NOTES ON THE PRINCIPLES OF PURE
AND APPLIED CALCULATION;
AND APPLICATIONS OF
MATHEMATICAL PRINCIPLES TO PHYSICS.
LIBRARY
.jjTJNIVK'nsiTY OF i
(MltFOENIA,
ffamtrtfcge:
PRINTED BY C. J. CLAY, M.A.
AT THE UNIVERSITY PRESS.
NOTESiJNfv,
ON THE ' PBJNC1PLES ' ; bfr ; 'ft
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PURE AND APPLIED CALCULATION;
AND
APPLICATIONS OF MATHEMATICAL PRINCIPLES
TO
THEORIES OF THE PHYSICAL FORCES.
BY THE
EEY. JAMES CHALLIS, M.A, F.R.S., F.RA.S.
NIVERSI
^^i
PLUMIAN PKOFE3SOE OF ASTEONOMT AND EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OP
CAMBRIDGE, AND LATE FBLLOW OF TRINITY COLLEGE.
AXXa Travra /x^r/ay Kal dpiO/j,^) Kal erra^y St^ra^as. WlSD. xi. 20.
CAMBRIDGE:
DEIGHTON, BELL, AND CO.
LONDON: BELL AND DALDY.
1869.
[All Rights reserved.}
THE AUTHOR desires to express Ms grateful acknowledg
ments to the SYNDICS OF THE UNIVERSITY PRESS for their
liberality in granting him the whole expense of the Printing
and Paper of this Volume,
TO
LIEUT.GEN. EDWAKD SABINE, R.A., D.C.L., LL.D.,
President of the Royal Society.
MY DEAR GENERAL SABINE,
I AM glad to be permitted to dedicate this Volume to
you, because an opportunity is thus given me of publicly ex
pressing the high opinion I entertain of the value of your
scientific labours, especially those relating to Terrestrial and
Cosmical Magnetism, which I have had peculiar means of
appreciating from the use I have made of them in my Theory
of Magnetic Force.
I feel also much gratified by the circumstance that in availing
myself of your permission I dedicate the work to the President of
the Royal Society. The special object of my theoretical researches
has been to arrive at a general physical theory by means of mathe
matical reasoning employed in accordance with principles and
rules laid down in the philosophical works of Newton. But I
am well aware that any success I may have had in this un
dertaking has altogether depended upon those experimental
inquiries into the facts and laws of natural phenomena which
it was the express object of the original institution of the
VI
Royal Society to promote, and in the prosecution of which the
Society has since borne so prominent a part. I may say, for
instance, that the complete exhibition of my general theory has
become possible only since the publication in the Philosophical
Transactions of your researches in Magnetism and those of
Faraday in Electricity. Under these circumstances I venture to
express the hope that this dedication to you on personal grounds
may, in consideration of the office you hold as President of the
Royal Society, be also regarded as a tribute of respect to the
Society itself.
I have the more reason for giving expression to this wish,
because during the twenty years I have been a Fellow of the
Society, I have contributed only one Memoir to the Transactions,
and may, therefore, seem* not to have shewn as much zeal for
promoting its objects as might be expected from the author of a
work like the present. This has happened, as I have explained
at the end of the Introduction, partly for reasons unconnected
with the Society as a body, and not inconsistent with a due
regard for its honour and interests, and partly because my theo
retical views eventually assumed a character which required that
the whole of them should be exhibited in connection by means of
a publication expressly devoted to the purpose.
I am,
My dear GENERAL SABINE,
Yours most truly,
J. CHALLIS.
CAMBRIDGE,
February 9, 1869.
LI l' RARY
r N i v
CONTENTS.
*,* In the absence of numerical indications of Articles, it is hoped that
the reader will be sufficiently directed to the particular subjects mentioned in
the following Table by ivords printed in Italics in the specified pages.
INTRODUCTION , . v Ixiii
Preliminary information respecting the origin, objects, and cha
racter of the work . v ix
*
EXPLANATIONS, historical notices, and occasional remarks, relating
to the subjects treated of under the head of " Principles of
Pure and Applied Calculation " ...... ix xliv
On the treatment of the subjects in logical order .... ix
Principles of Arithmetic and Algebra. On proving inductively the
law of the permanence of equivalent forms .... ix xii
On the Calculus of Functions, and derivation of the Differential
Calculus. Exact expression for the ratio of the increment of
a function to the increment of its variable . . . . xii xiv
Principles of Geometry. Eemarks on the discussion in pages
7088 relative to the application of abstract calculation to
Geometry. Distinction between geometrical reasoning and
analytical Geometry xiv xvi
Spherical Astronomy. The science of Time. Discrepancy of state
ments in pages 91 and 127 relative to the uniformity of the
earth's rotation accounted for xvi xvii
Eemarks on the explanation of the Aberration of Light . . xvii xviii
Principles of Statics. General equation of Virtual Velocities.
Principles of Hydrostatics xviii
Dynamics of bodies in motion. Fundamental facts of experience.
Necessity for the application of differential equations to calcu
late motion generally xviii xix
VIII CONTENTS.
Pages
Physical Astronomy. Whewell on the difference between Kepler's ,.
Laws and Newton's Law of Gravity. General statement of
the character of physical theory . . " . . . . xix xx
Correction of a misunderstanding in pages 120 124 relative to the
observations from which Newton first deduced the law of
gravity xx
On an inquiry into the signification of the occurrence of terms of
indefinite increase in the solution of the Problem of Three
Bodies. Narrative of the particulars of a discussion relating
to this point. On the inferior limit to the eccentricity of a
mean orbit . . . xxi xxviii
Dynamics of the motion of a Eigid System. Principle involved in
the mathematical theory of Foucault's Pendulum experi
ment xxix xxx
Hydrodynamics. Imperfect state of this department of applied
mathematics. The insufficiency of the received principles
proved by their giving absurd results. Eefusal of mathemati
cians to admit the logical consequence. To evade it recourse
had to conjectures. Importance, as regards physical theory,
of rectifying the principles of Hydrodynamics . . xxxi xxxvi
Discovery of the necessity for a third general equation based on
afprinciple of geometrical continuity. Consequences of arguing
from the new equation. No contradictions met with. Laplace's
method of treating hydrodynamical problems defective in
principle . xxxvi xl
On applications of the three general equations in the solutions of
various problems, and on the coexistence of steady motions . xl xliv
SUMMARY of the hypotheses and mathematical principles of the
Physical Theories contained in the second part of the work,
with historical notices, and references to different views . . xliv lix
Principles of the Undulatory Theory of Light. Hypotheses re
lating to the aether and to atoms. The same hypotheses, and
no others, adopted in the subsequent Theories . . . xlv li
Eefusal of physicists to accept the undulatory theory of light as
based on the hypothesis of a continuous ffither. Preference
given to a theory which refers the phenomena to the oscilla
tions of discrete atoms. Contradiction of the oscillatory theory
by facts. Arbitrary assumptions made to sustain it . xlvi xlvii
Historical notices. Fresnel's hypothesis of transverse vibrations.
Cauchy's hypothesis of an isotropic constitution of the tether.
Attempts to explain phenomena of Double Eefraction on these
CONTENTS. IX
Pages
principles unsuccessful. Better success of the theory ex
plained in pages 375 383j which rests on the hypothesis of a
continuous aether and finite spherical atoms. Unreasonable
rejection by physicists of Newton's views respecting finite
atoms . . xlviii liii
On the nature of Heat. Light and Heat different modes of force.
The principles of the Mathematical Theories of Heat and
Molecular Attraction. On a general formula for the transla
tory action of setherial waves on atoms. Its imperfection . liii lv
The mathematical Theory of the Force of Gravity, as deduced from
the general formula. Opinions of different physicists respect
ing the nature of gravitating force . . . . . lv Ivi
t
On steady motions of the aether, and on the origin of those to
which the theory attributes the attractions and repulsions of
Electricity, Galvanism, and Magnetism 4^i
The principles of the Theory of Electric Force . . . . Ivi Ivii
The principles of the Theory of Galvanic Force. The mathematical
conditions of this theory imperfectly known .... Ivii
The principles of the Theory of Magnetic and Diamagnetic Force.
On the origin and variations of Terrestrial and Cosmical Mag
netism. Publications from which the facts of Electricity,
Galvanism, and Magnetism have been drawn . . . Iviii lix
STATEMENTS in conclusion on the objects and contents of the Intro
duction, the actual state of theoretical physics, and the method
of philosophy advocated in this work lix Ixiii
Experimental physics in advance of theoretical. Necessity of
mathematical theory for determining particular correlations of
the physical forces. How the present theory does this. Its
imperfections and their correctibility. The adopted method
of philosophy opposed to that of Comte, and to methods
of deduction from general laws, accounts for Conservation of
Energy, is not speculative. Distinction between theory and
speculation . . . . .' . , / * . lix Ixii
The mathematical principles of hydrodynamics contained in this
Volume asserted to have the same relation to general physics
as Newton's mathematical principles to physical astronomy.
Conditions on which it is proposed to maintain this assertion
by argument * t . . Ixii
Explanation of the circumstance that no portion of these physical
theories has been communicated to the Eoyal Society . . Ixiii
Bearing of the contents of both parts of the work on the mathe
matical studies of the University of Cambridge . . . Ixiii
CONTENTS.
Pages
NOTES ON THE PRINCIPLES OF CALCULATION . . 1320
General principles of Pure Calculation . . . , . . 1 2
The different kinds of Applied Calculation. Their logical order, as
determined by the application of calculation to the ideas of
space, time, matter, and force . . . .... 3 4
NOTES ON THE PEINCIPLES OF PURE CALCULATION . . . . 4 59
The Principles of Arithmetic . . 420
Foundation of Arithmetic in number and ratio. The general arith
metic of ratios as contained in Euclid^ Book v. . . . 4 5
Quantitative measures. Quantity expressed generally as the ratio
of two numbers. Incommensurable quantity so expressible with
ad libitum approach to exactness . . . . . 5 8
Proportion denned to be the equality of two ratios. The ratio of
two given quantities of the same kind found (Prop. i.). De
duction of Def. v. of Eucl. v. from the definition of Proportion
(Prop. ii. and Corollary). Proof of the converse of Def. v.
Eemarks on Eucl. v . . . 8 14
Proofs of the rules for finding a given multiple, a given part, and a
given integral power of a given quantity . ' . . . 14 15
The principle on which fractional indices are necessary in a general
system of calculation. Proof that a x may have values as nearly
continuous as we please if x has such values. Meaning of a
logarithm . . ...... . . 1617
All forms of continuous expression of quantity derivable from the
two forms x m and a* by substitution and the usual rules of
arithmetical operation ... . . . 17 18
Bules of operation necessarily founded on arithmetical considera
tions. Investigations of the usual rules of general arithmetic.
A quotient and a fraction expressible by the same symbol as a
ratio 1820
The Principles of Algebra 21 37
Distinction between general arithmetic and algebra. Algebraic use
of the signs + and  . Principle which determines the rule
of signs t . , ' , f i . 21
Proofs of the rules of signs in algebraic addition, subtraction,
multiplication, and division . . ._,, ... ..'. . . 2224
Distinction between real and algebraic quantities. Origin of im
possible quantity .... . 2425
CONTENTS. XI
Pages
Proofs of the rules of operation with indices in general arithmetic.
Algebraic generalizations of the rules. Necessity of negative
and impossible indices arising out of such generalization.
Proofs of the rule of signs in the algebra of indices . . 25 28
Algebraic series, converging and diverging. Method of indeter
minate coefficients. Proposed mark for distinguishing identity
of value from special equality. The proof of the binomial
theorem dependent on 'ordinary rules of algebraic operation.
The expansion of a x dependent on the binomial theorem . 28 31
On the solution of numerical equations. (See Appendix, p. 688.) . 3137
Principles of the Calculus of Functions 37 59
All arithmetical and algebraical representations of quantity em
* braced by the Calculus of Functions. The variables of a
function. Different degrees of its generality according to the
number of the variables 37 38
(1) The Calculus of Functions of one variable . 3851
Explicit and implicit functions. Primitive and derived functions.
Principle and proof of Taylor's Theorem. Applications of the
Theorem 38 42
Derived equations. Elimination of constants. Successive orders
of derived equations. Eeverse operations for finding the
primitive equations. Particular solutions by primitives not con
taining arbitrary constants. Method of Variation of Parameters 42 46
The Differential and Integral Calculus founded on Taylor's Theorem.
Differential and differential coefficient defined. (See Errata,
and p. xiii of the Introduction.) Integration. Definite and
indefinite integrations between the limits zero and infinity . 46 51
(2) The Calculus of Functions of two or more variables . . 51 57
Expansion by Taylor's Theorem. Differentials of functions of two
variables. Proposed notation for partial differentials. Equa
tions of partial derived functions. Elimination of arbitrary
functions. (See Appendix, p. 691.) . ' .. . , . . 51 56
Miscellaneous notes on the calculus of functions of three or more
variables, on maximum and minimum values of algebraic
functions, and on the Calculus of Variations. (See Appendix,
p. 694.) 5658
Summary of results relative to Pure Calculation . . . . 58 59
NOTES ON THE PRINCIPLES OF APPLIED CALCULATION . . . 59 320
General remarks. Each department of Applied Mathematics dis
tinguished by definitions which are the basis of the calculation.
The results of abstract calculation to be regarded as axioms
with respect to any applied science 59 60
XII CONTENTS.
Pages
The Principles of Geometry ........ 6090
Different kinds of geometrical definitions. The definitions in
Euclid of a square, of parallel straight lines, and of similar
segments of circles, not strictly such. Proposed definition of
parallel straight lines 61 63
Definition of similarity of form. Proof of Eucl. Def. xi. Book in.
Postulates and Axioms. Proof of Axiom xn. of Book i., from
the proposed definition of parallels . . . . 63 65
Theorems and Problems. General remarks on the character of the
reasoning in Euclid. Arrangement in logical order of Pro
positions on which a system of Geometry might be founded.
Argument to shew that Def. v. of Book v. is not necessary for
the proofs of Propositions 1 and 33 of Book vi. 65*70
Necessity of measures of length in Trigonometry and Analytical
Geometry. . Principle of the calculation of areas. Eucl. i. 47
employed to calculate the hypothenuse, from the given sides,
of a rightangled triangle. The principle of measures necessary
for thjs purpose. Distinction between reasoning by geometrical
diagrams and by analysis. The latter alone proper for calcu
lation . ' . . . . . . . . . . 7074
Argument to shew that all the propositions of Trigonometry and
Analytical Geometry of two dimensions are deducible by
analytical calculation from the selfevident equality of two
triangles one of which has two angles and the included side
respectively equal to two angles and the included side of the
other 7488
Calculation, in part ; of the relations of the sides and angles of tri
angles. Calculation of areas of triangles and parallelograms.
Principles of algebraic geometry of two dimensions. Measures
of angles 7682
Application of the differential calculus to find the directionangle of
the tangent to a circle, or any curve, and of the integral calculus
to find the functions that the cosine and sine are of the arc.
Trigonometrical formulae. Complete calculation of the relations
of the sides and angles of triangles 82 88
General calculation of areas. Contacts. The essential principles
of Geometry of Three Dimensions not different from those of
Geometry of two dimensions 88 90
The Principles* of Plane Astronomy 9098
The problems of Plane, or Spherical, Astronomy essentially geo
metrical, but the determination of certain arcs involves the
element of time. The science of Time depends on Practical
Astronomy 9091
CONTENTS. XIII
Pages
Right Ascension and Declination, the former obtained by the
intervention of time. Corrections, instrumental and astro
nomical, required for measuring arcs and the uniform flow of
time. The sidereal time of any place. Bessel's formula for
the Sun's Mean Longitude at a given epoch. Relation between
sidereal time and mean solar time. Calculation of epochs and
intervals of time 91 96
Explanation of the aberration of light. The different amounts for
a fixed and a moving body 97 98
The Principles of Statics 98104
Measures of quantity of matter and of statical force. Weight. Pro
perties of a rigid body. Definitions of equilibrium . . . 98 99
Proofs of the Parallelogram of Forces. Lagrange's investigation
of a general equation of Virtual Velocities. The principles on
which it rests supplemented by a definition of equilibrium.
Inference of the Parallelogram of Forces from the Principle of
Virtual Velocities 100104
The Principles of Hydrostatics 104 108
A fluid defined by its properties of pressing and of easy separability
of parts. Measure of fluid pressure. The general law of
equality of pressure in all directions deduced from the defini
tions of properties . . 104107
Investigation of a general equation of the equilibrium of fluids . 107 108
The Principles of the Dynamics of Solid Bodies in Motion . 109 170
Definition of hiertia. Uniform velocity. Analytical expressions
for variable velocity in a given direction, and for the resolved
parts in three rectangular directions . . . . 109 111
Definition of constant accelerating force. Analytical expression
for variable accelerative force. Experimental Laws. Deduction
therefrom of the composition and resolution of constant and
variable accelerative forces .,..,,. Ill 117
Momentum and moving force defined and their analytical ex
pressions obtained 117119
Physical Astronomy, regarded as the Dynamics of the motion of a
free material particle 119152
Gravitation. Discovery of its law by Newton (see p. xx). Kepler's
Three Laws. Newton's dynamical explanation of them . . 119 134
Principles involved in the processes of solution of the Problem of
three or more bodies. The method of Variation of Parameters.
Remarks on the inverse Problem of Perturbations the solution
of which led to the discovery of Neptune, and on the con
XIV CONTENTS.
Pages
sequences of an exact mathematical determination of the
acceleration of the Moon's mean motion. Possible retardation
of the Earth's motion about its axis by the Moon's attraction
of the tidal waves 124127
Discussion of the meaning of terms of indefinite increase occurring
in certain cases of central forces, and in the Problem of three
Bodies. Inference from them that the motion is not neces
sarily periodic, and proof that every process which gets rid of
them introduces the hypothesis of a mean orbit .... . 128 138
Determination of the inferior limit to the eccentricity of a mean
orbit ; ... > .... . , . 138151
First approximation to the motion of the nodes of the Moon's mean
orbit . 151152
The Dynamics of the Motion of a Rigid System .... 153 170
Beason given for D'Alembert's Principle. Deduction from it of the.
Law of Vis Viva by means of the principle of virtual velocities.
Solution of a dynamical problem by an equation of virtual
velocities , . . . 153 157
Investigation of six general equations for determining the motion
of any rigid system acted upon by given forces. Application
to the case of a hoop rolling on a horizontal plane .'.'.. . . 157 166
Solution of the Problem of Foucault's Pendulum Experiment . 166 170
The Principles of the Dynamics of Fluids in Motion '. , . 170320
The definition of a perfect fluid at rest assumed to apply to the
fluid in motion. Prop. II. Proof of the law of equality of
pressure in all directions for fluid in motion .... 171 173
Axiom that the directions of motion are subject to the law of geo
metrical continuity. (Adopted rules of notation). Prop. III.
Investigation of the general equation (1) of geometrical conti
nuity 174175
Prop. IV. Investigation of the general equation (2) of constancy of
mass 176177
Prop. V. Investigation of the general dynamical equation (3) appli
cable to the motion of fluids. Remarks on the three general
equations. Proof that the direction of the motion of a given
element cannot change per saltum. Definition of a surface of
displacement 177181
Prop. VI. Deduction of a general equation (4) embracing the prin
ciples of (1) and (2) v .' , t : . , . . . . 181185
Prop. VII. Inference of rectilinear motion from (1) antecedently
to any given case of disturbance of the fluid, by supposing
udx+vdy + wdz, or \(d\[/), to be an exact differential . . 185188
CONTENTS. . XV
Pages
Prop. VIII. Investigation of a rule for calculating rate of propaga
tion. Lagrange's method involves a violation of principle . 188190
Prop. IX. General relation between velocity and density in uni
form propagation of density ... .... 190 192
Prop. X. General laws of the variation, with change of time and
distance, of the velocity and density in uniform propagation . 192 193
Example I. The problem of the propagation of planewaves
attempted without taking account of the equation (1). The
solution leads to absurd results, and a relation between the
velocity and density inconsistent with that from Prop. IX.
Details respecting a discussion of this question . . . 193 197
Example II. The problem of the propagation of spherical waves
similarly treated. A result obtained inconsistent with the law
given by Prop. X 197199
Course of the reasoning when the three general equations are used.
Supposition of a general law of rectilinear motion along an
axis independent of particular disturbances of the fluid . . 199 200
Prop. XI. The laws of motion along a rectilinear axis, due to the
mutual action of the parts of the fluid, \(d\f/) being an exact
differential. The motion found to be vibratory, and the pro
pagation of waves of all magnitudes to be uniform. Kelation
obtained to terms of the second order between the velocity and
the condensation ......... 200 207
Prop. XII. The determination to quantities of the first order of
the laws of the motion relative to an axis at any distance from
it, \(d\f>) being an exact differential. The motion found to
consist of vibrations partly direct and partly transverse . . 207 211
Prop. XIII. Proof of the coexistence of small vibrations relative
to the same axis, or to different axes 211 213
Prop. XIV. Determination of the numerical value of the rate of
propagation. The result of comparison with observation
inconsistent with attributing any effect to the development of
heat 214225
Prop. XV. Investigation of the laws of the spontaneous vibratory
motion relative to an axis, to the second approximation. Sig
nification of this use of the word ' spontaneous' . . . 225 228
Prop. XVI. Determination of the result of the composition of
spontaneous vibrations having a common^axis, to terms of the
first order .*.... 228230
Prop. XVII. The same problem solved to terms of the second
order. Extension of the last two solutions to the case of the
composition of vibrations relative to different axes . . 230239
XVI CONTENTS.
Definition of steady motion. Prop. XVIII. Determination of the
laws of the steady motion of a compressible fluid. Case of the
integrability of \(d\f/), and general formula for steady motion
independent of particular conditions. Proof of the law of the
coexistence of steady motions 239 243
Examples of the application of the foregoing Principles and Propo
sitions 243 316
Example I. Solution of the problem of the propagation of plane
waves to the first and second approximations. Application of
the principle that arbitrarily impressed motion is compounded
of spontaneous jnotions relative to axes. The state oj: the
fluid as to velocity and condensation propagated uniformly
and without undergoing change. The contradiction in page
195 got rid of. Proof of the possibility of the transmission of
a solitary condensed or rarefied wave . , . . .243 248
Example II. The problem of spherical waves to the first approxi
mation. Investigation, on the principle of the composition of
spontaneous motions, of a general equation (29) applicable to . *
given cases of motion. Expressions for the velocity and den
sity in spherical waves at any distance from the centre. The
condensation varies inversely as the square of the distance.
The contradiction in page 198 accounted for . . . . 249 254
Example III. Exact determination of the laws of the central
motion of an incompressible fluid . ' . ' . . . . 254256
Investigation of a general approximate equation (31) applicable to
motion produced under arbitrary circumstances. Formula
(32), (33), (34), for motion symmetrical about an axis. Proof
that \(d\f/) is an exact differential for vibratory motion pro
duced arbitrarily 256 260
Example IV. Problem of the resistance of an elastic fluid to the
vibrations of a small sphere. Approximate formulae for the
condensation and velocity of the fluid. Its backward and
forward flow compensatory at all times. Kelation between the
effective pressure (p'} and the condensation in composite mo
tion 260266
Expression for the acceleration of a ballpendulum in air or water,
resistance and buoyancy being taken into account. Compari
sons of the results with experiments by Du Buat, Bessel, and
Baily. The difference between the theoretical and experi
mental corrections of the coefficient of buoyancy mainly attri
butable to the resistance of the air to the motion of the sus
pending rods. Bemarks on the erroneous principle of a pre
vious attempt to solve the same problem. (See Introduction,
p. xli) . 266273
CONTENTS. XVII
Pages
Example V. Problem of the resistance of fluid to the vibrations
of a cylindrical rod. Approximate expressions for the con
densation and velocity of the fluid. The forward and back
ward flow compensatory. Expression for the acceleration of
the rod supposed to vibrate about one extremity. Comparison
of the result with experiments by Baily and Bessel , * . 273 279
Example VI. Approximate determination of the motion and pres
sure at any point of fluid the vibrations of which are incident
on a fixed 'sphere. The mean flow of the fluid is not altered.
To this approximation the accelerative action on the sphere is
the same for compressible as for incompressible fluid . . 279 287
Attempt to take account of the effect of compressibility by consi
derations respecting lateral divergence due to transverse
action. Consequent formula for the accelerative action of the
fluid on the sphere 288 296
Example VII. The same as Ex. VI. except that the sphefe is
moveable. Approximate formula. Attempt to include terms
of the second order. Inference of permanent motion of trans
lation from terms of that order. Argument to shew that the
motion of translation due to given waves will be uniformly
accelerated 296306
Circumstances under which the motion of translation might be
from or towards the origin of the waves. Independence of the
motions of translation produced by waves from different
sources. Variation of the accelerative action of the waves
according to the law of the inverse square. Further consi
derations relative to the simultaneous vibratory and tfansla
tory 'action of waves on spheres ....'.. 307 313
N.B. Examples VI. and VII. are more satisfactorily solved
in pages 441 452.
Example VIII. Accelerative action of the steady motion of an
elastic fluid on a small fixed sphere. Formula for the accele
ration. Applies very approximately to a moveable sphere.
Calculation of 'the accelerative effects of two or more steady
streams acting simultaneously . . . . . 313 316
General statement of the relation of the results obtained by ma
thematical reasoning under the head of Hydrodynamics to the
Physical Theories contained in the subsequent part of the ;
work . 316320
THE MATHEMATICAL PEINCIPLES OF THEOEIES OF
THE PHYSICAL FOKCES 320676
Intention to discuss the Theories with reference only to funda
mental principles, and those necessary for the explanation of
classes of phenomena . . . ... . .  320
XVIII CONTENTS.
Pages
The Theory of Light on the Hypothesis of Undulations . . . 320 436
The aether defined to be a continuous elastic fluid pressing propor
tionally to its density. Explanations of properties and pheno
mena of light deduced exclusively from the qualities of the
........... 320356
(1) i Kectilinear transmission. (2) A ray of common light. Its vibra
tions defined by formulae. (3) Uniform propagation in space.
(4) The nondependence of rate of propagation on intensity.
(5) Equality of the intensity of compound light to the sum of
the intensities of the components. (6) Variation of intensity
according to the law of the inverse square by divergence from
a centre. (7) Composite character of light and its resolvability
into parcels. (8) Distinction by colour. (9) Distinction by
phase. (10) Spectrum analysis, or resolvability into parcels of
different colours. (Eemarks on Transmutation of Eays).
(11) Coexistence of different parcels without interference.
(12) Interferences under particular circumstances . * . 321 329
Theory of transverse vibrations. Their mode of action on the
parts of the eye. (13) The nonpolarized character of light as
initially produced, the transverse vibrations in such light being
symmetrically disposed about an axis. (14) The resolution of
common light into equal portions polarized in rectangular
planes. (15) The noninterference of rays oppositely polar
ized. Proof, from the interference of the direct vibrations
under the same circumstances, that light is due to transverse
vibrations. (16) The nondependence of the combined inten
sities of two oppositely polarized rays having a common path
on the difference of their phases. (17) The proportion of the
intensities, under given circumstances, of the parts of a polar
ized ray resolved by a new polarization. (18) The distinguish
ing characters of planepolarized, ellipticallypolarized, and
circularlypolarized light , ....... 330 338
(19) The effects of compounding lights of different colours. The
mathematical theory of the composition of colours given at
considerable length, and compared with experiments, in the
sections numbered from 1 to 8. (20) Phenomena of Diffrac
tion. The usual mathematical treatment of diffraction prob
lems accords with the Undulatory Theory expounded in this
work. Evidence from the explanations of the phenomena
(1) (20) of the reality of the aether ..... 338356
EXPLANATIONS of phenomena of light depending on relations of the
motions of the eether to visible and tangible substances . .856421
CONTENTS. XIX
Pages
Foundations of the reasoning in the remainder of the Theory of
Light, and in all the subsequent Theories. Assumed atomic
constitution of substances. Assumed qualities of atoms. No
other kinds of force than the pressure of the aether and the
resistance to such pressure by the atoms. Force varying with
distance not inherent in matter. Newton's view of the nature
of gravity. Newton's and Locke's views of the quality of
atoms. Hypotheses respecting atoms and the aether necessary
foundations of physical science. Their truth established by
comparison of mathematical deductions from them with expe
riment. Imperfect verification of the hypotheses respecting
the ultimate constituency of substances ..... 3564^62
Problem I. Laws of transmission of light through noncrystalline
transparent media. General formula for the rate of trans
mission in a given substance. Modification of the formula by
the mobility of the atoms. Effect of the elasticity of the me
dium. Condition of transparency. Consequent formula (j3)
applicable to homogeneous light ...... 362 370
Modification of formula (/3) to adapt it to light of different refran
gibilities. Theoretical explanation of Dispersion. Formula
(7) for calculating the relation between /* and X. Numerical
comparison of results with experiment ..... 370 375
Problem II. Laws of transmission of light through crystalline
transparent media. Assumed difference of elasticity in differ
ent directions. Application of formula (j3). Equation of the
surface of elasticity. Only polarized light transmissible.
Equation of the wavesurface 375 382
Inference (1) that the rate of propagation is the same in every
plane through an optical axis, and in all directions in it, if the
transverse vibrations of the ray be perpendicular to the plane;
(2) that the transverse vibrations of a polarized ray are per
pendicular to the plane of polarization 382 383
Problem III. The laws of the reflection and refraction of light at
the surfaces of transparent substances . . . . . 383 415
Proof of the law of reflection. Loss of half an undulation by inter
nal reflection accounted for. Loss and gain of light by trans
mission through a plate. Explanation of the central dark
spot of Newton's rings. Polarization of common light by
reflection. Partial polarization. Generation of elliptically
polarized light by reflection. Eeflection of polarized rays.
Formula for the amount of reflected light .... 383 391
Theory of refraction at the surfaces of noncrystalline media.
Conditions of regular refraction. Auxiliary discussion of the
a 2
XXT CONTENTS.
Pages
character of composite rays. The precise action of the refrin
gent forces unknown. The law of refraction determined by
reference to a principle of least action : . . . . . 392 401
Laws of double refraction at the surfaces of crystalline media.
Bifurcation of an incident nonpolarized ray. Construction for
determining by means of the wavesurface, and the principle of
least action, the courses of the two rays. Refraction out of a
medium inferred from that into a medium by the law that
light can travel along the same path in opposite directions. A
hydrodynamical reason given for this law .... 401 404
Co$muation (from p. 391) of the theory of polarization by reflec
tion. Auxiliary investigation of the ratio of the condensations
of a given wave before and after intromittence, on the hypo
thesis that the proportionate space occupied by atoms is incon
siderable. Equality of the condensations when the tangent of
the angle of incidence is equal to the index of refraction, i. e.
for the polarizing angle . . . . . . . . 405 406
Incidence of common light on the surface of a crystallized medium.
Estimated quantities of reflected light. Comparisons of the
theory with Jamin's Experiments. The polarizing angle of
opaque bodies. Inference from the theoretical explanation of
the polarizing a'ngle "that \he proportionate space occupied by
the atoms of all known substances is very small . . . 406 410
Incidence of polarized light on the surface of a crystallized me
dium. Formulae for the quantities of reflected light. The
theory decides that the transverse vibrations of a ray polar
ized in the 1 plane of incidence a*re perpendicular to that plane.
Fresnel's empirical formulas for the intensities of reflected
rays especially adaptable to the proposed undulatory theory . 410 411
Theory of the total internal reflection of common light, and of
plane polarized light. Generation in the latter case of ellipti
cally polarized light. Fre&nel's Ehomb . . . . 411 412
Theory of the coloured rings formed by the passage of planepolar
ized light , through thin plates of crystal. Complete explana
tion of all the phenomena of this class by the proposed undu
latory theory. Failure in this respect of the " vibratory" (or
oscillatory) theory. (See Introduction, pp. xlvii xlix) . . 412 415
Additional explanations of phenomena. Colours produced by the
passage of light through glass in a state of mechanical con
straint. The colours of substances. Eegular and irregular
reflection. . Law of . brightness of bodies seen by irregular
reflection. Absorption. Epipolic dispersion, as due to change
of refrangibility . . . 415 421
CONTENTS. XX{
Pages
Addendum to the Theory of Light . .. . . . ' '421^436
More correct solution of Example VII. p. 296, to terms of the first
order. Correction of the expression in p. 298 for the accele
ration of the sphere. Corrected formulae (ft') and (7') for the
calculation of dispersion. Comparison of results from (7')
with experiment' . . . . . . . . .422 427
Calorific and chemical effects attributable to direct vibrations of
the rays of the solar spectrum. Chemical as well as luminous
effects produced by the transverse vibrations. The formula
for dispersion for a composite medium of the same form as
that for a simple medium. Rays of nearly the same refrangi
bility as (F) neutral as to calorific and chemical effects . . 427 432
The formula for dispersion applied to a gas. Bright spectrum
lines of an ignited gas. Theory of the dark lines of the solar
spectrum. Inference that the lines of a composite gas consist
of those of the components. Possible reversal of the order of
the spectrum colours of a gas. Imperfection of the Theory
of Dispersion 432 436
The Theory of Heat and Molecular Attraction .... 436 485
General principles of the'Theo'ry. The fact that lightproducing
rays are also heatproducing accounted for. Heatwaves pro
duce both vibratory and translatory motions of atoms. De
pendence of the mathematical theory of the translatory action
of waves on terms of the second order. An argument, apart
from symbolical reasoning, to shew that waves incident on a
small sphere necessarily cause a permanent motion of transla
tion 436439
Principles of a solution of Example VII. to terms of the second
order more correct than that given in pages 296 306. For
mula obtained for the' acceleration of the sphere involving two
unknown functions H l and ff s of m and X. Proof that a uni
formly accelerated motion results. Its direction from or
towards the origin of the waves dependent on the values of
HI and H. 2 . Waves of the smallest order always repulsive
from their origin. Theory of repulsive and attractive effects
of waves of different orders. Coexistence of the translatory
actions of waves from different origins 439 459
Theory of the forces by which discrete atoms form compact masses,
viz. caloric repulsion and a controlling molecular attraction.
Generation of secondary waves by the incidence of the prima
ries on atoms, and by the reaction at their surfaces. The
repulsion of heat attributed to the translatory action of secon
dary waves, supposed to emanate equally in all directions
XXII CONTENTS.
Pages
from each atom of a mass. The mutual action of neighbour
ing atoms always repulsive. Molecular attraction attributed
to waves of another order, resulting from the composition of
those of the first order emanating from a multitude of atoms.
Radiant heat distinguishable by the order of its waves from
caloric repulsion 459 465
Theory of the solid and liquid states of bodies. Increment of
density towards the interior in a thin superficial stratum of
every liquid and solid substance. Difference between the
liquid and solid states. The atomic repulsion of aeriform
bodies not controlled by molecular attraction. Theory of latent
heat. On the conservation of the caloric of large masses.
Collision of a,toms impossible 465 468
Investigation of the relation between pressure and density in
gaseous bodies. The measured pressure of a gas mechanically
equivalent to its atomic repulsion. Temperature of position
as determined by radiant heat. Relation between pressure
and density, inclusive of the effect of variation of temperature.
Change of temperature by sudden changes of density more
sensible in closed than unenclosed spaces. Theoretical ve
locity of sound 469474
Theory of the relation between pressure and density in liquid and
solid substances. Equilibrated action of atomic and mole
cular forces at and near their boundaries. Different from that
in the interior. Illustrative experiment. The general relation
between pressure and density the same in liquids as in solids.
Large masses to be regarded as liquid. Reason that the mean
figure of the earth corresponds to that of the ocean surf ace.
Formula obtained for the relation between pressure and density
in liquids and solids. Applied to determine the law of the
earth's interior density ........ 474 481
Considerations relating to the mechanical equivalent of atomic
and molecular forces. Independent pressures of different
gases in the same space. Reason for the different elasticities
of equal weights of different gases in equal spaces. Brief
notices of the bearing of the general theory on chemical and
crystallographical facts ........ 481 485
The Theory of the Force of Gravity 486 505
An explanation of the modus operandi of gravity necessarily in
cluded in a general theory of physical force. Reference of
gravitating force to pressure of the aether. Elucidation of a
point of analysis relating to the equation (e) in p. 443. Diffe
rent orders of molecules and of the waves emanating from them 486490
CONTENTS. XXII t
Pages
Bepulsive and attractive effects of waves of different breadths in
ferred from an equation (at the top of p. 455) involving the
factors H l and H 2 . Values of H l indicative of a repulsive
effect. Change of the repulsion into attraction by increase of
the value of H 2 Considerations, apart from the analytical
reasoning, of the signification of this factor. Inference that it
is always positive and greater than unity. Keasons given for
concluding that it is a function of X, and that for gravitywaves
it exceeds 1 490498
Comparison of the theory with known laws of gravity. The theory
too imperfect for demonstration of the laws. But primd faciz
evidence of its truth given by the comparisons (1) (6) with
ascertained facts. Nonretardation of masses (as planets)
moving in the aether with a velocity nearly uniform. Argu
ment shewing that the constant K in the formula for the
refractiveindex (p. 367) is very small for gravitywaves, and
inference that such waves are very little refracted. Evidence
that they undergo some degree of refraction deduced from
local irregularities of gravity. Magnitude. of the ultimate
gravitymolecules. Gravitymeasures of quantities of matter . 498 503
Instability of stellar systems if the action between the components
be solely attractive. Inference from the hydrodyraiiical
theory of gravity that the law of attraction may chf ng 3 by
distance, and that neighbouring stars may be mutually re
pellent. Control in that case of the repulsion by a n f IK class
of gravitywaves. Consequent explanation of periodic proper
motions of stars . . . . . . . . 504 505
The Theory of Electric Force . . . . . . . 505555
Extension given by modern experimentalists to the meaning of
"Electricity." Accounted for theoretically by the common
relation of the physical forces to the aether. Proposal to use
the terms Electric, Galvanic, and Magnetic, as distinctive of
classes of phenomena. The class generated by friction treated
of under the head of Electric Force . , . , ' . .505 507
Theory of molecular Forces (F) of the second order, attractive and
repulsive. Their equivalence to the mechanical forces in the
Statics and Dynamics of rigid bodies. Definition of the
electric state, as maintained by atomic and molecular forces.
Production of the electric state by friction al disturbance of
superficial atoms and the simultaneous generation of second
order molecular forces. Theory of two kinds of electricity,
vitreous and resinous, or positive and negative. Keason given
for the production by friction of equal quantities of the opposite
electricities . 607515
XXIV CONTENTS.
Case of the positive or negative electric state of a globe. Hypo
thesis that the secondorder molecular forces emanate equally
from all the elements of the interior, and vary inversely as the
square of the distance. Transition to the case of a spherical
shell. Explanations of facts from which it has been inferred
that electricity is confined to the surfaces of bodies. The con
ditions of electrical equilibrium in bodies of any form. Con
ductors distinguished from nonconductors by the property of
superficial conduction. Theory ef the superficial distribution
of electricity. Case of a cylinder with hemispherical ends.
Explanation of the accumulation of electricity at sharp points. 515 521
Theory of electricity by influence or induction. Induced electricity
attributable to the action of secondorder molecular forces.
Hypothesis that their setherial waves traverse substances
freely, the forces varying according to the law of the inverse
square. Proof that equal quantities of opposite electricities
are induced by a charged conductor on a neutral one. Reaction
of the induced electricity on the charged conductor, and neu
tralization of induced electricities by discharging or removing
this conductor. Effect of connecting the neutral conductor
with the ground in presence of the other charged. (Auxiliary
discussion of the distinction between primary and induced
electricity.) Theory of the observed effects of breaking the
connection with the ground and removing the charged con
ductor. Accumulation of primary electricity by an electric
machine 521532
Additional facts explained' by the theory of induced electricity.
(1) The neutral state o.f a sphere, after the, separation from it
of electrified hemispherical caps. (2) The phenomena of the
electrical condenser. (3) Electrifying a nonconductor by in
duction. (4) Induction by contact, and phenomena of the
Leyden Jar. (5) Influence of the air on electrical phenomena.
Discharge through the air accompanied by crepitations and
the electric spark. Loss of electricity by conduction through
the air. Theories of the brush discharge and the electric egg.
Conductiveness of moist air. Accompaniment of an electric
discharge by heat 539544
Theory of electrical attractions and repulsions. Not referable to the
translatory action of secondorder molecular forces. Hypo
thesis of the action of currents. Interior gradation of the
density of a body electrified inductively. Consequent genera
tion of secondary streams by the motions of the earth relatively
to the aether. Electrical attractions and repulsions ascribed to
the secondary streams . . . . . . . . 544 548
CONTENTS. XXV
Pages
The mutual attraction of two spheres, one electrified originally,
either positively or negatively, and the other electrified by it
inductively. The mutual attraction of two spheres, both elec
trified originally, and with opposite electricities, and their
mutual repulsion when electrified with the same electricities.
Explanation, of the attractions and repulsions in the goldleaf
electroscope. Mutual repulsion of two bodies after being
brought into contact by attraction. Explanation of the elec
tric wind observed to flow from a point connected with an
electrified conductor. Accordance of the theory with Coulomb's
experimental determination of the law of the inverse square
for the action of an electrified sphere on small bodies . . 548 555
The Theory of Galvanic Force. . 555604,
Difference between Electric and Galvanic disturbances of the equi
brium of superficial atoms. Galvanic disturbance produced
by atomic and molecular forces brought into action by the
chemical relation between a liquid and solid in contact. Dis
cussion of two fundamental experiments establishing this law.
Indication of galvanic electricity by the electroscope. Theory
of the galvanic battery and of the currents it generates.
Direction of the current shewn to be from the zinc plate to the
copper plate . 555 563
Preliminary considerations respecting the action of conducting
bodies as channels of galvanic currents. Mathematical treat'
ment of the case of a steady stream symmetrical about a
straight rheophore of wire. Inference that streams along
cylindrical wires move in spiral courses. Theory of the stop
page of a current at the terminals of a circuit not closed.
Explanation on the same principles of the confinement of the
current within conducting channels of irregular form. Theory
of the flow of the current upon closing the circuit . . 563 572
Definition of the intensity of a galvanic current. Proof that the
intensity is the same at all parts of the same circuit. Pre
sumptive evidence of the hydrodynamical character of galvanic
currents. Maintenance of the current by continuous galvanic
impulses. General formula for intensity involving the specific
conductivities of different portions of the current. Ohm's
Law. Inference from it that the resistance due to spiral
motion varies inversely as the square of the radius of the wire. 572575
Explanation of the heat and light emitted by a rheophore of fine
wire. The increment of temperature shewn to vary inversely
as the fourth power of the radius and to be uniform through
out wire of given radius. Mathematical argument to prove
XXVI CONTENTS.
Pages
that the developed heat varies as the square of the intensity
of the current 575_577
Inferences from the general formula for intensity that for a given
couple the intensity is less as the length of wire is greater, and
that if the circuit be short the intensity is nearly proportional
to the size of the zinc plate 577 578
Theory of the electric discharge produced by the approach of the
terminals of a galvanic circuit. Electric and galvanic dis
charges distinguished by the theory, and by experiment. The
theoretical conclusion that a galvanic current cannot flow in
vacuum confirmed experimentally. Discharges in Geissler's
tubes. Theoretical explanations of the coloured light, the
stratification, and the glow at the terminals .... 578 582
Theory of the voltaic arc. Volatilization of matter at the positive
terminal, and its transfer to the other. Greater heat at the
positive than at the negative terminal. Brightness of the
arc dependent on the size of the plates, its length on the num
ber of couples. Transfer of matter both ways in quantities
depending on the volatility of the terminals. The form of the
arc accounted for 582586
Theory of the analysis of liquids by galvanic currents. Decompo
sition of water. Analysis in definite proportions explained by
the theory on the hypothesis of Grotthus. Theory of Faraday's
law that[the decomposing action of a current is the same at
each of several sets of terminals. Theory of chemical decom
position by frictional electricity. Eeason given for its amount
being very small compared to that by a galvanic current. Ex
planation of an experiment by Faraday illustrative of galvanic
action at terminals. Theory of the maintenance of a galvanic
current when the rheophores terminate in the ground . . 586 592
Theory of the mutual action between galvanic currents. Attraction
or repulsion between two parallel rheophores according as the
currents are in the same or opposite directions. Eepulsion
between a fixed and a moveable rheophore placed end to end.
Motion along a rheophore of sinuous form inclusive of the
effect of centrifugal force. Case of the solenoid. Neutraliza
tion^ a sinuous rheophore of any form by a rectilinear rheo
phore. Mutual action between two of Ampere's solenoids as
due to the spiral motions about the axes of the wires. At
traction and repulsion between two solenoids placed end to end.
Inference from experiment that the spiral motion along a
cylindrical rheophore is always dextrorsum. Incapability of
the theory to give a reason for this law 592 599
Probable rate of propagation of limited currents, like those gene
CONTENTS. XXVII
rated in Bending messages by a Galvanic Telegraph. Theory
of Faraday's induced currents. Generation of an induced
current by' sudden interruptions, or sudden changes of inten
sity, of an existing current. Also by sudden changes of its
distance from a neutral rheophore. The case of the contiguity
of two coils. Augmentation of the inductive effect by increasing
the number of turns of the secondary coil and insulating them, .
and by producing the primary currents in rapid succession, as
by Kuhmkorff's apparatus. General conclusion that galvanic
phenomena are governed by hydrodynamical laws . . . 599 603
Theory of currents called thermoelectric. Their origin in grada
tion of interior density caused by heat. In other respects not
different from galvanic currents. Their phenomena shewn by
Matteucci to be connected with crystallization. Probable in
ference that the elementary circular motions of galvanism, and
their direction, are determined generally by disturbance of the
crystalline arrangement of atoms. Generation of a differential
current hi Seebeck's experiment. This class of facts peculiarly
indicative of the production of currents by gradation of interior
density 603604
The Theory of Magnetic Force 604 676
Hypothesis of the existence naturally in certain substances of gra
dation of interior density without disturbance of the state of
the superficial atoms. Direction of the gradation of density
dependent on the form of the body. Case of a magnetized
steel bar. Generation of secondary circulating streams. In
dication by the arrangement of attracted iron filings about the
bar that magnetic force is due to the dynamical action of such
streams. Inferences. (1) The magnetism is equal on the
opposite sides of a middle neutral position. (2) Each part of
a divided magnet becomes a magnet. (3) The intensity of the
current is as the size of the magnet and degree of its magneti
zation directly, and as the length of the circuit inversely.
(4) Positive and negative poles. Like poles repel and unlike
attract . 604608
Theory of the mutual action of a galvanic rheophore and a mag
netic needle. Oersted's experiment. Eeasons given for the
axis of one being transverse to that of the other in case of
equilibrium. Proof that stable rotatory equilibrium results
from the mutual action of two rheophores when their axes are
parallel. Laws bf angular currents ' 608 612
Theory of Terrestrial Magnetism. The earth's magnetism due to
secondary aetherial streams resulting from its motions relative
to the aether. Influence of the form and materials of the earth.
XX VIII CONTENTS.
Pages
The directions and intensities of the streams determined by
observations of magnetic declination, dip, and intensity.
Proof from the explanations of two experiments that the
earth's magnetic stream enters the north, or marked, end of
the needle, issuing consequently from the earth on the north
side of the magnetic equator, and entering it on the south
side. Theory of the directive action of terrestrial magnetism.
The south end of the needle the positive pole, or that from
which its own current and that of the earth both issue. Total
intensity of the magnetic force deduced from oscillations of
the needle about its mean position. Inference of total inten
sity and dip from measures of the horizontal and vertical
components. Theory of the action of a solenoid on a magnet,
and of the directive action of the earth's magnetism on a
solenoid 613618
Theory of magnetic induction, or magnetization, by natural mag
nets. By the earth's magnetism. By a galvanic current.
Change of the plane of polarization of light by the influence of
artificial magnets adduced in support of the theory. Differ
ence between the magnetization of Steel and soft iron. The
directions of the. magnetizing and induced currents coincident
in magnetism, opposed in diamagnetism. Explanation of the
transverse position of a bar of bismuth suspended between the
poles of a magnet. Consequent points 618 622
Theory of the attraction of iron filings by a magnet. Additional
mathematical investigation of the dynamical effects of com
posite steady motion. Case of the attraction of a small pris
matic bar (or iron filing) by a large magnetized bar, the direc
tions of their axes coinciding. Formula obtained for the
resulting moving forces. Inference from it that iron filings
are attracted at both poles. Application of the same formula
to account for the effects of diamagnetic action. Also to ex
plain why the earth's magnetism is solely directive, and why
nonmagnetic bodies are uninfluenced by magnetic attraction . 622 629
Explanations of experiments by Faraday shewing the influence of
magnets on ferruginous solutions. Supposed coincidence of
Faraday's lines of magnetic force with the curvilinear courses
of the magnetic streams. Theoretical reason for the diamag
netism of a piece of bismuth in a powdered state being nearly
the same as when it is whole 629 631
Attraction of iron filings by a galvanic current. Inference that a
galvanic current is capable of inducing magnetism. Theoreti
cal explanation of this action. Difference, according to the
CONTENTS. xxix
Pages
theory, of the attractions of iron filings by a Bolenoid and
a magnetized bar 631 634
Theory of magnetization by frictional electricity. Generation of a
feeble continuous current by an electric machine. Reason
given for the nonproduction of frictional electricity by gal
vanism or magnetism 634 636
Phenomena of the mutual action between a magnet and a mass of
copper. Proved experimentally by Faraday to be referable
to the induction of galvanic currents by magnetic currents.
Elementary experiment. The intensity of the induced current
proportional to the galvanic conductivity of the metal. Fara
day's" experimental results expresse'd in' a general law not .
deducible a priori. The phenomena explainable by the appli
cation of this law on hydrodynamical principles. Hypothesis
of elementary and composite circulating motions of the aether.
Consequent explanations of experiments of this class made by
Faraday, Gambey, Herschel and Babbage, and Arago . . 636 644
Theory of the variations of terrestrial magnetism. Mean solar
diurnal variation of declination. Induction by Sabine, from
observations in the north and south magnetic hemispheres, of
the occurrence of maximum and minimum deflections at the
same local hours. Hypothesis that the solardiurnal variation
is due to magnetism of the atmosphere generated by gradations
of its temperature and density caused by solar heat. Theore
tical explanations on this "hypothesis of the main features of
the diurnal variation of declination in mean and in high
latitudes.; , ;; 644650
Theory of the annual inequality of the solardiurnal variation of
declination. Attributed to the changes of distribution of the
solar heat, and of atmospheric magnetism, consequent upon
of the sun's declination . . . '".. ". . 650652
Disturbances of the declination by variations of atmospheric tem
perature and magnetism due to local causes. Earthcurrents.
Theory of the Aurora, so far as it is attributable to disturb
ances of a local character. Determination by Sabine of the
existence of a local hour of maximum disturbancevariations
of the declination. Indication by this fact of extraneous
magnetic action . . 652654
Regular diurnal variations of dip and intensity attributable, like
that of declination, to solar atmospheric magnetism. Evi
dence from the Greenwich observations of a diurnal variation
of vertical force. Annual inequalities of the diurnal range
of dip and intensity due to the changes of the Sun's declina
XXX CONTENTS.
Pages
tion. Reason that the atmospheric magnetic effects due to
solar heat are not greatest when the earth is nearest the Sun . 654 657
Detection by Sabine of variations of the magnetic elements obey
ing the same laws in both hemispheres. This class of varia
tions attributed by the theory to changes of terrestrial mag
netism due to the variable velocity of the earth in its orbit.
Mathematical argument in support of this view . . . 657661
Theory of the lunardiurnal variations of the magnetic elements.
Hypothesis that they are due to magnetism of the atmosphere
resulting from gradations of its density caused by the Moon's
gravitational attraction, Explanations of the phenomena on
this hypothesis 661665
Observed changes from year to year of the mean annual variations
of the magnetic elements. Necessity of referring such changes
to external or cosmical agency. Proposal of a theory of
cosmical variations. Hypothesis that the Sun, like the
earth, has its proper magnetism. Evidence supposed to be
given by the zodiacal light of the existence of solar mag
netic streams extending to the earth. The magnetic variation
called the nocturnal episode probably due to these streams.
Variations of the sun's proper magnetism ascribed to gravita
tional attraction of the solar atmosphere by the planets. A
solardiurnal disturbancevariation of declination referred to
this cause. Its law of periodicity different from that of the
regular solardiurnal variation . . . . . 665 671
Additional theoretical inferences. (1) Dependence of the amount
of disturbancevariation on the configuration of the Planets.
The cycle of about ten years, inferred by Sabine from observa
tion, probably referable to the fact that 13 semisynodic
periods of Venus are very nearly equal to 19 of Jupiter. (2)
Theory of the observed periodicity of solar spots. The coinci
dence of their period with that of the disturbancevariation
accounted for by supposing them to be generated by planetary
magnetic influence. (3) Magnetic storms considered to be
violent and transitory disturbancevariations due to solar local
causes. Observation of a remarkable phenomenon confirma
tory of this view. The larger displays of Aurora attributed to
these unsteady sunstreams. The local hours of maximum of
magnetic storms the same as those of the more regular dis
turbancevariations. (4) A possible cause of the secular
variations of the magnetic elements suggested . . . 671 676
GENERAL CONCLUSION. Remarks on the character and limits of
the proposed Physical Theory, and on the evidence for the
CONTENTS. XXXI
Pages
truth of its hypotheses. ' Objections to it answered. Con
siderations respecting the relation of the method.of philosophy
advocated in this work to metaphysical enquiry and to
Theology . . . , > . . . . . 677687
APPENDIX. I. Proof that every equation has as many roots as it
has dimensions, and method of finding them. II. Formation
of equations of partial derived functions by the elimination of
arbitrary functions. III. On the occurrence of discontinuity
in the solution of problems in the Calculus of Variations. . 688 696
The Diagrams referred to in pages 63 82, which the reader was re
quested to draw for himself, it has been thought better to add at the end of
the volume. All other requisite Diagrams and Figures will be found in the
Physical Treatises or Memoirs cited in the text or the notes.
EEEATA.
Page ixf, line 5, for only read mainly
xiii., line 11 from bottom, dele and the succeeding ones
Iviii., line 6 from bottom, read The periodic variations of the Sun's proper
magnetism are, &c.
11, last line but one, for ^ j read
' J cd cb
47, lines 13 and 14, for dx read 2dx, and for d .f(x) read 2d . f(x)
83, line 4, for dy read 2dy,&ndfor dx read 2dx
84, line 11 from bottom, for x read 
89, the formula in line 15 should have been obtained by Taylor's
Theorem used as in p. 47.
145, last line, for r' 4 read r 4
229, line 6, for d 2 in the second term read dz*
298, line 11 for c read C in both places
365, line 1, for read . This mistake of the author, and
i 'ii
the inferences from it, are corrected in page 501
372, the values of A, B, C should be 10,046655, 1,635638, 13,433268
373, lines 5, 7, and 8, the values of X by calculation for the rays
(D), (F), (G) should be respectively 2,1756, 1,7995, 1,5954
373, the values of A, B, C should be 4,569309, 0,660934, 4,483938,
and those of X by calculation for the rays ((7), (D), (F), (G),
2,4280, 2,1764, 1,7949, 1,5923. (The correct values, in both
cases, of "excess of calculation" are used in p. 427.)
378, line 8, in the expression for B, for /i 2 e 3 read /t 2 e 2
461, in the' running title, for LIGHT read HEAT
494, line 6 from bottom, for 5a' read 6a'
L! ':
INTRODUCTION.
IN order to account for the Title that has been given to this
Volume a few words of explanation will be necessary. The printing
of the work was commenced in 1857. I had then only the inten
tion of going through a revision of the principles of the different
departments of pure and applied mathematics, thinking that the
time was come when such revision was necessary as a preparation
for extending farther the application of mathematical reasoning to
physical questions. The extension I had principally in view had
reference to the existing state of the science of Hydrodynamics,
that is, to the processes of reasoning proper for the determination
of the motion and pressure of fluids, which, as is known, requires
an order of differential equations the solutions of which differ
altogether from those of equations appropriate to the dynamics of
rigid bodies. I had remarked that although by the labours of
Lagrange, Laplace, and others, great success had attended the
applications of differential equations containing in the final stage of
the analysis only two variables, the whole of Physical Astronomy is,
in fact, an instance of such application, the case was far different
with respect to the applications of equations containing three or
more variables. Here there was nothing but perplexity and un
certainty. After having laboured many years to overcome the
difficulties in which this department of applied mathematics is
involved, and to discover the necessary principles on which the
b
VI INTRODUCTION.
reasoning must be made to depend, I purposed adding to the dis
cussion of the principles of the other subjects, some new and spe
cial considerations respecting those of Hydrodynamics. The work,
as thus projected, was entitled " Notes on the Principles of Pure
and Applied Mathematics," the intention being to intimate by the
word " Notes" that it would contain no regular treatment of the
different mathematical subjects, but only such arguments and dis
cussions as might tend to elucidate fundamental principles.
After repeated efforts to prosecute this undertaking, I was
compelled by the pressure of my occupations at the Cambridge
Observatory, to desist from it in 1859, when 112 pages had been
printed. I had not, however, the least intention of abandoning
it. The very great advances that were being made in physics by
experiment and observation rendered it every day more necessary
that some one should meet the demand for theoretical investiga
tion which the establishment of facts and laws had created. For
I hold it to be indisputable that physical science is incomplete
till experimental inductions have been accounted for theoretically.
Also the completion of a physical theory especially demands mathe
matical reasoning, and can be accomplished by no other means.
When, according to the best judgment I could form respecting
the applications which the results of my hydrodynamical re
searches were capable of, I seemed to see that no one was as well
able as myself to undertake this necessary part in science, I gave
up (in 1861) my position at the Observatory, under the convic
tion, which I expressed at the time, that I could do more for the
honour of my University and the advancement of science by de
voting myself to theoretical investigations, than by continuing to
take and reduce astronomical observations after having been thus
occupied during twentyfive years. The publication of this work
will enable the cultivators of science to judge whether in coming
to this determination I acted wisely. Personally I have not for a
INTRODUCTION. Vll
moment regretted the course I took ; for although it' has been
attended with inconveniences arising from the sacrifice of income,
I felt that what I could best do, and no one else seemed capable
of undertaking, it was my duty to do.
It should, farther, be stated that after quitting the Obser
vatory, and before I entered upon my theoretical labours, I con
sidered that I was under the obligation to complete the publica
tion of the meridian observations taken during my superin
tendence of that Institution. This work occupied me till the
end of 1864, and thus it is only since the beginning of 1865
I have been able to give undivided attention to the composition of
the present volume. In April 1867, as soon as I was prepared
to furnish copy for the press, the printing was resumed, after I
had received assurance that I might expect assistance from the
Press Syndics in defraying the expense of completing the work.
In the mean while I had convinced myself that the hydrodyna
mical theorems which I had succeeded in demonstrating, admitted
of being applied in theoretical investigations of the laws of all
the different modes of physical force, that is, in theories of light,
heat, molecular attraction, gravity, electricity, galvanism, and
magnetism. It may well be conceived that it required no little
intellectual effort to think out and keep in mind the bearings and
applications of so extensive a physical theory, and probably, there
fore, I shall be judged to have acted prudently in at once pro
ducing, while I felt I had the ability to do so, the results of my
researches, although they thus appear in a somewhat crude form,
and in a work which in the first instance was simply designated
as " Notes." Had I waited to give them a more formal publi
cation, I might not, at my time of life, have been able to accom
plish my purpose. As it is, I have succeeded in laying a foun
dation of theoretical physics, which, although it has many imper
fections, as I am fully aware, and requires both correction and
Vlll INTRODUCTION.
extension, will not, I venture to say, be superseded. In order to
embrace in the Title page the second part of the work, the
original Title has been altered to the following : " Notes on the
Principles of pure and applied Calculation; and Applications of
mathematical principles to Theories of the Physical Forces."
The foregoing explanations will serve to shew how it has come
to pass that this work consists of two distinct parts, and takes
in a very wide range of subjects, so far as regards their mathe
matical principles. In the first part, the reasoning rests on defini
tions and selfevident axioms, and although the processes by which
the reasoning is to be conducted are subjects for enquiry, it is
presumed that there can be no question as to the character and
signification of definitions that are truly such. The first part
is not immediately subservient to the second excepting so far as
results obtained in it are applied in the latter. In the second
part the mathematical reasoning rests on hypotheses. It does not
concern me to enquire whether these hypotheses are accepted,
inasmuch as they are merely put upon trial. They are proved
to be true if they are capable of explaining all phenomena, and if
they are contradicted by a single one they are proved to be false.
From this general statement it will appear that in both portions
of the work the principles and processes of mathematical reasoning
are the matters of fundamental importance.
There are two general results of the arguments contained in
the first part which may be here announced, one of them relating
to pure calculation, and the other to applied calculation. (1) All
pure calculation consists of direct and reverse processes applied to
the fundamental ideas of number and ratio. (2) " All reasoning
upon concrete quantities is nothing but the application of the
principles and processes of abstract calculation to the definitions
of the qualities of those quantities." (p. 71.)
Having made these preliminary general remarks I shall pro
INTRODUCTION. IX
ceed to advert to the different subjects in the order in which they
occur in the body of the work, for the purpose of pointing out any
demonstrated results, or general views, which may be regarded as
accessions to scientific knowledge. I may as well say, at
that the work throughout lays claim to originality, consisting only
of results of independent thought and investigation on points
chiefly of a fundamental character. The first part is especially
directed towards the clearing up of difficulties which are still
to be met with both in the pure and the applied departments of
mathematics. Some of these had engaged my attention from the
very beginning of my mathematical career, and I now publish the
results of my most recent thoughts upon them. I take occasion
to state also that the commencements of the Physical Theories
which are contained in the second part of the volume were pub
lished from time to time in the Transactions of the Cambridge
Philosophical Society, and in the Philosophical Magazine. They
are now given in the most advanced stages to which my efforts
have availed to bring them, and being, as here exhibited, the
result of long and mature consideration, they are, I believe, free
from faults which, perhaps, were unavoidable in first attempts to
solve problems of so much novelty and difficulty.
In the treatment of the different subjects I have not sought to
systematize excepting so far as regards the order in which they
are taken. The order that I have adopted, as arising out of the
fundamental ideas of space, time, matter, and force, is, I believe,
the only one that is logically correct.
All that is said in pages 4 20 on the principles of general
arithmetic rests on the fundamental ideas of number and ratio.
As we can predicate of a ratio that it is greater or less than another
ratio, it follows that ratio is essentially quantity. But it is quan
tity independent of the magnitudes which are the antecedent and
the consequent of the ratio. Hence there may be the same ratio
X INTRODUCTION.
of two sets of antecedents and consequents, and the denomination
of one set is not necessarily the same as that of the other. This
constitutes proportion. Proportion, or equality of ratios, is a
fundamental conception of the human understanding, bound up
with its power of reasoning on quantity. Hence it cannot itself
be an induction from such reasoning. The Elements of Euclid are
remarkable for the nonrecognition of the definition of proportion
as the foundation of quantitative reasoning. The fifth definition
of Book v. is a monument of the ingenuity with which the Greek
mind evaded the admission of proportion as a fundamental idea.
By arguing from the definition of proportion, I have shewn (in
page 1 3) that Euclid's fifth definition may be demonstrated as if it
were a proposition, so that it cannot in any true sense be called a
definition. It is high time that the method of teaching general
arithmetic by the fifth Book of Euclid should be discontinued, the
logic of the method not being defensible.
In Peacock's Algebra (Preface, p. xvii.), mention is made of
" the principle of the permanence of equivalent forms." The word
"principle" is here used where "law" would have been more
appropriate. For it is certain that the permanence of equivalent
forms is not a selfevident property, nor did it become known by
intuition, but was rather a gradual induction from processes of
reasoning, the exact steps of which it might be difficult to trace
historically, but which nevertheless actually led to the knowledge
of the law. In the arguments which I have adduced in pages 15
20 I have endeavoured to shew how the law of the permanence
of equivalent forms was, or might have been, arrived at in
ductively.
In the rapid review of the principles of Algebra contained in
pp. 21 28, the point of chief importance is the distinction be
tween general arithmetic and algebra proper. In the former
certain general rules of operation are established by reasoning
INTKODUCTION. XI
involving considerations respecting the relative magnitudes con
cerned ; in the other these rules are simply adopted, and at the
same time are applied without respect to relative magnitude. In
order to make the reasoning good in that case the signs + and
are attached to the literal symbols. . The use of these signs in the
strictly algebraic sense is comparatively recent. It was imper
fectly apprehended by Vieta, who first used letters as general
designations of known quantities. The rules of signs were, I
believe, first systematically laid down by our countryman Ought
red. Regarded in its consequences the discovery of the algebraic
use of + and is perhaps the most fruitful that was ever made.
For my part I have never ceased to wonder how it was effected.
But the discovery being made, the rationale of the rules of signs
is simple enough. In pp. 22 24 I have strictly deduced the
rules for algebraic addition, subtraction, multiplication, and divi
sion, on the single principle of making these operations by the use
of the signs independent of the relative magnitudes of tJie quantities
represented by the letters. This principle is necessary and sufficient
for demonstrating the rules of signs in all cases. As far as I am
aware this demonstration had never been given before.
In p. 25 I have remarked that algebraic impossible quantities
necessarily arise out of algebraic negative quantities j the former
equally with the latter being indispensable for making algebra
an instrument of general reasoning on quantity. It would be
extremely illogical for any one to object to impossible quantities
in algebra without first objecting to negative quantities.
The rules of the arithmetic of indices are demonstrated in
pp. 25 27, on the principle that all modes of expressing quantity
with as near an approach to continuity of value as we please must
be included in a system of general arithmetic. It is then shewn
that an algebraic generalization of these rules gives rise to negative
and impossible indices, just as negative and impossible algebraic
Xll INTRODUCTION.
expressions resulted from the analogous generalization of the rules
of ordinary arithmetic.
In p. 28 I have proposed using the mark HI to signify that the
two sides of an equality are identical in value for all values whatever
of the literal symbols, the usual mark = being employed only in
cases of equality for particular values of an unknown quantity,
or particular forms of an unknown function. The former mark
contributes greatly towards distinctness in reasoning relating to
analytical principles, and I have accordingly used it systematically
in the subsequent part of the work.
The Calculus of Functions (p. 37) is regarded as a generaliza
tion of algebra analogous to the algebraic generalization of arith
metic. In the latter, theorems are obtained that are true for all
values of the literal symbols j in the other the theorems are equally
applicable to all forms of the functions. "Under the head of the
" Calculus of Functions of one Variable" I have given a proof of
Taylor's Theorem (p. 40), which is in fact a generalization of all
algebraic expansions of f(x + h) proceeding according to integral
powers of h, involving at the same time a general expression for
the remainder term. As the function and this expansion of it are
identical quantities, the sign IE is put between them. The co
efficients of k, h 2 , &c. in the expansion contain as factors the
derived functions f'(x), f" (#)> &c. It is important to remark
that the Calculus of Functions does not involve the consideration
of indefinitely small quantities, and that the derived functions
just mentioned are all obtainable by rules that may be established
on algebraic principles.
It is nevertheless true that by the consideration of indefinitely
small quantities the Differential Calculus is deducible from the
Calculus of Functions. The possibility of making this deduction
depends on that faculty of the human intellect by which, as already
remarked, it conceives of ratio as independent of the magnitudes
INTRODUCTION. Xlll
compared, which, the ratio remaining the same, may be as small
as we please, or as large as we please. This is Newton's founda
tion in Section i. of Book i. of a calculation which is virtually the
same as the differential calculus. Having fully treated of the
derivation of the differential calculus from the calculus of func
tions in pp. 47 49, I have occasion here to add only the fol
lowing remark.
In p. 47 I have shewn that the ratio of the excess of f(x + h)
above f(x Ji) to the excess of x + h above x h, that is, the
ratio of a finite increment of the function to the corresponding
finite increment of the variable, is equal tof (x)+f" (x) ^ +&c.,
in which there are no terms involving f"(x\ &c. Usually in
treatises on the Differential Calculus the expression for the same
ratio, in consequence of making x apply to a position at the begin
ning instead of at the middle of the increments, has/"" (x) h in the
second term. As far as regards the principles of the differential
calculus, the logic of the foregoing expression is much more exact
than that of the one generally given, because it shews that the
limit of the ratio of the increment of the function to that of the
variable is equal to the first derived function whatever be the value
of f" (x\ even if this second derived function and the succeeding
ones should be infinitely great. When the expression for that
ratio has a term containing f"(x)h y it is by no means evident that
that term vanishes on supposing h to be indefinitely small, if at
the same time the value of x makes f" (x) indefinitely great. For
this reason, in applications of the differential calculus to concrete
quantities, when an expression for a first derived function is to be
obtained by a consideration of indefinitely small increments, the
only logical course is to compare the increment f(x\}i) f(x A)
with 2h ; which, in fact, may always be done. This rule should
be attended to in finding the differentials of the area and the arc
XIV INTRODUCTION.
of any curve, and in all similar instances. It has been adopted in
the present work (as, I believe, had not been done in any other)
both in geometrical applications (pp. 83 and 89) and in dynamical
applications (pp. 110 and 112).
The differential calculus as applied to a function of two varia
bles is analogously derived (in pp. 51 55) from the calculus of
functions of two variables. In the course of making this deduc
tion I have expressed, for the sake of distinctness, the partial
differentials with respect to x and y of a function u of x and y by
the respective symbols du and d y u. This notation is particularly
applicable where every differential coefficient, whether partial or
complete, is regarded as the ultimate ratio of two indefinitely
small increments. I might have employed it with advantage in
my hydrodynamical researches ; but on the whole I have thought
it best to adopt the rules of notation stated in p. 174.
Under the head of "the principles of geometry," (p. 60),
I have discussed Euclid's definition of parallel straight lines and
its relation to Axiom xn. These points, as is well known, have
been very much litigated. I think I have correctly traced the
origin of all the difficulty to what I have already spoken of as the
nonrecognition in the Elements of Euclid of our perceptions of
equality, and equality of ratios, as the foundation of all quantita
tive reasoning. This foundation being admitted, there should be
no difficulty in accepting as the definition of parallel straight lines,
that "they are equally inclined, towards the same parts, to the
same straight line." (p. 62.) Equality is here predicated just as
when a right angle is defined by the equality of adjacent angles.
Euclid's definition, that parallel straight lines do not meet when
produced ever so far both ways, is objectionable for the reason
that it does not appeal to our perception of equality. Moreover,
if the proposed definition be adopted, the property of not meeting
is a logical sequence from Prop. xvi. of Book i. ; for, supposing
INTRODUCTION. XV
the lines to meet, a triangle would be formed, and the exterior
angle would be greater than the interior angle, which is contrary
to the definition. In p. 64 I have shewn that by means of the
same definition Axiom xn. may be proved as a proposition.
Another instance of a definition in Euclid being such as to
admit of being proved, is presented by Def. xi. of Book in., which
asserts that " similar segments of circles are those which contain
equal angles." This is in no sense a definition, because it is not
selfevident, nor does it appeal to our perception of proportion.
Def. i. of Book vi., inasmuch as it rests on equality of ratios is
strictly a definition of similarity of form, but applies only to recti
linear forms. By adopting (in p. 63) a definition which involves
only the perception of equality of ratios, and applies equally to
curvilinear and rectilinear figures, I have proved that " similar
segments of circles contain equal angles."
In p. 70 I maintain that the proportionalities asserted in Pro
positions i. and xxxin. of Book vi. are seen at once by an unaided
exercise of the reasoning faculty, and cannot be made more evident
by the complex reasoning founded on Def. v. of Book v. The use
made of that definition in proving the two Propositions is no
evidence that it is a necessary one.
The object of the discussion commencing in page 70 and
ending in page 88 is to shew that by the application of abstract
calculation all relations of space are deducible from geometrical
definitions, and from a few elementary Propositions the evidence
for which rests on an appeal to our primary conceptions of space.
This argument was, in fact, required for proving that the genera
lization announced in page viii is inclusive even of the relations of
pure space. In page 82 I have been careful to intimate that the
discussion was solely intended to elucidate the fundamental prin
ciples on which calculation is applied in geometry, and not to
inculcate a mode of teaching geometry different from that usually
XVI INTRODUCTION.
adopted. At the same time I have taken occasion to point out a
distinction, which appears not to have been generally recognised,
between geometrical reasoning, and analytical reasoning applied
to geometry. The former is reasoning respecting the relations of
lines, areas, and forms, necessarily conducted by means of diagrams,
on which account it is properly called " geometrical reasoning."
But it involves no measures of lines and angles, and in that respect
is essentially distinct from analytical reasoning, in which such
measures are indispensable. By many minds geometrical reason
ing is more readily apprehended than analytical, and on that
account it is better fitted than the latter to be a general instru
ment of education. Regarded, however, as a method of reasoning
on relations of space, it is incomplete, because it gives no means
of calculating such relations. The method of analytical geometry,
on the contrary, is not only capable, as I have endeavoured to
shew by the argument above referred to, of proving all geome
trical theorems, but also, by the intervention of the measures of
linos and angles, of calculating all geometrical relations. In short,
analytical geometry is the most perfect form of reasoning applied
to space*.
In page 90 I have employed the terms " Plane Astronomy"
as being in common use ; but I now think that " Spherical Astro
nomy" would have been more appropriate, inasmuch as applied
calculation in the department of Astronomy which those terms
designate consists mainly in finding relations between the arcs
* 1 quite assent to the propriety of that strict maiatenance of the distinction
"between geometrical reasoning and analytical geometry which is characteristic of
the Cambridge system of mathematical examinations ; but I am wholly unable to
see that this is a ground for the exclusion of analytical geometry to the extent
enjoined by the recently adopted scheme for the examinations. According to the
schedule the examiners have no opportunity, during the first three days of the
examination, of testing a candidate's knowledge of the application of algebra to
geometry, and it is consequently possible to obtain a mathematical honour without
knowing even the elementary equations of a straight line and a circle.
INTRODUCTION. XV11
and angles of spherical triangles. The arcs are such only as are
measured by astronomical instruments, either directly, or by the
intervention of time. The element of time makes a distinction
between the astronomical problems of this class and problems of
pure geometry. The purpose of the notes in pages 90" 96 on
the science of Time is to shew how measurements of the uniform
flow of time, and determinations of epochs, are effected by astro
nomical observation, and depend on the assumption of the uni
formity of the earth's rotation about its axis. In page 91 I say,
" there is no reason to doubt the fact that this rotation is per
fectly uniform." But in page 127 I have admitted the possibility
of a gradual retardation resulting from the moon's attraction of
the tidal waters. This inconsistency is attributable to the cir
cumstance that the reasons adduced in p. 127 for the latter view
became known in the interval from 1859 to 1867, during which
the printing of the work was suspended after it had proceeded to
p. 112.
The simple and satisfactory explanation of the Aberration of
Light given in pages 97 and 98 was first proposed by me in a
communication to the Phil. Mag. for January 1852, after attempts
made in 1845 and 1846 with only partial success. That Article
was followed by another in the Phil. Mag. for June 1855 referring
more especially to the effect of aberration on the apparent places
of planets. The explanation wholly turns on the facts that instru
mental direction is determined by the passage of the light from
an object through two points rigidly connected with the instru
ment, and that, by reason of the relative velocity of the earth and
light, the straight line joining the points is not coincident with
the direction in which the light travels. One of the points is
necessarily the optical centre of the objectglass of the Telescope.
Although this explanation has now been published a considerable
time, it has not yet found its way into the elementary Treatises
XV111 INTRODUCTION.
on Astronomy, which continue to give nothing more than vague
illustrations of the dependence of the phenomenon on the relative
motion of the earth and light. This being the case, I take the
opportunity to say, in order to draw attention to what is essential
in the explanation, that if the cause of the aberration of light
were set as a question in an examination, any answer which did
not make mention of the optical centre of the objectglass would
not deserve a single mark.
Under the head of the Principles of the Statics of rigid bodies
(pp. 98 104), I have shewn that Lagrange's beautiful proof of
the general equation of Virtual Velocities, after the correction at
one part of it of a logical fault (p. 102), rests (1) on the funda
mental property of a rigid body according to which the same
effect is produced by a given force in a given direction along a
straight line at whatever point of the line it be applied ; and (2)
on the definition of statical equilibrium. These are the funda
mental principles of Statics, whatever be the mode of treatment
of statical problems.
In stating the principles of Hydrostatics (p. 104), a fluid is
denned (1) by its property of pressing, and (2) by that of easy
separability of parts. The second of these definitions has been
adopted on account of its having important applications in Hy
drodynamics, as will be subsequently mentioned. The law of the
equality of pressure in all directions from a given fluid element is
rigidly deduced (in pages 105 107) from these two definitions.
In the statement of the principles of the Dynamics of solid
bodies in motion (pp. 109 119), I have adhered to the terms
which came into use at and after the Newtonian epoch of dyna
mical science, although I should be willing to admit that they
might in some respects be improved upon. But whatever terms
be adopted, all reasoning respecting velocity, accelerative force,
momentum, and moving force, is founded on certain elementary
INTRODUCTION. XIX
facts which have become known exclusively by observation and
experiment. These fundamental facts are the following : (1) in
uniform velocity equal spaces are described in equal times ; (2) a
constant force adds equal velocities in equal times j (3) the ve
locity added by a constant force in the direction in which it acts
is independent of the magnitude and direction of the acquired
velocity ; (4) the momentum is given if the product of the mass
and the velocity be given ; (5) the moving force is given if the
product of the mass and the accelerative force be given. It is
especially worthy of remark that although these facts were not
discoverable by any process of reasoning, it is possible by reason
ing to ascertain the function that the space is of the time in the
case of variable velocity, and the functions that the velocity and
space are of the time in the case of a variable accelerative force.
Since in these cases functions are to be found, it follows from the
principles of abstract calculation that we must for that purpose
obtain differential equations. The processes by which these are
deduced by the intervention of the facts (1), (2), and (3), are fully
detailed in pages 109 117. In this investigation Taylor's
Theorem has been used in the manner indicated in page xiii.
In the Notes on Physical Astronomy commencing in page
119, I have, in the first place, adverted to the essential distinction
which exists between the labours of Kepler and those of Newton
in this department of science. This distinction, which holds no
place in Comte's system of philosophy, is constantly maintained
in Whewell's History and Philosophy of the Inductive Sciences.
I select the following passage from the History (Vol. n. p. 181):
" Kepler's laws were merely formal rules, governing the celestial
motions according to the relations of space, time, and number;
Newton's was a causal law, referring these motions to mechanical
reasons. It is no doubt conceivable that future discoveries may
both extend and farther explain Newton's doctrines ; may make
XX INTRODUCTION.
gravitation a case of some wider law, and disclose something of
the mode in which it operates ; questions with which Newton
himself struggled." In accordance with these views I have noticed
that Kepler's observations and calculations do not involve the
consideration of force, and that the laws he discovered were really
only problems for solution. Newton solved these problems by
having found the means of calculating the effects of variable
forces. This was his greatest discovery. By calculations made
on the hypothesis that the force of gravity acts according to the
law of the inverse square, Newton gave dynamical reasons for
Kepler's laws, which may also be called causative reasons, inas
much as whatever causes is force, or power, as we know from
personal experience and consciousness. The principle which is
thus applied to physical astronomy I have extended in a subse
quent part of this work to all quantitative laws whatever. I
have maintained that all such laws, as discovered by observation
and experiment, are so many propositions, which admit of a priori
demonstration by calculations of the effects of force, founded on
appropriate hypotheses. This, in short, is Theory.
In making the remarks contained in pages 120 124 I was
under the impression that the first evidence obtained by Newton
for the law of gravity was derived from comparing the deflection
of the moon from a tangent to the orbit in a given time with the
descent of a falling body at the earth's surface at the same time,
and that he did not have recourse to Kepler's laws for that pur
pose. This, at least, might have been the course taken. But on
consulting Whewell's History of the Inductive Sciences, I find that
the inference of the law of gravity from the sesquiplicate ratio of
the periodic times to the mean distances, as given in Cor. 6 of
Prop, iv., Lib. i., and the converse inference of the sesquiplicate
ratio from the law, preceded historically those computations re
lative to the law of action of the Earth's gravity on the moon,
INTRODUCTION. xxi
which Newton finally made after obtaining a corrected value of
the earth's radius.
A discussion of considerable length (contained in pages 128
152) is devoted to the determination of the physical significance
of the occurrence, in the developments of radius vector and lati
tude, of terms which increase indefinitely with the time. The
consideration of this peculiarity of the Problem of Three Bodies
falls especially within the scope of the present work, inasmuch as
it is a question to be settled only by pure reasoning, and points
of principle are involved in the application of the reasoning. As
this question had not received the attention it deserves, and as
I could be certain that the clearing up of the obscurity surround
ing it demanded nothing but reasoning from the given conditions
of the problem, and would, if effected, be an important addition to
physical astronomy, I felt strongly impelled to make the attempt,
although my researches had previously been much more directed
to the applications of partial differential equations than to those
of differential equations between two variables. My first attempts
were far from being successful, and it was not till after repeated
and varied efforts that I at length ascertained the origin and
meaning of the terms of indefinite increase. As the decision of
this point is necessary for completing the solution of the Problem
of Three Bodies, I thought it might be regarded of sufficient in
terest to justify giving some historical details respecting the steps
by which it was arrived at.
My attention was first drawn to this question by a paragraph
in Mr Airy's Lunar Theory (Mathematical Tracts, art. 44*, p. 32,
3rd Ed.), where it is asserted that the form of the assumption for
the reciprocal of the radiusvector, viz. u = a{\+e cos (cO a)}, " is
in no degree left to our choice." It is then shewn how that form
may be obtained by assuming for u the general value a (1 + w) ;
but the principle on which this assumption is made is not ex
xxii INTRODUCTION.
plained. My first researches were directed towards finding out a
method of integrating the equations by which the above form of
u and the value of the factor c should be evolved by the usual
rules of integration without making any previous assumption.
Having, as I supposed, discovered such a method, I offered to the
Cambridge Philosophical Society a communication entitled " Proofs
of two new Theorems relating to the Moon's orbit," respecting
which an unfavourable report was made to the Council, and not
without reason; for it was a premature production, and had in it
much that was insufficiently developed, or entirely erroneous.
The paper, however, contained the important differential equation
at the bottom of page 145 of this Volume, arrived at, it is true,
by imperfect reasoning, and also the deduction from it of the
,. 2 , Ch* m* , . , . . , ,
equation e = 1 T + ~<r5 which is . equivalent to one near the
M *
top of page 147. This last equation, for reasons I shall presently
mention, forms an essential part of the solution of the Problem
of Three Bodies.
In this first essay I obtained the above mentioned differential
equation without distinctly shewing that it involves the hypo
thesis of a mean orbit. This defect is supplied by the argument
contained in pages 142 145, where the equation is arrived at by
suppressing terms containing explicitly the longitude of the dis
turbing body, which process is equivalent to making that hypo
thesis. Also, as is proved in pages 146 and 147, the integral of
the equation completely answers the purpose of obtaining the
proper form of u, and an approximation to the value of c, without
any previous assumption relative to that form. But it is import
ant to remark that the deduction of these results wholly depends
on the antecedent hypothesis of a mean orbit, which hypothesis is,
in fact, involved in every process made use of for getting rid of
terms of indefinite increase. This is the case in the method
INTRODUCTION. XX111
adopted in Pratt's Meclianical Philosophy (Art. 334), where the
substitution of u b for be cos (6  a) seems like employing for the
purpose a species of legerdemain, until it be understood that by
this step the hypothesis of a mean orbit is first introduced. The
same remark applies to the method already referred to as having
been proposed by Mr Airy, which in principle is the same as that
just mentioned.
It also appears that the differential equation in p. 145 is the
same that would be obtained on the supposition that the body is
acted upon by the force  2  ^^ tending to a fixed centre, and
consequently, as in Newton's Section ix., the radiusvector is
equal to that of an ellipse described by the action of a force
tending to the focus and varying inversely as the square of the
distance, the ellipse revolving at the same time uniformly about
its focus. This was my Theorem i., to which, after the explana
tion that it applies only to the Moon's mean orbit, there is nothing
to object. (See the discussion of this case in pages 149 151.)
But Theorem 11. was wholly erroneous, being deduced from
the foregoing value of e 2 by arguing on grounds which cannot be
m*
sustained that ju, 8 = (7/t 2 , and consequently that e 2 = = , m being
the ratio of the Moon's periodic time to the Earth's. I ought to
have inferred from that value, as is shewn by the course of rea
ra 2
soning concluded in page 148, that ^ is an inferior limit to the
square of the eccentricity of the mean orbit.
Having published the two Theorems in the Philosophical
Magazine for April 1854, in the June Number of that year I
invited Professor Adams, who was one of the reporters on my
paper, to discuss with me its merits. Accordingly, in a letter
addressed to the Editors of that Journal, contained in the July
Number, he gave in detail the reasons of his disapproval of the
xxiv INTRODUCTION.
new theorems. These reasons, I now willingly admit, proved that
I had no right to conclude from my arguments that Ch z = /A 2 , and
hence that the eccentricity of the Moon's orbit is j= . It was also
V 2
justly urged that the same theorem, applied to the orbit of Titan,
was contradicted by the actual eccentricity. Some of the objec
tions, which depended on my not carrying the approximations far
enough, are met by the more complete investigation contained in
this Volume. Professor Adams took no notice of the equation
e 2 = 1 5 + jr , which, as I thought, should have saved my
/A J
views from unqualified condemnation.
In my reply in the August Number I said much in the heat
of controversy that had better not have been said, and some things
also that were untrue. Still 1 claim the merit of having seen that
the question respecting the meaning of terms of indefinite in
crease was of so much importance, that till it was settled the
gravitational theory of the motions of the heavenly bodies was
incomplete. Professor Adams was precluded from adopting this
view by having taken up exclusively the position, that the solution
of the differential equations obtained by introducing the factors
usually called c and g is " the true and the only true solution,"
because it contains the proper number of arbitrary constants and
satisfies the equations. On the other hand I argued, but not as
logically as I might have done, that under those circumstances
" the constants a, e, e, w are not necessarily [i. e. absolutely] arbi
trary and independent of each other," inasmuch as the solution is
limited by the introduction of the factors c and g, and is therefore
not the complete, or most general, solution of the given equations.
The same argument, put in the form which longer consideration
of the subject has led me to adopt, may be stated as follows.
What is done by the insertion of the factors c and g is to im
pose arbitrarily the condition that there shall be a mean orbit,
INTRODUCTION. XXV
that is, an orbit in which the longitude shall oscillate about^that in
a revolving ellipse, and the latitude about that in a fixed plane.
That this is the case is demonstrated by conversely deducing the
values of c and g, as is done in pages 147 and 152, from equations
not containing explicitly the longitude of the disturbing body,
and, therefore, by arguing on the hypothesis of a mean orbit.
The fact that the differential equations can be satisfied after intro
ducing these factors, is the proper proof that a mean orbit is
possible. The analytical circumstance that before the introduction
of the factors the integration leads to terms of indefinite increase
proves that there is not always, or necessarily, a mean orbit. If
the differential equations could be exactly integrated, the integrals
would contain the same number of arbitrary constants as the
limited solution under discussion, but they would embrace non
periodic motion as well as periodic, and the constants introduced
by the integration would be absolutely arbitrary. Hence the
constants of the limited solution, although the same in number,
cannot be in the same degree arbitrary, but must have been sub
jected to limitations by the process which limited the solution. On
this point I have given the following direct evidence.
The equation (A) in page 139 is a first integral of the given
differential equations, obtained by usual processes, and its right
hand side contains exclusively the terms involving the disturbing
force. If in this side the elliptic values of a first approximation
be substituted for r and 0, all its terms will contain e as a factor.
The case would be the same if the approximate values of r and
belonging to a revolving ellipse were substituted. Hence it ap
SVJ2
pears (p. 139), since e* was put for 1 3, that if e=Q, the
equation (A) becomes
XXVI INTRODUCTION.
that is, the equation of a circle of radius ^,. Now this orbit
could not possibly be described so long as there is a disturbing
force, and we are thus driven to the conclusion that if there be
a disturbing force, e cannot be zero, but must have a certain
limiting value. Thus it is shewn that in this solution one of the
arbitrary constants is subject to limitation. I believe I may say
it was after discovering that e and the disturbing force vanish
together, that I fell, not unnaturally, into the error of supposing
that e 2 must contain the disturbing force as a factor. By sub
sequent research I found that this inference is not necessary.
The conclusion that e has a limiting value ought, I think, to
arrest the attention of mathematicians engaged on the theory of
gravitation. It had not been previously arrived at, because the
differential equation (A) of the first order, which is intermediate
to the given differential equations of the second order and their
complete integrals, has been singularly overlooked by investigators
in the theory of gravity. The determination of the limiting value
will be presently adverted to.
The foregoing narrative will have sufficiently indicated the
grounds of the divergence of my views from those of Professor
Adams. When I found that our discussion had not settled the
question as to the signification of terms of indefinite increase,
I pursued the enquiry in a series of communications contained in
the Numbers of the Philosophical Magazine for December 1854,
and February, March, and May 1855, which will, at least, attest
the diligence with which I laboured to get at the truth of this
question. These investigations, which referred mainly to the
Moon's orbit> were followed by a more elaborate paper on the
Problem of Three Bodies, read before the Royal Society on
May 22, 1856, and printed in thsir Transactions (1856, p. 525).
This treatment of the problem applies more especially to the
Planetary Theory.
INTRODUCTION. XXvii
The method of solution I adopted relative to the Moon's orbit
is characterized by successive approximations both to the mean
orbit and the actual orbit, proceeding pari passu. The former
approximations are made on the principle of omitting terms con
taining explicitly the longitude of the disturbing body, which is
the same as the principle of omitting in the Planetary Theory
periodic variations of short period in the investigation of secular
variations. The solution of the problem of three bodies in the
Philosophical Transactions is a direct determination of the actual
orbit only, peculiar in the respect that by making use of the
equation (A) the approximations are evolved without any initial
supposition as to the form of solution. The expressions for the
radiusvector and longitude are the same as those obtained by
Laplace. I may as well state here that I had no intention in my
researches in physical astronomy to furnish formulae for the cal
culation of Tables. I have perfect confidence in the principles on
which those that have been used for this purpose have been
investigated. My concern was solely with the logical deduction
of consequences from the analysis which, although they do not
affect the calculation of Tables, are important as regards the general
theory of gravitation.
In pages 128 152, I have collected from the abovementioned
papers, all the arguments which, after mature consideration, I
judged to be valid, (1) for explaining the nature of terms of
indefinite increase ; (2) for determining the limiting value of the
constant e.
On the first point, I have come to the conclusion that by
terms admitting of indefinite increase, the analysis indicates that
in the general problem of three bodies, the motion is not necessa
rily periodic, or stable, and that the motion of a particular planet,
or satellite, is proved to be stable by finding, after calculating on
the hypothesis of a mean orbit, that the resulting solution is
xxviii INTRODUCTION.
expressible in a series of convergent terms. This conclusion is,
however, more especially applicable to the Minor Planets, because
they are not embraced by the known general theorems which
prove that the stability of the motions of the larger planets is
secured by the smallness of the eccentricities and the inclinations.
With respect to the other point, by the approximations to the
actual orbit and to the mean orbit, and by determining (p. 147)
certain relations between their arbitrary constants, I have been
m*
finally conducted to the equation e s = e o 2 +  at the top of p. 148,
which, however, was obtained on the supposition that both e and
m are small quantities. Since e* is an arbitrary constant necessa
rily positive, this equation shews that if e 2 = 0, we have e 2 = and
m* = 0, the last result agreeing with that mentioned in p. xxvi.
The equation proves also that e 2 may have different arbitrary
THj
values, but all greater than the limiting value ^ .
In page 141 I have obtained the value of e lt the eccentricity
of the mean orbit, which is, in fact, what is called the mean
eccentricity, being independent of all particular values of the
longitude of the disturbing body. It is shewn also that e*=e*, if e
and m be small. Hence it may be inferred from the foregoing
limit to the value of e 2 , that p is an inferior limit to tJie mean
v ^
eccentricity. This theorem, which may, I think, be regarded as
an interesting addition to the theory of gravitation, has been
arrived at by patiently investigating the meaning of an unex
plained peculiarity of the analysis, in perfect confidence that an
explanation was possible, and could not fail to add something to
our theoretical knowledge. It should, however, be noticed that
the theorem is true only for the problem of three bodies. I have
not attempted to extend the reasoning to the case of the mutual
attractions of a greater number.
INTRODUCTION. XXIX
In the Notes on the Dynamics of the Motion of a Rigid
System (pp. 153 170), there are three points to which I think it
worth while to direct attention here.
(1) In page 153 I have endeavoured to state D'Alembert's
Principle in such manner that its truth may rest on a simple appeal
to our conception of an equality. It has already been remarked
(p. xiv.) that a principle or definition which satisfies this condition
is proper for being made the basis of quantitative reasoning.
(2) After deducing (in p. 154) the general equation of Yis
Viva by means of D'Alembert's Principle and the Principle of
Virtual Velocities, I have remarked (in p. 156) that there is
impropriety in speaking of the principle of the conservation of Vis
Viva, as expressed by that equation. For since the equation is a
general formula obtained by analytical reasoning from those two
principles, it is properly the expression of a law, it being the
special office of analysis to deduce laws from principles and
definitions. The distinction will not appear unimportant when
it is considered that the law of Vis Viva has been relied upon by
some mathematicians as if it were a principle of necessary and
universal application, whereas the applicability of a law is deter
mined and limited by the principles from which it is derived. To
speak of the principle of Virtual Velocities is not in the same
manner incorrect, because, for the reasons stated in page 102, the
general equation of Virtual Velocities rests only on the funda
mental principles of Statics, and may be regarded as the expression
of a single principle substituted for them.
(3) All problems in the Dynamics of Kigid Bodies admit of
being solved by means of the six equations given in page 157.
When the known values of the impressed moving forces for a
particular instance have been introduced into these equations, the
solution of the problem is a mere matter of reasoning conducted
according to the rules of abstract analysis. All circumstances
XXX INTRODUCTION.
whatever of the motion are necessarily embraced by this reasoning.
I have been induced to make these remarks because it is usual to
solve problems of this class by the initial consideration of angular
motions about rectangular axes. This method is, no doubt, correct
in principle, and is generally more convenient and elegant than
that of directly integrating the differential equations. But it
should be borne in mind that the latter method is comprehensive
of every other, and that all the equations involving angular mo
tions about rectangular axes are deducible from the integrations.
To illustrate these points I have attacked the problem of the
motion of a slender hoop (pp. 157 166), by first adapting the six
general equations to the particular instance, and then integrating
for the case in which the hoop has a uniform angular motion
about an axis through its centre perpendicular to its plane. I
have emphasized at the top of page 164 the inference that "when
a hoop rolls uniformly on a horizontal plane, it maintains a con
stant inclination to the plane and describes a circle," in con
sequence of having noticed that in the usual mode of solving the
problem, this inference, being regarded as selfevident, has not
been deduced by reasoning. Nothing that can be proved ought
to be taken for granted.
The mathematical theory of Foucault's Pendulum Experiment
(pp. 166 170) is prefaced by a remark which may serve to ex
plain why this problem had not been mathematically solved before
attention was drawn to it by experiment. By reason of the
earth's rotation about its axis, there is relatively to any given
position an equal motion of rotation of all points rigidly connected
with the earth about a parallel axis passing through that position.
This circumstance ought in strictness to be taken into account,
when it is required to refer motions, such as oscillations due to the
action of gravity, to directions fixed with respect to the earth.
This, it seems, no mathematician had thought of doing.
INTRODUCTION. XXXI
In the subjects that have hitherto been mentioned, I have
succeeded, I think, in shewing that in some few particulars they
admitted of additions to, or improvements upon, the processes of
reasoning that had been applied to them by my predecessors and
contemporaries in mathematical science. But in the subject of
Hydrodynamics, (which occupies the large portion of this work
extending from page 170 to page 316), I found the reasoning to
be altogether in a very unsatisfactory state. After accepting the
fundamental definitions on which the propositions of Hydro
dynamics are usually made to rest, I discovered that methods of
reasoning had been employed which were, for the most part, either
faulty or defective. The following statement relates to an in
stance of the prevalence of a faulty method of reasoning.
My first contribution to the science of Hydrodynamics was a
paper " On the theory of the small vibratory motions of elastic
fluids," read before the Cambridge Philosophical Society on March
30, 1829, and printed in Yol. in. of the Transactions. That
paper contains (in p. 276) the first instance, I believe, of the
determination of rate of propagation by differentiation, the prin
ciple of which method is insisted upon in pages 189 and 190 of
the present work. At the very commencement of my scientific
efforts I was unable to assent to Lagrange's method of deter
mining rate of propagation, although it appears to have been
accepted without hesitation by eminent mathematicians, and con
tinues to this day to hold a place in elementary treatises. I per
ceive, however, that Mr Airy in art. 24 of his recently published
work On Sound and Atmospheric Vibrations, has employed a
method equivalent to that of differentiation, and I have reason
to say that other mathematicians have now discarded Lagrange's
method. But no one except myself seems to have discerned that
as that method determines by arbitrary conditions a quantity that
is not arbitrary, it involves a violation of prfacipk. This, from
XXXH INTRODUCTION.
my point of view, is a very important consideration ; because if
principle has been violated in so simple a matter, what security is
there that the same thing has not been done in the more advanced
and more difficult parts of the subject 1 My researches have led
me to conclude that this has actually taken place.
The evidence on which I assert that reasoning has been usually
employed in Hydrodynamics which is defective in principle, and
requires to be supplemented, is in part given by the solution of
Example i., beginning in page 193. Without any departure from
the ordinary mode of reasoning the conclusion is there arrived at
that the same portion of the fluid may be at rest and in motion at
the same instant. "When I first published this reductio ad absur
dum, Professor Stokes attempted to meet it, (as I have mentioned
in page 196), by saying that the analysis indicated something like
a breaker or bore,' forgetting, so it seems to me, that as breakers
and bores are possible natural phenomena due to special circum
stances, they cannot be included in an investigation which takes
no account of those circumstances, which, besides, is found to
lead to an impossibility, or to what is per se a contradiction. I
have adverted also (p. 196) to similar views advanced by Mr Airy
in a communication which by his own admission " does not con
sist of strict mathematical reasoning, but of analogies and conjec
tures." It will suffice for pointing out the character of these
surmises to refer to the passage in the communication (p. 404) in
which Mr Airy speaks of " the probable sensational indications "
of the physical phenomenon "interruption of continuity of par
ticles of air," such as a hiss, a buzz, &c. Admitting the possible
applicability of these conceptions under circumstances which were
not taken into account in the antecedent investigation of the
differential equation, I deny altogether that the analysis in the
present case indicates any interruption of continuity of the par
ticles, inasmuch as, according to its strict meaning, after the
INTRODUCTION. Xxxiii
abovementioned contradiction is consummated, the motion goes
on just as smoothly as before ; which is only another phase of the
absurdity. Since, therefore, strict mathematical reasoning, which
neither of these two mathematicians has controverted, has shewn
that the differential equation on which their views are founded
leads to a reductio ad absurdum, it follows by necessary logical
sequence that the equation is a false one, and that analogies and
conjectures relating to it are misapplied.
That same equation is discussed by Mr Earnshaw in a paper
On the Mathematical Theory of Sound, contained in the Philoso
phical Transactions for 1860, p. 133. At the time of the pub
lication of his paper the author was well aware of the argument
by which I had concluded that the equation is an impossible one.
In the course of the discussion there occurs (p. 137) the singular
assertion that a wave, after assuming the form of a bore, "will
force its way in violation of our equations." Now the only in
terpretation that can possibly be given to this sentence is, that
Mr Earnshaw conceives he is justified in supplying by his imagi
nation what the equations fail to indicate, whereas it is unques
tionable that we can know nothing about what the wave does
except by direct indications of the equations. For the foregoing
reasons I think I may say that Mr Earnshaw has applied a false
method of reasoning to a false equation. It is not surprising that
his views are approved of by Mr Airy (Treatise on Sound, p. 48)
and by Professor Stokes (Phil. Trans, for 1868, p. 448), since they
are the same in principle as those which had been previously
advocated by themselves. But Mr Earnshaw in the sentence
above quoted has divulged the mental process by which the ex
istence of a bore, &c. is inferred, and has shewn that it involves
an exercise of the imaginative faculty *.
In a Lecture on " The Position and Prospects of Physical Science " delivered
by Professor Tait of Edinburgh, on November 7, 1860, mention is made of the
XXXIV INTRODUCTION.
The contradiction above discussed is not the only one that
results from reasoning founded upon the principles of Hydrody
namics as usually accepted. The solution of Example n. in page
197 leads to another contradiction. Perhaps the evidence in this
instance may be made more distinct by remarking, that in the
integral <rr=f(r at) in page 198, the arbitrary function may be
discontinuous in such manner that the values of tr in two spaces
separated by any indefinitely thin partition transverse to the direc
tion of the motion may be expressed by different algebraic func
tions, if only the condensations immediately contiguous to the
partition be equal on the opposite sides. This is a legitimate
consequence of the fundamental property of easy separability of
parts. It hence follows that the uniform propagation of a solitary
wave either of condensation or rarefaction from a centre is pos
sible. But in that case the condensation necessarily varies in
versely as the square of the distance r, whereas the formula shews
that it varies simply as the inverse of that distance. This is so
direct a contradiction that the principles on which the reasoning
was based must be pronounced to be either false or insufficient.
On the publication of Mr Airy's Treatise on Sound and At
mospheric Vibrations (in 1868), I naturally looked for some
attempt to meet the two foregoing arguments, which I had
urged as long since as 1849 in the Numbers of the Phil. Mag. for
February and April of that year. I found that by giving (in art.
32) only an approximate solution of an equation which is equi
valent to that in page 194 of this work, of which Poisson's exact
" meagre development " of Hydrodynamics, and the whole subject is represented
as having been "till lately in a very backward state." Two "very great im
provements " are then said to have been very recently made. On one of these I
shall have to speak afterwards ; the other was considered to be effected by Mr
Earnshaw's paper on Sound. The author of the Lecture had evidently not com
prehended the arguments by which 1 had proved that the equation on which Mr
Earnshaw relied was vitiated by defect of principle, nor the reasoning by which
already in 1851 and 1852 I had succeeded in supplying what was wanting in the
received principles of Hydrodynamics.
INTRODUCTION. XXXV
integral is well known, Mr Airy lias avoided the direct consi
deration of the first argument. He refers, however, in art. 34 to
a "conjectural" change of the character of the wave as "suggest
ed by Mr Earnshaw." On this point I have already said enough.
The exact integral logically treated leaves no room for conjecture.
With respect to the second argument, I have to remark that
Mr Airy has admitted (in Art. 31) the possibility of the trans
mission of a solitary wave ; but in the discussion (in Art. 50) of
the symmetrical divergent wave in air of three dimensions, no
notice is taken of the law of the variation of the condensation
inversely as the distance from the centre (the condensation itself
not being once mentioned), although the law is readily deducible
from the solution. Accordingly no attempt is made to shew how
the uniform transmission of a solitary condensed or rarefied wave,
as resulting from the principle of the discontinuity of the arbi
trary function, can be reconciled with the existence of that law ;
so that the consideration of the second argument is omitted. Thus
a reader of Mr Airy's book might be led to suppose that the
received principles of Hydrodynamics are not liable to the ob
jections which I have urged, which, to say the least, are not such
as can be overlooked. To keep difficulties out of sight is not
likely to advance philosophy.
It may surprise the nonmathematical world to find that on
a question the premises of which are not in dispute, mathema
ticians are not agreed as to the process of reasoning. This, in
fact, ought not to be ; for in such a case it is certain that some
rule of logic has been violated either on one side, or on both.
In the present instance the fault does not lie with me. By strict
reasoning from the received principles of Hydrodynamics I have
been led to contradictions, and have consequently concluded, ac
cording to an undoubted rule of logic, that the principles require
reforming. My opponents, without contesting either the reason
XXXVI INTRODUCTION.
ing or the contradictions, will not accept the logical consequence.
Rather than do this they have recourse to conjectures and to
imagined analogies and probabilities. This sort of unreasonable
ness is no new thing in the annals of science. There have been
epochs before in which argument has availed nothing against
prejudgment founded upon error of long standing.
I will not do the mathematical contemporaries whose names
I have had occasion to mention the injustice to think that they
will impute to me any other motive in making the foregoing
remarks than an honest contention for scientific truth. There
are special reasons for insisting on the arguments by which I have
inferred the insufficiency of the received principles of Hydro
dynamics. The whole future of theoretical philosophy, as this
work, I think, will shew, turns upon this point. As the great
discovery of the Newtonian epoch of theoretical science was the
method of calculating the motion of a single particle under the
action of given forces, so the existing state of theoretical physics
demands a knowledge of the method of calculating the motion
and pressure, under given conditions of force, of a congeries of
particles in juxtaposition. It was in researches for this object
that I encountered the failure of the usual principles of Hydro
dynamics, and subsequently discovered what was required for
making them good. Respecting this last question, on account
of its importance, a few subsidiary remarks may be appropriately
made h.ere.
Having no reason to suppose that the commonly received prin
ciples of Hydrodynamics were not true, I concluded, by a rule of
logic, that the aforesaid arguments only proved them to be insuf
ficient, and I took the course of endeavouring to supply what was
wanting. By slow degrees I arrived at the conviction that a new
equation of geometrical continuity was required. The investiga
tion of such an equation in pages 174 arid 175 is founded on
INTRODUCTION. XXXvii
Axiom II. in page 174, which is a perfectly intelligible expression
of a principle of geometrical continuity necessarily applicable to
the motion of a fluid mass, if only such motion admit of being
analytically calculated. On this account the axiom must be
granted. The equation (1) in page 175 is investigated so as to
secure that the above principle, viz. that the directions of motion
in a given element are normals to a continuous surface, shall
apply to all the elements of the fluid at all times, and is, therefore,
a veritable equation of continuity. This name has been improperly
given to the equation ,(2) in page 177, which only secures that the
quantity of matter shall be always the same. It should be noticed
that the continuity here considered is purely geometrical, and,
therefore, distinct from that mentioned in p. 181, according to
which the direction of the motion of a given particle is determined
by dynamical conditions to be so far continuous that it cannot
change per saltum.
The new general equation (1) introduces two variables \j/ and A,
for determining which in addition to the other variables there are
a sufficient number of equations, as is shewn in p. 179. Also
since A. (dty) = udx + vdy + ivdz, the same equation enables us to
reason generally, without being restricted to the cases in which
the righthand side of that equality is an exact differential, which
are usually treated by the intervention of the function <, or, as
it is called in Mr Airy's work, the characteristic function F. To
make the logic of analytical Hydrodynamics good, it is absolutely
necessary to be able to argue independently of that restriction ;
which the new equation gives the means of doing.
By so arguing I have, first, shewn (Prop. VIL, p. 186) that the
abstract supposition of the integrability of udx + vdy + wdz con
ducts to rectilinear motion, and then inferred from the reasoning
in pages 193200, and from the solution of Prop. XL (p. 201),
that the straight line along which the motion takes place is an
XXXV111 INTRODUCTION.
axis relative to the condensation, and to transverse motion, and
that both the direct and the transverse motions are vibratory.
As these results are obtained antecedently to any supposed case
of disturbance of the fluid, it is concluded, on principles carefully
stated in pages 180 and 188, that they indicate, with respect to
the mutual action of its parts, certain laws depending only on the
relations of its properties to time and space, and, therefore, not
arbitrary, which laws may yet coexist with the results of arbi
trarily imposed disturbances, in a manner analogous to the co
existence of the particular solution of a differential equation with
the complete integral (see p. 200). These principles are similar
to that which is adverted to (p. xxxi.) in connection with the
determination of rate of propagation. Considering how long
mathematicians persisted in determining rate of propagation by
Lagrange's method, notwithstanding its violation of principle, I
have, perhaps, little reason to expect that the abovementioned
arguments will be readily apprehended.
By the reasoning under Prop. xi. it is shewn that the rate of
propagation along the axis of the motion is the same quantity KO>
for every point of any given wave (p. 206), and consequently that
the waves undergo no alteration by propagation. This with regard
to future applications is a very important result. The analytical
expression for K, terms of the second order being neglected, is
( 1 H a) ) as found in p. 206, which, since e is always positive,
is greater than unity. Thus the rate of propagation, as deduced
exclusively on hydrodynamical principles, is greater than the con
stant a. Also this rate is independent of the maximum conden
sation of the waves ; but without determining the value of e ^ ,
7T
there is no reason to assert that it is independent of X their
breadth. It is necessary to find that value in order to calculate
theoretically the velocity of sound.
INTRODUCTION. XXXIX
For a long time I thought I had succeeded in solving this
question in a communication to the Phil. Mag. for February, 1853,
having relied too much on an accidental numerical coincidence.
But eventually I became convinced, by the expression in p. 289
which had been obtained by Sir W. R. Hamilton and Professor
Stokes for the values of f corresponding to large values of r in the
series (20), p. 210, that I had used erroneous values of that func
tion. (See Camb. Phil. Trans. Vol. ix., p. 182.) I then made
another attempt, in the Phil. Mag. for May, 1865, employing this
time the values of/ given by the abovementioned expression.
The value of K which resulted is the same as that obtained in p. 224
of this volume by the argument commencing in p. 216, which,
however, makes no use of that expression, the values of r belonging
only to points immediately contiguous to the axis of the motion.
This last is the best solution I have been able to give of a very
difficult problem, of which, possibly, a simpler or a truer one may
still be discoverable. The velocity of sound deduced from it
exceeds the experimental value by 17, 5 feet. (See the note in
p. 317.) Perhaps the difference may be owing to the hypothesis
of perfect fluidity, which cannot be supposed to be exactly satisfied
by the a,tmosphere, especially near the earth's surface.
It is unnecessary to add anything here to the reasons I have
adduced in pages 225 and 317 of this work, and elsewhere, for
concluding that the velocity of sound is not increased by the
developements of heat and cold accompanying the condensations
and rarefactions of a wave. I may, however, state that my diffi
culty in apprehending Laplace's theory was long anterior to the
investigations which led me to the inference that the excess of
the velocity above the value a might be accounted for hydrodynami
cally. The same kind of difficulty must, I think, have induced
Poisson to abandon Laplace's a priori views, and to substitute
for them the bare hypothesis, that the increments of temperature
xl INTRODUCTION.
by the developement of heat are at all points of a wave instanta
neously and exactly proportional to the increments of density.
The advocates of the usual theory are bound to shew in what
manner this entirely gratuitous hypothesis can be connected with
experiments made on air in closed spaces.
The two examples, the solutions of which on the received
principles of Hydrodynamics led, as before stated, to contradic
tions, are solved in pages 243 254 in accordance with the
reformed principles. No contradictions are met with in this
method, which conducts to the important results, (1) that plane
waves, or waves limited by a prismatic tube, whether they are
large or small, are transmitted to any distance without alteration,
either as to condensation, or velocity, or rate of propagation ;
(2) that a solitary condensed or rarefied wave can be propagated
uniformly from a centre, the condensation and velocity varying
inversely as the square of the distance from the centre. In both
cases the discontinuity of the condensation, and by consequence
that of the motion, is considered to be determined and limited by
the fundamental property of easy separability of parts, as explained
in page 248. It results, farther, from the new principles that the
limited method of treating hydrodynamical problems employed by
Laplace, and since extensively followed, is defective in principle.
There can, I think, be no doubt that the method of commencing
the reasoning by obtaining general equations on general principles,
as adopted by Euler, Lagrange, and Poisson, is logically exact,
and in other respects far preferable *.
* The question has been recently raised as to whether a fluid which when at
rest presses proportionally to its density, retains this property when in motion.
That it does so is simply an intelligible hypothesis, the truth of which can neither
be proved nor disproved by a priori reasoning. Already a presumption has been
established that the hypothesis is true, at least quam proxime, by comparison of
results deduced from it mathematically with facts of experience; such results, for
instance, as those relating to vibratory motions. Utterly absurd results obtained
from such an hypothesis do not prove that the hypothesis is untrue, but that some
fault has been committed in the reasoning.
INTRODUCTION. x
The solution of Example iv. in pages 200 272 consists of
a lengthened discussion of the problem of the motion of a ball
pendulum and the surrounding air, embracing both the applica
tion of the appropriate analysis, and a comparison of the results
with experimental facts. In page 272 I have stated that in my
first attempts to solve this problem, I erroneously supposed that
the prolongations of the radii of the vibrating sphere were lines
of motion of the fluid. Here again I relied too much on a numeri
cal coincidence, viz. that of the result obtained on this hypothesis
with Bessel's experimental correction of the coefficient of buoy
ancy. Subsequently I was confirmed in the error by a misappli
cation of the general law of rectilinear ity, which, as stated in page
xxxvii, I deduced from the new general equation, and which I
supposed to be applicable to the motion impressed by the moving
sphere. These views are corrected in the present volume in pages
256 259 (see particularly the note in page 259), and the differ
ential equations obtained for solving the problem, viz. the equa
tions (33) and (34) in page 258, are identical with those employed
in Poisson's solution, with the exception of having K 2 a 2 in the
place of a 2 . This difference has arisen from the circumstance that
all the antecedent reasoning takes account of the indications of the
general equation (1), which was clearly the only correct course of
investigation, the truth of that equation being supposed to be
admitted. On the ground of this admission I am entitled to say
that my solution is more exact, and rests on truer principles, than
any that had been given previously.
After effecting the above solution I have inferred (in p. 264),
what I believe had not been noticed by other mathematicians, that
a vibrating sphere causes no actual transfer of fluid in the direc
tion of its impulses, just as much flowing backwards at each in
stant as it urges forwards. (I convinced myself of the reality of
a backward flow by the experiments mentioned in page 272). Con
xlii INTRODUCTION.
versely it is shewn by the solution of Example vi. (p. 279), that
when planewaves are incident on a smooth sphere at rest, as much
fluid passes at each instant a transverse plane through the centre
of the sphere as would have passed a plane in the same position if
the sphere had been away (p. 284). These results, which I arrived
at only after extricating myself from misconception and error, are
applied in a very important manner in the part of the work de
voted to physical theories. It seems to be not uncommonly the
case, that those who possess the power of carrying on independent
research, and trouble themselves with exercising it, fall into error
before they succeed in advancing truth.
In pages 267 271 I have entered into experimental details
with the view of accounting for the difference between Bessel's
correction of the coefficient of the buoyancy of a vibrating sphere,
which is very nearly 2, and the theoretical value, which is 1,5.
The result of the enquiry is, that the difference is mainly to be
attributed to the effect of the resistance of the air to the motion
of the wire or rod by which the ball was suspended. The solution
of Example v. (p. 273), a problem which, as far as I know, had
not been before discussed, gives the means of calculating the
resistance of the air to the vibrations of a slender cylindrical
rod.
The object of the solution of Example vi. (p. 279) is to calcu
late the distribution of condensation about the surface of a smooth
fixed sphere, when a series of planewaves are incident upon it, and
considerations are adduced in pages 288 296 relative to the way
in which the distribution is modified by transverse action, or lateral
divergence, of the incident waves. In the solution of Example vn.
(p. 296) like considerations are applied to the case of waves inci
dent on a moveable sphere, and an attempt is, besides, made in
pages 298 306 to extend the reasoning so as to include terms of
the second order. The result of chief importance is, that when
INTRODUCTION. xliii
only terms of the first order are taken account of, the motion of
the sphere is simply vibratory, but when the calculation includes
terms of the second order, the vibrations are found to be accom
panied by a permanent motion of translation of the sphere. This
conclusion, and the inferences and Corollaries contained in pages
307 312, have important bearings on some of the subsequent
physical theories.
It must, however, be stated that on two points of much diffi
culty, the effect of lateral divergence, and the translatory action
due to terms of the second order, the solutions of Examples vi.
and vn. are neither complete nor accurate. While the work was
going through the press, I discovered a more exact mode of treat
ing Example vn., which is the more important problem of the
two, and this improved solution, as far as regards terms of the
first order, is given in pages 4=22 and 423, with reference to its
application in a theory of the Dispersion of Light. The more
complete solution, inclusive of all small quantities of the second
order, is taken up at page 441, and concluded in page 452, under
the head of " The Theory of Heat," the analytical determination of
the motion of translation forming a necessary part of that theory.
In this new solution some of the difficulties of the problem are
overcome, but others remain, as, especially, that mentioned in
page 453 relative to finding expressions for the constants H^ and
H z . The determination of these functions would, it seems to me,
require expressions to be obtained, to the second order of small
quantities, for the velocity and condensation at all points of the
fluid, whereas the investigation to that order of small quantities
which I have given is restricted to points on the surface of the
sphere. This generalization of the solution I have left (p. 453) to
be undertaken by more skilful analysts who may feel sufficient
confidence in the antecedent reasoning to be induced to cany it
on. It may, however, be here stated that from considerations
xliv INTRODUCTION.
entered into in the solution of Proposition xvn. (p. 230), I am of
opinion that it would be allowable to suppose udx + vdy + wdz
to be an exact differential, although the motions would not be
wholly vibratory, and that from the first approximation obtained
on that supposition it would be legitimate to proceed to the second
by the usual rules of approximating.
At the end of the solution of Prop. xvn. the remarkable con
clusion is arrived at that if udx + vdy + wdz be an exact differen
tial to terms of the second order, the total dynamical action of
simultaneous disturbances of the fluid, so far as regards the pro
duction of permanent motions of translation, is the sum of the effects
that would be produced by the disturbances acting separately.
Under Proposition xvm. (p. 240) a demonstration is given of
the coexistence of steady motions. This law had not, I think, been
noticed till I drew attention to it. It is an essential element in
some of the subsequent physical theories. The solution of Ex
ample VIIL (p. 313) serves to determine the dynamical action either
of a single steady motion on a small sphere, or that of two or
more steady motions acting upon it simultaneously. These results
also receive important physical applications.
I have now gone through all the particulars in the first portion
of the work which I thought it desirable to advert to in this
Introduction. As to the Physical Theories constituting the re
maining portion, the new investigations and new explanations of
phenomena which they contain are so many and various, that it
would be tedious, and occupy too much space, to speak of them
here in detail. I can only refer the reader to the Table of Contents
and hope that on all the physical subjects there indicated sufficient
explanations will be found in the body of the work. What I pro
pose to do in the remainder of the Introduction is, to sketch in
INTRODUCTION.
few words the leading principles of the several Theories of the
Physical Forces, and to take occasion at the same time to state
some facts and circumstances relating to theoretical physics, which
have come under my notice during a long course of devotion to
scientific pursuits, and which seem to me to be proper for illus
trating the modern progress and existing state of Natural Philo
sophy. My object in recording the facts and reminiscences I shall
have occasion to mention, will be to shew that a great deal of
misapprehension has prevailed respecting the true principles of
physical enquiry, and to endeavour to correct it, with the view of
gaining a hearing for the method of philosophy advocated in this
volume.
The Theory of Light, contained in pp. 320 436, rests on hypo
theses of two kinds, one relating, to the qualities of the aether, or
fluid medium, in which light is supposed to be generated and
transmitted, and the other to the qualities of the ultimate consti
tuents of the visible and tangible substances by the intervention of
which phenomena of light are either originally produced, or are
modified.
The hypothesis respecting the aether is simply that it is a con
tinuous elastic medium, perfectly fluid, and that it presses propor
tionally to its density. Out of this hypothesis, by sheer mathema
tical reasoning, I have extracted explanations of twenty different
classes of phenomena of light, namely, those enumerated in
pp. 321 354, which are all such as have no particular relations to
the qualities of visible and tangible substances. Among these are
the more notable phenomena of rectilinear and uniform propaga
tion, of composition and colour, of interferences, and of polariza
tion. It might have been supposed that to have to account for
the transmission of light all the distance from the fixed stars
without its undergoing any change of character, would have put in
peril the hypothesis of a continuous fluid. But the mathematical
xlvi INTRODUCTION.
reasoning above mentioned gives results completely accordant with
this fact. There is just reason, I think, to say that the number
and variety of the explanations of phenomena deduced by strict
reasoning from this simple hypothesis establish a very strong
presumption of its truth.
But my mathematical contemporaries will not allow of the
very reasonable hypothesis of a continuous fluid medium. This is
to be accounted for, in part, by the anterior refusal (p. xxxvi) to
admit the logical consequence on which I ground the necessity for
reforming the principles of hydrodynamics, and, as matter of
course, the nonacceptance of the reformed principles, on which,
in fact, the explanations which attest the reality of such a medium
depend. The opposition is, however, mainly due, I believe, to
another cause, with which certain historical details are connected,
which, as being illustrative of the course of scientific opinion on
this subject, I shall now proceed to give.
To Mr Airy is due the great merit of introducing by his Pro
fessorial Lectures the Undulatory Theory of Light as a subject of
study in the University of Cambridge. I had the advantage of
attending the lectures, and, from the first, felt no hesitation in
accepting that theory in preference to the theory of emission,
which still held its ground. In 1831 Mr Airy published the sub
stance of his Lectures as part of a volume of " Mathematical Tracts,"
and gave therein an able exposition of the merits of the Undula
tory Theory, accompanied by a fair statement of its difficulties
and defects. In the Preface he distinguishes between " the geo
metrical part" of the theory, which is considered to be certain,
and "the mechanical part" which is conceived to be far from
certain. This distinction I have difficulty in comprehending,
a physical theory, according to my view, being altogether mecha
nical, as having necessarily relation to force. My conclusion on
reading Mr Airy's Treatise rather was, that the theory was satis
INTRODUCTION. xlvii
factory so far as it was strictly undulatory, that is, rested on
hydrodynamical principles, and that the difficulties begin as soon
as the phenomena of light are referred to the vibrations of discrete
particles of the aether. After this modification is introduced into
the theory it ought to be called oscillatory rather than undulatory,
the latter word applying to a wave, or a congeries of particles in
vibration. I was quite confirmed in the above conclusion by
what is said at the end of the Treatise in Arts. 182 and 183
(editions of 1831 and 1842), where it is admitted that the oscil
latory theory does not distinguish beween common light and
elliptically polarized light, although they are proved by facts to be
distinguished by difference of qualities. In consequence of this
contradiction by fact, it follows, by an acknowledged rule of
philosophy, that the oscillatory (not the undulatory) theory of
light must be given up. I say this with the more confidence from
having proved (p. 338) that the undulatory theory, placed on a
hydrodynamical basis, does make the proper distinction between
the two kinds of light.
That the oscillatory theory is incapable of distinguishing
between these lights is only made more manifest by Mr Airy's
attempt to escape from the conclusion. To do this he assumes
that the transverse vibrations are subject periodically to sudden
transitions from one series to another accompanied by changes of
direction ; but as it is not pretended that these changes are dedu
cible from the antecedent hypotheses of the theory, and as no
attempt is made to account for them dynamically, the assumption
can only be regarded as a gratuitous personal conception. The
advocacy of similar ideas by Professor Stokes (Camb. Phil. Trans.
Vol. ix. p. 414), does not in any degree help us to conceive of a
cause for the transition from one series of vibrations to another.
I am not aware that such views have been adopted by continental
mathematicians.
Xlviii INTRODUCTION.
When in 1837 I commenced Professorial Lectures on Physical
Optics in continuation of those of Mr Airy, I judged it right to
point out the failure of the oscillatory theory, and to endeavour to
place the undulatory theory on a more extended basis of hydrody
namical principles. I was blamed at the time for goiDg against
the current of scientific opinion. But what else could I do 1
Whatever views others might hold, I felt that I could not dis
regard the consequences of the abovementioned application of a
rule of philosophy. All that has occurred relative to the Theory
of Light in the last thirty years has only convinced me that I
was right in the course I took, which will also, I think, be found
to be fully vindicated by the success with which the Theory is
treated on hydrodynamical principles in this Volume. Professor
Stokes, when he succeeded me in lecturing on Optics, recurred to
the oscillatory hypothesis. I must here be permitted to express
the opinion that the adoption of a different course might have
contributed towards forming at Cambridge an independent school
of philosophy on principles such as those which Newton inaugu
rated, which in recent times have been widely departed from both
in England and on the Continent.
When Fresnel first ventured to make the hypothesis of the
transverse vibrations of discrete particles, he stated that he did so
on account of " the incomplete notions respecting the vibrations of
elastic fluids that had been given by the calculations of geome
ters." (Memoires de VInstitut, Vol. vn. p. 53). Had it been
known in his time that transverse vibrations were deducible by
calculations properly applied to a continuous elastic fluid he
might, perhaps, not have had recourse to this method. As it
has happened, that hypothesis, together with the isotropic con
stitution of the aether, imagined by Cauchy, has obtained a very
firm footing in the theoretical science of the present day. I think,
however, that this remark applies in less degree to the mathe
INTRODUCTION. xlix
maticians of France than to those of other countries. It is well
known that Poisson did not accept these views. A very eminent
French geometer, in the course of a conversation I had with him
at the Cambridge Observatory, only said of Cauchy, " II ne con
clut rien." It is by British mathematicians especially that these
hypotheses have been unreservedly adopted and extensively ap
plied. It does not, however, appear, as far as regards the Theory
of Light, that the success in this line of research has been propor
tionate to the magnitude of the efforts. I say this on the autho
rity of Professor Stokes's elaborate and candid Report on Double
Refraction in the British Association Report for 1862. After
giving an account of the profound analytical processes applied to
that question by several eminent mathematicians, and of the use
made of Green's very comprehensive principle, he expresses the
opinion, that " the true dynamical theory of double refraction has
yet to be found." I think it must be allowed that from my point
of view there is reason to say, that the failure thus acknowledged,
which, in truth, is apparent from the whole tenour of the Report,
is attributable to the radical vice of an oscillatory theory.
The foregoing statements may sufficiently indicate the chief
cause that has operated to prevent the acceptance of the hypothesis
of a continuous sether. The contrary hypothesis of a discrete isotro
pic constitution of the medium, which was invented by Cauchy to
account for the polarization of light by transverse vibrations,
obtained such extensive recognition, that mathematicians, influ
enced by authority and current scientific opinion in greater degree,
perhaps, than they are themselves aware, are unwilling to sur
render it, although, as above stated, it has failed to explain pheno
mena, and is actually contradicted by fact. It will thus be seen
that I have been thrown into opposition to my scientific contem
poraries, first, by maintaining the consequences of applying a rule
of logic (p.xxxvi), and, again, by contending for the strict applica
1 INTRODUCTION.
tion of a rule of philosophy. I cannot forbear saying that under
these circumstances the opposition on their part is unreasonable,
and that, in my opinion, it very much resembles the opposition in
former times of the Aristotelians to Galileo, or that of the Carte
sians to Newton. History in this respect seems to repeat itself.
Cauchy's isotropic constitution of the aether is relied upon in the
theory of light, in the same manner as the vortices of Descartes
were relied upon for a theory of gravitation, and what Newton
said of the latter hypothesis, " multis premitur difficultatibus," is
equally true of the other. I hold myself justified in thus strenu
ously contesting the two points above mentioned, inasmuch as
they are like those strategic positions in warfare by gaining or
losing which all is gained or lost. If the rules of a strictly philo
sophic method be not maintained, philosophy will become just what
those who happen to have a scientific reputation may choose to
make it, which, I believe, is the case with respect to much that is
so called in the present day.
In page 354 it is stated that the explanation of the phenomena
of diffraction is incomplete, owing to mathematical difficulties not
overcome relative to lateral divergence, which, as mentioned in
page 292, I have left for the consideration of future investigators.
Poisson regarded the problem of the propagation of a line of light
(" une ligne de la lumiere ") as one of great physical importance.
(I remember to have heard this said by the late Mr Hopkins ;
but I have not myself met with the expression of this opinion in
Poisson's writings.). The possibility of such propagation appears to
be proved by the considerations entered into in pages 290 and 291,
the object of which is to shew that composite direct and transverse
vibrations contained within a cylindrical space of very small trans
verse section might be transmitted to any distance without lateral
divergence ; but they do not determine the law of the diminution of
the density towards the cylindrical boundary. The general deter
INTRODUCTION. li
mination of lateral diminution of condensation under given circum
stances, is a desideratum with respect to the complete explanation
of other physical phenomena as well as diffraction. There is
nothing, however, in these views opposed to the method in which
problems of diffraction are usually treated on the undulatory
hypothesis.
The explanations in pages 362 436 of phenomena of light
which depend on its relations to visible and tangible substances
are prefaced (in pages 357 and 358) by certain hypotheses respect
ing the qualities of the ultimate constituents of the substances.
These constituents are supposed to be inert spherical atoms, ex
tremely minute, and of different but constant magnitudes. Except
ing the spherical form, the qualities are those which were assigned
to the ultimate parts of bodies by Newton, and regarded by him
as " the foundation of all philosophy." According to hypothesis
v. (p. 358), no other kind of force is admissible than the pressure
of the sether, and the reaction to that pressure due to the constancy
of form of the atoms. Hence the sether at rest is everywhere of
the same density. I wish here to draw particular attention to the
circumstance that in the explanations of phenomena of light, and
in all the subsequent theories of the physical forces, no other
hypotheses than these, and the former ones relating to the aether,
are either admitted or required.
Although the evidence for the reality of the sether and its
supposed qualities, given by the explanations of the first class of
phenomena of light, adds much to the confidence with which those
of the second class may be attempted, the latter explanations do
not admit of the same degree of certainty as the others, on account
of the greater complexity of the problems, and our defective know
ledge of their precise mathematical conditions. The theory of Dis
persion is given in pages 362 375, and again in pages 422 427,
after introducing the correction spoken of in page xliii. The
Hi INTRODUCTION.
results by the two investigations differ very little (p. 427), shew
ing that numerical comparisons, in the case of this problem, afford
scarcely any test of the exactness of the formula.
The Theory of Double Refraction on the undulatory hypothesis
is briefly given in pages 375 383. It accounts satisfactorily for
the fact that "one of the rays of a doubly refracting medium, if
propagated in a principal plane, is subject to the ordinary law of
refraction *' (p. 382). In the Report on Double Refraction before
referred to Professor Stokes admits (p. 270) that "this simple
law " is not accounted for on the principles of the oscillatory
theory. It appears also from the same Report (pp. 256, 259, 264,
268) that on these principles inconclusive results are obtained as
to the direction of the transverse vibrations of a polarized ray
relative to the plane of polarization. The theory I have given
determines without ambiguity that the direction is perpendicular
to the plane of polarization (p. 383). These particulars are here .
mentioned because, while they confirm the assertion in page xlix,
that the oscillatory theory has failed, they shew that the pro
posed undulatory theory of double refraction is entitled to con
sideration.
The theories of reflection and refraction at the surfaces of
transparent bodies are given at great length in pages 383 415.
In page 411 it is found that the direction of the transverse motion
in a polarized ray is unequivocally determined to be perpendicular
to the plane of polarization, as was inferred from the theory of
double refraction.
The hypotheses respecting the qualities of the ultimate con
stituents of bodies have been as little accepted by my scientific
contemporaries as those relating to the aether. For instance, in
the Phil Mag. for July 1865 (note in p. 64), Professor W. Thom
son has expressed an opinion decidedly adverse to " finite atoms,"
and in the Number for July 1867, p. 15, has not hesitated to pro
INTRODUCTION. liii
nounce views admitted by Newton relative to the qualities of
atoms to be "monstrous." As 1 have already said (p. viii.), I
need not concern myself about a mere opinion, however strongly
expressed, respecting my hypotheses ; but I am, entitled to ask
for a fair consideration of the mathematical reasoning founded
upon them, and of the results to which it leads. These results
alone determine whether the hypotheses are true or false. All the
explanations of phenomena in this Yolume (the phenomena of
light of the first class being excepted) depend on the hypothesis
of finite atoms, the reality of which, when the number, variety,
and consistency of the explanations are taken into account, can
scarcely be regarded as doubtful.
Professor Thomson not only rejects Newton's atom, but puts
another in its place. He considers that results obtained by M.
Helmholtz in an elaborate mathematical investigation * respecting
vortexmotion (see Phil. Mag. vol. 33, p. 485) indicate motion of
such "an absolutely unalterable quality" as to suggest the idea
that " vortexrings are the only true atoms." From my point of
view I can readily grant that investigations of this kind, regarded
only as solutions of hydrodynamical questions, may admit of
important physical applications. I have, in fact, given the solu
tion, although by a very different process, of a problem of vortex
motion, which I had occasion to apply in the theory of galvanic
force. (See in pp. 563 569.) But I cannot see that there is
any reason for putting "the Helmholtz atom" in the place of
Newton's foundation of all philosophy.
The Theory of Heat in pp. 436 462 answers the question,
What is heat 1, by means of mathematical reasoning applied to the
aether of the same kind as that which applied to the air enables us
to answer the question, What is sound 1 The perceived effects are
* This is the other "great improvement" in Hydrodynamics referred to in the
note in p. xxxiii.
e
liv INTRODUCTION.
produced in the two cases by vibrations obeying the same laws,
but acting under different circumstances. Heat, accordingly, is
not a mode of motion only, as lias been recently said, but essenti
ally a mode of force. Light is also a mode of force, the dynamical
action which produces it being that of the transverse vibrations
accompanying the direct vibrations which are productive of heat.
For this reason I include light in the number of the physical forces.
With respect to the mathematical part of the theory it may be
stated that the reasoning contained in pp. 441 452 is much more
complete and satisfactory than any I had previously given. The
principal result is the expression in p. 452 for the constant accele
ration of an atom acted upon by setherial vibrations, the investiga
tion of which takes account of all terms of the second order, and
therefore embraces both vibratory motions and permanent motions
of translation of the atom. The general theory of the dynamical
action by which repulsive and attractive forces result from vibra
tions of the cether, depends on this formula. But the information
it gives is imperfect because, as the functions that H l and H a are
of m and X have not been determined, the values of the expression
for different values of these quantities cannot be calculated. It
can, however, be shewn that caloric repvtsion corresponds to waves
of the smallest order, and that these waves keep the atoms
asunder in such manner that collision between them is impossible.
(See pp. 458 and 468.)
In the Theory of Molecular Attraction, in pp. 462 468, the
attractive effect is supposed to be produced by waves of a new
order resulting from the composition of all the waves from a vast
number of atoms constituting a molecule. The values of m and X
resulting from the composition are assumed to be such as make the
abovementioned expression negative ; but the theory is not suffici
ently complete to determine the values for which the expression
changes sign.
INTRODUCTION. Iv
The theory of atomic and molecular forces is followed by an
investigation in pp. 469 485 of the relation between pressure and
density in gaseous, liquid, and solid substances, (particularly with
reference to the state of the interior of the earth), together with
some considerations respecting the different degrees of elasticity of
different gases.
The Theory oftJie Force of Gravity, in pp. 486 505, depends
on the same expression for the acceleration of an atom as that
applying to the forces of Heat and Molecular Attraction ; but
while in the case of the latter the excursion of a particle of the
sether may be supposed to be small compared to the diameter
of the atom, for waves producing the force of gravity the ex
cursions of the setherial particles must be large compared to the
diameter of any atom. For large values of X it appears that
HI = \ (p. 497) ; but since the function that H 2 is of m and /\ is
not ascertained, the theory is incomplete. Nevertheless several
inferences in accordance with the known laws of gravity are
deducible from antecedent hydrodynamical theorems. (See pp.
498 and 499.)
For a long time there has prevailed in the scientific world
a persuasion that it is unphilosophical to enquire into the modus
operandi of gravity. I think, however, it may be inferred from
the passage quoted in p. xix. that the author of the History of the
Inductive Sciences did not altogether share in this opinion. Not
long since Faraday called attention to the views held by Newton
on this question, and proposed speculations of his own as to the
conservation of force and mode of action of gravity, which, how
ever, he has not succeeded in making very intelligible. (Phil. Mag.
for April, 1857, p. 225.) Faraday's ideas were combated by
Professor Briicke of Vienna, who, in arguing for the actio in
distans, introduces abstract considerations respecting " the laws of
thought," such as German philosophers not unfrequently bring to
Ivi INTRODUCTION.
bear on physical subjects (Phil. Mag. for February, 1858, p. 81).
I have discussed Newton's views in p. 359. It would have been
a fatal objection to ray general physical theory if it had not
been capable of giving some account of the nature of the force of
gravity.
So far the aether has been supposed to act on atoms by means
of undulations, whether the effect be vibratory or translatory.
In the three remaining physical forces the motions of translation
are produced by variations of condensation accompanying steady
motions. The mathematical theory of this action on atoms, which
is given as the solution of Example vui. p. 313, is very much
simpler than that of the action of vibrations. It is necessary,
however, to account for the existence of the steady motions. Here
I wish it to be particularly noticed that this has been done, not
by any new hypothesis, but by what may be called a vera causa,
if the other hypotheses be admitted. It is proved in pp. 544
548, that whenever there is from any cause a regular gradation of
density in a considerable portion of any given substance, the
motion of the earth relative to the aether produces secondary
cetherial streams, in consequence of the occupation of space by the
substance of the atoms. These streams are steady because the
operation producing them is steady, and to their action on the
individual atoms the theory attributes the attractions and repul
sions in Electricity, Galvanism, and Magnetism, the distinctions
between the three kinds of force depending on the circumstances
under which the gradations of density are produced. In a sphere
the density of which is a function of the distance from the centre
the secondary streams are neutralized.
In the Theory of Electric Force, in pp. 505 555, the internal
gradation of density results from a disturbance by friction of the
atoms constituting a very thin superficial stratum of the substance.
The law of variation of the density of this stratum in the state of
INTRODUCTION. Ivii
equilibrium is discussed in p. 466 under the head of Molecular
Attraction. A large proportion of the theory of electricity,
extending from p. 507 to p. 544, is concerned with the circum
stances under which this equilibrium is disturbed, and new states
of equilibrium of more or less persistence are induced, and with
the explanations of electrical phenomena connected with these
changes of condition. In this part of the theory it is supposed
that attractionwaves and repulsionwaves intermediate to the
waves of molecular attraction and gravitywaves are concerned
in determining the state of the superficial strata, but not in
causing electrical attractions and repulsions, which are attributed
solely to the secondary streams due to the interior gradation of
density.
In The Theory of Galvanic Force, in pages 555 604, con
sideration is first given to the relation between the electric state and
galvanism. It is admitted that electricity not differing from that
generated by friction is produced by chemical affinity, or action,
between two substances, one a fluid, and the other a solid, and that
the interior gradation of density thence arising originates secondary
streams, as in ordinary electricity, but distinct in character in the
following respect. The galvanic currents, it is supposed (p. 598),
result from an unlimited number of elementary circular currents,
analogous to the elementary magnetic currents of Ampere, but
altogether setherial, and subject to hydrodynamical laws. These
resultants, after being conducted into a rheophore, are what are
usually called galvanic currents. The investigation in pages 563
569, already referred to, shews that the current along therheo
phore must fulfil the condition of vortexmotion, but it does not
account for the fact that the whirl is always dextrorsum (p. 598).
The explanation of this circumstance would probably require a
knowledge of the particular mode of generation of the elementary
currents.
Ivlii INTRODUCTION.
The above principles, together with the law of the coexistence
of steady motions, are applied in explanations of various galvanic
phenomena, for experimental details respecting which, as well as
respecting those of electricity, reference is made to the excellent
Treatises on Physics by M. Jamin and M. Ganot, and to the large
Treatise on Electricity by M. De La Eive.
The Theory of Magnetic Force, in pages 604 676, embraces
a large number of explanations of the phenomena of ordinary
magnetism, as well as of those of Terrestrial and Cosmical Magne
tism. With respect to all these explanations it may be said that
they depend upon principles and hypotheses the same in kind as
those already enunciated, the only distinguishing circumstances
being the conditions which determine the .interior gradations of
density. It is assumed that a bar of iron is susceptible of grada
tions of density in the direction of its length, with more or less
persistency, in virtue* of its peculiar atomic constitution, and in
dependently of such states of the superficial strata as those which
maintain the gradation of density in electrified bodies (p. 604). The
same supposition is made to account for the diamagnetism of a bar
of bismuth, only the gradation of density is temporary, and in the
transverse direction (p. 621). The proper magnetism of the Earth
is attributed to the mean effect of the asymmetry of the materials
of which it is composed relative to its equatorial plane (p. 613).
The diurnal and annual variations of terrestrial magnetism are
considered to be due for the most part to gradations of the density
of the atmosphere caused by solar lieat (pp. 645 651). The
Moon, and, in some degree, the Sun, generate magnetic streams by
the variation of density of the atmosphere due to unequal gravita
tional attraction of its different parts (p. 662). The Sun's proper
magnetism, and its periodical rariations, are in like manner pro
duced by unequal attractions of different parts of the solar at
mosphere by the Planets (p. 669).
INTRODUCTION. Hx
This theory of Magnetism is incomplete as far as regards the
generation of galvanic currents by magnetic currents, as men
tioned in pages 636 638. The reason is, that we are at present
unacquainted with the exact conditions under which the ele
mentary circular currents, which by their composition produce
galvanic currents, are hydrodynamically generated. The difficulty
is, therefore, the same as that before mentioned with respect to
galvanism.
The proposed theory of Terrestrial and Cosmical Magnetism
agrees in a remarkable manner with results obtained by General
Sabine from appropriate discussions of magnetic observations
taken at British Colonial Observatories, and at various other geo
graphical positions. In the treatment of this part of the subject
I have derived great assistance from Walker's AdamsPrize Essay
(cited in p. 645 and subsequently), which is a good specimen of
the way in which theory can be aided by a systematic exhibition
of the past history and actual state of a particular branch of ex
perimental science. For the facts of ordinary magnetism I have
referred to the works already mentioned, and to Faraday's ex
perimental Researches in Electricity.
In writing this long Introduction I have had two objects in
view. First, I wished to indicate, by what is said on the contents
of the first part of the work, the importance of a strictly logical
method of reasoning in pure and applied mathematics, with respect
both to their being studied for educational purposes, and to their
applications in the higher branches of physics. Again, in what
relates to the second part, I have endeavoured to convey some
idea of the existing state of theoretical physics, as well as to give
an account of the accessions to this department of knowledge
Ix INTRODUCTION.
which I claim to have made by my scientific researches as digested
and corrected in this Volume. On the state of physical science
much misconception has prevailed in the minds of most persons,
from not sufficiently discriminating between the experimental and
the theoretical departments, language which correctly describes
the great progress made in the former, being taken to apply to
the whole of the science. Certainly the advances made in recent
years in experimental physics have been wonderful. I can bear
personal testimony to the skill and discernment with which the
experiments have been made, and the clear and intelligible manner
in which they are described, by the extensive use I have made of
them in the composition of this work, many of the experiments
being such as I have never witnessed. During the same time, how
ever, theoretical philosophy arrived at little that was certain either
as to the principles or the results. This being the case it is not
to be wondered at that experimentalists began to think that theirs
is the only essential part of physics, and that mathematical theories
might be dispensed with. This, however, is not possible. Experi
ments are a necessary foundation of physical science j mathe
matical reasoning is equally necessary for making it completely
science. The existence of a "Correlation of the physical forces"
might be generally inferred from experiment alone. But the deter
mination of their particular mutual relations can be accomplished
only by mathematics. Hoc opus, hie labor est. This labour I
have undertaken, and the results of my endeavours, whatever may
be their value, are now given to the world. The conclusion my
theoretical researches point to is, that the physical forces are
mutually related because they are all modes of pressure of one and
the same medium, which has the property of pressing proportion
ally to its density just as the air does.
It is a point of wisdom to know how much one does know.
I have been very careful to mark in these researches the limits
INTRODUCTION. Ixi
to which I think I have gone securely, and to indicate, for the
sake of future investigators, what I have failed to accomplish.
Much, I know, remains to be done, and, very probably, much that
I suppose I have succeeded in, will require to be modified or cor
rected. But still an impartial survey of all that is here produced
relative to the Theories of the Physical Forces, must, I think,
lead to the conclusion that the right method of philosophy has been
employed. This is a great point gained. For in this case all
future corrections and extensions of the applications of the theory
will be accessions to scientific truth. To use an expression which
occurs in the Exploratio Philosophica of the late Professor Grote
(p. 206), "its fruitfulness is its correctibility." Some may think
that I have deferred too much to Newton's authority. I do not
feel that I have need of authority; but I have a distinct per
ception that no method of philosophy can be trustworthy which
disregards the rules and principles laid down in Newton's Prin
cipia.
The method of philosophy adopted in this work, inasmuch as
it accounts for laws by dynamical. causes, is directly opposed to
that of Comte, which rests satisfied with the knowledge of laws.
It is also opposed to systems of philosophy which deduce expla
nations of phenomena from general laws, such as a law of Vis
Viva, or that which is called the " Conservation of Energy." I do
not believe that human intelligence is capable of doing this. The
contrary method of reaching general laws by means of mathe
matical reasoning founded on necessary hypotheses, has conducted
to a meaning of Conservation of Energy not requiring to be quali
fied by any "dissipation of energy." From considerations like
those entered into in page 468 it follows that the Sun's heat, and
the heat of masses in general, are stable quantities, oscillating it
may be, like the planetary motions, about mean values, but never
permanently changing, so long as the "Upholder of the universe
Ixii INTRODUCTION.
conserves the force of the sether and the qualities of the atoms.
There is no law of dcstructibility; but the same Will that con
serves, can in a moment destroy.
In the philosophy I advocate there is nothing speculative.
Speculation, as I understand it, consists of personal conceptions
the truth of which does not admit of being tested by mathematical
reasoning ; whereas theory, properly so called, seeks to arrive at
results comparable with experience, by means of mathematical
reasoning applied to universal hypotheses intelligible from sensa
tion and experience.
After the foregoing statements I am entitled, I think, to found
upon the contents of the theoretical portion of this work the claim
that I have done for physical science in this day what Newton
did in his. To say this may appear presumptuous, but is not
really so, when it is understood that the claim refers exclusively
to points of reasoning. If I should be proved to be wrong by
other reasoning, I shall be glad to acknowledge it, being per
suaded that whatever tends towards right reasoning is a gain for
humanity. The point I most insist upon is the rectification I
have given to the principles of hydrodynamics, the consequences
thence arising as to the calculation of the effects of fluid pressure
having, as I have already said, the same relation to general physics,
as Newton's mathematical principles to Physical Astronomy. I am
far from expecting that this claim will be readily admitted, and
therefore, presuming that I may be called upon to maintain it,
I make the following statement, in order to limit as much as
possible the area of discussion. I shall decline to discuss the
principles of hydrodynamics with any one who does not previously
concede that the reasons I have urged prove the received prin
ciples to be insufficient. Neither will I discuss the theory of light
with any physicist who does not admit that the oscillatory theory
is contradicted by fact. There is no occasion to dispute about
INTRODUCTION. Ixiii
the hypotheses of my physical theories, since I am only bound to
maintain the reasonings based upon them. These conditions are
laid down because they seem to me to be adapted to bring to an
issue the question respecting the right method of philosophy. It is
much against my inclination that I am in a position of antagonism
towards my compatriots in matters of science, and that I have to
assert my own merits. It will be seen that the contention is about
principles of fundamental importance. Nothing but the feeling of
responsibility naturally accompanying the consciousness of ability
to deal with such principles has induced me to adopt and to
persevere in this course.
It may be proper to explain here why I have contributed
nothing in theoretical physics to the Transactions of the Royal
Society. This has happened, first, because I thought the Philo
sophical Magazine a better vehicle of communication while my
views were in a transition state, and then, as I received from
none of my mathematical contemporaries any expression of assent
to them, I was desirous of giving the opportunity for discussion
which is afforded by publication in that Journal. About two
years ago I drew up for presentation to the Royal Society a long
paper giving most of my views on theoretical subjects; but finding
that it necessarily contained much that would be included in this
publication, and might be therein treated more conveniently and
completely, I refrained from presenting it.
I have only, farther, to say that in the composition of this
work I have all along had in mind the mathematical studies in
the University of Cambridge, to the promotion of which the dis
cussion of principles which is contained in the first part may con
tribute something. The subjects of Heat, Electricity, and Mag
netism having, by the recently adopted scheme, been admitted
into the mathematical examinations, it seemed desirable that they
should be presented, at least to the higher class of students, not
Ixiv INTRODUCTION.
merely as collections of facts and laws, but as capable of being
brought within the domain of theory, and that in this respect the
Cambridge examinations should take the lead. It is hoped that
the contents of the second part of this volume may in some degree
answer this purpose. It was with this object in view that the
physical theories have been treated in greater detail than I had at
first intended, especially the theory of Magnetism.
CAMBRIDGE,
February 3, 1869.
A
ITY o
.
NOTES
PEINCIPLES OF CALCULATION.
CALCULATION is either pure and abstract, or is applied to
ideas which are derived from observation, or from experiment.
The general ideas to which calculation is applied are
space, time, matter and. force.
General principles of pure calculation.
Pure calculation rests on two fundamental ideas, number
and ratio.
By numbers we can answer the question, How many?
By number and ratio together, we answer the question, How
much?
The calculus of numbers may be performed antecedently
to the general idea of ratio. Under this branch of calculation
may be ranged, Numeration, Systems of Notation, Diophantine
Problems*, and, in great part, the Calculus of Finite Differences.
Arithmetic, Algebra, the Calculus of Functions and the
Differential and Integral Calculus, the Calculus of Variations
and the Calculus of partial Differentials, &c., are successive
generalizations of pure calculation. These different parts
constitute one system of calculation, in which quantitative
relations are expressed in all the ways in which they can
occur, and in different degrees of generality.
The leading principle in seeking for symbolical representa
tions, or expressions, of quantity, is that all quantities may be
conceived to consist of parts. This is a universal idea derived
from experience and from observation of concrete quantities.
* The Greek mathematicians made greater advances in calculations restricted
to integer numbers than in general calculation. One reason for this was probably
the want of a convenient system of notation, such as that now in use, in which the
place of a figure indicates its value.
1
2 GENERAL PRINCIPLES
Quantities may be altered in amount by addition and sub
traction, or by operations which in ultimate principle are the
same as these.
An equation consists of two symbolic expressions of equal
value. The equality is either identity under difference of
symbolic form, and therefore holds good for all values of the
symbols ; or it is only true for values of the symbols subject
to certain limitations*. The first kind arises out of operations
founded on the principles of pure calculation : the other arises
out of given quantitative relations.
The ultimate object of all pure calculation is to furnish the
means of finding unknown quantities from known quantities,
the conditions connecting them being given. Hence in pure
calculation there are two distinct enquiries. First, the in
vestigation of quantitative expressions with the view of forming
equations from given conditions : and then the investigation
of rules for solving the equations.
Arithmetic is employed to find unknown from known
quantities : but the given conditions are generally so simple
that it is not necessary to designate the unknown quantity
by a literal symbol.
The solution of an algebraic equation gives determinate
values, either numerical or literal, of the unknown quantity.
The solution of a differential equation containing two
variables is an algebraic relation between the variables, deter
minate in form, but generally involving arbitrary constants.
The solution of a differential equation containing three
variables is a relation between the variables involving func
tions of determinate algebraic expressions, but the forms of the
functions themselves are generally arbitrary.
Differential equations containing four variables occur in
the applications of analysis. Their solutions are still more
comprehensive, involving arbitrary functions of algebraic ex
pressions arbitrarily related.
* It would be a great advantage to learners if these two kinds of equations
were always distinguished by marks. I propose to indicate the former by the
mark = , and the other by the usual mark =.
OF CALCULATION. 3
The different kinds of applied calculation.
The application of pure calculation to the ideas of space,
time, matter, and force, gives rise to various branches of
mathematical science*.
1. Calculation applied to space is called Geometry. This
is the purest of all the branches of applied mathematics.
Calculation cannot be applied separately to either time, or
matter, or force, because time and matter cannot be numerically
measured independently of space, and force cannot be numeri
cally measured independently of matter, or space and time.
2. Calculation applied to space and time is the science
of motion and of measures of time. Under this head comes
Plane Astronomy, the application in this instance being
restricted to the motions, apparent or real, of the heavenly
bodies.
3. Calculation applied to space, matter, and force,
is the science of Equilibrium, or Statics. The Statics of
rigid bodies, and Hydrostatics, differ only in respect to pro
perties of the matter considered.
4. Calculation applied to space, time, matter and force,
is the science of the Dynamics f of Motion, or the science of
motion considered with reference to a producing cause. The
matter to which this science, as also that of equilibrium,
relates, may be rigid, elastic, or fluid. In the last instance,
it is called Hydrodynamics,
Physical Astronomy is the science of the motion of the
heavenly bodies, considered with reference to a producing
cause, gravitation.
* This science is properly called mathematical, because in every instance of
such applications of pure calculation, the object is to learn something respecting
the mutual relations of space, time, matter, and force.
t It is singular that a word which does not express motion, should have been
so generally employed to distinguish a branch of science necessarily involving
motion from one which is independent of motion.
12
4 THE PKINCIPLES OF
Physical Optics is the science of the phenomena of Light,
considered as resulting from some theoretical dynamical action.
Common optics is, for the most part, a special application
of pure calculation to the courses of rays of light, and may,
therefore, with propriety be called Geometrical Optics.
The Principles of Arithmetic.
A heap of stones (calculi) is formed by the addition of
single stones. Numbers result from the addition of units.
The first step towards a general system of calculation is to
give names to the different aggregations of units, and the next,
to represent to the eye by figures (figures, forms) the result of
the addition of any number of units. The figures now com
monly in use answer this purpose both by form and by position.
A figure represents a different amount according as it is in the
place of units, tens, hundreds, thousands, &c. The progression
by tens is arbitrary. Numeration might have proceeded by
any other gradation, as by fives, or by twelves, but the esta
blished numeration is sufficient for all purposes of calculation.
By numeration an amount of units of any magnitude may
be expressed either verbally or by figures. But for the
general purposes of calculation we require to express quantity
as well as quotity. To do this the idea of ratio* is necessary.
The fifth Book of Euclid treats especially of the arithmetic
of ratios. It would be incorrect to say that the reasoning in
that Book is Geometrical. It contains no relations of space.
Straight lines are there used to represent quantity in the
abstract, and independently of particular numbers. Towards
the close of it there is an approach to an algebraic repre
sentation of quantity by the substitution of the letters A , B,
C, &c., for straight lines. But the reasoning throughout,
* It is worthy of remark that this word also signifies reason. Probably the
adoption of the term in arithmetic may be accounted for by considering that ratio
and proportion are ideas derived from external objects by the exercise of the power
of reasoning. Without reason there would be no idea of proportion. A just
estimate of proportions indicates a high degree of cultivation of the reasoning faculty.
PURE CALCULATION. O
though independent of particular numbers, is essentially
arithmetical. It is universal arithmetic. When two quanti
ties are expressed numerically, we can say that one is greater
or less than the other, and how much : when expressed by
straight lines, we can still say that one is greater or less than
the other, although without the use of numbers we cannot say
how much. But when two quantities are expressed by
A and B, as in algebra, there is nothing to indicate which is
the greater.
As the subject of the fifth Book of Euclid is pure calcula
tion, logically it might have preceded all those which treat of
the relations of space. Probably the reason it is placed after
the fourth is, that the first four books require only the
arithmetic of integer numbers. The sixth Book is the first
that involves the application of the arithmetic of ratios.
Perfect exactness of reasoning from given definitions is the
characteristic feature of the Books of Euclid, which makes
them of so great value as means of intellectual training. But
after admitting this, it cannot be asserted that the definitions
themselves are in every instance the most elementary possible^
or such only as are absolutely necessary. An advanced stage
of mathematical science gives an advantage in looking back
upon elementary principles which the ancient geometers did
not possess, while at the same time their works have the
great value of indicating, and very much circumscribing, the
points that remained for future determination. One such point
is the following. Is the fifth Definition of the fifth Book of
Euclid a necessary, or an arbitrary, foundation of the doctrine
of proportions ? This question will receive an answer in the
sequel.
Let us now enquire in what manner quantity may be
generally expressed by means of numbers. For this purpose,
following the method of Euclid for the sake of distinctness of
conception, I take a straight line to be the general represen
tative of quantity. Although a particular kind of quantity
is thus employed to designate quantity of every kind, the
6 THE PRINCIPLES OF
generality of the reasoning will not be affected. For of the
four general ideas to which calculation is applied, space, time,
matter, and force, the last three do not admit of measures
independently of the first. Hence the principles on which
any portion of a straight line is quantitatively expressed by
means of numbers are applicable generally. Moreover as
space must be conceived of as infinitely divisible and infinitely
extended, every gradation of quantity, and every amount of
quantity, may be represented by a straight line.
In order to measure a straight line, it is first necessary to
fix upon a unit of length, that is, an arbitrary length repre
sented by unity. Then by the use of integer numbers we can
express twice, three times, &c., the unit of length, but no in
termediate values. To express every gradation of length it
is absolutely necessary to introduce the idea of ratio. Suppose
a straight line to be equal in length to the sum of an integral
number of units and part of a unit more. How is that addi
tional quantity to be expressed by figures ? First, it must be
regarded as being related to the unit by having a certain ratio
to it. Next, to express the ratio by numbers, the unit itself
must be regarded as consisting of as many equal parts as we
please. The possibility of conceiving of a continuous whole
as made up of any number of equal parts, which conception is
a general result of our experience of concrete quantities, is the
foundation of all numerical calculation of quantity. If then,
for example, the additional quantity contain seven parts,
fifteen of which make up the unit, the two numbers seven and
fifteen express by their ratio how much of a unit is contained
in that portion of the straight line which is additional to the
portion consisting of an integral number of units. Let the
number of units be 6. Then since by supposition each unit
contains fifteen parts, six units contain (by integer calculation)
90 parts. Then adding the 7 parts, the whole line contains
97 parts. Thus by the two numbers 97 and 15, the quantity
in question is exactly expressed : and it is necessary for this
purpose to exhibit the two numbers in juxtaposition, which
PURE CALCULATION. 7
97
is usually done thus, , the lower number indicating the
JLo
number of equal parts into which the unit is divided.
This example suffices to shew the necessary dependence of the
expression of quantity on the idea of ratio. The same symbol
97
expresses the ratio of the number 97 to the number 15.
15
Hence a numerical ratio is the expression of quantity with
reference to an arbitrary unit.
Generally if a quantity consist of a units and b equal parts
of the unit such that the unit contains c of those parts, the
quantity is symbolically expressed thus : , where ac + b
means the product of the integers a and c increased by the
integer b*.
Thus quotity is expressed by one number, but to express
quantity generally, two numbers are necessary and sufficient.
It might be objected to this mode of expressing quantity
generally, that space, time, matter and force are necessarily
conceived of as continuous in respect to quantity, and conse
quently may occur in quantities which do not admit of being
exactly expressed by any integral number of parts, such ex
pression proceeding gradatim. This, in fact, is the case in
such quantities as the diagonal of a square, the side being the
unit, the circumference of a circle, the radius being unit, &c. ;
that is, the ratios in these instances are incommensurable. The
answer to this objection is, that as the unit may be conceived
to be divided into an unlimited number of equal parts, we can
approach ad libitum to the value of a quantity which cannot
be exactly expressed by numbers. Thus in this early stage
of the subject, we meet with a peculiarity in the application of
calculation to concrete quantities, which perpetually recurs in
the subsequent stages. I shall not now enter upon considera
* It should be remarked that letters are here used in the place of numbers,
because the reasoning of the preceding paragraph applies whatever be the numbers.
This use of letters may be called general arithmetic, and is distinct from the use of
letters in algebra.
8 THE PRINCIPLES OF
tions from which it would appear that this is a necessary
circumstance in arithmetical calculation, and that it diminishes
in no respect the exactness and generality of its application.
At present it will suffice to say, that we can represent any
^
quantity whatever as nearly as we please by the symbol ^ ,
A and B being whole numbers. In other words, this symbol
is capable of expressing any amount of continuous quantity
with as much exactness as we please.
It has been shewn that if any quantity be exactly repre
j
sented by ^ , it contains A of the equal parts into which the
unit of measure is supposed to be divided, the unit containing
B of them. But each of these parts may be conceived to be
subdivided into any number (ri) of equal parts, in which case,
by the arithmetic of integers, the quantity will contain nA of
the smaller parts, and the unit will contain nB. Hence, by
what has gone before, the quantity is expressed by the symbol
^ . Thus it appears that while two numbers are necessary
to express quantity in general, the same quantity may be
expressed by different sets of two numbers. This result is
symbolically expressed thus :
A n A . .
B = nB (a) 
By parity of reasoning,
A _mA
B~mB'
Hence, because it may be assumed as an axiom that " things
equal to the same thing are equal to one another," it follows
that,
nA mA ~
The foregoing principle of the equality of ratios, or quantities,
expressed by different numbers constitutes proportion. The
PURE CALCULATION.
last equality represents the most general composition of any
proportion the terms of which consist of integer numbers.
PROPOSITION I. It is required to find the ratio of any two
given quantities of the same kind.
A C
Let the two quantities be ~ and ^ . Then by the reason
A
ing which conducted to the equality (a), the quantity ^ is the
AD , ^ G ^ CB u
same as 7 and the quantity  the same as  . Hence
conceiving the common unit to be divided into BD equal parts,
the first quantity contains AD of those parts, and the other
contains CB. Hence from the primitive idea of ratio, the two
numbers AD and CB determine the required ratio, which
AD
consequently must be expressed by the symbol ^75
Although the ratio of two quantities of the same kind is
independent of the quality of the quantities compared together,
it may still be regarded as quantity, because we may assert of
any ratio that it is greater or less than some other ratio, for
instance, a ratio of equality. This kind of quantity for dis
tinction may be called abstract quantity.
The foregoing expression for the ratio of two quantities of
the same kind informs us that if the second of the quantities
considered as a unit be divided into CB equal parts, the first
consists of AD of those parts. Thus the ratio of two quantities
of the same kind may be regarded as quantity relative to an
abstract unit; and the rules of operation which apply to
quantity of a given species, are true of abstract quantity.
The quantity^ is^> times the quantity , because, the
unit of both being divided into the same number of parts j5,
the first quantity consists of pA of those parts, that is, of p
times the number of parts that the other consists of.
10 THE PRINCIPLES OF
A A A
The quantity ^ is^ times the quantity =, because ^ is
equal to ^ , and consequently, the unit being divided into
A
pB equal parts, ^ consists of pA of those parts, that is, p times
^
the number of the parts that ^ consists of.
The foregoing conclusions will enable us to prove that the
equalities (a) and (j3) are true when A and B represent quan
C1 (*
tities instead of numbers. For j and ^ being any quantities,
u cL
we have by what has been shewn,
na nad
b bd nad __ ad
nc neb neb cb
~d ~U
a
But the ratio L = by Proposition I.
c cb
a na
Therefore I = L .
c nc
d ~d
Now the quantities j and r are respectively n times the
ft (*
quantities = and , . Hence if the latter quantities be repre
sented by single letters A and j5, the former may be represented
by the symbols nA and nB. Consequently
AnA
A tnA . . nA mA
80
PURE CALCULATION. 11
PROPOSITION II. If four quantities be proportionals, that
is, if the ratio of the first to the second be the same as
the ratio of the third to the fourth, then any equimultiples
whatever being taken of the first and third, and any whatever
of the second and fourth, the ratio of the first multiple to the
second is the same as the ratio of the third multiple to the
fourth.
ft f* P Ct
Let T > 7 > f and j be the quantities, and let
a e
1=1.
<L I
d h
Then, by Proposition I.
ad eh
and because " equimultiples of the same or of equal quantities
are equal to one another," (Eucl. Axiom I. Book v.)
pad peli
~*7s'
the two sides of this equality being, as already shewn, equi
multiples of r and ^ . Again, because " the quantities of
which the same or equal quantities are equimultiples, are equal
to one another," (Eucl. Axiom IT. Book v.)
pad peh
qcb qfg '
the two sides of this equality being, as already shewn, quanti
ties of which j and ^ are equimultiples.
pa pe
But ^ = land^=Z;
qcb qc qfg qg
1 ~h
12 THE PRINCIPLES OF
pa pe
7 /
therefore = J ,
6? A
which proves the Proposition.
COROLLARY. It follows from the very conception of ratio,
that if the first multiple be greater than the second, the third
is greater than the fourth ; and if equal, equal ; and if less,
less. This result coincides with Definition 5 of Euclid v.
Since the foregoing Proposition, (which is the same as
Eucl. Y. 4) was proved without reference to Definition 5, the
Definition results by a perfectly legitimate process. Conse
quently we have been conducted to that Definition by reasoning,
just as if it were a Proposition.
But a strict definition does not admit of being established
by reasoning, because reasoning is founded on definitions.
Therefore Definition 5 of Euclid v. is really a Proposition,
admitting of demonstration on principles which do not form a
part of the elements of Euclid.
The principles are, (1) That continuous quantity may be
conceived to be divided into as many equal parts as we please :
(2) That any portion of continuous quantity has to a certain
portion expressed by unity, a ratio expressible as nearly as we
please by two numbers. These are proved to be elementary
principles from their being indispensable in the various stages
and applications of pure calculation.
In the Elements of Euclid, the Definition 5 of Book v. is
used not only to draw the inference stated in the above
Corollary, but conversely to infer that the four quantities are
proportional. Both the uses occur in the demonstration of
Euclid V. 4. This is a legitimate manner of employing a
definition which is strictly such, because as neither a definition
nor its converse admits of proof, either may be used as a basis
of reasoning. But let us enquire whether on the principles
PURE CALCULATION. 13
employed to prove Proposition II. we can also prove the
following Proposition :
Any equimultiples whatever of the first and third of four
quantities being taken, and any whatever of the second and
fourth, if when the multiple of the first is greater than
that of the second, the multiple of the third is greater than
that of the fourth ; and when equal, equal ; and when less,
less ; then the first has to the second the same ratio as the
third to the fourth.
First, suppose that if v^ = ^ , we have also ^ = j? . Then
as it may be assumed as an axiom, that if one quantity be
equal to another, any multiples, or submultiples, of the first,
are equal to the same multiples or submultiples of the other,
it follows from the hypothesis, by the same reasoning as
that employed in the demonstration of Proposition II. that
.f. ad q A i T eh a
if j =  , then also   .
be p gf p
But there are values of q and p which satisfy the first equality
viz. q = ad and p = be. Hence it follows that
a e
ad eh , . b f
_. = _., and .'.=.
be gf c_ g_
d h
The proposition is, therefore, proved without reference to the
other two hypotheses, and consequently contains unnecessary
conditions. This is another reason why Definition 5 is not
an appropriate foundation of the doctrine of proportions.
The results of the foregoing discussion suggest a few
remarks. Admitting that the reasoning in Euclid v. is rigidly
exact, and that it is founded in perfect strictness on the defini
tions with which it sets out, it may yet be urged that the de
finition on which the doctrine of proportions is made to depend,
is not elementary, and, therefore, not a proper basis of reasoning.
The old geometers had not the advantage which the advanced
stages of calculation now afford for determining which are
14 THE PRINCIPLES OF
the most elementary of its principles. In this respect the fifth
Book of Euclid has little value ; and even the exactness of its
reasoning hardly justifies the expenditure of time and labor
required to master the demonstrations founded on the fifth
Definition, several of which are very long and intricate. The
Propositions thus proved, might with due care bestowed on
the arrangement of the steps of the reasoning, be deduced
from the elementary principles above enunciated with equal
strictness and greater brevity, and by processes which would
better prepare the student for the higher applications of pure
calculation. What then, it may be asked, is to be done with
the Propositions of Book vi. which depend on the fifth
Definition? To this question I propose to give an answer
when I come to consider the principles of Geometry. I pro
ceed now to other points relating to pure calculation.
The product of multiplying one integer by another is an
integer ascertainable by the rules of the arithmetic of whole
numbers. If A and B be the integers and G the product, the
operation is symbolized thus : AB = C. But this representation
is of no use for the purposes of general calculation considered
as an instrument of research, so long as A, B and C stand only
for whole numbers. For such purposes the equality must
have a meaning when A and B represent any quantities
whatever, inclusive of that which it has when they represent
numbers. Now when A and B are integers, G is also an
integer composed of the number B taken the number of times
A. When A is an integer and B any quantity, G is a
quantity composed of the quantity B taken the number of
times A ; and when A and B are both quantities, is a
quantity composed of the quantity B taken the quantity of
times A. But the terms of the statement in the last case are
inclusive of the other two cases, because a whole number may
be regarded as a quantity of units. For this reason the symboli
cal representation of the equality is the same in the three cases.
It remains to enquire by what operations the quantity G is
found in the last two cases.
PURE CALCULATION. 15
Let B stand for the quantity j and A for the integer n.
Then, by what has already been shewn, the quantity B taken
the number of times A is j , which accordingly is the quan
tity C. But if A stands for the quantity of times ^, then
because c times is the same as j times, that is, d times as
much as j, it follows that c times T is d times the same
d o
quantity taken j times. But c times j is = , and the quantity
d o o
of which this is d times is, as already proved, ^% . Hence ?
{* (1C*
taken the quantity of times ^ is jj , which is, therefore, the
cu ocL
value of C for this case.
Let A = B, A still representing any quantity whatever.
Then writing A 2 for AA, the index 2 indicating that this
symbol is the product of two factors, we have a new repre
sentation of quantity, which, since A is any quantity whatever,
satisfies the condition of admitting of every gradation of value.
Similarly AA* gives rise to the form A 3 , and so on. Generally
if m be a certain number, and x represent a quantity of any
magnitude, the form of function thus arrived at is x m .
If x stands for j , then by what was before shewn, xx
stands for ^ , or x* for ^ . So a? 8 stands for ^ 3 , and x m
a m
for ^r These results give the rule for calculating the quantity
which is any integral power (m) of a given quantity.
Let x m = y. Then x having every gradation of value, y
has also gradations of value increasing or decreasing by as
small differences as we please. The quantity y is called the
m th power of x, and the quantity x the m^ root of y.
16 THE PRINCIPLES OF
But if a be a certain quantity and in stand for whole
numbers, the symbol a m represents particular quantities, but
not quantities of every gradation, and is, therefore, inappro
priate to the purposes of general calculation. When, however,
in the place of m we put any quantity  , if the symbol a 9 can
receive such an interpretation that it may represent quantity
of every gradation inclusively of the quantities represented
when the index  is integral, we shall then have a new form
of the representation of quantity of every degree*. The fol
lowing considerations will shew that the symbol admits of
such interpretation.
Let, as above, x m y, m being a whole number. Then
n being any other whole number, x mn = y n , because it may
be assumed that the same powers of equal quantities are equal.
Let y n = z. Then y, which is the m th power of x, is the n ih
root of z, and z is the mn ih power of x. Therefore the n^
root of the mn ih power of any quantity is the w th power of that
mn
quantity. Hence putting x m under the form x n , the interpre
tation to be given to this symbol is, that it represents the 7i th
root of the mn ih power of x, and is equivalent in value to x m .
 V
If this interpretation be extended to a ? ,  being any quantity,
this symbol will signify the <? th root of the p^ power of a,
and has values ascertainable by rules of operation (involution
and evolution) which depend only on the principles of the
arithmetic of whole numbers and ratios. If ^ be equal to
<1
The discovery of fractional indices was one of the most important steps in
the generalization of calculation, and at the same time one of great difficulty.
On their being introduced by Wallis in Chap. LXXV. of his Algebra, he says,
" Eosdem indices sen exponentes retinet vir clarissimus Isaacus Newtonus in no
tatione sua." And subsequently he refers to Newton for the expansion of a bino
mial in the case of a fractional index. Certainly fractional indices were not in use
before Newton's time, and probably it required nothing short of his genius to
discover them.
PURE CALCULATION. 17
1, 2, 3, &c., the values of a q are integral powers of a. For
intermediate values of ^ , the values of a q are intermediate, as
may be thus shewn. Let  be greater than m by the proper frac
tion ^ , and, therefore, less than m + 1 by 1 3 . Then since
m ~*~ ~R ~ ~/3 ' a?= a ^ ~ *^ e ^ root ^ t ^ ie P ower m $ + a
of a. This quantity is greater than the /3 tb root of the power m/3
of a, that is, greater than a ? , or a m . And since  = m + 1
a\ /3 (m + 1) ({3 a) , ,., . ., ,
~i s Q ? j by like reasoning it may be
 p
shewn that a 9 is less than a r/m . Hence if the value of * lies
$
p
between m and m+ 1, the value of a ? lies between a w and

a wl+1 ; and by giving to L  every gradation of value, a q will have
every gradation of value. Consequently putting x for  , and
supposing a to have any fixed value, the symbol a* will express
quantity with as near an approach to continuity as we please.
On this account it must be included among the symbols of a
general system of calculation.
If a x N, the quantity x is called the Logarithm of the
quantity N. The series of logarithms is different for different
values of a, and if a be greater than unity, N may have any
amount of value. A Table of Logarithms is a practical illus
tration of the result arrived at by the foregoing reasoning.
We have found two distinct forms of expressing quantities
with as near an approach to continuity of value as we please,
viz., x m and a x , x being put for the primitive form ^ of ex
pressing quantity of any magnitude. It needs no demonstration
to perceive that the following forms, derived by substitution
2
18 THE PRINCIPLES OF
from the two simple forms, possess the same property of in
definite approach to continuity : a*" 1 , cf , d , &c. To these
may be added x nlx , a?"**"*, &c. It is remarkable that no other
simple forms of continuous expressions of quantity have
been discovered.
All other modes of expressing quantity are derived from
the foregoing by the operations of addition, subtraction, mul
tiplication, division, involution and evolution, indicated by
appropriate marks. There is an infinite variety of such ex
pressions, each of which derives from its component simple
expressions the property of varying from one value to another
by as small gradations as we please. The general symbol
used to embrace all such expressions is f(x), or function of a?,
the different values of the function being determined by the
different values of the variable x.
Although in the foregoing investigation of the different
modes of expressing quantity, letters have necessarily been
used for the purpose of obtaining general results, the conside
rations have been essentially arithmetical, it not being possible
to arrive at either rules of operation or forms of expression
without numbers.
Every general form of expression, when numbers are put
for the letters, is reducible by successive approximations to
j
the primitive form ^ > A an( i B being whole numbers. Un
less this were the case the value of the expression could not be
known. The reduction is effected by the rules of arithmetic.
It does not fall within the scope of these Notes to enter at
length upon the investigation of the rules of arithmetical
operation. To do this would require a formal Treatise. It
will suffice to remark that all the operations are derived from
the simple one of addition nearly as follows. The result of
adding two integers A and B being the integer C, the operation
is expressed thus : A + B = C. If A and B represent respec
tively the quantities and ^ , then since is the same
n ' n
PURE CALCULATION. 19
quantity as , and % the same quantity as , the sum
nq q qn
results from the addition of the parts mq and pn they respec
tively contain, nq of such parts making up the unit. Hence
the sum must be written ^ , which accordingly is the
nq
quantity that C represents in this case. On the principle that
what is added may be taken away, we may take from C the
quantity which was added to A, and the remainder will
plainly be A. This is subtraction, and being just the reverse
of the direct operation of addition, the rule for performing it is
thereby determined. By subtraction we answer the enquiry,
What is the result of taking a certain quantity from a certain
other quantity ?, and as the answer to this question must be
quantitative, for this reason alone the operation by which it
is obtained must be included in a general system of calcu
lation.
Similar considerations apply to the operations of multipli
cation and division. A quantity added to itself is taken twice,
added again, is taken thrice, and so on. This is strictly
multiplication. But the same term is employed when a
quantity is not taken an integral number of times, but a certain
quantity of times, and the symbolic representation of the
operation is the same, viz. AB = C. We have already had
occasion to investigate the rule for obtaining the product C
when A and B are given quantities. The reverse operation,
division, obtains A or B, when C and B, or C and A, are
given. Division answers the enquiry, How much of times
one quantity contains another ?, and as the answer is quanti
tative, the operation belongs to a general system of calculation.
In the case of whole numbers, the rule for obtaining the
quotient is immediately derived from the direct operation by
which the product was obtained, subtraction taking the place
of addition. As it was shewn that the product of the two
quantities j and , is TJ 5 it plainly follows that j is the pro
LIBRAE
20 THE PRINCIPLES OF
duct of jj and  . because ^^ = 7 . Hence ^ contains  the
bcL c oac o o c
quantity of times 75. This determines the rule for finding
the quantity of times that one quantity contains another.
As division determines generally the quantity of times one
quantity contains another, that is, from the primitive idea of
ratio, the ratio of the one quantity to the other, the operation of
division may be represented by the symbol that represents a
ratio. Thus ^ , the ratio of A to B, is also A divided by B.
It may also be remarked that & fraction, whether proper or
improper, is a ratio, and may be represented by the same
symbol.
The involution of integers is the multiplication of any
number by itself, the product by the same number again, and
so on. By evolution, we answer the question, What is the
number which by its involution a certain number of times will
produce a given number ? The rule for the operation, whicli
is virtually the reverse of the direct operation of involution, is
abbreviated by the aid of general arithmetic, the proposed
number being supposed to consist of parts indicated by the
involution of the general symbol a + b.
As it has been shewn that any integral power m of a
m n in
quantity 7 is 7^ , the m th root of 7^ is found by extracting
separately the m th root of a m and the w th root of b m , according
to the rule applicable to integers. The value of A q , A being
any quantity, is found either by extracting the q ih root of A and
raising the root to the p ih power, or extracting the ^ th root of
the /> th power of A.
Those operations of division and evolution which, not
being exactly the reverse of operations of multiplication and
involution, do not terminate, may be made the reverse of
direct operations as nearly as we please.
I proceed now to the next generalization of calculation.
PURE CALCULATION. 21
The Principles of Algebra.
In Algebra*, as in general arithmetic, quantities are
represented by letters, but for a different purpose. The
object in the former is not to investigate rules of operation or
forms of expression, but to answer questions which involve
quantitative relations. All such questions are answered by
means of equations. But till the quantity which answers the
proposed question is found, it is represented by a letter and
called the unknown quantity. This letter must be operated
upon according to previously established rules in order to form
and to solve the equation which by its solution gives the answer.
But being unknown, it cannot be affirmed of it that it is
greater or less than some other quantity from which, according
to the conditions of the question, it may have to be subtracted.
In the former case the operation would be impossible. But it
must be symbolically represented in the same manner, whether
it be possible or impossible. Some expedient is therefore
required to make the reasoning good in both cases, that is, to
make it independent of the relative magnitudes of the Quantities.
This is done by means of the signs + and . The symbol
+ a means that the quantity a is added: the symbol b that
the quantity b is subtracted. These symbols express, there
fore, both quantity and operation. Thus an algebraic expression
is not to be regarded simply as quantity, but as an exhibition
of operations upon quantity, and under this point of view the
expression holds good in general symbolical reasoning, whether
or not the operations indicated are arithmetically possible.
By convention the symbol a is called a negative quantity.
This is only a short way of saying that the quantity a has
been subtracted. All quantity is necessarily positive. As
the terms " negative quantity" are convenient, for the sake of
* This name was given to the science when it was almost exclusively directed
to the solution of numerical equations, and before a general system of symbolic
operations was known.
22 THE PRINCIPLES OF
distinction I shall call quantity regarded independently of the
signs + and , " real quantity."
Let c be the difference between two real quantities a and b.
Then if a be greater than b, a b is equivalent to b + c b or
f c. But if a be less than Z>, a b is equivalent to a a c,
that is, to c. In this manner the symbols + c and c are
defined, c being supposed to be a real quantity. It is next
required to ascertain the rules of operating on these symbols
by addition and subtraction.
Let d be any real quantity larger than the difference
between a and b. Then if a be greater than &, the result of
adding a biodisd + a bl>y general arithmetic. Also if
a be less than b the sum is represented in algebra by the same
expression.
Let a = b f c. The algebraic sum is then d + b + c b,
or d + c. But by the definition above + c has been added.
Hence the result of adding + c to d is written d + c.
Next let b = a f c. Then the algebraic sum d + a b
is d + a a c, or d c. But by definition c has been
added. Therefore the result of adding c to d is d c.
Hence algebraic addition is performed on the symbols
+ c and c by attaching them with their proper signs to
other such symbols.
If a b be subtracted from d, a being greater than &,
the remainder by general arithmetic is d a + b. And if a
be less than b the algebraic remainder is represented by
the same expression.
Let a = b + c. Then the value of the algebraic remainder
is d b c + b, or d c. But in this case, by definition, + c
has been subtracted. Therefore the result of subtracting + c
from d is d c.
Let b = a + c. Then the value of the algebraic remainder
d a + b is d a + a + c, or d + c. And as in this case < c
has been subtracted, it follows that the result of subtracting c
from d is d + c.
Hence algebraic subtraction is performed on the symbols
PURE CALCULATION. 23
+ c and c, by attaching them with signs changed to other
such symbols.
If the quantities to which the signs + and are attached
be not unknown quantities, but gixen quantities represented
generally by letters, the same rule of signs applies, because so
long as the quantities are represented generally, their relative
magnitudes are not expressed. The results of the reasoning
are thus made independent of the relative magnitudes, and
can be applied to particular cases.
The rule of signs in multiplication is established by
analogous reasoning. Let the difference between the real
quantities a and b be e, and the difference between the real
quantities c and d be/. Then if a be greater than b, and c be
greater than d, the result of multiplying a b by c d, is by
general arithmetic a times c d diminished by b times c d,
that is, a quantity less than ac by ad, diminished by a quantity
less than be by bd, which is written ac ad be + bd. The
algebraic expression for the product is the same whatever be
the relative magnitudes of a and b, and of c and d.
Let a = b + e and c = d +/. Then the symbols multiplied
are + e and +f, and the result is found by substituting b + e
for a and d+f for c in the expression ac ad be + bd, and
obtaining its value arithmetically. But by arithmetic
ca + bd=(d+f) (b + e) + bd = db + de + fb+fe + bd
sm&ad + bc=d(b + e)+b(d+f)=db+de +fb + bd.
Hence subtracting the latter quantity from the former the
result is +fe, which is thus shewn to be the product of the
symbols + e and +/ The symbols in this case being both
positive, this result might have been at once inferred from
general arithmetic.
Let b = a + e and c d +f. Then the symbols multiplied
are e and +/, and the result is found by substituting a + e
for b and d+f for c in the same expression as before. But by
arithmetic
24 THE PRINCIPLES OF
ca + bd=a(d +f) +d(a + e) = ad + af+ ad 4 de
ad+bc = ad + (a + e) (d+f) = ad + af+ ad+ de + ef.
Hence subtracting the latter quantity from the former the
result is ef subtracted, or ef, which is thus shewn to be the
algebraic product of the symbols e and +f.
By parity of reasoning the product of + e and / is ef.
Let b = a + e and d = c +f. Then the symbols multiplied
are by definition e and f, and the result is found by
substituting a + e for b and c +f for d in the expression
ac ad bc + bd. But by arithmetic,
ac + bd = ac + (a \e) (c f f) ac + ac\ af+ ec + ef
and ad + be = a (c +f) f c (a f e) = ac 4 ac + af+ ec.
Hence subtracting the latter quantity from the former the
result is + ef, which is thus shewn to be the product of the
symbols e and f.
Consequently in multiplication like signs produce + and
unlike . By means of this rule the operation of multiplica
tion may be extended to real quantities affected with the signs
+ and .
The rule of signs in the reverse operation of division
follows at once from that in multiplication.
In general arithmetic a letter always stands for a real
quantity, and if in the course of the reasoning a single letter
be put for a &, or any other literal expression, it still repre
sents a real quantity. In algebra it is necessary for the
purposes of the reasoning to put a single letter for a b and
like expressions ; but the letter will not now always represent
a real quantity, because in algebra a b may be equivalent to
a real quantity affected with a negative sign. Yet the letter
must be operated upon, and be affected with the signs + and
subject to the rules already established, just as if it repre
sented a real quantity. For it is the distinctive principle of
algebra to adopt without reference to relative magnitude, all
the rules and operations of general arithmetic which have
been established by numerical considerations. On this account
PURE CALCULATION. 25
in algebra such an extension must be given to the signification
of a letter, that + x may represent inclusively a negative
quantity, and y a real quantity. It may also be remarked
that if a letter be substituted for a negative quantity and after
any number of operations be replaced by the negative quantity
for which it was substituted, the result is the same as if the
negative quantity had all along been operated upon.
Impossible quantities. Since the product of + a and + a,
as well as that of a and a, is +a 2 , it follows that a
quantity affected with a negative sign has no square root.
Yet it is necessary to retain the symbol V b or ( &)*, be
cause as b may stand for a negative quantity, b may be a
real quantity. If b have a real arithmetical value, V b can
no longer be quantity, but merely expresses impossibility. By
convention it is called an impossible quantity.
Impossible quantities may be represented by single
letters and be operated upon as if they stood for real quantities.
p
Addition of indices. Let a 9 = N. Then cf = N q , and a np
np p nP
= N"*, n being an integer. Hence a nq = N, and /. a? = a*.
p r p gr
From this it follows that a? x a 8 = a? x 9 % which is equal to
.
a gs , or a q \
Negative indices. If m and n be whole numbers and a any
quantity, then by general arithmetic the ratio is a m ~ n if m
be greater than n, and ^ if n be greater than m. If the indices
OL
*) "7* 70
be  and  , p, q, r and s being any integers, and if * be
r a q cP 8 (o? 8 ^
greater than  , then the ratio = = ( ) = (a p8 ~ qr ) qs =
S e . \Ct /
p
ps qr p _ r ~. T dfl 1
a~**~ = a^~'. So if^ be less than , = __. Thus the
q s '' r j>_
a* as q
subtraction of one index from another originates in the
26 THE PRINCIPLES OF
principles of arithmetic, and in that science is always performed
so that the remaining index is real. But if we assume that
a
= a m ~*, without reference to the relative magnitudes of m
and n, we pass from arithmetical to algebraical indices, and
this generalization gives rise to negative indices in the same
manner that passing from arithmetical to algebraical subtrac
tion gave rise to negative quantities.
Since when the indices are algebraical = a m ~ n whatever
be the relative magnitudes of m and n, if in the course of
reasoning a single letter be substituted for m n, this letter
must be taken to represent either a real or a negative index.
This extension of the signification of a literal index is a
necessary consequence of the algebraic generalization of indices.
m in
Again, by previous notation a" x o7 l x &c. to p factors is
m mp
(a") p , and by addition of indices the same quantity is a"".
m mp m p
Hence (a^) p a^. Also by previous notation (a n )is the q ih
root of the p ih power of a", that is, by what has been just
mp mp 1
proved, the q ih root of a""", or (a""")*. But this quantity is
mp
equivalent to a, because each raised to the power qn gives
mp mpxqn mp 1
the same quantity. For (a*) gn =a * n =a mp ; and {(*)*}*
mp m P mp
= (a""")* = a mp . Consequently (a w )* = a n< * . This is multiplica
tion of indices in general arithmetic.
This operation being extended to the algebra of indices, a
rule of signs must be established; which may be done as follows:
^ _
"
fm\n _ _ n mn _ mxn
~(O^~a" w ~
Hence the rule of signs is the same as in common algebra.
PURE CALCULATION. 27
The rules for the division of indices follow from those
of multiplication. To perform the reverse operation to that of
affecting any quantity with the index  , or extracting the $ th
root of its j> th power, is to extract the p th root of its q ih power,
that is, to affect it with the index  . As the direct operation
m p_
was represented by the notation (a n )% let the reverse opera
t f~m
tion be represented by the notation // a n . Then from what
has been said,
i m mq
This operation being extended to the algebra of indices,
the rule of signs follows from that in multiplication of indices.
The involution and evolution of indices arise out of analogous
considerations. If a represent any numerical quantity, by
what has been shewn (a a ) a = a" 2 ; (a" 2 )" = a" 3 ; and so on.
These operations suggest the reverse one of finding an index
such that when a quantity is affected by a given power of it,
the result is the same as when the quantity is affected by a
given index. Let /3 be the given index, and k the given
power, and let of = aP. Then a* = 0, and a = /9*, the required
index. Thus the extraction of roots in the general arithmetic of
indices arises out of a re version of operations analogous to that
which led to extraction of roots in ordinary general arithmetic.
This part of the subject might be pursued farther, if the object
of these Notes required a more lengthened consideration of it.
The extension of the extraction of the roots of indices to
algebraic indices gives rise to impossible indices, for the same
reason that a like extension in common Algebra gave rise to
impossible quantities. In general calculation it is necessary to
have regard to such indices, because the symbol a^~ b may
represent either a real or an impossible quantity, b being an
algebraic symbol and therefore representing either a negative
28 THE PRINCIPLES OF
or a positive quantity. Also when b is positive, a combination
of such symbols may be equivalent to a real quantity.
In the form a x , the index x may now, for the sake of
generalization, be supposed to stand for an impossible quantity,
as well as for a positive, or a negative quantity, and with this
extension of its signification it must still be operated upon by
the rules that apply to a real index.
Algebraic series. If the trinomial A+Bx + Cx* be mul
tiplied by the quadrinomial a + bx + cx* + dx 3 , the terms being
arranged according to the powers of x, the operation is per
formed in a certain order, and although the same result would
be obtained by arranging the terms differently, it would not
be obtained so conveniently. This arrangement of the terms
according to the powers of a guiding letter, is more especially
requisite in the reverse operation of division in order to avoid
needless operations. If, for instance, it were proposed to divide
the product of the two polynomials above by one of them, the
arrangement of the terms according to the powers of x would
secure that the operation would be precisely the reverse of the
multiplication of this polynomial by the other according to the
same arrangement, and the quotient would thus be obtained in
the most direct manner. If the polynomials contained other
letters affected with indices, any one of them would answer
the same purpose.
But on the principle of extending and generalizing opera
tions it may be proposed to divide one algebraic polynomial
by another, although the former may not have resulted from
the multiplication of the latter by a third polynomial. In this
case the operation cannot terminate, and however performed
will leave a remainder. The truth of the operation depends
solely on the fact that the dividend is identically equal to the
product of the quotient and divisor, with the remainder added,
so that one side of the equality is equal to the other, whatever
real quantities be substituted for the same letters on both sides.
Thiskind of equality I have proposed to indicate by the mark =^
PURE CALCULATION. 29
The object of performing the operation may be, in the first
instance, to put the proposed ratio under another algebraic
form. But if the order of the successive steps be determined
by a selected letter, another object is answered. The proposed
quantity is thrown into a series, consisting of as many terms
as we please, arranged according to increasing or decreasing
powers of the guiding letter. The terms of the remainder
contain powers of the guiding letter higher by at least one
unit than the power contained in the last term of the series.
A series so formed may be useful for the purposes of
calculation. For suppose the guiding letter to represent a
very small quantity and its powers to increase : the terms of
the series will go on decreasing in value, and the remainder,
being multiplied by a high power of a small quantity, will on
that account be very small. By increasing the number of
terms we may dimmish the remainder as much as we please,
and the series deprived of it will for all purposes of calculation
be equivalent to the proposed ratio. This is a converging series.
In other cases the series is diverging and the equivalence of
the two sides of the equality does not hold good without
taking account of the remainder terms. A diverging series is
of no use for arithmetical calculation, unless it can be
converted into a converging series by transformations.
Like considerations apply to the extension of the extraction
of roots of polynomials to cases where the polynomials have
not resulted from involution. The operation being performed
by the same rule as if the polynomial were an exact power,
the root is thrown into a series the terms of which proceed
according to the powers of one of the letters. If P be the
proposed polynomial the n ih root of which is to be extracted,
and Q the sum of a certain number of extracted terms, then there
will be a remainder R such that P~=^ QT+R. The least power
of the guiding letter in R will be higher the farther the operation
is carried, and if the guiding letter represent a very small quan
tity, and the operation be carried far enough, the remainder
may for the purposes of numerical calculation be omitted.
30 THE PRINCIPLES OF
In diverging series, and in series non converging, the
remainder is necessary to constitute the identity of value of
the two sides of the equality, and cannot be left out of con
sideration. But a converging series consisting of an unlimited
number of terms, is identical in value without the remainder
with the quantity of which it is the expansion, the remainder
being indefinitely small.
By division it is found that
l+x ' l+x
n being any even number. If x 1, the lefthand side of the
equality = J, and the righthand side reduces itself to the
remainder term, which for this case becomes \. Thus it
appears that the identity of value of and its expansion
.1 "j SO
does not hold good when x = 1 without taking account of the
remainder term. This is also true if x have any value greater
than 1. But if x be less than 1 by any finite value however
small, by taking n large enough we may make the remainder
term less than any assigned quantity, and the identity of value
of the two sides of the equality may subsist as nearly as we
please when the remainder is omitted. The value for x = I is
the critical value between divergence and convergence. Such
critical values have no application in physical questions
unless the remainder can be calculated and is taken into
account.
The quantities in any series which multiply the powers of
the guiding letter are called coefficients.
There cannot be two converging expansions of the same
quantity, proceeding according to the same guiding letter,
the coefficients of which are not identical. For let
A + Bx + Cx* + &c. = a + Ix + ex* + &c.
whatever be x. Then if x = 0, A = a. Consequently
B+ Cx + &c. = b + cx + &c.,
whatever be x, and if x = 0, B=b. And so on.
PURE CALCULATION. 31
The above Theorem is the foundation of the method of
expansion by indeterminate coefficients. This method applies
whether the series be converging or diverging, because the
law of expansion is independent of the relative magnitudes of
the quantities represented by the letters. The value of the
remainder, after obtaining any number of terms of the
expansion, must in general be found by operating reversely
on the sum of those terms, and subtracting the result from the
quantity expanded.
The binomial and multinomial theorems, which are
methods of expressing generally the law of the coefficients of
an expansion, give the means of obtaining the expansions for
particular cases more readily than by employing the operations
of division and extraction of roots.
The investigation of the binomial theorem may be effected
by the method of indeterminate coefficients ; but for finding
the first and second terms of the expansion in the cases of
fractional and negative indices, it is necessary to have recourse
to the operations of division and extraction of roots performed
in the ordinary manner. The investigation of the multinomial
theorem may be made to depend on that of the binomial
theorem.
The expansion of a x in a series proceeding according to
the powers of x is effected by means of the binomial theorem
and the method of indeterminate coefficients.
Numerical Equations. Any question relating either to
abstract or to concrete quantities being proposed, the answer
to which may be obtained by the intervention of Algebra, the
given conditions of the question lead to an equation of this
form,
x n
n representing the dimensions of the equation, x the un
known quantity, and the coefficients p, q, &c. P, Q, being real
quantities positive or negative. Also if there are several
unknown quantities and as many different equations, the
32 THE PRINCIPLES OF
equation resulting from the elimination of all but one of them
is of the above form. Surd coefficients with the sign +, and
coefficients under an impossible form, are got rid of by involu
tion. Consequently the answer to the proposed question, if it
admits of a real quantitative answer, is obtained by extracting
from the final equation a real value of the unknown quantity
x which will satisfy the equation. It is found that negative
and impossible quantities, that is, numerical expressions under
an algebraical form, when substituted for x and operated
upon algebraically, will satisfy equations. Every value or
expression which satisfies an equation is called a root of the
equation. In a few instances roots of equations may be
found when the coefficients are literal. But in general only
equations with numerical coefficients can be solved, and these
for the most part require tentative or approximate processes.
The quadratic equation a? 2 +px + q = is equivalent to
(#+?) ^ + (7 = 0. and is satisfied if x be equal either to
\ 2/ 4
~~ 9 "*" \JtL ~ $ or * ~~ 9 ~~ \/A. ~~ ^' Calling tne se two quan
tities a and /3 we have the identical equation
This identity holds good whatever be the relative magnitudes
of p and q. But if q be a positive quantity greater than
2 , \ 2 2
^ , it is evident that the equation (x +j ^ f # = cannot
be true for any real value positive or negative of x, because
for such value the lefthand side will be the sum of two
positive quantities. In fact the roots a and fi are shewn
in this case to be impossible by containing the symbol
If
A/ j q, that is, the square root of a negative quantity.
If in the expression x* +px + q, q be positive and greater
than ^ , this expression possesses the property of not changing
PURE CALCULATION. 3B
sign whatever consecutive real values, positive or negative, be
substituted for x.
The general theory of the solution of equations rests on
principles analogous to those which apply to the simple case
of the quadratic. By direct multiplication it is known that
the product of n factors x a, x /S, x 7, &c., is an alge
braic polynomial of the form x n +px n ~ l + qx n ~ z + &c. + Px + Q,
and that if a, /3, 7, &c. be real quantities positive or negative,
the coefficients p, q, &c. P, Q, will also be real quantities
positive or negative. Any polynomial with numerical coeffi
cients which has actually resulted from such multiplication
being given, it is always possible by tentative methods to
arrive at the values of a, /S, 7, &c. In fact, if consecutive
numerical quantities separated by small differences, and ex
tending from a sufficiently large negative, to a sufficiently
large positive value, be substituted for x, among these must
be found the values of a, /3, 7, &c. nearly. Their actual values
may be approximated to as nearly as we please by interpolation.
If the number of such values be not equal to w, this circum
stance will indicate that two or more of them are equal. Let
the proposed polynomial contain /factors x 6. Then it may
be shewn by algebraic reasoning (given in Treatises on alge
braic equations) that the polynomial nx n ~ l + (n 1) px n ~* + &c.
+ P contains / 1 factors x 6. Consequently factors which
occur more than once are discoverable by the rule for finding
greatest common measures. Thus the reverse operation of
resolving a proposed polynomial which has resulted from
binomial factors into its component factors is practically
possible and complete.
But on the principle of extending algebraic operations for
the sake of generality in their applications, it may be proposed
to resolve into binomial factors a polynomial
x n +px n ~ l + qx n ~* + &c. + Px +Q,
which is not known to have resulted from the multiplication of
n binomial rational factors. The process of solution must be
3
34 THE PRINCIPLES OF
just the same as in the former case, and if after going through
it and finding the factors which occur more than once, the total
number of rational factors be some number m less than n, it
must be concluded that the proposed polynomial contains a
factor of n m dimensions which neither vanishes nor changes
sign whatever rational values positive or negative be put for x.
This polynomial must be of even dimensions, otherwise it
would vanish for a value of x between an infinitely large
negative and an infinitely large positive value.
In the manner above indicated it is shewn that the follow
ing identical equation is general, viz.
x n +px n ~ l + qx n ~* + &c. + Px+Q=s=: X(xa] (x /3) (x 7) &c.,
p, q, &c. and a, yS, 7, &c. being real numerical quantities
positive or negative. If the residual factor X be of two
dimensions with respect to a?, and be assumed to be identical
with the product (x k) (x l), then from the reasoning al
ready applied to a quadratic factor, we have & = + V &, and
l=a V b, b being a real positive quantity, and a being a
real quantity positive or negative. If X be of four dimensions
and be assumed to be identical with (x k) (x 1) (x m)
(x n), by the ordinary solutions of a biquadratic equation it
may be proved that k, I, m, and n are reducible to the forms
a 4. V J, a V b, a + V b', a V '; so that in this
case X is identical with {(x of + b] {(x a') 2 + b'}, a and a'
being real quantities positive or negative, and &, b' being real
positive quantities. The same direct reasoning cannot be
extended to a residual factor X of six dimensions, because no
general solution of an equation of six dimensions is known.
The above two instances, however, suggest the general
Theorem, that a rational polynomial which does not contain
any rational binomial factors, is resolvable into rational qua
dratic factors. It would be beyond the intention of these Notes
to attempt to give a general proof of this Theorem. Two
remarks may, however, be made. First, if a polynomial be
resolvable into quadratic factors of the kind above indicated, it
possesses the property of not vanishing or changing sign what
PURE CALCULATION. 35
ever real values positive or negative Ibe put for x, which is the
distinctive property of the residual factor X in the theory of
equations. And again, if there exist factors x k,x l, &c.,
the product of which is identical with a polynomial which
does not contain real binomial factors, then as &, I, &c., must
have impossible forms, it may be assumed h priori that the
forms are a V b ; because it can be shewn independently
of the theory of equations that every impossible expression is
reducible to those forms. In fact every algebraic expression,
when the 'letters are converted into numerical quantities, is
reducible to the forms A V B, A being any numerical
quantity affected with the positive or negative sign, and B
being a real positive quantity. If the expression be real B = 0.
According to the foregoing considerations, the method of
finding by the solution of an equation, an unknown quantity
subject to given quantitative relations, is in every respect
complete. It not only finds the unknown quantity if the con
ditions of the question be possible, but it also ascertains
whether proposed conditions are possible. When the condi
tions are possible the answer to the question is a real positive
root ; or if the question admits of several answers, there are at
least as many real positive roots of the equation. But if the
equation is found to contain only real negative roots, or, only
real negative roots having been found, if there remains a
residual factor incapable of vanishing or changing sign for
any real values of x, it must be concluded that the conditions
of the question are impossible. The same conclusion must
plainly be drawn if there are no real roots positive or
negative.
If it be enquired how negative and impossible roots
can result from the conditions of a question which are possible,
the answer is that the operations by which the equation is ob
tained in a rational form being algebraic, are necessarily per
formed on the symbol x not as representing quantity only,
but as representing quantity operated upon. Hence every
32
36 THE PRINCIPLES OF
numerical expression under an algebraic form which, operated
upon algebraically according to the conditions of the question,
satisfies the equation, must be represented by x. It some
times happens that the algebraic operations by which the
equation is formed introduce real positive roots which are not
answers to the proposed question.
The following instances are intended to illustrate the pre
ceding remarks.
(1) Let it be proposed to find a quantity which together
with its reciprocal makes up a given quantity greater than
unity ; the solution of the equation formed in accordance with
these conditions gives two real positive roots, because the
question admits of two answers. If the given quantity be less
than unity the roots are impossible, because the conditions of
the question are impossible.
(2) If the question be to find a quantity which together
with its square root makes up the number 6, the solution of
the equation gives two real positive roots 4 and 9. But the
latter number answers the proposed question only algebraically,
one of the algebraic square roots of 9 being 3.
(3) If it be required to find the number which multiplied
by a number less than itself by 2 gives the product 3, the
equation answers 3 and 1. The latter answer is algebraically
true, and for this reason was comprehended by the equation.
(4) Kequired the number which is exceeded by its cube
by 6 : that is, let x* x 6 = 0. The question is answered by
the number 2. Hence we have the identical equation
The factor x 2 + 2x+ 3, not changing sign nor vanishing
whatever real values positive or negative be put for x, shews
that there is but one real answer. The equation
x* + 2x + 3 = 0,
PURE CALCULATION. 37
gives "by its solution two impossible quantities 1 + V 2,
which, operated upon algebraically, must by substitution in
the equation x 3 x 6 = satisfy it, and for this reason are
symbolic roots of the equation.
The Calculus of Functions.
Any algebraic expression which contains a letter x is said
to be & function of x, and when this circumstance is to be
stated without reference to the particular form of the expres
sion, the symbol usually employed isf(x). Under this symbol
may be included all the forms which have their origin, in the
manner already indicated, in the principles of general arith
metic, and in the principles of algebra. Consequently any
reasoning which can be applied to such a symbol, will com
prehend all the forms of expressing quantitative relations
which we have hitherto discovered. This reasoning may be
called the Calculus of Functions. As the algebraic calculus
was independent of numerical values and relative magnitudes,
so the functional calculus is independent of particular algebraic
forms of expression.
It must be borne in mind that in seeking for literal and
general representations of quantity, the principle that deter
mined the forms of representation was that of expressing degrees
of quantity with as near an approach to continuity as we please.
Consequently as well the simple forms thus arrived at, as all
compound expressions resulting from operations upon them in
general arithmetic and algebra, must be regarded as susceptible
of values varying from one degree to the next by as small
differences as we please. The variation of value of any ex
pression may depend on the variation of value of one of the
letters which it contains, or of two, or of more. Accordingly
it may be & function of one variable, a function of two variables,
or a function of several variables. Each such function may
contain at the same time any number of constants. The Cal
38 THE PRINCIPLES OF
culus of Functions consists of parts rising in degrees of gene
rality and comprehensiveness according to the number of
variables which the functions are supposed to contain.
(1) The Calculus of Functions of one variable.
We shall designate a function of one variable by the sym
bol /(x). Putting y for any value of the function, we shall
have y=f(x), or yf(x) = 0. The sign = is here properly
used, because this is not an identity, but an equation. In
this instance y is an explicit function of x. But we might also
have such an equality as </> (x, y] = 0, the symbol on the left
hand side indicating that the function contains in some manner
both y and x. If this equation be regarded as solved according
to the theory of equations, y being the unknown quantity,
then we should have y =f(x] , or y an explicit function of x.
But prior to such solution y is called an implicit function of x.
The Calculus of Functions consists of two distinct parts,
analogous to the two parts into which, as we have seen,
algebra is divisible. The first part is concerned with properties
of functions and operations upon them : the other is analogous
to that part of the algebraic calculus, which relates to the
abstract formation of equations and the solution of them.
The Calculus of Functions, although it does not involve
indefinitely small quantities, is the foundation of the Dif
ferential and Integral Calculus, which, as the terms imply,
is essentially concerned with quantities regarded as admitting
of indefinitely small variations or increments. By the Dif
ferential Calculus, properly so called, an equation is formed
from certain data with the view of obtaining from it an un
known function: by the Integral Calculus the equation is
solved and the form of the unknown function obtained. The
solution of an algebraic equation gives a certain quantity:
the solution of a differential equation gives a certain algebraic
expression.
PURE CALCULATION. 39
On proceeding to reason generally on functions without
regard to their form, which is what is proposed to be done in
the Calculus of Functions, we may take for granted all the
results of the algebraic calculus.
A very general and important enquiry respecting any
function is the following : How may the value of it be ex
pressed when the variable receives any given increment? If
h be the given increment of the variable x, it might be
required to find a symbolical expression which shall be equi
valent to the new value f(x + h). In fact, from the prin
ciples of algebra we know that any such function may be
thrown into a series proceeding according to integral powers
of A, and that if a remainder term E be taken into account,
we shall have the identity,
f(x + A) =/() + Ah + Btf + Ch 5 + &c. + R,
A, B, 0, &c. being functions of x. The principles of algebra
furnish the means, in every particular instance, of deriving the
coefficient A from f(x). This coefficient is called the derived
function, and is expressed generally by the symbol f'(x). The
rules of obtaining f'(x) in particular cases from the primitive
function f(x), are often given in elementary Treatises under
the head of Differential Calculus. This is not logically cor
rect, because the dependence of f'(x) on /(a?) is simply a result
of algebraic analysis, without any reference whatever to differ
entials.
Rules for deriving the coefficients B, C, &c. from f(x) are
obtainable by algebra in some cases in which the forms of the
functions are given. But no general rule independent of the
forms of the functions can be deduced from algebraic principles
alone, although from algebra we may gather that these coeffi
cients are always in some manner dependent on the primitive
function f(x). This generalization is the proper office of the
Calculus of Functions, and the process by which it is effected
is next to be considered.
It is required, first, to find the derived function of the
40 THE PRINCIPLES OF
product of two functions. Let f(x) and $(x) be the functions.
Then
f(x + h) =/() +f(x)h + Bh z + &c.
<f>(x + h) =c <j)(x) + f () A +OT+ &c.
Hence
Consequently, by definition, the derived function of f(x) <f>(x) is
f(x)<f> (x) +f(x)(j>(x). I proceed now to find the general de
velopment of f(x + h).
For this purpose let us take the identical equation,
Putting z for x + h we shall have the following identical
equation containing z and x,
z ~~ x
substitute <j>(z, x), since that quantity may be
regarded as a function of z and x. Then
In this equation if z be considered constant, and x the only
variable, the identity of the two sides still remains for every
value of x. Hence any operation on one side will be equivalent
to the same operation on the other. Take the derived function
of each side with respect to the variable x. Then, having
regard to the value just obtained of the derived function of
the product of two functions, we shall have
=,=/() + f (*, x) (z  x) <f>(z, x).
Again, representing by dashes attached to the letters /and </>,
the order of the derived function, and taking successive de
rived functions with respect to x, we obtain
PURE CALCULATION. 41
=/" (x) + 4>"(g,x)(*x) 1$ (z  x)
=/"'(*) + f "(, *) (*) 3f (, ).
Hence substituting in succession the values of <f)(z, x), <j>'(z, x)
&c., it will be found that
This series, after putting x + k for z, may plainly be general
ized as follows,
f(x+h) =/(*) +f(x)+f'(x) + &c .
which is Taylor's Theorem*. The law of derivation of the
coefficients of A, A 2 , &c. from the primitive function, which it
was the object of the investigation to ascertain, is here plainly
exhibited. The last term is the representative of the remain
der term, which according to the principles of algebra was
found to be necessary in general to establish the identity
between the two sides of the equality. This term may be
assumed to be insignificant when k is very small, because the
smaller h is the nearer each side of the identity approaches to
f(x). It may also be remarked that the principle of investi
gating the above series by commencing with an identity is
strictly appropriate, because the algebraic operations, of which
the above process is a generalization, are all identical operations,
and the final result is an identity.
Taylor's Theorem has two important and extensive appli
cations. First, it is used to investigate JVIaclaurin's Theorem,
from which Lagrange's and Laplace's Theorems are deduced,
and accordingly it serves to generalize the developments of
functions of one variable whether explicit or implicit. Again,
it is the foundation of the method of forming differential
* This proof of Taylor's Theorem is given at length in Arts. 98100 of the
Treatise on the Differential Calculus by Baily and Lund.
42 THE PRINCIPLES OF
equations for the purpose of finding by their solutions unknown
functions which answer proposed questions. It is not necessary
for the object I have in view to speak of the first application,
and I shall, therefore, pass at once to the consideration of the
other.
Derived equations. If y =f(x) , or, more generally, if
</>(x,y) = 0, we have an equation containing two variables, and
if the form of the function be given and arbitrary values be
assigned to one of the variables as x, we can find corresponding
values of the other, real or symbolical. The number of such
coordinate values of x and y may be unlimited ; but the values
themselves are restricted by the condition of satisfying the
given equation. If, for the sake of illustration, x be repre
sented by a geometrical abscissa, and y by the corresponding
rectangular ordinate, the extremities of the ?/'s trace out a
curve. Values of x may be assumed corresponding to which
there are only impossible values of y. No point of the curve
answers to such values, which are only symbolically related to
each other.
Let y be equal to a given function of x, and let y ', y [ ', y"\
&c., represent the successive derived functions of y. Then by
previously established rules we can find the function of x
which is equal to any derived function of y, for instance, the
third. Let X be this function. Then we have the derived
equation y"' = X. As an example, let y = a + bx* + ex*.
Then y" 6b + 24cx. This is the simplest process for obtain
ing a derived equation, and gives the simplest form of such
equations. The reverse operation of remounting to the primi
tive equation from a derived equation of this form, is suggested
by the direct operation, and on this account, according to a
principle already stated, is to be included in a system of
general calculation. In fact, in the applications of analysis a
derived equation can be formed of which the primitive equation
is unknown and is required to be found. The reverse operation
is therefore a necessary part of calculation regarded as an
instrument of research. The rules for performing the reverse
PURE CALCULATION. 43
operation are known only by its being the reverse of the direct
operation. On this principle they have been investigated and
are given in elementary Treatises under the head of Integral
Calculus. It should further be remarked that as f(x) is
equally the derived function of f(x) and f(x) + c, c being a
constant of any value, in passing from any function to its
immediate primitive, an arbitrary constant should be added to
the latter for the sake of generality.
In the case in which y is an implicit function of x, that is,
when <f>(x, y) 0, let, as before, y', y", &c. represent the
successive derived functions of y regarded as a function of x.
Then if we take the derived function of <j>(x,y), it will in
general contain in some manner x, y and y', and may be
represented by ty(x, y, y\ so that (j)'(x, y) zx= ^(x, y, y). Now
it may be shewn as follows that the same corresponding values
of x and y that satisfy the equation <j>(x,y) = satisfy also the
equation <f)'(x, y) 0. For suppose that from the equation
<(#, y) = 0, y is obtained as an explicit function of x. By the
theory of equations there may be several such functions
according to the dimensions of the equation. By substituting
any one of them as %(x) in the equation <f>(x, y) = 0, so that
the equation becomes <f>{x, %(#)} = 0, we shall have an identical
equation. Hence the same operations on both sides of it will
give the same results. Consequently $'{x, x(x)} = 0. This
equation will be true if %(#) represent any of the other values
of y. Hence putting the general value y in place of %(#), we
have <f>'(x, y) 0, or ty(x, y, y') = 0. Similarly it may be
shewn that (j>"(x, y) =c <&(x, y, y', y") = ; and so on*.
In this manner from a given primitive equation may be
derived successive orders of derived equations. These ought
not to be called differential equations, because the formation of
them has required no consideration of differentials, or in
definitely small increments.
Since the corresponding values of x and y are the same in
* See Baily and Lund, Art. 42.
44 THE PRINCIPLES OF
all the equations thus derived as in the primitive equation, the
equations immediately derived may be combined with the
primitive in any manner consistent with algebraic rules, and
various other derived equations be formed all related to the
primitive equation. The object of forming and combining
such equations abstractedly, is to ascertain rules for remounting
from a derived equation to its primitive, when, as is usually
the case in the applications to concrete quantities, the derived
equation only is given. As these rules are essentially rules of
reverse operations, they must be found by first performing
direct operations and drawing inferences from them. Just in
the same manner abstract algebraic equations may be formed
ad libitum, and rules for solution be obtained to be afterwards
applied in solving equations formed according to the conditions
of proposed questions.
The primitive and its immediate derivatives may be em
ployed to eliminate constants. In general the number of
constants that may be eliminated is equal to the number of the
derived equations, or to the number indicating the order o the
resulting equation. The greater the number of constants thus
eliminated, the more the resulting equation is independent of
particular relations between the variables, and the farther is
the form of the primitive from being known. This process of
elimination is, therefore, the direct method of forming equa
tions containing two variables, one of which is an unknown
function of the other. In the reverse operation, by which the
solution of the equation is effected, the form of the unknown
function is ascertained, and the eliminated constants reappear
as arbitrary constants.
Another kind of elimination may be effected by means of
derived equations. Let the primitive equation be <(a?,y, u) = 0,
u being some function of x and y. Then u being the derived
function of u considered as a function of a?, the equation
immediately derived will be of the form M+ Ny' + Pu = 0,
Jf, N and P containing in general x, y and u. Now if u be
such that it makes P vanish, the elimination of u between the
PURE CALCULATION. 45
primitive <f>(x, y, u) = 0, and its immediate derivative M + Ny
= 0, will give the same resulting derived equation as if u had
been a constant. Hence it appears that in certain cases the
same derived equation of the first order may have two primi
tives, one of which contains an arbitrary constant, and by that
circumstance is distinguished from the other, which contains
no arbitrary constant. These primitives are so related that
they give the same value of y' for the same corresponding
values of x and y. The foregoing reasoning shews that when
the primitive <(#, y, c) = 0, which contains the arbitrary
constant c, is known, the other primitive may be obtained by
eliminating c between <(#, y, c) = 0, and the derivative taken
with respect to c only. In applications it often happens that
the equation containing the arbitrary constant or parameter, is
given by the given conditions of the proposed question, in
which case the relation between the variables which answers
the question is obtained by the direct process of elimination
just indicated.
If the elimination of c t and c 2 from the equation
<(#, y, c v C 2 ) = and its first and second derivatives, give the
same derived equation of the second order, whether c t and c 2
be constants, or be certain functions of x and y, that derived
equation has two primitives, one containing, and the other not
containing, arbitrary constants ; and these primitives give the
same values of y and y" for the same values of x and y. And
so on for derived equations of higher orders.
From the foregoing considerations it appears that by the
Calculus of Functions, the ultimate object of which is to
ascertain the forms of unknown functions, two kinds of
functions are obtainable, either definite functions containing
only given constants, or functions containing arbitrary con
stants. The arbitrary constants necessarily have their origin
in reverse processes ; but the definite functions, being in no
respect arbitrary, may be obtained by direct processes.
If there be n derived equations of the same order between
n + 1 variables, these may be reduced by direct processes of
46 THE PRINCIPLES OF
elimination to a single equation between two variables. The
function that one of these variables is of the other may then be
deduced by the solution of this resulting equation. Similarly
the function that any one of the other variables is of the same
variable may be found.
The method of obtaining in certain cases the primitive of
a derived equation by the Variation of Parameters rests upon
the foregoing conclusions. Let the known primitive of
^r(a:, y, y, y"} = 0, be <f>(x, y, c 19 c 2 ) = ; and R being a given
function of x and j/, let the primitive of ty(x,y,y',y") = R
be of the same form, c^ and c 2 being now variable. Then
assuming, in accordance with what is shewn above, that the
first derived equation may be the same whether c x and c 2 be
constant or variable, in the latter case let the derivative be
Then we must have P^\ + Q^\ = 0, and M^ + N^' = 0. Let
the derivative of this last equation be
*(*, y, y', y", ** O + PJ\ + QJ* = o.
By the elimination of y and y" between this equation and the
equations M t f N^y = 0, and *Jr(x, y, y, y") = jR, there will
result an equation of this form, Pc\ + Qc' z = JR, P and Q being
given functions of x, y, c^ and c 2 . Lastly, eliminating y from
this equation and from the equation P^c\ + Q\ = 0, by means
of <(x, y, c 1? cj = 0, we shall have two equations of the first
order between the variables c t , c 2 and x, which, as shewn in
the last paragraph, determine the functions that c^ and c 2 are
of x.
I proceed now to make use of Taylor's Theorem for laying
the foundation of the Differential and Integral Calculus.
By Taylor's Theorem, if the variable x of any function
f(x) receive an increment A, the consequent increment of the
function is given by the equality
f(x) =fWk +f'(x) +f'(x) + &c. + R, ;
PURE CALCULATION. 47
and this being an identical equation, we have also by putting
h for h,
fixh)f(x)=f(x)h+f'(x) f"(x) + &c
Hence by subtraction,
03
f(x + h) fix  h) =/'(*) 2*+/ ~ 4 &c. + JZ,  S 2 .
Now since the terms on the righthand side of this equality
after the first are multiplied by h 3 , h 5 , &c., and E v R z may be
multiplied by as high a power of h as we please, it follows
that h may be taken so small that the first term shall be in
comparably greater than the sum of all the other terms
inclusive of R t R 2 . This is true in certain cases even when
the values of /""(a?) and succeeding derived functions are in
definitely great. Hence representing by dx the indefinitely
small portion 2h of the variable #, and by d.f(x) the
corresponding portion of the function, we have as nearly as
we please,
d.fix) =f(x)dx.
The quantity d.f(x) is called the differential of the function
f(x), and dx is the differential of the variable x. Hence the
above result may be thus expressed : The differential of any
function of a variable is identically equal to the product of the
first derivative of the function and the differential of the variable.
This Theorem is the foundation of the DiiFerential Calculus,
and connects it with the Calculus of Functions. The Theorem
is true, as the reasoning by which it was arrived at proves,
with as near an approach to exactness as we please : and, as
already remarked, it is only in these terms that we can assert
of calculation in general that it is true*.
As d.f(x) and dx, however small they may be, must,
according to the principle of their derivation, be considered
quantities and treated as such, we have
* This general Theorem and its application in calculations relating to concrete
quantities, was the great discovery of the Newtonian epoch of mathematical science.
48 THE PRINCIPLES OF
Since it may be shewn by arithmetical reasoning that two
numerical quantities, taken as small as we please, have always
a ratio to each other, the lefthand side of the above equality
may be called the ratio of the differential of the function to the
differential of the variable ; and the equality proves that the
ultimate or limiting value of the ratio is the first derivative of
the function. For this reason the ratio is called a differential
coefficient, because it is equivalent to /'(a?), the coefficient
of h in the expansion off(x + A)*.
By the same reasoning as before,
d.f(x) =f'(x)dx.
Hence multiplying by dx,
dxd.f(x)=3=f"(x)dxdx.
Assuming now that dx is invariable, the differential off(x) dx
will be dx d.f'(x), which is consequently equal to the differen
tial of d.f(x), or d.d.f(x). Hence
d.d. t f(x}^f(x}dxdx.
Putting dx* for dx dx, and indicating the order of differentia
tion by a number attached to d,
dx 1
:/
^ x
dx n *** '
The above results will be seen to be of great importance
when it is considered, that the answers to questions relating to
concrete quantities are in a great variety of cases given by
functions of a variable, and that in order to find the unknown
functions it is necessary in general to form in the first instance
differential equations by reasoning upon indefinitely small
* It should be remarked that in the foregoing reasoning a distinction is made
between increment and differential.
PURE CALCULATION. 49
increments*. These equations are always convertible, by
means of the above equalities, into derivative equations, the
solutions of which may be effected by rules the investigation
of which belongs to the Calculus of Functions.
If we substitute a single letter y for /(a?), the successive
differential coefficients of f(x) will be written, , 
j~ . As the identity of differential coefficients and derived
functions has been proved, the notation for the former may be
used to express the latter. The differential notation is
especially appropriate in the applications of analysis, because
in them arises the necessity of reasoning upon differential
quantities.
In applications it often happens that an unknown function
of a variable may be expressed generally and explicitly in
terms of the differential coefficients of another function of the
same variable, so that when the latter function is given the
unknown function may be found by differentiation. Such ex
pressions, however, are obtained by reasoning upon indefinitely
small quantities. This remark is exemplified in geometry by
the theory of contacts.
Integration. We have seen that h may be taken so small
that
Substituting in this identical equation x + 2h for x, we have
f(x + 3A) f(x + h] = *hf(x + 2A),
so f(x + 5 A) f(x + 3 h) =t= 2hf(x + 4&)
f{x + (2w  l).h] f{x + (2w  3). h} 3= 2hf[x + (2w 
By adding all these equalities together,
* The modern history of applied mathematical science shews that this mode of
reasoning is indispensable.
4
50 THE PRINCIPLES OF
f{x+ (2w
If we suppose that x h = a, and x + (2n l)h = 5, we shall
have b a = 2w^ ; so that the difference between the values
b and a of the variable x is divided into n parts or increments
each equal to 2k. The n terms on the righthand side of the
above equality are the values of the n corresponding increments
of the function f(x) . Consequently f(b) f(a) is equal to the
sum of those increments, the number of which must be in
definitely great, because 2h is by hypothesis indefinitely small.
This result is expressed as follows :
that is, the sum of the differentials d.f(x) which lie between
the limiting values a and b of x is equal to the excess of the
function f(b) above the function /(a)*. Hence to find such a
sum between given limits, which in the applications of analysis
is a frequent and an important operation, it is only necessary
to obtain by the Calculus of Functions f(x) from its derivative
f'(x) supposed to be given.
The meaning of the term Integration, which is the reverse
of Differentiation, is in this manner apparent, when a differen
tial coefficient is given as an explicit function of the variable.
But in the different orders of differential equations, in which
the differential coefficients are implicit functions of the variable,
the applicability of the term is not so obvious. It may suffice
to say that in these cases the arbitrary constants evolved by
integrating the equation give the means of satisfying proposed
conditions.
There is often occasion to find the value of an integral be
tween the limits zero and infinity of the variable. As infinity
is an indefinite limit, this value can be obtained only in case
f(x) converges to zero in proportion as x is increased. Thus,
as is well known,
* See Todhunter's Integral Calculus. Chap. i. Arts. 19.
PURE CALCULATION. 51
/oo /oo
e~ ax cos xdx = and I e~ ax sin xdx =
1+a 2 J rt
1 + a 2 '
a being any positive quantity however small, if only ax
becomes eventually an infinite quantity when x is indefinitely
increased. For in this case e~ ax cos x and e~ ax sin x ultimately
vanish. But if a be absolutely zero, this is no longer true,
/ 00 / 00
and the integrals I cos xdx and I sin xdx become indefinite
J o J o
on account of the indefiniteness of co . Such integrals cannot,
therefore, have any application in physical questions.
(2) The Calculus of Functions of two variables.
The step from the Calculus of Functions of one variable to
the Calculus of Functions of two independent variables, is a
generalization of the same kind as that from the algebraic
calculus to the former.
The abstract questions to be answered respecting a function
of two variables are analogous to those already answered
respecting a function of one variable. Representing generally
a function of two variables by the symbol /(a?, y), we have first
to ascertain in what manner its value may be generally repre
sented when the variables receive given increments h and k.
This enquiry may be answered by means of Taylor's Theorem.
For, supposing at first that x changes to x + h, y remaining
constant, we have by Taylor's Theorem,
f(x + h, y)=f(x, y) +f(x, y)  +f"(x, y) + ...
But it must here be remarked that the above notation does
not indicate that f'(x, y), f"(x, y), &c., are derived functions
taken with respect to x only. If, therefore, the functional
notation be retained, it will be necessary to add some mark of
distinction, as Lagrange has done in his Galcul des Fonctions,
Legon xix. Since, however, we have proved that a derived
function and a differential coefficient are identical, it will be
42
52 THE PRINCIPLES OF
more convenient to adopt at once the differential notation.
For this purpose put u for f(x, y) for the sake of brevity of
expression. Then j may represent the derived function, or
differential coefficient, of u, taken with respect to x only, and
j that taken with respect to y only. These are called partial
e!/
differential coefficients of u. The above series thus becomes,
, ,x , du h , d 2 u ft d n u h n
f (x+ h,y)=u + ^ l + ^ T  2 + ...+_____ +ftA 
This being an identity we may change y into y + k on both sides,
and the identity still remains. But by this change, as above,
, du k d*u k 2
Becomes + + . +...+ <
, du 72 du
j 7 U . j j d . y 72
du , du dxk dx k
becomes
and so on. By substituting these values in the first equality,
we obtain the well known expansion of 'f(x + A, y + k) . It will
, du
~d~~
be seen that the coefficient of We in this expansion is _?.
If we had supposed that# first changed to y +k, and then x to
7 du
H~
x + h, the coefficient of hk would have been ^ . Hence as
dx
the expansions in the two cases must be identical, we have
, du j du
d.j d.j
dx ay
dy ~ dx
This equality is usually written for the sake of brevity,
dx dy ~~ c dy dx '
PURE CALCULATION. 53
The notation above employed is very generally adopted,
although as a differential notation it is defective, and is at
tended on that account with some obscurity and inconvenience.
It has been agreed that the ratio of one quantity to another
shall be represented by placing the former above the latter
//?/
with a line between. Consequently the symbol y must
mean the ratio of the differential of u to the differential of x y
and so long as it retains the form of a differential coefficient,
it may serve to indicate at the same time that the differential
of u is taken with respect to x only. But if the differential
dx be removed by any operation from the denominator, there
is nothing to indicate that du is taken with the above limita
tion. On this account solely, and not from any principle of
calculation, = must retain the form of a differential coefficient.
dx
But this restriction may easily be got rid of if we distinguish
by notation what in the calculus of differentials is actually
distinct. Having u a function of two variables x and y, we
may be required to distinguish the differential of u on the
supposition that x only varies, from its differential when y
only varies. In fact, a necessity for doing this often occurs
in the applications of analysis. I propose to represent the
former differential by the symbol d x u and the latter by the
symbol d y u. I am aware that the same notation has been
employed to signify differential coefficients, with the intention
of getting rid of representations of indefinitely small increments.
But a notation for this purpose is liable to the objection that
it tends to perpetuate a confusion between the principles of
the Calculus of Functions and those of the Differential Cal
culus. Lagrange has fully shewn that the consideration of
indefinitely small increments is not essential to the Calculus of
Functions, and in accordance with this view makes no use of
the letter d, which is the appropriate mark of a differential or
indefinitely small increment. There is inconsistency in using
this letter and at the same time excluding the consideration
of differentials.
54 THE PRINCIPLES OF
The proposed notation being adopted, ^ and j will be
quantities under a fractional form, which may be operated
upon according to the rules applicable to fractions. Hence,
since the variations of x and y are independent so that both
dx and dy may be constant, we shall have
and _^_
dy dx
Thus, since dy dx = dx dy, it follows that d v d x u = d x d v u, or
that the order of the differentiation is indifferent.
We might now go on to shew how the expansion of
f(x + h, y + k) by Taylor's Theorem may be employed to
establish rules for expanding functions of two variables : but
the purpose of these Notes rather requires us to deduce from
that expansion the differential of a function of two variables, as
we have already deduced from the expansion off(x f A) the
differential of a function of one variable.
Using the proposed notation, we have
d r u , d..u ,
d x u 1? d y d x u
dx* 1.2 +
+ &C.
This being an identity will remain such if h and k be changed
to h and k on both sides. Hence
d*u h 2 d y d x u , , dyU k z
i '~dx I ~l^ Jt "3yd~x +~df 172
&c.
PUKE CALCULATION. 55
Consequently by reasoning as in the case of a function of
one variable, h and k may be taken so small that we have as
nearly as we please,
Let d .f(x, y] represent the differential of the function corre
sponding to the differentials 2h and 2k of the variables, and
let the latter quantities be represented by dx and dy. Then
7 , N d x u 7 . d..u ,
or, as the equality may also be written,
du rrr d x u + d y u.
This result proves that the complete differential of a function of
two variables is the sum of the partial differentials taken with
respect to the variables separately.
By an extension of this rule,
and so on. If d x u ^x=pdx and d y u
du =apdx + qdy.
Also if dp ;3= rdx + sdy and dq n= sdx + tdy, by what has been
shewn, s = s. Hence
d 2 u re rdx* + Zsdxdy + tdy*,
and similarly the succeeding differentials may be formed.
By means of these equalities an equation resulting from the
consideration of partial differentials (such as frequently occurs
in the applications of analysis) is always convertible into an
equation between the partial differential coefficients p, q, r, 5, t,
&c., or, what is equivalent, into an equation of. partial derived
functions. The answer to the question proposed to be solved
by forming the partial differential equation, is then obtained
by finding the primitive of the equation of partial derived
functions according to rules established by the Calculus of
56 THE PRINCIPLES OF
Functions. We have, therefore, to enquire how these rules
are discovered.
Equations of partial derived functions. Let z =f(x, y), or
more generally let z be an implicit function of x and y, so that
</>(z, x, y) 0. Then, taking the partial derived function with
respect to x, and putting p for = , we shall have ^(z, x, y,p)
= ; for it may be shewn precisely as in the case of an equa
tion between two variables, that these two equations hold
good for the same corresponding values of z, x, and y. So
%(z, x, y, q) = 0, q being put for  . The two latter equations
y
may be employed to effect an elimination of a higher order of
generality than the elimination effected by derived equations
of two variables. By these, constants of arbitrary value were
eliminated. By the partial derivatives of an equation between
three variables, one of which is regarded as a function of the
other two, we may eliminate arbitrary functions, provided they
are functions of expressions containing the variables in a given
manner. Thus let z =zx* + *. Then
P =
So
Hence eliminating f'(x* + y 2 ), py qx 0. This is an equa
tion of partial derived functions arising from the elimination
of the arbitrary function /(a? 2 + y 2 ).
By proceeding to partial derived functions of the second
order, two arbitrary functions may be eliminated ; and so on.
It is clear that in this manner an unlimited number of partial
derived equations may be formed, and may be arranged in
orders and their composition be examined, with the view of
obtaining rules for performing the reverse operation of passing
from derivative equations to their primitives. These abstract
processes, not requiring the consideration of indefinitely small
quantities, form a part of the general Calculus of Functions.
Their use will be apparent by considering that in the applica
PURE CALCULATION. 57
tions of analysis equations of partial differentials are in the
first instance formed Iby reasoning upon indefinitely small
quantities, and that these, being converted into equations of
partial differential coefficients, are identical with equations of
derived functions. Their primitives, which answer the pro
posed questions, are consequently obtainable by the previously
established rules. The forms of the arbitrary functions in the
primitives are determined by satisfying given conditions ; and
as the solutions are more comprehensive than those of equations
of two variables, the conditions to be satisfied may embrace a
proportionately larger number of particulars. In fact, the
abstract processes by which the two kinds of differential equa
tions are formed, are determined by the principle of making the
one as independent as possible of particular values, and the
other of particular algebraic forms.
It should be especially noticed that the functions of arbi
trary form contained in the primitives of partial differential
equations of three variables, are functions of algebraic expres
sions containing the variables in a definite manner. The
forms of these expressions are determined by the solutions of
the equations, and are in no respect arbitrary.
If a differential equation contains four variables, the arbi
trary functions of its primitive are functions of two independent
expressions containing the variables. And so on to still higher
orders of generality.
Differential equations containing two variables, as well as
equations containing three or more variables, frequently do
not admit of exact solution, when formed according to the
conditions of proposed questions. An equation, however,
containing three variables, which does not admit of a general
exact solution, may sometimes be exactly satisfied by a parti
cular relation between the variables.
There are means of solving by approximation equations of
every class that cannot be solved by exact processes.
The maximum and minimum values of algebraic functions,
whether of one, or two, or more variables, are obtainable by
58 THE PRINCIPLES OF
known rules, the investigation of which, requiring the con
sideration of indefinitely small quantities, is properly put
under the head of Differential Calculus.
The object of the Calculus of Variations is to find, functions
whether of one, or two, or more variables, which possess
maximum or minimum properties. This Calculus is, therefore,
of a more comprehensive order than the calculus of maximum
and minimum values.
With these miscellaneous notes I conclude the consider
ation of the principles of Pure Calculation. It is not pretended
that the subject has been treated with any degree of complete
ness, but enough has been said to enable me to state the
principles of the application of calculation to the ideas of
space, time, matter and force, and the modes of investigating
relations between these quantities in answer to proposed ques
tions. But before proceeding to the head of Applied Mathema
tics, it will be proper to give a summary account of the main
results arrived at respecting Pure Calculation.
A system of Pure Calculation may be established by rea
soning on abstract quantity antecedently to the consideration
of properties of space, time, matter, and force. The results of
such reasoning may subsequently be applied to each of these
kinds of quantity. The leading principles on which Pure Cal
culation rests are: (1) The representation of quantity by
numbers with as much exactness as we please, and different
degrees of quantity with as near an approach to continuity as
we please. (2) Direct and reverse operations on such repre
sentations of quantity, analogous to, and arising out of, the fun
damental operations of addition and subtraction. (3) Symbo
lic generalization for the purpose of including in the reasoning
as many particulars as possible. From these principles are
derived numerical, algebraical, and functional forms of ex
pressing quantity. The rules of operating on quantity are
first established by numbers. The reasoning on literal, or
algebraical, expressions of quantity consists of two parts : in
the first, the arithmetical operations are generalized so as to
PURE CALCULATION. 59
be independent of particular values, and rules for expressing
the same quantity in various symbolic forms are investigated ;
in the other, these rules are applied to representatives of un
known quantities, or to general representatives of known quan
tities, for the purpose of forming equations by the solution of
which the unknown quantities become known. In the former,
the equality between two expressions of the same quantity is
indicated by the sign =0= ; in the latter, the equality of two
expressions involving the unknown quantity is indicated by
the sign =. The establishment of rules for operating upon
and transforming algebraic expressions necessarily precedes
the formation and solution of equations. The Calculus of
Functions of one variable is in like manner divisible into two
parts, the first treating of operations upon, and transformations
of, functional quantities, and the other containing the applica
tion of these rules of operation in the formation of equations,
with the object of finding by their solutions unknown functions.
And so on, with respect to the Calculus of Functions of two or
more variables.
The Differential Calculus, which is indispensable in the
applications of analysis, rests on the general proposition proved
by the Calculus of Functions, that the ultimate ratio of the
indefinitely small increment of a function of one or more
variables, to the corresponding indefinitely small increment,
of one of the variables, is identical with the first derivative of
the function with respect to that variable.
General Principles of Applied Calculation.
Each department of applied mathematics has its appro
priate definitions, by which it is distinguished from every
other. The definitions are the basis of applied calculation, or
reasoning. What is defined is simple matter of fact or expe
rience, and is not arrived at by reasoning ; although the case
may be that a definition admits of being deduced by reasoning
60 THE PRINCIPLES OF APPLIED CALCULATION.
from other more general principles as yet undiscovered. The
object in general of the applied reasoning is to trace the con
sequences of the definitions for the purpose of comparing the
mathematical results with observed facts, and referring the
latter to their elementary causes. By this means the facts are
explained, and brought under command for purposes of utility.
The reasoning is nothing else than the application of those
principles and rules of abstract calculation which have already
come under consideration, and which may be regarded as the
axioms of any applied science. It is admitted that we may
apply to concrete quantities the ideas of number and ratio
which have been shewn to be the basis of abstract calculation,
and that we may reason upon such quantities by the rules of
calculation which were abstractedly deduced from those fun
damental ideas. Thus, for instance, all the complex proper
ties and relations of space are deduced by calculation from
simple properties which are immediately perceived and must
be defined. And so with respect to time, matter, and force.
It is true that these three species of quantity are generally
considered in connection with space, and the reasoning applied
to them, taking for granted properties of space, is sometimes
said to be geometrical. But it must be borne in mind that
the geometrical properties have become known by calculation,
and that, consequently, there is no reasoning on concrete
quantities which does not virtually involve the principles and
the rules of abstract calculation* .
The principles of Geometry.
The science of Geometry embraces all relations of space
however ascertained, and must, therefore, be taken to include
not only the Propositions in the Books of Euclid ordinarily
* These considerations may serve to shew the propriety of naming by the word
ratio (reason), that which is the foundation of all calculation, the simplest form of
ratio, the ratio of equality, being included. Abstract numbers are collections of
equal units, and therefore involve the conception of a ratio of equality.
GEOMETRY. 61
read (exclusive of Book v.), but also Trigonometry, Analytical
Geometry of two and of three dimensions, and the various
properties of curves and curved surfaces commonly treated of
in the Differential and Integral Calculus. These different di
visions of the subject all consist really of deductions by calcu
lation from certain elementary principles which first of all
have to be stated in the form of Definitions. The initial
principles and definitions of Geometry will be best studied by
referring to the Elements of Euclid.
Geometrical definitions are of three kinds: (1) Those
which express our primary ideas of space, such as the defini
tions of a straight line, an angle, a plane, &c. (2) Those
which by means of the first class define certain simple forms,
the triangle, the square, and the circle, from the properties of
which all calculation of relative positions and superficial mag
nitudes is derived. (3) Definitions of other forms, as the
rhombus, trapezium, hexagon, ellipse, &c. the properties of
which are found by the application of theorems obtained from
the definitions of the simple forms.
A definition ought to exclude whatever differs from the
thing defined, and to include nothing that can be proved.
Euclid's definition of a square, viz. that it is " a foursided
figure which has all its sides equal and all its angles right
angles," is not exactly conformable to the second part of this
rule, because a figure of four equal sides which has one angle
a right angle may be proved to have all its angles right
angles.
Euclid's definition of parallel straight lines has up to the
present time been a subject of discussion. The questions that
have been raised respecting it appear to admit of answers on
the principles above maintained, as may be seen from the
following considerations. It has been argued that calculation
of abstract quantity rests on the ideas of equality and of ratio.
Hence definitions of concrete quantity embrace equalities and
ratios which being immediately perceived do not admit of
being inferred. Thus a right angle is defined by the equality
62 THE PRINCIPLES OP APPLIED CALCULATION.
of two adjacent angles. On the same principle, after defining
an angle to be " the inclination of two straight lines to one
another," parallel straight lines might be defined in these
terms : Two straight lines equally inclined to any the same
straight line towards the same parts are said to be parallel.
Or this form of the definition may be used : Parallel straight
lines are equally inclined towards the same parts to any the
same straight line.
The equality asserted in this definition is a simple concep
tion not requiring nor admitting of proof. Euclid's definition
of parallel straight lines, viz. " that being produced ever so far
both ways they do not meet," is rendered unnecessary by the
proposed definition. Besides, it expresses a property of pa
rallel straight lines which may be deduced as a corollary from
the demonstration of Proposition xvi. of Book I. For if in
case of parallelism according to our definition, the two lines
could meet, a triangle would be formed, and the exterior angle
would be greater than the interior, which is contrary to the
hypothesis. This property, admitting of being thus inferred,
cannot logically be made a definition of parallel straight lines.
But it may be urged that the proposed definition is itself
proved in the Prop. xxix. To this objection I reply, that the
proof of that Proposition is not effected without the interven
tion of Axiom xn. Now if that axiom be properly made the
basis of such reasoning, it should be included among the
definitions. But it does not profess to be a definition. It is,
in fact, a proposition capable of proof by means of the defini
tion of parallel straight lines for which I am arguing, as will
be presently shewn.
The definitions of a parallelogram and its diameter, usually
attached to Prop, xxxiv. of Book I, might apparently have
been placed with the definitions preceding that Book. Of the
definitions preceding the other geometrical Books, the greater
part are special, coming under the third of the classes above
mentioned. But there are some which are expressions of
general conceptions respecting rectilinear forms, and on that
GEOMETRY. 63
account rank with those of the first class ; as, for instance,
that which defines similar rectilinear figures to be "those
which have their several angles equal, each to each, and the
sides about the equal angles proportional." (Def. I. B. VI.)
There is nothing in this statement that admits of proof, the
definition merely giving expression to our conception of form
as being independent of actual magnitude. As ratio is a fun
damental idea applying to quantity generally, so it may be
applied specially to define similarity of form.
The definition of like forms may be generalized so as to
embrace curvilinear as well as rectilinear figures. For this
purpose it suffices to conceive of two similar figures as
similarly situated with respect to a common point. Then the
figures are such that a straight line drawn in any direction
from the point to the outer figure will be divided in a given
ratio by the inner figure. Definition xi. of Book in. asserts
that " similar segments of circles are those which contain
equal angles." This is not strictly a definition, because it ad
mits of being inferred from the above general definition of
like forms, by means of Prop. II. of Book vi, as may be thus
shewn. Place the bases AB, AC of the segments so that they
coincide in direction, and one extremity of each is at the point
A. Draw any straight line from A, cutting the arcs in D
and E, and join DB and CE. Then because by hypothesis
AD is to AE as AB to AC, by the Proposition cited DB
is parallel to EC. Hence the angle ADB in one segment is
equal to the angle AEC in the other.
The Postulates which are prefixed to Book I. require us to
admit that certain geometrical operations may be performed,
without respect to the manner of performing them. In fact
they appeal to our conceptions, and for all the purposes of
reasoning might be expressed tims :
Any two points may be conceived to be joined by a
straight line.
Any terminated straight line may be conceived to admit
of unlimited extension.
64 THE PRINCIPLES OF APPLIED CALCULATION.
A circle may be supposed to have any position for its
centre and a radius of any magnitude.
The following is another postulate of the same kind, which
we shall have occasion to refer to hereafter :
A straight line passing through any point may be con
ceived to be parallel to another straight line.
Although the words postulate and axiom do not differ in
signification, the former might, for the sake of distinction^
designate axioms that relate to space, while the word axiom
might be exclusively applied to abstract quantity. According
to this distinction, the axioms vin. x. and XI. of Book i,
which assert that " magnitudes which exactly fill the same
space are equal," that " two straight lines cannot enclose a
space," and that " all right angles are equal," would come
under the head of postulates. Like the other postulates they
require us to admit the existence of properties of space not
capable of demonstration, but of which, by experience, we
have distinct conceptions. Whether or not this division of
axioms into two classes be adopted, the two classes are really
separate, because the remaining axioms (excepting the twelfth)
relate to abstract quantity, and do not more belong to geo
metry than to any other department of applied mathematics.
The twelfth axiom of Book I. can neither be called a pos
tulate nor a definition, because it admits of demonstration on
principles which have been already stated, as I now proceed
to shew.
" If a straight line meets two straight lines, so as to make
the two interior angles on the same side of it taken together
less than two right angles, these straight lines being con
tinually produced, shall at length meet upon that side on
which are the angles which are less than two right angles."
Let the straight line ABC* meet the straight lines BD,
CE in the points B and (7, and let the angles DBG and ECB
be together less than two right angles. Let it be admitted,
according to a postulate previously enunciated, that a straight
* The reader is requested to form a figure for himself.
J, 1 i.i iv A
i UN i V K US 1 TY O
GEOMETRY. 65
line BF, passing through the point B, may be parallel to the
straight line CE. Then by the definition of parallels the
angle ABF is equal to the angle BCE. Hence adding the
angle CBF to each, the two ABF and FBC together are
equal to the two FBC and BCE together, and the former
being equal to two right angles, the latter are also equal to
two right angles. But the angles DBG and BCE are together
less than two right angles. Hence by Axiom v. the angle
FBC is greater than the angle DBG, and consequently the
straight line BD is inclined from BF towards the straight
line CE. But by the definition of parallels, BF and CE
are inclined by the same angle to any the same straight line
towards the same parts. Hence if BD be produced far enough
it will be inclined to CE by an angle equal to FBD and
towards the same parts. Hence BD produced must cross CE
produced.
Although the axioms of Euclid that relate to abstract
quantity, viz. I. VII. and IX., contain only affirmations of
the simplest kind, yet the principle upon which the terms
double and half occur in them, may be extended to quantitative
expressions of every kind, whether numerical, literal or func
tional. In fact, as already said, any part of abstract calcula
tion which admits of being applied in the determination of
relations of space, is axiomatic with respect to Geometry. On
this principle the Propositions of Book v. are applied in Book
VI. The reasoning in the Elements of Euclid is remarkable
for requiring the use of very few and very simple quantitative
expressions. As much as possible the reasoning is conducted
by means of equalities and ratios of graphic representations of
lines and figures, and the order and logical connection of the
different Propositions are chiefly determined by this circum
stance. This character of the ancient Geometry appears to
have been partly due to the rigid exactness which the cultiva
tors of it endeavoured to give to their reasoning, and partly to
the imperfect knowledge they had of symbolical and abstract
calculation.
5
66 THE PRINCIPLES OP APPLIED CALCULATION.
The Propositions of Euclid are divided into Theorems and
Problems. In the former properties of space are enunciated
and then proved to be true ; in the latter geometrical construc
tions are first described and are then shewn to be proper for
effecting what the Problem required to be done. After the
statement of the construction, the reasoning by which the
required conditions are proved to be satisfied is just like the
demonstration of a Theorem. In the Propositions of both
kinds, the reasoning is what is called Synthetical ; that is, the
enunciated property of space is shewn to be true, but is not
arrived at deductively, and the given construction is proved
to be the solution of the Problem, but by what steps it was,
or might be, discovered is not made apparent. The other
kind of reasoning, the Analytical, by which properties of space
are investigated, and solutions of Problems arrived at, is
necessarily conducted by quantitative symbols, and may be
rendered in a great measure independent of sensible represen
tations of lines and forms. The analytical method is especially
adapted for research, and for extending our knowledge of the
multifarious relations of space.
The truth of a geometrical Theorem and the demonstration
of its truth are not dependent upon our being able to perform
any geometrical construction. The solutions of geometrical
Problems by constructions are rather to be regarded as appli
cations of previously demonstrated Theorems to purposes of
utility and special research, and are analogous to solutions of
equations in abstract calculation. In fact, the solution by the
analytical method of a geometrical Problem is generally given
by an equation, from which an appropriate geometrical con
struction may be inferred.
Although the Propositions of Euclid, like all other geo
metrical propositions, are virtually only deductions from the
geometrical definitions by the application of the principles of
abstract calculation, yet this fact is not obviously exhibited in
the Elements of Euclid on two accounts : first, because the
reasoning is synthetical and ill adapted to present the process
GEOMETRY. 67
of deduction, and again, as no use is made of any symbolic
expression of quantity, the reasoning is necessarily conducted
by graphic representation to the eye of the quantities con
cerned. These two circumstances determine, for the most
part, the character of the reasoning and the order of the Pro
positions. Now it may be admitted without hesitation that in
point of strictness of reasoning the ancient geometers left
nothing to be desired, and that the Elements of Euclid must
ever be regarded as perfect examples of reasoning from given
principles, and the best possible illustration of the art of logic.
But when the question is concerning the intimate nature of
the processes by which the human mind has acquired in these
times so great a command over the complicated relations and
properties of space, the modern analytical methods cannot be
left out of consideration. By taking these into account it is
found that after establishing a few elementary propositions
by a direct appeal to definitions, all others are deducible by
analytical reasoning, and that the order of deduction is not the
same as that of Euclid. To illustrate this remark by an
instance, Proposition 8 of Book I. enunciates the equality
of two triangles under certain positive conditions. But the
demonstration of the equality is effected by means of a
reductio ad absurdum. This kind of proof, although con
vincing, cannot be regarded as indispensable for proving a
proposition of that kind. By the analytical method the same
equality is deduced from the given conditions in a direct
manner, but in a more advanced stage of the science, as I
shall have occasion to shew ^further on. The difference in the
order of deduction is due to difference in the process of rea
soning. It may here be remarked that the analytical method
never requires the introduction of the reductio ad absurdum
proof, and in this respect appears to be more complete than
that of Euclid. The proper office of that kind of proof is to
detect a false hypothesis, or false argument ; but for estab
lishing an actual property of space, it would seem that there
must always exist some direct process.
52
68 THE PRINCIPLES OF APPLIED CALCULATION.
I proceed now to indicate in the order of logical deduction
Propositions of Euclid on which a system of Analytical Geo
metry of two dimensions might be founded. I omit all
reference to constructions, on the principle that in proving
Theorems they may be regarded as Postulates.
Book I. Prop. 4. The proof of the equality of two tri
angles, one of which has two sides and the included angle
respectively equal to two sides and the included angle of the
other, depends on no previous proposition, and appeals only to
the simplest conceptions of space.
I. 5 and 6. These depend only on I. 4.
I. 26. The former part of this Proposition (to which alone
I refer) demonstrates the equality of two triangles, one of
which has two angles and the included side respectively equal
to two angles and the included side of the other, and might,
like I. 4, be proved by the principle of superposition. In
Euclid it is proved, with the help of I. 4, by a reductio ad ab
surdum. This proof can hardly be regarded as any thing
more than putting into formal evidence the impossibility of
not perceiving immediately the equality of the two triangles
when one is applied to the other.
I. 13 and 15. The equality of any two adjacent angles to
two right angles, proved in the former of these propositions,
is really a deduction, though of the simplest kind. But the
equality of opposite angles when two straight lines cross each
other, is perceived immediately from the very conception of
straight lines and angles, to which an appeal might at once
have been made without intermediate reasoning. We have
here an instance, like others that occur in the Elements of
Euclid, of superfluity of reasoning.
I. 29. If parallel straight lines be defined as proposed in
p. 62, the equality of the alternate angles follows from I. 15.
I. 32. The exterior angle of a triangle is proved to be
equal to the two opposite interior angles, and the three interior
angles are proved to be equal to two right angles, from the defi
nition of parallel straight lines and by I. 29, with the aid of
GEOMETRY. 69
the Postulate, that a straight line may pass through any point
parallel to another straight line.
I. 34. The proof that the opposite sides and angles of a
parallelogram are equal to one another, and that the diameter
bisects it, depends on I. 29, I. 26 and I. 4.
i. 35. The equality of parallelograms on the same base
and between the same parallels is proved by the definition of
parallels, and by I. 34, and I. 4.
I. 37. The equality of triangles on the same base and
between the same parallels is proved by I. 35", and I. 34.
1. 41. That a triangle is half a parallelogram on the
same base and between the same parallels is proved by I. 37,
and I. 34.
I. 43. Proved by I. 34.
I. 47. This is essentially an elementary Proposition of
Geometry, and such, consequently, are all those that are
necessary for the proof of it. The proof depends immediately
on I. 4, and I. 41, and on the definition of parallels.
II. 4. The Propositions employed in the proof are I. 29,
5, 6, 34, and 43.
II. 7. Depends on I. 43, and II. 4.
II. 12 and 13. These are proved by I. 47, II. 4, and II. 7.
On these two Propositions depend the mutual relations of the
sides and angles of a triangle, as treated of in Trigonometry.
in. 16. This is an elementary Proposition of a particular
kind. It ought, perhaps, in strictness to be regarded as a
definition of contact, involving considerations which are. appro
priate to the Differential Calculus. The reductio ad absurdum
proof applied to it serves to give distinctness to the conception
of the definition.
In addition to the above there are the elementary Proposi
tions VI. 1, and VI. 33, which are proved by the fifth Defini
tion of Book v. Having called in question the logic of that
definition, I shall now give reasons for concluding that the
use of it in the proof of these Propositions is unnecessary.
With respect to parallelograms between the same parallels, it
70 THE PRINCIPLES OF APPLIED CALCULATION.
has been shewn in Book I. that they are equal to rectangles
on the same bases and between the same parallels. But two
rectangles between the same parallels are to each other in the
ratio of their bases, as will be perceived immediately by con
ceiving them placed so that an extremity of the base of one
coincides with an extremity of the base of the other, and the
larger rectangle includes the less. This is a case in which the
same kind of appeal must be made to our conception of ratios
applied to space, as in the definition of similar rectilinear
figures. Any train of reasoning, like that founded on the fifth
Definition, is superfluous, seeing that the equality of the ratios
is as immediately perceived as any steps of such reasoning.
The rectangles being in the ratio of the bases, the parallelo
grams may be inferred to be in the same ratio. The same
argument applies, mutatis mutandis, to triangles between the
same parallels.
Similar remarks may be made on Prop. 33 of the same
Book. It is not possible to insert any argument between the
statement that two arcs of the same circle, or of equal circles,
are proportional to the central angles which they subtend, and
a rational perception of the truth of the statement. The pro
portionality is seen at once by an unaided exercise of the
reason, and consequently there is no room for the application
of reasoning such as that founded on Def. 5.
The above enumeration includes all the elementary Propo
sitions required for the foundation of Analytical Geometry of
two dimensions. If we except Proposition 16 of Book in.,
all the others may be divided into two classes, those relating
to the determination of the relative positions of two points,
and those relating to the determination of areas. .
It is now the place to make a remark which has an impor
tant bearing on a general enquiry into the principles of applied
calculation. The above Propositions, though usually referred
to as the foundation of Trigonometry and Analytical Geometry,
do not contain all the elements of these branches of Mathe
matics. If, for instance, it were required to find the length of
GEOMETRY. 71
the hypothenuse of a rightangled triangle, the lengths of the
sides being given, the Elements of Euclid do not enable us to
answer this question, except by mechanical construction. It
is no answer to say that the square standing on the hypothe
nuse is equal to the sum of the squares standing on the two
sides. To deduce the required quantity from this equality, it
is absolutely necessary to be able to express by numbers, both
the length of a straight line and the area of a square the
length of the side of which is given in numbers. Thus the
general application of calculation to space requires the intro
duction of a principle which holds no place in the ancient
geometry*. The necessity for this additional principle is an
important part of the evidence for the truth of the generalization
which it is the main object of these Notes to establish, viz.
that all reasoning upon concrete quantities is nothing but the
application of the principles and results of abstract calculation
to definitions of their qualities.
The manner in which the length of a straight line is ex
pressed in numbers by reference to an arbitrary unit of length
has already been sufficiently stated in page 6. The following
are the principles on which a rectangular area is expressed in
numbers by reference to an arbitrary unit of area. The re
ference unit of area must be a square, because it must involve
no other linear quantity than the unit of length. First, sup
pose two adjacent sides of the rectangle "to contain each an
integer number of units of length, as 5 and 9. Then conceiving
straight lines parallel to these sides to pass through the points
* The general use of a cumbrous system of notation by the Greeks and Romans
may possibly account for their not introducing into Geometry the principle of mea
sures. If we admit that they were acquainted with this principle, and if we also
admit with M. Chasles (Comptes Rendus Jan. 21, 1839), that the device of place in
numeration was not unknown to them, the facts still remain that the old notations
were not superseded, and numerical measures were not allowed to come within the
precincts of their Geometry. The rapid progress that geometrical science has made
since the time of Descartes, when the representation of lines by numerical mea
sures and algebraic symbols was fully recognised as an instrument of reasoning,
is in some sort a proof that this manner of reasoning is an essential principle of
applied calculation.
72 THE PRINCIPLES OF APPLIED CALCULATION.
which divide them into the aliquot parts, the rectangle will be
divided into spaces which may be shewn, by Propositions
already established, to be squares each equal to the unit of
area. The number of the squares is plainly 5x9, or 45,
which number consequently expresses the ratio of the super
ficies of the rectangle to that of the unit of area, or, as this
ratio is called, the area of the rectangle. If now the sides
containing 9 units be increased in length by the fractional
part f of a unit, and the dividing lines parallel to them be
equally extended, by completing the rectangle there will be
formed 5 additional spaces each of which has the ratio f to
the unit of area*. The whole area of the rectangle will thus
be 5 x 9 + 5 x f . Let now the sides containing 5 units be
increased in length by the fraction f of a unit. Then, for the
same reason as before, the area of the rectangle will be in
creased by 9 x , and, in addition, the column of fractional
spaces will be increased by the fractional part J of one of these
spaces, that is, by a space which has the ratio f to a space
which has the ratio f to the unit of area. But by Prop. I.
(p. 9), the quantity which has the ratio f to the quantity
3x5
whose value is f , has the value   . This last quantity
4: X I
being put, in accordance with the reasoning in p. 15, under
the form f x f , the whole area of the rectangle will be
which is what results by the rule of multiplication from
that is, from the multiplication of the quantities which express
the lengths of the sides of the rectangle f.
The same result is perhaps more simply arrived at thus.
* This is assumed on the principle stated in p. 70.
f This instance serves to explain the distinction which was made in abstract
calculation between taking any quantity a number of times and a quantity of times.
The last expression, which, taken abstractedly, is not very intelligible, here receives
a definite meaning.
GEOMETRY. 73
Conceive the unit of length to be divided into 28 equal parts,
that is, a number of parts equal to the product of the denomi
nators of the fractions f and f. Then one side of the rectangle
contains 9 x 28 +f x 28, or 9 x 28 + 5 x 4 of those parts, and
the other contains 5 x 28 + f x 28, or 5 x 28 + 3 x 7. Hence
by the same reasoning as that above, the whole rectangle
contains (9 x 28 + 5 x 4) x (5 x 28 + 3 x 7) small squares such
that the unit of area contains 28 x 28. Consequently the
ratio of these two numbers, which is the area of the rectangle, is
(9 x 28 + 5 x 4) x (5 x 28 + 3 x 7)
28x28 ~'
or, (9 + f) x (5 + 1), as above.
As the same reasoning might be employed whatever be
the ratios which express the lengths of the sides, the conclusion
may be stated generally in these terms. If a and b be the
lengths of the sides of a rectangle, expressed numerically by
reference to an assumed unit, then the numerical quantity ab
is the area of the rectangle referred to a square unit the side
of which is the unit of length.
In the case of any square area a = Z>, and the area = a*.
Hence, if , b, c be the lengths of the sides of a rightangled
triangle, by Euclid (i. 47) we have 2 = &' 2 +c 2 , a being the
length of the hypothenuse. When b and c are given in num
bers, the righthand side of this equality is a known numerical
quantity, by the extraction of the square root of which a is
found. To obtain this result it has been absolutely necessary
to make use of the principle of measures.
The opinion is held by some mathematicians that a dis
tinction should be scrupulously maintained between pure
Geometry, that is, the Geometry of the Elements of Euclid,
in which the reasoning is conducted by equalities, ratios, and
diagrams, and analytical Geometry, which employs symbols
of numerical measures of lines and areas. But though there
is this difference between the sensible means by which the
reasoning is carried on, there is no difference in ultimate
74 THE PRINCIPLES OF APPLIED CALCULATION.
principle between the two kinds of reasoning, the deductions
in both being made from the same definitions, and from a few
elementary Propositions the evidence for which requires a direct
appeal to our conceptions of space. It must, however, be
observed that the method of Euclid is essentially incomplete,
failing for want of the principle of measures, (as in the instance
just considered), to give answers to questions which must
necessarily be proposed. The analytical method, on the con
trary, is quite general, and is comprehensive of the other.
There is consequently no logical fault in the practice, frequently
adopted in mathematical Treatises, of joining reasoning con
ducted by geometrical diagrams and constructions, with reason
ing by symbolic representatives of lines and areas*. The
former kind of reasoning, except in the elementary Propositions
above referred to, is not indispensable ; but it frequently has
the advantage of aiding our conception of the process of de
monstration, and is capable of arriving at certain results with
much greater brevity than the general method of symbols.
As it appears that measures are indispensable in Geometry,
let us adopt this principle in limine, and enquire in that case
what are the elementary Propositions on which analytical
Geometry of two dimensions might be founded. I wish it,
however, to be understood that in entering on this enquiry my
object is not to propose a method of studying Geometry different
from that ordinarily taught. Excepting that, as I have already
urged, the reading of Book v. of Euclid might be dispensed
with, I see no reason to deviate from the usual practice of
teaching the elements of Geometry from Euclid. The sole
object I have in view in pointing out a course of demonstration
different from that of Euclid, is to ascertain the essential
principles of the application of calculation to Geometry.
The initial Propositions of Geometry relate either to the
determination of the relative positions of two points, or the
Excepting only that in giving the demonstrations of Euclid it would be im
proper to write AB 2 for the square of AS, because the Elements of Euclid contain
no numerical measures of lines.
GEOMETRY. 75
calculation of areas. The former depend on properties of the
triangle and circle, the other on properties of the square and
rectangle. The properties of the triangle are first to be consi
dered. As abstract calculation was founded on equalities and
ratios, let us commence the consideration of the triangle with
the application of these conceptions. It may be admitted as
selfevident, that two triangles, which, when applied one to the
other, are coincident in all respects, are equal. Also Euclid's
definition of similar rectilinear figures, viz. that they have their
several angles equal, each to each, and the sides about the
equal angles proportional, may be regarded as a necessary
and fundamental definition of Geometry. Being strictly a
definition, it is properly made the basis of reasoning.
There are various Propositions in Geometry, which relate
to the conditions of the equality of two triangles, but the
following, which is strictly elementary, is the only one which
is appropriate to the course of reasoning I propose to adopt :
If two angles and the included side of one triangle be equal,
each to each, to two angles and the included side of another
triangle, the two triangles are equal in all respects. This Pro
position is proved by the principle of superposition, neither re
quiring, nor admitting of, any other direct proof. For if one
triangle be placed on the other so that the equal sides and
equal angles are coincident, the coincidence and consequent
equality of the other parts may be perceived immediately.
By the aid of the foregoing Proposition we may find ele
mentary conditions under which two triangles are similar. Let
the triangles A and B have two angles of the one respectively
equal to two angles of the other, and let C be another triangle
similar to A. Then because A and C are similar, by definition,
the angles of C are severally equal to those of A. Hence G
has two angles respectively equal to two angles ofB. Also since
by the definition of similar rectilinear figures, the similarity
is independent of magnitude, it may be supposed that C has
a side equal to a side of B, and that the equal sides are
adjacent to the angles that are respectively equal. But in that
76 THE PRINCIPLES OF APPLIED CALCULATION.
case, by what is shewn above, C is equal to B in all respects.
And C is similar to A. Therefore B is similar to A. Hence it
follows that two triangles which have two angles of the one
respectively equal to two angles of the other, are similar.
We can now proceed to calculate the length of the hypo
thenuse of a rightangled triangle, the lengths of the other
two sides being given.
Let ABC be a triangle*, rightangled at A\ and conceive
another rightangled triangle, having its hypothenuse equal in
length to AB, and an angle equal to the angle ABC, to be so
placed that that angle coincides with the angle ABC, and the
hypothenuse with AB. Then its right angle being at D, BD
will be part of the straight line BC, and ADB being a right
angle, by the definition of right angles ADC will also be a right
angle. Hence each of the triangles ADB and AD C has two
angles respectively equal to two angles of the triangle ABC.
Consequently, by what has been proved, each of these triangles
is similar to the triangle ABC. Now let the lengths BC, AC
and AB be respectively represented by the symbols of quan
tity a, b, c. Then because the triangles ABC and ABD are
similar, by the definition of similar rectilinear figures BD is to
BA as BA to BC; or
BD c <?
 , and .'. BD =  .
c a a
SoDC=. Hence BD + DC =  +  = C ^^. But
a a a a
BD + DC=AC=a.
<? + b*
Hence a , and /. a 2 = b 2 + c 2 .
a
It thus appears that by commencing with the principle of
measures and symbolic representation of lines, this relation
between the sides of a rightangled triangle is deducible from
definitions by calculation, antecedently to any consideration
of areas.
* The reader is requested to draw a diagram, if one should be required.
GEOMETRY. 77
COROLLARY 1. It is evident that the two angles ABC and
ACB, being on account of the similarity of the triangles
respectively equal to DA G and DAB, are together equal to a
right angle, and that the three angles of the rightangled
triangle ABC are therefore equal to two right angles.
COROLLARY 2. Since if one of the acute angles of a
rightangled triangle be given, two angles are given, it follows
that the form of the triangle and the ratios of the sides are
given. Hence if the acute angle B of the rightangled triangle
ABC be given, the ratio of AB to BG is given. Hence this
ratio is a function of B, which we may designate by the usual
notation cos B. Thus we have
AB = A C cos B, or c = a cos B. So b = a cos C.
The foregoing Proposition and the two Corollaries deter
mine all the relations of the sides and angles of a rightangled
triangle, when the function that cos a? is of the angle x is
known. The process by which this function is found will be
considered hereafter. I proceed now to infer from the case of
the rightangled triangle, the relations of the parts of any tri
angle.
Let ABC be any triangle*, and let the angle ABC be
acute. Conceive a rightangled triangle ABD to be such and
so placed, that its hypothenuse is equal to and coincident with
AB, and one of its acute angles is coincident with the angle
ABC. Let
BC= a, AC= b, AB=c,
First, let BC be greater than BD. Then c 2 =/ + q\ and
V =/ + ( a  q) z =P* + f + * 2a0 = c 2 + a 2  2aq.
Next, let BD be greater than BC, in which case the tri
angle has an obtuse angle. Then c 2 =^? 2 + <f and
b*=p*+(q a) 2 = c 2 + a 2  2aq, as before.
* The diagram is the same as the preceding, excepting that the angle BA C is
not now a right angle.
78 THE PRINCIPLES OF APPLIED CALCULATION.
Again, in the second case let BD exceed SO by q, so that
q = a + q. Then by substitution,
tf = J + a 2 _ 2a(a + q) = c 2  a 2  2a^'.
Consequently c 2 = a 2 + 6 2 + 2^', c being the length of the side
opposite the obtuse angle.
According to previous notation q = c cos B. Hence
& 2 = c 2 + a 2  2ac cos .#,
the angle J5 being acute.
The Theorem that " the angles which one straight line
makes with another are together equal to two right angles,"
is inferred by very simple calculation from the definition of
right angles. Hence if the symbol TT represent two right
angles, and C be the obtuse angle of the triangle, the angle
adjacent to it is TT G. Consequently by the notation already
used, q T) cos (TT C) . Therefore c 2 = a 2 + & 2 + 2ab cos (TT G) ,
c being opposite the obtuse angle. The function of C ex
pressed by cos (TT C) will be found subsequently.
If a rightangled triangle be applied, in the manner before
stated, to any side of an acuteangled triangle, or to a side of
an obtuseangled triangle containing the obtuse angle, the
triangle will be divided into two rightangled triangles, and it
will be apparent that the sum of its angles is equal to the sum
of the acute angles of these two triangles. But the latter sum
has been proved to be equal to two right angles. Therefore
the three angles of every triangle are together equal to two
right angles.
Hence it follows, since an exterior angle of a triangle and
its adjacent interior angle are together equal to two right angles,
that the exterior angle is equal to the sum of the two interior
opposite angles.
The above series of deductions have all been made from
the rightangled triangle by the principle of superposition,
without geometrical constructions, and without reference to
parallels and areas. In Euclid (i. 32), the equality of the
GEOMETRY. 79
three angles of a triangle to two right angles is deduced from
parallels by a construction. It is, however, an important
illustration of the principles of Geometry to shew that this
method of proof is not indispensable, and that all relations
between the sides and angles of a triangle flow from the pro
perties of the rightangled triangle.
But when we come to the calculation of areas, parallels are
indispensable, as the following reasoning will shew. Parallels
being defined as stated in p. 62, the equality of alternate angles
is a consequence of the equality of the opposite angles made
by the intersection of two straight lines. A rectangular
parallelogram is divided by its diagonal into two rightangled
triangles which have a common side, the angles adjacent to
which, being alternate angles, are equal each to each. Conse
quently the triangles are equal in all respects. But by reasoning
which rests only on properties of the square and of parallels,
and on the selfevident Proposition that rectangles between the
same parallels are proportional to their bases, it has been shewn
(p. 72) that the area of a rectangle is equal to the product of
two adjacent sides. Hence the area of a rightangled triangle
is half the product of its base and altitude.
Every other triangle is shewn, by the process of applying a
rightangled triangle in the manner already employed several
times, to be the sum or the difference of two rightangled
triangles having the same vertex. Hence if p be the altitude,
and q, q the bases of the two rightangled triangles, the area
of the given triangle is f (q + q'), or ^ (q q). But its base
A &
a is equal to q + q, or q q. Hence its area =? .
2
By the same reasoning as that just now applied to a rect
angular parallelogram, it may be shewn that every parallelo
gram is double of one of the triangles into which its diagonal
divides it. Hence the area of every parallelogram is equal to
the product of its altitude and base.
These results might have been obtained consistently with
80 THE PRINCIPLES OF APPLIED CALCULATION.
the principles of our reasoning, just as in Euclid, by the aid
of Prop. 4 of Book I. since that Proposition is strictly elemen
tary, and is proved by the principle of superposition. But it
was of some importance to shew that the same condition of the
equality of two triangles, viz. the equality of a side and the
adjacent angles of one to a side and the adjacent angles of the
other, which was necessarily employed in the initial Proposi
tions relating to position, sufficed in those relating to area.
This condition has led in p. 78 to values of a side of a triangle
expressed in terms of the opposite angle and the sides includ
ing it, from which Prop. 4 might be inferred.
Algebraic Geometry of two dimensions. In this application
of calculation to space, the position of a point is determined by
its distances from two straight lines cutting each other at right
angles, and the distances are represented by literal symbols.
But although these distances fix the position of the point, they
do not immediately give all the information that might be re
quired respecting its position relative to other points. If, for
instance, it be asked, What is the distance, and what is the
bearing, or angular direction, of the point from the intersection
of the two reference lines?, the answer must be given by means
of the properties of the rightangled triangle proved above.
If x and y be the coordinates which determine the position of
the point, its distance (r) from the origin of coordinates is
Va? 2 + y z , and its angular direction with reference to the axis
sv* rv*
of a;, depends on the ratio , , or  . This answer is not
Vaj 2 + y 2 r
complete till we have shewn how to infer the angle from the
ratio.
Hitherto our reasoning with literal symbols of length has
been conducted according to the rules of general arithmetic.
But by availing ourselves of algebraic calculation, the reasoning
may be rendered more comprehensive. If two points are
situated on the same straight line, we cannot state how they
are posited with respect to each other, except by reference to
a third point in the same straight line. The position of the
GEOMETRY. 81
third point C being fixed, we may say of the other two A and
B, that A is more or less distant from C than B, and thus
indicate relative position. Let GA=a and CBb t and let
the two points be on the same side of C. Then if a be greater
than b, the distance of A from B in the direction from C,
which may be called the positive direction, is a b. But if a
be less than &, A is distant from B by b a in the direction
toioards C, or the negative direction. Consequently if the
letter c represent the difference between a and b, + c may in
dicate that A is distant from B by the length c, and that it
lies from B in the direction which it is agreed to call positive,
and then c will indicate that A is distant from B by the
length c, and that it lies from b in the opposite or negative
direction. Thus the use of signs dispenses with the reference
to the third point (7, although such reference is always implied.
Now in abstract algebraic reasoning, as we have seen (p. 24),
the symbol x was taken to be inclusive of negative as well as
positive quantities, and this, consequently, must be the case
in the application of Algebra to Geometry. The coordinates
of a point are represented generally by x and y, whether the
point be situated on the positive or negative side of either axis
of coordinates, and it is only when the signs of these symbols
are determined by special relations and conditions that they
indicate direction.
Measures of angles. To complete the application of calcu
lation to geometry of two dimensions, it is necessary to apply
the principle of measurement to angles. As an angle is not a
linear quantity, the measure is effected by means of a linear
arc to which the angle is always proportional. The angles
formed at the centre of a given circle by its radii, are propor
tional to the subtending arcs of the circle. This is a Propo
sition, which, as I have before said, admits of no demonstration,
the perception of its truth being simply an instance of that
perception of ratios which is the foundation of all calculation.
This Proposition being taken for granted, an angle may be
measured in two ways. Either the whole circumference of
6
82 THE PRINCIPLES OF APPLIED CALCULATION.
the circle may be conceived to be divided into a certain number
of equal parts, and the quantity of parts in the arc subtending
any angle, be taken as the measure of the angle : or, the radius
of the circle being the unit of length, the angle may be
measured by the length, referred to this unit, of the subtending
arc.
Recurring now to the triangle ABC, rightangled at A,
suppose BG to be the radius of a circle, the arc of which is cut
by BA produced in JS. Then if the arc CE be given in the
first kind o measure, in order to calculate the ratio ^^ , we
require to know the quantity of equal parts both in BC and
BA. But if the arc CE\>Q given in the other kind of measure,
then that ratio is equal to AB expressed in the same measure.
The latter case is first to be considered. Let s equal the
length of the arc CE, radius being the unit. Then representing
AB by the symbol cos s, because s is now the measure of the
angle A, we have to solve the following Problem : To find
the function that cos s is of s.
As the answer to this question is a function, according to
the previous explanation of the principles of abstract calcula
tion, the function is to be sought for by means of a differential
equation. The following will consequently be the course of
the reasoning*.
Let x and y be the coordinates of a point of any curve,
the form of which is determined by the relation between the
variables expressed by the equation y=f[x). And suppose
the curve to be cut by a straight line in two points whose
abscissae are x h and x + h. Then if y l and y z be the
ordinates of the points of section, by Taylor's Theorem,
y, = 
Although this process of reasoning would be altogether unfit for teaching
Geometry, it may yet be proper for elucidating the principles on which calcula
tion is applied, and might be advantageously attended to by those who have learnt
the science in the usual way.
GEOMETRY. 83
Hence
y,  y, =
Therefore putting dy for y^ y l and dx for 2A when A is
indefinitely diminished, we have
Now the ratio ^ 2 , ^ determines generally the angle of di
rection of the cutting line relatively to the axis of abscissae.
But when the points of section are indefinitely near, the secant
becomes a tangent, and the ultimate ratio ~ determines its
directionangle. It may be remarked that in the above
method of finding that ratio, the first of the omitted terms
contains A 2 , and consequently the equation is true even when
f'(x) and the other derived functions are infinitely great*.
Another remark may also be here made. The secant in its
ultimate position as tangent, must still be regarded as passing
through two points of the curve, otherwise its position is not
determined in any manner connected with the curve. Hence
it follows that for an indefinitely small portion, the curve is
ultimately coincident with the tangent and may be regarded
as rectilinear. Thfts although we may be able to conceive of the
curvature of a curve as absolutely continuous, so far as calcu
lation is concerned the curve must be treated as if it were
made up of indefinitely small rectilinear portions, approach
ing to continuity of curvature as nearly as we please. This is
an instance of that peculiarity of calculation alluded to at
the very beginning of the subject (p. 7), according to which
numerical values necessarily proceed gradatim.
* This is not shewn in the processes by which the value of ^ is usually obtained
in Treatises on the Differential Calculus.
62
84 THE PRINCIPLES OF APPLIED CALCULATION.
By these considerations it will be seen that if ds be the
differential of the curve, the ultimate relation between it and
the differentials of x and y, is
d?=d&*chf
Hence ds z = dx\l + [f (x)}*}
</>(#) representing the unknown function that 5 is of x. This
is the differential equation by the integration of which for any
given curve s becomes known.
In the instance of a circle of radius ,
and consequently
dx Va 2 a; 2 '
The integration of this equation according to rule gives
C
If the arbitrary constant c be determined on the supposition
that x = a when s = 0, it follows that c = a; and as  = cos s,
it may readily be shewn, if e be the bas of the Napierian
Logarithms, that
e*V~i + eV r ~i
cos s =    .
2
This result answers the proposed question. The function is
not, however, suited for numerical calculation ; but by alge
braic expansion of the exponentials we obtain
coss= !_
a series which is always eventually convergent.
GEOMETRY. 85
It suffices for making good the argument to have indicated
how by the application of pure calculation, the value of the
function cos s for any given value of s may be found, although
the Tables of cosines of arcs have been actually calculated by
processes different from this.
11 I H?
The ratio  , or \ / 1 5 is the function of the arc s which
a v a
/ / Y 
is called sin s. Since sin s = A/ 1 5 = Vl (cos s)*, it will
be found that
sin s =
By a reversion of this series according to the method of inde
terminate coefficients, s is obtainable in a converging series
proceeding according to powers of sins, and by putting
sin s = 1 , the numerical value of a quarter of the circumference
of a circle whose radius = 1 might be calculated.
Suppose that by this calculation the ratio of the circum
ference of a circle to its diameter were found to be 3,14159 &c.
Then dividing the circumference into 360 x 60 x 60 equal parts
the number of these parts in an arc of the circle equal in
length to the radius is ascertained by a proportion to be
206265 quamproxime. If, therefore, the measure of any angle
be given by a certain number of the equal parts into which
the whole circumference is divided, the ratio of the arc con
taining that number of parts to the radius of the circle is
known, and as that ratio is the quantity s, cos s may be calcu
lated as before. This completes the explanation of measures
of angles, and of the methods of calculating the ratio cos A.
The sole object of the foregoing reasoning has been to derive
from elementary principles, by a logical course of deduction,
the necessary processes of calculation applied to Geometry.
86 THE PKINCIPLES OF APPLIED CALCULATION.
Trigonometry. This part of Analytical Geometry is prin
cipally concerned, as the name implies, with the relations of
the parts of triangles. But under this title is also placed the
investigation of certain formulce relating to arcs, which are
useful not only in calculations applied to triangles, but also in
a great variety of other applications. These formulae may
be divided into two classes, the first of which consists of ex
pressions for the trigonometrical lines tan s, cotan s, sec s,
cosec s and versin s, in terms of the two lines sin s and cos s,
the relations of which to the arc s and to each other have
already been investigated. The value of versin s is radius
cos s. The values of the other lines in terms of sin s and
coss are obtained according to their definitions from right
angled triangles by simple proportions. These different functions
of the arc are all used, not from necessity, but for the sake of
brevity, both in symbolical and numerical calculation ; and to
expedite the use of them in obtaining numerical results, they
have been calculated and tabulated for arcs differing by one
/v nt
minute, or one second, of arc. Since cos s =  and sin s ^ , the
a a
signs of these functions in the four quadrants of the circle are
determined by the algebraic considerations which have been
already applied to the coordinates x and y. The signs of the
other trigonometrical lines are determined by their analytical
relations to these. The whole circumference being represented
by 27T, and the radius being = 1, the values of sin s for the
arcs 0, , TT and , are seen immediately to be 0, + 1, 0, 1,
and those of cos s to be +1, 0, 1, 0. The corresponding
values of the other trigonometrical lines are derived from
these by means of their analytical relations to sin s and cos s.
The other class of trigonometrical formulas are expressions
for the sines, cosines, &c. of the sums, differences, multiples,
and submultiples of arcs, values of the sums and differences
of sines and cosines, &c. These are all deducible from four
fundamental formulas, viz. those for the sines and cosines of
GEOMETRY. 87
the sum and difference of two arcs in terms of the sines and
cosines of the simple arcs, which are usually proved by the
intervention of a geometrical diagram. It is, however, a sig
nificant circumstance with reference to the principles of applied
calculation, that this method of deriving them is not indis
pensable. They admit of a strictly analytical deduction, as
may be thus shewn. If 6 and < be any two arcs of a circle
whose radius = 1 , then from what has been proved,
"^1 sin d = e^ l  e~^^ and 2 cos == e^
and so for the arc <. Also by putting 6 + < in the place of 0,
2\T^~T sin (6 + </>)= e( e
Hence by inference from the algebraic formula,
*
ab
we have
6 e\Ci e ^ ) vrT_g.0V^i e 0V~i =:= 2\/^] sin 6 cos $+2*J~l cos 6 sin <f) ;
.'. sin (@ + <t>) sin 0eos $> f cos $ sin <>.
The other three formulae may be obtained in an analogous
manner.
Since by the principle of the investigation of these
formulae the values of 6 and < are not restricted, we may sub
stitute for either of them the semicircumference 'TT, or any
multiple of it. Let TT be substituted for 6 in the formula for
the cosine of the difference of two arcs, viz.
cos (6 (j>) = cos 6 cos <> 4 sin sin fc.
Then taking account of the values
sin TT = and cos TT = 1 ,
we shall obtain cos (TT <f>) = cos <f>.
88 THE PRINCIPLES OF APPLIED CALCULATION.
We may now recur to the equations obtained in p. 78, ex
pressing relations between the sides and angles of a triangle.
The angle B being acute, it was found that
I 2 = a 2 + c 2  2ac cos B,
and the angle C being obtuse, that
c 2 = a 2 + I} 2 + 2ab cos (TT  C) .
But by what has just been shewn, cos (TT C) cos C. Hence
Consequently the forms of the expressions are the same in
both cases. We have thus finally arrived at an equation
which suffices for calculating all the relations of the sides and
angles of a triangle. If, for instance, from the equation
C S = ~
we obtain sin C, and from the analogous value of cos B we
obtain sin B, it will be shewn that sin B and sin C are to each
other in the ratio of the opposite sides b and c. Also we
might obtain sin (A + B) and sin C as functions of the sides,
and it would then appear that sin (A + B) = sin (7, and con
sequently that the sum of the three angles of a triangle are
equal to two right angles. As this relation between the
angles of a triangle has not been used in the previous course
of reasoning, it may be regarded as being strictly deduced in
this way by analytical calculation from elementary principles.
Calculation of areas. The general calculation of areas
might be made to depend, as is usually done, on the calcula
tion of the area of a rectangle. But as we have deduced by a
strictly elementary process the area of a rightangled triangle
from that of a rectangle, the former may be assumed to be
known in the investigation of a method of ca ] culating areas
generally. If the extremities of two ordinates of a curve be
joined by a straight line, the area bounded by this line, the
two ordinates, and the portion (h) of the axis of abscissae
GEOMETRY. 89
intercepted between the ordinates, will coincide, when h is
indefinitely diminished, with the corresponding area bounded
by the arc of the curve ; because, as we have before seen,
on passing to differentials the arc and chord must be treated
as coincident, or as having to each other a ratio of equality.
When h has a finite value, the first term of the series express
ing the difference between these two areas contains h z . Let
y, and y z be the two ordinates. Then the rectilinear area is
made up of a rectangular area hy v and the area of a right
angled triangle ^ ^ . The whole area is therefore
~~^ 1 ^ ; or, putting y for the mean between the values of
y l and ?/ 2 , the area = hy. Hence if dA represent the differential
of any area ^r (x) expressed as a function of the abscissa x,
and dx the corresponding differential of the abscissa, we have
dA ,. .
#H*fH
From this differential equation the function \j/ (x) is found
when y is a given function of x *.
Contacts. The simplest case of contact, that of the curve
and tangent, has already been considered. In this instance,
the value of the first differential coefficient of the ordinate y
of the point of contact, given by the equation of the tangent,
is the same as that given by the equation of the curve, and
the curve and tangent have two points in common indefinitely
near each other. The next order of contact is that of two
curves, the equations of which give the same values of the
first and second differential coefficients of the ordinate y of the
point of contact, the curves having three points in common
indefinitely near each other. And so on for higher orders.
The contact of the second order between a circle and any
* For the function y = the reasoning fails when x is indefinitely small because
h 2
in that case ty'"(x)  + &c. becomes indefinitely great. (See De Morgan's Diff.
and Int. Calc. Chap. XX. p. 571.)
90 THE PRINCIPLES OF APPLIED CALCULATION.
curve is of special interest, because the radius of the circle is
an inverse measure of the degree of curvature of the curve.
We might now proceed to apply like considerations to
Geometry of Three Dimensions, inclusive of Spherical Trigono
metry ; but as the reasoning would be analogous to that applied
to Geometry of two dimensions, and no new principle would
be evolved, for the sake of brevity I shall pass by these con
siderations and proceed to other applications of calculation.
The Principles of Plane Astronomy.
As the sole object of these Notes is to inquire into
principles of calculation, very little is required to be said on
Plane Astronomy, which, as these terms may be taken to
imply, consists mainly of Problems in Geometry, the solutions
of which are obtained by calculations the principles of which
have already been considered. It is, however, to be remarked
that the geometrical Problems of Plane Astronomy are founded
on actual observation, and that the science is eminently prac
tical. There is also another distinctive feature which separates
it from pure Geometry, namely, the introduction of the
element of time. If all the heavenly bodies maintained at
all times the same apparent relative positions, the consideration
of time might be dispensed with in assigning their positions,
although even in that case one of the spherical coordinates,
(Right Ascension,) might be most conveniently determined
by the intervention of the apparent uniform rotation of the
heavens. But since observation has shewn that all the
heavenly bodies undergo movements, apparent or real, by
which their relative positions are changed, it becomes ne
cessary when the position of a body is stated, to state also at
what time it had that position. The manner in which this
is done for the purposes of astronomical calculation deserves
particular attention, because astronomical measurements of
time and determinations of epochs are equally necessary in all
other calculations which involve the consideration of changes
GEOMETRY. 91
which depend on the lapse of time. The science of Time is
essentially a part of Plane Astronomy.
Right Ascension and Declination. The apparent positions
of the heavenly bodies are determined by two spherical co
ordinates, one being the arc of the Equator intercepted be
tween the first point of Aries and the great circle perpendicular
to the Equator which passes through the place of the heavenly
body, and the other the arc of this circle between the body
and the Equator. The latter coordinate, which is the De
clination^ is practically found by a Mural Circle, which mea
sures, first, angular distances on the Meridian from the Zenith
of the Observatory, and then, after ascertaining the latitude
of the Observatory, angular distances from the Pole of the
heavens, or from the Equator. The other coordinate, the
Right Ascension, is obtained by means of a Transit instrument,
which after being properly adjusted, is adapted to finding the
instant, as shewn by a Clock, of the passage of a heavenly
body across the Meridian of the Observatory. Now it is
presumed, and there is no reason to doubt the fact, that the
Earth's rotation about its axis is perfectly uniform. Con
sequently, the stars being supposed to have no motion real or
apparent, except the apparent diurnal motion, the returns of
the same star to the meridian will be separated by a constant
interval, that in which the Earth completes a revolution about
its axis. This interval being divided into 24 hours, and the
circle of Right Ascension into 360, one hour of time will
correspond to 15 of arc. Hence the interval between the
passages of two stars across the meridian being known, the
difference of their Right Ascensions is found by a simple
proportion. But the measurement of the timeinterval must
depend on an astronomical clock, and as no clock can be
mechanically made to go with perfect uniformity, it is ne
cessary to make use of some means of ascertaining the devia
tions from a uniform rate. The rating of the clock might,
in the first instance, be effected by noting the times of con
secutive transits of any star, or stars, across the meridian, the
92 THE PRINCIPLES OF APPLIED CALCULATION.
deviations of the noted intervals of consecutive transits of the
same star from 24 hours, being considered to be the clock's
rate. Repeated observations of this kind with a selected
number of stars would serve both to rate the clock and to tell
the differences of the Right Ascensions of these stars. If we
chose to fix the origin of Right Ascension at the point of the
Equator cut by the circle of declination passing through one of
the stars, the absolute Right Ascensions of all the others would
become known. By subsequent observations of these known
stars, not only might the clock be rated, but the unknown Right
Ascensions of all other celestial objects might also be obtained.
It must, however, be remarked that the foregoing sup
position of the permanence of the apparent relative positions
of stars is not strictly true. By continued and exact observa
tions it is found, that time as measured by their returns to
the meridian is not perfectly uniform. One of the disturbing
causes has been discovered to be a movement and nutation of
the Earth's axis, which has no effect upon the uniformity of
the Earth's rotation about the axis, but alters the apparent
positions of stars. Another cause is the aberration of light, in
consequence of which the measured angular direction of a
star differs by a small arc from the direction of the passage of
light to the eye of the observer, and to a different amount at
different times of the year. The law and the magnitude of
each of these disturbances have been well ascertained by ob
servations appropriate to the purpose, and the corrections they
render necessary can be calculated for any given observation.
After taking account of these corrections, by which the
apparent Right Ascensions of the known stars become more
exactly known, the observation of transits of these stars affords
a uniform standard for measuring time. There only remains a
possible source of error from any motions peculiar to the stars
themselves. Such proper motions have in fact been detected,
but as their amounts can be ascertained by comparisons of
observations made at distant epochs, their effect on the mea
sures of time may be taken into account.
GEOMETRY. 93
It is further to be remarked that for reasons which will be
presently stated, astronomers fix the origin of Eight Ascension,
not as supposed above by reference to a star, but by reference
to the first point of Aries, the direction of which is defined at
any time by the intersection of the plane of the Earth's Equa
tor with the plane of the Ecliptic. This line moves relatively
to stars on the plane of the Ecliptic, and, on account of
nutation, by an irregular motion. If, however, Right Ascen
sion be referred to the mean position of the first point of Aries,
this irregularity would not affect the uniformity of the sidereal
measures of time. But it has been agreed by astronomers to
call the sidereal time at any place, the arc intercepted between
the actual first point of Aries and the point of the Equator
which is on the meridian of the place, converted into time at
the rate of 15 to an hour. According to this reckoning as
tronomical sidereal time is not strictly uniform. No sensible
error, however, arises from this circumstance, because the
fluctuations of the first point of Aries about a mean position
(called the Equation of the Equinoxes) are very slow, and much
slower than the fluctuations to which the rate of the best con
structed timepiece is liable from extraneous causes. The
particular advantage of this conventional reckoning is that the
sidereal time at which a celestial object passes the meridian
becomes identical with its apparent Right Ascension. The
calculated apparent Right Ascensions of the known stars are
accordingly referred to the true Equinox. The error of the
clock being the difference between its indication and the cal
culated Apparent Right Ascension of ^a known star, it follows
that the timepiece is regulated to point to O h whenever the first
point of Aries (affected by aberration as a star) is apparently
on the meridian. Also the intersection of the Equator with
the Ecliptic is fixed upon for the origin of Right Ascension,
because the exact position of this point can be determined
from time to time by observation, as I now proceed to shew.
Suppose that for several days before and after the vernal
Equinox, the Sun were observed on the meridian both witli
94 THE PRINCIPLES OF APPLIED CALCULATION.
the Transit and the Mural Circle, the Transit clock being
regulated by known stars whose Right Ascensions are re
ferred to some arbitrary origin. Then the sidereal time at
which the Sun's declination was zero, might be ascertained by
interpolation. That sidereal time is to be subtracted from
the assumed Right Ascensions of all the known stars in order
that the position of the first point of Aries may be the origin
of Right Ascension at that time. The movement of the first
point of Aries in Longitude, and the Nutation of the Obliquity
of the Ecliptic, being known, by applying to the Right
Ascensions of the known stars corrections depending on these
variable quantities, the same point is made the origin of Right
Ascension at any subsequent time. It is evident that like
observations made near the Autumnal Equinox would de
termine the position of a point just 180 from the vernal
Equinox, and might, therefore, be employed to find the posi
tion of the first point of Aries. When this point has been found
very approximately in the manner just indicated, a more
exact determination might be made by comparing a large
number of meridian observations of the Sun with the Solar
Tables constructed on the theory of gravitation, such compa
rison furnishing data for correcting the Elements of the Tables,
and inferring the position of the origin of apparent Right
Ascension.
Bessel, the illustrious astronomer of Konigsberg, by a
comparison of his own observations of the Sun in 1820 1825,
and those of Bradley in 1753 and 1754, with Carlini's Solar
Tables, obtained the Sun's mean longitude at a given epoch,
from which the following element used in the computations of
the Nautical Almanac was derived * :
At the Greenwich Mean Noon of January 1, of the year
1800 + 1, the Sun's Mean Longitude (M) is
280 .53 / .32 // ,75f^.27 // ,605844+^ 2 .0 // ,0001221805/.14 / .47 // .083,
See the Astronomische Nachrichten, No 133, and the Nautical Almanac for
1857, p. vi.
GEOMETRY. 95
wh ere f denotes, for the nineteenth century, the number of years
from the year immediately preceding 1800 + tf, wjiich is
divisible by 4 without remainder. It is to be observed that
this value of the Mean Longitude includes the effect of
aberration.
A sidereal day is defined in Astronomy to be the interval
between two consecutive transits of the first point of Aries
across the meridian of any place. A mean solar day is the
interval between two consecutive transits of a fictitious Sun
supposed to move in the Equator with the Sun's mean motion
in Longitude, or Eight Ascension.
From the definition already given of sidereal time, the
following equation will be true :
The Sidereal Time at Mean Noon
= Sun's Mean K.A. 4 Nutation in E.A.
This equation serves to establish a relation between
sidereal time and mean solar time by means of the following
process. From the calculations of Bessel already referred to,
it was found that the mean motion of the Sun in 365,25 mean
solar days was less than 360 by 22",617656 : whence it follows
that the sidereal year, or complete revolution of the Sun with
regard to fixed space, is 365 d . 6*. 9 W . 10 8 ,7496, or 365,256374417
mean solar days. Taking the mean amount of the precession
of the equinoxes in the t years succeeding 1800 to be
t . 50",22350 + t*. 0,0001221805,
the mean length of the tropical year 1800 + 1 is
365 d . 5*. 48 m . 47,8091  t . O a ,00595
or 365 d ,242220013  1 . O d ,00000006686.
Dividing 360 by the length of the tropical year, the mean
motion of the Sun in Longitude in a mean solar day will be
found to be 59'. 8",3302, and consequently the mean motion
in Eight Ascension expressed in time, 3 m .56*,55548. Hence by
96 THE PRINCIPLES OF APPLIED CALCULATION.
the equation above we have for the Greenwich mean noon of
any day (n) of the year 1800 f t,
Sidereal Time = = + (n1) . 3 wl .56",55548 + Nutation in K. A.
lo
It appears by this equation that from one mean noon to the
next succeeding, the sidereal time increases by the mean
quantity 3 m . 56 8 ,55548, and consequently that 24 A of mean time
are equivalent to 24\ 3 m . 56 8 ,55548 of sidereal time.
By means of the above equation the sidereal time at the
mean noon of each day of the year may be readily found ;
whence by Tables of equivalents of the hours, minutes, and
seconds of the two kinds of time, the sidereal time correspond
ing to any given mean time, or the mean time corresponding
to any given sidereal time, may be calculated. The latter
operation is facilitated by first calculating for every day of the
year, (as is done in the Nautical Almanac for the meridian of
Greenwich) the mean time corresponding to 0* of sidereal time,
or the time of transit of the first point of Aries.
From the foregoing discussion of the calculation of time, it
appears that all measures of the uniform flow of time depend
on the uniformity of the earth's rotation, and that the current
of time is indicated by a clock regulated by the observation
of stars. The sidereal time thus reckoned serves in the first
instance to record the exact instant of any astronomical event
on any day. But when different events are to be referred
to a common epoch, the intervals from the epoch are most
conveniently expressed in time the divisions of which are years,
months, and mean solar days, these divisions, derived originally
from obvious celestial phenomena, being long established and
in general use. Accordingly it is the practice of astronomers to
change the record of an astronomical event in the sidereal time
of any day into the mean time of tlte day, and to add the year,
month, and day of the month*.
* As all calculation, whether in plane or physical Astronomy, depends on data
furnished by observation, the accuracy attainable by calculation must be limited
by the accuracy of the observations, and especially of those made with the
GEOMETRY. 97
The Aberration of Light. Much has been written to little
purpose about the cause of the aberration of light. The laws
of the phenomenon, so far as they are required to be known
for astronomical calculation, were long since ascertained ; but
the attempts to give the rationale of it have not been suc
cessful. This, I conceive, has arisen from not remarking,
that direction is determined by an astronomical instrument
by reference to two points rigidly connected with the instru
ment, through both of which the light by which the object is
seen at the instant of observation actually or virtually passes.
One of these points is the optical centre of the objectglass,
and the other an arbitrary point in the field of view of the
Telescope, which may be marked by the intersection of visible
lines. The instrumental arrangements are made so as to
determine the actual direction, relative to fixed planes, of the
line joining these points at the instant of any observation.
But the light does not travel in that direction, because the
first point, after the light has passed through it, is carried by
the earth's motion out of the path which the light must
traverse in order to pass through the other point at the instant
of observation. The angle which the straight line joining the
two points makes with the path of the ray is found by cal
culation founded on the known velocities of the earth and of
light, to be equal to the whole of the observed amount of
aberration, and consequently the phenomenon is sufficiently
accounted for by this explanation.
In addition to aberration from the above cause, which
applies to a star or fixed body, there is an aberration arising
from any motion of translation in space, by which a body
is carried out of the direction of the ray by which it is visible
Transit and Mural Circle. Hence the correction of instrumental errors and errors
of observation is essential to the advancement of astronomical science. Those
sources of error are most injurious, and, if uncorrected, most likely to affect
theoretical deductions, which always tend in the same direction, such as the wear
of the pivots of a Transit, and the flexure of a Mural Circle. The latest improve
ment in Practical Astronomy is the making use of optical means for correcting
instrumental errors of this nature.
7
98 THE PRINCIPLES OP APPLIED CALCULATION.
at any instant, in the interval the ray takes to pass from
it to the spectator. If the spectator and the body have
exactly equal and parallel movements in space, it is clear that
the body's motion will bring it into the direction of the line
joining the two points above spoken of, and that, consequently,
there will be no aberration, or the two kinds of aberration
destroy each other. In any case, therefore, of a moving body,
let us suppose the Earth's motion in space to be impressed on
the earth and the body in the direction contrary to that in
which it takes place. By what has just been proved, this
common motion produces no change of aberration. But on
this supposition the earth is reduced to rest, and there is no
aberration of the first kind. The aberration is wholly due
to the motion of the body relative to the earth's motion, and
is determined in amount and direction by the quantity and
direction of the relative motion in the interval light takes to
travel from the body to the earth. In other words, the
aberration is the change of the body's apparent position in
that interval. Hence is derived the rule, familiar to astro
nomers, by which the aberration of a planet is taken into
account, viz. to reckon the real direction at any given time
to be the apparent direction at a time later by the interval
light takes to travel from the planet to the spectator*.
The Principles of Statics.
The department of mathematics which next comes under
consideration is the science of the equilibrium of bodies.
Here time does not enter, the elementary ideas being space,
matter, and. force. The term Statics is restricted to the equi
librium of rigid bodies.
Matter has form and inertia, and being attracted to the
earth by the force of gravity, has weight. The force of gravity
being given, the weight of a body measures its quantity of
matter.
* See the Articles on the Aberration of Light which I communicated to the
Philosophical Magazine, N.S. 1852, Vol. HI. p. 53, and N.S. 1855, Vol. ix. p. 430.
STATICS. 99
It is not necessary for the purposes of calculation to define
force, but it is necessary to define measures of force. The
unit of the measure of force in Statics is the weight of a certain
size of a certain substance under given conditions. The
standard of weight in this country is called a pound. All
measures of force in Statics are numerical ratios to this unit.
A perfectly rigid body does not change form by the appli
cation of any force. It also possesses the following property,
which is perhaps only a consequence of perfect rigidity : A
given force acting along a given straight line, produces the
same effect, at whatever point of the line, rigidly connected
with the body, it be applied. Experience has shewn that
these properties exist in many bodies approximately. .,In
Statics they are assumed to be exact, for the purpose of ap
plying exact mathematical reasoning.
If two equal forces act along the same straight line in
opposite directions, they counteract each other. For according
to what has just been stated, the forces may be conceived to
be applied to the same point, and in that case there is no
reason from experience to conclude that one would in any
instance prevail over the other.
Definitions of Equilibrium. When any number of forces
are in equilibrium, the effect of each one is equal and op
posite to the resulting effect of all the others. Hence if any
one of the forces be changed in magnitude or direction in ever
so small a degree, the others remaining unchanged, the equi
librium is destroyed.
The following is another definition of equilibrium, the use
of which will be exemplified hereafter. When any number
of forces are in equilibrium, if ever so small an additional
force be applied in any direction, motion ensues.
In the latter definition it is supposed that the additional
force does not act against a fixed obstacle, or that for the
resistance of the fixed obstacle an equivalent force is sub
stituted, the point of the application of which is capable of
movement.
72
100 THE PRINCIPLES OF APPLIED CALCULATION.
The foregoing principles, combined with certain funda
mental equalities of the same kind as that above considered,
suffice for the basis of all calculation applied to the equilibrium
of rigid bodies.
The first Proposition of a general kind required to be
proved from these principles is that relating to the composition
of forces, usually called " The Parallelogram of Forces." The
proof of this Proposition by functional equalities, as given
in some Treatises on Statics, is ill adapted to shew what are
the elementary and essential principles of the science.
Perhaps the most elementary proof of the Proposition is
that which deduces it from the properties of the Lever*. The
fundamental equality from which the reasoning relating to
the Lever commences is, that equal weights suspended at the
extremities of the equal arms of a horizontal lever balance
each other. After deducing from this principle and from the
properties of a rigid body above stated, the general equation
applicable to the equilibrium of two forces acting on a lever,
the proof of the parallelogram of forces follows from a course
of reasoning which requires no other basis than the definition
of equilibrium.
Duchayla's proof of the Parallelogram of Forces f is not
perhaps as elementary as the foregoing, but with regard to
the reasoning is as unexceptionable, and equally shews that
the Proposition rests on that property of a rigid body ac
cording to which a force acting along a straight line has the
same effect at whatever point of the line it be applied. The
fundamental equality from which the reasoning commences is,
that the direction of the resultant of two equal forces acting
on a point, is equally inclined to the directions of the forces.
As the Proposition thus proved does not require the
antecedent demonstration of the properties of the Lever, it
may be employed to answer such a question as this: What
* Whewell's Mechanics, Second Edition, Chap. I. and n.
f Pratt's Mechanical Philosophy, p. 7, and Goodwin's Elementary Mechanics,
p. 71.
STATICS. 101
is the single force equivalent to two parallel forces acting
perpendicularly to a straight rod at its extremities and towards
the same parts ?
The answer is obtained by conceiving two equal forces to
be applied along the line of the rod at its extremities in
directions tending from its middle point. There will then be
two pairs of forces, the resultants of which will meet in a
point, and have a resultant through this point, which must be
the resultant of the two parallel forces, 'because the two
additional forces just counteract each other.
The Mecanique Analytique of Lagrange commences with a
general solution of all statical problems by means of the
Principle of Virtual Velocities. The virtual velocity of any
point to which a force is applied, is the projection on the
line of direction of the force of any movement of the point
which is consistent with its relation to the other points of the
system. If P be any force, and Sp the virtual velocity of its
point of application, the equation of Virtual Velocities is
2 . PBp = 0. Equal forces acting in opposite directions (such
as tensions) are excluded from this equation, because for
every + PSp there will be a PSp. The resistances of fixed
obstacles may be included if the points of resistance be con
ceived to be moveable, and the forces of resistance to remain
the same. If then P x be any applied force, and P 2 be any
resistance of a fixed obstacle, and if Sp 1 and Sp^ ^ e the re ~
spective Virtual Velocities, the general equation becomes
2 . P^ + 2 . Pp 2 = 0. But whenever there are movements
of the system consistent with the supposition that each Bp 2 0,
we shall have 2 . Pfp l = 0. In such cases there are two
equations of Virtual Velocities, one including, and the other
independent of, the resistances of fixed obstacles.
Lagrange arrives at the general equation of Virtual Velo
cities, by conceiving in the place of each force a compound
pully to act, consisting of two blocks between which a string
passes, in directions parallel to that of the force, a number of
times equal to the multiple that the force is of the tension of
102 THE PRINCIPLES OF APPLIED CALCULATION.
the string. One of the blocks is fixed and the other move
able. The same string is supposed to pass over all the com
pound pullies, and at the end of it a weight (w) is supported,
which measures its tension. An equation is obtained on the
principle that whatever movements the moveable pullies and
points of application of the forces undergo, the length of the
string remains the same ; or, I being its length, 1 = 0. It is
evident that this equation will be true when the movements
are wholly estimated in the directions of the forces, whatever
finite intervals there be between the blocks, provided that the
movements be indefinitely small. Hence by considering only
indefinitely small movements, the virtual velocities are inde
pendent of the intervals between the blocks, and thus the
principle is introduced, that forces have the same effect what
ever be the points of application along their lines of direction.
Again, as each force is a multiple of w, the forces are com
mensurable, and any alteration of w alters all the forces in the
same proportion. Hence w may have any magnitude what
ever without affecting the equilibrium. Another principle
necessary for establishing the equation of Virtual Velocities is
stated by Lagrange in these terms. " In order that the sys
tem drawn by the different forces may remain in equilibrium,
it is evidently necessary that the weight (w) should not de
scend by any infinitely small displacement of the points of
the system ; for as the weight always tends to descend, if
there be a displacement of the system which permits it to
descend, it will descend necessarily and produce this displace
ment of the system." Respecting the peculiar considerations
by which it is here inferred that w does not descend, it may
be said that they are not strictly physical, nor in accordance
with the principles of mathematical reasoning, which consists
entirely of deductions by calculation from definitions and fun
damental equalities. Also it does not appear by such con
siderations why w does not ascend. This logical fault may be
corrected by making use of the definition of equilibrium
already stated, viz. that when a system of forces is in equili
STATICS. 103
brium, any additional force, however small, produces motion.
The virtual velocities may accordingly be supposed to be the
effect of the application of an additional indefinitely small
force, on which supposition the other forces, and by conse
quence the tension of the string, will remain unchanged.
On this account the finite weight w neither ascends nor
descends *.
The Proposition being proved for commensurable forces
may be extended to incommensurable, on the general prin
ciple of abstract calculation, that incommensurable relations
may be approximated to by commensurable as nearly as we
please.
As the equation of Virtual Velocities may be considered
to be an a priori solution of all statical problems, and as we
have shewn that the principles on which it rests are the same
that were stated to be the foundation of the inductive method
of solving such problems, we have hence a proof that those
principles are both necessary and sufficient.
The following is the process by which the Parallelogram
of Forces is arrived at by the equation of Virtual Velocities.
Let three forces P, Q, R, acting in the same plane on a point,
be in equilibrium. Then the point may be caused to move in
any direction by an indefinitely small force acting in the same
plane. Let the directions of the forces make the "angles 0,
0', &' respectively with a fixed line, and let the arbitrary
direction in which the point is made to move, make the angle
a with the same line. Then $s being the amount of move
ment, the virtual velocities are respectively scos(0 a),
Ss cos (& a), and 8s cos (#" a). Hence by the general equa
tion of Virtual Velocities,
Pcos (0 a )+Q cos (ff a) + R cos (&' a) = 0.
As this equation is indeterminate with respect to a, we must
have Pcos e+Qcosff+fi cos 0"= 0,
and Psin + Q sin ff+ R sin 6"= 0.
* See on this subject an article which I communicated to the number of the
Philosophical Magazine for January, 1833, p. 16.
104 THE PRINCIPLES OP APPLIED CALCULATION.
These equations determine the direction and magnitude of one
of the forces when the directions and magnitudes of the other
two are given.
The equilibrium of elastic bodies may be treated in the
same manner as that of rigid bodies, because when the equi
librium is established they may be assumed to be rigid.
The object of these Notes does not require more to be said
on the principles of the Statics of rigid bodies.
The Principles of Hydrostatics.
The application of calculation to cases of the equilibrium
of fluid bodies, rests upon the following definitions of proper
ties by which such bodies are distinguished from solids.
Definition I. The parts of a fluid press against each
other, and against the surface of any solid with which they
are in contact.
Definition II. The parts of a fluid of perfect fluidity may
be separated by an indefinitely thin solid partition bounded
by plane faces, without the application of any assignable
force.
These definitions apply equally to an incompressible fluid,
as water, and to a compressible fluid, as air. The pressure of
a compressible fluid is generally a function of its density, the
temperature being given.
The first of the above definitions is the statement of a
general property of fluids known by common experience.
The other is equally drawn from experience, being at first
suggested by the facility with which it is found that the parts
of a fluid may be separated. As all known fluids possess
some degree of cohesiveness, none answer strictly to this defi
nition. The hypothesis of perfect fluidity is made the basis
of exact mathematical reasoning applied to the equilibrium
and motion of fluids, in the same way that the hypothesis of
HYDROSTATICS. 105
perfect rigidity is the basis of exact mathematical reasoning
applied to the equilibrium and motion of solids.
The numerical measure of the pressure at any point of a
fluid, is the weight which is equivalent to this pressure sup
posed to act equally upon all points of a unit of area. Thus,
if a barometer be taken to any position in the earth's atmo
sphere, the weight of the column of mercury, supposing its
transverse section to be the unit of area, is the measure of the
pressure at that position. This quantity is usually designated
by the letter p.
The first use to be made of the foregoing definition is to
investigate a certain law of pressure, which is common to all
perfect fluids, however they may be specifically distinguished.
The law is found as follows, the fluid being supposed to be
at rest.
Suppose an indefinitely small element of the fluid to be
separated from the surrounding fluid by indefinitely thin solid
plates, and let the form of the element be that of a prism, the
transverse section of which is a rightangled triangle. By
Definition II. the pressure is in no respect altered by insu
lating the element in this manner, since this may be done
without the application of any assignable force. Also by
Definition I. the element presses against the solid plates with
which it is in contact : and these pressures must be counter
acted by equal pressures against the element. But the plates,
being supposed to be indefinitely smooth, are incapable of
pressing in any other directions than those of normals to their
surfaces. Hence the directions of these mutual pressures are
perpendicular to the plane faces of the element. Conceive the
plates removed : the pressures will remain the same. Conse
quently the element is held in equilibrium by the pressures of
the surrounding fluid perpendicular to its surfaces, and by the
impressed accelerative forces.
Now let h be the length of the prism, a, /3, 7 the sides of
the triangular section, a and /3 including the right angle, and
let p^h, pfih, p 3 yh be the respective pressures on the three
106 THE PRINCIPLES OF APPLIED CALCULATION.
rectangular faces. The element being indefinitely small, the
pressure may be assumed to be uniform throughout each face.
Suppose the impressed accelerative forces*, resolved along the
sides a and /3 in the directions towards the right angle to be
2/J and 2f 2 . The impressed moving forces in the same di
rections are ftpafth and f 2 pa/3h, p being the density of the
element. These must be in equilibrium with the pressures on
the rectangular faces resolved in the opposite directions.
The pressure resolved in the direction of the side a and
tending from the right angle is
/?
P*fr P.lk x , or (p, p z }Ph.
The pressure resolved in the direction of the side /:?, and tend
ing from the right angle, is
Hence, (ftfl,)^/*^/^, or p, p a =f lP a,
and (Pipjah^ftpaph, or ^ ^ 3 =
Consequently, as a and /3 are indefinitely small, the right
hand sides of these equations are indefinitely small, unless f^
and f a be indefinitely great, which is assumed not to be the
case. Hence p 1 =p z =p s . By supposing the position of 7 to
be fixed, and those of a and ft to vary so as always to remain
perpendicular to each other, it may be inferred from the fore
going reasoning that the pressures in all directions from the
element in a given plane are the same. Supposing another
plane to pass through the element, it may be similarly shewn
that the pressures in all directions in this plane from the ele
ment are the same, and consequently the same as the pressures
* The terms accelerating force and moving force are here used by anticipation,
not having been yet defined. This apparently illogical use of them would be
avoided by treating Statics as a particular case of the Dynamics of Motion.
HYDROSTATICS. 107
in the former plane, because the two planes have two direc
tions in common. And as the second plane may have any
position whatever relatively to the first, it follows that the
pressures are the same in all directions from a given element,
or from a given point. This is the law of pressure which it
was required to investigate.
This law of equality of pressure has been taken by some
writers on Hydrostatics as a property by which the fluid is
defined. But as it has been shewn that the law is deducible
from another property, that of perfect separability, it can no
longer be regarded as a definition : for a definition which can
be deduced by reasoning, ceases to be such. Also it will be
shewn hereafter that the property of perfect separability is
necessarily referred to in the mathematical treatment of cer
tain hydrodynamical questions. The same property serves to
establish at once the following Theorem in Hydrostatics : If
any portion of a fluid mass in equilibrium be separated from
the rest by indefinitely thin partitions, and be removed, the
partitions remaining, the equilibrium will still subsist.
The above principles may be applied as follows in obtain
ing a general equation of the equilibrium of fluids.
Let the coordinates of the position of any element of the
fluid referred to three rectangular axes of coordinates be a?, y,
z, and be supposed all positive, and let the form of the element
be that of a rectangular parallelopipedon, its edges dx, dy, dz
being parallel to the axes of vcoordinates. Then if p be its
density, and X, Y, Z be the impressed accelerating forces
acting on the element in directions respectively parallel to the
axes of coordinates, and tending from the coordinate planes,
the impressed moving forces in the same directions are
Xpdxdydz, Ypdxdydz, Zpdxdydz.
These are counteracted by the excesses of the pressures on
the faces of the element farthest from the origin above the
pressures on the opposite faces. Let pdydz, qdxdz, rdxdy
be the pressures acting respectively parallel to the axes of
108 THE PRINCIPLES OF APPLIED CALCULATION.
x, y, z on the faces nearest the origin. Then the excesses of
pressure tending towards the coordinate planes are
J dxdy dz, ^ dx dy dz, = dx dy dz.
But by the law of equality of pressure just proved, p, q, r
differ from each other by infinitesimal quantities. Hence
substituting p for q and for r, and equating these pressures to
the impressed moving forces acting in the opposite directions,
the resulting equations are
dp v dp v dp
7 = A, 7 = JL , 7 = ZJ.
pdx pdy pdz
Hence, since (dp] = f dx + f dy +  dz,
dx dy ' dz
we have
\dp) \r i \r 7 rr ^
= Adx + J dy + Zidz.
This equation, being true of any element, is true of the
elements taken collectively, the mass of fluid being assumed
to be continuous. And although for the sake of simplicity in
the reasoning, the coordinates x, y, z were supposed positive,
by the principles of the algebraic representation of geometrical
quantity, the equation is true without this restriction. Also
as it was obtained prior to any supposed case of equilibrium,
it is perfectly general in its application.
This is all that need be said on the principles of calcula
tion applied to the equilibrium of fluids. We shall now
proceed to the consideration of the Dynamics of motion, that
is, to Problems which involve time as well as force. The
body whose movement is considered will first be supposed to
be solid and rigid.
DYNAMICS. 109
The principles of the Dynamics of solid bodies in motion.
The first step in this department of applied mathematics
is to define a universal property of matter called its inertia.
It is found by experience that all bodies maintain a state of
rest, or of uniform rectilinear motion, unless they are acted
upon by some force. This statement defines inertia suffi
ciently for our purpose. With respect to what is denominated
force in this definition, we may affirm that it is essentially
the same quality as force in Statics; but into its intrinsic
nature there is no need to inquire, because in treating of the
principles of the calculation appropriate to problems of equi
librium, or of motion, we are only concerned with measures of
force. In cases of equilibrium, as we have seen, force is
measured by weight : in those of motion the measure is of a
different kind, having reference to the property of inertia just
defined. In the Dynamics of motion, force is measured by the
quantity of motion of an inert body which it either generates
or destroys. This statement will become more explicit after
explanations have been given of the terms velocity, accelerat
ing force, momentum, and moving force.
Velocity, or rate of motion, when it is uniform, is the
space traversed by a body in a given time, which for the
purposes of calculation is the unit of time, for instance, one
second. Let F be this quantity expressed in linear measure.
Then we say that the velocity = F. But the velocity being
uniform, it is evident that if s be the space described in any
y
interval t referred to the same unit of time, the ratio is
s
1 F 1 s
equal to the ratio  ; or ==  , and consequently F=  .
v S v L
When, however, the velocity is not uniform, more general
considerations are necessary for obtaining a symbolical ex
pression of its value. In this case the space described in a
110 THE PRINCIPLES OF APPLIED CALCULATION.
given time is no longer proportional to the time, but must be
regarded as an unknown function of the time. That is, sym
bolically, s=f(t). Hence, s t and s 2 being respectively the
spaces described at the epochs t T and t + T, we shall have
by abstract calculation,
v ds (Ps T 2 d?S T 8
and o 2 j ^i/ p * / e T ~j.
s 2 s t _ ds d z s r 2 p
*' ~~^~~~dt + d?'~6 +
By what is said above, the lefthand side of this equality is
the rate of describing the space s 2 ^ with a uniform motion
in the interval 2r, however small r may be. But by taking
T indefinitely small, this mean velocity may be made to ap
proximate as nearly as we please to the actual velocity at
the intermediate epoch , the change of velocity being as
sumed to be continuous. And when T is indefinitely small
the righthand side of the equation ultimately reduces itself
to the first term. Consequently in variable motion the velo
city at any time t is expressed by the differential coefficient
of the space regarded as a function of the time. That is,
putting V for the velocity at the time t, we have
It may be remarked that this equality is true even if j be
Cbv
infinitely great, because the first omitted term of the series
contains r 2 .
If a point be conceived to move in a straight line in space
with the uniform velocity V, and a, /?, 7 be the angles which
the direction of motion makes with three axes at right angles
to each other, then the rates of motion with which the point
DYNAMICS. Ill
recedes from three planes at right angles to the axes are
Fcos a, Fcos ft, Fcos7; because these are the quantities by
which the distances from the plane are increased in the unit
of time. In the case supposed these expressions have the
same values for any length of time. But if the motion be
neither uniform nor rectilinear, it may still be conceived to
have a determinate rate and a determinate direction at each
instant, and the above quantities will express the rates of
motion from the planes at the particular epoch at which the
velocity is V and takes place" in the direction determined by
the angles a, /3, 7. Now the position of the point in space
being assumed to be a function of the time, it follows that the
coordinates x, y, z which determine its position must be
separately functions of the time. Hence by reasoning pre
cisely analogous to that by which we obtained a general
symbol for F, it may be proved that
Accelerating force. It has already been stated that a body
which moves from rest, or does not move uniformly in a
straight line, must, on account of its inertia, be acted upon by
some force, such as the force of gravity. The agent, as
experience shews, is extraneous to the body, and from the
observed effects is properly described as accelerating or re
tarding. But so far as regards calculation, "accelerating
force" always means the numerical measure of the action of
some force, and its symbolical expression includes both ac
celeration and retardation. For the sake of simplicity let us
first consider the case in < which the body moves in a straight
line, but with an increasing or decreasing velocity. In this
case the direction of the action of the force must be coincident
with the straight line of motion. The velocity, not being
uniform, may be regarded as a function of the space s passed
over, and as the space passed over in any case of continuous
motion is a function of time, the velocity may be assumed to
112 THE PRINCIPLES OF APPLIED CALCULATION.
be an unknown function of the time. Let therefore F=
Hence V l and V 2 being respectively the velocities at the
epochs t r and t + r, we shall have by abstract calculation,
T7 M* \ V dV
v = '*  T
d
2r
Now a constant, or uniformly accelerating, force is defined to
be a force which adds equal increments of velocity in equal
times t, and its numerical measure is the velocity added in
the unit of time, as one second. Hence if f be this measure,
and v be the velocity added in the interval t, by the definition
f I v
we shall have > or /=T, whatever be the magnitudes
.v t t
V V
of v and t. Consequently^  1 in the foregoing equation
is the numerical value of a constant accelerating force, which
acting during the interval 2r would add the velocity V z V lt
But suppose this velocity to be actually added by a variable
accelerating force. Then assuming that the force does not
vary per saltum, by taking T indefinitely small, the constant
or mean accelerating force may approach as near as we please
to the value of the variable accelerating force at the inter
mediate epoch t. But when T is indefinitely diminished, the
righthand side of the equation ultimately reduces itself to
the first term. Hence the value of a variable accelerating
* It may be remarked that this substitution for the purpose of obtaining a
differential equation the solution of which gives the form of an unknown function,
is analogous to the substitution of a letter for an unknown quantity, the value of
which is to be found by the solution of an algebraic equation.
f Galileo discovered that the descent of falling bodies at the earth's surface
presents an actual instance of this law. Prior to this discovery the process of
calculation applicable to forces could hardly have been imagined.
DYNAMICS. Ho
force at the time t being represented by F, we have
The reasoning shews that this symbol applies if the force
be indefinitely great, because the first of the omitted terms
contains r 2 .
ds
Since it has been proved that F= T , we have also
, ds_
"dttfs
'' dt ~ df
This is the general symbol of the measure of force by
space and time.
In the case in which the motion of a body is not in a
straight line, whether or not the velocity be uniform, the body
must be acted upon by some force. Now with respect to this
action a law has been ascertained by experiment, which it is
absolutely necessary to know prior to the application of cal
culation to the general case of variable motion. An experi
mental law relating to variable motion in a straight line has
already been announced, viz. that a constant accelerative
force adds the same velocity in the same time whatever be
the acquired velocity. When the motion is not in a straight
line, a constant accelerative force acting in a given direction
adds in a given time in the direction in which it acts a
velocity which is independent loth of the amount and the direc
tion of the actual velocity. It follows as a corollary from this
law that two or more constant forces acting simultaneously in
given directions add, in the directions in which they respec
tively act, the same velocities as if they acted separately.
Composition and resolution of accelerative forces. In the
reasoning which follows no account is required to be taken
of the dimensions of the accelerated body, which may, there
fore, be supposed to be an indefinitely small material particle.
Let us, first, consider the case of a material particle acted
8
114 THE PRINCIPLES OF APPLIED CALCULATION.
upon by two or more constant accelerative forces in a given
direction. Let V v F 2 , F 3 , &c. be the velocities which the
given forces F^ F^ F y &c. acting separately would add in
any interval t reckoned from a given epoch, and V be the
total velocity added. Then by the law of independent action
above enunciated,
dV dV. dV^ dV e
and therefore =  + +
But by what has already been proved,
dV, dV, dV,
' ^'* &'* ^df' &c "
and if x be the distance of the particle at the time t from a
fixed plane perpendicular to the direction of the motion,
^ dx , dV dx
V r > ancl .*. = = TS
dt ' dt df
Consequently,
This result proves that two or more constant accelerative
forces acting in a given direction have the same measure as
a single force equal to their sum acting in the same direction.
We proceed next to find the force equivalent to two con
stant accelerative forces acting simultaneously on a material
particle in a given plane and in directions at right angles to
each other. By the same law of independence of action, the
accelerative forces add in the directions in which they respec
tively act, in the interval from the time T to any time T+ t,
velocities which are independent of the magnitude and di
rection of the velocity at the time T. We may, therefore,
abstract from this velocity by conceiving an equal and oppo
site velocity to be impressed on the particle at that instant so
DYNAMICS. 115
as to bring it to rest*. Then if f l and / 2 be the given forces,
the velocities in the respective directions at the end of the
interval t, will be fj and fy. By the composition of velo
cities the resultant of these velocities is yj 2 + / 2 2 . t, and its
f
direction makes an angle whose tangent is ~ with the di
Ji
rection of the force . Hence the single force F, which is
equivalent to the two forces /j and f a acting in directions at
right angles to each other, is the force V/j a +f* acting in the
direction determined by the above angle. That is, the re
sultant equivalent force is represented in magnitude and di
rection by the diagonal of a rectangle the sides of which
represent in magnitude and direction the component forces.
If a third force f s be introduced, and be supposed to act
always in the direction perpendicular to the plane of the
other two, by the same reasoning the resultant of _Z^andj
is VP 2 +f 3 2 > and consequently the resultant of the three forces
is *Jfi+f*+f 3 2 ' This resultant is proportional to, and in
the direction of, the diagonal of the rectangular parallelopi
pedon the sides of which are proportional to, and in the
direction of, the forces f v / 2 , and / 3 .
The equivalence of three forces, acting in three directions
at right angles to each other, to a single force determined in
magnitude and direction by the magnitudes and directions of
the three forces, having been proved, we may conversely
resolve any given force into three forces acting in any rect
angular directions. The given force being F, and its direction
making the angles ct, /3, 7 with the three rectangular direc
tions, the resolved forces are plainly Fcos a, Fcos /?, and
Fcosy. It should be observed that a force strictly uncom
pounded may be legitimately resolved in this manner, the
* To shew the legitimacy of the process of abstracting from given velocities,
or accelerative forces, by conceiving to be impressed equal and opposite velocities,
or accelerative forces, it is sufficient to appeal to the experimental law of the
independent action of accelerative forces, from which the process is a direct
inference.
82
116 THE PRINCIPLES OF APPLIED CALCULATION.
resolution having no physical significance, but being merely a
step that may be taken on the principle of equivalence.
The preceding results give the means of finding the re
sultant of any number of constant accelerative forces acting
simultaneously on a material particle in given directions. For
each of the forces being resolved in the directions of three
rectangular axes, the sum of the resolved forces in the
direction of each axis is equivalent to a single force in that
direction, and the resultant of the three equivalent forces,
which is known by what is proved above, is the resultant in
magnitude and direction of the original forces.
If the accelerative forces acting on a material particle,
instead of being constant in magnitude and direction, as
supposed in all the forgoing reasoning, are variable with the
time, the same results still hold good; as may be shewn
by the following considerations. It will be assumed that the
forces do not vary either in direction or magnitude per saltum,
rnd that the law of independence of action is true as well for
variable forces as for constant. Then the velocity which each
variable force adds in the given interval r, in the direction of
its action at the middle of that interval, may be conceived to
be added by a constant force acting during the same interval
in that direction. Now the equivalent resultant of these
supposed constant forces is given by the rules already proved,
which are true however small the interval r may be. Let us,
therefore, suppose the time to be divided into an unlimited
number of very small intervals, and constant forces to act in
the manner above stated during each. In that case the
successive values of the constant forces may approach as nearly
as we please to continuity, and to coincidence with the values
of the actual forces both as to magnitude and direction. And
as by hypothesis they add the same velocities as the actual
forces, they may be regarded as ultimately equivalent to the
latter. Consequently the laws of the composition and resolu
tion of variable forces are the same as those of constant
forces.
DYNAMICS. 117
It will appear from the preceding discussion that the rules
for the composition and resolution of forces are the same in
the Dynamics of Motion as in Statics, although they are
deduced in the two cases from totally different principles. In
Statics the reasoning by which the rules were obtained had
reference to a body of finite dimensions, and depended on the
experimental fact, that a force acting on a rigid body produces
the same effect at whatever point of the line of its direction it
be applied. In the dynamics of variable motion the investi
gation of the resultant of given accelerative forces rests wholly
on the law of the mutual independence of action of the forces,
and that of their independence of acquired velocities. In fact,
these laws, known or suggested by experiment, are the basis
of all calculation applied to determine the motion of a material
particle acted upon by given forces.
The terms velocity and accelerative force having been de
fined, and symbolic expressions of their values obtained, we
may now proceed to treat similarly of momentum and moving
force.
Momentum. This term depends for its signification on the
general property of inertia, being employed exclusively with
reference, to an inert body in motion. "We have hitherto
regarded velocity and variation of velocity .apart from the
quality and dimensions of the moving body. But when we
perceive a body in motion, its essential inertia suggests the
* enquiry, How might it acquire velocity, or be deprived of it ?
From what has been said of the action of accelerative force, it
follows that the motion of a body may be both generated and
destroyed by such action. Also experience shews that velocity
may be suddenly communicated to a body, or taken from it,
by the impact of another moving body. There is reason to
conclude that even in this case the observed effect is due to
accelerative forces acting violently during a very short interval.
The term impact denotes this action apart from the considera
tion of time. The observed effect of impact is proper for
measuring momentum, that is, the efficacy of an inert body in
118 THE PEINCIPLES OF APPLIED CALCULATION.
motion. Now by experiment it is found that the measured
effect of the impact of a given body is doubled, trebled, &c., if
the velocity be doubled, trebled, &c., and that the different
measured effects of different bodies impinging with the same
velocity are in proportion to their masses. In this statement
the mass of a body is that quantity which is measured by its
weight, apart from magnitude, experiment shewing that bodies
of the same magnitude may have different weights.
Hence, regarding the effect of the impact of a body as
identical with its momentum, action and reaction being equal,
it follows from the foregoing experimental law that the
momentum of a body is proportional to the product of its mass
and velocity. Consequently if M be the mass referred to an
arbitrary unit of mass, (as the weight of a cubic inch of
distilled water of given temperature), and V be the velocity
referred to a unit as before stated, the numerical measure of
the momentum is the product of M and V. That is, for the
purposes of calculation, momentum = M V.
Moving force, in its scientific acceptation, has the same
relation to momentum that accelerating force has to velocity,
signifying the measure of the change of momentum. That
which moves a body from rest, or alters the velocity which it
has acquired in any manner, would in common parlance be
called a moving force. Thus gravity, inasmuch as it is
observed to produce such effects, might properly be called a
moving force. The same kind of effect is known to result
from another mode of action, viz. by the pressure of one body
against another. Conceive a perfectly smooth body to be
placed on a perfectly smooth horizontal table. Then by the
pressure of the hand, or other means, the body might be made
to move with a velocity either uniformly or variably ac
celerated. The effect in this case is of the same kind as in
the action of gravity, and possibly the modus operandi may
differ from that of gravity only in respect to being matter of
personal experience, or direct observation. But apart from
any consideration of the nature of the causes of motion, for
PHYSICAL ASTRONOMY. 119
the purposes of calculation moving force means conventionally
the measured effect of pressure, or some equivalent agency, in
producing change of momentum, as accelerating force is the
measured effect of the same kind of agency in producing
change of velocity*. The appropriate measure of moving
force is known only by experience and observation. By ex
periment it is ascertained that if the pressure against a given
mass be doubled, trebled, &c., the acceleration of the mass is
doubled, trebled, &c.; and that the pressures required to
accelerate to a given amount different masses are proportional
to the masses. Hence moving force is proportional to the
product of the mass and its acceleration ; and if M be the mass
referred to a known unit, and F be the acceleration numerically
estimated as already mentioned, then, for the purposes of
calculation, moving force = MF.
Physical Astronomy.
The principles of the Dynamics of motion thus far con
sidered, suffice for the solution of those problems of Physical
Astronomy which relate to the motions of translation in
space of the bodies of the Solar System. Problems of this
class generally allow of abstracting from the dimensions
of the moving body, and regarding it as a material
particle free to obey the impulses of an accelerative force.
The only force that comes under consideration in Physical
Astronomy is that of gravitation, which is assumed to have
the property of emanating from every portion of matter, to be
constantly the same from the same portion, and to be the
same from different portions having the same mass. The
accelerative force due to the gravitation from a small elemen
tary mass at the unit of distance from it is taken for a
* It would not be possible to reason upon moving force, i.e. cause of motion,
except by the intervention of its measured effect. It is on the ground of this
necessary relation that the terms accelerative force and moving force are applied
to the general symbolic expressions of the measured effects, in conformity with an
admitted use of language.
120 THE PRINCIPLES OF APPLIED CALCULATION.
measure of the mass. This measure is different in kind from
the measure of mass by weight which was before spoken of.
The latter measures the effect of the gravitation of an external
body assumed to attract every particle of the given body; the
other measures the effect of the body's own gravitation as
sumed to emanate from every one of its constituent particles
and to act on a given particle. These two measures must be
to each other in a fixed ratio, because each is proportional to
the number of particles of the given body.
Another characteristic of gravitation is its variation with the
distance from the body from which it emanates. Prior to any
knowledge of the cause of this variation, the law which it
obeys has been obtained by a combination of results from
observation with theoretical calculation. Newton, to whom
belongs the honour of this discovery, obtained the law in the
following manner. The space through which a body descends
from rest towards the earth's centre by the action of gravity
at the earth's surface during a given short interval, as one
second, is known by direct experiment. The distance of the
falling body from the earth's centre, that is, the earth's semi
diameter, is ascertained by measuring the actual length of a
certain number of degrees of a meridian arc. Also by obser
vation of the moon's apparent diameter it is found that her
orbit is guam proxime a circle having its centre coincident
with the earth's centre; the radius of the circle is deduced
from observations of the moon's parallax; and the time of
completing a revolution in the orbit is known from the results
of observations with the Transit instrument of an Observatory.
From data such as these Newton calculated the deflection of
the moon from a tangent to her orbit in the same interval of
one second. He then supposed, in accordance with dynamical
principles previously established, that this deflection might be
due to an attraction tending towards the earth's centre;
(ft*\
from the formula s = j that the
deflection in a given time is to the descent of a falling body at
PHYSICAL ASTRONOMY. 121
the earth's surface in the same time in the ratio of the force of
the attraction at the Moon to the force of the attraction at
the earth's surface. This ratio was found to be nearly that of
the inverse squares of the respective distances from the earth's
centre. Such calculation, though only roughly approximate,
thus gave a prima facie reason for supposing gravity to vary
inversely as the square of the distance from the points of ema
nation. The exactness of the law is proved by the accordance
of a vast number of results calculated on this assumption with
direct observation.
It thus appears that the law of the variation of gravity in
space is established by observation and calculation combined.
The law might be hypothetically assumed, but without ob
servation and appropriate calculation, it could not be proved
to be a reality. Although, as matter of fact, Newton verified
his hypothesis by means of determinations, by observation, of
the magnitude of the earth and the orbital motion of the moon,
it is yet interesting to enquire what means might have been
used if the earth had not been attended by a satellite. In
that case the observations of Kepler would have sufficed for
the purpose. Kepler's observations and calculations do not
involve the consideration of force; but the laws which they
establish furnish data from which the law of gravity might
have been inferred in his day, if the calculation proper for
enquiries relating to force had then been known. The follow
ing is the process, according to the Newtonian principles of
philosophy, by which the law of gravity is deduced from the
results of Kepler's observations*.
Kepler ascertained (1) that the planet Mars moves about
the sun in an ellipse, the sun's centre coinciding with a focus
of the ellipse ; (2) that it moves in such manner that the radius
vector drawn always from the sun's centre to the planet
sweeps over equal areas in equal times. The second law
symbolically expressed is
d . area = kclt,
* See Pratt's Mechanic^ Philosophy, Arts. 25G258.
122 THE PRINCIPLES OF APPLIED CALCULATION.
h being a certain constant. Referring the place of Mars at
the time t to rectangular axes drawn in the plane of the
motion through the sun's centre, and naming the coordinates
x and y, that differential equation becomes ,\ ,
xdy ydx = hdt.
Hence by differentiation, the increments of time being
constant,
d?y d*x
x rr yis = 0.
df J d?
Now making the hypothesis that the planet is acted upon
by some accelerative force and is free to obey its impulses,
this force, from what has been shewn (p. 115), may be re
solved into two forces X and Y acting parallel to the direc
tions of the axes of coordinates, and having values expressed
d*x d 2 y
by the differential coefficients ^ and ~ . Hence by substi
* nt nt *
U/l/ U/l/
tution in the above equation,
x_X
y~Y'
It is thus proved that the single equivalent force acts in a
direction passing through the origin of coordinates, or the
sun's centre. We have now to make use of Kepler's first
law, relating to the form of the orbit, to find the law of the force.
By calculation appropriate to forces emanating from a
centre it is shewn that if u =/(#) be the equation of the path
which a particle describes under the action of such a force,
u being the reciprocal of its distance r from the centre, the
expression for the force is
In the case of the ellipse, ua (1 e 2 ) = 1 + e cos (6 a).
Hence it will be found by the direct process of differentia
tion that the expression for the force becomes in this instance
tf I
PHYSICAL ASTRONOMY. 123
or that the force varies inversely as the square of the distance
from the centre.
This argument shews that the law of gravity was de
ducible from two of Kepler's laws, although it was no't
actually so deduced by Newton. In philosophical treatises
on the principles of Physical Astronomy great prominence is
usually given to the Three Laws of Kepler, as if the induction
of these laws from observation exemplified a principle of
scientific research*. The history of the progress of Physical
Astronomy would rather seem to indicate that it is the pro
vince of calculation to discover or demonstrate laws, while it
is the province of observation to furnish the data necessary for
applying the results of calculation to matter of fact, and to shew
that the laws deduced by calculation have a real and positive
existence. It is true that the law of the inverse square could
not have been discovered by observation alone, or by calcula
tion alone; but after it was proved to be at least approximately
true by a combination of calculation with observation in the
manner already stated, it required only the knowledge of the
proper rules of calculation to deduce by a brief process from this
hypothesis the three laws which cost Kepler so many years of
labour to establish. It was possible for Galileo to find the
ratio of the area of a cycloid to its circumscribing rectangle by
carefully weighing two pieces of lead which exactly covered
the two areas; but would he have adopted this method if he
had known how to calculate the area of the cycloid? So
Kepler might have been spared the trouble of deducing laws
from his observations, had it riot been the case that in his
time the science of observation was in advance of the science
of calculation. It is not intended by these remarks to depre
ciate in any degree the labours of Kepler; but rather to in
dicate the precise relation in which his three laws stand to the
discovery and the theory of gravitation. They were not, it is
* This is particularly the case in the Philosophy of Comte, who dwells much
more on the inference of "positive" laws by Kepler from observations, than oil
Newton's a priori deduction of the same laws by calculation.
THE PRINCIPLES OF APPLIED CALCULATION.
true, expressly used for inferring the law of gravity; but it
may be doubted whether cosmical gravity would have been
thought of, or its law sought for, unless the Laws of Kepler
had been proposed as problems for solution. The publication
of these laws naturally provoked enquiries as to their cause,
and various attempts were made to discover it; till at length
Newton succeeded in referring them by calculation to the
action of force, the force of gravity. The science of calcu
lation, as applied to the motions of the heavenly bodies, was
thus placed in advance of induction from observation, and
assumed its proper office of deducing and demonstrating laws.
Previously, not only Kepler's laws, but others relating to the
Moon's motion, as the Variation, Evection, Annual Equation,
&c. were inferred by astronomers from observation alone.
But from the date of the publication of Newton's Principia
there has been no need for the practical astronomer to do more
relatively to the moving bodies, than determine their apparent
positions as accurately as possible, and place his determina
tions in the hands of the theoretical calculator. These data
are by the latter used for calculating, (1) Elements of Orbits;
(2) Ephemerides for predicting the positions of the bodies from
day to day, that by comparisons of predicted with observed
places, data may be obtained for correcting assumed elements;
(3) if there be more than two bodies, the effect of their mutual
attractions in producing periodic and secular deviations of
their orbits from the mean orbits at a given epoch.
In the problem of the motions of three or more bodies
acted upon by their mutual attractions, it is usual to abstract,
in the first instance, the motion of one of them, and to calculate
the motions of the others relative to the motion of that one.
This is done by conceiving, first, that a velocity equal and
opposite to that which the selected body has at a given
instant is impressed upon it and upon the other bodies, and
that subsequently accelerative forces equal and opposite to
those by which the same body is acted upon are impressed
continually upon all. Under these operations the relative.
PHYSICAL ASTRONOMY. 125
motions will remain unaltered, the selected body will be at
rest and may be supposed to have a fixed position in space,
and the motions of the others may be referred to that position.
Then in order to calculate the actual motion of the body
conceived to be fixed, we may suppose the velocity of which
it was deprived at the given instant to be restored to it, and
the accelerative forces that were neutralized, to act upon it
in their proper directions. Now since from the previous cal
culations these accelerative forces and their directions become
known functions of the time, the position of the body at any
assigned time may be calculated, the velocity initially impress
ed being a datum of the calculation. Thus its absolute posi
tion in space will be known; and the positions of all the
others relative to it having been already found, the absolute
positions of all are known. The fixed body in the Lunar
Theory is the Earth, and in the Planetary Theory, the Sun.
In the case of the Sun it is not necessary to impress a
common velocity; because, as there is reason to conclude
that all the bodies of the Solar System are moving through
space at a certain uniform rate in a fixed direction, that
common velocity may be supposed to be compounded with
this uniform motion, and the resulting motion of translation
of the System, which is of unknown amount, may be left
out of consideration, or be abstracted by conceiving it im
pressed in the contrary direction. The above mentioned cal
culation will then determine the path described by the Sun's
centre, commencing at the position it occupied at the given
time. It has been found that this path is always confined
within narrow limits not exceeding the Sun's dimensions.
The relative positions of the bodies of the Solar System are
not affected by this orbital motion of the Sun.
The above considerations embrace all the fundamental
principles required for the calculation of the motions of
Planets and Satellites. The bodies are regarded as free
material particles, and at the same time as centres of force,
and the problem, stated generally, is to determine the motions
126 THE PRINCIPLES OF APPLIED CALCULATION.
produced by their mutual attractions, the attractive force of
each having a certain constant amount at a given distance,
and varying with distance according to the law of the inverse
square. After the formation of the differential equations of
the motion according to dynamical principles and given con
ditions, the solution of the problem is a process of pure
calculation, which, however, when the number of the bodies
exceeds two, is attended with considerable difficulties in the
details of the operations. As an exact solution is unattain
able when there are three or more bodies, methods of approxi
mation are employed requiring particular attention to the
magnitudes of the quantities involved, the values of 'the
coefficients of successive terms, and the augmentations of
these values produced in certain cases by integration; as is
fully explained in express Treatises on the Lunar and
Planetary Theories. On this part of the subject there is no
occasion for me to dwell: 1 will only remark farther, that the
employment of rectangular coordinates in the Lunar Theory*,
just as in the Planetary Theory, seems to be the simplest
mode of treatment, and that the method of variation of para
meters, which in principle is only a process of integration, is
equally applicable in both Theories. The separation of the
secular inequalities from the inequalities of short period is
allowable in the Planetary Theory, because the changes of
the former are so slow that the effects upon them of the
positive and negative fluctuations of the other inequalities
may be considered to be mutually destructive. This reason
does not equally apply in the Lunar Theory.
The reverse problem of perturbations by the solution of
which Adams and Leverrier detected the planet Neptune
from its disturbance of the orbit of Uranus, although it was
the first of its kind, and required for its successful treatment a
peculiar extension of theoretical calculation, did not involve
principles that were unknown to Newton. Also the question
* See a Memoir by Poisson in Tom. X. of the Memoires de I'lnstitut.
PHYSICAL ASTRONOMY. 127
raised by Professor Adams* relative to the calculation of the
acceleration of the moon's mean motion, is purely a mathema
tical one, involving no new physical principle; in which re
spect it resembles the old difficulty as to the theoretical amount
of the motion of the Moon's apse, and admits in like manner of
being settled by a strictly legitimate process of calculation.
As in such a case a permanent difference of opinion would
tend to throw discredit on theoretical calculation, it is a
satisfactory result of the discussion to which the question
gave rise among the most eminent theoretical astronomers of
the day, that the legitimacy of Professor Adams's process has
now been generally recognised. But the acceleration of rn^in
motion which the calculation gave, which was subsequently
confirmed by the researches of M. Delaunay, is only about
half the amount inferred from the records of ancient eclipses.
To what cause, then, is the other half due? Are we to attri
bute it to the action of a resisting medium ? M. Delaunay
has recently proposed to account for the difference by an
effect produced by the mutual attraction of the Moon and
the Tidal Wave. As observation shews that High Tide is
always behind the passage of the Moon across the meridian of
any place, since the opposite Tidal Waves are prominent on
opposite sides of the plane passing through the Moon's centre
and the meridian, it follows that the mutual attraction of the
Moon and the Tide acts as a kind of couple on the earth,
always tending to retard the motion about its axis. The
length of the day will thus be continually increasing, and the
moon's mean motion, supposed to be actually uniform, when
estimated by the angular motion in a given number of days
will be continually greater, and consequently be subject to an
apparent acceleration. The total observed acceleration might
thus be accounted for by the sole action of gravity, and
though it would be difficult to calculate exactly the amount
due to the Tides, it is possible to shew by approximate
* Philosophical Transactions, Vol. 143, Part III. p. 397.
128 THE PRINCIPLES OF APPLIED CALCULATION.
considerations that an adequate amount is quite within possi
ble limits*.
I proceed now to the consideration of another point in
Physical Astronomy, the discussion of which falls within the
scope of these notes, inasmuch as it involves an enquiry into
the physical signification of a certain peculiarity in the
analysis, namely, the occurrence in the developements for
radius vector and latitude of periodic terms having coefficients
that may increase indefinitely with the time. These terms
it may in the first place be remarked, arise out of a strict
application of the rules of approximating and integrating.
As they occur not only in approximating by series to the
solution of the Problem of Three Bodies, but also in like
approximations for the case of a central force varying as some
function of the distance from the centre, it will simplify the
enquiry into their origin to take, first, an instance of the
latter kind. Suppose the central force to be ^ yuV, r being the
distance from the centre. Then, putti
ential equation for finding the orbit is
distance from the centre. Then, putting u for  , the differ
To effect the integration of this equation by regular ap
proximation proceeding according to the powers of //, it is
necessary to begin by omittifig the last term. A first integra
tion will then give
A and B being the arbitrary constants. This value of u is
next to be substituted in the last term of the differential
equation, that term is to be expanded in a series proceeding
* See an Article by M. Delaunay in the Comptes Rendus of the Academy of
' Sciences of Paris, Tom. LXL, 11 Dec. 1865: also a discussion of the question in
the Monthly Notices of the Royal Astronomical Society, Vol. XXVI. p. 221235,
by the Astronomer Royal, who gives his assent to M. Delaunay's views.
PHYSICAL ASTRONOMY. 129
according to the powers of A, and the powers of the cosine
are to be transformed into cosines of multiple arcs. When
tMs has been done a second integration can be performed
however far the series may have been carried. The operation
may then be repeated with the new value of u\ and so on. It
is to be observed that we have here expanded strictly accord
ing to a rule which is independent of the relative magni
tudes of the quantities involved, and that consequently this
process gives the general form of the developement, although
it may not give a convergent form. It should also be noticed
that as no step in the process implies that pr is small
compared with  z , the former force, which is repulsive, might
be greater than the other, in which case the distance would
indefinitely increase, and the orbit have no resemblance to an
ellipse. The terms of the developement could not in that
case be exclusively periodic. If the analysis be restricted to
the first power of A, we have by the second integration
the last term increasing indefinitely with 0, so that this value
of u may diverge to any extent from that given by the first
integration.
There are various ways in which this form of an integration
that is convergent may be avoided ; among which I shall first
notice the following. Multiplying the differential equation
by 2du and integrating, we have
If the value of dd given by this equation be expanded accord
ing to the powers of fjf, and only the first power be retained,
the result is
_ hdu

 AVjTf '
9
130 THE PRINCIPLES OF APPLIED CALCULATION.
Here a step has been taken which is so much the more
accurate as the ratio of the force to the force fjuu 2 is smaller;
and this equation shews that if that ratio be very small the
value of ~~ cannot be very different from that which would be
due to the latter force acting alone. Accordingly on integrat
ing this equation to the same approximation as before, and
designating the arbitrary constants by the same letters, it will
be found that
As this result shews that the values of u are periodical and
restricted within limits, it may be regarded as a true approxi
mation to the orbit on the above supposition respecting the
ratio of the forces, it being also supposed that the orbit, so far
as it depends on the force /tw 2 , is an ellipse. The expression
for u may be made to consist of terms proceeding according
to the powers of // by expanding the cosine, and in that case
this form of solution ought to agree with that which is
obtained by approximating according to the general rule.
As far as is indicated by the expansion to the first power of
JJL'J the two expressions are clearly identical*. The fore
going reasoning shews that terms of indefinite increase are
got rid of in this instance by an operation which introduces
the condition of periodicity :
A method of avoiding terms containing the time (t) as a
factor, in principle the same as that of the preceding example,
I have employed with success in a general approximate
solution of the Problem of Three Bodies given in a communi
cation to the Eoyal Society (Phil. Trans, for 1856, p. 523).
In that solution, however, there appears in the expression for
* See on the subject here discussed two Articles on "The Theory of the
Moon's Motion" in the Numbers of the Philosophical Magazine for February
and March 1855.
PHYSICAL ASTRONOMY. 131
the eccentricity of the disturbed orbit a term containing t as
a factor, from which the periodicity of the variation of the
eccentricity has to be inferred by special considerations.
The method of the Variation of Parameters has the ad
vantage of entirely getting rid of the consideration of terms
of indefinite increase by the hypothesis of the instantaneous
ellipse, which secures the analysis against such terms, or
rather subjects it to the condition of periodicity. By that
method also, on the same hypothesis, the slow variations of
the elements are proved to be periodic*.
Another method of avoiding nonperiodic functions is to
introduce in the earliest stage of the investigation the factors
usually called c and #, on the ground that they are necessary
for satisfying the results of observation t. This process, which
has the appearance of being arbitrary, is proved to be legiti
mate by subsequently integrating the differential equations
of the motion so as to determine the functions which express
the values of these factors in terms of given quantities.
There is still another process which ensures the condition
of periodicity, and at the same time determines approximately
the values of c and g\. This method, the principle of which
is not satisfactorily explained in Treatises on Physical
Astronomy, is such as follows. After obtaining in the usual
manner the equations
* The Planetary Theory is throughout treated in this manner in Pratt's
Mechanical Philosophy (Arts. 349 392), and consequently no considerations like
those in Art. 334 ot his Lunar Theory are required. In the latter part of Airy's
Treatise (Arts. 102145) the Variation of Parameters is employed: but a different
method in the earlier part necessitates the consideration in Art. 91 of terms
involving an arc as a factor.
f Pontecoulant, Theorie du Movement de la Lune. Chap. I. No. 5.
t See Airy, Lunar Theory t Arts. 44 and 44*; and Pratt, Lunar Theory,
Art. 334.
92
132 THE PK1NCIPLES OF APPLIED CALCULATION.
the periodic terms involving the longitude of the disturbing
body being omitted, for ae cos (6 a) and k sin (6 7) are
substituted respectively u a and s, which are their equiva
lents by the first approximation. This being done, the re
sulting equations, since they contain no circular functions and
no terms indicative of the position of the disturbing body,
refer to a mean orbit. Hence integration of those equations
gives values of u and s which differ from the true values only
by periodic quantities, and are consequently real approxima
tions. There will presently be occasion to advert again to the
principle of this reasoning. It may here be remarked that
all the different methods of ensuring the periodicity of the
expressions for radiusvector and latitude lead to exactly the
same approximate solution of the Problem.
It is important to observe that as the processes of approxi
mation which conduct to terms of indefinite increase are
strictly legitimate and according to rule, the forms of solution
they give must have physical significance. With reference to
this point it is, first, to be remarked that these terms make
their appearance previous to introducing any limitations as to
the relative magnitudes and positions of the disturbing and
disturbed bodies. Consequently, since expansions containing
such terms are really more general in their application than
those which consist exclusively of periodic terms, they must
include the latter. In fact, as in the instance of central motion
above considered, so also when there are three or more bodies
mutually attracting, if the motion be wholly periodic, the non
periodic terms arise from expansions of periodic functions,
and from the former the functions may be arrived at by
certain analytical rules, the investigation of which has been
given by Laplace*. Now the application of such rules is
independent of the magnitudes of the quantities represented
by the symbols, inasmuch as the analytical form of expan
sion according to the powers of any symbol remains the same
whatever be the ratio of the quantity it represents to any
* Mecanique Celeste, Liv. n. No. 43.
PHYSICAL ASTRONOMY. 133
other quantity involved, the degree of convergency or
divergency of the expansion being alone affected by that ratio.
Hence it must not be inferred from the convertibility of the
expansion into one of which the terms are all periodic, that
the motion itself is in every case periodic. I am aware that
it has been the opinion of some mathematicians that the
Comet which is considered to have approached Jupiter to
within the orbits of his satellites, and to have suffered great
perturbation from its proximity to the Planet, will in the
course of ages be again in the same predicament. This idea
rests on the assumption that the developement of the general
analytical solution of the Problem of Three Bodies can
contain no other than periodic terms. But the terms now
under consideration contradict this assumption, their existence
constituting the analytical evidence that the motion is not
necessarily periodic. They may be taken as indicating, in
the instance just mentioned, that the motion of the Comet
might have ceased for a time to be periodic, and only after a
complete change of the orbit become periodic again. We have
no right to conclude, because in the usual approximate solu
tion of the Problem the arbitrary constants are equal in
number to those which would be contained in the exact solu
tion, that the approximate solution is of general application.
The criterion of its applicability is the convergency of the
series into which the integrations are thrown, and this can
only be tested by numerical calculation. It is true that in
the applications to bodies of the Solar System (such a case
as that just adverted to being excepted), the condition of con
vergency has been shewn by numerical calculation to be ful
filled. This amounts to a proof d posteriori of the legitimacy,
as far as regards the Lunar and Planetary Theories, of the
several processes by which, as we have seen, the condition of
periodicity is arbitrarily imposed. But in some of these very
applications there are cases of slow convergence (as in the
Lunar Theory and in the Theories of certain of the Minor
Planets), which point to the possible existence of circumstances
134 THE PRINCIPLES OF APPLIED CALCULATION.
under which the series would become divergent, and the mo
tions consequently be nonperiodic. As far as I am aware,
the solution of the Problem of Three Bodies has not hither
to been attempted by a method so general as to be capable
of determining the limits between periodic and nonperiodic
motions, or of indicating the character of the processes to
be adopted for computing the latter. Any method of suc
cessfully effecting the computations for the case of non
periodicity would, I conceive, involve the retention, without
alteration, of terms containing circular arcs as factors, or
some equivalent proceeding: but until an instance actually
occurs for which the usual expansions are found on trial
to be divergent, it is hardly worth while to endeavour to
ascertain the precise nature of the calculations which such an
instance would demand. What I am now contending for is,
that the occurrence of the nonperiodic factors proves that
the motion is not necessarily periodic, and that special opera
tions are required to adapt the expansions to periodic motions.
The following mathematical reasoning is here added for the
purpose of illustrating some points of the foregoing argument.
The exact differential equation, relative to the radius vector
(r) and the time (tf), for one of three bodies mutually attracting,
viz.
,
dt r j dt dr '
having been obtained in the usual way*, the first step in ap
proximating to the value of r is to integrate this equation
after omitting the terms which contain the disturbing function
R. By this integral, combined with that of the equation
r*d6 = hdt, the coordinates r and 9 of the disturbed body can
be expressed as functions of t thrown into series ; and like
expressions may be obtained for the coordinates of the disturb
ing body. The rule of approximation requires that these
values of the two sets of coordinates should be substituted in
the omitted terms containing R ; after which another integra
Airy's Planetary Theory, Arts. 7783.
PHYSICAL ASTRONOMY. 135
tion can be effected. This might be done by multiplying
by 2d.r*, and the integral thus obtained would be equivalent
to that which I have made use of in the Paper already refer
red to (Phil. Trans., 1856, p. 525), where it is shewn that
by this mode of integration nonperiodic functions are avoided,
because, in fact, it introduces the condition of periodic variation
of the radius vector. But the following process*, which is
also legitimate, for the opposite reason does not exclude such
functions. Let r = r l + v ; and as we have here two new
variables let us suppose that v and the disturbing force
vanish together, or that the value of v contains m as a factor.
Hence putting v = 0, we have for determining r t the equation
and r 1 is consequently the value of the radiusvector found by
the first approximation. On substituting r t + v for r, v* is to
be neglected, because by hypothesis it contains m' 2 as a factor,
and the second approximation only includes the first power
ofra'. The equation may consequently be put under this
form,
the usual mode of expressing the disturbing function being
adopted. The approximation, proceeding primarily accord
ing to the powers of m, is now made to proceed second
arily according to the powers of e the eccentricity of the un
disturbed orbit. In that case it is allowable to substitute in
the second term par 3 , or n*, for fwy 8 . Then putting the dis
turbing function under its general developed form, the equation
becomes
Now it is the integration of this equation for the purpose
of approximating to the value of v that gives rise to a term
* Airy's Planetary Theory, Arts. 8991.
136 THE PRINCIPLES OF APPLIED CALCULATION.
having t for a factor, one of the terms of the disturbing function
being of the form Pcos (nt + Q). On reviewing the foregoing
reasoning it will be seen, that while rules of developing have
been followed which are applicable independently of the
relative magnitudes of the quantities involved, no step has
been taken which ensures that 7 shall have small periodic
values, or that r shall have a mean value. This circumstance,
as already explained, accounts for the appearance of a term
that may increase indefinitely.
It may also be remarked that if we suppose v = and
/TOM
j~ = when t = 0, the integral of the foregoing equation will
be found to give, for determining the increment v of the
radiusvector in the small time t, the equation
v Pf
v =  2 . cos Q.
^ r i
This expression for v includes the term Pcos (nt + Q), and
may therefore be regarded as giving the true value of the
increment of the radius vector in the short interval t y whether
or not the motion be such as to make the variations of the
radiusvector periodic.
The foregoing discussion relative to the occurrence of
terms of indefinite increase in the solution of the Problem
of Three Bodies has been gone into, because it has an im
portant bearing on the interesting question of the stability of
the Solar System. The stability of the eccentricities and
inclinations of the planetary orbits has been usually inferred
from the known equations
2 . m Ja e* = c, 2 . m N /a tan 2 1 = c f .
But it is admitted by M. Le Verrier* that although such
an inference may be drawn from them for a planet the mass
* Recherches Astronomiques, Chap. IX. No. 6, in the Annals of the Paris
Observatory, Tom. II.
PHYSICAL ASTRONOMY. 137
of which " constitutes a considerable part of the sum of the
masses of the system of planets," an analogous conclusion
is not applicable to a planet whose mass is a small fraction
of that sum*. The general argument for the stability of the
planetary motions is of this kind. The analytical operations
which get rid of terms of indefinite increase consistently with
satisfying the differential equations of the motion prove the
possibility of expressing analytically the values of the radius
vector, longitude, and latitude in periodic terms. The method
of the Variation of Parameters does this in such manner as
to shew that even the slow variations of the elements of the
planetary orbits are expressible by periodic functions. But
the periodicity of these expressions, provided they are con
vergent, and therefore numerically, as well as analytically,
true, indicates fluctuation of value between restricted limits,
which is the proper evidence of the stability of the motions.
This reasoning, in short, establishes the abstract possibility
of a stable planetary system. In order to ascertain whether
the Solar System is stable, it would be necessary to substitute
the numerical data furnished by observation for each body,
in the system of equations from which the variations of the
elements are calculated, and to ascertain within what limits
the equations are satisfied by variations from the given values.
M. Le Verrier has, in fact, done this for all the Planets, except
Neptune and the Minor Planets, and has found that the actual
eccentricities and inclinations are subject to variations only
within narrow limits, so that being small at the present epoch,
they will always continue to be small t. M. Le Verrier con
cludes the investigation with these remarks : " This conse
quence, the importance of which is so considerable relative
to the stability of the planetary system, is, however, found to
* A proof of the truth of this statement by numerical calculation is given in
the Monthly Notices of the Royal Astronomical Society, Vol. XIII. p. 252, where
it is shewn that the above equations only ensure the stability of the orbits of the
four planets Jupiter, Saturn, Uranus, and Neptune.
f Recherches Astronomiques, Chap. IX. Nos. 1015.
138 THE PRINCIPLES OF APPLIED CALCULATION.
be established only for the ratios of the major axes which
have been considered, and we are ignorant of the conse
quences that might result from other mean distances of the
planets. It is to be regretted that we do not possess a general
expression for the limits of the eccentricities and the inclina
tions susceptible of an analytical discussion. Unhappily it
appears very difficult to form such an expression." It may
be noticed that these views are in accordance with the tenor
of some of the foregoing observations.
There is still another point in the Problem of Three
Bodies which demands explanation, although, as far as I am
aware, the difficulty it presents has not been noticed in express
treatises on the subject. The nature of the difficulty will be
best exhibited by reference to the mode of solving the problem
which I have adopted in the paper in the Philosophical Trans
actions already cited. At the beginning of that solution an
equation* which is necessary for the present purpose is ob
tained by the following investigation. Supposing, for sim
plicity, the three bodies to be in the plane of xy, we have
the usual equations
d*x fix dR_ d*y fiy dR _
++ ~^ + + ~~
d*y d*x dR dR .
x d/yw +x dj y ^="
By changing the coordinates x, y into the polar coordi
nates r, 9, and integrating the last equation,
dR d6 dR
dR.
d0 dt 
* The equation (7) in p. 525 of the Phil. Trans, for 1856.
PHYSICAL ASTRONOMY. 139
JO/2
Hence substituting for ^ in the first equation from the
second, and neglecting the square of the disturbing force,
dr* h* 2ji 2h dE tdR dO dR dr
7/3
But since on the righthand side of the equation 3 may
be put for 5 , it follows that
dt tr T J \dt \J dB ) dt dt
The approximate solution of this equation is to proceed
according to the powers of the disturbing force, and conse
quently the first step is to integrate after supposing R to
vanish. Let us assume that when this is done the values
obtained for r and 6 apply to elliptic motion, and let a be the
semiaxis major and e the eccentricity of the ellipse. Then
tfC
we shall have a and e 2 = 1 ^ > an ^ consequently that
Ch?
assumption imposes the conditions that C and 1 ^~ ^> e
positive quantities. When the known values of r and for
elliptic motion are substituted on the righthand side of the
equation to obtain a second approximation, it will be seen
that all the terms must have e for a factor. (See Art. 9 of
the paper referred to.) Consequently if e = 0, or /* 2 = W C,
the above equation becomes
Since C is positive, this equation can only be satisfied by
a circular orbit of which ^ is the radius ; in which case there
o
can be no disturbing force. Hence in the case of a disturbing
140 THE PRINCIPLES OF APPLIED CALCULATION.
force there must be a certain limit to the value of the arbi
trary constant e*, to find which is the object of the following
enquiry.
The radiusvector of the path of the disturbed body is
thus expressed in Art. 16 of my solution of the Problem of
Three Bodies :
4 terms involving the longitude of the disturbing body.
For the present purpose we may consider only terms in
volving the first power of e, and neglect the eccentricity of
the orbit of the disturbing body. Also for the sake of brevity
1 shall suppose the ratio of the arbitrary constant a to the
like constant a for the disturbing body to be very small.
Then for the calculation of A, E,f, N, and II in the above
expression for r, we have (in Art. 16)
A _l
2 '
_ __
n 2 ' da ' 4ft 2 ' da* ' ~2rcV da '
N= n +* d A* n = w/' ^ + 1 d * A } t
na' da ' \^a' c&* 2n' da*) '
11 .L 2 L f fl *i A r m' m'd*
in which equations n is put for 3 , and A Q for r ^
tt a 4a
terms involving higher powers of the ratio of a to a being
omitted. Hence if ri* = 75 and m=^ 7 , the following results
may be obtained :
da 2 ' da
* This remark is made in Art. 5 of the Paper in the Philosophical Trans
actions, which, however, contains no investigation of the limiting value. The
reference at the end of that Art. to note (A) is not to the purpose, because the
reasoning there relates to the eccentricity of the disturbed orbit solely as affected
by the eccentricity of the orbit of the disturbing body.
PHYSICAL ASTRONOMY. 141
w 2 \  3wi 2 e , T f1 2N
J, e/= , AT=w(lw 8 ),
Consequently for the part of r which does not contain the
longitude of the disturbing body, we have, to the first power
of the disturbing force,
r = a ( 1 + ~ }  ae ( 1 + rJ cos \Nt ( 1  ^ ) f e  & I + &c.
V * / \ o / l\ 4 / J
This value of r may be considered to belong to a mean
orbit. If j (1  ej and a t (1 + ej be the apsidal distances of
this orbit, the above expression gives
Hence = al+ and
( 1 + ^ J .
Since 6 X is the eccentricity of the mean orbit, the last
equation proves that e is proportional to that eccentricity.
By squaring we have nearly,
and if the product eW be omitted, e* e*. Now as far as
regards the expression for the complete value of r given by
this solution, which is the same, excepting the form, as that
given by Laplace's and other solutions, there appears to be no
reason why the constant e should not be zero. But the com
plete value of r consists partly of terms which do not contain
e, such, for instance, as that which in the Lunar Theory is the
exponent of the Variation. If, therefore, e = 0, the orbit will
not be an exact circle. This inference seems contradictory to
142 THE PRINCIPLES OF APPLIED CALCULATION.
that drawn above from the equation (A) ; and as no argu
ment, as far as I am able to discover, can be adduced against
the latter inference, we have here a difficulty which requires
to be cleared up, and which, probably, has not hitherto at
tracted attention, because, in fact, very little notice has been
taken by theoretical astronomers of the equation (A). But
to overlook the clear indications of that equation would be
nothing short of error, and it is, therefore, necessary to meet
the difficulty. This I propose to do by the following argu
ment.
It has already been shewn that the occurrence of non
periodic terms in the integrations may be got rid of by the
supposition of a mean orbit, that is, an orbit which is inde
pendent of particular values of the longitude of the disturbing
body. The following reasoning will, I think, shew that the
point now under consideration admits of being explained by
making the very same supposition. The masses of the central,
disturbed, and disturbing bodies being M, m, m', fi being put
for M+m, and P for (x  a;') 3 + (y  y'Y + (z z'}\ we have
the known equations,
As the object of the present investigation is not to obtain
an exact solution of the problem but to exhibit a course of
reasoning, it is allowable to make any supposition that will
not affect the validity of the reasoning. I shall accordingly
suppose, for the sake of simplicity, that the disturbing body
describes a circular orbit of radius a in the plane of xy with
the mean angular velocity v. Hence
PHYSICAL ASTRONOMY. 143
x' = a' cos (v't + e'), y' = a' sin (vt + e'), s' = 0,
dx' = v'y'dt, dy' = v'x'dt.
By taking account of these equations, and putting a' for r',
the following result is obtained :
ffx
Hence, representing by </> the angle between the radius
vectors of the two moving bodies, we get by integration
dx* dy* dz z _ , dy _ , dx n
It thus appears that the problem of three bodies admits
of an exact first integral on the supposition that the dis
turbing body moves uniformly in a circle, given in magnitude
and position, about the central body*.
To simplify the analysis farther, suppose the three bodies
to be in the plane of xy, and let 6 be the longitude of the
disturbed body. Then
and the above integral may be transformed into the follow
ing:
dr* tW .
2m f 2r r z \ *
+  1 j cos <i> + 72 .
a \ a a 2 J
* This theorem was first proved in a communication to the Philosophical
Magazine for December 1854.
144 THE PRINCIPLES OF APPLIED CALCULATION.
AI . dO d$ , ,
Also since j =  + v , and
at at
d.
dt
Jt
it follows that
w(I) / \
dt +V m'rsi
sn
a
r 2 ^)
^ h
a /
By expanding the trinomial affected with the indices 
and to terms including the fourth power of , the fol
lowing equations are found :
dr*
77 +
dt 2
' dt
2m
a
75 (3 cos + 5 cos 30)
t 4
+  ^ (9 + 20 cos 20 + 35 cos
' dt
dt
sin 20
v
r 74 (sin + 5 sin 30)
 r ( 2 sin 20 + 7 sin 40)
(C).
If the ratio of r to a' be not very small, it might be neces
sary to make use of all the terms of these equations. But in
PHYSICAL ASTKONOMY. 145
the present investigation it is not proposed to carry the
r*
approximation beyond the terms containing ^ , and accor
dingly, in order to use the equations for finding the mean
orbit, it is only required to obtain the values of r 2 cos 2<f> and
r 2 sin 2</> as functions of the time ; which is to be done by
successive approximations. The first approximation gives
elliptic values of r and as functions of t, which values, ex
panded as far as we please in terms proceeding according to the
powers of e, are to be substituted in r z cos 2<f> and r 2 sin 2</>.
Without actually performing the operations it will be seen
that the expressions for both quantities will consist wholly of
terms containing the longitude of the disturbing body. Also,
integrating the second equation, squaring the result, and
omitting the square of the disturbing force, we have
where, again, ir 2 sin 20 dt contains no terms that are inde
7/1
pendent of v't + e. Hence, eliminating ^ from the first equa
tion and suppressing the terms containing periodic functions
of v't + e', the result is
dr* W 2j, mr*
If we now alter the designations of the arbitrary constants
to indicate that they involve the hypothesis of a mean orbit,
we have for determining that orbit, and the motion in it, the
equations
dr> h mr* d0 ,
rdr
or, dt= j =7 .
s
146 THE PRINCIPLES OF APPLIED CALCULATION.
To integrate these equations put a 1 + (r a x ) for r in the
term containing ^ , and expand to the second power of r a r
Since a l may be taken for the mean radius, (r aj 2 will be of
the order of e*, and the approximation will consequently
embrace terms of the order of m\*. After the above operation
the equations will become
, rdr r*dO
dt = ^= _. _ . =?,
and we shall also have, putting ri z for 75 ,
By integrating the two equations the following results are
obtained :
a, (16,003 ^r), ,i3 (*+ TJ =^r + ^ sin 1/r,
,
Let us now put a Q for and e 2 for 1  V^ so that
These values of a and e belong to the first approximation
to the mean orbit, which, by hypothesis, is an ellipse. Hence,
for the same reasons as those adduced in the case of the first
approximation to the actual orbit, the arbitrary quantities
C and 1  5y 5  must both be positive. Now let
I*
/ 2 1 W
^ = 7i , and = m,
a* n
PHYSICAL ASTRONOMY. 147
Then it will be found that
/, * h ' i
= .(!+), f =
In obtaining these results terms involving m 2 e* are omit
ted for a reason which will be stated presently. To proceed
to another approximation to the mean orbit it would be neces
sary to substitute in the equations (B) and (C) for r and <
their values obtained by the second approximation to the
actual orbit. After this substitution new terms independent
of the longitude of the disturbing body make their appear
ance on the righthand sides of the equations (B) and (C),
and consequently on the righthand side of the equation
resulting from the elimination of between them. When
in this last equation, and in the equation for calculating
0, the terms containing vt + e' are suppressed, the integrals of
the resulting equations give a closer approximation to the
mean orbit. It is necessary to proceed to this new approxima
tion in order to find all the terms containing m z e* t on which
account such terms were not retained in the previous approxi
mation. This course of reasoning indicates that the determi
nations of the actual and the mean orbits proceed pari passu.
It is next required to find the relations of the arbitrary
constants a and e of the actual orbit to 'the arbitrary constants
a and e of the mean orbit. This may be very readily done
since we have already expressed the mean distance a l and the
mean eccentricity e L as functions of each set of constants. We
have, in fact,
j = a (l + ^ J and a t = a (1 + m z ).
Hence a =
Also 6 = 6* and e* = c* + .
102
148 THE PRINCIPLES OF APPLIED CALCULATION.
Hence e2 = e o 2 +i7'
(fi
Since hf = /A O (1  e 2 ), and ft = pa (1  e 2 ), it follows that
v .
v iv iv
Hence, omitting terms containing m* e*, h = h. The re
lation between the constants G and C follows from that be
tween a and a . For a = ^ and = ^ ; so that
Hence we have
Consequently, omitting ^^, e a = e 2 + ^ , as before*.
The foregoing results give the means of solving the diffi
culty stated at the commencement of this discussion. Since
e* is necessarily positive, if e 0, we must also have e = 0,
and m = 0. That is, the orbit is a circle, and there is no
disturbing force. Consequently, if there be a disturbing force
it is not allowable to suppose that the constant e can vanish.
As we have shewn that e is quam proxime the mean eccentri
city of the orbit, it follows that by reason of the action of the
disturbing body the mean eccentricity cannot be zero, but has
a limiting value obtained by putting e 0, namely, j= . It
is worthy of remark that the eccentricities of the Moon's orbit
* The equation e s = 1 5 +  was originally published in a communication
relative to the Moon's orbit in the Philosophical Magazine for April 1854. See
the Introduction.
PHYSICAL ASTRONOMY. 149
and of the orbits of Jupiter's satellites approach very closely to
the limiting values. For the Moon =. 0,0529, and the
known eccentricity of her orbit is 0,0548. The orbit of Titan,
however, which has a large angle of inclination to the plane
of Saturn's orbit, has an eccentricity nearly equal to 0,03,
which is much larger than the value of j= due to the Sun's
V2
perturbation. The approximations have hardly been carried
far enough to allow of application to the eccentricities of the
orbits of the planets. It may, however, be affirmed that the
77?
limiting value as expressed by the formula ^ will always be
very small for the planetary orbits. Supposing the disturbing
body to be a mass equal to the sum of the masses of Jupiter
and Saturn, and its distance from the Sun to be the mean
between the mean distances of these planets, if the disturbed
body be Venus, the value of = is 0,0024. M. Le Verrier has
V2
found 0,0034 for the minimum value of the eccentricity of the
orbit of Venus*.
Since, to the degree of approximation embraced by the
preceding reasoning, r 2 d6 = hdt, it follows that the motion of
the disturbed body is the same as if it were acted upon by a
/2
central force. In fact, supposing ^  to represent a cen
T 2i
tral force, the usual process gives
which is the equation that was employed above in the case of
disturbed motion. It may, therefore, be worth while to
enquire what results are obtained relative to the eccentricity
when the problem is simply one in which the force is central.
* Recherches Astronomiques, Tom. n. p. [29].
150 THE PRINCIPLES OF APPLIED CALCULATION.
In the first place we have, putting u for  ,
which can only be integrated by successive approximations.
If the steps of the approximation proceed according to powers
of ri'\ and if the term containing this quantity be very small
compared to the other terms under the radical, a true approxi
mation will be effected. But in that case the first step is to
integrate after putting n' 2 = 0, by which operation the first
approximation to the orbit will be found to be a conic section.
If we now assume that the conic section is an ellipse of which
the semiaxis major is a and eccentricity e, we shall have
a ^ and e z = 1 j The arbitrary constants C and h
CJi 2
will thus be subjected to the conditions that C and 1  z 
are positive quantities, which conditions they necessarily ^fulfil
through all the subsequent operations. The second approxi
mation may be effected so as to avoid nonperiodic factors by
substituting for u in the term involving ri* from the first
approximation, expanding to terms inclusive of e 2 , and elimi
nating the circular function by its elliptical value in terms of
u. When this is done the equation becomes
7/j hdu
do =  .
V  6" + 2^'w  #V
C', //, and h' having the same expressions as in the case of the
disturbed orbit. Hence, supposing a l (1 ej and a, (1 f e t ) to
be the two apsidal distances, and putting ?rafor j= , the results
V^
will be
PHYSICAL ASTRONOMY. 151
CJi 2
Here it is to be observed that since 1  g~ is a positive
quantity, e l cannot vanish unless m 2 vanishes, and that the
least value of ^ is p, omitting w 3 , &c. This limit to the
eccentricity is the same as that obtained for the mean dis
turbed orbit; which shews that the limitation of the eccentricity
of the disturbed orbit is so far due to the disturbing force
acting as a modification of the central force ^ . It is to be
noticed that, although the disturbed orbit can in no case be an
exact circle, such an orbit is always possible when the force is
central and attractive. This, however, is an isolated and
unstable case of motion, from which it cannot be inferred that
there may be gradations of eccentricity from zero to ~ . The
eccentricity of the disturbed orbit, as well as that of the orbit
described by the action of the central force, is arbitrary when
it exceeds the limiting value.
I propose to conclude the Notes on Physical Astronomy by
obtaining a first approximation to the mean motion of the
nodes of the Moon's orbit by a method somewhat resembling
in principle the above process for finding the mean value of
0* 2J7*
the eccentricity. If terms involving 4 and 4 be neglected,
the usual differential equations may be put under this form :
"
p being the projection of r on the plane of the ecliptic, and
O' t 6, being the true longitudes of the Sun and Moon. If
152 THE PRINCIPLES OF APPLIED CALCULATION.
we now omit the last term in each equation, and put for r, r'
their mean values a, a in the terms containing the disturbing
force, the Moon will move in a fixed plane, and be acted upon
by the central force z ( \ ^73) 5 so that the orbit will be
an ellipse, in which the periodic time will approximately be
27ray wV\
V ( 4/W V '
The forces expressed by the omitted terms of the first and
second equations have the effect of causing periodic variations
of this motion without permanently changing the plane of the
orbit. But the force  ^r produces a continual alteration of
2a
that plane, because by the action of that force the period of
the Moon's oscillation perpendicular to the plane of the
ecliptic is caused to be different from the period in the orbit.
After putting for r its mean value, or supposing the orbit to
be circular, the third equation becomes
and the mean period of the oscillation in latitude is therefore
2/ia'V '
which is less than the 'Moon's period by == . , . Hence.
v //. 4ytta
if p and P be respectively the periodic times of the Moon and
the Sun, the regression of the node in one revolution of the
Moon is the arc 2?r x ^ , which is the known first approxi
mation. Since, if the oscillation in latitude be small, its
period is independent of its extent, the regression of the node
is nearly the same for different small inclinations.
DYNAMICS OF A RIGID SYSTEM. 153
The Dynamics of the motion of a rigid system of points.
The dynamical principles hitherto considered are applicable
only to the motion of a single point acted upon by given
forces ; or to the motion of masses of finite dimensions sup
posed to be collected at single points. Such is the case with
respect to the masses whose motions are calculated in Physical
Astronomy, excepting that in the Problem of Precession and
Nutation it is necessary to regard the mass of the Earth as a
system of connected points. The class of problems in which
the motions of a system of points are to be determined, require
for their treatment, in addition to the principles on which the
motion of a single point is calculated, another which is called
D'Alembert's Principle. It would be beside the purpose of
these Notes to give an account in detail of particular applica
tions of this principle, such as those which form the subject
matter of express Treatises on Dynamics : but it will be proper
to discuss and exemplify its essential character, and to shew
how a general law of Vis viva is deducible from it.
The truth of D'Alembert's Principle may be made evident
by the following considerations. Suppose a system of points
constituting a machine to be moving in any manner in conse
quence of the, action of impressed forces, and at a given instant
the acceleration of the movement to be stopped by a sudden
suspension of the action of these forces. On account of the
acquired momentum every point will then continue to move
for a short interval with the velocity and in the direction it
had at the given instant. But the same effect would be pro
duced if at each point of the machine accelerative forces were
impressed just equal and opposite to the effective accelerative
forces. For such impressed forces would not alter the direction
of the motion, but would prevent its increment or decrement.
Since, therefore, these supposed impressed forces have the
same effect as a suspension of the actual impressed forces,
they must exactly counteract the latter, if both sets of forces
154 THE PRINCIPLES OP APPLIED CALCULATION.
act simultaneously. This counteraction can take place only
as a result of those laws of force and properties of rigid bodies
which are the foundation of statical equilibrium. Hence these
forces are in equilibrium according to the principles of Statics :
which, in fact, is D'Alembert's Principle.
On account of the statical equivalence of the two sets of
impressed forces, they must be such as to satisfy the general
equation of equilibrium given by the principle of Virtual
Velocities. In this case the actual motions of the several points
may be assumed to be their virtual velocities, being evidently
consistent with the connection of the parts of the machine.
Let us, therefore, suppose the effective accelerative forces of
d?x d*x f
the material particles m, m, &c. at the time t to be j^ , =5 ,
d*y d?y f
&c. in the direction of the axis of x, j^, fi> &c. in the
7g 72 /
direction of the axis of #, and j^, ^ , &c. in the direction
of the axis of z ; and let the resolved parts of the actual im
pressed forces acting on the same particles be X, X, &c.,
F, F, &c., Z, Z', &c. Then, the signs of the effective forces
being changed, the equation of virtual velocities is
d*x \dx
r \ dz
.,__
This equation gives by integration,
We have thus obtained, by the intervention of the prin
ciple of virtual velocities, the general equation which expresses
the law of Vis viva.
It may here be remarked that neither in discussing D'A
lembert's Principle, nor in deducing from it the law of Vis
DYNAMICS OF A RIGID SYSTEM. 155
viva, has any account been taken of the pressures on fixed
axes due to the rotation of masses about them. The centri
fugal force of each particle revolving about a fixed axis must
be counteracted by an equal force in the contrary direction,
depending on the reaction of the axis, and supplied by the
intervention of the rigidity of the mass. These forces tending
towards axes may be regarded as effective accelerative forces,
relative to which the reactions of the axes are impressed
forces. Consequently the forces of this kind are embraced
by D'Alembert's Principle, and might be introduced into
the general equation furnished by the principle of virtual
velocities. But it is clear that, as their virtual velocities are
always and in every case zero, they would disappear from
this equation. This is proof that the effects of centrifugal
force and of the reaction of fixed axes require separate con
sideration; which, however, they cannot in general receive
till the motions of the system have been previously deter
mined by means of the equation of Vis viva.
On reviewing the steps by which the general equation
which expresses the law of Vis viva has been obtained, it will
be seen that they involve, first, the usual principles of the
dynamics of the motion of a single particle; secondly, D'A
lembert's Principle, which, as is shewn above, is inclusive of
the property of vis inertias, or conservation of momentum;
thirdly, the principles on which the formation of the equation of
virtual velocities depends. It has been shewn in pages 101
103 that that equation rests (1) on a definition which expresses
the fundamental idea of the equilibrium of forces in Statics ;
and (2) on the property of rigid bodies according to which a
force acting along a straight line produces the same effect at
whatever point of the line, rigidly connected with the body, it
be applied. This property is to be regarded as a law of rigid
bodies, and as such capable of deduction from the anterior
principles which are proper for accounting generally for
rigidity. Thus an d priori theory of the rigidity of solids
would furnish an explanation of the whole class of facts
156 THE PRINCIPLES OF APPLIED CALCULATION.
embraced by the general equation of virtual velocities, and
besides these, as the foregoing argument shews, of the facts
embraced by the law of Vis viva. The process by which the
equation expressing that law was arrived at depends on no
other property of a rigid body than the one in question, in
addition to the property of vis inertias common to all bodies.
In Treatises on Dynamics it is usual to speak of the con
servation of Vis viva as a principle, and similarly of the con
servation of areas, &c. It seems preferable to designate as a
law whatever is expressed by a general formula obtained by
mathematical reasoning, and to apply the term principle
exclusively to the fundamental definitions or facts on which
the reasoning that conducted to the formula is based.
The solution of a problem may sometimes be conveniently
effected by employing immediately the equation of virtual
velocities; as in the following example. A given mass /,
suspended by a fine thread, and acted upon by gravity,
descends by the unwinding of the thread from a given
cylinder revolving about its axis, which is fixed, and the
centre of gravity of the cylinder is at a given distance from
the axis : it is required to determine the motion. Let a be
the radius of the cylinder, h the length of the perpendicular
on the axis from the centre of gravity, and a the angle which
this line makes with a horizontal line at the time t. Also let
the perpendicular on the axis from any element m make the
angle with a horizontal line, and its length be r. Then,
T being the angular velocity of the cylinder, the virtual
velocity of the particle m is r = , and its effective accelera
72
tive force r TT Relatively to the force of gravity the
virtual velocity of m is r j cos 6. Hence the equation given
d/t
by the principle of virtual velocities is
d 2 a den da. cfa. \ da
DYNAMICS OF A RIGID SYSTEM. 157
The mass of the cylinder being M, let 2 . mr z = M7c*. Then,
since 2 mr cos = Mh cos a, we have, after striking out the
P da.
common factor y ,
at
d*a _ Mgh cos a + figg*
W Mk* + /*a' '
by the integration of which equation the motion is determined.
By applying, in conformity with D'Alembert's Principle,
the laws of statical equilibrium to cases of the motion of a
rigid system acted upon by given forces, six general equations
are obtained, which suffice for the solution of every dynamical
problem. Let x, y, ~z be the coordinates of the centre of gra
vity of the system at the time t, referred to fixed rectangular
axes in arbitrary positions, and let a/, y, z be the coordinates
at the same time of any particle m referred to parallel axes
having their origin at the centre of gravity. Also let S . mX,
5 . m Y, 2 . mZ be the sums of the impressed moving forces
parallel respectively to the three axes. Then the six general
equations are conveniently expressed as follows*:
The following problem has been selected for solution for
the purpose of exhibiting a mode of applying these equations
directly, without the consideration of angular motions relative
to rectangular axes. A hoop in the form of a uniform circular
ring of very small transverse section, acted upon by gravity,
rolls on a horizontal plane the friction of which prevents
* See Pratt's Mechanical Philosophy, Arts. 428 and 429.
158 THE PRINCIPLES OF APPLIED CALCULATION.
sliding : required its motion and the path it describes under
given circumstances.
The axes of rectangular coordinates being taken so that
the axes of x and y are in arbitrary positions in the horizontal
plane, and the coordinates of the point of contact of the hoop
with the plane at the time t being x and y, let the normal to
the path of the hoop make at this point an angle a with the
plane of the hoop, and an angle {3 with the axis of #, the
latter angle being supposed to increase with the rolling. Also
let a be the radius of the hoop, and the angle which the
radius to any point makes with the radius to the lowest point.
Then it may be readily shewn that
x = x f a cos a cos /3,
y = y a cos a sin /?,
z = a sin a,
x a cos cos a cos /3 a sin sin {3,
y a cos cos cc sin ft a sin cos /?,
z' = a cos 6 sin a.
Again, if V be the rate of motion of the point xy of contact,
and s the arc described at the time t, we have, in consequence
of the rolling,
rr ds de
v =dt= a df
Also, the angle /3 increasing with the motion, and the curve
being concave towards the axis of x,
dx ds . Q dy ds n
JT = 77 sin p, f = j cos p.
dt at dt dt
Let F be the moving force of the friction acting in the di
rection of the normal towards the centre of curvature, and F'
that of the tangential friction acting in the direction contrary
to that of the motion of the centre of gravity ; and let P be
the pressure on the horizontal plane. Then, supposing the
moving forces F, F', P to be embraced by the sign 2,
DYNAMICS OF A RIGID SYSTEM. 159
S . m X= F cos 13 F sin /9,
M being the mass of the hoop. Consequently the first three
general equations become for this case,
df~ MV dt MV dt *
d*y_ F dx_J^_ dy_ / 2 x
de MV ' dt MV dt ' ' { h
+ ;
If now o/ , ?/ , s' be the values of a?', /, s' for the lowest
point of the hoop, we shall have, by putting 0=0,
x Q = a cos a cos /3, #' = a cos a sin /3, z' = a sin a.
Hence, since a/, #', ^' are referred to the centre of gravity of
the hoop, the following results are obtained, the moments of
the forces F, F', and P being supposed to be embraced by the
sign 2 :
2 . my'Z= Pa cos a sin /3,
2 . mz r Y=Fa sin a sin ft + F'a sin a cos /?,
2 . ws'JT Fa sin a cos + .F'a sin a sin /?,
2 . waj'^T = Pa cos a cos /?,
2 . wx' Y= (Fsin ft + F' cos /3) a cos a cos /3,
2 . my'X = (J^cos P  F' sin /3) a cos a sin /9.
Consequently the three equations of moments are,
(4),
(5),
160 THE PRINCIPLES OF APPLIED CALCULATION.
(6).
j j
Since the differential coefficients j and j^ , applying to a
given element, may be eliminated by the equation j = a 7 ,
Cvv Ctu
and the trigonometrical functions of 6 disappear by the inte
grations indicated by 2, it follows that the foregoing six equa
tions contain only the seven variables a, x, y, t, F, F', P.
They suffice, therefore, for obtaining, as functions of t, the
values of a, x, y, which determine the position of the hoop,
and the values of the forces F, F' P. Also by eliminating all
the variables except x and y, the differential equation of the
path of the hoop is found.
The eliminations required for completely effecting the
general solution of the problem become extremely complicated.
It is, however, to be observed that the six equations (1), (2),
(3), (4), (5), (6), take account of all the mechanical conditions
of the question, and that what remains to be done is merely an
application of the established rules of analysis. The conside
ration of revolutions about axes, which is usually employed
in problems of this class, does not involve any additional
mechanical principle, but is to be regarded as a means of
simplifying the analytical treatment of the differential equa
tions. To illustrate this point I shall now proceed with the
analytical processes required for the direct solution of the
problem, and after advancing so far as may be practicable in
the general case, shall apply the results under particular
restrictions.
In the first place, from the equations (1) and (2) we have
dyd z x dxd*y _FV
dt d? dt d'f~"W
dxtfx dd? F'V
If now we substitute in these equations the values of
DYNAMICS OF A RIGID SYSTEM. 161
T5, and r, deduced from the foregoing expressions for
x and y, the results will contain the angle /3 and its first and
second differential coefficients with respect to t. These may
be eliminated by means of the equations
dx Tr n dy

__
dt p ' at pdt p 2 <& '
p being the radius of curvature at the point xy of the path.
This having been done, the two equations give the following
values of F and F :
F V* / a \ da* . d*a
^= ( 1  cos a  a cos a 55 a sm a j ,
M p \ p J dr dr
F dVf^ a \ Fa/ . da dp
17=* ~ji U  cos a)  2sm a^+cos a
M dt \ p J p \ dt
Also by substituting a sin a for z we have from equation (3)
P . d^ d*a
_ = ^ slria _ + aC o S a^.
Again, from the equations (4) and (5) we get,
dx ^ ( ,tfz , d z y\ dy^ I , d*x , d?z'\
j* m (y j7T ~~ z TJT +* .miss TX x rjyr]
dt \ y dt' dt 2 J dt \ dt* d? J
= Va (P cos a ^ sin a),
=  Va F' sin a.
After substituting the preceding values of P, F, and F' in the
righthand sides' of these two equations and the equation (6),
we obtain a first set of values of the lefthand sides, viz.
11
1C2 THE PRINCIPLES OF APPLIED CALCULATION.
( F 2 / a \ d 2 a\
May \g cos a  1 1  cos a + a ^ \ ,
( P \ p <*t )
IT (f, <*> \ dV Va f . da. cos 8 adp\)
Ma\(l  cos a cos a j + sm 2a j +   )\ .
Up / at p \ dt p dtJ)
Another set of values of the same quantities is formed by
substituting in their expressions the values of x, y ', z' t and of
the second differential coefficients with respect to , which
substitutions give the quantities as functions of a, /3, 0, and
7/1
F, V being put for a j . These are simple operations, and
not very long if care be taken to suppress terms containing
sin 20 and cos 20, which will evidently be caused to disappear
by the integrations with respect to 6 from 6 = to = 2?r.
After performing these integrations and eliminating /3 by the
same means as before, the three quantities will be found to
have the following values :
a \ dV f a .
MaV4 '^ C<i "^ + r V
Vadp
a .

_
By equating the two sets of values the following equations
are immediately obtained :
*
0, ......... (a)
DYNAMICS OP A RIGID SYSTEM. 163
3a \ dV
sm a 2 cos a I rr
2 ; dt
dp
The last two equations are equivalent to the two following of
simpler form :
sin a dV 3 . Wa .
da a . dp
, , . ,
Smce 7. , _+ . , an d  = ^  ^ , it follows
that the equations (a), (/3), (7) contain only the variables
a, #, y, and ^, and are therefore proper for determining the
position of the hoop at any given time, and the curve which
it describes on the horizontal plane under given circumstances.
Also they are applicable whatever may be the initial circum
stances, the investigation having been perfectly general.
It will_now be supposed, as a particular case, that the
hoop has a uniform motion of rotation about the principal axis
perpendicular to its plane through the centre of gravity.
Then a ~r is constant, and consequently Fis constant. Hence
by the equations () and (7),
/_ 3a . 2 Wa da a . dp
I 2 cos a + sura }j = 0, 7 a sm a f = 0.
V 2p / dt dt 2p 2 dt
Since from the second of these equations p is obtainable as a
function of a, it follows that the first is of the form / (a) = =0.
112
164 THE PRINCIPLES OF APPLIED CALCULATION.
Consequently j 0, or a is constant. Hence also ~ = 0, or
dt at
p is constant. This reasoning proves that when a hoop rolls
uniformly on a horizontal plane, it maintains a constant incli
nation to the plane and describes a circle.
Again, let us suppose the inclination of the plane of the
hoop to the horizontal plane to be constant, or ^ = 0. Then it
follows from the equations (/3) and (7) that ^ = 0, and  = 0.
Hence the hoop rolls uniformly along a circle.
The equations (a), (/:?), (7) might be employed to deter
mine the conditions under which the hoop describes a given
curve. In that case one of the equations is superfluous, or
there would be a remaining equation of condition. Let, for
instance, the curve be a circle. Then = 0. Hence by (7)
at
^ = 0, and by (/3) ^=0; so that the differential of (a) is
identically satisfied. These results prove that the hoop de
scribes the circle uniformly with a constant inclination to the
horizontal plane,
In the case of each of the above suppositions let I be the
radius of the circular path. Then we have by (a), between
V, b, and a, the relation expressed by
F 2 / 7 3a \
g cot a = 75 ( 20 cos a J .
Since a is necessarily positive, this equation shews that 2b
must always be greater than cos a. Hence if b be very
small, a will be nearly  , and the hoop nearly vertical. In
this case the motion approximates to spinning about a vertical
axis.
DYNAMICS OF A RIGID SYSTEM. 165
By deducing from the above equation the value of b as a
function of V and a, there results
V\ L / 1 2a?cos 2 a\i)
& = tan a U + 1  * . p.
^r ( ~ \ 2 ^ sin a/ J
Hence for the same values of V and a there are two values of
, excepting when tan a sec a =*^T^ > in which case the two
values are equal. The value of b is impossible if tan a sec a
be less than
If a be considered the unknown quantity, and its value be
required as a function of V and Z>, we shall have, after putting
u for cos a, m for Q ^ T7 g, and w for , the following biquad
OOC r oCJ
ratic equation :
w 4  2nu* + (m 2 + ?i 2  1 ) w 2 + 2wu  ^ = 0.
The sign of the last term shews that the equation has at
least one positive and one negative root, and, whether m 2 + n* 1
be positive or negative, the signs of all the terms indicate
that there is but one negative root. The other two roots are
either impossible, or possible and both positive. Since it ap
peared that according to the dynamical conditions 2b must be
greater than   cos a, that is, cos a less than   or n, it
'i oCl
follows that the equation must have a positive root between
u = 0, and u = n. This, in fact, is found to be the case ; for
on substituting these values in the equation, the lefthand
side becomes respectively w 2 and mV.
The case of an indefinitely thin disk of radius a rolling
along a circle of radius b being treated in the same manner,
the equation applicable to the steady motion is found to be
3& 5a \*
3^ cosa j
* This result does not agree with that given in p. 200 of Mr Routh's Treatise
on the Dynamics of a System of Rigid Bodies. I am at present unable to account
for the discrepancy.
166 THE PRINCIPLES OF APPLIED CALCULATION.
It may here be remarked that if it were proposed to deter
mine under what conditions a hoop acted upon by gravity
might slide uniformly along a circle on a perfectly smooth
horizontal plane, the question would be one of Statics rather
than of the Dynamics of Motion, the action of gravity being
just counteracted by centrifugal force. It may, however, be
treated by the same process as that applied to the preceding
problem, but as the angle 6 for a given element would not
7/3
vary with the time we should have j 0. In this way it
CLii
might be shewn that the required conditions would be ex
pressed by the equation
V 2 (. 3a \
a = ^(b cos ccj ,
differing from that applicable to the rolling motion only in
having b in the brackets in the place of 2b.
Having sufficiently illustrated by the preceding discussion
the method of directly employing the general dynamical equa
tions for the solution of problems, I shall conclude the Notes
on this department of applied mathematics by the solution of
a problem the treatment of which requires a particular con
sideration, which appears to have received attention for the
first time only a few years ago. I refer to the problem of the
oscillations of a ball suspended from a fixed point by a cord,
and acted upon by the Earth's gravity, the motion of the
Earth about its axis being taken into account*. It is not
necessary in dynamical problems of motion to take account of
the movement of the Earth's centre of gravity, because all
points both of the Earth and the machine equally partake of
this motion, and we may conceive it to be got rid of by im
pressing an equal motion on all the points in the opposite
direction. But the case is not the same with respect to the
Earth's diurnal motion, by reason of which different points
* See an Article entitled "A Mathematical Theory of Foucault's Pendulum
Experiment," in the Philosophical Magazine for May 1852, p. 331.
DYNAMICS OF A RIGID SYSTEM. 167
move with different velocities and in different directions.
This circumstance ought in strictness to be included in any
reasoning relative to the action of gravity, whether the ques
tion be to determine motion relative to directions fixed with
respect to the Earth, or motion relative to fixed directions in
space. This may be done by the following process in the case
of the problem above enunciated.
Conceive a line to be drawn through the point of suspen
sion of the ball parallel to the axis of rotation of the Earth,
and a motion equal ajad opposite to that which this line has in
space at any instant to be impressed on all particles of the
Earth inclusive of the cord and ball. The relative motions of
the Earth and pendulum will thus remain unaltered, the line
will be brought to rest, and all points rigidly connected with
it will begin to move as if they were revolving about it with
the Earth's angular motion. Consequently, the direction of
the force of gravity, being always perpendicular to the Earth's
surface, will revolve about the same axis. Thus our problem
is identical in its dynamical conditions with the following :
To determine the motion of a ball suspended by a slender
cord from a point in a fixed axis, and acted upon by a con
stant force in the direction of a line making a given angle
with the axis and revolving about it with a given angular
velocity.
Suppose to be the point of suspension, and OX, Y,
OZ, to be rectangular axes fixed in space, of which OZ (drawn
downwards) coincides with the axis of rotation. OA is the
direction of the action of gravity, making a constant angle
AOZ(\) with OZ, viz. the colatitude of the place where the
pendulum oscillates. P is the position of the centre of the
ball, OP a the length of the cord, and #, y, z are the coor
dinates of P at the time t. Let o> = the Earth's velocity of
rotation, and consequently the angular velocity of the plane
A OZ about OZ; and let cot the angle which the plane A OZ
makes with the plane YOZ at the time t.
168 THE PRINCIPLES OF APPLIED CALCULATION.
The force of gravity being g, the resolved parts in the
directions OX, OY, OZ are
g . cos A OX, g . cos AOY, g . cos A OZ;
or, g . sin X sin cot, g . sin X cos cot, g . cos X
The accelerative force of the tension of the cord being T,
the resolved parts in the same directions are
Tx _Ty^ Tz
a ' " a ' ~ a
Consequently,
d*x Tx
jp = g sm \ sin cot  ,
~ s= g sin X cos cot   ,
(it/ Of
Tz
These are the differential equations of the motion referred
to fixed axes in space. In order to determine strictly the
motion relative to the Earth's surface, it is necessary to make
the investigation depend on these equations, and to transform
the coordinates x, y, z into others x, y', z fixed with reject
to the Earth. For this purpose it is convenient to take for
the origin of the new coordinates, the axis of x at right
angles to OA and in the plane A OZ, which is the plane of
the meridian of the place, the axis of y' perpendicular to that
plane, and the axis of z coincident with OA. Also it will
be supposed that for a place in North Latitude x is positive
towards the North, y positive towards the East, and z posi
tive towards the Nadir Point. Then, regard being had to the
direction of the Earth's rotation, the following will be found
to be the relation between the two systems of coordinates :
x (z sin X + x cos X) sin cot y cos cot,
y == (z sin X + x cos X) cos cot + y sin cot,
2 = 2' cos X x sin X.
DYNAMICS OF A RIGID SYSTEM. 169
These values of x, y, z are to be substituted in the fore
going differential equations in order to obtain differential equa
tions of the motion in which the variables are x, y, z. It
will now be supposed that the ball performs oscillations of
small extent, so that j is always very small ; and as co is
also a small quantity, terms involving the product a> x j and
dt
the square of co will be neglected. Thus the result of the sub
stitution will be as follows :
d*x' Tx' ^ dy'
T#  2o> cos \ f ,
df a dt '
d*z' Tz' . _ dy'
T~S a  2o> sm \ f .
df u a dt
Adding these equations together after multiplying them
respectively by 2dx f , 2dy', 2k', we get by integration, since
x'dx' + y'dy'+z'dz'^O,
dx* dy' 2 dz' 2
+ + * =
Again, multiplying the first of the three equations by y
and the second by x ', and subtracting, we have
, , x '' '
x
Hence by integration,
dtf AY
x' =  y' ^ = H + <o cos X (" + y").
Supposing that %, = tariff, and x" + '" = r", this equa
X
tion becomes
dO H
170 THE PRINCIPLES OF APPLIED CALCULATION.
TT
which shews that besides the angular velocity ^ , the ball
has a constant angular velocity a> cos X, by which the angle 6
is continually increased. Thus relatively to the Earth there is
a uniform angular motion of the ball from the axis OX' to
wards OY', that is, from North towards East, and conse
quently in the direction contrary to that of the Earth's rotation.
As a cos X is the resolved part of the Earth's angular motion
relative to a vertical axis, it follows that the oscillations of the
ball really take place in a plane fixed in space, or, if we regard
the actual motion of the point of suspension, in planes parallel
to a fixed plane.
The Principles of the Dynamics of Fluids in Motion.
The department of applied mathematics on which I now
enter differs essentially from the preceding one in the respect
that the parts of which the mass in motion is composed are
not rigidly connected, and are capable of moving inter se.
Under the condition of rigidity the differential equations to
which the dynamical principles conduct are all eventually
reducible to a single differential equation between two vari
ables. But when it is required to determine the simultaneous
motion of unconnected particles in juxtaposition, this is no
longer the case, and the investigation necessarily leads to dif
ferential equations containing three or more variables. Such
equations are as far removed in respect to comprehensiveness
and generality from differential equations between two vari
ables, as the latter are from ordinary algebraic equations. For
this reason their application in physical questions requires new
and peculiar processes, the logic of which demands very close
attention. I have, therefore, thought that the arguments
relating to this application of mathematics would be best
conducted by reference to express definitions and axioms,
and by the demonstrations of enunciated propositions, and
that by this means the character of the reasoning will be
HYDEO DYNAMICS. 171
clearly exhibited, and an opportunity be given for the discus
sion of points that may especially require .elucidation or con
firmation. Some of the propositions and their demonstrations
have been long established, and are given here, in conjunction
with others that are for the most part original, only for the
purpose of presenting the reasoning in a complete form.
The two following definitions of the qualities of a per
fect fluid are sufficient foundations of the subsequent mathe
matical reasoning applied to the motion of fluids*.
Definition I. The parts of a fluid of perfect fluidity in
motion may be separated, without the application of any
assignable force, by an infinitely thin solid partition having
smooth plane faces.
Definition II. The parts of a fluid in motion press against
each other, and against the surface of any solid with which
they are in contact.
The first of these definitions is the statement of a general
property of fluids, which, though not actually existing, is
suggested by the facility with which the parts of a fluid,
whether at rest or in motion, may be separated. As all known
fluids possess some degree of cohesiveness, strict conformity
to this definition is not an experimental fact. The hypothesis,
however, of perfect fluidity may be made the basis of exact
mathematical reasoning applied to the dynamics of the motion
of fluids, just as the hypothesis of perfect rigidity is the basis
of exact mathematical reasoning applied to the dynamics of
the motion of solids. A comparison of numerical results ob
tained by calculating on that hypothesis with corresponding
results deduced from direct experiments, would furnish a mea
sure of the effect of imperfect fluidity, or viscosity, such as
that which is found to exist to a sensible amount in water
and in air. The causes of imperfect fluidity are of such a
* These are the same Definitions as those which in p. 104 are made the foun
dation of Hydrostatics. They are assumed here to hold good for fluids in motion,
and are, therefore, reproduced in terms appropriate to the state of motion.
172 THE PRINCIPLES OF APPLIED CALCULATION.
nature that it does not seem possible, in the present state of
physical science, to bring them within the reach of d priori
investigation. Numerical measures obtained in the manner
above stated may contribute towards framing eventually a
theory to account for them.
The other definition is also a statement of a general pro
perty of fluids known by common experience. The pressure
of fluids is subject to a law, ascertained by experiment, ac
cording to which in fluid of invariable temperature the pres
sure is always a function of the density, so that whether the
fluid be at rest or in motion, the pressure is the same where
the density is the same. The relation between the pressure
and the density forms a specific distinction between one fluid
and another. In the case of water the variation of density
corresponding to a variation of pressure is so small as to be
practically inappreciable. This physical fact has suggested
the idea of an abstract fluid, which, in the mathematical
treatment of its pressures and motions, is regarded as incom
pressible. In fluids that are compressible, such as air of con
stant temperature, the variations of pressure are assumed on
experimental grounds to be exactly proportional to the varia
tions of density. I proceed now to the demonstration of the
law of pressure for fluids in motion.
Proposition I. The pressure at any point in the interior
of a perfect fluid at rest is the same in all directions from the
point.
The proof of this Proposition has already been given in
pages 105 107. The Proposition is enunciated here in order
to exhibit distinctly the steps of the reasoning by which the
law of pressure is proved for fluid in motion.
Axiom I. If a common velocity, or common increments
of velocity, be impressed on all the parts of a fluid mass,
and on the containing solids, in the same direction, the
density and pressure of the fluid remain unaltered.
HYDRODYNAMICS. 173
This axiom, the truth of which is selfevident, is used in
the proof of the next Proposition.
Proposition II. The pressure at any given instant at any
point in the interior of a perfect fluid in motion is the same in
all directions from the point.
Conceive the velocity which a fluid particle has at a given
point at a given time, to be impressed at that instant upon it
and upon all the parts of the fluid and the containing solids
in a direction opposite to that in which the motion takes
place. The particle is thus reduced to rest. If also its effec
tive accelerative forces at each succeeding instant be impressed
on all the parts of the fluid and the containing solids in the
directions contrary to the actual directions, the particle will
remain at rest. By Axiom I. the relative positions of the
particles of the fluid and the pressures at all points are in no
respect changed by thus impressing a common velocity and
common accelerative forces in common directions, the only
effect being that the motions of the fluid are no longer referred
to fixed space, but are relative to the motions of the selected
particle, and are referred to its position at the given time.
Since, then, the particle continues at rest, we may apply to
it the same reasoning as that employed in the proof of Pro
position I., the effects of the state of motion of the contiguous
parts, and of the variation, in time and space, of the density
of the particle being neglected, as being infinitesimal quanti
ties of the same order as the impressed moving forces. Hence,
the effective accelerative forces being assumed to be always
finite, the law of equal pressure results precisely as in the
case of fluid at rest. Being shewn to be true of any selected
particle at any time, it is true of all particles at all times.
Consequently the law of equal pressure in all directions
from a given position has been proved to hold generally both
in fluid at rest and in fluid in motion, having been deduced
with as much exactness for the one case as for the other from
the fundamental definitions of a perfect fluid.
174 THE PRINCIPLES OF APPLIED CALCULATION.
Axiom II. The directions of motion in each element of
a fluid mass in motion are such that a surface cutting them
at right angles is geometrically continuous.
The motion of a fluid mass differs from that of a rigid
body in the respect that the relative positions of its com
ponent parts are continually changing. The above axiom
asserts that consistently with such changes the directions of
the motion are subject to the law of geometrical continuity.
Unless this be the case, the motion is not within the reach
of analytical calculation : on which account the axiom must
be granted.
"N. B. The following rules of notation relative to differen
tials and differential coefficients have been adopted in all the
subsequent reasoning. A differential is put in brackets to
indicate that the differentiation is with respect to space only,
the time not varying. A differential coefficient with respect
to time is put in brackets when it is the complete differential
coefficient with respect to both space and time. Differential
coefficients not in brackets are partial.
Proposition III. To express by an equation that the
directions of motion in any given element are in successive
instants normals to continuous surfaces.
Let ijr be an unknown function of the coordinates and
the time such that (d^) is the differential equation of a
surface to which the directions of motion in a given element
are normals at a given time. By Axiom II. such a surface
exists. Hence ~ , ^ ,  are in the proportion of the
velocities u, v, w resolved in the directions of the axes of co
ordinates. Or, X being another unknown function of the co
ordinates and the time,
. dty dty dfy
u \ f , v = X ~ , w \ y .
ax ay dz
Hence, (dtyr) =  dx + r dy + dz 0.
A, A *" A,
HYDRODYNAMICS. 175
This equation expresses that the directions of motion in
the given element are normals to a continuous surface at one
instant. That the motion may be such as to satisfy this
condition at the succeeding instant it is necessary that the
equation
3 < =
should also be true, the symbol of variation 8 having reference
to change of position of the given element, and therefore to
change with respect to space and time. On account of the
independence of the symbols of operation 8 and d, that equa
tion is equivalent to (d . &/r) = 0. But
dt dx dy ' dz
and because the variation with respect to space has reference
to change of position of the given element,
&c = uSt, $y = vSt, Sz = wSt.
Hence,
and by integration,
Consequently, by substituting the foregoing values of
w, v, w t and supposing the arbitrary function of the time to
be included in we have
which is the equation it was required to find. It may be
remarked that although the reasoning applied to a single
element at a particular time t, since the element might be any
whatever, and the time any whatever, the above equation is
perfectly general. In fact the function \fr may be supposed
to embrace all the elements at all times. We have thus
176 THE PRINCIPLES OF APPLIED CALCULATION.
arrived at one of the general differential equations of Hydro
dynamics, the investigation of which, it will be seen, has only
taken into account space, time, and motion.
Axiom III. The motions of a fluid are consistent with
the physical condition that the mass of the fluid remains
constant.
This axiom must be conceded on the principle that matter
does not under any circumstances change as to quantity. By
the following investigation an equation is obtained, which
expresses that the motion of the fluid is at all points and at
all times consistent with this condition.
Proposition IV. It is required to express by an equation
that the motion of a fluid is consistent with the principle of
constancy of mass.
It is usual to obtain this equation on the supposition that
the mass of a given element remains the same from one instant
to the next ; and as the same reasoning applies whatever be
the element and the time, it is inferred, just as in the above
investigation of the first general equation, that the resulting
equation applies to the whole fluid mass. For the purpose of
varying the demonstration I shall here conduct it on the
principle that the sum of the elements remains constant from
one instant to the next.
The density being p at any point whose coordinates are
x, y, z at the time t, the whole mass is the sum of the elements
p DxDyDz, the variations Dx, Dy, Dz being independent
of each other and of the variation of time. Hence the con
dition to be satisfied is,
8 (p DxDyDz]  a constant, or S . S (p DxDyDz) = 0,
the symbol 8 having reference to change of time and position.
On account of the independence of the symbols of operation
B and 8 t the last equation is equivalent to
8 (S.p DxDyDz) = 0,
HYDRODYNAMICS. 177
which signifies that the sum of the variations of all the ele
ments by change of time and position is equal to zero.
Now
B . p DxDyDz=p(DyDzDSx+DxDzDSy+DxDyD$z)
And since &e, By, Bz are the variations of the coordinates
of any given element in the time &t, we have
Sx = uBt, By = v
Hence,
~
Consequently, by substituting in the foregoing equation,
This equation is satisfied if at every point of the fluid
which is the equation it was required to obtain.
The investigation of this second general equation has taken
into account space, time, motion, and mass, or quantity of
matter.
Proposition V. To obtain a general dynamical equation
applicable to the motion of a fluid.
Let x, y, z be the coordinates of the position of any
element at any time t } p the pressure and p the density at
that position at the same time; and let X, Y, Z be the
impressed accelerative forces. The form of the element being
supposed to be that of a rectangular parallelopipedon, and its
edges parallel to the axes of coordinates to be &c, By, $z,
conceive, for the sake of distinctness, the element to be in
that portion of space for which the coordinates .are all posi
12
178 THE PKINCIPLES OF APPLIED CALCULATION.
tive, and let x, y, z, p, and p strictly apply to that apex
of the parallelopipedon which is nearest the origin of co
ordinates. It is known that the generality of the analytical
reasoning is not affected by these particular assumptions. It
will further be supposed that the pressure is uniform through
out each of the faces which meet at the point xyz> because
any errors arising from this supposition are infinitesimal
quantities which in the ultimate analysis disappear. Let,
therefore, p Bx By be the pressure on that face of the element
which is turned towards the plane xy. Then by the law of
pressure demonstrated in Proposition I., the pressures on the
faces turned respectively towards the planes xz and yz are
p Bx Bz and p By Bz, the pressure p applying equally to the
three faces. Since p is a function of x, y, z, and t, the
pressure at the same instant on the face parallel to the plane
yz and turned from it, is
(,+J
Hence the moving force of the pressure in the direction
towards the plane yz is J Bx By Bs ; and the mass of the
element being p Bx By Bz, the accelerative force in the direction
fjff\
of the axis of x is ~ . So the accelerative forces of the
pax
pressures in the directions of the axes of y and z are re
spectively *r and  . Now by Axiom I. the element
pdy pdz
may be supposed to be brought to rest, and to be made to
continue at rest, by impressing, in the directions contrary to
the actual directions, the velocity it has at a given time, and
the increments of its velocity in successive instants, on all
parts of the fluid and the containing solids. In that case, by
the principles of Hydrostatics, the sum of the accelerative
forces in the direction of each of the axes of coordinates is
zero ; so that we have the following equations :
HYDRODYNAMICS. 179
dp fdu
These equations would evidently result from the im
mediate application of D'Alembert's Principle, the pressure
being considered an impressed force. By multiplying them
respectively by dx, dy, dz, and adding, we obtain
(dp) ( v (du\\ , ( v fdv\\ , , ( 7 AM) v /ON
^t^(
Although the reasoning referred to a particular element,
since the same reasoning is applicable to any element at any
time, the equation may be regarded as perfectly general.
This is the tfiird general equation of Hydrodynamics, the
investigation of which, it will be seen, has included all the
fundamental ideas appropriate to a dynamical enquiry, viz.
space, time, motion, quantity of matter, and. force.
The equations (l), (2), (3), with the equations
. ddr dty d^lr
U = \~, V = \Y, W = \j L  ,
dx ' dy dz
and a given relation between the pressure p and density p,
are equal in number to the seven variables ^, X, u, v, w, p,
and p, and therefore suffice for determining each of these
unknown quantities as functions of x, y, z, and t. It might
be possible to deduce from the seven equations a single dif
ferential equation containing the variables ty, x, y, z, and t,
ir being the principal variable; and this general equation
ought to embrace all the laws of the motion that are in
dependent of arbitrary conditions, and should also admit of
being applied to any case of arbitrary disturbance. But it
would be much too complicated for integration, and for being
122
180 THE PRINCIPLES OF APPLIED CALCULATION.
made available for application to specific instances; and
happily another course, not requiring the formation of this
equation in its most general form, may be followed, as I now
proceed to shew. But before entering upon this stage of the
reasoning it will be necessary to make some preliminary
remarks.
Assuming that the above mentioned equations are
necessary and sufficient for the determination of the motion
of a perfect fluid under any given circumstances, in applying
them for that purpose according to the method I am about to
explain, it will be important to bear in mind three considera
tions of a general character. (1) The indications of the
analysis are coextensive with the whole range of circumstances
of the motion that are possible, so that there is no possible
circumstance which has not its analytical expression, and no
analytical expression or deduction which does not admit of
interpretation relative to circumstances of the motion. (2) Any
definite analytical result obtained without taking into account
all the three general equations (1) (2) (3) must admit of
interpretation relative to the motion, although the application
of such interpretation may be limited by certain conditions.
(3) Analytical results which admit of interpretation relative
to the motion prior to the consideration of particular dis
turbances of the fluid, indicate circumstances of the motion
which are not arbitrary, depending only on the qualities of
the fluid and on necessary relations of its motions to time and
space. Such, for instance, is the uniform propagation of
motion in an elastic fluid the pressure of which is proportional
to its density. These three remarks will receive illustration
as we proceed.
There is also a general dynamical consideration which
may be properly introduced here, as it bears upon subsequent
investigations. The accelerative forces which act upon a
given particle at any time are the extraneous forces Jf, Y, Z,
and the force due to the pressure of the fluid, the components
HYDEODYNAMICS. 181
of which in the directions of the axes of coordinates are, as was
proved above, ~ , f> ^. Now all these forces
pax pay pdz
are by hypothesis finite, and consequently the direction of the
motion of a given particle cannot alter per saltum, since it
would require an infinite accelerative force to produce this
effect in an indefinitely short time. Thus although the course
of a given particle cannot be expressed generally except by
equations containing functions of the coordinates and the
time which change form with change of position of the
particle, the course must still be so far continuous that the
tangents at two consecutive points do not make a finite angle
with each other* Hence also the directions of the surfaces
which cut at right angles the lines of motion in a given element
in successive instants do not change per saltum.
It follows at the same time that any surface which cuts
at right angles the directions of the motions of the particles
through which it passeSj (which I have subsequently called
a surface of displacement), is subject to the limitation that no
two contiguous portions can ever make a finite angle with
each other. For if that were possible it is evident that the
directions of the motion of a given particle might alter per
saltum* The equation tyf(t)=0, which is the general
equation of surfaces of displacement, may be such as to change
form from one point of space to another, and from one instant
to another; but the tangent planes to two contiguous points
of any surface of displacement in no case make a finite angle
with each other.
Proposition VI. To obtain an equation which shall ex
press both that the motion is consistent with the principle
of constancy of mass, and that the directions of the motion
are normals to continuous surfaces.
This may be done either by independent elementary con
siderations, or by means of analytical deduction from formulas
already obtained. For the sake of distinctness of conception,
182 THE PKINCIPLES OF APPLIED CALCULATION.
I shall first give the former method, and then add the method
of analytical investigation.
Conceive two surfaces of displacement to be drawn at a
given instant indefinitely near each other, and let the interior
one pass through a point P given in position. On this surface
describe an indefinitely small rectangular area having P at its
centre, and having its sides in planes of greatest and least
curvature. Draw normals to the surface at the angular points
of the area, and produce them to meet the exterior surface.
By a known property of continuous surfaces these normals
will meet two and two in two focal lines, which are situated in
planes of greatest and least curvature, and intersect the normals
at right angles. Let the small area of which P is the centre
be w 2 , and let r, r be the distances of the focal lines from P.
Then if Sr be the given small interval between the surfaces,
the area on the exterior surface, formed by joining its points
of intersection by the normals, is ultimately
rr
But as the direction of the motion through P is in
general continually changing, the position of the surface of
displacement through that point will vary with the time.
Hence the positions of the focal lines and the magnitudes of
r and r' will change continually, whilst the area m 2 may be
supposed to be of constant magnitude. Let r and r represent
the values of the principal radii of curvature at the time ,
and let a and (B be the velocities of the focal lines resolved in
the directions of the radii of curvature, and considered positive
when the motion is towards P. Then at the time t + St the
values of r and r' become r a&t and r'ftSt, and the elemen
tary area on the exterior surface becomes
2 (r + $r  ogQ (r + Sr 
which, omitting small quantities of an order superior to the
second, is equal to
,
HYDRODYNAMICS. 1 83
(r+Sr)(r' + 8
This result shews that by rejecting small quantities of the
second order, a and /3 disappear, and the area is the same as
if the position of the focal lines had been supposed to be fixed.
If, therefore, V and p be the velocity and density of the fluid
which passes the area ra 2 , and V' and p be the velocity and
density of the fluid which simultaneously passes the other
area, since the differences of V and F', and of p and p, may
without sensible error be supposed constant during the small
interval St, the increment of matter between the two areas in
that interval is ultimately
But this quantity is also equal to m*Sr x  $t. Hence,
dt
since p'V' = pV+ 'j 8r, it follows that
+ .......... <
The other mode of investigating the equation (4) will be
sufficiently understood from the following indications of the
principal steps of the process.* The equation
^ <ty
w = Xy
ax
gives by differentiating with respect to x,
du _ d?ifr d\
dx dx 2 dx dx *
Putting, for the sake of brevity, L* for ~^ + jr* + ^ ,
y
we have X = y ; and by differentiating this equation,
* See the Philosophical Magazine for March, 1850, p. 173.
184 THE PRINCIPLES OF APPLIED CALCULATION.
d\_ !_ dV_ V_ fdjr dty djr dQ dty
dx " L dx L* (dx dx* + dy dy* + ~dz
Hence, substituting for X and y in the above value of
du , , . d^lr Lu .. ,, .
, and observing that ~ = ^ , the result is
du^u dV
dx V dx
 \
L*\ dx* dx* dx* dx Hy dxdy dx dz dxdz) '
T i L i ,. dv . dw
By obtaining analogous expressions for 7 and j ,
adding the three together, and having regard to the known
formula for  + ? in terms of partial differential coefficients
of 'v/r, and to the equality
dV__dVu dVv^ dVw
ds ~ dx V* dy V + dz V 9
the result is
=
dx ay dz ds
dV
It is to be noticed that y is the ratio at a given time
ds
of the increment of the velocity to the corresponding incre
ment of the line s drawn always in the direction of the motion,
and this differential coefficient is consequently the same as
dV
j in the former investigation, Now the general equation
(2) is equivalent to
.
dt dx d ~dz \&V dyV dzV
du dv , dw\ T7 dp
V =
HYDRODYNAMICS. 185
Hence, substituting for the quantity in brackets from the
r j ^ dV t. dV , dp dp x ,
equation above, and putting ^ tor j , and ~ lor  , the
equation (4) is readily obtained.
It may be noticed that the investigation of the equation
(4) is not immediately dependent on the general equation (1),
this equation not having been cited in the course of the proof.
It is, however, to be observed that both equations equally
take for granted the Axiom II., and that we should not have
been entitled to reason with the unknown function X in the
investigation of equation (4), unless the equation (1) had
shewn it to be a quantity that admits of determination.
The Propositions hitherto proved apply to fluids in
general, whatever be the relation between the pressure and
the density. Let us now suppose the fluid to be incom
pressible. Then since p is constant, the equation (2) is re
duced to
du dv dw _
dx dy dz ~
Hence the equation (4) becomes for an incompressible
fluid
The inferences to be drawn from the equations (4) and
(5) will be considered in a subsequent part of the argument.
It has been usual in hydrodynamical researches to sup
pose that udx + vdy + wdz is an exact differential (^<), and
to make <j> the principal variable in the differential equations
subsequently obtained. According to the principles on which
the general equation (1) was founded, a factor  always exists
A
by which that differential function may be made integrable ;
so that the supposition of its being integrable of itself in
troduces a limitation of the general problem. Now, as we
186 THE PRINCIPLES OF APPLIED CALCULATION.
have seen, X is determined by the solution of a partial dif
ferential equation, and its general expression involves arbi
trary functions of x, y, z, and t. The forms and values of
these functions must be derived from the given conditions of
the particular problem to be solved, and the integrability of
udx \ vdy + wdz will consequently depend on the arbitrary
circumstances of the motion. For 'instance, that quantity is
an exact differential if the motion be subject to the conditions
of being perpendicular to a fixed plane and a function of the
distance from the plane, or if it be in straight lines drawn
from a fixed point, and be a function of the distance from the
point If, however, definite results can be deduced from the
purely analytical supposition that udx + vdy + wdz is an
exact differential, made antecedently to any supposed case of
motion, such results, according to the preliminary remarks
(1) and (3), must admit of interpretation relative to the mo
tion, and indicate circumstances of the motion that are not
arbitrary. The solution of the next question conducts to an
inference of this kind.
Proposition VII. To obtain an integral of the first general
equation on the supposition Jhat udx + vdy + wdz is an exact
differential.
Since X (d^) = udx + vdy + wdz, if the righthand side of
this equation be assumed to be an exact differential, we must
have X a function of ^ and t. Let ~ represent the ratio
of corresponding increments at a given time of the function ^
and of a line s drawn always in the direction of the motions
of the particles through which it passes, and let x, y, z be
the coordinates of a point of this line at the given time.
Also let V be the velocity at that point at the same instant.
Then, since generally
HYDRODYNAMICS. 187
we have
dty _ dfy dx d^f dy d^r dz
ds ~~ dx ds dy ds dz ds
_dty u dty v dty w
~"dxT*~(hj ~V*~dz V
_ tf + v 9 + w* _ V
\V ~X*
But by the general equation (1),
t+?
Hence, substituting the above value of F,
d^r d^_
~dt* K ~ds*~"'
Making, now, the supposition that X is a function of ^
and tj the integration of this equation would give
f =/M).
Consequently,
The value of ^ obtained by this process is subject to the
limitation of being applicable only where udx + vdy + wdz is
an exact differential, but in other respects is perfectly general.
Hence the expression for (&r) given by the last equation is
in general the variation of ^ (under the same limitation) from
a given point to any contiguous point ; so that if we suppose
the variation to be from point to point of a surface of displace
ment, in which case (\/r) = 0, we shall have
But the multiplier of (85), being equal to ^, and there
fore proportional to F, does not vanish. Hence it follows
that (&?) = 0. This result proves that the lengths of the
188 THE PRINCIPLES OF APPLIED CALCULATION.
trajectories which at a given time commence at contiguous
points of a given surface of displacement, and terminate at
contiguous points of another given surface, are equal to each
other. Hence, so far as the condition of the integrability of
udx f vdy + wdz is satisfied, two surfaces of displacement,
whatever be the distance between them, are separated by the
same interval at all points. But this cannot be the case
unless the trajectories are straight lines, and the motion con
sequently rectilinear.
We have thus obtained a definite result, namely, recti li
nearity of the motion, solely by making the analytical suppo
sition that X (d^) is an integrable quantity, which supposition
does not involve any particular conditions under which the
fluid was put in motion. This result, according to the prin
ciples enunciated in p. 180, must admit of interpretation
relative to the motion; but inasmuch as it was arrived at
without employing all the fundamental equations, we are not
allowed to infer from it that the motion is necessarily recti
linear. Since the argument was conducted without reference
to arbitrary disturbances, the general inference to be drawn is,
that this integrability of udx + vdy + wdz is the analytical
exponent of rectilinear motion which takes place in the fluid
by reason of the mutual action of its parts. Motion of this
kind may be modified in any manner by the arbitrary con
ditions of particular instances ; but because it has been indi
cated by analysis antecedently to such conditions, it must
necessarily be taken account of in the application of the
general equations to specific cases of motion. This will be
more fully explained in a subsequent stage of the argument.
I advance now to propositions relating to the laws of the
propagation of velocity and density.
Definition. The rate of propagation of velocity and density
is the rate at which a given velocity or density travels through
space by reason of changes of the relative positions of the
particles due to changes of density.
HYDRODYNAMICS. 189
Proposition VIII. To obtain a rule for calculating rate
of propagation.
Let the total velocity F at any point be equal to F(p),
jj, being a function of the time t and the distance s reckoned
along a line of motion from an arbitrary origin. Then, ac
cording to the above definition, s and t must be made to vary
while F remains constant. Hence, since F=.F (//.),
Here & is evidently the space through which the velo
city F travels in the time St. Consequently, if o> be the rate
of propagation, we have
dp
Ss dt
"sr$
ds
This is the formula required for calculating the rate of
propagation of the velocity; and clearly an analogous rule
applies for calculating the rate of propagation of the density,
or any other circumstance of the fluid expressible as a func
tion of s and t. Let us suppose, for example, that the rate of
propagation is the constant co 1 . Then since the function /JL is
required to satisfy the partial differential equation
* + f =0 ,
1 ds dt
it follows that
fj, = <f> (* CBjtf).
Hence
FJF'Ww.O] /(,)
Conversely, if any process of reasoning conducts to an ex
pression of the form f (s at) for the velocity, or the density,
or any other unknown circumstance, by differentiating this
function with respect to s and #, the rate of propagation would
at once be determined to be the constant a.
190 THE PRINCIPLES OF APPLIED CALCULATION.
The above method of determining rate of propagation by
differentiation, the principle of which is obviously true, I have
indicated in a Paper dated March 30, 1829, contained in the
Transactions of the Cambridge Philosophical Society (Vol. in.
p. 276). A different method, given in the Mecanique Ana
lytique (Part II. Sect. xi. No. 14), and adopted by Poisson
(Traife de Mecanique, Tom. II. No. 661, Ed. of 1833), is em
ployed to this day in the Elementary Treatises on Hydro
dynamics. By this process the determination of rate of pro
pagation is made to depend on the arbitrary limits of the
initial disturbance ; that is, a circumstance which is not arbi
trary is attributed to arbitrary conditions. This is evidently
an erroneous principle, and I shall have occasion hereafter to
shew, that the adoption of it in hydrodynamical researches
has led to false conclusions.
Proposition IX. To find the relation between the velocity
and the density when the rate of propagation of the density
is constant.
For the sake of greater generality the proof of this Pro
position will take into account the convergence, or divergence,
of the lines of motion, and it will be assumed in conformity
with the principle of continuity already adopted, that for each
element of the fluid these lines are normals to a continuous
surface. Accordingly let us suppose the fluid to be contained,
through a very small extent, in a very slender tube whose
transverse section is quadrilateral, and whose bounding planes
produced pass through the two focal lines referable to the
geometrical properties of the surface. Let P, Q, E be three
positions on the axis of the tube separated by very small and
equal intervals. Then since the lines of motion are not sup
posed to be parallel, it is required to solve the following
general problem of propagation : viz. to express the rate at
which the excess of fluid in the space between Q and R above
that which would exist in the same space in the quiescent
state of the fluid, becomes the same as the excess in the space
HYDRODYNAMICS. 191
between P and Q. It is evident that the rate of propagation
determined on this principle is not the same as the rate of
propagation of a given density, unless the lines of motion are
parallel.
Let F be the mean velocity, and p the mean density, of
the fluid which in the small interval &t passes the section
at Q, and V ', p be the same quantities relative to the section
at R. Let the magnitude of the section at Q be m, and of
that at R be ra', and the interval between them be Ss. Then
the increment of matter in the time Bt in the space between Q
and R is ultimately
VpmBt  V'p'm'Bt,
the motion taking place from Q towards ft. Let this quan
tity be equal to the excess of the matter which is in the space
between P and Q in consequence of the state of motion, above
that in the space between Q and R, at the commencement of
the small interval St. The expression for this excess, sup
posing the density in the quiescent state of the fluid to be
represented by unity, is
(p 1) mBs (p 1) m'Bsj
small quantities of the second order being neglected. Hence,
passing to differentials, we have
d . Vpm _d.(p \}m Bs
ds ~ds '&'
fN
which equation gives the expression for the required rate .
ct
If this rate be supposed equal to a constant a', we obtain by
integration
V p = a > (p l) + m.
The principal radii of curvature of the surface of dis
placement at the given position being r and r, m will vary
192 THE PEINCIPLES OF APPLIED CALCULATION.
as the product rr. Hence the last equation may be thus
expressed :
Vp = a '( p l)+*ff .............. . ...... (6).
We have thus arrived at a general relation between V
and p on the hypothesis of uniform propagation of the kind
above enunciated. It will be seen that if p = 1 the expression
for the velocity V coincides with that which would be ob
tained by the integration of the equation (5), which applies
to an incompressible fluid. In this case, as there is no change
of density there is no finite rate of propagation either of den
sity or velocity.
If r and / be infinitely great, the motion is in parallel
lines, and we have
As this result shews that V is a function of p, V is propa
gated, as well as p, with the constant velocity a.
Proposition X. The lines of motion being supposed to be
normals to a continuous surface, and the rate of propagation
to be constant, it is required to find the laws of the variations
of the velocity and density due to the convergency of the lines
of motion.
Let a be the given rate of propagation. Then the solu
tion of the question may be effected as follows by means of
the equations (6) and (4). After obtaining ^ from the
ckir
former, and substituting in the other, it will be found, since
* =1
dt dr ' \r r
which, it may be remarked, is the same result as that which
HYDRODYNAMICS. 193
would be obtained if <f> (t) = 0. This equation admits of being
exactly integrated, the integral being
pi^S^ w
Hence y a.F(ra t ) ^
rr rr * '
These equations give the laws of the variations of F and p,
as resulting from the hypothesis of a constant rate of propaga
tion, and from the convergency of the lines of motion.
The proofs of the Propositions vi., VII., Till., IX. and X.
have not involved the consideration of force, having reference
only to laws of the velocity and density which depend on the
relations of space, time and matter, but are independent of the
action of pressure. I proceed now to the discussion of questions
in which force is concerned, and which consequently require
for their treatment the third general equation to be taken into
account. For the purpose of illustrating and confirming the
new hydrodynamical principles advanced in the foregoing
part of the reasoning, two examples will, in the first place, be
given of the treatment, in the usual manner, of problems in
volving pressure, no reference being made to the first general
equation, and subsequently it will be shewn that the results
thus obtained indicate the necessity of having recourse to that
equation.
Example I. Let the relation between the pressure and
the density be expressed by the equation p a?p, and let the
velocity be in directions perpendicular to a fixed plane, and
be a function of the time and of the distance from the plane :
it is required to determine the motion, the fluid being supposed
to be acted upon by no extraneous accelerative force.
Assuming that the fixed plane is parallel to the plane xy 9
we have
fdv\ d . pu d . pv A
dy
13
194 THE PRINCIPLES OF APPLIED CALCULATION.
Hence the equations (2) and (3) become for this case
a*, dp dw dw
r^ + ji + w J = >
pdz at dz
dp dp dw A
 J 3i + w% + T=0.
pat pdz dz
To obtain integrals of these two equations, substitute
7 If
~ for w. Then by integrating the first we get
dz
which, if $ = <' 1% {*)<&) and consequently w=~f ) be
Hence, eliminating p from the other equation by this last,
the result is
d*<j> ( , d<F\
dz* \ a dz*J
__^
dz dzdt de ~
This equation is not generally integrable, but is satisfied
by the particular integral
w=f{z(a + w)t}' J
whence it follows that between p and w there is the exact rela
tion p = e a .
Now although these results seem to have been arrived at
by a legitimate course of reasoning, and might be expected to
admit of interpretation consistent with the motion of a fluid,
yet upon trial this is not found to be the case. Let us sup
pose, for instance, since the form of the function / is arbi
trary, that
O
w = m sin  [z (a + w) t}.
A,
HYDRODYNAMICS. 195
Then if z = a t + , w = 0; and if
3 = ( + m) t + j , w m.
But these two values of s are the same if
at +  = (a + m) t +  ;
that is, if=: . Hence at the same distance from the
4m
origin the velocity of the fluid may be zero, and may have its
maximum value m, at the same moment. This result evi
dently admits of no interpretation, being a contradiction per se,
and therefore, according to an acknowledged rule of logic, it
indicates fault or defect in the premises, or fault in the reason
ing. It will subsequently appear that the argument which
accounts for this contradiction has an essential bearing on the
analytical theory of the motion of fluids.
It may here be mentioned that the above integral has been
discussed by Poisson in the Journal de VEcole Polytechniquej
Tom. vii., and that in p. 369 he comes to the following con
clusions : " The original disturbance will be transmitted
uniformly and with a velocity equal to a ; this velocity will
be independent of the original disturbance and of the magni
tude of the velocities of the molecules ; the duration of the
disturbance will be the same for all the molecules and equal
to  ; and finally the breadth of the moving wave, on each
side of the origin, will remain constantly equal to ." But
these conclusions are arrived at by the adoption of a principle
the error of which I have already pointed out, viz. that of
making the determination of the rate of propagation depend
on the arbitrary limits of the original disturbance. This fault
in the reasoning has the effect of concealing the signifi
cant reductio ad dbsurdum which I have pointed out above.
132
196 THE PRINCIPLES OF APPLIED CALCULATION.
"By applying to this instance the rule for calculating rate of
propagation demonstrated under Prop. VII. we shall have, since
yu, = z (a + w) t,
dp
rate of propagation = 7 = a + w.
dz
Thus whatever be the form of the arbitrary function which
expresses the initial disturbance, different parts of a wave are
propagated with different velocities. Also the relation between
w
p and w to which this reasoning conducts, viz. p = e a , is in
consistent with the equation wp = a (p 1), which was proved
(Prop. IX.) to be the true relation between the velocity and
the density when all the parts of the wave are propagated with
the velocity a.
After I had pointed out the above mentioned reductio ad
absurdum (in the Philosophical Magazine, Supplementary
Number of June, 1848, p. 496), the question was discussed
first by Prof. Stokes (Phil. Mag. for November, 1848, p. 349),
and afterwards by Mr Airy (Phil. Mag. for June, 1849, p. 401),
the former mathematician contending that at the point where,
according to the analysis, the velocity may be zero, and at
the same time have a maximum value, a physical condition
takes place analogous to that of a breaker or a bore; and the
other, that at this point a musical sound becomes unmusical.
Both mathematicians tacitly admitted the truth of the fore
going rule for determining rate of propagation. Against the
physical explanation proposed by Prof. Stokes it may be
urged that breakers and bores are observed phenomena, oc
curring under special and known circumstances, and that,
since in the case before us no such circumstances are taken
into account, the analogy fails. Mr Airy's conception of the
transition of sound from a musical to an unmusical state, is
merely* a gratuitous interpretation put upon the contradictory
HYDRODYNAMICS. 197
indications of the analysis, being unsupported by reference to
matter of fact. Besides, on logical grounds, neither of these
physical explanations is admissible. For if the analysis in
dicates that regular waves become breakers at any epoch, it
equally indicates that these breakers become regular waves
at a subsequent epoch. So if we are to conclude from the
analysis that at any position a musical sound passes into un
musical noise, we have equal reason for concluding that the
unmusical noise subsequently becomes musical. For it would
be doing violence to right reasoning to accept in part, and
reject in part, mathematical deductions from the same pre
mises. But these inferences are so utterly irreconcileable
with common sense, that there is no escape from the con
clusion that we have here a veritable reductio ad dbsurdum,
necessitating a different course of reasoning. To establish
this point is so important a step in the general argument,
that before proceeding farther I shall adduce another example
of the same kind of contradiction. The argument will be the
more confirmed if it should afterwards be made to appear that
the contradictions are in both instances got rid of by the
application of the same principles.
Example II. Let, as in the preceding example, p = a 2 /o,
and suppose the fluid to be disturbed in such a manner that
the velocity and density are functions of the distance from
a centre, and the lines of motion are radii drawn from the
centre: the fluid being acted upon by no extraneous force,
it is required to determine the motion.
In this instance, in order to obtain an integrable equation,
the velocity and condensation will be supposed to be very
small, and powers of the small quantities above the first will
be neglected. Let V be the velocity and 1 + a the density,
the condensation <r being very small ; and let the fixed centre
be the origin of the coordinates a?, y, z, and r be any distance
from it. Then
Vx Vy Vz
198 THE PRINCIPLES OF APPLIED CALCULATION.
Hence to the first order of small quantities,
fdu\ du_^dV
dt~ r dt '
(du\
(dt)
'dv\ y dV
~ji] =  ji ?
fdw\ z dV
v\_z
(di)~r ~dt*
Thus to the same order of approximation the equations
(2) and (3) become
The elimination of Ffrom these equations gives
d z . GT _ 2 d 2 . <rr
~w~ ~w
This equation is satisfied by the integral
<rr =f(r  at) ;
so that, by giving to the arbitrary function a particular form,
we may have
mb . 2?r , N
sm (r at).
r X x
deferring now to the value of p 1 obtained under Prop.
X., and supposing the lines of motion to radiate from a centre,
in which case r = r, we shall have, after changing for con
venience the notation,
which, on giving to the arbitrary function the same form as
before, becomes
ml* . 27T, .
o = ^sm (rat}>
HYDRODYNAMICS. 199
We have thus obtained by different courses of reasoning
two different values of the same quantity. As it is certain
that the second value results from necessary relations of space,
time, and matter, we must conclude that the former is incon
sistent with such relations*. We are consequently again
brought to a reductio ad absurdum. If it be objected to this
conclusion that the reasoning has not embraced the expression
for the velocity deducible from the given conditions of the
problem, the logical answer is, that the absurd result was
obtained by strict reasoning from admitted premises, and
cannot, therefore, be set aside by other reasoning from the
same premises.
The processes by which the solutions of the above two
hydrodynamical problems have been attempted, are in accord
ance with the principles that are usually applied to cases of
the motion of a fluid. What then, it may be asked, is the
reason that these processes have led to contradictions? To
this question I make, first of all, the general reply, that this
mode of treatment takes no account of the first fundamental
equation, and of the law of rectilinearity of the motion deduced
from it in Prop. VII. As that equation and the deduction
from it were shewn to be antecedently true, they cannot with
out error be excluded from consideration in subsequent ap
plications of the general reasoning. To establish fully the
validity of this answer, it is required to point out the course
of reasoning which is necessary when the three fundamental
equations are used conjointly. This part of the argument
I now enter upon.
In the first place it is to be observed that the law of
rectilinear motion inferred from the general equation (1), would
not be satisfied by the supposition that antecedently to the
imposition of arbitrary conditions the motion is in parallel
straight lines, or in waves having planefronts; for if such
* See the arguments relating to this point in the Philosophical Magazine for
December, 1848, p. 463, and in the Number for February, 1849, p. 90.
200 THE PEINCIPLES OF APPLIED CALCULATION.
were the case, no contradiction would result from the reason
ing employed in Example I. And similarly, the law is not
satisfied by supposing that the rectilinear motion takes place,
independently of the character of the disturbance, in straight
lines passing through a centre, or through focal lines ; for
then the solution of Example II. would not have led to a
contradiction. There is still another supposition that may be
made, viz. that the general law of rectilinearity applies to
motion along straight lines, which, with respect to the state
of the fluid as to velocity and density in their immediate
neighbourhood, may be regarded as axes. The consequences
of this supposition will be next investigated, the following
preliminary remarks being first made.
The reasoning is necessarily of an indirect character, be
cause the general equation of which ty is the principal variable
is so complicated, that it cannot be employed for drawing any
general inferences relative to the motion or the density. As,
however, the object of the present research is to determine
laws of the mutual action of the parts of the fluid that are
neither arbitrary nor indefinite, it is certain, if the research be
possible, that there must be a unique course of reasoning
appropriate to it, and that every other will lead to contradic
tions. Notwithstanding that the general equation cannot be
integrated, the investigation of laws that are not arbitrary
may be presumed to be possible for the following reason.
What is proposed to be done is to satisfy the general equa
tions by a solution between which and the complete inte
gration of the equations there shall be the same kind of
relation as that between the particular solution and complete
integral of a differential equation containing two variables.
As the particular solution is of a definite character, not in
volving arbitrary constants, so the solution with which we
are here concerned is definite in the respect that it can contain
no arbitrary functions, and should, therefore, admit of being
discovered without previously obtaining by integration the
complete value of ^. It is now proposed to conduct this
HYDRODYNAMICS. 201
research by making the hypothesis that the rectilinear motion
deduced from the general equation (1) is motion along a recti
linear axis, and taking into account the second and third
general equations.
Proposition XL Assuming that p = a?p, and that there is
no impressed force, it is required to determine the relation
between the velocity and the density, and the law of their
propagation, when the motion takes place along a rectilinear
axis.
As the hypothesis of a rectilinear axis is based on an in
ference drawn from the first general equation by supposing
udx + vdy + wdz
to be an exact differential, the same supposition must be made
in the present investigation. Also we are to express analyti
cally that the motion is along an axis. These conditions are
fulfilled by assuming that
(d.f<f>) = udx + vdy + wdz,
and that/ is a function of x and y, and <p a function of z and
t. For on these suppositions
df ,df ,d$
so that if the function / be such that /= 1, f 0,
7/ 1
7 =0 where #=0 and #=0, the axis of z will evidently
be an axis relative to the motion. It is, however, to be under
stood that the analysis applies only to points either on the
axis, or immediately contiguous to it, because the antecedent
hypothesis of the integrability of udx + vdy + wdz applies
only to such points.
After substituting the above values of u, v, w in the third
general equation (3), it becomes immediately integrable, and
supposing that X = 0, F=0, Z=0, the integration gives
202 THE PRINCIPLES OF APPLIED CALCULATION.
Combining, now, this equation with the second general
equation (2), and with the equations u = <j> j , v = $ j ,
w =f, and eliminating u, v, w, and p from the five equa
tions, the following result is obtained :
dx 2 da? dxdydx dy dtf
<>
Since these equations apply only to points on or con
tiguous to the axis, the terms involving 4 and J are in
finitely less than the other terms. Again, as the value f 1
results from the values x 0, y = 0, which make j and j
vanish, we may conclude that that value is either a maximum
or minimum. The supposition of a minimum would be found
to introduce subsequently logarithmic expressions inapplicable
to the present enquiry, and by that analytical circumstance it
is excluded. Since, therefore, /has a maximum value where
x and y = 0, it follows that for points on the axis
HYDRODYNAMICS. 203
6 2 being an unknown constant. Consequently, omitting in
equation (10) the terms involving  and jr , and putting
/= l, we have for determining the function ^> the equation
After obtaining the value of <f> by integrating this equa
tion, the velocity w along the axis is given by the equation
w = jT , and the density p by the equation
+ +J'(*)=0 ........ (12).
It should here be remarked that as the purpose of this
investigation is not to satisfy arbitrary conditions, but to
ascertain laws of the motion which are independent of all that
is arbitrary, if the investigation be possible no such arbitrary
function as F(f) can be involved, and consequently this func
tion is either zero or an arbitrary constant. The meaning of
this inference will be farther apparent at a subsequent stage
of the reasoning. Putting, therefore, F' (t) = 0, the equation
(11) is now to be employed for finding an expression for <.
It does not appear that an exact integral of this equation is
obtainable : but an integral applicable to the present research
is deducible as follows by successive approximations. Taking,
for a first approximation, the terms of the first order with
respect to <, we have
If now we put p, for z + at, v for z at, and e for %
this equation may be transformed into the following :
a
204 THE PRINCIPLES OF APPLIED CALCULATION.
The integral of this equation does not admit of being
expressed generally in a definite form ; but if we integrate
by successive approximations, regarding e as a small quan
tity, the complete integral will be obtained in a series as
follows :
G(v)
where
(v) dv, &c.
As the arbitrary functions F and G satisfy the equation
independently, it is allowable to make one of them vanish.
Let, therefore, F(jj)=0, so that
>= a +^ + + <? +&c.
By means of this form of the expansion of <f> we have to
ascertain whether it admits of a particular and exact ex
pression. Now this will plainly be the case if forms of the
function G can be found which satisfy the equation
J.g.M pg w
dv
for every value of n ; since for such forms the above series is
the expansion of exact functions of z and t. Now
and consequently by the above equality
dv*
The integration of this equation gives the required forms
of the function G. By taking the upper sign a logarithmic
HYDRODYNAMICS. 205
form is obtained, which is incompatible with any general law
of the motion of a fluid, and is therefore to be rejected. Taking
the lower sign and integrating, we have
6f n (z/) = A cos (kv + B\
which determines the form of the function G n for any one of
the values 0, 1, 2, 3, &c. of n. In conformity with this result
let G (v) = m^ cos (kv + c). Then it will be found that
<b = m, cos \k
(v ~j + c L
Or, putting r for k j , and substituting the values
of v and
We have thus been led, step after step, by the indications
of the analysis, to an exact and unique form of the function
$, without having made any supposition respecting the mode
of disturbance of the fluid. I cannot but regard this result
as a singular confirmation of the correctness of the foregoing
research as to principle, and as evidence of the possibility of
conducting it to a successful issue. If the integration had
given an exact expression for <f> containing arbitrary functions,
the argument would have fallen to the ground. As it is, the
above circular function is to be interpreted as indicating a
law of the mutual action of the parts of the fluid.
By means of this first approximate value of </> there is
no difficulty in deducing from the equation (11) successive
approximations. The result to the third approximation is
cos   sin      cos
i 2 1\
  J
206 THE PRINCIPLES OF APPLIED CALCULATION.
o
f being put for z aj, + c, and q for . If m be substituted
A
for gm^ , and KCL for o t , we shall have to the same approxi
mation,
The expression for the condensation may be derived from
the equation
which is what the equation (12) becomes when the arbitrary
function F() is supposed to vanish. Since it follows from
the foregoing value of $ that
if the velocity ^ = 0, we shall also have ~ = 0, and the
equation above is satisfied if p = 1, which is taken to be the
density of the fluid in its quiescent state. Hence it appears
that the vanishing of F(t) signifies that so far as regards the
mutual action of the parts of the fluid, the velocity along the
axis and the corresponding condensation vanish together.
This is the explanation of the vanishing of F'(t\ referred to
at a previous part of this argument. Supposing now that
p = 1 + <7, it will be found from the foregoing equations that
to the second approximation
= m K sing? * <x>s2tf+~( K *l) sin 2 gg...(16).
These results determine the laws and mutual relation of
the velocity and density along the axis, and shew that each
is propagated with the uniform velocity a x .
HYDRODYNAMICS. 207
Corollary, From the equations (14) and (16) the relation
between w and a to terms of the second order is found
to be
(17).
Since e, being put for $ , is necessarily positive, the
equation (15) shews that tc to the first approximation is
greater than unity. Hence the above equation informs us
that the condensation corresponding to a positive value of w
is greater than the rarefaction corresponding to an equal
w*
negative value by (/e 2 1) $ . The reason for this law will
be apparent by considering that as the motion is wholly
vibratory, the forward excursion of each particle must be
equal to its excursion backward, and that this cannot be the
case unless at each instant the variation of <r for a given
variation of z be greater at a point of condensation than at
the corresponding point of rarefaction.
Proposition XII. To obtain from the fundamental equa
tions expressions for the velocity and condensation on the
supposition that udx + vdy + wdz is an exact differential,
small quantities of an order superior to the first being neg
lected.
The reasoning of this Proposition, in so far as it involves
the three fundamental equations, and is restricted to quantities
of the first order, differs from that of Prop. VII. ; but because
the abstract analytical supposition that udx + vdy + wdz is an
exact differential is made in both, they have this in common,
that, for the reasons adduced under Prop. VII., the results
obtained relate to the mutual action of the parts of the fluid
irrespectively of arbitrary disturbances. The equations to be
employed here are
2 dcr du ^da dv A 2 da dw ..
a ^+^r = > a ^ + 3r = > a Tr+T77= 
dx dt dy dt dz dt
208 THE PRINCIPLES OF APPLIED CALCULATION.
Hence by integration,
d. ladt
a j dx i a
dx
adt d . I crdt
~dz~~>
where 0, (7, C" are in general arbitrary functions of x, y, z
not containing the time. Consequently, representing
ofjvdt
by 0, we have
= + = + =
e&e ~" cfoj dx ' c?y d/y efo/ ' dz dz dz '
It thus appears that udx + vdy + wdz is not an exact
differential independently of all that is arbitrary unless (7,
(7, and O' are constants ; that is, since we may always leave
out of consideration a uniform motion of the whole of the
fluid in a fixed direction, unless (7=0, C'=0, and <7" = 0.
Hence no part of the velocity is independent of the time. Now
this is the case if the motion be vibratory. The hypothesis,
therefore, of vibratory motion satisfies the condition of the
integrability of udx + vdy + wdz assumed in the enunciation
of the Proposition. Also this inference is in accordance
with the antecedent expressions for w and <j obtained under
Prop. XL
But the principle of the present research demands that the
precise modes of the vibrations should be ascertained. Now
since the vibrations, from what has already been proved,
must have reference to an axis, for the purpose pf carrying on
this enquiry let us adopt the supposition already made, viz.
(^ 'fit*) = udx + vdy + wdz,
and extend the application of this equality to points at any
HYDRODYNAMICS. 209
distance from the axis. The legitimacy of this procedure
will be proved if it leads to no contradiction and serves to
determine the function /. According to these considerations
we have to make use of the equation (10) to the first order of
small quantities, that is, the equation
Since <j) is independent of x and y, it has the same value
at all points of any plane perpendicular to the axis of 2, and,
therefore, the same as the value at the point of intersection of
this plane with the axis. But for points on the axis we have
seen (p. 203) that to the first approximation
Hence, by comparing the two equations, it follows that
We have thus arrived at an equation for determining/
which is consistent with the original supposition that this
function contains only the variables x and y. Also since it
has been shewn (Prop. XI.) that <f> is a circular function of z
and t, and since ^the velocities u 9 #, w are respectively <f> J ,
$ ~ , and f3 , it follows that the whole of the motion is
dy d/z
vibratory. Thus the supposition that udx f vdy + wdz is an
exact differential for points at any distance from the axis is
justified by finding vibratory motion, and the supposition
that the differential may be expressed as (d.f<j>), is justified
by obtaining an equation which determines fto be a function
of x and y.
To complete this investigation it is now required to find
the particular form of the function / appropriate to motion
14
210 THE PRINCIPLES OF APPLIED CALCULATION.
resulting from the mutual action of the parts of the fluid :
which may be done as follows. Since the equation (18) is
of exactly the same form as the equation (13), the same
process that conducted to a particular expression for </>, will
conduct to a particular expression for /. In fact, by this
process we obtain
which value of/ evidently satisfies the equation (18), if the
arbitrary quantities g and h be subject to the condition
f + h* = 4e.
If we substitute 2 Ve cos 6 for ^, we shall have h = 2^e sin 0,
and the above integral may be put under the form
/=acos{2 Ve^costff #sin0)} ............ (19).
By deriving from this equation Jr and ~ , and substi
tuting in the expressions <f> j and <f>~ for u and v, it will
be seen that the motion parallel to the. plane xy is parallel to
a direction in that plane depending on the arbitrary value
of 0. Consequently this value of f implies that 6 is deter
mined by some arbitrary condition. There is, however, an
integral of (18) which removes this arbitrariness from f by
embracing all directions corresponding to the different values
of 6. For since that equation is linear with constant coeffi
cients, it is satisfied by supposing that
/= 2 . a&d cos {2 */e (x cos + y sin 0)),
W being an infinitely small constant angle, and the summa
tion being taken from to = 2?r in order to include all
possible directions. By performing the integration, substi
tuting r* for a? a + y 8 , and determining a so as to satisfy the
condition that /= 1 where r = 0, the result is
i..a".8 + &a (20) '
HYDRODYNAMICS. 211
This value of f, containing no arbitrary quantity whatever,
expresses a law of the mutual action of the parts of the
fluid.
The equation which gives the condensation a to the first
order of approximation is
fe v + /f=o.
By substituting / from this equation in (18), striking out a
factor common to the three terms, and putting 4e for a , the
result is
From what was argued relative to the equation (18), the
particular integral of this equation appropriate to the present
investigation is
<r = S cos (2 *fe (x cos 6 + y sin 0)},
S being the condensation where x = and y = ; that is, on
the axis.
If we suppose that / in the equation (18) is a function of
the distance r from the axis, that equation becomes
d^^rtr
and is satisfied by the value of / expressed by the equa
tion (20).
Proposition XIII. To demonstrate the law of the co
existence of small vibrations in an ekstic fluid for which
p =. a?p.
This law is shewn as follows to be deducible, antecedently
to the consideration of particular disturbances, from the four
approximate equations
142
212 THE PRINCIPLES OF APPLIED CALCULATION.
dx dt
2 da dv _
z d(T dw
dcr du dv dw
dt dx dy dz ~
The last equation, differentiated with respect to t, gives
d z v d*w
= 0;
by substituting in which for the last three terms their values
derived from the first three equations, the result is
d (T 2 fd <T d <7 d <r\ . .
d = f \W^*W'
Hence if 0^, <r 2 , cr 3 , &c. be values of a which separately
satisfy this equation, and if 2 = ^ + <r z + & 3 + &c., since the
equation is linear with constant coefficients, 2 will also
satisfy it. To obtain this result no other condition has been
required than that the motion be small compared to a. It
has not been necessary to suppose that udx + vdy + wdz is an
exact differential, or that no part of a is independent of the
time. But on proceeding to consider the motion coexisting
with the condensation, no other than vibratory motion is ad
missible, because only this kind of motion has resulted from
the previous investigation under Prop. XI. Now it has been
shewn under Prop. XII. that for small vibratory motions the
above differential function is integrable. Supposing, therefore,
that
(d^r) = udx + vdy + wdz,
we have, to the first approximation,
du dv . dw
HYDRODYNAMICS. 213
Biit on the same supposition and to the same approximation,
a'Nap.
Consequently by substitution in the fourth of the original
equations, after obtaining y from this last equation, the
result is
_ }
 l + + .........
Let now ^^^ &c. be different values of ^ correspond
ing to different sets of vibrations, and let each value satisfy
this equation separately. Then since the equation is linear
with constant coefficients, it will be satisfied by a value <&
equal to ^ + ^ 2 + ^ 3 + &c. ; and we have also
d dilr. cfrlr dfa s
j = p+ f 2 + f 3 + &c. = Wj + u 2 + u 3 + &c.,
ax dx ax dx
This reasoning proves that the equation (23) is satisfied by
a total motion compounded of the separate motions, and con
sequently that the supposed sets of vibrations may coexist.
Thus the law of the coexistence of small vibrations in an
elastic fluid is completely demonstrated. Although the vibra
tory motions to which the reasoning refers can only be such
as the previous investigation defined, yet as the axes and
their positions were not explicitly involved in the argument,
it follows that with respect to these there is no limitation, and
that an indefinite number of sets of vibrations may coexist,
having their axes in perfectly arbitrary positions.
The foregoing argument holds good if while
2 = 0,+ <r,+ <r 8 + &c., 3> = k (^ + ^h + f 3+ &c )>
Jc being some constant.
214 THE PRINCIPLES OF APPLIED CALCULATION.
Proposition XIV. To find the velocity of the propagation
of vibrations in an elastic fluid the pressure of which varies as
the density.
Since the equations (14) and (16) prove that the velocity
and condensation on an axis are functions of f, or z aj + c,
and constants, it follows, by the rule proved in Prop. VIII.,
that each is propagated with the uniform velocity a,. Also K
being the ratio of a t to a, we have, by equation (15),
21 ^L
{ "
As j is a small quantity of the second order, and e is a
Cb
positive quantity, this equation shews that when quantities
of that order are omitted K* is greater than unity. Conse
quently, on proceeding to the next approximation, the third
term in the above equation is positive, and thus the rate of
propagation, as determined by purely hydrodynamical con
siderations, always exceeds the quantity a. It is the purpose
of the reasoning that follows to determine in what proportion
it is greater, and whether K be an abstract numerical quantity
independent of spatial relations. With reference to this last
point it may here be remarked, that the term in the above
expression which contains ra 2 would seem to indicate that the
rate of propagation depends in part on the maximum velocity,
or on the extent of the excursion, of a given particle. When,
however, it is considered that the present argument is wholly
independent of arbitrary disturbances of the fluid, there ap
pears to be no reason to affirm of m that it has degrees of
magnitude ; and accordingly the only appropriate supposition
is that it is an absolute constant of very small but finite mag
nitude. The mode in which vibrations of different magnitudes
are produced under different given circumstances will be dis
cussed in a subsequent Proposition. At present it will be
supposed that m has a fixed ratio to a, so that, as far as
regards that quantity, the value of K? is independent of linear
HYDRODYNAMICS. 215
*
magnitudes. Moreover it should be observed that the last
term of the expression for /e 2 is to be omitted if the investi
gation does not extend to small quantities of a higher order
than the second.
f &?
Thus we are required to calculate the quantity f 1 H ^
By referring to the proof of Prop. XI. it will be seen that the
6 a I 2
constant e, or $ , originated in putting ^ for the value of
*Ctf Cl
~ 2 + ~TT f r points on the axis. This constant, therefore,
has not an arbitrary character, but depends only on properties
of the fluid and independent laws of its motion; on which
account it should admit of determination on the principles em
ployed in the foregoing investigation. In short, the numerical
calculation of the rate of propagation resolves itself into the
discovery of the proper mode of determining the value of that
constant. This I have found to be a very difficult problem.
My first attempts to solve it were made on the principle of
comparing the transverse vibrations at a great distance from
tHe axis with vibrations along the axis resulting from two
equal sets propagated in opposite directions*. I afterwards
ascertained that erroneous values of the large roots of /=0
were employed in the investigation, and also that the compa
rison itself of the transverse with the direct vibrations was not
correctly made. These errors are rectified in a communication
to the Philosophical Magazine for May, 1865, and a new value
of the constant K is obtained. Subsequently it appeared to
me^ from a consideration of the way in which the constant e
originated, that the determination of its value should admit of
being effected by having regard only to the state of the fluid
on and very near the axis ; and accordingly the solution I am
* See the Philosophical Magazine for February, 1853, p. 86, and that for
August, 1862, p. 146.
t See at the beginning of an Article in the Philosophical Magazine for Jane,
1866.
216 THE PRINCIPLES OF APPLIED CALCULATION.
*
about to give is conducted on this principle. It leads to the
.Same numerical value of K as the method in the above
mentioned communication, but the reasoning is here more
direct, and in respect to details is more fully carried out.
From the results arrived at in the proofs of Propositions XL
and XII., it follows that the equations, to the first approxima
tion, applicable to the motion and condensation at small dis
tances from the axis are these :
/= 1 er* t u =
The vibrations defined by these equations are resolvable into
two equal sets in the same phase of vibration, having their
transverse motions parallel to two planes at right angles to
each other. The following is the proof of this property,
which has an important bearing on the subsequent reasoning.
Since the angle 6 in the equation (19) is arbitrary, the dif
ferential equation (18) is satisfied by
f l = QL cos 2 Je x, and^= a cos 2 Jey.
The former equation gives
7/
jrj = 2 Je sin 2 Je x aex nearly.
fjf
Hence < ~* =  4aeo^ = 2au ;
and supposing that 2a = 1, we have < ^ = u. So <f> j* = v.
ux dy
Also / 1 = cos2Ve^ = irf,
and /, = \ cos 2 Ve y = I  qf, nearly.
Hence /,+/, = 1  e (J + tf = ler* =/
HYDRODYNAMICS. 217
Again, let a\ +/ t = 0, and aV 2 +/ 2 = ; so that
But it has just been shewn that / 1 +j^=/ Consequently
0^ + 02 = <r. This reasoning proves that the vibrations de
fined by the functions <f> and f may be conceived to be com
posed of two equal sets defined respectively by the functions
<, f lt and <, f 2 ; and that each set satisfies the equations (13)
and (18). On this account it is allowable to take one into
consideration apart from the other, as is done in the succeed
ing part of this investigation. Since this resolvability of the
.original vibrations has been demonstrated by means of forms
of the functions 0, f, f^ / 2 , which were arrived at independ
ently of arbitrary conditions, we may conclude that it is a
general law or property of vibratory motion relative to an
axis, and may, therefore, be legitimately employed in the pre
sent enquiry. It should also be noticed that this resolution is
not possible if the value of f be taken to more than two
terms, and that consequently the application of the reasoning
is restricted to points very near the axis.
Supposing, therefore, the transverse vibrations to be pa
rallel to the axis of x, we have
w
! 3 cos 2 Je x sin q (z teat + c),
, df. m Je . t . .
u = <>  1 = sm %<Je cos q (z Kat + c),
; > WIK / , .
aa =  ~ cos 2 >Je x sm q (z icat + c).
QJ dt 2
Let, now, an exactly equal set of vibrations be propagated in
the contrary direction, and let w, u, v be the velocities and
condensation resulting from the two sets, their coexistence being
assumed from what is proved in Prop. XIII. Then measuring
218 THE PRINCIPLES OP APPLIED CALCULATION.
z from a point of no velocity, and substituting ^j , or q, for
A/
K\! C
2 tje t K for , and c for  , the following system of equations
A/ K
may be formed :
w = in cos q'x sin qz cos q/c (at c'),
u' = cos g sin ^'a? cos qx (at c'),
aa = 772/c cos q'x cos ^2 sin /e (at c'),
= cos gs cos ^'cc sin q/c' (at c).
Hence for points contiguous to the axis the direct and trans
verse velocities are expressible by analogous formulae, and the
condensation can be expressed by corresponding formulas. If
we substitute r^ for e in the value of K, we have
/c
so that /c' 2 = 5  . In order to determine K it is required to
obtain another relation between K and K which I propose to
do by the following considerations.
From the foregoing values of w and u it appears that the
ratio of the direct and transverse velocities at each point is in
dependent of the time (since qK=q'tc'), and that consequently
the lines of motion have fixed positions. To determine their
forms we have the equations
dz w a tan qz q*z ,
= = * ^_ J^_ nearly,
ax u q tan a; q x
the arcs qz and qx being by supposition very small. Hence
by integration,
HYDRODYNAMICS. 219
The different lines are obtained by giving different values to
the arbitrary constant G. They are all convex to the axis of
x if X' be greater than X, and convex to the axis of z in the
contrary case. It might easily be shewn that the trajectories
of these lines are similar ellipses having a common centre at
the origin of coordinates, and their axes coincident with the
axes of coordinates, those coincident with the axis of x hav
ing to the others the ratio of X' to X.
t '2 "\ 2
Since , = ^2~ = rTa , it will be seen, by putting x=z, that
XU Q Z X Z
X 2
the ratio of u to w at equal distances from the origin is ^.
A<
Designating by the ratio of the velocities subject to this
condition, we shall have tea = a 1 1 H ) . It is evident that
V w V
the ratio of u' to w' is that in which the transverse and direct
motions contribute to the changes of condensation at the
origin. This is also the ratio in which the transverse and
direct velocities contribute to the changes of density at any
point of the axis of z when a single series of vibrations, defined
by the foregoing values of w, u, and <r, is propagated along that
axis. For by comparing the values of j and 7 for the same
value of t, and supposing that dz dx, the ratio of du to dw
is found to be that of X 2 to X' 2 . By this reasoning it is proved
that the excess of the rate of propagation above the value a is
due to the transverse motion, and that if this motion had no
existence the rate of propagation would be exactly a.
Again, the foregoing values of w', u', and a' may be
expressed as follows, after putting, for brevity, at v for at c'i
772
w cos qx {sin q (z Kat^) + sin q (z + tcat^)},
u = 2. cos q z { sm ^ (x K 'at^ + sin q' (x+ rc'atj},
220 THE PKINCIPLES OF APPLIED CALCULATION.
a<r = cos q'x (sin q (z  icatj  sin q (z + Kat^},
= ?^L1 cos qz {sin q (x  Kat^  sin q' (x + KatJ}.
These equations shew that the transverse motion may be sup
posed to be compounded, like the direct motion, of two equal
series of vibrations propagated in opposite directions, and that
the direct and transverse vibrations are reciprocally related,
so that either set may be regarded as transverse to the
other. The rate of propagation, on this hypothesis, in the
direction of x is /c'a, which may be expressed under the
following forms:
Ka
The last form indicates, by what was shewn in the case of
direct propagation, that the rate of the transverse propagation
is greater than a because the direct and transverse motions
both contribute to the changes of density.
The foregoing values of w\ u, and c satisfy the general
hydrodynamical equations (2) and (3), and are, therefore, con
.sistent with the existence of an actual elasticity equal at all
points to a 2 . But although w and u are similarly expressed,
and <j admits of being put under two forms, one of which has
the same relation to w as the other to u, the velocities of
propagation in the directions of the two axes are not the
same, that in the direction of the axis of x being to the other
in the ratio of K to /c. Thus there are, apparently, different
elasticities in the two directions. If we call the elasticity
represented by the square of the velocity of propagation ap
parent elasticity, it will follow that the apparent elasticity in
the direction of the axis of x is to that in the direction of the
axis of z in the ratio of ' 2 to /c 2 . The next step is to obtain
another expression for the ratio of these apparent elasticities,
in order thence to infer a relation between K and K.
HYDRODYNAMICS. 221
Let us represent by / and /' the actual accelerative forces
at small equal distances from the origin in the directions of
the axes of z and x respectively, at any given time, the velo
cities and condensation being still represented by w', u, and cr'.
Then
f= a?^= mqtt cos q'x sin qz sin qic (at c'),
f = a 2 j mqn sin qsc cos qz sin qic (at c).
Consequently for small equal values of z and a?,
Hence the accelerative forces at equal distances are at each
instant in the ratio of V 2 to X 2 , and, therefore, the moving
forces of small equal columns of fluid along the axes are in
the same ratio. Now, X' being assumed to differ from X,
these two moving forces are unequal, and tend continually to
produce different condensations at the point of intersection of
the two axes. In consequence of this inequality there is a
mutual transverse action between the columns, causing an
apparent decrement, in a certain ratio, of the elasticity along
one axis, and an apparent increment, in the same ratio, of the
elasticity along the other. The effects must clearly be of
opposite kinds ; and the change of elasticity may be assumed
to be in the same ratio in both cases, because the defect of
the moving force of one column operates in the same degree
as the excess of the moving force of the other, small quantities
of the second order being left out of account. It is to be
observed that this mode of inferring apparent elasticity from
the mutual transverse action of two columns in fixed positions,
is independent of the inference of apparent elasticity from rate
of propagation. Also it should be kept in mind that the pro
pagation is actually along the axis of 2, and that that along
222 THE PRINCIPLES OP APPLIED CALCULATION.
the axis of x is only an apparent, or virtual, propagation.
This being understood, I proceed now to shew that the ap
parent decrement of elasticity takes place along the axis of z,
and to calculate its amount.
Conceive a single series of vibrations to be propagated in
the direction of the axis of z, and the motion to be subject to
the condition that the lines of motion are parallel to that axis.
Such a condition may be conceived to be the result of just
counteracting the effective transverse accelerative forces of
any given series of vibrations by impressed extraneous forces,
and then enclosing the fluid in a cylinder of small transverse
section, having its axis coincident with the axis of z. These
impressed forces, acting transversely, do not alter the rate of
propagation along the axis ; and by the property of the sepa
rability of the parts of a fluid, the portion within the cylinder
may be supposed to be separated from the rest. From these
considerations we may infer generally that vibrations in pa
rallel straight lines within a slender cylinder are propagated
at the same rate as vibrations along an axis in free space.
As this law is true independently of the magnitude of the
vibrations, we may assume that the velocity of the vibrating
particles and the value of X, are, as well as the rate of propaga
tion, the same in constrained motion within a cylinder as in
free motion along an axis. But the relation between the
velocity and the condensation in, the former is to be deter
mined by having recourse to Proposition IX. Let therefore
a be the condensation in an actual series of vibrations pro
pagated along an axis, and <7 t that in the assumed series
within a cylinder, and let w be the velocity common to both.
Then by Prop. IX.,
w = KCI<TI f(z /cat).
Hence
dw da do.
~jl = Ka JT  Ka ~T^
dt at dz
HYDRODYNAMICS. 223
But in the actual series,
n
Consequently
aa , dw a da s
, and rr= 77 = a*
K at K at
da^ _ z d<r
dz dz '
This result proves that the value of y for the direct motion
accompanied by transverse motion has to that of j 1 for equal
direct motion unaccompanied by transverse motion the ratio
of K* to 1. Hence the elasticity in the axis of z is apparently
changed by the transverse action in the ratio of 1 to /e 2 .
This reasoning may be considered to establish a general
law of the effect of transverse vibrations relative to apparent
elasticity in the direction of propagation. If another equal series
were propagated in the opposite direction, the same effect of
its transverse vibrations would be simultaneously produced,
and therefore the above law of the apparent change of elasticity
would apply to the compound series. We have thus been
brought to the case of motion considered in the last paragraph
but one, and the argument has shewn that in such motion the
elasticity in the axis of z is apparently altered in the ratio
1 /e' 2 1
expressed by 5 . Since 5 = nr~r *? is necessarily greater
than unity, and there is consequently an apparent decrement
of elasticity in the direction of that axis. But by the reason
ing in the same paragraph, as much as the elasticity is ap
parently diminished in the direction of one of the axes by the
mutual action of the direct and transverse vibrations, it is
increased in the direction of the other. Hence, the elastici
ties in the two directions being supposed to be cceteris paribus
the same, that in the direction of the axis of x is apparently
increased in the ratio of 2 to 1. Consequently the ratio of
224 THE PRINCIPLES OF APPLIED CALCULATION.
the latter apparent elasticity to the other is 4 . But it has
been shewn that the ratio of the apparent elasticities, as in
ferred from the different rates of propagation for the same
/e' 2 /c 2
actual elasticities, is 3 . Hence 5 = /e 4 , these ratios being
deductions by different processes from the same mutual action
/e' 2
of the parts of the fluid. Now it has been shewn that is
K
also equal to ,  . Hence the equation for determining K
K *" "* L
is K 6 K 4 =l. Thus K? has a fixed numerical value, to be
obtained by the solution of a cubic equation which has one
real positive root, and two impossible roots. The value of K
will be found to be 1,2106.
Taking for atmospheric air a = 916,32 feet, the velocity of
propagation given by this mathematical theory is 1109,3 feet
in one second. The value by observation, as calculated by
Sir J. Herschel in the Encyclopaedia Metropolitana (Vol. IT.
p. 750), is 1089,7 feet. The experiments of Dr Moll (Phil
Trans., 1824, p. 424), when Regnault's coefficient of expan
sion 0,00367 is used instead of 0,00375, give 1090,2 feet, which
is less than the theoretical value by 19,1 feet. With respect
to this difference between the theory and experiment, it should
be observed that the fluidity of the elastic medium has been
assumed to be perfect, and that we cannot assert that atmo
spheric air strictly satisfies this condition. Considering its
composition, and the mechanical suspension in it of foreign
ingredients, it seems reasonable that the theoretical value of
the rate of propagation should be found to exceed the experi
mental Possibly the difference between them may eventually
prove to be an exact datum for theoretically explaining the
causes of imperfect fluidity.
The hypothesis, as is well known, has been made that the
developement of heat by the condensations, and its absorption
by the rarefactions, of the aerial undulations, have the effect
of instantaneously changing the temperature from point to
HYDRODYNAMICS. 225
point in such manner that the increments or decrements of
temperature are always in exact proportion to the condensa
tions or rarefactions. In that case there would certainly be
an acceleration of the rate of propagation, which would have
to be added to the foregoing mathematical determination, and
thus increase the excess of the theoretical above the observed
value. But it does not appear that this assumed action of
heat has experimental grounds to rest upon. For the changes
of temperature produced by sudden compressions and dilata
tions of air in closed spaces cannot be said to establish the
fact of variations of temperature, according to a precise law,
from point to point of undulations taking place in unconfined
air. According to the conclusions that have been arrived at
by the previous reasoning relative to the mutual relation of
direct and transverse vibrations, the motion is the result of
free expansions and contractions successively generating and
filling a partial vacuum. As far as experimental evidence
goes, in such circumstances there is no perceptible change of
temperature. The above named hypothesis is consequently
without support, having, probably, been imagined only be
cause the mathematical determination of the rate of propa
gation, being made in an imperfect state of the theory of
hydrodynamics, was erroneously supposed to be the quan
tity a.
The reasoning hitherto employed has sufficed to determine
the laws of vibratory motion relative to an axis, so far as they
are capable of being expressed by terms involving only the
first power of the constant m. Some of the applications pro
posed to be made of these researches require the investigation
to be carried to terms involving m 2 . This, accordingly, is
the object of the next Proposition.
Proposition XV. To determine the laws of spontaneous
vibratory motion relative to an axis, to the second approxi
mation.
The word "spontaneous" is here used, for the sake of
15
226 THE PRINCIPLES OF APPLIED CALCULATION.
brevity, in the signification intended heretofore to be con
veyed by the expressions, " due to the mutual action of the
parts of the fluid," and " independent of arbitrary disturbances."
Since one analytical indication of this spontaneity is the in
tegrability per se of udx + vdy + wdz, this differential function
will still be represented by (dty). Hence the differential
equations applicable to the present enquiry are the following :
= a"
(24).
dx dxdt dy dydt dz dzdt
^ 'i/ " ^ t// "" 7"""
&' dy dz L _ (25)
As before, the axis of z coincides with the axis of motion.
Having regard, now, to the expression for the velocity along
the axis already obtained, viz. that given by the equation (14),
let us assume that for any distance from the axis we have
mf
* =  ~ cos qt  sin
9<T7
q being put for , f for z aj + c, and A for
and let us suppose that / and g are functions of r the distance
from the axis and constants. We have then to enquire on
what conditions this value of ty satisfies the equation (24).
On substituting it, the equation is found to be satisfied to
terms inclusive of the second power of m, if / and g are de
termined by the following equations :
rdr
HYDRODYNAMICS. 227
The first equation, and a series for / have already been ob
tained under the first approximation. Hence from the second
equation may be derived by the method of indeterminate
coefficients the following series for g :
2 3* 2 l 24 , 12* 2 +1 3 6
g \&r^   eV H  er + &c.
Since it has been shewn that /C G K* 1, it is easy to convert
the coefficients of this series that are functions of K into nu
merical quantities. The values of /, g, and ^ having been
thus determined, by using the equations (25), the following
values, to the second approximation, of the velocity w parallel
to the axis, the velocity a> in any direction transverse to the
axis, and the condensation o may be readily found :
j. . 2/cVa
w = mf sin q%    cos 2^f,
m df K 5 m z da .
6) =  f cos at   j~ sm 2qt.
dr * Ba dr
KW m* f ~ . 2 1 df*
0. =  h 4x I/ 2 sin qt ri
a 2/cV \ 4e dr 2
In these equations 4 and 4e have been substituted for 
and (/c 2 1) ^ 2 , to which they are respectively equal.
By assuming that
mf m 2 Ag .
B being put for V 2 2 ^ in accordance with equation (14),
and h being assumed to be a function of r, a series for A may
be similarly obtained, and the approximation thus be carried
to terms involving m 3 . I have found by this process that the
first two terms of the series for h, like those of the series for /
and g, are 1 er*. This result is confirmatory of the original
supposition that for points near the axis ^ =/</>, f being a
function of x and y, and </> a function of z and t. As the
152
228 THE PRINCIPLES OF APPLIED CALCULATION.
successive approximations may by like processes be carried
on ad libitum, we may conclude that for this kind of vibra
tory motion udx + vdy + wdz is a complete differential for the
exact values of u, v, and w ; and as this result has been ob
tained antecedently to the supposition of any disturbance of
the fluid, we may farther infer that the motion is of a sponta
neous character, or such as is determined by the mutual action
of the parts of the fluid.
The equations which express the laws of a single series
of vibrations relative to an axis having been found, we may
proceed next to investigate the laws of the composition of such
vibrations.
Proposition XVI. To determine the result of the com
position of different sets of vibrations having a common axis,
to terms of the first order.
The proof of the law of the coexistence of small motions
given under Prop. XIII., required that the motions should be
expressed by quantities of the first order, and also that they
should be vibratory. The spontaneous motions which have
been the subject of the preceding investigations were found
to be vibratory ; so that, to the first order of approximation,
the law of the coexistence of small motions is applicable to
them. Hence an unlimited number of sets of such vibrations,
having their axes in arbitrary positions in space, may coexist ;
and for each set the quantities / and </> which define the
motion are given by equations of the form
But here it is to be observed that since the quantity Z> 2 is
equal to ^ (/e 2 1), and /c 2 has been shewn to be a nume
A.
rical constant, that quantity has a different value for every
HYDRODYNAMICS. 229
different value of X, and therefore for every different set of
vibrations. Let us now suppose that there are any number
of different sets having a common axis. Then since the
vibrations coexist we shall have
s.sy .
~~ ( >
These equations prove that the composite motions are not of
the same character as the separate motions, except in the
particular case of , and therefore X, being the same in all the
components. In that case / will be the same for all, as it
contains only the constant e ; and assuming, for reasons al
ready alleged, that m has a fixed value, the values of <f> will
differ only in consequence of difference of values of the arbi
trary constant c. Thus we shall have
2 . w =  S . <r = w/*S . [sin (2 a t 4 c)],
fC
V 1 ^^/^
dr ' * 
But if Cj, c 2 , c a , &c. be the different values of c, by a known
trigonometrical formula,
= [n + 22 . cos q (C M  c,,)]*. q(z aj +
cos
n being the number of the components, and qO an arc such
that tan qd = ^ . Also the values of LU and v in the
2, . cos qc
difference C M c v are to be taken so as to comprehend all the
combinations, two and two, of c v c 2 , c s , &c. ; so that there will
be ^  such differences. Hence if n be a very large
230 THE PRINCIPLES OF APPLIED CALCULATION.
integer, the number of the terms represented by S. cos q (c^c^)
will be extremely large ; and supposing the values of c to be
wholly indeterminate, and that all values within limits large
compared to X are equally possible, there would be extremely
little probability that the sum of the positive values of
cos q (Cp c v ) would differ considerably from the sum of the
negative values. In that case 2S . [cos q (c^ c v )] is to be
neglected in comparison with the large number n, and we
have the results of the composition thus expressed :
5) . w =  2 . cr = n%?i/sin q (z aj + 6),
K
2 fjf
S . w = i cos q (z aj + 0).
q dr
These expressions for the composite vibrations are exactly
analogous to those for the simple vibrations, with the dif
ference that the maximum velocity n 2 m of the composite
vibrations may have different magnitudes according to the
different values of n, whereas the maximum velocity m of the
simple vibrations may be regarded as an absolute constant.
Notwithstanding this analogy, the compound vibrations always
retain their composite character.
If there were n sets of vibrations having the value \ of
X, n z sets having the value X 2 , and so on, the different parcels
would group themselves separately according to these values
of X, so as to form different sets of composite vibrations, it be
ing supposed that all the vibrations are relative to a common
axis. It is evident that the motion and propagation of each
composite set will be independent of the motions and propaga
tions of all the others. Also, by the law of the coexistence of
small vibrations, different composite sets may be propagated
simultaneously in opposite directions.
Proposition XVII. To determine the laws of the com
position of spontaneous vibrations to terms of the second
order.
HYDRODYNAMICS. 231
It will, first, be supposed that the component vibrations
have a common direction of propagation, and a common axis;
and, taking into account the laws of composition when the
approximation is of the first order, and also the expression for
ijr to terms of the second order for simple vibrations, it will
be assumed that
 cos 0?1  m*A 2 If sin
On substituting this value of n/r in (24), it is found that that
equation is satisfied if each value f s of f t and the corre
sponding value g s of g, be determined by the equations
and if at the same time the value of Q be such as to satisfy
the equation
df
a*
df, dj
\ ( d f* **' __
fsinfc&^')fe^)(^^
In the first two equations 5 represents all the integers 1, 2, 3,
&c. to n the number of the different vibrations ; and in the
third equation 5 and s represent all the combinations of these
integers taken two and two. The equation for determining Q
may evidently be satisfied by assuming that
Q 2 . [5 sin (j, + gy) + S sin (q 8 %s *&)]>
R and 8 being functions of r and constants, the expressions
for which may be obtained for each value of^ by the method
of indeterminate coefficients.
232 THE PRINCIPLES OF APPLIED CALCULATION.
Having found for ty an expression applicable to composite
vibrations relative to a common axis, we may deduce the
values of the direct and transverse composite velocities (w
and w'), and the composite condensation (cr'), by means of the
equations (25). The results will be as follows :
w' = mS [/sin j?]  2 [3 cos 2 2 r] + m 2 ,
cos  sm 2 + m
df
2

2 a
Here m z ^ is put for
dz
J 2 [/,// sin q, & sin g/ J>]
so that Q' is a quantity which may be expressed in the form
sin + + 8' sin
K and /S" being determinate functions of r and constants.
Reverting now to the values of w, w, and a obtained under
Prop XV., we have the following equations :
Hence we may infer that on proceeding to terms of the second
order with respect to m, the composite velocities and condensa
tions are no longer equal to the sums of the simple velocities
and condensations, but differ from such sums by quantities of
the second order involving the functions Q and Q'. Respect
ing these functions it is to be observed that they are periodic
HYDRODYNAMICS. 233
in such manner as to have as much positive as negative
value.
But it is chiefly important to remark that while w and a>'
are wholly periodic, the part of <r' which is expressed by 2 . <r
contains terms that do not change sign, viz.
  2
and that these condensations of the second order, corresponding
to the different terms which the symbol 2 embraces, coexist
in the same manner as the condensations of the first order.
A distinction should here be pointed out relative to terms
which have the same value of X, but different values of c. For
all such terms q, f, and are the same ; so that
(2 [/sin ^l) 2 =/ (2 [sin (z  aj + c)]) 2 ,
[jdr C S Z f ]) = p
Adopting, now, the hypothesis respecting the different values
of c for the same value of X which was made in Prop. XVI.,
and in consequence of which for a given group of n compo
nents we have
_cos *
it will be seen that the part of 2 . cr consisting of terms which
do not change sign, may be thus expressed :
Here stands for 2J aj 4 ^, ^ is different for every dif
ferent group, and n represents the number of the components
of any group.
Having thus ascertained to the second approximation the
result of the composition of vibrations that have a common
234 THE PRINCIPLES OF APPLIED CALCULATION.
axis, we may proceed to the more general problem of deter
mining to the same approximation the result of compounding
vibrations relative to different axes having any positions in
space. It has already been shewn, under Prop. XIII., that
the approximation of the first order indicates that spontaneous
vibrations relative to different axes may coexist. From the
general composite value of ty given by the first approximation
we may advance to the second approximation by first substi
tuting this value in the terms of (24) that are of the second
order, and then effecting a new integration. But in order to
obtain an expression for this composite value, it is necessary to
refer the expressions applying to the vibrations relative to the
different axes to a common origin, and common axes, of coordi
nates. Now for a single set of vibrations we obtained the equa
tion ir' =  cos ^ (z a^fc), the axis of z coinciding
with the axis of the vibrations, and f being a function of the
distance (a/ 2 + y*}* from this axis. (The dashes are here
used merely to indicate that the coordinates are subject to
these limitations). Let the coordinates x, y', z in this equa
tion be transformed, and let a?, y, z be the new coordinates
referred to rectangular axes whose positions and directions
are fixed upon arbitrarily; so that we have by the usual
formula,
x = x + <zx + /3y
Then if ^ represent the transformed expression of ^r', we
shall have
,
""
dx* dy* dz* a* df ~
because, as the equation was satisfied by the original function
Jr', it must be satisfied by the same function after the trans
formation, there being no limitation in the investigation of
HYDRODYNAMICS. 235
the equation (24) as to the origin and directions of the rect
angular coordinates. Similarly, for the vibrations relative
to another axis
,,
dtf dy* " dz*
and so on. By adding these equations it will be seen that
the equation (24) is satisfied to the first approximation if the
value of ty be equal to the sum ^ + ^ 2 + ^ 3 + &c. Hence
the general first integral of that equation, so far as it applies
to the spontaneous motions, may be thus expressed :
m 2 .  cos

the coordinates involved in f and being #, y, z. This is
the value of ty which is to be substituted in the terms of (24)
of the second order. By differentiating, we obtain, since
__
dx~dx~
cos ^ " a ' /sin
J 2 t =
dxdt
Hence
 cos ^ a'/sin
dx dxdt \_qdx
X X ^ sin q + qa'fcos q .
The multiplication indicated on the righthand side of this
equality will give rise to two kinds of terms, one consisting
of products of which the factors have the same values of
2, /, and f, and the other of the products of factors which
have different values of these quantities. It will be seen that
236
THE PRINCIPLES OF APPLIED CALCULATION.
the sum of the first kind admits of being expressed as
follows :
sn
These terms are consequently wholly periodic. Eepresenting
by q s ,fs, K* an d <?/> /*'? & any two sets of different values, the
sum of the other kind of terms will be found to be
 sn
cos
These terms are also periodic. It may be observed that both
kinds of terms may be supposed to be included in the last
expression, if for the case of s = s' the result be divided by 2.
If n be the number of the different sets of values, the number
of terms of both kinds will be 2n + 4 . '^ ' or 2w 2 . Also
n r
since the expressions for 2 f
n
and 2 
. ,
a^ ac
are
obtained in exactly the same manner, the whole number of
terms, expressing the value of that part of the equation (24)
which is of the second order, is 6n 2 .
From the foregoing reasoning it follows that the equation
(24) may be integrated to the second approximation by as
suming that
mS  cos q%
If sin 2^f + ZVcos 2^f
+ Psin (q + q&) + Q sin
+ E cos + + fif cos
HYDRODYNAMICS. 237
For since the terms of the second order in (24) have been
explicitly determined by the preceding investigation, by
substituting the above value of ifr in the equation, and
equating to zero the coefficients of the several circular func
tions, differential equations will be formed from which the
values of M, N, P, Q, R, S may be found by the method of
indeterminate coefficients.
On obtaining from the expression for ty the values of
Hie 9 ~dy> He'
it will be found that these velocities are periodic quantities,
having as much positive as negative value. Thus vibratory
motion results from the second approximation as well as from
the first.
We have now to obtain the condensation (cr) to the second
approximation by means of the equation
<fy 1 (dtf dtf d^\
2 Nap.Logp+ Xrh (T^ + T*+TT 1=0.
dt 2 \dx dy z dz z J
To quantities of the second order this equation gives
_ \
a 2 dt 2a 2 VaW dx* dy* dz* ) '
whence a may be calculated from the previous determination
of the value of AT. It is evident from the character of the
terms composing fy, that the condensation, so far as it is
given by the first term of the above expression, is wholly
periodic, having, like the velocity, equal positive and negative
values. But this is no longer the case when the quantity
within the brackets is taken into account, as is shewn by the
following reasoning. The value of ^ to the first approxima
tion being 2  cos ^u , we have to the same approxi
mation
238 THE PRINCIPLES OF APPLIED CALCULATION.
cos 2? ~ y " fa ' m 2?
Hence
= mV 2 [/ 2 sin 2 q% ] + periodic terms,
" 2 / 2 si n 2 U + periodic terms,
and similarly for  and  . Consequently, since
the value of or consists in part of the following terms which do
not change sign by change either as to space or time :
If r be the distance of the point whose coordinates are a?, y y z
from awy axis of vibration, f will be a function of r 2 , that is,
ofa;' 2 +y 2 . Hence
Ji&dxdf dtf J/o
dr
HYDRODYNAMICS. 239
Consequently, since a 2 + yS 2 + 7 2 =l, a' 2 + /3' 2 +y* = 1, and
oca' + 13/3' + 77' = 0, we have
Hence the terms which do not change sign are
This reasoning proves that the different sets of terms em
braced by 2 are the same that would apply to the different
component vibrations, supposing each in turn to exist
separately. So far, therefore, as regards the parts of the
condensations expressed by these terms, the law of co
existence holds good just as for the terms of the first order,
whatever be the number and relative positions of the axes of
vibrations. It may also be noticed that as the positions of
the axes may be any whatever, the foregoing reasoning in
cludes the case in which two sets of vibrations relative to
a common axis are propagated in opposite directions.
The argument that has conducted to the above extension
of the law of coexistence of vibrations has rested on the
hypothesis that udx + vdy + wdz is an exact differential to
terms of the second order for vibrations relative to different
axes. On the same hypothesis the equations (24) and (25)
have been satisfied to quantities of the second order ante
cedently to any supposed case of disturbance. Hence the
deduction of the law from that d priori analytical assumption,
indicates that it is a consequence of the mutual action of the
parts of the fluid, and that it is independent of particular
arbitrary disturbances.
The foregoing propositions embrace all that I proposed to
say on vibratory motion. I proceed now to the consideration
of motion of a different kind.
Definition. The steady motion of a fluid is motion which
240 THE PRINCIPLES OF APPLIED CALCULATION.
is a function of coordinates only, so that the velocity and
density at each point, and the direction of the velocity, are the
same at all times.
Proposition XVIII. To determine the laws of the steady
motion of an elastic fluid the pressure of which varies as the
density.
Resuming the equations
multiplying them respectively by dx, dy, dz, and adding, we
have
Suppose now the variation with respect to space to be from
point to point on a line drawn at a given instant in the
directions of the motions of the particles through which it
passes (which, for brevity, I call a line of motion) , and let s
represent a length reckoned on the line from an arbitrary
point to the point xyz. Then, ds being the differential of
the line, and V the velocity at the point xyz at the time t y
we shall have
T , ds j u , , v j j w ,
dt 1 TF?*> dy = ^ds, dz = j,ds.
Hence, since
a? (dp) _ (/udu\ fvdv\ fwdw\\ds
*jr " Jean + (dt) + \~~dr)} v
(d.V* d. V* ds\ ds^
~\ dt ds dt) 2V'
dV, Id.V*
= j ds   j ds.
dt 2 ds
HYDRODYNAMICS. 241
Therefore, by integration,
. Log p =/(a? , y , * , )
# , i/ , 2 being supposed to be the coordinates of a certain
point of the line of motion at the time t. Now in the case of
steady motion j = and j = for every line of motion.
Hence, the arbitrary function does not contain t, and is
determined by given values of p and V at the fixed point
2? y 2 . Thus in a case of steady motion taking place under
given circumstances, it is generally necessary to determine
the arbitrary function for each line of motion from the given
conditions. There is, however, a supposable case in which
the arbitrary function would be the same for all the lines of
motion, viz. that in which F=0 at some point of each line,
and p a constant p for each of these points. In that case
the relation between p and V would be
and this equation would be applicable to the whole of the
fluid in motion at all times. I now proceed to shew that the
case here supposed is that for which
udx + vdy + wdz
is an exact differential for the complete values of u, v,
and w.
a . dUf.dv.dw. *
Since j = 0, j = 0, j 0, we have
at at at
a*dp
pdx
du
+ U dx +
du
v j + w
dy
du
as '
a*dp
dv
dv
dv
Jay
~\ w 7 "T
ax
dy
~dz~ '
a*dp
dw
dw
dw
pdz
U ~dx
v j r W
dy
~dz ~
16
242 THE PEINCIPLES OF APPLIED CALCULATION.
But by the equation (26),
a? dp rjdV du dv dw
j == r 7  = wj  vj  w j J
pax ax ax ax ax
and similarly for J and y~. Hence by substitution in
pdy pdz
the above equations and adding them, the result is
, v (dv du\ . N (dw dv\ . N (du dw\
(uv) [T j ) + (vw) j r)4 fa *) TT~ H 
' Vtffo <%/ ' \d^ dk/ ' \^ oa?/
This equation is satisfied if
dv du dw dv du dw _
dx dy dy dz dz dx
that is, if udx + vdy + wdz be an exact differential. It may
hence be inferred, on the same principle as that applied to
II
vibratory motion, that the equation p = p^e 2 2 expresses a
general law of steady motion, so far as the motion is inde
pendent of particular conditions, such as those relating to the
limits of the fluid, and to containing surfaces.
Another general law of steady motions, relating to their
coexistence, may be demonstrated as follows. Putting (d%)
for udx 4 vdy + wdz, in order to distinguish this case of
integrability from that for vibratory motion, and proceeding
to form the general hydrodynamical equation of which ^ is
7 72
the principal variable, we shall have = 0, and ^ = ;
and also JF"(tf) = 0, since it has been shewn generally that
F(t] is zero or a constant when there are no arbitrary con
ditions. Thus the equation will become
the terms of the third order being omitted.
HYDRODYNAMICS. 243
If 2&, ^ 2 , ^ 3 , &c. be different values of % applicable
to different sets of steady motions taking place separately,
and if we suppose that % = %! + % 2 + %a + & c > ** * s ey id en t
that this value of % will satisfy the above equation, and that
we shall also have
&=&+&+&.+&,,,
ax ax ax ax
and analogous expressions for % and ~ . Hence it follows
that different sets of steady motions may . coexist, and that
the velocity of the compound motion is the resultant of the
velocities of the individual motions. It also appears, since
the resultant velocity and its direction are at each point
functions of coordinates only, that the compound motion,
like that of the components, is steady motion. Hence if p
represent the density, and V the velocity, for the composite
motion, we shall have by equation (26),
/ V'\
r, p' = p (1 ^fj nearly ......... (27).
The foregoing investigation determines sufficiently for my
purpose the laws of the steady motions of an elastic fluid.
The preceding eighteen Propositions, and the principles
and processes which the proofs of them have involved, are
necessary preliminaries to the application of Hydrodynamics
to specific cases of motion. Having carried these d priori
investigations as far as may be needful for future purposes,
I shall now give examples of the application of the results to
particular problems. The selection of the examples has
been made with reference to certain physical questions that
will come under consideration in a subsequent part of the
volume.
Example I. The relation between the pressure (p) and
162
244 THE PRINCIPLES OF APPLIED CALCULATION.
density (p) being ^> = a 2 p, and no extraneous force acting, let
the motion be subject to the condition of being in directions
perpendicular to a fixed plane, and the velocity and density
be functions of the time and the distance from the plane :
the circumstances of the initial disturbance of the fluid being
given, it is required to find the velocity and condensation at
any point and at any time.
It will be seen that this is the same example as that
following Prop. X., the attempted solution of which led to
contradictions on account of defect of principles. It will now
be treated in accordance with principles and theorems that
have been established by investigations subsequent to that
attempt. At first, for the sake of simplicity, only terms of
the first order will be taken into account. We may suppose
the fluid to be put in motion by a rigid plane of indefinite
extent caused to move in an arbitrary manner, but so as
always to be parallel to the fixed plane. The disturbing
plane is conceived to 'be indefinitely extended in order to
avoid the consideration of the mode in which the motion
would be affected near the boundaries of the plane if it were
limited; a problem of great difficulty, and requiring in
vestigations that I have hitherto not entered upon.
Since by the general preliminary argument the principle
is established that arbitrarily impressed motion must in
every "case be assumed to result from the composition of
primary or spontaneous motions, we must, in this instance,
suppose the motion to be compounded of an unlimited num
ber of spontaneous motions having their axes all perpendicu
lar to the plane, and distributed in such manner that the
transverse motions are destroyed. It is here assumed that
any arbitrary function of z icat + c may be expressed by
the sum of an unlimited number of terms such as
,  . 27T ,
m/sin (z /cat + c),
A<
HYDRODYNAMICS. 245
vri being put for mn* in accordance with what is proved
under Prop. XVI., and the three quantities m ', X, c being con
sequently all of arbitrary magnitude. This hypothesis may
be regarded as axiomatic, inasmuch as there is supposed to
be no limit to the number of arbitrary constants at disposal
for satisfying the required conditions. This being understood,
we may next infer from the analytical expressions of the
components, that the impressed velocity, independently of its
magnitude, is propagated at the uniform rate /ca, and that it
does not undergo alteration by the propagation, the lines of
motion being by the conditions of the problem straight and
parallel*. Also by reason of the same conditions the velocity
V l at any point and the condensation <r l have to each other
the relation V l = /caa v to the first approximation, as is proved
by Prop. IX. Thus we obtain by this reasoning, for any
values of z and t,
V l = icaar^ f(z /cat + cj,
the form of the arbitrary function being determined by the
successive values of the arbitrarily impressed velocity.
Again, on the principle of the coexistence of small vi
brations, contemporaneously with the propagation of F t and
oj in the positive direction, there may be propagated in the
contrary direction the velocity F 2 and condensation cr 2 , such
that F 2 =  /cr/(7 2 = F (z + teat + c,). Hence if F= F x + F 2 ,
and <j = 0 + cr we have
V=f(z  teat + CJ +F(z + Kat + c a )
=/ (z  /cat + cj F (z + Kat + cj.
It follows as a Corollary from these two equations that
v rt=
az at
* This law of propagation, depending only on properties of the fluid, and the
niutual action of its parts, is legitimately inferred from the results of the ante
cedent a priori investigation.
246 THE PRINCIPLES OF APPLIED CALCULATION.
This differential equation takes account of the composite
character of motion subject to arbitrary conditions, and for
this reason differs from the analogous equation applicable
to free motion. It appears, in fact, that under the conditional
or constrained motion, the effective elasticity of the fluid is
increased in the ratio of /t 2 to 1. This result is confirmatory
of the reasoning in p. 222, the extraneous transverse action
there assumed producing the same effect as the state of
composition which neutralizes the transverse motion.
If we now include those terms which involve m 2 in the
expressions for the component spontaneous motions, there
will be additional terms contributing to the value of the
composite velocity V 19 which by the demonstration of Prop.
XVII. will all be periodic functions of z /cat. Also, for
the same reason as in the first approximation, the components
may be such and so disposed, that the transverse motions
will be neutralized. Hence the velocity V l may still be
represented by such a function as f(z /cat + c), and its rate
of propagation will have the constant value /ca. At the
same time, since by the conditions of the problem the lines
of motion are perpendicular to the disturbing plane, we have
by Prop. IX. the relation V 1 p l = tca (p 1) between V and
p l on the hypothesis that the density is propagated with the
uniform velocity fca : or, to terms of the second order, the
equation
2
. (28).
/ca
Hence, a^ having been assumed to be a function of z /cat, it
follows from this equation that V l is a function of the same
quantity. But it has just been shewn that V l actually fulfils
this condition. That assumption is, therefore, justified, and
we may conclude that, as far as is indicated by terms to the
second order, both the velocity and the condensation are
propagated with the constant velocity /ca, and that the re
lation between them is expressed to the same approximation
HYDRODYNAMICS. 247
by the above equation. It should be noticed that while V l
in the case of vibratory motion is as often positive as negative,
oj contains terms that are always positive. This law, as
was remarked (p. 207), relatively to spontaneous motion, is
necessary in order that the forward and backward excursions
of a given particle may be exactly equal.
When the reasoning is extended to terms containing m 3 ,
a like result is obtained ; and so on to terms of any order.
It has thus been shewn that for the case in which the dis
turbance is such that the motion is everywhere in straight
lines perpendicular to a plane, the rate of propagation is tea
independently of the magnitude of the disturbance. In this
course of reasoning no contradiction has been met with like
that which was encountered in the treatment that was applied
to the same example (p. 194) before the laws of the com
ponent spontaneous vibrations were ascertained.
Since, from the supposed mode of the disturbance, there is
no motion parallel to the disturbing plane, we may conceive
a portion of the fluid, having the form of a cylinder or prism
of very small transverse section with its axis perpendicular
to the plane, to be insulated from the surrounding fluid by
infinitely thin rigid partitions. The divisibility, without as
signable force, of contiguous parts of a perfect fluid, which
is one of its fundamental properties, is here assumed to hold
good although the motion is of a composite character. In
fact, as experimentally ascertained, the property of easy divi
sibility is independent of motion, or of the kind of motion.
Hence in order that this principle of insulation may be legiti
mate, it suffices that the transverse motions admit of being
neutralized as nearly as we please. That this may be the
case an unlimited number of components must be at disposal,
and the value of the quantity m for each be inappreciably
small. These two conditions are consistent with the ante
cedent indications of the analysis ; but at the same time it is
to be understood that however small m may be, each com
ponent preserves its individuality under all circumstances.
248 THE PRINCIPLES OF APPLIED CALCULATION.
In short, the composition is not an abstract analytical con
ception, but a physical reality. From these considerations it
follows that motion propagated within a rigid cylindrical or
prismatic tube may be assumed, as far as calculation is con
cerned, to be exactly like motion taking place in directions
perpendicular to all points of a plane of indefinite extent.
Such motion in tubes is, in fact, matter, of experience.
The same principle may be employed to prove that the
functions expressing the gradations of the velocity and con
densation of planewaves may be discontinuous. For if a
very thin partition be made to divide parts of the fluid in a
state of condensation or rarefaction, and at the same time to
partake of the motion of the fluid in contact with it, since no
assignable force is thus introduced, it is evident that the
condensation is only required to satisfy the condition of being
equal on the opposite sides of the partition. It is not neces
sary that the changes of condensation from point to point at
a given instant on one side should be expressed by the same
function as those on the other. Consequently, supposing the
partition to be removed, ordinates drawn to represent the con
densations will have consecutive values, but the directions of
the tangents to the locus of their extremities may change per
saltum. Hence the motion of a given element is generally
expressible, not by a single function, but by different func
tions, in such manner, however, that the velocity always
changes continuously. The motion is, in fact, analogous to
that of a material particle acted upon by a central force
which from time to time changes abruptly both as to law and
amount. The path of the particle would in that case consist
of portions of different curves so joined together as to have
common tangents at the points of junction, and the velocities
in the different portions would be expressed by different func
tions. From this reasoning we may conclude that a solitary
planewave, consisting of arbitrary variations of condensation,
or of rarefaction, from one zero value to another, may be pro
pagated in the fluid without undergoing alteration.
HYDRODYNAMICS. 249
Example II. Let the fluid be disturbed in such manner
that the velocity and density are always functions of the
distance from a centre, and the lines of motion are radii drawn
from the centre : the velocity impressed at a given distance
being given, and no extraneous force acting, it is required to
determine, to the first approximation, the velocity and con
densation at any distance from the centre.
This, again, is a problem the solution of which was before
attempted (p. 197), but without success, because the attempt
was made at too early a stage of the general argument. The
method now about to be employed will take into account the
laws of composite motion subsequently established, and the
principle will be admitted that in any case of constrained
motion due to given arbitrary conditions, the motion at each
point is the result of free, or spontaneous, motions. The
general process for taking account of this principle rests on
the following reasoning.
It is shewn at the end of the solution of the preceding
example, that while regard is had to the effect of the com
position of spontaneous motions, we may at the same time
insulate from the rest of the fluid a portion contained in a
straight prismatic tube of indefinitely small transverse section.
It also appeared that, to the first approximation, the dyna
mical equation
9 <> da dV
K?O? = + T = 0,
dz dt
is applicable at any point of the tube, z being reckoned along
its axis from an arbitrary origin. The action of the sides of
the tube, which have the effect of neutralizing the tendency
to transverse motion, accounts for the factor /e 2 , by which this
equation is distinguished from the analogous one applicable
to free motion. This action, being transverse, leaves the rate
of propagation the same as for free motion, and simply re
places the transverse neutralizing effect of the composition.
Now if the axis of the tube, instead of being straight, were
250 THE PRINCIPLES OF APPLIED CALCULATION.
to become curvilinear, and if s be a line reckoned along it
from a fixed point to any other point, then, supposing the
transverse section still to be uniform, the above equation with
.9 in the place of z would remain true, because the sides
would, just as before, neutralize the tendency to transverse
motion, and would also have the effect of counteracting the
centrifugal force arising from the curvilinear motion. Again,
if instead of being uniform the transverse section varied from
point to point at a given instant, so, however, that the sides
of the tube may be inclined by indefinitely small angles to
its axis, the same equation would still hold good, provided
the curvature of the surfaces to which the lines of motion are
under these circumstances normal, be always and everywhere
finite. For we have seen that the composition of the motion
in effect changes the elasticity of the fluid from a 2 to # 2 a 2 when
the lines of motion are parallel, whether they be rectilinear
or curved. When they are not parallel, for the same reason
that in free motion the effective accelerative force in the direc
27
tion of a line of motion is 7 whatever be the curvature
ds
of the surface of displacement, in constrained motion the
effective accelerative force is j independently of the
same curvature, supposing always that it is finite. Now in
every instance of the constraint of motion by arbitrary cir
cumstances, the whole of the fluid may be assumed to be
composed of curved tubular portions of the kind above speci
fied, the axes and the sides of the tubes always following the
courses of the lines of motion. Also the axis of each tube,
while it may consist of any number of lines defined by dif
ferent equations, must at each instant be continuous so far as
not to vary in direction per saltum; for such a change could
only be produced by an infinite accelerative force. From the
foregoing reasoning I conclude that the equation
KV ^ + ^=0 (29)
ds dt v '
HYDRODYNAMICS. 251
applies at every point of the fluid, when caused to move
under given arbitrary circumstances, and that by this equation
the principle of the composition of spontaneous motions is
taken into account.
I proceed now to apply the above equation to the example
in hand. The disturbance of the fluid is supposed to be such
that the motion is constrained to take place equally in all
directions from the centre, so as to be a function of the dis
tance from the centre. It will suffice in this case to consider
the motion in a slender pyramidal tube bounded by planes
passing through the centre as its vertex ; and if F and a be
the velocity and condensation at the distance r from the ver
tex, we have by the equation (29),
<
Also for this case the equation of constancy of mass becomes
to the same approximation,
By eliminating <r from these equations the result is
d\Vr _ 2 (d\Vr _ 2Vr\
d?  Ka \ dr* '*3~r
of which Euler's known integral is
V _f'(rKat] F' (r + /cat) f (r  teat) F(r + icat}
The result obtained by eliminating F from the same equa
tions is
dt* dr* '
which by integration gives
err = <k (r  /cat) + ^(r + /cat).
On substituting these values of V and a in the equation (30)
252 THE PRINCIPLES OF APPLIED CALCULATION.
it will be found that tcafa is the same function as /', and
/cafa the same as F' ; so that
f(rfcat) F'
/caa =
r r
Supposing the disturbance to be such that propagation takes
place only in the direction from the centre, the arbitrary
function F will have no application, and must therefore be
made to disappear. In that case
TT f'(r /cat) f(rKat)
T/ / \ l__ J \ /
r r*
Kao . ^f( r t)
r
These equations, containing the arbitrary function^ are im
mediately applicable only to the parts of the fluid arbitrarily
disturbed. Let, for instance, the disturbance impress on the
fluid at the given distance b from the centre the velocity m% (t)
during any arbitrary interval, % (t) being a given function of
the time. We shall then have
which equation, by putting T for f(b /cat), is convertible
into
Thus we have a differential equation containing only two
variables, by the integration of which T, or/ (6 feat), is
determined. Hence the two parts
/'(****)
into which the impressed velocity m% (t) is resolved at each
instant by the dynamical action of the fluid, become known.
Calling these velocities and the condensation F 15 F 2 , ^ respec
tively, we have between them and the impressed velocity the
relations
HYDRODYNAMICS. 253
These relations are applicable only at the distance I from the
centre. To find what takes place at any other distance
recourse must be had to the equations (7) and (8) obtained
under Prop. X., which express the general relation between
velocity and density in uniform propagation. Adapted to the
present example these equations become
/car r
By comparison of the above equations with these it may be
inferred that the part V l of the impressed velocity, since its
relation to the condensation <r l is that which the law of uni
form propagation requires, gives rise to propagation at the
rate /ca both of velocity and condensation ; and that the other
part F 2 , not being accompanied by condensation, is trans
mitted instantaneously, just as if the fluid were incompressible.
It is, however, to be understood that this law, as being de
duced by an investigation carried only to terms of the first
order, must be regarded as approximative, and susceptible of
some modification by including terms of higher orders. Now
at the distance b we have
F(bieat)_f'(bieat) <(*)_
' 6"
so that F is the same function as bf, and < (t) is equal to
f(b icat). Accordingly the velocity and condensation at
the distance r from the centre at the time t are given by the
equations
F= /' (r  K at)  /(&*), o = ~f(r  rf),
the second term of the value of V being applicable only
during the disturbance, but at any distance from the centre,
and the other term, with the value of <r, applying during the
disturbance and subsequently, but only within the space
occupied by the uniformly propagated condensation. It will
be seen that the law of constancy of mass is satisfied by these
results, and that the course of the reasoning has accounted
254 THE PRINCIPLES OF APPLIED CALCULATION.
for the contradiction that was met with in the former
treatment (p. 198) of the same example.
To take a particular case, let % (t) sin 7rfca , and sup
A
pose the motion to continue for an indefinitely long interval.
Then, the integration for obtaining T being effected, it will be
found on the supposition that b is v very small compared to X,
that very nearly
M ,, . Zirnib* 2ir ,,
/ (6  /cat) = cos (b  teat),
A A
and f(l  /cat) = ml* sin (b  /cat) .
A
Since^is equivalent to If, the general function F(r icat) is
, 27T .
equal to  cos (r /cat). Hence
A A
2?r . . <mff . 27T /7
cos  (r /cat), V = /cacr  s sin  (b  /cat).
A x r \ ^
Corollary. Suppose a to be indefinitely great, which is
the case if the fluid be incompressible, and let ^ (t) sin atf, a
being some finite quantity. Then since a , X is also
A
indefinitely great. Hence the former of the above equations
shews that both <r and /cao vanish ; and since b will be indefi
2
nitely small compared to A, the value of V is sin at. This
result accords with the general law expressed by the equa
tion (5).
The case of the motion of an incompressible fluid towards
or from a fixed centre, being treated independently and strictly,
in the manner following, furnishes another example illustra
tive of hydrodynamical principles. For the sake of brevity
I shall call this kind of motion central motion.
Example III. To determine the laws of the central mo
tion of an incompressible fluid, no extraneous force acting.
HYDRODYNAMICS. 255
Let p be the pressure and V the velocity at any distance r
from the centre. Then we have for determining the velocity
and pressure at any point the equations

\ dt J
The first gives by integration, V=^~ . Here it may be re
marked that this result, although it depends on the particular
conditions of the problem, is the same that would be given by
integrating the general equation. (5) on the supposition that
r = r, and is, therefore, applicable at any distance from the
centre. For, in fact, the law of rectilinearity of the motion
deducible from the general equation (1), and implied in the
integration of (5) for this purpose, is identically satisfied by
those conditions. The second equation, after substituting the
value of Fand integrating, gives
p ~ '
r *,
If we suppose that where r is infinitely great p has the con
stant value II, we shall have
This result shews that if the velocity be constant at a given
distance from the centre, so that /' (t) = 0, the value of p will
F 2
be II , and the pressure will consequently be greater
as the distance from the centre is greater. Again, suppose
the fluid to be put in motion by being continuously impressed
at the distance b with the variable velocity m sin at. Then
the velocity at the distance r is 5 sin at, and/(tf) = mb* sin at.
Hence /' (t) mtfa cos at ; so that
... mb z a wfb 4 .
p n = cos at T sm 2 at.
* r 2r*
256 THE PRINCIPLES OF APPLIED CALCULATION.
When t = 0, and therefore V = 0, p = U + mba. at the distance
b. Thus although the fluid is assumed to be of infinite extent,
the initial pressure, supposed to take place when t = 0, exceeds
II by the finite quantity m&a. When at = ~ , and the velo
2
city is consequently a maximum,
ir m ^ *
5 ri
Hence the pressure in this case increases with the distance r ;
and as for the same value of t
dp 2m*b 4 _ _ (dV\
dr ~ r 5 \~dt) '
it follows that the acceleration of a given particle decreases in
the ratio of the fifth power of the distance. This is true at all
times if the velocity be constant at a given distance. When
V has the maximum value ^ , the total momentum of the
fluid is 4:7rmb*(r >), which is an infinite quantity, if r be
supposed infinite. Consequently an infinite amount of mo
mentum may be generated in a finite interval of time. This
peculiarity of incompressible fluid in motion appears to be
analogous to what is called " the hydrostatic paradox."
Before proceeding to the consideration of other examples,
it will be proper to introduce here the investigation of certain
equations applicable generally to instances of motion due to
arbitrary disturbances. It has been already proved that the
equation (29), viz.
da dV
applies generally to such instances. Now
da _ da dx da dy da dz
ds ~~ dx ds dy ds dz ds
da u da v do w
HYDKODYNAMICS. 257
and since F 2 = w 2 4 v 2 + w 2 ,
dV _du u du v dw w
Tt ~~ dt T + dt V + ~3JL ~V'
Hence by substituting in that equation,
da dw
This equation is as generally applicable as the equation (29).
If each of the terms be multiplied by Bt, the factors uSt, v&t,
w&t may be considered the virtual velocities of any element
the coordinates of which are x, y, z at the time t. Hence the
equation may be regarded as formed both on D'Alembert's
Principle and the Principle of Virtual Velocities. When it
is employed in a particular problem, it is necessary to intro
duce into it any relations between w, v, w, that may be deduci
ble from the given conditions of the problem. If the relations
between these velocities depend only on the mutual action of
the parts of the fluid not immediately disturbed, and must
consequently be determined by integration, the equation re
solves itself into the three following:
20 da du _ 22^;, ^" _ n 2 z da dw _
Tx*~dt~ l dy* dt~ l 'dz + 'dt~
If the given conditions furnish one relation between u, v, w,
there will be two residual equations, and if they furnish two
relations, there will be a single residual equation. The equa
tion, or equations, thus resulting will have to be employed,
together with the equation of constancy of mass, for obtaining
a partial differential equation by the integration of which the
solution of the problem is effected.
For instance, let the case of motion be that of Example II.
Then we have
_ Vx = Vy _Vz
w y y
furnishing the two relations
v = ^ w =
x ' x '
17
258 THE PRINCIPLES OF APPLIED CALCULATION.
Hence the equation (31) is equivalent to a single equation ;
which, since
du dV x , dcr da x
j =77  , &c. and r =   &c.,
dt dt, r dx dr r
is readily found to be the equation applicable to central
motion which was employed in the solution of that Example.
As another instance, let the motion and condensation be
symmetrically disposed about a rectilinear axis, and let U
and W be the resolved parts of the velocity along and per
pendicular to any radiusvector drawn from a fixed point in
the axis. In this case V 2 = U* + W*, and the condensation
a is a function of the. polar coordinates r and 6 referred
to the fixed point as origin, and to the axis of symmetry.
Hence
dV_dUU dWW
dt ~ dt V + ~di 7 '
do do dr da rdO
da U da^W
dr V + ri6'V'
Consequently by substituting in (29),
 a
dt J rdO dt
W=0 ...... (32).
The equation of constancy of mass to the same approxima
tion is
da dU 2U dW W /OON
77 +T +  +jn + cot0 = ......... (33).
dt dr r rdd r
If no relation between U and W be deducible from the con
ditions of the problem, we shall have to combine with this
last equation the two equations
2 2 da dU 2 2 da dW . .
/cV j+ :yr = 0, ic a * Tfl+'TT = ....... (34).
dr dt rdd dt
HYDRODYNAMICS. 259
From the three equations U and W may be eliminated, and
an equation be obtained containing the variables <7, r, #, and t,
a being the principal variable.
If the origin of coordinates instead of being fixed, be a
moving point on the axis of symmetry, we may still express
a, U t and W as functions of r, 6, and t. But since in this
case the coordinates r and 6 of a given position in space vary
with the time, the value of y will contain the additional
da dr , dcr rdO . . . . dr , rd6 , .
terms ^ r and ^ 7, the velocities = and  T  being
eft* eft tw dt dt dt
known from the given motion of the origin. And so with
dU , dW . . _
respect to , and 7 . bupposmg this motion to be a
UA, (it
quantity of the same order as the velocity and condensation
of the fluid, these additional terms will be of the second order,
and may, therefore, be neglected in a first approximation.
Hence the foregoing equations are equally applicable whether
the origin be fixed or moving, if the motion be small.
It is important to make here another general remark.
When there are no relations between u, v, w, given imme
diately by the conditions of the problem, and the equation (31)
consequently resolves itself into three equations, it may be in
ferred from these, just as was done in page 208 from the analo
gous equations for free motion, that udx 4 vdy + wdz is an
exact differential when the motion is exclusively vibratory.
In the reasoning referred to, vibratory motion of a particular
kind, partly longitudinal and partly transversal, was deduced
by an d priori investigation founded on the supposition of the
integrability of that differential quantity; but here the inference
is, that if the motion consist of vibrations having an arbitrary
origin, that differential is still exact*. This might, possibly,
* For a long time I maintained (in the Cambridge Philosophical Transactions,
and in Articles in the Philosophical Magazine) that the a priori proof of the inte
grability of ud# + vdy + wdz for the primary class of vibrations did not establish its
integrability for vibrations produced under arbitrary conditions. But the argument
172
260 THE PRINCIPLES OF APPLIED CALCULATION.
have been anticipated from the circumstance that the arbitrary
vibrations may be regarded as resulting from the composition
of primary, or spontaneous vibrations.
By the same argument, when the motion is symmetrical
with respect to an axis, and the arbitrary disturbance is such
as to cause vibratory motion, Udr + WrdO will be an exact
differential.
Example IV. A smooth sphere of very small magnitude
performs oscillations in an elastic fluid at rest, its centre
moving in a given manner in a straight line : it is required
to find the velocity and condensation of the fluid at any
point.
The equations to be employed for solving this problem
are (33) and the two equations (34). From what is shewn
in page 259, we may suppose the origin of the coordinates
r and 6 to be at the centre of the moving sphere, its vibra
tions being small. Then the elimination of U and W from
the three equations gives
1 d\ar d\crr 1 d\ ar d.<
A particular integral of this equation may be obtained by
supposing that err = ^ cos 0, and that ^ is a function of
r and t. For on substituting this value of or the equation
is satisfied if the function ^ be determined by integrating
the equation
<TA d'A , a
a'W di* r* ~
in which a' 2 is put for #V. The known integral of this
equation is
<k = i/( r _ a 't) f'( r a't) + F(r+at) 
above, which has not before been brought to bear on the question, proves that
this analytical condition is satisfied by both kinds of vibrations.
HYDRODYNAMICS. 261
Hence, representing for the sake of brevity the arbitrary
functions by /and F,
ff+F f
^ 
According to this equation <r = at all points for which
6 =  , and also if r be indefinitely great. By substituting
this value of or in the equations (34), and integrating on the
supposition that U and W contain no terms independent of
the time, it will be found that
r* r r
/ and FL being put respectively for Ifdr and \Fdr. Let
m sin at be the given velocity of the centre of the sphere.
Then since the velocity U of the fluid at any point of the
surface, supposed to be perfectly smooth, must be equal to
the velocity of that point resolved in the direction of the
radius, if we call the value of U for such points U , we shall
have
U Q = m cos sin at.
Consequently, putting in the general value of U the radius b
of the sphere for r, we obtain
/and Fnow standing for f(b at) and F(b + at). As the
general value of F indicates propagation towards the centre
of the sphere, and from the nature of the disturbance there is
no propagation in that direction, we must suppose F to
vanish. In fact, on supposing that
o__
/= m l sin (b  at + c t ),
and F= m^ sin ~ (b + at + cj,
A.
262 THE PRINCIPLES OF APPLIED CALCULATION.
I found that there were no conditions for determining w 2
and c 2 . Thus in order to determine f(b at] we have to
integrate the equation
? 2! mb .
t , . A ,
or, since/ =  , 4* , and / = ^ , the equation
ft (f CL dt>
The exact integral of this equation contains terms which have
an exponential factor of the form e'**, and on that account
disappear after a short interval, Jc being in this application
very large. It will therefore suffice to assume that
Then by substitution it will be found that the equation is
satisfied if the unknown constants /i, \, and c be determined
by the following equations :
2Tra' mb*
27T&
X 27T
Since the general value of j^ is the same function of r at
as the particular value thus obtained is of b at, we shall
clearly have for the general values of f lt f, and/',
/ = p sin (r  at + c) ,
2?r
HYDRODYNAMICS. 263
It will now be supposed that the oscillations of the sphere
are such that the value of X is extremely large compared to
b, and powers of  above the second will be neglected.
X
Then
mb s
Accordingly the general values of or, U, and W are given by
the equations,
, ("jrb 3 27T , , v 27T 2 & 3 , 2?r , , .}
a a = l j cos  (r a t) + ^ sm (r a t) > mcos 0,
I XT* X X T X J
W = ! s sin  (r at) + r 5 cos  (r at) [ m sin 0,
( 2r' X XT* X J
U=\( 5 + ^ jsin  (r a't)\ T cos (r a't)\ m cos 0.
I \ a X /* / A ^ * a /v ^ ^ I *
(_\ ? AT*/ A Ar A J
Again, it will be supposed that b is so extremely small com
pared to X, that values of r which are large multiples of b
are still very small compared to X. Thus ;~( =  x ;r) i g
X \ T X/
a small quantity of the second order. On these suppositions
the coefficients of the circular functions in the above equations
will all become of inappreciable magnitude where r is a
large multiple of &, although at the same time r is small
compared to X. On this account it is allowable to substitute
f . 2?rr ,, .
for sm  and cos  their expansions to one or two
X X
terms. When this has been done and terms incomparably
less than those retained have been omitted, the results are
Trmb* %7ra't ,_ mo 3 . %7rat .
a<r ^* cos  cos 0, \V g sm   sin 0,
Xr 2 X 2r 3 X
Tr .
U f sm cos 0.
r* X
264 THE PRINCIPLES OF APPLIED CALCULATION.
OTJ/*'
Or, since = a, if we put T for m sin at, we have finally
a'V = ^ ^cos 0, W= ^3 Ts'm 6, U= ^ Tcos 0.
These equations, with the exception of having a' 2 in place
of a 2 , are those usually obtained by solving approximately the
problem of the simultaneous movements of a ballpendulum
and the surrounding air on the supposition that udx+vdy + wdz
is an exact differential*. This supposition has not been
directly made in the foregoing reasoning; but since it was
antecedently proved (p. 259) that in every instance of vibra
tory motion arbitrarily produced that analytical condition
must be fulfilled, we ought to find it fulfilled by the above
values of W and U. And this, in fact, is the case; for
whether we take these approximate values, or those given im
mediately by the integrations, the integrability of Wrd6+ Udr
is verified. It may be remarked that the values of W and U
do not explicitly involve the elasticity of the fluid, or the rate
of propagation, being the same that would be obtained if the
rate of propagation were infinitely great, or the fluid incom
pressible.
It is worth noticing that the quantity of fluid which
passes at any instant the plane through the centre of the
sphere perpendicular to the axis of the motion, in the di
rection contrary to its motion, is just equal to the quantity
which the sphere displaces. For since for that plane Q ,
the quantity of fluid which passes it in the time St is
r tfT
Stl^Trr j dr, taken from r = b to r= infinity, which is TrffTSt.
This is plainly the amount of fluid displaced by the sphere
in the same time. There is, therefore, no actual transfer of
fluid in the direction of the impulse of the sphere. This
* Poisson's solution is in Tom. XT. of the Memoirs of the Paris Academy and
in the Connaissance des Terns for 1834
HYDRODYNAMICS. 265
would also be the case if the fluid were enclosed within
boundaries out of which none of it can pass; but as the
preceding investigation involves no such condition, we may
conclude that this equality between the displacement of the
fluid and the reciprocal flow, does not depend on the fluid
being enclosed. The law of the movement would seem to
be the same whether the mass of fluid be contained within
boundaries moderately distant from the vibrating sphere, or
be unlimited. I proceed next to calculate the resistance which
the fluid offers to the motions of the sphere, with the view
of comparing the result with experiment.
Before entering on this comparison it will be necessary
to make a preliminary remark relative to the equation (29).
It was shewn that this equation takes account, to the first
order of small quantities, of the action of the fluid when its
motion is compounded of the simple or primary vibrations.
But if p be the effective pressure of the fluid in composite
motion, we must have, to the same approximation,
at
7 I 7
Consequently the equation (29) shews that j K Z O? j
Hence by integration, p = o? (I + #V), a 2 being the pressure
at all points where the fluid is at rest and p = 1. As the
composite character of the motion has been shewn to be an
independent hydrodynamical law, this value of the pressure
is to be used instead of a 2 (1 + or) for calculating the effect
of given arbitrary disturbances. The same expression would
have to be used if the factor /c 2 , instead of having, as I have
argued, a purely hydrodynamical origin, were due to the
action of developed heat. In either case the fluid is virtually
acted upon by an extraneous force equal to (K? 1) a 2 y ,
which increases its effective elasticity when in motion in the
ratio of to 1. Also it is evident that the same value of
266 THE PRINCIPLES OF APPLIED CALCULATION.
the pressure which is used for calculating the mutual pressure
of the parts of the fluid, must be used for calculating the
pressure of the fluid against the surface of a solid. This
being admitted, the following calculation gives the resistance
of the air, or of any other fluid, whether highly elastic or
incompressible, to the motion of a ballpendulum.
Since p = a 8 (1 + /eV) the whole pressure on the ball
estimated in the direction for which = is
 27rZ> 2 fa 2 (1 +/cV) cos 6 sin 6 dQ,
taken from 6 = to 6 = TT. On substituting the foregoing
value of aVcr, this integral will be found to be j .
o cH
Hence if A be the ratio of the density of the ball to that
of the fluid in which it oscillates, the accelerative force in
the same direction is r ^ . Let x be the distance
2A at
of the centre of the ball from the lowest point, I the length
of the simple pendulum, and g the force of gravity, and let
the extent of the oscillations be so small that x is always
very small compared to I. Then since the accelerative
force, when buoyancy alone is taken into account, is
_gx
~ I
by adding to this the accelerative force of the resistance, we
obtain
^___ _ ___
dt ~ " ~ ;
dT
d 2 x _ gx
dt* = "T
HYDRODYNAMICS. 267
If L be the length of the pendulum which would oscillate in
the same time in vacuum, we shall have
In this formula A may have any value greater than unity.
In making a comparison of the above theoretical result
with experiment it must be borne in mind that in the theory
the fluid is supposed to be unlimited, whereas the experi
mental oscillations were almost necessarily performed in en
closed spaces, or in limited masses of fluid. But from the
considerations entered into in page 265, it is probable that
the comparisons with the experiments I am about to adduce
are little affected by that difference of circumstance. The
first I shall cite are those of Du Buat, contained in his
Principes d* Hydraulique (Tom. II. p. 236, Ed. of 1786). These
experiments were made with spheres of lead, glass, and wood,
of different weights and diameters, oscillating in water. The
diameters in inches* were 1,08, 2,82, 4,35, and 7,11, and the
time of oscillation varied from 1 second to 12 seconds, and in
one instance was 18 seconds. The vessel in which the
spheres oscillated was 54 inches long, 18 inches wide, and
15 inches deep, the spheres were entirely immersed to the
depth of about 3 inches below the surface, and the threads
by which they were suspended were as fine as the weights
would allow of. Although the dimensions of the vessel
and boundary of the fluid are smaller, relatively to the
magnitudes of the spheres, than is strictly compatible with
the theory, the law of the movement by which the fluid
that passes at any time the vertical plane through the
centre of the sphere fulfils the condition of being equal to
the quantity displaced by the sphere, might still be very
* In this, as in all other instances, foreign measures are converted into
English.
268 THE PRINCIPLES OF APPLIED CALCULATION.
nearly independent of those dimensions. On this account
it may be presumed that the results of the experiments ad
mit of comparison with the theory. Now Du Buat found
that a quantity which he calls n, for which he gives an
expression identical with (Al) (=  I J , had nearly the
same value under all the different circumstances above men
tioned. This is precisely the law which is indicated by the
theory. Also the mean value he gives for n is 1,585, which
differs little from the theoretical value 1,5.
In the same work (Tom. II. pp. 283 and 284) Du Buat
has recorded three experiments with spheres oscillating in
air. The diameters of two of the spheres, which were of
paper, were 4,31 in. and 7,07 in., and the lengths of the threads
by which they were suspended 78 in. and 102 in. respectively.
The smaller performed 100 oscillations in 151 seconds, and the
other 50 oscillations in 92 seconds. The third was a sphere
of bladder, its diameter 18.38 in., the length of the suspension
thread 92 in., and it performed 16 oscillations in 58 seconds.
The values of n obtained from the three experiments were
1,51, 1,63, and 1,54 respectively. The author has not stated
whether the spheres oscillated in an enclosed space ; but if, as
is probable, the experiments were made in a room of ordinary
dimensions, the value of n might not be affected by the
limited space, notwithstanding the large size of the spheres,
and the experiments may thus admit of comparison with the
theory. The mean value of n resulting from these experi
ments is 1,560, which agrees closely with that deduced from
the experiments in water. This, again, accords with an indi
cation of the theory, which gives the same value of n for air
as for an incompressible fluid.
The experiments I shall next adduce are those of Bessel
contained in his Untersuchungen uber die Ldnge des einfacJien
Secundenpendels (Berlin, 1828). These were made by noting
the times of oscillation of two spheres, one of brass, and the
other of ivory, each 2,14 in. in diameter. Two series of ob
HYDRODYNAMICS. 269
servations were taken with each sphere by attaching it in
succession to two suspensionwires of fine steel, one longer
than the other by the exact length of the Toise of Peru, and
the shorter one as nearly as possible of the same length as the
seconds' pendulum. The length of the longer pendulum was
therefore 11 6,1 in., and that of the shorter 39,2 in. Every
circumstance that might affect the accuracy of the determina
tion having been attended to, it was found that the experi
ments with the two spheres gave very nearly the same value
of the factor 1 +&, (the same as that we have called ,) arid that
the mean result was 1,9459. It is, however, to be noticed
that the calculation of this quantity was made on the assump
tion that n had the same value for the two pendulums.
In the Astronomische NacJirichten (Tom. x. col. 105)
Bessel has slightly corrected the above determination, and has
also given the results of a new set of experiments. In this
second series, instead of the spheres, a hollow brass cylinder,
two inches in height and diameter, was attached to the same
two lengths of wires, and was caused to oscillate both when it
was empty, and with three pieces of brass of different weights
enclosed in succession within it. Also various other sub
stances of different specific gravities were severally put into
the hollow cylinder, and the times of oscillation were noted.
Equations of condition, formed separately for the two pendu
lums, from the observations with all the substances, on the
suppositions that the value of n was independent of the spe
cific gravity of the oscillating system, but was different for the
two pendulums, gave results consistent with these suppositions.
It was found, by appropriately using all the equations given
by the two series of experiments, that by the earlier set the
value of n was 1,9557, and by the later set 1,9519 for the
longer pendulum and 1,7549 for the shorter. These results
seem to shew that the cylinder suffered nearly the same retar
dation as a sphere of equal diameter. (To this point I shall
recur after treating as a separate problem the case of the re
tardation of a cylindrical rod). But apart from the form of
270 THE PRINCIPLES OF APPLIED CALCULATION.
tlie attached body, the later experiments appear to indicate
that the suspensionwire suffers resistance to such an amount
that the time of oscillation is sensibly affected by it, and in
greater degree as the length of the wire is greater. It should
be observed that in all Bessel's experiments the oscillations
took place in an enclosed space, the horizontal dimensions of
which were comparatively small.
Bessel also observed the times of oscillation of the brass
ball in water, using the same two pendulumlengths. The
water vessel was cylindrical, and about 38 in. in diameter
and 11 in. deep, and the arc of oscillation was 2. The value
of n found for the longer pendulum was 1,648, and that for
the shorter 1,602. These numbers approach closely to those
of Du Buat.
It remains to mention the results of the experiments of
Baily contained in the Philosophical Transactions for 1832
(p. 399), so far, at least, as they bear on the object of the
present discussion. Pendulums consisting of spheres fastened
to the ends of wires, were swung within a brass cylinder
about five feet long and six inches and a half in diameter,
from which the air could be extracted by means of an at
tached airpump. The value of n was inferred from a com
parison of the times of oscillations in vacuum with those of
oscillations observed after admitting the air into the cylinder.
With spheres of platina, lead, brass, and ivory of 1^ inch
diameter, the mean value obtained for n was 1,864, and with
lead, brass, and ivory spheres of 2 inches diameter the mean
value was 1,748. The experiments shewed that this factor
depended on the form and magnitude of the oscillating body,
but not on its specific gravity. The length of the wire was
that of the seconds' pendulum, or about 39 inches, and, there
fore, the same as the length of Bessel's shorter pendulum.
The extent of the oscillations was always very small.
Baily also made additional experiments with three pendu
lum rods 58,8 in., 56,4 in., and 56,4 in. long, swinging them
first without attaching spheres, and then with spheres of the
HYDRODYNAMICS. 271
diameters 1,46 in., 2,06 in., and 3,03 in. attached successively
to each. The general expression he obtained for the quantity
of air dragged by a pendulum consisting of a sphere of dia
meter d, and a wire of length Z, is 0,002564Zf 0,123d 3 , I and d
being expressed in inches, and the mass of air in grains.
This formula proves that the air dragged by the wire may
have a sensible effect on the value of n, and that this effect is
cceteris paribus greater as the wire is longer. This inference
accords with the results obtained for the two pendulums in
Bessel's second series. In fact, if we assume the influence of
the wire on the value of n to be proportional to its length,
since the wires in these experiments were very nearly in the
ratio of 3 to 1, by subtracting half the difference of 1,9519 and
1,7529 from the latter, we get 1,653 for the value of n freed
from the effect of the wire. This result applies strictly only
to the experiments made with the hollow cylinder, but may
be taken as very approximately applicable to the experiments
with the spheres, when it is considered that for the longer
wire n was nearly the same in the two series. Also the above
result agrees very nearly with that obtained for oscillations
of spheres performed in water, in the case of which the re
sistance of the air on the wire would be comparatively very
small on account of the specific gravity of air being so much
less than that of water.
The general inference to be drawn from the preceding dis
cussion is, that the experimental value of n, after eliminating
the influence of the suspensionwire, approaches closely to the
theoretical value 1,5, but is still somewhat in excess. Accord
ing to Baily's experiments (Phil. Trans, for 1832, pp. 443 and
448) n is greater the less the spheres, the suspensionrods
being the same. This difference must be owing, in part at
least, to the comparative effect of the retardation of the wire
being greater the smaller the sphere; and it may also be
partly due to the confined dimensions of the cylindrical space
in which the pendulums oscillated, which would tend to faci
litate the backward flow of the air, and thus diminish the
272 THE PRINCIPLES OF APPLIED CALCULATION.
resistance, and the more so as the sphere is larger. What
remains of the excess of the experimental above the theoreti
cal value of n may be attributed to the neglect in the theory of
the effect of friction, and to the fluid having been considered
to be perfect.
In my original attempts* to solve the problem of the
simultaneous movements of a ballpendulum and the surround
ing fluid, I assumed that for vibratory motions produced under
arbitrary circumstances udx + vdy \wdz might be such as to be
only integrable by a factor, .and on the supposition that the
lines of motion in this instance are prolongations of the radii
of the sphere, I obtained the factor ^ . Having found by
this reasoning the correction of the coefficient of buoyancy to
be 2, I concluded that the solution was supported by the near
agreement of this result with Bessel's determination 1,956.
But it has now been shewn that this support fails, the preced
ing discussion having sufficiently accounted for the excess of
the experimental value of that coefficient above the value 1,5
given by Poisson's solution. Also, as was before intimated
(p. 260), I have for the first time in this work adduced an
analytical argument which proves that udx + vdy + wdz is an
exact differential, as for spontaneous vibratory motions, so
also for vibratory motions produced arbitrarily. In order to
test experimentally the course which, according to the theory,
the fluid takes in the neighbourhood of the sphere, I tried the
effect of causing a globe to pass quickly forwards and back
wards close to the flame of a candle, and found that the flame
decidedly indicated a rush of the air in the direction contrary
to that of the motion of the globe, in accordance with the
foregoing value of W (p. 264). The experiment was made
with globes of three inches and ten inches diameter, both in
the open air, and in rooms of different sizes, sometimes oppo
* The investigations here referred to are in the Cambridge Philosophical
Transactions, Vol. v. p. 200, and Vol. vii. p. 333; and in the Numbers of the
Philosophical Magazine for September, 1833, and December, 1840.
HYDRODYNAMICS. 273
site to an open window, and at other times with doors and
windows closed, and under all this variety of circumstances
the reverse movement of the fluid appeared to obey the same
law, and to be of the same amount, conforming in these
respects to the indications of the theory.
The next Problem, relating to the resistance of a fluid to
the oscillations of slender cylindrical rods, is one the solu
tion of which, as far as I am aware, has not been previously
attempted.
Example V. A slender cylindrical rod performs small
oscillations in a fluid in such manner that its axis moves
transversely to its length in a fixed plane : required the mo
tion communicated to the fluid by the rod, and the resistance
to the motion of the rod from the pressure of the fluid.
It will be supposed that the rod is of indefinite length in
order to avoid the consideration of the motion of the fluid
contiguous to its extremities. Let its axis be in the plane zx,
and, at first, let it always be parallel to the axis of z ; and let
a be its distance from that axis at any time t. In that case
w = 0, the motion being wholly parallel to the plane xy. Since
the relation between u and v depends only on the mutual
action of the parts of the fluid, the equations for finding to
the first approximation the pressure and motion are
" + =o, rf^+lo, t+ + ?a
dx dt dy dt dt dx dy
By eliminating u and v we obtain
dV = , 2 /dV dV\
de ~ \jbt**N
It will be convenient to transform this equation into one in
which the coordinates are r the distance of any point from
the axis of the rod, and 9 the angle which the line drawn
from the axis to the point makes with the plane zx. Thus we
shall have, putting x for x a,
x r cos 0, y = r sin 6, x' z + y* = r\
18
274 THE PRINCIPLES OF APPLIED CALCULATION.
After effecting the transformation by the usual rules, it will be
found that
Also, U and W being the velocities resolved along and per
pendicular to the radius vector, we have to the same ap
proximation
.& dU &r
a ~j~ + ~J7 ~ > a
tfr eft
As the diameter of the rod is supposed to be small, and its
motion extremely small compared to a', the motion of the fluid
will be very nearly the same as if it were incompressible. We
may, therefore, omit the term on the lefthand side of the first
of the above three equations, and we have then to integrate
the equation
d*o da 1 d*o _
dr* + rir + ^~d&~
It is, however, to be observed that in order to ascertain the
law of the motion as resulting from the mutual action of the
parts of the fluid, it is not the general integral of this equa
tion, but a particular solution of definite form that is required.
f/Q\
Let us, therefore, assume that a ~ . Then by substitu
tion in the equation it will be found that
Hence the following results are readily obtained :
f(ff) = Pcos (nd + Q), a = ^ cos (n0 + Q),
dU _ , 2 go _ na'*P
P and Q being generally functions of t. Now if m$ (t) repre
sent the velocity of the axis of the rod at any time t, we shall
have for any point of the surface,
HYDRODYNAMICS. 275
U = m<f> (t) cos 0, 7 = m(f>' (t) cos 6.
Hence, Z> being the radius of the rod,
/2 T)
ra<' (t) cos == rUr cos (nO + Q).
That this may be an identical equation we must have n = 1,
$ = 0, and P = ^ <' (t). Hence at any distance r from the
axis of the rod,
i t*\ a .,
*(*)<* ft ^
Hence, also,
7.2
and by integration, Z7= ^ < (^) cos
and by integration, W ^ <f> (0 sin 0.
In the above integrations no arbitrary functions of space have
been added, because by hypothesis the motion is wholly
vibratory. The above expressions for U and W evidently
make Udr + WrdQ an exact differential.
By putting 6 = , and r = 6, the value of TF becomes
2
m<f> (t) ; which shews that the motion of the fluid in contact
77*
with the rod at points for which 6  is just equal and oppo
2
site to that of the rod. The quantity of fluid which in the
small interval &t crosses a plane passing through the axis of
the rod at right angles to the direction of its motion is, for a
given length L of its axis, Lt I $ (t) dr taken from
r = I to r = infinity. This is ZLbm <f> (t) &t, which is clearly
the quantity of fluid which a portion of the rod of length L
displaces in the same indefinitely small interval. Thus the
182
276 THE PRINCIPLES OF APPLIED CALCULATION.
motion of the fluid caused by that of the rod satisfies the same
condition as that which was found to be satisfied in the case
of the vibrating sphere. It may also be remarked that al
though a particular form of expression was assumed for <r, we
may yet conclude, since it gives a definite result, that the pro
blem admits of no other solution for a first approximation.
The whole pressure in the direction contrary to that for
which 6 on a portion of the rod of length L is
Lb I a'*<r cos e dd
taken from 6 to 6 = 2ir ; which integration, after substi
tuting the value of a' 2 a and putting b for r, gives 7rb 2 Lm(f> f (t).
Hence A being the ratio of the specific gravity of the rod to
that of the fluid, the accelerative force of the resistance
m
Suppose now the cylindrical rod to be acted upon by
gravity, and to perform small oscillations in air about a hori
zontal axis passing through one extremity. In this case,
since the rod has an angular motion, the above investigation
does not immediately apply. But it may be presumed that
if we take an element of the rod of length 82 at the distance z
reckoned along the rod from the point of suspension, the
foregoing reasoning will give very approximately the resist
ance on this portion, supposing the oscillations to be of very
small angular extent. Hence if I' be the rod's length, and
m(f> (t) the velocity of its extremity, the accelerative force of
the resistance on the element at the distance z is ^
which, if be the angle made by the axis 'with the vertical,
72 c*
is equal to ^ 3^ . Consequently, tfSz being the elementary
mass of the rod, by D'Alembert's Principle,
HYDRODYNAMICS. 277
Hence integrating from 3 = to z = I', putting g ( 1 ^J for
2i'
g' on account of buoyancy, and substituting I for , which
o
is the distance of the centre of oscillation of the rod from the
point of suspension, the result is
It follows that for this case the theoretical value of the factor
n is 2.
This result admits of being tested by means of the ex
periments on vibrating cylinders recorded by Baily in the
Paper already referred to (Phil. Trans, for 1832). He has
there calculated (p. 433) the values of n for two cylinders
each 2 inches in diameter, one 2 inches and the other 4 inches
in length, which were made to vibrate by being attached to
the ends of rods 39 inches long. The value of n obtained
for the short cylinder is 1,86. We have seen (page 269)
that Bessel's determination for a cylinder of the same dimen
sions under the same conditions of vibration was 1,755. On
account of the short lengths of the cylinders, these results
can scarcely be compared with the theoretical value 2, ob
tained for a rod of indefinite length. When the effect of the
lateral action due to the abrupt terminations of the cylindrical
surface is considered, theory might lead us to expect that for
the shorter cylinder n would not differ much from its value
for a sphere of the same diameter ; and this, in fact, is found
to be the case. But there are no grounds from the theory to
conclude that the difference of form has no effect, and that n
has exactly the same value for the cylinder as for the sphere,
although the beforecited experiments of Bessel (page 269)
seem to indicate such an equality. In the case of the cylinder
4 inches long, the experimental result is 2,03 ; which agrees
more closely than that for the other cylinder with the theo
278 THE PEINCIPLES OF APPLIED CALCULATION.
retical value 2, apparently because by the increase of length
the conditions assumed in the theory are more nearly satisfied.
If, however, the effect of the suspending rod were eliminated,
it would probably be found that the experimental value of n
for the longer cylinder is really less than 2, owing to the
influence of the lateral action at its extremities.
For additional verification of the theory, I caused a cylin
der of about half an inch in diameter, and nine inches long, to
pass and repass the flame of a lamp, just as in the previous
experiments relative to the vibrating sphere, and I found that
the reverse movement of the air was indicated by the flame
even more decidedly than in the case of the globe.
Baily has also given the results of experiments made, in
the same apparatus, with plain cylindrical rods, the diameters
of which were l in ,500, O in ',410, O ln ',185, and O in ,072, and the
respective lengths 56 in> ,2, 58 in ,8, 56^,4, and 56^,4. The values
he finds for n are 2,29, 2,93, 4,08, and 7,53. Excepting the
first, these are much in excess of the value 2, and by a larger
quantity as the diameter of the rod is less. As the limited
dimensions of the apparatus would not be likely to produce
such effects, it seems that the excesses are to be attributed
to friction, or, rather, the dragging of the air by the rod in
consequence of capillary attraction. With respect to the
fourth rod Baily states that it was the finest steel wire he
could operate with, and that the vibrations of a pendulum of
this kind soon come to an end. If we suppose the quantity
of adhering air to be proportional to the surface of the rod,
the accelerative force of the retardation from this cause will
vary inversely as its radius. In fact, if we subtract 2 from
each of the above values of n, the remainders multiplied by
the respective diameters of the rods give the products 0,435,
0,381, 0,385, 0,398, which are so nearly equal as to afford
presumptive evidence of the reality of the cause assigned for
the excess of the experimental value of n above 2, and of the
exactness of the law it was supposed to follow.
Upon the whole the preceding comparisons of results of
HYDRODYNAMICS. 279
the theory with experimental facts may be regarded as satis
factory, the apparent differences between them having been
shewn to admit of explanations on admissible suppositions.
The next problem, which, relatively to the application pro
posed to be made of these researches, is of much importance,
is treated on the same principles.
Example VI. A given series of planewaves is incident
on a given smooth sphere at rest : it is required to find the
motion and condensation of the fluid at any point.
Since the motion, as in the case of the vibrating sphere
(Example IV.), is symmetrical about an axis, the equation
(35) in page 260 is again applicable. But the arbitrary con
ditions in the present problem require to be satisfied in a
different manner. I have found, in fact, that the equation
derived from (35) by differentiating it with respect to is
proper for this purpose, as will appear in the sequel of the
reasoning. The equation thus obtained, putting P for
da .
r d0>*
d z P d*P 1 d*P dP
By assuming that P= fa sin 6 + < 2 sin 6 cos 0, and that fa and
fa are each functions of r and t, it will be found that the
equation is satisfied if those functions be determined by inte
grating the equations
the former of which has already occurred in the solution of
Example IV.
It will be supposed that the incident waves are defined by
the equations
V = V = m sin ~ (a't + r cos + c )*,
Ai
* It should be observed that, excepting for the primary vibrations, the coeffici
ents designated as m, m, &c., have arbitrary values.
280 THE PRINCIPLES OF APPLIED CALCULATION.
the direction of incidence being contrary to that for which
6 = 0. As in the applications proposed to be made of these
researches, the sphere will always be extremely small, it will
be assumed that, while the distance r x from the centre of the
sphere within which its reaction on the fluid is of sensible
magnitude is very large compared to b the radius of the
sphere, it is very small compared to X the breadth of the
incident waves ; so that ~ x , or  is a small quantity
of the second order. Hence, since on that supposition the
values of r may be limited to those for which gr is very small,
it is allowable to expand the above sine in terms proceeding
according to the powers of r. We shall thus have to terms
of the second order,
V' = a'ff" = m sin q (at f c ) + mqr cos 0cos q (at + C Q )
gV cos 2 6 sin q (at + c ).
The conditions which the particular solution of the equa
tion (36) is required to fulfil are, (1) that these approximate
equations be satisfied where r is very large compared to b
and very small compared to X ; (2) that Z7= where r b, that
is, at the surface of the sphere. Since the equation (36) is
verified by supposing P to be either <f> t sin 6, or (f> 2 sin cos 0,
or the sum of these two quantities, let us first suppose that
P=(j> l sin 6. Then regard being had to the integral of the
equation (37), the following results are obtained :
cos 6,
W
E
a
>/i bein g P ut respectively for f(rdt], , and jfdr,
HYDRODYNAMICS. 281
7 77f r>
and F,F\F l for F(r + a't), , , and \Fdr. Since from
the conditions of the problem no part of <r can be a function
of r without 0, the arbitrary quantity ty (r, i) cannot contain
r, but must be a function of t only. To determine this
function let 6 . Then for all the corresponding values
of r, we shall have a = ty (t} = ar t suppose. But for the large
values of r corresponding to = ^ , ^ and a are identical.
Hence
m . r , .
0*!= sm (a t + c ).
As the forms of the functions / and F depend entirely on
that of the function which expresses the law of the velocity
and condensation of the incident waves r it will be assumed
that
f=m i sin q (r at + cj , F m z sin q (r + at + c 2 ) .
Both functions must now be retained, because, in consequence
of the incidence of the waves, there is propagation towards,
as well as from, the centre of the sphere. By the condition
that U= where r = b, we have
J
I? b
After substituting in this equation the above values of /and
F, and putting b for r, it will be found that the equation
is satisfied for all values of * if m z = m lf and c 2 = c^; and if
the arbitrary constant c t be determined by the equation
.
Also if we take another set of values of /and F y distinguish
ing them from the preceding by dashes attached to the con
stants, the same* equation will be satisfied if w 2 ' = m t ' and
c a ' = c/, and if c/ be determined by the equation
cot 6+= .
282 THE PRINCIPLES OP APPLIED CALCULATION.
As these two methods of satisfying the condition U0 are
equally entitled to consideration, both must be employed in
deducing the value of cr. Here it may be remarked that on
account of the linear form of the differential equation from
which <k is obtained, we might have 2/ and ^F in place of
jfand F. This being the case, it is allowable to substitute in
the expression for cr ^ the respective sums of the two
values of f and F. When this has been done, and the
relations between the constants are taken into account, the
result is
!2m 1 , N 2m t <7 . , J . , _
~ cos q (r + cj H  ** sin q (r + cj > sin qa t cos 9
 5 sin q (r + cj  cos q (r + c/) [ cos qat cos 0.
At the same time the foregoing equations for finding c x and
c/ give very approximately
cos qc^ r , sin qc^\ cos qc^ = , sm qc^ = *  .
By having regard to these values of c t and c/, expanding the
sines and cosines according to the powers of qr t and omitting
insignificant terms, the above equation becomes
<r <7 1 = (  h ^2 ) (w x sin qat + m^ cos qat} cos 0.
\ O oT /
When r is very large compared to b, the second term within
the first brackets may be omitted, and the consequent value
of <r a l must then satisfy the condition of being identical
with the term containing cos 6 in the expansion of a <r t .
Hence
f" ( m * 8 ^ n < ^ i + m * COS 2 a '^ == ~ TCOS ^ ( a ' t + c o)>
3m' . , 3m'
Substituting these values of m t and m/, we have for the con
densation at any point whose coordinates are r and 0,
HYDRODYNAMICS. 283
! + ^V (qr + pj cos q (at + c ) cos 0.
The first term within the brackets is due to the incident
waves, as may be seen by putting b = 0. The other term
expresses the law of the variation of the condensation pro
duced by the reaction of the sphere. For the condensation at
any point of the surface of the sphere, the equation gives
3m' 7 , \ /
cr = cr t f TTT go cos q (a t + cj cos 0.
2a
Also from the equation ,,,. + y = 0, we find for the
out/ dt
velocity along the surface,
Sm'
1^=, sin q (at + c ) sin 0.
With respect to these values of cr and Wit may be remarked,
that from them the values applicable to the case of a small
sphere oscillating in fluid at rest may be obtained as follows.
Let the incident vibrations of the fluid be counteracted by
impressing equal and opposite vibrations, and let the same
vibrations be impressed on the sphere. Then the fluid is
reduced to rest, excepting so far as it is agitated by the
oscillations of the sphere. But by these impressed velocities
W is diminished by m' sin q (a't + c ) sin 6, and or is diminished
by the amount of condensation due to the state of vibration
of the incident waves ; that is, by c^ +  b cos 6. After
subtracting these quantities the remaining values of cr and
W are those which were obtained in the solution of
Example IV.
The derivation of the general approximate values of U
and W from the equations
a*d<r dU_ a'*da dW _
' ~~ h ~ =
dr dt
gives
7=  m f sin q (a't + c ) (l  ^) cos 0,
284 THE PRINCIPLES OF APPLIED CALCULATION.
= m sin q (at + c ) (l + i ) sin 6.
It is to be noticed that these velocities are the same that
would be obtained on the supposition that the fluid is incom
pressible and that the whole mass is moved with the velocity
m sin q (at + c ). The parts of U and W which vary in
versely as r* give the law of the movement according to
which the fluid fills the space without change of density.
By reasoning analogous to that relative to the oscillating
sphere in page 265 it will be found that as much fluid passes
a plane through the centre of the sphere perpendicular to the
direction of incidence as would have passed the same plane if
the sphere had not been there.
We have now to trace the consequences of the particular
solution of the equation (36) which results from supposing
that P = </> 2 sin 6 cos 6. The integration of the equation (38)
by Euler's method gives
f+F f + F' f + F"
^75*T r + '
/ and F being any arbitrary functions respectively of r at
and r + at. Retaining both functions, the following results
are obtained by processes analogous to those applied to the
former value of P:
o = o _ (f+ F / + *" if"+*"'\ cos 2
W
'"} cos a
l^r
r* 3r 2 3r
As this integration is independent of the former one, it is not
necessary to suppose that f and F have the same values as
before. For this reason we may have U where r = b
without respect to the former value of U. Since from the
previous integration it may be presumed that two sets of
HYDRODYNAMICS. 285
values of / and F will be required to satisfy the given con
ditions, let us suppose that
f= m 3 sin q (r at + C 8 ) + m' 3 sin q (r at + c' 3 ),
F= m 4 sin q (r f at + c 4 ) + m\ sin ^ (r + a't + c' 4 ).
On substituting these functions in the above expression for U,
it will be found that the condition that U vanishes where
r = b is satisfied if m 4 = m 3 , c 4 = c 3 , w' 4 = w' 3 , and c' 4 = c' 8 ;
and if c 3 and c' 3 be determined by the equations
These equations give very approximately
By substituting in the foregoing expression for a the assumed
values of/ and F, and taking account of the relations between
the constants, the result will be
sn f+c +  ~  cos f+c m cos
3 ) + (3 ~ ? J cos q (f+c 8 ) [
' 8 ) ~ (^ ~ L) sin^ (r+c' 8 ) h
0.
After eliminating c 3 and c' 3 by the equations above, expanding
the sines and cosines of qr, and neglecting insignificant terms,
the equation is reduced to the following :
( ~= f ^ 3 ) (m 3 sin gat + m' s cos qa') cos 2
\4o loo/* /
Then, supposing r to be very much larger than 5, neglecting
in consequence the second term within the first brackets, and
equating the resulting value of a ^ to the term of <r' ^
which contains cos 2 0, we obtain
286 THE PRINCIPLES OF APPLIED CALCULATION.
, sin q (at + c ) = j (m 3 sin qa't + m' 3 cos qa't},
Hence, substituting these values of m s and m' 3 in the fore
going equation, we have for calculating the value of <r at any
point,
sin 2 ('* + c o)
where the first term in the brackets is evidently due to the
incident waves. The condensation at any point of the sur
face, obtained by putting b for r, is
Sm'cpb* . ff N 2/1
i  c 7 " sm 2 ( a t + c o) cos ^ >
and the velocity along the surface deduced from this value
of <r is
^qb cos q (at + c ) sin 6 cos 6.
Adding the results of the two integrations, and using now cr
and W to represent the total condensation and velocity at
any point of the surface, we have
cr oj = , qb cos q (a't+ c ) cos 6 , b* sin q (a't+ c ) cos 2 6,
W, =  sin 2 (a'< + c ) sin ^ + ^5 cos q (a't + c ) sin 2(9.
^5 b
The parts of <T O <7 t and W due exclusively to the incident
waves are respectively
qb cos 6 cos <? (at + c )   , q*b* cos 2 ^ sin q (a't + c ),
/
?7i' sin ^ sin g (a't + C ) + #& sin 2^ cos g' (a't + c ).
The resultant of the pressures at all points of the surface,
estimated in the direction of the incidence of the waves, is
sin cos ^,
HYDRODYNAMICS. 287
taken from 6 = to 6 = TT. Between these limits the integral
relative to the term containing cos 2 is evidently zero, and
the resultant pressure is therefore
27rma'qb s cos q (at + c ).
Hence, supposing the ratio of the density of the sphere to
that of the fluid to be A, the accelerative action of the fluid
on the sphere is
Sm'a . , .
which, it may be remarked, is independent of the magnitude
of the sphere.
The general approximate values of U and TF, as deduced
from that obtained above for cr, are as follows :
/ js\
Z7=  mq Ir  8  cos 2 6 cos q (at + c ),
W= m'q (r + , J sin 6 cos 6 cos q (at + C ) .
Since this additional term in the complete value of W
contains the factor sin 6 cos 6, it is always zero where 6 =  .
2
Hence the quantity of fluid which passes the plane through
the centre of the sphere perpendicular to the direction of in
cidence is unaffected by this part of TF", remaining, as before,
the same as if the sphere were removed. .
The pressure on the surface of the sphere represented by
the additional term in the general value of a cr^ being the
same at the same time at any point of the hemispherical sur
face on which the waves are incident as at the corresponding
point of the opposite surface, tends to produce no motion of
the sphere. The other part of the pressure, represented by
the first term, is equal with opposite signs at corresponding
points of the two hemispherical surfaces, and as at each point
it varies as cos q (a't + c ), it follows that .this part tends to
cause vibrations of the sphere, but no permanent motion of
translation.
288 THE PRINCIPLES OF APPLIED CALCULATION.
I now enter upon considerations which are supplementary
to those that have thus far been applied to the discussion of
this problem, and which appear to be necessary for its com
plete solution. We have seen that the condensation of the
fluid in contact with the surface of the sphere is partly due
to the condensation of the incident waves and partly to the
reaction of the sphere, and that, considered apart from its
phase, the quantity of the condensation is the same on each
side of the plane through the centre of the sphere perpen
dicular to the direction of incidence. But as far as regards
the part of the condensation which is not due to the reaction
of the sphere, there are reasons for concluding that the grada
tions of condensation from point to point of the surface are
not, as results from the foregoing reasoning, the same that
they would have been at the same points of space if the waves
had not been interrupted by the presence of the sphere. In
order to give these reasons it will be necessary to enter into
certain considerations relative to the lateral action of com
posite vibrations, which have been hitherto reserved, because
they will now receive their most important application.
It has been shewn (Prop. XI.) that the equations which
define the spontaneous simple vibrations relative to an axis
are, to the first approximation,
the factor / being given by the solution of the equation ob
tained in p. 211, viz.,
The exact integral of this equation can be expressed only
by the known series
4>*r* f> 3 r*
f 1 /> r 2 4 4 frr
jier +  2 + <xc.
HYDRODYNAMICS. 289
But for large values of r it may be put very approximately
under the finite form*
/= (47iT Ve)~^ cos ( 2 Ve r ^ j ,
this equation being, in fact, the exact integral of the equation
which evidently approximates to identity with the foregoing
equation in proportion as r is larger. From the above
expressions for the condensation and transverse velocity,
namely, , $ and 6 :, it may be shewn, by taking
a at ar
flf
account of the roots of the equations f and ~ = 0, that
there are positions of no condensation and of maximum con
densation, and like positions of maximum transverse velocity
and of no transverse velocity, in fixed cylindrical surfaces
about the axis, and that the number of such surfaces is un
limited. The maxima both of condensation and of transverse
velocity diminish at first very rapidly with the distance from
the axis, and afterwards more slowly, tending continually to
vanish as r increases ; and the intervals between their con
secutive positions, as also the intervals between the con
secutive positions of no condensation and no transverse velo
7T
city, go on decreasing till they reach the limiting value p ,
2 ve
\
2 /c
In treating of planewaves (page 244) regarded as com
posed of an unlimited number of simple vibrations having
their axes all parallel, and being in the same phase of vibra
tion, so that the transverse motions are neutralized, the plane
front was supposed to be of indefinite extent in order to avoid
* See a Paper by Professor Stokes in the Transactions of the Cambridge Philo
sophical Society (Vol. ix., Part i., p. 182).
19
THE PRINCIPLES OF APPLIED CALCULATION.
the consideration of the transverse motion and transverse
variation of condensation which would necessarily exist near
the borders of a limited planefront. But the argument now
requires that these circumstances should, as far as may be
practicable, be taken into consideration. Let us suppose that
the parallel axes of the component motions are included
within a limited space, for instance, a cylinder of given
radius, and consider what must be the kind of motion which
prevails under such circumstances at and near the boundary.
It is evident that since at these parts the transverse motion
is only partially destroyed, the total motion is there com
pounded of transverse and longitudinal vibrations. This
motion, however, does not spread laterally to an indefinite
extent, but is always confined within certain limits, as may
be inferred from the following considerations. The breadth
of the waves being supposed to be very minute, and the
velocities of the fluid particles to be extremely small com
pared to the velocity of propagation, by taking account of
the characteristics of the component vibrations above de
scribed, it will be seen that although the vibrations relative
to each axis are individually not limited as to distance from
the axis, a limit to the compound motion is imposed laterally
ty the composition of the vibrations. Admitting that the
number of the axes of the components within a given space,
the dimensions of which must be very large compared to X,
may be as great as we please, since the vibrations are by
hypothesis all in the same phase it will follow that beyond a
certain finite distance from the cylindrical surface, the sum of
the positive condensations at each point may be as nearly as
we please equal to the sum of the negative condensations.
Thus the resultant condensation will vanish, and there will
be neither transverse nor longitudinal motion. That distance
will be so much the less as the rapidity with which the mag
nitudes of the successive maxima of condensation and trans
verse velocity diminish is greater ; but in any case it must be
a large multiple of X, and cannot, therefore, be small unless X
HYDRODYNAMICS. 291
be extremely small. The magnitudes of the compound lon
gitudinal vibrations increase from zero at the limiting distance
till they acquire a maximum and uniform value at a certain
limit within the cylindrical surface ; and the transverse vibra
tions, increasing from zero at the exterior limit till they reach
a maximum near the cylindrical surface, afterwards diminish
till they disappear at the interior limit by the counteraction
of opposite vibrations. The thickness of the cylindrical shell
which within its interior and exterior surfaces includes the
whole of the transverse motion, will be less as the breadth
of the waves is less ; and if the waves be of extremely small
breadth, it is conceivable that that space, together with the
interior cylindrical space occupied by the motions which are
exclusively longitudinal, may be such as to make up a cy
linder of comparatively very small radius. Thus the motion
included within such a cylinder would be propagated to an
unlimited distance without lateral divergence.
It is supposed above that the condensation and transverse
motion of each set of simple vibrations are functions of the
distance from the axis, or that the component vibrations are
primary spontaneous vibrations. But in page 216 it is
shewn that each such set may be resolved into two sets in
which the vibrations are parallel to two planes at right angles
to each other. This resolution, however, can take place only
within distances from the axis which are extremely small
compared to X ; and the same is the case with respect to any
farther resolution that the resolved vibrations may undergo.
At other distances the laws of the motion and condensation
may be the same for resolved as for primary vibrations.
Moreover, motion compounded of an indefinite number of
resolved vibrations in the same phase of vibration and having
their axes all parallel, might still be such that the transverse
motion would be neutralized. For these reasons the above
conclusions respecting the nondivergence laterally of vibra
tions compounded of the primary vibrations may be extended
to those compounded of resolved vibrations.
192
292 THE PRINCIPLES OF APPLIED CALCULATION.
If within the same cylindrical space there were included
an unlimited number of parallel axes belonging to another
set of vibrations all having the same values of m, \, and c,
but values different from those of the first set, the resultants
of the two sets might, by the law of the coexistence of small
oscillations, exist simultaneously and be independent of each
other. In the same manner might any number of other sets
be added without necessarily increasing the extent of the
lateral divergence. And if at the same time the number of
axes in any given small space be at disposal, the transverse
velocity and condensation, as well as the longitudinal, might
be such as to satisfy arbitrary conditions. These considera
tions have reference to the proper method of determining the
laws and extent of lateral divergence of vibrations under
given circumstances; as, for instance, when a wavefront is
cut off abruptly in the lateral direction. But I do not profess
to have succeeded in discovering the principles appropriate to
the solution of the problem of the lateral divergence of waves,
the exact mathematical treatment of which is attended with
peculiar difficulties, which I must leave to be overcome by
future investigators*. My present object has simply been to
shew how limited lateral divergence may be a consequence of
the general law of the composition of vibrations, and having
done this, I have now only to explain in what manner this
view bears upon the problem under discussion.
The course of reasoning completed in page 287, led to the
conclusion that the accelerative action of a series of waves
incident on a sphere at rest is equal to
3m (t

This is the same expression as that which would be obtained
on the supposition that the fluid is incompressible, and that
the whole mass is moving with the velocity m sm# (a't + c ).
There is nothing, as far as I can perceive, in these hydrodynamical re
searches opposed to the method of calculating lateral divergence usually adopted
in explaining phenomena of Diffraction in the Undulatory Theory of Light.
HYDRODYNAMICS. 293
3 dV
If we call this velocity F, the expression becomes ^ ^ ,
which does not involve explicitly the elasticity of the fluid.
At this point of the reasoning an explanation is required in
order to remove an apparent discrepancy between the treat
ment of the present Example and that of Example II. In
the solution of the latter the principle is asserted (in page 252)
that the values of V and a given by the integrations, since
they contain arbitrary functions, are immediately applicable
only to the parts of the fluid arbitrarily disturbed, the velocity
and condensation at other parts being inferred from the laws
of propagation. But the above expression for the accelerative
action of waves on the sphere was obtained by supposing the
values of <7, F, and FT given by integration to be applicable
at any distance r from its centre. It is, however, to be con
sidered that if Example II. were solved on the same suppo
sitions as the present Example, namely, that the radius b of
b r
the sphere is so small compared to X that the ratios  and
are each extremely small where the motion is appreciable, the
results obtained would be equally applicable for any value of
r. For, in fact, on these suppositions the solution in each
case involves no property of a compressible fluid by which it
is distinguished from one that is incompressible.
But when the solution depends essentially on distinctive
properties of a compressible fluid, such as rate of propagation
and composition of vibrations., the above mentioned principle
must be applied, and those properties have to be taken into
account in determining the values of the velocity and con
densation at points where the disturbance is not immediately
impressed. JN T ow from the foregoing considerations respecting
the dependence of limited lateral divergence on the composite
character of the vibrations, we may infer that the law and
amount of lateral divergence have an effect in the present in
stance on the condensation and velocity at such points. In
consequence of the composition of the vibrations, as soon as
294 THE PRINCIPLES OF APPLIED CALCULATION.
they are propagated beyond the first hemisphere and direct
incidence ceases, the transverse action comes into play, being
no longer wholly neutralized, and the condensation is con
sequently modified in obedience to the law of this lateral
action. In the extreme case of vibrations so rapid that the
value of X is small compared to the radius of the sphere, the
limited lateral extent of the transverse action might cause
the waves to be of inappreciable magnitude beyond a limited
distance along the second hemispherical surface, so that the
fluid in contact with the remaining part would be at rest.
Although this case is far from being that of the present pro
blem, in which b has been assumed to be extremely small
compared to X, it may yet serve to indicate that in any case
the induced transverse vibrations will have the effect of modi
fying the condition, as to velocity and density, of the fluid
surrounding the sphere. The calculation of the exact amount
of this influence should be within the reach of analysis ; but
since, as before stated, the law of limited lateral divergence
has not yet been ascertained, we are not prepared to enter
upon an investigation for obtaining expressions for the velo
city and condensation applicable to points at any distance
from the sphere. It is, nevertheless, possible to arrive at
certain definite results relative to the condensation of the
fluid contiguous to the sphere, and the pressure by which the
sphere is solicited; as I propose to shew by the next argu
ment.
The state of the fluid contiguous to the spherical surface
is required to fulfil the following conditions : (1) being sym
metrical with respect to an axis it must be such as to satisfy
the equation (36) and the two equations (34) ; (2) the motion
being along the surface, y = where r = b. These condi
tions may be fulfilled in a unique manner by means of the
integral of the equation (36) which involves the function ^ ,
this function being now supposed to be limited in application
to the parts of the fluid contiguous to the spherical surface,
HYDRODYNAMICS. 295
but under that limitation to embrace the effect of transverse
action. That integral, as obtained in page 282, is
+ 2_ j ( TOI s i n qjt + Wi ' C os qat] cos 0,
in which b is to be put for r. According to this argument
the transverse action does not alter the law of the superficial
condensation a o~ i , so far as it is a function of 0, but alters
its amount. Hence the arbitrary quantities m l and w x ', which
cannot now be determined by supposing r to be very large,
are to be taken so that <r cr l shall have a constant ratio to
the value previously obtained. From this reasoning it follows
that
" ~
^) m ' cos 2 ( a ' t + c o) cos 0>
ii
1 h being an unknown constant factor depending on the
transverse action, the part h vanishing if the fluid be incom
pressible.
Precisely the same reasoning is applicable to that integral
of the equation (36) which involves the function 2 ; so that
from the result obtained in page 286 we may infer that when
transverse action is included, the superficial pressure indicated
by this integration is
5# 2 5 2 A , , . / , N 9/ .
 ^ am sm q (a t f c ) cos 0,
h' being an unknown constant factor, depending, as well as
1  h, on the transverse action. The corresponding velocity
along the surface is
^ m cos q (at + cj sin cos 6.
o
Now this integration is independent of the previous one
obtained by supposing that P = (^ sin 0, inasmuch as it only
satisfies the equation (36), whereas the first integration
satisfies (35) as well as (36). Hence the circumstances which
determine h' may be assumed to be different from those which
determine 1 h. Since the superficial velocity and conden
296 THE PRINCIPLES OF APPLIED CALCULATION.
sation given "by the seeond integration both vanish where
7T
= , it might "be allowable to suppose that the factor ti
applies exclusively to the transverse action relative to the
second hemispherical surface, and that there is no correspond
ing transverse action relative to the opposite surface. Until
a more complete investigation shall have determined whether
or not this be the case, we may, at least, assume that that
factor is not the same for the two hemispherical surfaces.
Taking, therefore, h' to represent its value for the first
surface, and h" that for the other, the pressure on the sphere
due to the condensations on both surfaces, and estimated in
the direction of incidence, will be found to be
57rq*b 4 a ,,, ,. , . / V, \
~ (h h ) m sin q (a t f c ).
12 v
Adding to this the resultant pressure deduced from the first
integration, namely,
27rb 3 qa (\h}m cos q(dt\ c ),
and dividing the sum by the mass of the sphere, the total
accelerative action of the fluid on the sphere is
(1  h) m' cos q (a't + c ) + (h 1  h") m'smq (dt + c ).
This result is necessary for effecting the solution of the next
Example.
Example VII. A given sphere is free to obey the im
pulses of the vibrations of an elastic fluid r it is required to
determine its motion.
I first called the attention of mathematicians to this pro
blem at the end of an Article in the Philosophical Magazine
for December 1840, and after a long series of investigations
relative to the principles of Hydrodynamics, I attempted the
solution of it in the Number of the Philosophical Magazine
for November 1859. I consider it to be a problem of special
interest on account of the physical applications it may pos
HYDRODYNAMICS. 297
sibly be capable of; but in respect to its mathematical treat
ment it presents great difficulties, which I do not profess to
have wholly overcome. The solution here proposed follows
as a Corollary from the foregoing expression for the accele
rative action of the vibrations of an elastic fluid incident on a
sphere at rest.
To make that expression applicable to the present Ex
ample, I adopt the principle that the action of the fluid on the
sphere in motion is the same as that of waves, the motion in
which is equal to the excess of the motion of the fluid above
that of the sphere. Let x be the distance of the centre of the
sphere at the time t from an arbitrary origin, and be reckoned
positive in the direction of incidence, and let the excess of the
velocity of the fluid at that distance above the velocity of the
sphere be
, dx
dt
According to the above principle this quantity holds the place
of m' sin q (at \ c ) in the former Example. The centre of
the sphere being supposed to perform small oscillations about
a mean position, if for x within the brackets we substitute its
mean value, or put for x f c the constant (7, only quantities
of the second and higher orders will be neglected. And since
the motion of the sphere is, by hypothesis, wholly vibratory,
and the vibrations are due to the action of the fluid, it follows
that 7 is a circular function having the same period as that
of the incident waves. We may, therefore, assume that
m sin q (at + c ) = m sin q (at + (7) 4 .
Hence, by differentiation,
mqa cos q (at + c ) = mqa cos q (at f C) y^ .
Now since ^ is here the acceleration of the sphere due to waves
298 THE PRINCIPLES OF APPLIED CALCULATION.
the relative velocity in which is expressed by m sin^
we may substitute for it the foregoing amount of accelerative
action of such waves on the fixed sphere, and the equation
must then be identically satisfied. These operations lead to
the following equations, qco being an auxiliary arc :
5ql (Ji  h") n , m 2A cos go)
After substituting the values of m and c given by these
equations in the lefthand side of the foregoing equation, and
neglecting terms involving the square of qco, which are of the
72
order of 2 , it will be found that
A.
d'x Sqa'(lh)
IF  3l
If, therefore, V= a S msin q (a't+ (7), V being the velocity
and S the condensation of the incident waves, and if H and K
represent numerical coefficients the values of which are known
if A and h be given, we have finally
= H(l  h) + Kfb (K A' V&
The acceleration of the sphere has thus been determined so far
as it depends on the terms of the first order in the values of
the velocity and condensation of the incident waves ; and it
will be seen that the above value of it is wholly periodic,
having just as much negative as positive value. Hence it
follows that the action of the fluid, as deduced from terms of
the first order, causes vibrations of the sphere, but no motion
of translation.
From this first approximation we might proceed to include
terms containing m 2 . But since these terms are of very small
magnitude compared to those which have been considered, we
may dispense with going through the details of the second
approximation by making use of a general analytical formula,
according to which if f(Q) be a first approximation to an
HYDRODYNAMICS. 299
unknown function of a variable quantity Q, the second ap
proximation is f(Q) +f (Q) BQ. By applying this formula
d*x
to the above expression for ^ , we have to the second ap
proximation, 1 h and h' h" being assumed to be constant,
It is next required to ascertain the values of the increments
.
at
It has been proved (p. 246) that for planewaves to the
second approximation
F F 2
S = ,+^ 2 .
a a
a'*dS ,dV jr dV
Hence, = = a r + 2 F j .
ax ax ax
But from the reasoning under Prop. XVII., combined with
that in p. 246, it may be inferred that for planewaves to the
second approximation V=f(x a'i), the propagation being
supposed to be in the positive direction. Hence
f =/(*_'<) __ L, **.
dx J v a dt
Consequently
__^d8_dVf 2F\
dx '" dt ( a )
a' 2 dS dVf, V\
and  f G = = 1 + nearly.
(l+S)ax dt \ a J
The lefthand side of the last equation is the effective accelera
tion of an elementary portion of the fluid of density !+>, the
constant a' 2 taking the place of a 2 because of the composition
d*x dV
of the motion. Now in the foregoing value of =^ , 8 . = is
dt at
dV
the increment of j~ for planewaves, consequent upon includ
300 THE PRINCIPLES OF APPLIED CALCULATION.
ing terms of the second order. And the above result proves
that in that case the accelerative force of an element of the
fluid is expressed to terms of the second order by adding
VdV
y to the expression of the first order. Hence
a ^
* dt a dt '
The increment SS of the condensation is that due to terms
of the second order for planewaves. Hence its composition
and value may be inferred from results obtained by the dis
cussions given under Prop. XVII. It is there shewn (pages
237 and 238) that in composite motion relative to a single
axis the condensation due to terms of the second order is partly
expressed by periodic terms having as much positive as nega
tive value, and partly by terms which do not change sign.
It is also proved that when there are any number of different
sets of vibrations relative either to the same axis, or to dif
ferent axes, the condensations expressed by the latter terms
may coexist ; so that the resultant of these condensations is
the sum of the separate condensations. Hence in the case
before us of plane waves assumed to result from the com
position of different sets of vibrations having parallel axes, the
value of &S consists partly of periodic terms, and partly of
terms which do not change sign, which, in fact, as appears
from the expression obtained in p. 239, are always positive.
dV
After this discussion of the values of 8 . j arid BS, we may
proceed to infer the motion of the sphere from the foregoing
. f d?x
expression for ^ T .
First, it is to be remarked that the two terms of which
that expression consists may be treated independently of each
other, inasmuch as the first term is derivable either from the
equation (35) or from (36), whereas the other can be obtained
only by means of the latter equation. Also the first term is
HYDRODYNAMICS. 301
independent of the magnitude of the sphere, whilst the other
contains the factor &, being of the order of the first multiplied
by  . Hence in case X were very large, we might have an
X
accelerative force of sensible amount expressed by the first
term, whilst that expressed by the second would be wholly
inappreciable. In short, the second part of the accelerative
force is especially applicable in cases for whicli X is so small
that the variation of condensation of the waves at a given time
in a linear space equal to the diameter of the sphere may be
considerable even when m is not large ; whereas the first
part is effective, if m be not very small, when X is so large
that the variation of the condensation of the waves in the
same space is extremely small, and the excursions of the fluid
particles are comparable with, or even exceed, the sphere's
diameter. For these reasons we may consider separately
the effect of the accelerative force expressed by the first term.
d 2 x*
Calling this force ~ , and substituting the value of
Cut
* dV T,
~~ ' WC 6
But since
fdV\dV
~" dx~ dt
dV\dV dVdV, _V\
L a')'
we have
V
and consequently by substitution in the foregoing equation,
Assuming that x has the mean value T O , it is supposed that
x  x = (*j  * ) + (x u  ar ),
and consequently that
d*x d?x\ (Pa*
5?" dP + dt* '
302 THE PRINCIPLES OF APPLIED CALCULATION.
Before applying this equation in the case of the incidence
of waves on the sphere, it will be proper to consider that of
the incidence of streams. Since the motion of the fluid in
a stream may be regarded as a case of vibratory motion for
which X, the breadth of the waves, is infinite, while m remains
finite, we may suppose this case to be embraced by the above
equation. And again, if the motion be in a uniformly acce
lerated stream, it may be regarded as a part of a vibration
for which X and m are as large as we please, and may for
this reason be included in the same equation. Let us, there
fore, suppose (1) that Fis constant. Then the equation shews
d z x
that y2 1 = 0, and that the velocity of the sphere is conse
quently uniform. Hence the distribution of condensation on
the hemispherical surface upon which the stream is incident,
as indicated by terms to the second order, must be similar
and equal to that on the other hemispherical surface*. Under
these circumstances we have also 1 h = 0. Consequently
the state of uniform motion, or of rest, of a sphere is not
altered by the action upon it of a uniform stream. And con
versely a sphere may move without suffering retardation, and
therefore move uniformly, in an elastic fluid "at rest. This
might also be inferred from the fact that when the motion of
the sphere is uniform the motion of the fluid is constantly the
same at points which have successively the same position
relative to the centre of the sphere, so that there is neither
loss nor gain of momentum.
Suppose (2) that Fis uniformly accelerated. Then f )
d*x
is constant ; and the equation (A) shews that ^ , the acce
di
leration of the sphere, is also constant if we omit the term of
the second order. This may be done in the case of a slowly
accelerated stream, to which the. result of this reasoning is
* See another method of obtaining this result in the Philosophical Maga
zine for November, 1859, p. 323.
HYDRODYNAMICS. 303
subsequently applied ; in which case also, the factor 1 h,
although it does not vanish, becomes extremely small. Thus
the effect of a stream uniformly but slowly accelerated is to
produce an acceleration of the sphere very nearly uniform ;
and conversely a sphere caused by any extraneous action to
move with a uniform but slow acceleration in the fluid at rest
is by the fluid uniformly retarded.
I proceed now to apply the equation (A) to determine the
motions of the sphere which are produced by the action of
waves. As that equation contains the complete differential
coefficient (T) , it admits of being immediately integrated,
giving by the integration
dt
f*f\rtcs4ar\'t ovvkvoaoTn o tli A train A f\t
dx
C is an arbitrary constant expressing the value of when
F 2
7=0. The factor F+ is F(l + 8) nearly, and by (28)
in p. 246, is equal to a (S + SS), if S represent the conden
sation to the first order of small terms, and SS the additional
condensation expressed by terms of the second order.
It may be here remarked that the quantity F(l + S) is
at each instant proportional to the momentum of a given
breadth, Ace, of the fluid (supposing the waves undisturbed)
at the position where the centre of the sphere is situated, and
that the above equation shews that the variable part of the
momentum of the sphere is always proportional to that part
of the momentum of the fluid. In the case of the first ap
proximation the momentum of the corresponding portion of
the fluid is proportional to V x 1. Hence the second ap
proximation is obtained by substituting for the latter mo
mentum of the first order that which is exact to quantities
of the second order. This process, as being antecedently
304 tHE PRINCIPLES OF APPLIED CALCULATION.
reasonable, tends to confirm the argument by which the
dx
second approximation to the value of r 1 was arrived at.
From what has been proved in pages 236 and 246 respect
ing the composition of vibrations to terms of the second order,
we may assume for the case in which the components have
all the same value of X, that
V= m sin q (at x + c) + Am 2 sin 2q (at x + c'),
A being a certain constant. In the present application of
this value of F, x is the coordinate (x^) of the centre of the
vibrating sphere at the time t. Consequently, leaving out of
account at present any nonperiodic motion the sphere may
have, x v will differ from a constant value by small periodic
quantities of the first order the values of which are known
by the first approximation. "Hence it will be found that V
may be thus expressed :
V= m sin q (at +0)4 Am 2 sin 2q (at + C'},
A, C, and C' being new constants. By means of this value
F 2
of Fwe have for that of V\ , ,
2
m sin q (at + C) + AW sin 2q (at + c) + ~ sin 2 q (at + (7),
which may evidently be put under the form
2
m sin q (at + C) + ^, + AW sin 2q (at + C").
Consequently by, substitution in the value of ^ ,
jjfc = C + H (1  h) f + periodic terms.
It thus appears that in addition to the arbitrary velo
city <7 , and the vibratory motion expressed by the periodic
terms, the sphere has the velocity H(lh) ~, due to the
HYDRODYNAMICS. 305
immediate action of the incident waves. This result proves
that the action of the waves has the effect of producing a
permanent motion of translation of the sphere, and that this
motion is in the direction of the incidence of the waves, or
the contrary direction, according as h is less or greater than
unity.
The following reasoning will, I think, shew that the
sphere actually receives, not a uniform, but an accelerated
motion of translation. First, it is to be observed that in the
preceding reasoning we assumed that the centre of the sphere
oscillates about a mean position without permanent motion of
translation ; whereas, according to the above result, the oscil
lations accompany a motion of translation expressed by
In order, therefore, to satisfy the assumed condition, it is
necessary to impress this motion both on the sphere and on
the fluid in the opposite direction. The motion of the sphere
will thus become wholly vibratory, and we shall have the
case of a uniform stream incident upon it, in addition to the
action of the waves. By the foregoing argument (p. 302)
relative to case (1), the state of rest, or uniform motion of the
sphere, will not be affected by the incidence of this stream.
Thus the action of the waves will remain the same as before,
and will operate independently of the impressed uniform velo
city in communicating to the sphere a motion of translation,
inasmuch as the action of the condensed portions of the waves
will still be more effective than that of the rarefied portions.
Hence to maintain the above mentioned condition the non
periodic velocity must be impressed on the sphere, not at one
instant only, but at successive instants, and the fluid will
consequently have an accelerated motion relative to the oscil
lating sphere. Hence actually the sphere will have an accele
rated motion of translation in space. It is plain that the
acceleration will be uniform, since the series of waves is
20
306 THE PRINCIPLES OF APPLIED CALCULATION.
uniform, and their action will be the same at one epoch as at
m 2
another. From this reasoning it follows that H (1 h) , is
not a velocity communicated once for all to the sphere, but
is equal, or proportional, to the rate at which the nonperiodic
part of the sphere's velocity is increased.
By reference to the discussion in p. 303 of the case (2)
of a uniformly accelerated stream, it will be seen that while
the sphere is uniformly accelerated by the action of the waves,
it is uniformly retarded by the resistance of the fluid, so that
the acceleration on the whole is equal, or proportional, to
.ZJj and /i x being new constants analogous to H and h, and the
latter such that 1 A t is exceedingly small.
Dx
If =* represent at any time the nonperiodic part of
dx
~ , we have according to the above results
Cut
j being an unknown constant factor. By integration
so that T is the interval, or unit of time, during which the
velocity of translation of the sphere is increased by
dx Dx
Since the values of  and ^r 1 do not involve the dimen
at JJt
sions of the sphere, both the vibratory motion and the motion
of translation are the same under the same circumstances for
spheres of different magnitudes,
HYDRODYNAMICS. 307
The origin of the factors 1 h and 1 \ has already been
discussed in pages 293 295. I propose to add here some
considerations respecting the magnitude of h, and the circum
stances which determine its value to be greater or less than
unity. Suppose m and X for the incident vibrations to be
very large. Then since the transverse vibrations are brought
into action by the disturbance which the planewaves undergo
by incidence on the sphere, the motion of the fluid will par
take of the character of direct and transverse vibrations rela
tive to an axis, the axis in this case being the prolongation
of a straight line through the centre of the sphere in the
direction of propagation. But for motion of that kind it has
been shewn that the transverse vibrations have the effect of
increasing the condensation on the axis, compared with that
for the same velocity when the motion is in parallel lines, in
the ratio of 2 to 1. By similar transverse action the con
densation on the farther side of the sphere might be so in
creased as to exceed that on the nearer side ; in which case li
would be greater than unity, arid the motion of translation of
the sphere would be towards the origin of the waves. On the
contrary, for very small values of m and X the defect of con
densation on the farther side might be only partially supplied
by the lateral confluence, so that h would be less than unity,
and the translation of the sphere would be from the origin
of the waves. The conditions under which the two effects
respectively take place cannot be determined in the present
imperfect state of the mathematical theory of the lateral
action.
Corollary I. Since it was proved (p. 233) that the con
densations of the second order to which the permanent mo
tions of translation of the sphere are to be attributed, may
coexist when there are different sets of vibrations originating
at different positions in space, it follows that simultaneous
undulations from different sources may independently produce
motions of translation of the sphere.
202
308 THE PRINCIPLES OF APPLIED CALCULATION.
Corollary II. If the sphere be acted upon by spherical
waves, that is, waves the axes of the components of which all
pass through a fixed point, the mode of action on a very
small sphere will be the same as that of composite plane
waves. But the amount of action which causes motion of
translation will be different at different distances from the
central point, varying with the distance according to a law
which may be thus determined. We have seen that the ac
celerated motion of translation of the sphere varies as the
nonperiodic part of the condensation of the composite waves,
which part, according to the reasoning concluded in p. 233, is
equal to the sum of the nonperiodic parts of the primary
component waves. Now this sum is cceteris paribus pro
portional to the number of the components, and therefore to
the number of their axes included within a given transverse
area. But when the axes diverge from a centre the number
within a given area at a certain distance from the centre
varies inversely as the square of the distance. Consequently
the accelerative action of the waves varies according to the law
of the inverse square.
This law seems to be also deducible in the following
manner. It is shewn in p. 230 that when an unlimited
number (n) of sets of primary vibrations have a common axis
and the same value of X, and are in all possible phases, we
have for points on or contiguous to the axis, to the first ap
proximation,
2.0' = n^m sin q (z at + 0),
/c
m being the constant maximum velocity common to all the
primary vibrations. If we suppose the n different sets of
vibrations, instead of having a common axis, to have their
axes uniformly distributed within a small area, whether the
axes be parallel or diverge from a centre the vibrations will
still coexist, and the value of S . cr will remain the same,
because for points very near an axis / is very nearly equal to
HYDKO DYNAMICS. 309
unity. By the uniform distribution of the axes transverse
motion will be neutralized within the small area in which
they are included, so that the direct motion will be the same
as that in composite planewaves. Hence if W and S be the
resultant velocity and condensation we shall have
W= tcaS = KC& . a = K?ntm sin q (z  at + 6}.
Now from what has already been proved the acceleration of
the sphere by these composite waves varies as (/c 2 n^m)*, that
is, as n, because K and m are constant. Hence since in central
waves the number n of the axes in a small given area varies
inversely as the square of the distance, the accelerative action
of the waves varies according to the same law.
Corollary III. If from the same centre another set of
waves were propagated having a different value of X, their
acceleration of the sphere would be independent of that pro
duced by the first set, and would in like manner vary in
versely as the square of the distance. Hence the sum of the
two accelerations would vary according to the same law ; and
so, by consequence, would the sum of any number of different
sets.
We have now to discuss the second term of the expres
d 2 x .
sion for ^ in page 298. Before drawing inferences from this
dt
term, I propose, for the sake of illustrating the course of the
reasoning, to refer back to some of the previous steps. In
the case of waves incident on a fixed sphere, the centre of the
sphere was taken for the origin of the polar coordinates, and
the equations giving the velocity and condensation of the
waves to the first approximation were
V= a'S msiuq (at + r cos + c ).
It being assumed that in the space within which the disturb
ance of the waves by the sphere is of sensible magnitude qr
Is very small, instead of the above value of a'S the approx
310 THE PRINCIPLES OF APPLIED CALCULATION.
mate value
~
was employed. The first two terms indicate that the excess
of the condensation above the value m sin q(at + c ) is nearly
proportional, at any given instant, to the distance r cos 6
reckoned from the centre of the sphere along the axis of the
motion. That excess is, therefore, equal with opposite signs
at corresponding points on the opposite sides of the centre.
The integration of the equation (36) obtained by supposing
that P = fa sin 6 only takes account of the dynamical action
of a variation of the condensation, arid of the accompanying
pressure, according to this law. It was found that this
variation of the pressure tends to produce an acceleration of
the sphere having the same period as that of the acceleration
of any given element of the waves. If instead of being fixed,
the sphere were free to move, the same kind of acceleration
results from the relative motion of the sphere and the waves,
and the consequent vibrations of the sphere were found to be
synchronous with those of the fluid. It was then argued
(p. 295) that the effect of transverse action, (which is not in
cluded in this reasoning), is taken account of by multiplying
the acceleration resulting as above stated, by an unknown
constant factor 1 h. Lastly, it was shewn (p. 304) that on
including terms of the second order in the relation between
V and $, the vibrations of the sphere were accompanied by a
permanent motion of translation, positive or negative accord
ing to the sign of 1 h.
But the effect of the third term in the foregoing ap
proximate value of aS is ascertained by that integration of
the equation (36) which was obtained by supposing that
P=fa sin 6 cos 0. Now that term has equal values at cor
responding points on opposite sides of the plane passing
through the centre of the sphere (supposed fixed), and con
sequently cannot give rise to any tendency to either accele
ration or motion of the sphere. This, in fact, is the result
HYDRODYNAMICS. 311
obtained by the reasoning concluded in page 287. But when
the effect of transverse action due to the disturbed state of the
waves is considered, the equality of the pressures on the
opposite hemispherical surfaces no longer subsists. It ap
pears from the reasoning in page 295, that the effect of trans
verse action is taken into account by multiplying the pressure
on the first hemispherical surface by a constant factor k r ;
and the equal pressure on the second by another constant
factor h", the two factors depending on the unknown law of
lateral divergence. Hence the expression for the resulting
pressure has the factor ti * h"; and as this factor originates
equally with 1 h in the transverse action, it may be pre
sumed that the two factors change sign under the same cir
cumstances, and that we may consequently suppose h' h"
to be equal to h' (I h), h r being always positive. This
being understood we may proceed to discuss the inferences
that may be drawn from the second term in the value of
Tg obtained in page 299.
d 2 x
Calling this part of the accelerative force p , and put
ting h' (1  h} for h'  h", we have
,72
^= Kfbh' (1  h) a' 2 (8+ 88).
Since the condensation S to the first approximation is wholly
periodic, if we omit $S the acceleration of the sphere is also
periodic, and its motion may consequently be wholly vibra
tory; as, in fact, it was assumed to .be when the relative
velocity of the fluid and sphere was expressed (in p. 297) by
a periodic function. But, as has been already remarked
(p. 300), 88, representing the terms of the second order, con
sists in part of terms that are nonperiodic and constant.
Hence the above equation shews at once that by reason of
these terms the sphere is constantly accelerated. It is, how
ever, here to be taken into consideration, just as in the dis
312 THE PRINCIPLES OF APPLIED CALCULATION.
cussion of the expression for * , that the relative motion of
the fluid and sphere in this case takes the place of the absolute
motion of the fluid in the case of the fixed sphere, and is there
fore supposed to be wholly vibratory. To maintain this con
dition it is consequently necessary to impress on the sphere
and the whole of the fluid in the contrary direction this acce
leration of the sphere ; which it is legitimate to do, because,
as was argued in p. 305 with reference to the first acceleration,
the action of the waves on the sphere will not thereby be
sensibly altered. By this impression of velocity the fluid is
accelerated in the reverse direction relatively to the mean
position of the sphere. Or, conversely, the mean position of
the sphere is uniformly accelerated relatively to the fluid.
d 2 x
Corollary I. Since the expression for ^ contains b as
a factor, it follows that the accelerations of different spheres
of the same density by the same waves are proportional
to their radii, so far as the motion results from the second
d*x
part of .
Corollary II. In the case of waves diverging from a
centre, the argument applied to the force , 2 J is equally
applicable in the present case, shewing that the force yy
also varies inversely as the square of the distance from the
centre. It is, however, to be observed that this law is no
longer exact if the constants h and h' should be found to be
susceptible of change from any cause depending on distance
from the centre. From considerations which I shall not now
dwell upon, I am led to expect that h would be slowly modi
fied by the decrement, at very large distances from the centre,
of the number of axes in a given area, even when X is very
large, and that for very small values of X, both h and h! may
HYDRODYNAMICS. 313
change with distance from the centre in such manner as con
siderably to alter the law of the inverse square.
Having thus carried as far as appears to be practicable in
the present state of the mathematical theory of fluids the in
vestigation of the dynamical action of undulations on small
spheres, it remains to consider in what manner they are acted
upon by steady motions of the fluid.
Example VIII. A small sphere is surrounded by elastic
fluid in steady motion : it is required to find the action of the
fluid upon it.
Conceive, at first, the sphere to be fixed. Then since the
motion of the fluid, taken apart from the disturbance by the
sphere, is constantly the same and in the same directions at the
same points of space, the circumstances will be identical with
those of a uniform stream impinging on a sphere at rest,
excepting that the lines of motion, instead of being parallel,
may be convergent or divergent. In the case, however, of a
very small sphere, to which alone this investigation applies,
the distribution of density on its surface, so far as it is caused
by the impact of the stream, will not be sensibly affected by
the nonparallelism of the lines of motion, provided the sur
faces of displacement of the fluid be always of finite curvature.
Hence from what is shewn in page 302, this distribution of
density will have no tendency to move the sphere. The only
cause tending to produce motion is the variation of density
and pressure from point to point of space due to the condition
of steady motion. It is true that this variation of density, the
effect of which is taken account of in the following investiga
tion, is partly dependent on the degree of convergence or di
vergence of the lines of motion.
It will be supposed that the fluid is of unlimited extent,
and that each line of motion may be traced to some point at
an indefinite distance where the density (p) is equal to the
constant p , and the velocity ( F) vanishes. Under these cir
cumstances the equation (26), obtained in page 241, viz.,
314 THE PRINCIPLES OF APPLIED CALCULATION.
is to be employed for calculating the accelerative action on the
sphere. As Fwill always be supposed to be very small com
pared to a, instead of this equation we may use
F 2
Conceive, now, the line of motion to be drawn whose di
rection passes through the centre of the sphere, and let s be
any length reckoned along this line from a given point. The
sphere being of very small magnitude, it will be assumed that
for all points of any transverse circular area the centre of
which is on the line of motion, and the radius of which is not
less than the radius of the sphere, we have with sufficient
approximation p =f(s). Let s^ be the value of s correspond
ing to the position of the centre of the sphere, and let 6 be the
angle which any radius of the sphere makes with the line of
motion. Then, the radius being equal to b, we have for any
point of the surface s = s l b cos 0, and
P = /( 5 i ^ cos 0) =/( s i) /' ( 5 i) ^ cos nearly.
The whole pressure on the sphere estimated in the direction
of the line of motion is
2?r la z pb* sin 6 cos Odd, from = to 6 = TT.
This integral, on substituting the above approximate value of
p y will be found to be
_47T&V
3 J W
Hence, A being the density of the sphere, the accelerative
force is
If /3 t and V l be the density and velocity corresponding to the
centre of the sphere,
HYDRODYNAMICS. 315
a s t
Hence by substituting for /' (sj in the above expression,
the accelerative force = Q ^  .
A ds l
If we assume that p Q = 1, A will be, as in previous for
mulae, the ratio of the density of the sphere to that of the fluid.
This expression proves that the accelerative action on the
sphere has a constant ratio to the acceleration of the fluid
where the sphere is situated.
If the sphere, instead of being fixed, be supposed to be
impressed with a uniform motion, its acceleration by the fluid
would, at each position, still have the same constant ratio to
that of the fluid in the same position. For, as has been shewn
(p. 302), the uniform motion does not alter the accelerative
action of the fluid on the sphere.
But the stream actually causes an acceleration of the mo
tion of the sphere, and from what is proved in p. 303, the
sphere suffers in consequence a retardation proportional to the
acceleration. But this retardation, the formula for which is of
the same kind as that in page 306, will, in the cases to which
it is proposed to apply these researches, be incomparably less
than the acceleration ; so that we may conclude that the ac
celerative action of fluid in steady motion upon a sphere free
to obey such action, is with sufficient approximation the same
as if the sphere were fixed.
The effect of two or more steady motions acting simul
taneously on a given sphere may be thus determined. It has
been shewn (p. 242) that different sets of steady motions may
coexist. Hence if the velocities which they would separately
produce at a given point of space, and the directions of these
velocities, be given, the resultant velocity and its direction
may be calculated in the usual manner. Then since the re
316 THE PRINCIPLES OF APPLIED CALCULATION.
sultant motion is also steady motion, if p and V be the
resultant density and velocity, we shall have
whence p may be calculated when V is known. This for
mula is to be applied in the case of a sphere acted upon by
several sets of steady motions at the same time, in the manner
indicated above with respect to the analogous formula for a
single steady motion.
For the sake o'f illustration, let the directions of the
velocities V^ and V z of two steady motions make the angle
a with each other at the position where the sphere is situated.
Then we have
r a =r i 8 + F a s + 2 7,7, cos a,
and
/ \
P ' = p (I  ,) very nearly.
From these equations it will be seen that the velocity V is
greatest, and the density and pressure of the fluid least, when
a = 0, or the two streams coincide in direction ; and that V
is least and the density and pressure greatest when a = TT,
or the two streams flow in opposite directions.
I have now completed the portion of these c Notes ' which
I proposed to devote exclusively to processes of reasoning. All
that precedes is reasoning founded on selfevident, or admitted
premises. This is not less true of the Propositions and Ex
amples in Hydrodynamics, by which so large a space in the
foregoing part of the work has been occupied, than of the
treatment of the other subjects. The properties of mobility,
divisibility, and pressure of two hypothetical fluids, one of
which is supposed to be wholly incompressible, and the other
to be susceptible of variations of density exactly proportional
to the variations of pressure, have been taken for granted. The
argumentation is in no manner concerned with any discussion
HYDKODYNAMICS. 317
of these properties, but only with the mathematical processes
proper for deducing from them conclusions relative to the
motion and pressure of the fluids under given circumstances.
Although there is no direct evidence of the existence of fluids
possessing these properties exactly, there is experimental
proof that water is compressed with extreme difficulty, that
the pressure of the air varies very nearly proportionally to its
density, and that both these fluids possess in a very high
degree the property of mobility. Consequently, conclusions
to which the mathematical reasoning leads relative to the
hypothetical fluids, admit of, at least, approximate comparison
with matter of fact, and such comparison may serve as a test
of the correctness of the mathematical reasoning. For in
stance, the near agreement of the velocity of propagation in
an elastic fluid, as determined by the solution of Proposition
XIV. (in pages 214 225), with the result of observations*,
may be regarded as giving evidence of the truth of the new
hydrodynamical principles by means of which that deter
mination was made. I do not admit that this inference can
be invalidated in any other way than by detecting a fallacy
in the course of the reasoning by which I have concluded,
first, that the theoretical value of the rate of propagation is
not the quantity a, and then that it is a quantity having
to a an ascertained ratio greater than unity. Till this reason
ing is set aside, any attempt to account by experiments for
the excess of the observed velocity of sound above the value
a is unnecessary. Besides, as I have urged in page 225, the
experiments hitherto made with this view have failed to
indicate the modus operand* by which development and ab
sorption of heat affects the rate of propagation. I have ad
verted to this question here, because it has an essential bear
ing on the applications that will subsequently be made of the
foregoing hydrodynamical theorems.
* Dr Schroder van der Kolk obtains 1091,8 feet per second, which is less than
the theoretical velocity by 17,5 feet. (See the Philosophical Magazine for July,
1865, p. 47.)
318 THE PRINCIPLES OF APPLIED CALCULATION.
Under the head of Hydrodynamics 1 endeavoured to
ascertain the true principles and processes required for the
mathematical determination of the motion and pressure of an
elastic fluid under given circumstances ; and for the purpose
of exemplifying the general reasoning, I added the solutions
of various problems, selecting them, as has already been inti
mated, with reference to subsequent physical researches. The
application, which I am now about to enter upon, of the
hydrodynamical theorems and problems, constitutes a dis
tinct part of the work, the object of which is, to account for
certain natural phenomena, and laws of phenomena, theo
retically. The reasoning it involves is therefore essentially
different from that in the preceding part, inasmuch as, having
reference to theory, it necessarily rests on hypotheses, and the
hypotheses are such that their truth can be established only
by the success with which the theories founded on them
explain phenomena. The theories that will come under con
sideration are those of Light, Heat and Molecular Attraction,
Force of Gravity, Electricity, Galvanism, and Magnetism,
respecting which I make the general hypothesis that their
phenomena all result from modes of action of an elastic fluid
the pressure of which is proportional to its density. The theo
retical researches are consequently wholly dependent on the
previously demonstrated hydrodynamical theorems.
For the establishment of a physical theory there is a
part which is necessarily performed by mathematical calcula
tion. This remark may be illustrated by reference to the
history of Physical Astronomy. Galileo's experimental dis
covery of the laws of the descent of a body acted upon by
terrestrial gravity was, it is true, a necessary step towards the
discovery of the mathematical calculation proper for deter
mining the motion and path of a particle acted upon by given
accelerative forces ; but the latter discovery, which was ef
fected by Newton, was indispensable for establishing the theory
of the motions of the moon and planets. (See the remarks on
this point in pages 123 arid 124). What Newton did, expressed
HYDRODYNAMICS. 319
in the language of modern analysis, was, to form the differ
ential equations proper for calculating the motion of a single
particle acted upon by given accelerative forces, to integrate
these equations, and to interpret the results relatively to the
motion and path of the particle. The problems of this class
are all solved by the integration of a differential equation of
the second order containing two variables, or a system of
differential equations reducible to a single one of that order
containing not more than two variables. This is the case
also with respect to the problems which relate to the motion
of a system of rigidly connected particles. The methods of
answering physical questions by the solution of differential
equations containing two variables characterized the epoch of
physical science which commenced with Newton.
What has since been required for the advancement of
Natural Philosophy is the farther discovery of the processes
of reasoning proper for ascertaining the motions and pressures
of a congeries of particles in juxtaposition forming an elastic
fluid. At least, the knowledge of such processes is necessary
for testing the truth of the abovementioned general hy
pothesis relative to the medium of action of the different
physical forces. The motions and pressures of a fluid require
for their determination the formation and integration of partial
differential equations, that is, of equations which in the final
analysis cannot contain fewer than three variables. This
greater number of variables, while it gives greater compre
hensiveness to the equations, increases the difficulty of draw
ing inferences from them. Having long since perceived that
the science of Hydrodynamics was in an incomplete and
unsatisfactory state, and being at the same time convinced
that the progress of Theoretical Physics, especially the theo
retical explanation of the phenomena of Light, absolutely
demanded a more exact and advanced knowledge of this de
partment of applied mathematics, I have during a long course
of years made efforts to overcome the difficulties that beset it.
The part of this work devoted to Hydrodynamics contains
320 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
such results of my labours as appeared to possess something
like certainty; but I am well aware that much remains to
be done in this direction, and that some parts of the reasoning,
especially where it relates to the extension of the calculation
to terms of the second order, are incomplete, and may require
modification or correction.
With, however, such materials for theoretical research as
I have been able to collect, I shall now attempt to give ex
planations of phenomena of the various kinds specified above,
and of laws which the phenomena are found by observation
and experiment to obey. For reasons which will appear in
the sequel, the subjects will be considered in the following
order: Light, Heat and Molecular Attraction, Gravity, Elec
tricity, Galvanism, Magnetism. Also as I am unable to re
gard any Theory as deserving that name, the hypotheses of
which do not form an intelligible basis for mathematical
calculation, the hypotheses which I shall have occasion to
propose will all be made to fulfil that condition: on which
account I entitle this section of my work
THE MATHEMATICAL PRINCIPLES OF PHYSICS.
This title has been adopted with reference to that of
Newton's Principia, the principles of the reasoning being
of the same kind as those of that work, although they com
prehend a wider range of subjects. It should, moreover, be
stated that the different Physical Theories will not be dis
cussed completely or in any detail, but solely with reference
to what is fundamental in principle, and necessary for the
explanation of classes of phenomena.
The Theory of Light.
The following Theory rests on the hypothesis that the
phenomena of Light are visible effects of the motions and
pressures of a continuous elastic fluid, the pressure of which is
proportional to its density, the effects being such only as are
cognisable by the sense of sight. This hypothesis brings the
THE THEORY OF LIGHT. 321
facts and laws to be accounted for into immediate connec
tion with hydrodynamical theorems demonstrated in the pre
ceding part of this work. In the instances of several of the
more common phenomena, the theoretical explanations are so
obvious that little more is required than merely referring to
the pages containing the appropriate theorems. With respect
to others, it will be necessary to introduce some special con
siderations. It is to be understood that since the hydro
dynamical theorems rest on principles and reasoning alto
gether independent of this application of them, the success
with which they explain phenomena is to be taken as evidence,
of the actuality of the hypothetical medium and of its assumed
properties. I shall, at first, confine myself to those pheno
mena which have no special relations to visible and tangible
substances, but depend only on qualities of the medium in
which the light is generated and transmitted. This medium
will be called the JEther. The phenomena of reflection, refrac
tion, dispersion, &c. are reserved for consideration after the
explanations of the other class of phenomena have tested the
reality of the aether and its supposed qualities.
(1) One of the most observable and general laws of light
is its transmission through space in straight lines independ
ently of the mode of its generation. This fact is theoretically
explained by the rectilinear axes of the free motion of the
aether, and by the circumstance that the motion resulting from
a given disturbance is, to the first power of the velocity, com
posed generally of vibratory motions relative to such axes.
The proof of the existence of rectilinear axes is given in
pages 186 188 under Proposition VII. The character and
composition of the vibrations result from the demonstrations
of Propositions XL, XII. , and XIII., and from the solution
of Example I. in pages 244 246.
(2) The law of rectilinear axes of free motion having
been deduced as above mentioned^ the mathematical reason
ing then conducted to specific analytical expressions for the
21
322 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
motions and condensations relative to these axes, antecedently
to the supposition of any arbitrary disturbance. This rea
soning is contained in pages 201 211. The axis of z being
supposed coincident with an axis of free motion, w being the
velocity transverse to the axis at the point xyz distant from
it by r, w being the velocity parallel to the axis, and or the
condensation at the same point, the approximate values of &>,
w, and o of the first order are given, for small values of r, by
the following equations :
~ = m sin ~ (z /cat + c), /= 1  er 2 , (pages 206 and 2 1 0)
dz A
. df e\r 2?r ,
w = 9 j = m cos (z /cat + c),
cti 77" A*
w =f  = (1 er 2 ) m sin ^(z /cat + c),
dz A
f d<b K , . 2?r , v
cr = ^27 =  (1 er ) m Sin (z /cat + c).
a dt a ^ \ ^
Assuming that the explanations of the phenomena of light
depend exclusively on terms of the first order with respect
to m, the sensation of light will be due to a vibratory action
of the fluid on the parts of the eye. The radius r has been
supposed above to be small, because, for reasons which will
be adduced subsequently, it may be concluded that the eye
is sensible only to vibrations that are very near the axis.
This circumstance appears to be of the same kind as that
which restricts the sensation of light within limiting values
of X, and to be equally due to physiological conditions. Under
these limitations of the values of r and X, the foregoing ex
pressions for w, w } and o may be regarded as the analytical
exponents of a ray of light. As these equations express laws
of the class of vibrations which have been denominated spon
taneous t we may infer that a ray of light as originally pro
duced, and before it has been subjected to arbitrary condi
tions, is symmetrically disposed about the axis, This is a
ray of common light.
THE THEORY OF LIGHT. 323
(3) Respecting the expressions for &>, w, and <r, it may,
now, be remarked that they are all functions of the quantity
z /cat f c. In consequence of this analytical circumstance,
the velocities and condensations of the undulations which
they represent are propagated through space with the con
stant velocity /ca. This is the theoretical explanation of the
ascertained fact that light is propagated through space with a
uniform velocity.
It is proper to state here that, by the reasoning in pages
205 and 206, the velocity of propagation is the constant xa
however far the approximation be carried. That reasoning
also shews that if the approximation be limited to terms of
the second order K is a numerical constant, but if it be ex
tended to terms of the third and higher orders, that the value
of K includes m, as is shewn by the equation (15) in page 206.
If, therefore, m has different values for different rays, the
rates of propagation will not be exactly the same for all. But
in page 214 reasons have been given for supposing that m
may be an absolute constant. That supposition being made,
if, instead of a single set of vibrations, an indefinitely large
number (n) be propagated along the same axis, by the rea
soning. contained in pages 229 and 230 it follows that the
resultant will be a composite ray defined by the equations
2 . w =  S . a = n*mfs'm ^(z /cat + 0),
K A.
df 27T , ...
it being supposed that the value of X is the same for all the
sets. These equations shew that the compound ray is exactly
like the component rays, excepting that it has nm in the
place of m. Since that coefficient depends on w, which may
be any very large number, the maximum velocity, which it
expresses, may be different in different composite vibrations.
But the rate of propagation, being the same quantity /ca for
all, is independent of this velocity, and is, moreover, abso
212
324 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
lutely constant, because by this argument the supposition
that m may be a constant is justified.
(4) If we assume that n^m is the exponent of the intensity
of a compound ray, it will follow from what is said above
that different composite rays may differ in intensity. The
existence of rays of different intensities is a fact of experience.
The present theory indicates that the fact is referable to the
greater or less number of the components of composite rays.
If we now suppose the term ray to be comprehensive of com
posite rays, we may conclude from the foregoing argument
that rays of different intensities are propagated with the same
velocity. This also is an experimental fact. It follows from
this theory that composite rays, originating at positions as
far distant as the fixed stars, may be propagated in a recti
linear course to the earth without undergoing change as to
form or intensity.
Since the intensity of light is only such as it is perceived
to be by the sense of sight, the assumption that n*m is the
measure of the intensity of a ray, is equivalent to making the
hypothesis that the sensible action of the aetherial undulations
on the parts of the eye is simply proportional to their maxi
mum condensation, or varies, cceteris paribus, as the pressure
corresponding to that condensation. This hypothesis will be
verified by subsequent considerations.
(5) Let us next assume the coexistence of an indefinite
number of composite vibrations all in exactly the same phase,
and having their axes parallel, equally distributed, and sepa
rated by indefinitely small intervals. Taking account only
of the transverse vibrations at small distances from the axes,
it will be supposed that these neutralize each other in such
manner that the transverse motions relative to a particular
axis are just equal and opposite to the resultant transverse
motions relative to all the other axes. Consequently if ^ be
the condensation proper to that axis, and ^ be the corre
sponding maximum velocity, which may be presumed to be
THE THEORY OF LIGHT. 325
proportional to n^m, we have by Proposition IX. (p. 192),
since the lines of motion are parallel,
tcacr^ = ^ sin (z /cat + ff) ;
At
and similarly for any other axis. Now since, according to
the reasoning employed in the solution of Example I. (p. 243),
this motion in parallel lines must satisfy the two linear dif
ferential equations
2 2 da dw da dw
K a T + ~J~> = > ^ + T = i
dz at dt dz
it follows that the law of coexistence holds good with respect
both to the condensation and the velocity. Therefore if <r lt
C7 2 , o 3 , &c. be the condensations, and /i 1? yu, 2 , ^t 3 , &c. the maxi
mum velocities, relative to all the axes contained withm. a
given small area, we have
0_
tea (oj + cr 2 + <7 3 + &c.) = (/ij + /^ + /* 3 + &c.) sin  (z Kat + &)
A
Consequently, by the above definition of intensity, the inten
sity of the compound light is equal to the sum of the intensities
of the separate lights.
(6) Suppose now that the axes of the composite rays,
instead of being parallel, are equally divergent from a centre,
that these rays are all equal, and that transverse motion is
neutralized. Then the resultant at any given distance from
the centre will be ultimately the same as if the axes were
parallel. But the number of the axes included within a given
small transverse area will vary inversely as the square of the
distance from the centre. Hence also the intensity of light
diverging from a centre, being proportional, by what is shewn
in (5), to that number of axes, varies inversely as the square
of the distance. This theoretical result is confirmed by ex
periment.
If at the same time from the same centre any number of
sets of rays diverge, each having a different value of X, the
326 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
same law would, by the law of the coexistence of vibrations,
hold good with respect to each set, and to the sum of all the
sets. Also, since it was proved generally (p. 253) that in
central vibratory motion the condensation varies inversely as
the square of the distance from the centre, it follows at once,
on the principle that the intensity of light is simply propor
tional to the condensation of the aetherial undulations, that
light diverging from a centre varies according to the same law.
(7) Since the observed differences of intensity of different
portions of light are referable, according to the theory ex
plained in (5) and (6), to the existence of compound vibrations
relative to a single axis, and to the composition of different
sets of such vibrations relative to different axes, it follows that
composition is a general characteristic of light, and that it
receives this quality by original generation. This theoretical
inference is confirmed by the fact that a parcel of light,
however generated, admits of being broken up into different
parcels having precisely the same qualities as the original
parcel. Additional evidence of the composite character of
light will be adduced farther on.
(8) The circular function which occurs as a factor in the
expressions for o>, w, and cr, indicates regular periodicity in
the dynamical effects of the undulations; and as we know
from experience that such periodicity in respect to sound
corresponds to the sensation of the pitch of a musical note,
there is reason to conclude analogically that regular periodic
vibrations of the sether have the effect of producing the sen
sation of colour. The kind of colour depends on the number
of vibrations in a given time, which again depends on the
relative values of X and the constant velocity tea. Conse
quently the linear quantity X, which had its origin in the
dj priori reasoning which conducted to the above mentioned
circular function, may be regarded as the exponent of colour.
This explanation is confirmatory of the adaptability of the
results of that reasoning to phenomena of light.
THE THEORY OF LIGHT. 327
(9) So also the linear quantity 6, which is known if the
velocity (wj at a given point of the axis be given at a given
time, and is usually named the phase of the vibrations, corre
sponds to a physical reality, as will appear from what will
shortly be said respecting the coalescence and interference of
different portions of light.
It should here be noticed that the phase of each compo
nent of a composite series of vibrations relative to an axis was
indicated (p. 229) by a quantity c analogous to 6, but that
observed phenomena do not depend on the phase of one of the
large number of components rather than on that of another,
and are, therefore, independent of the particular phases.
Hence when phase is spoken of, it is always to be understood
as relating to composite vibrations.
(10) According to the previously established hydrody
namical principles, any vibratory motion arbitrarily impressed
on the fluid may be assumed to be composed of vibrations of
the primary type, the number of the components, the direc
tions of their axes, and the values of /, X, and 6 being at
disposal for satisfying the given conditions of the disturbance.
Hence on applying this theorem to lightproducing disturb
ances of the asther, it may be inferred that the light may be
composed of rays not only differing in intensity and phase,
but also having different values of X, and, therefore, differing
in colour. The components may either have certain values
of X, or values of all gradations within the limits of vision,
the circumstances of the disturbance determining in which of
these ways the given conditions are satisfied. This theoretical
inference respecting the composition of light is confirmed by
the fact that a spectrum is produced when a beam of light is
refracted through a transparent prism. It is to be observed
that the separation, by this experiment, of light into parts
having different values of X, which is termed an analysis of it,
is distinct from the separation into parts mentioned in (7),
which was supposed to be unaccompanied by change of colour.
328 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
The two classes of facts are in strict agreement with the
indications of the theory respecting the composite character of
the getherial undulations.
Since the aether might be disturbed not only by the
original production of motion, but also by interruptions of
motion previously produced, it would not be inconsistent with
the theory if the breaking up of waves having values of X out
of the limits of those proper for vision were found to give rise
to luminous waves, and that too whether the breadths of the
original waves were larger or smaller than those adapted for
vision. The Drummond light produced by the incidence of
an oxyhydrogen flame on lime appears to be an instance of
such transmutation of rays, the change in this case being for
the most part into rays having values of X less than those of
the original rays. The experiment by which Professor Stokes
obtained visible rays from rays of the spectrum of too great
refrangibility for vision presents an instance of transmutation
of the opposite kind*. It does not belong to the part of the
theory of light now under consideration to enquire under what
circumstances the two kinds of transmutation might occur ;
but it is important to remark at present that each kind may
be conceived to be consistent with the antecedent mathematical
theory of the vibrations of an elastic fluid .
(11) The mutual independence of rays of light, exhibited
by the fact that the same parts of space may be simultaneously
traversed by rays from different origins without perceptible
disturbance of each other, is at once and satisfactorily ex
plained by the law of the coexistence of small vibrations
demonstrated by Proposition XIII. (p. 211). This law ap
plies to the setherial undulations of the present theory, be
cause the equations which express their properties were
Philosophical Transactions, 1852, Part 2, p. 463.
f Respecting the Theory of the Transmutation of Rajs see an Article in the
Supplementary Number of the Philosophical Magazine for December, 1856, p. 521,
and some remarks in that for May, 1865, p. 335.
THE THEORY OF LIGHT. 329
deduced from linear differential equations with constant co
efficients.
(12) The same law of the coexistence of small undula
tions serves to explain the observed interference of rays of the
same colour under certain circumstances. To take a simple
example, let two sets of composite undulations have coincident
axes and the same value of X. Then, according to that law,
the velocity at any point of the common axis at any time t
will be given by the expression
/* sin (z  feat + 0J + /t 2 sin  (z icat + 6).
A A
It will be seen from this expression that if the phases l and
2 be the same, or differ by an even multiple of  , the two
sets of undulations are in exact accordance, and the resulting
value of the maximum velocity is the sum of /^ and // 2 ; but
if the difference of phase be an odd multiple of  , that the
undulations are in complete discordance, and the resulting
maximum velocity is the difference of /^ and /* 2 . In the
latter case, if ^ = //, 2 , the velocity vanishes at all points of the
axis. Also the general values of co t w, and a shew that in the
same case the direct and transverse velocities and the conden
sation vanish at all distances from the axis included within
the limiting value of r. Consequently the combination of the
undulations under these circumstances produces darkness in
stead of light. Not only have these theoretical results been
verified experimentally by the combination of rays of light
traversing paths which differ in length by known multiples
of  , but experiment has also indicated the same interference
of undulations of the air under like circumstances, at least so
far as regards direct vibrations*.
* See a Paper by Mr Hopkins "On Aerial Vibrations in Cylindrical Tubes" in
the Cambridge Philosophical Transactions, Vol. v., Part n., p. 253.
330 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
Before proceeding to other comparisons of the theory with
facts, it must now be stated that from experiment we are led
to conclude, as will be fully shewn hereafter, that the sensa
tion of light is not due to the direct velocity w, but depends
exclusively on the transverse velocity &>, This must be
accepted as a fact resting only on experience, inasmuch as it
relates to the mode of action of the astherial undulations on
the constituent atoms of the eye, of which theory is at present
incapable of giving an account. Also we have reason from
experiment to conclude that light is produced by transverse
movements of the aether within distances from the axis very
small compared to X, it being a known fact that spaces very
much narrower than the breadth of an undulation have been
made visible by powerful microscopes ; which would not be
possible unless the effective transverse dimension of the ray
were much less than X.
If (T^ be the condensation and ^ the maximum velocity in
a given composite ray, from the expression for 2 . cr given
n__
under (3) we have, putting ffor  (z /cat + 6),
X
a\lcr df .
and j = Kfji^a sin = 2/c/^er sin f,
Hence the transverse accelerative force of the fluid varies
cceteris paribus as the distance r from the axis. Now in the
case of planewaves, in which the transverse motion is neutral
ized, there is no transverse accelerative action ; but when a
limited portion passes through the pupil of the eye and is
brought to a focus on the retina, the different axes of the com
ponents are made to converge to a point, and the transverse
action, being no longer neutralized, is brought into play,
causing the sensation of light. Also if the different axes
do not converge with mathematical exactness to a point,
since the separate transverse actions would in that case vary
THE THEORY OF LIGHT. 331
as the distances from the respective axes, it is readily seen
that the resultant would be a transverse action varying as the
distance from a mean axis passing through the centre of gra
vity of the component axes. Thus a bundle of rays would
act transversely like a single ray. This result appears to give
a physical reason for the above accelerative force being effec
tive for producing light only at small distances from the axis,
the distinctness with which images of external objects are de
picted on the retina being dependent on the fulfilment of 'that
condition.
(13) Hitherto we have had under consideration only such
undulations as are symmetrical with respect to axes, the ana
lytical expressions for which contain no constant quantities
that can be immediately satisfied by arbitrary conditions. It
may accordingly be supposed that this form of undulation is
always produced by an initial disturbance, independently of
the particular mode of the disturbance; for which reason I
have called it the primitive form. The characteristic of such
undulations, namely, the symmetrical arrangement of the
direct and transverse velocities and the condensation about
the axes, is at once explanatory of the term nonpolarized
applied by experimentalists to rays which have no sides, that
is, no relations to space in directions perpendicular to the
axes. In conformity with the theory experiment shews that
this quality belongs to all rays that have been subjected to no
other conditions than those of their original generation.
(14) But that this symmetry may be subsequently dis
turbed by arbitrary conditions is theoretically proved by the
analytical circumstance that the value of the factor / may
be determined by the integration of the partial differential
equation
In p. 210 I have obtained a particular solution of this equation
which indicates that the transverse motion is symmetrical
332 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
with respect to a plane the position of which depends on an
arbitrary angle (6) introduced by the integration, and that it
is perpendicular to this plane. Hence we may theoretically in
fer that to produce such transverse motion it is only necessary
to impress on undulations of the primitive type a disturbance
symmetrical with respect to a plane. It is found, in fact, that
rays of common light submitted to such disturbances are,
either wholly or in part, polarized, and the plane of symmetry
of the disturbance is the plane of polarization. Such modi
fication, for instance, light undergoes by reflection at polished
surfaces. Also it is shewn in pages 216 and 217 that when
primitive undulations are so modified, equal portions are po
larized in planes at right angles to each other. This theoreti
cal inference is confirmed by experiments.
(15) A polarized ray, the transverse vibrations of which
are parallel to the plane of xz, is defined by the equations
given in page 217, which, by expanding the sine and cosine
of 2 Ve x, omitting powers of x above the second, and substi
tuting f for q (z  Kat + c) become
f.
and the transverse accelerative force is
., d<T . c.
a j = 2m/caex sin ?.
dx
It is evident from these equations compared with those in
page 322 for a primitive ray, that the circumstances under
which two rays polarized in the same plane, and having coin
cident axes and the same value of X, coalesce or interfere,
are precisely the same as those already found for two primi
tive rays.
THE THEORY OF LIGHT. 333
But if the rays be polarized in planes at right angles to
each other, the results are different. Let the two rays be in
other respects exactly alike, and, first, let their phases be the
same, or differ by an even multiple of  . Then we have for
the transverse velocity of the ray polarized in the plane of yz t
. df 9 m\ey
v = $ 11:=  ^cosf.
fy *
Hence the resulting transverse velocity, or (u* + v*)^, is
m\er
cos f ;
7T
that is, it is the same as that for a primitive ray the maximum
velocity of which is m, and therefore double the maximum
velocity of each of the polarized rays. This will also be
the case with respect to the resulting values of w, cr, and
the transverse accelerative force ; so that the compound ray
will differ in no respect from a primitive ray. If, now, the
difference of phase be an odd multiple of , the value of
2
(u* + v*)^ and the resultant of the transverse accelerative forces
will be the same as in the former case ; but for the resulting
values of w and or we shall have
CL(T , z\ t,
w me (y x ) sm f.
/C
Hence w and a will each be extremely small, because
and the ratios of y and x to X are very small. Moreover,
the dynamic effect of the undulations in the direction of
z must be estimated by the sum of the values of a with
in a small circular space about the axis. But clearly within
such a space the sum 2 . #%&c is equal to the sum 2.x*SySx,
and consequently the total condensation is zero. Thus wa
334 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
may conclude that when the difference of phase is an odd
multiple of  the direct vibrations are wholly inoperative.
But experiment has shewn that the light perceived in this
case is exactly the same as when the difference of phase is an
even multiple of  . Hence it necessarily follows that the
2t
perception of light is in no degree affected by the direct vibra
tions, and that it is entirely due to transverse vibrations.
It is conceivable that a planewave may be compounded
of an unlimited number of polarized rays in the same phase of
vibration and so disposed that the transverse motions are all
parallel to a given direction in the planefront and neutralize
each other. Hence if a portion of such a wave enter the pupil
of the eye, so that a bundle of the component rays are made
to converge to a focus, the transverse motions severally re
appear, and produce a resultant transverse motion perpen
dicular to a certain plane, and exactly alike on the two sides
of it. This is the case even if the axes of the components do
not strictly converge to the same point. (See p. 331.) From
experience we have reason to conclude that the sensation of
light is produced as well by these resultant transverse mo
tions perpendicular to a plane, as by those perpendicular to
an axis.
By the reasoning in pages 216 and 217, the resolution of
primitive vibrations into two equal sets the transverse motions
of which are in rectangular directions is possible only for very
small distances from the axis. Hence, since the resolved
vibrations, equally with the primitive vibrations, produce the
sensation of light, it follows that light is due to the action of
transverse vibrations in the immediate vicinity of the axes.
This argument is referred to in paragraph (2), p. 322, and
justifies the limitation there given to the value of r.
(16) It being established that the sensation of light is
caused by transverse vibrations, we may hence infer that the
THE THEORY OF LIGHT. 335
undulations of two rays, having a common axis and polarized
in rectangular directions, produce independent luminous
effects, simply because their transverse accelerative forces
act independently. Also since, as is known by experience,
the luminous effect of a series of undulations is the same
whatever be their phase, it follows that the combined lu
minous effect of two oppositely polarized series is independent
of difference of phase. Thus the theory explains the expe
rimental fact that oppositely polarized rays having a com
mon path do not interfere whatever be the difference of their
phases.
(17) We have next to consider the effect of resolving a
polarized ray into two parts by a new polarization. There are
only two conditions which the resolved parts of a polarized
ray are required to satisfy in order that when recomposed
they may make up the original ray, namely, that the sum of
the condensations at corresponding points be equal to the
condensation at the corresponding point of the integral ray,
and that the velocities at corresponding points be the parts,
resolved in directions parallel and perpendicular to the new
plane of polarization, of the velocity in the integral ray at
the corresponding point. Let that plane make the angle 6
with the axis of x, and let s, <r^ a 2 be the condensations at
any corresponding points of the original ray and the resolved
rays, and /, f l , / 2 be the factors for the same points, which
must be such as to satisfy the differential equation in p. 209
already cited in paragraph (14). Then, if we assume that
cTj = s cos 2 6 and <r 2 = s sin 2 0, we have c^ + <r 2 = s, an3 the first
condition is fulfilled. Also it will appear from the following
considerations that the other condition is fulfilled by the same
suppositions. Let S, S 1? 2 2 be the condensations at the points
of intersection of the axes by the respective transverse planes
in which are the condensations s, <7 15 cr 2 ; so that s=fS,
2 = /^ 2 . Consequently
336 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
or, ?^=/cos 2 0, and ^ =/sin 2 6 ;
hence, by differentiating these equations with respect to x and
multiplying by </>,
&#,**, &$+&,.
S dx r dx >S dx Y dx
Now supposing the transverse velocity of the original ray,
(assumed to be parallel to the axis of x), to be expressed by
<f> f , the lefthand sides of the last two equations will
express the velocities in the bifurcated rays resolved parallel
to the same axis. But the righthand sides of the equations
7/
are the velocities resulting from resolving <f> f in the direc
(IX
tions parallel and perpendicular to the new plane of polariza
tion, and then resolving these parts in the direction of the
axis of x. Hence the velocities in the bifurcated rays, being
parallel and perpendicular to that plane, must be equal
7/1 JJ?
respectively to < ^ cos 6 and $ j sin 6. That is, they are
equal to the resolved parts of the velocity of the original ray
parallel and perpendicular to the new plane of polarization.
Thus the second condition is also satisfied by the equations
<r l = s cos 2 6 and <r 2 = s siri 2 6 ; and as there are no other con
ditions to be satisfied, we may conclude that these equations
give the true values of <r l and cr 2 .
On the principle that intensities are proportional to the
condensations, it appears from the above results that the
intensities of the resolved rays are in the ratio of cos 2 6 to
sin 2 #, and that the sum of their intensities is equal to the
intensity of the original ray. If 6 = 45, the two intensities
are equal, and we have also ^ra;=ij These theoretical
40
inferences accord exactly with known experimental result?.
THE THEORY OF LIGHT. 337
(18) The two rays of this second polarization, like those
of the first, produce independent luminous effects, because
their dynamical actions on the parts of the eye are in planes
at right angles to each other. Hence, although their phases
may be different by reason of difference of the lengths of
their paths, the total luminous effect of the rays combined
will always be the same. The compound ray is not, however,
identical in its properties with a ray of common light, the
resulting transverse vibrations not being of the same cha
racter, as will be seen by the following argument. Let the
plane of second polarization be now the plane of xz, and let
the transverse velocities of the two resolved rays, parallel
respectively to that plane and the plane of yz, be
f^<f> (x) sin (z feat + c) and fity (y) sin (z icat + c').
A A/
Then, supposing #, y, z to be the coordinates of a given
particle of the aether at the time t, we have
In obtaining from these equations the projection of the path
of the given particle on the plane of xy, the variations of
z may be neglected; and we may also leave out of con
sideration, since the reasoning embraces only quantities of the
first order, the changes of x and y in the coefficients p<t> (x)
and fju'^fr (y) due to the small changes of position of the
particle. By integrating the above equations, and eliminating
t from the results, an equation between x and y of the fol
lowing form will be obtained :
This equation shews that if c c' be zero, or any multiple of
 , the lefthand side of the equation is a complete square,
22
338 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
and the path of the particle is a straight line. For these par
ticular cases the compound raj is exactly equivalent to the
original polarized ray. It also appears that in general the
path is an ellipse, and that for the particular cases in which
c c= X, it is a circle. These theoretical results ex
4
plain the characters of the different kinds of light which have
been named planepolarized, ellipticallypolarized, and cir
cularlypolarized.
(19) There is still another class of facts the explanation
of which depends exclusively on properties of the setherial
medium, viz. the effects of compounding lights of different
colours*. In the following argument it is assumed that
simple colours, such as those presented by a pure spectrum of
sunlight, are functions of X only. Certain phenomena ac
companying the absorption of rays of light in their passage
through coloured media, which were thought at one time to
be opposed to this law, have been accounted for consistently
with it since the important discovery was made of the trans
mutability of rays into others of different refrangibility. (See
the remarks and references relative to this point in page 328.)
The theory of composition I am about to propose will, at
first, refer exclusively to the colours of the spectrum.
The analytical formula which expresses that the vibrations
of a ray are compounded of the vibrations of two or more
simple rays having different values of X, is the following :
v = fju sin (z /cat + 6} + fjf sin 7 (z teat + &} + &c.
A, X
Assuming, now, that the composition of colours corresponds
to the composition of aetherial undulations of different breadths,
the theory gives the following explanation of observed facts.
* See a Communication in the Report of the British Association for 1834
(p. 644), an Article in the Philosophical Magazine for November, 1856, p. 329,
and some remarks in the Number for May, 1865, p. 336.
THE THEORY OF LIGHT. 339
1. The general fact that colours admit of composition
and analysis is referable to the law of the coexistence of small
vibrations, on which the above formula depends.
2. The result of compounding any number of undula
tions for which X is the same is a series of undulations ex
pressible by the formula
,
A,
in which V is the algebraic sum of the separate velocities,
and M is a function of m, ra', &c., and of the phases c, c', &c.
of the component undulations. Hence the composition of
rays, or portions of light, of a given colour produces light of
the same colour, as is well known from experience to be the
case.
If fji = mn^j fjf = mri*, &c., and there be an unlimited
number of components, we have by the reasoning in page
229
M = m (n + ri + n" + &c.)* = (mV + mV + mV + &c.)*.
Hence in this case it results from the measure of intensity
previously adopted, that the square of the intensity of the
compound ray is equal to the sum of the squares of the in
tensities of the components. But in general M involves the
phases of the components.
3. If the values of v at a given time be represented
by the ordinates of a curve of which the abscissae are the
values of x, this curve will in general cut the axis of x
in a great number of points with irregular intervals between
them. When this is the case, the irregularity of the intervals
is incompatible with the sensation of colour, but does not
prevent the sensation of light ; so that the result of the com
position is white light, and the degree of whiteness, it may
be presumed, is greater the greater the irregularity. There is
here a strict analogy to soundsensations. As sounds are not
all musical, so light is not all coloured. It is reasonable to
222
340 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
suppose that as colour in a simple raj is due to regularity
of waveintervals, so in every instance of the production of
colour the sensation is due to some species of regularity of
recurrence in the waves. It may also be remarked that the
irregularity to which whiteness is due exists whatever epoch
(t) be selected, and independently of the particular phases
of the component undulations. This is known to be the case
from experience.
4. The effect of compounding two simple colours is ex
pressed in this theory by the formula
v = fj, sin  (z /cat f C) + JL sin 7 (z Kat + C').
X A
l \ j ! ! ft l \ r,
+ .), and 7 = (^). Then,
the time being given, the expression may be put under one
or the other of the two forms
. ~\ ftlTZ ~\
= 2fj, sin lj + (LJ cos lj + <7J
. , . /27T2! ^\ /27TZ ~\ . f%7TZ ~ ,\
v = <2p sm lj + A cos f y + C 2 J + z/ sin f + (7 S M .
Leaving out of consideration, at present, the term containing
v, the other term shews that the axis of z will be cut by the
curve at a series of points separated by the common interval
L, which is an harmonic mean between X and X', and at
another series of points separated by the common interval L
As the ratio of the greatest and least values of the breadths
of lightundulations is nearly that of 3 to 2, Z will be at least
equal to 6L. Hence the second series of recurrences will
always be slower than the first, and in case X' be not much
larger than X, they will be much slower. The effect of the
second trigonometrical factor is to cause the maximum velo
cities of the undulations expressed by the other factor to vary
THE THEORY OF LIGHT. 341
periodically from zero to 2m. This effect is analogous to the
production of beats, or discords, by the union of two series of
aerial vibrations. Now it is known from experience that if
a stream of light received by the eye be interrupted during
very short intervals, the sensation of light and colour is still
continuous, by reason, it may be presumed, of a temporary
persistence of the luminous impressions. It may hence be
inferred that when the vibrations, without being actually
interrupted, are subject to periodic variations of intensity, the
eye is insensible to such variations, and only perceives light
of the colour corresponding to the regular intervals between
the recurrences of maximum velocity. Accordingly we may
conclude from the above expressions relative to the compo
sition of two simple colours, that the eye will perceive the
colour corresponding to the wavelength L. As this length
is intermediate to \ and X', the theory accounts for a law
announced by Newton as a result of experiment, viz. that " if
any two colours be mixed, which in the series of those gene
rated by the prism are not too far distant from one another,
they, by their mutual allay, compound that colour which in
the said series appeareth in the midway between them."
M. Helmholtz states that "Newton's observations on the
combinations of every two prismatic colours coincide with
his own results." (Phil. Mag. for 1852, S. 4, Vol. 4, p. 528.)
5. The intermediate colour corresponding to the wave
length L is strictly produced only in case v 0, or p fju.
If /Ji = mn* and /jf = mri^, we shall have for this case n n;
so that the number of the rays of each kind, and consequently
the intensities of the two portions of light, will be equal.
Hence to produce the intermediate colour an adjustment of
the quantities of the components is required, as is known to
be the case from experience. If v does not vanish, the light
represented by the additional term will affect the tint of the
compound, and according to the value of v there may be every
gradation of tint from the colour corresponding to X, through
that for which v 0, to the colour corresponding to X'. The
342 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
production of such gradation of tints by varying the propor
tions of the components is matter of experience.
6. The following is the explanation, according to this
theory, of complementary colours of the solar spectrum. We
have seen that when the ratio of X to X' for two colours does
not differ much from a ratio of equality, the result of com
bining them is the intermediate colour corresponding to the
wavelength which is an harmonic mean between the wave
lengths of the components. But it is evident that this law
may cease to hold good when that ratio exceeds a certain
limit. For in proportion as X and X' differ from each other
the value of I becomes less, and the recurrence of the maxi
mum values of the factor cos f h Cu more frequent. As
suming that the beats thus produced have a tendency to
destroy the sensation of colour without destroying the per
ception of light, a limit will be reached at which the result
is white light, arid the colours become complementary to each
other. Since this limit depends on the particular conditions
required for^ the production of the sensations of light and
colour by the action of the aether on the particles of the eye,
it does not admit of a priori investigation, and must conse
quently be determined experimentally. This desideratum has
been furnished by the following experimental results obtained
by M. Helmholtz by an ingenious arrangement for viewing
the combinations, two and two, of the different gradations of
colour of a pure spectrum. (See Poggendorff's Annalen,
Vol. xciv.)
Colour. J9
Ked..,
'avelenj
2425
2244
2162
2120
2095
2085
2082
j.i.1. Complementary IT
Colour.
... Greenblue ...
... Blue ..
r aye length.
1 CM Q
Ratio of
wavelengths.
Orange ,
1809
1 24.0
Goldyellow ...
Goldyellow ...
Yellow
... Blue
1793
1 20fi
... Blue
1781 ...
. . 1 190
... Indigoblue...
... Indigoblue...
... Violet .
171 A
1 991
Yellow
1 70fi
1999
Greenyellow...
1600 .
, 1.301
THE THEORY OF LIGHT. 343
These results shew that the disappearance of the intermediate
colour takes place for ratios of the wavelengths varying from
about that of 4 to 3 for red and greenblue, to about that of
6 to 5 for goldyellow and blue. It is worthy of remark that
the ratio of the wavelengths is less for the brighter parts of
the spectrum than for the extreme and duller parts; appa
rently because increase of intensity tends to diminish the
perception of colour, as is known to be the case from inde
pendent experience. Whether it be for this reason or not,
goldyellow and blue are complementary for a ratio of wave
lengths less than the ratio for any of the other complementary
colours. This circumstance may be regarded as explanatory
of the fact, deduced by M. Helmholtz from his experiments,
that prismatic blue and yellow combined do not produce
green, or only a greenish white. The existence of green, in
however small a degree, is a phenomenon which the theory
has to account for, the sensation of green being so entirely
different from that of blue or yellow; and this, in fact, it
does account for by the foregoing formula for composition ;
but theory is incapable of determining the amount of the
sensation. It should, however, be observed that the above
ratios may depend in part on the particular circumstances of
the experiment, and in part also on the particular capabilities
of the observer's eye, it being a known fact that different
observers have different perceptions of colour.
Again, it appears from the above results that the colours
whose wavelengths lie between the numbers 2082 and 1818,
the difference of which is about onethird the difference of
the numbers for the extreme wavelengths, have no comple
mentary colour. This fact seems to admit of being explained
by the consideration that the ratios of their wavelengths to
the wavelengths of the other colours, might all, when in
tensity is taken into account, be too small for the neutraliza
tion of colour.
7. I enter now upon the theory of the composition of
344 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
impure colours, such as those of pigments and coloured pow
ders. With respect to these it is certain that green may be
produced by a mixture of yellow and blue. The following
passage occurs in Sir John Herschel's Treatise on Light in
the Encyclopaedia Metropolitana (Art. 516) : " Blue and yellow
combined produce green. The green thus arising is vivid
and rich ; and, when proper proportions of the elementary
colours are used, no way to be distinguished from the pris
matic green. Nothing can be 1 more striking, and even sur
prising, than the effect of mixing together a blue and yellow
powder, or covering a paper with blue and yellow lines drawn
close together, and alternating with eaeh other. The ele
mentary tints totally disappear, and cannot even be recalled
by the imagination*." According to this statement, which
I have verified by my own observations, a mixture of blue
and yellow powders has the same effect as a mixture of blue
and yellow lights ; for in the second mode of making the
experiment it is clear that the eye receives a mixture of blue
and yellow rays. Sir J. Herschel adds : " One of the most
marked facts in favour of the existence of three primary
colours, and of the possibility of an analysis of white light
distinct from that of tire spectrum, is to see the prismatic
green thus completely imitated by a mixture of adjacent rays
totally distinct from it both in refrangibility and colour."
The theory I am expounding rather tends to shew that there
would be no reason to conclude from the production of a per
fect sensation of green by a mixture of yellow and blue, that
the green of the spectrum is a compound colour. I admit,
however, that the theory ought to account for the great dif
ference, as to fulness and vividness, between the green ob
tained by the composition of pigments, and that resulting from
* It would seem that some eyes have a peculiar inaptitude for seeing green
when it is composed of yellow and blue. Mr Maxwell states generally that "blue
and 5 yellow do not make green, but a pinkish tint, when neither prevails in the
combination; " and in the particular instance of "viewing alternate stripes of blue
and yellow with a telescope out of focus," he finds the resultant tint te be "pink."
(Edinburgh Transactions, Vol. xxi. Part n. p. 291).
THE THEORY OF LIGHT. 345
the composition of the yellow and blue of a pure spectrum.
To this point I now propose to direct attention.
But I must first premise that I found the statements of
experimenters on the composition of colours so perplexing
and contradictory, and apparently so much influenced by an
anticipation of the resolvability of the colours of a pure spec
trum, that I had recourse to personal observation to satisfy
myself on certain points before comparing the theory with
experiment. The details of these observations are here sub
joined.
(a) Having painted on white paper a small circular space
with a mixture of ultramarine blue and chrome yellow form
ing a good green, I looked at the compound colour through
an ordinary equiangular prism at the angle of minimum de
viation. The green circle was seen to be resolved for the
most part into two circular images overlapping each other,
one yellow and the other blue. There was an admixture of
other coloured images, owing to the pigments not being pure
colours, but these were comparatively faint, and did not pre
vent the tracing of the outlines of the yellow and blue images.
It was readily perceived that the colour of the space common
to the two images was a bright green. The remaining spaces
were respectively yellow and blue. Consequently the green
effect could not be attributed to any absorbing action, but
must have been produced simply by the combination of yellow
and blue rays, each parcel of which was of nearly definite
refrangibility. The same effect resulted from using in the
same manner a mixture of ultramarine and gamboge; and
also when a circular green patch, formed by mixing blue and
yellow chalk powders, was viewed through the prism.
(/3) On white paper I placed in diffused daylight a rect
angular piece of nonreflecting black paper, and on the latter
a rectangular slip of the white paper onetwelfth of an inch
broad, with its longer edges parallel to edges of the black
paper. On viewing the two pieces through an equiangular
THE MATHEMATICAL PRINCIPLES OF PHYSICS.
prism at the angle of minimum deviation, with its edges
parallel to those of the papers, the usual internal and external
fringes were seen at the borders of the black paper, the former
consisting mainly of blue and violet light, and the other of
red and yellow, but neither exhibiting green. The same
fringes were formed in reverse order at the borders of the
white slip, and overlapped in such manner that the blue of
one fringe occupied the same space as the yellow of the other.
The total effect was a kind of spectrum consisting apparently
of only red, green, and violet rays. The green was very
vivid, and without doubt was produced by the mixture of the
yellow and blue rays.
(7) I marked on white paper by chalk pencils alternating
yellow and blue parallel spaces of not inconsiderable breadth,
and found that even when the eye was near enough to dis
tinguish the spaces easily, the whole appeared to be suffused
by a tinge of green. This effect, which was probably due to
the angular spreading of the lights by diffraction, shewed
that the eye was affected with the sensation of green by a
mixture of yellow and blue rays, quite independently of any
absorbing action on the daylight incident on the coloured
spaces. To make this more evident, I covered three quite
broad parallel spaces with alternate blue and yellow colours,
the yellow being in the middle, and looked at them after
retiring to a considerable distance. The green tinge was then
very apparent, but upon intercepting the light from the middle
space it wholly disappeared. The chalk pencils used in this
experiment furnished, by scraping, the coloured powders used
in experiment (a) ; whence it may be inferred that their
predominant colours were respectively prismatic yellow and
blue.
(8) I also tried the effect of combining colours by means
of revolving circular disks, the disks being divided into equal
sectors covered alternately with the two colours to be com
pounded. On using the same yellow and blue chalks as in
THE THEORY OF LIGHT. 347
experiments (a) and (7) I obtained a green colour, but the
green was not so vivid as in those two" experiments. The
colours of these chalks were far from being homogeneous, but
the predominance of prismatic blue and yellow, demonstrated
by experiment (a), seems to have determined the resulting
colour in this experiment*.
(e) On viewing in the same direction the yellow and blue
pigments and chalks employed in experiments (a), by trans
mission of one colour through plateglass, and by reflection
of the other at the same, according to the method employed
by M. Helmholtz (Phil. Mag. for 1852, Vol. 4, p. 530), I cer
tainly discerned green, but it was a very dull colour, and
could only be seen in strong daylight.
The foregoing series of experiments seem to justify the
conclusion that blue and yellow parcels of ordinary daylight,
not of prismatic purity, may produce green by simple com
bination, and independently of any modifications, by absorp
tion or otherwise, which they may have undergone since
their original generation at the Sun, and that this green is
much more conspicuous than any resulting from the com
bination of the blue and yellow of a pure spectrum. I shall
now endeavour to give a theoretical reason for this difference,
which is observable not only with respect to these two colours,
but, in less degree, in the composition of other colours of the
spectrum t. (Helmholtz, Phil. Mag. pages 525 and 526.)
* The colours on the revolving disk by which Mr Maxwell attempts to shew
that blue and yellow combined do not make green had scarcely any resemblance
to the colours which I employed. I suspect, therefore, that if analysed by the
prism they would exhibit no preponderance of blue and yellow, and that on this
account the result was a neutral tint.
f The theory of the composition of colours here given differs in some points
from that which I proposed in the Article contained in the Philosophical Magazine
for November, 1856. According to the present views the factor which is called m
is originally the same for all rays; so that unccmpounded rays do not differ from
each other in intrinsic intensity, and the difference of intensity of compound rays
depends on the number of the components. In consequence of these views, the
interpretation given in page 341 to the term in the formula for composition
which contains v> is different from that proposed in the Article referred to.
348 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
By means of the expression for v in page 340, it has been
shewn that if in the composition of two bundles of rays of
prismatic purity the quantities be so adjusted that v = 0, and
the difference between X and X' be small, the result is an
2XX'
intermediate colour corresponding to the wavelength  ry .
. A ~ X
Experiments confirm this theoretical result so far as regards
the production of an intermediate colour, but shew at the
same time that the colour becomes pale and diluted on in
creasing the difference between X and X', till for a certain
difference, depending on the positions of the components in
the scale of prismatic colours, it very nearly or wholly dis
appears, and the result is neutral or white light. (See page
342.) The limiting difference is least for the rays that are
in the brightest part of the spectrum, and appears, therefore,
to be determined in some degree by the intensity of the light.
But apart from the influence of intensity, the intermediate
colour is qualified by some cause operating alike on all the
different kinds of light ; and as the theory points to no other
qualifying circumstance than the frequency of recurrence of
the beats which are represented by the factor cos ( 1 C,
\ X
I shall for the present regard this as a vera causa.
This being understood, let us now consider the result of
compounding two impure parcels of light, that is, two parcels
each of which consists of simple rays having an unlimited
number of 'different wavelengths included within certain
limits. If fjb represent the maximum velocity resulting from
the composition of all the simple rays in one parcel having
the wavelength X, and fju that from the composition of the
simple rays of the other parcel having the wavelength X',
and if // = // + v, the result of compounding the two parcels
may be thus expressed, if // be greater than p,,
27T2
. /27T3
i'sm(
THE THEORY OF LIGHT. 349
or thus, if fi be less than ft,
S., = iSyrin^+C') cos( 2 f? + (7") + S.,sin( 2 ^ + 0"') ,
being put for i (I + ^ j , and ^ for  (  ,j , and X' being
supposed greater than X. First, it will be admitted that the
quantities of the two parcels may be so adjusted that the light
or colour corresponding to either aggregate of terms contain
ing v may be made to disappear ; that is, a distinct colour
may be produced free from any tinge of the colour of either
of the components. This adjustment would evidently be
effected if for every combination of two pure composite rays
the number of the simple rays is the same in each, so that
fjb = fi and v = Q. On this supposition the total number of
simple rays would be the same in the two parcels of light.
Again, it is possible and allowable under these conditions to
group the two series of values of X and X' (which, by hypo
thesis, are restricted within definite limits), so that the har
monic mean between a value of X from one series and a value
of X' from the other may be very nearly the same for every
set. Taking one such set, we have at any given time for the
resulting velocity, supposing v = 0,
Having regard, now, to only a limited portion AZ of the axis,
it is evident, since I is much larger than L, that within this
portion the changes of the first trigonometrical factor are much
more considerable than those of the other. Hence if z =* the
mean of the values of z in this space, and if r, represent any
one of the factors analogous to 2/j, cos f ^ + c" J , and <7/
any one of the arcs corresponding to <7', we shall have very
nearly
350 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
This expression, as is known, may be put under the form
(S . w. f + 22sw cos (0.' <y)} sin C~ +
in which the affixes 5 and s represent the series of integers
1, 2, 3, &c. to an indefinitely large integer n, and in the second
term the different values of s and s' are all the different com
binations, two and two, of these integers. By taking account
of the form of the functions which OT S represents, it will be
seen that the quantity in brackets consists of a part free from
circular functions, viz. 2 . 2//, g 2 , and of a part consisting en
tirely of the sum of sines or cosines of circular arcs with
coefficients attached. The former part is so much the greater
as the number of terms embraced by 2 is greater, while the
probability that the sums of the positive and negative terms
of the other part differ much from each other is less in pro
portion as the total number of terms is greater, there being
no antecedent reason why either sum should be in excess
when all values of the phases of the circular functions are
equally possible and probable. On this account, as the num
ber of the terms is not limited, we may neglect those con
taining circular functions in comparison with the others ; so
that, if /Ltj = mn^, ^ = mn^ &c., we have
2 . v = m (2n x + 2w 2 + &c.) 4 sin (^ +J)j .
The arc D is determined by the equation
2 . Tir 8 sin C,
tanZ> =
2 . VT 8 COS
the righthand side of which is constant at a given time for a
given value of . Consequently within the small interval
Az, and for a given value of z , the result of the composition
is equivalent to a pure ray the wavelength of which is L ;
and we have now to enquire what change D undergoes by a
THE THEORY OF LIGHT. 351
change of z . By differentiating the above equation with
respect to D and Z Q it will be found that
~ ^ 2 . SCT S sin Ca x 2 . izg cos C s ' 2 . ^ 8 cos C,' x 2 . sr 8 sin C 8 r
oU = T^ji 7T7\2 7^ ' /^"\ 2 *
Hence, since
f27TZ n ~,A T *. 47T//3 . /27T n ~,A ^
OT, =2/4, Cos ( j  + L> 8 1 , and 0^= j sin I ^  + O 8 I oz ,
it will readily be seen that the terms of the numerator of the
above fraction are all sines or cosines of arcs, with coeffi
cients attached, and that those of the denominator are of the
same kind, with the exception of the terms 2 . 2/it/. Now the
sum of these last terms is greater the greater their number,
while, for the reason given above, the probability that the sum
of the others in the denominator, or the sum of the terms of
the numerator, is of considerable magnitude, is less the greater
their number. Hence since the number of terms embraced by
2 is not limited, we may conclude without sensible error
that SZ> = 0, or D is a constant arc.
Consequently the above expression for 2 . v is true at a
given time for all values of z, and therefore true in successive
instants at a .position corresponding to a given value of z.
Thus the theory shews, conformably with experience, that two
impure parcels of light of different colours may combine to
produce an intermediate colour which is sensibly pure and of
uniform intensity. It is particularly to be noticed that the
resultant colour depends on the quality of impurity in the
component parcels. Since in this case there is no generation
of beats, as in the combination of two rays of prismatic purity,
the verification of the foregoing theoretical inference by expe
rience appears to justify the supposition made in page 348,
that the occurrence of beats is the cause of the diminution,
or destruction, of colour in the complementary combinations
of pure rays*.
* This theory seems to me to account for the green colour seen in the experi
ment described by Sir J. Herschel in the Proceedings of the Royal Society (Vol. x.
352 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
8. Various other phenomena may also be explained by
the mathematical theory of the composition of colours on the
hypothesis of undulations ; principally these which follow.
The extreme colours of the spectrum, red and violet, are
not obtainable by composition (Helmholtz, Phil. Mag. p. 532).
The theoretical reason for this fact is, that the wavelength of
the colour resulting from composition is necessarily inter
mediate to the wavelengths of the components, and, there
fore, cannot be the same as that of either of the extreme
colours.
Dr Young maintained that the three primitive colours, or
sensations, are red, green, and violet, by means of which, as
experiment shews, all the colours of the spectrum may with
more or less precision be imitated. The present theory ac
counts for the possibility of doing this, inasmuch as the inter
vals between the wavelengths of red and green, and of green
and violet, are not too great for the production of an interme
diate colour, especially if the experiments be made with pig
ments, or rather, parcels of light that are not of prismatic
purity.
Since all the spectrum colours may be imitated by mix
tures of red, green, and violet, from the fact that spectrum
colours combined make white it may be inferred, that white,
or neutral tint, is producible by mixtures of those three
colours : and by experiment this is found to be the case. On
the other hand, according to experiments mentioned by Mr
Maxwell (Edin. Trans. Vol. XXL, p. 291) the result of com
binations of red, yellow, and blue, could not be rendered
No. 35, p. 82). In a spectrum formed by two Fraunhofer flint prisms, and received,
after being concentrated by a lens, on a white screen, when looked at by reflection
at a black glass to diminish the intensity, the yellow was seen to be encroached
upon by " a full and undeniable green colour." This green, which, I presume,
was decomposable by a prism, might result from the composition of impure rays,
the effect of partial impurity of the spectrum, being increased by the concentration
of the rays by a lens. Possibly, also, the diffusion of the green may have been
caused to seme extent in the same manner as in the experiment (y) described in
p. 346.
THE THEORY OF LIGHT. 353
neutral. The reason seems to be that the spectrum colours
cannot all be imitated by these three, indigo and violet being
excluded.
It is found by experiment that yellow may be formed by
a combination of the less refrangible rays of the spectrum in
clusive of green, and blue by a combination of the remainder.
Hence by comparison with the theory it may be concluded
that each of these portions consists of two parts that are not
too impure to produce by their combination an intermediate
colour. The result, however, of combining the blue and
yellow thus produced is, as is known, white light ; most pro
bably because these components are too impure for producing
an intermediate colour*. It is evident that if from the more
refrangible portion we take away the indigo and violet, the
result of combining the two portions would not be a neutral
tint. (See the preceding paragraph.)
Judging from the analogy of colours to musical sounds,
the undulatory theory would lead to the expectation that the
sensation of colour would result from impulses that fulfil the
condition of regularity however produced. Now the ratio of
the wavelengths of red and violet is very nearly that of 3 to 2,
and the combination of wavelengths in this ratio gives rise,
as is known, not to beats, but to the regular recurrence of
maxima of the same magnitude. Accordingly it is found by
experience that mixtures of red and violet produce purple,
a decided colour in which the eye seems to distinguish the
components as the ear distinguishes the components of a
harmony. Possibly rose colour may be a harmonious result
* When, however, Sunlight is received on white paper so as to be contrasted
with the whiteness of the paper, it always appears, at least to my sight, to have a
tinge of.yellow. From this fact I should say that the result of combining all the
colours of the spectrum partakes in some degree of the colour of that component
which as to quantity is in excess, and which as to position divides the spectrum
into two parts of nearly equal intensity. Seen from a sufficient distance the Sun
might be classed among the yellow stars. To account for stars being of different
colours it is only necessary to suppose that the quantities of the components of
their spectra are in different proportions.
' ''
354 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
from a mixture of red and blue having wavelengths in the
ratio of 4 to 3.
On reviewing the foregoing comparison of the mathema
tical theory of the composition of colours with experiments, it
may be seen that the explanations which have been given all
rest on the hypothesis that the colours of a pure spectrum are
uncompounded. The number and variety of the explana
tions would seem therefore to have established the truth of
that hypothesis.
(20) The phenomena of Diffraction come under the same
category as those which have been hitherto considered, inas
much as experiments shew that they depend wholly on pro
perties of the medium which is the vehicle of light, not being
in any degree determined by the particular constitution or
intimate qualities of the diffracting body. But since the ex
planation of these phenomena rests on the law of limited
lateral divergence, and this law has not yet been mathemati
cally ascertained, I am not prepared to treat with strictness
this part of the Undulatory Theory of Light. It is, however,
to be said that the empirical principle usually adopted in the
theoretical calculation of the phenomena of diffraction, viz.,
that of dividing the front of a wave into elementary portions,
and attributing to each a limited amount of lateral divergence,
is (as I have intimated in p. 292) consistent with the laws
of composite motion to which my hydrodynamical researches
have conducted ; and, as far as I am aware, no other proposed
foundation of the theory of light is in the same manner and
degree compatible with that principle. I consider, therefore,
that I am entitled to regard the theoretical explanations of
phenomena of diffraction that have been given in the usual
manner according to Fresnel's views, as belonging exclusively
to the Undulatory Theory of Light expounded in this work.
I have now completed the comparison of the Theory with
the first of the two classes of phenomena mentioned in page
321, namely, those which are referable solely to the properties,
THE THEORY OF LIGHT. 355
as mathematically ascertained, of the aetherial medium. The
comparisons are comprised in the sections numbered (1) (20),
which include about as many different kinds of phenomena.
For the sake of distinctness and facility of reference, the facts
and laws which the theory has accounted for are indicated by
being expressed in Italics. The number and variety of the
explanations afford cumulative evidence of the truth of the
fundamental hypotheses. It is especially to be noticed that
together with the more obvious phenomena the theory has
accounted for the composite character of light, its polarization,
the transmutability of rays, and not less satisfactorily, 1 think,
for the effects of compounding colours. It should also be
remembered that these facts, so various and so peculiar, are
known to us only through the medium of the sense of sight,
and that prima" facie there would seem to be no probability of
any relation between such a sensation and the movements of
an elastic fluid. The case is, however, precisely the same
with the sensation of sound, which is something utterly di
verse from movements of the air ; and yet we know, as matter
of experience, that sound is generated by such movements.
This experience, without which it is scarcely possible that the
undulatory theory of light could have been imagined, sug
gested that as vibrations of the air acting dynamically on
the parts of the ear produce sound, so the vibrations of a
more subtle elastic medium, acting on the constituent parts of
the eye, might produce the sensation of light. Hence the hypo
thesis of an gether was adopted, and the necessity arose of
determining its movements by mathematical calculation, in
order to compare them with the observed phenomena of light.
The requisite mathematical reasoning having been gone
through under the head of Hydrodynamics, and the appropri
ate comparisons made in the foregoing sections (1) (20), the
points of analogy between the light sensations and the laws
of the movements of the aether are found to be so many and
of such particularity, that scarcely less than positive proof is
obtained of the actual existence of an elastic fluid such as the
232
356 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
aether was assumed to be. It is inconceivable that the
analogies can be accounted for in any other way. Resting,
therefore, on this argument, I shall, in subsequent physical
researches, regard the sether as a reality.
This position having been gained, we are prepared to
enter upon the consideration of phenomena of light of the
other class, those, namely, which depend on particular rela
tions of the motions of the aether to visible and tangible sub
stances. The theory of these phenomena necessarily rests on
hypotheses respecting the properties and constituency of such
substances, as well as on those that have been already made
relative to the aether. In framing hypotheses of the former
kind I shall adhere strictly to the principles enunciated by
Newton in his Eegula Tertia Philosophandi (Principia,
Lib. III.), and for the most part I shall adopt the views which
he has derived from them respecting the qualities of the ulti
mate parts of bodies. In Newton's Third Eule three distinct
principles of physical enquiry are embodied. First, that
hypotheses are not to be made arbitrarily, or from mere ima
gination*, but according to "the tenor of experiments;"
that is, as I understand the expression, they are to be such
only as are suggested by experience, or may be supported by
reasons drawn from the antecedent and actual state of experi
mental science. Secondly, that only such qualities are to be
attributed to the ultimate parts of bodies as are cognisable by
the senses, and by our experience of masses. Thirdly, that
the universal qualities of the ultimate parts of bodies admit of
no variation as to quantity ("intendi et remitti nequeunt"),
and are inseparable from them ("nonpossunt auferri"). The
following are the hypotheses which, guided by these rules or
principles, I have selected for the foundation of reasoning
both in the remaining part of the Theory of Light, and also
in all the subsequent Physical Theories. The reasons for
selecting them will be given at the same time.
* "Somnia temere confingenda non sunt." This rule has been very little
attended to by some theorists of the present day.
THE THEORY OF LIGHT. 357
I. It will be supposed that all visible and tangible sub
stances consist of extremely minute parts that are indivisible,
and are, therefore, properly called atoms. The adoption,
hypothetically, of this very ancient idea respecting the con
stituents of bodies, is justified by the facts of modern
chemical science, the ascertained laws of chemical combina
tions being very reasonably accounted for by supposing the
ultimate parts of bodies to be invariable and indivisible.
II. All atoms possess the quality of inertia. This hypo
thesis is made on the principle that the experienced inertia of
masses is due to the inertia of the constituent parts. I accept
the doctrine of Newton that inertia is not a quantitative, but
an essential quality. He calls it " vis insita," and affirms
that it is " immutabilis." In fact, it does not appear that
inertia is susceptible of measurement : there may be more or
less of inert matter, but not more or less of inertia. Accord
ingly all atoms have the same intrinsic inertia.
III. All atoms have magnitude and form. Since from
experience we have no conception of matter apart from mag
nitude and form, we necessarily attribute these properties to
the ultimate parts of matter. Both the magnitude and the
form of an atom must be supposed to be invariable, because
in the properties of ultimate parts no quality of variability
can enter, inasmuch a's these properties are fixed elements
from which the laws or modes of variation in masses are
to be determined by calculation. It may, however, be sup
posed that atoms differ in magnitude.
IV. To the above hypotheses I add another, not in
cluded among those of Newton, namely, that all atoms have
the spherical form. In adopting this hypothesis regard was
had to facts of experience, such as the following. The pro
perties of bodies in a fluid or gaseous state are in no respect
altered by any change of the relative positions of the parts,
358 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
This fact, which seems to indicate that the mutual action be
tween atoms has no relation to direction in the atoms, is, at
least, compatible with their being of a spherical form, but can
hardly be conceived to be consistent with any other form.
Again, light is found to traverse some substances without
undergoing any modification, or change of rate of propagation,
upon altering the direction of its passage through them ; and
although this is not the case with others, it is reasonable,
since the latter are known to be crystalline, to infer that the
changes are entirely attributable to the crystalline arrange
ment. Also the supposed spherical form will subsequently be
made the basis of calculation, by comparison of the results of
which with experiment the truth of the supposition may be
tested ; on which account it is the less necessary to sustain it
by antecedent considerations.
The fundamental ideas respecting matter embraced by the
foregoing hypotheses may be concisely expressed in the fol
lowing terms : All bodies consist of inert spherical atoms,
extremely small, and of different, but invariable, magni
tudes.
V. The fundamental and only admissible idea of force is
that of pressure, exerted either actively by the aether against
the surfaces of the atoms, or as reaction of the atoms on the
aether by resistance to that pressure. The principle of de
riving fundamental physical conceptions from the indications
of the senses, does not admit of regarding gravity, or any
other force varying with distance, as an essential quality of
matter, because, according to that principle, we must, in seek
ing for the simplest idea of physical force, have regard to the
sense of touch. Now by this sense we obtain a perception of
force as pressure, distinct and unique, and not involving the
variable element of distance which enters into the perception
of force as derived from the sense of sight alone. Thus on
the ground of simplicity, as well as of distinct perceptibility,
the fundamental idea of force is pressure. If it be urged that
THE THEORY OF LIGHT. 359
the progress of physical science has shewn that when the
hand touches any substance there is no actual contact of parts
of the hand with parts of the substance, I reply, after admit
ting this to be the case, that by touching we do in a certain
manner acquire a perception of contact as something distinctly
different from noncontact, and that as this is a common sensa
tion and universally experienced, it is proper for being placed
among the fundamentals of a system of philosophy which
rests on the indications of the senses. (This point will be
farther adverted to in a recapitulation of the general argu
ment, which will be given at the conclusion of the work.)
In conformity with the above views Newton says, at the
conclusion of Hegula III., that he by no means regards
gravity as being essential to bodies ("attamen gravitatem
corporibus essentialem esse minim e affirmo "), and assigns
as the reason, that gravity diminishes in quantity with in
crease of distance from the attracting body. This reason is
completely valid on the ground that the fundamental ideas
of philosophy are not quantitative, and that all quantitative
relations are determinable by mathematical calculation founded
on simple or primary ideas. Thus from the mere fact that
the expression of the law of gravity involves the word square,
it may be inferred that that law. is deducible from antece
dent principles. These considerations will sufficiently explain
why in the second part of the Theory of Light, as well as
in all the other Physical Theories, the aether is assumed to
be every where of the same density in its quiescent state.
All the different kinds of physical force being by hypothesis
modes of action of the pressure of the ajther, it follows that
the aether itself must be supposed to be incapable of being
acted upon by them.
When the aether is in a state of motion the variations of
the pressure are assumed to be exactly proportional to varia
tions of the density, because this law is suggested by the
relation known actually to exist between the pressure and
density of air of given temperature, and is besides the simplest
360 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
conceivable. With respect to the aether the law can be proved
to be true only in proportion as mathematical inferences drawn
from it shew that it is adequate to explain phenomena.
After Newton had inferred, from principles virtually the
same as those adopted above, the qualities of the ultimate
constituents of bodies, he added, " This is the foundation of
all philosophy*." Elsewhere in the Principia he disclaims
making hypotheses (" Hypotheses non fingo"). It is evident,
therefore, that he did not regard the qualities he assigned to
the ultimate parts of bodies as hypothetical in the usual sense
of that word, but as foundations necessary for physical research,
ascertainable by a priori reasoning, and necessarily true if
there be truth in philosophy. These ideas were also main
tained by Locke, and, in fact, characterized that epoch so
remarkable for advancement in science. Individually I have
never had any difficulty in giving them my assent, neither can
I imagine any reasons for objecting to them. Since, however,
some of my contemporaries, without giving reasons, have
expressed very strongly their dissent from these principles,
I have adopted the line of argument which follows, although
I do not allow that the a priori reasoning by which Newton
and Locke sustain their conceptions of the existence and
essential qualities of atoms is invalid or insufficient.
Waiving the reasons assigned in paragraphs I., II., III.,
and IV. for the qualities ascribed to atoms, as well as the
reasons subsequently given for the supposed properties of the
aither, I propose to regard the qualities of both kinds as
merely hypothetical; and I maintain that as such they cannot
reasonably, or logically, be objected to, inasmuch as, being
expressed in terms intelligible from sensation and experience,
and forming an appropriate foundation for mathematical cal
* " Hoc est fundamentum philosophise totius." See an Article on this dictum
in the Philosophical Magazine for October 1863, p. 280; also two Articles on the
" Principles of Theoretical Physics," one in the Supplementary Number of the
Phil. Mag. for June 1861, p. 504, and the other in the Number for April, 1862,
p. 313.
THE THEORY OF LIGHT. 361
culation, they fulfil every condition that can be demanded of
hypotheses. The only arguments that can be adduced against
such hypotheses are those which might be drawn from a
comparison of results obtained from them mathematically
with experimental facts. They would be proved to be false
by a single instance of contradiction by fact of any inference
strictly derived from them, or, on the other hand, they might
be verified by a large number of comparisons of facts with
such inferences. I take occasion to remark here, that the
evidence given by the reasoning in the first part of the
Theory of Light for the reality of the aether, would not be
invalidated by the failure of the second part to satisfy pheno
mena, as such failure would only involve the consequence
that the atoms or their supposed qualities must be abandoned.
But a perfectly successful comparison of the second part with
facts would confirm the previous evidence for the reality of
the supposed properties of the aether, and at the same time
establish the actual existence of the atoms and of the qualities
attributed to them.
Before proceeding to the second part of this Theory, it
will be right to draw a distinction as to kind and degree
between the verifications which the hypotheses relative to the
sether, and those relative to the ultimate constituents of bodies,
respectively admit of in the present state of physical science.
The verification of the former, as we have seen, is effected by
direct comparison of results deduced from them by rigid
mathematical reasoning with observed phenomena. But the
other class do not in the same manner or degree allow of this
kind of verification, because the theoretical explanation by
exact mathematical reasoning of phenomena depending on
the intimate constitution of bodies would require in general
the knowledge of the mutual action between the aether and
the atoms, and of the comparative numbers, magnitudes, and
arrangements of the latter. This knowledge cannot be imme
diately furnished by experimental physics, and ought rather
to be looked for as the final result of physical inquiry pursued
362 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
both experimentally and theoretically in different directions
and by all available means.
These preliminaries, the greater part of which apply to
physical theories in general, having been gone through, we
may now advance to the consideration of the other class of
questions relating to phenomena of light. It is intended to
enunciate these as separate Problems, and to attempt their
solutions by means of hydrodynamical Propositions and Ex
amples, which, mainly with reference to this application of
them, have already been under discussion.
Problem I. To account for the observed laws of trans
mission of light through noncrystalline transparent media.
Omitting at present the consideration of the circumstances
attending the incidence of light on the surfaces of transparent
media and its entrance into them, let us suppose that a portion
of homogeneous light has already entered into a certain medium,
and, for the sake of distinctness, that the entrance took place
by perpendicular incidence on a plane surface of the medium.
Under these conditions experiment has shewn that the intro
mitted light may differ in no respect from the same light
before intromittence, excepting that it is propagated with less
velocity. The theory has, accordingly, to account for these
two facts, the possibility of transmission of light in the
medium without change of quality, and the diminished rate
of propagation.
In consequence of the preliminary hypotheses the medium
must be supposed to consist of an unlimited number of minute
spherical atoms, and the eether in the spaces intermediate to
the atoms to be everywhere of the same density as in the
surrounding space outside the medium. Also the atoms must
be in such number and so arranged as to have the same effect
on the motion of the waves in whatever direction the light is
propagated. The retardation of the propagation may be
attributed to the obstacle which the presence of a vast number
of atoms opposes to the free motion of the aether, this being
THE THEORY OF LIGHT. 363
an obvious and perfectly intelligible cause of retardation ; and
that it operates in the manner supposed will appear from the
following considerations.
In the first place, supposing KO, to be the velocity of pro
pagation of the intromitted waves, it is plain that a certain
number of waves which out of the medium had the aggregate
KCL
breadth z t . would in the medium have the breadth z x ,
KO,
KZ
or l . Hence if X be the breadth of an individual wave
K
before entrance, and V be its breadth after, we shall have
n\ = z i and n\' = * ; so that ,  . Also by the hypo
K A/ K
thesis of uniform propagation, and the known relation in that
case between the velocity (V) and condensation (or) of the
waves, supposing them to be planewaves,
V /c'acr = m sin  (icat z + c).
A
Now if o be the condensation at any point of a wave out of
the medium, and a the corresponding condensation of the
same wave within, since in the two cases the variation of
condensation follows the same law of the circular sine, and
the total quantity of condensation of the wave remains under
the supposed circumstances the same, it follows that <r and <7
are to each other inversely as the breadths of the waves;
or a = ^7 = ~ . Consequently V /ea<r . This result shews
A. K
that the velocity of a particle of the aether is the same within
the medium as without; and as the times of oscillation are
V A
respectively  and , which are equal to each other, the
excursions must also be the same in the two cases. Thus
the motion of each particle is absolutely the same in the
medium as in free space, if we leave out of account the effect
of the loss of condensation caused by reflection at its surface.
364 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
Again, by differentiating the above equations,
dV , dV , 2 ,d<r
j = K a j~ = K a 7 .
at dz dz
These, together with the preceding equations, prove that the
setherial waves within the medium obey the same laws as
in sether free of atoms, provided the elasticity of the latter
/e'V
were ^ instead of a 2 ; and that the effective acceleration
of a particle of the aether in the medium, in the case of plane
waves, is to that in free space for the same variation of con
densation, as K* to K?. It is evident, since the action between
contiguous parts of the sether is diminished in this ratio, that
there must be the same diminution of moving force in the
action of the aether on an atom.
From these considerations it will be seen, that if we only
suppose the elasticity of the aether to be altered in the above
mentioned ratio, we may at once employ the solution of Ex
ample VI. (p. 279) for ascertaining the velocity at any point
due to the reaction of an individual atom of the medium.
First, it will be assumed that the atom is fixed. Let the
velocity ( V) in the incident wave at any time t, and at the
)__
position of the centre of the atom, be ??&sin T (/c'at + c }, and
A<
let U and W have the same significations as in the cited
Example. Then it will appear from the expressions for these
velocities in pages 283 and 284, that the parts along and
perpendicular to the radius vector, due to the reaction of the
atom, are respectively
VV , VV . a
 cos 6 and 5 sin 0,
r 3 2r 3
b being the radius of the atom, and the direction correspond
ing to 6 = being opposite to that of propagation. If these
velocities be resolved in the directions parallel and perpen
dicular to that of propagation, the two resultants will be
THE THEORY OF LIGHT. 365
.
m * C S '
The latter of these will have for the different values of
from to TT as many positive as negative values, whilst the
former has for all values of the same sign as V. This
reasoning proves that the mean of the reactionary velocities
due to the atom has a constant ratio to the velocity V and is
always opposed to it in direction. It is to be observed that
the expression in p. 283 for the condensation a v l due to
the reaction of the atom, since it contains the very small
factor qr, may be omitted in comparison with the terms that
have been taken into account ; for which reason also the parts
of the values of U and W obtained in page 287 may be left
out of consideration. The motion, in short, is the same in
this approximation as if the fluid were incompressible.
The effect of the reaction of a single atom of the medium
having been shewn to be such as this, the investigation of the
total effect of the reaction of the atoms may be conducted in
the following manner. Taking a slice of the medium bounded
c\
by planes parallel to the fronts of the waves, let z and
,
z \ be the distances of these planes from the origin of z,
$z being indefinitely small. Then the reactionary velocity of
the aether at any distance z a, produced by the reactions of
all the atoms of the slice, and assumed to be wholly perpen
dicular to its limiting planes, will, from what is proved above,
vary conjointly as the velocity of the sether at the distance z,
and an unknown function of a. It may, therefore, have the
following expression :
m(j) (a) Bz sin  (tc'at z + c ).
A/
To find the retarding effect of all such slices of the medium
at a given distance, let this distance be the constant z ; so
366 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
that Sz = Sa, and z z + a. Then the total velocity of re
tardation at that distance is
m
J <j> (a) Sa sin ~ (/cat  a  z + c ),
the integral being taken through all the values of a for which
<f> (a) has sensible magnitude. Now considering that the
velocities from which </> (a) is derived all have the multiplier
5 *, it is evident that for values of r which are large multiples
of 5, cj) (a) must be exceedingly small. Therefore, also, the
total value of / $ (a) Sa is obtained very approximately by
integrating between limits a t and + a t such that c^ is a
large multiple of b. But on account of the extremely small
size of the atoms, a large multiple of b may be very small
compared to X. Hence the integral would be very nearly the
same, if a in the trigonometrical function be supposed to have
its mean value, which is zero. Consequently putting K for
f<f) (a) Sa, which for a given medium will be absolutely con
stant, we have for the velocity of retardation KV\ that is, it
has a constant ratio to the actual velocity.
Since that ratio is constant, it follows that the retarding
force of the atoms has a constant ratio to the actual accele
rative force of the aether, and that this ratio is the quantity K.
But the actual accelerative force is the force due to the actual
variations of density diminished by the retarding force. Hence
if p be the density of the aether at any distance z from the
origin, and if /c'V be the apparent elasticity of the aether
within the medium, that without being V, we shall have
* In an investigation analogous to the present one, contained in the Philo
sophical Magazine for December 1863, p. 474, the argument rests on the sup
position that the multiplier is . This error, the origin of which has already
been pointed out in pages 259 and 272, vitiates the reasoning of that investigation
b 3
rather than the conclusions drawn from it. The factor ^ is evidently more
suitable to the tenor of the argument.
THE THEORY OF LIGHT. 367
ic" 2 a?dp

paz
whence K' Z (1 + K) = K\
Now for different degrees of density of the same substance,
the constant K will evidently vary proportionally to the
number of atoms in a given space, that is, to the density.
Hence calling the density 8, we have K=HS, H being a new
constant characterizing the medium. Consequently, substi
tuting this value of K, and putting p for the ratio of K to K ,
the resulting equation is
. or
This formula has been verified by M. Biot for atmospheric
air by a series of experiments in which the density of the air
ranged from zero to about the density corresponding to the
mean barometric pressure. (Traife de Physique, Tom. in.
p. 304).
We have now to determine what modification the above
value of $ I undergoes when the atom is supposed to be
moveable. The effect of the impulse of waves on a moveable
sphere has been considered in Example VII. (p. 296) : but in
that case the motion of the sphere was supposed to be wholly
due to the action of the waves. In the Problem before us
we must take into account that the atom is not a solitary one,
but a component of the medium which the light traverses,
and that it is consequently held in a position of rest by equi
librated attractive and repulsive forces*. These forces are
brought into play by the disturbance of the atom from its
normal position by the action of the waves, and have the
effect of modifying the motion of the atom. This effect I
propose to take account of in the following manner. Suppose
that by the impulse of the aetherial waves the mean interval
* A Theory of these forces is given subsequently under the head of " Heat
and Molecular .Attraction."
368 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
between consecutive atoms, estimated in the direction of pro
pagation, is diminished by a quantity e extremely small com
pared to that interval. Then since the resulting molecular
action is proportional to the relative displacement of the
atoms, the acceleration of an atom due to this cause is e 2 j ,
Ct
e being considered to be a function of 2, and e 2 being an un
known constant characteristic of the intrinsic elasticity of the
medium. Now it will be assumed to be a condition of trans
parency, that the movements of the atoms are determined by
the action of the waves in such manner that both these move
ments and the values of e are propagated through the medium
with the velocity ic'a of the propagation of the waves. Hence,
this being a case of uniform propagation of velocity and con
densation,
v = tc'ae f(z vat) ,
v being the velocity of any atom. Consequently
___ ___
dz ~ * V dt ~ K '*d 2 df
very nearly. It is to be observed that the constant e* may
be regarded as a measure of the force by which an atom dis
placed relatively to surrounding atoms tends to return to a
position of relative equilibrium. On account of the small
movements with which we are here concerned, which do not
sensibly alter the density of the medium, this force will be
very nearly the same as that by which any atom displaced
singly would tend to return to its position of absolute equi
librium ; so that e 2 may be taken as the measure of the mole
cular elasticity in the given direction of propagation. It is,
therefore, possible that this constant, as depending on the
immediate action of molecular forces, may be comparable in
magnitude with K*a*.
Let us now conceive, in accordance with the principle
adopted in the solution of Example VII., the actual accele
rative force of the atom to be impressed in the contrary direc
THE THEORY OF LIGHT. 369
tion botli upon it and upon the gether in such manner that
the action between them remains unaltered. The atom is
thus made to have a fixed position, and at the same time is
subjected to the action of virtual waves the velocity in which,
at the position of the atom's centre, is the excess of the
velocity in the actual waves above that of the atom, and the
condensation there is that which corresponds to this difference
of velocity. The Problem, therefore, becomes the same as
that of Example VI., if in place of the given velocity at
points for which ? cos = (p. 279), we suppose the velocity
to be
. 27T , dz
m sin "T
For, by hypothesis, the motion of the atom is wholly vibra
tory, and the period of its vibrations is the same as that of
the given waves, although, as we have seen, its motion is
modified by the proper elasticity of the medium. Also, just
as in the cited Example, the effect of lateral divergence is to
be taken into account by means of a like factor 1 h^
This being understood, by the formula in page 296, taking
only the first term (see p. 365), we have for the accelerative
force due exclusively to the action of the waves, the value
dv
dt*
o__
F being put for msm (tcat + cj. To this must be added,
A;
in order to get the total accelerative force, that which is due
to the elasticity of the medium, and we thus obtain
^_ 3 ( dV d * z \ e * d * z
"df ~ .2A ^ l ' ' ^ \dt ~ d?) + ^df'
By integration, supposing V and j to begin together,
dz 3 dz e * dz
'
24
370 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
which equation gives
r dz (*V6 2 )F
Cut ,00 o o
, , ,
(
This result shews that in order to take account of the mobi
lity of the atoms and the elasticity of the medium, the con
stant X, applicable to the case of the fixed atom, is to be
multiplied by the quantity which multiplies V in this last
equation. Accordingly, since K'[JL K, we have
In this equation IT, 8, and e 2 are constants for a given non
crystalline medium, a and A are absolutely constant, and h l
is a quantity involving X in a manner which will presently
come under consideration, but in other respects is absolutely
constant. From the reasoning thus far we may conclude
that the value of p is independent of the intensity of the light,
and is constant for a given medium and a given wavelength.
This result is confirmed by experiments.
It has already been stated (p. 294) that an exact expres
sion for the quantity 1 Ji is not obtainable, because the law
of lateral divergence has not "been ascertained. We may,
however, assume that its value depends on the function /,
which expresses the law of the diminution in the transverse
direction of simple vibrations relative to a given axis. Since
K = (l+ L and /c 4 (/c 2  1) = 1 (p. 224), it will readily be
found from the series for f in p. 210 that
7rr\ 2 1 /TrrV 1 /7rr\ 6
 &c 
From this expression it may be inferred that at any given
position the diminution of velocity and condensation due to
distance from the axes of simple vibrations, may, for any
THE THEORY OF LIGHT. 371
number of axes, whether or not they be parallel, be denoted
by such an expression as
the value of X being the same for all the sets of vibrations,
and the distance r from an axis being different for axes in
different positions. As on account of the extremely small
ratio of the radius of the atom to X, we are concerned here
only with values of r very small compared to X, it may be
presumed that the first term of the above series is much more
considerable than the remainder; so that h = ^.\^[ very
nearly, or h = . 2 , k being an unknown constant, always posi
X
tive because h is necessarily positive. But we have to ascer
tain the value (hj applicable to the case of propagation of the
waves in the medium. Relative to this point, it is first to be
remarked that since /is a function only of r, X, and the con
stant K, it is independent both of the elasticity of the fluid
and the magnitudes of the condensations. Now by entrance
into the medium, it is true that both the effective elasticity
and the condensations of the waves are altered ; but, as is
shewn in page 363, the velocity and the period of the vibra
tions remain the same. Thus there is reason to conclude that
the disturbed motion of the aether relative to any atom of the
medium is the same as for a single atom in free space, and
consequently that the law of distribution of condensation about
its surface is the same in the two cases. For, although X is
changed to X' in the medium, if the value of S . r 2 be changed
in the duplicate ratio, that of h does not alter. Guided by these
k
considerations I shall now suppose that \ = h = 3 , and that
A,
the constant k is independent of the particular medium. After
substituting this value of h v the relation between //, and X
given by the equation (/3) admits of being put under the
242
372 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
form
A, Bj C being respectively substituted for
Since it is known from experiment that //, 2 1 is always
positive*, we may suppose, in order that the equation (/3)
may satisfy this condition, that *V is always greater than
/u,V. In that <iase, according to the theory, A, J3, C will all
be positive quantities ; but it is not possible by theory alone
to determine their numerical values. I propose, therefore, to
give two instances of determining them from experimental
data, and at the same time to employ the results in testing
the truth of the formula (<y).
For this purpose I have adopted the values of X given by
Angstrom in Poggendorff's Annalen, 1864, Vol. 123, p. 493,
and for the first instance I have selected Fraunhofer's deter
mination of fju for Flint Glass, No. 13. (See Art. 437 of the
Treatise on Light in the Encyclopaedia Metropolitana.) To
calculate A, B, C for this case three equations were formed
by means of the values of // and X for the rays (5), (E), and
(H), the solution of which gave the following results :
A = 10,252642, B= 1,686649, C= 13,760015.
The values of \ for the other rays were then calculated by
the formula (7) from the corresponding values of /t, and com
pared as follows with observation t :
* It should be noticed that /a. is here assumed to be equal to the ratio of the
sine of the angle of incidence to the sine of the angle of refraction, this law not
having as yet been demonstrated theoretically.
f In an Article on the Dispersion of Light in the Philosophical Magazine for
December 1864 (Supplement), pp. 500, 501, I have made the same comparison by
means of Fraunhofer's values of X ; and in the Number for May, 1865, pp. 337, 338,
I repeated the calculation, using Angstrom's values, but the results are not as
accurate as those here given.
THE THE011Y OF LIGHT.
373
Xby
Xby
Excess of
Ray.
Value of /A.
observation.
calculation.
calculation.
(B)
1,62775
2,5397
(2,5397)
0,0000
(0)
1,62968
2,4263
2,4247
 0,0016
(D)
1,63504
2,1786
2,1758
0,0028
(E)
1,64202
1,9482
(1,9482)
0,0000
(F)
1,64826
1,7973
1,7996
+ 0,0023
(0)
1,66029
1,5923
1,5949
+ 0,0026
(S)
1,67106
1,4672
(1,4672)
0,0000
The values of X in brackets were used in determining the
constants. The adopted unit of X, which is arbitrary, was
chosen for convenience in calculating.
The following results were obtained by a like comparison
for oil of cassia, which was selected on account of its great
dispersive power. The values of fj, were taken from Baden
Powell's Paper in the Transactions of the Royal Society for
1837, Part L, p. 22. By calculating as in the former instance
it was found that .4 = 4,55574, =0,64905, (7=4,46624.
observation.
2,5397
2,4263
2,1786
1,9482
1,7973
1,5923
1,4672
In the first example the difference between the extreme
values of //, is 0,04331, and the difference between the corre
sponding values of X is 1,0725. Hence a difference of 0,0010
in X corresponds to an average difference of 0,00004 in //,.
In the other example the difference between the extreme
values of //, is 0,1117; so that a difference of 0,0010 in X
corresponds to a difference of 0,00010 in /t. It is, therefore,
probable that the excesses of calculation are scarcely greater
than those due to errors of observation. As, however, the
excesses in both instances seem to follow a law, the accord
Ray.
Value of fji.
/ T}\
1 Jj 1
1
,5885
(0}
1
,5918
(D)
1
,6017
(E)
1
,6155
(F)
1
,6295
( /y\
\ur)
1
,6607
(H)
1
,7002
X by
calculation.
Excess of
calculation.
(2,5397)
0,0000
2,4282
+ 0,0019
2,1771
0,0015
(1,9482)
0,0000
1,7958
0,0015
1,5929
+ 0,0006
(1,4672)
0,0000
374 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
ance might be made still closer by assuming h^ to be equal
k k'
to r 2 + ^i But as another unknown constant would have to
X X
be determined, the resulting equation would be little more
than a formula of interpolation.
Although, on account of the small range of the values
of fi, the foregoing comparisons are scarcely any direct test
of the truth of the theory, they are nevertheless important as
shewing that it is not incompatible with a certain class of
facts. If no such comparison had been possible, the theory
might be said to fail. This is the more to be remarked be
cause the possibility of making the comparisons has wholly
depended on the function of X which from a priori considera
tions was substituted for the constant h lt In short, since the
equation (7) and the mode in which it involves //, and X were
derived entirely from independent physical principles, the cir
cumstance of its being capable of giving results not at vari*
ance with experiments may be taken as presumptive evidence
of the correctness of the antecedent theoretical reasoning, and
in particular of the reasoning by which the quantity \ was
shewn to have its origin in the composite character of the
motion. (See pages 293 295.) I had previously obtained
other equations, which were found on trial not to admit of
satisfactory comparison with experiment. At the same time,
as the present investigation has shewn, they were not strictly
deduced from appropriate a priori principles*.
The foregoing theory requires as a condition of transparency
that the atoms of the medium should be susceptible of vibra
tions having the same periods as those of the vibrations of the
ogther which disturb them. It is not a necessary part of this
theory to determine under what physical circumstances this con
dition is fulfilled, and what are the causes of opacity. Relative
to this point it may, however, be remarked that it is quite
* See an Article in the Philosophical Magazine, Vol. vni., 1830, p. 169; also
the investigation of the equation (a) in Vol. xxvi., 1863, pp. 471476, with the
remarks in Vol. xxvu., 1864, p. 452.
THE THEORY OP LIGHT. 375
conceivable that the condition might be satisfied for certain
values of X and not for others, and that thus the fact may be
accounted for that some substances allow rays of certain
colours to pass through them and stop all others. It is
known that an ingredient which forms a very small portion of
a substance sometimes determines the colour of its transmitted
light, or even produces opacity; the reason probably being
that such ingredient imposes limitations on the periods of
the vibrations of the constituent atoms.
Problem II. To account for the observed laws of the
transmission of light through crystalline transparent media.
Reverting to the equation (/3) in page 370, and putting
^ for h lt it will be seen that, according to that equation, if
A,
the elasticity e* of the medium be the same in all directions,
the rate of transmission of light of a given colour will also be
independent of direction. But there is reason to conclude
from experimental evidence that the elasticity of certain
crystals is different in different directions ; and it may reason
ably be assumed that this is generally the case in regularly
crystallized substances. We have, therefore, now to inquire
what effect this circumstance may have on the transmission of
light in such bodies. In the first place, from the facts of
crystallography it may be presumed that the elasticity is in
some manner connected with atomic arrangement. It does
not seem possible to account for planes of cleavage on any
other principle. If the atomic arrangement should be such as
to be symmetrical about any straight line drawn parallel to a
fixed direction in the crystal, it seems to be a necessary con
sequence that the elasticity is the same in all directions per
pendicular to that line. For instance, in a uniaxal crystal, as
Iceland spar, the elasticities in directions perpendicular to the
crystallographical axis may be presumed to be all equal in
consequence of a symmetrical arrangement of the atoms about
that axis. But it is allowable to make a more comprehensive
supposition relative to atomic arrangement. We may sup
376 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
pose it to be symmetrical with respect to three planes drawn
always parallel to three fixed planes in the crystal at right
angles to each other. Taking the intersections of the planes
for the axes of coordinates, if a surface be described the
radius vector of which drawn from the origin is any function
of the atomic arrangement, it is plain that this surface must
satisfy the condition 2 2 =/(o? 2 , #*). Now, by hypothesis, the
elasticity in any direction is some function of the atomic
arrangement. Hence the radius vector (r) of a surface satis
fying the above condition may be taken to represent the
elasticity in the direction in which it is drawn. That is, e 2
representing the elasticity, and jB' 2 a certain constant,
This equation shews that e* may have a maximum or mini
mum value independently of the form of the function/. For
by differentiating
&*( d.f(x\y*}\
r \C' d.x* )>
_
dx '''' r C' d.x* > ~djj~~ r ' d.f
Hence the equations ^ = and jj = are respectively
satisfied by x and y = ; shewing that the elasticity in
the direction of the axis of z is a maximum or minimum.
The same is evidently the case with respect to the other two
axes. It thus appears that every crystal which satisfies the
assumed law of symmetrical atomic arrangement has three
axes of maximum or minimum elasticity at right angles to
each other.
Next let the elasticities in the directions of the three axes
of #, #, z be respectively e*, e 2 2 , e 3 2 , and suppose that the force
which is brought into action by a given displacement of an
atom in the direction of an axis, is equal to the elasticity in
that direction x displacement*. We have now to find the
* This is equivalent to the supposition made in p. 368, where the expression
for the force is e 8 . Also, as there shewn, e 2 is the same whether the dis
ax
placement be relative to the surrounding atoms in motion, or to the same atoms
fixed.
THE THEORY OF LIGHT. 377
elasticity in any direction making the angles a, /3, 7 with the
axes. For this purpose let us regard, as heretofore, the crys
talline medium as being composed of discrete atoms held in
positions of stable equilibrium by attractive and repulsive
forces, and assume that each atom, in accordance with the
law of the coexistence of small vibrations, can perform inde
pendently simultaneous oscillations in different directions. On
this principle a displacement (Br) in the given direction, (sup
posed for the sake of distinctness to take place relative to the
surrounding atoms fixed], may be considered to be the result
ant of the three displacements Sr cos a, Sr cos /3, $r cos 7 in
the directions of the axes. Now these displacements, by
hypothesis, give rise to forces in the directions of the axes
equal to e* x Sr cos a, e* x Sr cos /3,. e* x Br cos 7. But the
original displacement (8r) will riot generally be accompanied
by a force of restitution in the line of displacement, because,
excepting in the case of an axis, the resultant molecular
action of the surrounding atoms is not generally in that line.
It may, however, be presumed that so far as the force of resti
tution acts in the line of displacement, it is equal to the sum
of the parts of the above forces resolved in the direction of that
line ; that is, it is equal to
(e? cos 2 a + e* cos 2 /3 + e 3 2 cos 2 7) x Sr.
Hence since this force of restitution is wholly due to the
elasticity resulting from molecular action, if e 2 be the elasticity
in the given direction, we have
e 2 = 6* cos 2 a + 6 2 2 cos 2 ft + e* cos 2 7.
In this equality e 2 has the same signification as in the equa
tion (j3) in p. 370.
It will now be supposed, in conformity with the indica
tions of experiments, that for a given value of X the values of
//,* in crystals never differ much from a mean value. The
equation (0) shews that a like supposition must also be made
with respect to the values of e*. If then /z 2 and e* be the
378 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
respective mean values, and we assume that $ ^ + /j,' 2 and
e* = e* + e' z , p* and e' z will be small quantities the powers of
which above the first may be neglected. Accordingly by the
usual process of approximation it will be found that the equa
tion (ft) takes the form
A and B being put respectively for
 ,
'
3x?a 2 f k\
and Lj for shortness' sake, standing for r ( 1 ^ j . Hence
A and B are positive quantities if /cV be greater than /^ 2 e 2 .
At the same time from the foregoing value of e 2 we obtain
<? ~ *o 2 = Oi 2 ~ O cos2 + ~ O cos 2 ft + fe 2  e 2 ) cos 2 7 ,
or e* = e/ 2 cos 2 a + e 2 ' 2 cos 2 ft + e/ 2 cos 2 7.
By reverting to the reasoning in page 367, it will be seen that
fji 2 represents the ratio of the elasticity of free aether to the
apparent elasticity of the aether within a medium. If the elasti
city of the medium itself were the same in all directions, this
would also be the ratio of the rates of propagation of light
without and within the medium. But in the case we are
considering of a crystal having different elasticities in dif
ferent directions, these two ratios, for a reason that will be
stated shortly, are generally not identical. At present we are
only entitled to say that tf is the ratio of the elasticity of free
aether to the apparent elasticity of the a3ther within the crystal
in the direction of the transmission of the light, the latter
elasticity depending, as we have seen, both on the obstacles
presented to the motion of the aether by the atoms supposed
moveable, and on the proper elasticity of the medium in that
direction. Let, therefore, a' 2 , Z>' 2 , c' 2 be the apparent elasticities
of the aether within the medium in the directions of the axes
THE THEORY OF LIGHT. 379
of coordinates, and r 2 that in the given direction. Then for
light of a given colour we have the three equations
together with the equation
If the three equations be respectively multiplied by cos 2 a,
cos 2 /3, cos 2 7, and the sum of the results, after taking account
of the foregoing value of e' 2 , be compared with the last equa
tion, it will readily appear that
1 _cos 2 a cos 2 /3 cos 2 7
7~~a^ ~b^~ ~^~'
This may be called the equation of the surface of elasticity,
and will be subsequently cited by that appellation. It is
plain that if a surface be constructed the radius vectors of
which drawn from the origin of the rectangular coordinates
are proportional to r, the surface will be an ellipsoid the semi
axes of which are proportional to a', b r and c.
Although the above equation gives the effective elasticity
of the aether in any direction in the crystal, we cannot imme
diately infer from it velocity of propagation, because we must
take into account that the waves propagated in the crystal
are composed of rayundulations (which, for "brevity, I have
also called rays), and that we have to determine under what
conditions such undulations can be propagated in the medium.
First, it is evident that the transverse motions cannot be the
same in all directions from the axes, inasmuch as this con
dition cannot generally be fulfilled if the effective elasticity be
different in different directions. But rayundulations in which
the transverse motions are symmetrical about axes are the
exponents of common light. Hence it follows that common
light cannot be transmitted through any substance the elasti
city of which varies with the direction ; and it is, therefore,
380 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
incapable of transmission through a doubly refracting medium,
the doubly refracting property being assumed to be due to the
elasticity changing with direction. But a polarized ray is
found by experience to traverse such substances. This fact is,
therefore, to be accounted for by the theory ; which I propose
to do as follows.
For the basis of this enquiry, the principle will be adopted
that a polarized ray is unique in its character, and that under
all circumstances its rate of propagation is that due to the
effective elasticity of the medium in the direction of propaga
tion multiplied by the same constant K. In fact, it is only on
this principle that the motion in the ray satisfies the condition
of making udx + vdy + wdz an exact differential. Since, from
what has been previously shewn, it suffices to have regard
only to the motion contiguous to the axis of the undulations,
let that line be the axis of z, and let the transverse motion be
parallel to the axis of a?, so that there is no motion parallel to
the axis of y. Now it has been shewn (page 218) that for
points contiguous to the axis the direct and transverse velo
cities are expressible by similar formulae, and the condensa
tions in the two directions may also be expressed by analogous
formulae. Also, X and X' being respectively the breadths of
corresponding and simultaneous direct and transverse undula
/ X 2 \^
tions, it was found that K ( 1 + r 2 ) , the elasticities in the
two directions being the same. Suppose now that the elasti
cities in the directions of the axes of x and z are respectively
a* and c*. Then the change of elasticity from the value c x 2 to
a* in the transverse direction will change the rate of the
virtual propagation in that direction in the proportion of c^ to
a r But from what is shewn in page 363, the total condensa
tion of a given wave, and, in fact, the motion and time of
vibration of a given particle, are the same within the medium
as in free space. Hence if X ' be the value of X' for the case
of uniform elasticity, we shall have generally X' =  LJL . Also
THE THEORY OF LIGHT. 381
the foregoing expression for K shews that X will be altered in
the same proportion ; so that if X be its value when a x = c t ,
the general value is ^^ . Hence, since the time of the direct
vibration of a given particle remains the same, it follows that
the rate of propagation in that direction, which is the rate of
actual propagation, becomes KC^ x = /ca t . Thus it depends
entirely on the elasticity in the transverse direction.
It is now required to shew how a* may be calculated.
Conceive the surface of elasticity, the equation of which is
given in page 379, to be described about any point of the axis
of z as its centre, and to be cut by a diametral plane per
pendicular to that axis. Since the surface is an ellipsoid, the
section will be an ellipse, and the radius vectors drawn from
its centre will represent the elasticities in their respective di
rections. But on taking into account the condition of sym
metrical action which must be satisfied relative to a plane of
polarization (as indicated in section (14), page 331), it will be
apparent that the two directions coincident with the axes of
the ellipse are alone applicable to the present enquiry; for with
respect to these directions only are the elasticities symmetrically
disposed. There will, therefore, generally be two planes of
polarization at right angles to each other, and two values of
a, 2 . These values are the semiaxes of the above mentioned
elliptic section, and to obtain them from the equation of the
surface of elasticity is a geometrical problem, the wellknown
solution of which it is unnecessary to give here in detail.
The direction cosines being cos a, cos /8, cos 7, the quadratic
equation from which the two values of a* may be obtained
is the following :
_L 1 /sin 2 a sin 2 ft sin 2 7\ cos 2 a cos 2 /3 Cos2 7_
The positive values of a t derivable from this equation are
the two rates of propagation, in the given direction, of two
382 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
rays polarized in the planes of greatest and least trans
verse elasticity. By putting E*, or x z + y* + z 2 , for a* and
D > D > ~o respectively for cos a, cos /3, cos 7, there results
M Jtt M
the equation in rectangular coordinates of a surface, the two
radius vectors of which drawn from the centre in a given
direction represent the two rates of propagation in that di
rection. This is the known equation of the wavesurface. If
, ,, ,, , r l 1 , cos 2 7 sin 2 7
a = o , the two values of ^ are and ~ \ 7 .
R a a c
It is unnecessary to pursue this investigation farther, as it
will only lead to consequences which have been long esta
blished, although upon very different principles. I will only
add two obvious deductions from the theory*.
(1) An optical axis is defined to be such that the section
of the surface of elasticity by a plane at right angles to it
is a circle ; so that, according to a known property of an
ellipsoid, there are generally two such axes. A principal
plane is any plane passing through an optical axis. By the
theory, the effective elasticities in all directions perpendicular
to an optical axis are equal. Consequently if a ray be pro
pagated in any principal plane of a uniaxal, or biaxal, crystal,
and its transverse vibrations be perpendicular to the plane,
the velocity of propagation will be the same in all directions in
the plane, and the same also in every plane passing through an
optical axis. This result accords with the known fact that
one of the rays of a doubly refracting medium, if propagated
in a principal plane, is subject to the ordinary law of refrac
tion.
(2) If the principal plane of a uniaxal crystal be called
the plane of polarization of the ordinary ray, it follows from
* The law expressed in the first of these deductions has not, I believe, been
demonstrated in any previous theory, neither had it before been determined in an
unambiguous manner whether the vibrations of a polarized ray are perpendicular
or parallel to the plane of polarization. See Professor Stokes's " Report on Double
Refraction " in the Report of the British Association for 1862, pp. 258 and 270.
THE THEORY OF LIGHT. 383
the theory that the transverse motions of a polarized ray are
perpendicular to the plane of polarization* .
Problem III. To investigate the laws of the reflection
and refraction of light at the surfaces of transparent bodies.
It may be assumed that when a series of planewaves,
which obey the law V= Kacr, is incident on any medium, this
relation between the velocity and condensation is suddenly
changed by the obstacle which the atoms of the medium op
pose to the free motion of the aether. From the results of
the solution of Example VI. (page 279) it may be inferred
that the disturbing effect of the atoms extends to a very
minute distance (extremely small compared to X) from the
confines of the medium, and decreases very rapidly with the
increase of distance. Suppose, first, for the sake of simplicity,
that the waves are incident directly on a plane surface. Then
the effect of the retardation, at and very near the surface, will
be to increase suddenly the condensation of the condensed
part of a wave, and the rarefaction of the rarefied part. For
in the case of condensation, a particle of the gether just beyond
the sphere of retardation will move more freely towards the
medium than a particle within its influence ; and in the case
of rarefaction, a particle just beyond the same limit will move
more freely from the medium than one within the limit, the
retardation always acting in the direction contrary to that of
the motion. In the one case the mutual distances of the par
ticles are diminished, or the condensation made greater; in
the other the mutual distances are increased, and consequently
the rarefaction is also made greater. If, on the contrary, the
series of waves pass directly out of the medium into vacuum,
the effects will be reversed, acceleration taking the place of
retardation on account of the waves being suddenly released
* The foregoing theory of the transmission of light in crystallized media is
fundamentally the same as that contained in a Paper in the Transactions of the
Cambridge Philosophical Society, Vol. vin., Part iv., pp. 524 532, and in the
Philosophical Magazine, Vol. xxvr., 1863, pp. 466 483.
384 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
from the obstacles to the motion caused by the atoms. In
the case of condensation, a particle within the medium and
just beyond the limit of acceleration, will move less freely
towards its boundary than one within that limit, and con
sequently the condensation will be suddenly diminished; and
in the case of rarefaction, a particle in the medium just be
yond the influence of the acceleration, will move less freely
from the boundary than one within its influence, and con
sequently the rarefaction will also be diminished.
Analogous considerations are applicable when the direc
tion of incidence is not perpendicular to the reflecting surface.
As the atoms are only passively influential in producing such
effects as those described above, it may be assumed that the
change of condensation or rarefaction is always proportional,
at the virtual surface of reflection, to the condensation or
rarefaction that would have existed there if the waves had
been undisturbed. This is known to be the case when waves
of air are reflected at the plane surfaces of solids, or at the
closed or open ends of tubes. The mathematical solution of
the problem of reflection of light depends on the introduction
of this condition into the reasoning.
Let us now suppose that planewaves are incident in a
given direction on a plane reflecting surface. From the hydro
dynamical theory of the vibrations of an elastic fluid (Propo
sition XIII., page 211), it appears that when there is no im
pressed force, and the motion does not satisfy the relation V=K.acr,
it is composed of two or more sets of vibrations each of which
satisfies this law, and that the velocities and condensations of
the components coexist. In the instance before us there is
no impressed force, inasmuch as we are considering only the
effect which the medium produces on the motion of the aathe
rial particles as an obstacle acting or ceasing to act abruptly,
and not as a continuous cause of retardation. The effect is
supposed to take place at extremely small distances from the
reflecting surface, and before the waves have actually entered
or quitted the medium , and it is conceived to be independent
THE THEORY OF LIGHT. 385
of the particular action of the separate atoms of the medium
on the intromitted light. In short, this investigation applies
to the external reflection at the surfaces of opake bodies, as
well as to the external and internal reflections at the surfaces
of transparent bodies. In accordance with these views let
the state of density of the aether at or near the surface be
supposed to result from two sets of waves, whose directions
of propagation are in the same planes perpendicular to that
surface.
Let the origin of x be an arbitrary point of the intersection
of one of these planes with the surface of the medium. The
motion in every plane parallel to this will be the same. Then,
x being measured along the line of intersection, 6 and ff being
the angles which the directions of propagation make with that
line, and the respective condensations being <r and <j', we
have
Kao = mfi (feat x cos 6 + c),
Kacr ' = ?rif z (feat x cos & + c).
But by the above stated condition, er f &' = Jca, k being an
unknown constant. Hence
(Jc  1) mfi (/cat xcosO + c) m'f z (feat  x cos 0'+ c') = 0.
This equation is to be satisfied at all times at every point of
the reflecting surface, and, therefore, whatever be x and t.
Hence in the first place and^ must be the same functions,
and we shall have
Kat x cos 6 + c = /cat x cos 0' + c',
whatever be x. Hence c = c' and cos = cos &. This proves
that the two sets of waves are in the same phase at the point
whose abscissa is x, if m and m have the same sign, and in
opposite phases if they have different signs. The equation
cos 6 = cos & is satisfied either by = #', which indicates co
incidence in the directions of propagation, or by & %TT 0,
which shews that the directions of propagation may make
equal angles with the reflecting surface. The first case is
25
386 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
excluded by the nature of the question ; the other proves the
law of reflection. It is to be observed that as the conditions
of the problem of reflection are satisfied by two sets of undu
lations they cannot be satisfied by more.
From this theory the following inferences are deducible.
(1) Since j/j and f z are the same functions, we shall have
(k 1) m = m'. If k be greater than unity, which is the case
when the waves are passing from free space into a medium
and the reflection is outside the medium^ m is positive, and
the reflected and incident waves are in the same phase at the
point of reflection. When the passage is from the medium
into free space, and the reflection takes place within the
medium, Jc is less than unity ; so that m is negative, and the
incident and reflected waves are in opposite phases. The
latter inference explains what has been called " the loss of
half an undulation by reflection." As these terms merely
indicate the difference of the phases of the incident and re
flected light at the point of reflection when the waves pass
out of a medium into vacuum, the fact which they express is
fully accounted for by the foregoing theory. After placing
the Theory of Light on a hydrodynamical basis, it has been
found that the supposed "loss" is quite consistent with the
modes of the vibrations of an elastic fluid. A change of
phase of the very same kind occurs when aerial waves pro
pagated in a cylindrical tube are reflected at an open end,
the reflection being due to the sudden diminution which the
condensations and rarefactions undergo on passing out of the
mouth of the tube into the surrounding air. In this instance
Jc = and m m nearly*.
(2) Taking the case of perpendicular incidence on a me
dium bounded by parallel planes, the loss of condensation by
reflection at the first surface being proportional to (k 1) m,
the total condensation of a given wave after intromittence
* See an Article on "The loss of half an undulation in Physical Optics" in the
Philosophical Magazine, Vol. xvm. 1859, pp. 57 60.
THE THEORY OP LIGHT. 387
will be proportional to m (k 1) m, or (2 k) m. Supposing
the change of condensation by emergence at the second sur
face to be in the same ratio, since it will be additive instead
of subtractive, the condensation of the emergent waves will
be proportional to
(2 _ fc) m + (Jc  1) (2  k) m, or m {1  (k  l) 2 j.
Consequently the ratio of the loss of light by reflection to the
originally incident light is k I after the first reflection, and
(k I) 2 after the second. Hence if k be not much greater
than unity, which is the case for many transparent substances,
the latter ratio will be very small, and the light lost by the
first reflection is very nearly restored by the other.
(3) The same theory explains the formation of the central
dark spot in the experiment of Newton's Rings. Supposing
the rings to be produced by the perpendicular incidence of
light on a convex lens in contact with plateglass, the re
flections at the point of contact will be very nearly the same
and in opposite phases, and will, therefore, neutralize each
other. Or, perhaps, it is more correct to say that at and
immediately contiguous to the point of contact, the two pieces
of glass act as a continuous substance so far as regards the
retardation of the undulations, and that thus there is no abrupt
change of condensation at that point, and consequently no
reflection.
We have now to take into consideration the composite
character of the incident waves, and to inquire in what manner
the components are affected by the disturbance that causes
the reflection. It has already been explained (p. 230) that
rayundulations relative to a given axis may be composed of
an unlimited number of primary rayundulations relative to
the same axis in every variety of phase, but all having the
same value of X It is by reason of this composite quality
of the rayundulations constituting waves, that the waves are
separable into parts. The separation may be conceived to
252
388 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
take place in such manner that each composite ray is divided
into parts having to each other a certain ratio depending on
the circumstances which cause the separation, while the pri
mary rays are not similarly divided. In the present case,
the original waves are separated at the points of reflection
into two parts corresponding to the reflected light and the
transmitted light, and, according to these views, there are as
many axes of composite rays in one part as in the other ; but
the intensities of the two portions of light will depend both
on the respective numbers of the primary components in the
corresponding composite rays, and also on the quantities of
polarized light which the portions severally contain. This
last point I now proceed to consider.
It will, at first, be supposed that the incident waves con
sist of nonpolarized rays. Then, since the disturbance at
incidence is not symmetrical with respect to the axes of the
rays, but with respect to planes passing through the axes at
right angles to the plane of reflection, the circumstances are
those which, according to the theory, must give rise to polar
ization. (See section (14), page 331.) This theoretical in
ference is confirmed by experiment, by which also it is found,
as might from the theory be anticipated, that there is no
polarization when the incidence is perpendicular to the re
flecting surface, the disturbing action being in this case sym
metrical with respect to the axes of the rays. In every other
case the reflected light is more or less polarized, and the
polarization is clearly due to a modification, at the instant
of reflection, of the individual primary components of the
reflected waves, there being no reason why one should be
affected differently from another. ISTow the only kind of
modification a primary ray is susceptible of may be deter
mined by the following considerations. On referring to the
discussion in pages 209 211, it will be seen that the equa
tion
__
THE THEORY OF LIGHT. 389
is satisfied by giving to a the value
a i cos 2 Je x + <r 2 cos 2 Jey,
and that this is its most general exact integral. Hence, taking
<r 2 to be greater than ^ and putting r* for # 2 + y 2 , we have
for small values of x and
oj cos 2 Je x + o 2 cos 2 ^/e"^ = 2^ (1  er 2 ) + (<7 2  erj (1  2ey z ).
This equation proves that a ray generally consists of a non
polarized part, and a part completely polarized. This is
usually expressed by saying that the ray is partially polarized.
Thus the fact that reflected light is found by experiment to
be partially polarized is consistent with the theory, although
hitherto the theory has not determined the exact amount of
the polarization. Again, since the reflected and transmitted
rays together make up the original light, which is nonpolar
ized, it follows that there must be just as much transmitted
as reflected polarized light, and that these portions are polar
ized in planes at right angles to each other so as by their
combination to produce nonpolarized light These results,
it is well known, agree with experience.
From the above comparison of the theory with facts it
may be inferred that at the same instant that an original
composite ray is separated into parts by reflection, each pri
mary ray of the reflected portion also undergoes separation
into parts, one of which, completely polarized, accompanies
the transmitted light, and the other, partially polarized, is a
component of the reflected light*. If cr t + <r z be taken to
measure the intensity of the reflected light, and S on the same
scale measure that of the incident light, then S + (cr 1 + cr 2 ) will
be the intensity of the transmitted light according as the rays
enter into or emerge from the medium. Also since, as is
* If in consequence of this theory of the resolvability of primary rays it be
necessary to infer from the equation (15) in page 206, that the rate of propagation
is not absolutely constant, being altered by the change of value of m, no theo
retical explanations of phenomena would, as far as I am aware, be affected by this
conclusion.
\>
390 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
shewn above, the reflected light consists of a nonpolarized
part 20j and a polarized part <7 2 <r lf the transmitted light
consists of an equal polarized part a 2 <r l5 and the non
polarized part $ 2<7 2 or $+20^, according as it passes into
or out of the medium. The proportion of the reflected light
^ _j_ o 2 to the incident light S depends both on the angle of
incidence and on the reflective power of the medium, which
is some unknown function of its atomic constitution. To de
termine that proportion theoretically, and to ascertain also
the ratio of the polarized to the nonpolarized portion of the
reflected rays, are problems which hardly admit of solution
in the present state of physical science, since they involve
certain relations of the motions of the sether to the particular
atomic constitution of the medium. Some additional remarks,
bearing on these points, may, however, be made here.
First, it is to be observed that a nonpolarized incident
ray may be assumed to consist of two equal rays, one com
pletely polarized in the plane of incidence, and the other in
the plane perpendicular to this through the axis of the ray.
Also because the transverse motions are in rectangular planes,
the two parts may be supposed to be independently affected
by reflection at the surface of the medium. Again, because
in one of the polarized rays the transverse motion is parallel
to the reflecting surface, and in the other inclined to it at
angles varying with the direction of incidence, it may be pre
sumed that they will be differently affected by the reflection,
and that the respective amounts of reflected light will be dif
ferent. If, therefore, I be the angle of incidence, we may
suppose the quantities of reflected light polarized in and per
pendicular to the plane of incidence to be respectively 8$ (I)
and Sty (I), so that the total reflected light will be
Now although the two rays differ in intensity, if they be in
the same phase of vibration the effect of their combination
will be a ray consisting of nonpolarized light and plane
THE THEORY OF LIGHT. 391
polarized light. But on referring to the theory of reflection
in page 384, where it is shewn that the virtual surface of
reflection is at some very small finite distance from the sur
face of the medium, there will appear to be reason for con
cluding that this reflecting surface is different for the two
polarized rays, on account of the difference of the directions
of their transverse motions relative to the surface of the
medium. Accordingly we shall have in the reflected waves
oppositely polarized rays differing both in intensity and phase,
and at the same time travelling in a common direction. Under
these circumstances the light will be elliptically polarized.
(See page 337.) This theoretical result is confirmed by ex
periment*.
Let us now suppose the incident waves to consist of com
pletely polarized rays, and the plane of their polarization to
make a given angle 6 with the plane of incidence. In this
case, 8' representing the intensity of the incident light, we
may suppose it to consist of two parts 8' cos 2 and 8' sin 2 6,
the former polarized in the plane of incidence and the other
in the perpendicular plane. (See section (17) in pages 335
and 336.) t Hence, adopting the expressions in the preceding
paragraph for the quantities of reflected light polarized in
and perpendicular to the plane of incidence, the total reflected
light in the case of the incidence of polarized rays will be
8'$ (I) cos* 6 +8'^ (I)sin 2 0.
This theory of reflection will be subsequently extended after
consideration has been given to the theory of refraction ; which
I now proceed to enter upon.
* See M. Jamin's Cours de Physique, Tom. m. pp. 689, 690, and 695. It appears
from the experiments of M. Jamin that the degree of the elliptical polarization is
the same for metals as for glass.
f Sir J. Herschel asserts in Art. 850 of his Treatise on Light in the Encyclo
pedia Metropolitana that this resolution of a polarized ray " must be received as
an empirical law at present, for which any good theory of polarization ought to
be capable of assigning a reason a priori" I have shewn in the section cited
that the TJndulatory Theory of Light, established on hydrodynamical principles,
gives the reason for the law.
392 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
For the sake of simplicity I begin with the theory of
refraction at the surfaces of noncrystallized media, and for
the same reason it will at first be supposed that the light
passes out of vacuum into the medium. It has been shewn
in what manner a portion of the incident rays is reflected :
the remainder constitute a series of undulations, which, on
entering the medium, are diverted from their original direc
tion by the retarding effect of its atoms. It is evident that
the retardation begins to act at a very small finite distance
from the surface of the medium, and that it continually in
creases from this upper limit till at a certain lower limit
within the medium it becomes equal to the general internal
retardation the theory of which is given by the solution of
Problem I. It will be assumed, as a condition of regular
refraction, that the front of the waves remains continuous
while they are under the varying influence of this retardation.
Hence, because the retardation increases from the upper to
the lower limit, the continuous curve formed by the section
of the front by a plane of incidence will necessarily have its
convexity turned towards the medium. Consequently the
normal to the front of a given wave will pass by degrees from
coincidence with the direction of incidence to the final direc
tion of propagation within the medium. This theoretical
inference is confirmed by an experimental fact relative to the
phenomena of Newton's Rings. It is found that when the
incidences are very oblique the Rings are subject to only a
finite dilatation ; whereas if the course of the light changed
per saltum at the surface of the medium, the dilatation would
be unlimited*. The finite dilatation would most probably be
explained by taking into account that the course of the light
is curvilinear through a short space while it is undergoing
refraction, the effect of this circumstance in estimating the
length of the path being greater the more the incidence is
oblique.
Another condition of regular refraction is, that waves
* See Arts. 639 and 670 of Herschel's Treatise on Light.
THE THEORY OF LIGHT. 393
which have planefronts before incidence should have plane
fronts perpendicular to the plane of incidence after passing
within the medium beyond the influence of the refringent
action. This condition would evidently be satisfied by any
courses which are exactly alike and parallel for all the ele
ments of the intromitted waves, the rate of propagation in
the medium being the same for all. Clearly under these
circumstances the locus of all the points of an intromitted
wave which have the same condensation, would be a plane
perpendicular to the plane of incidence ; and that too, whether
or not the directions of propagation within the medium be in
the planes of incidence. But in the case under consideration
of a noncrystallized medium, there is no cause of deviation
of the refracted ray from the plane of incidence, and we have
to determine what direction in that plane is alone appropriate
to the problem of refraction.
The investigation for this purpose rests on the following
argument relative to composite rays. Hitherto it has been
supposed that a composite ray is the resultant of an unlimited
number of primary rays having a common axis and the same
wavelength, but every variety of phase. Let us now con
ceive the components, while they retain their phases, to have
separate axes, all parallel to a given direction, and very close
to each other. Also let the rays be supposed to be non
polarized, and their axes to be all included within a very
small transverse area. If then r be the distance of any point
within or without the area from any one of the axes, we shall
have the approximate relation a = cr^ (1 er 2 ) between the
condensation or due at the point to the vibrations relative to
that axis, and the corresponding condensation cr^ on the axis.
r z
Since the term er z is of the order of 5, and it suffices to
A.
M
restrict the reasoning to terms of the order of  , that term
\
may be neglected, and we have cr = o very nearly. Hence
within the small transverse area, and at distances from it that
394 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
are small compared to X, the components of the total con
densation are the same in phase, and quam proxime the same
in magnitude, as when the primary rays were supposed to
have a common axis. In order to calculate the resultant
transverse velocity at any point within the area, or at a small
distance from it, let x, y be the coordinates of the point, and
a?!, y l the coordinates of any one of the axes. Then since
we have relative to that axis
, U/ 1YI /c. \ / \
u = <p ~ = cos q (?+ cj x 2e (x og,
* =  =
cos (? + c i) x % e (y  &) >
n__
q being put for and for z Kat. Hence
2 . u = {cos q 2 . (x  ccj cos qc t sin ^J'S . (x x^ sin
If, therefore,
. x X
we obtain
S.M = [{2 . (x  aj sin ^} 2 + {S.
But since it has been shewn that the phase of the resultant
is the same as in the case of a common axis, q& is the same
arc as qQ in page 229 ; so that we have
S . (x Xj) sin qc t _ S . sin qc 1
2 . (x a?J cos qc\ ~ 'S, . cos qc^ '
Hence for any given value of a;,
2 . # sin 2 . sin c
. ^ cos qc t S . cos qc t '
THE THEORY OF LIGHT. 395
Let us now suppose that there is a value x of x for which
2 . u = 0, whatever be the values of z and t. Then from
the above expression for 2 . u it follows that
2 . (oJ xj sin qc^ = 0, 2 . (a? 05 t ) cos ^ = ;
and consequently that
_ 2 . as, sin c t _ S . a?, cos qc^
S . sin ^ 2 . cos ^Cj
This last equality, inasmuch as it is identical with the one
obtained above, proves the possibility of always satisfying the
condition 2 . u = by a certain value of x. In exactly the
same way it may be shewn that there is a value y Q of y which
satisfies the condition S . v = 0. Hence we may conclude
that # and y Q are the coordinates of a virtual axis of the
compound motion. By putting x 2 . sin qc^ for 2 . a5 t sin ^
and x S cos ^c t for S . a5 t cos qc^ it will be seen that
2 . u = (x X Q ) {(2 . sin qc^f + (S . cos gqj)*}? cos # (f + 0).
The analogous expression for S . v is evidently obtained by
putting y y Q for a; X Q in that for S . w. Hence
2.M X X
These results prove that the composite motion relative to the
virtual axis whose coordinates are X Q and y is just the same
as that which was before assumed to be relative to an actual
axis common to all the component rays. Hitherto the com
ponents have been supposed to be nonpolarized ; but the case
of polarized components is included in the above reasoning
and does not require a separate treatment. In fact, since it
was proved that there is a value X Q for which S . u vanishes,
it may be inferred that when the components are polarized
and have their planes of polarization all parallel to the plane
yz, and very close to each other, the resulting transverse
motion is relative to a virtual plane of polarization the position
of which is determined by that value of x.
396 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
As a consequence of the foregoing results we may now
give a more general definition of a composite ray than that
which is contained in page 230. We may consider it to be a
resultant ray composed of an indefinite number of primary
rays in every variety of phases, the axes of which are either
coincident, or, being confined within certain restricted limits,
are indefinitely near each other. If the axes, instead of being
parallel to each other, as is supposed above, are in the direc
tions of normals to a continuous surface, the foregoing argu
ment would remain the same, and we may, therefore, regard
the above definition as inclusive of the case of convergent or
divergent axes. A composite polarized ray may analogously
be defined to be the resultant of an indefinite number of
simple polarized rays in all possible phases, having their
planes of polarization either parallel to each other, or sepa
rated by indefinitely small angles of inclination, and restricted
within certain transverse limits.
Since a polarized ray is in every instance produced by
the bifurcation of a ray originally not polarized, it may
always be considered to have an axis ; about which, in fact,
the condensation is disposed in a manner depending on the
conditions under which the bifurcations take place. For
example, when a nonpolarized ray is divided into two equal
planepolarized rays, we may presume that in each of the
latter the condensation is so disposed at all distances from the
axis as to be symmetrical with respect to two planes at right
angles to each other, one of which is the plane of polarization.
To determine, however, in a general manner the condensation
at any point of a rayundulation that has been polarized
under given circumstances, is a problem of considerable
perplexity, the solution of which need not here be attempted,
because so far as regards phenomena of light we only require
to know the motions and condensations contiguous to the axis,
which, happily, can be ascertained without difficulty*. After
* In page 291 I have asserted that at remote distances from the axis " the
laws of the motion and condensation may be the same for resolved as for primary
THE THEORY OP LIGHT. 397
this discussion of the character of composite rays we may
resume the consideration of the theory of refraction.
Conceive the planefront of the incident waves to be cut
by two planes of incidence indefinitely near each other, and
the included portion of the wavefront to be divided into an
indefinite number of equal rectangular elements, containing
the same number of axes of rayundulations. Then, from
what is shewn in the last paragraph but one, the resultant of
all the transverse motions relative to the axes of any element,
will be transverse motion of the same kind relative to a
virtual axis situated at the mean of the positions of these
axes. It is evident that as the elements are incident in
succession on the refracting medium, they will all be affected
in precisely the same manner, and that their virtual axes will
be equally bent from the original direction and pursue parallel
courses. But by reason of the interruption of the plane
front caused by the refringent action, the wave will be broken
up into independent elementary parts, which we may suppose
to be the elements just mentioned. The physical reason for
the independence of these parts is, that the plane wave is
composed of simple and independent rayundulations (see
page 244), and is resolvable by disturbances into its compo
nents, or into particular combinations of them. The reasoning
here is of the same kind as that employed in the theoretical
calculation applied to phenomena of diffraction, in which the
front of a wave, after a portion has been abruptly cut off, is
in like manner conceived to be broken up into elements that
become independent centres of radiation within restricted
angular limits. According to the present hydrodynamical
theory, this lateral action simply consists in the production of
more or less divergence of the axes comprised in each inde
pendent element. In cases of diffraction the degree of diver
vibrations." This assertion is made conjecturally, not being supported by ante
cedent reasoning. At the beginning of a Theory of the Polarization of Light in
the Cambridge Philosophical Transactions (Vol. vin. p. 371), I have entered into
some considerations relative to the condensation and motion at any distance from
the axis of a plane polarized rayundulation.
398 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
gence is much greater than in those of refraction, because in
the former there is a complete interruption of the wavefront,
while in the latter the continuity of the front is maintained,
and there is a gradual, although rapid, transition laterally
from the condensation outside the medium to the augmented
condensation within. Also since this augmentation takes
place in the planes of incidence, and the condensation at a
given instant along any straight line perpendicular to these
planes is uniform, we may conclude that the divergence of the
axes is wholly in the planes of incidence.
These inferences being admitted, it follows that axes
belonging to different elements might meet at the same point
within the medium, and that this circumstance, just as in
cases of diffraction, must be taken into account in calculating
the total condensation at the point. Now that there must be
planefronts of the intromitted waves is evident from the
consideration that otherwise the medium is not transparent,
that is, does not allow of regular refraction at emergence,
which like that at entrance requires the incident waves to
have planefronts. We have, therefore, next to consider in
what manner this condition is satisfied under the above
described circumstances of divergence of the rayaxes. First,
it is to be observed that the effect of this divergence will be
taken account of by supposing each point of the plane which
limits the distance within the medium to which the refringent
action extends, to be an origin of divergent rayundulations in
the same phase, the angular extent of the divergence being
very small. For in that plane the axes have acquired their
final directions, and the divergence must take place in the
same manner and degree from all points of it.
Again, if the planefront of an incident wave and the
planesurface of the medium be cut by a plane of incidence,
and the lines of section meet at the point A at the given time
Jj, and at the point B at the subsequent time 2 , each point
from A to B will be in succession a centre of rayaxes. The
locus, at any time, of the positions at which the phases of the
THE THEORY OP LIGHT. 399
undulations are the same on the axes from a given centre,
will, in noncrystallized media, be a portion of a spherical
surface. If a rayundulation starting, from A at the time t l9
has reached the point C at the time 2 , the straight line BG
will be the locus of points in the same phase from different
centres : for another undulation starting in the same phase as
the first from an intermediate point P at the time , and pro
ceeding in a parallel course, will in the time t z t describe a
length of path which is to AC as BP is to BA. In a par
ticular case, namely, that in which BC is perpendicular to A (7,
and consequently a tangent to the abovementioned spherical
surfaces, the condensations along BG will have maximum
values, because in that case either the whole, or the greatest
possible number, of the undulations diverging from the points
of AB will reach that line in the same phase at the same time,
the arcs and tangents being considered for very small spaces
to be coincident. It is evident that under the same circum
stances the continuity of the wavefront is maintained. These
conclusions are independent of the distance between A and B,
and therefore hold good when that distance is supposed to be
indefinitely diminished. It remains to prove that the refracted
ray actually takes the course here supposed ; which I propose
to do by the following argument.
It is evident that the directions finally given to the
refracted rays depend entirely on the refringent forces which
operate in the small space within which the wavefront is
curved, and that these forces determine the amount of refrac
tion for a given angle of incidence and a given substance.
But this amount does not admit of exact a priori calculation,
because the particular modes of action of the forces are un
known, being dependent in part on the number, arrangement,
and magnitudes of the atoms of the refracting medium.
Experiment has, however, shewn that there is a certain law
of refraction for noncrystallized media, which is the same for
all angles of incidence and all such media, and which may,
therefore, be legitimately ascribed to a general mechanical
400 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
principle. Now the foregoing discussion points to a principle
of this kind, inasmuch as it has indicated circumstances under
which the refringent forces, whatever be their specific action,
modify the waves in such manner that after intromittence the
sum of the condensations of a given wave is a maximum, and
therefore differs by a minimum quantity from the sum of the
condensations of the same wave before incidence. This may
be regarded as a principle of least action, and as such may be
employed generally for determining the direction of a refracted
ray. In the case of a noncrystallized medium it has been
shewn above, that if this principle be adopted, the straight
line BG will be a tangent to the partial waves diverging from
the points of AJB, and that consequently BG and AC are at
right angles to each other. Whence the law of the constancy
of the ratio of the sine of the angle of incidence to the sine of the
angle of refraction may be inferred in the usual manner.
Perhaps the foregoing reasoning may be further elucidated
by the following considerations. Conceive the finite space in
which the refringent forces act to be divided into an indefinite
number of intervals by planes parallel to the surface of the
medium, and the retarding forces to be uniform through each
interval, but to vary abruptly from one interval to the next.
Then we may suppose that the direction of a ray changes per
saltum at each separating plane, the course through each
interval being rectilinear. In that case the total refraction
will be the sum of these differential refractions. Assuming
that the above stated principle governs the directions of the
refracted rays, if <^, < 2 , 3 ...< M+1 be the successive angles of
incidence, we shall have, by the same reasoning as that above,
sin (^ = m t sin < 2 , sin $ 2 = ??^ 2 sin < 8 , ... sin <f> n = m n sin < n+1 .
Consequently sin ^ = m^ m z m s . . .m n sin < n+1 = fi sin </> w+1 , which
proves the law of refraction. This reasoning would still be
applicable if the gradations of the refringent action should be
due in part to a gradual variation of density of the substance
in a very thin superficial stratum ; which variation, for reasons
THE THEOKY OF LIGHT. 401
that I shall subsequently adduce, may be supposed to exist at
the boundaries of all solid and fluid substances.
If the incident waves have a curved instead of a plane
front, and the surface of the medium be curved, the law of
refraction would still be proved in the same manner ; for since
it was shewn that the points A and B might be as near to
each other as we please, a very small portion of a curved front
might be treated as if it were a portion of a planefront, and
a small portion of a curved refracting surface as if it were a
plane.
I now proceed to investigate the laws of refraction at the
surfaces of crystallized media. At first it will be supposed, as
before, that the waves are composed of nonpolarized rays,
and that they pass out of vacuum into the medium. The
principles involved in this investigation are in several respects
the same as those for the case of noncrystallized media. The
incident waves being supposed to have planefronts, and the
surface of the medium to be a plane, let the intersection of the
surface by a planefront cut a certain plane of incidence at the
point A at the time ti and .at the point B at the time t 2 . Also
conceive to be described about A as centre the wavesurface
whose equation is obtained in page 381, and let its dimensions
be such that the radii from A are equal to the distances passed
over by propagation in the medium in their respective direc
tions during the interval 2 ^. In general there are two radii
in the same direction corresponding to the rates of propagation
of two rays oppositely polarized. Suppose & plane to pass
through that intersection of the refracting surface by ^ wave
front which contains 5, and let it revolve about this line till
it touches the surface described, as above stated, about A.
In general there will be two such planes touching the surface
in two points, which let us call C and G'.. Then AC and
AC' will both be directions of propagation in the medium
after the refraction of the portion of the wave incident at A,
and, for the same reason as in the case of ordinary refraction,
may be taken as the mean directions of two bundles of axes
26
402 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
diverging from A. The incident ray is separated by the
refraction into polarized rays, because, as is explained in
page 379, the medium is only capable of transmitting such
rays ; and the parts are equal and oppositely polarized because
they are derived from the bifurcation of a nonpolarized
primitive. In the instance of a uniaxal crystal one of the
lines AC, AC' is in the plane of incidence and obeys the
ordinary law of refraction, while the other is in general
inclined to that plane ; and in the case of a biaxal crystal
both lines are generally out of the plane of incidence. The
rays take the two directions A C and A C' in conformity with
the abovementioned principle of least action (or minimum
disturbing effect), the individual rays of each of the two
bundles whose axes are AC and AC' being always in the
same phase at the same time in the respective tangent planes,
which accordingly become planefronts of waves of maximum
condensation.
The refracted plane fronts are necessarily perpendicular
to the planes of incidence. Therefore, since, with the excep
tion of the ordinary refraction of a uniaxal crystal, the axes
of rays propagated in crystals are inclined to the planes of
incidence, they are not perpendicular to the planefronts. But
the transverse motions of the individual rays must in every
case be perpendicular to their planes of polarization ; for it
has been shewn (page 381) that the rates of propagation
wholly depend on the effective elasticities in these transverse
directions. Now when it is considered that there are an
unlimited number of axes parallel to a given direction of
propagation in the medium, it may be concluded that the
transverse motions in each plane at right angles to that
direction will neutralize each other, and that this will be the
case although the individual rays are not generally in the
same phase in that plane. For under these circumstances
there is just as much probability that the resulting transverse
motion at any point would be in one direction as in the
contrary direction, and we may therefore infer that there is no
THE THEORY OF LIGHT. 403
resulting motion in either. Thus there remains only the
motion in the direction of the axes, and consequently the
refracted waves differ from those in ordinary refraction in the
respect that the direction of the resultant vibratory motion is
not perpendicular to the planefronts of the waves.
Hitherto the waves have been supposed to be refracted by
entrance into a medium. The contrary case of refraction by
passage out of the medium might be treated, mutatis mutan
dis, according to the same principles. But it will suffice to
infer the explanation of the phenomena in the latter case from
that in the other, by referring to a general law which light is
found by experiment to obey ; namely, that any path which it
traverses it can traverse in the opposite direction. A hydro
dynamical reason for this law may be given in the present in
stance by making use of the general equation (29) in page 250.
Assuming that the retardation due to the medium is always
proportional, cceteris paribus, to the effective accelerative
force of the aether, and acts in the opposite direction, we may
represent the retarding force generally by the expression
dV
<j) (s) y . Then modifying the equation (29) in order to
include this force, we get
Combining with this the equation of constancy of mass,
~dt + ~ds = '
it will be found that
This equation remains the same when V and a both change
sign, since under these circumstances </> (s) retains the same
value and sign. Hence whatever be the successive values of
the velocity at any given point, the same values may occur in
262
404 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
the same succession at the same point when the velocity of the
fluid and the propagation of the velocity are in the opposite
direction. Consequently the law of refraction for internal in
cidence, whether the refraction be ordinary or extraordinary,
may be at once inferred from that for external incidence.
The reasoning in the three preceding paragraphs embraces
all that is necessary for accounting for the phenomenon of
double refraction on the hydrodynamical hypothesis of undu
lations. It should, however, be remarked that because in the
mathematical reasoning in page 378 only the first power of e*
was retained, the foregoing results are inapplicable if the
effective elasticity of the medium be very different in different
directions,
We are now prepared to resume the consideration of the
theory of polarization by reflection (suspended in page 391),
and to attempt the determination to some extent of the quan
tities of the reflected light under given circumstances. It will
be necessary, in the first place, to calculate the ratio of the
condensation of a given wave before incidence to that of the
same wave after intromittence. For this purpose conceive the
portion of an incident wave included between two planes of
incidence indefinitely near each other to be divided into small
rectangular elements of three dimensions, and let the lengths
of the edges of one of them be a and fi in the wavefront,
parallel respectively to the refracting surface and to the planes
of incidence, and 7 perpendicular to the wavefront. Cor
responding to every such element of any wave at the time t v
before intromittence, there will be an element of the intro
mitted wave at the time 2 , the edges of which have to the
edges a, /3, 7 ratios which may be found as follows.
Let otj, ^, 7 4 be the lengths of the analogous edges of the
intromitted element. Then we have evidently a = a t . To
obtain the other ratios, let A and B, as before, be the points
in a given plane of incidence where the wavefront meets the
surface of the medium at the times ^ and 2 , and let <f>
THE THEORY OF LIGHT. 405
and  fa be the angles made with the same surface by the
incident and refracted wavefronts. Then if D be the distance
between the points A and B, the length of wavefront which,
measured in a plane of incidence, is D sin <p out of the medium
at the time ^ , becomes D sin fa within the medium at the time
2 . And since the lengths of the corresponding elements in the
same directions are in the same ratio, it follows that /3 is to /^
as sin (f> to sin fa . Also the lengths of the perpendiculars from
B and A on the wavefronts passing respectively through A
and B at the times t t and t 2 , are D cos (f> and D cos ^ r But
these perpendiculars, being the aggregate breadths of the samQ
number of waves without and within the medium, are in the
ratio of the breadths, before and after entrance into the
medium, of a given wave, or of like portions of a given wave.
Consequently 7 is to y 1 as cos < to cos fa .
The required ratios having been ascertained, now let o and
cTj be the condensations of the waveelement before and after
the refraction. Then if we leave out of account the loss of
condensation by reflection, and the increment of condensation
due to the occupation of space by the atoms of the medium,
and suppose that in other respects the total condensation of
a given wave is not altered by the refraction, the conden
sations and magnitudes of the waveelement will be inversely
proportional to each other. Consequently
oj _ a/By sin $ cos </>
a , GCi/3,7! , . sin fa cos fa '
In the particular case in which o\ = <r, </> and fa are comple
mentary arcs ; and if m l be the particular value of the ratio of
sin </> to sin fa for that case, the corresponding value of <f> is
given by the equation tan <f> = m l . Thus there is generally a
value of </> for which the incident and intromitted waves (the
406 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
above suppositions being admitted) have the same condensa
tion, although they have not the same breadths. These
results apply both to single and to double refraction, if in the
case of the latter o^ is the condensation of either of the re
fracted waves, and cr half the condensation of the incident
wave.
Let us now consider more particularly the incidence of a
nonpolarized ray on the surface of a crystallized medium, and
let 7, I f , I" be respectively the angles which the incident ray
and the two refracted rays make with a perpendicular to the
surface at the point of incidence. Then, supposing the inci
dent ray to be represented by 2$, and to consist of two equal
parts completely polarized in planes parallel and perpen
dicular to the plane of incidence, the reflected ray, by the
same reasoning as that in page 390, will also consist of two
parts, which I shall call #< (/, 1') and fty (/, /"), and as
sume to be respectively polarized in the same planes. It is,
however, to be remarked that since the refracted rays are one
or both generally out of the plane of incidence, and the action
on the aether which produces the reflection cannot conse
quently be strictly symmetrical with respect to that plane, we
may not suppose that either the two parts composing the inci
dent ray, or the corresponding two parts of the reflected ray,
are accurately polarized in and at right angles to the plane of
incidence. In fact, Sir David Brewster has shewn experi
mentally that the position of the plane of polarization of the
reflected light may, under particular circumstances, depend
very much on the azimuth of the plane of incidence and on
the positions of the planes of polarization of the transmitted
rays. But in the usual circumstances of reflection, in which,
according to our theory, the retardation of the medium pro
duces the reflectent effect for the most part before the ray
has entered the medium, the deviations of the planes of polar
ization from the positions above assumed do not appear to be
of sensible magnitude. (See Philosophical Transactions, 1819,
p. 145).
THE THEORY OF LIGHT. 407
Since the above expressions for the reflected rays involve
/' and /", which vary with the azimuth of the plane of inci
dence, neither of the rays will be of constant intensity for a
given angle of incidence. But experiment has shewn that the
total quantity of reflected light is the same in all azimuths for
the same angle of incidence on a given surface ; that is
&/>(/,/') +#K/, n =2S X (i).
First, let S(f> (I, 7') be that reflected part in which the trans
verse motions are perpendicular to the plane of incidence.
Then in the corresponding incident part there is no alteration
of the transverse dimension of a given wave element by the
intromittence (since a= aj, and while the element changes its
dimensions in the other two directions, there is no angular
separation of the planes of polarization of individual rays,
these planes remaining parallel to the plane of incidence.
These circumstances appear to account for the observed fact
that the function <p (/, /') has no decided minimum value cor
responding to that value of / which satisfies the equation
tan/= m lt but increases continually as the angle of incidence
changes from to 90. For although, as we have seen, for
that angle of incidence the condensations of the incident and
refracted waves are the same (excepting the loss by reflection),
this circumstance influences only to a limited extent the
amount of reflection, inasmuch as the reflectent forces operate
chiefly at a sensible distance from the refracting surface, and
before the refraction is completed. When the angle of inci
dence exceeds that whose tangent is m l9 the condensation of
the refracted wave becomes less than that of the incident wave,
and assuming that the total reflection is partly dependent on
the difference of these condensations, there will be a certain
amount of reflection from this cause, but opposite in phase to
the reflection from the same cause when the tangent of the
angle of incidence is less than m*.
* These inferences accord with experimental results obtained by M. Jamin for
reflection at the surface of steel, and seem to account for the slow increment of
408 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
With respect to the part Sty (/, I") of the reflected ray,
which has its origin in the half of the incident ray which
is polarized in a plane perpendicular to that of incidence, the
same causes operate to change the dimensions of a given
waveelement as in the other part, with this additional circum
stance. Because the planes of polarization of the individual
rays are perpendicular to the plane" of incidence, and must also
be perpendicular to the wavefront, it follows that where the
wavefront is Curved the refraction produces an angular separa
tion of these planes. The consequence is that contemporane
ously with the refraction the number of component rays within
a given portion of the wavefront is diminished. It seems
evident that this effect is opposed to the tendency of the
retarding force of the medium to produce the sudden condensa
tion which causes the reflection, and by lessening the amount
of that condensation lessens also the amount of the reflection.
Thus the theory gives an intelligible reason for the observed
fact that for the same angle of incidence the amount of reflected
light is less when the incident ray is polarized perpendicu
larly to the plane of incidence than when polarized in that
plane*.
The effect of the angular separation of the planes of polar
ization of the individual rays will clearly be some function of
the lending of the wavefront by refraction, and from being
zero for perpendicular incidence will continually increase with
the angle of incidence. Hence while that reflection which is
produced, just as in the case of the ray polarized in the plane
of incidence, by the sudden change of effective elasticity, pre
ponderates for small angles of incidence, it might happen that
for larger angles the counter effect of the separation of the
planes of polarization would be in excess. In that case there
would be a certain angle of incidence for which the two effects
intensity for angles of incidence exceeding 75.' (Cours de Physique, Tom. m.
p. 688.)
* See the comparative amounts of the two reflections in the case of steel given
by M. Jamin in p. 688.
THE THEORY OF LIGHT. 409
neutralize each other, and the result for larger angles of inci
dence would then be light reflected in the opposite phase.
(The difference hence arising between the phases of the two
reflected rays is distinct from that considered in page 391,
which is due to difference of paths,) In the case of refraction
by entrance into transparent substances, it does not appear
that there can be a complete disappearance of reflected light
unless the abovementioned angle coincides with the one for
which the incident and intromitted rays have the same con
densation ; that is, the angle for which tan/=w 2 , usually
called the polarizing angle. For it is only under that condi
tion that there would be absolutely no cause for reflection.
But the theory does not point to any particular reason for
such coincidence, and experiment indicates that at the polar
izing angle, the reflection, although it is always a minimum,
does not generally vanish. According to the results of experi
ments made by M. Jamin, the ray is not completely extin
guished at the polarizing angle if the refractive index of the
substance exceeds 1,40, the minimum is very small for glass,
for alum it is zero, and for substances less refringent than alum,
it increases as the index of refraction diminishes. M. Jamin
has also given experimental results which shew that for glass
the difference of phase of the two reflected rays varies at the
polarizing angle very rapidly, and that at this angle of in
cidence, which is 565 / , the difference is 90, while for the
angle 58 it amounts to 172. (Cours de Physique, pp. 694 and
695). It may be seen that the above facts are not inconsistent
with the present theory.
The same considerations do not strictly apply to opaque
bodies, because with respect to them we do not know that the
intromitted wave maintains its continuity, nor in what manner
in other respects it comports itself. There is, however, reason
to conclude that within very minute distances from the sur
faces the laws of refraction are nearly the same for opaque as
for transparent substances ; and, in fact, it appears from expe
riment that the phenomenon of polarization by reflection, and
410 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
the existence of a polarizing angle, are common to the two
classes of substances. T he minimum of intensity is found to
be but slightly exhibited in the case of silver ; it is much more
marked for steel, and for certain metallic oxides it approaches
to zero. (Jamin, p. 687). The tangent of the polarizing
angle is assumed to be equal to the index of refraction, and
this law thus furnishes the means of obtaining the index of
refraction for an opaque substance.
According to the theory, the polarizing angle is always
that which fulfils the condition of making the sum of the
angles of incidence and refraction of the wave equal to 90.
Hence the polarizing angle will be different for rays of dif
ferent refrangibility, as is known from experiment to be
the case. Also for crystalline substances it will vary with
the position of the refracting surface relative to fixed di
rections in the crystal, and with the azimuth of the plane
of incidence on each such surface. This last inference from
the theory is experimentally confirmed by Brewster in the
Paper before cited in p. 406.
It is interesting to remark that as the law expressed by
the equation tan <t> = m 2 was theoretically deduced on the hy
pothesis that the space occupied by the atoms of the medium
is very small compared to the intervening spaces (p. 405),
the confirmation of the law by experiment justifies the con
clusion that this hypothesis is true even for substances of
great density.
Let us now take the case of the incidence on a crystalline
medium of a ray completely polarized in a plane making a
given angle (6) with the plane of incidence. Representing
by S the intensity of the incident ray, we may, by the same
reasoning as that in page 391, resolve this ray into $sin 2 6
and 8 cos 2 9 polarized in planes parallel and perpendicular to
the plane of incidence. Then the former will produce the
reflected ray $sm 2 0</> (I, /'), and the other the reflected ray
8 cos 2 0A/r (/, /"). If another equal ray completely polarized
in a plane at right angles to the plane of polarization of the
THE THEORY OF LIGHT. 411
former ray, be incident in the same direction the reflected
rays will be 8 cos 2 $ (/,/') and 8 sin 2 0f (/,/"). Hence
the total reflected light is
which is the "same quantity as that assumed in page 406 on
the supposition that the component incident rays are polar
ized in and perpendicularly to the plane of incidence. It is
to be understood that the two parts of the incident light are
in each case in the same phase.
The foregoing theory of reflection is consistent only with
the supposition that the transverse motion of a ray polarized
in the plane of incidence is perpendicular to that plane, and
therefore unequivocally determines the direction of the trans
verse motion to be the same as that inferred in page 382 from
the theory of double refraction.
I have not attempted to find by a priori investigation the
forms of the functions </> (/, /') and ty (7, 7"). The con
siderations by which Fresnel's formulae have been deduced,
being in great measure empirical, might as readily be adapted
to the present theory as to any other ; and in one respect no
other theory has equal claims to appropriate these formulae.
The polarizing angle, which is a constant and distinctive
feature in the phenomena of reflection, is in this theory re
ferred to the condition of equality between the condensations
of the incident and refracted waves, and the law that the
tangent of the polarizing angle is equal to the index of re
fraction is consequent upon this condition. No such distinct
physical explanation of the phenomenon has been given on
any other theory, because no theory, as I maintain, which
does not regard the aether as a continuous medium susceptible
of variations of density, is capable of explaining it.
The phenomenon of total internal reflection is referable to
the general law demonstrated in page 403, according to which
light can always traverse the same course in opposite direc
tions. Since the angle of refraction for external incidence
412 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
has a maximum limit, if the angle of internal incidence ex
ceed that limit, in consequence of that law the light cannot
after the incidence have its path exterior to the medium, and
must therefore be propagated wholly within. Hence the cir
cumstances which determine its course after incidence are the
same as those of ordinary external reflection, and the law of
reflection is proved by the same reasoning as that in page
385. If the incident light be completely polarized in the
plane of incidence, the whole will still be reflected ; and the
same will be the case if it be polarized perpendicularly to
that plane. But from the same considerations, mutatis mu
tandis, as those entered into in page 391, if these two polar
ized rays be in the same phase at incidence, a difference of
phase will be produced by the reflection. Consequently since
common light may always be supposed to consist of two
equal parts oppositely polarized, if in the present case the
incident light be common light, the reflected light will con
sist of two equal components, polarized in planes parallel and
perpendicular to that of incidence, but differing in phase.
But because the components are of equal intensity, they will
under all circumstances undergo complementary changes, and
their joint luminous effect, notwithstanding the difference of
phase, will not be perceived to be different from that of com
mon light. If, however, the incident light be planepolarized,
and the plane of polarization make an angle 6 with the plane
of incidence, it may, as usual, be supposed to consist of the
two parts $sin 2 # and $cos 2 # polarized in and perpendicularly
to the plane of incidence. In that case, as these two parts
are unequal, the alteration of phase produced by the reflection
will cause the reflected light to be elliptically polarized.
Fresnel's Rhomb is a wellknown exemplification of this
theoretical inference.
The coloured rings, formed by subjecting planepolarized
light which has passed through a thin plate of crystal to a
new polarization, are explained by this theory as follows.
For simplicity let us take the case of a plate of a uniaxal
THE THEORY OF LIGHT. 413
crystal bounded by planes perpendicular to the axis, and
suppose the planepolarized light to be incident in directions
either parallel, or nearly so, to the axis. Then if the light
be incident in planes parallel to the plane of its polarization,
the crystal produces no bifurcation, because only ordinary
rays are transmitted ; and if incident in planes perpendicular
to the same plane, there is also no bifurcation, because only
extraordinary rays are transmitted. In each case the trans
mitted ray, after incidence on a completely polarizing reflector
at its polarizing angle in a plane perpendicular to that of the
original polarization, is not reflected. When the incidence
on the crystal is in any other plane passing through the
crystallographical axis, making an angle with the plane of
original polarization, we may suppose the incident light to
consist of two parts Ssm 2 and Scos?0 polarized in and
perpendicularly to the plane of incidence. These parts re
spectively give rise to ordinary and extraordinary rays, which
traverse the crystal with different velocities, and issue from
it in different phases. For every ordinary ray proceeding,
after emergence, in a direction making a given angle with
the axis, there will be an extraordinary ray proceeding in the
same direction, but differing in phase to an amount which
depends only on that angle. If the difference of phase be an
exact multiple of  , it follows from the argument in pages
336 and 337, that the result of the composition of the two
rays is a planepolarized ray, equal in intensity to the original
ray (excepting loss by reflection), and polarized in the same
plane. Hence this compound ray, when incident on the
abovementioned polarizing reflector, gives rise to no reflec
tion. In the cases of all the other differences of phase, the
compound light will be elliptically polarized, and the two
components, each of which may be supposed to be resolved
into rays polarized in planes parallel and perpendicular to
that of original polarization, will be equivalent to the result
ants polarized in these two directions. The resultants polar
414 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
ized in planes parallel to that of incidence will be extinguished
by the reflector, and the others are more or less reflected.
The amount of this reflection is greatest, in a given principal
plane, when the difference of phase exceeds an exact multiple
of  by  , and the light is in consequence circularly polarized.
Also the maximum values of these different maxima are in
the principal planes inclined by angles + 45 and 45 to the
plane of original polarization.
The above theoretical results fully account for the phe
nomena witnessed in the case of the passage of homogeneous
light through a uniaxal plate, namely, alternate rings of com
parative brightness and darkness, intercepted by a dark cross
the axes of which are parallel and perpendicular to the plane
of first polarization. The effect produced when the light is
composed of rays of different refrangibilities may be inferred
from the superposition of the several effects that would be
produced if the components were employed separately. Con
siderations analogous to the foregoing may be applied to
explain the phenomena witnessed when the light is made to
pass through a thin plate of a biaxal crystal.
If the light, after passing through the crystal, were re
ceived by the eye before incidence on the reflector, no varia
tion of the intensity would be perceived, because the two
emergent parts, 8 sin 2 6 and S cos 2 0, being oppositely polar
ized, would act upon the eye independently, and produce a
total effect proportional to their sum $sin 2 #+ $cos 2 0, or 8.
Hence the intensity of the transmitted beam will be the same
at all points. Also if the incident beam were composed of
common light, no variation of intensity would result from in
cidence on the reflector, because the original light may be
assumed to consist of two equal beams of oppositely polarized
light, the effects of which after the incidence would be exactly
complementary, and the result of the combination would con
sequently be light of uniform intensity.
The foregoing argument may suffice to shew that the pre
THE THEORY OF LIGHT. 415
sent theory is capable of explaining all the phenomena of
polarized rings. The theoretical treatment of this problem in
Arts. 144174 of Mr Ahy's " Undulatory Theory of Optics "
(Mathematical Tracts, 2d Ed.), is, as far as regards the ma
thematical reasoning, as complete as can be desired. But the
attempt made in Arts. 181 183 to give the physical reasons
for the phenomena proves nothing so much as the inadequacy
for this purpose of the vibratory theory of light. (I designate
as "vibratory" the theory of light which takes account of
the vibrations of discrete particles of the sether, to distinguish
it from the one I have proposed, which, as resting exclusively
on hydrodynamical principles, and employing partial diffe
rential equations for calculating the motions, is alone entitled
to be called undulatory). The supposition made by Mr
Airy in Art. 183 to account for the phenomenal difference
between common light and elliptically polarized light is
arbitrary in the extreme, having no connection with ante
cedent principles, and the necessity for making a gratuitous
assertion respecting the character of the transverse motions
in order to prop up the vibratory theory, may legitimately be
regarded by an opponent of that theory as only giving evi
dence of its failure. The foregoing explanations, which
essentially depend on treating the sether as a continuous sub
stance, distinctly indicate the reason of the failure of the
vibratory theory.
Having discussed the chief problems in the second part of
the Undulatory Theory of Light, namely, those relating to
the transmission of light through noncrystallized and crys
tallized substances, and to its reflection and refraction at their
surfaces, I shall only give the explanations on the same prin
ciples of a few additional phenomena before I pass on to
another department of Physics.
(1) It is found that colours are produced when a beam
of polarized light, after being made to traverse a rectangular
piece of glass, unannealed, or otherwise put into a state of
mechanical constraint, is subjected to a second polarization.
416 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
The piece of glass is put in the place of the crystal in the
experiment which produces the polarized rings. To account
for the phenomena due to the state of constraint we may
suppose that in the ordinary state the arrangement of the
ultimate atoms of the glass is such as to have the same effect
on transmitted light in whatever direction the transmission
takes place, and that by the constraint the atomic arrange
ment is in such manner and degree altered as to become a
function of the direction. The most probable, and at the
same time most general, supposition that can be made re
specting this function is, that throughout a given very small
portion of the glass it satisfies with more or less exactness
the condition of symmetry attributed to crystals in page 376,
namely, that of being symmetrical with respect to three
planes at right angles to each other. On this hypothesis
each small portion of the glass will act upon light in the
same manner as a crystal, and the appearance of colours re
sembling those of the polarized rings will be accounted for.
There is, however, this difference between a crystal and
constrained glass, that whilst in the former the atomic ar
rangement is the same throughout, and the phenomena have
reference, not to position in the crystal, but solely to direction,
in the latter the atomic arrangement will in all probability
change in passing from one small portion of the glass to the
next, and consequently be a function of position relative to its
boundaries. Observation confirms this theoretical inference,
it being found that the polarized colours exhibited by con
strained glass are arranged in lines which have evident re
ference to its shape and dimensions.
(2) The theory gives the following account of the colours
of substances, and of the phenomena of absorption. We have
seen that reflection at the surfaces of bodies is produced by the
sudden retardation of the motion of the aether by the resist
ance it encounters from the atoms, and that this cause operates
before the incident waves have actually entered into the
medium, being the result of the aggregate resistance of the
THE THEORY OF LIGHT. 417
*
atoms, and therefore extending to a sensible distance from
the superficies of the medium. Hence the reflectent effect is
produced in the same manner and in the same proportion on
rays of all refrangibilities ; for which reason light of every
colour is regularly reflected at the planefacets of all bodies,
both black and white, or whatever may be their proper colour.
The nonreflected part of the incident wave enters into the
medium, whether it be an opake or a transparent substance,
but is differently affected afterwards, according as the sub
stance is of the one kind or the other.
Let us, first, suppose the medium to be transparent. In
that case the incident wave is regularly refracted and trans
mitted according to laws which we have already investigated.
There is no sensible reflection from the atoms of the medium
in its interior; because, as we have seen, the sole effect of
such reflection .is to convert the proper elasticity of the aether
into an apparent elasticity having to the former a given ratio.
Thus there is no propagation of secondary waves within the
medium so long as no change of interior constitution is en
countered by the original waves, and the number of atoms in
a given space and their arrangement remain the same. These
conditions must be satisfied in every perfectly transparent sub
stance, whether it be crystallized or noncrystallized, although
in the former the effective elasticity of the aether is different in
different directions. But the same conditions cannot be satis
fied at and very near the confines of the medium, as will
appear from the following considerations.
When an atom in the interior of a homogeneous medium
is held in equilibrium by attractive and repulsive forces, the
forces of each kind will be equal in opposite directions, there
being, by the hypothesis of homogeneity, no cause of in
equality. But this is no longer the case when the atom is
situated within a certain very small distance from the super
ficies. It is evident that here the resultant attractive force
acts in the direction perpendicular to the surface and towards
the interior, and must be just equal and opposite to the re
27
418 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
sultant repulsive force. The atomic conditions of this equi
librium will come under consideration in the subsequent
Theory of Heat and Molecular Attraction ; at present it
suffices to say that there will be a gradual increase of density
of the atoms through a small finite interval from the super
ficies towards the interior, analogous to the increment of
density of the Earth's atmosphere arising from the coun
teraction of the repulsive force of the air by the force of
terrestrial gravity. In consequence of this gradation of den
sity, besides the regular superficial reflection which we have
already discussed, there will be another kind of reflection
which for distinction may be called irregular, consisting of
nonneutralized reflections from individual atoms, and origi
nating at all those that are situated within a certain small
depth below the surface. Under these circumstances, when
the mode of reflection of condensation from an individual
atom (as determined by the solution of Example VI., p. 279)
is considered, the secondary waves reflected from the atoms at
different depths will evidently issue from the medium in all
possible directions. It is by means of this irregularly re
flected light that a body becomes visible from whatever quarter
it is looked at. For ( instance, when a transparent polished
substance is exposed to diffused daylight, so that waves are
incident upon it simultaneously from all surrounding objects,
at the same time that it sends to the eye by regular reflection
rays by which those objects may be seen, it is itself, as to
colour, shape, and contour, made visible by the irregular
reflection from a very thin superficial stratum of atoms.
Supposing that it is perfectly transparent, allowing of the
transmission of rays of all refrangibilities, since the rays of
irregular reflection proceed from points at sensible depths
below the surface, it may be assumed that these also will
consist of rays of all refrangibilities. In that case the sub
stance will appear to be white. If, however, a transparent
substance allows of the passage of rays of certain colours, and
stops all others, according to the same law the secondary
THE THEORY OF LIGHT. 419
rays that are of the same kind as the transmitted rays will be
either exclusively, or most copiously, reflected. Hence the
colour of a substance which allows of rays of certain ref Tangi
bilities to pass through it, is generally the same as the resultant
of the colours of these rays. This theoretical inference is con
firmed by experience*. Thus the blue colour of the sky,
which is perceived mainly by means of irregularly reflected
light, shews that the atmosphere transmits most readily blue
rays, and, similarly, the redness at sunset shews that the
vapour of water, suspended in an invisible form in the lower
regions of the atmosphere, transmits by preference red rays*
If the reflecting substance be opake, the theory of the
phenomena is such as follows. The laws of reflection, both
regular and irregular, and the laws of refraction, may be sup
posed, within a certain very small depth below the surface, to be
the same quam proxime as in the case of a transparent sub
stance. But if beyond that depth the continuity of the wave
fronts is not maintained, and the composition of the waves is
broken up, the result is opacity. Supposing that in this manner
rays of all refrangibilities are completely extinguished by a
very thin stratum of the substance, the same will be the case,
according to the law before assumed, with respect to the rays
of irregular reflection; and thus the substance will appear
completely black. But if the medium permits some waves to
penetrate to greater depths than others before being broken
up, we may suppose that like preference will be given to the
irregularly reflected rays of the same kind, and that these
will be allowed to issue from the medium while they are yet
in a form proper for vision. By this process the opake body
makes a selection of the secondary rays and appears coloured.
This theory of the dependance of the proper colours of bodies
on an action which is operative only within a very minute
superficial stratum, is supported by the fact that the inten
sities of the colours are perceptibly diminished when the
bodies are reduced to fine powders. The property of trans
* Herschel's Treatise on Light, Articles 498501.
272
420 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
mitting some rays in preference to others, which, according
to the theory, determines the proper colour of a body, depends
on the constituency and arrangement of its atoms in a manner
which, in the present state of science, does not appear to
admit of d priori investigation.
With respect to the emanation of irregularly reflected
light from the surfaces of bodies, rendering them visible in
all directions, it is matter of observation that the brightness of
an object thus seen is the same whatever be the inclination of
the direction of vision to the tangentplane of the surface.
From this fact it follows, as is known, that the intensity of
the emanating light varies as the sine of the angle of emana
tion. This law is clearly not inconsistent with the mode
of reflection of condensation, as theoretically determined,
from the surfaces of spherical atoms, and apparently might
admit on this principle of mathematical investigation. In fact,
supposing waves in the same phase to be incident equally
from all quarters on the outer hemisphere of an atom situated
at the boundary* of a medium, and the secondary condensation
at any given point of the surface of the atom, due to any
given wave, to vary as the cosine of the angular distance of
the point from a perpendicular to the wave through the atom's
centre (see p. 283), it may easily be shewn that the resulting
reflected condensation at any point the radius to which makes
the angle 6 with the surface of the medium varies as sin 6.
This is true if the incident waves are not in the same phase,
provided each series be compounded of simple waves in all
possible phases.
The phenomena of absorption are intermediate to those of
transparency and opacity, and are referable to causes which
differ only in degree from those which were adduced to
account for opacity and the colours of bodies. Certain sub
stances, which allow of the entrance and transmission of dif
ferent kinds of rays, extinguish them gradually, and the
" The law is probably modified by reflections from atoms situated a little below
the surface.
THE THEORY OP LIGHT. 421
absorption is at a quicker rate for some rays than for others.
The colours of such substances, as seen by transmitted light,
depend on the thicknesses traversed by the light*. In other
cases rays which have penetrated into the medium to a cer
tain small depth, there undergo a transformation by which
they are actually converted into others of such refrangibilities
that they are capable of traversing the medium without again
passing through a like change. This phenomenon, which
was called by Sir J. Herschel epipolic dispersion, has been
explained by Professor Stokes on the hypothesis of change of
refrangilility, by whom also the discovery has been made
that in this manner rays the wavelengths of which are much
too small for vision, may give rise to visible rays. It has
already been noticed that this transmutation of rays is con
sistent with the mathematical theory of the vibrations of an
elastic fluid as given in this workf. Farther, it may be
remarked that since condensations once generated are not
destroyed, except by regular interference, the condensations
of the luminous waves are not actually annihilated by absorp
tion, but rather they are so changed, and distributed in the
interior of the medium by the absorbing process, as to be
mixed up with the aggregate of undulations to which, as will
be subsequently explained, the forces of heat and molecular
attraction are due.
Addendum to the Theory of Light.
After nearly all that relates to the theory of light had
been printed, being obliged by other occupation to suspend
for a time the preparation of manuscript for the press, I took
occasion in the interval to review the propositions on which
the theory depends, and found that some parts of the mathe
matical reasoning might be made more complete, and others
required corrections. These amendments I propose to add
* See Articles 484504 of Herschel's Treatise on Light.
f See the remarks and references on this subject in page 328.
422 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
here before proceeding to the theory of heat and molecular
attraction,, on which, in fact, it will eventually be shewn that
they have an important bearing.
(a) The principle adopted in page 29T in order to pass
from the solution of Example VI., in which the waves are
supposed to be incident on a fixed sphere, to that of Example
VII., in which the sphere is moveable, was assumed hypo
thetically in default of exact reasoning. The following argu
ment dispenses, I think, with making any assumption, and at
the same time shews in what respects the one adopted is
inaccurate. A small sphere being caused by the impact of a
series of undulations to< perform small oscillations about a
mean position, conceive its actual acceleration to be impressed
at each instant both on itself and on the whole of the fluid.
Under these conditions the sphere is reduced to rest, and the
action between it and the fluid remains the same as when it
was in motion, because the circumstance that the fluid per
forms small oscillations bodily will not alter the relations of
its parts, nor affect the propagation of waves through its mass,
the only consequence being that a given condensation will
arrive a little sooner or later at a given point of space. The
effect of this inequality is a quantity of the second order and
may be neglected in a first approximation. Hence the imme
diate action of the waves on the sphere is the same as when
the sphere is fixed, and the expression for it is at once ob
tained from the solution of Example VI. But there is,
besides, to be taken into account the mutual action between
the vibrating mass and the sphere at rest. Now this is clearly
the same as when the sphere oscillates and the fluid is at rest,
the differences of momentum arising from different condensa
tions at different points of the mass being quantities of the
second order. Hence the expression for this retarding force,
to the first approximation, may be deduced from the solution
of Example IV. obtained in page 264.
(b) The expression in page 296 for the former of the
above mentioned forces contains in its first term the factor
THE THEORY OF LIGHT. 423
1 A, which depends on transverse action, and was assumed
to be of this form because the condensation on the first half of
the surface of the sphere was supposed to be unaffected by
that action. But as this supposition is not supported by rea
soning, and the composition of that factor is at present un
known, it will be preferable to call it \ A/ and to consider
\ to apply to the first hemispherical surface, and h t f to the
other. Also, for the sake of distinction, A 2 h z ' will be put in
the place of ti h" in the second term, A 2 and hj referring
respectively to the first and second hemispherical surfaces.
(c) These alterations being made, and V being put for
m sin q (at + c ), the expression for the first of the two forces
considered in paragraph (a) is
3 dV ^ 1 **
x
If 7Y be the acceleration of the sphere, the other force,
Cut
which is equal to the retardation due to the fluid deduced in
page 266 from the solution of Example IV., is   ^ esti
mated in the same direction. Consequently we have
d*x 3 , dV
, 2
r df = 1+2A dt ' ft
x
This value of ^ should take the place of that given in
page 298, which was obtained on the principle that the action
of waves on a moveable sphere i& the same as the action on a
fixed sphere of waves in which the velocity is equal to the
difference of the velocities of the actual waves and moving
sphere ; which principle is proved by the foregoing reasoning
to be not strictly true.
424 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
If a =TT2A' and *=a(TT2A)' "o that H and K
are functions of A only, we have, since Va'S,
+ Kfl (k.K^S.
This result does not differ in form from that given in page
298, but the values of H and K are now more correctly deter
mined. If the fluid be incompressible, the second term
vanishes because q = ; and at the same time h t h\ = 1
, or
(see p. 295) ; so that = j . Hence if A = 1
the fluid and sphere be of the same density, this equation be
d*x dV
comes jrj = TT , as evidently should be the case.
(d) With respect to the acceleration of an atom due to
the molecular forces of the medium of which it is a constituent,
I see no reason to depart from the principles adopted in page
368 to obtain an expression for the accelerative force brought
into play by the relative displacement of the atoms. By the
same reasoning as that in paragraph (a). it may be shewn that
the action of the setherial waves on the atom is unaffected by
the motion given to it by the action of the molecular force.
By this motion, however, the retardation of the asther is
changed. But if ^ be the actual acceleration of the atom,
the effect of molecular action being included, the retarding
1 d*x
force of the aether will still be ^r TJ Hence, adopting
the expression for the molecular force obtained in page 368,
we shall have
Now the condition of transparency, according to the reasoning
in pages 365370, is, that the ratio of  to F be constant,
THE THEORY OF LIGHT. 425
dx
or that V 7 have a constant ratio to V. But this condition
at
is not satisfied by the above equation unless the second term
on the right hand side be so small as to have no appreciable
effect. That term, which, since q'a' V is a quantity of the
dV
same order as j , and q'b has been assumed to be an ex
tremely small quantity, will in general be very small compared
to the preceding one, may possibly be the exponent of the
gradual absorption or extinction of light which is found to take
place in all substances, however transparent, when the spaces
traversed by the rays are very considerable. Neglecting,
therefore, the second term, so far as it relates to the theory of
dispersion, and integrating the equation, we have for a given
series of waves
It will now be supposed, regard being had to the considera
tions entered into in pages 370 and 371, that the factor \ h' t
( &'\
is equal to k f 1 ^J . The reasoning in page 371, from
which it was inferred that the quantity in brackets should
contain \ in the place of V appears to be invalid, inasmuch as
in the general series for 1 f, X is the actual wavelength
independently of the elasticity of the medium. Thus, since
Hence, admitting that the value of V 7 is accurately
(Jut
given by the above equation to the first approximation, and
that the apparent elasticity of the aether within the medium,
calculated as in pages 364 367 for the ease of fixed atoms, is
426 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
dx
to be altered in the ratio of V r to V when the atoms are
at
moveable, the formula for dispersion becomes
: it 2 1 dx
It may be remarked that if the equation ((3) in page 370 be
expanded to the first power of the factor l h lt and if that
factor be equal to k I 1 j J , the two equations become iden
tical, provided also be so small a quantity that it may be
neglected in comparison with unity,
The equation (?) may be put under the form
and if, for brevity, n* be substituted for (l + ^ , it will
be found that
w
In the instances of the two substances to which the calcula
tions in pages 372 and 373 refer, by employing, as there, the
values of M and X for the rays (J9), (E); and (H), the following
results were obtained :
For the Flint Glass, ^ = 14,54906, = 0,44611, 0=8,20984;
For Oil of Cassia, A= 9,35876, .5=0,33595, (7 = 6,28431.
With these constants I have calculated from the formula (7')
for each substance the value of X corresponding to the given
values of p for the other four rays> and compared, as follows,
the results with the observed values of X. The results given
by the formtda (7) in page 372 are similarly compared in
THE THEORY OF LIGHT. 427
order to furnish some means of estimating the weight due to
this numerical verification*.
Flint Glass No 13. Excess of the Oil of Cassia. Excess of the
calculated value of X. calculated value of X.
Bay. By formula 03). By formula (p). Kay. By formula ($. By formula (/?').
(0)... 0,0016. 0,0016 (0)... + 0,0017 + 0,0025
(D) . . .  0,0030  0,0028 (D) ...  0,0022  0,001 1
(F) ... + 0,0022 + 0,0021 (F) ...0,0024 0,0038
(G) ... + 0,0031 + 0,0029 (G) ... 0,0000 0,0028
It will be seen that the differences between the calculated
and observed values of X are in some degree less by the
second formula than by the first for the Flint Glass, while for
Oil of Cassia they are in greater degree greater. The dissimi
larity of the excesses for the two substances seems to point
to errors of data as the main cause of the differences between
calculation and observation, and as the given values of //, are
likely to be much more accurate for the Flint Glass than for
the Oil of Cassia, the more trustworthy comparisons may be
regarded as favourable to the second formula. When it is,
besides, considered that the above differences scarcely in any
case exceed amounts that may be attributed to erroneous data
(see p. 373), we shall, I think, be justified in concluding that
the foregoing comparisons are not inconsistent with the truth
of formula (/3'), and with its being deduced from exact
principles. This conclusion will receive confirmation from
certain physical consequences which I n<ow proceed to deduce
from the theory.
The unknown physical constants involved in the formula
(/3') are H$, e, A, k and &', of which the last two depend on
the hydrodynamical conditions of the problem, and might, by
a more complete solution of it than that here given, be ex
pressed in terms of known quantities. Those two constants
being at present unknown, the numerical values of A, B, and
* The excesses by the first formula are somewhat different from those in
page 373, owing to the correction of a mistake which was found to have been
made in the previous calculation. See the Errata.
428 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
G do not suffice to determine the other three. There are,
however, certain characteristics of the solar spectrum, by
means of which, as I am about to shew, some approximate
information may be obtained relative to the numerical values
of the three constants HS, n z , and k'. From experiment it is
known that the rays of the spectrum are capable of producing
heat and chemical action, as well as the sensation of light, and
that there are heatrays extending beyond the visible limits of
the red end, and chemical rays extending much beyond the
visible limits of the violet end. To account for these effects
being accompanied by light within the range of the sensibility
of the eye, it suffices to attribute them to the direct vibrations,
which, according to the hydrodynamical theory of undulations,
always coexist with the transverse vibrations ; and clearly the
same effects may be ascribed to direct vibrations which co
exist with transverse vibrations that are incapable of affecting
the sense of sight. What, then, is the explanation of the
transition from the calorific action to the chemical action?
The reply that the present theory gives to this question is
that the change may be supposed to correspond to a change of
k'u?
sign of the factor 1 ^ , which may be positive for the
A
larger values of X and negative for the smaller. For by the
mathematical theory of the dynamical action by which waves
produce a motion of translation of a small sphere (given in
pages 303 307), the setherial waves propagated in any sub
stance will tend to transfer the atoms in the direction of pro
'2
pagation, or the contrary direction, according as 1 ~~ is
A*
positive or negative; that is, their action will be repulsive,
or like that of heat, in the former case, and attractive, or
such as may be proper for producing chemical effects in the
latter.
It appears from experiments made by M. Edm. Becquerel
that in addition to the chemical action due to rays partly co
incident with, and partly extending beyond, the more refran
THE THEORY OF LIGHT. ^ 429
gible luminous rays, there is an action of the same kind
the intensity of which is very approximately represented by
Frauenhofer's curve of intensity for the luminous rays*.
This additional chemical action may, therefore, be reasonably
ascribed to the transverse luminous vibrations, and to their
being capable of producing permanent motion of translation of
the atoms, while the sensation of light is caused solely by
their vibratory action. Consequently through a portion of the
spectrum the rays will have the property of producing chemical
effects as well as heat, and there will be no point at which the
one kind of action ends and the other begins.
Again, it is to be considered that hitherto the reasoning
has applied only to a simple medium, consisting of atoms all
of the same kind. Let us now suppose the medium to be
composite, and to consist of atoms of n kinds, differing, how
ever, only in the magnitudes of their radii. Then for an
atom of each kind an equation such as (a') in p. 424 may be
formed; and if to satisfy the condition of transparency the
last term be omitted, the only constants depending on the
magnitude of the atom are e* and \ h^. Also in the ex
pression kfl ^1 , which has been substituted for the
latter quantity, p and X will be the same for all the atoms.
Hence if i/ 1? v 2 ... v n be the proportionate numbers of the dif
ferent kinds of atoms in a total number N 9 we shall have the
several equations
7J ,fJt?\dV
Suppose now that
z> d*x v d*x v d*x
* These experiments are cited in Jamin's Cours de Physique, Tom. m,
p. 430.
430 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
v& v n e n x n= jx
N "3F' N dt z ^ ' N df df
Then since the rate of propagation of the astherial waves in
the medium is affected independently by the different kinds
of atoms, and by each kind in proportion to their number
and mobility, it follows that the condition of transparency
dx
requires that y should be proportional to F, and therefore
jz proportional to j . But by adding together the several
do dt
equations applicable to the n different kinds of atoms, it will
be seen that this last condition is not satisfied unless e* in the
second of the above equations be absolutely constant. Such
a constant must therefore be regarded as .characteristic of a
composite medium which is transparent either with respect to
all rays of the spectrum, or to certain rays. These two con
ditions being fulfilled, if we substitute
Nk for vfa + vjc z + . ., + vje n ,
Nkk': for vjtfc + vjc&+ ... vjcje.',
and add together the foregoing n equations, we shall finally
obtain an equation of exactly the same farm as (/3'). That
equation may therefore be used whether the medium be
simple or ^compound.
Now since it may not be assumed that the beforemen
tioned change from a repulsive to an attractive action of the
setherial undulations takes place with respect to each kind of
atom for exactly the same value of X, we cannot affirm that
the calorific action of the direct vibrations in a composite
medium ends where the chemical action begins. Admitting,
however, that the mean, or aggregate, translating action of
setherial undulations propagated in such a medium must pass
through ser for some value of X, it may be presumed that
. < * ' ' / 7 ' 2\
this will fee the case when the quantity It (l ~J vanishes,
k' having the value appropriate to a compound medium, as
THE THEORY OF LIGHT.
431
determined by taking the ratio of the above expression for
Nick' to that for NJc.
These theoretical considerations are in accordance with
experimental results obtained by Becquerel, as exhibited by
means of a very instructive diagram in Jamin's GOUTS de
Physique (Tom. III. p. 428). From this diagram I gather,
as far as regards the direct vibrations, with which alone we
are concerned in a theory of dispersion, that the transition
from the calorific to the chemical action occurs where the
value of X is nearly equal to that for the ray (F). Although,
as already intimated, this transition may not take place for
a certain value of X independently of the composition and
intrinsic elasticity of the medium, yet as experience seems to
indicate that such is the case approximately, the truth of the
theory may in some degree be tested by tracing the conse
Ic'u?
quences of assuming that 1 ^ = when the value of X is
X
that for the ray (F). The following results were obtained
on this supposition in the two instances of the Flint Olass
No. 13 and Oil of Cassia, the values of /JL for the ray (F) being
taken from the data in page 373, and the adopted values of
A, B, and C being those given in page 426.
By the formula ('), when 1^ = 0, ^=l+?8.
Hence, since for the Flint Olass /*= 1,64826 for the ray (F),
it will be found that H8 = 1,71676, and that
n*(=0l ITS) =5,49308,
0,01326.
1 + 2A V 3&V
At the same time the value of X obtained from the equation
X 2 = &7fc 2 is 1,7994, the observed value for the ray (F) being
1,7973. The excess of the former is, as it ought to be, the
same as that given in page 427.
432 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
For Oil of Cassia, the value of //, for the raj (F) being
1,6295, like calculations give HS = 1,65527, n 2 = 3,62904,
k
k' 1,21147,   = 0,01539, and \= 1,7935. The excess
above the observed value of X is 0,0038 as in page 427.
These numerical results confirm by their consistency the
hypothesis that the change from the calorific to the chemical
action of the direct vibrations corresponds to a change of
k'u?
sign of 1 . It is to be observed that the quantity Jc is
A
not determined independently of A ; but since for an incom
pressible fluid its value is unity, it will not in any case differ
very much from unity for the aether. Hence we may infer
k
from the above numerical values of ^r that A is a large
quantity. With respect to the constant n 2 it is important to
remark that 7? //?, which is the denominator on the right
hand side of the equation (ft'), is positive in the case of the
Flint Glass for values of /t less than 2,3437, and in that of Oil
of Cassia for values less than 1,9050. These limits much ex
ceed the respective maximum values of //, for visible rays in
the two instances, and probably the same would be found to
be the case in any instance of a solid or fluid substance.
Let us now enquire what may happen with respect to the
value of w 2 p? when the formula (ft') is applied to a gaseous
body. By recent experiments it has been ascertained that
a large number of substances, when looked at in a vaporized
and ignited state with a spectroscope, exhibit, generally with
a faint continuous spectrum, certain bright lines of definite
refrangibility. On theoretical grounds it may be presumed
that these rays have their origin in the disturbance of the
aether caused by violent and rapid vibrations of the atoms of
the gas in its state of ignition. The number and positions of
these lines are constantly the same for the same substance,
and may be regarded as characteristic of it. It is a still more
remarkable circumstance, that many of the dark lines of the
THE THEORY OF LIGHT. 433
solar spectrum are found to have exactly the same refrangi
bilities as the bright lines of the aeriform bodies thus experi
mented upon. It appears from observation that certain of
the solar lines are produced by the passage of the Sun's rays
through the earth's atmosphere, and the remainder are with
much probability attributed to passage through a solar atmo
sphere. Hence it has been reasonably inferred from the
abovementioned coincidences of the refrangibilities of the
dark and bright lines, that the terrestrial and solar atmo
spheres contain the very same gases, or vapours, as those
employed in the experiments. But this view, in order to
account for the solar lines being dark, requires to be supple
mented by the hypothesis that a gas in its quiescent state
has the property of neutralizing those rays in their passage
through it which in its ignited state it is most capable of
emitting. Now although we may not be able with our present
knowledge to ascertain why the vibrating atoms of a gas
generate in the aether waves having particular periods of
vibration, it may yet be possible to explain theoretically in
what manner the solar rays which vibrate in the same periods
are caused by passing through the gas to disappear from the
spectrum. The explanation I am about to propose is founded
on the antecedent theory of dispersion.
Conceive an atom of the gaseous medium to perform
vibrations of a certain period about a mean position by the
action of its proper molecular forces, as brought into play by
the circumstances which cause the state of ignition ; and let
 be the molecular force at the distance x from the mean
position and tending towards it, e 2 being a constant of the
same signification as that we have already had in the fore
going investigations, and 1? another constant depending on the
period of the vibrations. Then, taking into account the resist
ance of the aether to the motion of the atom, we shall have
d^x c?x 1 d?x d^x $ \.t?
~rH = '1* ~2A d?' r ~df + (1 + 2A)J 2a!=:0 '
28,^
434 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
Also if x be the distance from its mean place of a particle of
the aether vibrating in the medium, wS have
ftiTKat \ d 2 x 4wVV , _
x = m cos , he; and .'. ^ \ ^ x Q
\ A. / CLi A
If, therefore, in accordance with the abovestated facts, the
period of vibration of the atom be the same as that of a par
ticle of the aether, it follows ( since 7 = ) that
\ A A /
4TrW
~^~
As it appears from the experiments that for the same gas,
even if it be simple, there may be several bright lines, we
must suppose that each atom is susceptible of complex mo
tions consisting of coexisting simple vibrations for each of
which the value of I is different. Hence, as the lefthand side
of the last equation is absolutely constant for a given simple
medium, it follows that for every such value of I there is
a corresponding value of X. Also, since the atom acting on
the sether by its vibrations generates setherial undulations
that produce light, its motion might be exactly like that of
a particle of the sether in lightproducing waves. Let us,
therefore, suppose that
271^ = ^, 27r/ 2 = X 2 , &c., so that
Hence n 2 = 1 for these particular values of X ; and since for
a gas fjb differs very little from unity, it follows that the deno
minator n 2 fj? in the equation (/3') becomes extremely small.
There is, in short, a breach of continuity in the values of /*,
given by that equation when X has these values. This result
1 take to be an indication that the rays corresponding to the
bright lines cannot be transmitted in the medium. Assuming
that the solar rays pass through various aeriform substances
either composing the solar and terrestrial atmospheres, or
suspended in them, the existence of dark lines in the spectrum
may in this manner be accounted for.
THE THEORY OF LIGHT. 435
If the aeriform body be composed of atoms of different
kinds, we may at first regard the atoms of one kind as con
stituting a simple medium capable of extinguishing rays of
certain refrangibilities in the manner above investigated.
The sether within this medium may then be treated as a fluid
like the actual sether, but of somewhat less elasticity, and as
being incapable of transmitting those particular rays ; and
the waves of this modified aether may be supposed to be pro
pagated in another simple medium, consisting of atoms of a
second kind, and having, , like the first, the property of extin
guishing certain rays; and so on. Thus we may account
for the observed fact that the fixed lines of a composite gas
consist of those which characterize the components.
It has already been .stated that the value of n 2 for liquid
and solid bodies is probably always greater than the greatest
value of [j? for the visible rays. Hence, according to this
theory, we should not expect dark lines to be generated by
the passage of light through such bodies ; and, as far as I am
aware, no lines have been ascertained to be generated under
these circumstances. So long, also, as n z exceeds //, 2 , the
order of the colours of the spectrum will be the same for all
substances. But we have no ground for asserting that n* /ji?
is always a positive quantity for vapours and gases, in which,
therefore, it is theoretically possible that the order of the
colours may be reversed. In fact, M. Jamin has cited experi
ments which shew that this is actually the case in the refrac
tion of vapour of iodine. (Cours de Physique, Tom. ill.
p. 440.)
The foregoing is the best solution I am able to give of the
difficult problem of Dispersion. 1 am aware that it is imper
fect, and that its complete verification requires an exact d
priori investigation of the expression for the factor h^ h{
depending on transverse action. Although the expression I
have employed was not strictly so deduced, it seems to be
verified, at least approximately, by experiment, and so far
may serve to indicate in what manner the Undulatory Theory
282
436 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
of Light bears upon the determination of the nature of the
forces which act on the ultimate atoms of matter. It was
with a view to this application that the theory of dispersion
has been so long dwelt upon. I proceed now to the theory
of those forces,
The Theory of Heat and Molecular Attraction.
The first part of the preceding theory of light may be
considered to have established with a very high degree of
probability the existence of an sether, which, so far as regards
phenomena of light, may be treated as a continuous medium
pressing proportionally to its density. In the second part
various phenomena were explained on certain additional
hypotheses respecting the ultimate parts and constituency of
visible and tangible substances, and these explanations, while
they strengthened the argument for the existence of the
aether, also rendered probable the supposed qualities of the
ultimate parts of bodies. The Theory of Heat and Molecular
Attraction, which are forces so related that they may be
included in the same investigation, will be made to rest on
the very same hypotheses.
It is proper to state at the commencement of this research
that its object is not to give explanations in detail of the
observed effects of heat and molecular attraction, but to
answer the questions, What are these two forces, and in what
manner do they counteract each other? I understand mole
cular attraction to be a force which has its origin in a
mass, or congeries of atoms, towards the centre of which the
attraction is directed. The general physical theory I am
propounding does not admit the existence of the action of
force through space without the intervention of a medium.
It assumes that atoms are incapable of change of form and
magnitude, and, therefore, passively resist any pressure on
their surfaces tending to produce such change ; but all active
forces are supposed to be modes of pressure of the setherial
THE THEOKY OF HEAT. 437
medium, subject to laws which may be deduced from the
mathematical principles of Hydrodynamics. The problem
proposed for solution is, accordingly, to ascertain in what
manner, and under what circumstances, the pressure of the
aether may act like the forces experimentally known as repul
sion of heat and attraction of aggregation, the reasoning being
conducted by means of hydrodynamical propositions demon
strated in the antecedent part of the work.
It is well ascertained that lightproducing rays may also
be heatproducing. This is so remarkable and significant a
fact, that a theory of light which does not account for it may
be said to fail in an essential particular. Since in the theory
I have proposed the transverse vibrations of rays always
accompany direct vibrations, and it was concluded (p. 334)
that the sensation of light is entirely due to the former, we
are at liberty to refer the action of heat, or other modes of
force, to the direct vibrations. There is, however, this dis
tinction to be made, that in the theory of light only terms of
the first order with respect to the velocity of the astherial
particles were taken into account, and the motion resulting
from the pressure of the aether on the atoms of substances was
found to be wholly vibratory ; whereas the forces of heat and
molecular attraction are known to produce permanent mo
tions of translation. Hence, taking into consideration the
hydrodynamical results obtained in pages 305 and 311, the
theory of these forces is to be inferred from terms of the
second order relative to the velocity and condensation. Be
fore proceeding to this enquiiy. it will be worth while to
introduce here an argument from which it follows, apart from
the results of the mathematical investigation, which is con
fessedly incomplete, that a spherical atom free to obey the
impulses of the setherial undulations necessarily receives a
permanent motion of translation.
It may be assumed that if a series of undulations be
incident on a small solid sphere in a fixed position, the
variation of condensation at any point of its surface obeys the
438 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
same law as the variation of condensation, at a given point,
of the original undulations ; and also that if the diameter of
the sphere be extremely small compared to the breadth of the
undulations, the phase of condensation will be quam proxime
the same at the same instant at all points of the surface of the
sphere. But the amount of condensation or rarefaction at
each instant will vary from point to point of the surface, and
in consequence of such variation the waves tend to move the
sphere. If at each point the sum of the successive con
densations be exactly equal to the sum of the successive
rarefactions, the waves will tend to give to the sphere only a
vibratory motion ; for the action of the condensed and rarefied
portions of each wave will produce equal and opposite effects.
But this equality between the condensation and rarefaction does
not strictly subsist in a wave of the sether, inasmuch as the
motions of its particles, as may be inferred from the equation
(14) in page 206, are wholly vibratory ; which could not be
the case unless the moving forces in the condensed part of the
wave were greater than those in the rarefied part, or the
condensations greater than the corresponding rarefactions.
(See the Corollary in page 207). It hence follows, the atom
not being susceptible, like the fluid, of variations of density,
that the accelerative forces due to the condensed portion of
a wave are more effective than those due to the rarefied
portion, and that thus there will be an excess of action in the
direction in which the condensation tends to move the sphere.
If the sphere be now supposed to be free to obey the
impulses of the waves, we may conceive its motion to be
impressed at each instant both on itself and on the whole
mass of fluid in the opposite direction, so that the sphere is
reduced to rest. The condensations are in no respect changed
by a motion which all the parts of the fluid partake of in
common, so that the waves are incident on the sphere, and
the condensation is distributed about it, just as when it was
supposed fixed. There is, however, the difference that the
times of incidence of the same condensation in the two cases
THE THEORY OF HEAT. 439
are separated by a small periodic interval, owing to the
vibratory motion of the mass. This inequality gives rise in
the case of the moveable sphere to a periodic condensation of
the second order, having as much positive as negative value,
and therefore incapable of producing permanent motion of
translation. Thus there remains an excess of accelerative
force due to the condensed part of the wave, in obedience
to which the sphere will perform larger excursions in one
direction than in the contrary direction. If, moreover, the
resistance of the fluid to the motion of the sphere be taken
into account, since its effect will be to diminish in the same
proportion the accelerations in the two directions, the ex
cursions will still be in excess in the direction of the action
of the condensed parts of the waves. Thus there will be
permanent motion of translation* .
I return now to the mathematical reasoning relating to
the motion of a small sphere acted upon by setherial undula
tions, with the view of ascertaining the conditions which
determine the direction of the permanent motion of transla
tion, this investigation being a necessary preliminary to a
theory of attractive and repulsive forces. Having found upon
reconsideration of the reasoning already devoted to this en
quiry that it may be extended with more exactness to
quantities of the second order, I shall here briefly recapitulate
the previous argument in order to introduce this modification
of it.
The equations (34) and (35) of the first order obtained in
pages 258 and 260, being applicable to motion symmetrical
about an axis, were first employed to find the motion and
pressure of the fluid caused by given rectilinear vibrations of
a small sphere, and also to find the motion and pressure
* It is desirable that this inference, which seems to be strictly deduced from
admitted dynamical principles, should be tested experimentally by means of the
action of rapid vibrations of the air on a small sphere. Although the effect
would in this instance be extremely small, modern experimental skill might suc
ceed in detecting it.
440 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
resulting from the incidence of a series of waves on a small
fixed sphere. For solving these two problems a particular
solution of the equation (35) was employed which satisfied
the given conditions to the first approximation. It was seen,
however, that although the e]asticity of the fluid was taken
into account, the resulting action on the sphere was the same
that would have been obtained if the fluid had been supposed
to be incompressible, all its parts, consequently, in the second
problem vibrating equally. Having discovered that the
equation (36) in page 279, derived from equation (35) by
differentiating with respect to 0, was satisfied both by the
same particular solution as (35), and also by an additional
one, I found on applying the latter to the second problem
that I could thereby embrace a term in the approximate
expression for the condensation of the incident waves which
was not included in the former integration. (See in pages
284 286). But it was still found, although the new term
has no existence unless the fluid be compressible, that the
action on the sphere did not differ from that of an incompres
sible fluid. The explanation of this result may be stated as
follows. The equations (34) and (35) are founded on the
equation (29) in page 250, which takes account of the prin
ciple of composition of spontaneous motions, and is true only
when the composition is such as to neutralize transverse
motion. Now when the regularity of a series of waves is
interrupted by incidence on a small sphere, transverse action
is necessarily induced, unless the fluid be either incompres
sible, in which case there is no transverse vibration, or so
extremely elastic that the transverse vibrations accompanying
direct vibrations of the order taken into account have no per
ceptible effect. Accordingly the equations (34) , (35) and (36)
are applicable only in these two cases, and when thus ap
plied they may be employed to determine the motion and
pressure at all points of the fluid.
It is, however, to be said with respect to the fluid that is
compelled to move along the surface of the sphere, that its
THE THEORY OF HEAT. 441
motion conforms to the conditions on which the equation (29)
was investigated, the sphere itself "by its reaction neutralizing
transverse motion. Hence if the .application of the three
equations be limited to the fluid immediately contiguous to
the sphere, they may be used to determine the pressure at
any point of the surface of the sphere. This has been done
to the first approximation by means of the reasoning com
mencing in page 294, according to which the value of the
first part of the superficial condensation is obtained by multi
plying the expression for it given in page 283 by a constant
factor L h, and that of the second part by multiplying its
expression in page 286 by another constant factor h' h".
These are the constants called h'^ h^ and h 2 h^ in page 423.
It is proper to state here that the reasoning referred to, while
it establishes the reality of these factors, does not prove that
they consist of parts applying separately to the first and
second halves of the spherical surface. I propose, therefore,
to designate them in future as H^ and H z , and to trace the
consequences of regarding each as applicable to the whole of
the surface. This being understood, I shall now attempt to
give a solution, inclusive of all small quantities of the second
order, of the problem of the motion of a small sphere acted
upon by a series of undulations. The accelerative force of
the fluid will, at first, be determined supposing the sphere to
be fixed.
It will be assumed, as in p. 279, that the incident waves
are defined to the first approximation by the equations
V a a' = m sin q (at + r cos + c ),
and that V ' = aV = m sin q(at\ c ). Also, in accordance
with what has just been stated, the expressions, to the first
approximation, that will be adopted for the superficial con
densation, and for the velocities along and perpendicular to
the radius vector r, are the following :
cosQ cos 6H+ ~ sin Qcos 2 0,
442 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
U=  m'H 1 (l   3 J sin $ cos  m'H a qr (l  5] cos Q cos 2 0,
Q being put for q(at + c ). Since these equations are to be
applied only to points for which r is very nearly equal to b,
U is an indefinitely small quantity.
It having been proved by the argument concluded in
page 239 that udx + vdy + wdz is an exact differential for the
resultant of any number of primary vibrations relative to dif
ferent axes, when expressed to terms of the second order, and
as the motion and pressure in the present example are to be
regarded as resulting from such vibrations, it follows that we
may suppose that differential to be exact on proceeding to the
second approximation. Let, therefore, (dfy = udx + vdy f wdz.
If we now assume, in accordance with principles already
advocated, that the dynamical equations applicable to com
posite motions in which transverse action is neutralized, are
the same as those applicable to simple motions, excepting
that a' 2 holds the place of a 2 , we shall have
a^dp (du\ _ a' 2 dp fdv\ _ a' 2 dp fdw\ _
pdx + (dt) 7^T + (df) ~ ' ~pdz~ + \dt) ~
provided these equations be applied only to the fluid con
tiguous to the sphere. Consequently, with that restriction,
the equations to be employed for the second approximation
are of exactly the same form as (24) and (25) in page 226 ;
and when adapted to the case of motion symmetrical about
an axis, and transformed from rectangular coordinates to the
polar coordinates r and 6, the centre of the sphere being
origin, they are changed to the following :
d\r4> l/f.rj,
.
"~d" * a )~ d*dt~ a"dt
d<f> d<f,
THE THEORY OF HEAT. 443
For the same reason as in the first approximation, that is,
for the purpose of taking into account the second term of the
expression for V V in page 280, I shall use, instead of the
first equation, that which results by differentiating it with
respect to 6. Thus, putting P for '^ , we have
d*P d\r
a'*d0dt 
in which equation Z7 2 , being indefinitely small in the present
application, is to be omitted. For the first approximation
an integral was used which satisfied this equation deprived
of the last two terms ; so that to proceed to the second ap
proximation it is required to express these terms as explicit
functions of r, 6, and t by means of the results of the first
d*P d* W
integration. Now observing that ^ = r z 2 , it will be
found by substituting for W its first approximation, that
' = ^ sin e + R * siri 26> + ^ sin 3(9 + ^ sin 46> >
JKj, B z , R^ RI being known functions of r and t. This ex
pression for the small terms being substituted in the differen
tial equation, an exact integral of it may be obtained by
supposing that
P= ^ sin 6 + >Jr 2 sin 20 4 ^ 3 sin 30 + fa sin 0.
In fact, on substituting this value of P the following dif
ferential equations result for determining T^, ^ 2 , \Jr 3 , >^ 4 :
dr*
_ 1 _ 3 _ ,
" a " a " "* * ' *
rfr" r a r 2 " 2> ofr 2 r 2
I have ascertained that these four equations admit of being
exactly integrated*. It is, however, to be observed that the
* The integrations may be effected by means of multipliers, as is shewn by
Euler in his Cafe. Integ. Tom. n., Art. 1226. See Peacock's Examples, p. 411.
444 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
expression for P will be required for no other purpose than
to calculate I j sin cos 6 dO, and since  =  \ j dO, it
J n dt dt r j dt
follows that the terms containing sin 20 and sin 40 disappear
by the integrations, and we have only to determine the values
of ^ and ijr 3 . By means of the first approximation to W
we get
R, =  m'Hrfr* (l + J) sin Q?*.
Hence taking account, at first, only of terms involving the
first power of in, we have to integrate
If jR 1 ' be put for the righthand side of the equation, the
integral is
which in its complete form contains two arbitrary functions
of the time. It is, however, unnecessary to introduce these,
as they may be considered to be included in the first ap
proximation ; so that the integration gives
Thus to terms containing the first power ofm,
According to the rules of approximation, new values of
<r and W should now be obtained from this value of P and
be substituted in the last term of the equation (e) ; but as this
operation would only give rise eventually to additional terms
of the order of <fl? x those resulting from the first values, it
may be omitted.
THE THEORY OP HEAT. 445
To proceed to terras containing m* we have first to in
tegrate
dr* r 2 2c
The righthand side of this equation being J? 3 , the exact in
tegral, omitting arbitrary functions of the time, is
3m' 2
=
Then the integration of the equation
gives for the part of fa containing w' 2 ,
3m' 2 r/1 165 5 \ r 211
These results shew that it is possible to calculate exactly the
terms involving m' 2 . It is, however, to be observed that
dd> 1 [dP 7
these terms, since ^ =  I ^ dv, give rise in the value of
~ to terms which, when b is put for r, contain the factor
(fb 3 , and may, therefore, in the present problem be omitted.
Hence with sufficient approximation, putting b for r, we
have
and consequently for our purpose
Now since to terms of the second order
446 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
we have next to calculate  (a'V  W 2 ) by employing the
first approximations to or and W, after substituting in them
b for r. We shall thus have, putting Ffor aV ,
J _. re,,,)'
1 /3#F . 5tf& <ZF . ,Y
   1 sin +  V T sin cos .
2\ 2 3a ' dt J
Consequently the moving force of the pressure of the fluid
f
on the sphere, viz. 2?rZ> 2 I a'V sin 6 cos 6 d6, is equal to
cos( > 7
J ) dt
terms which would disappear by the integration being omitted.
Hence, the mass of the sphere being ^ , the accelerative
o
action of the fluid will be found to be
&ff d v r ir ( /a ^\ i 7
d v r fir ( /a
^r I+ TO + I ^U
If we suppose the unknown constant H 1 to include as a factor
q*b* Vd V
\ + *r , and neglect quantities of the order <fl? x , . , the
J. \J CL Chit
expression for the accelerative action finally becomes
2A dt
It is now required to find the accelerative action of the
fluid when the sphere is supposed to be moveable. Since the
motion and condensation of the fluid are symmetrical with
respect to an axis, and we may still suppose udx + vdy + wdz
to be an exact differential, the same differential equations are
THE THEORY OF HEAT. 447
applicable to this case as to that of the fixed sphere. As
before, the process of solution will be, to obtain a first inte
gration by neglecting small terms, arid then after substituting
in the neglected terms values derived from the first approxi
mation, to effect a second integration. For calculating the
first approximate values of the velocity and condensation, the
principle enunciated in page 422 will be adopted, according
to which the fluid acts by propagated waves on the sphere in
motion just as if it were at rest, and the total action is the
sum of this action and that of the resistance of the fluid, sup
posed to be at rest, to the actual motion of the sphere. The
velocity and condensation due to this resistance may be at
once inferred from the results given at the top of page 264
for the case of the oscillating sphere, provided the sign of T
be changed, or T be the velocity of the sphere in the positive
direction, which is that of the propagation of the waves.
7/T7
Hence j is the acceleration of the sphere in the positive
direction to the first approximation, and is therefore the same
d 2 x
quantity as jg calculated in page 423. But it is to be
d*x
observed that the second term of the expression for ^ exists
only on the supposition that the factor H z has different values
for the two halves of the spherical surface. As we now sup
pose it to have the same value for the whole of the surface,
that term disappears by the integration with respect to 0, and
we have accordingly
d*x_ dT _ 3^ dV_ dx_^
~~~ ' ~ ~
dt ' dt~ ~2A+r
If therefore cr' } W' 9 U' represent the condensation and re
solved velocities due to the resistance of the fluid, it follows
that
dv ,
448 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
Adding these equations, on the principle of the coexistence
of small vibrations, severally to the analogous equations in
pages 441 and 442, we have to the first approximation the
following equations applicable to the problem of the moving
sphere :
Hj Al V\dV Q ffrff, 24 5 N rr
a ' ~ * = ' r+ 2A+T ?] ~dt cos0 ~ ~ r + = Fcos e >
These values of U, W, and P give the means of expressing as
functions of r, 0, and t the last two terms of the equation (e)
in page 443. Respecting that equation it is to be remarked
that the origin of the polar coordinates r and 6 is the centre
of the moving sphere. Hence if a be the variable distance of
the centre from a fixed origin of coordinates on the axis of x,
we have
so that (a being a function of t) r and 6 are each functions of
t as well as of x, y, z. But by the reasoning contained in
page 259, the equation (e), deprived of the last term, is equally
applicable, for a first approximation, whether the sphere be
moving or fixed. Also since the differentiation of r and 6
with respect to t in the last term of the equation (e) would
give rise to terms of the third order relatively to m', such dif
ferentiation may be omitted. Thus it is only in obtaining
the value of 75 to terms of the second order that r and 6 need
at
be regarded as involving t. This being understood, let us pro
ceed to the second approximation, commencing with substi
d 2 P
tuting at first for , 2 only terms containing the first power
QJ CLv
ofw'.
THE THEORY OF HEAT. 449
Since to terms of that order r and may be regarded as
independent of t, we have
Hence substituting in the equation (e), and assuming that
P = ^/s
it will be found that
_____ 2\
* ' ' +
_____
dr* r* ' 2a' \ 3rV dt
The integration of these equations gives
2

2a U 9/ dt
Consequently,
It is not necessary to obtain from this value of P new values
of cr, U, and W for substitution in the last term of the equation
(e), because, for the reason alleged in the case of the fixed
sphere, the additional terms containing gV may for that pur
pose be omitted.
To obtain the terms of >2 2 which contain m' 2 , we must
recur to the value of P to the first approximation, viz.,
and differentiate it with respect to r and as well as V,
29
450 THE MATHEMATICAL PRINCIPLES OP PHYSICS.
sidering that these coordinates change with the time for a
given position in space. It is readily seen that when r and
vary with the time under that condition, we have
dr ZHf rdO
Hence putting the expression for P under the form
dV
Bf t (r) Fsin 6 + H,F 2 (r) 2 sin aft
and taking account of the above differential coefficients, it will
tPP
be found that the terms of Tr which contain m' 2 are
If we represent these terms by
JB/ sin 6 + EJ sin 26 + E^ sin 30,
we may infer from the above reasoning that R t f and ^ 3 ' each
contain the factor gV.. Substituting now in the last term of the
equation (e) the values of U and W given by the first approxi
mation, the term in that of U containing the indefinitely small
Z> 5
factor r 5 being omitted, the result, as in the case of the
fixed sphere, will be of the form
E" sin + j? 2 " sin 26 + ^ 8 " sin 3d + R^' sin 40,
and, as in that case, jV is a factor of each of the coefficients
RI and R 8 ". Hence retaining only those terms the conse
quences of which are not subsequently cancelled by integra
tion, we have
THE THEORY OF HEAT. 451
From this result we might proceed to calculate by a second
integration the terms of P containing m* that do not eventu
ally disappear by integration. But, just as in the previous
case, these terms give rise in the final value of ^ to terms
which, as containing the very small factor <fb s , may be omit
ted. We have thus proved that in the value of P obtained to
the second approximation there are no significant terms con
taining m' 2 .
Kesuming, therefore, the foregoing expression for P of the
first order with respect to ra', omitting for the same reason as
before the term containing sin 20, and putting b for r, we have
relatively to points contiguous to the surface of the sphere
T} watjix MA \ + . *4 is I + &
2A+1
and consequently by integrating with respect to 6, and differ
entiating with respect to t,
it being unnecessary to add an arbitrary function of r and t.
Again, from the first approximations to <r, Z7, and W, after
V z
putting b for r, and V\ r for V (see formula (28) p. 246),
CL
we get, omitting terms of the third order,
3ff, A 2 1 I Sff.A IT a 5H * b dV
FCOS(?
2 1
) 2
Consequently, suppressing terms which would disappear by
the integration,
292
452 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
a'V sin d cos Odd =
sin cos BdO.
These integrations being effected, we get the moving force of
the fluid on the sphere; and dividing the result by the mass
of the sphere, we have finally for the accelerative force
If we put HI for H t \ 1 + * 1 1 r J [ , and reject terms con
7
taining 7 x ^^ the result may be expressed thus :
accelerative force =
Since q 2 b 2 is by hypothesis an extremely small quantity,
this force scarcely differs at all from that obtained in the case
of the fixed sphere, except in having 2A + 1 in the denomina
tor instead of 2A.
I have thus succeeded at length in obtaining an expression
for the acceleration of the moveable sphere by an investigation
which strictly takes into account all terms of the second order
so far, at least, as they have any appreciable effect*. This
important result, on which alone, according to the physical
principles advocated in this work, a theory of attractive and
repulsive forces can be based, was gradually arrived at after
* The attempt I made in the Philosophical Magazine for November, 1859, to
solve this problem rests fundamentally on the same principles as the solution now
given, but in respect to the mathematical reasoning is defective and inaccurate.
THE THEORY OF HEAT. 453
reconsideration of the solution of Example vii. given in pages
296306. In what relates to the second approximation the
reasoning of that solution was discovered to be not sufficiently
exact and complete. It is true that the problem cannot be
regarded as fully solved till expressions have been obtained
for the constants H t and H z ; but this undertaking I leave to
those who may think it worth while to follow out these re
searches, and shall only endeavour to indicate to some extent
the composition of these quantities by considerations apart
from an exact mathematical investigation.
It may, first, be observed that the above expression for
the acceleration of the sphere confirms the truth of the argu
ment in pages 437 439, by which it was antecedently con
cluded that the sphere must receive a permanent motion of
translation from the action of the waves. For by integrating,
and supposing the velocity 5 of the sphere and the velocity
V of the incident waves to be zero at the same time and posi
tion, we have
from which result it follows that an excursion of the sphere
estimated in the positive direction exceeds the excursion in
the opposite direction by the space
/ 2H 9 \ \m' z
1 V ' 3 ) 4a" '
2A +
This quantity divided by the interval occupied by the two
excursions gives the uniform velocity with which the space
would be described in that interval. The direction of the
motion of translation is the same as that of the propaga
tion of the waves, or the contrary direction, according as
HI ( 1  2 J is positive or negative.
454 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
dV
There is no acceleration of the sphere if = = 0, that
is, if Fbe either a maximum, or a constant. In the latter
case the sphere is acted upon by a uniform stream.
Again, the reasoning in pages 305 and 306, from which it
was inferred that the action of the waves on the sphere pro
duces, not a uniform motion of translation, but a uniformly
accelerated motion, may now be applied as follows*. The
velocity V was assumed to be always that which is due to
the incident waves at a distance from the origin of x equal to
that of the point at which the centre of the sphere is situated,
or about which it performs small oscillations. But since a
permanent motion of translation of the sphere is found to
result from the action of the waves, that condition is not ful
filled unless this motion be impressed on the sphere and on
the whole of the fluid in the contrary direction. Now such
impression of a common velocity, since it lias the effect of
producing a uniform stream incident on the sphere, by what
is shewn above has no tendency to accelerate its motion ;
neither does it affect the mode of action of the waves on the
sphere, nor to a sensible amount the quantity of action in a
given time. Thus the generation of motion of translation still
goes on, and to maintain the abovementioned condition it is
necessary to impress on the sphere and fluid an equal and
opposite motion continuously. Hence the fluid will be uni
formly accelerated relatively to the sphere supposed fixed, or
actually the sphere will be uniformly accelerated in the fluid.
Consequently, employing D for the symbol of differentia
tion with respect to the motion of translation, and taking
Dx
to represent the rate of this motion at the time t, referred
to an arbitrary unit, we shall have
* The force discussed in page 311 does not here come under consideration,
because on the supposition that H 2 applies to the whole of the spherical surface,
that force acts equally on the opposite hemispheres.
THE THEORY OP HEAT. 455
t D? 2A + 1V 3/4a"
x being measured from a fixed origin and reckoned positive
in the direction of the propagation of the waves.
We have next to enquire what are the circumstances which
determine the sign of this quantity, and, consequently, the
direction of the motion of translation. Reverting to the fore
going expression for the acceleration of the sphere, I observe
that it may be put, to the same approximation, under the
form
3H, w, v\f
TT~5T V a A
V z
But since aa Q V\ r , and V is a function of x at, it
follows, by the same reasoning as that in page 299, that to
the second approximation
dV ,
These results shew that 7 fl + 7 ] is the accelerative force
a V a /
of the fluid to the second approximation, and that the second
approximation is equal to the first multiplied by the ratio
1 + CT O of the density corresponding to the velocity V to the
mean density. Hence the foregoing expression for the accele
ration of the sphere proves that the second approximation is
deduced from the first by merely changing the moving force of
the waves in that ratio, in case H 2 be so small a quantity
that the term involving it may be neglected ; and that other
wise the factor 1 V has also to be taken into account.
It may also be remarked that if a' be extremely large so that
the fluid is almost incompressible, the terms of the second
order become inconsiderable, and the acceleration of the sphere
456 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
is very nearly the same as if it were acted upon by incompres
sible fluid moving bodily with the velocity V.
Respecting the composition and value of the constant H[
some information has already been gained from the theory of
Dispersion. It was found that by assuming the analogous
constant h^ h^ to be equal to Jc ( I } experimental facts
could be accounted for by that theory. (See page 425). Ac
cordingly, since in the present case /J, 1, let us suppose that
k
HI = k t f Then, leaving out of consideration at present
A<
the term involving JET 2 for reasons that will presently appear,
it follows that the translation of the sphere is from or to
wards the origin of the waves according as k^ is greater or less
Jc
than  . The theory of Dispersion appears to indicate that a
X
change of sign of H^ may occur when X passes through a
value included among those which belong to the phenomena
of light. (The reasoning bearing on this point is contained
in pages 428 432). Hence the translating action of waves
of that order may either be repulsive or neutral, or attractive.
From experience there is reason to infer that a class of
setherial vibrations exists for which the values of m and X are
very much smaller than those that are appropriate to the
phenomena of light. It is a known fact that parallel lines en
graven on glass, and separated by intervals much smaller
than the values of X for light, have been seen by microscopes,
the glass itself still appearing continuous in its structure.
Hence making the only hypothesis which is consistent with
the theoretical principles advocated in this work, namely, that
the ultimate atoms of the glass are kept asunder by the repul
sive action of aetherial undulations which have their origin at
individual atoms, it may be presumed that this atomic repul
sion is attributable to undulations incomparably smaller than
those which cause the sensation of light. This view is the
more admissible inasmuch as we have had reason to conclude
THE THEORY OF HEAT. 457
(p. 410) that the spaces occupied by the atoms are very small
compared to the intervening spaces.
In order to embrace the action that may be due to this
new class of undulations, let us now suppose, having regard
to the reasoning in pages 370 and 371, that
Making then the allowable supposition that the constants k l9
& 2 , & 3 are all positive, and of such values that H{ is positive
for very small values of X, negative for values approximating
to those for lightundulations, and again positive for still
larger values, it will follow that the action of the smallest
class of undulations is repulsive, that of a superior class of
greater breadth attractive, and that there is a transition from
these to another repulsive class as the values of A, increase.
Before proceeding to the physical applications of these
results, I propose, with the view of exhibiting with as much
precision as possible this essential part of the mathematical
theory, to advert again to the reasons for the abovementioned
variations of the undulatory action, and to add something to
what is said on this subject in pages 294 and 307. The
solution of Example VI. given in pages 279 286, in which
transverse action is not taken into account, may be regarded
as a first approximation to the complete solution which in
cludes that action. For this reason the first approximation
to each of the factors H t and H z is unity. The general effect
of the transverse action is to increase the condensations and
rarefactions contiguous to the second hemispherical surface
relatively to those contiguous to the first, and consequently to
diminish the effective acceleration of the sphere. In fact, the
result arrived at in page 295 may be expressed, so far as regards
the factor H v by saying that the transverse action has the
same effect as another series of undulations supposed to be
incident in the same phase as the first but in the opposite
direction ; so that, if the accelerative action of each series be
458 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
calculated without including transverse action, the excess of
the first result above the second gives the actual acceleration.
It hence follows that H t is always less than unity, and may
even be negative. The possibility of its becoming negative
may be seen from what is argued in page 307, namely, that
the motion of the fluid on the farther side of the sphere is in
such manner affected by the disturbance as to partake of the
character of spontaneous direct and transverse vibrations re
lative to an axis, in which, it has been shewn, the conden
sation for a given velocity is greater than that in planewaves
in the ratio of K? to 1. Such an effect, it may be presumed,
would not necessarily take place when the waves are of the
smallest class, because in the case of these the defect of con
densation on the farther side of the sphere might be only
partially supplied by lateral confluence. Hence for waves of
this class H is assumed to be less than unity and always
positive. For very large values of X the value of H ap
proaches to unity as its maximum, and the motion, so far as
this factor is concerned, is very nearly the same as in the case
of no transverse action.
The factor H z differs from H^ in being significant only in
terms of the second order with respect to V. It is not neces
sary to enter at present upon the consideration of the com
position of this factor, because, as will afterwards appear, so
far as regards the theory of heat and molecular attraction it
may be supposed to have its first approximate value, which,
as is argued above, is unity. Thus in the application of the
formula for the translatory force to that theory the factor
.
1  jp is always positive.
Hitherto the action of the fluid on the sphere has been
investigated to terms of the second order only for one series
of undulations defined by the equations
V 2
a'8= V+r, V=
THE THEORY OF HEAT. 459
If there be propagated in the same direction any number of
sets similarly defined, the investigation to the first approx
imation, since the differential equations are rectilinear, will
be precisely the same as for a single set, S . S and S . V being
respectively put for S and V.
On proceeding to the second approximation we may sub
stitute for the condensation and velocity in the terms of the
second order the values of 2 . 8 and 2 . V derived from the
first approximation. It will then be found, after suppressing
the terms indicative of vibratory action, that the remaining
terms indicating motion of translation are the sums of those
which would have been obtained if each component of the
composite series had been treated separately. For the general
argument respecting the law of independence of those terms
of the second order on which the translatory action of the
components of compound undulations depends, I may refer
to the discussion of Proposition xvii. contained in pages 231
239, where the independence of such terms is also proved
for the case in which any number of series are propagated in
any directions. In short, the translatory action of undula
tions in the most complicated cases may be resolved into
coexisting actions such as those which have been ascertained
by the foregoing investigations.
We are now prepared to enter upon the theory of the
forces by the action of which the ultimate parts of bodies are
kept asunder, and at the same time form compact and co
herent masses. The theory will be made to rest on the five
hypotheses contained in pages 357 and 358, which were there
enunciated with especial reference to the part of the theory of
Light that has relation to the properties of visible and tangible
substances, but are just as applicable to other physical
theories. The only additional hypothesis that will now be
made is, that there are undulations of the aether for which the
values of X are very much inferior in magnitude to those of
the undulations which produce the phenomena of light. The
origin of this class of undulations may, as well as that of all
460 THE MATHEMATICAL PRINCIPLES OF PHYSICS.
others, be ascribed to disturbances of the cether by the vibra
tions and motions of atoms. Although the periods of the
aetherial vibrations may under particular circumstances be
determined by the periods of the vibrations of the atoms, this
is not necessarily the case, but according to hydrodynamical
principles previously expounded (pp. 180 and 244), the cir
cumstances of initial disturbances will in general require to
be satisfied by the composition of undulations having different
values of m, X and c. To this principle is to be added that of
the transmutability of undulations (see p. 328), which allows
of the supposition that the undulations which are incapable
of regular transmission in any medium are broken up into
parts having various wavebreadths consistent with such
transmission. If also it be considered that condensations once
generated are not lost except by propagation into unlimited
space, there will, I think, appear to be sufficient ground for the
above additional hypothesis. Supposing, therefore, different
series of undulations of the assumed type to be propagated in
the medium in different directions, and to be incident upon a
given atom, each will give rise to a secondary series due to
the reaction of the atom, and having the same value of X
as the original series. From the solution of Example vi.
(p. 279) it would appear, since or cr Q is found to be expressed
as a function of r, 0, and t involving an arbitrary function of
r at, that initially at the surface, and therefore subse
quently, the condensation is propagated in the direction of
the radii produced, although its quantity is not the same in
all directions, being at the surface proportional to sin 0.
Thus secondary waves from each atom are incident on all the
surrounding atoms, and must be reckoned among those which
by their incidence on a given atom generate the aggregate of
the waves which emanate from that atom.
Now since this emanation from the atom results from the
incidence upon it of waves coming from an unlimited number
of sources and in all directions, so long as there are no known
circumstances which determine the quantity of the emanating
THE THEORY OF LIGHT. 461
waves to be greater in one direction than in another, we may
assume it to be the same in all, and to give rise in all direc
tions to the same amount of condensation, although not
necessarily in the same phase. The total emanation evidently
consists of a vast number of secondary waves having different
values of m, X, and c. These, by hypothesis, include all
waves of such breadths that, according to the foregoing theory,
they would have a repulsive action on surrounding atoms.
It is true that we must suppose to be also incident on each
atom, those waves of much larger breadth which, according to
the same theory, have an attractive effect. But these generate
secondary waves of much inferior magnitude, inasmuch as the
condensation of any secondary wave at the surface of the
sphere varies cceteris paribus as  very nearly (p. 286).
A*
Hence, even if the broader waves were equal in number to
the others, the repulsive action would greatly preponderate,
and we may accordingly conclude that the mutual action be
tween neighbouring atoms is always repulsive.
By employing the same reasoning as that contained in
Corollary n., page 308, it may be shewn that this repulsive
force varies inversely as the square of the distance from the
centre of the atom. It will, however, presently be shewn that
at distances which are large multiples of the mean interval
between the adjacent atoms this law is completely changed.
The first attempt I made to give a mechanical theory of
Heat (which is contained in the Philosophical Magazine for
March, 1859), was based on