (6) we have
U.A «,
cosa— oosa cosa
(e) Uaiiig tbrae in (4) Te find
" ~ 5Ei \~ir 1^ 3? df
..(9).
~y w) * '■
Usuig tills and the tvo BTmmetrical exprefniona in (3) and
adopting a common notation [App. B (;), § 491 (c), dco. ifa:.],
aocording to which Poisaon'a equation is written
i-U^-iwp (11),
11-2
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164 . PBELIMINART. [A, (e).
we find for the symbol ^ in tenna of the generalized coordinates
+ IP (cob a* cm a - ooe a') ^^ I
d irit"Jimii.'a' d „, , , , rf
*diQy S ^ + -ff(«»ao08«^'-CO8a)^,
+ 5" (cog a COB a' -008 a") ^ ,
•l-A(coBa'cas()i"-oosa)^ [ .--(IS)!
vhere for Q, ita value "by (8) in terms of a, a', a" is to be oaed,
and a, a', a", R, S", £" are all known Amotions (ji $, C, ^' when
the Byatetn of coordinates is completely defined.
Cutot (y) ^°^ ^^ c^ of rectangular co-ordinates whether plane
JJ^SC^ o^ curved a = a' = a" = A—A' = A" = 90° and Q = l, and therefore
SS&<r ■«• Iwve
"^^ , 1 (d/RR'd\ d /lt"Sd\ d /RltdVl „,.
' ^3afr\2£l-r m) *d^\~ff^de)*dr'\R'dr)r--<^^>-
which is the formula originally^ given by Iiim€ for exjvessing
in terms of his orthogonal curved oo-ordinate system the Fourier
equationa of the conduction of heat. The proof of Uie more
general formula (12) given above is an extension, in purely
anelytioal form, of a demonstratiou of I^m^'s fismnla (13) which
was ^ven in terms relating to thermal conduction in an article
" On the equations of Motion of Heat referred to curvilinear
co-ordinates " in the Cambridge JfathematiealJownal (1843).
(g) For the particular case of polar coordinates, r, $, ^
oonndering the rectangular parallelepiped corresponding to Sr,
S9,S^ we see in a moment that the lengths of it« edges are Sr,
rS6, r sin $ii^ Hence in the preceding notation S^l, K = t,
Jt" = rBia$, and Lam^s formula (13) gives
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A, (A).] KIHKMATICa 165
WAjbuu lot the oo-OTdinatee be of the kind vhich has i*pi«»'«,
been called " oolumnar " ; that u to eaj, distAuce from an ooiamnH
axis (r), angle from a plane of reference through this axis to diiiuu.
a plane through the axis and the specified point (0), and distanoe
&om a plane of reference perpaadioular to the axis (s). The
oo^rdinate sur&ces here are
coaxal circular cylinders (r = const),
planes through the axis (tft = const.),
planes perpendicular to the axis (s = oonst.).
The three edges of tiie infinitesim^ rectangular parallelepiped
are now dr, rd^, and <£>. Henoe £ = 1, &'=t, R'=\, and
I^m^'s fonnnla givee
. 1 df
m^{£)-*{£) o»).
which is Teiy useful for many physical problems, such as the
conduction of heat in a solid circular column, the magnetization
of a round bar or wire, the vibrations of air in a closed circular
cylinder, the vibrations of a vortex column, &c &a.
(i) For plane rectangular co-ordinates we have A>£'=^'; AlRebrato
so in this case (13) becomee (with x,y,e for $, f , 0!), mxioa '
trom plans
jt ji j« pixrtmgulmr
^■i*$*» w. SST-
which is Laplaoe's and Fourier's original form.
(j) Suppose now it be detdred to pass from plane reotangular
oo-ordinatee to the generalized co-ordinates.
Let X, y, « be expressed as functvms of (, t, f i tlien putting
for brevity , ,
-^' I-''' %'"■■ |=^'.*»-= f-^". fa .■•<");
8!,= TSf + r8f -fr'Sf',} (18);
8« = ^8(+ir'sr+r-si",)
B".J{X'-*T"-tZ-^ (1»).
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PBEUMDIABT. [A, (j).
and ihe dii^ction coeines of the three edges of ,the infjiitesuiial
m pAT&llelepiped conespottdiug to i(, Sf, 8£" aie
x'x"+F'r'+^'^"
jr"x+r"r+^"i
xjr'+rr'+^^'
jj/r
..(21).
{k) It IB important to remark that wben them expreesioiiB
for cos a, oos a', cob a", R, R, S", in terms of X, toj. are osed in
(^)f 6' becomes a complete square, so that QRR'R' is a lational
homcgeneouB function of the 3rd degree of X, T, Z, X', iic
For the ordinaiy process of finding from the direction ooaiiies
(20) of Uiree lines, the sine of the angle between one of them and
the plane of the other two gives
X, r, z
Bmp= X\ Y;Z' ^RRR'as.a (21);
X", r; z"
from this and (7) we see that QRR'R" is eqoal to the deter-
minant. From this and (8) we see that
«qt««o[. {x*+y+z^(X"+r''+z^{X''+7"*+z"^
""^ -(X'+T'+Z^{X'X"+T'r'+Z'Zy-{X''+T''+Z^{X"X+Y"Y+Z'Z)'
- (Z"*+ T"'+Z"^{XX'+ YT' + ZZ')'
•t-2{Z'X"+Y'Y" + ZZ"){X"X+r"Y+Z"ZXXX'+rY'+ZZ^
T, Z,
..(22),
Y', Z',
Y", Z"
an algebruo identity which may be verified by expanding both
members and comparing.
(l) Denoting now by T the complete determinant, we
have
«=bI«" m.
and OMDg this for § in (12) we have a formula for v* in which
only rational functions of X, F, Z, X', Ac. appear, and which
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A. wo
LB readily yerified by comparing with the following derived from AlsAnfo
(16) by direct transformation. matlon
where
L'=T"Z-7Z", M'=Z"X~ZX", N'=X"Y-XT\ \ (24).
L"=YZ'-Y'Z, M"=ZX'-Z'X, S"=XY'-X-Y, J
Henoe
dx~Tdi'^ Td^* T d^" dy'' •' dz" ''
and thus we have
*-(k^ EA. EA\' (^ '^ ^^ M" d\*
^ ~\Tdi'*' Tde'*' T d^'J *\Tdi* T d^* T d^'J
(N d N' d y d y ,„„,
*{T3i*Tde*i'de'). <''>•
^, &c. with those of the corresponding tenns of (12) with (21)
and (23) we find the two formulas, (12) and (25), identicaL
A. — Extension of Green's Theorem.
It U convenient that we Ghould here give the demoDstration
of a few theorems of pure analysis, of which we shall have
many and most important applications, not only in the subject
of spherical harmonics, which follows immediately, hut in the
general theories of attraction, of fluid motion, and of the con-
duction of heat, and in the most practical investigations r^ard-
ing electricity, and magnetic and electro-magnetic force.
(a) Let {/ and W denote two functions of three indq>eDdent
variables, x, jf, e, which we may conveniently regard as rect-
angalar co-ordinates of a point P, and let a denote a quantity
which may be either constant, or any arbitrary function of the
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168 PRELIMINAET. [A {a).
variAbles. t«t ffjdxdydz denote int^ration throughout a finite
tinffli/ eontimtoua space bounded by a close surface S; let JJ4S
denote integntttoii over the whole aurfaoe S; and let S, prefixed
to any function, denote itii rate of varUtion at any point of S,
per unit of length in the direction perpendicular to S outwards.
Then
--, ^fdUdW dUdXr dUdJTx , ^ J
^^^* U di^Ty^^-d^^j^y^
Gre^" =fJdS.U'a'&U-fjfU']
f-S) <-^"^
-Has. ws!r-///p| A^ + -i^ + 1^
..(I).
For, taking one term of the fiist member alone^ and int^mtang
" by parte," we have
i-dxdydz,
tiie firgt integral being between limits corrcBponding to the sar-
&ce S; that b to say, being from the n^^ative to the poeitiTe
end of the portion within S, or of each portion within S, of the
line X through the point (0, y, 2). Now if ■'1, and A^ denote the
inolination of the outward normal of the surface to this lin^ at
points where it enters and emerges from S respectively, and if
dS^ and dS, denote the elements of the surface in which it is cut
at these pointe by the roct«ngular prism standing on dydx, we
have
dyds = - coeAjiS, = cob A^dS,.
Thus the finit integral, between the proper limits, involvee the
elements Wt^-f-coeA.dS,, and — Wa* -i-coa A, dS,: the latter
(£» ' ' dx ' '
of which, as corresponding to the lower limit, is subtracted.
Hence, there being in the whole of iS* an element dS^ for each
element dS^, the first integral is simply
JfWa'^CMA dS,
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A (a).] KTNEKATICB. 169
for die whole enr&ce. Adding the corresponding terms for y icontiaDl
jind z, uid remarking that fbcorem or
dlT , dU „ dU . .„ ""'■
-J- COB J + -J- COB 5+ -j-cobC=8(7,
ax ay az
irhere B and C denote tiie InclinationB of the outward normal
through dS to lineB drawn throagb dS in the poaitive directions
panillel to y and z respectively, we perceive the truth of (1).
(£) Again, let U and W denote two fnnctions of x, y, z, which
have equal values at eveiy point of S, and <^ which the first
falfilfl the equation
<-f) ±t) <-f)_
dx
for every point within S.
Tbaa iiV-U=u, we have
■■(■*). tiODOtluM.
-///{(•f)'H"f)'*(-f)]-'-
^///{(•£)"H-I)'*(4")"}^^-
For the fint member is equal identically to tlie second member
with the addition of
But, by (1), thin is equal to
|if)/^.!(^|.
of which each term vanished; the first, or the double int^nil,
becausey by hypothesis, u is equal to nothing at eveiy point of iS',
and the necond, or the triple integral, because of (2),
(c) The second term of the second member of (3) is essentially pnipertrof
positive, provided a has a real value, whether positive, zero, or ^^'^'^
negative, for every point {x, y, ») within S. Hence the first g"""*"
member of (3) neceuarily exceeds the first term of the second
member. But the sole characteriBtio of J7 is that it satbfies (2). gninHan
Henoa U' cannot also satisfy (2). That is to say, P being any P«™""
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170 FRELIUNABT. [A (c).
ba drtwmi- one Bolutioii of (2), there can be no other solution agreeing with
it at ereiy point of S, but differing from it for some part of the
space within S.
mrai to (fQ One solution of (3) exists, satisfjing the ooudition that IT
'^^ has an arbitraiy value for every point of the sui&ce S. For let
U denote any fonction whatever which has the given arbitraiy
value at each point of jJj let u be any function whatever which
is equal to nothing at each point of S, and which is of any real
finite or infinitely small value, of the same fflgn as the value of
<-^^)/(-f),<-f)
dx dy dai
at each internal point, and therefbre, of course, equal to nothing
at every internal point, if any, for which the valae erf thia ex-
pT«sdoa is nothing; and let W = U+ 6u, where 6 denotes any
constant Then, uaing the formnln <^ (b), modified to suit the
altered drcnmBtancefi, and taking Q and Q" for brevity to denote
///{(•f)'H-f)'H4)'}-^^
and tlie corresponding integral for U', we have
-'■///{(•£)'H"S)"-(«£)'}^^
The coefficient of — 2l9 here is essentially podtive, in consequence
of the condition under which u is chosen, unless (2) is satisfied,
in which case it is nothing; and the coefficient <^6'ia essentially
poeicive, if not zero, because all the quantitiee involved are real.
Hence the equation may be written thus : —
where m and n are each positive. This shows that if any podtive
value less than n is assigned to 9, ^ is made smaller than Q •
that ia to say, unless (2) is satisfied, a fimctiou, having Uie same
value at >9 as U, may be found which shall make the Q inb^^ral
smaller than for U. In other words, a function V, which,
having any prescribed value over the surface S, makes the
int^ral Q for the interior as small as possible, must satisfy
equation (2). But the 1^ integral is essentially positive, and
therefore there is a limit than which it cannot be made smaller.
..Google
A {d).'] EIKEBU.TICS. 171
Hence there is a Bolution of (2) subject to the prescribed &arfiu» s<
conditioa, pc
(e) We have seen (e) that there ia, if one, only one, eolution
of (2) Bubject to the prescribed aar&oe condition, and now we
see that there ia on& To r©cftpittdate,^we conclude that, if
the Talne of ^ be glveii arbitrarily at every point of any cloeed ■ '
sur&M^ the equation ■
dx\ dx/ df/\ ay J dz\ ass/
determinefl ite Talue without ambiguity for every point within
that mrfaoe. That this important proposition holda also for the
whole infinite space without the stuface S, fullowa from the pre-
ceding demonstratioD, with only the precaation, that the difierent
functions dealt with must be bo taken as to render all the triple
laterals convergent. S need not be merely a single closed
Bor&ce, but it may be any number of surfaces enolaaing isolated
portions of space. The extreme case, too, of S, or any detached
part of S, an open shell, that is a finite unclosed sor&oe, is clearly
included. Or lastly, S, or any detached part of S, iaa,j be an
infinitely extended surface, provided the value of 1/ arbitrarily
assigned over it be so assigned as to render the triple and double
int^rals involved all convergent.
B. — Spherical Hakmonic Analysis.
The mathematical method which has been commonly referred '*lJ**^
to by English writers as that of " Laplace's Coefficients," but ^2?^°
which is here called spherical harmonic analysis, has for its
object the expression of an arbitrary periodic function of two
independent variables in the proper form for a large class of
physical problems involving arbitrary data over a spherical sur-
face, and the deduction of solutions for every point of space.
(a) A ipherical harm&niefwncUon is defined as a homogeneous DeBniUan
fimctdon, V, o(x,y,s, which satisfies the equation 2-!E.._..
dT <rr d*r . ,,,
. *?-^^*:^=** f*'-
Its degree may bo any podUve or negative integer; or it may
be feactional; or it may be imaginary.
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PHELDnKABT. \B (a),
EXAHFLES. The functiona written boloir are spldrical Iulp-
monics of the degrees noted; r repreeenting (s'+y'+s^": —
DeffTM Zero,
1 I *■ + '
tan-?; tan"' ?! log ^±5 ; '"(^; jj^.
OenersUy, in Tirtoe of (g) (15) and (13) below,
dV, dV, dV,
♦■-J-*! '"Tj t •'^I 1
if F, denote imy hannonlc of degree 0 : for instance, group TTT.
.below.
rx ex X /_ rx-tai\ _ 2»y _,y ar . r+s
iB*+y" a^+y" r+«\ ai'+y"/' ai'+y' at a?+y* ™r-s*
JTSL. _5L. JL. _2f^tan-'?+-^^lo«^^
at'+y" ic'+y" r+a' at'+y' x af+^^r-z'
Generally, in Tirtue of (g) (15), (13), below,
«.^_,(r"-'^'8.V;),
where Y, denotes anj spherical harmonic of int^^ral degree, j,
and $,, S,_,_| homogeneotiB int^^ functions of ^, j-, -j- ,
of degrees n and n—j—\ reapectiTelj : for instance, some of
group IL above, and groups T. and VL below,
,._,,_-,_,, d'tan"'?
^ (^ ') £.
«^'(^-)
-2V^1
rfy-
log-
=(-i)n.2...(t.-i)-
K+yO''
..Google
B (o).] KIITEMATICS. 173
BO the preceding yielda ^hT^'"
Bin
rf-'(f^-) cos"*
whM« ^ denotefl t&n~' - ,
Taking, in IV., j = -l,
„ 1 , r + z
K.= -log ,
where -7 denotes diflferentiation widi referraice to r on the aap-
poeition <rf z constant, and -r- differentiation with reference to
. z on aappoBition of x and y constant.
JDegree -i-1, or + i, ond (^fpe H{z, ^(x' + y*)} n^
ff denoting a homogeneooa functian; n any integer; and •
any positive integer.
Let U"' and F"" denote funciiona yielded by V. and VI. pre-
ceding. The following are the two* distinct functions of the
degrees and types now sought, and fonnd in virtue of (g) (15)
below : —
-*-i ^*' • -1-1 d:^*^ ' '
* Sea S (0 b«lo«.
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174
mpiNof or explicit!;
PBELIHINAfiT.
[B (a).
tbT'
ar"T-(4)"(N;4')](-/r=-.*
In the p&rticnl&rcBaeof »=0, these twoara notdiatdnct. EUtttor
p» =
'(')
IT-
Tlie o&er hBrmonic of Uie same dc^pree ftnd ^pe iB
^(N^
1 To obtain the harmonics of the same types, bat of degree t^
m, J multiply each of the preceding groups I. and IL by f"*', iu
I virtue of (tf) (13) below.
Deyree —1.
Qenerallj, in virtue of ig) (13) belov, any of the preceding
functions <^ d^ree zero divided by r; or, in virtue of {g) (15),
the differential coefficient of any of them with r^erence to x,
aty,oi». For instance^
- tan * ; - log : - tan ' ^ toe .
?T?' r(«' + y')' r(r+»)'
y . y* ■ y „
V+y" r(*' + y')' r(r + »)-
..Google
B (o).] KINEMATICS.
Degreeu -2 and +1.
fa y z
i.|?. p. ?; «. ». '■
( ? «
fa , r+s 2 , r+« „
_^-y_ toy rVrsQ _?r^iL
K+»0" (»■+?■)■' ("'+?■>■' («■ + »■)"
COS 3^ sin 3^ r*coa2^ r^sinS^
(the former being -r of III. 2 degree - 1, and the latter being
.-/(fe of VI. d^ree 0 wiUi n= 1).
The Rational Integral Harmonicg of Degrea 2.
I. Five distinct fimctioiu, for instance,
2«'-iB*-y'; a!*-!/*; y«; aa; asy.
' Or one fiinctjon with five arbitrary oonstants.
IL
(«e" + Jy* + ca^ + «y* +fai + gay,
where o + ft + o = 0.
Degree! — n — 1, and + n (n amy inttgor).
Wifli same notation and same references for proof as above for
Degree 0, group IV,
T- 8„,r„ J.r_„ or 8,„r,_..
n. s„w*.(»'"^'«.p;,.), and t**"«„^„(."^**'».r,^J.
..Google
luijriiioiiici.
176 PRELUDNAItT. [B (a).
Degreet e+rf, and — e — 1— ut
(v denoting V— 1, and « and/any real qnantitiea.)
i [(» + vy)*^"'* (* - vy)'*-^] ; ^„ [(« + ^Y^'^- <« - ^y+^H ^
or r*'^t^[{? + \fHl, 9**Wim[(e + ^)*],
where ? = %/{*' + *') ^i*^ ^ = taa"'-:
1
• j3»{*'»[coa{/l<«9'-e^) + uBin(/I(^y-e^)]
+ e-/>[coa{/ log y + e^) + i;ain(/ logy + e^)]};
■><«+^r)*]:
n. I the same with -^ -<- e^ instood of «^
^r-»- v|«j» [^coa (/log ? - e*) + V ain (/log J - 6+) j
+ <->[««(/log? + */.)+vBm{/log?+e*)]J.
(£) A tpherieal nurface hca-monic is the fiiaction of two
angular co-ordinates, or Bpherical Borface co-ordinates, which a.
clerical harmonic becomes at aaj spherical nurface described
from 0, the origin of co-ordinates, as centre. Sometimes a, func-
tion which, according to the definition (a), is simply a sfJierical
harmonic, will be called a gpheriaai gelid harmonic, when it
is desir«d to call attention to its not being confined to a nf^eHcal
Borfaoe.
(e) A compUfe tphtrical harmonie is one which is finite and
of single value for all finite values of the co-ordinates.
A partial harmonic a a spherical harmonic which either does
not continuously satisfy the fundamental equation (4) for space
completely surrounding the o^itre, or does not return to the
same value in going once round every closed curve. The
"partial" harmonic ia as it were a humonic for a part of the
spherical Bnrfiw}e: but it may be for a part which is greater than
the whole, or a part of which pottiona jointly and independently
occupy tlie same space.
..Google
B(d).-]
KUfEUATICS.
(d) It will be shown, later, § (A), ihat a complete spherical Aimbnic
h&rmouic la neceaoarily ^ther a rational integrftl function of the oampMe
oo-ordinates, or redudble.to one by a factor of the form °" '^
m being an integer.
(e) The genera] probli
naoet concisely stated thus :-
of finding hamionic functions Ib Ditfemittal
of dtKree ■.
To find the most general int^ra] of the equation
<Pu d*« cPtt „
tljr dif lis' * '
subject to the condition
'^ dx'^
,.(5),
the second of these equations being merely the analytical expres-
sion of the condition that u is a homogeneous function of x, y, s
of the degree n, which may be any whole number poeltive or
negative, any fraction, or any imaginary quantity.
Let P+vQ be a harmonic of degree t
real We have
■"/i", «, «,/b
(4*>';|*'a)<-P+>^)=- <•"/)(''■*■■«)■
dP dP dP
dQ dQ dQ
^ul*'7,-f^*'0
and
dx^
' ds dz
d d
•)'t/']« = 0.
..(5").
(f) Analytical expressions in Tatious forms for an absolutely Valoa of
general integration of these equations, may be found without vmboKcai
much difficulty ; but with us the only value or interest which
any such investigation can have, depends on the availability o
vol. I. 12
..Google
S PHELnOUART. [B if).
ite reeolte for solutiona falfilling the conditions at boDnding boi^
bitxa presented bj phyaioai problems. In a Tray large and most
important class of physical problems rf^arding space bounded by
a complete spherical surface, or by two complete conceu^c
spherical surfaces, or Inr closed surfiuwa differing very little from
spherical surfaces, the case of n any positiTe or oegataTe int^^,
integrated particularly under the restrictiim stated in {d), is of
paramount importance. It will be wcH'ked out thorotighly beloir.
Again, in similar problems regarding sections cut out of spherical
spaces by two diaiueti-al planes making uiy angle with one
another not a nA-midtiple of Ivo right aught, or regarding spaceB
bounded by two circular cones having a common vertex and
axis, and by the included portion of two e^iherical surfitceH
described from their vertex as centre, solutions for cases of
fractional and ima^naiy values of n are useful. lAstly, when
the subject is a solid <x fluid, shaped as a section cut from tiie
last-mentioned spaces by two planes through the axis of the
cones, inclined to one another at any angle, whether a snb-
multiple of ff or not, we meet with the case of n either int^ral
or not, but to be integrated under a restriction differing from
that specified in (i^. We shall accordingly, after investigating
general expreBsious for complete spherical harmonics, give some
indications as to the determination of the incomplete harmonics,
whether of fractional, of imaginary, or of integral degrees, which
are required for the solution of problems regarding such portions
of spherical spaces as we have just described.
A few formnl», which will be of constant use in what follows,
are brought tc^ether in the first place.
(^) Calling 0 the ori^ of co-ordinates, and P the point
X, y, z, let OP = r, so that le" + y* -i- e* = r*. Let 8, prefixed to
any function, denote its rate of variation per unit of space in
the direction OP ; so that
. X d y d z d ...
o = --i--^-■3-+- -T- (bi
r dx T dy r dz
If H^ denote any homogeneous function oi x,y,z of order n, we
have clearly
IH, = -H, (7);
whence x—r^ + y—r-^ + z^-f = nff_ (6) or (8>,
dx ay dz ' > / v />
..Google
B ((?).] KINEMATICS. 179
the wQlI-koown differmtial equation of a homogeneous funoUoa; T^^'"*
in which, of couTse, n may have any value, positive, int^ral,
negative, fractional, or imaginary. Again, denoting, for brevity,
-j-Tj + -j-j + -jj by v', we hare, by differentiation,
T'(0-»('»+l)''- (9).
Also, if u, u' denote any two functions,
• in II -fdudvi dudu' du du\ , , ,,..
'<"»>"^" + K5iS*5i;Si;*a,E-)*«vV (10);
whence, if u and u' are both solutions of (4),
• / <. „/eft*rfu' dudu' diidv,\ „,,
''<~'=He5j*^^*e&) <">'
or, by taking u = F^ a harmonic of degree n, and uf = t^,
or, by {8) and (9),
V'{r-V.) = »»(3« + m + l)T'-r, (12).
From this last it follows that r'""' r, is a harmonic ; which,
beioK of degree — »— 1, may be denoted by F_,_,, so that we
have
^■=? <•').
if n + f.' = -l j
a formula showing a reciprocal relation between two solid har-
monics which give the same form of surface harmonic at any
spherical surface described &om 0 as centre. Again, by taking
m = — 1, in (9), we have
V'l = 0 (H).
Hence - is a harmonic of degree — 1. We shall see later
g (A), tbat it is the only eompltU harmonie, <^ this degree.
If u be any solution of the equation v*it = 0, we have also
12-2
..Google
} PRELIMISARY. [B (^}.
liiid BO oa for an; number <^ differeDtUtiona, Hence if V, is
Ag&in, we have a most important theorem expreased hj thi?
following equations : —
JfSiSida=0 (16),
where dw denotes an element of a spherical sur&ce, described
from 0 as centre with radius nnfty ; // an integration over the
whole of this surface ; and Sf, Sg two complete surfiM» harraonics,
of which the degrees, i and i', are neither equal to one another,
nor such that ( + t' = — 1. For, denoting the solid harmonics
t^Sf and T^Sg by F( and T^ for any point (a:, y, a), we have, by
the general theorem (1) of A (a), above, applied to the space
between any two spherical surfaces having 0 for tbeir oommon
centre, and a and a^ their radii ; —
^^^\dx dx dy dy dz ds )^
But, acconling to (7), 8Ff=* Ki-, and SFiB-Pi. And for the
portions of the bounding surface coastitated by the two spherical
surfaces reapectively, da = oVw, and da = n,'dw. Hence the two
last equal members of the preceding double equations b
to satisfy which, when % differs fifom t", and a'*''*' from a,***'*',
(16) must hold.
The corresponding theorem for partial harmonics is this : —
Let Si, Si, denote any two different partial sur&ce harmonics
of d^rees t, t, having their sum different from — 1 ; and furtiicr,
fulfilling the condition that, at every point of tbe boundary of
some one port of the spherical surface either each of them
vanishes, or the rate of variation of each of them perpendicular
to this boundary vanishes, and that each is finit« and single in
its value at every point of the enclosed portion of surface; then,
with the integration // limited to the portion of surface in
..Google
B (jp).] KINEMATICS. 181
questioD, equation (16) holds. The proof differs fi-om the Eitmnion
preceding only in this, that instead of taking the whole space otCLsptacB
between two concentric spherical surfaces, we must now take lumioiiwi.
only the part of it enclosed by the cone having 0 for vertez, and
containing the boundary of the spherical area considered.
(A) Proceeding now to the inrestigation of complete harmonica, Jf "■J'*»- .
we shall firat prove that every such function id either rational and oompieto
integral in terms of the c»ordiaatea z, y, x, or is made so by
a factor of the form r".
Let Fbe any function of z, y, z, satisfying the equation
^"^=0 (17)
at every point within a spherical surface, S, described from 0 as
centre, with any radius a. Its value at this suriace, if a known
function of any arbitrary character, may be expanded according
to the general theorem of § 31, below, in the following series : —
(r = o), V=S,^S, + S,-i- +^i + etc (18)
where •$,, S^,...S, denote the surface valuee of solid spherical
harmonics of degrees 1, 2,...t, each a rational intend function
for evray point wiUiiQ S. But
iS. + jS,- + jS,t + ... +jS,-, + etc, (19) HBrmonio
Qreen'i pu-
is a function fulfilling these conditions, and therefore, as was blembrtbe
^vved above, A(c), V cannot differ from it. Now, as a parti- aspheriai
cular case, let T be a harmonic function of podtire d^ree t,
which may be denoted i)j S^ - : we mast have
This cannot be nnleeo i = ^ S, = iS„ and all the other functions
^,1 'S'l, S„ etc, vanish. Hence ttere can be no complete spheri-
cal harmonio of positive d«Free, which is not, as A — , , of integral Oomptot*
d^ree and an integral rational function of the co-ordinates. degrm
Again, let F be any function satisfying (17) for eveiy point ncioiud and
without the spherical suriace S, and Tanishing at an infinite dis-
tance in every direction; and let, as before, (18)ezpress its surface ^rmcmlo
value at S. We similarly prove that it cannot differ from Orem'ipny
^*^* ^* +-^j7T^+ew i-u;. ,p|„riwl
jiGoogle
2 PBELmiKABT. [B (&).
Hence if, as a particular case, V be xaj complete harmonic
— -, of negative degree k, we muat liave, for all pointa oat-
side^,
r'S. aS, a'S, <^S. a'*'S, ,
. = — !+ -5-!+ —.1+ +__j-' + etc,
wHch requires that K=-(i+l), ^i=jS,, and that all the otherfiuio-
tions S^, S^ , Sf, eto., vanish. Hence a complete spherical hannonic
of n^ative degree cannot be otlier than --^tr > or -^a+liS'^,
where S^ ix not only a rational intend function of the co-
ordinates, as asserted in the enunciatloD, but is itself a spherical
hannonic.
(i) Thus we have proved that a complete spherical harmonic,
if of positive, is necessarily of integral, degree, and is, beeides, a
rational integral function of the oo-ordinates, or if of negative
y
degree, -{i+ 1), ia necessarily of the form -^-\, where Fj is
a harmonic of positive degree, t. We shall therefore call the order
of a complete spherical harmonic of negative d^ree, the degree
or order of the complete harmonic of positive d^;ree allied to it;
and we shall call the order of a sui&oe harmonic, the d^ree or
order of the solid harmonic of positive degree, or the order of the
solid harmonic of negative degree which agrees with it at the
spherical surface.
(j) To obt&iu general expressions fbr complete spherical liar-
monies of all ordera, we may first remark that, inasmuch aa a
constant is iha only rational integral function of degree 0, a com-
plete harmonic of degree 0 is necessarily const&nt. Hence, by
what we have just seen, a complete harmonic of the degree — I
is necessarily of the form - . That this function ia a harmonic
we knew before, by (14).
Hence, by (15), we see that
di"" 1 1
-'-' d^dfdJ ^^^.Jf^^l I (21),
if jt» + i=i J
where r^,_i denotes a liarmonic, which is clearly a complete
harmonic, of degKO - (t + 1). The difiiiFeiitial coefficient hera in-
jiGoogle
B {j).] KINEMATICS, 18S
dicated, when worked out, is easily found to be a fractiun, of which B; difl^
'I' ' entiklion ot
the numerator is a rational int^;ral funotion of degree i, and the humonie ot
denominator is r^*'. Bj what we have jiutt seen, the nume-
rator must be a harmonic ; and, denoting it by F,, we thtu have
'■■-'*■ S^F m-
The number of independent harmonica of order i, which we Numbsr of
can thufl derive bj differentiation from -, is 2i + 1. For, althou^ Sn^'^
fj . a\/.- . i\ Jit±ti'
ther* are -
which j + k + l=i, only 2t + 1 of these are independent when -
is the subject of differentiation, inaamut^ sa
($*$^S)l">-- ("^
M being uij iutej^, nnd shows that
.(-l)i
ixH,fMT > ' ir'i/Vii' V/ ■• l...(24).
Hence, hy taking / = 0, and j + i = ^ in 4^e first place, we have
ti '™*1
j + iD» — 1, we have \ varieties of-j-
in all 21 + 1 varieties, and no more, when - U the subject. It ia
easily seen that these 2r + 1 varieties are in reality independent
We need not stup at present to show this, aa it will be apparent
in the actual expansions given below,
Now if Hf (a^ jr, £) deuoto any rational integral function of
z, y, 2 of d^:ree i, V'iT^ (a;, y, «) is (tf degree t — 2. Hence since
in JI^ there ai'e i— -ii '- terms, in Vifj there are *"'^— ' ,
DigilizedbyGOOgle
184 PRELIMINARY. [B (J).
Complete Hence i£v'S,=0, we have ^ — - equationit amone the constant
hMuionio of 2 ^
inrnti^ed coefficientB, and the munber of independent cxmstanta remaining in
consttuita in the general rational integral harmonic of d^p'ee i.
But we have seen that there are 2i + I distinct varieties of dif-
ferential coeffidents of -- of order i, and that the anmerator
of each ia a harmonic of d^ree i. Hence every complete har-
monic of order t ia expreeeible in terms of differential coeffidente
of - . It is impossible to form 2i + 1 functions aymmetrically
among three variables, except when 2t + 1 is divisible hj 3; that
is to say, when t = 3n + 1, n being any int^er. This class of
cases does not seem particularlj interesting or important, but
here are two examples of it
Example 1. i=l, St + 1 = 3.
The harmonics are obvionsly
^1 ^1 dl
dxr' rfy r ' dzr'
Formula (26) involves xsii^idarlj, and sand yBjrmmetricall;,
for every value of * greater than unity, bat for the case oi i=l
it is essentially symmetrical in respect to x, y, and 2, as In this
case it becomes
-(a ~ A - + S ^-
Example^. t = 4, 2t+l = 9.
Looking firvt for three diSerential ooeffidents of the 4th order,
singular with respect to at, and symmetrical with respect to
y and « ; and thence changing cyclically to yas and ctcy, we find
ff d* d*
di/'d^' dxdj^' d^^'
d* d* d*
d^^' dyd^' dydaf'
d* d* d*
da^djf" did^' dt^'
DigilizedbyGOOgle
BO)-]
KINE1U.TICS.
Them nine differentiatioiiB of — ore eaaentially distinct and '^
give ns therefore nine distinct harmonics of the 1th order formed ^'""'
^^nunetrically among x, y, z, Sy putting in them for -jj,
that it is - which is differentiated, and for ^ , its equivalent
"* rf~ (rf^ *" T>) ' *® ™*y P*™ irova them to (25).
But for every ralne of i the general harmonic may be exhibited
aa a fiinotion, with 2i + 1 constants, involving two out of the
three variablee aymmetrically. This may be done in a variety
of ways, of which we chooeo the two foEowing, as being the
moat useful : — First,
+■«■.
0«ti«ntlai-
DomphM
lunnonic [>f
(25).
Secondly, let x + i/v = (, x — yii = ti
where^ u formerly, v ia taken to denote V— 1.
ThiegiTe. «-Kf+* y=^(i-*'
• (26).
..(27):
r («, + «■)»•
where [x, y] and [(, ij] denote the same quantity, expressed in
terms of te, y, and c^ (,-ii reqtectively. From these we have,
farther,
..Google
PBGLDUNABT.
or, aocotrding to our abbreviated notation,
I
V' = 4;.
[BO")-
..(29).
Hence, u 7'*^= 0, if F denote - or anj other oolid harmonic,
i^'-'m'' w
TTsing (28) in (25) and taking ffl,, R,, »,, B^, to denote
another set of coefficients readily expressible in terms of
A^, J„ J„ J,, ... we and
{'•(i)'^«.ari^«.(ir(0--«.a)'}i i
{».(r-.(rri--(r(i)---"©"}|ir-'"'-
^D The differeatiations here are performed with gt«at esse, hj the
^ aid of Leibnitz's theorem. Thus we have
d$''di)' i
[«"-lT(^
=(-r-i.|.
^F
and
m(.n-l)..(»-l) ._ I
1.2.(oH-»-J)(,«.»^)'' « r'-e'cj
dfWSf"'"' i-5-5-('»+»+l).2>
[.M "•'» ■-i>.-i i m(M-l).«('n-l) _ .. , 1
This expression leads at once to a real development, in terms of
.(31)
polar co-ordinates, thuf
» = r<x»6, 3; = r sin^oos^ .ysrsintf
sothat £ = rsintf«"*, yj = r ain 0*-** .,
Then, since fij = !C*+y'=r*Bin'tf,
and
fV = (6))"f'=(f7)"(*"s™^*C«w*+v8in^)' = (r8infl)"*"(oo««^+vsin»0),
irhertf t = n-m; and if, farther, we take
-(32);
■ (33).
■ (34)
..Google
B m
L l.(m+»-J)
metHa]
■* ''(A. «w #^ + A,' ain »♦)
1.2.{»»+n-i)(/«+n-^)
■ di'drfiiz r " tii'di}'dz r
m(».-l). »(»-!)
(35)
Setting aode now constant fkotora, which Iiave been retained
hitherto to show the relatione of the expressions we have iavesti-
gated, to dlSerential coefficients of - ; taHng S to denote sum-
mation with respect to the arbitrary conatonts, Ai A't Bt B'i
and putting iaji0 = v, txe6 = fi; we have the following perfectly
geueral expression for a complete surface harmonic of order t : —
St=S (A.cos«^+A.'sia«*)»^j+S (B,«»»^B,'Bin»^)/i^t^j...(36)
where *= m ~ n, and
"*** M*t-t *»("»— !).«(«- 1)
1. {»»+»- J) 1.2.(nt+»-j)(m+»-|)
while Z^^ diffen from ©,_,^ only in having m+n+1 in place
of «* + n, in the denominators.
The formula most commonly given for a spherical harmonic
of order t (Laplace, Mecamique CeleeCe, livre in. chap, ii., or
Murphy's Eleetricity, Preliminary Prop, xi.) is somewhat simpler,
being as follows : —
--*^*"
■*-etc.
^t = S (A4 COB 9^ + B, sin «0}er
■ (37).
»_.L--(*::fHi-j-i)
'" L 2.(2t-l)
2.4.(2V-l)(2*-3) ''
..(38),
..Google
3 PIlEUMINJUtr. [B (j).
where it may be remarked that ^^ means tlie same as
(-1)~8^__^ if »n + n = t and m~n = t, or as (-1) " /t^„^,| if
m + n + l^i and m~n^(. Formula (38) may be derived
algebraically from (36) by putting ^(1-/1*) for v in ®[«.^-i->^
and in Z^^ mi'*'*'f^'- '^'^^ ™^7 ^^ obtained directly l^ the method
of differentiation followed above, varied suitably. But it may
also be obtained by assuming (wiUi a, and b, as arbitrary
constants)
whicb is obviously a prc^r form ; and determioing p, q, etc, by-
the differential equation v* F, = 0, with (39).
Another form may be obtained with even greater ease, thus :
- F, - %{a,e + 6,V)(*^ +?'.*'*'-'fl +p/-'-'(y + etc.),
and determining p^, p,, eta., by the difierential equation, we
have
r,.3{.^'*M) ['"-''r.'^,'l"i|.V '^'(1
,(i-.)(i-.-l)(i-.-i)(i-
Sa-^iV
I'-etel
(39)
4'.(.+ l)(. + 2).l.
which might also have beoi found easily by (he differentiation of
— . Hence, eliminating imaginary 8ymb(d8, and retaining the
notation of (37) and (38), we have
i.(,-
(i-.)(i-.-l)(i-.-2X.-— 3) ,.^.. . ^-j^
m
where (2.^.1)(2. 1 2)...(it.)
wlere "- -(2, + l)(2.+ S)...(«-l)-
This value of (7 is found by comparuig with (35). Thna ve aae
ibai C uiuGt be equal to the numerical coefficient of the laat
t«rm of (35), ineapectively of eign. Or C ia found by comparing
(40) with (38) : it ia equal to the coefficient of the laat tenn of
(38) divided by the coefficient of the Uat term within tha
b»cketB of (40). Or it ia found directly (that ia to aay,
jiGoogle
B ( j).] KIMEMATICS. 189
iadependentlj of other etjuiTalent formulas) thus : — We have, 1
by (29'), ■
d^-'d^ r
'=i-y^-
or =(-) ' 2'—' ,Z, ,«-i'. if t-« is odd.
Expanding the first member in terms of 2, (, yj, by successive
difTerentiittioii, with reference first to i;, a times, and then z,i-8
times, ve find
(-)'J.}...(.-J)(2,*l)(2.*2)(2,*3)...(i+,y-S-....(42),
for a term ia its numerator : comparing this with (39) and (40),
and the second numbei- of (41) with (35), we find C.
(A) It is very important to remark, first, that imponuit
}{u,u;d<,=o ("),^^'Jy°'
where U, and U! denote any two of the elements of which Fis tunciiniis.
composed in one of the preceding expreedons; and secondly, that
f M (1 .
»-0 (44),
the case of i-t' being of course excluded. For, taking r = a,
the radius of the spherical surface; and dir=a'din, as above;
we have dm - sin 9d6d^, etc., the limits cS 0 and ^, in the inte-
gration for the whole spherical surface, being 0 to ir, and 0 to 2*",
respectively. Thns, since I cos«^cob«'^ d^ = 0, we see the
tmth of the first remark; and from (16) and (36) we infer the
seccmd, which the reader may verify ^gebralcally, as an e
[I) Each one of the preceding series nay be taken by either Bipuuioos
end, and used with i or ^ either or IxAh of them negative bulLniici
or fractional or imaginary. Whether finite or infinite in its ^"^Hl^gn.
number of terms, any «eries thus obtuued expresses when
multiplied by r* a harmouic of degree t; since it is of degree i,
and satisfies 7' F| = 0. In any case in which one of the pre-
cediog series is not finite, the formula taken by one end gives
a convetging series; taken by the other end a diverging series.
Thus (40) taken in the orJcr shown above, converges when 6 is
between 0 and 45°, or between 135* and 180°; and taken with
the last term of that order first it converges when 6 > 45° and
..Google
)0 PBELMINART, [B {I).
-ciaS". ThuB, again, 0,^., and X,^,, of (36), being each of
diiBTBB a finite Dumber of terms when either m or n is a positive
^" iabegot, become when neither is so, iufinite series, which diverge
when f < 1 and converge when y>\. These two series, whether
both infinite or one finite and the other infinite, when convergent
are bo related that
/^(«-i.--o=y^Tei-.«> (36'),
as is easily verified for a few terms by multiplying ^{m-i, ■— jj
by the expansion of (l — t) in ascending powers of -^. But
expan»ons in ascending powers of -^ are of comparatively little
interest, as they are divergent for real values of B, and therefore
Finding or not available for the proposed physical ^tplicatJons. To find
ai|)4Didoiu. expansions which converge when r<l take the last t«mis
of (36) first. Thus, if we put
,._ ,_,. fn{m^l)...(m-n^2)(m-n+l)M^-l)...1iA .
^ ' 1.2....(n-lJn.(f«+n-i)(»»+n-|)...(m+5K™+l) ^ ''
supposing m to be > n, and » to be a positiTe int^ra-, we find
e,*---*-'^ L^ {S^::;rri).i *^^{m-«+i)(w»-n+2).i.2'^'^J -^^^ ^-
Writing down the corresponding expresdoa for ^(m-^B-u
from (36), and using (36'), we find
0 - r«v— Fl ("-iX'^+l).^, (>^i)(»-l).(*H-H)(m-t-2) 1
®'-''-^'"^ L^-(m-«+l).l'^^(m-n+l)(w=ir+2).0*"'*^J -^^^ ^
This expansion of 6^.| is derivable algebraically from (36'") by
multiplying the second member of (36'") by
-'(>H-'*0'--.'o-)
(which is equal to unity). Both expansions converge when
v'< 1, or, for all real values of 6 ; just failing when 6 = ^x,
In choosing between the two expansions (36'") and (36''), prefer
(36") when n difiers by IcBS than J from zero or some positive
integer, otherwise choose (36'"); but it is chiefly important to
have them both, because (36") is finite, bnt (36"') inSnite, when
2j-l
n= -w—; and (36'") is finite, but (36") infinite, when n=j- 1 ;
j being any positive integer.
..Google
B (0.] KINBMATICS. 191
FntBOT m + n = i, m—n = *, 1
or *» = i(* + '). «=i(t-')- (36*)
and denote b; f u" J
■wktA the second membera of (36'") tad (36") become vith these
Talnes for m and n. Again, put
..(36-)]
or m = l(t-«), n=i(i-i-.), I i^^")
and denote by Xf " J
what the second membeis of (36'") and (36") become with these
▼aluea for m and n. We thus have two eqnal convergent aeries
for u and two eqnal convergent aeries for v , and u , v are
functions <^ v (or of 6) snch that
«^ (^ cos 1^ + J sin »^) 1
and t>|^ ( J cos 1^ + ^ sin «^) |
are sorfaco harmonics of order t.
The first terms of u and v are t^ and y~', or /i»' and fir'*,
according as they are taken from (36'") or {36'"), and in general
u and V are distinct from one another.
Two distinct soIntionB are clearly needed for the physical
problems. But in the particular case of < an integer, u and «
are not distinct. For in this case each term of v after the first
t terms has the infinite factor — — ; thus if (7j denote the coeffi-
cient of the (j + l)"* term of «"*, the first » terms of . ' vanish
when e is an int^er, and those that follow constitute the same
serivs as that expressing m|", whether we take (36'") ov (36"^,
For the case of « an integer tiie wanting solution is to be found
by putting
«,«>^
iiL-
when cr = 0 :
. (36'")
thus foand is snch that
v>^ (A cos «^ + £ sin i-ft)
..Google
inwHvnding
pavenof fk
where,
192 PBELUflNABT. [B (0-
is a Burface harmonic of order t distinct from u'*'. The first
term of to , according to (36'"), ia v* log v, or fii^log v according
to (36'*), and Bnbsequent terms are of the form (a + blog v) v*, or
(a + 6 log f) /!>*, j being an integer. The cinminstanceB belcmg
to a well-known class of cases in the solution of linear dif-
ferential equations of the seoond order (see § (y*) below).
Again, lastly, remark that (38), nnlees it ia finite (trbich it ia
if and only if t - « is a powtive integer), diveigee when /» < 1
and converges when /<> 1, if taken in the order in which it is
given above. To obtain series which converge when >i<: 1
(that is to say, for real values of 6), reverse the order of (38)
for the cose of t — « a positive integer. IliaB, according as t — <
is even or odd, we find
l).(t+«4-l)fa»t-3) ,
"-{-)'
|iH-rt .
2.4...(i-
Jh'W-
-W
-l)(>-»-2)(i-«- 3).. .4.3.2.1
-»).(2i-l)(2i-3)...(i+.t3)(i+.+ l) I
I ...(38'),
. (38-).
where^ «-■ bdng odd,
- 1-' 2.4...(w-3)(w-lj.(2i-i)(2i-3)...(i«+ij(.«t2) J
Tiiea, whatever be t - «, or t, or s, integral or fractional, positive
or negative, real or imaginary, the formolas within the brackeU
I } are convergent scries when they are not finite integral func-
tions of p- Hence we see that if we put
n "'" 1.2 '^ 1.2.3.4 *" "®
-l).(«4-»^2) (i-»-l)(*-
3).(i4-a4-2)(f+«+4) ,
2.3.4.5
= A^ + A^n' + A^fi* + ius.,
- A,ii + A ^li' -i- A^fi.' -^^ ix.,
(38-)
where Ag=l, ^,= 1, and .^^,,= 1 — ^
illll).
(n+l)(n+2)
..Google
B (().] KINEMATICS. 193
the funotions p^, q^ thofl expressed, whether they be algebraic ^'^'^JJEfj^,
or traaBoendental, are Bucb that hS^SSSi^
p" (J COB «^ + 5 sin 8^)1^, \
■• (38"),
«... , „ . .. . , >'8')
and - ■
are the two sar&ce harmonicB of order t, and of the fonn
/(fl) t^. For example, if t - a be an even integer, pj"' is Uia
finite fnnction with which we are familiar as giving a rational
inb^^ Bolution of tlie form (38*), and ^ gives the solution of
the same form which is not intof^ or rationaL And if t - *
is odd, q^ gives the femiliar rational integral solution, and p^
the other solution of ihe some form but not integral or rationaL
The correBpondii^ solid harmonics of degrees i and — t — 1 are 9"
obtained by multiplying (SS*) by t* and r"'"'. Reducing the *"
latter from polar to rectangular co-ordinates, we find them f£ the
form
[r---"-'f.;"">--v.^.].g.fert
\r~'~'~'z - etc.] B, (x, y)
where IT, denotes a homogeneous function of degree «. Now {15)
— of any solid harmanio of d^^ree — t is a solid harmonic of
degree — t — 1. Hence
are sur&ce harmonica of order t, and they are clearly of the first
and second forms of (38'). Hence, putting into the forms
shown in (38") and performing the indicated differentiation for
the first term of the q function and the first and second terms of
the p fonction, so as to find the numerical coefficients of r~*~'~'
and r''~'~'z in the immediate results of the differentiation, and
then putting /ir for ^ we find
TOL. I, 13
..Google
PSBUHINART.
i['-"'"X-,l
[B(i).
...(38^.
To reduce back to polai* c<K>rdmateB pat for a moment
a^ + y' = a'. Then we have
i of (SS*"), we have
Pt =
and
r{.^-pr.
*).
[Compare § 762 (5) below.]
Supposing now t and t to be real qnantities, and going back
to (SS*^, to inrestigate the oonvei^ency of tbe aeries for p^ and
9^ , we Bee that, when n la infinitely great,
Now if (l-/)- = SSj.".
we hav^ h; the binomial theorem,
B,= l, B, = 0. and^'=I + ?i^.
Hence, when fi = ^ (1 - «), where a is ui infinitely small pocdtiTe
quantity,
p^^ = Q or =
and
Bocordingag
^"►''^O .
Hence if t > a, the quantities wi<iiiii the brackets under
— in (38'"') vanish when ^ = ^ 1 ; and sa tiiey Taiy con-
..Google
B(!).]
tinnouslr, and within finite limits, when a. is continnousl; Atqnintlnn
a from -\ to +1, it follows that p Taniehes one time 'I'eof*"'*"-
more than does 5' , and q one time more than does p^_^. Now
looking to (38'"), and supposing (as we clearly may without loss
of generality) that ( is positive, we see that every term of p
is positive if »<« + !. Hence if t is any quantity between jfj^f^"*"
a and s + 1, ^*'p'* vanishes ^rhen ^ = ^1, and is finite and^^^S'.'
positive for every intermediate value of ^
H^ice and &om the second formula of (SS""), q vanishes
just once as /t is increased continuously from — 1 to + 1 ; thence
and from the first of (38*^, p* vanishes twice ; hence and from
the second again, q vanishes thrice, and bo on. Again, as the
coefficient of every term of the series (38'") for 5"' ia positive te^°S^
▼hen t < s 4- 1, this is the case for q , and therefore this func- order hu '
tion vanishes only for fi = 0, as fi is increased from —I to + 1. <""*'
Hence p'^ vanish^ twice; and, then, continuing alternate ap-
plications of the second and first of (38'"), we see that q"^
vanishes thrice, p four times, and so on> l%ua, putting all
t<^ther, we see that 7'" has j or j+1 roots, and p"* has
j+1 or j roots, according as ^ is odd or even ; j being any
intf^^ and i, as defined above, any quantity between g and
« + 1. In other words, the number of roots of p"^ is the even Oeruni'or
' m roola of tai-
numbnr neit above i — »; and the number of roots of y, is the J^. '^j
odd number next above i-s. Farther, from (38'"') we see that "^ '"**■
the roots of p lie in order between those of y , and the roots
of y*" between those of p^^^, [Compare g (p) below.] These
properties of the p and q functions are of paramount importance,
not only in the theory of the developmeot of arbitrary fiinctions
by aid of them, bub in the physictd applications of the
fractional harmonic analysis. In each case of physical ap-
plication they belong to the foundation of the theory of the
ample and nodal modes of the action investigated. They
afibrd the principles for the determination of values of i— a,
which shall make Q or ^nd vanish for each of two stated
13—2
..Google
•phcrinl
two plans
mntlnRlD
I PBELIMINABT. [B (0-
valuea of 6. Tim ia an Analytical problem of lugh interest in oon-
nezion with those extensions of spherical harmonic analysis : it Ja
essentially involved in the physical application Teferred to above
where the spaces concerned are bounded partly by coaxal cones.
When the boundury is completed by the intercepted portions of
two concentric spherical surfoces, functions of the class desciibed
in (o) below also enter into the solution. When prepared to
take advantage of physical applications we shall return to the
subject; bnt it is necessary at present to restrict ooiselves to
these few obaervationa.
(m) If, in physical problems such as those already referred
to, the space considered is bounded by two planes meeting, at
any angle -, in a diameter, and the portion of spherical surface
in the angle between them (the case of a <: 1, that is to say, t^
case of angle exceeding two right angles, not being excluded) the
harmonics required are all of fractional degrees, but each a finite
algebraic function of the co-ordinates £, >;, s if a is any incom-
mensurable number. Thus, for instance, if the problem be to
find the internal temperature at any point of a solid of the shape
in question, when each point of the curved portion of it« sur&ce
is maintained permanently at any arbitrarily given temperature,
and its plane sides at one constant temperature, the forms and
the degrees of the banaonics referred to are as follows : —
» + 2,
<+3,
2s + 1,
2a + 2,
2> + 3,
HuraoDlc
D«gl».
i'
3.,
3.+ I,
dz r^*'
These harmonics are expressed, by various forraobe (36). ..(40),
eta, in terms of real co-ordinat«8, in what precedes.
(n) It is worthy of remark that t^ese, and every other spherical
harmonic, of whatever degree, integral, real but fractional, or
..Google
B (n).] KINEMATICS. 197
imagmaiy, are derivable by a general form of process, wbicb in-
dudes differentiation as a particalar case. Thus if | -r i denotes doiTed
'^ \rfij/ trom that at
an operation which, vhen a ie an int^er, constitntes taking the brninml-
<*^ differential coefficient, ve have clearly gutboion.
where P, denotea a function of a, which, when < is a real integer,
becomes (-)'i|-f-(«-i)-
The investigation of this generalized differentiatioa presents
difficulties which are confined to the evaluation of P„ and which
have formed the subject of higbly interesting mathematical In-
veeUgations by Liouville, Gregory, Kelland, and others.
If we set amde the factor P., and satisfy ourselves with deter- 1
minations of/orma of spheric^ hannonics, we have only to apply hi
Leibnitz's and other obvious formula for differentiation with any bf'sDramoii
fractional or imaginary number as index, to see that the equiva- vithKencr.
lent expressions above given for a complete spherical harmonic dice*.
of any degree, are derivable irom - by the process of generalized
differentiation now indicated, so as to include every possible
partial harmonic, of whatever d^ree, whether integral, or
fractional and real, or imaginary. But, as stated above, those
expressions may be used, in the manner explained, for partial
harmonics, whether finite algebr^c functioRS of f, ij, a, or tran-
scendento expressed by converging infinite series; quite irrespeo-
tively of the manner of derivation now remarked.
(o) To illustrate the use of spherical harmonics of imaginary bugtaiHr
d^rees, the problem regarding the conduction of heat specified roiiAai
above may be varied thus :— Let the solid be bonndod by two romitloiu
concentric spherical surfaues, of radii a and a', and by two ajfrtnti.
cones ta planes, and let every point of each of these flat or
conical sides be mainbuned with any arbitrarily given distribution
of temperaturo, and the whole spherical portion of Uie boundary
at one constant temperature. Harmonics will enter into the
solution, of degree
1 >V£I
jiGoogle
Iha ohume-
numbmor
S PRELIMINABT, [B (o).
where y denotea any integer. [Compare § (d") below.] Converg-
ing series for these and the otJiers required for the eolation
Bie included in our genor^ fbnuulas (36). ..(40), etc.
(p) The method of finding complete spherical harmonics by the
difierentiation of -, investigated abov^ has this great advantage,
that it shows immediately very important propertiee which they
possess with reference to the values of the variables for which
they vanish. Thus, inasmaoh as - and all its difierential coeffi-
cients vanish for a; = ■*■ oo , and for y = * oo , and for a = * co ,
it follows that
vanishes j times when x is increased from -
I
Ripraalon
tru7 fuiio-
[Compare with the investigation of the roots of p^ and q* in
g if) above.]
The reader who is not familiar with Fourier's theory of equations
will have no difficulty in verifying for himself the present appli-
cation of tlie principles developed in that admirable work. Its
interpretation for fractional or imaginary values of 3, k, I ia
wonderfully interesting, and of obvious value for the physical
applications of partial harmonics.
Thus it appears that spherical harmonics of large real degcws,
integral or fractional, or of imaginary degrees with lai^ real
parts {a + 0 tj— 1 , with a largo), belong to the general class, to
which Sir William it, Hamilton has applied the desdgnation
" Fluctiiating Functions." This property ia essentiaily involved
in their capacity for expressing arbitrary functions, to the
demonstration of which for the case of complete harmonics we
now proceed, in conclusion.
(r) Let 0 he the centre and a the radius of a spherical
sur&ce, which we shall denote by S. Let P be an; external or
internal point, and let / denote its distance from 0. Let t&r
(lenote an element of <?, at a point E, and let EP-D. Then,//
denoting aa integration extended over S, it is easily proved that
..Google
B (r).] KINEKATIC8. 199
eternal to S
(«).
and
js= ■? fi — li when F is extenxal t*
^w%
This is merely^ a particalar case of a very general theorem of
Green's, included in that of A (a), above, as will be shown when
we shall be particnlarlj occupied, later, with the general theory
of Attraction: a geometrical proof of a special theorem, of which
it is ft case, (§ 474, fig, 2, with P infinitely distant,) will oocnr
in connexion with elementary investigations regarding the dis- .
tribution of electricity on spherical condaotors: and, in the
meantime, tiie following direct evaluation of the int^i;ral itself
is given, in order that no part of the important inveat^tion
with which we are now engaged may be even temporarily
incomplete.
Choosing polar co-ordinates, 6 = EGP, and ^ the angle be-
tween the plane of ECP and a fixed plane through CP, we have
d<r= a* foaB d6 di^
Hence, by int^ration from ^ = 0 to ^ = 2ir,
il%''<^L
But 7)' = o'-2a/co8ff+/*;
and therefore sin 6dS = — -r- ;
af
the limiting values of 2> in the integral being
/—a,f+a, when/>o,
and a-/ o+/, when/<o.
Hence we have
in the two cases respectively, which proves (45).
(s) Let now F(S) denote any arbitrary fonction of the position SdntlMi al
otS oaS, and let raobian
«=//<^^=^^ (.e.g'l
When^is infinitely nearly equal to a, every element of this in- fnt"*™!-
tegtal will vanish except those for which D is infinitely small
..Google
) PREtnONABT. [B («).
Henoe the mtegral will have the same value aa it would have if
tor cue of F{E) had ererywhere the game value as it haa at the part at S
nearest to P ) and, therefore, denoting tliis value of the arbitral^'
function by FiP), we haTe
ff(/--«-)Ar
■phfliiokl
=^m//«^
when /differs infinitely little from a; or, by (4fi),
u=ivaF{P) ;. (46').
Now, if 6 denote any positive quantity lees thui onity, we
have, by ejq>ansion in a convergent series,
ard;?7?ji-'*«''*«-''*'*<' <"»•
Qii Q,i ^c-i denoting functions of $, for which expressions will be
investigated below. Each of them ia equal to + 1, when 0 = 0,
and they are alternately equal to — 1 and + 1, when 6 = w. It
is easily proved that each ia > - 1 and < + 1, for all values of
8 between 0 and w. Hence the series, which beoomes the
geometrical aeries 1 ^ « + e* -^ eta., in the extreme cases, con-
verges more rapidly than the geometrical series, except in tJioee
sof ^ = Oand e = T.
Henoe i =j (l + ^ + ^ + etc) when/;
and i = - 6 + ^+ ^+ etc.") when/* a
U a\ a a I ■"
..(48).
and therefore
Henoe by (46),
^-(4^4
^■-K'^'f"*-?^"- )^f.
^=^/=K-^*^v^^ )-"/<»]
D,g,i,„ab,Google
(49).
B (s).} KINEMATICS. SOI
Henoe^ for u {46), we h&ve the fallowing expomdons: — an«n-ipii>-
J K J J ) WlTSdeK
«.j{//fX«)<(,rt^//e,-F(i)di+^//e/'(£)<fa+...|, wheii/<<.
(51)-
These eeries being clearly conTergent, except in the case of /= a,
and, in this limiting case, the unexpanded value of w having been
proYed (46') to be finite and equal to iiraF(J'), it follows that the
Hum of each series aj^iroaches more and more nearly to this value
when/approacheB to equality with a. Hence, in the limit,
FlP)= ^hjF{E)dff + ZffQ,F{E)d<r + 6iJQ^{E)d<r + eta, } ... (52), ^g*
tapNifion at
■which is the celebrated development of an arbitrary function in t^SS^^
& series of "Laplace's coefficients," or, as we now call them,
tpherieal harmonicM,
{() ^le preceding investigation shows that when there is one
determinate value of the arbitrary function F for every point of
S, the series (52) converges to the value of this function at P.
The same reason shows that when there is an abrupt transition conTncenin
in the value of F, across any line on 8, the series cannot oon- omar itMt
verge when P is exactly on, but must still converge, however Sm?n**
near it may be to, this line. [Compare with last two paragraphs J^moI Sb
of { 77 above.] The degree of non-convergence is so slight thal^ Jj
as we see &om (51), the introduction of factors e, «*, e*, <&o. to
tlie BUDcessive terms e being < 1 by a very small difference, pro-
duces decided conveigeuoe for every position of P, and the value
of the series differs very little from F{P), passing very rapidly
through the finite difference when P is moved across the line <^
abrupt change in the value of F{P).
(u) In the development (47) of
(l-2ecoBfl + e*)l'
tjie coefficients of e, ef,...tf, are clearly rational integral functions
of cosff, of d^rees 1, 2...t, respectively. They are given ex-
pMtly below in (60) and (61), with ff = Q. Bu^ if ^ y, s and
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harmonio
oriheoo-
[B(.).
3!, y, ^ denote rectangular coordmates of P a&d of £ re-
spectively, we have
where r=(a^ + y' + a^l, and r' = (a!^ + y"+s")*. Henoe, de-
noting, as above, by Q, the coefficient of e' in tlie development,
we have
/f'*
(63).
I^i\ip< S< ")> (*'iy'i*')] denoting a eymmetrioal fiinction of (a^y, s)
and (x', y', «'), which ia homogeneous with reference to either set
alone. An explicit expression for thia function is of course found
from the expression for Q, in terms of cos $.
Viewed as a function of (a:, y, a), Q^t" is symmetrical
round OE ; and as a function of (x', y', z^ it ia symmetrical
round OP. We shall therefore call it the biaxal harmonic of
{x, y, z) (x', y', ^) of degree i ; and Q, the biaxal surface har-
monic of order t.
(v) But it is important to remark, that the coefficient of any
term, Bucb aa x''y''z', in it may be obtained alone, by meass of
Taylor's theorem, applied to a function of three variables, thus: —
1 _ r r
{l-2e cos tf+e')*"{r'-2rr'coefl4/')*"[(a:-!i;')'-l-(y- ?')•+(«-/)']*'
Now if F{x, y, z) denote any function of x, y, and z, we have
^\x+j,y+g,z + a, ^^ ^i_2...j.l.2...k.l.2...i dx'dy'tbf '
where it must be remarked tliat the interpretation of 1.2..J,
when^':^ 0, is unity, and so for k and I also. Hence, by taking
F(x,y,z)-
l
(3f + y'*s^)i
we have
[(«-»7+(j,-rt"+(»-»')T
(- ly'V's^-
"'1.2...j.\.2...i.l.2...l d^di^dJ ^^^.y•^.^i•
a development which, by comparing it with (48), above, we Bee
to be convei^nt whenever
w'*+ y+«"< as" + ?'+«■•
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B (n).] KINEMATICS.
Vrr/v, r- *■« ^i 2...j.l.2...i.l.2.../ di^dj/^tfe- (jc'+y*+a')i ^ ^'
the Bummatioii including all terms wMcli fulfil the indicated con-
dition {j + i + l=i). It is easy to Terify that the second member
is not only integral and homogeneous of the degree i, in x, t/, z,
as it is expressly in x, r/, ^ ; but that it is Bymmetrical with
reference to these two seta of variables. Arriving thus at the
oonclnsion expressed above by (53), we have now, for the function
there indicated, an explicit expression in terms of differential co-
efficients, which, fiirther, may be immediately expanded into an
algebraic form witii ease.
(v") In the particular case of x'=fi and ^"=0, (54) becomes
reduced to a single term, a function oi x,y, z symmetrical about
the axis OZ; and, dividing each member by /', or its equal, «'',
we have
* 1.2.3...i&'(^+j,.+ .')t '^'Sff,-
By actual differentiation it is easy to find the law of successive Anal Iw-
derivation of the numerators ; and thus we find, with about equal its co-ordl-
ease, either of the expansions (31)i (40), or (41), above, for the tormed
case m = n, or the trigonometrical formula, which are of course bUuL
obtained by putting a = rcosfl and a;'-Hy'=r'sin'tf.
(w) If now we put in tieae, cos tf = ^ , introducing
again, as in (u) above, the notation {pe, y, 2), (x', y', z'\ we arrive
at expansions of Q, in the terms indicated in (53).
(x) Some of the most useful expansions of Q, are very readily BinHiithnu
IS hBrmoDii^
J of order!.
obtained by introducing, aa before, the imaginary co-ordinates hi
(f, jj) instead of («, y), according to equations (26) of (J), and
BiniiUrly, (f, 1;') instead of 1^, y"). Thus we have
Hcnc«, as above,
1
=xsi
(-ir"'w»
1.2..j.l.2...*.1.2.,.( dfdrfdt! (f,+«")l'
jiGoogle
PBEUHINABT. [B (fr).
Of course we have in this case
and ooe9 = — — -' .
rr
And, just as above, we see that this expresaion, ohviouslf a homo-
geneous fuDctioQ of £', >)', s', of degree t, and also of ■g, (, z,
involvea these two systems of variables symmetrically.
Now, as ve have seen above, all the i* difiTereatial coefficieDts
of - are reducible to the 2t + 1 independent forms
/ d Y 1 \dz) dn T ' \ih) W r ' \tbiJ r '
\dz)r' /d\'-'d_l {£\"*/dyi fd\'l
W dir' W \di)r'- \d{)r-
Hence i^Q„ viewed as a function of s, f, 17, is expressed by
these 2i + l terms, each with a coefficient involving s', f, ij'.
And because of the symmetry we see that this coefficient must
be the same function of «', tf, (', into some &ctor involving
none of these variables (a, f, ij), («', yf, ('). Also, by the
symmetry with reference to f, >;' and ij, i', we see that the
numerical factor must be tho same for the terms similarly involv-
ing i, -q' on the one hand, and if, f ' on the other. Hence^
1 d' 1 J' 1 <i' in
■ r" d^dir ^ dz'^'dii'r' d^-'d^ rj J
K-
..(57).
1.2...tl.2...(i-.).f}...(.-J).(2.+l)(2,+2)...(,-+,) J
The value ot B^ ia obtuned thus : — Comparing the coefficient
of the term {zslf~'{^y in the numetator of t^e expression
whifA (56) becomes when the differential coefficient is expanded,
with the coefficient of the same term in (57), we have
l.-y^ V> ,58,
jiGoogle
B
C^)]
KINEMATICS.
20a
where
M denotea the coefficient of s'-'f in
***'
, A, 1
dz'-'de
, or,
Bianlbir<
, mcmio ™-
pnswdin
wUch
is the same.
the coefficient of a''-Sj'' ii
a r^
•r '
. dllTE^iitfal
From this, with the value (42) for M, we find E aa above.
(y) We are now readj to reduce the expansion of Q, to a real
trigonometrical form. Fixst, we have, by (33),
(fq')*+(f'ij)' = 2(rT'BiQeBintf')'oo8«(^-^') (59).
Let now
= 8iii'tf rcorf-#-^^^™li)ooa*-"tf Bin'«
(that is to Bay, C3 = d , in accordance with the previona no-
tation,) iind let the corresponding notation with accents apply
to **. Then, by the aid of (57), (58), and (59), we have
^ ^M-(*-|} (2-.l)(2,-.2)...(2.^i->) ^ ^^ (61)»eS
of which, however, the first term (that for which « = 0) must be J|^^
halved.
(z) Aj[ia(rapplementtothefimdftmentalpropomtion//5,5/rfw=0,
(16) of (y), and the corresponding propositions, (43) and (44),
regarding elementary terms of harmonics, we are now prepared to
evaluate fJS*dTii.
First, using the general expression (37) investigated above for
Sif and modifying the arbitrary constants to suit our present ^"^
notation, we have d«anite
5<=2J,oos(«*+<<.)a[ (62).
SJS^dw^irkA] hsf^y Bin Bd$ (63).
To evalnate the definite integrul In the second member, we have
only to apply the general theorem (52) for expansion, in terms of
surface harmonics, to the particular case in which the arbitrary
function F(£) is itself the harmonic, coated, . Thus, remem-
bering (16), we have
cai$<t,Srf = ^^^j' ion ede-Tdiji' COB si/b'^'^Q, (64).
DigilizedbyGOOgle
206 PBEXIMINABT. [B (z).
nnda- Using here for Q, ita trigoaometrical ezpanaion just inTestagKted,
iieintcKnU and performing tlie integration for tf>' between the stated limits,
we find that cos «0 <^/ may be divided out, and (omitting the
acoenta in the residual definite integrnl) we conclude,
This hotda without exception for the case « = 0, in which
the second member becomes -^-. — ■:^, It is convenient here to
recflil eqtiation (44), which, when expressed in terms of d,'
instead of 9„,^„,, becomes
^Bm$^'^Si^d$ = 0 (6G),
where t ^d t' most be different. The properties expressed hj
these two equations, (65) and (66), may be verified by direct
integration, from the explicit expression (60) for d, ; and to
do BO will be a good analytical exercise on the subject.
(a') Denote for brevity the second member of (65) by (t, <),
BO that
|'sinfl(9l")'(» = (t, «) (67).
Snppose the co-ordinates tf , ^ to be used in (53) ; so that a, 0,4'
are the three co-ordinates of P, and we may take dtr^a'mnBdBdtji.
Working out by aid of (61), (65), the processes indicated
symbolically in (52), we find
EJS'^t /'(fl,*)='2M.Sr+S(^;'cos»* + Si"sin«^)9r} (68),
mlheiii* of *"* •"'
incSn' where
mcluded. 9' 1 /» t*ir
A, = ^^j S'TBm$d$rF(e,it>)d^
A,^r. . I 3, sintfdfl I <XBs<jiF{S,if,)d<l,
^ = __L_ r a';' Bin ddfl p Bin 8^ i!'(#, ^) rf^
which is the explicit form most convenient for general use, of the
expansion of an arbitrary function of the co-ordinates $, ^ in
spherical surfiiice harmonics. It is most easily proved, [when
..Google
B (o").] KINEMATICS. 207
once the general theorem expressed by (66) and (65) has been in 8ph«ri<al
any Tray eatabluhed,] by assuming the form of expansion (68), Ki^r>t»ol
and tlien determining the coeffidenta by multiplying both mem- fonctioii. '
ben by d, coa s^ an. 9 d6d^, and again by b, eiov^ sin 6 d6d<p,
and integrating in each case over the whole spherical Burfitce.
{b') In what precedes the espansiona of surface harmonics, S^^^^'?
whether complete or not, have been obtained solely by the differ- ^^^'J^JJ.
entiation of - with reference to rectilineal rectangular co- propwtiBi.
ordinates x, y, z. The ezpansionB of the complete harmonica
have been found simply as expressions for differential ooeffi-
cients, or for linear functions of differential coefiSciEmts of — .
The expODsioiiB of harmonics of fractional and imaginary ordera
have been inferred from the expansions of the complete har-
monics merely by generaliziiig their algebraic forms. The pro-
perties of the harmonics have been investigated solely from the
difierential equatit^
dfV dTY d^r „
Sf^W^-d^'" <'«)■
in terms of the rectilineal rectangular co-ordinates. The original
investigations of Laplace, on the other hand, were founded
exclusively on the transformation of this equation into polar
co-ordinates. In our first edition this transformation was not
given — we now supply the omission, not only on account of the
historical interest attached to "Laplace's equation" in terms of
polar co-ordinates, but also because in this form it leads directly by
the ordinaty methods of treating differential equations, to every
possible expansion of surface harmonics in polar co-ordinates.
(y) By App. Z{g){H) we find for Laplace's equation (20)
transformed to polar co-ordinates,
d/T'dVS 1 df. .dV\ 1 dT -
^\-dr) *^^T6\^^-de) *^^w ^^ ^'
In this put
V=S/, or r=S,r-'-' (72).
We find
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■ailiicehw-
i PRELIMINAET. [B (c).
irhicli ia the celebrated formula commonly known in TJinglaml
as "Laplace's Equation" for determiniDg S,, &o "I^place's
coefficient" of order t; i being an integer, and the solatums
admitted or sought for being restricted to rational int^ral
fimctions of cos tf, sin ^ cos ^ and sin 9 aia ^.
((f) Doing away now with all anch TestrictionB, suppose t to
be any number, int«f;ral or fractional, real or imaginary, wily if
imaginaiy let it be such as to make i (t — 1) real [compare § (o)]
above. On the supposition that iS, is a rational integral func-
tion of cos 0, sin 9 COB ^ and sin d sin ^, it would be the sum of
terms such as 0,'*' #i^ Now, allowing g to have any valae
integral or fractional, real or imaginary, assome
«.e,-™^
..(74).
This will be a form of particular solution adapted for ^tplication
to problems snch as those referred to in §g {[), (m) above; and
(73) gives, for the determination of ©j"*,
.,(75).
=^^(-»^')^[^.^^<'-)]«- = »
^pi2?^ (O ^^«i * and » are both integets we know from §(A)
'nn«»ioD»." above, and we shall verify presently, by regular treatment
of it in its present form, that the differential equation (7G) has
for one solution a rational int^nral fonction of sin 0 and ooe &,
It is this solution that gives the "Laplac^s Function," or the
"complete surface harmonic" of the form B s^ . But b^g a
diSerential equation of the second order, (70) must have another
distinct solution, and from § (A) above it follows (iiat this second
solution cannot be a rational integral fimction of sin 0, oos 9. It
may of course be found by quadratures from the rational integral
solution according to the regular process for finding the second
particular solution of a differential eqaatlon of the second order
when one particular solution is known. Thus denoting by d,'*'
any solution, as for example the known rational int^ral sola-
tion expressed by equation (38), or (36) or (40) above, or
§ 782 (e) or (/) with (5) below, we have for the complete
..Google
B (c').] KINEMATICS. 209
solution,
..(76). 5«e"
plMe'i
huutlaiu."
For a direct inTestigadoQ of the complete solution in Unite
terms for the case i — att positive integer, see belov g (n").
Example 2 ; and for the case i an integer, and a either not an
integer ornot <t, see g (o') (III).
The ratioual int^ral Bolation alone can enter, and it. alone
Buffices, when tlie problem deals with the complete spherical
sur&ce. When there are boundarieB, whether by two planes
meeting in a diameter at an angle equal to a submuldple of
four right an^ee, or by coaxal oones corresponding to certain
particalar values of 6, or by planes and cones, both the rational
integral solution and the other are required. But when there
are coaxal conee for boundaries, the values of t required by the
boundary conditions [§ (/)] are not geneially integral, and it is
only when i — a is integral that either solution is a rational and
integral function of Bin 0 and cos 6. Hencc^ in general, for the
class of problems referred to, two solutions are required and
neither is a rational integral function of sin 0 and cos 6.
{f) The ordinary process for the solution of linear differential
equations in series of powera of the independent variable when
the multipliers of the differential coefficients are rational alge-
braic functions of the independent variable leads easily from the
equation (75) to any of the forms of rational integral solutions
referred to above, 4S well as to the second solution in a form
corresponding to each of them, when i and i are integers; and,
quite generally, to the two particular solutions in every case,
whether i and a be integral or fractional, real or imaginary.
Thus, potting as above, g (;!:),
coatf^^, siutf = v (77),
make >t the independent variable in the first place, in order to ^^h^'*'
find expansioiw in powen of /* : thus (75) becomes pendmi"'*"
|[a--'f]-[i^.-H<.i)]e," m. m
This is the form in which "Laplace's equation" has been most
commonly presented. To avoid the appearance of supposing
VOL I. 14
..Google
210 PBELtJnNAET. [B (/').
t and s to be integers or even real, put
e«=M», t(t + l) = o, »''h, {79).
Truing tlua notation, and mnltiplTing both membeia by (1 - /i*),
vre have, instead of (78),
<l-c')|;[(l-»'-)^]+[«(l-/'')-»]"=0 (80).
To int^rate tiiia equation, assume
Kiintiw in ^°*^ ^ ^^ series so found for its fiist member equate to tero tbe
Jj;^^^ J coefficient of /t". Thus we find
(n + l)(m-2)J-„.= [2n'-o + 6]^.-[(n-l)(n-2)-o]J,......(81).
Hie first member of this Tanisbes fur n = — 1, and f<a- n = — 2, if
f , and S^ be finite. Hence, we may put £,-0 for all negative
values of n, give arbitrary values to £^ and £*,, and then find
JT,, JT,, r,, Ac, by applications of {81 ) with n = 0, n = 1, N = 2,.. .
successively. Tbua if we first pnt fg^l, and ^, = 0; then
Bgaini f^, » 0, f , = 1 ; we find two series of the forms
1 + Jr,f«' + jr,|«* + Ac.
and It + Kj^ + -^gf*' ''' ^^t
each of which satisfies (80); and therefore the complete solu-
tionis
w-(7(l+^,/ + i'y + 4c) + (7V + -^,p'+-^,f'' + *<')"-(82).
From the form of (81) we see that for very great values of n we
have
jr,^j=2£', — X,., approximately,
and therefore
JT,^, - J, = J, — jr,_, approximately.
Hence each of the series in (82) converges for every value of ^
less than unity.
niwni.'' ^'^ ^"* ^^^ ^ * ^^^ unsatisfactory form of solution. It
gives in the form of an infinite series 1 + K^fi? + K^i^ + iic. or
/i 4- f ,^* + K^p.* + Aa, the finite solution which we know exists
in the form
..Google
B((r').] KINEMATICS. 211
, G
or (l-/i*)»(J,fi + V + ...J„^^), •*"'"»
-when h is the square of an odd int^er (a), and when a = t (t + 1),
t being an odd integer or an even integer; and, a minor defect^
bnt Htill a serious one, it does not show without elaborate veri-
fit»tion that one or other of its constituents \ + K^ii* + ka. or
fi + K^i^ + Ac, consists of a finite number, \i or J(t + 1), of terms
when 6 is the sqnare of an even integer and a=-i (i + 1), i being
an even integer or an odd integer.
{h') A form of solution which turns out to be much simpler
in everjr case is Bu^^ested byonr primary knowledge [g (j) above]
of integral solutions. Put
Vft
w = (l_^')ii« (83),
in (80) and divide the first member by (1 -fi') >. Thus we
find
(l-,.')0-2(^ + l);.g+[a-^t(V6 + l»' = O (84).' SS&iad.
Antune now
v = SA_p.' (85);
equating to zero the coefficient of /i* in the first member of
(84) gives
(«+l)(n+2)^,.,-[(«-l)»+2(V6+l)»-a+,yft(^+l)]J.=0...(86),}J;j
or (n+l)(n + 2)J.^,= (7. + J + » + a)(n + J+«-a)J (87),*''"
if wepnt a = ^(o + i), s = Jb (88),
and with this notation (84) becomes
(l-rt|^-2(.*l),.y°+[a'-(.*i)'l».0...-.(84-).
The second member of (87) shows that if the series (85) is in
descending powers of fi its first term must have eiUier
n=— J — » + «, or n = — J — s — a:
the expansion thus obtEuned would, if not finite, be convergent JSJI^uT'
when (1 > 1 and divei^nt when f< < 1 , and they are there- ^^3l,
fore not suited for the physical applications. On the other oIkmil
hand, the first member of (87) shows that if the series (85) is in
ascending powers of ft, its first term must have either n = 0 or
l*-2
..Google
212 PBELIMHTABT. [B (A').
Ono^lMa »=I: the expansions thus obtained are necessarily conT^^ent
uusral when >*< 1, and it is therefoi-e these that are suited for oar pur-
poBSB. Taking then ^, = 1 and A^ = 0, and denoting by p the
series so found, and again A^ = 0 and A, = 1, and f the series ; bo
that we have
and j = ^ + ii,(*' + J,p'+etc, / ^
J,, A^, etc. and A^, A,, eta. being found by two sets of suc-
cessive applications of (87); then the complete solntiou of (84) is
v = Cp + Cq (90).
This solution is identical with (Z&^ of § (I) above, as we see by
(88) and (79), which give
— i + i (91)-
Alternktira («') The ugn of either a or b may be changed, in virime of
ctunRinK (68). T^o variation hDwever is made m the solution by changing
the sign of a [which corresponda to changing i into — t— 1, and
verifies (13) (^) above]: but a very remarkable variation is made
by changing the sign of s, from which, looking to (88), (83), (87),
we infer that if p and q denote what p and q become when — a
is substituted for « in (89), we have
and q = {l-/*T?) * ^'
and the prescribed modification of (89) gives
p = !+«,/ + «
q = /* + «l,/ + a
n,, ft^, eta^snd ft,, ft,, etc. being found by sucoessive applica-
tions of
(»<-}-. ta)(.^i-.-.)
(j"f In tibe case of " complete harmonics" t is zero or an
integer, and the p or q solution expressing the result of multiply-
ing the already finite and integral p or q solution by the iiit«^p^
polynomial (I - /i*)', is only interesting on account of the way of
obtaining it from (87), etc. in virtue of (88). But when either
a— } or « is not an integer, the possession of the alternative solu-
tions, par f, g 01 n may come to be of gresit intrinsic importance,
in respect to obtaining resnlta in finite form. For, supposing a
and » to be both positive, it is impossible that both p and q can
be finite polyn<»nials, but one or botih of p and q may be so; or
..(93),
jiGoogle
B (J)'.] KINEMATICS. 213
one of the p, q forms and the otker of the p, q forms may be Cues
finite. This we see from (87) and (il4), which show as follows: — ilnbnio
1. If J + » — a is poflitive, p and q mmat eacli be an infinite
seriea ; but p w q will be finite if either J + « — oorJ + « + oifl
a positive int«ger*; and l>oth, p and q will be finite if \ + i — a,
and ^ + 8 + 0. are positive integers difiering by unity or any odd
number.
2. If a^ i+\, one of the two aeries p, q must be infinite ;
and if a — 8 - ^ is zero or a podtive integer, one of the two
, seriea p, q ia finite. If, lastly, a 4 s — ^ is zero or a positive
int^ier, one of die two |l, q is finite. It is p that is finite if
a-B — ^ is zero or even, $ if it is odd : and f that is finite if
a + «— ^iaz^xi or even, q if it is odd. Hence it is p and f, or
q and 4 that are finite if 2« be zero or even; but it is j> and q,
or q and y that are finite if 2s be odd. Hence in thia latter case
4he complete solution ia a finite algebraic function of fu
{k") Remembering tiiat by a and 8 we denote the positive
valnee of the square roots indicated in (88), we collect from (/)
1 and 2, that, if F denote a rational integral funddon of ft and
(1 — /t*}*") the cluuracter of tbe solution of (80) is as follows in
the sever^ cases indicated : —
A; ^<f*\> if' ^^ 1 - 1 ftre integers.
B; tt^' + i; if ' + J and a are int^jera.
The complete solution is p.
A; a<» + \i if S''=(a-i) is an int^^r, but a-^not an
int^er,
B; 0'3»*i> ifa-^->=« is an int€j[er, but » + ) not an
integer.
. A partionlar solution ts F ; but tbe complete solution u not F.
(f) "Complete Spherical Harmonica," or "lAplace'a Co-
efficients," are included in the particular solution F of Case II> Bt
{niT) Differentiate (81') and put
I- «•
' UnitjbungimdKrtoodaainaladedintheDlaaaof "pOBi(iyaiat4gerB,"
..Google
S14 PRELUnNABT. [B (>"')■
Ttinv We find immediately
model gl
SS™" (l-c')^.-2(«+2)/'J*K-('*l««-0 (»«)■
I" «'-c0*(-'<' + » + J)<' <")■
We have, as will be proved presently,
LuUj.let u".(l-rt^- ('••*•* J)/" (99)-
We have, as will ba proved presently,
(l-/)^-2(,+ l);.^+[(.-.l)'-(.tJ)l»".0...(100).
Tlie operation ^ performed on a Bolid harmonic of degree
— a—}, and lypef {a, iJi'^+y')] ^i >™d transformed to polar
co-ordioatea r, ft, ^, wiUi attention to (83), gives the transition
from t> to u", an ez^<eesed in (99), and thns (100) is proved by
to) (15)-
Similaiiy the opeiatioa
transformed to ocKirdinatea r, fi, ^ gives (97), oiid thus (98) b
proved by {g) (16).
I^us it was that (97) (98), and (99) (100) were fonnd. Bat,
ttaanming (97) and (99) arbitrarily as it wer^ ve prove (98) and
(100) most easily bs follows. Let
«'=SB'j*', and u"^-S,B"y (101).
Then, by (97) and (99), with (85), we find
■^.+.= (n+2*a + » + })J.„ \
J B-/ / ,.n+l + i + e*« . > (102).
^^ ^,+,=('«+''-J) — ^;-2 — ■^.«f '
I^ifitly, applying (87), we find that the conesponding equa-
tion is satisfied by J',^j-i-jB'_^ with a*l and »+\ instead of
a and a; and by ff'^^^-S'^, with a*l inatead of a, but with t
unchanged.
..Google
B(m').]
EINiaUTICS.
215
Aa to (96) and {96), they merely express far the generalized ^^**i>'
gui&ce harmonics the transition from s to «4 1 without change
of i ehown for complete harmonics by Murphy's formula,
§ 782 (6) below.
(n') Example* o/ {95) (96), and (99) (100).
Example 1. Let a=< + }.
of which the complete solution is
By (96) (90) we find
•dii.-
I-C,
,..{10S).
..(104).
(I-,.')^-2(» + . + l)^^-«(« + 2. + l)«-0
Tbia is the particular finite solution indicated in § (V) IL A>
The liberty we now hare to let n be negative as well as positive
allows OB now to include in our formula for u the oases repre-
sented by the doable sign ik in II. A of {!e).
Example 2. By m succesmve applications of (99) (100), with
the u|^>er idgn, to v of (103), we find for the complete intend of
P-^-)^-
(« + 1 )ft -T- + »»(»» + 2» + 1) w'= 0
«' = t?{/(^) l^j:^ + i'(^)} + cpo.)
.... (106), loritl^ta.
where /(fi), F{jx), F(/i) denote rational integral algebraic fiuu^
tions of /t.
Of this solution ^e part C'P(ji) is tlie particular finite solution
indicated in § (if) II, B> We now see that tlie complete solution
integer, tbig is reducible to the form
involveB no other transcendent than i
Wlien* is an
• log
*m.
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6 PRKLDIIKABT. [B (n').
a being a constant and f (^) a rational integral algebi«o function
of /I. In this caae, remembering that (105) is wliAt (84')
becomes when m + « + J ia put for o, we may recur to our
notation of ^ (y) {J), by putting t for m + s, which is now an
integer ; and going bact, by (83) to (80) or (78), put
w = (!-/*')'«' ^53^.
thus (106) is eqitivalent to
^[('-'•■'S]* [.---•<-')]«
(78>
He prooesBof Example 2, § (»-), gives the complete int^ral of
this equation when t - « ia a pOBitive integer. "When also #, and
therefore also t, is an int^er, the transcendent involTed be-
oomes log Y3^ : in this caae the algebraic part of the solution
[orC'P(/t)(l-^')f according to the notation of (105) and (TS")]
ia the ordinary " Laplace's Function" of order and type (L g) ■
the ®j , ^j , (to. of our previous notationa of ^ (J), (v). It ia
interesting to know that the other particular solution which we
now have, completing the solution of the differential equation
for these fimoti<ma, involves nothing of transcendent but
(oO ^«m;*. ^(99) (100), and (95) (96) «m(.'n«erf.
Example 3. Betuming to (»'), Example 2, lot a + 1 be an
integer : the integral j—^~ ia algebi-aio. Thus we have the
case of (A') I. R in -which the complete solution is algebraic
(p) Returning to {nT}, Example 1: let a = i and «=0;
(103) becomes
of which the complete integral is j- (lOS'i
As before, apply (95) (96) n times suooeaaively : we find
.. = 1.1.2...(„-l)c[(j-i-)--(j^)-] (,06)
D,g,i,„ab,Google
B(p')-] KINEHATICS. 217
as one solntioa of ibe other
deriTBd
jt„ a,, from tbl> by
a-/'')S-2('' + l)c^-»(»*'>»-'' (!>«■)■ SSff-,,
"r- "/» Bquinlant
To find the other : treat (106) hy {99) (100) with t*e lower ""^
sign ; the effect is to diminish a. from \ \q —\, and therefore to
make no change in the differential equation, but to derive from.
(106) another particular eolation, which iA as follows :
..-i.i.2...(,-i).„.c[(j4-J. (^)-] (ion
Giving any different values to (7 in (106) and (106'), and, using OompMa
K,K!U> denote two arbitrary constants, adding we have the com- tsHeral* of
plete eolation of (96'), which we may write as follows : "™*
K K'
(l-^)-"^(l+,ir'
. (107).
(ff) That (1 07) is the solation of (Oe*) we verify in a moment
by trial, and in bo doing we Bee fartlier that it is the complete
solution, whether » be int^^ or not
(/) Example 4. Apply (99) (100) with upper sign t" times to d.
(107) and successivv results. We get thas the complete solution both !«■-
of (81') for a — ^ =^ t aju.y integer, if n is nol an int^er. But if n vtvrj inte-
is an integer we get the complete solution only provided i<n: ^^ '
this is case I. A ^ h^y If ^^e take « = n-l, the result,
algebraic as it is, may be proved to be expressible in the form
(i-AT
which is therefore f or » an int^er the complete
int^p^of
bmng the case of (84*) for which o = s - }, and « ^ n an integer : °^J^
applying to this (99) (100) with upper sign, the constant C dis- ^°°^'"' **
appears, and we find u'= C aaa solation ot
Hence, for tpn one solution is lost The other, found by
..Google
S18 PBEUHINABT. [B (r*).
Bnmptaiat contmued applications of (99) (100) with upper ago, is the
*"* regular " Laplace's ftmction" growing &omCsm*d n^ which
is the case represented hf u'= C in (109). But in tikis oon-
tinuation we are only doing for the case of n an int^er, part
of what was done in § (»'), Example 2, where the other put,
from the other part of the solution of (109) now lost, gives the
other part of the complete solution of Laplace's equation subject
to the Umitation i— n (or is) a positive integer, but not to the
limitation td i an intfiger or n an int^^,
(a') Betuming to the commencement of § (r"), with » put
for n, we find a complete solution growing in the form
which mfty be immediately reduced to
(1 -»■•)•
/, denoting an integral algebrmc function of the t"* degree, readil;
found by the proper succesmve ^plicsti<nis of (99) (100).
Hence, by (83) (79), we have
^ _ -g/(M) (l^l^y* i-y^M-l^) (1 ' >■)• (jiij^
as the complete solution of lAplace'a equation
|;[<'---)g]*[i^.--('*'>]-=« ("^).
^10*8^*^ for the case of t an integer without any restriction as to the
P~5^"* vaJua of 8, which may be integral or fractional, real or ima^nary,
UimmiIiSl- with no fiulure except the case of s an integer and i > », of which
'U^l^o' the complete treatment is included in § (tn')> Example 2, above^
.. <I107;
jiGoogle
DTNAMICAL LA.WS AND FBIMCIFLES.
205. In the preceding chapter ve considered as a Bubject of ideu or
pure geometry the motion of points, lines, surfaces, and volumes, tan» intio-
■whether taking place with or without change of dimensions and
form ; and the results we there anived at are of course altogether
independent of the idea of matter, and of the forces which matter
exerts. We have heretofore assumed the existence merely of
motion, distortion, etc.; we now come to the consideration, not
of how we might consider such motions, etc., to be produced, but
of the actual causes which in the material world do produce
them. The axioms of the present chapter must therefore be
considered to be due to actual experience, in the shape either
of observation or experiment How this experience is to be
obtained will form the subject of a subsequent chapter.
306. We cannot do better, at all events in commencing, than
follow Kewton somewhat closely. Indeed the introduction to
the JVi?ictpia contains in a most lucid form the general founda-
tions of Dynamics. The Definitiones and Axtomata sive Legea
MotAs, there laid down, require only a few amplifications and
additional illustrations, su^ested by subsequent developments,
to suit them to the present state of science, and to make a much
better introduction to dynamics than we find in even some of
the best modem treatises,
207. We cannot, of course, give a definition of ifatterwhicfa Matter,
will saUsfy the metaphysician, but the naturalist may be con-
tent to know matter as that'wkich can be perceived by the eensee,
or as that which can be acted upon bif, or can exerit, force. The
..Google
220 PBEUMIKAKT. [207.
latter, and indeed tbe former also, of these defimtions involves
the idea of Farce, wbicb, in point of fact, is a direct object of
Bense; probably of all our Beosea, and certainlj of the "mus-
cular sense." To our chapter on Properties of Matter we must
refer for further discussion of the question. What is matter'
And T^e Shall then be in a position to discuss the question
of the subjectivity of Force.
208. The Quantity of Matter in a body, or, as we now call
" it, the Mom of a body, is proportional, according to Newton, to
the Volume and the Density conjcrintly. In reality, the deSni-
tion gives uB the meaning of density rather than of mass ; for
it shows us that if twice the original quantity of matter, air for
example, be forced into a vessel of given capacity, the density
will be doubled, and bo on. But it also shows us that, of matter
of uniform density, the mass or quantity is proportional to the
viflume or space it occupies.
Let Jfbe tiie mass, p tlie density, and Fthevolame^ of afaomo-
geneons body. Then
if we so take our units that unit of mass is that of unit yolume of
a body of unit density.
If the density vary from poi^t to point of the body, we have
evidently, by the above formula and the elementary notation of
the int^ral calculns,
M=S55pdxdyaz,
where p ie supposed to be a known function of x, y, e, and the
integration extends to the whole space occupied by the matter of
the body whether tlus be continuous or not.
It is worthy of particular notice that, in this definition,
Newton says, if there be anything which freely pervades the
interstices of all bodies, this is not taken account of in estimat-
ing their Mass or Density,
209. Newton further states, that a practical measure of the
mass of a body is its Weight. His experiments on pendulums,
by which be establishes this most important result, will be de-
scribed later, in our chapter on Properties of Hatter,
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209.] DYNAMICAL LAWS AND PRINCIPLES. 221
As will be presently explained, the unit maaa most convenient
for British measuremente \b an imperial pound of matter.
210. The Quantittf of Motion, or the Momentum, of a rig^d
body moving without rotation is proportional Co its mass and
velocity conjointly. The whole motion ia the sum of the motions
of its several parts. Thus a, doubled mass, or a doubled velocity,
would correspond to a double quantity of motion; and so on.
Heace, if we take ob unit of Biomentum the momentum ot
a unit of matter moving with unit velocity, the momentum of a
mass M moving with velocity v is Mv.
211. Change of Quantity of Motion, or Change of Momen-cbanr^ot
turn, is proportional to the mass moving asd the change of its
velocity conjointly.
Change of velocity is to be understood in the general sense
of § 27. Thus, in the figure of that section, if a velocity re-
presented by OA be changed to another represented by OC, the
change of velocity is represented in magnitude and direction
hy^a
212. Sate of Change of Momentum is proportional to the ^o' .
mass moving and the acceleration of its velocity conjointly, m * —
Thus (§ 35, b) the rate of change of momentum of a falling
body is constant, and in the vertical direction. Again (§ 35, a)
the rate of change of momentum of a mass M, describing a
R
directed to the centre of the circle ; that is to say, it is a
change of direction, not a change of speed, of the motion.
Hence if the mass be compelled to keep in the circle by a
cord attached to it and held fixed at the centre of the circle, the
MV*
force with which the cord is stretched is equal to — k- : this is
called the centrifugal force of the mass M moving with velocity
V in a circle of radi us R.
Generally ^ 29),. for a body of mass M moving anyhow in
..Google
222 PBELIUINIBT. [212.
tion of motion, and M— towards the centre of curvature of tte
n. P
path J and, if we choose, we may exhibit the whole acoeleration
of momentom by ita three rectangular components ^^^ > ^ 1% '
^TSt or, according to the Newtonian notation, Mx, My, Ms.
213. The Vis Viva, or Kinetic Energy, of a moving body is
proportional to the mass and the square of the velocity, con-
jointly. If we adopt the same units of mass and velocity as
before, there is particular advantage in defining kinetic energy
as half the product of the mass and the square of its velocity.
214.' Rate of Change of Kinetic Energy (when defined as
above) is the product of the velocity into the component of
rate of change of momentum in the direction of motion.
215. It is to be observed that, in what precedes, vrith tbe
exception of the definition of mass, we have taken no account
of the dimensions of tbe moving body. This is of no conse-
quence so long as it does not rotate, and so long as its parts
preserve the same relative positions amongst one another. In
this case we may suppose the whole of the matter in it to be
condensed in one point or particle. We thus speak of a material
particle, as distinguished from a geometrical point. If the body
rotate, or if its parts change their relative positions, then we
cannot choose any one point by whose motions alone we may
determine those of the other points. In such cases the momen-
tum and change of momentum of the whole body in any direc-
tion are, the sums of the momenta, and of the changes of
momentum, of its parts, in these directions ; while the kinetic
energy of the whole, being non-directional, is simply tbe sum
of tbe kinetic energies of the several parts or particles.
216. Matter has an innate power of resisting external in-
fluences, so that every body, as far as it can, remains at rest, or
moves uniformly in a straight line.
This, the Inertia of matter, is proportional to the quantity of
..Google
216.] DYNAMICAL LAWS AND PHINCIPI.ES. 223
matter in the body. And it follows that some cause is requiaita inertia,
to disturb a body's uDiformity of motion, or to change its direc-
tion from the natural rectilinear path.
217. Force is any cause which tends to alter a body's natural fotm.
state of rest, or of uniform motion in a straight line.
Force is wholly expended in the Action it produces; and the
body, after the force ceases to act, retains by its inertia the
direction of motion and the velocity which were given to it.
Force may be of divers kinds, as pressure, or gravity, or friction,
or any of the attractive or repulsive actions of electricity, mag-
netism, etc.
' S16. The tbree elements specifying a force, or the three sptdflw
elements which must be known, before a clear notion of the rant.
force under consideration can be formed, are, its place of appli-
cation, its direction, and its magnitude.
(a) The place of application of a force. The first case to be Piuwor
considered is that in which the place of application is a point.
It has been shown already in what sense the term "point"
is to be taken, and, therefore, in what way a force may be
imagined as acting at a point. In reality, however, the place of
application of a force is always either a surface or a space of
three dimensions occupied by matter. The point of the finest
needle, or the edge of the sharpest knife, is still a surface, and
acts by pressing over a finite area on bodies to which it may
be applied. Even the most rigid substances, when brought
together, do not touch at a point merely, but mould each other
so as to produce a surface of application. On the other hand,
gravity is a force of which the place of application is the whole
matter of the body whose weight is considered ; and the smallest
particle of matter that has weight occupies some finite portion
of space. Thus it is to be remarked, that there are two kinds
of force, distinguishable by their place of application — force,
whose place of application is a surface, and force, whose place
of application is a solid. "When a heavy body rests on the
ground, or on a table, force of the second character, acting
downwards, is balanced by force of the first character acting
upwards.
..Google
22+ PBELnCNABT. [218.
Dircotloi). (^) ^^ aecood elenutnt in the specificatioD of a force is its
dkection. The direction of a force is the line in which it acts.
If the place of application of a force he regarded as a point, a
line through that point, in the diiection in which the force
tends to move the body, is the direction of the force. In the
case of a force distributed over a surface, it is frequently pos-
sible aad convenient to assume a single point and a single line,
such that a certain force acting at that poiat In that hne would
produce sensibly the same effect as is really produced,
Hignitnde. (c) The third element in the specification of a force is its
magnitude. This involves a consideration of the method fol-
lowed in dynamics for measuring forces. Before measuring
anything, It is necessary to have a unit of measurement, or a
std:tidard to which to refer, and a principle of numerical specifi-
cation, or a mode of referring to the standard. These will be
supplied presently. See also § 258, below.
2X9. The Accelerative Effect of a Force is proportional to
the velocity which it produces in a given time, and is measured
by that which is, or would be, produced in unit of time; in
other words, the rate of change of velocity which it produces.
This is simply what we have already defined as acceleration, § 28.
r 220. The Measure of a Force is the quantity of motion which
it produces per unit of time.
The reader, who has been accustomed to speak of a force of
so many pounds, or so many tons, may be startled when he finds
tbat such expressions are not definite unless it be specified at
what part of the earth's surface the pound, or other definite
quantity of matter named, is to be weighed ; for the heaviness or
gravity of a given quantity of matter differs in different latitudes.
But the force required to produce a stated quantity of motion in
a pven time is perfectly definite, and independent of locality.
Thus, let W be the mass of a body, g the velocity it would
acquire in falling freely for a second, and P the force of gravity
upon it, measured in kinetic or absolute unite. We have
P = Wg.
..Google
221.] DTKAMICAL LAWS AKD PRHTCIPLES. 225
221. According to the system commonly followed in mathe- ^J^JJ^
matical treatises on dynamics till fourteen years ago, when a small ^^^^
instalment of the first edition of the present work was issued
for the use of our students, the unit of mass was g times the
mass of the standard or unit weight. This definition, giving a
varying and a very unnatural unit of mass, was exceedingly
inconvenient. By taking the gravity of a constant mass for ^ . : -
the unit of force it makes the unit of force greater in high than ^"JS*^
in low latitudes. In reality, standards of weight are masses, f^^^t—
not forces. They are employed primarily in commerce for the JJ^StST
purpose of measuring out a definite quantity of matter; not an "*"*■
amount of matter which shall be attracted hy the earth with a
given force,
A merchant, with a balance and a set of standard weights,
would give his customers the same quantity of the same kind of
matter however the earth's attraction might vary, depending as
he does upon weights for his measurement; another, using a
spring-balance, would defraud his customers iu high latitudes,
and himself in low, if his instrument (which depends on constant
forces and not on the gravity of constant masses) were correctly
adjusted in London.
It is a secondary application of our standards of weight to
employ them for the measurement ot/orees, such as steam pres-
sures, muscular power, etc. In all cases where great accuracy
is required, the results obtained by such a method have to be
reduced to what they would have been if the measurements of
force had been made by means of a perfect spring-balance,
graduated so as to indicate the forces of gravity on the standard
weights in some conventional locality.
It is therdbre very mudi simpler and better to take the
imperial pound, or other national or international standard
weight, as, for instance, the gramme (see the chapter on
Measures and Instruments), as the unit of mass, and to derive
from it, according to Newton's definition above, the unit of
force This is the method which Qaoss has adopted in bis
great improvement (§ 223 below) of the system of measurement
of forces.
VOL. I. 15
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228 PRELDONABT. [222.
^■frutifi 222. The formula, deduced by Clairault from obaervalion,
fbrtnola lor i j »
(baNDoimt and a certain theory regarding the figure and density of the
earth, may be employed to calculate the moat probable value
of the apparent force of gravity, being the resultant of true
gravitation and centrifugal force, in any locality vhere no
pendulum observation of sufficient accuracy has been made.
This formula, with the two coefGcienta which it involves,
corrected according to the best modem pendulum observations
(Airy, Ewyc. Metropolitana, Figure of the Earth),/^ as fol-
lows:—
I<et O be the apparent force of gravity on a unit mass at the
equator, and g that in any latitude \; then
y=G'(l+-0051338in*X).
The value of Q, in terms of the British absolute unit, to be
explained immediately, is
32-088.
According to this formula, therefor^ polar gravity will be
5r = 32088x l-003133 = 32-2527.
223. Gravity having failed to furnish a definite standard,
independent of locality, recourse must be had to something else.
The principle of measurement indicated as above by Newton,
^«^ but first introduced practically by Gauss, furnishes us with
Unit of what we want. Accoiiling to this principle, the unit force is
that force which, acting on a national standard unit of matter
during the unit of time, generates the unity of velocity.
This is known as Gauss's absolute unit ; absolute, because
it furnishes a standard force independent of the differing
amounts of gravity at different localities. It is however ter-
restrial and inconstant if the unit of time depends on the earth's
rotation, as it does in our present system of chronometry. The
period of vibration of a piece of quartz crystal of specified shape
and size and at a stated temperature (a tuning-fork, or bar, as
one of the bars of glass used in the "musical glasses") gives us
a unit of time which is constant through all space and all time,
and independent of the earth. A unit offeree founded on such
a unit of time would be better entitled to the designation abao-
..Google
223.] DTSAMICAL LA.VS AND PRINCIPLES. . 227
lute than is the "absolute unit" now generally adopted, which is Hnwdi-i
founded on the mean solar second. But this depends essentially ttooa x<a
on one particular piece of matter, and is therefore liable to all umt of
the accidents, etc. which- affect so-called National Standards
however carefully they may he preserved, as well as to the
almost insuperable practical difficulties which are experienced
when we attempt to make exact copies of them. Still, in the
present state of science, we are really confined to such approxi-
mations. The recent discoveries due to the Kinetic theory of
gases and to Spectrum analysis (especially when it is applied to
the light of the heavenly bodies) indicate to us natural standard
pieces of matter such as atoms of hydrogen, or sodium, ready made
in infinite numbers, all absolutely alike in every physical pro-
perty. The time of vibration of a sodium particle corresponding
to any one of its modes of vibration, is known to be absolutely
independent of its position in the universe, and it will probably
remain the same so long as the particle itself exists. The wave-
length for that particular ray, i. e. the space through which
light is propagated tn vacuo during the time of one complete
vibration of this period, gives a perfectly invariable unit of
length; and it is possible that at some not very dbtant day the
mass of such a sodium particle may be employed as a natural
standard for the remaining fundamental unit. This, the latest
improvement made upon our original su^estion of a P^etijoal
Spring (First edition, § 406J, is due to Clerk Maxwell*; who
haa also communicated to us another very important and in-
teresting suggestion for founding the unit of time upon physical
properties of a substance without the necessity of specifying any
particular quantity of it. It is this, water being chosen as the
snbstance of all others known to us which is most easily obtained
in perfect purity and in perfectly definite physical condition. —
Call the standard density of water the maximum density of
the liquid when under the pressure of its own vapour alone.
The time of revolution of an infinitesimal satellite close to the
surface of a globe of water at standard density (or of any kind
of matter at the same density) may he taken as the unit of
time ; for it is independent of the size of the globe. This has
* EUttrieilif and MagneUtm, 1S73.
..Google
■228 PKELiMlSAnr. [223.
Thiri wg- suggested to U3 still another unit, founded, however, still upon
S^d^te the same physical principle. The time of the gravest simple
nme. barmouic infinitesimal vibration of a globe of liquid, water at
Gtandard density, or of other perfect liquids at the same density,
may be taken as the unit of time ; for the time of the simple
harmooic vibration of any one of the fundamental modes of a
liquid sphere is independent of the size of the sphere.
Iiet f be the force of gravitational attraction between two
units of matter at unit distance. [Hie force of gnvity at the
surface of a globe of radios r, and density p, is -^f(fr. Hence
if (u be the angular velocity of an infinitesimal satellite, we
have, by tiie equilibrium of centrifugal force and gravity
(§§ 212, 477),
- in ,
and therefore if 7* be the satellite's period,
'wi'fp
(which is equal to the period of a mmple pendulum whose length
is the globe's radius, and weighted end iofinitely near the surface
of the globe). And it has been proved* that if a globe of liquid
be distorted infinitesimally according to a spherical harmonic of
order t, and left at rest, it will perform simple harmonic oscilla-
tions in a period equal to
Hrace if T' denote the period of the graveat, that, namely,
for which t = 2, we have
=Vl
The Bemi-period of an infinitesimal satellite round the earth is
equal, reckoned in seconds, to the square root of the number of
metres in die earth's radius, the metre being very approximately
* "DTTUunical Problems regarding Elastia Bpheioidal Bhelli uid Spheroids
of iDMmpresBible Liquid" (W. Thomson), Phil. Trant. Nov. S7, 1SS2.
..Google
223.] DYNAMICAL LAWS AND PBINCIPLES. 229
the length of the seconds pendulum, whcee period is two
Beconde. Hence taking the earth's mdiuB as 6,370,000 metres, tfnit cT"
and itB density as S^ times that of our standard globe,
r = 3 h. 17 m.
2" = 3 L 40 m.
324. The absolute unit depends on the unit of matter, the
unit of time, and the unit of velocity ; and es the unit of velo-
city depends on the unit of space and the unit of time, there is,
in the definition, a single reference to mass and space, but a
double reference to time; and this is a point that must be par-
ticularly attended to.
223. The unit of mass may be the British imperial pound;
the unit of space the British standard foot; and, accurately
enough for practical purposes for a few thousand years, the unit
of time may be the mean solar second.
We accordingly define the British absolute unit force as "the BrKish*)i
force which, acting on one pound of matter for one second,
generates a velocity of one foot per second." Prof. James
Thomson has suggested the name "Foundal" for this unit of
force.
226. To illustrate the reckoning of force in "absolute measure,"
find how many absolute units will produce, in any particular
locality, the same effect as the force of gravity on a given mass.
To do this, measure the effect of gravity in produciug accelera-
tion on a body unresisted in any way. The most accurate method
is indirect, by means of the pendulum. The result of pendulum
experiments made at Leith Fort, by Captain Kater, is, that the
velocity which would be acquired by a body falling unresiated
for one second is at that place 32207 feet per second. The
preceding formula gives exactly 322, for the latitude 55° 33',
which is approximately that of Edinburgh. The variation in
the force of gravity for one degree of difference of latitude about
the latitude of Edinburgh is only 0000832 of its own amount.
It is nearly the same, though somewhat more, for every degree
of latitude southwards, as far as the southern limits of the
British Isles, On the other hand, the variation per degree is sen-
sibly less, as far north as the Orkney and Shetland Isles. Hence
..Google
230 PBELDnNABY. - [226.
gf^^rf the augmentation of gravity per degree from south to north
j*™«in throughout the British Isles is at most about trimi °^ ••* whole
Enrtio amount in any locality. The average for the whole of Great
Britain and Ireland differs certainly but little from 32-2. Our
present application is, that the force of gravity at Edinburgh is
32'2 times the force which, acting on a pound for a second,
would generate a velocity of one foot per second; in other
words, 322 is the number of absolute units which measures the
weiglit of a pound in this Latitude. Thus, approximately, the
poundal is equal to the gravity of about half an ounce.
227. Forces (since tbey involve only direction and magni-
tude) may he represented, as velocities are, by straight lines in
their directions, and of lengths proportional to their magnitudes,
respectively.
Also the laws of composition and resolution of any number
of forces acting at the same point, are, ae we shall show later
(§ 255), the same as those which we have already proved to
hold for velocities; so that with the substitution of force for
velocity, §§ 26, 27, are still true.
jtototiTi 228. In rectangular resolution the Component of a force in
o(a ftra" any direction, (sometimes called the Effective Component in that
direction,) is therefore found by multiplying the magnitude of
the force by the cosine of the angle between the directions of
the force and the component. The remaining component in this
case is perpendicular to the other.
It is veiy generally convenient to resolve forces into com-
ponents parallel to three lines at right angles to each other;
each such resolution being effected by multiplying by the
cosine of the angle concerned.
G«oiiiftrii»i 229. The point whose distances from three planes at right
w-'ii^iniiTT angles to one another are respectively equal to the mean dis-
of oentn'of tauces of any group of points from these planes, is at a distance
from any plane whatever, equal to the mean distance of the
group from the same plane. Hence of course, if it is in motion,
its velocity perpendicular to that plane is the mean of the velo-
cities of the several points, in the same direction.
..Google
S29.] DTNAHICAL LAWS AND PRIKCIPLB3. 231
Let (x^, y„ e,), etc., be the pomts of the gronp in ntimber i ; 0
and i,§, zha the co-ordinates of a point at distances reepecdvetj pratiminuT
equal to their mean distances from the planes of reference: thatoiwntnof
", , , inertta.
IS to eaj, let
ThoB, if !>„;>„ eta, 63idp, denote the distances of the points in
question firom any pluie at a distance a from the origiti of co-
ordinates, perpendicular to the direction (I, m, n), the sum <^ a
and p, will make up the projectioD of the broken line aj„ y„ «,
on {I, m, n),'and therefore
Pi^&i + myj+jw,— a, etc.;
and similarly, p = l£ + nip + 7ii-a.
Substituting in this last the expressions for it, g, 5, we find
p.+p.+ etc
p=-i — ^ ,
vhich is the theorem to be proved. Hence, of course^
230. The CerUre of Iner^ of a ErjrBtem of equal material g
points (whether connected with one another or not) is the point
whose distance is equal to their average distance from any plane
whatever g 229).
A group of material points of unequal masses may always be
imagined as composed of a greater number of equal material
points, because we may imagine the given material points
divided into dififetent numbers of very small parts. In any
case in which the magnitudes of the given masses are incom-
mensurable, we may approach as near as we please to a rigorous
fulfilment of the preceding statement, by making the parts into
which we divide them sufficiently small.
On this understanding the preceding definition may be ap-
plied to define the centre of inertia of a system of material
points, whether given equal or not. The result is equivalent to
this: —
..Google
232 PRELIMINARY. [230.
The centre of inertia of aii; system of material points what-
ever (whether rigidly connected with one another, or connected
io any way, or quite detached), is a point whose distance from
any plane is equal to the sum of the products of each mass into
its distance from the same plane divided hy the sum of the
We also see, from the proposition stated above, that a point
whose distance from three rectangular planes fulfils this con-
dition, must fulfil this condition also for every other plane.
The co-ordinates of the centre of inertia, of maases 10,, to^
etc, at points {x^, y„ «,), (x^ y^ z^, etc., are given by the foUow-
ing fonnulffi : —
tOjX,-!- to^,+ etc. _ 'S.uxe * Sicy .ttoz
«>,+ Wj+etc. ~"5w' '"Stt' *~ Sw'
Theee formuhe are perfectly general, and can easily be put
into the particular shape required for any given case. Thus,
suppose that, iostaad of a set of detached material points, we
have a continuous distribution of matter through certain definite
portions of space ; the density at x, y, z being p, the elementary
principles of the integral calculus give us at once
jjjpdxdydz \ '
where the integrals extend through all the space occupied by the
mass in question, in which p has a value different &om zero.
The Centre of Inertia or Mass is thus a perfectly definite
point in every body, or group of bodies. The term Centre of
Gravity is often very inconveniently used for it. The theory
of the resultant action of gravity which will be given under
Abstract Dynamics shows that, except in a definite class of
distributions of matter, there is no one fixed point which caji
properly be called the Centre of Gravity of a rigid body. In
ordinary cases of terrestrial gravitation, however, an approxi-
mate solution is available, according to which, in common
parlance, the term "Centre of Gravity" may be used as equi-
valent to Centre of Inertia; but it must be carefully re-
membered that the fundamental ideas involved in the two
definitions are essentially different.
..Google
230.] DYNAMICAL LAWS AND PRINCIPLES. 233
The aecond proposition ia § 229 may now evidently becvDtnot
stated thus: — The sum of the momenta of the parte of the
system in any direction is equal to the momentum in the same
direction of a mass equal to the sum of the masses moving with
a velocity equal to the velocity of the centre of inertia.
231. The Moment of any physical agency is the numerical Homent
measure of its importance. Thus, the moment of a force round
a point or round a liae, signifies the measure of its importance
as regards producing or balancing rotation round that point or
round that line.
232. The Moment of a force about a point is defined aa the J*g,^' "*
product of the force into its perpendicular distance from thej^^'
point. It is numerically double the area of the triangle whose
vertex is the point, and whose base is a line representing the
force in magnitude and dii'ection. It is often convenient to
represent it by a line numerically equal to it, drawn through
the vertex of the triangle perpendicular to its plane, through
the front of a watch held in the plane with its centre at the
point, and facing so that the force tends to turn round thisj'g^^"'
point in a direction optJosite to the bands. The moment of a J^*"
force round any axis is the moment of its component in any
plane perpendicular to the asis, round the point in which the
plane is cut by the axis. Here we imagine the force resolved
into two components, one parallel to thS axis, which is ineffective
so far as rotation round the axis is concerned; the other perpen-
dicular to the axis (that is to say, having its line in any plane
perpendicular to the axis). This latter component may be called
the eflfective component of the force, with reference to rotation
round the axis. And its moment round the axis may be defined
as its moment round the nearest point of the axis, which is
equivalent to the preceding definition. It is clear that the
moment of a force round any axis, is equal to the area of the
projection on any plane perpendicular to the axis, of the figure
representing ite moment round any point of the axis.
233. The projection of an area, plane or curved, on any Dicnniat
plane, is the area included in the projection of its bounding tSnof
line.
..Google
234 PEELIMINART. [233.
If we imagine an area divided into any number of parta, the
projections of these parts on any plane make up the projection
of the vhole. But in this statement it must be understood that
the areas of partial projections are to be reckoned aa positive if
particular sides, which, for brevity, we may call the outside of
the projected area and the front of the plane of projection, face
the same way, and negative if they face oppositely.
Of course if the projected surface, or any part of it, be a plane
area at right angles to the plane of projection, the projection
vanishes. The projections of any two shells having a common
edge, on any plane, are equal, but with the same, or opposite,
signs as the case may be. Hence, by taking two such shells
facing opposite ways, we see that the projection of a closed
surface (or a shell with evanescent edge), on any plane, is
nothing.
Equal areas in one plane, or in paxaUel planes, have equal
projections on any plane, whatever may be their figures.
Hence the projection of any plane figure, or of any shell,
edged by a plane figure, on another plane, is equal to its area,
multiplied by the cosine of the angle at which its plane is in-
dined to the plane of projection. This angle is acute or obtuse,
according as the outside of the projected area, and the &ont of
plane of projection, face on the whole towards the same parts,
or oppositely. Hence lines representing, as above described,
moments about a point in different planes, are to be com-
pounded as forces are. — See an analogous theorem in § 96.
234. A Couple is a pair of equal forces acting in dissimilar
directions in parallel lines. The Moment of a couple is the
sum of the moments of its forces about any point in their plane,
and is therefore equal to the product of either force into the
shortest distance between their directiona This distance is called
the Arm of the couple.
The Aa:i3 of a Couple is a line drawn from any chosen point
of reference perpendicular to the plane of the couple, of such
magnitude and in such direction as to represent the magnitude
of the moment, and to indicate the direction in which the couple
tends to turn. The most convenient rule for fulfilUng the
latter condition is this: — Hold a watch with its centre at the
..Google
234.] DYNAMICAL LAWS AND FSINCIPLE9. 235
point of reference, and with its plane parallel to the plane ofceopk.
the couple. Then, accordiog as the motion of the hands ia
contrary to or along with the direction in which the couple
tends to turn, draw the axis of the couple through the f&ce
or through the back of the watch, from its centre. Thus a
couple ia completely represented by its axis ; and couples are to
be resolved and compounded by the same geometrical construc-
tions performed with reference to their axes as forces or velo-
cities, with reference to the lines directly representing them.
336. If we substitute, for the force in § 232, a velocity, we Uoment oi
have the moment of a velocity about a point ; and by intro-
ducing the mass of the moving body as a factor, we have an
important element of dynamical science, the Sfometii of MomeA- »
turn. The laws of composition and resolution are the same ''
as those already explained ; but for tbe sake of some simple
applications we give an elementary investigation.
The moment of a rectilineal motion ia the product of its Hommt of
length into the distance of its line from the point. dillpUal!
The moment of the resultant velocity of a particle about any
point in the plajie of tbe components is equal to the algebraic
sum of the moments of the components, the proper sign of each
moment being determined as above, § 233. The same is of
course true of moments of displacements, of moments of forces
and c^ moments of momentum.
First, consider two component motions, ASaxA AC, and let Tor two
AD be their resultant % 27). Their half moments round the motioiii,
point 0 are respectively the areas OAB, OCA. Now OC/jl,t* "no-
together with half the area of the parallelogram CABD, iscmepiMfl.
equal to OBJ), Hence the sum of the two half moments theif mo-
together with half the area of the parallelogram, is equal to''™Tt-,h
AOB together with BOD, that is to say, to the area of the{g™^'<^
whole figure OABD. But ABD, a part
of this figure, is equal to half tbe area of
the parallelc^ram; and therefore the re-
mainder, OAD, is equal to the sum of
the two half moments. But OAD is half
the moment of the resultant velocityround
the point 0. Hence the moment of the .
..Google
23G pEELiMmART. [235.
resultant ia equal to the Bum of the moments of the two com-
ponents.
If there are any number of component rectilineal moiions in
one plane, we may compound them in order, any two taken
together first, then a third, and eo on ; and it follows that the
sum of their moments is equal to the moment of their resultant.
. It follows, of course, that the sum of the moments of any number
of component velocities, all in one plane, into which the velo-
city of any point may be resolved, is equal to the moment of
their resultant, round any point in their plane. It follows also,
that if velocities, in different directions all in one plane, be
successively given to a moving point, eo that at any time ita
velocity ia their resultant, the moment of its velocity at any
time is the sum of the moments of all the velocities which have
been successively given to it.
Cor. — If one of the components always passes through the
point, its moment vanishes. This is the case of a motion in
which the acceleration is directed to a fixed point, and we thus
reproduce the theorem of § 36, a, that in this case the areas
described by the radius-vector are proportional to the times ;
for, as we have seen, the moment of velocity is double the area
traced out by the radius-vector in unit of time.
236. The moment of the velocity of a point round any axis
is the moment of the velocity of ita projection on a plane per-
pendicular to the axis, round the point in which the plane is cut
by the axis.
I The moment of the whole motion of a point during any
time, round any axis, ia twice the area described in that time
by the radius-vector of its projection on a plane perpendicular to
that axis.
If we consider the conical area traced by the radius-vector
drawn from any fixed point to a moving point whose motion is
not confined to one plane, we see that the projection of this area
on any plane through the fixed point is half of what we have
just defined as the moment of the whole motion round an axis
perpendicular to it through the fixed point. Of all these
pianos, there is one on which the projection of the area is greater
..Google
236.] DYNAMICAL LAWa AND PRINCIPLES. 237
than on any other ; and the projection of the conical area on Momeni at
any plane perpendicular to this plane, is equal to nothing, the motian,
proper interpretation of positive and negative projections being "^
used.
If any number of moving points are given, we may similarly
consider the conical surface described by the radius-vector of
each drawn from one fixed point. The same statement applies
to the projection of the many-sheeted conical surface, thus pre-
sented. The resultant axis of the whole motion in any finite Bnoitant
time, round the fixed point of the motions of all the moving ""'
points, is a line through the fixed point perpendicular to the
plane on which the area of the whole projection is greater than
on any other plane ; and the moment of the whole motion round
the resultant axis, is twice the area of this projection.
The resultant axis and moment of velocity, of any number of
moving points, relatively to any fixed point, are respectively the
resultant axis of the whole motion during an infinitely short
time, and its moment, divided by the time.
The moment of the whole motion round any axis, of the
motion of any number of points during any time, is equal
to the moment of the whole motion round the resultant axis
through any point of the former axis, multiplied into the cosine
of the angle between the two axes.
The resultant axis, relatively to any fixed point, of the whole
motion of any number of moving points, and the moment of
the whole motion round it, are deduced by the same elemen-
taiy constructions from the resultant axes and moments of the
individual points, or partial groups of points of the system, as
the direction and magnitude of a resultant displacement are
deduced firom any given lines and magnitudes of component Homuit of
displacements.
Corresponding statements apply, of course, to the moments of
velocity and of momentum.
237. If the point of application of a force be displaced J}J*^
through a small space, the resolved part of the displacement in
the direction of the force has been called its Virtual Velocity.
..Google
238 PBEUIUHART. [237.
Tliia 18 positive or oegative accordiog &a the virtual velocity is
in the same, or in the opposite, direction to that of the force.
The product of the force, into the virtual velocity of its point
of application, has been called the Virtual Moment of the force.
These terms ve have introduced since they stand in the histoiy
and developments of the science ; but, as we shall show further
on, they are inferior substitutes for a far more useful set of ideas
clearly laid down by Newton.
238. A force is said to do work if its place of application
has a positive component motion in its direction ; and the work
done by it is measured by the product of its amount into this
component motion.
Thus, in lifting coals from a pit, the amount of work done is
proportional to the weight of the coals lifted ; that is, to the
force overcome in rising them ; and also to the height through
which they are raised. The unit for the measurement of work
adopted in practice by British engineers, is that required to
overcome a force equal to the gravity of a pound through the
space of a foot; and is called a Foot-Pound.
In purely scientific measurements, the unit of work is not
the foot-pound, but the kinetic unit force (§ 226) acting through
unit of space. Thus, for ezample, as we shall show further on,
this unit is adopted in measuring the work done by an electric
current, the units for electric and m^netic measurements being
founded upon the kinetic unit force.
If the weight be raised obliquely, as, for instance, along a
smooth inclined plane, the space through which the force hae
to be overcome is increased in the ratio of the length to the
height of the plane ; but the force to be overcome is not the
whole gravity of the weight, but only the component of the
gravity parallel to the plane ; and this is less than the gravity
in the ratio of the height of the plane to its length. By
, multiplying these two expressions together, we find, as we
might expect, that the amount of work required is unchanged
by the substitution of the oblique for the vertical path.
239. Generally, for any force, the work done during an
infinitely small displacement of the point of application is the
..Google
239.] DTHAMICAL LAWS AND FIUNCIPLES. S39
virtual moment of the force (§ 237), or is the product of the ?"^*^ *
reBolved part of the force in the direction of the displacement
into the displacement.
From this it appears, that if the motion of the point of
application be always perpendicular to the direction in which
a force acts, such a force does no work. Thus the mutual
normal pressure between a fixed and moving body, as the
tension of the cord to which a pendulum bob is attached, or
the attraction of the sun on a planet if the planet describe a
circle with the son in the centre, is a case La which no work is
done by the force.
240. The work done by a force, or by a couple, upon a body ?Sjii^'
turning about an axis, is the product of the moment of the
force or couple into the angle (in radians, or fraction of a radian)
through which the body acted on turns, if the moment remains
the same in all portions of the body. If the moment be varia-
ble, the statement is only valid for infinitely small displace-
ments, but may be made accurate I7 employing the proper
average moment of the force or of the couple. The proof is
obvious.
If ^ be the moment of tbe force or couple for a position of
the body givrai by the angle 6,Q{6^-6^ i£ Q ia oonstftnt, or
I QM°q(fi^—9^ where q is the proper average valne of Q
when variable^ is the work done by the couple during the rotation
&om 0, to 0,.
241, Work done on a body by a force is always shown by a J^''?™-
corresponding increase of vis viva, or kinetic energy, if no other *ork.
forces act on the body which can do work or have work done
gainst them. K work be done against any forces, the increase
of kinetic energy is less thaQ in the former case by the amount
of work BO done. In virtue of this, however, the body possesses
an equivalent in the form of PoterUial Energy (§ 273), if its Pot«itf*l
physical conditions are such that these forces will act equally,
and in the same directions, if the motion of the system is
reversed. Thus there may be do change of kinetic energy pro-
..Google
240 PRELIMIHART. [241.
duced, and the work done may be wholly stored up as potential
energy.
Thus a weight requires work to raise it to a height, a spring
requires work to bend it, air requires work to compress it, etc;
but a raised weight, a bent spring, compressed air, etc., are
stores of energy which can be made use of at pleasure.
242. In what precedes we have ^ven some of Newton's
Definitiones nearly in his own words ; others have been enun-
ciated in a fonu more suitable to modem methods ; and some
terms have been introduced which were invented subsequent
to the publication of the Principia, But the Axiomata, give
Leges Mot&s, to which we now proceed, are given in Newton's
own words ; the two centuries which have nearly elapsed since
he first gave them have not shown a necessity for any addition
or modification. The first two, indeed, were discovered by
Galileo, and the third, in some of its many forms, was known
to Hooke, Huyghens, Wallis, Wren, and others; before the
publication of the Principia. Of late there has been a tendency
to split the second law into two, called respectively the second
and third, and to ignore the third entirely, though using it
directly in every dynamical problem ; but all who have done so
have been forced indirectly to acknowledge the completeness of
Newton's system, by introducing as an axiom what is called
D'Alembert's principle, which is really Newton's rejected third
law in another form. Newton's own interpretation of his third
law directly points out not only D'Alembert's principle, but also
the modem principles of Work and Energy.
243. An Axiom is a proposition, the truth of which must
be admitted as soon as the terms in which it is expressed are
clearly understood. But, as we shall show in our chapter on
" Experience," physical axioms are axiomatic to those only who
have sufficient knowledge of* the action of physical causes to
enable them to see their truth. Without further remark we
shall give Newton's Three Laws ; it being remembered that^ as
the properties of matter might have been such as to render a
totally different set of laws axiomatic, these laws must be con-
..Google
243.] DYNAMICAL LAWS AND PRINCIPLES. 24t
sidered as resting on convictions drawn from observation and
experiment, not on intuitive perception.
344. Lex I, Corpus omneperseverare in statu sua qu{esceadis»wton'i
vel movendi uniformtter in directum, nisi quatenus illud d virihas
impressis cogitur statum suum mtttare.
Every body continues in its state of rest or of uniform motion
in a araight line, except in so far ae it may he compelled by
force to change thai state.
245. The meaning of the term Rest, in physical science "Bft-
is essentially relative. Absolute rest is undefinable. If the
universe of matter were finite, its centre of inertia might fairly
be considered as absolutely at rest ; or it might be imagined to
be moving with any uniform velocity in any direction whatever
through infinite space. But it is remarkable that the first law
of motion enables us (§ 249, below) to explain what may be
called directional rest. As will soon be shown, § 267, the plane
.in which the moment of momentum of the universe (if finite)
round its centre of inertia is the greatest, which is clearly de-
terminable from the actual motions at any instant, is fixed in
direction in space.
24$. We may logically convert the assertion of the first law
of motion as to velocity into the following stateujents : —
The times during which any particular body, not compelled
by force to alter the speed of its motion, passes through equal
spaces, are equal. And, i^ain — Every other body in the uni-
verse, not compelled by force to alter the speed of ita motion,
moves over equal spaces in successive intervals, during which
the particular chosen body moves over equal spaces.
247. The first part merely expresses the convention uni- tidk-
versally adopted for the measurement of Time. The earth, in
its rotation about its axis, presents us with a case of motion in
which the coudition, of not being compelled by force to alter
its Efteed, is more nearly fulfilled than in any other which
we can easily or accurately observe. And the numerical
measurement of time practically rests on defining equal inter-
vats of time, as times during which the earth turns through equal
VOL. 1. 16
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242 FBELIXINABT. [2i7.
angles. This is, of course, a mere convention, and not a law of
nature ; and, aa we now see it, is a part of Newton's fiist lav.
it 248. The remainder of the law is not a convention, but a
"" "*' great truth of nature, which we may illustrate by referring to
small and trivial cases aa well as to the grandest phenomena we
can conceive.
A curling-stone, projected along a horizontal surface of ice,
travels equal distances, except in ho far as it is retarded by
friction and by the resistance of the air, in successive intervals
of time during which the earth turns through equal angles.
The sun moves through equal portions of interstellar space in
times during which the earth turns through equal angles, except
in so far as the resistance of interstellar matter, and the attrac-
tion of other bodies in the universe, alter his speed and that of
the earth's rotation,
DinetioTui 249. If two material points be projected from one position,
A,a.i the same instant with any velocities in any directions,
and each left to move uninfluenced by force, the line joining
them will be always parallel to a iixed direction. For the law
asserts, as we have seen, that AP :AP';: AQ lAQf, if P, Q, and
again P', Q' are simultaneous positions ; and therefore FQ is
parallel to PQ. Hence if four material points 0, P, Q, R are
all projected at one instant from one position, OP, OQ, OB
The'inn- are fixed directions of reference ever after. But, practically,
piKw" the determination of fixed directions in space, § 267, is made to
•ritcm. depend upon the rotation of groups of particles exerting forces
on each other, and thus involves the Third Law of Motion.
250. The whole taw is singularly at variance with the tenets
of the ancient philosophers who maintained that circular motion
is perfect.
The last clause, "niai quatenua," etc., admirably prepares for
the introduction of the second law, by conveying the idea that
it u force alone ivhich can produce a change of motion.. How,
we naturally inquire, does the change of motion produced
depend on the magnitude and direction of the force which
produces it ? And the answer is —
..Google
251.] DYNAMICAL LAWS AKD PRINCIPLES. 248
251. Lex II. Mutationem tnoti&a prtyporHonaUm esse vi n
motrid impresaw, et fieri secundum, lineam rectam qud vU ilia "
imprimitur.
Change of motion is proportional to fwce applied, and takes
place in the direction of the straight line in which Vie force acts.
262. If any force geoeratea motioo, a double force will
generate double motioo, and so on, whether simultaneously or
Buccessively, instantaneously, or gradually applied. And this
niotioQ, if the body was moving beforehand, is either added
to the previous motion if directly conspiring with it ; or is
subtracted if directly opposed ; or is geometrically compounded
with it, according to the kinematical principles already ex-
plained, if the line of previous motion and the direction of the
force are inclined to each other at an angle. (This is a para-
phrase of Newton's own comments on the second law.)
253. In Chapter i. we have considered change of velocity,
or acceleration, as a purely geometrical element, and have seen
how it may be at once inferred from the given initial and final
velocities of a body. By the definition of quantity of motion
(§ 210), we see that, if we multiply the chajige of velocity,
thus geometrically determined, by the mass of the body, we
have the change of motion referred to in Newton's law aa the
measure of the force which produces it.
It is to be particularly noticed, that in this statement there
is nothing said about the actual motion of the body before it
was acted on by the force : it is only the chaise of motion that
concerns us. Thus the same force will produce precisely the
same change of motion in a body, whether the body be at rest,
or in motion with any velocity whatever.
251, Again, it is to be noticed that nothing is said as to the
body being under the action of one force only ; so that we
may logic^ly put a part of the second law in the following
(apparently) amptt6ed form : —
When any forces whatever act on a body, Oien, whether the
body he oriffin<dly at rest or moving with any velocity and in any
direction, each force produces in the body the exact change of
16—2
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244 PiiELiHmABT. [254.
motion which it would have produced if it had acted aingty on
the body originally at rest.
255. A remarkable consequence follows immediately from
this view of the second law. Since forces are measured by the
cbaogeB of motion they produce, aod their directions assigned
by the directions in which these changes are produced; and
since the changes of motion of one and the same body are in
the directions of, and proportiooal to, the changes of velocity —
a single force, measured by the resultant chauge of velocity,
and in its direction, will be the equivalent of any number of
simultaneously acting forces. Hence
■ The resultant of any number offerees (applied at one point) is
tobefouJid by the same geometrical process as the resultant of any
number of simultaneous velocities.
256. From this follows at once (§ 27) the construction of
the Parallelogram of Forces for imdiug the resultant of two
forces, and the Polygon of Forces for the resultant of any num-
ber of forces, in lines nil through one point
The case of the equilibrium of a number of forces acting at
one point, is evidently deducible at once from this ; for if we
introduce one other force equal and opposite to their resultant,
this will produce a change of motion equal and opposite to the
resultant change of motion produced by the given forces ; that
is to say, will produce a condition in which the point expe-
riences no change of motion, which, aa we have already seen, is
the only kind of rest of which we can ever be conscious.
S67. Though Newton perceived that the Parallelogram of
Forces, or the fundamental principle of Statics, is essentially
involved in the second law of motion, and gave a proof which
is virtually the same as the preceding, subsequent writers on
Statics {especially in this country) have very generally ignored
the fact ; and the consequence has been the introduction of
various unnecessary Dynamical Axioms, more or less obvious,
but in reality included in or dependent upon Newton's laws
of motion. We have retained Newton's method, not only on
account of its admirable simplicity, but because we believe it
..Google
257.] DYNAMICAL LAWS AM) PEINCIPLES, 245
contains tlie most philosophical foundatioa for the static as well
as for the kinetic branch of the dynamic ecience.
25S. But the second lair gives us the means of measuring Monm.
force, and also of measuring the mass of a body. ^^'>1^
For, if we consider the actions of various forces upon the """^
same body for equal times, we evidently have changes of
velocity produced which are proportiojial to the forces. The
changes of velocity, then, give ua in this case the means of
comparing the magnitudes of different forces. Thus the velo-
cities acquired in one second by the same mass (falling freely]
at different parts of the earth's surface, give us the relative
amounts of the earth's attraction at these places.
Again, if equal forces be exerted on different bodies, the
changes of velocity produced in equal times must be inversely
as the masses of the various bodies. This is approximately the
case, for instance, with trains <^ various lengths started by the
same locomotive : it is exactly realized in such cases as
the action of an electrified body on a number of solid or hollow
spheres of the same external diameter, and of different metals
or of different thicknesses.
Again, if we find a case in which different bodies each acted
on by a force, acquire in the same time the same changes of
velocity, the forces must be proportional to the masses of the
bodies. This, when the resistance of the air is removed, is the
case of falling bodies ; and from it we conclnde that the weight
of a body in any given locality, or the force with which the
earth attracts it, is proportional to its mass; a most important
physical truth, which will be treated of more carefully in the
chapter devoted to " Properties of Matter."
259. It appears, lastly, from tbia law, that every theorem of Tt«iu>
Kinematics connected with acceleraticu has its counterpart in tb« kine-
A-inettCS. point.
For instance, suppose X, T, Z to be the componeate, parallel
to fixed axes ol x,y, g respectivelT', of the whole foroe acting on
a particle of mass Jif. We see by g 212 that
^t=^. <^^. ^s=^^
or Mx = Z, My=7, Mz^Z.
Google
246 PRELIMINABT. [259.
f'" ftum Also, from thoM, we m»y evidently write,
nutkacf M
P P^'*" P~''° P~'^' '
The second members of these equations are respectively tbe com-
ponents of the impressed force, along the tangent (§ 9), perpen-
didilar to the oscillating plane (§ 9), and towards the centre <^
currature, of the path described.
260. We hare, by means of the first two laws, arrired at a
definition and a measure of force ; and have also found how to
compoundj and therefore also how to resolve, forces ; and also
how to investigat* the motion of a single particle subjected to
given forces. But more is required before we can completely
uuderatand the more complex cases of motion, especially those
in which we have mutual actions between or amongst two or
more bodies ; such ae, for instance, attractions, or pressures, or
transference of energy in any form. This is perfectly supplied
by
261. Lex III. Actioni contrariam semper et (sgttatem esse
reactionem : sive corporum duorum actiones in te muf im) eemper
esse aquaies et in partes conta-arias dirigi.
To every action there is always an eqval and contrary re'
action: or, the mutual actions of avy ttoo bodies are always equal
and oppositely directed.
262. If one body presses or draws another, it is pressed or
drawn by this other with an equal force in the opposite direc-
tion. If any one presses a stone with his finger, bis finger is
pressed with the same force in the opposite direction by the
stone. A horse towing a boat on a canal is dragged back-
wards by a force equal to that which he impresses on the
towing-rope forwards. By whatever amount, and in whatever
direction, one body has its motion changed by impact upon
another, this other body has its motion changed l^ the same
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262.] DTNAinCAL LAWS AND PBIKCIPLE3. 247
amonnt in the opposite direction; for at each instaiit during
the impact the force between them was equal and opposite on
the two. When neither of the two bodies has any rotation,
whether before or after impact, the changes of velocity which
they experience are inversely as their masses.
When one body attracts another from a distance, this other
attracts it with aa equal and opposite force. This law holds
not only for the attraction of gravitation, but also, as Newton
himself remarked and verified by experiment, for magnetic
attractions : also for electric forces, as tested by Otto-Guericke.
263. What precedes is founded upon Newton's own com-
ments on the third law, and the actions and reactions con-
templated iu-e simple forces. In the scholium appended, he
makes the following remarkable statement, introducing another
description of actions and reactions subject to his third law,
the full meaning of which seems to have escaped the notice of
commentators : —
Si oBstimetur agmitCa actio ex ejus vi at velooitaie conjunct m ;
0t similiter reaietentis reactio cBsttmetur conjimctim ex ijvspartiwn
singvlarum velocitatibus et viribus resisiendi ah eanim aitritione,
cohcBsione, pondere, et accderatione oriundia; eruat actio et rmdio,
in omnt inatrumentorum u»u, nbi invicem semper CBquaUs.
In a previous discussion Newton has shown what is to be
understood by the velocity of a force or resistance ; %.e., that it
is the velocity of the point of application of the force resolved
in the direction of the force. Bearing this in mind, we may
read the above statement as follows : —
JftheA ctivity* of an agent be measured by its amount and its
vdocity conjointly; and if, similarly, the Counter-activity of the
resistance be measured by the velocities of its several parts and
their ^wral amovaUs conjointly, whetherthese arise from friction,
cohesion, weight, or acceleraiion ; — Activity and Cownier-acUvity,
in aU comMnations of machines, will be equal and opposite.
Farther on (§§ 264, 293) we shall give an account of the
* We tranekte Neirton's vord "Aetio"'bBTe b; " Activity "to avoid oonfttrion
with the wocd " Action" so QuiTerBollj used in modem dynamical treatiMa, ao-
oordiog to tha definition of g 326 belov, in lelation to Uftnpertnia' piineiide of
"Iicaet Aetion."
..Google
248 FBELIMINABT. [263.
splendid dynamical theory fouDded by D'Alembert and La-
grange on this most important remark.
D'&iem- 261. Newton, in the passage just quoted, points out that
oipia.**" forces of resistance against acceleration are to be reckoned as
reactions equal and opposite to the actions by which the ac-
celeration is produced. Thus, if we consider any one material
point of a system, its reaction against acceleration must be
equal and opposite to the resultant of the forces which that
point experiences, whether by the actions of other parts of the
eiystem upon it, or by the influence of matter not belonging to
the system. In other words, it must be in equilibrium with
these forces. Hence Newton's view amounts to this, that all the
forces of the system, with the reactions against acceleration of
the materia! points composing it, form groups of equilibrating
systems for these points considered individually. Hence, by
the principle of superposition of forces in equilibrium, all the
forces acting on points of the system form, with the reactions
against acceleration, an equilibrating set of forces on the whole
system. This is the celebrated principle first explicitly stated,
and very usefully applied, by D'Alembert in 1742, and still
known by his name. We have seen, however, that it is very
distinctly implied in Newton's own interpretation of his third
law of motion. As it is usual to investigate the general equa-
tions or conditions of equilibrium, in dynamical treatises, before
entering in detail on the kinetic branch of the subject, this
principle is found practically most useful in showing how we
may write down at once the equations of motion for any
system for which the equations of equilibrium have been in-
vestigated.
Hutini 266. Eveiy rigid body may be imagined to be divided into
t^I^puti. indefinitely small parts. Now, in whatever form we may
rigid'bSdy. eventually find a physical explanation of the origm of the forces
which act between these parts, it is certain that each such
small part may be considered to be held in its position
relatively to the others by mutual forces in lines joining them.
266. From this we have, as immediate consequences of the
second and third laws, and of the preceding theorems relating
..Google
2C6.] DYNAMICAL I^WS AND PBINCIPLES. 249
to Centre of Inertia and Moment of Momentum, a number of
important propositions such as the following : —
(o) The centre of inertia of a rigid body moving in any uatton of
3r, but free from external forces, moves uniformly in ainertiaori
straight line.
(&) When any forces whatever act on the body, the motion of
the centre of inei-tia is the same as it would have been had
these forces been applied with their proper magnitudes and
directions at that point itself.
(c) Since the moment of a force acting on a particle is the Woment of
same as the moment of momentum it produces in unit of time, oi a rigid
the changes of moment of momentum in any two parts of a
rigid body due to their mutual action are equal and opposite.
Hence the moment of momentum of a rigid body, about any axis
-which is fixed in direction, and passes through a point which
is either fixed in space or moves uniformly in a straight line, is
unaltered by the mutual actions of the parts of the body.
{d) The rate of increase of moment of momentum, when the
body iij acted on by external forces, is the sum of the moments
of these forces about the axis.
267. We shall for the present take for granted, that thec«u«Tra>
mutual action betweeu two rigid bodies may in every case be mamwtani,
imagined as composed of pairs of equal and opposite forces ment of
ID straight lines. From this it follows that the sum of the
quantities of motion, parallel to any fixed direction, of two
rigid bodies influencing one another in any possible way, re-
mains unchanged by their mutual action; also that the sum
of the moments of momentum of all the particles of the two
bodies, round any line in a fixed direction in space, and passing
through any point moving uniformly in a straight line in any
direction, remains constant. From the first of these propositions
we infer that the centre of inertia of any number of mutually
infiuencing bodies, if in motion, continues moving uniformly
in a strmght line, unless in so far as the direction or velocity
of its motion is changed by forces acting mutually between
them and some other matter not belonging to them ; also that
the centre of inertia of any body or system of bodies moves
..Google
250 pRELoimARr. [267.
Th«" Inn- just a3 all tlieir matter, if concentrated in a point, would move
Piuw"ii> under the influence of forces equal and parallel to the forces
through the really acting on its different parts. From the second we infer
'""di^ET" *''** ^^^ "^"^ '■'^ resultant rotation through the centre of inertia
■uitut'uii. **^ ^^y system of hodies, or through any point either at rest or
moving uniformly in a straight line, remains unchanged in
direction, and the sum of moments of momenta round it
remains constant if the system experiences no force from with-
out. This principle used to be called Conaervalion of Areas,
Temrtriiii a very ill-considered designation. From this principle it follows
that if by internal action such as geological upheavals or suh-
sidenceii, or pressure of the winds on the water, or by evapora-
tion and rain- or snow-fall, or by any influence not depending
on the attraction of sun or moon (even though dependent on
solar heat), the disposition of land and water becomes altered,
the component round any fixed axis of the moment of momen-
tum of the earth's rotation remains constant,
B^ of 368. The foundation of the abstract theory of energy is laid
by Newton in an admirably distinct and compact manner in the
sentence of his scholium already quoted (§ 263), in which he
points out its application to mechanics*. The actio ageiUis,
as he defines it, which is evidently equivalent to the product of
the effective component of the force, into the velocity of the
point on which it acts, is simply, in modem English phrase-
oli^g?) tlie rate at which the t^ent works. The subject for
measurement here is precisely the same as that for which Watt,
hundred years later, introduced the practical unit of a "Horse-
jwer," or the rate at which an agent works when overcoming
3,000 times the weight of a pound through the space of a foot
L a minute ; that is, producing 550 foot-pounds of work per
icond. The unit, however, which is most generally convenient
that which Newton's definition implies, namely, the rate of
oing work in which the unit of energy is produced in the unit
f time.
* Th« leader will remember that -ne use the word " mechanioB " in its true
asBioal seuse, the aaieoee of moebiaes, the BSDSe in irhiob Kewton himMlI
led it, when he dismissed the farther oondderation of it by Ba^g (in tb«
holiom ratened to), Catenm nuchanieim Iraetart mm Mt Am'iu tnitttNtf.
..Google
2G9.] DYNAMICAL LAWS AND PHINCIPLES. 251
269. Looking at Newton's worda (§ 263) in this light, we E
see that they may he logically converted into the following
form : —
Work done on any system of bodies (in Nemton's state-
ment, the parts of any machine) has its equivalent in work done
against friction, molecular forces, or gravity, if there be no
acceleration ; hut if there be acceleration, part of the work is
expended in overcoming tlie resistance to acceleration, and the
additional kinetic energy developed is equivalent to the work
80 spent. This is evident from § 214.
When part of the work is done against molecular forces, as
in bending a spring ; or against gravity, as in raising a weight ;
the recoil of the spring, and the fall of the weight, are capable
at any future time, of reproducing the work originally expended
(§ 241). But in Newton's day, and long afterwards, it was
supposed that work was absoluteti/ lost by friction ; and, indeed,
this statement is still to he found even in recent authoritative
treatises. But we must defer the examination of this point till
■we consider in its modem form the principle of Conaervaiion of
Energy.
270. If a system of bodies, ^ven either at rest or in
motion, he influenced by no forces from without, the sum of the
kinetic enei^es of all its parts is augmented in any time by an
amount equal to the whole work done in that time by the
mutual forces, which we may imagine as acting between its
points. When the lines in which these forces act remain all
unchanged in length, the forces do no work, and the sum of the
kinetic energies of the whole system remains constant. If, on
the other band, one of these lines varies in length during the
motion, the mutual forces in it will do work, or will consume
work, according as the distance varies with or against them.
271. A limited system of bodies is said to be dynamically
conservative (or simply conservative, when force is understood to
be the Subject), if the mutual forces between its parts always
perform, or always consume, the same amount of work during
any motion whatever, by which it can pass from one particular
configuration to miotber.
..Google
PRELWISABT. [272.
founded on the following proposition: —
If the mutual forces between the parts of a material system
are independent of their velocities, whether relative to one
another, or relative to any external matter, the system must be
dynamically conservative.
For if more work is done by the mutual forces on the
different parts of the system in passing from cue particnlar
phyiicd configuration to another, by one set of paths than hy another
^hTp^* set of paths, let the system be directed, by Mctiooless con-
Sotioiiii straint, to pass from the first configuration to the second by
introduced. One Set of paths and return by the other, over and over ag^n
for ever. It will be a continual source of energy witbout any
consumption of materials, vhicb is impossible.
Potcntud 273. The poteniial energy of a conservative system, in the
xm-' configuration which it has at any instant, is the amount of work
' required to bring it to that configuration against its mutual
forces during the passage of the system from any one chosen
configuration to the configuration at the time referred to. It
is generally, but not always, convenient to fix the particular
configuration chosen for the zero of reckoning of potential
enei^, so that the potential energy, in every other configuration
practically considered, shall be positive.
274. The potential energy of a conservative system, at any
instant, depends solely on its configuration at that instant,
being, according to definition, the same at all times when the
system is brought again and again to the same configuration.
It is therefore, in mathematical language, said to be a function
of the co-ordinates by which the positions of the different parts
of the system are specified. If, for example, we have a conser-
vative system consisting of two material points; or two rigid
bodies, acting upon one another with force dependent only on
the relative position of a point belonging to one of them, and a
point belonging to the other; the potential enei::gy of the
system depends upon the co-ordinates of one of these points
relatively to lines of reference in fixed directions through the
othef. It will therefore, in general, depend on three iodepen-
..Google
274.] DYNAMICAL LAWS AND PRINCIPLES. 253
dent co-ordinates, which we may conveniently take aa the dis- Potenttai
tance between the two points, and two angles specifying the amKi-vn-
absolute direction of the line joining them. Thus, for example,
let the bodies be two uniform metal globes, electrified with any
given quantities of electricity, and placed in an insulating
medium such as mr, in a region of space under the influence
of a vast distant electrified body. The mutual action between
these two spheres will depend solely on the relative position o£
their centres. It will consist partly of gravitation, depending
solely on the distance between their centres, and of electric
force, which will depend on the distance between them, but
also, in virtue of the inductive action of the distant body, will
depend on the absolute direction of the line joining their
centres. In our divisions devoted to gravitation and electricity
respectively, we shall investigate the portions of the mutual
potential energy of the two bodies depending on these two
f^encies separately. The former we shall find to be the pro-
duct of their masses divided by the distance between their
centres; the latter a somewhat complicated function of the
distance between the centres and the angle which this line
makes with the direction of the resultant electric force of the
distant electrified body. Or again, if the system consist of two
balls of soft iron, in any locality of the earth's surface, their
mutual action will be partly gravitation, and partly due to the
magnetism induced in them by terrestrial magnetic force. The
portion of the mutual potential energy depending on the latter
cause, will be a function of the distance hetween their centres
and the inclination of this line to the direction of the terrestrial
magnetic force. It will agree in mathematical expression with
the potential enei^ of electric action in the preceding case, so
far as the inclination is concerned, but the law of variation with
the distance will be less easily determined.
276. In nature the hypothetical condition of § 271 is appa- inarittbia
renily violated in all circumstances of motion. A material system men^ of
can never be brought tlirough any returning cycle of motion SJm. "^
without spending more work against the mutual forces of its
parts than is gained from these forces, because no relative
motion can take place without meeting with frictional or
..Google
251 PB£Ln[[IfABT. [S75.
other forms of resistaDce ; among which are included (1)
ansnoi mutual frictloD between aolids sliding upon one another; (S)
motion*, resistances due to the viscosity of fluids, or imperfect elasticity
of solids; (3) resistances due to the induction of electric cur-
rents; (4) resistances due to varying magnetization under the
influence of imperfect magnetic retentivenesa. No motion in
nature con take place without meeting resistance due to some,
if not to all, of these influences. It is matter of every day
experience that friction and imperfect elasticity of solids impede
the action of all artificial mechanisms; and that even when
bodies are detached, and left to move freely in the air, as falling
bodies, or as projectiles, they experience resistance owing to the
viscosity of the air.
The greater masses, planets and comets, moving in a less
resisting medium, show leas indications of resistance*. Indeed
it cannot be said that observation upon any one of these bodies,
with the exception of Encke's comet, has demonstrated resist-
ance. But the analogies of nature, and the ascertained facts of
physical science, forbid us to doubt that every one of them,
every star, and every body of any kind moving in any part of
space, has its relative motion impeded by the air, gas, vapour,
medium, or whatever we choose to call the substance occnpyiog
the space immediately round it; just as the motion of a rifle
bullet is impeded by the resistance of the air.
BflMtof 276. Hiere are also indirect resistances, owing to friction
Motkm. impeding the tidal motions, on all bodies (like the earth) par-
tially or wholly covered by liquid, which, as long as these bodies
move relatively to neighbouring bodies, must keep drawing off
energy from their relative motions. Thus, if we consider, in
the first place, the action of the moon alone, on the earth with
ita oceans, lakes, and rivers, we perceive that it must tend to
equalize the periods of the earth's rotation about its axis, and
of the revolution of the two bodies about their centre of inertia;
because as long as these periods differ, the tidal action on the
* Hewton, Prineipia. (Beraorke on th« first law of motion.) " H&jora ftntem
Fluwtaram et Cometanun corpom motuB sqob et pn^resaiTos et oLrenlarei, in
•patilB mlau iMiBtsoUbiiB tactos, oonaerrant diatius."
..Google
276.] DYNAMICAL LAWS AND PRINCIPLES. 255
earth's surface muBt keep subtracting energy from their motions, f^'*
To viev the subject more in detail, and, at the same time, to tHoUon.
avoid unnecessary complications, let us suppose the moon to be
a uniform spherical body. The mutual action and reaction of
gravitattoa between her mass and the earth's, will be equivalent
to a single force in some line through her centre; and must be
such 88 to impede the earth's rotation as long as this is per-
formed in a shorter period than the moon's motion round the
earth. It must therefore lie in some such direction as the line
MQ in the diagram, which represents, necessarily with enorraoua
exaggeration, its deviation, OQ, &om the
earth's centre. Now the actual force on
the moon in the line MQ, may be re-
garded as consisting of a force in the
line MO towards the earth's centre,
sensibly equal in amount to the whole
force, and a comparatively very small
force in the line MT perpendicular to
MO. This latter is very nearly tangential to the moon's path,
and is io the direction vnth her motion. Such a force, if sud-
denly commencing to act, would, iu the first place, increase the
moon's velocity; but after a certain time she would have moved
so much farther from the earth, in virtue of this acceleration, as
to have lost, by moving against the earth's attraction, as much
velocity as she had gained by the tangential accelerating force.
The effect of a continued tangential force, acting with the mo-
tion, but so small in amount as to make only a small deviation
at any moment from the circular form of the orbit, is to gra-
dually increase the distance from the central body, and to cause
as much i^in as its own amount of work to be done against
the attraction of the central mass, by the kinetic energy of
motion lost The circumstances will be readily understood, by
considering this motion round the central body in a very gradual
spiral path tending outwards. Provided the law of the central
force is the inverse square of the distance, the tangential
component of the central force against the motion will be twice
as great as the disturbing tangential force in the direction with
the motion ; and therefore one-half of the amount of work done
..Google
256 PEEmnNifiT. [276.
ineritabia against tlie former, is doae by the latter, and the other half by
i^l^of kinetic energy taken from the motioa. The int^^ effect on
m^uooi. the moon's motion, of the particular disturbing cause now under
irictko. consideration, is most easily found by using the principle of
moments of momenta. Thus we see that as much moment of
momentum is gfuned in any time by the motions of the centres
of inertia of the moon and earth relatively to their common
centre of inertia, as is lost by the earth's rotation about its axis.
The sum of the moments of momentum of the centres of inertia
of the moon and earth as moving at present, is about 4'45 times
the present moment of momentum of the earth's rotation. The
average plane of the former is the ecliptic ; and therefore the
axes of the two momenta are inclined to one another at the
average ai^le of 23° 27i^', which, as we are neglecting the sun's
influence on the plane of the moon's motion, may be taken as
the actual inclination of the two axes at present. The resultant,
or whole moment of momentum, is therefore 538 times that of
the earth's present rotation, and its axis is inclined 19° 13' to
the axis of the earth. Hence the ultimate tendency of the tides
is, to reduce the earth and moon to a simple uniform rotation
with this resultant moment round this resultant axis, as if they
were two parts of one rigid body: in which condition the moon's
distance would be increased (approximately) in the ratio 1 ; ViS,
being the ratio of the square of the present moment of momen-
tum of the centres of inertia to the square of the whole moment
of momentum ; and the period of revolution in the ratio 1 : 1'77,
being that of the cubes of the same quantities. The distance
would therefore be increased to 347,100 miles, and the period
lengthened to 48'36 days. Were there no other body in
the universe but the earth and the moon, these two bodies
might go on moving thus for ever, in circular orbits round their
common centre of inertia, and the earth rotating about its axis in
the same period, so as always to turn the same face to the moon,
and therefore to have all the liquids at its surface at rest rela-
tively to the solid. But the existence of the sun would pre-
vent any such state of things from being permanent. There
would be solar tides — twice high water and twice low water — in
the period of the earth's revolution relatively to the sun (that is
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276.] DYNAMICAL LAWS AND PRINCIPLES. 257
to Bay, twice in the solar day, or, which would be the same ineritabie
thing, the month). This could not go on without loss of energy *r?rj7 °*
by fluid friction. It ia easy to trace the whole course of the S,^?""-
disturhance in the earth's and moon's motions which thia cause frioBwi.
would produce*: its first effect' must he to bring the moon to
fall in to the earth, with compensation for loss of moment of
momentum of the two round their centre of inertia in increase of
its distance from the sun, and then to reduce the very rapid rota-
tion of the compound body, Earth-and-Moon, after the collision,
and farther increase its distance from the Sun till ultimately,
(corresponding action on liquid matter on the Sun having t^
effect also, and it being for our illustration supposed that there are
no other planets,) the two bodies shall rotate round their common
centre of inertia, like puis of one rigid body. It is remarkable
that the whole frictional effect of the lunar and solar tides
should be, first to augment the moon's distance from the earth
to a maximum, and then to diminish it, till ultimately the
moon falls in to the earth : and first to diminish, after that to
increase, and lastly to diminish the earth's rotational velocity.
We hope to return to the subject later, and to consider the
general problem of the motion of any number of rigid bodies
or material points acting on one another with mutual forces,
under any actual physical law, and therefore, as we shall see,
necessarily subject to loss of energy as long as any of their
mutual distances vaiy; that is to say, until all subside into
a state of motion in circles round an axis passing through their
centre of inertia, like parts of one rigid, body. It is probable
* Tba frietion ot thaw solar tidet on the earth wonld cause the earth to
rotate still slower; and then the moon's tnflnenoe, tending to keep the earth
rotating with alw^s the eune face towards harseU, would resist (his Inrther
reduction in the speed of the rotation. Thas (as explained above with refereuoe
to the moon) there wonld be from the sun a force oppoeing the earth's rotation,
and Erom (he moon a force promoting it. Hence according to the preceding
explanation appUed to the altered circnmstances, the line of the earth's at-
traction on the moon passes now as before, not through the centre of inertia of
the earth, bat now in a line slightlf behtTid it (instead of be/ore, as formerly).
It therefore now resists the moon's motion of rerolation. The combined effect
ot this resistance and of the earth's attraction on the moon is, lilce that of a
resisting medinm, to canse the moon to fall in towards the earth in a spiral patli
with gradoallr iaci«asing velooi^.
TOL. I. 17
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lonot
ThUl
258 TBXLnasAXT. [S76.
that the moon, in ancient times liquid or viscoos in its onter
^n ol layer if not throughout, was thuB brought to turn alirays the
same face to the earth.
277. We have no data in the present state of science for
estimating the relative importance of tidid friction, and of the
resistance of the resisting medium through whic^ the earth and
moon move ; hut -whatever it may be, there can be bat one
ultimate result for such a system as that of the sun and planets,
if continuing long enough under existing lavs, and not dis-
turbed by meeting with other moving masses in space. That
I result is theCalliQgt<^ether of all into one mass, which, although
rotating for a time, must in the end come to reet relatively to
the surrounding medium.
278. The theory of energy cannot be completed nntil we
are able to examine the physical influences which accompany
loss of energy in each of the classes of resistance mentioned
above, § 275. We shall then see that in every case in which
enei^ is lost by resistance, heat is generated; and we shall
leam from Joule's investigations that the quantity of heat so
generated is a perfecUy definite equivalent for the energy
lost. Also that in no natural action is there ever a develop-
ment of energy which cannot be accounted for by the dis-
appearance of an equal amount elsewhere by means of
some known physical agency. Thus we shall conclude that
if any limited portion of the material univerae could be per-
fectly isolated, so as to be prevented from either giving
energy to, or taking energy from, matter external to it, the
sum of its potential and kinetic energies would be the same at
all times : in other words, that every material system subject
to no other forces than actions and reactions between its parts,
is a dynamically conservative system, as defined above, § 271.
But it is only when the inscrutably minute motions among
small parts, possibly the ultimate molecules of matter, which
constitute light, beat, and mf^etism; and the intennolecular
forces of chemical affinity ; are taken into account, along with
the palpable motions and measurable forces of which we
become cognizant by direct observation, that we can recognise
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278.] DTNAHICAL LAWS AND PfilNCIPLES. 259
the universally coDservative character of all natural dynamic
action, and perceive the bearing of the principle of reversibility
on the whole class of natural actions involving resistance, which
Beem to violate it. In the meantime, in our studies of abstract
dynamice, it will be sufiBcient to introduce a special reckoning
for energy lost in working against, or gained from work done
by, forces not belonging palpably to the conservative class.
279. As of great importance in farther developments, we
prove a few propositions intimately connected with eneigy.
280. The kinetic energy of any system is equal to the sum Etnetio
of the kinetic energies of a mass equal to the sum of the maseee ' •Kum.
of the system, moving with a velocity equal to that of its centre
of inertia, and of the motions of the separate parts relatively to
the centre of inertia.
For if fc, y, X be tke co-ordinates of any partiole, m, of the
ByBbera; £, tj, C its co-ordinates relative to the centre of inertia;
and X, y, z, the co^rdlnaiteB of the centre of inertia iteelf; we have
for the whole kinetic energy
But by the properties of the centre of inertia, we have
2™ j; 77= jtSw»5- =0, etc. etc
at at at dt '
Hence the preceding is equal to
which proves the proposition.
281. The kinetic energy of rotation of a rigid system about
any axis is (§ 95) eipreased by J Smr'w', where m is the mass
of any part, r its distance from the axis, and a the angular
velocity of rotation. It may evidently he written in the form
J<»*Smr*. The factor tmr' is of very great importance in"
kinetic investigations, and has been called the Moment of^^^*'^
Inertia of the system about the axis in question. The moment
of inertia «bout any axis is therefore found by summing the
17-2
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260 PBELIMIKARr. [281.
Moment or producte of the masses of all the particles each into the square
of ita distance from the axis.
Momeittof It is importaDt to Dotice that the moment of momeDtom
^"iT'i^l?'" of any rigid system ahout an axts, being %mvr = Smr*o>, is the
boSr. product of the angular velocity into the moment of iuertia.
If we take a quantity k, such that
Badiuiof k is called the Badiua of Qyration about the axis from vfaicli
r is measured. The radius of gyration about any axis is there-
fore the distance irom that axis at which, if the whole mass
were placed, it would have the same moment of inertia as be-
Ky-wheeL fore. In a fly-wheel, where it is desirable to have as great a
moment of inertia with as small a mass as possible, within
certain limits of dimensions, the greater part of the mass is
formed into a ring of the lai^est admissible diameter, and the
radius of this risg is then approximately the radius of gyration
of the whole.
Kommter A rigid hoAj being referred to rectungnlar axes pasBing
■bout any through any point, it is requited to find tlie moment of inertia
about an axis through the origin m&lcing given angles with the
oo-ordinate axes.
Let X, ft, f be its direction-cosines. Then the distance (r) of
tlie point X, y, z from it is, by g 95,
7* = lji»-vy)* + {vx- A»)* + (Xy -/»«)',
and therefore
which may be written
JX' + if/ + <7i^- 2a^ - S/Si-X - 2yX/i,
where A, B, C are the moments of inertia about the axes, and
a = Imyas, j8 = Smsx, y = 'Stmxy. From its derivation we see tlisi
this quantity is ugentiaUt/ positive. Hence when, by a proper
linear transformation, it is deprived of the tenus oontaioing the
products of X, fi, V, it will be brought to the form
where A, B, C are essentially positive. They are evideutly the
ucBnenta of inertia about tlie new rectangular axes of oo^rdinatee,
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281.] DTNAUICAL LA.W3 AlfD PBINCIPLE3. 261
and X, fi, »• the oorreBponding directioa-coeines of the axis round Homoit at
which the moment of inertU is to be found. kbantwr
Let A>B>C,if the^ are nneqnaL Then
bLowb that Q cannot be greater than A, nor less than C. Alao,
if A, B,C he equal, Q ia eqn&l to each.
If a, &, c be tiie radii of gyration about the nev axes of a^ y, z,
A = Ma*, B=Mb\ C=Mc\
and the above equation givea
But if a^ y, s be any point in the line vhoae direction^codnes are
X,fL,v, and r its distance from the origin, we have
- = £ = - = f, and therefore
I^ therefore, we conidder the ellipaoid whose equation is
we see that it intercepts on the line whose direction-oodnea are
X, fi, *■ — and about which the radius of gyration iak,», length r
which ia given by the equation
or the rectangle nnder any radius-vector of tbis ellipeoid and
the radios <^ gyration about it is constant. Its semi-azee are
evidentiy ~ > t > — where « may have any value we may aaaigu.
Thns it is evident that
262. For every rigid body there may be described about "j™*^*^
any point as centre, an ellipsoid (called Poinsot's Momental
Ellipsoid*) which ia such that the length of any radius-vector is
* The definition is not Poiniot'B, but oqih. The momental ellipsoid ss we
define it ia tairlj called Foinaot'i, because ol tbe spleudid use he baa msde
ol it in his well-known kioeinatio repreientation of tbe solction of the problem
— to find tbe motion of a rigid body with one point held fixed bet otherwlss
inflnenced by no foieea — which, with Sylvester's beaatiliil theorem completing
it BO H to give ■ purely kinematical mechanism to show the time which tbe
body takes to attain any particnlai podtion, we Telnctsntly keep baok tor our
Seeond Vdnme,
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eUipuld.
262 PBEUUINABT. [2SS.
mverse]; proportional to the radius of gyration of the body
about that radius- vector as axis.
prindpBi The axes of this ellipsoid are, and might be defined as, the
Principal Axes of inertia of the body for the point in question:
but the best definition of principal axes of inertia is given
below. First take two preliminary lemmas : —
Baniiihn- (1) If a rigid body rotate round auy axis, the centrifugal
(JMitrifuffi forces are reducible to a single force perpendicular to the axis
of rotation, and to a couple (§ 23* above) having its axis parallel
to the line of this force.
(2) But in particular cases the couple may vanish, or both
eouple and force may vaniBh and the centrifugal forces be in
equilibrium. The force vanishes if, and only if, the axis of
rotation passes through the body's centre of inertia.
Definiiion Def. (1). Any axis is called a principal axis of a body's
AieToi inertia, or simply a principal axis of the body, if when the body
rotates round it the centrifugid forces either balance or are re-
ducible to a single force
Def. (2). A principal axis not through the centre of inertia
is called a principal axis of inertia for the point of itself through
which the resultant of centrifugal forces passes.
Def. (3). A principal axis which passes through the centre
of inertia is a principal axis for every point of itself.
The proofs of the lemmas may be safely left to the student as
exercises on § 559 below ; and from the proof the identification
of tbe principal axes as now defined with the principal axes of
Poinsot's momental elhpsoid is seen immediately by aid of the
analysis of § 281.
263. The proposition of § 280 shows that the moment of
inertia of a rigid body about any axis is equal to that which
the mass, if collected at the centre of inertia, would have about
this axis, together with that of the body about a parallel axis
through its centre of inertia. It leads us naturally to in-
vestigate the relation between principal axes for any point and
principal axes for the oeutre of inertia. The following investi-
gation proves the remarkable theorem of § 284, which was first
given in 1811 by Binet in the Journal de V£!cole Polyte<^mq%K.
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283.] DTNAincAL lAWS jLND PKIHaPLBS. S63
Let the orif^ 0, be tiie centre of uierti&, and the axes the Pi
|Hincipal kxea at that point Then, by gg 280, 261, we hare for
the moment of ioeitia about a line through the point P ((, i;, Oi
whose directioD-oosineB are A, ft, v;
Subatituting for Q, A, B, C their values, and dividing bj if,
«e have
Ljt it be required to find X, fL,ww> l^t the direction i^tecified
by them may be a principal axis. Let fX(+fi.ij + yt, i.e.
Jet 8 repreaeut the projection of OP on tlie axis Bought
The axes of the ellipeoid
(a' + y + P)a='+ -2(.?ft»+ )^H (a),
an found by means of the equations
~(^\-t-{b' + C' + e-p)l>^-vO"'0 } (i).
If, now, we take/ to denote Oi*, or (^+^ + f)i, these equation^
where p is clearly the square of the radius of gyration about
the axis to be found, may be written
<o-*/'-p)\-|(f» + „. + fr).0,
et&=etG;,
or (a'+y-f)X-i..O,
etc = ete.,
pt . (a'-r)l-f..O )
{t'-E)i.-v.!> I (c)
where K^p -f. Hence
Multiply, in order, by ^, i), {, add, and divide by a, and we get
^*v^*^-' w
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pomti
I FRELDflNART. [283.
B7 (c) we see that (X, ^, v) is tlie direction of tlie normal tlkFOUgh
the point P, {$, ij, Q of the sur&oe repreaented by the equation
_^+^+."" .1
J-K V-K. f-K
-w.
which is obTionaly a aurfaoe of the second d^ree oonfocal with
the ellipsoid
%44-' ^^'
and passiDg through F in virtue of (<Q, wliich determines K accord-
ingly. The three roots of this cnbic are clearly all real; one of
them is less than the least of a', 6*, c*, and positive or nt^tive
according as P \& within or without the ellipsoid (y). And if
a>b>e, the two others are between c* and b', and betwe«n ft* and
a', respectively. The addition off to each gives the square of Ihe
radius of gyration round the corresponding principal axis. Hence
Bimti 281. The principal axes for any poiot of a rigid body are
Bormals to the three surfaces of the second order through that
point, confocal Trith the ellipeoid, which has its centre at the
centre of inertia, and its three principal dianieteis co-incident
with the three principal axes for that point, and equal respec-
tively to the doubles of the radii of gyration round them.
cratr*! This ellipsoid is called the Central Ellipsoid.
Kinetia 285. A rigid bod; is said to be kinetically symmetrical
about ita centre of inertia when its moments of inertia about
three principal axes through that point are equal ; and there-
fore necessarily the moments of inertia about all axes through
that point equal, § 281, and all these axes principal axes. About
it uniform spheres, cubes, and in general any complete ciys-
talline solid of the first system (see chapter on Properties of
Matter), are kinetically sym metrical
A rigid body is kinetically symmetrical about an ojcu when
this axis is one of the principal axes through the centre of
inertia, and the momente of inertia about the other two, and
therefore about any line in their plane, are equal. A spheroid,
a Square or equilateral triangular priam or plate, a rircular ring,
disc, or cylinder, or any complete crystal of the second or
fourth system, is kinetically symmetrical about its axis.
..Google
286.] DINAMICAL LAWS AND PRINCIPLES. 265
386. The only actions and reactions between the parte of a ^"*^''<
system, not belongmg palpably to the conservative class, which djmmii*
we shall consider in abstract dynamics, are those of friction
between solids sliding on solids, except in a few instances in
which we shall consider the general character and ultimate
results of effects produced by viscosity of fluids, imperfect
elasticity of solids, imperfect electric conduction, or imperfect
magnetic retentiveness. We shall also, in abstract dynamics,
consider forces as applied to parts of a limited system arbitrarily
from without These we shall call, for brevity, the applied forces.
287. The law of energy may then, in abstract dynamics, be
expressed as follows : —
The whole work done in any time, on any limited material
system, by applied forces, is equal to the whole effect in the
forms of potential and kinetic energy produced in the system,
together with the work lost in friction.
288. This principle may be regarded as comprehending the
whole of abstract dynamics, because, as we now proceed to
show, the condirioQB of equilibrium and of motion, in eveij
possible case, may be immediately derived from it.
289. A material system, whose relative motions are unre-Bqniu-
Bisted by &tctioa, is in equilibrium in any particular configura-
tion if, and is not in equilibrium unless, the work done by
the applied forces is equal to the potential energy gained, in any
possible infinitely small displacement from that coniigu ration.
This U the celebrated principle of "virtual velocities" which
L^range made the basis of his Micanique Analytique. The ill-
chosen name "virtual velocities" is now falling into disuse.
290. To prove it, we have first to remark that the system Prinoipis
cannot possibly move away from any particular configuration tbIooIlIh.
except by work being done upon it by the forces to which it in
subject : it is therefore in equilibrium if the stated condition is
fulfilled. To ascertain that nothing less than this condition can
secure its equilibrium, let us first consider a system having
only one degree of freedom to move. Whatever forces act on
the whole system, we may always bold it in equilibrium by a
single force applied to any one point of the system in its line
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260 PRELUDNAar. [290,
of motion, opposite to the direction in vbich it tends to move,
and of such magnitude that, in any infinitely small motion in
either direction, it shall resist, <»* shall do, as much work as the
other forces, whether applied or internal, altc^etber do or resist.
Now, by the principle of superposition of forces in equilibrium,
we might, without altering their effect, apply to aay one point
of the system such a force as we have just seen would hold the
^stem in equilibrium, and another force equal and opposite
to it. All the other forces being balani^d by one of these two,
they and it might again, by the principle of auperpoaition of
forces in equilibrium, be removed; and therefore the whole set
of given forces would produce the same effect, whether for
equilibrium or for motion, as the single force which is left
acting alone. This single force, aince it is in a line in which
the point of its application is free to move, must move the
system. Hence the given forces, to which this single force has
been proved equivalent, cannot possibly be in equilibrium
unless their whole work for an infinitely small motion is
nothing, in which case the single equivalent force is reduced
to nothing. But whatever amount of freedom to move the
whole system may have, we may always, by the application of
frictionless constnunt, limit it to one degree of freedom only ;
— and this may be freedom to execute any particular motion
whatever, possible under the given conditions of the system.
If, therefore, in any such infinitely small motion, there is
variation of potential eneigy uncompensated by wo]^ of the
applied forces, constraint limiting the freedom of the ^stem to
only this motion will bring us to the case in which we hare
just demonstrated there cannot be equilibrium. But the appli-
cation of consljaintB limiting motion cannot possibly disturb
equilibrium, and therefore the given- system under the actual
conditions cannot be in equilibrium in any particular con-
figuration if there is more work done than resisted in any
possible infinitely small motion from that configuration by all
the forces to which it is subject.
291. If a material system, under the influence of internal
and applied forces, varying according to some definite law, is
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291.] DTNAUICAL LAWS AlfD PBINCIPLES. 267
balanced hy them in any portion in which it ma; be placed, Neqtni
ita equilibrium is said to be neutral. This is the case with any brinm.
spherical body of uniform material resting on a horizontal
plane. A right cylinder or cone, bounded by plane ends per-
pendicular to the axis, is also in neutral equilibrium on a
horizontal plane. Practically, &oy mass of moderate dimensions
is in neutral equilibrium when its centre of inertia only is
fixed, since, when its longest dimension is small in comparison
with the earth's radius, gravity is, as we shall see, approximately
equivalent to a single force through this point.
But if, when displaced infinitely little in any direction from suda
a particular position of equilibrium, and left to itself, it com- ^ul
mences and continues vibratang, without ever experiencing
more than infinitely small deviation in any of its parts, from
the position of equilibrium, the equilibrium in this position ia
Biud to be stable. A weight suspended by a string, a uniform
sphere in a hollow bowl, a loaded sphere resting on a horizontal
plane with the loaded side lowest, an oblate body resting with
one end of itn shortest diameter on a horizontal plane, a plank,
whose thickness is small compared with its length and breadth,
floating on water, etc. etc., are all cases of stable equilibrium; if
we neglect the motions of rotation about a vertical axis in the
second, third, and fourth cases, and horizontal motion in general,
in the fifth, for all of which the equilibrium is neutral
I^ on the other hand, the system can be displaced in any uiwtobis
way from a position of equilibrium, so that when left to itself tlnnm.
it will not vibrate within infinitely small limits about the posi-
tion of equilibrium, but will move farther and farther away from
it, the equilibrium in this position is s^d to be unstable. Thus
a loaded sphere resting on a horizontal plane with its load aa
bigb as possible, an egg-shaped body standing ou one end, a
board floating edgeways in water, etc. etc., would present, if
they could be realised in practice, cases of unstable equili-
brium.
When, as in many cases, the nature of tbe equilibrium varies
with the direction of displacement, if unstable for any possible
displacement it is practically unstable on the whole. Thus a
coin staoding on its edge, though in neutral equilibrium for
displacements in its plane, yet being in unstable equilibrium
..Google
S68 PEBLIHIKABT. [291 .
VniUUt for thoBe perpendicular to its plane, is practically unstable. A
brioDL sphere resting in equilibrium on a saddle presents a case in
which there is Btable, neutral, or unstable equilibrium, accord-
ing to the direction in which it may be displaced by rolling,
bat, practically, it would be unstable.
Tnteftha 292. The theory of enet^ shows a vety clear and simple
H^ii?"' test for discriminating these characters, or determining whether
bruun. ^j^g equilibrium is neutral, stable, or unstable, in any case. If
there is just as much work resisted as performed by the applied
and internal forces in any possible displacement the equilibrium
is neutral, but not unless. If in every possible infinitely small
displacement from a position of equilibrium they do less work
among them than they resist, the equilibrium is thoroughly
stable, and not unless. If in any or in every infinitely small
displacement &om a position of equilibrium they do more work
than the; resist, the equilibrium is unstable. It follows that
if the system is influenced only by internal forces, or if the
applied forces follow the law of doing always the same amount
of work upon the system passing &om one conSguratioQ to
another by all possible paths, the whole potential energy must
be constant, in all positions, for neutral equilibrium ; most
be a minimum for positions of thoroughly stable equilibrium ;
must be either an absolute maximum, or a maximum for some
displacements and a minimum for others when there is unstable
equilibrium.
DediMtiDn 293. We have seen that, according to D'Alembert's prin-
Siutioiu ciple, as ezpliuned above (5 264), forces acting on the different
motion tl K ' . ^ , . , ^^ J ^, ■ ^- ■ ^ ..
Mir vwem. points of a material system, and their reactions against the
accelerations which they actually experience in any case of
motion, are in equilibrium with one another. Hence in any actual
case of motion, not only is the actual work done by the forces
equal to the kinetic energy produced in any infinitely small time,
in virtue of the actual accelerations; hut so also is the work
which would be done by the forces, in any infinitely small time,
if the velocities of the points constituting the system, were at
any instant changed to any possible infinitely small velocities,
and the accelerations unchanged. This statement, when put in
..Google
293.] DTNAMICAL I.A.WS AND PRINCIPLES. 269
the concise language of mathematical analyeia, constitutes Dedsotioa
Lagrange's application of the " principle of virtual velocities " 'J'^?"
to express the conditions of D'Alembert's equilibrium hetneen wvium-
the forces acting, and the resistances of the masses to accelera-
tion. It comprehends, as we have seen, every possible condi-
tion of every case of motion. The "equations of motion" in
any particular case are, as Lagrange has shown, deduced from
it with great ease.
Let m be the mass of any one of the material pmnts of the
BjBtem; x, y, z ita rectangular co-ordinates at time I, relatively
to axes fixed in direction (g 249) through a point reckoned as
fixed (g 215) ; and X, T, Z the components, parallel b
axea^ of the whole force acting on it. Thus - m
And these, with X, F, Z, for the whole Byatem, mnet fulfil the
conditions of equilibrium. Hence if &i;, 8y, Ss denote any arbi-
trary variations of as, y, » oonsiBtent with the conditions of the
Efystem, we have
where 2 denotes summation to include all the particles of the "^ ■*■••"■
system. This may be called the indeterminate, or the Tariational,
equation of motion. I^grange used it as the foundation of his
whole kinetic systcon, deriving firom it all the common equations of
motion, and his own remarkable equations in generalized co-ordi-
nat«a (presently to be given). We may write it otherwise as follows ■.
Sm(i&B + ySy + »&5)=S(.ySa!+rBy + Z8«) (2),
where the first member denotes the work done by forces equal to
those required to produce 4^e real accelerations, acting through
the spaces of the arbitraiy displacements ; and the second member
tiie woi^ done by the actual forces through these imagined
spaces.
If the moving bodies ecnurtitute a conservative system, and if
F denote its potential energy in the configuration specified by
^ Vi ^ 0^)> we have of course (§§ 241, 273)
SF— S(^&«i + nytZ&) (3),
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) PBELDOKABT. [293.
and therefore the mdeterminate equation of motion becomes
aM(i&e + ySy4-s&!) = -Sr. (4),
where 8 F denotes the excess of the potenlJal energy in the eon-
figuration (x + 8x, y + Sy, « -»- Ss, etc.) above that in tlie configura-
tion (a!, y, z, etc.).
One immediate particular result must of coni«e be the common
eqnation <d energy, which must be obtained by suppodng Sx, Sy,
&, etc., to be the actual variations of the oo-ordinatea in an
infinitely amatl time St. Thus if we take &e = !i^, etc., and
divide both membere by St, we have
%(X£+ Y^ + 2i) = S,m{xi + if^ + B) (5).
Here the first member is composed of Newton's JctionM ^jr^n^um ,*
with his Reactionea ResUteniium so far as friction, gravity, and
molecular forces are concerned, subtracted : and tiie seocmd ctmsiaU
of the portion of the Reactumss doe to acceleration. Ah we have
seen above (§ 314), the second member is the rate ot increaM of
2 jm (ii' + j" T «*) per unit of tjme. Hence, denoting by r the
velocity of one of the particles, and by W the int^ral of the
fii^ member multiplied by dt, that is to say, the int^rol vrork
done by the working and resisting forces in any time, we have
Simw'=ff'+JF, (6),-
J?, being the initial kinetic energy. This is the integral equa-
tion of energy. In the particular case of a conservative system,
TT is a function of the co-ordinates, irrespectively of the time, or
of the paths which have been followed. According to the pre-
vious notation, with besides F, to denote the potential energy of
the system in its initial configuration, we have 0^= F, - F, and
the integral equation of energy becomes
or, if £ denote the sum of the 'potential and kinetic energies, a
constant, SJmp' = jff-F (7).
The general indeterminate equation gives immediately, for the
motion of a system of fi:ee particles,
Of these equations the three for each particle may d oonrse be
treated separately if there is no mutual influence between the
partioles: hot when tiiey exert force on one another, J,, F^, etc.,
will each in general be a function of all the co-ordinatesi
..Google
293.] DTHAUICAX LIWS AND PBINCIPLES. 271
From the iodetormmate equation (1) lAirraiise, by his meUiod Conitnint
- ... ,. , , . ... , . , mtrodnoed
of multiplien, deduceG the requisite number of equations for Into the in-
detenuiuiug the motion of a rigid body, or of any system of oon- eqnxioa.
nected particles or rigid bodies, thus : — Let the number of the
particles be t, and let the oonnexions between them be expressed
by » equations,
■^<(»..y,.«,.'«.. ■■■) = 0[ ^gj
r.,...) = 0l.
etc J
being the J:itMnaticai eqwUimu of the system. By taking the
variatiooa of these we find that every possible infinitely small dis-
placement fix,, ^,, Se, , fix,, ...must satisfy then linear equations
Multiplying the first of these by \, the second by X,, etc,
adding to the indeterminate equation, and then equating the co-
efGcienta of fix,, Sy,, etc., each to zero, we have
^dF ^ dF, - d**
A-i— + A, -i— ^ + ... + J , — m, ~T^ =
dy, ' rfy, ' ' cff
etc ete.
!niese are in all 3t equations to determine the n unknown iMannl-
quantities X, \,..., and the 3v-n independent variables to tiom^
which a;,, y,, ... are reduced by the kiuematical equations (8). deduced.
The same equations may be found synthetically in the following
manner, by which also we are helped to understand the precise
meaning of the terms containing the multipliers X, X,, etc
First let the particles be free fnmi consfraint, but acted on
both by the given foroeB X,, F,, etc, and by forces depending
cm mutual distances between the particles and up<m their
poaitionB relatively to fixed objects subject to the law of con-
servation, and having for their potential energy
-i (*/" + *,/■,' + etc.),
•0 that Gompon^its of the forces actually experienced by the
different partidee shall be
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272 PEEUMINAET. [293.
«""« ' dx, ' 'dx, *\ dx. • dx, J
motion ' ' • I 1
etc., etc.
Henoe the eqaatlons of motion are
'(ft* ' dxi ' 'dx *V dx ' dec^ /
<^y, _
(U).
Now Buppose k, k_, etc to be infinitely great: — in order ^at .the
Jbreet on the parHclei may not be infinitely grea^ we must hare
^=0, F, = 0, etc,
that is to say, the equations of condition (8) mnst be fulfilled ;
and the last groups of tenna in the second members of (II) now
disappear because they contain the squares of the infinitely small
quantities F, F^, ete. Put now kF=X, k^F^-\, etc., and we
hare equations (10). This second mode of proving Lagrange's
equations of motion of a constrained system corresponds pre-
cisely to the imperfect approach to the ideal case which can be
made by real mechanism. The levers and bars and guide-
Bur&cea cannot be infinitely rigid. Suppose then k, £,, etc to
be finite but very great quantities, and to be some functions of
the co-ordinates depending on the elastic qualities of the materials
of which the guiding mecbanism is composed: — equations (11)
will express the motion, and by supposing k, £,, etc to be
greater and greater we approach more and more nearly to the
ideal case of absolutely rigid mechanism constraining tlie predoe
fulfilment of equations (8).
The problem of finding the motion of a system subject to any
unvaryinff kinematical conditions whatever, under the action of
any given forces, ia thus reduced to a question of pure analysis.
In the still moregeneral problem of determining tiie motion when
certein parts of the system are constrained te move in a specified
manner, the equations of condition (8) involve not only the
co-ordinates, but also t, the time. It is easily seen however that
the equations (10) still hold, and with (8) fully determine the
motion. For : — consider the equations of equilibrium of the par-
ticles acted on by any forces JT/, F,', etc, and constrained by
..Google
293.] DYNAMICAL LAWS AND PBINCIPLES, 273
proper mechamsm to fulfil the eqoatioiis of condition (8) with DetermJ-
the actual T&lues of tho panuneterB for any particular value tioni^™"
of I. The equations of equilibrium will be imiafluenced doduoed.
by the fact that some of the parameters of the conditions
(8) have differrait Talnea at different times. Hence, -witli
_ <Px, _ (?«, . , .
'""*' de' '~'"''^' "^l^^aof Z,', F,', etc, according
to D'Alembert's principle, the equations of motion will still be
(8), (9), and (10) quite independently of whether the parameters
of (8) are all constant, or have values varying in any arbitrary
manner witii the time.
To find tho equation of energy multiply the first of equations
(10) by i„ the second by j?^, etc., and add. Then remarking
that in virtue of (8) we have
3- (*, + 3— tf, + eta + 1 -;- 1 = 0.
-*,^^S, + etc. + (-) = 0.
partial differential coefficients of F, F^, etc with reference to (
being denoted by (-j-A, (■57)1 «**'•: "^ denoting by T the
kinetic energy or j^Sm {if + ft-^^, we find
f-S(x..rs.^i)-x(f)-x,(f)-«„,=o,...(i2).
When the kinematic conditions are " vavoaryimg" that is to
say, when the equations of condition are equations among the
co-ordinates with constant parameters, we have
and tbe equation of energy becomes
(ft"
..(13),
showing that In this case the fulfilment of the equations of
condition involves neither gain nor loss of energy. On the
other hand, equation (12) shows how to find the work performed
or consumed in the iiilfibnent c^ the kinematical conditions when
they are not unvarying
VOL. I. 18
..Google
274 PBELDUNABT. [293.
r As a'umple example of Tarying constndnt, which will be veiy
easily worked out b;r equations (8) and (10), perfectly illustrating
the general principle, the student may take the case of a particle
acted on by any given forces and free to move anywhere in
a plane which is kept moving with any given uniform or vaiying
angular velocity round a fixed-axis.
When there are connexions between any parts of a ^'stem, the
motion is in general not the same as if all were free. If we con-
sider any particle during any infinitely anLall time of the motion,
and call the product of its masB into the equare of the distance
between its positiona at the end of this time, on the two sopposi-
tions, the eoiuiraint : the Blun of the constraints is a Tnininnim
This follows easily from (1).
291. When two bodies, in relative motion, come into con-
tact, pressure begins to act between them to prevent any parte
of them from jointly occupying the same spaoe. This force
commences from nothing at the first point of collision, and
gradually increases per unit of area on & gradually increasing
Buiface of contact. If, as is always the ease in nature, each
body possesses some degree of elasticity, and if they are not kept
together after the impact by coheffliom, or by some artificial
appliance, the mutual pressure between tbem will reach a
maximum, will begin to diminish, and in the end will come to
nothing, by gradiiaUy diminishing in amount per unit of area
on a gradually diminishing surfiace of contact. The whole pro-
cess would occupy not greatly more or less than an hour if
the bodies were of such dimensions as the earth, and such d^rees
of rigidity as copper, steel, or glass. It is finished, probably,
within a thousandth of a second if they are globes of any of
these substances not exceeding a yard in diameter.
296. The whole amount, and the direction, of the "Ivipact"
experienced by either body in any such case, are reckoned
according to the "change of momentum" which it experleDces.
The amount of the impact is measured by the amount, and its
direction by the direction, of the change of momentum which is
produced. The component of an impact in a direction parallel
to any fixed line is similarly reckoned according to the com-
ponent change of momentum in that direction.
..Google
296.J DYNAMICAL LAWS AND PEINCIPLE3. 275
296. If we imagine the whole time of an impact divided impwt.
into a very great number of equal intervals, each so short that
the force does not vary sensibly during it, the component
change of momentum in any direction during any. one of these
intervals will (§ 220) be equal to the force multiplied by
the measure of the interval Hence the component of tiie
impact is equal to the sum of the forces in all the intervals,
multiplied by the length of each interval.
Let P be the component foree in any direction atEmy mstant^
r, of the intorral, and let / be the amount of the corresponding
component of the whole impact. Then
/=/Prfr.
297. Any force in a constant direction acting in any cir- tidm'
cumstances, for any time great or small, may be reckoned on
the same principle ; so that what we may call its vrhole amount
during any time, or its " Ume-integral" will measure, or be
measured by, the whole momentum which it generates in the
time in question. But this reckoning is not often convenient
or useful except when the whole operation considered is over
before the position of the body, or configuration of the system
of bodies, involved, has altered to such a degree as to bring any
other forces into play, or alter forces j»*vioualy acting, to such
an extent as to produce any sensible effect on the momentum
measured. Thus if a person presses gently vrith his hand,
during a few seconds, upon a mass suspended by a cord or
chain, he produces an effect which, if we know the degree of
the force at each instant, may be thoroughly calculated on
elementary principles. No approximation to a full determina-
tion of the motion, or to answering such a partial question as
"how great will be the whole deflection produced!" can be
founded on a knowledge of the "time-integral" alone. If, for
instance, the force be at first very great and gradually diminish,
the effect will be very different from what it would be if the
force were to increase very gradually and to cease suddenly,
even although the time-integral were the same in the two
cases. But if the same body is " struck a blow," in a horizontal
direction, either by the hand, or by a mallet or other somewhat
18—2
..Google
276 PRELDJlNAaT. [297.
liard mass, the action of the force is finished before the sospend-
ing cord has experienced Einy sensible deflection from the ver-
ticaL Neither gravity nor any other force sensibly alters the
effect of the blow. And therefore the whole momentam at the
end of the blow is sensibly equal to the " amount of the impact,"
which ia, in this case, simply the time-int^raL
298. Such is the case of Robins' Ballistic Petidvlnm, a
massive cylindrical block of wood cased in a cylindrical sheath
of iron closed at one end and moveable about a horizontal axis
at a considerable distance above it — employed to measure the
velocity of a cannon or musket-shot. The shot is fired into the
block in a horizontal direction along the axis of the block and
perpendicular to the axis of sospenuon. The impul^ve
penetration is so nearly instantaneous, and the inertia of the
block so large compared with the momentum of the shot, that
the ball and pendulum are moving on as one mass before the
pendulum has been sensibly deflected from the vertical. This is
essential to the regular use of the apparatus. The iron sheath
with its flat end must be strong enough to guard agunat spliu-
ters of wood flying sidewise, and to keep in the bullet.
299. Other illustrations of the cases in which the time-
integral gives us the complete solution of the problem may be
given without limit. They include all cases in which the
direction of the force is always coincident with the direction
of motion of the moving body, and those special cases in which
the time of action of the force is so abort that the body's motion
does not, during its lapse, sensibly alter its relation to the direc-
tion of the force, or the action of any other forces to which it
may be subject Thus, in the vertical fall of a body, the time-
integral gives us at once the change of momentum; and the
same rule applies in most cases of forces of brief duration, as
in a " drive " in cricket or golf.
300. The simplest case which we can consider, and- the one
usually treated as an introduction to the subject, is that of the
collision of two smooth spherical bodies whose centres before
collision were moving in the same stnught line. The force
between them at each instant must be in this line, because of
..Google
300.] DYNAMICAL LAWB AND PBINCIPLBS. 277
the Efymmetry of circumatanceB round it ; and by the third £}^,'"
law it mnat be equal in amount on the two bodies. Hence "piw™.
(LbX u.) they mUBt experience changes of motion at equal rates
in contrary directions; and at any instant of the impact the
integral amounts of these changes of motion must be equal
Let us suppose, to fix the ideas, the two bodies to be moving
both before and after impact in the same direction in one line :
one of them gaining on the other before impact, (uid either
following it at a less speed, or moving aloi^ with it, as the
case may be, after the impact is completed. Cases in which
the former is driven backwards by the force of the collision,
or in which the two moving in opposite directions meet in
collision, are easily reduced to dependence on the same formula
by the ordinary algebraic convention with regard to positive
and n^ative signs.
In the standard case, then, the quantity of motion lost, up
to any instant of the impact, by one of the bodies, is equal to
that gained by the other. Hence at the instant when their
velocities are equalized they move as one mass with a momen-
tum equal to the sum of the momenta of the two before impact.
That is to say, if v denote the common velocity at this instant,
we have
"= M*M' '
if M, W denote the masses of the two bodies, and V, V their
Velocities before impact.
During this first period of the impact the bodies have been,
on the whole, coming into closer contact with one another,
through a compression or deformation expaienced by each,
and resulting, as remarked above, in a fitting together of the
two surfaces over a finite area. No body in nature is per-
fectly inelastic; and hence, at the instant of closest approxi-
mation, the mutual force called into action between the two
bodies continues, and tends to separate them. Unless pre-
vented by natural surfiice cohasion or welding (such aa is
always found, as we shall see later in our chapter on Properties
of Matter, however hard luid well polished the surfaces may
..Google
278 PBELDflHlItT. [300.
jHnotim* be), or bj artificial appliances (such aa a coating of wax, applied
■pbem. in one of the commoQ iUuetratire experiments; or the coupling
applied between two railway carriages when run together so as
to push in the springs, according to the usaal practice at rail-
Mtotri^ way etations), the two bodies are actually separated by this
force, and move away from one another. Xewton found that,
provided ike impact is not so violent as to make any sensible
permanent indentation in either body, the relative Telocity of
separation after the impact hears a proportion to their previous
relative velocity of approach, which is constant for the same
two bodies. This proportion, always leas than unity, ap-
proaches more and more nearly to it the harder the bodies are.
^1^* Thus with halls of compressed wool he found it J, iron nearly
"»*"•■ the same, glass ^. The results of more recent experiments on
the same subject have confirmed Newton's law. These will be
described later. In any case of the collision of two balls, let
e denote this proportion, to which we give the name Coefficient
of MesHtuiionj* and, with previous notation, let in addition
U, TT denote the velocities of the two bodies after the coDclnsioD
of the impact; in the standard case each being positive but
U >U. Then we have
V-U=e{7-7')
and, as before, since one has lost as much momentum as the
other has guned,
MU+M'U' = M7-¥M:r.
From these equations we find
{M+M')U.= M7-i-M'r-eM'{Y-r),
with a similar expression for TT.
Also we have, as above,
{M-¥AC)v = My+2fr.
Henc^ by subtraction,
{M-^M'){v~.U) = 6M'{7-r)^e{M'r~{M+3r)v + M7)
■ In most modem jrefttisee tius b sailed a " ooeffiaient of elMtici^," which
ia oIbstI; ft miglake; EOggested, it nifty be, by Newton's woids, bat ineonBieteut
with hl« laotB, and ntterly at variftnoe with modeni Iftneaage uid modem know-
ledge regarding elastieity.
..Google
300.] DYNAMICAL, LAWS AND PEINCIPLES. 279;
and therefore SlaS'""
v-U=e(V-v). •!*««*
Of course we have also
Theae results may be put in words thus : — The relative velocity
of either of the bodies, with regard to the centre of inertia of
the two is, after the completion of the impact, reversed in
direction, and diminished in the ratio e : 1.
801. Hence the loss of kinetic energy, being, according to
§§ 267, 280, due only to change of kinetic energy relative to
the centre of inertia, is to this part of the whole as 1 — e* : 1.
Thus
Initial kinetic energy = J {^+ JT) i»* + J Jf ( F - »)» + Ji^' (« - F)'-
5^»al « » -^^{M + M')^ + ^M{v-Uy + ^M'{U'-v)'.
Loss =J(l-«'){if(F-«)' + if'{t.-F7(.
302. When two elastic bodies, the two balls supposed above iHiMtm-
for instance, impinge, some portion of their previous kinetic Bnerts'ktter
energy will always remain in them as vibrations. A portion
of the loss of energy (miscalled the effect of imperfect elas-
ticity) is necessarily due to this cause in every real case.
Later, in our chapter on Properties of Matter, it will be
shown as a result of experiment, that forces of elasticity are,
to a very close degree of accuracy, simply propprtioual to the
strains (§ 154), within the limits of elasticity, in elastic solids
which, like metals, glass, etc., bear but small deformations with-
out permanent change. Hence when two such bodies come
into collision, sometimes with greater and sometimes with less
mutual velocity, but with all other oiroumstances similar, the
velocities of all particles of either body, at corresponding times
of tiie impacts, will be always in the same proportion. Hence
the velocity of separation of the centres of inertia after impact ^?^^,
will bear a constant proportion to the previous velocity ofg|^*J™^
approach; which agrees with the Newtonian Law. It is there- ^Jj?^^
fore probable that a very sensible portion, if not the whole, of
the loss of enezgy in the visible motions of two elastic bodies,
after impact, experimented on by Newton, may have been due
..Google
280 PRELTHIHABT. [302
to vibrationa; but nnlesa some other cause also vas lar^l;
■Aw operative, it is difficult to see how the loss was so much gTeat«r
with iron balls than with glass.
303. Id certain definite extreme cases, imaginable although
not realizable, no energy will be spent in vibratioDS, and the
two bodies will separate, each moving simply as a rigid body,
and having in this eimple motion the whole eneigy of work
done on it by elastic force during the collision. For instance,
let the two bodies be cylinders, or prismatic bars with fiat ends,
of the same kind of substance, and of equal and similar trans-
verse sections; and let this substance have the property of
compressibility with perfect elasticity, in the direction of the
length of tbo bar, and of absolute resistance to change in every
transverse dimension. Before impact, let the two bodies be
placed with their lengths in one line, and their transverse sec-
tions (if not circular) similarly atuated, and let one or both be
set in motion in this line. The result, as r^ards the motions
of the two bodies after the collision, will be sensibly the
same if they are of any real ordinary elastic solid material,
provided the greatest transverse diameter of each is very small
in comparison with its length. Then, if the lengths of the two
be equal, they will separate after impact with the same relative
velocity as that with which they approached, and neither will
retain any vibratory motion after the end of the collision.
801. If the two bars are of unequal length, the shorter will,
after the impact, be exactly in the same state as if it had
struck another of its own length, and it therefore will move as
a rigid body after the collision. But the other will, along with
a motion of its centre of gravity, calculable from the princiiJe
that its whole momentum must (§ 267) be chang^ by an
amount equal exactly to the momentum gained or lost by the
first, have also a vibratory motion, of which the whole kinetic
and potential energy will make up the deficiency of energy
which we shall presently calculate in the motions of the CMitres
of inertia. For simplicity, let the longer body be supposed to
be at rest before the collision. Then the shorter on striking it
will be left at rest ; this being clearly the result in the case of
..Google
304] DTNAMICAL LAWS AND PEINCIPLES. 281
e = 1 in the precedit^ formulte (§ 300) applied to the impact P'"*^-
of one hody striking another of equal mass previously at rest. «
The longer bar will move away with the same momeatum, and
therefore with less velocity of its centre of inertia, and less
kinetic energy of this motion, than the other body had before
impact, in the ratio of the smaller to the greater niass. It will
also have a very remarkable vibratory motion, which, when its
length is more than double of that of the other, will consist of
a, wave running backwards and forwards through its length, and
causing the motion of its ends, and, in fact, of every particle of
it, to take place by "Jits and starts," not continuously. The
full analysis of these circunuitances, though very simple, must
be reserved until we are especially occupied with waves, and
the kinetics of elastic solids. It is sufficient at present to
remark, that the motions of the centres of inertia of the two
bodies after impact, whatever they may have been previously,
are given by the preceding formulee with for e the value
~M ' ^'^^^ ^ ^"^^ ^ ^^ ^^^ smaller and the larger mass re-
spectively,
305. The mathematical theory of the vibrations of solid elastic
spheres has not yet been worked out; and its application to
the case of the vibrations produced by impact presents con- '
siderabie difficulty. Experinuent, however, renders it certain,
that but a small part of the whole kinetic energy of the pre-
vious motions can remain in the form of vibrations after the
impact of two equal spheres of glass or of ivory. This is
proved, for instance, by the common observation, that one of
them remfuns nearly motionless after striking the other pre-
viously at rest; since, the velocity of the common centre of
inertia of the two being necessarily unchanged by the impact,
we infer that the second ball acquires a velocity nearly equal
to that which the first had before striking it. But it is to be
expected that unequal balls of the same substance coining into
collision will, by impact, convert a very sensible proportion of
the kinetic energy of their previous motions into energy of
vibrations; and generally, that the same will be the case when
equfd or unequal masses of different substances come into colli-
..Google
282 PBEUHINABT. [305.
gjjf^ aion; although for one particular proportion of their diameters,
SiK.'''" depending on their densities and elastic quaJities, this eSect will
be a minimum, and possibly not much more sensible than it is
when the substances are the same and the diameters equaL
306. It need scarcely be said that in such cases of impact
as that of the tongue of a bell, or of a clock-hammer striking
its bell (or spiral spring as in the American clocks), or of piano'
forte hammers striking the strings, or of a drum struck with the
proper implement, a large part of the kinetic eneigy of the
blow is spent in generating vibrations.
uoment of 307. The Moment of an impact about any axis is derived
Sootau from the line and amount of the impact in the same way as the
moment of a velocity or force is determined firom the line and
amount of the velocity or force, §§ 235, 236. If a body is
struck, the change of its moment of momentum about any axis
is equal to the moment of the impact ronnd that axis. But,
without considering the measure of the impact^ we see (§ 267)
that the moment of momentum round any axis, lost by one
body in striking another, is, as in eveiy case of mutual action,
equal to that gained by the other.
BalllfHa Thus, to recur to the baUistic pendultun — ^the line of motion
''""^" of the ballet at impact may be in any direction whatever, but the
only part vhicb is elective is the component in a plane peipen-
dicolar to the axis. We may therefore, for simpUdty, coomder
the motion to be in a line perpendicular to the axis, though not
necessarily horizontal. Let m be the mass of the bullet, v ite
velocity, and p the distance of its line of motion from the axis.
Let Jf be the mass of the pendulum with the bullet lodged in it,
and k ita radius of gyration. Then if u be the angular velocity
of the pendulum when the impact is ccmiplete,
mvp — Jifi^a,
from which the solution of the question is easily determined.
For the kinetic energy after impact is changed (g 241) into
its equivalent in potential energy when the pendulimi reachce ita
position of greatest deflection. Let this be given by the angle
6 1 then the height to which the centre of inertia is isiaed is
A (1 - cos 0) if A be its distance from the axis. Thus
..Google
307.] DTNAinCAL LAira AlO) PHINCIPLES. ]
2Bm| = -^^
an expresuon for ihe ohord of the angle of deflection. In
practice the chord of th^ angle 6 is measured bj means of a
light tApe or cord attached to a point of the pendulum, and
slipping with Eonall fiiction through a clip fixed cloae to the posi-
tion occupied hy that point when the pendulum hangs at rest.
308. Work done hy an impact ia, in general, the product of ?
the impact into half the sum of the initial and final velocities
of the point at which it is applied, resolved in the direction of
the impact. In the case of direct impact, such as thdt treated
iQ § 300, the initial kinetic energy of the body is \MV, tlie
final ^MU*, and therefore the gain, by the impact, is
at, which is the same,
M{U~Y).^{U+7).
Bat M{ll~ V) is {§ 295) equal to the amount of the impact
Hence the proposition : the extension of which to the most
general circumstances is easily seen.
Let ( be the amount of the impulse up to time r, and / the
whole amount^ up to the end, T. Thus, —
t=rPdT, I=[^PdT; also-P=^.
Whatever may be tlie conditions to which the body struck is
Buljected, Ibe change of velocity in the point atruok is propor-
tional to the amount of the impulse up to any part of its whole
time, 80 that, if iA be a constant depending on the masses and
conditions of ooustraint involved, and if ^, v, Y denote the com-
ponent velocities of the point struck, in the direction of the
impulse, at the b^^iming, at the time r, and at the end, ro-
fipectively, we have
Hence, for the rate of the doing of work by the force P, at the
instant t, we have
jiGoogle
S84 . I>BBLIMINABT, [308.
iToriE dona Henoe tor the vhole work (W) done bv it,
iviniiMt. ^ ' ' ^
= P7+j7{r-cr)=j.j(p+r).
809. It is worthy of remark, that if any number of impacts
1)0 applied to a body, their whole effect will be the same whether
they be applied tc^ther or successively (provided that the
whole time occupied by them, be infinitely short), although
the work done by each particular impact is in general different
according to the order in which the several impacts are applied.
The whole amount of work is the sum of the products obtained
by multiplying each impact by half the sum of the components
of the initial and final velocities of the point to which it is
applied.
ihrntknu 310. The effect of any stated impulses, apphed to a rigid
nwtkia. body, or to a system of material points or rigid bodies con-
nected in any way, is to be found most readily by the aid of
D'Alembert's principle ; according to which the given impulses,
and the impulsive reaction against the generation of motion,
measured in amount by the momenta generated, are in equi-
librium ; and are therefore to be dealt with mathematically by
applying to them the equations of equilibrium of the system.
IJet F,, Q,t a, be the component impulses on the first particle,
nt,, and let i,, j^,, i, be the components of the velocity in-
Btontoneoiisly acquired hj this particle. ComponeDt forces equal
to {P^-m^±,), ($,-w»,3>,), ■.. must equilibrate the system,
and therefore we have (§ 290}
i{{P-mi)&e+{Q-mS).Blf+{S-mi)Sx] = 0 (a)
where Sx,, Si/,, ... denote the comjioneotB of any infinitely small
displacements of the particles possible under the conditions of
the system. Or, which amonnte to the same thing, noce any
possible infinitely small displacements are simply proportional to
any poaaible velocitieB in the same direotjons,
i{{P~mt)u + {Q-m$)v + {Q-nd)u>} = 0. (6)
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310.] DTBIHICAL LAWS AND PEINCIPLES. 285
where «,,«,, w, denote any posdble component Telodtiea of the BflnaMon*
^ '.,' ' odmpoliiw
firat partacl^ etc moiian.
One particular caw of this equation is of csurse had hj suppos-
ing u,, w,, ... to' be equal to the velocitieB A^, jjj, ... actually
acquired ; and, by halvTof^ etc, we find
3(^-4* + g.j3? + iJ.Ji) = iSm<*' + ^+iO (c).
This agrees with § 308 above.
311. Euler discovered that the kinetic 'enei^ acquired from Tboanmot
rest by a rigid body in virtue of an impulse fulfils a maximum- teudMi by
minimum conditioiL Lagrasge* extended this proposition to
a system of hodies connected by any invariable kinematic re-f^^*'
lations, and struck with any impulses. Delaunay found that maUoo.
it is really always a maximum when the wnptUses are given,
and when different motiotie poasihle vmder the conditiona of
the eystem, and fulJiUing the law of energy [§ 310 (c)], are
considered. Farther, B^rand shows that the energy actuaUy
acquired is not merely a'^maximam," but exceeds the energy
of any other motioa fulfilling these conditions; and that the
amount of the excess is equal to the energy of the motion which
mast be compounded with either to produce the other.
Let i£j', ^' ... be the component Telo^ties of any motion what-
eror fulfilling the equation (c), which becomes
it(Paf+Q^ + Ji^ = ^-Sjn{d^ + r + n = ^ W-
If, then, we takeii,'-i£, =«,, $i'-!^, = v,i etc., we have
r-r=}Sm{(2« + «)« + {2^+t.)w + (2a-t-w)w}
= Sm (*» + j?i> +*») + J Sm (»' + «* + »■) (fl).
But, by <(),
3m(ia + Sv + &w) = i(Pu+Qv + Bu)) (/);
and, by (c) and (d),
^(Pu+Qv + Sw) = 2T'-2T (g).
Hence (e) becomes
r-T=HT-T) + i%m{u* + t^ + vf),
whence r-J" = }3i»(t^+t>'+tO W» ,
which is Bertrand's result.
■ tUeanifue Analfiltqtu, V pattie, 8~* Motion, g V
..Google
286 PRELIKINAET. [312.
LiqnidMt 312. The energy of the inotion generated suddenly in a
inpuidTetr. mass of incompressible liquid given at rest completely filling
a Tessel of any shape, when the vessel is suddenly set in
motion, or when it is suddenly bent out of shape in any way
whatever, subject to the condition of not changing its volume,
is less than the energy of any other motion it can have with the
same motion of its hov/nding surface. The consideration of this
theorem, which, so far as we know, was first published in
the GantSirid^e and Dvilin Mathematicai Journal [Feb. 1849},
has led us to a general minimum property regarding motion
acquired by any system when any prescribed velocities are
generated suddenly in any of its parts; announced in tho
Proceedings of the Royal Society of Edinburgh for April, 1863.
It is, that provided impulsive forces are applied to the system
only at places where the velocities to be poduced are pre-
scribed, the kinetic energy is less in the actual motion than in
any other motion which the system can take, and which has
the same values for the prescribed velocities. The excess of
the energy of any possible motion above that of the actual
motion is (as in Bertrand's theorem) equal to the energy of the
motion which must be compounded with either to produce the
other. The proof is easy : — here it is : —
Equations (d), (e), and (/) hold as in § (311). But now each
velocify component, «,, f,, w,, «,, etc. vanishea for which the
component impulse /*,, Q^, H^, P^, etc does not vanish (because
*, + «,, S, +",, etc, fulfil the prescribed velocity conditions).
Hence every product P,u,, Q,v^, etc. Tamshes. Hence now
instead of (g) and (h) we have
%{±u+^+zta) = 0 (ffO,
and T-T=ilm{u' + i^ + io') (h').
We return to the subject in §§ 316, 317 aa an illustration of
the use of Lt^range's generalized co-ordinates ; to the introduc-
tion of which into Dynamics we now proceed.
impniiiTB 313. The method of generalized co-ordinates explained
■ - •-'^ above (§ 204) is extremely useful in its application to the
dynamics of a system; whether for expressing and working
out the details of any particular case in which there is any
..Google
313.] DTtflHICAL LAWS AND I^INCIPUS. 287
finite namber of degrees of freedom, or for proving general iinmiiiire
principiea applicable even to cases, such as that of a liquid, as '"™^{^
described in the preceding section, in which there may be an <=°^^
inflaite number of degrees of freedom. It leads us to generalize
the measure of inertia, and the resolution and composition of
forces, impulses, and momenta, on dynamical principles corre-
sponding with the kinematical principles explained in § 204,
-which gave us generalized component velocities: and, as we
shall see later, the generalized equations of continuouB motion
are not only very convenient for the solution of problems, but
most tnati-ucHve as to the nature of relations, however compli-
cated, between the motions of different parts of a system. In
the meantime we shall consider the generalised expressions for
the impulsive generation of motion. We have seen above
(§ 308) that the kinetic energy acquired by a system given at
rest and struck with any given impulses, is equal to half the
sum of the products of the component forces multiplied each
into the corresponding component of the velocity acquired by
its point of application, when the ordinary system of rectangular
co-ordinates is used. Precisely the same statement holds on
the generalized system, and if stated as the convention agreed
upon, it sufGces to define the generalized components of im- GmentinA
pulse, those of velocity having been fixed on kinematical or^^v£
principles (§ 204). Generalized components of momentum mcoinm.
of any specified motion are, of course, equal to the generalized
components of the impulse by which it could be generated from
reat,
(a) Let tf/, ^,$,...he the generalized co-ordinates of a material
system at any time; and let ^, ^, $, ... be the oorreepouding
geoeratized velocitynxtrnponents, that is to say, the rates at
which tp, ifi,0, ... increase per unit of time, at any instant, in
the actual motion. ^ x^t tfi' ''i denote the common rectangular
eo^>rdinatee of one particle of the system, and i^, ^,, e, ite com-
ponent velocities, we have
dxi , dsc, ,
..(I).
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3 PRELOnsABT. [313.
Hence the kinetic energy, whicti is 2 Jm (^ -f j/* ■>- i*), in terms
of rectangular co-ordinates, becomes a quadratic fimctioii of
^, ^ eto., wlien expressed in terms of generalized oo-ordinatcs,
80 that if we denote it by T we have
r=i{(.A,^)^' + (<^^)^' + ...+2(*,<^)W + -} <2),
where {ifr, iji), (<p, ifi), (^, ^), 'eta, denote Tarioua fnnctiona of the
oo-ordinatee, determinable according to the conditdons of the
Bjstem. The only condition eesentiaUy fulfilled by tiiese co-
efficients is, that th^ most give a finite positive value to T for
all values of the variables.
(&) Again let (X,, F,, Z^ (X„ T^,-Z^, etc., denote component
forces on the particles (iBj, y,, ej, (a=,) y,i £,)) fitc, respectively;
and let (Sz,, ^,, 82,), etc, denote the components of any in-
finitely small motions posdble without breaking the conditions of
the system. The work done hy tliose forces, t^pon the ^'stem
when BO disj^aced, will be
%{XSx+7^ + Z^ „....(3).
To tranafima this into an ezpreesion in tenns of graerallzed 00-
ordioates, we have
^^'%H*%H-<
etc.
..(i),
-i-«fi^+etc...
K^%*^%*'^.
-A'%*^%<:)
These qnantitiee, <t, O, eto., are clearly the genamtiud com-
ponent! qfthe/oree on the ayttem.
Let % f , etc denote component impulses, generalized on the
same principle ; that is to say, let
=|Vd<, 9^1'^
*(U, etc,
..Google
313.] DTNAUICAL LAWS AND FRIHCIPI.ES. 289
vhere %9, ■■■ denote generalized components of the continnons
force acting at amj inatant of the iaSmtelj short time r, within
Thich the impvdae is completed.
If this impolse is applied to the system, prerionaly in moUon ^"l*'''^^
in tiie manner specified above, and if &^, S^ ... denote the re- ^i^^^
suiting angmentationa of the components of velocitj, the means i
of the component velocities before and after the impulse Till be w
^ + J8i^, ^ + ii^
Hence, acccwding to the general prindple explained above for
calculating the work done by an impulse, the wbale Tork done
in tbiii case is
* ("f ■*■ iV) -^*(^ + H^) + etc.
To avoid unnecesaaiy complications, let ua suppose S^, S^, etc,
to be each infinitely smaU. The preceding ezpression for the
wort done becomes
*^+*(^+eto. ;
and, as the effect produced by this "work ia augmeDtati<m of
kinetic energy from 7* to 2* + ST, we must have
Sr=^'^ + *<^ + eto.
Hov let the impnlsea be such as to augment ^toiji + S^, and to
leave the other component velocities unchanged. We shall have
function of li, i, etc., we see that -^ , -- , etc., must be equal
(e) From this we se^ further, that the impulse required to pro-
duce the component velocity ^ from rest, or to generate it in
the system moving with any other possible velocity, has for its
Hence we conclude that to generate the whole resultant velodty
(ifr, ^ ...) from rest, requires an impulse, of which the com-
ponents, if denoted by ^, iji i, ... , are expressed as follows : —
vol.. L 18
..Google
PRELIMINABT.
[313.
f={V'. *)^ + (^, e)<i> + (6, 6)4*..
■m
where it tquEtt be remembered that, an seen in the origioal ex-
preaaion for T, from which they are derived, (0, f) means the
same thing as (^, ift), and bo on. The preceding expreedoos are
the differential coefficients of T with reference to the velodties ;
that is to Bay,
. dT dT ^ dT
(8).
d^ rf<^ d6
{d) The second members of these equations being Imenr fanc-
tionsof ^, ^..., we may, by ordinary elimination, find ^, ^, etc,
in termB of ^, tj, etc., and the expressions so obtained are of
course linear functions of Uie last-named elements. And, Binee
7* is a quadratio Ainction of ^, ^ etc., we hare
22'=6f + j,^ + £^ + etc (9).
From this, on the supposition that T, ^, ^ ... are exprased in
terms off, jj, ..., we have by differentiation
Now the algebraic process by wHcb ifr, ^, eta., are obtained in
tetma of (, ij, etc., sIiotb tliat, inaBmuch aa tlie coefBcient of ^ in
the expression, (7), for (, is equal to the coetSdent of ^, in the
expreeaion for ij, and ao od ; the coefficient of ,; in the expi«s-
sion for tp must be equal to the ooeffidont of £ in the expieaaion
for ^ and so on j that is to say,
tl<l>_d4 d4'_d6
d,,- dC di'dC "^
Hence the preceding espresaian heoomes
-dT . .di dj, ,dJt „,
di
d£
dT
'di-
..(10).
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313.] DYNAMICAL LAWS AND PBINCTPLES. 291
These expreeaiona solve the direct problem, — to find the velo- T*loeitlHi
city produced by a given impulse {(, ij, ...), when we have the momen.
kinetic energy, T, expressed as a quadratic function of the com-
ponents of the impulse.
(«) If we consider the motion simply, without reference to tho
impulse required to generate it from rest, or to stop it, the quanti'
ties (, 1), ... are clearly to be regarded as the components of tlie
momentum of the motion, according to the system of generalized
co-ordinates.
{/) The following algebraic relation will be usefal : — tt«i|irM*l
i,'^ + V,4-i-t6-t-etc = ^, + -nJ, +^6 +ete (11), mJ^
where, i, % ip, ^ ete., having the same signification as before, ("''"tJJ^
$^, ij_, l^, etc., denote the impulse-components corresponding to "«'''">*■
any other values, i^,, ^,, fl,, ete., of the velocity-components. It
is proved by observing that each member of the equation becomes
a symmetrical function of ^, \j/^; ^ 0j etc. ; when for (^, t)^, etc,
their values in terms of ^,, ^,, etc., and for ^, 17, etc., thdr values
in terms of ^, ^ etc., are substituted.
314. A material system of any kind, given at rest, and
, . ■<• 1 1- ■ 1 r Applliatton
subjected to an impulse m any apeciiied direction, and of any oTmnBr*!-
given ma^itude, moves off so as to take the greatest amount ordinata
of kinetic energy which the specified impulse can give it, ottni.
Bubject to § 308 or § 309 (c).
Let {, % ... be the components of the given impulse, and
<ir, ^, ... the components of the actual motion produced by it,
which are determined by the equations (10) above. Now let ua
suppose the ^^tem be guided, by means of merely directive
constraint, to take, from rent, nnder the influence of t^e given
impulse, some motion (^,, ^,, ...) different from the actual
motion; and let f,, tj^, ... be the impulse which, with this oon-
straint removed, would produce the motion {^,, ^„,..). Wa
nhall have, for this case, as above.
But ij — £, 17,-' 17... are the components of the impulse ex-
perienced in virtue of the constraint we have supposed introduced.
They neither perform nor consume work on the system when
moving as directed by this constraint ; that is to say,
(f,-«0,+ k-l)^, + {t-O^. + «t<^ = <» 02);
19—2
..Google
292 FREUKNABT. [314.
AppliaUion and therefore
ife^' . 2T=^,^r^, + ti + ^ (13).
UuDTMu of ' Hence we have
2(r-rj=«#-W+i(i-*.)+<*«-
=«-0(*-*,)+(i-iJW-W*«^
But, by (11) luid (12) above, we have
itf-W*i-W-^J+«i«--«-f,)^,+(i-5,)'f.+«'«--».
aad theiefore we bare finally
2(V'-rj.(f-f,)(*-«*(,-,J(i-i,) + etc. ...(U),
'^^^emi tJiat is to Bay, T eisoeeds 7, by the amoant of the kinetic e&ergj
lemi^* that ■would be generated by an impolse ((—£,, 1-1^ {-t' **"■)
00-ordl- applied simply to the system, which is essentially podtive.
In other words,
816. If the system is guided to take, under the action of a
given impulse, any motion (•^,, 0, , . . .) different from the natural
motion {^(r, 0, ...), it will have less kinetic energy than that of
the natural motion, by a difference equal to the kinetic ener^
of the motion (■^-■^,, ^ — 0,) ■■•)■
Cob. If a set of material points are struck independently
by impulses each given in amount, more kinetic energy is
generated if the points are perfectly free to move each iu-
dependently of all the others, than if they are connected in any
way. And the deficiency of eneigy in the latt«r case is equal
to the amount of the kinetic enei^ of the motion which
geometrically compounded with the motion of either case would
give that of the other.
Probtami (a) Hitherto we have either&upposed the motiontobefbllygiven,
inToira im- and the impnlses required to produce them, to be to be found ; or
TdmiirBi,* the impnises to be given and the mofaons produced by them to be
to be found, A not less important class of problems is presented
by suppooing as many linear equations of condition between the
impulses and components of motion to be given as there are de-
grees of freedom of the nystem to move (or independent coKtrdj-
nates). Hiese equations, and as many more supplied by (8)
or their equivalents (10), suflice for the oomplete solution of the
problem, to determine the impulses and the motitKL
..Google
315.] DTNAUICAL LAWS ASD PRIHCIPLES. 293
(() A veiy important case of this clasB la preeented by presoib- Piobtmu
ing, among the velocitiea alone, a number of linear equations with taToive Im-
constant terms, and snppoeiiig tiie impulses to be so directed and vaio^tiaa.
related aa to do no work on any Telocities Batisfying another pre-
scribed set of linear equations with no constant terms ; the whole
number of equations of course being equal to the number of ind&-
pendent co-ordinates of the system. The equations for solving
this problem need not be written down, as they are obvious ; but
the following reduction is useful, as affording the easiest proof of
the minimum jffopertf stated below.
(c) The gir^i equations among the Telocities may be reduced
to a set^ each homogeneous, except cms equation with a constant
term. Those bomogeneons equations diminish the number of de-
grees of freedom j and we may transform the co-ordinates so as
to have the number of independent co-ordinates diminished ac-
cordingly. Farther, we may choose the new co-ordinates, so
that the linear function of the Telocities in the single equation
with a constant term may be one of the new velocity-ooinponents ;
tmd the linwi.r functions of the velocities appearing in the equation
connected with the prescribed conditions as to the impulses may
be the remaining velocity-components. Thus the impulse will
fulfil the condition of doing no work on any other component
Telocity than the «ne which is given, and the general problem —
816. Given any xoaterial system at rest : let any parts of •*'S?i
it be set in motion suddenly with any specified velocities, pos- '<?'?.''^^
sible according to the conditions of the system; and let its
other parts be influenced only by its connexions with those;
req aired the motion:
takes the following veiy simple fonn : — An impulse of the cha-
racter specified as a particular component, according to the
generalized method of co-ordinates, acts on a material system ;
its amount being snob as to produce a given velocity -component
of the corresponding type. It is required to find tiie motion.
Tho solution of oourse is to be found from the equations
^:^A, ij = 0, i = 0 (Ifi)
(which are the special equations of condition of the problem) and
the general kinetic equations (7), or (10). Choosing the latter,
and denoting by [i, {], [{, i;], etc., the coefScients of J£*, ^ etc,,
..Google
PBEUUIHABV. . [3^6-
a T, we have
for th« result.
This result posseeaea the remarkable property, that the
kiiietic enei^ of the motion ezpioBsed by it is less than that of
any other motioa which fulfils the preecribed condition as to
velocity. For, if £,, >;,, ^, etc., denote the impulsee required to
produce any other motion, ^,, <j>^, tf,, etc., and 7*, the correapond-
ing kinetic enei:;g7, we have, by (9),
2r, = i,^, + j),<^, + i,^, + etc.
But by (11),
f> +\<^ + i,9 + etc. = f^,,
uace, by (16), we hare 1} = 0, f=0, etc. Heuoo
2r,= i^, + f,(^,-.^) + ,,(^,-^)+f,(^,-^)+...
Now let also this second case (^^ ^,<---) of motioa fulfil the pn-
Bcribed velodty-ooudition ip^^A. We ahall have
«,W,-*)+i, (*,-*)+{,(«,-«) + -
-«,-«tf,-*>*fe-i)('*,-*) + K-0(9,-i') + —
nnce ^,— <^~0, ))=0, {=0,.... Hence if S denote the kinetic
enei^cd the differential motioa (^,— ^, ^, -^...) we have
2r, = 2r+20 ■■■(17);
but IT is essentially positive and therefore T^, the kinetic energy
of any motion fulfilling the prescribed velocity -condition, but
differing from the actual motion, is greater than T the kinetic
energy of the actual motion ; and the aitLount, 9, of the diffei^
ence is given by the equation
2« = ij,(.^,-^) + f,(^,-^) + etc. (18),
OP in words,
817. The solution of the problem is this : — The motioa
actually takeo by the system is the motion which has less
kinetic energy thao any other fulfilling the prescribed velocity-
conditions. And the excess of the energy of any other sucli
motion, above that of the actual motion, is equal to the eoerg}'
of the motion which must be compounded with either to pro-
duce the other.
..Google
317.] DTHAMICAl lAWS AND PHINCIPLES. 293
In dealing with cases it mxy often happen that the use of the Kimtk
co-ordinate ajstem required for the application of the solution n
(16) is not convenient; but in all cases, erea in such as in
examples (2) and (3) below, whidi involve an infinite number
of degrees of fireedom, the Tninimum pn>j>erty now proved affords
an eaay solution.
Example (1). Let a smooth plane, constrained to keep moving impact qI
with a given normal velocity, g, come in contact witli a &ee rigid plane
inelastic rigid body at rest : to find the motion produced. The ^bmm «■ &
velocity-condition here is, that the motion ahall consist of any bodjM
motion whatever giving to the point of the body which is struck "*''
a stated velocity, q, perpendicular to the impinging plane, com-
pounded with any motion what«ver giving to the same point
any velocity pandlel to this plane. To express this condition, let
u, V, w be rectangular component linear velocities of the centre
of gravity, and let nr, p, tr be component angular velocities round
axes through the centre of gravity parallel to the liue of re-
ference. Thus, i£ X, y, z denote the co-ordinates of the point
struck relatively to these axes through the centre of gravity,
and il I, m, n bo the direction cosines of the normal to the im-
pinging plane, the preacribed velocity-condition becomes
{w + ps — <ry)i + {i) + i73! — ws)nt + (w + nry — px) ii = — g (o),
the negative sign being placed before q on the understanding
that the motion of the impinging plane is obliquely, if not directly,
lovrarda the centre of gravity, when /, m, n are each ptmtive.
If, now, we suppose the rectangular axes through the centre of
gravity to be principal axes of the body, and denote by Mf, Mff",
Mit' the moments of inertia round them, we have
y-ii/'{M'-n;' + K.'+/*w' + yV + AV) (6).
This must be made a minimam subject to the equation of oon-
dition (a). Hence, by the ordinary method of indeterminate
multipliers,
3f/*w + X{7iy-ine)^0, Mg'p+X{la-nx) = 0, JfAV+A(j7W!-Zy) = 0j'*'
These six equations give each of them explicitly the value of one
of the six unknown quantities u, v, -w, or, p, tr, in terms of X and
data. Using the values thus found in (a), we have an equation
to determine X ; and thus the solution is completed. The first
three of equations (c) show that A, which has entered as an
,w
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296 PRELDONABY. [317
iDdetenniiiate maltiplier, is to be interpreted as the measure of
the amount of the impulse.
^f^UI*" Example (2). A stated velocity in a stated directicm is com-
^^1™'" mimicated impnltdvelj' to each end of a flexible iiiexten«ble cord
Jjj^'*'' tonning any cnrvilineal arc: it is required to find the initial
otwin. motion of the whole cord.
Let 3^ ^, 2 be the co-ordinates of any point i' in it, and x, y, :
the components of the required initial velodty. Let abo a be
the length &om one end to the pdnt P.
If the cord were extensible, the rate per tinit of time of the
stretching per imlt of length which it would experience at P, in
Tirtne of the motion it, ^, z, would be
dxd± dffd$ dzdz
de de ds da dsds'
Hence, as the cord is inextensible, by hypothesis,
dxdi dyd^ did£_f. . .
d^di*didi*dtdi''^ ^'''■
Subject to Uiis, the kinematical condition of the t^stem, and
when » = 0, y = w" [ when e = l.
I denoting the length of the cord, and (u, v, t^), (u', e', vt"), the
componentB of the given velocities at its two ends : it is required
to find ^ 3^, 2 at every point, so as to midce
£j/' (*•+*■+ so* m
a minim mn, fi denoting the mass of the string per nnit of length,
at the point P, which need not be tmiform from pcont to point ;
and of course
A = ((£c' + dy'+&^S (c).
Multiplying (a) by A, an indeterminate multiplier, and proceeding
aa usual according to the method of variations, we have
in which we may regard x,y,za8 known functions of s, and this
it ifl convenient we should make independent variable. Inte-
..Google
317.] DYNAMICAL LAWS AND PRINCIPLES. 297
grating "hy parts" the portion of the first member which contains Genention
X, and attending to the terminal condidons, we find, according to br impnin
ike regular process, for the equations ooutaimug the solution ui^iibie
^ d /.dx\ , d /.dy\ . d/.dji\ /J, '*^"
'^=ds[''d.)' I'^-dsV'd^)' '"=di(.W ^'^^-
These three equations with (a) suffice to determine the four
nnknovn quantities, ±, y, i, and A. TTaing (d) to eliminate i,$,i
firom (a), we have
''4{tmy-H{smy-}-
Taking now a for independent Tariable, and perfomung the
differontiation here indicated, with attention to the following
relations : —
T3;+...=l, J--0+ .,.=0,
dt" ' da or '
dx<^x /d'xy
■■■*{-3f) *■-<>■
and the expression (§ 9) for p, the radius of corrature, vb fi
1(?X ''W^,
11 da* as da
a linear difforential equation of the second order to determine
A, when /i and p are given fiinctious d s.
The interpretation of (d) is veiy obviouB. It shows tliat \ is
the impulmve tension at the point P at the string ; and that the
velocity which this point acquires instantaneously is the retniltant
/ida
The differential equation (a) therefore shows the law of trans-
mission of the instantaneous tension along the string, and proves
tiiat it depends sdely on the mass of the cord per unit of lei^iih
in each pert, and the curvature &om. point to point, but not at
all on the plane of curvature^ of the initial form. Thus, for
instance, it will be the same along a helix as along a circle of
the same curvature.
..Google
8 PBELIMINiJlY. [317.
With reference to the fulfilling of the six terminAl equations,
a difficult occurs inoBiniich as :c, y, z ore expressed hj (d) imme-
diately, without the iutroduction of fresh arbitrary constants,
in terms of X, which, as the solution of a difierential equation of
the second degree, iuvoives only two arbitrary conslanta The
explanation is, that at any point of the oord, at any instant, any
velocity iu any direction perpendicular to the tatigeut mdy be
generated without at alt altering the condition of the cord eren
at points in&iitety near it. This, which seems clear enough
without proof, may be demonstrated analytically by transfomting
the kinematical equation (a) thus. Let y be the component tan-
gential velocity, q the oomponent velocity towards the centre of
curvature, and p the component velocity perpendicular to the
osculating plane. Using the elementary formulas fur the direc-
tion cosines of these lines (§ 9), and remembering that < is now
independent variable, we have
Substituting these in (a) and reducing, we find
<if_t
■ (/).
a form of the kinematical equation of a flexible line which will
be of much use to us later.
We see, therefore, that if the tangential components of the im-
pressed terminal velocities have any prescribed values, we may
give besides, to the ends, any velocities whatever perpendicular
to the tangente, without altering the motion acquired by any part
of the cord. From this it is clear also, that the directions of the
terminal impulses are necessarily tangential ; or, in other words,
that an impulse inclined to the tangent at either end, would
generate an intiiiite transverse velocity.
To express, then, the terminal conditions, let .F and I" be the
tangential velocities produced at the ends, which we suppose
known. We have, for any point, J*, as seen above &om (d),
^-^s ■■ <»'•
jiGoogle
317.] DYNAMICAL LAWS AND PRINCIPLES. !
uid hence when
and when t=l, - ~ -F'
which auffioe to determiiie the canstoDta of integratioD of (d\
Or if the datA are the tangential impulses, /, T, required at the
ends to produce the motion, to have
when » = 0, X = /,l ,.
and when t = l, X^/'J ^''"
Or if either end be free, we have X = 0 at it^ and any preecribed
condition as to impulse applied, or velocity generated, at the
other end.
The solution of thia problem is very interesting, as showing
how rapidly the propagation of the impulse falls otf with "change
of direction" along the cord. The reader will have no difficulty
in illustnting this by working it out In detail for the case of a
cord either uniform or sucb that t^-f- ^ constant, and ^ven ia
the form of a circle or helix. When /i and p are constant,
fur instance, the impulsive tension decreases in the proportion
of 1 to « per space along the curve equal to p. The results have
curious, and dynamically most interesting, bearings oa the mo-
tions of a whip lash, and of the rope in harpoouiug a whole.
Easymple (3). Let a mass of incompressible liquid be given at imenliiTe
rest completely filling a closed vessel of any shape ; and let, by inonniOTn^
suddenly commencing to change the shape of this vessel, any
arbitrarily prescribed normal velocities be suddenly produced in
the liquid at all points of its bounding surface, subject to the
condition of not altering the volume : It is required to find the
instantaneous velocity of any interior point of the fluid.
Let z, y, « be the co-ordinates of any point P of the space
occupied by the fluid, and let u, v, w be the components of the
required velocity of the fluid at this point. Then p being the
density of the fluid, and //j* denoting integration throughout the
space occupied by the fluid, we have
a"=///Jp(«' + «^+«^'£»V= («).
..Google
Biblal^nid.
»^)rfafy&..
0 PREXIMINABT. [317,
which, Bubject to the kmematical oondition (g 193),
du dv dv) „ „,
:j-+:j-+ T-=0 (6),
ax dy de * "
must be the least possible, with the given surface rallies of the
normal component Telocity. By the method of vamtJoa we have
///{p(rf„+**»s»)+x("^^ + ^ + '^?)}i<;W.-o....M.
But integrating by parts ve iuve
...(d),
and if 2, m, » denote the direction cosines of the normal at any
point of the anriace, dS an element of the surface, and J^ in-
tegration over the whole aurface, we have
f^\(pudydz+hvdidx + hu)da!dy) = jj\{Ku + m&v + T&w)dS = ^,
since the normal component of the velocity is given, which
requires that /Su -t- mSv + n&w = 0. Using this in going back
with the result to (c), (ij), and equating to zero the coefficients of
Su, Sr, 8«i, we find
dk d\ d\
f^-^' i^=d^' ^'^^ w-
These, need to eliminate u, v, to from (6), give
d<e\(>dx)* dsKpdy)* dzKpdz) ^^
an equation for the determination of A, whence by (a) the
solution is completed.
The condition to be fulfilled, bemdes the kinemaldcal equation
(&), amounts to this merely, — that p{udas+vdy + wdz^ia\uA be
a complete difiWrential. If the fluid is homogeneous, p is con-
stant, and vdx + vdy + wdt must be a complete differentia] ; in
other wcnrds, the motion suddenly generated must be of the
" non-rotational" tdiaracter [g 190, (*)] throughout the fluid maB&
The equation to detennine \ becomes, in this case,
tPk d'K <P\ ^
^*s?+^=o <^)-
..Google
317.] DYNAMICAL LAWS AND PRINCIPLES. 301
From tbe hydrodynamics] principles explained later it vUl IiimiBtn
^pear that A, the function of which p (udx + vdy + wdt) is inooniprw-
the differential, is the impulsiTe pressure at the point (x, y, z)
of the fluid. Hence we may iufer that the equation {J^, with
the condition that X shall have a given value at every point
of a certain closed surface, has a possible and a detenninate
solution for every point within that surface. This is preciaely
the same problem as the determination of the permanent tempe-
rature at any point within a heterogeneous solid of which Hno
Bur&ce is kept permanently with txaj non-uniform distribution
of temperature over it, (/) being Fourier's equation for the
uniform conduction of heat through a solid of which the conduct-
ing power at the point (x, y,z)M-. The possibility and the
P
determinat«ne8B of this problem (with an exception regarding
multiply continuous spaces, to be fully considered in Tol. II.)
were both proved above [Chap. i. App. A, («)] by a demonstra-
tion, the comparison of which with the present is instructive.
The other case of superficial condition — that with which we
have commenced here — shows that tJie equation {/), with
, dK dX dh. . , . ., _ . 1. ,
i-^ + n»-j- + »-5- given arbitrarily for every pomt of the sur-
face, has also (with like qualification respecting multiply con-
tinaous spaces) a possible and single solution for the whole
interior space. This, as we shall see in examining the mathe-
matical theory of magnetic induction, may also be inferred from
the general theorem (e) of App. A above, by supposing a to be
zero for all points without the given sur&ce, and to have the
value - for any internal point (x, y, z).
318. The equations of continued motion of a set of free i-Ri»nge>
particles acted on by any forces, or of a system connected in motion la
any manner and acted on by any forces, are readily obtained ^uniiMd
in terms of Lagrange's Qeneralized Co-ordinates by the regular
and direct process of analytical transformation, from the or-
dinary forms of the equations of motion in terms of Cartesian
(or rectilineal rectangular) co-ordinates. It is convenient first
to effect the transformation for a set of free particles acted
on by any forces. Tbe case of any system with invariable
connexions, or with connexions varied in a given manner, is
..Google
302 pREtranNART. pi8.
then to be dealt with by RuppoRing one or more of the gene-
mlized co-ordinates to be constant: or to be given functions
of the time. Thus the generalized equations of motion are
merely those for the reduced number of the co-ordinates re-
maining tin-given ; and their integration determines these
co-ordinates.
Let !»,, m„ etc. be the maeaea, ir,, ;/,, «,, «„ etc. be the co-
>- . ordinatea of die particles; and J!,, T^,Z^,X^, etc. the components
of the forces acting upon them. Let ^, ^, etc be other variables
(H]ual in number to the Cartesian co-ordinates, and let there be
the same number of relations given between the two sets of
variables ; so that we- may either regard ^, ^, etc as known
functions of x,, y^, etc., or a;,, y^, etc aa known functions of
i^, ^ etc. Proceeding on the latter supposition we have the
equations (a), (1), of § 313; and we have equations (&), (6), c^
the same section for the generalized components 4*, 4, etc. of the
force on the aystem.
For the Cartesian equations of motion we have
^'-"'df' ^■-'"■■3/' ^'-"'M- Jr.-m,^'efc...(19).
Multiplying the first by -j— ', the second by -j^, and so <m,
and adding all the products, we find by 313 (6)
lie d^'diy'dii)' 'dIdt'dlV'df/ '<(*
'di[i-df-i '^r <"'•
Using this and similar expressions with reference to the other
co-ordinates in (20), and remarking that
Jm,(i," + 3?,' + i,') + Jm,(etc.)+etc = 2' ...(22),
if, as before^ we put T for the kinetio enei^ of the system; we
find
, ddT dT
*= . -— (2S).
..Google
318.]
DYNAMICAL LAWS AND PRINCIPLES.
. d£,
dx.
«boT«, fmppose ;£, to be a function of the co-ordinates, and of the „
generalized Telocity-components, as shown in equations (1) of d>
§ 313. It is on this supposition [which makes T a qnadra-tic i,
function of the generalieed velocity-components with functions H
of the co-ordinates as coefficients as shown in § 31 3 (2)J that the ^
difTerentiations jj and -j- in (23) are performed. Proceeding ^
similarly with reference to <f>, etc, we find expressions similar to
(23) for 4>, etc, and thus we have for the equations of motion in
terms of the generalized co-ordinates
..(24).
ddTdT
d dT _dT
dtd^ di>~ '
It is to be remarked that there is nothing in the preceding
transformation which would be altered by supposing t to appear
JQ the relations between the Cartesian and the generalized oo-
: thus if we suppose these relations to be
F(x„3,„z„x„ ^,,l,,e, ()
0 = 0]
..(25),
..(26),
we now. Instead of § 313 (1), have
where (^) denotes what the velodty-oomponent i, would be
if ^, ^, etc were constant; being analytically the partial differ-
ential coefficient with reference to 1 of the formula derived from
(26) to express if, as a function <d t, tfr, ^, 0, etc
XTsing (26) in (22) we now find instead of a homogeneotu
quadratic function of ^, ^, etc, as in (2) of § 313, a mixed
..Google
S04 PRELIMINARY. [318.
"Ugntwfi fimctioQ of zero degree and first and second degrees, for the
motiim in Emetic energy, as foilows : —
SSSlr-^^W^+W**-'. +J!», ♦)♦'*(*, *).*'+-2»,*)«...l..(27),
I. j\ -e /<^ '^ dy du dz (fo\
motkm Id
tarmior
C»rtejl«i f/dx\dx
■f28);
K, (f), (it>), (^1 ff), {•!', fft)) ^^- being thus in general eact a kno«-n
fimctioQ of I, ^, <l>, etc.
Equationa (24) above are Lagrange's celebrated equations of
motion in terms of generalized co-ordinates. It -was first
pointed out by Viellle* that they are applicable not only when
^, 0, ete. are related to a;,, y^, «,, x^ etc, by invariable relations
as supposed in Lagrange's original demonstration, but also
when the relations involve t in the manner shown >n equa-
tions (25). Lagrange's original demonstration, to be found
in the Fourth Section of the Second Part of his M^cantque
Analytique, consisted of a transformation from Carte^an to
generalized co-ordinates of the indeterminate equation of
motion; and it is the same demonstration with unessential
variations that has been hitherto given, so &r as we know,
by all subsequent writers including ourselves in our first edition
(§ 329). It seems however an unnecessary complication to
introduce tlie indeterminate variations hx, Sy, etc ; and we find
it much simpler to deduce Lagrange's generalized equations
by direct transform&tton from the equations (A motion (19)
of a free particle.
* Snr leB fegnationi diffirentiellea de la dfruuniqiie, LiomiUt'* Journal,
1819, p. aoi.
..Google
318.]
DYNAMICAL LAWS AND FRINaPLES.
When the kinematic relationa are invariable, that is ta aay Lwranfls'*
jLngnrat
vhen t does not appear in the equations of condition (2fi), ve ^o^ti
find from (27) and (28),
2' = iU<f.^)^+2(V,*)if^+(^*)^*+...} (29),
\ df ^ rf* ^
**'-^**-}*
..(29'),
>")•
Qtm-i
Hence the ^-equation of motion expanded in this, the most
important class of cases, is as foUotrB :
where
.rf^^ d4, ^^ I dif, f^J }
(29"')-
Kemark that Q^ (T) is a quadratic function of the velocity-com-
poneata derived from that which expresses the kinetic enei^
(T) by the process indicated in the second of these equations,
in which ifi appears singularly, and the other co-ordinates sym-
metricallj' with one anotjier.
Multiply the ^-equation by i}i, the ^-equation by ^, and so Eqution
on ; and add. In what comes from Q^ {T) we find t«nus ennBj.
- ^ . ^, and —
+ iiM)^..
V.*i
which together yield
With thifl, and the rest simply as shown in (!
[(♦,«♦ + (*,*) + +■■■]*
jiGoogk'
PBELOnNAKT. [SIS.
dT ,
^d^''
^ + *^4- {29"),
or ^ = *^ + *^+ (29^),
'■ When the kinematical relations are inTEiriable, that is to sar,
when C does not appear in the equations of condition (25), the
equftlions of motion may be put under a slightly different fonu
fitst given by Hamilton, which is often convenient ; thus : — Let
T, \lr, 1^..., be expresBed in terms of (, t),..., the impulses re-
quired to produce the motion from rest at any instant [§313 (t/)] ;
so that T will now be a homc^neous quadratic function, and
iji, ^ ... each a linear function, of these element with coeffi-
cients— functions of ^, ^, etc., depending on the kinematical
conditions o£ the system, but not on the particular motion.
Thus, denoting, aa in g 322 (29), by d, partial difierentiation with
reference to f , >h --> ^i ^i-'-i considered as independent vari-
ables, w« have [§ 313 (10)]
^-'i- *'% (»>.
and, alloving d to denote, as in what precedes, the partial dif-
ferentiatJons with reference to the system ^, ^, ..., ^, ^ ..., we
have [§ 313 (8)J
(-rr "'n ("'•
The two expresdons for T being, as above, § 313,
2'=M(^.<f)^+-+2(^,^)^<^+...( = i{[^,^]f+,..4.2[^,^]f,+...l{32),
the second of these is to be obtained from the first by substitu-
ting for iji, ^..., their expressions in terms of |, i;, ... Hence
dTdT dTd4^ dTd4 _dT ,d_dT d^ dT
_dT d_/ dT dT \_^dr dT
df ^ d^ V di^^'dy,*- J-d^f^^d^-
From this we conclude
Hence Lagnnge's equatioiifl become
\ rff 9r , ,
31* dj-*'^
..(34).
jiGoogle
318.] DYNAMICAL LAWS AND PRINCIPLES. 307
Iq § 327 below a purely analytical proof will be given of HsmiiMn'*
LagraDge's geueralized equations of motion, establishing them
directly as a deduction from the principle of " Least Action,"
independently of any expres-sion either of this principle or of
the equations of motion in terms of Cartesian co-ordinates. In
their Hamiltonian form they are also deduced in § 330 (33) from
the principle of Least Action ultimately, but through the beau-
tiful " Characteristic Equation" of Hamilton.
319. Hamilton's form of Lagrange's equations of motion in
terms of generalized co-ordinates expresses that what is re-
quired to prevent any one of the components of momentum
from varying is a corresponding component force equal in
amount to the rate of change of the kinetic energy per unit
increase of the corresponding co-ordinate, with all components
of momentum constant : and that whatever is the amount of
the component force, its excess above this value measures the
rate of increase of the component momentum.
la the case of a conservative system, the same statement
takes the following form : — The rate at which any component
momentum increases per unit of time is equal to the rate, per
unit increase of the corresponding co-ordinate, at which the
sum of the potential energy, and the kinetic energy for con-
stant momentuma, diminishes. This is the celebrated "canonical
form " of the equations of motion of a system, though why it
has been so called it would be hard to say.
Let V denote the potential energy, so that [§ 293 (3)] "Cuioniaa
♦V + *S^+ ,.. --8F, ^SS"''
andtherdbre ' * — ~ , * = S' - SSSS^"
Let now 1/ denote the algebraic expression for the sum of tlie
potential energy, F, in terms of the co-ordinatee, ^, ^..., and the
kinetic energy, T, in terms of the co-ordinates and the oomponenta
of momentum, ^, 1^.... Then
# 3a\ f («>■
jiGoogle
308 PRELIKmAKT. [319.
the latter being eqalTalent to (30), since the potential energy doa
not contEUD £, if, etc
In the following examples ve shall adhere to Lagrange's i<^m
(24), as the most convenient for such applications,
Eninpiaot Example (A). — Motion of a single point (m) referred to pokr
l*Bnuije'» coordinates (r, 6, ^). From the well-known geometty of this
eqtutiuiior caso we see that Sr, rS0, and rsin^S^ are tho amouuts of lineai
poiuco- displacement corresponding to infinitely small increments, £r, Sd,
80, of the co-ordinates : also that these displacements are respec-
tively in the direction of r, of the arc rl6 (of a great circle)
in the plane of r and the pole, and of the arc rsin 68^ (of s
small circle in a plane perpendicular to the axis); and that ther
are therefore at right angles to nne another. Hence ii F, G, H
denote the components of the force experienced by the point, in
these three rectangular directions, we have
F=R, Gr=9, and iZr sin « = * ;
S, 0, 4 being what the generalized components of force (g 313)
become for this particular system of co-ordinatea. We also see
that t, r6, and rsinf)^ are three components of the velocity,
along the same rectangular directions. Hence
From this we have
dT . dT ,A dT _, . „.
~ = mr(^ + sin' 6^.% ^ = mr^sin tf cos tf^*, ^ = 0.
Hence the equations of motion become
or, ac<x>rtluig to tbe ordinaiy notAtion of the dJ£'eteiiti»l calcnlut.
(iPr fdff . ,„dA*\) _
jiGoogle
819.] DYNAMICAL LAWS ASD PKINCIPLE8.
Mi-t)-
■»s('"'^'»t)-^'"»-
ftenermlncd
If the motion is confined to one plane, that of r, 9, we have
^ = 0, and therefore S=0, and the two equatioua of motion
which remun are
These equations might haTe been written down at once in terms
of the second law of motion from the kinematical investigBtion of
§ 32, in which it was shown that -73— r -rs, and --,- ( r* — )
dr dr r dt\ at/
are the components of accelemtim along and perpendicular to
the radius-vector, when the motion of a point in a plane is ex-
pressed according to polar coordinates, r, 0.
The same equations, with ^ instead of 0, are obtained from the
polar equations in three dimensions by patting 0 - ^w, which
implies that ^ = 0, and confines the motion to the plane (r, <ft).
Example (B). — Two jwrtlcles are connected by a string ; one D]
of them, m, moves in any way on a smooth horizontal plane, and ^
the string, passing through a smooth infinitely small aperture in
this plane, bears ihe other particle m', hanging vertically down-
wards, and only moving in this vertical line : (the string re-
maining always stretched in any practical illustration, but, in
the problem, being of course supposed capable of transmitting
negative tension with its two parts straight.) Leti be the whole
length of the string, r that of the part of it from m to the aperture
in tJie plane, and let 6 be the angle between the direction of r
and a fixed line in the p]an& We have
dT , ,._^ dT .,
dr ' d0
Also, there being no other external force than ^tii, the weight
of the second particle,
R=~gm', 0=0.
..Google
the u« of
310 PBEUQCINART.
Hence the equations of motion are
(m + m")r - mr^ ^ - m'g,
[319.
Eunipleg
iiig door.
dt
The motion of tn' is of course tLat of a particle influenced onlr
by a force towards a fixed centre; but the law of ttis force, I'
(the tetudon of the string), is remarkable. To find it we hare
{§ 32), P = m(- r + r^). But, by the equations of the motion.
f-r(^ =
- (j + r^, and fl =
m + m' ^ ' " " ' w»r"
where h (according to the usual notation) denotes the moment
of momentum of the motion, being an arbitrary constant of in-
tegi'ation. Hence
'''£^'{'*^-'-)-
The particular case of projection which gives m a circular motion
and leayea m' at reet is interesting, inasmuch as (g 350, below)
the motion of m is stable, and therefore m' is in stable equi-
librium.
Example (C). — A rigid body m is supported on a fixed axis,
and another rigid body n is supported on the first, by another
axis ; the motion round each axis being pei-fectly free.
Case (a). — T/ie lecand axis parallel to the first. At any time,
t, let <f> and f be the inclinations of a fixed plane through the
first axis to the plane of it and the second axis, and to a
plane through the second axis and the centre of inertia <d the
second body. These two co-ordinates, ^ ^, it Is clear, completely
specify the configuration of the system. Now let a be the dis-
tance of the second axis from the first, and b that of the c«ntre
of inertia of the second body from the second axis. The velocity
of the second axis will be o^ ; and the velocity of the centre
of inertia of the second body will be the resultant of two velocities
a4, and b^,
in lines inclined to one another at an angle equal to ^ - ^ and
its square will therefore be equal to
aV -I' 2ahii4 cos (^ - ^) + b'lj/'.
Hence, if wi and n denote the masses, j the itkdiuB of gyration
of the first body aliout the fixed axis, and k that of the second
..Google
319.] DYNAMICil LAWS AKD PBIHCIPLES. 311
body about a parallel bti'h through its centre of inertia ; 'we hare, ^^'f'^
according to ^ 280, 281, oTni. roid-
ins door.
Hence we have,
— -=mj'*.^+jia''^+no6co8(0-^)^; — r=*«<icoB(^— ^)^+»(6'+A')^;
litj, dip
^=
difi~
The moBt general aappoeition we can make as to tlte applied forces,
is equivalent to ««aniniiig a couple, $, to act on the first body, and
a couple, % on the second, each in a plane perpendicular to the
axes ; and these are obvious] j what the generalized components of
strees become in this particular co-ordinate system, ^ ^. Hence
the equations of motion are
{mj' + na')'4 + nal '^l'^<^('^-^)]_^rin(^-^)^^^».
n«6 ^^^^"^^^ + « (6' + *■) ^ + naj sin (^ - 0) ;^.^ = *.
If there b no o&er applied force than gravity, and if, as ve may
suppose without losing generality, the two axes are horizontal, the
potential eneigy of the system will be
ymi (1 - cos ^) + ?« {o [1 - COB (^ + J)] + 6 [1 - ooB (-A + ^)]},
the distance of the centre of inertia of the first body &om the
fixed axis being denoted by h, the inclination of the plane
through the fixed axis and the centre of inertia of the first body,
to the plane of the two axes, being denoted by A, and the fixed
plane being so taken that ^ = 0 when the former plane is vertical.
By differentiating this, with reference to ^ and f, we therefore
— ^ = ffmh Bin ip + g^ia an {ifi + A), —'t-^nbma{>p + A).
We shall examine this case in some detail later, in connexion
with the interference of vibrations, a subject of much importance
When there are no applied or intrinsic working forces, we
hare ^ = 0 and 4' = 0 : or, if there are mutual forces between the
two bodies, but no forces applied from without, ♦ + * = 0. In
..Google
LaniMg*'!
ntrifugil
312 PBELmiNART. [319.
t either of these cases we hare the following fiist iut^nil : —
{mf + na') ij> + m'ah cob (^ - ^) (<^ + ^) +n (6' + i^ ^= C ;
obtained hj adding the two equations of motioa and int^rating.
This, which clearly ezpreaaes the constancf of the whole moment of
momentum, gives ^ and ^ in terms of (^ - ^) and (^ ~ 0). Udng
these in the integral equation of energy, provided the mutual forces
are funddons of ^-^ we have a single equation between
■ -, (^— ^), and constants, and thus the full solution of
the problem is reduced to quadratures. [It is worked out fullj
below, as Sub^ezample Q,.]
C (61. Ctue (6). — TJte second axil perpendicular to the JinL For
Koieraing simplicity suppose the pivoted axis of the aeoond body, n, to be
aprincipal axis relatively [§ 282 Def. (2)] to the point, N, ax
which it is cut by a plane perpeadicular to it through the fixed
imtMiioii axis of the first body,, m. Let ffE and NF bean's two other
o«r principal axes. Denote now by
h the distance from N to nia fixed axis ;
k, e,f the radii of gyratioa <^ n round its three principal
axes through N ;
j the radius of gyration of m round its fixed axis ;
$ the inclination, of iV£ to tn's fixed axis ;
ijr the inclination of the plane parallel to n's pivoted axis
through m's fixed axis, to a fixad plane through the
latter.
Bemacldag that the component angular velocities of n rouod
y£ and ^F are i^ cob 0 and ^ sin tf, we find immediately
T = J {[»«,■• + n {A' + 0* COB* «+/' sin* «)] ^ + ni" ^},
or, if we put
ffy*+n{A*+/^ = 0, »((«*-/•) = /);
r = J {((3 + Z) cob' fl) ^ + ni* ^(.
The farther working out of this case we -leaTe as a simple' but
most interesting exercise for the student. We may return
to it later, as its application to the theory of omtrifugal chrono-
metric regulators is very important.
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319.]
DTHAHICAL LAWS AND PRINCIPLES.
£xamph (C). Take the case G (b) and mount a third body M U
upoa au axis 00 fixed relatively b> n ia &ay poaitioD parallel to pi
If£. Suppose for Biiaplicity 0 to be the centre o£ inertia of if pi
and OC one of its principal axee ; and let OA, OB be its two ed
other principal azea relative to 0. The notation being in other El
respects the same as in Example C (b), denote now farther by
A, B, C the momenU of inertia of M round OA, OB, 00 ; 0 the
angle between the plane AOO and the plane through the fixed
axis of m perpendicular to the pivoted axis of n; w, p, tr the
component angular velocities of Jf round OA, OB, OC.
In the annexed diagram, taken from § 101 above, ZGZ' ia a
Letter 0 at cen-
tre of sphere
concealed by
Jf2' = .^ + ^
circle of unit radius having ita centre at 0 and its plane parallel
to the fixed axis of m and perpendicular to the pivoted axis
off*.
The component velocitiee of <7 in the direction of the arc ZO
and perpendicular to it are 6 and ^ ain 0 ; and the component
^.Tignlar velocity of the plane ZOZ' round OC a ijuxtaff. Hence
v={Jain^-^sin9cOB^
and <r = ^cos^ + ^
[CompoTO § 101.]
..Google
Motion of
> rigid bod)
piTDC«d on
pTincipal
ffimlM
Bo*L
BiKldbody
4 PSELIMIHAKT. [319.
The kinetic energy of the motion of M relatively to 0, ita
centre of inertia, is (g 381)
and (g 260) its whole kinetic energy is obtained by adding the
kinetic energy of a material point equal to ita mass moving with
the velocity of its centre of inertia. This latter part of the
kinetic energy of ^ is most simply taken into account by BDp-
posing n to include a material point equal to M placed at 0 ;
and using the previous notation h, e, f for radii of gyration c^ n
on the understanding tliat n nov includes tiiis addition. Hence
for the present exaniple, with the pi'eceding notation G, D, we
have
r=JUff + -D«08'tf)^ + nA'^'l
+ jf (^ sin ^ - ^ sin ^ cos ^)' -I- £ (^ cos ^ + ^ sin 0 sin ^)*
+ C(^COBfl+^)*}.
From this the three equations of motion are easily written down.
By putting 0 = 0, D = 0, and £ = 0, we have the case of the
motion of a free rigid body relatively to its centre of inertia.
By putting B = A we fall on a case which includes gyroscopes
and gyrostats of every variety ; and have the following much
simplified formula :*
T=^{{E + Fco^0)<}^ + (71^ + A)& + C{^Q<»e + ^y},
if weput ^=e + J, and F= D~A.
Example (D).— Gyroscopic pendulum. — A rigid body, P, 'a
attached to one axis of a universal flexure joint (§ 109), of which
the other is held fixed, and a second body, Q, is supported on P by
a fixed axis, in line with, or parallel to, the first-mentioned arm of
the joint. For simplicity, we shall suppose Q to be kinetlcally
symmetrical about ite bearing asjs, and OB to be a principal
axis of an ideal rigid body, PQ, composed of P and a mass so
distributed along the bearing axis of the actual body Q as to
have the same centre of inertia and the sune momenta of inertia
round axes perpendicular to it. Let AO be the fixed arm, 0 the
joint, OB the movable arm bearing the body P, and coindding
with, or parallel to, the axis of Q. Let BOA' = B; let ^ be Uie
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319.] DYNAMICAL lAWS AND KtlNCIPLES. 815
amde which the plane AOB makes with a fixed plane of reference, OyroKopia
through OA, chosen so as to contain a second
pi-indpal axis of the imagined rigid body, PQ,
when OB is placed in line with AO ; and let
^ be the angle between a plane of reference in
Q through its asis of syimnetr)' and the plane
of the two principal axes of PQ already men-
tioned. These three co-ordinates (f>, ^ ^)
clearly specify the configuration cf the system at
anytime, t. Let the moments of inertia of the
imagined rigid body PQ, round ita principal
axis OB, the other principal asis referred to above, and the
remaining one, be denoted by ft, 1&, 9t respectively; and let
ft' be the moment of inertia of Q ronnd ita bearing axis.
We have seen ^ 109) that, with the kind of joint we have anp-
posed at 0, every pos^ble motion of a body rigidly connected with
OB, ia resolvable into a rotation round 01, the line bisecting the
angle AOB, and a rotation round the line through 0 perpen-
dicular to tlie plane AOB. The angular velocity of the latter
is 6, according to our present notation. The former would give
to any point in OB the some absolute velocity by rotation round
01, that it has by rotation with angular velocity ^ round AA' ',
and is therefore equal to
waA'OB ,
'foaiOB ^'cosjfl"^
This may be resolved into Si^sin* J0= ^(l — cosf>) round OB,
and 2^ sin ^tf cos ^^ - ^ sin d round the perpendicular to OB, in
plane AOB. Again, in virtue of the symmetrical character of
the joint with reference ia the line 01, the angle ^ as defined
above, will be tii:[ael to the angle between the plane of the two
first-mentioned principal axes of body P, and the plane AOB.
Hence the axis of the angular velocity ^ sin d, is inclined to the
principal axis of moment V at an angle equal to 0. Resolving
therefore this angular velocity, and 6, into components round the
axes of 39 and 0, we find, for the whole component angular
velocities of the imagined rigid body PQ, round these axes,
0sin0coe^-H^sin^ MiA -i^sin0Bin<^-i-^cos^, respectively.
The whole kinetic energy, T, ia composed of that of the imnginetl
rigid lioily PQ, and that of Q about axes through its centre of
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316 PRELIHINA.RT. [319.
G^Touopto inertia : we derefore have
2r=ft(l-coa^)'<^'+V(^siit«coB^+^8iii^)'->-e(^Biii«siii^-^co8^)'
+ a'{^-^(l-coa«)}'.
Hence ^=a['{.^-.^(l-c<«tf)},^=0,
-rr = - 30 (^ sin tf cos 0 + ^ Bin ^) (^ sin d sin ^ - ^ cos i^)
+ &(^sin03in<^-^'coa<^)(<^sintfoo8^+^Bin^),
-,- = »(^ainffcos^+^8in0)Hin^-0(i^BinfiBin^-^cos^)ooBc^
and " = 9 (1 - cos tf) sin «0' + » COB fl cos ^.^ (^ sin ^ cos <^ + ^ ain <^)
+ ecos^sin^<^(^sindsin^-9cos^)-a'sintf^{^-(l-cos0)0}.
Now let a couple, G, act on the body Q, in a plane perpendi-
cular to its axis, and let £, J/, iV act on i*, in the plane perpen-
dicular to OBf in the plane A'OB, and in the plane through OB
perpendicular to the diagram. If f is kept constant, and ^
varied, the couple G will do or resist work in Bimple addition
with L. Hence, resolving Z + (? and N into components round
01, and perpendicular to it, rejecting the latter, and remembering
that 2 sin^9^ is the angular velocity round 01, we have
*=2ainJfl{-(Z;+C)8inJtf+jrcoaJfl}=|-(£+ff)(l-co8tf)+J^8intf}.
Also, obviously
Using these several expressions in Lagrange's general equations
(24), we have the equations of motion of the ^stem. They will
be <^ great use to us later, when we shall consider several parti-
cular cases of remarkable interest and of very gi-eat importance.
Excmiple (E). — Motion of a free parHder^erred to rotating axtt.
Let X, y,z\x the co-ordinates of a moving particle referred to
axes rotating with a constant or varying angular velocity round
the axis OZ. Let x^, y^, z, be ita co-ordinates referred to the
same axis, OZ, and two axes OX,, 0T^, fixed in the plane p^^
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(routing
319.] DTNikMICAL LAWS AND PRIK0IPLE3. 317
pendicular to it. We have B
£, = i£coaa— ^siiia-(a;Bina + ycosa)ii, ^, = etc.
vhere a, the angle XfiX, must be considered as a giyen ftmc- "
tion of t. Hence
dT ,, ., dT ,, ., rfr
ddT , . ... ddT ... ^. ...
and hence the equations of motion are
m(£-2ya — xa'^ya) = X, m(j/ + 2£a — Jfit* + iai)=Y, m'z = Z,
X, T, Z denoting simply the components of the force cm the
particle, parallel to the moving axes at any instftnt. In this
example t enters into the reUtioa between fixed rectaitgular axes
and the co-ordinate system to which the motion is referred ; but
there ia no constraint. The next is given as an example of vary-
ing, or kinetic, constraint.
Example (F). — A particle, infiueneed by any foreet, and at- B:
laehed to one end of a tiring oj which the oOier u moved with any
coTUlani or varying velocity in a straight line. Let 6 bo the kiHtia
inclination of the string at time (, to the given straight line, and
^ the angle between two planes through this line, one containing
the string at any instant, and the other fixed. These two co-
ordinates {B, ^) specify the position, P, of the particle at any
instant, the length of the string being a given constant, a, and
the distance OE, of its other end B, from a fixed pointy 0, of the
line in which it is moved, being & given function of t, which we
shall denote by w. Let x, j, « be the co-ordinates of the particle
referred to three fixed rectangular axes. Choosing OX as the given
straight line, and TOX the fixed plane from whidi ^ is measured,
we have
x=u*aoM$, y = a^n$coBA, s-asin0sin^
Bonitnint
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18 PKELIHIKABT. [319.
uid for ^, 2 we have the same ezpresaiona aa in Example (A).
Hence
where S deuotea the same as the T of Example (A), with
5* = 0, and r = 0. Hence, denoting as there, by O and H the two
comiKjaents of the force on the particle, perpendicular to EF,
respectively in the plane of $ and perpendioular to it, we find, for
the two required equations of motion,
m {« (d- Bin 6 coa*^-)- Bin *«}=(?, tnd ma"^^-^ = ff.
These show that the motion is the same aa i£ £ were fixed, and
a foi-ce equal to — mil wet« applied to the particle in a direction
parallel to EX ; a result that might have been arrived at at once
by superimposing on the whole system, an acceleration equal and
opposite to that of JE, to effect which on P the force ~mU is
required.
Example (F). Any case of varying relations such that in
318 (27) the coefficients {ij/, <p), (i^, ^) ... are independent of t.
Let ^ denote the quadratic part, L tJie linear part, and K [as
in g 318 (27)] the oonstant part of 7 in respect to the velocity
components, so that
L = (^)<f,*{4>)i> + ... I (a),
where (^, ^), (^, ^), (<^, 0) ... denote functions of the co-ordi-
nates without t, and (^), (^), ..., (tji, ^, $, ...) functions of the
co-ordinates and, may be also, of ( ; and
T = % + L-^-K. {b).
We have --=0.
Hence the contribution from K to the first member of the i^
equatitm of motion is simply —
dL
dl dip dtfi d-ft \ dt J
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319.] DYNAMICAL LAWS AND PRINCIPLES. 319
Fartlier we hare Bxample of
dZ _d(ip) . d (^) , relktioa
'^~'^''**~d<p **■■■• SiS^o
Hence the whole contribution from L to the i/^«quation of
motion is
/<7(^) d(<l,)\./d{i,) d(ff)\. ^/d{^)\
Lastly, the contribation from V is the same as the whole from
r in § 318 (29'") ; so that we have
and the completed ^hequation of motion U
d<m <m /djf) d(,^)\,rd(.t) dm\g,
dtd^ df, \ di, dp J \ de dp /
(dm\_dl
.(dm\_a_
\dl } dp
(.).
It is important to remark tiiat ^e coefficient of ^ in this ^
equation is equal but of opposite sign to the coefficient of ^ in
the ^.equation. [Compare Example G (19) below.]
Proceeding as in § 318 (29'') (29'), we have in respect to ® Eiimtion
precisely the same formulas as there in respect to T. The terms
involving first powers of 4^e velocities simply, balance in the
sum : and wo find finally
«,(-)_i.,..^^^,,,. C/),
where (2(^,^...) denotes difierentiatiou on the supposition of
^, ^, ... variable ; and { constant, where it appears explicitly.
Now with this notation we have
. dK (dK\ d„,,,...,K
Hence from (/) we have
T dm + Z + K) ,, ., rfrti. xL ,,. V ,,, ■■
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Inontion
9 PBELIHINABT. [319.
Take, for Olustratlon, Exuuples (E) and (F) from above; in
which we hare
[Example (E)]
and [Example (F)]
1[ = im«'(8m'fl^* + ^),
aee,
= h'»
Write out explicitly in each case equations {/) and (g), and
Terify them by direct work from the equations of motion f(»ining
the conclnsions of the examples as treated above (remembering
that d and tl are to be regarded as i^ven explidt functions of t).
ExampU (G), — Preliminary .to Oj/rotlatie eotmexiona and to
Fluid Motion. Let there be one or more co-ordinates x> x'> ^**-
which do not appear in the coefficients of velocitieB in the
dT dT
expression for T; that is to say let -3- =0, j-; =0i etc. The
equations corresponding to these co-ordinatea become
ddT ddT
did^=^ s^=^'«*«- (^>-
Farther let na suppose that the force-components X, X', etc.
corresponding to the co-ordinates Xi x'> ^^ ^^ ^^'^ zero: we
shall have
dT
dT
C, etc...
■■(2);
dx ' dp
or, expanded according to previous notation [318 (S9)],
('(■,)()\>+(*.x)*+- + 0<.x)x+(x.x')x + - =<?"
W,x')'f+(*.x')* + -+(>:'.x)x + C<',xlx'+-. -C" ...(3).
Hence, if we put
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319.]
DYNAMICAL LAWS AND PKINCIPLKEL
{x'.x)x + (x'a')x'+" =C'-
Kesolving thme for %)(, ... we find
(x'.x')' ^x'.x"). ■■
(x".x').{x".x")i"
{C-P)A
(x"'X')>(x'W)'-
<x\x')>(x"''X%-
(x>x)> (x-x')- (x.x")' ■■
<x'.x). (x'-x'). (x'-x"). ■■■
(x".x). (V'tx"). (x".x"). ■
and Bjminetrical exprmsions for )(, x", ..., or,
them short,
x-(c;c)(C-f)*{c;c-}{0'-r)
{(T-n*
as we may write
m.
where (C,C), {C,C}, (0',C), ■•• denote functions of the retained
co-ordinates ^, ^ $, .... It la to be remembered tliat, because
(X. X") = (x'. X)' (X. X") = (x". X)- "« «ee from (6) that
(C, C) = (C, C), {C. C") = (C", C), {C; C") ■= (C; C), and BO on. ..(8).
Ths following formulu for j(,^', .,., condensed in respect to
C, C, C" by aid of the notation (14) below, and expanded in
respect to ^, ^ ..., by (4), will also be useful.
^dC'
{iff + A'<i, + ...)
■-{J/''^ + i\''^ +
jf=(0,(7).(^,x)+(C',(r).(^,x')*-
jr =((7, (7). (^ x) + «^. <?')■(*. x") + -
*'-((7',C7).(^.x) + ((7',C).(^,x')-^...
,.(I0).
The elimination of x, x, ... fi-om T by these eipressions for
VOL. I. 21
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382 PBELIHINABT. [319. |
hnonttoa them is &cilitat«d b; remarking that, as it ia a quadratic fuiw-
tion of ^, ^, ..-x, Xt '■■) "^^ have
Hence by (3),
BO that ve have now only first powers of Xt x'l -■■ ^ eliminate.
Gleaning out Xi x't - ■ - ^^'''' ^^ ^'^ gfo^p of terms, and denoting
by r, the part of T not oontuning Xt X' —t '*'* ^*^
+ I.
or, acGOiding to the notation of (4),
THiminatiiig now ;^ ;^', ... by (7) we find
r. r. + } {(C, c) (C" - i^ + 2 (c, c) (OT - PiO + (ff', c) (C" - n
+ ■■■} (")■
It is remarkable that only second powers, and products, not
Jim powtrt, of the velodty-components i^, ^, ... appear in thii
expression. We may write it thus : —
r=« + jr (12),
where V denotes a quadratic function of ^, ^ ... , as follows: —
€== T,- i {{C, C) f + 3 {C,(r)PP' + {C',C)F* + ...}.... (13),
and f a quantity independent of ^, ^ ..., as follows: —
ir=} {(CO c* + s(c, c')cc'+(c;c-) c* ...} (U).
Next, to eliminate Xi x'> ■■■ ^'^ ^^ Lagrange's equations, we
have, in virtue of (12) and of the constitattcms of T, V, and E,
d^^^^dX^diaj,^,^ (15,,
d^ dx dtp dx diji difi
where -^ > -~ , etc are to be found by (7) or (9), and therefore
d^ dtji
are simply the coefficients <tf i^ in (9) ; bo that we hare
%-"■%-'■ "''-
where M,M' are functions of ^,0, ... explicitly expreased by
(10). Using (16) in (13) we find
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319.]
DTNAKICAL LAWS AHD PBINCIFLEa
?- + CJf + C-Jf + el« (17). gg.
A^n remarldiig that %-t-K contains ^, both aa it appeared
originallj in T, and as farther introduced in the expreasiona (7)
for x,)(, ..., we see that
And hj (9) we haTe
dT „dx „,dx
-"■ d^ ■'■'■
'J*d>lidC''
which, osed in the preceding, gives
^ i,w p-\ ■'^ r,(idM ,dN \ „,f.dir
.dir
,..)-«
dT M dK ,„l,dU jdS \ ,,.,
where 2 denotes summation with rc^^ord to the conatante C,
C, etc.
TTBing this and (1 7) in the LagiBnge'a ^-equation, we find finally
for the ^-equation of motion in terms of the non-ignored co-
ordinates alone, and conclude the symmetrical equations for ^
etc., as follows,
d /dB\ dt, _,^(/rf-V dAN - fdM dO'
dt \d^} " d^j,*^^ XKdi. ~ #; '^*\~d9
d fdZ\ dT, ^f.C(dN dM\ , ^ (dN
rf (dX\ dt _ - UdO dM\ , /dO dlf\ J
di\w~'iis-^^^\Kd;f~-der\d^--d^r
[Compare Example V (e) above. It is important to remark
that in each equation of motion tiie first power of the related
Telocity-component dis^q)ears ; and the coefficient of each oE the
other velocity-componentB in this equation is equal but of opposite
sign to the coefficient of the velocity-component corresponding to
this equation, in tlie equation corresponding to that other velocity-
component.]
21—2
4>
jiGoogk'
4 PBELTMINAET. [319.
The eqn&Uoii cA energy, fonnd as above [g 31S (SS**) utd
^(^:^=*^ + *^ + etc (20).
The luterpretatioii, conidderijig (12), la obvloas. Hie ooatntt
with Example V (ff)ia most instructive.
SvJhExamph (G,). — Take, from above, Example C, cue (a):
and put ^ = ^ + (1; also, for brevity, mf+na'=B,n{V+i^=A,
and totA = e. We have*
r = J { Ji^' + 2<^ {(J + ^) 008 « + .8 (^ + ^'1 ;
and from this find *
^=0, -,y=^i^ + c(2^ + ^)co8fl + 5(^ + ^;
o^ (^
— =-c.f(^ + ^)am«, ^-(^oo8fl + .B(^+^.
Here the coKirdinate 6 alone, and not tlie co-ordinate tfi, appMn
in the coefficients. Suppose now 4 = 0 [which is tihe case con-
dT
sidered at the end of C (a) above]. We have — , = C, and
di^
deduce
'^~ .i + B + 2ccoatf '
r=i(^^+(S^ = i{^ + e[(«cosfl + S).f + 2f^}
~*t J+^ + '2ccoetf + *^/ 3 j+j + 2ccoefl
Htnce
jr=}
t+2<:coBd
C
• Bemu^ tkkt, aoooiding to the altenti<ni bom V< i'l i>> 4> fo •}/, ^, t. f,
Aa indopendent variablM,
d^^Uf) * [di,)' da - \di>)'
d^ \df/ \d^/ d6 \d<^i
where { ) indicates the oiieiuAl notation ol C(a).
..Google
319.] DTKjUnCAL U.WS AND PBINCIPLES. 325
and thf one equation of the nLotion becomes ^g°«*ton at
dt\A+B + 2ccMe J~^ de\A*B + 2ccmej" dB'
which is to be fully integrated firat by multiplying by d$ and
integrating onoe ; and then solving for dl and integrating again
with respect to 6. The first int^ral, being simply the equation
of energy int^^iated, is [Example G (20)]
^ = jQd9-K;
and the £nal integral is
.{,. I ^-B-cos'g
J V 2(^ + ^ + 2ccostf)(/®ttf-i')'
In tiie particular case in which the motion oommeDces from iRnoratioa
rest, or is such that it can be brought to rest by proper applica- a '
tions of foi-oe-components, *, *, etc without any of the fopoe-
components X, X', etc., we have (7=0, C = 0, etc; and the
elimiuatioa of j^ ;^', etc. by (3) renders T a homogeneous quad-
ratic function (^ tfi, ^, etc. without C, C, etc ; and the equations
of motion become
±dT_dT_
dtdili d\ji~
ddT dT_
dld^~d^~^\- (21).
ddTdT^
dt d4 dO "
etc et«,
We conclude that on the suppositions made, the elimination of
the Teloctty-^wmponents corresponding to the non-appearing co-
ordinataa gives an expresBioa for the kinetic energy in terms
of the remaining Telocity-components and corresponding co-
ordinat-es which may be used in the geoeralised equations just
as if these were the sole co-ordinatea The reduced number of
equations of motion thus found suffices for the determinatioa
of the co-oi'dinates which they involve without the necessity
for knowing or finding the other co-ordinates. If the &rther
quesdon be put, — to determine the ignored co-ordinates. It is to
be answered by a simple integration of equations (7) with
£7 = 0, (7'=0, etc
One obvioos case of application for this example is a system in
which any number of fly wheels, that is to say, bodies which aro
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26 PBELIHmART'. [319.
kinetically sTiiimetiical round an axis (§ 285), are pirotod frio-
tionleaaly on saty moveable part of the BTStem. In this cue
with the particular suppoeitiou C = 0, C = 0, et«., the result is
simply that the motion is the same as if each £ty vh«el were
deprived of moment of ina^ round its bearing axis, that is to
say reduced to a line of matter fixed in the position of this axis
and having nnchanged moment of inertia round any axis per-
pendicular to it But if C, C, etc be not eaofa eero -we have a
case embradng a very iotereetlng class of dTnamical problemg
in which the motion of a system having what we may call
gyroatHtic links or connexions is the subject. Ejample (D)
above is an example, in which there is just one fly wheel and one
moveable body on which it is pivoted. The ignored cosHxliiuite
is ijf ; and auppoBing now t to be sero, we have
^-0(l-coafl) = (7 (o).
If we suppose C - 0 all the terms having 9' for a &ctor vanish
and the motion is the same as if the fly wheel were deprived of
inertia round its bearing axis, and we had umply the moti(m (^
the " ideal rigid body PQ" to consider. But when C does not
vanish we eliminate <p from the equations by means of (a). It
is important te remark that in every case of Example (O) in
which 0=0,0' -0, etc. the motion at each instent posBeeses the
property (§ 312 above) of having lees kinetic energy than any
other motion for which the velocity-componente of the non-ignored
co-ordinates have the same values.
> Take for another example the final form of Example C above^
putting £ for 0, and A for n^ + A. We have
T = ^{{S + Fco^'d),l^-i-B(4,<xm0+4,)' + Afi\ ...(22).
Here neither if/ nor ^ appears in the coefficients. Let us suppose
4 = 0, and eliminate ^ to let us ignore ^ We have
^=.B(^coafl + ^) = <7.
Hence ^ = -=-^coBfl (23),
« = J ((£ + /■ cos" tf)^ + J^} (24),
and K=i^ (25).
The place of ^ in (9) above is now taken by ^, and comparing
with (23) we find
M^oobS, Jir=0, 0=0.
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319.] DYNAsncAt x^yrs and pbinciples. 327
Hence, and aa K k constaiit, the equations of motion (19) ^f^^*"^
become
t?BmCtf =
dtdiff dip
d di^ dli „ . .J ^
and -, +U8in0Ji~9
dt d^ d$
ui<], using (24) and expanding
..(26);
■(27).
A most important caae for the "ignoration of co-ordinates" is
presented by a Urge claaa of problems regarding the motion of
frictionlesB incompressible fluid in which wf> can ignore the
infinite number of co-ordinates of individual portions of the fluid
and take into account only the co-ordinates which suffice to
specify the whole boandary of the fluid, including tiie bounding
eurfiices of any rigid or flexible solids immersed in the fluid.
The analytical working out of Elxample (G) shows in fact that when
the motion is such as could be produced from rest by merely
moving the boundary of the fluid without applying force to its
individual particles otherwise than by the transmitted fluid
pressure we have exactly the case of C = 0, C' = 0, etc; and
I^grange's generalized equations with the kinetic energy expressed
in terms of velocity-oomponeiits completely sped^ing the motion
of the boundary are available. Thus,
320. Frobkma in fluid motion of remarkable interest and Ki
importance, not hitherto attacked, are very readily solved by ^
the aid of Lagrange's generalized equations of motion. For
brevity we shall designate a mass which is absolutely incom-
pressible, and absolutely devoid of resistance to change of shape,
by the simple appellation of a liquid. We need scarcely say
that matter perfectly satisfying this definition does not exist
in nature : but we shall see (under properties of matter) how
nearly it is approached by water and other common real
liquids. And we shall find that much practical and interesting
information regarding their true motions is obtained by deduc-
tions from the principles of abstract dynamics applied to the
ideal perfect liquid of our definition. It follows from Example
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3S8 PEKLimNABT. [320.
Kiiutipaof (Q) above (and aevoral other proofs, some of them more
itqoid. synthetical in character, will be given in our Second Volume,)
that the motion of a homogeneous liquid, whether of infinite
eittent, or contained in a finite closed vessel of any form, with
any rigid or flexible bodies moving through it, if it has ever
been at rest, is the same at each instant as that determinate
motion (fulfilling, § 312, the condition of having the least
possible kinetic enei^) which would be impulsively produced
from rest by giving instantaneously to eveiy part irf" the
bouoding surface, and of the surface of each of the solids
within it, its actual velocity at that instant So that, for
example, however long it may have been moving, if all these
surfaces were suddenly or gradually brought to rest, the whole
fluid mass would come to rest at the same time. Hence, if
none of the surfaces is flexible, but we have one or more rigid
bodies moving in any way through the liquid, under the in-
fluence of any forces, the kinetic energy of the whole motion
at any instant will depend solely on the finite number of co-
ordinates and component velocities, specifying the position and
motion of those bodies, whatever may be the positioiis reached
by particles of the fluid (expressible only by an infinite number
of co-ordinates). And an expression for the whole kinetic
energy in terms of such elements, finite in number, is predsely
what is wanted, as we have seen, as the foundation of Lagrange's
equations in any particular case.
It will clearly, in the hydrodynomical, tia in all otiier cases,
be a homogeneous quadratic function of the components of velo-
city, if referred to an invariable oo-ordinate ^atem ; and the
coefficients of the several terms will in general be functions of
the co-ordinates, the determination of which follows immediately
from the solution of the minimum problem of Example (3) §317,
in each particular case.
Example (1). — A ball tei in jnotion through a mam of incom-
prettthle Jiuid extending infinitdy in all direetiona on one eide of
an infinite plane, and onginaily ul reel. Let X, y, zhe the co-
ordinates of the centre of the ball at time I, with reference to
i-ncttmgular axes through a fixed point 0 of the bounding plane,
with OS perpendicular to this plane. IE at any instant either
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320.] DTNAlnCAL lAffS AND PRINCIPLIS. S29
component j^ or S of the velocity be reversed, the kinetic energy Kinetlna ot
will clearly be unclianged, »nd hence no terms j^ 2i£, or £y can liquid.
kppear in the expreaaion for the kinetic energy ; ivhich, on this
acGOont, amd bec&use of Uie aymmetiy of cirouroatoncea with
reference to j/ and z, is
Also, ve see that P and Q are functiona of x dmply, tdnce the
circnmstanceB are similar for all valuee of y and x. Hence, by
differentiation,
■md the equations of motion are
Principles anffident for a practical Bolution of the problem of
determining P and Q will be given later. In the meantime, it
is obvious that each decreases as x increases. Hence the equa-
tiona of motion show that
321. A ball projected through a liquid perpendicularly BtiMoik
from an infinite plane boundary, and influenced by no other on tb? mo?
forces than those of fluid pressure, experienceB a gradual ac- ^"^ •
celeration, quickly approximating to a limiting velocity which
it sensibly reaches when its distance from the plane is many
times its diameter. But if projected parallel to the plane, it
experiences, as the resultant of fluid pressure, a resultant attrac-
tion towards the plane. The former of these reenlte is easily
proved by first considering projection towards the plane (in
which case the motion of the ball will obviously be retarded),
and by taking into account the general principle of reversibility
(§ 272) which has perfect application in the ideal case of a per-
fect liquid. The second result is less easily foreseen without
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330 PRELIHIITAHT. [321.
the aid of LagraTige's analysis ; but it ia an obrioas consequence
of the Hamiltonian form of his equations, as stated in words
^tS'Soo ^ § ^^^ above. In the precisely equivalent case, of a
two^imi ''l'''*^ extending infinitely in all directions, and given at rest ;
^MbI^" ^^^ *"'^ equal balls projected through it with equal velocities
d^^l perpendicular to the line joining their centres — the result that
the two balls will seem to attract one another is most re-
markable, and very suggestive.
Hjilro- Example (3). — A aolid aymTnetrical round an (tei*, moving
euiDplM through a liquid go ag to icetp itt axis altoaj/g in one plane.
Let (I) be the angular velocity of the body st any instant about
any axia perpendicular to the fixed plane, and let u and q be the
component velocities along and perpendicular to the axia of
figure, of any chosen point, C, of the body in this line. By the
general principle stated in § 330 (since changing the sign of
u cannot alter the kinetic energy), 'we have
r=i{^«' + ^y'+/«.' + 2^a,y) (a),
where A , B, fJ, and E are constants depending on the figiore of
the body, its iqeisb, and the density of the liquid. Kow let v
denote the velocity, perpendicular to tiie axis, of a point which
"'^1^''' ^^ shall call the centre of naoiion, being a point in the axis and
d^ued. K E
at a distance -„- from C, so that (§ 87) y = i»--giA Hen,
denoting /i'- -=^ by /i, we have T= \{Av.* + Btf + luo') (a").
Let X and y be the co-ordinates of the centre of reaction relatively
to any fixed i-ectaogular axes in the plane of motion of the axis
of figure, and let tf be the angle between this line and OX, at
any instant, so that
a = 6, u = ±coB6 + ^siii.$, r=-*Mnfl + ^oo8tf (6).
Substituting in T, difierentiating, and retiuning the notation
It, V where convenient for brevity, we have
■ »
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321.]
DTHAWCAL LAWS AND PRINCIPLSS.
Hence the eqiuttions of motion are
d<AucoB$-SvBva$) _
rf* '^'
d(Awaji6 + SvtxmO)
dt
= I'I
W,
There X, F are the ccnnpoiient forces in lines through C parallel
to OX and OY, and L the ecniple, applied to the body.
Denoting hy X,i,ri the impulsiTe couple, and the oomponents
of impuJsiTe force through C, required to produce the motion at
any instant, ire have of course [§ 313 (c)],
■■{-).
dT
dT
dT
d^ "
and thenfore \ij (e), and ((),
«-2(fcoBtf + i7mn^, «-^(-f8intf + i;coBfl), (i=-,
^ /cos'ff sin'tfN, /I 1\ . „ „ 1
■■■(A
^{(-f'4.^sm2ff + 2f,coii2tf} = Af=X,j7=r, (A).
and the equations of motion become
d'e A-B,,
'^df-'iAS^'
The simple case of X- 0, T= 0, Z = 0, is particularly interesting.
In it £ and n are each constuit; and we may therefore choose the
axes OX, 07, bo that i; shall vamsh. Thus we have, in (g), two
first int^rals of the equations of motion; and they become
*"Kt"*~s~;' '--US'
and the first of equations {h) becomes
Jia i »
^^sinSflsO.
j-fdn
..(*):
A-B
(0-
f^ghW. It becomes
In this let, for a moment, 3d -^ and
which is the equation of motion of a common pendnlonr, of
mass W, moment of inertia /i round its fixed axis, and lengldi
..Google
2 PBELOUNAAT. [3S1.
A from axis to centre of gravity; if ^ be the angle from
tli« positiun of equilibrium to the portion at time L As ve
shall see, under kinetics, the final int^ral of this equatioa
expreasea ^ in terms of t hj means of am elUptic function.
By using the valoe thus found for tf or ^0, in (i), we have
equations giving x and y in tenuB of t hj common int^ration ;
and thus the full solution of our pres»it problem is reduced to
quadraturee. The detailed worbdng out to exhibit both tlie actu&l
curve described by the centre of reaction, and the posititxi of
the axis of Ihe body at any instant, is highly intereeting. It is
very easily done approxinuUely for the case of very small angular
vibrations; that ia to say, when either j1 — £ is positive^ and
^ always very small, or A — B n^ative, and ^ very nearly
equal to ^r. But without attending at present to the final
integrals, rigorous or approximate, we see from (jfc) and (f) that
322. If a solid of revolution in an infinite liquid, be set in
motion round any axis perpendicular to its axis of figure, or
simply projected in any direction without rotation, it will move
with its axis always in one plane, and ereiy point of it moving
only parallel to this plane; and the strange evolutions which
it will, in general, perform, are perfectly defined by comparison
with the common pendulum thua First, for brevity, we shall
Qaairaittat call by the name of quadrantal pendvlam (which will be further
dsOiMd. exemplified in varioua cases described later, under electricity
and magnetism ; for instance, an elongated mass of soft iron
pivoted on a vertical axis, in a "uniform field of m^netic
force"), a body moving about an axis, according to the same
law with reference to a quadrant on each side of its position of
equilibrium, as the common pendulum with reference to a half
circle on each side.
Let now the body in question be set in motion by an im-
pulse, f, in any line through the centre of reaction, and an
impulsive couple \ iu the plane of that line and the axis. Thb
will (as will be proved later in the theory of statical couples)
have the same effect as a simple impulse ^ (applied to a point,
if not of the real body, connected with it by an imagiBary in-
finitely light framework) in a certain fixed line, which we shall
call the line of resultant impulse, or of resultant momentum.
..Google
S22.] DTHJUaCAL UW3 JUIB PRmCIFLES. S3S
beii^ parallel to tbe former line, and at a diatance from it equal to J'^^^'J'
5 . The whole momentum of the motion generated is of course ^^^ '
(§ 295) equal to f. The body will move ever a^rwards
according to tbd following conditions : — (1.) The angular velo-
city follows the law of the quadrantal pendulum. (2.) The
distance of the centre of reaction from the line of resultant
impulse varies ramply as the .angular velocity. (3.) The
velocity of the centre of reaction parallel to the line of
impulse is found by dividing the excess of the whole con-
stant energy of the motion above the part of it due to the
angular velocity round the centre of reaction, by half the
momentum. (4.) If A, B, and ^ denote constants, depending
on the mass of the solid and its distribution, the density of the
liquid, and the form and dimensions of the solid, such that
^, t, — are the linear velocities, and the angular velocity,
respectively produced by an impulse { along the axis, an' im-
pulse f in a line through the centre of reaction perpendicular
to the axis, and an impulsive couple \ in a plane through the
axis; the length of the simple gravitation pendulum, whose
motion would keep time with the periodic motion in question,
is &?Y^ nv. Bud, when the angular motion is vibratory, the
vibrations will, according aa A > B, ot ^ < .S, be of the
axis, or of a line perpendicular to the axis, vibrating on
each side of the line of impulsa The angular motion will
in fact be vibratory if the distance of the line of resultant
impulse from the centre of reaction is anything less than
= ^~- — — where a denotes the inclination of the im-
pulse to the initial position of the axis. In this case the path
of the centre of reaction will be a sinuous curve symmetrical on
the two sides of the line of impulse ; every time it cuts this line,
the angular motion will reverse, and the maximnm inclination
will be attained ; and every time the centre of reaction is at its
greatest distance on either side, the angular velocity will be at
its greatest, positive or negative, value, and the linear velocity of
/
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^/^
334 FBELDflNABT. [33S.
"Sw'rf *'''* c^"**™ of reaction will be at its least. If, on the other hand,
reroiHtbiD the line of the resultant impulse be at a greater distance than
/ - — —^.-jj — ~ from the centre of reaction, the angular motioD
will be always in one direction, but will increase and diminish
periodically, and the centre of reaction will describe a sinuons
curve on one side of that line; being at its greatest and least
deviations when the angular velocity is greatest and least. At
the same points the curvature of the path will be greatest and
least respectively, and the linear velocity of the describing
point will be least and greatest.
323. At any instant the component linear velocities along
and perpendicuhur to the axis of the solid will be — -^ — and
— -fi- respectively, if ^ be its inclination to Uie line of re-
sultant impulse ; and the angular velocity will be — if y be the
distance of the centre of reaction from that line. The vhole
kinetic energy of the motion will be
f cos'g rsin'tf fV
2A "^ 2S "^ 2^ '
and the last term is what we have referred to above as the
part due to rotation round the centre of reaction (defined in
§ 321). To stop the whole motion at any instant, a simple
impulse equal and opposite to ^ in the fixed "line of resultant
impulse" will suffice (or an equal and parallel impulse in any
line through the body, with the proper impulsive couple, accord-
ing to the principle already referred to).
331. From Lagrange's equations applied as above to the case
of a solid of revolution moving throu^ a liquid, the couple
which must be kept applied to it to prevent it from turning is
immediately found to be
jiGoogk'
324] DTNAHICAL LAWS AND FBINCIFLES. 335
if u and v be the coiDpoaent velocities along aad perpendicular *'|^^
to the axis, or [§ 321 (/)] S^aT
^M-g)sin 20 ^
^ 2AB
if, ae before, ^ be the generating iiupulae, and $ the angle be-
tween its line and the axis. The direction of this couple most
be such as to prevent 6 from dirainishiDg or from increasing,
according as A or £ is the greater. The former Trill clearly
be the case of a flat disc, or oblate spheroid ; the latter that of
an elongated, or oval-shaped body. The actual values of A
and B we shall learn how to calculate (hydrodynamics) for
several cases, including a body bounded by two spherical sur-
faces cutting one another at any angle a submultiple of two
right angles ; two complete spheres rigidly connected ; and an
oblate or a prolate spheroid,
326. The tendency of a body to turn its flat side, or its o'^^™'
length (as the case may be), across the direction of its motion
tbrough a liquid, to which the accelerations and retardations of
rotatory motion described in § 322 are due, and of which we
have now obtained the statical measure, is a remarkable iUus-
tration of the statement of § 319 ; and ia closely connected
with the dynamical explanation of many curious observations
well known in practical mechanics, among which may be men-
tioned : —
(1) That the course of a symmetrical square-ri^ed ship
sailing in the direction of the wind with rudder amidships is
unstable, and can only be kept by manipulating the rudder to
check infinitesimal deviations; — and that a child's toy-boat,
whether "square-rigged" or "fore-and-aft ri^ed*," cannot be
* "For«-Mid-aft" rig is aujiigin vrUeli (sa in "oatters" and " achoonen ")
tbfl ehief aaili eome into the plane of mart or maata and keel, b; the Mtioti of
the wind npon the sails when the TBsael'a head is to wind. Thia poatloii
of th« Mile ia nnstabls when the wind ie right astern. Aooordlngtr, In
"wearing" a fore-and-aft rigged Teaael (Out ii to aa^ turning her ronnd
aton to wind, from aailing with the wind on one aide to sailing with tJie
wind on the other aide) the mainsail most be hanled in at eloaely as m^ be ,
towards the middle poedtion before the wind is allowed to get on the other Ride
of the aail from that on which it bad been pressing, ao that when the wind
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336 PBELIHIKABT. [325.
got to sail permaoently before the wind b^ any pennaneut ad-
'' justment of rodder and sails, and that (without a wind vane, or
a weighted tiller, acting on the nidder to do the part of
steersman) it always, after running a few yards before the wind,
turns round till nearly in a direction perpendicular to the
wind (either "giWng" first, or "luffing" withont gibing if it
is a cutter or schooner) : —
(2) That the towing rope of a canal boat, when the rudder
is left straight, takes a position in a vertical plane cutting the
axis before its middle point : —
(3) That a boat sculled rapidly across the direction of the
wind, always (unless it is extraordinarily unaymmetrical in
its draught of water, and in the amounts of surface exposed
to the wind, towards its two ends) requires the weather oar
to he worked hardest to prevent it from running up on the
wind, and that for the same reason a Bailing vessel generally
"carries a weather helm*" or"gripes;" and that still more does
so a steamer with sail even if only in the forward half of her
length — griping so badly with any after canvass'f' that it is often
imposBible to steer : —
(4) That in a heavy gale it is exceedingly difficult, and
often found impossible, to get a ship out of " the trough of the
sea," and that it cannot he done at all without rapid motion
ahead, whether by steam or sails . —
(5) That in a smooth sea with moderate wind blowing
parallel to the shore, a sailing vessel heading towards the shore
with not enough of sail set can only be saved from creeping
ashore by setting more sail, and sailing rapidly towards the
shore, or the danger that is to be avoided, so as to allow her to
be steered away from it. The risk of going ashore in fulfilment
doea get on the othei side, uid vhen therelorfl the Mil daabes Mvoae throngfa
tbe mid-ship positioii to the othei aide, ourying munTe boom mud gfS with it,
tfa« nug« ol thla mdden motion, vhioh is oftUed "^Ung," shall be ■■ small
* The ireBthar side of any objeot ii the sids of it tontda the wind. A ship
Ifi said to "oaiTj a weather helm" when it is neoeasaij to hold the "helm" or
"tiller" permanently on the weather ride of He middle position (by whiah lh«
rodder 1b held towarda the lee aide) to keep the ship on her ooorae.
f Benoe miasn msstii are altogether coademoed in modem war-tliipB Itj
many oompetent nwtioal anthoritiet.
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325.] DTNAHICAI LAWS ASD PRINCIPLES. 337
of Lagrange's equations is a frequent incident of "getting
under way " while lifting anchor, or even after slipping from
moorings : —
(6) That an elongated ri6e-bullet requires rapid rotation >no tuti
about its axis to keep its point foremost.
(7) The curious moticHis of a fiat disc, c^ter-ehell, or the
like, when dropped obliquely into water, resemble, no doubt, to
some extent those described in § 322, But it must be re-
membered that the real circumstances differ greatly, because
of fluid friction, from those of the abstract problem, of which
we take leave for the present.
326. Maupertuis' celebrated principle of Least Action has i^*t
been, even up to the present time, regarded rather ae a curious
and somewhat perplexing property of motion, than as a useful
guide io kinetic investigations. We are strongly impressed
with the coQviction that a much more profound importance
will be attached to it, not only in abstract dynamics, but in the
theory of the several branches of physical science now beginning
to receive dynamic explanations. As an extension of it. Sir
W. R. Hamilton* has evolved his method of Varying Action,
which undoubtedly~mu8t become a most v^uable aid in future
generalizations.
What is meant by " Action " in these expressions is, unfor- Action,
tunately, something very different from the Actio Agentia de-
fined by Newtou'f, and, it must be admitted, is a much less
judiciously chosen word. Taking it, however, as we find it, J'^»,"'
row universally used by writers on dynamics, we define the "letr-
Action of a Moving System as proportional to the average
kinetic enei^, which the system has popsessed during the time
from any convenient epoch of reckoning, multiplied by the time.
According to the unit generally adopted, the action of a system
which has not varied in ite kinetic energy, is twice the amount
of the energy multipKed by the time from the epoch. Or if
the energy has been sometimes greater and sometimes less,
* Phil. Traiu. 1694—1836.
+ Wliich, haireT«i (§3&3), we hare tnuulaUd "actiTil;" toaToid oonfo^on.
VOL. L 22
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PEELIUmART.
'nowtTw the action at time t is the double of what we may call the
cDwiff tinu-inteffral of the eaergy, that is to say, it is what is de-
noted in the intend calculus by
^/:
Tdr,
-l>
where T denotes the kinetic eneigy at any time t, between
the epoch and t
Let ro be the mass, and v the velocity at time r, of any one <rf
die material points of which die Bystem is oompoaed. We have
7'=Simt^ (1),
and therefore, if A denote the action at time (,
f'Smp'rfr (2).
This may be put otherwise by taking da to denote the apace de-
scribed by a particle in time dr, so that vdr = da, and therefore
A=S^^vd» (3),
or, if (B, y, « be the rectangular co-ordinates of m at uny time,
A = l%m,{tdx + ^dy-*xd») (4).
Henoe we might, as many writers in fact have virtually done,
define action thus : —
The action of a system is equal to the sum of the average
vnomenlumt Jvr the spaces described by the particles from any
era each multiplied by the length of its path.
iMkt 327. The principle of Least Action is this: — Of all the
different sets of paths along which a conservative system may
be guided to move from one configuration to another, with the
sum of its potential and kinetic energies equal to a given cob-
Btant, that one for which the action is the least is such that
the system will require only to be started with the proper
oennai Velocities, to move along it uoguided. Consider the Problem ; —
aim. Given the whole initial kinetic energy ; find the initial velocities
wish ■im through one given configuration, which shall send the system
firnms unguidcd to another specified configuration. This problem is
Ta^ioo essentially determinate, but generally has multiple soluiioDS
nubcd. (§ 363 below) ; (or only imaginary solutions.)
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327.] DTKAMICAL LAWS AHD PRINCIPLES. 339
If there are any real solutions, there ia one of them for which u
the action is less than for any other real solution, and lean than *°
for any constrainedly guided motion with proper sum of po-
tential and kinetic energies. Compare §§ 346 — 36G helow.
Let X, jf, zhe the co-ordinates of a particle, m, of the aystem,
at time r, and V the potential energy of the system in its parti-
cular configuraboa at thia instant ; and let it be required to find
the way to pass from one given configuration to another with
velocities at each instant satisfying the condition
Sim(ie*+y' + i')+ r= ^, a constant (6),
BO that A, or
/5n* {±d!e + jWy + xdz)
may be the least possible.
By the method of variations we miist have &i » 0, where
iA-'f^m{±dSa + ydSy + id&e + &idx + S^l/ + iidg) (6). -
Taking in this dx = xdr, dy = ^dT, dz — idr, and remariung that
Sm{i&i+ySy+ jSi) = 82'. (7),
we have
i^m{Zidx-¥^dy*Udz)=i'iTdT (8).
Also by integration by parte,
/S»»(Af&«:+...)=JSm{A&+...)}-[5m(i&i3: + ...)]-/Sm(i&E+...)dr,
where [...] and {...} denote the values of the quantities enclosed,
at the beginning and end of the motion considered, and where,
further, it must be remembered that dA » iodr, etc. Hence,
from above,
8.i = (Sm (iSa; + ^Bff + «Ss)} - [Sot (*Ba; +^Sy + s&s)]
+ ldT\^T-^m{a^ + y^ + a&z)]..
..(9).
This, it may be obeerred, is a perfectly general kinematical expres-
sion, unrestricted by any terminal or kinetic conditions. Now
in the present problem we suppose the initial and final positions
to be invariable. Hence the terminal variations, &e, etc, most
all vanish, and therefore the integrated expressions {...}, [...] dis-
appear. Also, in the present problem 8r= — 8r, by the equation
of energy (6). Hence, to make hA = 0, since the intermediate
TariationB, hx, etc., are quite arbitrary, subject only to the con-
22—2
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340 PBELIHINAEr. [327.
^'■■t ditiona of the Bjstem, we must hftye
Sm(:B8a! + jf8y + »8e) + 8r=0 (10),
which [(4), § 293 above] is the general variational equation <£
motion of a oanservative B^tem. This proves the propoaition.
nut!^iaa ^^ ^ interesting and instractive as an illustration of l^e prinr
>pplM dpie of least action, to derive directly from It, without anjr use
L«cr*n(ie'i of Cartesian co-ordinates, Lagrange's equations in genentlized
co-ordinates, of the motion of a conservative syBt«m [§ 318 (24)]-
Wb have
A =./2rd(,
where T denotes the formula of g 3 1 3 (2). If now we put
so that di' -^ (ifi, ^) (f / + 2 (^, ^) dtjfd<i. + etc,
we have A = jjzdt.
^f
where Sf^.^.etc.) denotes variaUon dependent on the explicit ap-
pearance of ^, ^, etc in the coefficients of the quadratic func-
tion T. The second chief term in the formula for SA is clearly
equal to \—j <^, and this, integrated by parte, becomes
where [ ] denotes the difference of tiie values of the bracketed
expression, at the beginning and end of the time JcU. Thus we
have finally
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327.] DTNAMICAI, LAWS AND PMNCIPLE3. 341
80 &r we have » purely kinematical fonoula. Now introduce PriDdpieoi
the dynamical oondilion [§ 293 (7)] ^^t*^
T=G-V (10)". JS^SJS
fVom it we find muUdiu
/dV dV \ amottoo,
^'■-(%^*%^*'^) (lor-
Agtun, weliave
8c,,«,«2'-g'v+^!* + el«. (ion
Hence (10)' beoomes
To make thU a minitnnm we have
d dT dT dT ^ ^ ,,.,„
-di^*T^*df-''-'*^ (">>•
which are the required eqnationa [g 318 (24)].
From the propoaitioTi that hA=Q implies the equations of
motion, it follows that
328. In any unguided motion whatever, of a conservative why imiied
Rjstem, the Action from any one stated position to any other, ^™'^^
though not necessarily a minimum, fulfils the ^ationary txmdif- ""■
ti<m, that is to say, the condition that the variation vanishes,
which secures either a minimum or maximum, or nuudmum-
miaimum.
This can Bcaroely be made intelligible without mathematical Btatuonur
language. Let (a:,, y^, z^, (x^ y^ s,), etc., be the co-ordinates
of particles, m,, m,, etc., composing the system ; at any time t of
the actual motion. Let V be the potential enei^ of the system,
in this configuration ; and let E denote the given value of the
sum of the potential and kinetic energies. The equation at
energy is —
J {m, (±,'+#/+ i,") + i»,{je/+^/ + *,») + eta }+ r=^...(5)biB.
Choosing any part of the motion, for instance that from time 0
to time (, we have, for the action during it,
A=j'{E-V)dT = £:i-j'vdr (11).
..Google
i PBEUUINABY. [328.
Let noir tte Byatem be guided to move in any other way possible
for it, with any other velocitiea, from the same initial to the same
final oonfiguratioD as in the given motion, subject only to the
condition, that the sum of the kinetic and potential eiiergies shall
Btill be B. Let (a;,', y,', a,'), etc, be the co-ordinates, and Y'
Uie corresponding potential energy ; and let (*/, j?,', i,'), eta,
be the component velocities, at time r in this arbitraiy motion ;
equation (2) still holding, for the accented letters, with only £
unchanged. For the action we shall have
V^JSt'-frdr (12),
where f is the time oocnipied by this supposed motion. Let now
6 denote a small numerical quantity, and let $^, i),, etc., be finit«
lines such that
The "prindple of stationary action" is, that — ^ — vaniahes
when 0 is made infinitely small, for every possible deviation
(fj^i Vi^i ^^') from the natural way and velocities, subject only
to tike equation of energy and to the condition of passing Uirongh
the stated initial and final configurations : and conversely, that if
V- V
— ^— vanishes with 6 for every possible such deviation from a
certain way and velocities, specified by (oi,, y^, x^), etc., as the
co-ordinates at t, this way and Aeae velocities are such that the
system unguided will move accordingly if only started with
proper velocities from the initial configniatlou.
829. From this principle of statiouary action, founded, as
ve have seen, on a comparison between a natural motion, and
any other motion, arbitrarily guided and subject only to the
law of energy, the initial and final configurations of the
system being the same in each case, Hamilton passes to the
consideration of the variation of the action in a natural or
unguided motion of the system produced by varying the initial
and final configurations, and the sum of the potential and
kinetic euetgies. The result is, that
..Google
330.] DTSAMICAL LAWS AHD PRINCIPLE3. 313
830. The rate of decrease of the action per unit of increase ^|^^
of any one of the free (generalized) co-ordinates (§ 204) speci-
fying the initial configuration, is equal to the correspond-
ing (generalized) component momentum [§ 313, (c)] of the
actual motion from that configuration: the rate of increase of
the action per unit increase of any one of the free co-ordi-
nates specifying the final configuration, is equal to the corre-
spondiog compoaeat momentum of the actual motion towards
this second configuration : and the rate of increase of the action
per unit increase of the constant sum of the potential and kinetic
eoei^es, is equal to the time occupied by the motion of which
the action is reckoned.
To prove Uiis we must, in our preriona expression (9) for hA,
now nippose th« terminal co-ordmates to vary; hT to become
ZE — 8 F, in which ZE ia a oonatant during the motion ; and each Action
set of paths and velocities to belong to an ungnided motion of aikfuoo-
the system, which requires (10) to hold. Hence iniUkland
W = {Sm (iSa; + 3)8ff + «&)} - [Sot (i8aj + 3% + i8s)] + (SJf ...(13).
If, now, in the first place, we anppoae the particles oonstituting
the system to be all free from oonstraint, and therefore {x, y, e)
for each to be three independent variables, and if, for distinctness,
we denote by (a;,', y,', x^) and (x,, y„ s,) the co-ordinates of m^
in its initial and final pcsitiona, and by (:£,', y,', i^, {±^, ^ i)
the components of the velocity it has at those points, we have,
from the preceding, according to the ordinary notation of partial
differential coefficients.
dA ^, dA ., dA
dx^ ' " dy,' "" de^
^.m* — =mfl — -
"(I*)- to1S?ti2'
tion of the initial and final co-ordinates, in all six times as many
independent variables aa there are of particles ; and E, one more
variable^ the sum of the potential and kinetic energies.
If the system consist not of free particles, but of particles con-
nected in any way forming either one rigid body or any number
..Google
344 PBELIHIKABT. [330.
V'T^ins of rigid bodies connected with one another or not, we might, it ii
true, be contented to r^ard it stilt as a B^stem of &ee p&rticlea,
by taking into acooimt among the impressed forces, the foicea
necessary to compel the satisfaction of the conditions of con-
nexion. But although tliis method of dealing with a aystem of
connected particles is very simple, so far as the law of energy
merely ia concerned, Lagrange's methods, whether that of "equa-
tiona of condition," or, what for our present purposes is much
more convenient, his "generalized co^rdinat«3," relieve us from
very troublesome interpretations when we have to consider the
displacements of particles due to arbitrary variations in the con-
figuration of a system.
Let uB suppose then, for any particular configuration (x^, y,, z^)
(x^ y>> ^t) ■•■> ^6 expression
ffl,(£,8x, -H$,Sy,-H i,S«,)-f- etc, to become ^ + i}S^+(Stf + etc (15),
Samapro- when transformed into terms o£ ip, ^, 6..., generalized oo-ordi-
Iwger^ nates, as many in number as there are of degrees of freedom for
Udi^iSr the ^stem to move [§ 313, (c)].
The same transformation applied to the kinetic enei^ of the
system would obviously give
im,(i6,•■l■^/-^i,')■^etc. = i(^■^.lJ^ + ifl + etc) (16),
and hence f, 17, 4 ^'l^> ^^ those linear functions of the generalized
velocities which, in § 313 (e), we have designated as "gene-
ralized components of momentum ; " and which, when T, the
kinetic energy, is expressed as a quadratic function of the velo-
cities (of course witli, in general, functions of the co-ordinal«a
iji, ^ 6, etc, for the ooeffieiente) are derivable from it thus :
^ dF dT ^ dT _^
^=^' " = 5^* ^=T6'^ <"'■
Hence, taking as before non-accented letters for the second, and
accented letten for the initial, configurations of the system re-
Bpectively, ve have
dA „ dA , dA
d4'
-r.etc.
and} as before,
dl_
di~
..(18).
jiGoogle
330.] DTNIMICAL LAWS AKD FRIKCIPLGS. 345
The8eeqiUlions(18),iucladingof coiu:se(14)a8«pftrtionlar caae, YuTing
ezpreea in matlieaiatical terms the propoeition stated in vords
above, as the Principle of Varying Action.
The values of the momentums, thus, (14) and (IS), expressed
in terms of differential coefficients of A, must of course satisfy
the equation of energy. Hence, for the case of free particles,
_ 1 /rfj' dA' dA'\ -,„_,. ,,„. HunlKon'i
=^;;W*d7''w°^<^-'^^ *''>' '^S^"
Sif^+^^;4.^^ = 2(ff-r') (20). g5^
Or, in general, for a system of partidea or rigid bodies connected
is any way, Tre have, (16) and (18),
,|.,-..^....=.<.-n <., ^
where tfr, ^ eta, are ezpreasible as linear functions <^f jT i ~ri >
etc, by the solution of the equations
..(23),
, etc., hy
W',f)f +(*■,♦■) *'+»',»o »■+«»«•. ,'—^
etc. etc
(21).
where it must he remembered that (^, ^), {ifi, ift), etc., ore fdno-
tiona of the specifying elements, ^, ^ S, etc, depending on the
kinematical nature of the co-ordinate system alone, and quite
independent of the dynamical problem with which we are now
concerned ; being the coefScients of the half squares and the
ptWucts of the generalized velocities in the expression for tho
..Google
t! PKEUHINART. [330.
kinetic energy of any motion of the system ; and that (^', ^^,
{<)/, tj/), etc, ar« tha same functions with ip', t^', etc., writtrai for
ip, ^, 8, etc. ; liut, on the other hand, that Aia m function of all the
elements ^, ^, etc., \j/, ^', etc. Thua the first member of (31)
#'
known functions of ^, <ft, etc., depending merely on the Irine-
matic&l relations of the system, and Uie masses of its parts, bat
not at all on the actual forces or motions; while the second
member is a function of the co-ordinates ^, ^ ete., depeodlDg
on the forces in the dynamical problem, and a constant expressing
the particular value giren to the sum of the potential and kinetic
energies in the actual motion ; and so for (22), and ^', ^', etc
It is remarkable that the single linear partial differential equa-
tion (19) of the first order and second d^ree, for the case of
free particles, or its equivalent (21), is sufficient to determine a
function A, Bach that the equations (14) or (16) express the mo-
mentums in an actual motion of the system, sabject to the given
forces. For, taking the case of free particles first, and different
tiatiug (19) still on the Hamiltonian understanding that A is
expressed merely as a function of initial and final coordinates,
and of E, the Bum o£ the potential and kinetic energies, we hare
1 /dA (TA ^dA d'A ^dA <^^'\^_2 —
m \dx dx,dx dy dx^dy dz dx^de) ^ dx, '
But, by (U),
1 dJ
*..i
^-
elo.,
=".|
^A
<•
da:,*)!, * oat, ' (M;
TTsing these properly in the preceding and taking half; and
writing out for two particles to avoid confusion sfi to the mean-
ing of 2, we have
'\^-i
, da, . <&, ^ <&, , (te, , cte, . \ AY ,
Kow if we multiply the first member by dl, we hare cleariy the
change of the value of m,:^, due to vaiying, stall on tha Hamil-
..Google
330.] DTNAHTCAL LAWS AND PRINCIPLES. S47
tonian Bu[^>OBition, the oo-onUnatea of all tlie pointa, tlutt is to say, T'lrliw
the configuration of tl^ system, from wliat it is at any moment to p„^ j„^^
vhat it becomes at a time dt later ; and it is therefore the actual J^ rhuao-
chanse in the value of met,, in Uie natural motion, from tiie time, «i<iKtion
* '' ' ' dtiiiin the
(, when the oonfigoration is {x^, y,, «,, x^ ..., £), to the time p>°''oii. tor
t + dt. It is thereforo equal to mje^dt, and hence (25) becomes iwticioi.
simply m,*, — — -j— . Similarly we find
dV dV dV ^
Bat these are [% 293, (4)] the elementaiy differential equations
of the motions of a oonaerrative system composed of firee mutually
influencing particles.
If next we regard a;,, y,, z^, x„ etc, as oonstsnt, andgo
through precisely the same process with reference to x^',y^,z^, x',
eto^ we hare exactly the same equaticms among the accented
letters, with only the difierenoe that — A appears in place ot A;
and end with m,*,' = -p— , , from which we infer that, if (20)
Ui satisfied, the motion represented by (14) is a natural motion
through the configuration (jb,', y,', *,', «,', etc).
Hence if both (19) and (20) are satisfied, and if when x^^x,',
y, = V,> *■=«.» '"■ = '''#1 «*<^i ''« •'a'e 3—^-3—,, etc., the
motion represented by (14) is a natural motion through the
two oonfiguratione (a:,', y/, z,', aj/, etc.), and (a;,, y,, a,, x^,
etc). Although the signs in the preceding expressions hare been
fixed on the supposition that the motion is from the former, to the
latter configuration, it may clearly be from either towards the
other, since whichever way it is, the rereree is also a natural
motion (§ 271), according to the general property of a oonserva-
tive system.
To prove the same thing for a conservative system of particles Bamg pn>-
or rigid bodies connected in any way, we have, in the first place, KnI'™
from (18) ocmnoetod
dn di dt di (!«n^irt
d^ rf^' d^ dB' ^ " "«<*
where, on Uie Hamiltonian principle, we suppose ip, ^ etc, and
(, II, etc, to be expressed ss functions of ^, ^ etc, ^', if>', etc,
..Google
8 PBELDIINAST. [330.
&nd the aum of Uie potential and kinetic energies. On the aame
suppoution, differentiating (21), we have
But, by (26), and by the considerations above, we have
where i denotes the rate of variatioii of £ per unit of time in the
actual motion.
Again, we have
dip di d^i, d^i d>p dp
if, as in Hamilton's system of canonical equations of motion, we
suppose i}i, ^ etc., to be ezpressed as linear fiinctions of i, rj, etc,
with coeffidents involving^, ip, 6, etc., and if we take 3 to denote
the partial differentiation of these funotioDS with reference to the
Systran i, 17,-..^, ^,...) regarded as independent variables. Let
the coefficients be denoted by [1^, ip], etc., according to the plan
followed above; so that, if the formola for the kinetic energy be
2'- 4 {[*. i^] ^ + [*. *] 1' + - + 2 [,!>, ^] fi, + etc.) (30).
..(31),
^=5^ =[*,-/-]£+[*. ^I'J + E-^-^lt+etc.r
etc. etc J
where o£ couise [fji, ^], and [^ ip], mean tiie same.
Hence g- [*, fl, W,,;^ ^J ...;
and therefore, by (29),
dtp d^ dip '
-i. J. J, ' 4. Btj- J. 9 _ ■
Google
330.] DTlTAHICiLL LAWS AND PIttNCIPLES.
whence, by (28), we see that Hamiiton-
..(32).
Thifl, and (28), reduce the firet member of (27) to 2f + 2 ^ , "^^
and therefore, halving, we conclude
(*%'-%■ ""> ""'?■ '+|-i^'«'° ■■(")■
These, in all as many differential equations as there are of rari-
ables, ip, ^, etc, suffice for determining them in terms of t and
twice as many arbitrary constante. But every solution of the
dynamical problem, as has been demonstrated above, satisties
(21) and (23); and therefore it must satisff these (33), which we
have derived Itom them. These (33) are therefore iiie equations
of motion, of the system referred to generalized co-ordinates, as
many in number as it has of d^reee of freedom. They are the
Hamiltonian explicit equations of motion, of which a direct de-
monstration was given in § 318 abova Just as above, it appears
therefore, that if (21) and (22) ore satisfied, (18) expresses a
oatural motion of the system from one to another of the two con-
figurations (^, ffi, $, ...) (^', ^', ff, ...). Hence
331. The determiDation of the motioD of any conservative Benn
system from one to another of any two configurations, when the oi>«i«i-
sum of its potential and kinetic energies is given, depends on
the determination of a single function of the co-ordinates of
those coniigurations by solution of two quadratic partial differ-
ential equations of the first order, with reference to those two
sets of co-ordinateB respectively, with the condition that the
corresponding terms of the two differential equations become
separately equal when the values of the two sets of co-ordinates
agree. The function thus determined and employed to express
the solution of the kinetic problem was called the Ckaractertattc ctar»et«r»
Junction by Sir W. R Hamilton, to whom the method is dua two.
It is, as we have seen, the "action" from one of the configura-
tions to the other; but its peculiarity in Hamilton's system is,
that it is to be expressed as a function of the co-ordinates and
a constant, the whole enei^, as explained above. It is evi-
..Google
350 rRELIHINABT. [331.
dently symmetrical with respect to the two coDfignrations,
chaoging only in sign if their co-ordinates are interchaDged.
Chwicier- StDce not Only the complete solution of the probl^n of
tknor' motion gives a solution, A, of the partial differential equation
"°*'"- (19) or (31), but, as ire have just seen [g 330 (33), ete.],
eveij solution of this equation oorresponda to an actual pro-
blem relative to the motion, it becomes an object of mathe-
matical analysis, which could not be satisfiuitorilf avoided, to
find what character of completeneea a solution or integral oi
die differential equation must have in order that a complete in-
tegral of the dynamical equations may be derivable fronr it — a
question which seems to have been first noticed by JacobL What
Coopltte is called a " complete int^ral" of the differential equation ; that
ebanotcrii- is to Say, an expression,
SS^"" A = A^ + F{>{f, ^ e,...a, ft...) (34),
for A satisfTing it and involving the same number t, let ua sup*
pose, of independent arbitraiy constants, A^, a, /3,...as there are
of the independent variables, ^, ^ etc. ; leads, as he found, to a
ctKnpIete final int^ral of the equations of motion, expressed as
follows : —
dF dF
T« = «'^ = * (^>'
and,asaboTe, ^=* + « (3^)'
where c Is the constant depending on the epoch, or era of reckon-
ing, chosen, and fl, V,... are i—l other arbitrary oonatants, con-
stituting in all, with £, a, p,..., ilie proper number, 2i, c^ arbi-
trary constants. This is proved by remarking that (35) are the
equations of the " course " (or patha in the case of a system of
firee particles), which is obvious. For they give
. d dF,, d dF,^ d dF ^
""-T^SL^^^d^-di^^dBlk^
„ ddF^,_ ddF^^^ddJ^,^ \ (37),
~d<l,d^'"^^d<f.dp"
d6dfi
in allt-1 equations to determine the ratios <2^ : <^ : d0 ;,
these, and (21), we find
d^ _d^ d0
^"^""?
(38)
..Google
331.] DTNAMICAL UWS AND PRINCIPLES. 331
[idnce (37) are the same aa the equationH which we obtain bv CorapMs
differentiatiiig (21) and (23) with reference to a, P,... Bucces- chsncieru-
Bively, only tiiat they have rf^, dtft, dd,,.. in place of iji, 4, 0,--~\- tioi?'*'
A perfectly general solution of the partial difTerential equation, Genanl
that JB to say, an expression for A including every function of deriT«d
^, ^ 6,... which can satisfy (21), may of course be found, by the piMa
regnlar process, from the complete int^^ (34), hy eliminating '"t^s^-
A^ a, j3,... from it by means of an arbitraiy equation
/('<„•, ft..o-o,
and tiie (t- 1) equations
dF dF
1 da d0_
dA^ da dfi
irherey denotes an arbitrary function of the i elements A^ a, J3, . . .
now made to be variables depending on ^, ^,... But the full
meaning of the geoeral solution of (21) will be better understood
in connexion with the physical problem if we fiiat go back to the
Hamiltonian solution, and then from it to the general Thus,
first, let the equations (35) of the course be asaumed to be
satisfied for each of two sets ^, ^, $,..., and ^', ^', ff,..., of
the ocKtrdinates. They will give 2(t — I) equations for determin-
ing the 2(t-l) Constantsa, ^,..., % 30,..., in terms of ^, ^ ...,
^', 4''>---t ^ fulfil these conditions. Using the values of a, /3,...,
so foatid, and assigning A^ so that A shall vanish when ^= ^,
^= ^', etc., we have the Hamiltonian expression for A in terms
of ^, ^, ■•■, ^', ^', ■■; and E, which is therefore equivalent to a
"complete int^pral" of the partial differential equation (21).
Ifow let f, 0', ..., be connected by any single arbitrary equation
f(f,*',..,) = 0 (39),
and by means of this equation and tiie following (t— 1) equations,
let their values be determined in terms of ^, ^ ,.., and E ; —
dA dA dA
d^i' dif,' dS . ,,-.
i^'k^^ ^ ^
df d^ d&
Substituting the values thus found for ^', ^', ^, efas., in the
Hamiltonian J, we have an expression for ^, which ia the general
..Google
352 PBEUUINAST. [331.
GraRmi M^otion of (21). FoT we see immedS&tel; that (40) expresses
doivrd that the ralues of A are equal for all configni&tLons satisfyiag
piPte (39), that is to BB,j, we haye
dA .,, dA ,,,
when ^', ^', etc., satisfy (39)i a&d (40). Henee when, by means
of these equations, ^', ^', . . . , are eliminated &om tiie Hamiltonian
expression for A, the complete Hamiltonian difierential
^-©*KS)'^--V^*i^'* *">
beoomes merely
''■Q'**Q'^* <«).
where ( -jy j, etc., denote ti.e difibreutial coefficieula in the Hamil-
toniaa expression. Hence, A being now a function of ^, ^, etc.,
both as these appear in the Hamiltonian expresaitm and as they
are introduced by the elimination of <y, <ft, eto., we have
dA /dA\ dA /dA\ ^ ,,„.
and therefore the new expression satisfies the partial differential
equation (31). That it is a completely general solution we see,
because it satisfies the condition that the action is equal for all
coufiguratioDs fulfilling an absolutely arbitrary equation (39).
For the case of a mngle free particle, the interpretation of (39)
is that the point (af, ^, 2') is <m an arbitrary surface, and of (40)
that each line of motion cuts this sur&ce at right angles. Hence
P„ctiQki 332. The most general possible solution of the quadratic,
IH^^^' partial, differential equation of the first order, which Hamilton
pirtsiSu- showed to be satisfied by his Characteristic Function (either
<£^£" terminal con&guration alone varying), when interpreted for tbe
tioit'*'™" case of a single free particle, expresses the action up to any point
{x, y, 2), from some point of a certain arbitrarily given surface,
from which the particle has been projected, in the direction of
the normal, and with the proper velocity to make the sum of
the potential and actual energies have a given value. In other
..Google
.._,,»1
332.] DYNAMICAL LAWS AND PRINCIPLES. 353
words, the physical problem solved by the most general solu-
tion of that partial differential eqnatioD, is this ; —
Let free particles, not mutaaUy iafluencing one another, be Fiopcrtii
projected normally from all points of a certain arbitrarily given orequi
surface, each with the proper velocity to make the sum of
potential and kinetic energies have a given value. To find, for
the particle which passes through a given point (x, y, z), the
" action " in its course from the surface of projection to this
point. The Hamiltonian principles stated above, show that
the surfaces of equal action cut the paths of the particles at
right angles; and give also the following remarkable properties
of the motion ; —
If, from all points of an arbitrary surface, particles not
mutually influencing one another be projected with the proper
velocities in the directions of the normals ; points which they
reach with equal actions lie on a surface cutting the paths at
right angles. The infinitely small thickness of the space be-
tween any two such surfaces corresponding to amounts of
action differing by any infinitely small quantity, is inversely
proportional to the velocity of the particle traversing it ; being
equal to the infinitely small difference of action divided by the
whole momentum of the particle.
Let X, ^ V be the direction cosines of the normal to t&e sur-
face of equal action through (x, y, z). We have
But -j- = mA, etc., and, if g denote the resultant velod^,
/rfj* dA' dA*\i ,„.
Hence a = -, ii = -, y=-,
? ? ?
which proves the fii-st proposition. Again, if &A denote the in-
vr.T I 2S
VOL, L
..Google
i PKELIMINABT. [33i
finitely small difference of actirm from (z, y, z) to any otiier
point (a; + Saj, y + Sy, a + &), we have
dA. dA. dA.
-di^*d^^*^^
Let the second ptunt be at an infinitely small distance, e, fmm
the first, in the directioa of the normal to the surface of equal
action; that is to say, let
Sx=eX, Sy = e/t, Ss=ev.
U =
-(3);
Hence, by (1). U=.(^ ^ ^ ^^) ...
whence, by (2), <. = — (4),
which is the second proposition.
833. IrreBpectively of methods for finding the "character-
istic function" in kinetic problems, the fact that any case of
motion whatever can be represented by means of a single
function in the manner explained in § 331, is most remarkable,
and, when geometrically interpreted, leads to highly important
and interesting properties of motion, which have valuable
applications in various branches of Natural Philosophy. One
of the many applications of the general principle made by
Hamilton* led to a general theory of optical instruments, com-
prehending the whole in one expression.
Some of its most direct applications; to the motions of
planets, comets, etc., considered as free points, and to the cele-
brated problem of perturbations, known as the Problem of Three
Bodies, are worked out in considerable detwl by Hamilton
(Phil Trans., 1834-35), and in various memoirs by Jacobi,
Liouville, Bour, Bonkin, Cayley, Boole, etc. The now aban-
doned, but still interesting, corpuscular theory of light furnishes
a good and exceedingly simple illustration. In this theory light
is supposed to consist of material particles not mutually influenc-
ing one another, but subject to molecular forces from the par-
ticles of bodies — not sensible at sensible distances, and therefore
not causing any deviation from uniform rectilinear motion in a
homogeneous medium, except within an indefinitely small dis-
■ Ontht Tkeary of Sgitau of Ban: Trana. B.L A., ISU, 1830,1832.
..Google
333.] DTNAHICAL LAWS AND PRINCIPLEa • 355
tance from its boundary. The laws of reflection and of Bingle ^""fiS'
refraction follow correctly from this hypothesis, which therefore "cfMi-
suffices for what is called geometrical optics.
We hope to return to this subject, with sufficieat detail, Appiiotion
in treating of Optics. At present we limit ourselves to state optki,
a theorem comprehending the known rule for measuring the
magnifying power of a telescope or microscope (by comparing
the diameter of the object-glass with the diameter of pencil
of parallel rays ernerging from the eye-piece, when a point of
light is placed at a great distance in front of the object-glass),
as a particular case.
334. Let any number of attracting or repelling masses, ororkinetin
perfectly smooth elastic objects, be fixed in space. Let two ^^oS!''
stations, 0 and O*, be chosen. Let a shot be fired with a stated
velocity, V, &om 0, in such a direction as to pass through &.
There may clearly be more than one natural path by which this
may be done ; but, generally speaking, when one such path is
chosen, no other, not considerably diverging from it, can be
found ; and any infinitely small deviation in the line of fire from
0, will cause the bullet to pass infinitely near to, but not
through, 0'. Now let a circle, with infinitely small radius r, be
described round 0 as centre, in a plane perpendicular to the
line of fire from this point, and let — all with infinitely nearly the
same velocity, but fulfilling the condition that the sum of the
potential and kinetic energies is the same as that of the shot
from 0 — bullets be fired from all points of this circle, all directed
infinitely nearly parallel to the line of fire from 0, but each pre-
cisely so as to pass through (/. Let a target be held at an
infinitely small distance, a', beyond 0', in a plane perpendicular
to the line of the shot reaching it bom 0. The bullets fired
from the circumference of the circle round 0, will, after passing
through 0', strike this target in the circumference of an eiceed-
ingly small ellipse, each with a velocity (corresponding of course
to its position, under the law of enei^) differing infinitely
little from V, the common velocity with which they pas-t
through (y. Let now a circle, equal to the former, be described
round (/, in the plane perpendicular to the central path through
O', and let bullete be fired from points in its circumference, each
23-2
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S56 PREXIHIMART. [3^.
with the proper velocity, and m such a direction infiDitely
nearly p&rallel to the central path aa to make it pass through
0. These bullets, if a target is held to receive them perpen-
y
dicularly at a distance a = a' vj^ , beyond 0, will strike it along
the circumference of an ellipse equal to the former and placed
in a "corresponding" position ; and the points struck by the in-
dividual bullets will correspond ; according to the following law of
"correspondence": — Let Pand i* be points of the first and second
circles, and Q and Q' the points on the first and second targets
which bullets from them strike ; then if P' be in a plane con-
taining the central path through (X and the position which Q
would take if its ellipse were made circular by a pure strain
{§ 183) ; Q and ^ are similarly situated on the two ellipses.
For, let XOY be a plane perpendicular to the central path
through 0 ; and X'&T the correspoiiding plane through &. Let
A be the " action " &om 0 to C, and ^ the action from a point
P{x,y,z), in the neighbourhood of 0, specified with reference
to the former axes of coordinates, to a point P (x*, y', z"), in
the neighbourhood of (/, specified with reference to the latter.
Hie function ^ — A Ttutishes, of course, when x=0, y = 0,
s^O, a/^O, jr'— 0, s'^0. Also, for the same values of the
co-ordmatea, its differential coefficients -^ , -.- , and -X ,
dat dy dx
-,, , must vanish, and -^ , — ,-. i
dy dz dt
V and W, since, for any values whatever of the co-ordinatefi,
dif.
dx '
OXf OT, at the particle passing through F, when it comes from
P", and — -^ and - - -, are the components parallel to 0 A", 0 T,
of the velocity through P" directed so as to reach P. Hence by
Taylor's (or Maclaiuin's) theorem we have
^_J=_ Vz'-t- Vz
+ U{^r,X);>f + (T,r)y'+...+{X',T)x-^+.,.
4 2{r,^)ya + ... + 2(r,^)yV+...
+ 2{X,X')xaf-t-2{T,T')ffy' + S (Z, Z) «'
+ 2 (X, r)ay + 2 (X,ir)aa'+ ... + 2(^, y^ ^'1 + fl ...(I).
..Google
334.] DYNAMICAL LAWS AND PRINCIPLES. S57
irbere (X, X), (X, T), etc., denote conBtants, viz., the values of AppKoUan
the difierential coefficients ^-? , , ^ , etc., vhen each of the or klnatka
tur dxdy otBiinxie
six co-ordinates x, y, z, x', j/, 2' vanishes ; and Jt denotes the
remainder after the terms of the second d^ree. According to
Cauchf 'b principles regarding the oouvergence of Taylor's theorem,
we have a rigorous expression for il> — A ia the same form, with-
out R, if the coefficients {X, X), eta, denote the values of 1^
differential coeSicienta with some variable values intermediate
between 0 and the actual values of a;, y, etc., substituted for these
elements. Hence, provided the values of the differential co-
efficients are infinitely nearly the same for any infinitely small
values of the co-ordinates as for the vanishing values, £ becomes
infinitely smaller than the terms preceding it, when x, y, ete.,
are each infinitely small Hence when each of the variables
IB, y, e, af, y', / is infinitely small, we may omit S in the ez'
preeaion (1) for ^ — A. Now, as in the proposition to be proved,
let us suppose z and z' each to be rigorously zero : and we have
g-(x,^.+(x,r)j,+ (x,x)»'+(x,r)j^i
These expressions, if in them we make x=0, and jf°>0, be-
come the component velocities parallel to OX, OT, of a particle
passing through 0 having been projected from P'. Hence, if
(,t},i denote ito co-ordinates, an infinitely small time, -p , after
it passes ^-oogh 0, we have li=a, and
(-((x,x)«'+(x, r)!,-)2, ,.((r, jTjx't (7, r),')f ...(2).
Here f and -q are the rectangular co-ordinates of the point Q' in
which, in the second case, the supposed target is struck. And
by hypothesis
a^-Hy' = r' (3).
If we eliminate x', 1/ between these three equations, we have
clearly an ellipse ; and the former two express the relation of the
"corresponding" points. Corresponding equations with x and
y for *■ and y' ; with £', 1;' for f, >; ; and with - {X, X'),
-{K,X), ~{X, r), ~{Y, F), in place of (X, X), (X, T),
..Google
858 PREUMINART. [35*.
•n (F, X'), (F, T"), express the first case. Hence the propoEdtion,
as is moat eauily seen hy chooaing OX and (/S' bo that (J', i")
and {Y, X') may each be zero.
336. The moat obvious optical application of this remarkable
opHmT™" result is, tbat in the us& of any optical apparatus whatever, if
tbe eye and the object be interchanged without altering the
position of the instrument, the magnifying power is unaltered.
This is easily understood when, as in an ordinary telescope,
microscope, or opera-glass (Galilean telescope), the instrument
is symmetrical about an axis, and is curiously contradictory of
the common idea that a telescope " diminishes " when looked
through the wrong way, which no doubt is true if the telescope
is simply reversed about the middle of its length, eye and
object remaining fixed. But if the telescope be removed from
the eye till its eye-piece is close to the object, the part of the
object seen will be seen enlarged to the same extent as -when
viewed with the telescope held in the usual manner. This is
easily verified by looking from a distance of a few yards,
in through the object-glass of an opera-glass, at the eye of
another person holding it to his eye in the usual way.
The more general application may be illustrated thus : — Let
the points, 0, O (the centres of the two circles described in
the preceding enunciation), be the optic centres of the eyes of
two persons looking at one another through any set of lenses,
prisms, or transparent media arranged in any way between
them. If tb^ pupils are of equal sizes in reality, they will
be seen as similar ellipses of equal apparent dimensions by the
two observers. Here the imagined particles of light, projected
from the circumference of the pupil of either eye, are substituted
for the projectiles from the circumference of either circle, and
the retina of the other eye takes the place of the target receiv-
ing them, in the general kinetic statement.
Appiintion 336. If instead of one free particle we have a conservative
^m^."* system of any number of mutually influencing free particles, the
fliwDd'mc same statement may be applied with reference to the initial
position of one of the particles and the final position of another,
or with reference to the initial positions or to the fint^ positions
..Google
336.] DYNAMICAL LAWS AND PRINCIPLES. 859
of two of the particles. It serves to show how the influence of App«o«iioii
"I" ... low"**"'"'
an iofinitely small change in one of those positions, on the di- »«•"<""■
rection of the other particle passing through the other position, J^J^
is related to the influence on the direction of the former particle
passing through the former position produced by an infinitely
small change in the latter position. A corresponding statement, "idtDn-
in terms of generalized co-ordinatea, may of course be adapted ijriteiD.
to a system of rigid bodies or particles connected in any way.
All such statements are included in the following very general
proposition : —
The rate of increase of any one component momentum, corre-
sponding to any one of the co-ordinates, per unit of increase of
any other co-ordinate, is equal to the rate of increase of the com-
ponent momentum corresponding to the latter per unit increase
or diminution ot the former co-ordinate, according as the two co-
ordinates chosen heloDg to one configuration of the system, or
one of them belongs to the initial configuration and the other to
the final
Let t}i and x be two ont of the whole number of co-ordinates
constitnting ^e argument of the Hamiltonian characteriBtio
function A ; and £, t) the corrosponding momentoms. We have
[I 330 (18)]
dA , dl
the app«r or lower sign being used sccoiding as it is a final or
on initial coordinate that is concerned. Hence
and therefore T- - tt t
dx #'
if botli coordinates belong to one configuration, or
d-K "*(■'
if one belongs to the initial ccnfgnrBtion, and the other to the
final, which is the second propomtion. The gsometrical inter-
pretaUon of this statement i<jt the case of a free particle, and two
co.otdinatea botii belonging to one position, its final position, for
..Google
360 PRELIMINAST. [33C.
Applioaion iaatance, gives merely the propootion of g 332 above, for the
free mutu- oaae of particles projected from one point, with equal veloeitiM
du«idiis in all directions ; or, in other words, the case of tbe arlnbrary
and tpfi«- sur&ce of that enunciation, being reduced to a poinL To eom-
Kyaiem. plete the set of variatioual equations derired from § 330 we hare
-j-=*-^ which expresses another remarkable property ot eaor
eervative motion.
8iiish«y^ 387. By the help of L^range's form of the equations of
equilibrium, motion, § 318, we may now, as a preliminary to the considera-
ticai of stability of motion, investigate the motion of a system
infinitely little disturbed from a position of equilibrium, and
left free to move, the velocities of its parts being initially in-
iinitely small. The resulting equations give the values <^ the
independent co-ordinates at any future time, provided tbe dis-
placements continue infinitely small; and the mathematical
expressions fcs their values must of course show the nature of
the equilibrium, giving at tbe same time an interesting example
of the coexiapence of STnali moHofii, § 89. The method con-
sists simply in findir^ what the equations of motion, and their
integrals, become for co-ordinates which diflfer infinitely little
from values corresponding to a configuration of equihbrium —
and for an infinitely small initial kinetic energy. Tbe solution
of these diflFerential equations is always easy, as they are linear
and have constant coefGcients. If tbe solution indicates that
these differences remain infinitely small, the position is one tA
stable equilibrium ; if it shows that one or more of them may
increase indejlniiely, the result ef an infinitely small displace-
ment from or infinitely small velocity through the position of
equilibrium may be a finite departure from it — and thus the
eqoUibrium is unstable.
Since there is a position of equilibrium, the kinematic relations
must be invariable. As b^ore,
^= i Usfr. ^) '^ + (*. *) ^•+ 2 ("f, *) W + eto....i...(l),
which cannot be negative for any values of the oo-ordinatce.
Now, though the values of the coeffidents in this expression are
not generally constant, they are to be taken as couetant in the
approximate investigation, since their variations, depending on
..Google
337.] DYNAMICAL LAWS AND PIHNCIPLE3. 361
the infinitely small variations of ip, if>, etc, can only give rise to fl'f^^
terms of the thini or higher orders of small quantltiea. Hence equilibriom.
Lagninge's equations become aimply
^r^W*.etc. (2).
and the firat member of each of these equations is a linear func-
tion of ^, ^ etc., with constant coefficients.
Now, since we may take what origin we please for the gene-
mlized co-ordinates, it will he convenient to assume that tj/, ^ 0,
etc., are measured from the position of equilibrium considered;
and that their values are therefore always infinitely small.
Hence, infinitely small quantities of higher orders being
neglected, and the forces being supposed to be independent of the
velocities, we shall have linear expressions for ^I*, 4, etc, in
terms of ijr, ^ etc., which we may write as follows : —
* = a^ +&.
* = aV + 6> + c'tf+...|. (3).
+ &<^ +c$ +...1
+ b'<i> + e'0+...i .
etc J
Equations (2) consequently become linear differential equations
of the second order, with constant coefficients; as many in
number as there are variables ^, tf>, etc, to be determined.
The regular proc^raes explained in elementary treatises on dif-
ferential eqoations, lead of course, independently of any particu-
lar relation between the coefficients, to a general form of solution
(§ 343 below). But this form has very remarkable characteristics
in the case of a conservative system; which we therefore
examine particularly in the first place. In this case we have
dV ^ dV ,
where F is, in our approximation, a homogeneous quadratic
function of ^, ^, ... if we take the origin, or configuration of
equilibrium, as the configuration from which ^ 273) the poten-
tial energy is reckoned. Now, it is obvious*, &om the theory
* Foi in the first plaoe any snsh assumption as gl
<l'=Af, + Bits,+ ...
gives equations Un ip, ^, etc., in terms of ^^ ^,, etc., with the some ooeffloieuta,
A, B, ete., if theee are iadependeot of 1. Hence (th« co-imdiiiateB being i in
..Google
302 PKELnilNi3T. [337.
Blh*^ of the transformation of quadratic functions, tliat we may, by a
equiUbriDm. determinate linear ti'snalbrmation of the co-ordinates, i«duce tbe
8imu1t*ne- nnmber) ws hsve i* qnontities A. A', A", ... B, S, B", ... etc., to be determined
(anamtion ^7 '' eqnstiouB eipreuiog that ia ST the coeffioients of ^,', ^J, etc. are each
oTtwo eonsl to nnitj, and of -iii etc. each vuiish, and that in V the ooeScientH of
quAdrstio
runclloiu ^.^ii <to. each TaniBh. Bat, particnlailj in respect to onr dfnanxical problem,
Mu>nii!°' "** following proteaa in two steps is inEtmotive;^
(1) Let the qoadratio eipreseion for T in tenna of ^, ^, ^ ate., be
rsdnoad to the form ^,* + ^,'+ ... b; proper assignment of valoea to A, B, etc.
Tbis niay be done arbitrarily, in an infinite namber of iisj*, idthont the
sfJutioQ of onj algebraic equation of degree higher than the first; as «« maf
easily see b; working oat a synthetioal process algebraically according to the
analogy of finding first tbe conjugate diametral plane to any ohoeen dianmter of
an ellipsoid, and then the diameter of its elliptia sestion, eonjngate to any
chosen diameter of this ellipse. Thus, of the -L- — .' equations expressing that
the ooefScients of the piodnat* ^,^,, ^,0,, ^,S,, etc Tanish in T, tslte first the
one eipresBing that the ooeSftcient of ^^, -vanishes, and by it find tbe value of
one of the B'a, snpposing all the A't and aU the £'s but one to be knoim.
Then take tbe two eqnationa expreestng that tbe eoeffleienta of ^fi, and ^,tf,
vanish, and Xsj them find two of the Cs sopposing all the C% bnt two to be
known, as are now all tlie A'i and all the B't: and eo on. Thus; in trams of
aU the A'b, all the B's bnt one, all the Cs bnt two, all the D's bnt three, and so
on, saj^KMed known, we find by the solntion of linear eqnatioita the remaining
B't, Cs, ZJ's, eta. liastly, osing the valnes thns found tor tbe nnaBanned
qnautities, B, C, D, etc., and equating to onity the coefGoienta of ^,', ^,', 6*,
etc. in the transformed expression lor 3T, we have i eqnatiooe among the Bqnares
and produots of the ' '* assumed quantities, (i) A\, (t-lj fi, (i -3) Cs,
etc., by which anyone of the.il'i, any one of the B'c, anyone of thefts, and so
on, are glTsn immediately in terms of the — q— ' ratios of the others to then.
Thna the thing is done, and -^-^ — - disposable ratios ore left nndetemuned.
(2) These quantities may be determined by the -^-3 — ■ eqaations eipren-
ing that also in the transformed qoadratic V the ooefflaie&ts of 1^,^,, f^,,
i^fi,, eta. vanish.
Or, having made the first transformation as In (1) above^ with aesomed values
for ^ -' disposable ratios, make a seoond transtonnatioD detenninatety thns ;
eto., etc,
where the f quantities l,w, ..., V,ta', ... astis^ tbeii(f.(-l) eqaations
tf+Ba^+.-nO, rr+«'m"+...=0, eto.,
orth«OD*]
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337.] DYNAMICAL LAWS AND PRINCIPLES. 303
expression for 2T, which b essentially positive, to a sum of ^'"'g^^^
Miiarea of (reiieralized component velocities, and at the same '"^ "ib
? ,, , 1 ^ , ,. 1- kimttoand
time K to a sum of the squares of the corresponding co-ordi- potentna
nates, each multiplied by a constant, which may be either positive
or n^;ative, but la esseutially real. [In the case of an equality
or of any number of equalities among the values of these con-
stants (a, p, etc. in the notation below), roots as they are of a
detenninantal equation, the linear transformation ceases to be
wholly determinate ; but the degree or degrees of indeterminacy
which supervene is the reverse of embarrassing in respect to
either the process of obtaining the solution, or the interpretation
and use of it when obtained.] Hence ^, ^ ... may be so chosen
diat
r=J(^+^' + etc.) (4),
and F=J(ai/'" + ;3^' + eta) (5),
a, fi, etc., being real positive or negative constants. Henoe
Lagrange's equations become
^' — a^, * = -/»«, eta (6).
The solutions of these equations are
i,.Aa»(fJa-,), *.J'co8W;8-0, etc, (7), SXS"
Of mothm,
A, e, A', e", etc., being the arbitrary constants of integration. SS^mE
Hence we conclude the motion consists of a simple harmonic f^c^ot
variation of each co-ordinate, provided that a, /3, etc, are aU "'"•'i^-
positive. This condition is satisfied when F is a true minimum
at the configuration of equilibrium ; which, as we hav« seen
^ 292), is necessarily the case when the equilibrium is stable.
If any one or more of a, ^, ... vanishes, the equilibrium might
and P+B^ + ...»l, t''+in''+... = l,BtD., Blmultane-
Iea™g i i (i - 1) dispoMblss. lonution
We Bball Bid have, obriousl;, the same fonn for 2T, that U:-
And, aooordiiiB to the known theoiy of the braaatonoBtiDn of qiiBdnttia functions,
we ma; determine the ^{(i-l) dispoBsbles of I, m, ..., V, m', ... so as to make
the tl{i-]) prodnctB of the oo-oidinatee f„, #„, etc. disappear from the ex-
pression for V, and give
where a, p, y, etc., are the roots, necesBBril; real, of an equation of the ith
degree of which the ooeffioientB depend on the eoeffloienta of the sgnarec and
prodnota in the ezpresBion for F in terms ot ^„ ^„ eta. Later [{?'), (8) and (9)
of g SU/1, a *>«$!« I>roe«H tor oanTing ont this investigation will be worked out.
qoadimtla
nmctioiu
jiGoogle
of motion.
or ot filling
■nd Kinetic
eiprwanl bc
i PBELIMINAKT. [337.
be either stable or unstable, or neutral ; but terms of liigho'
orders in the ezpansion of F in ascending povers and prodads
of the co-ordinates vould faaTO to be eicamined to test it ; and if
it were stable, the period of an infinitely small oscillation in the
value of the correaponding co-ordinate or coordinates would be
infinitely great If any or all of a, j3, y, ... are n^ative, V is
not a ininimum, and the equilibrium is (§ 292) essentially nn-
Btable. The form (7) for the solution, for each co-ordinate for
which this is the case, becomes imaginary, and is to be changed
into the exponential form, thus; for instance, let ~a=3p, a |Meitire
quantity. Thus
^ = Ce+Wp + ^«-(^ (8),
which (unlees the disturbance is bo adjusted aa to make tbe
arbitrary constant C vanish) indicates an unlimited increase
in the deviation. This form of solution expresses the approxi-
mate law of falling away from a configuration of ucstable equili-
brium. In general, of course, the approximation becomes less
and less accurate as the deviation increases.
We have, by (5), (4), (7) and (8),
F=>l'[l-t-C0B2((V«-e)] + et«. 1 ,„,
or F--Jp[2C^-^-C"*«^^-^^.-aVp]-etc.| ^ ''
and r=W'[l-cos2(Va-e)] + etc. ]
or 7'=ip[-2(?J'+(7'««'Vj»+i',-«V*.]4.etc.| ^ ''
and, verifying the constancy of the sum of potential and kinetic
enei^ies,
r+r=}('^'+^^''+etc.) I
or T-f-r^-2{pC£+qC-£'+i>Us.)\ ^ ''
One example for the present will suffice. Let a solid, im-
mersed in an infinite liquid (§ 32*)), be prevented from any
motion of rotation, and left only freedom to move parallel to a
certain fixed plane, and let it be influenced by forces subject to
the conaervBtive law, which vanish in a particular position of
equilibrium. Taking any point of reference in the body, chooaing
its position when the body is in equilibrium, as origin of rect-
angular co-ordinates OX, OT, and reckoning the potential energy
from it, we shall have, as in general,
..Google
337.] DTNAMICAL LAWS AND PRINCIPLiS, 365
the principles stated In § 320 above, allowing hb to regard the Bamptant
oo-ordinatee x and y as fully specifying the system, provided ui modM.
always, that if the body is given at rest, or Ih brought to rest,
the whole liquid is at rest (§ 320) at the same time. By solving
the obviously determinate problem of finding that pair of coDJu-
gate diameters which ore in the same directions for the ellipse
Aa?+ B^ + 2Gxy = conatT
and the ellipse or hyperbola,
aa? + Sy* + 2ffsy = const.,
and choosing these as obUque axes of co-ordinates {x^, y^, we
shall have
2T=A ,*/ + £,1?,', and 2 r = a,a;,' + 6,i/,'.
And, as A^, S, are essentially positive, we may, to shorten our
expressions, take x^JA^=>^, y,JS, = c^; bo that we shall have
22"=^ + ^', 2r=af +^^',
the normal expressions, according to the general forma shown
above in (4) and (5).
The interpretation of the general solution is as follows : —
338. If a conservative system is infinitely little displaced aensmt
from a configuration of stable equilibrium, it will ever after runSTioen.
vibrate about this configuration, remaining infinitely near it ; iminita^
each particle of the system performing a motion which is com- moiion
posed of simple harmonic vibrations. If there' are t degrees of nminiiiou
freedom to move, and we consider any system (§ 202) of gene- liMum.
rolized co-ordinates specifying its position at any time, the
deviation of any one of these co-ordinates from its value for the
configuration of equilibrium will vary according to a complex
harmonic function {§ 68), composed of i simple harmonics gene-
rally of incommensurable periods, and therefore {§ 67) the whole
motion of the system will not in general recur periodically
through the same series of configurations. There are, however,
i distinct displacements, generally quite determinate, which we
^hall call the normal displacements, fulfilling the condition, that Nonmidu-
if any one of them he produced alone, and the system then lefl fiwT^ui-
to itself for an instant at rest, this displacement will diminish
and increase periodically according to a simple harmonic func-
..Google
366 PRBLuatfAitr. [SSii.
- tioD of the time, and consequently every particle of the system
Tibntioa. will cJtecQte a simple harmonic movement in the same periiwl.
This result, we shall see later (Vol ii,), includes cases in which
there are an infinite number of degrees of freedom ; as for in-
stance a stretched cord ; a mass of air in a closed vessel ; waves
in water, or oscillations of water in a vessel of limited extent, or
of an elastic solid ; and in these applications it gives the tbeory
of the so-called " fundamental vibration," and successive " har-
monics" of a cord or organ-pipe, and of all the different possible
simple modes of vibration in the other cases. In all these cafes
it is convenient to give the name "fundamental mode" to any
one of the possible simple harmonic vibrations, and not to
restrict it to the gravest simple harmonic mode, aa has been
hitherto usual in respect to vibrating cords and organ-pipes.
Thearnnirf The whole kinetic eneigy of any complex motion of the sys-
enetRTi tem IB |^ 337 (4)] equal to the sum of the kinetic energies of
ofpotentbi the fundamental constituents; and [§ 337 (5)] the potential
enei^ of any displacement is equal to the sum of the potential
energies of its normal components.
inflniMi- Corresponding theorems of normal constituents and fuoda-
in Deieh- mental modes of motion, and the summation of their kinetic
bemrhoodof , , , , . »• i
i»iin«iini- and potential energies m complex motions and displacements,
iiSri"* '^^' ^°'^ ^'*' motion in the neighbourhood of a configuration of un-
stable equilibrium. In this case, some or all of the constituent
motions are fallings away firom the position of equilibrium
(according as the potential energies of the constituent normal
vibrations are negative).
Cnenr 339. If, as may be in particular cases, the periods of the
^SjS vibrations for two or more of the normal displacements are equal,
any displacement compounded of them will also fulfil the condi-
tion of being a normal displacement. And if the system be dis-
placed according to any one such normal displacement, and
projected with velocity corresponding to another, it will execute
a movement, the resultaoit of two simple harmonic movements
Qnnhie in equal periods The graphic representation of the variation
'™™" " of the corresponding co-ordinates of the system, laid down as
two rectangular co-ordinates in a plane diagram, will conse-
quently (§ 65) be a circle or ah ellipse ; which will thei-eforc,
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S3d.] DTNAMICAL LAWS AND PRINCIPLES. 367
of course, bo the form of the orbit of any particle of the sjrstem anphio
which has a distinct direction of motion, for two of the displace- I^T"
mente in question. But it must be remembered that some of
. the principal parts [as for instance the body supported on the
fixed axis, in the illustration of § 319, Example (C)] may have
only one degree of freedom ; or even that each part of the
system may have only One d^ree of freedom, as for instance if
the system is composed of a set of particles each constrained to
remain on a given line, or of rigid bodies on fixed axes, mutually
influencing one another by elastic cords or otherwise. In such
a case as the last, no particle of the system can move otherwise
than in one line; and the elUpse, circle, or other graphical re-
preaentatioD of the composition of the harmonic motions of the
Rystera, is merely an aid to comprehension, and is not the orbit
of a motion actually taking place in any part of the system,
340. In nature, as has been said above (§ 278], every system
uninfluenced by matter external to it is conservative, vhen
the ultimate molecular motions constituting heat, light, and
magnetism, and the potential energy of chemical aflSnities,
are taken iuto account along with the palpable motions and
measurable forces. But (§ 275) practically we are obliged to Dintpatira
admit forces of friction, and resistances of the other classes ■'"*°°*-
there enumerated, as causing losses of energy, to be reckoned,
in abstract dynamics, without regard to the equivalents of heat
or other molecular actions which they generate. Hence when
such resistances are to be taken into account, forces opposed
to the motions of various parts of a system must be introduced
into the equations. According to the approximate knowledge
which we have from experiment, these forces are independent
of the velocities when due to the friction of solids: but are
simply proportional to the velocities when due to fluid viscosity
directly, or to electric or magnetic tnflaencea ; with corrections
depending on varying temperature, and on the Tatting con-
figuration of tbe system. In consequence of the last-mentioned
cause, tbe resistance of a real liquid (which is always more or
less viscous] against a body moving rapidly enough through it,
to leave a great deal of irregular motion, in the shape of
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368 PRELIMINARY. [340.
" eddiee," in ita wake, seems, when the motion of the solid has
been kept long enough uniform, to be nearly in proportioa to
the square of the velocity ; although, as Stokes has shown, at
the lowest speeds the resistance is probably in simple proportion
to the velocity, and for all speeds, after long enough time of
one speed, may, it is probable, be approximately expressed as
stnkra' pro- the Sum of two tenus, one simply as the velocity, and the
other as the square of the velocity. If a solid is started from
rest in an incompressible fluid, the initial law of resistance is
no doubt simple proportionality to velocity, (however great, if
suddenly enough given;) until by the gradual growth of eddies
the resistance is increased gradually till it comes to fulfil
Stokes' law.
Prieiionoi 341. The effect of friction of solids rubbing against one
another is simply to render impossible the injinitelff email
vibrations with which we are now particularly concerned ; and
to allow any system in which it is present, to rest balanced
when displaced, within certain finite limits, from a configuration
of frictionless equilibrium. In mechanics it is easy to estimate
ita effects with sufficient accuracy when any practical case of
finite oscillations is in question. But the other classes of dis-
sipative ^encies give rise to resistances simply as the velocities,
Kniituica without the Corrections referred to, when the motions are in-
TetodUM. finitely small; and can never balimoe the system in a con-
figuration deviating to any extent, however small, from a
configuration of equilibrinnt In the theory of infinitely small
vibrations, they are to be taken into account by adding to the
expressions for the generalized components of force, proper
(§ S4S a, below) linear functions of the generalized velocities,
which gives us equations still remarkably amenable to rigorous
mathematical treatment.
The result of the integration for the case of a single degree
of freedom is very simple; and it is of extreme importance,
both for the explanation of many natural phenomena, and for
iise in a large variety of experimental investigations in Natural
Philosophy. Partial conclusions from it are as follows : —
If the resistance per unit velocity is less than a certain
critical value, in any particular case, the motion is a simple
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341.] DYNAMICAL LAWS AND PRINCIPLES. 3C9
hannoDic oscillation, with amplitude decreawng in the same —-;—
ratio m equal successive intervals of time. But if the re- laiocWh.
sistance equals or exceeds the critical value, the sj^tem when
displaced from its position of equilibrium, and left to itself,
returns gradually towards its position of equilibrium, never os-
cillating through it to the other side, and only reaching it after
BJi infioite time.
In the unresisted motion, let »* be the rate of acceleration,
when the displacement ia unity ; so that (§ 07) we have
T= ^ : and let the rate of retardation .dfje.to tbe remstance
corresponding to unit velocity be k. Thea die motion is of the
oscillatory or non-oscillatory class according as i*<(2n)' orsiheiot
it* > (2n)*. In the first case, the period of the oscillation is nrring >■
increaaed by the reedstance from Tto T r, and the rate auM^n.
at which the Napierian logarithm of the amplitude diminishes
per unit of time is ^k. If a negative value be given to-i, the
case represented will be one in , which the motion is assisted,
instead of resisted, by force prc^rtional to the -velocity : but
this case is purely ideal.
The differential equation of motion for the case of one d^ree
of motion ia
of which the complete integral ia
^ = {^ Bin n'( + B COB «'*i«-*»', where n' = ^(n' - i*^,
or, which ia the same,
^ = (<7€-"/ + C'«-.0*" **. 'tere B, = Vii** - «0.
A and £ in one case, or C and C in the other, being the arbitrary
oonatontB of integration. Hence the propositions above. In the Ci» of
case of i'=(2n)'the general solution is i^ = (C + (?'()«"'*'. «i"i™*'
342. The general solution [§ 343 a (2) and § Sis'] of the infl..iteiy
problem, to find the motion of a system having any number, i, of tootion of ■
degrees of freedom, when infinitely little disturbed from a position ijitlim.'"
of stable equilibrium, and left to move subject to resistancea
proportional to velocities, shows that the whole motion may be
resolved, in general determinately, into 21 di&creot motions each
TOL. I. 24
S70 PREXmiKABT. [34^.
inOniMr either umple bannonic wiUi amplitude diminishiDg according
^■onofk to the law stated above, or nou-oscillatory and coosiatiiig of
ijitsm. equi-prc^rtionate diminutions of the components of displace-
ment in equal successive intervals of time.
343. It is now convenient to cease limiting our ideas to
infinitely small motions of an absolutely gener^ system through
configuiattons infinitely bttle difierent fi^m a configui^tioti of
equilibrium, and to coosider any motions large or small of a
t'T^^^ system so constituted that the positional* forces are proportional
dsflned. (o displacements and the motional* to velocities, and that the
kinetic eneigy is a quadratic function of the velocities with
constant coefficients. Such a system we shall call a cycloidalf
system ; and we shall call its motions oycloidal motions. A goo*!
and instructive Ulustration is presented in the motion of one
two or more weights in a vertical line, bung one from another,
and the highest from a fixed point, by spiral springs.
343 a. If now instead of ^, ^... we denote by ^,, ^„... the
generalized co-ordinates, and if we take 11, 12, 21,22..., ii, iz,
31, 33,... to signify constant ooeffidents (not nnmbers as in the
ordinary notation of arithmetic), the most general equations of
motions of a cycloidtd system may be written thus :
* Unoh tronUe and vetbiage Li to l>o avoided by the mtrodneUon of tfaeM
Motloii- adjEotivee, vhieh will hsnoatorth be in treqnsnt lue. Thoy teU thur own
meaning! ae clearly m any definition oonld.
■f A single adjective is needed to avoid a sea of tronbleo here. The adjective
'oyoloidal' is already olasdoal in respect to any motion mtb one d^ree ol
fraedom, anrvilineal or rectilineal, lineal or angnl&r (Conlomb-toTsional, lor ci-
amide), lolloping the Bame law as the oycloidal pendulom, that ii to tay: — the
ditplaeemenl a limple bamumie funtlUm of the time. The motion of a particle
on a oyoloid with vertex np may as properly be called eycloidal; and in it the
diaplaoement ia an imaginaiy limple harmonic, or a real exponential, or the
anm of two real e^onentiala of the time
{.:-JUc.-J\)-
In eyoloidal motion as defined in the text, each component of d
proved to be a snm of exponentials \Cc +Ct +et«.} real oi im^inaiy,
ledadble to a stun of pradnota of real eiponentialB and real simple taannoniM
[c.'^eoB (nt - «) + Cf'"''eos {»■( - O +8tc].
..Google
343 a.]
DTNAHICAIi LAWS AND PBINCIFLES.
. + 31^,+ aa^,+ ... = 0
etc.
■■(I).
Positional forces of the non-conserv&tiTe clasa are included hy
not aasuming iz = ai, 13 — 31, 23 = 3', etc.
Tbe theoi7 of Bunidtaneoas linear differential equations with
constant coefficients showa that the general solution for each
co-ordinate is the snm of particular solutions, and that eTerjr
particular solution is <£ the form
^,=a,f«, ifi, = a,t" (2).
ABsnming, then, this to be a solutian, and substituting in theTMrtoln-
differential equations, we ha*a
hX(llo, + 12o,+ ...)+iia,+ iao, + ... = 0
..(3),
X't— +\(2\a, + 22a.+ ...l + aio, + 120.+ ...=^
da, ^ ' ' ' ' '
vhere S denotes the same homi^neoua quadratic function of
a,, a,..., that T'is of 1^,,^,,.... TheM equations, i in number,
determine k by the detenninantal equation
(ri)X* + lU-(.ii. (iz)X*+12X + ri,,..
(ai)X' + 21X + ai, (js)X' + 22X+a3,...
= 0 W.
where (ii), (sa), (f a), (fi),otc. denote the coefficients of squares
and doubled products in the quadratic, 27* ; with identities
(iJ).(.i), (I3).(3.),elc (5).
The equation (4) is of the degree 2i, in X ; and if any one of its
roots be used fOT X in the * linear equations (3), these became
harmonized and give the i— 1 ratios <>,/'>,, <*,/<*,> etc.; and we
haTe then, in (2), a particular solution with one arbitrary con-
stant, a,. Thus, from the 2t roots, when unequal, we have 3i
distinct particular solutions, each with an arbitrary constant;
and the addition of these solutions, as explained abore, gives the
general solution.
24—2
DL3,;,;6:lbyG00gle
37S PB&LIMIKABT. [343 b.
BcdnttM or 343 b. To sbow explicitly the detenmnaticm of the ntioc
eqiBtlonl <>i/<*il '*■/<*!> ^^ P'**' ^^ brevi^
5dSS^ (li)X*+lU+ii = ri, (i2)X' + 12X + ia = r-a, eta,
"***■ (3a)A'+32X + 3a = 3-2,6tc (5)';
and geoerellj let j')t denote the coefficient of a„ in the ^ eqna-
tioD of (3), or die if* t«nn of the j^ line of the det«nninant (to
be called D for brerity) constituting the first member of (1).
Aj^o^ Let Jf(jA) denote the fiwtor of jA in i) so tb&t yk . M (Jk)
Hena. ia the Bnm of all the temu of D which oontun J-t, and
we have
7) = 1<^S*^/
'k.M{j-k)..
..{sy.
becanse in the stun SS each teim of D clearly occuis t times :
and taking different groapinga of terms, bat each one only oQce,
we have
= a-i M (j-i) + a-2 M (a-2) + a-3 M {a'i) + eto.
= 3-iJ«'(3-i) + 3aJf(3-a) + 3-3Jf(3-3) + etc.
- i-i if{vi) + a-i Jf (a-i) + 3-1 M(yt) + eto.
t: j-2M{f2) + 2-aM{3-a)-iyaM(ya) + eiba.
- i'3 J«"(i-3) + 2-3 i/"{2-3) + 3-3 Jf (3-3) + etc
,..(5r
in all 2i different expreBsionfi for D,
I^artiier, by tiie elementary law of formation of detenninanta
weaee tliat
JfO--i-4-.)-(-.)-
«+■)•*,
(J+»V*,
O'-")*. 0- =>■(»-
-(■')"
:;. Google
343 6.]
DYNAHICiX LAWS AND PBINCIPLES.
873
The quantities M{ii), Jf (ra) 3£{jk), thus i
are what are commonly called the first minora of the determi- nwL°"
nant D, with jnst this vamtion from ordimuy osage that the
proper sigaa are given to them by the foctor
in 6" so ttiat in the formation of Z) the ordinary ocmpUcation of
alternate positive and n^ative signs irhen t is even and all
signs positive vhen i is odd is avoided. In terms of the nota-
tion (9)' the linear equations (3) become
i-ia,+ i*2a,+ + i*Mi, = 0'
3'io, + 2'aa,+ +2'»o, = 0
»■!», +*'2a, + .
+ tiO(.=
..(5)',
and when V-0, which' Is reqnired'to harmonize them, they
Diay be put under any of the following i different but equivalent
fonns,
JtffTT) 3f(i-2) M{fs)
if(2i)-Jf(a-3)~JF(?55"
if (3-1) 2r^
from which we find
'Jr(3-3)
■■(sn
Jf(iv.) Jf(ra) Jf(3-a)
Jf(,>)"Jf(rO *(3-.)"
if(i-3) Jf(.-3) Jf(3-3).
-air.) y(?i>-if(3-.)~
J
The remarkable relations here shown anlong tlie minors, due Bvbti«u
to the evanescence of the major determinant D, are well known ^^ gt'
in algebra. They are all included in the following formula, ^Tdet^
MUk).M{l-n)^2f{j-n).M(lk) = 0 (6)"", '"'°"*-
which is given in Salmon's Higher Jlgebra (§ 33 Ex. 1), as a
consequence of the formula
M{j-k).M{l-n)-M{jn).M{l-k)=.D.M{j,l-k,n)...[5)'',
where M(j,l-k,n) denotes the second minor formed by sup-
pressing the/'' and P* columns and the H* and n*" lines.
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4 PHBLIUINAHT. [313 C.
343 c. When thei-9 are eqiulitiea among the roots tht
problem has generally solutions of the form
^, = (c,( + 6Ji«, v, = {V + *^«". «**=■ i^l
To ptftve thin let X, X' be two unequal roota which become
equal with some slight change of the values of some or all cj the
given oonstants (ir), 11, ii, (iz), 12, i3, etc.; and let
^, = ^V'-^..", ^,-=J,V''-J^, etc (6)'
be a partlcalar solution of (1) corresponding to these roots.
Now let
c,=.J,'(X'-X), o, = J,'(X'-X), etc
and bi = Ai-~A,, bt=A,' — Aj, eta
Unng tliese in (6)' we find
..(0)-.
*i=V
-<"
v-x
- + i,.», ^,-v
t*"*-."
+ 6,^, etc.. .(6)".
To find propereqoatioua for the relations tunong A„ &„.. .e, , c,, .. .
in order that (6)'" may be a solution of (I), proceed thus : — firat
writ« down equations (3) for the X' solution, with constants A^,Al,
etc; then subtract from these the corresponding equations for
tjie X solution : thus, and introducing the notation {6)", ve find
]{ii)X"+llX'+ii}c, + {(i3)X''+L2X' + i3f,<^ + eto.=0l
{{2i)X'"+2U'+3iie, + {j(a2)A'" + 22X' + 2s},<!i+eto. = oL.(6)'',
eta etc J
{(i,i)A"+llV + ii}fi, + {<i2)X'' + 12X'+r2};6, + etc
= -h-6,(X'-X)]H..)(X + X')tlll
-[c,-4,(X'-X)]H.,»)(XH.X')*12l-
{(ai.)X'' + 2U' + »i}6, + {<32)X" + 22X' +
-w
,(X'-X)]J<3ij(X + V) + 21f
-h-«,(X'-X)])l(3«)(X + V) + 22}4.eto.
...(6)-.
Equations (6)" require that X' be a loot of the detenninaat, and
t— 1 of bbem determines— 1 of the quantities e,,^, etc in terms
of one of them assumed arbitearily. Supposing now c,, c^, etc
to be thus all known, the i equations (6/ fail to determine the i
quantities fi„ i„ etc in terms of the right-hand memben
because X' is a root of the determinant. The two sets of
equations (6)'' and (6)' require that X be also a root of Uie de-
torminant ; and i — 1 of the equations (6)' determine t — I of the
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313 c] DYNAMICAL LAWS AND PBINCIPLE3. 875
qu&ntitiee ft,, &,, eto. in temu of o^, e,, otc. (sapposed already CiMef
known as above) and a properljr assumed value of one of the b'a.
343 d. When A' is infinitely nearly equal to X, (6)'" becomes
infinitely nearly the same as (6), and (6)" and (6)^ beoome in
terms of the notation (0)'
I'lo +i-2C,+ etc, = 0^
a-ic, + 2ac,+ eto. = ol (6)",
eto. crtal
dl-2
I J, + 2'aft- + etc =
rfi'i (fa '3
'"dX"'^ dK "
Theee, (6)", (6)*", are dearly the equations which we find
simply by trying if (6) is a solution of (1). (6)^ requires that \
be a root of the determinant D; and they give by (6)" with e
subetituted for a the valueS'Of i- 1 of die quantities c,, e,, etc.
in terms of one of them assumed arbitmrily. And by the way
we have found them we know that {6)'" superadded to (6)"
shows that X must be a dual root of the determinant. To verify
this multiply the first of them- by Jf(i-i), the seoond by
M{2'i), etc, and add. The coefficients of b„ b^, etc in tlie sum
are each identically zero in< virtue of the elementaiy oonstitntitm
of determinants, and the coeffioient of i, is the major determinant
D. Thus irrespectively of the vatUe of X we find in the first place,
/».-.. {if(.Ol^*if(.-.)^+.to.|
Now in virtue of (6)^ and (B)" we bare
Using sncoessively the several expressions given by (5)^ for
these ratios, in (6)*^, and putting Z> = 0, we find
which with 0 = 0 shows that X is a double root.
Suppose now that one of the e's has been assumed, and tiie
others found by (6)" : let one of the b'e be assumed ; the other
..Google
3 piiELiHiKAnr. [343 d.
i- 1 ('scire to be calculated by (-1 of the equations {6}'". Thus
for example take d, = 0. In the iirat phwe use all except the
fiiat of equations (6)" to determine &„ &,, eto^ : we thus
find
ilf(i-i)J,--|jf(i,2i
-(jf(i,s-r
,<fj-a
Jf{ii)i,.oto. Jf(ii)6..etc.
..(6f.
Secondly, use all except the second of (6)'" to find h„ 6„ etc :
we thus find
jr{2-i)6, = etc, Jr{a-i)ft, = etc, Jir(ri)6, =6tc. (6)-.
Thirdly, by utdng all of (e)*" except the tliird, fourthly, all
except the fourth, and so on, we find
J/(3-i)6, = etc., Jf(3-i)ft, = etc., itf(3-i)6,=etc {6)".
843 e. In certain cases of equality among the roota (343 m)
it is found that values of the coefficients (ii), 11, ji, etc.
differing infinitely little from particular values which give the
equality give values of a, and'a,', a, and a,', etc, which are
not infinitely nearly eqdaL to. such cases we see b;^ {€)" that
\, %, etc. ai^ finite, and e,, c„.etc. vanish : and so the solution
doesnot contain terms of the fonnff^: but the requisite number
of arbitraiy constants ia made up by a proper degree of inde-
terminatenesa in the residuary equations for tiie ratios &,/6,,
*./6„ etc.
Now when fl, = 0, e, = 0, etc. the second members of equa-
tions (6)'', (6)^, (6)^, etc. all taitisb, and as \, &„ \, eta do not all
vanish, it follows that we have
Jf(ri) = 0, Jf(a-i) = 0, J/-(3-i) = 0, etc (e)-".
Hence by (5)*" or (5)"" we infer that all the first minors are
Eero for any value of X which is doubly a root, and which yet
does not give terms of the form <«*' in the Bolatioo. This
important proposition is due to Bouth*, who, escaping the errors
of previous writers (g 343 m below), first gave the complete
theory of equal roots of the determinant in cycloidal motion.
■ SUMMI}! «/ Modon (AdamB Pme Eiwsy for 1877), oh»p, «. i «.
..Google
343 «,] DYNAMICAL LAWS AND PRIKCIPLBS. S77
He aUo remarked that the factor { does not necessarily imply South'*
instability, as terms of the form (c"^, or W^ cos (n< - e), when p
is positive, do not give inatftbility, but on the contrary corre-
spond to non-oscillatoiy or oscillatory Bubsidence to equilibrium.
313^ We fall back on the case of no motional forces by Ouaotno
taking 11 = 0, 12 = 0, etc., which reduces the oquationa (3) for iKS^
determiuing the ratios a, / a, , a, / a, , et«. to
a't— + iia, + i2ii.+ etc=0, a"^— + 2ia,-t- 22a, + eito. = 0,etc. (7),
oa, ' ' ew, ' ' I \ i>
or, expanded,
[(ri)V-ni]a, + [(i2)X'-i-i2]a. + etc = 0l
[(2i)X' + 2i]a, + [{22)A' + 2a]o,-i-etc. = 0/ ■■■^ ''
The determinantal equation (4) to harmonize these amplified
equations (7) or (7') becomes
|{ii)V + ii, (,2)V+i2,...| = 0 (8).
(2l)A' + 2I, (22)X*-f- 22, ...
This is of degree i, in X'; therefore X has t pairs of oppositely
signed equal values, vhich we may noir denote by
*X, -tV, *X", ... ;
and for each of these pairs the series of ratio-equations (7') are
the same. Hence the complete solation of the differential equa-
tions of motion may be written as follows, to show its arbitraries
explicitly : —
1^, = (^€« + Bt-»)+ (-1V'+ £'«-*'') + (4V'+J"f-*'')+etc.
^, = ^(J«« + J«-«)+^!('*'«*'* + B'«-*'<)+^!(^V*+5"«-*'')+etc.
where A, B; A', H; A", B"; etc. denote 2t arbitrary constants,
and
are i seta of i- 1 ratios each, the values of which, when all the
i roots of the determinantal equation in X' have different values,
are fully determined by ^ving Euccessively these i values to X'
jiGoogle
3'S PBEUinyABT. [343 ^.
OntfM S13;. When tbcn ue eqail roots, the solutint is to be
teas. eomidcied scconling to § 343 (2 or e, as the esse may be. The
caseofsc(Hiaervative8jstaD(343i) 06068881117 &l]saDder§ 343 «,
as is [HOTed in § 3(3 m. The same form, (9), still represenla the
complete stdntioDS when there are equalises (unong the roots, but
with dianged oonditioiis as to arbitrarioees of the el^nents appear-
ing in it, Snppoae X* = X'* for example. In this case any value
maj be choeai arfaitnriljr for i, / a, , and the remainder of the
set o, 'o,, s,/s, ... are then fully determined hy (7'); again
another value may be chosen for a^fa^', and with it a,'/a^\
a,' a^'f ... are determined by a fresh application of (7') with
the same raloe for X'; and the arbitraries now are A +A',
B + ff,
^A+^,A', '^B + ^ff, A", B", A"; £r\ ...J('-», and 5C-'t
a, a, a, a,
numbering still 2i in all. Similarly we see how, b^tudng
with the form (9), convenient for the genenU case of i different
roots, we have in it also the complete solution when X* is triply,
or quadruply, or any number of tames a root, and when aoy
other root or roots also ai» double or taultiple.
O^doidd 343 A. For the cose of a conservative ^stem, that ifi to say,
Cf»aiKnx' the case in which
JjJ^Jj;^ ia = ai, 13=31, 83 = 32, et<i, etc. (10),
the differential equations of motion, (I), become
KD^f-'S©^^'-.- w.
and the solving linear algebraic equations, (3), become
?-?-«. ?^?-« en
where
r-J(iH^/+3.i»^,lfr,+etc.),andf =J(ira,»+2.i2O,a,+etc.)...(10'").
In this case the i roots, X', of the determinantal equation ore the
negatives of the values of a, y3, . . . of our first investigation ; snd
thus in (10"), (6), and (9) we have the promised solution by one
completely expressed process. From § 337 and its footnote we .
infer that in the present case the roots X' aro all real, whether
negative or positive.
..Google
343 h.] DTNAHICAL LAWS AND PBINCIPLES. 37i*
la § 337 it vaa ezpresalj assumed that T (aa it must be in <^daidu
the dynamical problem) is essentially positive; but the inveBtigS' ConmrK-
tioQ was equally valid for any case in which either of the quad- tkaul, aod
r&tics Tot Via incapable of changing sign for real values of the at, timet.
variables (^,, ij/,, etc for T, or ^,, ip,, etc. for F). Thus wo
see that the roots X' are all real when the relations (5) and (9)
are satisfied, and when the magoitudea of the reeidual indepen-
dent coeffioientB (it), (aa), (12), ... and 11, 22, ii, ... are such
that of the resulting quadratics, 0, V, one or other is essen-
tially positive or essenti^y negative. This property of the
determinantal equation (7*) is very remarkable. A more direct
ivlgebraic proof is to be desired. Here is one : —
343 k. Writing out (7') for A', and for A", multiplying the
first for A' by Ju/, the second by }a,', and so on, and adding;
and again multiplying the first for A'' by Ja„ the second by Ja„
and so on, and adding, we find
X"J((a, .O + PK a'l.O)
and i."Zia, o') + ir(o, »').0/ '■"'•
wbere
«C(a, a')=i{(ii)«,<+(")(»i«.'+«.».') + et*^}) aS\.
and U(«, a')=J{ " «.».'+ " (<*,«,' + o,a,') + «*«■}/"
Remark that according to this (12) notation V (a, a) means
the same thing as S simply, according to the notation of (3) etc
above, and S (iff, ^) the same thing as T, Remark farther that
V (a, a') is a linear functioQ of a,, a,, ... with coefficients each
involving a,', a,', ... linearly; and that it is symmetrical with
reference to a,, a,', and o,, a,', etc.; and that we therefore
have
©(mp, p') = mK{p, j/) = ®(P> "*?") "^** ) (13)
Z(mp + ny,m'p+n'q) = mm'V{p,p)+{mn' + m'n)'Z{p,q)+nn"S{j,q))
Precisely cdmilar statements and formulas hold for V (a, a').
From (11) we infer that if A* and A" be unequal we must
have
B(a, (0 = 0, and V(a, a') = 0 (U).
Now if there can be imf^;inary roots, V, let \'=p+ir^~i
and A"=p— <r»/-l be a pair of them, p and cr being real. And,
Pi> l\tPt' ?!» ^^- being all real, letj>, 4-g'|y-l, p,-q,J-i, be
arbitntrily chosen values of a,, a/, and let
..Google
380 PRELIMINARY. [343 k.
be the determiiiately deduced valuea* of a,, a,, ..., a,', a,', ...
according to (7'); we have, by (13), with
i m = m'=l, n= J- ] , n' = - ^—1,
Z.{a,a-) = 'B(p,p) + €(q,q)) _
and V(a,a')=Yr{p,p)-t.V{q,q)i ^"'■
Kow hy hjrpothesis either V {x, x), or V(x,x) is essentially of
one sign for all real values of x^, a;,, etc Hence the second
member of one or other of equations (14') cannot be zero, because
P). P,i-> *•*'! ?ii 9,>— *™ ^ '^- ^'** ^7 (**) *^ fi™'
member of each of the equations (14') ia zero if A' and A" ue
unequal: hence they are equal: hence either ^, = 0, ^,=0, etc.,
or 7, = 0, 7, = 0, eta, that is to say the roots X' are all necesBarily
real, whether negative or positive.
343 I. Farther we now see by going back to (II) : —
(a) if for all real raluee of a:,, z,,.-- the values of S (ic, x)
and V {x, x) have the same unchanging sign, the roota X' are alt
nc^tive ;
{b) if for different real values of «,, x^ etc, one of the two
Vi{x, x), V{x, x) has different signs (the other by hypothesis
having always one sign), some of the roots A' are negative and
some positive;
(c) if the values of V and V have essentially opposite signs
(and each therefore according to hypothesis unchangeable in
uign), the roots X' are all positive-
The (a) and (c) of this tripartite conclusion we see by taking
X" = X' in (11), which reduces them to
K'Z{a,a) + V(a,a) = 0 (15),
and remarking that a„ a,, etc. are now all real if we please to
give a real value to a,. The (b) is proved in g 343 o below.
343 m. From (14) we see that when two roota X*, X", are
'' infinitely nearly equal there is no approacji to equality between
a, and a,', a, and a,', and therefore, when there are no motional
forces, and when the positional forces are conservative^ equality
of roots essentially falls under the case of § 343 e abova This
may be proved explicitly as follows ; — let
f, = (a/ + 6,)««, ^,= (o^ + &,)t«, etc (16)'
* Caaea of egnolitiea among the roots are diareguded for the moment merel.T
to Avoid drcumlocntiona, but (he; obTiousl; form no oicsptioa to the reaatming
aad oonclnsiou.
..Google
343 m.]
DTNAMICAL LAWS AND PEINCTPLES.
bo the complete solution coTres]x)ndiiig to the root X supposed to ^^^*'
be a dual root UsiDg this in equations (1) and equating to zero ^J^JJ^
in each equation so found the coefficients of t^ and of i^, with ''™^,|^
the notation of (12) we find aUlonM.
^,dV(b,b)^ ^^dZ(a,a) ^dV{b,b)
db da, t», *
= 0,ete.....(lfi)'',
,.<16)"'.
£W, da, di^ J
MoItiplTing the fiist^ second, third, etc. of (IS)" by 6,, bp bp etc
and adding we find
X'Z{a,b) + V{a,b) = 0 (1B)^
and nmilarly from (16)"' with multipliers a,, a^ etc.
k'Z{a,b) + tr {a,b) + 2XK{a,a) = 0 (15)'.
Subtracting (15)''from (15)' we see that C(«,o) = 0. Hence we
must have a, == 0, a, = 0, etc., that is to say there are no terms of
the form fc" in the solution. It is to be remarked that the in-
ference of a, =0, a^ = 0, etc. from V (a, a) = 0, is not limited to
real roots X because A.' in the present case is essentially real, and
whether it be poeitiYe or negative the ratios «,/«,. o^",! etc., are
essentially real.
It ia remarkable tliat botb Lagrauge and Laplace fell into
the error of supposing that equality among roots necessarily
implies terms iu the solution of the form ^(or tcoapt), and
therefore that for stability the roots must be all unequal. This
■we find in the M^nigue Analytique, Seconde Partie, eectioD VI.
Art 7 of the second edition of 1811 published three years before
Lagrange's death, and repeated without change in the posthu-
mous edition of 1853. It occurs in the course of a general
solntion of the problem of the infinitely small oscillations of a
system of bodies about their positions of equilibrium, with
conservative forces of position and no motional forces, which
from the ' Avertissement" (p. vi.) prefixed to the 1811 edition
seems to have been first published in the 1811 edition, and not
to have appeared in the original edition of 1788*. It would be
* E&iM this sUtement iras put in tjfe, the first edition of the MeeanigM
AmU^tique (which bad been inqnired for in yain in the UniTersity librarien of
Cambridge and Gbugow) has been loDnd in the UniverBitj library of EdiDbiirgli,
..Google
382 PREUMINART. [343 ni.
CydeMJ curious if such an error had remained for twenty-three years in
ooraiMi- Lagrange's mind. It could scarcely have eziEted even during
tiBoS^tni the writing and printing of the Article for his last edition if he
BLtoTOM. had been in the habit of considering particular applications of
his splendid analytical wort : if he had he would have seen that
a proposition which asserted that the equilibrium of a particle
in the bottom of a frictionless bowl is unstable if the bowl be
a figure of revolution with its axis vertical, cannot be true.
Ko such obvious illustration presents itself to suggest or prove
the error as Laplace has it in the Micanique Celeste (Premiere
Partie, Livre n. Art 67) in the course of an investigation of the
secular ineqitalities of the planetary syetem. But as [by a
peculiarly simple case of the process of § 345* (54)] he has
reduced bis analysis of this problem virtually to the same as
that of conservative osdllations about a configuration of equili-
brium, the physical illustrations which abound for this case
suffice to prove the error in Laplace's statement, different and
comparatively recondite as its dynamical subject is. An error
the converse of that of Laplace and Lagrange occurred in page
278 of our First Edition where it was s^d that "Cases in which
" there are equal roots leave a corresponding number of degrees
"of indeterminatenesa in the ratios I :m,l -.n, etc., and so allow
" the requisite number of arbitrary constants to be made up,"
without limiting this statement to the case of conservative
positional and no motional forces, for which its truth is obvious
from the nature of the problem, and for which alone it is obvious
at first sight; although for the cases of adynamic oscillations,
and of stable precessions, § 345^ it is also essentially true.
The correct theory of equal roots in the generalized problem
of cycloidal motion has been so far as we know first given by
Routh in his investigation referred to above (§ 343 e).
343 M. Ketuming to § 343 ^ to make more of (6), and to
underBtand the efficiency of the oppositely mgned roote, X*, as-
aerted in it, let (r*=-A" in any caae in which X' ia nqjative, and let
•^1 -*■,«« {erf -e), ^, = »-.coa{<Tt-e), etc (16),
be the OOTresponding particular solution in fully realised terma,
jmd it doea «,nWn the problem ol inBnitely nD«U owiUationa, with th«
renurkalile error referred to in ihe text.
..Google
343 n.] DTNAHICAL LAWB AND FBINCIPLES. SS3
as in S 337 (6> aibova but with aomewhat different notation. CycWitol
* , ^ motion.
By Bubstituting in (1) and mnltiplyiug the first of the resulting Comeir*-
equations by r, , the second by r,, and so on and adding, virtuaJly tioui, uid
aa we found (15), we now find ai.rDrMt
-o'Z{r,r) + T(r,r) = 0 (17).
Adopting now the notation of (9) for the real poeitLve ones of
the roots X* but taking, fca- brevity, a, = 1, o,' = 1, a" = 1, etc.,
we have for the complete Holntion when there are both negative
and positive roots of the determinantal equation (7');
^,= {A^+Br»)+ {dV*+-B'.-*''>+etc.-M-,cos(<rt-e)+r,'oo8(ff'i-e')+etc.\
V',=oj^««+J€-*0+a,V'«*''+^«'*^)+etc.+r,coB(<r(--«)+r,'coB(<r'i--e>etc.>-..(18).
ifr, = otc, it, = etc etc. etc )
343 0. TTaing this in the general expressions for T and V,
with the notation (13), and remarking that the products <^ xi*'',
etc. and i** x sin (iri — e), etc., and sin (vt— e) x sin (<r't — e'), etc.,
disappear from the terms in virtue of (II), we find
y= \'e(a, a)(^.*'- £<-«)• + X-^ («', a')(JV''-5'.-«)V etc^
+ a*Z (r. r) an' {<rt-e) + cr*^ {/, r') sin' (a't - a) + etc ) ^ ''
and
+ IT (r, r) cob' (<r(-«) + 17 (r', races' (</*-«') + etc j ^ ''
Hie factors which appear with
O (a, a), «t (»', aO, ■ ■ ■ C {r. r), ® (r', r^
in this expression (19) for 7 are all easentiimy positiTe; and the
same is true of 57" in (20) for V. Now for every set of real
co-ordinatefl and Telocity-components the potential and kinetic
eueigies are expressble by the formulas (30) and (19) because
(18) is the complete solution with 2t arbitraries. Hence if the
value of V can change sign with real valuee of the co-ordinates,
the quantities IT {a, a), V (a', oT), etc, and V (r, r). YT (/, r'),
etc, for the several roots mnst be some of them positive and
some of them negative; and if the value of T could change sign
with real valoea of the velocity-components, some of the quan-
tities ?C (a, o), O («', a% etc, and B (r, r), % (/, r'), etc. would
need to be positive and some negative. So much being learned
from (20) and (19) we must now ruoal to mind that according
to hypothesis one only of the two quadnitics T and 7 can change
logn, to oondude from (15) and (17) that there are bol;h positive
and negative roots A.' when either 7 or F can change sign. Thns
(6) of the tripartite oraiclurion above is rigorously proved.
..Google
384 PRELIHIHART. [343 p.
CroMdd 343 p. A ehort algebraic proof of (() could no doubt be easily
ConiOTn^ given ; but our Bomewliat elaborate discussion of the subject is im-
SSJ^TImi portant as sbowiug in (15). ..(20) the whole relation between
^tavMb' fi* previous short algebraic investigation, conducted in terms
inToIving quantities which are eeaentially imaginary for the
case of oscillations about a configuration of stable equilibrium,
and the fully realized solution, with fonnulas for the potential
and kinetic energies realized ixtCh for oaoillatitms and for
fallings away from unstable equilibrium.
We now see definitively by (15) and (17) that,infwa/dynamica
(that is to say T essentially positiTe) the factors V(a, a),
V(a', a"), et«., aro all negative, and B'(r, r), !?■(/, r^, etCL, all
BqatUflD o( positive in the expressioa (20) for the potential tiaeitgy. Adding
^^Sd" <20) to (19) and using (15) and (17) in the sum, we fiud
EJJS^a. T+r=-iA£X.'Z(a,a)-iA-B'k'^{a',a'),eU:. \
+ <r^(n r) + <r"E(/, O +eto. /-(-"J-
It is interesting to see in this formula how the constancy of
the sum of the potential and kinetic energies is attained in any
solution of the form At^ + Bt~^ [which, with \ = <rj—l,
includes the form r oos (cr( - e)], and to remark that for any single
Soluti<Ha (M^, or solution compounded of single solutions depend-
ing on unequal values of X' (whether real or imaginaty), the sum
of the potential and kinetic energies is essentially lero.
ArtiBcU or 3^^ When the positional forces of a system violate the law
li^nUiT* of conservatism, we have seen (§ 272) that energy without limit
"""^ may be drawn from it by guiding it perpetually through a
returning cycle of configurations, and we have inferred that in
every real system, not supplied with energy from without, the
positional forces fulfil the conservative law. But it is easy to ar-
range a system artificiaUy, in connexion with a source of energy,
so that its positional forces shall be non-conservative ; and tlie
consideration of the kinetic efTects of such an arrangement, es-
pecially of its oscillations about or motions round a configura-
tion of equilibrium, is moat instructive, by the contrasts which it
presents to the phenomena of a natural system. The preceding
formulas, (7).. .{9) of § 343/ and § 343 5, express the general
solution of the problem— to find the infinitely small motion of a
cycloidal system, when, without motional forces, there is devia-
tion from conservatism hy the character of the positional force*.
..Google
344.] DTNAMICAI. LAWS AND PRINCIPLES. 385
In this case [(10) not fulfilled,] just as in the case of motional ^''^^
forces fulfilling the conservative law (10), the character of the Bi"nui«a»8
equilibrium as to stability or instability is discriminated accord- criterion or
ing to the character of the roots of an algebraic equation of
degree equal to the number of degrees of freedom of the system.
If tlie roots (X') of the detenoiiumtal equation g 343 (8) are
all real and n^ativ^ the equilibrium is stable : in every other
case it is unstable,
346. But although, when the equilibrium is stable, no
possible infiaitely small displacement and velocity given to
the system can cause it, when left to itself, to go on moving
farther and farther away till either a finite displacement is
reached, or a finite velocity acquired ; it is very remarkable
that stability should be possible, considering that even in the
case of stability an endless increase of velocity may, as is easily
seen from § 272, be obtained merely by constraining the system
to a particular closed course, or circuit of configurations, no-
where deviating by more than an infinitely small amount from
the configuration of equilibrium, and leaving it at rest anywhere
in a certain part of this circuit This result, and the distinct
peculiarities of the cases of stebtlity and instability, will be
sufficiently illustrated hy the simplest possible example, that of
a material particle moving in a plane.
Let tb« mass be unity, and the components of force parallel
to two rectangular axes be fix + by, and a'x + h'y, when the
position of the particle is {x, y). The equations of motion
will be
i = ax^by, y = a'x + b'y ....(1).
Let J(a' + 6) = <!, and J(a'-ft) = e:
i tlie force become
+ cy-ty, and cx + b'y + ex,
dV
- -J- + ex,
dy
where F = - J (ax' + 6'/ + Icxy).
The terms - ey and -»■ ex are clearly the components of a force
«(3^ + y*)', perpendicular to the radius-vector of the particle.
Hence if we turn the axes of oo-ordinates through any angle, the
VOL I. 25
..Google
6 PBEUMIKART. [345.
cotreBpondiug t«rms in the transformed compouenta ore still
- ey and + ex. If, therefore, we chooee the axes so that
r=i(aa^ + /3y0 (2),
the equations of motion become, without loss of generality,
x^-ax-ey, y = -fy + ex.
To integrate these, assume, as in general [^ 343 (2)},
a: = i«", y = wm**.
Then, as before [§ 343 (7)],
{X' + a)i + «n = 0, and -rf+ (X* + ^)m = 0.
Whence (V + a)(X' + /3) = -e' (3),
which gives
*■ — } ("+«'■ {K—«' -<■!'■
This shows that the equilibrium is stable if both ajS-t-e* and
a + 0 are positive and e" < J (a - /9}* but unstable in every other
But let the particle be constrained to remain on a circle, of
radius r. Denoting by 0 its angle-vector fi'Om OX, and trans-
forming (§ 27) the equations of motion, we have
(l = ~(0-a)8in«cos*-l-e = -J(/3-a)ain2fl+8 (4).
If we had e = 0 (a conservative system of force) the positions of
equilibrium would be at 6 = 0, 6 — ^r, 9 — «-, and ^ = |x; and
the motion would be that of the quadrantal pendulum, Bnt
when e has any finite value less than } 03 — a) whidi, for conve-
nience, we may suppose positive, there are poaitlonB of equili-
brium at
tf = 9, # = ^-a, 0 = w + S, and 0 = — -^,
where b is half the acute angle whose sine is -= : the Erst and
third being positions of stable, and the second and fourth of un-
stable, equilibrium. Thus it appears that the effect of the con-
stant tangential force b to displace the positions of stable and
unstable equilibrium forwards and backwards on the circle
through angles each equal to b. And, by multiplying (4) by
26dt and integrating, we have as the int^pul equation of energy
^^C+i(^-a)co62e + 2e$ (5).
..Google
345.] DYNAMICAL LAWS AND PRINCIPLES. 387
From this we see that the valne of £7, to make the particle A^f**^ "
just reach the position of unstable equilibrium, is oumnUtiTe
and by equating to zero the expression (5) for $*, with this value
of C Bubfltituted, we have a transcendental equation in 0, of
which the least negative root, $^, givra the limit of vibrations on
the side reckoned backwards from a position of stable equilibrium.
If the particle be placed at rest on the circle at any distance less
than n ~ ^ before a position of stable equilibriimi, or less than
S - fi, behind it, it will vibrate. Bat if placed anywhere beyond
those limits and left either at rest or moving with any velocity
in either direction, it will end by flying round and round
forwards with a periodically increasing and diminishing velocity,
but increasing every half turn by equal additions to its squares.
If on the other hand e>^{fi-a), the positions both of stable
and unstable equilibrium are imaginary ; the tangential force
predominating in every position. If the particle be left at
rest in any part of the circle it will fly round with continually
increasing velocity, but periodically increasing and diminishing
acceleration.
345'. Leaving now the ideal case of positional forces violat-
ing the law of conservatisni, interestingly curioufi as it is, and
instructive in respect to the contrast it presents with the
positional forces of nature which are essentially conservative, let
us henceforth suppose the positional forces of our system to be
conservative and let us admit infringement of conservatism only
as in nature through motional forces. We shall soon see (§ SiS*
and ^) that we may have motional forces which do not violate
the law of conservatism. At present we make no restriction cjdtJd*!
■^ ... ijiwin with
upon the motional forces and no other restriction on the poai- ""•'Jj^-
tional forces than that they are conservative. mdun^'"'
THtTiclrd
The differential equations of motion, taken from (I) of 343n f^^^"*'
above, with the relations (10), and with V to denote the potential
energy, are,
25—2
..Google
PRELIMINAET. [345*.
ttonalfuniu
Hid nun-
(tricted
mottonal
fHoea. etc.
Multiplying tlie first of these by ^,, the seoond by ^„ adding
and tnmsposing, we find
^^=-« m.
G=ll^.' + (12+21)f^,+ 22f/ + {I3 + 31)\t,^, + eto. (3).
S46". The quadratic fuDctJon of the velocities here denoted
b; Q has beeD called by Lord Bayleigh* the Dissipation Fudc-
iHuipK- tion. We prefer to call it Disaipativity. It expresses the rate
tiuS! at which the palpable energy of our supposed cycloidal system is
lost, nut, as we now know, annihilated but (§§ 378, 340, 341,
342) dissipated away into other forms of energy, Itisesseniially
^wd positive when the assumed motional forces are such as can exist
glJ"™" -' in nature. That it is equal to a quadratic function of the velo-
tivtv. cities is an interesting and important theorem.
Int«m) Multiplying (2) by dt, and int^^ratiog, we find
T+r.
■K-j'^Qdt (4),
where E^ la a, congtont denoting the sum of the kinetic and
potential energies at the instant 1=0. Now F and Q are each
of them eesentially [Kwitive except when the system ia at rest,
and then each of them is zero. Therefore I Qdt muBt increase
to infinity nnless the system comes more and more nearly to rest
as time advances. Hence either tliis must b« the case, or I'
must diminish to — co . It follows that when V ia positive for all
real values of the coK>rdiuateB the system must as time advances
come more and more nearly to rest in its zero-configuration,
whatever may have been the initial values of the 4x>-ordinates
and velocities. Even if F is negative for some or for all values
of the co-ordinates, the system may be projected from tome giveti
* Proeetdingi of the London Mathematical Sorletg, Hajr, 1878; Thtoiy of
Sound, Vol. 1. 3 81,
..Google
3i5".] DYNAMICAL LAWS AND PRINCIPLES. 389
eonfigurafiont with such Telocitiee that when ( = oo it shall be ^rtmwith
at reat in its zero confignration : this we see by taking, as a ^J^JJ;
particular solution, the terms of (9) § 345'' below, for which m is ^ ''"°"
negative. But this equilibrium is essential!; unstable, unless ^^^^
is poeitive for all real values of the co-ordiuatea. To prove this tmaa.
imagine the system placed in any configuiation in which F is
negative, and left there either at rest or with any motion of
kinetic euei;gy less than or at the most equal to — F: thus £^
will be negative or aero; T+V will therefore have increasing
negative value as time advances; therefore V must always re-
main negative ; and therefore the system can never reach its
zero configuration. It is clear that — F and T must each on the
whole increase though there may be fluctuations, of T diminish-
ing for a time, during which — F must also diminish so as to
make the excess (- F) — T increase at the rate equal to Q per
unit of time according to formula (3).
34S'". To illustrate the circumstances of the several cases let
X=in-f-n^-l be a root of the determinantal equation, m and n
being both real The corresponding realised solntion of the
dynamical problem is
^i = r,t^cos(it( — ej, ^, = r,f"'cos («( — «,), etc (5),
where thedifierences of epochs >,-«,, «,-«,> etc. and the ratios
r, / r„ etc., in all 2t - 2 numerics *, are determined by the
2i simultaneoos linear equations (3) of § 343 harmonized by
taking for \='m^nj-l, and agcun \ = m-nj-l. Using
these expressions for i^,, tjr^ etc. in the expressions for F, Q, T,
F = ^{C + AcoB2nt + B^n2nt) \
Q = ^^{C'■^-A'coi2nt■^B'an2nt) V (6),
r = I*" (C + ^" cos 2n/ + 5" sin 2n() )
* The term nimieria ban been reoeutly introdiicad by ProfesBor James Tbom*
son to denote a nmuber, or a proper fraotion, or an imptoper traction, or an
ineommeDBDrable ratio (anch as r oi t). It miut also to be oaehil in mathe-
mnt'"^! analTaU ioelnde ima^ary ezpiesnons raoh as m+n i^-l, wheie
m and n aia real numerics. " Nnmerio" may be regarded as an abbreviation
lor " Qomerical expression." It lets na avoid the intolerable verbiagg of int^er
or proper or improper IractLon whieli ntatiiematiaal writers hitherto are so often
compellad to use; and is more appropriate for mere number or ratio than the
deitigiiatioa "qmmtit;," vhioh rather implies quantity of something than the
mere nnmerieal expresdon by nhioh quantities of any measurable things are
reokoned in teims of the unit of quantity.
..Google
0 PRELIHINABT. [345*^.
where C, A, B, C, A', ff, C", A", B", are determinate cocstanlB :
and in order th&t Q and T may be positive we have
C> + J{A'* + S").sndCr'=' + J(A"**B"') (7).
Substituting these in (2), and equating coefficients of corre-
sponding terms, we find
2m{C + C")^-C\
2{m(A+ A") +« (B*B")\--Ai (8).
2{m(B+B")~n{A*A")\^~B)
The first of these showa that C+C" and m must be of contraiy
signs. Hence if F be essentially positive [which requires that C
H^" be greater than +^(il' +5')], every value of m must be n^ativ&
346". If V have negative values ior some or all real values
of the coordinates, m must clearly be positive for some roots, but
there must still, and always, be roots for which m is n^ative.
To prove this last clause let us instead of (5) take sums of par-
ticular solutions correspondiug to different roots
\ = m*^nj-l, X' = m'*n' J- I, etc.,
m and n denoting real numerics. Thus we have
^1 = T^^ 008 (nt - e,) + /i*"'' cos («'*—«',) + etcj
^1 = '"i*^ cos (nf - e j + Z^""' oos {n't— e' j + etc.[ (9).
ttc. )
Suppose now m, m', etc. to be all positive ; then for ( " - oo , we
should have i^,=0, ^,=0, ^,=0, ^,=0, etc., and therefore F^O, T=Q.
Hence, for finite values of (, T would in virtue of (4) be less
than — V (which in this case ia eaeentially positive) : but we
may place the system in any configuration and project it with
any velocity we please, and therefore Uie amount of kinetic
energy we may give it is unlimited. Henoe, if (9) be the com-
plete solution, it must include some n^;ative value or values of
m, and therefore of all the roots X, \', etc there must be some of
which the real part ia negative. This conclusion is also obvious
on purely algebraic grounds, because the coefficient of X**-* in
the determinant is obviously 11 + 22+ 33+..., which is essentially
positive when Q is positive for all real values of the co-ordinates.
345'. It is an important subject for investigation, interesting
both in mere Algebra and in Dynamics, to find how many roots
there are viiix m positive, or how many with m negative in any
particuhtr case or class of cases; also to find under what con-
..Google
lUlingkwkr
345'.] DTNAMICAL LAWS AND PHINCIPLES, 391
ditions n disappears [or the motion non-oscUUtorf (compare Non-oKiil-
§ 341)], We hop© to return to it in our second volume, and
should be veiy glad to find it taken up and worked out fully by
mathematicianB in the mean time. At present it is obvious that irDrn'oa-
if Fbe negative for aU real values of iji„ ^„ etc, the motion must otdiiBtotr
be Qon-OBcillatory for every mode (or every value of X must be ™^^°°
real) if Q be but large enough : but as ve shall see immediately ^|""' „.
with Q not too larffe, n may appear in some or in all the roots, ^liiwavay
even though V be negative for all real co-ordinates, when there >t»bie.
are forces of the gyroscopic class [% 319, Examp. (G) above and »«» Som
§ 343' below). When the motional forces are wholly of the ^tobi/equi-
viscous class it is easily seen that » can only appear if F is
positive for some or all real valaes of the co-ordinates : n must I^'ttnK
disappear if F is negative for atl real values of the co-ordinates irtS^"^
(agwn compare § 341). 'l^™^
345". A chief part of the substance of §§ Sio" ...345'
above may be expressed shortly without symbols thus : — When
there is any dissipativity the equilibrium in the zero position is
stable or unstable according as the same system with no motional Bkbiut; at
forces, but with the same positional forces, is stable or unstable, vstem. *
The gyroscopic forces which we now proceed to consider may
convert instability into stability, as in the gyrostat § 345' below,
when there is no dtssipativity : — but when there ia any dissi-
pativity gyroscopic forces may convert rapid falling away from an
unstable configuration into falling by (aa it were) exceedingly
gradual spirals, but they cannot convert instability into stability
if there be any disaipativity.
The theorem of Dlssipativity [g 345', (2) and (3)] suggests the
following notation, —
i(12 + 21)=[i2]or[2i], H13-i-31) = [i3]or[3i],etc.^
and i(12-21)-ij]or-2i], ni3-31)=t3]or-3r],etcP ''
so that the symbols [12], [21J, [13], etc., and 12], 21], 13], etc.
denote quantities which respectively fulfil the following mutual
relations,
["l = [ai], [i3] = [3']. I23] = [3al, et«) ,„,
i3]=-2i], i3l = -3il. 23l = -3a]. «*«■/ ^ ''
Thus (3) of § 345' becomes
C=ll^,'+2[i2l^,^,+ 22^,'-i-2[i3l^,^,-»-etc. (12),
..Google
2 PBELIIONABT. [SiS^.
and going back to {1), with (10) aod (12) we hare
ddT dQ , , 7 . dF
d dT dQ
T — T + : +
dt dip, (Vi
: +2i]^, + a3]^, + et«. + -^
..(13).
In these equations the tertna 12]^^ 2i]^,, 13] -^j, 3i]'^„
JSjlSirof etc, reprcBent what we may call gyroscopic forces, because, as we
g^^"" have seen ia § 319, Ex. G, they occur when fly-wheels each ^ven
in a state of rapid rotation form part of the system by being
mounted on frictionless bearings connected through framework
with other parts of the system ; and because, as we have seen
in § 319, Ex. F, they occur when the motion considered is
motion of the given system relatively to a rigid body revolving
with a constrainedly constant angular velocity round a fixed
axis This last reason is especially interesting on account of
Laplace's dynamical theory of the tides at the foundation of
which it lies, and in which it is answerable for some of the most
curious and instructive results, such as the beautiful vortex
problem presented by what Laplace calls " Oscillations of the
First Species*."
3^"'. The gyrostatic terms disappear from the equation of
energy as we see by § 345', (2) and (3), and as we saw pre-
viously by § 319, Example G (19), and in § 319, Ex. F (/).
Comparing § 319 (/) and (3), we see that in the case of motion
tmm- relatively to a body revolving uniformly round a fixed aiis it is
not the equation of total absolute energy but the equation <^
cnei^ of the relative motion that the gyroscopic terms disappear
from, as (/) of § 319; and (2) and (3) of § 345' when the
subject of their application is to such relative motion.
* The integrated eqaation for tbia apeciee ot tidftl motiona, in an ideal OMtn
equally deep OTer the whole aolid rotating apberoid, ia given in a form read; for
nomerical oompatation ia " Mote on the ' Oscillationa ol the First SpeoiM' in
Laflace'« Tbeinj' ol the Tidea" (W. Thomson), Phil. Mag. Oet. ISTR.
..Google
SIS'".]
DTNAiaCAL LA.ira AND PBIKCIPLE3.
MS"*". To discover something of the character of the gjTO- ^J2*^
soopic influence an the motion of a RTatem, suppose there to be v>temi
no resistances (or viscous influences), that is to say let the
dissipativitf, Q, be zero. The detcnuinantal equation (4) beoomes
(ll)X' +11, (12)X'+i3]Xh
(21)X'+3i]X+21, {22)X' ^
12,.
Now bj the relations (12)^(21), etc., 12 = 21, etc, and 13] =- 21],
we see that if X be changed into —X the determinant becomes
altered merely by interchange of terms between columns and
rows, and hence the value of the determinant remains unchanged.
Henoe the first member of (14) cannot contain odd powers of X,
and therefore its roots must be in pairs of oppositely signed
equals. The condition for stability of equilibrium in the xero
configuration is therefore that the roots X' of the determinantal
equation be each real and negative.
34S''. The equations are simplified by transforming the oo- limpljncs.
ordinates (§ 337) so as to reduce 7 to a sum of squares with mi '
pocdtive coefficients and F to a sum of squares with positive or
negative coefficients as the case may be, or which is the same
thing to adopt for co-ordinatee those displacements which wonid
correspond to "fundamental modes" (§ 338), if the positional
forces were as they ore and there were no motional forces.
Suppose fiirther the unit values of the co-ordinates to be so
chosen that the coefficients of the squares of the velocities in
2T shall be each unity; and let us put w„ w,, w,, etc. instead of
the coefficients 11, 22, 33, etc., remaining in ST. Thus we have
r=H'A.' + ^i' + etc), and r="}{w,^,'+iif^/ + 6tc.) (15).
If now we omit the half brackets ] as no longer needed to avoid
ambiguity, and nnderstond that 12 = - 31, i3 = -3i, 23 — — 33,
etc., the equations of motion are
it, + "^,+ i3^,+ + w,^, = 0 ■
jfr,+ 2i^, + 23^^+ + in,^, = 0
i^» + 3'l^i + 3a^.+ +'"',^,= 0
..(16),
jiGoogle
1 PREtlMINART.
and tlie determiniuital eqiutioii becomes
.3X,...
.3A,...
[«5".
(")■
The detenmiiatit (vhich for brevity vre shall denote by D) in
this case is what has been called by Cayley a skew detemunant.
What it would become if zero were snbetituted for X'-fo-g
\'+iB„ etc in its principal diagonal ia what is called a skew
symmetric determinant. The known algebra of skew sitd skew
symmetric determinants gives
2).(V + wO(X" + .J...(V + ^,)
+ i*2(V + »J(X' + ii,)...(l' + «r,)ii"
+ X'2(i'+".)(A'+»J-(X'+"'0("- 34+31 .»4+23- •4)' I (18),
*i'2(l" + o,)(X' + <.J...(l"t<r.)(Sia.34.s6)" + el«.
+i'(si!. 34.36 •■-■, >y
when i is even. For example see (30) below. Wben X is odd
the last tenn is
X- 2(V+«J (Si.. 34. S«..- <-".•-■)'■
..(18^
and no otlier change in the formula is oeceesary. In each caae
the small 2 denotes the siun of the prodacts obtained by
making ereiy possible permutation of the numben in the line of
factors following it, with orders chosen acccording to a proper
rule to render the sign of each product poBitive (Salmon's Higher
Algebra, Lesson t. Art 40). This sum is in each case the square
root of a certain corresponding skew symmetric determinant.
An easy rule to find other products frpm any one given to
b^;in with is thia:— Invert the order in any one &ctor, and
make a simple interchange of any two numbers in different
factora. Thus, in the last S of (18) alter f- i, « to i, i- i, and
interchange i-i with 3; bo we find la.t— i, 4.56 t,3fop
a term : Bimilarly 12. 64.53 ... t, t-i, and 62.14.53...*— 1, ^
for two others. The same number must not occur more than
once in any one product. Two products differing only in the
orders of the two numbers in factors are not admitted. If n be
the number of factors in each term, the whole number of &ctors
is clearly i ■ 3 • 5 ... (sn - i)) and they may be found in regular
..Google
345".]
DTNAHICAL LAWS AND PRINCIPLES.
395
progresdoQ thus: Begin with a single factor and smgle term 13. ^g^.
Then apply to it the factor 34, and permute to Buit 34 instead ikewtj^-
of 34, and permute the result to suit 14 instead of 24. Thirdly,
apply to the sum thus found the factor 56, and permute suc-
cesaively from 56 to 46, from 46 to 36, from 36 to 26, and
from 36 to 16. Fourthly, introduce the factor 78; and so on.
Thus we find
J
y
4*, 42, 43. o
f. 34 + 31- 24 + 23- 14
13, 14, 15, 16
(13,
+ (12
+ {•
+ (3
+ (23.
.34+31-24+33-14)56
.53 + I3-52+23-5046
.45+41.53+43.51)36
.45+41.35 + 34.51)26
■45 + 34- 35 + 34. 25)16
(19).
o, 23, 24, 35, 36
32. o, 34, 35, 36
42, 43. o. 45. 46
52.53. 54. o. 56
-', 62. 63, 64, 65, o J
The second member of the last of these equations is That is
denoted by 513.34.56 in (18).
346*. Each term »f the determinant D except
(X' + or,) (X* + OT,) . . . (X' + w^
contains X' aa a factor. Hence, when all are expanded in powers *^SS*^!,,
of X', the term independent of X is w^w,. ..-aif If this betjrofTBe-
nqpitive there must be at least one real positive and one real
negative root X'. Hence for stability either must all of vSy,
t7„ .... v, be positive or an even number of them negative.
£x. : — Two modes of motion, x and y the co-ordinates. Let the
equations of motion be
Jy-gHFy = ls] ' "'•
and the determinantal equation is
(/X* + ^) (/X' + J^ +ff'X*=0.
If we put
"'ilJ', y-'ilJJ (21),
E~^I, F.tJ, mig.yJ{IJ) (22),
jiGoogle
1)1)
396 PEEUjnNAaT. [345\
equatiotu (20) and th« determiuiuital equation become
i*ri*'r(-0\ „,)
and (V + »)(i' + {) + /V«0 (24X
The eolntinn of this qnadratio in X' may be pnt under the
foUoving fonna, —
To make both values of - \' reel and positive rt and t must
be of the eame sign. If they are both positive no farther condi-
tion is necessary. If they are both negative we most have
y^J^^*sl~i (26).
These are the conditions that the zero configuration may be stable.
Remark that when (as practically in all the gyroabitic illnstra-
tiona) y" is very great in oompariflon with ^(w^), the greater
value of — X* is approximately equal to ■/, and therefore (as the
I product of the two roots is exactly wf), the less is approximately
equal to w{//- Remark also that 2ir/^nr and 2v j Jl are the
periods of the two fundamental vibrations of a system otherwise
the same as the given system, but with y = Q. Hence, using the
word irrotational to refer to the system with y = 0, and gyroecopic,
or gyrostatio, or gyrostat, to refer to the actual ^stem ;
From the preceding analysis we have the carious and in-
teresting result that, in a system with two freedoms, two
irrotational instabilities are converted into complete gyroatatic
» stability (each freedom stable) by euflSciently rapid rotation ;
but that with one irrotational stability the gyrostat is essentially
unstable, with one of its freedoms unstable and the other
stable, if there be one irrotational instability. Various good
illustrations of gyroatatic systems with two, three, and four free-
doms (§§ 345'," and "°) are afforded by the several different
modes of mounting shown in the accompanying sketches, ap-
plied to the ordinary gyrostat* (a rapidly rotating fly-wheel
pivoted as finely as possible within a rigid case, having a convex
curvilinear polygonal border, in the plane perpendicular to the
axis through the centre of gravity of the whole).
■ ^olure. No. S7S, Vol 15 (Fflbroaiy 1, 1877), page 397.
..Google
345'.] DYNAMICAL LAWS AND PRINClrLES.
3
SI
'I
b
ti
'I
El
It
1
SI
[i
I
SI
Oidinan
..Google
398 PEELmiNABT. [345*.
QjnMata,
on l(idf»«(tg» gliD\Ml •rilli fU
ofiHiD- ^« uid glmbftj-ring ve hmvv /-/ lu (SO), ftnd iDppoftiiig Ibe Jfireli of Uie Imiffrcdgq to ba tha
^^^)^^ Bxpreaed bj (he luno equstloiu of motion, li obUloed bj Bupportiog the ^jroibil on « Un]« elatfic
grroKoplc onlieruJ fleiure-Jotnt of, for eumpK (Un gieel plHofanv-wIre due or two cmtlmMn* long
(llOWlin
pLkCe oriHiD-
..Google
DYNAMICAL LAWS AND PRINCIPLES.
- (27),
345^. Take for another example a ayBtem having three G,
freedoms (that is to say, three independent oo^rdinates ^eo tnt-
|p^, ^,, ^^, (16) become
^1 ■•■ ?A - ffA + ^,fx = 0
where g^, g^, g^ denote the values of the three pairs of eqaals
23 or -32, 31 or - 13, la or —a I. Imagine ^,, ^„ ^, to be
rectangular coordinates of a material point, and let the co-
ordinates be transformed to other axes OX, 0 F, OZ, so chosen
that OZ coincides with the line whose direction cosines rela-
tively to the ^,-, ^,-, ^,- axes are proportional to g^ , g^, g^. The
equations become
- 2<o3> =
!/+2<-*=r \ (28),
..Google
400 PBELnaHABT. [345^.
OrtMtetta where " - ^{ff' + g' + sj)' ^^^ ^^ force-components par&lld
douu: to the fresh axes ara denoted by Z, Y, Z (instead of — -^ ,
_iV
dependent of the assumption 'we have been making latterly that
the positional forces ai-e oonserrative). These (28) are simply
the equations [g 319, Ex. (E)] of the motion of a particle rela-
tirelj to co-ordinates revolving with angular velocity u round
the axis OZ, if we suppose S, Y, Z to include the components
of the (Mntrifugal force due to this rotation.
redooedtd Hence the influence of the gyroscopic terms however ori-
l^jIS^ ginating in any system with three freedoms (and therefore alflo
""^ in any system with only two freedoms) may be represented by
the motion of a material particle supported by msssless springs
attached to a rigid body revolving uniformly round a Sxed axis.
It IB an interesting and instructive exercise to imagine or to
actually construct mechanical arrangements for the motion of a
material particle to illustrate the experiments described in
§ 345".
345*". Consider next the case of a ^stem with four free-
doms. The equations are
^1+ laij, + 13^,+ 14^,+ w,i/'j- 0
^,+ ai^, + 23^,+ 24\j/^+m,il/,= 0
^, + 4^'!', + 42"/-, + 43"^,+ w.^i =
- (29)-
Denoting by D tbe determinant we have, by (IS),
/)=(X'+ro,)(\'+iir,)(V4.w,)(A'+nr.)
+ X'{34'(A,'+nr,)(V+mJ+i2'(X*-HD-,)(X'+wJ-4-42'(X'+tir,)(X'+arJ
-H3'(X'-H«.)(X'+w,)+23'(XVt^,)(V+wJ+i4'(X*+i=rJ(V+«.J}
+ X*(i2 34'+ 13 41+ 14 23)'
.(30).
If tir,, or,, tir,, sr^ be each zero, D becomes
X'+(i2'+i3'+i4'+23'+43*+34*)X'+(i2 34+13 42 + 14 a3)'X*.
This equated to zero and viewed as an equation for X* has two
..Google
345"".] DYNAMICAL LAWS AND PRINCIPLEa 401
roots each equal to 0, and two othera eivon by the reeidual Qo^nipir
, . or ^ ireewro*
qnadratic Matio
■ratem
X*+(ia*+i3'+i4'+23*+a4V34')\' + (i2 34+i3 4J+i4 23)'=0...{31). too*
Now remarking that the solution of :^+pt-t-q''~0 may be
writtea
-»=llfVtl'+2j)Cp-2j))-jlV(P + 2«)*V(f-2«)l'.
we Lave from (31)
- \' = J ("'+ ■3'+ 14'+ =3'+ '4'+ 34" * .J') \ ,,.,
wken r = ,y|(i2 + 34)" + (i3*42)'4.(i4+j3)'[|
"Id •-v'l(i!-34)- + (-3-42r+<i4-23)'ll ' ''
At 12, 34, 13, eto. are essentially real, r and a are real, and
(uiUeftii 12 43 + 13 42 + 14 23 - 0, when one of the values of X* is ^J|^J[^,i.
Eero, a case which must be considered Bpecially, but la excluded ''v.p"^
for the present,) they are unequal. Hence the two values of doninMioe.
— X* given by (32) are real and positive. Hence two of the
four freedoms are stable. The other two (corresponding to
— A' = 0) are neutral
345'". Now suppose t^,, w,, w,, w, to be not zero, but each QiNdnini^
very smalL The determinantal equation will be a biquadratic ■! btwiu.
iu X', of which two roots (the two which vanish when m^, etc. aiiyduot'
vanish) are approziinately equal to the roots of the quadratic "
(12 34-*- 13 42+ 14 33)'A'4-(i2X"'4+i3X°^4+i4'w,°'.
+ 23'w,«r, + 24*w,w,+ 34V,w,)A'+ m,or,iir,tff^=0...(34),
and the other two roots are approximately equal to those of the
previous residual quadi-atic (31).
To solve equation (34), first write it thus :■-
(.')"^<'^
-i3'+r4'+z3''+24''4 34'')^+{i3'34'+i3'4a'+i4'23')' =
('■'),
V(».",)' ^'VK-J' '"VK-.)'
..(36).
TOL. I. 20
D,g,l,„ab,GOOgk'
PRELIMINART.
[345*
QnidraplT
tnr eycfoCd-
■1 ^t«m>
uMbl^br
tlnkt.
Thns, taken as a quadratic for X~', it has the same form as (31)
for X*, and so, u before in (32) and (33), ire find
■ (37),
..(38).
¥ = 1<*^''T
T^hete r'=s/{(i2'+34T+(i3'+42T+(i4'+23'ni
and «' = V{(i2'-340*+(i3'-42')'+(i4'-33'n I
Now if Wj , w,, or,, w, be all four positive or all four n^^tive,
ii', 34', 13', etc. are sll real, and therefore both the values of
- Tj giveu by (37) are real and positive (the excluded case
r«feTTed to at the end of § 345*", whioh makes
I3'34'+I3'42'+ r4'z3'=0,
and therefore the smaller value of — ^ = 0, being Aill excluded).
Hence the corresponding freedoms are stable. But it is not
tiecasary for stability that w^, m,, th^, w^ be all four of one
sign: it is neceasaiy that their product be positive: since if it
■were negative the values of X' given by (34) would both bo
real, but one only negative and the other positive. Suppose two
of them, m^ st, for example, be negative, and the other two,
nr,, n',, positive: this makes or, iir,,w,i?^, w,i7,,and ■nx^iB^ ne^tiv«,
and therefore 13', 14', 33', and 34' imaginaiy. Instead (tf four
of the six equations (36), put therefore
'3
,,^3 =
23
-.,'4'
(39).
V^w.w^'"' ^Ji-^.^^"" ~j{-^,^;>'"* "vF^^
Thus i3"etc. are real, and I3'=i3"»/--1 «*«■. "'I'l (38) become
»''"V{(i^' + 34r-(i3" + 40'-{'4"+23'7!)
•' = /!("'-34')'-(i3"-40'-<i4"-^3Ttr""
Hence for stability it is necessary and sufficient that
..(«).
(n'+347>(i3" + 42")' + (i4"
(■='-34l'>(i3"-4>'r*(i4"'
If these inequalitieR ai« reversed, the stabilities due to w,, w,
and 34' are undone by the gyrostatic connexions 13", 42", 14"
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345*.]
DYNAMICAL LAWS AN1> PRINCIPLES,
345"*. Going back to (29) we see th«t for the particular bfiibjtrro*
solution ^1 =«,«*'. ^, = a,t*', etc., given by the first pair of roote scnnier-
of (32), thej become approximately
..(42)i
licking in fact the linear algebraic equations for the solution in
tlie form c*' of the simple simultaneous differential equations
(53) below. And if we take
^, = -^««, ^, = -*'-€«,etc (43),
for cither particular approximate solution of (29) corresponding
to (37), we find from (29) approximately
-■6, + i2'i,+ i3'6.
"'ft, + 4i'6i + 42'6, + 43'6,
I4'i.=
24'6^ =
'6 =0
Hemark that in (42) the coefficients of the first terms are
imaginary and those of all the others real. Hence the ratios
■>,/<*,! '*i/<*si ^^> *^i^ imaginary. To realize the equations put
and let}),, q„ p,, etc. be real; we find, as equivalent to (42),
|-n?, + i3p,+ i3;),
+ i4 7,=0
..(46).
13?,-
— 115', + a I p, 4 23 Pj H
23?,-
HPt
-O]
'4 7,
= U
24 P,
= 0
= 0
Eliminating ?, , y„ etc. from the seconds by the firsts of these
{laint, we find
(n* + 11 )p, -I- 1! P, + 13 pj + u ;>, = 0 ■
»i;>,+(n* + M)p,+ »;-,+ M p, = 0
9ip,+ aa />,+(«'+ 33) p,+ »4pj = 0
*i p, + ■ « p, + « p, + («• + **)p, - 0 .
and by eliminating p^, p,, etc. similarljr we fiud similar equations
2G-2
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4 PRELIMNABY. [345*.
for the g^s; with the same coefficienta u, ii, efax, given b; the
following fonnulas : —
11=1321 + 1331 + 1441-
"=1333 + 1443
i»=ii 23+1443 (48).
11 = 3331 + 2441
etc etc._
Remember now that
II — ai, i3 = -3i. 32'=-a3.«*«- (*9).
and we see in (48) that
U — n, 13-si, B = si,etc. (^0)i
and fiuiher, that 11, 11, etc are the n^ativea of the coefficienta
of J a', a,a,, etc in the quadratic
i{(i3a,+ i3a,+ i4a,)*+(aia, + i3'», + 24o,)' + etc}...(51)
expanded. Hence if G{aa) denote this quadratic, and G(pp),
& (qq) the same of the p'a and the q'a, we may write (47) and
the corresponding equations for the q's as follows :
»?,*
= 0,
»?,-
.ojM.
'0,
etc
dG(qq)_
0
etc
<^.
^ese equations are harmonized by, and as is easily seen, onlj
by, Bsaigning to »' one or other of the two values of —A* giveD
in (33), above. Hence their detenninantol equation, a bi-
quadratic in n*, haa two pairs of equal real positive roots. Vfc
readily verify this by verifying that the square of the deter-
minant of (42), with A* replaced by — n', is equal to the detei'-
minant of (47) with 11, u, etc replaced by their values (48).
Hence (§ 343(/) there is for each root an indeterminacy in the
ratios p//*,, PjP,t PjPtt according to which one of them may be
assumed arbitrarily and the two others then determined by two
of the equations (47) ; so that with two of the jfa assomed '
arbi4z«rily the four are known : then the corresponding set of 1
four eft is determined explicitly by the firete of the pairs (46). |
Similarly the other root, n', of the determinantal equation gives
another solution with two fresh arbitraries. Thus we have the
complete solution of the four equations I
..Google
345*.] DYNAIUCAI, LAWS AND PEINCIPLES.
dtb
-^ + 13 if',+ 13 ^,+ 14^4 = 0
J.;
^ + ai ^, + 231^,+ 34^^ = 0
DvWlaaf
■olDikai.
. (63),
a'
ete.
with its four arbitrariea. The fonnnlas (46). ..(52) lire clearly
the game as we should hare found if ve hod commenced 'with
assmning
^^=p^a\nt^■¥q^tx»n^, ^, =j7, ainnf + g', coan^ etc.. ..(64),
as a paiidciilar solution of (63),
345".
thus: —
(a) Uultlpl; the fiists of (46) hj p,, p,, p^, p^ and add ;
or the Beoouds hj 9,, g„ 9,, 7, and add : either way we find twomm-
poiiBnli ot
p.?.+p.?.+ft?.+PA=o (S5)- i'S.SfS:
(6) Multiply the firets of (46) by y,, q„ ?„ q, and add:**"*^'
multiply the seconds by^,, p,,p„PtMid add: and compare the
reeolta : we find
nV = »Sg' = Sis(p^,-p,?J (56), "^gmiii/
where 2 of the last member denotes a sum of such doable *"""*'
terms as the sample without repetition <tf their equals, such as
" (p.S.-ft?,)-
(e) Let n* h" denote the two values of — X' pytai in (32), Oithcw^
and let (54) and glS?
iji^-p\e3nn't + q\coiin't, ^^=p\aiin't + q'^ooBn'l, etc. .,(57) {^"SjS'
he the two oorreaponding solutions of (53). Imagine (46) to be tkioa.
written out for ih* and call them (46') : multiply the finrta of (46)
hyp^^,p'^p^^p\taiiBdd:iimltipljth6&rBtaQi(i$')hyp^,p^,p^,p^
and add. Proceed correspondingly with the seconds. Proceed
Himilarly with multipliers q for the firsts and p for the seconds.
By comparisons of the sums we find that when n' is not equal
to n we must have
Sp'q='0, Si3 (p',p,-p'^,) = 0
t^P-0, Si3 (?',?. -g'rf,)=0
J^^=J} Si3(y'j,.-y'j>,) = 0, ti2(p\q,~p'.q,) = 0
(58).
..Google
406
PRBLIMINART.
[345".
345"'. Tho case of n = n' is intereflting. The equatjona
S9'y = 0, 5p'p = 0, S.p'g=0, 2?'p = 0, when n differe however
littlefrotan', sbow(a8 we saw in a correBpoading case in §343i»)
that equality of n to n' does not bring into the solution t«nna
of the form Ct con nt, and it must therefore come under § 343 e.
The condition to be fulfilled for the equality of the roots is seen
from (32) and (33) to be
"=34> 13 = 4^, and i4 = 23 (59):
and to give
«'=I2'+I3'+I4* (60)
for the common value of the roots. It is ea^ to Terify that
these relations reduce to zero each of the first minors of (42), as
they must according to Bouth's theoi-em (% 343«), because each
root. A, of (42) is a double root. According to the same theorem
all the first, second and third minors of (47) must vanish for
each root, because each root, n*, 1^ (47) is a quadruple root:
for this, as there are just four equations, it is neceasary and
sufficient that
n = £i=33 = « and is = 0, is = 0, u = 0, jb = 0, etc.. ..(60*),
which we see at once by (48) is the case when (59) are fulfilled.
In fact, these relations immediately reduce (51) to
G(a«) = l(i2'*i3*+'4')(V + «.'+«/ + »,') (61).
In this case one particular solution is readily seen from (62) and
(46) to be
^i = sinn<, i^, = cosni,
«.— ^. 1.—-,
^, = — ^ cos Tli, (fr, =
(62).
Hence tho general solution, with four arbitraries p,,p„P„Pt,i
V',=y, sinn( + -(i2p, + 13?,+ i4pJcosn<
^,=p,BinMi + -(- lap, + iyf,-i3Pi)cmtU
^,= p, sinnf + -(- i3P|- i4P,+ i2p,)ooant
^,=p^BinM( + -(- r4p,+ rjp,- izp,)ooi
It is easy to vciify that this satisfies tho four differentini
equations (53).
..(63).
jiGoogle
345"".] DYNAMICAL LAWS AND PRINCIPLES, 407
345"°. Quite tts we have dealt with (42), (45), (63), (54) in Two higher,
§ 345^, we ataj deal with (44) and the simple simultaneous equa-
tions f<^ the solution of which they serve, which are
diji, d<p^ diii, , „ 1
at at at I lo»H-.ortha
/fill . (our (iindi^
etc. eto. J
and all the formulas which we meet in bo doing are real when ■bniluir
tir,, w,, w,, vs^ are all of one sign, and therefore la', 13', etc., all b^aolaiioa
real In Uie case of some of the nr'a negative and some pOBitive aiiniUr
there is no difficulty in realining the formulas, but the con-'*
sideration of tiie simultaneoua reduction of the two quadratics,
*l "f, ^t )\ (66),
and J (w,*!* + ^fii * '"'t"'* + ''A*) ]
to which we are led when we go back from the notation 1 2', eto.
cS (36), is not completely instructive in reepect to stability, as
was our previous explicit working out of the two root« of the
determinantal equation in (37), (38), and (40).
345*^. The conditions to be fulfilled that the Q'stem may be proridtd
dominated by gyrostatio influence are that the smaller value of itatioiii-*
-X' found from (31) and the greater found from (34) be r&.(u^3toSS-
Bpectively very great in comparison with the greatest and very """^
email in comparison with the smallest, uf the four quantities
or,, ur,, nr,, w^ irrespectively of their signs. Supposing w, to be
the greatest and m^ the smallest, these conditions are easily
proved to be fulfilled when, and only when,
(12.34+13. 4J + I4-J3)' ^^^_ ,fiR^
— 1 = 1 : : 1 >;» * w (oDl,
I2'+I3'+I4 +34 +43 +23
(ia.34+'3-43 + '4-a3)' »*ot-' (67)
WjW,+ i3'itr^w,+ i4'w,OT,+ 34'w,OT,+ 4a'OT,nr,+ z^^w^w^ * \ h
where » denotes " very great in compariton wilh." When these
conditions are fulfilled, let 12, 13, 23, etc, be each increased in
the ratio ot iV to 1. The two greater values of n (otXJ— 1)
will be increased in the same ratio. A' to 1 j and the two smaller
..Google
Umlteof
i PEELMItJART. [845^.
will be diminiahed ea«h in the invene ratio, 1 to Jf. A^ain,
let V**!' J*^i> J^^n •J^'"* ^ ^""^ dinunisbed in the
ratio Jf to 1 ; the two lai^er values of n will be sensibly
unaltered ; and the two Biualler mil be dimiidshed in tlte ratio
Jf ' to 1.
Remark that
rhen (66) is satisfied the two greater values of n are
345'^.
(«) ■
each
and >— -"•34/^3-4^^t4.23 I (68);
^(12* + 13'+ 14'+ 34 + 42 + 23 ) ]
and that when they ara very unequal the greater ia approxi-
mately equal to the former limit and the leea to the latter.
(b) Wlien (67) is satisfied, and when the equilibrium is stabk-,
the two smalleF valnea of n are each
V(i2'cr,gf,+ I3X'°'.+ i4X°^i+34''g|"',+4'V'gf +'3'"
2 ■ 34 + 13 • 42 + I
W69).
Qndrnplj
d»1 (yateia
with non-
dominant
^{(i2'w,nrj+i3'w,Br,+ i4'w,Wj+34*(ir,iff,+4Z*OT,w,+a3'Wjf(rJ} j
and that when they are very unequal the greater of ihe two is
approximately equal to the former limits and the less to the
latter.
345°. Both (66) and (67) roust be satisfied in order that the
four periods may be found approximately by the solution of the
two quadratics (31), (34). If (66) ia satisfied but not (67),. the
biquadratic determinant still splits into two quadratics, of which
one is approximately (31) but the other is not approximately
(34). Similarly, if (67) is satisfied but not (66), the biquad-
ratic splits into two quadratics of which one is approximately
(34) but the other not approximately (31).
345^. When neither (66) nor (67) is fulfilled thne is not
generally any splitting of the biquadratic into two rational quad-
ratics; and the conditions of stability, the determination of the
fundamental periods, and the working out of the complete so-
lution depend essentially on the roots of a biquadratic equatiou.
When n,, nr„ nr„ tr, are all positive it is dear from the equation
..Google
345°'.] DTNAMICAl LAWS AND PRINCIPLES. 409
of energy [M5", (4), with Q=0] th&t the taotkm is etahle what- g^^^?;
ever be the values of the gyroatatic coeffioienta ij, 34, 13, etc. J||^^"^?
aad therefore in this case each of the four lYtote \' of the biijuaJ- dominant
itttio ill real and negative, a propoBition included in the general ^uanoM.
theorem of g 345"^ below. To illustrate the interesting questions
which occur when the m'ti are not all positive put
13 = ..S, 34 = 3*?, i3 = >tf. etc <70),
where », », 13, etc. denote any Qumerioa whatever subject only
to the condition that they do not make zero of
When ttr,, VI,, nr,, 0,, are all negative each root X' of tiie bi-
quadratic is as we have seen in % 345"" real and negative when
tlie gyroBtatic influences dominate. It becomes an interesting
question to be answered by treatment of the biquadratic, how
small may ? be to keep all the roots X' real and negative, and
how large may j be to render them other than real and positive
8S they are when ^ = 01 Similar questions occur in connexion
with the case of two of tbe fir's negative and two positive,
when the gyrostatic influences are so proportioned as to fulfil
345*" (41), 90 that when g is infinitely great there is complete
gyrostatic stabUity, though when £/=0 there are two instabilities
and two stabilities.
345"*. Returning now to 345" and 345", 345* and 346^ OyroiUiio
for a gyrostatic system with any number of freedoms, we see by with anj
345'' that the roots \' of the determinantal equation (14) or (17) ftZ^i*
are necessarily real and negative when m^, or,, ro,, 0^, eta are
all positive. This conclusion is founded on tbe reasoning of
g 345" regarding the equation of eoer^ (4) applied to the case
^ = 0, for which it becomes T + T=E^, or the same as for the
case of no motional forcea It is easy of course to eliminate
dynamical considerations from the reasoning and to give a purely
algebraic proof tbat the roots k' of the determinantal equation
(14) of 345**" are necessarily r«al and negative, provided both of
the two quadratic functions (ll)o/+ 2 (12)o,(i, + etc, and
ll0i'+a 12o,a,+ ete. are positive for all real values of o,, o^, etc.
But tbe equations (1 4) of g 343 (A), which we obtained and used
in the course of the corresponding demonstration for the case of
no motional forces, do not hold in our present case of gyrostatic
motional forces. Still for this present case we have the con-
..Google
410 PHELIMINARY. [345°".
cluBion of § 313 (m) that equality among the roots falls esaentJallj
Oiw or under the cose of g 343 (e) abova For we know from the can-
wiilh t™ sideration of euergy, as in § 345", that no particular solution
' ^' can be of the form l^ or t sin a-t, when the potential energy is
positive tor all diaplaoements : yet [though there cannot be
equal roots for the gyroBtatic Byatem of two freedoms (§ 345')
as we see from the solution (25) of the det«nninanlal equation
for thifl case] there obviously may be equality of roota* in a
quadruply free gyrostatic system, or in one with more than four
Amiiation freedoms. Hence^ if both the quadratic functions have the
thniRsuu same sign for all real values of a,, a^, etc., all the first minors
* EiuupleB ol this may be invented ad libitvm by oommeudng with fairs of
equaUons each m (23) and altering the variabloa by (generalized) ortbogooal
translormationa For oae very simple example put ^=vr and take (23) u one
pair of equaUons of motion, and as a second pair take
The seoond of (23) and the first of these multiplied respectively by eosa and
siu a, aud again by sin a and eoB a, and added and subtracted, give
and ^i + 7Bini]L^ + 7Coaa^ + c^i=0,
yrboK ^,-f edna + ijooBB,
aud i^j={'cosa-i|sina.
Eliminating {' and q by theee last equations, from the first and foartli of
the equations of motion, and for synunetry putting ^, instead of (, and ^^
instead of V< and fur simplicitr putting -fOosa^g, aud 7BiUB=A, and oollecUug
the equations of motion in order, we have the following, —
^t + Hi+S^fi + ^'I't^O,
•jr^-~ Alt, -0jtj + OT;/'4=O,
for the eqnations of motion of a quadruply free gyroitatia system having two
equalitias among its four fundamental periods. The two different periods are
the two values of the eipreseion
2v/Mis'+iA')*^Ul7' + lW + w)(.
When these two vatues are nneqnal the equalities among the roots do not
give rim to terms of the form ttU or tooairt in the solatian. But if
ffs -(^> + lA'), which makes these two values equal, and therefore all four
roots equal, terms of the form tcosfft dc appear ia the solution, and the equili.
brium is unstable in the tranBiUonal ease though it is stable if - w be less tbau
ia" + i'i'by ever BO small a difftreuco.
..Google
345°".] DYNAMICAL LAWS AND- PRINCIPLES. 411
of the determinautal equation (14), §345*"', must vaniah for each ipplicntion
double, triple, or multiple root of tlie equation, if it has any iheimini.
nach roots.
It will be interesting to find ft purely algebraic proof of this
theorem, and we leave it a^ an exercise to the student; remarking
only that, when the quadratic functions have contrary Bigna for
some real values of a^, a^, etc., there nmy be equality among the
roots without tbe evanescence of all the first minors; or, inBqulmat*
dynamical language, there may be terma of the form (t", or biiity ii"
^aino-^ in the solution expresaing tbe motion of a gyi"ostatic («»e« b..-
aystem, in transitional cases between stability and instability, uli^and'
It is easy to invent examples of such cases, taking for instance '"''•'"' 'W-
the quftdruply free gyroslatic system, whether gyros tatically
dominated as in g 345*"', but in this case with some of the four
quantities negative, and some positive; or, as in § StG"", not
gyrostatically dominated, with either some or all of the quantities
cr,, w,, ..., w, negative. All this we recommend to the student
as interesting and instructive exercise.
345*^. When all the quantities nr,, sr,, ..., m, are of the conditloni
same sign it is easy to find the conditions that must be fulfilled lullcari-
in order that the system may be gyrostaticully dominated. For ""'^""-
if p,, p,, ■.., p, are the roots of the equation
c/' + c,s— +...+C._,3 + C. = 0,
we have
Hence if —p, , -p„ ... -p^, be each poative, cjne^ is their arithmetic
mean, and iv:,j<:^_^ is their harmonic mean. Hence e^/nc^ is
greater than »<,/c,_,, and the greatest of — p,, -p,,..., -p_ is
greater than c, /nc„ and the least of them is less than nc,/c,_,.
Take now the two following equations :
X' + X'-'2i2'+V-'2(5"-34)* + V-2(Sia.34.56)'+e'«- = 0 (71),
(l)V©'"'2-"-(xy"'2(Si.'.54'A©'"'s(5»'.34'.56')Veta = 0(72),
' sf(^.^,y ■* '/{^.^>y^^ s/K«',)''"'' ' v/(w.„,o
..Google
O»dltloni
r. [345—.
Suppose for simplicity t to b« evea All the roote X* of (71)
SK (§ 3i5"^ below) essentially real aad negative. So are those of
(72) ptovidedia,,iEr|,..., ts, are all of one sign as we now suppose
them to bft Henoe the smallest root ~\* of (71) is less than
it's (" • 34 . S'..
i-.,.r
S(Si2.34.56,...,i-3, i-i)'
and the gnateat root — X' of (72) is greater thao
2(S"'.34'.S«',-.>-3.>-2')'
..(74).
li2(is'.34'.S''...., •
•')
,..(75).
Hence the conditions for gyroetntic domin&tion are that (74) must
be much greater thaji the greatest of the podtire qu&ntities '^■m^ ,
^w,,..., *w„ and that (75) must be very much less than tlie
least of these positiTe quantities. When these conditions are
fulfilled the t roots of (18) § 343" equated to zero are separable
into two groups of ^t roots which are infinitely nearly equal to
the roots of equations (71) and (72) respectively, conditions
of reality of which are investigated in § 345"^ below. The
interpretation leads to Uie following interesting oonclusious: — •
J 345"*'. Consider a cycloidal system provided with non-
rotating flywheels mounted on frames so connected with the
moving parts as to give infinitesimal angular motions to the
axes of the flywheels proportional to the motions of the system.
Let the number of freedoms of the system exclusive of the
ignored co-ordinates [§ 319, Ex. (Q)] of the flywheels relatively
to their frames be even. Let the forces of the system be such
that when the flywheels are given at rest, when the system is
at rest, the equilibrium is either stable for all the freedoms, or
unstable for all the freedoms Let the number and connexions
of the gyrostatic links be such as to permit gyrostatic domina-
tion (§ 345"^) when each of the flywheels is set into sufficiently
rapid rotation. Now let the flywheels be set each into suf-
flciently rapid rotation to fulfll the conditions of gyrostatic
domination (§ 345*^): the equilibrium of the system becomes
stable : with half the whole number i of its modes of vibration
exceedingly rapid, with frequencies equal to the roots of a cer-
taiu algebraic equation of the d^ree ^t; and the other half of
..Google
345"*.] DYNAMICAL LAWS AND PKINCIPLE8. 413
ita modes of vibratioD very slow, with frequencies given by the "^^r^""
roots of another algebraic equation of degree ^i. The first class "''J"^
of fundamenta] modes may be called adynamic because they
are the same as if no forces were applied to the system, or
acted between ita moring parts, except actions and reactions in ita »Ajn%.
the normals between mutually pressing parts (depending on the laiionHTery
inertias of the moving parts). The second class of fundamental
modes may be called precessional because the precession of the «nd m
equinoxes, and the slow precession of a rapidly spinning top o-oiiuiicni
supported on a veiy fine point, are familiar instances of it.
Remark however that the obliquity of the ecliptic should he
infioitely small to bring the precession of the equinoxes pre-
cisely within the scope of the equations of our "cycloidal
system."
345"". If the angular velocities of all the flywheels be
altered in the same proportion the frequencies of the adynamic
oscillations will be altered in the same proportion directly, and
those of the precessional modes in the same proportion in-
veniely. Now suppose there to be either no inertia in the
system except that of the flywheels round their pivoted axes
and round their equatorial diameters, or suppose the effective
inertia of the connecting parts to he comparable with that of
the flywheeb when given without rotation. The period of each ^^JJ]^*""
of the adynamic modes is comparable with the periods of the J^n™'^
flywheels. And the periods of the precessional modes are com- f^'Jue^I,
parable with a third proportional to a mean of the periods of ^^^'
the flywheels and a mean of the irrotational periods of the sys- K^^'^
tem, if the system be stable when the flywheels are deprived ^^tein,
of rotation. For the last mentioned term of the proportion we JU^^,
may, in the case of irrotational instainlity, substitute the time of "i^Sf^'u"
increasing a displacement a thousandfold, supposing the system wulTfly.
to be falling away from its configuration of equilibrium prm^ur'
according to one of its fundamental modes of motion (e*').
The reciprocal of this time we shall call, for brevity, the
rapidity of the system, for convenience of comparison with the
frequency of a vibrator or of a rotator, which is the name com-
monly given to the reciprocal of its period.
..Google
414 PRELIMINARY. [345°".
PrwFot 345"^, It reniaius to prove that the roots X' of (71), Bnd of
■dynmntc (73) also when izr,, ta,,..., -et, are lUl of one sign, are essentiallj
Mrion^ real and negative. (71) is the detcrminajital equation of
whiji § 34^)''' (42) with any even number of equations instead of only
^^tonal four. The treatment of ?S 345'" and 345" is all directly ap-
ffiSSrll'l'" plicable without change to this extension; and it proves that the
im^^iiu7. roots X* are real and negative hy bringing the problem to that of
the orthogonal reduction of the essentially positive quadratic
function
(?((Mi)=J{(i2a,+i3a^+etc.)'+(aia|+i3n,+etc.)'+(3ia,+32<T,+etc.)'+etc} (76):
it proves also the equalities of enei^es of (56), ^ 345", and tlie
orthogonalities of (55), (58) § 345": also the curious altiebraic
AiK^raio theorem that the dctenninantaj i-oote of the qoadralio function
consist of Jt pairs of equals.
Inasmuch as (72) is the same as (71) with X"' put for X and
la', 13', 23', etc. for 12, 13, 23, etc., all the formulas and proixv
sitions which we have proved for (71) hold correspondingly for
(72) when 12', 13', 23', etc. are all real, aa they are when.
or,, B?„...or,are all of one sign.
345""". Going back now to § 345'"', and taking advantage of
what we have learned in § 345" and the oongequent treatment of
the problem, particularly that in § 345"^, we see now how to
simplify eqnations (14) of § 345'"' otherwise than was done in
P 345", by a new method which has the advantage of being
applicable also to materially simplify the general equations (13)
of § 34-5''. Apply orthogonal transformation of the co-ordinates
to reduce to a sum of squai-es of simple coordinates, the quad-
ratic function (70). Thus denoting by &(Vn/') what (?(iui)
tiecomes when ^,, ^,, etc. are substituted for a^, a,, etc.; and
denoting by n', n',...,n^' the values of the pairs of ttwts of the
determinantal equation of degree i, which are simply the n^ative
of the roots X' of equation (71) of degree Ji in X'; and denoting
''y fi' 'Ji' ^i> '7(«'"fi(7t(i th6 fresh co-ordinat«s, we have
6'(W = Ji«X' + 0 + ".*(f.'^0 + -+V«i.* + TJH*)}...(77).
It is easy to see that the genei-al eqnations of cycloidal motion
(13) of g 345" transformed to the {-co-ordinates come out in Ji
|Hiirs OS follows :
..Google
345"".]
DYNAMICAL LAWS AND PRINCIPLES.
r ddT d(t ^d7 1
ddT dQ , dV
( d dT dQ rfF „
ddT dO , <ir „
f d dT dQ dV „
+ — f- + n.(nn + -iir ~ 0
d dT dQ dV
..(78).
346"
detailed treatment at prraent of gyroatatic eyatenw with odd
■lumbersofd^reeBof freedom, but it is obvious from §346""" and
• 345" that the general equationB (13) of § 345" may, when i the
nttmber of freedoms is odd, by proper transformation from go-
oi-dinates ^,, ^„ etc. to a set of co-ordinates t f,, Vn—t\{i-Ht
VUi-H be reduced to the following form:
rddT dQ dr „1
d dT dQ , dr ^
r d dT dQ dF „
d dT dQ t dr ^
d dT dQ dr
d dT dQ t . dr ^
lit ^■'-'i ^1^,0
dtdi di dt J
..(70).
.Google
416 PBEUMINART. [346.
346. Tliere is scarcely aoj question in dynamicB more im-
portant for Natural Philosophy than the stability or instability
of motion. We therefore, before concluding this chapter, pro-
pose to give some general explanations and leading principles
regarding it.
A "conservative disturbance of motion" is a disturbance
in the motion or configuration of a conservative system, not
altering the sum of the potential and kinetic energies. A
conservative disturbance of the motion through any particular
configuration is a change in velocities, or component velocities,
not altering the whole kinetic enet^. Thus, for example, a
conservative disturbance of the motion of a particle through
any point, is a change in the direction of its motion, unaccom-
panied by change of speed.
" 347, The actual motion of a system, from any particular
configuration, is said to be stable if every possible infinitely
"'*^- small conservative disturbance of its motion through that con-
figuration may be compounded of conservative disturbances,
any one of which would give rise to an alteration of motion
which would bring the system again to some configuration
belonging to the undisturbed path, in a finite time, and without
more than an infinitely small digression. If this condition is
not fulfilled, the motion is said to be unstable.
Eiwupio. 348_ For example, if a body. A, be supported on a fixed
vertical axis ; if a second, B, be supported on a parallel axis
belonging to the first; a third, C, similarly supported on B, and
so on ; and if B, C, etc., be so placed as to have each its centre
of inertia as far as possible from the fixed axis, and the whole
set in motion with a common angular velocity about this axis,
the motion will be stable, from every configuration, as is evi-
dent from the principles regarding the resultant centrifugal
force on a rigid body, to be proved later. If, for instance, each
of the bodies is a fiat rectangular board hinged on one edge, it
is obvious that the whole system will be kept stable by centri-
fugal force, when all are in one plane and as fax out &om the
axis as possibla But if A consist partly of a shaft and crank,
as a common spinning-wheel, or the fly-wheel and crank of a
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348.] DYNAMICAL LAWS AND PEMCIPLEB. 417
steam-engine, and if £ be supported on the crank-pin as axis,
and turned iowarda (towards the fixed axis, or across the fixed
axia), then, even although the centres of inertia of G, D, etc.,
are placed as far from the fixed axis ao possible, consistent with
this position of B, the motion of the system will be unstable.
349. The rectilinear motion of an elongated bodj lengthwise,
or of a flat disc edgewise, through a fluid is unstable. But the
motion of either body, with its length or its broadside perpen-
dicular to the direction of motion, is stable. This is demon*
strated for the ideal case of a perfect liquid {§ 320), in § 321,
Example (2); and the results explained in § 322 show, for aKinptioito-
solid of revolution, the precise character of the motion con- dro^'mmio
sequent upon an infinitely small disturbance in the direction"™"''
of the motion from being exactly along or exactly perpendicular
to the axis of figure ; whether the infinitely small oscillation,
in a definite period of time, when the rectilineal motion is
stable, or the swing round to an infinitely nearly inverted po-
sition when the rectilineal motion is unstable. Observation
proves the assertion we have just made, for real fluids, air and
water, and for a great variety of circumstances affecting the
motion. Several illustrations have been referred to in § 325 ;
and it is probable we shall return to the subject later, as being
not only of great practical importance, but profoundly interest-
ing although very difficult in theory.
360. The motion of a single particle affords simpler and
not less instructive illustrations of stability and instability.
Thus if a weight, hung from a fixed point by a light inexten- crrcnhr
aible cord, be set in motion ao as to describe a circle about a pEajuimu.
vertical line through its position of equilibrium, its motion is
stable. For, as we shall see later, if disturbed infinitely little
in direction without gain or loss of energy, it will describe a
sinuous path, cutting the undisturbed circle at points succes-
sively distant from one another by definite fractions of the cir-
cumference, depending upon the angle of inclination of the
string to the vertical. When this angle is very small, the
motion is sensibly the same as that of a particle confined to
one plane and moving under the influence of an attractive
VOL. I. 27
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4<18 PBELDCIKJlBT. [350
force towards a fixed point, simply proportional to the distance ;
and the disturbed path cuts the undisturbed circle four times
circQkr in a revolution. Or if a particle confined to one plane, move
under theinfluence of a centre in this plane, attracting with a
force inversely as the square of the distance, a path infinitely
little disturbed from a circle will cut the circle twice in a re-
volution. Or if the law of central force be the nth power
of the distance, and if n + 3 he positive, the disturbed path will
cut the undisturbed circular orbit at successive angular in-
tervals, each equal to irljn + 3. But the motion will be
unstable if n be negative, and — n > 3.
Kinmioita- The criteriou of stability b easily inveetigated for circular
euluoririt' motJoD round a centre of force from the differentia] equation of
the geoeral orbit (g 36),
Let tlie value of A be such that motion in. a circle of radius a~'
satisfies this equation. Thatistosay, let i'/A'u'=u, wlienu = a.
Let now u B a -<- p, p being infinitely small. We shall have
if a denotes the value of j-(«--t7-;1 when m = o: and therefore
rfit \ AV/
the differential equation for motion infinitely nearly circular is
The integral of tliix is moat coni'eniently written
when a is positive, and
when a ifl negative
Hence wo see that the circular motion is stable in the former
case, and nnstoble in the latter.
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350.] DYSAMICAL Tj^WS AND PRINCIPLES. 419
For instance, it P- /ir' = ^m"', we have K
p
and putting Tr~t - u •= i, in tliis we find a = n + 3 ; whence t)ie
i-psnit stated above.
Or, taking Example (B) of § 319, and putting mP for P, and
mh for h,
k'u' m + m'\h* J'
Sm \ U'u'J m + m
Hence, putting u = a, and making A' = gat'/tna' bo that motion
in a circle of radius a~' may be possible, ve find
Hence the circular motion is always stible ; and the j>eriod of
tite variation produced by an infinitely small disturbance from
V~3
3»i
351. The case of a particle moving on a smooth fixed surface K^;*^"*
under the influence of no other force than that of the con- pi^it
straint, and therefore always moving along a geodetic line of jjj?;™*
the surface, affords extremely simple illustrations of stability
and instability. For instance, a particle placed on the inner
circle of the surface of an anchor-ring, and projected in the
plane of the ring, would move perpetually in that circle, but
unstably, as the smallest disturbance would clearly send it
away from this path, never to return until after a digression
round the outer edge, (We suppose of couree that the particle
is held to the surface, as if it were placed in the infinitely
narrow space between a solid ring and a hollow one encloaing
it.) But if a particle is placoil on the outermost, or greatest,
27—2
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420 PRELDIINABT. [351
niuiiu o( ^^^^^ °f ^^^ ""g> ^^^ projected in ita plane, an infinitely small
JPJIJ^ dieturbance will cause it to describe a sinuous path cutting the
■iu«^ circle at points round it successively distant by angles each equal
to vjbja, or intervals of time, irjbjuja, where a denotes
the radius of that circle, a> the angular velocity in it, and b the
radius of the circular cross section of the ring. This is proved
by remarlciiig that an infinitely narrow band from the outer-
most part of the ring has, at each point, a and b for its principal
radii of curvature, and therefore {§ 150) has for its geodetic
lines the gtetft circles of a sphere of radius \'ai, upon which
(§ 152) it may be bent.
352. In all these cases the undisturbed motion has been
circular or rectilineal, and, when the motion has been stable, the
effect of a disturbance has been periodic, or recurring with the
same phases in equal successive intervals of time. An illus-
tration of thoroughly stable motion in which the effect of a
disturbance is not " periodic," is presented by a particle sliding
down an inclined groove under the action of gravity. To take
the simplest case, we may consider a particle sliding down
along the lowest straight Une of an inclined hollow cylinder.
If slightly disturbed from this straight line, it wiU oscillate
on each side of it perpetually in its descent, but not with a
uniform periodic motion, though the durations of its excursions
to each side of the straight line are all equal.
Kj"e*'<'»*»- 353. Avery curious case of stable motion is presented by
""hi°Mait * particle constrained to remain on the surface of an ancbor-
iMkiD*. ring fixed in a vertical plane, and projected along the great
circle from any point of it, with any velocity. An infinitely
small disturbance will give rise to a disturbed motion of which
the path will cut the vertical circle over and over again for
ever, at unequal intervals of time, and unequal angles of the
circle ; and obviously not recurring periodically in any cycle,
except with definite particular values for the whole energy,
K ime of which are less acd an infinite number are greater than
(bat which just suffices to bring the particle to the highest
point of the ring. The full mathematical investigation of these
..Google
S53.] DYNAMICAL LAW3 AND PRINCIPLES. 421
circumstancea would afford an excellent eierciae in the tteory
of differential equations, but it is not necessary for our present
illustrations.
834. In this case, as in all of stable motion with only two ^ii«1oit
degrees of freedom, which we have just considered, there has *'''i*r-
been stability throughout the motion ; and an iofiQitely small
disturbance from any point of the motion has given a disturbed
path which intersects the undisturbed path over and over again
at finite intervals of time. But, for the sake of simplicity at
present confining our attention to two degrees of freedom, we
have a limited stability in the motion of an unresisted Pi'o-M^^ta.
jectile, which satisfies the criterion of stability only at points ^^v-
of its upward, not of its downward, path. Tbud if MOPQ be
the path of a projectile, and if at 0 it be disturbed by an infi- Km^f^^
nitely small force either way perpendicular to its instantaneous » pRiJwUte.
direction of motion, the disturbed path will cut the undisturbed
infinitely near the point P where the direction of motion is per-
pendicular to that at 0 : as we easily see by considering that
the line joining two particles projected from one point at the
same instant with equal velocities in the directions of any two
lines, will always remain perpendicular to the line bisecting the
angle between these two lines.
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422 FBELIMIHAJCT. [355.
3%. The priudple of varying actiou gives a matheioatical
criterion for stability or instability in erery case <4 motion.
Thus in the first place it is obvious, and it will be proved below
(§§ 338, 361), tbat if the action is a true minimum in the motion
of a system from any one configuration to the configuration
reached at any other time, however much later, the motion is
thoroughly unstable. For instance, io the motion of a particle
constrained to remain on a smooth fixed surface, and unin-
fluenced by gravity, the action is simply the length of the path,
mtiltiplied by the constant velocity. Hence in the particular
case of a particle iminfluenced by gravity, moving round the
inner circle in the plane of an anchor-ring considered above, the
action, or length of path, is clearly a minimum from any one
point to the point reached at any subsequent time. (The action
is not merely a minimum, but is the smaller of two mioimums,
when the course is from any point of the circular path to any
other, through less than half a circumference of the circle.)
On the other hand, although the path from any point in the
greatest circle of the ring to any other at a diiitance from it
Eilong the circle, less than tfjab, is clearly least possible if along
the circumference ; the path of absolutely least length is not
along the circumference between two points at a greater circular
distance than irVad from one another, nor is the path along the
circumference between them a minimum at all in this latter
notion on Case. On any surface whatever which is everywhere anticlastic,
ikiiurtu* or along a geodetic of any surface which passes altogether
■table. through an anticlastic region, the motion is thoroughly un-
stable. For if it were stable from any point 0, we should have
the given undisturbed path, and the disturbed path from 0
cutting it at same point Q ; — two different geodetic lines join-
i(auna<<* ing two points ; which is impossible on an anticlastic surface,
inasmuch as the sum of the exterior angles of any closed
figure of geodetic lines exceeds four right angles (§ 13C)
when the integral curvature of the enclosed area is negative,
which {§§ 138, 128) is the case for every portion of surfact;
thoroughly anticlastic. But, on tlie other hand, it is easily
proved that if we have an endless rigid hand of curved surface
everywhere synclastic, with a geodetic line running through its
pirtielc
..Google
335.] DTNAHICAL LAWS AND PBIKCIFLEa 423
middle, the motion of a particle projected aloog this lioe vill on ■ »
be stable throughout, and an infinitely slight disturbance will t*^ ■>
give a disturbed path cutting the given undisturbed path again
and again for ever at successive distances differing according to
the different specific curvatures of the intermediate portions of
thesurface. Iffromany
point, N, of the undis-
turbed path, a perpen-
dicular be drawn to cut
the infinitely near dis-
turbed path in ^, the
angles 0£^" and iV"0£
must (§ 138) be toge-
ther greater than a right angle by an amount equal to the in- raihrsntiai
tegral curvature of the area EON. From this the differential dianirbed
equation of the disturbed path may be obtained immediately.
Let iHOy^a, OIf=B, and NE = u; and let 9, a known
function of », be the specific curvature (g 136) of the surface in
the neigbbourhood of N. Let also, for a moment, ^ denote the
complement of the angle OBN. We have
.-*=/W
Henoo
S—
But, obviously.
^-%
hence -7-, + 3u = 0.
d»''
When d is constant (as in the case of the equator of a snr&ce of
revolution considered above, § 351), this gives
« = -4co8{»^& + -ff),
agreeing with the result (§ 351) which we obt^ed by develop-
ment into a spherical surfaca.
The case of two or more bodies supported on parallel axes
in the manner explained above in § 348, and rotating with the
centre of inertia of the whole at the least possible distance from
the fixed axis, affords a very good illustration also of this pro-
position which may be safely left as an exercise to the student.
..Google
424 PRELIMINART. [356.
Qencnihf 366. To investigate the effect of ao infinitely small Con-
or diMurbod servative disturbance produced at any instant in the motion
of any conservative system, may be reduced to a practicable
problem {however complicated the required work may be) of
mathematical analysis, provided the undisturbed motion is
thoroughly known.
Q^^^ (o) First, for a system having but two degrees of freedom to
ssr»2 ■"»'■». '•'
SS"' 2r=/',j-+«-+2«w 0).
where P, Q, £ are fimctiiins of the oo-ordinatea not depending
on the actual motion. Then
and the Xiagrangian equations of motion [§ 316 (2i)] are
We shall suppose the system of co-ordinates so chosen that
none of the fimctions P, Q, R, nor their differential ooefficientB
jr , et43., can ever become infinite
iUf,
(6) To investigate the effects of an infinitely small disturbance,
we may consider a motion in which, at any time t, the co-ordi-
nates are ^ -^p and ^ + q, p and q being infinitely small; and, by
simply taking the variations of equations (3) in the usual manner,
we arrive at two simultaneous differential equations of the second
degree, linear with resjiect to
p7 ?. p. 9- P, ?,
but having variable coefficients which, when the undisturbed
motion ^, i^ is fully known, may be supposed to be known
functions of I. In these equations obviously none of the coeffi-
cients can at any time become infinite if the data correspond to
a real dynamical problem, provided the system of co-ordinates is
properly chosen («); and the cotfficients- of p and gr are the
...(3).
..Google
356 (5).] DYNAMICAL LATTO AND PRINCIPLES. 425
values, at the time t, of P, R, and R, Q, respectiv6!7, in the Got*™! in
order in which they appeal' in (3), P, Q, R being the coefficients of diiiurbe
of a homogeneous quadratic function (1) which is essentially
positive. These properties being taken into account, it may be
shown that in no case can an infinitely small interval of time be
the solution of the problem presented (§ 347) by the question of
kinetic stability or instability, which is as follows : —
(c) The component velocities ^, ^ are at any instant changed
to ^ + a, ^ + j3, subject to the condition of not changing the
value of T. Then, a and p being infinitely small, it is required
to find the interval of time until qjp first becomes equal to ^/^.
{d) The dtfierential equations in p and q reduce this problem,
and in fact the full problem of finding the disturbance in the
motion when the undisturbed motion is given, to a practicable
form. But, merely to prove the proposition that the disturbed
course cannot meet the undisturbed coune until after some finite
time, and to estimate a Hi" it which this time must exceed in any
particular case, it may be simpler to proceed thus : —
(e) To eliminate t from the general equations (3), let them
first be transformed so aa not to have t independent variable.
We mxoA put
■• dUPi^-d\li^t •• dtd*<^-d<ixPt
* d? • *= Jt'
And by the equation of energy we have
..(4).
^JPdfj^Qd^*jJR^d^^
{^{E- F)}i
it being assumed that the system is conservative. Eliminating
dt and (Pt between this and the two equations (3), we find a
dilferential equation of the second degree between i^ and ^,
which is the differential equation of the course. For simplicity,
let us suppose one of the co-ordinates, ^ for instance, to be inde-
pendent variable; that is, let t/'^= 0. We have, by {4),
..Google
4^6 PRELIMINARY. [35C («).
■ and the reaalt of the elimiriatioti becomea
4
(6),
■'' ( ii) ^^'^''"'8 * function of ^ of the third degree, with vari-
able coefficients, none of which can become infinite as long as
£— V, the kinetic enei^, ia finite.
{/) Taking the variation of thia equation on the snppositioQ
that ijf becomea ijf-i-p, where ^ ia infinitely small, we have
(P«-J?)^.tig + i«,.0 (7),
where Z and J/ denote known functiona of ift, neither of which
liaa any infiaitely great vulue. This detemiinea the deviation, p,
of the course. Inasmuch as the quadratic (1) is essentially
ulwaya positive, FQ- 1? must be always positive. Hence, if
for a particular value of ^, p vanishes, and —■ has a given value
which defines the disturbance we suppose made at any instant,
^ must increase by a finite amount (and therefore a finite time
, must elapse) before the value of }( can be again zero; that ia to
say, before the disturbed course can again cut the undisturbed
course.
(y) The same proposition consequently holds for a system
having any number of dt^reea of freedom. For the preceding
proof shows it to hold for the system subjected to any frictionless
constraint, leaving it only two degrees of freedom; including
that particular frictionless constraint which would not alter either
the undisturbed or the disturbed course. The full general inves-
tigation of the disturbed motion, with more than two degrees of
freedom, takes a necessaiily complicated form, but the principles
on which it ia to be carried out are sufficiently indicated by
what we have done.
(A) If for L/PQ-IP we substitute a constant 2a, leea than
ita least value, irrespectively of sign, and for Af/PQ - ^, a
..Google
B (A).] DYNAMICAL LAWS AND PRINCIPLES. 427
cOQStant /3 greater olgebrucally than ita greatest value, wo ^"taitttori"
have an equation ordiiturbed
■ l^^^'frf-o <«)■ "
Here the vslue of p vanishea for values of if> Buccessively ex-
ceeding one another by vj^li — a', which ia clearly less tliau
the increase titat ^ must have in the actual problem before p
vanialiea a necond time. Also, we see from this that if a*=- ^
the actual motion ia unstable. It might of course be unstable
even if a'-<fi; and the proper analytical methods for finding
either the rigorous solution of (7), or a sulliciently near practical
solution, would have to be used to close the criterion of stability
or instability, and to thoroughly determine the disturbance of
the course.
(t) When the system is only a single particle, confined to a Diatrenti«l
plane, the differential equation of the deviation may be put dTiturbed
under a remarkably simple form, useful for many practical tJIiBk p«r-
problems. Let N be the normal com|H>tieut of the force, per pimna.
unit of the ma8B, at any instant, v the velocity, and p the radius
of curvature of the path. We have {§ 259)
P
Let, in the diagram, 0^ be the undisturbed, and OS the
diaturbed path. Let
^iV; cutting Oy at
right angles, be de-
noted by «, and OjV
by t. If further we
denote by p' the
radius of curvature
in the disturbed path,
remembering that u is infinitely small, we easily find
i-Ugn." (1,,.
Hence, using S to denote variations from NtoE, we have
tif.^ttJS^^J^'^v-\ (10).
p p Vrfa* pV
..Google
[356 (.).
&-7--?=" (">■
or, if we denote by f the rate of vanation of N, per unit of dia-
taace irom the point ^ in the normal direction, bo that SA' = iu.
4
28 PBELniraAKT.
Diftnmttal
plHN.
and therefore
Hence (10) becomes
s?^(^-a-» w-
This includes, as a particular case, the equation of deviation
from a circular orbit, investigated above (§ 350).
857. If, from any one confignration, two courses differiDg
infinitely little from one another have i^in a coofiguratiou in
Kinitra common, this second configuration will be called a kinetic focus
relatively to the first : or (because of the reversibility of the
motion) these two configurations will be called conjugate kinetic
foci. Optic foci, if for a moment we adopt the corpuscular
theory of light, are included as a particular case of kinetic foci
in general. By § 356 (ff) we see that there must be finite in-
tervals of space and time between two conjugate foci in every
motion of every kind of system, only provided the kinetic
energy does not vanish.
368. Now it is obvious that, provided only a suflSciently
short course is considered, the action, in Any natural motion of
Tbaorem at a system, is less than for any other cour^ between its terminal
•otkw. configurations. It will be proved presently (§ 361) that the first
Action configuration up to which the action, reckoned from a given
minimum initial Configuration, ceases to be a minimum, is the first kinetic
incTudin?^ focus; aod conversely, that when the first kinetic focus is
passed, the action, reckoned from the initial configuration, ceases
to be a minimum ; aud therefore of course can never again be a
minimum, because a course of shorter action, deviating infi-
nitely little from it, can be found for a part, without altering the
remainder of the whole, natural course.
..Google
DTNAHICAL LAWS AND FBTNCIFLES.
429
359.]
359. In Buch statements as this it will frequently be cod-
venient to indicate particular configurations of the system by
single letters, as 0, P, Q, JB; and any particular course, in
wbich it mores through configurations thus indicated, will be
called the course O...P...Q...R The action in any natural
course will be denoted simply by the terminal letters, token in
the order of the motion. Thus OR will denote the action from
O to R; and therefore OR = —RO. When there are more
real natural courts from 0 to £ than one, the analytical
expression for OR will have more than one real value ; and it
may be necessary to specify for which of these courses the
action is reckoned. Thus we may have
OB torO...E...B,
OR for O...E'...R,
OS for 0..,E"...R,
three different values of one algebraic irrational expression.
860. In terms of this notation the preceding statement
(§ 358) may be expressed thus : — If, for a conservative system,
moving on a certain course O...P...O'...P', the first kinetic
focus conjugate to 0 be 0', the action OP", in this course, will
be less .than the action along any other course deviating in-i
finitely little from it: but, on the other hand. Of is greater than
the actions in some courses from 0 to f deviating infinitely
little from the specified natural course O...P...G,,.P',
361. It must not be supposed that the action along OP is
necessarily the least possible from 0 to P. There are, in fact,
cases in which the action ceases to be least of aU possible, before
llftantiam.
jiGoogle
430 rREUMINAIlT. [301.
Twoormora a kmelic focus 13 reached. Tlius if OEAPO'E'A' be a sinuous
j^niiaum geodetic liue ctitting tlie outer circle of an anchor-ring, or
putibie.' tlie equator of an ohlate eplieroid, in successive points 0,
A, A', it is easily seen that 0', the first kinetic focus
conjugate to O, must lie somewhat beyond A, But tlie
length OEAP, although a minimvm (a stable position for a
S^mo''" stretched string), is not the shortest distance on the surface
"V'iinniD°* f™™ 0 to 7*, as this must obviously be a line lying entirely on
(t^^Jj^ one side of the great circle. From 0, to any point, Q, short of
1^551.*'" -^< tlis distance along the geodetic OEQA is clearly the least
possible : but if (? be near enough to A (that is to say, between
A and the point in which the envelope of the geodetics dravn
from 0, cuts OEA), there will also be two other geodetics from
0 to Q. Tlie length of one of tlieso will be a minimum, and
that of the other not a minimum. If Q is moved forward to A,
the former becomes OE^A, equal and similar to OEA, but on the
other side of the great circle: and the latter becomes the great
circle from 0 to A. If now Q be moved on, to P, beyond A,
the minimum geodetic OEAP ceases to be the less of the two
minimiims, and the geodetic OFP lying altogether on the other
side of the great circle becomes the least possible line from 0 to /*.
But until P is advanced beyond the point, 0*, in which ft is cut
by another geodetic from 0 lying infinitely nearly along it, the
length OEAP remains a minimum, accortling to the general
proposition of § 3-58, which we now proceed to prove.
DilTerenwi (") Eefeniiig to tlie iioUition of | 360, lot /', be any configura-
iiid^°^d''° tion difiet'iiig intiiiitely little froia P, but not on the course
IVS**' O.-P.-.O'...!"; and let ,S' be a configuration on this courae.
tnuRie. i-cacUed at some finit« time after P is pnased. Let ^, 0,,.. be
tlie co-ordinates of y, ivnd i/i,, ^.,-.- those of/*,, nnrt t«t
^. ~ ^ ~ S^, '^, — 'l> — 8^ . . .
Thus, by Taylor's ilieovem,
jiGoogle
361 (a).] DYNAMICAL LAWS AND PHIKCIPLES. 431
But i£ (,-!],... denote tbe conipontjiits of momentuio at Pin the Diffemwo
course 0...P, which are the sHme oa those at P in the oontiniia- lideasnil
tion, P...S, of this courae, we have [§ 330 (18)] l!ki™'ii^ ""
. dOP dPS _dOP__dPS_
' ^^ dil^ " rfi^ ' ''" 'di> "^ ~i/^ ' ■■■
Hence the coefficients of the terms of the first degree of S^, S^
in the preceding expression vanish, and we Lave
+ etc. I
(6) Now, assuming
!P, = n,&i. + /3,4 + ...[ (2),
et«. etc. )
according to the known method of linear transformationti, let
a,, j3,,... a„ ^,,... be SO chosen that tbe preceding quadratic
function be reduced to the form
A^x' + A^x^-t- ... + A^',
tbe whole number of degrees of fi-eedom being i.
This may be done in nn infinite variety of ways; and, towards
fixing upon one ]iarticu1ar way, we may talce a, — tlf, A '= <^i ^^ >
and subject the others to the conditions
^, + ^^j + ... = 0, ^, + <^;8, + ... = 0, etc.
This will make A, = (i: for if fora moment we suppose /* to 1>o
on the course 0...P...0', we have
31, = ^(&;'' + H°+ .-■)■ '^1-1 = 0, ...*, = 0, fl;, = 0.
But in thin case OP, + P^S- OS; and therefore tbe value of the
quadratic must be stero; that is to soy, we must have A, = 0.
Hence we have
0P, + P^- OS=HA,x,' *A,x,'+ ... + A, .a;,..*)!
+ R i ^'
where Ji denotes a remainder consisting of terms of the thinl
itnd higher degreen in Infi, 6^ et«., or in x,, r-^, etc.
..Google
Z PRELIHIN'ART. [3G1 (c).
(e) Another form, whioh will be used below, m&j be giren to
tie same expression thus : — Let (f,, ij,, Z^,...) and (£/, ij/, {/,...)
be the compODenta of momentum at P^, in the courses OP, and
P,S respectively. By § 330 <18) we have
and therefore by Taylor's theorem
i-
dOP
d<l> *
d'OP
Similarly,
dPS d'PS.
dOP
H +
dPS
d-p '
and BO for %'~it,j ^^ Hence (1) la the same as
or,*p/i-os—i{{i;-()it*(,n;-i,)H*-
■■(5).
where S denotes a remainder consiating of terms of tiie third and
higher degrees. Also the bauaform&tioa ^m Si^, S^ ... to
*,i *,j ■■■1 gi^os clearly
f,' — £, = — (-^,0,*, +.i,a,a!j+ ... ■*- A,_jai_^x,_\
ij;-1, = -('*,^,a!,+-^A*.+ ■■ +A~A-,'i-i)\ (6)-
etc. etc }
{d) Now for any infinitely small time the velodtjee remain
sensibly ctmstant; as also do the coefficients (iji, tp), {ip, <ft), etc,
in the expression [§ 313 (2)] for Ti and therefore for the action
J2Tdt = j2ffj2fdt
= y:iT{(^, ,f) {^ - ^.)' + 2 {^, *) {^ - ^.) (*-*,) + etc. Ji
■where {^^, ^„ ...) are the co-ordinates of the configuration from
wliich the action is reckoned. Hence, if P, P", P' be aaiy three
configurations infinitely near one another, and if Q, with the
pmper diHerencea of co-ordinates written after it, be used to
denote square roots of quadratic functions such as that in the
pi-eceding expression, we have
..Google
361 (d).] DTKAHICAL LAWS Aim PBINCIPLES. 433
F'p"=^.Q{{>i'' -<!>"), {'f'' -r), -n (n Jliih-sS*
F"P =j2T.Q\{r-^), (*"-*'), ...}J tE^
la the partlcnlar case of a single free particle, these expresaiona
become simply proportional to the distances PP, FT", P"P;
and by Euclid we hare
rP+PP'<PP'
unless P is in the straight line PP'.
The Terificati<m of this propodtion by the pivoeding expresdcms
(7) is merely its proof by co-ordinate geometry with on oblique
rectilineal system of co-ordinatea, and is necessarily somewhat
eomplicated. If (^, ^) = (^, 9) = {6, ifi) = 0, the co-ordinates be-
come reotangolar and the algebraio proof is easy. There is no
difficulty, by following the analogies of these known processes to
proTB that, for any ntnnber of coK>rdiiiates, ^, ^ etc., we have
I'P + PP'^F'P",
unless
f -^ "" ^'-^~&'-&
(expressing that P is on the conise from P to P'), in which case
P'P-^Pi*" = P'P",
PP, eta, being given by (7). And fUrther, by the aid of (1),
it is easy to find the proper expression for PP^-FP' —PP',
when P is infinitely little off the oourae &om P to F": but it is
quite nnnecessaiy for ns hero to enter on such purely algebraic
investigations.
(e) It is obvious indeed, as has been already said (§ 358), that
the action along any natural course is Iha least pouible beUoeen
its terminal eonfiffuraiionM if ordy a sufficiently short oonrse is
included. Hence for all cases in which the time from 0 to j5 is
less than seme particular amonnt, the quadratic term in the ex-
pression (3) for OP, + P,S-OS is necessarily positive, for all
values of a;,, x,, etc; and therefore J,, A Jj_, must each be
positive.
(/) Let now S be removed further and further from 0, along amom on
the definite course 0...P...0', until it becomes (/. When it is SJ^'n-
ff, let P^ be taken on a natural course ihroogh 0 and ff, de- ^^'uci'tte
TOL. I. 28
..Google
thinl.
434 PBIXIMINAET. [361 (/).
bctwMDtwo ■riating infinitely little from the cooise OPQf, Hen, aa OP,(f
Uuit^lML ia a natura] courses
prDTHl ulU* .. , . , «
SSf f/-f,-V-^.--i>;
and tberefore (5) becomes
OP,^P,a-OOI = B,
wUcIi proves that the chief, or quadratic, term in the other ex-
pression (3) for the same, vanishes. Hence one at least of 'Uie
coeffidents A^, A,,... must vanish, and if one onlj, jj^_, = 0 for
instance, -we moat have
!r, = 0, !r, = 0,...!c,_,=0.
These equations express the condition that P, Ilea on a natural
course from 0 to O".
utvDiidM, (ff) Conversely if ons or more of the coefficients A^, A^, etc,
inBnitotr vanijBhee, if for instance A^^ -0,S must be a kinetic focoa. For
e take P, so that
?IS "« i*^«. by (6),
(A) Thus we have proved that at a kinetic focus conjugate to
0 the action from 0 is not a minimum of the first order*, and
tiiat the last configuration, iip to irhich the action from O m a
minimum of the first order, is a kinetic focus coqjugate to O.
(t) It remains to be proved that the action from 0 ceases to
be a minimum when the first kinetic focus ctmjugate to 0 ia
pascied. Let,aa above (§360), (?...P...O'...i^be anatural oonrse
extending beyond 0', the first kinetic focus cmjugate to 0. Let
P and i^ be so near one another that there is no focas conjugate
to either, between them; and let 0...P,...0' be a natural course
from OtoO' deviating infinitely little from O...P...Cf, By what
lTatiii»i we have just proved (e), the action OQ along O...P,...<y differs
"^^^ not *"^'y by ^ an infinitely small quantity of the third order, from
n"h^m' ^'^ "*'™ *^^ "W 0:P.:<y, and therefore
K^a. Ae.{0..,P...&...F^ = A<:.{p...P,...<y)^aF^R
toS2!° ^op,^p,<y^(yF^R.
* A nuoimmn or minimom "of the first order" of bdj fanction of one or
more variablee, is one in vhioh the diSereubal of the flnt degree vsnidieB, bat
not that of the second degree.
..Google
301 (0.] DYNAMICAL LAWS AND PRINCIPLES. 43S
But, by a proper applicattoa of (e) we aee that Naiuni
There Q denotefl an infiaitely Bmall quantity of the second order, ■ction.
whicb U easentially positive. Hence wwao
toon*.
Ac{O...P...O'...P) = OP,^F^ + Qa-R,
and tbere&ire, as £ is infinitely small in oompaiiaon with Q,
Ac{O...P...O'...P')>OP, + F^.
Hence the broken course O...P_, P,...P' has less action than
the natural course O...P...O'...P', and therefore, as the two
ai>e infinitely near one another, the latter is not a minimum.
862. As it has been proved that the action from any con- a oonne
figuration ceaseB to be a minimum at the first conjugate kinetic ciudn no
focus, we see immediately that if ff be the first kinetic fcKus J<>ipt« ^
conjugate to 0, reached after pa-ssing 0, no two configurations f"?^'^
on this course from 0 to (/ can be kinetic foci to one another, nopiirof
For, the action from 0 just ceasing to be a minimum when 0' ^
is reached, the action between any two intermediate configura-
tions of the same course is necessarily a mimmum.
363, When there are i degrees of freedom to move there HownMiy
are in general, on any natural course from any particular con- in w cow-
figuration, 0, at least i — 1 kinetic foci conjugate to 0, Thus,
for example, on the course of a ray of light emanating from
a luminous point 0, and passing through the centre of a con-
vex lens held obliquely to its path, there are two kinetic foci
conjugate to 0, as defined above, being the points in which the
line of the central ray is cut by the so-called " focal lines"* of
a pencil of rays diverging from 0 and made convei^ent after
passing through the lens. But some or all of these kinetic foci
may be on the course previous to 0 ; as for iastan6e in the
case of a common projectile when its course passes obliquely
downwards through 0. Or some or all may be lost ; as when,
in the optical illustration just referred to, the lens is only
strong enough to produce convei^nce in one of the principal
planes, or too weak to produce convergence in either. Thus
* In our BMODd Tolnme we hope to eive all nmeaiar; elameutar; ezplonationa
on this labjeet.
28—2
..Google
436 I>EELIMmABT. [363.
Howonix also in tbe case of the undisturbed rectilineal moUon of a
kiiHticCoa . . , , _ 111-
Inaarcan. point, or 10 the motion of a point ummluenced by force, on
an anticlastic surface (§ 355), there are no real kinetic foci.
In the motion of a projectile (ilot confined to one vertical plane)
there can only be one kinetic focus on each path, conjugate
to one given point; though there are three degrees of freedom.
Agftin^ there may be any number more than t — 1, of foci in
one course, all conjugate to one configuration, as for instance
OQ tbe course of a particle uninfluenced by force, moving round
the surface of an anchor-ring, along either the outer great
cirete, or along a sinuous geodetic such as we have considered
in § 361, in which clearly there are an infinite number of foci
each conjugate to aoy one point of the path, at ec[ttal successive
distances from one another.
Beferruig to the notation of § 361 (/), let 5' be gradoally
moved on until first one of the ooefficients, A,_, for instancy
vanishes; Uien another, A,_^, etc.; and so on. We have seen
that each of these positiona of iS^ is a kineUc focus: aad thus by
the BnccMBiTe vanishing dF the » — 1 coefficients we have i — 1
too. If none of the coefficients can ever nuush, there are no
kinetic focL If one or more of them, after vanishing, comes to
a minimum, and again vanishes, as iS^ is moved on, there may be
any number more than i— 1 of foci each conjugate to the same
configuration, 0.
•ntomott 364. If t — 1 distinct* courses from a configuration 0, each
Mtkm. differing infiuitely little from a certain natural course
O...S...O,...0,...O^....Q,
cut it in configurations 0,, 0,, 0^,...0^^, and if, besides these,
there are not on it any other kinetic foci conjugate to 0, between
0 and Q, and no focus at all, conjugate to E, between E and Q,
the action in this natural course from 0 to Q is the maximum
for all courses 0...P,, P,...Q; P, being a configuration infinitely
nearly agreeiog with some configuration between E and 0^ of
the standard course O...£...0,...0,...0^.,...(?, and 0...P, P.—Q
■ Two eonrsM «» not called distinet if they differ from one utotfaer only in
the atieolnte magnitnde, not in the proportions of the componenta, of the
deriations b; whieh the; diSer from the Btandard ooorse.
..Google
364.] DTNAMICAL LAWS AKD PBINCIPLE3. 137
denoting the natural cooTEes between 0 and P,, and P, and Q,
which deviate infiiiitely little from this standard coursa
In § 361 (*), let 0" be any one, 0„ of the foci 0„ 0,, ... 0,_„
and let P^ be called P^ in tliis case. The demonstration there
given BhowB that
OQ>OP, + P^Q.
Hence there are t- 1 different broken ooursea
0...P,, P,...Q; 0...P,, P,...Q; etc.,
in each of which the action is less than in the standard oonrse
fiom 0 W> Q. But whatever be the deviation of P,, it may
clearly be compounded of deviations P to i*,, P to P„ P to P„
..., PtoP,,,, corresponding to these » — 1 cases respectively ;
and it is easily seen from the analysis that
OP, + P,0- Oe= (OP, + P,« - OQ) + {OP^ + P,Q - OQ) + ...
Hence OP, + P,Q < OQ, whidi was to be proved.
363, Considering now, for simplicity, only cases in which Appih*.
there are but two degreea (§§ 195, 204) of freedom to move, dqfreMoi
we see that after any infinitely small conservative disturbance
of a system in passing through a certain configuration, the
system will first again pass through a configuration of the
xmdisturbed course, at the first configuration of the latter at
which the action in the undisturbed motion ceases to be a
minimum. For instance, in the case of a particle, confined to
a surface, and subject to any conservative system of force, an
infinitely small conservative disturbance of its motion through
any point, 0, produces a disturbed path, which cuts the un-
disturbed path at the first point, £7, at which the action in the
undisturbed path from 0 ceases to be a minimum. Or, if
projectiles, under the influence of gravity alone, be thrown irom
one point, 0, in all directions with equal velocities, in one
vertical plane, their paths, as is easily proved, intersect one
another consecutively in a parabola, of which the focus is 0,
and the vertex the point reached by the particle projected
directly upwards. The actual course of each particle from 0
is the course of least possible action to any point, P, reached
before the enveloping parabola, but is not a course of minimum
action to any point, Q, in its path after the envelope is passed.
..Google
43S PBELIHINART. [366.
Appuoi- 366. Or a?ain, if a particle slides rouad aloDe tbe ereatcst
ttonHotao » ' ' ,11, 1 .
tr^^mf' circle of the smooth inner surface of a hollow anchor-nng, the
" action," or simply the length of path, from point to point, will
be least possible for lengths (§ 351) less than ir Vat. Thus, if
a string be tied round outside on the greatest circle of a
perfectly smooth anchor-ring, it will slip off unless held in
position by staples, or checks of some kind, at distances of not
less than ir '/ah troro one another in succession round the circle.
With reference to this example, see also § 361, above.
Or, of a particle sliding down an inclined cylindrical groove,
the action from any point will be the least possible along the
strtught path to any other point reached in a time less than
that of the vibration one way of a simple pendulum of length
ec|ual to the radius of the groove, and influenced by a force
equal g cos r, instead of g the whole force of gravity. But the
action will not be a minimum from any point, along the straight
path, to any other point reached in a longer time than this.
The case in which tbe groove is horizontal (t = 0) and the par-
ticle is projected along it, is particularly simple and instructive,
and may be worked out in detail with great ease, without as-
suming any of the general theorems regarding action.
iiMniiion'i 367. In the preceding account of the Hamiltonian principle,
~™ and of developments and applications which it has received, we
have adhered to the system (§§ 328, 330) in which the initial
and fin^ co-ordinates and the constant sum of potential and
kinetic energies are the elements of which the action is supposed
to be a function. Another system was also given by HamiltoD,
according to which the action is expressed in terms of the initial
and final co-ordinates and the time prescribed for the motion;
and a set of expressions quite analogous to those with which
we have worked, are established. For practical applications
this method is generally less convenient than the other ; and
tbe analytical relations between the two are so obvious that we
need not devote any space to them here.
868. We conclude by calling attention to a very novel
analytical investigation of the motion of a conservative system,
by Liouville {Cotnptes Rmdue, June 16, 1856), which leads im-
..Google
368.] DYNAMICAL LAWS ASD PRINCIPLES. 439
mediately to the principle of least action, and the HamiltoniaD
principle with the developments by Jacob! and others; but
which also establishes a very remarkable and absolutely new
theorem regarding the amount of the action along any con-
Btrained course. For brevity we shall content ourselves witJi
giving it for a ungle free particle, referring the reader to the
original article for Liouville's complete investigation in terma
of generalized co-ordinates, applicable to any conservative
system whatever.
Let (x, y, z) be the co-ordinates of any point through whioh
tiie particle may move : V its potential energy in t^ poeition ;
S the Bum of the potential and kinetic energies of the motion in
question : A the action, from any position (x„, y„ 2J to (x, y, z)
along any course arbitrarily chosen (supposing, for instance, the
particle to be guided along it by a frictionless guiding tube).
Then @ 326), the mass of the particle being taken as unity,
Now let 9 be a fnnction ot x, y, x, which satisfies the partial
differential equation
But -y- dx + -^dy + ~rdx^dS,
dx dy ' OK
and, \li,-§,i denote the actual component velocities along the
arbitrary path, and ^ the rate at whioh d increases per unit of
time in this motion,
diB = &fi, dy^dt, dx = idi, £»=3dt
Hence the preceding becomes
A=>Sdd
n f dSi ,dSi\' f.dSi ^dSi\* fJSf .dSs\'\
jiGoogle
CHAPTER III.
KXPERIEHCE.
ot**™*!"" 389. Br the term Experience, in physical science, we desig-
""^ nate, according to a suggestion of Herschel's, our means of
becoming acquajnted with the material univerae and the laws
which regulate it In general the actions which we see ever
taking place around us are complex, or due to the simultaneous
action of many causes. When, as in astronomy, we endearour
to asceitfnn these causes by simply watching their effects, we
observe; when, as in our laboratories, we interfere arbitrarily
with the causes or circumstances of a phenomenon, we are said
tOB
370. For instance, supposing that we are possessed of instru-
mental means of measuring time and angles, we may trace out
by successive observations the relative position of the sun and
earth at different instants; and (the method is not susceptible
of any accuracy, but is alluded to here only for the sake
of illustration) from the variations in the apparent diameter
of the former we may calculate the ratios of our distances from
it at those instants. We have thus a set of observations in-
volving time, angular position with reference to the sun, and
ratios of distances &om it: sufficient (if numerous enough) to
enable us to discover the laws which connect the variations
of these co-ordinates.
Similar methods may be imagined as applicable to the
motion of any planet about the sun, of a satellite about its
primary, or of one star about another in a binary group.
871. In general all the data of Astronomy are determined
in this way, and the same may be said of such subjects as
..Google
371.] ESPEEIENCE. 441
Tides and Meteoroli^. laottormal Lines, Lines of Equal Dip, obKn»-
lines of Equal Intensity, Lines of Equal "Variation" (or "Decli-
nation" as it has Btill leisa happily been sometimee called),
the Connexion of Solar Spots witb Terrestrial Magnetism,
and a host of other data and phenomena, to be explained
under the proper heads in the course of the work, are thus
deducible from Observation merely. Id these cases the apparatus
for the gigantic experiments is found ready arranged in Nature,
and all that the philosopher has to do is to watch and measure
their pn^ress to its last details.
372. Even in the instance we have chosen above, that of
the planetary motions, the observed effects are complex^ because,
unless possibly in the case of a double star, we have no iustanoe
of the undisturbed action of one heavenly body ou another;
but to a first approximation the motion of a planet about the
sun is found to be the same as if no other bodies than these
two existed; and the approximation is sufficient to indicate
the probable law of mutual action, whose full confirmation is
obtained when, its truth being assumed, the disturbing effecta
thus calculated are allowed for, and found to account com-
pletely for the observed deviations from the consequences of
the first supposition. This may serve to give an idea of the
mode of obtaining the laws of phenomena, which can only be
observed in a complex form — and the method can always be
directly applied when one cause is known to be pre-eminent.
373. Let us take cases of the other kind — in which the effects Bipori-
are so complex that we cannot deduce the causes from the
observation of combinations arranged in Nature, but must en-
deavour to form for ourselves other combinations which may
enable us to study the effects of each cause separately, or at
least witb only slight modification from the interference of
other causes.
374; A stone, when dropped, falls to the ground; a brick
uid a boulder, if dropped from the top of a cliff at the same
moment, fall side by side, and reach the ground together. But
a brick and a slate do not; and while the former EelUb in a
nearly vertical direction, the latter describee a most complex
..Google
442 PBELDdKiBT. [374.
^JMi- path. A sheet of paper or a fragment of gold leaf presenta even
greater irregularities than the slate. But by a slight modifica-
tion of the circumstanceB, we gain a considerable insight into
the nature of the question. The paper and gold leaf, if rolled
into balls, fall nearly in a vertical line. Here, then, there are
evidently at least two causes at work, one which tends to make
all bodies fall, and fall vertically; and another which depends
on the form and substance of the body, and tends to retard
its fall and alter its course from the vertical direction. How
can we study the effects of the former on all bodies without
sensible complication from the latter? The effects of Wind,
etc., at once point out what the latter cause is, the air (whose
existence we may indeed suppose to have been discovered by
such effects) ; and to study the nature of the action of the fonner
it is necessary to get rid of the complications arising ftx>m the
presence of air. Hence the necessity for Experiment By means
of an apparatus to be afterwards described, we remove the
greater part of the air from the interior of a vessel, and in that
we try again our experiments on the fall of bodies; and now a
general law, simple in the extreme, though most important in
its consequences, is at once apparent — ^viz., that all bodies, of
whatever size, shape, or material, if dropped side by side at the
same instant, fall side by side in a space void of air. Before
experiment had thus separated the phenomena, hasty philo-
sophers bad rushed to the conclusion that some bodies possess
the quality of heaviness, others that of lightness, etc. Had this
state of confusion remained, the law of gravitation, vigorous
though its action be throughout the universe, could never have
been recognised as a general principle by the human mind.
Mere observation of lightning and its effects could never have
led to the discovery of their relation to the phenomena pre-
sented by rubbed amber, A modification of the course of
nature, such as the collecting of atmospheric electricity in
our laboratories, was necessary. Without experiment we could
never even have learned the existence of terrestrial magnetism.
^nitttiar 37^- When a particular agent or cause is to be studied,
^tf^?^ experiments should be arranged in such a way as to lead if
"^^ possible to results depending on it alone; or, if this cannot be
..Google
375.] EXPEEIENCE. 443
done, they sbould be arranged bo as to show differences pro- Baiufcr
, , , -^ . ., " 'the aondaat
duced by varying it ofeipcn-
376. Thus to determine the resistance of a wire against the
conduction of electricity through it, we may measure the whole
strength of current produced in it by electromotive force between
its ends when the amount of this electromotive force is given,
or can be ascertained. But when the wire is that of a submarine
telegraph cable there is always an unknown and ever varying
electromotive force between its ends, due to the earth (produc-
ing what is commonly called the "earth-current"), and to deter-
mine its resistance, the difference in the strength of the current
produced by suddenly adding to or subtracting from the terres-
trial electromotive force the electromotive force of a given
voltaic battery, is to be very quickly measured ; and this is to be
done over and over again, to eliminate the effect of variation of
the' earth-current during the few seconds of time which must
elapse before the electrostatic induction permits the current
due to the battery to reach nearly enough its full strength to
practically annul error on this score.
377. Endless patience and perseverance in designing and
trying different methods for investigation are necessary for
the advancement of science: and indeed, in discovery, he
is the most likely to succeed who, not allowing himself to bo
disheartened by the non-success of one form of experiment,
judiciously varies bb methods, and thus interrogates in every
conceivably useful manner the subject of his investigations.
878. A most important remark, due to Herschel, regards b
what are called residual phenomena, When, in an experiment,
all known causes being allowed for, there remain certain un-
explained effects (excessively slight it may be), these must
be carefully investigated, and every conceivable variation of
arrangement of apparatus, etc., tried; until, if possible, we
manage so to isolate the residual pheaomenon as to be able
to detect its cause. It is here, perhaps, that in the present
state of science we may most reasonably look for extensions
of our knowledge; at all events we are warranted by the recent
history of Natural Philosophy in so doing. Thus, to take only
..Google
444 PBEXDnKABIT. [37&
a very few ioatances, and to saj nothiag of the discoveiy of
electricity and magaetism by the ancieats, the peculiar amell
observed in a room in whioh an electrical machiDe is kept in
action, was long ago observed, but called the " smell of elec-
tricity," and thus left unexplained. The eagadty of Schonbein
led to the discovery that this is due to the formation of Ozone,
a most extraordinary body, of great chemical activity; whose
nature is still uncertaiu, though the attention of chemists has
for years been directed to it
879. Slight anomalies in the motion of Uranus led Adams
and Le Yerrier to the discovety of a new planet; and the fact
that the oscillations of a magnetized needle about its position
of equilibrium are "damped" by placing a plate of copper below
it, led Arago to his beautiful experiment showing a resistance to
relative motion between a magnet and a piece of copper; which
was first supposed to be due to nu^etiam in motion, but which
soon received its correct explanation from Faraday, and has since
been immensely extended, and applied to most important pur-
poses. In hict, from this accidental remark about the oscillation
of a needle was evolved the grand discovery of the Induction of
Electrical Currents by magnets or by other currents.
We need not enlarge upon this point, as in the following
p^es the proofs of the truth and usefulness of the principle will
continually recur. Our object has been not so much to give
applications as principles, and to show how to attack a new com-
bination, with the view of separating and studying in detail the
various causes which generally conspire to produce observed
phenomena, even those which are apparently the simplest.
CiuoiKeted 360. If on repetition several times, an experiment con-
tinusUy gives different results, it must either have been veiy
carelessly performed, or there must be some disturbing cause
not taken account of. And, on the other hand, in cases where
no very great coincidence is likely on repeated trials, aa unex-
pected degree of agreement between the results of various trials
should be regarded with the utmost suspicion, as probably due
to some unnoticed peculiarity of the apparatus employed. In
..Google
380.] EXPERIENCE. 445
either of these cases, however, careful observation cannot fail riieipoct*d
to detect the cause of the discrepancies or of the unexpected or di«ur.
■1 . , . >■ ■ ■ „ dwioeof
^p?eement, and may possibly lead to discovenes m a totally™!"^
unthought-of quarter. Instances of this kind may be given uiai^
without limit ; one or two must sufGce.
381. Thus, with a very good achromatic telescope a star
appears to have a sensible disc But, as it is observed that
the discs of all stars appear to be of equal angular diameter,
we of course suspect some common error. Limiting the aper-
ture of the object-glass increases the appearance in question,
which, on full iDvestigation, is found to have nothing to do with
discs at all. It ie, in fact, a diffraction phenomenon, and will
be explained in our chapters on light.
382. Again, in measuring the velocity of Sound by experi-
ments conducted at night with cannon, the results at one station
were never found to agree exactly with those at the other;
sometimes, indeed, the differences were very considerable. But
a little consideration led to the remark, that ou those nights in
which the discordance was greatest a strong wind was blowing
nearly from one station to the other. Allowing for the obvious
effect of this, or rather eliminating it altogether, the mean velo-
cities on different evenings were found to agree very closely.
383. It may perhaps be advisable to say a few words here
about the use of hypotheses, and especially those of very
different gradations of value which are promulgated in the
form of Mathematical Theories of different branches of Natural
Philosophy.
384. Where, as in the case of the planetary motions and
disturbances, the forces concerned are thoroughly known, the
mathematical theory is absolutely tru^ and requires only ana-
lysis to work out its remotest details. It is thus, in general, far
ahead of observation, and is competent to predict effects not yet
even observed — as, for instance. Lunar laequalities due to the
action of Venus upon the Earth, etc. etc., to which no amount
of observation, unaided by theory, could ever have enabled us
to assign the true cause. It may also, in such subjects as Geo-
metrical Optics, be carried to developments for beyond the reach
..Google
446 PEELIHIKART. [384.
of experiment ; but in this science the assumed bases (^ the
theory are only approximate ; and it fajle to explain in all theii
peculiarities even each comparatively simple phenomena ss
Halos and Rainbows — though it is perfectly auccessful for the
practical purposes of the maker of microscopes and telescopes,
and has enabled really scientific instrument-makers to cany tbe
conBtruction of optical apparatus to a d^^ree of perfection which
merely tentative processes never could have reached.
3B5. Another class of mathematical theories, based to some
extent on experiment, is at present useful, and has even in
certun cases pointed to new and important results, which ex-
periment has subsequently verified. Such are the Dynamical
Theory of Heat, the Undulatory Theory of Light, eta etc In
the former, which is based upon the conclusion from experi-
ment that heat ia a form of energy, many formulae are at pre-
sent obecure and uninterpretable, because we do not know the
mechanism of the motions or dititortions of the particles of
bodies. Besulte of the theoiy in which these are not involved,
are of course experimentally verified. The same di£Sculties exist
in the Theory of Light. But before this obscurity can be per-
fectly oleared up, we must know somethiog of the ultimate, or
molecular, constitution of the bodies, or groups of molecules,
at present known to us only in the a^regate.
Dftducifan 386. A third class is well represented by the Mathematical
taueimu Theories of Heat (Conduction), Electricity (Statical), and Mag-
bw o*X"" netism (Permanent). Although we do not know kow Heat ia
propagated in bodies, nor what Statical Electricity or Fenna-
Dfint Magnetism are — the laws of their fiuxes and forces are as
certainly known as that of Gravitation, and can therefore like
it be developed to their consequencea, by the application of
Mathematical Analysis. The works of Fourier* Green*}*, and
FoisBOn:^ areremarkable instances of such development An-
other good example is Amp^'a Theory of Electro-dynamics.
• Thiorie analytique de la Chaleur, Vuii, 1823,
t Etiay on Iht Applieatim of Mathematical AnalgtU to Iht Theoritt tf
^Uetrieity and Magnetum. Kottiiighun, 183B, Beprioted in Crelle'B Jomnd.
t Mlmoirtt tur U Kagnitimt. UCm. d« I'iMd. dM S«i«iieM, 1811.
..Google
387.] EZPEEIENCE. 447
387. When the most prohahle result is required from a DadmAtm
number of observations of the same quantity which do not ubienauit
exactly agree, we must appeal to the mathematical theory of beror
probabilities to guide us to a method of combining the results
of experience, so as to eliminate from them, as far as possible,
the inaccuracies of observation. Of course it is to be under-
stood that we do not here class as inaccuracies of obiervaHon
any errors which may affect alike every one of a series of
observations, such aa the inexact determination of a zero pointy
or of the essential units of time uid space, the personal equa-
tion of the observer, etc The process, whatever it may be,
which is to be employed in the elimination of errors, is ap-
plicable even to these, hut only when seeerai distinct series of
observations have been made, with a change of instrument, or
of observer, or of both.
368. We understand as inaccuracies of observation the
whole class of errors which are as likely to lie in one
direction as in another in successive trials, and which we may
fairly presume would, on the average of an infinite number of
repetitions, exactly balance each other in excess and defect.
Moreover, we consider only errors of snch a kind that their
probability is the less the greater they are ; so that such errors
as an accidental reading of a wrong number of whole de-
grees on a divided circle (which, by the way, can in genenJ be
"probably" corrected by comparison with other observations)
are not to be included.
369. Mathematically considered, the subject is by no means
an easy one, and many high authorities have asserted that the
reasoning employed by Laplace, Qauss, and others, is not well
founded ; although the results of their analysis have been
generally accepted. As an excellent treatise on the subject has
recently been published by Airy, it is not necessary for us to
do more than to sketch in the most cursory manner a simple and
apparenUy satisfactory method of arriving at what is called the
Method of Least Squares.
390. Supposing the zero-point and the graduation of an
instnunent (micrometer, mural circle, thermometer, electrometer,
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448 PBELIMINABY. [390.
Morton galvuiometer, etc) to be absolutely accarate, BuccesBiTe readings
ta^ n*i[t of the value of a quantity (linear distance, altitude of a star,
^°f?^ temperature, potential, strength of an electric current, eta) may,
and in general do, continually differ. What is moat probably
the tme value of the observed quantity ?
The most probable valae, in all such cases, if the observa-
tions are all equally tmatworthy, will evidently be the simple
mean; or if they are not equally trustworthy, the mean found by
attribating tae^hts to the several observations in proportion to
, their presumed exactness. But if several such means have
been taken, or several single observations, and if these several
means or observations have been di£r««ntly qualified for the
determination of the sought quantity (some of them being
likely to give a more exact value thui others), we must assign
theoretically the best practical method of combining them.
391. Inaccuracies of observation are, in general, as likely to
be in excess as in defect They are also (as before observed) more
likely to be small than great ; and (practically) large errors are
not to be expected at all, as such would come under the class
of avoidable mittakea. It follows that in any one of a series of
observations of the same quantity the probability of an error
of magnitude w must depend upon s^, and must be expressed
by some function whose value diminishes very rapidly as x
increases. The probability that the error lies betneen x and
x + Sx, where Bx is very small, must also be proportional to Sx.
Hence we may asanme the probability of an error of anj
nuiguitude included in the range ofxtox + Sxtobe
Now the error must be included between +ao and — eo.
Hence, as a first ooudition,
/^W"^"' w-
The consideration of a very simple case gives us the means of
determining the form of the function ^ involved in the preceding
expression*.
■ ConpBTB Boole, Tran*. S.S.E., 1667. See aim Tajt, Trwi$. B.S.E^ lSft4
..Google
391.] EXPERIENCE. 449
Suppose a stone to be let fall with the object of hitUng a mark Beduotion
on the ground. Let two horizont&l Imee be drawn through the b*biBra*ult
mark at right angles to one another, and take them as axes of x bmotab-
and y respectively. The chance of the st«ne falling at a distance
between x and x + Sx from the axis of ^ is ^ {«*} Sx.
Of its fidling between y and y+Sy from the axis of x the
chance is t^ (y*) 8y.
The chance of its falling on the elementary area SxBy, whose co-
ordinates are x, y, is therefore (since titese are independent events,
and it is to be observed that this is the assumption on which the
whole investigation depends)
*(^)*(y)My,ora^(*0*(A
if a denote the indefinitely small area about the point xy.
Had we taken any other set of rectangular axes with the same
origin, we should have found for the same probability the ex-
pression atft (a!™) iji (y^i
fc', y" bdng the new co-ordinates of a. Hence we must have
* ("^ * (yO -*{*")* (y^. if »'+»•=*''+ y"-
n^>m ihia functional equation we have at once
where A and m are constants. We see at once that m must be
negative (as the chance of a Iai;ge error is very small), and we
may write for it - ?( > so that h will indicate the degree of de-
licacy or coarseness of the system of measurement employed.
Substituting in (1) we have
"ifl<fc=l.
j-+«
'L
whence A = -i— r- , and the law of error is
JL -^^ ^
Jv' h'
Th9 law of error, as regards diUanee from the mark, imthout
r^erenee to Iha direction of error, is evidently
fj^{^)i.{y')dxdy,
taken through the space between concentric circles whose radii
are r and i-+ St", and is therefore
2 •*
jiGoogk'
450 PBELDCnUBT. [391.
Idiw of which is of the same form aa the law of eiroF to the right w left
of a line, with the additional factor t* for the greater space for
error at {greater dlatances from tlie oent3«. Aa a verification, we
Bee at ooce that
!■/."•"»""•■■
u was to be expected.
FrotoUa 392. The ProbcAte Error .of an obeerration is a namerical
*™' quantity such that the error of the obserratioD is as likely to
exceed as to fall short of it in magoitude
If we assume the law of error joat found, and call P the
probable error in one trial,
rP _«• ■■" *•
The Bolutjon of this equatiioi by trial and error leads to the
approximate result
P = 0-477 A.
■PjtitMit 893. The probable error of any given multiple of the value
SllXdfiftr- of an observed quantity is evidently the same multiple of the
m^iiSa. probable error of the quantity itself.
The probable error of the sum or difference of two quantities,
affected by independent errors, is the square root of the sum of
the squares of their separate probable errors.
To prove this, let ns Investigate the law of error of
where the lavs of error of X and T are
-7= « S — , and -7=, i^-r-t
J It a' J^ b'
respectively. The chance of an error in Z, of a magnitude in-
cluded between the limila t,x + &K,ia evidently
For, whatever value is assigned to x, the value of y is given by
the limits z-x and » + &»-x [or z + x, x + &t + x; but the
chances of * x are the same, and both are included in the limits
(* w ) of iutegration with respect to x].
..Google
)3.] EXPERIENCE. 451
The value of the abore integnil becomes, hj effecting the in- Prabable
tegratioa with reepect to y, nini.'diirtDr-
I'-J)* multiple.
*/-.■'■
and thiB is easily reduced to
1 -. -^ &
Thus the probable error is 0477^«'+ 6', whence the propoaitjos.
And the same theorem is evidently true for any number of ijaan-
titles.
891. As above remarked, the principal use of this theory is Fnuitini
• - . . »pplk»tioa,
ID the deduction, from a large series of obaervatioos, of the
values of the quautities sought in such a form as to be liable
to the smallest probable error. As an instance — b; the prin-
ciples of physical astronomy, the place of a planet is calculated
from assumed values of the elements of its orbit, and tabulated
ia the Xfautvxil Almanac. The observed places do not exactly
iigree with the predicted places, for two reaeous — first, the data
for calculation are not exact (and in fact the main object of the
ubservation is to correct their assumed values) ; second, each
observation is in error to some unknown amount. Now the
difference between the observed, and the calculated, places
depends on the errors of assumed elements and of observation.
The methods are applied to eliminate as far as possible the ,
second of these, and the resulting equations give the required
corrections of the elements.
Thus if f be the calculated B.A. of a planet : ia, ^, hn, etc.,
the corrections required for the assumed elements — the true
B. A. is 6 + Ala + Ehe + nSw + etc. Method of
wbeie A, E, II, eta, are approximately known. Suppose the 1^^^,^
observed B. A to be 0, then
$-\-Aha + Ehe + UliB+ ... =0
or Ala + Ehe+Utm--r ... = ®-6,
a known quantity, subject to error of obaemition. Every obser-
vation made gives us an equation of the same forfn as this, and
in geaeral the number of observationa greatly exoeeds that of the
quantities Sa, he, Snr, etc, to be found. But it will be sufficient to
consider tiie simile case where only one quantity is to bo found.
29—2
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452 PBEUMDtABT. [394.
Sapftooe a nnmber of otservatuina, of th« same qoautit; x, lead
to th« following equations : —
x = B„ x = B„ etc.,
and let tbe probable erron be £',, £,, ... Moltiplj the terms of
each eqofltioD bj nnmbors inTeraelf pn^rtional to £^, £,, ...
Iliis will make the probable erron of the second members of all
the equations ttie sam^ e suppose. The eqnataons hare now tbe
general form ax = b,
anil it is required to find a system of linear factors, hy whicb
these equations, being multiplied in order and added, shall lead
to a final equation giving the value of x with the probable error a
fniiiiinnin. Let them hof^,f^, etc Theo the final equation is
■nd therefore i" (So/)' = ^S (/^
by tbe tbeorams of § 393, if P denote the probable error of ai
Hence 7v^>/, ia a mifumam, and its difiereatial coefficients
with respect to each separate fttctor/most vanish.
This gives a series of equations, whose general t<mn is
/S(a/)-oS(/-).0,
which give evidently/ =(»,, /,=aj, et&
Henoe the following rule, which may eastly be eeen to bold for
any number of linear equations containing a smaller number of
aukm>wii quantities,
Make theprdxiUe error of the second member the same in eack
equation, hy the employment of a proper factor ; multiply each
equation by the coefficient of x in ii and add all, for one of the
final equations ; and so, with reference to y, z, etc, for the others.
The probable errors of the values of x, y, etc., found from these
gnal equations will be less than those of the values deiived
from any other linear method of combiaiDg the equations.
This process has been called tbe method of Leaei Squares,
because the values of the uokaown quantities found by it are
sucb as to render the sum of tbe squares of the errors of the
original equations a minimum.
That is, in the simple case ta&en above,
S {ax - h)' = minimum.
..Google
394.] EXFERIEHCE. 45S
For It is evident that this (pYea, on difTerentiatuig vith reepect ic^odof
to X, Sa{ax—bj = 0, tqion*.
which is the law above laid down for the formation of the siDgle
equation,
395. When a series of obserTationa of the name quantity JJ^Ji\^
has been made at different times, or under different circum- J^^l^itli."
stances, the law connecting the value of the quantity with tlie
time, or some other variable, may be dnived from the resulta
in several ways — all more or less approximate. Two of these
methods, however, are bo much more extensively used than the
others, that we shall devote a page or two here to a preliminary
Qotice of them, leaving detailed instances of their application
till we come to Heat, Electricity, etc. They consist in (1) a
Curve, giving a graphic representation of the relation between
the ordinate and abscissa, and (2) an Empirical Formula con-
necting the variables.
896. Thus if the abscissae represent intervals of time, and Cuttw.
the ordinates the corresponding height of the barometer, we
may construct curves which show at a glance the dependence
of barometric pressure apon the time of day; and so on. Such
curves may be accurately drawn by photographic processes on a
sheet of sensitive paper placed behind the mercurial column,
and made to move past it with a uniform horizontal velocity
by clockwork. A similar process is apphed to the Temperature
and Electrification of the atmosphere, and to the components
of terrestrial magnetism,
897, When the observations are not, as in the lost section,
continuous, they give us only a series of points in the curve,
from which, however, we may in general approximate very
closely to the result of continuous observation by drawing,
liberd manu, a curve passing through these points. This pro-
cess, however, must be employed with great caution ; because,
unless the observations are sufficiently close to each other,
most important fluctuations in the curve may escape notice. It
is applicable, with abundant accuracy, to all cases where the
quantity observed changes very slowly. Thus, for instance,
weekly observations of the temperature at depths of from 6 to
..Google
454 PBELnONABT. [397.
O'Tttt. 24 feet nnderground were found by Forbes suffideot for a very
accurate approximation to the law of the phenomenon,
inttrpoi^ 399, As an instance of the processes employed for obttuniog
™pincd an empirical formula, we may mention methods of Interpo-
lation, to which the problem can always be reduced. Thus from
sextant obeervatioiis, at known intervals, of the altitude of the
Bun, it is a common problem of astronomy to determine at what
instant the altitude is greatest, and what is that greatest alti-
tude. The first enables ns to fiud the true soltu- time at the
place; and the second, by the help of the KauHccU Almanac,
gives the latitude. The difTerential calculus, and the calculus
of finite differences, give us formulae for any required data ;
and Lagrange has shown bow to obtain a very useful one by
elementary algebra.
By Taylor's Theonm, ^y"f{x), we have
where 0 is a proper fifUTtdon, and x^ is any qoantdty whatever.
This formula is useful only wh^i the successive derived values
oi/{x^) diminish veiy rapidly.
In finite differences we have
/(« + *). D'/(.)-(l+4)'/(»)
./W +M/(.) + *<^y ay W H. (2);
a very useful formula when the higher differences are smalL
(1) siiggeeU the proper form for the required expression, but it
is only inrare cases that y (a:,),/" (aj, etc., are derivable directly
from observation. But {2) is useful, inasinach as the successtTo
differences, ^/(x), ^'/{r), etc., are easily calculated from the
tabulated results of observation, provided these have been taken
for equal successive increments of x.
If for values x,, a:,, ... «, a function takes the values y,, y„
y,, ... y,, Lagrange gives for it the obvious expression
.Google
398.] EXPEMEHCE. 455
Here it is of course aasumed that the function required is a ^^'^**
rational and integral one in x of tiie »— 1*^ degree ; and, in empinni
general, a similar limitation is in practice applied to the other
formaUe above; for in order to find the complete expressioii for
/(x) in either, it ia neceesary to determine the valaea of ^ (x^,
/" ("t)' — in the first, or of i/(a;), ^'/{x), ... in the seoond. If
n of the coeffidenta be reqiiired, so as to give the n chief temia
of the genera] valne of /(x), we must have n observed simul-
bmeoua Tslnee of x and /(x), and the expressions beoome deter-
minate and of the n — 1'^ degree in tc - x, and h respectively.
In practice it is usually sufficient to employ at most three terms
of either of the first two series Thus to express the length I
of a rod of metal as depending on its temperature t, vo may
assume &om (1)
l^ being the measured length at any temperature t^.
398'. These fonnuls are practically n&eful for calculating
the probable values of any observed element, for values of the
iodepeudent variable lying within the range for which observa-
tion has given values of the element. But except for values of
the independent variable either actually within this range, or
not far beyond it in either direction, these formulee express
functions which, in general, will differ more and more widely
from the truth the further their application is pushed beyond
the range of observation.
In a lai^e class of investigations the observed element is in T
its nature a periodic function of the independent variable. The
harmonic analysis (§ 77) is suitable for all such. When the
values of the independent variable for which the element has
been observed are not equidiflferent the coefficients, determined
according to the method of least squares, are found by a procesn
which is ■ necessarily very laborious ; but when they are equi-
different, and especially when the difference is a submultiple
of the period, the equation derived from the method of least
squares becomes greatly simplified. Thus, if $ denote an angle
increasing in proportion to t, the time, through four right angles
in tbe period, T, of the phenomenon ; so that
..Google
456 PRELIMINAHY. [398'.
let f{0) = Aa+A,coee + A,co6i0+...
+ B^^a$ + B^aiai9-i-...
where A^, A^, A^, ... 5,, 5„ ... are unknown coefiGcients, to be
determined bo that /(0) may express the most probable value
of the element, not merely at times between observations, but
through all time as long as the phenomenon is strictly periodic.
By taking as many of these coefiBcienta as there are oi disttnct
data by observation, the formula is made to agree jM-ecisely with
these data. But iu most applications of the method, the peri-
odically recurring part of the phenomenon is exprearible by a
small number of terms of the harmonic series, and the higher
terms, calculated from a great number of data, express either
irregularities of the phenomenon not likely to recur, or errors of
observation. Thus a comparatively small number of terms may
give values of the element even for the very times of observit-
tion, more probable than the values actually recorded as having
been observed, if the observations are numerous but not mi-
nutely accurate.
The student may exercise himself in writing out the equ»>
tions to determine five, or seven, or more of the coefficients
according to the method of least squares; and redu<»ng them
by proper fbrmuhe of analytical trigonometry to their amplest
and most easily calculated forms where the values of 0 for which
f{ff) is given are equidifferent. He will thus see that when the
difference is -:- , t being any integer, and when the number
of the data is i or any multiple of it, the equations conttun each
of them only one of the unknown quantities : so that the
method of least squares affords the most probable values of
tlie coefficients, by the easiest and most direct elimination.
..Google
CHAPTER IV.
MEASUBia AND INSTBUMENTa.
399. Having seen in the preceding chapter that for the N*oeHit7
investigatioD of the laws of nature we must carefullj watch m««re-
expcriments, either those gigantic ones which the universe
furnishes, or others devised and executed by man for special
objects — and having seen that in all such observations accurate
measurements of Time, Space, Force, etc., are absolutely neces-
sary, we may now appropriately describe a few of the more
useful of the instruments epployed for these purposes, and the
various standards or units which are employed iu them.
400. Before going into detail we may give a rapid r^tnd
of the principal Standards and Instmments to be described in
this chapter. Aa most, if not all, of them depend on physical
principles to he detailed in the course of this work — we shall
assume in anticipation the establishment of such ptinciples,
giving references to the future division or chapter in which the
experimental demonstrations are more particularly explained.
This course will entail a slight, hut unavoidable, confusion —
slight, because 0ocks, Balances, Screws, etc., are familiar even
to those who know nothing of Natural Philosophy; unavoid-
able, because it is in the very nature of our subject that no one
5)art can grow alone, each requiring for its full development the
utmost resources of all the others. But if one of our depart-
ments thus borrows from others, it is satisfactory to find that it
more than repays by the power which its improvement affords
them.
..Google
458 PBELIIOHABT. [401.
401, We may diride our more important and (midameatal
iastruments into four c
Those for tneasuriag Time ;
„ „ Space, linear or aaguLu:;
» » Force;
„ „ Mass.
Other instruments, adapted for special purposes such as the
measurement of Temperature, Light, Electric CurreDts, etc, will
come more naturally under the head of the particular physical
energies to whose measurement they are applicahle. Descrip-
tions of self-recording instruments such as tide-^uges, and
barometers, thermometers, electrometers, recording phott^raph-
ically or otherwise the continuously varying pressure, tempe-
rature, moisture, electric potential of the atmosphere, and
magnetometers recording photographically the continuously
varying direction and magnitude of the terrestrial magnetic
force, must likewise be kept for their proper places in our
work.
Calculating Machines have also important uses in assisting
physical research in a great variety of ways. They belong to
two classes : — ■
I. Purely Arithmetical, dealing with int^[ral numbers of
units. All of this class are evolved from the primitive use of
the calculuses or little atones for counters (from which are
derived the very names calculation and "The Calculus"),
through such mechanism as that of the Chinese Abacus, still
serving its original purpose well in infant schools, up to the
Arithmometer of Thomas of Colmar and the grand but partially
realized conceptions of calculating machines by Babbage.
XI, Continuous Calculating Machines. As these are not
only useful as auxiliaries for physical research but also involve
dynamical and kinematical principles belonging properly to
our subject, some of them have been described in the Appendix
to this Chapter, from which dynamical illustrations will be
taken in our chapters on Statics and Kinetics.
..Google
402.] HEASITBES Aim IK9TRUHENTS. 459
403. We eball consider in order the more prominent funda- ci»jjje« of
mental instnimente of the four classes, and some of their most i«oi*
important applications : —
Clock, Chronometer, Chronoscope, Applications to Obser-
vation and to self-registering Instruments.
Vernier and Screw-Micrometer, Cathetometer, Sphero-
meter. Dividing Engine, Theodolite, Sextant or Circle.
Common Balance, Bifilar Balance, Torsion Balance, Pen-
dulum, Ergometer.
Among Standards we may mention —
1. Tiine. — Day, Hour, Minute, Second, sidereal and solar.
2. Space. — Yard and M^tre: Radian, Degree, Minute, Second.
3. Force. — Weight of a Pound or Kilogramme, etc., in any
particular locality (gravitation unit) ; poundal, or dyne
(kinetic unit}.
4. Mass. Found, Kilogramme, etc.
403. Although without instruments it is impossible to pro-
cure or apply any standard, jet, as without the standards no
instrument could give us (^laolute meaaure, we may consider the
standards first — referring to the instruments as if we already
knew their principles and applications.
404. First we may notice the standards or units of angular Aiwml«
measure :
Radian, or angle whose arc is equal to radius ;
Degree, or ninetieth part of a right angle, and its successive
subdivisions into sixtieths called Minutes, Seconds, Thirds, etc
The division of the right angle into 90 degrees is convenient
because it makes the half-angle of an equilateral triangle
(sio'' i) an integral number (30) of degrees. It has long been
universally adopted by all Europe. The decimal division of the
right angle, decreed by the French Republic when it success-
fully introduced other more sweeping changes, utterly and
deservedly failed.
The division of the degree into 60 minutes and of the
minute into 60 seconds is not convenient; and tables of the
..Google
♦60 PRELIMIKAET. [404.
An«niu circular functiona for degrees aod hundredtlis of the degree are
much to be desired. Meantime, when reckouing to tenths of a
degree suffices for the accuracy desired, in any case the ordiuary
tables suffice, as 6' is ^ of a degree.
The decimal system is exclusively followed in reckoning by
radians. The value of two right angles in this reckoning is
3'14159..., or ■n-. Thus w radians is equal to 180". Hence
180° -Mr ia S?" -29578..., or 57° 17' 44"-8 is equal to one
radian. In mathematical analysis, angles are uniformly reck-
oned in terms of the radian.
Kfuora 403. The practical standard of time is the Sidereal Day,
being the period, nearly constant*, of the earth's rotation about
its axis {§ 247). From it is easily derived the Mean Solar Day,
or the mean interval which elapses between successive passages
of the sun across the meridian of any place. This is not so
nearly as the Sidereal Day, an absolute or invariable unit:
* In oaz fint edition it was stated in thit section tb«t Ltqilaoe had calonlatod
from ancient obseiratiDQB of eclipseH that the period of the aaitii's rotation ■boat
its ailB had not altered bj i,,^,,, of itself dnce TaOB.c In g 830 it was
pointed ont that this condiiBion is overthrown by farther infonnatiini from
Physical Aatronomy aognired in the interval between the printing of ths two
Eeotione, in virtoe of a correotion whloh Adams had made as earlj as 1663 opon
Laplace's dynamical investigation of an aoceleration of the moon's mean motion,
produced hj the Bnn's attraction, showing that only about half of the observed
acoeleration of the moon's mean motion relatively to the angular veloci^ of the
earth's rotation was accounted For by this oanse. [Quoting from the first edition,
% 830] " In 1359 Adams commnnioated to Delannay hie final result ; — that at
" the end of a century the moon is 5"-7 before the position abe wonld have,
' ' relatively to a meridian of the earth, according to the angular velocities of the
"two motions, at the beginning of the century, and the aooeleration of the
" moon's motion tmlj oalctdated from the varioiiB distnrbing caoeee then leeog-
" nized, Delatmey soon after verified this result : and about the banning irf
"1866 Buggeatad that the trne explanation maybe a retard^on of the earth's
" rotation by tidal fHction. Using this hy3>otheBia, and allowing for the oonsa-
■' qnent retardation of the moon's mean motion by tidal reaction (g 376), Adams,
*'iu an estimate which he has communicated to ns, founded on the rough as-
■• snmption that the parts of the earth's retaidation due to solar and Innai tides
"ore aa the sqnares of the respective tide.generating toroes, finds 23* as the
" error hy which the earth would in a oeatory get behind a perfect clock rated
■' at the b^tinning of the century. It the retardation of rate giving this integral
■' effect were uniform {% 36, &), the earth, as a timekeeper, wonld be going (dower
'■by -22 of a second per year in tlie middle, or -44 of a second per year at the
"end, than at the beginning of a centuiy."
..Google
405.] UEASDBES AND msTRUHENTS. 461
secular cbaDges in the period of the earth's rotation about the J^"™*^
BUD affect it, thoagh very slightly. It is divided into 24 hours,
and the hour, like the d^ree, is subdivided into succeBsive
sixtieths, called minutes and seconda The usual subdivision
of seconds is decimaL
It is well to observe that seconds and minutes of time
are distinguished from those of angular measure by notation.
Thus we have for time 13" 43" 27'58, but for angular measure
13* 43' 27"-58.
When long periods of time are to be measured, the mean solar
year, con^sting of 366*242203 sidereal days, or 365242242 mean
solar days, or the century consisting of 100 such years, may be
conveniently employed as the unit.
406. The ultimate standard of accurate chronometry must xv-miif
(if the human race live on the earth for a few million years) be perenniji
founded on the physical properties of some body of more con- awW^
Btaat character than the earth: for instance, a carefully arranged
metallic spring, hermetically sealed in an exhausted glass vessel.
The time of vibration of such a spring would be necessarily more
constant from day to day than that of the balance-spring of the
best possible chronometer, disturbed as this is by the train of
mechanism with which it is connected : and it would almost
certainly he more constant &om age to age than the time of
rotation of the earth (cooling and shrinking, as it certainly is,
to an extent that must he very considerable in fifty million
years).
407. The British standard of length is the Imperial Yard, H«nus of
defined as the distance between two marks on a certain metallic fimnded on
utifld*!
bar, preserved in the Tower of London, when the whole has a 2^^f,
temperature of 60° Fahrenheit. It was not directly derived
from any fixed quantity in nature, although some important
relations with such have been measured with great accuracy.
It has been carefully compared with the length of a seconds
pendulum vibrating at a certain station in the neighbourhood of
London, so that if it should again be destroyed, as it was at the
burning of the Houses of Parliament in 1834, and should all
exact copies of it, of which several are preserved in various
..Google
462 PRELIHINABT. [407.
places, be alao lost, it can be restored b; pendulam obserTa-
tions. A less accurate, but still (except in the eveut of
earthquake disturbance) a very good, means of reproduciiig it
exists ia the measured base-lines of the Ordnance Survey, aod
the thence calculated distances between definite station's in the
British Islands, nhich liave been ascertained in terms of it with
a degree of accuracy sometimes within an inch per mile, that ia
to say, within about ^^ni-
408. In scientific investigations, we endeavour as tnuch as
posBihle to keep to one unit at a time, and tbe foot, which is
defined to be one-tbird part of the yard, is, for British measure-
ment, generally the most convenient. Unfortunately the inch,
or one-twelfth of a foot, must sometimes be used. The statute
mile, or 1760 yards, is most unhappily often used when great
lengths are considered. The British measurements of area and
volume are infinitely inconvenient and wasteful of brain-wiergy,
and of plodding labour. Their contrast with the simple, uni-
form, metrical system of France, Germany, and Italy, is but
little creditable to English intelligence.
409. In the French metrical system the decimal division is
exclusively employed. The standard, (unhappily) called the
irm^ MUre,yia& defined originally as the ten-millionth part of the
i£'i(Kat length of the quadrant of the earth's meridian from the pole
to the equator; but it is now defined practically by the accurate
standard metres laid up in various national repositories in
Europe. It is somewhat longer than the yard, as the following
Table shows :
Inch = 25-39977 millimetres.
Foot= 3-047972 decimiitres.
British statute mile
= 1609 339 miitres.
Centimetre = '3937043 inch,
Mfetre = 3-280869 feet
Ki]om{itre= -6213767 British
statute mile.
iiwnir* cd 410. The unit of superficial measure is in Britain the square
•™**' yard, in France the m^tre carr^ Of course we may use square
inches, feet, or miles, as alao square millimMres, kilometres, eta,
or the Hectare = 10,000 square metres.
..Google
410.] HEASUBGS AND IHSTBUKENTS. 463
Sqtiare incli = 6-451483 square centimMres. Hw
„ foot= 9-290135 „ decimfetres. "^
„ yard ^ 8361121 „ decim^treB.
Acre = -4046792 of a hectare.
Square Britiah statute mile = 258 -9946 hectares.
Hectare = 2*471093 acres.
411. Similar remarks apply to the cubic measure in the two h«i
countries, and we have the following Table : — "'"^
Cubic inch= 16*38661 cubic centimfettea.
„ foot = 28*31606 „ decimfetraB or lAttres.
Gallon = 4-543808 litres.
„ =277-274 cubic inches, by Act of Parllam^t
now repealed.
Litre = -035315 cubic feet.
412. The British unit of mass is the Pound (defined by Me«i
standards only); the French is the Kilogramme, defined origi-
nally as a litre of water at its temperature of maximum density ;
but now practically defined by existing standards.
Grain - 64-79896 miUigrammes. I Gremme = 15-4323S grains.
Founds 4535927 grammes. | Kilogramme = 2-20462125 lbs.
Professor W. H. Miller finds (Phil. Trans. 1857) that the
" kilogramme des Archives " ia equal in mass to 15432*34(874
grains; and the "kilogramme type laiton," deposited in the
Miniature de I'lnt^rieure in Paris, as standard for French com-
merce, is 15432-344i grains.
413. The measurement of force, -whether in terms of the hs
weight of a stated mass in a stated locality, or in terms of the
absolute or kinetic unit, has been explained in Chap. iL (See
§§ 220 — 226). From the measures of force and length, we
derive at once the measure of work or mechanical effect. That
practically employed by engineers is founded on the gravita-
tion measure of force. Neglecting the difference of gravity at
London and Paris, we see from the above tables that the follow-
ing relations exist between the London and the Parisian reckon-
ing of work : —
Foot-pound = 0*13825 kilogramme-mbtre.
Eilogramme-m6tre= 7'2331 foot-pounda
..Google
464 PBELDONABT. [Hi.
OmI^ 414; A Clock is primarily &a instrument which, b; means
of a train of wheels, records the number of vibrations executed
by a pendulum ; a Chronometer or Watch performs the same duty
for the oscillations of a flat spiral spring — just as the train of
wheel-work in a gas-metre counts the number of revolutions of
the mtan shaft caused by the passage of the gas through the
machine. As, however, it is impossible to avoid Miction, re-
sistance of air, etc, a pendulum or spring, left to itself, would
not long continue its oscillations, and, while its motion con-
tinued, would perform each oscillation in less and less time as
the arc of vibration diminished: a continuous supply of enet^
is furnished by. the descent of a weight, or the uncoiling of
a powerful spring. This is so applied, through the train of
wheels, to the pendulum or balance-wheel by means of a
mechanical contrivance called an Escapement, that the oscilla^
tions are maintained of nearly aniform extent, and therefore
of nearly uniform duration. The construction of escapements,
as well as of trains of clock-wheels, is a matter of Mechanics,
with the details of which we are not concerned, although it may
easily be made the subject of mathematical investigation. The
means of avoiding errors introduced by changes of temperature,
which have been carried out in GompensaMon pendulums and
balances, will be more properly described in our chapters on
Heat. It is to be observed that there is little inconvenience
if a clock lose or gain regvlarlg; that can be easily and ac-
curately allowed for: irregular rate is &taL
xieatriautr 415. By means of a recent application of electricity to be
doaki. afterwards described, one good clock, carefully regulated from
time to time to ^ree with astronomical observations, may be
made (without injury to its own performance) to control any
number of other less-perfectly constructed clocks, so as to com-
pel their pendulums to vibrate, beat for beat, with its own.
onnif 416, In astronomical observations, time is estimated to
tenths of a second by a practised observer, who, while watching
the phenomena, counts the beats of the clock. But for the wry
accurate measurement of short intervals, many instruments have
been devised. Thus if a small orifice be opened in a large and
..Google
416.] MEASTJBE3 AND INSTEUMENTS. 465
deep vessel full of mercury, and if we know ty trial the weight cimoa-
of metal that escapes say in five minutes, a simple proportion
gives the interval which elapses during the escape of any given
weight It is easy to contrive an adjustment by which a vessel
may be placed under, and withdrawn from, the issuing stream
at the time of occurrence of any two successive phenomena.
417. Other contrivances, called Stop-watches, Chronoscopes,
etc., which can be read off at rest, started'on the occurrence of
any phenomenon, and stopped at the occurrence of a second,
then again read off; or which allow of the making (by pressing
a stud) a slight mark, on a dial revolving at a given rate,
at the instant of the occurrence of each phenomenon to be
noted, are common enough. But, of late, these have almost
entirely given place to the Electric Chronoscope, an instrument
which will be fully described later, when we shall have oc-
casion to refer to experiments in which it has been usefully
employed.
418. We now come to the measurement of space, and of
angles, and for these purposes the most important instruments
are the Vernier and the Screiu.
419. Elementary geometry, indeed, gives us the means ofnin^oiui
dividing any straight line into any assignable number of equal
parts ; but in practice this is by no
means an accurate or reliable method.
It was formerly used in the so-called
Diagonal Scale, of which the con-
struction is evident from the diagram.
The reading is effected by a sliding-
piece whose edge is perpendicular to
the length of the scale. Suppose
that it is PQ whose position on the
scale is required. This can evidently
cut only (M)« of the transverse lines, /te number gives the number
of tenths of an inch [4 in the figure], and the horizontal line
next above the point of intersection gives evidently the number
of hundredths [in the present case 4]. Hence the reading is
7'44. Ab an idea of the comparative uselessness of this
vol.. I. 30
..Google
466
PKELIMISARY.
[419.
metltCM], we ma; meDtion that a quadrant of 3 feet radius,
which belonged to Napier of Merchiston, aad is divided on
the limb by thia method, reads to mioutes of a degree ; no
higher accuracy than is now attainable by the pocket sextants
made by Troughton and Simms, the radiua of whose arc is
virtually little more than an inch. The latter inRtrument is
read by the help of a Vernier.
420. The Yemier is commonly employed for such instru-
ments aa the Barometer, Sextant, and Cathetometer, while the
Screw is micrometrically applied to the more delicate iustru-
ments, such as Astronomical Circles, and Micrometers, and the
Spherometer.
121. The yemier consists of a slip of metal which slides
along a divided scale, the edges of the two being coincident.
Hence, when it is applied to a divided circle, its edge is circular,
and it moves about an axis passing through the centre of the
divided limb.
In the sketch let 0, 1, 2,.. .10 be the divisions on the vernier,
A 1. 1 etc., any set of consecutive divisions on the limb or scale
along whose edge it slides. If, when 0 and o coin-
cide, 10 and 11 coincide also, then 10 divisions of
the vernier are equal in length to 11 on the limb;
and therefore each division on the vernier is -^ths
r 1^ of a division on the limb. If, then, the ver-
nier be moved till 1 coincides with i, 0 will be -^ih
of a division of the limb beyond o ; if 2 coincide
with £, 0 will be ^tbs beyond «; and so on.
Hence to read the vernier in any position, note
first the division next to 0, and behind it on
the limb- This is the integral number of divi-
sions to be read. For the fractional part, see
which division of the vernier is in a line with
one on the limb ; if it be the 4th {as in the
figure), that indicates an addition to the reading of ^ths of a
division of the limb; and so on. Thus, if the figure represent
a barometer scale divided into inches and tentilis, the reading
is S034, the zero line of the vernier being adjusted to the level
of the mercury.
z
30-
z
3
-w
29-
-
-
jiGoogle
*22.] MEASURES AND INSTRUMENTS. 467
423. If the limb of a sextant be divided, as it usually is, to Vemier.
third parts of a degree, and the vernier be formed by dividing
21 of these into 20 equal parts, the inBtrument can be read to
twentieths of divisions on the limb, that is, to minutes of arc.
If no line on the vernier coincide with one on the limb, then
since the divisions of the former are the longer there will be
one of the latter included between the two lines of the vernier,
and it is usual in practice to take the mean of the readings
which would be given by a coincidence of either pair of bound-
ing lines,
423. In the above sketch and description, the numbers on
the scale and vernier have been supposed to run opposite ways.
This is generally the case with British instruments. In some
foreign ones the divisions run in the same direction on vernier
and limb, and in that case it is easy to see that to read to
tenths of a scale division we must have ten divisions of the
vernier equal to nine of the scale.
In general, to read to the nth part of a scale division, n divi-
sions of the veruier must equal n-f- 1 or n— 1 divisions on the
limb, according as these ran in opposite or similar directions.
424. The principle of the Screw has been already noticed Screw.
(§ 102). It may be used in either of two ways, i.e., the nut
may be fixed, and the screw advance through it, or the screw
may be prevented from moving longitudinally by a fixed collar,
in which case the nut, if prevented by fixed guides from rotat-
ing, will move in the direction of the common axis. The
advance in either case is evidently proportional to the angle
through which the screw has turned about its axis, and this
may be measured by means of a divided head fixed perpendi-
cularly to the screw at one end, the divisions being read off by
a pointer or vernier attached to the frame of the instrument.
The nut carries with it either a tracing point (as in the divid-
ing engine) or a wire, thread, or half the object-glass of a tele-
scope (as in micrometers), the thread or wire, or the play of the
tracing point, being at right angles to the axis of the screw.
426. Suppose it be required to divide a line into any
number of equal parts. The line is placed parallel to the axis
30—2
..Google
468 PBELDIUTABr. [425.
of the screw vith one end exactly under the tracing point, or
under the fixed wire of a microscope earned by the nut, and
the Bciew-head is read o£ By turning the head, the tracing
point or microscope wire is brought to the other extremity of
the line ; and the number of turns tuid fractions of a turn re-
quired for the whole line is thus ascerttuned. Dividing this by
ihe number of equal parts required, we find at onoe the number
of turns and fractional parts corresponding to oiw of the
required divisions, and l^ giving that amount of rotation to
the screw over and over again, drawing a line after each rota-
tion, the required division is efTected.
126. In the Micrometer, tiiie movable wire carried by the
nut is parallel to a fixed wire. By bringing them into optical
contact the zero reading of the head is known ; hence when
another reading has been obtained, we have by subtraction the
number of turns corresponding to the length of the object to
be measured. The cAsolvte value of a turn of the screw is de-
termined by calculation firom the number of threads in an inch,
or by actually applying the micrometer to an object of known
dimensions,
127. For the measurement of the thickness of a plate, or
the curvature of a lens, the SpheroTneter is used. It consists of a
Bcrew nut rigidly fixed in the middle of a very rigid three-legged
table, with its axis perpendicular to the plane of the three feet
(or finely rounded ends of the legs), and an accurately cut screw
working in this nut. The lower extremity of the screw is also
finely rounded. The number of turns, whole or fractional, of
the screw, is read off by a divided head and a pointer fixed to
the stem. Suppose it be required to measure the thickness of
a plate of glass. The three feet of the instrument are placed
upon a nearly enough fiat surface of a hard body, and the screw
is gradually turned until its point touches and presses the sur-
face. The muscular sense of touch perceives resistance to the
turning of the screw wheu, after touching the hard body, it
presses on it with a force somewhat exceeding the weight of
the screw. The first effect of the contact is a diminution of
resistance to the turning, due to the weight of the screw coming
..Google
427.] HEASURES AlfD JHSTRtnCENTS. 469
to be borne oa its fine pointed end instead of on the thread of Bpboro-
the nut. The sudden increase of reedstance at the inStant when
the screw commences to bear part of the weight of the nut finds
the sense prepared to perceive it with remarkable delicacy on
account of its contrast with the immediately preceding diminu-
tion of resistance. The screw-head is now read off, and the screw
turned backwards until room is left for the insertion, beneath
its point, of the plate whose thickness is to be measured. The
screw is again turned until increase of resistance is again per-
<%iTed; and the ncrew-head is agiun read off. The difference of
the readings of the head is equal to the thickness of the plate,
reckoned in the proper unit of the screw and the diTisioD of its
head.
428. If the curvature of a lens is to be measured, the in-
strument is first placed, as before, on a plane surface, and the
reading for the contact is taken. The same operation is repeated
OD the spherical surface. The difference of the screw readings
is evidently the greatest thickness of the glass which would be
cut off by a plane passing through the three feet. This enables
us to calculate the radius of the spherical surface (the distance
from foot to foot of the instrument being known).
Let a be the diBtaace from foot to foot, I the length of screw
Gorreeponding to the difference pf the two readings, S the radius
of the spherical surface ; we have at once iS = ^ + 1, or, as f
is generally very small compared with a, the diameter is, very
approximat«ly, ^.
429. The Cathetojneter is used for the accurate determina- *£?*"
tion of differences of level — for instance, in measuring the
height to which a fluid rises in a capillary tube above the ex-
terior free surface. It consists of a long divided metallic stem,
turning round an axis as nearly as may be parallel to its length,
on a fixed tripod stand : and, attached to the stem, a spirit-level
Upon the stem slides a metallic piece bearing a telescope of
which the length is approximately enough perpendicular to the
axis. The telescope tube is as nearly as may be perpendicular
to the length of the stem. By levelling screws in two feet of the
..Google
470 PRELDUNABT. [4S9.
tripod the bubble of tbe spirit-level is brought to one position
of its glass when the stem is turned all round its axis. This
secures that the axis is verticaL In using the instniinent the
telescope is directed in succession to the two objects whose
difference of level is to be found, and in each case moved (gene-
rally by a delicate screw) up or down the stem, until a horizontal
wire in the focus of its eye-piece coincides with the image of
the object. The difference of readings on the vertical stem
{each taken generaUy by aid of a vernier sliding-piece) corre-
sponding to the two positions of the telescope gives the required
difference of level.
430. The common O-ravity Balance is an instrument for
testing the equality of the gravity of the masses placed in the
two pans. We may note here a Csw of the precautions adopted
in the best balances to guard against the various defects to
which the instrument is liable; and the chief points to be at-
tended to in its construction to secure delicat^, and rapidity of
weighing.
The balaoce-beam should be very stiff, and as light as possible
condstently with the requisite stiffness. For this purpose it is
generally formed either of tubes, or of a sort of lattice-framework.
To avoid friction, the axle consists of a knife-edge, as it is called ;
that ia> a wedge of hard steel, which, when the bf^ance is in use,
rests on horizontal plates of polished ^ate. A similar contri-
vance is applied in veiy delicate balances at the points of the
beam from which the scale-pans are suspended. When not in
use, and just before use, the beam with its knife-edge is lifted
by a lever arrangement from the agate plates. While thus
secured it is loaded with weights as nearly as possible equal
(this can be attained by previous trial with a coarser instru-
ment), and the accurate determination is then readily effected.
The last fraction of the required weight is determined by a rider,
a very small weight, generally formed of wire, which can be
worked (by a lever) from the outside of the glass case in which
the balance is enclosed, aud which may be placed in different
positions upon one arm of the beam. This arm is graduated to
tenths, etc., and thus shows at once the value of the rider in
any case as depending on its moment or leverage, § 2S2.
..Google
431.] MEASURES AND INSTRUMENTS. 471
431. Qualities of a balaoce : b
1. StaMlitif. — For stability of the beam aloDe without pans
and weights, its centre of gravity must be below ita beariag
knife-edge. For stability with the heaviest weights the line
joiDing the points at the ends of the beam from which the pans
are hung must be below the knife-edge bearing the whole.
2. SensibiKti/. — The beam should be sensibly deflected from
a horizontal position by the smallest difference between the
weights in the scale-pans. The definite measure of the sensi-
bility ia the angle through which the beam is deflected by a
stated difference between the loads in the pans.
3. Quichiess. — This means rapidity of oscillation, and con-
sequently speed in the performance of a weighing. It depends
raiunly upon the depth of the centre of gravity of the whole
below the knife-edge and the length of the beam.
In our Chapter on Statics we shall give the investigation.
The sensibility and quickness will there be calculated for any
given form and dimensions of the instrument.
A fine balance should turn with about a 500,000th of the
greatest load which can safely be placed in either pan. In
fact few measurements of any kind are correct to more than
six significant figures.
The process of Double Wetghiiig, which consists in counter-
poising a mass by shot, or sand, or pieces of fine wire, and then
substituting weights for it in the same pan till equilibrium is
attained, is more laborious, but more accurate, than single
weighing; as it eliminates all errors arising from unequal length
of the arms, etc.
Correction is required for the weights of air displaced by the
two bodies weighed gainst one another when their difi'erence
is too large to be negligible.
432. In the Toraitm-balance, invented and used with great ^
effect by Coulomb, a force is measured by the torsion of
a glass fibre, or of a metallic wire. The fibre or wire is
fixed at its upper end, or at both ends, according to circum-
stances. In general it carries a very light horizontal rod or
needle, to the extremities of which are attached the body on
..Google
47S PBEUHIITART. [132.
vliich is exerted the force to be measured, and a counterpoise.
The upper extremity of the torsion fibre is fixed to an index
passing through the centre of a divided disc, so that the angle
through which that extremity moves is directly measured. If,
at the same time, the angle through which the needle has
turned be measured, or, more simply, if the index be always
turned till the needle assumes a definite position determined
by marks or sights attached to the case of the instrument —
we bare the amount of torsion of the fibre, and it becomes a
simple statical problem to determine from the latt«r the force
to be measured; its direction, and point of application, and
the dimensions of the apparatus, being known. The force of
torsion as depending on the angle of torsion was found by Cou-
lomb to follow the law of simple proportion up to the limits of
perfect elasticity — as might have been expected from Hooke's
Law (see Properties o/3Iatter), and it only remains that we de-
termine the amount for a particular angle in absolute measure.
This determination is in general simple enough in theory; but
in practice requires considerable care and nicety. The torsion-
balance, however, being chiefly used for comparative, not
absolute, measure, this determination is often unnecessary.
More will be said about it when we come to its applications.
433. The ordinary spiral spring-balances used for roughly
comparing either small or large weights or forces, are, properly
speaking, only a modified form of torsion-balance*, as they act
almost entirely by the torsion of the wire, and not 1^ longi-
tudinal extension or by flexure. Spring-balances we believe
to be capable, if carefully constructed, of rivalling the ordinary
balance in accuracy, while, for some applications, they far sur-
pass it in sensibility and convenience. They measure directly
force, not mass; and therefore if used for determining masses
in different ports of the earth, a correction must be applied for
the varying force of gravity. The correction for temperature
must not be overlooked. These corrections may be avoided
by the method of double weighing.
* Binet, Jonnrnl tU V£eoXt Folytrelmiqut, x. IBIG : uid ]. ThomMn, Cam-
briig* and DiAUm Utah. Jbicnial (ISM).
..Google
434.] MEASURES AND OTSTRtTMENTS. 473
434. Perhaps the most delicate of all mstniments for the
measurement of force i^ the Pendulum. It is proved in kinetics
(Bee Dir. n.) that for any pendulum, whether oscillating about
a mean vertical position under the action of gravity, or in a
horizontal plane, under the action of magnetic force, or force
of torsion, the square of the number of sTnall oscillations in a
given time is proportional to the magnitude of the force under
which these oscillations take place.
For the estimation of the relative amonnts of gravity at
different places, this is by far the most perfect instrument.
The method of coincidences by wbich this process has been
rendered so excessively delicate will be described later.
435. The Bijiiar Suspension, an arraBgement for meaeur- B^iix
iug small horizontal forces, or couples in horizontal planes, in
terms of the weight of the suspended body, is due originally to
Sir 'William Snow Harris, who used it in one of his electro-
meters, aa a substitute for the simple torsion-balance of Coulomb.
It waa used also by Qauss in his bifilar magnetometer for mea- Biiitar Sair-
euring the horizontal component of the terrestrial magnetic
force*. In this instrument the bifilar suspension is adjusted to
keep a bar-mt^et in a positiou approximately perpendicular
to the magnetic meridian. The small natural augmentations
and diminutions of the horizontal component are shown by
small azimuthal motions of the bar. On account of some
obvious mechanical and dynamical difficulties this instrument
was not found very convenient for absolute determinations, but
from the time of its first practical introduction by Gauss and
Weber it has been in use in all M^;netic Observatories for
measuring the natural variations of the horizontal magnetic
component. It is now made with a much smaller mf^et than
the great bar weighing twenty-five pounds originaUy given with
it by QauBs; but the bars in actual use at the present day are
still enormously too Iarge*f for their duty. The weight of the
' OaOM, Kaullatt am dm BeohaeMmtgtn da magnetiiehni Vereiru (m
Jalire 18S7. Tnutlated In Tajlot'a BeienHfie Xemoin, Vol. II., Artlole ti.
f The miipended magnetH tued tot detenniniug the direction and the in.
tend^ ol the hoTicoutal nugnetM lone in the Ihiltliii Hagnetia Obeemtoiy,
..Google
474 FBIOJMIKABT. [433.
If Hag- bar with attached mirror ought not to exceed eight grammes,
so that two single silk fibres may su£Bce for the hearing threads.
The only substantial alteration, besides the diminution of its
magnitude, which has been made in the instrunient since Gauss
and Weber's time is the addition of photographic apparatus and
clockwork for automatic record of its motions. For absolute
determinations of the horizontal component force. Gauss's method
of deflecting a fireely suspended magnet by a magnetic bar brought
into proper positions in its neighbourhood, and again making
an independent set of obseirationa to determine the period of
oscillation of the same deflecting bar when suspended by a fine
,ti]te fibre and set to vibrate through a small horizontal angle on
("^ each side of the magnetic meridian, is the method which has
netia been uniformly in use both in magnetic observatories and in
travellers* observations with smaU portable apparatus mnce it
was first invented by Gauss*.
M In the bifilar balance the two threads may be of unequal
lengths, the line joining their upper fixed ends need not be hori-
zontal, and their other ends may be attached to any two points of
the suspended body : but for most purposes, and particularly for
regular instruments such as electrometers and ma^etometers
with bifilar suspension, it is convenient to have, as nearly as may
be, the two threads of equal length, their fixed ends at the same
level, and their other ends attached to the suspended body sym-
metrically with reference to its centre of gravity (as illustrated
in the last set of drawings of § 345*). Supposing the instrument-
maker to have fulfilled these conditions of symmetry as nearly
OS he can with reference to the four points of attachment of the
threads, we have still to adjust properly the lengths of the
threads. For this purpose remark that a small difference in the
lengths will throw the suspended body into an unsymmetrical
OB deecribed hj Dr Llo;d in his Treatue on ilagjutUm (London, 1871), ara eioh
of them 15 inehea long, £ ol no iuob brood, and J of an inch in tbicknesa, and
moat titerefore weigh abont a pannd each. The oorreBpoDding mi^etB used at
the Eew Obserrator; ore much Emaller, They are each 6'4 inctaeB long, 0'8
inch broad, and O'l inch thiolc, and theiefore the weii^t oF each U about 0-012
pound, CO' ueaily 66 grammes.
• Intmtitat Vit Magnetieae Terrettrit ad Menttrram Abmlulam Tevocata,
Gommeututkinea SooietatiB GoUingeoBis, 1632,
..Google
433.] MEASURES AXD iHSTBrifEirrs. 475
I>osition, in wHich, particularly if its ceDtfe of gravity be very Bji
low (aa it is ia Sir W. ThomaoD's Quadrant Electrometer), much
more of its -weight will be borne by one thread than by the
other. This will diminish very much the amount of the hori-
zont^ couple reqairod to produce a stated azimuthal deflection
in the regular use of the instmment, in other words will in-
crease its fiensibility above its proper amount, that is to say,
the amount which it would have if the conditions of symmetry
were folly realized. Hence the proper adjustment for equaliz-
ing the lengths of the threads in a symmetrical bifilar balance,
or for giving them their right difiference in an unsymmetrical
arrtuigement, in order to make the instrument as accurate as it
can be, is to alter the length of one or both of the tiireads, until
we attain to the condition of mimmmn $ejmbility, that is to
say minimum angle of deflection under the influence of a given
amount of couple.
The great merit of the bifilar balance over the simple toruon-
balance of Cottlomb for such applications as that to the hori-
zontal magnetometer in the continuous work of an observatory,
is the comparative smallness of the influence it experiences
from changes of temperature. The torsional rigidity of iron,
copper, and brass wires is diminished about J per cent, with 10*
elevation of temperature, while the linear expansions of the
same metals are each less than ^ per cent, with the same
elevation of temperature. Hence in the unifilar totsiou-
balance, if iron, copper, or brass (the only metals for which the
change of torsional rigidity with change of temperature has
hitherto been measured) is used for the material of the bearing
fibre, the sensibility is augmented ^ per cent, by 10^ elevation
of temperature.
On the other hand, in the bifilar balance, if torsional rigidity
does not contribute any sensible proportion to the whole direc-
tive couple (and this condition may be realized as nearly as we
please by making the bearing wires long enough and making
the distance between them great enough to give the requbite
amount of directive couple), the sensibility of the balance is
affected only l^ the linear expansions of the substances con-
cerned. If the equal distances between the two pairs-of points
..Google
476 FBEUHmART. [435.
of attachment, in the normal form of hifilar halance (or that in
which the two threads are vertical when the snapeniled body is
uninflueiiced hy horizontal force or couple), remained constant,
the sensihility would be augmented with elevation of tempera-
ture in simple proportion to the linear ezpaoBioos of the bearing
wires ; and this small influence might, if it were worth while
to make the requisite mechanical arrangements, be perfectly
compensated by chooBing materials for the &ames or bars bear-
ing the attachments of the wires so that the proportionate
augmentation of the distance between them should be just
half the elongation of either wire, because the sensibility, as
shown by the mathematical formula betow, is simply propor-
tional to the length of the wires and inversely proportional to
the square of the distance between them. But, even without any
such compensation, the temperature>error due to linearexpansions
of the materials of the biSlar balance is so small that in the most
accurate regular use of the instrument in magnetic observatories
it may be almost neglected ; and at most it is less than ^ of
the error of the unifilar torsion-balance, at all events if, as is
probably the case, the changes of rigidity with changes of tempe-
rature in other metals are of similar amounts to those for the
tiiree metals on which experiments have been made. In reality
the chief temperature^rror of the bifilar mi^etometer depends
on the change of the magnetic moment of the suspended magnet
with change of temperature. It seems that the magnetism of
a steel magnet diminishes with rise of temperature and aug-
ments with fall of temperature, but experimental information is
much wanted on this subject.
The amount of the effect is very different in different bars,
and it must be experimentally determined for each bar serving
in a bifilar magnetometer. The amount of ih.e change of mag*
netic moment in the bar which had been most used in the
Dublin Magnetic Observatory was found to be -000029 per de-
gree Fahrenheit or at the rate of '000052 per decree Centigrade,
being alraut the same amount as that of the change of torsional
rigidity with temperature of the three metals referred to above.
Let a be the half length of the bar between the points td
attachment of the wires, 6 the angle through which the bar has
..Google
435.] UEASUBES AND INSTBUHENTS. 477
been turned (in a horizontal plane) from iia poaition of equi- B
librium, I the length of one of the wires, i ito inclination to the
verticaL
Then 7 ooa ( is the difforenoe of levels between the ends of each
wire, and evidently, by the geometry of the cose,
}f sint=>aBin^A
Kow if Q be the couple tending to torn the bar, and W its weight,
the prindple of mechanical effect gives
Qde = -Wd (I ooBt)
But, by the geometrical condition above.
Hence ,■ a=l '
a Bin 0 (ooH (
-, Wa' Bin(9
which gives the conple in terms of the deflection A
If the torsion of the wires be taken into aocoont, it is
sensibly equal to $ (sinoe the greatest inclination to the vertical
is small), and therefore the conple resulting from it will be E6.
This mnst be added to the value of Q just foiwd in order to get
the whole deflecting couplet
436. Ergometeis are instrnmeDta for measurmg enei^.
White's friction brake meaaures the amount of work actoally
performed id any time by an engine or other "prime mover,"
by allowing it during the time of trial to waste all Its work on
Jriction. Morin's ergometer measures work without wasting
uiy <tf it, in the course of its transmission from the prime
mover to machines in which it is usefully employed. It con-
sists of a simple arrangemeut of springs, measuring at every
instant the couple with which the prime mover turns the shaft
that transmits its work, and an int^;rating machine from which
the work done by this conple during any time can be read oS.
Lot L be the conple at any instant, and ^ the whole angle
through which the shaft has turned &om the moment at which
the reduming commences. The int^^ting machine shows at
any moment the value of jLd^ which ({ 240) is the whole work
..Google
47S PBELaOHARY. [4S7.
■• 4S7. White's friction brake coasiiita of a lever clamped to
the ab^, but not aUowed to turn with iL The moment of the
force required to prevent the lever &om going round with the
shaft, multiplied b; the whole angle through which the shaft
turns, measures the whole work done against the frictioa of the
clamp. The same result ia much more easily obtained by
wrapping a rope or chain several times round the shaft, or
round a cylinder or drum carried round by the shaft, and
applying measured forces to its two ends in proper directions
to keep it nearly steady while the shaft turns round without it.
The difference of the moments of these two forces round the
axis, multiplied by the angle through which the shaft turns,
measures the whole work spent xin friction against the rope.
If we remove all other resistance to the shaft, and apply the
proper amount of force at each end of the dynamimetric rope
or chain (which is very easily done in practice), the prime
mover is kept running at the proper speed for the test, and
having ita whole work thus wasted for the time and measured.
..Google
APPENDIX B.
CONTINUOUS CALCULATING MACHINES.
I. TlDE-PBEDICTINQ MACHINE.
The object is to predict the tidea for any port for which the Jj'JI^^f''
tidal constituentB have been found from the harmonic analyeia UKhme.
from tide-gauge obBervations ; not merely to predict the times
and heights of high water, but the depths of water at any and
erety instant, showing them by a oontinnous curre, for a year, or
for any number of years in advance.
Tliis object requires the summation of the simple harmonic
functions representing the several constituents* to be taken into
account, which is performed by tiie machine in the following
manner : — For each tidal constituent to be taken into account
the machine has a abaft with an overhanging crank, which
carries a pulley pivoted on a parallel axis adjustable to a greater
or less distance from the shaft's aus, according to the greater or
lees, range of the particular tidal constituent for tlie different
porte for which the machine is to be used. The several shafts,
with their axes all parallel, are geared together so that their
periods are to a sufficient degree of ^tproximation proportional
to the periods of the tidal constituents. The crank on each
shaft can be turned ronnd on the shaft and clamped in any po-
sition : Uius it is set to the proper podtion for the epoch of the
particular tide which it is to produce. The axes of the several
shafta are horizontal, and their vertical planes are at sucocflsive
distances one from another, each equal to the diameter of one of
the pulleys (the diameters of these being equal). The shafts are
in two rows, an upper and a lower, and the grooves of the pulleys
are all in one plane perpendicular to th^ axes.
Suppose, now, the axes of the pulleya to be set each at zero
distance from the axis of ita shaft, and let a fine wire or chain,
* See Beport for 1BT6 ot the Conmiittee of the BritiBh AsBOciation appointed
tot the pnrpose ot promoting the Eitensioi], ImproTsment, and Hannonie
AntlyaiB ot Tidal Obaervatknu.
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dE^ns*' with ona end h&nging down and canying A w^gfat, pus ftlto-
natelj over and nnder the pulleys in order, and Torlically np-
warda or downwards {aooording aa the nnmber of pnllejs is eren
or odd) bxtia the last pulley to a fixed pcunt. The wo^t U
to be properly gnided for vertical motjon by a geomcitrical slide.
Tnm the machine now, and the wire will remain ondistorbed
with all ito free parts vertical and the liM.nging -wei^t unmoved.
But now set the axis of any one of the pulleys to a distuioe ^ T
from its shaft's axis and turn the machine. If the distance of
this pulley from the two on each side of it in the otJier row is a
ctHuiderable multiple of ^ T, the hanging weight will now (if the
machine is tamed umformly) move up and down with a simple
harmonic motion of amplitude (or semi-nwge) equal to 2* in the
period of its shsA, I^ next, a Becond pulley is di^daoed to a
distance ^ 2*, a third to a distance } T", and so on, the li«Tigiiig
weight will now perform a complex harmonic motion equal to
the sum of the_ several harmonic motions, each in its proper
period, which would be produced separately by tiie displace-
menta T, T', T". Thus, if the machine was made on a large
scale, with 7*, T',... equal respectively to the actual semi-ranges
of the several constituent tides, and if it was turned round
slowly (by clockwork, for example), each shaft going once round
in the actual period of the tide which it represente, the hanging
weight would rise and &11 exactly with the water-level as
aSected by the whole tidal action. This, of course, could be of
no use, and is only suggested by way of illustration. The actual
machine is made of such magnitude, that it can be set to give a
motion to the hanging weight equal to the actual motion of the
water-level reduced to any convenient scale : and provided the
whole range does not exceed about 30 centimetie^ the geo-
metrical error due to the deviation from perfect parallelism in
the successive free parts of the wire is not so great as to be
practically objectionablB. The proper order for the shafts is the
order of magnitude of the constituent tides which they produce,
the greatest next the hanging weight, and the least next the
fixed end of the wire : this so that the greatest constituent may
have only one pulley to move, the second in magnitude only two
pulleys, and so on.
One machine of this Hnd has already been constructed for the
British Association, and another (with a greater number of shafts
to include a greater number of tidal constituents) is being con-
..Google
CONTINIIOirS CAWniLiTINa MACHINIB. 481
stracted for the Indi&n GoTemment. The Britisli' AssocUtion Tide-pn-
maohine, which is kept available for general nse, under charge Haeh&s
of the Science and Art Department in South Kensington, has
ten shafts, vhich taken in order, from the hanging weight, gire
respedaTely the following tidal constituents*:
1. The mean lunar aemi-diumal.
2. The mean soUr semi-diurnal.
3. The larger elliptic aemi-diumal.
i. The luni-solar diurnal declinational
6. The lunar diurnal deelinationBl.
6. The luni-BoIor semi-diurnal declinationaL
7. The smaller elliptic semi-diurnal.
6. The solar dinmal declinatdonal.
9. The Innar quarter^umal, or first shallow-water tide of
mean lun&r semi-dinmaL
10. The lunl-solar quarter-dinmal, ehallow-water tid&
The hanging weight oooButs of an ink-bottle with a glass
tubular pen, which marks the tide level in a continuous curve
on a long band of paper, moved horizontally acroes the line of
motion of the pen, by a vertical cylinder geared to the revolving
shafts of the machine. One of the five sliding points of the
geometrical slide is the point of the pen sliding on the paper
stretched on the cylinder, and the couple formed by the normal
pressure on this point, and on another of the five, which is about
fonr centimetres above its level and one and a half centimetres
from the paper, balances the couple due to gravity of the ink-
bottle and the vertical component of the pull of the bearing wire,
which is in a line about a millimetre or two farther from the
paper than that in which the centre of gravity moves, . Thus is
ensured, notwithstanding small inequalities on the paper, a
pressure of the pen on the paper very approximately constant
and as small as is desired.
Hour marks are made on the curve by a small horizontal
movement of the ink-bottle's lateral guides, made once an hour ;
a somewhat greater movement, giving a deeper notch, serves to
mark the noon of every day.
The machine may be turned so rapidly as to run ofif a year's
tides for any port in about four hour^
Each crank should carry an adjustable counterpoise, to be
■ Bee BnpOTt for 1876 ol the BriiUh Asaocution's Tidal GommittM.
VOL. I. 81
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I APPENDIX B'. PX
adjusted so that ■whea. the cnmk ia not Terdcal the pnlls oS the
ftpproziiiiatelf vertical portions of wire aoting on it thnni^ the
pulley vhich it oanies shall, as exactly as may he, halance on
the axis of the shaft, and the motion of the shaft should he
reeisted by a slight weight han^ng on a tliread wmpped once
round it and attached at its other end to a fixed point. This
part of the design, planned to secure agunst "lost time" or
"back tash" in tie geaiings, and to preserve nnifonnlty of
pressure between teeth and teeth, teeth and screwB, and ends of
axles and "end-plates," wu not carried out in the British
IL Machinb fob the Solutioh or Sihdi.tamsoub
LiNUB EQUAnoifs*.
f^^S'"' I^ -^ti S^— -ff, be » bodies each supported <m a fixed axis
(in practice each is to be supported on knife-edges like the beam
^a balance).
liSt i*„, P^, P„, ... i*,, be » pulleys each pivoted cm B^
n ^n ^n ^* -•■ ^^ be « cords passing over the puU^;
„ i)„ P,„P,^ P^...P,j£„ be tJie course of C,;
„ J),, S^, D^ E^ ...D^ E^ be fixed points;
„ I,, /^ l^ ... I, be the leugtiis of the cords between 2),, E^,
and D^ E^ ... and D^ E^ along the oouiaee stated abore, when
B^, B^, . . . Bj are in particular positions which will be called
Hieir zero positiooii;
„ 2,-^e,> I,-t-e„ ...f, + 0, be their lengths between the same
fixed points, when B^, B^ ... B^ are turned throu^ angles x„
x^ ... a), from their zero porationa;
(11), (12), (13), ...(In),
(21), (22), (23), ...(2„),
(31), (32), (33), ...(3»),
' Sir W. nunason, Pnettdi»s* of tft. Sffgal SiKUt^, ToL n
jiGoogle
II.] coNTunrouB clLCULinira hachineb.
quAQtitififl Buch that
(ll)a!, + (12)a!.4-... + (l«)a. = fl,
{21)a!, + (22)a!,+ ... + (2n)a!. = e,
(31) «, + (32) 3!, + ... + (3n)a:. = e.
.(I).
(nl)iBj + («3)a!,+ ... + (wn)a!,=e..
We shall anppoee «,, a;,,...a;, to be each so small that (11),
(12),. ..(21), eto., do not vary senalbly from the valaea which
they have when a;,, x^...x_, are each infinitely smalL In
practice it will be convenient to bo place the axes of £„£„.. . B^
and the mountingB of the pulleys on 3^, B^, ... B^ and the fised
pointsZ',,£,,iP„ etc, that when x,,x„ ... a;, are infinitely BUiall,
the straight parts of each cord and the lines of infiniteeinial mo-
tion of the oentrea of the pulleys round which it passes shall be
all paralleL Then ^ (11), ^ (21), ... ^ (nl) will be simply equal to
the distajioeB of the centres of the pulleys P^^ , P„, . .. P,„ from the
ajdsof J,; i(12),J(22)...i(n2)thedistanoeBofP,^i'„,...P_,
from the axis of B^; and bo on.
In practice the mountings of the pulleys are to be adjuatable
by proper geometrical slides, to allow any prescribed podtive or
negative value to be given to each of the quantdtiea (11),
(12),. ..(31), etc
SuppoBe this to be done, and each of the bodies B^, B,, ... B^
to be placed in its zero position and held there. Attach now
the cords firmly to the fixed points D^, D^, ... D^ respectively;
and, passing them round their proper puUeya, bring them to the
other fixed points .£,, E^, ... E^, and pass them through infinitely
small smooth rings fixed at these points. I^^ow hold the bodiea
B^, B^, ... each fixed, and (in practice by welghte hung on their
ends, outside £,, E^, ... EJ pull the cords through E^, E^,... E^
with any given tensions* 2",, T„ ... 2",. Let ff„ ff„ ... ff, be
moments round the fixed axes of £,, B,, ... B^ of the forces re-
quired to hold the bodies fixed when acted on by the oorda thus
* The idea ot foroe hare flret introduced is not essentiB], indeed is not
technical]; admiMJbla to the purely kinematlo and algebnio part of the Babjeot
proposed. But it is not merely an ideal Idnematia oonstmoticm of the ftlgebrtio
problem that is intended; and the design of a kinematia machine, for snooeiB in
practice, caaentially involTea djnomioal considerationB. In the present eaae
some of the meet important ot the pniely algebraic qneetiona oonoeroed are very
inteieetingly illaetrated by these dyiutiucal wnsideratioiu,
31—2
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484. AITENDIX B'. [II.
Bqnatkui- Btretched. The principle of "virtual Tel(kiitae6,"jast as it came
'*^' from Lagrange (or the pricdple of "'work"), givea immediately,
in virtue of (I),
tf,= (11) r, + (21) T,+ ... + (nl) r.
e,= (12) r, + (22) r,+ . . . + {«2) r.
(?. =- (In) r, + (2)») r,+ ... + (n») r.
Apply and keep applied to each of the bodies, S^, B^, ... B^
(in practice by the veigbta of the pulleys, and by counter-pulling
Bpriugs), Buch forces as shall have for their moments the values
6„ G,-.0^, calculated from equations (II) with whatever values
seem desirable for the tensions 7*,, T,, ... 7*,, (In practice, the
straight parts of the cords are to be approximately vertical, and
the bodies £,, B,, are to be each balanced on its axis when the
pnlleys belonging to it are removed, and it is advisable to make
the tenuous each equal to half the weight of one of the pulleys
with its adjustable &ame.) 3^e machine is now ready for use.
To nse it, pull the cords simultaneously or snooessively till
lengths equal to «,, «„...«, are passed through the rings £^,
E^, ... E^, respectively,
The pull* required to do this may be positive or negative; in
practice, they will be infinitedmal downward or upward pressures
applied by hand to the stretching weights «hich remain per-
manently hanging on the cords.
Observe the angles tJirongh which the bodies 5,,B,, ... B, are
turned by this given movement of the cords. These angles are
the required values of the unknown tc,, x,, ... x^, satisfying the
simultaneous equations (I),
The actual construction of a practically useful machine for
calculating as maay as eight or ten or more of unknowns from
the same number of Unear equations does not promise to be either
difficult or over-elaborate. A fair approximation having been
found by a first application of the machine, a very moderate
amount of straightforward arithmetical work (aided very ad-
vantageously by Crelle's multiplication tables) suffices to calculate
the residual errors, and allow the machines (with the setting of
the pnlleys unchanged) to be reapplied to calculate the corrections
(which may be treated decimally, for convenience) : thus, 100
times the amount of the correction on each of the original un-
knowns may be made the new unknowns, if the magnitudes thni I
..Google
II.]
CONTINUOUS CAIX^ULATINQ UACHINES.
4«5
filing to be dealt with are oonvenient for the machine. There g^^
is, of (xrarae, no limit to the accuracy thus obtainable by succes-
sive approximations. The exceeding eaaioeBS of each aj^tlication
of the machine promises well for its real usefulness, whether for
caaes in which a single application suffices, or for others in which
the requisite accuracy is reached after two, three, or more, of
successive approximations.
The accompanying drawings represent a machine for finding
six* unknowns from six equations. Fig. 1 represents in eleva-
tion and plan one of the six bodies it,, £^ etc. Fig. 2 shows in
elevation and plan one of the thirty-six pulleys P, with He
cradle on geometrical slide (§ 198), Fig. 3 shows in frontele-
vation the general disposition of the instrument
Fid. I. One of the dz moveable bodies, B.
' This Qiimbei' has been ehosen for the Snt praeticsl maobine to be oou-
■tniated, because a chief application of tlie maohine ms; be to the oalonlation
of the oonectionB on approximate values aliead; found ol the six elements of
the oibit of a comet or asteroid.
..Google
ss*^^
pi
Fia. 9. One of the thiitr-dz pnlleys, P, with its eliding ondle.
Fall Size.
In Fig. 3 onl7 one of the six corda, and the dz pnlleTS over
which it pnnaoo, is shown, not any of the other thirty. The tiiree
pu]le3rs eeen at the top of the sketch are three out of eighteen
pivoted on immoveable bearings above the machine, for the pur-
pose of counterpoising the weights of the pulleys P, with their
sliding cradlee. Each of the coonterpoisee la eqnal to twice the
weight of one of the pulleys P with its sliding cradle. Thus if
the bodies B are balanced on their knife-edges with each sliding
cradle in its central position, they remain balanced when one
or all of the cradles are shiited to either tdde; and the tension
of each of the thirty-dx essential cords is exactly equal to half
tlie weiglit of one of the pulleys with its adjustable frame, as
sivecified above (the deviations from exact verticality of all the
free portions of the thirty-six essential cords and the eighteen
counterpoising cords being neglected).
..Google
n.]
COHTINUOnS OALCULATIKQ UIOHIKES.
Fta. 8. Oenenl dlepoutlon of mMihIne.
Sutver.
..Google
[lit
III. An iNTEORATiHa Machine ■hating a Mew Kine-
matic Peinciple*.
VU> The kiBematic principle for intq^ting j/dx, wliich is uaed in
CjiiDder- tbe instrumentB well known aa Moriu's Dynamometer f and
H^uT^ Bang's Planimeter J, admirable aa it is in many respects, inTolvea
one element of imperfection whicli cannot but prevent onr oon-
templating it with full satis&ction. This imperfection consiaU
in the sliding action which the edge wheel or roller is required
to take in conjunction with its rolling action, which alone is
desirable for exact communication of motion from tiie disk or
oone to the edge roller.
The very ingenious, simple, and practically ttsefol instrument
well known as Amsler's Polar Pliinimeter, although dlSerent in
ito main features of principle and mode of action from the instru-
ments just referred to, ranks along with them in involving the
like imperfection of requiring to have a aidewise sliding action
of its edge rolling wheel, besides the desirable rolling action on
the sui&ce which imparts to it its revolving motion — a surface
* 'ProteaaorJameBTbotatoii.PTvceedingto/ the Botjal Society, Y6Lxcir.,iSn(l,
p. 262.
+ InstmmentB of thla kind, and an; othera for measniuig mechwiioal work,
may bettei in fotoie be called EqjometerB tlian Dynamometers. Tbe ntma
" dynamomater" has been and continnea to be in common use for tigni^'iDg
a ipiing instromeDt for measurmg forct ; bnt an iQBtnuaent fen' meaenring
work, being dietinet in its uatnre and abject, ought to have a different and more
suitable dedgoation. The nuue >' dynamometer," beaidss, appears to be badly
formed from the Greek; and for designating an instrument tor mouurnRmf of
force, I voold BOggest that the name maj with admnt^e be changed lo
dynaminwtn-. In respect to ihe mode of forming words in inch ca»ea, referenee
may be made to CortiuB's arammar, Dr Smith's EnghBh edition, g 8fi4, p. 220. —
J. T„ 36th Febroary, 1BT6.
X Sang's Flanimeter is very clearly deeoribed and figured in a paper by its
inventor, in the Transactions of the Boyal Scottish Society of Arts, Yol. it,
January 12, 1852.
..Google
III."] CONTINUOUS CALCULATING MACHINES. 489
wliich in this case is not a disk or cone, but is the sur&c« of the di«Ii-,
paper, or any other plane face, on which the map or oUier plane Ofliiider-
diagr&m to be evaluated in area is drawn. Ha^lnc
Professor J. Clerk Maxwell, having seen Sang's Planimeter
in the Great Exhibition of 1851, and having become convinced
that the combination of slipping and rolling was a drawback on
the perfection of the instrument, began to search for some ar-
rangement bj which Hie motion should be that of perfect rolling
in every action of the instrument, corresponding to that of com-
bined slipping and rolling in previous instrnmonts. He suc-
ceeded in devising a new form of planimeter or integrating
machine with a quite new and very beautiful principle of kine-
matic action depending on the mutual rolling of two equal
spheres, each on the other. He described this in a paper sub-
mitted to the Royal Scottish Society of Arts in January 1855,
which iB published in Yol. iv. of the Traueactions of that Society.
In that paper he also offered a suggestion, which appears to be
both interesting and important, proposing the attainment of the
desired conditions of action by the mutual rolling of a oone and
cylinder with their Kxes at right angles.
The idea of using pure rotting instead of combined rolling
and slipping was oommunicated to me by Pro£ Maxwell, when
I had the pleaaiire of learning from himself some particulars as
to the nature of his contrivance. Afterwards (some time be-
tween the years 1861 and 1S64), while endeavouring to contrive
means for the attainment in meteorological observatories of
certain integrations in respect to the motions of the wind, and
also in endeavouring to devise a planimeter more satisfactory in
principle than either Sang's or Amsler's planimeter (even though,
on grounds of practical simplicity and convenience, nnlikely to
turn out preferable to Amsler's in ordinary cases of takii^
areas from maps or other diagrams, but something that I hoped
might possibly be attainable which, while having tlie merit of
working by pure rolling contact, might be simpler than the
instrument of Pro£ Maxwell and preferable to it in mechanism),
I succeeded in devising for the desired object a new kinematic
method, which has ever since appeared to me likely sometime
to prove valuable when occasion for its employment might be
found. Now, within the last few days, this principle, on being
suggested to my brother as perhaps capable of being usefully
emplc^ed towards the development of tide-calculating machines
..Google
490 APPEin>ix b'. [IIL
DWi', whicli ha IukI been devidng, has been foam] bj him to be capable
Cylinder- of being introdaced and combined in aeveral ways to prodnce
important results. On Iiis advice, Uierefiire, I now offer to the
Boyal Socie^ a brief description of tbe new princi{de as devised
The new principle oonsista primarily in the tnmsmissuMi of
motion from a disk or cone to a cylinder by the interrentum of
a looseball, which preeseal^ its gravity on the disk and ^linder,
or on the cone and cylinder, as the case may be, the preesnre
being sufficient to give the necessary frictional coherence at
each point of rolling contact; and the axis of the disk or cone
aad that of the cylinder being both held fixed in position by
bearings in stationary ftnmework, and the arrangement of tJieae
axes being snch that when the disk or the cone and the cylinder
are kept steady, or, in other words, withoat rotation on their
axes, the ball can roll along tliem in contact with both, so that
the point of rolling contact between the ball and the cylinder
shall traverM a Btiaight Hne on the cylindric sorfaoe parallel
necessarily to the axis of the cylinder — and so that, in the case
of a disk being used, the point of rolling contect of the ball
with the disk shall traverse a straight line passing throogh the
centre of the disk — or that, in case of a cone being nsed, the
line of rolling contact of the ball on the cone shall baverse a
straight line on the conical snrfiice, directed neoeesarily towards
the vertex of the cone. It will thna readily be seen that,
vhether the cylinder and the disk or cone be at rest or revolving
on their axes, the two lines of rolling contact of the ball, one
on the t^lindric surface and tJie otlier on the disk or cone, when
both ooDsidered as lines traced out in space fixed relatively to
the framing of the whole instromen^ will be two parallel straight
lines, and that tlie line of motion of the ball's centre will be
straight and parallel to them. For facilitating explanation^
the motion of the centre of the ball along its path parallel to
the axis of the cylinder may be called the ball's longitadinal
motion.
Now for the integration of t/dx : tiie distance of the point of
contact of tiie ball with the disk or cone &om the centre of the
disk or vertex of the cone in the ball's longitudinal motion ia
to represent tf, while the angular space turned by the disk or
cone from any initial position represents x; and then the angular
space tnmed by tita cylinder will, when multiplied hy a soitable
..Google
III.] COKTIKUOTTS OAICUlATINa MACBINES. 491
oonBtant numerical ooefficient, exprees the mt^^ in terms of Dbk-,
any required unit for its eraluation. Cjiindi
The longitudinal motion may be imparted to the ball by
having the framing of the whole instrument so placed that the
lines of longitudinal motion of the two pointe of contact and
of the ball's centre, which are three straight lines mutually
parallel, shall be inclined to the horizontal sufficiently to make
the ball tend decidedly to descend along the line of ita longitu-
dinal moti<m, and then regulating its motion by an abutting
controller, which may have at its point of contact, where it
presses on the ball, a plane face perpendicnlar to the line of the
ball's motion. Otherwise the longitudinal motion may, for some
cases, preferably be imparted to the ball by having the direction
of that motion horizontal, and having two controlling flat faces
acting in close contact withoat tightness at opposite extremities
of the ball's diameter, which at any moment is in tlie line of
the ball's motion or is paisUel to the axis of the cylinder.
It is worthy of notice that, in the case of the disk-, ball-, and
cylinder-integrator, no theoretical nor important pnCCtical fault
in the action of the instrument would be involved in any
defidency of perfect exactitude in the practical accomplishment
of the desired condition that the line of motion of the ball's
point of contact with the disk should pass through the centre of
the disk. The reason of this will be obvious enough on a little
consideration.
The plane of the disk may suitably be placed inclined to the
horizontal at some snch angle as 46°; and the accompanying
sketch, togethw with the model, which will be submitted to the
Society by my brother, will aid towards the clear nndeistanding
of the explanations which hsTe been given.
My brother hss pointed oat to me that an additional opera-
tion, important for some purposes, may be effected by arranging
that the machine shall give a continuous record of the growth
of the integral by introducing additional mechanisms suitable
for continually describing a curve such tiiat for each point of it
the abecisBa shall represent the value of x, and the ordinate
shall represent the intend attained ttom x=0 fmward to that
value of ic This, he has pmnted out, may be effected in practice
by having a cylinder axised on the axis of the disk, a roll of
paper covering this cylinder's surface and a straight bar situated
parallel to this cylinder's axis and resting with enough of pree-
..Google
492 - APPENDIX B". [Ill-
^k% rare on the surface of the pnmaiy registering or lA« indtealing
C^ifndn- cylinder (the one, namely, which is actuated by its contact with
the ball) to make it have sufficient frictdoaal coherence with that
SIDi ELEVATION.
Bar&ce, and by having this bar made to can; a pencil or other
tracing point which will mark the desired curve on the seoondary
registering or lAe recording ^linder. As, from the nature of
the apparatus, the axis of the disk and of the secondary register-
ing or recording cylinder ought to be steeply inclined to the
horizontal, and as, therefore, this bar, carrying the pencil, would
have the line of its length and of its motion alike steeply in-
clined with that axis, it seems that, to carry oat this idea, it
may be advisable to have a thread attached to the bar and
extending off in the line of the bar to a pulley, passing over the
pulley, and having suspended at its other end a weight which
will be just sufficient to counteract the tendency of the rod, in
virtue of gravity, to glide down along the line of its own slope,
so as to leave it perfectly free to be moved up or down by the
fictional coherence between itself and the moving surface of the
indicating cylinder worked directly by the ball.
..Google
ly.] cournraous cALcuiATiifa machines.
IV. An Instrument for CAicDLiTiNO (j4>(x)->^{a!)dx\ ,
THE Integral of the Product of two qiven Functions*.
In coDBequence of the recent meeting of the British ABsociation Htcblm to
ftt Bristol, I resumed on attempt to find an instrnment which int<s»i <^
should Bupereede the heavy arithmetical labour of cnlculating two Fudo-
the integrals required to analyze a function into its simple har-
monic conatituenta according to the method of Fourier. During
many years previously it had appeared to me that the object
ought to be accomplished by eome simple mechanical means ;
but it was not until recently that I succeeded in devising an
instrument approaching sufficiently to simplicity to promise
practically tiBefnl results. Having amved at this stag^ I de-
scribed my proposed machine a few days ago to my brother
Professor James Thomson, and he described to me in return a
kind of mechanical integrator which had occurred to him many
years ago, but of which he had never published any description.
I instantly saw that it gave me a much simpler means of attain-
ing my special object than anything I had been able to think ot
previously. An account of his int^rator is communicated to
the Boyal Society along with the present paper.
To calculate I <^ {x) ij/(x) dx, the rotating disk ia to be displaced
frtmi a zero or initial portion through an angle equal to
while the rolling globe is moved so as always to be at a distance
fromitszBropoBitione<{nalto^(2!). This being done, the cylinder
obviously turns through an angle equal to / ^(z) il/(x)dx, and
thus solves the problem.
Oue way of giving the required motionfl to Uie rotating disk
and roUing globe is aa follows : —
* Sir W, TboniBoii, Proettdingi of tht Royal Soeiely, Vol. nn. , 1876, p. SG6.
..Google
CAlcuUta
APPENDIX B". pV.
On two pieces of paper draw tie onrree
ttaam. ■' "
Attach these pieces of paper to the circamfeKnce of two cir-
cuIbt cylinders, (a to difierent parta of the circamfemnce <^ one
cylinder, with the axis of x in each in the direction perpendicular
to the azia of the cylinder. Let the two cylinders (if there are
two) be geared together bo as that their drcnmfereiices shall
move with equal velocities. Attached to the framework let
there be, close to the circumference of each cylinder, a slide or
guide-rod to guide a moveable poini^ moved by the hand <^ an
operator, so aa always to touch the curve on the BOiface of the
cylinder, while the two cylinders are moved round.
Two operators will be required, as one operator could not
move the two points so as to fulfil this condition — at all events
nnlesB the motion were very slow. One of theae points, by
proper mechanism, gives an ^iT^glll^^^ motion to the rotating disk
equal to its own linear motion, the other g^ves a linear motion
eqoal to its own to the centre of the rolling globe,
The machine thus described is immediately applicable to
calculate the values ff^, B^, H^, etc. of the harmonic constituents
of A fiinction ^ (x) in the splendid generalization of Fourier's
simple harmonic analyms, which he initiated himself in hia
.aolntionfl for the conduction of heat in the sphere and the
cylinder, and which was worked out so ably and beantifully by
Foisson*, and by Sturm and Liouville in their memorable
papers on this subject published in the fiiat volume of Lionville's
Jowmal dea Math&matiqaea. Thxa if
^ (a) = J7,^, {x) + ir,^, («) + jT,^, (aj) + etc
be the expression for an arbitraiy function i^ix, in terms of the
generalized harmonic functions ^, (x), ^, (^)i ^1(3;), etc., these
fonctiona being such that
[*,(*)*i(*)'*»'=0. /*.{«)*,(«)<i»=0, [^,(Jc)*,(a;) = 0,etc.,
■ Eis B^eral demonstration of the realit? of the roots of tianaceudentd
egnationa essential to this azuljiBU (an eioeedinglj important step in adTuice
from Fourier's position), whioh he flnt gsTs in the BuUeltn de la Soeiiti
Philomathiqut loi 1838, is reproduoed in hia ThiorU Mathimatiqat ie la
Chalem; i 90,
..Google
IT.] CONTniUOUS CAICDIATUTO MACHINES. 49S
/*,(.)^(.),fa RMS
"1
(*.W1'*»
In the physical applicatioiis of this theoij the integral!
whiok constitute the denominators of the formulasfor H^, H^, etc
are alwaja to be evaluated in finite terms \>j an extension of
Fourier's fomiula for the I sbm,' lia; of his problem of the cylinder*
made by Sturm in equation (10), g It. of his Memoire tw una
Clone ^Equaiumt d d^ereneei partuUtt in Liouville's Journal,
Tol. I. (1836). The integrals in the numerators are calculated
mth great ease bj aid of the machine wcrked in the manner
described above,
The great practical use of this machine vill be to perform
the umple harmonic Fourier-analysia for tidal, meteorological,
and perhaps even astronomioal, observations. It is the case in
irhich
^ (aj) = (nir)j
(» any int^er) that gives this application. In this caae the
addition of a simple crank mechanism, to give a simple harmonic
iwg'1^1^'' motion to the rotating disk in the proper period — ,
when the (^linder bearing tlie curve y = \fi(x) moves uniformly,
supersedes the necessity for a cylinder with the curve y = 't>{x)
traced on it, and an operator keeping a point always on this
curve in the manner described above. Thus one operator will be
enough to carry on the process ; and I believe that in the appli-
cation of it to the tidal harmonic analysis he will be able in an
■ Fooiier'i TJUoHt Anatytique dt la C\aUuif, § 819, p. S91 (Puis, 1833).
..Google
Inti«nlof
PndBOtor
J . APPENDIX B'. flV.
hour or two to find hy aid of tlie machine any one of the mm{^
hamuHuc elements of a year's tides recorded in curves in the
uanal manner by an ordinuj Iddfr^uge — a reenlt wluch hitherto
has required not less than twenty Uoura of calcolatjon by aldlled
arithmeticians. I believe this instrument will be of great value
also in determining the diurnal, semi-diumal, ter-diumal, and
quarter-diurnal constituents of the daUy variations of temperatore,
barometric pressure, east and west components of the velocity of
the vind, north and soutli components of the same ; also of the
three components ai the terrestrial magnetic force ; also of the
electric potential of the air at tiie point where the sb«am of
water breaks into drops in atmoepheric electrometers, and of
other snbjeotB of ordinary meteorological or magnetic observa-
tions ; also to estimate precisely the variaticm of terrestrial
magnetism in the eleven years sun-spot period, and of 8tiii-«pote
themselves in this period ; also to disprove (or prove, as the case
may be) supposed relations between sun-spots and planetary
positions and conjunctions; also to investigate lunar influence
on the hei^t of the barometer, and on the components of the
terrestrial nu^pietic force, and to find if lunar influ^iLO is
sensible on amy other meteorolc^cal phenomena — and if so, to
determine precisely its character and amount.
From the description given above it will be seen that tlie
mechanism required tot the instrument is exceedingly simple and
easy. Its accuracy will depend essentially on the accoiacy of the
circular cylinder, of the globe, and of tiie plane of the rotating
disk used in it. For each of the three satfaces a mnch less
elaborate application of the method of scraping than that by
which Sir Joseph Whitworth has g^ven a true plane with such
marvellous accuracy will no doubt' su£Sce for the practical re-
quirements of the instrument now proposed.
..Google
v.] CONTINUODS CALCOTATraO MACHINES.
y. MeCHANICAI, iNTBXiaATION OF LtNEAB DIFFEREN-
TIAL EqUATIONS OF THE SECOND ObDEB WITH VaKIABLE
Coefficients*.
Every linear differential eqaation of the second order may, as S'jjjj?^'
is known, be reduced to the form orLinau
L(^S)- <').
There F is any (pven function of x.
On account of the great importance of this equation in
mathematical physics (vibrations of a non-uniform stretched
cord, of a banging chain, of water in a canal of non-umform
breadth and depth, of air in a pipe of non-uniform sectional area,
conduction of heat along a bar of non-uniform section or nim-
uniform conductivity, Laplace's differential equation of the tides,
etc. etc.), I have long endeavoured to obbuin a means of facilitat-
ing its practical solution.
Methods <^ Calculation such as those used by Laplace him-
self are exceedingly valuable, but are very laboriona, too
laborious unless a serious object is to be attained by calculating
out results with minute accuracy. A ready means of obtaining
approximate results which shall show the general character of
the solutions, such as those so well worked out by Sturm t, has
always seemed to me a desideratum. Therefore I have made
many attempts to plan a mechanical integrator which should
give solutions by successive approximations. This is clearly done
now, when we have the instrument for calculating J4> (x) ip (x) dx,
founded on my brother's disk-, globe-, and cylinder-integrator,
and described in a previous communication to tiie Royal Society;
for it is easily proved} that if
■ BirW. Thojnion, Proeeediagi 0/ th* Soyal SoeUtji, Vol. xirv., 187S, p.269.
t Mimoirt lur Ui tqualiom d^iTentieUet linlairti du tteond t/firt, LiouTiUe'B
Jimnial, Vol. 1. 1836.
% Combtidge Senate-House Eiammation, Tharada; aftemcion, January 93Dd,
1874.
VOL. L 32
..Google
ifPEKDIX B .
». - I'P (C - J'«, (ir) <bs, I
(2)
etc., J
where u, is snj fonction of x, to b^:ui with, u for example
Uj = x; tbea u^ u^ etc. are succea^ve appro zimatians converg-
ing to that one of the solntions of (1) which TanisheB when x-0.
Kow let mj brother's integrator be applied to find 0 — I u,c/r,
and let its resalt feed, aa it were, contianoualy a second machine,
which shall find tb!e integral of the product of its result into
Pdx. The second machine will give out continuously the value
of u,. Use again the same process with u, instead of u„ and
then u,, and so on.
After thus altering, as it were, u, intou, bypassing it through
the machine, then u, into u, by a second passage through the
machine, and ao on, the thing will, as it were, become refined
into a solution which will be more and more nearly rigorously
correct the oftener we pass it through the machine. If u,^, dovs
not sensibly differ from u,, then each is sensibly a solution.
So far Z had gone and was satisfied, feeling I had done what
I wished to do for many years. But then came a pleasing
surprise. Compel agreement between the function fed into tlie
double machine and that given out by it. This is to be done by
establishing a connexion which shall cause the motion of tlic
centre of the globe of the first integrator of the double machine
to be the same as that of the surface of the second integrator's
cylinder. The motion of each will thus be necessarily a solution
of (1). Thus I was led to a conclusion which wss quite unex-
pected; and it seems to me very remarkable that the general
difierential equation of the second order with variable coefficients
may bo rigorously, continuously, and in a single process solved
by a machine.
Take up the whole matter ab initio : here it is. Take two of
my brother's disk-, globe-, and cylinder-integrators, and connect
the fork which guides the motion of the globe of each of tin*
integrators, by proper mechanical means, with the circuniferencc
of the other integrator's cylinder. Then move one integrator's
disk through as angle = as, and siinultaneously move the other
..Google
v.] CONTINTTOUS CALCULATINQ MACHINES. 499
rr HechMiicU
iategrator's disk through an angle always = I Pdx, a given inMs^iion
function of a^ The circumference of the second integrator's Equation*
cylinder and the centre of the first integrator's glohe move each order. .
of them through a space vhich satixfies the difTerential equa-
tion (1).
To prove this, let at any time g^, g^ be the displacements of
the centres of the two globes from the axial linen of the disks ;
and let dx, Pdx be infinitesimal angles tuiited through by the two
disks. The infinitesimal motions produced in the circumferences
of two cylinders will be
g^dx and g^Pdx.
But the connesions pull the second and first globes through spaces
respectively equal to those moved through by the circumferences
of the first and second cylinders. Hence
g^dx = dg„ and g,Pdx=dgj;
and eliminating g^
±(\dg\
dx\P dx)'
?..
which shows that 9, put for u satisfies the differential eqn^
tion (1).
' The machine gives the complete integral of the equation with
its two arbitrary constants. For, for any particular value of a,
give arbitrary values ff,, ff,. [That is to say mechanically; di»-
oonnect the forks from the cyhndera, shift the forts till the globes'
centres are at distances G^, G^ from the axial lines, then connect,
and move the machine.]
We have for this value of x,
g^ = ff, and ^?i = GP;
"' " dx ' '
that is, we secure arbitrary values for ^, and -^ by the arbitrari-
oess of the two initial positions G^, G^ of the globes.
..Google
[VI.
VI. Mechanical Ihteoratiok of the obneral Lineab
I>iffebi:mtial Equation of ant Oedeb with Vahiable
Coefficients*.
Take anj numl>er i of mj brother's disk-, globe-, &nd cjliader-
integratots, and make &n int^ratitig chain of them tbns :• —
Connect the ^linder of the first bo as to give a motion etpial to
its ownt to the fork of the aeoond. Sinulariy connect the
cylinder of the second with the fork ot die third, and so on.
htitg,g^g^vp tofobethepoaitionB} of the {^obea at any time.
Let iufiniteaimal motions P,dx, F,dx, P,dx, ... be given mmul-
taneously to all the disks (dx denoting an infiniteeimal motion of
some put of the mechanism whose displacement it is convenient
to take as independent variable). The motions {dx^, dx^ .'.. dx,)
of the cylindera thus prodnced are
dK^^g^P^dx, dK^ = g,P,dx,...dK,^g,P,dx (I).
But) by the connexions between the cylinders and fwks vhich
move the globes, dx^-dg^ dK^ = dg^ ...</«,_, = i^,; and there-
fore
^9, = 9,Pxdx, dg,=g,P,dx, ... dg,=g,,^P^^dx\
\P,dx, ...dK, = g,f
d 1 ,d I d>
=g^P^dx, dK^ = g,P^dx, ...dK, = g,P,dx.
Hence
]...
(2).
■(»)■
*'■ P,dxP^dx"P^_,dxP,dx'
Suppose, new, for the moment that we coaple the last cylinder
with the first fork, so that tlieir motions shall be eqn&l — that is
to say, K, = ;,. Then, putting w to denote the common value of
these variables, we have
-L^IA. I d 1 du
**~P,dxP,dx 'P,_^dxP,dx'
..(i).
' Sir W. Tbonuon, Procttdingt of tht Boyal Socitty, Vol xnr., 1876, p. 271.
f For brerit;, the motion ot the dicnmlerenoe at the ojlindar Is called the
cylinder's motion.
X ForbreTity, the term "poaition" ofsnjone of the glubee iansed to denote
its distsnce, positive or negative, from the axial line of the rotating diak on
which it pressei.
..Google
VI.] CONTINUOUS CALCULATma MACHINES. 601
Thus an endless chain or cycle of integrators with disks moved UKhsninl
as specified above gives to each fork 8 motion fulfilling a dif- orOeneni
ferential equation, which for the caae of the fork of the tth inte- Ditrerential
grator b equation (4). The differential equations of the displace- AnjMjnier.
menta of the second fork, third fork, ... (i — l)th fork may of
oonise be written out by inspection from equation (4).
This seemB to me an exceedingly interesting result; but
though /•„ i*,, P^ . . . P, may be any given funddcmB whatever of
X, the differential equations so solved by the simple cycle of inte-
grators cannot^ except for the case of t = 2, be regarded as the
general linear equation of the order i, because, so far m I know,
it has not been proved for any value of i greater than 2 that the
graiend equation, which in its usual form is as follows,
«.S-«.|?--«.s-"=<' w.
can be reduced to the form (4). The general equation of the
form (6), where Q,, Q^ ... Q, are any given forms of x, may be
integrated mechanically by a chain of connected integrators
First take an open chain of i simple integrators as described
above, and simplify the movement by taking
J',=i*, = P,= ...=P,= l,
BO that the speeds of all the disks are equal, and dx denotes an
infinitesimal angular motion of each. Then by (2) we have
Now establish conuexiona between the t forks and the tdi
cylinder, ao that
Q^9, + Q,3,+ ■■■ + G.-,P.., + 0.ff. = «. (7).
Putting in this for g^, g^ etc. their values by (6), we find an
equation the same as (5), except that k, appears instead of u.
Hence Uie mechanism, when moved so as to fulfil the condition
(7), performs by the motion of ite last cylinder an integration of
the equation (5). This mechanical solution is complete; for we
may give arbitrarily any initial values to «„ g„ ff,_^, ... g^, g^\
that is to say, to
rfw (Tw d'"'«
' dx' das" '" dx"''
..Google
lignatian at
Any Urdu.
2 APPENDIX b'. [VL
Until it is desired actually to conatmct a machine for thus
inte^'ating differential equations of the third or aa; Itiglier
order, it is not necessary^ to go into details as to plans for the
mechanical fulfilment of condition (7); it is enough to know
that it can be fulfilled bj pure mechanism working continnouslj
in connexion with the rotating disks of the train of integrators.
DilTurential
KqiMtion at
Anj Older.
ADDEKDUIf.
The intcgmtor may l>e applied to integrate any diflferential
equation of an; order. Let tbere bo t simple integrators; let
^i' 9,1 "i ^ ^*^ displacements of disk, globe, and cylinder of the
first, and so for the others. We have
Now by proper mechanism establish such relations between
(2t - 1 relations).
This will leave j ust one degree of freedom ; and thus we have
2i - 1 simultaneous equations solved. As one particular case
of relations take
and
BO tbat
= ...{» — 1 relations),
7, = K,, etc {i~\ relations);
Thus one relation is still avulable. Let it be
/(«.?..?.. -17., «J = 0-
Thus the machine solves the diiferential equation
dx''-
— , « 1 = 0 (putting u for k^.
Or again, take 2t doable integrators. Let the disks of all be
connected so as to move with the same speed, and let ( be the
..Google
VI.] CONTINUOUS CALCUt^TINO MACHINES. 503
dieplacemeut of any one o£ them from any particul&r position. HerhMint
Let o('«"j '™
, , „ „ „ ,, „ ,, Differentiar
a'l Vt ^. ^1 ^ , V ,.-.ar ', V ' Ennalionot
be the displB«ements of the second cylinders of the several
double integr&tors. Then (the second globe-frame of each being
connected to its first cylinder) the displacements of the first
globe-frames will be
^ ^ rfV rfy
df ' df de' d^'
Iiet now X, T, X', T', etc be each a given function of
X, y, a!, j/, ^', etc.
By proper mechanism moke the first globe of the first double
integrator-frame move so that its displacement shall be equal to
X, and BO on. The machine then Bolvea the equations
g = X, S=r, ^=X'.etc
di" ' df ' dt' '
For example, let
^-(«--=)/i(i"'-«)'+(y-jn
- (y - »)/!(»' -'»)■ + (y' - s)'\
J' = etc., r' = eto.,
wbere^denoteB an; function.
Construct m (Mctionless) Btecl the sur&ce wLose equation is
(and repetitions of it, for practical convenience, though on«
theoretically suffices). By aid of it (used as if it were a cam, but
for two independent variables) arrange that one moving auxiliai-y
piece (an a:-auxiliary I shall call it), capable of moving to and
fro in a straight line, shall have displacement always equal to
(«'-')/i<"='-«)'*(y-s)'i.
that another (a yauxiliaiy) shall have displacement always
equal to
jiGoogle
APPEinUX b'. [VI.
that another (an a^-aiuiliary) sliall liave dispUc«meat equal to
Then connect the first globe-frame of the first doable integra-
tor, BO that its displacement shall be equal to the sum of the
displacements of the avaiLxiliaries; that is to say, to
(^-.)/{(^-.)-t(y-yn
+ («'■- ^)/|(«"-.)V(!,"-y)'l
-I- etc.
This may be done by a cord passing over pulleys attached to
the avauxiliaries, with one end of it fixed and the other attached
to the globe-frame (as in my tide-predicting machine, or in
'Wbeatstone's alphabetic telegraph-sending instrument).
Then, to begin with, ac|just the second globe-frames and the
second cylinders to have their displacements equal to the initial
velocity-components and initial coordinates of ( particlea free
to more in one plane. Turn the machine, and the poedtions of
the partiolea at time t are shown by the second cylinders of the
Geveial double integrators, supposing Chem to be free particles
attracting or repelling one.another with forces varying according
to any function of the distance.
The same may clearly be done for particles moving in three
dimensions of space, since the componenta of force on each may
be mechanically constructed by aid of a cam-surface whose equa-
tion is
'-(Ai)
and titking ^ for tlie distance between any two partidea, and
or =y'-y
or -^a:"-*, etc.
Thus we have a complete mechanical integration of the pro-
blem of finding the free motions of any number of mutually
influencing particles, not restricted by any of the i^proximate
suppositions which the analytical treatment of the lunar and
planetary theories requires.
..Google
VII.] CONTINU^US CALCCLATINO MACHINES.
VII. Harmonic Analyzer*.
Thia is a realization of an instrument deaigne*} rudimentarilf Bsnuonic
in the anthor's communication to the Soyal Society (" Proceed-
ings," February 3rd, 1876), entitled "On an Instrument for
Calculating {Jip (x) ^ (x) dx), the Integral of the Product of two
given Functions."
It consists of five disk-, globe-, aud cylinder-integrators of the
kind described in Professor James l^omson's paper "On an
Integrating Machine having a new Kinematic Principle," of the
same date, and represented in the woodcuts of Appendix B*, iii.
The five disks are all in one plane, aud their centres in one
lina The axes of the cylinders are all in a line parallel to it.
The diameters of tlie five cylindera are all equal, so are those of
the globes ; hence the centres of the globes are in a line parallel
to the line of the centres of the disks, and to the line of the axes
of the cylinders.
One long woodeu rod, properly supported and guided, and
worked by a rack and pinion, carries five forks to move the five
globes and a pointer to trace the curve on the paper cylinder.
The shaft of the paper cylinder carries at its two ends cranks at
right angles to one another ; and a toothed wheel which turns a
pai-allel shaft, and a third shaft iu line with the first, by means
of three other toothed wheels. This third shaft carries at its
two ends two cranks at right angles to one another.
Another toothed wheel on the shaft of the paper drum turns
another parallel shaft, which, by a slightly oblique toothed wheel
working on a crown wheel with slightly oblique teeth, turns
one of the five disks uniformly (supposing to avoid circnmloou-
tion the paper drum to be turning uniformly). The cylinder of
the integi-ator, of which this one is the disk, gives the coatinu-
ously growing value of /^dc.
Each of the four cranks gives a simple harmouic angular
motion to one of the other four disks by means of a slide and
croBshead, carrying a rack which works a sector attached to the
disk. Heuce, the cylinders moved by the disks, driven by the
* Sir W. TbomeoD, Proettdtngt of llu Eoj/al Society, Vol IZVIL. 187S, p.BTl.
VOL I. 33
..Google
J APPENDIX B'. [VII.
iint mentdtmed pur ot enaks, give tbe contiiuiotuljr growing
valoeBof
ly OOB— — ax, and lyam —
where e denotes the cdnnmLfereiioe of the psper dnua : and the
two remaming cylinders give
lycoB ax, and jyau ax;
where u denolee the angular velodly of the shaft carrying die
second pair of ahafts, that of the first being onitf .
The machine, with the toothed wheels actaally monnted on it
when shown to tbe Boyal Societj, gave w = 2, and wsa tiierBfore
adopted for the meteorological application. By removal of two
of the wheeU and substitntion of two others, which were laid on
the table of the Royal Society, the value of a beoome* ^ Yin*
(according to factors found by Mr E. Roberts, and supplied by
him to the author, for the ratio of the mean lunar to the mean
solar periods relatively to the earth's rotation). Thus, the same
machine can serve for analyaug out dmultaneously the mean
lunar and mean solar semi-diurnal tides &om a tide-gauge curve.
But the dimensions of the actual machine do not allow range
enough of motion for the m^'ority of tide-gaoge curves, and thej
are perfectly sufficient and suitable for meteorological woik. The
machine, with the train giving w = 2, is therefore handed over to
the Meteorological Office to be brought immediately into prac-
tical work by Mr Scott (as soon as a brass cylinder of proper
diameter to suit the 21A length of his curves ia substituted for
the wooden model cylinder in the machine as shown to tlie
Royal Society) : and the constmction of a new machine for the
tidal analysis, to have eleven disk-, globe-, and cylinder-int^rators
in line, and four crank shafts having their axes in line with the
paper drum, according to the preceding description, in proper
periods to analyse a Ude curve by one process for mean level, and
for the two components of each of the five chief tidal ctm-
stitueQte — that is to say,
..Google
VII.] CONTlSUOnS CALCULATISa MACHINES. 507
(1) The meAn BoUr aemi-diamal ;
(2) „ „ Innar „
(3) „ „ lunar quarter-diurnal, oballow-'water tide ;
(i) „ „ Innar deolinational diumat ;
(5) II » Iniii-solar declinational diurnal ;
ia to be inunediatel; commenced. It is hoped tliat it ma^ be
completed vithont need to apply for any addition to the gruit
already made by the Royal Society for harmonic analycers.
Counterpoises are applied to the crank shafts to fulfil the con-
dition that gravity on cranks, and sliding pieces, and sectors, is
in equilibrium. Error from "back lash" or "lost time" is thus
prevented simply by fictional resiHtance against the rotation alt
the uniformly rotating disk and of the tertiary shafts, and 1^
tiie weights of the sectois attached to the oscillating disks.
Addition, Apiiil, 1.879. The machine promised in the pre-
ceding paper has now been completed with one important modi-
fication : — Two of the eleven constituent integratois, instead of
being devoted, as proposed in No. 3 of t£e preceding schedule,
to evaluate the lunar quarter-diumal shallow-water tide, are
arranged to evaluate the aolar declinational diuroal tide, this
being a constituent of great practical importance in all other
seas than the North Atlantic, and of very great scientific interest
For the evaluation of quarter-diurnal tidee, whether lunar or
solar, and of semi-diurnal tides of periods the halves of those of
the diurnal tides, that is to say of all tidal constituents whose
periods are the halves of those of the five main constituents for
which die machine is primarily designed, an extra papcr-t^linder,
of half the diameter of the one used in the primsiy application
of the maohiue, is construoted. By putting in this secondary g,
ir^linder and repassing the tidal curve through the machine'tiie JSmSutx,
secondary tidal constituents (corresponding to the first "over- S^loh^
tones " or secondary harmonic constitaente <^ musical sounds) ^^^
are to be evaluated. Similarly tertiary, quaternary, eto. tides w"*"".—
(corresponding to the second and higher overtones in musical uaiogom
sounds) may be evaluated by passing the curve over cylinders of Jj^lj^
one-third and of smaller sub-multiples of the diameter of the
primary cylinder. These secondary and tertiaty tidal consti-
tuents are only perceptible at places where the rise and fall is
influenced by a laige area of sea, or a oonsidentble length of
..Google
3 APPESDIX B'. [VII.
channel through which the wlude amoaiit of the rise and fall is
notable in proportion to the mean depth. They are very percep-
tible at almost all commerdal ports, except in the Mediterranean,
and to them are due sncli carious and piactically impm-tant
tidal characteristics aa the double high vatets at Southampton
and in the Solent and on ihe south coast of En^and from the
Isle of Wight to Portland, and the |»otncted duration (A high
water at Havre;
END OP PAET ]
cutBNDOM; patMTUi Bi 0. t, ouir, m.a., a laa cxirasstri pbxbs.
..Google
University Pkbss, Caubudgb,
April, 1879.
CATALOGUE OF
WORKS
PUBLISHED FOR THE SYNDICS
OF TUB
loninn:
CAMBRIDGE WAREHOUSE, 17 PATERNOSTER ROW.
CfilArilrgt: DEIGHTON, BELL, AND CO.
lti»ig: F. A. BROCKHAUIJ.
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PUBUCATIONS OF
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Editor of the Greek Testament, Codex Augiensis, &c., and one of
the Revisers of the Authoriied Version. Crown Quarto, cloth, gilt, ii^r.
From the Timit. copr of ibc Bible, which pi
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lariy gnteful to (Iht Cimbiidgt Ui
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•ilUncc oT Dr ScriTvneT, a complete
•dioonoftbeAnthoriicdVf-- '■'
lUh Bible. Id edi'
iBtroductioD. Hi
the hiUDTT - ■
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ihgcnphr.
K of Ihe Oteik T(
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be (^mbridgo Univenily Prc^
the Engliih B" '
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e best pMagraj* BiMe tier pnbUMwd,
Dr Scrivener npiy be congrituUtcd on n
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From the LomJen Quaritrlj Rttvw^
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4 PUBLICATIONS OF
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TRANSLATION OF THE FOURTH BOOK of EZRA,
discovered, and edited with an Introduction and Notes, and a
facsimile of the MS., by Robert 1_ Benslv, M.A. Sub-Librarian
of the University Library, and Reader in Hebrew, GonTille and Caini
College, Cambridge. Demy Quarto. Qoth, loj.
"Editd wilh true icholulT OHBplcle- added > sew dupta to the Bible, Bid, aut-
Tien.'—U'rilminilrr Rnrirm. lingiu the (UluaEiii may at Gnt ri^ >p>
"Wer uch je mil deal 4 Buche Eha P^^p ^^ " na eiaggetatioD of the actual Tsic^
einEebcnder beuhKfcJKt lut, wmi durch die tf by Ibe Bible we underitand thu of die
oblE#, ID jeder Beziehuag miuterhafte Pub- laner Aire whieb cmlaiiu tlie ApocTTplH,
licalion iu heudigCs ErUauDen reneut Hci- ud if the Snund Book of Etdnu can b«
den."— rAioiifiKAr LiliralBmiliaig. fairly called a part of the Apoaypha."—
*■ It hu bcea laid of Ibii book that it has Saturday Rtviev.
THEOLOGTHANdENT).
SAYINGS OF THE JEWISH FATHERS,
comprising Pirqe Aboth and Pereq R. Melr in Hebrew and En^^,
with Critical and Illustrative Notes. By Charles Taylor, MA
Fellow and Divinity Lecttirer of St John's College, Cambridge^ and
Honorary Fellow of King's College, London. Demy 8vo. doth. los.
„ ;1 Mauefcelh Abolh
or Pirque Aboth, which title hepuaphraiei
■■ " Sayings of tlie Fithen." iliac fathen
■re Ribbii who estaUiihed xAaoli sod Bught
in the period from two centuriei before to
Dt, tlu ii the fintire*
. . the Eu^iih lauguif*
accompaaied by tcholajjy ncie^ of any pcr-
tion at the Tahnod. In other wmdi, 11 h
Ihe fini iastaucE of thai most TnJiiaUF ud
Tiegltcted porlioQ of Jewish Hieral
OB that Scripture at a
L the
. Their
Bed, we fcreiei will be Ihe
It aiiU of the future for the proper 1
" w of the Bitfc. . . The Sufbip
■rir
._ _jr Lord Himseifand 10 the
leaned Fhaiiiee, St Paul. To a large ea-
lent ll waa aecepud in the eaily aget of the
Chriuiaa Chureh, and, through the authariiy
nnaded Co the Fathen of Ox Church, be-
came the uoqueitioDed and orthodox lyttem
il peculiarly in
.nlhov
10 look
bead of Hebrew n
3^ Mmelhing'of their jS^i^
The New Tesumenl oboundi
lUi^blcTlhoie of"the' Je^ih
1 (heu latter probably would
Ealiafactory and frequent illui-
I text than Ihe Old TeiUmeut.''
uekelh Aboth' atandi it tbc
dentasdbw of the Bitfc. ,
thi JrwitSFaiitn uy c
ly, aiHl. moreonr, of a icholiinhip nnunallr
ihorougb Bad fimtfaed. It U greatly to M
hoped that ihif ioKEalmenl ii an earnest of
future work in the lame direction; the Tal-
■nud i< a mine Ibal will take yean to wok
oat."—Dit£IiK UHtvrrrH^ MagoMimt,
"A careful and Ihoraugh edition whid
doei credit lo Engliih Kholanliip. oif ■ >hcR
treatiie from the Hishna, coauiam^ a laio
Jewish teac^iers iaxaiedlalelT preceding, cr
immedialely fdlowing the Qvitfiin oa. , •
Mr Tayloc has his traunre-bouie replela
wilh RabUaie lore, and the cadre nhnw
(espedally Ibe "Eicunuiei'^iiriillaf auH
inierniing matter. . . . We would alio call
special attentioD to the fieqiient f"
Te»tanenc.'~C«»/™>mHy Xm
Zendon: Cambridge Warehoust, 17 Paiemesttr Row,
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THEODORE OF MOPSUESTIA.
Tbe Latin version of the Commentary on St Paul's Epistles, with the
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libros quinque adversus Hsereses, versionc Lalina cum Codicibus
Claromonlano ac Anindcliano denuo collata, prsmissa de placitis
Gnosticorum prolusione, fragmenta necnon Grxce, Syriace, Armeniace,
conunentattone perpetua et indicibus variis edidit W. WiGAN Harvey,
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The tert newly revised from the original MS,, with an English Com.
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H. A. HOLDEN, LL.D. Head Master of Ipswich School, late Fellow
of Trinity Collie, Cambridge. Crown Oilavo. TS.fad.
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LIBRI TRES AD AUTOLYCUM
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GiLSON Humphry, S.T.B. Collegii Sanfliss. Trin. apud CanUbri-
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TERTULLIANUS DE CORONA MILITia DE
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Crown Oflavo. s*-
THEOLOGT-(ENGLISH).
WORKS OF ISAAC BARROW,
compared with the Original MSS., enlarged with Materials hitherto
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Vicar of Holkham, Norfolk. 9 Vols. Demy Oftavo. ^3. y.
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and a Discoime concerning the Unity oi the Churdi, by ISAAC
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PEARSON'S EXPOSITION OF THE CREED.
edited by Temple Chevallier, B.D. late Felloir and Tutor of
St Cathaiine's Ct^egc, Cambridge. Second Editioo. Demy O^vo.
7t.6ii.
AN ANALYSIS OF THE EXPOSITION OF
THE CREED
written by the Right Rer. Faiher in God, John Peakson, D.D.
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CESAR MORGAN'S INVESTIGATION OF THE
TRINITY OF PLATO,
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TWO FORMS OF PRAYER OF THE TIME OF
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— .-, kiblo niluia^ in tltc kindr™
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iSt^ by W- Canw Haxlitt (p. 340), wcUam been lo«t si^t of for x» yean.' Bt ii
nluable Tolunit, DoiiiuniDE in >]l : '
,.. .. ^ ., nublHaliani. I am Htsbt^ la mn
;h widi th« utOEnpii of Hum^irey DnoA,
.......,_■.__ !_.. .i. 1 — Hi* of my friend
Df the rcHD of EJuabeih. puUica
^pboTHuiD^ireyDrHa, fDlloirii.., <—•-- - — — a-
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THE CAMBRIDGE UNIVERSITY PRESS. 7
SELECT DISCOURSES,
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Sdaol [the OmbridfE Pliloaists). Tbey ofthcir time, beiweea whom theysoi^ht la
hue ■ li^t to a plies in Enalish litouy ■icdialc...,Wbi>i ihey really did lot the auie
UMary."— Ur MArTusw Abhulu. in the aC niigioui (housht has never been ade-
CntUm^tntrir Rminf. quattfly appncialec. lliey worked with too
"Of ill ttK products of the Cambridce biile comUnation and eonustency. But it b
5dHwl» the 'Select Diseaunes' are perhuA impossible in any leal uudy of the aie wM to
the highest, as they ate the toou accessible recognise tbe ligiuficancB of their laEotm, or
and the nnit widely appreciated... and indeed to fail to use how mvch the hiBher movenent
no ■piritualiyihDughtfuI mind can read tliem of the national mind was due to them, while
immoTed. They cany us lo directly into an othets earned the religious and dvil stninle
"-ere of divine pUloiophT, luminoui forward In ill slemer ■snues."— Principal
richest liahl* of meditative lenlus... IVlloch, Kallimti Tietitty in England
OH of tboae tare ihinken m whoa rn IIk i;«i CtnlHr,.
•aisncaa of view, and deplh, and wealth of "We may insianoe Mr Henry GriBhl
po«ic and ipecalative insight, only lerved to Wiliams'i revised edition of Mr John Smith'i
mbe onn fidly the reUeioiB spirit, and 'Select Discounet,' which have won Mr
while he drew the mouM ol^his thought from Matthew Arnold-^ adminilon, as aa eumple
Plolinos, he vivified the lubitanoe of it From of worthy work for an Univenity Fran n
St PauL'" Uiiderlalte."— n>w«,
THE HOMILIES,
with Various Readings, md the Quotations from the Fatheirs given
at length in the Original Languages. Edited by G. E. CORRiE, D.D.
Master of Jesus College. Demy 0<;Uvo. 7s.f>d.
DE OBLIGATIONE CONSCIENTI.^ PILELEC-
TIONES decern Oxonii in Schola Theologica habits a Roberto
Sakdeksok, SS. Theologiae ibidem Professore Regio. With English
Notes, including an abridged Translation, by W. Whewell, D.D.
late Master of Trinity College. Demy Oftavo. ^s. td.
ARCHBISHOP USHER'S ANSWER TO A JESUIT,
with other Trails on Popery. Edited by J, Scholefield, M.A. late
Regius Professor of Greek in the University. Demy Oflavo, "jt, 6</,
WILSON'S ILLUSTRATION OF THE METHOD
of exfdaining the New Testament, by the early opinions of Jews and
Christians concerning Christ Edited by T. TURTON, D.D. late Lord
Bishop of Ely. Demy Oflavo. 5^.
LECTURES ON DIVINITY
delivered in the University of Cambridge, by John Hey, D.D.
Third Edition, revised by T. Turton, D.D. late Lord Bishop of Ely.
3 vols. DemyO^vo. iff.
London: Cambridge Warehouse, 17 PcUemoster Row.
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8 PUBLICATIONS OF
aSEEE AUD LATIN CLASSIOS, ftc. (See also pp. I6-2OJ
THE AGAMEMNON OF AESCHYLUS.
With a Translation in English Rhythm, and Notes Critical and Ex-
planatory. By Benjauin Hall Kennedy, D.D., Regius Professor
of Greek. Crown Octavo, doth. &r.
" OfK di Ihe Imu edidom of the oaitcr- IntnsUlor, the criticBl idKJar, ud ike etlucid
pliaot'Gfnktnicsdy.'— ^(4mniM. ludenL We must be conteuled to thulE
" By asuberleM ecber like tuppr o^d PnCeuor Kennedy im hit udminble execa-
weigfaty helps to a coheretil and cunuiEent tioo of x great ivdenokiiii^''-.-
:^r&"— Cin/iMfcrarv hie i.
_ ,__.- Pn&ce
*'Il is Deedleis lo mnllipl^ proofs of the pOkDL" — Pzofe
nhw of thtl roJume alike to the poedcal
HEPI AIKAI02TNH2.
THE FIFTH BOOK OF THE NtCOMACHEAN ETHICS OF
ARISTOTLE. Edited by Henby Jackson, M.A., FeUow of Trinity
College, Cambridge. Demy Octavo, cloth. &r.
"It ii not loo much to aj that ume of Sctaolui iriU hope dul All b ant the coljr
th« poiDti he discunea hare Derrr hod «o portion 04* the AriitDtelian writing which h«
■ucn light thnrtrn vpoB them hefore. ... u Likely to ediL" — AiMtrntntm,
PINDAR.
OLYMPIAN AND PYTHIAN ODES. With Notes Explanatory
and Critical, Introductions and Introductory Essays. Edited by
C. A. M. Fennell, M.A., late Fellow of Jesus College. Crown Oc-
tavo, cloth. 9^.
"Mr Fennell deienret die thinki of all Jndfnest, ind, la pirticuUr, copieai lad
edition of the Olympun and Pythian odei. To hi< qualiJicaliQni in thii lau re
He bridgt lo hii tAAk the necessary eothu- page h^n " '"-—^ — —
tiud for hii author, great tndiutryt a wund
THE NEMEAN AND ISTHMIAN ODES. [Preparing.
PRIVATE ORATIONS OF DEMOSTHENES,
with Introductions and English Notes, by F. A. Paley, MA. Editor
of Aeschylus, etc. and J. E. Sandys, M.A. F«11ow and Tutor of St
John's College, and Public Orator in the University of Cambridge.
Fart I. containing Contra Pbormionem, Lacritum, Pantaenetum,
Boeotum de Nomine, Boeotum de Dote, Dionysodorum. Crown
Oflavo, cloth. 6j.
"Mr Paley'i uJwliRhlp ii Kund and the elncddallOBafinanenDr daily UIc. iDlhs
■camte, hii enperieoce of editing "ide, and delineation of which Demotlhens i* M nch,
if he ii content to devote hia teaming and obtaini full justice at hii hand*. .... We
abilitiei to the production of luch manual! bope that thii edition may lead iha way
as thoc, Ihvv will bo receiTcd with giatihide to a more general atudy ot these apec<^H
thnyugluitittliehLgherKhODlanf the country. in achooli ihan haa hitherto heen pouibla.
Mr Sandyi ii deeply read in the Gcrmin .... The indei ik aitRiBcly corapleig, and
litenttnrc which bean upon hit author, and of great service to Icamcn." — Acadtmy^
Part II. containing Pro Phonnione, Contra Stephanum I. II.;
Nicostratum, Cononem, Calliclem. ys. fid.
"To give even a brief ikelch of th«e cax It ii long nnco wa bars coBa
iptechct [Prv Pksrmitnt and CtMtm SU- upon a work evincing more point, achola^
Jj^mtm] would bfl incmnpaLible with our ship, and varied research and illustratialk than
limiti,tluniEh we can hardly conceive I talk Mr Suidyi'i contiihulion to the 'Private
moni uteful to the datsicu or pmfeuional Orationt of I>«notlheneiV — Saimrdajr
tcholar tban to make one for bmielf. .... Rnim,
It i> a great boon lo Ihoie who Kt them- " the edhion reflect* oedir ca
■civet to unravel the thread of aivumcnti Cambridge scholarahip, and ought to be a-
proaiHl eon in have the aid of Ur Sandys'i teotivcly u^d." — AtAeiuewm.
ini" an"»y"'ita™e"ii iva
in the needlul help which ena_.__
utfully tiati-'SfKlaU
Loudon : CamhrUge Warehouse, 1 7 PalemosUr Raw,
.,j,.,.ib;,Goot^lc
THE CAMBRIDGE UNIVERSITY PRESS. 9
THE BACCHAE OF EURIPIDES,
with Introduction, Critical Notes, and Arch ecoI epical Illustrations,
by 1. E. Sandys, M.A., Fellow and Tutor of St John's College, Cam-
bridge, and Public Orator. [Preparing,
PLATO'S PH.^DO,
literally translated, by the late E. M. Cope, Fellow of Trinity College,
Cambridge. Denty Oflavo. ^i.
ARISTOTLE.
THE RHETORIC. With a Commentary by the late E. M. Cope,
Fellow of Trinity College, Cambridge, revised and edited for the
Syndics of the University Press by J. E. Sandys, M.A., Fellow and
Tutor of St John's College, Cambridge, and Public Orator. With
a biographical Memoir by H. A. J. MuNRO, M.A. Three Volumes,
Demy Oflavo. £i. 1 is. 6rf.
"ThiiworitiiiDuany wayscredilable to to his Uiboiiri When the oriniia] Com-
ciIeniJveuudiliDnDf MrU<>i>ehiiiueirbe>n fun (he end oT Ihe third bonk. Mr Sandra
none (he leu speaking evidence to (ho value carefuJly Hippliea the de6cicncy, tfbilciwinE
of the tradiiion which he continued, if it ii Mr Ccrpc^i Kcnerai plan and the ftligh(e»(
notequaUyaccoDipanied by (hooe Qiuliticiof av-jilabJe indications of his uiUnded treat.
■peculalive origiDaLity and lOflepenaeDt ju^g' ment. In Appendiovri he iias reprin(ed from
writer than to his ichooL, And while it niut Cope's; and. what is lielter, he has given the
ever be regretted that a work so laborioui best of the late Mr Shilleto's 'Advenaria.'
riiould not hava received the last toiKface of In every pan of his work — revising, siu>ple-
Mr Sandy*, lor the minly, unielGih, and un- cecdingj'y wtAV'—Sxamiarr.
moxdiffinliaiiddelicaietuk. If nnEngliih that the high ej
the crowning merit of our best EngLith e,
- j< diuppointed. Mr Cope'i
* wide nod minute ncquaintAnce with ail the
Aristotelian vritingi,' to which Mr Sandyt
justly bean teitinony, hii thorough know
ledge of the important contributiou of mo-
dem Ceman tcholan, hit ripe imd ttcciinu
KhoLarship, and above all, that sound judg-
' Una g. ■
, >f the anssics. «i.
reference 10 impailanl works that have ap- knowledge oT Gt«k literature which wt have
peared unce Hi Cope's illnets put a period had for many yan.'S/a'ntBr.
P. VERGILI MARONIS OPERA
cum Prolegomenis et Commentario Critico pro Syndicis Preli
Academici edidit Benjamin Hall Kennedy, S.T.P,, Graecae
Linguae Professor Regius. Extra Fcap. Oflavo, cloth, ^s.
M. T. CICERONIS DE OFFICIIS LIBRI TRES.
with Marginal Analysis, an English Commentary, and copious Indices,
by H. A. HOLDEN, LL.D. Head Master of Ipswich School, late Fellow
of Trinity College, Cambridge, Classical Examiner to the University
of London. Hew Edition. Crown O^vo. 7s. 6d.
M. TULLII CICERONIS DE NATURA DEORUM
Libri Tres, with Introduction and Commentary by Joseph B. Mayor,
MJL, Professor of Classical Literature at King's College, London,
fonnerly Fellow and Tutor of St John's College, Cambridge, together
with a new collation of several of the English MSS. by J. H. Swain-
SON, M.A., formerly Fellow of Trinity College, Cambridge.
[Nearly Ready.
London : Cambridge Warehouse, 1 7 Patemosler Row.
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PUBLICATIONS OF
AKABIC Am) SAK3KBIT.
POEMS OF BEHA ED DIN ZOHEIR OF EGYPT.
With a Metrical Translation, Notes attd Introduction, by E. H.
Palmer, M.A., Bamster-at-Law of the Middle Temple, Lord
Almoner's Professor of Arabic and Fellow of St John's College
in the University of Cambridge. 3 vols. Crown Quarto.
Vol I. The Arabic Text. itw. f>d. ; Ooth extra, 15/.
VoL n. English Translation, 10*. 6d.; Ooth extra, 15*.
" PnfeiKr Palmei'i iLCDTitT m advuicinrt
Aivbic KhgUnhip has fDrmaijr ihown ilscif
Gianiur, ud hi* I>Eicripdvt ciulcigiK of
Anbic MSS. in tbt Ubniy of Thniiy Ccl-
IcfC, Canbtidcc He hu mnr prnduced an
■diiunble mt, irhidi illuMnui u a rouuk-
"' ' ribilitv uid cFmcci of the
H well, and of which he
_ ID he perfect inaMn....Tfa«SyBd>caia
tt CoiabHdge UniireriiiT nun not pom with-
out the rKO^Dilion of theb- libeialiiy in
bnDHing out, id a hcmliv fotm, » tinporlaiit
Oricnul idialmhip hu thiu been tr'ttiAl
«tlb«idi«ed by Camhridge."— /lerfiM «ai7.
out an exprcbion of adtninttion for tbe pfT-
fiWioD to vhkfa Atabic lypogiaphy has Iki^h
hmuhl in Englaad in ibu ougnifkcnt Ori-
vBtu work, the productiofi of which redouodfl
bythela;
langiUK^ ha la
M the ImperiiboMo credit <
Hily fair to add that the book.
nfl«u gnai credit en the Cambridge Uni-
dming-rcMoi"— "-BWJ. * " "
"For eaie and facility, for wiety of
OMtArm, of'the i^k^Df Knraf^ ou^owii
Prof. Palmer hai made .
nadficia
An^iini
«rUimbndge. Itiaaii beproDou
of the Ei£i» worthily rinlt Ihe technical
if liUm, the Kudy
.!y of the original, his Engliib
ro diitinfuiahed by renatility, commam or
inguBso, rhythmical cadence, and, as we
ave remarked, by not UD^ilfnl imitaliocuof
ic uvlef of several of our own favourite
Dclsjiving and Acfd."— Saturday Sreinm.
"Thi! iumptuuuj edition of the poeml of
lehied-dln Zaheii i> a very welcome addi-
on ID the small xries dT EasLem poeia
ccesiible Id readen who are not Oiienlal-
ilL ... la all there ii ihal eniuiiiite finiih of
'hich AiaUc poetry n luiocpiible in » rare
degm. The foim ii a)moit always bean-
iful, be the thought what it may. But this,
f course, can only be fully appncialed by
Irientalists. And this brings vs to the mns-
iiian. It is enccllently well done. Mr
'aimer hai tried id imitate the (all of the
liginal in his leleciion of tbe Eng litb men
is Ihe rarioua pieceit and thus coitTiva to
onvey a faint idea of the graceful Bow of
leAiabic Altogcther.heinsideoflhe
ook ii worth* of tbe beautiful arabeviuc
inding chat rej«ces the eye of tbe lorcr of
NALOPAKHYANAM, OR. THE TALE OF NALA ;
containing the Sanskrit Text in Romaji Characters, followed by a
Vocabulary in which each word is placed under its root, with references
to derived words in Cognate Languages, and a sketch of Sanskrit
Grammar. By the Rev. Thomas Jarrett, M.A, Trinity College,
Regius Professor of Hebrew, late Professor of Arabic, and formerly
Fellow of St Catharine's College, Cambridge. Demy Oiflavo. tcts.
boKiu of Ihe poetry of the Arab*.
Ant we make the acquaintance of a po
hlng better than monD
cd a> supeHar in^an
MvTHOLOcvAiiaHari
TttHuL).t- '9i-
■aimer has pmduced th<
SeKTipllODS of
London: Cambridge Wareheute, i-j Faiemoster Row.
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THE CAMBRIDGE UNIVERSITY PRESS.
HATHEHATICS, FH73I0AL SCIENCE, &c
A TREATISE ON NATURAL PHILOSOPHY.
By Sir W. Thomson, LL.D., D.C.L., F.R.S., Professor of Natural
Philosophy in the University of Glasgow, Fellow of St Peter's College,
Cambridge, and P. G. Tait, M.A., Professor of Natural Philosophy
in the University of EdinbuE^h; formerly Fellow of St Peter's College,
Cambridge. VoL 1. Part 1. Demy Octavo. \bs.
ELEMENTS OF NATURAL PHILOSOPHY.
By Professors Sir W. THOMSON and P. G. Tait. Part I. 8vo. cloth,
Second Edition. ^.
"Thii woA ii duiciicd apedally Tor Ihf trigoDomelry. Tyru in NatunI PUlcHci|>h*
lue of ichwls ud Junioi (rasM» in ihe Uoi- caimol be bitwrdirgcted Ihan by beini; tali
THE ELECTRICAL RESEARCHES OF THE
HONOURABLE HENRY CAVENDISH, F.R.S.
Written between 1771 and 1781, Edited from the original manuscripts
in the piossessian of the Duke of Devonshire, K. G., by J. Clerk
Maxwell, F.R.S. [Ntarly ready.
HYDRODYNAMICS.
A Treatise on the Mathematical Theory of Fluid Motion, by HORACE
Lamb, M. A., formerly Fellow of Trinity College, Cambridge ; Professor
of Mathematics in the University of Adelaide. [/« Ihe Press,
THE ANALYTICAL THEORY OF HEAT.
By Joseph Fourier. Translated, with Notes, by A. Freeman, M.A.,
Fellow of St John's College, Cambridge. Demy Octavo. i6j.
anliquALcd hy the pnwTHs of idencc. It it i« a model ofmalheinalicat RAvninjE applie<d
Dot only (lie lirA and Ihe gteamt book on ro physical phenomena, and in rcTnarkable for
the phyaicJ luhject of the conduclion of the ineeniiily of the analylical proccu em-
Heal, but in every Chaplei new vicwi an ployeiT by the author.''— CM<imA>ra>7
opened up into •tut fields of maihemaiical Srvuvi, Oeiobet, iS;!.
n-booki nay
" There cannot be two opinions a> to the
I proofA of CAitifHT. it has been called *an eaquiuta
LndepcDdend/, l^ malhematidansaf di^
Anafylicai Theory pf Heat, trans- seems Ijltle prescDt pro^Kct of its being
by Mr Alex. Freeman, should be in- Aupeneded, iFionoh it is already more than
■elf will in all lime coming retain h» unique uidepeDdeniiy, by malhemalicianiaf diHcmit
pnr(^tive of being the guide of hia reader schools. Manf of the very erentest of me-
tni tmtnr—EnlratlJrBm UtUrrf Prt- key which iint opened to Iheni the litainie-
JttitT CUr* Majrarll. house of nuthematical phyHCV ltisstiU<4<
1.1. :. .t — .v., !■ — ;_■. — .. — ; — — '.booli of Heat Conduciioo,_ and theTE
AN ELEMENTARY TREATISE ON
QUATERNIONS.
By P. G. Tait, M. A., Professor of Natural Philosophy in the Univer-
sity of Edinburgh; formerly Fellow of St Peter's College, Cambridge.
Second Edition. Demy Svo. t4.r.
London: Cambri^e Warehouse, ij Paiemosler Row.
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13 PUBUCATIONS OF
COUNTERPOINT.
A Practical Course of Study, by Professor G, A. MaCFArren, M.A,
Mus. Uoc. Demy Quarto, cloth, ^i. f>d.
A CATALOGUE OF AUSTRALIAN FOSSILS
(including Tasmania and the Island of Timor), Stratigraphically and
illoologically arranged, by ROBERT Etheridge, Jun., F.G.S., Acdog
Palaeontologist, H..M. GeoL Survey of Scotland, (formerly Assistant-
Geologist, GeoL Survey of Victoria),
"The wDilc ■> amngnt with gmt clear- pip«i CDUnIled b« Ihe intbor. md n bdta
■KU, and coBUimi a rull liu of ibe b^M^ mi » ihc gCKia. "— Saturdrnj XoKm.
ILLUSTRATIONS OF COMPARATIVE ANA-
TOMY, VERTEBRATE AND INVERTEBRATE,
for the Use of Students in the Museum of Zoology and ComparatiTe
Anatomy. Second Edition. Demy Octavo, cloth, 2x. 6d.
A SYNOPSIS OF THE CLASSIFICATION OF
THE BRITISH PALEOZOIC ROCKS,
by the Rev, Aoau Sedgwick, M.A., F,R.S,, and Frederick
M'COY, F.G.S. One voL, Royal Quarto, Plates, £i. it.
A CATALOGUE OF THE COLLECTION OF
CAMBRIAN AND SILURIAN FOSSILS
contained in the Geological Museum of the University of Cambridge^
by J. W. Salter, F.C.S, With a Portrait of Professor Skdgwick.
Royal Quarto, cloth, yt.fid.
CATALOGUE OF OSTEOLOGICAL SPECIMENS
contained in the Anatomical Museum of the University of Cam-
bridge. Demy Oflava 2x. 6d.
THE MATHEMATICAL WORKS OF
ISAAC BARROW, D.D.
Edited by W. WHEWELL, D.D, Demy Octavo, js. 6rf.
ASTRONOMICAL OBSERVATIONS
made at the Observatory of Cambridge by the Rev. JaHES Challis,
M.A., F.R.S., F.R.A.S., Plumian Professor of Astronomy and Experi-
mental Philosophy in the University of Cambridge, and Fellow of
Trinity College, For various Years, from 1846 to i86a
Loadm : Cambridge Warthoust, 17 Fatemosier Row.
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THE CAAfBRIDGE UNIVERSITY PRESS. 13
LAW.
THE FRAGMENTS OF THE PERPETUAL
EDICT OF SALVIUS JULIANUS,
collected, arranged, and annotated by Brvan Walker, M.A. LL.D.,
Law Lecturer of St John's College, and late Fellow of Corpus Christi
College, Cambridge. Crown Bvo., Cloth, Price 6j.
"Thii b one oC ihe lateic, we believe menuriei and llie Initilutei . . . Hithena
legal ichoUrship by Ihu revived ttudy of Ibe oidlDary EDglisb itudent, and 4uch a
■he Ranun Law u Ciunbridge which is DOW uudeni will be inlecetted ai well as perhapi
10 Durked a feaiure ia the induitrial life ■uipriieil la find how abundantly Ihe exlanl
of the University. ... Ill ibe present book fi-aginents iUiuttateandcleiruppainii which
we bave cbe fruitt of the same kind of have aiuuled his auemion ID the Cumcnen-
ihoiough and well-ordered study which was taries, at the Iiulllules. ot Iht Digest'—
brought to bear upon the notes to the Cum- Lttu/ Tirtux,
THE COMMENTARIES OF GAIUS AND RULES
OF ULPIAN. (New Edition, revised and enlarged.)
With a Translation and Notes, by J. T. Abdy, LL.D., Judge of County
Courts, late Regius Professor of Laws in the University of Cambridge,
and Bryan Walker, M.A., LL.D., Law Lecturer of St John's
College, Cambridge, fonneriy Law Student of Trinity Hall and
Chancellor's Medallist for Legal Studies. Crown O^vo, i6j.
;dl>on Mesm AbdT "Tbenumber of ImxAioh varknu nibjecti
e Iheir worlc well. oTihe dvil law, which have latelyJBiiedfnnii
Che ediion dcKIve the Preu, ahewa that Ihe nvivalof Ihe mtlidT
>hich have lately!
that Ihe nvivalo
ipecial cammendaiion. They have pTBenled of Roman iurisi
_.._-«; _
Howed to vpcak for himself, and Ih
kIi that he ii really iludying Roc
1 the original, ud not a fanciful rep
lOBOfil.*-.^ ■
THE INSTITUTES OF JUSTINIAN,
translated with Notes by J. T. ABDV, LL.D., Judge of County Courts,
late Regius Professor of Laws in the University of Cambridge, and
fonneriy Fellow of Trinity Hall ; and BRYAN WALKER, M.A., LL.D.,
Law Lecturer of St John's College, Cambridge ; late Fellow and
Lecturer of Corpus Christi College ; and formerly Law Student of
Trinity Hall. Crown Oflavo, i6j-.
the study of jurisprudence. The iHtt of
by the difficulty
of stiuggling Ihrough the
language in whilh
^it,i^tain=d, j. Allb.
pnaiicd scholar^ wh^ie l^owledge of
uical models does not alwayi avail ihem
"The notes are
learned and carefully aim-
dealing wiih Ibe cecbnicat'tties of leial
[»kd, and this cd
ition wiU be found Uieful
raseologV- Xor can the ordinary dicjtion.
esbe expecied Co furniphall Ihe help IhaC
.CJflYjri
Dr Walker have prodiaed
SELECTED TITLES FROM THE DIGEST,
annotated by B. Walker, M.A., LL.D. Fart 1. Mandati vel
Contra. Digest xvii. i. [In the Press.
GROTIUS DE JURE BELLI ET PACIS,
with the Notes of Barbeyrac and others ; accompanied by an abridged
Translation of the Text, by W. Whewell, D.D. late Master of Trinity
College. 3 Vols. Demy O^avo, \2s. The translation separate, 6t,
London: Cambriiige Warehouse, 17 Paternoster Row,
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U PUBUCATIONS OF
mSTOET.
LIFE AND TIMES OF STEIN, OR GERMANY
AND PRUSSIA IN THE NAPOLEONIC AGE,
by J, R, Seelev, M.A-, Regius Professor of Modem History ia
the University of Cambridge, with Portraits and Maps. 3 Vols.
Demy 8vo. 48/.
"If i» cou^d coDCcifC anrthiDt limilar dihnf Tor Gcraivi at well m Eqglisb read«n
Inna vnce in Carlyle uid L«wrs bioeraphcn n«kc that ii onuitcreilii^ To UDtkr-
SrjuUpuL Prafc«wr S«leT of Cambi-id^ ituT tiudf hu ben made tacy bf ihk vwki
prctcdted (U wilh a biography of Stem lo whicb no ooe can hesitalc to avi^ a WT
which, ihouffh IE inodeslTv (leclinA comptti- hi|!Eh pJacT amooE thoK RCvnt hismviea nrlikfl
■ilh^i
lu inu Ihc iludc by iu biil-
—At*e-
y all thai
kitltcrtp wnRen about Stein.... In five Lang hlcraiurt, and bndfe< over ihe ^ul™? '^
dupuit Scdey eapoooda the legisUtive uio Pruuia from tht time of Fredendi the Greal
the pervn and the hiU the befliDnin^ of
tia( iDiighl, than
wghiKu, vith piore pene< erandaie Cambridge and )kt ProJcnor of
"Dr Bu^h'fl ■i>lui1ie h^ nude people u umeOilag upon which we may CDa^rattitata
think and talk even nioie than miial of Prince England that on the especial field of tlie Gcr*
Uisnurck,aTidPit>le»gr5eetey'iTeryleanled mam, hiUoiyi on the huiory of iheir OWB
workcn SleJD will lumatleDtioD toanearlier country, by the uie of tbeir own Litenry
and an almott equally emiDent German itatet- weapons^ an Englnhnun has produced a his-
man. ...... It it iDothing to the national tory of GcnnaDr in the Napc4eonic afe far
fetf-reipect to 6ad a few EnEliUunen. such Hiperior to any (hat costs in GcrmaiL"-^
as Ibe l^le Mr Lew« and I^ofeuoc Seeley, Ejtamimr.
THE UNIVERSITY OF CAMBRIDGE FROM
THE EARLIEST TIMES TO THE ROYAL
INJUNCTIONS OF 1535,
by James Bass Mullinger, M.A. Demy 8vo. doth (734 pp.), lu.
-' We inisl Mr MulUnier will yet conlmus
** He has brought together a mass of in-
smiclive details respecting the rise and pro-
Agei Wehope some day thai he may mend this book to our readetv ">-?/fE<a/ar.
continue hislaboursi and give us a history of
HISTORY OF THE COLLEGE OF ST JOHN
THE EVANGELIST,
by Thomas Baker, B.D., Ejected Fellow. Edited by John E. B.
Mayor, M.A., Fellow of St John's, Two Vols. Demy Svo. i\s.
e* the book wi
" " nl with 'Dye;
'—Atlumrum. with St John's College, Cambridge; they
membenof the by a fat wider circle ..The index wiiE! which
lotiinsitwill be found a work of considerable "It may be thought that the history of
merrice oo questions respecting out social eollegecannot beparliculai-lyirtractivc. Th
charged his editorial functions are creditable those who have been in any way con
to his leaminf and industry- ■- ......... _ . ..
" The work displays ve
ud it will be of p«*l use to members of the by a fai
Gotten and of the uniTenity, and, perhaps, Mr Uayui nas lumiinca mis uieiui wor
of sfiU greater us* to students of English leaves nothing to be desired. "-^/K/ater.
history. ecclcsiBslicat, political, social, literary
London: Cambridge Warehouse, 17 Paternoster Holt.
THE CAMBEIDGE UNIVERSITY PRESS. is
HISTORY OF NEPAL,
translated by MUNSHf Shew Shunker Sfnch and Pandit ShrT
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People by Dr D. Wright, late Residency Surgeon at KaihmandQ,
and with facsimiles of native drawings, and portraits of Sir Jung
Bahadur, the King of Nepal, &C. Super-royal 8vo. Price 2ij.
"The Cimbridee Uniwniiy P«" h»vt "Von nichigeringemWerthidMiainiind
done urell in puhHshing Ihii work. Such die Beinbcit.weLche Wrighc all 'Appendix'
DTAn&lBCions ore valuable not only ED Ihe hifl- hinler der 'hisiory' folften liist, AuffAh-
lotiu but elsa to ihe ethnotDgiit; Dr lungen nlmlkh der in NepU Cblichen Muiik-
Wiifbt'i iDtroduciion ii bued on penoosl liutnunenK^ Ackeitedihe, Mflnten, Ge-
geiitly and landidly, and iddi much to the VucibulirlD PvballyS und Neirlit, einige
value of the wolune, Uht coloured liiho- Ncwflil fongb mit Iqicrliaear-UebeneUunE.
graphic plates lu-e interesting." — Nalurt. eine KBntgtlistei und, last Dot least, «ii
nCTi.. The volume.. .iibeautifulK San>ktil-I^.. «]c™jelj"n de?lJni™i?
d supplied with pmliaiu of Sii till- Bibliothe): in CambHdge deponiit sind. "
ing Bahadoor and oihen, and wiih e>cel- —A. Wkbir, LiliratmmitKiif, jahrgaag
, =lory chapters coo- que vicnt de publict Mr Daniel Wright
tribuled b» Dl Wright himielf, *to uw ai kus le titrt di ' Hiuory of Nepal, IraniJated
much of Nepal during his ten yewn' Kijounl fiom the Parballya. elc'"— ij. GiSciN DE
■1 the strict rules enforced against foieignerj T«ssv in /.a Langiutt la LilUralun Hin-
even by Jung Bahadur would let him see."— duuslanui iit 1B7J. Pans, iBjS.
SCHOLAE ACADEMICAE:
Some Account of the Studies 3t the English Universities in the
Eighteenth Century. By Christopher Wordsworth, M.A^
Fellow of Peierhouse ; Author of " Social Life at the English
Universities in the Eighteenth Century." Demy octavo, cloth, i^.
"The general object of Mr Wordsworth's teresiing, and instcucllve. Amonj; the mat-
book is tuifidently apparent from its title. ten ittnched upon are Lltinries, Lecturea,
He has collecied a great quantity of minula Ihe Tripoi, the Trivium, ih- ■-■ "
■nd^ curious informalicHi about the working the Schools, tevt-bookA, «
lung
religion."— £mm(«.'.
"Inpleaiinitconlnutwith the native his- de Sir Jang Bahadur.
of Cambridge inititutiaas in the lau century, fore^ opuiiani, interior life. We li
with an oceanonal compsrison of the cone- even of tlie vi ' - " '
•I«ndiiia ital* of Ihbgs at Oxford. It Is of
eeuna impotuble that a book of ihii kind
tore. To a great ejtteni it ii purely a book
"inlhewoiVbeforeus,wh;chil«rict]ywhBt bouri will be able fully
din, we obtain authentic information upon the discernible In every paae.
1 ... . -i-i 1.:^ UKPUjhl volume it may be laidt)
indee student to John Strype, giving
d idea of hre as an undetgraduaie and
rarda, as the wrirer became a graduate
^ly those who have enga^d in hke la-
id ciianges of philosophical ui
>f Ihe whp''^
of Tettera, upoo die relations of doctrine and history, aiid that the hahiu of thought of any
■cience, upon the range and thoroughness of writer educated at either seat of learning in
education, and we may add, upon the cat- the last century will, in oaany cases, be far
like tenacity of life of ancient forms.... The belter undeufood after a consideration of Lhc
THE ARCHITECTURAL HISTORY OF THE
UNIVERSITY AND COLLEGES OF CAMBRIDGE,
By the late Professor Willis, M.A. With numerous Maps, Plans,
and Illustrations, Continued to the present time, and edited
by John Willis Clark, M.A, formerly Fellow
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TEE CAIKBBIDOE BIBLE FOB SCHOOLS.
The want of an Annotated Edition of the Bible, ia handy porlions,
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with introductions and explanatory notes.
The Very Reverend J, J. S. Pekowne, D.D., Dean of Peter-
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and will be assisted by a staff of eminent coadjutors. Some of the
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Rev. A. Cark, M.A., late Fellma ef Oritl Collie, Oxford, AuistanI
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Rev. T. K. CiiEVNE, Ftllme of Baltiol College, Oxford.
Rev. S. Cox, NolHngiam.
Rev. A. B. Davidson, D.D., Preftssor of Hebrew, Edinburgh.
Rev. F. W. FaRRAR, D.D., Canon of Wtslminitir.
Rev. A. E. Hi;uPHHEYS, M.A., Felimo of Trimly ColUgt, Cambridp.
Rev, A. F. Kirkfatrick, M.A., Fellow of Trinity Coliege.
Kev. J. J. Lias, M.A., Prafossor at Si DatMt Collrge, Lampeter.
Rev.J.R. LUMBY, D.D., FeltoTaefSI CalAarine'i Coll^.
Rev. G. F. Maclear, D.D., Head Mailer of King's Coll. Seiaol, London.
Rev. H. C. G. MoDLE, M.A., Feilow of Trinity ColUgt.
Rtv.W. F.Moui.TOH,jy.Ty., //eaJ Maiiera/lAeLefsSciaol, Camiridge.
Rev. E. H. PerownK, D,D., Master af Corpus Christi College, Cam-
■ bridge, Examining Chaflatit to the Bishop of St Asafh.
The Ven. T. T. Perowne, M.A., latt Fellow of Corpus ChriiH Collegi,
Cambrid^, Arehdeacon ef Noraiich.
Rev, E. H. Plumptre, D.D., Profeiior of Biilieal Exegait, Kin^i
Coll^, London.
Rev. W. Sanday, M.A,, Prindpal of Bishop Uaifidd Hall, Lhirham.
Rev. W. SiMCOX, M.A., Ralor of WeyhUl, Hants.
Rev, RoBERTsuH Smith, M.A., Professor of HArea, Aberdeen.
Rev. A. W. Streane, M.A., FellmB af Corpus ChHsti Coll. .Cambridge.
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Rev. G. H. Whitaker, M.A., Fdlom of St fohn't Cotl^, CamMdge.
Now Beady. Olotli, Extra Fcap. 8vo.
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THE BOOK OF JONAH. By Archdn. Pkrowne. is. M.
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A BOOK OF BALLADS ON GERMAN HIS-
TORV. Arratigcd and Annotated by Wilhelm Wagnbk,
Ph.D., Professor at the Johanneum, Hambarg. lyicets.
" It omo the reiderniiiclly ihraufh Kune at ttu: iwsl inporlul ineidaU
cnuTMCted vilii the Gcmun wax taA naac» from the uvuion of lUlp bv the
ViiiEDthl under their Kin* Alaric, dnwn to (he FnncD-Genun War and tlie
initallaliDD of ihc pnscDt Empcrar. Thv nol««ip|jly irery wctl the conaecEinB
Hnki between Ihe succeoiw periods, and exhibit in ia vkrioiu phaae* of crowth
DER STAAT FRIEDRICHS DES GROSSEN.
ByG. FRBlrTAO. With Noles. By Wilhblm Wagnkr, Ph.D.,
Professor at the JohanDcum, HamtMUg. Ji-ia u.
"Theie ue recent vlditinu to ihe handy repriuta pTm tn tSw 'Fftt Pm
Seriei.' In both the intenilDB i> to combine the uudiei of litennre and his-
tnry, , . In the second ofthete little hoolu, the editor gives, with aoHe altets-
tiona, a 1ah\y written eway OD Mt CariyLe*! hera The nolea appeDded m At
•my, lika aaee roUowuis the bKlUda, are nouly egacue ud ueM.'
jfOimnm.
"PniHiiBBdirFtederiefcthe Great, and FriBce under the Diioclmy, trtig
Di face (o Cue reqiectivehr with petiodi of hiatsry whkh it b ri(hi ihoiM be
known tborougtily, and which are well treated in ue Pitt Preb vJvrneL~
(Sottfit'8 JCnatenia^te. (1749—1759-) GOETHE'S
BOYHOOD : being the Fiist Three Books of his AutoUt^iaphy.
Arranged and Annotated by WtLKELM Wacner, Ph. D., Pio-
fessoc at the Johanneam, Hamburg. /Yict u.
GOETHE'S HERMANN AND DOROTHEA.
, With an Introduction and Notet. By the tame Editor. J^e 31.
Dfl« 3a^r 1813 (The Year i8i3),by F. Kohlracsch.
With English Notes. By Ihe same Editor, jyict *i.
Lwtdon! Cambridp Warehouse, 17 Pafemcster Rom.
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THE CAMBRIDGE UNIVBRSITY PRESS. 2
PITT PRESS SERIES (continued).
V. ENGLISH.
THE TWO NOBLE KINSMEN, edited with
Inlroduclion and Notes bj the Kev. Prcifess<n 9keat, M.A.,
formerl}' Fdlaw of Christ's Collie, Cambridge. Piiet y, id.
" This cdHlini of ■ riar '1»1 h well ntrth iludy, for mor
by » csrcrul M idiolar ■> Mr Slinl. dacivn ■ hearty oelcsn
"Mr Sliot is B convientinK editor, w>d hH left PD diA
<hher oTtCDH of Uiisiuse."^7'i''ruf.
BACON'S HISTORY OF THE REIGN OF
KING HENRY VII. Wilh Notes by the Rer. J. Rawson
LUMBY, D.D., Fellow of St Catharine't College, Camtmdge.
Price y.
SIR THOMAS MORE'S UTOPIA. With Notes
by the Rev. J. Rawson LuM»r, D.D., Fellow of St Cathirine'i
Collie, Cwnbridge. [JVearfy rfodjt.
\Olher Volumes are in preparation?^
CAHBBIDaE TJinVEBSITT BSFOBTES,
Publithei by Authority,
Containing all th& Official Notices of the Univenity, Report! of
Discussions in the Schools, and Proceedings of the Cambridge Philo-
■ophjcal, Antiquarian, and Philological Societies, yi, weekly.
CllCBBISaE UNIVEBSIT7 EXAKINATIOK
FAPESS.
PuUisbed in occasional niimben erery Tenn, and in volumes for the
Acadentical year.
London .- Cambridge Warehouse, i J Paternoster Row.
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