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Full text of "Notes on the principles of pure and applied calculation; and applications of mathematical principles to theories of the physical forces"

NOTES ON THE PRINCIPLES OF PURE 
AND APPLIED CALCULATION; 

AND APPLICATIONS OF 
MATHEMATICAL PRINCIPLES TO PHYSICS. 



LIBRARY 

.jj-TJNIVK'nsiTY OF i| 

(MltFOENIA, 



ffamtrtfcge: 

PRINTED BY C. J. CLAY, M.A. 
AT THE UNIVERSITY PRESS. 



NOTESiJNfv, 

ON THE ' PBJNC1PLES ' ; bfr ; 'ft 

ii-n d 'ptx -u s, i c..s 
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 co-existence 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 right-angled 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 self-evident 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 direction-angle 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 ; ... > .... .- , . 138-151 

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 plane-waves 
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 
co-existence 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 ball-pendulum 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 non-dependence 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) Co-existence 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 non-polarized 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 non-interference 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 non-dependence 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 plane-polarized, elliptically-polarized, and 
circularly-polarized 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 ..... 356-4^62 

Problem I. Laws of transmission of light through non-crystalline 
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 wave-surface 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 non-crystalline 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 non-polarized ray. Construction for 
determining by means of the wave-surface, 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 plane-polar- 
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 light-producing 
rays are also heat-producing accounted for. Heat-waves 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 gravity-waves 
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. Non-retardation 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 
refractive-index (p. 367) is very small for gravity-waves, 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 
gravity-molecules. Gravity-measures 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 gravity-waves. 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 . 607-515 



XXIV CONTENTS. 



Case of the positive or negative electric state of a globe. Hypo- 
thesis that the second-order 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 non-conductors 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 second-order 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 non-conductor 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 second-order 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 gold-leaf 
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 thermo-electric. 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 
non-magnetic 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 non-production 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 solar-diurnal 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 solar-diurnal 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. Earth-currents. 
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 disturbance-variations 
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 lunar-diurnal 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 
solar-diurnal disturbance-variation of declination referred to 
this cause. Its law of periodicity different from that of the 
regular solar-diurnal variation . . . . . 665 671 

Additional theoretical inferences. (1) Dependence of the amount 
of disturbance-variation 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 semi-synodic 
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 disturbance-variation 
accounted for by supposing them to be generated by planetary 
magnetic influence. (3) Magnetic storms considered to be 
violent and transitory disturbance-variations due to solar local 
causes. Observation of a remarkable phenomenon confirma- 
tory of this view. The larger displays of Aurora attributed to 
these unsteady sun-streams. The local hours of maximum of 
magnetic storms the same as those of the more regular dis- 
turbance-variations. (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 twenty-five 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 self-evident 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 non-recognition 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 self-evident 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 
non-recognition 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 
self-evident, 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 object-glass 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 object-glass 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 radius-vector, 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 radius-vector 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 
radius-vector 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 above-mentioned 
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 self-evident, 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 

above-mentioned 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 non-mathematical 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 right-hand 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 above-mentioned 
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 above-mentioned 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 plane-waves 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 plane-waves 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 non-acceptance 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 above-mentioned 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 
vortex-motion (see Phil. Mag. vol. 33, p. 485) indicate motion of 
such "an absolutely unalterable quality" as to suggest the idea 
that " vortex-rings 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 
above-mentioned 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 attraction-waves and repulsion-waves intermediate to the 
waves of molecular attraction and gravity-waves 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 the-rheo- 
phore must fulfil the condition of vortex-motion, 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 Adams-Prize 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 juxta-position, 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 j-j , 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 T-J 5 it plainly follows that j- is the pro- 




LIBRAE 



20 THE PRINCIPLES OF 



duct of j-j and - . because ^^ = 7 . Hence ^ contains - the 
bcL c oac o o c 

quantity of times 7-5. 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: 






^ _ 

" 



f-m\-n _ _ n mn _ -mx-n 

~(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 left-hand side of the 
equality = J, and the right-hand 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 left-hand 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(x-a] (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 
co-ordinate 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, 



fix-h)-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 right-hand 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 left-hand 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 right-hand 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 four-sided 
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 right-angled 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 right-angled 
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 right-hand 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 
self-evident, 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 right-angled triangle, the lengths of the other 
two sides being given. 

Let ABC be a triangle*, right-angled at A\ and conceive 
another right-angled 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 right-angled 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 right-angled 
triangle ABC are therefore equal to two right angles. 

COROLLARY 2. Since if one of the acute angles of a 
right-angled 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 right-angled 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 right-angled 
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 right-angled triangle, the relations of the parts of any tri- 
angle. 

Let ABC be any triangle*, and let the angle ABC be 
acute. Conceive a right-angled 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 right-angled triangle be applied, in the manner before 
stated, to any side of an acute-angled triangle, or to a side of 
an obtuse-angled triangle containing the obtuse angle, the 
triangle will be divided into two right-angled 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 right-angled 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 right-angled 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 right-angled 
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 self-evident 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 right-angled triangle 
is half the product of its base and altitude. 

Every other triangle is shewn, by the process of applying a 
right-angled triangle in the manner already employed several 
times, to be the sum or the difference of two right-angled 
triangles having the same vertex. Hence if p be the altitude, 
and q, q the bases of the two right-angled 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 right-angled triangle proved above. 
If x and y be the co-ordinates which determine the position of 
the point, its distance (r) from the origin of co-ordinates 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 co-ordinates 
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 co-ordinates, 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, right-angled 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 co-ordinates 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 

direction-angle. 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 + e-V 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 co-ordinates x and y. The signs of the 
other trigonometrical lines are determined by their analytical 
relations to these. The whole circumference being represented 
by 2-7T, 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 right-angled 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 co-ordinates, 
(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 co-ordinate, 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 co-ordinate, 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 time-interval 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 time-piece 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 time-piece 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 // ,75-f^.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 = -= + (n-1) . 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 object-glass, 
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 right-angled 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, (ft-fl,)^/*^/^, or p, -p a =f lP a, 
and (Pi-pjah^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 co-ordinates of the position of any element of the 
fluid referred to three rectangular axes of co-ordinates 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 vco-ordinates. 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 co-ordinates, and tending from the co-ordinate 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 co-ordinate 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 co-ordinates 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 left-hand 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 right-hand 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 
co-ordinates 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 
right-hand 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 .*. =- = -T-S 

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. 

8-2 



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. 25G-258. 



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 co-ordinates 
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 yi-s = 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 co-ordinates, 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 co-ordinates, 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 co-ordinates 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 ^ yu-V, 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 non-periodic 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 radius-vector 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 non-periodic. 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 non-periodic 
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 non-periodic 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 co-ordinates 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 non-periodic 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 radius-vector 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 
radius-vector 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 
radius-vector 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 co-ordinates x, y into the polar co-ordi 
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 right-hand 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 
semi-axis 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 right-hand 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 radius-vector 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*) ' 

1-1 .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(l-w 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 + -r-J 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, (1-6,003 ^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 -- 5-y 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 right-hand sides of the equations (B) and (C), 
and consequently on the right-hand 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 i-v i-v 

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 semi-axis 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 non-periodic 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 
right-hand sides' of these two equations and the equation (6), 
we obtain a first set of values of the left-hand 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 left-hand 
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 co-latitude 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 co-or- 
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 co-ordinates 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 co-ordinates, 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 co-ordinates : 
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 self-evident, 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 co-ordinates 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 co-ordinates 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 co-ordinates 
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 co-ordinates 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 co-ordinates to be &c, By, $z, 
conceive, for the sake of distinctness, the element to be in 
that portion of space for which the co-ordinates .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 co-ordinates 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 

. d-dr 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, 
i|r 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 co-extensive 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 co-ordinates 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 co-ordinates 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 = X-y- 

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 right-hand 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 co-ordinates 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 _ d-fy 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 

F-JF'W-w.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 

p-i-^S^ w- 

Hence y a.F(r-a 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 co-ordinates 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* . 2-7T, . 
o- = -^-sm (r-at}> 



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 plane-fronts; 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. la-dt 



a j dx i a 




dx 

a-dt 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 f-3- , 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^costf-f #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), 

2-1 ^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^ = i-rf, 

and /, = \ cos 2 Ve y = I - qf, nearly. 



Hence /,+/, = 1 - e (J + tf = l-er* =/ 



HYDRODYNAMICS. 217 

Again, let a\ +/ t = 0, and aV 2 +/ 2 = ; so that 



But it has just been shewn that / 1 +j^=/ Consequently 
0^ + 0-2 = <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 co-ordinates, and their axes coincident with the 
axes of co-ordinates, 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 4-x 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 co-ordi- 
nates. Now for a single set of vibrations we obtained the equa- 

tion i|r' = -- cos ^ (z a^-f-c), 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 co-ordinates are subject to 
these limitations). Let the co-ordinates x, y', z in this equa- 
tion be transformed, and let a?, y, z be the new co-ordinates 
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 co-ordinates. 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 co-ordinates 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 right-hand 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+ -X-rh- (-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 A|T. 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 co-ordinates 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 co-existence 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 co-ordinates 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 co-ordinates 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 -rj-dV du dv dw 

j == -r- 7 - = -w-j -- v-j -- 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\ 
(u-v) [-T- --j- ) + (v-w) -j -r)4- fa *) T--T~ 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 co-ordinates 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 
o-j 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. 24-7 

by the above equation. It should be noticed that while V l 
in the case of vibratory motion is as often positive as negative, 
o-j 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 plane-waves 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 
plane-wave, 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'(r-Kat] 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(r-fcat) 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(r-Kat) 

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(b-ieat)_f'(b-ieat) <(*)_ 
' 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 . 2-7T /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 co-ordinates 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 radius-vector 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 co-ordinates 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 co-ordinates 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 co-ordinates 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 co-ordinates 
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 : 

2-Tra' mb* 



27T& 
X 2-7T 





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 2-7T , , 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 ball-pendulum 
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 ball-pendulum. 

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 (A-l) (-= - 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 suspension-wires 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 suspension-wire 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 pendulum-lengths. 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 air-pump. 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 0,002564Z-f 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 suspension-wire, 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 suspension-rods 
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 ball-pendulum 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-^+l-o, 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 co-ordinates 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 left-hand 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 before-cited 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 plane-waves 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(r-dt], -, 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 co-ordinates 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 o-j = , 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 

j-i-er + - 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 

fl-f 
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 plane-waves (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 plane-front. 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 non-divergence 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 wave-front 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 hydro-dynamical 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 left-hand 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'(l-h) 

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 plane-waves 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 plane-waves 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 left-hand 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 plane-waves, 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 plane-waves. 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 -y-2 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\-rtcs4-ar\'t- o-vvkvoaoTn 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 co-ordinate (x^) of the centre of the 
vibrating sphere at the time t. Consequently, leaving out of 
account at present any non-periodic 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(l-h) ~, 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 non-periodic 
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 non-periodic 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 plane-waves 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 
non-periodic 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 non-periodic 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 plane-waves. 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 co-ordinates, 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 non-periodic 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 non-parallelism 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 self-evident, 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 above-mentioned 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 = ^2-7- = - (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 2-7T , ... 



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 n-m 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 (o-j + 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 light-producing 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 oxy-hydrogen 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 plane-waves, 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 non-polarized 
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 plane-wave 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 plane-front 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 left-hand sides of the last two equations will 

express the velocities in the bifurcated rays resolved parallel 
to the same axis. But the right-hand 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 co-ordinates 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 left-hand 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 plane-polarized, elliptically-polarized, and cir- 
cularly-polarized. 

(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 
sun-light, 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 sound-sensations. 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 wave-intervals, 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 l-j- + (LJ cos l-j- + <7J 

. , . /2-7T2! ^\ /27TZ ~\ . f%7TZ ~ ,\ 

v = <2p sm l-j- + 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 light-undulations 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 wave-length 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 
wave-length 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.., 


'ave-lenj 

2425 
2244 
2162 
2120 
2095 
2085 
2082 


j.i.1. Complementary -IT 
Colour. 

... Green-blue ... 
... Blue .. 


r aye length. 

1 CM Q 


Ratio of 
wave-lengths. 


Orange , 


1809 


1 24.0 


Gold-yellow ... 
Gold-yellow ... 
Yellow 


... Blue 


1793 


1 20fi 


... Blue 


1781 ... 


. . 1 190 


... Indigo-blue... 
... Indigo-blue... 
... Violet . 


171 A 


1 991 


Yellow 


1 70fi 


1999 


Green-yellow... 


1600 . 


, 1.301 



THE THEORY OF LIGHT. 343 

These results shew that the disappearance of the intermediate 
colour takes place for ratios of the wave-lengths varying from 
about that of 4 to 3 for red and green-blue, to about that of 
6 to 5 for gold-yellow and blue. It is worthy of remark that 
the ratio of the wave-lengths 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, 
gold-yellow 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 wave-lengths lie between the numbers 2082 and 1818, 
the difference of which is about one-third the difference of 
the numbers for the extreme wave-lengths, have no comple- 
mentary colour. This fact seems to admit of being explained 
by the consideration that the ratios of their wave-lengths to 
the wave-lengths 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 day-light a rect- 
angular piece of non-reflecting black paper, and on the latter 
a rectangular slip of the white paper one-twelfth 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 day-light 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 plate-glass, 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 day-light. 

The foregoing series of experiments seem to justify the 
conclusion that blue and yellow parcels of ordinary day-light, 
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 wave-length - r-y . 

. 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 wave-lengths included within certain 
limits. If fjb represent the maximum velocity resulting from 
the composition of all the simple rays in one parcel having 
the wave-length X, and fju that from the composition of the 
simple rays of the other parcel having the wave-length 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 right-hand 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 wave-length 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 . iz-g 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 wave-length of 
the colour resulting from composition is necessarily inter- 
mediate to the wave-lengths 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 wave-lengths 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 wave-lengths of red and violet is very nearly that of 3 to 2, 
and the combination of wave-lengths 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, Sun-light 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 wave-lengths 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 non-contact, 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 non-crystalline 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 plane-waves, 

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 

. 2-7T , 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 (*V-6 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 wave-length. 
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 co-ordinates, 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 co-ordinates, 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 co-ordinates 
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 ray-undulations (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 ray-undulations 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 semi-axes of the above mentioned 
elliptic section, and to obtain them from the equation of the 
surface of elasticity is a geometrical problem, the well-known 
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 co-ordinates 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 wave-surface. 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 plane-waves, 
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 plane-waves 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 plate-glass, 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 
ray-undulations relative to a given axis may be composed of 
an unlimited number of primary ray-undulations 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 ray-undulations 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 non-polarized 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 

o-j 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 non-polar- 
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 non-polarized 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 non-polarized 
part 20-j 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 non-polarized 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 non-polarized 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 non-polarized 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 non-crystallized 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 plane-fronts 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 non-crystallized 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 
wave-length, 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 co-ordinates of the point, and 
a?!, y l the co-ordinates 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 co-ordinates 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 co-ordinates 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 non-polarized ; 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 non-polarized ray is divided into two equal 
plane-polarized 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 ray-undulation 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 plane-front of the incident waves to be cut 
by two planes of incidence indefinitely near each other, and 
the included portion of the wave-front to be divided into an 
indefinite number of equal rectangular elements, containing 
the same number of axes of ray-undulations. 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 ray-undulations (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 ray-undulation. 



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 wave-front, 
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 
plane-fronts 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 plane-fronts. We have, therefore, next to consider in 
what manner this condition is satisfied under the above 
described circumstances of divergence of the ray-axes. 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 ray-undulations 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 plane-front of an incident wave and the 
plane-surface 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 ray-axes. 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 non-crystallized media, be a portion of a spherical 
surface. If a ray-undulation 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 above-mentioned 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 wave-front 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 wave-front 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 non-crystallized 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 non-crystallized 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 plane-front, 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 non-polarized 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 non-crystallized media. The 
incident waves being supposed to have plane-fronts, and the 
surface of the medium to be a plane, let the intersection of the 
surface by a plane-front 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 wave-surface 
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 non-polarized 
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 above-mentioned 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 plane-fronts 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 plane-fronts. 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 plane-fronts 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 wave-front, 
parallel respectively to the refracting surface and to the planes 
of incidence, and 7 perpendicular to the wave-front. 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 wave-front 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 wave-fronts. Then if D be the distance 
between the points A and B, the length of wave-front 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 wave-fronts 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 wave-element 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 wave-element will be inversely 
proportional to each other. Consequently 

o-j _ 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 
non-polarized 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 
wave-element 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 wave-front, it follows that where the 
wave-front 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 wave-front 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 wave-front 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 above-mentioned 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 plane-polarized, 
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 well-known exemplification of this 
theoretical inference. 

The coloured rings, formed by subjecting plane-polarized 
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 plane-polarized 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 plane-polarized 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 
above-mentioned 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 non-crystallized 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 plane-facets of all bodies, 
both black and white, or whatever may be their proper colour. 
The non-reflected 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 non-crystallized, 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 
non-neutralized 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 day-light, 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 sun-set 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 tangent-plane 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 wave-lengths 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 wave-length 
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 heat-rays 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 co-exist 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 before-men- 
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*(=0-l -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 
above-mentioned 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 above-stated 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 / 

4-TrW 
~^~ 

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 co-existing simple vibrations for each of 
which the value of I is different. Hence, as the left-hand 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 light-producing 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 light-producing rays may also 
be heat-producing. 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 6-H+ ~ 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 co-ordinates to the 
polar co-ordinates 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 right-hand 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 right-hand 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 -j-g 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 A-l 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 co-ordinates 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 co-ordinates 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 -7-5 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 co-ordinates 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} w-a-tjix 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 S-ff.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 above-mentioned 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 light-undulations, 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 above-mentioned 
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 plane-waves 
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 wave-breadths 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